Diffusion-Free Scaling in Rotating Spherical Rayleigh-Bénard Convection (original) (raw)

Rapidly rotating turbulent Rayleigh-Bénard convection

Journal of Fluid Mechanics, 1996

Turbulent Boussinesq convection under the influence of rapid rotation (i.e. with comparable characteristic rotation and convection timescales) is studied. The transition to turbulence proceeds through a relatively simple bifurcation sequence, starting with unstable convection rolls at moderate Rayleigh (Ra) and Taylor numbers (Ta) and culminating in a state dominated by coherent plume structures at high Ra and Ta. Like non-rotating turbulent convection, the rapidly rotating state exhibits a simple power-law dependence on Ra for all statistical properties of the flow. When the fluid layer is bounded by no-slip surfaces, the convective heat transport (Nu -1, where Nu is the Nusselt number) exhibits scaling with Ra'17 similar to non-rotating laboratory experiments. When the boundaries are stress free, the heat transport obeys 'classical' scaling (Ra'13) for a limited range in Ra, then appears to undergo a transition to a different law at Ra = 4 x lo7. Important dynamical differences between rotating and non-rotating convection are observed: aside from the (expected) differences in the boundary layers due to Ekman pumping effects, angular momentum conservation forces all plume structures created at flow-convergent sites of the heated and cooled boundaries to spin-up cyclonically; the resulting plume/cyclones undergo strong vortex-vortex interactions which dramatically alter the mean state of the flow and result in a finite background temperature gradient as Ra -+ 00, holding Ra/Ta fixed.

Heat transport and flow structure in rotating Rayleigh–Bénard convection

European Journal of Mechanics - B/Fluids, 2013

Here we summarize the results from our direct numerical simulations (DNS) and experimental measurements on rotating Rayleigh-Bénard (RB) convection. Our experiments and simulations are performed in a cylindrical samples with aspect ratio of 0.5 ≤ Γ ≤ 2.0. Here Γ = D/L with D and L are the diameter and height of the sample, respectively. When the rotation rate is increased, while a fixed temperature difference between the hot bottom and cold top plate is maintained, a sharp increase in the heat transfer is observed before the heat transfer drops drastically at stronger rotation rates. Here we focus on the question of how the heat transfer enhancement with respect to the non-rotating case depends on the Rayleigh number Ra, the Prandtl number Pr, and the rotation rate, indicated by the Rossby number Ro. Special attention will be given to influence of the aspect ratio on rotation rate that is required to get heat transport enhancement. In addition, we will discuss the relation between the heat transfer and the large scale flow structures that are formed in the different regimes of rotating RB convection and how the different regimes can be identified in experiments and simulations.

Numerical simulations of rotating Rayleigh-Benard convection

Chemometrics and Intelligent Laboratory Systems, 2011

The Rayleigh-Bénard (RB) system is relevant to astro-and geophysical phenomena, including convection in the ocean, the Earth's outer core, and the outer layer of the Sun. The dimensionless heat transfer (the Nusselt number Nu) in the system depends on the Rayleigh number Ra = β gΔ L 3 /(νκ) and the Prandtl number Pr = ν/κ. Here, β is the thermal expansion coefficient, g the gravitational acceleration, Δ the temperature difference between the bottom and top, and ν and κ the kinematic viscosity and the thermal diffusivity, respectively. The rotation rate H is used in the form of the Rossby number Ro = (β gΔ /L)/(2H). The key question is: How does the heat transfer depend on rotation and the other two control parameters: Nu(Ra, Pr, Ro)? Here we will answer this question by giving a summary of our results presented in [1, 2, 3].

Dynamics of large-scale quantities in Rayleigh-Bénard convection

Physical Review E, 2016

In this paper we estimate the relative strengths of various terms of the Rayleigh-Bénard equations. Based on these estimates and scaling analysis, we derive a general formula for the large-scale velocity, U , or the Péclet number that is applicable for arbitrary Rayleigh number Ra and Prandtl number Pr. Our formula fits reasonably well with the earlier simulation and experimental results. Our analysis also shows that the wall-bounded convection has enhanced viscous force compared to free turbulence. We also demonstrate how correlations deviate the Nusselt number scaling from the theoretical prediction of Ra 1/2 to the experimentally observed scaling of nearly Ra 0.3 .

Scaling of large-scale quantities in Rayleigh-Bénard convection

2016

We derive a formula for the Péclet number (Pe) by estimating the relative strengths of various terms of the momentum equation. Using direct numerical simulations in three dimensions we show that in the turbulent regime, the fluid acceleration is dominated by the pressure gradient, with relatively small contributions arising from the buoyancy and the viscous term, in the viscous regime, acceleration is very small due to a balance between the buoyancy and the viscous term. Our formula for Pe describes the past experiments and numerical data quite well. We also show that the ratio of the nonlinear term and the viscous term is ReRa^-0.14, where Re and Ra are Reynolds and Rayleigh numbers respectively, and that the viscous dissipation rate ϵ_u = (U^3/d) Ra^-0.21, where U is the root mean square velocity and d is the distance between the two horizontal plates. The aforementioned decrease in nonlinearity compared to free turbulence arises due to the wall effects.

What rotation rate maximizes heat transport in rotating Rayleigh-Bénard convection

APS Division of Fluid Dynamics Meeting Abstracts, 2020

The heat transfer and flow structure in rotating Rayleigh-Bénard convection are strongly influenced by the Rayleigh (Ra), Prandtl (P r), and Rossby (Ro) number. For P r 1 and intermediate rotation rates, the heat transfer is increased compared to the non-rotating case. We find that the regime of increased heat transfer is subdivided into a low and a high Ra number regime. For Ra 5 × 10 8 the heat transfer at a given Ra and P r is highest at an optimal rotation rate, at which the thickness of the viscous and thermal boundary layer is about equal. From the scaling relations of the thermal and viscous boundary layer thicknesses, we derive that the optimal rotation rate scales as 1/Roopt ≈ 0.12P r 1/2 Ra 1/6. In the low Ra regime the heat transfer is similar in a periodic domain and cylindrical cells with different aspect ratios, i.e. the ratio of diameter to height. This is consistent with the view that the vertically aligned vortices are the dominant flow structure. For Ra 5 × 10 8 the above scaling for the optimal rotation rate does not hold anymore. It turns out that in the high Ra regime, the flow structures at the optimal rotation rate are very different than for lower Ra. Surprisingly, the heat transfer in the high Ra regime differs significantly for a periodic domain and cylindrical cells with different aspect ratios, which originates from the sidewall boundary layer dynamics and the corresponding secondary circulation.

Heat Transport in Low-Rossby-Number Rayleigh-Bénard Convection

Physical Review Letters, 2012

We demonstrate, via simulations of asymptotically reduced equations describing rotationally constrained Rayleigh-Bénard convection, that efficiency of turbulent motion in the fluid bulk limits overall heat transport and determines the scaling of the nondimensional Nusselt number Nu with the Rayleigh number Ra, the Ekman number E and the Prandtl number σ. For E ≪ 1 inviscid scaling theory predicts and simulations confirm the large Ra scaling law Nu − 1 ≈ C1 σ − 1 2 Ra 3 2 E 2 , where C1 is a constant, estimated as C1 ≈ 0.04 ± 0.0025. In contrast, the corresponding result for nonrotating convection, Nu − 1 ≈ C2Ra α is determined by the efficiency of the thermal boundary layers (laminar: 0.28 α 0.31, turbulent: α ∼ 0.38). The 3/2 scaling law breaks down at Rayleigh numbers at which the thermal boundary layer loses rotational constraint, i.e., when the local Rossby number ≈ 1. The breakdown takes place while the bulk Rossby number is still small and results in a gradual transition to the nonrotating scaling law. For low Ekman numbers the location of this transition is independent of the mechanical boundary conditions.

Breakdown of the large-scale circulation in Γ=1/2 rotating Rayleigh-Bénard flow

Physical Review E, 2012

Experiments and simulations of rotating Rayleigh-Bénard convection in cylindrical samples have revealed an increase in heat transport with increasing rotation rate. This heat transport enhancement is intimately related to a transition in the turbulent flow structure from a regime dominated by a large-scale circulation (LSC), consisting of a single convection roll, at no or weak rotation to a regime dominated by vertically-aligned vortices at strong rotation. For a sample with an aspect ratio Γ = D/L = 1 (D is the sample diameter and L its height) the transition between the two regimes is indicated by a strong decrease in the LSC strength. In contrast, for Γ = 1/2 Weiss and Ahlers [J. Fluid Mech. 688, 461 (2011)] revealed the presence of a LSC-like sidewall temperature signature beyond the critical rotation rate. They suggested that this might be due to the formation of a two-vortex state, in which one vortex extends vertically from the bottom into the sample interior and brings up warm fluid, while another vortex brings down cold fluid from the top; this flow field would yield a sidewall temperature signature similar to that of the LSC. Here we show by direct numerical simulations for Γ = 1/2 and parameters that allow direct comparison with experiment that the spatial organization of the vertically-aligned vortical structures in the convection cell do indeed yield (for the time average) a sinusoidal variation of the temperature near the sidewall, as found in the experiment. This is also the essential and non-trivial difference with the Γ = 1 sample, where the vertically-aligned vortices are distributed randomly.

Traveling concentric-roll patterns in Rayleigh-Bénard convection with modulated rotation

Physical Review E, 2002

We present experimental results for pattern formation in Rayleigh-Bénard convection with modulated rotation about a vertical axis. The dimensionless rotation rate ⍀ was varied as ⍀ m ϭ⍀͓1ϩ␦ cos(⍀t)͔ ͑time is scaled by the vertical viscous diffusion time of the cell͒. We used a cylindrical cell of aspect ratio ͑radius/ height͒ ⌫ϭ11.8 and varied ⍀, ␦, , and ⑀ϵR/R c (⍀)Ϫ1 (R is the Rayleigh number͒. The fluid was water with a Prandtl number of 4.5. Sufficiently far above onset even a small ␦տ0.02 stabilized a concentric-roll ͑target͒ pattern. Multiarmed spirals were observed close to onset. The rolls of the target patterns traveled radially inward independent of the sense of rotation. The radial speed v was nearly independent of ⑀ for fixed ⍀, ␦, and. However, v increased with any one of ⍀, ␦, and when all the other parameters were held fixed.

Turbulent Rotating Rayleigh–Benard Convection: Spatiotemporal and Statistical Study

Journal of Heat Transfer-transactions of The Asme, 2009

The present study involves a 3D numerical investigation of rotating Rayleigh-Benard convection in a large aspect-ratio (8:8:1) rectangular enclosure. The rectangular cavity is rotated about a vertical axis passing through the center of the cavity. The governing equations of mass, momentum, and energy for a frame rotating with the enclosure, subject to generalized Boussinesq approximation applied to the body and centrifugal force terms, have been solved on a collocated grid using a semi-implicit finite difference technique. The simulations have been carried out for liquid metal flows having a fixed Prandtl number Prϭ 0.01 and fixed Rayleigh number Raϭ 10 7 while rotational Rayleigh number Ra w and Taylor number Ta are varied through nondimensional rotation rate ͑⍀͒ ranging from 0 to 10 4 . Generation of large-scale structures is observed at low-rotation ͑⍀ ϭ 10͒ rates though at higher-rotation rates ͑⍀ ϭ 10 4 ͒ the increase in magnitude of Coriolis forces leads to redistribution of buoyancy-induced vertical kinetic energy to horizontal kinetic energy. This brings about inhibition of vertical fluid transport, thereby leading to reduced vertical heat transfer. The magnitude of rms velocities remains unaffected with an increase in Coriolis forces from ⍀ ϭ 0 to 10 4 . An increase in rotational buoyancy ͑Ra w ͒, at constant rotation rate ͑⍀ ϭ 10 4 ͒, on variation in Ra w / Ta from 10 Ϫ3 to 10 Ϫ2 results in enhanced breakup of large-scale structures with a consequent decrease in rms velocities but with negligible reduction in vertical heat transport.