Motion through spherical droplet with non-homogenous porous layer in spherical container

The problem of the creeping flow through a spherical droplet with a nonhomogenous porous layer in a spherical container has been studied analytically. Darcy's model for the flow inside the porous annular region and the Stokes equation for the flow inside the spherical cavity and container are used to analyze the flow. The drag force is exerted on the porous spherical particles enclosing a cavity, and the hydrodynamic permeability of the spherical droplet with a non-homogeneous porous layer is calculated. Emphasis is placed on the spatially varying permeability of a porous medium, which is not covered in all the previous works related to spherical containers. The variation of hydrodynamic permeability and the wall effect with respect to various flow parameters are presented and discussed graphically. The streamlines are presented to discuss the kinematics of the flow. Some previous results for hydrodynamic permeability and drag forces have been verified as special limiting cases.     

Lattice Boltzmann simulations of turbulent shear flow between parallel porous walls

The effects of two parallel porous walls are investigated, consisting of the Darcy number and the porosity of a porous medium, on the behavior of turbulent shear flows as well as skin-friction drag. The turbulent channel flow with a porous surface is directly simulated by the lattice Boltzmann method （LBM）. The Darcy-Brinkman- Forcheimer （DBF） acting force term is added in the lattice Boltzmann equation to simu- late the turbulent flow bounded by porous walls. It is found that there are two opposite trends （enhancement or reduction） for the porous medium to modify the intensities of the velocity fluctuations and the Reynolds stresses in the near wall region. The parametric study shows that flow modification depends on the Darcy number and the porosity of the porous medium. The results show that, with respect to the conventional impermeable wall, the degree of turbulence modification does not depend on any simple set of param- eters obviously. Moreover, the drag in porous wall-bounded turbulent flow decreases if the Darcy number is smaller than the order of O（10-4） and the porosity of porous walls is up to 0.4.     

Effect of momentum transfer condition at the interface of a model of creeping flow past a spherical permeable aggregate

Creeping flow past an isolated, spherical and permeable aggregate has been studied adopting the Stokes equation to model the fluid external to the aggregate and the Brinkman equation for the internal flow. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used along with the continuity of velocity components and continuity of the normal stress. Using Faxen’s laws, drag and torque are calculated for different flow conditions and it is observed that drag and torque not only change with the permeability of the porous region, but as stress jump coefficient increases, the rate of change in behavior of drag and torque increases.     

Contrôle passif d'écoulements incompressibles autour d'obstacles à l'aide de milieux poreux

Passive control of the flow behind a bluff-body is obtained by integrating porous area on the body. The penalisation method is used to modelize the flow in three different media. In fact each medium can be considered as a porous medium. The fluid is identified as a porous medium of infinite permeability and the solid is identified as a porous medium of zero permeability. This way, it is easy to compute the flow in each medium using the same parameter. Some benefical effects are due to the porous interface: the flow is smoothed, and the enstrophy and drag are significantly reduced.     

Drag reduction of turbulent channel flows over an anisotropic porous wall with reduced spanwise permeability

The direct numerical simulation(DNS) is carried out for the incompressible viscous turbulent flows over an anisotropic porous wall. Effects of the anisotropic porous wall on turbulence modifications as well as on the turbulent drag reduction are investigated. The simulation is carried out at a friction Reynolds number of 180, which is based on the averaged friction velocity at the interface between the porous medium and the clear fluid domain. The depth of the porous layer ranges from 0.9 to 54 viscous units. The permeability in the spanwise direction is set to be lower than the other directions in the present simulation. The maximum drag reduction obtained is about 15.3% which occurs for a depth of 9 viscous units. The increasing of drag is addressed when the depth of the porous layer is more than 25 wall units. The thinner porous layer restricts the spanwise extension of the streamwise vortices which suppresses the bursting events near the wall. However, for the thicker porous layer, the wall-normal fluctuations are enhanced due to the weakening of the wall-blocking effect which can trigger strong turbulent structures near the wall.     

Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity

This paper concerns the Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid in their stream function formulations are used. The hydrodynamic drag force acting on each porous cylindrical particle in a cell and permeability of membrane built up by cylindrical particles with a porous shell are evaluated. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified. Variation of the drag coefficient and dimensionless hydrodynamic permeability with permeability parameter σ, particle volume fraction γ has been studied and some new results are reported. The flow patterns through the regions have been analyzed by stream lines. Effect of particle volume fraction γ and permeability parameter σ on flow pattern is also discussed. In our opinion, these results will have significant contributions in studying, Stokes flow through cylindrical swarms.     

Slow Motion of a Porous Sphere Translating Along the Axis of a Circular Cylindrical Pore Subject to a Stress Jump Condition

The coupled flow problem of an incompressible axisymmetrical quasisteady motion of a porous sphere translating in a viscous fluid along the axis of a circular cylindrical pore is discussed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stress along with continuity of normal stress and velocity components are employed. The flow through the porous particle is governed by the Brinkman model and the flow in the outside porous region is governed by Stokes equations. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The boundary conditions are satisfied first at the cylindrical pore wall by the Fourier transforms and then on the surface of the porous particle by a collocation method. The collocation solutions for the normalized hydrodynamic drag force exerted by the clear fluid on the porous particle is calculated with good convergence for various values of the ratio of radii of the porous sphere and pore, the stress jump coefficient, and a coefficient that is proportional to the permeability. The shape effect of the cylindrical pore on the axial translation of the porous sphere is compared with that of the particle in a spherical cavity; it found that the porous particle in a circular cylindrical pore in general attains a lower hydrodynamic drag than in a spherical envelope.     

Effects of hall current on MHD Couette flow in a channel partially filled with a porous medium in a rotating system

MHD Couette flow in a channel with non-conducting walls, partially filled with a porous medium, is investigated in the presence of an inclined magnetic field in a rotating system. It is observed that the MHD flow behaviour in the channel has been influenced significantly by the Coriolis force, the hydromagnetic force with an inclusion of Hall current and the permeability of the porous medium. Effects of the parameters of these forces on the velocity distributions, induced magnetic field distributions and the skin friction have been depicted graphically and discussed.     

Reduction in drag and vortex shedding frequency through porous sheath around a circular cylinder

A numerical study on the laminar vortex shedding and wake flow due to a porous‐wrapped solid circular cylinder has been made in this paper. The cylinder is horizontally placed, and is subjected to a uniform cross flow. The aim is to control the vortex shedding and drag force through a thin porous wrapper around a solid cylinder. The flow field is investigated for a wide range of Reynolds number in the laminar regime. The flow in the porous zone is governed by the Darcy–Brinkman–Forchheimer extended model and the Navier–Stokes equations in the fluid region. A control volume approach is adopted for computation of the governing equations along with a second‐order upwind scheme, which is used to discretize the convective terms inside the fluid region. The inclusion of a thin porous wrapper produces a significant reduction in drag and damps the oscillation compared with a solid cylinder. Dependence of Strouhal number and drag coefficient on porous layer thickness at different Reynolds number is analyzed. The dependence of Strouhal number and drag on the permeability of the medium is also examined. Copyright © 2009 John Wiley & Sons, Ltd.     

Free Flow at the Interface of Porous Surfaces: A Generalization of the Taylor Brush Configuration

A solution to the problem of shallow laminar water flow above a porous surface is essential when modeling phenomena such as erosion, resuspension, and mass transfer between the porous media and the flow above it. Previous studies proposed theoretical, experimental, and numerical insight with no single general solution to the problem. Many studies have used the Brinkman equation, while others showed that it does not represent the actual interface flow conditions. In this paper we show that the interface macroscopic velocity can be accurately modeled by introducing a modification to the Brinkman equation. A moving average approach was proved to be successful when choosing the correct representative elementary volume and comparing the macroscopic solution with the average microscopic flow. As the size of the representative elementary volume was found to be equal to the product of the square root of the permeability and an exponential function of the porosity, a general solution is now available for any brush configuration. Given the properties of the porous media (porosity and permeability), the flow height and its driving force, a complete macroscopic solution of the interface flow is obtained.     

Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous concentric spherical shell-in-cell model is studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin and Cunningham/Mehta-Morse models are considered. At the fluid-porous interfaces, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. The hydrodynamic drag force acting on the porous shell by the external fluid in each of the four boundary conditions on the cell surface is evaluated. It is found that the normalized mobility of the particles (the hydrodynamic interaction among the porous shell particles) depends not only on the permeability of the porous shells and volume fraction of the porous shell particles, but also on the stress jump coefficient. As a limiting case, the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres with jump.     

Evaluation of the Darcy's law performance for two-fluid flow hydrodynamics in a particle debris bed using a lattice-Boltzmann model

In the present paper, multiphase flow dynamics in a porous medium are analyzed by employing the lattice-Boltzmann modeling approach. A two-dimensional formulation of a lattice-Boltzmann model, using a D2Q9 scheme, is used. Results of the FlowLab code simulation for single phase flow in porous media and for two-phase flow in a channel are compared with analytical solutions. Excellent agreement is obtained. Additionally, fluid-fluid interaction and fluid-solid interaction (wettability) are modeled and examined. Calculations are performed to simulate two-fluid dynamics in porous media, in a wide range of physical parameters of porous media and flowing fluids. It is shown that the model is capable of determining the minimum body force needed for the nonwetting fluid to percolate through the porous medium. Dependence of the force on the pore size, and geometry, as well as on the saturation of the nonwetting fluid is predicted by the model. In these simulations, the results obtained for the relative permeability coefficients indicate the validity of the reciprocity for the two coupling terms in the modified Darcy's law equations. Implication of the simulation results on two-fluid flow hydrodynamics in a decay-heated particle debris bed is discussed. Received on 1 December 1999     

No‐slip consistent immersed boundary particle tracking to simulate impaction filtration in porous media

In this paper, we present a new method for simulating the motion of a disperse particle phase in a carrier gas through porous media. We assume a sufficiently dilute particle‐laden flow and compute, independently of the disperse phase, the steady laminar fluid velocity using the immersed boundary method. Given the velocity of the carrier gas, the equations of motion for the particles experiencing the Stokes drag force are solved to determine their trajectories. The ‘no‐slip consistent’ particle tracking algorithm avoids possible numerical filtration of very small particles due to the nonzero velocity field at the solid–fluid interface introduced by the immersed boundary method. This physically consistent tracking allows a reliable estimation of the filtration efficiency of porous filters due to inertial impaction. We illustrate and test our new approach for model porous media consisting of a structured array of aligned rectangular fibers, arranged in line and staggered. In the staggered geometry, the effect of the residual velocity at the solid–fluid interface is significant for particles with low inertia. Without adopting the developed no‐slip consistent numerical method, an artificial numerical filtration is observed, which becomes dominant for small enough particles. For both the in line and the staggered geometries, the filtration rate depends quite strongly and non monotonically on the particle inertia. This is expressed most clearly in the staggered arrangement in which a very strong increase in the filtration efficiency is observed at a well‐defined critical droplet size, corresponding to a qualitative change in the dominant particle paths in the porous medium. Copyright © 2013 John Wiley & Sons, Ltd.     

