Page 196 - Computational Fluid Dynamics for Engineers

184 6. Inviscid Flow Equations for Incompressible Flows

and write Eqs. (6.3.1) and (6.3.4) as
2
d j> l c t y 1 d<j>
(6.3.6)
2
2
2
dr f dr r d6
r = 1, — - 0 (6.3.7a)
or
f —> oo, (j)—>f cos9 (6.3.7b)
Since the solution of an elliptic equation requires the specification of bound-
ary conditions along the entire boundary of the domain, additional boundary
conditions are needed other than those given by Eq. (6.3.7). Since the flow is
symmetric about a horizontal line through the center of the circle, it is sufficient
to compute the flow only in the upper half of the physical plane and set

86
or
^ = 0 (6.3.
86
along the symmetry line at 0 — 0 and 6 = TT (Fig. 6.4).

""•^^ Free stream
infinity
>N
N 0 =rcos0

4=o /

cL 0 = 7t 0 = 0 B
symmetric line symmetric line

Fig. 6.4. Physical plane for flow over a circular cylinder.

The boundary condition in Eq. (6.3.7b) requires that the velocity be uniform
at infinity. While this requirement presents no difficulties for the analytical
solution in Eq. (6.3.2), it can cause problems for the numerical solutions since
the numerical solution must be carried out in a finite computational plane. To
partially resolve this dilemna, we introduce a transformation which transforms
infinity into zero; we define a new independent variable rj by

(6.3.9a)