Page 119 - Basic Structured Grid Generation

108 Basic Structured Grid Generation

then called to adjust the positions of grid-nodes on the intermediate curves, denoted
by r 2 and r 3 , so as to enforce orthogonality of the grid at the inner and outer
boundaries.
If we consider a typical grid node R with co-ordinates (x 1 ,y 1 ) on the inner boundary
r 1 , at which the gradient of the boundary is calculated to be m 1 , then the straight line
normal to the boundary through R has equation

1
y − y 1 =− (x − x 1 ). (4.118)
m 1
i
Suppose that the corresponding node S on the linearly interpolated curve r has
2
i
i
i
co-ordinates (x ,y ) and that we can estimate the gradient of the curve r at this point
2 2 2
i
as m 2 . The tangent to the curve r at this node has equation
2
y − y 2 = m 2 (x − x 2 ). (4.119)
A better position for S should be at the intersection of these two straight lines; the
subroutine calculates the new co-ordinates. A similar procedure is applied to points on
the outer boundary r 4 and the linearly interpolated curve r 3 . Having now constructed
new intermediate curves r 2 and r 3 such that grid-lines are orthogonal at inner and
outer boundaries, we can employ the multisurface transformation formula (4.114),
with a chosen value of a, in the form
2
2
r(ξ, η) = (1 − h) (1 − a 1 h)r 1 (ξ) + (2 + a 1 )h(1 − h) r 2 (ξ)
2
2
+h (1 − h)(2 + a 2 )r 3 (ξ) + h (1 − a 2 + a 2 h)r 4 (ξ), (4.120)
where we define h in terms of a linear interpolation between the two stretching
functions,
h = h AF + ξ(h CD − h AF ). (4.121)
The program Two-boundary.f is essentially a modification of Multisurface.f.The
subroutine orthogonality is removed, and instead Hermite interpolation is used in a
modified form of eqn (4.105),


dy AB dy DC
x(ξ, η) = 1 (h)x AB (ξ) + 2 (h)x DC (ξ) + T 1 3 (h) + T 2 4 (h)

dξ dξ

 dx AB dx DC
y(ξ, η) = 1 (h)y AB (ξ) + 2 (h)y DC (ξ) − T 1 3 (h) − T 2 4 (h) ,


dξ dξ
(4.122)
with h given by (4.121).
4.6 Website programs

Here we list the programs in the directory Book contained on the companion website
(www.bh.com/companions/0750650583) that are relevant to the material in this chapter.
First the subdirectories are specified and then the names of the files with a description
are given.