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Exercise 1
Laminar Pipe Flow
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 uin = 1 m/s (the conditions from the FLUENT case considered in class). 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 points × radial points). The mesh files can be downloaded from Blackboard.
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 τxy at the wall in the fully-developed region for the three meshes. Calculate the corresponding value from fully-developed pipe analysis in HW6. 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 uc is the centerline value from FLUENT and uexact is the corresponding exact (analytical) value. We expect the error to take the form
!error2_120.png!
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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