Numerical Simulation of Rarefied Flow Instabilities Using Kinetic Approach

Anton A. Shershnev; Alexey N. Kudryavtsev; Alexander V. Kashkovsky; Semyon P. Borisov; Sergey O. Poleshkin

doi:10.14529/jsfi240205

Authors

Anton A. Shershnev Khristianovich Institute of Theoretical and Applied Mechanics Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russian Federation
Alexey N. Kudryavtsev Khristianovich Institute of Theoretical and Applied Mechanics Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russian Federation
Alexander V. Kashkovsky Khristianovich Institute of Theoretical and Applied Mechanics Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russian Federation
Semyon P. Borisov Khristianovich Institute of Theoretical and Applied Mechanics Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russian Federation
Sergey O. Poleshkin Khristianovich Institute of Theoretical and Applied Mechanics Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russian Federation

DOI:

https://doi.org/10.14529/jsfi240205

Keywords:

DSMC method, the Boltzmann transport equation, model kinetic equations, mixing layer, plane jet, the Rayleigh–Taylor instability, the Richtmyer–Meshkov instability

Abstract

The results of numerical investigations of hydrodynamic instabilities in rarefied flows obtained using the kinetic approach are presented and discussed. The rarefied flow instabilities are simulated using both statistical and deterministic methods to solve kinetic equations for the velocity distribution function. The Direct Simulation Monte Carlo method is employed for statistical modeling of rarefied flows while the deterministic methods used are high-order finite-difference schemes for solving the Boltzmann transport equation and model kinetic equations in a multidimensional phase space. Numerical solvers used for the simulations are run on multiple-GPU clusters using CUDA platform and MPI communication interface. The development of instabilities in compressible free shear flows such as a mixing layer between two parallel streams and a plane jet in a coflow is considered. The Rayleigh–Taylor and Richtmyer–Meshkov instabilities induced by external body forces are also simulated. The results of kinetic simulations are compared with data from the linear stability theory and Navier–Stokes computations and it is shown that good agreement is observed between kinetic and continuum approaches.

References

Rapaport, D.C.: Molecular-dynamics study of Rayleigh–Bénard convection. Phys. Rev. Lett. 60(24), 2480–2483 (1988). https://doi.org/10.1103/physrevlett.60.2480

Puhl, A., Malek Mansour, M., Marechal, M.: Quantitative comparison of molecular dynamics with hydrodynamics in Rayleigh–Bénard convection. Phys. Rev. A 40(4), 1999–2012 (1989). https://doi.org/10.1103/PhysRevA.40.1999

Stefanov, S., Cercignani, C.: Monte Carlo simulation of Bénard’s instability in a rarefied gas. Eur. J. Mech. B/Fluids 11(5), 543–554 (1992).

Golshtein, E., Elperin, T.: Convective instabilities in rarefied gases by Direct Simulation Monte Carlo method. J. Thermophys. Heat Transfer 10(2), 250–256 (1996). https://doi.org/10.2514/3.782

Stefanov, S., Cercignani, C.: Monte Carlo simulation of the Taylor–Couette flow of a rarefied gas. J. Fluid Mech. 256, 199–213 (1993). https://doi.org/10.1017/S0022112093002769

Reichelman, D., Nanbu, K.: Monte Carlo direct simulation of the Taylor instability in a rarefied gas. Phys. Fluids A 5, 2585–2587 (1993). https://doi.org/10.1063/1.858775

Hirshfeld, D., Rapaport, D.C.: Molecular dynamics simulation of Taylor–Couette vortex formation. Phys. Rev. Lett. 80(24), 5337–5340 (1998). https://doi.org/10.1103/PhysRevLett.80.5337

Aristov, V.V., Zabelok, S.A.: Vortex structure of the unstable jet studied on the basis of the Boltzmann equation. Matem. Modelirovanie 13(6), 87–92 (2001, in Russian).

Rovenskaya, O.I., Aristov, V.V., Farkhutdinov, T.I.: Invesigation of the Taylor–Görtler instability in jets with help of kinetic approach. Matematika i Matem. Modelirovanie 01, 38–54 (2016, in Russian). https://doi.org/10.7463/mathm.0116.0833621

Yang, J.-Y., Chang, J.-W.: Rarefied flow instability simulation using model Boltzmann equations. AIAA Paper No. 97-2017 (1997). https://doi.org/10.2514/6.1997-2017

Kadau, K., Germann, T.C., Hadjiconstantinou, N.G., et al.: Nanohydrodynamics simulations: an atomistic view of the Rayleigh–Taylor instability. Proc. Nat. Acad. Sci. 101(6), 5851–5855 (2004). https://doi.org/10.1073/pnas.0401228101

Kadau, K., Barber, J.L., Germann, T.C, et al.: Atomistic methods in fluid simulation. Phil. Trans. R. Soc. A (368), 1547–1560 (2010). https://doi.org/10.1098/rsta.2009.0218

Gallis, M.A., Koehler, T.P., Torczynski, J.R., Plimpton, S.J.: Direct simulation Monte Carlo investigation of the Richtmyer–Meshkov instability. Phys. Fluids 27(8), 084105 (2015). https://doi.org/10.1063/1.4928338

Gallis, M.A., Koehler, T.P., Torczynski, J.R., Plimpton, S.J.: Direct simulation Monte Carlo investigation of the Rayleigh–Taylor instability. Phys. Rev. Fluids 1(4), 043403 (2016). https://doi.org/10.1103/PhysRevFluids.1.043403

Kudryavtsev, A.N., Kashkovsky, A.V., Shershnev, A.A.: Investigation of Kelvin–Helmholtz instability with the DSMC method. AIP Conf. Proc. 2125, 030028 (2019). https://doi.org/10.1063/1.5117410

