# As5-all - ME 608 Numerical Methods in Heat Mass and Momentum Transfer Assignment No 5 Due Date April 4 2011 Instructor J Murthy 1 Consider the use

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Unformatted text preview: ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer Assignment No: 5 Due Date: April 4, 2011 Instructor: J. Murthy 1. Consider the use of higher-order convection schemes with limiters on the 1-D uniform mesh shown in Fig. 1. Find φe using: ∆x φP − φW φe = φP + Ψ (re ) 2 ∆x φE −φP where re = φP −φW . Compute φe for the four φ patterns given below using (a) the superbee limiter, and (b) the minmod limiter. In each case, explain how φe is related to (φE − φP ) or (φP − φW ). Assume u is positive in the x-direction. (a) φW = 300, φP = 200, φE = 100. (b) φW = 100, φP = 50, φE = 200. (c) φW = 300, φP = 150, φE = 100. (d) φW = 300, φP = 250, φE = 100. WW W P E e ∆x Figure 1: Computational Domain for Problem 1 2. Consider a 1-D convection equation with a source term: 2π A d 2π (ρ uφ ) = − sin x dx L L Consider a 1D uniform mesh on a domain of length L = 10 with ρ = 1 and u = 1, as shown in Fig. 2. You are given that φ (0) = 100 and A = 100. Find the discrete values of φ at the cell centroids using a higher-order interpolation for the face values of φ . For φe , for example, use: φe = φP + Ψ (re ) with re = φE −φP φP −φW ∆x φP − φW 2 ∆x . Use the quadratic limiter discussed in class for Ψ (re ). For faces f1 and fN in Fig. 2 use a pure ﬁrst-order upwind scheme: φe = φP Start with the following initial guess for φ : φ (x) = 100 − 50 1 x L (a) Use a deferred correction strategy and the TDMA to ﬁnd the converged values of φ for any N of your choice. (b) Find the exact solution to the problem. (c) Explore the truncation error behavior of the method by plotting the RMS error E versus ∆x. The RMS error is deﬁned as: E= 1 2 1 φi,computed − φi,exact N∑ N 2 2 I where N is the number of cells. x 1 Φ=100 f 1 f 3 2 f3 f N I ∆x u=1 Figure 2: Computational Domain for Problem 2 3. Problem 6.3 from Patankar’s book 4. Problem 6.5 from Patankar’s book 5. Problem 6.6 from Patankar’s book 2 f N ...
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## This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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As5-all - ME 608 Numerical Methods in Heat Mass and Momentum Transfer Assignment No 5 Due Date April 4 2011 Instructor J Murthy 1 Consider the use

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