Institute for Complex Systems - Sapienza - CNR

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ISC Sapienza Turbulence and Transport

Turbulence and Transport

Statistical and dynamical properties of transported substances in turbulent flows

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Introduction to Turbulence

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"What is Turbulence? Turbulence is like pornography. It is hard to define but if you see it, you recognize it immediately." [G.K. Vallis (1999)]

Turbulence is ubiquitous in nature and encompasses phenomena taking place over an extremely wide range of scales from a few millimiters to thousands or hundred of thousands kilmometers, from laboratory to galaxies.

Although the term turbulence is often used to denote very irregular motions taking place in strongly nonlinear systems, fluid turbulence has a more precise meaning being the state of motion of a fluid which is characterized by chaotic, stochastic changes in both its spatial and temporal properties. Fully developed turbulence establishes when the Reynolds number (i.e the ratio between the nonlinear and linear -- dissipative -- terms of the Navier-Stokes equation, describing fluid motion) becomes very high. In such a condition a nonlinear cascade of energy takes place from the scale where motion is excited (the forcing scale, which is typically large) to that where energy is dissipated (at a molecular level), and this inertial range of scales is characterized by nontrivial scale invariance properties. In particular the probability of observing large fluctuations of velocity increments (v(x+r)-v(x)) over a scale r becomes higher and higher as the scale r decreases. This is the intermittency of turbulence which stands still at the frontiers of our understanding, and links to the presence of anomalous scaling laws in the statistics of the velocity field. The only hope to theoretically cope with turbulence is, from a physicists point of view, to assess the universtality of such scaling laws, which would imply that the possibility to understand them should be hidden in Navier-Stokes equations. However, at a mathematical level turbulence, actually the Navier-Stokes equations, constitutes a --literally speaking-- million dollar problem being one of the millenium problems at the Clay Mathematics Institute.

 


 

Scalar Turbulence

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The ability of efficiently mixing transported substances is one of the most distinctive properties of turbulence. For instance, it is turbulence (induced by the spoon) that allows cream to rapidly invade a cup of coffee, indeed if only molecular diffusion would be at play in the coffee at rest the same process would require many hours! Given the statistical complexity of a turbulent velocity field, it is natural to wonder about the resulting complexity in the statistical features of the transported concentration field of a substance (e.g. a fluorescent dye, as in the figure below on the right, the temperature o magnetic field in a star, etc.). Extensive experimental and numerical studies have indeed demonstrated that scalar substances transported by turbulent velocity fields share with the turbulent velocity field many common properties such as intermittency and anomalous scaling laws, with the associated strongly non Gaussian statistics. Therefore, naively one would conclude that as for turbulence the problem cannot be solved.

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Lagrangian Turbulence

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Recently, part of the research activity on turbulence has focused on temporal properties of turbulent statistics which are much less known than the equivalent spatial properties, and are expected to bring information on some of the mechanisms responsible for intermittency in turbulence, for example lagrangian motion is strongly affected by the presence of vortical motion around vortex filaments (see Figure 1). The idea is to study correlations of the fluid velocity along the path of fluid elements (Lagrangian point of view), this is indeed necessary to eliminate spurious correlations induced by the so-called sweeping (i.e. the advection of velocity fluctuations on a given scale due to fluctuations taking place over larger scales). Several independent studies demonstrated that Lagrangian statistics is highly nontrivial expecially the role of vortex entraping seems to be crucial in determining the short time (below and around the Kolmogorov time) behavior of the velocity correlation function

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Inertial Particles in Turbulent Flows

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We already mentioned that enhanced mixing is probably one of the most distinguishing feature of turbulence. When a turbulent flow is seeded with particulate matter having a finite size and/or density different from that of the carrier fluid, new features appear. The figure on the left show the instantaneous position particles which are heavier (e.g. water drops in air) resp. lighter (e.g. air bubbles in water) than the carrier fluid. As one can see two features can be identified: heavy/light particles distribute in a very inhomogeneous way (even if the flow is incompressible) forming clusters and voids; heavy and light particles spontaneously segregate visiting different regions of the flow. Both these phenomena find their roots in the presence of inertia (due to the density difference between particles and fluid and to their finite size) -- hence the name inertial particles-- and they are both very important.

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Transport in binary mixtures

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When a binary fluid mixture at the critical concentration is cooled from a high temperature to a sufficiently low temperature (below a critical one), the original homogeneous phase becomes unstable and spontaneously evolves into two phases separated by an interface. As time advances, an out-of-equilibrium process of phase ordering takes place through the formation of domains of a single phase that grows algebraically in time as L(t)~t1/3. In fluids, the presence of a hydrodynamic velocity field makes this process morecomplicated than the corresponding one in solid alloys. For instance, it is well known that hydrodynamics may accelerate the domain growth to L(t)~t2/3 (see Fig-left, red line). Phase ordering dynamics becomes even more complex and less understood when the fluid mixture is externally driven; beyond their theoretical interest, phase separating binary fluids under flow embody a great technological interest for their distinctive rheological properties.
Last Updated on Wednesday, 27 July 2011 12:45 Read more...