Showing papers on "Rayleigh number published in 1993"

Bénard cells and Taylor vortices

Abstract: Part I. Benard Convection and Rayleigh-Benard Convection: 1. Benard's experiments 2. Linear theory of Rayleigh-Benard convection 3. Theory of surface tension driven Benard convection 4. Surface tension driven Benard convection experiments 5. Linear Rayleigh-Benard convection experiments 6. Supercritical Rayleigh-Benard convection experiments 7. Nonlinear theory of Rayleigh-Benard convection 8. Miscellaneous topics Part II. Taylor Vortex Flow: 9. Circular Couette flow 10. Rayleigh's stability criterion 11. G. I. Taylor's work 12. Other early experiments 13. Supercritical Taylor vortex experiments 14. Experiments with two independently rotating cylinders 15. Nonlinear theory of Taylor vortices 16. Miscellaneous topics.

Transient high-Rayleigh-number thermal convection with large viscosity variations

Abstract: The characteristics of thermal convection in a fluid whose viscosity varies strongly with temperature are studied in the laboratory. At the start of an experiment, the upper boundary of an isothermal layer of Golden Syrup is cooled rapidly and maintained at a fixed temperature. The fluid layer is insulated at the bottom and cools continuously. Rayleigh numbers calculated with the viscosity of the well-mixed interior are between 106 and 108 and viscosity contrasts are up to 106. Thermal convection develops only in the lower part of the thermal boundary layer, and the upper part remains stagnant. Vertical profiles of temperature are measured with precision, allowing deduction of the thickness of the stagnant lid and the convective heat flux. At the onset of convection, the viscosity contrast across the unstable boundary layer has a value of about 3. In fully developed convection, this viscosity contrast is higher, with a typical value of 10. The heat flux through the top of the layer depends solely on local conditions in the unstable boundary layer and may be written \[Q_{\rm s} = - CK_{\rm m} (\alpha g/\kappa u_{\rm m})^{\frac{1}{3}} \Delta T^{\frac{4}{3}}_{\rm v}\], where km and νm are thermal conductivity and kinematic viscosity at the temperature of the well-mixed interior, κ thermal diffusivity, α the coefficient of thermal expansion, g the acceleration due to gravity. ΔTv, is the ‘viscous’ temperature scale defined by \[\Delta T_{\rm v} = - \frac{\mu (T_{\rm m})}{({\rm d}\mu /{\rm d}T)(T_{\rm m})}\] where μ(T) is the fluid viscosity and Tm the temperature of the well-mixed interior. Constant C takes a value of 0.47 ± 0.03. Using these relations, the magnitude of temperature fluctuations and the thickness of the stagnant lid are calculated to be in excellent agreement with the experimental data. One condition for the existence of a stagnant lid is that the applied temperature difference exceeds a threshold value equal to (2ΔTv).

Order parameter equations for patterns

Abstract: Patterns of an almost periodic nature appear all over the place. One sees them in cloud streets, in sand ripples on flat beaches and desert dunes, in the morphology of plants and animals, in chemically reacting media, in boundary layers, on weather maps, in geological formations, in interacting laser beams in wide gainband lasers, on the surface of thin buckling shells, and in the grid scale instabilities of numerical algorithms. This review deals with the class of problems into which these examples fall, namely with pattern formation in spatially extended, continuous, dissipative systems which are driven far from equilibrium by an external stress. Under the influence of this stress, the system can undergo a series of symmetry breaking bifurcations or phase transitions and the resulting patterns become more and more complicated, both temporally and spatially, as the stress is increased. Figures 1 through 3 show examples of patterns in lasers, binary and ordinary fluids, and liquid crystals. The goal of theory is to provide a means of understanding and explaining these patterns from a macroscopic viewpoint that both simplifies and unifies classes of problems which are seemingly unrelated at the microscopic level. Convection in a large aspect ratio horizontal layer of fluid heated from below is the granddaddy of canonical examples used to study pattern formation and behavior in spatially extended systems. For low values of the vertical temperature difference, which is the external stress parameter in this case and whose non-dimensional measure is called the Rayleigh

High Rayleigh number β-convection

Abstract: High resolution numerical simulations of thermal convection in a rapidly rotating channel with gravity perpendicular to the rotation vector are described. The convecting columns are subject to a β-effect resulting from topographic vortex stretching. The symmetries of the problem allow many invariant wavenumber sets, and this property is associated with the existence of stable multiple-equilibria at modest supercriticality. The transition to chaotic behavior involves the production of intermittent unstable orbits off a two-torus in energy space. At very high Rayleigh number (of order 106 to 107) the motion can be turbulent, depending on the size of β. However, the turbulence is usually characterized by an almost-periodic formation of patches of small scale convection that cause regular pulsations in the accompanying strong zonal jets. The processes maintaining these flows may be related to those responsible for the zonal currents on Jupiter and for cyclic variability on the Sun.

