Hypothèses de mélange de Prandtl

vignette|La longueur de mélange est la distance sur laquelle une parcelle de fluide conservera ses caractéristiques d'origine avant de se disperser dans le fluide environnant. Ici, la barre située à gauche de la figure représente la longueur de mélange. L’existence d’une longueur de mélange dans un écoulement fluide est une hypothèse importante formulée en 1925 par L. Prandtl, afin de calculer les inconnues supplémentaires qui apparaissent dans les équations de Navier-Stokes d'un fluide en régime turbulent. Principe En mécanique des fluides, les tourbillons peuvent être représentés comme le produit d’une vitesse et d'une longueur caractéristique, dite longueur de mélange. Dans un écoulement à deux fluides miscibles, par exemple, cette longueur pourra être la distance entre le point d'impact de deux jets fluides, et le point où le champ de vitesse de l'écoulement devient uniforme (jet homocinétique). Elle est également appelée « longueur de cohérence », puisqu'on peut aussi

Approximation numérique des écoulements turbulents dans des cuves d'électrolyse de l'aluminium

Jonathan Rochat

Aluminium is a metal extracted from bauxite ore using electrolysis process in cells of big size. A huge electric current goes in the cell through an electrolytic bath and aluminium liquid. These currents generate strong magnetic forces that allow the bath and the aluminium to move. A good knowledge of these turbulent flows is very important to optimize the process. The purpose of this thesis is to study and simulate turbulent flows in the aluminium smelting process. These flows are solved numerically with a finite element method. In particular, the Navier-Stokes equations for bifluid flows with free moving interface are solved numerically. In the first part of this work, we develop some mixing-length models that take into account the effects of the wall. A theoretical mathematical study shows the validity of these models and we give some recommendation on the choice of the parameters of the computation. In the second part we study the resolution of the Navier-Stokes equations. The study focuses on algorithms that decouple the computation of the speed and pressure, commonly called projection method or Chorin-Temam algorithm. The final section provides answers on the relevance of wall modelling and projection methods in numerical simulation of turbulent flows in the aluminium smelting process. In particular, we obtain a numerical model that produces a realistic flow with a reasonable CPU time and we discuss the choice of certain parameters involved in the different models.

EPFL2016

On Von Karman Modeling For Turbulent Flow Near A Wall

Jacques Rappaz, Jonathan Rochat

Mixing-length models are often used by engineers in order to take into account turbulence phenomena in a flow. This kind of model is obtained by adding a turbulent viscosity to the laminar one in Navier-Stokes equations. When the flow is confined between two close walls, von Karman's model consists of adding a viscosity which depends on the rate of strain multiplied by the square of distance to the wall. In this short paper, we present a mathematical analysis of such modeling. In particular, we explain why von Karman's model is numerically ill-conditioned when using a finite element method with a small laminar viscosity. Details of analysis can be found in [1], [2].

2019

Systematic errors of skin-friction measurements by oil-film interferometry

Peter Monkewitz, Jean-Daniel Rüedi

In recent years, the independent measurement of wall shear stress with oil-film or oil-drop interferometry has become a cornerstone of turbulent-boundary-layer research as many arguments depend critically on a precise knowledge of the skin friction tau(w). To our knowledge, all practitioners of oil-drop interferometry have so far used the leading-order similarity solution for asymptotically thin, wedge-shaped, two-dimensional oil films established by Tanner & Blows (J. Phys. E: Sci. Instrum., vol. 9, 1976, pp. 194202) to relate the evolution of drop thickness to tau(w). It is generally believed that this procedure, if carefully implemented, yields the true time-averaged tau(w)* within +/- 1% or possibly better, but the systematic errors due to the finite thickness of the oil film have never been determined. They are analysed here for oil films with a thickness of the order of a viscous unit in a zero-pressure-gradient turbulent boundary layer. Neglecting spanwise surface curvature and surface tension effects, corrections due to the secondary air boundary layer above the oil film are derived with a linearised triple-layer approach that accounts for the turbulent shear-stress perturbation by means of modified van-Driest-type closure models. In addition, the correction due to processing oil drops with a slight streamwise surface curvature as if they were exact wedges is quantified. Both corrections are evaluated for oil-drop interferograms acquired in a zero-pressure-gradient turbulent boundary layer at a Reynolds number of around 3500, based on displacement thickness, and are shown to produce a reduction of the friction velocity relative to the basic Tanner and Blows theory of between -0.1% and -1.5 %, depending on the mixing-length model. Despite the uncertainty about the true correction, the analysis allows the formulation of some guidelines on where and when to analyse interference fringes in order to minimise the error on the measured wall shear stress.

Cambridge University Press2015

Systematic errors of skin-friction measurements by oil-film interferometry

Peter Monkewitz, Jean-Daniel Rüedi

Cambridge University Press2015