A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

40 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation
As noted previously, the strategy in defining the boundary stresses is to relate them to the velocity at a point above the bound-
ary. For this purpose, the well-known logarithmic velocity law generally is adequate so long as the bottom topography is not highly
nonuniform and the boundary layer is not stratified. 19 It is assumed that the boundary layer is unaffected by wave-generated cur-
rents. Gross and Nowell (1983) found that the dynamic effects from temporal variability (unsteadiness) in a tidal boundary layer
did not measurably affect the accuracy of predictions with the logarithmic law in the near-bed region. The appropriate logarithmic
law for rough boundary flow is (Tennekes and Lumley, 1972, p. 165, eq. 5.2.49)
where
u
res
-------- =
u
*
1
--ln
κ
z'
---z0
, (2.63)
z′ is a vertical distance measured outward from an average position among the roughness elements,
u res is the resultant velocity parallel to the boundary at a height z' ,
u * is the resultant shear velocity defined as τb ⁄ ρ0 ,
z 0 is a characteristic length scale related to the height of the roughness elements and corresponding with the point
of zero velocity (u res = 0 at z′ = z 0 ), and
κ is von Kármán’s constant (~ 0.41).
Equation 2.63 is theoretically valid outside the region of boundary roughness and within a layer of constant (Reynolds) stress.
Although a “constant stress layer” exists in channel flow only in the immediate neighborhood of the boundary, the logarithmic
profile typically represents the flow well outside this layer. In fact, for steady, uniform channel flow, the logarithmic law
traditionally has been used to describe the velocity profile over the entire depth of flow (Keulegan, 1938). Under many ordinary
conditions of tidal flow in estuaries, the logarithmic law is a reasonable representation of the velocity profile for the first few meters
above the bed.
Equation 2.63 can be solved for the boundary shear stress to obtain a quadratic stress law of the form
τ b
= ρ0Cd u
res
u
res
, (2.64)
where Cd is a dimensionless drag coefficient. The quadratic term in 2.64 involving the velocity is written with an absolute value
sign so the frictional stress always will oppose the flow. In practice, equation 2.64 is rearranged to represent the two components
of shear stress with
τ
xb
ρ
0
C ⎛ 2 2
d
u
b
+ v ⎞
⎝ b⎠
1⁄ 2
ub and τ
yb
ρ
0
C ⎛ 2 2
d
u
b
+ v ⎞
⎝ b⎠
1⁄ 2
= =
vb , (2.65)
where u b, v b are the horizontal velocity components at a point z' b above the bottom, and
1981).
C d
κ 2 ln z' ⎛ b
----- ⎞
⎝ z ⎠
0
2 –
= . (2.66)
19 Refinements of the logarithmic law that incorporate the effect of density stratification are available (Turner, 1973, Chapter 5; Adams and Weatherly,