Heat transport and flow structure in rotating Rayleigh–Bénard convection
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 for the velocity field , the kinematic pressure field , and the temperature field 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 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 , , and . 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 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 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 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.
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Present address: Department of Mechanical Engineering, Johns Hopkins University, Baltimore, Maryland 21218, USA.