Simple Sweep Theory
Lifting line theory works only for unswept wings.
Weissinger theory provides a means for computing the distribution of lift
on swept wings, but not the chordwise distribution of pressures.
Vortex lattice models, panel methods, and nonlinear CFD provide pressure
distributions on swept-back wings, but do not provide some of the insight
that we can obtain with the simpler models.
It is mostly for this reason, and partly for historical reasons that simple
sweep theory is interesting. It was invented by Buseman around 1935 and
independently by R.T. Jones.
Consider an infinite wing as shown below.
If we ignore the effects of viscosity, and the wing is painted white so
there is nothing to distinguish one section from another, we can slide the
wing sideways and we could not tell that it was moving sideways. The air
could not tell either, so the pressure distribution would remain unchanged.
We have just created an infinite, obliquely-swept wing that is moving with
respect to the air at a speed:
We can use this fact to design a wing which can fly at a high speed with
a pressure distribution associated with a lower speed.
The main idea behind sweeping the wing is to reduce the effects of compressibility.
The component of the flow parallel to the wing is not effected by the presence
of the wing; the normal component is decoupled from the tangential component.
This is true not only according to linear flow theory, but also in the case
of nonlinear compressible flow with shock waves. It is an interesting exercise
to show how the full potential equations decompose into a normal term and
a tangential term when one asserts that nothing changes in the tangential
direction. This idea is called simple sweep theory. We can consider sections
normal to the wing edges as operating in a flow with lower Mach number and
dynamic pressure. The effective normal Mach number is then:
but because of the reduced normal dynamic pressure, the section lift coefficient
based on this component of the freestream velocity must be increased if
the total lift is fixed:
Furthermore, at a given angle of attack, the lift is reduced. The reduction
of lift at a given angle of attack for swept wings has important implications:
the airplane incidence angle must be higher, causing several problems for
some aircraft on landing approach (e.g. Concorde's drooped nose and long
nose gear, F-8 variable incidence wing). The reduced lift curve slope due
to sweep can improve the ride quality in gusty air, however.
This basic idea permits subsonic sections to be used at supersonic freestream
Mach numbers or transonic airfoils to be used at Mach numbers higher than
they would otherwise be able to operate. This works quite well even up to
Mach numbers well over 1.0. A recent airplane design that I worked on had
a design Mach number of 1.4. The airfoils were designed to operate at a
Mach number of 0.7 (normal Mach number) with the wing swept 60°. Although
the airplane lift coefficient was not meant to exceed 0.25, the airfoils
had a design Cl of 1.0 because of the reduced normal dynamic pressure from
simple sweep theory.
The reason for designing a supersonic wing with sweep and subsonic airfoil
sections can be seen in the results of thin airfoil theory which predicted
no drag for subsonic sections, but did indicate that supersonic airfoil
sections would produce drag due to thickness, camber, and lift.
Since the effect varies with cosine of the sweep angle, we expect that either
forward, or aft swept wings would realize similar benefits. This is basically
true, although, as discussed in the section on wing design and forward-swept wings, there are some important differences.
Similarly, wings with oblique sweep have been designed and tested. Further
discussions of oblique wings are given in the
section on supersonic wings.