Interacting convection modes in a saturated porous medium of nearly square planform: four modes

2017 ◽  
Vol 82 (3) ◽  
pp. 526-547 ◽  
Author(s):  
Brendan J. Florio ◽  
Andrew P. Bassom ◽  
Konstantinos Sakellariou ◽  
Thomas Stemler
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Basins Of Attraction ◽  
Stable States ◽  
Pitchfork Bifurcations ◽  
Multiple Stable States ◽  
Box Dimensions

Abstract Convection can occur in a confined saturated porous box when the associated Rayleigh number exceeds a threshold critical value: the identity of the preferred onset convection mode depends sensitively on the geometry of the box. Here we discuss examples for which the box dimensions are such that four modes share a common critical Rayleigh number. Perturbation theory is used to derive a system of coupled ordinary differential equations that governs the nonlinear interaction of the four modes and an analysis of this set is undertaken. In particular, it is demonstrated that as the Rayleigh number is increased beyond critical so a series of pitchfork bifurcations occur and multiple stable states are identified that correspond to the survival of just one of the four modes. The basins of attraction for each mode in the 4D phase space are described by a reduction to a suitable 3D counterpart.

Thermal Convective Instabilities in a Porous Medium

1984 ◽  
Vol 106 (1) ◽  
pp. 137-142 ◽  
Author(s):  
M. Kaviany
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Temperature Distribution ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Onset Of Convection ◽  
Long Time ◽  
Nonlinear Temperature

The onset of convection due to a nonlinear and time-dependent temperature stratification in a saturated porous medium with upper and lower free surfaces is considered. The initial parabolic temperature distribution is due to uniform internal heating. The medium is then cooled by decreasing the upper surface temperature linearly with time. Linear stability theory is applied to the more formally developed governing equations. In order to obtain an asymptotic solution for transient problems involving very long time scales, the critical Rayleigh number for steady-state, nonlinear temperature distribution is also obtained. The effects of porosity, permeability, and Prandtl number on the time of the onset of convection are examined. The steady-state results show that the critical Rayleigh number depends only on the ratio of porosity to permeability and when this ratio exceeds a value of one thousand, the critical Rayleigh number is directly proportional to this ratio.

Onset of Double Diffusive Convection in a Thermally Modulated Fluid-Saturated Porous Medium

2008 ◽  
Vol 63 (5-6) ◽  
pp. 291-300 ◽  
Author(s):  
Beer S. Bhadauria ◽  
Aalam Sherani
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Periodic Part ◽  
Double Diffusive Convection ◽  
Diffusive Convection ◽  
Fluid Saturated Porous Medium ◽  
The Galerkin Method ◽  
Double Diffusive

The onset of double diffusive convection in a sparsely packed porous medium was studied under modulated temperature at the boundaries, and a linear stability analysis has been made. The primary temperature field between the walls of the porous layer consisted of a steady part and a timedependent periodic part and the Galerkin method and the Floquet were used. The critical Rayleigh number was found to be a function of frequency and amplitude of modulation, Prandtl number, porous parameter, diffusivity ratio and solute Rayleigh number.

Finite Amplitude Convection in a Rotating Porous Medium

1987 ◽  
Vol 42 (1) ◽  
pp. 13-20
Author(s):  
B. S. Dandapat
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Interstitial Fluid ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Vertical Axis ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Linear Stability Theory ◽  
Onset Of Convection

The onset of convection in a horizontal layer of a saturated porous medium heated from below and rotating about a vertical axis with uniform angular velocity is investigated. It is shown that when S ∈ σ >1, overstability cannot occur, where ε is the porosity, σ the Prandtl number and S is related to the heat capacities of the solid and the interstitial fluid. It is also shown that for small values of the rotation parameter T1, finite amplitude motion with subcritical values of Rayleigh number R (i.e. R < Re, where Re is the critical Rayleigh number according to linear stability theory) is possible. For large values of T1, overstability is the preferred mode.

Effects of Nonuniform Thermal Gradient and Adiabatic Boundaries on Convection in Porous Media

1980 ◽  
Vol 102 (2) ◽  
pp. 254-260 ◽  
Author(s):  
N. Rudraiah ◽  
B. Veerappa ◽  
S. Balachandra Rao
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Heat Source ◽  
Thermal Gradient ◽  
Saturated Porous Medium ◽  
Critical Rayleigh Number ◽  
Single Term ◽  
Special Cases ◽  
Basic Temperature ◽  
Bounding Surfaces

