Showing papers on "Rayleigh number published in 2006"

Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields

Abstract: SUMMARY We study numerically an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude. Convection is driven by a fixed temperature contrast between rigid boundaries. There are two distinct classes of solutions with strong and weak dipole contributions to the magnetic field, respectively. Non-dipolar dynamos are found when inertia plays a significant role in the force balance. In the dipolar regime the critical magnetic Reynolds number for self-sustained dynamos is of order 50, independent of the magnetic Prandtl number Pm. However, dynamos at low Pm exist only at sufficiently low Ekman number E. For dynamos in the dipolar regime we attempt to establish scaling laws that fit our numerical results. Assuming that diffusive effects do not play a primary role, we introduce non-dimensional parameters that are independent of any diffusivity. These are a modified Rayleigh number based on heat (or buoyancy) flux Ra ∗ , the Rossby number Ro measuring the flow velocity, the Lorentz number Lo measuring magnetic field strength, and a modified Nusselt number Nu ∗ for the advected heat flow. To first approximation, all our dynamo results can be collapsed into simple power-law dependencies on the modified Rayleigh number, with approximate exponents of 2/5, 1/2 and 1/3 for the Rossby number, modified Nusselt number and Lorentz number, respectively. Residual dependencies on the parameters related to diffusion (E, Pm, Prandtl number Pr) are weak. Our scaling laws are in agreement with the assumption that the magnetic field strength is controlled by the available power and not necessarily by a force balance. The Elsasser number � , which is the conventional measure for the ratio of Lorentz force to Coriolis force, is found to vary widely. We try to assess the relative importance of the various forces by studying sources and sinks of enstrophy (squared vorticity). In general Coriolis and buoyancy forces are of the same order, inertia and viscous forces make smaller and variable contributions, and the Lorentz force is highly variable. Ignoring a possible weak dependence on the Prandtl numbers or the Ekman number, a surprising prediction is that the magnetic field strength is independent both of conductivity and of rotation rate and is basically controlled by the buoyancy flux. Estimating the buoyancy flux in the Earth’s core using our Rossby number scaling and a typical velocity inferred from geomagnetic secular variations, we predict a small growth rate and old age of the inner core and obtain a reasonable magnetic field strength of order 1 mT inside the core. From the observed heat flow in Jupiter, we predict an internal field of 8 mT, in agreement with Jupiter’s external field being 10 times stronger than that of the Earth.

Numerical simulation of an asymptotically reduced system for rotationally constrained convection

Abstract: For rotationally constrained convection, the Taylor–Proudman theorem enforces an organization of nonlinear flows into tall columnar or compact plume structures. While coherent structures in convection under moderate rotation are exclusively cyclonic, recent experiments for rapid rotation have revealed a transition to equal populations of cyclonic and anticyclonic structures. Direct numerical simulation (DNS) of this regime is expensive, however, and existing simulations have yet to reveal anticyclonic vortical structures. In this paper, we use a reduced system of equations for rotationally constrained convection valid in the asymptotic limit of thin columnar structures and rapid rotation to perform numerical simulation of Rayleigh–B´ enard convection in an infinite layer rotating uniformly about the vertical axis. Visualization indicates the existence of cyclonic and anticyclonic vortical populations for all parameters examined. Moreover, it is found that the flow evolves through three distinct regimes with increasing Rayleigh number (Ra). For small, but supercritical Ra ,t he fl ow is dominated by a cellular system of hot and cold columns spanning the fluid layer. As Ra increases, the number density of these columns decreases, the up- and downdrafts within them strengthen and the columns develop opposite-signed ‘sleeves’ in all fields. The resulting columns are highly efficient at transporting heat across the fluid layer. In the final regime, lateral mixing plays a dominant role in the interior and the columnar structure is destroyed. However, thermal plumes are still injected and rejected from the thermal boundary layers. We identify the latter two regimes with the vortex-grid and geostrophic turbulence regimes, respectively. Within these regimes, we investigate convective heat transport (measured by the Nusselt number), mean temperature profiles, and root-mean-square profiles of the temperature, vertical velocity and vertical vorticity anomalies. For all Prandtl numbers investigated, the mean temperature saturates in a non-isothermal profile as Ra increases owing to intense lateral mixing.

Turbulent convection at high Rayleigh numbers and aspect ratio 4

Abstract: We report measurements of the Nusselt number, Nu, in turbulent thermal convection in a cylindrical container of aspect ratio 4. The highest Rayleigh number achieved was Ra=2×10 13 . Except for the last half a decade or so of Ra, experimental conditions obey the Boussinesq approximation accurately. For these conditions, the data show that the log Nu-log Ra slope saturates at a value close to 1/3, as observed previously by us in experiments with smaller aspect ratios. The increasing slope over the last half a decade of Ra is inconclusive because the corresponding conditions are non-Boussinesq. Finally, we report a modified scaling relation between the plume advection frequency and Ra that collapses data for different aspect ratios.

Convective instability of electrokinetic flows in a cross-shaped microchannel

Abstract: We present a parametric experimental study of convective electrokinetic instability (EKI) in an isotropically etched, cross-shaped microchannel using quantitative epifluorescence imaging. The base state is a three-inlet, one-outlet electrokinetic focusing flow configuration where the centre sample stream and sheath flows have mismatched ionic conductivities. Electrokinetic flows with conductivity gradients become unstable when the electroviscous stretching and folding of conductivity interfaces grows faster than the dissipative effect of molecular diffusion. Scalar images, critical applied fields required for instability, and temporal and spatial scalar energy are presented for flows with a wide range of applied d.c. electric field and centre-tosheath conductivity ratios. These parameters impose variations of the electric Rayleigh number across four orders of magnitude. We introduce a scaling for charge density in the bulk fluid as a function of local maximum conductivity gradients in the flow. This scaling shows that the flow becomes unstable at a critical electric Rayleigh number (Ra e,l =205) and applies to a wide range of applied field and centre-to-sheath conductivity ratios. This work is relevant to on-chip electrokinetic flows with conductivity gradients such as field amplified sample stacking, flow at the intersections of multi-dimensional assays, electrokinetic control and separation of sample streams with poorly specified chemistry, and low-Reynolds number micromixing.

The quasi-geostrophic model for rapidly rotating spherical convection outside the tangent cylinder

Abstract: Rapidly rotating convection in spherical geometry outside the tangent cylinder is investigated using the quasi-geostrophic approximation. The validity of the approximation is discussed, and numerical simulations using these equations are performed, reaching Ekman numbers, E, down to 10 -6 . The results are compared with experiments and fully three-dimensional numerical simulations. We find that the inertial scaling developed to study rapidly rotating convection does not represent the Prandtl-number dependence of our results adequately. Instead, we find that even in strongly supercritical situations the dominant wavenumbers at the onset of convection still have a strong influence on the behaviour. We find that the local Peclet number, the product of the typical convective velocity and local convective length scale divided by the thermal diffusivity, is helpful for understanding the dynamics of rapidly rotating convection. We explore the zonal flows driven by Reynolds stresses with no-slip boundaries and explore their Prandtl-number dependence. We also study the convective heat transport at low E, and consider the boundary layer structures that can form at large Rayleigh number, slowing down the rate of growth of the Nusselt number with Rayleigh number.

The onset of thermo-bioconvection in a shallow fluid saturated porous layer heated from below in a suspension of oxytactic microorganisms

Abstract: The aim of this paper is to present a continuum model for thermo-bioconvection of oxytactic bacteria in a porous medium and investigate the combined effects of microorganisms' upswimming and heating from below on the stability of bioconvection in a horizontal layer filled with a fluid saturated porous medium. Different from traditional bioconvection, thermo-bioconvection has two destabilizing mechanisms that contribute to creating the unstable density stratification. This problem may be relevant to a number of geophysical applications, such as the investigation of the dynamics of oxytactic species of thermophiles (heat loving microorganisms) living in hot springs, microbial-enhanced oil recovery, and modeling oil- and gas-bearing sedimentary basins. The utilization of the Galerkin method to solve a linear stability problem leads to a correlation between the critical value of the bioconvection Rayleigh number and the traditional “thermal” Rayleigh number.

Spatially localized binary-fluid convection

Abstract: Multiple states of spatially localized steady convection are found in numerical simulations of water–ethanol mixtures in two dimensions. Realistic boundary conditions at the top and bottom are used, with periodic boundary conditions in the horizontal. The states form by a mechanism similar to the pinning region around a Maxwell point in variational systems, but are located in a parameter regime in which the conduction state is overstable. Despite this the localized states can be stable. The properties of the localized states are described in detail, and the mechanism of their destruction with increasing or decreasing Rayleigh number is elucidated. When the Rayleigh number becomes too large the fronts bounding the state at either end unpin and move apart, allowing steady convection to invade the domain. In contrast, when the Rayleigh number is too small the fronts move inwards, and eliminate the localized state which decays into dispersive chaos. Out of this state spatially localized states re-emerge at irregular times before decaying again. Thus an interval of Rayleigh numbers exists that is characterized by relaxation oscillations between localized convection and dispersive chaos.

Azimuthal motion of the mean wind in turbulent thermal convection.

Abstract: We present an experimental study of the azimuthal motion of the mean wind in turbulent thermal convection. The experiments were conducted with cylindrical convection cells of unity aspect ratio and over the range of the Rayleigh number from $1\ifmmode\times\else\texttimes\fi{}{10}^{9}$ to $1\ifmmode\times\else\texttimes\fi{}{10}^{10}$. The azimuthal angle of the circulation plane of the mean wind was measured using both the particle image velocimetry and flow-visualization techniques. It is found that the azimuthal motion consists of erratic fluctuations and a time-periodic oscillation. The orientation of the wind is found to be ``locked,'' i.e., it fluctuates about a preferred direction most of the time with all other orientations appearing as ``transient states,'' and large excursions of the azimuthal angle often result in a net rotation which takes the wind back to the preferred orientation. The rate of erratic rotation of the circulation plane is found to have a strong dependence on Ra. Our result suggests that the oscillatory motion of the wind in its vertically oriented circulation plane and the orientational oscillation of the circulation plane itself have the same dynamic origin.

Effect of length and inclination of a thin fin on natural convection in a square enclosure

Abstract: A numerical investigation of steady, laminar, natural convective fluid flow in a square enclosure with an inclined heated thin fin of arbitrary length attached to the hot wall is considered. A transverse temperature gradient is applied on two opposing walls of the enclosure, while the other two walls are adiabatic. Attachment of highly conductive inclined thin fins with lengths equal to 20%, 35%, and 50% of the side, positioned in the middle of the hot left wall of the enclosure, is examined. The problem is formulated in terms of the vorticity–stream function procedure. A numerical solution based on the finite-volume method is obtained. Representative results illustrating the effects of the thin-fin inclination angle and length on the streamlines and temperature contours within the enclosure are reported. In addition, results for the local and average Nusselt numbers at the heated wall of the enclosure are presented and discussed for various parametric conditions. It is found that the Rayleigh number and ...

Turbulent convection at high Rayleigh numbers and aspect ratio 4

Abstract: We report measurements of the Nusselt number, Nu, in turbulent thermal convection in a cylindrical container of aspect ratio 4. The highest Rayleigh number achieved was Ra=2×10 13 . Except for the last half a decade or so of Ra, experimental conditions obey the Boussinesq approximation accurately. For these conditions, the data show that the log Nu-log Ra slope saturates at a value close to 1/3, as observed previously by us in experiments with smaller aspect ratios. The increasing slope over the last half a decade of Ra is inconclusive because the corresponding conditions are non-Boussinesq. Finally, we report a modified scaling relation between the plume advection frequency and Ra that collapses data for different aspect ratios.

Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh–Bénard convection

Abstract: For the infinite-Prandtl-number limit of the Boussinesq equations, the enhancement of vertical heat transport in Rayleigh-Benard convection, the Nusselt number Nu, is bounded above in terms of the Rayleigh number Ra according to Nu≤0.644× Ra 1/3 [log R a ] 1/3 as R a → ∞. This result follows from the utilization of a novel logarithmic profile in the background method for producing bounds on bulk transport, together with new estimates for the bi-Laplacian in a weighted L 2 space. It is a quantitative improvement of the best currently available analytic result, and it comes within the logarithmic factor of the pure 1/3 scaling anticipated by both the classical marginally stable boundary layer argument and the most recent high-resolution numerical computations of the optimal bound on Nu using the background method.

Transitional Heat Transfer in Plain Horizontal Tubes

Abstract: In this study, the heat transfer behavior in the transition region for plain horizontal tubes under a uniform wall heat flux boundary condition is discussed in detail. In particular, the influence of inlet configuration and free convection superimposed on the forced convection (or mixed convection) at the start and end of the transition region and the magnitude of heat transfer are addressed. The available correlations to predict the heat transfer coefficient in the transition region are reviewed, and their performance are evaluated based on 1290 experimental data points obtained under a wide range of experimental conditions. Appropriate correlations for the mixed and forced convection transition regions are recommended. Finally, a flow regime map for determination of the boundary between forced and mixed convection in horizontal tubes with different inlets is presented.

Global nonlinear stability in porous convection with a thermal non-equilibrium model

Abstract: We show that the global nonlinear stability threshold for convection with a thermal non-equilibrium model is exactly the same as the linear instability boundary. This result is shown to hold for the porous medium equations of Darcy, Forchheimer or Brinkman. This optimal result is important because it shows that linearized instability theory has captured completely the physics of the onset of convection. The equivalence of the linear instability and nonlinear stability boundaries is also demonstrated for thermal convection in a non-equilibrium model with the Darcy law, when the layer rotates with a constant angular velocity about an axis in the same direction as gravity.

An improved MPS method for numerical simulations of convective heat transfer problems

Abstract: An improved moving-particle semi-implicit (MPS) method was developed for numerical simulations of convective heat transfer problems. The MPS method, which is based on particles and their interactions, is a fully Lagrangian particle method for incompressible flows. A new Laplacian model and a new method for treating boundary conditions were proposed to solve numerical difficulties resulting from the original MPS method. Results of several numerical tests show the validity of the improved MPS method with the proposed model and method. The application of the present MPS method to Rayleigh–Benard convection phenomena demonstrated the effectiveness of the proposed model and method on the numerical simulation of convective heat transfer problems. The dependence of the Nusselt number on the Rayleigh number was in good agreement with an empirical formula. The temperature contour and velocity distribution also agree well with the simulation results obtained with other methods. The roll pattern developed in the horizontal fluid layer as well as the convective heat transfer was successfully simulated with three-dimensional MPS calculations. Copyright © 2005 John Wiley & Sons, Ltd.

