Element contents
A Unified Computational Fluid Dynamics Framework from Rarefied to Continuum Regimes
Published online by Cambridge University Press: 13 May 2021
Summary
This Element presents a unified computational fluid dynamics framework from rarefied to continuum regimes. The framework is based on the direct modelling of flow physics in a discretized space. The mesh size and time step are used as modelling scales in the construction of discretized governing equations. With the variation-of-cell Knudsen number, continuous modelling equations in different regimes have been obtained, and the Boltzmann and Navier-Stokes equations become two limiting equations in the kinetic and hydrodynamic scales. The unified algorithms include the discrete velocity method (DVM)–based unified gas-kinetic scheme (UGKS), the particlebased unified gas-kinetic particle method (UGKP), and the wave and particle–based unified gas-kinetic wave-particle method (UGKWP). The UGKWP is a multi-scale method with the particle for non-equilibrium transport and wave for equilibrium evolution. The particle dynamics in the rarefied regime and the hydrodynamic flow solver in the continuum regime have been unified according to the cell's Knudsen number.
Keywords
- Type
- Element
- Information
- Online ISBN: 9781108877534Publisher: Cambridge University PressPrint publication: 10 June 2021
References
Alexander, F.J., Garcia, A.L., & Alder, B.J. (1998). Cell size dependence of transport coefficients in stochastic particle algorithm. Phys. Fluids, 10, 1540–1542.CrossRefGoogle Scholar
Aristov, V. (2012). Direct methods for solving the Boltzmann equation and study of nonequilibrium flows. Springer Science & Business Media.Google Scholar
Bhatnagar, P.L., Gross, E.P., & Krook, M. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 (3), 511.Google Scholar
Bird, G. (1970). Aspects of the structure of strong shock waves. Phys. Fluids 13, 1172–1177.CrossRefGoogle Scholar
Bird, G. (1994). Molecular gas dynamics and the direct simulation of gas flows. Oxford Science Publications.CrossRefGoogle Scholar
Burt, J., Josyula, E., Deschenes, T., & Boyd, I. (2011). Evaluation of a hybrid Boltzmann-continuum method for high-speed nonequilibrium flows. J. Thermophys. Heat Trans. 25, 500–515.CrossRefGoogle Scholar
Chacon, L., Chen, G., Knoll, D., Newman, C., Park, H., Taitano, W., Willert, J., & Womeldorff, G. (2017). Multiscale high-order/low-order (HOLO) algorithms and applications. J. Comput. Phys. 330, 21–45.CrossRefGoogle Scholar
Chapman, S., Cowling, T.G., & Burnett, D. (1990). The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press.Google Scholar
Chen, S., & Doolen, G. (1998). Lattice Boltzmann method for fluid flows. Ann. Rev. Fluid Mech. 30, 329–364.CrossRefGoogle Scholar
Chen, S.Z., Xu, K., & Cai, Q. (2015). A comparison and unification of ellipsoidal statistical and Shakhov BGK. Adv. Appl. Math. Mech. 7, 245–266.Google Scholar
Chen, S.Z., Xu, K., Lee, C.B., & Cai, Q. (2012). A unified gas kinetic scheme with moving mesh and velocity space adaptation. J. Comput. Phys. 231 (20), 6643–6664.CrossRefGoogle Scholar
Chen, S.Z, Zhang, C., Zhu, L., & Guo, Z. (2017). A unified implicit scheme for kinetic model equations. part I. memory reduction technique. Science Bull. 62 (2), 119–129.Google Scholar
Chen, Y.P., Zhu, Y.J., & Xu, K. (2020). A three-dimensional unified gas-kinetic wave-particle solver for flow computation in all regimes. Phys. Fluids 32, 096108.CrossRefGoogle Scholar
Chou, S., & Baganoff, D. (1997). Kinetic flux-vector splitting for the Navier-Stokes equations. J. Comput. Phys. 130, 217–230.Google Scholar
Chu, C.K. (1965). Kinetic-theoretic description of the formation of a shock wave. Phys. Fluids 8 (1), 12–22.CrossRefGoogle Scholar
