Scaling relations in large-Prandtl-number natural thermal convection

Abstract

In this study, we follow Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)], who derived various scalings regimes for the dependence of the Nusselt number Nu and the Reynolds number Re on the Rayleigh number Ra and the Prandtl number Pr. We focus on theoretical arguments as well as on numerical simulations for the case of large-Pr natural thermal convection. Based on an analysis of self-similarity of the boundary layer equations, we derive that in this case the limiting large-Pr boundary-layer dominated regime is I-infinity(<), introduced and defined by Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)], with the scaling relations Nu similar to Pr-0 Ra-1/3 and Re similar to Pr-1 Ra-2/3. Our direct numerical simulations for Ra from 10(4) to 10(9) and Pr from 0.1 to 200 showthat the regime I-infinity(<) is almost indistinguishable from the regime III infinity, where the kinetic dissipation is bulk-dominated. With increasing Ra, the scaling relations undergo a transition to those in IVu of Grossmann and Lohse [Phys. Rev. Lett. 86, 3316 (2001)], where the thermal dissipation is determined by its bulk contribution

From zonal flow to convection rolls in Rayleigh–Bénard convection with free-slip plates

“…For the convection roll states we find that the effective scaling exponent γ Nu in Nu ∼ Ra γ Nu is approximately 0.3. It increases with increasing Γ r , reaching approximately 1/3 for the largest Γ r = 8, which is the value predicated by the Grossmann-Lohse (GL) theory for the I < ∞ and III ∞ regimes (Grossmann & Lohse 2000Shishkina et al 2017) for the no-slip case. For zonal flow γ Nu is much smaller, namely only 0.17.…”

Flow organisation in laterally unconfined Rayleigh–Bénard turbulence

“…So far, the LSC and BL properties have mainly been studied in cells featuring a small aspect ratio Γ , typically Γ = 1/2 or Γ = 1. Various studies have shown that the BLs indeed follow the laminar Prandtl-Blasius (PB) type predictions in the classical regime (Ahlers et al 2009;Shi, Emran & Schumacher 2012;Stevens et al 2012;Shishkina et al 2015;Schumacher et al 2016;Shishkina et al 2017a). Previous studies by, for example, Wagner, Shishkina & Wagner (2012) and , have used results from direct numerical simulations (DNS) in aspect ratio Γ = 1 cells to study the properties of the BLs in detail.…”

On the role of the Prandtl number in convection driven by heat sources and sinks

“…Direct numerical simulations (DNS) exhibit much more flexibility from the standpoint of diffusivity coefficients and have proved invaluable for understanding the role on the Prandtl number for both RBC (Shishkina etal. 2017) and HC (Shishkina & Wagner 2016). In the present study, we employ DNS to explore the parameter space of CISS.…”