Effects of strongly temperature‐dependent viscosity on time‐dependent, three‐dimensional models of mantle convection

Abstract: Numerical simulations of thermal convection in a wide (8×8×1) Cartesian box heated from below with temperature-dependent viscosity contrasts of 1000, and Rayleigh number 105 show that boundary conditions and aspect ratio have an enormous effect on the preferred flow pattern. With rigid upper and lower boundaries, spoke-pattern flow with small (diameter ∼ 1.5) cells is obtained, consistent with laboratory experiments and previous numerical results. However, with the arguably more realistic stress-free boundaries, the flow chooses the largest possible wavelength, forming a single square cell of aspect ratio 8, with one huge cylindrical downwelling surrounded by upwelling sheets. The addition of stress-dependence to the rheology weakens the stiff upper boundary layer, resulting in smaller cells, though still with upwelling sheets and downwelling plumes.

Dynamic feedback between a continentlike raft and thermal convection

Abstract: Seismic observations of the mantle, which include long-wavelength structure, a k^(−1) dependence of heterogeneity on harmonic k, and a heterogeneous upper boundary layer, and supercontinent kinematics may be explained by the dynamic interaction between a continent like raft and thermal convection. We have formulated finite element models of convection with rafts simulating continental plates in a cylindrical geometry. The azimuthal interconnectivity of this geometry is vital to resolve the two-way dynamics between rafts and convection. Computations show that (1) raft motion is periodic, (2) long-wavelength thermal structure is significant within both thermal boundary layers and the fluid interior, and (3) the large-scale thermal structure with a wavelength longer than the width of raft is responsible for raft motion. These three results, which are observed for a range of Rayleigh numbers, internal heating rates, and raft sizes, are a direct consequence of the dynamic interaction between the raft and convection. The physical processes for a model with a Rayleigh number of 10^5 are representative: when the raft is stationary, due to the less efficient heat transfer through the raft and instabilities from the bottom boundary layer, heat accumulates beneath the raft and results in long-wavelength thermal anomalies. The long-wavelength thermal anomalies enhance raft motion. Accompanying the enhanced raft movement, the long-wavelength thermal anomalies diminish and the raft velocity decreases or the raft comes to rest. Since convection models without rafts generate less long-wavelength heterogeneity compared to the models with rafts, or continental plates, we suspect that continental plates may play a crucial role in mantle dynamics. Interestingly, raft motion with a period of about 10 transit times is usually significant; 10 transit times is about 600 m.y. if scaled to the Earth. This is close to the observed 300–500 m.y. period of supercontinent aggregation and dispersal.

The effects of a composite non-Newtonian and Newtonian rheology on mantle convection

Abstract: Summary A finite element method based on a primitive variables formulation is used to model both steady-state and time-dependent mantle convection with a composite Newtonian and non-Newtonian (power-law) rheology. The rheological model employs the transition stress as a means of partitioning the relative importance of the two rheologies. Results show that there is no direct correlation between viscosity and temperature anomalies. Fluctuations of the velocity fields are much greater and faster than for Newtonian flows. Fluctuations with amplitudes several times the background velocity are quite common. Intermittency effects with quiescent periods punctuated by chaotic bursts are observed. From scaling arguments temporal fluctuations of the volume-averaged viscosity are comparable in magnitude to the variations in the surface heat flow for the non-Newtonian flows, but are smaller than the variations in the velocity field. At larger transition stress the Newtonian behaviour becomes dominant and the temporal variations of the viscosity diminish. Both steady-state and time-dependent results show that for a given transition stress the non-Newtonian behaviour prevails to a greater extent with increasing Rayleigh number. Implications of this non-Newtonian tendency for Archaean tectonics are discussed.

