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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 & 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, (1953).   Google Scholar

[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,, \emph{Discr.} & \emph{Cont. Dyn. Systems A}, 17 (2007), 481.   Google Scholar

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A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence,, \emph{Royal Soc. A, 461 (2005), 629.  doi: 10.1098/rspa.2004.1373.  Google Scholar

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

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

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C. Doering, The 3D Navier-Stokes problem,, \emph{Annu. Rev. Fluid Mech}, 41 (2009), 109.  doi: 10.1146/annurev.fluid.010908.165218.  Google Scholar

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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,, \emph{J. Statist. Phys.}, 108 (2002), 591.  doi: 10.1023/A:1015782025005.  Google Scholar

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

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

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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,, \emph{J. Dynam. Differential Equations}, 14 (2002), 1.  doi: 10.1023/A:1012984210582.  Google Scholar

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

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

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

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C. Foias and G. Prodi, Sur les solutions statistiques des equations de Navier-Stokes,, \emph{Ann. Mat. Pura Appl.}, 111 (2001), 307.   Google Scholar

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

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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, ().   Google Scholar

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

[19]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,, \emph{Proc. Roy. Soc. London Ser. A}, 434 (1991), 9.  doi: 10.1098/rspa.1991.0075.  Google Scholar

[20]

R. H. Kraichnan, Inertial ranges in two-dimensional turbulence,, \emph{Phys. Fluids}, 5 (1962), 1374.   Google Scholar

[21]

C. E. Leith, Diffusion approximation for two-dimensional turbulence,, \emph{Phys. Fluids}, 11 (1968), 671.   Google Scholar

[22]

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

[23]

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

[24]

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

[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,, \emph{Mathematicheskii Sbornik}, 198 (2007), 3.  doi: 10.1070/SM2007v198n12ABEH003902.  Google Scholar

show all references

References:
[1]

G. K. Batchelor, The Theory of Homogeneous Turbulence,, Cambridge Monographs on Mechanics and Applied Mathematics, (1953).   Google Scholar

[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,, \emph{Discr.} & \emph{Cont. Dyn. Systems A}, 17 (2007), 481.   Google Scholar

[3]

A. Cheskidov, D. D. Holm, E. Olson and E. S. Titi, On a Leray-$\alpha$ model of turbulence,, \emph{Royal Soc. A, 461 (2005), 629.  doi: 10.1098/rspa.2004.1373.  Google Scholar

[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,, \emph{J. Dynam. Differential Equations}, 17 (2005), 643.  doi: 10.1007/s10884-005-8269-6.  Google Scholar

[5]

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

[6]

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

[7]

C. Doering, The 3D Navier-Stokes problem,, \emph{Annu. Rev. Fluid Mech}, 41 (2009), 109.  doi: 10.1146/annurev.fluid.010908.165218.  Google Scholar

[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,, \emph{J. Statist. Phys.}, 108 (2002), 591.  doi: 10.1023/A:1015782025005.  Google Scholar

[9]

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

[10]

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

[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,, \emph{J. Dynam. Differential Equations}, 14 (2002), 1.  doi: 10.1023/A:1012984210582.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

C. Foias and G. Prodi, Sur les solutions statistiques des equations de Navier-Stokes,, \emph{Ann. Mat. Pura Appl.}, 111 (2001), 307.   Google Scholar

[16]

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

[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, ().   Google Scholar

[18]

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

[19]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,, \emph{Proc. Roy. Soc. London Ser. A}, 434 (1991), 9.  doi: 10.1098/rspa.1991.0075.  Google Scholar

[20]

R. H. Kraichnan, Inertial ranges in two-dimensional turbulence,, \emph{Phys. Fluids}, 5 (1962), 1374.   Google Scholar

[21]

C. E. Leith, Diffusion approximation for two-dimensional turbulence,, \emph{Phys. Fluids}, 11 (1968), 671.   Google Scholar

[22]

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

[23]

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

[24]

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

[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,, \emph{Mathematicheskii Sbornik}, 198 (2007), 3.  doi: 10.1070/SM2007v198n12ABEH003902.  Google Scholar

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