7 2dh Equation

8/10/2019 7 2dh Equation

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Basic Equation 7Eka O. N.

Derivation 2-DH Depth Averaged

byEka Oktariyanto Nugroho

Derivation 2-DH Depth Averaged Page - 51




7.1. GENERAL CONSIDERATIONS

Basic assumptions to derivate the 2 Dimensional Horizontal Depth Averaged for shallow water problem are :

• Incompressible

• Turbulent time averaged• A ke assumption for depth averaging is that the flow in the vertical direction is small

• This implies that all terms in the !"direction #e nolds $%uation are small compared to thegravit and pressure terms& Thus the !"direction #e nolds $%uation reduces to

p g

z ρ ∂ = −∂

This implies that the pressure distribution over the vertical is h drostatic

• 'cetch (onditions

ig!re 7. 1 Illustration for depth averaged velocity distribution.

• (onsider the geoid to be defined at 0 z = ) the free surface *water"air interface+ at z η = ) and

the bottom *water"sediment interface+ at z h= −• Depth Averaged velocities are defined as:

h h

1 1u U udz v V vdz

h h

η η

− −= = = =+ η + η∫ ∫ &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

• The total water column height is defined as:

H h η = +

• .low rate over the vertical is defined as:

% x

h

q udz uH η

−= =∫ and y

h

q vdz vH η

−= =∫ % &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

• The (ontinuit e%uation is:

0u v w x y z

∂ ∂ ∂+ + =∂ ∂ ∂ ............................................................................................................................................................ */"2+

• The 0omentum e%uation is:

0 0 0 0

1 1 1 1 yx xx zxu u u u pu v wt x y z x x y z

τ τ τ ρ ρ ρ ρ

∂∂ ∂∂ ∂ ∂ ∂ ∂+ + + = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ......................... */"1+ or */" +





• Boundar conditions are:

1. Free surface condition

S !"y"z"t#$ % !"y"t#-z $ 0" &ada z $ %

dS0

dt=

'ydS ! z

0dt t ! t y t t

∂∂η ∂η ∂η∂ ∂= + + − =∂ ∂ ∂ ∂ ∂ ∂

u v ( 0t ! y

∂η ∂η ∂η+ + − =∂ ∂ ∂

z z z z

Dw u v

Dt t x yη η η η

η η η η = = ==

∂ ∂ ∂= = + +∂ ∂ ∂ ................................................................. )-*#

z u v w x y t η

η η η

=

∂ ∂ ∂− + − = ∂ ∂ ∂ ..................................................................................... )-+#

,. otto surface condition

' 3 z 4 5 z 3 4 ) at z 3 z 4

'*6) )z)t+3 "z 4 *6) )t+"z 3 4

(ithdS

0dt

= '

0 0 0z z z ydS ! z0

dt t ! t y t t∂ ∂ ∂ ∂∂ ∂= + + + =∂ ∂ ∂ ∂ ∂ ∂

0 0 00 0 0

z z zu !" y" z # v !" y" z # ( !" y" z # 0

t ! y∂ ∂ ∂+ + + =∂ ∂ ∂ at z $ z 0

The velocit in the bottom is: 7*"h+ 3 8*"h+ 3 9*"h+ 3 4) thenh

0t

∂ =∂( ) ( ) ( ) ( )

z h z h z h z h

D h h h hw u v

Dt t x y=− =− =−=−

− ∂ − ∂ − ∂ −= = + +∂ ∂ ∂ &&&&&&&&&&&&&&&&&&&&&&&&&&&&&

( ) ( )0

z h

h hu v w

x y =−

∂ − ∂ −+ − = ∂ ∂

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&





7.2. CONTIN"IT# DE$TH A%ERAGED E&"ATION

0u v w x y z

∂ ∂ ∂+ + =∂ ∂ ∂ ............................................................................................................................................................ */"2+

8erticall average

10

h

u v wdz

H x y z

η

−

∂ ∂ ∂+ + = ÷∂ ∂ ∂ ∫ 0ultipl ing through b H and evaluating the last integral:

( )

( )

( )

( )" " " "

" " " "

0 x y t x y t

h h x y t h x y t h

u v w u v wdz dz dz dz

x y z x y z

η η η η

− − − −

∂ ∂ ∂ ∂ ∂ ∂ + + = + + = ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ ∫ ∫

( )

( )

( )

( )

( ) ( )" " " "

