Self-similarity and Reynolds number invariance in Froude modelling

This review aims to improve the reliability of Froude modelling in fluid flows where both the Froude number and Reynolds number are a priori relevant. Two concepts may help to exclude significant Reynolds number scale effects under these conditions: (i) self-similarity and (ii) Reynolds number invariance. Both concepts relate herein to turbulent flows, thereby excluding self-similarity observed in laminar flows and in non-fluid phenomena. These two concepts are illustrated with a wide range of examples: (i) irrotational vortices, wakes, jets and plumes, shear-driven entrainment, high-velocity open channel flows, sediment transport and homogeneous isotropic turbulence and (ii) tidal energy converters, complete mixing in contact tanks and gravity currents. The limitations of self-similarity and Reynolds number invariance are also highlighted. Many fluid phenomena with the limitations under which self-similarity and Reynolds number invariance are observed are summarised in tables, aimed at excluding significant Reynolds number scale effects in physical Froude-based models

A nonextensive approach to Bose-Einstein condensation of trapped interacting boson gas

In the Bose-Einstein condensation of interacting atoms or molecules such as 87Rb, 23Na and 7Li, the theoretical understanding of the transition temperature is not always obvious due to the interactions or zero point energy which cannot be exactly taken into account. The S-wave collision model fails sometimes to account for the condensation temperatures. In this work, we look at the problem within the nonextensive statistics which is considered as a possible theory describing interacting systems. The generalized energy Uq and the particle number Nq of boson gas are given in terms of the nonextensive parameter q. q>1 (q<1) implies repulsive (attractive) interaction with respect to the perfect gas. The generalized condensation temperature Tcq is derived versus Tc given by the perfect gas theory. Thanks to the observed condensation temperatures, we find q ~ 0.1 for 87Rb atomic gas, q ~ 0.95 for 7Li and q ~ 0.62 for 23Na. It is concluded that the effective interactions are essentially attractive for the three considered atoms, which is consistent with the observed temperatures higher than those predicted by the conventional theory

Understanding heavy Fermion from generalized statistics

Heavy electrons in superconducting materials are widely studied with the Kondo lattice t - J model. Numerical results have shown that the Fermi surface of these correlated particles undergoes a flattening effect according to the coupling degree J. This behaviour is not easy to understand from the theoretical point of view within standard Fermi - Dirac statistics and non-standard theories such as fractional exclusion statistics for anyons and Tsallis nonextensive statistics. The present work is an attempt to account for the heavy electron distribution within incomplete statistics (IS) which is developed for complex systems with interactions which make the statistics incomplete such that Σ i=1 w P i q=1. The parameter q, when different from unity, characterizes the incompleteness of the statistics. It is shown that the correlated electrons can be described with the help of IS with q related to the coupling constant J in the context of Kondo model. © Springer Science+Business Media, LLC 2007

Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte

On Froude-Cauchy similitude

It is common engineering practice in a hydraulic model study involving both gravity waves and a solid structure to measure the hydrodynamic forces on the model of the structure and then calculate the resulting internal structural stresses. Because of the large variety of available elastic material, and the latest development in solid state physics, it is now feasible to measure directly these structural stresses in a scale model study. It is shown how the similitude of elastic force s in structure s subjected to wave action can b e made compatible with the Froude similitude valid for hydraulic motion. Several examples are presented to illustrate the method . These include the study of the elastic response of an ice floe, the motion of an underwater membrane-type oil storage tank, the behavior of the Mohole riser, and the motion of a Texas Tower type of structure under wave action. Results obtained in the NESCO wave tank are also presented