Engineering and Construction

Mechanical Engineering

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solve the homework

Ansys Exercises (2-D Pipe!)

Consider developing ?ow in a pipe of length L = 8 m, diameter D = 0.2 m, ? = 1 kg/m3 , µ =2 × 10^-3 kg/m s, and entrance velocity u_in = 1 m/s. Use FLUENT with the “second-order upwind” scheme for momentum to solve for the ?ow?eld on meshes of 100 × 5, 100 × 10 and 100 × 20 (axial divisions × radial divisions).

1. Plot the axial velocity pro?les at the exit obtained from the three meshes. Also, plot the corresponding velocity pro?le obtained from fully-developed pipe analysis. Indicate the equation you used to generate this pro?le. In all, you should have four curves in a single plot. Use a legend to identify the various curves. Axial velocity u should be on the abscissa and r on the ordinate.

2. Calculate the shear stress Tau_xy at the wall in the fully-developed region for the three meshes. Calculate the corresponding value from fully-developed pipe analysis. For each mesh, calculate the % error relative to the analytical value. Include your results as a table:

3. At the exit of the pipe where the ?ow is fully-developed, we can define the error in the centerline velocity as

where u_c is the centerline value from FLUENT and u_exact is the corresponding exact (analytical) value. We expect the error to take the form

where the coefficient K and power p depend upon the order of accuracy of the discretization. Using MATLAB, perform a linear least squares ?t of

to obtain the coe?cients p and K. Plot ? vs. ?r (using symbols) on a log-log plot. Add a line corresponding to the least-squares ?t to this plot.

Hint: In FLUENT, you can write out the data in any “XY” plot to a ?le by selecting the “Write to File” option in the Solution XY Plot menu. Then click on Write and enter a ?lename. You can strip the headers and footers in this ?le and read this into MATLAB as column data using the load function in MATLAB.

4. Let’s see how p changes when using a ?rst-order accurate discretization. In FLUENT, use “?rst-order upwind” scheme for momentum to solve for the ?ow?eld on the three meshes. Repeat the calculation of coe?cients p and K as above. Add this ? vs. ?r data (using symbols) to the above log-log plot. Add a line corresponding to the least-squares ?t to this plot. In all, you should have four curves on this plot (two each for second- and ?rst-order discretization). Make sure you include an appropriate legend in the ?gure.

If you have trouble doing this without a tutorial — see:

https://confluence.cornell.edu/display/SIMULATION/FLUENT+-+Laminar+Pipe+Flow+-+Problem+Specification

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