Degond, P., Dimarco, G., & Mieussens, L. (2010). A multiscale kinetic–fluid solver with dynamic localization of kinetic effects. J. Comput. Phys. 229 (13), 4907–4933.CrossRefGoogle Scholar
Degond, P., Liu, J., & Mieussens, L. (2006). Macroscopic fluid models with localized kinetic upscale effects. Multiscale Model. Simul. 5, 940–979.CrossRefGoogle Scholar
Deshpande, S. (1986). A second order accurate, kinetic-theory based method for inviscid compressible flows. NASA Langley Tech. Paper No. 2613.Google Scholar
Dimarco, G., & Pareschi, L. (2013). Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal. 51, 1064–1087.CrossRefGoogle Scholar
Eu, B. (2016). Kinetic theory of nonequilibrium ensembles, irreversible thermodynamics, and generalized hydrodynamics: Volume 1. Nonrelativistic theories. Springer.Google Scholar
Fei, F., Zhang, J., Li, J., & Liu, Z. (2020). A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows. J. Comput. Phys. 400, 108972.Google Scholar
Filbet, F., & Jin, S. (2010). A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229 (20), 7625–7648.CrossRefGoogle Scholar
Gallis, M., & Torczynski, J. (2000). The application of the BGK model in particle simulations. In 34th Thermophysics Conference, 2360.Google Scholar
Ghia, U., Ghia, K., & Shin, C. (1982). High-Resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411.CrossRefGoogle Scholar
Gorji, M.H., & Jenny, P. (2015). Fokker-Planck-DSMC algorithm for simulations of rarefied gas flows. J. Comput. Phys. 287, 110–129.Google Scholar
Grad, H. (1949). On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 325.Google Scholar
Gu, X., & Emerson, D. (2009). A high-order moment approach for capturing non-equilibrium phenomena in the transition regime. J. Fluid Mech. 636, 177–216.CrossRefGoogle Scholar
Guo, Z., Li, J., & Xu, K. (2020). On unified preserving properties of kinetic schemes. arXiv: 1909.04923v4.Google Scholar
Guo, Z., Wang, R., & Xu, K. (2015). Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case. Phys. Rev. E 91 (3), 033313.CrossRefGoogle ScholarPubMed
Guo, Z., Xu, K., & Wang, R. (2013). Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case. Phys. Rev. E 88 (3), 033305.CrossRefGoogle ScholarPubMed
Huang, J.C., Xu, K., & Yu, P. (2012). A unified gas-kinetic scheme for continuum and rarefied flows II: Multidimensional cases. Comm. Comput. Phys. 12 (3), 662–690.Google Scholar
Huang, J.C., Xu, K., & Yu, P. (2013). A unified gas-kinetic scheme for continuum and rarefied flows III: Microflow simulations. Comm. Comput. Phys. 14 (5), 1147–1173.Google Scholar
Jenny, P., Torrilhon, M., & Heinz, S. (2010). A solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion. J. Comput. Phys. 229 (4) 1077–1098.CrossRefGoogle Scholar
Ji, X., Pan, L., Shyy, W., & Xu, K. (2018b). A compact fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations. J. Comput. Phys. 372, 446–472.Google Scholar
Ji, X., Zhao, F., Shyy, W., & Xu, K. (2018a). A family of high-order gas-kinetic schemes and its comparison with Riemann solver based high-order methods. J. Comput. Phys. 356, 150–173.Google Scholar
Ji, X., Zhao, F., Shyy, W., & Xu, K. (2020). A HWENO reconstruction based high-order compact gas-kinetic scheme on unstructured meshes. J. Comput. Phys. 410, 109367.Google Scholar
Jiang, D.W., Mao, M.L., & Deng, X.G. (2019a). An implicit parallel UGKS solver for flows covering various regimes. Adv. Aerodynamics 1:8, https://doi.org/10.1186/s42774-019-0008-5.CrossRefGoogle Scholar
Jiang, Z., Zhao, W., Yuan, Z., Chen, W., & Myong, R. (2019b). Computation of hypersonic flows over flying configurations using a nonlinear constitutive model. AIAA J. 57 (12), 5252–5268.CrossRefGoogle Scholar
