Partial Differential Equations
On the existence of weak solutions to a model problem for the unsteady turbulent pipe-flow
Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 451-456.

We consider a coupled system of PDEs for the scalar functions u and k in a cylinder Ω×]0,T[ (ΩR2 bounded domain, 0<T<+). This system represents a simplified version of Prandtlʼs (1945) model of turbulence in the case of an unsteady motion of a fluid through a pipe with cross-section Ω (u = one-dimensional velocity, k = turbulent kinetic energy). We prove the existence of weak solutions to the problem under consideration with homogeneous Dirichlet conditions on u and homogeneous Neumann conditions on k along Ω×]0,T[, and initial conditions on u and k in Ω×{0}.

On considère un système couplé dʼéquations aux dérivées partielles pour des fonctions scalaires u et k dans un cylindre Ω×]0,T[ (ΩR2 domaine borné, 0<T<+). Ce système représente une version simplifiée du modèle de turbulence de Prandtl (1945) dans le cas de lʼécoulement non stationnaire dʼun liquide dans une conduite de section Ω (u=vitesse à une dimension, k = énergie cinétique de la turbulence). Nous démontrons lʼexistence de solutions faibles pour le système envisagé avec des conditions homogènes de Dirichlet pour u et des conditions de Neumann pour k sur Ω×]0,T[, et des conditions initiales pour des fonctions u et k dans Ω×{0}.

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DOI: 10.1016/j.crma.2013.06.011
Naumann, Joachim 1

1 Department of Mathematics, Humboldt University Berlin, Unter den Linden 6, 10099 Berlin, Germany
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Naumann, Joachim. On the existence of weak solutions to a model problem for the unsteady turbulent pipe-flow. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 451-456. doi : 10.1016/j.crma.2013.06.011. http://www.numdam.org/articles/10.1016/j.crma.2013.06.011/

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