On the Clark–α model of turbulence: global regularity and long-time dynamics

In this paper we study a well-known three-dimensional turbulence model, the filtered Clark model, or Clark–α model. This is a large eddy simulation (LES) tensor-diffusivity model of turbulent flows with an explicit spatial filter of width α. We show the global well-posedness of this model with constant Navier–Stokes viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an analytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to (L/l d )3, where L is the integral spatial scale and l d is the viscous dissipation length scale. This explicit bound is consistent with the physical estimate for the number of degrees of freedom based on heuristic arguments. Using semi-rigorous physical arguments we show that the inertial range of the energy spectrum for the Clark–α model has the usual k −5/3 Kolmogorov power law for wave numbers kα ≪ 1 and k −3 decay power law for kα ≫ 1. This is an evidence that the Clark–α model parameterizes efficiently the large wave numbers within the inertial range, kα ≫ 1, so that they contain much less translational kinetic energy than their counterparts in the Navier–Stokes equations.

[1]  Darryl D. Holm,et al.  The Navier–Stokes-alpha model of fluid turbulence , 2001, nlin/0103037.

[2]  Richard E. Mortensen,et al.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam) , 1991, SIAM Rev..

[3]  G. Gustafson,et al.  Boundary Value Problems of Mathematical Physics , 1998 .

[4]  Darryl D. Holm,et al.  A connection between the Camassa–Holm equations and turbulent flows in channels and pipes , 1999, chao-dyn/9903033.

[5]  Darryl D. Holm,et al.  On a Leray–α model of turbulence , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  Peter Constantin,et al.  Global Lyapunov Exponents, Kaplan-Yorke Formulas and the Dimension of the Attractors for 2D Navier-Stokes Equations , 1985 .

[7]  Edriss S. Titi,et al.  GEVREY REGULARITY FOR NONLINEAR ANALYTIC PARABOLIC EQUATIONS , 1998 .

[8]  J. Ferziger,et al.  Evaluation of subgrid-scale models using an accurately simulated turbulent flow , 1979, Journal of Fluid Mechanics.

[9]  R. Kraichnan.  Inertial Ranges in Two‐Dimensional Turbulence , 1967 .

[10]  N. Mansour.  Large-eddy simulation of a turbulent mixing layer , 1978 .

[11]  R. Temam.  Navier-Stokes Equations , 1977 .

[12]  Darryl D. Holm,et al.  Camassa-Holm Equations as a Closure Model for Turbulent Channel and Pipe Flow , 1998, chao-dyn/9804026.

[13]  A. Eden,et al.  Exponential Attractors for Dissipative Evolution Equations , 1995 .

[14]  A. Leonard.  Energy Cascade in Large-Eddy Simulations of Turbulent Fluid Flows , 1975 .

[15]  Darryl D. Holm,et al.  Direct numerical simulations of the Navier–Stokes alpha model , 1999, Physica D: Nonlinear Phenomena.

[16]  R. Temam,et al.  Gevrey class regularity for the solutions of the Navier-Stokes equations , 1989 .

[17]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[18]  O. Ladyzhenskaya,et al.  Attractors for Semigroups and Evolution Equations , 1991 .

[19]  Darryl D. Holm,et al.  The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory , 2001, nlin/0103039.

[20]  Hans Kuerten,et al.  Large-eddy simulation of the temporal mixing layer using the Clark model , 1996 .

[21]  Darryl D. Holm,et al.  The Camassa-Holm equations and turbulence , 1999 .

[22]  Meinhard E. Mayer,et al.  Navier-Stokes Equations and Turbulence , 2008 .

[23]  Hervé Jeanmart,et al.  Explicit-filtering large-eddy simulation using the tensor-diffusivity model supplemented by a dynami , 2001 .

[24]  Roger Temam,et al.  Navier-Stokes Equations and Turbulence by C. Foias , 2001 .

[25]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[26]  C. Foias.  What do the Navier-Stokes equations tell us about turbulence , 1995 .

[27]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .