Skip to main content Accessibility help
×
×
Home

Spatial structure of shock formation

  • J. Eggers (a1), T. Grava (a1) (a2), M. A. Herrada (a1) (a3) and G. Pitton (a2)

Abstract

The formation of a singularity in a compressible gas, as described by the Euler equation, is characterized by the steepening and eventual overturning of a wave. Using self-similar variables in two space dimensions and a power series expansion based on powers of $|t_{0}-t|^{1/2}$ , $t_{0}$ being the singularity time, we show that the spatial structure of this process, which starts at a point, is equivalent to the formation of a caustic, i.e. to a cusp catastrophe. The lines along which the profile has infinite slope correspond to the caustic lines, from which we construct the position of the shock. By solving the similarity equation, we obtain a complete local description of wave steepening and of the spreading of the shock from a point. The shock spreads in the transversal direction as $|t_{0}-t|^{1/2}$ and in the direction of propagation as $|t_{0}-t|^{3/2}$ , as also found in a one-dimensional model problem.

Copyright

Corresponding author

Email address for correspondence: jens.eggers@bris.ac.uk

References

Hide All
Alinhac, S. 1993 Temps de vie des solutions régulières des équations d’Euler compressibles. Invent. Math. 111, 627670.
Arnold, V. I. 1989 Mathematical Methods of Classical Mechanics, 2nd edn. Springer.
Arnold, V. I. 1990 Singularities of Caustics and Wave Fronts. Kluwer.
Berry, M. V. 1981 Singularities in waves and rays. In Les Houches, Session XXXV (ed. Balian, R., Kleman, M. & Poirier, J.-P.), pp. 453543. North-Holland.
Bianchini, S. & Bressan, A. 2005 Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Maths 161, 223342.
Bordemann, M. & Hoppe, J. 1993 The dynamics of relativistic membranes. Reduction to 2-dimensional fluid dynamics. Phys. Lett. B 317, 315320.
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zhang, T. 2006 Spectral Methods. vol. 1. Springer.
Cates, A. T. & Crighton, D. G. 1990 Nonlinear diffraction and caustic formation. Proc. R. Soc. Lond. A 430, 6988.
Cates, A. T. & Sturtevant, B. 1997 Shock wave focusing using geometrical shock dynamics. Phys. Fluids 9, 30583068.
Chiodaroli, E. & De Lellis, C. 2015 Global ill-posedness of the isentropic system of gas dynamics. Commun. Pure Appl. Maths 68, 11571190.
Christodoulou, D. 2007 The Formation of Shocks in 3-dimensional Fluids. EMS Monographs in Mathematics.
Cole, J. D. 1951 On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Maths 9, 225236.
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Springer.
Cramer, M. S. & Seebass, A. R. 1978 Focusing of weak shock waves at an arête. J. Fluid Mech. 88, 209222.
Dubrovin, B., Grava, T., Klein, C. & Moro, A. 2015 On critical behaviour in systems of hamiltonian partial differential equations. J. Nonlinear Sci. 25, 631707.
Dubrovin, B., Grava, T. & Klein, C. 2016 On critical behaviour in generalized Kadomtsev-Petviashvili equations. Physica D 333, 157170.
Eggers, J. & Fontelos, M. A. 2009 The role of self-similarity in singularities of partial differential equations. Nonlinearity 22, R1R44.
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation. Cambridge University Press.
Eggers, J., Hoppe, J., Hynek, M. & Suramlishvili, N. 2014 Singularities of relativistic membranes. Geom. Flows 1, 1733.
Elling, V. 2006 A possible counterexample to well posdness of entropy solutions and to Godunov scheme convergence. Maths Comput. 75, 17211733.
Grava, T., Klein, C. & Eggers, J. 2016 Shock formation in the dispersionless Kadomtsev–Petviashvili equation. Nonlinearity 29, 13841416.
Hopf, E. 1950 The partial differential equation u t + uu x =𝜇 u xx . Commun. Pure Appl. Maths 3, 201230.
Hou, T. Y. 2009 Blow-up or no blow-up? A unified computational and analytic approach to 3D incompressible Euler and Navier–Stokes equations. Acta Numerica 18, 277346.
Kruzkov, S. N. 1969 Generalized solutions of the Cauchy problem in the large for first order nonlinear equations. Dokl. Akad. Nauk SSSR 187, 2932.
Kurganov, A. & Levy, D. 2002 Central-upwind schemes for the Saint-Venant system. Math. Modelling Numer. Anal. 36, 397425.
Landau, L. D. & Lifshitz, E. M. 1984 Fluid Mechanics. Pergamon.
Lax, P. D. 1973 Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. (CBMS Regional Conf. Ser. in Appl. Math.) , vol. 11. SIAM.
van Leer, B. 1979 Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101136.
Majda, A. 1984 Smooth solutions for the equations of compressible and incompressible fluid flow. In Fluid Dynamics (ed. Beirão da Veiga, H.), vol. 1047, pp. 75126. Springer.
Manakov, S. V. & Santini, P. M. 2008 On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking. Nonlinearity 41, 055204.
Manakov, S. V. & Santini, P. M. 2012 Wave breaking in the solutions of the dispersionless Kadomtsev–Petviashvili equation at a finite time. Theor. Math. Phys. 172, 11181126.
Nye, J. 1999 Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations. Institute of Physics Publishing.
Pomeau, Y., Jamin, T., Le Bars, M., Le Gal, P. & Audoly, B. 2008a Law of spreading of the crest of a breaking wave. Proc. R. Soc. Lond. A 464, 18511866.
Pomeau, Y., Le Berre, M., Guyenne, P. & Grilli, S. 2008b Wave-breaking and generic singularities of nonlinear hyperbolic equations. Nonlinearity 21, T61T79.
Popinet, S. 2011 Quadtree-adaptive tsunami modelling. Ocean Dyn. 61, 12611285.
Poston, T. & Stewart, I. 1978 Catastrophe Theory and Its Applications. Dover.
Riemann, B. 1860 Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 8, 4366.
Rosenblum, S., Bechler, O., Shomroni, I., Kaner, R., Arusi-Parpar, T., Raz, O. & Dayan, B. 2014 Demonstration of fold and cusp catastrophes in an atomic cloud reflected from an optical barrier in the presence of gravity. Phys. Rev. Lett. 112, 120403.
Sturtevant, B. & Kulkarny, V. A. 1976 The focusing of weak shock waves. J. Fluid Mech. 73, 651671.
Thom, R. 1976 The two-fold way of catastrophe theory. In Structural Stability, the Theory of Catastrophes, and Applications in the Sciences (ed. Hilton, P. J.), pp. 235252. Springer.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed