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Lecture notes Numerical Hydraulics 4-1

4 Computation of the free surface level in open channel flow4.1

Basic equations

The computation of the free surface level in open channels is based on the continuity equation (4-1)

and the Navier Stokes equation (4-2).

0=u (4-1)

{

( ){ { 32143421

kraftibungs

kraftSchwer

kraftDruck

gungBeschleuniadvektive

gungBeschleunilokale

ugpuut

u

++=

+

Re

(4-2)

We take the x-axis parallel to the channel axis and so we have uux

and the velocity components

normal to the channel axis are neglected ( 0, zy

uu ). In this way, Eq. (4-2) simplifies to:

ux

zg

x

p

x

uu

t

u+

=

+

(4-3)

Integration over the cross sectional area (with uv = ) leads to:

A

l

x

h

gx

v

vt

v bp 0'

=

+

(4-4)

is a correction factor, which takes the velocity profile into account (note that22 vv ). However,

in the following we assume for simplicity that =1. This is also justified by the logarithmic profile.The level hp is defined as the sum of the channel bed level and the flow depth (z + h).

The internal friction cancels out through integration. Only the friction at the channel bed remains,

analogous to pipe flow, where only the friction at the wall remained after integration.

When integrating the continuity equation over the cross section, one has to keep in mind that the

free surface is a moveable boundary:

=

+

qtA

xQ 0 (4-5)

and for the case of a rectangular cross section with A = bh, b = const, q = 0:

0)(

=

+

t

h

x

hv(4-6)

The friction force per unit volume is defined as (Fig. 4-1):

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Lecture notes Numerical Hydraulics4-2

Fig. 4-1: Calculation of the friction force

hy

b

frictionRA

l

volume

forcef 00

=== , (Rhy = hydraulic radius) (4-7)

with:

force = 0lbx (4-8)

volume = Ax (4-9)

For the frictional shear stress one can use the following empirical relation:o

2

08

v

= , with

=

hyR

KRe, (Darcy Weisbach) (4-10)

The loss of energy E can be interpreted as work done by the friction forces:

xfV

Efriction =

=>

hyRx

VE 0/ =

(4-11)

The slope of the energy line IE is then:

( ) 20/ 14 2

E

hy hy

E V g vI

x gR R g

= = =

(4-12)

For the sake of clarity, in the following we consider a rectangular cross section. Using xzIS = ,the continuity and momentum equations become:

0

)(

=

+

+

=

+

x

vh

x

hv

t

h

x

hgIIg

x

vv

t

vES

(4-13)

These equations are known as the St.-Venant Equations (or shallow water wave equations). They

can also be written in a more general form for generic cross sections as a function of Q and A.

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Lecture notes Numerical Hydraulics 4-3

4.2 Steady state solution4.2.1 ApproximationFor the approximation we assume that SI

x

hg

x

vv

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Lecture notes Numerical Hydraulics4-4

32

2

gb

Qhgr = (4-20)

The Froude-number can also be written as a function of the critical flow depth as:

3

2

=

h

hFr

gr(4-21)

Using Eq. (4-19) it is not difficult to calculate the free surface level numerically. When the flow is

subcritical, the calculation is done from downstream towards the upstream direction, while for

supercritical conditions it is the opposite, that is, one integrates the equation from upstream to

downstream (i.e. in streamwise direction).

Analyzing Eq. (4-19) it is possible to deduce the main evolution of the free surface level along the

channel. As an example we consider a backwater flow in steady state conditions:

Fig. 4-2: Example of a free surface level profile (backwater upstream of a weir)

We start at a location in the upstream and choose a water depth that is somewhat greater than the

uniform flow depth. (uniform flow = Normalabfluss)

Moreover we consider the case hN > hgr(subcritical flow, mild channel slope).

We obtain: 0>dx

dh,because both enumerator and denominator in Eq. (4-19) are greater than zero:

1 - Fr2 > 0, because the flow is subcritical (h > hgr and uniform flow is subcritical).

IE < IS , because h > hN (IE = IS, when h = hN => IE > or < IS, when h < or > hN, respectively).

Therefore, the flow depth increases in flow direction. In upstream direction the flow depth

approaches the uniform flow depth.

0,: dx

dhIIhh ESN

Thus, a typical backwater curve is obtained in this case. The numerical integration has to be done

appropriately, from the backwater side towards the upstream direction.

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Lecture notes Numerical Hydraulics 4-5

4.3 Non-steady state solutionThe time-varying momentum and continuity equations are given in (4-13) for the case of a

rectangular cross section. It is possible to apply the following approximations to the momentum

equation:

444444 3444444 21

444 3444 21

43421

WelleDynamische:Exakt

WelleDiffusive:2.Nherung

WelleheKinematisc:1.Nherung

)( ES IIgx

hg

x

vv

t

v+

=

(4-22)

4.3.1 1. Approximation: Kinematic waveFor the kinematic wave approximation we assume that SgI

x

vv

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Lecture notes Numerical Hydraulics4-6

The water wave (pressure wave) is thus faster than a wave of dissolved matter, which propagates

with the mean water velocity.

A constant wave velocity is a valid approximation only for small amplitudes (wave heights). Ingeneral, the wave velocity is a function of water depth c = c(h). In particular, c is greater for greater

flow depth. This leads to an overtaking and to a steepening of the wave front. Since the wave

equation 'h hct x

+ =

0 is of first order, only one boundary condition (inflow condition) is required.

The propagation occurs only in positive x-direction and therefore no influence from the downstream

side is possible. To take backwater into account a partial differential equation of at least second

order is needed.

4.3.2 2. approximation: Diffusive waveWe use again the assumptions:

SgI

x

vv

t

v

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