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Nonlinear reflection of grazing acoustic shock waves: unsteady transition from von Neumann to Mach to Snell–Descartes reflections

Published online by Cambridge University Press:  07 March 2007

SAMBANDAM BASKAR ,
FRANÇOIS COULOUVRAT  and
RÉEGIS MARCHIANO
SAMBANDAM BASKAR
Affiliation:
Laboratoire de Modélisation en Méecanique, Université Pierre et Marie Curie–Paris 6 & CNRS (UMR 7607), 4 place Jussieu, 75252 Paris cedex 05, France
FRANÇOIS COULOUVRAT*
Affiliation:
Laboratoire de Modélisation en Méecanique, Université Pierre et Marie Curie–Paris 6 & CNRS (UMR 7607), 4 place Jussieu, 75252 Paris cedex 05, France
RÉEGIS MARCHIANO
Affiliation:
Laboratoire de Modélisation en Méecanique, Université Pierre et Marie Curie–Paris 6 & CNRS (UMR 7607), 4 place Jussieu, 75252 Paris cedex 05, France
*
†Author to whom correspondence should be addressed: [email protected]
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Abstract

We study the reflection of acoustic shock waves grazing at a small angle over a rigid surface. Depending on the incidence angle and the Mach number, the reflection patterns are mainly categorized into two types, namely regular reflection and irregular reflection. In the present work, using the nonlinear KZ equation, this reflection problem is investigated for extremely weak shocks as encountered in acoustics. A critical parameter, defined as the ratio of the sine of the incidence angle and the square root of the acoustic Mach number, is introduced in a natural way. For step shocks, we recover the self-similar (pseudo-steady) nature of the reflection, which is well known from von Neumann's work. Four types of reflection as a function of the critical parameter can be categorized. Thus, we describe the continuous but nonlinear and non-monotonic transition from linear reflection (according to the Snell–Descartes laws) to the weak von-Neumann-type reflection observed for almost perfectly grazing incidence. This last regime is a new, one-shock regime, in contrast with the other, already known, two-shock (regular reflection) or three-shock (von Neumann-type reflection) regimes. Hence, the transition also resolves another paradox on acoustic shock waves addressed by von Neumann in his classical paper. However, step shocks are quite unrealistic in acoustics. Therefore, we investigate the generalization of this transition for N-waves or periodic sawtooth waves, which are more appropriate for acoustics. Our results show an unsteady reflection effect necessarily associated with the energy decay of the incident wave. This effect is the counterpart of step-shock propagation over a concave surface. For a given value of the critical parameter, all the patterns categorized for the step shock may successively appear when the shock is propagating along the surface, starting from weak von-Neumann-type reflection, then gradually turning to von Neumann reflection and finally evolving into nonlinear regular reflection. This last one will asymptotically result in linear regular reflection (Snell–Descartes). The transition back to regular reflection is one of two types, depending on whether a secondary reflected shock is observed. The latter case, here described for the first time, appears to be related to the non-constant state behind the incident shock, which prevents secondary reflection.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 575 , March 2007 , pp. 27 - 55
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Auger, T. & Coulouvrat, F. 2002 Numerical simulation of sonic boom focusing. AIAA J. 40, 1726–1734.CrossRefGoogle Scholar
Baskar, S., Coulouvrat, F. & Marchiano, R. 2006 Irregular reflection of acoustic shock waves and von Neumann paradox. In Innovations in Nonlinear Acoustics (Proceedings of 17th International Symposium on Nonlinear Acoustics) (ed. Atchley, A. A., Sparrow, V. W. & Keolian, R. M.), pp. 536–539. AIP.Google Scholar
Ben-Dor, G. 1987 A reconsideration of the three-shock theory for a pseudo-steady Mach reflection. J. Fluid Mech. 181, 467–484.CrossRefGoogle Scholar
Ben-Dor, G. 1992 Shock Wave Reflection Phenomena. Springer.CrossRefGoogle Scholar
Ben-Dor, G. & Takayama, K. 1985 Analytical prediction of the transition from Mach to regular reflection over cylindrical concave wedges. J. Fluid Mech. 158, 365–380.Google Scholar
Ben-Dor, G. & Takayama, K. 1992 The phenomena of shock wave reflection – a review of unsolved problems and future research needs. Shock Waves 2, 211–223.CrossRefGoogle Scholar
Birkhoff, G. 1950 Hydrodynamics, A Study in Logic, Fact and Similitude. Princeton University Press.Google Scholar
Brio, M. & Hunter, J. K. 1992 Mach reflection for the two-dimensional Burgers' equation. Physica D 60, 194–207.CrossRefGoogle Scholar
Colella, P. & Henderson, L. F. 1990 The von Neumann paradox for the diffraction of a weak shock waves. J. Fluid Mech. 213, 71–94.CrossRefGoogle Scholar
Coulouvrat, F. & Marchiano, R. 2003 Nonlinear Fresnel diffraction of weak shock waves. J. Acoust. Soc. Am. 114, 1749–1757.CrossRefGoogle ScholarPubMed
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flows and Shock Waves. Interscience.Google Scholar
Guderley, K. G. 1962 The Theory of Transonic Flow. Pergamon.Google Scholar
Hamilton, M. F. & Blackstock, D. T. 1998 Nonlinear Acoustics. Academic Press.Google Scholar
Henderson, L. F. 1987 Region and boundaries for diffracting shock wave systems. Z. Angew. Math. Mech. 67, 73–86.CrossRefGoogle Scholar
Henderson, L. F., Crutchfield, W. Y. & Virgona, R. J. 1997 The effects of thermal conductivity and viscosity of argon on shock waves diffraction over rigid ramps. J. Fluid Mech. 331, 1–36.CrossRefGoogle Scholar
Hunter, J. K. 1991 Nonlinear geometrical optics. In Multidimensional hyperbolic problems and computations (ed. Majda, A. J. & Glimm, J.), pp. 179–197. IMA Volumes in Mathematics and its Applications Vol. 29, Springer.CrossRefGoogle Scholar
Hunter, J. K. & Brio, M. 2000 Weak shock reflection. J. Fluid Mech. 410, 235–261.CrossRefGoogle Scholar
Kobayashi, S., Adachi, T. & Suzuki, T. 1995 Examination of the von Neumann paradox for a weak shock wave. Fluid Dyn. Res. 17, 13–25.CrossRefGoogle Scholar
Kobayashi, S., Adachi, T. & Suzuki, T. 2004 Non-self-similar characteristics of weak Mach reflection: the von Neumann paradox. Fluid Dyn. Res. 35, 275–286.Google Scholar
Lee, Y. S. & Hamilton, M. F. 1995 Time-domain modeling of pulsed finite-amplitude sound beams. J. Acoust. Soc. Am. 97, 906–917.CrossRefGoogle Scholar
Mach, E. 1878 Über den Verlauf von Funkenwellen in der Ebene und im Raume. Sitzungsbr. Akad. Wiss. Wien 78, 819–838.Google Scholar
Marchiano, R., Coulouvrat, F. & Grenon, R. 2003 Numerical simulation of shock wave focusing at fold caustics, with application to sonic boom. J. Acoust. Soc. Am. 114, 1758–1771.CrossRefGoogle ScholarPubMed
Marchiano, R., Coulouvrat, F. & Thomas, J. L. 2005 Nonlinear focusing of acoustic shock waves at a caustic cusp. J. Acoust. Soc. Am. 117, 566–577.CrossRefGoogle Scholar
McDonald, B. E. & Kuperman, W. A. 1987 Time domain formulation for pulse propagation including nonlinear behavior at a caustic. J. Acoust. Soc. Am. 81, 1406–1417.CrossRefGoogle Scholar
vonNeumann, J. Neumann, J. 1943 Oblique reflection of shocks. In John von Neumann Collected Work, vol. 6 (1963) (ed. Taub, A. H.), pp. 238–299. Pergamon.Google Scholar
Skews, B. W. 1972 The flow in the vicinity of the three shock interaction. CASI Trans. 4, 99–107.Google Scholar
Skews, B. W. & Ashworth, J. T. 2005 The physical nature of weak shock wave reflection. J. Fluid Mech. 542, 105–114.CrossRefGoogle Scholar
Sternberg, J. 1959 Triple-shock-wave intersections. Phys. Fluids 2, 179–206.CrossRefGoogle Scholar
Tabak, E. G. & Rosales, R. R. 1994 Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection. Phys. Fluids 6, 1874–1892.CrossRefGoogle Scholar
Takayama, K. & Ben-Dor, G. 1985 The inverse Mach reflection. AIAA J. 23, 1853–1859.Google Scholar
Tesdall, A. M. & Hunter, J. K. 2002 Self-similar solutions for weak shock reflection. SIAM. J. Appl. Maths 63, 42–61.CrossRefGoogle Scholar
Vasil'ev, E. & Kraiko, A. 1999 Numerical simulation of weak shock diffraction over a wedge under the von Neumann paradox conditions. Comput. Maths. Math. Phys. 39, 1335–1345.Google Scholar
Zabolotskaya, E. A. & Khokhlov, R. V. 1969 Quasi-plane waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoust. 15, 35–40.Google Scholar
Zakharian, A. R., Brio, M., Hunter, J. K. & Webb, G. M. 2000 The von Neumann paradox in weak shock reflection. J. Fluid Mech. 422, 193–205.CrossRefGoogle Scholar