A discrete unified gas kinetic scheme on unstructured grids for viscid compressible flows and its parallel algorithm

Lei Xu; Zhengzheng Yan; Rongliang Chen; Lei Xu; Zhengzheng Yan; Rongliang Chen

doi:10.3934/math.2023443

AIMS Mathematics

2023, Volume 8, Issue 4: 8829-8846. doi: 10.3934/math.2023443

Previous Article Next Article

Research article Special Issues

A discrete unified gas kinetic scheme on unstructured grids for viscid compressible flows and its parallel algorithm

Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China

Received: 10 October 2022 Revised: 09 January 2023 Accepted: 12 January 2023 Published: 08 February 2023
MSC : 26A33, 34B10, 34B15

In this paper, we present a discrete unified gas kinetic scheme (DUGKS) on unstructured grids for high-speed viscid compressible flows on the basis of double distribution function (the density and the total energy distribution functions) Boltzmann-BGK equations. In the DUGKS, the discrete equilibrium distribution functions are constructed based on a D2Q17 circular function. In order to accelerate the simulation, we also illustrate a corresponding parallel algorithm. The DUGKS is validated by two benchmark problems, i.e., flows around the NACA0012 airfoil and flows past a circular cylinder with the Mach numbers range from 0.5 to 2.5. Good agreements with the referenced results are observed from the numerical results. The results of parallel test indicate that the DUGKS is highly parallel scalable, in which the parallel efficiency achieves $ 93.88\% $ on a supercomputer using up to $ 4800 $ processors. The proposed method can be utilized for high-resolution numerical simulation of complex and high Mach number flows.
- discrete unified gas kinetic scheme,
- unstructured grid,
- parallel algorithm,
- viscid compressible flow,
- Boltzmann equation
Citation: Lei Xu, Zhengzheng Yan, Rongliang Chen. A discrete unified gas kinetic scheme on unstructured grids for viscid compressible flows and its parallel algorithm[J]. AIMS Mathematics, 2023, 8(4): 8829-8846. doi: 10.3934/math.2023443

Related Papers:

Abstract

In this paper, we present a discrete unified gas kinetic scheme (DUGKS) on unstructured grids for high-speed viscid compressible flows on the basis of double distribution function (the density and the total energy distribution functions) Boltzmann-BGK equations. In the DUGKS, the discrete equilibrium distribution functions are constructed based on a D2Q17 circular function. In order to accelerate the simulation, we also illustrate a corresponding parallel algorithm. The DUGKS is validated by two benchmark problems, i.e., flows around the NACA0012 airfoil and flows past a circular cylinder with the Mach numbers range from 0.5 to 2.5. Good agreements with the referenced results are observed from the numerical results. The results of parallel test indicate that the DUGKS is highly parallel scalable, in which the parallel efficiency achieves $ 93.88\% $ on a supercomputer using up to $ 4800 $ processors. The proposed method can be utilized for high-resolution numerical simulation of complex and high Mach number flows.

References

[1]	Y. Deng, J. Li, H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Rational Mech. Anal., 231 (2019), 153–187. https://doi.org/10.1007/s00205-018-1276-7 doi: 10.1007/s00205-018-1276-7
[2]	Y. Deng, J. Li, H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Rational Mech. Anal., 235 (2020), 691–721. https://doi.org/10.1007/s00205-019-01429-x doi: 10.1007/s00205-019-01429-x
[3]	Y. Deng, H. Liu, W.-Y. Tsui, Identifying varying magnetic anomalies using geomagnetic monitoring, Discrete Contin. Dyn. Syst., 40 (2020), 6411–6440. https://doi.org/10.3934/dcds.2020285 doi: 10.3934/dcds.2020285
[4]	Y. Deng, Y. Gao, J. Li, H. Liu, R. Chen, Locating multiple magnetized anomalies by geomagnetic monitoring, unpublished work.
[5]	W. Li, S. Liu, S. Osher, Controlling conservation laws II: Compressible Navier-Stokes equations, J. Comput. Phys., 463 (2022), 111264. https://doi.org/10.1016/j.jcp.2022.111264 doi: 10.1016/j.jcp.2022.111264
[6]	M. Natarajan, R. Grout, W. Zhang, M. Day, A moving embedded boundary approach for the compressible Navier-Stokes equations in a block-structured adaptive refinement framework, J. Comput. Phys., 465 (2022), 111315. https://doi.org/10.1016/j.jcp.2022.111315 doi: 10.1016/j.jcp.2022.111315
[7]	G. Ju, C. Chen, R. Chen, J. Li, K. Li, S. Zhang, Numerical simulation for 3D flow in flow channel of aeroengine turbine fan based on dimension splitting method, Electron. Res. Arch., 28 (2020), 837–851. https://doi.org/10.3934/era.2020043 doi: 10.3934/era.2020043
[8]	M. E. Danis, J. Yan, A new direct discontinuous Galerkin method with interface correction for two-dimensional compressible Navier-Stokes equations, J. Comput. Phys., 452 (2022), 110904. https://doi.org/10.1016/j.jcp.2021.110904 doi: 10.1016/j.jcp.2021.110904
[9]	Z. Qiao, X. Yang, A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model, Electron. Res. Arch., 28 (2020), 1207–1225. https://doi.org/10.3934/era.2020066 doi: 10.3934/era.2020066
[10]	K. Xu, A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (2001), 289–335. https://doi.org/10.1006/jcph.2001.6790 doi: 10.1006/jcph.2001.6790
[11]	C. Shu, Y. Wang, C. J. Teo, J. Wu, Development of lattice Boltzmann flux solver for simulation of incompressible flows, Adv. Appl. Math. Mech., 6 (2014), 436–460. https://doi.org/10.4208/aamm.2014.4.s2 doi: 10.4208/aamm.2014.4.s2
[12]	Z. H. Li, H. X. Zhang, Gas-kinetic numerical studies of three-dimensional complex flows on spacecraft re-entry, J. Comput. Phys., 228 (2009), 1116–1138. https://doi.org/10.1016/j.jcp.2008.10.013 doi: 10.1016/j.jcp.2008.10.013
[13]	K. Xu, J. C. Huang, A unified gas-kinetic scheme for continuum and rarefied flows, J. Comput. Phys., 229 (2010), 7747–7764. https://doi.org/10.1016/j.jcp.2010.06.032 doi: 10.1016/j.jcp.2010.06.032
[14]	Z. Guo, K. Xu, R. Wang, Discrete unified gas kinetic scheme for all Knudsen number flows: low-speed isothermal case, Phys. Rev. E, 88 (2013), 033305. https://doi.org/10.1103/PhysRevE.88.033305 doi: 10.1103/PhysRevE.88.033305
[15]	A. U. Shirsat, S. G. Nayak, D. V. Patil, Simulation of high-Mach-number inviscid flows using a third-order Runge-Kutta and fifth-order WENO-based finite-difference lattice Boltzmann method, Phys. Rev. E, 106 (2022), 025314. https://doi.org/10.1103/PhysRevE.106.025314 doi: 10.1103/PhysRevE.106.025314
[16]	J. Huang, X.-C. Cai, C. Yang, A fully implicit method for lattice Boltzmann equations, SIAM J. Sci. Comput., 37 (2015), S291–S313. https://doi.org/10.1137/140975346 doi: 10.1137/140975346
[17]	J. Huang, C. Yao, X.-C. Cai, A nonlinearly preconditioned inexact Newton algorithm for steady state lattice Boltzmann equations, SIAM J. Sci. Comput., 38 (2015), A1701–A1724. https://doi.org/10.1137/15M1028078 doi: 10.1137/15M1028078
[18]	R. Matin, M. K. Misztal, A. Hernandez-Garcia, J. Mathiesen, Finite element lattice Boltzmann simulations of contact line dynamics, Phys. Rev. E, 97 (2018), 013307. https://doi.org/10.1103/PhysRevE.97.013307 doi: 10.1103/PhysRevE.97.013307
[19]	J. Wu, M. Shen, C. Liu, Study of flow over object problems by a nodal discontinuous Galerkin-lattice Boltzmann method, Phys. Fluids, 30 (2018), 040903. https://doi.org/10.1063/1.5010964 doi: 10.1063/1.5010964
[20]	L. Xu, J. Li, R. Chen, A scalable parallel unstructured finite volume lattice Boltzmann method for three-dimensional incompressible flow simulations, Int. J. Numer. Methods Fluids, 93 (2021), 2744–2762. https://doi.org/10.1002/fld.4996 doi: 10.1002/fld.4996
[21]	L. Xu, R. Chen, Scalable parallel finite volume lattice Boltzmann method for thermal incompressible flows on unstructured grids, Int. J. Heat Mass Tran., 160 (2020), 120156. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120156 doi: 10.1016/j.ijheatmasstransfer.2020.120156
[22]	L. Xu, R. Chen, X.-C. Cai, Parallel finite-volume discrete Boltzmann method for inviscid compressible flows on unstructured grids, Phys. Rev. E, 103 (2021), 023306. https://doi.org/10.1103/PhysRevE.103.023306 doi: 10.1103/PhysRevE.103.023306
[23]	C. Sun, A. T. Hsu, Three-dimensional lattice Boltzmann model for compressible flows, Phys. Rev. E, 68 (2003), 016303. https://doi.org/10.1103/PhysRevE.68.016303 doi: 10.1103/PhysRevE.68.016303
[24]	K. Qu, C. Shu, Y. T. Chew, Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number, Phys. Rev. E, 75 (2007), 036706. https://doi.org/10.1103/PhysRevE.75.036706 doi: 10.1103/PhysRevE.75.036706
[25]	K. Li, C. Zhong, A lattice Boltzmann model for simulation of compressible flows, Int. J. Numer. Methods Fluids, 77 (2015), 334–357. https://doi.org/10.1002/fld.3984 doi: 10.1002/fld.3984
[26]	K. Qu, Development of lattice Boltzmann method for compressible flows, Ph.D thesis, National University of Singapore, Singapore, 2008.
[27]	R.-F. Qiu, C.-X. Zhu, R.-Q. Chen, J.-F. Zhu, Y.-C. You, A double-distribution-function lattice Boltzmann model for high-speed compressible viscous flows, Comput. Fluids, 166 (2018), 24–31. https://doi.org/10.1016/j.compfluid.2018.01.039 doi: 10.1016/j.compfluid.2018.01.039
[28]	Z. Liu, R. Chen, L. Xu, Parallel unstructured finite volume lattice Boltzmann method for high-speed viscid compressible flows, International Journal of Modern Physics C, 33 (2022), 2250066. https://doi.org/10.1142/S0129183122500668 doi: 10.1142/S0129183122500668
[29]	L. Zhu, S. Chen, Z. Guo, dugksFoam: An open source OpenFOAM solver for the Boltzmann model equation, Comput. Phys. Commun., 213 (2017), 155–164. https://doi.org/10.1016/j.cpc.2016.11.010 doi: 10.1016/j.cpc.2016.11.010
[30]	Z. Guo, K. Xu, Discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon Boltzmann transport equation, Int. J. Heat Mass Tran., 102 (2016), 944–958. https://doi.org/10.1016/j.ijheatmasstransfer.2016.06.088 doi: 10.1016/j.ijheatmasstransfer.2016.06.088
[31]	P. Wang, S. Tao, Z. Guo, A coupled discrete unified gas-kinetic scheme for Boussinesq flows, Comput. Fluids, 120 (2015), 70–81. https://doi.org/10.1016/j.compfluid.2015.07.012 doi: 10.1016/j.compfluid.2015.07.012
[32]	P. Wang, Y. Zhang, Z. Guo, Numerical study of three-dimensional natural convection in a cubical cavity at high Rayleigh numbers, Int. J. Heat Mass Tran., 113 (2017), 217–228. https://doi.org/10.1016/j.ijheatmasstransfer.2017.05.057 doi: 10.1016/j.ijheatmasstransfer.2017.05.057
[33]	H. Liu, M. Kong, Q. Chen, L. Zheng, Y. Cao, Coupled discrete unified gas kinetic scheme for the thermal compressible flows in all Knudsen number regimes, Phys. Rev. E, 98 (2018), 053310. https://doi.org/10.1103/PhysRevE.98.053310 doi: 10.1103/PhysRevE.98.053310
[34]	Y. Wang, C. Zhong, S. Liu, Arbitrary Lagrangian-Eulerian-type discrete unified gas kinetic scheme for low-speed continuum and rarefied flow simulations with moving boundaries, Phys. Rev. E, 100 (2019), 063310. https://doi.org/10.1103/PhysRevE.100.063310 doi: 10.1103/PhysRevE.100.063310
[35]	Y. Zhang, L. Zhu, P. Wang, Z. Guo, Discrete unified gas kinetic scheme for flows of binary gas mixture based on the McCormark model, Phys. Fluids, 31 (2019), 017101. https://doi.org/10.1063/1.5063846 doi: 10.1063/1.5063846
[36]	J. Shang, Z. Chai, H. Wang, B. Shi, Discrete unified gas kinetic scheme for nonlinear convection-diffusion equations, Phys. Rev. E, 101 (2020), 023306. https://doi.org/10.1103/PhysRevE.101.023306 doi: 10.1103/PhysRevE.101.023306
[37]	M. Zhong, S. Zou, D. Pan, C. Zhuo, C. Zhong, A simplified discrete unified gas kinetic scheme for incompressible flow, Phys. Fluids, 32 (2020), 093601. https://doi.org/10.1063/5.0021332 doi: 10.1063/5.0021332
[38]	M. Zhong, S. Zou, D. Pan, C. Zhuo, C. Zhong, A simplified discrete unified gas–kinetic scheme for compressible flow, Phys. Fluids, 33 (2021), 036103. https://doi.org/10.1063/5.0033911 doi: 10.1063/5.0033911
[39]	G. Karypis, K. Schloegel, PARMETIS: Parallel graph partitioning and sparse matrix ordering library version 4.0, Technical Report, 97-060.
[40]	S. Balay, S. Abhyankar, M. F. Adams, S. Benson, J. Brown, P. Brune, et al., PETSc/TAO users manual, Argonne National Laboratory, ANL-21/39 - Revision 3.17, 2022. Available from: https://petsc.org/
[41]	Z. Guo, C. Zheng, B. Shi, T. S. Zhao, Thermal lattice Boltzmann equation for low Mach number flows: decoupling model, Phys. Rev. E, 75 (2007), 036704. https://doi.org/10.1103/PhysRevE.75.036704 doi: 10.1103/PhysRevE.75.036704
[42]	V. Venkatakrishnan, Convergence to steady-state solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys., 118 (1995), 120–130. https://doi.org/10.1006/jcph.1995.1084 doi: 10.1006/jcph.1995.1084
[43]	Z. Guo, C. Zheng, B. Shi, Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method, Chinese Phys., 11 (2002), 366–375. https://doi.org/10.1088/1009-1963/11/4/310 doi: 10.1088/1009-1963/11/4/310
[44]	F. Bassi, S. Rebay, A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations, J. Comput. Phys., 131 (1997), 267–279. https://doi.org/10.1006/jcph.1996.5572 doi: 10.1006/jcph.1996.5572
[45]	P. A. Forsyth, H. Jiang, Nonlinear iteration methods for high speed laminar compressible Navier-Stokes equations, Comput. Fluids, 26 (1997), 249–279. https://doi.org/10.1016/S0045-7930(96)00041-2 doi: 10.1016/S0045-7930(96)00041-2
[46]	Q. Li, Y. L. He, Y. J. Gao, Implementation of finite-difference lattice Boltzmann method on general body-fitted curvilinear coordinates, International Journal of Modern Physics C, 19 (2008), 1581–1595. https://doi.org/10.1142/S0129183108013126 doi: 10.1142/S0129183108013126
[47]	M. D. De Tullio, P. De Palma, G. Iaccarino, G. Pascazio, M. Napolitano, An immersed boundary method for compressible flows using local grid refinement, J. Comput. Phys., 225 (2007), 2098–2117. https://doi.org/10.1016/j.jcp.2007.03.008 doi: 10.1016/j.jcp.2007.03.008
[48]	X. Liao, L. Xiao, C. Yang, Y. Lu, MilkyWay-2 supercomputer: system and application, Front. Comput. Sci., 8 (2014), 345–356. https://doi.org/10.1007/s11704-014-3501-3 doi: 10.1007/s11704-014-3501-3
[49]	Y. T. Chow, Y. Deng, Y. He, H. Liu, X. Wang, Surface-localized transmission eigenstates, super-resolution imaging, and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), 946–975. https://doi.org/10.1137/20M1388498 doi: 10.1137/20M1388498
[50]	Z. Bai, H. Diao, H. Liu, Q. Meng, Stable determination of an elastic medium scatterer by a single far-field measurement and beyond, Calc. Var., 61 (2022), 170. https://doi.org/10.1007/s00526-022-02278-5 doi: 10.1007/s00526-022-02278-5
[51]	H. Liu, On local and global structures of transmission eigenfunctions and beyond, J. Inverse Ill-Posed Probl., 30 (2022), 287–305. https://doi.org/10.1515/jiip-2020-0099 doi: 10.1515/jiip-2020-0099
[52]	Y. Gao, H. Liu, X. Wang, K. Zhang, On an artificial neural network for inverse scattering problems, J. Comput. Phys., 448 (2022), 110771. https://doi.org/10.1016/j.jcp.2021.110771 doi: 10.1016/j.jcp.2021.110771
[53]	E. L. K. Blasten, H. Liu, Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems, SIAM J. Math. Anal., 53 (2021), 3801–3837. https://doi.org/10.1137/20M1384002 doi: 10.1137/20M1384002

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)