4/98
SED2D-WES Version 4.3 Beta
{C}
[T ]
+ [ K]{C} - { F} = 0
(3)
t
where each element in the computation mesh contributes the following terms to the global matrix
[ T] = ∫∫ [ N] T [ N]dxdy ;
D
C
C
$+
$
$
$
$
Nj
Nj
C
C
[ K] = ∫∫ K j u + v - α1 C
dxdy
+
D
D
x
y y y
x x x
y
D
(the steady-state system coefficient matrix);
{F} = - ∫∫ [ N]T {α 2}dxdy + ∫ [ N]T {q}dζ
ζ
D
(the boundary loadings vector);
Applying the Crank-Nicholson scheme, where θ is the implicitness coefficient, gives the following equation, where
n refers to the present, n+1 to the future time - step, and ∆t the computation time interval.
[ T]
[ T]
+ θ[ K] n + 1 {C} n + 1 =
- (1 - θ)[K] n {C} n + θ{ F} n +1 + (1 - θ){F}n
(4)
∆t
∆t
Bed shear stress. Several options are available for computing bed shear stresses using
b.
τb = ρ (u*)
2
(5)
where
ρ = water density
u* = shear velocity
a. Smooth-wall log velocity profile,
u* D
u
= 5.75 log 3.32
(6)
ν
u*
9
WORKING DRAFT
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