Page 284 - Computational Fluid Dynamics for Engineers

274 9. Grid Generation

(the subscript i denotes differentiation with respect to i)
To construct the hyperbolic tangent distribution in this case, first let:

A =^M and B =
VAST Iy/ASi.AS2
Then solve the following non-linear equation for 8:

sinh 8
(9.4.17)

The arc length distribution along the curve is then given by:
u(i)
(9.4.18)
5 W
A + (1-A).u(i)
With
(9.4.19)
~ k tanh(f) j

Spacing specified at one end of the curve

When the spacing AS is specified only at i = 0, the distribution is computed
as follows. First B is computed from:
1
B
IAS
Equation (9.4.17) is used to compute 6. The arc length distribution is then given
by:
tanh[f | - 1)1 ,
(
s(i) = 1 + [2lL -r—- 9.4.20
V V
tanh(|) '
When the spacing AS is specified only at i = I, the procedure is the same except
that equation (9.4.20) is replaced by:

tanh[f(f)]
=
(9.4.21)
tanh(f)

Spacing specified at an interior point of the curve
If the spacing AS is specified only at an interior point S = p, B is again calcu-
lated as above but 6 is determined as the solution of:

1 cosh 8 — 1 +
1 + (9.4.22)
[B.p.6\ sinh 8
The value of % at which S — p is obtained by solving the non-linear equation: