Yvan Peter
email: yvan.peter@univ-lille1.fr

Thomas Vantroys
email: thomas.vantroys@univ-lille1.fr

Platform Support for Pedagogical Scenarios

Keywords: Pedagogical scenario, Unit of study engineering, Scenario execution, Workflow engine

This article deals with providing support for the execution of pedagogical scenarios in Learning Management Systems. It takes an engineering point of view to identifies actors, design and use processes. Next it defines the necessary capabilities of a platform so that actors can manage or use pedagogical scenarios. The second part of the article is dedicated to the presentation of the design and properties of a component we have developed for the execution of scenarios: a flexible Workflow engine and a user interface providing indicators about the student progress within a unit of study.

Introduction

E-Learning has been spotted as the solution to the growing needs of education. Many institutions and enterprises have set up Learning Management Systems (LMS) and organized their work around these new technologies. However, when the number of learners grow, it becomes difficult for tutors to support them correctly while still being aware of individuals that need some special care because they experience difficulties. Thus, LMS should provide some level of automation for the management of the activities within units of study while keeping tutors aware of students having difficulties and permitting an adaptation of the activities to these special cases. One way to define the activities that will take place within a unit of study is to describe them in a pedagogical scenario. Such a scenario defines the activities which must be done by the learners and the tutors, the sequencing of these activities as well as the learning objects and tools that should be provided to the different actors. For instance, the emerging standard IMS-LD [START_REF] Ims | IMS Learning Design Information Model -version 1.0[END_REF] uses a theatrical metaphor where the activities take place in different acts that define the sequencing. The activities are associated to roles corresponding to users and to an environment composed of learning objects and tools. The benefit of the use of pedagogical scenarios, as stated in [START_REF] Hummel | Educational modelling language and learning design: new opportunities for instructional reusability and personalised learning[END_REF], is that the focus is put on the learning activities that should be done to achieve a learning objective rather than the learning objects. In this approach, the activities provide a context for the use of the learning objects rather than having a "passive" consumption of them. This article presents our work towards providing support for pedagogical scenarios at both organizational and technical level. Indeed, the production and use of pedagogical scenarios should be backed by a correct organization as it is the case for producing and using learning objects. For this reason, we have defined the life cycle of a pedagogical scenario and identified the different actors involved in it. Considering the execution, we have taken care of flexibility issues to increase adaptability to the learners and reusability, i.e., with have the possibility to modify the scenario at run time. This is useful because it is sometimes difficult to know beforehand the activities that will take place (e.g., in life long learning where the learning process lasts a long time) or because an adaptation of the course is needed (e.g., to suit the learners' level).

The first part of the article presents the design and use of pedagogical scenarios from a platform engineering point of view. In the second part, we review the requirements towards platforms support for scenario execution. The third part presents the basis we have used to build our solution. The fourth part is dedicated to the Cooperative Open Workflow, the solution we have developed based on Workflow technology and shows how we meet the defined requirements with an illustration on a sample scenario. The solution is compared to existing alternatives. Finally, we present a user interface taking benefit of the Workflow engine to give learners indicators about their progress within a unit of study to enable better planning of their work. The conclusion summarizes the work done and presents ongoing work.

Engineering of pedagogical scenarios

The architecture and use of Learning Management Systems is getting more and more complex. The design and delivery of a new course implies many tasks which rely on highly specialized actors during the whole life cycle.

In the next sections, we will present the design process, the use process as well as the actors involved in the definition and operation of a course in the form of a pedagogical scenario.

Design process

The design process of a course shown in the figure 1 is composed of 6 main phases. This is inspired from the Rational Unified Process (Jacobson et al., 2000). Each phase enables to go to the next or to go back to the previous one if problems related to it appear.

Figure 1. Design process of a unit of study

Initial expression of needs and models. This phase corresponds to the initial definition of a course. A teacher will describe in an informal manner the activities of the course and will define the objectives, the prerequisites ... He will also define the type of learning objects and tools which will be necessary. RUP relies on UML use cases for this but there is not any equivalent formalism in the field of Technology Enhanced Learning. Analysis and design. Following the informal description, a pedagogical engineer, which is aware of the technical characteristics of the target platform, will collaborate with the teacher to translate the scenario into a more formal one where the different phases are scheduled. Some of the activities may have to be adapted or replaced depending on the capabilities of the platform. This design phase may require interactions with pedagogical resource providers and component developers. The former will define the learning objects which are available or have to be created. The latter is more concerned with the development of new software components on the platform to provide the needed tools. Implementation. This phase corresponds to the development of the software components and pedagogical resources needed for the course. It is mainly done by the component developer in collaboration with the pedagogical engineer. It may also involve a component assembler if a functionality can be provided by an existing set of components. Deployment. All elements developed or assembled during the previous phase are installed on the target platform. This phase concerns the component assembler and the platform administrator. At the end of this phase the new course can be referenced.

Test. This phase corresponds to the control of the behaviour of the platform and resources and the validation of the coherence of the model. The tests are done by the pedagogical engineer who is qualified for the control at both the pedagogical and technical level. The component developer and assembler are responsible for the control of the respect of the specifications at the technical level. Evaluation. During the use of the course model, it is important to monitor the behaviour of the learners and tutors to see if the initial model is correctly executed. If not, this is the opportunity to reengineer the model to better suit the real use. At a higher level, there is also a phase for building a learning path composed of many courses which is at the limit between the design and use processes. In this phase, the pedagogical engineer in collaboration with teachers responsible for the units and a learner orientation adviser will create a complete learning path.

Use process

The use process describes the different phases corresponding to the life cycle of a course model. These phases are described hereafter:

Instantiation. This phase determines which people will actually take part in the course according to the model. This corresponds to the following roles: learner, tutor and referent. The learner orientation adviser may be involved in the definition of the groups of learners while the tutor is responsible for checking that resources (i.e., learning objects and tools) are available and that the course can begin.

Execution. This phase corresponds to the most visible part where users are involved in a learning process through the platform. The learners will process through the different activities of the course while the tutor checks the correct execution of the model. The referent is responsible for a group and will serve as central point to solve learners' problems be it technical, pedagogical or administrative.

Roles involved in distance learning platforms

So as to define the different tasks implied in operating a course on a Learning Management System, we have defined roles which are related to these tasks. This is done in the same spirit as in the J2EE roles (component developer, assembler ...) [START_REF] Matena | Applying Enterprise JavaBeans: Component-Based Development for the J2EE platform[END_REF], in design processes (Jacobson et al., 2000) and in application development [START_REF] Schmidt-Wesche | IBM WebSphere Platform User Roles[END_REF]. The role list may not be exhaustive but rises from our experience and research in the field of distance learning. The different roles are shown figure 2 with their type of responsibility: administration, modelling, technical, use. It must be noted that multiple roles may be taken by the same person even though it is likely that many people will be required due to the increasing complexity of e-learning platforms. We will give a brief description of each role and their relations hereafter.

