Abstract
A novel approach for the study of turbulent Rayleigh–Bénard convection (RBC) in the compound physical/scale space domain is presented. All data come from direct numerical simulations of turbulent RBC in a laterally unbounded domain confined between two horizontal walls, for Prandtl number $0.7$ and Rayleigh numbers $1.7\times 10^{5}$, $1.0\times 10^{6}$ and $1.0\times 10^{7}$. A preliminary analysis of the flow topology focuses on the events of impingement and emission of thermal plumes, which are identified here in terms of the horizontal divergence of the instantaneous velocity field. The flow dynamics is then described in more detail in terms of turbulent kinetic energy and temperature variance budgets. Three distinct regions where turbulent fluctuations are produced, transferred and finally dissipated are identified: a bulk region, a transitional layer and a boundary layer. A description of turbulent RBC dynamics in both physical and scale space is finally presented, completing the classic single-point balances. Detailed scale-by-scale budgets for the second-order velocity and temperature structure functions are shown for different geometrical locations. An unexpected behaviour is observed in both the viscous and thermal transitional layers consisting of a diffusive reverse transfer from small to large scales of velocity and temperature fluctuations. Through the analysis of the instantaneous field in terms of the horizontal divergence, it is found that the enlargement of thermal plumes following the impingement represents the triggering mechanism which entails the reverse transfer. The coupling of this reverse transfer with the spatial transport towards the wall is an interesting mechanism found at the basis of some peculiar aspects of the flow. As an example, it is found that, during the impingement, the presence of the wall is felt by the plumes through the pressure field mainly at large scales. These and other peculiar aspects shed light on the role of thermal plumes in the self-sustained cycle of turbulence in RBC, and may have strong repercussions on both theoretical and modelling approaches to convective turbulence.
Highlights
Driven turbulence plays a major role in several natural phenomena, from oceanic and atmospheric circulations to convection in stars and planets
Apparent is the importance of turbulent thermal convection in industrial processes, which range from passive cooling of nuclear reactors to the controlled growth of crystals from the melt (Ahlers, Grossmann & Lohse 2009)
A canonical flow for approaching all the cases above is Rayleigh–Bénard convection (RBC), which consists of a fluid layer heated from below and cooled from above in a vertically bounded domain (Siggia 1994)
Summary
Driven turbulence plays a major role in several natural phenomena, from oceanic and atmospheric circulations to convection in stars and planets. The current literature shows many useful analyses of turbulent RBC in terms of the budgets of turbulent kinetic and temperature variance (Deardorff & Willis 1967; Wörner & Grötzbach 1998) and, more recently, of mechanical energy budgets (Gayen, Hughes & Griffiths 2013) These approaches are limited to a description in physical space alone. The present work completes the above-mentioned approaches by analysing, for the first time, turbulent RBC in the compound physical/scale space domain To this end, generalized forms of both the Kolmogorov (Hill 2002) and Yaglom equations are employed. Equations and numerical method The governing equations for the RBC are the momentum, continuity and temperature equations in the Boussinesq approximation,
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