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TABLE OF CONTENTS

CHAPTER 1: INTRODUCTION

1.1 OPEN-CHANNEL FLOW

1.2 TYPES OF FLOW

1.3 STATE OF FLOW

1.4 FLOW REGIMES

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 2: PROPERTIES OF OPEN CHANNELS

2.1 KINDS OF OPEN CHANNELS

2.2 CHANNEL GEOMETRY

2.3 VELOCITY DISTRIBUTION

2.4 MEASUREMENTS OF VELOCITY

2.5 VELOCITY DISTRIBUTION COEFFICIENTS

2.6 PRESSURE DISTRIBUTION

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 3: ENERGY AND MOMENTUM PRINCIPLES

3.1 ENERGY PRINCIPLE

3.2 SPECIFIC ENERGY

3.3 LOCAL PHENOMENA

3.4 MOMENTUM PRINCIPLE

3.5 SPECIFIC FORCE

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 4: CRITICAL FLOW

4.1 CRITICAL FLOW

4.2 COMPUTATION OF CRITICAL FLOW

4.3 CRITICAL FLOW CONTROL

4.3 SHARP-CRESTED WEIRS

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 5: UNIFORM FLOW

5.1 UNIFORM FLOW

5.2 CHEZY FORMULA

5.3 MANNING FORMULA

5.4 MANNING ROUGHNESS

5.5 COMPUTATION OF UNIFORM FLOW

5.6 COMPUTATION OF FLOOD DISCHARGE

5.7 UNIFORM SURFACE FLOW

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 6: CHANNEL DESIGN

6.1 NONERODIBLE CHANNELS

6.2 ERODIBLE CHANNELS

6.3 PERMISSIBLE VELOCITY

6.4 PERMISSIBLE TRACTIVE FORCE

6.4 OTHER FEATURES

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 7: STEADY GRADUALLY VARIED FLOW

7.1 EQUATION OF GRADUALLY VARIED FLOW

7.2 CHARACTERISTICS OF FLOW PROFILES

7.3 LIMITS TO WATER SURFACE PROFILES

7.4 METHODOLOGIES

7.5 DIRECT STEP METHOD EXAMPLE

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 8: CULVERT HYDRAULICS

8.1 CULVERTS

8.2 INLET CONTROL

8.3 OUTLET CONTROL

8.4 CULVERT DESIGN

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 9: STEADY RAPIDLY VARIED FLOW

9.1 THE SHARP-CRESTED WEIR

9.2 CREST SHAPE OF OVERFLOW SPILLWAY

9.3 RATING OF SPILLWAYS

9.4 THE HYDRAULIC JUMP

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 10: UNSTEADY GRADUALLY VARIED FLOW

10.1 GOVERNING EQUATIONS

10.2 LINEAR SOLUTION

10.3 KINEMATIC WAVES

10.4 DIFFUSION WAVES

10.5 MUSKINGUM METHOD

10.6 MUSKINGUM-CUNGE METHOD

10.7 DYNAMIC WAVES

QUESTIONS

PROBLEMS

REFERENCES

CHAPTER 11: UNSTEADY RAPIDLY VARIED FLOW

11.1 SURFACE WAVES

11.2 SURGES

11.3 KINEMATIC SHOCKS

11.4 ROLL WAVES

11.5 TIDAL WAVES

11.6 DEBRIS FLOWS

11.7 LAHARS

QUESTIONS

PROBLEMS

REFERENCES

APPENDIX

[Open-channel Flow] [Types of Flow] [State of Flow] [Flow Regimes] [Questions] [Problems] [References] •

CHAPTER 1: INTRODUCTION

1.1 OPEN-CHANNEL FLOW

[Types of Flow] [State of Flow] [Flow Regimes] [Questions] [Problems] [References] • [Top]

Open-channel flow has a free surface and it is, therefore, subject to atmospheric

pressure (Fig. 1-1). In contrast, closed-conduit flow, or pipe flow, does not have a free

surface, being subject only to a certain hydraulic pressure. Due to its free surface, the

analysis of open-channel flow is generally more complex than that of closed-conduit

flow. The free surface is likely to vary in space and time. When the free surface varies

in space, the flow is referred to as varied; when it varies in time, it is referred to

as unsteady.

Fig. 1-1 An open channel used to convey irrigation water (irrigation canal).

In closed-conduit flow, the cross section is fixed by the pipe boundaries. On the other

hand, in open-channel flow, the flow cross section is not fixed, varying with the flow. In

closed-conduit flow, the roughness varies from smooth brass to corroded pipes; in

open-channel flow, it varies from acrylic glass or lucite® (a very smooth type of plastic),

to that of natural stream channels and their neighboring flood plains.

In closed-conduit flow, the hydraulic pressure at the center of the pipe defines the

hydraulic grade line (HGL in Fig. 1-2). The hydraulic pressure (head of water) above

the centerline of the pipe is referred to as the piezometric head. The energy grade line

includes the velocity head hv = V 2

/(2g) at the cross section. The head loss, or friction

head loss, from Section 1 to Section 2, is hf.

Fig. 1-2 Schematic of closed-conduit flow.

In open-channel flow, the flow depth measured above the channel bottom defines the

water surface elevation, which is equivalent to the hydraulic grade line of closed-

conduit flow; see Fig. 1-3. The total energy grade line includes the velocity

head hv = V 2/(2g) at the cross section. The head loss, or friction head loss, from

Section 1 to Section 2, is hf.

Fig. 1-3 Schematic of open-channel flow.

Note the difference between closed-conduit and open-channel flow. In closed-conduit

flow, water will rise in a piezometer tube up the level where it defines the hydraulic

grade line associated with the hydraulic pressure in the conduit. On the other hand, in

open-channel flow, the water surface is the hydraulic grade line, which is at

atmospheric pressure.

1.2 TYPES OF FLOW

[State of Flow] [Flow Regimes] [Questions] [Problems] [References] • [Top] [Open-channel Flow]

There are two general types of open-channel cross sections:

1. Prismatic, and 2. Nonprismatic.

Artificial, or human-made channels, are usually prismatic, featuring a constant shape

and size, at least for a certain length of channel. Conversely, natural channels are

typically nonprismatic, i.e., the shape and size of the cross section varies along the

channel. Artificial channels are also referred to as canals.

Several geometric and hydraulic properties help describe an open channel (Fig. 1-4).

These are:

Discharge Q, Flow area A, Mean velocity V, with V = Q /A, Wetted perimeter P, Top width T, Hydraulic radius R, with R = A /P, Hydraulic depth D, with D = A /T, Channel width B, Channel bottom elevation, or bed elevation z, Flow depth d, Stage, or water surface elevation, with y = z + d.

Fig. 1-4 Definition sketch for open-channel flow.

In prismatic channels, the flow depth d is often referred to as y, particularly when it

cannot be confused with stage. Also, the channel side slope is often referred to as z H :

1 V, particularly when it cannot be confused with bed elevation.

Classification of open-channel flow

Open-channel flow may be classified as follows:

A. Steady or unsteady. B. Uniform or equilibrium. C. Gradually varied or rapidly varied. D. Spatially varied.

The flow is steady when the hydraulic variables (discharge, flow area, mean velocity,

flow depth, and so on) do not vary in time. Conversely, the flow is unsteady when the

hydraulic variables vary in time and space. Steady flow is relatively simple to calculate,

compared to unsteady flow.

The flow is uniform when the channel is prismatic and the hydraulic variables (Q, A, V,

d, and so on) areconstant in time and space. The flow is in equilibrium when the

channel is nonprismatic and the hydraulic variables are approximately constant in time

and space. The calculation of uniform flow is relatively straight forward when compared

to that of other states of flow.

The flow is gradually varied when the discharge Q is constant but the other hydraulic

variables (A, V, d,and so on) vary gradually in space. Under gradually varied flow, the

pressure distribution in the vertical direction, normal to the flow, is very close to

hydrostatic, i.e., proportional to the flow depth.

The flow is rapidly varied when the discharge is constant but the other hydraulic

variables (A, V, d, and so on) vary rapidly in space, in such a way that a hydrostatic

pressure distribution cannot be assumed in the vertical direction normal to the flow.

While the calculation of gradually varied flow is somewhat involved but doable, the

calculation of rapidly varied flow is generally more complex, in practice being based on

empirical formulas, for lack of a theoretical solution.

The flow is spatially varied when the discharge Q varies in space only, i.e., along the

channel, typically due to lateral inflow or outflow.

Occurrence of various types of flow

Steady uniform flow occurs in a prismatic channel (Fig. 1-5); steady equilibrium flow

occurs in a nonprismatic channel. Unsteady uniform flow does not exist in nature,

because the flow cannot be uniform and unsteady at the same time. The word

"unsteady" implies nonequilibrium; thus, unsteady equilibrium flow does not exist.

Fig. 1-5 Uniform flow in an irrigation canal, Wellton-Mohawk project, Wellton, Arizona.

Steady gradually varied flow is represented by the water surface profiles, also referred

to as backwater (or drawdown) computations (Chapter 7). Unsteady gradually varied

flow is the calculation of flood flows, or flood routing (Chapter 10).

Steady rapidly varied flow is represented by the flow over spillways or the hydraulic

jump. Unsteady rapidly varied flow is represented by the moving hydraulic jump,

surges, roll waves, kinematic shocks, and tidal bores. Figure 1-6 shows a train of roll

waves in a steep irrigation canal.

Spatially varied flow occurs in an artificial canal when the discharge is varying along the

channel, due to lateral water extractions or channel overflow.

Fig. 1-6 Roll waves in a steep irrigation canal, Cabana-Mañazo project, Puno, Peru.

1.3 STATE OF FLOW

[Flow Regimes] [Questions] [Problems] [References] • [Top] [Open-channel Flow] [Types of Flow]

The state of open-channel flow may be described in terms of certain characteristic

velocities and viscosities. Velocity is the ratio of length (distance) over time, with units L

T -1

. Viscosity is the first moment of the velocity, with units L2 T

-1. In open-channel

hydraulics, the term diffusivity is used to refer to viscosity. Two velocity and two

diffusivity ratios are defined to complete the characterization of open-channel flow.

Velocity ratios

There are three characteristic velocities in open-channel flow:

1. The mean velocity u of the steady uniform flow;

2. The velocity, or celerity ck of kinematic waves; and

3. The velocity, or celerity (actually, two celerities) cd of dynamic waves.

The mean velocity of the steady uniform flow using the Manning equation (SI units) is:

1

u = _____

R 2/3

S 1/2

n

(1-1)

in which n = Manning's friction coefficient, R = hydraulic radius, and S = friction slope.

The mean velocity of the steady uniform flow using the Chezy equation is:

u = C R 1/2

S 1/2

(1-2)

in which C = Chezy friction coefficient.

In general, four forces are active in a control volume in open-channel flow. These

forces are due to friction, gravity, the pressure (flow depth) gradient, and inertia.

Kinematic waves are those where the momentum balance is expressed in terms of the

frictional and gravitational forces only (Lighthill and Whitham, 1955).

The celerity of kinematic waves, or Seddon celerity, is (Seddon, 1990; Chow,

1959; Ponce, 1989):

ck = β u

(1-3)

in which β = exponent of the discharge-flow area rating, defined as follows:

Q = α Aβ

(1-4)

Dynamic waves are those where the momentum balance is expressed in terms of the

pressure-gradient and inertial forces only. The celerity of dynamic waves is:

cd = u ± (g D )1/2

(1-5)

in which g = gravitational acceleration, and D = hydraulic depth, D = A /T.

From Eq. 1-3, the relative celerity of kinematic waves is:

v = (β - 1) u

(1-6)

From Eq. 1-5, the (absolute value of the) relative celerity of dynamic waves is:

w = (g D )1/2

(1-7)

For rectangular channels, for which D = d, or for hydraulically wide channels, for

which D ≅ d, the relative celerity of dynamic waves is:

w = (g d )1/2

(1-8)

Equation 1-8 is known as the Lagrange (relative) celerity equation, after Lagrange

(1788), who first derived it.

The Froude Number

The Froude number is defined as follows (Chow, 1959):

u

F = _____

w

(1-9)

The Froude number characterizes the condition of:

F < 1: Subcritical flow, or u < w, F = 1: Critical flow, or u = w, F > 1: Supercritical flow, or u > w.

Under subcritical flow, surface waves (perturbations) can travel upstream, because

their upstream celerity -w is greater than the mean flow velocity u.

Under critical flow, surface waves (perturbations) remain stationary, because their

(absolute) celerity w is equal to the mean flow velocity u.

Under supercritical flow, surface waves (perturbations) can travel downstream only,

because their upstream celerity -w is smaller than the mean flow velocity u.

The Vedernikov Number

The Vedernikov number is defined as follows (Vedernikov, 1945; 1946; Powell,

1948; Craya, 1952):

v

V = _____

w

(1-10)

The Vedernikov number characterizes the following states of flow:

V < 1: Stable flow, or v < w, V = 1: Neutrally stable flow, or v = w, V > 1: Unstable flow, or v > w.

Under stable flow, the relative kinematic wave celerity v is smaller than the relative

dynamic wave celerityw and, therefore, surface waves (perturbations) are able to

attenuate (dissipate).

Under neutrally stable flow, the relative kinematic wave celerity v is equal to the

relative dynamic wave celerity w and, therefore, surface waves (perturbations) neither

attenuate nor amplify. Amplification amounts to negative dissipation.

Under unstable flow, the relative kinematic wave celerity v is greater than the relative

dynamic wave celerity w. Therefore, surface waves (perturbations) are subject to

amplification. In practice, the conditionV ≥ 1 leads to the development of roll waves, a

train of waves that travel downstream, typically in artificial channels of steep slope (Fig.

1-7).

Unknown

Fig. 1-7 Roll waves in a masonry-lined channel in the Swiss Alps (c. 1910).

History of the Froude Number

William Froude was born in Dartington, Devon, England on November 28, 1810 and died from a

stroke on a cruise to Simonstown, South Africa at age 69. He was an engineer, hydrodynamicist

and naval architect.

He acquired his education in mathematics at Oxford in 1832. Immediately after his graduation,

he worked for Isambard Kingdom Burnel, the famed developer of railways, as a surveyor on the

Great Western Railway, in England. In 1857, Brunel consulted him on the behavior of the Great

Eastern ship at sea. Based on Froude's recommendations, Brunel modified the design of the ship

to avoid rolling.

Starting in 1859, Froude built the first towing tank, using his own resources. He carried out ship

model experiments using the tank, first at his home in Paignton, and later, in his other home,

called Chelston Cross, in Torquay.

In 1861, he wrote a paper on the design of ship stability in a seaway. The paper was published in

the Proceedings of the Institution of Naval Architects. Later, between 1863 and 1867, he showed

that scaling between model and prototype (the actual ship) applied (i.e., the frictional resistance

was equal) when the speed (V) was proportional to the square root of the length (L). He called

this concept the "Law of Comparison."

V = k (L)1/2

in which k is the number that applies to both model and prototype. This law is known as

Froude's law, even though Froude himself recognized that Ferdinand Reech (1805-1850) has

presented the concept twenty years earlier.

Froude's work was pioneering because he was the first to identify the most efficient shape for the

hull of ships, as well as to predict ship stability with reference to reduced-scale models.

In open-channel hydraulics, Froude's Law is embodied in the Froude number, defined as:

F = V / (gD) 1/2

in which V = mean velocity, D = hydraulic depth, and g = gravitational acceleration. Unlike

Froude's original relation (k), the Froude number F is dimensionless. The L has been replaced

by D to better represent the force of gravity in open-channel flow.

The exponent β of the discharge-area rating

The three velocities u, v, and w lead to only two independent velocity ratios, the

Froude (Eq. 1-8) and Vedernikov (Eq. 1-9) numbers. The third ratio:

v V

β - 1 = _____

= _____

u F

(1-11)

is the dimensionless relative kinematic wave celerity, equal to the exponent of the

discharge-area rating minus 1. Thus, it is seen that the exponent β in Eq. 1-4 is a

function of both the Froude and Vedernikov numbers.

The value of β varies with the type of friction regime (laminar, transitional, or turbulent;

and turbulent Manning or Chezy) and cross-sectional shape. For laminar flow, β = 3.

For turbulent flow, under Manning friction: 1 ≤ β ≤ 5/3, depending on the shape of the

cross section. For turbulent flow, under Chezy friction: 1 ≤ β ≤ 3/2, depending on the

shape of the cross section.

Types of cross-sectional shape

There are three asymptotic cross-sectional shapes in open channels:

1. The hydraulically wide channel, for which the wetted perimeter P is a constant

(Ponce and Porras, 1995). In this case, β = 5/3 for Manning friction and β = 3/2

for Chezy friction. Generally, a channel cross-section is considered to be

hydraulically wide for values of the ratio of top width to flow depthT /d > 10. In

practice, most natural channels are hydraulically wide (Fig. 1-8).

Nuccitelli

Fig. 1-8 Mississippi river at Mud Island, Memphis, Tennessee.

2. The triangular channel, in which the top width T is proportional to the flow

depth d (Fig. 1-9). In this case, β = 4/3 for Manning friction and β = 5/4 for Chezy

friction. Roadside drainage (gutters) often feature a triangular cross section.

Fig. 1-9 The triangular cross section.

3. The inherently stable channel, for which the hydraulic radius R is a constant

(Ponce and Porras, 1995). In this case, β ≡ 1 (Fig. 1-10).

Fig. 1-10 The inherently stable channel.

Neutrally stable flow

For neutral stability: V = 1. Therefore, from Eq. 1-11, the Froude number corresponding

to neutrally stable flow is:

1

Fns = ________

β - 1

(1-12)

Table 1-1 shows values of Fns for selected values of β. It is seen that as β varies

from β = 3 (laminar flow) to β = 1 (inherently stable channel), the value of Fns varies

from Fns = 1/2 to Fns = ∞. In other words, as β⇒ 1, Fns ⇒ ∞.

In practice, since friction has effectively a lower bound, the Froude number is restricted

to an upper bound, which is seldom likely to exceed F = 25. Therefore, in most cases, a

value of β = 1.04 would be already stable for practical purposes.

Table 1-1 Values of Fns for selected values of β.

β Type of friction Shape of cross section Fns

3 Laminar Hydraulically wide 1/2

8/3 Mixed laminar-turbulent (25% turbulent Manning)

Hydraulically wide 3/5

21/8 Mixed laminar-turbulent (25% turbulent Chezy)

Hydraulically wide 8/13

7/3 Mixed laminar-turbulent (50% turbulent Manning)

Hydraulically wide 3/4

9/4 Mixed laminar-turbulent (50% turbulent Chezy)

Hydraulically wide 4/5

2 Mixed laminar-turbulent (75% turbulent Manning)

Hydraulically wide 1

15/8 Mixed laminar-turbulent (75% turbulent Chezy)

Hydraulically wide 8/7

5/3 Turbulent Manning Hydraulically wide 3/2

3/2 Turbulent Chezy Hydraulically wide 2

4/3 Turbulent Manning Triangular 3

5/4 Turbulent Chezy Triangular 4

1 Any Inherently stable ∞

As shown in Table 1-1, values of β in open-channel and overland flow are limited in the

range 1 ≤ β ≤ 3.However, β for a circular culvert flowing nearly full may actually attain

values less than 1 (Chow, 1959).

Viscosity ratios

There are three characteristic viscosities in open-channel flow:

1. The internal viscosity, or kinematic viscosity ν of the fluid (Appendix A),

2. The external viscosity (or hydraulic diffusivity νh) of the steady flow; and

3. The external viscosity (or wave diffusivity νw) of the unsteady flow.

The kinematic viscosity ν of the fluid varies as a function of temperature (Appendix A).

The concept of hydraulic diffusivity νh is due to Hayami (1951). Hayami combined the

governing equations of open-channel flow (Chapter 10) to develop a single convection-

diffusion equation, i.e., an equation describing the convection (first-order process) and

diffusion (second-order process) of a flood wave. The hydraulic diffusivity is defined as

follows:

qo

νh = _______

2 So

(1-13)

in which qo = equilibrium unit-width discharge, and So = friction (energy) slope. It is

seen that flood wave diffusion is directly proportional to unit-width discharge and

inversely proportional to friction (energy) slope.

Equation 1-13 can be expressed in terms of velocity and flow depth as follows:

uo do

νh = _________

2 So

(1-14)

A related value of diffusivity, which is independent of slope, is:

νh' = uo do

(1-15)

In general, for an arbitrary cross-sectional shape:

νh' = uo Ro

(1-16)

in which Ro = hydraulic radius.

In kinematic wave theory, the characteristic reach length is defined as follows (Lighthill

and Whitham, 1955):

do

Lo = ______

So

(1-17)

in which Lo is the length of channel in which the equilibrium flow drops a head equal to

its depth. Thus, in terms of the characteristic reach length, the hydraulic diffusivity is:

uo Lo

νh = _______

2

(1-18)

In a manner resembling the hydraulic diffusivity, the wave diffusivity is conveniently

defined as follows:

uo L

νw = _______

4 π

(1-19)

in which L = wavelength of the disturbance.

The Reynolds Number

The Reynolds number R is (Chow, 1959):

vh' uo Ro

R = ______

= ________

ν ν

(1-20)

The Reynolds number R describes the flow regime as either:

1. Laminar, 2. Transitional, or 3. Turbulent.

Under steady flow conditions in open-channel flow, laminar flow occurs for R ≤ 500 and

turbulent flow forR > 2000. Transitional flow occurs in the intermediate range: 500 < R ≤

2000. Under unsteady flow, themixed laminar-turbulent flow described in Table 1-1 is

akin to transitional flow, featuring a comparable range of Reynolds numbers.

In practice, most open-channel flow cases are in the turbulent regime. Conversely,

most overland flow cases (i.e., free-surface plane flow) are in the laminar or mixed

laminar-turbulent regime.

The dimensionless wavenumber

The dimensionless wavenumber σ is defined as follows (Ponce and Simons, 1977):

νh 2 π

σ = ______

= _____

Lo

νw L

(1-21)

The wavenumber σ describes the dimensionless length scale of the wave, as shown in

Fig. 1-11, in terms of: (a) kinematic waves, (b) dyamic waves, and (c) mixed kinematic-

dynamic waves. Figure 11 is applicable for the case of Chezy friction in hydraulically

wide channels.

Under kinematic flow, depicted on the left side of Fig. 1-11:

a. The momentum balance is described only in terms of the frictional and

gravitational forces,

b. The dimensionless relative kinematic wave celerity (Eq. 1-3), a constant for all

wavenumbers and Froude numbers, prevails, and

c. Wave attenuation is theoretically zero.

Under dynamic flow, depicted on the right side of Fig. 1-11:

a. The momentum balance is described only in terms of the pressure-gradient and

inertial forces,

b. The dimensionless relative dynamic wave celerity (Eq. 1-5), a constant for all

wavenumbers and varying with Froude number, prevails, and

c. Wave attenuation is theoretically zero.

Under mixed kinematic-dynamic flow, depicted by the middle section of Fig. 1-11:

a. The momentum balance incorporates all four forces present in unsteady open-

channel flow (frictional, gravitational, pressure-gradient, and inertial),

b. There is no characteristic celerity and, therefore, the wave is subject to very

strong attenuation in the midrange of dimensionless wavenumbers,

c. For each curve depicted in Fig. 1-11, the attenuation rate is a maximum at the

point of inflexion (See Chapter 10) (Ponce and Simons, 1977).

Fig. 1-11 Celerity of wave propagation in open-channel flow (Ponce and Simons, 1977).

Dynamic hydraulic diffusivity

The dynamic hydraulic diffusivity, which unlike the Hayami hydraulic diffusivity of Eq. 1-

18, considers the complete momentum balance, is (Dooge et al., 1982; Ponce,

1991a; 1991b):

uo Lo

νh = _______

(1 - V 2)

2

(1-22)

For low Vedernikov numbers, V ⇒ 0, the dynamic hydraulic diffusivity reduces to the

expression forkinematic hydraulic diffusivity, i.e., Eq. 1-18. Conversely, for high

Vedernikov numbers, V ⇒ 1, and the dynamic hydraulic diffusivity vanishes. Under this

flow condition, the total absence of wave attenuation is conducive in the development

of roll waves (Figs. 1-6 and 1-7).

A word of caution regarding the usage of the term "dynamic wave"

In hydraulic engineering practice, the mixed kinematic-dynamic wave, which includes all

the terms in the momentum equation--friction, gravity, pressure gradient, and inertia--is

commonly referred to as "dynamic wave," following the work of Fread (1993). However, the

original dynamic wave of Lagrange (1788) accounted only for pressure gradient and inertia.

To add to the confusion, the dynamic wave of Lagrange has often been referred to as "gravity

wave." Clearly, the term "gravity wave" is not appropriate, because the gravity force is

ostensibly missing from its formulation.

Thus, it appears better to reserve the term "kinematic" for waves governed by friction and

gravity (following Lighthill and Whitham), "dynamic" for waves governed by pressure

gradient and inertia (following Lagrange), and to use "mixed kinematic-dynamic" for waves

governed by the complete momentum statement (all four forces).

1.4 FLOW REGIMES

[Questions] [Problems] [References] • [Top] [Open-channel Flow] [Types of Flow] [State of Flow]

The flow regimes in open-channel flow are:

1. Laminar, 2. Transitional, and 3. Turbulent.

The flow regimes are characterized by the Reynolds number R, Eq. 1-20. In open-

channel flow, the laminar regime prevails for R ≤ 500, the transitional regime for 500

< R ≤ 2000, and the turbulent regime for R > 2000.

The flow regimes vary with roughness of the channel surface. Figure 1-12 shows the

relation between Reynolds number R and Darcy-Weisbach friction factor f for flow in

smooth channels. Figure 1-13 shows the relation between Reynolds number R and

Darcy-Weisbach friction factor f for flow in rough channels.

The Darcy-Weisbach friction formula, developed in connection with flow in pipes, is:

L V 2

hf = f _____

______

do 2 g

(1-23)

in which hf = frictional head loss; f = Darcy-Weisbach friction factor; L = length of the

pipe; do = pipe diameter; V = mean flow velocity in the pipe; and g = gravitational

acceleration.

Fig. 1-12 The f-R relation for flow in smooth channels (Chow, 1959).

Fig. 1-13 The f-R relation for flow in rough channels (Chow, 1959).

The examination of Figs. 1-12 and 1-13 enables the following conclusions:

A. Laminar flow prevails under low Reynolds numbers. Under laminar flow, the

Darcy-Weisbach friction factor is inversely proportional to the Reynolds number.

In general:

K

f = _____

R

(1-24)

B. in which K is a constant varying between 14 (for triangular channels) and 24 (for

rectangular channels) for smooth channel surfaces, and between 33 and 60 for

rough channel surfaces.

C. The transitional range is not well defined, depending to some extent on channel

shape. For practical purposes, the transitional range for open-channel flow may

be assumed to be:500 ≤ R ≤ 2000.

D. Under turbulent flow, the relation between f and R follows the Blasius formula

(Chow, 1959):

0.223

f = _________

R 0.25

(1-25)

E. This equation is valid for Reynolds numbers in the range: 750 ≤ R ≤ 25,000.

F. A general expression for the relation between f and R was developed by von

Karman and later modified by Prandlt to agree more closely with the applicable

data. The resulting Prandtl-von Karman equation is (Chow, 1959):

1 _______

= 2 log (R f 1/2

) + 0.4

f 1/2

(1-26)

Note that the Prandtl-von Karman equation can be expressed in explicit form as

follows:

1

R = _______ 10 [ ( 1 - 0.4 f 1/2 ) / ( 2 f 1/2 ) ]

f 1/2

(1-27)

The Darcy-Weisbach friction factor for open-channel flow

The Darcy-Weisbach formula, Eq. 1-23, is strictly applicable to closed-conduit (pipe)

flow. In pipe flow, the characteristic frictional length is the pipe diameter do. On the

other hand, in open-channel flow, the characteristic frictional length is the hydraulic

radius R, i.e., the ratio of flow area to wetted perimeter:

A

R = ______

P

(1-28)

Since the flow area of a circular pipe (flowing full) is A = π do2/4, and the wetted

perimeter is P = π do, it follows that the hydraulic radius is equal to 1/4 of the pipe

diameter or, conversely, that the diameter is equal to 4 times the hydraulic radius.

Therefore, the Darcy-Weisbach formula applicable to open-channel flow is:

L V 2

hf = f ______

______

(1-29)

4R 2g

in which V = mean flow velocity in the channel.

In open-channel flow, the energy slope, which under steady flow is the same as friction,

bed, or bottom slope, is:

hf V 2

f V 2

S = ______

= f _______

= ___

_______

L 8gR 8 gR

(1-30)

For any arbitrary cross-sectional shape, the Froude number is:

V

F = __________

(gD)1/2

(1-31)

in which D = hydraulic depth: D = A /T.

Equation 1-30 can be expressed in terms of the Froude number as follows:

f D

S = ___

_____

F 2

8 R

(1-32)

Equation 1-32 states the proportionality between energy slope and Froude number,

with the proportionality factor being a function of the Darcy-Weisbach friction factor and

the shape factor D /R.

For a hydraulically wide channel, for which D ≅ R, Eq. 1-32 reduces to:

f

S = ___

F 2

8

(1-33)

Thus, for a hydraulically wide channel, the proportionality factor between energy slope

and Froude number is only a function of the Darcy-Weisbach friction factor. In essence,

for application to open-channel flow, a modified Darcy-Weisbach friction factor f, equal

to 1/8 of the conventional Darcy-Weisbach friction factor f is applicable. The modified

Darcy-Weisbach equation for open-channel flow is:

S = f F 2

(1-34)

Table 1-2 shows approximate values of Darcy-Weisbach friction factor f and

corresponding modified friction factor f for selected values of R in the turbulent range.

Table 1-2 Approximate values of friction factors f and f for selected values of R in the turbulent range.

R f f

2000 0.036 0.0045

4000 0.032 0.004

7000 0.028 0.0035

10000 0.024 0.003

15000 0.020 0.0025

60000 0.016 0.002

On the proportionality between energy slope and Froude number

Equation 1-33 reveals a fundamental property of open-channel flow in the turbulent

range: The proportionality between energy slope and Froude number. The following

relations hold:

For f constant, the energy slope S will increase proportionally with an

increase in Froude number F, and vice versa.

For F constant, the energy slope S will increase proportionally with an

increase in friction factor f, and vice versa.

For S constant, the Froude number F will increase proportionally with

adecrease in friction factor f, and vice versa.

QUESTIONS

[Problems] [References] • [Top] [Open-channel Flow] [Types of Flow] [State of Flow] [Flow Regimes]

1. What is the difference between flow depth d and hydraulic depth D?

2. Is unsteady uniform flow possible? Why not?

3. What four forces are acting in the momentum balance in open-channel flow?

4. What is a kinematic wave?

5. What is a dynamic wave according to Lagrange?

6. What is the Froude number?

7. What is the Vedernikov number?

8. What is neutrally stable flow?

9. What is the normal range in the exponent β of the discharge-flow area rating in

open-channel and free-surface flow?

10. What type of friction is described by β = 2?

11. For what condition can the value of β be less than 1?

12. What is the Reynolds number?

13. What is the commonly estimated range of Reynolds numbers for the

transitional regime in open-channel flow?

14. What is the dimensionless wavenumber?

15. Do kinematic waves attenuate?

16. Do dynamic waves attenuate?

17. Do mixed kinematic-dynamic waves attenuate?

18. Which type of open-channel flow wave attenuates the most?

19. For what value of dimensionless wavenumber is the attenuation rate of

mixed kinematic-dynamic waves a maximum?

20. What is the dimensionless relative kinematic wave celerity under Chezy

friction for a hydraulically wide channel?

21. What is the characteristic reach length?

22. What is the hydraulic diffusivity?

23. For what value of channel slope is the hydraulic diffusivity a maximum?

24. What is the dynamic hydraulic diffusivity?

25. Under what flow condition does the dynamic hydraulic diffusivity vanish?

26. In hydraulic engineering practice, what does the term "dynamic wave"

commonly refer to?

27. What is the modified Darcy-Weisbach formula, applicable to open-channel

flow?

28. What is the typical range of Darcy-Weisbach friction factor f in the turbulent

range?

29. What is the typical range of the modified Darcy-Weisbach friction factor f in

the turbulent range?

30. Is there a theoretical upper limit to the Froude number? How can it be

calculated?

PROBLEMS

[References] • [Top] [Open-channel Flow] [Types of Flow] [State of Flow] [Flow Regimes] [Questions]

1. Derive the expression for the angle θ in a circular channel (culvert) as a function

of flow depth y and diameter D, where D = 2r (Fig. 1-14).

Fig. 1-14 Definition sketch for a circular channel.

2. Show that the maximum discharge in a circular channel (Fig. 1-14) is attained

at y = 0.94 D. UseONLINE CHANNEL 03. Explain the reason for this behavior.

3. Derive the formula for flow area A, wetted perimeter P, and top width T for a

trapezoidal channel, in terms of flow depth y, bottom width b and side slope z H:

1 V (Fig. 1-15).

Fig. 1-15 Definition sketch for a rectangular channel.

4. Assume a trapezoidal channel of flow depth y, bottom width b, and size

slope z H: 1 V (Fig. 1-15).Derive an expression for the width-to-depth ratio b/y as

a function of α, the ratio of hydraulic depthD to flow depth y . Given z = 1, and α =

0.99, what is the value of b/y?

5. Calculate the dimensionless relative kinematic wave celerity under Manning

friction for a hydraulically wide channel.

6. What is the value of β for a hydraulically wide channel at Froude number F =

1.8 and Vedernikov number V = 0.9?

7. A hydraulically wide channel has flow depth = 1 m, and flow velocity = 1.5 m/s.

Calculate the Froude number. Confirm with ONLINE FROUDE.

8. A hydraulically wide channel has an exponent of the rating β = 1.6. The flow

depth is 1 m, and the flow velocity 1.5 m/s. Calculate the Vedernikov number.

Confirm with ONLINE VEDERNIKOV.

9. Given a hydraulically wide channel with Manning friction, and mean velocity u = 1

m/s, flow depth d= 1 m, and bottom slope S = 0.001. Determine the kinematic

and dynamic hydraulic diffusivities.

10. Given a hydraulically wide channel with Chezy friction, and flow depth d = 1

m. What mean velocity will make the dynamic hydraulic diffusivity vanish?

11. According to the Blasius formula, what is the Reynolds

number R corresponding to a Darcy-Weisbach friction factor f = 0.03?

12. According to the Prandt-von Karman formula, what is the Reynolds

number R corresponding to a Darcy-Weisbach friction factor f = 0.03?

13. A WES spillway downstream slope is 0.6 H to 1 V. The Darcy-Weisbach

friction factor is f = 0.03.Calculate the maximum possible Froude number for

these conditions.

14. Assuming a maximum possible Froude number F = 25, calculate the value

of β that will assure neutrally stable flow.

15. A prismatic channel is flowing close to critical flow. The bottom slope is S =

0.004. What is the value of the Darcy-Weisbach's modified friction factor f?

REFERENCES

• [Top] [Open-channel Flow] [Types of Flow] [State of Flow] [Flow Regimes] [Questions] [Problems]

Chow, V. T. 1959. Open-channel Hydraulics. Mc-Graw Hill, New York.

Craya, A. 1952. The criterion for the possibility of roll wave formation. Gravity

Waves, Circular 521, 141-151, National Institute of Standards and Technology,

Gaithersburg, Md.

Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkoswki. 1982. Hydrodynamic

derivation of storage parameters in the Muskingum model. Journal of Hydrology,

54, 371-387.

Fread, D. 1993. "Flow Routing," Chapter 10 in Handbook of Hydrology, D. R.

Maidment, editor, McGraw-Hill, New York.

Hayami, I. 1951. On the propagation of flood waves. Bulletin, Disaster Prevention

Research Institute,No. 1, December.

Lagrange, J. L. de. 1788. Mécanique analytique, Paris, part 2, section II, article 2, p

192.

Lighthill, M. J., and G. B. Whitham. 1955. On kinematic waves: I. Flood movement in

long rivers.Proceedings, Royal Society of London, Series A, 229, 281-316.

Ponce, V, M., and D. B. Simons. 1977. Shallow wave propagation in open-channel

flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, December, 1461-

1476.

Ponce, V. M. 1989. Engineering Hydrology: Principles and Practices. Prentice-Hall,

Englewood Cliffs, New Jersey.

Ponce, V. M. 1991a. The kinematic wave controversy. Journal of Hydraulic

Engineering, ASCE, Vol. 117, No. 4, April, 511-525.

Ponce, V. M. 1991b. New perspective on the Vedernikov number. Water Resources

Research, Vol. 27, No. 7, 1777-1779, July.

Ponce, V. M., and P. J. Porras. 1995. Effect of cross-sectional shape on free-surface

instability. Journal of Hydraulic Engineering, ASCE, Vol. 121, No. 4, April, 376-380.

Powell, R. W. 1948. Vedernikov's criterion for ultra-rapid flow. Transactions, American

Geophysical Union, Vol. 29, No. 6, 882-886.

Seddon, J. A. 1900. River hydraulics. Transactions, ASCE, Vol. XLIII, 179-243, June.

Vedernikov, V. V. 1945. Conditions at the front of a translation wave disturbing a steady

motion of a real fluid, Dokl. Akad. Nauk SSSR, 48(4), 239-242.

Vedernikov, V. V. 1946. Characteristic features of a liquid flow in an open

channel, Dokl. Akad. Nauk SSSR, 52(3), 207-210.

http://openchannelhydraulics.sdsu.edu

Appendices:

Craya, A. 1952. The criterion for the possibility of roll-wave formation. Gravity Waves, National

Bureau of Standards Circular No. 521, National Bureau of Standards, Washington, D.C. 141-

151.

Excerpt of:

Seddon, J. A. 1900. River Hydraulics. Transactions, American Society of Civil Engineers, Vol.

XLIII, 179-243, June.

Excerpt of:

Seddon, J. A. 1900. River Hydraulics. Transactions, American Society of Civil Engineers, Vol.

XLIII, 179-243, June.

Excerpt of:

Seddon, J. A. 1900. River Hydraulics. Transactions, American Society of Civil Engineers, Vol.

XLIII, 179-243, June.

[Kinds of Open Channels] [Channel Geometry] [Velocity Distribution] [Measurements of Velocity] [Velocity

Distribution Coefficients] [Pressure Distribution] [Questions] [Problems] [References] •

CHAPTER 2: PROPERTIES OF OPEN CHANNELS

2.1 KINDS OF OPEN CHANNELS

[Channel Geometry] [Velocity Distribution] [Measurements of Velocity] [Velocity Distribution

Coefficients] [Pressure Distribution] [Questions] [Problems] [References] • [Top]

In general, there are two kinds of open channels: (1) natural, and (2) artificial. Natural

channels are formed by Nature through geologic, geomorphologic, and hydrologic

action. They include all natural watercourses, from the smallest to the largest. An

example of a small watercourse is the small stream or rivulet shown in Fig. 2-1. An

example of a large watercourse is the large river shown in Fig. 2-2. The largest natural

channel is the Amazon river at its mouth, shown in Fig. 2-3. Applications of open-

channel hydraulics in natural streams are typically in the fields of flood control,

navigation, and stream restoration.

Fig. 2-1 A small stream, or rivulet, upstream of Sheep Creek Barrier Dam, Utah.

Nuccitelli

Fig. 2-2 The Mississippi river at Mud Island, Memphis, Tennessee.

Fig. 2-3 The Amazon river near its mouth, at Macapa, Amapa, Brazil.

Artificial channels are also referred to as canals. They are made by humans for a

specific purpose, usually to transfer water from a place where it exists in ample supply

to a place where it is in short supply, or to transfer water between two locations to

satisfy daily, monthly, seasonal, or annual requirements. Artificial channels are also

built for flood control and navigation.

Artificial canals have been built by ancient societies, usually to transfer water for

irrigation or water supply. For instance, Fig. 2-4 shows a small canal built about 3,500

years ago by the ancient people of Cajamarca, Peru.

Fig. 2-4 The Cumbemayo Canal, near Cajamarca, Peru.

Artificial channels can be referred to with various names, depending on their features

and intended use:

Canal: Usually long, generally of mild slope, lined or unlined, ground supported.

The lining can be of various materials, such as concrete, masonry, asphalt, or

wood (Fig. 2-5).

Fig. 2-5 Irrigation canal, Wellton-Mohawk Irrigation District, Wellton, Arizona.

Flume: An open-channel conduit, supported above ground, either full scale or

laboratory scale. Llining can be of various materials, such as metal, wood, or

acrylic plastic (Fig. 2-6).

Fig. 2-6 The Dulzura conduit, in San Diego County, California,

overflowing after heavy rain, on March 5, 2005.

Chute: A canal of very steep slope (Fig. 2-7).

Fig. 2-7 Chute at Taymi Canal, Lambayeque, Peru.

Drop: A chute within a very short distance, usually to follow the topography (Fig.

2-8).

Fig. 2-8 Drop in an irrigation canal, Arequipa, Peru.

Aqueduct: A canal to transport water for a specific use, typically over terrain, or

elevated over a valley, stream or road (Fig. 2-9).

Fig. 2-9 Wari aqueduct at Sumaq Tika, Cuzco, Peru, dated c. 1000 A.D.

Culvert: A covered conduit (or conduits) of relatively short length, usually flowing

partially full, to enable a stream to cross a highway or other embankment (Fig. 2-

10).

Fig. 2-10 Culvert crossing the U.S.-Mexico border at Yoghurt Canyon, California.

Uses of Open Channels

Open channels, both natural and artificial, are used in various water resources fields,

including:

Irrigation and drainage, Flood control, Urban drainage, Hydropower generation, River navigation, Urban water supply, Wastewater disposal, and Stream restoration.

Figure 2-11 shows a fish-passage channel at a pond-and-plug stream restoration site in

Ferris Creek, Plumas County, California.

Fig. 2-11 Fish-passage channel at restored Ferris Creek, Plumas County, California.

Related Fields

Other related fields that benefit directly from open channel hydraulics knowledge

include:

Hydrology: The study of water in the hydrologic cycle.

Fluvial geomorphology: The study of the genesis and shape of streams and

rivers.

River mechanics: The study of the mechanical properties and behavior or

rivers.

Fluvial sedimentology: The study of fluvial sediments, including their source,

transport, and fate.

Hydroclimatology: The study of climate and the hydrologic cycle.

Ecohydrology: The study of the interaction between water and vegetation in

the hydrologic cycle.

Potamology: The scientific study of rivers.

2.2 CHANNEL GEOMETRY

[Velocity Distribution] [Measurements of Velocity] [Velocity Distribution Coefficients] [Pressure

Distribution] [Questions] [Problems] [References] • [Top] [Kinds of Open Channels]

Natural channels refer to the great number of streams and rivers around the world. With

respect to longitudinal orientation, natural channels are classified as: (a) straight, (b)

meandering, or (c) braided.

A straight channel is one that follows more or less a straight path. Conversely, a

meandering channel curves around in plan view, as shown in Fig. 2-12. The ratio of

stream length to valley length is referred to assinuosity. The sinuosity of the stream

shown is about 5.

Fig. 2-12 A meandering channel: Humea river, Meta department, Colombia.

A braided channel features multiple interconnected meandering subchannels (Fig. 2-

13). The flow and sediment conditions under which a channel assumes a certain

longitudinal shape is a subject of fluvial geomorphology.

Fig. 2-13 A braided channel: Guatiquia river, Meta department, Colombia.

During floods, natural channels may overflow their banks and temporarily take up

significant portions of the adjacent floodplain (Fig. 2-14). With respect to flood stage,

flow in natural channels is classified as: (1) inbank, when the flow remains in the main

channel, or (2) overbank, when the flow spills over to the adjacent floodplains.

Fig. 2-14 A river at flood stage, overflowing its banks.

Figure 2-15 shows the typical variation of stage and flood wave travel time K with

extent of inundation. These variations are due to relative increases or decreases of

surface bottom friction as the stage increases.

Fig. 2-15 Schematic of the typical variation of flood wave travel time with flood stage.

Artificial channels are classified as either: (1) prismatic, or (2) nonprismatic. Prismatic

channels have a constant cross section, either trapezoidal, rectangular, or triangular.

Trapezoidal channels are the most common type (Fig. 2-16). Rectangular channels are

often used in sanitary flows and laboratory applications. Triangular channels are

normally used in road and urban drainage.

Fig. 2-16 Tinajones feeder canal, Lambayeque, Peru.

Nonprismatic channels are those for which the cross section is not constant.

Nonprismatic channels are used in the transition from a cross section of one size to

another of a different size, usually to account for the variability in bottom slope.

Hydraulically wide channel

There are three asymptotic cross-sectional shapes in prismatic channels (Section 1.3):

1. Hydraulically wide, for which the wetted perimeter is a constant, 2. Triangular, for which the top width is proportional to the flow depth (Fig. 1-9), and 3. Inherently stable, for which the hydraulic radius is a constant (Fig. 1-10).

A channel is hydraulically wide when the wetted perimeter P can be approximated by

the top width T. By definition, the hydraulic radius R is equal to the flow area A divided

by the wetted perimeter P. Likewise, the hydraulic depth D is equal to the flow

area A divided by the top width T. Therefore, the following relations hold for a

hydraulically wide channel:

P ≅ T

(2-1)

R ≅ D

(2-2)

In practice, a channel may be considered hydraulically wide if the top width T is greater

than or equal to 10 times the hydraulic depth D (Chow, 1959):

T _____

≥ 10

D

(2-3)

In general:

Q = V A

(2-4)

in which Q = discharge, A = flow area, and V = mean velocity. Since by definition:

A = D T

(2-5)

it follows that:

Q = V D T

(2-6)

and:

q = V D

(2-7)

in which q = discharge per unit of channel width, or unit-width discharge [L2 T

-1].

For simplicity, a hydraulically wide channel may be analyzed in terms of its unit-width

discharge. For a hypothetical unit-width channel, the wetted perimeter is a

constant: P = T = 1. Therefore, a hydraulically wide channel is alternatively defined as

that for which the wetted perimeter is a constant. In practice, most natural channels are

hydraulically wide (Fig. 2-17).

Fig. 2-17 The Apa river, border between Brazil and Paraguay.

Hydraulics of the cross section

The discharge-flow area rating in open-channel flow is:

Q = α Aβ

(2-8)

The asymptotic values of β are (Section 1.3):

β = 5/3 for a hydraulically wide channel with turbulent Manning friction,

β = 3/2 for a hydraulically wide channel with turbulent Chezy friction,

β = 4/3 for a triangular channel with turbulent Manning friction,

β = 5/2 for a triangular channel with turbulent Chezy friction, and

β = 1 for an inherently stable channel.

For trapezoidal channels, values of β lie between the values for hydraulically wide and

triangular channels. Typical values of β for trapezoidal channels lie approximately in the

range 1.35 ≤ β ≤ 1.65 (Fig. 2-18) (Ponce and Windingland, 1985).

Fig. 2-18 The San Luis Canal, near Coalinga, in the Central Valley, California.

Depth of flow and depth-of-flow section

There are two related depths of open-channel flow (Fig. 2-19):

1. The depth of flow y, and

2. The depth-of-flow section d.

The depth of flow y is the vertical distance from the free surface to the lowest point in

the channel cross section. The depth-of-flow section d is the distance, normal to the

direction of flow, measured from the free surface to the lowest point in the channel

cross section.

Fig. 2-19 Definition sketch for depth of flow y and depth of flow section d.

The relation between depth of flow y and depth of flow section d is:

d = y cos θ

(2-9)

Geometric elements

Table 2-1 shows geometric elements for four commonly used cross-sectional shapes.

Table 2-1 Geometric elements of four commonly used cross-sectional shapes.

Shape (Click on figure

to enlarge)

Area A

Wetted perimeter

P

Top width

T

Hydraulic radius

R

Hydraulic depth

D

by b + 2y b by

__________

b + 2y

y

(b + zy )y b + 2y (1 + z 2)1/2 b + 2zy

(b + zy )y ____________________

b + 2y (1 + z

2)1/2

(b + zy )y ____________

b + 2zy

zy2 2y (1 + z

2)1/2 2zy

zy _______________

2 (1 + z

2)1/2

y ___

2

(θ - sin θ) do2

________________

8

θ do ______

2

do sin (θ/2) (θ - sin θ) do

________________

4 θ

(θ - sin θ) do ________________

8 sin (θ/2)

2.3 VELOCITY DISTRIBUTION

[Measurements of Velocity] [Velocity Distribution Coefficients] [Pressure

Distribution] [Questions] [Problems] [References] • [Top] [Kinds of Open Channels] [Channel Geometry]

The mean velocity in open-channel flow is:

Q

V = _____

A

(2-10)

Local velocities, however, are likely to vary widely. Vertical velocity profiles vary from

zero at the boundary (the no-slip condition at the channel bottom) to values slightly

greater than the mean near the water surface (Fig. 2-20).

Fig. 2-20 Shape of the vertical velocity profile.

Transversal velocities are also likely to vary widely, from zero at the side boundaries to

maximum values at or near the middle of the channel (Fig. 2-21). In meandering

channels, or channels of curved alignment, velocities are larger along the outside of the

bend and smaller on the inside. However, for hydraulically wide channels, the sides

have a negligible influence on the flow. In this case, the flow is effectively two-

dimensional (in the longitudinal and vertical directions).

Fig. 2-21 Typical curves of equal velocity (isovels) for various channel shapes (Chow, 1959).

The assumption of mean velocity V (Eq. 2-10) further reduces the flow to a statement of

one-dimensionality (in the longitudinal direction). This assumption is strictly applicable

to channels that are relatively straight and for which flows remain inbank. As a

convenient approximation, the one-dimensional assumption has been applied to

channels with mild meandering tendencies and limited overbank flows. However, the

one-dimensional assumption generally breaks down for channels of large sinuosity

and/or substantial overbank flows (Fig. 2-22).

Fig. 2-22 A sinuous channel, Meta river, Colombia.

2.4 MEASUREMENTS OF VELOCITY

[Velocity Distribution Coefficients] [Pressure Distribution] [Questions] [Problems] [References] • [Top] [Kinds

of Open Channels] [Channel Geometry] [Velocity Distribution]

Measurements of velocity are commonly performed using a current meter. Current

meters measure flow velocity by counting the number of revolutions per second of the

meter assembly. The rotation can be around a vertical axis, leading to the cup meter, or

around a horizontal axis, leading to the propeller meter.

Cup meters are widely used in the United States. The most common type of cup meter

is the Price current meter, which has six cups mounted on a vertical axis (Fig. 2-23).

The flow velocity is proportional to the angular velocity of the meter rotor. The flow

velocity is determined by counting the number of revolutions per second of the rotor

and consulting the meter calibration table.

U.S. Geological Survey

Fig. 2-23 The Price current meter.

Discharge measurements. A discharge measurement at a stream cross section

requires the determination of flow area and mean velocity for a given stage. The cross

section should be perpendicular to the flow, and mean velocity should be based on a

sufficient number of velocity measurements across the section.

In a typical stream-gaging procedure, each of several depth soundings, usually 20 to

30, defines the position of a vertical (Fig. 2-24). Each depth sounding is associated with

a partial section of the stream. A partial section is a rectangle of depth equal to the

sounding and of width equal to half the difference of the distances to adjacent verticals.

At each vertical, the following observations are made:

1. The flow depth, and 2. The velocity as measured by a current meter at one or two points along the

vertical.

U.S. Geological Survey

Fig. 2-24 Streamgaging procedure.

In the two-point method, the current meter is positioned at 0.2 and 0.8 of the flow depth.

In the one point method, the current meter is positioned at 0.6 of the flow depth,

measured from the water surface. The average of the velocities at 0.2 and 0.8 depth or

the single velocity at 0.6 depth is taken as the mean velocity in the vertical. Where a

two-point measurement is impractical (e.g., in very shallow streams), the one-point

method is recommended.

For each partial section, the discharge is calculated as:

q = v a (2-11)

in which q = discharge, v = mean velocity, and a = flow area. The total stream

discharge Q is the sum of the discharges of each partial section.

Price current meters types AA and A are used for two-point velocity measurements in

streams with flow depths above 0.75 m and for one-point measurements in streams

with depths ranging from 0.45 to 0.75 m. The dwarf (or pigmy) current meter is used for

one-point measurements in shallow streams or laboratory flumes with depths in the

range 0.10 to 0.45 m.

Techniques for measuring stream velocity with a current meter vary with stream size. If

the stream is wadable, the meter is affixed to a graduated depth rod. If the stream is too

deep to wade, the meter is suspended on a cable and is held in the water with a

sounding weight. The weights are made of various sizes, from 6.8 kg to 135 kg.

Measurements using cable suspension are made from bridges, cableways, or boats

(Fig. 2-25). For the heavier sounding weights or when using a boat, a sounding reel

may be required.

Fig. 2-25 USGS streamgaging station, Campo Creek at Campo Road,

San Diego County, California.

2.5 VELOCITY DISTRIBUTION COEFFICIENTS

[Pressure Distribution] [Questions] [Problems] [References] • [Top] [Kinds of Open Channels] [Channel

Geometry] [Velocity Distribution] [Measurements of Velocity]

Due to the nonuniform velocity distribution over a cross section, the true velocity

head hv is greater than the velocity head computed based on the mean velocity V.

Generally, the true velocity head is:

V 2

hv = α ____

2g

(2-12)

where in this case α refers to the energy coefficient, or Coriolis coefficient. Values

of α range from about 1.03 to 1.36 for fairly straight prismatic channels. The higher

values correspond to smaller channels and the lower values to larger channels.

Similarly to the case of energy, momentum also requires a correction due to the

nonuniform velocities (Fig. 2-26). The true expression for momentum flux (force) is:

F = β ρ Q V

(2-13)

where in this case β refers to the momentum coefficient, or Boussinesq coefficient;

and ρ = mass density of water. Values of β range from about 1.01 to 1.12 for fairly

straight prismatic channels.

Values of α and β are slightly greater than 1, with α being always greater than β. For

channels of composite cross section, values of α and β can easily get as great as 1.6

and 1.2, respectively (Figs. 2-26 and 2-27). In channels of irregular

alignment, α and β may vary widely. In extreme cases, values of β > 2 have been

observed (Chow, 1959).

Fig. 2-26 A compound channel cross section, featuring a high value of Coriolis coefficient α.

The Coriolis coefficient

Assume:

A = total area of the cross section ... [L2]

V = mean velocity of the cross section ... [L T -1

] ΔA = incremental area ... [L

2]

v = velocity through incremental area ΔA ... [L T -1

]

The mean velocity is defined as:

Σ v ΔA

V = ____________

... [L T -1

]

Σ ΔA

(2-14)

In general, energy is equal to a force integrated over a distance, or force times length:

E = ∫ F ds = F L ... [F L]

(2-15)

The kinetic energy for the total area A is equal to mass M times acceleration times

distance:

1

E = ____

M V 2 ... [M L

2 T

-2]

2

(2-16)

The kinetic energy for the incremental area ΔA is:

1

ΔE = ____

m v 2

... [M L 2

T -2

]

2

(2-17)

The velocity head through ΔA is the kinetic energy divided by the weight, where weight

= mg :

v 2

hv = ______

... [L]

2 g

(2-18)

The volumetric flux through ΔA is:

ΔQ = v ΔA ... [L 3

T -1

]

(2-19)

The weight flux through ΔA is:

Δ(WF) = γ v ΔA ... [F T -1

]

(2-20)

in which γ = unit weight of water.

The incremental kinetic energy flux through ΔA is equal to the kinetic energy per

weight hv (Eq. 2-18) times the incremental weight flux Δ(WF) (Eq. 2-20):

γ v 3

ΔA

Δ(EF) = ___________

... [F L T -1

]

2 g

(2-21)

The sum of kinetic energy fluxes for all incremental areas is:

γ v 3

ΔA

∑ Δ(EF) = ∑ ___________

... [F L T -1

]

2 g

(2-22)

The kinetic energy flux through the total area A, based on mean velocity V, is:

γ V 3

A

(EF) = α ___________

... [F L T -1

]

2 g

(2-23)

Equating Eqs. 2-22 and 2-23, and solving for the Coriolis energy coefficient:

Σ v 3

ΔA

α = _____________

V 3

A

(2-24)

The Boussinesq coefficient

The mass flux J through the total area A is equal to the mass density ρ times the

volumetric flux Q = V A:

J = ρ V A ... [M T -1

]

(2-25)

The mass flux through the incremental area ΔA is equal to the mass density ρ times the

incremental volumetric flux ΔQ = v ΔA:

ΔJ = ρ v ΔA ... [M T -1

]

(2-26)

In general, momentum M is equal to a force integrated over a period of time, or mass

times velocity:

M = ∫ F dt = m V ... [M L T -1

]

(2-27)

The momentum flux, or force F, is equal to the mass flux times the velocity. Thus, the

momentum flux through the incremental area ΔA is:

ΔF = ρ v 2

ΔA ... [F = M L T -2

]

(2-28)

The sum of all momentum fluxes, or sum of forces for all the incremental areas is:

∑ ΔF = ∑ ρ v 2

ΔA ... [F]

(2-29)

The momentum flux, or force F, through the total area is:

F = β ρ V 2

A ... [F]

(2-30)

Equating Eqs. 2-29 and 2-30, and solving for the Boussinesq momentum coefficient:

Σ v 2

ΔA

β = _____________

V 2

A

(2-31)

Fig. 2-27 A channel cross section at high flood stage,

featuring a large value of Boussinesq coefficient β.

Approximate formulations of α and β

While Eqs. 2-24 and 2-31 are accurate, they require detailed measurement of the

velocity distribution in the cross section of interest. Alternatively, when only the mean

velocity V and maximum velocity Vmax are known, the following formulas may be used

to calculate approximate values of the Coriolis and Boussinesq coefficients (Chow,

1959). Defining:

Vmax

ε = ________

- 1

V

(2-32)

Assuming a logarithmic velocity distribution, the coefficients may be approximated as:

α = 1 + 3ε 2

- 2ε 3

(2-33)

β = 1 + ε 2

(2-34)

On the other hand, assuming a linear velocity distribution:

α = 1 + ε 2

(2-35)

ε 2

β = 1 + _____

3

(2-36)

2.6 PRESSURE DISTRIBUTION

[Questions] [Problems] [References] • [Top] [Kinds of Open Channels] [Channel Geometry] [Velocity

Distribution] [Measurements of Velocity] [Velocity Distribution Coefficients]

The pressure distribution along the flow depth in an open channel varies with the shape

of the channel bottom relative to the shape of the water surface, measured along the

direction of flow. There are three possible cases: (1) parallel flow, (2) convex curvilinear

flow, and (3) concave curvilinear flow.

Parallel flow

Under parallel flow, the channel bottom is relatively straight and it is either: (a) parallel

to the water surface, as with uniform flow (Fig. 2-28), or (b) approximately parallel to the

water surface, as with gradually varied flow. In this case, the pressure distribution is

essentially hydrostatic, with the pressure varying as a linear function of partial flow

depth (line AB in Fig. 2-28). At the partial flow depth hs, the pressure is: ps = γ hs. In

other words, a piezometer located at the partial depth hs below the water surface would

rise to the elevation of the water surface.

Fig. 2-28 Parallel flow.

In practice, uniform flow and gradually varied flow feature a hydrostatic pressure

distribution (Fig. 2-29). This distribution is characterized by velocities and accelerations

which are either zero or negligible in the (almost vertical) direction normal to the flow.

Fig. 2-29 Uniform flow in an open channel.

Convex curvilinear flow

Under convex curvilinear flow, the channel bottom is substantially curved, featuring a

convex profile in the vertical plane (Fig. 2-30). In this case, the pressure distribution

is nonhydrostatic, with the pressure along the flow depth varying as a nonlinear

function of partial flow depth (curve AB' in Fig. 2-30). A piezometer located at the partial

depth hs below the water surface would rise to an elevation which is lower than the

water surface elevation. The true piezometric head is: h = hs - c. The value of c is

negative because the centrifugal forces are pointing upward, reducing the action of

gravity .

Fig. 2-30 Convex curvilinear flow.

In practice, rapidly varied flow, e.g., flow over the crest of an ogee spillway, features a

nonhydrostatic pressure distribution (Fig. 2-31). This distribution is characterized by

substantial velocity components and accelerations in the direction normal to the flow.

Fig. 2-31 Ogee emergency spillway, El Capitan Dam, San Diego County, California.

Concave curvilinear flow

Under concave curvilinear flow, the channel bottom is substantially curved, featuring a

concave profile in the vertical plane (Fig. 2-32). In this case, the pressure distribution

is nonhydrostatic, with the pressure along the flow depth varying as a nonlinear

function of partial flow depth (curve AB' in Fig. 2-32). A piezometer located at the partial

depth hs below the water surface would rise to an elevation which is higher than the

water surface elevation. The true piezometric head is: h = hs + c. The value of c is

positive because the centrifugal forces are pointing downward, adding to the action of

gravity .

Fig. 2-32 Concave curvilinear flow.

In practice, rapidly varied flow, e.g., flow at or near the toe of a spillway, features a

nonhydrostatic pressure distribution (Fig. 2-33). This distribution is characterized by

substantial velocity components and accelerations in the direction normal to the flow.

Fig. 2-33 Concave curvilinear flow at spillway toe, Cresta dam,

North Fork Feather River, Northern California.

Centrifugal pressure head

The centrifugal pressure head c can be approximately calculated as follows:

The centrifugal force is equal to the mass times the centrifugal acceleration

(f = m × a).

The centrifugal pressure pc is equal to the mass per unit of area times the

centrifugal acceleration.

The centrifugal pressure head c is equal to the centrifugal pressure divided by the

unit weight of water (γ).

The mass per unit of area is:

Mass/Area = ρ (Volume/Area) = ρ d = (γ/g) d ... [M L -2

= F T 2

L -3

]

(2-37)

in which d = flow depth.

The centrifugal acceleration is:

V 2

a = ______

... [L T -2

]

r

(2-38)

in which r = radius of curvature of the streamlines.

Thus, the centrifugal pressure is Eq. 2-37 times Eq. 2-38:

γd V 2

pc = _____

_____

... [F L -2

]

g r

(2-39)

and the centrifugal pressure head is:

d V 2

c = ____

_____

... [L]

g r

(2-40)

The rise is negative for convex curvilinear flow and positive for concave curvilinear flow

(Fig. 2-34).

Fig. 2-34 Concave curvilinear flow (ski jump), Tucurui spillway, Para, Brazil.

Effect of slope on pressure distribution

Figure 2-35 shows a schematic of the pressure distribution in a channel of steep

slope θ. Note the following:

The weight of the shaded element is: γ d [dL (1)] = γ y cosθ [dL (1)]

The pressure exerted by the shaded element is: γ d cosθ = γ y cos2θ

The pressure rise is:

h = d cos θ = y cos2θ ... [L]

(2-41)

For a channel of small slope, h ≅ y, and the vertical depth y may be used to estimate

the pressure rise h. However, for a channel of large slope, the difference may be

substantial.

A 10% slope (i.e., a slope equal to 0.1) corresponds to an angle θ = 5.7°, for which

cos2 (5.7°) = 0.99. Thus, for a slope greater than 10%, the error in taking the vertical

depth y in lieu of the pressure rise h is more than 1%. A channel with slope greater than

10% is referred to as a channel of large slope.

Fig. 2-35 Pressure distribution in a channel of steep slope.

In practice, a channel of large slope features high velocities, which may lead to air

entrainment, volume swell, and consequent increase in depth and decrease in density.

Thus, the pressure calculated by Eq. 2-41 may be somewhat higher that the actual

value.

QUESTIONS

[Problems] [References] • [Top] [Kinds of Open Channels] [Channel Geometry] [Velocity

Distribution] [Measurements of Velocity] [Velocity Distribution Coefficients] [Pressure Distribution]

1. What is a chute?

2. What is an aqueduct?

3. What is a culvert?

4. What is potamology?

5. What is sinuosity in connection with rivers?

6. What is a hydraulically wide channel?

7. What ratio of top width to hydraulic depth can be considered hydraulically wide?

8. What is the typical range of the exponent β for trapezoidal channels?

9. What is the most common type of current meter in the United States?

10. Why is the true velocity head greater than the velocity head computed

based on mean velocity?

11. How does energy differ from momentum?

12. How does the pressure distribution in the vertical vary under parallel flow?

13. How does the pressure distribution in the vertical vary under convex

curvilinear flow?

14. How does the pressure distribution in the vertical vary under concave

curvilinear flow?

15. What is a channel of large slope?

PROBLEMS

[References] • [Top] [Kinds of Open Channels] [Channel Geometry] [Velocity Distribution] [Measurements of

Velocity] [Velocity Distribution Coefficients] [Pressure Distribution] [Questions]

1. What is the discharge per unit of width if the discharge is 24 m3/s and the channel

top width isT = 8 m?

2. Given: coefficient of the rating α = 0.4; exponent β = 1.55; flow area A = 45.6 m2.

What is the discharge Q?

3. Given culvert diameter do = 1 m; what is the flow area for θ = 300°? (See Table 2-

1).

4. What is the force F developed by a discharge Q = 10 ft3/s at a velocity V = 1 ft/s,

at a cross section with a Boussinesq coefficient β = 1.05?

5. A stream of flow area A = 100 m2 is divided into three sections: (1) left overbank,

with 20% of the flow area, and velocity 0.2 m/s; (2) inbank center, with 70% of the

flow area, and velocity 1 m/s; and (3) right overbank, with 10% of the flow area,

and velocity 0.1 m/s. Calculate the Coriolis coefficient α and the Boussinesq

coefficient β.

Fig. 2-36 A composite cross section.

6. Calculate the velocity distribution coefficients for the following canal data, with

flow area A = 2768 ft2.

Increment Velocity v

(ft/s)

Incremental

flow area ΔA (%)

1 3.5 0.5

2 4.0 2.9

3 4.5 10.3

4 5.0 23.5

5 5.5 52.7

6 6.0 10.1

7. Compare the results with those of approximate logarithmic formulas.

8. A spillway is designed for a discharge q = 5 m2/s with a flow depth d = 0.5 m.

What is the minimum radius of curvature of the spillway cross section to ensure

that the pressure does not fall below 50% of hydrostatic?

REFERENCES

• [Top] [Kinds of Open Channels] [Channel Geometry] [Velocity Distribution] [Measurements of

Velocity] [Velocity Distribution Coefficients] [Pressure Distribution] [Questions] [Problems]

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

Ponce, V. M., and D. Windingland. 1985. Kinematic shock: Sensitivity analysis. Journal of

Hydraulic Engineering, ASCE, Vol. 111, No. 4, April, 600-611.

http://openchannelhydraulics.sdsu.edu

[Energy Principle] [Specific Energy] [Local Phenomena] [Momentum Principle] [Specific

Force] [Questions] [Problems] [References] •

CHAPTER 3: ENERGY AND MOMENTUM

PRINCIPLES

3.1 ENERGY PRINCIPLE

[Specific Energy] [Local Phenomena] [Momentum Principle] [Specific

Force] [Questions] [Problems] [References] • [Top]

Under steady flow, mass and energy are conserved in open-channel flow. On the other

hand, under unsteady flow, mass and momentum are conserved. Energy is expressed

in FL units(energy = force × distance). Energy per unit of weight is expressed in length

units [L]. Potential and kinetic energy may be expressed in terms of head, in units of

vertical depth [L].

With reference to the channel of large slope of Fig. 3-1, the total head H at a section 0

containing point Aon a streamline may be written as follows:

VA2

H = zA + dA cos θ + ______

2g

(3-1)

in which zA = elevation of point A above a datum plane, dA = depth of point A below the

water surface, θ= slope angle of the channel bottom, and VA2/(2g) = velocity head of

the flow in the streamline passing through point A.

Fig. 3-1 Definition sketch for energy in gradually varied open-channel flow (Chow, 1959).

Due to cross-sectional nonuniformity, the velocity head is likely to vary along the flow

depth and width. In practice, at a given cross section, the velocity head is based on the

mean velocity V for that cross section, and the Coriolis coefficient (Section 2.2) is used

to account for the nonuniformity in the velocity distribution. This leads to the total

energy for the channel section:

V 2

H = z + d cos θ + α _____

2g

(3-2)

For a channel of small slope, cos θ ≅ 1. Thus, the total energy reduces to:

V 2

H = z + d + α _____

2g

(3-3)

The statement of conservation of energy between cross sections 1 and 2 leads to (Fig.

3-1):

V12 V2

2

z1 + d1 cos θ + α1 _____

= z2 + d2 cos θ + α2 _____

+ hf

2g 2g

(3-4)

in which hf = energy loss.

The line representing the total head is the energy grade line (EGL) or energy line. The

slope of the line is the energy gradient, or Sf. The slope of the water surface is denoted

by Sw and the slope of the channel bottom is So, where So = tan θ. Under uniform flow

conditions, all three slopes are the same: Sf = Sw =So.

For a channel of small slope, Eq. 3-4 reduces to:

V12 V2

2

z1 + y1 + α1 _____

= z2 + y2 + α2 _____

+ hf

2g 2g

(3-5)

When α1 = α2 ≅ 1 (a hydraulically wide channel), and hf = 0, Eq. 3-4 reduces to the

statement of conservation of energy, or Bernoulli equation:

V12 V2

2

z1 + y1 + _____

= z2 + y2 + _____

= constant

2g 2g

(3-6)

3.2 SPECIFIC ENERGY

[Local Phenomena] [Momentum Principle] [Specific

Force] [Questions] [Problems] [References] • [Top] [Energy Principle]

Equation 3-6 is applicable to a channel of small slope. In the limit, when the channel is

horizontal, or else, as an approximation, when the channel is sufficiently short, Eq. 3-6

simplifies to a statement of conservation of specific energy. In effect, for z1 ≅ z2, Eq. 3-

6 reduces to:

V12 V2

2

y1 + _____

= y2 + _____

= constant

2g 2g

(3-7)

Equation 3-7 states the constancy of specific energy in a hydraulically wide horizontal

channel. The specific energy is:

V 2

E = y + _____

2g

(3-8)

where E is the energy per (unit of) weight, measured with respect to the channel

bottom, defined in terms of the flow depth y and the velocity head V 2

/(2g).

Since V = Q /A, the specific energy in terms of discharge is:

Q 2

E = y + _________

2g A 2

(3-9)

Since Q is a constant under steady flow, and flow area A is always a unique function of

flow depth y, the specific energy for a given Q is only a function of y.

For a given cross section and discharge Q, a plot of flow depth y versus specific

energy E leads to the specific energy curve. A typical curve is shown in Fig. 3-2. For a

given Q, the specific energy curve is a type of hyperbolic function, featuring a minimum

value of E and two limbs, a lower limb AC and an upper limb BC.

Fig. 3-2 Specific energy curve (Chow, 1959).

The minimum value of E (at point C) characterizes the critical state of flow, with critical

depth. The lower limb AC, with smaller depths, describes supercritical flow. This limb is

asymptotic to the horizontal axis. The upper limb BC, with larger depths, describes

subcritical flow. This limb is asymptotic to a 45° line OD that originates at the origin (O).

At any point P on the curve, the abscissa represents the specific energy E and the

ordinate the flow depth y.

For a given specific energy, there are two points on the curve, one corresponding to

low stage (small depth) and another corresponding to high stage (large depth). These

depths are referred to as alternatedepths. At point C, the alternate depths coalesce

into one depth, termed the critical depth; at that point, the specific energy is a minimum.

When the depth of flow is greater than the critical depth, the flow is termed subcritical.

When the depth of flow is smaller than the critical depth, the flow is termed

supercritical. For a discharge Q, the specific energy curve is curve AB in Fig. 3-2. For a

smaller discharge, it is A'B'; for a larger discharge, it is A"B".

Critical flow criterion

From Fig. 3-2, critical depth (critical flow) corresponds to minimum specific energy. To

prove this assertion mathematically, differentiate Eq. 3-9 with respect to y and equate

to zero, to yield:

dE Q 2

dA ______

= 1 - _______

______

= 0

dy g A 3

dy

(3-10)

By definition, the differential change in flow area dA is equal to the top width T times the

differential change in flow depth dy (Fig. 3-2):

dA = T dy

(3-11)

Thus, Eq. 3-10 reduces to:

Q 2

Tc ________

= 1

g Ac 3

(3-12)

Since Dc = Ac /Tc , and Vc = Q /Ac , Eq. 3-12 reduces to:

Vc 2

______

= 1

g Dc

(3-13)

The left-hand side of Eq. 3-13 is effectively the square of the Froude number (Section

1-3). Thus, the condition F 2

= 1, or better yet, F = 1, properly describes the condition of

critical flow, for which the specific energy is a minimum. Alternatively, Eq. 3-13 may be

expressed as follows:

Vc 2

Dc ______

= _____

2g 2

(3-14)

which states that, under critical flow, the velocity head is equal to one-half of the

hydraulic depth. Figure 3-3 shows a channel operating at near critical flow.

Fig. 3-3 A canal operating at near critical flow.

Equation 3-14 describes the critical flow condition for a channel of small slope that is

hydraulically wide (α = 1). In general, for a channel of large slope and arbitrary cross-

sectional shape, the critical flow condition is:

Vc 2

Dc cos θ

α ______

= __________

2g 2

(3-15)

in which Dc is the hydraulic depth corresponding to critical flow, measured normal to the

channel bottom. Thus, the general critical flow condition is:

Vc 2

_________________

= 1

(g Dc cos θ) / α

(3-16)

and the general definition for Froude number, applicable to any channel, regardless of

slope and cross-sectional shape, is:

V

F = _____________________

[ (g D cos θ) / α ]1/2

(3-17)

3.3 LOCAL PHENOMENA

[Momentum Principle] [Specific Force] [Questions] [Problems] [References] • [Top] [Energy

Principle] [Specific Energy]

Changes from supercritical to subcritical flow, or from subcritical to supercritical flow,

occur frequently in open-channel flow, depending on the prevailing channel slope and

cross section. If the changes occur within a relatively short distance, they constitutes

local phenomena. The hydraulic drop and the hydraulic jump are examples of local

phenomena.

The hydraulic drop

The hydraulic drop is triggered by a sharp depression in the water surface, usually

caused by an abrupt change in bottom elevation. The free overfall shown in Fig. 3-4 is

an example of a hydraulic drop.

Fig. 3-4 A free overfall.

Flow in the proximity of a free overfall is usually rapidly varied; therefore, the brink

depth is somewhat smaller than the critical depth computed by the theory of parallel

flow. Figure 3-5 shows a schematic of the flow near the brink. The actual drawdown

curve is shown with a solid line, while the theoretical water-surface curve, assuming

parallel flow, is shown with a dashed line. For channels of small slope, the computed

critical depth is about 1.4 times the brink depth; i.e., yc ≅ 1.4 yo. The computed critical

depth is located upstream of the brink, a distance of about 3 to 4 times the critical depth

(Chow, 1959).

Fig. 3-5 Critical depth near a free overfall (Chow, 1959).

The hydraulic jump

The hydraulic jump is triggered by an abrupt rise in the water surface as the flow

progresses downstream. In a hydraulic jump the flow changes from supercritical

upstream to subcritical downstream, accompanied by an appreciable loss of energy.

The amount of energy loss depends on the upstream and downstream flow conditions.

The jump occurs frequently under one of the following situations:

1. Below a regulating sluice gate (Fig. 3-6),

2. At the toe (or foot) of a spillway, or

3. At the location along the channel where the bottom slope suddenly changes from

steep to mild.

Fig. 3-6 A hydraulic jump.

The flow depth before the jump is called the initial depth y1, while the flow after the

jump is called thesequent depth y2. The initial and sequent depths y1 and y2 are shown

on the specific energy curve of Fig. 3-7. They should be distinguished from the

alternate depths y1 and y2', which are two possible depths for the same specific energy.

The specific energy E1 at the initial depth y1 is greater than the specific energy E2 at the

sequent depthy2. The energy loss due to the hydraulic jump is the difference between

specific energies for initial and sequent depths:

ΔE = E1 - E2

(3-18)

Also shown, to the right of Fig. 3-7, is the specific force curve (Section 3.5), where the

sequent depths y1and y2 have the same specific force: F1 = F2.

Fig. 3-7 Hydraulic jump interpreted by specific energy and specific force curves (Chow, 1959).

3.4 MOMENTUM PRINCIPLE

[Specific Force] [Questions] [Problems] [References] • [Top] [Energy Principle] [Specific Energy] [Local

Phenomena]

Momentum M is equal to a force integrated over a period of time, or mass times

velocity, Eq. 2-27, repeated here for convenience:

M = ∫ F dt = m V ... [M L T -1

]

(3-19)

The momentum flux, or force F, of a flow with velocity V through a cross section of

area A, Eq. 2-30, repeated here:

F = β ρ V 2

A ... [F]

(3-20)

Since Q = VA, the momentum flux, or force F, exerted by a flow of discharge Q and

velocity V is:

F = β ρ Q V ... [F]

(3-21)

According to Newton's second law of motion, the change ΔF in momentum flux through

a control volume is equal to the resultant of all the external forces acting on the control

volume. The external forces are: (1) body force, and (2) surface forces. The body force

is the gravitational force, resolved along the direction of motion (the force

labeled W sinθ in Fig. 3-8). In general, there is a nonzero channel bottom slope θ;

otherwise, the channel bottom would be horizontal and the gravitational component

along the direction of motion would vanish.

The surface forces on the control volume are of three kinds: (1) on the bottom, (2) on

the sides, and (3) on the top. The bottom surface force is due to friction, which is

always acting in the direction opposite to the flow (the force labeled Ff in Fig. 3-8).

There is no such thing as zero friction; under certain conditions, friction may be

neglected, but it is never zero.

The side surface forces are two: one upstream, the force labeled P1 in Fig. 3-8, and

another downstream, the force labeled P2. These forces are due to the water pressure,

which is either hydrostatic under parallel flow, or nonhydrostatic under convex or

concave curvilinear flow (Section 2.6).

The top surface force is due to wind. Under the scales normally considered in open-

channel flow, wind forces are small and are usually neglected. However, wind forces

may not be negligible in cases on free-surface flow in reservoirs or flow in a wide open

space such as the ocean.

The statement of momentum (flux) conservation in a control volume is (Fig. 3-8):

ρ Q (β2V2 - β1V1) = P1 - P2 + W sin θ - Ff ... [F]

(3-22)

Or, in terms of unit weight:

(γ/g) Q (β2V2 - β1V1) = P1 - P2 + W sin θ - Ff ... [F]

(3-23)

Fig. 3-8 Definition sketch for the surface forces acting in a control volume (Chow, 1959).

Following the usual convention of mechanics, the momentum flux difference is equal to

the flux at the downstream section 2 minus the flux at the upstream section 1. The

forces acting on the control volume are positive in the flow direction and negative

against it. Equation 3-23 is known as the momentum flux balance equation or, for short,

the momentum equation.

For parallel flow in a rectangular channel of small slope and width b, the force P1 is:

P1 = (1/2) γ b y12 ... [F]

(3-24)

Similarly, the force P2 is:

P2 = (1/2) γ b y22 ... [F]

(3-25)

The weight W of the control volume (Fig. 3-8) is:

W = (1/2) γ b L (y1 + y2) ... [F]

(3-26)

The weight of the control volume, resolved along the direction of motion (Fig. 3-8), is:

W sin θ = (1/2) γ b (y1 + y2) (z1 - z2) ... [F]

(3-27)

Generally, under typical gradually varied flow conditions, the friction force Ff along the

channel bottom is about equal and opposite in sign to the gravitational

force W sin θ (Eq. 3-27). Thus, the friction force may be expressed as follows:

Ff = (1/2) γ b (y1 + y2) hf' ... [F]

(3-28)

in which hf' = head loss due to friction.

The discharge Q is:

Q = (1/2) (V1 + V2) b (1/2) (y1 + y2) ... [L3 T

-1]

(3-29)

Substituting Eqs. 3-24 to 3-29 into Eq. 3-23 and simplifying terms, the following

equation is obtained:

V12 V2

2

z1 + y1 + β1 _____

= z2 + y2 + β2 _____

+ hf'

2g 2g

(3-30)

Equation 3-30 differs from Eq. 3-5 in several important respects:

Energy is a scalar, therefore, Eq, 3-5 is a scalar equation; on the other hand,

momentum is a vector, therefore, Eq. 3-30 is derived from a vector equation.

The velocity distribution coefficients are not the same; while the Coriolis

coefficients apply to Eq. 3-5, the Boussinesq coefficients apply to Eq. 3-30.

The energy head loss hf of Eq. 3-5 measures internal losses throughout the

control volume, while the friction head loss hf' of Eq. 3-30

measures external losses due to the action of surface forces on the control

volume.

Thus, while Eqs. 3-5 and 3-30 look similar, they are not equivalent. Equation 3-5

applies to steady gradually varied flow, while Eq. 3-30 applies for unsteady gradually

varied flow. In other words, Eq. 3-30does account for the inertial forces which are

present in unsteady flow, while Eq. 3-5 does not.

In practice, the momentum principle applies to problems where forces can be shown to

play a significant role. Typically, problems of steady gradually varied flow use

conservation of energy, while problems ofunsteady gradually varied flow use

conservation of momentum. The exception is the hydraulic jump, which is rapidly

varied. Both energy and momentum principles are used in the solution of the hydraulic

jump.

3.5 SPECIFIC FORCE

[Questions] [Problems] [References] • [Top] [Energy Principle] [Specific Energy] [Local

Phenomena] [Momentum Principle]

In horizontal channels, the gravitational force resolved along the direction of motion is

effectively zero. As a convenient approximation, for nearly horizontal channels, the

gravitational force may be considered small and be neglected on practical grounds.

The frictional force develops along the channel bottom; the longer the channel, the

greater the frictional force. Thus, for a short channel, the frictional force may be taken

as sufficiently small and neglected on practical grounds. Note that the frictional force is

never zero; its neglect is only justified as an approximation, when compared with the

other forces that are present in open-channel flow.

The neglect or elimination of the gravitational and frictional forces in the momentum flux

balance reduces it to:

(γ/g) Q (β2V2 - β1V1) = P1 - P2 ... [F]

(3-31)

Assuming β1 = β2 = 1, Eq. 3-31 reduces to:

(γ/g) Q (V2 - V1) = P1 - P2 ... [F]

(3-32)

Since V1 = Q / A1, and V2 = Q / A2:

Q 2

Q 2

(γ/g) ( ______

- ______

) = P1 - P2 ... [F]

A2 A1

(3-33)

The pressure force P acting on a cross-section of area A and distance from the

centroid of the area to the water surface [Fig. 3-9 (b)] is:

P = γ A ... [F]

(3-34)

Fig. 3-9 Definition sketch for a specific force curve (Chow, 1959).

Thus:

P1 = γ 1 A1 ... [F]

(3-35)

P2 = γ 2 A2 ... [F]

(3-36)

Substituting Eqs. 3-35 and 3-36 into Eq. 3-33, and dividing by unit weight γ :

Q 2

Q 2

______ + 1 A1 =

______ + 2 A2 ... [L

3]

gA1 gA2

(3-37)

In general, the specific force is defined as follows:

Q 2

F = ______

+ A ... [L 3

]

gA

(3-38)

Equation 3-37 states that specific force is conserved in open-channel flow in a short

horizontal channel, i.e., F1 = F2. Note that specific force is a force per unit of γ, the

weight per unit of volume; therefore, specific force has the units of volume [L3].

The specific force curve shown in Fig. 3-9 (c) is obtained by plotting F in the abscissas

and y in the ordinates. This curve is similar to the specific energy curve [Fig. 3-9 (a)],

but with significant differences. The limb AC approaches the horizontal axis

asymptotically toward the right. The limb BC rises upward and extends without limit to

the right.

For a given value of specific force F1, the curve has two possible depths: y1 and y2.

These are the initial and sequent depths of a hydraulic jump, respectively. At point C

[Fig. 3-9 (c)], the sequent depths coalesce into one depth, termed the critical depth; at

that point, the specific force is a minimum. Note that this is the same critical depth

obtained by specific energy considerations; see Fig. 3-9 (a).

Critical flow criterion for specific force

As in the case of specific energy, to prove that minimum specific force corresponds to

the critical flow criterion, differentiate Eq. 3-38 with respect to y to yield:

dF Q 2

dA d( ) ______

= - _______

______

+ ________

= 0

dy g A 2

dy dy

(3-39)

For a change in depth dy, the corresponding change d( ) in the static moment of the

water area about the free surface is equal to:

d( ) = [A ( + dy) + (1/2) T (dy)2] -

(3-40)

As usual in differential calculus, the second-order term in Eq. 3-40 is neglected, to

yield:

d( ) = A dy

(3-41)

Therefore, Eq. 3-39 reduces to:

dF Q 2

dA ______

= - _______

______

+ A = 0

dy g A 2

dy

(3-42)

Simplifying Eq. 3-42:

Q 2

dA ______

______

= A

g A 2

dy

(3-43)

Since dA /dy = T, Q /A = V, and A /T = D, Eq. 3-43 reduces to:

V 2

______

= 1

g D

(3-44)

which is the square of the Froude number:

Vc 2

F 2

= ______

= 1

g Dc

(3-45)

Equation 3-45 is the criterion for critical flow, applicable to both specific energy and

specific force (specific momentum) curves.

Note that the sequent depth y2 is always smaller than the high alternate depth y2' (Fig.

3-7). Furthermore, the energy E2 is always smaller than the energy E1, while the

specific force F2 remains equal to the specific force F1 [Fig. 3-7 and Fig. 3-9 (c)]. In

order to maintain a constant specific force, the flow depth must increase from y1 to y2 at

the cost of losing a certain amount of energy. The energy loss is equal to: ΔE = E1 - E2.

This situation occurs in the hydraulic jump, where the specific forces before and after

the jump are equal, but with a consequent loss of energy (Fig. 3-10).

Fig. 3-10 Hydraulic jump roller.

Specific force per unit of channel width

In a hydraulically wide channel, the specific force per unit of channel width b is:

F q 2

y 2

___

= _____

+ _____

... [L 2

]

b gy 2

(3-46)

where q = Q /b.

In terms of mean velocity V = q/y, the specific force per unit of channel width b is:

F V 2

y y 2

___

= _______

+ _____

... [L 2

]

b g 2

(3-47)

Specific force in units of force

In units of force, the specific force is:

γQ 2

γF = _______

+ γ A ... [F]

gA

(3-48)

In units of force, the specific force per unit of channel width b is:

γF γq 2

γy 2

____

= ______

+ ______

... [F L -1

]

b gy 2

(3-49)

In units of force, and in terms of mean velocity V, the specific force per unit of channel

width b is:

γF γV 2

y γy 2

____

= _________

+ _____

... [F L -1

]

b g 2

(3-50)

Example 3-1: The hydraulic jump equation

Derive the relation between the initial (y1) and sequent (y2) depth in a hydraulic jump (Eq. 9-13) on a

horizontal floor in a rectangular channel (Fig. 3-11).

Fig. 3-11 Definition sketch for a hydraulic jump.

From Eq. 3-47, the specific force balance between upstream (1) and downstream (2) of the hydraulic

jump:

V1 2y1 y1

2 V2

2y2 y2

2

_________ +

_____ =

_________ +

_____

g 2 g 2

(3-51)

Replacing F1 = V1 /(gy1)1/2

:

y2 y2

( ____

)3 - (2 F1

2 + 1) (

____ ) + 2 F1

2 = 0

y1 y1

(3-52)

Factoring:

y2 y2 y2

[ ( ____

)2 - (

____ ) + 2 F1

2 ] (

____ - 1 ) = 0

y1 y1 y1

(3-53)

The solution of the quadratic equation within brackets is:

y2 1 ____

= ____

[ (1 + 8 F1 2 )

1/2 - 1 ]

y1 2

(3-54)

which is the same as Eq. 9-13.

QUESTIONS

[Problems] [References] • [Top] [Energy Principle] [Specific Energy] [Local Phenomena] [Momentum

Principle] [Specific Force]

1. What is the difference between total energy and specific energy?

2. What is critical flow in terms of specific energy?

3. What causes a hydraulic drop?

4. Why is the flow depth at a free overfall less than critical?

5. What causes a hydraulic jump?

6. Why is momentum unsteady?

7. What is the body force considered in open-channel flow?

8. What is the magnitude of the body force in a horizontal channel?

9. What is the difference between alternate depths and sequent depths?

10. Where does specific force apply?

11. For what type of problems is specific force used?

PROBLEMS

[References] • [Top] [Energy Principle] [Specific Energy] [Local Phenomena] [Momentum Principle] [Specific

Force] [Questions]

1. Derive the relation for the discharge per unit of width q under a sluice gate, as a

function of upstream depth y1 and downstream depth y3 (Fig. 3-12).

Fig. 3-12 Discharge under a sluice gate.

2. Prove that the equation derived in the previous problem (Problem 1) is

mathematically equivalent to the equation based on y1 and y2 used in ONLINE

CHANNEL 13.

3. Using the specific energy principle, derive the formula for the dimensionless

throat width of a channel constriction that forces critical flow through it (Fig. 3-13)

[Henderson (1966), p. 267].

bc (27)1/2

F1

σ = _____

= _______________

b1 (2 + F12)

3/2

(3-55)

Fig. 3-13 Critical width constriction using specific energy.

5. Use ONLINE CHANNEL 17 to calculate the required throat width bc for the

following upstream conditions: y1 = 2.2 m, v1 = 1.2 m/s, and b1 = 3.2 m. What

would be the required throat width if the upstream channel width is b1 = 2.2 m?

6. Using the specific force principle, show that the force fo (in kN/m) exerted by a

blunt obstruction at the bottom of a wide rectangular channel is:

1 - α 2

1

fo = - γ y12 [

________ + ( 1 -

_____ ) F1

2 ]

2 α

(3-56)

7. where γ = unit weight of water, F1 = upstream Froude number, y1 = upstream flow

depth, andα = y2/y1, where y2 = downstream flow depth (after the obstruction).

Given q = 1.5 m2/s,v1 = 1.0 m/s, and α = 0.91, calculate the force fo. Assuming

the flow is from left to right, in what direction is the force fo acting?

8. Using the specific force principle, derive the formula for the dimensionless throat

width of a channel constriction that forces critical flow through it (Fig. 3-14)

[modified from Henderson (1966), p. 267].

bc (3)3/4

F3

σ = _____

= _________________

b3 (1 + 2 F32)

3/4

(3-57)

Fig. 3-14 Critical width constriction using specific force.

10. Using ONLINE LIMITING CONTRACTION, calculate the limiting

contraction ratios using both energy and momentum principles, for a Froude

number F = 0.3.

11. Using ONLINE LIMITING CONTRACTION SET, calculate the limiting

contraction ratios using both energy and momentum principles, for Froude

numbers is the range 0.1 ≤ F ≤ 2.0, at intervals of 0.1. Discuss the results.

12. A submerged hydraulic jump occurs immediately downstream of a sluice

outlet in a rectangular channel. Using the momentum principle, prove that the

ratio of submerged depth ys to tailwater depth y2 is:

ys y2 _____

= [ 1 + 2 F22 ( 1 -

_____ ) ] 1/2

y2 y1

(3-58)

Fig. 3-15 A submerged hydraulic jump at a sluice outlet.

REFERENCES

• [Top] [Energy Principle] [Specific Energy] [Local Phenomena] [Momentum Principle] [Specific

Force] [Questions] [Problems]

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

Henderson, F. M. 1966. Open channel flow. Macmillan, New York.

http://openchannelhydraulics.sdsu.edu

[Critical Flow] [Computation of Critical Flow] [Critical Flow Control] [Sharp-crested

Weirs] [Questions] [Problems] [References] •

CHAPTER 4: CRITICAL FLOW

4.1 CRITICAL FLOW

[Computation of Critical Flow] [Critical Flow Control] [Sharp-crested

Weirs] [Questions] [Problems] [References] • [Top]

In open-channel flow, two characteristic thresholds describe the state of flow (Section

1.3):

1. Critical flow, which describes steady flow at Froude number F = 1, and

2. Neutrally stable flow, which describes the onset of flow instability, which occurs at

Vedernikov number V = 1.

The ratio V/F embodies two important properties:

1. The type of friction, and

2. The type of cross-sectional geometry, or shape.

The discharge-flow area rating, Eq. 1-4, is repeated here for convenience:

Q = α Aβ (4-1)

in which α = coefficient of the rating, and β = exponent. The latter is defined as follows

(Section 1.3):

V

β = 1 + _____

F

(4-2)

Values of Froude number F corresponding to values of Vedernikov number V = 1, i.e.,

values of Fns, are listed in Table 1-1.

Critical flow criterion

Critical flow occurs under the following conditions (Fig. 4-1):

For a given discharge Q, when the specific energy E is a minimum.

For a given specific energy E, when discharge Q is a maximum.

For a given discharge Q, when the specific force F is a minimum.

For a given specific force F, when discharge Q is a maximum.

When the mean flow velocity u is equal to the relative celerity w of small surface

perturbations, i.e., dynamic waves (Section 1.3).

When the velocity head hv is equal to one-half of the hydraulic depth D.

When the Froude number F is equal to 1.

Fig. 4-1 Critical flow over emergency spillway, Turner reservoir,

San Diego County, California.

The condition of critical flow represents a threshold between subcritical flow, for

which F < 1, and supercritical flow, for which F > 1.

Dynamic waves in open-channel flow have two components: (1) primary, and (2)

secondary (Ponce and Simons, 1977). Primary waves travel with absolute velocity:

v1 = u + w

(4-3)

Secondary wave travels with absolute velocity:

v2 = u - w

(4-4)

While primary waves always travel downstream, secondary waves may travel

upstream or downstream, depending on the flow conditions. In subcritical flow, w > u,

and secondary waves are able to travel upstream. In supercritical flow, w < u, and

secondary waves cannot travel upstream, instead traveling downstream. In practice,

this means that subcritical flow is controlled from downstream, because surface

perturbations are able to travel upstream. Conversely, supercritical flow cannot be

controlled from downstream, because surface perturbations are not able to travel

upstream. Instead, supercritical flow can be controlled only from upstream.

Critical flow may occur in one of two distinct modes:

1. Along the channel, under uniform flow, in what is referred to as critical uniform

flow. The slope of the channel that sustains critical uniform flow is referred to as

the critical slope.

2. At a specific cross section, under gradually varied flow upstream and

downstream, in what is referred to as the critical cross section. [Actually, in the

proximity of the critical flow cross section, the flow may be rapidly varied (Section

3.3)]. At a critical cross section, the flow depth is referred to as the critical depth

(Fig. 4-2).

Fig. 4-2 Critical flow over a natural weir formed after stream degradation to bedrock,

Aguaje de la Tuna, Tijuana, Baja California, Mexico.

The Darcy-Weisbach equation for open channel flow

The Darcy-Weisbach equation for open channel flow, Eq. 1-32, is repeated here for

convenience (Section 1.4):

f D

S = ___

_____

F 2

8 R

(4-5)

in which the Froude number is:

V

F = __________

(gD)1/2

(4-6)

For a hydraulically wide channel, for which D ≅ R, Eq. 4-5 reduces to:

f (4-7)

S = ___

F 2

8

For application to open-channel flow, a modified Darcy-Weisbach friction factor f, equal

to 1/8 of the usual Darcy-Weisbach friction factor f is applicable. The modified Darcy-

Weisbach equation for open-channel flow is:

S = f F 2

(4-8)

Physical meaning of the critical slope

The critical slope is that for which F = 1. In Eq. 4-5, for F = 1:

f D

Sc = ___

_____

8 R

(4-9)

in which Sc = critical slope. In Eq. 4-7, for F = 1:

f

Sc = ___

8

(4-10)

Furthermore, in Eq. 4-8, for F = 1:

Sc = f

(4-11)

Equations 4-9 to 4-11 confirm that the friction factor and the critical slope are indeed

closely related. For a channel of arbitrary cross section, Eq. 4-9 is applicable; for a

hydraulically wide channel, Eq. 4-10 is applicable. When the modified Darcy-Weisbach

friction factor f is used, Eq. 4-11 is applicable.

In general:

S = Sc F 2

(4-12)

in which Sc is appropriately defined by either Eqs. 4-9, 4-10, or 4-11.

Equation 4-12 constitutes a type of Darcy-Weisbach equation applicable to open-

channel flow. In uniform flow, this equation provides an enhanced physical meaning to

the concept of critical slope. In gradually varied flow, Eq. 4-12 enables an increased

understanding of the asymptotic limits to the water surface profiles (Section 7.3).

The following conclusions are drawn from Eq. 4-12:

Relationship between bottom slope, critical slope and Froude number

1. The bottom slope is directly proportional to the Froude number, when the constant of proportionality

is the critical slope.

2. The bottom slope is directly proportional to the critical slope, when the constant of proportionality is

the Froude number.

3. The critical slope is inversely proportional to the Froude number, when the constant of

proportionality is the bottom slope.

Occurrence of critical flow

As Eq. 4-12 indicates, critical flow occurs when the discharge is such that the critical

slope (friction slope) equals the bottom slope. This is possible in a lined channel of

fixed geometry, where as discharge increases, the friction slope eventually decreases

to match the bottom slope. Thus, in an artificial prismatic channel, it is possible to

achieve critical, and even supercritical, flow depths. Supercritical flows are rare, but not

unheard of. For example, under Chezy friction in hydraulically wide channels, roll

waves form under Vedernikov number V = 1, which corresponds to Froude number F =

2 (Fig. 1-6 and Table 1-1).

The situation is quite different in a natural channel, where the flow is able to freely

interact with the boundary, increasing the "actual" or effective friction slope. In practice,

the friction slope is not able to decrease to match the bottom slope, with the actual flow

remaining subcritical, at least through most of its domain. Thus, as argued by Jarrett

(1982), it is extremely rare to have critical or supercritical flow in a natural channel (Fig.

4-3).

ASCE

Fig. 4-3 Arkansas River near Buena Vista, Colorado, with F < 1.

4.2 COMPUTATION OF CRITICAL FLOW

[Critical Flow Control] [Sharp-crested Weirs] [Questions] [Problems] [References] • [Top] [Critical Flow]

From Eq. 4-6, the square of the Froude number is:

V 2

F 2

= _______

gD

(4-13)

In Eq. 4-13, replacing V = Q /A and D = A /T leads to:

Q 2

T

F 2

= _______

g A 3

(4-14)

At critical flow, F = 1, and Eq. 4-14 is conveniently expressed in the following form:

Q 2

Tc ________

- Ac 3

= 0

g

(4-15)

With reference to Fig 4-4, the top width T is:

T = b + 2zy

(4-16)

The flow area A is:

A = (b + x ) y = (b + zy ) y

(4-17)

Fig. 4-4 Definition sketch for a trapezoidal cross section.

Replacing Eqs. 4-16 and 4-17 into Eq. 4-15:

Q 2

(b + 2zyc) ________________

- [ (b + zyc ) yc ] 3 = 0

g

(4-18)

Given g = gravitational acceleration, and input data consisting of discharge Q, bottom

width b, and side slope z [z:H to 1:V, Fig. 4-3], Eq. 4-18 is solved for the critical

depth yc . Then:

Tc = b + 2zyc

(4-19)

Ac = (b + zyc ) yc

(4-20)

Ac

Dc = ______

Tc

(4-21)

Q

Vc = ______

Ac

(4-22)

Equation 4-18 is expressed as follows:

Q 2

(b + 2zyc)

f (yc) = _________________

- [ (b + zyc ) yc ] 3

g

(4-23)

Equation 4-23 is the general formula for critical flow, applicable to trapezoidal channels.

For a rectangular channel: z = 0. Likewise, for a triangular channel of symmetrical

section: b = 0.

In Eq. 4-23, making the change of variable x = yc for simplicity:

Q 2

(b + 2zx)

f (x) = ________________

- [ (b + zx ) x ] 3

g

(4-24)

The solution of the cubic Eq. 4-24 may be accomplished by a trial-and-error procedure.

A time-tested algorithm for the approximation to the first root of the equation is

described below.

Critical flow algorithm

1. Assume an initial value of x = 0. Then f(0) = Q2b/g, a large positive number. It is confirmed that the

initial value of the function is greater than zero.

2. Assume an initial value of the trial interval Δx = 1.

3. Set x = x + Δx

4. Calculate f (x)

5. Stop when Δx < Δx TOL. A typical value of Δx TOL is 0.0001.

6. If f (x) > 0, return to Step 3.

7. if f (x) < 0, set Δx = 0.1 Δx

8. Set x = x - 9 Δx

9. Return to Step 4.

http://onlinecalc.sdsu.edu/onlinechannel02.php

Critical depth in a hydraulically wide channel

For a hydraulically wide channel, or else, a rectangular channel: Q = qb, and z = 1.

Replacing these values is Eq. 4-18 leads to an expression for critical depth yc in terms

only of unit-width discharge q :

q2

yc = ( ____

) 1/3

g

(4-25)

Example 4-1.

Using ONLINE CHANNEL 02, calculate the critical flow depth for the following flow conditions: Q = 10 m3/s;b = 5 m; z = 1.

ONLINE CALCULATION. Using the ONLINE CHANNEL 02 calculator, the critical depth isyc = 0.706 m; the

critical velocity is vc = 2.482 m/s.

Example 4-2.

Using ONLINE CHANNEL 02, calculate the critical flow depth for the following flow conditions: Q = 100 ft3/s;b = 12 ft; z = 0.

ONLINE CALCULATION. Using the ONLINE CHANNEL 02 calculator, the critical depth isyc = 1.292 ft; the

critical velocity is vc = 6.448 ft/s.

4.3 CRITICAL FLOW CONTROL

[Sharp-crested Weirs] [Questions] [Problems] [References] • [Top] [Critical Flow] [Computation of Critical

Flow]

In Eq. 4-23, the critical flow depth is a function only of Q, b, and z. Thus, the critical flow

depth, and for that matter, critical flow, is independent of the bottom slope and channel

friction. Note that in Eq. 4-12, forF = 1, the critical slope effectively cancels with the

bottom slope. Thus, in critical flow, the flow depth is independent of the slope and is,

therefore, a constant for a given discharge.

The unique discharge-flow area (and unique discharge-stage rating), i.e., only one flow

depth for every discharge, and vice versa, is referred to as control. Control can be

either of types: (1) section control (transversal control), acting at a cross section, or (2)

channel control (longitudinal control), acting alongthe channel.

Figures 4-5 to 4-7 shows three cases of critical flow control in a prismatic channel,

under the following slopes: (1) subcritical, (2) critical, and (3) supercritical.

The subcritical flow condition depicted on Fig. 4-5 shows the existence of critical

section control only at the downstream dam crest. Thus, the flow is controlled at the

downstream end.

Fig. 4-5 Location of critical control section in subcritical flow (Chow, 1959).

The critical flow condition of Fig. 4-6 shows critical control in two places: (1) section

control at the downstream dam crest, and (2) channel control along the upstream

critical slope channel. As shown, a backwater profile, which is a nearly horizontal pool

level, connects the two flow control sections. The flow is controlled at the downstream

end.

Fig. 4-6 Location of critical control section in critical flow (Chow, 1959).

The supercritical flow condition of Fig. 4-7 shows critical control in two places: (1)

section control at the downstream dam crest, and (2) section control at the very

upstream section of the supercritical channel. The flow is controlled at both

downstream and upstream ends, with a hydraulic jump occurring somewhere in the

middle.

Fig. 4-7 Location of critical control sections in supercritical flow (Chow, 1959).

The concept of uniqueness of the rating qualifies critical flow as a (section or channel)

control. This provides an expedient way of determining discharge from stage, or stage

from discharge, if one or the other is known. This property of critical flow is useful in

flow measurements.

In practice, critical control for flow measurement is accomplished in two ways: (1) weir

flow (section control), and (2) critical flow flume (channel control). Weir flow is

described in Section 4-4. A typical example of a critical flow flume is the Parshall flume

(Fig. 4-8). Under free-flowing conditions (low tailwater depth), only one gage

measurement is required to determine the discharge. However, under submerged

conditions (with high tailwater depth), two gage measurements are required.

Fig. 4-8 Parshall flume, Cucuchucho constructed wetland, Michoacan, Mexico.