Convection in a rotating cylinder. Part 1. Linear theory for moderate Prandtl numbers

Abstract: The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios τ ≡ d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow. The precession velocity of the slow modes vanishes as τ → ∞. Thus it is these modes which provide the connection between the results for a finite-aspect-ratio System and the unbounded layer in which the instability is a steady-state one, except in low Prandtl number fluids.The linear stability problem is solved for several different sets of boundary conditions, and the results compared with recent experiments. Results are presented for Prandtl numbers σ in the range 6.7 ≤ σ ≤ 7.0 as a function of both the rotation rate and the aspect ratio. The results for rigid walls, thermally conducting top and bottom and an insulating sidewall agree well with the measured critical Rayleigh numbers and precession frequencies for water in a τ = 1 cylinder. A conducting sidewall raises the critical Rayleigh number, while free-slip boundary conditions lower it. The difference between the critical Rayleigh numbers with no-slip and free-slip boundaries becomes small for dimensionless rotation rates Ωh2/v ≥ 200, where v is the kinematic viscosity.

Temperature and velocity profiles of turbulent convection in water

Abstract: Measurements of temperature and velocity profiles in a convection cell of aspect ratio 1 at a Rayleigh number of 1.1\ifmmode\times\else\texttimes\fi{}${10}^{9}$ and a Prandtl number (Pr) of 6.6 are presented. A large-scale flow persists in the cell. The temperature boundary layer is entirely contained in the viscous boundary layer and the velocity field is smooth on the characteristic length scale of temperature variations. The core of the cell is stably stratified. Most of the heat transported between the plates is carried by the general circulation and does not traverse the core of the cell. Comparison is made with observations at Pr=0.7.

Rotating Rayleigh–Bénard convection: asymmetric modes and vortex states

Abstract: We present optical shadowgraph flow visualization and heat transport measurements of Rayleigh–Benard convection with rotation about a vertical axis. The fluid, water with Prandtl number 6.4, is confined in a cylindrical convection cell with radius-to-height ratio Γ = 1. For dimensionless rotation rates 150 < Ω < 8800, the onset of convection occurs at critical Rayleigh numbers Rc(Ω) much less than those predicted by linear stability analysis for a laterally infinite system and qualitatively consistent with finite-aspect-ratio, linear-stability calculations of Buell & Catton (1983). As in the calculations, the forward bifurcation at onset is to states of localized flow near the lateral walls with azimuthal periodicity of 3 < m < 8. These states precess in the rotating frame, contrary to the assumptions of Buell & Catton (1983) but in quantitative agreement with recent calculations of Goldstein et al. (1992), with a frequency that is finite at onset but goes to zero as Ω goes to zero. At Ω = 2145 we find primary and secondary stability boundaries for states with m = 4, 5, 6, and 7. Further, we show that at higher Rayleigh number, there is a transition to a vortex state where the vortices form with the symmetry of the existing azimuthal periodicity of the sidewall state. Aperiodic, time-dependent heat transport begins for Rayleigh numbers at or slightly above the first appearance of vortices. Visualization of the formation and interactions of thermal vortices is presented, and the behaviour of the Nusselt number at high Rayleigh numbers is discussed.

Rayleigh-Bénard convection near the gas-liquid critical point.

Abstract: We present experimental results on Rayleigh-Benard convection in SF 6 near the gas-liquid critical point. We measured the critical temperature difference for the onset of convection, ΔT 0 , as a function of the reduced average temperature τ=(T-T c )/T c and found ΔT 0 =525×τ 1.89 , which is close to the expected power law behavior. The strong temperature dependence of the physical properties is used to scan the Prandtl number in a wide range. A new, many «target» pattern state, initiated by a defect instability, was observed

Instabilities from phase transitions and the timescales of mantle thermal evolution

Abstract: We have investigated the effects of mantle phase-transitions on the Earth's thermal evolution by numerical modelling Our results show that the tendency of layering grows with increasing vigor of convection A self-consistent model which couples the convective dynamics of the mantle to the secular cooling of the core is constructed For a temperature-dependent effective viscosity, we found that the thermal history of the Earth is characterized by two timescales, one is associated with layered convection in the early period, which is followed by a transition period with dramatic overturns As the Rayleigh number decreases sufficiently low, whole mantle convection prevails and a faster cooling timescale ensues

Time-dependent thermocapillary convection in a rectangular cavity: numerical results for a moderate Prandtl number fluid

