Numerical stability and numerical viscosity in certain meshless vortex methods as applied to the Navier–Stokes and heat equations

Ya. A. Dynnikov   ; G. Ya. Dynnikova

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 10, pp. 1905-1917

Résumé

The stability and numerical viscosity of certain Lagrangian methods for flow simulation are analyzed in numerical experiments. Most attention is given to the viscous vortex domains method and its generalization, the viscous vortex thermal domains method.

Bibliographie

[1] Rosenhead L., “Formation of vortices from a surface of discontinuity”, Proc. Roy. Soc. Ser. A, 134 (1931), 170–192 | DOI

[2] Belotserkovskii S. M., Nisht M. I., Otryvnoe i bezotryvnoe obtekanie tonkikh krylev idealnoi zhidkostyu, Nauka, M., 1978

[3] Belotserkovskii S. M., Kotovskii V. N., Nisht M. I., Fedorov R. M., Matematicheskoe modelirovanie ploskoparallelnogo otryvnogo obtekaniya tel, Nauka, M., 1988 | MR

[4] Chorin A. J., “Numerical study of slightly viscous flow”, J. Fluid. Mech., 57:4 (1973), 785–796 | DOI | MR

[5] Cottet G.-H., Mas-Gallic S., “A particle method to solve the Navier–Stokes system”, Numer. Math., 57 (1990), 805–827 | DOI | MR | Zbl

[6] Raviart P.-A., “An analysis of particle methods”, Numer. Meth. Fluid Dynamics., Lect. Notes Math., Springer, N. Y.–Berlin, 1985, 243–324 | DOI | MR

[7] Degond P., Mas-Gallic S., “The weighted particle method for convection-diffusion equations, I: The case of an isotropic viscosity”, Math. Comput., 53 (1989), 485–507 | MR | Zbl

[8] Cottet G.-H., Koumoutsakos P., Vortex methods. Theory and practice, Cambridge Univ. Press, 2000 | MR

[9] Shankar S., van Dommelen L., “A new diffusion procedure for vortex methods”, J. Comput. Phys., 127 (1996), 88–109 | DOI | MR | Zbl

[10] Fishelov D., “A new vortex scheme for viscous flows”, J. Comput. Phys., 86 (1990), 211–224 | DOI | MR | Zbl

[11] Barba L. A., Leonard A., Allen C. B., “Advances in viscous vortex methods-meshless spatial adaption based on radial basis function interpolation”, Int. J. Numer. Meth. Fluids, 47 (2005), 387–421 | DOI | MR | Zbl

[12] Ogami Y., Akamatsu T., “Viscous flow simulation using the discrete vortex model. The diffusion velocity method”, Comput. Fluids, 19:3–4 (1991), 433–441 | DOI | Zbl

[13] Takeda K., Tutty O. R., Fitt A. D., A comparison of four viscous models for the discrete vortex method, AIAA Paper 9741997, 1997, 11 pp.

[14] Dynnikova G. Ya., “Lagranzhev podkhod k resheniyu nestatsionarnykh uravnenii Nave–Stoksa”, Dokl. RAN, 399:1 (2004), 42–46 | MR

[15] Guvernyuk S. V., Dynnikova G. Ya., “Modelirovanie obtekaniya koleblyuschegosya profilya metodom vyazkikh vikhrevykh domenov”, Izv. RAN. Mekhan. zhidkosti i gaza, 2007, no. 1, 3–14 | MR | Zbl

[16] Andronov P. R., Guvernyuk S. V., Dynnikova G. Ya., Vikhrevye metody rascheta nestatsionarnykh gidrodinamicheskikh nagruzok, MGU, M., 2006

[17] Dynnikova G. Ya., “Raschet obtekaniya krugovogo tsilindra na osnove dvumernykh uravnenii Nave–Stoksa pri bolshikh chislakh Re s vysokim razresheniem v pogranichnom sloe”, Dokl. RAN, 422:6 (2008), 755–757 | Zbl

