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

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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

Abstract
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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)*, (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

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