Page 263 - Introduction to Computational Fluid Dynamics

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NUMERICAL GRID GENERATION
Notethat s 0 isthephysicaldistancebetweenboundarynode1anditsneighbouring
interior node 2 in the ξ 2 direction. Similar definitions are introduced at the north
boundary ξ 2 = ξ 2max .
Now, let θ 0 be the angle of intersection between ξ 1 and ξ 2 grid lines at ξ 2 = 0.
Then,
1
2
∇ ξ 1 ·∇ ξ 2 =| a || a | cos θ 0 . (8.31)
Using Equation 8.19, however, it follows that

∂ξ 1 ∂ξ 2 ∂ξ 1 ∂ξ 2
∇ ξ 1 ·∇ ξ 2 = +
∂x 1 ∂x 1 ∂x 2 ∂x 2
ξ 2 =0
⎡ ⎤
0.5
2 2 2 2

0.5
∂ξ 1 ∂ξ 1 ∂ξ 2 ∂ξ 2
= ⎣ + + cos θ 0 ⎦ ,
∂x 1 ∂x 2 ∂x 1 ∂x 2
ξ 2 =0
(8.32)
j
but, from the definitions of β introduced in Chapter 6,
i
∂ξ 1 1 ∂x 2 ∂ξ 2 1 ∂x 2
= , =− ,
∂x 1 J ∂ξ 2 ∂x 1 J ∂ξ 1
∂ξ 1 1 ∂x 1 ∂ξ 2 1 ∂x 1
=− , = . (8.33)
∂x 2 J ∂ξ 2 ∂x 2 J ∂ξ 1
Substituting these definitions and using Equation 8.30, we can write Equation 8.32
as
⎡ ⎤
0.5
2 2

∂x 2 ∂x 2 ∂x 1 ∂x 1 ds 0 ∂x 2 ∂x 1
− + = ⎣ + cos θ 0 ⎦ .
∂ξ 1 ∂ξ 2 ∂ξ 1 ∂ξ 2 d ξ 2 ∂ξ 1 ∂ξ 1
ξ 2 =0
ξ 2 =0
(8.34)
Evaluation of ∂x 1 /∂ξ and ∂x 2 /∂ξ 2
2
To make further progress, we must evaluate ∂x 1 /∂ξ 2 and ∂x 2 /∂ξ 2 at ξ 2 = 0. This
can be done using Equation 8.34. Thus,
⎡ ⎤ −1
0.5
2 2

∂x 2 ∂x 2 ∂x 1 ∂x 1 ds 0 ∂x 2 ∂x 1
cos θ 0 =− + ⎣ + ⎦ .
∂ξ 1 ∂ξ 2 ∂ξ 1 ∂ξ 2 d ξ 2 ∂ξ 1 ∂ξ 1
ξ 2 =0
(8.35)
$ 2
Therefore, since sin θ 0 = 1 − cos θ 0 ,
⎡ ⎤ −1
0.5

2 2

∂x 2 ∂x 1 ∂x 1 ∂x 2 ds 0 ∂x 2 ∂x 1
sin θ 0 = − ⎣ + ⎦ .
∂ξ 2 ∂ξ 1 ∂ξ 2 ∂ξ 1 d ξ 2 ∂ξ 1 ∂ξ 1
ξ 2 =0
(8.36)