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Basic Equation 7Eka O. N.

Derivation 2-DH Depth Averaged

byEka Oktariyanto Nugroho

Derivation 2-DH Depth Averaged Page - 51

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Basic Equation 7Eka O. N.

7.1. GENERAL CONSIDERATIONS

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

β€’ Incompressible

β€’ Turbulent time averagedβ€’ A ke assumption for depth averaging is that the flow in the vertical direction is small

β€’ 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 βˆ‚ = βˆ’βˆ‚

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

β€’ 'cetch (onditions

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

β€’ (onsider the geoid to be defined at 0 z = ) the free surface *water"air interface+ at z Ξ· = ) and

the bottom *water"sediment interface+ at z h= βˆ’β€’ Depth Averaged velocities are defined as:

h h

1 1u U udz v V vdz

h h

Ξ· Ξ·

βˆ’ βˆ’= = = =+ Ξ· + η∫ ∫ &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

β€’ The total water column height is defined as:

H h Ξ· = +

β€’ .low rate over the vertical is defined as:

% x

h

q udz uH Ξ·

βˆ’= =∫ and y

h

q vdz vH Ξ·

βˆ’= =∫ % &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

β€’ The (ontinuit e%uation is:

0u v w x y z

βˆ‚ βˆ‚ βˆ‚+ + =βˆ‚ βˆ‚ βˆ‚ ............................................................................................................................................................ */"2+

β€’ 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

Ο„ Ο„ Ο„ ρ ρ ρ ρ

βˆ‚βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + +βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ......................... */"1+ or */" +

Derivation 2-DH Depth Averaged Page - 52

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Basic Equation 7Eka O. N.

β€’ 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

βˆ‚βˆ‚Ξ· βˆ‚Ξ· βˆ‚Ξ·βˆ‚ βˆ‚= + + βˆ’ =βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

u v ( 0t ! y

βˆ‚Ξ· βˆ‚Ξ· βˆ‚Ξ·+ + βˆ’ =βˆ‚ βˆ‚ βˆ‚

z z z z

Dw u v

Dt t x yΞ· Ξ· Ξ· Ξ·

Ξ· Ξ· Ξ· Ξ· = = ==

βˆ‚ βˆ‚ βˆ‚= = + +βˆ‚ βˆ‚ βˆ‚ ................................................................. )-*#

z u v w x y t Ξ·

Ξ· Ξ· Ξ·

=

βˆ‚ βˆ‚ βˆ‚βˆ’ + βˆ’ = βˆ‚ βˆ‚ βˆ‚ ..................................................................................... )-+#

,. 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 tβˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ‚ βˆ‚= + + + =βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

0 0 00 0 0

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

t ! yβˆ‚ βˆ‚ βˆ‚+ + + =βˆ‚ βˆ‚ βˆ‚ at z $ z 0

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

0t

βˆ‚ =βˆ‚( ) ( ) ( ) ( )

z h z h z h z h

D h h h hw u v

Dt t x y=βˆ’ =βˆ’ =βˆ’=βˆ’

βˆ’ βˆ‚ βˆ’ βˆ‚ βˆ’ βˆ‚ βˆ’= = + +βˆ‚ βˆ‚ βˆ‚ &&&&&&&&&&&&&&&&&&&&&&&&&&&&&

( ) ( )0

z h

h hu v w

x y =βˆ’

βˆ‚ βˆ’ βˆ‚ βˆ’+ βˆ’ = βˆ‚ βˆ‚

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

Derivation 2-DH Depth Averaged Page - 53

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Basic Equation 7Eka O. N.

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

0u v w x y z

βˆ‚ βˆ‚ βˆ‚+ + =βˆ‚ βˆ‚ βˆ‚ ............................................................................................................................................................ */"2+

8erticall average

10

h

u v wdz

H x y z

Ξ·

βˆ’

βˆ‚ βˆ‚ βˆ‚+ + = Γ·βˆ‚ βˆ‚ βˆ‚ ∫ 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

Ξ· Ξ· Ξ· Ξ·

βˆ’ βˆ’ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ + + = + + = Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ ∫ ∫

( )

( )

( )

( )

( ) ( )" " " "

" " " "0

x y t x y t

h x y t h x y t

u vdz dz w w h x y

Ξ· Ξ·

Ξ· βˆ’ βˆ’

