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March  2016, 9(1): 51-74. doi: 10.3934/krm.2016.9.51

Asymptotic preserving scheme for a kinetic model describing incompressible fluids

1. 

Inria Rennes Bretagne Atlantique (team IPSO) and IRMAR, University of Rennes 1, Campus de Beaulieu, 35042 Rennes, France

2. 

CNRS and IRMAR, University of Rennes 1, Campus de Beaulieu, 35042 Rennes, France

3. 

Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India

4. 

National Mathematics Initiative, Indian Institute of Science, Bangalore, India

5. 

Department of Civil Engineering, Indian Institute of Science, Bangalore, India

Received  September 2014 Revised  July 2015 Published  October 2015

The kinetic theory of fluid turbulence modeling developed by Degond and Lemou in [7] is considered for further study, analysis and simulation. Starting with the Boltzmann like equation representation for turbulence modeling, a relaxation type collision term is introduced for isotropic turbulence. In order to describe some important turbulence phenomenology, the relaxation time incorporates a dependency on the turbulent microscopic energy and this makes difficult the construction of efficient numerical methods. To investigate this problem, we focus here on a multi-dimensional prototype model and first propose an appropriate change of frame that makes the numerical study simpler. Then, a numerical strategy to tackle the stiff relaxation source term is introduced in the spirit of Asymptotic Preserving Schemes. Numerical tests are performed in a one-dimensional framework on the basis of the developed strategy to confirm its efficiency.
Citation: Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic & Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51
References:
[1]

M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics,, J. Comput. Phys., 227 (2008), 3781.  doi: 10.1016/j.jcp.2007.11.032.  Google Scholar

[2]

F. Bouchut and B. Perthame, The BGK model for small Prandtl number in the Navier-Stokes approximation,, J. Stat. Phys., 71 (1993), 191.  doi: 10.1007/BF01048094.  Google Scholar

[3]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations., Comm. Partial Differential Equations, 25 (2000), 737.  doi: 10.1080/03605300008821529.  Google Scholar

[4]

H. Chen, S. Kandasamy, S. Orzag, R. Shock, S. Succi and V. Yakhot, Extended Boltzmann kinetic equation for turbulent flows,, Science Magazine, 301 (2003), 633.  doi: 10.1126/science.1085048.  Google Scholar

[5]

F. Coron and B. Perthame, Numerical passage from kinetic to fluid equations,, SIAM J. Numer. Anal., 28 (1991), 26.  doi: 10.1137/0728002.  Google Scholar

[6]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits,, Kinetic Related Models, 4 (2011), 441.  doi: 10.3934/krm.2011.4.441.  Google Scholar

[7]

P. Degond and M. Lemou, Turbulence models for incompressible fluids derived from kinetic theory,, J. Math. Fluid Mech., 4 (2002), 257.  doi: 10.1007/s00021-002-8545-8.  Google Scholar

[8]

P. Degond and M. Lemou, On viscosity and termal conduction of fluids with multivalued internal energy,, Eur. J. Mech. B-Fluids, 20 (2001), 303.  doi: 10.1016/S0997-7546(00)01095-5.  Google Scholar

[9]

P. Degond and P. F. Peyrard, Un modèle de collisions ondes-particules en physique des plasmas: Application la dynamique des gaz,, C. R. Acad. Sci. Paris, 323 (1996), 209.   Google Scholar

[10]

P. Degond, J. L. López and P. F. Peyrard, On the macroscopic dynamics induced by a model wave-particle collision operator,, Continuum Mechanics and Thermodynamics, 10 (1998), 153.  doi: 10.1007/s001610050087.  Google Scholar

[11]

P. Degond, J.L. López, F. Poupaud and C. Schmeiser, Existence of solutions of a kinetic equation modeling cometary flows,, J. Stat. Phys., 96 (1999), 361.  doi: 10.1023/A:1004584719071.  Google Scholar

[12]

G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for non-linear kinetic equations,, SIAM Journal of Numerical Analysis, 51 (2013), 1064.  doi: 10.1137/12087606X.  Google Scholar

[13]

G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations,, SIAM Journal of Numerical Analysis, 49 (2011), 2057.  doi: 10.1137/100811052.  Google Scholar

[14]

J. Earl, J. R. Jokipii and G. Morfill, Cosmic ray viscosity,, Astrophysical Journal, 331 (1988).  doi: 10.1086/185242.  Google Scholar

[15]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources,, J. of Comput. Phys., 229 (2010), 7625.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[16]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations,, SIAM Journal on Scientific Computing, 21 (1999), 441.  doi: 10.1137/S1064827598334599.  Google Scholar

[17]

A. Klar and C. Schmeiser, Numerical passage from radiative heat transfer to nonlinear diffusion models,, Math. Models Methods Appl. Sci., 11 (2001), 749.  doi: 10.1142/S0218202501001082.  Google Scholar

[18]

M. Lemou, Relaxed micro-macro schemes for kinetic equations,, C.R. Acad. Sci. Paris, 348 (2010), 455.  doi: 10.1016/j.crma.2010.02.017.  Google Scholar

[19]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit,, SIAM J. Sci. Comput., 31 (2008), 334.  doi: 10.1137/07069479X.  Google Scholar

[20]

L. Mieussens and H. Struchtrup, Numerical comparison of BGK-models with proper Prandtl number,, Phys. Fluids, 16 (2004), 2797.   Google Scholar

[21]

S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations,, Journal of Scientific Computing, 32 (2007), 1.  doi: 10.1007/s10915-006-9116-6.  Google Scholar

[22]

S. B. Pope, Turbulent Flows,, Cambridge University Press, (2000).  doi: 10.1017/CBO9780511840531.  Google Scholar

[23]

H. Struchtrup, The BGK-model with velocity-dependent collision time,, Cont. Mech. Thermodyn., 9 (1997), 23.  doi: 10.1007/s001610050053.  Google Scholar

[24]

D. C. Wilcox, Turbulence Modeling for CFD,, D.C.W. Industries Inc., (1994).   Google Scholar

show all references

References:
[1]

M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving compressible Navier-Stokes asymptotics,, J. Comput. Phys., 227 (2008), 3781.  doi: 10.1016/j.jcp.2007.11.032.  Google Scholar

[2]

F. Bouchut and B. Perthame, The BGK model for small Prandtl number in the Navier-Stokes approximation,, J. Stat. Phys., 71 (1993), 191.  doi: 10.1007/BF01048094.  Google Scholar

[3]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations., Comm. Partial Differential Equations, 25 (2000), 737.  doi: 10.1080/03605300008821529.  Google Scholar

[4]

H. Chen, S. Kandasamy, S. Orzag, R. Shock, S. Succi and V. Yakhot, Extended Boltzmann kinetic equation for turbulent flows,, Science Magazine, 301 (2003), 633.  doi: 10.1126/science.1085048.  Google Scholar

[5]

F. Coron and B. Perthame, Numerical passage from kinetic to fluid equations,, SIAM J. Numer. Anal., 28 (1991), 26.  doi: 10.1137/0728002.  Google Scholar

[6]

N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits,, Kinetic Related Models, 4 (2011), 441.  doi: 10.3934/krm.2011.4.441.  Google Scholar

[7]

P. Degond and M. Lemou, Turbulence models for incompressible fluids derived from kinetic theory,, J. Math. Fluid Mech., 4 (2002), 257.  doi: 10.1007/s00021-002-8545-8.  Google Scholar

[8]

P. Degond and M. Lemou, On viscosity and termal conduction of fluids with multivalued internal energy,, Eur. J. Mech. B-Fluids, 20 (2001), 303.  doi: 10.1016/S0997-7546(00)01095-5.  Google Scholar

[9]

P. Degond and P. F. Peyrard, Un modèle de collisions ondes-particules en physique des plasmas: Application la dynamique des gaz,, C. R. Acad. Sci. Paris, 323 (1996), 209.   Google Scholar

[10]

P. Degond, J. L. López and P. F. Peyrard, On the macroscopic dynamics induced by a model wave-particle collision operator,, Continuum Mechanics and Thermodynamics, 10 (1998), 153.  doi: 10.1007/s001610050087.  Google Scholar

[11]

P. Degond, J.L. López, F. Poupaud and C. Schmeiser, Existence of solutions of a kinetic equation modeling cometary flows,, J. Stat. Phys., 96 (1999), 361.  doi: 10.1023/A:1004584719071.  Google Scholar

[12]

G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for non-linear kinetic equations,, SIAM Journal of Numerical Analysis, 51 (2013), 1064.  doi: 10.1137/12087606X.  Google Scholar

[13]

G. Dimarco and L. Pareschi, Exponential Runge-Kutta methods for stiff kinetic equations,, SIAM Journal of Numerical Analysis, 49 (2011), 2057.  doi: 10.1137/100811052.  Google Scholar

[14]

J. Earl, J. R. Jokipii and G. Morfill, Cosmic ray viscosity,, Astrophysical Journal, 331 (1988).  doi: 10.1086/185242.  Google Scholar

[15]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources,, J. of Comput. Phys., 229 (2010), 7625.  doi: 10.1016/j.jcp.2010.06.017.  Google Scholar

[16]

S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations,, SIAM Journal on Scientific Computing, 21 (1999), 441.  doi: 10.1137/S1064827598334599.  Google Scholar

[17]

A. Klar and C. Schmeiser, Numerical passage from radiative heat transfer to nonlinear diffusion models,, Math. Models Methods Appl. Sci., 11 (2001), 749.  doi: 10.1142/S0218202501001082.  Google Scholar

[18]

M. Lemou, Relaxed micro-macro schemes for kinetic equations,, C.R. Acad. Sci. Paris, 348 (2010), 455.  doi: 10.1016/j.crma.2010.02.017.  Google Scholar

[19]

M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit,, SIAM J. Sci. Comput., 31 (2008), 334.  doi: 10.1137/07069479X.  Google Scholar

[20]

L. Mieussens and H. Struchtrup, Numerical comparison of BGK-models with proper Prandtl number,, Phys. Fluids, 16 (2004), 2797.   Google Scholar

[21]

S. Pieraccini and G. Puppo, Implicit-explicit schemes for BGK kinetic equations,, Journal of Scientific Computing, 32 (2007), 1.  doi: 10.1007/s10915-006-9116-6.  Google Scholar

[22]

S. B. Pope, Turbulent Flows,, Cambridge University Press, (2000).  doi: 10.1017/CBO9780511840531.  Google Scholar

[23]

H. Struchtrup, The BGK-model with velocity-dependent collision time,, Cont. Mech. Thermodyn., 9 (1997), 23.  doi: 10.1007/s001610050053.  Google Scholar

[24]

D. C. Wilcox, Turbulence Modeling for CFD,, D.C.W. Industries Inc., (1994).   Google Scholar

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