The unifying theory of scaling in thermal convection (Grossmann & Lohse (2000)) (henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (2001) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (2001) in the regime 3 × 10 7 ≤ Ra ≤ 3 × 10 9 and 4 ≤ P r ≤ 34 and Grossmann & Lohse (2002) used the experimental data point from Qiu & Tong (2001) and the fact that N u(Ra, P r) is independent of the parameter a, which relates the dimensionless kinetic boundary thickness with the square root of the wind Reynolds number, to fix the Reynolds number dependence. Meanwhile the theory is on one hand well confirmed through various new experiments and numerical simulations. On the other hand these new data points provide the basis for an updated fit in a much larger parameter space. Here we pick four well established (and sufficiently distant) Nu(Ra,Pr) data points and show that the resulting N u(Ra, P r) function is in agreement with almost all established experimental and numerical data up to the ultimate regime of thermal convection, whose onset also follows from the theory. One extra Re(Ra, P r) data point is used to fix Re(Ra, P r). As Re can depend on the definition and the aspect ratio the transformation properties of the GL equations are discussed in order to show how the GL coefficients can easily be adapted to new Reynolds number data while keeping N u(Ra, P r) unchanged.

Two dimensional (2D) and three dimensional (3D) Rayleigh-Bénard convection is compared using results from direct numerical simulations and prior experiments. The explored phase diagrams for both cases are reviewed. The differences and similarities between 2D and 3D are studied using Nu(Ra) for Pr = 4.38 and Pr = 0.7 and Nu(Pr) for Ra up to 10 8 . In the Nu(Ra) scaling at higher Pr, 2D and 3D are very similar; differing only by a constant factor up to Ra = 10 10 . In contrast, the difference is large at lower Pr, due to the strong roll state dependence of Nu in 2D. The behaviour of Nu(Pr) is similar in 2D and 3D at large Pr. However, it differs significantly around Pr = 1. The Reynolds number values are consistently higher in 2D and additionally converge at large Pr. Finally, the thermal boundary layer profiles are compared in 2D and 3D.

The double diffusive convection between two parallel plates is numerically studied for a series of parameters. The flow is driven by the salinity difference and stabilised by the thermal field. Our simulations are directly compared to experiments by Hage and Tilgner (Phys. Fluids 22, 076603 (2010)) for several sets of parameters and reasonable agreement is found. This in particular holds for the salinity flux and its dependence on the salinity Rayleigh number. Salt fingers are present in all simulations and extend through the entire height. The thermal Rayleigh number seems to have minor influence on salinity flux but affects the Reynolds number and the morphology of the flow. Next to the numerical calculation, we apply the Grossmann-Lohse theory for Rayleigh-Bénard flow to the current problem without introducing any new coefficients. The theory successfully predicts the salinity flux both with respect to the scaling and even with respect to the absolute value for the numerical and experimental results.