94 research outputs found

### 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

### 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

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