We review the properties of the nonlinearly dispersive Navier–Stokes-alpha (NS-α) model of incompressible fluid turbulence — also called the viscous Camassa–Holm equations in the literature. We first re-derive the NS-α model by filtering the velocity of the fluid loop in Kelvin’s circulation theorem for the Navier–Stokes equations. Then we show that this filtering causes the wavenumber spectrum of the translational kinetic energy for the NS-α model to roll off as k −3 for kα> 1 in three dimensions, instead of continuing along the slower Kolmogorov scaling law, k −5/3, that it follows for kα < 1. This roll off at higher wavenumbers shortens the inertial range for the NS-α model and thereby makes it more computable. We also explain how the NS-α model is related to large eddy simulation (LES) turbulence modeling and to the stress tensor for second-grade fluids. We close by surveying recent results in the literature for the NS-α model and its inviscid limit (the Euler-α model).

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Abstract. In this paper we present analytical studies of three-dimensional viscous and inviscid simplified Bardina turbulence models with periodic boundary conditions. The global existence and uniqueness of weak solutions to the viscous model has already been established by Layton and Lewandowski. However, we prove here the global well-posedness of this model for weaker initial conditions. We also establish an upper bound to the dimension of its global attractor and identify this dimension with the number of degrees of freedom for this model. We show that the number of degrees of freedom of the long-time dynamics of the solution is of the order of (L/ld) 12/5, where L is the size of the periodic box and ld is the dissipation length scale – believed and defined to be the smallest length scale actively participating in the dynamics of the flow. This upper bound estimate is smaller than those established for Navier-Stokes-α, Clark-α and Modified-Leray-α turbulence models which are of the order (L/ld) 3. Finally, we establish the global existence and uniqueness of weak solutions to the inviscid model. This result has an important application in computational fluid dynamics when the inviscid simplified Bardina model is considered as a regularizing model of the three-dimensional Euler equations.

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The purpose of this paper is twofold. First, we give a derivation of the Lagrangian averaged Euler (LAE-α) and Navier-Stokes (LANS-α) equations. This theory involves a spatial scale α and the equations are designed to accurately capture the dynamics of the Euler and Navier-Stokes equations at length scales larger than α, while averaging the motion at scales smaller than α. The derivation involves an averaging procedure that combines ideas from both the material (Lagrangian) and spatial (Eulerian) viewpoints. This framework allows the use of a variant of G. I. Taylor’s “frozen turbulence ” hypothesis as the foundation for the model equations; more precisely, the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow. In this article, we use this hypothesis to derive the averaged Lagrangian for the theory, and all the terms up to and including order α² are accounted for. The equations come in both an isotropic and anisotropic version. The anisotropic equations are a coupled system of PDEs (partial differential equations) for the mean velocity field and the Lagrangian covariance tensor. In earlier works by Foias, Holm &Titi [10], and ourselves [16], an analysis of the isotropic equations has been given. In the second part of this paper, we establish local in time well-posedness of the anisotropic LANS-α equations using quasilinear PDE type methods.

Abstract. In this paper we study a well-known three–dimensional turbulence model, the filtered Clark model, or Clark−α model [7]. This is Large Eddy Simulation (LES) tensor-diffusivity model of turbulent flows with an additional spatial filter of width α. We show the global well-posedness of this model with constant Navier-Stokes (eddy) viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an anaytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to (L/ld) 3, where L is the integral spatial scale and ld 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 semirigorous 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 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.

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..

We present a framework for discussing LES equations with nonlinear dispersion. In this framework, we discuss the properties of the nonlinearly dispersive Navier-Stokes-alpha (NS−α) model of incompressible fluid turbulence — also called the viscous Camassa-Holm equations in the literature — in comparison with the corresponding properties of large eddy simulation (LES) equations obtained via the approximate-inverse approach. In this comparison, we identify the spatially filtered NS−α equations with a class of generalized LES similarity models. Applying a certain approximate inverse to this class of LES models restores the Kelvin circulation theorem for the defiltered velocity and shows that the NS−α model describes the dynamics of the defiltered velocity for this class of generalized LES similarity models. We also show that the subgrid scale forces in the NS−α model transform covariantly under Galilean transformations and under a change to a uniformly rotating reference frame. Finally, we discuss in the spectral formulation how the NS−α model retains the local interactions among the large scales; retains the nonlocal sweeping effects of large scales on small scales; yet attenuates the local interactions of the small scales amongst themselves. 1

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. The Navier-Stokes-Voigt (NSV) model of viscoelastic incompressible fluid has been recently proposed as a regularization of the 3D Navier-Stokes equations for the purpose of direct numerical simulations. In this work we investigate its statistical properties by employing phenomenological heuristic arguments, in combination with Sabra shell model simulations of the analogue of the NSV model. For large values of the regularizing parameter, compared to the Kolmogorov length scale, simulations exhibit multiscaling inertial range, and the dissipation range displaying low intermittency. These facts provide evidence that the NSV regularization may reduce the stiffness of direct numerical simulations of turbulent flows, with a small impact on the energy containing scales. AMS subject classification: 35Q30, 35Q35, 76F20, 76F55 1 Keywords:Navier-Stokes-Voigt equations, Navier-Stokes-Voight equations, Navier-Stokes equations, regularization of the Navier-Stokes equations, turbulence models, viscoelastic models, Shell models, Dynamic models. 1