Page 135 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – CARTESIAN GRIDS
2. In incompressible flows, density is independent of pressure. Therefore, ρ l+1 =
l
ρ = ρ (say). Derivation of the pressure-correction equation for compressible
flow is left to the reader as an exercise (see Date [15, 17]).
3. On staggered grids, the momentum equations are solved at the cell faces and,
therefore, residuals R uf1 and R uf2 must vanish at full convergence, rendering
˙ m R = 0. Although this state of affairs will prevail only at convergence, one
may ignore ˙ m R even during iterative solution. Thus, effectively, the pressure-
correction equation applicable to computations on staggered grids is
l+1
l+1
1 ∂ ρ r α V ∂p 1 ∂ ρ r α V ∂p
m m
+
r ∂x 1 AP u f1 ∂x 1 r ∂x 2 AP u f2 ∂x 2
∂(ρ l+1 ) 1 ∂ l+1 l 1 ∂ l+1 l
= + r ρ u f1 + r ρ u f2 . (5.32)
∂t r ∂x 1 r ∂x 2
This equation is derived in [51] via an alternative route. It is solved with the
boundary condition

∂p
= 0. (5.33)
m
∂n
b
The explanation for this boundary condition is given in a later section.
4. On collocated grids, cell-face velocities must be evaluated by interpolation
to complete evaluation of ˙ m P because only nodal velocities u i are computed
through momentum equations. Thus, ˙ m P in Equation 5.30 is evaluated as

l+1 l+1
˙ m P = ρ r u l − ρ r u l x 2
1 e 1 w
V

l+1 l+1
l+1 o
+ ρ r u l − ρ r u l x 1 + ρ − ρ . (5.34)
2 n 2 s P P
t
Now, to evaluate u i ,weuse multidimensional averaging rather than simple one-
dimensional averaging. Thus, for example,

l l
1 1
x 2,n u 1,se + x 2,s u 1,ne
u l = u l + u l + ,
1,e 1,P 1,E
2 2 x 2,n + x 2,s
u l = 1
u l + u l + u l + u l ,
1,se 1,P 1,E 1,S 1,SE
4
u l 1,ne = 1
u l 1,P + u l 1,E + u l 1,N + u l 1,NE . (5.35)
4
Similar expressions can be derived for other interpolated cell-face velocities.
5. On collocated grids, we do not explicitly satisfy momentum equations at the
cell-face locations. Therefore, there is no guarantee that ˙ m R will vanish even at