Showing papers on "Rayleigh number published in 1986"
TL;DR: In this paper, the authors reviewed advances made during the last seven years in the application of fluid dynamics to problems of igneous petrology, with emphasis on the laboratory work with which the authors have been particularly involved.
265 citations
TL;DR: In this paper, the effect of morphological and double-diffusive instabilities during directional solidification on the interface of a phase melt and phase change interface is investigated, and various applications for melt-flow interactions with solid-liquid interface to engineering and welding are discussed.
Abstract: The coupling between a crystal-melt interface and fluid flow is investigated. The solidification boundary conditions at the crystal-melt interface are described. The influences of morphological and double-diffusive instabilities during directional solidification on the interface are studied. The experiments by Glicksman and Mickalonis (1982) and Fang et al. (1985) which examine the relationship between the hydrodynamic state of the phase melt and phase-change interface are analyzed. The Rayleigh-Benard problem, and the effect of crystal-melt interaction on the Rayleigh number are examined. Various applications for melt-flow interactions with solid-liquid interface to engineering and welding are discussed.
204 citations
TL;DR: In this article, the authors investigated thermal convection for flows in which the production of turbulence energy is due solely to buoyancy, and the statistics of the flow are homogeneous in horizontal planes.
Abstract: Turbulence in thermal convection is investigated for flows in which the production of turbulence energy is due solely to buoyancy, and the statistics of the flow are homogeneous in horizontal planes. New experimental results for high Rayleigh number unsteady turbulent convection in a horizontal layer heated from below and insulated from above are presented and compared to turbulent Rayleigh convection, convection in the planetary boundary layer, and laboratory penetrative convection. Mean temperature fields are correlated in terms of wall layer scales and convection scales. Joint statistics of turbulent temperature and horizontal velocity and vertical velocity through fourth order are presented for the core region of the convection layer.
166 citations
TL;DR: In this paper, an analytical and numerical study of natural convection heat and mass transfer through a vertical porous layer subjected to uniform fluxes of heat from the side is presented, which is driven by the combined buoyancy effect due to temperature and concentration variations through the porous medium.
163 citations
TL;DR: In this article, a series of laboratory experiments were conducted to study convective structures in rotating fluids (distilled water) in ranges of Rayleigh flux number Raf from 106 to 2 × 1011 and of Taylor number Ta from 106-1012.
Abstract: We describe a series of laboratory experiments to study convective structures in rotating fluids (distilled water) in ranges of Rayleigh flux number Raf from 106 to 2 × 1011 and of Taylor number Ta from 106 to 1012. An intermediate quasi-stationary ring pattern of convection was found to arise from the interaction of the onset of convection with the fluid spin-up, for which we determined the times of origin and destruction, the distances between the rings, and the diameter of the central ring in terms of Raf and Ta. The ring structure evolves into a vortex grid which can be regular or irregular. In terms of Raf and Ta the regular grid exists in the linear regime, when the number of vortices N is in accord with the linear theory, when , or in the nonlinear regime when N ∝ h−2Ta½Raf−⅙ ∝ where Ω is the angular velocity and h is the fluid depth. In the irregular regime we always have N ∝ Ω. The transition from the regular regime to the irregular one is rather gradual and is determined by the value of the ordinary Rayleigh number, which we found to be greater than the first critical number Ra ∝ Ta2/3 by a factor about 25–40. In the transition region vortex interactions are observed, which start with rotation of two adjacent vortices around a common axis, then the vortices come closer and rotation accelerates, following which the vortices form a double helix and then coalesce into one stronger vortex.Some other qualitative experiments show that if the rotating vessel with the convective fluid is inclined to the horizontal, the vortex grid is formed along the rotation axis in accordance with the Proudman–Taylor theorem.
162 citations
TL;DR: In this paper, a mathematical model of convection, obtained by truncation from the two-dimensional Boussinesq equations, is shown to exhibit a bifurcation from symmetrical cells to tilted non-symmetrical ones.
