Scaling in thermal convection: A unifying theory

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 versus Prandtl number phase space, defined by whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively. The crossover between the regimes is calculated. In the regime which has most frequently been studied in experiment (Ra smaller than 10^{11}) the leading terms are $Nu\sim Ra^{1/4}Pr^{1/8}$, $Re \sim Ra^{1/2} Pr^{-3/4}$ for $Pr < 1$ and $Nu\sim Ra^{1/4}Pr^{-1/12}$, $Re \sim Ra^{1/2} Pr^{-5/6}$ for $Pr > 1$. In most measurements these laws are modified by additive corrections from the neighboring regimes so that the impression of a slightly larger (effective) Nu vs Ra scaling exponent can arise. -- The presented theory is best summarized in the phase diagram figure 1.Comment: 30 pages, latex, 7 figures, under review at Journal of Fluid Mec

On geometry effects in Rayleigh-Benard convection

Various recent experiments hint at a geometry dependence of scaling relations in Rayleigh-B\'enard convection. Aspect ratio and shape dependences have been found. In this paper a mechanism is offered which can account for such dependences. It is based on Prandtl's theory for laminar boundary layers and on the conservation of volume flux of the large scale wind. The mechanism implies the possibility of different thicknesses of the kinetic boundary layers at the sidewalls and the top/bottom plates, just as experimentally found by Qiu and Xia (Phys. Rev. E58, 486 (1998)), and also different $Ra$-scaling of the wind measured over the plates and at the sidewalls. In the second part of the paper a scaling argument for the velocity and temperature fluctuations in the bulk is developeVarious recent experiments hint at a geometry dependence of scaling relations in Rayleigh-Benard convection. Aspect ratio and shape dependences have been found. In this paper a mechanism is offered which can account for such dependences. It is based on Prandtl's theory for laminar boundary layers and on the conservation of volume flux of the large scale wind. The mechanism implies the possibility of different thicknesses of the kinetic boundary layers at the sidewalls and the top/bottom plates, just as experimentally found by Qiu and Xia (Phys. Rev. E58, 486 (1998)), and also different $Ra$-scaling of the wind measured over the plates and at the sidewalls. In the second part of the paper a scaling argument for the velocity and temperature fluctuations in the bulk is developeComment: 4 pages, 1 figur

Maxwell's Demon at work: Two types of Bose condensate fluctuations in power-law traps

After discussing the key idea underlying the Maxwell's Demon ensemble, we employ this idea for calculating fluctuations of ideal Bose gas condensates in traps with power-law single-particle energy spectra. Two essentially different cases have to be distinguished. If the heat capacity remains continuous at the condensation point in the large-N-limit, the fluctuations of the number of condensate particles vanish linearly with temperature, independent of the trap characteristics. If the heat capacity becomes discontinuous, the fluctuations vanish algebraically with temperature, with an exponent determined by the trap. Our results are based on an integral representation that yields the solution to both the canonical and the microcanonical fluctuation problem in a singularly transparent manner.Comment: 10 pages LaTeX and 3 eps-figures, published versio

Scale resolved intermittency in turbulence

The deviations $\delta\zeta_m$ ("intermittency corrections") from classical ("K41") scaling $\zeta_m=m/3$ of the $m^{th}$ moments of the velocity differences in high Reynolds number turbulence are calculated, extending a method to approximately solve the Navier-Stokes equation described earlier. We suggest to introduce the notion of scale resolved intermittency corrections $\delta\zeta_m(p)$, because we find that these $\delta\zeta_m(p)$ are large in the viscous subrange, moderate in the nonuniversal stirring subrange but, surprisingly, extremely small if not zero in the inertial subrange. If ISR intermittency corrections persisted in experiment up to the large Reynolds number limit, our calculation would show, that this could be due to the opening of phase space for larger wave vectors. In the higher order velocity moment $\langle|u(p)|^m\rangle$ the crossover between inertial and viscous subrange is $(10\eta m/2)^{-1}$, thus the inertial subrange is {\it smaller} for higher moments.Comment: 12 pages, Latex, 2 tables, 7 figure

Scaling of global momentum transport in Taylor-Couette and pipe flow

We interpret measurements of the Reynolds number dependence of the torque in Taylor-Couette flow by Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] and of the pressure drop in pipe flow by Smits and Zagarola, [Phys. Fluids 10, 1045 (1998)] within the scaling theory of Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)], developed in the context of thermal convection. The main idea is to split the energy dissipation into contributions from a boundary layer and the turbulent bulk. This ansatz can account for the observed scaling in both cases if it is assumed that the internal wind velocity $U_w$ introduced through the rotational or pressure forcing is related to the the external (imposed) velocity U, by $U_w/U \propto Re^\xi$ with xi = -0.051 and xi = -0.041 for the Taylor-Couette (U inner cylinder velocity) and pipe flow (U mean flow velocity) case, respectively. In contrast to the Rayleigh-Benard case the scaling exponents cannot (yet) be derived from the dynamical equations.Comment: 7 pages, 4 ps figures with 4 program files included in the source. European Physical Journal B, accepte

Scaling of the irreducible SO(3)-invariants of velocity correlations in turbulence

The scaling behavior of the SO(3) irreducible amplitudes $d_n^l(r)$ of velocity structure tensors (see L'vov, Podivilov, and Procaccia, Phys. Rev. Lett. (1997)) is numerically examined for Navier-Stokes turbulence. Here, l characterizes the irreducible representation by the index of the corresponding Legendre polynomial, and n denotes the tensorial rank, i.e., the order of the moment. For moments of different order n but with the same representation index l extended self similarity (ESS) towards large scales is found. Intermittency seems to increase with l. We estimate that a crossover behavior between different inertial subrange scaling regimes in the longitudinal and transversal structure functions will hardly be detectable for achievable Reynolds numbers.Comment: 4 pages, 3 eps-figure

Application of extended self similarity in turbulence

From Navier-Stokes turbulence numerical simulations we show that for the extended self similarity (ESS) method it is essential to take the third order structure function taken with the modulus and called $D_3^*(r)$, rather than the standard third order structure function $D_3(r)$ itself. If done so, we find ESS towards scales larger than roughly 10 eta, where eta is the Kolmogorov scale. If $D_3(r)$ is used, there is no ESS. We also analyze ESS within the Batchelor parametrization of the second and third order longitudinal structure function and focus on the scaling of the transversal structure function. The Re-asymptotic inertial range scaling develops only beyond a Taylor-Reynolds number of about 500.Comment: 12 pages, 7 eps-figures, replaces version from April 11th, 1997; paper now in press at Phys. Rev.

Logarithmic temperature profiles in the ultimate regime of thermal convection

We report on the theory of logarithmic temperature profiles in very strongly developed thermal convection in the geometry of a Rayleigh-Benard cell with aspect ratio one and discuss the degree of agreement with the recently measured profiles in the ultimate state of very large Rayleigh number flow. The parameters of the log-profile are calculated and compared with the measure ones. Their physical interpretation as well as their dependence on the radial position are discussed.Comment: 14 pages, no figur