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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 $%uation reduces to
p g
z Ο β = ββ
This implies that the pressure distribution over the vertical is h drostatic
β’ 'cetch (onditions
ig!re 7. 1 Illustration for depth averaged velocity distribution.
β’ (onsider the geoid to be defined at 0 z = ) the free surface *water"air interface+ at z Ξ· = ) and
the bottom *water"sediment interface+ at z h= ββ’ Depth Averaged velocities are defined as:
h h
1 1u U udz v V vdz
h h
Ξ· Ξ·
β β= = = =+ Ξ· + Ξ·β« β« &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
β’ The total water column height is defined as:
H h Ξ· = +
β’ .low rate over the vertical is defined as:
% x
h
q udz uH Ξ·
β= =β« and y
h
q vdz vH Ξ·
β= =β« % &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
β’ The (ontinuit e%uation is:
0u v w x y z
β β β+ + =β β β ............................................................................................................................................................ */"2+
β’ The 0omentum e%uation is:
0 0 0 0
1 1 1 1 yx xx zxu u u u pu v wt x y z x x y z
Ο Ο Ο Ο Ο Ο Ο
ββ ββ β β β β+ + + = β + + +β β β β β β β β ......................... */"1+ or */" +
Derivation 2-DH Depth Averaged Page - 52
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Basic Equation 7Eka O. N.
β’ Boundar conditions are:
1. Free surface condition
S !"y"z"t#$ % !"y"t#-z $ 0" &ada z $ %
dS0
dt=
'ydS ! z
0dt t ! t y t t
ββΞ· βΞ· βΞ·β β= + + β =β β β β β β
u v ( 0t ! y
βΞ· βΞ· βΞ·+ + β =β β β
z z z z
Dw u v
Dt t x yΞ· Ξ· Ξ· Ξ·
Ξ· Ξ· Ξ· Ξ· = = ==
β β β= = + +β β β ................................................................. )-*#
z u v w x y t Ξ·
Ξ· Ξ· Ξ·
=
β β ββ + β = β β β ..................................................................................... )-+#
,. 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 tβ β β ββ β= + + + =β β β β β β
0 0 00 0 0
z z zu !" y" z # v !" y" z # ( !" y" z # 0
t ! yβ β β+ + + =β β β at z $ z 0
The velocit in the bottom is: 7*"h+ 3 8*"h+ 3 9*"h+ 3 4) thenh
0t
β =β( ) ( ) ( ) ( )
z h z h z h z h
D h h h hw u v
Dt t x y=β =β =β=β
β β β β β β β= = + +β β β &&&&&&&&&&&&&&&&&&&&&&&&&&&&&
( ) ( )0
z h
h hu v w
x y =β
β β β β+ β = β β
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
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
β β β+ + =β β β ............................................................................................................................................................ */"2+
8erticall average
10
h
u v wdz
H x y z
Ξ·
β
β β β+ + = Γ·β β β β« 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·
β β β β
β β β β β β + + = + + = Γ· Γ·β β β β β β β« β« β« β«
( )
( )
( )
( )
( ) ( )" " " "
" " " "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· β β
β β+ + β β =β ββ« β«
7sing Leibnitzβs 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= =β β β β= + ββ β β ββ« β« ................................................................................................. *,",+
Hence e%& */&2+ become
z h z
h h
u hdz udz u u
x x x x
Ξ· Ξ·
Ξ·
Ξ· =β =
β β
β β β β= + ββ β β ββ« β« ( )
z h z h h
hvdz vdz v v
y y y y
Ξ· Ξ·
Ξ·
Ξ· =β =
β β
β ββ β β= + ββ β β ββ« β«
z z hh
wdz w w
z
Ξ·
Ξ· = =ββ
β = βββ«
( ) ( )
( )
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·
β β = = β
β β β β β β β β+ β + β + + β = β β β β β β
β« β« *,";+
9ith bottom and free surface condition) e% *," + and *,"/+
0h h
udz vdz t x y
Ξ· Ξ· Ξ·
β β
β β β+ + =β β ββ« β« *,"<+
Termsh
udzΞ·
ββ« and
h
vdzΞ·
ββ« 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·
β
β ββ β β β+ + = + + = Γ·β β β β β β β« ............................................................................................................. *,"-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
β =β &
Depth Averaged (ontinuit $%uation :
%( ) ( )0
uH d vH H
t x dy
ββ + + =β β
%................................................................................................................................................ *,"--+
?r
%( ) ( )0
uH d vH
t x dyΞ· ββ + + =β β
%................................................................................................................................................. *,"-2+
7.'. (O(ENT"( DE$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 Ο Ο Ο Ο ββ ββ β β β β+ + + = β + + +β β β β β β β β
.................................. */"1+ or */" +
// /ij i j i
j
uu u
x
Ο Ο
Ο β= ββ
0/ / /iij j i
j
uu u
xΟ Β΅ Ο β= ββ
0 / / xx
uu u
xΟ Β΅ Ο β= ββ
" 0 / / yx
uu v
yΟ Β΅ Ο β= ββ " 0 / / zx
uu w
z Ο Β΅ Ο β= ββ
And p
g z
Ο β = ββ) 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·
Ο =
β = β ββ« β« where s p 3 pressure at the free surface
And assuming that densit is constant:
( ) s p p gz g Ο Ο Ξ· β = β β
s p p g gz Ο Ξ· Ο = + β1 1 s p p z
g g x x x x
Ξ· Ο Ο
ββ β ββ = β β +β β β βThe surface pressure does not var spatiall :
1 p g
x xΞ·
Ο β ββ = ββ β
or 00
1 z p g g g gs
x x x xΞ· Ξ·
Ο ββ β ββ = β β = β +β β β β
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·
Ο Ο Ο
ββ ββ β β β β+ + + = β + + +β β β β β β β β
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· Ο Ο Ο
ββ ββ β β β β β β β+ + + + + + = β + + +β β β β β β β β β β β#e"arranging
( ) ( ),1 yx xx zx
uv uwu u g
t x y z x x y z
Ο Ο Ο Ξ· Ο
β β β β ββ β β+ + + = β + + + β β β β β β β β
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 Ο Ο Ξ· Ο β β β β β β
β β ββ β β β+ + + = β + + + Γ·β β β β β β β β« β« β« β« β« β« 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 Leibnitzβs 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· = =β
β β
β β ββ β= + ββ β β ββ« β«
................................................................................. *,",+
9here,
" " " " " xx xy f u u uv Ξ· Ο Ο = and
z z hh h
g dz g g g
z
Ξ· Ξ·
Ξ· = =ββ β
β = β = βββ« β« 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· = =β
β β
ββ β= β +β ββ« β«
Ter) 2( ) ( ), , ,
z z hh h
d d huudz u z u u
x x dx dx
Ξ· Ξ·
Ξ·
Ξ· = =ββ β
ββ β= β β +β ββ« β«
Ter) '
( ) ( ) z z h
h h
d d huv dz uv z uv uv x x dx dx
Ξ· Ξ·
Ξ· Ξ·
= =ββ β
ββ β= β β +β ββ« β« Ter) *
hh
z z h
uwdz uw
z
w w
Ξ· Ξ·
Ξ·
ββ
= =β
β =β= β
β«
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· = =β
β β
β ββ β ββ = β β +β β β ββ« β«
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 Ο Ο Ο Ο Ο Ο Ο
Ξ· Ξ· Ο Ο Ο Ο Ο Ο
Ο Ο
β β β
= =β
β β β β β+ + = + Γ·β β β β β β β β β β ββ + β + + β β β β β
β« β« β«
#ewrite $%uation ,&-1 becomes:
( ) ( ) ( )
( ) 1 1
1 1
h h h z
z h
xx yxh h h
xx yx zx xx
z
du uu uvdz dz dz u u v w
dt x y t x y
h h h
u v wt x y
h g dz g g dz dz
x x x x y
x y
Ξ· Ξ· Ξ·
Ξ·
Ξ· Ξ· Ξ·
Ξ·
Ξ· Ξ· Ξ·
Ξ· Ξ· Ξ· Ξ· Ο Ο
Ο Ο
Ξ· Ξ· Ο Ο Ο Ο
Ο Ο
β β β =
=β
β β β
=
β β β β β+ + β + + β Γ·β β β β β β β β β β β
+ + + β β β β β ββ β β β= β + β + +β β β β β
β β β ββ + β + β β
β« β« β«
β« β« β« ( ) ( )
yx zx
z h
h h
x yΟ Ο
=β
β β+ β β β
