Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T17:05:08.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear analysis of shock–vortex interaction: Mach stem formation

Published online by Cambridge University Press:  13 March 2013

Paul Clavin
Paul Clavin*
Affiliation:
Aix-Marseille Université, CNRS, IRPHE, F-13013 Marseille, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Shock–vortex interaction is analysed for strong gaseous shock waves and a ratio of specific heats close to unity. A nonlinear wave equation for the wrinkles of the shock front is obtained for weak vortices. The solution breaks down after a finite time and the slope of the front develops jump discontinuities, indicating the formation of Mach stems. Shock–turbulence interactions are also briefly discussed.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 721 , 25 April 2013 , pp. 324 - 339
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agui, J. H., Briassulis, G. & Andreopoulos, Y. 2005 Studies of interactions of propagating shock wave with decaying grid turbulence: velocity and vorticity fields. J. Fluid Mech. 524, 143–195.CrossRefGoogle Scholar
Bates, J. W. 2012 On the theory of shock wave driven by a corrugated piston in a non-ideal fluid. J. Fluid Mech. 691, 146–164.Google Scholar
Briscoe, M. G. & Kovitz, A. A. 1968 Experimental and theoretical study of the stability of plane shock waves reflected normally from perturbed flat walls. J. Fluid Mech. 31 (3), 529–546.Google Scholar
Clavin, P. 2002a Instabilities and nonlinear patterns of overdriven detonations in gases. In Nonlinear PDE’s in Condensed Matter and Reactive Flows (ed. Berestycki, H. & Pomeau, Y.), pp. 49–97. Kluwer Academic.Google Scholar
Clavin, P. 2002b Self-sustained mean streaming motion in diamond patterns of gaseous detonation. Intl J. Bifurcation Chaos 12 (11), 2535–2546.Google Scholar
Clavin, P & Denet, B. 2002 Diamond patterns in the cellular front of an overdriven detonation. Phys. Rev. Lett. 88 (4)044502–1–4.Google Scholar
Clavin, P. & Williams, F. A. 2012 Analytical studies of the dynamics of gaseous detonations. Phil. Trans. R. Soc A 370, 597–624.Google Scholar
D’Yakov, S. P. 1954 The stabiliy of shockwaves: investigation of the problem of stability of shock waves in arbritary media. Zh. Eksp. Teor. Fiz. 27, 288.Google Scholar
Ellzey, J. L., Henneke, M. R., Picone, J. M. & Oran, E. S. 1995 The interaction of a shock with a vortex: Shock distortion and the production of acoustic waves. Phys. Fluids 7 (1), 172–184.Google Scholar
Guichard, L., Vervich, L. & Domingo, P. 1995 Two-dimensional weak shock–vortex interaction in a mixing zone. AIAA J. 33 (10), 1797–1802.Google Scholar
Howe, M. S. 2003 Theory of Vortex Sound. Cambridge University Press.Google Scholar
Inoue, O. 2000 Propagation of sound generated by weak shock–vortex interaction. Phys. Fluids 12 (5), 1258–1261.Google Scholar
Inoue, O. & Hattory, Y. 1999 Sound generation by shock–vortex interactions. J. Fluid. Mech. 380, 81–116.CrossRefGoogle Scholar
Kontorovich, V. M. 1957 Concerning the stability of shock waves. Zh. Eksp. Teor. Fiz. 33, 1525.Google Scholar
Lapworth, K. C. 1959 An experimental investigation of the stability of planar shock waves. J. Fluid Mech. 6, 469–480.CrossRefGoogle Scholar
Larsson, J & Lele, S. K. 2009 Direct numerical simulation of canonical shock/turbulence interaction. Phys. Fluids 21, 126101–1–12.Google Scholar
Lele, S. K. & Larsson, J. 2009 Shock–turbulence interaction: what we know and what we can learn from peta-scale simulations. J. Phys.: Conf. Ser. 180, 012032.Google Scholar
Huete Ruiz de Lira, C. 2012 Interaction of planar shock waves with non uniforme flows. PhD thesis, Universidad Nacional De Education A Distancia (UNED).Google Scholar
Majda, A. & Rosales, R. 1983 A theory for spontaneous mach stem formation in reacting fronts, I The basic perturbation analysis. SIAM J. Appl. Maths 43 (6), 1310–1334.Google Scholar
Ribner, S. S. 1985 Cylindrical sound wave generated by shock–vortex interaction. AIAA J. 23 (11), 1708–1715.Google Scholar
Shchelkin, K. I. & Troshin, Ya. K. 1965 Gasdynamics of combustion. Mono Book Corp.Google Scholar
Strehlow, R. A. 1979 Fundamentals of Combustion. Kreiger.Google Scholar
Van-Moorhem, K. & George, A. R. 1975 On the stability of plane shock. J. Fluid Mech. 68 (1), 97–108.Google Scholar
Whitham, G. B. 1957 A new approach to problem of shock dynamics. Part I two-dimensional problem. J. Fluid Mech. 2 (02), 145–171.Google Scholar
Wouchuk, J. G. & Huete Ruiz de Lira, C. 2009 Analytical linear theory of planar shock wave with isotropic turbulent flow field. Phys. Rev. E 79, 06315–1–35.Google Scholar
Zhang, S., Zhang, Y-T. & Shu, C-W. 2005 Multistage interaction of a shock wave and a strong vortex. Phys. Fluids 17 (116101), 1–13.CrossRefGoogle Scholar