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Consider developing flow in a pipe of length L = 8 m, diameter D = 0.2 m, ρ = 1 kg/m3 , µ =2 × 10^−3 kg/m s, and entrance velocity u_in = 1 m/s (the conditions specified in the Problem Specification section). Use FLUENT with the "second-order upwind" scheme for momentum to solve for the flowfield on meshes of 100 × 5, 100 × 10 and 100 × 20 (axial divisions × radial divisions).
1. Plot the axial velocity profiles at the exit obtained from the three meshes. Also, plot the corresponding velocity profile obtained from fully-developed pipe analysis. Indicate the equation you used to generate this profile. 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 flow 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 fit of
to obtain the coefficients p and K. Plot ϵ vs. ∆r (using symbols) on a log-log plot. Add a line corresponding to the least-squares fit to this plot.
Hint: In FLUENT, you can write out the data in any "XY" plot to a file by selecting the "Write to File" option in the Solution XY Plot menu. Then click on Write and enter a filename. You can strip the headers and footers in this file and read this into MATLAB as column data using the load function in MATLAB.
4. Let's see how p changes when using a first-order accurate discretization. In FLUENT, use "first-order upwind" scheme for momentum to solve for the flowfield on the three meshes. Repeat the calculation of coefficients p and K as above. Add this ϵ vs. ∆r data (using symbols) to the above log-log plot. Add a line corresponding to the least-squares fit to this plot. In all, you should have four curves on this plot (two each for second- and first-order discretization). Make sure you include an appropriate legend in the figure.
Contrast the value of p obtained in the two cases and briefly explain your results (2-3sentences).
Hint: To interpret your results, you should keep in mind that the first or second-order upwind discretization applies only to the inertia terms in the momentum equation. The discretization of the viscous terms is always second-order accurate.
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