Jin, C., & Xu, K. (2007). A unified moving grid gas-kinetic method in Eulerian space for viscous flow computation. J. Comput. Phys. 222, 155–175.CrossRefGoogle Scholar
Jin, S. (1999). Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comp. 21 (2), 441–454.Google Scholar
John, B., Gu, X., & Emerson, D. (2011). Effects of incomplete surface accommodation on non-equilibrium heat transfer in cavity flow: A parallel DSMC study. Comput. Fluids 45, 197–201.CrossRefGoogle Scholar
Kolobov, V., Arslanbekov, R., Aristov, V., Frolova, A., & Zabelok, S. (2007). Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement. J. Comput. Phys. 223, 589–608.Google Scholar
Kumar, G., Girimaji, S., & Kerimo, J. (2013). WENO-enhanced gas-kinetic scheme for direct simulations of compressible transition and turbulence. J. Comput. Phys. 234, 499–523.CrossRefGoogle Scholar
Larsen, A., Morel, J., & Miller, W. (1987). Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. J. Comput. Phys. 69(2),283–324.CrossRefGoogle Scholar
Lele, S. (1992). Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42.CrossRefGoogle Scholar
Levermore, C. (1996). Moment closure hierarchies for kinetic theories. J. Statistical Phys. 83, 1021.CrossRefGoogle Scholar
Li, J., & Du, Z. (2016). A two-stage fourth order time-accurate discretization for Lax–Wendroff type flow solvers I: Hyperbolic conservation laws. SIAM J. Sci. Comput. 38 (5), A3046–A3069.Google Scholar
Li, Q., Xu, K., & Fu, S. (2010). A high-order gas-kinetic Navier-Stokes solver. J. Comput. Phys. 229, 6715–6731.Google Scholar
Li, S., Li, Q., Fu, S., & Xu, K. (2018). A unified gas-kinetic scheme for axisymmetric flow in all Knudsen number regimes. J. Comput. Phys. 366, 144–169.Google Scholar
Li, W., Liu, C., Zhu, Y., Zhang, J., & Xu, K. (2020). Unified gas-kinetic wave-particle methods III: Multiscale photon transport. J. Comput. Phys. 408, 109280.CrossRefGoogle Scholar
Li, Z.H., & Zhang, H.X. (2004). Study on gas kinetic unified algorithm for flows from rarefied transition to continuum. J. Comput. Phys. 193 (2), 708–738.CrossRefGoogle Scholar
Liu, C. (2016). Unified gas-kinetic scheme for the study of multi-scale flows. PhD thesis, Hong Kong University of Science and Technology.Google Scholar
Liu, C., Wang, Z., & Xu, K. (2019a). A unified gas-kinetic scheme for continuum and rarefied flows VI: Dilute disperse gas-particle multiphase system. J. Comput. Phys. 386, 264–295.CrossRefGoogle Scholar
Liu, C., & Xu, K. (2017). A unified gas kinetic scheme for continuum and rarefied flows V: Multiscale and multicomponent plasma transport. Comm. Comput. Phys. 22 (5), 1175–1223.CrossRefGoogle Scholar
Liu, C., & Xu, K. (2020a). A unified gas-kinetic scheme for micro flow simulation based on linearized kinetic equation. Adv. Aerodynamics 2:21, https://doi.org/10.1186/s42774-020-00045-8.Google Scholar
Liu, C., & Xu, K. (2021). Unified gas-kinetic wave-particle methods IV: Multi-species gas mixture and plasma transport. Adv. Aerodynamics 3:9, https://doi.org/10.1186/s42774-021-00062-1Google Scholar
Liu, C., Xu, K., Sun, Q., & Cai, Q. (2016). A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations. J. Comput. Phys. 314, 305–40.Google Scholar
Liu, C., Zhou, G., Shyy, W., & Xu, K. (2019b). Limitation principle for computational fluid dynamics. Shock Waves 29:1083–1102.Google Scholar
Liu, C., Zhu, Y., & Xu, K. (2020). Unified gas-kinetic wave-particle methods I: Continuum and rarefied gas flow. J. Comput. Phys. 401, 108977.Google Scholar
Liu, S., Yu, P., Xu, K., & Zhong, C. (2014). Unified gas-kinetic scheme for diatomic molecular simulations in all flow regimes. J. Comput. Phys. 259, 96–113.CrossRefGoogle Scholar
