2023Activity reportProject-TeamCALISTO

3.2.1 Stand-alone Lagrangian simulations in atmospheric boundary layer (ABL)

The turbulent nature of the atmospheric boundary layer (ABL) contributes to the uncertainty of the wind energy estimation. This has to be taken into account in the modeling approach when assessing the wind power production. The purpose of the Stochastic Downscaling Model (SDM) is to compute the wind at a refined scale in the ABL, from a coarse wind computation obtained with a mesoscale meteorological solver. The main features of SDM reside in the choice of a fully Lagrangian viewpoint for the turbulent flow modeling. This is allowed by stochastic Lagrangian modeling approaches that adopt the viewpoint of a fluid-particle dynamics in a flow. Such particle methods have more flexible numerical convergence constraints, allowing a freer choice of time step size relative to the grid in space, in contrast to the limits imposed (as the Courant-Friedrichs-Lewy condition) on the convergence of many explicit time-marching numerical methods. This makes SDM advantageous in terms of computational cost when finer spatial resolutions are required.

Special emphasis is given to enhancing stand-alone Lagrangian numerical models within the Atmospheric Boundary Layer (ABL), including additional buoyancy models and canopy models. Furthermore, the coupling of fluid particle modeling with phase particle models opens up new modeling tools for some of some of the applications we are considering.

3.2.2 Advanced stochastic models for discrete particle dispersion and resuspension

As a particle nears a surface, deposition can occur depending on the interactions between the two objects. Deposits formed on a surface can then be resuspended, i.e. detached from the surface and brought back in the bulk of the fluid. Resuspension results from a subtle coupling between forces acting to move a particle (including hydrodynamic forces) and forces preventing its motion (such as adhesive forces, gravity). In the last decades, significant progresses have been achieved in the understanding and modeling of these processes within the multiphase flow community. Despite these recent progresses, particle resuspension is still often studied in a specific context and cross-sectoral or cross-disciplinary exchange are scarce. Indeed, resuspension depends on a number of processes making it very difficult to come up with a general formulation that takes all these processes into account.

Our goal here is to improve deposition law and resuspension law for more complex deposits in turbulent flows, especially towards multilayered deposits. For that purpose, we are improving existing Lagrangian stochastic models while resorting to meta-modeling to develop tailored resuspension law from experimental measurements and fine-scale numerical simulations. We are targeting practical applications such as pollutants in the atmosphere and plastic in marine systems.

3.2.3 Coherent descriptions for fluid and particle phases

Simulations of two-phase flows require the coupling of solvers for the fluid and particle phases. Numerical Weather Prediction (NWP) software usually rely on an Eulerian solver to solve Navier-Stokes equations. Solid particles are often treated using a Lagrangian point of view, i.e. their motion is explicitly tracked by solving Newton's equation of motion, the key difficulty being then to couple these intrinsically different approaches together. In line with the models and numerical methods developed in Sections 3.2.1 and 3.2.2, as an alternative to Eulerian-Lagragian approaches, Calisto is developing a new Lagrange-Lagrange formulation that remains tractable to perform simulations for two-phase turbulent flows. We are particularly interested in Lagrange-Lagrange models for interactions with surfaces, as turbulence and collisions with surfaces can significantly affect the concentration of particles in the near-wall region.

3.2.4 Active particles near boundary

Surface effects can lead to the trapping of micro-swimmers near boundaries, as the presence of a boundary breaks both the symmetry of the fluid (leading to strong anisotropy) and the symmetry of the fluid-swimmer system. The better understanding of fluid-particle interactions near boundaries are expected here to help in the design of new control actuation for driving artificial swimmers in confined environments (developed in Axis C).

Modeling approach

The equations of motion of the swimmer derive from its hydrodynamical interactions with the fluid through Newton laws. At a high level of description, this can be described by coupling the Navier-Stokes equations with the hyper-elastic equations describing the swimmer's deformation (in the case of elastic bodies). In the case of artificial magnetic swimmers, additional contribution representing the action of an external magnetic field on the swimmer needs to be added in the equations of motion. Solving the resulting system of PDEs is a challenging task, since it combines a set of equations deemed to be numerically difficult to solve even when they are decoupled. To overcome these difficulties, Calisto considers various types of models, ranging from simpler but rough models to more realistic but complex models.

Control and optimal control for swimmers displacement

Calisto investigates the controllability issues and the optimal control problems related in particular to two situations: the displacement of (i) real self-propelled swimmer by assuming that the control is the deformation of its body (ii) artificial bio-inspired swimmers that are able to swim using an external magnetic field.

Reinforcement learning

Another line of research concerns optimal path planning in turbulent flow. As a microswimmer swims towards a target in a dynamically evolving turbulent fluid, it is buffeted by the flow or it gets trapped in whirlpools. The general question we want to address is whether such a microswimmer can develop an optimal strategy that reduces the average time or energy it needs to reach a target at a fixed distance. In this context, the use of methods borrowed from artificial intelligence has been mainly used for navigation problems in which locomotion mechanisms are oversimplified. Based on recent smart-swimming approach developed in Calisto,   21  26, we plan to extend this work to more realistic cases, including for instance other control parameters for the swimmers, the presence of boundaries, of a feedback of the swimmers on the fluid flow, or interactions between swimmers.

Fundamental aspects of turbulence and turbulence transport

This research line has the scope of providing a unified description of turbulent flows in the limit of large Reynolds numbers and thus will be applicable to a large range of physical applications. It is conjectured since Kolmogorov and Onsager that the flow develops a sufficiently singular structure to provide a finite dissipation of kinetic energy when the viscosity vanishes. This dissipative anomaly gives a consistent framework to select physically acceptable solutions of the limiting inviscid dynamics. However, recent mathematical constructions of weak dissipative solutions face the problem of non-uniqueness, raising new questions on the relevance to turbulence and on the notion of physical admissibility.

On the one hand, the conservation of kinetic energy is actually not the only symmetry that is broken by turbulence. Various experimental and numerical measurements show significant deviations from simple scaling, time-irreversible fluctuations along fluid elements trajectories, and possibly other broken inviscid symmetries, such as circulation. Still, these anomalies may have a universal nature and, as such, provide new constraints for the design of physically admissible solutions. On the other hand, non-uniqueness could be an intrinsic feature of turbulence. Singular solutions to non-linear problems have an explosive sensitivity leading to spontaneously stochastic behaviors, thus questioning the pertinence of uniqueness and providing a framework to interpret solutions at a probabilistic level. To address such issues and provide unified appreciation, we simultaneously develop three strongly interrelated viewpoints: a) numerical approach, exploiting relevant and efficient fully-resolved simulations; b) new theoretical approaches based on the statistical physics of turbulent flow; c) mathematical construction of "very weak" flows, such as measure-valued solutions to the Euler equations.

Stochastic dynamics : analysis and numerics

Variability in wind/hydro simulation at small scale: application to wind/hydro energy

The turbulent nature of the atmospheric boundary layer (ABL) contributes to the uncertainty of the wind energy estimation. This has to be taken into account in the modeling approach when assessing the wind power production. The stochastic nature of the SDM approach developed in Axis B offers some rich perspectives to assess variability and uncertainty quantification issues in the particular context of environmental flows and power extraction evaluation. In particular, as a PDF method, SDM delivers a probability distribution field of the computed entities. Merging such numerical strategy with Sensitivity Analysis (SA)/Uncertainty Quantification (UQ) are potentially fruitful in terms of computational efficiency.

Metamodeling and uncertainty

While building and using computational fluid dynamics (CFD) simulation models, sensitivity analysis and uncertainty quantification methods allow to study how the uncertainty in the output of a model can be apportioned to different sources of uncertainty in the model input. UQ approaches allow to model verification and factor prioritization. It is a precious aid in the validation of a computer code, guidance research efforts, and in terms of system design safety in dedicated application. As CFD code users, we aim at applying UQ tools in our dedicated modeling and workflow simulation. As Stochastic Lagrangian CFD developers, we aim at developing dedicated SA and UQ tools as Stochastic solvers have the ability to support cross Monte Carlo strategy at the basis of SA methodology.

Another goal is to address some control and optimization problems associated with the displacement of swimmers through metamodeling, such as Gaussian process regression model, proved to be efficient for solving optimization of PDEs systems in other contexts.

6.1.1 CDM-Sci

• Name:
  Common Data Model for Scientific Workflows
• Keywords:
  Databases, Multiphysics modelling
• Scientific Description:

  CDM-Sci provides a unique framework to facilitate coupling/chaining of physical models, which operate at different scales or at different levels of description ranging from molecular dynamics to fluid mechanics, to build multi-physics and multi-scale computational workflows that can be automatically executed. Hence, CDM-Sci is composed of two key ingredients:

  1) A common data model for scientific computing: CDM-Sci offers a solution that collects information in a standardized method in the form a query tree designed to identify the physical and numerical models needed in a given situation. This means that each situation to be simulated is first decomposed into a set of physical components, which are defined as either discrete or continuum systems (quantum systems are not addressed in the present version). Then, a physical description is detailed for each component to fully characterize its nature as well as the physical processes at play. This can lead to introducing a number of sub-components which play a specific physical role. Once the physics at stake (variables, laws, evolution equations) is clarified, a numerical description is chosen either in terms of a set of interacting particles or continuous fields for every component or sub-component. By resorting to this combined physical and numerical points of view, the proposed solution offers naturally a way to specify the various numerical softwares needed to simulate the systems of interest. In addition, regardless of whether the workflow requires a single software or to couple several ones, the information exchanged between components, or sub-components, is clearly identified thanks to specific queries. Finally, when the workflow combines several softwares, similar queries allow to identify the information exchanged between them (inputs/outputs identification), as well as the transformations required (mappers) and the format of the data to be exchanged (wrappers).

  2) Computational sciences tools for automatic code generation and execution In order to facilitate coding of new models and execution of calculation workflows coupling different models (like discrete and continuum models), new tools for automatic code generation are developed and validated. For that purpose, CDM-Sci relies on a Model Driven Architecture (MDA) development approach, which is based on the assumption that codes can be produced from abstract, human-elaborated modelling diagrams. It allows to generate standardized data to describe models, such as XSD (XML Schema Definition).

  Targeted audience: CDM-Sci is of great interest for physics-oriented users as well as computer scientists.

  - From the perspective of physics-oriented users, CDM-Sci allows engineers and scientists to go beyond the simulation set-ups they are familiar with by offering them the possibility to explore new multi-scale physics by linking/coupling with models developed in nearby areas.

  - From a computer science perspective, CDM-Sci can help experts to understand the needs of engineers/scientists, so that they can adapt existing computational techniques or orient new developments to make them accessible and relevant to engineers/scientists. In short, CDM-Sci bridges the gap between physics-oriented users and computer scientists.

• Functional Description:
  CDM-Sci stands for "Common Data Model for Scientific Simulation Workflows". As a data model, CDM-Sci is a tool that organizes elements of data and standardizes how they relate to one another and to the properties of real-world entities. As applied to scientific simulations, CDM-Sci is specifically designed to organize elements of data corresponding to the physical and numerical information required to set-up scientific simulations. CDM-Sci has been developed to address multi-physics and multi-scale problems (e.g. multiphase flows involving fluid mechanics, surface sciences, material sciences and/or chemistry). CDM-Sci allows naturally such multi-scale models to be mapped into computational workflows, which link/chain scientific softwares operating at different scales. These computational workflows can then be executed on a set of computational resources following an automatic procedure. In short, CDM-Sci provides a common standardized structure (Common Data Model) for the set-up and execution of scientific simulation workflows.
• Release Contributions:
  CDM-Sci has been developed first conceptually (through a documentation in the form of a list, which gives the structure of the query tree). Second, an operational version of this query tree has been implemented in an Eficas catalog. It currently contains all the information to set-up simulations of multiphase flows and, to a lesser extent, some Molecular Dynamics simulation. Since the structure is generic and open, extra information can be added to account for other physical phenomena (e.g. material science, chemistry, electro-magnetism, quantum effects).
• URL:
  https://­gitlab.­inria.­fr/­cdm/­cdm-sci
• Contact:
  Christophe Henry
• Participants:
  Christophe Henry, Mireille Bossy
• Partner:
  Edf

7.1.1 Lagrangian stochastic model for the orientation of non-spherical particles in turbulent flow: an efficient numerical method for CFD approach

Participants: Mireille Bossy, Christophe Henry.

Suspension of anisotropic particles can be found in various industrial applications. Microscopic ellipsoidal bodies suspended in a turbulent fluid flow rotate in response to the velocity gradient of the flow. Understanding their orientation is important since it can affect the optical or rheological properties of the suspension. The equations of motion for the orientation of microscopic ellipsoidal particles were obtained by Jeffery 34. But so far, this description has always been investigated in the framework of direct numerical simulations (DNS) and experimental measurements. In particular, inertia-free particles, with sizes smaller than the Kolmogorov length, follow the fluid motion with an orientation generally defined by the local turbulent velocity gradient.

In this work, in collaboration with Lorenzo Campana former PhD of the team, our focus is to characterize the dynamics of these objects in turbulence by means of a stochastic Lagrangian approach. The development of a model that can be used as a predictive computational tool in industrial computational fluid dynamics (CFD) codes is highly valuable for practical applications. Models that reach an acceptable compromise between simplicity and accuracy are needed for progressing in the field of medical, environmental and industrial processes. A large part of our activity dealt with the dynamics of ellipsoidal particles in turbulent flows, with the main motivation of improving models used in applications. Actually, the translational and rotational dynamics of such non-spherical particles is directly related to their orientation, which has to be properly calculated. Micro- and macroscopic viewpoints were addressed simultaneously.

More precisely, we have introduced a stochastic Lagrangian model for the angular dynamics of inertialess spheroids. The model consists of superimposing a short-correlated random component to the large-scale average fluid-velocity gradient. This Lagrangian stochastic approach is implemented in a practical 3D macroscopic model for turbulent transport (currently available in the open-source CFD software Code_Saturne) and is published in Computers & Fluids 4.

7.1.2 Approximating the van der Waals interaction potentials between agglomerates of nanoparticles

Participant: Christophe Henry.

Studying the agglomeration of small nanoparticles (a few nanometers in size) or atomic clusters has remarkable importance for the synthesis of nanoparticles at industrial scale. However, this is a challenge since different physical phenomena acting at various scales have to be considered: this includes the transport across large distances and short-range attractive forces. The adhesion between complex fractal-like agglomerates is currently still poorly understood due to the complex role of the aggregate morphology.

This study focuses on evaluating the adhesion forces between raspberry-like and fractal-like agglomerates of nanoparticles. For that purpose, we consider only the van der Waals (vdW) interactions, since they are probably one of the most important interaction potentials or forces between nanoparticles and interfaces found in many applications of powder technology (like filtration, impaction, coagulation, agglomerates restructuring, and resuspension). We have developed both analytical and semi-analytical models to predict the agglomerate-molecule and agglomerate-agglomerate vdW interactions. These new models can find multiple applications in practical situations.

This work is the result of a collaboration with José Moran (who did his PhD in Coria, later Post-doc in Carleton University and University of Minnesota), Jérôme Yon (from Coria) and Reza Kholgy (Carleton University), which started in 2021 and has been pursued in 2023. It is published in Advanced Powder Technology in 2023 9.

7.1.3 Review article on the dynamics of discrete particles in turbulent flows

Participant: Christophe Henry.

The purpose of this work is to review the wide range of situations where small discrete elements, either bubbles, droplets or solid particles, are embedded in turbulent flows. Occurring often at a human scale and in our daily environments, these turbulent dispersed two-phase flows display complex behavior due to the interplay of two fundamental interactions, the fluid-particle and particle-particle interactions, compounded by the turbulence of the carrier flow. This is not a domain where the basic laws are unknown but where the huge number of degrees of freedom involved call for reduced, or coarse-grained, statistical descriptions to be developed. Since we are considering transport and collision phenomena or relaxation processes, it would seem that they can be handled by kinetic theory. In the general case of non-fully resolved turbulent flows, we are however dealing with particles influenced by random media with non-zero time and space correlations. The second aspect of this work is to recognize the limitations of kinetic-based descriptions and to address the challenges driving us to extend the classical framework, for fluid-particle as well as particle-particle interactions. Taking the standpoint provided by the modern formulation of stochastic processes and focusing on the description of the particle phase, this review proposes a step-by-step pedagogical presentation of current models while pointing out new directions and remaining uncharted territories. This is done to provide answers to the question `why?' as much as `how?' and to try to kindle interest into these open and fascinating issues.

This work is a collaboration with Jean-Pierre Minier (EDF R&D, MFEE, Chatou, France). It has been submitted to Review of Modern Physics in 2023 16.

7.1.4 Clusters of heavy particles in two-dimensional Keplerian turbulence

Participant: Jérémie Bec.

Protoplanetary disks are gaseous systems in Keplerian rotation around young stars, known to be turbulent. They include a small fraction of dust from which planets form. In the incremental scenario for planet growth, the formation of kilometer-size objects (planetesimals) from pebbles is a major open question. Clustering of particles is necessary for solids to undergo a local gravitational collapse. To address this question, we have studied in this work the dynamics of inertial particles in turbulent flows with Keplerian rotation and shear. Two-dimensional direct numerical simulations have been performed to explore systematically two physical parameters: the rotation rate, which depends on the distance to the star, and the particle response time, which relates to their size. Shear was found to drastically affect the characteristics of the turbulent flow destroying cyclones and favoring the survival of anticyclones. Faster Keplerian rotation enhances clustering of particles. For intermediate sizes, particles concentrate in anticyclones. These clusters form in a hierarchical manner and merge together with time. For other parameter values, solids concentrate on fractal sets that get more singular with rotation. The mass distribution of particles was then found to be multifractal with small dimensions at large orders, intriguing for triggering their gravitational collapse. Such results are promising for a precise description and better understanding of planetesimal formation.

This work is a collaboration with Héloïse Méheut (Laboratoire Lagrange, Observatoire de la Côte d'Azur), and Fabiola Gerosa (Laboratoire Lagrange, Observatoire de la Côte d'Azur), PhD student co-advised by J. Bec and H. Méheut. It has been published in the European Physical Journal Plus in 2023 6.

7.1.5 Velocity and acceleration statistics of heavy spheroidal particles in turbulence

Participant: Jérémie Bec.

Non-spherical particles transported by turbulent flow have a rich dynamics that combines their translational and rotational motions. In this work, the focus was on small, heavy, inertial particles with a spheroidal shape fully prescribed by their aspect ratio. Such particles undergo an anisotropic, orientation-dependent viscous drag with the carrier fluid flow whose associated torque is given by the Jeffery equations. Direct numerical simulations of homogeneous, isotropic turbulence have been performed to study systematically how the translational motion of such spheroidal particles depends on their shape and size. Surprisingly, it was found that the Lagrangian statistics of both velocity and acceleration can be thoroughly described in terms of an effective Stokes number obtained as an isotropic average over angles of the particle's orientation. Corrections to the translational motion of particles due to their non-sphericity and rotation can hence be fully recast as an effective radius obtained from such a mean.

This work is a collaboration with Sofía Allende (Earth and Climate Research, Earth and Life Institute, Université catholique de Louvain, Belgium). It has been published in the Journal of Fluid Mechanics in 2023 1.

7.1.6 Review article on statistical models for the dynamics of heavy particles in turbulence

Participant: Jérémie Bec.

When very small particles are suspended in a fluid in motion, they tend to follow the flow. How such tracer particles are mixed, transported, and dispersed by turbulent flow has been successfully described by statistical models. Heavy particles, with mass densities larger than that of the carrying fluid, can detach from the flow. This results in preferential sampling, small-scale fractal clustering, and large collision velocities. To describe these effects of particle inertia, it is necessary to consider both particle positions and velocities in phase space. In recent years, statistical phase-space models have significantly contributed to our understanding of inertial-particle dynamics in turbulence. These models help to identify the key mechanisms and non-dimensional parameters governing the particle dynamics, and have made qualitative, and in some cases quantitative predictions. This article reviews statistical phase-space models for the dynamics of small, yet heavy, spherical particles in turbulence. We evaluate their effectiveness by comparing their predictions with results from numerical simulations and laboratory experiments, and summarize their successes and failures.

This review was written in collaboration with Kristian Gustavsson and Bernhard Mehlig from the University of Gothenburg (Sweden). It will soon appear in the Annual Review of Fluid Mechanics 3.

7.2.1 Influence of engineered roughness microstructures on adhesion and turbulent resuspension of microparticles

Participants: Mireille Bossy, Christophe Henry.

From microplastics resuspending into the atmosphere to earth particles left behind during extraterrestrial explorations, the resuspension of microparticles by a turbulent gas flow occurs in many natural, environmental and industrial systems. Wall surfaces, onto which particles initially adhere, are rarely smooth and this surface roughness affects particle resuspension. Available experimental data on particle resuspension have been obtained with substrates, whose surfaces are either unaltered or manually abraded with, for instance, sand blasting. However, the precise role of surface roughness is poorly understood due to the chaotic nature of surface roughness.

This study explores the role of surface roughness using engineered rough surfaces. For that purpose, we rely on surface functionalization, a modern technique allowing the precise fabrication of a wall surface with well-characterized microstructures, hence reducing the asperity randomness associated with conventional abrasion techniques. We present here a new set of reference data, characterizing various functionalized surfaces as well as adhesion force measurements with microparticles and observations of particle resuspension (when particles deposited on these surfaces are exposed to a gas flow). The presented results will primarily contribute to the improvement of resuspension models, which until now rely on a simplified representation of the surface roughness elements.

This is a collaborative work with researchers from the Technical University of Dresden in Germany (especially Grégory Lécrivain and Uwe Hampel) as well as their colleagues from the Helmholtz Center in Friburg. These results are published in Journal of Aerosol Science 2.

7.2.2 Robust algorithms for particle trajectories in complex 3D unstructured meshes

Participants: Christophe Henry.

This work proposes a new algorithm to compute more efficiently and accurately the trajectories of particles in the context of stochastic Lagrangian approaches. The challenges are two-fold: first, the algorithm should remain tractable in complex 3D unstructured meshes; second, given the stochastic nature of the models, a key aspect is to derive estimations of the residence times that do not anticipate the future of the Wiener process. For that purpose, we propose a time-step-robust cell-to-cell integration of particle trajectories, where the main idea is to dynamically update the mean fields used in the time integration by splitting, for each particle, the time step into sub-steps such that each of these sub-steps corresponds to particle cell residence times. This reduces the spatial discretization error. In line with the constraints imposed by stochastic models, the new algorithm relies on a virtual particle, attached to each stochastic one, whose mean conditional behavior provides free-of-statistical-bias predictions of residence times. This new algorithm has been validated on two representative test cases: particle dispersion in a statistically uniform flow and particle dynamics in a non-uniform flow.

This is a collaborative work with researchers from EDF R&D (Martin Ferrand, Guilhem Balvet, Jean-Pierre Minier) and CEREA (Yelva Roustan). The results are published in Monte-Carlo Methods and Applications 12.

7.2.3 Enhanced transport of flexible fibers by pole vaulting in turbulent wall-bounded flow

Participants: Jérémie Bec, Christophe Henry.

Long, flexible fibers transported by a turbulent channel flow sample non-linear variations of the fluid velocity along their length. One of the challenges of the dynamics of flexible fibers in confined flows is related to their interactions with boundaries, which can lead to a range of behavior. In this work, we explore how such fibers tumble and collide with the boundaries and the resulting effect on their near-wall dynamics. Using fine microscopic simulations (based on a Direct Numerical Simulations of the turbulent flow coupled to an explicit tracking of inertialess flexible fibers described with the slender-body theory), we show that as fibers bounce off surfaces, an impulse propels them toward the center of the flow, similar to pole vaulting. As a result, the fibers migrate away from the walls, leading to depleted regions near the boundaries and more concentrated regions in the bulk. These higher concentrations in the center of the channel result in a greater net flux of fibers than what was initially imposed by the fluid. This effect becomes more pronounced as fiber length increases, especially when it approaches the channel height.

This is a collaborative work with Christophe Brouzet (Institute of Physics in Nice). An article has been submitted to Physical Review Letters 13.

7.3.1 Mathematical and computation framework for moving and colliding rigid bodies in a Newtonian fluid

Participants: Laetitia Giraldi.

This work is done in collaboration, with Celine Van-Landeghem (IRMA, Université de Strasbourg), PhD student coadvised by L. Giraldi.

We studied numerically the dynamics of colliding rigid bodies in a Newtonian fluid. The finite element method is used to solve the fluid-body interaction and the fluid motion is described in the Arbitrary-Lagrangian-Eulerian framework. To model the interactions between bodies, we consider a repulsive collision-avoidance model, defined by R. Glowinski. The main emphasis in this work is the generalization of this collision model to multiple rigid bodies of arbitrary shape. Our model first uses a narrow-band fast marching method to detect the set of colliding bodies. Then, collision forces and torques are computed for these bodies via a general expression, which does not depend on their shape. Numerical experiments examining the performance of the narrow-band fast marching method and the parallel execution of the collision algorithm are discussed. We validate our model with literature results and show various applications of colliding bodies in two and three dimensions. In these applications, the bodies either move due to gravity, a flow, or can actuate themselves. Finally, we present a tool to create arbitrary shaped bodies in discretized fluid domains, enabling conforming body-fluid interface and allowing to perform simulations of fluid-body interactions with collision treatment in these realistic environments. All simulations are conducted with the Feel++ open source library.

This is collaborative work with Christophe Prud'Homme, Luca Berti, Yannick Hoarau, Vincent Chabannes and Agathe Chouippe. The paper 15 is accepted in Annals of Mathematical Sciences and Applications.

7.3.2 Necessary conditions for local controllability of a particular class of systems with two scalar controls

Participant: Laetitia Giraldi.

In this paper 7, in collaboration with Pierre Lissy (Ceremade, Paris), Jean-Baptiste Pomet (Inria, McTAO) and Clement Moreau (LS2N, University of Nantes), we consider control-afflne systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions for local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in ${\mathrm{L}}^{\infty }$ or small in ${\mathrm{W}}^{1,\infty }$. These results are illustrated with several examples. This paper is now accepted in ESAIM: Control, Optimisation and Calculus of Variations (ESAIM:COCV).

7.3.3 Reinforcement learning for the locomotion and navigation of undulatory micro-swimmers in chaotic flow

Participants: Zakarya El Khiyati, Raphaël Chesneaux, Laetitia Giraldi, Jérémie Bec.

This study aimed at finding optimal navigation policies for thin, deformable microswimmers that progress in a viscous fluid by propagating a sinusoidal undulation along their slender body. These active filaments are embedded in a prescribed, non-homogeneous flow, in which their swimming undulations have to compete with the drifts, strains, and deformations inflicted by the outer velocity field. Such an intricate situation, where swimming and navigation are tightly bonded, has been addressed using various methods of reinforcement learning. Each swimmer has only access to restricted information on its configuration and has to select accordingly an action among a limited set. The optimization problem then consists in finding the policy leading to the most efficient displacement in a given direction. It was found that usual methods do not converge and this pitfall has been interpreted as a combined consequence of the non-Markovianity of the decision process, together with the highly chaotic nature of the dynamics, which is responsible for high variability in learning efficiencies. Still, we provided an alternative method to construct efficient policies, which is based on running several independent realizations of Q-learning. This allowed the construction of a set of admissible policies whose properties have been studied in detail and compared to assess their efficiency and robustness. This work has been published in the European Physical Journal E 5.

7.4.1 Anomalous dissipation and spontaneous stochasticity in deterministic surface quasi-geostrophic flow

Participants: Jérémie Bec, Simon Thalabard, Nicolas Valade.

Surface quasi geostrophy (SQG) describes the two-dimensional active transport of a temperature field in a strongly stratified and rotating environment. Besides its relevance to geophysics, SQG bears formal resemblance with various flows of interest for turbulence studies, from passive scalar and Burgers to incompressible fluids in two and three dimensions. In the article 10, published in Annales Henri Poincaré in 2023, we substantiated this analogy by considering the turbulent SQG regime emerging from deterministic and smooth initial data prescribed by the superposition of a few Fourier modes. While still unsettled in the inviscid case, the initial value problem is known to be mathematically well-posed when regularized by a small viscosity. In practice, our numerics revealed that in the presence of viscosity, a turbulent regime appears in finite time, which features three of the distinctive anomalies usually observed in three-dimensional developed turbulence: (i) dissipative anomaly, (ii) multifractal scaling, and (iii) super-diffusive separation of fluid particles, both backward and forward in time. These three anomalies point towards three spontaneously broken symmetries in the vanishing viscosity limit: scale invariance, time reversal and uniqueness of the Lagrangian flow, a fascinating phenomenon dubbed spontaneous stochasticity. In the light of previous work on the passive scalar problem, we argued that spontaneous stochasticity and irreversibility are intertwined in SQG, and provided numerical evidence for this connection. Our numerics, though, revealed that the deterministic SQG setting only features a tempered version of spontaneous stochasticity, characterized in particular by non-universal statistics.

7.4.2 Numerical analysis of stochastic system

7.4.3 Finite-time Lyapunov exponents of deep neural networks

Participants: Jérémie Bec.

For deep neural networks trained on different classification problems, we explored geometrical structures of finite-time Lyapunov exponents (FTLE) in input space. In fluid mechanics, such Lagrangian coherent structures appear as ridges of large exponents, and they are used with great success to organize the phase space of complex spatiotemporal flow patterns. The same is true for deep neural networks: FTLE ridges partition input space into different regions associated with different classes. Our analysis showed how the network exploits its exponential expressivity to form the ridges. Their sharpness determines how quickly classification errors and prediction uncertainty decreases as one moves away from the ridge. As the width of the network increases, the contrast between ridge and background disappears, leading to a different learning mechanism, random embedding, with qualitative differences regarding classification errors and predictive uncertainties. The transition to this lazy-learning regime occurs for very wide networks. The transition may explain why wider networks are more robust against adversarial attacks: the less important the ridges are for representing the relevant data structures, the harder it is to realize adversarial attacks. The geometrical method presented in our work may extend to other network architectures, such as resnets or transformers, and could help to visualize and understand the mechanisms that allow such neural networks to learn. This study is the subject of a collaboration with Kristian Gustavsson, Hampus Linander, Bernhard Mehlig, and Ludvig Storm (University of Gothenburg) and will soon appear in Physical Review Letters 17.

7.5.1 Short term predictive models with times-serie based on Lagrangian stochastic approach in CFD and application to wind energy

Participants: Mireille Bossy.

The need of statistical information on the wind, at a given location and on large time period, is of major importance in many applications such as the structural safety of large construction projects or the economy of a wind farm, whether it concerns an investment project, a wind farm operation or its repowering. The evaluation of the local wind is expressed on different time scales: monthly, annually or over several decades for resource assessment, daily, hourly or even less for dynamical forecasting (these scales being addressed with an increasing panel of methodologies). In the literature, wind forecasting models are generally classified into physical models (numerical weather prediction models), statistical approaches (time-series models, machine learning models, and more recently deep learning methods), and hybrid physical and statistical models. At a given site and height in the atmospheric boundary layer, measuring instruments record time series of characteristics of the wind, such as wind speed characterizing load conditions, wind direction, kinetic energy and possibly power production. Such observations should feed into forecasting, but also uncertainty modeling. In this context, probabilistic or statistical approaches are widely used, helping to characterize uncertainty through quantile indicators.

In 28 in collaboration with Kerlyns Martinez (University of Valparaiso), we construct an original stochastic model for the instantaneous turbulent kinetic energy at a given point of a flow, and we validate estimator methods on this model with observational data examples. Motivated by the need for wind energy industry of acquiring relevant statistical information of air motion at a local place, we adopt the Lagrangian stochastic description of fluid flows to derive, from the 3D+time equations of the physics, a 0D+time-stochastic model for the time series of the instantaneous turbulent kinetic energy at a given position.

We pursue our work by now focusing on a reduced model for the short term prediction of wind gust risk.

7.5.2 Methodology to quantify uncertainties in dispersed two-phase flows

Participants: Christophe Henry, Mireille Bossy.

Within the framework of the VIMMP EU project (Virtual Materials Market Place) that took place between 2017 and 2022, one objective was to set up a methodology to analyze the sensitivity (SA) and then quantify uncertainty (UQ) in numerical simulations of multiphase flows to a number of input variables. For that purpose, a Common Data Model (CDM) has been developed to provide a comprehensive, adequate and standardized structure to collect all the physical and numerical information required to set-up scientific simulation workflows (possibly including various software) while allowing for a systematic analysis of the results obtained using UQ&SA tools.

This collaborative work with partners involved in the VIMMP project (Pascale Noyret, Eric Fayolle and Jean-Pierre Minier from EDF R&D) resulted in the development of the software CDM-Sci (see Section 6). To complement this software, we are in the process of writing a book proposal for Specials Issues in Springer to describe the CDM (both in terms of physics and in terms of implementation) and illustrate it on a few selected examples.

Indo-French Center for Applied Mathematics

Participants: Jérémie Bec, Ciro Campolina, Nicolas Valade.

The Indo-French Center for Applied Mathematics (IFCAM) is a CNRS IRL that provides support for a collaboration on “Turbulence in classical and quantum fluids” with teams at the Indian Institute of Science and the International Center for Theoretical Science in Bangalore. Visits of Calisto members to Bangalore supported by this initiative took place in January 2023. It is however unclear whether this initiative will be renewed or not.

8.1.1 CEFIPRA project “Polymers in turbulent flows”

Participants: Jérémie Bec.

This bilateral project aims at studying the dynamics of polymers in turbulent flows, with the idea to use Lagrangian approaches to find links between microscopic scales and the rheology of polymer suspensions and macroscopic continuum models. The french PI is Dario Vincenzi (CNRS-LJAD) and the Indian PI is Jason Picardo (IIT Mumbai). The Calisto team is a partner of the project and received funding to support bilateral visits.

SDE: Theory, Numerics and Applications (RSE Saltire Facilitation Network)

Participants: Mireille Bossy.

Started in 2022, Calisto takes part to the Royal Society of Edinburgh (RSE) Saltire Facilitation Network on SDEs: Theory, Numerics and Applications. This 2 years network project involves mathematicians of 9 European countries on the theme of (numerical) analysis for highly nonlinear SDEs. On the same themes, a submission of an European COST action was submitted this fall.

8.2.1 Visits of international scientists

8.3.1 ANR PRC TILT

Participant: Jérémie Bec.

The ANR PRC project TILT (Time Irreversibility in Lagrangian Turbulence) started on Jan. 1, 2021. It is devoted to the study and modeling of the fine structure of fluid turbulence, as it is observed in experiments and numerical simulations. In particular, recall that the finite amount of dissipation of kinetic energy in turbulent fluid, where viscosity seemingly plays a vanishing role, is one of the main properties of turbulence, known as the dissipative anomaly. This property rests on the singular nature and deep irreversibility of turbulent flows, and is the source of difficulties in applying concepts developed in equilibrium statistical mechanics. The TILT project aims at exploring the influence of irreversibility on the motion of tracers transported by the flow. The consortium consists of 3 groups with complementary numerical and theoretical expertise, in statistical mechanics and fluid turbulence. They are located in Saclay, at CEA (Bérengère Dubrulle), in Lyon, at ENSL (Laurent Chevillard, Alain Pumir), and in Sophia Antipolis (Jérémie Bec). Within TILT, a new postdoc joined Calisto in January 2023.

8.3.2 ANR JCJC NEMO

Participant: Laetitia Giraldi.

The JCJC project NEMO (controlliNg a magnEtic Micro-swimmer in cOnfined and complex environments) was selected by ANR in 2021, and started on Jan. 1, 2022 for four years. NEMO team is composed of Laetitia Giraldi, Mickael Binois (Inria, Acumes) and Laurent Monasse (Inria, Coffee).

NEMO aims at developing numerical methods to control a micro-robot swimmer in the arteries of the human body. These robots could deliver drugs specifically to cancer cells before they form new tumors, thus avoiding metastasis and the traditional chemotherapy side effects.

NEMO will focus on micro-robots, called Magnetozoons, composed of a magnetic head and an elastic tail immersed into a laminar fluid possibly non-Newtonian. These robots imitate the propulsion of spermatozoa by propagating a wave along their tail. Their movement is controlled by an external magnetic field that produces a torque on the head of the robot, producing a deformation of the tail. The tail then pushes the surrounding fluid and the robot moves forward. The advantage of such a deformable swimmer is its aptness to carry out a large set of swimming strategies, which could be selected according to the geometry or the rheology of the biological media where the swimmer evolves (blood, eye retina, or other body tissues).

Although the control of such micro-robots has mostly focused on simple unconfined environments, the main challenge is today to design external magnetic fields that allow them to navigate efficiently in complex realistic environments.

NEMO aims at elaborating efficient controls, which will be designed by tuning the external magnetic field, through a combination of Bayesian optimization and accurate simulations of the swimmer's dynamics with Newtonian or non-Newtonian fluids. Then, the resulting magnetic fields will be validated experimentally in a range of confined environments. In such an intricate situation, where the surrounding fluid is bounded laminar and possibly non-Newtonian, optimization of a strongly nonlinear, and possibly chaotic, high-dimensional dynamical system will lead to new paradigms.

8.3.3 ANR PRC NETFLEX

Participant: Jérémie Bec, Mireille Bossy, Laetitia Giraldi, Christophe Henry, Paul Maurer.

The ANR PRC project NEFFLEX (Tangles, knots, and breakups of flexible fibers in turbulent fluids) started on January 1, 2022. NETFLEX is a four-years project that aims at advancing our knowledge on the dynamics of long, flexible, macroscopic fibres in turbulent flows, and to understand and model the processes of fibre fragmentation and aggregation. NETFLEX brings together Université Côte d'Azur (INPHYNI, LJAD), Inria (Calisto) and Aix-Marseille University (IRPHE). NETFLEX approach combines three levels of description (micro, meso, and macroscopic) and relies on a synergy between mathematical modeling, numerical simulations, and laboratory experiments. It relies on the development of newly designed experiments and a substantial improvement of the mathematical and numerical tools currently used in the study of fibre dynamics. An overall aim is to develop a new framework able to cope with such intricate effects of turbulence and to reproduce the significant observable features in a macroscopic approach.

Improved modeling of turbulent fluctuations and effective transport models for aggregates are among the key issues to be addressed in order to extend the macroscopic models.

Starting in October 2022, Paul Maurer's thesis work on analysis and simulation for temporal intermittency and long-range correlation models is part of this project, and will be applied to the dynamics of flexible particles.

9.1.1 Scientific events: organisation

9.1.2 Journal

9.1.3 Invited talks

• Mireille Bossy was invited speaker to the following events: RSE Saltire Facilitation Network on SDEs Seminar; MASCOT-NUM Conference (Le Croisic); "Stochastic Computation" workshop at Foundations of Computational Mathematics (FoCM 2023) Conference; workshop on "Stability, mixing and fluid dynamics" at the University of Münster ; RSE Saltire Facilitation Network on SDEs Workshop at University of Strathclyde ; Workshop on "SDEs with low-regularity coefficients: theory and numerics" at the University of Torino;

  She also gave a talk to the following conference: NASPDE 2023 at Eindhoven University of Technology; 14th International Conference on Monte Carlo Methods and Applications MCM2023 (Paris).

• Jérémie Bec was an invited speaker at the conferences “Frontiers in Non-equilibrium Physics” (Chennai, India), “Turbulence: Problems at the Interface of Mathematics and Physics” (Bangalore, India), and at the workshop “Particle Growth in Turbulence” (Stockholm, Sweden).
• Laetitia Giraldi was invited speaker to the Jacques-Louis Lions Spanish-French School on Numerical Simulations in Physics and Engineering at Barcelone in July 2023.

9.1.4 Leadership within the scientific community

• Jérémie Bec is in charge of the Academy of excellence "Complex Systems" of the IDEX Université Côte d'Azur (Decision-making role for funding; Coordination and animation of federative actions; Participation in the IDEX evaluation).
• Mireille Bossy is Chairing of the Scientific Council of the Academy of excellence "Complex Systems" of the IDEX Université Côte d'Azur.

Calisto team members are involved in the scientific/steering committees of several national research networks. This includes the following GdR (CNRS Research network):

• GdR Mascot-NUM on stochastic methods for the analysis of numerical codes,
• GdR Défis théoriques pour les sciences du climat on theoretical aspects for climate science
• GDR Calcul that promotes communication and exchange within the computing community in France.

Calisto team members are also involved as partners in other networks including: GdR Navier-Stokes 2.0 on turbulence, and ASFERA, the French Association on Studies and Research on Aerosols.

9.1.5 Scientific expertise

• Christophe Henry reviewed project propositions from the PAZY foundation (an Israeli research fund that supports scientific projects in the areas of chemistry and materials, engineering and physics).

9.1.6 Research administration

• Since September 2023, Christophe Henry is acting as the local correspondent for the yearly activity reports of all Teams within Inria Center at Université Côte d'Azur.
• Jérémie Bec and Laëtitia Giraldi are members of the “Comité Nice” at Inria Center at Université Côte d'Azur.
• Mireille Bossy is member of the "Bureau du comité des projets" at Inria Center at Université Côte d'Azur.

9.2.1 Teaching

Calisto scientific staff have been involved in the following teaching activities:

• Assignments (Khôlles) in preparatory schools MPSI, MP* (Laetitia Giraldi, 2h weekly, Centre International de Valbonne).
• Advanced models for hydrology (C. Henry): master students in their final (5th) year within the program “Génie de l'eau” (47h) at the engineering school Polytech'Nice.

9.2.2 Supervision

• PhD in progress: Paul Maurer, “Stochastic models for deformable particle dynamics in turbulence: mathematical analysis and simulation for temporal intermittency and long-range correlation models” started in October 2022; supervised by Mireille Bossy.
• PhD in progress: Nicolas Valade, “Spontaneous stochasticity of quasi-geostrophic surface turbulence” started in October 2022; supervised by Jérémie Bec and Simon Thalabard (INPHYNI, Université Côte d'Azur).
• PhD in progress: Zakarya El Khiyati, “Reinforcement learning for the optimal locomotion of micro-swimmers in a complex chaotic environment” started in October 2021; supervised by Jérémie Bec and Laetitia Giraldi.
• PhD in progress: Celine Van-Landeghem, “numerical methods for locomotion of micro-swimmers in a confined environment” started in October 2021; supervised by Christophe Prud'Homme (IRMA, Strasbourg), Yannick Hoarau and Laetitia Giraldi.
• PhD in progress: Fabiola Gerosa, “Turbulent fluid-particles coupling and applications to planet formation” started in October 2021; supervised by Jérémie Bec and Héloïse Méheut (Lagrange, Observatoire de la Côte d'Azur).
• PhD in progress: Luca Palazzolo, "Numerical methods for optimising the locomotion of flagellate microswimmers" started in October 2023; supervised by Laetitia Giraldi, Mickael Binois (Inria EPC Acumes) and Christophe Prudhomme (Université de Strasbourg).
• M2 internship : Hao Liu (Technology University Darmstad), "Particle transport in the environment", supervised by Christophe Henry and Mireille Bossy; Luca Palazzolo supervised by Laetitia Giraldi ; Antoine Serres (ENS Paris Saclay) supervised by Jérémie Bec.

9.2.3 Juries

• Jérémie Bec served as an examiner for the Ph.D. thesis of Sumithra Reddy Yerasi (U. Côte d'Azur).
• Laetitia Giraldi served as an examiner for the Ph.D. thesis of Xiao Xie (Nantes university, IMT Atlantique/ LS2N).

9.3.1 Education

Calisto team members took part in various education activities related to popularization.

