The computation of turbulent natural convection flows - Turbulence ...

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Pe = P ∗ +P ′ ′
1 +P 2
(5.26)
As a first approximation, the SIMPLE scheme assumes that U ′ ′
e,2 = 0 V n,2 =
0. Then from equations 5.25 and 5.26:
U ∗∗
e = U∗ e +
P ′ ′
P,1 −P
V ∗∗
n = V ∗ n +
E,1
P ′
P,1 −P ′
N,1
DU
DV
(5.27)
The velocitiesU ∗∗
e ,U ∗∗
w ,V ∗∗
n ,V ∗∗
s must satisfy the discretized continuity equation,
requiring that:
[(ρU ∗∗ ) w −(ρU ∗∗ ) e ]∆x+[(ρV ∗∗ ) n −(ρV ∗∗ ) s ]∆y = 0 (5.28)
Substituting forU ∗∗ andV ∗∗ from equation 5.27 leads to a discretized equa-
tion for the pressure corrections P ′
1 , of the form:
where
a P′
′
P P
P,1 = a P′
i
P ′
i,1 +b1
(5.29)
b1 = [(ρU ∗ ) w −(ρU ∗ ) e ]∆x+[(ρV ∗ ) n −(ρV ∗ ) s ]∆y (5.30)
Solution of this equation produces a discretized pressure correction field,
which is used to update the pressure field and also to correct the velocity field,
through eqn 5.27, so that it can satisfy continuity. For a converged velocity
field, solution of eqn 5.29 would result in zero P ′
values at all nodal locations.
In the SIMPLE algorithm, only P ′
1 is used for the pressure field correction.
In the PISO algorithm, a second correction P ′
2 for pressure field is included.
Therefore U ′
e,2
and V ′
n,2
which are assumed zero in the SIMPLE scheme are
non-zero in the PISO scheme and can be calculated by equation 5.24. The sum
of neighbouring corrections is based only on the first stage velocity corrections
(so that is effectively a known quantity).