Flow Patterns in a Lake (COMSOL) Fig. E7.4.1 shows a bird’s-eye view of a lake whose shape is approximated by merging two ellipses and two rectangles. These rectangles protrude to the “west” at AB and to the “south” at CD, and can serve as an inlet and an outlet to and from the lake, respectively.

Question

Flow Patterns in a Lake (COMSOL)

Fig. E7.4.1 shows a bird's-eye view of a lake whose shape is approximated by merging two ellipses and two rectangles. These rectangles protrude to the "west" at [latex]\mathrm{AB}[/latex] and to the "south" at [latex]\mathrm{CD}[/latex], and can serve as an inlet and an outlet to and from the lake, respectively.

Note that the inviscid flow (possibly with rotation) is governed by Poisson's equation:

[latex]- abla^{2} \psi=-\left(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}} ight)=f,[/latex] (E7.4.1)

in which [latex]\psi[/latex] is the stream function. In COMSOL, Eqn. (E7.4.1) is represented as [latex] abla \cdot(-c abla u)=f[/latex], where [latex]u=\psi[/latex] and [latex]c=1[/latex] in our case.

For two-dimensional flow, it can readily be shown that [latex]- abla^{2} \psi=\zeta[/latex], where [latex]\zeta=\left(\partial v_{y} / \partial x-\partial v_{x} / \partial y ight)[/latex] is the vorticity, so that [latex]f[/latex] acts as a source term for the vorticity. If [latex]f=0[/latex], the vorticity will be zero and there will be no rotation. But a positive value of [latex]f[/latex] would mean that the vorticity is positive, corresponding to a counterclockwise rotation. And a negative value of [latex]f[/latex] would indicate a clockwise rotation. Such rotations could result from the shear stresses induced on the surface of the lake by a nonuniform wind, whose velocity [latex]v_{y}[/latex] varies with [latex]x[/latex] as shown in Fig. E7.4.2. Note that in the eastern region [latex](x>0)[/latex] the intensity increases toward the east, and in the western region [latex](x<0)[/latex] the intensity increases toward the west.

Solve for the pattern of streamlines for each of the following three cases, using SI units throughout:

The stream function equals [latex]\psi=0[/latex] along the near shore between points B and C and equals [latex]\psi=100[/latex] along the far shore between points D and A. From B to A and from C to D, [latex]\psi[/latex] varies linearly from zero to 100 . Everywhere, [latex]f=0[/latex].
The rivers are "turned off," so the lake is now isolated and its boundary becomes the single streamline [latex]\psi=0[/latex]. However, the vorticity source term is now [latex]f=2.5 imes 10^{-6} x \mathrm{~s}^{-1}[/latex]. Thus, we would expect a counterclockwise rotation for [latex]x>0[/latex] and a clockwise rotation for [latex]x<0[/latex].
Finally, consider the case of both river flow and the nonuniform wind. The boundary conditions will be the same as for Case 1, but the expression for [latex]f[/latex] will be the same as for Case 2 .

Accepted Answer

Chapter 14 gives more details about COMSOL Multiphysics. All mouse-clicks are left-clicks (the same as Select), unless specifically denoted as right-click (R).

Select the Physics

Open COMSOL. Select Model Wizard, 2D, Mathematics (the little rotating triangle is called a "glyph"), Classical PDEs, Poisson's Equation, Add. In the Unit section, scroll down to set the Dependent variable to Velocity potential [latex]\left(\mathrm{m}^{2} / \mathrm{s}\right)[/latex] and the Source term quantity to Fluid conductance [latex](1 / \mathrm{s})[/latex].
L-click Study, Stationary, Done.

Establish the Geometry

R-click Geometry and select Rectangle. Enter the values for Width and Height as 400 and 200 (all lengths are in meters, m). Note the default is based on the lower left corner being at [latex](0,0)[/latex]. Update the corner position values to [latex]-1300[/latex] and [latex]100 .[/latex] Build Selected.
Repeat Step 3 with values for Width and Height of 200 and 300 and corner position [latex](100,-800)[/latex]. Build Selected. Select Zoom Extents at the top of the Graphics window to display both geometries.
R-click Geometry and select Ellipse. Enter values of 1000 and 500 for the [latex]a[/latex] semi-axis and [latex]b[/latex] semi-axis. Ensure that the Position option Base is Center and enter values of 0 and 200 for the [latex]x[/latex] and [latex]y[/latex] positions.
R-click Geometry and select Circle. Enter values of 400 for the radius. Ensure that the Position option Base is Center and enter values of 200 and [latex]-200[/latex] for the [latex]x[/latex] and [latex]y[/latex] positions.
L-click Build All Objects in the Settings window and L-click Zoom Extents at the top of the Graphics window.
R-click Geometry and select Boolean and Partitions and then Union. Add all geometries to the union by L-clicking each in the Graphics window. Ensure that the Keep interior boundaries option is not selected in the Settings panel.

Build Selected.

Table E7.4.1 Subdomain and Boundary Settings for the Three Cases
Case	Subdomain Coefficients	Dirichlet Boundary Settings, in which u is called r in COMSOL
1	c = 1 (default value),	Near shore (B to C): r = 0
	f = 0	Far shore (D to A): r = 100
		AB: r = 0.5(y − 100)
		CD: r = 0.5(x − 100)
2	c = 1, f = 2.5 × [latex]10^{-6}x [/latex]	All boundary segments: r = 0
3	c = 1, f = 2.5 × [latex]10^{-6}x [/latex]	Same as for Case 1

Define the Boundary Conditions and Source Term (Case 1)

L-click the glyph at the left of Poisson's Equation, select Source, return to Poisson's Equation 1 and set [latex]f[/latex] to 0 in the Settings window.
R-click Poisson's Equation, select Dirichlet Boundary Condition. R-click the newly created Dirichlet Boundary Condition 1. Select Rename and enter North Shore, OK. In the Settings window, enter a value of 100 for [latex]r[/latex], the Prescribed value of [latex]u[/latex]. Add the boundaries defining AD by successively L-clicking each of the six segments in the Graphics panel, each segment color changing from red to blue.
Repeat the process for the South Shore, for the four segments of [latex]\mathrm{BC}[/latex] and a Prescribed value for [latex]u[/latex] of zero.
R-click Poisson's Equation and create a Dirichlet Boundary Condition named Inlet. Enter a prescribed value for [latex]u[/latex] of [latex]0.5[\mathrm{~m} / \mathrm{s}] *(y-100)[[/latex] note that the units of [latex](y-100)[/latex] are implied to be [latex]\mathrm{m}[/latex], so overall it becomes [latex]\left.\mathrm{m}^{2} / \mathrm{s}\right][/latex] and [latex]\mathrm{L}-\mathrm{click}[/latex] segment [latex]\mathrm{AB}[/latex] in the Graphics window. Note: If there is a mismatch in units, the boundary condition will appear in light brown. If there is a syntax error, the boundary condition will appear in red.
Repeat for the Outlet segment CD with a prescribed value of [latex]0.5[\mathrm{~m} / \mathrm{s}] *(x-[/latex] [latex]100)[/latex].

Create the Mesh and Solve the Problem

Select Mesh 1 and Finer under Element size. Build All.
R-click the Study node and select Compute. A surface plot, with a color bar or "legend" on the right, gives the value of the stream function. If you click on any point, the coordinates of that point and the interpolated value of the stream function will appear in a window just below the Graphics window.

