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[Problem Specification|FLUENT - Laminar Pipe Flow - Problem Specification]\\
[1. |FLUENT - Laminar Pipe Flow - Pre-Analysis & Start-Up]\\
[ Geometry|FLUENT - Laminar Pipe Flow - Geometry]\\
[ Mesh|FLUENT - Laminar Pipe Flow - Mesh]\\
[ |FLUENT - Laminar Pipe Flow - Setup (Physics)]\\
[ Solution|FLUENT - Laminar Pipe Flow - Solution]\\
[ Results|FLUENT - Laminar Pipe Flow - Results]\\
[ Validation|FLUENT - Laminar Pipe Flow - Verification & Validation]\\
{color:#ff0000}{*}Exercise 1{*}{color}\\
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h2. Problem 1
h4. Problem
a) Consider the problem solved in this tutorial. At the exit of the pipe, we can define the error in the calculation of the centerline velocity as:
{latex}
\large
$$
{\varepsilon} = {\mid U_c - U_{exact} \mid}
$$
{latex}
where _U{_}{_}{~}c{~}_ is the centerline value from FLUENT and _U{_}{_}{~}exact{~}_ is the exact analytical value for fully-developed laminar pipe flow. We expect the error to take the form:
\\
{latex}
\large
$$
{\varepsilon} = {K \Delta r^p }
$$
{latex}
where the coefficient _K_ and the power _p_ depend upon the method . Consider the solutions obtained on the 100x5, 100x10, and 100x20 meshes. Using MATLAB, perform a linear least squares fit of:
{latex}
\large
$$
{\ln \varepsilon} = {\ln K + p \ln \Delta r}
$$
{latex}
to obtain the coefficients _K_ and _p_. You can look up the value of _U{_}{_}exact_ from any introductory textbook in fluid mechanics such as _Fluid Mechanics_ by F. White. Explain why your values make sense.
b) Repeat the above exercise using the "first-order upwind" scheme for the momentum equation. Contrast the value of _p_ obtained in this case with the previous one and explain your results briefly (2-3 sentences).
h4. Hints
Note that the first or second order discretization applies only to the convective terms in the Navier-Stokes equations. The viscous terms are always second order accurate.
[Go to Problem 2|FLUENT - Laminar Pipe Flow - Problem 2]\\
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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:
Mesh
τxy
% error
3. At the exit of the pipe where the flow is fully-developed, we can define the error in the
centerline velocity as
uc − uexact |
ϵ=
uexact
where uc is the centerline value from FLUENT and uexact is the corresponding exact
(analytical) value. We expect the error to take the form
ϵ = K∆rp
where the coefficient K and power p depend upon the order of accuracy of the dis-
cretization. Using MATLAB, perform a linear least squares fit of
ln ϵ = ln K + p ln ∆r
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
sentences).
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.
Go to Problem 2