Page 133 - Basic Structured Grid Generation

122 Basic Structured Grid Generation

where a r maybecalledthe aspect ratio at a point, since we can write

2 2
∂x ∂y
+
∂σ ∂σ
,
a r =
2 2

∂x ∂y
+
∂χ ∂χ
which by eqns (1.42) and (1.158) implies that it is the ratio of the lengths of sides of an
infinitesimal grid element in physical space corresponding to an infinitesimal element
of a uniform grid in parameter space.
If we now apply stretching functions of the form (5.9), we easily obtain

∂y f (η) ∂x ∂x f (η) ∂y
2 2
= a r , and =−a r . (5.17)

∂η f (ξ) ∂ξ ∂η f (ξ) ∂ξ
1 1
It follows that
∂x ∂x ∂y ∂y
+ = 0, (5.18)
∂ξ ∂η ∂ξ ∂η
and hence orthogonality still holds for the ξ, η co-ordinate curves, i.e. it has not been
affected by the stretching transformation.
A differential model of grid generation which aims at orthogonality can be written
down immediately by regarding the g 12 terms in eqns (5.14) as zero everywhere. Thus
we have
2
2
∂ x ∂ x f 1 ∂x f 2 ∂x
g 22 + g 11 = g 22 + g 11 ,
∂ξ 2 ∂η 2 f 1 ∂ξ f 2 ∂η
2
2
∂ y ∂ y f 1 ∂y f 2 ∂y
g 22 2 + g 11 2 = g 22 + g 11 . (5.19)
∂ξ ∂η f ∂ξ f ∂η
1 2
These equations can be solved numerically in the computational domain subject to
Dirichlet boundary conditions on x and y. However, we can also seek to satisfy the
orthogonality condition (5.18) on the boundary. The numerical techniques are pursued
in Section 5.6.2.

5.4 Conformal and quasi-conformal mapping

In considering mappings ξ(x, y), η(x, y) from plane physical domains to computational
domains, it is sometimes mathematically advantageous to introduce complex variables
w = ξ + iη and z = x + iy, the domains then becoming part of complex w and
z planes. Since
x = (z + z)/2, y = (z − z)/2i, (5.20)

where the conjugate of z is given by z = x − iy, a mapping can be expressed as
w = ξ(x, y) + iη(x, y) = f(z, z) (5.21)