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
which some section (the critical section) reaches its 2-D maximum Cl. In
the example below the outboard sections have Clmax = 1.5 so
the wing begins
to stall near the tip when CL = 1.24
The effects of wing sweep must be taken into account when using critical section theory as the outboard flow of the boundary layer acts to reduce the maximum Cl available over the outboard sections.
When the sweep is very large, 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.
However, 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 α, and the full-suction drag: CL2 / π AR
where α is the angle of attack.
The total suction force coefficient, Cs, is then:
Cs = (Cn sin α - CL2/π AR) / cos Λ
where Λ is the leading edge sweep angle. If this acts as an additional normal force then:
Cn' = Cn + (Cn tan α- CL2/π AR) / cos Λ
= Cn + (Cn sin α - CL2/π AR) / cos Λ
so, Cn' = Cn + (Cn sin α - CL2/π AR) / cos Λ
and in attached flow:
CL = CLa sin α with Cn = CL cos α
Thus, Cn' = CL cos α + (CL cos α sin α - CL2/π AR) / cos Λ
= CLa sin α cos α + (CLa sin α cos α sin α - (CLa sin α)2/π AR) / cos Λ
= CLa sin α cos α + CLa/ cos Λ sin2 α cos α - CLa2/(π AR cos Λ) sin2 a
CL' = CLa [sin α cos2 α + sin2 α cos2 α /cos Λ - CLa/(π AR cos Λ) cos α sin2 α]
= CLa sin α cos α (cos α + sin α cos α/ cos Λ - CLa sin α /(π AR cos Λ))
If we take the low aspect ratio result: CLa = π AR/2, then:
CL '= π AR/2 sin α cos α (cos α + sin α cos α/ cos Λ - sin α /(2 cos Λ) )
The plot below shows this computation compared with experiment for a 80° delta wing (AR = 0.705)
Attached flow computations, Polhamus suction analogy, and experiment for lift on a 80° delta wing.
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 and interaction with downstream surfaces is often important and hard to predict.
Vortices generated by the fuselage and leading edge stakes of an F-18 are visible in the photo below and this QuickTime video clip of NASA's High Alpha Research Vehicle, used to investigate these phenomena and ways to control them.