Showing papers on "Rayleigh number published in 1988"
TL;DR: In this paper, the complex nature of the natural convection phenomena in enclosures is discussed and the boundary value problem is formulated, assuming that the motion is 2D and steady, the fluid is incompressible and frictional heating is negligible, and the difference between the hot wall and cold wall temperatures is small relative to the absolute temperatures of the cold wall.
Abstract: Publisher Summary This chapter discusses the complex nature of the natural convection phenomena in enclosures It discusses the two basic configurations of natural convection— that is, a rectangular cavity and a horizontal circular cylinder In rectangular cavities, consideration is given to the two-dimensional convective motion generated by the buoyancy force on the fluid in a rectangle and to the associated heat transfer The two long sides are vertical boundaries held at different temperatures and the short sides can either be heat conducting or insulated Particular attention is given to the different flow regimes that can occur and the heat transfer across the fluid space between the two plane parallel vertical boundaries Although heat transfer by radiation may not be negligible it is independent of the other types of heat transfer and can be fairly accurately calculated separately To formulate the boundary value problem that describes this phenomena it is assumed that: (a) the motion is two-dimensional and steady, (b) the fluid is incompressible and frictional heating is negligible, and (c) the difference between the hot wall and cold wall temperatures is small relative to the absolute temperatures of the cold wall In horizontal circular cylinder, consideration is given to the large Rayleigh number flow with the Prandtl number large and the Grashof number of unit order of the magnitude
973 citations
TL;DR: In this paper, the transport of passive impurities in nearly two-dimensional, time-periodic Rayleigh-B\'enard convection is studied experimentally and numerically, and the transport may be described as a one-dimensional diffusive process with a local effective diffusion constant that is found to depend linearly on the local amplitude of the roll oscillation.
Abstract: The transport of passive impurities in nearly two-dimensional, time-periodic Rayleigh-B\'enard convection is studied experimentally and numerically. The transport may be described as a one-dimensional diffusive process with a local effective diffusion constant ${D}^{\mathrm{*}}$(x) that is found to depend linearly on the local amplitude of the roll oscillation. The transport is independent of the molecular diffusion coefficient and is enhanced by 1--3 orders of magnitude over that for steady convective flows. The local amplitude of oscillation shows strong spatial variations, causing ${D}^{\mathrm{*}}$(x) to be highly nonuniform. Computer simulations of a simplified model show that the basic mechanism of transport is chaotic advection in the vicinity of oscillating roll boundaries. Numerical estimates of ${D}^{\mathrm{*}}$ are found to agree semiquantitatively with the experimental results. Chaotic advection is shown to provide a well-defined transition from the slow, diffusion-limited transport of time-independent cellular flows to the rapid transport of turbulent flows.
243 citations
TL;DR: In this paper, the authors identify the most basic scales and regimes of the phenomenon of melting with natural convection in an enclosure heated from the side and show that the phenomenon consists of a sequence of four regimes: pure conduction regime, mixed regime, convection regime and shrinking solid regime.
209 citations
TL;DR: In this article, the inner core of an iron hexagonal hexagonal closest packed (ϵ) phase is estimated to be convecting, such that compressional-wave velocities would be greater in the axial relative to the equatorial direction.
Abstract: Estimation of the Rayleigh number of the inner core suggests that this region is convecting. The flow pattern is likely to produce crystallographic preferred orientation of the elastically anisotropic hexagonal closest packed (ϵ) phase of iron, such that compressional-wave velocities would be greater in the axial relative to the equatorial direction by roughly 1 percent. This result is in agreement with seismological evidence that the inner core is elastically anisotropic.
197 citations
TL;DR: In this paper, the problem of the onset of finger convection in a porous layer underlying a fluid layer is considered using linear stability analysis, and the linear stability equations for the porous layer are formulated for temperature and salinity gradients existing in both layers.
Abstract: The problem of the onset of finger convection in a porous layer underlying a fluid layer is considered using linear stability analysis. The linear stability equations for the porous layer are formulated for temperature and salinity gradients existing in both layers. The eigenvalue problem is solved by a shooting method. The solution method and associated computer program are validated by comparison with the results of Sun (1973) for the thermal convection case. Results are also presented for the onset of salt-finger convection.
188 citations
Abstract: An experimental investigation was made of convection in a fluid with a strongly temperature-dependent viscosity. The determination of the critical Rayleigh number, Rc, using the appearance of convection to define onset, was complicated by the occurrence of subcritical instabilities initiated by horizontal temperature gradients at the side boundaries. The increase in Rc over the expected value was less than predicted by linear theory, probably owing to the effect of finite conductivity boundaries and the temperature dependence of other fluid properties.The stability of various convective planforms was studied as a function of Rayleigh number, wavenumber and viscosity variation using controlled initial conditions to specify the wavenumber and pattern, Rayleigh numbers of up to 63000 and viscosity variations of up to 1000. In addition to the rolls and hexagons seen in constant- and weakly temperature-dependent-viscosity fluid, a new planform of squares was observed at large viscosity variations.Experiments with viscosity variations of 50 and 1000 showed that hexagons and squares were stable at Rayleigh numbers less than 25000 over a limited range of wavenumbers, which was shifted to higher values with increasing viscosity variation. Temperature profiles through the layer revealed that this shift in wavenumber was associated with the development of a thick, stagnant, cold boundary layer which reduced the effective depth of the layer.Experiments with a fixed wavenumber showed that rolls were unstable at all Rayleigh numbers for a viscosity contrast greater than 40, whereas squares did not become stable until the viscosity contrast exceeded 6. At low viscosity variations and high Rayleigh numbers rolls became unstable to a bimodal pattern, but at high viscosity variations and a Rayleigh number of 25000 squares broke down into the spoke pattern, a convective flow not observed until Rayleigh numbers of around 100000 in a constant-viscosity fluid.
172 citations
TL;DR: In this paper, a numerical and experimental study of natural convection in a vertical rectangular fluid enclosure that is partially filled with a fluid-saturated porous medium is reported. But the authors did not consider the effect of the porous layer geometry on the degree of penetration of fluid into the medium.
Abstract: A numerical and experimental study is reported of natural convection in a vertical rectangular fluid enclosure that is partially filled with a fluid-saturated porous medium. Velocities, stresses, temperatures, and heat fluxes are assumed to be continuous across the fluid/porous-medium interface, and the conservation equations for the fluid and the porous regions are combined into a single set of equations for numerical solution. Thermocouples as well as a Mach-Zehnder interferometer are used to measure temperature distributions and infer fluid flow patterns within the fluid and the porous medium. For various test cells, porous-layer configurations and fluid-solid combinations, the model predictions show excellent agreement with the experimental measurements. It is found that the intensity of natural convection is always much stronger in the fluid regions, while the amount of fluid penetrating into the porous medium increases with increasing Darcy and Rayleigh numbers. The degree of penetration of fluid into the porous medium depends strongly on the porous-layer geometry and is less for a horizontal porous layer occupying the lower half of the test cell. If penetration takes place, the flow patterns in the fluid regions are significantly altered and the streamlines show cusps at the fluid/porous-medium interfaces. For a high effective-thermal-conductivity porous medium, natural convection in the medium is suppressed, while the isotherms bend sharply at the fluid/porous-medium interface.
166 citations
TL;DR: In this article, the authors considered the problem of free convection flow of a non-Newtonian power law fluid along an isothermal vertical flat plate embedded in the porous medium.
Abstract: The problem of free convection flow of a non-Newtonian power law fluid along an isothermal vertical flat plate embedded in the porous medium is considered in the present study. The physical coordinate system is shown schematically in Fig 1. In the present study, it is assumed that the modified Darcy law and the boundary layer approximation are applicable. This implies that the present solutions are valid at a high Rayleigh number. With these simplifications, the governing partial nonlinear differential equations can be transformed into a set of coupled ordinary differential equations which can be solved by the fourth-order Runge-Kutta method. Algebraic equations for heat transfer rate and boundary layer thickness as a function of the prescribed wall temperature and physical properties of liquid-porous medium are obtained. The similarity solutions can be applied to problems in geophysics and engineering. The primary purpose of the present study is to predict the characteristics of steady natural convection heat transfer using the model of the flow of a non-Newtonian power law fluid in a porous medium given by Dharmadhikari and Kale (1985). Secondly, the effects of the new power law index n on heat transfer are investigated.
155 citations
TL;DR: In this paper, the surface heat flux, topography, gravity, and geoid (but not plate velocities or stresses) were derived for the case of whole mantle convection, and the calculated surface signatures were in first-order agreement with observations.
Abstract: Plate geometry and kinematics generally reflect the mechanical properties of the solid lithosphere rather than those of the fluid mantle underneath, and plate formation and subduction account for most of the heat transport from the Earth's interior. Correspondingly, mantle convection models must incorporate a stiff but mobile boundary layer, like the lithosphere, before they can reproduce the main features of mantle convection. A relatively easy way to accomplish this in numerical models is to combine a temperature-dependent viscosity with an imposed, piecewise constant surface velocity boundary condition. It is shown how surface heat flux, topography, gravity, and geoid (but not plate velocities or stresses) can then be derived. Numerical models confirm that a lithosphere has a first-order effect on the underlying flow structure. For internally heated models, approx-imating the case of whole mantle convection, the calculated surface signatures are in first-order agreement with observations, a level of empirical success which hitherto has not been approached by models of mantle convection. Companion papers exploit the observations more fully to constrain the main features of mantle convection.
134 citations
TL;DR: In this paper, numerical experiments of convection at high Rayleigh number were conducted to show a strong dependence of planform on heating mode, and the preferred planform consists of an array of hot axial plumes and elongated cold sheets when half the heat is generated within the box and the other half is input through the base.
Abstract: A fundamental property of a convecting fluid is its planform—the distribution in the horizontal plane of hot rising regions and cold sinking regions. For the Earth's mantle the planform might he visualized as a map of subduction zones, hotspots and possibly ocean ridges. Here I report numerical experiments of convection at high Rayleigh number which show a strong dependence of planform on heating mode. When heat generation is distributed uniformly through the box the preferred planform consists of an ensemble of time-dependent cold axial sinkers distributed in a hot diffuse upward flow. When half of the heat is generated within the box and the other half is input through the base, the preferred planform consists of an array of hot axial plumes and elongated cold sheets. In the former case the mean horizontal wavelength is about equal to the layer depth; for the latter it is about twice the layer depth.
125 citations
01 Jan 1988
TL;DR: In this paper, a review of the various origins of inhomogeneities occuring during crystal growth from the melt is given, and it is shown that convection is the major source of the non-uniformities in the technically used growth configurations, e.g. Czochralski-, zone-and Bridgman-methods, because the growth rate is controlled by the heat transport.
Abstract: The bulk single crystab of semiconductors (e.g. Si, GaAs) and oxides which are at present commercially produced have mostly non-uniform properties in the microscale (e.g. doping striations) and in the macroscale (longitudinal and lateral segregation). Such inhomogeneities are deleterious for the performance of the devices produced from these crystals. This book gives a review of the various origins of inhomogeneities occuring during crystal growth from the melt. It is shown that convection is the major source of the non-uniformities in the technically used growth configurations, e.g. Czochralski-, zone- and Bridgman-methods, because the growth rate is controlled by the heat transport. The formalism of hydrodynamics, especially dimensionless numbers, is used for a modeling of melt growth, giving a correlation between the occurrence of inhomogeneities and relevant growth parameters.
TL;DR: In this article, the authors investigated the effect of stratification on the temperature of the heated and unheated sections in a tall vertical cavity with one isothermal vertical cold wall, and eleven alternately un-heated and flush-mounted sections of equal height on the opposing vertical wall.
Abstract: Natural convection heat transfer in a tall vertical cavity (aspect ratio = 16.5), with one isothermal vertical cold wall, and eleven alternately unheated and flush-heated sections of equal height on the opposing vertical wall, is experimentally investigated. The flow visualization pictures for the ethylene glycol-filled cavity reveal a flow pattern consisting of primary, secondary, and tertiary flows. The heat transfer data and the flow visualization photographs indicate that the stratification is the primary factor influencing the temperature of the heated sections. This behavior persists for all the runs where the secondary flow cells cover a large vertical extend of the cavity. Based on the analysis of the photographs it is suggested that the turbulent flow should be expected when the local modified Rayleigh number is in the range of 9.3 {times} 10 {sup 11} to 1.9 {times} 10{sup 12}. It is found that discrete flush-mounted heating in the enclosure results in local Nusselt numbers that are nearly the same as those reported for a wide flush-mounted heater on a vertical plate. This is believed to be due to the fact that the present problems in inherently unstable, and the smallest temperature difference between a heated section and the cold wallmore » results in the onset of convection motion.« less
TL;DR: In this article, the inner wall temperature is a function of diameter ratio and Rayleigh number, and a crescent-shaped eddy dominates for small diameter ratios and a kidney-shaped flow pattern appears for large diameter ratios.
TL;DR: In this paper, the entire thermo-fluid-dynamic field resulting from the coupling of natural convection along and conduction inside a heated flat plate is studied by means of two expansions.
TL;DR: In this paper, the problem of natural convection of a non-Newtonian fluid about a horizontal isothermal cylinder and an isothermal sphere in the porous medium is considered.
TL;DR: In this article, two-dimensional numerical simulations are used to study fully compressible convection in the presence of an imposed magnetic field, where highly nonlinear flows are considered that span multiple density scale heights.
Abstract: Two-dimensional numerical simulations are used to study fully compressible convection in the presence of an imposed magnetic field. Highly nonlinear flows are considered that span multiple density scale heights. The convection tends to sweep the initially uniform vertical magnetic field into concentrated flux sheets with significant magnetic pressures. These flux sheets are partially evacuated, and effects of buoyancy and Lorentz forces there can serve to suppress motions. The flux sheets can be surrounded by a sheath of descending flow. If the imposed magnetic field is sufficiently strong, the convection can become oscillatory. The unstably stratified fluid layer has an initial density ratio (bottom to top of layer) of 11. Surveys of solutions at fixed Rayleigh number sample Chandrasekhar numbers from 1 to 1000 and magnetic Prandtl numbers from 1/16 to 1. These nonlinear simulations utilize a two-dimensional numerical scheme based on a modified two-step Lax-Wendroff method.
TL;DR: In this paper, the temporal evolution of thermal convection in stress-free, base-heated boxes is investigated by means of a finite-element model, and it is shown that the aspect ratio and also the initial conditions have a tremendous influence on the evolution.
Abstract: SUMMARY The temporal evolution of thermal convection in stress-free, base-heated boxes is investigated by means of a finite-element model. It is shown that the aspect ratio and also the initial conditions have a tremendous influence on the evolution. In boxes of aspect ratio A, significantly greater than unity (1.8 < A < 3), the onset of time-dependence occurs at much lower values of the Rayleigh number Ra than predicted from studies which assumed square boxes (A = 1). While steady-state solutions can be obtained by a particular choice of initial conditions, stationary convection breaks down for less restrictive conditions. It is also demonstrated, that the long held view, that convection cells with A = 1 would break down into smaller units, is not valid. At Ra = lo6 elongated convection cells of A=3 with superimposed boundary-layer instabilities are found in the long-term range of the temporal evolution. Regarding the Earth’s mantle, the model of a time-dependent multiscale flow can basically explain the coexistence of different scales of convection in the mantle.
TL;DR: In this paper, the interaction between shear and buoyancy effects for Benard convection in plane Couette flow is studied by performing direct numerical simulations, and the energy balance in the flow is analyzed.
Abstract: The interaction between shear and buoyancy effects for Benard convection in plane Couette flow is studied by performing direct numerical simulations. At moderate Rayleigh number (≈10000−50000), shear tends to organize the flow into quasi-two-dimensional rolls parallel to the mean flow and can enhance heat transfer, while at higher Rayleigh number (>150000), shear tends to disrupt the formation of convective plumes and can reduce heat transfer. A significant temporal oscillation in the local Nusselt number was consistently observed at high Rayleigh numbers, a factor that may contribute to the scatter seen in experimental data. This effect, plus the time-varying reversal of the mean temperature gradient in the middle of the channel, is consistent with a flow model in which the dynamics of large-scale, quasi-two-dimensional, counter-rotating vortical cells are alternately driven by buoyancy and inertial effects. An analysis of the energy balance in the flow shows that the conservative pressure diffusion term, which has been frequently neglected in turbulence models, plays a very important dynamical role in the flow evolution and should be more carefully modelled. Most of the turbulent energy production due to mean shear is generated in the boundary layers, while the buoyant production occurs mainly in the relatively uniform convective core. The simulations and the laboratory experiments of Deardorff & Willis (1967) are in very reasonable qualitative agreement, suggesting that the basic dynamics of the flow are being accurately simulated.
TL;DR: In this paper, the effects caused by the sublayer thickness ratio, permeability contrast and non-uniform conductivity in a system comprising two sublayers were investigated for steady-state natural convection in a two-dimensional layered porous cavity heated from the side wall.
TL;DR: In this article, two-dimensional convection in a Boussinesq fluid with infinite Prandtl number, confined between rigid horizontal boundaries and stress-free lateral boundaries, has been investigated in a series of numerical experiments.
Abstract: Two-dimensional convection in a Boussinesq fluid with infinite Prandtl number, confined between rigid horizontal boundaries and stress-free lateral boundaries, has been investigated in a series of numerical experiments. In a layer heated from below steady convection becomes unstable to oscillatory modes caused by the formation of hot or cold blobs in thermal boundary layers. Convection driven by internal heating shows a transition from steady motion through periodic oscillations to a chaotic regime, owing to the formation of cold blobs which plunge downwards and eventually split the roll. The interesting feature of this idealized problem is the interaction between constraints imposed by nonlinear dynamics and the obvious spatial structures associated with the sinking sheets and changes in the preferred cell size. These spatial structures modify the bifurcation patterns that are familiar from transitions to chaos in low-order systems. On the other hand, even large-amplitude disturbances are constrained to show periodic or quasi-periodic behaviour, and the bifurcation sequences can be followed in considerable detail. There are examples of quasi-periodic behaviour followed by intermittency, of period-doubling cascades and of transitions from quasi-periodicity to chaos, associated with a preference for narrower rolls as the Rayleigh number is increased.
TL;DR: In this paper, the scaling laws for the heat transfer rate and the effectiveness (energy storage fraction) are determined based on scale analysis, where the heating is applied suddenly along one of the side walls, while the remaining three walls are maintained insulated.
TL;DR: On compare une simulation microscopique d'un fluide, fait de 500 disques durs et maintenu a un nombre de Rayleigh supercritique, a l'hydrodynamique macroscopique correspondante.
Abstract: We compare a microscopic simulation of a fluid made up of 5000 hard disks and maintained at a supercritical Rayleigh number to the corresponding macroscopic hydrodynamics. Very good quantitative agreement is demonstrated.
TL;DR: In this paper, a finite element numerical method is used to analyze the effect of a low-viscosity zone on convection driven by heating from below in the upper mantle, in particular on the formation of midplate swells.
Abstract: A finite-element numerical method is used here to analyze the effect of a low-viscosity zone on convection driven by heating from below in the upper mantle, in particular on the formation of midplate swells. The convective temperature and velocity solutions are calculated for different combinations of the viscosity in the top layer, the fluid layer thicknesses, and the Rayleigh number based on the viscosity in the bottom layer. The temperature solutions are used to calculate the geoid, topography, and heat flow anomalies, the elastic plate thickness, the depth of compensation, and an upper bound on the uplift time that result from the flow. The results are compared to data at the Hawaii, Bermuda, Cape Verde, and Marquesas swells. The magnitudes and the trend with age are consistent with theoretical and other estimates of the viscosity variation in the shallow upper mantle. Convective models can therefore explain the uplift and observed anomalies at midplate swells.
TL;DR: In this article, a new approach for detecting the oscillatory instability is presented in which the transition from steady to periodic flow is identified with a Hopf bifurcation in the solution of the steady equations.
Abstract: When growing semiconductor crystals from the melt, the free convection induced in the liquid phase by the imposed thermal boundary conditions can become periodic in time. Recently, the onset of this oscillatory convection has been simulated directly through solution of the time-dependent governing equations for the particular case of an imposed horizontal temperature gradient. In this paper a new approach for detecting the oscillatory instability is presented in which the transition from steady to periodic flow is identified with a Hopf bifurcation in the solution of the steady equations. The critical Grashof number and frequency are predicted by solving an extended system of steady equations that locates exactly the Hopf bifurcation point, and the variation with aspect ratio and Prandtl number of the threshold for oscillations is obtained through continuation methods. By introducing a homotopy parameter into the boundary conditions the variation of the critical Grashof number is computed as the thermal and viscous conditions on the upper surface vary.
Book•
28 Jul 1988
TL;DR: In this paper, the equations of heat conduction, convection, and radiation were introduced, and the laws of black and grey body radiation were discussed. But they did not consider the effects of heat exchanges.
Abstract: Introduction to conduction, convection, radiation the equations of heat conduction one-dimensional steady state conduction two-dimensional steady state conduction transient conduction forced convection - boundary layer principles forced convection - Reynolds analogy and dimensional analysis natural convection separated flow convection convection with phase change extended surfaces heat exchanges the laws of black and grey body radiation.
TL;DR: In this paper, a two-dimensional, steady mixed convection in a vertical porous layer has been numerically studied for the case when a finite isothermal heat source is located on one vertical wall which is otherwise adiabatic and the other vertical wall is isothermally cooled.
TL;DR: The parameters of the linear instability to oscillatory convection (critical Rayleigh number, onset frequency, and others) are calculated for the experimentally common situation of rigid, impermeable boundaries, both near and away from the degenerate (codimension-2) bifurcation with stationary convection.
Abstract: The parameters of the linear instability to oscillatory convection (critical Rayleigh number, onset frequency, and others) are calculated for the experimentally common situation of rigid, impermeable boundaries, both near and away from the degenerate (codimension-2) bifurcation with stationary convection. This gives all linear coefficients of the standard and degenerate-amplitude equations. The small-Lewis-number limit is explicitly calculated. Wave-number and frequency jumps are confirmed in the vicinity of the codimension-2 point.
TL;DR: In this paper, a numerical study of a buoyancy-induced flow generated by a finite-size heat source located on a vertical wall of an enclosure with a single opening is carried out.
Abstract: A numerical study of a buoyancy-induced flow generated by a finite-size heat source located on a vertical wall of an enclosure with a single opening is carried out. A two-dimensional laminar flow is assumed. Employing a stream function vorticity formulation within the framework of the Boussinesq approximations, the temperature and the flow fields are computed. The effect of the opening on the induced flow is investigated in terms of the size and the location of the opening. Numerical results are obtained for a wide range of governing parameters, particularly the Rayleigh number. The Prandtl number is taken as that corresponding to air at normal con ditions, Pr = 0.72, and the aspect ratio of the enclosure is varied. Of particular interest were the flow generated in the vicinity of the opening, the flow adjacent to the heated surface, and any stratification that might arise in the enclosure. All these and several other relevant aspects are considered in this study. The numerical formulation for the boundar...
TL;DR: In this paper, an experiment was carried out to study two-dimensional laminar natural convection within an inclined square enclosure containing fluid with internal energy sources bounded by four rigid planes of constant equal temperature.
Abstract: An experiment was carried out to study two-dimensional laminar natural convection within an inclined square enclosure containing fluid with internal energy sources bounded by four rigid planes of constant equal temperature. Inclination angles, from the horizontal, of 0, 15, 30, and 45 deg for Rayleigh numbers from 1.0 {times} 10{sup 4} to 1.5 {times} 10{sup 5} were studied. At inclined angles of 0 and 15 deg, there are two extreme values of temperature and temperature gradient within the fluid, while there is only one at 30 and 45 deg. Local and average Nusselt numbers are obtained on all four walls. As the inclination angles increases, the average Nusselt number increases on the right (upper) and bottom walls, decreases on the left (lower) wall and stays almost constant on the top wall.
TL;DR: In this article, an analysis of the combined forced and free convection for laminar flow in the entrance region of isothermal, inclined tubes is made, where three independent parameters are introduced: the Prandtl number Pr, a modified Rayleigh number Ra*, and {Omega}, a parameter that measures the relative importance of free and forced convection.
Abstract: An analysis is made of the combined forced and free convection for laminar flow in the entrance region of isothermal, inclined tubes. This involves the numerical calculation of the developing flow with significant buoyancy effects. Three independent parameters are introduced: the Prandtl number Pr, a modified Rayleigh number Ra*, and {Omega}, a parameter that measures the relative importance of free and forced convection. The inclination angle does not appear explicitly in the formulation. Numerical results are obtained for Pr = 0.7, 5, and 10, and representative values of Ra* and {Omega}. The axial development of the velocity profiles, temperature field, local pressure gradient, and the Nusselt number are presented. These results reveal that the buoyancy effects have a considerable influence on the fluid flow and heat transfer characteristics of the development flow. A comparison of the numerical results with the available experimental data is also presented.