QUANTUM MACHINE LEARNING @ QuDIT:
methods traditionally developed for classical and quantum complex systems will be applied to optimize and characterize complex quantum computing machines and complex topological quantum states possibly establishing a relation between quantum computational complexity and quantum complex behavior.
We use a neural network variational ansatz to compute Gaussian quantum discrete solitons in an array of waveguides described by the quantum discrete nonlinear Schroedinger equation. By training the quantum machine learning model in the phase space, we find different quantum soliton solutions varying the number of particles and interaction strength. The use of Gaussian states enables measuring the degree of entanglement and the boson sampling patterns. We compute the probability of generating different particle pairs when varying the soliton features and unveil that bound states of discrete solitons emit correlated pairs of photons. These results may have a role in boson sampling experiments with nonlinear systems and in developing quantum processors to generate entangled many-photon nonlinear states.
Claudio Conti, arxv:2110.12379 (2021)
Training Gaussian boson sampling by quantum machine learning
We use neural networks to represent the characteristic function of many-body Gaussian states in the quantum phase space. By a pullback mechanism, we model transformations due to unitary operators as linear layers that can be cascaded to simulate complex multi-particle processes. We use the layered neural networks for non-classical light propagation in random interferometers, and compute boson pattern probabilities by automatic differentiation. This is a viable strategy for training Gaussian boson sampling. We demonstrate that multi-particle events in Gaussian boson sampling can be optimized by a proper design and training of the neural network weights. The results are potentially useful to the creation of new sources and complex circuits for quantum technologies.
C. Conti, Quantum Machine Intelligence (2021) 3:26
WHAT QUANTUM OPTICS, QUANTUM INFORMATION AND QUANTUM MATERIAL THEORY CAN TEACH US ABOUT GRAVITY AND FUNDAMENTAL QUANTUM MECHANICS
There is only one time
We draw a picture of physical systems that allows us to recognize what “time” is by requiring consistency with the way that time enters the fundamental laws of Physics. Elements of the picture are two non-interacting and yet entangled quantum systems, one of which acting as a clock. The setting is based on the Page and Wootters mechanism, with tools from large-N quantum approaches. Starting from an overall quantum description, we first take the classical limit of the clock only, and then of the clock and the evolving system altogether; we thus derive the Schrödinger equation in the first case, and the Hamilton equations of motion in the second. This work shows that there is not a “quantum time”, possibly opposed to a “classical” one; there is only one time, and it is a manifestation of entanglement.
C. Foti, A. Coppo, G. Barni and A Cuccoli, Nature Communications 12, 1787 (2021)
ULTRASTRONG LIGHT-MATTER COUPLING AND QUANTUM SENSING
Critical Quantum Metrology with a Finite-Component Quantum Phase Transition
Physical systems close to a quantum phase transition exhibit a divergent susceptibility, suggesting that an arbitrarily high precision may be achieved by exploiting quantum critical systems as probes to estimate a physical parameter. However, such an improvement in sensitivity is counterbalanced by the closing of the energy gap, which implies a critical slowing down and an inevitable growth of the protocol duration. Here, we design different metrological protocols that exploit the superradiant phase transition of the quantum Rabi model, a finite-component system composed of a single two-level atom interacting with a single bosonic mode. We show that, in spite of the critical slowing down, critical quantum optical probes can achieve a quantum-enhanced time scaling of the sensitivity in frequency-estimation protocols.
Universal Spectral Features of Ultrastrongly Coupled Systems
IWe identify universal properties of the low-energy subspace of a wide class of quantum optical models in the ultrastrong coupling limit, where the coupling strength dominates over all other energy scales in the system. We show that the symmetry of the light-matter interaction is at the origin of a twofold degeneracy in the spectrum. We prove analytically this result for bounded Hamiltonians and extend it to a class of models with unbounded operators. As a consequence, we show that the emergence of superradiant phases previously investigated in the context of critical phenomena, is a general property of the ultrastrong coupling limit. The set of models we consider encompasses different scenarios of possible interplay between critical behavior and superradiance.
QUANTUM DISSIPATIVE DYNAMICS IN THE BARDEEN-COOPER-SCHRIEFFER THEORY OF SUPERCONDUCTIVITY
Emergent parametric resonances and time-crystal phases in driven Bardeen-Cooper-Schrieffer systems
We study the out-of-equilibrium dynamics of a Bardeen-Cooper-Schrieffer condensate subject to a periodic drive. We demonstrate that the combined effect of drive and interactions results in emerging parametric resonances, analogous to a vertically driving pendulum. In particular, Arnold tongues appear when the driving frequency matches 2Δ0/n, with n a natural number, and Δ0 the equilibrium gap parameter. Inside the Arnold tongues we find a commensurate time-crystal condensate which retains the U(1) symmetry breaking of the parent superfluid/superconducting phase and shows an additional time-translational symmetry breaking. Outside these tongues, the synchronized collective Higgs mode found in quench protocols is stabilized without the need of a strong perturbation. Our results are directly relevant to cold-atom and condensed-matter systems and do not require very long energy relaxation times to be observed.
Nonlinear Dynamics of Driven Superconductors with Dissipation
In the absence of dissipation a periodically driven BCS superconductor can enter a coherent nonlinear regime of collective Rabi oscillations which last for arbitrary long times [Ojeda Collado et al., Phys. Rev. B 98, 214519 (2018)]. Here we show that dissipation effects introduce dramatic changes: (i) The collective Rabi mode becomes a transient. (ii) At long times a steady state is reached showing strong nonlinear effects for large enough drive strength. We identify the physical parameters governing the various crossovers and present a detailed computation of time- and angle-resolved photoemission spectroscopy (tr-ARPES) and time-resolved tunneling spectra aiming at detecting the collective Rabi oscillations and the steady-state nonlinearities. We show also that second harmonic generation is allowed for a drive which acts on the BCS coupling constant.
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Scalar Turbulence
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.
Fortunately, this is not the case. As recognized after an ispiring work due to the Dirac medalist R. Kraichnan, at least in some “theoretical circumstances”, when the velocity field statistics can be prescribed and mimic that of a true turbulent flow, it is possible to understand much of the scalar statistics and, in particular, to identifythe mechanism at the basis of intermittency. The main reason underlying the possibility to understand scalar turbulence is the fact that this is a linear problem. Indeed, unlike Navier-Stokes equations, which are non linear in the velocity field, the advection diffusion equation is linear in the scalar field. In a nutshell, we can say that the problem of scalar turbulence, after a suitable procedure of averaging of the velocity field statistcs can be reduced to a generalization of heat transport and thus can be solved.
However, this is possible only when the transported scalar field is passive, i.e. it does not influence (modify) the carrier flow, as e.g. the dye in the figure. The situation is completely different when the transported field has some feedback on the fluid as, e.g., temperature (acting on the fluid via bouyancy) or the magnetic potential (which in two dimensions is also a scalar field acting on the fluid via the Lorentz force). In this case we speak of active scalars and the problem is again fully nonlinear as the Navier-Stokes equations and we are back to all the difficulties of turbulence.
The research activity undertaken by us has focused on several aspects of turbulent scalar transport. In particular, active and passive scalar fields (whose evolution rules are the same) evolving in the same realization of a turbulent velocity field have been numerically and theoretically investigated in order to elucidate the differences and similarities between them. The problem has been brought back to the statistical properties of the trajectories of fluid elements and their correlation with the mechanism of excitation of the scalar field fluctuations. Extensive studies have shown that under different circumstances (different flow settings) active and passive scalars may or may not share statistical features. For instance, it has been found that there are cases in which the two fields blatantly differ not only in terms of statistics but also in their dynamical behavior as, e.g., illustrated in the movie below. For details on these studies see a short review[1].
The left and right panels illustrate the evolution of active and passive scalar fields, respectively which evolve in the same velocity field. The problem here illustrated is magnetohydrodynamics in two dimensions, where the active field is the magnetic potential. Note that while the active scalar is characterized by non stationary structures which grow in scale, the passive scalar structures soon become statistically steady. Technically speaking the former is performing an inverse cascade while the latter a direct one. Such huge dynamical difference originate from the strong correlations between the trajectories of fluid elements and the active field. Note indeed the strong similarity between Lagrangian propagator (middle) and the active scalar structures. (note that time is going backward)
In the framework of passive transport it has been also investigated the Lagrangian (i.e. based on the properties of particle trajectories) origin of the formation of very strong and very weak fluctuations of the concetration field — i.e. of those structures which are dubbed plateaus or fronts in atmospheric science—. This was possible thanks to the development of a new algorithmic strategy allowing the back in time integration of the Navier-Stokes equation to be efficiently performed [2]. Moreover, the universality of passive scalar statistics with respect to the energy injection mechanism has been explored in comparison with fluid (and modeled) turbulence by considering power law forcing which stand at the basis of Renormalization Group approaches to turbulence [3], while an explanation has been proposed for some non-universal aspects of scalar statistics in the presence of a mean shear superimposed to turbulent flows [4].
Relevant Publications
[1] Active and passive fields face to face
A. Celani, M.Cencini, A. Mazzino and M. Vergassola
New J. Phys. 6 , 72 (2004)
[2] Going forth and back in time: a fast and parsimonious algorithm for mixed initial/final-value problems
A. Celani, M. Cencini and A. Noullez
Physica D 195 , 283 (2004)
[3] Anomalous scaling and universality in hydrodynamic systems with power-law forcing
L. Biferale, M. Cencini, A. Lanotte, M. Sbragaglia and F. Toschi
New J. Phys. 6, 37 (2004)
[4] Shear effects in passive scalars spectra
A. Celani, M. Cencini, M. Vergassola, D. Vincenzi and E. Villermaux
J. Fluid Mech. 523 , 99 (2005)
Inertial Particles in Turbulent Flows
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.
Heavy (red) and Light (blue) particles in a slice of the simulation box, obtained from a direct numerical simulation of turbulence. Note that the two particle classes are segregated and organized in complex clusterized structures. (RIGHT) Fractal dimension of particle clusters as a function of both the response time and the density ratio, as estimated from the computation of the Lyapunov dimension. Note that light (B>1) particles tend to clusterize much more than heavy (B<1) ones, and that there is an optimal Stokes number for observing maximal clustering.
For instance, the presence of clusters enhances the probability of two particles to be at interaction distance for e.g. collisions or chemical reactions. Collisions rates are further enhanced by the velocity difference among two particles, which can be rather large thanks to inertia which makes particle velocities uncorrelated with that of the fluid. The one just described seems to be an important mechanism responsible for the formation of rain drops in warm clouds, which is an important still open problem. Understanding the statistical and dynamical properties of inertial particles together with their clustering is relevant also to aerosol physics whose importance in climatic issues, pollution or in human health cannot be understimated.
Typically inertial particles are characterized in terms of two “control paramenter”: the Stokes number (St) measuring the response time of particles in unit of the smallest time scale of the flow, and the mas density ratio with respect to the fluid (B). Therefore, the main issue is to understand the behavior of the dynamics and statistics particles as a function of both St and B.
Fractal dimension of particle clusters as a function of both the response time and the density ratio, as estimated from the computation of the Lyapunov dimension. Note that light (B>1) particles tend to clusterize much more than heavy (B<1) ones, and that there is an optimal Stokes number for observing maximal clustering.
Further, properly characterizing the segregation among particles with different inertia (i.e. different density) would help both in devising new mass separation strategies and also in developing new tools for probing specific regions/structures of turbulent flows.
For their importance in enviromental and industrial applications inertial particles are subject of an ongoing research activity since the last five years. Our reasearch focused on several aspects of the problem. Theoretical studies in which the properties of the velocity field are prescribed and well under control allowed us to achieve a better understanding of the mechanisms at the basis of clustering and to develop effective models for the collision rate among particles (click here for details) [1]. In particular, the problem of polydisperse particle suspensions have been approached borrowing tools and ideas from dynamical systems theory –such a strange attractors with multifractal measures–, allowing us to achieve a suitable theoretical language for the description of inertial particles [1].
Still considering stochastic flows mimiking true turbulent ones, but with some simplification expecially concerning time correlations, it was possible to obtain an (even analytical) understanding of many features of clustering expecially in the very large and very small St asymptotics in both differentiable [2] (mimiking turbulence dissipative scales) and rough [3](mimiking turbulence inertial range) stochastic flows. These results have been extended and summarized in a compact review [4]. In parallel with the investigation of simplified random flow it has been carried on an extensive program of large scale direct numerical simulations of moderately high Reynolds number flows transporting millions of particles having different density and response times this allowed us to achieve a rather systematic characterization of particle clustering [9], acceleration [5,6], Lyapunov exponents [8] and segregation [10] as a function of St and B. For instance, the figure on the right depicts the Lyapunov dimension as a function of the density ratio and of St for particles in a DNS with resolution 1283. Some of these results were also subject of successful comparison with independent experimental results.
Relevant Publications
Stochastic model flows
[1] Clustering and collisions of heavy particles in random smooth flows
J. Bec, A. Celani, M. Cencini and S. Musacchio Phys. Fluids 17 073301 (2005)
[2] Heavy particles in incompressible flows: the large Stokes number asymptotics
J. Bec, M. Cencini and R. Hillenbrand Physica D 226, 11 (2007)
[3] Clustering of Heavy particles in random self-similar flows
J. Bec, M. Cencini and R. Hillenbrand Phys. Rev. E (Rapid Comm.) 75, 025301 (2007)
[4] Stochastic suspensions of heavy particles
J. Bec, M. Cencini, R. Hillerbrand and K. Turitsyn Physica D 237 2037 (2008)
Turbulent realistic flows
[5] Acceleration statistics of heavy particles in turbulence
J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini, A. Lanotte, S. Musacchio, and F. Toschi J. Fluid Mech. 550, 349 (2006)
[6] Dynamics and statistics of heavy particles in turbulent flows
M. Cencini, J. Bec, L. Biferale, G. Boffetta, A. Celani, A. Lanotte, S. Musacchio and F. Toschi J.Turb. 7 , 1 (2006)
[7] On the effects of vortex trapping on the velocity statistics of tracers and heavy particle in turbulent flows
J. Bec, L. Biferale, M. Cencini, A. Lanotte and F. Toschi Phys. Fluids (Letter to the Editor) 18 , 081702 (2006)
[8] Lyapunov exponents of heavy particles in turbulence
J. Bec, L. Biferale, G. Boffetta, M. Cencini, S. Musacchio and F. Toschi Phys. Fluids (Letter to the Editor) 18 , 091702 (2006)
[9] Heavy particle concentration in turbulence at dissipative and inertial scales
J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio and F. Toschi Phys. Rev. Lett. 98, 084502 (2007)
[10] Quantifying turbulence induced segregation of inertial particles
E. Calzavarini, M. Cencini, D. Lohse and F. Toschi Phys. Rev. Lett. 101, 084504 (2008)
Introduction to Turbulence
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
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.
Fortunately, this is not the case. As recognized after an ispiring work due to the Dirac medalist R. Kraichnan, at least in some “theoretical circumstances”, when the velocity field statistics can be prescribed and mimic that of a true turbulent flow, it is possible to understand much of the scalar statistics and, in particular, to identifythe mechanism at the basis of intermittency. The main reason underlying the possibility to understand scalar turbulence is the fact that this is a linear problem. Indeed, unlike Navier-Stokes equations, which are non linear in the velocity field, the advection diffusion equation is linear in the scalar field. In a nutshell, we can say that the problem of scalar turbulence, after a suitable procedure of averaging of the velocity field statistcs can be reduced to a generalization of heat transport and thus can be solved.
However, this is possible only when the transported scalar field is passive, i.e. it does not influence (modify) the carrier flow, as e.g. the dye in the figure. The situation is completely different when the transported field has some feedback on the fluid as, e.g., temperature (acting on the fluid via bouyancy) or the magnetic potential (which in two dimensions is also a scalar field acting on the fluid via the Lorentz force). In this case we speak of active scalars and the problem is again fully nonlinear as the Navier-Stokes equations and we are back to all the difficulties of turbulence.
The research activity undertaken by us has focused on several aspects of turbulent scalar transport. In particular, active and passive scalar fields (whose evolution rules are the same) evolving in the same realization of a turbulent velocity field have been numerically and theoretically investigated in order to elucidate the differences and similarities between them. The problem has been brought back to the statistical properties of the trajectories of fluid elements and their correlation with the mechanism of excitation of the scalar field fluctuations. Extensive studies have shown that under different circumstances (different flow settings) active and passive scalars may or may not share statistical features. For instance, it has been found that there are cases in which the two fields blatantly differ not only in terms of statistics but also in their dynamical behavior as, e.g., illustrated in the movie below. For details on these studies see a short review[1].
The left and right panels illustrate the evolution of active and passive scalar fields, respectively which evolve in the same velocity field. The problem here illustrated is magnetohydrodynamics in two dimensions, where the active field is the magnetic potential. Note that while the active scalar is characterized by non stationary structures which grow in scale, the passive scalar structures soon become statistically steady. Technically speaking the former is performing an inverse cascade while the latter a direct one. Such huge dynamical difference originate from the strong correlations between the trajectories of fluid elements and the active field. Note indeed the strong similarity between Lagrangian propagator (middle) and the active scalar structures. (note that time is going backward)
In the framework of passive transport it has been also investigated the Lagrangian (i.e. based on the properties of particle trajectories) origin of the formation of very strong and very weak fluctuations of the concetration field — i.e. of those structures which are dubbed plateaus or fronts in atmospheric science—. This was possible thanks to the development of a new algorithmic strategy allowing the back in time integration of the Navier-Stokes equation to be efficiently performed [2]. Moreover, the universality of passive scalar statistics with respect to the energy injection mechanism has been explored in comparison with fluid (and modeled) turbulence by considering power law forcing which stand at the basis of Renormalization Group approaches to turbulence [3], while an explanation has been proposed for some non-universal aspects of scalar statistics in the presence of a mean shear superimposed to turbulent flows [4].
Relevant Publications
[1] Active and passive fields face to face
A. Celani, M.Cencini, A. Mazzino and M. Vergassola
New J. Phys. 6 , 72 (2004)
[2] Going forth and back in time: a fast and parsimonious algorithm for mixed initial/final-value problems
A. Celani, M. Cencini and A. Noullez
Physica D 195 , 283 (2004)
[3] Anomalous scaling and universality in hydrodynamic systems with power-law forcing
L. Biferale, M. Cencini, A. Lanotte, M. Sbragaglia and F. Toschi
New J. Phys. 6, 37 (2004)
[4] Shear effects in passive scalars spectra
A. Celani, M. Cencini, M. Vergassola, D. Vincenzi and E. Villermaux
J. Fluid Mech. 523 , 99 (2005)
Inertial Particles in Turbulent Flows
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.
Heavy (red) and Light (blue) particles in a slice of the simulation box, obtained from a direct numerical simulation of turbulence. Note that the two particle classes are segregated and organized in complex clusterized structures. (RIGHT) Fractal dimension of particle clusters as a function of both the response time and the density ratio, as estimated from the computation of the Lyapunov dimension. Note that light (B>1) particles tend to clusterize much more than heavy (B<1) ones, and that there is an optimal Stokes number for observing maximal clustering.
For instance, the presence of clusters enhances the probability of two particles to be at interaction distance for e.g. collisions or chemical reactions. Collisions rates are further enhanced by the velocity difference among two particles, which can be rather large thanks to inertia which makes particle velocities uncorrelated with that of the fluid. The one just described seems to be an important mechanism responsible for the formation of rain drops in warm clouds, which is an important still open problem. Understanding the statistical and dynamical properties of inertial particles together with their clustering is relevant also to aerosol physics whose importance in climatic issues, pollution or in human health cannot be understimated.
Typically inertial particles are characterized in terms of two “control paramenter”: the Stokes number (St) measuring the response time of particles in unit of the smallest time scale of the flow, and the mas density ratio with respect to the fluid (B). Therefore, the main issue is to understand the behavior of the dynamics and statistics particles as a function of both St and B.
Fractal dimension of particle clusters as a function of both the response time and the density ratio, as estimated from the computation of the Lyapunov dimension. Note that light (B>1) particles tend to clusterize much more than heavy (B<1) ones, and that there is an optimal Stokes number for observing maximal clustering.
Further, properly characterizing the segregation among particles with different inertia (i.e. different density) would help both in devising new mass separation strategies and also in developing new tools for probing specific regions/structures of turbulent flows.
For their importance in enviromental and industrial applications inertial particles are subject of an ongoing research activity since the last five years. Our reasearch focused on several aspects of the problem. Theoretical studies in which the properties of the velocity field are prescribed and well under control allowed us to achieve a better understanding of the mechanisms at the basis of clustering and to develop effective models for the collision rate among particles (click here for details) [1]. In particular, the problem of polydisperse particle suspensions have been approached borrowing tools and ideas from dynamical systems theory –such a strange attractors with multifractal measures–, allowing us to achieve a suitable theoretical language for the description of inertial particles [1].
Still considering stochastic flows mimiking true turbulent ones, but with some simplification expecially concerning time correlations, it was possible to obtain an (even analytical) understanding of many features of clustering expecially in the very large and very small St asymptotics in both differentiable [2] (mimiking turbulence dissipative scales) and rough [3](mimiking turbulence inertial range) stochastic flows. These results have been extended and summarized in a compact review [4]. In parallel with the investigation of simplified random flow it has been carried on an extensive program of large scale direct numerical simulations of moderately high Reynolds number flows transporting millions of particles having different density and response times this allowed us to achieve a rather systematic characterization of particle clustering [9], acceleration [5,6], Lyapunov exponents [8] and segregation [10] as a function of St and B. For instance, the figure on the right depicts the Lyapunov dimension as a function of the density ratio and of St for particles in a DNS with resolution 1283. Some of these results were also subject of successful comparison with independent experimental results.
Relevant Publications
Stochastic model flows
[1] Clustering and collisions of heavy particles in random smooth flows
J. Bec, A. Celani, M. Cencini and S. Musacchio Phys. Fluids 17 073301 (2005)
[2] Heavy particles in incompressible flows: the large Stokes number asymptotics
J. Bec, M. Cencini and R. Hillenbrand Physica D 226, 11 (2007)
[3] Clustering of Heavy particles in random self-similar flows
J. Bec, M. Cencini and R. Hillenbrand Phys. Rev. E (Rapid Comm.) 75, 025301 (2007)
[4] Stochastic suspensions of heavy particles
J. Bec, M. Cencini, R. Hillerbrand and K. Turitsyn Physica D 237 2037 (2008)
Turbulent realistic flows
[5] Acceleration statistics of heavy particles in turbulence
J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini, A. Lanotte, S. Musacchio, and F. Toschi J. Fluid Mech. 550, 349 (2006)
[6] Dynamics and statistics of heavy particles in turbulent flows
M. Cencini, J. Bec, L. Biferale, G. Boffetta, A. Celani, A. Lanotte, S. Musacchio and F. Toschi J.Turb. 7 , 1 (2006)
[7] On the effects of vortex trapping on the velocity statistics of tracers and heavy particle in turbulent flows
J. Bec, L. Biferale, M. Cencini, A. Lanotte and F. Toschi Phys. Fluids (Letter to the Editor) 18 , 081702 (2006)
[8] Lyapunov exponents of heavy particles in turbulence
J. Bec, L. Biferale, G. Boffetta, M. Cencini, S. Musacchio and F. Toschi Phys. Fluids (Letter to the Editor) 18 , 091702 (2006)
[9] Heavy particle concentration in turbulence at dissipative and inertial scales
J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio and F. Toschi Phys. Rev. Lett. 98, 084502 (2007)
[10] Quantifying turbulence induced segregation of inertial particles
E. Calzavarini, M. Cencini, D. Lohse and F. Toschi Phys. Rev. Lett. 101, 084504 (2008)
Introduction to Turbulence
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
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.
Fortunately, this is not the case. As recognized after an ispiring work due to the Dirac medalist R. Kraichnan, at least in some “theoretical circumstances”, when the velocity field statistics can be prescribed and mimic that of a true turbulent flow, it is possible to understand much of the scalar statistics and, in particular, to identifythe mechanism at the basis of intermittency. The main reason underlying the possibility to understand scalar turbulence is the fact that this is a linear problem. Indeed, unlike Navier-Stokes equations, which are non linear in the velocity field, the advection diffusion equation is linear in the scalar field. In a nutshell, we can say that the problem of scalar turbulence, after a suitable procedure of averaging of the velocity field statistcs can be reduced to a generalization of heat transport and thus can be solved.
However, this is possible only when the transported scalar field is passive, i.e. it does not influence (modify) the carrier flow, as e.g. the dye in the figure. The situation is completely different when the transported field has some feedback on the fluid as, e.g., temperature (acting on the fluid via bouyancy) or the magnetic potential (which in two dimensions is also a scalar field acting on the fluid via the Lorentz force). In this case we speak of active scalars and the problem is again fully nonlinear as the Navier-Stokes equations and we are back to all the difficulties of turbulence.
The research activity undertaken by us has focused on several aspects of turbulent scalar transport. In particular, active and passive scalar fields (whose evolution rules are the same) evolving in the same realization of a turbulent velocity field have been numerically and theoretically investigated in order to elucidate the differences and similarities between them. The problem has been brought back to the statistical properties of the trajectories of fluid elements and their correlation with the mechanism of excitation of the scalar field fluctuations. Extensive studies have shown that under different circumstances (different flow settings) active and passive scalars may or may not share statistical features. For instance, it has been found that there are cases in which the two fields blatantly differ not only in terms of statistics but also in their dynamical behavior as, e.g., illustrated in the movie below. For details on these studies see a short review[1].
The left and right panels illustrate the evolution of active and passive scalar fields, respectively which evolve in the same velocity field. The problem here illustrated is magnetohydrodynamics in two dimensions, where the active field is the magnetic potential. Note that while the active scalar is characterized by non stationary structures which grow in scale, the passive scalar structures soon become statistically steady. Technically speaking the former is performing an inverse cascade while the latter a direct one. Such huge dynamical difference originate from the strong correlations between the trajectories of fluid elements and the active field. Note indeed the strong similarity between Lagrangian propagator (middle) and the active scalar structures. (note that time is going backward)
In the framework of passive transport it has been also investigated the Lagrangian (i.e. based on the properties of particle trajectories) origin of the formation of very strong and very weak fluctuations of the concetration field — i.e. of those structures which are dubbed plateaus or fronts in atmospheric science—. This was possible thanks to the development of a new algorithmic strategy allowing the back in time integration of the Navier-Stokes equation to be efficiently performed [2]. Moreover, the universality of passive scalar statistics with respect to the energy injection mechanism has been explored in comparison with fluid (and modeled) turbulence by considering power law forcing which stand at the basis of Renormalization Group approaches to turbulence [3], while an explanation has been proposed for some non-universal aspects of scalar statistics in the presence of a mean shear superimposed to turbulent flows [4].
Relevant Publications
[1] Active and passive fields face to face
A. Celani, M.Cencini, A. Mazzino and M. Vergassola
New J. Phys. 6 , 72 (2004)
[2] Going forth and back in time: a fast and parsimonious algorithm for mixed initial/final-value problems
A. Celani, M. Cencini and A. Noullez
Physica D 195 , 283 (2004)
[3] Anomalous scaling and universality in hydrodynamic systems with power-law forcing
L. Biferale, M. Cencini, A. Lanotte, M. Sbragaglia and F. Toschi
New J. Phys. 6, 37 (2004)
[4] Shear effects in passive scalars spectra
A. Celani, M. Cencini, M. Vergassola, D. Vincenzi and E. Villermaux
J. Fluid Mech. 523 , 99 (2005)
Inertial Particles in Turbulent Flows
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.
Heavy (red) and Light (blue) particles in a slice of the simulation box, obtained from a direct numerical simulation of turbulence. Note that the two particle classes are segregated and organized in complex clusterized structures. (RIGHT) Fractal dimension of particle clusters as a function of both the response time and the density ratio, as estimated from the computation of the Lyapunov dimension. Note that light (B>1) particles tend to clusterize much more than heavy (B<1) ones, and that there is an optimal Stokes number for observing maximal clustering.
For instance, the presence of clusters enhances the probability of two particles to be at interaction distance for e.g. collisions or chemical reactions. Collisions rates are further enhanced by the velocity difference among two particles, which can be rather large thanks to inertia which makes particle velocities uncorrelated with that of the fluid. The one just described seems to be an important mechanism responsible for the formation of rain drops in warm clouds, which is an important still open problem. Understanding the statistical and dynamical properties of inertial particles together with their clustering is relevant also to aerosol physics whose importance in climatic issues, pollution or in human health cannot be understimated.
Typically inertial particles are characterized in terms of two “control paramenter”: the Stokes number (St) measuring the response time of particles in unit of the smallest time scale of the flow, and the mas density ratio with respect to the fluid (B). Therefore, the main issue is to understand the behavior of the dynamics and statistics particles as a function of both St and B.
Fractal dimension of particle clusters as a function of both the response time and the density ratio, as estimated from the computation of the Lyapunov dimension. Note that light (B>1) particles tend to clusterize much more than heavy (B<1) ones, and that there is an optimal Stokes number for observing maximal clustering.
Further, properly characterizing the segregation among particles with different inertia (i.e. different density) would help both in devising new mass separation strategies and also in developing new tools for probing specific regions/structures of turbulent flows.
For their importance in enviromental and industrial applications inertial particles are subject of an ongoing research activity since the last five years. Our reasearch focused on several aspects of the problem. Theoretical studies in which the properties of the velocity field are prescribed and well under control allowed us to achieve a better understanding of the mechanisms at the basis of clustering and to develop effective models for the collision rate among particles (click here for details) [1]. In particular, the problem of polydisperse particle suspensions have been approached borrowing tools and ideas from dynamical systems theory –such a strange attractors with multifractal measures–, allowing us to achieve a suitable theoretical language for the description of inertial particles [1].
Still considering stochastic flows mimiking true turbulent ones, but with some simplification expecially concerning time correlations, it was possible to obtain an (even analytical) understanding of many features of clustering expecially in the very large and very small St asymptotics in both differentiable [2] (mimiking turbulence dissipative scales) and rough [3](mimiking turbulence inertial range) stochastic flows. These results have been extended and summarized in a compact review [4]. In parallel with the investigation of simplified random flow it has been carried on an extensive program of large scale direct numerical simulations of moderately high Reynolds number flows transporting millions of particles having different density and response times this allowed us to achieve a rather systematic characterization of particle clustering [9], acceleration [5,6], Lyapunov exponents [8] and segregation [10] as a function of St and B. For instance, the figure on the right depicts the Lyapunov dimension as a function of the density ratio and of St for particles in a DNS with resolution 1283. Some of these results were also subject of successful comparison with independent experimental results.
Relevant Publications
Stochastic model flows
[1] Clustering and collisions of heavy particles in random smooth flows
J. Bec, A. Celani, M. Cencini and S. Musacchio Phys. Fluids 17 073301 (2005)
[2] Heavy particles in incompressible flows: the large Stokes number asymptotics
J. Bec, M. Cencini and R. Hillenbrand Physica D 226, 11 (2007)
[3] Clustering of Heavy particles in random self-similar flows
J. Bec, M. Cencini and R. Hillenbrand Phys. Rev. E (Rapid Comm.) 75, 025301 (2007)
[4] Stochastic suspensions of heavy particles
J. Bec, M. Cencini, R. Hillerbrand and K. Turitsyn Physica D 237 2037 (2008)
Turbulent realistic flows
[5] Acceleration statistics of heavy particles in turbulence
J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini, A. Lanotte, S. Musacchio, and F. Toschi J. Fluid Mech. 550, 349 (2006)
[6] Dynamics and statistics of heavy particles in turbulent flows
M. Cencini, J. Bec, L. Biferale, G. Boffetta, A. Celani, A. Lanotte, S. Musacchio and F. Toschi J.Turb. 7 , 1 (2006)
[7] On the effects of vortex trapping on the velocity statistics of tracers and heavy particle in turbulent flows
J. Bec, L. Biferale, M. Cencini, A. Lanotte and F. Toschi Phys. Fluids (Letter to the Editor) 18 , 081702 (2006)
[8] Lyapunov exponents of heavy particles in turbulence
J. Bec, L. Biferale, G. Boffetta, M. Cencini, S. Musacchio and F. Toschi Phys. Fluids (Letter to the Editor) 18 , 091702 (2006)
[9] Heavy particle concentration in turbulence at dissipative and inertial scales
J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio and F. Toschi Phys. Rev. Lett. 98, 084502 (2007)
[10] Quantifying turbulence induced segregation of inertial particles
E. Calzavarini, M. Cencini, D. Lohse and F. Toschi Phys. Rev. Lett. 101, 084504 (2008)
Introduction to Turbulence
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
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.
Fortunately, this is not the case. As recognized after an ispiring work due to the Dirac medalist R. Kraichnan, at least in some “theoretical circumstances”, when the velocity field statistics can be prescribed and mimic that of a true turbulent flow, it is possible to understand much of the scalar statistics and, in particular, to identifythe mechanism at the basis of intermittency. The main reason underlying the possibility to understand scalar turbulence is the fact that this is a linear problem. Indeed, unlike Navier-Stokes equations, which are non linear in the velocity field, the advection diffusion equation is linear in the scalar field. In a nutshell, we can say that the problem of scalar turbulence, after a suitable procedure of averaging of the velocity field statistcs can be reduced to a generalization of heat transport and thus can be solved.
However, this is possible only when the transported scalar field is passive, i.e. it does not influence (modify) the carrier flow, as e.g. the dye in the figure. The situation is completely different when the transported field has some feedback on the fluid as, e.g., temperature (acting on the fluid via bouyancy) or the magnetic potential (which in two dimensions is also a scalar field acting on the fluid via the Lorentz force). In this case we speak of active scalars and the problem is again fully nonlinear as the Navier-Stokes equations and we are back to all the difficulties of turbulence.
The research activity undertaken by us has focused on several aspects of turbulent scalar transport. In particular, active and passive scalar fields (whose evolution rules are the same) evolving in the same realization of a turbulent velocity field have been numerically and theoretically investigated in order to elucidate the differences and similarities between them. The problem has been brought back to the statistical properties of the trajectories of fluid elements and their correlation with the mechanism of excitation of the scalar field fluctuations. Extensive studies have shown that under different circumstances (different flow settings) active and passive scalars may or may not share statistical features. For instance, it has been found that there are cases in which the two fields blatantly differ not only in terms of statistics but also in their dynamical behavior as, e.g., illustrated in the movie below. For details on these studies see a short review[1].
The left and right panels illustrate the evolution of active and passive scalar fields, respectively which evolve in the same velocity field. The problem here illustrated is magnetohydrodynamics in two dimensions, where the active field is the magnetic potential. Note that while the active scalar is characterized by non stationary structures which grow in scale, the passive scalar structures soon become statistically steady. Technically speaking the former is performing an inverse cascade while the latter a direct one. Such huge dynamical difference originate from the strong correlations between the trajectories of fluid elements and the active field. Note indeed the strong similarity between Lagrangian propagator (middle) and the active scalar structures. (note that time is going backward)
In the framework of passive transport it has been also investigated the Lagrangian (i.e. based on the properties of particle trajectories) origin of the formation of very strong and very weak fluctuations of the concetration field — i.e. of those structures which are dubbed plateaus or fronts in atmospheric science—. This was possible thanks to the development of a new algorithmic strategy allowing the back in time integration of the Navier-Stokes equation to be efficiently performed [2]. Moreover, the universality of passive scalar statistics with respect to the energy injection mechanism has been explored in comparison with fluid (and modeled) turbulence by considering power law forcing which stand at the basis of Renormalization Group approaches to turbulence [3], while an explanation has been proposed for some non-universal aspects of scalar statistics in the presence of a mean shear superimposed to turbulent flows [4].
Relevant Publications
[1] Active and passive fields face to face
A. Celani, M.Cencini, A. Mazzino and M. Vergassola
New J. Phys. 6 , 72 (2004)
[2] Going forth and back in time: a fast and parsimonious algorithm for mixed initial/final-value problems
A. Celani, M. Cencini and A. Noullez
Physica D 195 , 283 (2004)
[3] Anomalous scaling and universality in hydrodynamic systems with power-law forcing
L. Biferale, M. Cencini, A. Lanotte, M. Sbragaglia and F. Toschi
New J. Phys. 6, 37 (2004)
[4] Shear effects in passive scalars spectra
A. Celani, M. Cencini, M. Vergassola, D. Vincenzi and E. Villermaux
J. Fluid Mech. 523 , 99 (2005)
Inertial Particles in Turbulent Flows
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.