................................................. *,"- +
The boundar condition is done with e%uation ," and ,"/&
B performing a stress balance at the surface) it can be shown that s xΟ =applied surface stress in
the 6 direction and parallel to the surface&
1 s x xx yx zx
z x y Ξ·
Ξ· Ξ· Ο Ο Ο Ο
Ο =
β β= β + β β β 'imilarl at the bottom:
( ) ( )1b x xx yx zx
z h
h h
x yΟ Ο Ο Ο
Ο =β
β β β β= β + β β β 'ubstituting reduces the @"momentum e%uation to:
( ) 1 1
h h h
s b x x
xx yxh h h
du uu uvdz dz dz
dt x y
h g dz g g dz dz
x x x x y
Ξ· Ξ· Ξ·
Ξ· Ξ· Ξ· Ο Ο Ξ· Ξ· Ξ· Ξ· Ο Ο
Ο Ο Ο Ο
β β β
β β β
β β+ + =β ββ ββ β β ββ + β + + + ββ β β β β
β« β« β«
β« β« β« .................... *,"- +
9e define the depth averaged variable as:
Β° 1
h
dz H
Ξ·
Ξ± Ξ± β
β‘ β« The #e nolds averaged %uantit is then defined as the sum of the depth averaged variable and thedeviation from the depth averaged variable
Β° 2Ξ± Ξ± Ξ± β‘ +Thus we define velocities in terms of the depth averaged %uantit and the deviation from the depthaveraged %uantit
Derivation 2-DH Depth Averaged Page - 57
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Basic Equation 7Eka O. N.
The spatial averaging is applied as:%
h
udz HuΞ·
ββ‘β«
h
vdz HvΞ·
ββ‘β«
.urthermore we let:
( ) %( ) ( )2" " " " " " " " "u x y z t u x y z t u x y z t = +( ) ( ) ( )2" " " " " " " " "v x y z t v x y z t v x y z t = +%
This implies that:
2 0h
udz Ξ·
β=β« and 2 0
h
vdz Ξ·
β=β«
Hence
% %$ $( ),,,
h h
uudz u uu u z Ξ· Ξ·
β β= + + ββ« β«
% % $ $,,,
h h h h
uudz u z u u z u z Ξ· Ξ· Ξ· Ξ·
β β β β= β + β + ββ« β« β« β«
% $,,
h h
uudz H u u z Ξ· Ξ·
β β= + ββ« β«
'imilarl
% % $ $( ) % $
h h h
uvdz uv uv uv uv dz uvH uvdz Ξ· Ξ· Ξ·
β β β= + + + = +β« β« β« $ $ $% % %
h h
dz dz H Ξ· Ξ·
Ξ· Ξ· Ξ· β β
= =β« β« '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· Ο Ο Ο Ξ· Ξ· Ξ·
Ο Ο Ο Ο β β
ββ β+ + =β β β
β β β β ββ + + + β + β + β Γ· Γ·β β β β β β« β«
%
$
...... *,"-/+
$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·
β β +β β β ββ + + = β β + β β β β β β ................................................................ *,"-,+
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·
β β β ββ + + = ββ β β β9e also note that the acceleration terms can be e6panded as:
%( ) % % Hu H u
u H t t t
β β β= +β β β
%( ) % ( ) % Hu h uu H
t t t
Ξ· β β + β= +β β β%( ) %
% Hu uu H
t t t Ξ· β β β= +
β β β%( ) %
%( ) % %
, Hu Hu u
u H u x x x
β β β= +β β β%( )
% ( ) % Huv H v u
u H v y y y
β β β= +β β β
% %
%
'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 Ο Ο Ο Ξ· Ο Ο Ο Ο β β
β ββ β β β + + + + + =β β β β β β β β ββ + β + β + β Γ· Γ·β β β β« β«
%%
$
................................................. *,"-;+
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 Ο Ο Ο Ξ· Ο Ο Ο Ο β β
β β β β β β+ + + = β + β + β + β Γ· Γ·β β β β β β β« β« $%
or
%%
% %$ $,
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 Ο Ο Ο Ξ· Ο Ο Ο Ο β β
β β β β β β+ + + = β + + β + β + β Γ· Γ·β β β β β β β« β« $%
............................................................................................................................................................................................................. *,&-<+
% $ ,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 Ο Ο Ο Ξ· Ο Ο Ο Ο β β
β β β β β β+ + + = β + β + β + β Γ· Γ·β β β β β β β« β« % % %
$ $%
............................................................................................................................................................................................................. *,&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 Ο Ο Ο Ο