A New Application of the Differential Quadrature Element-Incremental Method in Moving-Boundary Problems

Upper bounds on the permeability of random porous media are presented, which improve significantly on existing bounds. The derived bounds rely on a variational formulation of the upscaling problem from a viscous flow at the pore scale, described by Stokes equation, to a Darcy formulation at the macroscopic scale. A systematic strategy to derive upper bounds based on trial force fields is proposed. Earlier results based on uniform void or interface force fields are presented within this unified framework, together with a new proposal of surface force field and a combination of them. The obtained bounds feature detailed statistical information on the pore morphology, including two- and three-point correlation functions of the pore phase, the solid–fluid interface and its local orientation. The required spatial correlation functions are explicitly derived for the Boolean model of spheres, in which the solid phase is modelled as the union of penetrable spheres. Existing and new bounds are evaluated for this model and compared to full-field simulations on representative volume elements. For the first time, bounds allow to retrieve the correct order of magnitude of permeability for a wide range of porosity and even improve on some estimates. However, none of the bounds reproduces the non-analytic behaviour of the permeability–porosity curve at low solid concentration.     

Drag and Separation Flow Past Solid Sphere with Porous Shell at Moderate Reynolds Number

Axisymmetric viscous, two-dimensional steady and incompressible fluid flow past a solid sphere with porous shell at moderate Reynolds numbers is investigated numerically. There are two dimensionless parameters that govern the flow in this study: the Reynolds number based on the free stream fluid velocity and the diameter of the solid core, and the ratio of the porous shell thickness to the square root of its permeability. The flow in the free fluid region outside the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by a Darcy model. Using a commercially available computational fluid dynamics (CFD) package, drag coefficient and separation angle have been computed for flow past a solid sphere with a porous shell for Reynolds numbers of 50, 100, and 200, and for porous parameter in the range of 0.025–2.5. In all simulation cases, the ratio of b/a was fixed at 1.5; i.e., the ratio of outer shell radius to the inner core radius. A parametric equation relating the drag coefficient and separation point with the Reynolds number and porosity parameter were obtained by multiple linear regression. In the limit of very high permeability, the computed drag coefficient as well as the separation angle approaches that for a solid sphere of radius a, as expected. In the limit of very low permeability, the computed total drag coefficient approaches that for a solid sphere of radius b, as expected. The simulation results are presented in terms of viscous drag coefficient, separation angles and total drag coefficient. It was found that the total drag coefficient around the solid sphere as well as the separation angle are strongly governed by the porous shell permeability as well as the Reynolds number. The separation point shifts toward the rear stagnation point as the shell permeability is increased. Separation angle and drag coefficient for the special case of a solid sphere of radius r = a was found to be in good agreement with previous experimental results and with the standard drag curve.     

Analysis of a benchmark solution for non-Newtonian radial displacement in porous media

We present an analytical formulation useful to interpret the key phenomena involved in non-Newtonian displacement in porous media and an analysis of the results obtained by considering the uncertainty associated with relevant problem parameters. To derive a benchmark solution, we consider the radial dynamics of a moving stable interface in a porous domain saturated by two fluids, displacing and displaced, both non-Newtonian of shear-thinning power-law behavior, assuming the pressure and velocity to be continuous at the interface, and constant initial pressure. The flow law for both fluids is a modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation, and taking into account compressibility effects, yields a set of nonlinear second-order partial differential equations. Considering two fluids with the same flow behavior index n allows transformation of the latter equations via a self-similar variable; further transformation of the equations incorporating the conditions at the interface shows for n<1 the existence of a compression front ahead of the moving interface. Solving the resulting set of nonlinear equations yields the positions of the moving interface and compression front, and the pressure distributions; the latter are derived in closed form for any value of n. A sensitivity analysis of the model responses is conducted both in a deterministic and a stochastic framework. In the latter case, Global Sensitivity Analysis (GSA) of the benchmark analytical model is adopted to study how the effects of uncertainty affecting selected parameters: (a) the fluids flow behavior index, (b) the relative total compressibility and mobility in the displaced and displacing fluid domains, and (c) the domain permeability and porosity, propagate to state variables. The relative influence of input parameters on model outputs is evaluated by means of associated Sobol indices, calculated via the Polynomial Chaos Expansion (PCE) technique. The goodness of the results obtained by the PCE is assessed by comparison against a traditional Monte Carlo (MC) approach.     

