Abstract:
The report is dedicated to the life and work of a scientist in the field of mathematics and mechanics, academician L.V. Ovsyannikov and the main stages of development of the theory of transonic gas flows.
L.V. Ovsyannikov started his scientific research in the second half of the 1940s, dur-ing the development of high-speed aviation, approaching transonic speeds, determined the in-itial range of his interests: transonic flow around profiles of the simplest forms, transonic flows in channels and jet flows, flows with the simplest form of sound line, definition fea-tures and the development of a mathematical methods for calculating such flows and their justification.
Defined by S.A. Chaplygin [1] methods for studying flows in the transonic range of velocities made it possible to take into account the very important role of flow compressibility in these problems. The methods used by him were distinguished by fundamental mathe-matical rigor. However, this path did not in any way allow for further progress in the follow-ing stages - the creation of a model of unsteady transonic flow and the inclusion of viscous effects. The scientific interests of L.V. Ovsyannikov subsequently went mainly into the field of group analysis of partial differential equations.
For an unsteady transonic flow, the governing equation was derived in 1948 by the method of asymptotic expansions of Ñ. Lin, E. Reissner and H.S. Tsien [2], (Lin- Reissner-Tsien equation, LRT). Since the late 1960s, the theory of nonclassical boundary layers has been intensively developed - for supersonic flows V.Ya. Neyland (Neiland), 1968 [3], in transonic mode O.S. Ryzhov [4], 1977.
The resulting wealth of solutions for transonic mode overshadowed, however, a very serious problem. Subsequently, it turned out [5] that the direction of research followed a path leading away from reality — the classical model was singular, but this circumstance remained ignored. The LRT equation, which has undoubted advantages (describes both the supersonic and subsonic regions of the transonic flow, its one-dimensional, non-stationary and non-linear nature), however, has drawbacks that do not allow us to correctly describe the propagation of non-stationary disturbances in the flow: this equation is a degenerate hyperbolic equation and describes the propagation of unsteady disturbances in the flow field is only partially (only upstream). In this regard, to study the problems of the theory of non-classical transonic boundary layer, a modified model was proposed [5]. Modification of the model consists in regularizing the LRT equation - storing in it during asymptotic calculations of the term with the second time derivative.
As a member of program conferences on fluid and gas mechanics L.V. Ovsyannikov invariably stayed up to date on the ongoing studies of transonic flows and comprehensively supported them, not being their direct co-executor.
References
1. Yur'yev I.M. Znacheniye issledovaniya S.A. Chaplygina «O gazovyh struyah» dlya gazovoj dinamiki. ( The value of S.A. Chaplygin's research "On gas jets" for gas dynamics) Izd-vo CAGI. 1969. 22 s.
2. Lin C.C., Reissner E., Tsien H.S. On two-dimensional non-steady motion of a slender body in a compressible fluid // J. of Mathematics and Physics. 1948. V.27. ¹ 3. P. 220-231.
3. Neiland V.YA. Sverhzvukovoye techeniye vyazkogo gaza vblizi tochki otryva/ III Vsesoyuznyj s"yezd po teoreticheskoj i prikladnoj mekhanike. (Supersonic viscous gas flow near the separation point / III All-Union Congress on Theoretical and Applied Mechanics) 25.01 – 01.02.1968 g. Sb. annotacij dokladov. M.: Nauka. 1968. S. 224.
4. Ryzhov O.S. O nestacionarnom pogranichnom sloye s samoinducirovannym davleniyem pri okolozvukovyh skorostyah vneshnego potoka (On a non-stationary boundary layer with self-induced pressure at transonic external flow velocities) // Dokl. AN SSSR. 1977. T.236. ¹ 5. S. 1091-1094.
5. Bogdanov A.N., Diyesperov V.N. Modelirovaniye nestacionarnogo transzvukovogo techeniya i ustojchivost' transzvukovogo pogranichnogo sloya (Modeling of unsteady transonic flow and stability of a transonic boundary layer) // PMM. 2005. T. 69. Vyp. 3. S. 394-403.
