519.63:532.5 Numerical-analytical method of solving two-dimensional problems of natural convection in a closed cavity

Basarab M. A. (Bauman Moscow State Technical University)

NATURAL CONVECTION, THE METHOD OF R-FUNCTIONS, PETROV – GALERKIN METHOD.


doi: 10.18698/2309-3684-2014-1-1835


The author offers a method (PGRM) of numerical-analytical solving the equation system in partial derivatives describing the natural thermal convection in the complicated-shaped dimensional cavity with arbitrary boundary conditions. The new approach is based on a combination of Petrov – Galerkin method and R-functions (Rvachev functions) and makes it possible to obtain temperature, vortex and current functions satisfying the boundary condi-tions in the form of expansions in certain bases. The coordinated choice of bases provides a natural way to approximate the boundary conditions for the flow function. Unsteady convec-tion problems are solved by combining PGRM and Rothe method.


[1] Rouch P. Vychislitel'naya gidrodinamika [Computational hydrodynamics]. Moscow, Mir Publ., 1980, 618 p.
[2] Patankar S.V. Chislennoe reshenie zadach teploprovodnosti i konvektivnogo teploobmena pri techenii v kanalakh [Numerical solution of problems of heat conductivity and convectional heat exchange in the canal current]. Moscow, MPEI Publ., 2003, 312 p.
[3] Samarsky A.A., Vabishchevich P.N. Chislennye metody resheniya zadach konvektsii-diffuzii [Numerical methods for solving problems of convection-diffusion]. Moscow, URSS, 2009, 248 p.
[4] Shi D. Chislennye metody v zadachakh teploobmena [Numerical methods in problems of heat exchange]. Moscow, Mir Publ., 1988, 544 p.
[5] De Vahl Davis, G. Finite Difference Methods for Natural and Mixed Convec-tion in Enclosures in Heat Transfer. C.L. Tien, V.P. Carey and J.K. Ferrell, eds. Washington, Hemisphere Publ. Corp., 1986, vol. 1, pp. 101–109.
[6] Fletcher K. Chislennye metody na osnove metoda Galerkina [Numerical meth-ods based on Galerkin method]. Moscow, Mir Publ., 1988, 352 p.
[7] Volkov K.N., Emelyanov V.N. Vychislitel'nye tekhnologii v zadachakh mekhaniki zhidkosti i gaza [Computational technologies in problems of liquid and gas mechanics]. Moscow, Fizmatlit Publ., 2012, 468 p.
[8] Lui G.R. Mesh Free Methods: Moving Beyond the Finite Element Method. CRC Press, 2003, 792 p.
[9] Katz A.J. Meshless Methods for Computational Fluid Dynamics. Ph.D. Thesis. Dept. of Aeronautics and Astronautics, Stanford University, 2009, 131 p.
[10] Rvachev V.L. Teoriya R-funktsii i nekotorye ee prilozheniya [Theory of R-function and some of its applications]. Kiev, Naukova dumka Publ., 1982, 552 p.
[11] Kravchenko V.F., Basarab M.A. Buleva algebra i metody approksimatsii v kraevykh zadachakh elektrodinamiki [Boolean algebra and approximation methods in boundary problems of electrodynamics]. Moscow, Fizmatlit Publ., 2004, 308 p.
[12] Tsukanov I., Shapiro V., Zhang S.A. International Journal for Numerical Methods in Engineering, 2003, vol. 58, pp. 127–158. doi: 10.1002/nme.760.
[13] Basarab M.A. Inzhenernyi zhurnal: nauka i innovatsii — Engineering journal: science and innovations, 2013, no. 11 (23). URL: http://engjournal.ru/ catalog/mathmodel/hidden/1071.html
[14] Kantorovich L.V., Krylov V.I. Priblizhennye metody vysshego analiza [Ap-proximate methods of higher analysis]. Moscow–Leningrad, Fizmatlit Publ., 1962, 708 p.
[15] Basarab M.A., Kravchenko V.F., Matveev V.A. Matematicheskoe modelirovanie fizicheskikh protsessov v giroskopii [Mathematical modeling of physical processes in gyroscopy]. Moscow, Radiotekhnika Publ., 2005, 176 p.
[16] De Vahl Davis G. International Journal of Numerical Methods in Fluids, 1983, vol. 3, pp. 249–264.


Basarab M. Numerical-analytical method of solving two-dimensional problems of natural convection in a closed cavity. Маthematical Modeling and Coтputational Methods, 2014, №1 (1), pp. 18-35



Download article

Колличество скачиваний: 256