as5-sol

as5-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 Assignment No: 5 Due Date: April 4, 2011 Instructor: J. Murthy 1. We are asked to reconstruct φ e from φ e = φ P + Ψ ( r e ) ∆ x 2 φ P − φ W ∆ x with r e = φ E − φ P φ P − φ W using the superbee and minmod limiters. The superbee limiter is given by: Ψ ( r ) = max [ , min ( 2 r , 1 ) , min ( r , 2 )] The minmod limiter is given by: Ψ ( r ) = min ( r , 1 ) if r > Ψ ( r ) = if r ≤ The values of r e and the corresponding values of Ψ and φ e are given in Tables 2 and 1. Case φ (W,P,E) r e Ψ φ e φ e value Comment a 300,200,100 1 1 φ p + φ P − φ W 2 150 linear extrapolation b 100,50,200-3 φ P 50 extremum, therefore use upwinding c 300,150,100 1 3 r e (= 1 3 ) φ P + φ E − φ P 2 125 use downwind gradient d 300,250,100 3 1 φ p + φ P − φ W 2 225 use upwind gradient Table 1: Problem 1 - φ e Values Using Minmod Limiter Case φ (W,P,E) r e Ψ φ e φ e value Comment a 300,200,100 1 1 φ p + φ P − φ W 2 150 linear extrapolation b 100,50,200-3 φ P 50 extremum, therefore use upwinding c 300,150,100 1 3 2 r e (= 2 3 ) φ P + ( φ E − φ P ) = φ E 100 use full downwind difference d 300,250,100 3 2 φ p + ( φ P − φ W ) 200 use full upwind differnece Table 2: Problem 1 - φ e Values Using Superbee Limiter 2. We are given: d dx ( ρ u φ ) = − 2 π A L sin parenleftbigg 2 π L x parenrightbigg with φ ( ) = 100 and A = 100. 1 (a) Integrating the governing equation over the control volume and using higher-order upwinding for face values yields the following discrete equation for an interior point: a E = ....
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as5-sol - ME 608 Numerical Methods in Heat Mass and...

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