Perfect gas convection in a porous medium between two coaxial horizontal cylinders of large length

Natural convection of a perfect gas in a porous medium between two coaxial horizontal cylinders of large length located in heat-conducting space is considered. The two-dimensional problem (thin porous ring) is investigated in a plane orthogonal of the axis of the cylinders. The dependence of the criterion of the onset of convection on the non-Boussinesq parameters is studied. In the steady-state case an analytic solution of the nonlinear problem is obtained and its asymptotic behavior is considered for large Rayleigh numbers and when the compressibility criterion tends to zero. The gas flow rate in the ring and other characteristics of convection are studied as functions of the gas compressibility criterion and a constant temperature gradient given far away from the contour.     

The Onset and Nonlinear Regimes of Convection in a Two-Layer System of Fluid and Porous Medium Saturated by the Fluid

The onset of convection and its nonlinear regimes in a heated from below two-layer system consisting of a horizontal pure fluid layer and porous medium saturated by the same fluid is studied under the conditions of static gravitational field. The problem is solved numerically by the finite-difference method. The competition between the long-wave and short-wave convective modes at various ratios of the porous layer to the fluid layer thicknesses is analyzed. The data on the nature of convective motion excitation and flow structure transformation are obtained for the range of the Rayleigh numbers up to quintuple supercriticality. It has been found that in the case of a thick porous layer the steady-state convective regime occurring after the establishment of the mechanical equilibrium becomes unstable and gives way to the oscillatory regime at some value of the Rayleigh number. As the Rayleigh number grows further the oscillatory regime of convection is again replaced by the steady-state convective regime.     

Combined free and forced convection in a porous medium between two vertical walls with viscous dissipation

Viscous dissipation effects in the problem of a fully-developed combined free and forced convection flow between two symmetrically and asymmetrically heated vertical parallel walls filled with a porous medium is analyzed. The equation of motion contains the modified Rayleigh number for a porous medium and the small-order viscous dissipation parameter. Particular attention is given to the solutions near the critical Rayleigh numbers at which infinite flow rates are predicted. Information concerning the multiplicity of solutions at critical Rayleigh numbers is also deduced from perturbation solutions of the governing equation.     

Free Fluid Convection in a Closed Contour in a Heat-Conducting Half-Space

The steady-state convection of a fluid in a thin porous vertical ring located in a heat-conducting half-plane is considered. For this problem approximate equations are derived. For a circular ring an analytic solution is obtained. For an elliptic ring a numerical-analytic solution is found. The Nusselt number and the fluid flow rate as functions of the Rayleigh number, the aspect ratio, and the contour depth are investigated.Many studies have been devoted to fluid convection in a porous ring [1–3]. In [1] two-dimensional convection with an isothermal internal boundary was considered when a temperature stratification is given on the outer boundary. A feature of this problem is the fact that the ring is located inside an impermeable heat-conducting medium in which a thermal gradient directed vertically downward is specified at a large distance from the ring. In [2, 3] two-dimensional convection in an annular ring occupied by a porous medium was investigated. From the results obtained in these studies it follows that in the formulation considered the hydraulic approximation can be used with satisfactory accuracy. In the present study this question is discussed more concretely and the necessary estimates are found. The results obtained could be useful for investigating hydrothermal convection in the Earth's crust, which has important geophysical applications [4–6].     

Anisotropy effect on the convection of a heat-conducting fluid in a porous medium and cosymmetry of the Darcy problem

The onset of convection in a porous rectangle is analyzed with account for the anisotropy of the thermal parameters and the permeability. For the Darcy–Boussinesq equations the conditions under which the problem pertains to the class of cosymmetric systems are established and explicit formulas for the critical Rayleigh numbers corresponding to the loss of stability of the mechanical equilibrium are derived. The critical numbers and the branching stationary convection regimes are calculated using a finite difference method conserving the problem cosymmetry.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Penetrative convection in sublayer of water including density inversion

Numerical solutions of stability and convective flow in an infinite horizontal water layer, including density inversion, have been obtained using a finite element code. The evolution of the temperature field and flow pattern near the onset of convection are studied in detail. It is known that natural convection develops primarily in the lower unstably stratified layer. Of interest is the penetration of the convection rolls into the upper stably stratified layer and concurrent liquid entrainment as a function of the increasing Rayleigh number at different aspect ratios. Individual convection rolls may grow and expand before splitting up into two roll cells. It is shown that changing the aspect ratio influences critical Rayleigh number, flow symmetry, flow pattern, and transitions between flow patterns. Numerical results on heating from above or from below, agree well with available results in the literature. A correlation to predict critical Rayleigh numbers is given for the case of heating from above.     

Stability of Steady Flow in a Vertical Layer in the Microconvection Model

The stability of steady flow in a vertical gap is analyzed using the perturbation method within the framework of the microconvection model. The resulting spectral problem is not self-conjugate. The stability of the flow to long-wave perturbations is established. It is shown that if the Boussinesq parameter is small, the spectrum of this problem approximates the spectra of the corresponding problems for a viscous heat-conducting fluid and thermogravitational convection with a finite Rayleigh number. The numerical calculations indicate that in the microconvection model the instability develops at smaller wave numbers.     

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.     

Effect of modulation on onset of thermal convection of a second-grade fluid layer

This study examines the stability of a horizontally extended second-grade fluid layer heated from below, when a steady temperature difference between the walls is superimposed on sinusoidal temperature perturbations. A linear stability analysis proposed by Venezian (J. Fluid Mech. 35 (1969) 243) is employed to obtain the critical Rayleigh numbers for different types of temperature modulation. The free–free and isothermal boundary conditions are considered so as to allow analytic solutions. The stability characterized by the shift in critical Rayleigh number R_2c is calculated as a function of the modulation frequency ω, the Prandtl number Pr, and the viscoelastic parameter Q. It is found that the onset of convection can be delayed or advanced by these parameters.     

Effect of a Standing Acoustic Wave on the Development of Large-Scale Convection in a Horizontal Layer

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.     

The Prandtl number effect near the onset of Bénard convection in a porous medium

This note focuses on Kladias and Prasad's claim that the critical Rayleigh number for the onset of Bénard convection in an infinite horizontal porous layer increases as the Prandtl number decreases, and argues that the critical Rayleigh number (Ra_c) depends only on the Darcy number (Da), as linear stability analysis indicates. The two-dimensional steady-convection problem is then solved numerically to document the convection heat transfer effect of the Rayleigh number, Darcy number, Prandtl number, and porosity. The note concludes with an empirical correlation for the overall Nusselt number, which shows the effect of Prandtl number at above-critical Rayleigh numbers. The correlation is consistent with the corresponding correlation known for Bénard convection in a pure fluid.     

Convective heat transfer and temperature stratification in a sphere completely filled with a liquid,with a given heat flux

A number of articles have been devoted to the theoretical and experimental investigation of natural convection in spherical vessels completely filled with a liquid [1–6]. Analytical solutions are known, obtained by the expansion of the sought function in series in powers of the Rayleigh number (see, for example, [1]), valid for very small values of this number. A numerical solution of the nonlinear Boussinesq equations can be used to obtain solutions with larger Rayleigh numbers, but the existing data for spherical regions [2, 3] embrace a relatively narrow range of Rayleigh numbers. The experimental data with a given heat flux, published in [4–6], were obtained with relatively large Rayleigh numbers (Ra*=10⁹?10¹¹) and Prandtl numbers (P= 3?1500). Data on the characteristics of convection in spherical vessels are still not very numerous and, in a number of cases, contradictory. This relates, in particular, to the boundaries of unsteady-state conditions. The present article, continuing [7–9], expounds a method and gives the results of a calculation of convection in a sphere with a thinwalled shell, in a range of Rayleigh and Fourier numbers embracing the principal conditions of unsteady-state laminar convection with a given heat flux.     

On the criteria of the absolute convective stability for compressible fluids

The conditions under which natural convection is absent from compressible fluids are investigated. It is shown that in the parameter “Rayleigh number-given temperature difference” plane there is a domain in which convection occurs for neither Rayleigh numbers. It is proposed to refer to this domain as the absolute convective stability region and to name the criterion determining the boundary of this region the absolute convective stability criterion. The necessary, sufficient, and necessary and sufficient conditions of the absolute convective stability for a viscous compressible fluid are derived. It is shown that in the particular case in which the thermal properties of the fluid and the adiabatic gradient are constant, these conditions coincide with the Schwarzschild criterion.     

