Prandtl–Glauert singularity

F/A-18 with vapor cone

The Prandtl–Glauert singularity is the prediction by the Prandtl–Glauert transformation that infinite pressure conditions would be experienced by an aircraft as it approaches the speed of sound. Because it is invalid to apply the transformation at these speeds, the predicted singularity does not emerge. This is related to the early 20th century misconception of the impenetrability of the sound barrier.

Prandtl-Glauert Transformation

Main article: Prandtl–Glauert transformation

Plot of the Prandtl-Glauert transformation as a function of Mach number. Notice the infinite limit at Mach 1.

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]

$c_{p} = \frac {c_{p0}} {\sqrt {|1-{M_{\infty}}^2|}}$

where

• c_p is the compressible pressure coefficient
• c_p0 is the incompressible pressure coefficient
• M_∞ is the freestream Mach number.

This correction factor works well up to low-transonic Mach numbers (M < ~0.7). However, note the limit:

$\lim_{M_{\infty} \to 1 }c_p = \infty$

This obviously nonphysical result (of an infinite pressure) is known as the Prandtl-Glauert singularity.

See also

• Prandtl-Glauert transformation
• Compressible flow
• Sonic boom

References

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