Page 112 - Basic Structured Grid Generation

Structured grid generation – algebraic methods 101

The inverse transformation is given by

x = ξ

sinh[r(η − B)] . (4.92)
y = y 0 1 +
sinh(rB)
Other possible stretching functions are given in Section 4.6.1.

The Eriksson function
The following stretching transformation, due to Eriksson (1982), also involves expo-
nential functions, but has the simpler form

e − 1
αη
y = h (4.93)
α
e − 1
for some constant α, with inverse
1 y α
η = ln 1 + (e − 1) . (4.94)
α h
Here dy/dη takes its lowest values, thereby increasing grid density, as we approach
y = 0. Denoting the function in (4.93) by f(η), we can move the clustering to y = h
by forming the function {h − f(1 − η)}, which gives
α α(1−η)
e − e
y = h . (4.95)
α
e − 1
It is straightforward to verify that, if we wish to create a clustering of grid lines
near the interior line y = y 0 (corresponding to η = η 0 = y 0 /h in the computational
plane), the above functions can be fitted together after appropriate scaling, with
α α(1−η/η 0 ) α
hη 0 [(e − e )/(e − 1)], 0 η η 0
y = α(η−η 0 )/(1−η 0 ) α (4.96)
hη 0 + h(1 − η 0 )[(e − 1)/(e − 1)],η 0 η 1.

This function is monotonically increasing, and has a continuous derivative at η = η 0 .
The functions (4.93) and (4.95), suitably scaled, can also be fitted together in the
opposite order at an arbitrary interior value y = y 1 (with corresponding η = η 1 =
y 1 /h) to give a stretching function which gives grid clustering near both boundaries
y = 0and y = h. With f(η) again as given by eqn (4.93), the two parts of the function
are y = y 1 f(η/η 1 ) for 0 η η 1 and y = h − (h − y 1 )f ((1 − η)/(1 − η 1 )) for
η 1 η 1.

Exercise 2. Show that the required function is given by

α
hη 1 (e αη/η 1 − 1)/(e − 1), 0 η η 1
y = (4.97)
α
h − h(1 − η 1 )(e α(1−η)/(1−η 1 ) − 1)/(e − 1), η 1 η 1,
which has a continuous first derivative at y = y 1 .
Example – Flow in a Divergent Nozzle: