September  2021, 26(9): 4613-4643. doi: 10.3934/dcdsb.2020305

Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus

1. 

Università di Pisa, Dipartimento di Matematica, Via Buonarroti 1/c, I-56127 Pisa, Italy

2. 

Department of Mathematics and Informatics, University Politehnica of Bucharest, Bucharest, Romania

3. 

IRMAR, UMR CNRS 6625, University of Rennes 1 and FLUMINANCE Team, INRIA, Rennes, France

* Corresponding author

Received  March 2020 Revised  August 2020 Published  September 2021 Early access  October 2020

This paper is devoted to study the rate of convergence of the weak solutions $ {\bf u}_\alpha $ of $ \alpha $-regularization models to the weak solution $ {\bf u} $ of the Navier-Stokes equations in the two-dimensional periodic case, as the regularization parameter $ \alpha $ goes to zero. More specifically, we will consider the Leray-$ \alpha $, the simplified Bardina, and the modified Leray-$ \alpha $ models. Our aim is to improve known results in terms of convergence rates and also to show estimates valid over long-time intervals. The results also hold in the case of bounded domain with homogeneous Dirichlet boundary conditions.

Citation: Luigi C. Berselli, Argus Adrian Dunca, Roger Lewandowski, Dinh Duong Nguyen. Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4613-4643. doi: 10.3934/dcdsb.2020305
References:
[1]

J. Bardina, J. Ferziger and W. Reynolds, Improved subgrid scale models for large eddy simulation, AIAA Paper, 80 (1980), 1357. doi: 10.2514/6.1980-1357.

[2]

L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer-Verlag, Berlin, 2006.

[3]

L. C. Berselli and R. Lewandowski, Convergence of approximate deconvolution models to the mean Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 171-198.  doi: 10.1016/j.anihpc.2011.10.001.

[4]

A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28.  doi: 10.1007/BF02097055.

[5]

H. Brézis and T. Gallouët, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1.

[6]

Y. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[7]

Y. Cao and E. S. Titi, On the rate of convergence of the two-dimensional $\alpha$-models of turbulence to the Navier-Stokes equations, Numer. Funct. Anal. Optim., 30 (2009), 1231-1271.  doi: 10.1080/01630560903439189.

[8]

M. J. CastroJ. Macías and C. Parés, A multi-layer shallow-water model, The Mathematics of Models for Climatology and Environment, NATO ASI Ser. Ser. I Glob. Environ. Change, Springer, Berlin, 48 (1997), 367-394. 

[9]

T. Chacón-Rebollo and R. Lewandowski, Mathematical and Numerical Foundations of Turbulence Models and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2014. doi: 10.1007/978-1-4939-0455-6.

[10]

L. ChenR. B. GuentherS.-C. KimE. A. Thomann and E. C. Waymire, A rate of convergence for the LANS$\alpha$ regularization of Navier-Stokes equations, J. Math. Anal. Appl., 348 (2008), 637-649.  doi: 10.1016/j.jmaa.2008.07.051.

[11]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.

[12]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Predictability: Quantifying Uncertainty in Models of Complex Phenomena, Phys. D, 133 (1999), 49-65.  doi: 10.1016/S0167-2789(99)00098-6.

[13]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, The International Conference on Turbulence (Los Alamos, NM, 1998), Phys. Fluids, 11 (1999), 2343-2353.  doi: 10.1063/1.870096.

[14]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.

[15]

C. R. Doering and C. Foias, Energy dissipation in body-forced turbulence, J. Fluid Mech., 467 (2002), 289-306.  doi: 10.1017/S0022112002001386.

[16]

A. A. Dunca, Estimates of the modelling error of the alpha-models of turbulence in two and three space dimensions, J. Math. Fluid Mech., 20 (2018), 1123-1135.  doi: 10.1007/s00021-017-0357-y.

[17]

A. Dunca and V. John, Finite element error analysis of space averaged flow fields defined by a differential filter, Math. Models Methods Appl. Sci., 14 (2004), 603-618.  doi: 10.1142/S0218202504003374.

[18]

R. H. Dyer and D. E. Edmunds, Lower bounds for solutions of the Navier-Stokes equations, Proc. London Math. Soc. (3), 18 (1968), 169-178.  doi: 10.1112/plms/s3-18.1.169.

[19] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.
[20]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Advances in Nonlinear Mathematics and Science, Phys. D, 152/153 (2001), 505–519. doi: 10.1016/S0167-2789(01)00191-9.

[21]

C. FoiasD. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[22]

G. P. Galdi, An introduction to the Navier-Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser, Basel, (2000), 1–70.

[23]

M. Germano, Differential filters of elliptic type, Phys. Fluids, 29 (1986), 1757-1758.  doi: 10.1063/1.865650.

[24]

B. J. Geurts, A. K. Kuczaj and E. S. Titi, Regularization modeling for large-eddy simulation of homogeneous isotropic decaying turbulence, J. Phys. A, 41 (2008), 344008, 29 pp. doi: 10.1088/1751-8113/41/34/344008.

[25]

D. D. Holm and E. S. Titi, Computational models of turbulence: The LANS-$\alpha$ model and the role of global analysis, SIAM News, 38 (2005), 1-5. 

[26]

A. A. IlyinE. M. Lunasin and E. S. Titi, A modified-Leray-$\alpha$ subgrid scale model of turbulence, Nonlinearity, 19 (2006), 879-897.  doi: 10.1088/0951-7715/19/4/006.

[27]

T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal., 25 (1967), 188-200.  doi: 10.1007/BF00251588.

[28]

T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28.  doi: 10.1016/0022-1236(84)90024-7.

[29]

O. A. Ladyžhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, Revised and Enlarged, Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris 1969.

[30]

W. Layton and R. Lewandowski, A simple and stable scale-similarity model for large eddy simulation: Energy balance and existence of weak solutions, Appl. Math. Lett., 16 (2003), 1205-1209.  doi: 10.1016/S0893-9659(03)90118-2.

[31]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128.  doi: 10.3934/dcdsb.2006.6.111.

[32]

W. Layton and R. Lewandowski, A high accuracy Leray-deconvolution model of turbulence and its limiting behavior, Anal. Appl. (Singap.), 6 (2008), 23-49.  doi: 10.1142/S0219530508001043.

[33]

J. Leray, Essai sur les mouvements plans d'une liquide visqueux que limitent des parois, J. Math. Pures Appl. (9), 13 (1934), 331-418. 

[34]

J. Leray, Sur les mouvements d'une liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[35]

R. Lewandowski, Analyse Mathématique et Océanographie, Recherches en Mathématiques Appliquées, 39. Masson, Paris, 1997.

[36]

R. Lewandowski and L. C. Berselli, On the Bardina's model in the whole space, J. Math. Fluid Mech., 20 (2018), 1335-1351.  doi: 10.1007/s00021-018-0369-2.

[37]

M. C. Lopes FilhoH. J. Nussenzveig LopesE. S. Titi and A. Zang, Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: Indifference to boundary layers, Phys. D, 292/293 (2015), 51-61.  doi: 10.1016/j.physd.2014.11.001.

[38]

G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.  doi: 10.1007/BF02410664.

[39]

V. Scheffer, Turbulence and hausdorff dimension, Turbulence and Navier-Stokes equations, Lecture Notes in Math., Springer, Berlin, 565 (1976), 174-183. 

[40]

J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, Wis., (1963), 69–98.

[41]

R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43.  doi: 10.1016/0022-1236(75)90052-X.

[42]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.

[43]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

show all references

References:
[1]

J. Bardina, J. Ferziger and W. Reynolds, Improved subgrid scale models for large eddy simulation, AIAA Paper, 80 (1980), 1357. doi: 10.2514/6.1980-1357.

[2]

L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer-Verlag, Berlin, 2006.

[3]

L. C. Berselli and R. Lewandowski, Convergence of approximate deconvolution models to the mean Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 171-198.  doi: 10.1016/j.anihpc.2011.10.001.

[4]

A. L. Bertozzi and P. Constantin, Global regularity for vortex patches, Comm. Math. Phys., 152 (1993), 19-28.  doi: 10.1007/BF02097055.

[5]

H. Brézis and T. Gallouët, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1.

[6]

Y. CaoE. M. Lunasin and E. S. Titi, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun. Math. Sci., 4 (2006), 823-848.  doi: 10.4310/CMS.2006.v4.n4.a8.

[7]

Y. Cao and E. S. Titi, On the rate of convergence of the two-dimensional $\alpha$-models of turbulence to the Navier-Stokes equations, Numer. Funct. Anal. Optim., 30 (2009), 1231-1271.  doi: 10.1080/01630560903439189.

[8]

M. J. CastroJ. Macías and C. Parés, A multi-layer shallow-water model, The Mathematics of Models for Climatology and Environment, NATO ASI Ser. Ser. I Glob. Environ. Change, Springer, Berlin, 48 (1997), 367-394. 

[9]

T. Chacón-Rebollo and R. Lewandowski, Mathematical and Numerical Foundations of Turbulence Models and Applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2014. doi: 10.1007/978-1-4939-0455-6.

[10]

L. ChenR. B. GuentherS.-C. KimE. A. Thomann and E. C. Waymire, A rate of convergence for the LANS$\alpha$ regularization of Navier-Stokes equations, J. Math. Anal. Appl., 348 (2008), 637-649.  doi: 10.1016/j.jmaa.2008.07.051.

[11]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.

[12]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Predictability: Quantifying Uncertainty in Models of Complex Phenomena, Phys. D, 133 (1999), 49-65.  doi: 10.1016/S0167-2789(99)00098-6.

