Hostname: page-component-857557d7f7-8wkb5 Total loading time: 0 Render date: 2025-11-21T23:17:45.362Z Has data issue: false hasContentIssue false
  • English
  • Français

The kinetic Shakhov–Enskog model for non-equilibrium flow of dense gases

Published online by Cambridge University Press:  28 November 2019

Peng Wang Open the ORCID record for Peng Wang [Opens in a new window] ,
Lei Wu Open the ORCID record for Lei Wu [Opens in a new window] ,
Minh Tuan Ho ,
Jun Li Open the ORCID record for Jun Li [Opens in a new window] ,
Zhi-Hui Li  and
Yonghao Zhang Open the ORCID record for Yonghao Zhang [Opens in a new window]
Peng Wang
Affiliation:
James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering,University of Strathclyde, GlasgowG1 1XJ, UK
Lei Wu
Affiliation:
James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering,University of Strathclyde, GlasgowG1 1XJ, UK
Minh Tuan Ho
Affiliation:
James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering,University of Strathclyde, GlasgowG1 1XJ, UK
Jun Li
Affiliation:
Center for Integrative Petroleum Research, College of Petroleum Engineering and Geosciences,King Fahd University of Petroleum and Minerals, Saudi Arabia
Zhi-Hui Li
Affiliation:
Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang621000, PR China
Yonghao Zhang*
Affiliation:
James Weir Fluids Laboratory, Department of Mechanical and Aerospace Engineering,University of Strathclyde, GlasgowG1 1XJ, UK
*
†Email address for correspondence: yonghao.zhang@strath.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

When the average intermolecular distance is comparable to the size of gas molecules, the Boltzmann equation, based on the dilute gas assumption, becomes invalid. The Enskog equation was developed to account for this finite size effect that makes the collision non-local and increases the collision frequency. However, it is time-consuming to solve the Enskog equation due to its complicated structure of collision operator and high dimensionality. In this work, on the basis of the Shakhov model, a gas kinetic model is proposed to simplify the Enskog equation for non-ideal monatomic dense gases. The accuracy of the proposed Shakhov–Enskog model is assessed by comparing its solutions of the normal shock wave structures with the results of the Enskog equation obtained by the fast spectral method. It is shown that the Shakhov–Enskog model is able to describe non-equilibrium flow of dense gases, when the maximum local mean free path of gas molecules is still greater than the size of a molecular diameter. The accuracy and efficiency of the present model enable simulations of non-equilibrium flow of dense gases for practical applications.

