Page 222 - Aerodynamics for Engineering Students

Two-dimensional wing theory 205

Equation (4.115) is now replaced by the above two equations so that M in Eqn
(4.116) is now a (N + 2) x (N + 1) matrix. The problem is now overdetermined, i.e.
there is one more equation than the number of unknowns, and Eqn (4.116) can no
longer be solved for the vector a, i.e. for the source and vortex strengths.
The calculation of the influence coefficients is at the heart of a panel method. In
Section 3.5 a computational routine in FORTRAN 77 is given for computing the
influence coefficients for the non-lifting case. It is shown below how this routine can
be extended to include the calculation of the influence coefficients due to the vortices
required for a lifting flow.
Two modifications to SUBROUTINE INFLU in Section 3.5 are required to
extend it to the lifting case.

(1) The first two execution statements i.e.
DO101=1,N
10 READ(7,x) XP(1) ,YP(I)
should be replaced by
NP1=N+1
DO 10 I = l,N
AN(1, NP1) = 0.0
AT(I,NP?) =PI
10 READ(7,+)XP(I) ,YP(I)
The additional lines initialize the values of the influence coefficients, Ni,~+l and
Ti, - 1 in preparation for their calculation later in the program. Note that the initial
~
value of Ti,~+l is set at 7r because in Eqn (4.1 13)

that is the tangential velocity induced on a panel by vortices of unit strength per unit
length distributed over the same panel is, from Eqn (4.112), the same as the normal
velocity induced by sources of unit strength per unit length distributed over the panel.
This was shown to take the value 7r in Eqn (3.100b).
(2) It remains to insert the two lines of instruction that calculate the additional
influence coefficients according to Eqn (4.11 3). This is accomplished by inserting
two additional lines below the last two execution statements in the routine, as shown
AN(1, J) =VX*NTIJ+W*NNIJ Existingline
AT(1, J) =VX*TTIJ+VY*TNIJ Existingline
AN(1, NP1) =AN(1, NP1) +W*NTIJ-VX*NNIJ New line
AT(I, NP1) =AT(I, NP1) +W*TTIJ-VX*TNIJ New line
As with the original routine presented in Section 3.5 this modified routine is
primarily intended for educational purposes. Nevertheless, as is shown by the exam-
ple computation for a NACA 4412 aerofoil presented below, a computer program
based on this routine and LU decomposition gives accurate results for the pressure
distribution and coefficients of lift and pitching moment. The computation times
required are typically a few seconds using a modern personal computer.
The NACA 4412 wing section has been chosen to illustrate the use of the panel
method. The corresponding aerofoil profile is shown inset in Fig. 4.25. As can be
seen it is a moderately thick aerofoil with moderate camber. The variation of the
pressure coefficient around a NACA 4412 wing section at an angle of attack of 8
degrees is presented in Fig. 4.25. Experimental data are compared with the computed