Page 260 - Introduction to Computational Fluid Dynamics

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8.3 DIFFERENTIAL GRID GENERATION
direction ξ i ) is defined as May 10, 2005 16:28 239
r
d ∂x 1 ∂x 2 ∂x 3
a i = = i + j + k . (8.17)
d ξ i ∂ξ i ∂ξ i ∂ξ i
Similarly, the contravariant base vector (normal to coordinate surface ξ i = constant)
is defined as
i
a =∇ ξ i = i ∂ξ i + j ∂ξ i + k ∂ξ i = j × k /J, (8.18)
a
a
∂x 1 ∂x 2 ∂x 3
where J is the Jacobian. Now, from Green’s theorem [70], for any quantity (vector
or scalar) ,

3 3 3
1 ∂ ∂ i i ∂
∇ = a j × a k · = a = a (8.19)
J ∂ξ i ∂ξ i ∂ξ i
i=1 i=1 i=1
i
since ∂ /∂ξ i = 0. Therefore,
a

3 3
2 i ∂ l ∂
∇ =∇ · ∇ = a . a
∂ξ i ∂ξ l
i=1 l=1
3 3 3 3 l
a
∂ ∂ ∂ ∂
i l i
a
= a · + a .
∂ξ i ∂ξ l ∂ξ i ∂ξ l
i=1 l=1 i=1 l=1
(8.20)
If we now set = ξ l (a scalar), then
3 l
a
2 i ∂
∇ ξ l = a . (8.21)
∂ξ i
i=1
Substituting Equation 8.21 in Equation 8.20 gives,

3 3 3
∂ ∂ ∂
2 i l 2
a
∇ = a · + ∇ ξ l . (8.22)
i=1 l=1 ∂ξ i ∂ξ l l=1 ∂ξ l
In two dimensions (∂/∂x 3 = ∂/∂ξ 3 = 0), Equation 8.22 will read as
2
2
2
∂ ∂ ∂
1
2
2
1
a
a
∇ = a · a 1 + 2 · 2 + · 2
a
a
∂ξ 2 ∂ξ 1 ∂ξ 2 ∂ξ 2
1 2
∂ ∂
2 2
+∇ ξ 1 +∇ ξ 2 . (8.23)
∂ξ 1 ∂ξ 2