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Gil'manov A.N.

Doctor of Phys. and Math. Sciences Gil'manov Anvar N.

The main research direction is:

One of the greatest difficulties at numerical simulation of high-speed flows of viscous gas at large Reynolds numbers is represented by areas of viscous-inviscid interaction. These zones are characterized by a combination of discontinuous distributions of gas flow variables on shocks and high gradients of gas variables in boundary layers. Frequently influence of shock waves on the boundary layers results in a separation of a flow and essential reorganization of an external inviscid flow.

When the use of small spatial grid steps, which arise, in particular, in the case of large Reynolds number is necessary, there is preferable to use the dynamically adaptive grids allowing to improve the solution locally. Along with this it is important to utilize schemes of high-order accuracy which do not lead to numerical oscillations on shocks.

The methods of dynamically adaptive grids are the most effective approaches for increase of accuracy of the numerical solution in areas with strongly differing spatial scales due to non-uniform structure of a flow.

In this work an adaptive-moving and adaptive-embedding grids are presented. The adaptive-moving grids consist of fixed number of cells which are redistributed from their initial position moving in zones with large gradients of gas variables. The adaptive-embedding grids assume "embedding" of additional cells in those zones of the computational domain where gas flow variables vary significantly.

The developed method for solving problems of an inviscid or viscous gas flow past obstacles is a combination of methods of dynamically adaptive grids and a scheme of second order of accuracy in zones with continuous variation of gasdynamic variables.

TEST PROBLEMS. To estimate reliability of numerical results obtained by the presented method its wide and all-round testing on known problems of gas dynamics was conducted. The obtained data compare with the solutions derived by approximate methods by other authors.

A problem of a supersonic viscous gas flow past a flat heat-insulated plate is considered.

Fig.1a presents isochores obtained on the adaptive-embedded grid. One can clearly see the boundary layer and shock wave which resulted from the interaction of undisturbed supersonic gas flow and the area of displacement formed by the boundary layer. Temperature T as a function of self-similar variable is given on Fig.1b. The curves 1-4 represent the temperature distributions along the corresponding normal coordinate curves to the surface of the plate (Fig.1a).

The coincidence of this temperature distributions along various sections of the boundary layer show that this solution is self-similar. The results were compared with data in (Shlichting 1974). Comparisons of data obtained by the represented method with the solutions derived by other authors for problems of supersonic viscous gas flow past a plate show their good agreement. Obtained results indicate also that the numerical viscosity influences on the numerical solution significantly less compared with the real viscosity.

Interaction of a boundary layer with shock. An oblique shock act on the boundary layer formed on the plate by a viscous gas flow. The computation was performed on a grid 60x30. The method of adaptive-moving mesh was employed in such a manner, that no less than three-five cells was placed across the boundary layer. The further calculation was realized using the algorithm of adaptive-embedded grids (Gil'manov et al. 1995). Izohores of the steady-state solution are presented on Fig.2a.

All the gas dynamic features characterizing the considered problem are clearly seen on this figure: shock waves, the rarefaction wave and the boundary layer. Total number of the cells in the grid of a level 4 was approximately 12200. Fig.2b shows a fragment of the field of the velocity vector directions with their origins in cell centers. This fragment contains the part of computational domain in the location of the separation zone. A conclusion that complex gasdynamic problems with flow separation are accurately simulated by the numerical technique proposed in this work can be drawn.

Viscous gas flow in a supersonic inlet. This problem is much more complicated than problem considered above. In this case the multiple interaction of a boundary layer with the shock waves and the rarefaction waves occur, whereas in the previous problem there was only single interaction of boundary layer with shock. For the clearness of the presentation the real sizes in the figures related to this problem are changed . The pictures of the inlet are stretched three times in a vertical direction. Fig.3a presents the grid used in computations which have 23800 cells. To get solution of the same accuracy the uniform grid of more than 300000 (!) cells would be required.

23800 cells
Fig.3bp, Fig.3br shows isobars and isochores, respectively, obtained in computations. The same problem was solved for the case of inviscid gas. It was found from the analysis and comparison of the viscous and inviscid solutions that they differ both numerically and qualitatively.

Thus, the present results of the computations show, that the proposed approach enables effectively and economically to obtain the solutions of the complex problems of aerodynamics including the phenomena of the boundary layer separation.


[1] Gil'manov, A.N., Kulachkova, N.A. (1995). "A method TVD on adaptive-embedded grids in problems of supersonic gas dynamics," Voprosi Atomnoy Nauki i Techniki. Serija: Matematicheskoe Modelirovanie Physicheskich Processov, Vipusk 1-2, 72-79 (In Russian).
[2] Gil'manov, A.N., Kulachkova, N.A. (1995). "Numerical research of two dimensional gas flow with shocks by a method TVD on a physically adaptive grids," Matematicheskoe Modelirovanie," Vol.7, N.3, 97-106 (In Russian).
[3] Shlichting, G. (1974). The Theory of Boundary Layer, Nauka, Moskva (In Russian).


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