Identification of the wind in Rayleigh–Bénard convection

Abstract

Using a symmetry-accounting ensemble-averaging method, we have identified the wind in unbounded Rayleigh-Bénard convection. This makes it possible to distinguish the wind from fluctuations and to identify dynamic features of each. We present some results from processing five independent three-dimensional direct numerical simulations of a ⌫ = 4 aspect-ratio domain with periodic side boundaries at Ra=10 7 and Pr= 1. It is found that the wind boundary layer scales linearly very close to the wall and has a logarithmic region further away. Despite the still noticeable molecular effects, the identification of log scaling and significant velocity and temperature fluctuations well within the thermal boundary layer clearly indicate that the boundary layer cannot be classified as laminar. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1920350͔Recent theories for the prediction of the scaling behavior of the nondimensional heat flux Nu and the Reynolds number Re in the Rayleigh-Bénard convection presume the existence of "wind," 1,2 a large scale circulation that autonomously arises in the system. Indeed, this circulation has been observed by various groups and received quite some attention over the last years. [3][4][5][6][7][8][9] The theories use wind primarily as a conceptual notion; little is known about the actual magnitude of the wind as compared to velocity fluctuations and if it is present for all Rayleigh numbers. 9 But most important, a straightforward definition of wind seems to be missing. The natural candidates are the ensemble average, a time average over a sufficiently long period, a spatial average over a sufficiently large area, and a combination of the three. However, the problem is that all these averages yield a vanishing wind ū = 0. For the ensemble average and the spatial average this may be obvious from symmetry considerations ͑see below͒, but for the time average it is more subtle. In this context the experiments 5,9 are most instructive. Here it is shown that the wind erratically reverses its direction on time scales far exceeding the convective turnover time, Sreenivasan, Bershadskii, and Niemela 9 propose a simple dynamical picture to explain these wind reversals, with two metastable states, i.e., clockwise and counterclockwise mean wind motions; only extreme events, such as ͑a combination of͒ energetic plumes, can flip the system from one state to another. The relevant point here is that the reversals actually conserve the ergodicity of the system: the ͑long͒ time average becomes equal to the ensemble average-which is zero. The purpose of this letter is therefore to come up with a suitable and useful definition of wind. Based on this definition we then determine the wind based on five direct numerical simulations ͑DNS͒ of moderate aspect ratio and with periodic side boundaries. An analysis of the wind field and fluctuations shows evidence of a boundary layer that is not laminar.The Rayleigh-Bénard convection is generated when a layer of fluid with thickness H is subjected to a positive te...

Large-scale patterns in Rayleigh–Bénard convection

“…An interesting issue concerns the possible presence of energy in the Krishnamurti-Howard flow at k = 0, which is allowed by periodic boundary conditions. In our previous work [21], as in Hartlep et al [20,22] and van Reeuwijk et al [23], we found no evidence for the excitation of the k = 0 mode. The same is observed here, even with much larger values of the aspect ratio.…”

“…These structures have approximately circular horizontal cross-sections with diameters comparable to the boundary layer thickness and they typically traverse the full body of the fluid [1,2]. When the aspect ratio of the container is moderately large (horizontal domain size of about 4 to 6 times the layer depth) the plumes are found to congregate into patterns associated with large-scale circulations that have been studied experimentally [3][4][5][6][7][8][9][10], analytically [11][12][13][14] and numerically [15][16][17][18][19][20][21][22][23]. The dependence of the properties of turbulent convection and of large-scale circulations on aspect * Corresponding author.…”

Physical and scale-by-scale analysis of Rayleigh–Bénard convection

“…The fact that both the Kolmogorov and Yaglom equations are satisfied is also evidence of the fact that the terms associated with the possible presence of a mean flow are negligible. Indeed, it should be pointed out that a mean velocity field would exist locally in time and space due to the presence of large-scale circulations which erratically reverse their directions on time scales exceeding the large eddy turnover time as reported in Van Reeuwijk, Jonker & Hanjalić (2005), Brown & Ahlers (2006) and Xi, Zhou & Xia (2006). It is well known that by increasing the Rayleigh number the characteristic length scales of the problem decrease.…”

Probing the energy cascade of convective turbulence

“…There is an inhomogeneous flow structure that causes the discrepancy: just as in Rayleigh-Bénard convection in cylindrical domains, convection in a horizontally periodic domain is known to organize into a domain-filling circulation pattern, e.g., Refs. [42,45,46]. This circulation forms with a random orientation and then changes its orientation only very slowly.…”