January  2015, 20(1): 59-75. doi: 10.3934/dcdsb.2015.20.59

Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering

1. 

Università degli Studi di Firenze, Dipartimento di Matematica e Informatica "U. Dini", Via S. Marta 3, I-50139 Firenze, Italy

2. 

Università degli Studi di Brescia, Sezione Matematica (DICATAM), Via Valotti 9, I-25133 Brescia, Italy

Received  February 2014 Revised  June 2014 Published  November 2014

We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator.
    It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiflow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the flow under periodicity in all directions and discuss the possibility to introduce a solution semiflow.
Citation: Luca Bisconti, Davide Catania. Remarks on global attractors for the 3D Navier--Stokes equations with horizontal filtering. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 59-75. doi: 10.3934/dcdsb.2015.20.59
References:
[1]

N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation,, in Modern Simulation Strategies for Turbulent Flow (ed. B. J. Geurts), (2001), 21.   Google Scholar

[2]

H. Ali, Approximate deconvolution model in a bounded domain with vertical regularization,, J. Math. Anal. App., 408 (2013), 355.  doi: 10.1016/j.jmaa.2013.06.023.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).   Google Scholar

[4]

L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows,, Springer, (2006).   Google Scholar

[5]

L. C. Berselli, Analysis of a Large Eddy Simulation model based on anisotropic filtering,, J. Math. Anal. Appl., 386 (2012), 149.  doi: 10.1016/j.jmaa.2011.07.044.  Google Scholar

[6]

L. C. Berselli and D. Catania, On the well-posedness of the Boussinesq equations with horizontal filter for turbulent flows,, to appear in Z. Anal. Anwend., ().   Google Scholar

[7]

L. C. Berselli and D. Catania, On the equations with anisotropic filter in a vertical pipe,, to appear in Dyn. Partial Differ. Equ., ().   Google Scholar

[8]

L. C. Berselli, D. Catania and R. Lewandowski, Convergence of approximate deconvolution models to the mean magnetohydrodynamics equations: Analysis of two models,, J. Math. Anal. Appl., 401 (2013), 864.  doi: 10.1016/j.jmaa.2012.12.051.  Google Scholar

[9]

H. Bessaih and F. Flandoli, Weak attractor for a dissipative Euler equation,, J. Dynamics Diff. Equations, 12 (2000), 713.  doi: 10.1023/A:1009042520953.  Google Scholar

[10]

L. Bisconti, On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations,, to appear in Mathematical Methods in the Applied Sciences, (2014).  doi: 10.1002/mma.3160.  Google Scholar

[11]

D. Catania and P. Secchi, Global existence and finite dimensional global attractor for a 3D double viscous MHD-$\alpha$ model,, Commun. Math. Sci., 8 (2010), 1021.  doi: 10.4310/CMS.2010.v8.n4.a12.  Google Scholar

[12]

J. W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers,, J. Fluid Mech., 41 (1970), 453.  doi: 10.1017/S0022112070000691.  Google Scholar

[13]

C. Foias, 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.  doi: 10.1023/A:1012984210582.  Google Scholar

[14]

M. Germano, Differential filters for the large eddy simulation of turbulent flows,, Phys. Fluids, 29 (1986), 1755.  doi: 10.1063/1.865649.  Google Scholar

[15]

W. Layton and R. Lewandowski, A simple and stable scale-similarity model for large Eddy simulation: Energy balance and existence of weak solutions,, App. Math. Letters, 16 (2003), 1205.  doi: 10.1016/S0893-9659(03)90118-2.  Google Scholar

[16]

J. L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications,, Dunod, (1968).   Google Scholar

[17]

G. R. Sell, Global attractor for the three-dimensional Navier-Stokes equations,, J. Dynam. Differential Equations, 8 (1996), 1.  doi: 10.1007/BF02218613.  Google Scholar

[18]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[19]

A. Scotti, C. Meneveau and D. K. Lilly, Generalized Smagorinsky model for anisotropic grids,, Phys. Fluids, 5 (1993).  doi: 10.1063/1.858537.  Google Scholar

[20]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

show all references

References:
[1]

N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation,, in Modern Simulation Strategies for Turbulent Flow (ed. B. J. Geurts), (2001), 21.   Google Scholar

[2]

H. Ali, Approximate deconvolution model in a bounded domain with vertical regularization,, J. Math. Anal. App., 408 (2013), 355.  doi: 10.1016/j.jmaa.2013.06.023.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).   Google Scholar

