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Unformatted text preview: ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer Solution to Final Examination Date: May 4, 2011 1:00 3:00 PM Instructor: J. Murthy Open Book, Open Notes Total: 100 points Use the finite volume method in all problems. 1. Consider steady convection and diffusion of a scalar in a square domain, as shown in Fig. ?? . The governing equation is given by: ( V ) = ( )+( 1 ) The left and bottom boundaries are at = 0 and = 1 respectively, while the other two boundaries are outflow boundaries. The flow field is given by V = i + j . The density = 1 and = 1 . 0. The side of the square is given by L = 2. (a) Derive the discrete algebraic equations for for the four cell centroids shown at the finest mesh level in Fig. ?? . Use a firstorder upwind difference scheme for the convective operator. (b) Now derive the discrete algebraic equations for the corrections for the coarse levels 1 and 2 in Fig. ?? using the algebraic multigrid procedure developed in class. (c) Assuming an initial guess at the finest level of = . 5, execute one V cycle with 1 = 0 and 2 = 1. Show your answers at each level clearly. All units may be assumed to be consistent, so that no unit conversion is required. 1 V = i + j =1 =0 Outflow Outflow x y 1 2 3 4 L=2 1 2 1 Level Level 1 Level 2 Figure 1: Computational Domain for Problem 1 2 (a) Discrete Equations at Level 0 Cell 1 y x ( 2 1 )+( u y ) e 1 + 2 y x ( 1 le f t ) ( u y ) w le f t x y ( 4 1 )+( v x ) n 1 + 2 x y ( 1 bot ) ( v x ) s bot = ( 1 1 ) x y Cell 2 ( u y ) e 2 + y x ( 2 1 ) ( u y ) w 1 x y ( 3 2 )+ ( v x ) n 2 + 2 x y ( 2 bot ) ( v x ) s bot = ( 1 2 )...
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