Page 83 - Basic Structured Grid Generation

72 Basic Structured Grid Generation

or, more explicitly, making use of eqn (3.31), and writing (x, y, z) for cartesians, (u, v)
for surface co-ordinates, and partial derivatives x u ,etc.
 
1 a 22 x u − a 12 x v −a 12 x u + a 11 x v
C = √  a 22 y u − a 12 y v −a 12 y v + a 11 y v  . (3.143)
a
a 22 z u − a 12 z v −a 12 z u + a 11 z v
Equation (3.141) represents a non-conservative expression for ∇ϕ. Summation is
implied over α from 1 to 2, although we have been a little less than rigorous here in
not writing the first α as a superscript.
Re-writing eqn (3.136) as
∂ ∂ √
(a 2 × N) + (N × a 1 ) = 2κ m aN, (3.144)
∂u 1 ∂u 2
it follows that we can express eqn (3.140) in the conservative form
1 ∂ ∂
∇ϕ = √ ((a 2 × N)ϕ) + ((N × a 1 )ϕ) − 2κ m ϕN, (3.145)
a ∂u 1 ∂u 2
giving cartesian components

1 ∂
(∇ϕ) i = √ (C iα ϕ) − 2κ m ϕN i . (3.146)
a ∂u α
Although C is not a square matrix, we can effectively define an inverse, at least
on the left. To see this, consider the 3 × 2 (Jacobian) matrix J of the transformation
(u, v) → (x, y, z),given by
 
x u x v

J =  y u y v  = a 1 a 2 . (3.147)
z u z v
Since eqn (3.27) implies the matrix identity

a 1 1 2
10
a a = ,
a 2 01

T 1
or, otherwise expressed, J √ C = I 2 ,the 2 × 2 identity matrix, it follows that C
a
has a left inverse
1 1
−1 T x u y u z u
C = √ J = √ , (3.148)
a a x v y v z v
1
α
−
or (C −1 ) αi = (a) 2 ∂y i /∂u .
We can now, by multiplication on the left, invert eqn (3.141) to obtain
∂φ √ −1
= a(C ) αi (∇ϕ) i (3.149)
∂u α
with summation over i from 1 to 3. This equation in fact gives the general relationship
between the covariant components and the background cartesian components of a
surface vector.