Elsevier

European Journal of Mechanics - B/Fluids

Volume 40, July–August 2013, Pages 41-49
European Journal of Mechanics - B/Fluids

Heat transport and flow structure in rotating Rayleigh–Bénard convection

https://doi.org/10.1016/j.euromechflu.2013.01.004Get rights and content

Abstract

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 cylindrical samples with an aspect ratio Γ varying from 1/2 to 2. Here Γ=D/L, where 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 the influence of the aspect ratio on the 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.

Introduction

Rayleigh–Bénard (RB) convection, i.e. the flow of a fluid heated from below and cooled from above, is the classical system to study thermally driven turbulence in confined space [1], [2]. Buoyancy-driven flows play a role in many natural phenomena and technological applications. In many cases the fluid flow is also affected by rotation, for example, in geophysical flows, astrophysical flows, and flows in technology [3]. On Earth, many large-scale fluid motions are driven by temperature-induced buoyancy, while the length scales of these phenomena are large enough to be influenced by the Earth’s rotation. Key examples include the convection in the atmosphere [4] and oceans [5], including the global thermohaline circulation [6]. These natural phenomena are crucial for the Earth’s climate. Rotating thermal convection also plays a significant role in the spontaneous reversals of the Earth’s magnetic field [7]. Rotating RB convection is the relevant model to study the fundamental influence of rotation on thermal convection in order to better understand the basic physics of these problems.

In this paper we discuss the recent progress that has been made in the field of rotating RB convection. First we discuss the dimensionless parameters that are used to describe the system. Subsequently, we give an overview of the parameter regimes in which the heat transport in rotating RB is measured in experiments and direct numerical simulations (DNS). This will be followed by a description of the characteristics of the Nusselt number measurements and a description of the flow structures in the different regimes of rotating RB. Finally, we address how the different turbulent states are identified in experiments and simulations by flow visualization, detection of vortices, and from sidewall temperature measurements.

Section snippets

Rotating RB convection

When a classical RB sample is rotated around its center axis, it is called rotating RB convection. For not too large temperature gradients, this system can be described with the Boussinesq approximation ut+uu+2Ω×u=p+ν2u+βgθzˆ,θt+uθ=κ2θ, for the velocity field u, the kinematic pressure field p, and the temperature field θ relative to some reference temperature. In the Boussinesq approximation it is assumed that the material properties of the fluid such as the thermal expansion

Parameter regimes covered

In Fig. 1 we present the explored RaPrRo parameter space for rotating RB convection.2 Here we emphasize that numerical simulations and experiments on rotating RB convection are complementary, because different aspects of the problem can be addressed. Namely, in accurate experimental measurements of the heat transfer a completely insulated system is needed. Therefore, one cannot visualize the flow while

Nusselt number measurements

Early linear stability analysis, see e.g. Chandrasekhar [30], revealed that rotation has a stabilizing effect due to which the onset of heat transfer is delayed. This can be understood from the thermal wind balance, which implies that convective heat transport parallel to the rotation axis is suppressed. Experimental and numerical investigations concerning the onset of convective heat transfer and the pattern formation in cylindrical cells just above the onset under the influence of rotation

Different turbulent states

When the heat transport enhancement as a function of the rotation rate is considered, a division in three regimes is possible [63], [49], [11]. Here we will call these regimes: regime I (weak rotation), regime II (moderate rotation), and regime III (strong rotation), see Fig. 2. It is well known that without rotation a large scale circulation (LSC) is the dominant flow structure in RB convection, see e.g. Ref. [1] and Fig. 6a. This has motivated Hart, Kittelman & Ohlsen [64], Kunnen et al. [26]

Sidewall temperature measurements

In recent high precision heat transport measurements of rotating RB convection the samples are equipped with 24 thermistors embedded in the sidewall [15], [12], [18], [20], [11]. These thermistors are divided over 3 rings of 8 thermistors that are placed at 0.25z/L, 0.50z/L, and 0.75z/L. In non-rotating convection this arrangement of thermistors is used to determine the orientation of the LSC as the thermistors can detect the location of the upflow (downflow) by registering a relatively high

Determination of vortex statistics

In regime II and III, see Fig. 2, the flow is dominated by vertically-aligned vortices. The experiments of Boubnov & Golitsyn [63], Zhong, Ecke & Steinberg [36], and Sakai [77] first showed with flow visualization experiments that there is a typical spatial ordering of vertically-aligned vortices under the influence of rotation. In general the vertically-aligned vortices prefer to arrange themselves in a checkerboard pattern. This is nicely visualized in Fig. 1 of Ref. [70]. A three-dimensional

Influence of the aspect ratio

In a Γ=1 sample the onset of heat transport enhancement is visible in temperature measurements at the sidewall by a strong decrease of the relative LSC strength, see Fig. 8a. However, this is not the case for all aspect ratio samples. Namely, it was revealed by Weiss & Ahlers [20] that in a Γ=1/2 sample no strong decrease in the relative LSC strength is observed at the moment that the heat transport enhancement sets in, see Fig. 8b. These authors discuss that the relative LSC strength according

Conclusions

We have summarized and discussed recent works on rotating Rayleigh–Bénard (RB) convection and discussed some of our work in more detail. We have seen that a combination of experimental, numerical, and theoretical work has greatly increased our understanding of this problem. As is shown in the rotating RB parameter diagrams, see Fig. 1, some parts of the parameter space are still relatively unexplored. We especially note that up to now the rotating high Ra number regime has only been achieved in

Acknowledgments

We benefited form numerous stimulating discussions with Guenter Ahlers, GertJan van Heijst, Rudie Kunnen, Jim Overkamp, Roberto Verzicco, Stephan Weiss, and Jin-Qiang Zhong over the last years. RJAMS was financially supported by the Foundation for Fundamental Research on Matter (FOM), which is part of NWO.

References (83)

  • J. Marshall et al.

    Open-ocean convection: observations, theory, and models

    Rev. Geophys.

    (1999)
  • S. Rahmstorf

    The thermohaline ocean circulation: a system with dangerous thresholds?

    Clim. Change

    (2000)
  • D. Hassler et al.

    Solar wind outflow and the chromospheric magnetic network

    Science

    (1999)
  • M. Lappa

    Rayleigh–Bénard Convection with Rotation

    (2012)
  • R.J.A.M. Stevens et al.

    Effect of aspect-ratio on vortex distribution and heat transfer in rotating Rayleigh–Bénard

    Phys. Rev. E

    (2011)
  • R.P.J. Kunnen et al.

    Enhanced vertical inhomogeneity in turbulent rotating convection

    Phys. Rev. Lett.

    (2008)
  • R.P.J. Kunnen et al.

    The role of Stewartson and Ekman layers in turbulent rotating Rayleigh–Bénard convection

    J. Fluid Mech.

    (2011)
  • J.-Q. Zhong et al.

    Heat transport and the large-scale circulation in rotating turbulent Rayleigh–Bénard convection

    J. Fluid Mech.

    (2010)
  • S. Weiss et al.

    Heat transport by turbulent rotating Rayleigh–Bénard convection and its dependence on the aspect ratio

    J. Fluid Mech.

    (2011)
  • J.E. Hart

    On the influence of centrifugal buoyancy on rotating convection

    J. Fluid Mech.

    (2000)
  • J.-Q. Zhong et al.

    Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent rotating Rayleigh–Bénard convection

    Phys. Rev. Lett.

    (2009)
  • R.J.A.M. Stevens et al.

    Boundary layers in rotating weakly turbulent Rayleigh–Bénard convection

    Phys. Fluids

    (2010)
  • R.J.A.M. Stevens et al.

    Transitions between turbulent states in rotating Rayleigh–Bénard convection

    Phys. Rev. Lett.

    (2009)
  • S. Weiss et al.

    Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh–Bénard convection

    Phys. Rev. Lett.

    (2010)
  • R.J.A.M. Stevens et al.

    Optimal Prandtl number for heat transfer in rotating Rayleigh–Bénard convection

    New J. Phys.

    (2010)
  • S. Weiss et al.

    The large-scale flow structure in turbulent rotating Rayleigh–Bénard convection

    J. Fluid Mech.

    (2011)
  • J. Niemela et al.

    Turbulent rotating convection at high Rayleigh and Taylor numbers

    J. Fluid Mech.

    (2010)
  • S. Schmitz et al.

    Heat transport in rotating convection without Ekman layers

    Phys. Rev. E

    (2009)
  • E.M. King et al.

    Boundary layer control of rotating convection systems

    Nature

    (2009)
  • Y. Liu et al.

    Heat transport measurements in turbulent rotating Rayleigh–Bénard convection

    Phys. Rev. E

    (2009)
  • Y. Liu et al.

    Heat transport scaling in turbulent Rayleigh–Bénard convection: effects of rotation and Prandtl number

    Phys. Rev. Lett.

    (1997)
  • R.P.J. Kunnen et al.

    Breakdown of large-scale circulation in turbulent rotating convection

    Europhys. Lett.

    (2008)
  • R.P.J. Kunnen et al.

    Heat flux intensification by vortical flow localization in rotating convection

    Phys. Rev. E

    (2006)
  • K. Julien et al.

    Rapidly rotating Rayleigh–Bénard convection

    J. Fluid Mech.

    (1996)
  • S. Chandrasekhar

    Hydrodynamic and Hydromagnetic Stability

    (1981)
  • Y. Nakagawa et al.

    A theoretical and experimental study of cellular convection in rotating fluids

    Tellus

    (1955)
  • P.J. Lucas et al.

    Stability and heat transfer of rotating cryogens. Part 1. Influence of rotation on the onset of convection in liquid 4He

    J. Fluid Mech.

    (1983)
  • J.M. Pfotenhauer et al.

    Stability and heat transfer of rotating cryogens. Part 2. Effects of rotation on heat-transfer properties of convection in liquid He

    J. Fluid Mech.

    (1984)
  • J. Pfotenhauer et al.

    Stability and heat transfer of rotating cryogens. Part 3. Effects of finite cylindrical geometry and rotation on the onset of convection

    J. Fluid Mech.

    (1987)
  • F. Zhong et al.

    Asymmetric modes and the transition to vortex structures in rotating Rayleigh–Bénard convection

    Phys. Rev. Lett.

    (1991)
  • F. Zhong et al.

    Rotating Rayleigh–Bénard convection: asymmetrix modes and vortex states

    J. Fluid Mech.

    (1993)
  • Cited by (105)

    View all citing articles on Scopus
    1

    Present address: Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA.

    View full text