Page 190 - Introduction to Computational Fluid Dynamics

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6.2 CURVILINEAR GRIDS
where the convective coefficients are given by May 25, 2005 11:10 169
1 2
C e = ρ e r e U f1,e = ρ e r e β u 1e + β u 2e ,
1e 1e
1 2
C w = ρ w r w U f1,w = ρ w r w β u 1w + β u 2w ,
1w 1w
2 2
C n = ρ n r n U f2,n = ρ n r n β u 1n + β u 2n ,
1n 2n
2 2
C s = ρ s r s U f2,s = ρ s r s β u 1s + β u 2s , (6.33)
1s 2s
and the diffusion coefficients are

r
eff dA 2 r
eff dA 2
1 1
d e = , d w = ,
J J
e w

r
eff dA 2 r
eff dA 2
2 2
d n = , d s = ,
J J
n s

(r
eff dA 12 ) (r
eff dA 12 )
AC e = , AC w = ,
J J
e w

(r
eff dA 12 ) (r
eff dA 12 )
AC n = , AC s = . (6.34)
J J
n s
In evaluating the convective coefficients (or the mass fluxes at the cell faces),
the u at the cell faces are evaluated by linear interpolation from neighbouring
nodal velocities. For example, u 1e = 0.5(u 1P + u 1E ). Similarly, the values of
at the control-volume corners are also linearly interpolated. For example, ne =
0.25( P + E + NE + N ). Finally, we note that the diffusion coefficients again
have dimensions of conductance.
Equation 6.32 applies to = u 1 , u 2 and all other scalar variables. When = 1,
however, we recover the mass-conservation equation. Thus,
r P J P
o
ρ P − ρ P + C e − C w + C n − C s = 0. (6.35)
t
Now, making use of this equation, we can recast Equation 6.32 again in the following
familiar form
AP P = AE E + AW W + AN N + AS S + D, (6.36)
where the convective–diffusive coefficients AE, AW, AN, and AS are given by
, 0)], = C e /d e ,
AE = d e [A + max(−P c e P c e
, 0)], = C w /d w ,
AW = d w [A + max(P c w P c w
, 0)], = C n /d n ,
AN = d n [A + max(−P c n P c n
, 0)], = C s /d s ,
AS = d s [A + max(P c s P c s
o
r ρ J
AP = AE + AW + AN + AS + . (6.37)
t P