Teacher

The teacher is the starting point in the creation of a course. He defines, in terms of competencies, the prerequisites and objectives of the pedagogical activities. He may associate specific services (e.g. asynchronous communication), tools (e.g., simulator) or learning objects. The teacher is an expert in a specific domain. He collaborates with the pedagogical engineer to adapt his pedagogical scenario to the capabilities of the platform. He may assist the learner orientation adviser in assessing the competencies of a learner for a particular course.

Pedagogical engineer

The pedagogical engineer is concerned with the whole design process since he has both pedagogical competencies and knowledge of the used platform. He can manage the link between the design of courses, associated developments and use. He helps the teacher formalize his pedagogical scenario according to the capabilities of the platform and can interact with the component developer to introduce new tools or services into the platform. He can guide the learner orientation adviser in designing a new learning path. He can interact with the referent and tutor to handle modifications of a course model.

Pedagogical resource provider

He designs and develops learning objects for the teacher and pedagogical engineer.

Learner

The learner uses the platform to get new competences. He may work alone or in a group. The learner uses the platform tools and accesses the learning objects in the scope of the assigned activities to learn. He can interact with the tutor to get help on specific topics of the course and with the referent for more general problems. He can build his learning path with the help of the learner orientation adviser.

Tutor

The tutor takes care of the good unfolding of the course and handles the animation of the groups and activities.

He follows and assists the learners so as to meet the learning objectives. He informs the referent of any particular difficulties. The tutor can make local modifications to the course model in collaboration with the pedagogical engineer and the referent.

Referent

The referent manages one or more groups. He serves as a unique access point during the whole learning path. He can act as a mediator between the tutors or tutors and learners. He can ask for modifications of the model to the tutors.

Administrative manager

This role is more concerned with administrative management of learning path: enrolment, control of the presence ... He is responsible for the respect of organizational rules and laws.

Learner orientation adviser

The learner orientation adviser welcomes learners and guides them towards the best courses and learning paths according to their competencies and wishes.

Component assembler

The component assembler gathers existing components to create new services or tools for the platform.

Component developer

The component developer is an expert in the development for the target platform and provides basic elements for the component assembler and pedagogical resource provider.

Platform administrator

The platform administrator is responsible for the correct operation of the platform. He manages user accounts in collaboration with the administrative manager, the pedagogical engineer and the referent.

Platform support for pedagogical scenarios

When it comes to the point of deploying a pedagogical scenario on a specific platform, most of the time it comes down to organizing the access to the pedagogical resources and most of the scenario is lost. For instance, if an order has been defined for accessing the resources, this may not be enforced by the platform. This is the case for example with the OpenUSS platform [START_REF] Grob | Developing, Deploying, Using and Evaluating an Open Source Learning Management System[END_REF] where one can organize access to learning materials into semesters, and courses but would have to manually deliver materials if he wishes a better control on the order of access. This is mainly due to the fact that the focus has been directed to accessing the resources rather that scripting of the access. In this part, we will describe the properties needed to fully support pedagogical scenarios and how they relate to the roles involved in the operation of a distance learning platform.

Formal model for the pedagogical scenarios

To take advantage from a platform support for pedagogical scenarios, there must be a formal description that can be handled by the platform. Educational Modelling Languages such as IMS-LD (IMS, 2003) can be a good candidate for this. This formal language will be used by the pedagogical engineer to translate the ideas of the teachers and learner orientation advisers while taking into account the concerns of the administrative manager.

To this end, a general modelling language should support: Individual and group scheduling of activities. People can engage in a course or a learning path either individually or within a group. In the latter case, it is reasonable to allow for individual progression between the activities so that each one can progress at his own rhythm. Group synchronization will be enforced only for collaborative activities or to respect time constraints. The definition of the individual parts is done by the teacher and pedagogical engineer. Time management. It is important to be able to enforce time constraints on a unit of study or learning path. Firstly to support the directives from the administrative manager (e.g., to respect academic years) and secondly for pedagogical reasons (e.g., to limit the duration of an online test). Collaboration support. Collaborative learning is a means to enhance the learning experience and a way to sustain the learners' motivation [START_REF] Eales | The Motivation is the Message: Comparing CSCL in different Settings[END_REF]. As such it should be supported.

Platform support for model execution

Platform support for model execution is tightly related to what can be expressed in the model. In a general manner, the platform is responsible for automating the progress of the learners from activity to activity within a course while providing the right tools and learning objects. This capability is of paramount importance to relieve the tutors from a daunting task in case of large groups of students.

Monitoring of the model execution

Automation of the learning process is a great help in the operation of distance learning. However, it should be accompanied with monitoring facilities to assist the tutors and referents in the follow-up of the learners. The platform should provide the means to raise awareness of the potential problems that learners may have (e.g., slow progression within a group, violation of a time constraint ...).

Enhanced management of the models

Having identified a learner or group problem thanks to the monitoring facilities, the tutor and pedagogical engineer should be able to modify the model at run time to achieve the pedagogical objectives. Moreover, if the reengineered model is better, the modified model should be available for the next execution. This ensures that the pedagogical scenario takes benefit from the previous experiences. One can also find easier to build courses from existing activities or scenarios.

Planning support

From the learner point of view, the engine should provide means to assess his progression within a course and relative to the group. This is another kind of monitoring facility. It should also assist the learner in planning his activities by showing the activities to perform to complete the course.

Building flexible support for pedagogical scenario

In our research work on supporting the execution of pedagogical scenarios, we have decided not to focus on a specific platform but rather to develop a dedicated component to be embedded in existing Learning Management Systems. This has raised some technical issues to bear in mind during the design of this component:

Standard based: Since we are designing a technical component to be embedded in different platforms, the first requirement is that we have is to keep up with existing standards. Integration support: Integration within a platform should be easy, for this Web Services standards and event based communication are good solutions. Persistence: Learning scenarios may correspond to long running processes. For this reason, we have to take care of persistence issues.

To tackle the last two issues, we have chosen to rely on the J2EE standards. The component has been developed using Enterprise JavaBeans components [START_REF] Eberhart | Java Tools: Using XML, EJB, CORBA, Servlets and SOAP[END_REF] that deal with the persistence issues. These components can be accessed through RMI/IIOP or as Web Services so we are able to support integration with RMI, CORBA and SOAP protocols.

Considering the first issue, standards, we have headed towards a general solution rather than using a specific pedagogical language. Executing a pedagogical scenario can be compared to the execution of a traditional process (e.g., in business, administration ...). The natural solution for process execution is a Workflow engine. However, traditional Workflow engines are too strict to support our requirement for model modification at run time. In general we have not found any that can handle all the requirements we have identified in the previous section. For this reason we have started to build up our own solution starting from existing Workflow standards: Workflow Management Coalition standards and the Object Management Group's Workflow Management Facility. An additional benefit of this approach is that we are not restricted to the description of pedagogical scenarios but we can handle more administrative parts of the platforms such as registration, group creation... In the next section we will describe the standards we have used before presenting how we have fulfilled our requirements.

Workflow standards

The Workflow Reference Model and XML Process Definition Language

The Workflow Reference Model [START_REF] Hollingsworth | The Workflow Reference Model -version 1.1[END_REF] is a standard from the Workflow Management Coalition (WfMC) whose aim is to promote the use of Workflow through standardization and interoperability. This standard does not define the Workflow engine itself but rather its interfaces. In this article, we are more concerned with interface 1 which defines an XML language called XPDL (XML Process Definition Language) [START_REF] Wfmc | Workflow Process Definition Interface -XML Process Definition Language -version 1.0[END_REF] to define process models regardless of the enacting platform. Interface 5 is also of interest since it defines how administration and monitoring tools can interact with the Workflow engine. Application corresponds to the applications which can be used to perform the activities. There, we can define the tools used by the learners and support staff (e.g., mail, forum, text editor ...); Relevant Data are the data used and produced by the process and the activities. This data can be linked with the data managed by the learning management system; Participant represents a human, role or group to who work items are assigned. Participant can be linked with the user accounts and groups as defined on the LMS.

We have used the XPDL language with extensions to support the definition of group activities and time constraints. We have also separated process and activity models into two parts so as to be able to build a scenario from existing activities.

Workflow Management Facility

The Workflow Management Facility (WMF) [START_REF] Omg | Workflow Management Facility -Version 1.2[END_REF] standardizes the architecture of the Workflow engine. It has been defined by the Object Management Group (OMG) in accordance with the reference model of the WfMC. This standard defines the interfaces of objects such as WfProcess and WfActivity or WfResource which correspond to the elements described in a XPDL process model. There is also a hierarchy of *EventAudit interfaces which corresponds to the events tracked by the Workflow engine. This is useful to build a monitoring facility. We have made an implementation of the interfaces proposed in the WMF with extensions to support collaborative activities, group management and to enhance the reuse of models both at the process and activity levels.

Cooperative Open Workflow

In this section we will describe the Workflow engine we have developed based on these standards which is called Cooperative Open Workflow (COW), [START_REF] Vantroys | COW, a Flexible Platform for the Enactment of Learning Scenarios[END_REF]. We will first illustrate how we can model a sample pedagogical scenario. We will next focus on collaborative activities and time constraints. Then we will show how the model is handled by the Workflow engine at run time. Finally, we give some details about the implementation before presenting related work.

Unit of study modelling

The main function of the Workflow system is to schedule the activities of a course. Such a course is attended by a group of learners (individual learning being a special case since the "group" is reduced to one learner). In the sequel, we will take a course in physics as an example of the modelling of a course to illustrate the management of models and instance inside the Workflow and the management of both individual and group activities. The scenario corresponding to the course is composed of four activities described hereafter:

Course learning activity associated to the role learner which will be given access to a set of learning objects related to the subject. We can set a maximum duration to this activity so that learners do not take too long on this activity; Self test activity associated to the role learner which will have to pass a multiple-choice questionnaire. Since this is a test, we would like to state that it has to be performed in a limited time;

Test correction activity associated to the role tutor which will review the results of the tests and look at the errors made by the students; Discussion about the unit of study activity associated to the role learner and tutor. Here the tutor can initiate a discussion about the concepts studied based on the tests results. We can set a deadline for starting this activity to help synchronize everyone.

Since some parts of the course can be realized at his own rhythm by each learner, one has to take it into account so as to enable flexibility in the schedule of the activities. There are two ways to manage the schedule of the activities for a group of learners:

In the first mode, an activity is terminated only when all the learners have terminated it. In such a way, the activities of a whole group are synchronized. Even though it is very close to traditional face to face learning, it does not take benefit of distance learning mode;

The second mode identifies the parts of a course that can be realized autonomously. This way each learner can progress at his own rhythm inside a group with some activities giving a synchronization point to the group.

These two modes are supported in COW by means of sub-processes. In our scenario, the teacher decides that the three first activities can be realized individually by each learner. These activities are then modelled into a single process. The process corresponding to the whole course is then composed of two sequential activities (see figure 4), the first one being in fact a reference to the individual work sub-process and the second one corresponding to a synchronous discussion between the members of the group. Figure 5 shows the model of the global course model in the XML format (derived from XPDL). We have separated the process and activity models. This way, we can easily change and replace an activity by another or modify an activity within a process without changing the whole schedule. We can also build a new process starting from individual existing activities. The model presented defines the identifiers of the activity models that compose the process and the transitions that order these activities. Figure 6 shows the activity model of the discussion referenced in the global process.

<WorkflowProcess Id="physics" name="physics"> ... <Activities> <ActivityLink Id="START" URL="start" /> <ActivityLink Id="A1" URL="read_and_self_test"/> <ActivityLink Id="A2" URL="chat"/> <ActivityLink Id="END" URL="end"/> </Activities> <Transitions> <Transition Id="T0" From="START" To="A1" /> <Transition Id="T1"

From="A1" To="A2" /> <Transition Id="T2"

From="A2" To="END" /> </WorkflowProcess> To handle collaborative activities, we have made some modifications to XPDL and the WMF to introduce the notion of workitem. A workitem is an atomic piece of work and an activity is composed of workitems and defines the execution context for the inner workitems. In the simplest case, there is only one workitem in an activity. However, within a collaborative activity, there can be more workitems. A workitem is attributed to a role. So if multiple actors have the same role, there will be an instance of the workitem for each of them in the activity. Resources are allocated to the workitems rather than the activity.

<Activity Id="chat" Name="chat" /> ... <Implementation> <Tool Id="chat_tool" /> </Implementation> <Group><YES/></Group> <Performer> <Participant Id="R1" Name="Learner" ParticipantType="ROLE" /> <Participant Id="R2 Name="Tutor" ParticipantType="ROLE" /> </Performer> ...

</Activity>

Figure 6. XML model of a collaborative activity

In our example the discussion activity (figure 6) could be done with a chat tool. During this activity, each people having the role learner or tutor will have a workitem corresponding to the work to do. Here, everyone has the same task since no specific workitems have been defined. However, the learners and the tutor may not have the same rights on the tool since they do not have the same roles. There is only one tool defined for this activity which is not actually identified in the model to enable late binding depending on the platform and available tools.

Time constraints

Management of time constraints is an important aspect of learning activities. This is particularly true for group based learning where there must not be too much lag between the learners. COW supports the notion of deadlines which correspond to the time at which an activity must be started or completed. It also supports the notion of limit which defines the minimal or maximal duration of an activity. When a stop deadline or a maximum duration limit is reached, the Workflow engine suspends the activity. It can then use different policies to handle this case. For example, the system can terminate the activity authoritatively or notify the tutor who will take a decision about it. These behaviours can be dynamically changed at run-time by the tutor to be adapted to a specific context.

Our scenario defines two types of time constraints. The first one is an end deadline for accessing learning objects which is illustrated figure 7. Here the end deadline is set to four days (not including week ends) and the strategy is to send a mail to the student to warn him that he should proceed to the next activity. The second time constraint is defined in the self-test activity, the maximum duration will be 3 hours as illustrated in figure 8 and when this duration is reached, the strategy will be to automatically complete the activity.

