Vibrational Convection in an Inclined Fluid Layer Heated from Below

The stability of mechanical equilibrium of an inclined fluid layer with respect to three-dimensional perturbations under the action of high-frequency vibration is studied. It is shown that under heating from below the spiral perturbations are always the most dangerous for vibration transverse to the layer. For vertical vibration the stability limit is determined by three-dimensional perturbations whose shape depends in a complicated way on the angle of inclination of the layer and the vibrational Rayleigh number. In the limiting case of a thin vertical layer supercritical vibrational-convective motions are calculated numerically and analytically and scenarios of transition from quasi-equilibrium to irregular motions are studied.     

Stability of Convection in a Gravity Modulated Porous Layer Heated from Below

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogenous porous layer heated from below. The gravitational field consists of a constant part and a sinusoidally varying part, which is tantamount to a vertically oscillating porous layer subjected to constant gravity. The linear stability results are presented for the specific case of low amplitude vibration for which it is shown that increasing the frequency of vibration stabilises the convection.     

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.     

Numerical Simulation of Evolution of Helical-Vortex Instability in a Fluid Layer Heated from Below

Large-scale helical-vortex instability is simulated numerically for laminar convection regimes. The simulation is based on the nonlinear Boussinesq equations supplemented by an external force of special form, whose structure is identical to the tensor structure of the generative term in the equation for the hydrodynamic alpha-effect. Introducing the external force makes it possible to model the average effect produced by small-scale helical turbulence: generation of a positive feedback between the toroidal and poloidal components of the velocity vector field. The structure and energetics of helical-vortex flows developing in a convectively unstable medium are analyzed. The parameters of helical and non-helical convection regimes are compared. Certain novel effects related to spiral feedback are discussed: generation of a toroidal velocity field in structures with the poloidal circulation typical of ordinary convection; enlargement of the horizontal scale of supercritical motions as a result of the merging of vortex cells, accompanied by a significant increase in the circulation strength in the larger helical vortices formed; qualitative changes in the energetics.     

周期加热的Rayleigh-Benard对流时空结构

通过二维流体力学基本方程组的数值模拟,研究了普朗特数Pr=6.99时矩形渠槽周期加热对Rayleigh-Benard对流时空结构的影响．当水平流动强度Re=0时,发现稳定的由周期加热引起的局部定常对流．当Re比较小时,对流滚动抑制水平流动,获得了由周期加热引起的局部行波对流．当水平流动强度比较大时,由于周期加热与水平流动相互作用,水平流动抑制部分对流滚动,导致对流区域上游附近出现传导区域,对流区域减小,从而形成一种新的局部行波对流结构．并进一步讨论了Rayleigh-Benard对流时空结构的动力学特性．     

Linear Stability and Convection in a Gravity Modulated Porous Layer Heated from Below: Transition from Synchronous to Subharmonic Solutions

The linear stability theory is used to investigate analytically the effects of gravity modulation on convection in a homogenous porous layer heated from below. The linear stability results are presented for both the synchronous and subharmonic solutions and the exact point for the transition from synchronous to subharmonic solutions is computed. It is also demonstrated that increasing the excitation frequency rapidly stabilizes the convection up to the transition point from synchronous to subharmonic convection. Beyond the transition point, the effect of increasing the frequency is to slowly destabilize the convection.     

Mixed Convection in a Horizontal Channel with Local Heating from Below

Mixed convection induced in the entrance region of a horizontal plane channel by a bottom heat source of finite dimensions is considered. The calculations were performed for the Prandtl number Pr = 1, Grashof numbers ranging from 4 · 10³ to 3.2 · 10⁴, and Reynolds numbers varying from 0 to 10. The dimensions of the heat source and its location were also varied. The results were obtained from a numerical solution of the 2D unsteady Navier-Stokes equations in the Boussinesq approximation, written in vorticity – stream function – temperature variables. The solution was found by the Galerkin finite element method.     

Onset of Convection in a Porous Layer with Continuous Periodic Horizontal Stratification. Part I. Two-Dimensional Convection

The onset of convection in a horizontal porous layer is investigated theoretically. The permeability of the porous medium is a continuous periodic function of the horizontal x coordinate. Floquet theory has been employed to determine the favoured two-dimensional mode of convection. For a wide range of periods of the permeability variation, a matrix eigenvalue technique with eighth order accuracy has been employed to find the critical Darcy– Rayleigh number. This is supplemented by a multiple-scales analysis of the large-period limit, and a brief consideration of the anisotropic limit for very short periods.     

Non-Darcy Free Convection Along a Horizontal Heated Surface

In this paper we analyse how the presence of inertia (Forchheimerform-drag) affects the steady free convective boundary layer flow over anupward-facing horizontal surface embedded in a porous medium. The surfacetemperature is assumed to display a power-law variation,x ⁿ with distance from the leading edge, x. It is shown thatthere are three distinct cases to consider: n<0.5, n=0.5 and0.5相似文献   

The Effect of Conducting Boundaries on the Onset of Convection in a Porous Layer Which is Heated from Below by Internal Heating

The onset of convection in a porous layer heated from below is considered, and we determine how the presence of two solid but heat-conducting bounding plates of finite thickness alters the manner in which convection ensues. Heat is generated by the lower plate (with an insulating lower boundary), but the upper one is passive with a fixed upper boundary temperature. It is shown that this composite layer may mimic in turn one of the three different types of classical single-layer onset problems which are well-known in the literature. The type which is selected (or indeed whether it corresponds to a transitional case) depends quite critically on the precise values of the relative thickness of the solid layers and their conductivity ratio. It is also shown that care needs to be taken over declaring that the solid plates are thin: extreme values of the conductivity ratio can yield a stability criterion which appears to be different from that suggested by the imposed boundary conditions.     

Stability of Thermal Convection in a Fluid-Porous System Saturated with an Oldroyd-B Fluid Heated from Below

A linear stability analysis is conducted for thermal convection in a two-layer system composed of a fluid layer overlying a porous medium saturated with an Oldroyd-B fluid heated from below. It is found that the convection pattern in the system is controlled by the porous medium when the ratio of the depth of the fluid layer to that of the porous medium is small. However, the fluid layer takes a predominant role if the depth ratio exceeds a critical value. Compared with stationary convection, the switching point from a porous-dominated mode to a fluid-dominated mode for oscillatory convection is located at a lower depth ratio. The effects of different parameters on stationary convection and oscillatory convection are also investigated in detail.     

The Onset of Darcy–Forchheimer Convection in Inclined Porous Layers Heated from Below

It is well known that the onset of convection in an inclined porous layer heated from below takes the form of longitudinal vortices when Darcy’s law is valid. In this paper we consider briefly how the onset criterion alters when form drag, as modelled by the Forchheimer terms, is significant. In general, the critical Rayleigh number increases substantially as form drag effects strengthen, but the wavenumber rises by only a small amount. This numerical study is supplemented by a brief asymptotic analysis of the case when the Forchheimer terms dominate and it is shown that the critical Rayleigh number increases in direct proportion with the form drag parameter.     

Local and Global Transitions to Chaos and Hysteresis in a Porous Layer Heated from Below

The routes to chaos in a fluid saturated porous layer heated from below are investigated by using the weak nonlinear theory as well as Adomian's decomposition method to solve a system of ordinary differential equations which result from a truncated Galerkin representation of the governing equations. This representation is equivalent to the familiar Lorenz equations with different coefficients which correspond to the porous media convection. While the weak nonlinear method of solution provides significant insight to the problem, to its solution and corresponding bifurcations and other transitions, it is limited because of its local domain of validity, which in the present case is in the neighbourhood of any one of the two steady state convective solutions. On the other hand, the Adomian's decomposition method provides an analytical solution to the problem 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 finite accuracy. The transition from the steady solution to chaos is analysed by using both methods and their results are compared, showing a very good agreement in the neighbourhood of the convective steady solutions. The analysis explains previously obtained computational results for low Prandtl number convection in porous media suggesting a transition from steady convection to chaos via a Hopf bifurcation, represented by a solitary limit cycle at a sub-critical value of Rayleigh number. A simple explanation of the well known experimental phenomenon of Hysteresis in the transition from steady convection to chaos and backwards from chaos to steady state is provided in terms of the present analysis results.     

Convection in a Horizontal Layer in a Rotating Heat Field

An asymptotic laminar-convection pattern in a plane horizontal liquid layer with a radially nonuniform temperature gradient on its boundaries is investigated. The problem arises in applications connected with modified Czochralski crystal growth technology using the heat field rotation method. An analytical model of the flow is compared with the results of experiments, specially carried out using model fluids and a technological melt. The conditions of adequacy of the model are analyzed and the restrictions on the parameter values and fluid thermophysical properties that ensure the validity of the model are found. The range of variation of the heat field rotation velocity for which the mixing of the melt in the crucible is maximum is determined.     

Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number $10^{-5}\le \hbox {Da}\le 10^{-3}$ , porous layer height ratio $0\le d/L\le 1$ , thermal conductivity ratio $1\le k_{1,3}\le 20$ , and dimensionless time $0\le \tau \le 1000$ on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.     

Convective Instability of Oldroyd-B Fluid Saturated Porous Layer Heated from Below using a Thermal Non-equilibrium Model

The linear stability of a viscoelastic fluid saturated densely packed horizontal porous layer heated from below and cooled from above is investigated by considering the Oldroyd-B type fluid. A generalized Darcy model, which takes into account the viscoelastic properties, is employed as momentum equation and a two-field model is used for energy equation each representing solid and fluid phases separately. Linear stability analysis suggests that, there is a competition between the processes of viscous relaxation and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Analytical expression for the occurrence of oscillatory onset is obtained, and it is found that the necessary condition for the existence of the same is Λ < 1. Besides, the effect of viscoelastic parameters and the thermal non-equilibrium on the stability of the system is analyzed.     

Thermal Convection in a Plane Layer Rotating About a Horizontal Axis

The thermal convection of a fluid in a plane vertical layer with a cylindrical lateral boundary, which rotates uniformly about a horizontal symmetry axis, is investigated experimentally. The structure and excitation limit of the convective flows are studied as functions of the rotation frequency, the temperature difference between the layer boundaries, and the layer thickness. The determining dimensionless parameters are found. It is shown that the period-average gravity action produces convection in the form of hexagonally distributed cells stationary in the reference system tied to the cavity.     

Free Convection Boundary Layer Flow from a Heated Upward Facing Horizontal Flat Plate Embedded in a Porous Medium Filled by a Nanofluid with Convective Boundary Condition

The steady laminar incompressible free convective flow of a nanofluid over a permeable upward facing horizontal plate located in porous medium taking into account the thermal convective boundary condition is studied numerically. The nanofluid model used involves the effect of Brownian motion and the thermophoresis. Using similarity transformations the continuity, the momentum, the energy, and the nanoparticle volume fraction equations are transformed into a set of coupled similarity equations, before being solved numerically, by an implicit finite difference numerical method. Our analysis reveals that for a true similarity solution, the convective heat transfer coefficient related with the hot fluid and the mass transfer velocity must be proportional to x ^−2/3, where x is the horizontal distance along the plate from the origin. Effects of the various parameters on the dimensionless longitudinal velocity, the temperature, the nanoparticle volume fraction, as well as on the rate of heat transfer and the rate of nanoparticle volume fraction have been presented graphically and discussed. It is found that Lewis number, the Brownian motion, and the convective heat transfer parameters increase the heat transfer rate whilst the thermophoresis decreases the heat transfer rate. It is also found that Lewis number and the convective heat transfer parameter enhance the nanoparticle volume fraction rate whilst the thermophoresis parameter decreases nanoparticle volume fraction rate. A very good agreement is found between numerical results of the present article for special case and published results. This close agreement supports the validity of our analysis and the accuracy of the numerical computations.     

Convection in a Horizontal Fluid Layer with a Uniform Internal Heat Source

On the basis of a numerical simulation of convection in a horizontal fluid layer with a uniform heat source it is concluded that the convective heat flux is constant over the entire convection layer not only in the case of steady-state external conditions but also in the case of heating (cooling) of the fluid layer at a constant rate. The convective heat flux is mainly determined by the Rayleigh number and depends only slightly on the layer heating (cooling) rate.     

Mixed Convection Adjacent to a Suddenly Heated Horizontal Circular Cylinder Embedded in a Porous Medium

The mixed convection caused when a horizontal circular cylinder is suddenly heated is investigated in the situation when the initial flow past the cylinder is uniform and its direction either upwards or downwards. An analytical series solution, which is valid at small times, is obtained using the matched asymptotic expansions technique. A numerical solution, which is valid at all times and for any values of the Rayleigh and Péclet numbers, is also obtained using a fully implicit finite-difference method. Three different regimes, when either the free or forced convection is dominant or when they have the same order of magnitude, are considered. In the free convection dominated regime, two vortices develop near the sides of the cylinder in both situations of an upward or downward external flow. Comparisons between the analytical and numerical results at small times, as well as a detailed discussion of the evolution of the numerical solution are presented. The numerical results obtained for large Rayleigh, Ra, and Péclet Pe, numbers show that a thermal boundary-layer forms adjacent to the cylinder for any value of the ratio Ra/e. The steady state boundary-layer analysis, similar to that performed by Cheng and Merkin, is analysed in comparison to the numerical solution obtained for large values of Ra and Pe at very large times.