TRENDS IN THE DEVELOPMENT OF DETONATION ENGINES FOR HIGH-SPEED AEROSPACE AIRCRAFTS AND THE PROBLEM OF TRIPLE CONFIGURATIONS OF SHOCK WAVES. Part II - Research of counterpropagating shock waves and triple shock wave configurations

The paper deals with current issues of the interference theory development of gas-dynamic discontinuities as applied to a problem of propulsion refinement for the air-spacecrafts, designed for hypersonic flight speeds. In the first part of the review we have presented the history of detonation study and different concepts of detonation engines, as well as air intakes designed for hypersonic flight speeds. The second part provides an overview of works on the interference theory development for gas-dynamic discontinuities. We report about classification of the gas-dynamic discontinuities, shock wave propagation, shock-wave structures and triple configurations of shock waves. We have shown that many of these processes are accompanied by a hysteresis phenomenon, there are areas of ambiguity; therefore, in the design of engines and air intakes optimal shock-wave structures should be provided and their sustainability should be ensured. Much attention has recently been given to the use of the air intakes in the shock-wave structures with the rereflection of shock waves and the interference of shock waves in the opposite directions. This review provides increased focus on it, contains references to landmark works, the last calculated and experimental results. Unfortunately, foreign surveys missed many landmark works of the Soviet and Russian researchers, as they were not published in English. At the same time, it was the Soviet school of gas dynamics that has formulated the interference theory of gas-dynamic discontinuities in its present form. To fill this gap is one of this review scopes. The review may be recommended for professionals, engineers and scientists working in the field of aerospace engineering.

 1. Uskov V.N. Nauka o Stiikhiyakh [Science about Elements]. St. Petersburg, BSTU Publ., 2007, 140 p.
2. Uskov V.N. Stages of Aerogasdynamics Becoming. In Aerodynamics. Ed. R.N. Mitroshin. St. Petersburg, VVM Publ., 2006, pp. 153–211. (In Russian)
3. Uskov V.N. Historical review of research development of gas-dynamic gaps. In Aerodynamics. Ed. R.N. Mitroshin. St. Petersburg, SPbSU Publ., 2013, pp. 131–156. (In Russian)
4. Matveev S.K., Uskov V.N. Isaac Pavlovich Ginzburg. To the 100-th anniversary. Vestnik of the St. Petersburg University: Mathematics, 2010, no. 3, pp. 132–137.
5. Matveev S.K., Uskov V.N. Viktor Georgievich Dulov. To the 80-th anniversary. Vestnik of the St. Petersburg University: Mathematics, 2010, no. 3, pp. 138–141.
6. Handbook of Shock Waves. Academic Press, 2001, vol. 1-3.
7. Deich M.E. Tekhnicheskaya Gazodinamika [Technical Gasdynamics]. 3rd ed. Moscow, Energiya Publ., 1974, 592 p.
8. Mach E. Weitere ballistisch-photografische Versuche. Akad.Wiss.Wien, 1889, vol. 98, pp. 1310–1326.
9. Lagrange J.L. Mecanique Analytique. Paris, 1788.
10. Poisson S.D. Memoire sur la theorie du son. Journal de l’Ecole Polytechnique, 1808, vol. VII, no. 14, pp. 319–392.
11. Stokes G.G. On a difficulty in the theory of sound. In Classic Papers in Stock Compression Science. Eds. J.N. Johnson, R. Cheret. Springer, 1998, pp. 71–81. doi: 10.1007/978-1-4612-2218-7_2
12. Arnol'd V.I. Geometricheskie Metody v Teorii Obyknovennykh Differentsial'nykh Uravnenii [Geometric Methods in the Theory of Ordinary Differential Equations]. Izhevsk, MTsNMO Publ., 2000, 384 p.
13. Arnol'd V.I. Dopolnitel'nye Glavy Teorii Obyknovennykh Differentsial'nykh Uravnenii [Additional Chapters of the Ordinary Differential Equations Theory]. Moscow, Nauka Publ., 1978, 304 p.
