Thermal Convection In Porous Medium Research Articles

On the modelling of thermal convection in porous media through rate-type equations

AbstractThe paper investigates current models of flows in porous media from the viewpoint of the mixture theory. The constitutive equations are investigated for compressible, viscous, heat-conducting fluids subject to relaxation phenomena. The thermodynamic analysis is performed via the Clausius-Duhem inequality based directly on the peculiar fields of the mixture. The detailed analysis so developed involves the peculiar heat fluxes and stresses per se while the balance equations for energy and entropy of the whole body would involve also diffusion effects. Following the objectivity principle, the constitutive equations for stresses and heat fluxes are taken to be governed by objective rate equations.

A Bio-Thermal Convection in a Porous Medium Saturated by Nanofluid Containing Gyrotactic Microorganisms Under an External Magnetic Field

The study of thermal convection in porous media saturated by nanofluid and microorganisms is an important problem for many geophysical and engineering applications. The concept of a mixture of nanofluids and microorganisms has attracted the interest of many researchers due to its ability to improve thermal properties and, as a result, heat transfer rates. This property is actively used both in electronic cooling systems and biological applications. Thus, the purpose of this research is to study biothermal instability in a porous medium saturated by a water-based nanofluid containing gyrotactic microorganisms in the presence of a vertical magnetic field. Given the presence of an external magnetic field in both natural and technological situations, we were motivated to perform this theoretical research. Using the Darcy-Brinkman model, a linear analysis of the convective instability has been considered for both-free boundaries, taking into account the effects of Brownian diffusion and thermophoresis. The Galerkin method was used to perform this analytical study. We have established that heat transfer is accomplished by stationary convection without oscillatory movements. In stationary convection regimes, metal oxide nanofluids (Al2O3), metallic nanofluids (Cu, Ag), and semiconductor nanofluids (TiO2, SiO2) are analyzed. Increasing the Chandrasekhar and Darcy numbers improve system stability significantly, but increasing porosity and modified bioconvection Rayleigh-Darcy number speed up the beginning of instability. To determine the transient behavior of heat and mass transports, a non-linear theory based on the representation of the Fourier series method is applied. In small time intervals, the transitional Nusselt and Sherwood numbers exhibit an oscillatory character. The Sherwood numbers (mass transfer) in the time interval reach stationary values faster than the Nusselt numbers (heat transfer). This research might help with seawater convection in the oceanic crust as well as the construction of biosensors.

Nonlinear stability analysis for the rheology of Oldroyd-B nanofluids embedded by Darcy–Brinkman porous media using a two-phase model

An alternative to the Darcy model called the Darcy–Brinkman is employed to investigate a linear and nonlinear analysis of thermal convection in porous media saturated by Oldroyd-B nanofluids. The two-phase model is utilized for a nanofluid which is based on the theory of nanoparticles–fluid relative velocity. The analytical expressions for oscillatory and stationary thermal Rayleigh numbers have been derived using the Galerkin scheme and normal mode technique to study the impacts of viscoelastic and other non-dimensional parameters governing the stability of the system. Nonlinear analysis is made using minimal double Fourier series with a weekly period. Both time-dependent and independent variations of the Nusselt number and Sherwood number for emerging parameters are represented to study heat and mass transport. The novelty of the paper lies in the fact that due to the viscoelasticity of the fluid oscillatory mode of convection came into existence for top-heavy nanoparticle arrangement which was otherwise not possible for nanofluids with Newtonian base fluids. The results of the mathematical model revealed that heat and mass transfer is enhanced with the higher values of viscoelastic parameters, namely stress relaxation and strain retardation numbers. The increasing Darcy number is observed to have substantial control over the stability as it inhibits heat and mass transfer rate. Graphical interpretations reveal that viscoelastic non-Newtonian nanofluids have higher heat and mass transfer rates than Newtonian nanofluids for the system under consideration.

Effects of pore scale and conjugate heat transfer on thermal convection in porous media

The study of thermal convection in porous media is of both fundamental and practical interest. Typically, numerical studies have relied on the volume-averaged Darcy–Oberbeck–Boussinesq (DOB) equations, where convection dynamics are assumed to be controlled solely by the Rayleigh number (Ra). Nusselt numbers (Nu) from these models predict Nu–Ra scaling exponents of 0.9–0.95. However, experiments and direct numerical simulations (DNS) have suggested scaling exponents as low as 0.319. Recent findings for solutal convection between DNS and DOB models have demonstrated that the ‘pore-scale parameters’ not captured by the DOB equations greatly influence convection. Thermal convection also has the additional complication of different thermal transport properties (e.g. solid-to-fluid thermal conductivity ratio ks/kf and heat capacity ratio σ) in different phases. Thus, in this work we compare results for thermal convection from the DNS and DOB equations. On the effects of pore size, DNS results show that Nu increases as pore size decreases. Mega-plumes are also found to be more frequent and smaller for reduced pore sizes. On the effects of conjugate heat transfer, two groups of cases (Group 1 with varying ks/kf at σ = 1 and Group 2 with varying σ at ks/kf = 1) are examined to compare the Nu–Ra relations at different porosity (ϕ) and ks/kf and σ values. Furthermore, we report that the boundary layer thickness is determined by the pore size in DNS results, while by both the Rayleigh number and the effective heat capacity ratio, $\bar{\phi } = \phi + (1 - \phi )\sigma$ , in the DOB model.