Abstract: The present numerical simulation explores a thermal–convective mechanism for oscillatory thermocapillary convection in a shallow rectangular cavity for a Prandtl number 6.78 fluid. The computer program developed for this simulation integrates the two-dimensional, time-dependent Navier–Stokes equations and the energy equation by a time-accurate method on a stretched, staggered mesh. Flat free surfaces are assumed. The instability is shown to depend upon temporal coupling between large-scale thermal structures within the flow field and the temperature sensitive free surface. A primary result of this study is the development of a stability diagram presenting the critical Marangoni number separating the steady from the time-dependent flow states as a function of aspect ratio for the range of values between 2.3 and 3.8. Within this range, a minimum critical aspect ratio near 2.3 and a minimum critical Marangoni number near 20 000 are predicted, below which steady convection is found.

Stabilization of the no-motion state in Rayleigh-Bénard convection through the use of feedback control.

Abstract: It is demonstrated theoretically that the critical Rayleigh number for transition from the no-motion (conduction) to the motion state in the Rayleigh-B\'enard problem of an infinite fluid layer heated from below and cooled from above can be significantly increased through the use of feedback control strategies effecting small perturbations in the boundary data.

Convection in a rotating spherical fluid shell with an inhomogeneous temperature boundary condition at infinite Prandtl number

Abstract: We examine thermal convection in a rotating spherical shell with a spatially non-uniformly heated outer surface, concentrating on three distinct heating modes: first, with wavelength and symmetry corresponding to the most unstable mode of the uniformly heated problem; secondly, with the critical wavelength but opposite equatorial symmetry; and thirdly, with wavelength much larger than that of the most unstable mode. Analysis is focused on boundary-locked convection, the associated spatial resonance phenomena, the stability properties of the resonance solution, and time-dependent secondary convection. A number of new forms of instability and convection are found: the most interesting is perhaps the saddle-node bifurcation, which is the first to be found for realistic fluid systems governed by partial differential equations. An analogous Landau amplitude equation is also analysed, providing an important mathematical framework for understanding the complicated numerical solutions.

Boundary layer and scaling properties in turbulent thermal convection

Abstract: We describe an experiment which has been designed to measure both spatial and temporal features of turbulent thermal convection in a fluid layer heated from below. Specifically we have studied the dependence of the heat flowvs. the Rayleigh number, the thermal boundary layer profile, the temperature probability distribution function, the frequency and wave number power spectra. All the results have been compared with recent theories. The relevant scales of the problem, the Bolgiano and dissipative lengths, are also computed as a function of control parameters.

Lbe simulations of rayleigh-bénard convection on the ape100 parallel processor

Abstract: In this paper we describe an implementation of the Lattice Boltzmann Equation method for fluid-dynamics simulations on the APE100 parallel computer. We have performed a simulation of a two-dimensional Rayleigh-Benard convection cell. We have tested the theory proposed by Shraiman and Siggia for the scaling of the Nusselt number vs. Rayleigh number.

Hard turbulent thermal convection and thermal evolution of the mantle

Abstract: Hard turbulent convection is investigated using laboratory experiments and numerical simulations. In Newtonian mantle convection, the appearance of disconnected plumes marks the transition from soft to hard turbulence. For non-Newtonian rheology, the transition to hard turbulence takes place at much lower Nusselt numbers than it does for Newtonian rheology. This has important ramifications for the mantle. Large curvatures are developed in the trajectories of non-Newtonian plumes in the hard turbulent regime, in contrast to the trajectories of Newtonian plumes. When phase transitions are considered, mantle convection tends to become more layered with increasing Rayleigh numbers. The manner of mantle convection might have changed with time from a layered to a more whole mantle type of flow. Superplume events could have been caused by catastrophic overturns associated with strong gravitational instabilities in the transition zone.

The crystallization of lava lakes

Abstract: Thermal convection driven by the kinetic undercooling of solidification can produce compositional and textural variations in an initially uniform lava flow that is cooled predominantly from above. We show that such kinetic undercooling is sufficient to drive convection at high Rayleigh numbers even in shallow lava lakes. We then employ a theoretical model that includes the effects of convection at high Rayleigh number to make predictions of the complete evolution of the lava. We show that predictions of the rate of growth of solid crust at the surface of the lake do not differ much from predictions made by ignoring effects due to convection. However, convective motions do shorten the time for complete solidification because, when they are driven by kinetic undercooling, there is internal crystallization in addition to that occurring near the cooled upper boundary of the lava. This important effect, which is absent from purely conductive models, predicts stratification and zonation of the lava, in agreement with field observations. We also examine the effect of taking into account the heat transfer to the country rock below the lava lake, and determine how the evolution of the lava lake varies with the viscosity of the lava.