[4]

L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows,, Springer, (2006).   Google Scholar

[5]

L. C. Berselli, Analysis of a Large Eddy Simulation model based on anisotropic filtering,, J. Math. Anal. Appl., 386 (2012), 149.  doi: 10.1016/j.jmaa.2011.07.044.  Google Scholar

[6]

L. C. Berselli and D. Catania, On the well-posedness of the Boussinesq equations with horizontal filter for turbulent flows,, to appear in Z. Anal. Anwend., ().   Google Scholar

[7]

L. C. Berselli and D. Catania, On the equations with anisotropic filter in a vertical pipe,, to appear in Dyn. Partial Differ. Equ., ().   Google Scholar

[8]

L. C. Berselli, D. Catania and R. Lewandowski, Convergence of approximate deconvolution models to the mean magnetohydrodynamics equations: Analysis of two models,, J. Math. Anal. Appl., 401 (2013), 864.  doi: 10.1016/j.jmaa.2012.12.051.  Google Scholar

[9]

H. Bessaih and F. Flandoli, Weak attractor for a dissipative Euler equation,, J. Dynamics Diff. Equations, 12 (2000), 713.  doi: 10.1023/A:1009042520953.  Google Scholar

[10]

L. Bisconti, On the convergence of an approximate deconvolution model to the 3D mean Boussinesq equations,, to appear in Mathematical Methods in the Applied Sciences, (2014).  doi: 10.1002/mma.3160.  Google Scholar

[11]

D. Catania and P. Secchi, Global existence and finite dimensional global attractor for a 3D double viscous MHD-$\alpha$ model,, Commun. Math. Sci., 8 (2010), 1021.  doi: 10.4310/CMS.2010.v8.n4.a12.  Google Scholar

[12]

J. W. Deardorff, A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers,, J. Fluid Mech., 41 (1970), 453.  doi: 10.1017/S0022112070000691.  Google Scholar

[13]

C. Foias, 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.  doi: 10.1023/A:1012984210582.  Google Scholar

[14]

M. Germano, Differential filters for the large eddy simulation of turbulent flows,, Phys. Fluids, 29 (1986), 1755.  doi: 10.1063/1.865649.  Google Scholar

[15]

W. Layton and R. Lewandowski, A simple and stable scale-similarity model for large Eddy simulation: Energy balance and existence of weak solutions,, App. Math. Letters, 16 (2003), 1205.  doi: 10.1016/S0893-9659(03)90118-2.  Google Scholar

[16]

J. L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications,, Dunod, (1968).   Google Scholar

[17]

G. R. Sell, Global attractor for the three-dimensional Navier-Stokes equations,, J. Dynam. Differential Equations, 8 (1996), 1.  doi: 10.1007/BF02218613.  Google Scholar

[18]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[19]

A. Scotti, C. Meneveau and D. K. Lilly, Generalized Smagorinsky model for anisotropic grids,, Phys. Fluids, 5 (1993).  doi: 10.1063/1.858537.  Google Scholar

[20]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Applied Mathematical Sciences, (1988).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[1]

Patrick Penel, Milan Pokorný. Improvement of some anisotropic regularity criteria for the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1401-1407. doi: 10.3934/dcdss.2013.6.1401

[2]

Yutaka Tsuzuki. Solvability of generalized nonlinear heat equations with constraints coupled with Navier--Stokes equations in 2D domains. Conference Publications, 2015, 2015 (special) : 1079-1088. doi: 10.3934/proc.2015.1079

[3]

D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1251-1259. doi: 10.3934/dcdsb.2015.20.1251

[4]

Mustafa A. H. Al-Jaboori, D. Wirosoetisno. Navier--Stokes equations on the $\beta$-plane. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 687-701. doi: 10.3934/dcdsb.2011.16.687

[5]

Tian Ma, Shouhong Wang. Asymptotic structure for solutions of the Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 189-204. doi: 10.3934/dcds.2004.11.189

[6]

Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1

[7]

Luigi C. Berselli, Tae-Yeon Kim, Leo G. Rebholz. Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027

[8]

Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234

[9]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355

[10]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045

[11]

C. Foias, M. S Jolly, O. P. Manley. Recurrence in the 2-$D$ Navier--Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 253-268. doi: 10.3934/dcds.2004.10.253

[12]

Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151

[13]

Konstantinos Chrysafinos. Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077

[14]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209

[15]

Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611

[16]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101

[17]

Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122

[18]

Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521

[19]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[20]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]