In this paper we introduce and study a new model for three-dimensional turbulence, the Leray-α model. This model is inspired by the Lagrangian averaged Navier–Stokesα model of turbulence (also known Navier–Stokes-α model or the viscous Camassa– Holm equations). As in the case of the Lagrangian averaged Navier–Stokes-α model, the Leray-α model compares successfully with empirical data from turbulent channel and pipe flows, for a wide range of Reynolds numbers. We establish here an upper bound for the dimension of the global attractor (the number of degrees of freedom) of the Leray-α model of the order of (L/ld) 12/7, where L is the size of the domain and ld is the dissipation length-scale. This upper bound is much smaller than what one would expect for three-dimensional models, i.e. (L/ld) 3. This remarkable result suggests that the Leray-α model has a great potential to become a good sub-gridscale large-eddy simulation model of turbulence. We support this observation by studying, analytically and computationally, the energy spectrum and show that in addition to the usual k −5/3 Kolmogorov power law the inertial range has a steeper power-law spectrum for wavenumbers larger than 1/α. Finally, we propose a Prandtllike

The two-dimensional Navier–Stokes-a model is considered on the torus and on the sphere. Upper and lower bounds for the dimension of the global attractors are given. The dependence of the dimension of the global attractors on a is studied. Special attention is given for the limiting cases when a Q 0, that is, when the Navier–Stokes-a model tends to the Navier–Stokes equations, and to the case when a Q..

The α-modeling strategy is followed to derive a new subgrid parameterization of the turbulent stress tensor in large-eddy simulation (LES). The LES-α modeling yields an explicitly filtered subgrid parameterization which contains the filtered nonlinear gradient model as well as a model which represents ‘Leray-regularization’. The LES-α model is compared with similarity and eddy-viscosity models that also use the dynamic procedure. Numerical simulations of a turbulent mixing layer are performed using both a second order, and a fourth order accurate finite volume discretization. The Leray model emerges as the most accurate, robust and computationally efficient among the three LES-α subgrid parameterizations for the turbulent mixing layer. The evolution of the resolved kinetic energy is analyzed and the various subgrid-model contributions to it are identified. By comparing LES-α at different subgrid resolutions, an impression of finite volume discretization error dynamics is obtained.

We provide a treatment of real-valued, smooth, and bounded algebro-geometric solutions of the Camassa-Holm (CH) hierarchy and describe the associated isospectral torus. We also discuss real-valued algebro-geometric solutions with a cusp behavior.

Abstract. We explore a level set representation of vorticity in the study of the singularity problems for incompressible fluid models. This representation exists for all initial vorticity fields. We further apply it to study the 3D Lagrangian averaged Euler equations and the 3D Euler equations, and obtain new global existence conditions.

Abstract. We study the global well-posedness of the Lagrangian averaged Euler equations in three dimensions. We show that a necessary and sufficient condition for the global existence is that the bounded mean oscillation of the stream function is integrable in time. We also derive a sufficient condition in terms of the total variation of certain level set functions, which guarantees the global existence. Furthermore, we obtain the global existence of the averaged two-dimensional (2D) Boussinesq equations and the Lagrangian averaged 2D quasi-geostrophic equations in finite Sobolev space in the absence of viscosity or dissipation.

ABSTRACT. We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n � 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-α model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-α-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. This family of models has become particularly important in the development of mathematical and computational models of turbulence. We give a unified analysis of the entire three-parameter family of models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish some results for singular perturbations,

Abstract. We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatly improves accuracy in simulations. Standard finite element schemes for NS-α suffer from two major sources of error if their solutions are considered approximations to true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure error on the velocity error that arises from the (necessary) use of the rotational form nonlinearity. The proposed scheme “fixes ” these two numerical issues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We prove the scheme is stable, optimally convergent, and the effect of the pressure error on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate the effectiveness of the method.

Cosserat fluids and the continuum mechanics of turbulence: a generalized Navier–Stokes-α

Abstract. We consider the viscous n-dimensional Camassa-Holm equations, with n = 2, 3,4 in the whole space. We establish existence and regularity of the solutions and study the large time behavior of the solutions in several Sobolev spaces. We first show that if the data is only in L 2 then the solution decays without a rate and that this is the best that can be expected for data in L 2. For solutions with data in H m ∩ L 1 we obtain decay at an algebraic rate which is optimal in the sense that it coincides with the rate of the underlying linear part. Quelques questions de decroissance et existence pour les equations visqueuses de Camassa-Holm. Résumé: On considère les équations visqueuses de Camassa–Holm dans R n, n = 2, 3, 4. Nous établissons l’existence et regularité des solutuions. Nous étudions le comportament asymptotique des solutions dans plusieurs espaces de Sobolev quand le temps tend vers l’infini. On montre que si la donnée est seulement dans L 2 la solution decroît vers zero, mais la decroissance ne peux être uniforme. Pour les solutions avec de donnée dans L 1 ∩ H m on obtient une decroissance algébrique avec une vitesse qui est optimale dans le sens que c’est la même que pour les solutions correspondant a l’équation linéaire. 1.