14. Zeldovich Ya.B. Gravitational instability: an approximate theory for large density perturbations. Astronomy and Astrophysics, 1970, vol. 5, no. 1, pp. 84–89.
15. von Karman Th., Burgers J.M. General Aerodynamic Theory. Perfect Fluids. Springer, 1935.
16. Earnshaw S. On the mathematical theory of sound. Proceedings of Royal Society of London, 1858, pp. 590–591.
17. Earnshaw S. On the mathematical theory of sound. Philosophical Transactions of the Royal Society of London, 1860, vol. 150, no. 8, pp. 133–148. doi: 10.1098/rstl.1860.0009
18. Bulat P.V., Bulat M.P. Gas-dynamic variable relation on opposite sides of the gas-dynamic discontinuity. Research Journal of Applied Sciences, Engineering and Technology, 2015, vol. 9, no. 12, pp. 1097–1104.
19. Uskov V.N. Interferentsiya statsionarnykh gazodinamicheskikh razryvov [Interference stationary gasdynamic discontinuities]. Sbornik Statei "Sverkhzvukovye Gazovye Strui" [Proc. Supersonic Gas Jet]. Novosibirsk, Nauka Publ., 1983, pp. 22–46.
20. Uskov V.N., Tao Gan, Omel'chenko A.V. O povedenii gazodinamicheskikh peremennykh za kosoi udarnoi volnoi [On the behavior of gas-dynamic variables for the oblique shock wave]. Sbornik Statei "Sovremennye Problemy Neravnovesnoi Gazo- i Termodinamiki" [Proc. Modern Problems of
Non-Equilibrium Thermodynamics and Gas], 2002, pp. 179–191.
21. Uskov V.N. Udarnye Volny i Ikh Vzaimodeistvie. Uchebnoe Posobie [Shock Waves and Their Interaction. Textbook]. Leningrad, Len. Mekh. Inst. Publ., 1980, 88 p.
22. Riemann B. Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingweite. Abhandlungen der koniglichen Gesellschaft der Wissenschaften zu Gottingen, 1860, vol. 8, p. 43.
23. Rankine W.J.M. On the thermodynamic theory of waves of finite longitudinal disturbance. Proceedings of the Royal Society of London, 1869, vol. 18, pp. 80–83.
24. Rankine W.J.M. On the thermodynamic theory of waves of finite longitudinal disturbance. Philosophical Magazine, 1870, vol. 39, no. 4, pp. 306–309.
25. Rankine W.J.M. On the thermodynamic theory of waves of finite longitudinal disturbance. Philosophical Transactions of the Royal Society of London, 1870, vol. 160, pp. 277–288.
26. Hugoniot H. Propagation du mouvement dans les corps. Chapitre V. Sur les discontinuités qui se manifestent dans la propagation du movement. Journal de l’École Polytechnique, 1889, vol. LVIII, pp. 68–125.
27. Meyer Th. Ueber zweidimensionale Bewegungsvorg ä nge in einem Gas, dasmit Ueberschallgeschwindigkeit strömt. Forschungsheft des Vereins deutcher Ingenieure, 1908, vol. 62, pp. 31–67.
28. Uskov V.N. Begushchie Odnomernye Volny [Running One-Dimensional Waves]. St. Petersburg, BSTU "VOENMEH" Publ., 2000, 224 p.
29. Adrianov A.L., Starykh A.L., Uskov V.N. Interferentsiya Statsionarnykh Gazodinamicheskikh Razryvov [Interference Stationary Gasdynamic Discontinuities]. Novosibirsk, Nauka Publ., 1995, 180 p.
