Alexis Evain
email: alexis.evain@mines-albir.fr

Imt Mines

Albi Aurélie Congès

Frederick Benaben
email: frederick.benaben@mines-albi.fr

Management of crisis exercises in Virtual Reality

Keywords: Virtual Reality, training, teaching, collaboration K.6.1 [Management of Computing and Information Systems]: Project and People Management-Life Cycle; K.7.m [The Computing Profession]: Miscellaneous-Ethics

When we consider the question of crisis management, training and practising exercises are regularly requested to be ready for any possible crisis. Unfortunately, these exercises come with many disadvantages, especially the cost, the complexity of implementation, the lack of replicability, and agility. The project aims to create customizable Virtual Reality simulations in existing environments. In this article, we detail our approach dedicated to using VR for a fire in a University Research Lab involving a game master managing the scenario, five people, interacting and acting conjunctly within this environment, and several coordinators the rescue process.

INTRODUCTION

In crisis management, training and preparation are essential: "exercises are a fundamental part of building the necessary capacity and required capabilities to effectively deal with information during and after disasters" [START_REF] Meesters | Information Management in Largescale Disaster Exercises: An Integrated Perspective[END_REF]. Training exercises are often mandatory in critical sites, which must set up exercises several times a year, depending on the legislation and the site, to be ready to face any kind of crisis. In this article, "critical sites" refers to sites in which the users' life can be endangered by dysfunctions or incidents. However, despite the technological advances, crisis management training continues to be "paper-based, collective group dynamic exercises" [START_REF] Bacon | Developing a smart environment for crisis management training[END_REF], which presents limitations.

Paper-based or classroom exercises even for table exercises have little retention potential: as presented in [START_REF] Wood | Problem-Based Learning: Exploiting Knowledge of how People Learn to Promote Effective Learning[END_REF], the average retention rate of information following teaching is only 5% when students just listen, and 10% when they read. On the contrary, the retention rate of information is much higher if students are active rather than passive: 75% of information is retained when trainees practise by doing. However, simulation-based exercises in real size also have drawbacks. As explained in [START_REF] Conges | On the usage of Virtual Reality for Crisis Management exercises in Critical Industrial Sites[END_REF], exercises are expensive, they require a lot of time and resources, and they lack realism and flexibility.

Mobilizing actors for exercises is difficult, it is hard to predict their implications and how impacted they will be. Immobilizing a site for the duration of the exercise is costly, and thus it cannot be done often enough to cover any type of crisis that might occur. As explained in [START_REF] Ardila | LVC Training Environment for Strategic and Tactical Emergency Operations[END_REF], "traditional training simulated-based systems are expensive and offer little flexibility", and "real training in the field of emergency management is very expensive and complicated regarding the harmonization of procedures between agencies". Consequently, the frequency and coverage of the crisis scenario are not high enough. Besides, the realism of the scenario is hard to get, as, for instance, it is not possible to burn a building every time we want to practice putting out a fire.

To overpass those limitations, virtual exercises are already used in the aviation, military, or medical fields [START_REF] Kwok | A Virtual Collaborative Simulation-based Training System[END_REF]. Those virtual exercises can be performed through keyboard and mouse, controllers, or immersive Virtual Reality, and they allow students to gain skills through a safe and guided practice [START_REF] Lateef | Simulation-based learning: Just like the real thing[END_REF]. Based on those observations, this research project proposes to use Virtual Reality to build a training platform dedicated to crisis management. This article is structured as follows: first, an overview of the use of Virtual Reality as a crisis management training tool. Then, the presentation of the project Virtual Reality training platform and its architecture, as well as a use case showing how it can be used and an evaluation of the benefits. Finally, the presentation of the project's next steps and perspectives.

CONTEXT

According to the Oxford dictionary, learning is "the acquisition of knowledge or skills through study, experience, or being taught" [START_REF]Learning | Definition of Learning by Oxford Dictionary on Lexico.com also meaning of Learning[END_REF]. In [START_REF] Gerard | Savoir… oui mais encore ?[END_REF], Gerard proposes a decomposition of knowledge in three activities: know how to reproduce (being able to reproduce or repeat a message, a gesture, an action, without transformation and out of any context), know how to do (being able to perform a learned action in a new and appropriate context), and know-how to be (being able to apprehend situations and other actors and adapt one's actions and reactions).

