Natural Convection In Porous Media Research Articles

A posteriori error analysis of a semi‐augmented finite element method for double‐diffusive natural convection in porous media

AbstractThis paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi‐augmented mixed‐primal finite element method previously developed by us to numerically solve double‐diffusive natural convection problem in porous media. The model combines Brinkman‐Navier‐Stokes equations for velocity and pressure coupled to a vector advection‐diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo‐stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart‐Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual‐based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart‐Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.

ANN-based deep collocation method for natural convection in porous media

ANN-based deep collocation method for natural convection in porous media

Modeling transient natural convection in heterogeneous porous media with Convolutional Neural Networks

Convolutional Neural Networks (CNNs) are gaining significant attention in applications related to coupled flow and transfer processes in porous media, especially when dealing with image-like data. In this context, the most important applications are related to surrogate modeling, where data obtained from simulators is used to train a CNN model. CNNs are also used as optimizers for inverse modeling or parameter estimation. For natural convection in porous media, applications of CNNs are scarce and limited to steady-state data. The main goal of this paper is to extend the applications of CNNs to transient data, by developing new CNN models that allow for integrating time-variant images. Thus, we suggest using an Encoder-Decoder CNN (ED-CNN) for surrogate modeling and a 3D-CNN for inverse modeling. Besides surrogate and inverse modeling, we suggest using CNN for time prediction by coupling it with long short-term memory (LSTM). The performances of these suggested approaches are investigated by applying them to the benchmark of natural convection in porous cavity with heterogeneous property fields, and by comparing the suggested approaches to other alternatives such as standard deep neural network (DNN) and 2D-CNN trained on steady-state data. The results show that, for surrogate modeling, with the same amount of data and equivalent training times, ED-CNN is more practical than DNN because it provides spatially distributed prediction while DNN is limited to local data. The transient data allows for improving the performance of CNN in inverse modeling because it provides more information about heat transfer across the different material zones and thus heterogeneity. The 3D-CNN approach is more efficient than 2D-CNN as it allows for considering the time progress in the training. For instance, the error with 3D-CNN and transient data is about 11 %, while it is about 18 % with 2D-CNN. Coupling CNN with LSTM allows for improving the performance of CNN in time series prediction.

Numerical Study of Entropy Generation with Magnetohydrodynamics on Natural Convection in a Porous Medium using Lattice Boltzmann Method

The study seeks to presents numerically the heat transfer phenomenon using natural convection in porous media. A square enclosure (cavity) filled with a porous medium simulated through Lattice Boltzmann Method which was used to analysed its effect on Fluid flows. The Darcy Forcheimer model will be adopted to show transfer of heat on the porous media. Magnetic effect is considered by including a magnetic field in they-momentum equation. The effects by various dimensionless parameters on generation of entropy are efficiently analyzed. Results from the investigations show that the Darcy-Forcheimer and magnetic effect both have relevant impact on entropy in a porous medium. The Eckert number affects the energy conversion and dissipation mechanisms, which in turns affect the rate of entropy generation.

Reduced order modeling for flow and transport problems with Barlow Twins self-supervised learning

We propose a unified data-driven reduced order model (ROM) that bridges the performance gap between linear and nonlinear manifold approaches. Deep learning ROM (DL-ROM) using deep-convolutional autoencoders (DC–AE) has been shown to capture nonlinear solution manifolds but fails to perform adequately when linear subspace approaches such as proper orthogonal decomposition (POD) would be optimal. Besides, most DL-ROM models rely on convolutional layers, which might limit its application to only a structured mesh. The proposed framework in this study relies on the combination of an autoencoder (AE) and Barlow Twins (BT) self-supervised learning, where BT maximizes the information content of the embedding with the latent space through a joint embedding architecture. Through a series of benchmark problems of natural convection in porous media, BT–AE performs better than the previous DL-ROM framework by providing comparable results to POD-based approaches for problems where the solution lies within a linear subspace as well as DL-ROM autoencoder-based techniques where the solution lies on a nonlinear manifold; consequently, bridges the gap between linear and nonlinear reduced manifolds. We illustrate that a proficient construction of the latent space is key to achieving these results, enabling us to map these latent spaces using regression models. The proposed framework achieves a relative error of 2% on average and 12% in the worst-case scenario (i.e., the training data is small, but the parameter space is large.). We also show that our framework provides a speed-up of 7 times 10^{6} times, in the best case, and 7 times 10^{3} times on average compared to a finite element solver. Furthermore, this BT–AE framework can operate on unstructured meshes, which provides flexibility in its application to standard numerical solvers, on-site measurements, experimental data, or a combination of these sources.

Experimental characterization of shear-enhanced dispersion in porous media

The shear flow generated by natural convection in porous media considerably influences the dispersion of contaminants in various geological formations. Previous studies have revealed that the dispersion coefficient depends on the Péclet number (Pe), shear rate, and complex pore structures. In this study, the dispersion behavior of a NaI tracer cloud was investigated under different shear rates ranging from 0.001 to 0.04 s−1 and Pe ranging from 10 to 200 using an innovative experimental apparatus via an X-ray technique. In contrast to uniform flow, a considerably enhanced dispersion was observed in shear flow along the longitudinal direction, and spreading in the longitudinal and transverse directions grew with time as ∼t3 and ∼t, respectively. In the power-law regime, the dispersion behaviors in the transverse and longitudinal directions were strengthened by shear action, with a larger dependence of the power law on the shear rates and Pe compared with that in the uniform flow. The strengthened dispersion promoted the formation of a hyper-mixing state, which was evaluated using the temporal evolution of the maximum concentration of the tracer plume, dilution index, and scalar dissipation rate. The results indicate that the dispersion behavior in a two-dimensional shear flow is quantitatively equivalent to that in a four-dimensional uniform-flow field. The findings of this study can provide a deeper understanding of the influence of shear flow on the hydrodynamic dispersion in porous media.

Prediction of pore-scale-property dependent natural convection in porous media at high Rayleigh numbers

Natural convection in porous media has received increasing attention in recent years due to its significance in engineering applications. This process is traditionally analyzed by the solution of the classical Darcy-Oberbeck-Boussinesq (DOB) equations. According to the DOB equations, natural convection in porous media is exclusively dependent on the Rayleigh-Darcy number, Ra, while the Sherwood number, Sh, has a linear relationship with Ra at high Rayleigh numbers. However, these predictions conflict with experimental observations. In this study, we have performed a pore-scale resolved direct numerical simulation (DNS) study of natural convection in periodic porous media composed of two-dimensional square and circular obstacles. Based on our analysis, a new correlation of Sh for large Rayleigh numbers (Ra≥1000), low Darcy numbers Da, and high Schmidt numbers Sc (Da/Sc≤2×10−8) has been proposed, expressed as Sh=aRa1−0.2φ2+1, where a=0.011±0.002 is a pore-scale geometric parameter. The new correlation has been validated over a wide range of Rayleigh numbers, porosity values, and pore-scale geometries. Our DNS results also show that, with a decrease of porosity, it becomes more difficult for mega-plumes with low wavenumbers to enter the boundary layer. Low wavenumber motions decay much faster with a decrease of Da than the pore-scale motions near the wall. The volume-averaged dissipation rate nondimensionalized using the pore size εˆi has the scaling law εˆi∼Da in the internal region and εˆi∼Da1/2 in the near-wall region. We expect that these characteristics obtained from DNS also apply to natural convection in porous media with much lower Darcy numbers.

Analysis of a semi-augmented mixed finite element method for double-diffusive natural convection in porous media

In this paper we study a stationary double-diffusive natural convection problem in porous media given by a Navier-Stokes/Brinkman type system, for describing the velocity and the pressure, coupled to a vector advection-diffusion equation relate to the heat and substance concentration, of a viscous fluid in a porous media with physical boundary conditions. The model problem is rewritten in terms of a first-order system, without the pressure, based on the introduction of the strain tensor and a nonlinear pseudo-stress tensor in the fluid equations. After a variational approach, the resulting weak model is then augmented using appropriate redundant penalization terms for the fluid equations along with a standard primal formulation for the heat and substance concentration. Then, it is rewritten as an equivalent fixed-point problem. Well-posedness results for both the continuous and the discrete schemes are stated, as well as the respective convergence result under certain regularity assumptions combined with the Lax-Milgram theorem, and the Banach and Brouwer fixed-point theorems. In particular, Raviart-Thomas elements of order k are used for approximating the pseudo-stress tensor, piecewise polynomials of degree ≤k and ≤k+1 are utilized for approximating the strain tensor and the velocity, respectively, and the heat and substance concentration are approximated by means of Lagrange finite elements of order ≤k+1. Optimal a priori error estimates are derived and confirmed through some numerical examples that illustrate the performance of the proposed semi-augmented mixed-primal scheme.

Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: Comparison with linear subspace techniques

Natural convection in porous media is a highly nonlinear multiphysical problem relevant to many engineering applications (e.g., the process of CO2 sequestration). Here, we extend and present a non-intrusive reduced order model of natural convection in porous media employing deep convolutional autoencoders for the compression and reconstruction and either radial basis function (RBF) interpolation or artificial neural networks (ANNs) for mapping parameters of partial differential equations (PDEs) on the corresponding nonlinear manifolds. To benchmark our approach, we also describe linear compression and reconstruction processes relying on proper orthogonal decomposition (POD) and ANNs. We present comprehensive comparisons among different models through three benchmark problems. The reduced order models, linear and nonlinear approaches, are much faster than the finite element model, obtaining a maximum speed-up of 7×106 because our framework is not bound by the Courant–Friedrichs–Lewy condition; hence, it could deliver quantities of interest at any given time contrary to the finite element model. Our model’s accuracy still lies within a relative error of 7% in the worst-case scenario. We illustrate that, in specific settings, the nonlinear approach outperforms its linear counterpart and vice versa. We hypothesize that a visual comparison between principal component analysis (PCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE) could indicate which method will perform better prior to employing any specific compression strategy.

Analyzing the efficiency and robustness of deep convolutional neural networks for modeling natural convection in heterogeneous porous media

Natural convection in porous media (NCPM) is governed by highly nonlinear dynamics due to the dependence of fluid density and viscosity to temperature. Given the high computational cost associated with numerical modeling of NCPM, data-driven metamodels are commonly used to reduce the computational time in applications that require repetitive model runs. However, to various degrees, all metamodels previously used in this context struggle at scaling to high dimensional input and output problems. This work aims at investigating the performance of encoder-decoder convolutional neural networks (ED-CNNs), as a specialized deep learning architecture, in assisting the procedure of numerical modeling of NCPM. Our interest is directed to image-to-image regression tasks in which both the inputs and outputs of the model are high-dimensional, often high resolution, spatial distributions of the features. Large datasets of images (for e.g. heat map) can be generated by numerical modeling of NCPM, and also through high-resolution imaging and non-destructive scanning techniques. Hence, we apply ED-CNNs to develop a methodology for image-to-image regression. The goals are twofold: (1) to assess the robustness of ED-CNNs in metamodeling and uncertainty propagation analysis, and (2) to evaluate the performance of ED-CNNs as optimizer in input parameter estimation. To do so, we apply the ED-CNNs to the common benchmark of natural convection in a porous cavity. Numerical experiments highlight the robustness and efficiency of the ED-CNNs in handling heterogeneous domains.

A macroscopic two-length-scale model for natural convection in porous media driven by a species-concentration gradient

The modelling of natural convection in porous media is receiving increased interest due to its significance in environmental and engineering problems. State-of-the-art simulations are based on the classic macroscopic Darcy–Oberbeck–Boussinesq (DOB) equations, which are widely accepted to capture the underlying physics of convection in porous media provided the Darcy number, $Da$ , is small. In this paper we analyse and extend the recent pore-resolved direct numerical simulations (DNS) of Gasow et al. (J. Fluid Mech, vol. 891, 2020, p. A25) and show that the macroscopic diffusion, which is neglected in DOB, is of the same order (with respect to $Da$ ) as the buoyancy force and the Darcy drag. Consequently, the macroscopic diffusion must be modelled even if the value of $Da$ is small. We propose a ‘two-length-scale diffusion’ model, in which the effect of the pore scale on the momentum transport is approximated with a macroscopic diffusion term. This term is determined by both the macroscopic length scale and the pore scale. It includes a transport coefficient that solely depends on the pore-scale geometry. Simulations of our model render a more accurate Sherwood number, root mean square (r.m.s.) of the mass concentration and r.m.s. of the velocity than simulations that employ the DOB equations. In particular, we find that the Sherwood number $Sh$ increases with decreasing porosity and with increasing Schmidt number $(Sc)$ . In addition, for high values of $Ra$ and high porosities, $Sh$ scales nonlinearly. These trends agree with the DNS, but are not captured in the DOB simulations.

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.

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.

Effects of pore scale on the macroscopic properties of natural convection in porous media

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 Rayleigh number ( $Ra$ ) for high values of $Ra$ ( $Ra\gg 1300$ ). Here, we perform direct numerical simulations (DNS), fully resolving the flow field within the pores. We show that the boundary layer thickness is determined by the pore size instead of the Rayleigh number, as previously assumed. The mega- and proto-plume sizes increase with the pore size. Our DNS results exhibit a nonlinear scaling of the Sherwood number at high porosity, and for the same Rayleigh number, higher Sherwood numbers are predicted by DNS at lower porosities. It can be concluded that the scaling of the Sherwood number depends on the porosity and the pore-scale parameters, which is consistent with experimental studies.

Numerical solution and Taguchi experimental method for variable viscosity and non-Newtonian fluids effects on heat and mass transfer by natural convection in porous media

Numerical solution and Taguchi experimental method for variable viscosity and non-Newtonian fluids effects on heat and mass transfer by natural convection in porous media

Numerical solution and Taguchi experimental method for variable viscosity and non-Newtonian fluids effects on heat and mass transfer by natural convection in porous media

In this article, both numerical solution and Taguchi method are presented to study the variable viscosity and non-Newtonian fluids effects on coupled heat and mass transfer by free convection over a vertical permeable plate in porous media. The surface temperature, concentration of the plate, and blowing/suction velocity are uniform. The viscosity of the fluid varies inversely as a linear function of the temperature. The partial differential equations are transformed into non-similar equations and solved by Keller box method. Numerical results of the local Nusselt number and local Sherwood number are expressed in the five parameters: 1) blowing/suction parameter ξ; 2) the power-law index of non-Newtonian fluid n; 3) buoyancy ratio N; 4) Lewis number Le; 5) viscosity-variation parameter θr. The best value for confirming the maximum of the local Nusselt (Sherwood) number by the Taguchi method is 6.6328 (10.3056).

