RIVER CONSERVATION AND REHABILITATION

EAH 416PUAY HOW TION

ROOM 117 AEROSPACE

AFTER COMPLETING THIS COURSE, YOU SHOULD BE ABLE TO

• WRITE/UNDERSTAND THE GOVERNING EQUATIONS FOR STEADY FLOW

• PREDICT WATER SURFACE PROFILE (WSP)

• UNDERSTAND THE REQUIRED BOUNDARY CONDITIONS TO SOLVE FOR WSP

• SOLVE NUMERICALLY FOR WATER SURFACE PROFILE

• BY SPREADSHEETS

• BY SIMPLE FORTRAN/MATLAB (OR OTHER LANGUAGE)

• BY USING HEC-RAS

TEACHING PLAN

Week Date Teaching plan

6 13th Oct 2015 Revisiting basic governing equations

7 20th Oct 2015 Water surface profile tracing for steady flow

8 27th Oct 2015 Solving WSP numerically

9 3rd Nov 2015 Solving WSP numerically

10 10th Nov 2015 Mid semester break

11 17th Nov 2015 Test 2

12 24th Nov 2015 Hands on 1

13 1st Dec 2015 Hands on 2

14 8th Dec 2015 Hands on 3

15 15th Dec 2015 Mini Project Presentation

16 22nd Dec 2015 Revision week

Governing Laws

Conservation of mass Continuity equation

Conservation of energy Bernoulli equation

Conservation of momentum Momentum equation

State of material Equation of state

Flow classification

Based on space

Uniform flow

Non uniform flow

Based on time

Steady flow

Unsteady flow

0t

0t

0x

0x

flow variables, e.g. velocity, depth, cross section areas..

THE MANY FORMS OF GOVERNING EQS..

0u v wx y z

0A Qt x

22 2

2 2 2

1 1 xyxx xzx

u uu uv uw P gt x y z x x y z

Continuity equation (per unit volume) 1-D depth averaged continuity equation

Momentum equation (per unit volume)

, hydraulic radiusbxQ uQ hgA gA Rt x x gR

1-D depth averaged momentum equation

Some examples…………

HOW DO WE CHOOSE THE SUITABLE FORM OF GOV. EQ ?

• The flow direction :

• Mainly in one direction – use 1D equation

• Mainly in two-directions – use 2D equation

• Flow is complex, all directions – use 3D equation

• What phenomena are we interested in?

• Hydraulic jump- at least 2D equation

• Turbulent flow structure – 3D equation

• Wave breaking on surface – 3D equation

• Nature of the flow

• Steady flow : use steady equation

• Unsteady flow : use unsteady equation

• Study purpose and accuracy needed

• To reduce complexity of the solution, assumptions are made and governing equations are simplified. Assumptions made must not sacrifice accuracy and cause lost of important flow characteristics.

OPEN- CHANNEL STEADY FLOW

• Characteristics of open channel flow

• Existence of surface

• If the flow changes gradually, hydrostatic pressure is adequate

• Basic equation

• Continuity equation, energy equation, momentum equation

cosP gh

Flow classification

Based on space

Uniform flow(a)

Non uniform flow(b)

Based on time

Steady flow(c)

Unsteady flow(d)

Steady non-uniform flow

(b)+(c)

Gradually varied flow

Rapidly varied flow

Unsteady non uniform flow

(b)+(d)

Gradually varied unsteady flow

Rapidly varied unsteady flow

GRADUALLY VARIED FLOW

• DERIVATION OF CONTINUITY AND ENERGY EQUATION

• FLOW SURFACE PROFILE CLASSIFICATION

• UNIFORM FLOW RESISTANCE LAWS

Please refer to notes as well

DERIVATION OF CONTINUITY EQUATION FOR OPEN CHANNEL FLOW

From derivation (see handout), we have the continuity equation for unsteady flow,

0A Qt x

For steady flow, the first term is zero

0

constant

Qx

Q

MOMENTUM EQUATION

Momentum equation (unsteady flow)

Neglecting Reynold shear stress,

sin Reynold's stressess byQ uQ gA gAt x x gR

2

2u

so f

yQ uQ gA gA S St x x

Derivation skipped

ENERGY EQUATION

Derivation skipped

UNSTEADY FLOW

STEADY FLOW

By energy principle

Simplifying

Change in total head

frictionloss

divide by x

2 2 2

2 2 2 fV V Vz d z dz d d d d hg g g

2

2 f fVd z d h S xg

2 2

Specific energy ,

cos2 2

o

o

H

V VH d hg g

Channel slope o

o

SdzSdx

Friction slope f

f

S

SgR

2 2

Specific energy ,

cos2 2

o

o

H

V VH d hg g

22

2 22 2

2 2 3

2

3

2

3

2 2

3 3

cos cos2 2

1 2

2

2cos2

2cos2

o

o

dH d V dh dh Vdx dx g dx g dx

d d Q d Q dAV Qdx dx A dx A A dx

Q A A dhA x h dx

dH dh Q A A dhdx dx g A x h dx

dh Q A Q Adx g A h g A x

constant constant

Because,

h x

dA A dx Adx x dx hdh A A dhdx x h dx

rearrange

2 2

3 3

2

3

2

3

cos

cos

oo f

o f

o f

dH S Sdx

dh Q A Q A S Sdx gA h gA x

Q AS Sdh gA x

Q AdxgA h

Determination of flow profile of steady flow

Case 1 : uniform channel

Assume uniform channel :

