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.     

Onset of unsteady axi-symmetric laminar natural convection in a vertical cylindrical enclosure heated at the wall

In the present study laminar transition to oscillatory convection of fluids having different Prandtl numbers in a laterally heated vertical cylindrical enclosure for different aspect ratios (melt height to crucible radius) of 2–4 is investigated numerically for 0.01 ≤ Pr ≤ 10. Numerical solution to two-dimensional axisymmetric transient Navier Stokes equations and energy equation were solved by finite volume method using SIMPLE algorithm. Numerical results illustrate that there exists a critical Rayleigh number for each Prandtl number beyond which sustained laminar oscillatory flow sets in. The oscillatory regime was characterised by the oscillation of the average kinetic energy and average thermal energy of the melt. For a given aspect ratio, critical Rayleigh number increases with Pr upto 1 and then flattens. It was observed that for low Prandtl number fluids, Pr < 1.0, critical Rayleigh number is found to increase with increase in aspect ratio while for high Prandtl number fluids, Pr ≥ 1.0, it is found to decrease with increase in aspect ratio. The influence of aspect ratio on the transient behaviour of the melt volume below and above the critical Rayleigh number was studied.     

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.      

Instability of Hadley–Prats Flow with Viscous Heating in a Horizontal Porous Layer

The onset of convective rolls instability in a horizontal porous layer subject to a basic temperature gradient inclined with respect to gravity is investigated. The basic velocity has a linear profile with a non-vanishing mass flow rate, i.e., it is the superposition of a Hadley-type flow and a uniform flow. The influence of the viscous heating contribution on the critical conditions for the onset of the instability is assessed. There are four governing parameters: a horizontal Rayleigh number and a vertical Rayleigh number defining the intensity of the inclined temperature gradient, a Péclet number associated with the basic horizontal flow rate, and a Gebhart number associated with the viscous dissipation effect. The critical wave number and the critical vertical Rayleigh number are evaluated for assigned values of the horizontal Rayleigh number, of the Péclet number, and of the Gebhart number. The linear stability analysis is performed with reference either to transverse or to longitudinal roll disturbances. It is shown that generally the longitudinal rolls represent the preferred mode of instability.     

On pattern selection in mixed convection in rectangular ducts

Mixed convection in a horizontal rectangular duct has the same critical Rayleigh number as natural convection in a rectangular cavity for the onset of convection. The linear stability analysis predicts either an odd or an even number of convective rolls to appear depending on the aspect ratio of the cross section. However, it has been shown both experimentally and numerically that an even number of convective rolls appears under supercritical conditions for fully developed mixed convection. The paper first presents an analytical solution for the buoyancy-induced mainstream velocity, w ^b , at the onset of buoyancy-induced motion in a forced convective flow. Then, a comparison in the initial growth rate of w ^b is made between the case of an odd and an even number of rolls; which shows the selection of an even number of rolls over an odd number in mixed convection except for low aspect ratio ducts.     

矩形倾斜腔体中的两种对流斑图和分区

为了研究矩形倾斜腔体中普朗特数Pr=0.72的流体对流斑图和分区,本文基于流体力学方程组进行了数值模拟。在相对瑞利数r=6.0的情况下,观察了倾角θ=10°和θ=60°时对流斑图随着时间的发展,发现系统存在单圈型对流和多圈型对流两种斑图。流线随着倾角的变化说明:随着倾角增加,对流圈数逐渐减少,对流波长逐渐增加,对流波数减小;然后,随着对流圈数显著地减少,系统由多圈型对流过渡到单圈型对流。根据模拟计算结果,给出了多圈型对流过渡到单圈型对流的临界倾角θc随着相对瑞利数r变化的关系曲线。对流在θ-r平面上分为两个区域:θ<θc时系统是单圈型对流,θ>θc时系统是多圈型对流。对流最大振幅A和努塞尔数Nu随着倾角θ的变化曲线被临界倾角θc分成两段,它们有不同的变化规律。因此,临界倾角也可以由对流最大振幅A或努塞尔数Nu的变化曲线来确定。     

Convection mixte en fluide binaire avec effet Soret : étude analytique de la transition vers les rouleaux transversaux 2D

This Note deals with mixed convection in binary fluid with Soret effect in a rectangular duct heated from below. In particular, we study the transition towards transverse 2D rolls appearing at low Reynolds and Rayleigh numbers. The linear stability analysis of Poiseuille flow, with linearly stratified temperature and concentration fields, shows the influence of the separation ratio on the critical Rayleigh number at the transition towards the transversal 2D convective patterns. It highlights the presence, at low Reynolds numbers, of propagating transverse rolls in the downwards as well as in the upwards direction. Finally, we point out that, under these conditions, the propagating frequency of the rolls is the sum of two well defined frequencies: the first related to the Reynolds, the second to the separation ratio. To cite this article: E. Piquer et al., C. R. Mecanique 333 (2005).     

Efficient computation of natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder

In this work, the natural convection in a concentric annulus between a cold outer square cylinder and a heated inner circular cylinder is simulated using the differential quadrature (DQ) method. The vorticity‐stream function formulation is used as the governing equation, and the coordinate transformation technique is introduced in the DQ computation. It is shown in this paper that the outer square boundary can be approximated by a super elliptic function. As a result, the coordinate transformation from the physical domain to the computational domain is set up by an analytical expression, and all the geometrical parameters can be computed exactly. Numerical results for Rayleigh numbers range from 104 to 106 and aspect ratios between 1.67 and 5.0 are presented, which are in a good agreement with available data in the literature. It is found that both the aspect ratio and the Rayleigh number are critical to the patterns of flow and thermal fields. The present study suggests that a critical aspect ratio may exist at high Rayleigh number to distinguish the flow and thermal patterns. Copyright © 2002 John Wiley & Sons, Ltd.     

Overstability Analysis of Thermo-bioconvection Saturating a Porous Medium in a Suspension of Gyrotactic Microorganisms

A linear stability analysis is performed to analyze bioconvection in a dilute suspension of gyrotactic microorganisms in horizontal shallow fluid layer cooling from below and saturated by a porous medium, in the rigid boundary case. It is established that due to cooling from below thermally stratified layer is stabilized, which opposes the development of bioconvection and the situations for oscillatory convection may take place. The stability criterion is obtained in terms of thermal Rayleigh number, bioconvection Rayleigh number, gyrotactic number, bioconvection Peclet number, measure of cell eccentricity, Prandtl number, and Lewis number. It is observed that the presence of porous medium results in decrease of the magnitude of critical bioconvection Rayleigh number in comparison with its non-existence; hence due to porous effect, the system becomes less stable.     

Rayleigh–Bénard convection driven by a long wavelength heating

Natural convection in a two-dimensional horizontal layer has been investigated. The layer is confined between two parallel horizontal plates. The upper plate is kept isothermal, while the lower plate has an externally imposed, long wavelength, spatially sinusoidal heating with the amplitude expressed in terms of the Rayleigh number Ra and the wavelength characterized by the wave number α. Only steady-state flow structures and their bifurcations have been considered. The detailed analysis has been carried out for two Prandtl numbers, i.e. Pr = 0.7 and Pr = 7, and only small differences in the bifurcation diagrams have been observed. When Ra < Ra _cr = 427, convection has a simple topology consisting of one pair of counter-rotating rolls per heating period. Secondary motion in the form of rolls aligned in the direction of the primary rolls and concentrated around the hot spots occurs for Ra > 427. When 427 < Ra < ～470 and α < ～0.14, the secondary motion is described by the supercritical pitchfork bifurcation. One of the branches of this bifurcation is associated with an odd number of secondary rolls per half wavelength, with rolls above the hot spots rotating in the direction opposite to the primary rolls. The other branch is associated with an even number of secondary rolls per half wavelength, with the rolls above the hot spots co-rotating with the primary rolls. The new rolls are pinched off in pairs when α decreases. When Ra > ～470 and α > ~0.14, bifurcation assumes the form of “bifurcation from infinity”. The main branch is associated with one pair of rolls per heating period for α > 0.25. Decrease in α along this branch results in the formation of secondary rolls, with the rolls at the hot spot co-rotating with the primary rolls. The lower part of the other branch is associated with one pair of rolls per heating period in the limit α → 0. Increase in α results in pinching off a single roll which counter-rotates with respect to the primary roll at the hot spot.     

