Numerical simulations of thermal convection at high Prandtl numbers

In this thesis we present the results of an extensive campaign of direct numerical simulations of Rayleigh-B\\'enard convection at high Prandtl numbers (10^{-1}\\leq Pr \\leq 10^4) and moderate Rayleigh numbers (10^{5}\\leq Pr \\leq 10^9). The computational domain is a cylindrical cell of aspect-ratio (diameter over cell height) \\Gamma=1/2, with the no-slip condition imposed to the boundaries. By scaling the results, we find a 1/\\sqrt{Pr} correction to apply to the free-fall velocity, obtaining a more appropriate representation of the large scale velocity at high Pr. We investigate the Nusselt and the Reynolds number dependence on Ra and Pr, comparing the results to previous numerical and experimental work. At high Pr the scaling behavior of the Nusselt number with respect to Ra is generally consistent with the power-law exponent 0.309. The Nusselt number is independent of Pr, even at the highest Ra simulated. The Reynolds number scales as Re\\sim \\sqrt{Ra}/Pr, neglecting logarithmic corrections. We analyze the global and local features of viscous and thermal boundary layers and their scaling behavior with respect to Rayleigh and Prandtl numbers, and with respect to Reynolds and Peclet numbers. We find that the flow approaches a saturation regime when Reynolds number decreases below the critical value Re_s\\simeq 40. The thermal boundary layer thickness turns out to increase slightly even when the Peclet number increases. We explain this behavior as a combined effect of the Peclet number and the viscous boundary layer influences. The range of Ra and Pr simulated contains steady, periodic and turbulent solutions. A rough estimate of the transition from steady to unsteady flow is obtained by monitoring the time-evolution of the system until it reaches stationary solutions (Ra_U\\simeq 7.5 \\times 10^6 at Pr=10^3). We find multiple solutions as long-term phenomena at Ra=10^8 and Pr=10^3 which, however, do not result in significantly different Nusselt number. One of these multiple solutions, even if stable for a long time interval, shows a break in the mid-plane symmetry of the temperature profile. The result is similar to that of some non-Boussinesq effects. We analyze the flow structures through the transitional phases by direct visualizations of the temperature and velocity fields. We also describe how the behavior of the flow structures changes for increasing Pr. A wide variety of large-scale circulations and plumes structures are found. The single-roll circulation is characteristic only of the steady and periodic solutions. For other solutions, at lower Pr, the mean flow generally consists of two opposite toroidal structures; at higher Pr, the flow is organized in multi-cell structures extending mostly in the vertical direction. At high Pr, plumes detach from sheet-like structures. The different large-scale-structure signatures are generally reflected in the data trends with respect to Ra, but not in those with respect to Pr. In particular, the Nusselt number is independent of Pr, even when the flow structures appear strongly different varying Pr. In order to assess the reliability of the data-set we perform a systematic analysis of the error affecting the data. Refinement grid analysis is extensively applied.