Page 155 - Introduction to Computational Fluid Dynamics

P1: IWV
0 521 85326 5
12:28
May 20, 2005
0521853265c05
CB908/Date
134
U-LID 2D CONVECTION – CARTESIAN GRIDS
X 2

L X 1 Figure 5.9. Square cavity with a moving lid.

L

the predicted pressure on the collocated grid is zigzag. Note that the zigzagness
is most pronounced in regions where the staggered-grid pressure distribution con-
siderably departs from linearity. Figure 5.10(b) shows the results obtained with a
41 × 41 grid. Notice that the pressures predicted on both grids are nearly identical
and smooth. This suggests that pressure smoothing is in fact not required when fine
grids are used. In Figure 5.10(c), the coarse-grid solutions are repeated but now
the smoothing pressure correction is applied. It is seen that the predicted pressure
distribution on collocated grids is now smooth though not in exact agreement with
the staggered-grid pressure distribution because of the coarseness of the grid and
also because p is evaluated by multidimensional averaging.
Then, what is the role of the smoothing pressure correction? This can be under-
stood from definition (5.49). The smoothing correction represents the difference
between the point value of pressure p and the control-volume-averaged pressure
p. The latter is defined by Equation 5.43 as the average of linearly interpolated
pressures in the x 1 and x 2 directions. Thus, p sm can be finite only when spatial
variation of pressure p multidimensionally departs from linearity. This is the case
at the midplane of the square cavity. On coarse grids, we observe zigzagness if
smoothing is not applied. However, when grids are refined, p → 0. That is, as a
sm
continuum is approached, no smoothing should be required. The role of smoothing
pressure correction is thus simply to predict smooth pressure distribution on coarse
grids.
We now recall the quantity λ 1 (p − p) introduced in the normal stress expression
in Chapter 1. It was stated in that chapter that λ 1 is trivially zero in a continuum but
is finite in discretised space. We have recovered λ 1 = 0.5 in our definition of p .

sm
But, as the grid size is refined, one approaches a continuum and, therefore, λ 1 can
be set to zero to predict smooth pressure distributions as shown in Figure 5.10(b).
As a corollary, we may now view pressure zigzagness as a spatial counterpart of
the oscillating compressible sphere of isothermal gas explained by Schlichting [65].