Page 39 - Basic Structured Grid Generation

28 Basic Structured Grid Generation

∂h i
[ii, j]=−h i j ,
∂x
j h i ∂h i
=− ,(i, j different, no summation over i) (1.186)
ii 2 j
(h j ) ∂x
∂h i
[ij, i]= [ji, i]= h i j ,
∂x
i i 1 ∂h i
= ji = ,(i, j different, no summation over i) (1.187)
ij
h i ∂x j
k
[ij, k]= 0, = 0,(i, j, k all different). (1.188)
ij
2 m
m
ij
Exercise 16. Use the identity ∇ x =−g (see eqn (1.111)) to prove the identities
ij
1 ∂ h 2 h 3 1 ∂ h 3 h 1 1 ∂ h 1 h 2
2 2 2
∇ ξ = √ , ∇ η = √ , ∇ ς = √ .
g ∂ξ h 1 g ∂η h 2 g ∂ς h 3
(1.189)
Note that these formulas can be substituted directly into eqn (1.150) to obtain an
2
expression for the Laplacian ∇ φ of a general scalar field φ in an orthogonal curvilinear
co-ordinate system.
Exercise 17. Use eqn (1.189) together with eqn (1.114) to show that there is a rela-
tionship between the orthogonal curvilinear co-ordinates and cartesian co-ordinates y k ,
k = 1, 2, 3, given by

∂ h 2 h 3 ∂y k ∂ h 3 h 1 ∂y k ∂ h 1 h 2 ∂y k
+ + = 0, k = 1, 2, 3. (1.190)
∂ξ h 1 ∂ξ ∂η h 2 ∂η ∂ς h 3 ∂ς

More identities may be obtained from the vanishing of the six independent compo-
nents of the curvature tensor given by eqn (1.181). These are the six Lam´e Equations:
∂ 1 ∂h k ∂ 1 ∂h j 1 ∂h j ∂h k
+ + = 0,
i
2
∂x j h j ∂x j ∂x k h k ∂x k (h i ) ∂x ∂x i
2
1 ∂h i ∂h j 1 ∂h i ∂h k ∂ h i
+ − = 0, (1.191)
k
j
j
h j ∂x ∂x k h k ∂x ∂x j ∂x ∂x k
where in each equation i, j, k must all be different and taken in the cyclic order 1, 2, 3.
So the three scale parameters must satisfy six compatibility equations.
1.11 Tangential and normal derivatives –
an introduction

The rates of change of scalar functions in directions tangential to co-ordinate curves and
normal to co-ordinate surfaces are often needed in grid-generation work in connection
with the formulation of boundary conditions. These derivatives may be obtained by