1. PARTICLES AND FIELDS IN TURBULENT FLOWS



The research proposed in this Iniziativa Specifica (IS) is focused on the problem of "Particles and Fields" transported by --and reacting with-- turbulent flows [Frisch 1995]. We want to understand the behavior of the flow once seeded with polymers, surfactants, bubbles, with reactive quantities, as in combustion, or with active scalars like Rayleigh-Benard convection, to cite just some. These are real problems prompting interesting questions for theoretical physics.

The objective of this research effort is to focus on these problems with the methods developed in the context of fundamental theoretical research. In the last 20 years the situation has become even more complex by the recognition of the ubiquity of Lagrangian chaos (chaotic advection) even in the presence of simple Eulerian fields.



Roughly speaking one can distinguish three different classes of problems according to properties of the transported field:



(i) The passive problem, with particles and fields which are transported by the flow without affecting on it [see Falkovich et al 2001, and references therein]



(ii) The reacting problem, given by quantities which are still transported by the flow without affecting on it, but developing chemical or biological reactions [see Livi and Vulpiani 2003, and refrences therein].



(iii) The active problem, where fields and particles modify the flow while being transported [Lumley 1967; Kadanoff 2001].



We propose here to continue a recently initiated research activity, combining theory, simulations and experimental data analysis to advance these fields. The combination of expertise of the groups represented in this IS offers good prospects for rapid progress on realistic and challenging objectives.



From a methodological point of view, we expect to attack two different situations:



(1) The first is when a large-scale and large-time dynamics for the transported quantity can be defined (existence of a scale separation between the underlying flow structures and the advected/reacting quantities).



(2) The second, more common, is when turbulence affects all scales, no scale-separation is any more present, transport, diffusion and feed-back on the flow are intimately related on all relevant scale of motion and on all relevant frequencies.



Concerning the first situation, one among the most challenging issues of modern turbulence theories is to find the way through which ultraviolet degrees of freedom can be averaged out to give rise to large scale effective equations. Solve this problem amounts to solving the long-standing closure problem which characterizes all the known turbulent systems including those which are governed by linear stochastic partial differential equations.

For that purpose, many field theoretic methods have been applied in the last decades, including the Renormalization Group (RG) [Adzheyman 1999] and other renormalized strategies among which the most popular is probably the so-called Multiple Scale Expansion (MSE) [Majda 1999]. All such techniques lead to large scale effective equations where the bare parameters (e.g., bare

viscosity, bare diffusivity or bare resistivity) are replaced by renormalized parameters. The goal is to find such parameters, a task that, in general, requires an interplay between analytical (e.g. RG and MSE) and numerical tools (e.g, Direct Numerical Simulations, DNS). Researchers of this IS have developed in the past optimal analytical and numerical tools to attack such problems. We expect important advancements in the field trying to extend MSE to the case of non-Newtonian flows and active tracers, cases where the long-time, large-scale asymptotic is not always diffusive.



In the absence of scale separation --the second scenario-- one is often forced to resort to Direct Numerical Simulations (DNS). Numerical studies of turbulent systems is a scientific challenge by itself. Benchmarks of all massive parallel machines are usually made on numerical code integrating Navier-Stokes equations. A huge computational power is not sufficient to make long lasting research in the field. New statistical and theoretical tools to analyze and probe physical observables have been developed. Recently, theoretical and numerical tools focused on the Lagrangian properties of the advected particles have been proposed [Falkovich et al. 2001]. Tracking particles statistics allow for a direct control on both the advected quantity and on the turbulent velocity statistics along their trajectories. Recently, the experimental side has been revolutionized by the application of particles detecting techniques borrowed from high energy physics [La Porta et al, 2001]. We believe that the application of Lagrangian techniques, based on the statistics of first passage times of groups of particles to a targeted inter-particle distance and on the shape-to-shape evolution of the particle clouds may dramatically improve the numerical and theoretical understanding of many applied problems. Such techniques have already been extremely useful for the analysis of passive scalar dynamics in isotropic and homogeneous models. The introduction of similar techniques in other flows would dramatically change the subject, with important ramifications in the analysis of field data and numerical simulations. Extending the application of these statistical tools to the velocity components of the particle phase-space may allow us to study also the statistical properties of the carrying fluid.

Let us stress that researchers of this IS have recently performed state-of-the-art DNS of turbulent flows with passive particles advection [Biferale et al 2004].

The extension of these expertises to the cases of ``inertial particles'' and two-phase flows with and without feed-back must be seen as a natural consequence.



Objective & State of the Art



Most known turbulence phenomena in the real world can be classified to belong to one of the three classes cited above. In many cases a small change in a few constituent parameters may let the system switch from one class to another. The scholarly example usually presented is the case of local temperature fluctuations of a mono-phase fluid in the presence of gravity. Here, depending on the temperature imposed at the boundaries, and on the thermal expansion coefficient of the fluid, one may switch from a purely passive case, class (i), where temperature is advected and diffused by the flow without affecting it to the celebrated Rayleigh-Benard convection where temperature drives the whole fluid motion --an example of class (iii). Similarly, macroscopic magnetic fields can affect or not the motion of a conducting fluid depending on the intensity of the Lorentz force. An important astrophysical-geophysical example behind the already cited case of temperature in class (i) is the so called Fermi acceleration of charged particles originally proposed by Fermi for the origin of the cosmic ray. The original proposal in which charged particles could be accelerated by reflection from magnetic propagating disturbances had enormous impact both in the context of dynamical systems theory and astrophysical applications [Lichtenberg 1991]. Another very interesting case belonging to class (i) is given by the passive transport of inertial particles, i.e.

particles with a different density from the carrying flow [Maxey and Riley, 1983]. In most cases, the densities mismatch leads to a transport problem in an ``effective'' compressible flows with interesting properties of particles clusterization in high (low) pressure region if the particles are heavier (lighter) than the carrying fluid. The behavior may strongly depend on the parameters of the two-phases, leading to different collective behavior ranging from fully sticky phases (with collapse of all material in a few attractive blobs) to almost homogeneous concentration -- with intermediate situations of fractal and multifractal clustering. When the reaction of the particle on the flow cannot be neglected one speaks about two-phase flows in the two-way coupling regime, now a problem in class (iii). This is the realm of bubbly laden flows, turbulent or laminar. Here, the problem, intensively studied experimentally and numerically since the beginning of the last century has recently attracted the interest of theoretical physicists. Modelization is still in its infancy. To give the idea, a general agreement on the optimal continuous modelization of bubble feed-back on the flow is still lacking. In other words, scientists in the field are forced to treat the interaction with the flow "bubble by bubble", as descriptions in terms of a bubble concentration field are still inadequate. In class (ii) we have the advection-reaction-diffusion (ARD) which have received in the last decades some attention due to their relevance for spatially extended ecological communities, mixing in reacting flows environmental processes in atmosphere such as ozone reactions. The most general ARD contains 3 terms: an advection term, accounting for the effect of the velocity; a diffusive term; a reaction term, representing the chemical or biological aspects. The analytical treatment of the complete case is not trivial at all. So that for the complete ARD only the limit cases (even if non trivial) have been studied, some simple laminar velocity fields, and velocity field assumed to be random processes in space and/or time.

Another problem which attracts our attention is connected to the spectacular drag reduction effect. The case of turbulent flows that after the injection of infinitesimal polymer concentration (a few parts over million) enjoys a macroscopic reduction in the drag (up to 80 %). This effect, known for more than 50 years ago, has been studied with the proper tools of many-body statistical mechanics only recently.





Expertises



The groups participating in this IS have initiated and led the Italian and European research effort in many general and detailed topics related to the research project. The group of Rome II is formed of three permanent scientists plus a few post-docs and PhD (2 at the present moment). It has a leading experience in numerical and theoretical aspects of turbulence. The group has access to different massively parallel computers (APE, PC-farm of 32 double processors; Ibm-SP4 architectures at CINECA;). In the past, members of the group have performed state-of-the art numerical simulations in three and two dimensional turbulence. They have been pioneering numerical algorithms based on Lattice-Boltzmann theory [Benzi et al. 1992]. Recently, in collaboration with the group of Torino and Ferrara, they have performed a state-of-the-art DNS at 1024^3 resolution, of 3d turbulent flows seeded with millions of particles, collecting the widest data-base of Lagrangian turbulence available worldwide with that resolution [Biferale et al. 2004]. They have also pioneered the use of SO(3) decomposition to disentangle isotropic from anisotropic contributions in turbulent flows [see Biferale and Procaccia 2004, and refrences therein].



The group of Rome I is formed of one permanent scientist and of about 3-4 PhD and/or Post-docs. It has been among the leading group, worldwide, about the application of modern dynamical system theory in hydrodynamical problems [Paladin and Vulpiani, 1987; Crisanti et al. 1991]. Among them we cite the application of multifractals to turbulence and dynamical systems and the development (in collaboration with the group of Torino) of the concept of Finite Size Lyapunov Exponents [Boffetta et al, 2002a].



The group of Torino is formed of 1 permanent scientist plus a few PhD and Post-docs. It has a leading experience in DNS of two dimensional turbulence in both fluid and in plasma systems. The group has also numerically demonstrate, for the first time, the essential validity of Richardson law for relative dispersion in turbulence [Boffetta & Sokolov 2002b] .



The group of Genova is formed of two permanent scientists plus from 3 to 4 PhD and/or post-docs. It has been among the first team that have applied (also in collaboration with the group of Roma I and Torino) modern techniques for Lagrangian simulations to flows with geophysical interest. They also have been pioneering the application of Renormalization Group techniques and Instantonic calculus to turbulence (in collaboration with the group of Firenze) [Collina et al, 2004].



The group of Ferrara is formed of 2 permanent scientists plus 2 among PhD and post-docs. It has pioneered the use of Lattice Boltzmann schemes in SIMD architecture [Tripiccione 2001]. The machines employed has been, over time, Ape 100, ApeMille and nowadays the group is in process of porting all programs to the brand new Ape NEXT platform. Among the physical systems that have been studied in the past, we just cite the turbulent flow in a channel and the convective turbulence in a Rayleigh-Benard cell [Calzavarini et al. 2002].



Scientific Goals



Drag-Reduction: It is well known that the addition of few part per million of long chain polymers to water reduce the

turbulent friction factor in a pipe flow up to 80% [see for example Sreenivasan and White (2000) and reference therein]. Despite the vast amount of studies on the subject, the physical mechanism of polymer drag reduction remains obscure. Recent studies have shown that it is possible to reproduce the drag reduction phenomenology by numerical simulations of viscoelastic fluid in turbulent channel flow and in homogeneous and isotropic turbulence. The proposed research will investigate polymer drag reduction by means of theoretical development of simplified models and DNS in simplified geometry. The general objective is to point out the physical ingredients necessary to observe turbulent drag reduction. Another interesting issue is connected to the possibility that drag-reduction can be reproduced by a suitable scale (position) dependent effective viscosity for homogeneous (bounded) flow induced by the polymer fields, as recently proposed [Benzi et al 2004]. We intend to follow this path by performing DNS of pure turbulent flows with different eddy-viscosity modeled to reproduce the drag-reduction in the presence of diluted polymers.

Tools: DNS in 2d and 3d; low-dimensional modelization (Shell Models); Multiple Scale Expansions; linear-stability analysis (transition to chaos in the presence/absence of polymers).

Sections: Roma II, Genova, Torino, Ferrara.



Two-Phase Flows:

The second problem we want to address is related to active bubble dynamics in fully developed turbulence. The problem has many physical applications, ranging from chemical industry (gas-liquid reactors rely on bubbles to increase the area between the phases), to friction drag reduction in turbulence (with applications for navigation). One part of the project consists in studying numerically the effects of bubble feed-back in turbulent flows with a mean profile. To simplify the problem we will start from a flow without boundaries. We intend to perform DNS of Kolmogorov Flows coupled to small concentration of micro-bubbles. Here the effects on the mean velocity profile and the importance of the lift force will be investigated in detail. Another part of the project is the development of a suitable continuous model for bubble-fluid interaction. Available models are based on a Lagrangian approach, in which single bubble trajectories are resolved. This approach in not fully efficient for numerical implementations and the development of a fluid model, which considers bubble concentration, may results very important.

