%0 Journal Article
%T Analysis of a Singular Convection Diffusion System Arising in Turbulence Modelling
%A P. Dreyfuss
%J International Journal of Partial Differential Equations
%D 2013
%R 10.1155/2013/940924
%X We shall study some singular stationary convection diffusion system governing the steady state of a turbulence model closely related to the one. We shall establish existence, positivity, and regularity results in a very general framework. 1. Introduction We shall first recall some basic ideas concerning the statistical turbulence modelling for fluids. The reader can consult [1, 2] for a more detailed introduction. Let , , , and be the velocity, pressure, density, and temperature of a Newtonien compressible fluid. Let also be a domain which is assumed to be bounded. Then the motion of the flow in at a time can be described by the compressible Navier Stokes equations (see system (C) page 8 in [3]). It is well known that direct simulation based on such a model is harder or even impossible at high Reynolds numbers. The reason is that too many points of discretization are necessary, and so only very simple configurations can be handled. Thus, engineers and physicists have proposed new sets of equations to describe the average of a turbulent flow. The most famous one is the model, introduced by Kolmogorov [4]. We shall briefly present its basic principles in the following. Let denote a generic physical quantity subject to turbulent (i.e., unpredictable at the macroscopic scale); we introduce its mean part (or its esperance) by setting: where the integral is taken in a probablistic context which we shall not detail any more here. Note, however, that the operation is more generally called a filter. The probablistic meaning is one but not the only possible filter (see, for instance, [1] chapter 3). We shall then consider the decomposition: , where is referred to the noncomputable or the nonrelevant part and is called the mean part (i.e., the macroscopic part). The principle of the model is to describe the mean flow in terms of the mean quantities , , , and together with two scalar functions and , which contains relevant information about the small scales processes (or the turbulent processes). The variable (SI: [m2/s2]) is called the turbulent kinetic energy, and [m2/s2] is the rate of dissipation of the kinetic energy. They are defined by where is the molecular viscosity of the fluid. The model is then constructed by averaging (i.e., by appling the operator on) the Navier-Stokes equations. Under appropriate assumptions (i.e., the Reynolds hypothesis in the incompressible case and the Favre average in the compressible case), we obtain a closed system of equations for the variables , , , , , and (see [1] pages 61-62 for the incompressible case and pages 116-117
%U http://www.hindawi.com/journals/ijpde/2013/940924/