Three dimensional finite element analysis for Rayleight-Benard convection in rectangular enclosures

The pattern of Rayleigh-Benard convection of air in a rectangular box heated-from-below is studied by numerically solving the three-dimensional time-dependent Navier-Stokes equations under the Boussinesq approximation. Slightly supercritical Rayleigh number was adopted to track the evolutions of flow structure as a function of enclosure's aspect ratio (A=L/H). The flow will asymptotically evolve to different patterns, among which, two possible types of flow pattern are found. One consists of the pair of straight vortex rolls and the other appears as closed vortex rings. The transition between the flow patterns indicates that there exists a flow bifurcation with the variation of container's aspect ratio. In addition, both steady and oscillatory flows have been observed, corresponding to the pair of straight vortex rolls and the vortex ring, respectively. The complexity of flow structure tends to increase with the increasing aspect ratio of the rectangular enclosure. The project supported by the National Natural Science Foundation of China (19889210), the National Distinguished Young Fund (10125210), the Hundred Talents Program of CAS, and the Training Program for the Trans-Century Outstanding Young of MOE     

Convective Roll Instabilities of Vertical Throughflow with Viscous Dissipation in a Horizontal Porous Layer

The vertical throughflow with viscous dissipation in a horizontal porous layer is studied. The horizontal plane boundaries are assumed to be isothermal with unequal temperatures and bottom heating. A basic stationary solution of the governing equations with a uniform vertical velocity field (throughflow) is determined. The temperature field in the basic solution depends only on the vertical coordinate. Departures from the linear heat conduction profile are displayed by the temperature distribution due to the forced convection effect and to the viscous dissipation effect. A linear stability analysis of the basic solution is carried out in order to determine the conditions for the onset of convective rolls. The critical values of the wave number and of the Darcy–Rayleigh number are determined numerically by the fourth-order Runge–Kutta method. It is shown that, although generally weak, the effect of viscous dissipation yields an increase of the critical value of the Darcy–Rayleigh number for downward throughflow and a decrease in the case of upward throughflow. Finally, the limiting case of a vanishing boundary temperature difference is discussed.     

Convection in a low prandtl number fluid with internal heating

This paper studies the problem of non-linear thermal convection in a horizontal layer of a low Prandtl number fluid with nearly insulating boundaries and in the presence of horizontally uniform internal heat sources. Two-dimensional rolls and hexagonal cells are found to be the only possible stable convection cells. Subcritical instability associated with the hexagons can occur for a range of the amplitude of convection. It is found that non-uniform internal heating can affect various flow features and the stability of the convective motion. A new subcritical instability which exists even in a symmetric layer with arbitrary Prandtl number is also found for the case where the variations of the internal heating with respect to the vertical variable is sufficiently high.     

Free thermoconcentration convection above a localized heat source

Shadowgraph and probe techniques are used to investigate the free convective flow pattern above various localized heat sources (electrolytic cell — point source, short vertical cylinder, sphere) in an exponentially stratified fluid. The characteristic types of structure are classified with respect to the appearance of new forms of instability. The critical values of the global Rayleigh number, at which the flow pattern is restructured, are determined. The dependence of the height of the convection zone and the height of the individual cells on the governing parameters of the problem is investigated and the results are reduced to universal form. Laboratory experiments and oceanic observations are qualitatively compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–34, March–April, 1989.     

Fully developed magneto convection flow in a vertical rectangular duct

An analysis is performed to study the MHD free convection flow in a vertical rectangular duct for laminar and fully developed regime taking into consideration the effects of Ohmic heating and viscous dissipation. Numerical solutions are found using finite difference method of second-order accuracy. The effects of various physical parameters such as Hartmann number, aspect ratio, buoyancy parameter and circuit parameter are presented graphically. It is found that as Hartmann number, buoyancy parameter and aspect ratio increase, the upward and downward flow rates are increased for open circuit but decrease for short circuit.     

Turbulent penetrative convection with an internal heat source

An infinite fluid with a vertical cubic temperature profile in the absence of fluid motion is considered as a model for penetrative convection in which a central unstably stratified fluid layer is bounded above and below by stably stratified layers. Turbulence statistics from direct and large eddy numerical simulations for the mean temperature gradient, the velocity and temperature variances and the heat flux are presented for Rayleigh numbers R up to four orders of magnitude above critical. By means of a simplified second-moment closure, analytical scaling laws for the statistics are determined. For high Rayleigh numbers, the mean temperature gradient approaches zero in a central well-mixed layer, a reduced positive (stable) value in upper and lower partially mixed layers, and an unmixed value far above and below. The temperature variance is a factor of R^1/3 larger in the partially mixed layers compared to the well-mixed layer; the velocity variance and heat flux scales the same in both layers. Approximation of the three layers by a two layer model yields an estimate for the height of the mixed layer: the height decreases slowly with increasing Rayleigh number and at the highest Rayleigh number simulated is approximately 30% longer than the unstable layer in the absence of fluid motion.     

