Siegfried Grossmann

Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection

The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large scale convection roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.

Scaling in thermal convection: a unifying theory

A systematic theory for the scaling of the Nusselt number Nu and of the A systematic theory for the scaling of the Nusselt number Nu and of the Reynolds number Re in strong Rayleigh–Benard convection is suggested and shown to be compatible with recent experiments. It assumes a coherent large-scale convection roll (‘wind of turbulence’) and is based on the dynamical equations both in the bulk and in the boundary layers. Several regimes are identified in the Rayleigh number Ra versus Prandtl number Pr phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the kinetic boundary layer is thicker. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra [less, similar] 1011) the leading terms are Nu [similar] Ra1/4Pr1/8, Re [similar] Ra1/2Pr[minus sign]3/4 for Pr [less, similar] 1 and Nu [similar] Ra1/4Pr[minus sign]1/12, Re [similar] Ra1/2Pr[minus sign]5/6 for Pr [greater, similar] 1. In most measurements these laws are modified by additive corrections from the neighbouring regimes so that the impression of a slightly larger (effective) Nu vs. Ra scaling exponent can arise. The most important of the neighbouring regimes towards large Ra are a regime with scaling Nu [similar] Ra1/2Pr1/2, Re [similar] Ra1/2Pr[minus sign]1/2 for medium Pr (‘Kraichnan regime’), a regime with scaling Nu [similar] Ra1/5Pr1/5, Re [similar] Ra2/5Pr[minus sign]3/5 for small Pr, a regime with Nu [similar] Ra1/3, Re [similar] Ra4/9Pr[minus sign]2/3 for larger Pr, and a regime with scaling Nu [similar] Ra3/7Pr[minus sign]1/7, Re [similar] Ra4/7Pr[minus sign]6/7 for even larger Pr. In particular, a linear combination of the ¼ and the 1/3 power laws for Nu with Ra, Nu = 0.27Ra1/4 + 0.038Ra1/3 (the prefactors follow from experiment), mimics a 2/7 power-law exponent in a regime as large as ten decades. For very large Ra the laminar shear boundary layer is speculated to break down through the non-normal-nonlinear transition to turbulence and another regime emerges.

Invariant Distributions and Stationary Correlation Functions of One-Dimensional Discrete Processes

Abstract The connection between one-dimensional dynamical laws generating discrete processes and their invariant densities as well as their stationary correlaton functions is discussed. In particular the changes occuring under a special equivalence transformation are considered. Correlation functions are used to describe the gradual transition from periodic states to chaotic states via periodic motions with superimposed nonlinearity noise.

A simple explanation of light emission in sonoluminescence

Ultrasonically driven gas bubbles in liquids can emit intense bursts of light when they collapse. The physical mechanism for single-bubble sonoluminescence has been much debated,. The conditions required for, and generated by, bubble collapse can be deduced within the framework of a hydrodynamic (Rayleigh–Plesset) analysis of bubble dynamics and stability,, and by considering the dissociation and outward diffusion of gases under the extreme conditions induced by collapse,. We show here that by extending this hydrodynamic/chemical picture in a simple way, the light emission can be explained too. The additional elements that we add are a model for the volume dependence of the bubbles temperature, and allowance for the small emissivity of a weakly ionized gas. Despite its simplicity, our approach can account quantitatively for the observed parameter dependences of the light intensity and pulse width, as well as for the spectral shape and wavelength independence of the pulses.

Fluctuations in turbulent Rayleigh-Bénard convection: The role of plumes

Our unifying theory of turbulent thermal convection [Grossmann and Lohse, J. Fluid. Mech. 407, 27 (2000); Phys. Rev. Lett. 86, 3316 (2001); Phys. Rev. E 66, 016305 (2002)] is revisited, considering the role of thermal plumes for the thermal dissipation rate and addressing the local distribution of the thermal dissipation rate, which had numerically been calculated by Verzicco and Camussi [J. Fluid Mech. 477, 19 (2003); Eur. Phys. J. B 35, 133 (2003)]. Predictions for the local heat flux and for the temperature and velocity fluctuations as functions of the Rayleigh and Prandtl numbers are offered. We conclude with a list of suggestions for measurements that seem suitable to verify or falsify our present understanding of heat transport and fluctuations in turbulent thermal convection.

Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution

Results on the Prandtl–Blasius-type kinetic and thermal boundary layer (BL) thicknesses in turbulent Rayleigh–Benard (RB) convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl–Blasius BL equations, we calculate the ratio between the thermal and kinetic BL thicknesses, which depends on the Prandtl number only. It is approximated as for and as for , with . Comparison of the Prandtl–Blasius velocity BL thickness with that evaluated in the direct numerical simulations by Stevens et al (2010 J. Fluid Mech. 643 495) shows very good agreement between them. Based on the Prandtl–Blasius-type considerations, we derive a lower-bound estimate for the minimum number of computational mesh nodes required to conduct accurate numerical simulations of moderately high (BL-dominated) turbulent RB convection, in the thermal and kinetic BLs close to the bottom and top plates. It is shown that the number of required nodes within each BL depends on and and grows with the Rayleigh number not slower than . This estimate is in excellent agreement with empirical results, which were based on the convergence of the Nusselt number in numerical simulations

Multiple scaling in the ultimate regime of thermal convection

Very different types of scaling of the Nusselt number Nu with the Rayleigh number Ra have experimentally been found in the very large Ra regime beyond 1011. We understand and interpret these results by extending the unifying theory of thermal convection [Grossmann and Lohse, Phys. Rev. Lett. 86, 3316 (2001)] to the very large Ra regime where the kinetic boundary-layer is turbulent. The central idea is that the spatial extension of this turbulent boundary-layer with a logarithmic velocity profile is comparable to the size of the cell. Depending on whether the thermal transport is plume dominated, dominated by the background thermal fluctuations, or whether also the thermal boundary-layer is fully turbulent (leading to a logarithmic temperature profile), we obtain effective scaling laws of about Nu∝Ra0.14, Nu∝Ra0.22, and Nu∝Ra0.38, respectively. Depending on the initial conditions or random fluctuations, one or the other of these states may be realized. Since the theory is for both the heat flux Nu and the ...

Analysis of Rayleigh Plesset dynamics for sonoluminescing bubbles

Recent work on single-bubble sonoluminescence (SBSL) has shown that many features of this phenomenon, especially the dependence of SBSL intensity and stability on experimental parameters, can be explained within a hydrodynamic approach. More specifically, many important properties can be derived from an analysis of bubble wall dynamics. This dynamics is conveniently described by the Rayleigh-Plesset (RP) equation. Here we derive analytical approximations for RP dynamics and subsequent analytical laws for parameter dependences. These results include (i) an expression for the onset threshold of SL, (ii) an analytical explanation of the transition from diffusively unstable to stable equilibria for the bubble ambient radius (unstable and stable sonoluminescence), and (iii) a detailed understanding of the resonance structure of the RP equation. It is found that the threshold for SL emission is shifted to larger bubble radii and larger driving pressures if surface tension is increased, whereas even a considerable change in liquid viscosity leaves this threshold virtually unaltered. As an enhanced viscosity stabilizes the bubbles to surface oscillations, we conclude that the ideal liquid for violently collapsing, surface-stable SL bubbles should have small surface tension and large viscosity, although too large viscosity ([eta]l[gt-or-equal, slanted]40[eta]water) will again preclude collapses.

ON BOSE-EINSTEIN CONDENSATION IN HARMONIC TRAPS

Abstract Assuming the validity of grand canonical statistics, we study Bose-Einstein condensation of relatively small numbers of particles confined by a harmonic potential. Corrections to the case of large particle numbers appear as a downward shift of the condensation temperature T C and an enhancement of the specific heat capacity below T C . Even if the particle number is merely of order 10 4 , the specific heat capacity exhibits a sharp drop at the onset of condensation, reminiscent of the heat capacity of liquid 4 He at the λ-point.

Torque scaling in turbulent Taylor–Couette flow between independently rotating cylinders

Turbulent Taylor–Couette flow with arbitrary rotation frequencies ω1, ω2 of the two coaxial cylinders with radii r1 < r2 is analysed theoretically. The current Jω of the angular velocity ω(x,t) = u(r,,z,t)/r across the cylinder gap and and the excess energy dissipation rate w due to the turbulent, convective fluctuations (the ‘wind’) are derived and their dependence on the control parameters analysed. The very close correspondence of Taylor–Couette flow with thermal Rayleigh–Benard convection is elaborated, using these basic quantities and the exact relations among them to calculate the torque as a function of the rotation frequencies and the radius ratio η = r1/r2 or the gap width d = r2 − r1 between the cylinders. A quantity σ corresponding to the Prandtl number in Rayleigh–Benard flow can be introduced, . In Taylor–Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh–Benard flow. The analogue of the Rayleigh number is the Taylor number, defined as Ta (ω1 − ω2)2 times a specific geometrical factor. The experimental data show no pure power law, but the exponent α of the torque versus the rotation frequency ω1 depends on the driving frequency ω1. An explanation for the physical origin of the ω1-dependence of the measured local power-law exponents α(ω1) is put forward. Also, the dependence of the torque on the gap width η is discussed and, in particular its strong increase for η → 1.