Modeling error of <inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-models of turbulence on a two-dimensional torus

Luigi C. Berselli; Argus Adrian Dunca; Roger Lewandowski; Dinh Duong Nguyen

doi:10.3934/dcdsb.2020305

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

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

Luigi C. Berselli ^1, , Argus Adrian Dunca ^2, , Roger Lewandowski ^3, and Dinh Duong Nguyen ^3,,

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.

Keywords: Rate of convergence, $ \alpha $-turbulence models, Navier-Stokes equations.

Mathematics Subject Classification: Primary:35Q30, 35Q35;Secondary:65M15, 76D05, 76F65.

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. Cao, E. 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. Castro, J. 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. Chen, R. B. Guenther, S.-C. Kim, E. 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. Chen, C. Foias, D. D. Holm, E. Olson, E. 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. Chen, C. Foias, D. D. Holm, E. Olson, E. 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. Chen, C. Foias, D. D. Holm, E. Olson, E. 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. Cheskidov, D. D. Holm, E. 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. Foias, O. Manley, R. 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. Foias, D. 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. Ilyin, E. 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 Filho, H. J. Nussenzveig Lopes, E. 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. Cao, E. 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. Castro, J. 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. Chen, R. B. Guenther, S.-C. Kim, E. 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. Chen, C. Foias, D. D. Holm, E. Olson, E. 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. Chen, C. Foias, D. D. Holm, E. Olson, E. 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. Chen, C. Foias, D. D. Holm, E. Olson, E. 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. Cheskidov, D. D. Holm, E. 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. Foias, O. Manley, R. 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. Foias, D. 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. Ilyin, E. 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 Filho, H. J. Nussenzveig Lopes, E. 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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Modeling error of $ \alpha $-models of turbulence on a two-dimensional torus

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

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