Abstract: A mathematical model of convection, obtained by truncation from the two-dimensional Boussinesq equations, is shown to exhibit a bifurcation from symmetrical cells to tilted non-symmetrical ones. A subsequent bifurcation leads to time-dependent flow with similarly tilted transient plumes and a large-scale Lagrangian mean flow. This change of symmetry is similar to that occurring with the advent of a large-scale flow and transient tilted plumes seen in laboratory experiments on turbulent convection at high Rayleigh number. Though not intended as a description of turbulent convection, the model does bring out in a theoretically tractable context the possibility of the spontaneous change of symmetry suggested by the experiments.Further bifurcations of the model lead to stable chaotic phenomena as well. These are numerically found to occur in association with heteroclinic orbits. Some mathematical results clarifying this association are also presented.
154 citations
TL;DR: In this paper, a height and azimuth independent steady-state solution of the Navier-Stokes and cell conservation equations is presented, and the growth rate of a concentration fluctuation is governed by a parameter similar to a Rayleigh number.
Abstract: Gravitational and viscous torques acting on swimming micro-organisms orient their trajectories. The horizontal component of the swimming velocity of individuals of the many algal genera having a centre of mass displaced toward the rear of the cell is therefore in the direction g × ([dtri ] × u), where g is the acceleration due to gravity. This phenomenon, called gyrotaxis, results in the cells swimming toward downward-flowing regions of their environment. Since the cells’ density is greater than that of water, regions of high (low) cell concentration sink (rise). The horizontal component of gyrotaxis reinforces this type of buoyant convection, whilst the vertical one maintains it. Gyrotactic buoyant convection results in the spontaneous generation of descending plumes containing high cell concentration, in spatially regular concentration/convection patterns, and in the perturbation of initially well-defined flow fields. This paper presents a height- and azimuth-independent steady-state solution of the Navier-Stokes and cell conservation equations. This solution, and the growth rate of a concentration fluctuation, are shown to be governed by a parameter similar to a Rayleigh number.
149 citations
TL;DR: In this article, the effect of conduction in the wall on the natural convection flow in a square enclosure has been analyzed numerically, and three separate models to account for the wall conduction are investigated: (i) the complete conjugate case in which conduction was assumed to be fully two-dimensional, (ii) a one-dimensional model in which the wall convection in the horizontal direction only, and (iii) a lumped parameter approach which assumed the solid-fluid interface temperature to be uniform.
138 citations
TL;DR: In this article, a model to describe growth rates in laminar flow systems on the basis of concentration profiles under diffusion controlled conditions has been developed, which includes the definition of an entrance length for the concentration profile to developed.
131 citations
TL;DR: In this article, small-amplitude nonlinear solutions in the form of standing and traveling waves are found and their relative stability is established. And the interaction of the two types of wave with steady overturning convection is also studied.
Abstract: Two-dimensional oscillatory convection in a binary fluid mixture in an infinite plane porous layer heated from below is studied. Small-amplitude nonlinear solutions in the form of standing and traveling waves are found and their relative stability is established. Stable traveling waves are preferred near onset. The interaction of the two types of wave with steady overturning convection is also studied. As the Rayleigh number is increased the period of each type of wave approaches infinity, standing waves as -ln(${R}_{c}^{\mathrm{SW}\mathrm{\ensuremath{-}}\mathrm{R}}$) and traveling waves as 1/(${R}_{c}^{\mathrm{TW}\mathrm{\ensuremath{-}}\mathrm{R}}$), where ${R}_{c}$ is the critical Rayleigh number at which the transition to finite amplitude overturning convection occurs. This transition is hysteretic. The presence of modulated traveling waves (i.e., waves with two distinct frequencies) is also predicted. These predictions are made on the basis of analyses of multiple bifurcations in the presence of O(2) symmetry. This symmetry is present in two-dimensional problems with periodic boundary conditions and a reflection symmetry in a vertical plane. The relevance of the results to recent experiments on binary fluids, both in bulk mixtures and in a porous medium, is discussed.
