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:[1]
where
- cp is the compressible pressure coefficient
- cp0 is the incompressible pressure coefficient
- M is the Mach number.
This correction factor works well for all Mach numbers 0.3 < M < 0.7
It should be noted that since this correction factor is derived from linearized equations, the pressures calculated is always less in magnitude than the actual pressures within the fluid.
Aerodynamic Coefficients
Since the aerodynamic coefficients of lift and drag are simply integrals of the pressure coefficient, the Prandtl-Glauert transformation is valid for these coefficients as well. The correction factor is not valid for wall shear stress or other viscous based interactions, however, as the correction is based on inviscid flow theory. This compressibility correction factor also affects lift slope curve for the lift coefficient with respect to angle of attack for aerodynamic structures.
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 calculated to approach infinity. In reality aerodynamic and thermodynamic perturbations get amplified strongly near the sonic speed, however a singularity does not occur. An explanation for this is that the Prandtl-Glauert transformation is a linearized approximation of compressible, inviscid potential flow. As the flow approaches sonic (M=1), the nonlinear phenomena dominate within the flow, which this transformation completely ignores for the sake of simplicity.
Despite this the theoretical singularity is often - however not correctly (see above) – used to explain phenomena near the sonic speed.
See also
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