βˆ‚ βˆ‚+ + βˆ’ βˆ’ =βˆ‚ βˆ‚βˆ« ∫

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= =βˆ‚ βˆ‚ βˆ‚ βˆ‚= + βˆ’βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ« ∫ ................................................................................................. *,",+

Hence e%& */&2+ become

z h z

h h

u hdz udz u u

x x x x

Ξ· Ξ·

Ξ·

Ξ· =βˆ’ =

βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚= + βˆ’βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ« ∫ ( )

z h z h h

hvdz vdz v v

y y y y

Ξ· Ξ·

Ξ·

Ξ· =βˆ’ =

βˆ’ βˆ’

βˆ‚ βˆ’βˆ‚ βˆ‚ βˆ‚= + βˆ’βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ« ∫

z z hh

wdz w w

z

Ξ·

Ξ· = =βˆ’βˆ’

βˆ‚ = βˆ’βˆ‚βˆ«

( ) ( )

( )

0h h z z h

h hudz vdz u v w u v w

x y x y x y

Ξ· Ξ·

Ξ·

Ξ· Ξ·

βˆ’ βˆ’ = = βˆ’

βˆ‚ βˆ’ βˆ‚ βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ βˆ’ + βˆ’ + + βˆ’ = βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

∫ ∫ *,";+

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

0h h

udz vdz t x y

Ξ· Ξ· Ξ·

βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚+ + =βˆ‚ βˆ‚ βˆ‚βˆ« ∫ *,"<+

Termsh

udzΞ·

βˆ’βˆ« and

h

vdzΞ·

βˆ’βˆ« 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

Ξ· Ξ·

βˆ’

βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + = + + = Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ............................................................................................................. *,"-4+

Derivation 2-DH Depth Averaged Page - 54

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Basic Equation 7Eka O. N.

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

0t

βˆ‚ =βˆ‚ &

Depth Averaged (ontinuit $%uation :

%( ) ( )0

uH d vH H

t x dy

βˆ‚βˆ‚ + + =βˆ‚ βˆ‚

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

?r

%( ) ( )0

uH d vH

t x dyΞ· βˆ‚βˆ‚ + + =βˆ‚ βˆ‚

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

Ο„ Ο„ Ο„ ρ ρ ρ ρ βˆ‚βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + +βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

// /ij i j i

j

uu u

x

Ο„ Ο…

ρ βˆ‚= βˆ’βˆ‚

0/ / /iij j i

j

uu u

xΟ„ Β΅ ρ βˆ‚= βˆ’βˆ‚

0 / / xx

uu u

xΟ„ Β΅ ρ βˆ‚= βˆ’βˆ‚

" 0 / / yx

uu v

yΟ„ Β΅ ρ βˆ‚= βˆ’βˆ‚ " 0 / / zx

uu w

z Ο„ Β΅ ρ βˆ‚= βˆ’βˆ‚

And p

g z

ρ βˆ‚ = βˆ’βˆ‚) integrating this e%uation between the free surface at z Ξ· = and some level z

( )

( )" "

" s

p x y z z

p x y z

p g z Ξ·

ρ =

βˆ‚ = βˆ’ βˆ‚βˆ« ∫ where s p 3 pressure at the free surface

And assuming that densit is constant:

( ) s p p gz g ρ ρ Ξ· βˆ’ = βˆ’ βˆ’

s p p g gz ρ Ξ· ρ = + βˆ’1 1 s p p z

g g x x x x

η ρ ρ

βˆ‚βˆ‚ βˆ‚ βˆ‚βˆ’ = βˆ’ βˆ’ +βˆ‚ βˆ‚ βˆ‚ βˆ‚The surface pressure does not var spatiall :

1 p g

x xΞ·

ρ βˆ‚ βˆ‚βˆ’ = βˆ’βˆ‚ βˆ‚

or 00

1 z p g g g gs

x x x xΞ· Ξ·

ρ βˆ‚βˆ‚ βˆ‚ βˆ‚βˆ’ = βˆ’ βˆ’ = βˆ’ +βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

Ο„ Ο„ Ο„ Ξ·

ρ ρ ρ

βˆ‚βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + +βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

Derivation 2-DH Depth Averaged Page - 55

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Basic Equation 7Eka O. N.

