Page 220 - Aerodynamics for Engineering Students

Two-dimensional wing theory 203

To obtain the corresponding velocity components at P due to all the vortices on
the panel, integration along the length of the panel is carried out to give

(4.1 10)

Following the basic method described in Section 3.5 normal and tangential influ-
ence coefficients, Ni and Tb are introduced, the primes are used to distinguish these
coefficients from those introduced in Section 3.5 for the source panels. Nb and Ti
represent the normal and tangential velocity components at collocation point i due to
vortices of unit strength per unit length distributed on panel j. Let ii and
ki(i = 1 , 2, . . . , N) denote the unit tangent and normal vectors for each of the panels,
and let the point P correspond to collocation point i, then in vector form the velocity
at collocation point i is given by
+
VpQ = VxQ 4 + VyQfij
To obtain the components of this velocity vector perpendicular and tangential to
panel i take the scalar product of the velocity vector with ki and fi respectively. If
furthermore -1 is set equal to 1 in Eqns (4.109) and (4.1 10) the following expressions
-
are obtained for the influence coefficients
N!. = vpQ .hi = V XQ i.. i. + V YQ &. fi. (4.11 la)
J
J
2
2/ -..
;..fi.
T!.=VpQ.fi=V XQ'J t^..i.+V YQ'J (4.1 1 lb)
2/
Making a comparison between Eqns (4.109) and (4.1 11) for the vortices and the
corresponding expressions (3.97) and (3.99) for the source panels shows that
and (4.112)
[VxQIVOflkXS [V.QISOLUL%S [VYQIVOI'tiL%S = -[vxQISOllrCeS
With the results given above it is now possible to describe how the basic panel
method of Section 3.5 may be extended to lifting aerofoils. Each of the N panels now
carries a source distribution of strength q per unit length and a vortex distribution of
strength y per unit length. Thus there are now N + 1 unknown quantities. The N x N
influence coefficient matrices Nu and Tu corresponding to the sources must now be
expanded to N x (N + 1) matrices. The (N + 1)th column now contains the velocities
induced at the collocation points by vortices of unit strength per unit length on all
the panels. Thus N~,N+I represents the normal velocity at the ith collocation point
induced by the vortices over all the panels and similarly for Ti,~+l. Thus using
Eqns (4.11 1)
N N
(4.113)
j=l j= 1