Page 32 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 21

The gradient operator was defined in eqns (1.12) and (1.13). We can also write
3
1 ∂ϕ
∇ϕ = √ (g j × g k ) i (1.140)
g ∂x
i=1
in non-conservative form, using eqn (1.32), or, by eqn (1.137),

3
1 ∂ 1 ∂ √ i
∇ϕ = √ {(g j × g k )ϕ}= √ ( gg ϕ) (1.141)
g ∂x i g ∂x i
i=1
in conservative form, where again i, j, k when they appear together are always in cyclic
order 1, 2, 3.
The curl of the vector with cartesian components in eqn (1.132) is the vector

i
1 i 2 i 3

curlu = ∂/∂y 1 ∂/∂y 2 ∂/∂y 3 ,

U 1 U 2 U 3
i i with summation
∂U k
otherwise denoted by ∇× u; this expression is equivalent to e ijk ∂y j
over i, j, k, which, making use of (1.93), generalizes to
1
ijk ijk
∇× u = ε u k,j g i = √ e u k,j g i (1.142)
g
j
in curvilinear co-ordinates. Since u k,j = g k · ∂u/x , we have, writing out eqn (1.142)
in full,
1
∇× u = √ (u 2,3 g 1 − u 3,2 g 1 + u 3,1 g 2 − u 1,3 g 2 + u 1,2 g 3 − u 2,1 g 3 ).
g
The expression in the brackets is

∂u ∂u ∂u ∂u
g 2 · g 1 − g 1 · g 2 + g 3 · g 2 − g 2 · g 3
∂x 3 ∂x 3 ∂x 1 ∂x 1

∂u ∂u
+ g 1 · g 3 − g 3 · g 1
∂x 2 ∂x 2
after some re-arrangement, so ∇× u can be written as

1 ∂u
3
∇× u = √ (g j × g k ) × i , (1.143)
g ∂x
i=1
after using well-known identities for vector triple products. Here again j and k are
constrained, given any value of i, such that i, j, k are always in cyclic order 1, 2, 3.
Equation (1.143) is a non-conservative form for ∇× u. However, by eqn (1.137) we
immediately have the conservative forms

3
1 ∂
∇× u = √ i {(g j × g k ) × u} (1.144)
g ∂x
i=1