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

Ο„ Ο„ Ο„ Ξ· ρ ρ ρ

βˆ‚βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + + + + = βˆ’ + + +βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚#e"arranging

( ) ( ),1 yx xx zx

uv uwu u g

t x y z x x y z

Ο„ Ο„ Ο„ Ξ· ρ

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + + βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

Ξ· Ξ· Ξ· Ξ· Ξ· Ξ· Ο„ Οƒ Ο„ Ξ· ρ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + + Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ ∫ ∫ ∫ ∫ 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

Ξ· Ξ·

Ξ·

Ξ· = =βˆ’

βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ’βˆ‚ βˆ‚= + βˆ’βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ« ∫

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

9here,

" " " " " xx xy f u u uv Ξ· Ο„ Ο„ = and

z z hh h

g dz g g g

z

Ξ· Ξ·

Ξ· = =βˆ’βˆ’ βˆ’

βˆ‚ = βˆ‚ = βˆ’βˆ‚βˆ« ∫ 9here " zx g uw Ο„ =

Ter) 1( ) ( )

z z hh h

d d hudz udz u u

t t dt dt

Ξ· Ξ·

Ξ·

Ξ· = =βˆ’

βˆ’ βˆ’

βˆ’βˆ‚ βˆ‚= βˆ’ +βˆ‚ βˆ‚βˆ« ∫

Ter) 2( ) ( ), , ,

z z hh h

d d huudz u z u u

x x dx dx

Ξ· Ξ·

Ξ·

Ξ· = =βˆ’βˆ’ βˆ’

βˆ’βˆ‚ βˆ‚= βˆ‚ βˆ’ +βˆ‚ βˆ‚βˆ« ∫

Ter) '

( ) ( ) z z h

h h

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

Ξ· Ξ·

Ξ· Ξ·

= =βˆ’βˆ’ βˆ’

βˆ’βˆ‚ βˆ‚= βˆ‚ βˆ’ +βˆ‚ βˆ‚βˆ« ∫ Ter) *

hh

z z h

uwdz uw

z

w w

Ξ· Ξ·

Ξ·

βˆ’βˆ’

= =βˆ’

βˆ‚ =βˆ‚= βˆ’

∫

Ter) +

Derivation 2-DH Depth Averaged Page - 56

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Basic Equation 7Eka O. N.

( ) z z h

h h

h g dz g dz g g

x x x x

Ξ· Ξ·

Ξ·

Ξ· Ξ· Ξ· Ξ· Ξ· = =βˆ’

βˆ’ βˆ’

βˆ‚ βˆ’βˆ‚ βˆ‚ βˆ‚βˆ’ = βˆ’ βˆ’ +βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ« ∫

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

Ξ· Ξ· Ξ·

Ξ·

Ο„ Οƒ Ο„ Οƒ Ο„ ρ ρ ρ

Ξ· Ξ· Οƒ Ο„ Ο„ Οƒ Ο„ Ο„

ρ ρ

βˆ’ βˆ’ βˆ’

= =βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + = + Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ’ βˆ‚ βˆ’ βˆ‚ βˆ‚βˆ’ + βˆ’ + + βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚

∫ ∫ ∫

#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

Ξ· Ξ· Ξ·

Ξ·

Ξ· Ξ· Ξ·

Ξ·

Ξ· Ξ· Ξ·

Ξ· Ξ· Ξ· Ξ· Οƒ Ο„

ρ ρ

Ξ· Ξ· Οƒ Ο„ Ο„ Οƒ

ρ ρ

βˆ’ βˆ’ βˆ’ =

=βˆ’

βˆ’ βˆ’ βˆ’

=

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + βˆ’ + + βˆ’ Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ’ βˆ‚ βˆ’ βˆ‚ βˆ’

+ + + βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ’βˆ‚ βˆ‚ βˆ‚ βˆ‚= βˆ’ + βˆ’ + +βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

βˆ‚ βˆ’ βˆ‚ βˆ‚βˆ’ + βˆ’ + βˆ‚ βˆ‚

∫ ∫ ∫

∫ ∫ ∫ ( ) ( )

yx zx

z h

h h

x yΟ„ Ο„

=βˆ’

