This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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 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...
View
Full
Document
This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue.
 Fall '10
 NA

Click to edit the document details