Page 36 - Basic Structured Grid Generation

Mathematical preliminaries – vector and tensor analysis 25

From eqns (1.138) and (1.161) we deduce a conservative form for the diver-
gence ∇· u:
1 ∂ √ 1 ∂ √ 2
∇· u = √ ( gg · u) + ( gg · u)
g ∂ξ ∂η
1 ∂ ∂
= √ (y η U 1 − x η U 2 ) + (−y ξ U 1 + x ξ U 2 ) , (1.168)
g ∂ξ ∂η
where u = U 1 i + U 2 j. Again by further differentiation, or directly from eqn (1.134),
we deduce the non-conservative form:
1 ∂U 1 ∂U 2 ∂U 1 ∂U 2
∇· u = √ y η − x η − y ξ + x ξ . (1.169)
g ∂ξ ∂ξ ∂η ∂η
A similar treatment of eqns (1.145) and (1.146) yields a conservative form for ∇×u:

1 ∂ ∂
∇× u = k√ (x η U 1 + y η U 2 ) − (x ξ U 1 + y ξ U 2 ) (1.170)
g ∂ξ ∂η
and the non-conservative form

1 ∂U 1 ∂U 2 ∂U 1 ∂U 2
∇× u = k√ x η + y η − x ξ − y ξ . (1.171)
g ∂ξ ∂ξ ∂η ∂η
Making use of both eqns (1.166) and (1.168), we obtain the two-dimensional Lapla-
2
cian ∇ ϕ =∇ · (∇ϕ) in conservative form:
1 ∂ y η ∂ ∂ x η ∂ ∂
2
∇ ϕ = √ √ [y η ϕ]− [y ξ ϕ] − √ − [x η ϕ]+ [x ξ ϕ]
g ∂ξ g ∂ξ ∂η g ∂ξ ∂η
1 ∂ y ξ ∂ ∂ x ξ ∂ ∂
+√ −√ [y η ϕ]− [y ξ ϕ] + √ − [x η ϕ]+ [x ξ ϕ] .
g ∂η g ∂ξ ∂η g ∂ξ ∂η
(1.172)
Note that eqn (1.152) also gives, making use of eqn (1.163), the form
1 ∂ 2 ∂ 2 ∂ 2
2 g 22 g 12 g 11
∇ ϕ = √ √ ϕ − 2 √ ϕ + √ ϕ
g ∂ξ 2 g ∂ξ∂η g ∂η 2 g
∂ √ 2 ∂ √ 2
− [( g∇ ξ)ϕ]− [( g∇ η)ϕ] . (1.173)
∂ξ ∂η

Exercise 15. Using eqns (1.111), (1.162), and (1.163), show that

2 g 22 g 12 g 11
∇ ξ = (x η y ξξ − y η x ξξ ) − 2 (x η y ξη − y η x ξη ) + (x η y ηη − y η x ηη ),
g 3/2 g 3/2 g 3/2
(1.174)
2 g 22 g 12 g 11
∇ η = (y ξ x ξξ − x ξ y ξξ ) − 2 (y ξ x ξη − x ξ y ξη ) + (y ξ x ηη − x ξ y ηη ).
g 3/2 g 3/2 g 3/2
(1.175)