Page 288 - Aerodynamics for Engineering Students

270 Aerodynamics for Engineering Students

Segment i trailing edge
Fig. 5.46 Panel method applied to a wing-body combination

For a given application there is no unique mix of sources and doublets. For many
methods* in common use each panel of the lifting surface is assigned a distribution
of constant-strength sources. The doublet distribution must now be such that it
provides one additional independent parameter for each segment of the trailing edge.
Once the doublet strength is known at the trailing edge then the doublet strength on the
panels comprising the trailing vorticity is determined. The initially unknown doublet
strength at the trailing edge segments represents the spanwise load distribution of the
wing. With this arrangement each chordwise segment of wing comprises N (say) panels
and 1 trailing-edge segment. There are therefore N unknown source strengths and one
unknown doublet parameter. Thus for each chordwise segment the N + 1 unknowns
are determined by satisfying the N zero-normal-velocity conditions at the collocation
points of the panels on the wing, plus the Kutta condition.
As in Section 4.10 the Kutta condition may be implemented either by adding an
additional panel at the trailing edge or by requiring that the pressure be the same for
the upper and lower panels defining the trailing edge - see Fig. 4.23. The former
method is much less accurate since in the three-dimensional case the streamline
leaving the trailing edge does not, in general, follow the bisector of the trailing edge.
On the other hand, in the three-dimensional case equating the pressures on the two
trailing-edge panels leads to a nonlinear system of equations because the pressure is
related by Bernoulli equation to the square of the velocity. Nevertheless this method
is still to be preferred if computational inaccuracy is to be avoided.

Exercises

1 An aeroplane weighing 73.6 kN has elliptic wings 15.23 m in span. For a speed of
90m/s in straight and level flight at low altitude find (a) the induced drag; (b) the
circulation around sections halfway along the wings. (Answer: 1.37 kN, 44m2/s)

* See B. Hunt (1978) ‘The panel method for subsonic aerodynamic flows: A survey of mathematical for-
mulations and numerical models and an outline of the new British Aerospace scheme’, in Computational
FZuidDynumics, ed. by W. Kollmann, Hemisphere Pub. Corp., 100-165; and a review by J.L. Hess (1990)
‘Panel methods in computational fluid dynamics’, Ann. Rev. Fluid Mech, 22, 255-274.