Page 289 - Computational Fluid Dynamics for Engineers

9.5 Differential Equation Methods 279

(9.5.3b)
(3 = x^x v + y^y v
9 5 3c
7 = 4 + y\ ( - - )
J = x^y v + x^y v (9.5.3d)
The boundary conditions for the above equations are the values of the (x, y)
positions of the grid points on the boundaries of the physical region in the
(x,y) plane when they are transformed to the corresponding locations along
the boundaries of the region in the computational plane. Therefore, once (x, y)
values are specified along the boundary segments of the region in the computa-
tional plane, Eqs. (9.5.2) can be solved by finite-difference methods to determine
the values of (x, y) at each grid point at the interior of the region. For further
details, the reader is referred to Thompson [1].
The Poisson equations (9.5.2) can be written in generalized form as:

#22(% + Pf^) + gnifrjrj + Qr v) - 2g 12f^ = 0 (9.5.4)

where, r is the coordinate vector in physical space r = xi + yj and the gij are
the covariant metric components:

011 = z | + 2/f =?£.?£
922 =x^y^ = r v.f v (9.5.5)

#12 = X£X V + y^y v = f^.frj

Two-Dimensional Grid Generation Using the Poisson Equations

Let us consider the generation of a C-grid around an isolated airfoil. An impor-
tant feature of grid generation programs is the inclusion of control functions in
order to optimize grid stretching and orthogonality. This leads to the solution
of the Poisson equations (9.5.2). There are several types of control functions.
Some control only the stretching, others only the orthogonality, some others
both. Some control functions will cluster grid points or lines in a local area,
without altering the overall grid structure. There are different forms of control
functions that perform the same task [1]. One example of control functions is
the form employed by Sorenson [5]:
6
ar
P(f, n) = p(£).e- > + (C).e- ^ m a x "^ (9.5.6a)
r
Qfo V) = q{0-e~ av + s(0-e~ 6(r?max_r?) (9.5.6b)
where a and b are positive constants. The first terms in the above expressions
control the grid characteristics at the inner boundary (rj = 0), and the second
terms control the characteristics at the outer boundary (77 = r/ maaj ). The con-
stants a and b, specified by the user, control the degree of propagation of these
control functions away from these two boundaries. In all cases, a and b must be