On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow

Joachim Naumann; Jörg Wolf

doi:10.3934/dcdss.2013.6.1371

Article Contents

2013, Volume 6, Issue 5: 1371-1390. Doi: 10.3934/dcdss.2013.6.1371

This issue Previous Article Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains Next Article A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem

On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow

Joachim Naumann^1, and
Jörg Wolf^1,

1.
Departement of Mathematics, Humboldt University Berlin, Unter den Linden 6, 10099 Berlin

Received: October 31, 2011

Revised: February 29, 2012

Published: September 2013

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Abstract

Starting from Prandtl's (1945) turbulence model, we consider two systems of PDEs for the scalar functions $u$ and $k$ which characterize the stationary turbulent pipe-flow. This system is completed by a homogeneous Dirichlet condition on $u$, and homogeneuos Neumann or mixed boundary conditions on $k$, respectively. For these boundary value problems we prove the existence of weak solutions $(u,k)$ such that $k>0$ on a set of positive measure.

Keywords:

Prandtl's turbulence model,
stationary uni-directional turbulent pipe-flow,
eddy viscosity,
mean turbulent kinetic energy.

Mathematics Subject Classification: Primary: 35J70, 76F02; Secondary: 35Q35, 76F25.

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