Page 157 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – CARTESIAN GRIDS
On collocated grids, when density is constant and steady state prevails (as in our
calculation of the square cavity problem), ˙ m P = ρ ∇V and thus ˙ m P = 0, as was

recognized in Section 5.3.2. Now, as our control volume is fixed, ∇V = 0 (which

implies rate of volume change) creates dissipation in the system. It is this dissi-
pation that generates p different from p. We had anticipated this result in Chapter
1. The need for p sm = 0.5(p − p) discovered through our discretisation of equa-
tions applicable to a continuum is therefore not surprising. In summary, therefore,
introduction of p sm simply accounts for the dissipation introduced in the system.
Further discussion of smoothing pressure correction can be found in [16, 17].
Finally, we note that equation 5.41 suggests that p is a solution to the dis-
x 1 ,P
cretised version of
2
∂ p
= 0, (5.111)
∂x 2
1 P
and, similarly, p (Equation 5.42) is a solution to the discretised version of
x 2 ,P
2
∂ p
= 0. (5.112)

∂x 2
2 P
These deductions were also anticipated in Chapter 1.
Before considering applications of our SIMPLE-collocated procedure, it would
be of interest to examine the effect of introduction of p sm on the convergence rate
of the solution procedure. To do this, we plot variation of momentum and mass
residuals with iteration number l for the case of 41 × 41 grid solutions shown in
Figure 5.10(b). Figure 5.11 shows these variations for staggered and collocated
grids. The initial guess and the underrelaxation factors are identical in the two
computations. The figure shows that the convergence histories are almost identi-
cal on both types of grids. Further, computations were stopped when momentum
−5
residuals fell below 10 . At this stage of convergence, the mass residuals are seen
to be smaller by an order of magnitude. Thus, we may conclude that our SIMPLE-
collocated grid procedure is successful in mimicking the SIMPLE-staggered grid
procedure in all respects.
The convergence rate of an iterative procedure greatly depends on the ini-
tial guess for the relevant variables. Among the different variables, the initial
guess for pressure is perhaps the most difficult to provide. Further, in deriving
the pressure-correction equation, quantities A i u and A i v are set to zero.

i i
Thus, the pressure-correction equation is only an approximate one. In spite of this,
computational experience shows that the predicted pressure-correction distribution
provides very good velocity corrections, which are proportional to the pressure-
correction gradient (see Equations 5.63 and 5.64), but a rather poor correction of
pressure itself.