I am desigining an aircraft, which shall fly up to Mach=0.9 in cruise flight in an altitude of 10 km.

So when I think of which airfoil to choose, I have to consider aspects like

  • max. lift coefficient required
  • max. necessary/allowed angle of attack
  • ...

But what comes first in my mind is, that I have to prevent shock waves forming on my airfoil, because drag will dramatically increase. So actually, this is my first demand for the airfoil.

NACA 64006 is a supercritical airfoil. In order to prevent Mach=1 on the airfoil, I would need a swept wing with an angle of more than 50° to reach Ma_crit=0.88 .

I am wondering now, if this is normal that I need such a high sweep angle? Or did I choose the wrong airfoil. So normal commerical aircraft have angles between 30 to max. 40° ?

So the only way to have less sweep angle: find another airfoil (I think 64006 is already one of the best free available airfoils) or I accept higher values of drag coefficient ?

How do airliners tackle the problem? I think I read that they allow mach numbers a little bit higher until the drag divergence velocity?

  • $\begingroup$ For those high Mach numbers I would look at business jets. Commercial airliners rarely reach speeds of M0.9. In addition, consider that commercial airliners don't use a standard NACA airfoil, but along the span of the wing the airfoil changes and the shape is quite a secret $\endgroup$
    – Afe
    Mar 18, 2021 at 12:43
  • 1
    $\begingroup$ NACA 64006 is not a supercritical airfoil. There are better choices. While not many have been published, any of those will be better than the old NACA. For a collection see here $\endgroup$ Mar 18, 2021 at 13:41

1 Answer 1


Yes, you chose the wrong airfoil.

While the NACA 6-digit series was among the first set of airfoils computed from a design pressure distribution, they will suffer from shocks when operated above their critical Mach number just as any other airfoil.

Comparison of drag rise over Mach for 6-series and supercritical airfoils

Comparison of drag rise over Mach for 6-series and early supercritical airfoils from NASA Technical Paper 2969. It should be obvious which one is to prefer.

When transsonic research started, inverted airfoils paradoxically turned out to perform better at moderate lift coefficients and high Mach numbers than regular airfoils. Key is the low curvature on the suction side which makes a shock-free pressure rise possible. Practical designs aim for a weak shock over a range of lift coefficients. In oder to produce the most lift at a given Mach number, the pressure difference between upper and lower side can be maximized where thickness is low, i.e. in the rear area of the airfoil. This is called "rear loading". Supersonic flow on the upper side allows to create more lift over the full chord. On the most recent designs another small lift contribution results from a slower pressure drop on the forward edge of the lower side pressure distribution (the forward lower-surface undercutting of Phase 3 airfoils).

Effect of forward lower surface undercutting

Effect of forward lower surface undercutting, from NASA Technical Paper 2969. The added lift at the forward part also lowers the pitching moment of supercritical airfoils.

Supercritical airfoils have negative camber over most of their forward part and strong positive camber only in the rearmost 20 - 30% of their chord. This NASA report (PDF!) lists a range of supercritical airfoils with different thicknesses and design lift coefficients. But it can only serve as a starting point since wing sweep will change the pressure distribution locally and demands a shift in the point of maximum thickness (forward at the root, backward at the tip) and more negative camber at the root in combination with a higher local incidence.

Mach 0.9 is only used by business jets where owners care more about bragging rights than fuel consumption. Airliners normally fly at Mach numbers ranging between 0.78 (short range) and 0.84 (long range) and for best efficiency should stay below Mach 0.82. They fly a bit above their drag divergence Mach until the gains in speed are balanced by the losses in drag. Of course, they can be operated at higher speeds but do so only to make up for a delay.

  • $\begingroup$ Thank you Mr. Peter Kämpf. $\endgroup$
    – Lucas
    Mar 19, 2021 at 14:12
  • $\begingroup$ Thank you Mr. Peter Kämpf. So airliners don't use critical airfoils? And what if I want to fly at Ma=0.9 ? Then I should choose as "beginner" without any access to company secret details the mentioned supercritical airfoils of you? Is this the main design driver to avoid drag? $\endgroup$
    – Lucas
    Mar 19, 2021 at 14:22
  • $\begingroup$ @Lucas Airliners use supercritical airfoils since the A310. Still, they prefer to fly a bit slower than Mach 0.9. Supercritical only allows to raise Mach a bit, or to use a thicker airfoil at the same speed. Or use less sweep at the same speed. Read the linked answers and all should be clear. $\endgroup$ Mar 19, 2021 at 16:18
  • $\begingroup$ Dear Mr Kämpf, so I looked up : ntrs.nasa.gov/api/citations/19900007394/downloads/… On page 70 (72 in pdf) the pressure distribution is plotted. So for alpha=0° the cp_min = -1,1 what is way lower than approx. cp_sonic=-0.35 . So the velocity on the airfoil is higher than Ma_crit . So why should this be a supercritical airfoil? Can it only used together with negative angle of attack? $\endgroup$
    – Lucas
    Mar 23, 2021 at 14:22
  • $\begingroup$ @Lucas It is a supercritical airfoil because it can tolerate higher than sonic (= critical) speeds without a shock. The very fact that velocity on the airfoil is higher than Ma$_{crit}$ without a strong shock makes it supercritical. Besides, $c_{p\;crit}$ is -0.65 if you talk of Fig. 36. Don't look at the figures only, please also read the text in the PDF. $\endgroup$ Mar 24, 2021 at 0:47

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