Page 265 - Introduction to Computational Fluid Dynamics

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NUMERICAL GRID GENERATION
Identical expressions again emerge for P (ξ 1 ,ξ 2max ) and Q (ξ 1 ,ξ 2max ). One can
thus prescribe P and Q functions over the whole domain using Equations 8.39 and
8.40.
8.4.3 Discretisation
Equations 8.26 and 8.27 can be written in the following general form:
2
2
∂ ∂ ∂ ∂ ∂ ∂
P ∗ − α 2 + Q ∗ − γ 2 =−2β , (8.45)
∂ξ 1 ∂ξ 1 ∂ξ 2 ∂ξ 2 ∂ξ 1 ∂ξ 2
2
2
∗
∗
where = x 1 , x 2 , P =−PJ , and Q =−QJ . Equation 8.45, being of the
conduction–convection type, can be discretised using the UDS to yield
AP P = AE E + AW W + AN N + AS S + S, (8.46)
where
1
∗
∗
AE = α P + (| P |− P ),
P
P
2
1
∗
∗
AW = α P + (| P |+ P ),
P
P
2
1
∗
∗
AN = γ P + (| Q |− Q ),
P
P
2
1
∗
∗
AS = γ P + (| Q |+ Q ),
P
P
2
AP = AE + AW + AN + AS,
S =−2β P ( ne − nw − se + sw ). (8.47)
Equation 8.46 can be solved using the ADI method.
8.4.4 Solution Procedure
Sorenson’s method can be implemented through the following steps.
Initialisation
1.Choose coordinates x 1 (ξ 1 , 0), x 2 (ξ 1 , 0), x 1 (ξ 1 ,ξ 2max ), and x 2 (ξ 1 ,ξ 2max ) on the
south and north boundaries, respectively. Also specify x 1 (0,ξ 2 ) (west) and
x 1 (ξ 1max ,ξ 2 ) (east).
1
2.Specify s 0 and s max and θ 0 and θ max . For orthogonal intersection, θ = π/2.
3.Let P (ξ 1 ,ξ 2 ) = Q (ξ 1 ,ξ 2 ) = 0.

1 It will be appreciated that this liberty to specify s 0 and s max can be very useful when south
and north boundaries are walls and the HRE e– turbulence model is employed. One can therefore
place the first node away from the wall in the range 30 < y < 100.
+