\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 76D05; Secondary: 35Q30, 76F65, 76D03.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(129) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return