30. Uskov V.N., Chernyshov M. V. Extreme shockwave systems in problems of external supersonic aerodynamics. Thermophysics and Aeromechanics, 2014, vol. 21, no. 1, pp. 15–30. doi: 10.1134/S086986431401003X
31. Uskov V.N. Optimum one-dimensional waves traveling in gas flow. XV Sessiya Mezhdunarodnoi Shkoly po Modelyam Mekhaniki Sploshnoi Sredy [XV Session of Int. School on Models of Continuum Mechanics]. St. Petersburg, 2000, pp. 63–78. (In Russian)
32. Uskov V.N. Mostovykh P.S. Extreme properties of an oblique shock wave traveling along the gas flow. Chetvertye Polyakhovskie Chteniya: Izbrannye Trudy Mezhdunarodnoi Nauchnoi Konferentsii po Mekhanike. St. Petersburg, 2006, pp. 444–454. (In Russian)
33. Vieille P. Sur les discontinuiti és produites par la det énte brusque de gaz comprim és. Comptes Rendus, 1899, vol. CXXIX, pp. 1228–1230.
34. Stodola A. Beitrag zur Stromung von Gasen und D ä mpfen durch Rohre mit veranderlichem Querschnitt. Zeitschrift des Vereins deutcher Ingenieure, 1903, vol. 47, pp. 1787–1788.
35. Uskov V.N. Interference of gasdynamic and tangential discontinuities. Fluid Dynamics, 1979, no. 4, pp. 191–197.
36. Schardin H. Bemerkungen zum Druckausgleichsvorgang in einer Rohrleitung. Physik. Zeits, 1932, vol. 33, pp. 60–64.
37. Bitondo D., Glass I.I., Patterson G.N. One Dimensional Theory of Absorption and Amplification of a Plane Shock Wave by a Gaseous Layer. University of Toronto Institute of Aerophysics (UTIA), 1950, report no. 5.
38. Bitondo D., Lobb R.K. Experiments on the Amplification of a Plane Shock Wave. University of Toronto Institute of Aerophysics (UTIA), 1959, report no. 7.
39. Ford C.A., Glass I.I. An experimental study of one–dimensional shock wave refraction. Journal of Aerospace Sciences, 1956, vol. 23, no. 2, pp. 189–191.
40. Gould D.G. The Head–On Collision of Two Shock Waves and a Shock and Rarefaction Wave in One–Dimensional Flow. University of Toronto Institute for Aerospace Studies (UTIA), 1952, report no. 17.
41. Nicholl C.I.H. The Head–On Collision of Shock and Rarefaction Waves. University of Toronto Institute of Aerophysics (UTIA), 1951, report no. 10.
42. Billington I.I., Glass I.I. On the One–Dimensional Refraction of a Rarefaction Wave at a Contact Surface. University of Toronto Institute of Aerophysics (UTIA), 1955, report no. 31.
43. Billington I.I. An experimental study of one–dimensional refraction of a rarefaction wave at a contact surface. Journal of Aeronautical Sciences, 1956, vol. 23, no. 11, pp. 997–1006.
44. Taub A.H. Refraction of plane shock waves. Physical Review, 1947, vol. 72, no. 1, p. 51. doi: 10.1103/PhysRev.72.51
45. Arkhipova L.P., Uskov V.N. Universal'noe reshenie zadachi ob otrazhenii odnomernykh begushchikh voln ot tverdoi stenki i ego analiz dlya voln uplotneniya [Universal solution of problem of 1D moving waves reflection from a solid wall and analysis for waves seal]. Vestnik of the St. Petersburg University: Mathematics, 2013, no. 2, pp. 77–81.
46. Arkhipova L.P., Uskov V.N. Refraction of the rarefaction Riemann wave from the vertical solid wall. Vestnik of the St. Petersburg University: Mathematics, 2012, no. 4, pp. 62–65. (In Russian)
47. Uskov V.N., Chernyshov M.V. The interaction of Prandtl-Meyer wave with the oblique shock of the same direction. Journal of Energy and Power Engineering, 2014, vol. 8, pp. 121–136.
48. Uskov V.N., Chernyshov M.V. Analytical solutions for overtaking Prandtl-Meyer wave-oblique shock. Proc. 19th International Shock Interaction Symposium, ISIS. Moscow, 2010.
