Non-Oberbeck–Boussinesq effects in strongly turbulent Rayleigh–Bénard convection

Abstract

Non-Oberbeck-Boussinesq (NOB) effects on the Nusselt number Nu and Reynolds number Re in strongly turbulent Rayleigh-Bénard (RB) convection in liquids were investigated both experimentally and theoretically. In the experiments the heat current, the temperature difference, and the temperature at the horizontal midplane were measured. Three cells of different heights L, all filled with water and all with aspect ratio Γ close to 1, were used. For each L, about 1.5 decades in Ra were covered, together spanning the range 10 8 6 Ra 6 10 11 . For the largest temperature difference between the bottom and top plates, ∆ = 40 K, the kinematic viscosity and the thermal expansion coefficient, owing to their temperature dependence, varied by more than a factor of 2. The Oberbeck-Boussinesq (OB) approximation of temperature-independent material parameters thus was no longer valid. The ratio χ of the temperature drops across the bottom and top thermal boundary layers became as small as χ = 0.83, which may be compared with the ratio χ = 1 in the OB case. Nevertheless, the Nusselt number Nu was found to be only slightly smaller (by at most 1.4%) than in the next larger cell with the same Rayleigh number, where the material parameters were still nearly height independent. The Reynolds numbers in the OB and NOB case agreed with each other within the experimental resolution of about 2%, showing that NOB effects for this parameter were small as well. Thus Nu and Re are rather insensitive against even significant deviations from OB conditions. Theoretically, we first account for the robustness of Nu with respect to NOB corrections: the NOB effects in the top boundary layer cancel those which arise in the bottom boundary layer as long as they are linear in the temperature difference ∆. The net effects on Nu are proportional to ∆ 2 and thus increase only slowly and still remain minor despite drastic materialparameter changes. We then extend the Prandtl-Blasius boundary-layer theory to NOB Rayleigh-Bénard flow with temperature-dependent viscosity and thermal diffusivity. This allows calculation of the shift in the bulk temperature, the temperature drops across the boundary layers, and the ratio χ without the introduction of any fitting parameter. The calculated quantities are in very good agreement with experiment. When in addition we use the experimental finding that for water the sum of the top and bottom thermal boundary-layer widths (based on the slopes of the temperature profiles at the plates) remains unchanged under NOB effects within the experimental resolution, the theory also gives the measured small Nusselt-number reduction for the NOB case. In addition, it predicts an increase by about 0.5%

Horizontal structures of velocity and temperature boundary layers in two-dimensional numerical turbulent Rayleigh-Bénard convection

“…As the GL theory is based on the assumption that the BL thickness scales inversely proportional to the square root of the Reynolds number according to Prandtl's 1904 theory, the validity of Prandtl-Blasius BL flow needs to be tested also locally. Note that comparison of the mean bulk temperature calculated using the Prandtl-Blasius theory with that measured in both liquid and gaseous nonOberbeck-Boussinesq RB convection shows very good agreement [16][17][18] . In addition, the kinematic BL thickness evaluated by solving the laminar Prandtl-Blasius BL equations was found to agree well with that obtained in the direct numerical simulation (DNS) 19 .…”

Experimental velocity study of non-Boussinesq Rayleigh-Bénard convection

“…This is an experimental verification on a 3D domain and with accurate direct velocity measurements of the several numerical and theoretical [3][4][5][6] studies on NB effects in the velocity fields of RB convection.…”

“…In water, fluorescent particles were used with the purpose of avoiding the effect of reflections off the glass walls of the laser light in the images and improving the quality of the measurements close to the wall. The fluorescent particles (Fluostar) have an average diameter of 13 μm and a density of 1.1 g/cm 3 . The fluorescent dye is rhodamine B, which is excited by a wavelength of 550 nm and emits at 580 nm.…”

“…In Ahlers et al [3] an experimental and theoretical study on the effects of NB conditions on the Nusselt and Reynolds number is presented, with heat flux and temperature data. Their Reynolds number measurements are based on thermal plumes transient times calculated from temperature auto-and cross-correlation functions [3].…”

“…In Ahlers et al [3] an experimental and theoretical study on the effects of NB conditions on the Nusselt and Reynolds number is presented, with heat flux and temperature data. Their Reynolds number measurements are based on thermal plumes transient times calculated from temperature auto-and cross-correlation functions [3]. The working fluid in their study is water at atmospheric pressure, which shows significant changes of the kinematic viscosity and the volumetric thermal expansion coefficient in the range of temperatures considered.…”