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Existence for turbulent flows through permeable media with unbounded turbulent-depending coefficients

Hermenegildo Borges de Oliveira

Abstract

A mathematical model that governs turbulent flows through permeable media is considered in this work. The model under consideration is based on a double-averaging concept which in turn is described by the time-averaging technique characteristic of the turbulence k-epsilon model and by the volume-averaging methodology that is used to model unstable flows through porous media. The functions of turbulence viscosity, turbulence diffusion and turbulence production are assumed to be unbounded with respect to the turbulent kinetic energy. For the associated initial-and boundary-value problem, we prove the existence of suitable weak solutions.

Existence for turbulent flows through permeable media with unbounded turbulent-depending coefficients

Abstract

A mathematical model that governs turbulent flows through permeable media is considered in this work. The model under consideration is based on a double-averaging concept which in turn is described by the time-averaging technique characteristic of the turbulence k-epsilon model and by the volume-averaging methodology that is used to model unstable flows through porous media. The functions of turbulence viscosity, turbulence diffusion and turbulence production are assumed to be unbounded with respect to the turbulent kinetic energy. For the associated initial-and boundary-value problem, we prove the existence of suitable weak solutions.
Paper Structure (8 sections, 8 theorems, 180 equations)

This paper contains 8 sections, 8 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Truncated problem
  4. Galerkin approximations for the truncated problem
  5. Estimates independent of $j$
  6. Passing to the limit as $j\to\infty$
  7. Passing to the limit as $n\to\infty$
  8. Attainment of the initial conditions

Key Result

Theorem 1

Let $\Omega$ be a bounded domain of $\mathbb{R}^d$, where it is supposed that $2\leq d\leq 4$ and $\partial\Omega$ is Lipschitz-continuous. Assume (e-visc-Carath), (f:visc-turb)-(f:dissip-turb), (g:V'), eq:cond_ini_1-eq:cond_ini_2 and (k0>C), and eq:Cond1-Hyp:theta:gamma hold true. Then, there exist Moreover,

Theorems & Definitions (22)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 12 more