Page 33 - Basic Structured Grid Generation

22 Basic Structured Grid Generation

and, using eqn (1.32),
1 ∂ √ i
∇× u = √ ( gg × u). (1.145)
g ∂x i
Note that, again by eqn (1.32), eqn (1.143) can be written as

∂u
i
∇× u = g × (1.146)
∂x i
with summation over i, which may be directly compared with eqn (1.134).
2 2
To obtain an expression for the Laplacian ∇ ϕ of a scalar field ϕ,where ∇ ϕ =
∇·(∇ϕ), using eqns (1.133) or (1.135), the contravariant component of ∇ϕ is needed.
This is just
∂ϕ
ij
i
(∇ϕ) = g ,
∂x j
j
where the effect of the g ij term is to ‘raise the index’ of the covariant vector ∂ϕ/∂x .
Then eqn (1.135) gives
1 ∂ √ ∂ϕ
2 ij
∇ ϕ = √ gg . (1.147)
g ∂x i ∂x j
Alternatively, we have, using the expressions for div and grad in eqns (1.13)
and (1.134),

∂ ∂ϕ
2 i j
∇ ϕ = g · g (1.148)
∂x i ∂x j
with summation over both i and j.Hence
j
2
∂ ϕ ∂g ∂ϕ
2 i j i
∇ ϕ = g · g + g · . (1.149)
i
i
∂x ∂x j ∂x ∂x j
But since, from eqn (1.148),
∂ ∂x k ∂ ∂g k
2 k i j i j k i
∇ x = g · g = g · (g δ ) = g · ,
j
∂x i ∂x j ∂x i ∂x i
the identity (1.149) may be written in the form
2
∂ ϕ ∂ϕ
2 ij 2 j
∇ ϕ = g + (∇ x ) . (1.150)
i
∂x ∂x j ∂x j
k
Substituting ϕ = x in eqn (1.147) gives another formula
1 ∂ √ 1 ∂ √
2 k ij k ik
∇ x = √ ( gg δ ) = √ ( gg ). (1.151)
j
g ∂x i g ∂x i
Thus, eqn (1.147) also gives
1 ∂ ∂ √ ∂ √
2 ij ij
∇ ϕ = √ i j ( gg ϕ) − ϕ j ( gg )
g ∂x ∂x ∂x
1 ∂ 2 √ ij 1 ∂ √ 2 i
= √ i j ( gg ϕ) − √ i (ϕ g(∇ x )). (1.152)
g ∂x ∂x g ∂x