Convection In Porous Medium Research Articles

Effect of The Aspect Ratio on The Natural Convection in a Rectangular Porous Medium.(Dept.M)

The effect of the aspect ratio on natural convection in a two dimensional rectangular cavity filled with fluid eaturated porous media is examined numerically. The vertical walls of the cavity are adiabatic and the horizontal walls are isothermal with lower hotter one and upper colder one. The test is done for aspect ratios 1 to 8 and wide range of Darcy-Rayleigh number Ra up to 800. Results are presented. In terms of the streamlines and isotherms, maximum velocity and stream function, the local and global Nusselt number as well as the number of flow cells. By the increase of the aspect ratio a two dimensional cellular pattern appears with vigorous motion as the Darcy-Rayleigh number increases. The number of flow cells increases with the aspect ratio. The non-dimensional average flow velocity is independent on the aspect ratio and increases linearly with the increase of Ra. The extremun value of non-dimensional stream function decreases with the increase of aspect ratio. The turning points in the local Nu at the isothermal walls occurs at the boundaries of the cells. Its number equals to the cells number and increases as the aspect ratio increases. The global Nu at the isothermal walls is independent on the aspect ratio and increases with Ra due to the relation Nu = 1.59344 in Ra – 5.186 .

Porous medium convection in a chemically reacting ferrofluid with lower boundary subjected to constant heat flux

The effect of exothermic chemical reaction of zero-order on Bénard-Darcy ferroconvection is investigated using the technique of small perturbation. The eigenvalues associated with an adiabatic lower wall are determined by employing the Galerkin method. The Darcy-Rayleigh number is computed in terms of the parameters pertaining to chemical reaction and ferromagnetic fluid. It is established that, when chemical reaction escalates, there is a considerable shift from linearity and occurrence of asymmetry in the basic temperature profiles. It is ascertained that the threshold of Bénard-Darcy ferroconvection is augmented through the stresses of both mechanisms due to chemical reaction and magnetization, and the ferroconvective instability due to nonlinearity of magnetization is rather inconsequential when chemical reaction is present. It is also shown that the destabilizing feature of magnetic forces resulting from the fluid magnetization is less pronounced when chemical reaction is present.

Lagrangian dynamics and heat transfer in porous-media convection

Abstract

Darcy–Carreau–Yasuda rheological model and onset of inelastic non-Newtonian mixed convection in porous media

An extension of Carreau and Carreau–Yasuda rheological models to porous media is proposed to study the onset of mixed convection of both pseudoplastic fluids (PF) and dilatant fluids (DF) in a porous layer heated from below in the presence of a horizontal throughflow. In comparison with Newtonian fluids, three more dimensionless parameters are introduced, namely, the Darcy–Weissenberg number Wi, the power–law index n, and the Yasuda parameter a. Temporal stability analysis of the basic state showed that in the absence of a throughflow (Wi = 0), the critical Rayleigh number and the critical wavenumber at the onset of convection are the same as for Newtonian fluids, namely, Rac=4π2 and kc=π, respectively. When the throughflow is added (Wi > 0), it is found that moving transverse rolls (stationary longitudinal rolls) are the dominant mode of the instability for PF (for DF). Furthermore, depending on Wi, two regimes of instability were identified. In the weakly non-Newtonian regime (i.e., Wi<Wit≈1), a destabilizing effect is observed for PF, while the reverse occurs for DF. These effects are more intense by reducing (increasing) the index n for PF (for DF). In this regime, a significant qualitative difference is found between the Darcy–Carreau model and the power–law model. However, in the strongly non-Newtonian regime, the two models lead to similar results. A mechanical energy budget analysis is performed to understand the physical effects of the interaction between the basic throughflow and the disturbances. It is also shown that the intrinsic macroscale properties of the porous medium may play a key role in the stabilizing/destabilizing effect. Finally, a comparison is made between the present theoretical predictions and recent mixed convection experiments in a Hele–Shaw cell.

