Implementation of the Particle Strength Exchange Method for Fragmentons to Account for Viscosity in Vortex Element Method

This paper focuses on the modification of the vortex element method for viscous incompressible flow simulations. Viscosity is accounted for using the Particle Strength Exchange (PSE) method by means of approximation of the viscous term in the vorticity evolution equation with an integral operator. Resolution equations for the parameters of the vortex elements (fragmentons) have been deduced: their marker position, direction and intensity. All the derivations are made by the assumption of noneffect of the vortex element additional vorticity. It has been shown that in the scope of this assumption, viscosity affects only the change of intensity of vortex elements. In such a case, in PSE method the elements follow trajectories of fluid particles, unlike in the Diffusive Velocity Method (DVM). Special focus is given to the discretization of initial vorticity onto the distributed system of vortex elements. Finally, the results of the diffusion problem of infinite vortex tube are described. Comparison of the numerical and analytical results confirms the correctness of the viscosity model for the considered problem

The study was supported by a grant from the Russian Foundation for Basic Research (project no. 17-08-01468 А)

References

[1] Patankar S. Numerical heat transfer and fluid flow. Hemisphere Publishing Corporation, 1980. 197 p.

[2] Winckelmans G.S., Leonard A. Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows. Journal of Computational Physics, 1993, vol. 109, iss. 2, pp. 247–273. DOI: 10.1006/jcph.1993.1216

[3] Cottet G.H., Koumoutsakos P.D. Vortex methods: theory and practice. Cambridge University Press, 2000. 328 p.

[4] Kotsur O., Scheglov G., Leyland P. Verification of modelling of fluid-structure interaction (FSI) problems based on experimental research of bluff body oscillations in fluids. 29th Congress of the International Council of the Aeronautical Sciences, 2014, pp. 987–996.

[5] Raviart P.-A. An analysis of particle methods. Numerical methods in fluid dynamics. Springer, 1985. Pp. 243–324.

[6] Chorin A.J. Numerical study of slightly viscous flow. Journal of Fluid Mechanics, 1973, vol. 57, iss. 4, pp. 785–796. DOI: 10.1017/S0022112073002016

[7] Sizykh G.B. Vorticity evolution in swirling axisimmetrical flows of viscous incompressible fluid. Uchenye zapiski TsAGI, 2015, no. 3, pp. 14–20 (in Russ.).

[8] Lacombe G., Mas-Gallic S. Presentation and analysis of a diffusion-velocity method. ESAIM: Proc., 1999, vol. 7, pp. 225–233. DOI: 10.1051/proc:1999021

[9] Dynnikova G.Ya. Vortex motion in two-dimensional viscous fluid flows. Fluid Dynamics, 2003, vol. 38, iss. 5, pp. 670–678. DOI: 10.1023/B:FLUI.0000007829.78673.01

[10] Degond P., Mas-Gallic S. The weighted particle method for convection-diffusion equations. Part 1: the case of an isotropic viscosity. Mathematics of Computation, 1989, vol. 53, no. 188, pp. 485–507. DOI: 10.2307/2008716

[11] Mycek P., Pinon G., Germain G., Rivoalen E. Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations. Computational and Applied Mathematics, 2016, vol. 35, iss. 2, pp. 447–473. DOI: 10.1007/s40314-014-0200-5

[12] Bogomolov D.V., Marchevskiy I.K., Setukha A.V., Shcheglov G.A. Numerical modelling of movement of pair vortical rings in the ideal liquid methods of discrete vortical elements. Inzhenernaya fizika [Engineering Physics], 2008, no. 4, pp. 8–14 (in Russ.).

[13] Eldredge J.D., Leonard A., Colonius T. A general deterministic treatment of derivatives in particle methods. Journal of Computational Physics, 2002, vol. 180, iss. 2, pp. 686–709. DOI: 10.1006/jcph.2002.7112

[14] Brutyan M.A. Point vortex singularity in N-dimensional space. Fundamentalnye i prikladnye issledovaniya: problemy i rezultaty, 2014, no. 10, pp. 192–197 (in Russ.).

[15] Beale J.T., Majda A. Vortex methods. I. Convergence in three dimensions. Mathematics of Computation, 1982, vol. 39, no. 159, pp. 1–27. DOI: 10.1090/S0025-5718-1982-0658212-5

[16] Novikov E.A. Generalized dynamics of three-dimensional vertical singularities (vortons). JETP, 1983, vol. 57, no. 2, pp. 566–569.

[17] Basin M.A., Kornev N.V. Approximation of vorticity field in unbounded volume. Zhurnal tekhnicheskoy fiziki, 1994, vol. 64, no. 11, pp. 179–185 (in Russ.).

[18] Marchevsky I.K., Shcheglov G.A. 3D vortex structures dynamics simulation using vortex fragmentons. Proc. of ECCOMAS, 2012, pp. 5716–5735.

[19] Alkemade A.J.Q. On vortex atoms and vortons. PhD Thesis. Delft, Netherlands, 1994. 209 p.

[20] Saffman P.G. Vortex dynamics. Cambridge University Press, 1992, 311 p.

[21] Loytsyanskiy L.G. Mekhanika zhidkosti i gaza [Fluid mechanics]. Moscow, Drofa Publ., 2003. 840 p.

[22] Samarskiy A.A. Vvedenie v teoriyu raznostnykh skhem [Introduction into the theory of difference schemes]. Moscow, Nauka Publ., 1989. 616 p.