• Version
• 76 Downloads
• 1.16 MB File Size
• 1 File Count
• December 8, 2016 Create Date
• December 8, 2016 Last Updated

National Advisory Committee for Aeronautics, Technical Notes - Study by the Prandtl Glauert Method of Compressibility Effects and Critical Mach Number for Ellipsoids of Various Aspect Ratios and Thickness Ratios

By the use of a form of the Prandtl-Glauert method that is valid
for three-dimensional flow problems, the value of the maximum incremental
velocity for compressible flow about thin ellipsoids at zero angle of
attack is calculated as a function of the Mach number for various aspect
ratios and thickness ratios. The critical Mach numbers (within the
accuracy of the Prandtl-Glauert method) of the various ellipsoids are
also determined. The results indicate an increase in critical mach
number with decrease in aspect ratio which is large enough to explain
experimental results on low-aspect—ratio wings at zero lift-

Recent tests (references 1 and 2) have shown that an appreciable
increase in the critical Mach number, tOgether with other improvements
of the aerodynamic characteristics at supercritical Mach numbers, results
from the use of wings of very low aspect ratio. These improved'charac-
teristics have been somewhat qualitatively ascribed to "three-dimensional
relief," although no quantitative theoretical discussion has yet been
provided.

In the present paper an effort is made to provide such a study by
considering the flow, at zero angle of attack, about a series of thin
ellipsoids of various aspect ratios and thickness ratios. Ellipsoids
were chosen because they are amenable to calculation. Although they
differ appreciably from the wings of reference 1, which had an NACA
0012 airfoil section and rectangular plan form, ellipsoids should never-
theless show similar aspect-ratio effects- The calculations were made
for ellipsoids of thickness ratios 0.10, 0-15, and 0-20, and for the
entire range of aspect ratios from the elliptic cylinder to the
ellipsoid of revolution.

The compressibility effects were computed by the use of a form of
the Prandtl-Glauert method that is valid for three-dimensional flow
problems. The method has been given by Gothert (reference 3) without,
however, very explicit mathematical proof. Another correct statement
of the three-dimensional form of the Prandtl-Glauert method was given
earlier by A. Busemann in reference 1+, where, however, no formulas were
given.