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

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 × 10 . 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.

Resolved energy budget of superstructures in Rayleigh–Bénard convection

“…Therefore it effectively increases the dissipation, as it does for the global balance. However, there is an inverse energy transfer from the unresolved to the resolved scales near the wall in agreement with the results on RBC (Togni et al 2015(Togni et al , 2017(Togni et al , 2019 and other wallbounded flows (Domaradzki et al 1994;Marati et al 2004;Cimarelli & De Angelis 2011Cimarelli et al 2015). This shows that the boundary layers are important for the dynamics of the superstructures since additional energy from the unresolved scales is provided there.…”

“…We then apply a filtering approach (Germano 1992) to isolate the superstructure dynamics. The scale-resolved energy and thermal variance budgets of convective flows have previously been studied by Kimmel & Domaradzki (2000); Togni et al (2017Togni et al ( , 2019 with respect to large eddy simulation models for small scales and by Togni et al (2015) using velocity and temperature increment statistics. These studies revealed an inverse energy transfer from smaller to larger scales close to the wall, which is closely connected to the enlargement of plumes during impinging.…”

Weak formulation and scaling properties of energy fluxes in three-dimensional numerical turbulent Rayleigh–Bénard convection

“…Instead, in the homogeneous case it is found the contrary which may lead to the impossibility to observe a Bolgiano-Oboukhov scaling (Verma et al 2017). (c) The contribution of the buoyancy coupling term is found to be important at all scales as reported previously (Togni et al 2015), however its relative importance with respect to other terms differs. Notably, it is found to be dominant in the temperature budget at almost all scales, whereas the inertial term of the kinetic energy budget is is the most important term at small scales.…”

“…The way how these limits are achieved is however informative about the scaling properties of the flow, as is shown in Section 3.4. When averaged, equations (3.2)-(3.3) give the general forms of the mean energy and temperature budgets, and they are interesting since they provide a scale-byscale way to analyse turbulent flows, as highlighted in several recent works focused on anisotropic turbulent flows (Hill 1997;Danaila et al 1999;Rincon 2006;Cimarelli et al 2013;Gauding et al 2014;Togni et al 2015;Mollicone et al 2018).…”

Coherence of temperature and velocity superstructures in turbulent Rayleigh–Bénard flow

“…In order to uncover the origin of the different spectral distribution of temperature and vertical velocity that became apparent in figure 2a,b, we now study the variance production terms of the respective variance budgets. These production terms are (Deardorff & Willis 1967;Kerr 2001;Togni et al 2015) S θ = −2 θw dΘ dz (3.3)…”