" " " "0

x y t x y t

h x y t h x y t

u vdz dz w w h x y

η η

η − −

∂ ∂+ + − − =∂ ∂∫ ∫

7sing Leibnitz’s Rule :

( )

( )" "

" "

B x y t b

z A z B A x y t a

F A Bdz Fdz F F

x x x x= =∂ ∂ ∂ ∂= + −∂ ∂ ∂ ∂∫ ∫ ................................................................................................. *,",+

Hence e%& */&2+ become

z h z

h h

u hdz udz u u

x x x x

η η

η

η =− =

− −

∂ ∂ ∂ ∂= + −∂ ∂ ∂ ∂∫ ∫ ( )

z h z h h

hvdz vdz v v

y y y y

η η

η

η =− =

− −

∂ −∂ ∂ ∂= + −∂ ∂ ∂ ∂∫ ∫

z z hh

wdz w w

z

η

η = =−−

∂ = −∂∫

( ) ( )

( )

0h h z z h

h hudz vdz u v w u v w

x y x y x y

η η

η

η η

− − = = −

∂ − ∂ − ∂ ∂ ∂ ∂+ − + − + + − = ∂ ∂ ∂ ∂ ∂ ∂

∫ ∫ *,";+

9ith bottom and free surface condition) e% *," + and *,"/+

0h h

udz vdz t x y

η η η

− −

∂ ∂ ∂+ + =∂ ∂ ∂∫ ∫ *,"<+

Termsh

udzη

−∫ and

h

vdzη

−∫ in e%& *,"<+ are called depth averaged velocit ) U and V ) substitute b

e%uation *,&2+ hence

0 y x

h

qqu v wdz

x y z t x y

η η

−

∂ ∂∂ ∂ ∂ ∂+ + = + + = ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ............................................................................................................. *,"-4+





where H 3 h = >& Because h is contant) thenh

0t

∂ =∂ &

Depth Averaged (ontinuit $%uation :

%( ) ( )0

uH d vH H

t x dy

∂∂ + + =∂ ∂

%................................................................................................................................................ *,"--+

?r

%( ) ( )0

uH d vH

t x dyη ∂∂ + + =∂ ∂

%................................................................................................................................................. *,"-2+

7.'. (O(ENT"( DE$TH A%ERAGED E&"ATION

(onsider the @"direction #e nolds e%uation:

0 0 0 0

1 1 1 1 yx xx zxu u u u pu v wt x y z x x y z

τ τ τ ρ ρ ρ ρ ∂∂ ∂∂ ∂ ∂ ∂ ∂+ + + = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

.................................. */"1+ or */" +

// /ij i j i

j

uu u

x

τ υ

ρ ∂= −∂

0/ / /iij j i

j

uu u

xτ µ ρ ∂= −∂

0 / / xx

uu u

xτ µ ρ ∂= −∂

" 0 / / yx

uu v

yτ µ ρ ∂= −∂ " 0 / / zx

uu w

z τ µ ρ ∂= −∂

And p

g z

ρ ∂ = −∂) integrating this e%uation between the free surface at z η = and some level z

( )

( )" "

" s

p x y z z

p x y z

p g z η

ρ =

∂ = − ∂∫ ∫ where s p 3 pressure at the free surface

And assuming that densit is constant:

( ) s p p gz g ρ ρ η − = − −

s p p g gz ρ η ρ = + −1 1 s p p z

g g x x x x

η ρ ρ

∂∂ ∂ ∂− = − − +∂ ∂ ∂ ∂The surface pressure does not var spatiall :

1 p g

x xη

ρ ∂ ∂− = −∂ ∂

or 00

1 z p g g g gs

x x x xη η

ρ ∂∂ ∂ ∂− = − − = − +∂ ∂ ∂ ∂

9here s 43 channel bottom slopeDepth averaged e%uation form for the above e%uation is:

1 1 1 yx xx zxu u u uu v w g

t x y z x x y z

τ τ τ η

ρ ρ ρ

∂∂ ∂∂ ∂ ∂ ∂ ∂+ + + = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂





B adding u times the continuit e%uation to the above e%uation:

1 1 1 yx xx zxu u u u u v wu v w u u u g

t x y z x y z x x y z

τ τ τ η ρ ρ ρ

∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + + + + + = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂#e"arranging

( ) ( ),1 yx xx zx

uv uwu u g

t x y z x x y z

τ τ τ η ρ

∂ ∂ ∂ ∂ ∂∂ ∂ ∂+ + + = − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

ter. 1 ter. , ter. * ter. + ter. ter.

