Axisymmetric convection in a spherical layer: Transitions between steady-state regimes

The steady-state convective motions of a viscous fluid occupying a spherical layer R₁ r R₂, R₂/R₁=1.2 are studied. The non-deformable boundaries of the layer are assumed to be free of shear stresses. At the outer boundary the constant temperature and at the inner boundary the constant heat flux are given. The system of equations in the Boussinesq approximation is solved by the Galerkin method with time stabilization on the assumption of axial and equatorial symmetry. It is shown that at the point Ra=Ra_c the state of mechanical equilibrium loses stability and steady symmetrical supercritical bifurcation is observed. The modes most unstable in the linear sense determine the form of convection when Ra > Ra_c and the supercriticality is not too great. At Rayleigh numbers Ra_c < Ra < 200Ra_c there exists a set of steady-state solutions with different spatial structures. The realization of solutions of a particular type depends on the supercriticality and the initial conditions. The evolution of the solutions with variation of the Rayleigh number is investigated. The changes in the spatial kinetic energy spectra and the integral heat fluxes upon transition from one branch of the solutions to another and with variation of the supercriticality are analyzed. As the supercriticality increases, despite the excitation of more and more new small-scale modes, the large-scale motions begin to make an ever greater contribution to the total energy. The results obtained can be used for constructing hydrodynamic models of the global motions in the atmospheres of giant planets, the convective envelopes of stars, and in the depths of the earth's mantle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–24, November–December, 1989.     

Instability of small-amplitude convective flows in a rotating layer with free boundaries

The stability of steady convective flows in a horizontal layer with free boundaries, heated from below and rotating about a vertical axis, is studied in the Boussinesq approximation (Rayleigh-Bénard convection). The flows considered are convective rolls or square cells that are sums of two perpendicular rolls with equal wave numbers k. It is assumed that the Rayleigh number is almost critical in order for convective flows with a wave number k: R = R _c (k) + ε² to arise, the amplitude of the supercritical states being of the order of ε. It is shown that the flows are always unstable relative to perturbations that are the sum of one long-and two short-wave modes corresponding to linear rolls turned through small angles in opposite directions.     

Non-linear convection in a porous medium with internal heat sources

The paper studies non-linear thermal convection in a horizontal porous layer of fluid with nearly insulating boundaries and in the presence of internal heat sources. Square and hexagonal cells are found to be the only possible stable convection cells. Finite amplitude instability could exist for some particular forms of an internal heat source Q. For a uniform Q, the preferred flow pattern is that of hexagons for amplitude ε smaller than some critical value ε_c, while both squares and hexagonal cells are stable for ε ? ε_c. The convective motion is downward at the hexagonal cell's centers. For a non-uniform Q, the qualitative features of thermal convection depend on the actual form Q. In particular, a non-uniform Q can increase or decrease the cell's size and the critical Rayleigh number at the onset of convection, and stabilize or destabilize the convective motion in the form of hexagonal cells with either upward or downward motion at the cell's centers.     

Convection in a closed cavity heated from below when equilibrium conditions are disrupted

The equilibrium of a fluid is possible in a closed cavity in the presence of a strictly vertical temperature gradient (heating from below) [1]. There is a distinct sequence of critical Rayleigh numbers R_i at which this equilibrium loses its stability relative to low characteristic perturbations. The presence of different finite perturbations, unavoidable in an experiment, leads to the absence of a strict equilibrium when R < R₁. The problem of the influence of the perturbation on the convection conditions near the critical points arises in this context [2, 3]. The case in which the cavity is heated not strictly from below is investigated in [2] and the case in which the perturbation of the equilibrium is due to the slow movement of the upper boundary of the region is investigated in [3]. In [2, 3] the perturbation has the structure of a first critical motion and thus the results of these papers coincide qualitatively. The perturbation of the temperature in the horizontal sections of the boundary, which creates a perturbation with a two-vortex structure corresponding to the second critical point R₂, is examined in this paper. A similar type of perturbation is characteristic for experiments in which the thermal conductivity properties of the fluid and the cavity walls are different. The nonlinear convection conditions are investigated numerically by the net-point method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 203–207, March–April, 1977.The author wishes to thank D. B. Lyubimova, V. I. Chernatynskii, and A. A, Nepomnyashchii for their helpful comments.     

