Page 140 - Introduction to Computational Fluid Dynamics

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5.2 SIMPLE – COLLOCATED GRIDS
to Equation 5.32, which is applicable to staggered grids, although the dependent 12:28 119
variables have different meanings.
5.2.5 Overall Calculation Procedure
The sequence of calculations on collocated grids is as follows.

l
1. At a given time step, guess the pressure field p . This may be the pressure field
i, j
from the previous time step.
2. Solve (see the next section) the momentum equations (5.12) once each for
= u 1 and u 2 with problem-dependent boundary conditions. Designate the
l
l
velocity fields so generated by u and u .
1 2
7
3. Form ˙ m i, j (Equation 5.34) using multidimensional interpolations of cell-face
velocity. Now, solve Equation 5.60 with boundary condition (5.58) iteratively
to yield the total pressure-correction p field. The number of iterations may
i, j
not exceed 5 to 10.
4. Recover the mass-conserving pressure correction via Equation 5.59. Thus,
p = p − p = p − 1
p l − p l , (5.61)
m,i, j i, j sm,i, j i, j i, j i, j
2
where p l is evaluated from Equation 5.43.
i, j
5. Correct the pressure and velocity fields according to
l+1 l
p = p + β p , 0 <β < 1, (5.62)
i, j i, j m,i, j

u l+1 = u l 1,i, j − r α x 2 (p m,i+1/2, j − p m,i−1/2, j ), (5.63)
1,i, j
AP u1
i, j

u l+1 = u l 2,i, j − r α x 1 (p m,i, j+1/2 − p m,i, j−1/2 ). (5.64)

2,i, j
AP u2
i, j
u2
Note that AP u1 = AP .
6. Solve the discretised equations (5.12) for all other scalar i, j relevant to the
problem at hand.
7. Check convergence through evaluation of residuals (see the next section) for
momentum and scalar equations. Care is, however, required in calculation of
mass residuals as will be discussed shortly.
l
l
8. If the convergence criterion is not satisfied, treat p l+1 = p , l+1 = and
return to step 2
o
9. To execute the next time step, set all = l+1 and return to step 1.

7 Although multidimensional interpolation is prescribed, in actual computations, one-dimensional
interpolations suffice in most applications.