Study of the Effect of Thermal Dispersion on Internal Natural Convection in Porous Media Using Fourier Series

Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to solve the coupled equations of Darcy’s flow and heat transfer with a full velocity-dependent dispersion tensor, employing the stream function formulation. A sound implementation of the FG method is developed to obtain accurate solutions within affordable computational costs. In the spectral space, the stream function is expressed analytically in terms of temperature, and the spectral system is solved using temperature as the primary unknown. The FG method is compared to finite element solutions obtained using an in-house code (TRACES), OpenGeoSys and COMSOL Multiphysics®. These comparisons show the high accuracy of the FG solution which avoids numerical artifacts related to time and spatial discretization. Several examples having different dispersion coefficients and Rayleigh numbers are tested to analyze the solution behavior and to gain physical insight into the thermal dispersion processes. The effect of thermal dispersion coefficients on heat transfer and convective flow in a porous square cavity has not been investigated previously. Here, taking advantage of the developed FG solution, a detailed parameter sensitivity analysis is carried out to address this gap. In the presence of thermal dispersion, the Rayleigh number mainly affects the convective velocity and the heat flux to the domain. At high Rayleigh numbers, the temperature distribution is mainly controlled by the longitudinal dispersion coefficient. Longitudinal dispersion flux is important along the adiabatic walls while transverse dispersion dominates the heat flux toward the isothermal walls. Correlations between the average Nusselt number and dispersion coefficients are derived for three Rayleigh number regimes.

Stability and finite element approximation of phase change models for natural convection in porous media

Abstract In this paper we study a phase change problem for non-isothermal incompressible viscous flows. The underlying continuum is modelled as a viscous Newtonian fluid where the change of phase is either encoded in the viscosity itself, or in the Brinkman–Boussinesq approximation where the solidification process influences the drag directly. We address these and other modelling assumptions and their consequences in the simulation of differentially heated cavity flows of diverse type. A second order finite element method for the primal formulation of the problem in terms of velocity, temperature, and pressure is constructed, and we provide conditions for its stability. We finally present several numerical tests in 2D and 3D, corroborating the accuracy of the numerical scheme as well as illustrating key properties of the model.

Natural Convection in Porous Media and the Collapse of the Wave Function

The problem of nonlinear natural convection in a fluid saturated porous layer heated from below is reviewed focusing on the specific result of a collapse of the wave function. When the conditions for the onset of convection are met, a wave function is obtained as the solution of the linearized equations expressed in terms of a Fourier expansion. Only one mode of this expansion survives at the onset of convection, a result that can be seen as the “collapse of the wave function” in a very similar fashion as in quantum mechanics, although the explanations of the latter are very distinct from the ones in quantum mechanics. The reasons behind the “collapse of the wave function” result in natural convection are discussed and the analysis is extended into the nonlinear domain of convection, by using a weak nonlinear analysis.

Investigation of MHD natural convection in a porous media by double MRT lattice Boltzmann method utilizing MWCNT–Fe3O4/water hybrid nanofluid

In this paper, a new double multi relaxation time (MRT) Lattice Boltzmann method (LBM) has been used to simulate magnetohydrodynamics (MHD) natural convection in a porous media. A new nanofluid named; multi-walled carbon nanotubes–iron oxide/water nanofluid (MWCNT–Fe3O4/water hybrid nanofluid) has been utilized to investigate the effect of nanoparticle on heat transfer. The thermo-physical properties of the nanofluid have been extracted from experimental results. D2Q9 and D2Q5 lattices are used to solve the flow and temperature fields respectively, and the effect of various parameters such as; Darcy number (Da) (10−2–10−1), Rayleigh number (103 ≤ Ra ≤ 105), porosity (0.4 ≤ ε ≤ 0.9), volume fraction of nanoparticles (0≤ Ø ≤ 0.003) and Hartmann number (0 ≤ Ha ≤ 50) have been investigated. Based on the present results, the new double MRT LBM is a proper method to solve the complex flows such as MHD natural convection in porous media. The results show that augmentation of Rayleigh number increases the heat transfer rate for all cases however by increasing the Hartmann number the effect of Rayleigh number decreases. Also, adding the nanoparticles enhances the average Nusselt number as increases 4.9% for Ra = 105, Da = 10−1, Ha = 50, and ε = 0.9 when the volume fraction of nanoparticles rises from 0 to 0.003. Results indicate that by enhancing Darcy number heat transfer rate increases and average Nusselt number improves by porosity.