Page 35 - Basic Structured Grid Generation

24 Basic Structured Grid Generation

The contravariant base vectors are
1 1
1
g = √ g 2 × k = √ (iy η − jx η ),
g g
1 1
2
g = √ k × g = √ (−iy ξ + jx ξ ), (1.161)
1
g g
3
g = k,
and in comparison with eqn (1.38) we have
√ √ √ √
ξ x = y η / g, ξ y =−x η / g, η x =−y ξ / g, η y = x ξ / g. (1.162)
The components of the contravariant metric tensor are given by
 11 12 13   
g g g g 22 /g −g 12 /g 0
 g 12 g 22 g 23  =  −g 12 /g g 11 /g 0  . (1.163)
g 13 g 23 g 33 0 0 1
Note that the two-dimensional version of eqn (1.136) is
∂ √ 1 ∂ √ 2 ∂ 2 ∂ 1
( gg ) + ( gg ) = (g × k) + (k × g ) = 0. (1.164)
∂ξ ∂η ∂ξ ∂η
From eqn (1.141) we have the two-dimensional form for ∇ϕ:

1 ∂ √ 1 ∂ √ 2
∇ϕ = √ ( gg ϕ) + ( gg ϕ)
g ∂ξ ∂η
1 ∂ 1 ∂
= √ [(iy η − jx η )ϕ] + √ [−iy ξ + jx ξ ]ϕ
g ∂ξ g ∂η
1 ∂ ∂ 1 ∂ ∂
= i√ (y η ϕ) − (y ξ ϕ) + j√ − (x η ϕ) + (x ξ ϕ) (1.165)
g ∂ξ ∂η g ∂ξ ∂η
in conservative form. Thus the cartesian components of ∇ϕ are given by
∂ϕ 1 ∂ ∂
= √ (y η ϕ) − (y ξ ϕ)
∂x g ∂ξ ∂η

and
∂ϕ 1 ∂ ∂
= √ − (x η ϕ) + (x ξ ϕ) (1.166)
∂y g ∂ξ ∂η
in conservative form. By further differentiation, or directly from eqn (1.13), we obtain
the non-conservative forms
∂ϕ 1 ∂ϕ ∂ϕ
= √ y η − y ξ
∂x g ∂ξ ∂η
and
∂ϕ 1 ∂ϕ ∂ϕ
= √ −x η + x ξ . (1.167)
∂y g ∂ξ ∂η