Asymptotic theory of wall-attached convection in a rotating fluid layer

Abstract: Asymptotic expressions for the onset of convection in a horizontal fluid layer of finite extent heated from below and rotating about a vertical axis are derived in the limit of large rotation rates in the case of stress-free upper and lower boundaries. In the presence of vertical sidewalls, the critical Rayleigh number Rc is much lower than the classical value for an infinitely extended layer. In particular, we find that Rc grows in proportion to τ when the sidewall is insulating, where τ is the dimensionless rotation rate. When the sidewall is infinitely conducting, Rc grows in proportion to as in the case of an infinitely extended layer but with a lower coefficient of proportionality. Numerical results obtained at finite values of τ show good agreement with the asymptotic formulae.

Vortical nature of thermal plumes in turbulent convection

Abstract: Thermal plumes in Rayleigh–Benard convection have been observed to occur with strong vertical component of vorticity, resulting in spiraling hot updrafts and cold downdrafts. Results from two different simulations at Rayleigh numbers 9800 and 33 000 times the critical Rayleigh number close to the boundaries show rapid increase in the conditional averages of vertical velocity and temperature perturbation at large values of vertical vorticity. This result along with the spatial distribution of vertical vorticity indicates a strong correlation between narrow regions of upmoving hot fluid (or downmoving cold fluid) and local vertical vorticity. The horizontal dimension of these vortical plumes is an order of magnitude smaller than the layer depth, but they extend up to half the convective layer in the vertical direction.

Nonequilibrium fluctuations in liquid mixtures under the influence of gravity

Abstract: The theory of fluctuations in liquid mixtures that are subjected to a stationary temperature gradient, while still being kept in a quiescent state, is extended to include the influence of gravity. Specifically, we evaluate the effects of a temperature gradient combined with gravity on the Rayleigh component of the light-scattering spectrum of liquid mixtures. For these case that the liquid system is heated from below we analyze the crossover from one-component-like to mixture-like behavior of the critical slowing down of the fluctuations near the convective instability. When the liquid mixture is heated from above, propagating modes appear for sufficiently large temperature gradients; a phenomenon similar to that previously encountered in one-component liquids. By retaining all the couplings between the temperature and concentration fluctuations in the hydrodynamic equations, we have also derived a new expression for the Rayleigh number R in liquid mixtures.

Plume Formation and Resonant Bifurcations in Porous Media Convection

Abstract: The classical boundary layer scaling laws proposd by Howard for Rayleigh-Benard convection at high Rayleigh number extend to the analogous case of convection in saturated porous media. We computationally study two-dimensional porous media convection near the onset of this scaling behavior. The main result of the paper is the observation of instabilities that lead to deviations from the scaling relations. At Rayleigh numbers below the scaling regime, boundary layer fluctuations born at a Hopf bifurcation strengthen and eventually develop into thermal plumes. The appearance of plumes corresponds to the onset of the boundary layer scaling behavior of the oscillation frequency and mean Nusselt number, in agreement with the classical theory. As the Rayleigh number increases further, the flow undergoes instabilities that lead to "bubbles" in parameter space of quasiperiodic flow, and eventually to weakly chaotic flow. The instabilities disturb the plume formation process, effectively leading to a phase modulation of the process and to deviations from the scaling laws. We argue that these instabilities correspond to parametric resonances between the time scale for plume formation and the characteristic convective time scale of the flow.

On the Angular Momentum Transport Associated with Convective Eddies in Accretion Disks

Abstract: Protostellar disks are intrinsically unstable against thermal convection in the direction normal to the plane of the disk. Because the origin of effective disk viscosity is unknown it is of great theoretical interest to investigate the effect of convection on the transport of angular momentum. To study this effect we present here results of direct two-dimensional simulations of compressible convection in axisymmetric accretion disks. In order to make the problem tractable an underlying viscosity (possibly due to shear instabilities and/or smaller scale convection) is used such that the calculations are performed for a Rayleigh number 10 times the critical value for marginal stability