The Kolmogorov Law of turbulence: what can rigorously be proved?

Conferencia plenaria por invitaciónWe define a mathematical framework in which we can specify the Reynolds decomposition and the correlation tensors of an incompressible locally homogeneous and isotropic turbulent flow. After having fixed the technical background and some probabilistic tools, we focus on the 2-order correlation tensor, which is the covariance matrix of the velocity vectors at two different points of the flow. We perform a Taylor expansion of this matrix when the two points are close to one another. We characterize the principal part of this expansion, for which we prove the law of the 2/3 by a mathematical similarity principle.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Conferencias plan propio de investigación de la UM

A high accuracy Leray-deconvolution model of turbulence and its limiting behavior

In 1934 J. Leray proposed a regularization of the Navier-Stokes equations whose limits were weak solutions of the NSE. Recently, a modification of the Leray model, called the Leray-alpha model, has atracted study for turbulent flow simulation. One common drawback of Leray type regularizations is their low accuracy. Increasing the accuracy of a simulation based on a Leray regularization requires cutting the averaging radius, i.e., remeshing and resolving on finer meshes. This report analyzes a family of Leray type models of arbitrarily high orders of accuracy for fixed averaging radius. We establish the basic theory of the entire family including limiting behavior as the averaging radius decreases to zero, (a simple extension of results known for the Leray model). We also give a more technically interesting result on the limit as the order of the models increases with fixed averaging radius. Because of this property, increasing accuracy of the model is potentially cheaper than decreasing the averaging radius (or meshwidth) and high order models are doubly interesting

Convergence of approximate deconvolution models to the mean Magnetohydrodynamics Equations: Analysis of two models

We consider two Large Eddy Simulation (LES) models for the approximation of large scales of the equations of Magnetohydrodynamics (MHD in the sequel). We study two $\alpha$-models, which are obtained adapting to the MHD the approach by Stolz and Adams with van Cittert approximate deconvolution operators. First, we prove existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations. Then we study the behavior of solutions as the deconvolution parameter goes to infinity. The main result of this paper is the convergence to a solution of the filtered MHD equations. In the final section we study also the problem with filtering acting only on the velocity equation

Numerical simulation of water flow around a rigid fishing net

This paper is devoted to the simulation of the flow around and inside a rigid axisymmetric net. We describe first how experimental data have been obtained. We show in detail the modelization. The model is based on a Reynolds Averaged Navier-Stokes turbulence model penalized by a term based on the Brinkman law. At the out-boundary of the computational box, we have used a "ghost" boundary condition. We show that the corresponding variational problem has a solution. Then the numerical scheme is given and the paper finishes with numerical simulations compared with the experimental data.Comment: 39 page

The Kolmogorov Law of turbulence, What can rigorously be proved ? Part II

International audienceWe recall what are the different known solutions for the incompressible Navier-Stokes Equations, in order to fix a suitable functional setting for the probabilistic frame that we use to derive turbulence models, in particular to define the mean velocity and pressure fields, the Reynolds stress and eddy viscosities. Homogeneity and isotropy are discussed within this framework and we give a mathematical proof of the famous −5/3 Kolmogorov law, which is discussed in a numerical simulation performed in a numerical box with a non trivial topography on the ground

Attractors for a deconvolution model of turbulence

We consider a deconvolution model for 3D periodic flows. We show the existence of a global attractor for the model

Error estimates in approximate deconvolution models

International audienceWe consider general Approximate Deconvolution Models (ADM). We estimate the error modeling as a function of the residual stress $\tau_N$ and we compute the rate of convergence to the mean Navier-Stokes Equations in terms of the deconvolution order $N$

Consistency and feasibility of approximate deconvolution models of turbulence

We prove that the time averaged consistency error of the Nth approximate deconvolution LES model converges to zero uniformly in the kinematic viscosity and in the Reynolds number as the cube root of the averaging radius. We also give a higher order but non-uniform consistency error bound for the zeroth order model directly from the Navier-Stokes equations