Page 154 - Introduction to Computational Fluid Dynamics

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5.5 NOTION OF SMOOTHING PRESSURE CORRECTION
and recover T b from Equation 5.105. Similarly, if the wall temperature T b is specified 12:28 133
then

eff x 1
eff x 1
Su T = Su T + T b , Sp T = Sp T + , (5.108)
y P y P
and q w is recovered from Equation 5.105. For further refinements of the wall-
function approach, see references [41, 69].

= ω k
It is not clear if universal mass transfer laws exist for all mass transfer rates. Fol-
lowing theory developed by Spalding [73], however, it is possible to show that

eff ρ u τ ln(1 + B)
= , (5.109)
y P Pr t (u + + PF) B
1P
where the Spalding number B is given by

ω k,P − ω k,b
B = , (5.110)
ω k,b − ω k,T
and ω k,T is the mass fraction deep inside the wall from where mass transfer is taking
place. Note that as B → 0, ln (1 + B) → B. Further, PF is still given by Equation
5.88 but with Pr replaced by Schmidt number Sc. All other adjustments are the
same as those for the temperature variable.

5.5 Notion of Smoothing Pressure Correction

It is important to consider the notion of smoothing pressure correction introduced
in our analysis of the collocated-grid calculation procedure. This is because, in
the original SIMPLE-staggered grid procedure, such a smoothing correction is not
required. However, its introduction is vital if zigzag pressure prediction is to be
avoided on collocated grids, particularly when coarse grids are used. To understand
the importance of smoothing correction, we consider computation of laminar flow
in a square cavity (see Figure 5.9) of side L that is infinitely long in the x 3 direction.
The top side (the lid) of this cavity is moving in the positive x 1 direction with
velocity U lid (say). Because of the no-slip condition, the linear lid movement sets
up fluid circulation in the clockwise direction. In this case, steady-state equations
for = u 1 , u 2 , and p need to be solved.

Figure 5.10 shows the computed distribution of pressure for Re = U lid L/ν =
100. In Figure 5.10(a), solutions obtained with a 15 × 15 grid are shown at the ver-
tical midplane (x 1 /L = 0.5). The solutions are obtained using both staggered and
collocated grids with identical grid dispositions. However, in the latter, smoothing
pressure correction is not applied (see step 4 of the calculation procedure). It is clear
that whereas the staggered-grid procedure produces a smooth pressure distribution,