Transitions and bifurcations of Darcy-Brinkman-Marangoni convection

<p style='text-indent:20px;'>This study examines dynamic transitions of Brinkman equation coupled with the thermal diffusion equation modeling the surface tension driven convection in porous media. First, we show that the equilibrium of the equation loses its linear stability if the Marangoni number is greater than a threshold, and the corresponding principle of exchange stability (PES) condition is then verified. Second, we establish the nonlinear transition theorems describing the possible transition types associated with the linear instability of the equilibrium by applying the center manifold theory to reduce the infinite dynamical system to a finite dimensional one together with several non-dimensional transition numbers. Finally, careful numerical computations are performed to determine the sign of these transition numbers as well as related transition types. Our result indicates that the system favors all three types of transitions. Unlike the buoyancy forces driven convections, jump and mixed type transition can occur at certain parameter regimes.</p>

Three-Dimensional Imaging of Dissolution-Driven Convection in Simple Porous Media Using X-ray CT Scanning

Motivated by geological carbon dioxide (CO2) storage and the associated process of CO2 convective mixing in brine, we present an experimental method to visualize dissolution-driven convection in a three-dimensional porous medium. Owing to the difficulty of imaging the time-dependent convective process within an opaque medium, experiments so far have largely focused on two-dimensional systems. However, this imposes a strong restriction on the lateral spreading of the plumes. To explore the flow pattern within a three-dimensional medium, we present an experimental procedure employing X-ray computed tomography that enables the visualization of the spatial and temporal evolution of the mixing process non-invasively. To mimic the dissolution of CO2 in brine, we use potassium iodide (KI) in water. Here, we show the results with high spatial and temporal resolution for an experiment with a homogeneous packing of glass beads (Ra=5300). We discuss the evolution of the average solute mass fraction, compare it to numerical results of a purely diffusive process and report a strong enhancement of the dissolution process when convection dominates over diffusive mass transfer effects.

Visual investigation of CO2 dissolution and convection in heterogeneous porous media at reservoir temperature and pressure conditions

AbstractCO2 convective mixing in saline aquifers has been widely studied numerically and experimentally. Reservoir heterogeneity is significant for CO2 convective mixing and experimental studies are still limited. In this study, we have conducted a visualization of CO2 convective mixing experiments in heterogeneous porous media at reservoir conditions using CO2 and water. We have used a two‐dimension Hele–Shaw cell, different glass beads of different permeability at porous media, and water solution with pH indicator. Glass beads were packed in a different way (horizontally and vertically) to generate the heterogeneity inside the test cell. We have studied transport velocity deviation due to the heterogeneity and effects of permeability transition zone together with the effects of boundary conditions. It was found out that having a low permeable layer below a high permeable layer restructure the flow of CO2 fingers and dampens the CO2 transport velocity. With the vertical permeability zones, having a high permeability zone accelerates CO2 gravity transport through that zone which is a good representation for a fracture or a fault in the reservoir. CO2 convection onset is governed by the vertical high‐permeable layer. Boundary conditions have been dominant with the presence of high permeable zones. It also found out that the experimental results presented in this study match with the simulation studies that are available in the literature. © 2021 The Authors. Greenhouse Gases: Science and Technology published by Society of Chemical Industry and John Wiley & Sons Ltd.

Towards the ultimate regime in Rayleigh–Darcy convection

Abstract

Pore-scale simulation of salt fingers in porous media using a coupled iterative source-correction immersed boundary-lattice Boltzmann solver

In the conventional representative elementary volume-scale models, details of fluid movement, thermal transfer and mass transport in the porous media are relatively hard to capture, while the understanding of them is crucial in many diverse engineering and environmental applications such as running nuclear power facilities or pollutant diffusion in soil environment. In this paper, a pore-scale lattice Boltzmann method (LBM) modeling of double diffusion systems in saturated porous media is developed to investigate the role played by pore structure in the behavior of salt finger phenomena together with their fluid flow mechanism. An iterative source-correction immersed boundary method is developed to describe the solid skeleton with different boundary conditions for velocity field, heat transfer, and mass transport, respectively. The natural convection in a vertical annulus and the conjugate heat transfer problem in a two-layer horizontal annulus are employed to verify the accuracy and the applicability of the immersed boundary method for Dirichlet-type, Neumann-type and conjugate heat transfer boundary conditions, respectively. Comparisons between the present results and existing investigation show a very good agreement and demonstrate that the double diffusive process in porous media can be predicted with a good accuracy by the proposed model. Subsequently, the instability evolution process of salt finger-type of double diffusive convection in porous media is investigated, with a focus on the structure and behavior of salt fingers and the kinetic energy evolution. The obtained simulation results show that in saturated porous media, relatively large vertical mass fluxes can be generated even if the density distribution is of dynamical stability.

