Small and Moderate Prandtl Number Chaotic Convection in Porous Media in the Presence of Feedback Control

Feedback control on thermal convection in a fluid-saturated porous medium is investigated based on the dynamical systems approach. A low dimensional Lorenz-like model was obtained using the Galerkin-truncated approximation. The possible suppression or enhancement of chaotic convection is demonstrated when the fluid layer is subjected to feedback control in a low-dimensional framework.     

Weak Turbulence and Chaos for Low Prandtl Number Gravity Driven Convection in Porous Media

Low Prandtl number convection in porous media is relevant to modern applications of transport phenomena in porous media such as the process of solidification of binary alloys. The transition from steady convection to chaos is analysed by using Adomian's decomposition method to obtain an analytical solution in terms of infinite power series. The practical need to evaluate the solution and obtain numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the analytical results into a computational solution evaluated up to a finite accuracy. The solution shows a transition from steady convection to chaos via a Hopf bifurcation producing a 'solitary limit cycle which may be associated with an homoclinic explosion. This occurs at a slightly subcritical value of Rayleigh number, the critical value being associated with the loss of linear stability of the steady convection solution. Periodic windows within the broad band of parameter regime where the chaotic solution persists are identified and analysed. It is evident that the further transition from chaos to a high Rayleigh number periodic convection occurs via a period halving sequence of bifurcations.     

Multidimensional Observations of Dissolution-Driven Convection in Simple Porous Media Using X-ray CT Scanning

Transport in Porous Media - We present an experimental study of dissolution-driven convection in a three-dimensional porous medium formed from a dense random packing of glass beads. Measurements...     

Backward Free Convection Boundary Layers in Porous Media

The well known steady free convection forward boundary layer (FBL) flows ascending over a heated upwards projecting semi-infinite flat plate embedded in a fluid saturated porous medium are compared in this paper to their less well known backward (BBL) counterparts descending over a cooled (also upwards projecting!) semi-infinite flat plate. The circumstance that the definite edge of the plate (x = 0) in the former case is a leading edge and in the latter one a trailing edge, leads to substantially different mathematical and physical features of the FBL and BBL flows, respectively. The paper considers under this aspect the case of similar flows corresponding to surface temperature distributions which are power-law functions of the distance x from the definite edge. For permeable plates the effect of an adequate lateral suction and injection of the fluid is also taken into account. The detailed investigation, however, is restricted to the particular values m = +1 and m = –1/3 of the power-law exponent m, where both FBL and BBL solutions are available in exact analytic form. For each of these values, both exponentially and algebraically decaying BBL solutions were found. In addition, the existence of an exact algebraic BBL solution valid for any value of m is reported.     

A Brief Introduction to Convection in Porous Media

Transport in Porous Media - This paper serves as a brief introduction to the longer introduction provided by the book by Nield and Bejan (NB). Attention is focussed on the modelling of the...     

Radiative Effect on Natural Convection Flows in Porous Media

A regular two-parameter perturbation analysis based upon the boundary layer approximation is presented here to study the radiative effects of both first- and second-order resistances due to a solid matrix on the natural convection flows in porous media. Four different flows have been studied, those adjacent to an isothermal surface, a uniform heat flux surface, a plane plume and the flow generated from a horizontal line energy source on a vertical adiabatic surface. The first-order perturbation quantities are presented for all these flows. Numerical results for the four conditions with various radiation parameters are tabulated.     

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.     

A Two-Equation Analysis of Convection Heat Transfer in Porous Media

This paper presents a two-equation analysis on the convection heat transfer in porous media based on the modeling developed by Carbonell and Whitaker (1984). The porous system under consideration is bounded by two parallel walls and heated uniformly from one side surface. The Darcy flow is imposed and the fully developed heat transfer is assumed. General solutions, which take into account the additional convective and conductive terms, are obtained for the temperature fields and the Nusselt number. The detailed studies are presented for the porous systems characterized by consolidated and unconsolidated circular unit cells. The results show that, for the consolidated unit cell case, a prediction without the additional convective term overestimates the heat transfer, while for the unconsolidated unit cell case, this effect is negligible. The additional conductive terms are also examined and found to act conventionally as part of the conductive terms.     

On the Nield-Kuznetsov Theory for Convection in Bidispersive Porous Media

We revisit the problem of thermal convection in a bidispersive porous medium, first addressed by Nield and Kuznetsov (Int. J. Heat Mass Transfer, 49: 3068–3074, 2006). We investigate the possibility of oscillatory convection by using a highly accurate Chebyshev tau numerical method. We also develop a nonlinear energy stability theory for the same problem. This yields a global stability threshold below which instabilities cannot arise. These thresholds together with the linear instability boundaries yield a zone where thermal instability may be found. The results and theory of Nield and Kuznetsov (Int. J. Heat Mass Transfer, 49: 3068–3074, 2006) are thus proven to be a highly important development in the modern theory of designer porous materials, cf. Nield and Bejan (Convection in Porous Media, Springer, New York, 2006), pp. 94–97. This work was supported in part by a Research Project Grant of the Leverhulme Trust—Grant Number F/00128/AK.     

Transition to Chaos and Flow Dynamics of Thermochemical Porous Medium Convection

In a fluid-saturated porous medium, dissolved species advect at the pore velocity, while thermal retardation causes heat to move at the Darcy velocity. The Darcy model with the Boussinesq approximation in a square medium with a porosity of = 0.01 subject to two sources of buoyancy is used, to study numerically the dynamics of this so-called double-advective instability. The vertical walls of the medium are impermeable and adiabatic, while Dirichlet boundary conditions are imposed on the horizontal walls such that the medium is heated and salted from below. For an increasing ratio between chemical and thermal buoyancy, while keeping the thermal buoyancy fixed, a transition from a steady to a chaotic convective solution is observed. At the transition a stable limit cycle is found, suggesting that the transition takes the form of a Hopf bifurcation. The dynamics of the chaotic flow is characterized by irregular transitions between nonlayered and layered flow patterns, as a result of the spontaneous formation and disappearance of gravitationally stable interfaces. These interfaces temporarily divide the domain in layers of distinct solute concentration and lead to a significant reduction of kinetic energy and vertical heat and solute fluxes. The stability of an interface is described by a balance between the viscous drag forces in the convective layers and the buoyancy force associated with the density interface.     

Combined Free and Forced Convection in Vertical Channels of Porous Media

In this paper we investigate the combined free and forced convection of a fully developed Newtonian fluid within a vertical channel composed of porous media when viscous dissipation effects are taken into consideration. The flow is analysed in the region of a first critical Rayleigh number in order to interpret the multiple-valued solutions and discuss their validity. The governing fourth-order, ordinary differential equation, which contains the Darcy and the viscous dissipation terms, is solved analytically using perturbation techniques and numerically using D02HBF NAG Library. A detailed investigation of the governing O.D.E. is performed on both clear fluid and porous medium for various values of the viscous dissipation parameter, , when the wall temperature decreases linearly with height, and the pressure gradient is both above and below its hydrostatic value. Although mathematically the results in all cases show that there are two solution branches, producing four possible solutions, the study of the velocity and buoyancy profiles together with the Darcy effect indicate that only one of the two solutions at any value of the Rayleigh number appears to be physically acceptable. It is shown that the effect of the Darcy number decreases as the critical Rayleigh numbers increase.     

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

Transport in Porous Media - Natural convection in a porous enclosure in the presence of thermal dispersion is investigated. The Fourier–Galerkin (FG) spectral element method is adapted to...     

A Conceptual Study of Cavity Aeroacoustics Control Using Porous Media Inserts

In this study, an integrated flow simulation and aeroacoustics prediction methodology is applied to testing a sound control technique using porous inserts in an open cavity. Large eddy simulation (LES) combined with a three-dimensional Ffowcs Williams–Hawkings (FW–H) acoustic analogy is employed to predict the flow field, the acoustic sources and the sound radiation. The Darcy pressure – velocity law is applied to conceptually mimic the effect of porous media placed on the cavity floor and/or rear wall. Consequently, flow in the cavity could locally move in or out through these porous walls, depending on the local pressure differences. LES with “standard” subgrid-scale models for compressible flow is carried out to simulate the flow field covering the sound source and near fields, and the fully three-dimensional FW–H acoustic analogy is used to predict the sound field. The numerical results show that applying the conceptual porous media on cavity floor and/or rear wall could decrease the pressure fluctuations in the cavity and the sound pressure level in the far field. The amplitudes of the dominant oscillations (Rossiter modes) are suppressed and their frequencies are slightly modified. The dominant sound source is the transverse dipole term, which is significantly reduced due to the porous walls. As a result, the sound pressure in the far field is also suppressed. The preliminary study reveals that using porous-inserts is a promising technology for flow and sound radiation control.     

Transient Free Convection Fluid Flow in Domains Partially Filled with Porous Media

The problem of transient free convection in domains partly filled with porous substrates is investigated analytically using Laplace transformation technique. Four configurations are considered which are subject to an isothermal heating boundary condition. The Brinkman-extended Darcy model is adopted to describe the hydrodynamics behavior of the porous domain.     

The Effect of Thermal Expansion on Porous Media Convection. Part 2: Thermal Convection Solution

The impact of thermal expansion and the corresponding non-Boussinesq effects on porous media convection are considered. The results show that the non-Boussinesq effects decouple from the rest, and solving the Boussinesq system separately is needed even when non-Boussinesq effects are being investigated. The thermal expansion is shown to have a lasting impact on the post-transient convection only for values of Rayleigh number substantially beyond the convection threshold, where the amplitude of convection is not small. In the neighbourhood of the convection threshold the thermal expansion has only a transient impact on the solution. It is also evident from the results that the neglect of the time derivative term in the extended Darcy equation might introduce a significant error when oscillatory effects are present.     

Route to Chaos for Moderate Prandtl Number Convection in a Porous Layer Heated from Below

The route to chaos for moderate Prandtl number gravity driven convection in porous media is analysed by using Adomian's decomposition method which provides an accurate analytical solution in terms of infinite power series. The practical need to evaluate numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the otherwise analytical results into a computational solution achieved up to a desired but finite accuracy. The solution shows a transition to chaos via a period doubling sequence of bifurcations at a Rayleigh number value far beyond the critical value associated with the loss of stability of the convection steady solution. This result is extremely distinct from the sequence of events leading to chaos in low Prandtl number convection in porous media, where a sudden transition from steady convection to chaos associated with an homoclinic explosion occurs in the neighbourhood of the critical Rayleigh number (unless mentioned otherwise by 'the critical Rayleigh number' we mean the value associated with the loss of stability of the convection steady solution). In the present case of moderate Prandtl number convection the homoclinic explosion leads to a transition from steady convection to a period-2 periodic solution in the neighbourhood of the critical Rayleigh number. This occurs at a slightly sub-critical value of Rayleigh number via a transition associated with a period-1 limit cycle which seem to belong to the sub-critical Hopf bifurcation around the point where the convection steady solution looses its stability. The different regimes are analysed and periodic windows within the chaotic regime are identified. The significance of including a time derivative term in Darcy's equation when wave phenomena are being investigated becomes evident from the results.     

Nonsimilar Solutions for Mixed Convection in Non-Newtonian Fluids Along Horizontal Surfaces in Porous Media

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law type non-Newtonian fluids along horizontal surfaces with variable heat flux distribution. The mixed convection regime is divided into two regions, namely, the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

On the Effect of Anisotropy on the Stability of Convection in Rotating Porous Media

We investigate natural convection in an anisotropic porous layer subjected to centrifugal body forces. The Darcy model (including centrifugal and permeability anisotropy effects) is used to describe the flow and a modified energy equation (including the effects of thermal anisotropy) is used in the current analysis. The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection in the presence of thermal and mechanical anisotropy. It is found that the convection is stabilized when the thermal anisotropy ratio (which is a function of the thermal and mechanical anisotropy parameters) is increased in magnitude.     

Onset of Buoyancy-Driven Convection in Porous Media Saturated with Cold Water Cooled from Above

When porous media saturated with initially stagnant cold water around the density maximum temperature are cooled from above, convection may be induced in an unstable lower layer. In this study, the onset of buoyancy-driven convection during time-dependent cooling is investigated using the propagation theory, which transforms disturbance equations similarly, and also considering the density inversion effect. The critical Darcy–Rayleigh number Ra _D,c is found as a function of the dimensionless density maximum temperature θ _max. For Ra _D > Ra _D,c the dimensionless critical time τ _c to mark the onset of instability is presented as a function of Ra _D and θ _max. These critical conditions are compared with previous theoretical results.