**Nikiforov**

Vladimir O.

D.Sc., Prof.

Vladimir O.

D.Sc., Prof.

doi: 10.17586/2226-1494-2015-15-6-1155-1168

doi: 10.17586/2226-1494-2015-15-6-1155-1168

# ARBITRARY INTERACTION OF PLANE SUPERSONIC FLOWS

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**Article in**Russian

**For citation:**Bulat P.V., Volkov K.N. Arbitrary interaction of plane supersonic flows.

*Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2015, vol. 15, no. 6, pp. 1155-1168.

**Abstract**

**Subject of study.**We consider the Riemann problem for parameters at collision of two plane flows at a certain angle. The problem is solved in the exact statement. Most cases of interference, both stationary and non-stationary gas-dynamic discontinuities, followed by supersonic flows can be reduced to the problem of random interaction of two supersonic flows. Depending on the ratio of the parameters in the flows, outgoing discontinuities turn out to be shock waves, or rarefactionwaves. In some cases, there is no solution at all. It is important to know how to find the domain of existence for the relevant decisions, as the type of shock-wave structures in these domains is known in advance. The Riemann problem is used in numerical methods such as the method of Godunov. As a rule, approximate solution is used, known as the Osher solution, but for a number of problems with a high precision required, solution of this problem needs to be in the exact statement. **Main results.**Domains of existence for solutions with different types of shock-wave structure have been considered. Boundaries of existence for solutions with two outgoing shock waves are analytically defined, as well as with the outgoing shock wave and rarefaction wave. We identify the area of Mach numbers and angles at which the flows interact and there is no solution. Specific flows with two outgoing rarefaction waves are not considered. **Practical significance.** The results supplement interference theory of stationary gas-dynamic discontinuities and can be used to develop new methods of numerical calculation with extraction of discontinuities.

**Keywords:**rarefaction wave, Riemann problem, shock wave, shock-wave structures.

**Acknowledgements.**The study has been financially supported by the Ministry of Education and Science of the Russian Federation (agreement № 14.575.21.0057, the unique identifier of applied research RFMEFI57514X0057.

**References**

1. Uskov V.N., Bulat P.V., Arkhipova L.P. Gas-dynamic discontinuity conception. Research Journal of Applied Sciences, Engineering and Technology, 2014, vol. 8, no. 22, pp. 2255–2259.

2. Uskov V.N., Bulat P.V. Shock and detonation wave in terms of view of the theory of interaction gasdynamic discontinuities. Life Science Journal, 2014, vol. 11, no. 8, pp. 307–310.

3. Bulat P.V. Shock and detonation wave in terms of view of the theory of interference gasdynamic discontinuities. Part I. The geometric meaning of the equations of gas dynamics of supersonic flows. Fundamental'nye Issledovaniya, 2013, no. 10–9, pp. 1951–1954. (In Russian)

4. Kozhemyakin A.O., Omel'chenko A.V., Uskov V.N. Oblique interaction of supersonic flows. Fluid Dynamics, 1999, vol. 34, no. 4, pp. 551–557.

5. Kochine N. Determination rigoureuse des ondes permanentes d'ampleur finie a la surface de separation de deux liquides de profodeur finie. Mathematische Annalen, 1927, vol. 98, pp. 582–615.

6. Landau L.D., Lifshits E.M. Mechanics of Continua. Moscow, Fizmatlit Publ., 2003, 736 p. (In Russian)

7. Kobzeva T.A., Moiseev N.Ya. A method of undetermined coefficients for solving Riemann problem. VANT. Ser.: Mat. Mod. Fiz. Proc., 2003, no 1, pp. 3–9.

8. Zapryagaev V.I., Uskov V.N., Gaponov S.A., Maslov A.A., Glaznev V.N. Struinye i Nestatsionarnye Techeniya v Gazovoi Dinamike [Jet and unsteady flow in gas dynamics]. Novosibirsk, SO RAN Publ., 2000.

9. Uskov V.N. Begushchie Odnomernye Volny [Running One-Dimensional Waves]. St. Petersburg, BGTU "Voenmekh" Publ., 2000, 220 p.

10. Igra О. One-dimensional interactions. In Handbook of Shock Waves. V2. Shock wave interactions and propagation. Eds. G. Ben-Dor, O. Igra., T. Elperin. Academic Press, 2001, 924 p.

11. Godunov S.K. A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Sbornik: Mathematics, 1959, vol. 47, no. 8–9, pp. 357–393.

12. Godunov S.K., Zabrodin A.V., Ivanov M.Ya., Kraiko A.N. Chislennoe Reshenie Mnogomernykh Zadach Gazovoi Dinamiki [Numerical Solution of Multi-Dimensional Problems in Gas Dynamics]. Moscow, Nauka Publ., 1976, 400 p.

13. Kulikovskii A.G., Pogorelov N.V., Semenov A.Yu. Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Chapman & Hall/CRC Press, 2001, 560 p.

14. Volkov K.N. Application and implementation of high-resolution difference schemes for solution of gas dynamics problems on unstructured meshes. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2014, no. 6, pp. 153–162. (In Russian)

15. Volkov K.N. Raznostnye skhemy rascheta potokov povyshennoi razreshayushchei sposobnosti i ikh primenenie dlya resheniya zadach gazovoi dinamiki [High-resolution difference schemes of flux calculation and their application to solving gas dynamics problems]. Vychislitel'nye Metody i Programmirovanie, 2005, vol. 6, no. 1, pp. 146–167.

16. Bulat P.V., Volkov K.N., Sil'nikov M.V., Chernyshev M.V. Analysis of finite-difference schemes based on exact and approximate solution of Riemann problem. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2015, no. 1, p. 139–148.

17. Mach E. Uber den verlauf von funkenwellen in der ebene und im raume [Over the course of radio waves in the plane and in space]. Sitzungsbr. Akad. Wiss. Wien, 1878, vol. 78, pp. 819–838.

18. Uskov V.N. Interference gas dynamic and tangential discontinuities. Fluid Dynamics, 1979, no. 4, pp.

191–197. (In Russian)

19. Uskov V.N., Bulat P.V., Arkhipova L.P. Classification of gas-dynamic discontinuities and their interference problems. Research Journal of Applied Sciences, 2014, vol. 8, no. 22, pp. 2248–2254.

20. Bulat P.V., Bulat M.P., Discontinuity of gas-dynamic variables in the center of the compression wave. Research Journal of Applied Sciences, Engineering and Technology, 2014, vol. 8, no. 23, pp. 2343–2349.

21. Uskov V.N. Analysis of the shock-wave structures in a non-uniform steady flow. In Fundamental'nye Problemy Fiziki Udarnykh Voln [Fundamental Problems of Shock Waves Physics], 1987, vol. 2, 166–169.

22. Bulat P.V., Bulat M.P. Definition of the existence region of the solution of the problem of an arbitrary gas-dynamic discontinuity breakdown at interaction of flat supersonic jets with formation of two outgoing compression shocks. Research Journal of Applied Sciences, Engineering and Technology, 2015, vol. 9, no. 1, pp. 65–70.

23. Uskov V.N., Chernyshov M.V. Theoretical analysis of the aerodynamic coefficients polygonal profiles in supersonic flow. Mekhanika i Protsessy Upravleniya: Trudy XXXI Ural'skogo Seminara [Proc. XXXI Ural Workshop on Mechanics and Control]. Ekaterinburg, 2001, pp. 187–191. (In Russian)

24. Uskov V.N., Shakhova O.A. K raschetu troinoi konfiguratsii udarnykh voln [The calculation of a triple configuration shock waves]. Gidroaeromekhanika i Teoriya Uprugosti, 1976, vol. 21, no. 7, pp. 13–18.

25. Uskov V.N., Starykh A.L. Analiz oblastei sushchestvovaniya reshenii uravnenii interferentsii statsionarnykh gazodinamicheskikh razryvov Analiz oblastey sushchestvovaniya resheniy uravneniy interferentsii statsionarnykh gazodinamicheskikh razryvov [Analysis of the regions of solutions existence of the interference of stationary gas-dynamic discontinuities]. Nestatsionarnye Techeniya Gazov s Udarnymi Volnami, 1990, pp. 359–372.