Mobility of Power-law and Carreau Fluids through Fibrous Media

The flow of generalized Newtonian fluids with a rate-dependent viscosity through fibrous media is studied with a focus on developing relationships for evaluating the effective fluid mobility. Three different methods have been used here: i) a numerical solution of the Cauchy momentum equation with the Carreau or power-law constitutive equations for pressure-driven flow in a fiber bed consisting of a periodic array of cylindrical fibers, ii) an analytical solution for a unit cell model representing the flow characteristics of a periodic fibrous medium, and iii) a scaling analysis of characteristic bulk parameters such as the effective shear rate, the effective viscosity, geometrical parameters of the system, and the fluid rheology. Our scaling analysis yields simple expressions for evaluating the transverse mobility functions for each model, which can be used for a wide range of medium porosity and fluid rheological parameters. While the dimensionless mobility is, in general, a function of the Carreau number and the medium porosity, our results show that for porosities less than $\varepsilon\simeq0.65$, the dimensionless mobility becomes independent of the Carreau number and the mobility function exhibits power-law characteristics as a result of high shear rates at the pore scale. We derive a suitable criterion for determining the flow regime and the transition from a constant viscosity Newtonian response to a power-law regime in terms of a new Carreau number rescaled with a dimensionless function which incorporates the medium porosity and the arrangement of fibers

Forchheimer Model for Non-Darcy Flow in Porous Media and Fractures

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Erosion of a granular bed driven by laminar fluid flow

Motivated by examples of erosive incision of channels in sand, we investigate the motion of individual grains in a granular bed driven by a laminar fluid to give us new insights into the relationship between hydrodynamic stress and surface granular flow. A closed cell of rectangular cross-section is partially filled with glass beads and a constant fluid flux $Q$ flows through the cell. The refractive indices of the fluid and the glass beads are matched and the cell is illuminated with a laser sheet, allowing us to image individual beads. The bed erodes to a rest height $h_r$ which depends on $Q$. The Shields threshold criterion assumes that the non-dimensional ratio $\theta$ of the viscous stress on the bed to the hydrostatic pressure difference across a grain is sufficient to predict the granular flux. Furthermore, the Shields criterion states that the granular flux is non-zero only for $\theta >\theta_c$. We find that the Shields criterion describes the observed relationship $h_r \propto Q^{1/2}$ when the bed height is offset by approximately half a grain diameter. Introducing this offset in the estimation of $\theta$ yields a collapse of the measured Einstein number $q^*$ to a power-law function of $\theta - \theta_c$ with exponent $1.75 \pm 0.25$. The dynamics of the bed height relaxation are well described by the power law relationship between the granular flux and the bed stress.Comment: 12 pages, 5 figure

Pressure jump interface law for the Stokes-Darcy coupling: Confirmation by direct numerical simulations

It is generally accepted that the effective velocity of a viscous flow over a porous bed satisfies the Beavers-Joseph slip law. To the contrary, interface law for the effective stress has been a subject of controversy. Recently, a pressure jump interface law has been rigorously derived by Marciniak-Czochra and Mikeli\'c. In this paper, we provide a confirmation of the analytical result using direct numerical simulation of the flow at the microscopic level.Comment: 25 pages, preprin

Direct numerical simulation of turbulent channel flow over porous walls

We perform direct numerical simulations (DNS) of a turbulent channel flow over porous walls. In the fluid region the flow is governed by the incompressible Navier--Stokes (NS) equations, while in the porous layers the Volume-Averaged Navier--Stokes (VANS) equations are used, which are obtained by volume-averaging the microscopic flow field over a small volume that is larger than the typical dimensions of the pores. In this way the porous medium has a continuum description, and can be specified without the need of a detailed knowledge of the pore microstructure by indipendently assigning permeability and porosity. At the interface between the porous material and the fluid region, momentum-transfer conditions are applied, in which an available coefficient related to the unknown structure of the interface can be used as an error estimate. To set up the numerical problem, the velocity-vorticity formulation of the coupled NS and VANS equations is derived and implemented in a pseudo-spectral DNS solver. Most of the simulations are carried out at $Re_\tau=180$ and consider low-permeability materials; a parameter study is used to describe the role played by permeability, porosity, thickness of the porous material, and the coefficient of the momentum-transfer interface conditions. Among them permeability, even when very small, is shown to play a major role in determining the response of the channel flow to the permeable wall. Turbulence statistics and instantaneous flow fields, in comparative form to the flow over a smooth impermeable wall, are used to understand the main changes introduced by the porous material. A simulations at higher Reynolds number is used to illustrate the main scaling quantities.Comment: Revised version, with additional data and more in-depth analysi