Natural Convection in a Cavity Filled with Porous Layers on the Top and Bottom Walls

Natural convection in a partially filled porous square cavity is numerically investigated using SIMPLEC method. The Brinkman-Forchheimer extended model was used to govern the flow in the porous medium region. At the porous-fluid interface, the flow boundary condition imposed is a shear stress jump, which includes both the viscous and inertial effects, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The results are presented with flow configurations and isotherms, local and average Nusselt number along the cold wall for different Darcy numbers from 10⁻¹ to 10⁻⁶, porosity values from 0.2 to 0.8, Rayleigh numbers from 10³ to 10⁷, and the ratio of porous layer thickness to cavity height from 0 to 0.50. The flow pattern inside the cavity is affected with these parameters and hence the local and global heat transfer. A modified Darcy–Rayleigh number is proposed for the heat convection intensity in porous/fluid filled domains. When its value is less than unit, global heat transfer keeps unchanged. The interfacial stress jump coefficients β ₁ and β ₂ were varied from  −1 to +1, and their effects on the local and average Nusselt numbers, velocity and temperature profiles in the mid-width of the cavity are investigated.     

Motion of a porous sphere in a spherical container

The creeping motion of a porous sphere at the instant it passes the center of a spherical container has been investigated. The Brinkman's model for the flow inside the porous sphere and the Stokes equation for the flow in the spherical container were used to study the motion. The stream function (and thus the velocity) and pressure (both for the flow inside the porous sphere and inside the spherical container) are calculated. The drag force experienced by the porous spherical particle and wall correction factor is determined. To cite this article: D. Srinivasacharya, C. R. Mecanique 333 (2005).     

Modeling Fluid Flow in Fractured Porous Media with the Interfacial Conditions Between Porous Medium and Fracture

One of the most popular models that has been applied to predict the fluid velocity inside the fracture with impermeable walls is the cubic law. It highlights that the mean flux along the fracture is proportional to the cubic of fracture aperture. However, for a fractured porous medium, the normal and tangential interface conditions between the fracture and porous matrix can change the velocity profile inside the fracture. In this paper, a correction factor is introduced for flow equation along the fracture by imposing the continuity of normal and tangential components of velocity at the interface between the fracture and porous matrix. As a result, the mean velocity inside the fracture depends not only on the fracture aperture, but also on a set of non-dimensional numbers, including the matrix porosity, the ratio of intrinsic permeability of fracture to that of matrix, the wall Reynolds number, and the ratio of normal velocity on one wall to the other. Finally, the introduced correction factor is employed within the extended finite element method, which is widely used for numerical simulation of fluid flow within the fractured porous media. Several numerical results are presented for the fluid flow through a specimen containing single fracture, in order to investigate the deviation from the cubic law in different case studies.

     

Peristaltic Hemodynamic Flow of Couple-Stress Fluids Through a Porous Medium with Slip Effect

The present investigation deals with a theoretical study of the peristaltic hemodynamic flow of couple-stress fluids through a porous medium under the influence of wall slip condition. This study is motivated towards the physiological flow of blood in the micro-circulatory system, by taking account of the particle size effect. Reynolds number is small enough and the wavelength to diameter ratio is large enough to negate inertial effects. Analytical solutions for axial velocity, pressure gradient, frictional force, stream function and mechanical efficiency are obtained. Effects of different physical parameters reflecting couple-stress parameter, permeability parameter, slip parameter, as well as amplitude ratio on pumping characteristics and frictional force, streamlines pattern and trapping of peristaltic flow pattern are studied with particular emphasis. The computational results are presented in graphical form. This study puts forward an important observation that pressure reduces by increasing the magnitude of couple-stress parameter, permeability parameter, slip parameter, whereas it enhances by increasing the amplitude ratio.