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# The influence of aspect ratio on flow states in the buoyancy-driven turbulence with free slip boundaries

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## Abstract

Buoyancy driven turbulence with free-slip top and bottom plates in a confined box is studied via direct numerical simulations (DNS). The Rayleigh number is fixed to Ra=108 and the Prandtl number is Pr=10. The length/height aspect ratio Γy is fixed to 2, while the depth/height aspect ratio Γx varies from 0.125 to 0.3. With the variation of Γx, the hysteresis-like behavior of the heat and momentum transfer is found due to the emergence of two stable states of flow organization. For small Γx, the sheared flow occurs with disordered plumes occupying the entire bulk region, while it is replaced by convection rolls with increasing Γx. The striking feature is that the heat and momentum transport is greatly suppressed in the shear flow (SF) state, compared with that in the convection roll (CR) state. The turbulent kinetic energy budget is performed to delineate the relative contributions of physical processes that govern energy transport. It is found that the kinetic energy in the SF state is mainly produced by the buoyancy force, while it is dominated by the shear production near the plates in the CR state. During energy transport, it seems that the buoyancy force plays a more important role in the SF state than in the CR state, particularly in the bulk. Furthermore, through the analysis of dynamic mode decomposition of temperature fields, we illustrate that the flow in the SF state is dominated by the cloud-like spatially coherent structures with low frequencies, while the low-frequency roll structures play a leading role in the CR state. The flow state transition from SF to CR corresponds to the process that the cloud-like coherent structures form roll structures.

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§1. We shall denote by u α ( P ) = u α ( x 1 , x 2 , x 3 , t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 , x 2 , x 3 . In considering the turbulence it is natural to assume the components of the velocity u α ( P ) at every point P = ( x 1 , x 2 , x 3 , t ) of the considered domain G of the four-dimensional space ( x 1 , x 2 , x 3 , t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ ² α and (d u α /d x β ) ² ― are finite and bounded in every bounded subdomain of the domain G .
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We report on a numerical study of the aspect-ratio dependency of Rayleigh-Bénard convection, using direct numerical simulations. The investigated domains have equal height and width while the aspect ratio Γ of depth per height is varied between 1/10 and 1. The Rayleigh numbers Ra for this study variate between 105 and 109, while the Prandtl number is Pr = 0.786. The main focus of the study concerns the dependency of the Nusselt number Nu and the Reynolds number Re on Ra and Γ. It turns out that due to Γ, differences to the cubic case (i.e., Γ = 1) in Nu of up to 55% and in Re of up to 97% occur, which decrease for increasing Ra. In particular for small Γ sudden drops in the Ra-scaling of Nu and Re appear for Ra ≈ 106. Further analysis reveals that these correspond to the onset of unsteady motion accompanied by changes in the global flow structure. The latter is investigated by statistical analysis of the heat flux distribution on the bottom and top plates and a decomposition of the instantaneous flow fields into two-dimensional modes. For Ra slightly above the onset of unsteady motion (i.e., Ra ≈ 106) for all considered Γ ⩽ 1/3 a four-roll structure is present, which corresponds to thermal plumes moving vertically through the domain's center. For Ra ≥ 107, also for small Γ, a single-roll structure is dominant, in agreement with two-dimensional simulations and experiments at larger Ra and Pr.
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