1

Numerical Hydraulics Open channel flow 2

Wolfgang Kinzelbach with

Marc Wolf and

Cornel Beffa

2

Approximations

• Kinematic wave

• Diffusive wave

• Dynamic wave (Full equations)

0

S R

v v hv g g I I

t x xh h vv h

t x x

Example:Rectangular channel

3

Approximations

1. Approximation: Kinematic wave

0S R

v v hv g g I I

t x x

2. Approximation: Diffusive wave

Complete solution: Dynamic wave

0S R

hg g I Ix

0S Rg I I

In the different approximations different terms in the equation of motion are neglected against the term gIS:

4

Kinematic wave

• Normal flow depth. Energy slope is equal to channel bottom slope. Therefore Q is only a function of water depth. E.g. using the Strickler/Manning equation:

• Inserting into the continuity equation yields

• This is the form of a wave equation (see pressure surge) with wave velocity w = v+c

2/3 1/ 2 ( )str hy SQ Ak R I Q h

Q dQ h

x dh x

0h hw

t x

dQ dhw

b

Instead of using Q=Q(h) the equation can be derived using v=v(h)

0`

t

A

x

Q

5

Kinematic wave

• With the Strickler/Manning equation we get:

• For a broad channel the hydraulic radius is approximately equal to the water depth. The wave velocity then becomes

2/31/ 2

2str S

hbQ hbk I

h b

2/3 1/3 21/ 2 1/ 2

2

2

2 3 2 ( 2 )str S str S

dQ dh bh bh bw k I k hI

b b h b h b h

and

1/2 2/35 5 2

3 3 3str Sw k I h v or c v

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Kinematic wave• The wave velocity is not constant as v is a

function of water depth h. • Varying velocities for different water depth

lead to self-sharpening of wave front• Pressure propagates faster than the average

flow.• Advantage of approximation: PDE of first

order, only one upstream boundary condition required.

• Disadvantage of approximation: Not applicable for bottom slope 0. No backwater feasible as there is no downstream boundary condition.

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Diffusive wave

• Now Q is not only a function of h but also of h/ x.

• Insertion into the continuity equation

yields:

S R

hI I

x

with IR = IR(Q/A) from Strickler or Darcy-Weisbach

10

h Q

t b x

2

2

1 ( , / ) 1 ( , / )0

( / )

h Q h h x h Q h h x h

t b h x b h x x

8

Diffusive wave

• This equation has the form of an advection-diffusion equation with a wave velocity w=v+c and a diffusion coefficient D:

2

20

h h hw D

t x x

dQ dhw

b

( / )dQ d h xD

b

with

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Diffusive wave

• Using the Strickler/Manning equation and assuming a broad rectangular channel (h = Rhy) one obtains:

2

20

h h hw D

t x x

5

3w v

3

10 ( / )S

c hD

I h x

5/3 /str SQ vhb k bh I h x 2

2

5

3 2( / )S

Q h vbh hbv

x x I h x x

and

Insertion into the continuity equation yields

with and

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Diffusive wave• D is always positive, as the energy slope

is always positive in flow direction.

• The wave moves downstream and flattens out diffusively. A lower boundary condition is necessary because of the second derivative. This allows the implementation of a backwater effect.

/R SI I h x

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St. Venant equation as wave equation

0

S R

v v hv g g I I

t x xh h vv h

t x x

Linear combinations: Multiply second equation with ± and add to first equation

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St. Venant equation as wave equation

0S R

v v v h h hv h v g g I I

t x x t x x

Write derivatives of h and v as total derivatives alonga characteristic line:

( ) ( ) 0S R

v v h g hv h v g I I

t x t x

Choosingg

h the two characteristics have the same relative

wave velocity c (with respect to average water velocity v).

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St. Venant equation as wave equation

( ) ( ) 0S R

v v g h hv gh v gh g I I

t x t xgh

and the relative wave velocity for shallow water waves is c gh

The characteristics are therefore:

dxw v gh v c

dt

In contrast to the surge in pipes, v cannot be neglected in comparison to c!

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St. Venant equation as wave equation: alternative view

Comparison with the pipe flow case shows the equivalence

c gh

using h=p/g:

Continuity channel flow Continuity pipe flow

0

x

vh

x

hv

t

h0

'

1

x

v

x

pv

t

p

E

01

x

v

x

pv

t

p

gh

'

11

Egh

and with

'E

c