Page 90 - Introduction to Computational Fluid Dynamics

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EXERCISES
6. Consider the steady 1D conduction–convection problem discussed in this chap- 10:54 69
ter. Assume a nonuniform grid (i.e., x e = x w ). Hence, derive expressions for
AE, AW, and AP using the power-law scheme. If E = 1 and W = 0, calcu-
= u x e /α =−10, −5, −1, 0, 1, 5, and 10 when x e / x w =
late P for P c e
).]
1.2. [Hint: Start with Equation 3.20 with ψ e = F(P c e ) and ψ w = F(P c w
7. Show that if f in Equation 3.40 equals its associated ξ, Equations 3.38 and 3.39
will yield the UDS formula. Hence, derive Equations 3.42 and 3.43 and the
expression for the S TVD term in Equation 3.44.
8. Use E = 1 and W = 0 and determine the variation of P with P c for the
TVD scheme when EE = 5 and WW =−0.1. Assume −200 < P c < 200
and use the Lin–Lin and HLPA schemes. Assume a uniform grid. Compare
your results with those given in Table 3.1 and comment on the result. (Hint:
Iterations are required.)

9. Show that for a general differencing scheme, the false conductivity is given
by k false = ρ C p u x (ψ − 0.5), where ψ is defined by Equation 3.20. Hence,
compare k false for UDS and HDS and comment on the result. Assume a uniform
grid.
10. Runchal [61] developed a controlled numerical diffusion with internal feed-
back (CONDIF) scheme capable of sensing the shape of the local profile.
According to this scheme, AE and AW in Equation 3.14 are given by
1 |P c |− P c |P c |+ P c

AE = 1 + 1 + , AW = 1 + (1 + R) ,
R 4 4
where

∂ /∂ X | e ( E − P ) X w
R = = .
∂ /∂ X | w ( P − W ) X e
Further, the values of R are constrained as follows: If R < 1/R max then R =
1/R max ;if R > R max then R = R max . Typical values assigned to R max vary
between 4 and 10. Assuming a uniform grid, show that

(a) If R = 1, the CONDIF scheme is the same as the UDS.
(b) CONDIF represents both convection and diffusion terms to second-order
accuracy irrespective of the sign and the magnitude of the Peclet number.
(c) Taking W = 0 and E = 1, compare values of P for | P c | < 20 with the
exact solution given in Table 3.1. Carry out this comparison for R max = 4
and 10.

11. Derive Equations 3.55, 3.57, 3.63, and 3.64.
12. Starting with Equation 3.59, show the correctness of Equations 3.60.
13. Verify that T = T 0 exp(−τ)sin(X) is an exact solution to the unsteady heat
2
2
conduction equation ∂T /∂τ = ∂ T /∂ X .