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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:

enter image description here

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?

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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.

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  • $\begingroup$ Thank you Peter! Are you aware of any instance when an aircraft was designed badly enough to make this effect actually manifest? $\endgroup$ – Akerai May 15 at 1:31
  • $\begingroup$ @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. $\endgroup$ – Peter Kämpf May 15 at 5:44
  • $\begingroup$ most excellent exposition! $\endgroup$ – niels nielsen May 16 at 4:16
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Here's another drag coefficient graphic with more information:

enter image description here

This is noticeable in practice: Concorde used its afterburners to accelerate from Mach 0.9 to 1.7 (while climbing), and then shut down the afterburners for its Mach 2 cruise flight.

Whether the total drag force drops, I don't know. But the difference in Cd is a factor of 3, while total drag has a v2 term, which would mean a 4x increase going from Mach 1 to M2, so total drag still increases.

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  • $\begingroup$ In the wikipedia of the plot you've shown it states that it is a "Qualitative variation in Cd factor with Mach number for aircraft", so I am not sure if you should depend on the numbers shown. At the same time, I understand that even if the force was to drop, it would eventually rise again due to the quadratic speed term. I am just curious if the slope of the drag coefficient is sufficiently negative to outdo the quadratic increase, at least just after reaching the maximum. $\endgroup$ – Akerai May 14 at 10:34
  • $\begingroup$ @Akerai this phenomena seems to be similar to the reduction in drag when the stern of a boat breaks free from its wake. There is a reduction in overall drag and increase in speed. However airflow cooling if the engines may not be a serious issue. Increased speed and better ram efficiency approaching Mach 2 would help cancel increased overall drag. Concorde would like to go faster and higher, but frictional heating of leading edges limit it to Mach 2 at around 60,000 feet. Fuel consumption per mile for a given weight will tell the true story. $\endgroup$ – Robert DiGiovanni May 14 at 13:53
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The non-engineering answer to your question is yes, the plane will accelerate more rapidly after it goes supersonic.

Cruising at 0.98 mach, for example is a little risky. The power required to hold 0.98 mach is much less than to hold >1.0. So if you nose over, or accidentally accelerate from cruising at 0.98 mach, you'll be sustaining supersonic flight. Which is frowned upon.

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