Two-point velocity average of turbulence: statistics and their implications

For turbulence, although the two-point velocity difference u(x+r)-u(x) at each scale r has been studied in detail, the velocity average [u(x+r)+u(x)]/2 has not thus far. Theoretically or experimentally, we find interesting features of the velocity average. It satisfies an exact scale-by-scale energy budget equation. The flatness factor varies with the scale r in a universal manner. These features are not consistent with the existing assumption that the velocity average is independent of r and represents energy-containing large-scale motions alone. We accordingly propose that it represents motions over scales >= r as long as the velocity difference represents motions at the scale r.Comment: 8 pages, accepted by Physics of Fluids (see http://pof.aip.org/

"Locally homogeneous turbulence" Is it an inconsistent framework?

In his first 1941 paper Kolmogorov assumed that the velocity has increments which are homogeneous and independent of the velocity at a suitable reference point. This assumption of local homogeneity is consistent with the nonlinear dynamics only in an asymptotic sense when the reference point is far away. The inconsistency is illustrated numerically using the Burgers equation. Kolmogorov's derivation of the four-fifths law for the third-order structure function and its anisotropic generalization are actually valid only for homogeneous turbulence, but a local version due to Duchon and Robert still holds. A Kolomogorov--Landau approach is proposed to handle the effect of fluctuations in the large-scale velocity on small-scale statistical properties; it is is only a mild extension of the 1941 theory and does not incorporate intermittency effects.Comment: 4 pages, 2 figure

Entropy and Area in Loop Quantum Gravity

Black hole thermodynamics suggests that the maximum entropy that can be contained in a region of space is proportional to the area enclosing it rather than its volume. I argue that this follows naturally from loop quantum gravity and a result of Kolmogorov and Bardzin' on the the realizability of networks in three dimensions. This represents an alternative to other approaches in which some sort of correlation between field configurations helps limit the degrees of freedom within a region. It also provides an approach to thinking about black hole entropy in terms of states inside rather than on its surface. Intuitively, a spin network complicated enough to imbue a region with volume only lets that volume grow as quickly as the area bounding it.Comment: 7 pages, this essay received an Honourable Mention in the Gravity Research Foundation Essay Competition 2005; reformatted for IJMP (accepted for publication) with minor typographical corrections and some extended discussio

Relation between shear parameter and Reynolds number in statistically stationary turbulent shear flows

Studies of the relation between the shear parameter S^* and the Reynolds number Re are presented for a nearly homogeneous and statistically stationary turbulent shear flow. The parametric investigations are in line with a generalized perspective on the return to local isotropy in shear flows that was outlined recently [Schumacher, Sreenivasan and Yeung, Phys. Fluids, vol.15, 84 (2003)]. Therefore, two parameters, the constant shear rate S and the level of initial turbulent fluctuations as prescribed by an energy injection rate epsilon_{in}, are varied systematically. The investigations suggest that the shear parameter levels off for larger Reynolds numbers which is supported by dimensional arguments. It is found that the skewness of the transverse derivative shows a different decay behavior with respect to Reynolds number when the sequence of simulation runs follows different pathways across the two-parameter plane. The study can shed new light on different interpretations of the decay of odd order moments in high-Reynolds number experiments.Comment: 9 pages, 9 Postscript figure

Analogy between turbulence and quantum gravity: beyond Kolmogorov's 1941 theory

Simple arguments based on the general properties of quantum fluctuations have been recently shown to imply that quantum fluctuations of spacetime obey the same scaling laws of the velocity fluctuations in a homogeneous incompressible turbulent flow, as described by Kolmogorov 1941 (K41) scaling theory. Less noted, however, is the fact that this analogy rules out the possibility of a fractal quantum spacetime, in contradiction with growing evidence in quantum gravity research. In this Note, we show that the notion of a fractal quantum spacetime can be restored by extending the analogy between turbulence and quantum gravity beyond the realm of K41 theory. In particular, it is shown that compatibility of a fractal quantum-space time with the recent Horava-Lifshitz scenario for quantum gravity, implies singular quantum wavefunctions. Finally, we propose an operational procedure, based on Extended Self-Similarity techniques, to inspect the (multi)-scaling properties of quantum gravitational fluctuations.Comment: Sliglty modified version of the article about to appear in IJMP

Shear Effects in Non-Homogeneous Turbulence

Motivated by recent experimental and numerical results, a simple unifying picture of intermittency in turbulent shear flows is suggested. Integral Structure Functions (ISF), taking into account explicitly the shear intensity, are introduced on phenomenological grounds. ISF can exhibit a universal scaling behavior, independent of the shear intensity. This picture is in satisfactory agreement with both experimental and numerical data. Possible extension to convective turbulence and implication on closure conditions for Large-Eddy Simulation of non-homogeneous flows are briefly discussed.Comment: 4 pages, 5 figure

Scaling Relations of Compressible MHD Turbulence

We study scaling relations of compressible strongly magnetized turbulence. We find a good correspondence of our results with the Fleck (1996) model of compressible hydrodynamic turbulence. In particular, we find that the density-weighted velocity, i.e. $u \equiv \rho^{1/3} v$, proposed in Kritsuk et al. (2007) obeys the Kolmogorov scaling, i.e. $E_{u}(k)\sim k^{-5/3}$ for the high Mach number turbulence. Similarly, we find that the exponents of the third order structure functions for $u$ stay equal to unity for the all the Mach numbers studied. The scaling of higher order correlations obeys the She-Leveque (1994) scalings corresponding to the two-dimensional dissipative structures, and this result does not change with the Mach number either. In contrast to $v$ which exhibits different scaling parallel and perpendicular to the local magnetic field, the scaling of $u$ is similar in both directions. In addition, we find that the peaks of density create a hierarchy in which both physical and column densities decrease with the scale in accordance to the Fleck (1996) predictions. This hierarchy can be related ubiquitous small ionized and neutral structures (SINS) in the interstellar gas. We believe that studies of statistics of the column density peaks can provide both consistency check for the turbulence velocity studies and insight into supersonic turbulence, when the velocity information is not available.Comment: 4 pages, 5 figure

The Kelvin-wave cascade in the vortex filament model

The energy transfer mechanism in zero temperature superfluid turbulence of helium-4 is still a widely debated topic. Currently, the main hypothesis is that weakly nonlinear interacting Kelvin waves transfer energy to sufficiently small scales such that energy is dissipated as heat via phonon excitations. Theoretically, there are at least two proposed theories for Kelvin-wave interactions. We perform the most comprehensive numerical simulation of weakly nonlinear interacting Kelvin-waves to date and show, using a specially designed numerical algorithm incorporating the full Biot-Savart equation, that our results are consistent with nonlocal six-wave Kelvin wave interactions as proposed by L'vov and Nazarenko.Comment: 6 pages, 6 figure

Highly turbulent solutions of LANS-alpha and their LES potential

We compute solutions of the Lagrangian-Averaged Navier-Stokes alpha-model (LANS) for significantly higher Reynolds numbers (up to Re 8300) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes (NS) inertial range followed by a 2nd LANS inertial range. The analysis of the third-order structure function scaling supports the predicted l^3 scaling; it corresponds to a k^(-1) scaling of the energy spectrum. The energy spectrum itself shows a different scaling which goes as k^1. This latter spectrum is consistent with the absence of stretching in the sub-filter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of LANS. These two scalings are conjectured to coexist in different spatial portions of the flow. The l^3 (E(k) k^(-1)) scaling is subdominant to k^1 in the energy spectrum, but the l^3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We verify the prediction for the size of the LANS attractor resulting from this scaling. From this, we give a methodology either for arriving at grid-independent solutions for LANS, or for obtaining a formulation of a LES optimal in the context of the alpha models. The fully converged grid-independent LANS may not be the best approximation to a direct numerical simulation of the NS equations since the minimum error is a balance between truncation errors and the approximation error due to using LANS instead of the primitive equations. Furthermore, the small-scale behavior of LANS contributes to a reduction of flux at constant energy, leading to a shallower energy spectrum for large alpha. These small-scale features, do not preclude LANS to reproduce correctly the intermittency properties of high Re flow.Comment: 37 pages, 17 figure

Self-organization in turbulence as a route to order in plasma and fluids

Transitions from turbulence to order are studied experimentally in thin fluid layers and magnetically confined toroidal plasma. It is shown that turbulence self-organizes through the mechanism of spectral condensation. The spectral redistribution of the turbulent energy leads to the reduction in the turbulence level, generation of coherent flow, reduction in the particle diffusion and increase in the system's energy. The higher order state is sustained via the nonlocal spectral coupling of the linearly unstable spectral range to the large-scale mean flow. The similarity of self-organization in two-dimensional fluids and low-to-high confinement transitions in plasma suggests the universality of the mechanism.Comment: 5 pages, 4 figure