Analytic construction of the near field in a~transonic flow near a~thin axisymmetric body

K. V. Kurmaeva ; S. S. Titov

Sibirskij žurnal industrialʹnoj matematiki, Tome 8 (2005) no. 3, pp. 93-101.

Résumé

A solution of the axisymmetric problem of unsteady transonic flow around thin bodies of revolution is proposed in the form of a double series expansion in powers of the distance to the axis of symmetry and its logarithm in a neighborhood of a given point at the symmetry axis. Chains of recurrence equations are obtained for the coefficients of the series. The convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution to the boundary-value problem for a nonlinear partial differential equation with a singularity at the symmetry axis is obtained in the asymptotic model of unsteady transonic flow under consideration. Thereby the application of the proposed series is justified to the problems of unsteady transonic flow around thin axisymmetric bodies with a drift of the nonpenetration condition onto the symmetry axis. Hence, these series can be used in numerical-analytical methods and model computations.

Bibliographie

[1] Sidorov A. F., Izbrannye trudy. Matematika. Mekhanika, Fizmatlit, M., 2001 | MR

[2] Ovsyannikov L. V., “O skhodimosti ryada Meiera dlya osesimmetrichnogo sopla”, Dopolnenie 1 v kn.: Martesen E., Zengbush R. fon, Raschet okolozvukovoi chasti ploskikh i osesimmetrichnykh sopel s krivolineinoi liniei perekhoda, Izd-vo SO AN SSSR, Novosibirsk, 1962, 41–43

[3] Tzyan Kh. Sh., Lin Ts. Ts., Reissner E., “O dvumernom neustanovivshemsya dvizhenii tonkogo tela v szhimaemoi zhidkosti”, Gazovaya dinamika, Izd-vo inostr. lit., M., 1950, 183–196

[4] Teshukov V. M., “Prostranstvennaya zadacha o rasprostranenii kontaktnogo razryva v idealnom gaze”, Dinamika sploshnoi spedy, no. 32, 1977, 82–94

[5] Teshukov V. M., “Postroenie fronta udarnoi volny v prostranstvennoi zadache o porshne”, Dinamika sploshnoi spedy, no. 33, 1978, 114–133 | MR

[6] Teshukov V. M., “Tsentrirovannye volny v prostranstvennykh techeniyakh gaza”, Dinamika sploshnoi spedy, no. 39, 1979, 102–118 | MR

[7] Teshukov V. M., “Raspad proizvolnogo razryva na krivolineinoi poverkhnosti”, Zhurn. prikl. mekhaniki i tekhn. fiziki, 1980, no. 2, 126–133 | MR

[8] Teshukov V. M., “Prostranstvennyi analog tsentrirovannykh voln Rimana i Prandtlya–Maiera”, Zhurn. prikl. mekhaniki i tekhn. fiziki, 1982, no. 4, 98–106 | MR

[9] Bautin S. P., Analiticheskaya teplovaya volna, Fizmatlit, M., 2003

[10] Titov S. S., “Ob okolozvukovom obtekanii gazom tonkikh tel vrascheniya”, Analiticheskie metody mekhaniki sploshnoi sredy, no. 33, izd. IMM UNTs AN SSSR, Sverdlovsk, 1979, 65–72

[11] Ovsyannikov L. V., Lektsii po osnovam gazovoi dinamiki, izd. In-ta kompyuternykh issledovanii, M., Izhevsk, 2003

[12] Titov S. S., Reshenie uravnenii s osobennostyami v analiticheskikh shkalakh banakhovykh prostranstv, Preprint UralGAKhA, Ekaterinburg, 1999

[13] Lipman G. V., Roshko A., Elementy gazovoi dinamiki, Izd-vo inostr. lit-ry, M., 1960 | MR

[14] Guderlei K. G., Teoriya okolozvukovykh techenii, Izd-vo inostr. lit-ry, M., 1960

[15] Van Dyke M., “Supersonic flow past bodies of revolution”, Aeronaut. Sci., 18 (1951)

[16] Blinkov N. A., Chernov I. A., “Reduktsiya nestatsionarnogo transzvukovogo uravneniya k kvazistatsionarnomu”, Aerodinamika. Ploskie i osesimmetrichnye techeniya zhidkosti, Mezhvuz. nauch. sb., Izd-vo Sarat. gos. un-ta, Saratov, 1988, 110–131

[17] Larkin N. A., Gladkie resheniya uravnenii transzvukovoi gazodinamiki, Nauka, Novosibirsk, 1991 | MR

[18] Prudnikov A. P. i dr., Integraly i ryady. Spetsialnye funktsii, eds. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Nauka, M., 1983 | MR | Zbl