Dynamics of reorientations and reversals of large scale flow in Rayleigh-Benard convection

arXiv:1003.2102v4 [physics.flu-dyn] 26 Aug 2010 Under consideration for publication in J. Fluid Mech. 1 Dynamics of reorientations and reversals of large scale flow in Rayleigh-B´enard convection P. K. M I S H R A1, A. K. D E2, M. K. V E R M A1AND V. E S W A R A N3 1Department of Physics, Indian Institute of Technology, Kanpur 208 016, India 2Department of Mechanical Engineering, Indian Institute of Technology Guwahati 781039, India 3Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016, India (Received 13 October 2018) We present a numerical study of the reversals and reorientations of the large scale cir- culation (LSC) of convective fluid in a cylindrical container of aspect ratio one. We take Prandtl number to be 0.7 and Rayleigh numbers in the range from 6 × 105 to 3 × 107. It is observed that the reversals of the LSC are induced by its reorientation along the azimuthal direction, which are quantified using the phases of the first Fourier mode of the vertical velocity measured near the lateral surface in the mid plane. During a “complete reversal”, the above phase changes by around 1800 leading to reversals of the vertical velocity at all the probes. On the contrary, the vertical velocity reverses only at some of the probes during a “partial reversal” with phase change other than 1800. Numeri- cally we observe rotation-led and cessation-led reorientations, in agreement with earlier experimental results. The ratio of the amplitude of the second Fourier mode and the first Fourier mode rises sharply during the cessation-led reorientations. This observation is consistent with the quadrupolar dominant temperature profile observed during the cessations. We also observe reorientations involving double cessation. Key Words: Rayleigh-B´enard Convection, Convective Turbulence, Direct Numerical Simulation. 1. Introduction Turbulent convection is ubiquitous in nature and in many engineering applications. Rayleigh-B´enard Convection (RBC) in which fluid confined between two plates is heated from below and cooled on the top is an idealized yet an important paradigm to understand convective turbulence. The dynamics of RBC is governed by the two non-dimensional parameters: the Rayleigh number R = α∆T d3g/νκ and the Prandtl number P = ν/κ, where d is the vertical height of the container, g is the acceleration due to gravity, ∆T is the temperature difference between the bottom and top plates, and α, κ, and ν are the thermal heat expansion coefficient, thermal diffusivity, and kinematic viscosity, respectively, of the fluid. Krishnamurti & Howard (1981) performed experiments on water (P ≃7.0) and silicon oil (P ∼860) and observed coherent roll structures, also known as “large scale circula- tion” (LSC), in the turbulent regime. Subsequently, Castaing et al. (1989) ascertained the 2 P. K. Mishra, A. K. De, M. K. Verma, and V. Eswaran existence of LSC in Helium (P ≃0.65-1.5) contained in a cylindrical container. They pro- posed that coherent large scale structures exist statistically only above a certain Rayleigh Number (R ≃108). They also observed a low frequency peak in the power spectrum of the temperature field. Xi, Lam, & Xia (2004) studied the onset of large-scale coherent mean flow in RBC using shadowgraph and particle image velocimetry techniques and showed that LSC is a result of the organization of plume motion. Cioni, Ciliberto, & Sommeria (1997) performed RBC experiments on mercury (0.021 < P < 0.026) and placed several thermistors along the azimuth of the cylinder. They de- duced the presence of global circulation from the dipolar temperature distribution mea- sured by the probes. The temperature fluctuations (after subtracting the mean) switched sign randomly in their experiment. Since the warmer fluid ascends from one side, and the cooler fluid descends from the other side of the apparatus, Cioni, Ciliberto, & Sommeria (1997) deduced that the vertical velocity would also exhibit random “reversals” in phase with the temperature fluctuations. This feature of convection has been studied extensively using theoretical, experimental, and computational tools (see reviews by Ahlers, Grossmann, & Lohse 2009; Kadanoff2001). In this paper we perform computational investigation of the re- versal dynamics of LSC in a cylindrical geometry. Cioni, Ciliberto, & Sommeria (1997) computed the first Fourier mode of the measured temperature field. They observed that the amplitude of the first Fourier mode never vanishes, but the phase of the Fourier mode is highly variable. A phase change of π cor- responds to the reversal of the flow. Cioni, Ciliberto, & Sommeria (1997) also observed a low-frequency peak in the energy spectrum that corresponds to the circulation frequency of the large-scale flow inside the cylinder. Tsuji et al. (2005) performed an experiment with mercury contained in cylindrical container for aspect ratios 1/2, 1, and 2 and found that the low frequency peak is absent for aspect ratio 1/2. Niemela et al.

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