Course instance

The creation of a process instance requires an instance model which describes the mapping of roles to actual users and of resource identifiers to actual learning objects and tools. The mappings can be global to the model or defined on an activity basis. This separation between process model and instance data allows a better reuse of models. Figure 9 illustrates the Workflow operation for a group of three learners. Taking the process and activities models and instance data, the Workflow engine will create a sub process for each learner. These sub processes contain the three activities that can be performed individually by each learner and each activity contains only one workitem. The third activity performed by the tutor role will have three workitems assigned from different processes. When all sub processes are terminated, the engine will create a collaborative activity with one workitem for each learner and one for the tutor.

Implementation of the Workflow engine

The implementation of the Workflow has been done following a micro-kernel architecture. The basic functions related to the scheduling of the activities constitute the kernel and corresponds to the implementation of the WMF. We can then add more elaborated services on top of the kernel following a facade pattern [START_REF] Gamma | Design Patterns: Elements of reusable Object-Oriented Software[END_REF]. For instance one facade can build the workitem list of a user. This type of facade is dedicated to the "normal" use of the engine.

Flexibility is provided following the Meta-Object Protocol [START_REF] Kiczales | The Art of the Metaobject Protocol[END_REF]. This means that some facades are dedicated to the consultation and modification of the process model and engine behaviour: Process modification consists in adding/deleting activities and transitions or modifying the activity model (e.g., changing tools or learning objects, modifying workitem assignment). These changes can be realized for one learner or for a group of learners.

Behaviour modification changes the way the model is interpreted. It has been particularly used for time constraints and exception handling. For instance duration can be interpreted in absolute time or taking work days into account. One can define different strategies and use them following the strategy pattern [START_REF] Gamma | Design Patterns: Elements of reusable Object-Oriented Software[END_REF].

The integration within Learning Management Systems is realized with a web-services approach. Each interface of our system is accessible by using the SOAP protocol [START_REF] Eberhart | Java Tools: Using XML, EJB, CORBA, Servlets and SOAP[END_REF].

Related work

A kind of scheduling has been added lately in platforms like Blackboard or WebCT by the means of timed release of the resources. However, this cannot be compared to the use of pedagogical scenarios since the definition of the release time is done only for learning resources and does not take into account activities. Moreover, the time is defined beforehand and does not take into account the unfolding of the activities while COW provides activities and resources when they are needed according to the scenario and can manage time constraints relative to the start of the process or activities.

There are few platforms that provide the capability to enact pedagogical scenarios the Flexible e-learning system (Flex-eL) [START_REF] Lin | On Workflow Enabled e-Learning Services[END_REF] from Distributed Systems Technology Centre (DSTC) and the university of Queensland in Australia, the Virtual Campus Project developed at Politecnico di Milano [START_REF] Cesarini | Carrying on the elearning process with a workflow management engine[END_REF] and CopperCore from Open Universiteit Nederland (http://www.coppercore.org/).

Flex-eL consists of a distance education platform supported by a flexible workflow system. There is a learning process for each student. This allows an easy way to adapt the learning path to the real need. Groups of students are dynamically constructed. Managed Workflows are sequential and time constraints are not taken into account.

The Virtual Campus project relies on the BizTalk business process engine from Microsoft. Hence scenario designers can take benefit from the capabilities of the XLANG language [START_REF] Thatte | XLANG -Web Services for Business Process Design[END_REF] to express the learning activities organization.

Coppercore is an implementation of an IMS-LD interpretation engine. It is based on similar technological basis as COW. COW is halfway between CopperCore and the other projects since it is based on Workflow rather than a dedicated Educational Modelling Language but we are working on an import mechanism to support IMS-LD.

Interactive Study Guide

So as to illustrate the potential of the Workflow engine as a means to support pedagogical scenarios and show how we can provide indicators about the learners' progress within a course, we have built a user interface called the Interactive Study Guide. This learner oriented environment provides the list of the activities he has to perform according to the pedagogical scenario as well as indicators of is progression relative to the allocated time and to the group progression. The next sections show how we can give indicators to help the learner plan his work.

Global view and indicators

Figure 10 shows the global view presented to the learner. It is composed of indicators and elements for easy navigation through the activities.

Figure 10. Global view and indicators

The log book which shows the activities terminated and in progress so as to easily check if the work plan is respected. Indicators to know the progression within the course, relative to the group and relative to the next deadlines.

Recent works to access directly the last activities without having to navigate through the scenario.

Unit of study indicators and navigation

Figure 11 shows how the course can be presented to the learner. The interface is composed of the following elements.

Header which gives the title of the course , a description of the pedagogical objectives , deadline for the termination of the course as well as an indicator of the progression within the course . Activity sections which compose the course. Sections can be further broken up into arbitrary levels. For each activity there is a description, the estimated time and actual time spent on the activity and the possibility to start/suspend/stop the activity .

Building indicators

To be able to construct the indicators, we have added a notion of estimated time for completion of the activity in the model. By this way, we are able to compare the actual time spent with the estimated time so as to tell the learner if he is late or in advance at the level of the course or the single activity. The group indicator is built by situating the learner relative to the average progression of the group. Finally for each activity for which there is a deadline, we calculate the remaining time so as to raise attention of the learner towards these activities.

Conclusion

Pedagogical scenarios raise great interest within the Technology Enhanced Learning field. They can be at the centre of the engineering of the courses and are a means to define the use of tools and learning objects during the course as well as the tasks in which people are involved. Introducing pedagogical scenarios and activities in Learning Management Systems must be handled at both organisational level and technical level. Multiple actors and phases are implied in the definition and operation of scenarios. The pedagogical engineer is a key actor since he has both pedagogical and technical competencies and can bridge the gap between the two worlds. Indeed, the production of pedagogical scenarios is not an easy task as no clear methodology exists at the moment. For this reason, initiatives such as the UNFOLD European project are created to support communities of interest and share knowledge (see http://www.unfold-project.net/UNFOLD). To lower the cost of this production, the scenarios must have a high reusability and be easily adapted. For this reason the scenario must not be too much tied to actual learning objects and tools which should be chosen at deployment and/or instantiation time according to the platform and learners. Continuous adaptation of the learning scenarios (i.e., during enactment of the scenario) is also important to provide the best result for a specific group of learners.

Starting from the idea that executing a pedagogical scenario is not so far from executing a process on a Workflow engine, we have developed a flexible Workflow engine suited to the flexible execution of pedagogical scenarios. Based on a sample course in physics, we show how we can model a pedagogical scenario using a modified version of the standard XML Process Definition Language. This modified version enables the modelling of individual and group work within a process and the definition of time constraints. The engine allows the modification of the model at run-time for smooth adaptation and keeps track of the modified models for easy reuse of the enhanced models.