Display the Results (Case 1)

R-click Results and L-click 2D Plot Group. R-click the new 2D Plot Group 2 and select Contour. Plot. Within the Settings panel (for ease in reproduction on a black-and-white printer), set Coloring to Uniform, Color to Black, and deselect Color legend. Plot, giving Fig. E7.4.3.
For ease in reproduction, use R-click and hold to "drag" the image closer to the [latex]x[/latex] - and [latex]y[/latex]-axes.
In the File pull-down menu, Save as Ex-7.4-Case 1.mph or similar. Repeat for Case 3 (Easier to do this before Case 2)
L-click the Poisson's Equation 1 node under the Poisson's Equation physics tree and change the value of the source term, [latex]f[/latex], to [latex]2.5 \mathrm{e}-6\left[1 /\left(\mathrm{s}^{*} \mathrm{~m}\right)\right]^{*} x[/latex], with overall units of [latex]\mathrm{s}^{-1}[/latex].
R-click the Study node and select Compute.
R-click Results and L-click 2D Plot Group. R-click the new 2D Plot Group 3 and select Contour. Plot. For black-and-white printing, proceed as in Step [latex]16 .[/latex]
In the File pull-down menu, Save as Ex-7.4-Case-3.mph or similar.Repeat for Case 2
L-click each of the North Shore, South Shore, Inlet, and Outlet boundary conditions and set the Prescribed value to zero in the Settings window.
R-click the Study node and select Compute.
R-click Results and L-click 2D Plot Group. R-click the new 2D Plot Group 4 and select Contour. Plot. For black-and-white printing, proceed as in Step 16 .
In the File pull-down menu, Save as Ex-7.4-Case-2.mph or similar.

Discussion of Results

Case 1. The streamlines are shown in Fig. E7.4.3. The COMSOL results do not show the direction of flow, but it can easily be ascertained as follows. Observe over most of the lake that [latex]v_{x}=\partial \psi / \partial y[/latex] is positive and that [latex]v_{y}=-\partial \psi / \partial x[/latex] is negative. Clearly, the flow enters at [latex]\mathrm{AB}[/latex] and leaves at [latex]\mathrm{CD}[/latex], and arrows to that effect have been added.

The direction of flow could easily be reversed, by assigning [latex]\psi=100[/latex] to the near shore BC and [latex]\psi=0[/latex] to the far shore DA. Note that a Dirichlet boundary condition was used for the inlet, with the stream function varying linearly from zero at [latex]\mathrm{B}[/latex] to 100 at [latex]\mathrm{A}[/latex].

Case 2. The streamlines are shown in Fig. E7.4.4. The lake is now completely closed, with no river flow, so the entire shoreline is the single streamline [latex]\psi=0[/latex]. However, there is now a nonuniform wind blowing, corresponding to Fig. E7.4.2, resulting in counter-rotating circulatory patterns in the two halves of the lake. Arrows have been added to indicate the direction of motion. Recall that the interpolated value of the dependent variable [latex]u(=\psi)[/latex] can be obtained by clicking on any point in the lake. With this feature in mind, the COMSOL results show that the stream function is approximately [latex]\psi=-68.9 \mathrm{~m}^{2} / \mathrm{s}[/latex] at the "eye" of the left-hand half and [latex]\psi=73.7 \mathrm{~m}^{2} / \mathrm{s}[/latex] at the eye of the right-hand half.

Case 3. The streamlines are shown in Fig. E7.4.5. The river has been switched "on" again, and the nonuniform wind still blows. The results are essentially a combination of Cases 1 and 2. The river still enters at AB and leaves at CD, but has to follow a somewhat more tortuous path as it wends its way between the two circulating patterns, which are now diminished in size as a result. The stream function is approximately [latex]\psi=-27.5 \mathrm{~m}^{2} / \mathrm{s}[/latex] at the left-hand eye and [latex]\psi=165.9[/latex] [latex]\mathrm{m}^{2} / \mathrm{s}[/latex] at the right-hand eye.

Question 7.4: Flow Patterns in a Lake (COMSOL) Fig. E7.4.1 shows a bird’s-...

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