Roger Lewandowski

Mathematical and numerical foundations of turbulence models and applications

Introduction.- Incompressible Navier-Stokes Equations.- Mathematical Basis of Turbulence Modeling.- The k - epsilon Model.- Laws of the Turbulence by Similarity Principles.- Steady Navier-Stokes Equations with Wall Laws and Fixed Eddy Viscosities.- Analysis of the Continuous Steady NS-TKE Model.- Evolutionary NS-TKE Model.- Finite Element Approximation of Steady Smagorinsky Model.- Finite Element Approximation of Evolution Smagorinsky Model.- A Projection-based Variational Multi-Scale Model.- Numerical Approximation of NS-TKE Model.- Numerical Experiments.- Appendix A: Tool Box.

On a turbulent system with unbounded Eddy viscosities

In this system, the functions ν and a are real valued functions of k which represent eddy viscosities. This system is a mathematical subproduct of the large scale one degree closure Reynolds system used by engineers, oceanographs, meteorologists and others for simulating turbulent flows. (The reader can find details concerning modelization in [10], [13] and [3].) The variable k is the turbulent kinetic energy and the variable u an “idealization” of the velocity of the flow. The interest in studying the mathematical system (S) lies in a better understanding of the interaction between an eddy diffusion

A Model for Two Coupled Turbulent Fluids Part II: Numerical Analysis of a Spectral Discretization

We consider a system of equations that models the stationary flow of two immiscible turbulent fluids on adjacent subdomains. The equations are coupled by nonlinear boundary conditions on the interface which is here a fixed given surface. We propose a spectral discretization of this problem and perform its numerical analysis. The convergence of the method is proven in the two-dimensional case, together with optimal error estimates.

Automatic insertion of a turbulence model in the finite element discretization of the Navier-Stokes equations

We consider the finite element discretization of the Navier–Stokes equations locally coupled with the equation for the turbulent kinetic energy through an eddy viscosity. We prove a posteriori error estimates which allow to automatically determine the zone where the turbulent kinetic energy must be inserted in the Navier–Stokes equations and also to perform mesh adaptivity in order to optimize the discretization of these equations. Numerical results confirm the interest of such an approach.

Stability of some turbulent vertical models for the ocean mixing boundary layer

We consider four turbulent models for simulating the boundary mixing layer of the ocean. We show the existence of solutions to these models in the steady state case and then we study the mathematical linear stability of these solutions.

On the Bardina's model in the whole space

We consider the Bardina’s model for turbulent incompressible flows in the whole space with a cut-off frequency of order

Evolutionary NS-TKE Model

\alpha ^{-1} >0

Steady Navier–Stokes Equations with Wall Laws and Fixed Eddy Viscosities

α-1>0. We show that for any