Temperature gradients, and search for non-Boussinesq effects, in the interior of turbulent Rayleigh-Bénard convection

We report temperature measurements for a cylindrical sample of turbulent RayleighBénard convection (RBC) at points in the interior, well away from the thermal boundary layers near the top and bottom of the sample. The aspect ratio was equal to 1.00 and the Prandtl number σ was equal to 4.4 or 5.5. The data are in the range 5× 10 <R< 10, where R is the Rayleigh number. Measurements of the temperatures T (r, z, θ) at the side wall (r=L/2) at eight equally spaced azimuthal positions θ and on three horizontal planes located at vertical positions z =−L/4, 0, L/4 (the sample height and diameter are equal to L and z = 0 is located at half height) are reported. An analysis of the harmonic contents of T (L/2, 0, θ) did not reveal any symmetry-breaking deviations from the Oberbeck-Boussinesq approximation even under conditions where the azimuthal average of the center temperature Tw(z) = 〈T (L/2, z, θ)〉θ at z = 0 differed appreciably from the mean temperature Tm = (Tt +Tb)/2 (Tt and Tb are the top and bottom temperatures, respectively). The azimuthal average of the vertical temperature variation 2[Tw(−L/4)−Tw(L/4)]/(Tb −Tt) at the side wall, presumably dominated by plume activity, was found to be destabilizing and quite large, ranging from about 0.2 at R= 5× 10 to about 0.06 at R= 10. We also report data for the temperature T0(z) along the center line (r= 0) at z =−L/4, 0, L/4. In contrast to Tw(z), T0(z) revealed a small stabilizing gradient 2[T0(−L/4)−T0(L/4)]/(Tb −Tt) that depended only weakly on R and was about equal to −0.007 for σ= 4.4 and −0.013 for σ= 5.5. Copyright c © EPLA, 2007 Introduction. – Turbulence in a fluid heated from below (Rayleigh-Bénard convection [1,2] or RBC) has provided the opportunity to study a number of interesting physical phenomena, [3–5] including the nature of viscous [6,7] and thermal boundary layers, [8–10] the interaction between these layers and a large-scale circulation (LSC) [11–14] in the interior or bulk of the system, interactions between the scale of order the sample height L of the LSC and the small scales of turbulent fluctuations [15–17], and interactions between the LSC and an external field due to rotation provided either by Earth’s Coriolis force [18] or deliberately in the laboratory [19]. In models of this system [12,20–25] it is usually assumed that part of the temperature drop ∆T = Tb −Tt (Tb and Tt are the temperatures of the bottom and top of the sample) occurs across a thermal boundary layer near the bottom of the sample, and that the remainder of ∆T is found across another such layer near the top. In this approximation the interior, even though fluctuating vigorously, in the time average has a uniform temperature. The purpose of the present paper is to contribute to our experimental knowledge about the extent of the deviations from these necessary but over-simplified model assumptions. In this system the temperature difference is usually expressed in terms of the dimensionless combination of parameters [26]