**Financial support through the following projects:**

UID/MAT/00297/2019

UID/MAT/00297/2013

PEst-OE/MAT/UI0297/2014

PEst-OE/MAT/UI0297/2011

Analisys Seminar

Speaker: Hermenegildo Borges de Oliveira, FCT - Universidade do Algarve

Date: 18/10/2017

Time: 14:00

Place: Room 1.6, building. VII

Abstract:

A general one-equation turbulent model is studied in this seminar in the steady-state and with homogeneous Dirichlet boundary conditions.

We begin by making a historical review of the equations governing laminar flows in porous media, from Darcy's law to Darcy-Brinkman-Forchheimer's more general model.

Then, using the double averaging concept (in time and in space), we explain how to obtain the most general equations governing turbulent flows through porous media.

The novelty of the problem studied here relies on the consideration of the classical Navier-Stokes equations with a feedback forces field that accounts for drag due to the solid obstacles inside the porous medium.

The presence of this forces field 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 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.

If there is time, we will address the question of existence when both the feedback forces field 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 seminar is mostly based on the follwoing reference: H.B. de Oliveira and A. Paiva, A stationary one-equation turbulent model with applications in porous media, J. Math. Fluid Mechanics, First Online: 12 May 2017.