Page 133 - Introduction to Computational Fluid Dynamics

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5.2.3 Pressure-Correction Equation 2D CONVECTION – CARTESIAN GRIDS
In the collocated-grid SIMPLE algorithm, the nodal velocities are determined using
Equations 5.12 written for = u 1 and u 2 . The pressure gradients appearing in the
source terms of these equations are of course evaluated by central difference [for
l
l
l
example, ∂p/∂x 1 | P = (p − p )/(2 x 1 ), where p is the guessed pressure field
W
E
l
l
and l is the iteration number]. The task now is to correct the u and p fields such
i
that mass conservation over the control volume surrounding node P is satisfied. To
do this, and to remain consistent with the SIMPLE-staggered grid, we imagine that
l
the momentum equations are also being solved for the cell-face velocities u . The
fi
discretised versions of these imagined equations with underrelaxation will appear as

α l+1 ∂p l+1 l l
l+1
u = A k u − V + D + (1 − α)u , (5.20)
f1 f1,k u 1 f1
AP u f1 ∂x 1
k

α l+1 ∂p l+1 l l
l+1
u = A k u − V + D + (1 − α)u , (5.21)
f2 f2,k u 2 f2
AP u f2 ∂x 2
k
l
l
where D and D contain source terms (if any) other than the pressure gradient,
u 1 u 2
α is the underrelaxation factor, and the summation symbol indicates summation
over all immediate neighbours of the cell-face location under consideration. Thus,
when Equation 5.20 is written for cell face e, for example, running counter k refers
to locations ee, Ne, w, and Se. Now, at iteration level l + 1, it is expected that
∂(ρ l+1 ) 1 ∂ l+1 l+1 1 ∂ l+1 l+1
+ r ρ u f1 + r ρ u f2 = 0. (5.22)
∂t r ∂x 1 r ∂x 2
Substituting Equations 5.20 and 5.21 in Equation 5.22 we can show that
∂(ρ l+1 ) 1 ∂ l+1 l 1 ∂ l+1 l
+ r ρ u f1 + r ρ u f2
∂t r ∂x 1 r ∂x 2

1 ∂ r ρ l+1 α l+1 ∂p l+1
l
= AP u f1 u − A k u f1,k + V − D l
f1
r ∂x 1 AP u f1 ∂x 1 u 1
k

1 ∂ r ρ l+1 α l l+1 ∂p l+1 l
+ AP u f2 u − A k u f2,k + V − D .
f2
r ∂x 2 AP u f2 ∂x 2 u 2
k
(5.23)
To develop the pressure-correction equation, we introduce the following
substitutions:
l
l
l

u l+1 = u + u , u l+1 = u + u , p l+1 = p + p , (5.24)

f1 f1 f1 f2 f2 f2 m