• Christophe Henry took part in some of the activities organized by Terra Numerica for young students, including:
  • The event "Stage Math C2+" in June 2023, giving a 1h30 intervention on the "Physics and Mathematics of sand castles" to an audience of 40 students in the first year of upper secondary education (classe de 2nde au Lycée).
  • The event "Stage 3e" in December 2023, giving a 1h intervention on the "Physics and Mathematics of sand castles" to an audience of 11 students in the last year of lower secondary education (classe de 3ème au collège).

9.3.2 Interventions

• Fête de la Science: Every year, a science festival is held across France in autumn where researchers present their activities to the public (especially for kids and young students).
  • Christophe Henry took part in the festival held in Villeneuve-Loubet, in Mouans Sartoux and in Antibes-Juan Les Pins in October 2023, presenting a workshop on the “Physics of sand castles”.
• Complex Days: the Academy “Complex Systems” of Université Côte d'Azur promotes interdisciplinary research on the analysis and applications of complex systems. Since 4 years, the Academy organises an annual event dedicated to its community, called Complex Days. The aim is to bring together young and senior researchers from different backgrounds to exchange ideas and promote new developments in complex systems. In February 2023, Calisto team members made two contributions:
  • Christophe Henry gave a talk on "Particle and flow dynamics in wall-bounded environments: Modeling & simulation of particle agglomeration" 18.
  • Paul Maurer gave a talk on "Stochastic models driven by a Lévy noise: Application to rods orientation in turbulence" 19.

International journals

• 1 articleS.Sofia Allende and J.Jérémie Bec. Velocity and acceleration statistics of heavy spheroidal particles in turbulence.Journal of Fluid Mechanics967July 2023, R4HALDOIback to text
• 2 articleA.Amir Banari, K.Klaus Graebe, M.Martin Rudolph, E.Ehsan Mohseni, P.Pierre Lorenz, K.Klaus Zimmer, R.René Hübner, C.Christophe Henry, M.Mireille Bossy, U.Uwe Hampel and G.Gregory Lecrivain. Influence of engineered roughness microstructures on adhesion and turbulent resuspension of microparticles.Journal of Aerosol Science174November 2023, 106258HALDOIback to text
• 3 articleJ.J. Bec, K.K. Gustavsson and B.B. Mehlig. Statistical models for the dynamics of heavy particles in turbulence.Annual Review of Fluid Mechanics562024HALDOIback to text
• 4 articleL.Lorenzo Campana, M.Mireille Bossy and C.Christophe Henry. Lagrangian stochastic model for the orientation of inertialess non spherical particles in turbulent flows: an efficient numerical method for CFD approach.Computers and Fluids257May 2023, 105870HALDOIback to text
• 5 articleZ.Zakarya El Khiyati, R.Raphaël Chesneaux, L.Laëtitia Giraldi and J.Jérémie Bec. Steering undulatory micro-swimmers in a fluid flow through reinforcement learning.European Physical Journal E: Soft matter and biological physics466June 2023, 43HALDOIback to text
• 6 articleF. A.Fabiola Antonietta Gerosa, H.Héloïse Meheut and J.Jérémie Bec. Clusters of heavy particles in two-dimensional Keplerian turbulence.The European Physical Journal Plus1382023, 9HALDOIback to text
• 7 articleL.Laetitia Giraldi, P.Pierre Lissy, C.Clément Moreau and J.-B.Jean-Baptiste Pomet. Necessary conditions for local controllability of a particular class of systems with two scalar controls.ESAIM: Control, Optimisation and Calculus of Variations3042024HALDOIback to text
• 8 articleC.Christophe Henry, J.-P.Jean-Pierre Minier and S.Sara Brambilla. Particle resuspension: challenges and perspectives for future models.Physics Reports1007January 2023, 1-98HALDOI
• 9 articleJ.José Morán, J.Jérôme Yon, C.Christophe Henry and M. R.M. Reza Kholghy. Approximating the van der Waals interaction potentials between agglomerates of nanoparticles.Advanced Powder Technology3412December 2023, 104269HALDOIback to text
• 10 articleN.Nicolas Valade, S.Simon Thalabard and J.Jérémie Bec. Anomalous dissipation and spontaneous stochasticity in deterministic surface quasi-geostrophic flow.Annales Henri PoincaréFebruary 2023HALDOIback to text

Doctoral dissertations and habilitation theses

• 11 thesisC.Christophe Henry. Modeling particle-laden flows.Université Cote d'Azur; ED SFAMarch 2023HALback to text

Reports & preprints

• 12 miscG.Guilhem Balvet, J.-P.Jean-Pierre Minier, C.Christophe Henry, Y.Yelva Roustan and M.Martin Ferrand. A time-step-robust algorithm to compute particle trajectories in 3-D unstructured meshes for Lagrangian stochastic methods.March 2023HALDOIback to text
• 13 miscJ.J Bec, C.Christophe Brouzet and C.Christophe Henry. Enhanced transport of flexible fibers by pole vaulting in turbulent wall-bounded flow.November 2023HALback to text
• 14 miscM.Mireille Bossy and P.Paul Maurer. On the $$-Euler-Maruyama scheme for time inhomogeneous jump-driven SDEs.January 2024HALback to text
• 15 miscC. V.Céline Van Landeghem, L.Luca Berti, V.Vincent Chabannes, C.Christophe Prud'Homme, A.Agathe Chouippe, Y.Yannick Hoarau and L.Laëtitia Giraldi. Mathematical and computation framework for moving and colliding rigid bodies in a Newtonian fluid.2023HALback to text
• 16 miscJ.-P.Jean-Pierre Minier and C.Christophe Henry. The dynamics of discrete particles in turbulent flows: open issues and current challenges in statistical modeling.2023HALDOIback to text
• 17 miscL.L. Storm, H.H. Linander, J.J. Bec, K.K. Gustavsson and B.B. Mehlig. Finite-time Lyapunov exponents of deep neural networks.June 2023HALback to text

Scientific popularization

• 18 inproceedingsC.Christophe Henry. Particle and flow dynamics in wall-bounded environments: Modeling & simulation of particle agglomeration.Complex DaysComplex DaysComplex Days3Nice, FranceFebruary 2023HALback to text
• 19 inproceedingsP.Paul Maurer. Stochastic models driven by a Lévy noise: Application to rods orientation in turbulence.Complex DaysComplex DaysComplex Days3Nice, France2023HALback to text