128 citations
TL;DR: The effect of finite geometry on the competition between traveling waves and standing waves in systems with a Hopf bifurcation to a state with spatial structure is considered in the linear and weakly nonlinear regimes.
Abstract: The effect of finite geometry on the competition between traveling waves and standing waves in systems with a Hopf bifurcation to a state with spatial structure is considered in the linear and weakly nonlinear regimes. The spatial structure observed by Kolodner et al. in binary-fluid convection is explained in terms of the reflection of the linear traveling waves. The reflection coefficient is calculated, and is found to go to zero as the frequency of the waves becomes small. The pattern expected in a saturated nonlinear state is discussed.
TL;DR: In this article, an experimental investigation of the onset of convection in shallow fluid layers heated uniformly from below and cooled from above by an air layer has been made, and it has been shown that if the depth of the silicone layer is smaller than 2 mm, then convection takes place in two stages.
Abstract: An experimental investigation of the onset of convection in shallow fluid layers heated uniformly from below and cooled from above by an air layer has been made. If the depth of the silicone layer is smaller than 2 mm the onset of convection takes place in two stages. There is first a weak pattern, which is characterized by its appearance at ever smaller temperature gradients as the depth of the fluid is decreased. When the temperature difference across the fluid is increased a second strong pattern forms near the predicted critical Marangoni number. The cells in this pattern are hexagonal and seem to be what one has always referred to as Benard cells. The temperature gradient at which this pattern appears increases with decreased depth. The heat transfer through the fluid has been measured. The critical temperature gradient for the formation of the hexagonal pattern has been determined from the break of the heat transfer curve.
TL;DR: In this paper, a self-similar model is developed based on a boundary-layer analysis of a thin diffusive layer surrounding a spherical thermal for which the flow field is given by the exact solution for non-diffusive Stokes' flow.
Abstract: The flow induced by injection of a given amount of buoyancy or hot fluid from a localized source in a viscous fluid is investigated for conditions under which the Reynolds number Re is small compared with one, and the dimensionless buoyancy or Rayleigh number Ra is large compared with one. Laboratory experiments show that the buoyant fluid rises in the form of an extremely viscous ‘thermal’ which enlarges with time as a result of entrainment of surrounding fluid. The formation of a stable ‘chemical ring’ or torus of passive tracer similar in appearance to high Reynolds-number vortex rings is a notable feature of the creeping flow for high Rayleigh numbers. The possibility of large variations of viscosity due to temperature differences is included. A self-similar model is developed based on a boundary-layer analysis of a thin diffusive layer surrounding a spherical thermal for which the flow field is given by the exact solution for non-diffusive Stokes’ flow. Experiments at 2.5 × 102 < Ra < 2.5 × 104 and Re < 10−2 demonstrate the nature of extremely viscous thermals, support the similarity solution and enable evaluation of a proportionality constant. Possible applications of the results to dispersion by viscous drops and particularly to thermal convection in the Earth's solid mantle are mentioned.
01 Jan 1986
TL;DR: In this article, a finite difference method based on flux-corrected transport (FCT) is applied to natural convective flow in a porous medium at large Rayleigh number.
Abstract: A numerical technique designed to solve a wide class of convectively dominated flow problems is applied to natural convective flow in a porous medium at large Rayleigh number. The technique is a finite difference method based on flux-corrected transport (FCT) and possesses four desirable numerical properties: stability, accuracy, monotonicity, and conservation. Steady natural convection is investigated for Rayleigh numbers as large as 10,000. An efficient methodology for obtaining steady state solutions is illustrated. A simulation is performed for transient thermal convection at a Rayleigh number of 2500. Transient natural convection involving both heat and mass transfer is illustrated for a Rayleigh number of 2500, Lewis number of 2, and buoyancy ratio of 0.1. All simulations are performed in a square cavity with heated vertical side walls. 17 refs.
