Page 201 - Computational Fluid Dynamics for Engineers

6.4 Hess-Smith Panel Method 189

cylinder. The solid and dashed lines correspond to the numerical results and the
symbols denote the analytical results. For example, from the definition of VQ in
Eq. (6.3.3) and from the exact solution given by Eq. (6.3.2), it follows that

VQ = - 2 sin 9 (6.3.20a)

and the pressure coefficient C p is
= 1 - V£ = 1 - 4 sin 2 9 (6.3.20b)
C p
Since the numerical solutions are obtained in terms of the dimensionless velocity
potential, the circumferential velocity component VQ is computed by a second-
order central difference,

\Ye)i+i,j ~ 2(A£) (6.3.21)

Overall, the numerical results are in good agreement with the analytical results.
An error of less than one percent is largely due to the accuracy of the numerical
scheme.

6.4 Hess-Smith Panel Method

We consider an airfoil at rest in an onset flow of velocity VQQ . We assume that
the airfoil is at an angle of attack, a (the angle between its chord line and the
onset velocity), and that the upper and lower surfaces are given by functions
Y u(x) and Y[(x), respectively. These functions can be defined analytically, or (as
often is the case) by a set of (x, y) values of the airfoil coordinates. We denote
the distance of any field point (x,y) measured from an arbitrary point, 6, on
the airfoil surface by r, as shown in Fig. 6.7. Let ft also denote the unit vector
normal to the airfoil surface and directed from the body into the fluid and t,
a unit vector tangential to the surface, and assume that its inclination to the
x-axis is given by 9. It follows from Fig. 6.7 that with i and j denoting unit
vectors in the x- and ^/-directions, respectively,

ft — — sin 9 i + cos 9 j
(6.4.1)
t — cos 0i + sin 9 j
If the airfoil contour is divided into a large number of small segments, ds, then
we can write
dx = cos 9 ds
(6.4.2)
dy = sin 9 ds

We next assume that the airfoil geometry is represented by a finite number (N)
of short straight-line elements called panels, defined by (N + l)(xj,yj) pairs