Instability of the equilibrium state of a liquid with a free surface in the presence of volume heat sources

The problem of the convection of a weakly compressible fluid is considered. In the free convection equations a heat source function is taken into account. The stability of the equilibrium state of a horizontal layer relative to small perturbations is studied using the linearization method. On the basis of numerical calculations it is shown that the mechanical equilibrium state of the fluid is unstable. The neutral curves are plotted and the critical Rayleigh numbers are found. In the calculations values of the physical parameters typical of Lake Baikal were used.     

Stability of Moderate Vadasz Number Solutal Convection in a Cylindrical Mushy Layer Subjected to Vertical Vibration

We consider vibration effects on the stability of solutal convection in a mushy layer being cast in a cylindrical geometry. The near eutectic limit is applied and moderate Vadasz numbers are considered to retain the second-order time derivative in the Darcy equation. Since small to moderate radii casting crucibles are the current area of interest, only synchronous modes are analyzed. The results indicate that the presence of vibration in solidifying mushy layers stabilizes the convection, and provides a quantification of the Rayleigh number associated with solutal convection. Of particular interest is the fact that in solidifying systems, the Rayleigh numbers are significantly smaller than that of a passive porous layer.     

Convective stability analysis of a micropolar fluid layer by variational method

This paper studies Rayleigh-Bénard convection of micropolar fluid layer heated from below with realistic boundary conditions. A specific approach for stability analysis of a convective problem based on variational principle is applied to characterize the Rayleigh number for quite general nature of bounding surfaces. The analysis consists of replacing the set of field equations by a variational principle and the expressions for Rayleigh number are then obtained by using trial function satisfying the essential boundary conditions. Further, the values of the Rayleigh number for particular cases of large and small values of the microrotation coefficient have been obtained. The effects of wave number and micropolar parameter on the Rayleigh numbers for onset of stationary instability for each possible combination of the bounding surfaces are discussed and illustrated graphically. The present analysis establishes that the nature of bounding surfaces combination and microrotation have significant effect on the onset of convection.     

Rayleigh-Benard Convection in a Near-Critical Fluid in the Neighborhood of the Stability Threshold

Steady-state Rayleigh-Benard convection in a medium with parameters close to the thermodynamic critical point is simulated within the framework of the complete Navier-Stokes equations with a two-scale representation of the pressure and the Van-der-Waals equation of state. A calibration relation is obtained for a realistic Rayleigh number in a compressible stratified medium. The parameters of the numerical simulation are determined from experimental data for near-critical helium on the basis of the calibration relation. The threshold Rayleigh numbers are found without and with allowance for stratification and a comparison with the experimental and theoretical data is carried out. The effect of compressibility of the near-critical fluid on steady-state convection flows is investigated beyond the stability threshold and the effect of adiabatic compression of the medium is analyzed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, 2005, pp. 48–61.Original Russian Text Copyright © 2005 by Polezhaev and Soboleva.     

Average stationary binary-mixture convection in connected channels in the presence of high-frequency vibration

The effect of vertical high-frequency vibration on steady-state binary-mixture flows in connected channels is studied theoretically. Mixtures with both positive and negative thermal diffusion are considered. It is shown that the convection excitation mode changes with the sign of the thermal diffusion. The dependence of the flow amplitude on the supercriticality is analytically obtained for various vibrational Rayleigh numbers.     

非定常流函数涡量方程的一种数值解法的研究

对非定常流函数涡量方程的数值求解方法进行了改进,其中流函数一阶导数即速度项采用四阶精度的Hermitian公式,对流项由一般二阶精度的中心差分提高到四阶精度离散差分,包含温度方程在内的离散方程组采用ADI迭代方法求得定常解．以无内热体及有一内热体的封闭方腔内自然对流为例,进行了不同瑞利数（Ra）条件下的数值研究．结果表明,该方法推导简单,求解精度高且计算稳定,适用于封闭腔内高瑞利数复杂混合对流的数值模拟．