Acceleration of a supersonic aircraft after breaking the sound barrier

People on the physics stackexchange recommended I ask here, so I am pasting my question regarding supersonic aircraft.

I stumbled upon an interesting plot; in particular, the dependence of wave drag on the Mach number:

It is curious to see that the drag coefficient drops so abruptly in the supersonic regime, but I am even more curious if the total drag force acting on the airplane also drops, i.e. if the plane starts accelerating in the supersonic regime.

I did some quick analysis of the problem. The drag force is defined as $$F_{\text{drag}} = C(v) v^2 ,$$ where $$C(v) = A C_D(v) \rho(v)/2$$ is the newly introduced drag constant with the drag coefficient $$C_D(v)$$, flight medium density $$\rho(v)$$, and cross-sectional area $$A$$.

Assuming these relations, one can taylor expand the equation for the drag force to find that

$$\Delta F_{\text{drag}} = C'(v)v^2 \Delta v + 2Cv \Delta v.$$

As $$C'(v)$$ is clearly negative in the supersonic regime, this means that the total drag force drops if $$C'(v) < -\frac{2C(v)}{v}.$$

It seems that this criterion could indeed be satisfied in air, as the right hand term in the inequality is a fairly small number.

Is there anyone that could elaborate further on what actually happens to the drag force as the plane breaks the sound barrier? Do the plane engines lower their power in order to maintain reasonable supersonic cruising speed, and if they didn't, would they be under too much heat load?

Your Taylor expansion only uses the first two terms, so it is only a crude approximation. But still, your observation is correct, however, whether it applies depends on the aerodynamic quality of the particular aircraft.

Practical supersonic aircraft have been designed to minimize the Mach drag peak. The ways to do this should be familiar:

• Stretch the aircraft lengthwise so it becomes long and thin.
• Smooth out the cross sectional area distribution over length ("area ruling")
• Use swept wings with thin airfoils

If this is properly done, the drag peak will be small enough to have overall drag increase with Mach number above Mach 1. In case of the F-16 the drag coefficient rises from 0.02 (subsonic) to 0.045 (Mach 1.1) and stays roughly constant with increasing Mach number, so the absolute drag still grows with speed squared. No significant decrease in the drag coefficient occurs because of the complex flow around the entire aircraft. Only when you have a poor design will drag actually become lower at low supersonic Mach numbers.

Another factor is thrust: Since ram pressure at the intake increases, so does thrust of the same engine as speed increases. This is the main reason why Concorde could already super-cruise 20 years prior to Lockheed marketing inventing this term.

• Thank you Peter! Are you aware of any instance when an aircraft was designed badly enough to make this effect actually manifest? May 15 '19 at 1:31
• @Akerai: I have no numbers but am sure that many of the early designs which transversed the Mach barrier by brute force are good candidates for a search. May 15 '19 at 5:44
• most excellent exposition! May 16 '19 at 4:16