domain: math sde spi

text:
A Beaudoin
email: anthony.beaudoin@univ-poitiers.fr

M Farhad

Impact of fracture contact area on macro-dispersion in single rough fractures

Keywords: Single rough fractures, Fracture contact area, Macro-dispersion, Non-Fickian transport, Monte Carlo simulations, RW model

The study of the impact of fracture contact area on macro-dispersion in single rough fractures is still an open question. In this work, we study numerically the combined effects of the fracture roughness and the fracture contact area on the transport. This objective is achieved by estimating the macro-dispersion coefficient from Monte Carlo numerical simulations. When the fractional void is equal to 1, the Monte Carlo simulations show that macro-dispersion results of two contributions, dispersion caused by the heterogeneity of fracture apertures and molecular diffusion. When the fraction void is different from 1, a third mechanism plays in macro-dispersion. The presence of contacts or obstacles causes a disruption of flow paths. This mechanism is identical to that induced by the fracture roughness with a lower amplitude, depending on the fractional void.

Introduction

In the scientific literature, it was shown that the particle transport has a non-Fickian behavior in a single rough fracture. The particle transport cannot then be adequately described by the advectiondispersion equation [START_REF] Neuman | Perspective on theories of non-Fickian transport in heterogeneous media[END_REF]. This was explained by the heterogeneity of fracture apertures that induces complex flow fields. Numerous numerical studies have been made to investigate particle transport in single fractures with different physical complexities, solutes or dense particles transport in an uniform fracture or in a rough fracture with spatially variable apertures [START_REF] Wang | An efficient quasi-3D particle tracking-based approach for transport through fractures with application to dynamic dispersion calculation[END_REF]. However, the study of the impact of the fracture contact area on macrodispersion in single rough fractures is still an open question. The researchers have observed that the increase of the fracture contact area with normal stresses or normal deformations induces a channeling of the flow which in turn leads to a non-Fickian transport [START_REF] Hu | Non-Fickian solute transport in rough-walled fractures: the effect of contact area[END_REF]. In this work, we study numerically the combined effects of the fracture roughness and the fracture contact area on the non-Fickian transport. In particular, we quantify the contribution of the fracture contact area on macro-dispersion. In order to reach our objective, we use a methodology to determine the macro-dispersion coefficient from Monte Carlo numerical simulations in pure advection and advection-diffusion cases [START_REF] De Dreuzy | Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations[END_REF]. Our paper is organized according to two sections. In section 2, the numerical model used is described. The numerical results are presented in section 3.

Numerical model

The generation of a single rough fracture is based on the work presented by Mourzenko et al. in 1996 [5]. The single rough fracture is delimited by two surfaces z ± (x) = h ± o + h ± (x) with h ± (x) representing the fluctuations of the surface around average planes. The geometrical properties of the single rough fracture studied are given by:

where bm is the distance between the two average planes, ω is the separation between the two random surfaces, and b is the fracture aperture. 〈•〉 denotes the statistical expectation. The fluctuations h ± are characterized by a Gaussian probability distribution φ and a spatial covariance function C:

where σh is the standard deviation of h ± , λ is the correlation length, and H is the Hurst exponent. The mean of h ± is assumed to be zero. H is fixed here to 1. The fracture contact area can be quantified by the fractional void defined by the ratio between the total surface of the void volume of the fracture and the cross-sectional area of the fracture plane. The fracture aperture field is generated within a computational domain of dimensions Lx×Ly in the x and y directions with a spatial step identical in all the directions. The Fourier method is applied here for evaluating the fracture aperture field [START_REF] Gutjahr | Fast fourier transforms for random field generation[END_REF].

When the Reynolds number Re is lower than 1 and the mean plane of the single rough fracture presents weak fluctuations, the local cubic law can be used, in lieu of the full Navier-Stokes equations, to describe the flow in fractures [START_REF] Zimmerman | Hydromechanical behavior of fractured rocks[END_REF]:

where H is the hydraulic head, Txx and Tyy are the fracture transmissivities in the x and y directions, respectively. The transmissivity is assumed to be isotropic and is defined by the Cubic law:

where ρ is the fluid density, µ the dynamic viscosity of the fluid and g the gravity acceleration. To ensure the uniqueness of the solution, the boundary conditions are fixed hydraulic heads on the vertical sides of computational domain and periodic conditions on the horizontal sides of computational domain. Thus the main flow direction is parallel to the x direction because of these boundary conditions. The value of hydraulic heads is chosen in such a way as to keep the Reynolds number below 1. The components of the mean flow velocity are calculated by :

The Reynolds equations are discretized by means of a finite volume scheme with a harmonic composition rule for the transmissivity between adjacent mesh cells. The linear system, giving the hydraulic head H, is solved by using the algebraic multigrid iterative method implemented in HYPRE.

The two components of the mean velocity are then computed on each grid face [START_REF] Erhel | A paralle scientific software for heterogeneous hydrogeology[END_REF].

The RWPT method is used to simulate the solute transport in the single rough fracture [START_REF] Zheng | Influence of aperture field heterogeneity and anisotropy on dispersion regimes and dispersivity in single fractures[END_REF]. Considering only the mechanisms of advection and diffusion in the single rough fracture and neglecting the matrix diffusion in the rock formations, the solute transport equation is given by: where (x n , y n ) are the particle location in the Cartesian space at time nδt, n refers to the current time step, n+1 is the succeeding time step, N(0,1) is a random selection from the standard normal distribution with zero mean and unit variance, and Dm is the molecular diffusion coefficient. Within grid cells, the mean flow velocity v is derived from linear interpolations in both directions [START_REF] Prevost | Streamline tracing on curvilinear structured and unstructured grids[END_REF]. Np particles are injected in an injection window of size 0.8Ly, orthogonal to the flow direction and located at least five correlation lengths λ downstream from the side of the system. The ratio between the mean advection rate and the molecular diffusion is called the Péclet number Pe given by Pe = bm〈|v|〉 Dm. In the numerical simulations performed, we have studied two values of the diffusion coefficient, Dm = 0 and 0.01 mm²/s. It allows to study the purely advective regime (Pe = ∞) and the advective-diffusive regime (Pe < 100). As in the scientific literature, the Courant-Friedrichs-Levy condition is used to maintain the stability and accuracy of the Euler scheme by adapting the time step δt. The estimation of dispersivities is performed in two steps [START_REF] De Dreuzy | Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations[END_REF]. First, the longitudinal and transverse dispersivities are evaluated on one Monte Carlo simulation by means of the following expressions:

Second, the average over a number MC of Monte Carlo simulations is performed.

Numerical results

This asymptotic value α R Lasymp is presented in Figures 1 and2 as a function of σlnb for bm = 4 mm and 12 mm, respectively. In the two figures, the analytical solution proposed by Gelhar in 1993 has been plotted [START_REF] Gelhar | Stochastic Subsurface Hydrology[END_REF]. This analytical solution is based on two assumptions. The logarithm of the fracture aperture is a statistically stationary, isotropic, 2D Gaussian random field. The flow velocity through a single rough fracture can be modeled by the Reynolds equation. These two assumptions are verified by the present numerical model, making comparison possible. In Figures 1 and2, we can observe that the numerical results have a good agreement with the analytical solution for σlnb < 0.25. In the contrary case, a difference between the numerical results and the analytical solution increases with σlnb. This shows that the macro-dispersion does indeed result from the sum of effects of the fracture roughness and molecular diffusion up to σlnb < 0.25, where the fraction void remains equal to 1 [START_REF] Beaudoin | Impact of the fracture contact area on macro-dispersion in single rough fractures[END_REF].

Figure 1 .

 1 Figure 1. Asymptotic value α R Lasymp of the longitudinal dispersivity and analytical solution αanal as functions of σlnb for bm = 4 mm and two values of number Péclet Pe = ∞ and 97.6.

Figure 2 .

 2 Figure 2. Asymptotic value α R Lasymp of the longitudinal dispersivity and analytical solution αanal as functions of σlnb for bm = 12 mm and two values of number Péclet Pe = ∞ and 79.2.


Claudia-Lavinia Ignat
email: ignatcla@loria.fr

Gérald Oster
email: oster@loria.fr

Pascal Molli
email: molli@loria.fr

DooSo6: Easy collaboration over shared projects

Keywords: collaborative systems, version control, capability-based security

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Collaboration is a key requirement of teams of individuals working together towards some common goal and hence of importance to any organisation -be it business, science, education, administration or social. A great deal of information central to the operation of an organisation is held in documents and therefore support for sharing and collaboration over documents has to be offered.

In spite of the existence of specialised collaborative writing tools, very often when people want to collaboratively author a document they edit locally the document and then they send emails to the other members of the group transmitting them their version of the document as an attachment [START_REF] Noël | Empirical study on collaborative writing: What do co-authors do, use, and like?[END_REF]. Users have then to manually integrate the changes done in parallel, which might become a difficult task.

Studies have shown that the most desired features users need for supporting their group work are control over document versions and concurrent access to the shared documents [START_REF] Noël | Empirical study on collaborative writing: What do co-authors do, use, and like?[END_REF]. Concurrent access means that while a user is editing a document, the other users have the possibility to consult the document or edit it. However, existing tools that offer support for collaborative parallel work feature some complexities of use. The software must be correctly installed at the site of all users that collaborate independently of users operating system, users need to accept this software and it must be usable by all users. As shown in [START_REF] Grasso | Distributed coordination and workflow on the world wide web[END_REF], installing and using software can pose several problems. Installation of software might be a difficult task for certain users. Moreover, very often learning how to use a tool requires a great deal of effort and users might easily abandon that tool. Furthermore, for certain users it might be a problem renouncing at their favourite word processors in order to use a coauthorship application. Moreover, existing tools for parallel work require that either each client opens an account or the project administrator creates accounts for each client and assigns clients to the collaboration group.

In this paper, we propose a collaboration tool over shared projects that offers the basic functionalities of a version control system for supporting parallel work and overcomes the problems mentioned above. Each user maintains locally a copy of the shared project. Users can perform modifications on their local copy and publish their changes on the repository containing the shared project at a later time. The collaboration tool requires no installation and no accounts for the users. Using the tool is very easy and users do not need learning complex commands necessary for tool functionality. Moreover, users can edit their local copy of the project with their favourite editor and are not obliged to use an imposed editor. Furthermore, as group awareness is a very important feature of a collaborative system [START_REF] Dourish | Awareness and coordination in shared workspaces[END_REF], we integrate some basic aspects that provide information about group members activity.

Concerning the aspect of easiness of use of our tool we were inspired by the shared agenda called Doodle [START_REF] Doodle | Easy scheduling[END_REF] that helps finding suitable dates and times for group events such as an appointment, a conference call or a family reunion. Doodle is an online tool that does not require any software installation and no account creation for using a shared agenda. However, Doodle is limited to the use of a shared agenda and cannot be used for collaboration over shared projects.

For dealing with authentication issues we use a capability-based security [START_REF] Levy | Capability-Based Computer Systems[END_REF] for restricting actions of updating the local workspace and publishing changes to the shared repository only to the members of the group that possess these rights.

The paper is structured as follows. In section 2 we start by presenting by means of an example the limitations of existing systems that offer support for sharing and collaboration over a project. In section 3 we present our DooSo6 collaborative system for supporting parallel work and we show that it is very easy to use. In section 4 we give some details about our synchronisation mechanism. Concluding remarks and directions for future work are presented in section 5.

Motivating Example and Related Work

Consider the example of two researchers of two different universities that wish to collaborate on a paper. They have several approaches for sharing the source documents of the paper. A first solution would be to ask the system administrators of their universities to create accounts for the remote user. However, this solution imposes an administrative burden that does not scale with a large number of users and projects and needs a careful analysis of the rights given to the external user. Often, the latency for opening an account is large and users cannot accept it. Moreover, sometimes institutions do not allow opening guest user accounts for people that do not work for that institution.

The two users could use individual word processors for editing the document and then sending each other emails containing as attachment versions of the document that integrate their changes. This work mode requires a good planning of activities. Integration raises no difficulties if people work sequentially. Manual integration in the case of parallel work can be easily performed by a coordinator if the document is well segmented and the different document segments are assigned to different authors. However, if the document decomposition is not possible and the number of collaborators is large, manual integration of parallel changes becomes complex. The number of exchanged document versions grows proportionally with the number of collaborators and therefore user changes in the exchanged versions are more difficult to be tracked.

Another solution would be that users put their documents on the web which is unacceptable as documents are often confidential. There exist some online document sharing services that offer solutions for a secured space which can be used as a virtual drive. Unfortunately, these solutions do not deal with concurrent modifications of the same document. Therefore, users have to manually perform versioning and merging of their shared documents.

Another option for the two users is to use real-time collaborative editors such as SubEthaEdit [START_REF]Collaborative text editing[END_REF] to edit shared documents. These editors offer an awareness mechanism about the activity of the other group members as it shows the users that are editing a document at the same time. However, this solution requires that the two users install these tools and use these editors that they are not familiar with. Very often projects contain documents other than textual ones for which the real-time editors might not offer support such as latex, code source or XML documents. These documents need to be compiled with special tools and they need to be kept in a stable state that prevents continuous integration of concurrent changes. Therefore, users need to make a copy of the shared documents each time they need to compile them. If, for resolving issues at compilation time, the users modify the local documents, they have to manually re-integrate their changes into the real-time editor. Manual integration is necessary as realtime editors do not offer support for work in isolation, i.e. users cannot work in their private workspaces and synchronise their changes with a shared repository. Moreover, these editors synchronise in real-time changes done on the same document, but do not synchronise changes targeting shared directories.

The burden of tool installation is eliminated in the case of web-based collaborative editors such as GoogleDocs [START_REF] Googledocs | Create and share your work online[END_REF] and ZohoWriter [START_REF] Zohowriter | Easy Online Word Processor[END_REF] at the price of necessity of creation of user accounts. All the other disadvantages previously mentioned for real-time editors hold for web-based collaborative editors.

Version control systems are popular systems for supporting parallel work. Centralised version control systems such as CVS [START_REF] Berliner | CVS II: Parallelizing Software Development[END_REF] and Subversion [START_REF] Collins-Sussman | Version control with Subversion[END_REF] require that a server is installed by an administrator which has to create also user accounts for all project collaborators. Users have then to install the client applications. Distributed version control systems such as Git [START_REF]Git. Fast version control system[END_REF], Darcs [START_REF] Darcs | Interactive. Smart[END_REF] and Mercurial [START_REF] Mercurial | A fast and lightweight Source Control Management system[END_REF] eliminate the need of a server, but require that users install locally the client applications. In distributed version control systems users maintain locally their data and their changes and can push their changes to different channels. Other users that have granted rights may pull these changes from these channels. These channels may be hosted on users' hardware. However, for maintaining channels permanently available for allowing authorized users to pull changes from those channels and for avoiding firewall problems, users of distributed version control systems often push their changes on dedicated servers that have to be installed for hosting their projects. Therefore, all version control systems require an installation process that might be difficult to be achieved by certain people. Moreover, these tools are quite difficult to be used by non developers as they require an understanding of how the system functions and a good knowledge of the set of commands.

This section proved the need of a collaborative system that offers support for parallel work, requires no installation and no burden for users to create accounts and log on for using the system. Moreover, the collaborative system should be easy to use. Users should be able to use their favourite editors for editing changes on the project and do not be obliged to use an imposed editor. Furthermore, group awareness mechanism about the activity of the other members of the group should be offered. In the next sections we present our approach that offers all these features.

DooSo6: easy collaborative system

The basic methods that have to be supplied by a collaborative system that supports the parallel work of a set of users are checkout, commit and update. A checkout operation creates a local working copy of the project from the repository. A commit operation creates in the repository a new version of the project based on the local copy, assuming that the repository does not contain a more recent version of the project than the local copy. An update operation performs the merging of the local copy of the project with the last version of that project stored in the repository. In what follows we describe all the phases necessary for setting up and using the DooSo6 framework.

Project setup

For setting up a project, the initiator of the collaboration has to create first a repository as shown in the Figure 1. The project initiator is then required to fill in a title of the project together with a short description. As shown in Figure 2, the user is then provided with a link for the administration of the project such as deletion of the project, a link for committing changes to the repository and a link for updating changes from the repository. These links represent system capabilities. A capability is defined to be a protected object reference which, by virtue of its possession by a user, grants that user the capability to interact with the object in certain ways. For instance, the Fig. 1. Creation of a repository Fig. 2. Links that represent system capabilities Administration link in Figure 2 is a key that gives the initiator the right of administration of the project. The initiator is supposed to do not distribute the key to other users. On the contrary, the initiator should forward the key for committing to the repository and updating changes from the repository to the other members involved in the project. The commit and update keys offer users that are in their possession the capabilities of updating and committing respectively to the repository. It is possible that the key for updating is distributed to some users and these users do not receive the key for committing. It means that these members of the group have only the right to update changes from the repository and do not have the right of committing changes to the repository.

Check-out/Update of the project in the local workspace

For performing a check-out or an update a user has to follow the update link, being provided with the interface shown in Figure 3.

The user can see the list of the other members of the group that already checked out the project and started working on it. Each member is associated with his up-to-date status, i.e. the percentage of the number of commits in the repository that he integrated in his local workspace. If the user has not yet performed a checkout, his name does not appear among the members of the group. If, however, the user previously performed a checkout, the name that he Fig. 3. Updating from the repository provided appears in the list of names together with his up-to-date status. Note that user names are used only for awareness purposes and not for authentication.

When the Update button is clicked, the client application is downloaded and launched. This is achieved by using the Java web start technology. Server sends the update capability to the client application. If Update button is clicked for the first time, i.e. a checkout is performed, user is asked to provide a name and a path to a local directory where his copy of the project will be stored. This path is stored by the local file system and will be used for next updates. Afterwards, the local workspace is synchronised with the latest version of the project from the repository.

Committing changes to the repository

If the user wants to commit changes that he performed locally he is presented with almost the same interface as the one shown in Figure 3 with an additional Commit button. If Commit button is pressed and the user is not up-to-date, he is advised to perform first an update. If, however, the user is up-to-date, he can commit the changes he performed locally. Afterwards, the client application is started and commit procedure executed. The system automatically detects the patch of operations performed locally since the last update and sends this patch to the repository.

Why collaboration is easy with DooSo6?

As we use a capability-based security mechanism according to which users that own a key have the right of updating or committing to the repository, we do not need to authenticate users by means of logins and passwords. Therefore, users do not need to create accounts and logins for using the system.

The facility of non-installation of software is achieved by means of technology. The system is written in Java which is a platform independent language and therefore we do not have problems of incompatibility between operating systems. By using the Java Web Start technology, our Java software application can be started by clients directly from the Internet simply by using a web browser.

Our collaborative system is easy to be used as users do not need to learn and type commands for publishing and synchronising their changes. Users have basically two buttons for activating the actions of updating their local version of the project and for publishing the changes to the repository.

Changes on the local copies of the shared project can be edited using any editor. At the moment when synchronisation is performed between the local project and the shared one in the repository, directories and files belonging to the project are synchronised according to their content (textual, XML or binary).

An awareness mechanism is offered for tracking group activity -users can see their up-to-date status of the project and the delay with which users follow the project. From our first experimentation we observed that, due to this awareness visualisation, users try to provide a good impression to the group by contributing and keeping their local copies of the shared projects up-to-date.

The DooSo6 synchronisation mechanism

In this section we present a short overview of the DooSo6 synchronisation mechanism. The DooSo6 system contains two main components: the DooSo6 repository and several DooSo6 workspaces associated to different clients. The DooSo6 repository hosts a timestamper and a sequence of patches of operations. A patch of operations contains a list of operations representing changes performed by a certain user. When a user executes a commit, the list of all operations locally performed is sent as a single patch to the repository. The list of operations is obtained by executing a diff [START_REF] Myers | An O(ND) Difference Algorithm and its Variations[END_REF] algorithm between the last updated and the current local version of the project.

A DooSo6 workspace stores all documents shared by a user. Operations locally performed by that user are saved in a log of operations. The DooSo6 workspace also keeps the timestamp of the last delivered/received patch to/from the repository. Initially the local timestamp is set to 0. When a user wants to publish his changes into the DooSo6 repository, he needs first to request a timestamp. According to this received timestamp the user is allowed or not to save his changes. If the timestamp received equals to the value of the local timestamp, the user is allowed to save his changes. A patch of operations containing the operations locally generated since the last time the user committed is sent to the DooSo6 repository. If, however, the timestamp received from the DooSo6 repository is greater than the value of the local timestamp, the user needs first to update his local workspace. The DooSo6 repository sends the workspace the list of unconsumed patches. The client then merges the list of operations representing the unconsumed patches with the local operations and applies them to the workspace.

The merge algorithm used by DooSo6 is adapted from SOCT4 [START_REF] Vidot | Copies Convergence in a Distributed Real-Time Collaborative Environment[END_REF]. The algorithmic description of the checkout, update and commit procedures is similar to the one of the same procedures presented in the So6 system [START_REF] Oster | Supporting Collaborative Writing of XML Documents[END_REF]. The main difference between the DooSo6 merge mechanism and the So6 merge mechanism is that DooSo6 works with patches of operations while So6 works with operations.

Conclusion

In this paper we presented a very easy to use system for the collaboration over a shared project. Users do not need to open accounts for having access to the shared projects and do not need to learn any commands for using the system. Moreover, they can use their favourite editor for performing changes to the project. We use a capability-based security for restricting actions of updating and committing changes to the shared repository only to members of the group. Currently, our system uses a centralised repository. Besides issues of limited scalability, lack of shared administration costs and limited fault tolerance, data centralisation is an inherent threat to privacy. In order to overcome these issues, we investigate to store repositories containing patches of operations on a distributed hash table.


Jules Dallant

Éric Thiébaut

Maud Langlois

Olivier Flasseur

Fusion de données par filtrage adapté pour la détection d'exoplanètes en imagerie directe

Détecter des exoplanètes par imagerie directe est une tâche extrêmement ardue en raison du très haut contraste entre ces sources et leur étoile hôte. L'algorithme PACOME combine des observations à des époques différentes d'un même système extrasolaire en prenant en compte le mouvement Képlérien des sources qu'il cherche à détecter et permet d'atteindre une sensibilité encore inégalée. Dans ce papier, nous étendons le formalisme de cette méthode à une approche par filtrage adapté pour en dériver un critère optimal interprétable en terme de rapport signal sur bruit. Nous illustrons l'efficacité de la méthode sur des simulations réalistes de sources ajoutées dans des données réelles.

Abstract -Detecting exoplanets by direct imaging is an extremely difficult task because of the extremely high contrast between them and their host star. The PACOME algorithm combines observations at different epochs of the same extrasolar system by taking into account the Keplerian motion of the sought sources and achieves unprecedented sensitivity. In this paper, we extend the formalism of this method to a matched filter approach to derive an optimal criterion interpretable in terms of signal-to-noise ratio. We illustrate the efficiency of the method on realistic simulations of sources added in real data.

Introduction

Seulement une douzaine de planètes extrasolaires ont à ce jour été détectées par imagerie directe. Ceci s'explique par la présence de termes de nuisance (bruits et fuites de lumière stellaire spatialement corrélées) qui dominent les signaux des objets d'intérêt. Il est donc nécessaire de recourir à des méthodes de démélange de sources afin d'extraire le signal utile des exoplanètes. Les méthodes les plus courantes sont basées sur l'imagerie différentielle angulaire (ADI) et spectrale (ASDI) qui font usage de la rotation apparente du champ de vue lors de la séquence d'observations (et/ou de la variété spectrale des données) pour construire un modèle des termes de nuisance qui est soustrait aux données, les sources faibles étant détectées dans les résidus ainsi obtenus. En particulier, l'algorithme PACO [START_REF] Flasseur | Exoplanet detection in angular differential imaging by statistical learning of the nonstationary patch covariances[END_REF] apprend localement la statistique des nuisances à partir des corrélations spatio-spectrales. Afin d'améliorer la sensibilité de détection, nous avons développé l'algorithme PACOME [START_REF] Dallant | Optimal multi-epoch combination of direct imaging observations for improved exoplanet detection[END_REF] qui tire partie du formalisme statistique de PACO, basée sur le maximum de vraisemblance. PACOME combine différentes observations d'un même système extrasolaire en maximisant une fonction de coût tenant compte du mouvement orbital des sources qu'il cherche à détecter.

Dans ce papier, nous étendons l'approche de PACOME au formalisme du filtrage adapté qui permet l'obtention d'un score de détection interprétable en terme de rapport signal à bruit (RSB), nous discutons des garanties statistiques de cette approche et nous démontrons l'efficacité de la méthode sur des sources simulées dans des données réelles.

Dérivation d'un RSB multiépoque

Nous présentons le formalisme du filtrage adapté permettant de combiner linéairement différentes observations ASDI de manière optimale. Il permet simultanément de détecter des compagnons orbitant autour d'étoiles et de dériver les paramètres décrivant leurs orbites.

Modèle direct des observations

Les données ASDI classiques sont des séries temporelles d'images coronographiques, réparties sur plusieurs canaux spectraux, et observées à différentes dates appelées époques. La position apparente θ t (µ) sur le ciel d'une source ponctuelle décrivant un mouvement képlérien à l'époque d'observation t dépend seulement de ses éléments orbitaux µ et de la masse totale du système. Nous modélisons la trame k d'un jeu de données r t,ℓ,k ∈ R N prise à l'époque t, au canal spectral ℓ comme dans [START_REF] Dallant | Optimal multi-epoch combination of direct imaging observations for improved exoplanet detection[END_REF] :

r t,ℓ,k = α t,ℓ h t,ℓ (θ t (µ)) + f t,ℓ,k , (1)
avec h t,ℓ (θ) ∈ R N la fonction d'étalement de point (FEP) hors axe centrée à la position θ et identique pour toutes les trames k de la même époque, α t,ℓ l'intensité de la source (valant 0 en l'absence de source) à l'époque t et au canal spectral ℓ et f t,ℓ,k les termes de nuisance. Le nombre de photons collectés par chaque pixel du détecteur étant suffisant, nous supposons que les termes de nuisance suivent une distribution gaussienne multivariée [START_REF] Flasseur | Robustness to bad frames in angular differential imaging: a local weighting approach[END_REF][START_REF] Flasseur | ASDI: an algorithm for exoplanet detection and characterization in direct imaging with integral field spectrographs[END_REF]. Les images des différents canaux spectraux et époques sont mutuellement indépendantes et quasi-identiquement distribuées :

f t,ℓ,k ∼ N ( ft,ℓ , σ 2 t,ℓ,k C t,ℓ ) , (2)
avec ft,ℓ et C t,ℓ l'espérance et la matrice de covariance typique du terme de nuisance f t,ℓ,k , et où σ 2 t,ℓ,k sont des facteurs de pondérations temporo-spectraux rendant compte localement de la qualité inégale des images [START_REF] Flasseur | Robustness to bad frames in angular differential imaging: a local weighting approach[END_REF]. Il est montré dans [START_REF] Thiébaut | Fast and robust exo-planet detection in multi-spectral, multi-temporal data[END_REF] et [START_REF] Flasseur | Exoplanet detection in angular differential imaging by statistical learning of the nonstationary patch covariances[END_REF] qu'à partir de l'estimation de la matrice de précision W t,ℓ,k = σ -2 t,ℓ,k C -1 t,ℓ (évaluée localement par PACO à l'échelle de petits patchs) et des termes :

a t,ℓ (θ) = k h t,ℓ (θ) T W t,ℓ,k h t,ℓ (θ) , b t,ℓ (θ) = k h t,ℓ (θ) T W t,ℓ,k r t,ℓ,k -ft,ℓ , (3)
l'estimateur du maximum de vraisemblance de α t,ℓ s'exprime :

α + t,ℓ (µ) = arg max α t,ℓ ≥0 L (α, µ) = b t,ℓ (θ t (µ)) + a t,ℓ (θ t (µ)) , (4)
où [x] + dénote la partie positive de x et L (α, µ) la logvraisemblance des données connaissant les flux α et les paramètres orbitaux µ de la source. En effet, le flux étant nécessairement positif (ou nul si la source est absente), il est avantageux de lui imposer une contrainte de positivité [START_REF] Mugnier | Optimal method for exoplanet detection by angular differential imaging[END_REF]. On peut montrer que l'estimateur, au sens du maximum de vraisemblance, des éléments orbitaux µ permettant de combiner optimalement les signaux d'un point source potentiel le long de son orbite dans différentes données s'écrit [START_REF] Dallant | Optimal multi-epoch combination of direct imaging observations for improved exoplanet detection[END_REF] :

µ = arg max µ C (µ) = t,ℓ b t,ℓ (θ t (µ)) + 2 a t,ℓ (θ t (µ)) . (5)

Formalisme du filtrage adapté

Dans le cas général où des données d ∈ R n sont mesurées dans des conditions reproductibles paramétrées par ϕ ∈ R p (e.g. conditions expérimentales, paramètres de l'objet, etc.), il est possible d'écrire que :

d = m + z , (6)
avec m = E{d | ϕ} l'espérance des mesures dans les conditions données ϕ et z ∈ R n un terme stochastique. Considérons une quantité mesurable η ∈ R dépendant linéairement des données tel que :

η = q T d , (7)
avec q ∈ R n le vecteur des coefficients de la combinaison linéaire. On appelle filtre adapté [START_REF] Kay | Fundamentals of statistical signal processing : Estimation theory[END_REF] le filtre linéaire q maximisant le RSB de cette combinaison linéaire. Il s'écrit :

q = ξ S -1 m , (8)
avec S = E{z z T | ϕ} = Cov{d | ϕ} la covariance du terme stochastique z dans des conditions données ϕ et ξ ∈ R ⋆ un facteur arbitraire. Dans ce cas, le meilleur RSB possible de η dans les conditions ϕ s'écrit [START_REF] Kay | Fundamentals of statistical signal processing : Estimation theory[END_REF][START_REF] Dallant | Optimal multi-epoch combination of direct imaging observations for joint exoplanet detection and orbit estimation[END_REF] :

S/N (η | ϕ) = max q E{η | ϕ} Var{η | ϕ} = √ m T S -1 m . (9)

Application à la détection multiépoque

Nous appliquons le formalisme du filtre adapté décrit cidessus au modèle direct des observations ASDI en considérant les données résiduelles :

d t,ℓ,k = r t,ℓ,k -ft,ℓ , (10)
où d t,ℓ,k est supposé avoir une distribution gaussienne d'espérance α t,ℓ h t,ℓ (θ t (µ)) et de variance σ 2 t,ℓ,k C t,ℓ . En supposant les données résiduelles provenant de différentes époques, trames et canaux spectraux mutuellement indépendantes, la quantité maximisant le RSB vaut [START_REF] Kay | Fundamentals of statistical signal processing : Estimation theory[END_REF][START_REF] Dallant | Optimal multi-epoch combination of direct imaging observations for joint exoplanet detection and orbit estimation[END_REF] :

η = ξ t,ℓ α t,ℓ k h t,ℓ (θ t (µ)) T W t,ℓ,k d t,ℓ,k = ξ t,ℓ α t,ℓ b t,ℓ (θ t (µ)) . (11)
Compte tenu des données et des indépendances supposées, le RSB maximal correspondant s'obtient par l'Éq. [START_REF] Thiébaut | Optimization issues in blind deconvolution algorithms[END_REF] :

S/N (µ) = t,ℓ,k α 2 t,ℓ h t,ℓ (θ t (µ)) T W t,ℓ,k h t,ℓ (θ t (µ)) = t,ℓ α 2 t,ℓ a t,ℓ (θ t (µ)) . (12)
Si l'on considère l'estimateur du maximum de vraisemblance de α t,ℓ de l'Éq. ( 4), cette valeur théorique peut être approximée par l'estimateur suivant :

S/N (µ) ≃ t,ℓ b t,ℓ (θ t (µ)) + 2 a t,ℓ (θ t (µ)) . ( 13
)
En présence d'une source, ce RSB multiépoque combine optimalement les informations contenues dans les données le long de la trajectoire apparente du compagnon. Nous retrouvons que ce RSB multiépoque optimal est exactement la racine carrée du critère dérivé par approche du maximum de vraisemblance en Éq. ( 5) [START_REF] Dallant | Optimal multi-epoch combination of direct imaging observations for improved exoplanet detection[END_REF]. Maximiser l'un ou l'autre par rapport à µ donne ainsi exactement la même solution. Le critère de combinaison est donc optimal à la fois dans le sens du maximum de vraisemblance et en terme de RSB. Il permet la détection de sources encore indétectables à chaque époque individuelle et fournit simultanément une estimation de leurs éléments orbitaux. Ce critère est non convexe et présente de nombreux extrema locaux, le maximiser nécessite l'utilisation de méthodes d'optimisation globale. L'algorithme PACOME (voir Sec. 3) a été développé dans ce but.

Garanties statistiques

Dans l'expression de l'estimateur de l'intensité de la source α + t,ℓ , le terme a t,ℓ est déterministe et seul le terme b t,ℓ fluctue. Étant donné notre modèle statistique et l'indépendance mutuelle des trames, la variance de b t,ℓ vaut :

Var{b t,ℓ (θ)} = a t,ℓ (θ). (14)
De façon remarquable, b t,ℓ (θ) et a t,ℓ (θ) constituent une statistique suffisante pour détecter une source potentielle. En approximant la variance de α + t,ℓ donnée dans l'Éq. (4) par celle de l'estimateur non contraint, il vient :

Var{ α + t,ℓ (µ)} ≈ Var{b t,ℓ (θ t (µ))} a 2 t,ℓ (θ t (µ) = 1 a t,ℓ (θ t (µ)) . (15)
Étant donné l'estimateur du flux et de sa variance [START_REF] Flasseur | Exoplanet detection in angular differential imaging by statistical learning of the nonstationary patch covariances[END_REF], [START_REF] Flasseur | ASDI: an algorithm for exoplanet detection and characterization in direct imaging with integral field spectrographs[END_REF], [START_REF] Flasseur | Robustness to bad frames in angular differential imaging: a local weighting approach[END_REF], le RSB temporo-spectral est calculé indépendamment pour chaque époque aux positions θ comme :

S/N t,ℓ (θ) = E α t,ℓ (θ)} Var{ α + t,ℓ (θ) ≈ b t,ℓ (θ) a t,ℓ (θ) , (16)
et suit une distribution normale en l'absence de source. Grâce à cette propriété, ce RSB monoépoque est directement interprétable en termes de probabilité de détection et de probabilité de fausse alarme. En remarquant que le RSB multiépoque de l'Éq. (13) est la racine de la somme quadratique des RSB monoépoques positifs, il peut lui aussi être interprété en termes de probabilité de détection multiépoque. En effet, la distribution statistique du critère correspond à la somme des carrés de T × L = M distributions gaussiennes rectifiées et indépendantes, où T et L représentent respectivement le nombre total d'époques et de canaux spectraux. À notre connaissance, cette distribution n'a pas d'expression analytique, nous recourons donc à une approche de type Monte-Carlo pour construire la distribution du critère et nous en déduisons empiriquement, à niveau de confiance 1-ρ ∈ [0, 1] donné, la valeur de la fonction quantile Q M (1 -ρ) [START_REF] Hyndman | Sample Quantiles in Statistical Packages[END_REF] associée. Cette dernière représente la valeur du seuil de détection pour le RSB multiépoque de l'Éq. (13) en dessous duquel se situe une proportion 1 -ρ des données en l'absence de compagnon. Les fonctions de densité de probabilité du RSB multiépoque calculées par cette procédure sont présentées dans la Fig. 1. Quelques valeurs de la fonction quantile évaluée à différents niveau de confiance et pour plusieurs degrés de liberté sont données en Fig. 2.

Valeur du RSB multiépoque (sans source)

M = 15 M = 25 M = 50 M = 100 M = 75 M = 10

L'algorithme PACOME

L'algorithme PACOME [START_REF] Dallant | Optimal multi-epoch combination of direct imaging observations for improved exoplanet detection[END_REF] échantillonne le RSB multiépoque sur une grille multidimensionnelle représentant l'espace de recherche des éléments orbitaux. Pour cela, les positions projetées 2-D subpixeliques θ t sont calculées pour chaque noeud de l'espace des paramètres et pour toutes les époques. Les termes a t,ℓ et b t,ℓ sont interpolés à ces positions afin d'évaluer le RSB multiépoque. Les quelques N opt meilleures solutions orbitales sont ensuite raffinées par optimisation locale via la méthode VMLMB [START_REF] Thiébaut | Optimization issues in blind deconvolution algorithms[END_REF]. Le RSB multiépoque maximal obtenu est finalement comparé à la limite de détection multiépoque (à un niveau de confiance donné 1 -ρ) estimée numériquement via la distribution statistique empirique du critère pour décider d'une éventuelle détection.

Application sur données simulées

Nous évaluons les performances de PACOME sur un jeu de données semi-synthétiques construit par la simulation réaliste de sources synthétiques ajoutées aux données réelles du système exoplanétaire HR 8799 observé le 2016-11-18 avec l'instrument SPHERE-IRDIS [START_REF] Beuzit | SPHERE: the exoplanet imager for the Very Large Telescope[END_REF] opérant au Very Large Telescope. Nous avons simulés 9 fausses époques totalisant 5.12 heures d'observation espacées sur 59 ans. Quatre sources (désignées par (1), ( 2), ( 3) et ( 4)) y ont été ajoutées selon le modèle direct de l'Éq. (1) sur des orbites connues aux époques correspondantes en utilisant la FEP hors axe mesurée lors de l'observation. Le flux injecté des sources est choisi tel qu'elles restent indétectables (aux probabilités de fausses alarmes usuelles dans le domaine) par le traitement non-conjoint de chaque époque d'observation. Seule la source (1) a un RSB monoépoque légèrement supérieure à cette limite parmi 5 époques sur 18.

Nous cherchons à détecter en aveugle les quatre sources avec PACOME. Au total, N orb ≈ 43 × 10 10 orbites différentes ont été explorées afin de couvrir au mieux l'espace des paramètres des éléments orbitaux. Le niveau de confiance de détection multiépoque a été fixé à ρ = 10 -7 ce qui correspond, à 18 degrés de liberté, au seuil de Q 18 (1 -ρ) ≈ 7.4.

Les estimées de µ trouvées par PACOME pour chaque source sont donnés dans la Tab. 1 et comparés à la vérité terrain, au sens de la racine de l'erreur quadratique moyenne (REQM) entre les positions projetées 2-D sur le capteur.

Les solutions orbitales obtenues sont très proches de la vérité terrain (REQM < 0.75 pixels pour chacune des quatre sources, voir Tab. 1) à l'exception de l'élément orbital ω de la source (4), qui est significativement différent. Ceci révèle une dégénérescence connue du paramètre. L'orbite projetée estimée est en revanche bien identique à celle simulée puisque leurs positions projetées sur le détecteur aux époques données sont quasi-égales (REQM = 0.2 pixels).

Leurs valeurs du RSB multiépoque sont toutes significativement plus élevées que la limite de détection empirique multiépoque à ρ = 10 -7 tandis que la grande majorité des points sources ne sont pas détectées individuellement (S/N t,ℓ < 5). Plus précisément, les S/N t,ℓ monoépoques moyens des solutions optimales sont de 4.0, 2.2, 2.4, 2.3 tandis que leurs RSB multiépoques sont de 18.5, 9.9, 10.5, 10.2 (de la source (1) à (4) respectivement). Cela signifie que passer de mono à multiépoque entraîne un gain moyen en RSB d'environ 4.5, facteur quasi égal à la racine carrée du nombre de degrés de liberté ( √ T × L = √ 18 ≈ 4.2) ce qui illustre bien l'optimalité de PACOME. La Fig. 3 met en valeur ce gain en comparant différentes cartes de RSB monoépoque et multiépoque centrées autour des solutions orbitales estimées par PACOME. Tous les signaux dans le champ qui ne se trouvent pas aux positions de la source se combinent faiblement (des couleurs bleues aux couleurs blanches, en dessous du niveau de confiance fixé). Enfin, la Fig. 4 représente les solutions orbitales trouvées par PACOME (en rouge) ainsi que les 10 3 autres meilleures orbites optimisées (en orange).

Conclusions

Nous avons montré que le formalisme du filtrage adapté appliqué à PACOME permet de dériver un critère de détection interprétable en terme de RSB, qui plus est optimal. Ce critère permet de contrôler de manière fiable la confiance de détection.

L'application à des données réelles montre un gain en RSB de l'ordre de la racine carrée du nombre d'époques prises en compte. Ce gain permet de détecter des sources indécelables dans les observations individuelles et d'en caractériser les éléments orbitaux.

Plusieurs améliorations sont à prévoir concernant l'algorithme, notamment sur la stratégie d'échantillonnage des éléments orbitaux ainsi que sur la robustesse de la méthode vis à vis de la qualité très variable des observations, se traduisant par des variations temporelles de la réponse impulsionnelle des sources recherchées.

FIGURE 1 :

 1 FIGURE 1 : Densités de probabilité théoriques du RSB multiépoque en l'absence de source.

FIGURE 2 :

 2 FIGURE 2 : Seuil de détection du RSB multiépoque pour différents degrés de liberté M et niveaux de confiance 1 -ρ.

FIGURE 3 :

 3 FIGURE 3 : Zoom sur les régions d'intérêts dans les cartes de RSB autour des sources considérées pour (a) le plus faible RSB de détection monoépoque, (b) le meilleur RSB monoépoque, (c) le RSB multiépoque obtenu par l'approche proposée.

FIGURE 4 :

 4 FIGURE 4 : Solutions orbitales trouvées pour chacune des quatre sources ainsi que les 10 3 meilleures autres orbites.

TABLE 1 :

 1 Éléments orbitaux, intervalles de confiance à 95% associés et RSB multiépoque des quatre orbites simulées et des solutions orbitales estimées par PACOME.

	Élém.	Unité	Source (1) Estimation Simulation	Source (2) Estimation Simulation	Source (3) Estimation Simulation	Source (4) Estimation Simulation
	a	mas	976.26 +5.22 -4.09	976.23	542.71 +5.86 -6.24	543.75	334.88 +3.62 -3.89	331.66	229.68 +3.86 -5.04	229.01
	e	-	0.28 +0.01 -0.01	0.28	0.15 +0.01 -0.01	0.15	0.01 +0.02 -0.01	0.01	0.00 +0.02 -0.00	0.0
	i	deg	52.19 +0.63 -0.65	52.19	49.86 +1.51 -1.57	50.45	48.19 +2.51 -2.38	48.13	44.62 +2.96 -3.56	45.33
	t0	yr	1958.88 +0.42 -0.42	1958.90	1986.05 +0.34 -0.30	1986.02	2004.99 +3.68 -1.87	2004.52	2011.71 +7.43 -1.29	2014.27
	ω	deg	146.51 +2.41 -1.86	146.12	63.93 +2.59 -2.96	63.39	145.79 +61.60 -43.61	134.21	259.95 +46.78 -112.67	350.32
	Ω	deg	298.92 +0.79 -0.96	298.99	295.17 +1.96 -1.90	295.76	298.62 +2.94 -2.77	299.11	302.97 +5.93 -5.91	302.02
	K (×10 4 )	mas 3 /yr 2	11.44 +0.25 -0.27	11.50	11.48 +0.38 -0.39	11.50	11.86 +0.38 -0.41	11.50	11.62 +0.62 -0.73	11.50
	REQM [pix]	0.12		0.34		0.61		0.20
	S/N multiépoque	18.50	18.46	9.84	9.66	10.52	9.86	10.17	10.12
	Seuil de détec. Q18(1 -ρ)	7.37		7.37		7.37		7.37


domain: sde math spi

text:
Alaa Ghzayel

Anthony Beaudoin

Sebastien Jarny

Modelling of scour formation using SedFoam, continuum approach

SedFOAM is a two-phase flow solver built upon consecutive laws, based on the CFD toolbox Open-FOAM. The sediment body is considered as a continuum phase. The intergranular and fluid stresses are modeled with the dense granular flow rheology and the k-turbulent model, respectively. The model setup will be based on an experimental study on the scour due to a water jet subjected to lateral confinement. A comparison study will be made based on precise experimental data by Martino et al. (2019) that will show the advantages and defaults of SedFoam. The objective of this work is to reproduce the digging and filling cycle of the scour formation due to the water jet in a confined canal. The first numerical results show that it needs to use 3D numerical simulations because of the fluctuation of the jet direction induced by the presence of a driven flow cavity.

Introduction

Two phase flow models for sediment transport have been widely developed in the past decades, in which dynamical equations are applied on both the fluid and particles phases where the particles are considered as a continuous body within the fluid. This approach enables us to take into consideration most of the physical processes such as particles interactions, turbulence modulation and particle interaction with another. Only very recently, a 3D simulation two-phase flow model, called SedFOAM, has been applied for sheet-flow sediment transport [1]. The intergranular and fluid stresses are modeled with the dense granular flow rheology [2] and the kturbulent model [3]. In this work, we will use SedFOAM to simulate a case scenario of a hydro-sedimentary canal proposed by Martino et al. in 2019 [4]. Benchmarking will be done based on the experimental study of Martino et al. on the development of a scour and mound formation. For a certain flow rate, a digging and refilling cycle of the sediments inside the scour is observed. The objective of this work is to be able to reproduce numerically the same cycle of digging and refilling and to be able to conclude the phenomena behind this type of sediment erosion cycle. This numerical study will provide the advantages and defaults of SedFOAM.

SedFoam

Mathematical approach

To construct the model based on the Eulerian two-phase flow approach, the mass and momentum conservation equations have to be averaged over the fluid and particles phases. In order to be able to apply this formulation to turbulent flows, turbulence averaging is required. The mass conservation equations are as following: * e-mail: Alaa.ghzayel@univ-poitiers.fr

∂α ∂t + ∂αu a i ∂x i = 0 and ∂β ∂t + ∂βu b i ∂x i = 0 (1)
where α and β (β = 1 -α) are the particles and fluid volume concentrations, u a i and u b i are the particles and fluid velocities. The momentum equations for the particles and fluid phases are expressed as following:

∂ρ a αu a i ∂t + ∂ρ a αu a i u a j ∂x j = -α ∂p ∂xi + α f i - ∂ pa ∂xi + ∂τ a i j ∂x j + αρ a g i α f i + αβK(u b i -u a i ) -S U S βKν b t ∂α ∂xi , (2)
∂ρ b αu b i ∂t + ∂ρ b αu b i u b j ∂x j = -β ∂p ∂xi + β f i + ∂τ b i j ∂x j + βρ b g i + β f i -αβK(u b i -u a i ) -S U S βKν b t ∂α ∂xi (3)
where ρ a and ρ b are the particles and fluid densities, g i is the gravitational acceleration, f i is an external force, p is the pressure of the fluid, τ b i j is the fluid stress, pa and τ a i j are the particles normal and shear stresses, k is the drag parameter, ν b t is the turbulent viscosity and the inverse of the Schmidt number S U S = 1/σ c . The Kmodel is implemented into the solver. It is a two equation model that gives a general description of turbulence by means of two transport equations that take into account the turbulent kinetic energy (TKE) and the rate of dissipation of the turbulent kinetic energy ( ).

Model solution procedure

The previous mathematical formulation is solved with the following procedure:

1. Solve the sediment concentration α according to equation (1).

2. Update the concentration volume β =1 -α.

3. Calculate K, the drag parameter:

K = 0.75C d ρ b d e f f u b -u a β -h E x p (4)
4. Solve the fluid turbulence parameter based on the chosen model and then calculate the eddy and fluid effective viscosities.

∂k ∂t + u b j ∂k ∂x j = R bt i j ρ b ∂u b i ∂x j + ∂ ∂x j [(ν b + ν b t σ k ∂k ∂x j ] - - 2K(1 -t m f )αK ρ b - S U S β ν b t ∂α ∂x j ( ρ a ρ b -1)g j (5)
5. Solve the particles phase stress based on the chosen model, the dense granular-flow rheology:

τa i j = µ(I) pa S a i j 2S a i j .S a i j (6)
6. Solve the velocity-pressure coupling.

3 Scour due to a water jet

Dynamics of the scour process

To understand the process, the scour formation submits three different phases (see Fig. 2). In the first phase, the particles are extracted and transported forming the hole and settled downstream to the right forming a mound which is a very quick process. This process barely took few seconds for the formation of the hole. In a development phase, the process slows down and the rate of particles transport decreases. The formation changes from a triangular shape to a trapizoidal shape, the more the particles accumulate the weaker the flow jet can transport particles which leads to accumulation of particles on the left side of the mound until the scour reaches a maximum depth and the grains on the left side of the mound migrate downslope. In a long term phase, after reaching a critical value of the resistance of the mound system and when the jets is not able anymore to support this mound, the left side of the mound collapses into the scour due to gravity in a steady recirculation regime and the hole will be partially refilled and this process will be repeated causing a digging and refilling cycles.

Criteria for comparison

To analyse this process, we had to take into account the variables and compare them with constant values to check if there is a specific pattern. From this perspective, we consider the critical time t c where the mound submits a morphological change from a triangular to a trapizoidal shape as mentioned in the previous section. In the experiment done by Martino et al., t c varies from 8 to 15 seconds for the tested values of the flow rate Q. t c decreases with Q. Furthermore, an evolution of the maximum depth of the scour Y 0 with Q should also be taken into account, to be able to understand the scour process in both the steady regime and the digging and refilling cycle. Figure 4 presents the time evolution of Y 0 for different value of Q. It has been observed that for a flow rate value lower than a certain critical flow rate, a transaction between the steady regime (SR) and the digging and refilling regime (DR) takes place. During the process, all the depths of the scour have deviations around the mean maximum value Y 0 . The amplitude of deviations increases with the flow rate Q. 4 Preliminary results

Numerical setup and Mesh construction

The flow conditions and particles properties are set according to the experiment done by Martino et al.. The particle diameter is set to be 1.1 mm with a density of 2650 kg/m 3 .

In the dense granular flow rheology, the coefficients of friction are fixed to µ s = 0.4 and µ 2 = 0.6, as it has been proposed in the literature for glass beads. The water flow velocity profile was introduced at the inlet boundary condition using the equation proposed by Martino et al.:

U Um = A 1 y y 1/2 1/7 1 -er f A 2 y y 1/2 , (7)
U m U 0 = 2 x b 0 -1/3 (8)
where U m and U o are the maximum water jet velocity and the mean water velocity, A 1 and A 2 are non-dimensional constant fit parameters that are equal to 1.48 and 0.68 respectively and y 1/2 is the vertical position of the velocity where U = U m /2. The results of the numerical water velocity profile of both the 2D and 3D models are in good agreement with the proposed equation based on the measured experimental velocity profile (see Fig. 4). The computational domain has the dimensions with z = -0.1m and a water column of z = 0.15m and a canal length of x = 0.3m. BlockMesh, a meshing utility that is part of extend-Foam in OpenFOAM, was used to construct the finite volume mesh. In 2D, the mesh is a quasi 3D mesh with only one layer in the third direction. The mesh is defined with a cell size in the x-axis direction (stream-wise direction) of a value ∆ x = 0.0012 m and in the vertical direction with a value of ∆ y = 0.0005 m for the particles body and ∆ y = 0.0023 m in the water column. In 3D, the same setup was used but the mesh was extended 20 cells in the third direction, making the cells number 20 times more than the 2D mesh. It will influence dramatically the CPU time. The CPU time will increase from few days in 2D to about 2 weeks in 3D for a real time simulated of 20 seconds. The time steps were controlled by the Courant number C 0 = 0.1 that was taken smaller than 1 to avoid numerical instability.

2D numerical simulations

Four different flow rate Q were used in the 2D numerical simulation on SedFOAM, Q = 6.5, 7.5, 8.5 and 11.5 l/min. The scour and dune formation were observed, but the dune stabilized with a triangular shape, no change in shape was observed (see Fig. 5). The maximum scour depth was recorded. As the experiment showed, it increases with increasing flow rate. The numerical and experimental results have approximately the same trend, but the scour depth was higher in the experiment as shown in figure 5. As for the water flow, the jet was always following the morphological change, and there was no fluctuation on the water. This behavior of water flow (see Fig. 6) is not in good agreement with the literature, thus a 3D simulation has to be made, to take into account the turbulent flow fluctuation inside the scour.

3D numerical simulations

The 2D numerical simulations couldn't reproduce the fluctuation flow pattern caused by the scour and mound formation process observed in the experiment. Thus the 3D numerical simulations were more appropriate to solve this issue. The water flow in the 3D numerical simulations is described as following, at first, the water jet is flowing with the particle bed until the scour starts to form, at this point, a fluctuation of the flow in the scour will be observed with a backward re-circulation, that leads to move the jet slightly upward, at the same time, the dune will cause jet flipping towards the surface (see Fig 7). This description in the literature is also observed.

Conclusion and perspectives

To be able to take into account the water flow fluctuation, a 3D multi phase flow model has to be used in the sense where the flow fluctuation pattern is not observed in simple 2D numerical simulations as well as the change of the form of the mound as shown in the previous sections. The phenomena of jet flipping and formation of a cavity reversed flow and change of the mound form was observed in the 3D numerical simulations. In contrast, the maximum depth in the 3D numerical simulations was far less than the experimental results while in the 2D simulations, the maximum scour depth was almost in good agreement with the numerical data. Furthermore, the cycle of digging and refiling was not observed in any of the numerical simulations and a difference of the scour depth between the 2D and 3D numerical simulation raise questions on the calibration of the numerical model's rheology parameters of the inter-granular stresses as well as the hydrodynamic of the model, as in the proposed equation only the x-axis direction is considered and there is no enough data on the turbulence model used. This work is still a work in progress. More improved numerical simulations will be performed during the period of the study to assure the compatibility of the physical models parameters with the experimental data. To take into account the missing water velocities of the water flow in the other directions as well as the parameters of the turbulent model, interFoam, a hydrodynamic solver of OpenFoam, will be used to simulate the flow downstream of a sluice gate to be able to produce the missing parameters numerically and introduce them inside the SedFoam simulations instead of the water jet equation used previously.

3. 1

 1 Experimental setupSedFOAM is used to simulate the hydro-sedimentary canal setup used in the experimental study proposed by Martino et al.. The hydro-sedimentary canal setup is a narrow channel of dimensions 2 cm wide and 140 cm long, with a glass side walls and flow rate Q controlled by a pump. In the bottom of the canal, there is a flat granular layer of 10 cm depth and 60 cm length composed of glass beads of a median diameter of 1.1 mm and density of 2.65 g/cm 3 . The water depth was 130 mm. A sluice gate is located at the upstream of the system with an aperture b 0 . A rotameter measures the flow rate Q that can range from 2.5 l/min up to 17 l/min. Thus the Reynolds number ranges between 3790 and 16200. The benchmark experiment consists of an aperture b 0 = 1cm and Q = 6.5, 7.5, 8.5 and 11.5 l/min . The setup is shown in figure1.

Figure 1 :

 1 Figure 1: Experience setup by Martino et al. (2019).

Figure 2 :

 2 Figure 2: The digging-refilling (DR) regime, for b 0 = 1cm and Q = 10l/min.

Figure 3 :

 3 Figure 3: Time evolution of the maximum depth Y 0 for different rate Q and b 0 = 1cm, SR is the steady regime and DR is the digging-refilling regime (Martino et al., 2019).

Figure 4 :

 4 Figure 4: Measured experimental water velocity profile in comparison with the numerical velocity profiles for Q = 11.5l/min in 2D and 3D.

Figure 5 :

 5 Figure 5: 2D numerical results of the scour and dune formation with Q = 8.5l/min (right) and Y max the maximum erosion depth for different flow rate Q compared with the experimental data (right).

Figure 6 :

 6 Figure 6: 2D numerical results of the water flow jet downstream of the sluice gate of aperture b 0 = 1cm (left: U x and right: U y ).

Figure 7 :

 7 Figure 7: 3D numerical results of the water flow jet streamlines downstream of the sluice gate of aperture b 0 = 1cm with Q = 11.5l/min at two times t = 1s (top) and t = 10s (bottom).As for the scour and dune formation, a change in the shape of the dune was observed in the 3D numerical simulations (see Fig 8)unlike the 2D numerical simulations while the scour depth was less than the experimental measurements,


Abdeldjelil Nehari

Alain Brenier

Gerard Panzer

Kheireddine Lebbou
email: kheirreddine.lebbou@univ-lyon1.fr.

Jerome Godfroy

Serge Labor

Herve Legal

Gilles Cheriaux

Gilles Ch

Jean Paul Chambaret

Theirry Duffar

Richard Moncorg

Ti-Doped Sapphire (Al(2

come

I. Introduction

Sapphire crystals are used for a wide range of applications: the clock and watch industry, synthetic jewelry, and optical components for microelectronics and laser systems. Today, there is an increasing demand for undoped sapphire wafers as substrates for GaN and ZnO thin films and for Ti 3þ doped sapphire for ultrashort and high power laser systems. Sapphire doped with Ti 3þ ions is now recognized as the solid state laser material which allows the direct generation of the shortest laser pulses and as the basic amplifier medium for most of the future high power laser chains at the petawatt level. 1,2 Undoped and titanium-doped sapphire crystals are now considered as very strategic materials. Concerning the latter, a lot of work remains to be done to grow and provide highquality large-size highly doped single crystals with reproducible performance. Improvements have been obtained here by working on the construction of the crystal growth equipment and by optimizing the growth process throughout different parameters. The numerical simulation and modeling of the equipment and processing have been an important tool for developing sapphire crystal growth in the industry. [3][4][5] Today, the main sapphire growth techniques are the Verneuil, Czochralski, Stepanov, GOI (Musatov method), HEM (heat exchanger method), HDC (horizontal directional crystallization or Bagdasarov method), and Kyropoulos (KT) techniques. Although large undoped sapphire crystals have been grown by Khattak et al. 6 by using the HEM technique and by Barish et al. 7 by the temperature gradient technique (TGT), growth of large Ti 3þ -doped Al 2 O 3 crystals with a homogeneous titanium distribution and good optical quality is still a challenge. The main problem in Ti-sapphire crystal growth is strong segregation of the Ti 3þ ions, because of their size difference with the substituted Al 3þ ions, which generates irremediable defects during the growth process. Moreover, the laser performance is often limited by optical loss infrared absorption bands of Ti 3þ -Ti 4þ ion pairs 8 that precisely occur in the laser emission wavelength. Annealing improves the quality of the as grown crystals. It allows changing partially Ti 4þ into Ti 3þ and improving the so-called figure of merit (FOM), which measures the ratio of the absorption coefficient at the pump wavelength of about 500 nm over the absorption coefficient associated with the parasitic Ti 3þ -Ti 4þ absorption band around 800 nm. In the present paper, large size and highly Ti 3þ -doped Al 2 O 3 single crystals were successfully grown by the Kyropoulos technique (KT) at the RSA le Rubis Company and optical characterizations of the as grown crystals were performed. II-2. Crystal Growth. The Ti-sapphire crystals were grown by the Kyropoulos technique in a molybdenum (Mo) crucible with the diameter of the required crystal. Figure 1 shows a schematic illustration of the melt solidification inside the crucible. The crystal quality strongly depends on the hot zone of the Kyropoulos furnace. According to the thermal behavior during the growth process, this technique is considered within the framework of a 2D axisymmetric approach. 9 The radiative heat exchange in the crystal and melt convection can be well controlled. Because of molybdenum crucible oxidation, high vacuum is necessary to avoid MoO 2 formation during the growth process. Two seed types were used, and the growth was initiated along the a-and c-axes. The total process including installation, heating up, the growth operation, and crystal cooling took about two weeks. Compared to the Czochralski and Bridgman techniques, this method uses a low thermal gradient in the solidification region. The crystals do not cross a high gradient region above the crucible during the growth process, which is a good condition to eliminate the thermal stresses and to reduce the dislocation density.

II. Experimental Section

III. Results

III-1. Grown Crystals. Figure 2 shows the grown crystals. The crystals were transparent and colored because of titanium doping. The diameter of the crystals was 100 mm, and their weight was about 5 kg, depending on the size of the crucibles and of the starting raw materials. Because of the sapphire high melting temperature (2050 °C) and the crystal size (diameter 100 mm), special care was taken to reduce the thermal losses from the crucible. The obtained crystals were exempt from macroscopic defects such as cracks, subgrain boundaries, and precipitates. For the laser applications, it is necessary to have the Ti 3þ concentration in the crystals with the minimum level of Ti 4þ and without titanium segregation. In order to reach the opti mumtitaniumcontent,th ecry stalswe regrownfromthe melts with different Ti concentrations (0.15-0.35 atom %). According to our results, 0.25 atom % Ti in the melt was the good concentration to control the growth process (interface stability) and to obtain good quality transparent Ti-doped Al 2 O 3 crystals under stationary stable conditions. At greater content of TiO 2 ,t h eA l 2 TiO 5 phase was formed. During the growth of large Ti-sapphire crystals by the Kyropoulos technique, high pulling rates and high rotation speeds induce vibrations and temperature variations of the crystallization interface, causing high thermal stresses. These factors decrease the optical quality of the crystals, and macroscopic defects such as bubbles and cracks, particular for large Ti 3þ -Al 2 O 3 crystals, were observed. The distribution of the Ti 3þ ion along the crystals was inspected in different parts of the ingots by the ICP method. The unit cell volume increased in proportion to titanium concentration (Figure 3). Thus, the titanium concentration (X) can be estimated from the measurement of the unit cell volume (A ˚3) using the approximation V = 254.7141 þ 0.8636[X]. The segregation coefficient was calculated according to the Pfann 10 equation, C s = kC 0 (1 -g) k-1 ,whereC s is the Ti-dopant concentration in the crystal, C 0 is the starting dopant (Ti) concentration in the liquid, g is the melt solidification fraction and varies between 0 for the beginning of the growth and 1 when the entire melt was crystallized, and k is the segregation coefficient. The titanium segregation coefficient for the crystals grown by the Kyropoulos technique (this work) was measured to be 0.20. As was expected, due to the ionic radii difference between Ti 3þ (60.5 pm) and Al 3þ (53.3 pm), the segregation coefficient was small. III-2. Optical Characterization. In order to estimate the intrinsic quality of the as grown Ti-sapphire crystals, we have used a nondestructive method based on the utilization of an He/Ne laser emitting at 633 nm. [START_REF] Lebbou | Advances in Materials Research[END_REF] A focused beam from the laser was sent through the crystal (φ = 100 mm, e =15mm). The output intensity was recorded with a CCD digital camera connected to a computer. Whatever the growth direction (a-axis or c-axis), the best crystals were found exempt of defects, such as scattering, bubbles, and cracks, and demonstrated a near Gaussian profile of the transmitted laser beam (Figure 4), which is a first confirmation that the crystals were defect-free and of high optical grade. On the other hand, for higher titanium concentration, because of the presence of microscopic and macroscopic defects (bubbles, scattering), the beam profile of the grown crystals was non-Gaussian, with a strong disturbance (Figure 5). The absorption spectra (Figure 6) of Ti 3þ -Al 2 O 3 at room temperature for different crystal orientations show two overlapping broad bands peaking around 490 and 525 nm, which correspond to two phonon side bands of the same Ti 3þ optical transition from level 2 T 2 to level 2 E but split into two sublevels according to the Jahn-Teller effect. [START_REF] Ham | [END_REF] In the R-sapphire lattice, Ti 3þ (3d 1 ) substitutes Al 3þ and is surrounded by an octahedron of oxygens with a slight trigonal distortion. The measured luminescence lifetime of Ti 3þ (0.25 atom %) in Al 2 O 3 was 3 μs at room temperature. Ah i g hc o n c e n t r a t i o no fT i 4þ ions could be responsible for some UV absorption, but this was not observed in our crystals. Moreover, crystals containing other impurities, such as iron (Fe), may show a broad absorption band and a long tail, extending well beyond 800 nm, and this band is usually related to a charge transfer transition between coupled iron-titanium ion pairs (Fe 3þ Ti 3þ -Fe 2þ Ti 4þ ). 13 A qualitative measurement of the Ti 3þ spatial distribution inside the crystals was inspected by microluminescence analysis. The spectra were recorded on the crystals of 100 mm in diameter and 10 mm in thickness. For the measurement, the crystal was placed under a microscope (lens Â100) and was directly excited with the focused beam of a 532 nm frequency-doubled and Q-switched Nd:YAG laser. The luminescence was recorded through the microscope by an optical fiber and transmitted to a monochromator that dispersed the emitted light onto a CCD detector cooled to 0 °C. The measurement was performed in different transversal sections of the crystal (Figure7). The curves indicated higher luminescence intensity and higher Ti 3þ ion concentration at the periphery of the crystal. The radial distribution of Ti 3þ in the Ti-doped Al 2 O 3 crystal expressed as luminescence at 730 nm is illustrated in Figure 8. There is a relatively large concentration gradient of Ti 3þ at the periphery of the crystal, but the radial Ti distribution is rather homogeneous at the central part of the crystal. 80% of the crystal (80 mm) had homogeneous radial titanium concentration and allowed exploring a large part of the grown ingot. Following these results, it is possible to cut large laser rods or slabs along the plane perpendicular to the growth direction. III-3. Optical Amplification. In order to validate the performance of the crystal, optical amplification experiment has been realized on a crystal of diameter 100 mm and thickness 50 mm. The crystal has been optically pumped by a frequency-doubled Nd:YAG laser, emitting 800 mJ of 532 nm light at a repetition rate of 10 Hz. The titanium doped sapphire crystal was positioned in a multipass amplifier (2 passes) of a classical chirped pulse amplification (CPA) laser chain. The seed pulses to be amplified have an energy of 8 mJ. An amplified energy of 65 mJ after the two passes has been obtained. The result is consistent with numerical calculations of amplification. In order to validate the spectroscopic characteristics of this crystal, we also measured the amplified spectrum and compared it to the input one. These two measurements are presented in Figure 9. This confirms that the doping of the sapphire host was completely adequate and in good agreement with the crystal quality.

IV. Conclusion

Large size and high optical quality (diameter 100 mm, weight 5 kg) Ti 3þ -doped sapphire crystals have been successfully grown by the Kyropoulos technique. The crystals were transparent and exempt of cracks, inclusions, and other scattering centers. The crystals discussed in this work amplify the energy without any damaging. Without post growth annealing, the absorption and luminescence did not show the presence of unwanted impurities. The obtained results indicate homogeneous Ti 3þ ion concentrations in large sections (80%).

II- 1 .

 1 Starting Raw Materials. The starting raw materials were alumina and TiO 2 powders. The chemical components and the particle size of the starting raw materials were high purity TiO 2 (rutile) powders and undoped and Ti-doped R-alumina (Al 2 O 3 )of spherical micrometric geometry, in good agreement with JSPDS File No. 46-1212 (a = 4.7570 A ˚, c = 12.9898 A ˚), but traces of θ-Al 2 O 3 (JSPDS File No. 86-1410) were also observed.

Figure 2 .

 2 Figure 2. Sapphire ingots grown by the Kyropoulos technique: (a) a-axis; (b) c-axis; (c) wafer thickness 20 mm.

Figure 1 .

 1 Figure 1. Schematic the Kyropoulos technique.

Figure 3 .

 3 Figure 3. Evolution of volume as a function of titanium concentration in Ti-doped Al 2 O 3 . Different fragments of the crystals were cut from the ingots and analyzed by X-ray diffraction and ICP.

Figure 4 .

 4 Figure 4. He/Ne laser beam shape (P = 0.95 mW) in air (a) and as passed through the crystal (φ = 100 mm) in regions 1 (b), 2 (c), and 3 (d). The Ti starting concentration in the melt was 0.25 atom %.

Figure 6 .

 6 Figure 6. Room temperature absorption spectra of the as-grown Ti-sapphire single crystals grown by KT: (a) c-axis; (b) a-axis.

Figure 5 .

 5 Figure 5. He/Ne laser beam shape (P = 0.95 mW) in air (a) and as passed through the crystal (φ = 100 mm) regions (b, c) with defects.

Figure 8 .

 8 Figure 8. Profile of the Ti 3þ ion distribution in the as grown Ti-sapphire crystal (starting titanium concentration in the melt was 0.25 atom %).

Figure 7 .

 7 Figure 7. Microluminescence spectra (uncorrected for monochromator and detector response) recorded in different regions of the grown crystals (excitation of the Ti 3þ luminescence at 532 nm).Figure 9. Input and amplified spectra of Ti-Al 2 O 3 sapphire.

Figure 9 .

 9 Figure 7. Microluminescence spectra (uncorrected for monochromator and detector response) recorded in different regions of the grown crystals (excitation of the Ti 3þ luminescence at 532 nm).Figure 9. Input and amplified spectra of Ti-Al 2 O 3 sapphire.