Existence for a one-equation turbulent model with strong nonlinearities

  • H. B. de OliveiraEmail author
  • A. Paiva


The purpose of this article is to improve the existence theory for the steady problem of an one-equation turbulent model. For this study, we consider a very general model that encompasses distinct situations of turbulent flows described by the k-epsilon model. Although the boundary-value problem we consider here is motivated by the modelling of turbulent flows through porous media, the importance of our results goes beyond this application. In particular, our results are suited for any turbulent flows described by the k-epsilon model whose mean flow equation incorporates a feedback term, as the Coriolis force, the Lorentz force or the Darcy–Forchheimer’s drag force. The consideration of feedback forces in the mean flow equation will affect the equation for the turbulent kinetic energy (TKE) with a new term that is known as the production and represents the rate at which TKE is transferred from the mean flow to the turbulence. For the associated boundary-value problem, we prove the existence of weak solutions by assuming that the feedback force and the turbulent dissipation are strong nonlinearities, i.e. when no upper restrictions on the growth of these functions with respect to the mean velocity and to the turbulent kinetic energy, respectively, are required. This result improves, in particular, the existence theory for the classical turbulent k-epsilon model which corresponds to assume that both the feedback force and the production term are absent in our model.


Turbulence k-epsilon modelling Strong nonlinearities Existence 

Mathematics Subject Classification

76F60 76S05 35J57 35D30 76D03 


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© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.FCT, Universidade do AlgarveFaroPortugal
  2. 2.CMAFCIO, Universidade de LisboaLisbonPortugal

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