49. Uskov V.N., Meshkov V.R., Omel'chenko A.V. Vzaimodeistvie skachka uplotneniya so vstrechnoi volnoi razrezheniya [Shock wave interaction with counter rarefaction wave]. Vestnik of the St. Petersburg University: Mathematics, 2002, no. 2.
50. Silnikov M.V., Chernyshov M.V., Uskov V.N. Analytical solutions for Prandtl-Meyer wave-oblique shock overtaking interaction. Acta Astronautica, 2014, vol. 99, no. 1, pp. 175–183. doi: 10.1016/j.actaastro.2014.02.025
51. Uskov V.N., Karasev K.A. Criterion intensity of the interference of the direct shock wave and shock wave counte. In Nauka i Tekhnologii [Science and Technology], 2003, pp. 4–11.
52. Omel'chenko A.V., Uskov V.N. Interference of nonstationary oblique shock waves. Technical Physics Letters, 2002, vol. 28, no. 6, pp. 491–493. doi: 10.1134/1.1490969
53. Uskov V.N., Mostovykh P.S. Triple configurations of traveling shock waves in inviscid gas flows. Journal of Applied Mechanics and Technical Physics, 2008, vol. 49, no. 3, pp. 347–353.
54. Oswatitsch K., Schwarzenberger R. Ubungen Zur Gasdynamik. Wien, Springer, 1963.
55. Uskov V.N., Omel'chenko A.V. Optimal shock-wave systems. Fluid Dynamics, 1995, vol. 30, no. 6, pp. 905–911. doi: 10.1007/BF02078208
56. Uskov V.N., Omel'chenko A.V. Supersonic Flow Control. Report to 90th birthday of prof. N.N. Polyahov, 1997. (In Russian)
57. Omel'chenko A.V., Uskov V.N. Geometry of optimal shock-wave systems. Journal of Applied Mathematics and Mechanics, 1997, vol. 38, no. 5, pp. 679–684. doi: 10.1007/BF02467878
58. Malozemov V.N., Omel'chenko A.V., Uskov V.N. The minimization of the total pressure loss accompanying the breakdown of a supersonic flow. Journal of Applied Mathematics and Mechanics, 1998, vol. 62, no. 6, pp. 939–944.
59. Omel'chenko A.V., Uskov V.N. Optimal shock-wave systems under constraints on the total flow turning angle. Fluid Dynamics, 1996, vol. 31, no. 4, pp. 597–603. doi: 10.1007/BF02031768
60. Omel'chenko A.V., Uskov V.N. Maximum turning angles of a supersonic flow in shock-wave systems. Fluid Dynamics, 1998, vol. 33, no. 3, pp. 419–426. doi: 10.1007/BF02698194
61. Omel'chenko A.V., Uskov V.N. Optimum overtaking compression shocks with restrictions imposed on the total flow-deflection angle. Journal of Applied Mechanics and Technical Physics, 1999, vol. 40, no. 4, pp. 638–646. doi: 10.1007/BF02468438
62. Omel'chenko A.V., Uskov V.N. Optimal shock-expansion system in a steady gas flow. Journal of Applied Mechanics and Technical Physics, 1997, vol. 38, no. 2, pp. 204–210.
63. Uskov V.N., Chernyshov M.V., Erofeev V.K., Genkin P. Optimal shock-wave structures and new ideas about supersonic gas jet noise generation. Proc. 13th Int. Congress on Sound and Vibration. Vienna, Austria, 2006, pp. 1439–1446.
64. Uskov V.N., Chernyshov M.V. Theoretical analysis of the aerodynamic coefficients of polygonal profiles in supersonic flow, mechanics and control processes. Trudy XXXI Ural'skogo Seminara [Proc. XXXI Ural Seminar]. Ekaterinburg, 2001, pp. 187–191. (In Russian)
65. Uskov V.N., Chernyshov M.V. Analysis of aerodynamic coefficients of polygonal profiles in supersonic flow. Sbornik Trudov X Vserossiiskogo Seminara po Upravleniyu Dvizheniem i Navigatsii Letatel'nykh Apparatov [Proc. X All-Russian Seminar on Traffic Control and Aircraft Navigation]. Samara, Russia, 2002, pp. 322–326. (In Russian)
66. Uskov V.N., Chernyshov M.V. Extreme load on construction elements, delivered by a shock-wave systems. Trudy 8i Vserossiiskoi Konferentsii "Problemy Obespecheniya Vzryvobezopasnosti i Protivodeistviya Terrorizmu [Proc. 8th All-Russian Conference on Problems of Explosion Protection and Counter-Terrorism]. St. Petersburg, 2013, pp. 203–226.
