Page 266 - Introduction to Computational Fluid Dynamics

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8.4 SORENSON’S METHOD
4.From the known coordinates on the south and the north boundaries, interpolate 16:28 245
x 1 (ξ 1 ,ξ 2 ) and x 2 (ξ 1 ,ξ 2 ) to serve as the initial guess. Usually, linear interpolation
between corresponding points on the south and north boundary for each ξ 1
suffices.
5.Now evaluate ∂x 1 /∂ξ 1 , ∂x 2 /∂ξ 1 , ∂x 1 /∂ξ 2 , and ∂x 2 /∂ξ 2 at ξ 2 = 0 and ξ 2 = ξ 2max .
These remain fixed for all subsequent operations.

Iterations Begin
6.Evaluate L 1 and L 2 at ξ 2 = 0 and ξ 2 = ξ 2max . In these evaluations, the second-
order derivatives at ξ 2 = 0, for example, are represented as follows:
2
∂ 1
= (−7 i,1 + 8 i,2 − i,3 ) − 3( i,2 − i,1 ), (8.48)
∂ξ 2 2
2
2
∂
= i+1,1 − 2 i,1 − i−1,1 , (8.49)
∂ξ 1 2

∂ ∂ 1 ∂ ∂

= − . (8.50)
∂ξ 1 ∂ξ 2 2 ∂ξ 2 i+1 ∂ξ 2

i−1
7.Useequationssuchas8.43and8.44toevaluate P (ξ 1 , 0), Q (ξ 1 , 0), P (ξ 1 ,ξ 2max ),
and Q (ξ 1 ,ξ 2max ).
2
8.Using the preceding information and already chosen constants a, b, c, and d,
evaluate P (ξ 1 ,ξ 2 ) and Q (ξ 1 ,ξ 2 ) at all nodes in the domain. Between iterations,
underrelaxation in evaluation of P and Q is advised.
9.Specify boundary conditions for x 2 at the west and east boundaries. Here, care
must be taken to take account of the type of grid being generated. If an H- or
C-type grid is being generated, one must specify the x 2 from known equations
of the west and east boundaries since x 1 values are already known (see step 1).
Alternatively, one may specify the ∂x 2 /∂ξ 1 condition to let the ξ 2 = constant line
intersect the boundary at a desired angle. If an O-type grid is being generated
then one specifies periodic condition (0,ξ 2 ) = (ξ 1max ,ξ 2 ).
10.Solve Equation 8.45 for = x 1 , x 2 and check convergence.
11.If the convergence criterion is not met, go to step 6.

8.4.5 Applications

H Grid
Figure 8.6 shows the grid for a flow between parallel plates with a constriction.
South (x 2 = 0) is the axis of symmetry, north is a wall, west (x 1 =−8) is the

2 Typically, a = b = c = d = 0.7. If too small a value is used (0.2, say), the effect of the constants
decays slowly away from the south/north boundaries. If too large a value is chosen, the effect decays
very rapidly.