Existence for a one-equation turbulent model with strong nonlinearities

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.info:eu-repo/semantics/publishedVersio

Modeling the pore level fluid flow in porous media using the immersed boundary method

This chapter demonstrates the potential of the immersed boundary method for the direct numerical simulation of the flow through porous media. A 2D compact finite differences method was employed to solve the unsteady incompressible Navier-Stokes equations with fourth-order Runge-Kutta temporal discretization and fourth-order compact schemes for spatial discretization. The solutions were obtained in a Cartesian grid, with all the associated advantages. The porous media is made of equal size square cylinders in a staggered arrangement and is bounded by solid walls. The transverse and longitudinal distances between cylinders are equal to two cylinder diameters and at the inlet a fully developed velocity profile is specified. The Reynolds number based on the cylinder diameter and maximum inlet velocity ranges from 40 to 80. The different flow regimes are identified and characterised, along with the prediction of the Reynolds number at which transition from steady to unsteady flow takes place. Additionally, the average drag and lift coefficients are presented as a function of the Reynolds number