Page 204 - Computational Fluid Dynamics for Engineers

192 6. Inviscid Flow Equations for Incompressible Flows

sin(^ -- 6j) In ^ ± 1 + c o s ( ^ - Orffiij i+3
2TT T i,j
*ij 1 (6.4.9)
i = j

sin(0j -- OjWij - cos(6i - 6j) In ^ ± i ] i +3
M (6.4.10)
ij
i=j
n 1 1 n
R — — A Ft — A (6.4.11)
ij ij ij ij
Here
1/2
2
r (x m . - x j+1) + - yj+i)
ij+l {ym x
1/2
T -Xj) 2 + -Vjf
iJ ~ (x mz (y mi
x (6.4.12)
m z = 7T (Xi + Xi+i), ym z = ^{Vi + Vi+l)
- l f Vi+l ~ Vi - l (Vj+i ~Vj
0i — tan tan
Xi+i — Xi Xj+l Xj
1 ( ym l - Vj+1 Vn Vj
A tan tan
ij
rrii x j+l,
Regardless of the nature of qj(s) and TJ(S), Eq. (6.4.8) satisfies the irrota-
tionality condition and the infinity boundary condition, Eq. (6.2.9). To satisfy
the requirements given by Eq. (6.4.7) and the condition related to the circu-
lation, it is necessary to adjust these functions. In the approach adopted by
Hess and Smith [1], the source strength qj(s) is assumed to be constant over
the j-th panel and is adjusted to give zero normal velocity over the airfoil, and
the vorticity strength Tj is taken to be constant on all panels (TJ = r) and its
single value is adjusted to satisfy the condition associated with the specification
of circulation. Since the specification of the circulation renders the solution to
be unique, a rational way to determine the solution is required.
The best approach is to adjust the circulation to give the correct force on the
body as determined by experiment. However, this requires advance knowledge
of that force, and one of the principal aims of a flow calculation method is
to calculate the force and not to take it as given. Thus, another criterion for
determining circulation is needed.
For smooth bodies such as ellipses, the problem of rationally determining
the circulation has yet to be solved. Such bodies have circulation associated
with them and resulting lift forces, but there is no rule for calculating these
forces. If, on the other hand, we deal with an airfoil having a sharp trailing
edge, we can apply the Kutta condition [2,4]. It turns out that for every value
of circulation except one, the inviscid velocity is infinite at the trailing edge. The
Kutta condition states that the particular value of circulation that gives a finite