βˆ‚ βˆ’+ βˆ’ βˆ‚ βˆ‚

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

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

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

the 6 direction and parallel to the surface&

1 s x xx yx zx

z x y Ξ·

Ξ· Ξ· Ο„ Οƒ Ο„ Ο„

ρ =

βˆ‚ βˆ‚= βˆ’ + βˆ’ βˆ‚ βˆ‚ 'imilarl at the bottom:

( ) ( )1b x xx yx zx

z h

h h

x yΟ„ Οƒ Ο„ Ο„

ρ =βˆ’

βˆ‚ βˆ’ βˆ‚ βˆ’= βˆ’ + βˆ’ βˆ‚ βˆ‚ '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

Ξ· Ξ· Ξ·

Ξ· Ξ· Ξ· Ο„ Ο„ Ξ· Ξ· Ξ· Ξ· Οƒ Ο„

ρ ρ ρ ρ

βˆ’ βˆ’ βˆ’

βˆ’ βˆ’ βˆ’

βˆ‚ βˆ‚+ + =βˆ‚ βˆ‚βˆ‚ βˆ’βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ’ + βˆ’ + + + βˆ’βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

∫ ∫ ∫

∫ ∫ ∫ .................... *,"- +

9e define the depth averaged variable as:

Β° 1

h

dz H

Ξ·

Ξ± Ξ± βˆ’

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

Β° 2Ξ± Ξ± Ξ± ≑ +Thus we define velocities in terms of the depth averaged %uantit and the deviation from the depthaveraged %uantit

Derivation 2-DH Depth Averaged Page - 57

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Basic Equation 7Eka O. N.

The spatial averaging is applied as:%

h

udz HuΞ·

βˆ’β‰‘βˆ«

h

vdz HvΞ·

βˆ’β‰‘βˆ«

.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 Ξ·

βˆ’=∫ and 2 0

h

vdz Ξ·

βˆ’=∫

Hence

% %$ $( ),,,

h h

uudz u uu u z Ξ· Ξ·

βˆ’ βˆ’= + + βˆ‚βˆ« ∫

% % $ $,,,

h h h h

uudz u z u u z u z Ξ· Ξ· Ξ· Ξ·

βˆ’ βˆ’ βˆ’ βˆ’= βˆ‚ + βˆ‚ + βˆ‚βˆ« ∫ ∫ ∫

% $,,

h h

uudz H u u z Ξ· Ξ·

βˆ’ βˆ’= + βˆ‚βˆ« ∫

'imilarl

% % $ $( ) % $

h h h

uvdz uv uv uv uv dz uvH uvdz Ξ· Ξ· Ξ·

βˆ’ βˆ’ βˆ’= + + + = +∫ ∫ ∫ $ $ $% % %

h h

dz dz H Ξ· Ξ·

Ξ· Ξ· Ξ· βˆ’ βˆ’

= =∫ ∫ '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

Ξ· Ξ· Ο„ Ξ· Οƒ Ο„ Ο„ Ξ· Ξ· Ξ·

ρ ρ ρ ρ βˆ’ βˆ’

βˆ‚βˆ‚ βˆ‚+ + =βˆ‚ βˆ‚ βˆ‚

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ’ + + + βˆ’ + βˆ’ + βˆ’ Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫

%

$

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

$6panding the terms involving gravit

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

x x x x x x

Ξ· Ξ· Ξ· Ξ· Ξ· Ξ· Ξ· Ξ·

βˆ‚ βˆ‚ +βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ’ + + = βˆ’ βˆ’ + βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ................................................................ *,"-,+

Derivation 2-DH Depth Averaged Page - 58

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Basic Equation 7Eka O. N.