1 yx xx zx

h h h h h h

du uu uv uwdz dz dz dz g dz dz

dt x y z x x y z

η η η η η η τ σ τ η ρ − − − − − −

∂ ∂ ∂∂ ∂ ∂ ∂+ + + = − + + + ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ ∫ ∫ ∫ ∫ 142 43 14 2 43 14 2 43 142 43 142 43 1 4 4 4 44 2 4 4 4 4 43

.........*,"-1+

7sing Leibnitz’s Rule :

( )

( ) ( ) ( )" "

" "

x y t

z z hh x y t h

h f dz f dz f f

t t t t

η η

η

η = =−

− −

∂ ∂ −∂ ∂= + −∂ ∂ ∂ ∂∫ ∫

................................................................................. *,",+

9here,

" " " " " xx xy f u u uv η τ τ = and

z z hh h

g dz g g g

z

η η

η = =−− −

∂ = ∂ = −∂∫ ∫ 9here " zx g uw τ =

Ter) 1( ) ( )

z z hh h

d d hudz udz u u

t t dt dt

η η

η

η = =−

− −

−∂ ∂= − +∂ ∂∫ ∫

Ter) 2( ) ( ), , ,

z z hh h

d d huudz u z u u

x x dx dx

η η

η

η = =−− −

−∂ ∂= ∂ − +∂ ∂∫ ∫

Ter) '

( ) ( ) z z h

h h

d d huv dz uv z uv uv x x dx dx

η η

η η

= =−− −

−∂ ∂= ∂ − +∂ ∂∫ ∫ Ter) *

hh

z z h

uwdz uw

z

w w

η η

η

−−

= =−

∂ =∂= −

∫

Ter) +





( ) z z h

h h

h g dz g dz g g

x x x x

η η

η

η η η η η = =−

− −

∂ −∂ ∂ ∂− = − − +∂ ∂ ∂ ∂∫ ∫

Ter) ,

( ) ( )

1 1 1

1 1

yx xx zx xx yx

h h h

xx yx zx xx yx zx

z z h

dz dz dz x y z x y

h h

x y x y

η η η

η

τ σ τ σ τ ρ ρ ρ

η η σ τ τ σ τ τ

ρ ρ

− − −

= =−

∂ ∂ ∂ ∂ ∂+ + = + ÷∂ ∂ ∂ ∂ ∂ ∂ − ∂ − ∂ ∂− + − + + − ∂ ∂ ∂ ∂

∫ ∫ ∫

#ewrite $%uation ,&-1 becomes:

( ) ( ) ( )

( ) 1 1

1 1

h h h z

z h

xx yxh h h

xx yx zx xx

z

du uu uvdz dz dz u u v w

dt x y t x y

h h h

u v wt x y

h g dz g g dz dz

x x x x y

x y

η η η

η

η η η

η

η η η

η η η η σ τ

ρ ρ

η η σ τ τ σ

ρ ρ

− − − =

=−

− − −

=

∂ ∂ ∂ ∂ ∂+ + − + + − ÷∂ ∂ ∂ ∂ ∂ ∂ − ∂ − ∂ −

+ + + − ∂ ∂ ∂ ∂ −∂ ∂ ∂ ∂= − + − + +∂ ∂ ∂ ∂ ∂

∂ − ∂ ∂− + − + ∂ ∂

∫ ∫ ∫

∫ ∫ ∫ ( ) ( )

yx zx

z h

h h

x yτ τ

=−

∂ −+ − ∂ ∂

................................................. *,"- +

The boundar condition is done with e%uation ," and ,"/&

B performing a stress balance at the surface) it can be shown that s xτ =applied surface stress in

the 6 direction and parallel to the surface&

1 s x xx yx zx

z x y η

η η τ σ τ τ

ρ =

∂ ∂= − + − ∂ ∂ 'imilarl at the bottom:

( ) ( )1b x xx yx zx

z h

h h

x yτ σ τ τ

ρ =−

∂ − ∂ −= − + − ∂ ∂ 'ubstituting reduces the @"momentum e%uation to:

( ) 1 1

h h h

s b x x

xx yxh h h

du uu uvdz dz dz

dt x y

h g dz g g dz dz

x x x x y

η η η

η η η τ τ η η η η σ τ

ρ ρ ρ ρ

− − −

− − −

∂ ∂+ + =∂ ∂∂ −∂ ∂ ∂ ∂− + − + + + −∂ ∂ ∂ ∂ ∂

∫ ∫ ∫

∫ ∫ ∫ .................... *,"- +

9e define the depth averaged variable as:

° 1

h

dz H

η

α α −

≡ ∫ The #e nolds averaged %uantit is then defined as the sum of the depth averaged variable and thedeviation from the depth averaged variable

° 2α α α ≡ +Thus we define velocities in terms of the depth averaged %uantit and the deviation from the depthaveraged %uantit





The spatial averaging is applied as:%

h

udz Huη

−≡∫

h

vdz Hvη

−≡∫

.urthermore we let:

( ) %( ) ( )2" " " " " " " " "u x y z t u x y z t u x y z t = +( ) ( ) ( )2" " " " " " " " "v x y z t v x y z t v x y z t = +%

This implies that:

2 0h

udz η

−=∫ and 2 0

h

vdz η

−=∫

Hence

% %$ $( ),,,

h h

uudz u uu u z η η

− −= + + ∂∫ ∫

% % $ $,,,

h h h h

uudz u z u u z u z η η η η

− − − −= ∂ + ∂ + ∂∫ ∫ ∫ ∫

% $,,

h h

uudz H u u z η η

− −= + ∂∫ ∫

'imilarl

% % $ $( ) % $

h h h

uvdz uv uv uv uv dz uvH uvdz η η η

− − −= + + + = +∫ ∫ ∫ $ $ $% % %

h h

dz dz H η η

η η η − −

= =∫ ∫ 'ubstituting and re"arranging:

%( ) %( ) %( )

( ) $ $

,

,

s b yx xx x x

h h

H u Hu H uv

t x y

H h g g g u dz uv dz

x x x x y

η η τ η σ τ τ η η η

ρ ρ ρ ρ − −

∂∂ ∂+ + =∂ ∂ ∂

∂ ∂ ∂ ∂ ∂− + + + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∫ ∫

%

$

...... *,"-/+

$6panding the terms involving gravit

( ) ( ) H hh H g g g g H

x x x x x x

η η η η η η η η

∂ ∂ +∂ ∂ ∂ ∂− + + = − − + ∂ ∂ ∂ ∂ ∂ ∂ ................................................................ *,"-,+





( ) H h g g g gH

x x x x

η η η η η

∂ ∂ ∂ ∂− + + = −∂ ∂ ∂ ∂9e also note that the acceleration terms can be e6panded as:

%( ) % % Hu H u

u H t t t

∂ ∂ ∂= +∂ ∂ ∂

%( ) % ( ) % Hu h uu H

t t t

η ∂ ∂ + ∂= +∂ ∂ ∂%( ) %

% Hu uu H

t t t η ∂ ∂ ∂= +

∂ ∂ ∂%( ) %

%( ) % %

, Hu Hu u

u H u x x x

∂ ∂ ∂= +∂ ∂ ∂%( )

% ( ) % Huv H v u

u H v y y y

∂ ∂ ∂= +∂ ∂ ∂

% %

%

'ubstituting in the gravit and acceleration term re"arrangements:

%%

% %%

%( ) ( )

$ $,

s b yx xx x x

h h

Hu H vu u u H Hu H v u

t x x t x y

gH u dz uv dz x x y

η η

η

τ σ τ τ η ρ ρ ρ ρ − −

∂ ∂∂ ∂ ∂ ∂ + + + + + =∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂− + − + − + − ÷ ÷∂ ∂ ∂ ∫ ∫

%%

$

................................................. *,"-;+

It is clear that the depth averaged continuit e%uation is embedded in the previous e%uation and

therefore drops out& Dividing through b H will result in the depth averaged conservation of momentum e%uation in non"conservative form&

%%

% %$ $,1 1 1 1 s b

yx xx x x

h h

u u uu v g u dz uv dz

t x x x H x H y H H

η η τ σ τ τ η ρ ρ ρ ρ − −

∂ ∂ ∂ ∂ ∂ ∂+ + + = − + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ $%

or

%%

% %$ $,

0

1 1 1 1 s b yx xx x x

h h

u u uu v g gs u dz uv dz

t x x x H x H y H H


∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ $%

............................................................................................................................................................................................................. *,&-<+

% $ ,1 1 1 1 s b

xy yy y y

h h

v v vu v g uv dz v dz

t x x y H x H y H H


∂ ∂ ∂ ∂ ∂ ∂+ + + = − + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ % % %

$ $%

............................................................................................................................................................................................................. *,&24+





?therwise.or @ direction) e%& */"1 or /" +:

1 1 yx xx zxdu uu uv uw pdt x y z x x y z

τ σ τ ρ ρ

∂ ∂ ∂∂ ∂ ∂ ∂+ + + = − + + + ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ Depth averaged e%uation form for the above e%uation is:

1 1 1 1

h h h h

yx xx zx

h h h h

du uu uv uwdz dz dz dz

dt x y z

pdz dz dz dz

x x y z

η η η η

η η η η τ σ τ ρ ρ ρ ρ

− − − −

− − − −

∂ ∂ ∂+ + +∂ ∂ ∂∂∂ ∂∂= − + + +∂ ∂ ∂ ∂

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ...................................................... *,"2-+

with ( ) & g z= ρ η −

( )dp g g z dx x xη ρ ρ η ∂ ∂= + −∂ ∂ ........................................................................................................................................... *,"22+

'ubstitute e%& *,&22+ to e%& *,&2-+) obtain:

1 1 yx xx zx

h h h h h h


dt x y z x x y z

η η η η η η τ σ τ η ρ

ρ ρ − − − − − −

∂ ∂ ∂∂ ∂ ∂ ∂ + + + = − + + + ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ ∫ ∫ ∫ ∫ 1 yx xx zx

h h h h h h


dt x y z x x y z

η η η η η η τ σ τ η ρ − − − − − −

∂ ∂ ∂∂ ∂ ∂ ∂+ + + = − + + + ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ ∫ ∫ ∫ ∫

( )1 , * +

)

1

h h h h

term term term term

yx xx zx

h h h

term term term

du uu uv uwdz dz dz dz dt x y z

g g dz z dz dz

x x x y z

η η η η

η η η τ σ τ η ρ η

ρ ρ

− − − −

− − −

∂ ∂ ∂+ + +∂ ∂ ∂

∂ ∂ ∂∂ ∂= − − − + + + ÷∂ ∂ ∂ ∂ ∂

∫ ∫ ∫ ∫

∫ ∫ ∫

14 2 43 14 2 43 14 2 43 14 2 43

14 2 43 1 4 44 2 4 4 43 1 4 4 4 442 4 4 4 4 3 4

............................................... *,"21+

7sing eibniz #ule s:

Ter) 1

( ) ( )

( ) ( )

( ) ( ) ( )

( )( ) ( ) ( )

0h h

h

d h d du d dz udz u h udt dt dt dt

d d u u

dt dt d d

u h udt dt

η η

η

η η

η η

η η η

− −=

−

−= + − −

= −

= + −

∫ ∫ 1 442 4 43





Ter) 2

( ) ( ) ( ) ( )

( ) ( )( )

( ) ( ) ( )

, , ,

0

, , ,,

, ,

1- instance $

h h

xxh h

xx

d h d uu d dz u dz u h u

x dx dx dx

d d u dz u u dz

dx dx h u

d d h u u

dx dx

η η

η η

η η

η η β

η

η β η η

− −=

− −

−∂ = + − −∂

= +

= + −

∫ ∫

∫ ∫

1 44 2 4 43

Ter) '

( ) ( ) ( ) ( )

( ) ( )( )

( ) ( ) ( )

0

1 instance $

h h

yxh h

yx

d h d uv d dz uvdz uv h uv

y dy dy dy

d d uvdz uv uvdz

dy dy h uv

d d h uv uv

dy dy

η η

η η

η η

η η β

η

η β η η

− −=

− −

−∂ = + − −∂

= − +

= + −

∫ ∫

∫ ∫

1 44 2 4 43

Ter) *

( ) ( )0

hh

uwdz uw

z

uw uw h

η η

η

−−

∂ =∂

= − −=

∫

Ter) +

( )hh

g dz g x x

g h x

η η η η

η η

−−

∂ ∂− = −∂ ∂

∂= − +∂

∫

Ter) ,

( ) ( )

( )

,

,

1,

1,

h h

h

g g z dz z dz

x x

g z z

x

g h

x

η η

η

ρ ρ η η

ρ ρ

ρ η

ρ

ρ η

ρ

− −

−

∂ ∂− − = − −∂ ∂ ∂= − − = ÷ ÷∂

∂= − + ∂

∫ ∫





Ter) 7

( ) ( )

( ) ( )