Quantitative study of density-driven convection mass transfer in porous media by MRI

CO2 geologic storage in deep saline aquifers is a promising technology to reduce greenhouse gas emissions and maintain the pace of economic development. The dissolution and convection of CO2 injected into brines are the primary trapping mechanisms to enhance storage efficiency and provide long-term storage security. The spatial mass transfer characteristics of convection in porous media remain poorly understood. In this study, two kinds of analogous fluid pairs were used as the equivalents of CO2-saturated brine and in situ brine. Density-driven convection experiments using magnetic resonance imaging (MRI) technology in heterogeneous porous media were performed at various stages, including the diffusion stage, the unstable trigger stage and the convection development stage. Combined with the spatial evolution of the fingers and the change in MRI mean intensity, the local fluid concentration and overall mass transfer behavior could be quantified. Physical quantities associated with mass transfer by convection, such as the mixing layer thickness and scalar dissipation rate at the diffusion stages, the onset time at the unstable trigger stage, the average velocity of the front finger, the finger number, the transverse concentration gradient of the finger, and the sweep area of the finger at the convection development stage, were also analyzed systematically. We found that convective mixing substantially promotes mass transfer over pure diffusion, and the dissolution rate can be further enhanced by improving the injection method at each stage. The range of Ra in this study covers most CO2 capture and storage (CCS) sites and can provide basic data for engineering applications.

A general numerical procedure for solidification and melting in porous media and free fluids

A numerical model of the solid/liquid phase change problem in porous media and free fluids is presented in this paper for the first time by employing a modified version of the generalized porous medium model, in 2D and 3D, solved by means of the Characteristic-Based-Split (CBS) scheme. The proposed procedure is based on the solution of one set of equations to describe different phenomena: natural convection in porous media, solidification and melting both in porous media and in free fluids. The generalized model has been modified to incorporate the apparent heat capacity, source term and pseudo-viscosity methods in order to reproduce phase change phenomena.Two versions of the algorithm, the Semi-Implicit (SI) and the Quasi-Implicit (QI) schemes, have been developed to solve the governing equations, and validated against numerical and experimental data available in the literature. The two schemes have been compared in terms of accuracy and efficiency for the solution of the benchmark problems.

A Fourier Series Solution for Transient Three‐Dimensional Thermohaline Convection in Porous Enclosures

AbstractThermohaline convection (THC) in porous media is frequently investigated using the problem of porous enclosure. Most of the existing modeling‐based studies are limited to 2‐D simulations, because 2‐D assumption is widely used to deal with computational requirement of 3‐D numerical solutions. Analytical solutions serve as an alternative to deal with computational requirement of numerical solutions. Existing analytical solutions of THC are mostly limited to 2‐D and also under steady‐state regime. In this work, we develop a meshless 3‐D semianalytical solution for the problem of THC in a porous box under crossed thermal and solute gradients, for both steady state and transient regimes. The semianalytical solution is developed using the Fourier series (FS) method applied to the vector potential form of the governing equations. The extension to transient solutions represents an important technical feature of this work, as the applications of the FS method to density‐driven problems have been limited to steady‐state conditions. The FS solution is validated against a finite element solution obtained using COMSOL Multiphysics. Numerical experiments show the worthiness of the developed FS solution as a benchmark because it clearly allows making distinction between different numerical techniques. The effects of governing parameters on three‐dimensional THC have not been investigated previously. We perform a detailed parameter sensitivity analysis to address this gap. A vortex convective flow is observed and the orientation and intensity of the flow is sensitive to the gravity number. The increase in the temperature gradient reduces the salinity flux.

A Fully-Mixed Formulation for the Steady Double-Diffusive Convection System Based upon Brinkman–Forchheimer Equations

We propose and analyze a new mixed finite element method for the problem of steady double-diffusive convection in a fluid-saturated porous medium. More precisely, the model is described by the coupling of the Brinkman–Forchheimer and double-diffusion equations, in which the originally sought variables are the velocity and pressure of the fluid, and the temperature and concentration of a solute. Our approach is based on the introduction of the further unknowns given by the fluid pseudostress tensor, and the pseudoheat and pseudodiffusive vectors, thus yielding a fully-mixed formulation. Furthermore, since the nonlinear term in the Brinkman–Forchheimer equation requires the velocity to live in a smaller space than usual, we partially augment the variational formulation with suitable Galerkin type terms, which forces both the temperature and concentration scalar fields to live in $$\mathrm {L}^4$$ . As a consequence, the aforementioned pseudoheat and pseudodiffusive vectors live in a suitable $$\mathrm {H}(\mathrm {div})$$ -type Banach space. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram and Banach–Necas–Babuska theorems, allow to prove the unique solvability of the continuous problem. As for the associated Galerkin scheme we utilize Raviart–Thomas spaces of order $$k\ge 0$$ for approximating the pseudostress tensor, as well as the pseudoheat and pseudodiffusive vectors, whereas continuous piecewise polynomials of degree $$\le k + 1$$ are employed for the velocity, and piecewise polynomials of degree $$\le k$$ for the temperature and concentration fields. In turn, the existence and uniqueness of the discrete solution is established similarly to its continuous counterpart, applying in this case the Brouwer and Banach fixed-point theorems, respectively. Finally, we derive optimal a priori error estimates and provide several numerical results confirming the theoretical rates of convergence and illustrating the performance and flexibility of the method.

Time-dependent solution for natural convection in a porous enclosure using the Darcy–Lapwood–Brinkman model

Natural convection (NC) in high permeable porous media is usually investigated using the Darcy–Lapwood–Brinkman model (DLB). The problem of the porous squared cavity is widely used as a common benchmark case for NC in porous media. The solutions to this problem with the DLB model are limited to steady-state conditions. In this paper, we developed a time-dependent high accurate solution based on the Fourier–Galerkin method (FG). The solution is derived considering two configurations dealing with unsteady and transient modes. The governing equations are reformulated using the stream function. The Temperature and the stream functions are expended as unknowns in space using Fourier series which are appropriately substituted in the equations. The equations are then projected to the spectral space using Fourier trigonometric trial functions. The obtained developed equations form a nonlinear differential algebraic system of equations. An appropriate technique is used to integrate the spectral system in time and to ensure high accuracy. The results of the FG method are compared to a finite element solution for different Rayleigh and Darcy numbers values. The transient and unsteady solutions are obtained with a feasible and low computational cost. The paper provides high accurate time-dependent solutions useful for benchmarking numerical models dealing with NC in porous media. The results of the developed solutions are efficient to gain physical insight into the time-dependent NC processes.

On three-dimensional stable long-wavelength convection in the presence of Dirichlet thermal boundary conditions

It is a well-known fact that the onset of Rayleigh–Benard convection occurs via a long-wavelength instability when the horizontal boundaries are thermally insulated. The aim of this paper is to quantify the exact dimensions of a cylinder of rectangular cross-section wherein stable three-dimensional Rayleigh–Benard convection sets in via a long-wavelength instability from the motionless state at the same value of the critical Rayleigh number as the corresponding horizontally unbounded problem when the bounding horizontal walls have infinite thermal conductance. Hence, we consider three-dimensional Rayleigh–Benard convection in a cell of infinite extent in the x-direction, confined between two vertical walls located at $$y= \pm H$$ and horizontal boundaries located at $$z=0$$ and $$z=d$$ . Our analysis predicts the existence of the sought stable state for experimental velocity boundary conditions at the vertical walls provided the aspect ratio $$\delta = H/d$$ takes a certain value. In the limit $$H \rightarrow \infty $$ , we retrieve the stability characteristics of the horizontally unbounded problem. As expected, the analysis predicts two counter-rotating rolls aligned along the y-direction of period $$2 \pi /\delta $$ equal to the period of the roll in the y-direction of the corresponding unbounded problem. A long-scale asymptotic analysis leads to the derivation of an evolution partial differential equation (PDE) that is fourth order in space and contains a single bifurcation parameter. The PDE, valid for a specific value of $$\delta $$ , is analyzed analytically and numerically as function of the bifurcation parameter and for a variety of velocity boundary conditions at the vertical walls to seek the stable steady-state solutions. The same analysis is also extended to the case of convection in a fluid-saturated porous medium.