Turbulent thermal convection in wide horizontal fluid layers

Turbulence in thermal convection is investigated for flows in which the production of turbulence energy is due solely to buoyancy, and the statistics of the flow are homogeneous in horizontal planes. New experimental results for high Rayleigh number unsteady turbulent convection in a horizontal layer heated from below and insulated from above are presented and compared to turbulent Rayleigh convection, convection in the planetary boundary layer, and laboratory penetrative convection. Mean temperature fields are correlated in terms of wall layer scales and convection scales. Joint statistics of turbulent temperature and horizontal velocity and vertical velocity through fourth order are presented for the core region of the convection layer.This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Linear Stability of the Double-Diffusive Convection in a Horizontal Porous Layer with Open Top: Soret and Viscous Dissipation Effects

The linear stability of the double-diffusive convection in a horizontal porous layer is studied considering the upper boundary to be open. A horizontal temperature gradient is applied along the upper boundary. It is assumed that the viscous dissipation and Soret effect are significant in the medium. The governing parameters are horizontal Rayleigh number (\(Ra_\mathrm{H}\)), solutal Rayleigh number (\(Ra_\mathrm{S}\)), Lewis number (Le), Gebhart number (Ge) and Soret parameter (Sr). The Rayleigh number (Ra) corresponding to the applied heat flux at the bottom boundary is considered as the eigenvalue. The influence of the solutal gradient caused due to the thermal diffusion on the double-diffusive instability is investigated by varying the Soret parameter. A horizontal basic flow is induced by the applied horizontal temperature gradient. The stability of this basic flow is analyzed by calculating the critical Rayleigh number (\(Ra_\mathrm{cr}\)) using the Runge–Kutta scheme accompanied by the Shooting method. The longitudinal rolls are more unstable except for some special cases. The Soret parameter has a significant effect on the stability of the flow when the upper boundary is at constant pressure. The critical Rayleigh number is decreasing in the presence of viscous dissipation except for some positive values of the Soret parameter. How a change in Soret parameter is attributing to the convective rolls is presented.     

Aspect ratio effect on natural convection in water near its density maximum temperature

The effect of the aspect ratio on natural convection in water subjected to density inversion has been investigated in this study. Numerical simulations of the two-dimensional, steady state, incompressible flow in a rectangular enclosure with a variety of aspect ratios, ranging from 0.125 to 100, have been accomplished using a finite element model. Computations cover Rayleigh numbers from 10³ to 10⁶. Results reveal that the aspect ratio, A, the Rayleigh number, Ra, and the density distribution parameter, R, are the key parameters to determine the heat transfer and fluid flow characteristics for density inversion fluids in an enclosure. A new correlation for predicting the maximum mean Nusselt number is proposed in the form of , with the constants a and b depending on density distribution number R. It is demonstrated that the aspect ratio has a strong impact on flow patterns and temperature distributions in rectangular enclosures. The stream function ratio Ψ_inv/|Ψ_reg| is introduced to describe quantitatively the interaction between inversional and regular convection. For R=0.33, the density inversion enhancement is observed in the regime near A=3.     

Double-diffusive parallel flow induced in a horizontal Brinkman porous layer subjected to constant heat and mass fluxes: analytical and numerical studies

Fluid flow and heat and mass transfer induced by double-diffusive natural convection in a horizontal porous layer subjected to vertical gradients of temperature and concentration are studied analytically and numerically using the Brinkman-extended Darcy model. Both cases of rigid and free horizontal boundaries are examined in the present study. The parameters governing the problem are the Rayleigh number R_T, the Lewis number Le, the buoyancy ratio N, the Darcy number Da and the aspect ratio Ar. The analytical solution is based on the parallel flow approximation. The critical Rayleigh number corresponding to the onset of the parallel flow in this system is determined analytically as a function of Le, N and Da. For sufficiently small Da, both free and rigid boundaries yield results which are identical to those predicted by the Darcy model. The present investigation shows that there exists a region in the plane (N, Le) where the convective flow is not possible in the layer regardless of the Rayleigh and Darcy numbers considered. Received on 21 December 1998