Boundary conditions for simulation of compressible gas by SPH method

A. S. Vasyura ; M. A. Butenko ; N. M. Kuzmin

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 4 (2014), pp. 53-67.

Résumé

The establishing of boundary conditions during numerical hydrodynamic modeling with use of Smoothed Particles Hydrodynamics (SPH) method is described. Three types of boundary conditions are described: solid wall, flowing and boundary substance — vacuum. For solid wall modeling at the border of area Monaghan virtual particles and Morris virtual particles are used. Monaghan virtual particles are particles placed at the border of the computational area in a single layer. Their parameters are unchangeable with time. And between virtual particles and gas particles some interaction potential is set, the gradient of which is included into rate equation as additional repulsive force. Morris particles are placed outside the computational area and at its border in several layers. These particles change their properties and participate in calculations together with gas particles but their location is unchangeable. If the task presumes occurrence liquid or gas flow over the border, then it is necessary to set boundary condition of type «flowing». To simulate flowing it is necessary to calculate the particle's behavior at the moment when it leaves the boundaries of computation area. If a particle leaves the border of computational area for a distance of $\delta x$, then this particle should be initialized by the initial parameters values and it should be located at the area of particles inflow at the distance of $\delta x$ from the border of computational area. This method allows simulating liquid flow and does not allow particles to leave the boundaries of computational area. Boundary substance — vacuum in computing by SPH method is not needed in special description unlike Eulerian methods. Because of the ideology of the method such boundary occurs naturally: particles at the border lack neighbor particles and interaction between them stops naturally — they leave continuous medium. Because of this peculiarity SPH-method is widely used for solving a wide class of astrophysical tasks, in which the method does not need additional calculation of boundary conditions and at the same time provides proper result. Two-dimensional SPH algorithm was implemented and tested. On basis of it we carried out modeling for the purpose of examination of correctness of setting of boundary conditions of different types. The new result of our research is a nearest neighbor particles search algorithm for simulating of gas flowing over the border in that specific case of equality of gas inflow into computational area and gas outflow from it.

Détail

Keywords: smoothed particles hydrodynamics, numerical simulation, boundary conditions, search algorithm of neighboring particles.
Mots-clés : hydrodynamic

Bibliographie

[1] N.\;S. Bahvalov, N.\;P. Zhidkov, G.\;M. Kobelkov, The numerical methods, Nauka Publ., M., 1987, 640 pp.

[2] A.\;G. Kulikovsky, N.\;V. Pogorelov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Fizmatlit Publ., M., 2001, 608 pp.

[3] L.\;G. Loytsyanskiy, Fluid Mechanics: a textbook for high schools, Ucheb. dlya vuzov, Nauka Publ., M., 1987, 153 pp.

[4] A.\;G. Morozov, A.\;V. Khoperskov, Physics disks, Izd-vo VolGU Publ., Volgograd, 2005, 423 pp.

[5] L.\;I. Sedov, Continuum Mechanics, Nauka Publ., M., 1978, 560 pp.

[6] A.\;V. Khoperskov, M.\;A. Eremin, S.\;A. Khoperskov, M.\;A. Butenko, A.\;G. Morozov, “Dynamics of the gas disk in the dark halo axisymmetrical”, Astronomy Reports, 89:1 (2012), 19–31

[7] G. Falkovich, Fluid Mechanics: A Short Course for Physicists, Cambridge University Press, Cambridge, 2011, 32 pp.

[8] R.\;A. Gingold, J.\;J. Monaghan, “Smoothed Particle Hydrodynamics: Theory and Application to Non-Spherical Stars”, Monthly Notices of the Royal Astronomical Society, 181:3 (1977), 375–389

[9] M.\;B. Liu, G.\;R. Liu, K.\;Y. Lam, “Constructing smoothing functions in smoothed particle hydrodynamics with applications”, Journal of Computational and Applied Mathematics, 155:2 (2003), 263–284

[10] G.\;R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, 2003, 388–406

[11] J.\;J. Monaghan, “Smoothed Particles Hydrodynamics”, Annual Review of Astronomy and Astrophysics, 30 (1992), 543–574

[12] J.\;J. Monaghan, J.\;C. Lattanzio, “A refined particle method for astrophysical problems”, Annual Review of Astronomy and Astrophysics, 149 (1985), 135–143

[13] J.\;P. Morris, P.\;J. Fox, Y. Zhu, “Modelling Low Reynolds Number Incompressible Flows Using SPH”, Comp. Physics, 136:1 (1997), 214–266

[14] M. Müller, D. Charypar, M. Gross, “Particle-Based Fluid Simulation for Interactive Applications”, Proceedings of the 2003 ACM SIGGRAPH/Eurographics symposium on Computer animation, 2003, 154–159

[15] J. von Neumann, R.\;D. Richtmyer, “A method for the numerical calculation of hydrodynamics shocks”, Journal of Applied Physics, 21:3 (1950), 232–237

[16] Harada Takahiro, Koshizuka Seiichi, Kawaguchi Yoichiro, “Improvement of the boundary conditions in Smoothed Particle Hydrodynamics”, Computer Graphics and Geometry, 9:3 (2007), 2–15