( ) H h g g g gH

x x x x

Ξ· Ξ· Ξ· Ξ· Ξ·

βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ’ + + = βˆ’βˆ‚ βˆ‚ βˆ‚ βˆ‚9e also note that the acceleration terms can be e6panded as:

%( ) % % Hu H u

u H t t t

βˆ‚ βˆ‚ βˆ‚= +βˆ‚ βˆ‚ βˆ‚

%( ) % ( ) % Hu h uu H

t t t

Ξ· βˆ‚ βˆ‚ + βˆ‚= +βˆ‚ βˆ‚ βˆ‚%( ) %

% Hu uu H

t t t Ξ· βˆ‚ βˆ‚ βˆ‚= +

βˆ‚ βˆ‚ βˆ‚%( ) %

%( ) % %

, Hu Hu u

u H u x x x

βˆ‚ βˆ‚ βˆ‚= +βˆ‚ βˆ‚ βˆ‚%( )

% ( ) % Huv H v u

u H v y y y

βˆ‚ βˆ‚ βˆ‚= +βˆ‚ βˆ‚ βˆ‚

% %

%

'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

Ξ· Ξ·

Ξ·

Ο„ Οƒ Ο„ Ο„ Ξ· ρ ρ ρ ρ βˆ’ βˆ’

βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚ + + + + + =βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚βˆ’ + βˆ’ + βˆ’ + βˆ’ Γ· Γ·βˆ‚ βˆ‚ βˆ‚ ∫ ∫

%%

$

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

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

Ξ· Ξ· Ο„ Οƒ Ο„ Ο„ Ξ· ρ ρ ρ ρ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + βˆ’ + βˆ’ + βˆ’ Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ $%

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

Ξ· Ξ· Ο„ Οƒ Ο„ Ο„ Ξ· ρ ρ ρ ρ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + βˆ’ + βˆ’ + βˆ’ Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ $%

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

% $ ,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

Ξ· Ξ· Ο„ Οƒ Ο„ Ο„ Ξ· ρ ρ ρ ρ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + βˆ’ + βˆ’ + βˆ’ Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ % % %

$ $%

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

Derivation 2-DH Depth Averaged Page - 59

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Basic Equation 7Eka O. N.

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

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

Ο„ Οƒ Ο„ ρ ρ

βˆ‚ βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + + Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ 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

Ξ· Ξ· Ξ· Ξ·

Ξ· Ξ· Ξ· Ξ· Ο„ Οƒ Ο„ ρ ρ ρ ρ

βˆ’ βˆ’ βˆ’ βˆ’

βˆ’ βˆ’ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚+ + +βˆ‚ βˆ‚ βˆ‚βˆ‚βˆ‚ βˆ‚βˆ‚= βˆ’ + + +βˆ‚ βˆ‚ βˆ‚ βˆ‚

∫ ∫ ∫ ∫

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

with ( ) & g z= ρ Ξ· βˆ’

( )dp g g z dx x xΞ· ρ ρ Ξ· βˆ‚ βˆ‚= + βˆ’βˆ‚ βˆ‚ ........................................................................................................................................... *,"22+

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

1 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

Ξ· Ξ· Ξ· Ξ· Ξ· Ξ· Ο„ Οƒ Ο„ Ξ· ρ

ρ ρ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚ + + + = βˆ’ + + + Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ ∫ ∫ ∫ ∫ 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

Ξ· Ξ· Ξ· Ξ· Ξ· Ξ· Ο„ Οƒ Ο„ Ξ· ρ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + + Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ ∫ ∫ ∫ ∫

( )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

Ξ· Ξ· Ξ· Ξ·

Ξ· Ξ· Ξ· Ο„ Οƒ Ο„ Ξ· ρ Ξ·

ρ ρ

βˆ’ βˆ’ βˆ’ βˆ’

βˆ’ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚+ + +βˆ‚ βˆ‚ βˆ‚

βˆ‚ βˆ‚ βˆ‚βˆ‚ βˆ‚= βˆ’ βˆ’ βˆ’ + + + Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

∫ ∫ ∫ ∫

∫ ∫ ∫

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

Ξ· Ξ·

Ξ·

Ξ· Ξ·

Ξ· Ξ·

Ξ· Ξ· Ξ·

βˆ’ βˆ’=

βˆ’

βˆ’= + βˆ’ βˆ’

= βˆ’

= + βˆ’

∫ ∫ 1 442 4 43

Derivation 2-DH Depth Averaged Page - 60

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Basic Equation 7Eka O. N.

