Linear problem of shock wave disturbance analysis. Part~1: General solution, incidence, refraction and reflection in general case

E. V. Semenko

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 451-472.

Résumé

This article is devoted to the linear problem of shock wave disturbance, where a number of questions related to this problem are considered. A new representation of problem's solution, having completely algebraic form in spectral variables, is found, which allows us to scrutinize the problem, obtain new results and refine known ones. The analytical results are approved and illustrated by numerical calculations. A whole article is divided into three parts because of a large volume. In first part, the basic representation of initial-value problem's solution is established, and the basic techniques of its analysis — singular and regular terms detachment, incident, refracted and reflected waves separation — is described. On this basic, the incidence upon the shock, refraction and reflection of waves in general form is inspected. The peculiarity of refraction, which haven't been noted before, is found: any incident wave may be decomposed into the sum of waves with physically different interaction with shock, namely, one summand interacts with shock, i.e. generates shock disturbance, but doesn't generate any transmitted waves; other summands don't interact with shock, i.e. don't generate shock disturbance, but generate different kinds of transmitted waves. A post-shock incidence of different kinds of waves and its reflection is inspected, in particular a four-wave configuration at reflection is stated.

Détail

Keywords: shock wave, shock disturbance, entropy-vorticity wave, acoustic wave, incident wave, transmitted wave, reflection, reflected wave, stability, neutral stability, spontaneous emission
Mots-clés : refraction, Fourier transform.

Bibliographie

[1] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics, v. 6, Fluid Mechanics, 2nd edition, Elsevier, 1987 | MR

[2] S. P. D'yakov, “On the stability of shock waves”, Sov. Phys. JETP, 27 (1954), 288–295 | MR | Zbl

[3] S. P. D'yakov, “The interaction of shock waves with small perturbations”, Sov. Phys. JETP, 33 (1957), 948–974 | MR

[4] V. M. Kontorovitch, “Reflection and refraction of sound by shock waves”, Sov. Phys. JETP, 33 (1957), 1527–1528

[5] J. F. McKenzie, K. O. Westphal, “Interaction of linear waves with oblique shock waves”, Phys. Fluids, 11 (1968), 2350–2362 | DOI | Zbl

[6] W. R. Johnson, O. Laporte, “Interaction of Cylindrical Sound Waves with a Stationary Shock Wave”, Phys. Fluids, 1 (1958), 82–94 | DOI | MR | Zbl

[7] G. R. Fowles, G. W. Swan, “Stability of plane shock waves”, Phys. Rev. Lett., 30 (1973), 1023–1025 | DOI

[8] W. K. Van Moorhem, A. R. George, “On the stability of plane shocks”, Journal of Fluid Mechanics, 68 (1975), 97–108 | DOI | Zbl

[9] G. W. Swan, G. R. Fowles, “Shock wave stability”, Phys. Fluids, 18 (1975), 28–35 | DOI | Zbl

[10] G. R. Fowles, “Conditional stability of shock waves — a criterion for detonation”, Phys. Fluids, 19 (1976), 227–238 | DOI | Zbl

[11] G. R. Fowles, “Stimulated and spontaneous emission of acoustic waves from shock fronts”, Phys. Fluids, 24 (1981), 220–227 | DOI | Zbl

[12] M. Mond, I. M. Rutkevich, “Spontaneous acoustic emission from strong ionizing shocks”, Journal of Fluid Mechanics, 275 (1994), 121–146 | DOI | MR | Zbl

[13] J.-C. Robinet, G. Casalis, “Critical interaction of a shock wave with an acoustic wave”, Phys. Fluids, 13 (2001), 1047–1059 | DOI | MR

[14] I. Men'shov, Y. Nakamura, Abnormal amplification of sound waves refracted by an oblique shock wave, JAXA Special Publication, SP-03-002, 2004, 23–28

[15] S. V. Iordanski, “On stability of a plane shock wave”, Journal of Applied Mathematics and Mechanics, 21 (1957), 465–472 | MR

[16] J. J. Erpenbeck, “Stability of Steady-State Equilibrium Detonations”, Phys. Fluids, 5 (1962), 604–614 | DOI | MR

[17] J. J. Erpenbeck, “Stability of Step Shocks”, Phys. Fluids, 5 (1962), 1181–1187 | DOI | MR | Zbl

[18] R. M. Zaidel, “Shock wave from a slightly curved piston”, Journal of Applied Mathematics and Mechanics, 24 (1960), 316–320 | DOI | MR

[19] R. M. Zaidel, “The perturbations propagation in plane shock waves”, Journal of Applied Mechanics and Technical Physics, 4 (1967), 30–39

[20] J. W. Bates, “Initial-value-problem solution for isolated rippled shock fronts in arbitrary fluid media”, Phys. Rev. E, 69 (2004), 056313-1–056313-16 | DOI | MR

[21] J. W. Bates, “Instability of isolated planar shock waves”, Phys. Fluids, 19 (2007), 1–15 | Zbl

[22] A. Tumin, “Initial-value problem for small disturbances in an idealized one-dimensional detonation”, Phys. Fluids, 19 (2007), 106105-1–106105-12 | DOI | Zbl

[23] G. R. Fowles, A. F. P. Houwing, “Instabilities of shock and detonation waves”, Phys. Fluids, 27 (1984), 1982–1990 | DOI | MR | Zbl

[24] N. M. Kuznetsov, “Contribution to shock-wave stability theory”, Zh. Eksp. Teor. Fiz., 88 (1985), 470–486

[25] N. M. Kuznetsov, “Stability of shock waves”, Usp. Fiz. Nauk, 159 (1989), 493–527 | DOI

[26] J. W. Bates, D. C. Montgomery, “The D'yakov–Kontorovich Instability of Shock Waves in Real Gases”, Phys. Rev. Lett., 84 (2000), 1180–1183 | DOI

[27] A. V. Konyukhov, A. P. Likhachev, V. E. Fortov, S. I. Anisimov, A. M. Oparin, “Stability and ambiguous representation of shock wave discontinuity in thermodynamically nonideal media”, Pis'ma Zh. Eksp. Teor. Fiz., 90 (2009), 28–34

[28] G. F. Carrier, M. Krook, C. E. Pearson, Functions of a complex variable, Hod Books, Ithaka, N. Y., 1983 | MR | Zbl

[29] E. Zauderer, Partial Differential Equations, 3rd edition, Wiley, 2006 | MR | Zbl