Non-Oberbeck-Boussinesq effects in two-dimensional Rayleigh- Bénard convection in glycerol

We numerically analyze Non-Oberbeck-Boussinesq (NOB) effects in two-dimensional Rayleigh-Bénard flow in glycerol, which shows a dramatic change in the viscosity with temperature. The results are presented both as functions of the Rayleigh number Ra up to 10 (for fixed temperature difference ∆ between the top and bottom plates) and as functions of ∆ (“nonOberbeck-Boussinesqness” or “NOBness”) up to 50K (for fixed Ra). For this large NOBness the center temperature Tc is more than 5K larger than the arithmetic mean temperature Tm between top and bottom plate and only weakly depends on Ra. To physically account for the NOB deviations of the Nusselt numbers from its Oberbeck-Boussinesq values, we apply the decomposition of NuNOB/NuOB into the product of two effects, namely first the change in the sum of the top and bottom thermal BL thicknesses, and second the shift of the center temperature Tc as compared to Tm. While for water the origin of the Nu deviation is totally dominated by the second effect (cf. Ahlers et al. J. Fluid Mech. 569, 409 (2006)) for glycerol the first effect is dominating, in spite of the large increase of Tc as compared to Tm. Introduction. – In most theoretical and numerical studies on Rayleigh-Bénard (RB) convection, the Oberbeck-Boussinesq (OB) approximation [1, 2] is employed, i.e., the fluid material properties are assumed to be independent of temperature T except for the density in the buoyancy term which is taken to be linear in T . The problem has two control parameters, namely the Rayleigh number Ra = βgL∆/(κν) (here β is the thermal expansion coefficient, g the gravitational acceleration, L the height, ∆ the temperature difference between bottom and top plates, κ the thermal diffusivity, and ν the kinematic viscosity), and the Prandtl number Pr = ν/κ. For the OB case the mean temperature profile shows top-bottom symmetry. However, in real fluids, if ∆ is large, this symmetry no longer holds due to the temperature dependences of the material properties. Thus, for given fluid, ∆ appears as an additional control parameter, which characterizes the deviations from OB conditions, leading to so called Non-Oberbeck-Boussinesq (NOB) effects. The NOB signatures can be quantified by (i) a shift Tc − Tm of the bulk (or center) temperature Tc from the arithmetic mean temperature Tm between the bottom and top plates) and (ii) by the ratio of the Nusselt numbers NuNOB/NuOB in the NOB and OB cases, which deviates from one. Both quantities have been measured in the large Ra regime for helium [3], glycerol [4], ethane [5], and water [6] as functions of the NOB-ness ∆. As shown in Ahlers et al. [6] the Nusselt number ratio NuNOB/NuOB can be connected to Tc by the identity NuNOB NuOB = 2λ OB λsl t + λ sl b · κt∆t + κb∆b κm∆ =: Fλ · F∆. (1) Here the labels on material properties indicate the temperature at which they are taken, e.g. κt = κ(Tt) etc. ∆t = Tc − Tt and ∆b = Tb − Tc denote the temperature drops over the top and bottom thermal boundary layers, and λ t and λ sl b indicate their thicknesses, based on the temperature slopes at the top and bottom plates, respectively. λ OB is the thermal BL thickness in the OB case, both at top and at bottom. The factor F∆ can be calculated from the temperature dependences of the material properties immediately, once Tc is known. Remarkably,