Luo, J., & Xu, K. (2013). A high-order multidimensional gas-kinetic scheme for hydrodynamic equations. Science China, Technological Sciences 56, 2370–2384.Google Scholar
Luo, J., Xuan, L., & Xu, K. (2013). Comparison of fifth-order WENO scheme and WENO-gas-kinetic scheme for inviscid and viscous flow simulation. Commun. Comput. Phys. 14, 599–620.Google Scholar
Macrossan, M.N. (2001). A particle simulation method for the BGK equation. In AIP Conference Proceedings, Vol. 585, AIP, 426–433.Google Scholar
Mieussens, L. (2000). Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Methods Applied Sci. 10 (8), 1121–1149.Google Scholar
Mieussens, L. (2013). On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic model. J. Comput. Phys. 253, 138–156.Google Scholar
Mouhot, C., & Pareschi, L. (2006). Fast algorithms for computing the Boltzmann collision operator. Math. Comp. 75 (256), 1833–1852.Google Scholar
Myong, R. (2001). A computational method for Eu’s generalized hydrodynamic equations of rarefied and microscale gas dynamics. J. Comput. Phys. 168, 47–72.CrossRefGoogle Scholar
Ohwada, T., Adachi, R., Xu, K., & Luo, J. (2013). On the remedy against shock anomalies in kinetic schemes. J. Comput. Phys. 255, 106–129.CrossRefGoogle Scholar
Ohwada, T., & Kobayashi, S. (2004). Management of discontinuous reconstruction in kinetic schemes. J. Comput. Phys. 197, 116–138.CrossRefGoogle Scholar
Oran, E., Oh, C., & Cybyk, B. (1998). Direct simulation Monte Carlo: Recent advances and applications. Ann. Rev. Fluid Mech. 30 (1), 403–441.Google Scholar
Pan, L., & Xu, K. (2016). A third-order compact gas-kinetic scheme on unstructured meshes for compressible Navier–Stokes solutions. J. Comput. Phys. 318, 327–348.CrossRefGoogle Scholar
Pan, L., Xu, K., Li, Q., & Li, J. (2016). An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations. J. Comput. Phys. 326, 197–221.Google Scholar
Pareschi, L., & Russo, G. (2000). Asymptotic preserving Monte Carlo methods for the Boltzmann equation. Trans. Theory Stat. Phys. 29 (3–5) 415–430.CrossRefGoogle Scholar
Perthame, B. (1992). Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29, 1–19.Google Scholar
Pieraccini, S., & Puppo, G. (2007). Implicit-explicit schemes for BGK kinetic equations. J. Sci. Comput. 32, 1–28.Google Scholar
Pullin, D. (1980). Direct simulation methods for compressible inviscid ideal gas flow. J. Comput. Phys. 34, 231–244.Google Scholar
Ren, W., Liu, H., & Jin, S. (2014). An asymptotic-preserving Monte Carlo method for the Boltzmann equation. J. Comput. Phys. 276, 380–404.Google Scholar
Ren, X.D., Xu, K., & Shyy, W. (2016). A multidimensional high-order DG-ALE method based on gas-kinetic theory with application to oscillating bodies. J. Comput. Phys. 316, 700–720.CrossRefGoogle Scholar
Schwartzentruber, T.E., & Boyd, I. (2007). A molecular particle-continuum method for hypersonic nonequilibrium gas flows. J. Comput. Phys. 225, 1159–1174.Google Scholar
Schwartzentruber, T.E., Scalabrin, L.C., & Boyd, I. (2008). Hybrid particle-continuum simulations of nonequilibrium hypersonic blunt-body flowfields. J Thermophys. Heat Trans. 22 (1), 29–37.CrossRefGoogle Scholar
Shakhov, E. (1968). Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 3 (5), 95–96.Google Scholar
Shi, Y., Song, P., & Sun, W. (2020). An asymptotic preserving unified gas kinetic particle method for radiative transfer equations. J. Comput. Phys. 420, 109687.Google Scholar
Shi, Y., Sun, W., Li, L., & Song, P. (2021). An improved unified gas kinetic particle method for radiative transfer equations. J. Quant. Spectrosc. Radiat. Transfer 261,107428.Google Scholar
Shizgal, B. (1981). A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems. J. Comput. Phys. 41, 309–328.CrossRefGoogle Scholar