TL;DR: In this article, a pseudo-spectral numerical scheme is used to study two-dimensional, single-cell, time-dependent convection in a square cross-section of fluid saturated porous material heated from below.
Abstract: A pseudo-spectral numerical scheme is used to study two-dimensional, single-cell, time-dependent convection in a square cross-section of fluid saturated porous material heated from below. With increasing Rayleigh number R convection evolves from steady S to chaotic NP through the sequence of bifurcations S→P(1)→QP2→P(2)→NP, where P(1) and P(2) are simply periodic regimes and QP2 is a quasi-periodic state with two basic frequencies. The transitions (from onset of convection to chaos) occur at Rayleigh numbers of 4π2, 380–400, 500–520, 560–570, and 850–1000. In the first simply periodic regime the fundamental frequency f1 varies as . The chaotic states are characterized by spectral peaks with at least 3 fundamental frequencies superimposed on a broadband background noise. The time dependence of these states arises from the random generation of tongue-like disturbances within the horizontal thermal boundary layers. Transition to the chaotic regime is accompanied by the growth of spectral components that destroy the centre-symmetry of convection in the other states. Over-truncation can lead to spurious transitions and bifurcation sequences; in general it produces overly complex flows.
TL;DR: In this paper, the authors studied the transition from nonlinear periodic oscillations to temporal chaos and revealed sequences of period-doubling bifurcations, leading to aperiodicity, as the thermal Rayleigh number R(T) is increased.
Abstract: The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity tau is less than 1 and the solutal Rayleigh number Rs is sufficiently large. Solutions have been obtained for two representative values of tau. For tau = 0.316, R(s) = 10,000, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number R(T) is increased. At higher values of R(T), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As R(T) is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for tau = 0.1 by varying R(T) for several different values of R(s). When R(s) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As R(s) is increased, chaos appears in the neighborhood of these heteroclinic orbits. At higher values of R(s), chaos is found for a broader range in R(T). A truncated fifth-order model suggest that the appearance of chaos is associated with heteroclinic bifurcations.
TL;DR: Etude experimentale realisee par anemometrie laser Doppler. La cavite est rectangulaire avec une paroi laterale ouverte as discussed by the authors.
Abstract: Etude experimentale realisee par anemometrie laser Doppler. La cavite est rectangulaire avec une paroi laterale ouverte
TL;DR: In this article, it is shown that the overall range of the Rayleigh number, Ra, can be divided into two subregions, called low and high, in each of which the Nusselt number behaves differently.
TL;DR: In this paper, the authors discuss the problems of convection in anneau cylindrique en rotation with a chauffeur de l'exterieur and refroidi de the interieur.
Abstract: Etude du probleme de convection dans un anneau cylindrique en rotation chauffe de l'exterieur et refroidi de l'interieur
TL;DR: In this article, the onset Rayleigh number, frequency of oscillation, the linear growth rate of the instability, and the spatial pattern of the flow are studied as functions of the parameters of the fluid mixtures.
Abstract: Flow-visualization experiments are described which study the onset of convection in ethanolwater mixtures. It is shown that the conducting state becomes unstable to oscillatory convection, and the onset Rayleigh number, the frequency of oscillation, the linear growth rate of the instability, and the spatial pattern of the flow are studied as functions of the parameters of the fluid mixtures. This oscillatory fluid motion does not stabilize at a finite amplitude but grows exponentially with time until large-amplitude, overturning convection is triggered.
TL;DR: In this paper, the equations of the three-dimensional convective motion of an infinite Prandtl number fluid are solved in spherical geometry, for Rayleigh numbers up to 15 times the critical number.
