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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

Luca Bisconti 1, and  Davide Catania 2,

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.
Keywords: Approximate Deconvolution Methods (ADM)., Navier--Stokes equations, anisotropic filters, global attractor, Large Eddy Simulation (LES), turbulent flows in domains with boundary, time-shift semiflow.
Mathematics Subject Classification: Primary: 76D05; Secondary: 35Q30, 76F65, 76D0.
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), R. T. Edwards, 2001, 21-44. Google Scholar

[2]

H. Ali, Approximate deconvolution model in a bounded domain with vertical regularization, J. Math. Anal. App., 408 (2013), 355-363. doi: 10.1016/j.jmaa.2013.06.023.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 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), 149-170. 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), 864-880. 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), 713-732. 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, published online, (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), 1021-1040. 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), 453-480. 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), 1-35. doi: 10.1023/A:1012984210582.  Google Scholar

[14]

M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757. 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), 1205-1209. 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, Paris, 1968.  Google Scholar

[17]

G. R. Sell, Global attractor for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33. doi: 10.1007/BF02218613.  Google Scholar

[18]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, 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), 2306. doi: 10.1063/1.858537.  Google Scholar

[20]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer, 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), R. T. Edwards, 2001, 21-44. Google Scholar

[2]

H. Ali, Approximate deconvolution model in a bounded domain with vertical regularization, J. Math. Anal. App., 408 (2013), 355-363. doi: 10.1016/j.jmaa.2013.06.023.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 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), 149-170. 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), 864-880. 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), 713-732. 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, published online, (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), 1021-1040. 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), 453-480. 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), 1-35. doi: 10.1023/A:1012984210582.  Google Scholar

[14]

M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757. 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), 1205-1209. 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, Paris, 1968.  Google Scholar

[17]

G. R. Sell, Global attractor for the three-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 8 (1996), 1-33. doi: 10.1007/BF02218613.  Google Scholar

[18]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, 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), 2306. doi: 10.1063/1.858537.  Google Scholar

[20]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

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