[13]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, The International Conference on Turbulence (Los Alamos, NM, 1998), Phys. Fluids, 11 (1999), 2343-2353.  doi: 10.1063/1.870096.

[14]

A. CheskidovD. D. HolmE. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 629-649.  doi: 10.1098/rspa.2004.1373.

[15]

C. R. Doering and C. Foias, Energy dissipation in body-forced turbulence, J. Fluid Mech., 467 (2002), 289-306.  doi: 10.1017/S0022112002001386.

[16]

A. A. Dunca, Estimates of the modelling error of the alpha-models of turbulence in two and three space dimensions, J. Math. Fluid Mech., 20 (2018), 1123-1135.  doi: 10.1007/s00021-017-0357-y.

[17]

A. Dunca and V. John, Finite element error analysis of space averaged flow fields defined by a differential filter, Math. Models Methods Appl. Sci., 14 (2004), 603-618.  doi: 10.1142/S0218202504003374.

[18]

R. H. Dyer and D. E. Edmunds, Lower bounds for solutions of the Navier-Stokes equations, Proc. London Math. Soc. (3), 18 (1968), 169-178.  doi: 10.1112/plms/s3-18.1.169.

[19] C. FoiasO. ManleyR. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511546754.
[20]

C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Advances in Nonlinear Mathematics and Science, Phys. D, 152/153 (2001), 505–519. doi: 10.1016/S0167-2789(01)00191-9.

[21]

C. FoiasD. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.

[22]

G. P. Galdi, An introduction to the Navier-Stokes initial-boundary value problem, Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser, Basel, (2000), 1–70.

[23]

M. Germano, Differential filters of elliptic type, Phys. Fluids, 29 (1986), 1757-1758.  doi: 10.1063/1.865650.

[24]

B. J. Geurts, A. K. Kuczaj and E. S. Titi, Regularization modeling for large-eddy simulation of homogeneous isotropic decaying turbulence, J. Phys. A, 41 (2008), 344008, 29 pp. doi: 10.1088/1751-8113/41/34/344008.

[25]

D. D. Holm and E. S. Titi, Computational models of turbulence: The LANS-$\alpha$ model and the role of global analysis, SIAM News, 38 (2005), 1-5. 

[26]

A. A. IlyinE. M. Lunasin and E. S. Titi, A modified-Leray-$\alpha$ subgrid scale model of turbulence, Nonlinearity, 19 (2006), 879-897.  doi: 10.1088/0951-7715/19/4/006.

[27]

T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal., 25 (1967), 188-200.  doi: 10.1007/BF00251588.

[28]

T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28.  doi: 10.1016/0022-1236(84)90024-7.

[29]

O. A. Ladyžhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, Revised and Enlarged, Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris 1969.

[30]

W. Layton and R. Lewandowski, A simple and stable scale-similarity model for large eddy simulation: Energy balance and existence of weak solutions, Appl. Math. Lett., 16 (2003), 1205-1209.  doi: 10.1016/S0893-9659(03)90118-2.

[31]

W. Layton and R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 111-128.  doi: 10.3934/dcdsb.2006.6.111.

[32]

W. Layton and R. Lewandowski, A high accuracy Leray-deconvolution model of turbulence and its limiting behavior, Anal. Appl. (Singap.), 6 (2008), 23-49.  doi: 10.1142/S0219530508001043.

[33]

J. Leray, Essai sur les mouvements plans d'une liquide visqueux que limitent des parois, J. Math. Pures Appl. (9), 13 (1934), 331-418. 

[34]

J. Leray, Sur les mouvements d'une liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.

[35]

R. Lewandowski, Analyse Mathématique et Océanographie, Recherches en Mathématiques Appliquées, 39. Masson, Paris, 1997.

[36]

R. Lewandowski and L. C. Berselli, On the Bardina's model in the whole space, J. Math. Fluid Mech., 20 (2018), 1335-1351.  doi: 10.1007/s00021-018-0369-2.

[37]

M. C. Lopes FilhoH. J. Nussenzveig LopesE. S. Titi and A. Zang, Convergence of the 2D Euler-$\alpha$ to Euler equations in the Dirichlet case: Indifference to boundary layers, Phys. D, 292/293 (2015), 51-61.  doi: 10.1016/j.physd.2014.11.001.

[38]

G. Prodi, Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48 (1959), 173-182.  doi: 10.1007/BF02410664.

[39]

V. Scheffer, Turbulence and hausdorff dimension, Turbulence and Navier-Stokes equations, Lecture Notes in Math., Springer, Berlin, 565 (1976), 174-183. 

[40]

J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear Problems, Univ. of Wisconsin Press, Madison, Wis., (1963), 69–98.

[41]

R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43.  doi: 10.1016/0022-1236(75)90052-X.

[42]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050.

[43]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001. doi: 10.1090/chel/343.

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