Broad-crested weirs

In a broad-crested weir, critical flow occurs in the vicinity of the crest. The discharge per

unit of width is:

q = Vc yc

(4-26)

q = (gyc)1/2

yc

(4-27)

q = (g)1/2

(yc)3/2

(4-28)

By definition, the critical depth is 2/3 of the total head H measured above the weir crest:

yc = (2/3) H

(4-29)

Replacing Eq. 4-29 in Eq. 4-28:

q = C H 3/2

(4-30)

in which C is a discharge coefficient defined as follows:

C = (2/3)3/2

g1/2

(4-31)

In SI units, C = 1.704; in U.S. Customary units, C = 3.087. For various reasons, an

actual design value Cdmay differ from the theoretical value C. Experience has shown

that the approximate range is: 0.8 ≤ Cd /C≤ 1.3. Figure 4-9 shows a broad-crested weir

for which Cd = 1.45 (SI units).

In practice, H is taken as the elevation of the water surface above the weir crest. This

assumes that the approach velocity Va at a section sufficiently upstream from the weir

is zero: Va = 0.

Fig. 4-9 The 8,000-ft long overflow spillway weir at the Boerasirie Conservancy, Guyana.

4.4 SHARP-CRESTED WEIRS

[Questions] [Problems] [References] • [Top] [Critical Flow] [Computation of Critical Flow] [Critical Flow

Control] [Sharp-crested Weirs]

Sharp-crested weirs are used to force section control in an open channel, for purposes

of flow measurement. In practice, sharp-crested weirs have been built using: (1)

triangular, (2) trapezoidal, or (3) rectangular sections.

Triangular weirs

Triangular weirs can be of two types: (a) fully contracted, or (b) partially contracted.

Contraction refers to the size of the weir flow area as compared to the size of the

approach channel flow area. Depending on size and design, a weir may be subject to

both vertical and horizontal contraction. For a weir to be fully contracted, the ends of

the weir should be sufficiently far from the sides and bottom of the approach channel

(Fig. 4-10). Full contraction increases the measurement accuracy by providing a more

precise channel control (a unique discharge-stage relation) in the immediate vinicity of

the weir.

Fully contracted V-notch weir. The V-notch weir is a commonly used type of

triangular sharp-crested weir. For V-notch weirs, full contraction is produced when the

distance b from each side of the weir notch to each side of the weir pool is greater than

2H. For a 90° V-notch weir, the flow width at head level is equal to 2H. Therefore, the

weir may be considered to be fully contracted when the ratio B/H > 6, i.e., forH/B <

0.167. For a 60° notch weir, the requirement for a fully contracted weir is: H/B < 0.194.

In USBR practice, this translates into the practical criterion for a fully contracted V-

notch weir: H/B ≤ 0.2.

A weir not satisfying the above criterion is partially contracted, i.e., the approach

channel width B is too narrow relative to the head H. In USBR practice, the practical

criterion for a partially contracted V-notch weir is: H/B ≤ 0.4.

Fig. 4-10 V-notch weir schematic.

The fully contracted V-notch weir formula, in U.S. Customary units (Q in cfs, H in ft), is

(USBR Water Manual):

Q = 4.28 Ce tan (θ/2) (H + k)5/2

(4-32)

In Eq. 4-32, the discharge Q is a function of hydraulic head H and angle θ. The

discharge coefficient Ceand head correction coefficient k are a funtion of θ. The width of

the approach channel B is used to check to see if the weir is fully contracted: H/B ≤

0.2.

The formula (polinomial fit) for Ce, with θ in degrees, is:

Ce = 0.607165052 - 0.000874466963 θ + 0.0000061039334 θ2

(4-33)

The formula (polinomial fit) for k, with θ in degrees, is (LMNO Engineering):

k = 0.0144902648 - 0.00033955535 θ + 0.00000329819003 θ2 - 0.0000000106215442 θ3

(4-34)

The fully contracted V-notch weir is restricted to the following conditions:

1. Head H < 1.25 ft (38 cm). 2. Width B > 3 ft (91 cm). 3. Height P > 1.5 ft (46 cm). 4. Ratio b/H ≥ 2.0. 5. Head/width ratio H/B ≤ 0.2.

Partially contracted V-notch weir. For the partially contracted 90° V-notch weir, Eq.

4-32 reduces to:

Q = 4.28 Ce (H + 0.0029)5/2

(4-35)

The discharge coefficient Ce is a function of the ratios H/P and P/B, as shown in Fig. 4-

11.

Fig. 4-11 Variation of Ce as a function of H/P and P/B.

The formula for the partially contracted 90° V-notch weir is subject to the following

restrictions:

1. Head H < 2 ft (61 cm). 2. Width B > 2 ft (61 cm). 3. Height P > 0.33 ft (10 cm). 4. Head/width ratio H/B ≤ 0.4.

Example 4-3.

Using ONLINE VEE NOTCH 1, calculate the discharge over a fully contracted V-notch weir for the following flow conditions:

Head H = 1 ft; width B = 6 ft; width b = 2 ft; angle θ = 90°.

ONLINE CALCULATION. Using the ONLINE VEE NOTCH 1 calculator, the discharge is:Q = 2.49 ft3/s.

Trapezoidal weir: Cipolletti weir

A standard Cipolletti weir is trapezoidal in shape (Fig. 4-12). The crest and sides of the

weir plate are placed far enough from the bottom and sides of the approach channel to

produce full contraction. The sides incline outwardly at a slope of 1 horizontal to 4

vertical. The computation procedure follows Section 12 of Chapter 7 of the USBR

Water Measurement Manual.

Fig. 4-12 Cipolletti weir schematic.

The formula for the Cipolletti weir, in U.S. Customary units, is:

Q = 3.367 L H 3/2

(4-36)

in which L = length of the weir crest, in ft, and H = head on the weir crest, in ft.

The accuracy of measurements obtained by Eq. 4-36 is considerably less than that

obtainable with V-notch weirs. The accuracy of the discharge coefficient is ± 5 percent.

The Cipolletti weir is subject to the following restrictions:

1. The head H ≥ 0.2 ft (6.1 cm). 2. The ratio P/H ≥ 2. 3. The ratio b/H ≥ 2.

The head H is measured at a distance of at least 4H upstream from the crest.

Rectangular weirs

A rectangular weir has a rectangular shape, as shown in Fig. 4-13. To produce full

contraction, the crest and sides of the weir plate are placed sufficiently far enough from

the bottom and sides of the approach channel. The computation procedure follows

Section 6 of Chapter 7 of the USBR Water Measurement Manual.

Fig. 4-13 Three-dimensional perspective of a rectangular weir.

The Kindsvater-Carter formula for a rectangular weir, in U.S. Customary units, is:

Q = Ce (L + kb) (H + 0.003)3/2

(4-37)

in which Ce = effective coefficient of discharge; L = length of the weir crest, in ft, kb = a

correction factor to obtain effective weir length, in ft; H = head measured above the weir

crest, in ft; and Q = discharge, in cfs. The value B is the average width of the approach

channel.

The correction factor kb is a function of the ratio L/B, as shown in Fig. 4-14.

Fig. 4-14 Correction factor kb to obtain effective weir length.

The effective coefficient of discharge Ce includes effects of relative depth and relative

width of the approach channel. It is a function of H/P and L/B, as shown in Fig. 4-15.

Fig. 4-15 Values of Ce as a function of H/P and L/B.

Given H, L, B and P, and the ratios H/P and L/B, the computation proceeds with the

following steps:

1. The correction factor kb is calculated using Fig. 4-14.

2. The effective coefficient of discharge Ce is calculated using Fig. 4-15.

3. The discharge Q is calculated using Eq. 4-37.

The rectangular weir equation (Eq. 4-37) is subject to the following restrictions:

1. The calibration relationships (Figs. 4-14 and 4-15) were developed with

rectangular approach flow. For applications with other flow section shapes, the

average width of the flow section for each head should be used as B to calculate

discharges.

2. The head H should be at least 0.2 ft (0.061 m).

3. The crest height P ≥ 4 in (0.3333 ft, or 0.1015 m).

4. The crest length L ≥ 6 in (0.5 ft, or 0.1524 m).

5. The ratio H/P ≤ 2.4.

6. The water surface elevation in the downstream channel should be at least 2 in (5

cm, or 0.05 m) below the weir crest (Fig. 4-16).

Gary Player

Fig. 4-16 Rectangular weir on Ash Creek, near New Harmony, Utah.

Standard contracted rectangular weir. The standard contracted rectangular weir is

shown inFig. 4-17. To be fully contracted, the overflow plate sides and ends must be

located at a distance of at least 2H from the approach flow boundaries. The head is

measured at a distance upstream of at least 4H from the weir. The computation

procedure follows Section 9 of Chapter 7 of the USBR Water Measurement Manual.

Fig. 4-17 Definition sketch for a standard contracted rectangular weir.

The formula for the standard contracted rectangular weir is the Francis equation. In

U.S. Customary units, this equation is:

Q = 3.33 (L - 0.2H ) H 3/2

(4-38)

in which Q = discharge, in cfs, L = length of the weir crest, in ft, and H = head on the

weir crest, in ft.

The accuracy of measurements obtained by Eq. 4-38 is considerably less than that

obtainable with V-notch weirs. The accuracy of the discharge coefficient is ± 5 percent.

Equation 4-38 has a constant discharge coefficient (3.33), which facilitates

computations. However, the coefficient does not remain constant for a ratio of head-to-

crest H/L > 1/3, and the actual discharge exceeds that given by the equation. Francis'

experiments were made on comparatively long weirs, most of them with crest

length L = 10 ft and heads in the range 0.4 ft ≤ H ≤ 1.6 ft. Thus, the equation applies

particularly to such conditions. USBR experiments on 6-in, 1-ft, and 2-ft weirs has

shown that the Francis equation also applies fairly well to shorter crest lengths L,

provided H/L ≤ 1/3.

Equation 4-38 is subject to the following restrictions:

1. Height-to-head ratio P/H ≥ 2. 2. Ratio b/H ≥ 2. 3. Length-to-head ratio: L/H ≥ 3.

Standard suppressed rectangular weir. A standard suppressed rectangular weir has

a horizontal crest that crosses the full channel width (Fig. 4-18). The elevation of the

crest is high enough to assure full bottom crest contraction of the nappe. The vertical

sidewalls of the approach channel continue downstream past the weir plate, to prevent

side contraction or lateral expansion of the overflow jet. The computation procedure

follows Section 10 of Chapter 7 of the USBR Water Measurement Manual.

Special care must be taken to secure proper aeration beneath the overflowing sheet at

the crest. Aeration is usually accomplished by placing vents on both sides of the weir

box under the nappe. Other conditions for accuracy of measurement for this type of

weir are generally the same as for those of the contracted rectangular weir, except for

the absence of side contraction. However, the crest height Pshould be at least 3H.

Fig. 4-18 Definition sketch for a standard suppressed rectangular weir.

The formula for the standard suppressed rectangular weir is the Francis equation. In

U.S. Customary units, this equation is:

Q = 3.33 (L - 0.2H ) H 3/2

(4-39)

in which Q = discharge, in cfs, L = length of the weir crest, in ft, and H = head on the

weir crest, in ft.

The accuracy of the discharge coefficient is ± 5 percent. Equation 4-39 should not be

used for heads H < 0.2 ft. These small heads do not give accurate flow measurements

because the nappe of water going over the crest may not spring free of the crest.

Equation 4-39 is subject to the following restrictions:

1. Head H ≥ 0.2 ft. 2. Length L ≥ 4 ft. 3. Height-to-head ratio P/H ≥ 3. 4. Length-to-head ratio: L/H ≥ 3.

The head H is measured at an upstream distance of at least 4H from the weir. The

sidewalls must extend a distance of at least 0.3H downstream from the crest. The

overflow jet must be adequately ventilated to the atmosphere.

Example 4-4.

Using ONLINE STANDARD SUPPRESSED RECTANGULAR, calculate the discharge over a standard suppressed rectangular

weir for the following flow conditions: Head H = 0.5 m; length L = 5 m; height P = 2 m.

ONLINE CALCULATION. Using the ONLINE STANDARD SUPPRESSED RECTANGULARcalculator, the

discharge is: Q = 3,250 L/s.

QUESTIONS

[Problems] [References] • [Top] [Critical Flow] [Computation of Critical Flow] [Critical Flow Control] [Sharp-

crested Weirs]

1. Why is β, the exponent of the rating, important in open-channel flow?

2. For a given discharge Q, what is the value of specific energy at critical flow?

3. What is the relation between velocity head and hydraulic depth under critical

flow?

4. What is the difference bewteen primary and secondary waves in unsteady open-

channel flow?

5. How are bottom slope, critical slope, and Froude number related in uniform flow?

6. Why is it extremely rare to have critical or supercritical flow in a natural channel?

7. In a hydraulically wide channel, critical depth is only a function of what hydraulic

variable?

8. How many types of control are there in open-channel flow?

9. Under what condition are two gage measurements required in a Parshall flume?

10. Why is full contraction necessary in flow measurement using a sharp-

crested weir?

11. Is the Cipolletti weir fully or partially contracted?

12. How is full contraction implemented in a standard suppressed rectangular

weir?

PROBLEMS

[References] • [Top] [Critical Flow] [Computation of Critical Flow] [Critical Flow Control] [Sharp-crested

Weirs] [Questions]

1. Using the Chezy equation (Eq. 5-10), derive the formula for the critical slope Sc in

a prismatic channel.

2. Using the Manning equation (Eq. 5-17 or Eq. 5-18), derive the formula for the

critical slope Sc in a prismatic channel.

3. Calculate the theoretical discharge over a broad-crested weir of length L = 22 ft,

when the total head above the weir is H = 1.2 ft. Verify with ONLINE CHANNEL

14.

4. Calculate the theoretical discharge over a broad-crested weir of length L = 8 m,

when the total head above the weir is H = 0.5 m. Verify with ONLINE CHANNEL

14.

5. Calculate the critical depth corresponding to a unit-width discharge q = 1.36 m2/s.

6. Given a modified Darcy-Weisbach friction factor f = 0.003 and bottom slope S =

0.002. Calculate the Froude number.

7. Use ONLINE CHANNEL 04 to calculate the critical slope for the following canal

data: Q = 11 m3/s,b = 1.5 m, z = 1, S = 0.0006, n = 0.013.

8. Use ONLINE CHANNEL 04 to calculate the critical slope for the following canal

data: Q = 35 ft3/s, b= 3 ft, z = 1, S = 0.001, n = 0.015.

9. Use ONLINE CIPOLLETTI to calculate the weir discharge through a Cipolletti

weir (Fig. 4-19), given head H = 1 ft, length L = 3.2 ft, height P = 4 ft, and

width B = 6 ft.

Fig. 4-19 Definition sketch for a Cipolletti weir.

10. Use ONLINE STANDARD CONTRACTED RECTANGULAR to calculate

the weir discharge through a standard contracted rectangular weir (Fig. 4-20),

given head H = 0.5 m, length L = 3 m, height P = 3 m, and width b = 1.5 m.

Fig. 4-20 Definition sketch for a standard contracted rectangular weir.

REFERENCES

• [Top] [Critical Flow] [Computation of Critical Flow] [Critical Flow Control] [Sharp-crested

Weirs] [Questions] [Problems]

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

Jarrett, R. D., 1984. Hydraulics of High-Gradient Streams. ASCE Journal of Hydraulic Engineering, Vol. 110, No. 11, November, 1519-1539.

Ponce, V, M., and D. B. Simons. 1977. Shallow wave propagation in open-channel

flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, December, 1461-

1476.

http://openchannelhydraulics.sdsu.edu

Extract of

Jarrett, R. D., 1984. "Hydraulics of High-Gradient Streams,"

ASCE Journal of Hydraulic Engineering, 110(11), 1519-1539

[Uniform Flow] [Chézy Formula] [Manning Formula] [Manning Roughness] [Computation of Uniform

Flow] [Computation of Flood Discharge] [Uniform Surface Flow] [Questions] [Problems] [References] •

CHAPTER 5: UNIFORM FLOW

5.1 UNIFORM FLOW

[Chézy Formula] [Manning Formula] [Manning Roughness] [Computation of Uniform Flow] [Computation of Flood

Discharge] [Uniform Surface Flow] [Questions] [Problems] [References] • [Top]

Uniform flow is strictly applicable only to a prismatic channel. In uniform flow, flow

depth, flow area, mean velocity, and discharge are constant along the channel. For

nonprismatic (natural) channels of nearly uniform cross section, the term equilibrium

flow is often used to describe the flow condition that approximates or resembles the

uniform flow of prismatic channels.

In uniform flow, all slopes, friction slope Sf, energy slope Se, water-surface slope Sw,

and bottom slopeSo, are constant and equal to one slope: S.

Sf = Se = Sw = So = S (5-1)

There is no such thing as unsteady uniform flow. If the flow is unsteady, it is simply not

uniform. However, under a Vedernikov number V = 1, uniform flow becomes neutrally

stable and this is conducive to the development of roll waves (Section 1.3). This is the

"instability of uniform flow" described by Chow (1959). At V < 1, flow disturbances are

attenuated, and roll waves do not develop.

Establishment of uniform flow

From a mechanical standpoint, uniform flow occurs in a control volume when the

frictional force is equal to the gravitational force. In the absence of section controls

(Section 4.3), all open-channel flows tend toward uniform flow. In essence, Mother

Nature "likes uniform flow."

In uniform flow, the property of uniqueness of the discharge-flow area rating (or

discharge-flow depth rating) qualifies uniform flow as a channel control Thus, uniform

critical flow is a very strong type of open-channel control.

The depth of uniform flow is referred to as normal depth. A depth smaller than normal is

referred to as supercritical; conversely, a depth larger than normal is referred to as

subcritical. Figure 5-1 shows the establishment of uniform flow in a sufficiently long

channel. The top figure depicts subcritical normal flow, with upstream and downstream

section controls. The center figure depicts critical flow, with upstream and downstream

section controls. The bottom figure depicts supercritical normal flow, with upstream

section control only.

Fig. 5-1 Establishment of uniform flow (Chow, 1959).

Velocity of uniform flow

In general, the mean velocity of uniform flow is described by the following formula:

V = C R x S y (5-2)

in which C = friction coefficient, R = hydraulic radius, defined as R = A/P,

and x and y are exponents of Rand S, respectively. The exponents vary with type of

roughness (laminar, turbulent, transitional, or mixed laminar-turbulent) and cross-

sectional shape (arbitrary, hydraulically wide, rectangular, trapezoidal, triangular, or

inherently stable).

In practice, there are two established uniform flow formulas: (1) the Chézy formula, and

(2) the Manning formula. Variations of these formulas are in current use (the

dimensionless Chézy and the Manning-Strickler).

5.2 CHÉZY FORMULA

[Manning Formula] [Manning Roughness] [Computation of Uniform Flow] [Computation of Flood Discharge] [Uniform

Surface Flow] [Questions] [Problems] [References] • [Top] [Uniform Flow]

To derive the Chézy formula, the shear stress τb developed along the channel bottom is

modeled as a quadratic friction law:

τb = ρ f V 2 (5-3)

in which ρ = mass density, f = a type of friction factor (drag coefficient), and V = mean

velocity. This equation is dimensionless; therefore, it has a strong theoretical basis.

The shear force developed along the wetted perimeter of a control volume of length L is

(Fig. 5-2):

Fs = τb PL = ρ f V 2 PL (5-4)

Fig. 5-2 The control volume for uniform flow (Chow, 1959).

The weight of the water in the control volume is W. This gravitational force is resolved

along the direction of motion to give:

Fg = W sin θ (5-5)

For a channel of small slope: sin θ ≅ tan θ = S. Therefore:

Fg = W tan θ = W S = γ A L S (5-6)

Equating frictional (Eq. 5-4) and gravitational forces (Eq. 5-6):

ρ f V 2 P = γ A S (5-7)

which reduces to:

f V 2 = g (A /P ) S = g R S (5-8)

where R = hydraulic radius.

Solving for V:

V = (g/f )1/2 (R S ) 1/2 (5-9)

V = C (R S )1/2 (5-10)

in which C = Chézy coefficient, defined as follows:

C = (g/f )1/2 (5-11)

Thus, the friction factor f in Eq. 5-3 is:

g

f = ______

C 2

(5-12)

Equation 5-10 is the Chézy formula.

A variation of the Chézy formula may be derived by solving for bottom slope S from Eq.

5-8:

V 2

S = f _____

g R

(5-13)

which is equivalent to:

D V 2

S = f _____ _____

R g D

(5-14)

Given Eq. 4-6, Eq. 5-14 reduces to:

D

S = f _____ F 2

R

(5-15)

Equation 5-15 is basically the same as Eq. 4-5, which was derived from the Darcy-

Weisbach equation applied to open-channel flow. Thus, the dimensionless Chézy

equation (Eq. 5-15) and the modified Darcy-Weisbach equation (Eq. 4-5) are the same.

For a hydraulically wide channel, D ≅ R, and Eq. 5-15 reduces to:

S = f F 2

(5-16)

which is the same as Eq. 4-8.

Table 5-1 shows corresponding values of f, Darcy-Weisbach friction factor f, and Chézy

coefficients.

[See also Lab video: Establishment of uniform flow].

Table 5-1 Corresponding values of f, f, and C.

Friction

factor

f

Darcy-

Weisbach

f

Chézy C

SI

units

U.S.

Customary

units

0.002 0.016 70.02 126.83

0.003 0.024 57.17 103.55

0.004 0.032 49.51 89.68

0.005 0.040 44.29 80.21

History of the Chézy Formula

Antoine Chézy was born at Chalon-sur-Marne, France, on September 1, 1718, and died on

October 4, 1798.

In 1749, working in Amsterdam, Cornelius Velsen stated:

"The velocity should be proportional to the square root of the slope."

In 1757, in Hannover, Germany, Albert Brahms wrote:

"The decelerative action of the bed in uniform flow was not only equal to the accelerative action of

gravity but also proportional to the square of the velocity."

Velsen and Brahms were working on the general laws and theories of Torricelli and Bernoulli.

Chézy used some of these ideas to develop his formula.

Chézy was given the task to determine the cross section and the related discharge for a proposed

canal on the river Yvette, which is close to Paris, but at a higher elevation. Since 1769, he was

collecting experimental data from the canal of Courpalet and from the river Seine. His studies

and conclusions are contained in a report to Mr. Perronet dated October 21, 1775. The original

document, written in French, is titled "Thesis on the velocity of the flow in a given ditch," and it

is signed by Mr. Chézy, General Inspector of des Ponts et Chaussées. It resides in file No. 847,

Ms. 1915 of the collection of manuscripts in the library of the École.

In 1776, Chézy wrote another paper, entitled: "Formula to find the uniform velocity that the

water will have in a ditch or in a canal of which the slope is known." This document resides in

the same file [No. 847, Ms. 1915]. It contains the famous Chézy formula:

V = 272 (ah/p)1/2

in which h is the slope, a is the area, and p is the wetted perimeter. The coefficient 272 is given

for the canal of Courpalet in an old system of units. In the metric system, the equivalent value is:

V = 31 (ah/p)1/2

For the river Seine, the value of the coefficient is 44.

Clemens translated into English the two Chézy papers. Riche de Prony, one of Chézy's former

students, was the first to use the Chézy formula. Later, in 1801, in Germany, Eytelwein used

Chézy's and De Prony's ideas to further the development of the formula.

5.3 MANNING FORMULA

[Manning Roughness] [Computation of Uniform Flow] [Computation of Flood Discharge] [Uniform Surface

Flow] [Questions] [Problems] [References] • [Top] [Uniform Flow] [Chézy Formula]

The Manning formula, in SI units, is:

1

V = ____ R 2/3 S 1/2

n

(5-17)

where n = Manning's friction coefficient, friction factor, or simply Manning's n.

In U.S. Customary units, the Manning formula is:

1.486

V = ________ R 2/3 S 1/2

n

(5-18)

The quantity 1.486 is a conversion factor arising from the equivalence 1.486 =

1/(0.3048)1/3

. The factor is required to express the original Manning equation (Eq. 5-17)

in U.S. Customary units.

In order to compare with the Chézy formula, the Manning equation is expressed as

follows:

1.486

V = ________ R 1/6 R 1/2 S 1/2

n

(5-19)

Comparing Eqs. 5-10 and 5-19, the relation between Manning and Chézy coefficients is

obtained:

1.486

C = ________ R 1/6

n

(5-20)

Equation 5-20 implies that while C varies with the hydraulic radius, the value of n does

not. This may be approximately true for (artificial) prismatic channels, but it is generally

not true for natural channels (Barnes, 1967).

In natural channels, the value of n may vary with stage and flow depth. This is

attributed to:

1. Natural variations in channel roughness with increasing stage, including the effect

of overbank flows (Fig. 2-15), or

2. Morphological changes in total bottom friction, composed of skin and form

friction, as the flow rises from low stage, through intermediate stage, to high

stage (Simons and Richardson, 1966).

Empirical formulas for Manning's n

Several correlations between Manning's n and particle size (grain diameter) have been

developed. Williamson (1951) correlated the Darcy-Weisbach friction factor f with

relative roughness to yield the following relation (Henderson, 1966):

ks

f = 0.113 ( ____ ) 1/3

R

(5-21)

in which ks = grain roughness, in length units, and R = hydraulic radius.

Since f = 8 (g /C 2), Eq. 5-21 reduces to:

8g R

C = ( ________ )1/2 ( _____ ) 1/6

0.113 ks

(5-22)

In U.S. Customary units, Eq. 5-22 may be conveniently reduced to:

1.486 R1/6

C = ________________

0.0311 ks1/6

(5-23)

Comparing Eq. 5-23 with Eq. 5-20, n can be expressed in terms of boundary roughness

as follows (ks in ft):

n = 0.0311 ks1/6 (5-24)

A general expression for Manning's n in terms of relative roughness and absolute

roughness is (Chow, 1959):

n = [f (R/ks)] ks1/6 (5-25)

which implies that in Eq. 5-24 the relative roughness is a constant (0.0311).

Assuming that boundary roughness may be represented by the d84 particle size, i.e.,

that for which 84% of the grains (by weight) are finer, Eq. 5-24 converts to:

n = 0.0311 d841/6 (5-26)

Strickler used a constant (0.0342) for the function of relative roughness f(R/ks), and the

median particle size d50 as the representative grain diameter, to yield:

n = 0.0342 d501/6 (5-27)

Since d84 > d50, it is seen that the Strickler and Williamson equations are mutually

consistent.

Table 5-2 shows values of Manning's n calculated with the Strickler formula (Eq. 5-27).

Table 5-2 Values of Manning's n calculated with the

Strickler formula (Eq. 5-27).

Mean particle size d50

(ft) Manning's n

0.0001 0.007

0.001 0.011

0.01 0.016

0.1 0.023

1 0.034

History of the Manning Formula

Robert Manning was born in Normandy, France, in 1816, and died in 1897. In 1826, he moved

to Waterford, Ireland, and worked as an accountant.

In 1846, during the year of the great famine, Manning was recruited into the Arterial Drainage

Division of the Irish Office of Public Works. After working as a draftsman for a while, he was

promoted to assistant engineer. In 1848, he became district engineer, a position he held until

1855. As a district engineer, he read "Traité d'Hydraulique" by d'Aubisson des Voissons, after

which he developed a great interest in hydraulics.

From 1855 to 1869, Manning was employed by the Marquis of Downshire, while he supervised

the construction of the Dundrum Bay Harbor in Ireland and designed a water supply system for

Belfast. After the Marquis' death in 1869, Manning returned to the Irish Office of Public Works

as assistant to the chief engineer. He became chief engineer in 1874, a position he held it until

his retirement in 1891.

Manning did not receive any education or formal training in fluid mechanics or engineering. His

accounting background and pragmatism influenced his work and drove him to reduce problems

to their simplest form. He compared and evaluated seven best known formulas of the time: Du

Buat (1786), Eyelwein (1814), Weisbach (1845), St. Venant (1851), Neville (1860), Darcy and

Bazin (1865), and Ganguillet and Kutter (1869). He calculated the velocity obtained from each

formula for a given slope and for hydraulic radius varying from 0.25 m to 30 m. Then, for each

condition, he found the mean value of the seven velocities and developed a formula that best

fitted the data.

Manning's original best-fit formula was the following:

V = 32 [RS (1 + R1/3)]1/2

which he later simplified to:

V = C Rx S1/2

In 1885, Manning gave x the value of 2/3 and wrote his formula as follows:

V = C R2/3 S1/2

In a letter to Flamant, Manning stated: "The reciprocal ofC corresponds closely with that of n, as

determined by Ganguillet and Kutter; both C and n being constant for the same channel."

On December 4, 1889, at the age of 73, Manning first proposed his formula to the Institution of

Civil Engineers (Ireland). This formula saw the light in 1891, in a paper written by him entitled

"On the flow of water in open channels and pipes," published in the Transactions of the

Institution of Civil Engineers (Ireland).

Manning did not like his own equation for two reasons: First, it was difficult in those days to

determine the cube root of a number and then square it to arrive at a number to the 2/3 power. In

addition, the equation was dimensionally incorrect, and so to obtain dimensional correctness he

developed the following equation:

V = C (gS)1/2 [R1/2 + (0.22/m1/2 )(R - 0.15 m)]

where m = "height of a column of mercury which balances the atmosphere," and C was a

dimensionless number "which varies with the nature of the surface."

However, in late 19th century textbooks, the Manning formula was written as follows:

V = (1/n) R2/3 S1/2

Through his "Handbook of Hydraulics," King (1918) led to the widespread use of the Manning

formula as it is known today, as well as to the acceptance that the Manning coefficient C should

be the reciprocal of Kutter's n.

In the United States, n is referred to as Manning's friction factor, or simply Manning's n. In

Europe, the Strickler K is the same as Manning's C, i.e., the reciprocal of n. When Kis used in

lieu of n, the Manning equation is referred to as the Manning-Strickler or Strickler equation.

5.4 MANNING ROUGHNESS

[Computation of Uniform Flow] [Computation of Flood Discharge] [Uniform Surface

Flow] [Questions] [Problems] [References] • [Top] [Uniform Flow] [Chézy Formula] [Manning Formula]

Given Eq. 5-17 (or 5-18), once three of the variables are known, the fourth one can be

calculated. Typically, R and S are known, and n is estimated, from which V can be

calculated. This is the direct method, the most typical way of using the Manning

equation.

When increased accuracy is required, or else, when n cannot be estimated with

complete certainty, a measurement of velocity V, together with the measurement of

hydraulic radius R and channel slope S, is recommended to calculate n. This procedure

is referred to as the inverse method, or the calibration method. In practice, most

applications have used the direct method.

Estimation of Manning's n

There is no exact method or procedure to estimate Manning's n. A proven set of

recommendations is given below.

Recommendations for the estimation of Manning's n

1. To understand the factors that affect the value of Manning's n and proceed accordingly.

2. To consult a table of typical values, and to base the estimation on judgment and experience.

3. To consult several pictorial collections for which the value of Manning's n has been documented

with sufficient accuracy.

4. To become acquainted with the appearance of typical channels for which the Manning's nvalues are

known.

Chow (1959) presented a pictorial collection of twenty-four (24) typical channels for

which the Manning'sn has been established. The values documented by Chow range

from n = 0.012 (a canal lined with concrete slabs, with very smooth surface) to n =

0.150 (a natural river in sand clay soil, irregular sides slopes and uneven bottom).

Chow (1959) listed values of Manning's coefficient as low as n = 0.008 (lucite, acryclic

plastic) to as high as n = 0.200 (flood plains of natural streams, with dense willows, in

the summer) (Table 5-4). These values are applicable to channel flow in the turbulent

regime.

Barnes (1967) presented a full-color pictorial collection of fifty (50) typical stream

channels across the United States, for which the Manning's n had been calculated by

calibration. The Barnes collection can be viewed online at Roughness Characteristics

of Natural Channels. The lowest value of Manning's ndocumented by Barnes is n =

0.024, for the Columbia River at Vernita, Washington (Fig. 5-3). The highest value of

Manning's n is n = 0.075, for Rock Creek near Darby, Montana (Fig. 5-4).

Fig. 5-3 The Columbia River at Vernita, Washington.

Fig. 5-4 Rock Creek near Darby, Montana.

Arcement and Schneider (1989) presented a full-color pictorial collection of fifteen (15)

typical flood plains in the Southestern United States, for which the Manning's n had

been calculated by calibration. The Arcement and Schneider collection can be viewed

online at Manning's Roughness Coefficients for Natural Channels and Flood Plains.

The lowest value of Manning's n documented by Arcement and Schneider is n =

0.100, corresponding to Cypress Creek near Downsville, Louisiana (Fig. 5-5). The

highest value is n = 0.200, corresponding to Thompson Creek near Clara, Mississippi

(Fig. 5-6).

Fig. 5-5 Cypress Creek near Downsville, Louisiana.

Fig. 5-5 Thompson Creek near Clara, Mississippi.

Factors affecting Manning's n

In practice, the value of Manning's n is highly variable. In natural stream channels it can

range from slightly lower than 0.020 for some very large rivers featuring a relatively

smooth boundary (Fig. 5-7), to higher than 0.200 for small creeks in steep mountain

streams (Fig. 5-8). The various factors affecting Manning's roughness coefficient are

listed in Table 5-3.

Fig. 5-7 Paraguay River at Forte Coimbra, Mato Grosso do Sul, Brazil.

Fig. 5-8 Rachichuela Creek, La Leche river basin, Lambayeque, Peru.

Table 5-3 Factors affecting Manning's roughness coefficient.

Factor Description

Surface

roughness

Fine grain sizes lead to low values, while coarse grain sizes lead to high values.

Vegetation

Type, height, density, and spatial distribution of vegetation have a definite role in affecting flow velocity. Values of n in vegetated channels may exceed 0.250, and in some cases, rise to 0.400 or greater.

Channel

irregularities

Sand bars, ridges and depressions, and holes/humps in the channel bed create additional roughness in the form of local energy losses.

Channel

alignment

Generally, a straight channel will feature a lower n, while a sinous channel will have a larger n. Sinuosity may increase channel roughness by as much as 30% (Chow, 1959).

Aggradation and

degradation

Changes in channel morphology will increase/decrease roughness in unpredictable ways. The effect will depend on the type of material forming the bed, the width-to-depth ratio (aspect ratio), and the quantity of sediment being transported (sediment load).

Channel

obstructions Log jams, bridge piers, and other obtructions tend to increase channel roughness. The effect will depend on the type of

obstructions, their relative size, shape, number, and spatial distribution.

Size and shape

of the channel

Generally, smaller channels have larger roughness, while larger channels have smaller roughness (compare Fig. 5-7 with Fig. 5-8 above). The typically higher aspect ratio of larger channels tends to decrease roughness.

Stage and

discharge

Roughness varies with stage and discharge in largely unpredictable ways. Mean velocities vary from very low stage to very high stage in complex patterns. A typical sketch is shown in Fig. 2-15.

Season of the

year

For vegetated channels, or channels lines with vegetation, surface roughness increases during the growing season, and decreases during the dormant season, subject to a latitudinal effect.

Suspended load

and bedload

Sediment transport, either as suspended load or bed load, will consume additional energy and lead to increases in overall channel friction.

Cowan (1956) has developed a rational methodology for estimating Manning's n.

Cowan's equation is:

n = (no + n1 + n2 + n3 + n4 ) m5

(5-28)

where:

no = basic n value for a straight, uniform, smooth channel

n1 = addition to account for surface irregularities

n2 = addition to account for variations in the size and shape of the cross section

n3 = addition to account for obstructions

n4 = addition to account for the effect of vegetation on flow conditions

m5 = factor to account for channel sinuosity (meandering).

Table 5-3 lists the appropriate values to be used in Eq. 5-28.

Table 5-3 Corrections to Manning's n (Eq. 5-21).

Channel conditions Values

Type of

material on

channel

boundary

Earth Sand, silt and clay boundary

no

0.020

Rock cut Rock outcrop or rock boundary 0.025

Fine gravel Gravel up to 8 mm diameter 0.024

Coarse

gravel Gravel of more than 8 mm diameter 0.028

Degree of

surface

irregularities

Smooth Best regular condition

n1

0.000

Minor Good dredged channels, slightly eroded

side slopes 0.005

Moderate Fair to poor dredged channels, moderately

eroded side slopes 0.010

Severe Badly eroded canals and channels, highly

irregular or jagged surfaces of channels

excavated in rock 0.020

Variations in

shape and

size of

channel

cross section

Gradual Smooth, or small variations

n2

0.000

Alternating

occasionally

Large and small sections alternate

occasionally, occasional shifting of main

flow from side to side 0.005

Alternating

frequently

Large and small sections alternate

frequently, frequent shifting of main flow

from side to side

0.010-

0.015

Effect of

obstructions

Negligible

(a) The extent to which the obstructions

occupy or reduce the flow area, (b) the

character of the obstructions (sharp-edged

or angular objects induce greater turbulence

than curved, smooth-surface objects),

and (c) the positioning and spacing of the

obstructions, transversally and

longitudinally, in the channel reachunder

consideration

n3

0.000

Minor 0.010-

0.015

Appreciable 0.015-

0.030

Severe 0.030-

0.060

Effect of

vegetation Low Turf grasses or weeds, where the flow

depth is 2 to 3 times the height of the

n4 0.005-

0.010

vegetation

Medium Turf grasses or weeds, where the flow

depth is 1 to 2 times the height of the

vegetation

0.010-

0.025

High Turf grasses or weeds, where the flow

depth is about equal to the height of the

vegetation

0.025-

0.050

Very high Turf grasses or weeds, where the flow

depth is less than one-half (1/2) the height

of the vegetation

0.050-

0.100

Channel

sinuosity

Low Sinuosity less than 1.2

m5

1.00

Medium Sinuosity between 1.2 and 1.5 1.15

High Sinuosity greater than 1.5 1.30

Table 5-4 lists values of Manning's n for channels of various kinds, compiled by Chow

(1959). For each kind of channel, the minimum, normal, and maximum values of n are

shown. The normal values are recommended only for channels with good maintenance.

Values generally recommended for design are shown in bold.

Table 5-4 Recommended range of values of Manning's n. 1

1 2 3 4 Type of channel and description Minimum Normal Maximum

A Closed conduits flowing partly full

A-

1 Metal

a. Brass, smooth 0.009 0.010 0.013

b. Steel

1. Lockbar and welded 0.010 0.012 0.014

2. Riveted and spiral 0.013 0.016 0.017

c. Cast iron

1. Coated 0.010 0.013 0.014

2. Uncoated 0.011 0.014 0.016

d. Wrought iron

1. Black 0.012 0.014 0.015

2. Galvanized 0.013 0.016 0.017

e. Corrugated metal

1. Subdrain 0.017 0.019 0.021

2. Storm drain 0.021 0.024 0.030

1 Chow, V. T. 1959. Open-channel hydraulics. McGraw-Hill, New York.

Click -here- to display the complete Table 5-4.

5.5 COMPUTATION OF UNIFORM FLOW

[Computation of Flood Discharge] [Uniform Surface Flow] [Questions] [Problems] [References] • [Top] [Uniform

Flow] [Chézy Formula] [Manning Formula] [Manning Roughness]

From Eq. 5-2, the discharge in open-channel flow is:

Q = V A = C R x S y A (5-29)

Equation 5-29 can be expressed as follows:

Q = K S y = K S 1/2 (5-30)

in which K = conveyance, defined as:

K = C R x A (5-31)

Or, alternatively:

Q

K = ______

S 1/2

(5-32)

Following Chézy:

K = C A R 1/2 (5-33)

Following Manning, in SI units:

1

K = ____ A R 2/3

n

(5-34)

In U.S. Customary units:

1.486

K = ________ A R 2/3

n

(5-35)

The conveyance K contains information on friction and cross-sectional size and shape,

and is independent of the channel slope.

Channels with composite roughness

A channel that overflows its banks usually has more than one value of Manning's n,

one for inbank flow, and two additional ones, one for left overbank and another for right

overbank (Fig. 5-9). A composite value of Manning's n may be calculated under the

assumption that the velocities for all three subsections, inbank, left overbank, and right

overbank, remain the same. While this assumption is convenient, it sidesteps the issue

of possible flow nonuniformity across the composite cross section.

Fig. 5-9 A composite channel cross section.

Assume a channel of varying roughness along its wetted perimeter, with N being the

number of subsections. The wetted perimeters are: P1, P2, P3, ..., PN. The

corresponding values of roughness are:n1, n2, n3, ..., nN. Assuming that all velocities are

equal:

V1 = V2 = V3 = VN = V (5-36)

For any subsection i :

1

Vi = ____ Ri 2/3 S 1/2

ni

(5-37)

1

Vi = ____ (Ai / Pi ) 2/3 S 1/2

ni

(5-38)

The flow area for subsection i is:

Vi 3/2 ni

3/2 Pi

Ai = _________________

S 3/4

(5-39)

The total flow area of the channel is:

V 3/2 n 3/2 P

A = _________________

S 3/4

(5-40)

The total flow area is equal to the sum of the subareas. Therefore:

V 3/2 n 3/2 P = ∑ (Vi 3/2 ni

3/2 Pi ) (5-41)

N

From Eq. 5-36, all the velocities are equal. Thus, Eq. 5-41 reduces to:

n 3/2 P = ∑ (ni 3/2 Pi )

(5-42)

N

Thus, the value of Manning's n for a channel of composite cross section is:

∑ (ni 3/2 Pi )

N

n = [ _________________ ] 2/3

P

(5-43)

Computation of uniform flow

With reference to Fig. 5-10, the following proportion holds: z /1 = x /y. Then, the top

width T is:

T = b + 2x = b + 2zy

(5-44)

The flow area A is:

A = (b + x ) y = (b + zy ) y

(5-45)

Fig. 5-10 Definition sketch for a trapezoidal cross section.

The wetted perimeter P is:

P = b + 2 (y 2 + z 2y 2 )1/2

(5-46)

Simplifying:

P = b + 2 y ( 1 + z 2 )1/2

(5-47)

From the Manning equation, the discharge Q is:

k

Q = _____ A R 2/3 S 1/2

n

(5-48)

in which k = 1 in SI units, and k = 1.486 in U.S. Customary units.

Since R = A /P, Eq. 5-48 reduces to:

Q n A 5/3 _________ = _______

k S 1/2 P 2/3

(5-49)

Replacing Eqs. 5-45 and 5-47 into Eq. 5-49:

Q n [ (b + zy ) y ] 5/3 _________ = _____________________________

k S 1/2 [b + 2 y ( 1 + z 2 )1/2] 2/3

(5-50)

Simplifying:

Q n Q n

[ (b + zy ) y ] 5/2 - ( ________ ) 3/2 [ 2 y ( 1 + z 2 )1/2 ] - ( ________ ) 3/2 b = 0

k S 1/2 k S 1/2

(5-51)

The input data consisting of: (1) discharge Q, (2) bottom width b, (3) size slope z [z:H to

1:V, Fig. 5-10], (4) bottom slope S, and (5) Manning's n. With given input data, Eq. 5-51

is solved for the normal depth y .Then, with Eqs. 5-44 and 5-45:

A

D = ______

T

(5-52)

Q

V = ______

A

(5-53)

Equation 5-51 is the general formula for uniform or normal flow, applicable to prismatic

channels of trapezoidal cross section. For a rectangular channel: z = 0. Likewise, for a

triangular channel of symmetrical section: b = 0.

To solve Eq. 5-51, it is expressed in the following form:

Q n Q n

f (y) = [ (b + zy ) y ] 5/2 - ( ________ ) 3/2 [ 2 y ( 1 + z 2 )1/2 ] - ( ________ ) 3/2 b

k S 1/2 k S 1/2

(5-54)

Making the change of variable x = y for simplicity:

Q n Q n

f (x) = [ (b + zx ) x ] 5/2 - ( ________ ) 3/2 [ 2 x ( 1 + z 2 )1/2 ] - ( ________ ) 3/2 b

k S 1/2 k S 1/2

(5-55)

The solution of Eq. 5-55 is accomplished by a trial-and-error procedure. An iterative

algorithm based on function value is described below. An example of normal flow is

shown in Fig. 5-11.

Normal depth algorithm based on function value

1. Assume an initial value of x = 0. Then:

f (0) = - [ ( Qn ) / (k So ) ] 3/2 b

which is a large negative number. It is confirmed that the initial value of the function is

smaller than zero.

2. Assume an initial value of the trial interval Δx = 1.

3. Set x = x + Δx

4. Calculate f (x)

5. Stop when Δx < Δx TOL. A typical value of Δx TOL is 0.0001.

6. If f (x) < 0, return to Step 3.

7. if f (x) > 0, set Δx = 0.1 Δx

8. Set x = x - 9 Δx

9. Return to Step 4.

Fig. 5-11 Normal flow at Wellton-Mohawk canal, Wellton, Arizona.

Newton's approximation to the root

The above iteration uses solely the value of the function to approximate to the root. A

faster algorithm makes use of Newton's approximation, which is based on the tangent.

Note that for Newton's iteration to work well, it is first necessary get close to the root

using the function iteration described above. Otherwise, Newton's tangent method may

not converge.

With reference to Fig. 5-12, the value of the tangent at xo is:

f(xo)

f '(xo) = _________

xo - xr

(5-56)

where xo = current trial value of x, f(xo) = value of the function at xo, xr = new value of x,

which gets closer to the root.

Fig. 5-12 Definition sketch for Newton's iteration.

From Eq. 5-56, solving for xr :

f (xo)

xr = xo - ________

f '(xo)

(5-57)

As shown in Fig. 5-12, when f (xo) increases with xo (as is the case for Eq. 5-55), as the

root is passed, the function value and the tangent value are positive; therefore, the

denominator of Eq. 5-56 is also positive, and xr lies to the left of xo. With each

subsequent iteration, the root is approximated in a sawtooth fashion, until the specified

tolerance is satisfied.

It is readily shown that Eq. 5-57 applies also when f (xo) decreases as xo increases, i.e.,

as in the case of critical flow, see Section 4.2.

The value of f '(x) is:

Q n

f ' (x) = x 5/2 (5/2) (b + zx ) 3/2 z + (b + zx ) 5/2 (5/2) x 3/2 - ( ________ ) 3/2 [ 2 ( 1 + z 2 )1/2 ]

k S 1/2

(5-58)

Simplifying Eq. 5-58:

Q n

f ' (x) = 5 z x 5/2 (b + zx ) 3/2 + 5 x 3/2 (b + zx ) 5/2 - ( ________ ) 3/2 ( 1 + z 2 )1/2

k S 1/2

(5-59)

The procedure for Newton's approximation to the root of Eq. 5-55 is described below.

Normal depth algorithm: Newton's approximation

1. Assume an initial value of xo = 0.

2. Assume an initial value of the trial interval Δx = 1.

3. Set xo = xo + Δx

4. Calculate f (xo)

5. If f (xo) < 0, return to Step 3.

6. If f (xo) > 0, calculate the root xr using Eqs. 5-55 and 5-59:

f (xo)

xr = xo - ________

f '(xo)

7. Stop when (xr - xo) is small enough. A typical value of the difference is 0.0001.

8. Otherwise, set xo = xr and return to Step 6.

Example 5-1.

Using ONLINE CHANNEL 01, calculate the critical flow depth for the following flow conditions: Q = 3 m3/s;b = 5 m; z = 1; S =

0.001; n = 0.015.

ONLINE CALCULATION. Using the ONLINE CHANNEL 01 calculator, the normal depth isyn = 0.473 m; the

normal velocity is vn = 1.16 m/s. the normal Froude number is Fn = 0.562.

Example 5-2.

Using ONLINE CHANNEL 01, calculate the critical flow depth for the following flow conditions: Q = 20 ft3/s;b = 13 ft; z =

2; S = 0.0008; n = 0.013.

ONLINE CALCULATION. Using the ONLINE CHANNEL 01 calculator, the normal depth isyn = 0.631 ft; the

normal velocity is vn = 2.221 ft/s. the normal Froude number is Fn = 0.514.

5.6 COMPUTATION OF FLOOD DISCHARGE

[Uniform Surface Flow] [Questions] [Problems] [References] • [Top] [Uniform Flow] [Chézy Formula] [Manning

Formula] [Manning Roughness] [Computation of Uniform Flow]

The high stages and swift currents that prevail during floods combine to increase the

risk of accident and bodily harm (Fig. 5-13). Therefore, it is generally not possible to

measure discharge during the passage of a flood. An estimate of peak discharge can

be obtained indirectly by the use of open channel flow formulas. This is the basis of

the slope-area method.

Fig. 5-13 Flood stage in a tropical river.

To apply the slope-area method for a given river reach, the following data are required:

1. The reach length,

2. The fall, i.e., the mean change in water surface elevation through the reach,

3. The flow area, wetted perimeter, and velocity head coefficients at upstream and

downstream cross sections, and

4. The average value of Manning's n for the reach.

The following guidelines are used in selecting a suitable reach:

1. High-water marks should be readily recognizable (Fig. 5-14).

2. The reach should be sufficiently long so that fall can be measured accurately.

3. The cross-sectional shape and channel dimensions should be relatively constant.

4. The reach should be relatively straight, although a contracting reach is preferred

over an expanding reach.

5. Bridges, channel bends, waterfalls, and other features causing flow nonuniformity

should be avoided.

Fig. 5-14 Local man showing water level reached by flood, Karnataka, India (1991).

The accuracy of the slope-area method improves as the reach length increases (Fig. 5-

15). A suitable reach should satisfy one or more of the following criteria:

1. The ratio of reach length to hydraulic depth should be greater than 75,

2. The fall should be greater than or equal to 0.15 m: F ≥ 0.15 m, and

3. The fall should be greater than either of the velocity heads computed at the

upstream and downstream cross sections.

Fig. 5-15 Slope-area method schematic.

The procedure consists of the following steps:

1. Calculate the conveyance K at upstream and downstream sections:

1

K1 = ( __ ) A1 R1 2/3

n

(5-59a)

1

K2 = ( __ ) A2 R2 2/3

n

(5-59b)

3. in which K = conveyance; A = flow area; R = hydraulic radius; n = average reach

Manning's n; and 1 and 2 denote the upstream and downstream sections,

respectively (Eq. 5-59 is in SI units).

4. Calculate the reach conveyance, equal to the geometric mean of upstream and

downstream conveyances:

K = ( K1 K2 )1/2 (5-60)

5. in which K = reach conveyance.

6. Calculate the first approximation to the energy slope:

F

S = ___

L

(5-61)

7. in which S = first approximation to the energy slope; F = fall; and L = reach

length.

8. Calculate the first approximation to the peak discharge:

Qi = K S 1/2 (5-62)

9. in which Qi = first approximation to the peak discharge.

10. Calculate the velocity heads:

α1 ( Qi /A1 ) 2

hv1 = ______________

2g

(5-63a)

α2 ( Qi /A2 ) 2

hv2 = ______________

2g

(5-63b)

12. in which hv1 and hv2 = velocity heads at upstream and downtream sections,

respectively; α1 and α2= velocity head coefficients at upstream and downstream

cross sections, respectively; and g =gravitational acceleration.

13. Calculate an updated value of energy slope:

F + k ( hv1 - hv2 )

Si = ___________________

L

(5-64)

14. in which Si = updated value of energy slope, and k = loss coefficient. For

expanding flow, i.e.,A2 > A1, k = 0.5; for contracting flow , i.e., A1 > A2, k = 1.

15. Calculate an updated value of peak discharge:

Qi = K Si 1/2 (5-65)

16. Return to step 5 and repeat steps 5 to 7. The procedure is terminated when

the difference between two successive values of peak discharge Q obtained in

step 7 is negligible. In practice, this is usually accomplished within three

iterations.

5.7 UNIFORM SURFACE FLOW

[Questions] [Problems] [References] • [Top] [Uniform Flow] [Chézy Formula] [Manning Formula] [Manning

Roughness] [Computation of Uniform Flow] [Computation of Flow Discharge]

Overland flow over a catchment surface is referred to as surface flow. Typically, in

overland flow, the depth of flow is very small compared to the width. Under these

conditions, the flow may be either laminar or turbulent, depending on the absolute and

relative roughnesses. If velocities and depths of flow are sufficiently small, the flow may

be laminar; otherwise, the flow may be transitional or turbulent, depending on the

Reynolds number (Section 1.4).

In overland flow, a mixed laminar-turbulent regime is commonly present. This type of

flow is characterized by changes from laminar to turbulent regime under the spatially

varying flow conditions normally encountered in catchment/watershed/basin surface

flow. Under laminar flow conditions, the exponent of the rating for surface flow is β = 3.

Under turbulent flow conditions, the exponent is β = 3/2 for Chézy friction, and β =

5/3 for Manning friction. Mixed laminar-turbulent flow regimes feature values

of β varying between laminar and turbulent.