67. Uskov V.N., Chernyshov M.V. Troinye konfiguratsii statsionarnykh udarnykh voln [Triple configuration of stationary shock waves]. XII Vserossiiskaya Nauchno-Prakticheskaya Konferentsiya Aktual'nye Problemy Zashchity i Bezopasnosti [XII All-Russian Scientific-Practical Conference on Actual Problems of Protection and Security]. St. Petersburg, 2009, pp. 420–435.
68. Uskov V.N., Shakhova O.A. On the calculation of a triple configuration of shock waves. Gidroaeromekhanika i Teoriya Uprugosti, 1976, no. 21, pp. 13–18. (In Russian)
69. Mel’nikov D.A. Otrazhenie skachkov uplotneniya ot osi simmetrii. Izv. AN SSSR. Mehanika i Mashinostroenie, 1962, no. 3, pp. 24–30.
70. Rylov A.I. To a question about the impossibility of regular reflection stationary shock wave from the axis of symmetry. Journal of Applied Mathematics and Mechanics, 1990, vol. 25, no. 2, pp. 245–249.
71. Isakova N.P., Kraiko A.N., P'yankov K.S., Tillyayeva N.I. The amplification of weak shock waves in axisymmetric supersonic flow and their reflection from an axis of symmetry. Journal of Applied Mathematics and Mechanics, 2012, vol. 76, no. 4, pp. 451–465. doi: 10.1016/j.jappmathmech.2012.09.013
72. von Neumann J. Oblique reflection of shocks. In: Collected Works, Pergamon, 1943, vol. 6, pp. 239–299.
73. Courant R., Friedrichs K. O. Supersonic Flow and Shock Waves. NY, Interscience, 1948.
74. Bleakney W., Fletcher C.H., Weimer D.K. The density field in mach reflection of shock waves. Physical Review, 1949, vol. 76, no. 2, pp. 323–324. doi: 10.1103/PhysRev.76.323.2
75. Breed B.R. Impossibility of three confluent shocks in two-dimensional irrotational flow. Physics of Fluids, 1967, vol. 10, no. 1, pp. 21–23. doi: 10.1063/1.1761977
76. Sternberg J. Triple-shock-wave intersections. Physics of Fluids, 1959, vol. 2, no. 2, pp. 179–206. doi: 10.1063/1.1705909
77. Sakurai A. On the problem of weak Mach reflection. Journal of the Physical Society of Japan, 1964, vol. 19, no. 8, pp. 1440–1450. doi: 10.1143/JPSJ.19.1440
78. Dulov V.G. Motion of triple configuration of shock waves with formation of wake behind branching point. Journal of Applied Mechanics and Technical Physics, 1973, vol. 14, no. 6, pp. 791–797. doi: 10.1007/BF00853193
79. Dulov V.G., Luk'yanov G.A. Gazodinamika Protsessov Istecheniya [Gasdynamics of the Expiration Processes]. Novosibirsk, Nauka Publ., 1984, 236 p.
