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

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136 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation u VELOCITY COMPONENT, IN CENTIMETERS PER SECOND 30 20 10 0 -10 -20 -30 30 20 10 0 -10 -20 -30 30 20 10 0 -10 -20 -30 Node i = 13, j = 5 Node i = 13, j = 5 Node i = 13, j = 5 Computed Analytical Δ t = 3 minutes Δ t = 5 minutes Δ t = 10 minutes 0 5 10 15 20 TIME, IN HOURS Figure 5.27. Mid-basin solutions for the 3-dimensional seiching problem using an ideal fluid and three different time steps (Δt) for the semi-implicit leapfrog-trapezoidal scheme. The analytical solution is shown for comparison.
5. Numerical Experiments 137 After testing the 3-D scheme for the seiching problem using a non-viscous (ideal) fluid, it was tested for a viscous (real) fluid. For the viscous fluid problem, all terms in the finite-difference equations were included except the Coriolis term. 51 The fluid was assumed to be of homogeneous density, and the bottom drag coefficient (used in eq. 2.64) was set to C d = 0.008. The horizontal eddy viscosity A H was set to zero. 52 The vertical turbulence parameterization used to simulate the seiching problem by Leendertse and Liu (1977) was based on a one-equation turbulence model (Launder and Spaulding, 1972; Rodi, 1980a). In the one-equation model, a differential transport equation is solved for the distribution of the turbulent kinetic energy per unit mass (K), and a simple empirical relation (such as eq. 2.32) is used to specify the distribution of the turbulent mixing length (Λ); the turbulent eddy viscosity is computed from K and Λ using the Kolmogorov-Prandtl expression: AV = Λ K . (5.3) As an independent check on the full nonlinear calculations of the semi-implicit, 3-D scheme, this code was used to reproduce as nearly as possible the predictions for the viscous seiching problem presented by Leendertse and Liu (1977). For this purpose the mixing length turbulence parameterization of the semi-implicit model was modified to more closely represent the one-equation parameterization used in Leendertse and Liu (1977). As discussed in Rodi (1980a, p. 22), the mixing length parameterization is a special case of the one-equation parameterization if the turbulence is in a state of local equilibrium. Local equilibrium means that the production and dissipation of the turbulence energy are in balance at each point in the flow and that the transport influence, by advection or diffusion, of turbulent kinetic energy from other points in the flow is negligible. By equating the production and dissipation terms used by Leendertse and Liu (1977), the following mixing length formula is obtained: A V0 2 u 3.16Λ ∂ = ⎛----- ⎞ ⎝∂z⎠ . (5.4) This equation is identical to the form of equation 2.31 in this report (assuming ∂v / ∂z = 0), except for the introduction of the coefficient 3.16. Thus, in the simulation of the seiching problem, the coefficient 3.16 was inserted into the turbulence coding of the semi-implicit scheme. The initial value of the turbulent kinetic energy per unit mass was chosen to be 0.4 cm 2 /s 2 (Leendertse and Liu, 1977) and was used in equation 5.4 to estimate a background value for the eddy viscosity; the background value was then added to the value calculated by equation 5.4. The value for the horizontal eddy viscosity used by Leendertse and Liu (1977) was zero. A sample velocity distribution from the seiching problem is shown in figure 5.28 after 0.6 hour of simulation. The gradual development of the vertical velocity profile at the center of the basin is shown in figure 5.29A. The viscous effects in the computed solution cause these velocities to be lower than those in the analytical solution for a non-viscous fluid. The computed water surface elevation for the viscous fluid is shown at the west end of the basin in figure 5.29B and is damped slightly from the ideal fluid solution. The computed results in figure 5.29 are in close agreement with those presented by Leendertse and Liu (1977, figures 7 and 8). The 20-hour time history of the viscous solution is plotted in figure 5.30, illustrating the solution damping. The solution was checked for mass preservation in the surface layer of the model and the error was virtually zero. is large. 51 The Coriolis term was set to zero so that there was no y-direction variation in the computed velocity field. 52 Several test simulations were done using increasing values for AH . No noticeable diffusion appeared in solutions for A H less than about 100 m 2 /s, which
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In cooperation with the Interagency
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U.S. Department of the Interior Dir
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vi 4.2 Semi-Implicit One-Dimensiona
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viii Figure 5.6. Graph showing solu
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x Tables Table 2.1. Dimensionless n
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2 A Semi-Implicit, Three-Dimensiona
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2.4.1.2 Dynamic Surface Condition 2
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Water surface Water surface Level p
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h 1 h 2 h 3 h 4 h 5 h 6 z 1 ⁄ 2 z
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A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

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