A Stationary One-Equation Turbulent Model with Applications in Porous Media
- 52 Downloads
A one-equation turbulent model is studied in this work in the steady-state and with homogeneous Dirichlet boundary conditions. The considered problem generalizes two distinct approaches that are being used with success in the applications to model different flows through porous media. The novelty of the problem relies on the consideration of the classical Navier–Stokes equations with a feedback forces field, whose presence in the momentum 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. By assuming suitable growth conditions on the feedback forces field and on the function that describes the rate of dissipation of the TKE, as well as on the production term, we will prove the existence of the velocity field and of the TKE. The proof of their uniqueness is made by assuming monotonicity conditions on the feedback forces field and on the turbulent dissipation function, together with a condition of Lipschitz continuity on the production term. The existence of a unique pressure, will follow by the application of a standard version of de Rham’s lemma.
KeywordsTurbulence k-epsilon modelling Porous media Existence Uniqueness
Mathematics Subject Classification76F60 76S05 35J57 35D30 76D03
Unable to display preview. Download preview PDF.
- 6.Antontsev, S.N., Díaz, J.I., de Oliveira, H.B.: Stopping a viscous fluid by a feedback dissipative field: thermal effects without phase changing. In: Progr. Nonlinear Differential Equations Appl., vol. 61, pp. 1–14. Birkhäuser (2005)Google Scholar
- 12.de Lemos, M.J.S.: Turbulence in Porous Media, 2nd edn. Elsevier, Waltham (2012)Google Scholar
- 13.de Oliveira, H.B., Paiva, A.: On a one equation turbulent model with feedbacks. In: Pinelas, S., et al. (eds.) Differential and Difference Equations with Applications, Springer Proc. Math. Stat., vol. 164 (2016)Google Scholar
- 15.Druet, P.-E.: On existence and regularity of solutions for a stationary Navier–Stokes system coupled to an equation for the turbulent kinetic energy. Weierstrass Institut für Angewandte Analysis und Stochastik, Preprint 2007–13Google Scholar
- 17.Evans, L.C.: Partial Differential Equations. Graduate Studies in Math., vol. 19. American Mathematical Society, Providence (1998)Google Scholar
- 27.Mohammadi, B., Pironneau, O.: Analysis of the K-Epsilon Turbulence Model. Wiley-Masson, Paris (1993)Google Scholar
- 28.Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/71)Google Scholar
- 30.Naumann, J.: Existence of weak solutions to the equations of stationary motion of heat-conducting incompressible viscous fluids. In: Progr. Nonlinear Differential Equations Appl., vol. 64, pp. 373–390, Birkhäuser, Basel (2005)Google Scholar