80. Uskov V.N., Chernyshov M.V. Theoretical analysis of the characteristics of triple configurations shocks. In Sovremennye Problemy Neravnovesnoi Gazo- i Termodinamiki [Modern Problems of Nonequilibrium Thermodynamics and Gas]. St. Petersburg, 2002, pp. 75–99. (In Russian)
81. Roslyakov G.S., Starykh A.L., Uskov V.N. Interference of steady shock waves traveling in the same direction. Fluid Dynamics, 1987, vol. 22, no. 4, pp. 614–622. doi: 10.1007/BF01051429
82. Uskov V.N., Starykh A.L. An analysis of solutions existence domains for equations of stationary interference gasdynamic discontinuities. In Nestatsionarnye Techeniya Gazov s Udarnymi Volnami [Unsteady Gas Flows with Shock Waves]. Leningrad, 1990, pp. 359–372. (In Russian)
83. Uskov V.N., Adrianov A.L. The complex routines for the simulation of two-dimensional non-stationary supersonic flows with allocation of gaps to inconsistent grid. In Kompleksy Programm Matematicheskoi Fiziki i Arkhitektura EVM [Complexes of Mathematical Physics Software and Computer Architecture], 1988, pp. 3–7.
84. Uskov V.N., Chernyshov M.V. Special and extreme triple shock-wave configurations. Journal of Applied Mechanics and Technical Physics, 2006, vol. 47, no. 4, pp. 492–504. doi: 10.1007/s10808-006-0081-5
85. Uskov V.N., Chernyshev M.V. Analiz i optimizatsiya udarno-volnovoi struktury sverkhzvukovykh gazovykh strui. IX Vserossiiskii S"ezd po Teoreticheskoi i Prikladnoi Mekhanike. Nizhnii Novgorod, 2006.
86. Uskov V.N., Chernyshov M.V. Special and optimal properties of stationary Mach configurations. Izvestija TulGU, 2001, no. 4–1, pp. 216–220. (In Russian)
87. Uskov V.N., Mostovykh P.S., Chernyshov M.V. Special and extreme structures of stationary and non-stationary shocks. Proc. 18th Int. Shock Interaction Symposium. Rouen, 2008, pp. 71–74.
88. Uskov V.N., Mostovykh P.S. Triple configurations of traveling shock waves in inviscid gas flows. Journal of Applied Mechanics and Technical Physics, 2008, vol. 49, no. 3, pp. 347–353. doi: 10.1007/s10808-008-0048-9
89. Law C.K., Glass I.I. Diffraction of strong shock waves by a sharp compressive corner. CASI Trans, 1971, vol. 4, no. 1, pp. 2–12.
90. Ben-Dor G. Regions and transitions of nonstationary oblique shock-waves diffractions in perfect and imperfect gases. Toronto University Institute for Aerospace Studies Report, 1978, no. 232.
91. Lee J.-H., Glass I.I. Pseudo-stationary oblique-shock-wave reflections in frozen and equilibrium air. Progress in Aerospace Sciences, 1984, vol. 21, pp. 33–80. doi: 10.1016/0376-0421(84)90003-4
92. Mostovykh P.S., Uskov V.N. Triple-shock-wave configurations: comparison of different thermodynamic models for diatomic gases. Proc. 28th International Symposium on Shock Waves, 2012, vol. 2, pp. 945–951. doi: 10.1007/978-3-642-25685-1_144
93. Mach E. Uber den verlauf von funkenwellen in der ebene und im raume. Sitzungsber. Akad. Wiss. Wien, 1878, vol. 78, pp. 819–838.
94. Neumann J. Collected Works. Pergamon Press, 1963, 795 p.
95. Smith L.G. Photographic investigations of the reflection of plane shocks in air. Office of Scientific Research and Development, 6271, NDRC Rep. A-350, 1945.
96. White D.R. An Experimental Survey of the Mach Reflection of Shock Waves. Princeton University, Department of Physics, 1951.
97. Ben-Dor G. Shock Wave Reflection Phenomena. 2nd ed. Springer, 2007, 342 p. doi: 10.1007/978-3-540-71382-1
98. Bleakney W., Taub A.H. Interaction of shock waves. Reviews of Modern Physics, 1949, vol. 21, no. 4, pp. 584–605. doi: 10.1103/RevModPhys.21.584
99. Bargmann V. On nearly glancing reflection of shocks. Office Sci. Res. Develop. Rep. no. 5117, 1945.
100. Lighthill M.J. The diffraction of blast. Proceedings of the Royal Society of London, Series A, 1949, vol. 198, pp. 454–470.
