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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 higherorder convection schemes with limiters on the 1D 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 xdirection. (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 1D 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 higherorder 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 ﬁrstorder 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 UniversityWest Lafayette.
 Fall '10
 NA

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