The effects of different combinations of thermally insulated boundaries and nonuniform thermal gradient caused by either sudden heating or cooling at the boundaries or by distributed heat sources on convective stability in a fluid saturated porous medium are investigated using linear theory by considering the Brinkman model. In the case of sudden heating or cooling, solutions are obtained using single-term Galerkin expansion and attention is focused on the situation where the critical Rayleigh number is less than that for uniform temperature gradient and the convection is not maintained. Numerical values are obtained for various basic temperature profiles and some general conclusions about their destabilizing effects are presented. In particular, it is shown that the results of viscous fluid (σ = 0) and the usual Darcy porous medium (σ → ∞) emerge from our analysis as special cases. In the case of convection caused by heat source, since the effect of heat source is not brought out by the single-term Galerkin expansion, the critical internal Rayleigh number is determined using higher order expansion by specifying the external Rayleigh number. It is shown that, for values of σ2 ≥ 2.45 × 105, the different combinations of bounding surfaces give almost the same Rayleigh number and an explanation, following Lapwood, for this surprising behavior is given. It is found that the heat source’s effect on convection decreases for wave numbers up to the value 2.2 and drops suddenly around the critical value of 2.4 and then increases up to 2.5.

Non-Darcian Effects in Open-Ended Cavities Filled With a Porous Medium

1991 ◽  
Vol 113 (3) ◽  
pp. 747-756 ◽  
Author(s):  
J. Ettefagh ◽  
K. Vafai ◽  
S. J. Kim
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Saturated Porous Medium ◽  
Critical Value ◽  
Inertial Parameter ◽  
Flow Models ◽  
Flow And Heat Transfer ◽  
Buoyancy Driven Convection

The importance and relevance of non-Darcian effects associated with the buoyancy driven convection in open-ended cavities filled with fluid-saturated porous medium is analyzed in this work. Several different flow models for porous media, such as Brinkman-extended Darcy, Forchheimer-extended Darcy, and generalized flow models, are considered. The significance of inertia and boundary effects, and their crucial influence on the prediction of buouancy-induced flow and heat transfer in open-ended cavities, are investigated. Analysis is made on the proper choice of parameters that can fully determine the criteria for the range of validity of Darcy’s law in this type of configuration. Critical values of the inertial parameter, Λcrit, below which, for any given modified Rayleigh number, the Darcy flow model breaks down, have been investigated. It is shown that the critical value of the inertial parameter depends on the modified Rayleigh number and that this critical value increases as Ra* increases. It is also observed that for higher modified Rayleigh number, the deviation from a Darcian formulation appears at Darcy numbers greater than 1×10−4. The Prandtl number effects on convective flow and heat transfer are shown to be quite significant for small values of Pr. The Prandtl number effects are reduced significantly for higher values of the Prandtl number.

scholarly journals Nonlinear feedback in a six-dimensional Lorenz Model: impact of an additional heating term

2015 ◽  
Vol 2 (2) ◽  
pp. 475-512
Author(s):  
B.-W. Shen
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Critical Value ◽  
Small Scale ◽  
Solution Stability ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Additional Heating ◽  
The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the steamfunction is referred to as a secondary streamfunction mode, while the two additional modes, that appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): If the flap of a butterfly's wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado. The implications of this and previous work, as well as future work, are also discussed.

Stability and Convection in Impulsively Heated Porous Layers

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
M. J. Kohl ◽  
M. Kristoffersen ◽  
F. A. Kulacki
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Heated Surface ◽  
Critical Rayleigh Number ◽  
Axisymmetric Flow ◽  
Temperature Fluctuations ◽  
Lower Surface ◽  
Onset Of Convection ◽  
Upward Moving ◽  
The Stability

Experiments are reported on initial instability, turbulence, and overall heat transfer in a porous medium heated from below. The porous medium comprises either water or a water-glycerin solution and randomly stacked glass spheres in an insulated cylinder of height:diameter ratio of 1.9. Heating is with a constant flux lower surface and a constant temperature upper surface, and the stability criterion is determined for a step heat input. The critical Rayleigh number for the onset of convection is obtained in terms of a length scale normalized to the thermal penetration depth as Rac=83/(1.08η−0.08η2) for 0.02<η<0.18. Steady convection in terms of the Nusselt and Rayleigh numbers is Nu=0.047Ra0.91Pr0.11(μ/μ0)0.72 for 100<Ra<5000. Time-averaged temperatures suggest the existence of a unicellular axisymmetric flow dominated by upflow over the central region of the heated surface. When turbulence is present, the magnitude and frequency of temperature fluctuations increase weakly with increasing Rayleigh number. Analysis of temperature fluctuations in the fluid provides an estimate of the speed of the upward moving thermals, which decreases with distance from the heated surface.