101. Ting L., Ludloff H.F. Aerodynamics of blasts. Journal of Aeronautical Sciences, 1951, vol. 18, pp. 143.
102. Fletcher C.H., Bleakney W. The Mach reflection of shock waves at nearly glancing incidence. Reviews of Modern Physics, 1951, vol. 23, no. 3, pp. 271–286. doi: 10.1103/RevModPhys.23.271
103. Henderson L.F., Siegenthaler A. Experiments on the diffraction of weak blast waves: the von Neumann paradox. Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 1980, vol. 369, no. 1739, pp. 537–555. doi: 10.1098/rspa.1980.0015
104. Colella P., Henderson L.F. The von Neumann paradox for the diffraction of weak shock waves. Journal of Fluid Mechanics, 1990, vol. 213, pp. 71–94. doi: 10.1017/S0022112090002221
105. Adachi Т., Suzuki Т., Kobayashi S. Mach reflection of a weak shock waves. Transactions of the Japan Society of Mechanical Engineers, Part B, 1994, vol. 60, no. 575, pp. 2281–2286.
106. Olim M., Dewey J.M. A revised three-shock solution for the Mach reflection of weak shocks. Shock Waves, 1992, vol. 2, no. 3, pp. 167–176. doi: 10.1007/BF01414639
107. Guderley K.G. Considerations on the structure of mixed subsonic supersonic flow patterns, air materiel command. Technical Report F-TR-2168-ND. Wright Field, Dayton, 1947.
108. Guderley K.G. The Theory of Transonic Flow. Oxford, Pergamon Press, 1962, 344 p.
109. Vasil'ev E.I., Kraiko A.N. Numerical simulation of weak shock diffraction over a wedge under the von neumann paradox conditions. Computational Mathematics and Mathematical Physics, 1999, vol. 39, no. 8, pp. 1335–1345.
110. Vasil'ev E.I. Four-wave scheme of weak Mach shock waves interaction under the von Neumann paradox conditions. Fluid Dynamics, 1999, vol. 34, no. 3, pp. 421–427.
111. Vasilev E., Olkhovsky M. Proc. 27th Int. Symposium on Shock Waves. St. Petersburg, 2009.
112. Bulat P.V., Denissenko P.V. Interference of unidirectional shock waves. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol.15, no. 3, pp. 500–508. doi: 10.17586/2226-1494-2015-15-3-500-508
113. Busemann A. Verdichtungsstоße in ebenen Gasstrоmungen. Berlin, Julius Springer, 1930, pp. 162–169.
114. Busemann A. Gasdynamik, Handbuch der experimentellen Physik. Leipzig: Akademischer Verlag, 1931, vol. 4(1), p. 394.
115. Busemann A. Hodographenmethode der Gasdynamik. ZAMM, 1937, vol. 17, no. 2, pp. 73–79.
116. Kawamura R., Saito H. Reflection of shock waves-1 pseudo-stationary case. Journal of the Physical Society of Japan, 1956, vol. 11, no. 5, pp. 584–592.
117. Bulat P.V., Uskov V.N. Mach reflection of a shock wave from the symmetry axis of the supersonic nonisobaric jet. Research Journal of Applied Sciences, Engineering and Technology, 2014, vol. 8, no. 1, pp. 135–142.
118. Molder S. Head-on interaction of oblique shock waves. University of Toronto Institute of Aerophysics Technical Note, 1960, no. 38.
119. Bulat P.V., Denissenko P.V., Upyrev V.V. Asymmetrical interference of counter oblique shock waves. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 5, pp. 942–949.
120. Bulat P.V., Upyrev V.V., Denisenko P.V. Oblique shock wave reflection from the wall. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, vol. 15, no. 2, pp. 338–345. doi: 10.17586/2226-1494-2015-15-2-338-345
121. Starykh A.L. Neregulyarnoe vzaimodeistvie skachkov uplotneniya mezhdu soboi i s tangentsial'nymi razryvami [Irregular interaction of shocks between themselves and with tangential discontinuities]. Chislennye Metody Mekhaniki Sploshnoi Sredy, 1986, vol. 17, no. 6, pp. 119–124.
