In a range of physical systems, the first instability in Rayleigh-Bernard convection between nearly thermally insulating horizontal plates is large scale. This holds for thermal convection of fluids saturating porous media. Large-scale thermal convection in a horizontal layer is governed by remarkably similar equations both in the presence of a porous matrix and without it, with only one additional term for the latter case, which, however, vanishes under certain conditions (e.g., two-dimensional flows or infinite Prandtl number). We provide a rigorous derivation of long-wavelength equations for a porous layer with inhomogeneous heating and possible pumping.
In the above paper the authors treat the natural convection boundary layer flow in a Darcy-Brinkman-Forchheimer porous medium. In the energy equation the radiation effect has been taken into account. A two-parameter perturbation method is used for the solution of the equations. The first order results are presented in tables and figures. This is an interesting work but there are some weak points which are presented below:
We present a numerical study of convection in a horizontal layer comprising a fluid-saturated porous bed overlain by an unconfined fluid layer. Convection is driven by a vertical, destabilising temperature difference applied across the whole system, as in the canonical Rayleigh-Bénard problem. Numerical simulations are carried out using a single-domain formulation of the two-layer problem based on the Darcy-Brinkman equations. We explore the dynamics and heat flux through the system in the limit of large Rayleigh number, but small Darcy number, such that the flow exhibits vigorous convection i…
The problem of steady mixed convection boundary-layer flow on a cooled vertical permeable circular cylinder embedded in a fluid-saturated porous medium is studied. Here, we evaluate the flow and heat transfer characteristics numerically for various values of the governing parameters and demonstrate the existence of dual solutions beyond a critical point.
Convection heat transfer in porous media is a universal phenomenon in nature, and it is also frequently encountered in scientific and engineering fields. An in-depth understanding of the fundamental mechanism of convection heat transfer in porous media requires efficient and powerful numerical tools. In this paper, a cascaded lattice Boltzmann (CLB) method for convection heat transfer in porous media at the representative elementary volume (REV) scale is presented. In the CLB method, the flow field is solved by an isothermal CLB model with the D2Q9 lattice based on the generalized non-Darcy mo…
We report a numerical study of Rayleigh--Bénard convection through random porous media using pore-scale modelling, focusing on the Lagrangian dynamics of fluid particles and heat transfer for varied porosities $ϕ$. Due to the interaction between the porous medium and the coherent flow structures, the flow is found to be highly heterogeneous, consisting of convection channels with strong flow strength and stagnant regions with low velocities. The modifications of flow field due to porous structure have a significant influence on the dynamics of fluid particles. Evaluation of the particle displa…
A ubiquitous arrangement in nature is a free-flowing fluid coupled to a porous medium, for example a river or lake lying above a porous bed. Depending on the environmental conditions, thermal convection can occur and may be confined to the clear fluid region, forming shallow convection cells, or it can penetrate into the porous medium, forming deep cells. Here, we combine three complementary approaches -- linear stability analysis, fully nonlinear numerical simulations, and a coarse-grained model -- to determine the circumstances that lead to each configuration. The coarse-grained model yields…
We perform a numerical study of the heat transfer and flow structure of Rayleigh-Bénard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity $ϕ$ in the flow properties during the transition process from the traditional RB convection with $ϕ=1$ (so no obstacles included) to Darcy-type porous-media convection with $ϕ$ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for the Rayleigh number $Ra$ from $10^5$ to $10^{10}$ and varying porosities…
The paper presents a theoretical study using the Karman-Pohlhausen method for describing the transient heat exchange between the boundary-layer free convection and a vertical flat plate embedded in a porous medium. The unsteady behavior is developed after the generation of an impulsive heat flux step at the right-hand side of the plate. Two cases are considered according to whether the plate has a finite thickness or no thickness. The time and space evolution of the interface temperature is evidenced.
In the above paper the author treats the boundary layer flow along a vertical flat plate, immersed in a Darcy Brinkman Forchheimer porous medium. The porosity and the permeability of the porous medium are variable across the boundary layer. In addition a magnetic field with constant strength is applied normal to the plate. The fluid temperature at the plate is constant and different from that of the ambient fluid. This temperature difference creates a buoyancy force and the flow is characterized as mixed convection. The partial differential equations of the boundary layer flow are transformed …
In this paper, a double multiple-relaxation-time lattice Boltzmann model is developed for simulating transient solid-liquid phase change problems in porous media at the representative elementary volume scale. The model uses two different multiple-relaxation-time lattice Boltzmann equations, one for the flow field and the other for the temperature field with nonlinear latent heat source term. The model is based on the generalized non-Darcy formulation, and the solid-liquid phase change interface is traced through the liquid fraction which is determined by the enthalpy method. The model is valid…
We perform linear and nonlinear stability analysis for thermal convection in a fluid overlying a saturated porous medium. We use a coupled system, with the Navier-Stokes equations and Darcy's equation governing the free-flow and the porous regions respectively. Incorporating a dynamic pressure term in the Lions interface condition (which specifies the normal force balance across the fluid-medium interface) permits an energy bound on the typically uncooperative nonlinear advection term, enabling new nonlinear stability results. Within certain regimes, the nonlinear stability thresholds agree cl…
This investigation deals with the analysis of stagnation point heat transfer and corresponding flow features of hydromagnetic viscous incompressible fluid over a vertical shrinking sheet. The considered sheet is assumed to be permeable and subject to addition of stagnation point to control the generated vorticity in the boundary layer. The sheet is placed on the right side of the fluid saturated porous medium which is having permeability of specified form. Nonlinear convection waves in the flow field are realized due to the envisaged nonlinear relation between density and temperature. The equa…
Natural convection in porous media is a fundamental process for the long-term storage of CO2 in deep saline aquifers. Typically, details of mass transfer in porous media are inferred from the numerical solution of the volume-averaged Darcy-Oberbeck-Boussinesq (DOB) equations, even though these equations do not account for the microscopic properties of a porous medium. According to the DOB equations, natural convection in a porous medium is uniquely determined by the Rayleigh number. However, in contrast with experiments, DOB simulations yield a linear scaling of the Sherwood number with the Ra…
We study the fractionation of two components between a well-mixed gas and a saturated convecting porous layer. Motivated by geological carbon dioxide (CO$_2$) storage we assume that convection is driven only by the dissolved concentration of the first component, while the second acts as a tracer with increased diffusivity. Direct numerical simulations for convection at high Rayleigh numbers reveal that the partitioning of the components, in general, does not follow a Rayleigh fractionation trend, as commonly assumed. Initially, increases in tracer diffusivity also increase its flux, because th…
New insight into the contribution of the microscale vortex evolution towards convection heat transfer in porous media is presented in this paper. The objective is to determine how the microscale vortices influence convection heat transfer in turbulent flow inside porous media. The microscale temperature distribution is analyzed using flow visualization in 2D using streamlines and in 3D using Q structures. The pertinent observations are supplemented with the comparison of surface skin friction and heat transfer using: (1) surface skin friction lines and (2) joint PDF of pressure and skin fricti…
Ortiz-Pérez and Dávalos investigated the stability of parallel convective flow. From the article it seems that this equations of flow obtained by the authors. Meanwhile, it is well studied theoretically and experimentally(exact solution obtained more than half a century ago.