JFM classification

Compressible Flows: Shock waves

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A48
Copyright
© 2019 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alder, B. J. & Wainwright, T. E. 1960 Studies in molecular dynamics. II. Behavior of a small number of elastic spheres. J. Chem. Phys. 33 (5), 1439–1451.CrossRefGoogle Scholar
Alexander, F. J., Garcia, A. L. & Alder, B. J. 1995 A consistent Boltzmann algorithm. Phys. Rev. Lett. 74 (26), 5212–5215.CrossRefGoogle ScholarPubMed
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), 511–525.CrossRefGoogle Scholar
Brenner, M. P., Hilgenfeldt, S. & Lohse, D. 2002 Single-bubble sonoluminescence. Rev. Mod. Phys. 74 (2), 425–484.CrossRefGoogle Scholar
Carnahan, N. F. & Starling, K. E. 1969 Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51 (2), 635–636.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1970 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., Xu, K. & Cai, Q. 2015 A comparison and unification of ellipsoidal statistical and Shakhov BGK models. Adv. Appl. Maths Mech. 7 (2), 245–266.CrossRefGoogle Scholar
Dahms, R. N., Manin, J., Pickett, L. M. & Oefelein, J. C. 2013 Understanding high-pressure gas–liquid interface phenomena in diesel engines. Proc. Combust. Inst. 34 (1), 1667–1675.CrossRefGoogle Scholar
Dahms, R. N. & Oefelein, J. C. 2015 Non-equilibrium gas–liquid interface dynamics in high-pressure liquid injection systems. Proc. Combust. Inst. 35 (2), 1587–1594.CrossRefGoogle Scholar
Enskog, D. 1922 Kinetische Theorie der Wärmeleitung: Reibung und Selbst-diffusion in Gewissen verdichteten gasen und flüssigkeiten. Almqvist & Wiksells.Google Scholar
Ferziger, J. H. & Kaper, H. G. 1972 Mathematical Theory of Transport Processes in Gases. North-Holland.Google Scholar
Frezzotti, A. 1997a Molecular dynamics and Enskog theory calculation of one dimensional problems in the dynamics of dense gases. Physica A: Stat. Mech. Appl. 240 (1–2), 202–211.CrossRefGoogle Scholar
Frezzotti, A. 1997b A particle scheme for the numerical solution of the Enskog equation. Phys. Fluids 9 (5), 1329–1335.CrossRefGoogle Scholar
Frezzotti, A. 1998 Molecular dynamics and enskog theory calculation of shock profiles in a dense hard sphere gas. Comput. Math. Appl. 35 (1–2), 103–112.CrossRefGoogle Scholar
Frezzotti, A. & Sgarra, C. 1993 Numerical analysis of a shock-wave solution of the Enskog equation obtained via a Monte Carlo method. J. Stat. Phys. 73 (1–2), 193–207.CrossRefGoogle Scholar
Guo, Z. & Shu, C. 2013 Lattice Boltzmann Method and its Applications in Engineering. World Scientific.CrossRefGoogle Scholar
Guo, Z., Zhao, T. S. & Shi, Y. 2005 Simple kinetic model for fluid flows in the nanometer scale. Phys. Rev. E 71 (3), 035301.CrossRefGoogle ScholarPubMed
Guo, Z., Zhao, T. S. & Shi, Y. 2006 Generalized hydrodynamic model for fluid flows: from nanoscale to macroscale. Phys. Fluids 18 (6), 067107.CrossRefGoogle Scholar
de Haro, M. L. & Garzó, V. 1995 Shock waves in a dense gas. Phys. Rev. E 52 (5), 5688–5691.CrossRefGoogle Scholar
He, X. & Doolen, G. D. 2002 Thermodynamic foundations of kinetic theory and lattice Boltzmann models for multiphase flows. J. Stat. Phys. 107 (1-2), 309–328.CrossRefGoogle Scholar
Kremer, G. M. 2010 An Introduction to the Boltzmann Equation and Transport Processes in Gases. Springer.CrossRefGoogle Scholar
Li, Z.-H., Peng, A.-P., Zhang, H.-X. & Yang, J.-Y. 2015 Rarefied gas flow simulations using high-order gas-kinetic unified algorithms for Boltzmann model equations. Prog. Aerosp. Sci. 74, 81–113.CrossRefGoogle Scholar
Li, Z.-H. & Zhang, H.-X. 2008 Gas-kinetic description of shock wave structures by solving Boltzmann model equation. Intl J. Comput. Fluid Dyn. 22 (9), 623–638.CrossRefGoogle Scholar
Luo, L.-S. 1998 Unified theory of lattice Boltzmann models for nonideal gases. Phys. Rev. Lett. 81 (8), 1618–1621.CrossRefGoogle Scholar
Luo, L.-S. 2000 Theory of the lattice Boltzmann method: Lattice Boltzmann models for nonideal gases. Phys. Rev. E 62 (4), 4982–4996.CrossRefGoogle ScholarPubMed
Montanero, J. M., de Haro, M. L., Garzo, V. & Santos, A. 1998 Strong shock waves in a dense gas: Burnett theory versus Monte Carlo simulation. Phys. Rev. E 58 (6), 7319–7324.CrossRefGoogle Scholar
Montanero, J. M., de Haro, M. L., Santos, A. & Garzo, V. 1999 Simple and accurate theory for strong shock waves in a dense hard-sphere fluid. Phys. Rev. E 60 (6), 7592–7595.CrossRefGoogle Scholar
Montanero, J. M. & Santos, A. 1996 Monte Carlo simulation method for the Enskog equation. Phys. Rev. E 54 (1), 438–444.CrossRefGoogle ScholarPubMed
Petersen, E. L. & Hanson, R. K. 2001 Nonideal effects behind reflected shock waves in a high-pressure shock tube. Shock Waves 10 (6), 405–420.CrossRefGoogle Scholar
Sadr, M. & Gorji, M. H. 2017 A continuous stochastic model for non-equilibrium dense gases. Phys. Fluids 29 (12), 122007.CrossRefGoogle Scholar
Sadr, M. & Gorji, M. H. 2019 Treatment of long-range interactions arising in the Enskog–Vlasov description of dense fluids. J. Comput. Phys. 378, 129–142.CrossRefGoogle Scholar
Sander, R., Pan, Z. & Connell, L. D. 2017 Laboratory measurement of low permeability unconventional gas reservoir rocks: a review of experimental methods. J. Nat. Gas Sci. Engng 37, 248–279.CrossRefGoogle Scholar
Shakhov, E. M. 1968 Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 3 (5), 95–96.CrossRefGoogle Scholar
Titarev, V. A. 2018 Application of model kinetic equations to hypersonic rarefied gas flows. Comput. Fluids 169, 62–70.CrossRefGoogle Scholar
Venugopal, V., Praturi, D. S. & Girimaji, S. S. 2019 Non-equilibrium thermal transport and entropy analyses in rarefied cavity flows. J. Fluid Mech. 864, 995–1025.CrossRefGoogle Scholar
Wu, L., Liu, H., Reese, J. M. & Zhang, Y. 2016 Non–equilibrium dynamics of dense gas under tight confinement. J. Fluid Mech. 794, 252–266.CrossRefGoogle 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, Y. & Reese, J. M. 2015 Fast spectral solution of the generalized Enskog equation for dense gases. J. Comput. Phys. 303, 66–79.CrossRefGoogle Scholar
Xu, K. & Huang, J.-C. 2011 An improved unified gas-kinetic scheme and the study of shock structures. IMA J. Appl. Maths 76 (5), 698–711.CrossRefGoogle Scholar
Yang, J. Y. & Huang, J. C. 1995 Rarefied flow computations using nonlinear model Boltzmann equations. J. Comput. Phys. 120 (2), 323–339.CrossRefGoogle Scholar
Zhu, Y., Zhong, C. & Xu, K. 2019 An implicit unified gas-kinetic scheme for unsteady flow in all Knudsen regimes. J. Comput. Phys. 386, 190–217.CrossRefGoogle Scholar