Laminar surface flow

With reference to Fig. 5-16, the acting shear stress at level P is:

τa = γ (ym - y ) S

(5-66)

According to Newton's law of viscosity, the resisting shear stress at P is proportional to

the vertical velocity gradient:

dv

τr = μ _____

dy

(5-67)

in which μ = a constant of proportionality referred to as the dynamic viscosity.

Fig. 5-16 Definition sketch for uniform surface flow.

Equating acting and resisting stresses:

dv

μ _____ = γ (ym - y ) S

dy

(5-68)

In differential form:

μ dv = γ (ym - y ) S dy

(5-69)

The mass density γ = ρg, and the dynamic viscosity μ = ρν, in which ν = kinematic

viscosity. Thus, Eq. 5-69 reduces to:

gS

dv = _____ (ym - y ) dy

ν

(5-70)

Integrating Eq. 5-70:

gS

v = ∫ _____ (ym - y ) dy

(5-71)

ν

gS y 2

v = _____ [ ym y - _____ ] + C

ν 2

(5-72)

where C is a constant of integration. For v = 0: y = 0; therefore: C = 0, and the mean

velocity-flow depth relation is:

gS y 2

v = _____ [ ym y - _____ ]

ν 2

(5-73)

Equation 5-73 reveals that the velocity profile of uniform surface flow has a parabolic

distribution.

The discharge-depth rating is obtained by integrating Eq. 5-73 between the limits of 0

and ym, i.e., from bottom to surface, to yield:

gS y 2

q = ∫ v dy = _____ ∫ [ ym y - _____ ] dy

ν 2

(5-74)

gS ym 2 ym

3

q = _____ [ _____ - _____ ]

ν 2 6

(5-75)

which reduces to:

gS

q = ______ ym 3

3ν

(5-76)

Or:

q = CL ym 3

(5-77)

where CL = laminar discharge rating coefficient, defined as:

gS

CL = ______

3ν

(5-78)

Note that under laminar flow, the exponent of the discharge rating is β = 3 (Eq. 5-77),

and the rating is a function of the internal friction, or internal viscosity, represented by

the kinematic viscosity ν. Thus, laminar flow is a function of temperture.

Given Eq. 5-77, the mean velocity in laminar flow, v = q /ym, is:

v = CL ym 2

(5-79)

The turbulent Chézy discharge rating is:

q = C S 1/2 ym 3/2

(5-80)

The turbulent Manning discharge rating in SI units is:

q = (1/n) S 1/2 ym 5/3

(5-81a)

Likewise, in U.S. Customary units is:

q = (1.486 / n) S 1/2 ym 5/3

(5-81b)

It is seen that the exponent of the rating varies from β = 3 for laminar flow (Eq. 5-77),

to β = 3/2 for turbulent Chézy friction (Eq. 5-80), or β = 5/3 for turbulent Manning friction

(Eq. 5-81). In uniform surface flow, values of β in the range between laminar and

turbulent represent the condition of mixed laminar-turbulent flow (Section 1.3).

The Vedernikov number is:

(β - 1) v

V = ____________

(g y )1/2

(5-82)

Under V = 1, the flow is neutrally stable, promoting the development of roll waves (Fig.

5-17). The relation between the exponent β and the Vedernikov number V is described

below.

Relation between exponent β and Vedernikov number

Under laminar conditions: β = 3. Thus, V = 2 F. Therefore, under laminar conditions, the flow

becomes unstable when F = 0.5.

Under turbulent Chézy conditions: β = 1.5. Thus, V = 0.5 F. Therefore, under turbulent Chézy

conditions, the flow becomes unstable when F = 2.

Under turbulent Manning conditions: β = 5/3. Thus, V = (2/3) F. Therefore, under turbulent

Manning conditions, the flow becomes unstable when F = 3/2, i.e., F = 1.5.

Fig. 5-17 Roll waves on the spillway at Turner reservoir, San Diego County, California.

QUESTIONS

[Problems] [References] • [Top] [Uniform Flow] [Chézy Formula] [Manning Formula] [Manning

Roughness] [Computation of Uniform Flow] [Computation of Flow Discharge] [Uniform Surface Flow]

1. When does uniform flow become unstable?

2. What is the Chezy formula based on?

3. What is the difference between the Manning and Chezy formulas?

4. What is the minimum value of Manning's n that can be achieved in practice?

5. What is the range of values of Manning's n measured by Barnes?

6. What is the range of values of Manning's n measured by Arcement and

Schneider for flood plains?

7. Why is the calculation of composite roughness using Eq. 5-43 only an

approximation?

8. What five input variables are used in the computation of uniform flow in a

trapezoidal channel?

9. Why is it better to use Newton's approximation to the root rather than relying

solely on the function approximation to solve the normal depth problem?

10. What is the minimum ratio of reach length to hydraulic depth in the slope-

area method?

11. What is the exponent of the discharge-depth rating under laminar flow

conditions?

12. What is the exponent of the discharge-depth rating under turbulent Chezy

friction in hydraulically wide channels?

13. What is the exponent of the discharge-depth rating under turbulent

Manning friction in hydraulically wide channels?

14. Under what value of Froude number is the flow likely to become unstable

under laminar flow conditions?

PROBLEMS

[References] • [Top] [Uniform Flow] [Chézy Formula] [Manning Formula] [Manning Roughness] [Computation of

Uniform Flow] [Computation of Flow Discharge] [Uniform Surface Flow] [Questions]

1. Prove that the Darcy-Weisbach friction factor is related to Manning's n by the

following relation:

fD = 8 g n 2

/ (k 2 R

1/3)

in which fD = Darcy-Weisbach friction factor, g = gravitational acceleration, R =

hydraulic radius, and k = constant specific for the system of units, equal to 1 in SI

units and 1.486 in U.S. Customary Units. Express the relation in SI and U.S.

Customary units.

2. Calculate the discharge Q using the Manning equation, given: flow area A = 23.5

ft2; hydraulic radius R = 5.6 ft; channel slope S = 0.0025; Manning's n = 0.035.

3. Calculate the discharge Q using the Manning equation, given: flow area A = 45

m2; hydraulic radiusR = 6 m; channel slope S = 0.003; Manning's n = 0.04.

4. Given f = 0.0025, calculate the discharge Q for a flow area A = 12.4 m2, hydraulic

radiusR = 2.1 m; and channel slope S = 0.0015.

5. Given f = 0.0035, calculate the discharge Q for a flow area A = 18 ft2, hydraulic

radius R = 4.5 ft; and channel slope S = 0.0018.

6. Use ONLINE CHANNEL 01 to calculate normal depth, velocity, and Froude

number for the following case: Q = 150 m3/s, b = 10 m, z = 2, So = 0.0005, n =

0.025.

7. Use ONLINE CHANNEL 01 to calculate normal depth, velocity, and Froude

number for the following case: Q = 250 cfs, b = 20 ft, z = 1, So = 0.001, n = 0.030.

8. Using ONLINE CHANNEL 15, calculate the discharge for a prismatic channel

with b = 20 ft, y = 3 ft,z = 2, n = 0.025, S = 0.0016.

Fig. 5-18 Definition sketch for a trapezoidal channel.

9. Using ONLINE CHANNEL 15, calculate the discharge for a prismatic channel

with b = 6 m, y = 1 m,z = 1.5, n = 0.015, S = 0.0002.

10. A recent flood on Clearwater Creek has left observable water marks on a

certain river reach. To estimate the flood magnitude, hydraulic data has been

measured at two cross sections A and B, a distance of 1,850 ft apart. The reach

fall between the cross sections is 9.1 ft and the average Manning's n is 0.035.

The upstream flow area, wetted perimeter, and Coriolis α coefficient are 550 ft2,

55 ft, and 1.17; the downstream flow area, wetted perimeter, and α coefficient are

620 ft2, 52 ft, and 1.10. Use SLOPE AREA to calculate the flood discharge.

11. Calculate the unit-width discharge in an overland flow plane, under laminar

flow, with mean depth of 1.5 cm and slope of 0.001. Assume water

temperature T = 20oC. Report discharge in L/s/m.

REFERENCES

• [Top] [Uniform Flow] [Chézy Formula] [Manning Formula] [Manning Roughness] [Computation of Uniform

Flow] [Computation of Flow Discharge] [Uniform Surface Flow] [Questions] [Problems]

Barnes, H. A. 1967. Roughness characteristics of natural channels. U.S. Geological Survey Water-Supply Paper 1849, Washington, D.C.

Cowan, W. L. 1956. Estimating hydraulic roughness coefficients. Agricultural Engineering, Vol. 37, No. 7, pp. 473-475, July.

Chow, V. T. 1959. Open-channel hydraulics. McGraw-Hill, New York.

Henderson, M. H. 1966. Open-channel flow. Macmillan, New York.

Simons, D. B., and E. V. Richardson. 1966. Resistance to flow in alluvial channels. U.S. Geological Survey Professional Paper 422-J, Washington, D.C.

Williamson, J. 1951. The laws of flow in rough pipes. La Houille Blanche, Vol. 6, No. 5, September-October, p. 738.

[Nonerodible Channels] [Erodible Channels] [Tractive Force] [Permissible Tractive Force] [Other

Features] [Questions] [Problems] [References] •

CHAPTER 6: CHANNEL DESIGN

6.1 NONERODIBLE CHANNELS

[Erodible Channels] [Tractive Force] [Permissible Tractive Force] [Other

Features] [Questions] [Problems] [References] • [Top]

There are two types of channels: (1) nonerodible, and (2) erodible. Nonerodible

channels are typically lined with a hard construction material, such as concrete or stone

masonry. Erodible channels are dug on the ground surface and, therefore, directly in

contact with the underlying soil. The design of erodible channels is much more complex

than that of nonerodible channels.

The design of nonerodible channels begins with the choice of a uniform flow formula

(Chapter 5). The following factors are considered in the design of nonerodible

channels:

Surface roughness

The surface roughness (boundary friction) is determined by the type and finish of

the material lining the channel boundary. Generally, coarser surfaces have higher

friction than smoother surfaces (Fig. 6-1).

Fig. 6-1 A masonry-lined channel.

Minimum permissible velocity

All channels carry a certain amount of sediments (sand, silt, clay). Therefore, flow

in channels of very mild bottom slope may lead to excessive sediment deposition.

Thus, a minimum permissible velocity, i.e., a minimum bottom slope, is needed to

avoid clogging.

Maximum bottom slope

All nonerodible channels will have a tendency to develop roll waves if the

Vedernikov V > 1 (Chapter 1). To avoid roll waves, the bottom slope must be kept

below a maximum value which is a function of the hydraulics of the flow. For

instance, Fig. 6-2 shows a series of drops implemented in a channel to reduce

the bottom slope.

Fig. 6-2 Channel drops implemented in a channel to decrease the channel slope,

La Joya Canal, Arequipa, Peru.

Shape of the cross section

The channel section of maximum conveyance and, therefore, maximum

discharge, is that which has the minimum wetted perimeter for a given flow area.

This shape of cross section is referred to as the best hydraulic section. The best

hydraulic circular section is a semicircle. The best hydraulic trapezoidal section is

a half regular hexagon inscribed in a circle (Chow, 1959) (Fig. 6-3). In practice,

however, other design considerations are usually more important than that of

maximum conveyance.

Fig. 6-3 Best hydraulic section of trapezoidal shape.

Side slopes

The side slope is z H : 1 V, wherein z = 0 for a rectangular section and z > 0 for a

trapezoidal section, is a design decision that varies with local conditions (Fig. 6-

4). Typical side slopes vary from z = 0 to z = 3.

Fig. 6-4 Channel transition from rectangular to trapezoidal shape.

Freeboard

The freeboard is the vertical depth measured above the design channel depth, up

to the total channel depth (Fig. 6-5). It is intended to account for a safety factor

and to minimize wave overtopping.

Fig. 6-5 Freeboard in the San Luis Canal, California.

Slide risk reduction

In hillslope alignments, care should be taken to reduce the risk of local slides,

which, depending on their extent, may render the channel inoperable. Some

hillslopes are prone to slides, while others are not. A thorough

geological/geotechnical site investigation is required to reduce the risk of slides.

Figure 6-6 shows a slide-generated breach in the La Cano Canal, Arequipa,

Peru, which occurred on November 4, 2010 (see related video).

Fig. 6-6 Slide-generated breach, La Cano Canal, Arequipa, Peru,

which occurred on November 4, 2010.

Channel lining

Channels or canals may be lined with one of several materials, including concrete,

stone masonry, steel, cast iron, timber, glass, and plastic. Channel lining reduces

boundary friction and minimizes channel maintenance costs. Figure 6-7 shows the

Santa Ana river, at Huntington Beach, California, paved with concrete to reduce flood

stage and control floods.

Fig. 6-7 Paved Santa Ana river, Huntington Beach, California.

Figure 6-8 shows Alamar Creek, in Tijuana, Mexico, recently lined up with concrete

(2013). It is noted that current environmental design practices strongly discourage the

lining of natural channels with concrete.

Fig. 6-8 Alamar Creek, Tijuana, Baja California, Mexico.

Minimum permissible velocity

All water carries of certain amount of suspended solids, in the form of sediments that

have entrained somewhere upstream and are being transported by the flow. Typical

values of suspended load are 200-300 parts per million (or ppm, equivalent, at this

range, to mg/L) (Ponce, 1989). The no-slip condition at the channel boundary (i.e., the

local longitudinal velocity is zero at the channel boundary) produces shear stresses and

results in sediment entrainment. Once entrained, this sediment needs to be transported

through the channel; otherwise, it will settle and eventually clog the channel. The

minimum velocity to avoid settling/clogging is the minimum permissible velocity.

The evaluation of minimum permissible velocity may be accomplished with the Shields

criterion for initiation of motion. This criterion is embodied in the Shields diagram,

shown in Fig. 6-9, which relates the dimensionless shear stress τ* with the boundary

Reynolds number R*. The solid curve in Fig. 6-9 separates no motion (below the curve)

from motion (above the curve).

Fig. 6-9 Shields diagram for initiation of motion (American Society of Civil Engineers, 1975).

Minimum Froude number and minimum permissible velocity

The quadratic friction law, Eq. 5-3, is recast here with τo as the bottom shear stress:

τo = ρ f V 2

(6-1)

The Shields criterion for initiation of motion is:

τo

τ* = _____________ ≥ τ*c

(γs - γ ) ds

(6-2)

in which τ* = dimensionless shear stress, γs = specific weight of sediment particles, γ = specific weight

of water, ds = sediment particle diameter, and τ*c = dimensionless critical shear stress.

Figure 6-9 shows the Shields curve, i.e., the variation of dimensionless critical shear

stressτ*c with the boundary Reynolds number R*:

U* ds

R* = _________

ν

(6-3)

in which U* = shear velocity = (τo /ρ)1/2 = ( f )1/2 V, and ν = kinematic viscosity.

The Froude number F is:

V

F = ___________

(g D )1/2 (6-4)

in which V = mean velocity, D = hydraulic depth, and g = gravitational acceleration.

Replacing Eqs. 6-1 and 6-4 into Eq. 6-2:

f D F 2

_____________________ ≥ τ*c

[ (γs / γ ) - 1 ] ds

(6-5)

In most cases of practical interest, the ratio of specific weights of sediment and water γs /γ is

equal to 2.65. Therefore, in terms of Froude number, the Shields criterion for initiation of

motion reduces to:

1.65 τ*c (ds / D )

F ≥ [ __________________ ] 1/2

f

(6-6)

For practical applications, a constant value of dimensionless critical shear stress τ*c = 0.04

may be considered. From Fig. 6-9, it is seen that this value encompasses the range: 4 ≤ R* ≤

60. Therefore:

0.066 (ds / D )

F ≥ [ ________________ ] 1/2

f

(6-7)

Equation 6-7 is the practical Shields-based Froude criterion for initiation of motion, applicable

through a wide range of boundary Reynolds numbers.

As a practical example of the use of Eq. 6-7, assume ds = 0.4 mm, D = 1 m, and a midrange

dimensionless Chezy friction factor f = 0.0035 (equivalent to a Darcy-Weisbach friction

factorf = 0.028). The application of Eq. 6-7 leads to:

F ≥ 0.087

(6-8)

Combining Eqs. 6-4 and 6-7:

0.066 (g ds)

V ≥ [ ______________ ] 1/2

f

(6-9)

From Eq. 6-9, the minimum permissible velocity for this flow condition is: Vmin = 0.27 m/s.

A somewhat more precise value of minimum Froude number and minimum permissible

velocity may be obtained by using the actual value of dimensionless critical shear stress in

Eq. 6-7, in lieu of the convenient approximation τ*c = 0.04. This procedure, however, requires

an iteration. For this purpose, the following algorithm is suggested:

1. Assume R*

2. Using Fig. 6-9, find τ*c

3. Using Eq. 6-2, calculate τo

4. Using Eq. 6-1, calculate the shear velocity U* = (τo /ρ)1/2

5. Using Eq. 6-3, calculate the new value of R*

6. Stop if the new value of R* is the same as that assumed in Step 1 (within a certain

small tolerance), and use the last value of τ*c (calculated in Step 2) in Eq. 6-6;

7. Otherwise, return to Step 1, using the new value of R* as the assumed value, and

repeat the iteration.

Example 6-1.

Assume the following channel data: particle diameter ds = 0.4 mm, hydraulic depth D = 1 m, dimensionless Darcy-Weisbach

friction factor f = 0.0035, and water temperature T = 20°C. Using the Shields criterion for initiation of motion, calculate the

minimum Froude number and minimum permissible velocity.

ONLINE CALCULATION. Using ONLINE SHIELDS VELOCITY, the minimum Froude number isFmin = 0.081,

and the minimum permissible velocity is Vmin = 0.25 m/s. Note that these results are slightly less conservative than

those obtained using Eqs. 6-7 and 6-9.

Channel slope

The design channel slope is usually governed by the chosen alignment and prevailing

topography. The actual design channel slope may depend on the purpose of the

channel. For example, channels used for irrigation and hydropower require small

slopes, in order not to lose too much head during conveyance.

Side slopes depend on local soil and construction conditions, usually as steep as

practicable. For lined canals, a side slope of 1.5 H : 1 V is recommended as a standard

by the U.S. Bureau of Reclamation, for use in most canals.

For lined canals and steep slopes, when the Vedernikov number exceeds 1, the

possibility arises for the development of roll waves (Chapter 1).

Freeboard

Freeboard is the vertical distance from the top of the channel to the water surface at the

design condition. This distance should be sufficient to prevent waves or fluctuations in

the water surface from overflowing the channel sides. This feature becomes important,

particularly in the design of elevated flumes, for the flume substructure may be

endangered by an overflow (Fig. 6-10).

Fig. 6-10 The Dulzura conduit, in San Diego County, California,

overflowing after heavy rain, on March 5, 2005.

There is no universally accepted rule for the determination of freeboard. Wind-

generated waves or tidal action may induce high waves, which would need to be kept

within channel bounds. Freeboards ranging from less than 5% to more than 30% of the

design flow depth are in common practice. For smooth, interior, semicircular flumes on

tangents, carrying water at velocities no greater than the critical velocity, with a

maximum of 8 ft/s (2.4 m/s), experience has indicated that a freeboard of 6% of the

flume diameter is appropriate. For flumes on curves, freeboard must be increased to

prevent water from sloping over (Chow, 1959). Note that hydraulic design criteria

commonly allow for the use of all or part of the freeboard to accomodate the Probable

Maximum Flood (Ponce, 1989).

According to the U.S. Bureau of Reclamation, the approximate range of freeboard

extends from 1 ft (0.3 m) for small laterals with shallow depths, to 4 ft (1.2 m) in canals

up to 3,000 cfs (85 m3/s) or more. The following formula is applicable:

Fb = ( C y ) 1/2

(6-10)

in which Fb = freeboard, in ft; y = depth of water in the canal, in ft; and C = coefficient,

varying from C = 1.5for a canal capacity of 20 cfs (0.57 m3/s) to C = 2.5 for a canal

capacity of 3,000 cfs (85 m3/s) or more. Figure 6-11 shows U.S. Bureau of Reclamation

recommended height of lining and height of bank, applicable for the design of

freeboard.

Fig. 6-11 U.S. Bureau of Reclamation recommended height of lining

and height of bank (Chow, 1959).

Section dimensions

The following steps are taken to choose the dimensions of a channel or canal cross

section:

1. Select the design discharge Q

2. Select the bottom width b

3. Select the side slope z [z H : 1 V] (Fig. 6-12)

Fig. 6-12 Definition sketch for bottom width b and side slope z.

4. Select the bottom slope S

5. Estimate the value of Manning's n (Chapter 5: Manning's n)

6. Using Q, b, z, S, and n [Steps 1 to 5], compute the normal depth yn, normal

velocity vn, and normal Froude number Fn (Chapter 5: Computation)

7. Check to see if the normal velocity vn and normal Froude number Fn are large

enough to avoid clogging of the canal with sediment (see Minimum permissible

velocity in this Chapter)

8. Select an appropriate freeboard Fb .

Figure 6-13 shows U.S. Bureau of Reclamation recommended bottom width and water

depth of lined channels.

Fig. 6-13 U.S. Bureau of Reclamation recommended bottom width and water depth

of lined channels (Chow, 1959).

Example 6-2.

Design a channel for the following flow conditions: Q = 10 m3/s; b = 5 m; z = 2; S = 0.0016; n = 0.025.

Using Eq. 5-55, by iteration, the normal depth is yn = 1.049 m and the normal velocity is vn = 1.342 m/s. The Froude

number based on normal depth is Fn = 0.418. The calculated Froude number is shown to be much larger than 0.087

(Eq. 6-8), the minimum value required to avoid sediment deposition. A freeboard Fb = 0.6 m is assumed.

ONLINE CALCULATION. Using the ONLINECHANNEL01 calculator, the normal depth is yn = 1.049 m; the

normal velocity is vn = 1.342 m/s; the Froude number (based on hydraulic depth D = 0.81 m) is: Fn = 0.476.

Note that the true Froude number is that based on hydraulic depth (Eq. 4-6). For a hydraulically wide channel, for

which D ≅ R, both Froude numbers are approximately the same.

Figure 6-14 shows a major canal built in 1993 in Ceara, Brazil, for the purpose of

combating a regional drought. The hydraulic characteristics are: Q = 8.32 m3/s, b = 5

m, z = 1.5, S = 0.00005, and n = 0.015. Using the ONLINECHANNEL01 calculator, the

normal depth is yn = 1.915 m, the normal velocity is vn = 0.552 m/s, and the Froude

number is Fn = 0.149.

Cogerh

Fig. 6-14 The Workers' Channel (Canal do Trabalhador), Ceara, Brazil (1993).

6.2 ERODIBLE CHANNELS

[Tractive Force] [Permissible Tractive Force] [Other

Features] [Questions] [Problems] [References] • [Top] [Nonerodible Channels]

The design of an erodible channel is much more complex than that of a nonerodible channel. The long-term

stability of the channel boundary depends on the properties of the material lining the boundary (sand, silt,

clay, and combinations thereof), and of the properties of the sediment being transported by the flow.

Changes in water and sediment discharge result in cross-sectional changes, either channel depth, channel

width, or both.

The stability of the channel boundary is normally assessed in terms of two criteria:

1. Maximum permissible velocity, and

2. Maximum permissible tractive stress.

Given Eq. 6-1, these two methods are related as follows:

τo max = ρ f (Vmax )2

(6-11)

Example 6-3.

Calculate the maximum permissible velocity corresponding to a maximum permissible tractive stress of20 N/m2. Assume f =

0.0035, i.e., Darcy-Weisbach f = 0.028.

Using Eq. 6-11:

Vmax = [τo max / (ρ f ) ] 1/2

= [ 20 N/m2 / (1000 N s

2/m

4 × 0.0035 ) ]

1/2 = 2.39 m/s.

Example 6-4.

Calculate the maximum permissible velocity corresponding to a maximum permissible tractive stress of0.1 lb/ft2. Assume f =

0.005, i.e., Darcy-Weisbach f = 0.04.

Using Eq. 6-11:

Vmax = [τo max / (ρ f ) ] 1/2

= [ 0.1 lb/ft2 / (1.94 lbs-s

2/ft

4 × 0.005 ) ]

1/2 = 3.21 ft/s.

Vegetative channels

Table 6-1 shows typical values of maximum permissible tractive stress for various types of vegetative

channel linings.

Table 6-1 Typical values of maximum

permissible tractive stress.

Channel lining

Maximum

permissible

tractive stress

(N/m2)

Lawn (short-time loaded) 20-30

Lawn (long-time loaded) 15-18

Fascine sausage 60-70

Fascine roll 100-150

Weighted fascine 60-100

Brush mattress 150-300

Live staking in rip rap > 140

Willows/alder 80-140

Gabions 80-140

Figure 6-15 shows details of the design of a channel lining using a brush mattress.

Fig. 6-15 (a) Design of brush mattress: Top view.

Fig. 6-15 (b) Design of brush mattress: Side view.

Fig. 6-15 (c) Design of brush mattress: Side view after a few months.

Fig. 6-15 (d) Design of brush mattress: View after project completion.

Figure 6-16 shows details of a channel lining using live staking in riprap.

Fig. 6-16 (a) Live staking: Schematic.

Fig. 6-16 (b) Live staking: Soon after installation.

Fig. 6-16 (c) Live staking: Sometime after installation.

Fig. 6-16 (d) Live staking: 2-5 years after installation.

Fig. 6-16 (e) Live staking: Established project.

Gabion-lined channels

A gabion system is wire-enclosed riprap consisting of mats or baskets fabricated with wire mesh, filled with

small riprap, and anchored to a slope (Fig. 6-17). Wrapping the riprap enables the use of smaller stone sizes

for the same resistance to displacement by water energy. This is a particular advantage when constructing

rock lining in areas of difficult access. The wire basket also allows steeper (up to vertical) channel linings to

be constructed from commercially available wire units or from wire-fencing material.

Unknown

Fig. 6-17 Placement of gabion mattress.

Gabion systems provide an effective way to control erosion in streams, rivers and canals. They are normally

designed to sustain large channel velocities [5 m/s (15 ft/s) or higher]. Gabions are constructed by individual

units that vary in length from 2 m (6 ft) to nearly 30 m (100 ft); therefore, applications can range anywhere

from small ditches to large canals (Fig. 6-18).

Unknown

Fig. 6-18 Dimensions of gabions boxes and mattresses.

Gabion channels are a compromise between riprap and concrete channels. When the same-size rocks are

used in gabions and riprap, the acceptable velocity for gabions is three to four times that of riprap. Unlike

concrete, gabions can be vegetated to blend into the natural landscape (Fig. 6-19).

Gabion channels with vegetation have the following advantages:

Allow infiltration and exfiltration

Filter out contaminants

More flexible than paved channels

Provide greater energy dissipation than concrete channels

Improve habitat for flora and fauna

More aesthetically pleasing

Lower cost to install, although some maintenance is required.

Values of Manning's n for gabion-lined channels are normally in the range 0.025 ≤ n ≤ 0.030.

Fig. 6-19 Established channel lining with gabions.

6.3 TRACTIVE FORCE

[Permissible Tractive Force] [Other Features] [Questions] [Problems] [References] • [Top] [Nonerodible

Channels] [Erodible Channels]

The tractive force is the summation of the (tractive) shear stresses over (an area of) the channel boundary.

Under equilibrium conditions, to develop the tractive stress equation, the acting gravitational force is made

equal to the resisting frictional force. The gravitational force is (Fig. 6-20):

W sin θ ≅ W tan θ = W (ΔH /Δx )

(6-12)

in which W = weight of the control volume of area A and length Δx, ΔH = drop within the control volume,

and θ = channel bottom slope.

Fig. 6-20 Definition sketch for derivation of the tractive force equation.

Since So = ΔH /Δx, it follows that the acting gravitational force is:

Fg = γ A Δx So

(6-13)

The resisting frictional force is:

Ff = τo P Δx

(6-14)

in which τo = bottom shear stress, and P = wetted perimeter.

Equating gravitational and frictional forces:

τo = γ R So

(6-15)

in which R = A /P = hydraulic radius.

For hydraulically wide channels, for which R ≅ D ≅ y,

τo = γ y So

(6-16)

The tractive stress varies along the wetted perimeter, reaching a peak value along the middle of the channel.

Figure 6-21 shows a typical variation for a trapezoidal cross section, in which b/y = 4 andz = 1.5.

Fig. 6-21 Variation of tractive stress in a trapezoidal channel, with b/y = 4 and z = 1.5 (Chow, 1959).

At the bottom center, the tractive stress approaches asymptotically its full (maximum) value: τb = γySo ,as

shown in Fig. 6-22. Along the channel sides, the tractive stress approaches asymptotically a fraction of the

full value: τs = 0.78 γySo , as shown in Fig. 6-23.

Fig. 6-22 Variation of maximum tractive stress on channel bottom

with aspect ratio b/y (Chow, 1959).

Fig. 6-23 Variation of maximum tractive stress on channel sides

with aspect ratio b/y (Chow, 1959).

Tractive force ratio

Figure 6-24 shows three particles on the channel boundary, one on the left side,

another on the right side, and the third one on level ground (bottom). The tractive force

ratio K is defined as:

a τs τs

K = _______ = ______

a τL τL (6-17)

in which a = effective area of the particle, τs = shear stress on the side, and τL = shear

stress on level ground.

Fig. 6-24 Definition sketch for the forces acting on a particle resting on the surface

of a channel bed (Chow, 1959).

The ratio K is a function of the channel side slope υ and of the angle of repose θ of the

material forming the channel bed. To derive K, consider the two forces that are acting

on a particle of submerged weight Ws resting on the side of the channel, with tractive

stress τs (Fig. 6-20):

1. The tractive force: aτs

2. The gravitational force component along the side: Ws sin υ

The resultant force acting along the side plane is:

Fa = (Ws2 sin2φ + a2τs

2)1/2

(6-18)

At equilibrium, the resisting force is equal to the acting force. From solid mechanics, the

resisting force is equal to the normal force (Ws cosυ ) times the coefficient of friction

(tanθ ):

Fr = Ws cosφ tanθ

(6-19)

Therefore:

Ws cosφ tanθ = (Ws2 sin2φ + a2τs

2)1/2

(6-20)

Squaring both sides:

Ws2 cos2φ tan2θ = Ws

2 sin2φ + a2τs2

(6-21)

a2τs2 = Ws

2 cos2φ tan2θ - Ws2 sin2φ

(6-22)

τs2 = (Ws /a)2 cos2φ tan2θ - (Ws /a)2 sin2φ

(6-23)

tan2φ

τs2 = (Ws /a)2 cos2φ tan2θ [ 1 - _________ ]

tan2θ

(6-24)

tan2φ

τs = (Ws /a) cosφ tanθ [ 1 - _________ ] 1/2

tan2θ

(6-25)

Equation 6-25 is the shear stress on the side of a channel with side slope angle υ and

repose angleθ. For a level surface, with υ = 0: sinυ = 0 and cosυ = 1. Thus, the force

balance of Eq. 6-20 reduces to:

Ws tanθ = aτL

(6-26)

Therefore, the shear stress that causes impending motion on a level surface is:

τL = (Ws /a) tanθ

(6-27)

Combining Eqs. 6-17, 6-25, and 6-27, the tractive force ratio is:

tan2φ

K = cosφ [ 1 - _________ ] 1/2

tan2θ

(6-28)

It is seen that K is only a function of υ and θ. Equation 6-28 is equivalent to (See box

below):

sin2φ

K = [ 1 - __________ ] 1/2

sin2θ

(6-29)

Derivation of Eq. 6-29

Equation 6-28 is simplified by resorting to established trigonometric identities:

tan2φ

K = cosφ [ 1 - _________

] 1/2

tan2θ

(6-28a)

cos2φ tan

2φ

K = [ cos2φ -

________________ ]

1/2

tan2θ

(6-28b)

sin2φ (6-28c)

K = [ cos2φ -

_________ ]

1/2

tan2θ

sin2φ cos

2θ

K = [ cos2φ -

________________ ]

1/2

sin2θ

(6-28d)

sin2φ (1 - sin

2θ)

K = [ 1 - sin2φ -

_____________________ ]

1/2

sin2θ

(6-28e)

sin2φ

K = [ 1 - __________

] 1/2

sin2θ

(6-29)

Angle of repose

Figure 6-25 shows the angle of repose of noncohesive materials, i.e., the angle θ in Eq.

6-29. The particle size (in inches) is that for which 25% of the material, by weight, is

larger.

Fig. 6-25 Angle of repose of noncohesive materials (Chow, 1959).

6.4 PERMISSIBLE TRACTIVE FORCE

[Other Features] [Questions] [Problems] [References] • [Top] [Nonerodible Channels] [Erodible Channels] [Tractive

Force]

The permissible tractive force (unit force, or stress) method is used in the design of

erodible channels and canals. The permissible unit tractive force is the maximum shear

stress that will not cause erosion of the material forming the channel bed on a level

surface. The value, obtained from laboratory experiments, is referred to as the critical

unit tractive force, or critical shear stress.

Figure 6-26 shows permissible unit tractive force (unit tractive stress) for canals in

noncohesive materials. The figure applies through a range of (average) particle

diameter from 0.1 to 100 mm. The left side of the graph applies for fine noncohesive

material (sand), with diameter ranging from 0.1 to5 mm. Three curves are shown:

1. The upper curve, applicable for canals with high content of fine sediment (silt) in

the water;

2. The middle curve, applicable for canals with low content of fine sediment (silt) in

the water;

3. The lower curve, applicable for canals with clear water.

Note that the cleaner the water, the more likely it is to pick up sediment from the

boundary and, therefore, the lower the value of the permissible unit tractive force.

The right side of Fig. 6-26 applies for canals in coarse noncohesive material, with

diameter ranging from 5 to 100 mm (fine to very coarse gravel, to small cobbles). The

line shown in Fig. 6-26 is equal to 0.4 times the particle diameter (in inches) for which

25% of the material, by weight, i.e., permissible tractive stress = 0.4 × d25 (in).

Fig. 6-26 Permissible unit tractive force for canals in noncohesive materials (Chow, 1959).

For cohesive materials, the critical shear stress is a function of the type of soil (sandy

clays to lean clays) and the voids ratio, as shown in Fig. 6-27.

Fig. 6-27 Permissible unit tractive force for canals in cohesive materials (Chow, 1959).

The values of Figs. 6-26 and 6-27 are applicable to straight channels. For sinuous

channels, the values shown in the figures should be reduced to account for bank scour.

Table 6-2 shows recommended percentage reduction in critical shear stress to account

for channel sinuosity.

Table 6-2 Reduction in critical shear stress

to account for channel sinuosity.

Sinuosity Reduction

(%)

Low 10

Moderate 25

High 40

Permissible tractive force method

The permissible tractive force method is a method to design erodible channels based

on the estimated critical shear stress for the material(s) forming the channel boundary.

The value of critical shear stress is determined experimentally or from established

experience. The computational procedure is described below, along with a worked

example, for two possible cases:

A. Use of the same material on sides and bottom, and

B. Use of one material on the sides and another on the bottom.

A. The same material on sides and bottom

Input data: Discharge Q, side slope z, channel slope S,

and Manning's n; particle d25 and grain shape; material

on sides and bottom are the same.

1. Assume b/y = 6, i.e., a reasonable value to start.

2. Assume that the tractive force on the sides is

controlling the design. This is the usually case

when the material on the sides and bottom are

the same.

3. With b/y and z, enter Fig. 6-23 to determine the

value of Cs (Cs is the ordinate of Fig. 6-23) on

the expression for the acting unit tractive force

on the sides: Ts = Cs γ y S.

4. With d25 and grain shape, use Fig. 6-25 to find

the angle of repose θ.

5. Calculate side slope: φ = tan-1

(1/z)

6. Use Eq. 6-29 to calculate K.

7. Use Fig. 6-26 to determine the permissible unit

tractive force on level ground τL.

8. Calculate the permissible unit tractive force on

the sides: τs = K τL.

9. Set the permissible unit tractive force greater

than or equal to the acting unit tractive

A. Worked example

Input data: Q = 600 cfs; z = 2, channel slope S =

0.001, and Manning's n = 0.022; particle d25 = 0.7 in,

and slightly angular grain shape; material on sides and

bottom are the same.

1. Assume b/y = 6.

2. Assume that the tractive force on the sides is

controlling the design.

3. With b/y = 6 and z = 2, enter Fig. 6-23 to

determine the value of Cs = 0.78.

4. With d25 = 0.7 in and slightly angular grain

shape, use Fig. 6-25 to find θ = 34°.

5. φ = tan-1

(1/z) = 26.565°

6. Use Eq. 6-29 to calculate K = 0.6.

force:τs ≥ Ts; i.e., τs ≥ Cs γ y S

10. Solve for flow depth: y = τs / (Cs γ S)

11. Calculate b: b = (b/y) y

12. With Q, b, z, n, and S known, solve for normal

depth yn.

13. Test to confirm that yn ≤ y. If not, assumed

value of b/y is too small. Assume a larger value

and return to Step 11 and iterate. Once

satisfied, set y = yn and go to next step.

14. With last b/y and z, enter Fig. 6-22 to determine

the value of Cb (the ordinate of Fig. 6-22) on

the expression for the acting unit tractive force

on level ground TL =Cb γ y S.

15. Calculate TL = Cb γ yn S

16. Compare TL calculated in Step 15

with τLcalculated in Step 7.

17. If TL ≤ τL, the sides control the design, as

initially assumed. Stop here.

7. Use d25 = 0.7 inches in Fig. 6-26 to

determine τL = 0.28 lbs/ft2.

8. τs = K τL = 0.6 × 0.28 = 0.168 lbs/ft2.

9. Set the permissible unit tractive force greater

than or equal to the acting unit tractive

force:τs = 0.168 ≥ Cs γ y S = (0.78)

(62.4) y (0.001)

10. Solve for flow depth: y ≤ 3.45 ft. Set y = 3.45

as the target flow depth.

11. b = (6) (3.45) = 20.7. Round to b = 21 ft.

12. With Q = 600, b = 21, z = 2, n = 0.022, andS =

0.001, solve for normal depth: yn = 4.536 ft.

13. yn = 4.536 > y = 3.45. The width b is too short.

Assume (by trial and error) b/y = 10. b= (10)

(3.45) = 34.5. Round to b = 35 ft. With new b,

solve for normal depth: yn = 3.376 ft.

Now yn < y. Set y = 3.376 ft and go to next step.

14. With b/y = 10 and z = 2, enter Fig. 6-22 to

determine the value of Cb = 1.0.

15. Calculate TL = (1.0) (62.4) (3.376) (0.001) =

0.211 lbs/ft2.

16. Compare TL = 0.211 calculated in Step 15

with τL = 0.28 calculated in Step 7.

17. TL = 0.211 < τL = 0.28; therefore, the sides

control the design, as initially assumed. Stop

here.

B. One material on the sides and another on the

bottom

Input data: Discharge Q, side slope z, channel slope S,

and Manning's n; material on sides and bottom are

different, specify type of material, particle size and grain

shape, and fine sediment content if required.

1. Assume b/y = 6, i.e., a reasonable value to start.

2. Assume that the tractive force on the sides is

controlling the design. This is the usually case

when the material on the sides and bottom are

the same.

3. With b/y and z, enter Fig. 6-23 to determine the

value of Cs (Cs is the ordinate of Fig. 6-23) on

the expression for the acting unit tractive force

B. Worked example

Input data: Q = 600 cfs; z = 2, channel slope S =

0.001, and Manning's n = 0.022; material on the

sides: noncohesive, d25 = 0.7 in, slightly angular grain

shape; material on the bottom: noncohesive,d60 = 0.8

mm, with high content of fine sediment in the water.

1. Assume b/y = 6.

2. Assume that the tractive force on the sides is

controlling the design.

on the sides: Ts = Cs γ y S.

4. With d25 and grain shape, use Fig. 6-25 to find

the angle of repose θ.

5. Calculate side slope: φ = tan-1

(1/z)

6. Use Eq. 6-29 to calculate K.

7. Use Fig. 6-26 to determine the permissible unit

tractive force on level ground based on the

material from the sides τLs (right of Fig. 6-26),

and the permissible unit tractive force on level

ground based on the material from the

bottom τLb (left of Fig. 6-26)

8. Calculate the permissible unit tractive force on

the sides: τs = K τLs.

9. Set the permissible unit tractive force greater

than or equal to the acting unit tractive

force:τs ≥ Ts; i.e., τs ≥ Cs γ y S

10. Solve for flow depth: y = τs / (Cs γ S)

11. Calculate b: b = (b/y) y

12. With Q, b, z, n, and S known, solve for normal

depth yn.

13. Test to confirm that yn ≤ y. If not, assumed

value of b/y is too small. Assume a larger value

and return to Step 11 and iterate. Once

satisfied, set y = yn and go to next step.

14. With last b/y and z, enter Fig. 6-22 to determine

the value of Cb (the ordinate of Fig. 6-22) on

the expression for the acting unit tractive force

on level ground TL =Cb γ y S.

15. Calculate TL = Cb γ yn S

16. Compare TL calculated in Step 15

with τLbcalculated in Step 7.

17. If TL ≤ τL, the sides control the design, as

initially assumed. Instead, if TL > τL, the bottom

controls the design.

18. If TL > τL, force TL = τL.

19. Solve for new y, confirming bottom control.

3. With b/y = 6 and z = 2, enter Fig. 6-23 to

determine the value of Cs = 0.78.

4. With d25 = 0.7 in and slightly angular grain

shape, use Fig. 6-25 to find θ = 34°.

5. φ = tan-1

(1/z) = 26.565°

6. Use Eq. 6-29 to calculate K = 0.6.

7. Use Fig. 6-26 (right) to determineτLs = 0.28

lbs/ft2; use Fig. 6-26 (left) to determine τLb =

0.09 lbs/ft2.

8. τs = K τLs = 0.6 × 0.28 = 0.168 lbs/ft2.

9. Set the permissible unit tractive force greater

than or equal to the acting unit tractive

force:τs = 0.168 ≥ Cs γ y S = (0.78)

(62.4) y (0.001)

10. Solve for flow depth: y ≤ 3.45 ft.

11. b = (6) (3.45) = 20.7. Round to b = 21 ft.

12. With Q = 600, b = 21, z = 2, n = 0.022, andS =

0.001, solve for normal depth: yn = 4.536 ft.

13. yn = 4.536 > y = 3.45. The width b is too short.

Assume (by trial and error) b/y = 10. b= (10)

(3.45) = 34.5. Round to b = 35 ft. With new b,

solve for normal depth: yn = 3.376 ft.

Now yn < y. Set y = 3.376 ft and go to next step.

14. With b/y = 10 and z = 2, enter Fig. 6-22 to

determine the value of Cb = 1.0.

15. Calculate TL = (1.0) (62.4) (3.376) (0.001) =

0.211 lbs/ft2.

16. Compare TL = 0.211 calculated in Step 15

with τLb = 0.09 calculated in Step 7.

17. TL = 0.211 > τL = 0.09; therefore, the bottom

controls the design.

18. Force TL = Cb γ yn S =(1.0) (62.4) y (0.001) =

0.09. Therefore: y = 1.44 ft. Assume (by trial

and error) b/y = 106. Therefore: b = 154 ft.

19. With Q = 600, b = 152, z = 2, S = 0.001, andn =

0.022, calculate y = 1.44 ft. Design now OK!

6.5 OTHER FEATURES

[Questions] [Problems] [References] • [Top] [Nonerodible Channels] [Erodible Channels] [Tractive

Force] [Permissible Tractive Force]

Other features in hydraulic channel design include canal drops and creek crossings,

dissipation structures, and grade control structures. Figures 6-28 to 6-33 show some

illustrative examples.

Fig. 6-28 A series of canal drops for the purpose of controlling flow instability,

Cabana-Ma�azo irrigation project, Puno, Peru.

Fig. 6-29 Crossing of Tinajones feeder canal through Chiriquipe Creek, Lambayeque, Peru.

Fig. 6-30 A small creek bypass, Wellton-Mohawk Canal, Wellton, Arizona.

Fig. 6-31 A canal crossing a stream by means of a siphon,

Cabana-Ma�azo irrigation project, Puno, Peru.

Fig. 6-32 Dissipation structure, Mashcon river, Cajamarca, Peru.

Fig. 6-33 Grade control structure, Caqueza river, Cundinamarca, Colombia.

QUESTIONS

[Problems] [References] • [Top] [Nonerodible Channels] [Erodible Channels] [Tractive Force] [Permissible Tractive

Force] [Other Features]

1. What determines the surface roughness in an artificial canal?

2. What is the freeboard in the design of a canal?

3. What is the abscissa in the Shields diagram?

4. What is the ordinate in the Shields diagram?

5. What Froude number will generally assure initiation of motion?

6. How are the minimum and maximum permissible velocities reconciled in a

channel design?

7. How are the maximum permissible velocities and shear stresses related?

8. What is the maximum value of the coeeficient Cs for shear stress on the channel

sides?

9. What is the range of angle of repose of noncohesive materials?

10. What is the tractive force ratio?

11. How does the content of fine sediment in the water affect the value of

permissible unit tractive force?

12. When is a grade control structure justified?

PROBLEMS

[References] • [Top] [Nonerodible Channels] [Erodible Channels] [Tractive Force] [Permissible Tractive Force] [Other

Features] [Questions]

1. What is the minimum permissible velocity for a sediment particle diameter ds =

0.6 mm and a dimensionless Chezy (modified Darcy-Weisbach) friction factor f =

0.004?

2. What is the minimum Froude number and minimum permissible velocity for a

sediment particle diameter ds = 0.6 mm, dimensionless Chezy friction factor f =

0.004, hydraulic depth D = 1 m, and water temperature T = 20°C?

3. What is the minimum Froude number and minimum permissible velocity for a

sediment particle diameter ds = 0.3 mm, dimensionless Chezy friction factor f =

0.003, hydraulic depth D = 3 ft, and water temperature T = 68°F?

4. A channel has the following data: Q = 330 cfs, z = 2, n = 0.025, and S = 0.0018.

Use the tractive force method to calculate the bottom width and depth under the

following conditions: a. The particles are the same in sides and bottom; they are moderately

angular and of sized25 = 0.9 in. b. Same as in (a), but the particles are moderately rounded.

Discuss how the particle shape affects the design. Verify with ONLINE

TRACTIVE FORCE.

5. A channel has the following data: Q = 220 m3/s, z = 2, n = 0.03, and S = 0.0006.

Use the tractive force method to calculate the bottom width and depth under the

following conditions: a. The particles on the sides are slightly angular and of size d25 = 35 mm; the

particles on the bottom are of size d50 = 5 mm. b. The particles on the sides are the same as in (a), but the particles on the

bottom are smaller, of size d50 = 4 mm.

Both cases (a) and (b) have low content of fine sediment in the water and the

channel sinuosity is negligible. Discuss how the bottom particle size affects the

design. Verify with ONLINE TRACTIVE FORCE.

6. A certain type of lawn has a critical shear stress τc = 30 N/m2. The dimensionless

Chezy friction factor f = 0.0075. What is a good estimate of the critical velocity?

REFERENCES

• [Top] [Nonerodible Channels] [Erodible Channels] [Tractive Force] [Permissible Tractive Force] [Other

Features] [Questions] [Problems]

American Society of Civil Engineers, 1975. Sedimentation Engineering. Manual of

Practice No. 54.

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

Ponce, V. M. 1989. Engineering Hydrology, Principles and Practices. Prentice Hall,

Englewood Cliffs, New Jersey.

http://openchannelhydraulics.sdsu.edu

140902 09:15

[Equation of Gradually Varied Flow] [Characteristics of Flow Profiles] [Limits to Water Surface

Profiles] [Methodologies] [Direct Step Method Example] [Questions] [Problems] [References] •

CHAPTER 7:

STEADY GRADUALLY VARIED

FLOW

7.1 EQUATION OF GRADUALLY VARIED FLOW

[Characteristics of Flow Profiles] [Limits to Water Surface Profiles] [Methodologies] [Direct Step Method

Example] [Questions] [Problems] [References] • [Top]

The flow is gradually varied when the discharge Q is constant but the other hydraulic

variables (A,V, D, R, P, and so on) vary gradually in space. The basic assumptions of

gradually varied flow are:

1. The flow is steady, i.e., none of the hydraulic variables vary in time.

2. The streamlines are essentially parallel; thus, the pressure distribution in the

vertical direction is hydrostatic, i.e., proportional to the flow depth.

3. The head loss is the same as that corresponding to uniform flow; therefore, the

uniform flow formula may be used to evaluate the energy slope.

4. The value of Manning's n is the same as that of uniform flow.

Other assumptions of gradually varied flow are:

1. The slope of the channel is small.

2. The pressure correction factor cosθ ≅ 1.

3. There is negligible air entrainment.

4. The conveyance is an exponential function of the flow depth (except for the case

of circular culverts).

5. The roughness (Manning's n) is independent of the flow depth (only an

approximation) and is constant throughout the reach under consideration.

Fig. 7-1 Definition sketch for energy in open-channel flow.

Under gradually varied flow, the gradient of hydraulic head is (Fig. 7-1):

dH d V 2 _____ = ___ ( z + y + _____ ) = - Sf

dx dx 2g

(7-1)

The negative sign in front of the friction slope Sf is required because the flow direction

is from left to right, while the derivative is taken from right to left, by convention. By

definition, the friction slope is:

hf

Sf = _____

ΔL

(7-2)

in which ΔL = length of the channel reach under consideration.

The gradient of specific energy is:

dE d V 2 dz _____ = ___ ( y + _____ ) = - ____ - Sf

dx dx 2g dx

(7-3)

The gradient of the channel bed, or channel slope (bottom slope), is:

dz z2 - z1 _____ = ________

dx ΔL

(7-4)

dz z1 - z2

- _____ = ________ = So

dx ΔL

(7-5)

Therefore, the gradient of specific energy is:

dE d V 2 _____ = ___ ( y + _____ ) = So - Sf

dx dx 2g

(7-6)

Under steady flow: Q = V A = constant. Therefore:

d Q 2 ____ ( y + _______ ) = So - Sf

dx 2g A2

(7-7)

dy d Q 2 _____ + _____ ( _______ ) = So - Sf

dx dx 2g A2

(7-8)

dy Q 2 dA _____ - ( ______ ) _____ = So - Sf

dx g A3 dx

(7-9)

dy Q 2 dA dy _____ - ( ______ ) _____ _____ = So - Sf

dx g A3 dy dx

(7-10)

Using Eq. 3-11:

dy Q 2 T dy _____ - ( ________ ) _____ = So - Sf

dx g A3 dx

(7-11)

Therefore, the flow-depth gradient is:

dy So - Sf _____ = _______________________

dx 1 - [(Q 2 T ) / (g A3)]

(7-12)

The friction slope based on the Chezy equation (Eqs. 5-10 and 2-4) is:

Q 2

Sf = ____________

C 2 A2 R

(7-13)

Since R = A / P :

Q 2 P

Sf = _________

C 2 A3

(7-14)

Substituting Eq. 7-14 into Eq. 7-12, the flow-depth gradient is:

dy So - [(Q 2 P ) / (C 2 A3)] _____ = ___________________________

dx 1 - [(Q 2 T ) / (g A3)]

(7-15)

dy So - (g/C 2) (P / T ) [(Q 2 T ) / (g A3)] _____ = _______________________________________

dx 1 - [(Q 2 T ) / (g A3)]

(7-16)

The square of the Froude number is (Eq. 3-12):

Q 2 T

F 2 = _________

g A3

(7-17)

Substituting Eq. 7-17 into Eq. 7-16:

dy So - (g/C 2) (P / T ) F 2 _____ = _________________________

dx 1 - F 2

(7-18)

Substituting Eq. 5-12 into Eq. 7-18:

dy So - f (P / T ) F 2 _____ = _____________________

dx 1 - F 2

(7-19)

Therefore, the depth gradient (dy/dx) is a function of:

1. Channel slope So,

2. Friction coefficient f,

3. Ratio of wetted perimeter to top width P / T, and

4. Froude number.

For dy/dx = 0, Eq. 7-19 reduces to a statement of uniform flow:

So = f (P / T ) F 2

(7-20)

For F = 1, Eq. 7-20 reduces to a statement of critical uniform flow:

So = f (Pc / Tc ) = Sc

(7-21)

in which Sc = critical slope, i.e., the channel slope for which the flow is critical.

