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Distortion of sonic bangs by atmospheric turbulence

Published online by Cambridge University Press:  29 March 2006

S. C. Crow
S. C. Crow
Affiliation:
National Physical Laboratory, Teddington, England Present address: Boeing Scientific Research Laboratories, Seattle, Washington.
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Abstract

Recorded pressure signatures of supersonic aircraft often show intense, spiky perturbations superimposed on a basic N-shaped pattern. A first-order scattering theory, incorporating both inertial and thermal interactions, is developed to explain the spikes. Scattering from a weak shock is studied first. The solution of the scattering equation is derived as a sum of three terms: a phase shift corresponding to the singularity found by Lighthill; a small local compression or rarefaction; a surface integral over a paraboloid of dependence, whose focus is the observation point and whose directrix is the shock. The solution is found to degenerate at the shock into the result given by ray acoustics, and the surface integral is identified with the scattered waves that make up the spikes. The solution is generalized for arbitrary wave-forms by means of a superposition integral. Eddies in the Kolmogorov inertial subrange are found to be the main source of spikes, and Kolmogorov's similarity theory is used to show that, for almost all times t after a sonic-bang shock passes an observation point, the mean-square pressure perturbation equals $(\Delta p)^2 (t_c/t)^{\frac{7}{6}}$, where Δp is the pressure jump across the shock and tc is a critical time predicted in terms of meteorological conditions. For an idealized model of the atmospheric boundary layer, tc is calculated to be about 1 ms, a figure consistent with the qualitative data currently available. The mean-square pressure perturbation just behind the shock itself is found to be finite but enormous, according to first-order scattering theory. It is conjectured that a second-order theory might explain the shock thickening that actually occurs.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 37 , Issue 3 , 10 July 1969 , pp. 529 - 563
Copyright
© 1969 Cambridge University Press

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References

Batchelor, G. K. 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K. 1957 Wave scattering due to turbulence. Symposium on Naval Hydrodynamics, ch. XVI, publication 515, National Academy of Science—National Research Council. Washington, D.C.
Bradshaw, P., Ferriss, D. H. & Atwell, N. P. 1967 Calculation of boundary-layer development using the turbulent energy equation. J. Fluid Mech. 28, 593.Google Scholar
Carlson, H. W. 1967 Experimental and analytical research on sonic boom generation at NASA. Sonic Boom Research. NASA SP-147.Google Scholar
Chernov, L. A. 1960 Wave Propagation in a Random Medium. New York: McGraw-Hill.
Courant, R. & Hilbert, D. 1965 Methods of Mathematical Physics, vol. II. New York: Interscience.
Crow, S. C. 1967 Visco-elastic character of fine-grained isotropic turbulence. Phys. Fluids, 10, 1587.Google Scholar
Crow, S. C. 1968 Distortion of sonic bangs by atmospheric turbulence. NPL Aero Report 1260.Google Scholar
Ellison, T. H. 1962 The universal small-scale spectrum of turbulence at high Reynolds number. Mecanique de la Turbulence. Paris: CNRS.
Friedman, M. P., Kane, E. J. & Sigalla, A. 1963 Effects of atmosphere and aircraft motion on the location and intensity of a sonic boom. AIAA J. 1, 1327.Google Scholar
Garrick, I. E. & Maglieri, D. J. 1968 A summary of results on sonic-boom pressure-signature variations associated with atmospheric conditions. NASA TN D-4588.Google Scholar
Grant, H. L., Stewart, R. W. & Moilliet, A. 1962 Turbulence spectra from a tidal channel. J. Fluid Mech. 12, 241.Google Scholar
Johnson, D. R. & Robinson, D. W. 1967 The subjective evaluation of sonic bangs. Acustica, 18, 241.Google Scholar
Kane, E. J. & Palmer, T. Y. 1964 Meteorological aspects of the sonic boom. Boeing SRDS Report no. RD 64-160.Google Scholar
Lansing, D. L. 1964 Application of acoustic theory to prediction of sonic-boom ground patterns from maneuvering aircraft. NASA TN D-1860.Google Scholar
Lighthill, M. J. 1953 Interaction of turbulence with sound or shock waves. Proc. Camb. Phil. Soc. 49, 531.Google Scholar
Lighthill, M. J. 1956 Viscosity effects in sound waves of finite amplitude. Surveys in Mechanics. Cambridge University Press.
Lilley, G. M. 1965 The structure of shock waves at large distances from bodies travelling at high speeds. Rapports du 5e Congres International d'Acoustique, vol. II. Liege.
Lumley, J. L. & Panofsky, H. A. 1964 The Structure of Atmospheric Turbulence. New York: Interscience.
Phillips, H. B. 1933 Vector Analysis. New York: John Wiley.
Tatarski, V. I. 1961 Wave Propagation in a Turbulent Medium New York: McGraw-Hill.
Townsend, A. A. 1957 Turbulent flow in a stably stratified medium. J. Fluid Mech. 3, 261.Google Scholar
Warren, C. H. E. 1965 The propagation of sonic bangs in a non-homogeneous still atmosphere. Aero. Res. Council Report no. 26774.Google Scholar
Webb, D. R. B. & Warren, C. H. E. 1965 Physical characteristics of the sonic bangs and other events at Exercise Westminster. Royal Aircraft Establishment Tech. Rep. no. 65248.Google Scholar
Whitham, G. B. 1952 The flow pattern of a supersonic projectile. Comm. on Pure and Appl. Math. 5, 301.Google Scholar
Whitham, G. B. 1956 On the propagation of weak shock waves. J. Fluid Mech. 1, 290.Google Scholar
Zilitinkevich, S. S., Laiktman, D. L. & Monin, A. S. 1967 Dynamics of the atmospheric boundary layer. Izv., Atmos. Oceanic Phys. 3, 170.Google Scholar