[18] Dynnikova G. Ya., “Ispolzovanie bystrogo metoda resheniya "zadachi $N$ tel" pri vikhrevom modelirovanii techenii”, Zh. vychisl. matem. i matem. fiz., 49:8 (2009), 1458–1465 | MR | Zbl

[19] Ivanova O. A., “Numerical simulation of wind induced conductor motion”, Proc. ICVFM 2010 (Nov. 8–10, San Leucio(CE), Italy), 63

[20] Dynnikova G. Ya., “The viscous vortex domains (VVD) method for non-stationary viscous incompressible flow simulation”, Proc. IV European Conf. Comput. Mech. (Paris, May 16–21, 2010) http://www.eccm2010.org/complet/fullpaper_193.pdf

[21] Guvernyuk S. V., Dynnikova G. Ya., Dynnikov Ya. A., Malakhova T. V., “O stabilizatsii sleda za krugovym tsilindrom, sovershayuschim vysokochastotnye vraschatelnye kolebaniya”, Dokl. RAN, 432:1 (2010), 45–49

[22] Andronov P. R., Grigorenko D. A., Guvernyuk S. V., Dynnikova G. Ya., “Chislennoe modelirovanie samovrascheniya plastin v potoke vyazkoi zhidkosti”, Izv. RAN. Mekhan. zhidkosti i gaza, 2007, no. 5, 47–60 | Zbl

[23] Andronov P. R., Dosaev M. Z., Dynnikova G. Ya., “Modelirovanie vetrodvigatelya volnovogo tipa”, Probl. mashinostr. i nadezhnosti mashin, 2009, no. 4, 86–91

[24] Dynnikov Y., Andronov P., Guvernyuk S., Dynnikova G., “Numerical simulation of the flow-structure interaction by the VVD method”, Proc. ICVFM 2010 (Nov. 8–10, San Leucio (CE), Italy), 31 | Zbl

[25] Andronov P. R., Guvernyuk S. V., Dynnikova G. Ya., “Lagranzhev chislennyi metod resheniya dvumernykh zadach svobodnoi konvektsii”, Tr. IV Ros. nats. konf. po teploobmenu, v. 3, Izdat. dom MEI, M., 2006, 38–41

[26] Dynnikov Ya. A., Malakhova T. V., “Chislennoe modelirovanie nestatsionarnoi teplootdachi tsilindra v potoke vyazkoi zhidkosti metodom vyazkikh vikhre-teplovykh domenov”, Tr. XIV Mezhdunar. simp. “Metody diskretnykh osobennostei v zadachakh matem. fiz.” (MDOZMF42009), 2009, 95–98

[27] Malakhova T., Dynnikova G., Guvernyuk S., “Investigation of the heat transfer from oscillating cylinders by the VVTD method”, Proc. of ICVFM 2010 (Nov. 8–10, San Leucio (CE), Italy), 30 | Zbl

[28] Dynnikova G. Ya., “Raschet trekhmernykh techenii neszhimaemoi zhidkosti na osnove dipolnogo predstavleniya zavikhrennosti”, Dokl. RAN, 437:1 (2011), 35–38 | MR

[29] Ogamy Y., “Simulation of heat-vortex interaction by the diffusion velocity method”, ESAIM: Proc., Third Internat. Workshop Vortex Flows and Related Numer. Meth., 7, 1999, 314–324

[30] Loitsyanskii L. G., Mekhanika zhidkosti i gaza, Nauka, M., 1978 | MR

[31] Seffmen F. Dzh., Dinamika vikhrei, Nauchn. mir, M., 2000

[32] Dynnikov Ya. A., “Analiz effektov skhemnoi dissipatsii v lagranzhevykh vikhrevykh metodakh”, Tr. konf. konkursa molodykh uchenykh, 2008 g., MGU, 2009, 85–91

[33] Dynnikov Ya. A., Dynnikova G. Ya., “Otsenka effektov skhemnoi vyazkosti pri raschete techenii neszhimaemoi zhidkosti vikhrevymi metodami”, Tr. XIV Mezhdunar. simp. “Metody diskretnykh osobennostei v zadachakh matem. fiz.” (MDOZMF42009), Kharkov–Kherson, 2009, 291–294