Having a definition of a pedagogical scenario with the different activities to perform can be used to provide support for the learners in their planning and to understand how they will reach a pedagogical objective of a unit of study. For this, we have developed an Interactive Study Guide, a user interface which helps tutors and learners plan their work with the help of indicators of the advancement within a unit of study and relative to the group.

Since the beginning of this work, IMS-LD has emerged as a standard language for describing pedagogical scenarios. Recently a first engine, CopperCore (CopperCore), has become available to interpret IMS-LD models and some platforms like EduPlone (http://eduplone.net) are starting to support it. This language seems more comprehensible by pedagogical engineer to formalize the scenarios. However, IMS-LD suffers some deficiencies like the scheduling model which is almost sequential [START_REF] Caeiro | A Critical Analysis of IMS Learning Design[END_REF] or the missing management of the data-flow. We aim now to use the experience we have gained on Workflow systems to support in a flexible way an augmented version of IMS-LD.

Figure 2 .

 2 Figure 2. Roles and types of responsibility

Figure 3 .

 3 Figure3. XPDL Meta-model from[START_REF] Wfmc | Workflow Process Definition Interface -XML Process Definition Language -version 1.0[END_REF]

Figure 4 .

 4 Figure 4. Pedagogical scenario model

Figure 5 .

 5 XML model for the course 131 Collaborative activities

  <Strategy id="sendMail"> </EndDeadLine> </Deadline> Figure 7. XML model of deadline constraint

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B. Favier A Gay

G Verhille

Characterisation of flexible fibre deformations in turbulence

Keywords: 47, 27, -i -Turbulent flows 47, 55, Kf -Particle-laden flows 47, 57, Ng -Polymers and polymer solutions

Characterisation of flexible fibre deformations in turbulence

Introduction. Particle advection in turbulence is common in industrial processes, e.g. papermaking [1], and in natural phenomena with the transport of plankton in ocean [2] or the dispersion of pollen species in the atmosphere [3]. The Lagrangian dynamics of particles was extensively studied, exploring the properties of isotropic or anisotropic rigid objects [4,5]. Recent studies highlighted the importance of particle shape on the transport. For instance, the rotation rate of small axisymmetric anisotropic particles depends on their aspect ratio [6,7]. One can then address the question of the transport of flexible objects, such as plankton in the ocean or pulp fibres in the papermaking industry, for which the shape may evolve with time depending on the external stresses. Several studies considered the case of flexible fibres at low Reynolds numbers [8,9] but very little is known about the motion of such objects when the flow is turbulent.

Inspired by flexible objects in nature or industrial processes which can often be considered as one-dimensional elongated objects, this study is focused on flexible fibres in Homogeneous and Isotropic Turbulence (HIT). Their dynamics should be determined by their time-dependent shape which is related to their deformations due to the turbulent viscous stresses. This letter is focused on the characterisation of the fibre deformations which is a first step to understand their dynamics. As the fibre extensibility and torsion are second order effects with respect to the bending [10], we only consider the bending deformations.

(a) E-mail: gautier.verhille@irphe.univ-mrs.fr Following Kirchhoff-Love theory and Euler-Bernoulli elasticity, their dynamics is then given by [START_REF] Antman | Nonlinear Problems of Elasticity[END_REF][START_REF] Lindner | Elastic fibers in flows in Fluid-Structure Interactions in Low-Reynolds-Number Flows[END_REF][13]

σ∂ 2 t r -∂ s (T ∂ s r) + B∂ 4 s r = ξ , (1)
where r is the position of the fibre centre line, ξ an external forcing and s the curvilinear coordinate. σ and B are, respectively, the linear density and the bending modulus of the fibre, all assumed to be constant along the fibre. T is a tension term acting as a Lagrange multiplier in order to ensure the fibre inextensibility (|∂ s r| = 1). It is expected that this system shares some similarities with worm-like chain polymers as their idealised equations of motion are similar [14,15]. The crucial difference between polymers in a thermal bath and fibres in a turbulent flow lies in the nature of the linear forcing term ξ. Assuming that the particle Reynolds number is small and ignoring the anisotropic nature of the viscous drag for slender bodies, we have ξ 4πη(ln 2λ) -1 u slip [START_REF] Guazzelli | A physical introduction to suspension dynamics[END_REF] where η is the fluid dynamical viscosity, λ = L/d the aspect ratio between the fibre length L and its diameter d, and u slip = u -∂ t r the slip velocity between the fluid and the fibre. For the polymer regime, we assume that the forcing is effectively delta correlated in space and time

u(x + s, t + τ ) • u(x, t) = u 2 rms δ(τ )δ(s) , (2)
where • corresponds to an ensemble average. For stationary HIT, the fluid velocity u is spatially and temporally correlated in the inertial range. This result is at the origin of the classical Kolmogorov 4/5 law [START_REF] Frisch | Turbulence[END_REF]:

[(u(x + χ) -u(x)) • e χ ] 3 = - 4 5 χ, (3)
where is the dissipation rate of the turbulent kinetic energy per unit mass and e χ = χ/|χ|. Using the integral length scale L I and the root mean square velocity of the turbulence U as references, a dimensionless version of equation ( 1) is

∂ 2 t r -∂ s (T ∂ s r) + γ∂ 4 s r + 1 St ∂ t r = 1 St u . ( 4
)
The fibre dynamics depends on two dimensionless parameters: a Stokes number

St = σU/ 4πη(ln(2λ)) -1 L I , (5)
which compares the inertial term σ∂ tt r and the viscous term 4πη(ln 2λ) -1 ∂ t r and a dimensionless rigidity γ which compares the inertial term and the bending term B∂ 4 s r:

γ = B/ σU 2 L 2 I . (6)
In polymer science, the deformations of worm-like chain polymers are classically characterised by the persistence length p defined by C( ) ≡ t(s + ) • t(s) = e -/ p where t = ∂ s r/|∂ s r| is the unit tangent vector. This length is generally related to a balance between the elastic energy and the thermal energy B/ p ∼ k B T where k B is the Boltzmann constant and T the bath temperature [START_REF] Yamakawa | Modern Theory of Polymer Solutions[END_REF]. By measuring the distance between fibre extremities, Brouzet et al. [START_REF] Brouzet | [END_REF] showed that the same approach may be applied to fibres in turbulence. However, in the inertial range, the persistence length cannot be modelled from an energy balance since the fibre dynamics and the forcing occur on different timescales. Thus, they used a power balance to define a new characteristic length called the elastic length

e = B 1/4 /(ρ f η ) 1/8 , (7)
where ρ f is the fluid density. The transition from rigid to flexible fibres is then governed by the ratio of the fibre length L to the elastic length e : for L < e (resp. L > e ) the fibre deformations are weak (resp. large), i.e. the averaged distance between the fibre extremities is nearly equal to (resp. much smaller than) the fibre length L. To investigate this analogy, Verhille and Bartoli [20] performed a 3D experimental reconstruction analysis to measure the correlation function C( ). They observed that the correlation function is well described by an exponential law C( ) ∼ e -/ p for > e . However, in turbulence, and contrary to polymers, the persistence length p depends on the fibre length L itself and cannot be assimilated to the elastic length. The main objective of this letter is to understand this phenomenon by investigating the role of the spatial correlations of the forcing.

Several regimes are expected depending on the ratio between the fibre length scales (its length L and elastic length e ) and the flow length scales (the integral scale L I and the Kolmogorov scale η K ). In the following, we assume that all fibre scales are much larger than the Kolmogorov scale and that fibres are flexible (L > e ). Therefore, three different regimes can be distinguished (see fig. 1). In a first regime, the integral scale is much smaller than the elastic length (L > e L I ), so that the forcing is effectively uncorrelated at the scale of the fibre deformations (neglecting for simplicity spatial correlations at scales larger than L I ). This is called the polymer regime, for which the turbulent fluctuations can be seen as a thermal bath since it is only correlated at scales much smaller than the relevant elastic scale. Note that physically the elastic length e , derived in the inertial range, is not pertinent for this regime and should be replaced by a persistence length p as done in polymer science [START_REF] Yamakawa | Modern Theory of Polymer Solutions[END_REF]. In a second case, the fibre length lies in the inertial range (L I L > e ) and the forcing is now correlated along the fibre. This case is called the turbulent regime. Finally the last case is an intermediate regime (L > L I > e ) for which the fibre experiences a forcing correlated at small scale but uncorrelated at the larger scale. Both turbulent and intermediate regimes were largely unexplored up to now. In this letter, we investigate experimentally and numerically the evolution of the local curvature κ(s) = |∂ s t(s)| of a flexible but inextensible fibre. It is natural to focus on the local curvature since it gives access to the elastic energy stored in the fibre

E B = (1/2)B κ 2 (s)ds and is also related to the correla- tion function C( ) 1 -κ 2 2 /2 + O( 3 ).
Materials and methods. We now describe our experimental setup and numerical model. Experimentally, the turbulence is generated within a von Kármán setup similar to the one used in our previous experiments [20] (see fig. 2). It is a water-filled cubic tank (side length 20 cm) with two impellers (R = 8.6 cm in radius fitted with blades) Since the turbulent regime cannot be achieved experimentally using our current setup, we also rely on an idealised numerical model. For the elastic part of the problem, we solve equation (4) using a numerical scheme directly inspired from previous studies on flexible fibres [22][23][24]. Spatial derivatives are approximated using sixth-order finite differences on a staggered grid and we use a semi-implicit backward difference temporal scheme of third order [25], the bending term being solved implicitly while the other terms are solved explicitly. We typically use between 128 and 2048 grid points regularly distributed along the fibre. The inexensibility of the fibre leads to a Poisson-type equation on the tension T , which is solved at each time step. We have checked that the relative error on the fibre length is typically smaller than 10 -5 for all cases discussed here. Concerning the fluid flow, and contrary to previous methods considering Stokes flows [22] or using Immersed Boundary Methods [26], our fluid velocity u is obtained analytically as a superposition of incompressible random Fourier modes. This method, often called Kinematic Simulation [27], has a long history [28] and was used in various Lagrangian studies [29,30]. Although the flow is not a solution of the Navier-Stokes equations, it is rigorously incompressible and its secondorder statistics can be finely tuned to mimic those of realistic turbulence. All the results shown in this paper are obtained using 10 3 incompressible Fourier modes whose two-point correlations are derived from an energy spectrum given by E(k) = C k 2/3 k -5/3 and using a typical eddy turnover time ω(k) = 1/2 k 3 E(k) [31]. In all cases, our synthetic turbulent flow is defined over three decades of wave numbers, thus ensuring ample scale separation between the integral and Kolmogorov scales. For simplicity, we neglect correlations at scales larger than the integral scale, thus idealising the transition between the turbulent and polymer regimes. Note that the results do not qualitatively depend on these particular choices, only the spatial correlations and therefore the amplitude and slope of the energy spectrum matter. As mentioned in the introduction, the viscous forcing is assumed isotropic (which is not the case for a slender body, see [10,32]). We have checked that this assumption does not change qualitatively the different results presented here. Note that solving the full two-way coupling between the elastic fibre and the turbulent flow using Direct Numerical Simulations (DNS) would be a tremendous task, which we leave for future works, whereas our simple approach allows us to consider many realisations of the same input parameters. Finally, the dimensionless elastic length can be defined as:

( e /L I ) 8 = (σ/ρ f L 2 I )γ 2 St ≈ γ 2 St , (8)
where the prefactor σ/ρ f L 2 I is constant and of order unity for all the numerical simulations discussed below (the fibre is assumed to have the same density of the fluid and in the experiment (σ/ρ f L 2 I ) 1/8 ∼ 0.3 justifying the approximation made here). We used St = 0.1 and different fibre rigidities γ from 10 -6 to 10 6 , noting that different values of the Stokes number do not qualitatively change the results discussed in this letter. In the following, we particularly focus on three cases: γ = 10 3 which corresponds to the polymer regime ( e /L I ≈ 4.2), γ = 0.2 which corresponds to the intermediate regime ( e /L I ≈ 0.5) and γ = 10 -5 which corresponds to the turbulent regime ( e /L I ≈ 0.042). All the numerical results are obtained after time averaging during the quasi-steady state over tens of turnover times and over 10 2 independent realisations. An example of a fibre conformation from a numerical simulation in the turbulent regime can be seen in Fig. 2.

Results. We first discuss the evolution of the mean squared distance between the fibre extremities R e 2 = |r(s = 0)r(s = L)| 2 . Within the worm-like chain assumption, this quantity is related to the fibre length L and the persistence length p [START_REF] Yamakawa | Modern Theory of Polymer Solutions[END_REF]:

R 2 e = 2 2 p -2L p 1 -e -L/ p . ( 9
)
Figure 3 represents the evolution of the mean squared distance between the fibre extremities R 2 e as a function of the fibre length L both normalised by the persistence length, determined by fitting expression (9) to our data. We can observe that all the experimental and numerical points are very close to the worm-like chain prediction, as already observed by [START_REF] Brouzet | [END_REF]. However, for the longest fibres in the turbulent regime, equation ( 9) tends to underestimate the values observed in our numerical model. This indicates that the forcing correlations play a role in the deformations. Our observations show that the global quantity R e is not sensitive enough to investigate the influence of the correlations of the forcing, as mentioned by [20]. This is why we now focus on the evolution of the mean curvature as a function of the fibre length in the different regimes. Since the boundary conditions impose a curvature κ = 0 at the fibre extremities, we compute the mean value of the curvature as:

κ = 1/(L -2 e )
Le e κ(s) ds .

