Flat Basin: Difference between revisions

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The model is initialized from zero current velocities and water
levels. The simulation is then allowed to reach steady state over
48 hr.
''Results and Discussion''
The calculated wind setup (water surface elevation) is shown in
Figure 2 for the case of wind from the north (left) and from west
(right). For both cases, the calculated wind setup is symmetric and
has straight contour lines, which is consistent with the analytical
solution. Figure 3 shows the wind setup along the center line of the
domain for the case with wind from the north compared to the
analytical solution. The goodness-of-fit statistics along this
transect include the Normalized Root Mean Square Error (NRMSE), Mean
Absolute Error (MAE), squared correlation coefficient,<math>R^2</math>, and
Bias as given in Table 3.

Revision as of 19:48, 15 April 2014

Test C1-Ex1: Wind Setup in a Flat Basin

Purpose

This verification test is designed to test the most basic model capabilities by solving the most reduced or simplified form of the governing equations in which only the water level gradient balances the wind surface drag. The specific model features/aspects to be tested are (1) spatially constant wind fields, (2) water surface gradient implementation, and (3) land-water boundary condition.

Problem and Analytical Solution

Assuming a closed basin with a spatially constant, steady state wind in one direction, no advection, diffusion, bottom friction, waves or Coriolis force, the momentum equations reduce to

  (1)


where is the total water depth, is the still water depth, is the water surface elevation (water level) with respect to the still water level, is the wind drag coefficient, is the coordinate in the direction of the wind, is the gravitational acceleration, is the water density, is the air density, and is the wind speed. Assuming a constant wind drag coefficient, the following analytical expression for the water level may be obtained by integrating the above equation (Dean and Dalrymple 1984)

  (2)

where is a constant of integration.

Model Setup

A computational grid with constant water depth of 5 m and irregular boundaries is used to verify the numerical methods. The computational grid has 60 columns and 70 rows and a constant resolution of 500 m. The irregular geometry is intentionally used to check for any discontinuities in processes near the land-water boundaries. The solution should be perfectly symmetric and independent of the geometry of the closed basin. The steady state solution is reached by increasing the wind speed over a 3 hr ramp period and allowing the solution to reach steady state over a 48 hr time period. During the ramp period, all model forcing is slowly increased from the initial condition (not necessarily zero), to the specified boundary condition time series. The purpose of the ramp period is to allow the model to slowly adjust to the forcing conditions without “shocking” it with a step function. In CMS, a cosine ramp function of the form

  (3)
Table 2. CMS-Flow settings for the wind setup test case.
Parameter Value
Solution scheme Implicit
Time step 10 min
Simulation duration 48 hr
Ramp period duration 3 hr
Wind speed 10 m/s
Drag coefficient 0.0016
Advection terms Off
Mixing terms Off
Bottom friction Off
Wall friction Off
Coriolis force Off

The model is initialized from zero current velocities and water levels. The simulation is then allowed to reach steady state over 48 hr.

Results and Discussion

The calculated wind setup (water surface elevation) is shown in Figure 2 for the case of wind from the north (left) and from west (right). For both cases, the calculated wind setup is symmetric and has straight contour lines, which is consistent with the analytical solution. Figure 3 shows the wind setup along the center line of the domain for the case with wind from the north compared to the analytical solution. The goodness-of-fit statistics along this transect include the Normalized Root Mean Square Error (NRMSE), Mean Absolute Error (MAE), squared correlation coefficient,, and Bias as given in Table 3.