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September  2014, 13(5): 2127-2140. doi: 10.3934/cpaa.2014.13.2127

Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models

Aseel Farhat 1, , M. S Jolly 2, and  Evelyn Lunasin 3,

1. 

Indiana University, Department of Mathematics, Bloomington, IN 47405, United States
2. 

Department of Mathematics, Indiana University, Bloomington, IN, 47405, United States
3. 

Department of Mathematics, U.S. Naval Academy, Annapolis, MD 21402-5002, United States

Received  September 2013 Revised  February 2014 Published  June 2014

We construct semi-integral curves which bound the projections of the global attractors of the 3D NS-$\alpha$ and 3D Leray-$\alpha$ sub-grid scale turbulence models in the plane spanned by their energy and enstrophy. We note the dependence of these bounds on the filter width parameter $\alpha$, and determine subregions where each quantity, energy and enstrophy, must decrease, while isolating one which is recurrent.
Keywords: NS-$\alpha$ turbulence model, Leray-$\alpha$ turbulence model, enstrophy., Global attractor, energy.
Mathematics Subject Classification: Primary: 35Q30, 76F0.
Citation: Aseel Farhat, M. S Jolly, Evelyn Lunasin. Bounds on energy and enstrophy for the 3D Navier-Stokes-$\alpha$ and Leray-$\alpha$ models. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2127-2140. doi: 10.3934/cpaa.2014.13.2127
References:
[1]

G. K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge at the University Press, 1953.

[2]

V. Chepyzhov, E. S. Titi and M. Vishik, On the convergence of solutions of the 3D Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discr. & Cont. Dyn. Systems A, 17 (2007), 481-500.

[3]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Royal Soc. A, Mathematical, Physical and Engineering Sciences, 461 (2005), 629-649. doi: 10.1098/rspa.2004.1373.

[4]

R. Dascaliuc, C. Foias and M. S. Jolly, Relations between energy and enstrophy on the global attractor of the 2-D Navier-Stokes equations, J. Dynam. Differential Equations, 17 (2005), 643-736. doi: 10.1007/s10884-005-8269-6.

[5]

R. Dascaliuc, C. Foias and M. S. Jolly, Universal bounds on the attractor of the Navier-Stokes equation in the energy, enstrophy plane, J. Math. Phys., 48 (2007), 065201. doi: 10.1063/1.2710349.

[6]

R. Dascaliuc, C. Foias and M. Jolly, Estimates on enstrophy, palinstrophy, and invariant measures for 2-d turbulence, J. Differential Eqns, 248 (2010), 792-819. doi: 10.1016/j.jde.2009.11.020.

[7]

C. Doering, The 3D Navier-Stokes problem, Annu. Rev. Fluid Mech, 41 (2009), 109-128. doi: 10.1146/annurev.fluid.010908.165218.

[8]

C. Foias, M. S. Jolly, O. P. Manley and R. Rosa, Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence, J. Statist. Phys., 108 (2002), 591-645. doi: 10.1023/A:1015782025005.

[9]

C. Foias, M. S. Jolly, O. P. Manley, R. Rosa and R. Temam, Kolmogorov theory via finite-time averages, Phys. D, 212 (2005), 245-270. doi: 10.1016/j.physd.2005.10.002.

[10]

C. Foias, M. S. Jolly and M. Yang, On single mode forcing of the 2D-NSE, J. Dynam. Diff. Eqns., 25 (2013), 393-433. doi: 10.1007/s10884-013-9301-x.

[11]

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.

[12]

C. Foias, O. Manley, R. Rosa and R. Temam, Cascade of energy in turbulent flows, Comptes Rendus Acad. Sci. Paris, Serie I, 332 (2001), 509-514. doi: 10.1016/S0764-4442(01)01831-6.

[13]

C. Foias, O. Manley, R. Rosa and R. Temam, Estimates for the energy cascade in three-dimensional turbulent flows, Comptes Rendus Acad. Sci. Paris, Serie I, 333 (2001), 499-504. doi: 10.1016/S0764-4442(01)02008-0.

[14]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.

[15]

C. Foias and G. Prodi, Sur les solutions statistiques des equations de Navier-Stokes, Ann. Mat. Pura Appl., 111 (2001), 307-330.

[16]

D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid, Physica D, 133 (1999), 215-269. doi: 10.1016/S0167-2789(99)00093-7.

[17]

D. D. Holm, J. E. Marsden and T. Ratiu, Euler-Poincaré Equations in Geophysical Fluid Dynamics,, \emph{In Proceedings of the Isaac Newton Institute Programme on the Mathematics of Atmospheric and Ocean Dynamics, (). 

[18]

M. Holst, E. Lunasin and G. Tsotgtgerel, Analytical study of generalized $\alpha$-models of turbulence, Journal of Nonlinear Science, 20 (2010), 523-567. doi: 10.1007/s00332-010-9066-x.

[19]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proc. Roy. Soc. London Ser. A, 434 (1890, 1991), 9-13. Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov's ideas 50 years on. doi: 10.1098/rspa.1991.0075.

[20]

R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids, 5 (1962), 1374-1389.

[21]

C. E. Leith, Diffusion approximation for two-dimensional turbulence, Phys. Fluids, 11 (1968), 671-673.

[22]

L. Lu and C. Doering, Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations, Indiana Univ. Math J., 57 (2008), 2693-2727. doi: 10.1512/iumj.2008.57.3716.

[23]

E. Lunasin, S. Kurien, M. Taylor and E. S. Titi, A study of the Navier-Stokes-$\alpha$ model for two-dimensional turbulence, Journal of Turbulence, 8 (2007), 751-778. doi: 10.1080/14685240701439403.

