Page 203 - Computational Fluid Dynamics for Engineers

6.4 Hess-Smith Panel Method 191

v(x,y) = / v sqj(s)dsj + / v vTj(s)dsj (6.4.6)

Here qjdsj is the source strength for the element dsj on the -th panel. Similarly,
j
Tjdsj is the vorticity strength for the element dsj on the same panel.
Each of the N panels are represented by similar sources and vortices dis-
tributed on the airfoil surface. The induced velocities in Eq. (6.4.6) satisfy the
irrotationality condition and the boundary condition at infinity, Eq. (6.2.9). For
uniqueness of the solutions, it is also necessary to specify the magnitude of the
circulation around the body. To satisfy the boundary conditions on the body,
which correspond to the requirement that the surface of the body is a stream-
line of the flow given by Eq. (6.2.8). It is customary to choose the control points
to numerically satisfy the requirement that the resultant flow is tangent to the
surface. If the tangential and normal components of the total velocity at the
n
l
control point of the i-th panel are denoted by {V )i and (V )^, respectively, the
flow tangency conditions are then satisfied at panel control points by requiring
t
that the resultant velocity at each control point has only (V )i 1 and
n
l
(V )i = 0 i = ,2,...,iV (6.4.7)
Thus, to solve the Laplace equation with this approach, at the z-th panel con-
n
trol point we compute the normal (V )i and tangential (V*)j, (z = 1,2,..., N)
velocity components induced by the source and vorticity distributions on all
2
panels, j (j = 1, , . . . , TV), including the z'-th panel itself, and separately sum all
the induced velocities for the normal and tangential components together with
the freestream velocity components. The resulting expressions, which satisfy the
irrotationality condition, must also satisfy the boundary conditions discussed
above. Before discussing this aspect of the problem, it is convenient to write
t
n
Eq. (6.4.6) expressed in terms of its velocity components (V )i and (V )i by
TV N
( O i = £ AijQj + E Bfri + Kx, sin(a - 0-) (6.4.8a)
3 = 1 3 = 1

N N
A
v
B r
( -
(y*)i = E %^ + E h J + °° cos a °i) ( 6A8b )
3 = 1 3 = 1
7
where A?A< B ?-, A*--, B\- are known as influence coefficients defined as the ve-
IJ i l J CJ IJ
locities induced at a control point (x mi, y mi); more specifically, Af- and A\-
denote the normal and tangential velocity components, respectively, induced at
the i-th panel control point by a unit strength source distribution on the - t h
j
panel, and Bf- and Bj- are those induced by unit strength vorticity distribution
j
on the -th panel. The influence coefficients are related to the airfoil geometry
and the panel arrangement; they are given by the following expressions: