Page 167 - Aerodynamics for Engineering Students

150 Aerodynamics for Engineering Students

point i are given by the scalar (or dot) products i$ . rii and 6, . & respectively. Both of
these quantities are proportional to the strength of the sources on panel j and
therefore they can be written in the forms
cy.A. -0.N.. and 9.. . I-J'J -g.T.. (3.90)
;.
V
J
Ll
1-
Ng and Tg are the perpendicular and tangential velocities induced at the collocation
point of panel i by sources of unit strength distributed over panelj; they are known as
the normal and tangential influence coefficients.
The actual velocity perpendicular to the surface at collocation point i is the sum of
the perpendicular velocities induced by each of the N panels plus the contribution
due to the free stream. It is given by
N
(3.91)
j= 1
In a similar fashion the tangential velocity at collocation point i is given by

(3.92)

If the surface represented by the panels is to correspond to a solid surface then the
actual perpendicular velocity at each collocation point must be zero. This condition
may be expressed mathematically as vni = 0 so that Eqn (3.91) becomes
N +
CgjNij= -U.&(i= 172: ..., N) (3.93)
j=l
Equation (3.93) is a system of linear algebraic equations for the N unknown source
strengths, aj(i = 1,2,. . . , N). It takes the form of a matrix equation
NO=b (3.94)
where N is an N x N matrix composed of the elements Nu, o is a column matrix
coxnposed of the N elements gi, and b is a column matrix composed of the N elements
-
- U rii. Assuming for the moment that the perpendicular influence coefficients Nu have
been calculated and that the elements of the right-hand column matrix b have also been
calculated, then Eqn (3.94) may, in principle at least, be solved for the source strengths
comprising the elements of the column matrix cr. Systems of linear equations like (3.94)
can be readily solved numerically using standard methods. For the results presented here
the LU decomposition was used to solve for the source strengths. This method is
described by Press et al.* who also give listings for the necessary computational routines.
Once the influence coefficients Nu have been calculated the source strengths can be
determined by solving the system of Eqn (3.93) by some standard numerical technique.
If the tangential influence coefficients To have also been calculated then, once the
source strengths have been determined, the tangential velocities may be obtained from
Eqn (3.92). The Bernoulli equation can then be used to calculate the pressure acting at
collocation point i, in particular the coefficient of pressure is given by Eqn (2.24) as:

2
c, = 1 - ($) (3.95)

* W.H. Press etal. (1992) Numerical Recipes. The Art of Scientific Computing. 2nd ed. Cambridge Uni-
versity Press.