Page 91 - Introduction to Computational Fluid Dynamics

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1D CONDUCTION–CONVECTION
14. It is desired to investigate stability of the equation in Exercise 13 for differ-
ent values of weighting factor ψ (see Equation 2.6) so that the equation will
read as
2
2
∂T ∂ T ∂ T o
= ψ + (1 − ψ) .
∂τ ∂ X 2 ∂ X 2
(a) Obtain a discretised analogue of this equation and substitute the exact
solution for temperatures at P, E, and W. Set X P = π/2 and show that
2
1 − 4 A (1 − ψ)sin ( X/2)
exp(− τ) = 2 ,
1 + 4 A ψ sin ( X/2)
2
where A = AE = AW = τ/( X) .
(b) Hence, show that AR for any X P is given by
T P
AR = = exp(− τ).
T o
P
(c) For stability, |AR| < 1. Hence, show that for ψ< 0.5, the solution is sta-
ble when A < 0.5/(1 − 2ψ) whereas, for 0.5 ≤ ψ ≤ 1, the solution is
unconditionally stable.