In terms of critical slope (Eq. 7-21), the flow-depth gradient is:

dy So - (P / T ) (Tc / Pc ) Sc F 2

_____ = ________________________________

dx 1 - F 2

(7-22)

For (P / T ) ≅ (Pc / Tc ), i.e., for a constant ratio (P / T) with flow depth, Eq. 7-22 reduces

to:

dy So - Sc F 2

_____ = _______________ (7-23)

dx 1 - F 2

For conciseness, the flow-depth gradient can be written as:

dy

Sy = _____

dx

(7-24)

Substituting Eq. 7-24 into Eq. 7-23, the flow-depth gradient is:

Sy (So / Sc) - F 2

____ = __________________

Sc 1 - F 2

(7-25)

Equation 7-25 (or 7-23) is the steady gradually varied flow equation (Fig. 7-2). The

depth gradient Syis a function only of: (1) channel slope So, (2) critical slope Sc, and (3)

Froude number F.

Fig. 7-2 Steady gradually varied flow.

Note on the applicability of Eq. 7-25

Strictly speaking, Eq. 7-25 applies only for the case (P / T ) (Tc / Pc ) = 1, which is the same as (P / T )

= (Pc / Tc ); that is, for a constant ratio (P / T ), regardless of flow depth. This condition is less

restrictive that the (asymptotic) hydraulically wide channel condition, for which (P / T ) = 1.

Therefore, for a hydraulically wide channel, for which P ≅ T, it follows that: (P / T ) (Tc / Pc ) ≅

1. Thus, it is concluded that Eq. 7-25 applies for a hydraulically wide channel.

7.2 CHARACTERISTICS OF FLOW PROFILES

[Limits to Water Surface Profiles] [Methodologies] [Direct Step Method

Example] [Questions] [Problems] [References] • [Top] [Equation of Gradually Varied Flow]

In Eq. 7-25, the sign of the left-hand side (LHS) is that of Sy (numerator),

since Sc (denominator) is always positive (friction is always positive). The sign of Sy (i.e,

the sign of the LHS) may be one of three possibilities:

A positive value, leading to RETARDED FLOW (BACKWATER), A zero value, leading to UNIFORM FLOW (NORMAL), or A negative value, leading to ACCELERATED FLOW (DRAWDOWN).

In the right-hand side (RHS) of Eq. 7-25, there are three possibilities for the numerator

(USDA Soil Conservation Service, 1971):

So / Sc > F 2

, leading to SUBNORMAL FLOW, So / Sc = F

2, leading to NORMAL FLOW, or

So / Sc < F 2

, leading to SUPERNORMAL FLOW.

There are three possibilities for the denominator:

1 > F 2

, leading to SUBCRITICAL FLOW, 1 = F

2, leading to CRITICAL FLOW, or

1 < F 2

, leading to SUPERCRITICAL FLOW.

Given the above inequalities, there arise three types (or families) of water-surface

profiles, shown in Table 7-1. The total number of profiles is 12. A summary is shown in

Table 7-2.

Table 7-1 Types of water-surface profiles.

Type Description Numerator and denominator of RHS

of Eq. 7-25

Sign

of

LHS

Flow

profile

I Subnormal/subcritical flow Both numerator and denominator are positive + Retarded

II

A Subnormal/supercritical flow Numerator is positive and denominator is negative - Accelerated

B Supernormal/subcritical flow Numerator is negative and denominator is positive - Accelerated

III Supernormal/supercritical flow Both numerator and denominator are negative + Retarded

Table 7-2 Summary of water-surface profiles.

Family Character Rule So > Sc So = Sc So < Sc So = 0 So < 0

I Retarded

(Backwater) 1 > F

2 < (So / Sc) S1 C1 M1 - -

II

A Accelerated

(Drawdown) 1 < F

2 < (So / Sc) S2 - - - -

B Accelerated

(Drawdown) 1 > F

2 > (So / Sc) - - M2 H2 A2

III Retarded

(Backwater) 1 < F

2 > (So / Sc) S3 C3 M3 H3 A3

Type I

In the Type I family, the flow is subnormal/subcritical. Therefore, the rule is:

1 > F 2 < (So / Sc )

(7-26)

which is the same as:

(So / Sc ) > < 1

(7-27)

Equation 7-27 states that So may be lesser than, equal to, or greater than Sc . This gives rise to three types of

profiles:

M1: So < Sc

Fig. 7-3 Definition sketch for M1 water-surface profile.

The depth gradient Sy varies from asymptotic to So (i.e., asymptotic to the horizontal) at the

downstream end to asymptotic to zero (i.e., asymptotic to the normal depth) at the upstream end.

C1: So = Sc

Fig. 7-4 Definition sketch for C1 water-surface profile.

The depth gradient Sy varies from Sc at the downstream end to Sc at the upstream end, i.e., the water-

surface profile is a horizontal line.

S1: So > Sc

Fig. 7-5 Definition sketch for S1 water-surface profile.

The depth gradient Sy varies from asymptotic to So (i.e., asymptotic to the horizontal) at the

downstream end to asymptotic to + ∞ (i.e., asymptotic to the hydraulic jump) at the upstream end.

Since

(So / Sc ) > F 2

(7-28)

and

F 2 > 0

(7-29)

it follows that

(So / Sc ) > 0

(7-30)

Thus:

So > 0

(7-31)

Therefore, no horizontal (H) or adverse (A) profiles are possible in the Type I family of water-surface

profiles.

Type II A

In the Type II A family, the flow is subnormal/supercritical. Therefore, the rule is:

1 < F 2 < (So / Sc )

(7-32)

which is the same as:

(7-33)

(So / Sc ) > 1

Equation 7-33 states that So may only be greater than Sc . This gives rise to only one profile:

S2: So > Sc

Fig. 7-6 Definition sketch for S2 water-surface profile.

The depth gradient Sy varies from - ∞ (i.e., asymptotic to an abrupt slope break) at the upstream end

to asymptotic to zero (i.e., asymptotic to the normal depth) at the downstream end.

Since

(So / Sc ) > F 2

(7-34)

and

F 2 > 0

(7-35)

it follows that

(So / Sc ) > 0

(7-36)

Thus:

So > 0

(7-37)

Therefore, no horizontal (H) or adverse (A) profiles are possible in the Type II A family of water-surface

profiles.

Type II B

In the Type II B family, the flow is supernormal/subcritical. Therefore, the rule is:

1 > F 2 > (So / Sc )

(7-38)

which is the same as:

(So / Sc ) < 1

(7-39)

Equation 7-39 states that So may be lesser than Sc , equal to 0, or lesser than 0. This gives rise to three types

of profiles:

M2: 0 < So < Sc

Fig. 7-7 Definition sketch for M2 water-surface profile.

The depth gradient Sy varies from - ∞ (i.e., abrupt slope break) at the downstream end to asymptotic

to zero (i.e., asymptotic to the normal depth) at the upstream end.

H2: 0 = So < Sc

Fig. 7-8 Definition sketch for H2 water-surface profile.

The depth gradient Sy varies from - ∞ (i.e., abrupt slope break) at the downstream end to asymptotic

to 0 (to headwater) at the upstream end.

A2: So < 0 < Sc

Fig. 7-9 Definition sketch for A2 water-surface profile.

The depth gradient Sy varies from - ∞ (i.e., abrupt slope break) at the downstream end to asymptotic

to < 0 (to headwater) at the upstream end.

Type III

In the Type III family, the flow is supernormal/supercritical. Therefore, the rule is:

1 < F 2 > (So / Sc )

(7-44)

which is the same as:

(So / Sc ) > < 1

(7-45)

Equation 7-45 states that So may be lesser than, equal to, or greater than Sc . This gives rise to five types of

profiles:

S3: So > Sc

Fig. 7-10 Definition sketch for S3 water-surface profile.

The depth gradient Sy varies from asymptotic to Sc at the upstream end to asymptotic to zero (i.e.,

asymptotic to the normal depth) at the downstream end.

C3: So = Sc

Fig. 7-11 Definition sketch for C3 water-surface profile.

The depth gradient Sy varies from Sc at the downstream end to Sc at the upstream end, i.e., the water-

surface profile is a horizontal line.

M3: 0 < So < Sc

Fig. 7-12 Definition sketch for M3 water-surface profile.

The depth gradient Sy varies from asymptotic to Sc at the upstream end to asymptotic to + ∞ (i.e.,

asymptotic to the hydraulic jump) at the downstream end.

H3: 0 = So < Sc

Fig. 7-13 Definition sketch for H3 water-surface profile.

The depth gradient Sy varies from asymptotic to Sc at the upstream end to asymptotic to + ∞ (i.e.,

asymptotic to the hydraulic jump) at the downstream end.

A3: So = Sc

Fig. 7-14 Definition sketch for A3 water-surface profile.

The depth gradient Sy varies from asymptotic to Sc at the upstream end to asymptotic to + ∞ (i.e.,

asymptotic to the hydraulic jump) at the downstream end.

Figure 7-15 shows a graphical representation of flow-depth gradients in water-surface

profile computations. The arrows indicate the direction of computation.

Fig. 7-15 Graphical representation of flow-depth gradient ranges

in water-surface profile computations.

7.3 LIMITS TO WATER SURFACE PROFILES

[Methodologies] [Direct Step Method Example] [Questions] [Problems] [References] • [Top] [Equation of Gradually

Varied Flow] [Characteristics of Flow Profiles]

The flow-depth gradients vary between five (5) limits (Fig. 7-15):

1. The channel slope So

2. The critical slope Sc

3. Zero.

4. + ∞

5. - ∞

The theoretical limits to the water-surface profiles may be analyzed using Eq. 7-25,

repeated here in a slightly different form:

So - Sc F 2

Sy = ______________

1 - F 2

(7-46)

Operating in Eq. 7-46:

Sy (1 - F 2) = So - Sc F 2

(7-47)

So - Sy

F 2 = ____________

Sc - Sy

(7-48)

For uniform (normal) flow: Sy = 0, and Eq. 7-48 reduces to:

So = Sc F 2

(7-49)

For gradually varied flow: Sy ≠ 0, and Eq. 7-48 is subject to three (3) cases:

ONE

F 2

> 0

a. So > Sy and Sc > Sy

The following inequality is satisfied: So > Sy < Sc

b. So < Sy and Sc < Sy

The following inequality is satisfied: So < Sy > Sc

c. It is concluded that Sy has to be either less than both So and Sc, or greater than both.

TWO

F 2

= 0

This leads to: So = Sy

z1 - z2

So = _________

L

(7-51)

y2 - y1

Sy = _________

L

(7-52)

Combining Eqs. 7-51 and 7-52:

z1 + y1 = z2 + y2

(7-53)

Equation 7-53 depicts a true reservoir (Fig. 7-15).

Fig. 7-16 True reservoir condition.

1.

THREE

2. F 2

< 0: Since F ≥ 0, this condition is impossible.

3. The following inequality is NOT satisfied: So > Sy > Sc

4. The following inequality is NOT satisfied: So < Sy < Sc

5. It is concluded that Sy cannot be less than So and greater than Sc , or less than Sc and greater

than So . Thus, Sy has to be less than both So AND Sc, or greater than both (See Case 1).

Uses of water-surface profiles

Table 7-3 and Fig. 7-17 show the typical occurrence of mild water-surface profiles.

Table 7-3 Occurrence of mild water-surface profiles.

M1 Flow in a mild channel, upstream of a reservoir.

M2 Flow in a mild channel, upstream of an abrupt change in grade or a steep channel carrying supercritical

flow.

M3 Flow in a mild channel, downstream of a steep channel carrying supercritical flow.

Fig. 7-17 Typical occurrence of mild profiles.

Table 7-4 and Fig. 7-18 show the typical occurrence of steep water-surface profiles.

Table 7-4 Occurrence of steep water-surface profiles.

S1 Flow in a steep channel, upstream of a reservoir.

S2 Flow in a steep channel, downstream of a mild channel carrying subcritical flow.

S3 Flow in a steep channel, downstream of a steeper channel carrying supercritical flow.

Fig. 7-18 Typical occurrence of steep profiles.

7.4 METHODOLOGIES

[Direct Step Method Example] [Questions] [Problems] [References] • [Top] [Equation of Gradually Varied

Flow] [Characteristics of Flow Profiles] [Limits to Water Surface Profiles]

There are two ways to calculate water-surface profiles:

1. The direct step method.

2. The standard step method.

The direct step method is applicable to prismatic channels, while the standard step

method is applicable to any channel, prismatic and nonprismatic (Table 7-5). The direct

step method is direct, readily amenable to use with a spreadsheet, and relatively

straight forward in its solution. The standard step method is iterative and complex in its

solution. In practice, the standard step method is represented by the Hydrologic

Engineering Center's River Analysis System, referred to as HEC-RAS (U.S. Army

Corps of Engineers, 2014).

The direct step method applies particularly where data is scarce and resources are

limited. The standard step method applies for comprehensive projects. The use of a

widely accepted government program such as HEC-RAS enhances credibility.

The required number of cross sections in the standard step method increases with the

channel slope. Steeper channels may require more cross sections. Lesser cross

sectional variability results in more reliable and accurate results. Note that extensive

two- and three-dimensional flow features may not be accurately represented in the one-

dimensional water-surface profile model.

Table 7-5 Comparison between direct step and standard step methods.

No. Characteristic Direct step method Standard step method

1 Cross-sectional

shape Prismatic Any (prismatic or nonprismatic)

2 Ease of computation Easy (hours) Difficult (months)

3 Calculation advances

⇒ Directly By iteration (trial and error)

4 Type of cross-section

input One typical cross section

(prismatic) Several cross sections (nonprismatic)

5 Data needs Minimal Extensive

6 Accuracy increases

with ↠ A smaller flow depth

increment More cross sections and/or lesser cross-sectional

variability

7 Independent variable Flow depth Length of channel

8 Dependent variable Length of channel Flow depth

9 Tools Spreadsheet or

HEC-RAS

programming

10 Reliability Answer is always possible Answer is sometimes not possible, depending on the

type of cross-sectional input data

11 Cost Comparatively small Comparatively large

12 Public acceptance Average High

7.5 DIRECT STEP METHOD EXAMPLE

[Questions] [Problems] [References] • [Top] [Equation of Gradually Varied Flow] [Characteristics of Flow

Profiles] [Limits to Water Surface Profiles] [Methodologies]

Calculation of M2 and S2 profiles, upstream and downstream of a change in

grade, from mild to steep

Input data:

Discharge Q = 2000 m3

Bottom width b = 100 m. Side slope 2 H : 1 V Upstream channel slope = 0.0001 Upstream channel Manning's n = 0.025 Downstream channel slope = 0.03 Downstream channel Manning's n = 0.045

Solution

Calculate the normal depth and velocity, and critical depth and velocity, in the upstream

and downstream channels. Use ONLINE CHANNEL 05.

For the upstream channel:

Normal depth = 10.098 m

Normal velocity = 1.648 m/s

Normal Froude number = 0.179

Critical depth = 3.364 m

Critical velocity = 5.571 m/s

For the downstream channel:

Normal depth = 2.669 m

Normal velocity = 7.113 m/s

Normal Froude number = 1.425

Critical depth = 3.364 m

Critical velocity = 5.571 m/s

Calculation of critical slope for the upstream channel:

yc = 3.364 m

Vc = 5.571 m/s

Ac = (b + zyc ) yc = 359.033 m2

P = b + 2 yc (1 + z 2)1/2 = 115.044 m

Rc = Ac / Pc = 3.121 m

Sc = n 2 Vc2 / Rc

4/3 = (0.025)2 (5.571)2 (3.121)4/3 = 0.00425

Verify the critical slope with ONLINE CHANNEL 04: Sc = 0.004254.

Calculation of critical slope for the downstream channel:

Sc = n 2 Vc2 / Rc

4/3 = (0.045)2 (5.571)2 (3.121)4/3 = 0.0138

Verify the critical slope with ONLINE CHANNEL 04: Sc = 0.01378.

The calculation of the M2 water-surface profile is shown in Table 7-6. The following instructions are

indicated:

The calculation moves in the upstream direction, starting at critical depth at the downstream end.

Column [1]: The second depth is set at 4 m; subsequently, the depth interval is set at 1 m.

Column [2]: The flow area is: A = (b + zy ) y . . . (1)

Column [3]: The mean velocity is: V = Q / A . . . (2)

Column [4]: The velocity head is: V 2 / (2g) . . . (3)

Column [5]: The specific head is: H = y + V 2 / (2g) . . . (4)

Column [6]: The wetted perimeter is: P = b + 2 y (1 + z 2)1/2 . . . (5)

Column [7]: The hydraulic radius is: R = A / P . . . (6)

Column [8]: The friction slope is: Sf = n 2 V 2 / R 4/3 . . . (7)

Column [9]: The average friction slope is: Sf ave = 0.5 (Sf 1 + Sf 2) . . . (8)

Column [10]: The specific head difference is: ΔH = H2 - H1 . . . (9)

Column [11]: The channel length increment ΔL, explained in the box below.

Column [12]: The cumulative channel length, i.e., the cumulative sum of ΔL increments.

Derivation of the formula for the channel length increment ΔL

With reference to Fig. 7-19, the average friction slope is:

hf

Sf ave = ______

ΔL

(7-54)

Fig. 7-19 Definition sketch for the calculation of channel length increment ΔL.

H1 + z1 = H2 + z2 + Sf ave ΔL

(7-55)

z1 - z2 = H2 - H1 + Sf ave ΔL

(7-56)

z1 - z2 - Sf ave ΔL = H2 - H1

(7-57)

z1 - z2

So = _________

ΔL

(7-58)

So ΔL - Sf ave ΔL = H2 - H1

(7-59)

(So - Sf ave) ΔL = H2 - H1

(7-60)

H2 - H1

ΔL = ______________

(7-61)

So - Sf ave

ΔH

ΔL = ______________

So - Sf ave

(7-62)

Equation 7-62 enables the calculation of the channel length increment ΔL, i.e., Col. 11 of Table 7-6.

When ΔL is negative, the calculation moves upstream, as in the M2 profile (Table 7-6). Conversely,

when ΔL is positive, the calculation moves downstream, as in the S2 profile (Table 7-7).

Table 7-6 Calculation of M2 water-surface profile (So = 0.0001).

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

y A V V 2/(2g ) H P R Sf Sf ave ΔH ΔL ∑ ΔL

(1) (2) (3) (4) (5) (6) (7) (8) (9) Box

3.364 359.033 5.570 1.581 4.945 115.044 3.1208 0.00425 --- --- --- 0

4.000 432.000 4.630 1.092 5.092 117.888 3.664 0.00237 0.00331 0.147 -45.794 -45.794

5.000 550.000 3.636 0.674 5.674 122.360 4.495 0.00111 0.00174 0.528 -354.878 -400.672

6.000 672.000 2.976 0.451 6.451 126.833 5.298 0.00060 0.000855 0.777 -1029.139 -1429.811

7.000

8.000

9.000

10.000

10.097

0.0001

In the direct step method, the accuracy of the computation depends on the size of depth interval(Col. [1] of

Table 7-6). The smaller the interval, the more accurate the computation will be. This is because the average

friction slope for a subreach (Col. [9]) is the arithmetic average of the friction slopes at the grid points

(linear assumption). In practice, a computation using a computer would have a much finer interval than that

shown in Table 7-6.

The calculation of the S2 water-surface profile is shown in Table 7-7.

Table 7-7 Calculation of S2 water-surface profile (So = 0.03).

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

y A V V 2/(2g ) H P R Sf Sf ave ΔH ΔL ∑ ΔL

(1) (2) (3) (4) (5) (6) (7) (8) (9) Box

3.364 359.033 5.570 1.581 4.945 115.044 3.1208 0.00425 --- --- --- 0

3.300 351.780 5.685 1.647 4.9475 114.758 3.0654 0.0147 0.01425 0.002 0.127 0.127

3.200 340.48 5.874 1.759 4.9586 114.311 2.979 0.0163 0.0155 0.0111 0.765 0.892

3.100 329.22 6.075 1.881 4.981 113.864 2.891 0.0181 0.0172 0.0224 1.750 2.642

3.000

2.900

2.800

2.700

2.669

0.03

Online calculations

The M2 water-surface profiles may be calculated online

using ONLINE_WSPROFILES_22 (Fig. 7-20). Using the number of computational

intervals n = 100, and number of tabular output intervals m = 100, the length of the

M2 water-surface profile is calculated to be: ∑ ΔL = 147,691.5 m.

Fig. 7-20 Definition sketch for M2 water-surface profile.

The S2 water-surface profiles may be calculated online

using ONLINE_WSPROFILES_25 (Fig. 7-21). Using the number of computational

intervals n = 100, and number of tabular output intervals m = 100, the length of the

S2 water-surface profile is calculated to be: ∑ ΔL = 152.02 m.

Fig. 7-21 Definition sketch for S2 water-surface profile.

QUESTIONS

[Problems] [References] • [Top] [Equation of Gradually Varied Flow] [Characteristics of Flow Profiles] [Limits to Water

Surface Profiles] [Methodologies] [Direct Step Method Example]

1. What is steady gradually varied flow?

2. What is retarded flow?

3. What is accelerated flow?

4. What is subnormal flow?

5. What is supernormal flow?

6. How many profiles are possible in steady gradually varied flow?

7. What is the rule for the Type I family of water-surface profiles?

8. What is the rule for the Type III family of water-surface profiles?

9. Which water-surface profiles are completely horizontal?

10. What are the five limits to the water surface profiles?

11. What is the typical application of the M1 water-surface profile?

12. What is the typical application of the S1 water-surface profile?

13. What is the typical application of the S3 water-surface profile?

14. What is the main difference between the direct-step and standard-step

methods of water-surface profile computations?

PROBLEMS

[References] • [Top] [Equation of Gradually Varied Flow] [Characteristics of Flow Profiles] [Limits to Water Surface

Profiles] [Methodologies] [Direct Step Method Example] [Questions]

1. A perennial stream has the following properties: discharge Q = 30 m3/s, bottom

width b = 55 m, side slope z = 2, bottom slope So = 0.0004, and Manning's n =

0.035. A 2-m high diversion dam is planned on the stream to raise the head for

an irrigation canal (Fig. 7-22). a. Calculate the normal depth. b. Calculate the total length of the M1 water-surface profile. Use n = 400

and m = 400. c. Calculate the partial length of the M1 profile, from the dam to a location

upstream where the normal depth is exceeded by 1%.

Use ONLINE CHANNEL 01 and ONLINE WSPROFILES 21.

Fig. 7-22 A diversion dam.

2. A perennial stream has the following properties: discharge Q = 1000 ft3/s,

bottom width b = 150 m, side slope z = 2, bottom slope So = 0.00038, and

Manning's n = 0.035. A 6-ft high diversion dam is planned on the stream to raise

the head for an irrigation canal. a. Calculate the normal depth. b. Calculate the total length of the M1 water-surface profile. Use n = 400

and m = 400. c. Calculate the partial length of the M1 profile, from the dam to a location

upstream where the normal depth is exceeded by 1%.

Use ONLINE CHANNEL 01 and ONLINE WSPROFILES 21.

3. A mild stream flows into a steep channel, producing an M2 upstream of the brink.

The hydraulic conditions in the channel are: Q = 28 m3/s, bottom width b = 12

m, side slope z = 2.5, bottom slope So = 0.0007, and Manning's n = 0.04.

Assume critical depth near the change in slope. a. Calculate critical and normal depth. b. Calculate the length of the M2 water-surface profile using n = 100 and m =

100. c. Calculate the length of the M2 water-surface profile using n = 200 and m =

200. d. Calculate the length of the M2 water-surface profile using n = 400 and m =

400. e. Comment on the results of b, c, and d. f. Determine the design channel length at 99% of the normal depth. Use n =

400.

Use ONLINE CHANNEL 05 and ONLINE WSPROFILES 22.

4. A mild stream flows into a steep channel, producing an M2 upstream of the brink.

The hydraulic conditions in the channel are: Q = 500 ft3/s, bottom width b = 30

ft, side slope z = 1.5, bottom slope So = 0.00075, and Manning's n = 0.04.

Assume critical depth near the change in slope. a. Calculate critical and normal depth. b. Calculate the length of the M2 water-surface profile using n = 100 and m =

100. c. Calculate the length of the M2 water-surface profile using n = 200 and m =

200. d. Calculate the length of the M2 water-surface profile using n = 400 and m =

400. e. Comment on the results of b, c, and d. f. Determine the design channel length at 99% of the normal depth. Use n =

400.

Use ONLINE CHANNEL 05 and ONLINE WSPROFILES 22.

5. An overflow spillway flows into a mild channel, producing a hydraulic jump. The

channel is rectangular, with Q = 3 m3/s, bottom width b = 8 m, and Manning's n =

0.015. The flow depth at the toe of the spillway is 0.1 m and the approximate

slope at the toe of the spillway is 0.1. The slope of the [mild] downstream

channel, which functions as a stilling basin, is 0.0001. Calculate: a. the critical depth, b. the length Ltc from the toe of the spillway to critical depth downstream, c. the Froude number at the toe of the spillway, d. the Froude number downstream of the hydraulic jump, e. the sequent depth y2 (the normal depth downstream of the hydraulic jump), f. the length Lj of the jump, assuming Lj = 6.2 y2 g. the minimum length of the stilling basin Lsb = Ltc + Lj

Use ONLINE CHANNEL 02 and ONLINE WSPROFILES 23. In the latter, use n = 100 and m = 100.

6. An overflow spillway flows into a mild channel, producing a hydraulic jump. The

channel is rectangular, with Q = 100 ft3/s, bottom width b = 20 ft, and

Manning's n = 0.015. The flow depth at the toe of the spillway is 0.4 ft and the

approximate slope at the toe of the spillway is 0.1. The slope of the [mild]

downstream channel, which functions as a stilling basin, is 0.0001. Calculate: a. the critical depth, b. the length Ltc from the toe of the spillway to critical depth downstream, c. the Froude number at the toe of the spillway, d. the Froude number downstream of the hydraulic jump, e. the sequent depth y2 (the normal depth downstream of the hydraulic jump), f. the length Lj of the jump, assuming Lj = 6.2 y2 g. the minimum length of the stilling basin Lsb = Ltc + Lj

Use ONLINE CHANNEL 02 and ONLINE WSPROFILES 23. In the latter, use n = 100 and m = 100.

7. A diversion dam of height H = 1.8 m is planned on a steep stream with bottom

slope So = 0.035. A hydraulic jump is expected upstream of the dam. Identify the

type of water surface profile. Using ONLINE CALC, calculate the length of the

water surface profile, from the location of the diversion dam, in the upstream

direction, to the [downstream end of the] hydraulic jump. The channel has Q = 4

m3/s, bottom width b = 3 m, side slope z = 1, and Manning's n = 0.03. What are

the sequent depths? What is the Froude number of the upstream flow? Use m =

100 and n= 100. Verify the sequent depth y2 using ONLINE CHANNEL 11.

8. A mild channel enters into a steep channel of slope So = 0.03. Identify the type of

water surface profile in the steep channel. Using ONLINE CALC, calculate the

normal depth in the steep channel, and the length of the water surface profile to

within 2% of normal depth. The channel has Q = 3 m3/s, bottom width b = 5 m,

side slope z = 0, and Manning's n = 0.015. What is the normal-depth Froude

number in the steep channel? Use m = 100 and n = 100.

9. A steep channel of slope So = 0.035 enters into a milder steep channel of

slope So = 0.012. Identify the type of water surface profile in the milder steep

channel. Using ONLINE CALC, calculate the normal depth in the downstream

channel, and the length [to normal depth] of the water surface profile. The

channel has Q = 3.2 m3/s, bottom width b = 4 m, side slope z = 2, and

Manning's n = 0.015. What are the normal-depth Froude numbers? What is the

normal depth in the upstream channel? What would be the length of the water

surface profile if Manning's n was instead estimated at 0.013? Use m = 100

and n = 100.

10. A perennial stream has the following properties: discharge Q = 15 m3/s,

bottom width b = 8 m, side slope z = 2, bottom slope So = 0.0025, and

Manning's n = 0.035. A 2.0-m high diversion dam is planned on the stream to

raise the head for an irrigation canal (Fig. 7-23). a. Calculate the normal depth. b. Calculate the total length of the M1 water-surface profile. Use three

resolutions: (a) n = 100 and m = 100, (b) n = 200 and m = 200, and (c) n = 400 and m = 400. Comment on the results.

c. Using the higher resolution results, calculate the partial length of the M1 profile, from the dam location to a point upstream where the normal depth is exceeded by 1%.

Use ONLINE CHANNEL 01 and ONLINE WSPROFILES 21.

Fig. 7-23 A diversion dam.

11. An overflow spillway flows into a mild channel, producing a hydraulic jump.

The channel is rectangular, with Q = 3.6 m3/s, bottom width b = 5 m, and

Manning's n = 0.015. The flow depth at the toe of the spillway is 0.15 m and the

approximate slope at the toe of the spillway is 0.1. The slope of the downstream

mild channel, which functions as a stilling basin, is 0.00016. Use n= 100 and m =

100. Calculate: a. the critical depth, b. the length Ltc from the toe of the spillway to critical depth downstream, c. the Froude number at the toe of the spillway, d. the Froude number downstream of the hydraulic jump, e. the sequent depth y2 (the normal depth downstream of the hydraulic jump),

f. the length Lj of the jump, assuming Lj = 6.2 y2 g. the minimum length of the stilling basin Lsb = Ltc + Lj

What would be the length of the stilling basin if the bottom width were to be increased to 7 m?

Use ONLINE CHANNEL 02 and ONLINE WSPROFILES 23. In the latter, use n = 100 and m = 100.

12. A 5-m high diversion dam is planned on a channel operating at critical flow.

The channel is rectangular, with Q = 100 m3/s, bottom width b = 4.7 m, bottom

slope So = 0.01, and Manning'sn = 0.023. Calculate the length of the C1 water-

surface profile. Assume n = 100 and m = 100.

13. A steep channel, with bottom slope So = 0.03, flows into a channel

operating at critical flow. The channel is rectangular, with Q = 100 m3/s, bottom

width b = 4.7 m, bottom slope So = 0.01, and Manning's n = 0.024. Calculate the

length of the C3 water-surface profile. Assume n = 100 andm = 100.

14. A horizontal channel of length L = 500 m is designed to convey Q = 5 m3/s

from a reservoir to a free overfall. The bottom width is b = 2 m, and side slope z =

1.5. The channel is lined with gabions and the Manning's n recommended by the

manufacturer is n = 0.028. a. What is the headwater depth (accuracy to 1 cm) required to pass the

design discharge? b. What is the tailwater depth, that is, the critical flow depth at the downstream

boundary?

Use ONLINE_WSPROFILES_32; assume n = 100 and m = 100.

15. A sluice gate is designed to release supercritical flow into a stilling basin,

where a hydraulic jump will occur. The basin channel bottom is horizontal, of

rectangular cross section. The design discharge is Q = 5 m3/s, the bottom

width b = 5 m, and Manning's n = 0.015.

a. What should be the gate opening (accuracy to 1 mm) to ensure that the

length of the downstream water surface profile is not greater than 10 m?

b. What is the critical depth?

Use ONLINE_WSPROFILES_35; assume n = 100 and m = 100. Assume So,u/s = 0.05 so that the flow through the sluice gate remains supercritical.

REFERENCES

• [Top] [Equation of Gradually Varied Flow] [Characteristics of Flow Profiles] [Limits to Water Surface

Profiles] [Methodologies] [Direct Step Method Example] [Questions] [Problems]

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

U.S. Army Corps of Engineers. (2014). HEC-RAS: Hydrologic Engineering Center River

Analysis System.

USDA Soil Conservation Service. (1971). Classification system for varied flow in

prismatic channels. Technical Release No. 47 (TR-47), Washington, D.C.

http://openchannelhydraulics.sdsu.edu

140830 08:00

[Culverts] [Inlet Control] [Outlet Control] [Culvert Design] [Questions] [Problems] [References] •

CHAPTER 8:

CULVERT HYDRAULICS

8.1 CULVERTS

[Inlet Control] [Outlet Control] [Culvert Design] [Questions] [Problems] [References] • [Top]

Culverts are hydraulically short drainage conduits placed in locations where the

drainage network intersects the transportation network (roads, railroad tracks). Culverts

differ from bridges in that they are much smaller; thus, there are many more culverts

than bridges. Culverts are designed to operate under gradually varied flow; therefore,

the principles of Chapter 7 are applicable.

Culverts are designed to pass the design discharge without overtopping of the

superstructure. The design discharge is derived from the design storm, which is based

on hydrologic considerations. The return period (the reciprocal of the frequency)

typically varies from 10 to 50 years. The longer the return period, the greater the design

discharge and, therefore, the larger the required culvert size(Fig. 8-1).

Fig. 8-1 Large culvert at Yogurt Canyon, U.S.-Mexico international border, California.

The flow in a culvert is a function of:

1. Cross-sectional size and shape,

2. Bottom slope,

3. Barrel length,

4. Roughness, and

5. Entrance and exit characteristics.

The flow in a culvert may be either (a) completely free-surface (open-channel flow), (b)

completely closed-conduit (pipe flow), or (c) partially free-surface and closed-conduit

flow. Headwater (HW) is the depth of water above the culvert invert at the inlet.

Tailwater (TW) is the depth of water above the culvert invert at the outlet. The design

headwater and tailwater elevations are major factors in determining whether the culvert

flows partially full or completely full.

The design objective is to find the most economical design (i.e., the smallest culvert

size) that will pass the design discharge without exceeding a specified headwater

elevation (Fig. 8-2). The design depends on whether the culvert flow is under: (a) inlet

control, or (b) outlet control.

Fig. 8-2 Large culvert crossing a railroad embankment, Tecate, Baja California, Mexico.

8.2 INLET CONTROL

[Outlet Control] [Culvert Design] [Questions] [Problems] [References] • [Top] [Culverts]

Culvert flow is under inlet control when the discharge depends only on the conditions at

the inlet. Assume a circular culvert of diameter D, length L, slope S, headwater depth

HW, and tailwater depth TW. Calculate the normal depth yn and the critical flow

depth yc . Then, the following conditions hold:

If yn < yc , the flow in the culvert barrel is supercritical and the tailwater has no

influence on the upstream conditions (Fig. 8-3). Therefore, the headwater is

solely controlled by the conditions at the inlet.

Fig. 8-3 Culvert flow under supercritical conditions, with inlet submerged

and outlet unsubmerged.

If TW > yn, a hydraulic jump may form at or near the culvert outlet (Fig. 8-4).

Fig. 8-4 Culvert flow under supercritical conditions, with inlet unsubmerged

and outlet submerged due to high tailwater.

Occurrence of inlet control

Inlet control occurs when the culvert barrel is capable of conveying more discharge

than the inlet will allow. The control section is located just inside the entrance of the

culvert. The flow passes through critical depth at the control section and becomes

supercritical downstream of the inlet.

Under inlet control, the culvert acts as an orifice or weir. If the inlet is submerged (a

common design situation), the flow condition resembles that of an orifice. If the inlet is

unsubmerged (i.e., open to the atmosphere), the flow condition resembles that of a

weir. [If HW < 1.2 D, the inlet will be unsubmerged]. If the inlet is unsubmerged but the

outlet is submerged, a hydraulic jump will form inside the culvert barrel.

8.3 OUTLET CONTROL

[Culvert Design] [Questions] [Problems] [References] • [Top] [Culverts] [Inlet Control]

Outlet control occurs when TW > 1.2 D, i.e., for high tailwater. In this case, the culvert

barrel will be completely full of water, resembling closed-conduit flow. The headwater

may be computed by applying the energy equation from the upstream (u/s) pool

elevation to the downstream (d/s) pool elevation. The headwater is directly controlled

by the tailwater elevation and the frictional characteristics of the culvert barrel.

Occurrence of outlet control

Outlet control occurs when inlet and outlet are submerged. Outlet control also occurs

when the culvert slope is mild (subcritical flow) and both the headwater and tailwater

are less than the culvert diameter (HW < D; TW < D). In this case, the best approach is

to calculate the water-surface profile. Figure 8-5 shows a schematic portrayal of flow

rate as a function of headwater energy under inlet and outlet control (U.S. Army Corps

of Engineers, 2014).

Fig. 8-5 Discharge as a function of headwater energy under inlet and outlet control

(U.S. Army Corps of Engineers, 2014).

8.4 CULVERT DESIGN

[Questions] [Problems] [References] • [Top] [Culverts] [Inlet Control] [Outlet Control]

The following steps are followed in culvert design:

1. Assemble design data Discharge, Tailwater elevation, and Slope of culvert barrel.

2. Choose the culvert characteristics Cross-sectional shape (circular, square, rectangular, arch), Dimensions (diameter, if circular), Barrel length,

Kind of material (Figs. 8-6 and 8-7) (concrete, corrugated steel, corrugated

aluminum, stone masonry), and

Type of entrance (square-edged or rounded).

Fig. 8-6 A set of two culverts made with corrugated steel.

3. Ascertain the prevailing type of control (inlet or outlet), based on (a) headwater

elevation, (b) tailwater elevation, (c) diameter, and (d) slope.

4. If inlet control prevails, calculate the required headwater elevation to pass the

design discharge.

5. If outlet control prevails, calculate the required headwater elevation using (a) the

energy equation or (b) water-surface profile computations.

6. If the calculated headwater elevation is greater than allowed, choose a larger-

size culvert and repeat the process.

7. In some cases, it may not be possible to determine the type of control a priori. In

this case, both calculations are advised. The design type of control is that which

results in the greatest headwater elevation.

8. Other design considerations in culvert design

Piping in the embankment surrounding the culvert,

Local scour at culvert outlet,

Erosion of fill material near the inlet,

Clogging with excessive debris, and

Provision for fish passage.

Fig. 8-7 A highway underpass featuring a rectangular stone-masonry culvert.

Design Example

Design a culvert for the following conditions:

Design discharge: Q = 200 cfs.

Return period: T = 25 yr.

Barrel length: L = 200 ft.

Bottom slope: So = 0.01.

Culvert material: Concrete.

Manning's n = 0.013.

Inlet invert elevation: z1 = 100 ft.

Roadway shoulder elevation: Es = 110 ft.

Tailwater depth above outlet invert: TW = y2 = 3.5 ft.

Freeboard: Fb = 2 ft.

Solution

The design elevation for the upstream pool is: Es - Fb = 100 - 2 = 108 ft.

Assume a circular concrete pipe, with square edge with headwalls.

Assume outlet control.

Assume that the hydraulic grade line (HGL) is at the elevation of the downstream pool.

Calculate the outlet invert elevation: z2 = z1 - (So L) = 100 - (0.01 × 200) = 98 ft.

Calculate the downstream pool elevation: z2 + y2 = 98 + 3.5 = 101.5 ft.

Set up the energy balance (Fig. 8-8):

V12 V2

2

z1 + y1 + _____ = z2 + y2 + _____ + ∑hL

2g 2g

(8-1)

Fig. 8-8 Energy balance in culvert flow.

Assume V1 = 0, i.e., the velocity is zero in the upstream pool.

Assume V2 = 0, i.e., the velocity dissipates to zero in the downstream pool.

The head loss ∑hL is equal to the sum of entrance losses (with loss coefficient Ke), exit losses (with

loss coefficient KE), and barrel losses.

Using the Darcy-Weisbach equation, the head loss is:

V 2

∑hL = [ (Ke + KE + f (L / D ) ] _____

2g

(8-2)

From Table 8-1, assume Ke = 0.5 and KE = 1 (Roberson et al., 1985).

Table 8-1 Loss coefficients in pipe entrance, contraction, and expansion.

Description Sketch (Click on figure to display)

Additional

data Loss

coefficient K

Pipe entrance

hL = Ke [V 2/(2g)]

r /d Ke

0.0 0.50

0.1 0.12

> 0.2 0.03

1.0

Contraction

hL = KC [V22/(2g)]

D2 /D1 KC

θ = 60° KC

θ = 180°

0.0 0.08 0.50

0.2 0.08 0.49

0.4 0.07 0.42

0.6 0.06 0.32

0.8 0.05 0.18

0.9 0.04 0.10

Expansion

hL = KE [V12/(2g)]

D1 /D2 KE

θ = 10° KE

θ = 180°

0.0 1.00

0.2 0.13 0.92

0.4 0.11 0.72

0.6 0.06 0.42

0.8 0.03 0.16

The relation between Darcy-Weisbach friction factor f and Manning's n is (Chapter 5):

8 g n 2

f = __________

k 2R 1/3

(8-3)

in which k = 1 in SI units, and k = 1.486 in U.S. Customary units.

In U.S. Customary units, with k = 1.486, and g = 32.17 ft/s2:

116.55 n 2

f = ____________

R 1/3

(8-4)

For a circular pipe: R = D / 4. Therefore:

185.01 n 2

f = ____________

D 1/3

(8-5)

From Eq. 8-1, the energy balance reduces to:

z1 + y1 = z2 + y2 + ∑hL

(8-6)

108 = 101.5 + ∑hL

(8-7)

The head loss equation (Eq. 8-2) is repeated here for convenience:

V 2

∑hL = [ (Ke + KE + f (L / D ) ] _____

2g

(8-2)

Replacing Eq. 8-5 in Eq. 8-2:

V 2

∑hL = [ 0.5 + 1.0 + (185.01 n 2 L / D 4/3 ) ] _____

2g (8-8)

Combining Eqs. 8-7 and 8-8:

V 2

6.5 = [ 1.5 + (6.253 / D 4/3 ) ] _____

2g

(8-9)

The flow velocity is: V = Q / A.

Therefore: V = 200 / A = 200 / [ (π/4) D 2

]

The velocity head is: V 2

/ (2g) = { 2002 / [ (π/4)

2 D

4 ] } / (2g) = 1008 / D

4

Replacing the velocity head in Eq. 8-9:

1008

6.5 = [ 1.5 + (6.253 / D 4/3 ) ] _______

D 4

(8-10)

Solving Eq. 8-10 by iteration: D = 4.38 ft. For design purposes, assume the next larger size: D = 4.5

ft.

With Q = 200 cfs, D = 4.5 ft = 54 in, enter Fig. 8-9 to find the ratio of headwater depth to diameter

HW/D = 2.2, for the case of square edge with headwalls (1).

Fig. 8-9 Headwater depth for concrete culverts with inlet control.

The headwater depth is: HW = (HW/D) × D = 2.2 × 4.5 = 9.9 ft.

The upstream pool elevation is: 100 + 9.9 = 109.9 ft. This upstream pool elevation is greater than

108 ft; therefore, it is too large. The chosen D = 4.5 ft is too small. Try the next size: D = 5 ft.

With Q = 200 cfs, D = 5.0 ft = 60 in, enter Fig. 8-7 to find the ratio of headwater depth to diameter

HW/D = 1.6, for the case of square edge with headwalls (1).

The headwater depth is: HW = HW/D × D = 1.6 × 5.0 = 8.0 ft.

The upstream pool elevation is: 100 + 8.0 = 108.0 ft. This upstream pool elevation is the same as

the design elevation; therefore, the design is now OK.

Calculate the normal depth using ONLINE CHANNEL 06: yn = 3.284 ft.

Calculate the critical depth using ONLINE CHANNEL 07: yc = 4.037 ft.

Since yn < yc, the flow is supercritical.

Since TW = 3.5 > yn = 3.284, there will be a small hydraulic jump at or near the outlet.

Since the flow is supercritical for most of the culvert length, it is concluded that inlet control

prevails.

The design diameter is: D = 5 ft = 60 in. ANSWER.

QUESTIONS

[Problems] [References] • [Top] [Culverts] [Inlet Control] [Outlet Control] [Culvert Design]

1. What is a culvert?

2. What is the typical return period for culvert design?

3. When is a culvert under inlet control?

4. When is a culvert under outlet control?

5. List the hydraulic variables affecting culvert flow.

6. List other considerations in culvert design.

PROBLEMS

[References] • [Top] [Culverts] [Inlet Control] [Outlet Control] [Culvert Design] [Questions]

1. Design a circular concrete culvert with the following data: Q = 300 cfs; inlet invert

elevation z1 = 100 ft; tailwater depth y2 = 4 ft; barrel slope So = 0.02; barrel

length L = 200 ft; Manning's n = 0.013; roadway shoulder elevation Er = 112 ft;

upstream freeboard Fb = 2 ft. The entrance type is square edge with headwalls

(Fig. 8-10). Use <="" a="" style="text-decoration: none;">ONLINECHANNEL

06 to calculate normal depth and <="" a="" style="text-decoration:

none;">ONLINECHANNEL 07 to calculate critical depth in the culvert.

Fig. 8-10 Typical culvert underpass.

2. Design a circular concrete culvert with the following data: Q = 500 cfs; inlet invert

elevation z1 = 100 ft; tailwater depth y2 = 4 ft; barrel slope So = 0.01; barrel

length L = 200 ft; Manning's n = 0.013; roadway shoulder elevation Er = 115 ft;

upstream freeboard Fb = 2 ft. The entrance type is square edge with headwalls.

Use <="" a="" style="text-decoration: none;">ONLINECHANNEL 06 to calculate

normal depth and<="" a="" style="text-decoration: none;">ONLINECHANNEL

07 to calculate critical depth in the culvert. Verify the culvert design using<=""

a="" style="text-decoration: none;">ONLINECULVERT.

REFERENCES

• [Top] [Culverts] [Inlet Control] [Outlet Control] [Culvert Design] [Questions] [Problems]

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

U.S. Army Corps of Engineers. 2014. HEC-RAS River Analysis System. Version 4.1,

Hydrologic Engineering Center, Davis, California.

http://openchannelhydraulics.sdsu.edu

140905 21:30

[The Sharp-Crested Weir] [Crest Shape of Overflow Spillway] [Rating of Spillways] [The hydraulic

Jump] [Questions] [Problems] [References] •

CHAPTER 9:

STEADY RAPIDLY VARIED FLOW

9.1 THE SHARP-CRESTED WEIR

[Crest Shape of Overflow Spillway] [Labyrinth Spillways] [The hydraulic

Jump] [Questions] [Problems] [References] • [Top]

Rapidly varied flow differs from gradually varied flow (Chapter 7) in the greater

curvature of the streamlines. In extreme cases, the flow is virtually broken, resulting in

high levels of turbulence and associated energy loss.

The following characteristics describe rapidly varied flow:

The curvature of the streamlines is so pronounced that the pressure distribution

cannot be assumed to be hydrostatic.

Unlike gradually varied flow, changes in flow variables take place within a

relatively short distance (Fig. 9-1).

The boundary friction is small compared to the other forces, and often negligible.

The flow characteristics are fixed by the usually rigid boundary geometry.

The velocity distribution coefficients α and β are much greater than unity (1) and

cannot be determined accurately.

Separation zones, eddies, and rollers tend to complicate the flow pattern, with

flow often confined to separation zones.

To date, there is no theoretical solution for rapidly varied flow. In lieu of theory,

empirical relations are used in many applications.

Fig. 9-1 A canal drop, a typical example of rapidly varied flow.

The sharp-crested weir

The sharp-crested weir is a measuring device in open-channel flow (Chapter 4). It is

also the simplest form of overflow spillway. The profile of the spillway can be made to

match the shape of the lower surface of the flow nappe. The shape of the flow nappe

may be interpreted by the principle of the projectile (Chow, 1959) (Fig. 9-2).

According to this principle, the horizontal velocity component of the flow is constant,

and the only force acting on the nappe is the gravitational force. In time t, a particle of

water in the lower surface of the nappe will travel a horizontal distance x (from the face

of the weir) equal to:

Fig. 9-2 Nappe profiles over sharp-crested weir by the principle of the projectile (Chow, 1959).

x = vo t cos θ (9-1)

In the same time, the particle will travel a vertical distance y equal to:

y = - vo t sin θ + (1/2) g t 2 + C' (9-2)

in which g = gravitational acceleration, and C' is the value of y at x = 0. The

constant C' may be taken as the vertical distance between the highest point of the

nappe and the elevation of the crest.

Eliminating t from Eqs. 9-1 and 9-2:

y x x ____ = A (____)2 + B (____) + C

H H H

(9-3)

in which:

g H

A = ________________

2 vo2 cos2 θ

(9-4)

B = - tan θ

(9-5)

C'

C = ______

H

(9-6)

Assuming the vertical thickness of the nappe T, an additional term may be added to Eq.

9-5 to yield the nappe equation:

y x x ____ = A (____)2 + B (____) + C + D

H H H

(9-7)

in which D = T / H .

Discharge of the sharp-crested weir

A common formula for the discharge over a sharp-crested weir is:

Q = C L H 3/2 (9-8)

in which C = discharge coefficient, L = effective length of the weir crest, and H =

measured head above the crest, excluding the velocity head. Discharge coefficients for

various types of sharp-crested weirs are given in Chapter 4.

9.2 CREST SHAPE OF OVERFLOW SPILLWAY

[Rating of Spillways] [The hydraulic Jump] [Questions] [Problems] [References] • [Top] [The Sharp-Crested Weir]

The spillway shape shown in Fig. 9-3 is referred to as an ogee shape, because it

resembles an ogee curve, shaped somewhat like an S, consisting of two archs that

curve in opposite sides, so that the ends are approximately parallel. In practice,

notwithstanding the weir shape, negative pressures may develop in an ogee spiillway.

Excessive negative pressures may lead to cavitation damage and are, therefore, to be

controlled.

Fig. 9-3 (a) Emergency spillway of Turner reservoir,

in San Diego, California, not spilling.

Fig. 9-3 (b) Emergency spillway of Turner reservoir, in San Diego, California,

spilling on February 24, 2005.

WES spillway shapes

The U.S. Army Corps of Engineers has developed several standard shapes at its

Waterways Experiment Station. These shapes, shown in Fig. 9-4, are referred to as the

WES standard spillway shapes.

Fig. 9-4 The WES standard spillway shapes (Chow, 1959).

The WES shapes follow the equation:

X n = K Hd n-1 Y (9-9)

in which X and Y are the coordinates of the crest profile, with the origin of coordinates

at the highest point of the crest; Hd = design head excluding the velocity head of the

approach flow; and K and n are parameters which depend on the slope of the upstream

face. The values of K and n are shown in Table 9-1. For intermediate values,

interpolation is possible.

Table 9-1 Values of K and n (Eq. 9-9).

Slope on

upstream face K n

Vertical 2.000 1.850

3 V : 1 H 1.936 1.836

3 V : 2 H 1.939 1.810

3 V : 3 H 1.873 1.776

The upstream face of the spillway may sometimes be designed to set back, as shown

by the dashed lines of Fig. 9.4. The set back will not affect the shape of the crest,

provided the modification begins with at least one-half of the total head He below the

origin. Below this depth, the vertical velocities are small, and the corresponding effect

on the nappe profile is negligible.

For WES shapes, the discharge over the spillway is:

Q = C L He 3/2 (9-10)

in which He = total energy head on the crest, including the velocity head in the

approach channel. Model test have shown that the effect of the approach velocity is

negligible when the height h of the spillway is greater than 1.33 Hd, where Hd = design

head, excluding the approach velocity. Under this condition, the coefficient of

discharge, in U.S. Customary units, is Cd = 4.03.

For low spillways, where h/Hd < 1.33, the approach velocity will have appreciable effect

on the discharge and, therefore, on the nappe profile. Figure 9-5 shows the effect of the

approach velocity on the relationship between (He /Hd) and (C /Cd) for WES spillways

with a vertical upstream face. From Fig. 9-5, for h /Hd ≥ 1.33 and He > Hd, the

ratio C/Cd > 1. It is seen that the ratio C /Cd varies from 0.70 near crest level (He /Hd ≅

0) to 1.05 for He /Hd > 1.3. Thus, effectively, C varies from 2.82 to 4.23 (U.S. Customary

units).

For a sloping upstream face, C is multiplied by the correction factor shown in the top-

left side ofFig. 9-5.

Fig. 9-5 Head-discharge relation for WES standard spillway shapes (Chow, 1959).

WES spillway: Design example

Determine the crest elevation and the shape of an overflow spillway section (WES standard spillway

shape) having a vertical upstream face and a crest length L = 250 ft. The design discharge is Q =

75,000 cfs. The upstream water surface, at the design discharge, is at elevation 1,000 ft, while the

channel floor is at elevation 880 ft (Fig. 9-6).

Fig. 9-6 Design of an overflow spillway section (Chow, 1959).

