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pset_4_sol - 2.29 Numerical Fluid Mechanics Solution of...

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2.29: Numerical Fluid Mechanics Solution of Problem Set 4 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.29 NUMERICAL FLUID MECHANICS— SPRING 2007 Solution of Problem Set 4 Totally 120 points Posted 04/21/07, due Thursday 4 p.m. 05/08/07, Focused on Lecture 18 to 22 Problem 4.1 (120 points): Steady state unidirectional fully developed incompressible laminar flow in a rectangular pipe a) Write down the Navier-Stokes equation for the fully developed incompressible laminar flow in a pipe with arbitrary constant cross section and simplify the equations 1 . Furthermore simplify the equation for the case when the pressure gradient is fixed. Explain when your assumption holds. b) What is the name of this equation? Categorize that and give at least five examples of other cases (possibly from other physical domains) where we encounter the same equation. 2b 2a x y Now consider the upper rectangular pipe where . c) Find an analytical solution by separation of variables. d) (EXTRA CREDIT 10 Points) Explain how your solution (method) changes if we had a pulsating flow. You do not need to solve it thoroughly. 1 If you have difficulty in deriving the equations you can look at different basic textbooks. In particular “Analysis of Transport Phenomena” by W. M. Deen can be suggested. a ! b 1
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2.29: Numerical Fluid Mechanics Solution of Problem Set 4 e) Compute the maximum shear stress and the net volumetric flow rate. f) Compare the maximum shear stress and the pressure gradient in a square pipe with a circular pipe with the same area and the same flow rate. By now we have solved the equation analytically for our simple cross section, but in general it will be very hard or almost impossible to solve it analytically for arbitrary cross sections. Indeed we usually have to rely on numerical methods and here we will develop a finite difference method to solve the equation. g) Develop a finite difference scheme to solve the equation for the square pipe. Start by a mesh consisting of 2 elements (3 nodes) in each directions and refine it in each step by a factor of two until your net flow rate computed has a relative error of less than 0.1% compared to the previous step. Use the proper integration for computing the flow rate. h) Compare the flow rate computed above with the analytical solution. i) Plot the flow rate as a function of mesh size and discuss the curve slope. j) Plot the numerical and analytical velocity contours. k) Due to Laplacian operator in the equations it sounds appealing to use a uniform grid. However, it is not always possible to use a uniform grid, especially in extreme cases like when a b ! 0 . To manipulate those cases it is very good to nondimensionalize the equation. So nondimensionalize the equation with ! = x a , " = y b ( ! , " ) .
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