Natural convective heat transfer in a hemispherical cavity filled with ZnO–H2O nanofluid saturated porous medium

2018 ◽  
Vol 29 (10) ◽  
pp. 1850097 ◽  
Author(s):  
Abderrahmane Baïri ◽  
Najib Laraqi
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Volume Fraction ◽  
Saturated Porous Medium ◽  
Physical Parameters ◽  
Numerical Work

This three-dimensional (3D) numerical work based on the volume control method quantifies the convective heat transfer occurring in a hemispherical cavity filled with a ZnO–H2O nanofluid saturated porous medium. Its main objective is to improve the cooling of an electronic component contained in this enclosure. The volume fraction of the considered monophasic nanofluid varies between 0% (pure water) and 10%, while the cupola is maintained isothermal at cold temperature. During operation, the active device generates a heat flux leading to high Rayleigh number reaching [Formula: see text] and may be inclined with respect to the horizontal plane at an angle ranging from 0[Formula: see text] to 180[Formula: see text] (horizontal position with cupola facing upwards and downwards, respectively) by steps of 15[Formula: see text]. The natural convective heat transfer represented by the average Nusselt number has been quantified for many configurations obtained by combining the tilt angle, the Rayleigh number, the nanofluid volume fraction and the ratio between the thermal conductivity of the porous medium’s solid matrix and that of the base fluid. This ratio has a significant influence on the free convective heat transfer and ranges from 0 (without porous media) to 70 in this work. The influence of the four physical parameters is analyzed and commented. An empirical correlation between the Nusselt number and these parameters is proposed, allowing determination of the average natural convective heat transfer occurring in the hemispherical cavity.

Stability of Bioconvection in Suspensions of Gyrotactic Microorganisms in Porous Media

2002 ◽  
Author(s):  
A. V. Kuznetsov ◽  
A. A. Avramenko
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Saturated Porous Medium ◽  
Critical Value ◽  
Numerical Results ◽  
Velocity Fluctuations ◽  
Gyrotactic Microorganisms ◽  
Upward Direction ◽  
Upward Velocity ◽  
Fluid Saturated Porous Medium

In this paper, a model of bioconvection in a suspension of gyrotactic motile microorganisms in a fluid saturated porous medium is suggested. The microorganisms considered in this paper are heavier than water and gyrotactic behavior results in their swimming towards the regions of most rapid downflow. Because of that, the regions of downflow become denser than the regions of upflow. Buoyancy increases the upward velocity in the regions of upflow and downward velocity in the regions of downflow, thus enhancing the velocity fluctuations. The experiments performed by Kessler (1986) and the numerical results of Kuznetsov and Jiang (2001) indicate that if the permeability of porous medium is sufficiently small it will prevent the development of convection instability. However, for practical purposes, in order to maximize the flux of the cells in the upward direction it is desirable to have the permeability of the porous medium as high as possible. The aim of this paper is to investigate the value of critical permeability. If permeability is smaller than this critical value bioconvection does not occur and microorganisms simply swim in the upward direction.

Stability of Gravity Driven Convection in a Cylindrical Porous Layer Subjected to Vibration

2006 ◽  
Author(s):  
Saneshan Govender
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Density Gradient ◽  
Porous Layer ◽  
Binary Alloy ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Direct Result ◽  
Porous Cylinder

In both pure fluids and porous media, the density gradient becomes unstable and fluid motion (convection) occurs when the critical Rayleigh number is exceeded. The classical stability analysis no longer applies if the Rayleigh number is time dependant, as found in systems where the density gradient is subjected to vibration. The influence of vibrations on thermal convection depends on the orientation of the time dependant acceleration with respect to the thermal stratification. The problem of a vibrating porous cylinder has numerous important engineering applications, the most important one being in the field of binary alloy solidification. In particular we may extend the above results to understanding the dynamics in the mushy layer (essentially a reactive porous medium) that is sandwiched between the underlying solid and overlying melt regions. Alloyed components are widely used in demanding and critical applications, such as turbine blades, and a consistent internal structure is paramount to the performance and integrity of the component. Alloys are susceptible to the formation of vertical channels which are a direct result of the presence convection, so any technique that suppresses convection/the formation of channels would be welcomed by the plant metallurgical engineer. In the current study, the linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogeneous cylindrical porous layer heated from below. The linear stability results show that increasing the frequency of vibration stabilizes the convection. In addition the aspect ratio of the porous cylinder is shown to influence the stability of convection for all frequencies analysed. It was also observed that only synchronous solutions are possible in cylindrical porous layers, with no transition to sub harmonic solutions as was the case in Govender (2005a) for rectangular layers or cavities. The results of the current analysis will be used in the formulation of a model for binary alloy systems that includes the reactive porous medium model.

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