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

Even in subsonic flow the compressibility of the fluid (often air) is getting more and more influential with increasing velocity. Thus characteristic values of the flow can be multiplied with a correction factor to account for this influence; this is called the Prandtl–Glauert transformation. As an example following equation shows this for the pressure coefficient c_p, given as a function of the c_p0 of an incompressible flow and the Mach number "M" [Erich Truckenbrodt: Fluidmechanik Band 2, 4. Auflage, Springer Verlag, 1996, p. 178-179 ] :

:$c\_\{p\}\; =\; frac\; \{c\_\{p0\; \{sqrt\; \{|1-\{M\}^2|$

The equation is not valid for velocities between "M=0.7" 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. [H. Glauert, The Effect of Compressibility on the Lift of an Airfoil. Proc. Roy. Soc. London. VOL. CXVIII, 1928, p. 113–119. ] The introduction of this relation allowed the design of aircraft which were able to operate in higher subsonic speed areas [ [http://www.dglr.de/literatur/publikationen/pfeilfluegel/Kapitel1.pdf 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] ] . 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 correctlyFact|date=October 2008 – used to explain phenomena near the sonic speed, e.g. see Prandtl–Glauert singularity.

References