Isolated buoyant convection in a two-layered porous medium with an inclined permeability jump

Abstract

A Pore-Scale Investigation of the Transient Response of Forced Convection in Porous Media to Inlet Ramp Inputs

Abstract This paper investigates the transient response of forced convection of heat in a reticulated porous medium through taking a pore-scale approach. The thermal system is subject to a ramp disturbance superimposed on the entrance flow temperature/velocity. The developed model consisted of ten cylindrical obstacles aligned in a staggered arrangement with set isothermal boundary conditions. A few types of fluids, along with different values of porosity and Reynolds number, are considered. Assuming a laminar flow, the unsteady Navier Stokes and energy equations are solved numerically. The temporally developing flow and temperature fields as well as the surface-averaged Nusselt numbers are used to explore the transient response of the system. Also, a response lag ratio (RLR) is defined to further characterize the transient response of the system. The results reveal that an increase in amplitude increases the RLR. Nonetheless, an increase in ramp duration decreases the RLR, particularly for high-density fluids. Interestingly, it is found that the Reynolds number has almost negligible effects upon RLR. This study clearly reflects the importance of conducting pore-scale analyses for understanding the transient response of heat convection in porous media.

Lattice Boltzmann Simulation of MHD Rayleigh–Bénard Convection in Porous Media

Lattice Boltzmann method is used to investigate the Rayleigh–Benard convection of magnetohydrodynamic fluid flow inside a rectangular cavity filled by porous media. The Brinkman–Forchheimer model is considered in the simulation to formulate a porous medium mathematically, and the multi-distribution function model is considered to include the magnetic field effect with different inclination angles. The water is considered as the working fluid, which is electrically conducting. A comprehensive analysis of the impact of governing dimensionless parameters is performed by varying Rayleigh (Ra), Hartmann (Ha), and Darcy (Da) numbers, porosity ( $$\epsilon $$ ), and inclination angles ( $$\phi $$ ) of the applied magnetic field. Numerical results are evaluated in the form of streamlines, isotherms, and the rate of heat transfer in terms of the local and average Nusselt number as well as the entropy generation due to the irreversibility of the fluid friction, temperature gradient, and magnetic field effects. The results imply that increasing Ha and decreasing Da reduce the rate of heat transfer. The average Bejan number $$\hbox {Be}_\mathrm{{avg}}$$ increases for increasing the Hartmann number. On the other hand, augmenting Ra and $$\epsilon $$ improves the heat transfer rate. It is also found that the change of the magnetic field inclination angle $$\phi $$ changes the rate of heat transfer and entropy generation.

Centrifugal filtration convection in bidisperse media

This work is concerned with thermal dispersion through a rotating fissured porous medium. In particular, linear and nonlinear analyses are made to investigate centrifugation driven thermal convection in a bidisperse porous medium. The presence of micro-pores in addition to the usual ones in a porous medium is considered, which allows the possibility of momentum exchange between the two families of pores. The linear analysis is performed through the normal modes, whereas the nonlinear one is based on a suitably defined generalized energy functional. Sharp and unconditional nonlinear L2 stability limits are obtained through the variational principles. Compound matrix method based numerical solutions of the resulting eigenvalue problems are obtained, and the usefulness of nonlinear results is established in most parts of the parameter space. It is found that an increase in momentum exchange delays convection, however, depending on the ratio of permeabilities in the two families of pores.

Unsteady natural convection in a liquid-saturated porous enclosure with local thermal non-equilibrium effect

Stability analysis of free convection in a liquid-saturated sparsely-packed porous medium with local-thermal-non-equilibrium (LTNE) effect is presented. For the vertical boundaries free–free, adiabatic and rigid–rigid, adiabatic are considered while for horizontal boundaries it is the stress-free, isothermal and rigid–rigid, isothermal boundary combinations we consider. From the linear theory, it is apparent that there is advanced onset of convection in a shallow enclosure followed by that in square and tall enclosures. Asymptotic analysis of the thermal Rayleigh number for small and large values of the inter-phase heat transfer coefficient is reported. Results of Darcy–Benard convection (DBC) and Rayleigh–Benard convection can be obtained as limiting cases of the study. LTNE effect is prominent in the case of Brinkman–Benard convection compared to that in DBC. Using a multi-scale method and by performing a non-linear stability analysis the Ginzburg–Landau equation is derived from the five-mode Lorenz modal. Heat transport is estimated at the lower plate of the channel. The effect of the Brinkman number, the porous parameter and the inter-phase heat transfer coefficient is to favour delayed onset of convection and thereby enhanced heat transport while the porosity-modified ratio of thermal conductivities shows the opposite effect.