{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T14:37:42Z","timestamp":1762353462914,"version":"build-2065373602"},"reference-count":61,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T00:00:00Z","timestamp":1708905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62376234","11902281","11972136","2020Z006068001","6142905213104"],"award-info":[{"award-number":["62376234","11902281","11972136","2020Z006068001","6142905213104"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Aeronautical Science Foundation","award":["62376234","11902281","11972136","2020Z006068001","6142905213104"],"award-info":[{"award-number":["62376234","11902281","11972136","2020Z006068001","6142905213104"]}]},{"name":"National key laboratory of science and technology on advanced composites in special environments","award":["62376234","11902281","11972136","2020Z006068001","6142905213104"],"award-info":[{"award-number":["62376234","11902281","11972136","2020Z006068001","6142905213104"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A shock wave is a flow phenomenon that needs to be considered in the development of high-speed aircraft and engines. The traditional computational fluid dynamics (CFD) method describes it from the perspective of macroscopic variables, such as the Mach number, pressure, density, and temperature. The thickness of the shock wave is close to the level of the molecular free path, and molecular motion has a strong influence on the shock wave. According to the analysis of the Chapman-Enskog approach, the nonequilibrium effect is the source term that causes the fluid system to deviate from the equilibrium state. The nonequilibrium effect can be used to obtain a description of the physical characteristics of shock waves that are different from the macroscopic variables. The basic idea of the nonequilibrium effect approach is to obtain the nonequilibrium moment of the molecular velocity distribution function by solving the Boltzmann\u2013Bhatnagar\u2013Gross\u2013Krook (Boltzmann BGK) equations or multiple relaxation times Boltzmann (MRT-Boltzmann) equations and to explore the nonequilibrium effect near the shock wave from the molecular motion level. This article introduces the theory and understanding of the nonequilibrium effect approach and reviews the research progress of nonequilibrium behavior in shock-related flow phenomena. The role of nonequilibrium moments played on the macroscopic governing equations of fluids is discussed, the physical meaning of nonequilibrium moments is given from the perspective of molecular motion, and the relationship between nonequilibrium moments and equilibrium moments is analyzed. Studies on the nonequilibrium effects of shock problems, such as the Riemann problem, shock reflection, shock wave\/boundary layer interaction, and detonation wave, are introduced. It reveals the nonequilibrium behavior of the shock wave from the mesoscopic level, which is different from the traditional macro perspective and shows the application potential of the mesoscopic kinetic approach of the nonequilibrium effect in the shock problem.<\/jats:p>","DOI":"10.3390\/e26030200","type":"journal-article","created":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T11:31:21Z","timestamp":1708947081000},"page":"200","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Mesoscopic Kinetic Approach of Nonequilibrium Effects for Shock Waves"],"prefix":"10.3390","volume":"26","author":[{"given":"Ruofan","family":"Qiu","sequence":"first","affiliation":[{"name":"School of Aerospace Engineering, Xiamen University, Xiamen 361005, China"}]},{"given":"Xinyuan","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Aerospace Engineering, Xiamen University, Xiamen 361005, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4946-8336","authenticated-orcid":false,"given":"Yue","family":"Bao","sequence":"additional","affiliation":[{"name":"School of Aerospace Engineering, Xiamen University, Xiamen 361005, China"}]},{"given":"Yancheng","family":"You","sequence":"additional","affiliation":[{"name":"School of Aerospace Engineering, Xiamen University, Xiamen 361005, China"}]},{"given":"Hua","family":"Jin","sequence":"additional","affiliation":[{"name":"School of Aerospace Engineering, Xiamen University, Xiamen 361005, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1126\/science.245.4918.624","article-title":"Nonequilibrium molecularmotion in a hypersonic shock-wave","volume":"245","author":"Erwin","year":"1989","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1103\/PhysRev.94.511","article-title":"A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems","volume":"94","author":"Bhatnagar","year":"1954","journal-title":"Phys. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"043306","DOI":"10.1103\/PhysRevE.91.043306","article-title":"Multiple-relaxation-time lattice Boltzmann kinetic model for combustion","volume":"91","author":"Xu","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1209\/0295-5075\/17\/6\/001","article-title":"Lattice BGK model for Navier-Stokes equation","volume":"17","author":"Qian","year":"1992","journal-title":"Europhys. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1146\/annurev.fluid.30.1.329","article-title":"Lattice Boltzmann method for fluid flows","volume":"30","author":"Chen","year":"1998","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1126\/science.1085048","article-title":"Extended Boltzmann Kinetic Equation for Turbulent Flows","volume":"301","author":"Chen","year":"2003","journal-title":"Science"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1017\/jfm.2018.441","article-title":"Lattice Boltzmann simulation of resolved oblate spheroids in wall turbulence","volume":"849","author":"Eshghinejadfard","year":"2018","journal-title":"J. Fluid Mech."},{"key":"ref_8","first-page":"138","article-title":"Progress of discrete Boltzmann study on multiphase complex flows","volume":"39","author":"Xu","year":"2021","journal-title":"Acta Aerodyn. Sin."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1017\/jfm.2017.859","article-title":"A hybrid lattice Boltzmann and finite difference method for droplet dynamics with insoluble surfactants","volume":"837","author":"Liu","year":"2018","journal-title":"J. Fluid Mech."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1016\/j.ijheatmasstransfer.2019.03.057","article-title":"Numerical simulation of mixed convection heat transfer of fluid in a cavity driven by an oscillating lid using lattice Boltzmann method","volume":"137","author":"Lamarti","year":"2019","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"043309","DOI":"10.1103\/PhysRevE.98.043309","article-title":"Lattice Boltzmann method for conjugated heat and mass transfer with general interfacial conditions","volume":"98","author":"Mu","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.jcp.2018.08.024","article-title":"A novel geometry-adaptive Cartesian grid based immersed boundary\u2013lattice Boltzmann method for fluid\u2013structure interactions at moderate and high Reynolds numbers","volume":"375","author":"Xu","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"047109","DOI":"10.1063\/1.5144752","article-title":"Fluid\u2013structure interaction simulation based on immersed boundary-lattice Boltzmann flux solver and absolute nodal coordinate formula","volume":"32","author":"Liu","year":"2020","journal-title":"Phys. Fluids"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.molliq.2017.03.104","article-title":"Influence of magnetic field on nanofluid free convection in an open porous cavity by means of Lattice Boltzmann method","volume":"234","author":"Sheikholeslami","year":"2017","journal-title":"J. Mol. Liq."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.jcp.2017.01.023","article-title":"Low-and high-order accurate boundary conditions: From Stokes to Darcy porous flow modeled with standard and improved Brinkman lattice Boltzmann schemes","volume":"335","author":"Silva","year":"2017","journal-title":"J. Comput. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"035701","DOI":"10.1103\/PhysRevE.69.035701","article-title":"Lattice Boltzmann model for the compressible Navier-Stokes equations with flexible specific-heat ratio","volume":"69","author":"Kataoka","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"24003","DOI":"10.1209\/0295-5075\/103\/24003","article-title":"Lattice BGK kinetic model for high-speed compressible flows: Hydrodynamic and nonequilibrium behaviors","volume":"103","author":"Gan","year":"2013","journal-title":"Europhys. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1017\/S0022112005008153","article-title":"Kinetic theory representation of hydrodynamics: A way beyond the Navier\u2013Stokes equation","volume":"550","author":"Shan","year":"2006","journal-title":"J. Fluid Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"036706","DOI":"10.1103\/PhysRevE.75.036706","article-title":"Alternative method to construct equilibrium distribution functions in lattice-Boltzmann method simulation of inviscid compressible flows at high Mach number","volume":"75","author":"Qu","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"056705","DOI":"10.1103\/PhysRevE.76.056705","article-title":"Coupled double-distribution-function lattice Boltzmann method for the compressible Navier-Stokes equations","volume":"76","author":"Li","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"061301","DOI":"10.1103\/PhysRevE.92.061301","article-title":"Entropic lattice Boltzmann model for compressible flows","volume":"92","author":"Frapolli","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"20190559","DOI":"10.1098\/rsta.2019.0559","article-title":"Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria","volume":"378","author":"Latt","year":"2020","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhang, D., Xu, A., Song, J., Gan, Y., Zhang, Y., and Li, Y. (2023). Specific-heat ratio effects on the interaction between shock wave and heavy-cylindrical bubble: Based on discrete Boltzmann method. arXiv.","DOI":"10.1016\/j.compfluid.2023.106021"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"5336","DOI":"10.1039\/C5SM01125F","article-title":"Discrete Boltzmann modeling of multiphase flows: Hydrodynamic and thermodynamic nonequilibrium effects","volume":"11","author":"Gan","year":"2015","journal-title":"Soft Matter"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"023106","DOI":"10.1103\/PhysRevE.94.023106","article-title":"Nonequilibrium thermohydrodynamic effects on the Rayleigh-Taylor instability in compressible flows","volume":"94","author":"Lai","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"A8","DOI":"10.1017\/jfm.2022.844","article-title":"Discrete Boltzmann multi-scale modelling of non-equilibrium multiphase flows","volume":"951","author":"Gan","year":"2022","journal-title":"J. Fluid Mech."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"104111","DOI":"10.1063\/5.0023364","article-title":"Morphological and nonequilibrium analysis of coupled Rayleigh\u2013Taylor\u2013Kelvin\u2013Helmholtz instability","volume":"32","author":"Chen","year":"2020","journal-title":"Phys. Fluids"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Lin, C., Su, X., and Zhang, Y. (2020). Hydrodynamic and Thermodynamic Nonequilibrium Effects around Shock Waves: Based on a Discrete Boltzmann Method. Entropy, 22.","DOI":"10.3390\/e22121397"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.combustflame.2016.04.003","article-title":"Kinetic modeling of detonation and effects of negative temperature coefficient","volume":"173","author":"Zhang","year":"2016","journal-title":"Combust. Flame"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1016\/j.combustflame.2018.09.027","article-title":"Mesoscopic simulation of nonequilibrium detonation with discrete Boltzmann method","volume":"198","author":"Lin","year":"2018","journal-title":"Combust. Flame"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"012142","DOI":"10.1103\/PhysRevE.99.012142","article-title":"Discrete Boltzmann modeling of unsteady reactive flows with nonequilibrium effects","volume":"99","author":"Lin","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_32","unstructured":"Chapman, S., and Cowling, T. (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."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Struchtrup, H. (2005). Macroscopic Transport Equations for Rarefied Gas Flows, Springer.","DOI":"10.1007\/3-540-32386-4"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0898-1221(97)00254-X","article-title":"Recent advances and current challenges for DSMC","volume":"35","author":"Bird","year":"1998","journal-title":"Comput. Math. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1017\/S0022112064000817","article-title":"Study of rarefied shear flow by the discrete velocity method","volume":"19","author":"Broadwell","year":"1964","journal-title":"J. Fluid Mech."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1016\/S0021-9991(03)00099-8","article-title":"Statistical error in particle simulations of hydrodynamic phenomena","volume":"187","author":"Hadjiconstantinou","year":"2003","journal-title":"J. Comput. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"A25","DOI":"10.1017\/jfm.2020.813","article-title":"Accuracy of high-order lattice Boltzmann method for non-equilibrium gas flow","volume":"907","author":"Shi","year":"2021","journal-title":"J. Fluid Mech."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.compfluid.2017.01.014","article-title":"Comparative study of discrete velocity method and high-order lattice Boltzmann method for simulation of rarefied flows","volume":"146","author":"Yang","year":"2017","journal-title":"Comput. Fluids"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"109570","DOI":"10.1016\/j.jcp.2020.109570","article-title":"An efficient lattice Boltzmann method for compressible aerodynamics on D3Q19 lattice","volume":"418","author":"Guo","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"086104","DOI":"10.1063\/5.0100873","article-title":"Discrete Boltzmann modeling of high-speed compressible flows with various depths of non-equilibrium","volume":"34","author":"Zhang","year":"2022","journal-title":"Phys. Fluids"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"053312","DOI":"10.1103\/PhysRevE.97.053312","article-title":"Discrete Boltzmann trans-scale modeling of high-speed compressible flows","volume":"97","author":"Gan","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1007\/s11467-012-0269-5","article-title":"Lattice Boltzmann modeling and simulation of compressible flows","volume":"7","author":"Xu","year":"2012","journal-title":"Front. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"110010","DOI":"10.1063\/1.5130855","article-title":"A direct simulation Monte Carlo approach on the Riemann problem for gas mixtures","volume":"2164","author":"Meskos","year":"2019","journal-title":"AIP Conf. Proc."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"106106","DOI":"10.1063\/5.0024801","article-title":"Study of regular reflection shock waves using a mesoscopic kinetic approach: Curvature pattern and effects of viscosity","volume":"32","author":"Qiu","year":"2020","journal-title":"Phys. Fluids"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"046109","DOI":"10.1063\/5.0085570","article-title":"Study of shock wave\/boundary layer interaction from the perspective of nonequilibrium effects","volume":"34","author":"Bao","year":"2022","journal-title":"Phys. Fluids"},{"key":"ref_46","first-page":"528520","article-title":"Entropy increase characteristics of shock wave\/plate laminar boundary layer interaction","volume":"44","author":"Song","year":"2023","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2517","DOI":"10.1177\/09544062221096254","article-title":"Discrete Boltzmann modeling of detonation: Based on the Shakhov model","volume":"237","author":"Shan","year":"2023","journal-title":"Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"053113","DOI":"10.1103\/PhysRevE.103.053113","article-title":"Mesoscopic kinetic approach for studying nonequilibrium hydrodynamic and thermodynamic effects of shock wave, contact discontinuity, and rarefaction wave in the unsteady shock tube","volume":"103","author":"Qiu","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/S0045-7930(00)00006-2","article-title":"Evaluation of TVD high resolution schemes for unsteady viscous shocked flows","volume":"30","author":"Daru","year":"2000","journal-title":"Comput. Fluids"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1007\/s00193-015-0589-9","article-title":"Curved shock theory","volume":"26","year":"2016","journal-title":"Shock. Waves"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"A21","DOI":"10.1017\/jfm.2020.158","article-title":"Second-order curved shock theory","volume":"891","author":"Shi","year":"2020","journal-title":"J. Fluid Mech."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"A36","DOI":"10.1017\/jfm.2021.454","article-title":"Method of curved-shock characteristics with application to inverse design of supersonic flowfields","volume":"920","author":"Shi","year":"2021","journal-title":"J. Fluid Mech."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.jcp.2016.03.014","article-title":"A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations","volume":"314","author":"Liu","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"145","DOI":"10.2514\/2.1933","article-title":"Recent developments in the research on pulse detonation engines","volume":"41","author":"Kailasanath","year":"2003","journal-title":"AIAA J."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Bussing, T., and Pappas, G. (1994, January 10\u201313). An introduction to pulse detonation engines. Proceedings of the 32nd Aerospace Sciences Meeting and Exhibit, Reno, NV, USA.","DOI":"10.2514\/6.1994-263"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"131","DOI":"10.2514\/1.B36303","article-title":"Overview of performance, application, and analysis of rotating detonation engine technologies","volume":"33","author":"Rankin","year":"2017","journal-title":"J. Propuls. Power"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.cja.2015.12.006","article-title":"Progress of continuously rotating detonation engines","volume":"29","author":"Zhou","year":"2016","journal-title":"Chin. J. Aeronaut."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"322","DOI":"10.2514\/3.24031","article-title":"Oblique detonation wave engine performance prediction","volume":"12","author":"Ashford","year":"1996","journal-title":"J. Propuls. Power"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"106054","DOI":"10.1016\/j.ast.2020.106054","article-title":"Numerical investigation of a Mach 9 oblique detonation engine with fuel pre-injection","volume":"105","author":"Zhang","year":"2020","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"087119","DOI":"10.1063\/5.0157789","article-title":"Investigating the flow characteristics and thermodynamic performance of curved detonation waves","volume":"35","author":"Xiong","year":"2023","journal-title":"Phys. Fluids"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1939","DOI":"10.1016\/j.proci.2012.06.012","article-title":"Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channels","volume":"34","author":"Nakayama","year":"2013","journal-title":"Proc. Combust. Inst."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/3\/200\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:05:19Z","timestamp":1760105119000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/3\/200"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,26]]},"references-count":61,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["e26030200"],"URL":"https:\/\/doi.org\/10.3390\/e26030200","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,2,26]]}}}