why lift coefficient decreases at supersonic flow?

In Anderson's performance book, he wrote that the higher the speed, the greater the pressure difference between two points, and as a result, the lift coefficient is greater. But when we reach the speed of sound, due to the interference of the shock wave and the boundary layer and the separation of the boundary layer, the lift coefficient decreases.

My question is: in supersonic speed, the speed behind the shock is still supersonic, so the pressure difference between two points in supersonic speed is more than subsonic speed, and the separation of the boundary layer is less than the separation that exists in the speed of sound(transonic speed). So why does the lift coefficient decrease at supersonic speed?

• "in supersonic speed, the speed behind the shock is still supersonic" Not necessarily: the component of speed perpendicular to the shock is always subsonic, so the total speed after it might actually be subsonic Oct 7, 2023 at 20:18
• but we always use M2 that is superconic and not Mn2 that is subsonic ! Oct 7, 2023 at 21:03
• Could it be that you mean to ask why the lift curve slope decreases in supersonic flow? Oct 7, 2023 at 23:40
• @PeterKämpf yeah Oct 8, 2023 at 0:11

As you probably know already, the 2D lift curve slope in supersonic flow is $$C_{L\alpha} = \frac{4\cdot\alpha}{\sqrt{Ma^2 - 1}}$$ when calculated by potential flow theory. But I guess you would like to understand better why that is.

In supersonic flow the only zone of influence which a wing can have is within its Mach cone. With increasing Mach number this cone gets narrower, so less air can be "reached" by the wing in order to be accelerated downwards and to create lift. This makes the wing less efficient the higher its Mach number becomes.

This might be a very simplified view, but it captures the essential mechanism well. At the same air density, a higher Mach number also means a higher dynamic pressure, so even as the coefficient of lift decreases for the same angle of attack at higher Mach numbers, the absolute lift will still grow.

Let's consider a flat plate and let it fly both at subsonic and at supersonic speeds.

Let's start with the subsonic case and let's see how the airflow looks like around the flat plate (picture source)

At subsonic speed the airflow has enough time to move out of the way and "get around" the flat plate. As depicted in the picture, the airflow starts to bend well in front of the flat plate and basically all around and far from it. The bending of the streamlines generates a relevant pressure field (according to Bernoulli) which is always globally pointing upward i.e. it generates a total aerodynamic force which is always a pure lift. The fact that at subsonic speed no drag is generated around an aerodynamic body is called D'Alembert paradox.

Drag obviously does exist at subsonic speed and is mainly due to the fluid's viscosity but as a first approximation we can simply ignore it and live with the fact that the airflow "encircles" the aerodynamic body (our flat plate) generating a pure lift.

What happens now at the flat plate at supersonic speed? The flow is now so fast that it simply cannot realise the presence of the flat plate until it just crashes against its leading edge. At that point the flow suddenly changes its curvature to follow the flat plate's surface. And when it reaches the leading edge, it just changes (suddenly again) its curvature to become again parallel to the freestream (picture source):

Also at supersonic speed this sudden change in speed comes with a relevant change in pressure (lower on the upper surface; higher on the bottom) but this time the pressure acts everywhere perpendicular to the flat plate's surface i.e. it always possesses a component pointing backward: at supersonic speed, even considering an inviscid fluid, the aerodynamic force (R in the previous picture) is always composed by a lift and by a drag. This latter is called wave drag.

For that reason alone the same flat plate flying at supersonic speed generates a lower $$C_l$$ than at subsonic speed. Anyway at supersonic speed there's much more going on, with changes in density and temperature which cannot be neglected and which globally contributes to increasing dissipation and to reducing lift in respect to the subsonic case.