122. Fomin V.M., Hornung H.G., Ivanov M.S., Kharitonov A.M., Klemenkov G.P., Kudryavtsev A.N., Pavlov A.A. The study of transition between regular and Mach reflection of shock waves in different wind tunnels. Proc. 12th Int. Mach Reflection Symposium. Pilanesberg, South Africa, 1996, pp. 137–151.
123. Ivanov M.S., Ben-Dor G., Elperin T., Kudryavtsev A.N., Khotyanovsky D.V. Mach-number-variation-induced hysteresis in steady flow shock wave reflections. AIAA Journal, 2001, vol. 39, no. 5, pp. 972–974. doi: 10.2514/2.1406
124. Ben-Dor G. Two-dimensional interactions. In Handbook of Shock Waves. Eds G. Ben-Dor, O. Igra, T. Elperin. Boston, Academic Press, 2001, 824 p.
125. Ivanov M.S., Ben-Dor G., Elperin Т., Kudryavtsev A.N., Khotyanovsky D.V. The reflection of asymmetric shock waves in steady flows: a numerical investigation. Journal of Fluid Mechanics, 2002, vol. 469, pp. 71–87. doi: 10.1017/S0022112002001799
126. Vasil'ev E.I. W-Modifikatsiya Metoda Godunova i ee Prilozheniya v Modelirovanii Gazodinamicheskikh Techenii s Udarnymi Volnami. Dis. d-ra fiz.-mat. nauk [W-Modification of Godunov's Method and its Applications in Modeling Gasdynamic Flows with Shock Waves. Diss. Dr. Phys.-Math. Sci.]. Volgograd, 1999, 213 p.
127. Khotyanovskii D.V. Chislennyi Analiz Sverkhzvukovykh Techenii so Slozhnymi Udarno-Volnovymi Strukturami. Dis. kand. fiz.-mat. nauk [Numerical Analysis of Supersonic Flows with Complicated Shock-Wave Structures. PhD Diss. Phys.-Math. Sci.]. Novosibirsk, 2007, 148 p.
128. Kudryavtsev A.N. Vychislitel'naya Aerodinamika Sverkhzvukovykh Techenii s Udarnymi Volnami. Dis. d-ra fiz.-mat. nauk [Computational Dynamics of Supersonic Flows with Shock Waves Diss. Dr. Phys.-Math. Sci.]. Novosibirsk, 2014, 337 p.
129. Shoev G.V. Chislennoe Issledovanie Vliyaniya Vyazkosti na Protsessy Vzaimodeistviya i Rasprostraneniya Udarnykh Voln. Dis. kand. fiz.-mat. nauk [Numerical Study of the Effect of Viscosity on the Processes of Interaction and Propagation of Shock Waves. PhD Diss. Phys.-Math. Sci.]. Novosibirsk, 2013, 134 p.
130. Bulat P.V., Upyrev V.V. Boundaries of the ambiguity area upon reflection of compression shock waves. Technical Physics Letters, 2016, vol. 42, no. 1, pp. 15–18. doi: 10.1134/S106378501601003X
131. Ilina E.E., Ilina T.E., Bulat P.V. Analysis of the application of turbulence models in the calculation of supersonic gas jet. American Journal of Applied Sciences, 2014, vol. 11, no. 11, pp. 1914–1920. doi: 10.3844/ajassp.2014.1914.1920
132. Bulat M.P., Bulat P.V. Comparison of turbulence models in the calculation of supersonic separated flows. World Applied Sciences Journal, 2013, vol. 27, no. 10, pp. 1263–1266. doi: 10.5829/idosi.wasj.2013.27.10.13715
133. Uskov V.N., Mostovykh P.S. Differential characteristics of shock waves and triple shock wave configurations. Proc. 20th Int. Shock Interaction Symposium, ISIS 2012. Stockholm, 2012, pp. 211–214.
 

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