Page 205 - Computational Fluid Dynamics for Engineers

6.4 Hess-Smith Panel Method 193

velocity at the trailing edge is the proper one to choose. This condition does not
include bodies with nonsharp trailing edges and bodies on which the viscous
effects have been simulated by, for example, surface blowing, as discussed in
detail in [4]. Thus, the classical Kutta condition is of strictly limited validity. It is
customary to apply a "Kutta condition" to bodies outside its narrow definition,
but this is an approximation; nevertheless the calculations are often in close
accord with experiment.
In the panel method, the Kutta condition is indirectly applied by deducing
another property of the flow at the trailing edge that is a direct consequence
of the finiteness of velocity; this property is used as "the Kutta condition."
Properties that have been used in lieu of "the Kutta condition" in panel methods
include the following:
(a) A streamline of the flow leaves the trailing edge along the bisector of the
trailing-edge angle.
(b) Upper and lower surface total velocities approach a common limit at the
trailing edge. The limiting value is zero if the trailing-edge angle is nonzero.
(c) Source and/or vorticity strengths at the trailing edge must satisfy conditions
to allow finite velocity.
Of the above, property (b) is more widely used. At first it may be thought
that this property requires setting both the upper and lower surface velocities
equal to zero. This gives two conditions, which cannot be satisfied by adjusting
a single parameter. The most reasonable choice is to make these two total veloc-
ities in the downstream direction at the 1st and 7V-th panel control points equal
so that the flow leaves the trailing edge smoothly. Since the normal velocity on
the surface is zero according to Eq. (6.4.7), the magnitude of the two tangential
velocities at the trailing edge must be equal to each other, that is,

-
(V^N = (V*)i (6.4.13)

Introducing the flow tangency condition, Eq. (6.4.7), into Eq. (6.4.8a) and
noting that TJ = r, we get

N N
A T B V s i n a
E ^J + E ij + ™ ( - °i) = °> * = 1, 2,..., W (6.4.14)
3=1 3=1

2
In terms of the unknowns, q 3•, (j = 1, , . . . , N) and , the Kutta condition of Eq.
r
(6.4.13) and Eq. (6.4.14) form a system of algebraic equations whose solution
can be obtained by the Gaussian elimination method discussed in Section 4.5.
The details of the solution procedure and the computer program are given in
the following section.