The transition to a new turbulent regime in horizontal convection in the case of low Prandtl numbers is analyzed using the Shishkina, Grossmann & Lohse (SGL) theory. The flow driven by the horizontal gradient along a horizontal surface, perpendicular to the acceleration of gravity is shown to transition to turbulence in the plume and the core. This transition to turbulence sets a sequence of heat transfer and momentum transport scalings which are found to follow the SGL prediction for the scaling factors and the prediction of Hughes, Griffith & Mullarney (HGM) for larger forcing amplitudes. Th…
We investigate theoretically the nonlinear state of ideal straight rolls in the Rayleigh-Bénard system of a fluid layer heated from below with a porous medium using a Galerkin method. Applying the Oberbeck-Boussinesq approximation, binary mixtures with positive separation ratio are studied and compared to one-component fluids. Our results for the structural properties of roll convection resemble qualitatively the situation in the Rayleigh--Bénard system without porous medium except for the fact that the streamlines of binary mixtures are deformed in the so-called Soret regime. The deformation …
In Rayleigh-Bénard convection, it has been found that the amount of heat passing through the fluid has a power law dependence on the imposed temperature difference. Modifying this dependence, either enhancing or reducing the heat transfer capability of fluids, is important in many scientific and practical applications. Here, we present a simple means to control the vertical heat transfer in Rayleigh-Bénard convection by injecting heat through one lateral side of the fluid domain and extracting the same amount of heat from the opposite side. This horizontal heat flux regulates the large-scale c…
We numerically examine solutal convection in porous media, driven by the dissolution of carbon dioxide (CO2) into water---an effective mechanism for CO2 storage in saline aquifers. Dissolution is associated with slow diffusion of free-phase CO2 into the underlying aqueous phase followed by density-driven convective mixing of CO2 throughout the water-saturated layer. We study the fluid dynamics of CO2 convection in the single aqueous-phase region. A comparison is made between two different boundary conditions in the top of the formation: (i) a constant, maximum aqueous-phase concentration of CO…
Gravity-driven convection of fluid at parameters near its thermodynamic critical point inside a porous layer heated from below (Rayleigh-Darcy convection) is studied. The fluid having the temperature slightly above the critical one is one-component. The hydrodynamic model describing a high compressible fluid phase at variable physical properties inside a solid matrix at a uniform porosity is analyzed. A near-critical fluid is assumed to be the van der Waals gas. In the limit of small variations in the density and thermodynamic coefficients where the Oberbeck-Boussinesq approximation is applica…
Fully developed forced convective flow in a channel filled with porous material bounded by two impermeable walls subject to constant heat flux is considered. We consider Brinkman-Forchhimer equation to govern the flow inside the porous medium which accounts for the presence of inertial term. We assume that the porous medium is anisotropic in nature and permeability is varying along all the directions, correspondingly, permeability will be a matrix which appear as a positive semi-definite matrix in the momentum equation. The anisotropic nature of the porous medium is characterized by anisotropi…
Oscillatory instability of buoyancy convection in a laterally heated cube with perfectly thermally conducting horizontal boundaries is studied. The effect of the spanwise boundaries on the oscillatory instability onset is studied. The problem is treated by Krylov-subspace-iteration based Newton and Arnoldi methods. The Krylov basis vectors are calculated by a novel approach that involves the SIMPLE iteration and a projection onto a space of functions satisfying all linearized and homogeneous boundary conditions. The finite volume grid is gradually refined from 1003 to 2563 finite volumes. A se…
We report the existence of two new limiting turbulent regimes in horizontal convection (HC) using direct numerical simulations at intermediate to low Prandtl numbers. The flow driven by a horizontal gradient along a horizontal surface, perpendicular to the acceleration of gravity is shown to transition to turbulence in the plume and the core, modifying the rate of heat and momentum transport. These transitions set a sequence of scaling laws blending the theoretical arguments from both the Shishkina, Grossmann & Lohse (SGL) theory with the Hughes, Griffiths & Mullarney (HGM) regime. These resul…
In this paper, global sensitivity analysis (GSA) and uncertainty quantification (UQ) have been applied to the problem of natural convection (NC) in a porous square cavity. This problem is widely used to provide physical insights into the processes of fluid flow and heat transfer in porous media. It introduces however several parameters whose values are usually uncertain. We herein explore the effect of the imperfect knowledge of the system parameters and their variability on the model quantities of interest (QoIs) characterizing the NC mechanisms. To this end, we use GSA in conjunction with th…
