Darcy Convection Research Articles

Bifurcations phenomena and route to chaos via the cosymmetry breaking in the Darcy convection problem

The purpose of this paper is to examine and exhibit bifurcations arising from cosymmetry breaking in dynamical systems. Curves of equilibria are typical of cosymmetric dynamical systems. Perturbations violating cosymmetry destroy these curves, leading to different attractors. An example of a cosymmetric problem is the Darcy convection in a rectangular vessel with a linear temperature profile on its boundary. Low-intensity internal heat sources or weak through-flow violate the cosymmetry. The article presented numerical results of bifurcations in the convection problem caused by these perturbations. It achieved the discretization of partial differential equations using the Bubnov–Galerkin method. It has developed an algorithm based on cosymmetry theory to calculate equilibrium curves and analyse their destruction. A long-time integration using high-order methods applies for the searching of attractors. The study reveals that both perturbations breaking cosymmetry have similar effects on system dynamics. When all equilibria on a curve are stable, the curve can break into finite sets of equilibria or exhibit slow periodic motions. If arcs of unstable equilibria exist, then new quasi-periodic and chaotic attractors may arise. The dynamics on these attractors include slow movements close to destroyed arcs of equilibria and fast transitions between arcs. We may interpret this scenario as the route to chaos via the cosymmetry breaking. The results also show the typicality of multistability in cosymmetric and near-cosymmetric problems.

The Prandtl–Darcy Convection in a Vertical Porous Layer may be Unstable with Internal Heating

The Prandtl–Darcy Convection in a Vertical Porous Layer may be Unstable with Internal Heating

Internal Heat Modulation on Darcy Convection in a Porous Media Saturated by Nanofluid

In this paper we investigate the effect of internal heat modulation over a nanofluid saturated porous medium. We consider a small variation in time dependant heat source and vary sinusoidally with slow time. An energy equation will be altered by adding time dependant internal heat source. This internal heat source has its time dependent and independent parts. Time dependent part shows that the internal heat modulation over a porous media and defines controls on heat/mass transfer in the layer. We have performed a nonlinear stability analysis to investigate heat/mass transfer in the system. The nonlinear system of partial differential equations are transformed into nonlinear ordinary differential equations under similarity transforms up to the second term. This system has different system parameters and they have been investigated on heat and mass transfer graphically. The dual nature, stabilize or destabilize is due to the significant effect of internal heating modulation of the system. Further, the effect of internal heating is to destabilize the system, as a consequence heat/mass transfer enhances. It is found that internal heating modulation can be used effectively to regulate heat/mass transfer in the system.

An iterative scheme for solving a coupled Darcy–convection–diffusion model

We present an iterative scheme for the numerical analysis of a class of coupled Darcy-convection-diffusion problems modelling flow and heat transfer in porous media. The governing equations consist of the Darcy equations for the flow coupled to a convection-diffusion equation for heat transfer with nonlinear viscosity and diffusion coefficient depending on the temperature. Existence and uniqueness of the weak solution for the considered problem are first analyzed using a fixed-point method and then convergence study of two iterative schemes for the fixed-point algorithm is presented. Two numerical examples are selected to validate the theoretical estimates and to demonstrate the performance of the proposed algorithm. The obtained results support our theoretical expectations for a good numerical convergence with the developed estimates.

Strong Rayleigh–Darcy convection regime in three-dimensional porous media

We perform large-scale numerical simulations to study Rayleigh–Darcy convection in three-dimensional fluid-saturated porous media up to Rayleigh–Darcy number $Ra=8\times 10^4$ . At these large values of $Ra$ , the flow is dominated by large columnar structures – called megaplumes – which span the entire height of the domain. Near the boundaries, the flow is hierarchically organized, with fine-scale structures interacting and nesting to form larger-scale structures called supercells. We observe that the correlation between the flow structure in the core of the domain and at the boundaries decreases only slightly for increasing $Ra$ , and remains rather high even at the largest $Ra$ considered here. This confirms that supercells are the boundary footprint of megaplumes dominating the core of the domain. In agreement with available literature predictions, we show that the thickness of the thermal boundary layer scales very well with the Nusselt number as $\delta \sim 1/ Nu$ . Measurements of the mean wavenumber – inverse of the mean length scale – in the core of the flow support the scaling $\bar {k} \sim Ra^{0.49}$ , in very good agreement with theoretical and numerical predictions. Interestingly, the behaviour of the mean wavenumber near the boundaries scales as $\bar {k} \sim Ra^{0.81}$ , which is distinguishably different from the presumed linear behaviour. We hypothesize that a linear behaviour can only be observed in the ultimate regime, which we argue to set in only at $Ra$ in excess of $5\times 10^5$ , whereas a sublinear behaviour is recovered at more modest $Ra$ . The present results are expected to help the development of long desired reliable models to predict the large- and fine-scale structure of Rayleigh–Darcy convection in the high- $Ra$ regime typically encountered in geophysical processes, such as for instance in geological carbon dioxide sequestration.

Towards the ultimate regime in Rayleigh–Darcy convection

Abstract

Darcy Convection Anisotropic Problem with Multi Stability of Stationary Motions for Rectangle

Darcy Convection Anisotropic Problem with Multi Stability of Stationary Motions for Rectangle

The anisotropic problem of Darcy convection: a family of stationary motions and its decay when cosymmetry is destroyed

The anisotropic problem of Darcy convection: a family of stationary motions and its decay when cosymmetry is destroyed

Anisotropic Problem of Darcy Convection: Family of Steady Flows and Its Disintegration during the Destruction of Cosymmetry

Fluid convection in a porous rectangle is analyzed on the basis of the Darcy model with regard to anisotropy of the thermal characteristics and the permeability. Relations between the parameters for which the problem belongs to the class of cosymmetric systems are obtained. In this case explicit formulas for the critical numbers of the loss of stability of mechanical equilibrium are derived. Using a finite-difference method that retains the cosymmetry of the problem, families of steady convective regimes are calculated. The destruction of these families is demonstrated by means of the computational experiment in the case of violation of the cosymmetry conditions leading to appearance of a finite number of stationary regimes.

Dissolution-driven convection in a heterogeneous porous medium

Motivated by subsurface carbon sequestration, an experimental investigation of dissolution-driven Rayleigh–Darcy convection using two miscible fluids in a Hele-Shaw cell is conducted. A thin horizontal layer of circular impermeable discs is inserted to create an environment with heterogeneous and anisotropic permeability. The Sherwood number that measures the convective mass transfer rate between two fluids at the interface is linked to different parameters of the disc layer, including the disc size, spacing, layer permeability and its relative height with respect to the fluid interface. It is surprising that the convective mass transfer rate in our configuration is dominated by the disc spacing, but almost independent of either the disc size or the mean permeability of the layer. To explain this dependence, the convective mass transfer rate is decomposed into the number, velocity and density contrast of fingers travelling through the disc layer. Both the number and density contrast of fingers show dependences on the disc layer permeability, even though the product of them, the mass transfer rate, does not. In addition, the density contrast also shows a non-monotonic dependence on the disc spacing. The transition point is at a spacing that is close to the finger width. Based on this observation, a simple model based on mixing and scale competition is proposed, and it shows an excellent agreement with the experimental results.

Nonlinear throughflow effects on thermally modulated porous medium

Abstract Effect of vertical throughflow on Darcy convection has been investigated subject to time-periodic temperature modulation of the boundaries. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small, and the disturbances are expanded in terms of power series of amplitude of convection. A weak nonlinear stability analysis has been performed for the stationary mode of convection, and heat transport in terms of the Nusselt number, which is governed by the non-autonomous Ginzburg–Landau equation, is calculated. The effect of vertical throughflow is found to be either to destabilize or stabilize the system for downward or upward throughflows in the case of impermeable boundary conditions. The effect of amplitude and frequency of modulation, Prandtl–Darcy number on heat transport has been analyzed and depicted graphically. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system.

