Page 192 - Introduction to Computational Fluid Dynamics

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6.2 CURVILINEAR GRIDS
l
Therefore, separating the solution for p ,weget May 25, 2005 11:10 171
P
A
l
p x1,P = p = ,
P
B
1 1 1 1 &
A = β β p E + β β (p ne − p se ) J e
1,e 1,e 2,e 1,e
1 1 1 1 &
+ β β p W − β β (p nw − p sw ) J w
1,w 1,w 2,w 1,w
1 1 1 1 &
+ β β p N + β β (p ne − p nw ) J n
2,n 2,n 2,n 1,n
1 1 1 1 &
+ β β p S − β β (p se − p sw ) J s ,
2,s 2,s 2,s 1,s
β 1 β 1 β 1 β 1 β 1 β 1 β 1 β 1
1,e 1,e 1,w 1,w 2,n 2,n 2,s 2,s
B = + + + . (6.42)
J e J w J n J s
2
2 l
Similarly, evaluation of p is accomplished from ∂ p /∂x = 0 and evaluation
x 2 2
of p is completed.
6.2.6 Overall Calculation Procedure
The overall calculation procedure on curvilinear grids is nearly the same as that on
Cartesian grids. Some important features are highlighted in the following:
1.Read coordinates x 1 (i, j) and x 2 (i, j) for i = 1, 2,..., IN and j =
i
1, 2,..., JN. Hence calculate the geometric coefficients β and areas and vol-
j
umes once and for all.
l
2.At a given time step, guess the pressure field p . This may be the pressure field
i, j
from the previous time step.
3.Solve, using ADI, Equation 6.20 for Cartesian velocity components = u l
1
l
and u with appropriate boundary conditions (see next subsection).
2
4.Evaluate U f1 and U f2 from Equations 6.22 and 6.23. In these evaluations, the
cell-face velocities u f1 and u f2 are evaluated by arithmetic averaging. Hence,
evaluate the source term of the total pressure-correction equation (6.39). Solve
Equation 6.39 to obtain the p field.
i, j
5.Evaluate p i, j as described in the previous subsection. Hence recover p m,i, j to
correct pressure as p l+1 = p l + β p .
i, j i, j m,i, j
6.Correct Cartesian velocities as

1
1

u l+1 = u l 1,P − ρ r α
β 1 P (p m,e − p m,w ) + β 2 P (p m,n − p m,s ) , (6.43)
1,P
AP u 1
2
2

u l+1 = u l 2,P − ρ r α
β 1 P (p m,e − p m,w ) + β 2 P (p m,n − p m,s ) . (6.44)

2,P
AP u 2
u2
Note that AP u1 = AP .
7.Solve for other relevant scalar s.