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[Problem Specification|FLUENT - Laminar Pipe Flow - Problem Specification]\\
[1. Pre-Analysis & Start-up|FLUENT - Laminar Pipe Flow - PreAnalysis]\\
[2. Geometry|FLUENT - Laminar Pipe Flow - Geometry]\\
[3. Mesh|FLUENT - Laminar Pipe Flow - Mesh]\\
[4. Setup (Physics)|FLUENT - Laminar Pipe Flow - Setup (Physics)]\\
[5. Solution|FLUENT - Laminar Pipe Flow - Solution]\\
[6. Results|FLUENT - Laminar Pipe Flow - Results]\\
[7. Verification & Validation|FLUENT - Laminar Pipe Flow - Verification & Validation]\\ {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:
{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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