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

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140 A Semi-Implicit, Three-Dimensional Model for Estuarine Circulation Continuing with comparisons of test simulations with those of Leendertse and Liu (1977), a second simulation of the viscous seiching problem was computed using a slightly shorter basin length of 44 km. The simulation covered 20 hours using the same model parameters as those used to simulate the longer basin. The calculations used the semi-implicit, leapfrog scheme with a time step of 45 seconds. The vertical profiles of the u velocity component at mid-basin were saved to a file at a constant time interval of 3.125 hours (250 time steps). A plot of these profiles is shown in figure 5.31. The same solution profiles, digitized from the report by Leendertse and Liu (1977, fig. 12), are also shown. Although the two sets of profiles are in reasonably close agreement, the semi-implicit solution predicts larger absolute magnitudes for the velocities in each profile than was computed by Leendertse and Liu. It was suspected that the cause of the difference was that equation 5.4 underestimated the magnitude of the eddy viscosities computed by the turbulence model of Leendertse and Liu. Leendertse and Liu (1977, fig. 6D) graphically present a distribution of the magnitudes of the turbulent kinetic energy per unit mass calculated from their turbulence model after 60 time steps of simulation; using equation 5.3 to translate these values into a distribution of the eddy viscosity, a depth-averaged eddy viscosity at mid-basin was calculated as approximately 300 cm 2 /s. 53 The depth-averaged eddy viscosity calculated during the semi-implicit simulation after 60 time steps (using eq. 5.4) was only 200 cm 2 /s. To decrease the differences between the two sets of velocity profiles in figure 5.31, another simulation with the semi-implicit, leapfrog scheme was made using a constant value of 300 cm 2 /s for the eddy viscosity. The two sets of profiles are in closer agreement (fig. 5.32); some differences still exist but are most likely attributable to the difference in turbulence parameterizations or to inexact digitizing of profiles from the report by Leendertse and Liu. Note that the computed profiles using the constant eddy viscosity exhibit less vertical variation in velocity (shear) than in the previous solution in which the eddy viscosity varied over the depth. One final simulation of the seiching problem was made to test the semi-implicit scheme for conditions using variable density. A linear vertical density gradient of Δρ / Δz = 0.035 kg/m 3 /cm was introduced with freshwater at the free surface. Equation 5.4 was used to estimate the eddy viscosity under homogeneous density conditions, and the effects of stratification were incorporated using the stability functions (eqs. 2.27 and 2.36) proposed by Munk and Anderson (1948). The longer basin length of 46 km was used. The oscillation in this test showed greater vertical shear in the velocity profile than the test using the homogeneous density because the vertical mixing was reduced by the stratification (fig. 5.33). The results were very similar to those of Leendertse and Liu (1977, fig. 9), but they were not identical because of differences in the turbulence parameterizations and stability functions used in the two models. 53 A depth-averaged eddy viscosity of 300 cm 2 /s is quite high for a basin 12 meters deep. A standard estimate for the depth-averaged eddy viscosity in constant density flow in which the turbulence is generated by bottom shear stress is 0.067Hu* (Fischer and others, 1979). This estimate corresponds to a maximum of about AV0 = 100 cm 2 /s. A simulation using A = 100 cm V0 2 /s did not agree well with the solution of Leendertse and Liu (1977).
DISTANCE FROM BOTTOM, IN METERS 12 10 8 6 4 2 0 Bottom Node i = 13, j = 5 (mid-basin) t = 3.125 hours t = 6.250 hours t = 12.500 hours t = 15.625 hours t = 9.375 hours t = 18.750 hours Leendertse and Liu (1977) Semi-implicit, leapfrog scheme -12 -9 -6 -3 0 3 6 9 12 u VELOCITY COMPONENT, IN CENTIMETERS PER SECOND 5. Numerical Experiments 141 Surface Figure 5.31. Velocity profiles for the 3-dimensional (3-D) seiching problem using a viscous, homogeneous fluid in a 44-kilometer basin and a vertical eddy viscosity defined by a mixing length formula. The profiles were computed using the 3-D, semi-implicit, leapfrog scheme and a vertical eddy viscosity defined by equation 5.4. For comparison, the velocity profiles from the report by Leendertse and Liu (1977, fig. 12) also are shown. DISTANCE FROM BOTTOM, IN METERS 12 10 8 6 4 2 0 Bottom Node i = 13, j = 5 (mid-basin) t = 3.125 hours t = 12.500 hours t = 6.250 hours t = 15.625 hours t = 9.375 hours t = 18.750 hours Leendertse and Liu (1977) Semi-implicit, leapfrog scheme Surface -12 -9 -6 -3 0 3 6 9 12 u VELOCITY COMPONENT, IN CENTIMETERS PER SECOND Figure 5.32. Velocity profiles for the 3-dimensional (3-D) seiching problem using a viscous, homogeneous fluid in a 44-kilometer basin and a constant vertical eddy viscosity. The profiles were computed using the 3-D, semi-implicit, leapfrog scheme and a vertical eddy viscosity of 300 cm 2 /sec. For comparison, the velocity profiles from the report by Leendertse and Liu (1977, fig. 12) also are shown.
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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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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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