1 1 1 1

1 1

1 1

yx yx xx zx xx zx

h h hh

yx xx

zx zx

dz x y z x y

h h x y

h

η η η η τ τ σ τ σ τ

ρ ρ ρ ρ

τ σ η η ρ ρ

τ η τ ρ ρ

− − −−

∂ ∂ ∂ ∂ ∂+ + = + + ÷∂ ∂ ∂ ∂ ∂

∂∂= + + +∂ ∂

+ − −

∫

#e"arranging e%uation *,&21+

( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) (

, , ,

,

1 1 1 1 1

,

xx

yx

yx xx zx zx

d d d d u h u h u u

dt dt dx dxd d

h uv uvdy dy

g g h h h h h

x x x y

η η η η β η η

η β η η

τ σ η ρ η η η η τ η τ

ρ ρ ρ ρ ρ

+ − + + − + + −

∂ ∂∂ ∂= − + − + + + + + + − − ∂ ∂ ∂ ∂ ............................................................................................................................................................................................................. *,"2 +

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

,

,1 1 1 1 1

,

xx yx

yx xx zx zx

d d d d d d u h u h u u u h uv u v

dt dt dx dx dy dy

g g h h h h h x x x y

η η η η η β η η β η η

τ σ η ρ η η η η τ η τ ρ ρ ρ ρ ρ

+ − + + − + + − ÷ ÷ ÷ ∂ ∂∂ ∂

= − + − + + + + + + − − ∂ ∂ ∂ ∂

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

,

,1 1 1 1 1

,

xx yx

yx xx zx zx

d d d d d d u h h u h uv u u v

dt dx dy dt dx dy

g g h h h h h

x x x y

η η η η β η β η η η η


ρ ρ ρ ρ ρ

+ + + + + − + + ∂ ∂∂ ∂= − + − + + + + + + − − ∂ ∂ ∂ ∂

3ith ( ) ( ) ( ) ( ) ( ) ( )d d du v

dt d! dy

η η η η + η + η

$0

( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

,

,1 1 1 1 1

,

xx yx

yx xx zx zx

d d d u h h u h uv

dt dx dy

g g h h h h h

x x x y

η β η β η


ρ ρ ρ ρ ρ

+ + + + + ∂ ∂∂ ∂= − + − + + + + + + − − ∂ ∂ ∂ ∂

( )hη+ 3H) h constant) h

0!

∂ =∂ hence( )h 4

! ! !

∂ η+∂η ∂= =∂ ∂ ∂&

Boussines% coefisien C 3- and 4

! !∂η ∂=∂ ∂

hence:





[ ] [ ]

( ) ( )

,

, 1 1 1 1,

yx xx zx zx

d d d uH u H uvH

dt dx dy

H g gH H H H h

x x x y

τ σ ρ τ η τ

ρ ρ ρ ρ ρ

+ + ∂ ∂∂ ∂= − − + + + − − ∂ ∂ ∂ ∂

............................ *,"2 +

( )z!1 τ ηρ is define to surface shear stress) this stress is caused b wind:

( )z! a ! s!563 3τ η ρ τ= =ρ ρ ρ

where:a 3 water densit

(E 3 $kman coefficient 3 4&42/9 6 3 wind velocit in @ direction

9 3 wind velocit in F direction

! y3 3 3= +

( )z!1

hτ −ρ is define to bottom shear stress) this stress is caused b roughness effect of bottom

channel) with (hez $%uation:

( )z! h b!,

g U U

5

−τ τ= =ρ ρ

$%& *,&2 + rewrite to:

[ ] [ ],

,,

61 1,

yx xx a x

d d d uH u H uvH dt dx dy

g U U C W W H g gH H H H

x x x y C

τ σ ρ ρ ρ ρ ρ ρ

+ + ∂ ∂∂ ∂= − − + + + − ∂ ∂ ∂ ∂

............................. *,"2/+

'imilarl for F direction&

[ ] [ ] ,

,,61 1

, xy yy a y

d d d vH vuH v H

dt dx dy

C W W g V V H g gH H H H y y x y C

τ σ ρ ρ ρ ρ ρ ρ

+ +

∂ ∂ ∂ ∂= − − + + + − ∂ ∂ ∂ ∂

............................. *,"2,+

3ith iji j

t ji hk!

"#!