If we assume friction is negligible, Sf equals 0

The above eq. can be transformed into :

Definition of normal depth ho

Specific energy

Definition of critical depth hc

or

or Denominator =0numerator =0

Definition of critical slope ic

for a rectangular channel where physical shape and roughness n are known, foran arbitrary discharge Q, there exists a slope where the flow is normal flow(dh/dx=0) and the depth is equivalent to critical depth, this slope is called the“critical slope”

in the case wide rectangular channel , R=h

Classification of slope

wide rectangular channel, using Manning’s n

10/3

3

1 /1 /

oo

c

h hdh Sdx h h

Steep slope

Mild slope

Critical slope

steep slope mild slope

steep slopemild slope

METHOD OF CALCULATION

1) Standard Step Method - all kinds of channel : varying width (river), fixed width (prismatic channel)

For river : - the bottom bed of a river is often uneven, so it is not practical to measure water depth, h. So, we measure the total value of z+d from the datum that we set (here d=h cos theta). Theta is the angle between the bed and the datum line

By energy principle

2 2 2

2

2

2 2 2

2

where cos2

f

f f

f

V V Vz d z dz d d d d hg g g

Vd z d h S xg

Vd Z S x Z z d z hg

2 2

1 22 1

1 where 2 2 2

f f f fV VZ Z S x S S Sg g

V not U

2 2

2 12 2f

V VZ Z S xg g

Known (obtained from control section)

4) Make a guess for Z25) Calculate V2 using continuity equation

2) Calculate from Manning’s eq.

2 2 2 2

1 1 2 21 2 4/3 4/3

1 2

1 12 2

f f fV n V nS S SR R

1 2

1 1 2 2

Q QV A V A

Cross section data A2 is needed !

1) Standard Step Method – for non prismatic channel

Weakness !?When standard step method is applied on river,

a) With non-prismatic channels, the value of n, A and R vary and have to be found from cross-sectional data obtained from survey

b) The solution doesn’t give information on flow depth ! Only the elevation of water surface above a datum , which is the value of Z=z+dTherefore, R and A data for all sections

have to be provided from survey data

unknown

If guess is incorrect, re-guess

If guess is correct, LEFT=RIGHT

LEFT

RIGHT

1) Standard Step Method – for prismatic channel

2 2 2

2

2

2 2 2

2

where is called the specific energy2

where is the slope of the bed

Therefo

f

ff

f

f fo o

V V Vz d z dz d d d d hg g g

Vd z d h S xg

Vd z H S x H dg

dH dz dzS S S Sdx dx dx

2 1

2 1

re,

fo

fo

H H S S x

H H S S x

2 1 foH H S S x

Given data Calculate from Manning’s eq.

2 22 2

2 2 2 cos2 2V VH d hg g

1) Guess h22) We can calculate A23) Calculate V2 since V2=Q/A24) So, H2 can be estimated

Known from control section

1) Standard Step Method – for prismatic channel

2 2 2 2

1 1 2 21 2 4/3 4/3

1 2

1 12 2

f f fV n V nS S SR R

V2 from (3) is usedR2 can be calculated from (1)

RIGHTLEFT

If guess is in correct re- guess h2

If guess is correct LEFT=RIGHT

2) Direct Step Method – for prismatic channel only

21o fS Sdh

dx Fr

Prismatic channel

2) Direct Step Method – for prismatic channel only

2

2

11

o f

ave

o f ave

S Sdhdx Fr

Frx h

S S

1) Predetermine delta h

A Bh hhn

n= number of calculation point Between h1* and h2*

2) Terms here calculated using h(i) and h(i+1)

( 1) (i)h i h h

3) Location of h(i+1) can be determined

PRACTICE 1• PREDICT THE FLOW PROFILE FOR STEADY FLOW

USING STANDARD STEP METHOD

A prismatic, rectangular channel is 6m wide.The channel has bed slope of 1 in 800 and Manning’s n is 0.017 s m-1/3.

The channel terminates in a vertical drop so that the flow falls freely into a lower reservoir. The discharge in the channel is 35m3s-1

Assume the flow passes the critical depth at the drop. Calculate the elevation of the water surface in the channel using specific energy head (use standard step method).

And determine:a) The distance at which the water surface

is within 10mm of the normal depth.

Hint : use dx=100m, since theoretically the water surface approaches the normal depth at infinity.