Tools: DNS in 2d and 3d; low-dimensional modelization (Shell Models); phenomenological modelization of the bubble-flow interaction;

Sections: Rome II, Roma I, Genova, Torino, Ferrara.





DNS + Algorithms: Direct Numerical Simulations are a necessary tool in many of the scientific issues here proposed. We are continuously investing in improving the computational power and expertise of our groups. We are interested in finding new algorithms and/or new architectures where to probe and perform our simulations. The Lattice Boltzmann algorithms on the APE machines has played a particular key role in this field. Nowadays we are in the process of porting our programs to the brand new Ape NEXT platform. The future Ape NEXT machine that we plan to employ has 512 nodes with 256Mbytes/node for a total memory of about 131 Gbytes. Typical communication bandwidth will be of the order of 200Mbyte/sec with a latency of 0.1 mu s. This machine may allow us to perform high resolution numerical simulations of the turbulent full 3D velocity field plus an additional scalar field (which may be a passive or active scalar) up to resolutions of the order of 1024^3. A challenging problem for this architecture would be the implementation of an efficient Fast Fourier Transform. We have already some experience in implementing it on both APE 100 and ApeMille architecture [Lippert 1997]. Here we just underline that with pseudo-spectral numerical codes it may be reasonable to obtain on ``small'' machine configuration (i.e. 16 nodes) reasonable performances comparable with the ones of commercial super-computers. In our experience on 64 processors of an IBM SP4 a 1024^3 simulation with pseudo-spectral code is running at a sustained performance of 135Mflops/proc for a total of 8.6 Gflops. On Ape Next it may be reasonable to expect performances of the order of 2.5 Gflops on 16 nodes.

Sections: Roma II, Roma I, Ferrara, Torino, Firenze, Genova.



MSE: Multiple Scale Expansion is the most advanced analytical tool able to highlight the asymptotic behavior of PDE's. The other monument is the Renormalization Group. We intend to investigate their limits and advantages also when applied to less ideal situations, as the case of dynamics with no clear ``small parameters'' and large scale-separations. To be more specific, as far as the large scale limit is concerned, we aim at investigating the large scale dynamics of non-newtonian flows where the Navier-Stokes equations are actively coupled to elastic degrees of freedom mimicking polymers, membranes, heavy particles etc. We shall use MSE to study both the linear stability analysis and the weakly nonlinear regime. DNS will be exploited both to verify the perturbative predictions and to explore nonperturbative regimes.

Tools: singular perturbative techniques; DNS.

Sections: Firenze, Genova, Roma I, Torino.



MHD: Magnetohydrodynamics. We intend to perform DNS of a conducting flow in the regime of strong Lorentz feedback. The main issue we want to understand is connected to the presence of strong anisotropic magnetic fluctuations induced by propagation of Alfven waves. We intend to exploit modern methods of data-analysis based on the

SO(3) decomposition to quantitatively disentangle the

isotropic from anisotropic fluctuations at all scales. Another different problem of interest will be the motion of charged particles in localized electromagnetic fields, here we intend to study the presence of standard/anomalous diffusion in the particle's phase space.

Tools: DNS; SO(3) decomposition; stochastic calculus.

Sections: Roma II, Roma I, Ferrara.



The developments and achievements described before will strongly exploit the already well established scientific and friendly connections between all group's members. Also, all groups have strong and long lasting scientific collaboration with some of the most active foreign groups and scientific research centers. Among them we cite the well established connections with the group of Prof. N. Antonov at the University of San Petersburg (Russia), Prof. U. Frisch in Nice (France), the group of Prof. A. Kupiainen andProf. Y. Honkonen at the University of Helsinki (Finland), the group of Prof. D. Lohse at the University of Twente(The Netherlands), the group of Prof. I. Procaccia at the Weizmann Institute of Science (Israel), the group of Prof. M.H. Jensen at Niels Bohr Institute in Copenhagen (Denmark). Scientific and fruitful connections exist also with the groups of Prof. K.R. Sreenivasan, Director of the ICTP (Italy); of Dr. A. Celani (INLN, Nice, France) and of Prof. I. Sokolov (Berlin University, Germany).



Short Bibliography



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