Page 188 - Introduction to Computational Fluid Dynamics

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6.2 CURVILINEAR GRIDS
Similarly, it can be shown that May 25, 2005 11:10 167
%

1 2
2 2
dA 2 = β + β dξ 1 . (6.26)
2 2
Comparison of the last two equations with Equations 6.21 shows that dA 1 and
dA 2 represent areas with dξ 1 = dξ 2 = 1.
Elemental Volume
The volume element in curvilinear coordinates is given by
r
r
r
∂ ∂ ∂
dV = · × dξ i dξ j dξ k . (6.27)
∂ξ i ∂ξ j ∂ξ k
Thus, taking i = 1, j = 2, and k = 3, it follows that
r
r
∂ ∂
1 2 2 1
dV = · dξ 1 dξ 2 = β β − β β 2 dξ 1 dξ 2 . (6.28)
2
1
1
∂ξ 1 ∂ξ 2
Comparison of Equation 6.28 with Equation 6.14 shows that the Jacobian J is
nothing but element volume dV with dξ 1 = dξ 2 = 1.
The Normal Fluxes
Note that Equation 6.20 can be written in the following form:

∂(ρ ) ∂ ∂
rJ + [rq ξ 1 ] + [rq ξ 2 ] = rJ S, (6.29)
∂t ∂ξ 1 ∂ξ 2
are given by
where q ξ 1 and q ξ 2

eff 2 ∂ ∂
= ρ U f1 − dA , (6.30)
q ξ 1 1 + dA 12
J ∂ξ 1 ∂ξ 2

eff 2 ∂ ∂
= ρ U f2 − dA . (6.31)
q ξ 2 2 + dA 12
J ∂ξ 2 ∂ξ 1
With reference to Figure 6.4, these expressions represent total (convective + dif-
fusive) transport of normal to the two curvilinear directions, respectively. The
convective transport ρ U fi is thus directed normal to the constant-ξ i lines. In other
i
words, U fi is directed along the contravariant base vector a . Note that, in general,
lines of constant ξ 1 and ξ 2 do not intersect orthogonally. Thus, the total normal
2
diffusive contribution is made up of two components. The first, containing dA ,is
i
a
due to the property gradient along the covariant base vector direction i , the second,
containing dA 12 , is due to the property gradient along the direction ξ j , j = i.If
the intersection of coordinate lines were to be orthogonal, dA 12 = 0. Also, from
Equations 6.13, it is clear that dA 12 can be both positive as well as negative.