Page 221 - Aerodynamics for Engineering Students

204 Aerodynamics for Engineering Students

In a similar fashion as for the non-lifting case described in Section 3.5 the total
normal velocity at each collocation point, due to the net effect of all the sources, the
vortices and the oncoming flow, is required to be zero. This requirement can be
written in the form:

Sources
These Nequations are supplemented by imposing Condition (b). The simplest way to
do this is to equate the magnitudes of the tangential velocities at the collocation point
of the two panels defining the trailing edge (see Fig. 4.23b). Remembering that the
unit tangent vectors it and &+I are in opposite directions Condition (b) can be
expressed mathematically as

Equations (4.1 14) and (4.11 5) combine to form a matrix equation that can be written
as

Ma=b (4.116)

where M is an (N + 1) x (N + 1) matrix and a and b are (N + 1) column vectors. The
elements of the matrix and vectors are as follows:

M..=N.. i=1,2 ,..., N j=1,2 ,..., N+l
1J
1J
MN+l,j = T,,j + Tt+l,j j = 1,2,. . . , N + 1
ai=ui i= 1,2, ..., N and ~ + =y
1
-+
bi=-U.Ai i= 1,2, ..., N
-+
bN+1 = -u. (2, + &+I)
Systems of linear equations like (4.11 6) can be readily solved numerically for the
unknowns ai using standard methods (see Section 3.5). Also it is now possible to see
why the Condition (c), requiring that the tangential velocities on the upper and lower
surfaces both tend to zero at the trailing edge, cannot be satisfied in this sort of
numerical scheme. Condition (c) could be imposed approximately by requiring,
say, that the tangential velocities on panels t and t + 1 are both zero. Referring to
Eqn (4.1 15) this approximate condition can be expressed mathematically as