Defining the averaging volume for porous media in inertial to transitional regimes

Porous media are intrinsically multi-scale in nature, and we can therefore separate the velocity and pressure fields into a mean and a fluctuating component e.g.: u = <u>+ u’, allowing us to separate the small-scale effects -represented by the fluctuating components- and the larger scale effects by studying the averaged field. This averaged field is obtained by spatially smoothing the quantities involved in the fluid phase over a representative volume that involves both the fluid and solid phases, and some questions that naturally arise are: what is the appropriate size of the volume taken into account into the averaging process? and what are the limitations of the method for the different volumes?

We aim to explore this subject both numerically and experimentally in a transitional regime (from laminar to turbulent) by studying the pore-scale hydrodynamics of fixed beds of spherical particles in random arrangements. To achieve this,  an experimental set up was designed to use the Particle Tracking Velocimetry (PTV) technique to obtain the local velocity field. Velocity statistics were computed, such as the Reynolds stress tensor and velocity correlation functions which allow, for instance, to obtain a characteristic length representative of the length-scale needed for averaging the equations. We also study the local porosity of the medium and if there is any preferential distribution in the passage of the flow. The experimental results are to be then compared with those obtained by Direct Numerical Simulations in three-periodic domains, where we can also explore further effects, such as varying the volume of the unit cell.