Low Aspect Ratio Wings at High Angles of Attack

At high angles of attack, several phenomena usually distinct from the cruise flow appear. Usually part of the wing begins to stall (separation occurs and the lift over that section is reduced). An approximate way to predict when this will occur on well-designed high aspect ratio wings is to look at the Cl distribution over the wing and determine the wing CL at which some section (the critical section) reaches its 2-D maximum Cl.

When the sweep is very large, or aspect ratio low, this approach does not work. Separation tends to occur near the leading edge of the wing, but unlike in the low sweep situation, the separated region is not large and does not reduce the lift. Instead, the flow rolls up into a vortex that lies just above the wing surface.

Rather than reducing the lift of the wing, the leading edge vortices, increase the wing lift in a nonlinear manner. The vortex can be viewed as reducing the upper surface pressures by inducing higher velocities on the upper surface.

The net result can be large as seen on the plot here.

The effect can be predicted quantitatively by computing the motion of the separated vortices using a nonlinear panel code or an Euler or Navier-Stokes solver.

This figure shows computations from an unsteady non-linear panel method. Wakes are shed from leading and trailing edges and allowed to roll-up with the local flow field. Results are quite good for thin wings until the vortices become unstable and "burst" - a phenomenon that is not well predicted by these methods. Even these simple methods are computation-intensive.

Polhamus Suction Analogy

A simple method of estimating the so-called "vortex lift" was given by Polhamus in 1971. The Polhamus suction analogy states that the extra normal force that is produced by a highly swept wing at high angles of attack is equal to the loss of leading edge suction associated with the separated flow. The figure below shows how, according to this idea, the leading edge suction force present in attached flow (upper figure) is transformed to a lifting force when the flow separates and forms a leading edge vortex (lower figure).

The suction force includes a component of force in the drag direction. This component is the difference between the no-suction drag:
CDi = Cn sin a, and the full-suction drag: CL2 / p AR
where a is the angle of attack.

The total suction force coefficient, Cs, is then:
Cs = (Cn sin a - CL2/p AR) / cos L

where L is the leading edge sweep angle. If this acts as an additional normal force then:
Cn' = Cn + (Cn sin a - CL2/p AR) / cos L

and in attached flow:
CL = CLa sin a with Cn = CL cos a

Thus, Cn' = CL cos a + (CL cos a sin a - CL2/p AR) / cos L
= CLa sin a cos a + (CLa sin a cos a sin a - (CLa sin a)2/p AR) / cos L
= CLa sin a cos a + CLa/ cos L sin2 a cos a - CLa2/(p AR cos L) sin2 a

CL' = CLa [sin a cos2 a + sin2 a cos2 a /cos L - CLa/(p AR cos L) cos a sin2 a]
= CLa sin a cos a (cos a + sin a cos a/ cos L - CLa sin a /(p AR cos L))

If we take the low aspect ratio result: CLa = p AR/2, then:
CL '= p AR/2 sin a cos a (cos a + sin a cos a/ cos L - sin a /(2 cos L) )

Cross-Flow Drag Analogy

An even simpler method of computing the nonlinear lift is to use the cross-flow drag analogy. The idea is to add the drag force that would be associated with the normal component of the freestream velocity and resolve it in the lift direction. The increment in lift is then simply: D CL = CDx sin2a cosa.

The plot below shows each of these computations compared with experiment for a 80° delta wing (AR = 0.705). In these calculations a cross-flow drag coefficient of 2.0 was used.

Another case with much higher aspect ratio is shown below. Note that the very simple model seems to do nearly as well as the more involved suction analogy.

The maximum lift of a low aspect ratio wing is significantly increased by the presence of these vortices and is limited either by vortex bursting or by allowable angle of attack. Vortex bursting is a phenomenon in which the structured character of the vortex is destroyed resulting in a loss of most of the vortex lift. It occurs due to adverse pressure gradients acting on the vortex. When the vortex burst occurs on the wing (as opposed to downstream of the wing) the lift drops substantially. Although there are some empirical methods for predicting vortex burst, the phenomenon is quite complex and difficult to predict accurately. For many SST designs, however, the maximum CL may be predicted by assuming that the vortex does not burst at the maximum permissible angle of attack. Because of the length of the fuselage, this angle may be restricted to a value of 10-13 degrees. Using this value in the above expression for CL leads to a reasonable estimate for maximum lift on such designs.

A flow pattern, similar to that of the highly swept delta wing, is found at the tips of low aspect ratio wings and over fuselages. The vortex formation significantly increases the lift in these cases as well. Especially in the case of fuselage vortices, the airplane stability is affected. Interaction with downstream surfaces is often important, but hard to predict. Computations of lift at a specified angle using the cross-flow drag analogy can easily include the component associated with fuselage lift as well.

Flaps are often not used on SST designs due to difficulties with longitudinal trim. Designs with tail surfaces or canards can employ some flaps, increasing the effective alpha limit by 2-3 degrees. Clearly, conventional slats do not help these designs as they produce little change in CL at a given angle of attack. However, studies have shown that some types of leading edge vortex flaps, intended to strengthen the leading edge vortices can be used to further increase the maximum usable CL.