Page 187 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – COMPLEX DOMAINS
Using Equation 5.2, it is now possible to replace Cartesian fluxes q 1 and q 2 . After
some algebra, it can be shown that

∂(ρ ) ∂
eff 2 ∂
rJ + ρ rU f1 − r dA 1
∂t ∂ξ 1 J ∂ξ 1
∂
eff ∂
+ ρ rU f2 − r dA 2 2
∂ξ 2 J ∂ξ 2
∂
eff ∂ ∂
eff ∂
= r dA 12 + r dA 12 + rJ S,
∂ξ 1 J ∂ξ 2 ∂ξ 2 J ∂ξ 1
(6.20)
where
2

dA = β 1 1 2 + β 2 1 2 ,
1
2

dA = β 1 2 + β 2 2 ,
2 2 2
1 1 2 2
dA 12 = β β + β β (6.21)
1 2 1 2
and the contravariant flow velocities are given by
1 2 ∂x 2 ∂x 2
U f1 = β u f1 + β u f2 = u f1 − u f2 , (6.22)
1
1
∂ξ 2 ∂ξ 1
1 2 ∂x 1 ∂x 1
U f2 = β u f1 + β u f2 = u f2 − u f1 , (6.23)
2 2
∂ξ 1 ∂ξ 2
where u f1 and u f2 are the Cartesian velocity components.
6.2.3 Interpretation of Terms
Several new terms appearing in Equation 6.20 can be interpreted using vector
mathematics.

Elemental Area
The elemental area dA i normal to the (ξ j ,ξ k ) plane is given by

r
r
∂ ∂

d A i = × dξ j dξ k , (6.24)
∂ξ j ∂ξ k
where the position vector = ix 1 + jx 2 + kx 3 . For our 2D case, if we set i = 1,

r

r
j = 2, and k = 3 then ∂ /∂ξ 3 = ∂x 3 /∂ξ 3 = 1 because the x 3 and ξ 3 directions
coincide and are normal to the (ξ 1 ,ξ 2 ) plane. Thus, taking unit dimension in the x 3
direction gives

∂x 2
∂ r ∂x 1 1

dA 1 = dξ 2 = i − j 2
dξ 2 = i β + j β dξ 2
1 1

∂ξ 2 ∂ξ 2 ∂ξ 2
%

1 2
2 2
= β + β dξ 2 . (6.25)
1 1