Degenerate Parabolic Problems in Turbulence Modelling

Abstract

In this paper, we consider one-equation models of turbulence with turbulent-viscosity \( \nu_T = \ell \sqrt{k} \) (\( \ell \) = length scale, \( k \) = mean turbulent kinetic energy). The following system of two parabolic equations represents a simplified model for the turbulent flow of an incompressible fluid through a pipe with cross-section \( \Omega \subset \mathbb{R}^2 \): \[ \frac{\partial u}{\partial t} - \mathop{\mathrm{div}} \left( \sqrt{k} \nabla u \right) = 0, \] \[ \frac{\partial k}{\partial t} - \mathop{\mathrm{div}} \left( (\mu + \sqrt{k}) \nabla k \right) = \sqrt{k} | \nabla u |^2 -k \sqrt{k} , \quad \mbox{in \( \Omega \times ]0,T[ \)} , \] where \( \mu = \mathrm{const} > 0 \). Here, the differential equation on the left is degenerate due to the coefficient \( \sqrt{k} \). We prove the existence of a weak solution \( (u, k) \) of this system under homogeneous boundary conditions and initial conditions \( u(0) = u_0 \) and \( k(0) = k_0 \). The pair \( (u, k) \) exhibits the phenomenon of turbulence as follows. If \[ \int_0^T \int_\Omega | \nabla u |^2 dx dt > 0, \] then there exists a set \( Q^\ast \subset \Omega \times ]0,T[ \) such that \[ \mathop{\mathrm{mes}} Q^\ast > 0, \quad k > 0, \quad \mbox{a.e. in \( Q^\ast \).} \]