Exercises

Exercise 1: Vertical Channel Flow

Problem Specification (pdf file)

Exercise 2: Laminar Flow within Two Rotating Concentric Cylinders

Contributed by Prof. John Cimbala and Matthew Erdman, The Pennsylvania State University

Problem Specification (pdf file)

The video below shows how to use ANSYS Fluent to set up and solve a problem like this.

<iframe width="560" height="315" src="https://www.youtube.com/embed/3DnLP9-UruA?rel=0" frameborder="0" allowfullscreen></iframe>

Exercise 3:

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 10^−3 kg/m s, and entrance velocity uin u_in = 1 m/s (the conditions from the FLUENT case
considered in classspecified 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 points divisions × radial
points). The mesh files can be downloaded from Blackboarddivisions).

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 Tau_xy 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:

Mesh

τxy

Image Added% error

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

Image Addedϵ=
uexact

where uc u_c is the centerline value from FLUENT and uexact u_exact is the corresponding exact
(analytical) value. We expect the error to take the form
ϵ = K∆rp
Image Added

where the coefficient coefficient K and power p depend upon the order of accuracy of the dis-
cretizationdiscretization. Using MATLAB, perform a linear least squares fit of
ln ϵ = ln K + p ln ∆r
Image Added

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 (2-3
sentences3sentences).

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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