Horizontal Convection (HC) at large Rayleigh and Prandtl numbers, is studied experimentally in a regime up to seven orders of magnitude larger in terms of Rayleigh numbers than previously achieved. To reach Rayleigh up to $10^{17}$, the horizontal density gradient is generated using differential solutal convection by a differential input of salt and fresh water controlled by diffusion in a novel experiment where the zero-net mass flux of water is ensured through permeable membranes. This setup allows for measuring accurately the Nusselt number in solutal convection by carefully controlling the…
Laminar natural convection from two horizontally aligned isothermal cylinders in unconfined Power-law fluids has been investigated numerically. The effect of horizontal spacing (0<=(S/D)<=10) on both momentum and heat transfer characteristics has been delineated under the following pertinent parameters: Grashof number (10<=Gr<=1e3), Prandtl number (0.71<=n<=100), and Power-law index (0.4<=n<=1.6). The heat transfer characteristics are elucidated in terms of isotherms, local Nusselt number (Nu) distributions and average Nusselt number values, whereas the flow characteristics are interpreted in …
We study the velocity field in a horizontal (x-y) plane 1.5 mm above the hot plate in turbulent natural convection using PIV at a Rayleigh number Raw=106 and Prandtl number Pr=5.2. The plane of measurement is inside the velocity boundary layer estimated from the natural convection boundary layer equations[7] as well as inside the velocity boundary layer due to the large scale flow[2, 5].The boundary layer comprises of line plumes with sinking fluid between them. The instantaneous velocity variation from the center of the sinking fluid to the line plumes is found to deviate with the classical P…
Direct numerical simulations (DNS) indicate that at large values of the Rayleigh number ($Ra$) convection in porous media self-organizes into narrowly-spaced columnar flows, with more complex spatiotemporal features being confined to boundary layers near the top and bottom walls. In this investigation of high-$Ra$ porous media convection in a minimal flow unit, two reduced modeling strategies are proposed that exploit these specific flow characteristics. Both approaches utilize the idea of decomposition since the flow exhibits different dynamics in different regions of the domain: small-scale …
Double-diffusive convection in porous media is a common phenomenon in nature, and has received considerable attention in a wide variety of engineering applications. In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is developed for simulating double-diffusive convection in porous media at the representative elementary volume scale. The MRT-LB model is constructed in the framework of the triple-distribution-function approach: the velocity field, the temperature and concentration fields are solved separately by three different MRT-LB equations. The present model has tw…
A bidisperse porous medium is one with two porosity scales. There are the usual pores known as macro pores but also cracks or fissures in the skeleton which give rise to micro pores. In this article we develop and analyse a model for thermal convection where a layer of viscous incompressible fluid overlies a layer of bidisperse porous medium. Care has to be taken with the boundary conditions at the interface of the fluid and the porous material and this aspect is investigated. The situation is one in a layer which is heated from below and under appropriate conditions bimodal neutral curves are…
Recent planetary data and geophysical modelling suggest that hydrothermal activity is ongoing under the ice crust of Enceladus, one of Saturn's moons. According to these models, hydrothermal flow in the porous, rocky core of the satellite is driven by tidal deformation that induces dissipation and volumetric internal heating. Despite the effort in the modelling of Enceladus' interior, systematic understanding---and even basic scaling laws---of internally-heated porous convection and hydrothermal activity are still lacking. In this article, using an idealised model of an internally-heated porou…
The study investigates the problem of magnetohydrodynamic (MHD) free convection over a non-isothermal axisymmetric body under the action of transverse magnetic field. The body is embedded in a porous medium saturated with electrically conducting non-Newtonian power law fluid. In order to obtain similarity solution, it is assumed that the viscosity of the fluid decays exponentially with temperature. The qualitative results are illustrated for a vertical flat plate, horizontal cylinder and sphere.
We investigate the effect of dispersion on convection in porous media by performing direct numerical simulations (DNS) in a two-dimensional Rayleigh-Darcy domain. Scaling analysis of the governing equations shows that the dynamics of this system are not only controlled by the classical Rayleigh-Darcy number based on molecular diffusion, $Ra_m$, and the domain aspect ratio, but also controlled by two other dimensionless parameters: the dispersive Rayleigh number $Ra_d = H/α_t$ and the dispersivity ratio $r = α_l/α_t$, where $H$ is the domain height, $α_t$ and $α_l$ are the transverse and longit…
Three dimensional roll-type double-diffusive convection in a horizontally infinite layer of an uncompressible liquid is considered in the neighborhood of Hopf bifurcation points. A system of amplitude equations for the variations of convective rolls amplitude is derived by multiple-scaled method. An attention is paid to an interaction of convection and horizontal vortex. Different cases of the derived equations are discussed.
