Page 290 - Computational Fluid Dynamics for Engineers

280 9. Grid Generation

such that the values of the inner boundary control functions become vanishingly
small at the outer boundary, and the outer boundary control functions become
vanishingly small at the inner boundary.
Thus, we have:
^,o)= (0
P
P&r knax)=r(Z)
9.5.7)

<5(£,%iax) = *(0
These components of the control functions are derived as follows. Two conditions
are imposed at the boundaries, i.e. orthogonality and specified grid spacing.
Since conditions are imposed at the r] = 0 and rj = rjmax boundaries, the spacing
is along the r\ coordinate. If S^ is the desired spacing along this coordinate, the
condition can be written as:

= xl + y% = S% (9.5.8)
r v.r 7l
The orthogonality condition is expressed as follows:

= x ^ + = 0 (9.5.9)
^,f v y^y v
On the rj = 0 and rj — rimax boundaries, all the derivatives with respect to £
are known, since they involve only boundary points. Hence, in the equations
(9.5.8) and (9.5.9) above, the x^ and y^ terms are known, and we thus have two
equations in two unknowns, yielding the following expressions for the x^ and y^
terms:
= S riy i/(xl + y ^ 2 (9.5.10a)
x v
1 2
Vr, = S r}x i/{x\ + yl) l (9.5.10b)
and
= Xrji + y^j (9.5.10c)
r v
Thus, r v is known on the boundary. Furthermore, it can be shown that if equa-
tions (9.5.5) and (9.5.9) are combined with equation (9.5.4), the latter reduces
to the following on the boundary:

( V ^ ) ( % + Pf^) + (f^f^ifrfrj + Q ^ ) ! boundary = 0 (9.5.11)

Multiplying the above equation by f^ or by r^ and again using the condition of
orthogonality, yields the following two equations for the control functions:

^boundary = - ( ^ . r ^ ) / ( f f .7^) - ( ^ . f ^ ) / ^ . ^ ) [boundary (9.5.12a)

Qlboundary = - ( V ^ / ^ . T ^ ) - ( r ^ . f ^ ) / ^ . ^ ) [boundary (9.5.12b)
By evaluating the above equations on the boundaries, the p, q, r, and s terms
described in equations (5.5.7) can be computed. All quantities in equations