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[Problem Specification|FLUENT - Laminar Pipe Flow Problem Specification]\\
[1. Create Geometry in GAMBIT|FLUENT - Laminar Pipe Flow Step 1]\\
[2. Mesh Geometry in GAMBIT|FLUENT - Laminar Pipe Flow Step 2]\\
[3. Specify Boundary Types in GAMBIT|FLUENT - Laminar Pipe Flow Step 3]\\
[4. Set Up Problem in FLUENT|FLUENT - Laminar Pipe Flow Step 4]\\
[5. Solve\!|FLUENT - Laminar Pipe Flow Step 5]\\
[6. Analyze Results|FLUENT - Laminar Pipe Flow Step 6]\\
[7. Refine Mesh|FLUENT - Laminar Pipe Flow Step 7]\\ {color:#ff0000}{*}Problem 1{*}{color}\\
[Problem 2|FLUENT - Laminar Pipe Flow Problem 2]
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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:
!ps1eq3.jpg!
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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Problem 1
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:
where Uc is the centerline value from FLUENT and Uexact is the exact analytical value for fully-developed laminar pipe flow. We expect the error to take the form:
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:
to obtain the coefficients K and p. You can look up the value of Uexact 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).
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
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