chpt10examp.doc

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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 steady-state 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 cats

eyes.

Problem Setting: 419 nodes 775 elements


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Solutions: (velocity contour and velocity vectors)

Isotherms:

Example 10.2 In this second example, we examine flow over a backward-

facing step. The Reynolds number is 800, and the flow enters from the left

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and exits from the right hand boundary. We use a 20 x 40 mesh consisting

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

flow at Re = 100 steady or transitory?

Problem Setting for both Re=10 and Re=100

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Solution for Re=10

Solution for Re=100, Downstream of the cylinder the

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Karman path is clearly visible in the plot of the last

time step.


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.

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Problem Setting (half)1434 nodes and 2688 elements

Solution Isotherm(half)

Problem Setting (2831 nodes, 5346 elements):

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Solutions Isotherm:


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