Solution

Assume a high overflow spillway, for which the effect of the approach velocity is negligible.

Then: Hd = He , and C = 4.03.

h + Hd = 1000 - 880 = 120 ft.

From Eq. 9-10:

Q = C L He3/2

75,000 = (4.03) (250) He3/2

Therefore, the total energy head, including the velocity head, is: He = 17.7 ft.

The approach velocity is:

Va = Q / [L (h + Hd ) ] = 75,000 / [(250) (120)] = 2.5 fps.

The velocity head is:

Ha = Va 2 / (2 g ) = 2.5

2 / (2 × 32.17) = 0.097 ≅ 0.1

The design head, excluding the velocity head, is:

Hd = He - Ha = 17.7 - 0.1 = 17.6 ft.

The height of the dam is:

h = 120 - 17.6 = 102.4 ft.

h = 102.4 > 1.33 Hd = 23.4

Therefore, the effect of the approach velocity is negligible.

The crest elevation is: 1000 - 17.6 = 982.4 ft.

From Eq. 9-9 and Table 9-1:

X 1.85

= 2 Hd 0.85

Y

Therefore, the nappe equation is:

Y = X 1.85

/ (22.89)

To calculate the coordinates of the point of tangency (Fig. 9-6):

Y' = (1.85 / 22.89) X 0.85

= 0.081 X 0.85

Y' = = ΔY /ΔX = 1 / 0.6 = 1.667

1.667 = 0.081 X 0.85

Solving for the abscissa of the point of tangency:

X = 35.09 ft. ANSWER.

Solving for the ordinate of the point of tangency (Fig. 9-6):

Y = X 1.85

/ (22.89) = 31.54 ft. ANSWER.

The length of the base, projected from the point of tangency at a slope 1 V : 0.6 H to the base

of the dam (see Figs. 9-4 and 9-6) is:

Lb = 0.282 Hd + X + (0.6) (h - Y )

Lb = [ 0.282 (17.6) ] + 35.09 + [ (0.6) (102.4 - 31.54) ]

Lb = 82.6 ft. ANSWER.

0.282 Hd = 4.96 ft. ANSWER.

0.175 Hd = 3.08 ft. ANSWER.

0.2 Hd = 3.52 ft. ANSWER.

0.5 Hd = 8.80 ft. ANSWER.

Ogee-type spillways

Figure 9-7 shows several ogee-type spillways. This type of spillway is preferred for high

dams, because their discharge coefficient (C = 4.03 in U.S. Customary units and C =

2.22 in SI units) is substantially higher than the theoretical value for broad-crested weirs

(C = 3.087 in U.S. Customary units and C = 1.704 in SI units) (Chapter 4 Critical Flow

Control).

Fig. 9-7 (a) Emergency spillway at El Capitan Dam,

on the San Diego river, San Diego, California.

Fig. 9-7 (b) Emergency spillway at Mangla Dam,

on the Jhelum river, Pakistan.

Fig. 9-7 (c) Emergency spillway at Oroville Dam, on the Feather river,

near Oroville, California.

Broad-crested spillways

Figure 9-8 shows two broad-crested spillways. The 8000-ft long relief weir (emergency

spillway) of the Boerasirie Conservancy, shown in Fig. 9-8 (a), has been designed with

discharge coefficient varying from C = 1.45 at spill level to C = 1.78 at the design

head Hd = 0.215 m (discharge coefficient in SI units). The Valle Grande dam, shown in

Fig. 9-8 (b), is an off-stream storage reservoir, with a small contributing drainage area;

thus, no major floods are expected and the broad-crested weir is considered sufficient

to handle the design peak flow.

Fig. 9-8 (a) The 8000-ft long relief weir, Boerasirie Conservancy, Guyana.

Fig. 9-8 (b) Overflow spillway at Villa Grande Dam, Cuajone, Peru.

Labyrinth spillways

The labyrinth spillway is used to increase the effective length of the weir crest, when

warranted due to an increased risk of hydrologic failure. Figure 9-9 shows three

examples of labyrinth spillways. The labyrinth works well at the design stage above

spillway crest. However, for stages that are higher than the design stage, the effective

length eventually reduces to the actual length and the labyrinth ceases to provide the

desired advantage.

Fig. 9-9 (a) Valentine Mill pond,

Valentine, Nebraska.

Fig. 9-9 (b) Lower Las Vegas Wash detention basin,

Las Vegas, Nevada.

Fig. 9-9 (c) Ute Dam, New Mexico.

Fuse spillways

A fuse spillway or fuse plug is an embankment designed to wash out in a predictable

and controlled manner when a capacity in excess of the normal capacity of the service

spillway and outlet works is required (Pugh and Gray, 1984). According to the U.S.

Bureau of Reclamation, fuse plug designs have been selected for the Bartlett and

Horseshoe dams, on the Verde River, Arizona. The fuse plug for Bartlett Dam is

designed with an erosion resistant invert and abutment structure to pass 10,100 m3/s.

Three erodible embankment sections will operate in sequence. The Horseshoe Dam

fuse plug is designed to pass 6,850 m3/s through three 44- to 52-m-long openings 6.0-

m to 7.9-m high.

Fig. 9-10 Fuse spillway, La Leche river, Lambayeque, Peru.

Other spillways features

Other spillway features include rubber-gated spillways (Fig. 9-11), sky-jump spillways

(Fig. 9-12), and baffle-dissipation structures (Fig. 9-13).

Fig. 9-11 Rubber-gated emergency spillway, Arroyo Pasajero retention basin,

near Coalinga, California.

Fig. 9-12 Sky-jump spillway, Oros dam, Jaguaribe river, Ceara, Brazil.

Fig. 9-13 Baffle-dissipation structure, Gallito Ciego dam, La Libertad, Peru.

9.3 RATING OF SPILLWAYS

[The hydraulic Jump] [Questions] [Problems] [References] • [Top] [The Sharp-Crested Weir] [Crest Shape of Overflow

Spillway]

At the design discharge, the pressure on the spillway crest is close to atmospheric. The

design head HD, which includes the velocity head, corresponds to the design discharge.

Flow rates less than the design flow rate will produce pressures on the spillway face

above the atmospheric pressure, while flow rates greater than the design flow rate will

produce subatmospheric pressures.

The discharge over an ungated ogee spillway may be given as follows (Roberson et al.,

1998):

QD = CD (2 g)1/2 L HD 3/2 (9-11)

in which QD = design discharge, CD = dimensionless design discharge coefficient, L =

crest length perpendicular to the flow, and HD = total design head on the

crest, including the approach velocity head ha.

Figure 9-14 shows values of CD as a function of P/HD, in which P = height of the

spillway crest measured from the channel bed. Note that for large values of P/HD, the

value of CD approaches asymptotically a value of CD = 0.492.

In general, the discharge over an ogee-type spillways is:

Q = C (2 g)1/2 L H 3/2 (9-12)

in which Q = discharge, C = dimensionless discharge coefficient, L = crest length

perpendicular to the flow, and H = head on the crest, including the approach velocity

head.

Fig. 9-14 Variation of discharge coefficient CD with relative dam height P/HD

(Roberson et al., 1998).

The ratio C/CD varies as a function of H/HD as shown in Fig. 9-15. Figures 9-14 and 9-

15 may be used to develop a spillway rating curve, i.e., a relation between flow

rate Q and head above spillway crest H. If the actual head H exceeds the design

head HD, subatmospheric pressures will develop on the spillway, and this may lead to

cavitation damage. To prevent cavitation damage, the negative pressure head should

be kept within -20 ft.

Fig. 9-15 Variation of dimensionless discharge coefficient C/CD with relative stage

above crest level H /HD (Roberson et al., 1998).

Spillway rating: Example

Determine the rating curve for an ogee-type spillway with length L = 30 m. The design head is: HD =

10 m. The freeboard is: Fb = 5 m. The spillway crest elevation is atElev. 1,000 m. The riverbed

elevation is at Elev. 970 m. Neglect the approach velocity.

Solution

L = 30 m.

P = 1000 - 970 = 30 m.

P /HD = 30 / 10 = 3

From Fig. 9-14, for P /HD = 3: CD = 0.493

The spillway rating equation is (Eq. 9-12):

Q = C (2 g )1/2

L H 3/2

Q = C (2 × 9.806)1/2

(30) H 3/2

Q = 132.856 C H 3/2

The spillway rating equation is:

Q = 132.856 C H 3/2

ANSWER.

The calculations are summarized in Table 9-2.

Column 1 shows postulated water-surface elevations, from crest elevation (1,000 m) through

design water-surface elevation (1,010 m) to a water-surface elevation that includes the

freeboard (1,015 m).

Column 2 shows the actual head over the spillway crest.

Column 3 shows the H /HD ratios.

Column 4 shows C /CD ratios, obtained from Fig. 9-15.

Column 5 shows values of C = CD (C /CD). = 0.493 (C /CD).

Column 6 shows the values of Q calculated with the spillway rating equation:

Q = 132.856 C H 3/2

The spillway rating curve is shown in Fig. 9-16.

Table 9-2 Calculation of spillway rating.

[1] [2] [3] [4] [5] [6]

Elevation

(m) H

(m) H /HD C /CD C

Q

(m3/s)

1000 0 0.0 0.780 0.385 0.000

1001 1 0.1 0.811 0.400 53.119

1002 2 0.2 0.842 0.415 155.986

1003 3 0.3 0.869 0.428 295.584

1004 4 0.4 0.895 0.441 468.987

1005 5 0.5 0.915 0.451 670.047

1006 6 0.6 0.935 0.461 900.052

1007 7 0.7 0.951 0.469 1153.604

1008 8 0.8 0.967 0.477 1433.146

1009 9 0.9 0.984 0.485 1739.271

1010 10 1.0 1.000 0.493 2071.234

1011 11 1.1 1.013 0.499 2419.431

1012 12 1.2 1.025 0.505 2790.775

1013 13 1.3 1.037 0.511 3182.097

1014 14 1.4 1.048 0.517 3595.692

1015 15 1.5 1.059 0.522 4029.600

Fig. 9-16 Calculated spillway rating curve.

Example 9-1

Solve the spillway rating example using an online calculator.

ONLINE CALCULATION. Use ONLINE OGEE RATING, with L = 30 m, HD = 10 m, approach velocity Va = 0,

dam height P = 30 m, freeboard Fb = 5 m, and spillway crest elevationE = 1000 m. The result of ONLINE OGEE

RATING is the same of Table 9-2.

9.4 THE HYDRAULIC JUMP

[Questions] [Problems] [References] • [Top] [The Sharp-Crested Weir] [Crest Shape of Overflow Spillway] [Rating of

Spillways]

The hydraulic jump is an open-channel flow phenomenon where the flow changes

suddenly from supercritical to subcritical (Fig. 9-17). The equation for the hydraulic

jump is developed for a horizontal channel, for which the jump becomes stationary, i.e.,

it occurs in a specific place in the channel. A nonstationary hydraulic jump may occur in

a channel of finite (nonzero) slope. In practice, the stationary hydraulic jump is

preferred over the nonstationary or "moving" jump.

Fig. 9-17 A hydraulic jump downstream of a sluice gate.

The hydraulic jump is used in the following applications:

To dissipate energy in water flowing over dams, weirs, and other hydraulic

structures, to prevent downstream scouring of the structures.

To recover hydraulic head on the downstream side of a measuring flume.

To increase weight on an apron and reduce uplift pressures under a masonry

structure.

To aerate water for purposes of water purification.

Hydraulic jump equation

A hydraulic jump will form in a rectangular channel if the following equation is satisfied:

y2 1 ____ = ____ [ (1 + 8 F1

2 )1/2 - 1 ]

y1 2

(9-13)

where y1 = upstream flow depth (supercritical), y2 = downstream flow depth

(subcritical), and F1 = Froude number of the upstream flow. The depth y1 is the initial

depth and y2 is the sequent depth.

The hydraulic jump relation (Eq. 9-13) is shown in Fig. 9-18. Note that for Froude

number F1 > 2 the equation is almost linear.

Fig. 9-18 The hydraulic jump relation.

Example 9-2

Given y1 = 0.1 m and v1 = 5 m/s, calculate the sequent depth y2. Confirm the result using ONLINE CHANNEL 11.

The upstream flow Froude number is: F1 = 5.049. Using Eq. 9-13: y2 = 0.665 m.

ONLINE CALCULATION. Using ONLINE CHANNEL 11, with y1 = 0.1 m, and v1 = 5 m/s, the Froude number

is: F1 = 5.049. The sequent depth is y2 = 0.665 m, confirming the hand calculation.

Types of hydraulic jump

Hydraulic jumps are classified as shown in Table 9-3 (Chow, 1959).

Table 9-3 Types of hydraulic jump.

Upstream

Froude number

F1 Type of jump Graphical description

1.0 - 1.7 Undular jump

1.7 - 2.5 Weak jump

2.5 - 4.5 Oscillating jump

4.5 - 9.0 Steady jump

> 9.0 Strong jump

Characteristics of the hydraulic jump

The energy loss is:

(y2 - y1)3

ΔE = E1 - E2 = ______________

4 y1 y2

(9-14)

[See also Lab video: The hydraulic jump].

The relative energy loss is:

ΔE E2 ______ = 1 - ______

E1 E1

(9-15)

The efficiency of the jump is (see box below):

E2 ( 1 + 8 F12 )3/2 - 4 F1

2 + 1 ______ = _______________________________

E1 8 F12 ( 2 + F1

2 )

(9-16)

The height of the jump is:

hj = y2 - y1

(9-17)

The relative height of the jump is:

hj y2 y1 _____ = _____ - _____

E1 E1 E1

(9-18)

where y1 /E1 = relative initial depth, and y2 /E1 = relative sequent depth.

In terms of the Froude number of the upstream flow, the relative height of the jump is:

hj ( 1 + 8 F12 )1/2 - 3

______ = _______________________

E1 2 + F12

(9-19)

Figure 9-19 is a graphical portrayal of the characteristics of the hydraulic jump. Note the

following:

The relative sequent depth reaches a maximum value y2 /E1 = 0.8 for F1 = 1.73.

The relative height of the jump reaches a maximum value hj /E1 = 0.507 for F1 =

2.77.

It is confirmed that for F1 = 1, the initial depth y1 is equal to 2/3 (0.667) of the

specific energyE1.

For F > 3, the changes in all characteristics become gradual.

Fig. 9-19 Characteristics of the hydraulic jump.

Efficiency of the hydraulic jump

With reference to Fig. 9-20, the efficiency of the hydraulic jump is:

E2 ( 1 + 8 F12 )3/2 - 4 F1

2 + 1 ______ = _______________________________

E1 8 F12 ( 2 + F1

2 )

(9-16)

Fig. 9-20 Definition sketch for a hydraulic jump.

F1 = v1 / (gy1)1/2

F2 = v2 / (gy2)1/2

v1y1 = v2 y2

v12y1

2 = v22 y2

2

F12 y1

3 = F22 y2

3

F22 = F1

2 / (y2 / y1)3

The hydraulic jump equation is (Eq. 9-13):

y2 / y1 = (1/2) [ (1 + 8 F12)1/2 - 1 ]

N 2 = 1 + 8 F12

y2 / y1 = (1/2) [ N - 1]

2 (y2 / y1) = N - 1

(y2 / y1)3 = (1/8) [ N - 1]3

2 (y2 / y1)3 = (1/4) [ N - 1]3

N = ( 1 + 8 F12)1/2

N 3 = ( 1 + 8 F12)3/2

N 2 - 1 = 8 F12

F12 = ( N 2 - 1) / 8

4 F12 = ( N 2 - 1) / 2

The efficiency of the hydraulic jump is:

E2/E1 = [ y2 + v22/(2g) ] / [ y1 + v1

2/(2g) ]

E2/E1 = [ y2(1 + F22/2) ] / [ y1(1 + F1

2/2) ]

E2/E1 = 2 (y2/y1) {1 + F12 / [ 2 (y2/y1)

3] } / (2 + F12)

E2/E1 = (N - 1) {1 + (N 2 - 1) / [ 2 (N - 1)3 ] } / (2 + F12)

E2/E1 = (N 2 - 1)(N - 1) {1 + (N 2 - 1) / [ 2 (N - 1)3 ] } / [8 F12(2 + F1

2) ]

E2/E1 = [ (N 2 - 1)(N - 1) + (1/2)(N + 1)2 ] / [ 8 F12(2 + F1

2) ]

E2/E1 = { (N 3 - N 2 - N - 1) + [ (N 2/2) + N + (1/2)] } / [ 8 F12(2 + F1

2) ]

E2/E1 = { (N 3 - [(N 2 - 1)/2] + 1} / [ 8 F12 (2 + F1

2) ]

E2/E1 = [ (1 + 8 F12)3/2 - 4F1

2 + 1] / [ 8 F12(2 + F1

2) ] ANSWER.

Example 9-3

Given q = 0.5 m2/s and ΔE = 0.678 m, calculate the sequent depths y1 and y2. Confirm the result using ONLINE CHANNEL 16.

The calculation proceeds by trial and error.

1. Assume a low value of F1, say 2.

2. Use Eq. 9-13 to calculate y2 / y1.

3. Use Eq. 9-14 to solve for y1.

4. Solve for the usptream velocity: v1 = q / y1

5. Calculate the value of F1.

6. If the value calculated in Step 5 is the same as that assumed in Step 1 (within a certain tolerance), stop and report the

sequent depths.

7. Otherwise, increase the value of F1 by a suitable increment, and return to Step 1.

ONLINE CALCULATION. Use ONLINE CHANNEL 16, with q = 0.5 m2/s and ΔE = 0.678 m. The sequent depths

are: y1 = 0.1 m and y2 = 0.665 m.

Example 9-4

Given y1 = 0.1 m and v1 = 5 m/s, calculate the efficiency of the hydraulic jump. Confirm the result using ONLINE CHANNEL 18.

The upstream flow Froude number is: F1 = 5.049. Using Eq. 9-16: E2 / E1 = 0.505.

ONLINE CALCULATION. Using ONLINE CHANNEL 18, with y1 = 0.1 m, and v1 = 5 m/s, the Froude number

is: F1 = 5.049. The efficiency of the jump is: E2 / E1 = 0.505, confirming the hand calculation.

Length of the hydraulic jump

The length of the hydraulic jump is defined as the distance measured from the front

face to a point (on the water surface) located immediately downstream of the roller (see

insert of Fig. 9-21). The relative length of the jump L/y2 has been determined

experimentally. It varies from about 5.0 to 6.15 for a wide range of upstream Froude

numbers. Within the range 4.5 ≤ F1 ≤ 13, the relative length of the jump is slightly

greater than 6 (Fig. 9-21).

Fig. 9-21 Length of the hydraulic jump (U.S. Bureau of Reclamation).

Profile of the hydraulic jump

Experimental measurements have shown that the actual surface profile of the hydraulic

jump varies somewhat with the upstream Froude number. The profiles may be plotted

as dimensionless curves, as shown in Fig. 9-22. A closeup of a hydraulic jump is shown

in Fig. 9-23.

Fig. 9-22 Dimensionless profiles of the hydraulic jump (Chow, 1959).

Fig. 9-23 Closeup of a hydraulic jump.

QUESTIONS

[Problems] [References] • [Top] [The Sharp-Crested Weir] [Crest Shape of Overflow Spillway] [Rating of

Spillways] [The Hydraulic Jump]

1. How does rapidly varied flow differ from gradually varied flow?

2. Is there a theoretical solution of one-dimensional rapidly varied flow?

3. What is the exponent of the rating in a broad-crested weir?

4. What is the coefficient of the rating for high spillways in U.S. Customary units?

5. What does "ogee" stand for?

6. What is the rationale for the use of a labyrinth spillway?

7. What is the risk when flow over a spillway exceeds the design stage?

8. What flow condition produces a hydraulic jump?

9. Is the hydraulic jump equation linear or nonlinear?

10. How is the length of the hydraulic jump measured?

PROBLEMS

[References] • [Top] [The Sharp-Crested Weir] [Crest Shape of Overflow Spillway] [Rating of Spillways] [The Hydraulic

Jump] [Questions]

1. An emergency spillway is being considered for the Demerera Water Conservancy

to safeguard the integrity of the dam under conditions of climate change (similar

to Fig. 9-21). Assume that the existing relief sluices would be inoperable during a

major flood due to high tailwater. Determine the length of the free-overflow

spillway required to pass the Probable Maximum Flood (PMF). The following data

is applicable: a. PMF 1-day duration: 428 mm b. Hydrologic abstraction: 18 mm c. Contributing drainage area: 582 km

2

d. Time base of the flood hydrograph: 3 days e. Embankment crest elevation: 18.288 m f. Spillway crest elevation: 17.526 m g. Freeboard: 0.3 m h. Weir discharge coefficient: 1.45

For simplicity, assume a triangular-shaped flood hydrograph. Use all the

freeboard to contain the PMF.

Fig. 9-21 The 8000-ft weir of the Boerasirie Water Conservancy.

2. Design an overflow-spillway section having a vertical upstream face and a crest

lengthL = 150 ft. The design discharge is Q = 50,000 ft3/s. The upstream water

surface at design discharge is at Elev. 750 ft, and the average channel floor is at

Elev. 650 ft (see Fig. 9-6 for graphical example).

3. Use ONLINE OGEE RATING to determine the rating for an ogee spillway with

length L = 15 m, design head Hd = 2 m, spillway crest elevation = 1045 m, river

bed elevation = 1000 m, and freeboard Fb = 1 m. Neglect the approach velocity.

What should be the spillway length to pass the Probable Maximum Flood QPMF =

250 m3/s while taking all the freeboard? Express spillway length to nearest 0.1 m

by excess.

4. Prove Eq. 9-14.

5. Prove Eq. 9-19.

6. Calculate the energy loss in a hydraulic jump, given the sequent depths y1 = 0.58

m andy2 = 2.688 m.

7. Calculate the relative height of the hydraulic jump hj / E1 for F1 = 3.

8. Use ONLINE CHANNEL 11 to calculate the sequent depth y2 through a hydraulic

jump when the discharge is q = 5 m2/s and the initial depth y1 = 0.58 m.

9. Use ONLINE CHANNEL 12 to determine the sequent depth y2 and energy loss

ΔE through a hydraulic jump when the discharge is q = 10 ft2/s and the upstream

flow depth is y1 = 0.5 ft.

10. Use ONLINE CHANNEL 16 to calculate the sequent depths through a

hydraulic jump when the discharge is q = 10 ft2/s and the energy loss is ΔE =

3.287 ft.

11. Use ONLINE CHANNEL 16 to calculate the sequent depths through a

hydraulic jump when the discharge is q = 5 m2/s and the energy loss is ΔE = 1.5

m.

12. Use ONLINE CHANNEL 18 to calculate the efficiency of the hydraulic

jump E2 / E2 forq = 10 ft2/s and y1 = 0.5 ft.

REFERENCES

• [Top] [The Sharp-Crested Weir] [Crest Shape of Overflow Spillway] [Rating of Spillways] [The Hydraulic

Jump] [Questions] [Problems]

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

Roberson, J. A., J. J. Cassidy, and M. H. Chaudhry. 1998. Hydraulic Engineering. John

Wiley and Sons, New York, Second edition.

Pugh, C. A., and E. W. Gray. 1984. Fuse Plug Embankments in Auxiliary Spillways -

Developing Design Guidelines and Parameters, United States Committee on Large

Dams (as cited in U.S. Bureau of Reclamation).

http://openchannelhydraulics.sdsu.edu

140902 19:30

Hydraulic Research in Transition

The Evolving Role of Hydraulic Structures - From Development to Management of Water

by Philip H. Burgi, Manager, Water Resources Research Laboratory Issues and Directions in Hydraulics - An Iowa Hydraulics Colloquium, Iowa City, Iowa, May 23-25, 1995

(continued)

PROTECTING THE INFRASTRUCTURE - DAM SAFETY

The Department of the Interior's (DOI) dam safety program is administered by Reclamation. A component of the program is the development of new technologies (applied research) to cost-effectively solve dam safety problems. Inadequate spillway capacity is one of the primary reasons dams fail. Reclamation's dam safety research focuses on hydraulic investigations performed in the Water Resources Research Laboratory (WRRL). Alternative spillway designs, fuse plug concepts, and overtopping protection concepts are investigated. The safety improvements, innovative new design concepts and construction cost savings realized from this research have been significant.

INCREASED SPILLWAY CAPACITY - LABYRINTH SPILLWAYS

Reclamation has used labyrinth spillways on existing dams where the discharge capacity of a spillway is insufficient or where a reservoir must be enlarged. Research on the labyrinth spillway concept produced design criteria that were applied to augment the spillway capacity at Ute Dam on the Canadian River, New Mexico, and generated significant savings in field construction cost. Ute Dam spillway was constructed for $10 million, a $24 million cost savings over the estimated $34 million cost for a traditional gated structure. Houston (1982) described laboratory sectional flume studies and a 1:80 scale model of the full labyrinth spillway, figure 1. The 9-m-high, 14-cycle spillway had a length magnification ratio of 4 and a flow magnification ratio of 2.4.

Figure 1. View of the 1:80 scale model an prototype of the 14-cycle, 9-m-high Ute Dam Labyrinth crest passing 15,500 CMS.

EMERGENCY SPILLWAY CONCEPTS - FUSE PLUGS

Pugh (1984) defines a fuse plug as ". . . an embankment designed to wash out in a predictable and controlled manner when the capacity in excess of the normal capacity of the service spillway and outlet works is needed." A number of laboratory embankments, 0.15- to 0.38-m high and 2.7-m long at scales of 1:10 and 1:25, were tested in the WRRL to develop fuse plug spillway design criteria. Figure 2 illustrates a typical laboratory fuse plug investigation. Fuse plug designs have been selected for the dam safety Corrective Action Plan on the Verde River in Arizona: Horseshoe and Bartlett Dams. The fuse plug for Bartlett Dam is designed with an erosion resistant invert and abutment structure and will pass 10 100 CMS. Three erodible embankment sections will operate in sequence. The Horseshoe Dam fuse plug is designed to pass 6850 CMS through three 44- to 52-m-long openings 6.0-m to 7.9-m high, figure 3.

Figure 2. Laboratory photo showing fuse plug washout process.

Figure 3. Flow of 4250 CMS through the first two of the Horseshoe Dam Fuse Plug Model. Scale 1:60.

The documented construction cost savings of $150 to $300 million on the Verde River Dams are an example of the significant benefits resulting from this hydraulic research.

OVERTOPPING EMBANKMENT PROTECTION CONCEPTS

Flood flow overtopping an embankment is considered unacceptable. However, hydraulic research in recent years has greatly advanced the concept of embankment protection systems, not only for low dams (under 15-m high), but now for high dams as well. Frizell and others (1994) have reported on recent cooperative research funded by DOI dam safety, the Electric Power Research Institute (EPRI), and Colorado State University (CSU), which has resulted in design criteria development for concrete step overlay protection for embankment dams. Studies were completed in Reclamation's WRRL as well as tests performed in a large-scale, outdoor overtopping facility at CSU. The 1.5-m-wide, 15-m-high outdoor test facility subjected tapered blocks to unit discharges as high as 3.2 CMS/M, figure 4. The 35-cm-long by 5-cm-high by 60-cm-wide blocks are placed in an overlapping pattern on filter material. The blocks are designed to aspirate water from the filter layer through small drainage slots formed in each block. The block shapes developed through these studies are effective for a range of embankment slopes.

Figure 4A. View of flow starting over the 2:1 slope. Blocks are placed in an overlapping fashion on 15 cm of gravel.

Figure 4B. View of the 1.5-m-wide 15-m-high test facility at CSU discharging 3.0 CMS.

EVALUATING THE EXTENT OF DAM FOUNDATION EROSION

Dam failure can occur when the foundation is undercut or the downstream slope is eroded, eventually causing slumping of the dam crest and failure by overtopping. There are numerous documented instances of extensive erosion damage to spillways, stilling basins, and riverbeds that threaten dam stability. Predicting flow patterns, velocity distributions, material erosion, and other factors of impinging jets in prototype situations is largely beyond the capability of conventional physical and numerical modeling techniques. Reclamation has entered into a multi-year cooperative research and development agreement to develop new technologies for predicting the extent of erosion and scour at dam foundations, specifically at the downstream toe and abutments of dams. The study approach couples hydraulics and geomechanical index concepts and will utilize a laboratory scale model as well as a near- prototype test facility to develop and evaluate a predictive numerical code to estimate the extent of dam foundation erosion. The variety of materials and material properties present in dam foundations indicates the need for an erodibility index which is a product of mass strength, block size, inter-particle strength and relative orientation.

The 3.0-m-wide by 4.0-m-long by 1.8-m-high laboratory model is located in the hydraulic laboratory at the Engineering Research Center, Colorado State University, figure 5. The model is a 1:3 scale representation of the near-prototype facility. A 4.1-cm by 1.0-m-long orifice slot located approximately 1.5 m above the basin tailwater is capable of discharging 18.4 L/s. The near-prototype outdoor facility will have an 11.0-m-wide by 14.6-m-long by 4.6-m-deep basin. A large 12.2-cm by 3.0-m-long orifice located approximately 4.6 m over the basin will be capable of discharging 2.83 CMS into the plunge pool basin.

Figure 5. View of 1:3 scale laboratory model with orifice jet discharging into basin.

The laboratory and near-prototype studies will scientifically investigate the erosion processes of jacking, dislodging, and transporting foundation material. The test results will provide reliable data to verify the numerical code under development.

ENVIRONMENTAL RESTORATION

Water development and environmental interests are striving to coexist in providing today's society with a higher standard of living, while fully protecting environmental resources. In recent years hydropower production and agricultural water supply have been cut back substantially in the U.S. to meet regulatory agency requirements. Rivers regulated for hydropower development, urban and agriculture water supply, and flood control are complicated systems; operational decisions must consider fish behavior and environmental resources as well as engineering design. A bioengineering focus has led to new, innovative concepts for using hydraulic structures to manage regulated aquatic ecosystems in the West. A look at fishery and stream restoration issues in the West illustrates these new technological approaches.

RESERVOIR SELECTIVE WITHDRAWAL

The winter-run Chinook salmon population in the Sacramento River, California, has declined over the past two decades. A contributing cause of this decline is thought to be the mortality of eggs and fry caused by elevated water temperatures during the late summer and fall incubation and rearing season. Water temperatures exceeding 12.0 degrees C can cause significant egg mortality. Reclamation's hydraulic research in the late 1980's developed flexible curtain barriers to manage and control reservoir-release water temperatures for structures in the Trinity and Sacramento River drainages.

A sophisticated temperature stratified test facility (9-m by 9-m by 2.4-m deep) was built in Reclamation's WRRL to develop and test various temperature control device concepts for reservoir release. A refrigeration system was used to create temperature profiles in the range from 7 degrees C to 24 degrees C in the facility. Scaling laws allow research engineers to simulate releases from temperature-stratified reservoirs in the model facility. Flow in reservoirs approaching the outlets is significantly affected by water density which is directly related to temperature; therefore, it is important to properly simulate the water temperature in the laboratory test facility.

A model scale of 1:72 was used to simulate a 91-m-deep and 396-m-long, flexible curtain to control releases through the powerplant at Shasta Dam. The curtain could be lowered from the reservoir surface to permit withdrawal of warm surface water or could be raised off the bottom to access the cold bottom water. The 1.2-hectare curtain was to be made of 32-mil Hypalon reinforced with nylon. Lack of historic reference and field experience using underwater curtains of this size prompted the decision to use a "more traditional" steel structure. The steel structure to be installed on Shasta Dam in 1996 will extend 15.2 m out into the reservoir, run 122 m horizontally and plunge as deep as 107 m into the deeper part of the reservoir permitting access to near-bottom cold water. Field construction costs for the steel structure are estimated to be four times the cost of a flexible reservoir curtain.

The recent drought (1988-1992) in northern California resulted in limited volumes of stored cold water deep in reservoirs. Because of the urgent need to reduce reservoir outflow temperatures, Reclamation initiated an aggressive research program to study and install temperature control curtains in more shallow waters, such as those present at Lewiston Lake. Two curtains were designed and installed in Lewiston Lake in August 1992. The primary (reservoir) curtain, figure 6, was designed to hold back the warm water while colder water was released through Clear Creek Tunnel to the Sacramento River. The second curtain was designed to provide temperature control for water supplied to a nearby fish hatchery. Laboratory results indicated the reservoir curtain would reduce the water temperatures released from Lewiston Lake to the Clear Creek Tunnel by about 1.5 degrees C. Actual temperature measurements made at Clear Creek Tunnel intake after the August 1992 installation of the curtain showed a 1 to 1.5 degrees C temperature reduction, Vermeyen and Johnson (1993). Although this seems a small change in temperature, every degree reduction can significantly decrease the salmon egg mortality rate.

Figure 6. Illustration of the Lewiston Lake temperature control curtain.

In a continuing multi-agency effort, two additional flexible curtains were laboratory tested, designed, and installed in Whiskeytown Lake in 1993. The use of this new temperature control technology, as well as the steel shutter structure at Shasta Dam, will increase the selective withdrawal capability within the Sacramento River basin and provide improved management by selective withdrawal of the limited cold water resource in the reservoirs. Table 1 summarizes retrofit selective withdrawal structure technologies used at five Reclamation power stations.

FISH PASSAGE

Considerable effort has been placed on improving fish passage technologies in recent years, including new designs for fishways, improved spawning facilities, fish barriers with associated bypass designs for canals and various screening and fish behavioral control concepts. Most recently, efforts have centered on returning the Sacramento River near Red Bluff Diversion Dam to a "run of the river" condition. Several alternatives are under study to improve the fishery. One alternative proposed by Liston and Johnson (1992) will evaluate the feasibility of replacing the diversion dam with a pumping station utilizing fish-friendly pumps. The plant will deliver 76.5 CMS, with a lift of 4.3 m to the Tehama Colusa Canal, while incurring minimal fish mortality. Every effort will be made to minimize fish entrainment at the pump intakes. A pilot pumping plant has been built that will initially pump up to 9.5 CMS. The pilot plant is designed to evaluate and monitor the mechanical performance of two fish-friendly pump concepts as well as evaluate fishery issues associated with pumping. An Archimedes screw pump and a screw-centrifugal (helical) pump will be evaluated, figures 7 and 8. Two 3.0-m-diameter, 8.0-m-long Archimedes pumps, placed on a 38ø angle, will deliver a total of 4.5 CMS at a rotational speed of 28 r/min. One 1.2-m helical pump will deliver 5.0 CMS at 400 to 600 r/min. Construction of the pilot pumping plant will be completed in the spring of 1995. Fishery and mechanical issues will be evaluated over several years before

construction of the larger, permanent pumping plant. Figure 7. Schematic views of helical pump.

Figure 8. Schematic view of Archimedes pump.

STREAM RESTORATION

Reclamation's water development projects have altered the charter of rivers and watersheds. We now refer to these systems as regulated aquatic ecosystems. Restoring a watershed or ecosystem damaged by physical alterations to the natural flow regime requires multidisciplinary research involving engineers, biologists, geomorphologists and others.

Muddy Creek, near Great Falls, Montana, is an example of Reclamation's stream restoration efforts. This creek has been drastically altered by irrigation return flows. Muddy Creek experienced flows of 12.3 million cubic meters/year before irrigation. The creek now sustains runoff of 98.7 million cubic meters/year. This has resulted in a 15-m incised channel in the glacially deposited silty soil since the early 1900's. Active measures to restore stream gradient and reduce erosion are underway in a cooperative project involving Reclamation, the Natural Resources Conservation Service and the local irrigation district. The erosion control demonstration project includes 19 rock ramps and 3 barbs placed along a 6.6-km reach of the creek in 1994 to control 8.2 m of stream gradient. The demonstration project will look at long term performance of the in-place technology with monitoring programs to track stream response in the short and long term.

ENVIRONMENTALLY ACCEPTABLE LUBRICANTS

Hydropower stations have historically used oil and grease for lubrication. Some of this grease inevitably enters the water, creating a source of contamination of potable water and aquatic habitat. Blum (1994) describes a test apparatus developed by the WRRL specifically to test the mechanical performance of environmentally safe greases for wicket gate bushing applications, figure 9. A 1:4 scale model of a prototype wicket gate at Mt. Elbert Powerplant, Colorado, is enclosed in a rectangular conduit with flow through the facility scaled to represent flow through one wicket gate passage. A computer-controlled, motor-driven operator is used to cycle the gate through a model test time of 20 hours. This involves 1,370 opening and closing cycles. As a result of these tests, we are able to identify several environmentally safe lubricants, which meet both water quality standards and mechanical performance requirements.

Figure 9. Sectional view through the wicket gate lubricant test facility.

WATER QUALITY IMPROVEMENTS - DISSOLVED GASES

Hydraulic structures often dramatically affect water quality and aquatic life by changing the spatial and temporal distribution of dissolved gases within reservoirs and natural streams. Reclamation is using research to address problems of both nitrogen supersaturation and reduced dissolved oxygen concentrations downstream of energy dissipation structures and powerplants.

Spillways and outlet works associated with hydraulic structures affect the dissolved gas content of the released flow. Depending on the structure and conditions, there may be positive or negative impacts to the water quality. Releases may aerate flows depleted in dissolved gas, create supersaturated dissolved gas levels, or reduce supersaturated levels in the flow. Johnson (1975) presents an analysis to predict the effect of a wide variety of hydraulic structures on the dissolved gas content of the flow.

The Tennessee Valley Authority (TVA) has led efforts in recent years to improve dissolved oxygen conditions in reservoirs and downstream of powerplants, Bohac and Ruane (1990). Reclamation is now assisting TVA, the U.S. Army Corps of Engineers, the Iowa Institute of Hydraulic Research, and CSU with efforts to develop auto-venting turbine technologies that use aeration of powerplant flows to improve dissolved oxygen (DO) concentrations of powerplant releases. Reclamation has also retrofitted turbines at Deer Creek Powerplant on the Provo River, Utah, to improved DO concentrations through turbine aeration, Wahl (1995).

FUTURE NEEDS

There are numerous practical problems associated with hydraulic structures on water resource projects that will serve to motivate applied hydraulic research into the next century. Water use in the future will, of necessity, require continued use of hydraulic structures to effectively manage the water resource. Crucial topics for future research include:

Improved understanding of dam breach phenomena - Research is needed to provide better data and methods for selection of breach parameters and/or algorithms typically used in programs, such as BREACH and DAMBRK.

Removal of sediment deposition in reservoirs - Development of sediment transport bypass methods are needed for extension of reservoir life and environmental benefits.

Development of criteria for flushing flows - Flushing flows are now used to clean silts from gravel and substrate to provide improved spawning gravels. Development of a numerical model is needed to evaluate water routing schemes and provide management options for protection and restoration of riverine habitats.

Improved fish passage alternatives - Synergistic effort of hydraulic engineers and fishery biologists is needed to develop and evaluate new technologies for fish passage at dams. This is particularly important for the hundreds of small dams that are barriers to native species habitat.

Improved management of regulated aquatic ecosystems - Studies are needed to improve the application of computer and communication technologies to better regulate systems in real time and thus provide for demand management.

Some of our past successes in water resource development have produced new problems in water management as societal values have changed. The challenge is to keep our hydraulic research contemporary by clearly identifying the problems and identifying those social and scientific disciplines needed as partners in developing holistic solutions.

REFERENCES

Blum, Leslie J., and Pugh, Clifford A., (1994) "Environmentally Safe 'Green' Lubricants for Wicket Gates," Concepts for the Future, HydroVision 94.

Bohac, Charles E., and Ruane, Richard J., (1990) "Solving the Dissolved Oxygen Problem," Hydro Review, February 1990.

Bureau of Reclamation, (1960) Design of Small Dams, A Water Resources Technical Publication.

Frizell, Kathleen H., Smith, D. H., Ruff, J. F., (1994) "Stepped Overlays Proven for Use in Protecting Overtopped Embankment Dams," 11th Annual ASDSO Conference, Boston.

Houston, Kathleen L., (1982) "Hydraulic Model Study of Ute Dam Labyrinth Spillway," Bureau of Reclamation, GR-82-7.

Johnson, Perry, (1975) "Prediction of Dissolved Gas at Hydraulic Structures," Bureau of Reclamation Report GR-8-75.

Liston, Charles R. and Johnson, Perry L., (1992) "Design Criteria and Environmental Setting For a Pilot Pumping Installation at Red Bluff Diversion Dam on the Sacramento River, California," Internal Report.

Peterka, A. J., (1958) Engineering Monograph No. 25, "Hydraulic Design of Stilling Basins and Energy Dissipators."

Pugh, C. A., and Gray, E. W., (1984) "Fuse Plug Embankments in Auxiliary Spillways - Developing Design Guidelines and Parameters," United States Committee on Large Dams.

Rhone, Thomas J., (1990) "Development of Hydraulic Structures," 50th Anniversary of the Hydraulics Division, ASCE, pp 132-147.

Vermeyen, Tracy B., and Perry L. Johnson, (1993), "Hydraulic Performance of a Flexible Curtain Used for Selective Withdrawal - A Physical Model and Prototype Comparison," Proceedings ASCE National Hydraulic Engineering Conference.

Wahl, Tony L., and Young, Doug, (1995) "Dissolved Oxygen Enhancement on the Provo River," Waterpower '95, San Francisco, California.

[Governing Equations] [Linear Solutions] [Kinematic Waves] [Diffusion Waves] [Muskingum Method] [Muskingum-

Cunge Method] [Dynamic Waves] [Questions] [Problems] [References] •

CHAPTER 10:

UNSTEADY GRADUALLY VARIED

FLOW

10.1 GOVERNING EQUATIONS

[Linear Solutions] [Kinematic Waves] [Diffusion Waves] [Muskingum Method] [Muskingum-Cunge Method] [Dynamic

Waves] [Questions] [Problems] [References] • [Top]

Three conservation principles are applicable in open-channel flow:

1. Conservation of mass, which could be either: (a) steady, or (b) unsteady,

2. Conservation of energy, which is steady (recall that energy is equal to the integral

of all forces, excluding inertia, in space, Eq. 2-15), and

3. Conservation of momentum, which is unsteady (recall that momentum is equal to

the integral of the inertia force in time, Eq. 2-27).

Steady gradually varied flow combines the statements of steady conservation of mass

and conservation of energy (Chapter 7). Unsteady gradually varied flow combines the

statements of unsteady conservation of mass and conservation of momentum (Table

10-1). Thus, unsteady gradually varied flow differs from steady gradually varied flow in

its description of the temporal variation of the flow variables (discharge, stage, flow

depth, mean velocity, and so on).

In practice, steady gradually varied flow is simply referred to as "gradually varied flow"

(GVF), while unsteady gradually varied flow is commonly referred to as "unsteady flow"

(UF).

Table 10-1 Conservation laws and types of gradually varied flow.

Model

First equation:

Conservation of mass, either → Steady mass Unsteady mass

Second equation:

Conservation of → Energy Momentum

Flow

Type (name) → Steady GVF Unsteady GVF

Commonly referred to as → Gradually varied flow Unsteady flow

Treated in → Chapter 7 Chapter 10

(this chapter)

Figure 10-1 depicts the forces acting on a control volume. One body force and two

surface forces are shown. The body force is the component of the gravitational force

resolved along the direction of motion (W sin θ). The surface forces are: (1) the force

due to the pressure gradient (due to the difference in flow depths), ΔP = P2 - P1, and (2)

the force developed along the bottom boundary due to friction (Ff ). When these three

forces are in equilibrium along the direction of motion, the flow is steady from the force

standpoint. When the three forces are NOT in equilibrium along the direction of motion,

the flow is unsteady and a fourth force arises (the inertia force) to produce a balance.

Fig. 10-1 Body and surface forces in a control volume.

Governing equations

The derivation of the governing equations of unsteady flow (Fig. 10-2) (unsteady

gradually varied flow) considers the statements of mass and momentum conservation

in a control volume (Fig. 10-3).

Fig. 10-2 Definition sketch for flow depth h and velocity u

under unsteady flow.

Fig. 10-3 Gradients of flow depth and velocity in a control volume.

The statement of conservation of mass is:

Inflow - outflow = change in storage

For a unit-width channel (Liggett, 1975):

∂u Δx ∂h Δx ∂u Δx ∂h Δx ∂h

(u - ____ ____ ) (h - ____ ____ ) - (u + ____ ____ ) (h + ____ ____ ) = ____ Δx

∂x 2 ∂x 2 ∂x 2 ∂x 2 ∂t

(10-1)

Simplifying, and neglecting second-order terms as Δx → 0:

∂h ∂h ∂u ____ + u ____ + h ____ = 0

∂t ∂x ∂x

(10-2)

∂h ∂ ____ + ____ (uh) = 0

∂t ∂x

(10-3)

Equation 10-3 is the unsteady conservation of mass equation, commonly referred to as

the equation of continuity. However, its complete name is: the differential equation of

water continuity.

For steady flow: ∂h/∂t = 0, and Eq. 10-3 reduces to: q = uh = constant. In general, for a

channel of flow area A, the steady water continuity equation is: Q = uA = constant.

The statement of conservation of momentum is:

The net rate of momentum entering the control

volume + the sum of the forces acting on it = the

rate of accumulation of momentum.

The momentum, per unit of channel width, is: ρ u (uh). Therefore, the net rate of

momentum entering the control volume FΔ (force per unit of width) is:

∂ Δx ∂ Δx

FΔ = ρ { u (u h) - ____ [ u (u h) ] ____ } - ρ { u (u h) + ____ [ u (u h) ] ____ }

∂x 2 ∂x 2

(10-4)

The forces acting on the control volume, resolved along the direction of motion, are: (1)

gravitational force, (2) pressure-gradient force, and (3) frictional force (Fig. 10-1).

The gravitational force, per unit of width, is:

Fg = ρ g h Δx sin θ ≅ ρ g h Δx tan θ = ρ g h Δx So

(10-5)

The pressure-gradient force, per unit of width, developed along the sides of the control

volume, is:

h h 1

Fp = ∫ p dz = ρ g ∫ (h - z ) dz = = ____ ρ g h 2

0 0 2

(10-6)

The frictional force, per unit of width, developed along the channel bottom, resembles

the gravitational force, but it is opposite in sign:

Ff = - ρ g h Δx Sf

(10-7)

in which Sf = friction slope. [As an exception, for channels of adverse bottom slope, for

which So < 0, the gravitational and frictional forces are of the same sign].

The size of the control volume, per unit of width, is: (h Δx). The mass, per unit of width,

is: (ρ h Δx).Thus, the rate of accumulation of momentum Fm (force per unit of width) is:

∂

Fm = ____ (ρ u h) Δx

∂t

(10-8)

The conservation of momentum is:

FΔ + Fg + Fp + Ff = Fm

(10-9)

Replacing Eqs. 10-4 to 10-8 into Eq. 10-9:

∂ Δx ∂ Δx

ρ { (u 2 h ) - ____ (u 2 h ) ____ } - ρ { (u 2 h ) + ____ (u 2 h ) ____ } + ρ g h Δx So

∂x 2 ∂x 2

1 ∂h 2 Δx ∂h 2 Δx

+ ____ ρ g [ (h 2 - _____ ____ ) - (h 2 + _____ ____ ) ] + ρ g h Δx Sf

2 ∂x 2 ∂x 2

∂

= ____ (ρ u h) Δx

∂t

(10-10)

Simplifying Eq. 10-10:

∂ ∂ g ∂h 2 ____ (u h ) + ____ (u 2 h ) + ____ ______ = g h (So - Sf )

∂t ∂x 2 ∂x

(10-11)

Equation 10-11 is in conservation form. For certain applications, it must remain in this

form. However, it is often expressed in reduced form, by operating on the derivatives:

∂u ∂h ∂ ∂u ∂h

h ____ + u ____ + u ____ (u h ) + (u h ) ____ + g h ____ = g h (So - Sf )

∂t ∂t ∂x ∂x ∂x

(10-12)

The second and third terms of Eq. 10-12 have implicit in them the continuity equation

(Eq. 10-3). Thus, Eq. 10-12 reduces to:

∂u ∂u ∂h ____ + u ____ + g ____ + g (Sf - So ) = 0

∂t ∂x ∂x

(10-13)

By dividing by g, Eq. 10-13 is expressed in slope units:

1 ∂u u ∂u ∂h ____ _____ + ____ _____ + _____ + (Sf - So ) = 0

g ∂t g ∂x ∂x

(10-14)

Equation 10-14 is referred to as the equation of motion. It is expressed in terms of

slopes as follows:

Sa + Sc + Sp + Sf - So = 0

(10-15)

in which Sa = local acceleration slope, Sc = convective acceleration slope, Sp =

pressure-gradient slope, Sf = friction slope, and So = bottom slope.

Wave types

Equation 10-15 indicates that the momentum balance is essentially a balance of

slopes. In the general case, when all forces are present, all slopes are acting and the

solution is the most general. In certain cases, however, one or more slopes may be

reduced to zero, or assumed to be negligible (compared to the remaining slopes). This

simplification gives rise to several types of waves, described in Table 10-2.

Table 10-2 Types of waves in unsteady open-channel flow.

No. Wave type

Slopes

Common name Applications

Sa Sc Sp Sf So

1 Kinematic wave ✓ ✓ Kinematic wave Overland flow

2 Kinematic-with-

diffusionwave ✓ ✓ ✓ Diffusion wave Flood routing

3 Dynamic wave ✓ ✓ ✓ Gravity

wave Laboratory flumes,

small canals

4 Steady-dynamic wave ✓ ✓ ✓ ✓ Steady-dynamic

wave Special cases

5 Mixedkinematic-

dynamicwave ✓ ✓ ✓ ✓ ✓ Dynamic wave

Dam-breachflood

routing

Applicability of wave types

Kinematic waves [1] apply to overland flow, where the bottom slopes are steep, typically greater

than So > 0.01.

Diffusion waves [2] apply to flood routing in streams and rivers, with intermediate bottom slopes

(0.01 > So > 0.0001).

Dynamic waves [3] apply to the short waves of laboratory flumes and small canals.

Steady-dynamic waves [4] apply to the few special cases where the neglect of the local acceleration

term (Sa) is justified on practical grounds. In general, however, it is more accurate to

neglect both Sa and Sc (i.e., a diffusion wave [2]) than just Sa (Ponce, 1990).

Mixed kinematic-dynamic waves [5] apply to a sudden wave such as that originating in a dam

breach, and may also apply to channels of very mild slope (So < 0.0001). Note that, in practice, there is

some confusion in the use of the term "dynamic wave" for the waves that include both inertia terms

(Sa and Sc), i.e., waves types [3] and [5].

10.2 LINEAR SOLUTION

[Kinematic Waves] [Diffusion Waves] [Muskingum Method] [Muskingum-Cunge Method] [Dynamic

Waves] [Questions] [Problems] [References] • [Top] [Governing Equations]

Equations 10-2 and 10-14 are the governing equations of continuity and motion, also

referred to as the Saint-Venant equations (Saint-Venant, 1871). They are repeated

here for convenience, as Eqs. 10-16 and 10-17, respectively.

∂h ∂h ∂u ____ + u ____ + h ____ = 0

∂t ∂x ∂x

(10-16)

1 ∂u u ∂u ∂h ____ _____ + ____ _____ + _____ + (Sf - So ) = 0

g ∂t g ∂x ∂x

(10-17)

These equations constitute a set of two nonlinear (actually, quasilinear) partial

differential equations, which when appropriately combined, result in a second-order

partial differential equation of the hyperbolic type, featuring two solutions. To this date,

there is no closed-form analytical solution of the set of Eqs. 10-16 and 10-17. An

approximate solution may be obtained by linearizing the equation set and using the

tools of linear stability analysis (Ponce and Simons, 1977).

The friction slope Sf is directly related to the bottom shear stress τ by the expression

(similar to Eq. 6-16):

τ

Sf = _____

γ h

(10-18)

In the usual manner of stability calculations, Eqs. 10-16 and 10-17 must satisfy the

unperturbed flow, for which u = uo , h = ho , and τ = τo . They must also satisfy the

perturbed flow, for which u = uo + u' ,h = ho + h' , and τ = τo + τ' . The

superscript ' represents a small perturbation to the steady uniform flow. Thus, all

quadratic terms in the fluctuating components may be negledted due to an order-of-

magnitude reasoning.

