Circ Basin
Test C1-Ex2: Wind-driven Flow in a Circular Basin
Purpose
The purpose of this test is to verify the steady state linear hydrodynamics when forced by spatially variable winds, a linear bottom friction, and with and without Coriolis force. Specific model features evaluated in this problem are spatially variable winds and Coriolis force.
Problem
Assuming steady state conditions and no advection, diffusion, waves or spatial gradients in atmospheric pressure, the governing equations reduce to
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \left(hU_j\right)}{\partial x_j} = 0 } | (1) |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ij}f_chU_j = -gh \frac{\partial \eta}{\partial x_i} + \frac{1}{\rho}\left(\tau^s_i-\tau^b_i\right)} | (2) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon_{ij}}
is the permutation operator equal to
1 for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i,j=1,2}
; -1 for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i,j=2,1}
; and 0 for
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i=j}
, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_c}
is the Coriolis parameter,
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_j}
is the depth-averaged current velocity in the
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j}
th direction, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau^s_i}
is the wind driving
force per unit water surface area, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau^b_i}
is the
bottom friction. The problem is further simplified by assuming a flat
bed and deep water conditions Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h>>\eta}
, so that
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h}
may be considered constant. A linear bottom friction is
specified as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau^b_i=\epsilon_{i2}x_jW/R }
where
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_c}
is a linear bottom friction coefficient. The wind
stress is given by where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle W}
is the gradient or slope of
the wind speed and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R}
is the radius of the circular basin.
Analytical Solution
Dupont (2001) presented the analytical solution to the problem above. The water surface elevation is given by
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \eta = \begin{array}{cc} \frac{W x_1 x_2}{2ghR} & f_c=0 \\ \frac{W f_c}{Rgh\kappa} \left[ \frac{R^2}{8} + \frac{1}{4}\left(\frac{2\kappa x_1 x_2}{f_c}-x_1^2-x_2^2\right) \right] & f_c \neq 0 \end{array} } | (3) |
The current velocities are independent of the Coriolis coefficient and are given by
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_i = \frac{\epsilon_{ij}x_jW}{2Rh\kappa} } | (4) |
Model Setup
The computational grid, shown in Figure 4, has 5 levels of refinement
from 2km to 125m and a total of 15,272 computational cells.
Figure 4. CMS-Flow computational grid used for the wind-driven flow in
a circular basin.
The CMS model is run to steady state from zero current and water level as initial conditions. The relevant model parameters are summarized in Table 4. Two cases are run: one with Coriolis and one without.
| Parameter | Value |
|---|---|
| Time step | 1 hr |
| Simulation duration | 72 hr |
| Ramp period duration | 24 hr |
| Water depth | 100 m |
| Mixing terms | Off |
| Wall friction | Off |
| Linear bottom friction | On |
| coefficient | 0.001 |
| Coriolis | 0.0, 0.0001 rad/s |
| Wind gradient | 0.0001Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m^2/s^2} |
Results and Discussion
Without Coriolis Force
A comparison of the calculated and analytical current velocities and water levels in the case without Coriolis force are shown in Figure 5. The goodness-of-fit statistics for the velocity components and water level are shown in Table 5. The calculated water level shows excellent agreement with the analytical solution, even along the outer boundary, and is demonstrated by the NMAE of 0.02%. The current velocities agree well with the analytical solution. The largest errors for the current velocities occur near the outer boundary due to the staircase representation of the curved boundary. The error is reduced by increasing the refinement at the boundary.
Figure 5. Analytical (a) and calculated (b) current velocities
and water levels without Coriolis force.
- defined in Appendix A
| Variable | NRMSE, % | NMAE, % | Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^2} | Bias |
|---|---|---|---|---|
| U-velocity | 1.88 | 0.30 | 0.999 | -4.5e-7 m/s |
| V-velocity | 2.51 | 0.37 | 0.998 | 9.7e-9 m/s |
| Water level | 0.03 | 0.02 | 0.999 | 3.14e-8 m |
With Coriolis Force
A comparison of the calculated and analytical solutions of current velocities and water levels for the case with Coriolis force is shown in Figure 6. The goodness-of-fit statistics for the velocity components and water level are shown in Table 6. Similar to the case without Coriolis force, agreement between the calculated water elevation and current velocity and the analytical solutions resulted in a NRMSE of 0.03% for water level and 2.53% for current velocities. The largest errors in current velocity occur adjacent to the outer boundary where the curved boundary exists. A positive Coriolis parameter corresponds to the northern hemisphere where ocean currents are deflected to the right. For this case, the Coriolis force has the net effect of pushing water towards the center of the circular basin creating higher water levels and lower water levels around the perimeter.
From Equation (4), it can be seen that the water level, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \eta} , at the basin center Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_1=x_2=0m} , is equal to Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Wf_cR/\left(8gh\kappa\right) } for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_c \neq 0/s} and equal to 0m for 0/s.
Figure 6. Analytical (a) and calculated (b) current velocities
and water levels with Coriolis force.
- defined in Appendix A
| Variable | NRMSE, % | NMAE, % | R2 | Bias |
|---|---|---|---|---|
| U-Velocity | 1.90 | 0.30 | 0.999 | -4.5e-7 m/s |
| V-Velocity | 2.51 | 0.37 | 0.998 | 4.6e-9 m/s |
| Water level | 0.03 | 0.02 | 0.999 | 9.5e-8 m |
Conclusions and Recommendations
The analytical solution for the steady-state wind-induced linear hydrodynamics in a closed circular basin was simulated. Computed water levels were accurate within 0.03% NRMSE, and showed little influence from the staircase representation of the curved outer boundary. Current velocities were less accurate with a NRMSE of 2.53% due to errors near the outer boundary. For most coastal applications open boundaries are represented by straight boundaries so the staircase boundary does not exist. Curved boundaries usually occur along the wet-dry interface in very shallow water where the current velocities are usually small due to the increased bottom friction. However, if the curved boundary occurs in deep water or in areas where the current velocities are strong, then errors will be incurred due to the staircase representation of the boundary. Nevertheless, the errors may be reduced by increasing the grid refinement along the specific boundary. In the future, this problem can be eliminated by implementing a boundary fitting method, such as a cut-cell or embedded boundary, or quadrilateral mesh.