Interaction of rigid body motion and rarefied gas dynamics based on the BGK model

Sudarshan Tiwari; Axel Klar; Giovanni Russo; Sudarshan Tiwari; Axel Klar; Giovanni Russo

doi:10.3934/mine.2020010

Mathematics in Engineering

2020, Volume 2, Issue 2: 203-229. doi: 10.3934/mine.2020010

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Interaction of rigid body motion and rarefied gas dynamics based on the BGK model

1.
Department of Mathematics, Technische Universität Kaiserslautern, Erwin-Schrödinger-Straße, 67663 Kaiserslautern, Germany
2.
Fraunhofer ITWM, Fraunhoferplatz 1, 67663 Kaiserslautern, Germany
3.
Department of Mathematics and Computer Science, University of Catania, Italy

Received: 22 July 2019 Accepted: 04 December 2019 Published: 31 December 2019

In this paper we present simulations of moving rigid bodies immersed in a rarefied gas. The rarefied gas is simulated by solving the Bhatnager-Gross-Krook (BGK) model for the Boltzmann equation. The Newton-Euler equations are solved to simulate the rigid body motion. The force and the torque on the rigid body is computed from the surrounded gas. An explicit Euler scheme is used for the time integration of the Newton-Euler equations. The BGK model is solved by the semi-Lagrangian method suggested by Russo & Filbet ^[22]. Due to the motion of the rigid body, the computational domain for the rarefied gas (and the interface between the rigid body and the gas domain) changes with respect to time. To allow a simpler handling of the interface motion we have used a meshfree method for the interpolation procedure in the semi-Lagrangian scheme. We have considered a one way, as well as a two-way coupling of rigid body and gas flow. We use diffuse reflection boundary conditions on the rigid body and also on the boundary of the computational domain. In one space dimension the numerical results are compared with analytical as well as with Direct Simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. In the two-dimensional case results are compared with DSMC simulations for the Boltzmann equation and with results obtained by other researchers. Several test problems and applications illustrate the versatility of the approach.
- rigid body motion,
- rarefied gas,
- kinetic equation,
- BGK model,
- meshfree method,
- semi-Lagrangian method
Citation: Sudarshan Tiwari, Axel Klar, Giovanni Russo. Interaction of rigid body motion and rarefied gas dynamics based on the BGK model[J]. Mathematics in Engineering, 2020, 2(2): 203-229. doi: 10.3934/mine.2020010

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Abstract

In this paper we present simulations of moving rigid bodies immersed in a rarefied gas. The rarefied gas is simulated by solving the Bhatnager-Gross-Krook (BGK) model for the Boltzmann equation. The Newton-Euler equations are solved to simulate the rigid body motion. The force and the torque on the rigid body is computed from the surrounded gas. An explicit Euler scheme is used for the time integration of the Newton-Euler equations. The BGK model is solved by the semi-Lagrangian method suggested by Russo & Filbet ^[22]. Due to the motion of the rigid body, the computational domain for the rarefied gas (and the interface between the rigid body and the gas domain) changes with respect to time. To allow a simpler handling of the interface motion we have used a meshfree method for the interpolation procedure in the semi-Lagrangian scheme. We have considered a one way, as well as a two-way coupling of rigid body and gas flow. We use diffuse reflection boundary conditions on the rigid body and also on the boundary of the computational domain. In one space dimension the numerical results are compared with analytical as well as with Direct Simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. In the two-dimensional case results are compared with DSMC simulations for the Boltzmann equation and with results obtained by other researchers. Several test problems and applications illustrate the versatility of the approach.

