Resolved and subgrid dynamics of Rayleigh–Bénard convection

Abstract

In this work we present and demonstrate the reliability of a theoretical framework for the study of thermally driven turbulence. It consists of scale-by-scale budget equations for the second-order velocity and temperature structure functions and their limiting cases, represented by the turbulent kinetic energy and temperature variance budgets. This framework represents an extension of the classical Kolmogorov and Yaglom equations to inhomogeneous and anisotropic flows, and allows for a novel assessment of the turbulent processes occurring at different scales and locations in the fluid domain. Two relevant characteristic scales, $\ell _{c}^{u}$ for the velocity field and $\ell _{c}^{\unicode[STIX]{x1D703}}$ for the temperature field, are identified. These variables separate the space of scales into a quasi-homogeneous range, characterized by turbulent kinetic energy and temperature variance cascades towards dissipation, and an inhomogeneity-dominated range, where the production and the transport in physical space are important. This theoretical framework is then extended to the context of large-eddy simulation to quantify the effect of a low-pass filtering operation on both resolved and subgrid dynamics of turbulent Rayleigh–Bénard convection. It consists of single-point and scale-by-scale budget equations for the filtered velocity and temperature fields. To evaluate the effect of the filter length $\ell _{F}$ on the resolved and subgrid dynamics, the velocity and temperature fields obtained from a direct numerical simulation are split into filtered and residual components using a spectral cutoff filter. It is found that when $\ell _{F}$ is smaller than the minimum values of the cross-over scales given by $\ell _{c,min}^{\unicode[STIX]{x1D703}\ast }=\ell _{c,min}^{\unicode[STIX]{x1D703}}Nu/H=0.8$, the resolved processes correspond to the exact ones, except for a depletion of viscous and thermal dissipations, and the only role of the subgrid scales is to drain turbulent kinetic energy and temperature variance to dissipate them. On the other hand, the resolved dynamics is much poorer in the near-wall region and the effects of the subgrid scales are more complex for filter lengths of the order of $\ell _{F}\approx 3\ell _{c,min}^{\unicode[STIX]{x1D703}}$ or larger. This study suggests that classic eddy-viscosity/diffusivity models employed in large-eddy simulation may suffer from some limitations for large filter lengths, and that alternative closures should be considered to account for the inhomogeneous processes at subgrid level. Moreover, the theoretical framework based on the filtered Kolmogorov and Yaglom equations may represent a valuable tool for future assessments of the subgrid-scale models.

Resolved energy budget of superstructures in Rayleigh–Bénard convection

“…The inverse energy transfer close to the wall only occurs for large filter widths. For small filter widths the profiles are consistent with the ones reported by Togni et al (2017Togni et al ( , 2019 for a similar Rayleigh number. In Togni et al (2017Togni et al ( , 2019) the height-dependent budgets have been considered for small filter width (l < 0.25)…”

“…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.…”

“…At Ra = 10 7 the layer's structure becomes more complicated with the emergence of a thin dissipative layer even closer to the wall. These findings complement the results from Togni et al (2017Togni et al ( , 2019 by considering superstructures and their energetic interaction with turbulent fluctuations. † We shall make the first attempt to link the scale-resolved layer structure revealed in figure 4c with the boundary layer structure of RBC.…”

Spatially evolving cascades in temporal planar jets

“…(2012) for turbulent channel flows, Mollicone et al. (2018) for separated flows, Rincon (2006), Togni, Cimarelli & De Angelis (2015, 2019) for thermally driven turbulence, Gomes-Fernandes, Ganapathisubramani & Vassilicos (2015); Portela, Papadakis & Vassilicos (2017) for turbulent wakes and Burattini, Antonia & Danaila (2005), Sadeghi, Lavoie & Pollard (2016) for round jets. Here, we extend the use of the generalized Kolmogorov equation to the symmetries of a turbulent planar temporal jet.…”

Flow topology dynamics in a three-dimensional phase space for turbulent Rayleigh-Bénard convection

“…Therefore, a detailed analysis of the small scale motions during their lifetime, in terms of these non-linearities, can provide fundamental perspectives about how the thermal turbulence is evolving with Ra in RBC. Moreover, apart from improving our understanding of buoyancy-driven turbulence, it can also provide highly valuable information needed to build up better turbulence models [16,17].…”