Flat Basin

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

$\rho gh{\frac {\partial \eta }{\partial y}}=\rho _{a}C_{d}\left|W\right|W$

(1)

where $h=\zeta +\eta$ is the total water depth, $\zeta$ is the still water depth, $\eta$ is the water surface elevation (water level) with respect to the still water level, $C_{d}$ is the wind drag coefficient, $y$ is the coordinate in the direction of the wind, $g$ is the gravitational acceleration, $\rho$ is the water density, $\rho _{a}$ is the air density, and $W$ 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)

$\eta ={\sqrt {{\frac {2\rho _{a}C_{d}\left|W\right|W}{\rho g}}\left(y+C\right)+\zeta ^{2}}}-\zeta$

(2)

where $C$ 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

$f_{R}=0.5-0.5\cos \left[\pi \min \left(t,T_{R}\right)/T_{R}\right]$

(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,$R^{2}$, and Bias as given in Table 3.

Figure 2. Calculated water levels in an irregular domain with a flat bed for the cases of wind from the north (left) and from the west (right).

Figure 3. Analytical and calculated water level along the vertical centerline of an irregular basin with flat bed and winds from the south. The calculated results are shown on every 10th grid point for better visualization.

Table 3. Water level goodness-of-fit statistics* for in the idealized wind setup test case.

• defined in Appendix A

Table 2. CMS-Flow settings for the wind setup test case.

Statistic	Value
NRMSE, %	0.01
NMAE, %	0.02
$R^{2}$	0.999
Bias, m	0.000