We consider the axisymmetric displacement of an ambient fluid by a second input fluid of lower density and lower viscosity in a horizontal porous layer. If the two fluids have been vertically segregated by buoyancy, the flow becomes self-similar with the input fluid preferentially flowing near the upper boundary. We show that this axisymmetric self-similar flow is stable to angular-dependent perturbations for any viscosity ratio. The Saffman-Taylor instability is suppressed due to the buoyancy segregation of the fluids. The radial extent of the segregated current is inversely proportional to t…
Laminar natural convection heat transfer from two horizontally aligned square cylinders has been investigated numerically using a finite-volume method (FVM) approach. Computations were performed to delineate the momentum and heat transfer characteristics under the following ranges of parameters: horizontal spacing between the cylinders (0 <= S/W <= 10), tilt angle of the square cylinder (0^0 <= δ <= 60^0), and Grashof number (10 <= Gr <= 10^5) for some specific Newtonian fluids having Prandtl number (0.71 <= Pr <= 7). The comprehensive results are represented in terms of temperature contours a…
A sequence of three steady - oscillatory transitions of buoyancy convection of air in a laterally heated cube with perfectly thermally insulated horizontal and spanwise boundaries is studied. The problem is treated by Newton and Arnoldi methods based on Krylov subspace iteration. The finite volume grid is gradually refined from 100^3 to 256^3 finite volumes. It is shown that the primary instability is characterized by two competing eigenmodes, whose temporal development results in two different oscillatory states that differ by their symmetries. Bifurcations due to both modes are subcritical. …
Unlike the power-law model, the Ellis model describes the apparent viscosity of a shear-thinning fluid with no singularity in the limit of a vanishingly small shear stress. In particular, this model matches the Newtonian behaviour when the shear stresses are very small. The emergence of the Rayleigh-Bénard instability is studied when a horizontal pressure gradient, yielding a basic throughflow, is prescribed in a horizontal porous layer. The threshold conditions for the linear instability of this system are obtained both analytically and numerically. In the case of a negligible flow rate, the …
Natural convection is of great importance in many engineering applications. This paper presents a smoothed particle hydrodynamics (SPH) method for natural convection. The conservation equations of mass, momentum and energy of fluid are discretized into SPH equations. The body force due to the change of density in a temperature field is considered by the Boussinesq approximation. The numerical method is validated by comparing numerical results with experimental results from literature. The numerical and experimental results reach a good agreement. Then the SPH method is applied to study the nat…
We simulate three-dimensional, horizontally periodic Rayleigh-Bénard convection between free-slip horizontal plates, rotating about a distant horizontal axis. When both the temperature difference between the plates and the rotation rate are sufficiently large, a strong horizontal wind is generated that is perpendicular to both the rotation vector and the gravity vector. The wind is turbulent, large-scale, and vertically sheared. Horizontal anisotropy, engendered here by rotation, appears necessary for such wind generation. Most of the kinetic energy of the flow resides in the wind, and the ver…
In this study we consider the effect of a horizontal magnetic field on the Rayleigh-Bénard convection in a finite liquid metal layer contained in a cuboid vessel (200 \times 200 \times 40 mm^3). Laboratory experiments are performed for measuring temperature and flow field in the alloy GaInSn at Prandtl number Pr = 0.03 and in a Rayleigh number range 2.3 \times 10^4 < Ra < 2.6 \times 10^5. The field strength is varied up to a maximum value of 320 mT (Ha = 2470, Q = 6.11 \times 10^6). The magnetic field forces the flow to form two-dimensional rolls parallel to the direction of the field lines. T…
The onset of thermal convection in a uniformly rotating and horizontally isotropic bi-disperse porous medium, taking into account the Vadasz term, is investigated. Via linear instability analysis, it has been proven that the Vadasz term allows the onset of convection via an oscillatory state but does not affect convection via a stationary motion.
We investigate the structures of the near-plate velocity and temperature profiles at different horizontal positions along the conducting bottom (and top) plate of a Rayleigh-Bénard convection cell, using two-dimensional (2D) numerical data obtained at the Rayleigh number Ra=10^8 and the Prandtl number Pr=4.4 of an Oberbeck-Boussinesq flow with constant material parameters. The results show that most of the time, and for both velocity and temperature, the instantaneous profiles scaled by the dynamical frame method [Q. Zhou and K.-Q. Xia, Phys. Rev. Lett. 104, 104301 (2010) agree well with the c…
We report theoretical and experimental investigations of the flow instability responsible for the mushy-layer mode of convection and the formation of chimneys, drainage channels devoid of solid, during steady-state solidification of aqueous ammonium chloride. Under certain growth conditions a state of steady mushy-layer growth with no flow is unstable to the onset of convection, resulting in the formation of chimneys. We present regime diagrams to quantify the state of the flow as a function of the initial liquid concentration, the porous-medium Rayleigh number, and the sample width. For a giv…
