Page 181 - Basic Structured Grid Generation

170 Basic Structured Grid Generation

the integrand represents the square of the scalar product of unit tangent vectors to the ξ
and η co-ordinate curves. Thus this functional controls the angle between intersecting
co-ordinate curves.

Orthogonality three functional
Another functional related to orthogonality is

√
I O,3 = g 11 g 22 dξ dη. (6.64)
R
Exercise 6. Show that the Euler-Lagrange equations for the functional (6.64) may be
expressed as

g 22 g 11
r ξ + r η = 0. (6.65)
g 11 g 22
ξ η
The connection between these equations and orthogonality may be seen by con-
sidering eqns (1.164). For an orthogonal grid it is clear that we must have g 12 = 0
√ 1 2
everywhere, g = g 11 g 22 , g = g 1 /g 11 ,and g = g 2 /g 22 .
Thus it is necessary (but not sufficient) for a grid to be orthogonal that, by (1.164),
∂ √ 1 ∂ √ 2 ∂ g 22 ∂ g 11
( g g ) + ( g g ) = g 1 + g 2 = 0,
∂ξ ∂η ∂ξ g 11 ∂η g 22
which is equivalent to eqn (6.65). Equation (6.65) has been much used in the context
of generating orthogonal grids – see, for example, Warsi and Thompson (1980) and
Ryskin and Leal (1983).

6.4.7 The Liao functionals

Liao and Liu (1993) proposed the grid generation functional

2 2 2 2
I li = [(g 11 + g 22 ) − 2g] dξ dη = [(g 11 ) + (g 22 ) + 2(g 12 ) ] dξ dη, (6.66)
R R
but this has a tendency to produce folded grids. Experience has shown that this tendency
is apparently diminished by taking the same functional with the integrand divided by
g. This gives, after discarding a constant, the modified Liao functional
2

g 11 + g 22
I ml = √ dξ dη (6.67)
R g
which is similar to the Winslow functional (6.44).
The following table shows a list of unweighted functionals that are in use.

Functional Symbol Integrand

Length I L g 11 + g 22
Orthogonality-1 I O (g 12 ) 2
√
Area I A ( g) 2
2
Orthogonality-2 I O,2 (g 12 ) /g 11 g 22