(

) 10
This allows to remove effects from fibre extremities where the curvature rapidly increases (see fig. 4). Otherwise, for short rigid fibres with L < e , the mean value is taken as the maximum of the mean curvature along the fibre, i.e. at the center of the fibre s = L/2. This quantity is represented in fig. 3 for both experiments and numerical simulations as a function of the fibre length normalised by the elastic length e defined by the power budget. In the polymer regime ( , γ = 10 3 , e /L I ≈ 4.2), the numerical results are compatible with worm-like chain prediction. The mean curvature increases with the fibre length until it saturates to a constant value independent of the fibre length. The transition length between the two tendencies is nearly the persistence length p independent of the fibre length L as it is defined in polymer science. For the turbulent regime ( , γ = 10 curvature increases with the fibre length L up to a maximum after which it decreases. This effect is also observed in the experimental data (•) which are in the intermediate regime. This shows that the forcing correlation is at the origin of a surprising result: long fibres are statistically less curved than shorter ones. As will be demonstrated below, this unexpected straightening of flexible fibres as their length increases is due to two physical ingredients: inextensibility and cumulative viscous effects.

In the following, we derive a scaling to describe more quantitatively this decrease thanks to a power budget similar to the one used to determine the elastic length e [START_REF] Brouzet | [END_REF]. For the derivation of the elastic length, an eddy with a size comparable to the fibre length is considered. Here, the deformation is local meaning that κL 1 and, hence, the only characteristic length remaining is the characteristic size of the deformation κ -1 . Therefore in the power budget the turbulent power available scales as P turb ∼ ρ f κ -3 . As the fibre is inextensible, the local deformation at the curvilinear coordinate s is accompanied by a global rearrangement of the fibre conformation. This rearrangement is at the origin of an additional viscous dissipation. For an infinite fibre, two scenarios are possible. For the first one, the creation of the deformation of size κ -1 is fast, meaning that it evolves on a timescale smaller than the bending relaxation time of a fibre of length s. In that case, the global conformation can be considered as frozen and the fibre displacement induced by the local deformation is everywhere parallel to the tangent vector. In the second case, the creation of the deformation of size κ -1 is slow, then the global conformation can change and a sweeping phenomenon may occur. The characteristic timescale of the formation of the deformation κ is given by τ v (κ -1 ) ∼ ηκ -4 /B and the one of the global rearrangement is given by τ v (s) ∼ ηs 4 /B [10]. Then, for curvature high enough (such that τ v (κ -1 ) τ v (s)), the conformation can indeed be considered as frozen. For fibre with finite length the reconfiguration should occur on the small-est part of the fibre to minimise the dissipation ([0 : s] if s < L/2 or [s : L] if s > L/2). For a deformation occurring at a curvilinear coordinate s (s < L/2), the average viscous dissipation P v due to the global rearrangement can be written as

P v (s) = s 0 ηu slip • ∂ t r(s , t)) ds , (11)
where ∂ t r(s , t)) = (∂ t r(s , t) • t) t is the tangent velocity of the center line. Given that the fibre is inextensible, the norm of the tangent velocity | ∂ t r(s , t)) || | is constant along the fibre and scales with 1/(κτ v (κ -1 ). Thus, the average dissipated power can be rewritten as

P v (s) η κτ v s 0 u (s )ds , (12)
where u = u slip •t. In a turbulent flow, it was shown for a sphere of diameter d in the inertial range that the slipping velocity scales as ( d) 1/3 [33]. By analogy, the integral (12) can be approximated by u s s where u s ∼ ( s) 1/3 is the typical velocity of an eddy of size s. The contribution of the smallest eddies cancels out as they are not correlated over the distance s. Hence, the dissipative power at scale s scales as P v ∼ η/(κτ v )su s . So for long fibres and s e , the turbulent power P turb = ρ f κ -3 , is balanced by the cumulative viscous power P v , the bending power being negligible. Thus, the curvature κ(s) should scale as

κ(s) ∝ s -2/9 . (13)
Note that even though equation ( 1) is local, the cumulative effect of the viscous drag along the fibre is present through the tension term ∂ s (T ∂ s r) which ensures the fibre inextensibility.

For the polymer regime where the elastic length is larger than the integral scale, this additional term also exists. However, the slipping velocity is here a random delta correlated variable. So, the integral [START_REF] Lindner | Elastic fibers in flows in Fluid-Structure Interactions in Low-Reynolds-Number Flows[END_REF] vanishes and only the bending term in the power balance remains. ). The power law in the turbulent regime corresponds to equation [START_REF] Guazzelli | A physical introduction to suspension dynamics[END_REF].

The power law in the polymer regime corresponds to a balance between thermal and bending energies.

The influence of this additional friction term is highlighted in fig. 4 which represents, for the three regimes, the evolution of the local mean curvature κ(s) as a function of the curvilinear coordinate s from the numerical model only. While the experimental data also shows a global straightening of the fibres as their lengths increases, equation ( 13) is only valid when there is clear scale separation between the fibre length and the integral scale which is not the case experimentally. The boundary conditions (no torque and no force at the extremities) impose that the curvature is zero at the extremities. For all the regimes, the curvature increases with the curvilinear coordinate from the extremities until it reaches a maximum for s ∼ e . In the turbulent case (fig. 4 on the right), for fibres long enough, the mean curvature decreases for e s L/2 due to the additional dissipation term that dominates the bending term in the power budget. The scaling proposed in equation ( 13) is in good agreement for the longest fibre L = 23.71 L I . For the polymer regime, the delta correlation of the forcing leads to no additional dissipation and the mean local curvature is constant along the as shown in fig. 4 on the left. This behaviour is similar to the one observed for worm-like chain polymers in ideal solvent [34]. For the intermediate case (fig. 4 in the middle), the evolution of the local curvature is a mix of these two regimes. For long fibres, the mean local curvature decreases for e s L I and then saturates as the forcing is no more correlated at larger scales.

We finally focus on the maximal amplitude of the mean local curvature κmax that can be reached for fibres with a length L greater than e . We notice in fig. 3 that the elastic length e properly estimates the typical length at which the maximum of the curvature is reached. However, the amplitude of this maximum does not scale with the elastic length. In fig. 5, we report the evolution of the maximum of the dimensionless mean curvature κmax L I as a function of the dimensionless rigidity γ. We varied numerically γ from 10 -6 to 10 6 , leading to e /L I ∈ [0.02, 23.7] in order to continuously transit from the turbulent regime to the polymer regime. As expected, two different power laws are observed, one for each asymptotic regime. We first analyse the turbulent regime. We have seen that the fibre inextensibility, and so the lineic tension along the fiber, plays a key role to understand the decrease of the curvature profile for long fibres. We now show that the fibre inextensibility also determines the maximum of the amplitude of the curvature. The tension term ensuring the fibre inextensibility in equation ( 1) is actually composed of two terms [13]:

∂ s (T ∂ s r) = ∂ s (T t -Bκ 2 )∂ s r , (14)
where T t is the classical tangential force whereas Bκ 2 is necessary to satisfy the inextensibility constraint when the fibre is curved. The term ∂ s Bκ 2 ∂ s r has one component parallel to the tangent vector, B∂ s (κ 2 )∂ s r, and one parallel to the curvature vector, Bκ 3 e κ , where e κ = ∂ 2 s r/|∂ 2 s r|. This last term is parallel to the classical bending term B∂ 4 s r and may be larger than the linear term for large amplitudes, typically κ > 1/ e . In that case, the saturation occurs when the nonlinear term Bκ 3 is of the order of the forcing term ηu ∼ η( ) 1/3 . The length scale to consider corresponds to the size of the eddy that is responsible for the deformation at s = e , i.e. e . Thus the following equilibrium can be written

