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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 :
• 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 $%uation reduces to
p g
z ρ ∂ = −∂
This implies that the pressure distribution over the vertical is h drostatic
• 'cetch (onditions
ig!re 7. 1 Illustration for depth averaged velocity distribution.
• (onsider the geoid to be defined at 0 z = ) the free surface *water"air interface+ at z η = ) and
the bottom *water"sediment interface+ at z h= −• Depth Averaged velocities are defined as:
h h
1 1u U udz v V vdz
h h
η η
− −= = = =+ η + η∫ ∫ &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
• The total water column height is defined as:
H h η = +
• .low rate over the vertical is defined as:
% x
h
q udz uH η
−= =∫ and y
h
q vdz vH η
−= =∫ % &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
• The (ontinuit e%uation is:
0u v w x y z
∂ ∂ ∂+ + =∂ ∂ ∂ ............................................................................................................................................................ */"2+
• The 0omentum e%uation is:
0 0 0 0
1 1 1 1 yx xx zxu u u u pu v wt x y z x x y z
τ τ τ ρ ρ ρ ρ
∂∂ ∂∂ ∂ ∂ ∂ ∂+ + + = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ......................... */"1+ or */" +
Derivation 2-DH Depth Averaged Page - 52
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Basic Equation 7Eka O. N.
• Boundar conditions are:
1. Free surface condition
S !"y"z"t#$ % !"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 =−
∂ − ∂ −+ − = ∂ ∂
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
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
∂ ∂ ∂+ + =∂ ∂ ∂ ............................................................................................................................................................ */"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+
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
∂ =∂ &
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
∂∂ ∂∂ ∂ ∂ ∂ ∂+ + + = − + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
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
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) +
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
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 $%uation ,&-1 becomes:
( ) ( ) ( )
( ) 1 1
1 1
h h h z
z h
xx yxh h h
xx yx zx xx
z
du uu uvdz dz dz u u v w
dt x y t x y
h h h
u v wt x y
h g dz g g dz dz
x x x x y
x y
η η η
η
η η η
η
η η η
η η η η σ τ
ρ ρ
η η σ τ τ σ
ρ ρ
− − − =
=−
− − −
=
∂ ∂ ∂ ∂ ∂+ + − + + − ÷∂ ∂ ∂ ∂ ∂ ∂ − ∂ − ∂ −
+ + + − ∂ ∂ ∂ ∂ −∂ ∂ ∂ ∂= − + − + +∂ ∂ ∂ ∂ ∂
∂ − ∂ ∂− + − + ∂ ∂
∫ ∫ ∫
∫ ∫ ∫ ( ) ( )
yx zx
z h
h h
x yτ τ
=−
∂ −+ − ∂ ∂
................................................. *,"- +
The boundar condition is done with e%uation ," and ,"/&
B performing a stress balance at the surface) it can be shown that s xτ =applied surface stress in
the 6 direction and parallel to the surface&
1 s x xx yx zx
z x y η
η η τ σ τ τ
ρ =
∂ ∂= − + − ∂ ∂ 'imilarl at the bottom:
( ) ( )1b x xx yx zx
z h
h h
x yτ σ τ τ
ρ =−
∂ − ∂ −= − + − ∂ ∂ 'ubstituting reduces the @"momentum e%uation to:
( ) 1 1
h h h
s b x x
xx yxh h h
du uu uvdz dz dz
dt x y
h g dz g g dz dz
x x x x y
η η η
η η η τ τ η η η η σ τ
ρ ρ ρ ρ
− − −
− − −
∂ ∂+ + =∂ ∂∂ −∂ ∂ ∂ ∂− + − + + + −∂ ∂ ∂ ∂ ∂
∫ ∫ ∫
∫ ∫ ∫ .................... *,"- +
9e define the depth averaged variable as:
° 1
h
dz H
η
α α −
≡ ∫ The #e nolds averaged %uantit is then defined as the sum of the depth averaged variable and thedeviation from the depth averaged variable
° 2α α α ≡ +Thus we define velocities in terms of the depth averaged %uantit and the deviation from the depthaveraged %uantit
Derivation 2-DH Depth Averaged Page - 57
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Basic Equation 7Eka O. N.
