Scaling laws in turbulent Rayleigh-Bénard convection under different geometry

A systematic study of turbulent Rayleigh-Bénard convection is carried out in two horizontal cylindrical cells of different lengths filled with water. Global heat transport and local temperature and velocity measurements are made over varying Rayleigh numbers Ra. The scaling behavior of the measured Nusselt number Nu(Ra) and the Reynolds number Re(Ra) associated with the large-scale circulation remains the same as that in the upright cylinders. The scaling exponent for the rms value of local temperature fluctuations, however, is strongly influenced by the aspect ratio and shape of the convection cell. The experiment clearly reveals the important roles played by the cell geometry in determining the scaling properties of convective turbulence. Copyright c © EPLA, 2010 Introduction. – The discovery of scaling laws in the heat flux and temperature statistics [1] in turbulent Rayleigh-Bénard convection, where a fluid layer of thickness H is heated from below and cooled from the top, has stimulated considerable experimental and theoretical efforts [2–5], aimed at accurately determining and explaining the effective power laws of the global and local quantities in turbulent convection. Among them the global quantities include the Nusselt number Nu(Ra, Pr), which is a normalized total heat flux, and the Reynolds number Re(Ra, Pr) associated with the large-scale circulation speed U across the convection cell. There are two experimental control parameters in Rayleigh-Bénard convection. One is the Rayleigh number Ra= αg∆TH/(νκ), where g is the gravitational acceleration, ∆T is the temperature difference across the fluid layer, and α, ν, and κ are, respectively, the thermal expansion coefficient, the kinematic viscosity, and the thermal diffusivity of the convecting fluid. The other control parameter is the Prandtl number, which is defined as Pr= ν/κ. The theory of Grossmann and Lohse (GL) [3,6] explains the scaling behavior of Nu(Ra, Pr) and Re(Ra, Pr) by a decomposition of the thermal dissipation field ǫT (r) into two parts. In one scenario [3], ǫT (r) is decomposed into the boundary layer and bulk contributions, which have different scaling behavior with varying Ra and Pr. More (a)E-mail: [email protected] recently, a second scenario was proposed [6] with ǫT (r) being decomposed into two different contributions: thermal plumes (including the boundary layers) and turbulent background. The latter scenario considered fluctuations in turbulent convection and gave predictions on the statistics of various local quantities, such as the rms value of local temperature fluctuations. While the two scenarios involve different physical pictures about the local dynamics of turbulent convection, the calculated scaling ofNu(Ra, Pr) and Re(Ra, Pr) using the two different models turns out to be of the same form. The GL theory is capable of providing a correct functional form of Nu(Ra, Pr) and Re(Ra, Pr) for a large number of transport and velocity measurements [1,7–18]. Up to now many of the convection experiments were conducted in upright cylindrical cells with the cylinder diameter D being comparable to its height H. These experiments have resulted in a large body of knowledge about the global heat transport [1,7–16], structure [17,18] and oscillations [18–21] of the large-scale circulation, boundary layer properties [22–25], local temperature [1,26] and velocity [18,27] fluctuations, structure and statistics of the local convective heat flux [28,29] and thermal dissipation rate [30,31]. Our current theoretical understanding of convective turbulence is largely built upon this body of experimental results [5,32]. While the use of small aspect-ratio cylinders has the advantages of simple cell geometry, better experimental