Homogeneous Venturi-effect concentrators for creeping flows: Magnifying flow velocities and heat fluxes simultaneously

Although thermal metamaterials have received considerable attention for approximately a decade, previous studies have so far focused on heat conduction or thermal convection in porous media, hampering applications related to moving in media. Based on transformation heat transfer, here we have analytically designed Venturi-effect concentrators by simultaneously manipulating the dynamic viscosity and the thermal conductivity. Further based on this theory and integral median theorem, inhomogeneous concentrators metamaterials simplified to homogeneous counterparts are proposed. Noteworthily, the proposed method can be employed to simplify not only thermal metamaterials, but also other inhomogeneous metamaterials, such as optical metamaterials, electromagnetic metamaterials, acoustic metamaterials, among others. Numerical simulations demonstrate that these Venturi-effect concentrators can not only amplify both flow velocities and heat fluxes in thermal-flow fields, but also achieve cloaking effects in thermal-flow systems. Since these Venturi-effect concentrators do not interfere with the thermal-flow field outside the concentrators, they secure advantages over conventional Venturi tubes and will facilitate potential applications related to Venturi effects. • Homogeneous Venturi-effect concentrators for creeping flows are fabricated. • Analytic solutions of concentrators are obtained by transformation heat transfer. • A general way to simplify inhomogeneous metamaterials to homogeneous ones is built. • Our study paves a way to further explore applications related to Venturi effects.

Onset of fractional-order thermal convection in porous media.

The macroscopic description of buoyancy-driven thermal convection in porous media is governed by advection-diffusion processes, which in the presence of thermophysical heterogeneities fail to predict the onset of thermal convection and the average rate of heat transfer. This work extends the classical model of heat transfer in porous media by including a fractional-order advective-dispersive term to account for the role of thermophysical heterogeneities in shifting the thermal instability point. The proposed fractional-order model overcomes limitations of the common closure approaches for the thermal dispersion term by replacing the diffusive assumption with a fractional-order model. Through a linear stability analysis and Galerkin procedure, we derive an analytical formula for the critical Rayleigh number as a function of the fractional model parameters. The resulting critical Rayleigh number reduces to the classical value in the absence of thermophysical heterogeneities when solid and fluid phases have similar thermal conductivities. Numerical simulations of the coupled flow equation with the fractional-order energy model near the primary bifurcation point confirm our analytical results. Moreover, data from pore-scale simulations are used to examine the potential of the proposed fractional-order model in predicting the amount of heat transfer across the porous enclosure. The linear stability and numerical results show that, unlike the classical thermal advection-dispersion models, the fractional-order model captures the advance and delay in the onset of convection in porous media and provides correct scalings for the average heat transfer in a thermophysically heterogeneous medium.

Multiple equilibria, bifurcations and selection scenarios in cosymmetric problem of thermal convection in porous medium

We study convection in a two-dimensional container of porous material saturated with fluid and heated from below. This problem belongs to the class of dynamical systems with nontrivial cosymmetry. The cosymmetry gives rise to a hidden parameter in the system and continuous families of infinitely many equilibria, and leads to non-trivial bifurcations. In this article we present our numerical studies that demonstrate nonlinear phenomena resulting from the existence of cosymmetry. We give a comprehensive picture of different bifurcations which occur in cosymmetric dynamical systems and in the convection problem. It includes internal and external (as an invariant set) bifurcations of one-parameter families of equilibria, as well as bifurcations leading to periodic, quasiperiodic and chaotic behaviour. The existence of infinite number of stable steady-state regimes begs the important question as to which of them can realize in physical experiments. In this paper, this question (known as the selection problem) is studied in detail. In particular, we show that the selection scenarios strongly depend on the initial temperature distribution of the fluid. The calculations are carried out by the global cosymmetry-preserving Galerkin method, and numerical methods used to analyse cosymmetric systems are also described.