Bκ 3 max ∼ η( e ) 1/3 (15)
Using equation (7), the maximum of curvature should scale as:

κmax ∼ γ -11/36 St -23/72 ∼ γ -0.3 St -0.3 . (16)
This power law is in good agreement with our observation as shown in fig. 5. For the polymer regime, the forcing is not correlated at the scale of the deformation. Hence, the amplitude of the curvature is expected to be weak, as observed in fig. 3. The saturation is then expected to be related to the balance between the "thermal energy" ρ f (L/L I )L 3 I U 2 with the bending energy BLκ 2 leading to κmax ∼ γ -1/2 . The "thermal energy" ρ f (L/L I )L 3 I U 2 is the sum of the energy brought by each eddy of size L I along the fibre. This scaling seems compatible for the largest values of γ we have tested (γ > 10 4 ). Note that the case presented in fig. 5 is not in this asymptotic regime yet. This is due to the scale separation between the elastic length and the integral scale which is not large enough, as mentioned previously. Note also that in polymer science the elastic energy stored in the fibre is generally related to the persistence length: E el = Bκ 2 ds ∼ B/ p and the persistence length to the fibre curvature p ∼ κ-1 [35], should lead to κ ∼ γ -1 . Understanding the discrepancy between this simple argument and our measurement is beyond the scope of this paper and will be investigated later.

Conclusion.

We performed a detailed study focusing on the mean curvature of deformable fibres in turbulent flows. Three different regimes were identified: the polymer regime (L I e < L), the intermediate regime ( e L I < L) and the turbulent regime ( e < L < L I ). We showed that the differences between these regimes are related to the temporal and spatial correlations of the turbulence. In a first regime, a fibre in a turbulent flow can be analogous to a polymer in a thermal bath, leading to a maximum curvature independent of the fibre length. In the turbulent regime however, which is the main focus of this paper, we observe a straightening of the fibre as the mean curvature decreases when the fibre length increases, an effect ultimately due to a combination of the fiber inextensibility and the spatial correlation of the forcing through the cumulative viscous forces along the fibre. Finally, the intermediate regime is a combination of these two previous cases in which the flow is correlated at small scales and uncorrelated for the largest deformations.

In the future, the importance of the fibre inertia, measured by the Stokes number St here, needs to be addressed. Due to the many simplifying assumptions used to derive the numerical model used here, further comparative studies using DNS of the Navier-Stokes equations, fully coupled with the elastic dynamics of the fibre, are a natural next step. Improving the experimental apparatus in order to reach the turbulent regime, by using a more flexible material for the fibres and a larger tank, would also be valuable. Finally, the statistics of the deformation should be investigated further to see how the different quantities discussed in this letter fluctuate around their mean values. These fluctuations can have a major impact on the transport of particles, especially for brittle objects for which large curvatures may be responsible for their fragmentation. Once the deformations are fully characterised in each regime, studying the fibre dynamics is the next step to model the transport of flexible objects in turbulence.

Fig. 1 :

 1 Fig. 1: Schematic comparing the fibre length L with the different length scales in the problem from largest fibres behaving like polymers (L LI ) down to small fibres in the Stokes regime (L ηK ). ηK is the Kolmogorov length scale and LI the integral scale of the homogeneous isotropic turbulent flow. We focus on a new turbulent regime where ηK L LI . The vertical axis shows the fibre length versus l d , the typical length above which elastic deformations become significant. In the turbulent regime of interest here, l d = e as discussed in the text. The regimes covered by both experiments and numerical models are qualitatively indicated by the dashed boxes.

Fig. 2 :

 2 Fig. 2: Left: image of the experimental setup. Middle: example of 3D reconstruction of a fibre conformation from experimental data, the colour codes the curvature κ(s). Right: example of a fibre conformation from a numerical simulation in the turbulent regime (the fibre length is smaller than the integral scale), the colour shows the local value of the curvature κ(s) normalised by the integral length scale LI .

Fig. 3 :

 3 Fig. 3: Top: Evolution of the mean squared distance between fibre extremities R 2 e as a function of the fibre length L, both quantities being normalised by the persistence length p. The dot-dashed line represents the worm-like chain prediction and the dashed line the evolution for rigid fibres. Bottom: Evolution of the fibre mean curvature κ as a function of the fibre length L both normalised by e for the experiments (•, the error bars indicate the uncertainties associated with the fitting process from the 3D reconstructed points), the turbulent regime ( , γ = 10 -5 and e/LI = 4.3×10 -2 ), the intermediate regime ( , γ = 0.2 and e/LI = 0.5) and the polymer regime ( , γ = 10 3 and e/LI = 4.5).

Fig. 4 :

 4 Fig. 4: Evolution of the local mean curvature κ(s) for fibres of increasing length L as a function of the normalised curvilinear coordinate s/L. From left to right, the polymer regime ( e ≈ 4.2LI ), intermediate regime ( e ≈ 0.5LI ) and turbulent regime ( e ≈ 0.042LI ). The fiber lengths L/ e for each curve are given in the legends. The dashed line in the third graph represents the scaling law (13).

6 Fig. 5 :

 65 Fig. 5: Evolution of the maximum of the mean curvature κmax as a function of the fibre stiffness γ. The grey area corresponds roughly to the intermediate regime ( e/LI ∈ [10 -1 , 10]). The power law in the turbulent regime corresponds to equation[START_REF] Guazzelli | A physical introduction to suspension dynamics[END_REF]. The power law in the polymer regime corresponds to a balance between thermal and bending energies.

  -5 , e /L I ≈ 0.042) and the intermediate one ( , γ = 0.2, e /L I ≈ 0.5), the mean

				Polymer regime γ = 10 3						Intermediate regime γ = 0.2					Turbulent regime γ = 10 -5
		0.04								1.6
										1.4										25
		0.03								1.2										20
										1
	κ LI	0 0.01 0.02	0	0.2	0.4	0.6 L/le = 0.47 s/L 1.19 2.37 4.74 7.11 9.48 18.97 37.94	0.8	1	κ LI	0 0.2 0.4 0.6 0.8	0	0.2	0.4	s/L	0.6	1 2 3 4 6 8 16 20 30	0.8	1	κ LI	15 10 5 0	0	0.2	0.4	s/L	0.6	0.47 1.19 1.78 2.37 3.56 4.74 7.11 11.86 23.71	0.8	1

This work was carried out in the framework of FlexFiT Project (ANR-17-CE30-0005-01) funded by the French National Research Agency (ANR), the Labex MEC Project (No. ANR-10-LABX-0092) and of the A*MIDEX Project (No. ANR-11-IDEX-0001-02), funded by the Investissements d'Avenir French Government program managed by the French National Research Agency (ANR).