# How does supersonic speed affect vortex lift of a delta wing?

Recent questions have asked why deltas are preferred as commercial passenger supersonic aircraft airfoils.

Sweeping the wings is known to delay shockwave formation as transonic speeds are approached, but what happens to the upper wing vortices of a delta under these conditions?

With straight or swept wings, a "tuck" phenomena is produced by shockwaves forming in different places on the upper and lower wing, resulting in a nose down pitching moment.

The Concorde was known to transfer fuel to adjust to the change of center of lift, but what exactly happens to the airflow when a delta goes supersonic?

Is the "tuck" effect stronger or weaker than a swept or straight wing?

From an historical perspective, the transonic flow has always been considered tough.

Back in the beginning of the transonic/supersonic era (let's say the '40s) the mathematical equations of the aerodynamics had been already known for quite a few centuries. The simplified version of those equations, that is their manually solvable version, quite correctly predict(ed) the aerodynamic behaviour for the subsonic and supersonic (and hypersonic) flows, but they failed to represent the transonic flow where they just gave (give) a solution tending toward infinity. Since computers to resolve the full version of the equations had yet to come, all that was left to do was to come up with a good expression to describe this impossible-to-reach flight state: the sound barrier.

Thanks to the progress both in the computer and in the fligh test field, the sound barrier proved to be not such a destroying effect as predicted by the simplified version of the equations. Anyway even for the today's standards the transonic regime still remains a complicated matter.

The transonic state starts when somewhere on the aircraft a Mach number greater or equal to 1 is reached and this condition is reached even if the whole aircraft has a speed lower than Mach 1 due to local accelerations of the fluid. Tipically this happens on the upper part of the wing, as visible in this picture taken from Wikipedia:

The shockwave which forms could lead to a detachment of the flow behind it with a sudden loss of lift and rise of drag. Anyway modern airfoils for jet airliners are designed to develop and tolerate a mild shockwave, which basically gives no issues.

This event happens also on a delta wing but due to the typical vortices detaching from the leading edge, this phenomenon is a bit more complicated in that 2 sets of shockwaves actually develop.

This picture taken from Schiavetta, Boelens and Fritz (and in turn taken from AIAA Journal, Vol. 35, No. 10, 1997, pp. 1568–1573) depicts the situation quite well:

The first shockwave (left figure, the wing is seen from the back) develops between the upper surface of the wing and the "lower part" of the leading edge vortex and is called embedded cross-flow shock wave. The second shock wave (right figure, dark gray area) develops towards the trailing edge and parallel to it and it is called terminating shock wave.

The behaviour of the aircraft (with or without a delta wing) during the transonic flight or, more generally, during the transition from subsonic to supersonic, is more or less predictable only via complicated and time-consuming simulations and/or flight/wind tunnel tests.

What is generally well known and understood is the overall trend of the aerodynamic characteristics in the subsonic and in the supersonic flow (that is just before and just after the transonic regime): just before, $$C_l$$ is (more or less) proportional to $$2 \pi \alpha$$ and the aerodynamic center is at 25% of the airfoil's chord; just after, $$C_l$$ is (more or less) proportional to $$4 \alpha$$ and the aerodynamic center is at 50% of the airfoil's chord.

This means that in the transonic regime the lift becomes lower at the same pitch angle and it migrates to the back of the wing. This migration has to be somehow compensated for and in the case of the Concorde it was done pumping fuel in the tail i.e. shifting the CG backward too to make up for the migration of the lift.

• Interesting that the migration of Cp is very similar to stall. This may also help explain supercritical design. With so much forward speed, one can "surf" the air (Newtonian bottom lift) and minimize drag on top. Sort of opens the door to low AR supersonic wings, and, lo and behold, we have deltas and trapezoidal designs flying around today (the XS-1 wings were straight). Aug 6, 2022 at 11:30