Prandtl-Glauert transformation

The Prandtl-Glauert transformation or Prandtl-Glauert rule (also Prandtl-Glauert-Ackeret's rule ) describes an approximation function that can be used to compare aerodynamic processes that take place with different Mach numbers .

Mathematical formulation

Since compressibility effects of the air become noticeable with increasing speed in subsonic flows , the aerodynamic parameters must be multiplied by a factor. The following formula shows this _using the example of the pressure coefficient c _p as a function of the pressure coefficient c _{p0 of} an incompressible flow:

{\ displaystyle c_ {p} = {\ frac {c_ {p0}} {\ sqrt {| 1-Ma ^ {2} |}}}}

where Ma is the Mach number.

This rule does not apply in the speed range between Ma = 0.7 and Ma = 1.3 as well as with hypersonic currents (Ma> 3).

history

Ludwig Prandtl had presented such a correction several times in lectures, but the first real publication came in 1928 by Hermann Glauert . Therefore this rule is called the Prandtl-Glauert rule.

The introduction of these equations made it possible to design aircraft for higher speeds in the subsonic range. This rule was later extended by Jakob Ackeret to the form common today, which also applies in the supersonic range.

Singularity

In the above-mentioned invalid range around the speed of sound, the equation shows a singularity at which the flow resistance theoretically approaches infinity. This singularity point is also referred to as the Prandtl-Glauert singularity . In reality, aerodynamic and thermal disturbances are increased disproportionately near the speed of sound and spread very far across the direction of flow, but a singularity does not occur.

Nevertheless, this theoretical singularity is wrongly used to explain real phenomena at the speed of sound (see cloud disk effect ).

Individual evidence

↑ Erich Truckenbrodt: Fluidmechanik Volume 2, 4th edition, Springer Verlag, 1996, pages 178-179
↑ H. Glauert, The Effect of Compressibility on the Lift of an Airfoil. Proc. Roy. Soc. London. VOL. CXVIII, 1928, pp. 113-119.
↑ Meier, H.-U .: The development of the swept wing, a technical challenge, Ludwig Prandtl memory lecture, GAMM annual conference 2005, March 28 to April 1, 2005, University of Luxembourg, Chapter 1 (PDF; 2.7 MB )

[1] Erich Truckenbrodt: Fluidmechanik Volume 2, 4th edition, Springer Verlag, 1996, pages 178-179

[2] H. Glauert, The Effect of Compressibility on the Lift of an Airfoil. Proc. Roy. Soc. London. VOL. CXVIII, 1928, pp. 113-119.

[3] Meier, H.-U .: The development of the swept wing, a technical challenge, Ludwig Prandtl memory lecture, GAMM annual conference 2005, March 28 to April 1, 2005, University of Luxembourg, Chapter 1 (PDF; 2.7 MB )