Yield stress effects on Rayleigh–Bénard convection

Abstract: We examine the effects of a fluid yield stress on the classical Rayleigh-Benard instability between heated parallel plates. The focus is on a qualitative characterization of these flows, by theoretical and computational means. In contrast to Newtonian fluids, we show that these flows are linearly stable at all Rayleigh numbers, Ra, although the usual linear modal stability analysis cannot be performed. Below the critical Rayleigh number for energy stability of a Newtonian fluid, Ra E , the Bingham fluid is also globally asymptotically stable. Above Ra E , we provide stability bounds that are conditional on Ra - Ra E , as well as on the Bingham number B, the Prandtl number Pr, and the magnitude of the initial perturbation. The stability characteristics therefore differ considerably from those for a Newtonian fluid. A second important way in which the yield stress affects the flow is that when the flow is asymptotically stable, the velocity perturbation decays to zero in a finite time. We are able to provide estimates for the stopping time for the various types of stability. A consequence of the finite time decay is that the temperature perturbation decays on two distinctly different time scales, i.e. before/after natural convection stops. The two decay time scales are clearly observed in our computational results. We are also able to determine approximate marginal stability parameters via computation, when in the conditional stability regime, although computation is not ideal for this purpose. When just above the marginal stability limits, perturbations grow into a self-sustained cellular motion that appears to resemble closely the Newtonian secondary motion, i.e. Rayleigh-Benard cells. When stable, however, the decaying flow pattern is distinctly different to that of a Newtonian perturbation. As t → ∞, a stable Newtonian perturbation decays exponentially and asymptotically resembles the least stable eigenfunction of the linearized problem. By contrast, as t approaches its stopping value, the Bingham fluid is characterized by growth of a slowly rotating (almost) unyielded core within each convection cell, with fully yielded fluid contained in a progressively narrow layer surrounding the core. Finally, preliminary analyses and remarks are made concerning extension of our results to inclined channels, stability of three-dimensional flows and the inclusion of residual stresses in the analysis.

Free convection and radiation characteristics for a vertical plate embedded in a porous medium

Abstract: Steady two-dimensional free convection flow due to combined effect of radiation and convection through a porous medium bounded by a vertical infinite plate is considered. The behaviour of Darcy and non-Darcy flow is investigated. The flow of water through different porous media under different environmental conditions is discussed. Effect of four non-dimensional parameters, i.e. Prandtl number (Pr), modified Grashof number (G), permeability parameter (K) and radiation parameter (N) has been studied. Effect of these parameters on Nusselt number is analysed. Copyright © 2005 John Wiley & Sons, Ltd.

Rayleigh-Taylor turbulence in two dimensions.

Abstract: The first consistent phenomenological theory for two- and three-dimensional Rayleigh-Taylor (RT) turbulence has recently been presented by Chertkov [Phys. Rev. Lett. 91, 115001 (2003)]. By means of direct numerical simulations, we confirm the spatiotemporal prediction of the theory in two dimensions and explore the breakdown of the phenomenological description due to intermittency effects. We show that small-scale statistics of velocity and temperature follow Bolgiano-Obukhov scaling. At the level of global observables, we show that the time-dependent Nusselt and Reynolds numbers scale as the square root of the Rayleigh number. These results point to the conclusion that RT turbulence in two and three dimensions, thanks to the absence of boundaries, provides a natural physical realization of the Kraichnan scaling regime hitherto associated with the elusive ``ultimate state of thermal convection.''

Rayleigh-Bénard Convection: Thirty Years of Experimental, Theoretical, and Modeling Work

Abstract: A brief review of Rayleigh-Benard studies performed during the twentieth century is presented, with an emphasis on the transition to turbulence and the appropriate theoretical framework, relying on the strength of confinement effects and the distance to threshold, either dynamical systems for temporal chaos in the strongly confined case, or models of space-time chaos when confinement effects are weak.

Exponentially growing solutions in homogeneous Rayleigh-Bénard convection.

Abstract: It is shown that homogeneous Rayleigh-Benard flow, i.e., Rayleigh-Benard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient, has a family of exact, exponentially growing, separable solutions of the full nonlinear system of equations. These solutions are clearly manifest in numerical simulations above a computable critical value of the Rayleigh number. In our numerical simulations they are subject to secondary numerical noise and resolution dependent instabilities that limit their growth to produce statistically steady turbulent transport.