Prandtl–Glauert transformation

The Prandtl–Glauert transformation or Prandtl–Glauert rule (also Prandtl–Glauert–Ackeret rule) is an approximation function which allows to compare aerodynamical processes occurring at different Mach numbers.

Mathematical expression

In subsonic flow the compressibility of the fluid (often air) becomes more and more influential with increasing velocity. Thus, characteristic values of the flow, as found from incompressible, inviscid potential flow theory, can be multiplied with a correction factor to account for the influence of compressibility. The Prandtl–Glauert transformation is one such correction factor. The Prandtl-Glauert transformation is found by linearizing the potential equations associated with compressible, inviscid flow. It was discovered that the linearized pressures in such a flow were equal to those found from incompressible flow theory multiplied by a correction factor. This correction factor is given below. c_p, given as a function of the c_p0 of an incompressible flow and the Mach number M ^[1]:

c_{p}={\frac {c_{p0}}{\sqrt {|1-{M}^{2}|}}}.

This correction factor works well for all Mach numbers M<.7 and M>1.3.

History

Ludwig Prandtl had been teaching this transformation in his lectures for a while, however the first publication was in 1928 by Hermann Glauert.^[2] The introduction of this relation allowed the design of aircraft which were able to operate in higher subsonic speed areas ^[3]. Subsequently the equation was extended by Jakob Ackeret to the common form used today, which is also valid in the supersonic region.

Singularity

Near the sonic speed (M=1) the discussed equation features a singularity, although this point is not within the area of validity. The singularity is also called the Prandtl–Glauert singularity, and the flow resistance is approaching infinity here. In reality aerodynamic and thermodynamic perturbations get amplified strongly near the sonic speed, however a singularity does not occur.

Besides this the theoretical singularity is often - however not correctly (see above) – used to explain phenomena near the sonic speed.

Notes

^ Erich Truckenbrodt: Fluidmechanik Band 2, 4. Auflage, Springer Verlag, 1996, p. 178-179
^ H. Glauert, The Effect of Compressibility on the Lift of an Airfoil. Proc. Roy. Soc. London. VOL. CXVIII, 1928, p. 113–119.
^ Meier, H.-U.: Die Entwicklung des Pfeilflügels, eine technische Herausforderung, Ludwig Prandtl memorial lecture, GAMM 2005, March 28th - April 1st 2005, Universität Luxemburg, Kapitel 1

[1] Erich Truckenbrodt: Fluidmechanik Band 2, 4. Auflage, Springer Verlag, 1996, p. 178-179

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

[3] Meier, H.-U.: Die Entwicklung des Pfeilflügels, eine technische Herausforderung, Ludwig Prandtl memorial lecture, GAMM 2005, March 28th - April 1st 2005, Universität Luxemburg, Kapitel 1

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