Abstract: In this study, the equations of the three-dimensional convective motion of an infinite Prandtl number fluid are solved in spherical geometry, for Rayleigh numbers up to 15 times the critical number. An iterative method is used to find stationary solutions. The spherical parts of the operators are treated using a Galerkin collocation method while the radial and time dependences are expressed using finite difference methods. A systematic search for stationary solutions has led to eight different stream patterns for a low Rayleigh number (1.28 times the critical number). They can be classified as: I) Axisymmetrical solutions, analogous to rolls in plane geometry. II) Solutions which have several ascending plumes within a large area of ascending current, and also several descending plumes within an area of descending current. This type of flow is analogous to bimodal circulation in plane geometry. III) Solutions characterized by isolated ascending (or descending) plumes separated from each other by a...
TL;DR: In this article, a correlation equation for the local and average mixed convection Nusselt numbers is developed, which are found to agree well with the numerically predicted values and available experimental data for both buoyancy assisting and opposing flow conditions.
Abstract: Local Nusselt numbers for laminar mixed convection flows along isothermal vertical, inclined, and horizontal flat plates are presented for the entire mixed convection regime for a wide range of Prandtl numbers, 0.1 ≤ Pr ≤ 100. Simple correlation equations for the local and average mixed convection Nusselt numbers are developed, which are found to agree well with the numerically predicted values and available experimental data for both buoyancy assisting and opposing flow conditions. The threshold values of significant buoyancy effects on forced convection and forced flow effects on free convection, as well as the maximum increase in the local mixed convection Nusselt number from the respective pure convection limits, are also presented for all flow configurations. It is found that the buoyancy or forced flow effect can increase the surface heat transfer rate from pure forced or pure free convection by about 20 percent.
TL;DR: In this article, the equations of motion are solved numerically for a Boussinesq fluid with infinite Prandtl number in a square 2D box where the viscosity increases with depth.
Abstract: The equations of motion are solved numerically for a Boussinesq fluid with infinite Prandtl number in a square 2-D box where the viscosity increases with depth. Three heating modes are employed: bottom heating, internal heating, and half bottom and half internal heating. In all cases the boundaries are free slip. The range of Rayleigh numbers employed is 10^4-10^7. The viscosity increases as 10^(β(1-y)), where y is distance measured from the bottom upwards and β is a free parameter. In the bottom heated cases, the convective velocities slow near the bottom and result in a large temperature drop between the bottom boundary and interior compared with the top boundary and the interior. This results in increased buoyancy in the ascending limb. In the internally heated case, the flow in the top half of the box resembles Rayleigh-Benard convection and in the bottom half it approaches a conductive thermal regime for β greater than about 2. In this case the top surface heat flux decays from ascending to descending limb and the ascending and descending limbs become more equal in their buoyancy. Increasing β decreases the efficiency of heat transport, but has little effect on the exponents of Nu-Ra and Pe-Ra relations. There is a larger decrease in heat transport efficiency for a given β in the bottom heated case compared to the internally heated case.
TL;DR: In this paper, the authors studied the flow patterns associated with Rayleigh convection in rectangular containers of approximate proportions 10 × 5 × 1 at Prandtl numbers σ between 2 and 20.
Abstract: We report a study of the flow patterns associated with Rayleigh—Benard convection in rectangular containers of approximate proportions 10 × 5 × 1 at Prandtl numbers σ between 2 and 20. The flow is studied at Rayleigh numbers ranging from the onset of convective flow to the onset of time dependence; Nusselt-number measurements are also presented. The results are discussed in the content of the theory for the stability of a laterally infinite system of parallel rolls. We observed transitions between time-independent flow patterns which depend on roll wavenumber, Rayleigh number and Prandtl number in a manner that is reasonably well described by this theory. For σ [lsim ] 10, the skewed-varicose instability (which leads directly to time dependence in much larger containers) is found to initiate transitions between time-independent patterns. We are then able to study the approach to time dependence in a regime of larger Rayleigh number where the instabilities in the flow are found to have an intrinsic time dependence. In this regime, the onset of time dependence appears to be explained by the recent predictions of Bolton, Busse & Clever for a new set of time-dependent instabilities.
