American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

W. Layton, Iuliana Stanculescu, Catalin Trenchea. Theory of the NS-$\overline{\omega}$ model: A complement to the NS-$\alpha$ model. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1763-1777. doi: 10.3934/cpaa.2011.10.1763
[2]

Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481
[3]

Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781
[4]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[5]

Jishan Fan, Tohru Ozawa. Regularity criteria for the magnetohydrodynamic equations with partial viscous terms and the Leray-$\alpha$-MHD model. Kinetic & Related Models, 2009, 2 (2) : 293-305. doi: 10.3934/krm.2009.2.293
[6]

Paolo Secchi. An alpha model for compressible fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 351-359. doi: 10.3934/dcdss.2010.3.351
[7]

W. Layton, R. Lewandowski. On a well-posed turbulence model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 111-128. doi: 10.3934/dcdsb.2006.6.111
[8]

T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure & Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415
[9]

S. Danilov. Non-universal features of forced 2D turbulence in the energy and enstrophy ranges. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 67-78. doi: 10.3934/dcdsb.2005.5.67
[10]

Eleftherios Gkioulekas, Ka Kit Tung. On the double cascades of energy and enstrophy in two dimensional turbulence. Part 1. Theoretical formulation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 79-102. doi: 10.3934/dcdsb.2005.5.79
[11]

Peter R. Kramer, Joseph A. Biello, Yuri Lvov. Application of weak turbulence theory to FPU model. Conference Publications, 2003, 2003 (Special) : 482-491. doi: 10.3934/proc.2003.2003.482
[12]

Eleftherios Gkioulekas, Ka Kit Tung. On the double cascades of energy and enstrophy in two dimensional turbulence. Part 2. Approach to the KLB limit and interpretation of experimental evidence. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 103-124. doi: 10.3934/dcdsb.2005.5.103
[13]

Yong Zhou, Jishan Fan. Regularity criteria for a magnetohydrodynamic-$\alpha$ model. Communications on Pure & Applied Analysis, 2011, 10 (1) : 309-326. doi: 10.3934/cpaa.2011.10.309
[14]

Jishan Fan, Tohru Ozawa. Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model. Conference Publications, 2011, 2011 (Special) : 400-409. doi: 10.3934/proc.2011.2011.400
[15]

Xiaojing Xu, Zhuan Ye. Note on global regularity of 3D generalized magnetohydrodynamic-$\alpha$ model with zero diffusivity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 585-595. doi: 10.3934/cpaa.2015.14.585
[16]

Tania Biswas, Sheetal Dharmatti. Control problems and invariant subspaces for sabra shell model of turbulence. Evolution Equations & Control Theory, 2018, 7 (3) : 417-445. doi: 10.3934/eect.2018021
[17]

T. Gallouët, J.-C. Latché. Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2371-2391. doi: 10.3934/cpaa.2012.11.2371
[18]

Luca Bisconti, Davide Catania. Global well-posedness of the two-dimensional horizontally filtered simplified Bardina turbulence model on a strip-like region. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1861-1881. doi: 10.3934/cpaa.2017090
[19]

Anne Bronzi, Ricardo Rosa. On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 19-49. doi: 10.3934/dcds.2014.34.19
[20]

Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences