# Does lift-to-drag ratio asymptote to 4 (or 6) at high Mach numbers?

The Kuchemann equations for L/D at high mach numbers, approximately verified by wind tunnel tests, were:

$$\left(\frac{L}{D}\right)_{max} = \frac{4\cdot(M+3)}{M}$$

and, for wave riders

$$\left(\frac{L}{D}\right)_{max} = \frac{6\cdot(M+2)}{M}$$

As you can see, these equations asymptote to 4 and 6 respectively.

If my understanding of the matter is correct, at ultra high Mach numbers, air begins to behave more like rays of radiation. Hence, L/D would simply be:

L/D = 1/tanA


where A is angle of attack.

Hence, the corresponding angles of attack for the two asymptotes would be 14 degrees and 7.5 degrees, respectively.

Is there any physical evidence for the above asymptotes (and perhaps the AoA values)?

My source is STUDIES OF HIGH LIFT/DRAG RATIO HYPERSONIC CONFIGURATIONS by John V. Becker, to be found online here. It is from 1964, so more than 50 years old, but since a lot of research had been performed already before that date, it might still be relevant.

In short: It depends. On thickness ratio and Reynolds number, for example, as can be seen in this plot for flat plates from the above linked report:

Note the grey areas noting the maximum for glide rsp. reentry vehicles. Their performance is roughly half as good as that of a flat plate.

Most of the report uses the volumetric efficiency parameter $$\frac{V^⅔}{S}$$ for comparisons. This is the volume $$V$$ to the power of ⅔ divided by the reference area $$S$$. Below is a carpet plot for a variety of wedge shapes, valid for Mach 6.8 (same source as above).

An infinitely thin wing will achieve an L/D of 6.5, irrespective of its leading edge sweep, and a more realistic t/c of 0.09 will not get above an L/D of 5.5 at 85° sweep, dropping below 5 for 70° sweep.

It seems the Küchemann approximation holds up well for single-digit Mach numbers but becomes too optimistic above this speed. And impact theory alone is not sufficient to predict L/D with any confidence.

The plot below explains why the Shuttle is a low wing configuration: Above Mach 10 it is cleary advantageous to put the flat side on the bottom:

The index FB represents the flat bottom version of the vehicle.

Curiously, when we plug some numbers, L/D comparison of the "waverider" get better and better as Mach increases:

M $$\frac{4 \cdot (M+3)}{M}$$ $$\frac{6 \cdot (M+2)}{M}$$
1 16 18
2 10 12
3 8 10
4 7 9
5 6.4 8.4

This confirms that supersonic aircraft do better as "bottom lifters", essentially an extrapolation of supercritical wing design.

The XB-70 prototypes took advantage of this by folding its wingtips down in flight, and riding its own shock wave.

Supersonic wings are very thin and diamond shaped. Add a little AOA and one can see a parallelogram, with very little "classical" top lift being produced.

In the supersonic realm, the same lift with less drag is achieved with the waverider.

• It should be no surprise that hypersonic lift is mostly created on the lower surface. With the low density in the hypersonic flight altitude, the pressure difference on the top surface cannot exceed that between a vacuum and atmospheric pressure, while on the bottom the pressure can add up to multiples of atmospheric pressure. Commented Apr 29, 2021 at 11:09
• Thanks Peter Kampf and ROIMaison Commented Apr 29, 2021 at 12:15