High Rayleigh number thermal convection

Experimental and numerical studies of Rayleigh-Bénard convection are complementary. Experiments provide direct heat-transfer measurements with controlled uncertainties, while direct numerical simulations provide the heat transfer and the complete modeled flow field simultaneously. Experimental data can therefore validate simulations, while simulations can help explain effects observed in experiments. In DNS it is essential to resolve all relevant flow scales; we showed excellent agreement between experiments and simulations when the resolution is sufficient [1], see figure 1a. Figure 2 shows what was, when published in 2011, the highest-Rayleigh-number DNS of Rayleigh-Bénard convection [3]. The simulation has Ra = 2 x 10¹² and Pr = 0.7 in a cylindrical Γ = 0.5 cell, using a computational mesh of 2701 x 671 x 2501 grid points. In later work we used DNS to study how the sidewall temperature boundary condition influences heat transport [4].

Figure 1: Compensated Nusselt number vs the Rayleigh number for Pr=0.7. (a) Purple stars are experimental data from Niemela et al. (2000) and the green squares are from Chavanne et al. (2001). The DNS results from Verzicco & Camussi (2003) and Amati et al. (2005) are indicated in red and high resolution results by R.J.A.M. Stevens, R. Verzicco, D. Lohse, J. Fluid Mech. 643, 495-507 (2010) are indicated by the black dots. When the vertical errorbar is not visible the error is smaller than the dot size. The results of underresolved simulations of this study are indicated by the blue dots. (b) Comparison between simulations, experiments, and the latest version of the Grossmann-Lohse theory published in the paper The unifying theory of scaling in thermal convection: The updated prefactors, J. Fluid Mech. 730 , 295-308 (2013). Good agreement is found up to Ra=10¹¹, but still unexplained differences are observed for higher Ra.

Movie 1: Movie of the temperature in a vertical cut for the simulation at Ra=2x10¹², Pr=0.7, and Γ=0.5. The dimensionless time is indicated in the top of the movie. This movie is rendered with a lower resolution for the website. Additional movies of this simulation are available on the PhD thesis supplemental material page.