References

[1]	Avesani D, Dumbser M, Bellin A (2014) A new class of Moving-Least-Squares WENO-SPH schemes. J Comput Phys 270: 278-299. doi: 10.1016/j.jcp.2014.03.041
[2]	Arslanbekov RR, Kolobov VI, Frolova AA (2011) Immersed Boundary Method for Boltzmann and Navier-Stokes Solvers with Adaptive Cartesian Mesh. AIP Conference Proceedings 1333: 873-877.
[3]	Babovsky H (1989) A convergence proof for Nanbu's Boltzmann simulation scheme. Eur J Mech 8: 41-55.
[4]	Baier T, Tiwari S, Shrestha S, et al. (2018) Thermophoresis of Janus particles at large Knudsen numbers. Phys Rev Fluids 3: 094202. doi: 10.1103/PhysRevFluids.3.094202
[5]	Bird G (1995) Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford: Clarendon Press.
[6]	Cercignani C, Illner R, Pulvirenti M (2013) The Mathematical Theory of Dilute Gases, Springer Science & Business Media.
[7]	Chapman S, Cowling TW (1970) The Mathematical Theory of Non-Uniform Gases, England: Cambridge University Press.
[8]	Chertock A, Coco A, Kurganov A, et al. (2018) A second-order finite-difference method for compressible fluids in domains with moving boundaries. Commun Comput Phys 23: 230-263.
[9]	Chu CK (1965) Kinetic-theoretic description of the formation of a shock wave. Phys Fluids 8: 12-22. doi: 10.1063/1.1761077
[10]	Dechristé G, Mieussens L (2012) Numerical simulation of micro flows with moving obstacles. Journal of Physics: Conference Series 362: 012030. doi: 10.1088/1742-6596/362/1/012030
[11]	Dechristé G, Mieussens L (2016) A Cartesian cut cell method for rarefied flow simulations around moving obstacles. J Comput Phys 314: 454-488.
[12]	Degond P, Dimarco G, Pareshi L (2011) The moment guided Monte Carlo method. Int J Numer Meth Fluids 67: 189-213. doi: 10.1002/fld.2345
[13]	Frangi A, Frezzotti A, Lorenzani S (2007) On the application of the BGK kinetic model to the analysis of gas-structure interactions in MEMS. Comput Struct 85: 810-817. doi: 10.1016/j.compstruc.2007.01.011
[14]	Groppi M, Russo G, Stracquadanio G (2007) High order semi-Lagrangian methods for the BGK equation. Commun Math Sci 14: 289-417.
[15]	Groppi M, Russo G, Stracquadanio G (2018) Semi-Lagrangian Approximation of BGK Models for Inert and Reactive Gas Mixtures, In: Goncalves P, Soares AJ, From Particle Systems to Partial Differential Equations, Springer.
[16]	Karniadakis G, Beskok A, Aluru N (2005) Microflows and Nanoflows: Fundamentals and Simulations, New York: Springer.
[17]	Kuhnert J (1999) General smoothed particle hydrodynamics, PhD Thesis, University of Kaiserslautern, Germany.
[18]	Li Q, Luo KH, Li XJ (2012) Forcing scheme in pseudopotential lattice Boltzmann model for multiphase flows. Phys Rev E 86: 278-299.
[19]	Neunzert H, Struckmeier J (1995) Particle methods for the Boltzmann equation. Acta Numer 4: 417-457. doi: 10.1017/S0962492900002579
[20]	Pareshi L, Russo G (2000) Numerical solution of the Boltzmann equation I: Spectrally accurate accurate approximation of the collision operator. SIAM J Numer Anal 37: 1217-1245. doi: 10.1137/S0036142998343300
[21]	Peskin CS (1972) Flow patterns around heart valves: A digital computer method for solving the equations of motion, PhD thesis, Albert Einstein College of Medicine.
[22]	Russo G, Filbet F (2009) Semi-Lagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics. Kinet Relat Mod, AIMS 2: 231-250. doi: 10.3934/krm.2009.2.231
[23]	Shan X, Chen H (1993) Latticeemph Boltzmann model for simulating flows with multiple phases and components. Phys Rev E 47: 1815-1819. doi: 10.1103/PhysRevE.47.1815
[24]	Shrestha S, Tiwari S, Klar A, et al. (2015) Numerical simulation of a moving rigid body in a rarefied gas. J Comput Phys 292: 239-252. doi: 10.1016/j.jcp.2015.03.030
[25]	Shrestha S, Tiwari S, Klar A (2015) Comparison of numerical simulations of the Boltzmann and the Navier-Stokes equations for a moving rigid circular body in a micro scaled cavity. Int J Adv Eng Sci Appl Math 7: 38-50. doi: 10.1007/s12572-015-0125-2
[26]	Sonar T (2005) Difference operators from interpolating moving least squares and their deviation from optimality. ESAIM: Math Model Num 39: 883-908. doi: 10.1051/m2an:2005039
[27]	Tiwari S, Klar A, Russo G (2019) A meshfree method for solving BGK model of rarefied gas dynamics. Int J Adv Eng Sci Appl Math 11: 187-197. doi: 10.1007/s12572-019-00254-5
[28]	Tiwari S, Klar A, Hardt S (2009) A particle-particle hybrid method for kinetic and continuum equations. J Comput Phys 228: 7109-7124. doi: 10.1016/j.jcp.2009.06.019
[29]	Tiwari S, Klar A, Hardt S (2013) Coupled solution of the Boltzmann and Navier-Stokes equations in gas-liquid two phase flow. Comput Fluids 71: 283-296. doi: 10.1016/j.compfluid.2012.10.018
[30]	Tsuji T, Aoki K (2013) Moving boundary problems for a rarefied gas: Spatially one dimensional case. J Comput Phys 250: 574-600. doi: 10.1016/j.jcp.2013.05.017
[31]	Tsuji T, Aoki K (2014) Gas motion in a microgap between a stationary plate and a plate oscillating in its normal direction. Microfluid Nanofluid 16: 1033-1045. doi: 10.1007/s10404-014-1374-2
[32]	Xiong T, Russo G, Qiu JM (2019) Conservative multi-dimensional semi-Lagrangian finite difference scheme: Stability and applications to the kinetic and fluid simulations, J Sci Comput 79: 1241-1270.
[33]	Yomsatieanku W (2010) High-order non-oscillatory schemes using meshfree interpolating moving least squares reconstruction for hyperbolic conservation laws, PhD Thesis, TU CaroloWilhelmina zu Braunschweig, Germany.

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