"#uu δ−

∂∂

+∂

∂ ν=′′−

1277

Gi (here

=δ≠=δ=

δ4

-

$ ji$ ji

ij ................................................................. *,"2;+





,

,

,

,,

*,

,*

,,

*

xx

yy

zz

ukh u u u

xv

kh v v v y

wkh w w w

z

σ ρν

σ ρν

σ ρν

∂ ′ ′ ′= − = − = −∂∂ ′ ′ ′= − = − = −∂∂ ′ ′ ′= − = − = −∂

xy yx

xz zx

yz zy

u vu v v u

y x

u wu w w u

z x

v wv w w v

z y

τ τ ρν

τ τ ρν

τ τ ρν

∂ ∂ ′ ′ ′ ′= = + = − = − ÷∂ ∂ ∂ ∂ ′ ′ ′ ′= = + = − = − ÷∂ ∂ ∂ ∂ ′ ′ ′ ′= = + = − = − ÷∂ ∂

[ ] [ ] [ ]

,,

6,,

, *a x

d d d uH uuH uvH dt dx dy

g U U C W W H g u u v gH H H kH H

x x x x y y x C ρ ρ

ν ν ρ ρ

+ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ = − − + − + + + − ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∂

............................................................................................................................................................................................................. *,"2<+

[ ] [ ] [ ]

, ,*

s b x x

d d d uH uuH uvH

dt dx dy

H u u v gH H kH H x x x y y x

τ τ ν ν ρ ρ

+ +

∂ ∂ ∂ ∂ ∂ ∂ = − + − + + + − ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ∂

................................ *,"14+

[ ] [ ] [ ]

1 1

s b yx xx x x

d d d uH uuH uvH

dt dx dy

H gH H H

x x y

τ σ τ τ ρ ρ ρ ρ

+ + ∂ ∂∂= − + + + − ∂ ∂ ∂

................................................................................... *,"1-+

or

[ ] [ ] [ ]

0

, ,

*

s b x x

d d d uH uuH uvH

dt dx dy

H u u v gH Hgs H kH H

x x x y y xτ τ

ν ν ρ ρ

+ + ∂ ∂ ∂ ∂ ∂ ∂ = − + + − + + + − ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ∂

............... *,"12+

[ ] [ ] [ ]

0

1 1

s b yx xx x x

d d d uH uuH uvH

dt dx dy

H gH Hgs H H x x y


+ + ∂ ∂∂

= − + + + + − ∂ ∂ ∂

................................................................. *,"11+





or

0

1 , 1 1 1 ,

*

s b x x

d d d u uu uv

dt dx dy

H u u v g gs H kH H

x H x x H y y x H H τ τ

ν ν ρ ρ

+ + ∂ ∂ ∂ ∂ ∂ ∂ = − + + − + + + − ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ∂

. *,"1 +

0

1 1 1 1 s b yx xx x xd d d H

u uu uv g gsdt dx dy x x y H H


∂ ∂∂+ + = − + + + + − ∂ ∂ ∂ .......................... *,"1 +

'imilarl for F"direction:

1 1 1 1 s b

xy yy y yd d d H v uv vv g

dt dx dy y x y H H


∂ ∂ ∂+ + = − + + + − ∂ ∂ ∂ ........................................ *,"1/+





7.*. RES"(E O SHALLO ATER E&"ATION

#esume of the governing e%uation for 'hallow 9ater $%uation are:

%( ) ( )0

uH d vH

t x dyη ∂∂ + + =∂ ∂

%

%%

% %$ $,

0

1 1 1 1 s b yx xx x x

h h

u u uu v g gs u dz uv dz

t x x x H x H y H H


∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ $%

% $ ,1 1 1 1 s b

xy yy y y

h h

v v vu v g uv dz v dz t x x y H x H y H H

η η

τ σ τ τ η ρ ρ ρ ρ − −

∂ ∂ ∂ ∂ ∂ ∂+ + + = − + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ % % % $ $%

The 'hallow 9ater $%uations were established in -,, b aplace&The momentum conservation statements are %uite similar to the #e nolds e%uations with thefollowing e6ceptions:

• 8ariables are now in depth averaged %uantities&

• The !"dimension has been eliminated&

• There are convective inertia forces caused b the flow deviation from the depth averaged

velocities %"u v &

These e%uations have built into then 1 levels of averaging:

• Averaging over the molecular time space scale&

• Averaging over the turbulent time space scale&

• Averaging over the depth space scale&

• The latter two produce momentum transport terms that are intimatel related to the convectiveterms&

There are now three mechanisms of momentum transfer built into these e%uations:

"h

udz

x

η

υ −

∂∂∫ t pe terms are the 8iscous 'tresses and represent the averaged effect of

molecular motions& These terms are necessar since we are not directl simulating

momentum transfer via molecular level collisions&

" / /h

u u dz η

−∫ t pe terms are the Turbulent #e nolds 'tresses and represent the averaged

effect of momentum transfer due to turbulent fluctuations& These terms are necessarwhen using turbulent time averaged variables since we are not directl simulatingmomentum transfer via turbulent fluctuations&

