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August  2020, 13(4): 739-758. doi: 10.3934/krm.2020025

Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations

CEA-DAM Ile-de-France, France

* Corresponding author: Sébastien Guisset

Received  August 2019 Revised  January 2020 Published  May 2020

Angular moments models based on a minimum entropy problem have been largely used to describe the transport of photons [14] or charged particles [18]. In this communication the $ M_1 $ and $ M_2 $ angular moments models are presented for rarefied gas dynamics applications. After introducing the models studied, numerical simulations carried out in various collisional regimes are presented and illustrate the interest in considering angular moments models for rarefied gas dynamics applications. For each numerical test cases, the differences observed between the angular moments models and the well-known Navier-Stokes equations are discussed and compared with reference kinetic solutions.

Citation: Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025
References:
[1]

G. W. Alldredge, C. D. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry Ⅱ: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), B361-B391. doi: 10.1137/11084772X.

[2]

C. BerthonP. Charrier and B. Dubroca, An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions, J. Sci. Comput., 31 (2007), 347-389.  doi: 10.1007/s10915-006-9108-6.

[3]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Birkhäuser Verlag, Basel, (2004). doi: 10.1007/b93802.

[4]

S. Brull and L. Mieussens, Local discrete velocity grids for deterministic rarefied flow simulations, J. Comput. Phys., 266 (2014), 22-46.  doi: 10.1016/j.jcp.2014.01.050.

[5]

S. Chapman, On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas, Phil. Trans. Roy. Soc. London Ser. A, 216 (1916), 538-548.  doi: 10.1098/rsta.1916.0006.

[6] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939. 
[7]

P. Charrier, B. Dubroca, G. Duffa, and R. Turpault, Multigroup model for radiating flows during atmospheric hypersonic re-entry, in Proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Lisbonne, Portugal, 2003,103–110.

[8]

A. Decoster, Personal communication, (2018).,

[9]

B. Dubroca, J.-L. Feugeas and M. Frank, Angular moment model for the Fokker-Planck equation, Eur. Phys. J. D, 60, (2010) 302–307. doi: 10.1140/epjd/e2010-00190-8.

[10]

B. Dubroca and J. L. Feugeas, Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.

[11]

R. DuclousB. Dubroca and M. Frank, Deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843-3857.  doi: 10.1088/0031-9155/55/13/018.

[12]

D. Enskog, Kinetische Theorie der Vorgänge in Mässig Verdünnten Gasen, Uppsala, 1917.

[13]

F. Filbet and T. Rey, A hierarchy of hybrid numerical methods for multi-scale kinetic equation, SIAM J. Sci. Computing, 37 (2015), A1218–A1247. doi: 10.1137/140958773.

[14]

M. GonzálezE. Audit and P. Huynh, Heracles: A three-dimensional radiation hydrodynamics code, A and A, 464 (2007), 429-435. 

[15]

H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[16]

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

[17]

S. GuissetD. AregbaS. Brull and B. Dubroca, The M1 angular moments model in a velocity-adaptive frame for rarefied gas dynamics applications, Multiscale Model. Simul., 15 (2017), 1719-1747.  doi: 10.1137/16M1099327.

[18]

S. GuissetS. BrullE. d'HumièresB. Dubroca and V. Tikhonchuk, Classical transport theory for the collisional electronic M1 model, Phys. A, 446 (2016), 182-194.  doi: 10.1016/j.physa.2015.12.001.

[19]

S. GuissetS. BrullB. DubrocaE. d'HumièresS. Karpov and I. Potapenko, Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Commun. Comput. Phys., 19 (2016), 301-328.  doi: 10.4208/cicp.131014.030615a.

[20]

S. Guisset, J. G. Moreau, R. Nuter, S. Brull, E. d'Humieres, B. Dubroca and V. T. Tikhonchuk, Limits of the M1 and M2 angular moments models for kinetic plasma physics studies, J. Phys. A, 48 (2015), 335501. doi: 10.1088/1751-8113/48/33/335501.

[21]

A. HartenP. D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35-61.  doi: 10.1137/1025002.

[22]

P. Helluy, M. Massaro, L. Navoret, N. Pham and T. Strub, Reduced Vlasov-Maxwell modeling, PIERS Proceedings, (2014), 2622–2627.

[23]

F. Hermeline, A finite volume method for the approximation of the PN Boltzmann and similar systems of equations on general meshes, CEA Report, (2015).

[24]

M. Junk and A. Unterreiter, Maximum entropy moment systems and Galilean invariance, Contin. Mech. Thermodyn., 14 (2002), 563-576.  doi: 10.1007/s00161-002-0096-y.

[25]

B. Van Leer, Towards the ultimate conservative difference scheme Ⅲ. Upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys., 23 (1977), 263-275.  doi: 10.1016/0021-9991(77)90094-8.

[26]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[27]

J. MalletS. Brull and B. Dubroca, General moment system for plasma physics based on minimum entropy principle, Kinet. Relat. Models, 8 (2015), 533-558.  doi: 10.3934/krm.2015.8.533.

[28]

J. McDonald and M. Torrilhon, An affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.  doi: 10.1016/j.jcp.2013.05.046.

[29]

J. G. McDonald and C. P. T. Groth, Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution, Contin. Mech. Thermodyn., 25 (2012), 573-603.  doi: 10.1007/s00161-012-0252-y.

[30]

L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci., 10 (2000), 1121-1149.  doi: 10.1142/S0218202500000562.

[31]

G. N. Minerbo, Maximum entropy eddington factors, J. Quant. Spectrosc. Radiat. Trans., 20 (1978), 541-545.  doi: 10.1016/0022-4073(78)90024-9.

[32]

I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-2210-1.

[33]

G. C. Pomraning, Maximum entropy Eddington factors and flux limited diffusion theory, J. Quant. Spectrosc. Radiat. Trans., 26 (1981), 385-388.  doi: 10.1016/0022-4073(81)90101-1.

[34]

J.-F. Ripoll, An averaged formulation of the M1 radiation model with presumed probability density function for turbulent flows, J. Quant. Spectrosc. Radiat. Trans., 83 (2004), 493-517. 

[35]

J.-F. RipollB. Dubroca and E. Audit, A factored operator method for solving coupled radiation-hydrodynamics models, Transport Theory Statist. Phys., 31 (2002), 531-557.  doi: 10.1081/TT-120015513.

[36]

J. Schneider, Entropic approximation in kinetic theory, M2AN Math. Model. Numer. Anal., 38 (2004), 541-561.  doi: 10.1051/m2an:2004025.

[37]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin, (2005).

[38]

B. Su, More on boundary conditions for differential approximations, J. Quant. Spectrosc. Radiat. Trans., 64 (2000), 409-419.  doi: 10.1016/S0022-4073(99)00128-4.

[39]

T. Pichard, PhD thesis, 2016.,

[40]

M. Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annu. Rev. Fluid Mech., 48 (2016), 429-458. 

[41]

R. Turpault, A consistent multigroup model for radiative transfer and its underlying mean opacity, J. Quant. Spectrosc. Radiat. Trans., 94 (2005), 357-371.  doi: 10.1016/j.jqsrt.2004.09.042.

[42]

R. TurpaultM. FrankB. Dubroca and A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371. 

show all references

References:
[1]

G. W. Alldredge, C. D. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry Ⅱ: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), B361-B391. doi: 10.1137/11084772X.

[2]

C. BerthonP. Charrier and B. Dubroca, An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions, J. Sci. Comput., 31 (2007), 347-389.  doi: 10.1007/s10915-006-9108-6.

[3]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Birkhäuser Verlag, Basel, (2004). doi: 10.1007/b93802.

[4]

S. Brull and L. Mieussens, Local discrete velocity grids for deterministic rarefied flow simulations, J. Comput. Phys., 266 (2014), 22-46.  doi: 10.1016/j.jcp.2014.01.050.

[5]

S. Chapman, On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas, Phil. Trans. Roy. Soc. London Ser. A, 216 (1916), 538-548.  doi: 10.1098/rsta.1916.0006.

[6] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939. 
[7]

P. Charrier, B. Dubroca, G. Duffa, and R. Turpault, Multigroup model for radiating flows during atmospheric hypersonic re-entry, in Proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Lisbonne, Portugal, 2003,103–110.

[8]

A. Decoster, Personal communication, (2018).,

[9]

B. Dubroca, J.-L. Feugeas and M. Frank, Angular moment model for the Fokker-Planck equation, Eur. Phys. J. D, 60, (2010) 302–307. doi: 10.1140/epjd/e2010-00190-8.

[10]

B. Dubroca and J. L. Feugeas, Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.

[11]

R. DuclousB. Dubroca and M. Frank, Deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843-3857.  doi: 10.1088/0031-9155/55/13/018.

[12]

D. Enskog, Kinetische Theorie der Vorgänge in Mässig Verdünnten Gasen, Uppsala, 1917.

[13]

F. Filbet and T. Rey, A hierarchy of hybrid numerical methods for multi-scale kinetic equation, SIAM J. Sci. Computing, 37 (2015), A1218–A1247. doi: 10.1137/140958773.

[14]

M. GonzálezE. Audit and P. Huynh, Heracles: A three-dimensional radiation hydrodynamics code, A and A, 464 (2007), 429-435. 

[15]

H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[16]

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

[17]

S. GuissetD. AregbaS. Brull and B. Dubroca, The M1 angular moments model in a velocity-adaptive frame for rarefied gas dynamics applications, Multiscale Model. Simul., 15 (2017), 1719-1747.  doi: 10.1137/16M1099327.

[18]

S. GuissetS. BrullE. d'HumièresB. Dubroca and V. Tikhonchuk, Classical transport theory for the collisional electronic M1 model, Phys. A, 446 (2016), 182-194.  doi: 10.1016/j.physa.2015.12.001.

[19]

S. GuissetS. BrullB. DubrocaE. d'HumièresS. Karpov and I. Potapenko, Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Commun. Comput. Phys., 19 (2016), 301-328.  doi: 10.4208/cicp.131014.030615a.

[20]

S. Guisset, J. G. Moreau, R. Nuter, S. Brull, E. d'Humieres, B. Dubroca and V. T. Tikhonchuk, Limits of the M1 and M2 angular moments models for kinetic plasma physics studies, J. Phys. A, 48 (2015), 335501. doi: 10.1088/1751-8113/48/33/335501.

[21]

A. HartenP. D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35-61.  doi: 10.1137/1025002.

[22]

P. Helluy, M. Massaro, L. Navoret, N. Pham and T. Strub, Reduced Vlasov-Maxwell modeling, PIERS Proceedings, (2014), 2622–2627.

[23]

F. Hermeline, A finite volume method for the approximation of the PN Boltzmann and similar systems of equations on general meshes, CEA Report, (2015).

[24]

M. Junk and A. Unterreiter, Maximum entropy moment systems and Galilean invariance, Contin. Mech. Thermodyn., 14 (2002), 563-576.  doi: 10.1007/s00161-002-0096-y.

[25]

B. Van Leer, Towards the ultimate conservative difference scheme Ⅲ. Upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys., 23 (1977), 263-275.  doi: 10.1016/0021-9991(77)90094-8.

[26]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[27]

J. MalletS. Brull and B. Dubroca, General moment system for plasma physics based on minimum entropy principle, Kinet. Relat. Models, 8 (2015), 533-558.  doi: 10.3934/krm.2015.8.533.

[28]

J. McDonald and M. Torrilhon, An affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.  doi: 10.1016/j.jcp.2013.05.046.

[29]

J. G. McDonald and C. P. T. Groth, Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution, Contin. Mech. Thermodyn., 25 (2012), 573-603.  doi: 10.1007/s00161-012-0252-y.

[30]

L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci., 10 (2000), 1121-1149.  doi: 10.1142/S0218202500000562.

[31]

G. N. Minerbo, Maximum entropy eddington factors, J. Quant. Spectrosc. Radiat. Trans., 20 (1978), 541-545.  doi: 10.1016/0022-4073(78)90024-9.

[32]

I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-2210-1.

[33]

G. C. Pomraning, Maximum entropy Eddington factors and flux limited diffusion theory, J. Quant. Spectrosc. Radiat. Trans., 26 (1981), 385-388.  doi: 10.1016/0022-4073(81)90101-1.

[34]

J.-F. Ripoll, An averaged formulation of the M1 radiation model with presumed probability density function for turbulent flows, J. Quant. Spectrosc. Radiat. Trans., 83 (2004), 493-517. 

[35]

J.-F. RipollB. Dubroca and E. Audit, A factored operator method for solving coupled radiation-hydrodynamics models, Transport Theory Statist. Phys., 31 (2002), 531-557.  doi: 10.1081/TT-120015513.

[36]

J. Schneider, Entropic approximation in kinetic theory, M2AN Math. Model. Numer. Anal., 38 (2004), 541-561.  doi: 10.1051/m2an:2004025.

[37]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin, (2005).

[38]

B. Su, More on boundary conditions for differential approximations, J. Quant. Spectrosc. Radiat. Trans., 64 (2000), 409-419.  doi: 10.1016/S0022-4073(99)00128-4.

[39]

T. Pichard, PhD thesis, 2016.,

[40]

M. Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annu. Rev. Fluid Mech., 48 (2016), 429-458. 

[41]

R. Turpault, A consistent multigroup model for radiative transfer and its underlying mean opacity, J. Quant. Spectrosc. Radiat. Trans., 94 (2005), 357-371.  doi: 10.1016/j.jqsrt.2004.09.042.

[42]

R. TurpaultM. FrankB. Dubroca and A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371. 

Figure 1.  Density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) profiles in the case $ \tau = 1/\nu = 0 $ (fluid regime) at time $ t = 15 $
Figure 2.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 1 $ (close fluid regime) at time $ t = 5 $
Figure 3.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 100 $ (rarefied regime) at time $ t = 5 $
Figure 4.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 10^{4} $ (strongly rarefied regime) at time $ t = 5 $
Figure 5.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 0 $ (fluid regime) at time $ t = 0.25 $
Figure 6.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 10^{-2} $ (close of fluid regime) at time $ t = 0.1 $
Figure 7.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 10^{-1} $ at time $ t = 0.1 $
Figure 8.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 1 $ (intermediate regimes) at time $ t = 0.1 $
Figure 9.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 0 $ (fluid regime) at time $ t = 0.2 $
Figure 10.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 10^{-1} $ at time $ t = 0.1 $
Figure 11.  Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 1 $ at time $ t = 0.1 $
Figure 12.  Representation of the inverse of the shock thickness as function of the Mach number (speed of the in-going flow)
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