TL;DR: In this paper, the authors investigated the stability of rotating double-diffusive convection in a sparsely packed porous medium considering a non-Darcy equation and showed that the effect of rotation and porous parameter is to decrease the region of instabilities.
TL;DR: In this paper, a computational methodology is presented for the problem of melting of a pure metal from an isothermal vertical wall, where the governing conservation equations of mass, momentum, and energy are solved with a control volume-based discretization scheme adapted for irregular geometries.
Abstract: A computational methodology is presented for the problem of melting of a pure metal from an isothermal vertical wall. The governing conservation equations of mass, momentum, and energy are solved with a control volume-based discretization scheme adapted for irregular geometries. The moving boundary is immobilized by employing the quasi-steady assumption with an algebraically generated nonorthogonal grid. All terms in the governing equations arising from the nonorthogonatity of the computational grid are retained in the solution algorithm. The model is validated by comparison with available experimental data for the effect of natural convection on melting heat transfer in a pure metal system. The influence of Rayleigh number on velocity and temperature fields is investigated, and sample results are presented for overall heat transfer and melting rate. The predictions indicate that buoyancy-induced fluid motion may have a less dominant role in energy transport in the transition to quasi-steady melting in th...
TL;DR: In this article, a numerical investigation of laminar mixed convection of air in a vertical channel is made, where the thermal boundary conditions considered are symmetric heating and asymmetric heating, where one plate is heated and the other is adiabatic.
Abstract: A numerical investigation is made of laminar mixed convection of air in a vertical channel. The thermal boundary conditions considered are symmetric heating, where both plates are heated, and asymmetric heating, where one plate is heated and the other is adiabatic. Results are obtained for Rayleigh numbers of 10 3 , 10 5 , and 10 6 and Gr/Re 2 values of 0.1, 1, 3, and 5. The temperatures are observed to increase with increasing Gr/Re 2 and decreasing Rayleigh number. The velocity profile peaks near the hot plates and exhibits a concavity, which, in the symmetric heating case, occurs around the centerline. With increasing Gr/Re 2 the velocity near the hot wall increases while the velocity near the centerline decreases. The Nusselt number attains its maximum value near the inlet of the channel and increases with decreasing Gr/Re 2 values.
TL;DR: In this paper, a series of numerical experiments concerning the phenomenon of natural convection in a two-layer composite system heated from below is reported, and the numerical solutions are based on the full equations for time-dependent flow, and document the main features of the flow at Rayleigh numbers several orders of magnitude above critical value.
TL;DR: In this article, the spectral analysis of known temperature fields represents the forward problem corresponding to the inverse problem of inferring the thermal structure of the mantle from the spectral components of its lateral heterogeneity as determined by seismic tomography.
Abstract: Thermally induced lateral heterogeneity in the earth's mantle has been studied in terms of idealized two-dimensional, constant viscosity, numerical models of mantle convection. Steady model solutions of the temperature and velocity fields were analyzed with respect to both the spatial and spectral signatures of their inherent lateral heterogeneity. By examining the changes in spectral signature produced by varying the Rayleigh number, the degree of internal heating, and the depth in the convection cell, it has been possible to identify those features of the spectra of lateral heterogeneity which are most indicative of these variations. Particular attention has been paid to the spectral characteristics of thermal boundary layers within the convecting layer. The spectral analysis of known temperature fields represents the forward problem corresponding to the inverse problem of inferring the thermal structure of the mantle from the spectral components of its lateral heterogeneity as determined by seismic tomography. The forward problem presented here illustrates a new approach in convection modeling which may ultimately provide important diagnostic criteria for the interpretation of recent tomographic data.