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final-exam-sol

# final-exam-sol - ME 608 Numerical Methods in Heat Mass and...

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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 φ in a square domain, as shown in Fig. ?? . The governing equation is given by: ∇ · ( ρ V φ ) = ∇ · ( Γ∇ φ )+( 1- φ ) The left and bottom boundaries are at φ = 0 and φ = 1 respectively, while the other two boundaries are outflow boundaries. The flow field is given by V = i + j . The density ρ = 1 and Γ = 1 . 0. The side of the square is given by L = 2. (a) Derive the discrete algebraic equations for φ for the four cell centroids shown at the finest mesh level in Fig. ?? . Use a first-order 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 φ = . 5, execute one V cycle with ν 1 = 0 and ν 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- ΓΔ y Δ x ( φ 2- φ 1 )+( ρ u Δ y ) e φ 1 + 2 ΓΔ y Δ x ( φ 1- φ le f t )- ( ρ u Δ y ) w φ le f t- ΓΔ x Δ y ( φ 4- φ 1 )+( ρ v Δ x ) n φ 1 + 2 ΓΔ x Δ y ( φ 1- φ bot )- ( ρ v Δ x ) s φ bot = ( 1- φ 1 ) Δ x Δ y Cell 2 ( ρ u Δ y ) e φ 2 + ΓΔ y Δ x ( φ 2- φ 1 )- ( ρ u Δ y ) w φ 1- ΓΔ x Δ y ( φ 3- φ 2 )+ ( ρ v Δ x ) n φ 2 + 2 ΓΔ x Δ y ( φ 2- φ bot )- ( ρ v Δ x ) s φ bot = ( 1- φ 2 ) Δ...
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final-exam-sol - ME 608 Numerical Methods in Heat Mass and...

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