Substitution of the perturbed variables in Eqs. 10-16, 10-17, and 10-18, yields, after

linearization (Lighthill and Whitham, 1955):

∂h' ∂h' ∂u' ____ + uo ____ + ho ____ = 0

∂t ∂x ∂x

(10-19)

1 ∂u' uo ∂u' ∂h' τ' h' ____ _____ + ____ _____ + _____ + So ( _____ - _____ ) = 0

g ∂t g ∂x ∂x τo ho

(10-20)

in which:

τo

So = _____

γ ho

(10-21)

The boundary shear stress τ can be related to the mean velocity u as follows (Eq. 5-3):

τ = f ρ u 2

(10-22)

in which the friction factor f is (Eq. 5-12):

g

f = ______

C 2

(10-23)

In view of Eq. 10-22, Eq. 10-20 is converted to:

1 ∂u' uo ∂u' ∂h' u' h' ____ _____ + ____ _____ + _____ + - So ( 2 _____ - _____ ) = 0

g ∂t g ∂x ∂x uo ho

(10-24)

Small perturbation analysis

The solution for a small perturbation in the depth of flow is postulated in the following

exponential form (Ponce and Simons, 1977):

h' ____ = h* e i (ς*

x*

- β*

t*

)

ho

(10-25)

in which the subscript * indicates dimensionless variables, and i = (-1)1/2

. The

quantity σ* = dimensionless wavenumber, β* = dimensionless complex propagation

factor, and x* and t* are dimensionless space and time coordinates, such that:

2 π

ς* = ( _____ ) Lo

L

(10-26)

2 π Lo

βR* = ( _____ ) _____

T uo

(10-27)

x

x* = _____

Lo

(10-28)

uo

t* = t _____

Lo

(10-29)

and βI* is an amplitude propagation factor. The quantity Lo is the length of channel in

which the uniform flow drops a head equal to its depth:

ho

Lo = _____

So

(10-30)

The depth disturbance is associated with a velocity disturbance of the form:

u' ____ = u* e i (ς*

x*

- β*

t*

)

uo

(10-31)

The substitution of Eqs. 10-25 and 10-31 into Eqs. 10-19 and 10-24 yields the set:

ς* u* + (ς* - β* ) d* = 0

(10-32)

[ 2 + i Fo2 (ς* - β* ) ] u* + (i ς* - 1 ) d* = 0

(10-33)

in which

uo 2

Fo2 = ______

g ho

(10-34)

Equations 10-32 and 10-33 constitute a homogeneous system of linear equations in the

unknowns u*and h*. For the solution to be nontrivial, the determinant of the coefficient

matrix must vanish. Therefore, the following relation holds:

i β* 2 Fo

2 - i ς* 2 (1 - Fo

2 ) + 3 ς* - 2 β* - 2 i ς* β* Fo2 = 0

(10-35)

Equation 10-35 is the characteristic equation governing the propagation of small

amplitude water waves. Through algebraic manipulation, Eq. 10-35 reduces to:

Fo2 β*

2 - 2 (ς* Fo2 - i ) β* - [ ς*

2 (1 - Fo2 ) + 3 ς* i ] = 0

(10-36)

The solution of Eq. 10-36 is (Ponce and Simons, 1977):

1

β* = ς* ( 1 - i ζ ) + ς* [ ( ______ - ζ 2 ) + i ζ ] 1/2

Fo2

(10-37)

in which

1

ζ = _________

ς* Fo2

(10-38)

The equations for dimensionless celerity and attenuation for the primary and secondary

waves are:

C + A

c1* = 1 + ( _________ )1/2

2

(10-39)

B - E

δ1 = - 2π _____________

| 1 + D |

(10-40)

C + A

c2* = 1 - ( _________ )1/2

2

(10-41)

B + E

δ2 = - 2π _____________

| 1 - D |

(10-42)

in which

1

A = ______ - ζ 2

Fo2

(10-43)

B = ζ

(10-44)

1

C = [ ( ______ - ζ 2 ) 2 + ζ 2 ] 1/2

Fo2

(10-45)

C + A

D = ( _________ )1/2

2

(10-46)

C - A

E = ( _________ )1/2

2

(10-47)

The dimensionless relative wave celerity is:

C + A (10-48)

cr* = ( _________ )1/2 = D

2

Figure 10-4 shows a plot of the dimensionless relative wave celerity cr* versus the

dimensionless wavenumber σ*. Figure 10-5 shows a plot of the primary wave

logarithmic decrement -δ1 versus the dimensionless wavenumber σ*, applicable for

Froude numbers F < 2. Figure 10-6 shows a plot of the primary wave logarithmic

increment +δ1 versus the dimensionless wavenumber σ*, applicable for Froude

numbers F > 2. Based on these figures, the characteristics of shallow waves are

described in the box below.

Fig. 10-4 Dimensionless relative wave celerity vs dimensionless wavenumber

in unsteady open-channel flow.

Fig. 10-5 Primary wave logarithmic decrement

for Froude numbers F < 2.

Fig. 10-6 Primary wave logarithmic increment

for Froude numbers F > 2.

Shallow wave propagation in open-channel flow 1

Kinematic waves lie to the extreme left of the wavenumber spectrum, with dimensionless

relative celerity cr* = 0.5 (under Chezy friction), and have very little, if any, attenuation

under F < 2 (very small logarithmic decrement).

Diffusion waves lie to the left of the wavenumber spectrum, with dimensionless relative

celerity approximately equal to cr* = 0.5 (under Chezy friction), and small but measurable

attenuation under F < 2 (small logarithmic decrement).

Dynamic waves lie to the extreme right of the wavenumber spectrum, with dimensionless

relative celerity cr* = 1/F, and very little, if any, attenuation underF < 2 (very small

logarithmic decrement).

Mixed kinematic-dynamic waves lie toward the middle of the wavenumber spectrum, with

varying dimensionless relative celerity and very strong attenuation under F < 2 (large

logarithmic decrement).

For Vedernikov number V < 1, which corresponds to F < 2 (Chezy friction in hydraulically

wide channels), the dimensionless relative celerity varies widely across the wavenumber

spectrum. Maximum attenuation occurs at the point of inflexion of the dimensionless celerity

curve.

For V = 1, which corresponds to F = 2, the dimensionless relative celerity is a constant across

the wavenumber spectrum. This flow condition is the threshold for the development of roll

waves, which occur for F ≥ 2 (Craya, 1952).

For V > 1, which corresponds to F > 2, the dimensionless relative celerity varies across the

wavenumber spectrum. Maximum amplification (i.e., negative attenuation) occurs at the point

of inflexion of the dimensionless celerity curve.

1 See Ponce and Simons (1977) for a detailed treatment of shallow wave propagation in open-channel flow.

10.3 KINEMATIC WAVES

[Diffusion Waves] [Muskingum Method] [Muskingum-Cunge Method] [Dynamic

Waves] [Questions] [Problems] [References] • [Top] [Governing Equations] [Linear Solution]

A kinematic wave is an idealization (of gradually varied unsteady open-channel flow)

that neglectsboth acceleration terms (local and convective) and the pressure-gradient

term (Table 10-2). By neglecting these terms, the equation of motion (Eq. 10-14) is

reduced to a statement of steady uniform flow:

Sf = So

(10-49)

The unsteadiness of the phenomenon, however, is preserved through the time-varying

term in the continuity equation (Eq. 10-3). The combination of Eqs. 10-3 and 10-49

gives rise to the kinematic wave equation.

Since q = uh, Eq. 10-3 may be expressed in terms of the unit-width discharge:

∂h ∂q ____ + ____ = 0

∂t ∂x

(10-50)

In terms of discharge Q, the continuity equation is:

∂A ∂Q _____ + _____ = 0

∂t ∂x

(10-51)

A statement of uniform flow (Eq. 10-49) may be properly represented by the discharge-

area rating:

Q = α A β

(10-52)

in which α and β are coefficient and exponent, respectively. The coefficient α varies as

a function of type of friction, cross-sectional shape, and bottom slope. The

exponent β varies as a function of type of friction and cross-sectional shape.

Assuming for the sake of simplicity that α and β are independent of A, Eq. 10-52 yields:

dQ _____ = α β A β - 1

dA

(10-53)

dQ Q _____ = β ____

dA A

(10-54)

dQ _____ = β V

dA

(10-55)

in which V = Q / A = mean velocity.

The kinematic wave equation is obtained by combining Eqs. 10-51 and 10-55 to yield:

∂Q ∂Q _____ + β V _____ = 0

∂t ∂x

(10-56)

In terms of unit-width discharge q :

∂q ∂q _____ + β V _____ = 0

∂t ∂x

(10-57)

Convective celerity

Equation 10-56 (or Eq. 10-57) is a first-order partial differential equation. It describes

the convection of the quantity Q (or q) with the convective velocity or celerity ck,

where ck is:

ck = β V

(10-58)

Given Eq. 10-55, the convective velocity may also be expressed as:

dQ

ck = _____

dA

(10-59)

Given that dA = T dy (Eq. 3-11), where T = channel top width, and y = stage, the

convective velocity may also be expressed as follows:

1 dQ

ck = ____ _____

T dy

(10-60)

Equation 10-60 was originally derived by Kleitz (1877) and was later discovered from

actual field observations by Seddon (1900). It often referred to as the Kleitz-Seddon

Law, or simply Seddon's Law. Equation 10-58 is used when β is known with certainty,

Eq. 10-59 in theoretical formulations, and Eq. 10-60 in practical applications.

Since Eq. 10-56 is a first-order partial differential equation, it does not allow for wave

diffusion (wave attenuation, or wave dissipation). Diffusion can only be obtained

through the agency of a second-order term. Under the assumption of linearity (constant

convective celerity), a kinematic wave will convect its discharge with no wave diffusion;

that is, the discharge will retain its shape and remain constant in space and time upon

propagation.

Absence of wave diffusion in the kinematic wave

The absence of wave diffusion can be further demonstrated by a mathematical argument. The

total derivative for Q is:

∂Q ∂Q

dQ = _____ dt + _____ dx

∂t ∂x

(10-61)

Therefore:

dQ ∂Q dx ∂Q _____ = _____ + _____ _____

dt ∂t dt ∂x

(10-62)

The kinematic wave equation is repeated here:

∂Q ∂Q _____ + β V _____ = 0

∂t ∂x

(10-56)

Comparing Eq. 10-61 with Eq. 10-55, it follows that dQ/dt = 0, that is, Q remains constant in

time for waves traveling with the convective celerity β V.

Kinematic shock

When the linearity assumption is relaxed, the kinematic wave may change its shape by

becoming either (a) steeper [Fig. 10-7 (a)], or (b) flatter [Fig. 10-7 (b)]. Whether a wave

will steepen or flatten out will depend largely on the channel cross-sectional shape.

Two asymptotic limits are recognized: (1) waves propagating in hydraulically wide

channels, while (2) waves propagating in inherently stable channels (Chapter 1). In

hydraulically wide channels, the waves will steepen, while in inherently stable channels,

they will flatten out (Ponce and Windingland, 1985).

Fig. 10-7 (a) Kinematic wave steepening.

Fig. 10-7 (b) Kinematic wave flattening.

When allowed to proceed unchecked, the steepening will eventually result in a

kinematic wave becoming a kinematic shock. Thus, a kinematic shock is an unsteady

open-channel flow feature intrinsically related to the kinematic wave: A wave must be

kinematic before it can develop into a kinematic shock (Lighthill and Whitham, 1955).

Kibler and Woolhiser (1970) sought to clarify the occurrence of kinematic shock

phenomena by stating:

"While the shock wave development may arise under certain highly

selective physical circumstances, it is looked upon in this study as

a property of the mathematical equations used to explore the

overland flow problem, rather than an observable feature of this

hydrodynamic process..."

Thus, the kinematic shock is real but rare in the physical world, where spatial

irregularities manifest themselves as diffusion, with the net effect of arresting shock

development. On the other hand, the computational world is likely to be much more

regular, thereby inhibiting diffusion and promoting "numerical" shock development.

Kinematic wave celerity

The relative kinematic wave celerity, ie., the kinematic wave celerity taken relative to

the flow velocity, is:

crk = (β - 1 ) V

(10-63)

Furthermore, the dimensionless relative kinematic wave celerity is:

crk

cdrk = _____ = β - 1

V

(10-64)

According to Eq. 1-11, the relative dimensionless kinematic wave celerity is:

V

cdrk = β - 1 = ____

F

(10-65)

Thus, for V = 1, i.e., for neutrally stable flow, the Froude number is:

1 1

Fns = _______ = _______ (10-66)

β - 1 cdrk

Table 10-3 shows the variation of: (a) the exponent β, (b) the dimensionless relative

kinematic wave celerity cdrk, and (c) the neutral-stability Froude number, with selected

types of friction and cross-sectional shape.

Table 10-3 Variation of β as a function of type of friction and cross-sectional shape.

[1] [2] [3] [4] [5] [6]

β 24 β Type of friction Cross-sectional shape cdrk Fns

3 72 Laminar Hydraulically wide 2 1/2

8/3 64 Mixed laminar-turbulent

(25% turbulent Manning) Hydraulically wide 5/3 3/5

21/8 63 Mixed laminar-turbulent

(25% turbulent Chezy) Hydraulically wide 13/8 8/13

7/3 56 Mixed laminar-turbulent

(50% turbulent Manning) Hydraulically wide 4/3 3/4

9/4 54 Mixed laminar-turbulent

(50% turbulent Chezy) Hydraulically wide 5/4 4/5

2 48 Mixed laminar-turbulent

(75% turbulent Manning) Hydraulically wide 1 1

15/8 45 Mixed laminar-turbulent

(75% turbulent Chezy) Hydraulically wide 7/8 8/7

5/3 40 Turbulent Manning Hydraulically wide 2/3 3/2

3/2 36 Turbulent Chezy Hydraulically wide 1/2 2

4/3 32 Turbulent Manning Triangular 1/3 3

5/4 30 Turbulent Chezy Triangular 1/4 4

1 24 Any Inherently stable 0 ∞

The following conclusions can be drawn from Table 10-3:

1. The value of the rating exponent varies from as high as β = 3 for laminar flow in a

hydraulically wide channel, to as low as β = 1 for an inherently stable channel.

2. The value of the dimensionless relative kinematic wave celerity varies from as

high as cdrk = 2 for laminar flow in a hydraulically wide channel, to as low as cdrk =

0 for an inherently stable channel.

3. The Froude number for neutral stability varies from as low as Fns = 0.5 for laminar

flow in a hydraulically wide channel (i.e., laminar overland flow), to as high

as Fns = ∞ for an inherently stable channel under any type of friction (though

usually turbulent).

4. The turbulent Chezy hydraulically wide dimensionless relative kinematic wave

celerity iscdrk = 0.5, confirming the results shown on (the left side of) Fig. 10-5.

Thus, kinematic waves feature long wavelengths L and correspondingly "short"

dimensionless wavenumbers σ* .

5. The turbulent Manning hydraulically wide dimensionless relative kinematic wave

celerity iscdrk = 2/3, i.e., the Manning value exceeds the Chezy value by 1/6.

Note that the value of β may be less than 1 for cases other than those shown in Table

10-3; for example, when the cross section does not grow monotonically with stage, as

in circular culvert flow. Also, note that since the Froude number has an upper limit

(corresponding to a realistically achievable lower limit on the bottom friction), the

value Fns = ∞ is of limited practical value. If the maximum attainable Froude number is

conservatively assumed to be Fmax = 25 (Chow, 1959), the shape of the cross-section

could be designed accordingly, assuring stability (Ponce and Porras, 1995).

In summary, kinematic waves have the following properties:

1. Kinematic waves travel with dimensionless relative wave celerity equal to 0.5,

under Chezy friction; under Manning friction, the value is 2/3.

2. Kinematic waves do not attenuate, but they can undergo changes in shape due

to nonlinearities; in extreme cases in hydraulically wide channels, the kinematic

wave may steepen to the point where it becomes a kinematic shock.

A word of caution regarding kinematic wave modeling 1

Despite the fact that wave attenuation is disavowed by kinematic wave theory, certain numerical

solutions of Eq. 10-56 do exhibit a distinct attenuation. Cunge (1969) traced this apparent

contradiction to the fact that numerical solutions, by virtue of their discrete grid size, introduce an

error which expresses itself as numerical diffusion. As the grid is refined, the numerical effect

decreases; however, a finite numerical grid (and all numerical grids are finite) will always have a

residual error.

The dilemma is resolved by making the numerical diffusion simulate the physical diffusion, if any, of

the physical problem. This procedure is embodied in the Muskingum-Cunge method of flood routing,

described in Section 10.6.

1 See Ponce (1991) for a detailed treatment of the controversy regarding kinematic waves.

Kinematic wave rating

Kinematic waves are based on a single-valued discharge-area rating, Eq. 10-51. Thus,

a kinematic wave rating is single-valued, exhibiting a one-to-one correspondence

between (a) discharge, and (b) flow area, depth, or stage. A kinematic wave rating is

calculated by using a uniform flow formula such as Manning or Chezy, for a range of (a)

flow depths, in artificial channels, or (b) flow stages, in natural channels.

Applicability of kinematic waves

A kinematic wave is a simplified type of wave, wherein three terms in the equation of

motion (Table 10-2) have been either neglected or assumed to be too small to be of

any practical significance. Thus, the kinematic wave does not apply to the general

case. Its use is recommended for cases where the flow unsteadiness is relatively small.

In practice, a kinematic wave will apply provided the following dimensionless inequality

is satisfied (Ponce, 1989; Ponce, 2014):

tr So Vo ___________ ≥ 85

do

(10-67)

in which tr = time-of-rise of the hydrograph, So = bottom slope, Vo = average flow

velocity, and do = average flow depth.

10.4 DIFFUSION WAVES

[Muskingum Method] [Muskingum-Cunge Method] [Dynamic

Waves] [Questions] [Problems] [References] • [Top] [Governing Equations] [Linear Solution] [Kinematic Waves]

A diffusion wave is an idealization that neglects both acceleration terms in the equation

of motion (Table 10-2). By neglecting these terms, Eq. 10-14 is reduced to the following

statement:

∂h

Sf = So - _____

∂x

(10-68)

The unsteadiness of the phenomenon, however, is preserved through the time-varying

term in the continuity equation (Eq. 10-3). The combination of Eqs. 10-3 and 10-68

gives rise to the diffusion wave equation.

The kinematic wave equation was derived by using a statement of steady uniform flow

in lieu of the equation of motion (Section 10.4). In deriving the diffusion wave equation,

a statement of steadynonuniform flow (friction slope is equal to water-surface slope) is

used instead (Fig. 10-8). In this case, the discharge-area rating, using the Manning

formula in SI Units (Eq. 5-17), is:

1 dh

Q = ___ A R 2/3 (So - ____ ) 1/2

n dx

(10-69)

in which the term within parentheses (...) is the water-surface slope Sw.

Figure 10-8 Diffusion wave assumption.

The difference between kinematic and diffusion waves lies in the pressure-gradient

term (dh/dx). When this term is included in the formulation, the resulting equation is of

second order and, therefore, it is able to simulate diffusion. Lighthill and Whitham

(1955) referred to this situation as the "diffusion of kinematic waves," i.e., a type of

kinematic wave, still with no inertia in its formulation, that is nevertheless able to

diffuse.

To derive the diffusion wave equation, Eq. 10-51 is repeated here in a slightly different

form:

∂Q ∂A _____ + _____ = 0

∂x ∂t

(10-70)

Equation 10-69 is expressed in a more convenient form (Cunge, 1969):

dh

m Q 2 = So - ____

dx

(10-71)

in which m is the reciprocal of the square of the channel conveyance K (Eq. 5-34),

repeated here for convenience:

1

K = ___ A R 2/3

n

(5-34)

With dA = T dh, in which T = top width, Eq. 10-71 changes to:

1 dA _____ ______ + m Q 2 - So = 0

T dx

(10-72)

Equations 10-70 and 10-72 constitute a set of two partial differential equations

describing diffusion waves. These equations can be combined into one equation

with Q as dependent variable. However, it is first necessary to linearize the equations

around reference flow values. For simplicity, a constant top width is assumed (i.e., a

wide channel assumption).

The linearization of Eqs. 10-70 and 10-72 is accomplished by small perturbation theory

(Cunge, 1969). The variables Q, A, and m can be expressed in terms of the sum of a

reference value (with subscript o) and a small perturbation to the reference value (with

superscript ' ): Q = Qo + Q' ; A = Ao+ A' ; m = mo + m'. Substituting these into Eqs. 10-

70 and 10-72, neglecting squared perturbations and subtracting the reference flow,

leads to:

∂Q' ∂A' ____ + ____ = 0

∂x ∂t

(10-73)

and

1 ∂A' _____ ______ + Qo

2 m' + 2 mo Qo Q' = 0

T ∂x

(10-74)

Differentiating Eq. 10-73 with respect to x and Eq. 10-74 with respect to t gives:

∂2Q' ∂2A' ______ + _______ = 0

∂x 2 ∂x ∂t

(10-75)

1 ∂2A' ∂m' ∂Q' _____ _________ + Qo

2 _____ + 2 mo Qo ______ = 0

T ∂x ∂t ∂t ∂t

(10-76)

Using the chain rule and Eq. 10-73 yields:

∂m' ∂m' ∂A' ∂m' ∂Q' _____ = _______ _______ = - ______ ______

(10-77)

∂t ∂A' ∂t ∂A' ∂x

Combining Eq. 10-76 with Eq. 10-77:

1 ∂2A' ∂m' ∂Q' ∂Q' _____ ________ - Qo

2 ______ ______ + 2 mo Qo ______ = 0

T ∂x ∂t ∂A' ∂x ∂t

(10-78)

Combining Eqs. 10-75 and 10-78 and rearranging terms, yields:

∂Q' Qo ∂m' ∂Q' 1 ∂Q'2 ______ - _______ ______ ______ = _____________ _______

∂t 2 mo ∂A' ∂x 2 T mo Qo ∂x2

(10-79)

By definition: mQ 2 = Sf (Eq. 10-70). Therefore:

∂Q' ∂Q Qo _____ = _______ = - _______

∂m' ∂m 2 mo

(10-80)

and also

So

mo Qo = ______

Qo

(10-81)

Substituting Eqs. 10-80 and 10-81 into Eq. 10-79, using the chain rule, and dropping

the superscripts for simplicity, the following equation is obtained:

∂Q ∂Q ∂Q Qo ∂2Q

______ + ( ______ ) ______ = ( _________ ) _______

∂t ∂A ∂x 2 T So ∂x2

(10-82)

The left side of Eq. 10-82 is recognized as the kinematic wave equation, with ∂Q/∂A as

the kinematic wave celerity. The right side is a second-order (partial differential) term

that accounts for the physical diffusion effect. The coefficient of the second-order term

has the units of diffusivity [L2 T

-1], being referred to as the hydraulic diffusivity, or

channel diffusivity.

The hydraulic diffusivity, a characteristic of the flow and channel, is defined as follows:

Qo qo

νh = _________ = _______

2 T So 2 So

(10-83)

in which qo = Qo /T is the reference discharge per unit of channel width. From Eq. 10-

83, it is concluded that the hydraulic diffusivity is small for steep bottom slopes (e.g.,

those of mountain streams), and large for mild bottom slopes (e.g., those of large rivers

near their mouths).

Equation 10-82 describes the movement of flood waves in a better way than Eq. 10-50.

While it falls short from describing the full inertial effects, it does physically account for

wave attenuation.

Equation 10-82 is a second-order parabolic partial differential equation. It can be solved

analytically, leading to the diffusion analogy solution for flood waves (Hayami, 1951), or

numerically with the aid of a numerical scheme for parabolic equations. An alternative

approach is to match the hydraulic diffusivity (Eq. 10-83) with the numerical diffusion

coefficient of the Muskingum flood routing method (Section 10.5). This approach is the

basis of the Muskingum-Cunge method (Section 10.6).

Diffusion wave rating

Diffusion waves are not based on a single-valued discharge-area rating. Thus, a

diffusion wave rating is not single-valued, exhibiting a loop. In general, however, the

loop is relatively small and may be neglected on practical grounds. A kinematic wave

rating may be used as an approximation in diffusion wave routing.

Diffusion wave celerity

According to Eq. 10-82, the diffusion wave celerity should be the same as the kinematic

wave celerity (Ponce and Simons, 1977). However, diffusion waves attenuate;

therefore, the actual discharge-area rating is not exactly single-valued. In practice, the

difusion wave celerity equals the kinematic wave celerity only as an approximation.

Applicability of diffusion waves

A diffusion wave is a simplified type of wave, wherein two terms in the equation of

motion (Table 10-2) have been either neglected or assumed to be too small to be of

any practical significance. Thus, while the diffusion wave applies for a wider range of

cases than the kinematic wave, it is still not suited to the general case. Its use is

recommended for cases where the flow unsteadiness is small to medium size (where

the wave remains within 30% of its original strength, within one period of propagation).

A diffusion wave will apply provided the following dimensionless inequality is satisfied

(Ponce, 1989; Ponce, 2014):

g

tr So ( ____ )1/2 ≥ 15

do

(10-84)

in which tr = time-of-rise of the hydrograph, So = bottom slope, g = gravitational

acceleration, and do = average flow depth.

Diffusion waves apply to problems of flood wave propagation (see Hayami's diffusion

analogy of flood waves in the box below). While kinematic waves apply to flood waves

that do not diffuse, diffusion waves apply to flood waves that attenuate appreciably.

Where the diffusion wave fails to account for the wave propagation, only the mixed

kinematic-dynamic wave (read "dynamic wave", Table 10-2) is able to solve the

problem correctly. In practice, however, diffusion waves apply to a wide range of flood

propagation problems.

Hayami's diffusion analogy of flood waves

In 1951, Hayami published a paper entitled "On the propagation of flood waves." In it, he argued that

flood waves could be modeled with a convection-diffusion equation similar to Eq. 10-82. To put it in

Hayami's words:

In natural rivers, the form of the channels, the bed slopes, the breadth, the form of

the cross section, etc. are all very irregular and incessantly changing... Yet the flow

in rivers is steady and nearly uniform in the broad means. The disturbances on the

flow caused by these irregularities damp away within a few kilometers and have

certain limited dimensions and durations. The stochastic character of the collective

of these elementary disturbances causes a large-scale longitudinal mixing...

Introducing the effect of longitudinal diffusion caused by the mixing into the

equation of continuity and assuming the mean flow taken over a suitable range to

be steady and uniform, the differential equation of flood waves was derived. It is an

equation of diffusion containing a term of advection.

Figure 10-9 Upper Paraguay river near Porto Murtinho, Mato Grosso do Sul,

Brazil, featuring a diffusion wave. The annual rise of the river lasts about six

months. The bottom slope at this point is So = 0.00002.

Dynamic hydraulic diffusivity

The hydraulic diffusivity (Eq. 10-83) is a fundamental property of diffusion waves. It

states that the coefficient of diffusion is directly proportional to the unit-width discharge

and inversely proportional to the channel slope. This conclusion ia applicable to

diffusion waves, which are governed by the convection-diffusion equation represented

by Eq. 10-82.

Using concepts of linear theory, Dooge (1973) has developed a convection-diffusion

equation using the complete equation of motion. Dooge's approach extends the

concept of diffusion wave to the realm of dynamic waves (Table 10-2). When all terms

are included in the formulation, the hydraulic diffusivity is essentially

a dynamic hydraulic diffusivity, expressed, for hydraulically wide channels, as follows:

qo F 2

νhF = _______ ( 1 - ______ )

2 So 4

(10-85)

Ponce (1991) has expressed the dynamic hydraulic diffusivity in terms of the

Vedernikov number, as follows:

qo

νhV = _______ ( 1 - V 2 )

2 So

(10-86)

Unlike Eq. 10-85, Eq. 10-86 is not limited to hydraulically wide channels, being

applicable to channels of any cross-sectional shape.

Equation 10-86 is altogether better than Eq. 10-83. They are equivalent only if V = 0,

i.e., for very small Froude-number flows. For V = 1, using Eq. 10-86, the hydraulic

diffusivity vanishes, while this is not the case for Eq. 10-83, for which the hydraulic

diffusivity remains finite.

10.5 MUSKINGUM METHOD

[Muskingum-Cunge Method] [Dynamic Waves] [Questions] [Problems] [References] • [Top] [Governing

Equations] [Linear Solution] [Kinematic Waves] [Diffusion Waves]

The Muskingum method of flood routing was developed in connection with the design

of flood protection schemes in the Muskingum River Basin, Ohio (Fig. 10-10)

(McCarthy, 1938). It is the most widely used method of flood routing, with numerous

applications in the United States and throughout the world.

Figure 10-10 The Muskingum river near Marietta, Ohio.

The method is based on the differential equation of storage (Fig. 10-11):

dS

I - O = _____

dt

(10-87)

in which I = inflow, O = outflow, and S = storage.

Figure 10-11 Definition sketch for inflow, outflow, and storage in a reservoir.

In an ideal channel, storage is a function of inflow and outflow. This is in constrast with

an ideal reservoir, in which storage is solely a function of outflow. In the Muskingum

method, storage is a linear function of inflow and outflow:

S = K [ X I + ( 1 - X ) O ]

(10-88)

in which S = storage volume; I = inflow; O = outflow; K = a time constant or storage

coefficient; andX = a dimensionless weighting factor. With inflow and outflow in cubic

meters per second, and K in hours, storage volume is in (cubic meters per second)-

hour. Alternatively, K could be expressed in seconds, in which case storage volume is

in cubic meters.

To derive the Muskingum routing equation, Eq. 10-87 is discretized on the x-t plane

(Fig. 10-12), to yield:

I1 + I2 O1 + O2 S2 - S1 __________ - ___________ = ___________

2 2 Δt

(10-89)

Figure 10-12 Discretization on the x-t plane.

Equation 10-88 is expressed at time levels 1 and 2:

S1 = K [ X I1 + ( 1 - X ) O1 ] (10-90)

S2 = K [ X I2 + ( 1 - X ) O2 ] (10-91)

Substituting Eqs. 10-90 and 10-91 into Eq. 10-89 and solving for O2 yields:

O2 = C0 I2 + C1 I1 + C2 I1

(10-92)

in which C0, C1 and C2 are routing coefficients defined in terms of Δt, K, and X as

follows:

( Δt / K ) - 2X

C0 = _______________________

2(1 - X) + ( Δt / K )

(10-93a)

( Δt / K ) + 2X

C1 = _______________________

2(1 - X) + ( Δt / K )

(10-93b)

2(1 - X) - ( Δt / K )

C2 = _______________________

2(1 - X) + ( Δt / K )

(10-93c)

Since C0 + C1 + C2 = 1, the routing coefficients may be interpreted as weighting

coefficients.

Given an inflow hydrograph, an initial flow condition, a chosen time interval Δt, and

routing parameters X and K, the routing coefficients can be calculated with Eq. 10-93,

and the outflow hydrograph with Eq. 10-92. The routing parameters K and K are related

to flow and channel characteristics, K being interpreted as the travel time of the flood

wave from upstream end to downstream end of the channel reach.

Therefore, K accounts for the translation portion of the routing (Fig. 10-12).

The parameter X accounts for the storage portion of the routing. For a given flood

event, there is a value of X for which the storage in the calculated outflow hydrograph

matches that of the measured outflow hydrograph. The effect of storage is to reduce

the peak flow and spread the hydrograph in time (Fig. 10-13). Therefore, it is often used

interchangeably with the terms diffusion and peak attenuation.

Figure 10-13 Translation and storage processes in stream channel routing.

The routing parameter K is a function of channel reach length and flood wave speed;

conversely, the parameter X is a function of the flow and channel characteristics that

cause runoff diffusion. In the Muskingum method, X is interpreted as a weighting factor

and restricted in the range 0 ≤ X ≤ 0.5. Values of X greater than 0.5 produce

hydrograph amplification (i.e., negative diffusion), which does not correspond with

reality (under the Froude numbers applicable to flood flows). With K = Δt and X = 0.5,

flow conditions are such that the outflow hydrograph retains the same shape as the

inflow hydrograph, but it is translated downstream a time equal to K. For X = 0,

Muskingum routing reduces to linear reservoir routing.

In the Muskingum method, the parameters K and X are determined by calibration using

streamflow records. Simultaneous inflow-outflow discharge measurements for a given

channel reach are coupled with a trial-and-error procedure, leading to the determination

of K and X (see Example 10-1). The procedure is time-consuming and lacks predictive

capability. Values of K and X determined in this way are valid only for the given reach

and flood event used in the calibration. Extrapolation to other reaches or to other flood

events (of different magnitude) within the same reach is usually unwarranted.

When sufficient data are available, a calibration can be performed for several flood

events, each of different magnitude, to cover a wide range of flood levels. In this way,

the variation of K and X as a function of flood level can be ascertained. In practice, K is

more sensitive to flood level than X. A sketch of the variation of K with stage and

discharge is shown in Fig. 10-14.

Figure 10-14 Sketch of travel time as a function of discharge and stage.

Example 10-1.

An inflow hydrograph to a channel reach is shown in Col. 2 of Table 10-4. Assume baseflow is 352 m3/s. Using the Muskingum

method, route this hydrograph through a channel reach with K = 2 d and X = 0.1 to calculate an outflow hydrograph.

First, it is necessary to select a time interval Δt. In this case, it is convenient to choose Δt = 1 d. As with reservoir routing, the ratio

of time-to-peak to time interval (tp /Δt ) should be greater than or equal to 5. In addition, the chosen time interval should be such

that the routing coefficients remain positive. With Δt = 1 d,K = 2 d, and X = 0.1, the routing coefficients (Eq. 10-93) are: C0 =

0.1304; C1 = 0.3044; and C2 = 0.5652. It is verified that C0 + C1 + C2 = 1. The routing calculations are shown in Table 10-4.

Column 1 shows the time in days.

Column 2 shows the inflow hydrograph ordinates in cubic meters per second.

Columns 3-5 show the partial flows.

Following Eq. 10-91, Cols. 3-5 are summed to obtain Col. 6, the outflow hydrograph ordinates in cubic meters per

second.

To explain the procedure briefly, the outflow at the start (day 0) is assumed to be equal to the inflow at the start: 352 m3/s. The

inflow at day 1 multiplied by C0 is entered in Col. 3, day 1: 76.6 m3/s. The inflow at day 0 multiplied by C1 is entered in Col. 4,

day 1: 107.1 m3/s. The outflow at day 0 multiplied by C2 is entered in Col. 5, day 1: 199 m

3/s. Columns 3-5 of day 1 are summed

to obtain Col. 6 of day 1: 76.6 + 107.1 + 199.0 = 382.7 m3/s. The calculations proceed in a recursive manner until all outflows in

Col. 6 have been obtained. Inflow and outflow hydrographs are plotted in Fig. 10-15. The outflow peak is 6352.6 m3/s, which

shows that the inflow peak, 6951 m3/s, has attenuated to about 91 % of its initial value. The peak outflow occurs at day 9, 2 d after

the peak inflow, which occurs at day 7. The time elapsed between the occurrence of peak inflow and peak outflow is generally

equal to K, the travel time.

Table 10-4 Channel Routing by the Muskingum Method: Example 10-1.

(1) (2) (3) (4) (5) (6)

Time

(h) Inflow

(m3/s)

Partial Flows (m3/s)

Outflow

(m3/s)

C0I2 C1I1 C2I1

0 352.0 ___ ___ ___ 352.0

1 587.0 76.6 107.1 199.0 382.7

2 1353.0 176.5 178.6 216.3 571.4

3 2725.0 355.4 411.8 323.0 1090.2

4 4408.5 575.0 829.4 616.2 2020.6

5 5987.0 780.9 1341.7 1142.1 3264.7

6 6704.0 874.4 1822.1 1845.3 4541.8

7 6951.0 906.7 2040.3 2567.1 5514.1

8 6839.0 892.0 2115.5 3116.7 6124.2

9 6207.0 809.6 2081.5 3461.5 6352.6

10 5346.0 697.3 1889.1 3590.6 6177.0

11 4560.0 594.8 1627.0 3491.4 5713.2

12 3861.5 503.7 1387.8 3229.2 5120.7

13 3007.0 392.2 1175.2 2894.3 4461.7

14 2357.5 307.5 915.2 2521.8 3744.5

15 1779.0 232.0 717.5 2116.5 3066.0

16 1405.0 183.3 541.4 1733.0 2457.7

17 1123.0 146.5 427.6 1389.1 1963.2

18 952.5 124.2 341.8 1109.6 1575.6

19 730.0 95.2 289.9 890.6 1275.7

20 605.0 78.9 222.2 721.0 1022.1

21 514.0 67.1 184.1 577.7 828.9

22 422.0 55.1 156.4 468.5 680.0

23 352.0 45.9 128.4 384.4 558.7

24 352.0 45.9 107.1 315.8 468.8

25 352.0 45.9 107.1 265.0 418.0

Figure 10-15 Stream channel routing by Muskingum method:

Example 10-1.

ONLINE CALCULATION. Using ONLINE ROUTING04, the answer is essentially the same as that of Col. 6, Table

10-4.

Example 10-1 has illustrated the predictive stage of the Muskingum method, in which

the routing parameters are known in advance of the routing. If the parameters are not

known, it is first necessary to perform a calibration. The trial-and-error procedure to

calibrate the routing parameters is illustrated by Example 10-2.

Example 10-2.

Use the outflow hydrograph calculated in the previous example together with the given inflow hydrograph to calibrate

the Muskingum method, that is, to find the routing parameters K and X.

The procedure is summarized in Table 10-5.

Column 1 shows the time in days.

Column 2 shows the inflow hydrograph in cubic meters per second.

Column 3 shows the outflow hydrograph in cubic meters per second.

Column 4 shows the channel storage in (cubic meters per second)-days.

Channel storage at the start is assumed to be 0, and this value is entered in Col. 4, day 0.

Channel storage is calculated by solving Eq. 10-89 for S2:

S2 = S1 + ( Δt / 2 ) ( I1 + I2 - O1 - O2 )

Several values of X are tried, within the range 0.0 to 0.5, for example, 0.1, 0.2 and 0.3.

For each trial value of X, the weighted flows [ XI + ( 1 - X ) O ] are calculated, as shown in Cols. 5-7.

Each of the weighted flows is plotted against channel storage (Col. 4), as shown in Fig. 10-16.

The value of X for which the storage versus weighted flow data plots closest to a line is taken as the correct

value of X. In this case, Fig. 10-16 (a): X = 0.1 is chosen.

Following Eq. 10-88, the value of K is obtained from Fig. 10-16 (a) by calculating the slope of the storage vs

weighted outflow curve.

In this case, the value of K = [2000 (m3/s)-d]/(1000 m

3/s) = 2 d.

Thus, it is shown that K = 2 days and X = 0.1 are the Muskingum routing parameters for the given inflow and

outflow hydrographs.

Table 10-5 Calibration of Muskingum Routing Parameters: Example 10-2.

(1) (2) (3) (4) (5) (6) (7)

Time

(h) Inflow

(m3/s)

Outflow

(m3/s)

Storage

(m3/s)-d

Weighted Flow (m3/s)

X = 0.1 X = 0.2 X = 0.3

0 352.0 352.0 0 ___ ___ ___

1 587.0 382.7 102.2 403.0 423.5 443.9

2 1353.0 571.4 595.2 649.6 727.7 805.9

3 2725.0 1090.2 1803.4 1253.7 1417.2 1580.6

4 4408.5 2020.6 3814.7 2259.4 2498.2 2737.0

5 5987.0 3264.7 6369.8 3536.9 3809.2 4081.4

6 6704.0 4541.8 8812.1 4758.0 4974.2 5190.5

7 6951.0 5514.1 10611.6 5657.8 5801.5 5945.2

8 6839.0 6124.2 11687.5 6195.7 6267.2 6338.6

9 6207.0 6352.6 11972.1 6338.0 6323.5 6308.9

10 5346.0 6177.0 11483.8 6093.9 6010.8 5927.7

11 4560.0 5713.2 10491.7 5597.9 5482.6 5367.2

12 3861.5 5120.7 9285.5 4994.8 4868.9 4742.9

13 3007.0 4461.7 7928.5 4316.2 4170.8 4025.3

14 2357.5 3744.5 6507.7 3605.8 3467.1 3328.4

15 1779.0 3066.0 5170.7 2937.3 2808.6 2679.9

16 1405.0 2457.7 4000.8 2352.4 2247.2 2141.9

17 1123.0 1963.2 3054.4 1879.2 1795.2 1711.1

18 952.5 1575.6 2322.7 1513.4 1451.1 1388.7

19 730.0 1275.7 1738.2 1221.1 1166.6 1112.0

20 605.0 1022.1 1256.8 980.4 938.7 897.0

21 514.0 828.9 890.8 797.4 765.9 734.4

22 422.0 680.0 604.4 654.2 628.4 602.6

23 352.0 558.7 372.0 537.9 517.3 496.6

24 352.0 468.8 210.3 457.1 445.4 433.8

25 352.0 418.0 118.9 411.4 404.8 398.2

Figure 10-16 Calibration of Muskingum routing parameters: Example 10-2.

The estimation of routing parameters is crucial to the application of the Muskingum

method. The parameters are not constant, tending to vary with flow rate. If the routing

parameters can be related to flow and channel characteristics, the need for trial-and-

error calibration would be eliminated. Parameter K could be related to reach length and

flood wave velocity, whereas X could be related to the diffusivity characteristics of flow

and channel. These propositions are the basis of the Muskingum-Cunge method.

10.6 MUSKINGUM-CUNGE METHOD

[Dynamic Waves] [Questions] [Problems] [References] • [Top] [Governing Equations] [Linear Solution] [Kinematic

Waves] [Diffusion Waves] [Muskingum Method]

The Muskingum method requires a calibration in order to identify the routing

parameters (Example 10-2). The procedure is based on measured input and output

hydrographs, as shown in Cols. 2 and 3 of Table 10-5. Therefore, a gaging station is an

absolute necessity for the Muskingum method to be properly used. This limits the

applicability of the method to reaches which have a streamgaging station.

Unlike the Muskingum method, the Muskingum-Cunge method requires hydraulic rather

than hydrologic data in order to calculate the routing parameters. The hydraulic data

consists of geomorphic data such as channel slope and cross-sectional characteristics.

Thus, the Muskingum-Cunge method does not explicitly require a streamgaging station,

being applicable to any channel reach as long as the geomorphic data is available.

The Muskingum method is derived by combining the differential equation of storage,

i.e., the continuity equation expressed in total differential form (Eq. 10-87) with a linear

inflow-outflow-storage relation (Eq. 10-88). This leads to the Muskingum routing

equation (Eq. 10-92) with appropriate routing coefficients (Eq. 10-93). The Muskingum-

Cunge method is derived by discretizing the kinematic wave equation (Eq. 10-55) in a

linear mode, in a manner similar to that of the Muskingum method, which leads to the

same routing equation. The coefficients, however, are defined based on measurable

channel characteristics.

The similarities appear to end there. The Muskingum method is lumped across the

channel reach, based on storage, and able to describe flood wave diffusion (Example

10-1). The Muskingum-Cunge method is distributed, with data specified at cross

sections, and based on a discretization of the kinematic wave equation, which

ostensibly does not diffuse. Yet actual calculations using the Muskingum-Cunge

method shows that it is able to describe wave diffusion, in a manner similar to the

Muskingum method.

Cunge (1969) traced the diffusion of the discretized analog of the kinematic wave

equation to the numerical diffusion of the scheme itself (see box). Thus, he was able to

explain the paradox. The Muskingum and Muskingum-Cunge methods have the same

theoretical basis. The routing parameters of the Muskingum method are hydrologic,

based on storage, and determined by calibration using streamflow data. The routing

parameters of the Muskingum-Cunge method are hydraulic, distributed (at a cross

section), and based exclusively on geomorphic data.

A word of explanation on the concept of numerical diffusion 1

The existence of diffusion in a numerical scheme needs further elaboration. As Hayami (1951) pointed

out, in nature irregularities produce physical diffusion. The same process extends to the computational

world: Irregularities produce numerical diffusion. The irregularities are interpreted as the errors

inherent to the numerical computation, among which the truncation errors stand out primordially. The

replacement of a partial differential equation, based on ∂, with a finite difference equation, based on Δ,

results in errors, which, when taken in the aggregate, manifest themselves as diffusion. Thus,

numerical diffusion is in the computational world what physical diffusion is in the real world.

1 See Cunge (1969) for a detailed treatment of numerical diffusion in connection with the Muskingum method.

Muskingum-Cunge routing equation

To derive the Muskingum-Cunge routing equation, the kinematic wave equation (Eq.

10-55) is discretized on the x-t plane (Fig. 10-17) in a way that parallels the Muskingum

method, centering the spatial derivative and off-centering the temporal derivative by

means of a weighting factor X:

X (Q j n+1 - Q j

n ) + (1 - X) (Q j+1n+1 - Q j+1

n ) ________________________________________________ +

Δt

(Q j+1 n - Q j

n ) + (Q j+1n+1 - Q j

n+1 )

c _______________________________________ = 0

2 Δx

(10-94)

in which c = βV is the kinematic wave celerity.

Figure 10-17 Space-time discretization of the kinematic wave equation

paralleling the Muskingum method.

Solving Eq. 10-94 for the unknown discharge leads to the following routing equation:

Q j+1 n+1 = C0 Q j

n+1 + C1 Q j n + C2 Q j+1

n

(10-95)

The routing coefficients are:

c ( Δt / Δx ) - 2X

C0 = ________________________

2(1 - X) + c ( Δt / Δx )

(10-96a)

c ( Δt / Δx ) + 2X

C1 = ________________________

2(1 - X) + c ( Δt / Δx )

(10-96b)

2(1 - X) - c ( Δt / Δx )

C2 = ________________________

2(1 - X) + c ( Δt / Δx )

(10-96c)

By defining the travel time as

Δx

K = ______

c

(10-97)

it is seen that the sets of Eq. 10-93 and Eq. 10-96 are the same.

Equation 10-97 confirms that K is in fact the flood wave travel time, i.e., the time it takes

a given discharge to travel the reach length Δx with the kinematic wave celerity c.

Numerical properties

The Courant number is defined as the ratio of physical celerity (c) to the numerical, or

grid, celerityΔx /Δt. Therefore:

Δt

C = c ______

Δx

(10-98)

Equation 10-95 is a numerical analog of Eq. 10-55 and, therefore, subject to numerical

diffusion and dispersion. Numerical diffusion is the second-order error; numerical

dispersion is the third-order error. The following conditions hold:

For X = 0.5 and C = 1, the routing equation is third-order accurate, i.e., the

numerical solution is equal to the analytical solution (of the kinematic wave

equation).

For X = 0.5 and C ≠ 1, the routing equation is second-order accurate, exhibiting

only numerical dispersion.

For X < 0.5 and C ≠ 1, the routing equation is first-order accurate,

exhibiting both numerical diffusion and dispersion.

For X < 0.5 and C = 1, the routing equation is first-order accurate, exhibiting only

numerical diffusion.

These relations are summarized in Table 10-6.

Table 10-6 Numerical properties of the Muskingum-Cunge method.

Parameter

X Parameter

C Order of

Accuracy Numerical

Diffusion Numerical

Dispersion

0.5 1 Third No No

0.5 ≠ 1 Second No Yes

< 0.5 ≠ 1 First Yes Yes

< 0.5 1 First Yes No

In practice, the numerical diffusion can be used to simulate the physical diffusion of the

actual flood wave. By expanding the discrete function Q (jΔx, n Δt) in Taylor series

about grid point (jΔx, n Δt), the numerical diffusion coefficient of the Muskingum

scheme is derived (Cunge, 1969) (Appendix B):

1

νn = c Δx ( ____ - X )

2

(10-99)

in which νn is the numerical diffusion coefficient of the Muskingum scheme. This

equation reveals the following:

For X = 0.5 there is no numerical diffusion, although there is some numerical

dispersion forC ≠ 1;

For X > 0.5, the numerical diffusion coefficient is negative, i.e., numerical

amplification, which explains the behavior of the Muskingum method for this

range of X values;

For Δx = 0, the numerical diffusion coefficient is zero, clearly the trivial case.

A predictive equation for X can be obtained by matching the hydraulic diffusivity νh (Eq.

10-83) with the numerical diffusion coefficient of the Muskingum scheme νn (Eq. 10-99).

This leads to the following expression for X:

1 qo

X = ___ ( 1 - __________ )

2 So c Δx

(10-100)

With X calculated by Eq. 10-100, the Muskingum method is referred to as Muskingum-

Cunge method. Thus, the routing parameter X can be calculated as a function of the

following numerical and physical properties:

1. Reach length Δx,

2. Reference discharge per unit width qo,

3. Kinematic wave celerity c, and

4. Bottom slope So.

It should be noted that Eq. 10-100 was derived by matching physical and numerical

diffusion (a second-order processes), and does not account for dispersion (a third-order

process). Therefore, in order to simulate wave diffusion properly with the Muskingum-

Cunge method, it is necessary to optimize numerical diffusion (with Eq. 10-100) while at

the same time minimizing numerical dispersion by keeping the value of C ≅ 1.

Advantage of the Muskingum-Cunge method

A unique feature of the Muskingum-Cunge method, which sets it apart from other numerical kinematic

wave models, is the grid independence of the calculated outflow hydrograph. If numerical dispersion is

minimized, the calculated outflow at the downstream end of a channel reach will be essentially the

same, regardless of how many subreaches are used in the computation. This is because X is a function

of Δx, and the routing coefficients C0, C1, and C2 vary accordingly.

An improved version of the Muskingum-Cunge method is due to Ponce and Yevjevich

(1978). The grid diffusivity is defined as the numerical diffusivity for the case of X = 0.

From Eq. 10-99, the grid diffusivity is:

Δx

νg = c _____

2

(10-101)

The cell Reynolds number D is defined as the ratio of hydraulic diffusivity (Eq. 10-83) to

grid diffusivity (Eq. 10-101). This leads to:

qo

D = __________

So c Δx

(10-102)

Therefore:

1

X = ___ ( 1 - D )

2

(10-103)

Equations 10-101 and 10-102 imply that for very small values of Δx, D may be greater

than 1, leading to negative values of X. In fact, for the characteristic reach length

qo

Δxc = ________

So c

(10-104)

the cell Reynolds number is D = 1, and X = 0. Therefore, in the Muskingum-Cunge

method, reach lengths shorter than the characteristic reach length result in negative

values of X. This should be contrasted with the classical Muskingum method (Section

10.4), in which X is restricted in the range 0.0 ≤ X ≤ 0.5. In the classical

Muskingum, X is interpreted as a weighting factor. As shown by Eqs. 10-101 and 10-

102, nonnegative values of X are associated with long reaches, typical of the manual

computation used in the development and early application of the Muskingum method.

In the Muskingum-Cunge method, however, X is interpreted in a moment-matching

sense or diffusion-matching factor. Therefore, negative values of X are entirely

possible. This feature allows the use of shorter reaches than would otherwise be

possible if X were restricted to nonnegative values.

Routing coefficients

The substitution of Eqs. 10-98 and 10-100 into Eq. 10-96 leads to routing coefficients

expressed in terms of Courant and cell Reynolds numbers:

-1 + C + D

C0 = ______________

1 + C + D

(10-105a)

1 + C - D

C1 = ______________

1 + C + D

(10-105b)

1 - C + D

C2 = ______________

1 + C + D

(10-105c)

The calculation of routing parameters C and D can be performed in several ways. The

wave celerity can be calculated with either Eq. 10-57 or Eq. 10-59. With Eq. 10-57, c =

βV; with Eq. 10-59, c = (1/T)dQ/dy. Theoretically, these two equations are the same.

For practical applications, if a stage-discharge rating and cross-sectional geometry are

available (i.e., stage-discharge-top width tables), Eq. 10-59 is preferred over Eq. 10-57

because it accounts directly for cross-sectional shape. In the absence of a stage-

discharge rating and cross-sectional data, Eq. 10-57 can be used to estimate flood

wave celerity.

With the aid of Eqs. 10-98 and 10-102, the routing parameters may be based on flow

characteristics. The calculations can proceed in a linear or nonlinear mode. In the linear

mode, the routing parameters are based on reference flow values and kept constant

throughout the computation in time. The choice of reference flow has a bearing on the

calculated results, although the overall effect is likely to be small (Ponce and Yevjevich,

1978). For practical applications, either an average or peak flow value can be used as

reference flow. The peak flow value has the advantage that it can be readily

ascertained, although a better approximation may be obtained by using an average

value. The linear mode of computation is referred to as the constant-parameter

Muskingum-Cunge method to distinguish it from the variable-parameter Muskingum-

Cunge method, in which the routing parameters are allowed to vary with the flow. The

constant parameter method resembles the Muskingum method, with the difference that

the routing parameters are based on measurable flow and channel characteristics

instead of on historical streamflow data.