Rock dissolution patterns and geochemical shutdown of –brine–carbonate reactions during convective mixing in porous media

AbstractMotivated by the process of $\text{CO}_{2}$ convective mixing in porous media, here we study the formation of rock-dissolution patterns that arise from geochemical reactions during Rayleigh–Bénard–Darcy convection. Under the assumption of instantaneous chemical equilibrium, we adopt a formulation of the local reaction rate as a function of scalar dissipation rate, a measure that depends solely on flow and transport, and chemical speciation, which is a measure that depends only on the equilibrium thermodynamics of the chemical system. We use high-resolution simulations to examine the interplay between the density-driven hydrodynamic instability and the rock dissolution reactions, and analyse the impact of geochemical reactions on the macroscopic mass exchange rate. We find that dissolution of carbonate rock initiates in regions of locally high mixing, but that the geochemical reaction shuts down significantly earlier than shutdown of convective mixing. This early shutdown feature reflects the important role that chemical speciation plays in this hydrodynamics–reaction coupled process. Finally, we extend our analysis to three dimensions and explore the morphology of dissolution patterns in three dimensions.

Nonlinear Thermal Darcy Convection in a Nanofluid Saturated Porous Medium Under Gravity Modulation

Nonlinear Thermal Darcy Convection in a Nanofluid Saturated Porous Medium Under Gravity Modulation

Synchronous and asynchronous boundary temperature modulations of Bénard–Darcy convection

A theoretical analysis of thermo-convective instability in a densely packed porous medium is carried out when the boundary temperatures vary with time in a sinusoidal manner. By performing a weakly non-linear stability analysis, the Nusselt number is obtained as a function of amplitude of convection which is governed by a non-autonomous Ginzburg–Landau equation derived for the stationary mode of convection. The paper succeeds in unifying the modulated Bénard–Darcy, Bénard–Rayleigh, Bénard–Brinkman and Bénard–Chandrasekhar convection problems and hence precludes the study of these individual problems in isolation. A new result that shows that asynchronous temperature modulation may be effectively used to either enhance or reduce heat transport by suitably adjusting the frequency and phase-difference of the modulated temperature is presented.

Destruction of the family of steady states in the planar problem of Darcy convection

We consider natural convection of an incompressible fluid in a porous medium described by the planar Darcy equation. For some boundary conditions, Darcy problem may have non-unique solutions in form of a continuous family of steady states. We are interested in the situation when these boundary conditions are violated. The resulting destruction of the family of steady states is studied via computer experiments based on a mimetic finite-difference approach. Convection in a rectangular enclosure is considered under different perturbations of boundary conditions (heat sources, infiltration). Two scenario of the family of equilibria are found: the transformation to a limit cycle and the formation of isolated convective patterns.

Staggered grids discretization in three-dimensional Darcy convection

We consider three-dimensional convection of an incompressible fluid saturated in a parallelepiped with a porous medium. A mimetic finite-difference scheme for the Darcy convection problem in the primitive variables is developed. It consists of staggered nonuniform grids with five types of nodes, differencing and averaging operators on a two-nodes stencil. The nonlinear terms are approximated using special schemes. Two problems with different boundary conditions are considered to study scenarios of instability of the state of rest. Branching off of a continuous family of steady states was detected for the problem with zero heat fluxes on two opposite lateral planes.

Effect of Temperature Modulation on the onset of Darcy Convection in a Rotating Porous Medium

Effect of Temperature Modulation on the onset of Darcy Convection in a Rotating Porous Medium

Cosymmetric families of steady states in 3D convection of incompressible fluid in a porous medium

AbstractWe consider three‐dimensional convection of an incompressible fluid saturated in a parallelepiped with a porous medium. A mimetic finite‐difference scheme for the Darcy convection problem in the primitive variables is developed. Two problems with different boundary conditions are considered to study scenarios of instability of the state of rest. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Selection of steady states in planar Darcy convection

AbstractThe planar natural convection of incompressible fluid in a porous medium is considered. We study selection of steady states under temperature perturbations on the boundary. A selection map is introduced to analyze the selection of steady state from a continuous family of equilibria which exists under zero boundary conditions. The results of finite‐difference modeling for the rectangular enclosure are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Selection of steady states in planar Darcy convection

The planar natural convection of an incompressible fluid in a porous medium is considered. We study the selection of steady states under temperature perturbations on the boundary. A selection map is introduced in order to analyze the selection of a steady state from a continuous family of equilibria which exists under zero boundary conditions. The results of finite-difference modeling for a rectangular enclosure are presented.