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Example 10.1 We will discretize the domain in Figure 10.1 using a
uniform mesh of 40 by 40 square bilinear elements. The parameters are
chosen as Ra = 105, Pr = 1.0 and the time step t = 10-3. Starting from
uniform initial conditions T(x,y,0) = u(x,y,0) = v(x,y,0) = 0.0, the steadystate solution is obtained in approximately 400 time steps. Figure 10.2
shows the calculated steady-state solution. In Figure 10.2a only every
other velocity vector has been plotted. In Figure 10.2b the isotherms are
shown at intervals T = 0.1. We observe a boundary layer flow with two
recirculation cells. These were first observed experimentally by Elder
(1965) who called them “cat’s eyes.”
Problem Setting: 419 nodes 775 elements
Solutions: (velocity contour and velocity vectors)
Isotherms:
Example 10.2 In this second example, we examine flow over a backwardfacing step. The Reynolds number is 800, and the flow enters from the left
and exits from the right hand boundary. We use a 20 x 40 mesh consisting
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of bilinear elements. Figure 10.3 shows the initial mesh for the problem
domain, and Fig. 10.4 shows the streamlines overlaid on the velocity
vectors. Notice the recirculation bubble on the top wall, and the
recirculation length behind the step on the bottom wall.
Problem Setting:
Total 12769 elements and 6731 nodes for the mesh
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Solutions:
Streamlines overlaid on the velocity vectors
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10.1
Calculate the flow over the square cavity shown below for Re =
100 using FEMLAB. Assume a nondimensional horizontal
velocity of u = 1 for the top moving from left to right; the
remaining walls are no-slip surfaces. Plot the streamlines, pressure
contours, and velocity vectors.
(insert problem Figure 10.1 here)
Problem Setting:
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Streamlines:
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Pressure Contour:
10.2
Calculate the flow over a circular cylinder shown in the figure
below for Re = 10 and Re = 100 using FEMLAB. What happens to
the wake behind the cylinder as the flow increases in speed? Is the
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flow at Re = 100 steady or transitory?
Problem Setting for both Re=10 and Re=100
Solution for Re=10
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Solution for Re=100, Downstream of the cylinder the Karman path is
clearly visible in the plot of the last time step.
(insert problem Figure 10.2 here)
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10.3
Calculate the flow and temperature distribution within the
eccentric annulus shown below for Ra = 104 using FEMLAB.
Assume a nondimensional temperature of T = 1 for the inner
cylinder and T = 0 for the outer cylinder. You may wish to employ
vertical symmetry to model the domain - be sure to incorporate the
proper boundary conditions along the line of symmetry.
Problem Setting (half)1434 nodes and 2688 elements
Solution Isotherm(half)
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Problem Setting (2831 nodes, 5346 elements):
Solutions Isotherm:
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(insert problem Figure 10.3 here)
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