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]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020397
[2]

Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021004
[3]

Mohamed Dellal, Bachir Bar. Global analysis of a model of competition in the chemostat with internal inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1129-1148. doi: 10.3934/dcdsb.2020156
[4]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[5]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[6]

Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021011
[7]

Shujing Shi, Jicai Huang, Yang Kuang. Global dynamics in a tumor-immune model with an immune checkpoint inhibitor. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1149-1170. doi: 10.3934/dcdsb.2020157
[8]

Yanhong Zhang. Global attractors of two layer baroclinic quasi-geostrophic model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021023
[9]

Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147
[10]

Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020396
[11]

Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046
[12]

Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021024
[13]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[14]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032
[15]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012
[16]

M. Dambrine, B. Puig, G. Vallet. A mathematical model for marine dinoflagellates blooms. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 615-633. doi: 10.3934/dcdss.2020424
[17]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457
[18]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464
[19]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[20]

Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020362

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences