The research group that we have entitled fluminance from a contraction between the words “Fluid” and “Luminance” is dedicated to the extraction of information on fluid flows from image sequences and to the development of tools for the analysis and control of these flows. The objectives of the group are at the frontiers of several important domains that range from fluid mechanics to geophysics. One of the main originality of the fluminance group is to combine cutting-edge researches on data-assimilation and flow numerical modeling with an ability to conduct proper intensive experimental validations on prototype flows mastered in laboratory. The scientific objectives decompose in four main themes:

Fluid flows characterization from images

In this first axis, we aim at providing accurate measurements and consistent analysis of complex fluid flows through image analysis techniques.The application domain ranges from industrial processes and experimental fluid mechanics to environmental sciences. This theme includes also the use of non-conventional imaging techniques such as Schlieren techniques, Shadowgraphs, holography. The objective will be here to go towards 3D dense velocity measurements.

Coupling dynamical model and image data

We focus here on the study, through image data, of complex and partially known fluid flows involving complex boundary conditions, multi-phase fluids, fluids and structures interaction problems. Our credo is that image analysis can provide sufficiently fine observations on small and medium scales to construct models which, applied at medium and large scale, account accurately for a wider range of the dynamics scales. The image data and a sound modeling of the dynamical uncertainty at the observation scale should allow us to reconstruct the observed flow and to provide efficient real flows (experimental or natural) based dynamical modeling. Our final goal will be to go towards a 3D reconstruction of real flows, or to operate large motion scales simulations that fit real world flow data and incorporate an appropriate uncertainty modeling.

Control and optimization of turbulent flows

We are interested on active control and more precisely on closed-loop control. The main idea is to extract reliable image features to act on the flow. This approach is well known in the robot control community, it is called visual servoing. More generally, it is a technique to control a dynamic system from image features. We plan to apply this approach on flows involved in various domains such as environment, transport, microfluidic, industrial chemistry, pharmacy, food industry, agriculture, etc.

The measurement of fluid representative features such as vector fields, potential functions or vorticity maps, enables physicists to have better understanding of experimental or geophysical fluid flows. Such measurements date back to one century and more but became an intensive subject of research since the emergence of correlation techniques to track fluid movements in pairs of images of a particles laden fluid or by the way of clouds photometric pattern identification in meteorological images. In computer vision, the estimation of the projection of the apparent motion of a 3D scene onto the image plane, referred to in the literature as optical-flow, is an intensive subject of researches since the 80's and the seminal work of B. Horn and B. Schunk . Unlike to dense optical flow estimators, the former approach provides techniques that supply only sparse velocity fields. These methods have demonstrated to be robust and to provide accurate measurements for flows seeded with particles. These restrictions and their inherent discrete local nature limit too much their use and prevent any evolutions of these techniques towards the devising of methods supplying physically consistent results and small scale velocity measurements. It does not authorize also the use of scalar images exploited in numerous situations to visualize flows (image showing the diffusion of a scalar such as dye, p ollutant, light index refraction, fluorescein,...). At the opposite, variational techniques enable in a well-established mathematical framework to estimate spatially continuous velocity fields, which should allow more properly to go towards the measurement of smaller motion scales. As these methods are defined through PDE's systems they allow quite naturally constraints to be included such as kinematic properties or dynamic laws governing the observed fluid flows. Besides, within this framework it is also much easier to define characteristic features estimation procedures on the basis of physically grounded data model that describes the relation linking the observed luminance function and some state variables of the observed flow. The Fluminance group has allowed a substantial progress in this direction with the design of dedicated dense estimation techniques to estimate dense fluid motion fields. See for a detailed review. More recently problems related to scale measurement and uncertainty estimation have been investigated . Dynamically consistent and highly robust techniques have been also proposed for the recovery of surface oceanic streams from satellite images . Very recently parameter-free approaches relying on uncertainty concept has been devised . This technique outperforms the state of the art.

Real flows have an extent of complexity, even in carefully controlled experimental conditions, which prevents any set of sensors from providing enough information to describe them completely. Even with the highest levels of accuracy, space-time coverage and grid refinement, there will always remain at least a lack of resolution and some missing input about the actual boundary conditions. This is obviously true for the complex flows encountered in industrial and natural conditions, but remains also an obstacle even for standard academic flows thoroughly investigated in research conditions.

This unavoidable deficiency of the experimental techniques is nevertheless more and more compensated by numerical simulations. The parallel advances in sensors, acquisition, treatment and computer efficiency allow the mixing of experimental and simulated data produced at compatible scales in space and time. The inclusion of dynamical models as constraints of the data analysis process brings a guaranty of coherency based on fundamental equations known to correctly represent the dynamics of the flow (e.g. Navier Stokes equations) . Conversely, the injection of experimental data into simulations ensures some fitting of the model with reality.

To enable data and models coupling to achieve its potential, some difficulties have to be tackled. It is in particular important to outline the fact that the coupling of dynamical models and image data are far from being straightforward. The first difficulty is related to the space of the physical model. As a matter of fact, physical models describe generally the phenomenon evolution in a 3D Cartesian space whereas images provides generally only 2D tomographic views or projections of the 3D space on the 2D image plane. Furthermore, these views are sometimes incomplete because of partial occlusions and the relations between the model state variables and the image intensity function are otherwise often intricate and only partially known. Besides, the dynamical model and the image data may be related to spatio-temporal scale spaces of very different natures which increases the complexity of an eventual multiscale coupling. As a consequence of these difficulties, it is necessary generally to define simpler dynamical models in order to assimilate image data. This redefinition can be done for instance on an uncertainty analysis basis, through physical considerations or by the way of data based empirical specifications. Such modeling comes to define inexact evolution laws and leads to the handling of stochastic dynamical models. The necessity to make use and define sound approximate models, the dimension of the state variables of interest and the complex relations linking the state variables and the intensity function, together with the potential applications described earlier constitute very stimulating issues for the design of efficient data-model coupling techniques based on image sequences.

On top of the problems mentioned above, the models exploited in assimilation techniques often suffer from some uncertainties on the parameters which define them. Hence, a new emerging field of research focuses on the characterization of the set of achievable solutions as a function of these uncertainties. This sort of characterization indeed turns out to be crucial for the relevant analysis of any simulation outputs or the correct interpretation of operational forecasting schemes. In this context, stochastic modeling play a crucial role to model and process uncertainty evolution along time. As a consequence, stochastic parameterization of flow dynamics has already been present in many contributions of the Fluminance group in the last years and will remain a cornerstone of the new methodologies investigated by the team in the domain of uncertainty characterization.

This wide theme of research problems is a central topic in our research group. As a matter of fact, such a coupling may rely on adequate instantaneous motion descriptors extracted with the help of the techniques studied in the first research axis of the fluminance group. In the same time, this coupling is also essential with respect to visual flow control studies explored in the third theme.
The coupling between a dynamics and data, designated in the literature as a Data Assimilation issue, can be either conducted with optimal control techniques , or through stochastic filtering approaches , . These two frameworks have their own advantages and deficiencies. We rely indifferently on both approaches.

Fluid flow control is a recent and active research domain. A significant part of the work carried out so far in that field has been dedicated to the control of the transition from laminarity to turbulence. Delaying, accelerating or modifying this transition is of great economical interest for industrial applications. For instance, it has been shown that for an aircraft, a drag reduction can be obtained while enhancing the lift, leading consequently to limit fuel consumption. In contrast, in other application domains such as industrial chemistry, turbulence phenomena are encouraged to improve heat exchange, increase the mixing of chemical components and enhance chemical reactions. Similarly, in military and civilians applications where combustion is involved, the control of mixing by means of turbulence handling rouses a great interest, for example to limit infra-red signatures of fighter aircraft.

Flow control can be achieved in two different ways: passive or active control. Passive control provides a permanent action on a system. Most often it consists in optimizing shapes or in choosing suitable surfacing (see for example where longitudinal riblets are used to reduce the drag caused by turbulence). The main problem with such an approach is that the control is, of course, inoperative when the system changes. Conversely, in active control the action is time varying and adapted to the current system's state. This approach requires an external energy to act on the system through actuators enabling a forcing on the flow through for instance blowing and suction actions , . A closed-loop problem can be formulated as an optimal control issue where a control law minimizing an objective cost function (minimization of the drag, minimization of the actuators power, etc.) must be applied to the actuators . Most of the works of the literature indeed comes back to open-loop control approaches , , or to forcing approaches with control laws acting without any feedback information on the flow actual state. In order for these methods to be operative, the model used to derive the control law must describe as accurately as possible the flow and all the eventual perturbations of the surrounding environment, which is very unlikely in real situations. In addition, as such approaches rely on a perfect model, a high computational costs is usually required. This inescapable pitfall has motivated a strong interest on model reduction. Their key advantage being that they can be specified empirically from the data and represent quite accurately, with only few modes, complex flows' dynamics. This motivates an important research axis in the Fluminance group.

The team is strongly involved in numerical models for hydrogeology and geophysics. There are many scientific challenges in the area of groundwater simulations. This interdisciplinary research is very fruitful with cross-fertilizing subjects.

In geophysics, a main concern is to solve inverse problems in order to fit the measured data with the model. Generally, this amounts to solve a linear or nonlinear least-squares problem.

Models of geophysics are in general coupled and multi-physics. For example, reactive transport couples advection-diffusion with chemistry. Here, the mathematical model is a set of nonlinear Partial Differential Algebraic Equations. At each timestep of an implicit scheme, a large nonlinear system of equations arise. The challenge is to solve efficiently and accurately these large nonlinear systems.

Linear algebra is at the kernel of most scientific applications, in particular in physical or chemical engineering. The objectives are to analyze the complexity of these different methods, to accelerate convergence of iterative methods, to measure and improve the efficiency on parallel architectures, to define criteria of choice.

By designing new approaches for the analysis of fluid image sequences, data -model coupling and stochastic representation of fluid flows the Fluminance group contrinutes to several application domains of great interest for the community and in which the anaysis of complex turbulent flow is key. The group focuses on two broad application domains:

More recently a focus on ocean dynamics and indoor environmental flow has been operated.

The team dedicates entirely its research effort to environmental problems related to the climate change issue and its consequences for the future of our planet. Our activities concerns mainly Mathematics for planet Earth.

The team's research results are principally published in journals dedicated to Mathematics for planet Earth and environmental sciences issues. As such we intend to contribute with our stength and skills to these questions.

The goal is to design a new image-based flow measurement method for large-scale industrial applications. From this point of view, providing in situ measurement technique requires: (i) the development of precise models relating the large-scale flow observations to the velocity; (ii) appropriate large-scale regularization strategies; and (iii) adapted seeding and lighting systems, like Hellium Filled Soap Bubles (HFSB) and led ramp lighting. This work conducted within the PhD of Romain Schuster in collaboration with the compagny ITGA has started in february 2016. The first step has been to evaluate the performances of a stochastic uncertainty motion estimator when using large scale scalar images, like those obtained when seeding a flow with smoke. The PIV characterization of flows on large fields of view requires an adaptation of the motion estimation method from image sequences. The backward shift of the camera coupled to a dense scalar seeding involves a large scale observation of the flow, thereby producing uncertainty about the observed phenomena. By introducing a stochastic term related to this uncertainty into the observation term, we obtained a significant improvement of the estimated velocity field accuracy. The technique was validated on a mixing layer in a wind tunnel for HFSB and smoke tracers and applied on a laboratory fume-hood.

Our work focuses on the design of new tools for the estimation of 3D turbulent flow motion in the experimental setup of Tomo-PIV. This task includes both the study of physically-sound models on the observations and the fluid motion, and the design of low-complexity and accurate estimation algorithms. We have proposed a novel method for volumetric velocity reconstruction exploring the locality of 3D object space. Under this formulation the velocity of local patch was sought to match the projection of the particles within the local patch in image space to the image recorded by camera. The core algorithm to solve the matching problem is an instance-based estimation scheme that can overcome the difficulties of optimization originated from the nonlinear relationship between the image intensity residual and the volumetric velocity. The proposed method labeled as Lagrangian Particle Image Velocimetry (LaPIV) is quantitatively evaluated with synthetic particle image data. The promising results that have been obtained indicates the potential application of LaPIV to a large variety of volumetric velocity reconstruction problems .

In collaboration with the CSTB Nantes centre and within the PhD of Yacine Ben Ali we explored the definition of efficient data assimilation schemes for wind engineering. The goal is here to couple Reynolds average model to pressure data at the surface of buildings. Several techniques have been proposed to that end. We show in particular that optimisation conducted in a Sobolev space is highly beneficial as it brings natural smoothing to the sought solutions and avoids the use of regularization penalty. The techniques proposed consists in correcting the equations related to turbulent kinetic energy and dissipation. This work is thoroughly detailed in the PhD manuscript of Yacine Ben Ali . Two journal papers are currently under submission.

In another line of work, we addressed the study of variational data assimilation from a learning point of view. Data assimilation aims to reconstruct the time evolution of some state given a series of observations, possibly noisy and irregularly-sampled. Using automatic differentiation tools embedded in deep learning frameworks, we introduce end-to-end neural network architectures for data assimilation. It comprises two key components: a variational model and a gradient-based solver both implemented as neural networks. A key feature of the proposed end-to-end learning architecture is that we may train the NN models using both supervised and unsupervised strategies. This work has been published in

We investigated the application of a physically relevant stochastic dynamical model in ensemble Kalman filter methods. Ensemble Kalman filters are very popular in data assimilation because of their ability to handle the filtering of high-dimensional systems with reasonably small ensembles (especially when they are accompanied with so called localization techniques). The stochastic framework used in this study relies on Location Uncertainty (LU) principles which model the effects of the model errors on the large-scale flow components. The experiments carried out on the Surface Quasi Geostrophic (SQG) model with the localized square root filter demonstrate two significant improvements compared to the deterministic framework. Firstly, as the uncertainty is a priori built into the model through the stochastic parametrization, there is no need for ad-hoc variance inflation or perturbation of the initial condition. Secondly, it yields better MSE results than the deterministic ones. This work is currently under submission in a journal.

The uncertainty based representation of Navier-Stokes equations proposed in

has been applied in the context of POD-Galerkin methods to devise stochastic reduced order models. This uncertainty modeling methodology provides a theoretically grounded technique to define an appropriate subgrid tensor as well as drift correction terms. This reduced order stochastic system has been evaluated on wake flow at moderate Reynolds number. For this flow the system has shown to provide very good uncertainty quantification properties as well as meaningful physical behavior with respect to the simulation of the neutral modes of the dynamics. This study is pursued within a strong collaboration with the industrial partner: SCALIAN

.

A methodological framework for ensemble-base estimation and simulation of high dimensional dynamical systems such as the oceanic or atmospheric flows is proposed. To that end, the dynamical system is embedded in a manifold of reproducible kernel Hilbert spaces with kernel functions driven by the dynamics. This manifold is nicknamed Wonderland for its appealing properties. In Wonderland the Koopman and Perron-Frobenius operator (also referred to in the literature as the composition and transfer operators, respectively) are unitary and uniformly continuous. They can be safely expressed in exponential series of diagonalizable bounded infinitesimal generators. Access to Lyapunov exponents and to exact ensemble based expressions of the tangent linear dynamics are directly available as well. Wonderland enables us the devise of strikingly simple ensemble data assimilation methods for trajectory reconstructions in terms of constant-in-time linear combinations of trajectory samples. Such an embarrassingly simple strategy is made possible through a fully justified superposition principle ensuing from several fundamental theorems. Numerical proofs of concept for data assimilation and trajectory recovery have been performed with a quasi-geostrophic flow model.

A class of flows, denoted "oscillator flows", are characterised by unstable modes of the linearised operator. A consequence is the dominance of relatively regular oscillations associated with a nonlinear saturation. Despite its non-linear behaviour, the associated structures and dynamical evolution are relatively easy to predict. Canonical configurations of this type of flows are the cylinder wake flow or the flow over an open cavity.

By opposition to that, "amplifier flows" are linearly stable with regard to the linearised operator. However, due to their convective nature, a wide range of perturbations are amplified in time and convected away such that it vanishes at long time. As a consequence there is a high sensitivity to perturbations together with a broad band response that forbid any low rank representation. Jets and mixing layers show this behaviour and a wide range of industrial applications and geophysical flows are affected by these broad band perturbations. It constitutes then a class of problems that are worth to treat separately since it is one of the scientific locks that make the estimation of observation-driven flow reconstruction by data assimilation in realistic configurations.

The stochastic framework of the modelling under location uncertainty (LU) considers a separation between a resolved velocity field and unresolved incoherent turbulent fluctuations. Amplifier flows are particularly sensitive to these perturbations and we believe that LU would improve significantly the predictions of turbulent amplifier flows.

In a deterministic context, there exists a type of models, termed as "parabolised", that enable to efficiently represent amplifier flows. These models, such as parabolised stability equations and one-way Navier-Stokes propagate, in the frequency domain, hydrodynamic instability waves over a given turbulent mean flow. The extension of parabolised models to a stochastic version is a promising approach which is perfectly adapted to represent the evolution and the variability of an instability propagating within a turbulent flow.

The dynamics of upper-ocean high-frequency motions — internal waves and, more specifically, internal tides, has been addressed using different strategies. As part of F. Le Guillou PhD work (IGE, Grenoble) and in collaboration with J. Le Sommer, E. Cosme, S. Metref (MEOM team, IGE, Grenoble), C. Ubelmann (Ocean Next, Grenoble), E. Blayo, A. Vidard (Inria, Grenoble) and A. Ponte (LOPS, Brest), a joint algorithm for estimating the vortices as well as first mode internal tide from future wide-swath altimeter data has been proposed and tested in an idealized framework . This work is being pursued in the Fluminance team. In the context of Z. Caspar-Cohen PhD work, the signature of internal tides collected by surface drifters has been investigated using idealized numerical simulations. It has been shown that an apparent loss of coherence (time regularity of the oscillations at the tidal frequencies) arises in the Lagrangian perspective, resulting from the drifters being advected by the slow flow , . A statistical model based on the statistical properties of the slow flow has been proposed and validated against the data in this framework. It allows to predict this « apparent » loss of coherence.

In this task, the dynamics of coherent structures in reduced models of the ocean upper layer was addressed. First, in collaboration with V. Zeitlin and T. Dubos (LMD, Paris), we investigated the dynamics of coherent dipolar analytical solutions in the Thermal Rotating Shallow Water model. This model is a vertically averaged model of the hydrostatic Navier-Stokes equations under the Boussinesq approximation and with Coriolis term, and retains the horizontal variations of density (or temperature). In the strong rotation regime, a low Rossby number asymptotic model can be derived : the Thermal Quasi-Geostrophic model. It exhibited a surprising small scale instability, leading to mixing of the thermal anomaly carried by the dipole. This research activity is pursued to better understand the dynamics of these models, in connexion with the STUOD project, which involves such models. In a second part of this task, in collaboration with A. Paci and S. Llewellyn Smith, we investigated the destabilization of surface eddies in a two layer with outcropping isopycnal layer — a different approach for representing horizontal variations of temperature and density in the ocean. This work highlighted the stabilization of surface eddies by a weak co-rotating lower-layer flow, which was shown to result from the suppression of one of the mode triggering instability through wave resonance

.

This study focuses on the interactions between internal tides and oceanic currents. The goal is to develop new modelling strategies to extract, understand and predict physical features involved in the interactions between internal waves and flow structures. The phenomenon is isolated in a simplified numerical simulation of the rotating shallow water model, where a single wave propagates through a jet flow. Spectral proper orthogonal decomposition (SPOD) allows to identify from data purely coherent structures evolving at a given frequency. It allows to perform physical analyses and it constitutes reference observations. Linearising the model over the mean flow, resolvent analysis allows to predict the most responsive non-linearities and the associated responses. The framework will allow to predict coherence decay and to explicitly compute triadic interactions responsible of wave scattering by the flow. On this basis, extension to stochastic modelling under location uncertainty will refine the predictions. A potential direction is the development of data assimilation procedures defined in the frequency domain.

This research focuses on physical analysis of internal tides from very large databases issued from high fidelity numerical simulations. After a projection onto vertical modes, the energy exchanges at the tidal frequencies are quantified. The role of the topography and the currents in the exchange between modes are explored. In a second step, loss of coherence of internal tides due to currents will be explored. Three regions increasing in complexity are targeted: the Açores where the topography plays a major role, a region where the gulf stream and internal tildes interact strongly, and equatorial regions where very strong non-linearities occur.

To predict the tracer deformations by ocean eddies and the evolution of their 2nd-order statistics, an efficient proxy has been proposed in collaboration with Bertrand Chapron (Ifremer) and Valentin Resseguier (Scalian)

. Applied to a single velocity snapshot, this proxy extends the Okubo-Weiss criterion. For the Lagrangian-advection-based downscaling methods, it successfully predicts the evolution of tracer spectral energy density after a finite time, and the optimal time to stop the downscaling operation. A practical estimation can then be proposed to define an effective parameterization of the horizontal eddy diffusivity.

In this research axis we have devised a principle to derive representation of flow dynamics under location uncertainty

. Such an uncertainty is formalized through the introduction of a random term that enables taking into account large-scale approximations or truncation effects performed within the dynamics analytical constitution steps. Rigorously derived from a stochastic version of the Reynolds transport theorem, this framework, referred to as modeling under location uncertainty (LU), encompasses several meaningful mechanisms for turbulence modeling. It indeed introduces without any supplementary assumption the following pertinent mechanisms: (i) a dissipative operator related to the mixing effect of the large-scale components by the small-scale velocity;(ii) a multiplicative noise representing small-scale energy backscattering; and (iii) a modified advection term related to the so-called

phenomena, attached to the migration of inertial particles in regions of lower turbulent diffusivity.

Within the PhDs of Long Li and Valentin Resseguier , , we have shown how LU modeling can be applied to provide stochastic representations of a variety of classical geophysical flows dynamics. Numerical simulations and uncertainty quantification have been performed on Quasi Geostophic approximation (QG) of oceanic models. It has been shown that LU leads to remarkable estimation of the unresolved errors opposite to classical eddy viscosity based models. The noise brings also an additional degree of freedom in the modeling step and pertinent diagnostic relations and variations of the model can be obtained with different scaling assumptions of the turbulent kinetic energy (i.e. of the noise amplitude). For a wind forced QG model in a square box, which is an idealized model of north-Atlantic circulation, we have shown that for different versions of the noise the QG LU model leads to improve long-terms statistics when compared to classical large-eddies simulation strategies. For a QG model we have demonstrated that the LU model allows conserving the global energy. We have also shown numerically that Rosby waves were conserved and that inhomogeneity of the random component triggers secondary circulations. This feature enabled us to draw a formal bridge between a classical system describing the interactions between the mean current and the surface waves and the LU model in which the turbophoresis advection term plays the role of the classical Stokes drift.

In collaboration with Ruediger Brecht, PhD student at Memorial University of Newfoundland, we worked on the incorporation of a stochastic representation of the small-scale velocity component of a fluid flow in a variational integrator for the rotating shallow-water equations on the sphere, already developed within the first part of its PhD work. This work has been published in JAMES .

A study of a stochastic version of the primitive equations model is currently investigated within the PhD of Francesco Tucciarone. An investigation of such model in a realistic ocean numerical code with various noise assumptions is undergoing.

We study here a Hamiltonian stochastic formulation of the water wave problem in the setting of the modelling under location uncertainty. Starting from reduction of the stochastic fluid motion equations to the free surface, we show how one can naturally deduce Hamiltonian structure under a small noise assumption. Moreover, as in the classical water wave theory, the non-local Dirichlet-Neumann operator appears explicitly in the energy functional. This, in particular, allows us to conduct in a natural way the systematic approximation of the Dirichlet-Neumann operator and to devise different simplified wave models including noise. Well posedness analysis of some of such models is currently under investigation.

We work on simple parameterization for coarse-resolution oceanic models to replace computationally expensive high-resolution ocean models. We focus on the eddy-permitting scale (grid step Rossby radius) and computationally cheap parameterization. We are currently investigating the modification of the diffusion (friction) operator to reproduce the mean velocity observed via measurements or a high-resolution reference solution. To test this new parameterization on a double-gyre quasi-geostrophic model, we are implementing a fast and portable python implementation of the multilayer quasi-geostrophic model. Preliminary results are encouraging as they show that we can fairly reproduce the reference mean velocity and significantly improve energy statistics. This method shall serve as a deterministic basis for future coarse-resolution stochastic parameterizations.

In collaboration with Rudiger Brecht (MPI, Hamburg) and Werner Bauer (Imperial College) we introduced in

a physically relevant stochastic representation of the rotating shallow water equations. This derivation relies mainly on a stochastic transport principle and on a decomposition of the fluid flow into a large-scale component and a noise term that models the unresolved flow components. As for the classical (deterministic) system, this scheme, referred to as modeling under location uncertainty (LU), conserves the global energy of any realization and provides the possibility to generate an ensemble of physically relevant random simulations with a good trade-off between the model error representation and the ensemble's spread. To maintain numerically the energy conservation feature, we combine an energy (in space) preserving discretization of the underlying deterministic model with approximations of the stochastic terms that are based on standard finite volume/difference operators. The LU derivation, built from the very same conservation principles as the usual geophysical models, together with the numerical scheme proposed can be directly used in existing dynamical cores of global numerical weather prediction models. The capabilities of the proposed framework is demonstrated for an inviscid test case on the f-plane and for a barotropically unstable jet on the sphere.

In this work we studied Milstein-type schemes for models under location uncertainty. We focus in particular on study the surface quasi-geostrophic (SQG) system under location uncertainty (LU). For this model we devised efficient scheme based on a second order Runge-Kutta scheme with the Levy area term set to zero. This scheme has been numerically compared to Milstein scheme in which the Levy area was fully taken into account. It is shown that the proposed scheme leads to better results with a comparable computational cost.

We are currently working on the extension of the stochastic formulation under location uncertainty to compressible flows. The interest is to extend the formulation on the one hand to compressible fluids (for instability mechanisms involved in areoacoustics for instance, or for thermal effects in mixing layers) and on the other hand to geophysical flows where the Boussinesq equation is not valid anymore (density variations due to temperature or salinity gradients). A theoretical study has been performed that opens the door to numerical validations. In particular a baroclinic torque term has been identified that could have major effects in some situations.

In order to predict instability waves propagating within turbulent flows, eigenmodes of the linearised operator is not well suited since it neglects the effect of turbulent fluctuations on the wave dynamics. To cope this difficulty, resolvent analysis has become popular since it represents the response of the linearised operator to any forcing representing the generalised stress tensors. The absence of information on the non-linearity is a strong limitation of the method. In order to refine these models, we propose to consider a stochastic model under location uncertainty expressed in the Fourier domain, to linearise it around the corrected mean-flow and to study resulting eigenmodes. The stochastic part represents the effect of the turbulent field onto the instability wave. It allows to specify a structure of the noise and then to improve existing models. Improvements compared to the resolvent analysis have been found for turbulent channel flow data at

,

and

. A paper has been published in Journal of Fluid Mechanics

, and a second one is in preparation for Journal of Fluid Mechanics. This work is in collaboration with André Cavalieri (Instituto Tecnologico de Aeronautica, SP, Brésil).

The common thread of this work is the problem set by J. Leray in 1934 : does a regular solution of the Navier- Stokes equations (NSE) with a smooth initial data develop a singularity in finite time, what is the precise structure of a global weak solution to the Navier-Stokes equations, and are we able to prove any uniqueness result of such a solution. This is a very hard problem for which there is for the moment no answer. Nevertheless, this question leads us to reconsider the theory of Leray for the study of the Navier-Stokes equations in the whole space with an additional eddy viscosity term that models the Reynolds stress in the context of large- scale flow modelling. It appears that Leray's theory cannot be generalized turnkey for this problem, and must be reconsidered from the beginning. This problem is approached by a regularization process using mollifiers, and particular attention must be paid to the eddy viscosity term. For this regularized problem and when the eddy viscosity has enough regularity, we have been able to prove the existence of a global unique solution that is of class C1 in time and space and that satisfies the energy balance. Moreover, when the eddy viscosity is of compact support in space, uniformly in time, we recently shown that this solution converges to a turbulent solution of the corresponding Navier-Stokes equations carried when the regularizing parameter goes to 0. These results are described in a paper published in JMAA.

In the framework of the collaboration with the University of Pisa (Italy), namely with Luigi Berselli collaboration, we considered the three dimensional incompressible Navier-Stokes equations with non stationary source terms chosen in a suitable space. We proved the existence of Leray-Hopf weak solutions and that it is possible to characterize (up to sub-sequences) their long-time averages, which satisfy the Reynolds averaged equations, involving a Reynolds stress. Moreover, we showed that the turbulent dissipation is bounded by the sum of the Reynolds stress work and of the external turbulent fluxes, without any additional assumption, than that of dealing with Leray-Hopf weak solutions. This is a very nice generalisation to non stationnary source terms of a famous results by Foais. In the same work, we also considered ensemble averages of solutions, associated with a set of different forces and we proved that the fluctuations continue to have a dissipative effect on the mean flow. These results have been published in Nonlinearity. We have studied in the rate of convergence of the weak solutions

Another study in collaboration with B. Pinier, P. Chandramouli and E. M/'emin has been undertaken. This work takes place within the context of the PhD work of B. Pinier. We have tested the performances of an incompressible turbulence Reynolds-Averaged Navier-Stokes one-closure equation model in a boundary layer, which requires the determination of the mixing length

Perforated liners are a technology implanted in the nacelles of aircraft engines, in order to absorb noise coming from the fan and the combustion chamber. It is constituted by honeycomb cavities covered by a perforated plate. The cavities produce resonance, thus inducing a flow through the perforations where viscous dissipation occurs, necessary for sound absorption. These perforations cause drag in the air intake, which can be reduced by considering micro perforates. However, existing models are not able to predict correctly the impedance of such plates. Understanding the dissipation mechanisms and improving the impedance predictions for microperforated plates was the objective of the thesis of Robin Billard, successfully defended this year and leaded in collaboration with Gwénaël Gabard (LAUM – laboratoire d'acoustique de l'université du Mans) and Safran Nacelles. The challenge was to account for non-linear effects and grazing flow in the developed models. Resolvent analysis has been explored to identify relevant non-linearities that impact the impedance, with and without flow. Part of this study has been pubished in

.

Groundwater resources are essential for life and society, and should be preserved from contamination. Pollutants are transported through the porous medium and a plume can propagate. Reactive transport models aims at simulating this dynamic contamination by coupling advection dispersion equations with chemistry equations. If chemistry is at thermodynamic equilibrium, then the system is a set of partial differential and algebraic equations (PDAE). Space discretization leads to a semi-discrete DAE system which should be discretized in time. An explicit time scheme allows an easy decoupling of transport and chemistry, but very small timesteps should be taken, leading to a very large CPU time. Therefore, an implicit time scheme is preferred, coupling transport and chemistry in a nonlinear system. The special structure of linearized systems can be used in preconditioned Newton-Krylov methods in order to improve efficiency. Some experiments illustrate the methodology and show also the need for an adaptive timestep and a control of convergence in Newton's iterations.

Geochemistry at thermodynamic equilibrium involves aqueous reactions and mineral precipitation or dissolution. Quantities of solute species are assumed to be strictly positive, whereas those of minerals can vanish. The mathematical model is expressed as the minimization of Gibbs energy subject to positivity of mineral quantities and conservation of mass. Optimality conditions lead to a complementarity problem. We show that, in the case of a dilute solution, this problem can also be considered as optimality conditions of another minimization problem, subject to inequality constraints. This new problem is easier to handle, both from a theoretical and a practical point of view. Then we define a partition of the total quantities in the mass conservation equation. This partition builds a precipitation diagram such that a mineral is either precipitated or dissolved in each subset. We propose a symbolic algorithm to compute this diagram. Simple numerical examples illustrate our methodology.

In geochemistry, kinetic reactions can lead to the appearance or disappearance of minerals or gas. We defined two mathematical models based first on a differential inclusion system and second on a projected dynamical system. We proposed a regularization process for the first model and a projection algorithm for the second one.

This work, supported by IFPEN, has been published in .

Sparse linear systems

This partnership between Inria, Irstea and ITGA funds the PhD of Romain Schuster. The goal of this PhD is to design new image-based flow measurement methods for the study of industrial fluid flows. Those techniques will be used in particular to calibrate industrial fume hood.

This partnership between Inria, Irstea and CSTB funds the PhD of Yacine Ben Ali. This PhD aims to design new data assimilation scheme for Reynolds Average Simulation (RANS) of flows involved in wind engineering and buildings construction. The goal pursued here consists to couple RANS models and surface pressure data in order to define data driven models with accurate turbulent parameterization.

Noé Lahaye and Etienne Mémin are part of the SWOT Science Team. They participate to the DIEGO project (CNES-NASA) headed by Aurélien Ponté (Ifremer). Noé Lahaye is leader of WP4: Internal tide mapping and predictability.

Etienne Mémin is Visiting Professor at the Mathematics department of Imperial College London.

PI: Etienne Mémin

Duration: 01/03/2020 - 01/03/2026

Partners:
IFREMER, IMPERIAL COLLEGE London ((UK))

Summary: 71 percent of Earth is covered by ocean. The ocean has absorbed 93 percent of the heat trapped by human’s greenhouse gas emissions. The ocean’s future responses to continued warming are uncertain. Our project will deliver new capabilities for assessing variability and uncertainty in upper ocean dynamics. It will provide decision makers a means of quantifying the effects of local patterns of sea level rise, heat uptake, carbon storage and change of oxygen content and pH in the ocean. Its multimodal monitoring will enhance the scientific understanding of marine debris transport, tracking of oil spills and accumulation of plastic in the sea. Our approach accounts for transport on scales that are currently unresolvable in computer simulations, yet are observable by satellites, drifters and floats. Four scientific capabilities will be engaged: (i) observations at high resolution of upper ocean properties such as temperature, salinity, topography, wind, waves and velocity; (ii) large scale numerical simulations; (iii) data-based stochastic equations for upper ocean dynamics that quantify simulation error; and (iv) stochastic data assimilation to reduce uncertainty. These four scientific capabilities will tackle a network of joint tasks achieved through cooperation of three world-calibre institutions: IFREMER (ocean observations, reanalysis); INRIA (computational science); and Imperial College (mathematics, data assimilation). Our complementary skill sets comprise a single systemic effort: (1) Coordinate and interpret high-resolution satellite and in situ upper ocean observations (2) Extract correlations from data needed for the mathematical model (3) Perform an ensemble of computer simulations using our new stochastic partial differential equations (SPDE) which are derived by matching the observed statistical properties (4) Apply advanced data assimilation and computer simulations to reduce model uncertainty The key to achieving these goals will be synergy in our combined expertise.

duration 48 months.
The SEACS project whose acronym stands for: “Stochastic modEl-dAta-Coupled representationS
for the analysis, simulation and reconstruction of upper ocean dynamics” is a Joint Research Initiative between the three Britanny clusters of excellence of the "Laboratoires d'Excellence" program: Cominlabs, Lebesgue and LabexMer centered on numerical sciences, mathematics and oceanography respectively. Within this project we aim at studying the potential of large-scale oceanic dynamics modeling under uncertainty for ensemble forecasting and satellite image data assimilation.

The BECOSE project aims to extend the scope of sparsity techniques much beyond the academic setting of random and well-conditioned dictionaries. In particular, one goal of the project is to step back from the popular L1-convexification of the sparse representation problem and consider more involved nonconvex formulations, both from a methodological and theoretical point of view. The algorithms will be assessed in the context of tomographic Particle Image Velocimetry (PIV), a rapidly growing imaging technique in fluid mechanics that will have strong impact in several industrial sectors including environment, automotive and aeronautical industries. The consortium gathers the Fluminance and Panama Inria research teams, the Research Center for Automatic Control of Nancy (CRAN), The Research Institute of Communication and Cybernetics of Nantes (IRCCyN), and ONERA, the French Aerospace Lab.

Contract with IFPEN (Institut Français du Pétrole et Energies Nouvelles) Duration: three years from October 2016. Title: Fully implicit Formulations for the Simulation of Multiphase Flow and Reactive Transport Coordination: Jocelyne Erhel. Contract with IFPEN (Institut FranÃ§ais du Pétrole et Energies Nouvelles). Duration: three years October 2016-September 2019. Title: Fully implicit Formulations for the Simulation of Multiphase Flow and Reactive Transport. Coordination: Jocelyne Erhel. Abstract: Modeling multiphase flow in porous media coupled with fluid-rock chemical reactions is essential in order to understand the origin of sub-surface natural resources and optimize their use. This project focused on chemistry models, with kinetic reactions. We developed a mathematical tool, which can be embedded into a reactive transport code.

Title: Multiple Scale Ocean Model

Duration: From 2018 to 2021

Coordination: Bruno Deremble (CNRS LMD/ENS Paris)

Abstract: The objective of this project is to propose a numerical framework of a multiscale ocean model and to demonstrate its utility in the understanding of the interaction between the mean current and eddies.

Title: Apprentissage de la Dynamique de modèles Océaniques par des approches issues de la Théorie des Semi-groupes pour l’Assimilation de Données (ADOTSAD). Duration: From 2019 to 2021

The Britany ARED project "COmpréhension et Modélisation de mécanismes non-linéaires dans l'océan : les Interactions entre Ondes internes et Ecoulement" funds 50 percent of the PhD thesis of Adrien Bela.

Dominique Heitz

Roger Lewandowski

Etienne Mémin

Dominique Heitz

Etienne Mémin

Jocelyne Erhel