" $$h

uudz η

−∫ t pe terms represent the spreading of momentum over the water column& This

process is known as momentum dispersion& These terms are necessar since we are nolonger directl simulating this process via the actual depth var ing velocit profiles& Thespread momentum laterall &

The shallow water e%uations greatl simplif flow computation in free surface water bodies&





• #educe the number of p&d&e& s from to 1&

• #educe the comple6it of the variables

%( ) ( ) ( )" " " " " " " "u x y t v x y t x y t η % instead of ( ) ( ) ( ) ( )" " " " " " " " " " " " " " "u x y z t v x y z t w x y z t p x y z t

• Built"in positioning of the free surface boundar which is t picall unknown when appl ing the

#e nolds e%uations&The shallow water e%uations include -4 additional unknowns as compared to the avier"'tokese%autions&

• ( )( )( )/ / " / / " / /u u u v v v ateral turbulent momentum diffusion

• $$& $& °" "uu uv vv$ $$ ateral momentum dispersion related to vertical velocit profile

• " s s x yτ τ Applied free surface stress

• "b b x yτ τ Applied bottom stress& It is related to the vertical velocit profile) momentum transport

through the water column) bottom roughness&

These -4 additional unknown re%uire that -4 constitutive relationships are provided in order toclose the s stem& Aver simple model for the combined lateral momentum diffusion *due toturbulence+ and dispersion *due to averaging out vertical velocit profile+ is:

$%( ),

xx xx

h

Huu dz

x x

η σ ρ −

∂ ∂ − = ÷∂ ∂ ∫ ( ), yy

yyh

H vv dz

x y

η σ ρ −

∂ ∂ − == ÷∂ ∂ ∫ %

$

$%( ) ( ) xy

xyh

Hu H vuv dz

x y x

η τ

ρ −

∂ ∂ ∂ − == + ÷∂ ∂ ∂ ∫

%$

" " xx yy xy are called the edd dispersion coefficients& This model assumes that the dispersion

process dominates the turbulent momentum diffusion process which dominates the molecular diffusion process& In a t pical gravit "driven open channel flow) the lateral momentum dispersionterms do not pla a maGor role in the momentum balance e%uations) the can be neglected&Bottom stress is closed b the empirical relationships:

%( ) %1 ,, ,

b x

f ! u v uτ ρ

= +%

%

( )1 ,, ,

b y

f ! u v v

τ

ρ = +%

9here:

f ! 3 friction factor

17 f DW ! f = Darc 9eisbach

, f

g !

!= (hez

,

1 * f

" g !

h= 0anning





SDS-2DH LES

Fro1. 8urbu9ent Shear F9o(s in Sha99o( O&en 5hanne9s (ith training Structures"

:issertation 5hen" Fei-;ong,. Organized 4orizonta9 Vortices and <atera9 Sedi ent 8rans&ort in 5o &ound

5hanne9 F9o(s" =keda S." Sano 8." Fuku oto >." and ?a(a ura ?.

0h uh vh

t x y

∂ ∂ ∂+ + =∂ ∂ ∂

, ,0

1 , 1,

* f

x t t

!u u u h u v uu v g gs f u u v h kh h

t x y x h h x x h y x yυ υ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + = − + − − + + − + + ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ , , 1 , 1

,*

f y t t

!v v v h v v uu v g f v u v h kh h

t x y y h h y y h x x yυ υ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ + = − − − + + − + + ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

, ,d x

a! f u u v

h= +

, ,d y

a! f v u v

h= +

1 1t t kh kv

k k

k k k k k h h p p

t x y h x x h y yυ υ

ε σ σ

∂ ∂ ∂ ∂ ∂ ∂ ∂+ + = + + + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ ,

t

k ! µ υ

ε =

* ,* + k

!# µ ε =

# hα =, ,,

, ,kh t

u v u v p

x y y xυ

∂ ∂ ∂ ∂ = + + + ÷ ÷ ÷∂ ∂ ∂ ∂

( ) * ,, ,

, f d

kv

! a! p u v

h = + + ÷

1. 3hat <ES,. Sche e for Nu erica9*. E!&9icit or = &9icit+. oundary 5ondition

. Fortran or other

. 5onvergent criteria





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7 2dh Equation