Enhancement of synthetic schlieren image resolution using total variation optical flow: application to thermal experiments in a Hele-Shaw cell

We present an improvement to the standard synthetic schlieren technique to obtain the temperature distribution of a fluid inside of a Hele-Shaw cell. We aim to use the total variation \(L^1\)-norm optical flow method to treat experimental images and to obtain quantitative results of the development of thermal convection inside a cell, by detecting the gradients of the optical refractive index. We present a simple algorithm to set the optical flow parameters, which is based on the comparison between the optical flow output and the result obtained by digital PIV using the structural index metric. As an example of the application of the proposed method, we analyze laboratory experiments of thermal convection in porous media using a Hele-Shaw cell. We demonstrate that the application of the proposed method produces important improvements versus digital PIV, for the quantification of the gradients of the refractive index including the detection of small-scale convective structures. In comparison with correlation-based digital methods, we demonstrate the advantages of the proposed method, such as denoising and edge capture. These features allow us to obtain the temperature, for this experimental setting, with better image resolution than other techniques reported in the literature.

Nonlinear thermal convection in a viscoelastic nanofluid saturated porous medium under gravity modulation

Abstract This paper carried out a nonlinear thermal convection in a porous medium saturated with viscoelastic nanofluid under vibrations. The Darcy model has been used for the porous medium, while the nanofluid layer incorporates the effect of Brownian motion along with thermophoresis. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The non-uniform vertical vibrations of the system, which can be realized by oscillating the system vertically, is considered to vary sinusoidally with time. In order to find the heat and mass transports for unsteady state, a nonlinear analysis, using a minimal representation of the truncated Fourier series of two terms, has been performed. Effect of various parameters has been investigated on heat and mass transport and then presented graphically. It is found that gravity modulation can be used effectively to regulate either heat or mass transports in the system.

Thermal Convection in Porous Media at High Rayleigh Numbers

An experimental study of thermal convection in a porous medium investigates the heat transfer across a horizontal layer heated from below at high Rayleigh number. Using a packed bed of polypropylene spheres in a cubic enclosure saturated with compressed argon, the pressure was varied between 5.6 bar and 77 bar to obtain fluid Rayleigh numbers between 1.68 × 109 and 3.86 × 1011, corresponding to Rayleigh–Darcy numbers between 7.47 × 103 and 2.03 × 106. From the present and earlier studies of Rayleigh–Benard convection in both porous media and homogeneous fluid systems, the existence and importance of a thin thermal boundary layer are clearly demonstrated. In addition to identifying the governing role of the thermal boundary layer at high Rayleigh numbers, the successful correlation of data using homogeneous fluid dimensionless groups when the thermal boundary layer thickness becomes smaller than the length scale associated with the pore features is shown.

Experimental Investigation of Transient Thermal Convection in Porous Media

A laboratory experiment of transient thermal convection in a 1-m-high cell was conducted to compare the length and time scales of plume development to theory. The temperature field was resolved to less than 1 mm and was measured by dissolving a solution of thermochromic crystals into the water–glycerin working fluid. The time-dependent experiment was run by applying heat at the bottom boundary that eventually was $$6\,^\circ $$ C above the background temperature of the fluid. After development of a thermal boundary layer, the instability became visible at 26 min, with the development of 11, 3–4 cm width plumes growing from the boundary layer. The initially rapid growth rate reached a limiting velocity of approximately 0.5 cm min $$^{-1}$$ , and then decelerated throughout the experiment. Plumes interacted primarily by merging together; by the end of the experiment only three plumes were present. The Nusselt number at the onset of convection was 10, although it dropped to 4 after 45 min, which would be expected of a barely unstable system.

Evaluation of melting process of the permafrost on Mars: Its implication for surface features

For supplying massive liquid water to the outflow channels, igneous melting of the permafrost layer could have played a significant role. We numerically simulate the melting process of the permafrost layer induced by magmatic intrusion. The point of our simulation is incorporation of thermal convection in porous media, which has not been modeled well in previous studies of the melting of the permafrost. Our results show that convection in the melted zone causes drastic change in heat transfer, which results in focusing in the growth of the melt region and enhancement of water generation. The resulting melt zone extends vertically up just next to the surface, like a plume with a single column (mushroom structure). The volume of meltwater is considerably more than that expected in the conduction case. These characteristics suggest that a substantial amount of water may exist very near the surface. In response to compaction, segregated liquid water may have erupted out of the ground to form the fluvial features. Such an event would certainly be accompanied by surface destruction, which we can see as chaotic terrain. We propose a consistent scenario of forming surface features around the outflow channels.

Rayleigh-Bénard percolation transition of thermal convection in porous media: computational fluid dynamics, NMR velocity mapping, NMR temperature mapping.

Stationary thermal convection and heat conduction were studied in random-site percolation clusters in a Rayleigh-Bénard configuration experimentally by NMR microscopy techniques and numerically with the aid of a finite volume method. The porosity of the percolation clusters, and the temperature difference applied to the convection cell were varied. Two-dimensional percolation networks were generated with the aid of a random-number algorithm. The resulting clusters were used as templates for the fabrication of model objects. The convective velocity distribution of silicon oil or ethylene glycol filled into the pore space was mapped and evaluated in the form of histograms. The flow patterns visualized in the simulated and the measured velocity maps show good coincidence. In the histograms, two velocity regimes can be distinguished and attributed to local convection rolls responsible for the low-velocity part, and cluster-spanning flow loops characterized by a high-velocity cut-off, respectively. The maximum velocity as a function of the porosity and the overall temperature difference is shown to be indicative for the hydro-thermodynamic Rayleigh-Bénard instability and the geometrical percolation threshold. The coincidence of the Rayleigh-Bénard instability (modified for porous media) and the percolation transition (modified for closed loops) gives rise to a new critical phenomenon termed the Rayleigh-Bénard percolation transition. It occurs at a certain combination of the porosity and the overall temperature difference in the cell. Temperature maps were recorded with the aid of a relaxation-based NMR technique. The consequence of different thermal conductivities in the matrix and in the fluid is that horizontal-temperature gradients arise even in the absence of flow. This leads to a superposition of uncritical convective flow driven by the horizontal-temperature gradients whenever closed-loop pathways are possible and the critical Rayleigh-Bénard convection based on vertical-temperature gradients.

Rayleigh-Bénard percolation transition study of thermal convection in porous media: numerical simulation and NMR experiments.

Thermal convection was studied as a function of the porosity in random-site percolation model objects in a Rayleigh-Bénard configuration. NMR velocity mapping experiments and numerical simulations using the finite-volume method are compared. Velocity histograms were evaluated and can be described by power laws in a wide range. The maximum velocity as a function of the porosity indicates a combined percolation/Rayleigh-Bénard transition.

Pattern formation in an inhomogeneous environment

A major step in understanding pattern formation has been achieved by studying idealized systems. However, in nature inhomogeneous systems are much more abundant than their idealized homogeneous counterparts. Here we report about experimental results on pattern formation in two inhomogeneous systems, thermal convection in porous media and Taylor-vortex flow between a rough and a smooth cylinder. Several aspects of heterogeneity effects in pattern formation are theoretically investigated for model equations and analytical descriptions are given for a few phenomena.

Co-symmetry breakdown in problems of thermal convection in porous medium

We investigate two-dimensional thermal convection of saturating incompressible fluid in a horizontal cylinder filled with porous medium. The temperature distribution on the boundaries is time-independent and corresponds to the heating from below. At supercritical parameter values the problem has infinite number of stationary solutions for arbitrary shape of the region. This degeneracy is connected with the so-called co-symmetry property: the existence of the vector field which is orthogonal to the considered one. Non-coincidence of zeroes of these two fields leads generally speaking, to the degeneracy of the solutions. To destroy the degeneracy we add weak fluid seeping of the fluid through the boundaries either in vertical or in the horizontal direction. The breakdown of the family of the stationary solutions at high supercritical values of the Rayleigh number is studied in detail with the help of the corresponding normal form. Several limit cycles with the twisted leading manifolds appear as a result of the family destruction. To investigate the dynamical behavior the finite-dimensional models of the convection which maintain the breakdown of co-symmetry, are constructed on the base of the Galerkin approximation. The same scenario of the transition to chaos which seems to be connected with the co-symmetry breakdown, is recovered for both kinds of seeping. The quasi-periodic solution branches from the limit cycle. The further increase of the Peclet number leads to mode-locking, which is followed by the appearance of the homoclinic surface formed by the unstable manifold of the saddle periodic orbit; destruction of the latter surface leaves in the phase space the object with torus-like shape and non-integer fractal dimension.

Asymptotic solutions for nonlinear thermal convection in porous media

Nonlinear thermal convection in a saturated porous medium is studied under different approximations. Asymptotic solutions for free boundaries are obtained for two limiting cases: Ra large, σ 1 = 0(1) and 0( Ra) using the method of matched asymptotic expansions. The analytical approximation method is also used to study the convection problem over a wide range of Ra and σ 1. The results of the analysis show that the heat transfer rate varies as Ra 1 2 for a given σ 1. The results are in agreement with the experimental results of Schneider.

Dispersion effects on thermal convection in porous media

The influence of hydrodynamic dispersion on thermal convection in porous media is studied theoretically. The fluid-saturated porous layer is homogeneous, isotropic and bounded by two infinite horizontal planes kept at constant temperatures. The supercritical, steady two-dimensional motion, the heat transport and the stability of the motion are investigated. The dispersion effects depend strongly on the Rayleigh number and on the ratio of grain diameter to layer depth. The present results provide new and closer approximations to experimental data of the heat transport.