Page 130 - Introduction to Computational Fluid Dynamics

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5.2 SIMPLE – COLLOCATED GRIDS
where V = r P x 1 x 2 and superscript o represents values at the old time. 12:28 109
Superscript n is dropped for convenience.
Equation 5.6 indicates that the velocities with suffix f appear at the cell faces
of the control volume surrounding node P. Therefore, in SIMPLE-staggered, mo-
mentum equations, Equation (5.4) is solved over control volume n-nW-sW-s and
Equation 5.6 is solved over the control volume w-wS-eS-e without explicit commit-
ment to satisfy mass conservation over these control volumes. The overall strategy
for solution of the flow equations is as follows:
l
1. Guess a p field and solve momentum equations (5.4) and (5.6) over control
l
l
volumes surrounding cell faces to yield u and u fields.
f1 f2
2. These fields, in general, will not satisfy the mass conservation equation (5.6).
3. Derive a mass-conserving pressure-correction equation to satisfy mass conser-
vation over the control volume surrounding node P.
4. Use the pressure correction p so determined to correct the guessed pressure

l
l
l
p and velocities u and u .
f1 f2
For a complete description of the SIMPLE-staggered method, the reader is
referred to [49, 51].

5.2 SIMPLE – Collocated Grids

5.2.1 Main Idea

Although the SIMPLE-staggered grid method enjoyed considerable success par-
ticularly when Cartesian grids were employed, the procedure was found to be in-
convenient when curvilinear or unstructured grids were to be employed to compute
over ever more complex domains. Further, even on Cartesian grids, the process of
discretisation required considerable book keeping because the dimensions of the
control volumes of vector and scalar variables were different.
Since the early 1980s, therefore, researchers began to explore the possibility
4
of implementing the SIMPLE procedure using collocated variables. That is, the
velocity and the scalar variables were to be stored/defined at the same node P (i, j).
This, it was felt, would permit attention to be directed to a single transport equation
(5.1), thereby reducing the book-keeping requirements considerably.
Although convenient, this departure also brought within its wake a major diffi-
culty with respect to the pressure-field prediction. It was found that if the pressure-
correction equation as derived for staggered grids was used to predict pressure on
collocated grids, the predicted pressure distribution showed zigzagness. Depending
on the identified cause of this problem, different researchers (see, for example, [59])

4 In the literature, the procedure with collocated variables is sometimes referred to as a procedure
employing nonstaggered or collocated grids.