Example 10-3.

Use the constant-parameter Muskingum-Cunge method to route a flood wave with the following flood and channel characteristics:

peak flow Qp = 1000 m3/s; baseflow Qb, = 0 m

3/s; channel bottom slope So = 0.000868; flow area at peak discharge Ap = 400 m

2;

top width at peak discharge Tp = 100 m; rating exponent β = 1.6; reach length Δx = 14.4 km; time interval Δt = 1 h.

Time (h) 0 1 2 3 4 5 6 7 8 9 10

Flow (m3/s 0 200 400 600 800 1000 800 600 400 200 0

The mean velocity (based on the peak discharge) is V = Qp/Ap = 2.5 m/s. The wave celerity is c = βV =4 m/s. The flow per unit

width (based on the peak discharge) is qo = Qp/Tp = 10 m2/s. The Courant number (Eq. 10-98) is C = 1. The cell Reynolds number

(Eq. 10-102) is D = 0.2. The routing coefficients (Eq. 10-105) are C0 = 0.091; C1 = 0.818; and C2 = 0.091. It is confirmed that the

sum of routing coefficients is equal to 1. The routing calculations are shown in Table 10-7.

Table 10-7 Channel Routing by Muskingum-Cunge Method, Example 10-2.

(1) (2) (3) (4) (5) (6)

Time Inflow Partial Flows (m

3/s)

Outflow

(h) (m3/s) C0 I2 C1 I1 C2 I1 (m

3/s)

0 0 ___ ___ ___ 0.0

1 200 18.2 0.0 0.0 18.20

2 400 36.4 163.6 1.66 201.66

3 600 54.6 327.2 18.35 400.15

4 800 72.8 490.8 36.41 600.01

5 1000 91.0 654.4 54.60 800.00

6 800 72.8 818.0 72.80 963.60

7 600 54.6 654.4 87.69 796.69

8 400 36.4 490.8 72.50 599.70

9 200 18.2 327.2 54.57 399.97

10 0 0.0 163.6 36.40 200.00

11 0 0.0 0.0 18.20 18.20

12 0 0.0 0.0 1.66 1.66

13 0 0.0 00.0 0.16 0.16

ONLINE CALCULATION. Using ONLINE ROUTING05, the answer is essentially the same as that of Col. 6, Table

10-7.

Resolution requirements

When using the Muskingum-Cunge method, care should be taken to ensure that the

values of Δx and Δt are sufficiently small to approximate closely the actual shape of the

hydrograph. For smoothly rising hydrographs, a minimum value of tp /Δt = 5 is

recommended. This requirement usually results in the hydrograph time base being

resolved into at least 15 to 25 discrete points, considered adequate for Muskingum

routing.

Unlike temporal resolution, there is no definite criteria for spatial resolution. A criterion

borne out by experience is based on the fact that Courant and cell Reynolds numbers

are inversely related to reach length Δx. Therefore, to keep Δx sufficiently small,

Courant and cell Reynolds numbers should be kept sufficiently large. This leads to the

practical criterion (Ponce and Theurer, 1982):

C + D ≥ 1 (10-106)

which can be written as follows: -1 + C + D ≥ 0. This confirms the necessity of avoiding

negative values of C0 in Muskingum-Cunge routing (see Eq. 10-105a). Experience has

shown that negative values of either C1 or C2 do not adversely affect the method's

overall accuracy.

Notwithstanding Eq. 10-106, the Muskingum-Cunge method works best when the

numerical dispersion is minimized, that is, when C ≅ 1. Values of C substantially less

than 1 are likely to cause the notorious dips, or negative outflows, in portions of the

calculated hydrograph. This computational anomaly is attributed to excessive numerical

dispersion and should be avoided.

Nonlinear Muskingum-Cunge method

The kinematic wave equation, Eq. 10-55, is nonlinear because the kinematic wave

celerity varies with discharge. The nonlinearity is mild, among other things because the

wave celerity variation is usually restricted within a narrow range. However, in certain

cases it may be necessary to account for this nonlinearity. This can be done in two

ways: (1) during the discretization, by allowing the wave celerity to vary, resulting in a

nonlinear numerical scheme to be solved by iterative means; and (2) after the

discretization, by varying the routing parameters, as in the variable-parameter

Muskingum-Cunge method (Ponce and Yevjevich, 1978). The latter approach is

particularly useful if the overall nonlinear effect is small, which is often the case.

The variable parameter Muskingum-Cunge method represents a small yet sometimes

perceptible improvement over the constant parameter method. The differences are

likely to be more marked for very long reaches and/or wide variations in flow levels.

Flood hydrographs calculated with variable parameters show a certain amount of

distortion, either wave steepening in the case of flows contained inbank or wave

attenuation (flattening) in the case of typical overbank flows. This is a physical

manifestation of the nonlinear effect, i.e., different flow levels traveling with different

celerities. On the other hand, flood hydrographs calculated using constant parameters

do not show wave distortion.

Assessment of Muskingum-Cunge method

The Muskingum-Cunge method is a physically based alternative to the Muskingum

method. Unlike the Muskingum method where the parameters are calibrated using

streamflow data, in the Muskingum-Cunge method the parameters are calculated

based on flow and channel characteristics. This makes possible channel routing without

the need for time-consuming and cumbersome parameter calibration. More importantly,

it makes possible extensive channel routing in ungaged streams with a reasonable

expectation of accuracy. With the variable-parameter feature, nonlinear properties of

flood waves (which could otherwise only be obtained by more elaborate numerical

procedures) can be described within the context of the Muskingum formulation.

Like the Muskingum method, the Muskingum-Cunge method is limited to diffusion

waves. Furthermore, the Muskingum-Cunge method is based on a single-valued rating

and does not take into account strong flow non-uniformity or unsteady flows exhibiting

substantial loops in discharge-stage rating (i.e., dynamic waves). Thus, the

Muskingum-Cunge method is suited for channel routing in natural streams without

significant backwater effects and for unsteady flows that classify under the diffusion

wave criterion (Eq. 10-66).

An important difference between the Muskingum and Muskingum-Cunge methods

should be noted. The Muskingum method is based on the storage concept (Eq. 10-87)

and, therefore, it is lumped, with the parameters K and X being reach averages. The

Muskingum-Cunge method, however, is distributed in nature, with the

parameters C and D being based on values evaluated at channel cross sections.

Therefore, for the Muskingum-Cunge method to improve on the Muskingum method, it

is necessary that the routing parameters evaluated at channel cross sections be

representative of the channel reach under consideration (Fig. 10-18).

Historically, the Muskingum method has been calibrated using streamflow data. On the

contrary, the Muskingum-Cunge method relies on physical characteristics such as

rating curves, cross-sectional data and channel slope. The different data requirements

reflect the different theoretical bases of the methods, i.e., lumped storage concept in

the Muskingum method, and distributed kinematic/diffusion wave theory in the

Muskingum-Cunge method.

Figure 10-18 Moyan river, Lambayeque, Peru.

10.7 DYNAMIC WAVES

[Questions] [Problems] [References] • [Top] [Governing Equations] [Linear Solution] [Kinematic Waves] [Diffusion

Waves] [Muskingum Method] [Muskingum-Cunge Method]

In unsteady open-channel flow, the term dynamic wave is used to refer to two different

types of waves:

1. A wave that neglects the friction and bottom slope, that is, wave [3] in Table 10-2,

and

2. A wave that includes all terms in the equation of motion, that is, wave [5] in Table

10-2.

To avoid confusion, the first type of wave [3] is referred here as true dynamic wave.

The second type [5] is referred to as mixed kinematic-dynamic wave, for

short, mixed dynamic wave.

True dynamic waves

Conceptually, true dynamic waves are the exact opposite of kinematic waves. While

kinematic waves lie to the left side of the wavenumber spectrum, true dynamic waves

lie to the right (Fig. 10-3). Thus, their dimensionless wavenumber is long, that is, the

wavelength L is short relative to the reference channel length Lo (Eq. 10-30).

While the dimensionless relative celerity of a kinematic wave is constant and equal to

0.5, that of the true dynamic wave is equal to the reciprocal of the Froude number (Fig.

10-3):

1 (gho)1/2

cdrd = ____ = _________

F uo

(10-107)

The relative celerity of a true dynamic wave is:

crd = (gho)1/2

(10-108)

The celerity of a true dynamic wave is:

crd = uo ± (gho)1/2

(10-109)

Therefore, a true dynamic wave has two components, with celerities:

crd1 = uo + (gho)1/2

(10-110)

crd2 = uo - (gho)1/2

(10-111)

Kinematic and true dynamic waves share a distinct property: They do not attenuate.

This is due to the constancy of the dimensionless relative wave celerity within the

applicable range of dimensionless wavenumbers (Fig. 10-3).

In practice, true dynamic waves apply to the "short" waves that may be present in

laboratory flumes and small canals. They do not apply for flood waves, which lie on the

left side of the dimensionless wavenumber spectrum.

Mixed dynamic waves

Mixed kinematic-dynamic waves lie toward the middle of the wavenumber spectrum

(Fig. 10-3). Conceptually, they are the most complete type of wave, because they

consider all the terms in the equation of motion (Table 10-2). However, for Vedernikov

numbers V < 1, (corresponding to Froude numbers F < 2 under Chezy friction in

hydraulically wide channels), the mixed waves are subject to strong attenuation. The

attenuation is strongest at the point of inflexion of the dimensionless relative celerity

versus dimensionless wavenumber function (Fig. 10-3). Figure 10-4 shows the

attenuation rates as described by the logarithmic decrement δ.

Lighthill and Whitham (1955) described the impermanence of dynamic waves in the

following terms:

"In the case of flood waves, kinematic and dynamic waves are both possible

together. However, the dynamic waves have both a much higher wave velocity

and also a rapid attenuation. Hence, although any disturbance sends some

signal downstream at the ordinary wave velocity for long gravity waves (sic),

this signal is too weak to be noticed at any considerable distance downstream,

and the main signal arrives in the form of a kinematic wave at a much slower

velocity."

They followed up with this statement (op. cit., page 291):

"We have thought it desirable to give a mathematical treatment of the

'competition' between kinematic and dynamic waves in river flow, in order to

show how completely the dynamic waves are subordinated in the case of

greatest interest, that is, when the speed of the river is well subcritical. This

demonstrates the unsuitability of the characteristics of the dynamic wave as

the basis for computation."

Thus, in general, mixed dynamic waves do not apply to flood flows in natural streams.

Once generated, dynamic waves tend to dissipate very quickly, with their mass going to

join the predominant underlying kinematic or kinematic-with-diffusion (diffusion) wave.

The exception may be a flood wave generated by a dam breach, which is typically so

sudden that it may actually be a mixed dynamic wave. These waves attenuate very

quickly, confirming the correctness of the theory. For example, consider the failure of

Teton Dam, in Idaho, on June 5, 1976 (Fig. 10-19). The flood wave released at the

damsite attenuated to a small fraction of its initial strength within a relatively short

distance downstream. Many other examples of actual dam breaches have confirmed

that dam-breach flood waves tend to dissipate rather quickly.

USBR

Figure 10-19 Failure of Teton Dam, Idaho, on June 5, 1976.

Modeling of mixed dynamic waves

In a dynamic wave solution, the equations of continuity and motion are solved by a

numerical procedure, either (a) the method of finite differences, (b) the method of

characteristics, or (c) the finite element method. In the method of finite differences, the

partial differential equations are discretized following a chosen numerical scheme. The

method of characteristics is based on the conversion of the set of partial differential

equations into a related set of ordinary differential equations, and the solution along a

characteristic grid, i.e. a grid that follows characteristic directions. The method of finite

elements solves a set of integral equations over a chosen grid of finite elements.

In the past four decades, the method of finite differences has come to be regarded as

the most expedient way of obtaining a (mixed) dynamic wave solution for practical

applications. Among several numerical schemes that have been used in connection

with the dynamic wave, the Preissmann scheme is perhaps the most popular (Liggett

and Cunge, 1975). This is a four-point scheme, centered in the temporal derivatives

and slightly off-centered in the spatial derivatives, by use of a weighting factor θ. The

off-centering in the spatial derivatives introduces a small amount of numerical diffusion

necessary to control the numerical stability of the nonlinear scheme. This produces a

workable yet sufficiently accurate scheme.

The independent variables used in (mixed) dynamic wave routing are usually

discharge Q andstage y. The stream channel is divided into several subreaches for

computational purposes(Fig. 10-19). The application of the Preissmann scheme to the

governing equations for the various subreaches results in a matrix solution requiring a

double sweep algorithm, i.e., one that accounts only for the nonzero entries of the

coefficient matrix, which are located within a narrow band surrounding the main

diagonal (Liggett and Cunge, 1975). This technique leads to a considerable savings in

storage and execution time. With the appropriate upstream and downstream boundary

conditions (Fig. 10-20), the solution of the set of hyperbolic equations marches in time

until a specified number of time intervals is completed.

Figure 10-20 Reach subdivision for dynamic wave routing.

In practice, a (mixed) dynamic wave solution represents an order-of-magnitude

increase in complexity and associated data requirements when compared to either

kinematic or diffusion wave solutions. Its use is recommended in situations where

neither kinematic nor diffusion wave solutions are likely to adequately represent the

physical phenomena. In particular, (mixed) dynamic wave solutions are applicable to

dam-breach flood waves, flow over very flat slopes, flow into large reservoirs, strong

backwater conditions and flow reversals. In general, the (mixed) dynamic wave is

recommended for cases warranting a precise determination of the unsteady variation of

river stages.

The current version (Version 4.1) of the U.S. Army Corps of Engineers' HEC-RAS

model (U.S. Army Corps of Engineers, 2010) contains a dynamic wave module suited

for practical applications.

Mixed dynamic rating curve

Mixed dynamic wave solutions are often referred to as hydraulic river routing. As such,

they have the capability to calculate unsteady discharges and stages when presented

with the appropriate geometric channel data and initial and boundary conditions. Their

importance in unsteady flow is examined here by comparing them to kinematic and

diffusion waves.

Kinematic waves calculate unsteady discharges; the corresponding stages are

subsequently obtained from the appropriate rating curves. Equilibrium (steady, uniform)

rating curves are used for this purpose. Diffusion waves may or may not use

equilibrium rating curves to calculate stages. Some methods, e.g., Muskingum-Cunge,

use equilibrium ratings, but more elaborate diffusion wave solutions may not.

Mixed dynamic waves rely on the physics of the phenomena as built into the governing

equations to generate their own unsteady flow rating. A looped rating curve is produced

at every cross section, as shown in Fig. 10-21. For any given stage, the discharge is

higher in the rising limb of the hydrograph and lower in the receding limb. This loop is

due to hydrodynamic reasons and should not be confused with other loops, which may

be due to erosion, sedimentation, or changes in bed configuration.

Figure 10-21 Sketch of the looped rating of dynamic waves.

The width of the loop is a measure of the flow unsteadiness, with wider loops

corresponding to highly unsteady flow. If the loop is narrow, it implies that the flow is

mildly unsteady, perhaps a diffusion wave. If the loop is practically nonexistent, the flow

can be approximated as kinematic flow. In fact, the basic assumption of kinematic flow

is that momentum can be simulated as steady uniform flow, i.e., that the rating curve is

single-valued.

The preceding observations lead to the conclusion that the importance of mixed

dynamic waves is directly related to the flow unsteadiness and the associated loop in

the rating curve. For highly unsteady flows such as dam-breach flood waves, it may

well be the only way to properly account for the looped rating. For other less unsteady

flows, kinematic and diffusion waves are a viable alternative, provided their applicability

can be clearly demonstrated.

Downstream boundary condition

Modeling of a mixed dynamic wave presents an interesting paradox: In order to solve

the problem correctly, a dynamic downstream boundary condition (usually a rating

curve) must be specified. However, a dynamic downstream boundary condition is not

known a priori. Abbott (1976) put it in the right perspective when he stated:

"A common source of error is the use of a boundary condition in a dynamic

model that is really only proper to a pure kinematic model... but in a dynamic

model the solution itself generates different discharge-stage relations according

to the variabilities of the flows, so that a unique relation at the boundary

contradicts the solution within the boundary. A finely balanced model will

usually go unstable in this situation, but models with heavy numerical

damping may survive, only to give erroneous results."

A way out of this difficulty is to artificially extend the channel several subreaches

downstream, and to specify a kinematic rating at the newly defined downstream

boundary, while giving the loop a chance to develop at the upstream cross section of

interest (Fig. 10-22). Ponce and Lugo (2001) have used sensitivity analysis to show

that the artificial extension of the channel by an amount equal to the channel length

(i.e., doubling the channel length) may be sufficient to produce an accurate looped

rating at the cross section of interest.

Figure 10-22 Artificial extension of the computational domain, for use in mixed dynamic wave

models to properly specify the downstream boundary.

QUESTIONS

[Problems] [References] • [Top] [Governing Equations] [Linear Solution] [Kinematic Waves] [Diffusion

Waves] [Muskingum Method] [Muskingum-Cunge Method] [Dynamic Waves]

1. What conservation laws are used in the description of steady gradually varied

flow?

2. What conservation laws are used in the description of unsteady gradually varied

flow?

3. What forces are acting in a control volume in unsteady gradually varied flow?

4. What waves types are common in unsteady open-channel flow?

5. To what problems do kinematic waves apply?

6. To what problems do diffusion waves apply?

7. How is Lo defined?

8. What is the dimensionless wavenumber?

9. What is the dimensionless relative celerity of kinematic waves under Chezy

friction in hydraulically wide channels?

10. What is the dimensionless relative celerity of dynamic waves under Chezy

friction in hydraulically wide channels?

11. What are three ways of expressing the convective celerity of kinematic

waves?

12. What is the neutral-stability Froude number for triangular channels with

Chezy friction?

13. What is the neutral-stability Froude number for hydraulically wide channels

with Manning friction?

14. What is the basis of the Muskingum-Cunge method?

15. What is Hayami's contribution to flood routing?

16. For what value of Vedernikov number are kinematic and dynamic hydraulic

diffusivities the same?

17. For what value of Vedernikov number does the dynamic hydraulic diffusivity

vanish?

18. What part of the flood routing does the Muskingum parameter X account

for?

19. What is numerical diffusion?

20. What is numerical dispersion?

21. What is the advantage of the Muskingum-Cunge method?

22. When will the Muskigum-Cunge method fail to give good results?

23. How is the downstream boundary specified in a mixed kinematic-dynamic

model if increased accuracy is desired?

PROBLEMS

[References] • [Top] [Governing Equations] [Linear Solution] [Kinematic Waves] [Diffusion Waves] [Muskingum

Method] [Muskingum-Cunge Method] [Dynamic Waves] [Questions]

1. You are observing the rise of a river during flood. The width of the river at the

observation point, and for some distance upstream is 65 m, and according to the

gage reading, the discharge is Q= 70 m3/s. Estimate the discharge at a point

located 15.875 km upstream, if the water surface is rising at the rate of 9 mm/hr

at your location and at 12 mm/hr at the upstream cross section.

2. A hydraulically wide channel is operating at Froude number F = 0.22. The unit-

width discharge is q = 2.8 m2/s. What are the two absolute Lagrange celerities?

3. A flood wave is traveling inbank through a straight river reach of top width T =

320 m wide and length L = 5625 m. For every 1 cm of flood rise, the discharge

goes up 10 m3/s. How long will it take for a given discharge to travel the length of

the reach?

4. Calculate the exponent β of the discharge-area rating for a triangular channel

with Chezy friction.

Fig. 10-23 Definition sketch for a triangular channel.

5. Calculate the exponent β of the discharge-area rating for a triangular channel

with Manning friction.

6. A flood hydrograph has the following data: time-of-rise tr = 2 hr, reference

velocity Vo = 2 ft/s, reference flow depth do = 6 ft, bottom slope So = 0.004.

Determine is this wave is a kinematic wave.

7. A flood hydrograph has the following data: time-of-rise tr = 2 hr, reference

velocity Vo = 2 ft/s, reference flow depth do = 6 ft, bottom slope So = 0.004.

Determine is this wave is a diffusion wave.

8. A flood hydrograph has the following data: time-of-rise tr = 1 hr, reference

velocity Vo = 2 m/s, reference flow depth do = 2 m, bottom slope So = 0.0004.

Determine is this wave is a diffusion wave.

9. Using ONLINE ROUTING 04, route a flood wave using the Muskingum method.

The inflow hydrograph ordinates are [25 ordinates, starting at time = 0, to time =

24 hr]:

100, 130, 150, 180, 220, 250, 300, 360, 450, 550, 700, 550, 490, 370, 330, 310, 280, 230, 170, 150, 130, 120, 110, 105, 100.

Assume Δt = 1 hr, K = 1 hr, X = 0.3. Report the peak outflow and time of occurrence.

10. Using ONLINE ROUTING 05, route the same flood wave as the previous

problem using the Muskingum-Cunge method. Use the following input data: Qp =

700 m3/s, Ap = 400 m

2,Tp = 88 m, Δt = 1 hr, Δx = 9.6 km, β = 1.65, So = 0.0007.

Report the peak outflow and time of occurrence.

REFERENCES

• [Top] [Governing Equations] [Linear Solution] [Kinematic Waves] [Diffusion Waves] [Muskingum

Method] [Muskingum-Cunge Method] [Dynamic Waves] [Questions] [Problems]

Abbott, M. A. 1976. Computational hydraulics: A short pathology. Journal of Hydraulic

Research, Vol. 14, No. 4, p. 276.

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

Craya, A. 1952. The criterion for the possibility of roll-wave formation. Gravity

Waves, Circular No. 521, National Bureau of Standards, Washington, D.C. 141-151.

Cunge, J. A. (1969). On the Subject of a Flood Propagation Computation Method

(Muskingum Method), Journal of Hydraulic Research, Vol. 7, No. 2, 205-230.

Dooge, J . C. I. 1973. Linear theory of hydrologic systems. Technical Bulletin No.

1468, U.S. Department of Agriculture, Washington, D.C.

Hayami, I. 1951. On the propagation of flood waves. Bulletin, Disaster Prevention

Research Institute,No. 1, December.

Kibler, D. F., and D. A. Woolhiser. 1970. The kinematic cascade as a hydrologic

model. Hydrology Paper No. 39, Colorado State University, Ft. Collins, Colorado.

Kleitz, M. 1877. Note sur la théorie du mouvement non permanent des liquides et sur

application à la propagation des crues des rivières, (Note on the theory of unsteady

flow of liquids and on application to flood propagation in rivers), Annals des Ponts et

Chaussées, ser. 5, Vol. 16, 2e semestre, 133-196.

Liggett, J. A. 1975. Basic equations of unsteady flow. Chapter 2 in Unsteady flow in

open channels, K. Mahmood and V. Yevjevich, eds., Water Resources Publications, Ft.

Collins, Colorado.

Liggett, J. A., and J. A. Cunge. 1975. Numerical methods of solution of the unsteady

flow equations. Chapter 4 In Unsteady flow in open channels, K. Mahmood and V.

Yevjevich, eds., Water Resources Publications, Ft. Collins, Colorado.

Lighthill, M. J., and G. B. Whitham. 1955. On kinematic waves: I. Flood movement in

long rivers.Proceedings, Royal Society of London, Series A, 229, 281-316.

McCarthy, G. T. (1938). "The Unit Hydrograph and Flood Routing," unpublished

manuscript, presented at a Conference of the North Atlantic Division, U.S. Army Corps

of Engineers, June 24.

Ponce, V, M., and D. B. Simons. 1977. Shallow wave propagation in open-channel

flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, December, 1461-

1476.

Ponce, V. M., and V. Yevjevich. 1978. Muskingum-Cunge method with variable

parameters. Journal of the Hydraulics Division, ASCE, Vol. 104, No. HY12, December,

1663-1667.

Ponce, V. M., and F. D. Theurer. 1982. Accuracy criteria in diffusion routing. Journal of

the Hydraulics Division, ASCE, Vol. 108, No. HY6, June, 747-757.

Ponce, V, M., and D. Windingland. 1985. Kinematic shock: Sensitivity analysis. Journal

of Hydraulic Engineering, ASCE, Vol. 111, No. 4, April, 600-611.

Ponce, V. M. 1989. Engineering hydrology: Principles and practices. Prentice Hall,

Englewood Cliffs, New Jersey.

Ponce, V. M. 1990. Generalized diffusion wave equation with inertial effects. Water

Resources Research, Vol. 26, No. 5, May, 1099-1101.

Ponce, V. M. 1991. New perspective on the Vedernikov number. Water Resources

Research, Vol. 27, No. 7, 1777-1779, July.

Ponce, V, M., and P. J. Porras. 1995. Effect of cross-sectional shape of free-surface

instability.Journal of Hydraulic Engineering, ASCE, Vol. 121, No. 4, April, 376-380.

Ponce, V, M., and A. Lugo. 2001. Modeling looped ratings in Muskingum-Cunge

routing. Journal of Hydrologic Engineering, ASCE, Vol. 6, No. 2, March-April, 119-124.

Saint-Venant, B. de. 1871. Theorie du mouvement non-permanent des eaux avec

application aux crues des rivieres et l' introduction des varees dans leur lit, Competes

Rendus Hebdomadaires des Seances de l'Academie des Science, Paris, France, Vol.

73, 1871, 148-154.

Seddon, J. A. 1900. River hydraulics. Transactions, ASCE, Vol. XLIII, 179-243, June.

U.S. Army Corps of Engineers. 2010. River Analysis System: HEC-RAS, Version 4.1,

Hydrologic Engineering Center, Davis, California.

http://openchannelhydraulics.sdsu.edu

140901 20:45

REFERENCES

[Surface waves] [Surges] [Kinematic shocks] [Roll waves] [Tidal waves] [Debris

flows] [Lahars] [Questions] [Problems] [References] •

CHAPTER 11:

UNSTEADY RAPIDLY VARIED

FLOW

11.1 SURFACE WAVES

[Surges] [Kinematic shocks] [Roll waves] [Tidal waves] [Debris

flows] [Lahars] [Questions] [Problems] [References] • [Top]

Surface waves are unsteady open-channel flow features typically occurring under high

Froude-number flows. Figure 11-1 shows a surface wave on the Hassayampa river,

near Morristown, Arizona (Phillips and Ingersoll, 1998). The disturbance is a wave

propagating slowly downstream, and it is indicative of hydraulic characteristics in

alluvial channels under upper-regime flow conditions(Fig. 11-2) (Simons and

Richardson, 1966).

U.S. Geological Survey

Fig. 11-1 Hassayampa river, near Morristown, Arizona: View from cableway,

looking upstream, during the flood of February 9, 1993.

U.S. Geological Survey

Fig. 11-2 Channel bedforms under upper regime.

Figure 11-3 shows the Santa Catarina river, at Monterrey, Nuevo Leon, Mexico, during

the passage of Hurricane Gilbert, on September 17, 1988. The observed large surface

waves are an indication of the high Froude-number flows which most likely prevailed

during the passage of the flood.

Unknown

Fig. 11-3 The Santa Catarina river, at Monterrey, Nuevo Leon, Mexico, during the passage

of Hurricane Gilbert, on September 17, 1988.

11.2 SURGES

[Kinematic shocks] [Roll waves] [Tidal waves] [Debris

flows] [Lahars] [Questions] [Problems] [References] • [Top] [Surface waves]

Surges are perturbations created by sudden gate closures or rapid changes in stage or

flow depth. Typically, a surge does not attenuate readily, traveling along the channel for

a considerable distance. Surges are avoided by opening gates slowly, to minimize the

possibility of sudden stage or depth changes. Surge attenuation increases as the

dimensionless wavenumber decreases from that corresponding to a true dynamic wave

(with constant celerity, i.e, to the right side of the wavenumber spectrum) to that

corresponding to a mixed dynamic wave (with variable celerity, i.e., along the upper

falling side of the S-curve) (Fig. 11-4).

Fig. 11-4 Dimensionless relative wave celerity versus dimensionless wavenumber

in open-channel flow.

Based on the above criterion, a first approximation to the time of opening To of a canal

gate, to avoid a surge, is (Ponce et al., 1999):

k uo

To ≥ _______

g So

(11-1)

in which uo = flow velocity, So = channel slope, g = gravitational acceleration, and k = a

constant varying with Froude number as shown in Table 11-1.

Table 11-1 Time of opening of canal gate

to avoid surges.

Froude number Fo Constant k

0.1 1.2

0.2 1.3

0.3 1.4

0.4 1.5

0.5 1.7

11.3 KINEMATIC SHOCKS

[Roll waves] [Tidal waves] [Debris flows] [Lahars] [Questions] [Problems] [References] • [Top] [Surface

waves] [Surges]

Kinematic shocks are kinematic waves that have steepened to the point where they

become, for all practical purposes, a wall of water, with a near vertical face. Actually,

the shock is not a perfect discontinuity; however, its thickness is relatively small

compared to the wavelength of the disturbance (Lighthill and Whitham, 1955).

Kinematic waves travel downstream, and may either steepen or flatten out, depending

on their interaction with the cross-sectional geometry. Only kinematic waves that

steepen can develop into kinematic shocks. While kinematic waves are gradually

varied, kinematic shocks are rapidly varied.

The development of a kinematic shock is a function of the following flow conditions

(Ponce and Windingland, 1985):

1. The type of wave

The more kinematic a wave is (the smaller the dimensionless wavenumber), the

less its tendency to attenuate and, therefore, the more its tendency to steepen.

2. The amount of baseflow

The smaller the base-to-peak flow ratio, the greater the tendency for the wave to

steepen. This is due to wave nonlinearities (inbank larger flows travel with greater

celerities).

3. The flow regime

High-Froude number flows have less tendency to attenuate disturbances;

therefore, the higher the Froude number, the lesser the attenuation. In the limit,

as the flow approaches the Vedernikov number V = 1 (i.e., F = 2 for Chezy

friction in hydraulically wide channels), the flow becomes neutrally stable and

attenuation is reduced to zero.

4. The type of cross section

Hydraulically wide channels, for which the rating exponent β = 1.5 (Chezy

friction), have a greater tendency to steepen that triangular channels, for

which β = 1.25. Inherently stable channels, for which β = 1, have no tendency to

steepen (See Table 10-3 for other applicable values of β).

Thus, kinematic shocks will develop in field situations involving: (a) a kinematic wave,

(b) an ephemeral channel (zero baseflow), (c) a high flow rate (a flood), and (d) a

hydraulically wide channel. This may be the case of a major cloudburst rapidly

concentrating flow through a canyon in a semiarid region.

A well-documented flash flood, which was in all probability a kinematic shock, occurred

July 26, 1981 in Tanque Verde Creek, a tributary of the Santa Cruz river, in eastern

Arizona. The flood, which killed eight people, was, by all accounts, only a 2-yr flood.

However, the suddenness of the flood which, according to survivors' reports amounted

to a "wall of water," resulted in substantial loss of life (Hjalmarson, 1984).

Kinematic shocks have been observed with certain frequency in laboratory overland

flow computations. These computations are known to be conducive to kinematic shock

development (Kibler and Woolhiser, 1970). The shock presence is attributed to the

prescribed spatial regularity which is necessary to make the problem more tractable.

11.4 ROLL WAVES

[Tidal waves] [Debris flows] [Lahars] [Questions] [Problems] [References] • [Top] [Surface

waves] [Surges] [Kinematic shocks]

Roll waves develop in open-channel flow when the Vedernikov number V > 1 (Craya,

1952; Chow, 1959). In natural channels, the condition V > 1 is seldom, if ever, met

(Jarrett, 1982). Thus, roll waves are restricted to artificial channels lined with either

concrete or masonry. Typically, roll waves appear as a train of waves that move

downstream, as shown in Fig. 1-7 (Chapter 1).

The Vedernikov is defined as the ratio of relative kinematic wave celerity to relative

dynamic wave celerity (Ponce, 1991):

(β - 1) u

V = __________

(g h )1/2

(11-2)

in which β = exponent of the discharge-area rating (Eq. 10-52), u = mean velocity, h =

flow depth, andg = gravitational acceleration. Roll waves occur when the relative

kinematic wave celerity exceeds the relative dynamic wave celerity. Since kinematic

waves transport mass, and (true) dynamic waves transport energy, roll waves occur at

the threshold where mass and energy are being transported at the same speed. In

practice, roll waves occur in steep artificial canals when the Vedernikov numberV >

1 (Fig. 11-5).

Fig. 11-5 Roll waves in a steep irrigation canal, Cabana-Ma�azo project, Puno, Peru.

The condition V > 1 is necessary, but may not be sufficient. As shown in Fig. 11-6,

certain dimensionless wavenumbers have stronger amplification tendencies than

others. For F > 2 (V > 1), the logarithmic increment peaks at the point of inflexion of the

celerity-wavenumber function(Fig. 11-4). Thus, certain scales of disturbances are more

likely to be amplified than others, as verified by Ponce and Maisner (1993) using the

Brock (1967) data.

Fig. 11-6 Primary wave logarithmic increment for Froude numbers F > 2.

11.5 TIDAL WAVES

[Debris flows] [Lahars] [Questions] [Problems] [References] • [Top] [Surface waves] [Surges] [Kinematic

shocks] [Roll waves]

Tidal bores are rapidly varied unsteady free-surface flow features which take place in

certain rivers in the proximity of their estuaries. A tidal waves occurs in an estuary for a

large tidal range on or around either of the equinoxes (March 20 and September 22).

Whether the tidal wave is able to move upstream into the river proper and develop into

a recognizable bore of finite depth, depends largely on the cross-sectional geometry of

the estuary. Tidal bores are more likely to form in smooth, hydraulically wide channels

of relatively constant depth.

Fig. 11-7 Tidal bore on the Araguari river, Amapa, Brazil, at 8:00 am, on January 22, 1989.

Large tidal bores have been observed on the Araguari river, in Brazil (Fig. 11-7), on the

Chien Tang river, in China (Fig. 11-8), and in other selected estuaries around the world.

Chow (1959) described the Hangchow bore at Haining on the Chien Tang river, China.

The wavefront was about 16 ft high, traveling at high velocity. Seven miles after it could

first be distinguished on the horizon, the wave had passed. The water reached a final

height of about 28 ft within 30 minutes. The width of the river at the observation point

was about 1 mile.

Chow

Fig. 11-8 Tidal bore on the Chien Tang river, at Haining, China.

11.6 DEBRIS FLOWS

[Lahars] [Questions] [Problems] [References] • [Top] [Surface waves] [Surges] [Kinematic shocks] [Roll

waves] [Tidal waves]

Debris flows are sudden accumulations of runoff containing great quantities of sediment

particles, usually boulder size and above. Debris flows travel downstream at great

speeds, destroying everything in their path and threatening life and property (Fig. 11-9).

Unknown

Fig. 11-9 Debris flow in Jiangjiaguo, Yunnan, China.

Debris flows are induced by intense rains, but they can also be triggered by

earthquakes. In Southern California, along the base of the San Gabriel Mountains, east

of Los Angeles, rain-induced debris flows recur with predictable regularity. The factors

leading to the formation of these debris flows are:

1. Tectonism

The San Gabriel Mountains are among the most tectonically active ranges in the

United States(Fig. 11-10). The uplift leads to slope steepening, which increases

gravitational forces and maximizes erosion potential.

Wikimedia Commons

Fig. 11-10 The San Gabriel Mountains, Southern California.

2. Chaparral ecosystem

The chaparral ecosystem is endemic to the region, surviving the long droughts by

developing singular adaptations which rely on wax protective surface coating to

suppress evapotranspiration (Fig. 11-11) (McPhee, 1989).

Fig. 11-11 Chaparral ecosystem, Tierra del Sol, San Diego County, California.

3. Wind

The region is home to the Santa Ana winds, which provide instant fire weather by

featuring dry winds which can readily reach more than 40 miles per hour.

4. Fire

Fire follows wind after a long drought, with the chaparral not having burned for 30

years of more (Fig. 11-12).

Fig. 11-12 Aftermath of the Shockey fire, Tierra del Sol, California, on September 27, 2012.

5. Rain

Rain follows fire due to ash particles in the air promoting coalescence and

raindrop formation.

The steep slopes covered with chaparral vegetation, followed by a sequence of wind,

fire, and rain, is what triggers the debris flows in the Southern California region. During

fire, the waxlike substances vaporize at the surface and recondense at a certain depth

below the surface, producing the non-wettable layer, of 10 to 50 mm thickness. The

accumulation of intense rainfall, exceeding 25 mm per hour, below the surface and

above the non-wettable layer leads to the entrainment of large quantities of sediment

which go on to constitute the debris flows. A typical rain-induced debris flow in

Southern California may carry away 10-50 mm of soil in a few hours. By way of

comparison, normal erosion rates are typically less than 1 mm per year.

Earthquake-triggered slides

Massive debris flows can also be triggered by earthquakes. Such was the case of the

Huascaran slide on May 30, 1970, in Peru, which buried the town of Yungay, killing

more than 20,000 people (Fig. 11-13). The town has since been rebuilt at a location just

north of the ill-fated site.

Fig. 11-13 Remnant of the church in the central square of Yungay, Peru,

buried by a landslide on May 30, 1970.

11.7 LAHARS

[Questions] [Problems] [References] • [Top] [Surface waves] [Surges] [Kinematic shocks] [Roll waves] [Tidal

waves] [Debris flows]

Lahars are debris flows triggered by snowmelt, following a volcanic eruption and

subsequent sudden melting of the snowcap. The word lahar originated in Indonesia,

where the phenomenon is common. Lahars have the consistency, viscosity, and

approximately the same density as concrete; fluid when moving, and solid when

stopped (Fig. 11-14).

Wikimedia Commons

Fig. 11-14 A lahar on the Lower Sacobia river, Phillipines,

on July 22, 1993.

Lahars can be massive and deadly, as shown by the November 13, 1985 eruption of

the Nevado del Ruiz volcano, in Colombia. Four lahars came down the river valleys on

the volcano flanks. The largest of them virtually destroyed the town of Armero, burying

it under 5 m of mud and debris, and killing more than 75% of its 28,700 inhabitants (Fig.

11-15).

Wikimedia Commons

Fig. 11-15 Site of the town of Armero, Colombia, buried by a lahar on November 13, 1985.

QUESTIONS

[Problems] [References] • [Top] [Surface waves] [Surges] [Kinematic shocks] [Roll waves] [Tidal waves] [Debris

flows] [Lahars]

1. What bedforms develop under upper regime in alluvial channel flow?

2. When do surges form in open-channel flow?

3. What is a kinematic shock?

4. What conditions are necessary for a kinematic wave to steepen into a kinematic

shock?

5. When do roll waves develop?

6. Do roll waves develop always in steep channels?

7. Where are tidal bores are more likely to form?

8. What is a debris flow?

9. What conditions are conducive to the development of debris flows in the

Southern California climate?

10. What is a lahar?

PROBLEMS

[References] • [Top] [Surface waves] [Surges] [Kinematic shocks] [Roll waves] [Tidal waves] [Debris

flows] [Lahars] [Questions]

1. Calculate the time of opening of a canal gate under the following flow conditions:

flow depthdo = 1.2 m, flow velocity uo = 1.5 m/s, and channel slope So = 0.001.

2. A certain channel has a dimensionless Darcy-Weisbach friction factor f = 0.0035.

The channel slope is So = 0.014. Determine if this flow condition is stable for

Chezy friction in a hydraulically wide channel.

REFERENCES

• [Top] [Surface waves] [Surges] [Kinematic shocks] [Roll waves] [Tidal waves] [Debris

flows] [Lahars] [Questions] [Problems]

Brock, R. R. 1967. Development of roll waves in open channels. Report No. KH-R-

16, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of

Technology, Pasadena, California,

Chow, V. T. 1959. Open-channel Hydraulics. McGraw Hill, New York.

Cornish, V. 1907. Progressive waves in rivers. The Geographical Journal. Vol. 29, No.

1, January, 23-31.

Craya, A. 1952. The criterion for the possibility of roll-wave formation. Gravity

Waves, Circular No. 521, National Bureau of Standards, Washington, D.C. 141-151.

Hjalmarson, H. W. 1984. Flash flood in Tanque Verde Creek, Tucson, Arizona. Journal

of Hydraulic Engineering, Vol. 110, No. 12, 1841-1852.

Jarrett, R. D. 1984. Hydraulics of high-gradient streams. Journal of Hydraulic

Engineering, Vol. 110, No. 11, 1519-1539.

Kibler, D. F., and D. A. Woolhiser. 1970. The kinematic cascade as a hydrologic

model. Hydrology Paper No. 39, Colorado State University, Ft. Collins, Colorado.

Lighthill, M. J., and G. B. Whitham. 1955. On kinematic waves: I. Flood movement in

long rivers.Proceedings, Royal Society of London, Series A, 229, 281-316.

McPhee, J. 1989. The Control of Nature. Farrar Straus Giroux, New York.

Ponce, V, M., and D. Windingland. 1985. Kinematic shock: Sensitivity analysis. Journal

of Hydraulic Engineering, ASCE, Vol. 111, No. 4, April, 600-611.

Ponce, V. M. 1991. New perspective on the Vedernikov number. Water Resources

Research, Vol. 27, No. 7, 1777-1779, July.

Ponce, V, M., and M. P. Maisner. 1993. Verification of theory of roll wave

formation. Journal of Hydraulic Engineering, ASCE, Vol. 119, No. 6, June, 768-773.

Ponce, V, M., Y. R. S.Rao, and N. M. Mansury. 1999. Time of opening of irrigation

canal gates.Journal of Hydraulic Engineering, ASCE, Vol. 125, No. 9, September, 979-

980.

Phillips, J. V., and T. L. Ingersoll. 1998. Verification of roughness coefficients for

selected natural and constructed stream channels in Arizona. U.S. Geological Survey

Professional Paper 1584,Washington, D.C.

Simons, D. B., and E. V. Richardson. 1966. Resistance to flow in alluvial channels. U.S.

Geological Survey Professional Paper 422-J, Washington, D.C.

http://openchannelhydraulics.sdsu.edu

140902 19:45

Extract of

Jarrett, R. D., 1984. "Hydraulics of High-Gradient Streams,"

ASCE Journal of Hydraulic Engineering, 110(11), 1519-1539.

APPENDIX A: TABLES

TABLE A-1 PROPERTIES OF WATER IN SI UNITS

Temperature (°C)

Specific Gravity

Density (g/cm

3)

Heat of Vaporization

(cal/g)

Viscosity Vapor Pressure

Absolute (cp)

Kinematic (cs)

(mm Hg) (mb) (g/cm2)

0 0.99987 0.99984 597.3 1.790 1.790 4.58 6.11 6.23

5 0.99999 0.99996 594.5 1.520 1.520 6.54 8.72 8.89

10 0.99973 0.99970 591.7 1.310 1.310 9.20 12.27 12.51

15 0.99913 0.99910 588.9 1.140 1.140 12.78 17.04 17.38

20 0.99824 0.998211 586.0 1.000 1.000 17.53 23.37 23.83

25 0.99708 0.99705 583.2 0.890 0.893 23.76 31.67 32.20

30 0.99568 0.99565 580.4 0.798 0.801 31.83 42.43 43.27

35 0.99407 0.99404 577.6 0.719 0.723 42.18 56.24 57.34

40 0.99225 0.99222 574.7 0.653 0.658 55.34 73.78 75.23

50 0.98807 0.98804 569.0 0.547 0.554 92.56 123.40 125.83

60 0.98323 0.98320 563.2 0.466 0.474 149.46 199.26 203.19

70 0.97780 0.97777 557.4 0.404 0.413 233.79 311.69 317.84

80 0.97182 0.97179 551.4 0.355 0.365 355.28 473.67 483.01

90 0.96534 0.96531 545.3 0.315 0.326 525.89 701.13 714.95

100 0.95839 0.95836 539.1 0.282 0.294 760.00 1013.25 1033.23

Source: Linsley, R. K. et al. (1982). Hydrology for Engineers. 3d. ed., New York: McGraw-Hill.

TABLE A-2 PROPERTIES OF WATER IN U.S. CUSTOMARY UNITS

Temperature (°F)

Specific Gravity

Density (lb/ft

3)

Heat of Vaporization

(Btu/lb)

Viscosity 1 Vapor Pressure

Absolute (lbs/ft

2)

Kinematic (ft

2/s)

(in Hg) (mb) (lb/in2)

32 0.99986 62.418 1075.5 3.746 1.931 0.180 6.11 0.089

40 0.99998 62.426 1071.0 3.229 1.664 0.248 8.39 0.122

50 0.99971 62.409 1065.3 2.735 1.410 0.362 12.27 0.178

60 0.99902 62.366 1059.7 2.359 1.217 0.522 17.66 0.256

70 0.99798 62.301 1054.0 2.050 1.058 0.739 25.03 0.363

80 0.99662 62.216 1048.4 1.799 0.930 1.032 34.96 0.507

90 0.99497 62.113 1042.7 1.595 0.826 1.422 48.15 0.698

100 0.99306 61.994 1037.1 1.424 0.739 1.933 65.47 0.950

120 0..98856 61.713 1025.6 1.168 0.609 3.448 116.75 1.693

140 0.98321 61.379 1014.0 0.981 0.514 5.884 199.26 2.890

160 0.97714 61.000 1002.2 0.838 0.442 9.656 326.98 4.742

180 0.97041 60.580 990.2 0.726 0.386 15.295 517.95 7.512

200 0.96306 60.121 977.9 0.637 0.341 23.468 794.72 11.526

212 0.95837 59.828 970.3 0.593 0.319 29.921 1013.25 14.696

1 To obtain values of viscosity, multiply values shown in Table by 10-5. Source: Linsley, R. K. et al. (1982). Hydrology for Engineers. 3d. ed., New York: McGraw-Hill.

APPENDIX B: DERIVATION OF THE NUMERICAL DIFFUSION COEFFICIENT

OF THE MUSKINGUM-CUNGE METHOD

Figure B-1 Space-time discretization of kinematic wave equation.

Expanding the grid function Q( jΔx,nΔt ) (Fig. B-1) in Taylor series about point ( jΔx,nΔt ) leads to:

∂Q 1 ∂2Q

Q j n+1

= Q j n + [

_____ ] j Δt +

___ [

______ ] j Δt

2 + o (Δt

3)

∂t 2 ∂t 2

(B.1)

∂Q 1 ∂2Q

Q j+1n+1

= Q j+1 n + [

_____ ] j+1 Δt +

___ [

______ ] j+1 Δt

2 + o (Δt

3)

∂t 2 ∂t 2

(B.2)

∂Q 1 ∂2Q

Q j+1n = Q j

n + [

_____ ] n Δx +

___ [

______ ] n Δx

2 + o (Δx

3)

∂x 2 ∂x 2

(B.3)

∂Q 1 ∂2Q

Q j+1n+1

= Q j n+1

+ [ _____

] n+1 Δx + ___

[ ______

] n+1 Δx 2

+ o (Δx 3)

∂x 2 ∂x 2

(B.4)

Substituting Eqs. B.1 to B.4 into Eq. 9-61 and neglecting third-order terms yields:

∂Q 1 ∂2Q

X { [ ____

] j Δt + ____

[ ______

] j Δt 2 }

∂t 2 ∂t 2

∂Q 1 ∂2Q

+ (1 - X ) { [ _____

] j+1 Δt + ____

[ _____

] j+1 Δt 2 }

∂t 2 ∂t 2

C ∂Q 1 ∂2Q

+ ____

{ [ _____

] n Δx + ____

[ ______

] n Δx 2 }

2 ∂x 2 ∂x 2

C ∂Q 1 ∂2Q

+ ____

{ [ _____

] n+1 Δx + ____

[ ______

] n+1 Δx 2 } = 0

2 ∂x 2 ∂x 2

(B.5)

in which C = c (Δt /Δx) is the Courant number.

Expressing the derivatives at grid point [( j + 1)Δx, (n + 1)Δt ] in terms of the derivatives at grid

point( jΔx, nΔt ) by means of Taylor series:

∂Q ∂Q ∂2Q

[ _____

] j+1 = [ _____

] j + [ ______

] j,n Δx + o (Δx 2)

∂t ∂t ∂x ∂t

(B.6)

∂Q ∂Q ∂2Q

[ _____

] n+1 = [ _____

] n + [ _______

] j,n Δt + o (Δt 2)

∂x ∂x ∂x ∂t

(B.7)

∂2Q ∂

2Q ∂

3Q (B.8)

[ ______

] j+1 = [ ______

] j + [ _______

] j Δx + o (Δx 2)

∂t 2 ∂t

2 ∂t

2∂x

∂2Q ∂

2Q ∂

3Q

[ ______

] n+1 = [ ______

] n + [ _______

] n Δt + o (Δt 2)

∂x 2 ∂x

2 ∂x

2∂t

(B.9)

Substituting Eqs. B.6 to B.9 into B.5 and neglecting third-order terms:

∂Q 1 ∂2Q

X { [ _____

] j Δt + ____

[ ______

] j Δt 2 }

∂t 2 ∂t 2

∂Q ∂2Q 1 ∂

2Q

+ (1 - X ) { [ _____

] j Δt + [ ______

] j,n Δx Δt + ____

[ ______

] j Δt 2 }

∂t ∂x ∂t 2 ∂t 2

C ∂Q 1 ∂2Q

+ ____

{ [ _____

] n Δx + ____

[ ______

] n Δx 2 }

2 ∂x 2 ∂x 2

C ∂Q ∂2Q 1 ∂

2Q

+ ____

{ [ _____

] n Δx + [ ______

] j,n Δx Δt + ____

[ ______

] n Δx 2 } = 0

2 ∂x ∂x ∂t 2 ∂x 2

(B.10)

In Eq. B.10, dividing by Δt and simplifying:

∂Q ∂Q Δt ∂2Q c Δx ∂

2Q

[ _____

] j + c [ _____

] n + ____

[ ______

] j + ______

[ ______

] n

∂t ∂x 2 ∂t 2 2 ∂x

2

C ∂2Q

+ Δx { ( 1 - X ) + ____

} [ ______

] j,n = 0

2 ∂x ∂t

(B.11)

The first two terms of Eq. B.11 constitute the kinematic wave equation, Eq. 9-18. The remaining terms

are the error R of the first-order-accurate numerical scheme:

Δt ∂2Q c Δx ∂

2Q C ∂

2Q

R = ____

[ ______

] j + ______

[ ______

] n + Δx { ( 1 - X ) + _____

} [ ______

] j,n = 0

2 ∂t 2 2 ∂x

2 2 ∂x ∂t

(B.12)

From Eq. 9-18:

∂Q ∂Q ____

= - c ____

∂t ∂x

(B.13)

Therefore:

∂2Q ∂

2Q

______ = - c

______

∂x ∂t ∂x 2

(B.14)

∂2Q ∂

2Q

______ = c

2

______

∂t 2 ∂x

2

(B.15)

Substituting Eqs. B.14 and B.15 into B.12 and simplifying:

1 ∂2Q

R = c Δx ( X - ___

) _____

2 ∂x 2

(B.16)

Comparing Eq. B.16 with the right-hand side of the diffusion wave equation, repeated here:

∂Q ∂Q ∂2Q

____

+ c _____

= νh _______

∂t ∂x ∂x 2

(B.17)

it follows that the numerical diffusion coefficient of the Muskingum-Cunge method is:

1

νh = c Δx ( ___

- X )

2

(B.18)

140815

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ABOUT THE AUTHOR

Dr. VICTOR MIGUEL PONCE has been on the civil engineering faculty at San Diego State

University, California, since 1980. His areas of specialty are hydraulics, hydrology, and water

resources. He is the author of Engineering Hydrology: Principles and Practices (Prentice Hall,

1989; second edition online, 2014), and Fundamentals of Open-channel Hydraulics(published online,

2014). Since 1977, Professor Ponce has published numerous papers in the hydraulic and hydrologic

engineering literature. He is the recipient of the 1979 ASCE Karl E. Hilgard Hydraulics Prize. Since

1999, Professor Ponce has actively supported the development of http://ponce.sdsu.edu. His Visualab is

a world-class facility for online teaching and research, featuring more than 200 free online

calculators and more than 200 online videos.