VORTEX-DOMINATED FLOWS AT HIGH ANGLES OF ATTACK

 

Why would any craft fly at high angles of attack? Three applications come to mind:

1. Fighter planes going through sharp maneuvers.

2. Supersonic aircraft (fighters, bombers, the Concorde, the Space Shuttle, the HSCT) coming in to land

3. The Space Shuttle and other lifting bodies (X-38, X-33) re-entering the atmosphere and decelerating.

 

 

At high angles of attack on an airfoil, the flow "separates" from the surface, leaving a region of slow-moving, or "recirculating" flow below it. The pressure on the upper surface is not as low as it was with attached flow near the surface. Hence the lift is lost. This is called "stall".

However, if the leading edge is swept, the vortex sheet from more downstream locations keeps adding to the vorticity in the recirculation zone, and the vorticity in this zone forms a strong, tight, "leading edge vortex", which lies close to the surface. Very high velocity values occur near the core of this vortex. The static pressure is low in the vortex core, and on the surface immediately below the vortex. This causes "vortex-induced lift".

Note that as the core strengthens, a strong helical flow develops along the core axis. This is like a swirling jet. The jet velocity can reach values as high as 3 times the freestream velocity. However, this does not mean that these vortices produce thrust! The stagnation pressure in the core of the vortex is below the freestream value.

As the angle of attack increases, the pressure gradient becomes adverse over the aft portions of the wing. The vorticity addition from the leading edge thus ceases at these locations. As the angle of attack increases further, the flow in the core encounters an adverse pressure gradient which it cannot overcome. The axial flow comes to a sudden stop. By conservation of mass, we see that the core diameter must increase when this happens. Conservation of angular momentum dictates that the peak swirl velocity must come down. This is "vortex breakdown", or "vortex bursting". It sounds simple and obvious, but the precise mechanisms by which it occurs are the subjects of intense debate.

There are good reasons for interest in the precise mechanisms of vortex breakdown. Often, as an airplane goes through a maneuver where the angle of attack increases continuously, vortex breakdown occurs at different chordwise stations on the left and right sides of the wing, even when the geometry of the wing appears to be quite symmetric. This causes severe differences in lift, a rolling moment. The rate at which vortex breakdown propagates up the wing with increasing angle of attack is a matter of intense interest. This rate is sometimes so slow that it cannot keep up with the rate of the maneuver.

A second reason for interest in vortex breakdown mechanisms is that the vertical tails of several modern fighter planes are directly in the path of the bursting vortex. When vortex breakdown occurs, the tails are subjected to violent "buffeting", with unfortunate consequences. Again, one would like to be able to predict the frequencies and amplitudes of these fluctuations, so that the tail structures can be designed to take advantage of these characteristics.

When the angle of attack is increased still further, the vortex breakdown becomes asymmetric, and eventually reaches the very apex of the wing. Beyond this angle of attack, the flow over the wings consists of a pair of large, swirling flows with no strong core. The turbulence level is high.

 

These changes are summarized in the table below:

 

The above features are seen in low-speed flow, and well into the "subsonic" regime. As the Mach number increases further, shocks form, and other features are seen.

Effects of Compressibility

Lets consider a slender, sharp-edged delta wing at high angles of attack.

 

The Mach normal to the leading edge is given by the following:

If the wing leading edge is subsonic; i.e., the normal Mach number is below 1, then the flow over the wing resembles subsonic vortex flow. If the wing leading edge is supersonic, i.e., the normal Mach number is above 1, then the upper-surface flow is dominated by Prandtl-Meyer expansions, shocks, and separation.

 

The nose and forebody of a missile or high-speed aircraft is a sharp-pointed body. Flows over such bodies are similar to those over highly swept wings. Flow separates along a line on each side of the body, and the vortex sheet rolls up into a tight vortex on each side. These vortices are very hard to keep symmetric because they are very sensitive to geometry. As a result, the vortex-induced suction is different on the 2 sides. Because the nose is a long way from the center of gravity, this causes large yawing moments. Also, rolling moments on the wing because one vortex creates lift on its wing than the other vortex.

 

Linear Model: Horseshoe vortices lie in the plane of the wing. Nonlinear: vortice from panel edges take off from surface at an angle approximately half the angle of attack.

The nonlinear variation of lift with increasing angle of attack can be modeled by the velocities induced by the free vortices which are released from the surface and side edges, and flow over the wing. Free vortices generally go at roughly half the angle of attack. For example, the correlation developed by Garner & Lehrian (1965) uses the linear lift computed from Multhopp's linear lifting surface method, and the nonlinear lift from methods where free vortices are included.

Betz(1935) postulated that the aerodynamic force on a low AR wing is due mainly to the drag force caused by the normal component of velocity in the cross-plane. Normal velocity  .

Normal Force 

Lift is  .

is the drag coefficient for a flat rectangular wing in cross-flow:  .

. This gives a reasonable first-order estimation of the nonlinear lift for small Aspect Ratio wings. Note: At small angle of attack,  . So, 

 

Assumes that total lift = potential-flow lift + lift due to separated leading-edge vortices.

The potential-flow lift can be calculated using linear lifting-surface theory, with attached flow over the leading edge. The vortex lift is equated to the suction developed along the leading edge, as calculated by thin-wing linear lifting surface theory.

Thus,  . Kp and Kv can be obtained as functions of Aspect Ratio from Figures 6.14, 1.16 and 6.17 (p. 168-170 of Rom).

Good agreement with experiment for sharp-edged delta wings of AR0.5 to 2.0, angles of attack less than 25 degrees.