The spatial averaging is applied as:%
h
udz Huη
−≡∫
h
vdz Hvη
−≡∫
.urthermore we let:
( ) %( ) ( )2" " " " " " " " "u x y z t u x y z t u x y z t = +( ) ( ) ( )2" " " " " " " " "v x y z t v x y z t v x y z t = +%
This implies that:
2 0h
udz η
−=∫ and 2 0
h
vdz η
−=∫
Hence
% %$ $( ),,,
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
∂ ∂ +∂ ∂ ∂ ∂− + + = − − + ∂ ∂ ∂ ∂ ∂ ∂ ................................................................ *,"-,+
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
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
η η τ σ τ τ η ρ ρ ρ ρ − −
∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ $%
............................................................................................................................................................................................................. *,&-<+
% $ ,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 − −
∂ ∂ ∂ ∂ ∂ ∂+ + + = − + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ % % %
$ $%
............................................................................................................................................................................................................. *,&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
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
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
ρ ρ − − − − − −
∂ ∂ ∂∂ ∂ ∂ ∂ + + + = − + + + ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ ∫ ∫ ∫ ∫ 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 − − − − − −
∂ ∂ ∂∂ ∂ ∂ ∂+ + + = − + + + ÷∂ ∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ ∫ ∫ ∫ ∫
( )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
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
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
− −
−
∂ ∂− − = − −∂ ∂ ∂= − − = ÷ ÷∂
∂= − + ∂
∫ ∫
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
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
τ σ η ρ η η η η τ η τ
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
τ σ η ρ η η η η τ η τ
ρ ρ ρ ρ ρ
+ + + + + ∂ ∂∂ ∂= − + − + + + + + + − − ∂ ∂ ∂ ∂
( )hη+ 3H) h constant) h
0!
∂ =∂ hence( )h 4
! ! !
∂ η+∂η ∂= =∂ ∂ ∂&
Boussines% coefisien C 3- and 4
! !∂η ∂=∂ ∂
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
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;+
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
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
τ σ τ τ ρ ρ ρ ρ
+ + ∂ ∂∂
= − + + + + − ∂ ∂ ∂
................................................................. *,"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 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
τ σ τ τ ρ ρ ρ ρ
∂ ∂∂+ + = − + + + + − ∂ ∂ ∂ .......................... *,"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
τ σ τ τ ρ ρ ρ ρ
∂ ∂ ∂+ + = − + + + − ∂ ∂ ∂ ........................................ *,"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 $%uation are:
%( ) ( )0
uH d vH
t x dyη ∂∂ + + =∂ ∂
%
%%
% %$ $,
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
η η τ σ τ τ η ρ ρ ρ ρ − −
∂ ∂ ∂ ∂ ∂ ∂+ + + = − + + − + − + − ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ ∫ ∫ $%
% $ ,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&
" $$h
uudz η
−∫ t pe terms represent the spreading of momentum over the water column& This
process is known as momentum dispersion& These terms are necessar since we are nolonger directl simulating this process via the actual depth var ing velocit profiles& Thespread momentum laterall &
The shallow water e%uations greatl simplif flow computation in free surface water bodies&
Derivation 2-DH Depth Averaged Page - 66
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Basic Equation 7Eka O. N.
• #educe the number of p&d&e& s from to 1&
• #educe the comple6it of the variables
%( ) ( ) ( )" " " " " " " "u x y t v x y t x y t η % instead of ( ) ( ) ( ) ( )" " " " " " " " " " " " " " "u x y z t v x y z t w x y z t p x y z t
• Built"in positioning of the free surface boundar which is t picall unknown when appl ing the
#e nolds e%uations&The shallow water e%uations include -4 additional unknowns as compared to the avier"'tokese%autions&
• ( )( )( )/ / " / / " / /u u u v v v ateral turbulent momentum diffusion
• $$& $& °" "uu uv vv$ $$ 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
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
∂ ∂ ∂+ + =∂ ∂ ∂
, ,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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Basic Equation 7Eka O. N.
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