β β ββ β β β+ + + = β + + + Γ·β β β β β β β 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 Ο Ο Ο Ο Ο Ο
β β β β
β β β β
β β β+ + +β β βββ ββ= β + + +β β β β
β« β« β« β«
β« β« β« β« ...................................................... *,"2-+
with ( ) & g z= Ο Ξ· β
( )dp g g z dx x xΞ· Ο Ο Ξ· β β= + ββ β ........................................................................................................................................... *,"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 Ο Ο Ξ· Ο
Ο Ο β β β β β β
β β ββ β β β + + + = β + + + Γ· Γ·β β β β β β β β« β« β« β« β« β« 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 Ο Ο Ξ· Ο β β β β β β
β β ββ β β β+ + + = β + + + Γ·β β β β β β β β« β« β« β« β« β«
( )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 Ο Ο Ξ· Ο Ξ·
Ο Ο
β β β β
β β β
β β β+ + +β β β
β β ββ β= β β β + + + Γ·β β β β β
β« β« β« β«
β« β« β«
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·
β β=
β
β= + β β
= β
= + β
β« β« 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·
β β=
β β
ββ = + β ββ
= +
= + β
β« β«
β« β«
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·
β β=
β β
ββ = + β ββ
= β +
= + β
β« β«
β« β«
1 44 2 4 43
Ter) *
( ) ( )0
hh
uwdz uw
z
uw uw h
Ξ· Ξ·
Ξ·
ββ
β =β
= β β=
β«
Ter) +
( )hh
g dz g x x
g h x
Ξ· Ξ· Ξ· Ξ·
Ξ· Ξ·
ββ
β ββ = ββ β
β= β +β
β«
Ter) ,
( ) ( )
( )
,
,
1,
1,
h h
h
g g z dz z dz
x x
g z z
x
g h
x
Ξ· Ξ·
Ξ·
Ο Ο Ξ· Ξ·
Ο Ο
Ο Ξ·
Ο
Ο Ξ·
Ο
β β
β
β ββ β = β ββ β β= β β = Γ· Γ·β
β= β + β
β« β«
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 Ο Ο Ο Ο
Ο Ο Ο Ο
Ο Ο Ξ· Ξ· Ο Ο
Ο Ξ· Ο Ο Ο
β β ββ
β β β β β+ + = + + Γ·β β β β β
ββ= + + +β β
+ β β
β«
#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 Ο Ξ· Ο Ξ· Ξ· Ξ· Ξ· Ο Ξ· Ο
Ο Ο Ο Ο Ο
+ β + + β + + β
β ββ β= β + β + + + + + + β β β β β β ............................................................................................................................................................................................................. *,"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 Ο Ξ· Ο Ξ· Ξ· Ξ· Ξ· Ο Ξ· Ο Ο Ο Ο Ο Ο
+ β + + β + + β Γ· Γ· Γ· β ββ β
= β + β + + + + + + β β β β β β
( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
,
,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 Ο Ξ· Ο Ξ· Ξ· Ξ· Ξ· Ο Ξ· Ο
Ο Ο Ο Ο Ο
+ + + + + β + + β ββ β= β + β + + + + + + β β β β β β
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 Ο Ξ· Ο Ξ· Ξ· Ξ· Ξ· Ο Ξ· Ο
Ο Ο Ο Ο Ο
+ + + + + β ββ β= β + β + + + + + + β β β β β β
( )hΞ·+ 3H) h constant) h
0!
β =β hence( )h 4
! ! !
β Ξ·+βΞ· β= =β β β&
Boussines% coefisien C 3- and 4
! !βΞ· β=β β
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 Ο Ο Ο Ξ· Ο
Ο Ο Ο Ο Ο
+ + β ββ β= β β + + + β β β β β β
............................ *,"2 +
( )z!1 Ο Ξ·Ο is define to surface shear stress) this stress is caused b wind:
( )z! a ! s!563 3Ο Ξ· Ο Ο= =Ο Ο Ο
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 βΟ 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
βΟ Ο= =Ο Ο
$%& *,&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 Ο Ο Ο Ο Ο Ο Ο
+ + β ββ β= β β + + + β β β β β
............................. *,"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 Ο Ο Ο Ο Ο Ο Ο
+ +
β β β β= β β + + + β β β β β
............................. *,"2,+
3ith iji j
t ji hk!
"#!
"#uu Ξ΄β
ββ
+β
β Ξ½=β²β²β
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
Ο ΟΞ½
Ο ΟΞ½
Ο ΟΞ½
β β² β² β²= β = β = βββ β² β² β²= β = β = βββ β² β² β²= β = β = ββ
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 ΟΞ½
Ο Ο ΟΞ½
Ο Ο ΟΞ½
β β β² β² β² β²= = + = β = β Γ·β β β β β² β² β² β²= = + = β = β Γ·β β β β β² β² β² β²= = + = β = β Γ·β β
[ ] [ ] [ ]
,,
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 Ο Ο
Ξ½ Ξ½ Ο Ο
+ + β β β β β β β = β β + β + + + β Γ· Γ· Γ·β β β β β β β
............................................................................................................................................................................................................. *,"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 Ο Ο
+ +
β β β β β β = β + β + + + β Γ· Γ· Γ·β β β β β β
................................ *,"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 Ο Ο Ο Ο Ο Ο Ο
+ + β ββ= β + + + β β β β
................................................................................... *,"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
Ξ½ Ξ½ Ο Ο
+ + β β β β β β = β + + β + + + β Γ· Γ· Γ·β β β β β β
............... *,"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 Ο Ο Ο Ο Ο Ο Ο
+ + β ββ
= β + + + + β β β β
................................................................. *,"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
Ξ½ Ξ½ Ο Ο
+ + β β β β β β = β + + β + + + β Γ· Γ· Γ·β β β β β β
. *,"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 Ο Ο Ο Ο Ο Ο Ο
β ββ+ + = β + + + + β β β β .......................... *,"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 Ο Ο Ο Ο Ο Ο Ο
β β β+ + = β + + + β β β β ........................................ *,"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 $%uation are:
%( ) ( )0
uH d vH
t x dyΞ· ββ + + =β β
%
%%
% %$ $,
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 Ο Ο Ο Ξ· Ο Ο Ο Ο β β
β β β β β β+ + + = β + + β + β + β Γ· Γ·β β β β β β β« β« $%
% $ ,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 Ο Ο Ο Ξ· Ο Ο Ο Ο β β
β β β β β β+ + + = β + β + β + β Γ· Γ·β β β β β β β« β« % % % $ $%
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 β
βββ« 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·
ββ« 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&
" $$h
uudz Ξ·
ββ« t pe terms represent the spreading of momentum over the water column& This
process is known as momentum dispersion& These terms are necessar since we are nolonger directl simulating this process via the actual depth var ing velocit profiles& Thespread momentum laterall &
The shallow water e%uations greatl simplif flow computation in free surface water bodies&
Derivation 2-DH Depth Averaged Page - 66
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Basic Equation 7Eka O. N.
β’ #educe the number of p&d&e& s from to 1&
β’ #educe the comple6it of the variables
%( ) ( ) ( )" " " " " " " "u x y t v x y t x y t Ξ· % instead of ( ) ( ) ( ) ( )" " " " " " " " " " " " " " "u x y z t v x y z t w x y z t p x y z t
β’ Built"in positioning of the free surface boundar which is t picall unknown when appl ing the
#e nolds e%uations&The shallow water e%uations include -4 additional unknowns as compared to the avier"'tokese%autions&
β’ ( )( )( )/ / " / / " / /u u u v v v ateral turbulent momentum diffusion
β’ $$& $& Β°" "uu uv vv$ $$ 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· Ο Ο β
β β β = Γ·β β β« ( ), yy
yyh
H vv dz
x y
Ξ· Ο Ο β
β β β == Γ·β β β« %
$
$%( ) ( ) xy
xyh
Hu H vuv dz
x y x
Ξ· Ο
Ο β
β β β β == + Γ·β β β β«
%$
" " 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 Ο
= +%
%
( )1 ,, ,
b y
f ! u v v
Ο
Ο = +%
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
β β β+ + =β β β
, ,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
β β β β β β β β β + + = β + β β + + β + + Γ· Γ·β β β β β β β β β , , 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
β β β β β β β β β+ + = β β β + + β + + Γ· Γ·β β β β β β β β β
, ,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΅ Ο Ο
β β β β β β β+ + = + + + β Γ· Γ·β β β β β β β ,
t
k ! Β΅ Ο
Ξ΅ =
* ,* + k
!# Β΅ Ξ΅ =
# hΞ± =, ,,
, ,kh t
u v u v p
x y y xΟ
β β β β = + + + Γ· Γ· Γ·β β β β
( ) * ,, ,
, 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.