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{panel} [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] {panel} 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}\\ !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]\\ [See and rate the complete Learning Module|FLUENT - Laminar Pipe Flow] [Go to all FLUENT Learning Modules|FLUENT Learning Modules] |
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