Struchtrup, H. (2005). Macroscopic transport equations for rarefied gas flows: Approximation methods in kinetic theory. Springer.Google Scholar
Struchtrup, H., & Torrihon, M. (2003). Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids 15, 2668.Google Scholar
Su, W., Zhu, L., Wang, P., Zhang, Y., & Wu, L. (2020). Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations? J. Comput. Phys. 407, 109245.Google Scholar
Sun, W., Jiang, S., & Xu, K. (2015a). An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations. J. Comput. Phys. 285, 265–279.CrossRefGoogle Scholar
Sun, W., Jiang, S., & Xu, K. (2017). A multidimensional unified gas-kinetic scheme for radiative transfer equations on unstructured mesh. J. Comput. Phys. 351, 455–472.Google Scholar
Sun, W., Jiang, S., Xu, K., & Li, S. (2015b). An asymptotic preserving unified gas kinetic scheme for frequency dependent radiative transfer equations. J. Comput. Phys. 302, 222–238.CrossRefGoogle Scholar
Tan, S., Sun, W., Wei, J., & Ni, G. (2019). A parallel unified gas kinetic scheme for three-dimensional multi-group neutron transport. J. Comput. Phys. 391, 37–58.Google Scholar
Tcheremissine, F. (2005). Direct numerical solution of the Boltzmann equation. In AIP Conference Proceedings, Vol. 762, AIP, 677–685.Google Scholar
Tiwari, S. (1998). Coupling of the Boltzmann and Euler equations with automatic domain decomposition. J. Comput. Phys. 144, 710–726.Google Scholar
Tumuklu, O., Li, Z., & Levin, D.A. (2016). Particle ellipsoidal statistical Bhatnagar-Gross-Krook approach for simulation of hypersonic shocks. AIAA J. 3701–3716.Google Scholar
van Leer, B. (1977). Towards the ultimate conservative difference scheme IV: a new approach to numerical convection. J. Comput. Phys. 23, 276–299.Google Scholar
van Leer, B. (1979). Towards the ultimate conservative difference scheme V: A second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136.CrossRefGoogle Scholar
Wagner, W. (1992). A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation. J. Statistical Phys. 66, 1011–1044.Google Scholar
Wang, Z., Yan, H., Li, Q., & Xu, K. (2017). Unified gas-kinetic scheme for diatomic molecular flow with translational, rotational, and vibrational modes. J. Comput. Phys. 350, 237–259.Google Scholar
Wijesinghe, H.S., & Hadjiconstantinou, N. (2004). Three-dimensional hybrid continuum-atomistic simulations for multiscale hydrodynamics. J. Fluid Engineering 126, 768–777.Google Scholar
Woodward, P., & Colella, P. (1984). Numerical simulations of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173.CrossRefGoogle Scholar
Wu, L., Reese, J.M., & Zhang, Y. (2014). Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows. J. Fluid Mech. 746, 53–84.Google Scholar
Wu, L., White, C., Scanlon, T.J., Reese, J.M., & Zhang, Y. (2013). Deterministic numerical solutions of the Boltzmann equation using the fast spectral method. J. Comput. Phys. 250, 27–52.CrossRefGoogle Scholar
Wu, L., Zhang, J., Reese, J.M., & Zhang, Y. (2015). A fast spectral method for the Boltzmann equation for monatomic gas mixtures. J. Comput. Phys. 298, 602–621.Google Scholar
Xiao, T., Cai, Q., & Xu, K. (2017). A well-balanced unified gas-kinetic scheme for multiscale flow transport under gravitational field. J. Comput. Phys. 332, 475–491.Google Scholar
Xiao, T., Liu, C., Xu, K., & Cai, Q. (2020). A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows. J. Comput. Phys. 415, 109535.Google Scholar
Xu, K. (2001). A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171 (1), 289–335.CrossRefGoogle Scholar
Xu, K. (2015). Direct modelling for computational fluid dynamics: Construction and application of unified gas-kinetic scheme. World Scientific.Google Scholar
