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

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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 boundary. 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,
2. Governing Equations and Boundary Conditions 41 Equations 2.65 allow the shear stress components (τ xb, τ yb) to be estimated from the velocities (u b, v b) determined by the model and an estimate of either z 0 or C d. The value of z 0 depends greatly on the grain size, bottom shape, and near-bed sediment transport (Grant and Madsen, 1982) and can vary widely. Pritchard (1956) suggested the values z 0 = 0.02 cm for a mud bottom, z 0 = 0.13 cm for a sand-gravel bottom, and z 0 = 0.16 cm for a sand-mud bottom. The higher value suggested for the sand-mud bottom was attributed to the possible presence of ripples. In the presence of high sediment bedload transport, much higher values of z 0 are often used, such as 1 cm (Adams and Weatherly, 1981). Some modelers prefer to specify the drag coefficient C d instead of z 0 because it does not vary as widely as z 0 . Values of C d derived from fitting the logarithmic law to field data are usually reported using z' b = 100 cm. In 3-D model studies, the vertical location of the lowest model grid point above the boundary normally defines the value for z' b . In 2-D model studies, C d values are sometimes reported that are based on the use of depth-averaged velocity components in equations 2.65. In cases where the bed roughness z 0 is uniform, C d should not remain constant if the value of z' b is varying significantly. In general, the use of C d in the range 0.0025 to 0.004 is fairly typical among 3-D coastal and estuarine modelers; this range of C d corresponds to a range in z 0, for z' b = 100 cm, of 0.027 to 0.15 cm. Walters (1992), however, required values for Cd in excess of 0.03 when modeling the tides in the upper Delaware Estuary where channel constrictions and bars are prevalent. 2.4.3 Shoreline Along a shoreline boundary, in theory, either a no-slip or quadratic-stress-law boundary condition can be applied. When modeling large water bodies, however, the typical horizontal grid spacing is so large (>0.25 km) that these boundary conditions yield a distorted horizontal velocity distribution near the boundary. Instead, a “perfect-slip” boundary condition is normally applied that permits the flow to move parallel to the boundary without any boundary frictional resistance. The perfect-slip condition is a reasonable choice so long as the shoreline boundary layer has little influence on horizontal velocity distributions, which is true in most estuaries. The perfect-slip condition can be implemented by adding a line of model nodes outside the boundary and defining the velocity at those outside nodes to be equal to the velocity at the interior nodes adjacent to the boundary. The mass flux of salt at a shoreline boundary must be set to zero. This is normally achieved by letting no flow cross the boundary, although care must be exercised so that the numerical method does not inadvertently transport salt across the boundary by diffusion.
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A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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