[24]

E. Lunasin, S. Kurien and E. S. Titi, Spectral scaling of the Leray-$\alpha$ model for two-dimensional turbulence, Journal of Physics A: Math. Theor., 41 (2008), 344014. doi: 10.1088/1751-8113/41/34/344014.

[25]

M. Vishik, E. S. Titi and V. Chepyzhov, On convergence of trajectory attractors of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ approaches 0, Mathematicheskii Sbornik, 198 (2007), 3-36. doi: 10.1070/SM2007v198n12ABEH003902.

show all references

References:
[1]

G. K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge at the University Press, 1953.

[2]

V. Chepyzhov, E. S. Titi and M. Vishik, On the convergence of solutions of the 3D Leray-$\alpha$ model to the trajectory attractor of the 3D Navier-Stokes system, Discr. & Cont. Dyn. Systems A, 17 (2007), 481-500.

[3]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence, Royal Soc. A, Mathematical, Physical and Engineering Sciences, 461 (2005), 629-649. doi: 10.1098/rspa.2004.1373.

[4]

R. Dascaliuc, C. Foias and M. S. Jolly, Relations between energy and enstrophy on the global attractor of the 2-D Navier-Stokes equations, J. Dynam. Differential Equations, 17 (2005), 643-736. doi: 10.1007/s10884-005-8269-6.

[5]

R. Dascaliuc, C. Foias and M. S. Jolly, Universal bounds on the attractor of the Navier-Stokes equation in the energy, enstrophy plane, J. Math. Phys., 48 (2007), 065201. doi: 10.1063/1.2710349.

[6]

R. Dascaliuc, C. Foias and M. Jolly, Estimates on enstrophy, palinstrophy, and invariant measures for 2-d turbulence, J. Differential Eqns, 248 (2010), 792-819. doi: 10.1016/j.jde.2009.11.020.

[7]

C. Doering, The 3D Navier-Stokes problem, Annu. Rev. Fluid Mech, 41 (2009), 109-128. doi: 10.1146/annurev.fluid.010908.165218.

[8]

C. Foias, M. S. Jolly, O. P. Manley and R. Rosa, Statistical estimates for the Navier-Stokes equations and the Kraichnan theory of 2-D fully developed turbulence, J. Statist. Phys., 108 (2002), 591-645. doi: 10.1023/A:1015782025005.

[9]

C. Foias, M. S. Jolly, O. P. Manley, R. Rosa and R. Temam, Kolmogorov theory via finite-time averages, Phys. D, 212 (2005), 245-270. doi: 10.1016/j.physd.2005.10.002.

[10]

C. Foias, M. S. Jolly and M. Yang, On single mode forcing of the 2D-NSE, J. Dynam. Diff. Eqns., 25 (2013), 393-433. doi: 10.1007/s10884-013-9301-x.

[11]

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.

[12]

C. Foias, O. Manley, R. Rosa and R. Temam, Cascade of energy in turbulent flows, Comptes Rendus Acad. Sci. Paris, Serie I, 332 (2001), 509-514. doi: 10.1016/S0764-4442(01)01831-6.

[13]

C. Foias, O. Manley, R. Rosa and R. Temam, Estimates for the energy cascade in three-dimensional turbulent flows, Comptes Rendus Acad. Sci. Paris, Serie I, 333 (2001), 499-504. doi: 10.1016/S0764-4442(01)02008-0.

[14]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.

[15]

C. Foias and G. Prodi, Sur les solutions statistiques des equations de Navier-Stokes, Ann. Mat. Pura Appl., 111 (2001), 307-330.

[16]

D. D. Holm, Fluctuation effects on 3D Lagrangian mean and Eulerian mean fluid, Physica D, 133 (1999), 215-269. doi: 10.1016/S0167-2789(99)00093-7.

[17]

D. D. Holm, J. E. Marsden and T. Ratiu, Euler-Poincaré Equations in Geophysical Fluid Dynamics,, \emph{In Proceedings of the Isaac Newton Institute Programme on the Mathematics of Atmospheric and Ocean Dynamics, (). 

[18]

M. Holst, E. Lunasin and G. Tsotgtgerel, Analytical study of generalized $\alpha$-models of turbulence, Journal of Nonlinear Science, 20 (2010), 523-567. doi: 10.1007/s00332-010-9066-x.

[19]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proc. Roy. Soc. London Ser. A, 434 (1890, 1991), 9-13. Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov's ideas 50 years on. doi: 10.1098/rspa.1991.0075.

[20]

R. H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids, 5 (1962), 1374-1389.

[21]

C. E. Leith, Diffusion approximation for two-dimensional turbulence, Phys. Fluids, 11 (1968), 671-673.

[22]

L. Lu and C. Doering, Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations, Indiana Univ. Math J., 57 (2008), 2693-2727. doi: 10.1512/iumj.2008.57.3716.

[23]

E. Lunasin, S. Kurien, M. Taylor and E. S. Titi, A study of the Navier-Stokes-$\alpha$ model for two-dimensional turbulence, Journal of Turbulence, 8 (2007), 751-778. doi: 10.1080/14685240701439403.

[24]

E. Lunasin, S. Kurien and E. S. Titi, Spectral scaling of the Leray-$\alpha$ model for two-dimensional turbulence, Journal of Physics A: Math. Theor., 41 (2008), 344014. doi: 10.1088/1751-8113/41/34/344014.

[25]

M. Vishik, E. S. Titi and V. Chepyzhov, On convergence of trajectory attractors of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ approaches 0, Mathematicheskii Sbornik, 198 (2007), 3-36. doi: 10.1070/SM2007v198n12ABEH003902.

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