Special features of difference schemes for solving the two-dimensional Navier–Stokes equations, connected with the formulation of the boundary conditions on the solid surface

M. N. Zakharenkov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 30 (1990) no. 8, pp. 1224-1236

Résumé

In the context of the numerical treatment of the two-dimensional Navier–Stokes equations, the boundary conditionsatasolid surface may be realized in different ways, some of which are examined. The approach considered here assumes that the equations of the system are solved separately. It is shown that then, irrespective of the specific formulation of the Navier–Stokes equations for an incompressible viscous liquid – in terms of velocity-pressure, velocity-vorticity or vorticity-stream function – the boundary conditions can be realized in an algorithmically universal way, based on a two-parameter formula previously proposed to approximate vorticity on a wall.

Bibliographie

[1] Rouch P., Vychislitelnaya gidrodinamika, Mir, M., 1980 | Zbl

[2] Tarunin E. L., “Analiz approksimatsionnykh formul dlya vikhrya skorosti na tverdoi granitse”, Gidrodinamika, Uch. zap. PGPI, 152, no. 9, 1976, 167–178

[3] Zakharenkov M. N., “Approksimatsiya granichnogo usloviya dlya zavikhrennosti na poverkhnosti tverdogo tela pri reshenii uravnenii Nave–Stoksa v peremennykh funktsiyakh toka i zavikhrennost”, Chisl. metody mekhan. sploshnoi sredy, 11, no. 7, ITPM SO AN SSSR, Novosibirsk, 1980, 56–74 | MR

[4] Zakharenkov M. N., “Ob approksimatsii granichnogo usloviya dlya zavikhrennosti”, Chisl. metody mekhan. sploshnoi sredy, 13, no. 2, ITPM SO AN SSSR, Novosibirsk, 1982, 64–81

[5] Zakharenkov M. N., “Raschet obtekaniya vraschayuschegosya i koleblyuschegosya ellipticheskogo tsilindra potokom vyazkoi neszhimaemoi zhidkosti”, Chisl. metody mekhan. sploshnoi sredy, 15, no. 1, ITPM SO AN SSSR, Novosibirsk, 1984, 45–59

[6] Mehta U. B., “Dynamic stall of an oscillating airfoil”, AGARD Conf. Proc. CP-227, Symp. Unsteady Aerodynamics (Ottawa, Canada, 1977), 23

[7] Mazhorova O. S., Popov Yu. P., Ob odnom algoritme chislennogo resheniya dvumernykh uravnenii Nave–Stoksa, Preprint No 37, IPMatem. AN SSSR, M., 1979

[8] Lyumkis E. D., “Ob uvelichenii shaga po vremeni pri integrirovanii uravnenii Nave–Stoksa v peremennykh vikhr-funktsiya toka”, Differents. ur-niya, 21:7 (1985), 1208–1217 | MR

[9] Goncharov A. L., Fryazinov I. V., “Ob odnom setochnom metode resheniya uravnenii Nave–Stoksa v peremennykh vikhr-funktsiya toka”, Differents. ur-niya, 21:7 (1985), 1269–1273 | MR

[10] Slezkin N. A., Dinamika vyazkoi zhidkosti, Gostekhteorizdat, M., 1985

[11] Rubin S. G., Khosla P. K., “Polynomial interpolation methods for viscous flow calculations”, J. Comput. Phys., 24:3 (1977), 217–244 | DOI | MR | Zbl

[12] Kleiser L., Schumann U., “Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows”, Notes Numer. Fluid Mech., v. 2, Vieweg, Braunschweig, 1980, 165–173

[13] Gresho P. M., Sani R. L., “On pressure boundary conditions for the incompressible Navier–Stokes equations”, Internat. J. Numer. Meth. in Fluids, 7:10 (1987), 1111–1145 | DOI | Zbl

[14] Belotserkovskii O. M., Chislennoe modelirovanie v mekhanike sploshnykh sred, Nauka, M., 1984 | MR

[15] Fortin M., Peyret R., Temam R., “Resolution numerique des equations de Navier–Stokes pour un fluid incompressible”, J. mecan., 10:3 (1971), 357–390 | MR | Zbl

[16] Polezhaev V. I., Gryaznov V. L., “Metod rascheta granichnykh uslovii dlya uravnenii Nave–Stoksa v peremennykh “vikhr–funktsiya toka””, Dokl. AN SSSR, 219:2 (1974), 301–304 | Zbl

[17] Daikovskii A. G., Polezhaev V. I., Fedoseev A. N., Chislennoe modelirovanie perekhodnogo i turbulentnogo rezhimov konvektsii na osnove nestatsionarnykh uravnenii Nave–Stoksa, Preprint No 101, IPMekhan. AN SSSR, M., 1978

[18] Faruk B., Fusegi T., “A coupled solution of the vorticity-velocity formulation of the incompressible Navier–Stokes equations”, Internat. J. Numer. Meth. in Fluids, 5 (1985), 1017–1034 | DOI | MR

[19] Orlandi P., “Vorticity-velocity formulation for high Re flows”, Internat. J. Comput. and Fluids, 15:2 (1987), 137–149 | DOI | Zbl

[20] Zakharenkov M. H., “Raschet nestatsionarnogo obtekaniya vyazkoi zhidkostyu tsilindra, sovershayuschego vraschatelnye kolebaniya v ravnomernom potoke”, Izv. AN SSSR. Mekhan. zhidkosti i gaza, 1989, no. 5, 32–38

[21] Zakharenkov M. N., Mikhailova N. I., “Vikhrevye struktury v slede za koleblyuschimsya i nepodvizhnym profilem”, Turbulentnye techeniya i tekhn. eksperimenta, Tezisy dokl. VI Vses. sovesch. po teor. i prikl. aspektam turbulentnykh techenii, In-t termofiz. i elektrofiz. AN SSSR, Tallinn, 1989, 110–113