Page 185 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – COMPLEX DOMAINS
Chapter 8, some simpler approaches will be introduced. Most CFD practitioners,
however, employ commercially available packages such as ANSYS for unstructured
grid generation.
6.2 Curvilinear Grids
6.2.1 Coordinate Transformation
Our first task is to transform the transport equations in Cartesian coordinates to
those in curvilinear coordinates. Thus, employing the chain rule, we can write the
first-order derivatives as
∂ ∂ξ 1 ∂ ∂ξ 2 ∂
= + , (6.2)
∂x 1 ∂x 1 ∂ξ 1 ∂x 1 ∂ξ 2

∂ ∂ξ 1 ∂ ∂ξ 2 ∂
= + . (6.3)
∂x 2 ∂x 2 ∂ξ 1 ∂x 2 ∂ξ 2

The next task is to determine derivatives of ξ 1 and ξ 2 with respect to x 1 and x 2
knowing functions (6.1). To do this, we note that

∂x 1 ∂x 1
dx 1 = d ξ 1 + d ξ 2 , (6.4)
∂ξ 1 ∂ξ 2

∂x 2 ∂x 2
dx 2 = d ξ 1 + d ξ 2 . (6.5)
∂ξ 1 ∂ξ 2
These relations can be written in matrix form as |dx|=|A||dξ|,or

dx 1 ∂x 1 /∂ξ 1 ∂x 1 /∂ξ 2 d ξ 1

= . (6.6)

dx 2 ∂x 2 /∂ξ 1 ∂x 2 /∂ξ 2 d ξ 2
Now, manipulation of Equations 6.4 and 6.5 will show that
1 ∂x 1 ∂x 2
d ξ 1 = cof dx 1 + cof dx 2 , (6.7)
Det A ∂ξ 1 ∂ξ 1

1 ∂x 1 ∂x 2
d ξ 2 = cof dx 1 + cof dx 2 , (6.8)
Det A ∂ξ 2 ∂ξ 2
where cof denotes cofactor of and Det A stands for determinant of A. Thus, from
the last two equations, it is easy to deduce that

∂ξ 1 1 ∂x 1 1 ∂x 2 β 1 1
= cof = = , (6.9)
∂x 1 Det A ∂ξ 1 Det A ∂ξ 2 Det A

2
∂ξ 1 1 ∂x 2 1 ∂x 1 β 1
= cof =− = , (6.10)
∂x 2 Det A ∂ξ 1 Det A ∂ξ 2 Det A