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

Previous Article
On the existence of solutions for a drift-diffusion system arising in corrosion modeling
DCDS-B Home
This Issue
Next Article
Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc

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
Related Papers

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 and 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.
[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.
[3]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
[4]	L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Springer, 2006.
[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.
[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.
[7]	L. C. Berselli and D. Catania, On the equations with anisotropic filter in a vertical pipe, to appear in Dyn. Partial Differ. Equ.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757. doi: 10.1063/1.865649.
[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.
[16]	J. L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Dunod, Paris, 1968.
[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.
[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.
[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.
[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.

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.
[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.
[3]	A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
[4]	L. C. Berselli, T. Iliescu and W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Springer, 2006.
[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.
[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.
[7]	L. C. Berselli and D. Catania, On the equations with anisotropic filter in a vertical pipe, to appear in Dyn. Partial Differ. Equ.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	M. Germano, Differential filters for the large eddy simulation of turbulent flows, Phys. Fluids, 29 (1986), 1755-1757. doi: 10.1063/1.865649.
[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.
[16]	J. L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Dunod, Paris, 1968.
[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.
[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.
[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.
[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.

[1]	Patrick Penel, Milan Pokorný. Improvement of some anisotropic regularity criteria for the Navier--Stokes equations. Discrete and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027
[8]	Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080
[9]	Milan Pokorný, Piotr B. Mucha. 3D steady compressible Navier--Stokes equations. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 151-163. doi: 10.3934/dcdss.2008.1.151
[10]	C. Foias, M. S Jolly, O. P. Manley. Recurrence in the 2-$D$ Navier--Stokes equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 253-268. doi: 10.3934/dcds.2004.10.253
[11]	Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[12]	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
[13]	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 and Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[14]	Konstantinos Chrysafinos. Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1077-1096. doi: 10.3934/dcdsb.2006.6.1077
[15]	Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151
[16]	Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[17]	Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611
[18]	Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101
[19]	Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203
[20]	Yuming Qin, T. F. Ma, M. M. Cavalcanti, D. Andrade. Exponential stability in $H^4$ for the Navier--Stokes equations of compressible and heat conductive fluid. Communications on Pure and Applied Analysis, 2005, 4 (3) : 635-664. doi: 10.3934/cpaa.2005.4.635

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Luca Bisconti Davide Catania
on Google Scholar
Luca Bisconti Davide Catania

[Back to Top]