Urbanet al.Reply:

Dynamics of large-scale quantities in Rayleigh-Bénard convection

“…According to GL scaling, for small and moderate Pr's, Pe ∼ √ RaPr, but for large Pr, Pe ∼ Ra 3/5 . The above scaling have been verified in many experiments [14][15][16][17][18][19][20][21] and numerical simulations [22][23][24][25][26][27][28][29][30]. In this paper we derive a general formula for Pe, of which the aforementioned relations are limiting cases, by comparing the relative strengths of the nonlinear, pressure, buoyancy, and viscous terms, and quantifying them using the numerical data.…”

“…The Nusselt number Nu ∼ C uθres u Table II we list the Ra dependence for the above quantities that provide the appropriate corrections to the Nusselt number from the Kraichnan's predictions [38] for the ultimate regime (Nu ∼ Ra 1/2 ) to the experimentally observed Nu ∼ Ra 0.30 . It is tempting to connect our findings to the ultimate regime of turbulent convection [20,21]. We conjecture that in the ultimate regime, C uθres and θ res , as well as the coefficients c i 's, would become independent of Ra due to boundary layer detachment, and hence yield Nu ∼ Ra 1/2 .…”

Thermal convection over fractal surfaces

“…In a wide range of studies at O(1) Prandtl number, the exponent β is found to vary between 2/7 (Verzicco & Camussi 2003;Johnston & Doering 2009;Stevens, Verzicco & Lohse 2010;Urban, Musilová & Skrbek 2011;Urban et al 2012;Doering, Toppaladoddi & Wettlaufer 2019;Iyer et al 2020) and 1/3 (Niemela et al 2000;Niemela & Sreenivasan 2003;Verzicco & Camussi 2003;Niemela & Sreenivasan 2006;Stevens et al 2010;Urban et al 2011Urban et al , 2012Doering et al 2019;Iyer et al 2020). Several experiments have reported β > 1/3 (Chavanne et al 1997;He et al 2012); however, because of the diversity of scalings reported for overlapping ranges of Ra, those findings await independent confirmation (Urban et al 2012;He et al 2013;Urban et al 2013;Skrbek & Urban 2015;He, Bodenschatz & Ahlers 2016).…”

Thermal Convection over Fractal Surfaces

“…(For flat no-slip boundaries at infinite Prandtl number the upper bound corresponds to the classical scaling N u < ∼ Ra 1/3 within logarithmic corrections [16][17][18].) In a wide range of studies at O(1) Prandtl number, the exponent β is found to vary between 2/7 [19][20][21][22][23] and 1/3 [19][20][21][23][24][25][26]. Several experiments have reported β > 1/3 [12,27] but those findings await independent confirmation [20,[28][29][30][31].…”