Lift Distributions and Performance
Wing design has several goals related to the wing performance and lift
distribution. One would like to have a distribution of Cl(y) that is relatively
flat so that the airfoil sections in one area are not "working too
hard" while others are at low Cl. In such a case, the airfoils with
Cl much higher than the average will likely develop shocks sooner or will
start stalling prematurely.
The induced drag depends solely on the lift distribution, so one would like
to achieve a nearly elliptical distribution of section lift. On the other
hand structural weight is affected by the lift distribution also so that
the ideal shape depends on the relative importance of induced drag and wing
weight.
With taper, sweep, and twist to "play with", these goals can be
easily achieved at a given design point. The difficulty appears when the
wing must perform well over a range of conditions.
One of the more interesting tradeoffs that is often required in the design
of a wing is that between drag and structural weight. This may be done
in several ways. Some problems that have been solved include:
- Minimum induced drag with given span -- Prandtl
- Minimum induced drag with given root bending moment -- Jones, Lamar,
and others
- Minimum induced drag with fixed wing weight and constant thickness --
Prandtl, Jones
- Minimum induced drag with given wing weight and specified thickness-to-chord
ratio -- Ward, McGeer, Kroo
- Minimum total drag with given wing span and planform -- Kuhlman
... there are many problems of this sort left to solve and many approaches
to the solution of such problems. These include some closed-form analytic
results, analytic results together with iteration, and finally numerical
optimization.
The best wing design will depend on the construction materials, the arrangement
of the high-lift devices, the flight conditions (CL, Re, M) and the relative
importance of drag and weight. All of this is just to say that it is difficult
to design just a wing without designing the entire airplane. If we were
just given the job of minimizing cruise drag the wing would have a very
high aspect ratio. If we add a constraint on the wing's structural weight
based on a trade-off between cost and fuel savings then the problem is somewhat
better posed but we would still select a wing with very small taper ratio.
High t/c and high sweep are often suggested by studies that include only
weight and drag.
The high lift characteristics of the design force the taper ratio and sweep
to more usual values and therefore must be a fundamental consideration at
the early stages of wing design. Unfortunately the estimation of CLmax is
one of the more difficult parts of the preliminary design process. An example
of this sensitivity is shown in the figure below.
The effect of a high lift constraint on optimal wing designs. Wing sweep,
area, span, and twist, chord, and t/c distributions were optimized for minimum
drag with a structural weight constraint. (Results from work of Sean Wakayama.)