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

Ξ· Ξ·

Ξ· Ξ·

Ξ· Ξ·

Ξ· Ξ· Ξ²

Ξ·

Ξ· Ξ² Ξ· Ξ·

βˆ’ βˆ’=

βˆ’ βˆ’

βˆ’βˆ‚ = + βˆ’ βˆ’βˆ‚

= +

= + βˆ’

∫ ∫

∫ ∫

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

Ξ· Ξ·

Ξ· Ξ·

Ξ· Ξ·

Ξ· Ξ· Ξ²

Ξ·

Ξ· Ξ² Ξ· Ξ·

βˆ’ βˆ’=

βˆ’ βˆ’

βˆ’βˆ‚ = + βˆ’ βˆ’βˆ‚

= βˆ’ +

= + βˆ’

∫ ∫

∫ ∫

1 44 2 4 43

Ter) *

( ) ( )0

hh

uwdz uw

z

uw uw h

Ξ· Ξ·

Ξ·

βˆ’βˆ’

βˆ‚ =βˆ‚

= βˆ’ βˆ’=

∫

Ter) +

( )hh

g dz g x x

g h x

Ξ· Ξ· Ξ· Ξ·

Ξ· Ξ·

βˆ’βˆ’

βˆ‚ βˆ‚βˆ’ = βˆ’βˆ‚ βˆ‚

βˆ‚= βˆ’ +βˆ‚

∫

Ter) ,

( ) ( )

( )

,

,

1,

1,

h h

h

g g z dz z dz

x x

g z z

x

g h

x

Ξ· Ξ·

Ξ·

ρ ρ η η

ρ ρ

ρ η

ρ

ρ η

ρ

βˆ’ βˆ’

βˆ’

βˆ‚ βˆ‚βˆ’ βˆ’ = βˆ’ βˆ’βˆ‚ βˆ‚ βˆ‚= βˆ’ βˆ’ = Γ· Γ·βˆ‚

βˆ‚= βˆ’ + βˆ‚

∫ ∫

Derivation 2-DH Depth Averaged Page - 61

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Basic Equation 7Eka O. N.

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

Ξ· Ξ· Ξ· Ξ· Ο„ Ο„ Οƒ Ο„ Οƒ Ο„

ρ ρ ρ ρ

Ο„ Οƒ Ξ· Ξ· ρ ρ

Ο„ Ξ· Ο„ ρ ρ

βˆ’ βˆ’ βˆ’βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + = + + Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

βˆ‚βˆ‚= + + +βˆ‚ βˆ‚

+ βˆ’ βˆ’

∫

#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

Ξ· Ξ· Ξ· Ξ· Ξ² Ξ· Ξ·

Ξ· Ξ² Ξ· Ξ·

Ο„ Οƒ Ξ· ρ Ξ· Ξ· Ξ· Ξ· Ο„ Ξ· Ο„

ρ ρ ρ ρ ρ

+ βˆ’ + + βˆ’ + + βˆ’

βˆ‚ βˆ‚βˆ‚ βˆ‚= βˆ’ + βˆ’ + + + + + + βˆ’ βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ............................................................................................................................................................................................................. *,"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

Ξ· Ξ· Ξ· Ξ· Ξ· Ξ² Ξ· Ξ· Ξ² Ξ· Ξ·

Ο„ Οƒ Ξ· ρ Ξ· Ξ· Ξ· Ξ· Ο„ Ξ· Ο„ ρ ρ ρ ρ ρ

+ βˆ’ + + βˆ’ + + βˆ’ Γ· Γ· Γ· βˆ‚ βˆ‚βˆ‚ βˆ‚

= βˆ’ + βˆ’ + + + + + + βˆ’ βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

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

,

,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

Ξ· Ξ· Ξ· Ξ· Ξ² Ξ· Ξ² Ξ· Ξ· Ξ· Ξ·

Ο„ Οƒ Ξ· ρ Ξ· Ξ· Ξ· Ξ· Ο„ Ξ· Ο„

ρ ρ ρ ρ ρ

+ + + + + βˆ’ + + βˆ‚ βˆ‚βˆ‚ βˆ‚= βˆ’ + βˆ’ + + + + + + βˆ’ βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

dt d! dy

Ξ· Ξ· Ξ· Ξ· + Ξ· + Ξ·

$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

Ξ· Ξ² Ξ· Ξ² Ξ·

Ο„ Οƒ Ξ· ρ Ξ· Ξ· Ξ· Ξ· Ο„ Ξ· Ο„

ρ ρ ρ ρ ρ

+ + + + + βˆ‚ βˆ‚βˆ‚ βˆ‚= βˆ’ + βˆ’ + + + + + + βˆ’ βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚

( )hΞ·+ 3H) h constant) h

0!

βˆ‚ =βˆ‚ hence( )h 4

! ! !

βˆ‚ Ξ·+βˆ‚Ξ· βˆ‚= =βˆ‚ βˆ‚ βˆ‚&

Boussines% coefisien C 3- and 4

! !βˆ‚Ξ· βˆ‚=βˆ‚ βˆ‚

hence:

Derivation 2-DH Depth Averaged Page - 62

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Basic Equation 7Eka O. N.

[ ] [ ]

( ) ( )

,

, 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

Ο„ Οƒ ρ Ο„ Ξ· Ο„

ρ ρ ρ ρ ρ

+ + βˆ‚ βˆ‚βˆ‚ βˆ‚= βˆ’ βˆ’ + + + βˆ’ βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

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

( )z! a ! s!563 3Ο„ Ξ· ρ Ο„= =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

hΟ„ βˆ’Ο 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

βˆ’Ο„ Ο„= =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

Ο„ Οƒ ρ ρ ρ ρ ρ ρ

+ + βˆ‚ βˆ‚βˆ‚ βˆ‚= βˆ’ βˆ’ + + + βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

Ο„ Οƒ ρ ρ ρ ρ ρ ρ

+ +

βˆ‚ βˆ‚ βˆ‚ βˆ‚= βˆ’ βˆ’ + + + βˆ’ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

3ith iji j

t ji hk!

"#!

"#uu Ξ΄βˆ’

βˆ‚βˆ‚

+βˆ‚

βˆ‚ Ξ½=β€²β€²βˆ’

1277

Gi (here

=Ξ΄β‰ =Ξ΄=

Ξ΄4

-

$ ji$ ji

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

Derivation 2-DH Depth Averaged Page - 63

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Basic Equation 7Eka O. N.

,

,

,

,,

*,

,*

,,

*

xx

yy

zz

ukh u u u

xv

kh v v v y

wkh w w w

z

Οƒ ρν

Οƒ ρν

Οƒ ρν

βˆ‚ β€² β€² β€²= βˆ’ = βˆ’ = βˆ’βˆ‚βˆ‚ β€² β€² β€²= βˆ’ = βˆ’ = βˆ’βˆ‚βˆ‚ β€² β€² β€²= βˆ’ = βˆ’ = βˆ’βˆ‚

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

Ο„ Ο„ ρν

Ο„ Ο„ ρν

Ο„ Ο„ ρν

βˆ‚ βˆ‚ β€² β€² β€² β€²= = + = βˆ’ = βˆ’ Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ β€² β€² β€² β€²= = + = βˆ’ = βˆ’ Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ β€² β€² β€² β€²= = + = βˆ’ = βˆ’ Γ·βˆ‚ βˆ‚

[ ] [ ] [ ]

,,

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

+ + βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ = βˆ’ βˆ’ + βˆ’ + + + βˆ’ Γ· Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

Ο„ Ο„ Ξ½ Ξ½ ρ ρ

+ +

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ = βˆ’ + βˆ’ + + + βˆ’ Γ· Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

Ο„ Οƒ Ο„ Ο„ ρ ρ ρ ρ

+ + βˆ‚ βˆ‚βˆ‚= βˆ’ + + + βˆ’ βˆ‚ βˆ‚ βˆ‚

................................................................................... *,"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 xΟ„ Ο„

ν ν ρ ρ

+ + βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ = βˆ’ + + βˆ’ + + + βˆ’ Γ· Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

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

Ο„ Οƒ Ο„ Ο„ ρ ρ ρ ρ

+ + βˆ‚ βˆ‚βˆ‚

= βˆ’ + + + + βˆ’ βˆ‚ βˆ‚ βˆ‚

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

Derivation 2-DH Depth Averaged Page - 64

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Basic Equation 7Eka O. N.

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 Ο„ Ο„

ν ν ρ ρ

+ + βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ = βˆ’ + + βˆ’ + + + βˆ’ Γ· Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

. *,"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

Ο„ Οƒ Ο„ Ο„ ρ ρ ρ ρ

βˆ‚ βˆ‚βˆ‚+ + = βˆ’ + + + + βˆ’ βˆ‚ βˆ‚ βˆ‚ .......................... *,"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

Ο„ Οƒ Ο„ Ο„ ρ ρ ρ ρ

βˆ‚ βˆ‚ βˆ‚+ + = βˆ’ + + + βˆ’ βˆ‚ βˆ‚ βˆ‚ ........................................ *,"1/+

Derivation 2-DH Depth Averaged Page - 65

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Basic Equation 7Eka O. N.

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 dyΞ· βˆ‚βˆ‚ + + =βˆ‚ βˆ‚

%

%%

% %$ $,

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

Ξ· Ξ· Ο„ Οƒ Ο„ Ο„ Ξ· ρ ρ ρ ρ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + + βˆ’ + βˆ’ + βˆ’ Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ $%

% $ ,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

Ξ· Ξ·

Ο„ Οƒ Ο„ Ο„ Ξ· ρ ρ ρ ρ βˆ’ βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + + = βˆ’ + βˆ’ + βˆ’ + βˆ’ Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ∫ ∫ % % % $ $%

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:

β€’ 8ariables are now in depth averaged %uantities&

β€’ The !"dimension has been eliminated&

β€’ 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:

β€’ Averaging over the molecular time space scale&

β€’ Averaging over the turbulent time space scale&

β€’ Averaging over the depth space scale&

β€’ 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

Ξ·

Ο… βˆ’

βˆ‚βˆ‚βˆ« 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 Ξ·

βˆ’βˆ« 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 Ξ·

βˆ’βˆ« 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&

Derivation 2-DH Depth Averaged Page - 66

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Basic Equation 7Eka O. N.

β€’ #educe the number of p&d&e& s from to 1&

β€’ #educe the comple6it of the variables

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

β€’ 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&

β€’ ( )( )( )/ / " / / " / /u u u v v v ateral turbulent momentum diffusion

β€’ $$& $& Β°" "uu uv vv$ $$ ateral momentum dispersion related to vertical velocit profile

β€’ " s s x yΟ„ Ο„ Applied free surface stress

β€’ "b b x yΟ„ Ο„ 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

Ξ· Οƒ ρ βˆ’

βˆ‚ βˆ‚ βˆ’ = Γ·βˆ‚ βˆ‚ ∫ ( ), yy

yyh

H vv dz

x y

Ξ· Οƒ ρ βˆ’

βˆ‚ βˆ‚ βˆ’ == Γ·βˆ‚ βˆ‚ ∫ %

$

$%( ) ( ) xy

xyh

Hu H vuv dz

x y x

Ξ· Ο„

ρ βˆ’

βˆ‚ βˆ‚ βˆ‚ βˆ’ == + Γ·βˆ‚ βˆ‚ βˆ‚ ∫

%$

" " 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 uΟ„ ρ

= +%

%

( )1 ,, ,

b y

f ! u v v

Ο„

ρ = +%

9here:

f ! 3 friction factor

17 f DW ! f = Darc 9eisbach

, f

g !

!= (hez

,

1 * f

" g !

h= 0anning

Derivation 2-DH Depth Averaged Page - 67

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Basic Equation 7Eka O. N.

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

βˆ‚ βˆ‚ βˆ‚+ + =βˆ‚ βˆ‚ βˆ‚

, ,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 yΟ… Ο…

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ + + = βˆ’ + βˆ’ βˆ’ + + βˆ’ + + Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ , , 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 yΟ… Ο…

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + = βˆ’ βˆ’ βˆ’ + + βˆ’ + + Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚

, ,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 yΟ… Ο…

Ξ΅ Οƒ Οƒ

βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚+ + = + + + βˆ’ Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ βˆ‚ ,

t

k ! Β΅ Ο…

Ξ΅ =

* ,* + k

!# Β΅ Ξ΅ =

# hΞ± =, ,,

, ,kh t

u v u v p

x y y xΟ…

βˆ‚ βˆ‚ βˆ‚ βˆ‚ = + + + Γ· Γ· Γ·βˆ‚ βˆ‚ βˆ‚ βˆ‚

( ) * ,, ,

, f d

kv

! a! p u v

h = + + Γ·

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

. Fortran or other

. 5onvergent criteria

Derivation 2-DH Depth Averaged Page - 68

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Basic Equation 7Eka O. N.


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