Kashkovsky, A.V., Kudryavtsev, A.N., Shershnev, A.A.: Numerical simulation of the Rayleigh–Taylor instability in rarefied Ar/He mixture using the Direct Simulation Monte Carlo method. J. Phys. Conf. Ser. 1382, 012154 (2019). https://doi.org/10.1088/1742-6596/1382/1/012154

Kashkovsky, A.V., Kudryavtsev, A.N., Shershnev, A.A.: Direct Monte Carlo simulation of development of the Richtmyer–Meshkov on the Ar/He interface. J. Phys. Conf. Ser. 1404, 012109 (2019). https://doi.org/10.1088/1742-6596/1404/1/012109

Kashkovsky, A.V., Kudryavtsev, A.N., Shershnev, A.A.: DSMC simulation of instability of plane supersonic jet in coflow. AIP Conf. Proceedings 2504, 030087 (2023). https://doi.org/10.1063/5.0132264

Kashkovsky, A.V., Kudryavtsev, A.N., Shershnev, A.A.: Numerical simulation of the Rayleigh–Taylor instability in rarefied mixture of monatomic gases using continuum and kinetic approaches. AIP Conf. Proceedings 2504, 030111 (2023). https://doi.org/10.1063/5.0132266

Kudryavtsev, A.N, Epstein, D.B.: Richtmyer–Meshkov instability in a rarefied gas, results of continuum and kinetic numerical simulations. AIP Conf. Proceedings 2125, 020010 (2019). https://doi.org/10.1063/1.5117370

Kudryavtsev, A.N., Poleshkin, S.O., Shershnev, A.A.: Numerical simulation of the instability development in a compressible mixing layer using kinetic and continuum approaches. AIP Conf. Proc. 2125, 030033 (2019). https://doi.org/10.1063/1.5117415

Poleshkin, S.O., Kudryavtsev, A.N.: Kinetic simulation of the Rayleigh–Taylor instability. AIP Conf. Proc. 2288, 030010 (2020). https://doi.org/10.1063/5.0028881

Malkov, E.A., Kudryavtsev, A.N.: Non-stationary Antonov self-gravitating layer. MNRAS 491(3), 3952–3966 (2020). https://doi.org/10.1093/mnras/stz3276

Ivanov, M.S., Kashkovsky, A.V., Vashchenkov, P.V., Bondar, Ye.A.: Parallel object-oriented software system for DSMC modeling of high-altitude aerothermodynamic problems. AIP Conf. Proceedings 1333, 211–218 (2011). https://doi.org/10.1063/1.3562650

Kashkovsky, A.V.: 3D DSMC computations on a heterogeneous CPU-GPU cluster with a large number of GPUs. AIP Conf. Proceedings 1628(1), 192–198 (2014) https://doi.org/10.1063/1.4902592

Malkov, E.A., Bondar, Ye.A., Kokhanchik, A.A., et al.: High-accuracy deterministic solution of the Boltzmann equation for the shock wave structure. Shock Waves 25(4), 387–397 (2015). https://doi.org/10.1007/s00193-015-0563-6

Malkov, E.A., Poleshkin, S.O., Kudryavtsev, A.N., Shershnev, A.A.: Numerical approach for solving kinetic equations in two-dimensional case on hybrid computational clusters. AIP Conf. Proceedings 1770, 030077 (2016). https://doi.org/10.1063/1.4964019

Bhantnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94(3), 511–525 (1954). https://doi.org/10.1103/PhysRev.94.511

Holway, L.H.: New statistical models for kinetic theory: methods of construction. Phys. Fluids 9(9), 1658–1673 (1966). http://doi.org/10.1063/1.1761920

Shakhov, E.M.: Approximate kinetic equations in rarefied gas theory. Fluid Dynamics 3(1), 112–115 (1968). http://doi.org/10.1007/BF01016254

Kudryavtsev, A.N., Shershnev, A.A.: Numerical simulation of microflows using direct solving of kinetic equations with WENO schemes. J. Sci. Comp. 57(1), 42–73 (2013). http://doi.org/10.1007/s10915-013-9694-z

Shershnev, A.A., Kudryavtsev, A.N., Kashkovsky, A.V., et al.: A numerical code for a wide range of compressible flows on hybrid computational architectures. Supercomputing Frontiers and Innovations 9(4), 85–99 (2022). https://doi.org/10.14529/jsfi220408

Ho, C.-M., Huerre, P.: Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424 (1984). https://doi.org/10.1146/annurev.fl.16.010184.002053

Kashkovsky, A.V., Kudryavtsev, A.N., Shershnev, A.A.: Development of compressible mixing layer instability simulated using the Direct Simulation Monte Carlo method. Supercomputing Frontiers and Innovations 11(2), 48–64 (2024). https://doi.org/10.14529/jsfi240204

Kelly, R.E.: On the stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27(4), 657–689 (1967). https://doi.org/10.1017/S0022112067002538

Watanabe, D., Maekawa, H.: Transition of supersonic plane jet due to symmetric/antisymmetric unstable modes. J. Turbulence 3, 047 (2002). https://doi.org/10.1088/1468-5248/3/1/047

Zhou, Y.: Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720–722, 1–136 (2017). https://doi.org/10.1016/j.physrep.2017.07.005

Zhou, Y.: Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723–725, 1–160 (2017). https://doi.org/10.1016/j.physrep.2017.07.008

Brouillette, M.: The Richtmyer–Meshkov instability. Ann. Rev. Fluid Mech. 34, 445–468 (2002). https://doi.org/10.1146/annurev.fluid.34.090101.162238