Heavy (red) and Light (blue) particles in a slice of the simulation box, obtained from a direct numerical simulation of turbulence. Note that the two particle classes are segregated and organized in complex clusterized structures. (RIGHT) Fractal dimension of particle clusters as a function of both the response time and the density ratio, as estimated from the computation of the Lyapunov dimension. Note that light (B>1) particles tend to clusterize much more than heavy (B<1) ones, and that there is an optimal Stokes number for observing maximal clustering.
For instance, the presence of clusters enhances the probability of two particles to be at interaction distance for e.g. collisions or chemical reactions. Collisions rates are further enhanced by the velocity difference among two particles, which can be rather large thanks to inertia which makes particle velocities uncorrelated with that of the fluid. The one just described seems to be an important mechanism responsible for the formation of rain drops in warm clouds, which is an important still open problem. Understanding the statistical and dynamical properties of inertial particles together with their clustering is relevant also to aerosol physics whose importance in climatic issues, pollution or in human health cannot be understimated.
Typically inertial particles are characterized in terms of two “control paramenter”: the Stokes number (St) measuring the response time of particles in unit of the smallest time scale of the flow, and the mas density ratio with respect to the fluid (B). Therefore, the main issue is to understand the behavior of the dynamics and statistics particles as a function of both St and B.
Fractal dimension of particle clusters as a function of both the response time and the density ratio, as estimated from the computation of the Lyapunov dimension. Note that light (B>1) particles tend to clusterize much more than heavy (B<1) ones, and that there is an optimal Stokes number for observing maximal clustering.
Further, properly characterizing the segregation among particles with different inertia (i.e. different density) would help both in devising new mass separation strategies and also in developing new tools for probing specific regions/structures of turbulent flows.
For their importance in enviromental and industrial applications inertial particles are subject of an ongoing research activity since the last five years. Our reasearch focused on several aspects of the problem. Theoretical studies in which the properties of the velocity field are prescribed and well under control allowed us to achieve a better understanding of the mechanisms at the basis of clustering and to develop effective models for the collision rate among particles (click here for details) [1]. In particular, the problem of polydisperse particle suspensions have been approached borrowing tools and ideas from dynamical systems theory –such a strange attractors with multifractal measures–, allowing us to achieve a suitable theoretical language for the description of inertial particles [1].
Still considering stochastic flows mimiking true turbulent ones, but with some simplification expecially concerning time correlations, it was possible to obtain an (even analytical) understanding of many features of clustering expecially in the very large and very small St asymptotics in both differentiable [2] (mimiking turbulence dissipative scales) and rough [3](mimiking turbulence inertial range) stochastic flows. These results have been extended and summarized in a compact review [4]. In parallel with the investigation of simplified random flow it has been carried on an extensive program of large scale direct numerical simulations of moderately high Reynolds number flows transporting millions of particles having different density and response times this allowed us to achieve a rather systematic characterization of particle clustering [9], acceleration [5,6], Lyapunov exponents [8] and segregation [10] as a function of St and B. For instance, the figure on the right depicts the Lyapunov dimension as a function of the density ratio and of St for particles in a DNS with resolution 1283. Some of these results were also subject of successful comparison with independent experimental results.
Relevant Publications
Stochastic model flows
[1] Clustering and collisions of heavy particles in random smooth flows
J. Bec, A. Celani, M. Cencini and S. Musacchio Phys. Fluids 17 073301 (2005)
[2] Heavy particles in incompressible flows: the large Stokes number asymptotics
J. Bec, M. Cencini and R. Hillenbrand Physica D 226, 11 (2007)
[3] Clustering of Heavy particles in random self-similar flows
J. Bec, M. Cencini and R. Hillenbrand Phys. Rev. E (Rapid Comm.) 75, 025301 (2007)
[4] Stochastic suspensions of heavy particles
J. Bec, M. Cencini, R. Hillerbrand and K. Turitsyn Physica D 237 2037 (2008)
Turbulent realistic flows
[5] Acceleration statistics of heavy particles in turbulence
J. Bec, L. Biferale, G. Boffetta, A. Celani, M. Cencini, A. Lanotte, S. Musacchio, and F. Toschi J. Fluid Mech. 550, 349 (2006)
[6] Dynamics and statistics of heavy particles in turbulent flows
M. Cencini, J. Bec, L. Biferale, G. Boffetta, A. Celani, A. Lanotte, S. Musacchio and F. Toschi J.Turb. 7 , 1 (2006)
[7] On the effects of vortex trapping on the velocity statistics of tracers and heavy particle in turbulent flows
J. Bec, L. Biferale, M. Cencini, A. Lanotte and F. Toschi Phys. Fluids (Letter to the Editor) 18 , 081702 (2006)
[8] Lyapunov exponents of heavy particles in turbulence
J. Bec, L. Biferale, G. Boffetta, M. Cencini, S. Musacchio and F. Toschi Phys. Fluids (Letter to the Editor) 18 , 091702 (2006)
[9] Heavy particle concentration in turbulence at dissipative and inertial scales
J. Bec, L. Biferale, M. Cencini, A. Lanotte, S. Musacchio and F. Toschi Phys. Rev. Lett. 98, 084502 (2007)
[10] Quantifying turbulence induced segregation of inertial particles
E. Calzavarini, M. Cencini, D. Lohse and F. Toschi Phys. Rev. Lett. 101, 084504 (2008)
Introduction to Turbulence
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.
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