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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Antohe, B.V., Lage, J.L.: A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int. J. Heat Mass Transfer 40, 3013–3024 (1997)
Antontsev, S.N., Díaz, J.I., de Oliveira, H.B.: Stopping a viscous fluid by a feedback dissipative field: I. The stationary Stokes problem. J. Math. Fluid Mech. 6, 439–461 (2004)
Antontsev, S.N., Diaz, J.I., de Oliveira, H.B.: Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem. Rend. Mat. Acc. Lincei 15, 257–270 (2004)
Bernis, F.: Elliptic and parabolic semilinear problems without conditions at infinity. Arch. Ration. Mech. Anal. 106, 217–241 (1989)
Boccardo, L., Gallouët, T.: Strongly nonlinear elliptic equations having natural growth terms and \(L^1\) data. Nonlinear Anal. 19(6), 573–579 (1992)
Brezis, H., Browder, F.E.: Strongly non-linear elliptic boundary value problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, 587–603 (1978)
Chacón-Rebollo, T., Lewandowski, R.: Mathematical and numerical foundations of turbulence models and applications. Springer, New York (2014)
Ciarlet, P.: The finite element method for elliptic problems. Elsevier, Amsterdam (1978)
Dreyfuss, P.: Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness and smoothness. Nonlinear Anal. 68, 1462–1478 (2008)
Druet, P.-É., Naumann, J.: On the existence of weak solutions to a stationary one-equation RANS model with unbounded eddy viscosities. Ann. Univ. Ferrara 55, 67–87 (2009)
Gallouët, T., Lederer, J., Lewandowski, R., Murat, F., Tartar, L.: On a turbulent system with unbounded eddy viscosities. Nonlinear Anal. 52, 1051–1068 (2003)
Landes, R.: On the existence of weak solutions of perturbated systems with critical growth. J. Reine Angew. Math. 393, 21–38 (1989)
Lederer, J., Lewandowski, R.: A RANS 3D model with unbounded eddy viscosities. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 413–441 (2007)
de Lemos, M.J.S.: Turbulence in Porous Media, 2nd edn. Elsevier, Waltham (2012)
Lewandowski, R.: The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity. Nonlinear Anal. 28, 393–417 (1997)
Mohammadi, B., Pironneau, O.: Analysis of the K-Epsilon Turbulence Model. Wiley-Masson, Paris (1993)
Nakayama, A., Kuwahara, F.: A macroscopic turbulence model for flow in a porous medium. J. Fluid Eng. 121, 427–433 (1999)
Naumann, J.: Existence of weak solutions to the equations of stationary motion of heat-conducting incompressible viscous fluids. Progr. Nonlinear Differ. Equ. Appl. 64, 373–390 (2005)
Naumann, J., Wolf, J.: On Prandtl’s turbulence model: existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete Contin. Dyn. Syst. Ser. S 6(5), 1371–1390 (2013)
H.B. de Oliveira and A. Paiva. On a one equation turbulent model with feedbacks. In Differential and Difference Equations with Applications, S. Pinelas et al. (eds.), Springer Proc. Math. Stat. 164 (2016), 51–61
H.B. de Oliveira and A. Paiva. A stationary turbulent one-equation model with applications in porous media. J. Math. Fluid Mech. First Online: 12 May 2017
Pedras, M.H.J., de Lemos, M.J.S.: On the definition of turbulent kinetic energy for flow in porous media. Int. Commun. Heat Mass Transfer 27(2), 211–220 (2000)
Rakotoson, J.-M.: Quasilinear elliptic problems with measures as data. Differ. Integral Equ. 4(3), 449–457 (1991)
Temam, R.: Navier-Stokes equations. Elsevier, Amsterdam (1979)
About this article
Cite this article
Oliveira, H.B.d., Paiva, A. Existence for a one-equation turbulent model with strong nonlinearities. J Elliptic Parabol Equ 3, 65–91 (2017). https://doi.org/10.1007/s41808-017-0005-y
- k-epsilon modelling
- Strong nonlinearities
Mathematics Subject Classification