On the existence of weak solutions to a model problem for the unsteady turbulent pipe-flow

Abstract

We consider a coupled system of PDEs for the scalar functions u and k in a cylinder Ω×]0,T[ (Ω⊂R2 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}.