Convective instability of a rotating fluid

A physical system may be in thermodynamic equilibrium when participating as a whole in uniform rotational motion [1]. In particular, mechanical equilibrium of a liquid in a cavity rotating about a stationary axis with the constant angular velocity (solid-body rotation of the liquid) is possible. If the liquid is uniform in composition and isothermal, then such equilibrium, as shown in [2], is stable for all . However, in the case of a nonuniformly heated liquid, stability of the solid-state rotation is, generally speaking, impossible.The appearance of two steady-state force fields is associated with uniform rotation: the centrifugal field and the Coriolis force field. The former field forces the liquid elements which are less heated and therefore more dense to move away from the axis of rotation, displacing the less dense liquid layers (centrifugation). If we maintain in the liquid a temperature gradient which prevents the establishment of equilibrium stratification of the liquid, then with a suitable value of this gradient (the magnitude obviously depending on ) undamped flows—convection—will develop in the liquid. Thus, while in conventional gravitational convection the gravity field is the reason for the appearance of the Archimedes buoyant forces, in the rotating cavity the mixing of the nonuniformly heated liquid is caused by the centrifugal field. As soon as the convective flows arise the Coriolis forces come into play. Account for the latter, as is shown below, prevents reducing in a trivial fashion the study of convective stability of rotating liquid to the well-studied problems of gravitational convection.     

Variable permeability effect on convection in binary mixtures saturating a porous layer

The Darcy Model with the Boussinesq approximation is used to study natural convection in a shallow porous layer, with variable permeability, filled with a binary fluid. The permeability of the medium is assumed to vary exponentially with the depth of the layer. The two horizontal walls of the cavity are subject to constant fluxes of heat and solute while the two vertical ones are impermeable and adiabatic. The governing parameters for the problem are the thermal Rayleigh number, R _T, the Lewis number, Le, the buoyancy ratio, φ, the aspect ratio of the cavity, A, the normalized porosity, ε, the variable permeability constant, c, and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained on the basis of the parallel flow approximation. The onset of supercritical convection, or subcritical, convection are predicted by the present theory. A linear stability analysis of the parallel flow model is conducted and the critical Rayleigh number for the onset of Hopf’s bifurcation is predicted numerically. Numerical solutions of the full governing equations are found to be in excellent agreement with the analytical predictions.     

Unsteady Natural Convection Flow in a Square Cavity Filled with a Porous Medium Due to Impulsive Change in Wall Temperature

Unsteady natural convection flow in a two-dimensional square cavity filled with a porous material has been studied. The flow is initially steady where the left-hand vertical wall has temperature T _h and the right-hand vertical wall is maintained at temperature T _c (T _h > T _c) and the horizontal walls are insulated. At time t > 0, the left-hand vertical wall temperature is suddenly raised to which introduces unsteadiness in the flow field. The partial differential equations governing the unsteady natural convection flow have been solved numerically using a finite control volume method. The computation has been carried out until the final steady state is reached. It is found that the average Nusselt number attains a minimum during the transient period and that the time required to reach the final steady state is longer for low Rayleigh number and shorter for high Rayleigh number.     

On the convective instability of a thermal boundary layer

When the surface temperature of a liquid is a harmonic function of time with a frequency, a temperature wave propagates into the liquid. The amplitude of this wave decreases exponentially with distance from the surface. The temperature oscillation is essentially concentrated in a layer of the order of (2/)^1/2, where x is the thermal conductivity of the liquid (thermal boundary layer). Depending on the phase, at certain positions below the surface the temperature gradient is directed downwards and if its magnitude is sufficiently large (the magnitude is a function of the amplitude and frequency of the surface oscillations) the liquid can become unstable with respect to the onset of convection. In that case the convective motion may spread beyond the initial unstable layer. For low frequencies the stability condition can be derived from the usual static Rayleigh criterion, on the basis of the Rayleigh number and the average temperature gradient of the unstable layer. This quasi-static approach, used by Sal'nikov [1], is appropriate to those cases in which the period of the temperature oscillations is much larger than the characteristic time of the perturbations. But when these times are of the same order, the problem must be analyzed in dynamic terms. The stability problem must then be formulated as a problem of parametricresonance excitation of velocity oscillations due to the action of a variable parameter-the temperature gradient.In an earlier work [2] we considered the problem of the stability of a horizontal layer of liquid with a periodically varying temperature gradient. It was assumed that the thickness of the layer was much smaller than the penetration depth of the thermal wave, so that the temperature gradient could be assumed to be independent of position. In the present work we consider the opposite case, in which the liquid layer is assumed to be much larger than the penetration depth, i. e., a thermal boundary layer can be defined. The temperature gradient at equilibrium, which is a parameter in the equations determining the onset of perturbations, is here a periodic function of time and a relatively complicated function of the depth coordinate z. The periodic oscillations are solved by the Fourier method; the equations for the amplitudes are solved by the approximate method of KarmanPohlhausen.The authors are grateful to L. G. Loitsyanskii for helpful criticism.     

Rayleigh-Benard convection: experimental study of time-dependent instabilities

The transition from steady thermal convection to turbulent thermal convection in a horizontal layer of water (Prandtl number=5.8) contained by a square cavity of large aspect ratio (48.5) has been studied using laser Doppler velocimetry. Power spectra of the horizontal velocity fluctuations were measured in the Rayleigh number range from 30,000 and 99,000, wherein periodic, quasi-periodic, and broad-band time-dependent instabilities coexist. At Rayleigh numbers greater than 32,000 a narrow-band spectrum emerges. The frequency of this motion scales with x/d ² modified by a Prandtl number factor for intermediate values of the Prandtl number. Between 10 Ra _c and 30 Ra _i the frequency undergoes three abrupt jumps while increasing along an Ra ^2/3 power law. A different frequency mode that occurs above 30 Ra _c appears to be associated with fully turbulent convection.List of symbols c heat capacity - d depth of fluid layer - f frequency - g gravitational acceleration - H heat flux - k thermal conductivity - Nu Nusselt number=Hd/k T - Pr Prandtl number=v/x - Ra Rayleigh number - Ra _c critical Rayleigh number=1708 - T temperature difference across the fluid layer - thermal coefficient of expansion - _v aspect ratio=width/depth - x thermometric conductivity=k/ - kinematic viscosity - density A version of this paper was presented at the Tenth Symposium of Turbulence, University of Missouri-Rolla, September 22–24, 1986     

Influence of boundary conditions on the convective stability of a rotating horizontal layer of liquid

A theoretical study is made of the critical curves for the onset of convection in a plane horizontal layer of liquid rotating with constant angular velocity for different conditions on the boundary of the layer. It is shown that, in contrast to Chandresekhar's curves [1] obtained under the condition of constancy of the temperature on the boundaries, the curves for a constant heat flux lie significantly lower, so that convection occurs earlier for all Taylor numbers. At large Taylor numbers all the stability curves, as in [1], tend to the asymptotes R_C Ta^2/3, where Ta is the Taylor number and R_C is the critical Rayleigh number. A similar investigation for a nonrotating liquid was made in [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 124–129, May–June, 1988.We thank G. S. Golitsyn for the proposed subject and constant interest in the work.     

Numerical investigation of convective motion in a closed cavity

The investigation of thermal convection in a closed cavity is of considerable interest in connection with the problem of heat transfer. The problem may be solved comparatively simply in the case of small characteristic temperature difference with heating from the side, when equilibrium is not possible and when slow movement is initiated for an arbitrarily small horizontal temperature gradient. In this case the motion may be studied using the small parameter method, based on expanding the velocity, temperature, and pressure in series in powers of the Grashof number—the dimensionless parameter which characterizes the intensity of the convection [1–4]. In the problems considered it has been possible to find only two or three terms of these series. The solutions obtained in this approximation describe only weak nonlinear effects and the region of their applicability is limited, naturally, to small values of the Grashof number (no larger than 10³).With increase of the temperature difference the nature of the motion gradually changes—at the boundaries of the cavity a convective boundary layer is formed, in which the primary temperature and velocity gradients are concentrated; the remaining portion of the liquid forms the flow core. On the basis of an analysis of the equations of motion for the plane case, Batchelor [4] suggested that the core is isothermal and rotates with constant and uniform vorticity. The value of the vorticity in the core must be determined as the eigenvalue of the problem of a closed boundary layer. A closed convective boundary layer in a horizontal cylinder and in a plane vertical stratum was considered in [5, 6] using the Batchelor scheme. The boundary layer parameters and the vorticity in the core were determined with the aid of an integral method. An attempt to solve the boundary layer equations analytically for a horizontal cylinder using the Oseen linearization method was made in [7].However, the results of experiments in which a study was made of the structure of the convective motion of various liquids and gases in closed cavities of different shapes [8–13] definitely contradict the Batchelor hypothesis. The measurements show that the core is not isothermal; on the contrary, there is a constant vertical temperature gradient directed upward in the core. Further, the core is practically motionless. In the core there are found retrograde motions with velocities much smaller than the velocities in the boundary layer.The use of numerical methods may be of assistance in clarifying the laws governing the convective motion in a closed cavity with large temperature differences. In [14] the two-dimensional problem of steady air convection in a square cavity was solved by expansion in orthogonal polynomials. The author was able to progress in the calculation only to a value of the Grashof numberG=10⁴. At these values of the Grashof numberG the formation of the boundary layer and the core has really only started, therefore the author's conclusion on the agreement of the numerical results with the Batchelor hypothesis is not justified. In addition, the bifurcation of the central isotherm (Fig. 3 of [14]), on the basis of which the conclusion was drawn concerning the formation of the isothermal core, is apparently the result of a misunderstanding, since an isotherm of this form obviously contradicts the symmetry of the solution.In [5] the method of finite differences is used to obtain the solution of the problem of strong convection of a gas in a horizontal cylinder whose lateral sides have different temperatures. According to the results of the calculation and in accordance with the experimental data [9], in the cavity there is a practically stationary core. However, since the authors started from the convection equations in the boundary layer approximation they did not obtain any detailed information on the core structure, in particular on the distribution of the temperature in the core.In the following we present the results of a finite difference solution of the complete nonlinear problem of plane convective motion in a square cavity. The vertical boundaries of the cavity are held at constant temperatures; the temperature varies linearly on the horizontal boundaries. The velocity and temperature distributions are obtained for values of the Grashof number in the range 0<G4·10⁵ and for a value of the Prandtl number P=1. The results of the calculation permit following the formation of the closed boundary layer and the very slowly moving core with a constant vertical temperature gradient. The heat flux through the cavity is found as a function of the Grashof number.     

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part I. Normal Modes

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number R _h < 49, the instability is absolute implying that, as the vertical Rayleigh number, R _v, increases passing through its critical value, R _vc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as R _v, increases beyond the critical value, while the horizontal Rayleigh number, R _h, and the Péclet number, Q _v, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, R _v, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient.     

Application of the Galerkin method to the solution of the problem of combined (forced and free) turbulent convection in a vertical circular channel in a uniform solid mass

We consider steady-state combined (forced and free) turbulent convection in a vertical circular channel in a uniform solid medium for the case in which a constant vertical temperature gradientis maintained in the solid mass, far from the channel. The velocity and temperature distributions are found, and the critical values of the Rayleigh number for axisymmetric and antisymmetric fluid motions are calculated. The problem is solved by the Galerkin method.Notation v⁽⁰⁾ forced convection velocity - v⁽¹⁾ free convection velocity - v velocity with combination of forced and free convection - v average velocity across channel section - T temperature with combined forced and free convection - T_w channel wall temperature - y distance from channel wall - y_* dimensionless distance from wall - r₀ channel radius - r distance from centerline - v_t turbulent viscosity - _t turbulent thermal diffusivity - P₀ averaged pressure corresponding to constant fluid ternperature - z coordinate along channel axis, directed upward - Q quantity of heat released by internal sources per unit fluid volume per unit time - fluid thermal conductivity (_e for the surrounding mass) - R Reynolds number - R^* Rayleigh number - P Prandtl number - G Grashof number - V_* dynamic viscosity     

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。     

Transition from Convective to Absolute Instability in a Porous Layer with Either Horizontal or Vertical Solutal and Inclined Thermal Gradients,and Horizontal Throughflow

Recently, in Diaz and Brevdo (J Fluid Mech 681: 567–596, 2011), further in the text referred to as D&B, we found an absolute/convective instability dichotomy at the onset of convection in a flow in a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. The control parameter in D&B triggering the destabilization is the vertical thermal Rayleigh number, R _v. In this article, we treat the parameter cases considered in D&B in which the onset of convection has the character of convective instability and occurs through longitudinal modes. By increasing the vertical thermal Rayleigh number starting from its critical value, R _vc, we determine the value R _vt of R _v at which the transition from convective to absolute instability takes place and compute the physical characteristics of the emerging absolutely unstable wave packet. In some cases, the value of the transitional vertical thermal Rayleigh number, R _vt, is only slightly greater than the critical value, R _vc, meaning that at the onset of convection the base convectively unstable state can be viewed as marginally absolutely unstable. However, in several cases considered, the value of R _vt is significantly greater than the critical value, R _vc, implying that the base state is not marginally but essentially absolutely stable at the point of destabilization.     

Kinematic regimes of convection at high Prandtl number in a shallow cavityRégimes cinématiques de la convection à haut nombre de Prandtl dans une cavité allongée.

We consider free convection in a horizontal shallow cavity with different end temperatures, filled with a high Prandtl number fluid. From scaling analysis, we find two kinematic regimes resulting from the competition of heat transfer by conduction and by convection. Numerical simulations realized for a large range of Rayleigh number and aspect ratio confirm the phenomenological analysis and provide the threshold between the two regimes. The conductive and convective regimes occur at $\mathit{Ra}{A}^2$ smaller and larger than 443 respectively, where Ra is the Rayleigh number and A is the aspect ratio. In the convective regime, the characteristic velocity is independent of depth of the cavity. To cite this article: J.-M. Flesselles, F. Pigeonneau, C. R. Mecanique 332 (2004).     

Boundary effects on the Bénard-Marangoni instability under an electric field

This article theoretically studies the Bénard-Marangoni instability problem for a liquid layer with a free upper surface, which is heated from below by a heating coil through a solid plate in ana.c. electric field. The boundary effects of the solid plate, which include its thermal conductivity, electric conductivity and thickness, have great influences on the onset of convective instability in the liquid layer. The stability analysis in this study is based on the linear stability theory. The eigenvalue equations obtained from the analysis are solved by using the fourth order Runge-Kutta-Gill's method with the shooting technique. The results indicate that the solid plate with a higher thermal or electric conductivity and a bigger thickness tends to stabilize the system. It is also found that the critical Rayleigh numberR _c, the critical Marangoni numberM _c, and the criticala.c. Rayleigh numberE _ac become smaller as the intensity of thea.c. electric field increases.     

Experimental determination of the boundaries of the region of existence of anomalous convection flow in an tilted cube

This paper presents an experimentally study of the bifurcation of steady-state air convection in a cubic cavity heated from below under controlled deviations from equilibrium heating conditions due to a slow quasisteady-state tilt of the cavity at a predetermined angle α. It is found that in the supercritical range of Rayleigh numbers Ra at a tilt of the cavity not exceeding 7°, the existence of two stable steady-state convection regimes (normal and anomalous) with circulation in opposite directions is possible. A study is made of the transformations of the temperature distribution in the middle (with respect to the planes in which heat exchangers are located) plane during transition from the anomalous flow regime to the normal regime by instantaneous rotation of the entire mass of air in the cavity around the vertical axis by an angle of 90 to 135°. It is shown that this rotation occurs when the tilt of the cavity exceeds a critical value α _cr(Ra), which was determined experimentally for Rayleigh numbers 0 < Ra < 25Racr, where Racr is the critical Rayleigh number for stability of mechanical equilibrium for heating from below.     

Convection in a horizontal fluid layer with specific-volume inversion

The effect of the position of the inversion point within the layer on the critical values of the Rayleigh number and the amplitudes of the rectangular-cell convective flows is numerically investigated. The monotonic instability of the mechanical equilibrium of the fluid with respect to small perturbations periodic along the layer is studied by the linearization method. The Lyapunov-Schmidt method is used to construct the secondary steady convective flows. The applicability of these methods in incompressible fluid stability problems was demonstrated in [8–10]. The calculations show that, starting from a certain value of the parameter , the branching is subcritical for any cell side ratio and a fixed wave vector modulus. For smaller values of the nature of the branching depends on the cell side ratio. This points to subcritical branching and hysteresis effects in those cases in which the periodicity of the perturbations is determined by external factors (corrugation of the boundary, spatially periodic temperature modulation, etc.). It is noted that the rectangular convection amplitude tends to zero when the cell side ratio tends to 3, the value at which hexagonal cellular convection is possible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 43–49, January–February, 1989.The author wishes to thank V. I. Yudovich for his interest and useful advice and the participants in the Rostov State University Computational Mathematics Department's Scientific Seminar for discussing the results.