Xu, K., He, X., & Cai, C. (2008). Multiple temperature kinetic model and gas-kinetic method for hypersonic nonequilibrium flow computations. J. Comput. Phys. 227, 6779–6794.CrossRefGoogle Scholar
Xu, K., & Huang, J.C. (2010). A unified gas-kinetic scheme for continuum and rarefied flows. J. Comput. Phys. 229 (20), 7747–7764.Google Scholar
Xu, K., & Huang, J.C. (2011). An improved unified gas-kinetic scheme and the study of shock structures. IMA J. App. Math. 76, 698−711.Google Scholar
Xu, K., & Li, Z. (2001). Dissipative mechanism in Godunov-type schemes. Int. J. Numer. Meth. Fluids 37, 1–22.Google Scholar
Xu, K., & Liu, C. (2017). A paradigm for modelling and computation of gas dynamics. Phys. Fluids 29, 026101.Google Scholar
Xu, K., Mao, M., & Tang, L. (2005). A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow. J. Comput. Phys. 203, 405–421.Google Scholar
Xu, X., Chen, Y., Liu, C., Li, Z., & Xu, K. (2020). Unified gas-kinetic wave-particle methods V: Diatomic molecular flow. arXiv:2010.07195v1 [physics.comp-ph] 14 Oct.Google Scholar
Xu, X., Chen, Y., & Xu, K. (2021). Modelling and computation of non-equilibrium gas dynamics: Beyond single relaxation time kinetic models. Phys. Fluids 33, 011703; doi: 10.1063/5.0036203.CrossRefGoogle Scholar
Yang, J., & Huang, J. (1995). Rarefied flow computations using nonlinear model Boltzmann equations. J. Comput. Phys. 120 (2), 323–339.Google Scholar
Yang, L., Shu, C., Yang, W., & Wu, J. (2018). An implicit scheme with memory reduction technique for steady state solutions of DVBE in all flow regimes. Phys. Fluids 30 (4), 040901.Google Scholar
Yu, P. (2013). A unified gas kinetic scheme for all Knudsen number flows. PhD thesis, Hong Kong University of Science and Technology.Google Scholar
Zhao, F., Ji, X., Shyy, W., & Xu, K. (2019). Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations. Adv. Aerodynamics 1:13, https://doi.org/10.1186/s42774-019-0015-6.Google Scholar
Zhao, F., Ji, X., Shyy, W., & Xu, K. (2020a). An acoustic and shock wave capturing compact high-order gas-kinetic scheme with spectral-like resolution. Int. J. Comput. Fluid. Dyn. https://doi.org/10.1080/10618562.2020.1821879.CrossRefGoogle Scholar
Zhao, F., Ji, X., Shyy, W., & Xu, K. (2020b). Compact high-order gas-kinetic scheme on unstructured mesh for acoustic and shock wave computations. arXiv:2010.05717v2 [math.NA] 19 Oct 2020.Google Scholar
Zhong, X.L., MacCormack, R.W., & Chapman, D. (1993). Stabilization of the Burnett equations and application to hypersonic flows. AIAA J. 31 (1), 1036.Google Scholar
Zhu, L., Guo, Z., & Xu, K. (2016). Discrete unified gas kinetic scheme on unstructured meshes. Comp. Fluids 127, 211–225.Google Scholar
Zhu, L., Pi, X., Su, W., Li, Z.H., Zhang, Y., & Wu, L. (2021). General synthetic iteration scheme for non-linear gas kinetic simulation of multi-scale rarefied gas flows. J. Comput. Phys. 430, 110091.Google Scholar
Zhu, Y.J., Liu, C., Zhong, C.W., & Xu, K. (2019b). Unified gas-kinetic wave-particle methods II: Multiscale simulation on unstructured mesh. Phys. Fluids 31, 067105.Google Scholar
Zhu, Y.J., Zhong, C., & Xu, K. (2017a). Implicit unified gas-kinetic scheme for steady state solution in all flow regimes. J. Comput. Phys. 315, 16–38.Google Scholar
Zhu, Y.J., Zhong, C., & Xu, K. (2017b). Unified gas-kinetic scheme with multigrid convergence for rarefied flow study. Phys. Fluids 29, 096102.Google Scholar
Zhu, Y.J., Zhong, C., & Xu, K. (2019a). An implicit unified gas-kinetic scheme for unsteady flow in all Knudsen regimes. J. Comput. Phys. 386, 190–217.Google Scholar
Zhu, Y.J., Zhong, C.W., & Xu, K. (2020). Ray effect in rarefied flow simulation. J. Comput. Phys. 422, 109751.Google Scholar
- 14
- Cited by