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.

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

Problem
Use FLUENT to resolve the developing flow in a pipe (same configuration as was done in class) for a pipe Reynolds number of 10,000 on the following meshes: 100×5, 100×20 with uniform spacing in the radial direction. Plot the skin friction cf as a function of axial location for each grid. Compare the exit value with the expected value for fully developed flow (e.g., see White pgs. 345-346). Recall that a key question for the integrity of the mesh is the non-dimensional value of the first nodal point:

This should be either less than 4 (so that you resolve down into the viscous sublayer) or greater than 30 (where wall functions can accurately compensate for the poorly resolved viscous sublayer). Intermediate values can lead to greater errors. Calculate the value of y1+ for each mesh; use that to help explain (briefly) the trends in the agreement that you observe.

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Ansys Exercises (2-D Pipe!)