Novosibirsk, Russia, May, 30 – June, 4, 2011

International Conference
"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

Воеводин А.Ф.   Гончарова О.Н.  

Method of computation of the problems of convection: realization of a splitting into physical processes

Reporter: Воеводин А.Ф.

     For investigation of the problems of viscous fluid dynamics the numerical methods, relating to the splitting methods, are developed actively (Kovenya V.M., Slunyaev A.Yu, 2010). For the problems of convection in the closed domains and by the long-time processes the no-slip and no-flow conditions on the fixed walls should be fulfilled exactly. Conservation of the solenoidality of the velocity field and its energetic neutrality should be also guarantied. For computation of the convective fluid flows in the three-dimensional domains (in the parallelepipeds) a method is proposed, where an idea of splitting into physical processes is realized (Voevodin A.F., Goncharova O.N., 2009). The proposed splitting scheme is a physically legitimate scheme which is characterized by a property of stability in the linear approximation.
     Splitting into convective and diffusive transfer is performed in the Oberbeck-Boussinesq equations of convection written in the physical variables. Separation of the stage of convection allows to avoid calculation of the pressure gradient and to provide correctness of the splitting for the fulfillment of the boundary conditions. The stage of convection is realized for the components of a tentative velocity (auxiliary function) on the basis of the elementary Crank-Nicholson schemes. On the stage of diffusion a transition to the functions “rotor of velocity – vector potential” is carried out. To realize the stage of diffusion a variant of the sweep methods with parameters is constructed. A second order finite difference scheme is presented. Adherence of the order of calculations relative to the directions is established. A problem of statement of the boundary conditions for the auxiliary functions on the inner stages of the finite difference scheme is solved. The method of splitting into physical processes is generalized to the domains with sufficiently smooth boundaries and also to the case of a dependence of the transfer coefficients on temperature.
     Testing of the method is carried out with the help of the benchmarks of convection in a cubic cavity and in a parallelepiped by heating of one of the walls.

The research has been supported by the Siberian Branch of Russian Academy of Sciences (Integrated project No. 116) and by the Russian Foundation for Basic Research (No. 10-01-00007).

Abstracts file: VoGo+ENG.doc
Full text file: VG.pdf

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