Page 186 - Introduction to Computational Fluid Dynamics

P1: IWV
CB908/Date
0521853265c06
6.2 CURVILINEAR GRIDS
1
∂ξ 2 0 521 85326 5 ∂x 1 1 ∂x 2 β 2 1 May 25, 2005 11:10 165
= cof =− = , (6.11)
∂x 1 Det A ∂ξ 2 Det A ∂ξ 1 Det A
1 1 β 2
∂ξ 2 ∂x 2 ∂x 1 2
= cof = = , (6.12)
∂x 2 Det A ∂ξ 2 Det A ∂ξ 1 Det A
where the βs are called the geometric coefficients and are given by
1 ∂x 2 2 ∂x 1 1 ∂x 2 2 ∂x 1
β = , β =− , β =− , β = . (6.13)
1 1 2 2
∂ξ 2 ∂ξ 2 ∂ξ 1 ∂ξ 1
Further, it follows that
∂x 1 ∂x 2 ∂x 1 ∂x 2 1 2 1 2
Det A = − = β β − β β = J, (6.14)
1
1
2
2
∂ξ 1 ∂ξ 2 ∂ξ 2 ∂ξ 1
where symbol J stands for the Jacobian of the matrix A. We can now rewrite
Equations 6.2 and 6.3 as
∂ 1 1 ∂ 1 ∂
= β 1 + β 2 , (6.15)
∂x 1 J ∂ξ 1 ∂ξ 2
∂ 1 2 ∂ 2 ∂
= β 1 + β 2 . (6.16)
J
∂x 2 ∂ξ 1 ∂ξ 2

6.2.2 Transport Equation

The first task is to transform the general transport equation (5.1) from the (x 1 , x 2 )
coordinate system to the (ξ 1 ,ξ 2 ) coordinate system using relations (6.15) and (6.16).
Thus,
∂(ρ ) 1 ∂(rq 1 ) ∂(rq 1 ) ∂(rq 2 ) ∂(rq 2 )
r + β 1 1 + β 2 1 + β 1 2 + β 2 2 = rS.
∂t J ∂ξ 1 ∂ξ 2 ∂ξ 1 ∂ξ 2
(6.17)

This equation can also be written as

1
1
2
2
∂(ρ ) ∂ β rq 1 ∂ β rq 1 ∂ β rq 2 ∂ β rq 2
2
1
2
1
rJ + + + +
∂t ∂ξ 1 ∂ξ 2 ∂ξ 1 ∂ξ 2
1 1 2 2
∂β ∂β ∂β ∂β
1 2 1 2
= rq 1 + + rq 2 + + rJ S. (6.18)
∂ξ 1 ∂ξ 2 ∂ξ 1 ∂ξ 2
Using definitions (6.13), however, we can show that the terms in the square brackets
are identically zero. Hence, Equation 6.18 can be written as
∂(ρ ) ∂
∂
1
2
1
2
rJ + β rq 1 + β rq 2 + β rq 1 + β rq 2 = rJ S.
1
2
2
1
∂t ∂ξ 1 ∂ξ 2
(6.19)