Convective Instability in Slip Flow in a Vertical Circular Porous Microchannel

The paper represents an analysis of convective instability in a vertical cylindrical porous microchannel performed using the Galerkin method. The dependence of the critical Rayleigh number on the Darcy, Knudsen, and Prandtl numbers, as well as on the ratio of the thermal conductivities of the fluid and the wall, was obtained. It was shown that a decrease in permeability of the porous medium (in other words, increase in its porosity) causes an increase in flow stability. This effect is substantially nonlinear. Under the condition Da?>?0.1, the effect of the porosity on the critical Rayleigh number practically vanishes. Strengthening of the slippage effects leads to an increase in the instability of the entire system. The slippage effect on the critical Rayleigh number is nonlinear. The level of nonlinearity depends on the Prandtl number. With an increase in the Prandtl number, the effect of slippage on the onset of convection weakens. With an increase in the ratio of the thermal conductivities of the fluid and the wall, the influence of the Prandtl number decreases. At high values of the Prandtl numbers (Pr?>?10), its influence practically vanishes.

     

The Onset of Prandtl–Darcy–Prats Convection in a Horizontal Porous Layer

We consider the effect of finite Prandtl–Darcy numbers of the onset of convection in a porous layer heated isothermally from below and which is subject to a horizontal pressure gradient. A dispersion relation is found which relates the critical Darcy–Rayleigh number and the induced phase speed of the cells to the wavenumber and the imposed Péclet and Prandtl–Darcy numbers. Exact numerical solutions are given and these are supplemented by asymptotic solutions for both large and small values of the governing nondimensional parameters. The classical value of the critical Darcy–Rayleigh number is $4\pi ^2$ 4 π 2 , and we show that this value increases whenever the Péclet number is nonzero and the Prandtl–Darcy number is finite simultaneously. The corresponding wavenumber is always less than $\pi $ π and the phase speed of the convection cells is always smaller than the background flux velocity.     

Onset of Rayleigh–Bénard convection in cold water near its density maximum in vertical annular containers

A set of three-dimensional numerical simulations of Rayleigh–Bénard convection in cold water near its density maximum in vertical annular containers is performed with the aim of determining the critical Rayleigh number at the onset of convection and the primary flow patterns for different geometric dimensions and density inversion parameters. The Prandtl number of cold water is about 11.57. The annular container is heated from below and cooled from above. The inner and outer sidewalls are considered to be perfectly adiabatic. The results obtained show that the critical Rayleigh number at the onset of convection increases with increase in the density inversion parameter and the radius ratio and with decrease in the aspect ratio. When the radius ratio is small, the flow patterns in vertical annular containers are similar to those in cylindrical containers. At large radius ratios the flow pattern is relatively simple, with several convective rolls observable along the azimuthal direction and similar with those characteristic of Rayleigh–Bénard convection in the Boussinesq fluid. The stratified flow phenomenon is found to exist at moderate values of the density inversion parameter. The results are compared with those obtained in the Boussinesq fluid to reveal the effect of the density inversion parameter.     

Unsteady natural convection in a liquid-saturated porous enclosure with local thermal non-equilibrium effect

Stability analysis of free convection in a liquid-saturated sparsely-packed porous medium with local-thermal-non-equilibrium (LTNE) effect is presented. For the vertical boundaries free–free, adiabatic and rigid–rigid, adiabatic are considered while for horizontal boundaries it is the stress-free, isothermal and rigid–rigid, isothermal boundary combinations we consider. From the linear theory, it is apparent that there is advanced onset of convection in a shallow enclosure followed by that in square and tall enclosures. Asymptotic analysis of the thermal Rayleigh number for small and large values of the inter-phase heat transfer coefficient is reported. Results of Darcy–Bénard convection (DBC) and Rayleigh–Bénard convection can be obtained as limiting cases of the study. LTNE effect is prominent in the case of Brinkman–Bénard convection compared to that in DBC. Using a multi-scale method and by performing a non-linear stability analysis the Ginzburg–Landau equation is derived from the five-mode Lorenz modal. Heat transport is estimated at the lower plate of the channel. The effect of the Brinkman number, the porous parameter and the inter-phase heat transfer coefficient is to favour delayed onset of convection and thereby enhanced heat transport while the porosity-modified ratio of thermal conductivities shows the opposite effect.

     

FST方法分析二维Rayleigh-Bénard湍流热对流系统的能量输运

本文应用空间滤波方法：FST(Filter-space technique)方法，研究二维Rayleigh-Bénard(RB)湍流热对流系统中湍动能、热能和拟涡能的能量输运．研究中Rayleigh数(Ra)选取为1x10^8、1x10^9和1x10^10，Prandtl数(Pr)固定为4.38．我们展示了的结果表明，在二维RB系统中，三个Ra数下全场的平均湍动能和平均拟涡能在不同滤波尺度下的能量输运与Kraichnan在1967年预测的二维湍流中的级串理论有所偏差，而中心区域的能量都是向小尺度输运的．结果还揭示了瞬时能量输运的一些局部特性，包括它们在小尺度上不对称的分布．     

Onset of Darcy–Brinkman convection in a binary viscoelastic fluid saturated porous layer

The onset of Darcy–Brinkman double-diffusive convection in a binary viscoelastic fluid-saturated porous layer is studied using both linear and weakly nonlinear stability analyses. The Oldroyd-B model is employed to describe the rheological behavior of the fluid. An extended form of Darcy–Oldroyd law incorporating the Brinkman’s correction and time derivative is used to describe the fluid flow and the Oberbeck–Boussinesq approximation is invoked. The onset criterion for stationary and oscillatory convection is derived analytically. The effects of rheological parameters, Darcy number, normalized porosity, Lewis number, solute Rayleigh number, and Darcy–Prandtl number on the stability of the system is investigated. The results indicated that there is a competition among the processes of thermal, solute diffusions and viscoelasticity that causes the convection to set in through the oscillatory modes rather than the stationary. The Darcy–Prandtl number has a dual effect on the threshold of oscillatory convection. The nonlinear theory based on the method of truncated representation of Fourier series is used to find the transient heat and mass transfer. Some existing results are reproduced as the particular cases of present study.     

多孔介质内黏弹性流体的热对流稳定性研究

基于修正的Darcy模型, 介绍了多孔介质内黏弹性流体热对流稳定性研究的现状和主要进展. 通过线性稳定性理论, 分析计算多孔介质几何形状(水平多孔介质层、多孔圆柱以及多孔方腔)、热边界条件(底部等温加热、底部等热流加热、底部对流换热以及顶部自由开口边界)、黏弹性流体的流动模型(Darcy-Jeffrey, Darcy-Brinkman-Oldroyd以及Darcy-Brinkman -Maxwell模型)、局部热非平衡效应以及旋转效应对黏弹性流体热对流失稳的临界Rayleigh数的影响. 利用弱非线性分析方法, 揭示失稳临界点附近热对流流动的分叉情况, 以及失稳临界点附近黏弹性流体换热Nusselt数的解析表达式. 采用数值模拟方法, 研究高Rayleigh数下黏弹性流体换热Nusselt数和流场的演化规律,分析各参数对黏弹性流体热对流失稳和对流换热速率的影响.主要结果:(1)流体的黏弹性能够促进振荡对流的发生;(2)旋转效应、流体与多孔介质间的传热能够抑制黏弹性流体的热对流失稳;(3)在临界Rayleigh数附近,静态对流分叉解是超临界稳定的, 而振荡对流分叉可能是超临界或者亚临界的,主要取决于流体的黏弹性参数、Prandtl数以及Darcy数;(4)随着Rayleigh数的增加,热对流的流场从单个涡胞逐渐演化为多个不规则单元涡胞, 最后发展为混沌状态.      

Mechanism of mode selection in Rayleigh—Bénard convection with free—rigid boundaries

Higher harmonic resonances with wavenumber ratio of 1:2, 1:2:3 and so on are shown to take place in Rayleigh—Bénard convection under free—rigid boundary condition. Bifurcation diagrams for two-dimensional motion are obtained for the Prandtl number P = 7. The subharmonic instability is explained by a couple of amplitude equations obtained from weakly nonlinear stability theory. A straightforward extension of the coupled amplitude equations leads to a model which consists of n amplitude equations. The mechanism of mode selection is illustrated by numerical simulations of the model equations.     

A comparative study on low‐order,amplitude‐equation and perturbation approaches in thermal convection

A comparison among three weakly nonlinear approaches for thermo‐gravitational instability in a Newtonian fluid layer heated from below is presented. First, the dynamical systems describing the time evolution of the problem from different weakly nonlinear approaches, namely, the Lorenz model, the amplitude equations and the perturbation expansion approaches are obtained. Next, the steady states and their stability, as well as the transient behaviour are obtained from each dynamical system. The similarity and difference among the three models are emphasized. The role of each of the nondimensional groups, the Rayleigh number and the Prandtl number is compared for the three models. The different approaches lead to similar behaviours when the Rayleigh number is just above its critical value and Prandtl number is high. However, only the dynamical system obtained from the amplitude equations is able to reflect the role of the Prandtl number. On the other hand, the amplitude equations and perturbation expansion techniques are not suitable for predicting the uniform oscillatory behaviour observed frequently in Rayleigh–Bénard convection. The novelty of the current work lies in studying the critical differences in the findings of the three popular approaches to investigate weakly nonlinear thermal convection for the first time. Copyright © 2011 John Wiley & Sons, Ltd.     

The Effect of Rotation on the Onset of Double Diffusive Convection in a Sparsely Packed Anisotropic Porous Layer

The effect of rotation on the onset of double diffusive convection in a sparsely packed anisotropic porous layer, which is heated and salted from below, is investigated analytically using the linear and nonlinear theories. The Brinkman model that includes the Coriolis term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and a dispersion relation are obtained analytically using linear theory. The effect of anisotropy parameters, Taylor number, Darcy number, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory and finite amplitude convection is shown graphically. It is found that contrary to its usual influence on the onset of convection in the absence of rotation, the mechanical anisotropy parameter show contrasting effect on the onset criterion at moderate and high rotation rates. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The effect of various parameters on heat and mass transfer is shown graphically. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer

The stability of a fluid-saturated horizontal rotating porous layer subjected to time-periodic temperature modulation is investigated when the condition for the principle of exchange of stabilities is valid. The linear stability analysis is used to study the effect of infinitesimal disturbances. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Darcy–Rayleigh number and wavenumber. The shift in critical Darcy–Rayleigh number is calculated as a function of frequency of modulation, Taylor number, and Darcy–Prandtl number. It is established that the convection can be advanced by the low frequency in-phase and lower-wall temperature modulation, where as delayed by the out-of-phase modulation. The effect of Taylor number and Darcy–Prandtl number on the stability of the system is also discussed. We found that by proper tuning of modulation frequency, Taylor number, and Darcy–Prandtl number it is possible to advance or delay the onset of convection.     

Numerical investigation of roll convection stability

This study is devoted to the mathematical modeling of Rayleigh-Bénard convection in a rectangular cavity with rigid boundaries. The stability of the roll motion induced by an initial disturbance of special form is studied on the basis of two-dimensional and three-dimensional calculations. Different patterns of flow restructuring with respect to the wavenumber are analyzed on the Rayleigh number range Ra = 1708–7000 for the Prandtl numbers Pr = 1 and 0.71.     

Oscillatory Convection Onset in a Porous Rectangle with Non-analytical Corners

The analytical theory on Darcy–Bénard convection is dominated by normal-mode approaches, which essentially reduce the spatial order from four to two. This paper goes beyond the normal-mode paradigm of convection onset in a porous rectangle. A handpicked case where all four corners of the rectangle are non-analytical is therefore investigated. The marginal state is oscillatory with one-way horizontal wave propagation. The time-periodic convection pattern has no spatial periodicity and requires heavy numerical computation by the finite element method. The critical Rayleigh number at convection onset is computed, with its associated frequency of oscillation. Snapshots of the 2D eigenfunctions for the flow field and temperature field are plotted. Detailed local gradient analyses near two corners indicate that they hide logarithmic singularities, where the displayed eigenfunctions may represent outer solutions in matched asymptotic expansions. The results are validated with respect to the asymptotic limit of Nield (Water Resour Res 11:553–560, 1968).

     

THE BNARD CONVECTION IN A LAYER OF FLUID WITH A TIME-DEPENDENT MEAN TEMPERATURE

The onset of Bénard convection, or the critical Rayleigh number in a layer of fluid with a time-dependent mean temperature has been investigated theoretically. The critical Rayleigh number is regarded as a function of time and is expanded in series of a small parameter. Up to second approximation a simple expression of critical Rayleigh number is obtained for the time region for away from the point of zero.     

Double diffusive convection in a porous medium with modulated temperature on the boundaries

The effect of temperature modulation on the onset of double diffusive convection in a sparsely packed porous medium is studied by making linear stability analysis, and using Brinkman-Forchheimer extended Darcy model. The temperature field between the walls of the porous layer consists of a steady part and a time dependent periodic part that oscillates with time. Only infinitesimal disturbances are considered. The effect of permeability and thermal modulation on the onset of double diffusive convection has been studied using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. The effects of other parameters are also discussed on the stability of the system. Some results as the particular cases of the present study have also been obtained. Also the results corresponding to the Brinkman model and Darcy model have been compared.     

The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium

We investigate Rayleigh–Benard convection in a porous layer subjected to gravitational and Coriolis body forces, when the fluid and solid phases are not in local thermodynamic equilibrium. The Darcy model (extended to include Coriolis effects and anisotropic permeability) is used to describe the flow, whilst the two-equation model is used for the energy equation (for the solid and fluid phases separately). The linear stability theory is used to evaluate the critical Rayleigh number for the onset of convection and the effect of both thermal and mechanical anisotropy on the critical Rayleigh number is discussed.     

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.     

Effects of Time-Periodic Thermal Boundary Conditions and Internal Heating on Heat Transport in a Porous Medium

The effects of time-periodic boundary temperatures and internal heating on Nusselt number in the Bénard–Darcy convective problem has been considered. The amplitudes of temperature modulation at the lower and upper surfaces are considered to be very small. By performing a weakly non-linear stability analysis, the Nusselt number is obtained in terms of the amplitude of convection, which is governed by the non-autonomous Ginzburg–Landau equation, derived for the stationary mode of convection. The effects of internal Rayleigh number, amplitude and frequency of modulation, thermo-mechanical anisotropies, and Vadasz number on heat transport have been analyzed and depicted graphically. Increasing values of internal Rayleigh number results in the enhancement of heat transport in the system. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system.     

Transverse-roll global modes in a Rayleigh–Bénard–Poiseuille system with streamwise variable heating

The effect of a spatially inhomogeneous heating of the bottom wall in Rayleigh–Bénard–Poiseuille convection is studied for slow streamwise variations of the temperature profile. The problem is defined by the constant Reynolds number of the Poiseuille through flow, assumed to be low (typically 10), the constant Prandtl number, and the spatial evolution of the Rayleigh number , assumed to be subcritical everywhere except in a limited region around its single maximum . In this initial study, all spanwise inhomogeneities such as side walls or spanwise variable heating are neglected to obtain two-dimensional (transverse roll) global mode solutions by means of WKBJ asymptotics. The resulting frequency selection yields, at leading order, a global mode frequency equal to the local absolute frequency ω^t at the streamwise location where the Rayleigh number is maximum, with higher-order corrections for non-parallelism. These allow the determination of critical values of for global instability as a function of the profile of the local Rayleigh number and of Prandtl and Reynolds numbers.     

Convection in a porous medium with velocity slip and temperature jump boundary conditions

The onset of convection in a rarefield gas saturating a horizontal layer of a porous medium has been investigated using both Darcy and Brinkman models. It is assumed that due to rarefaction both velocity slip and temperature jump exist at the boundaries. The results show that (i) when the degree of rarefaction increases the critical Rayleigh number as well as the critical wave number for the onset of convection increases, (ii) stabilizing effect of temperature jump is more than that of velocity slip, (iii) Darcy model is seen to be the most stable one when compared to Brinkman model or the pure gaseous layer (i.e. in the absence of porous medium).