We propose to add a first and more basic step to those activities: know (mastering the theory before trying to repeat a message or an action). Before he can operate, a surgeon needs to master several skills. First, he needs to gain knowledge of the human body and the organs he will have to operate (know). Then, he needs to master the necessary actions to operate: how to use a scalpel, how to sew skin, etc. (know how to reproduce). He also needs to know the order in which he has to perform the different tasks during the operation: the process he must follow (know how to do). And finally, the surgeon needs to know how to work with the people present in the operating room with him: nurses, anaesthetists, other doctors, etc. That is the collaborative aspect (know how to be). Those four steps (knowledge, action, process, and collaboration) will constitute our decomposition of learning levels, that we will use in this article. To master those steps, class-room exercises are not enough, and trainees need to practice in a realistic situation. Virtual Reality presents several characteristics that make it a great tool for learning and practising within those four steps and offer solutions to the drawbacks of classical exercises. In this section, we are going to explain why. Then, we will see how those characteristics can be used to improve training in crisis management.

Virtual Reality: a great learning tool

Virtual Reality can help trainees gain theoretical knowledge thanks to its ability to faithfully represent things in 3D on the one hand, and the other hand to its ability to "extend the user's capability to feel data that would normally be beyond the range of their sensed or experiences" [START_REF] Mikropoulos | Educational virtual environments: A ten-year review of empirical research (1999-2009[END_REF]. As stated in [START_REF] Lester | Pathfinder Linden's guide to getting started in Second Life[END_REF], if you want to teach biology, there is no point in building a virtual classroom if you can build a realistic and interactive human cell. Virtual Reality allows scalability, which gives "the users the ability to 'change' their physical size so that they can navigate and interact in macro and micro worlds" [START_REF] Mikropoulos | Educational virtual environments: A ten-year review of empirical research (1999-2009[END_REF]. This possibility can also be used in data visualization or analysis, which can also help to gain theoretical knowledge. For instance, the AR bone puzzle [START_REF] Stefan | An AR edutainment system supporting bone anatomy learning[END_REF] uses augmented reality to help medical students to learn human bone anatomy. The system uses interactions based on intuitive gestures (grasping, moving, releasing) and the 3D aspects offered by augmented reality to help students to learn. Even if that particular game uses augmented reality, it is based on features that also exist in Virtual Reality. This could also bring more possibilities to help students visualize and manipulate objects to better understand how they are made or how they work. The VR Anatomy System [START_REF] Falah | Virtual Reality medical training system for anatomy education[END_REF] is another example of an educational tool in Virtual Reality. This system aims at teaching medical students the anatomy and functioning of a human heart through a realistic and interactive 3D model and was positively evaluated by medical professionals.

The practical knowledge (action) is probably the one that benefits the most from the advantages of Virtual Reality. According to [START_REF] Gallerati | The Use of Virtual Reality and Simulators for Emergency Management Training Increases Employees' Role Awareness, Commitment and Information Retention, While Helping Companies to Stay Compliant[END_REF], "Virtual Reality is a three-dimensional user interface in which the user can perform actions and experience their consequences". Immersion and interaction with the environment are key features of Virtual Reality that allows trainees to practice specific gestures and actions. As stated in the introduction, the retention potential for students is much higher when they can practise by doing, and Virtual Reality offers that possibility: "interactions are central to all Virtual Reality training simulations. They are what define the user experience and allow the simulation to be an effective training tool" [START_REF] Ruthenbeck | Virtual reality for medical training: the state-of-the-art[END_REF]. This possibility to interact with the environment, along with the feeling of immersion induced by Virtual Reality, allow a bigger involvement of the trainees, which are then more committed to the exercise and abler to learn, as shown in [START_REF] Mikropoulos | Educational virtual environments: A ten-year review of empirical research (1999-2009[END_REF]. Practising gestures in Virtual Reality can for instance be used in sports: ballet dancers can 1 https://immersivefactory.com/ practice movements and receive feedbacks [START_REF] Kyan | An Approach to Ballet Dance Training through MS Kinect and Visualization in a CAVE Virtual Reality Environment[END_REF]. In the medical field, [START_REF] Heng | A Virtual-Reality Training System for Knee Arthroscopic Surgery[END_REF] proposes a system to train surgeons for knee arthroscopic surgery. This system provides realistic visual and haptic feedback that allows students to train efficiently for this surgery, and thus limit the potential damages done to the patient.

One of the weaknesses of classical exercises is that they are not frequent enough for trainees to be prepared for any kind of crisis. Not all the conditions or possibilities can be recreated or studied in a classroom exercise, and they lack realism. For instance, "the acquisition of clinical skills by medical trainees can be haphazard, relying on the chance presentation of patients with specific diseases or surgical conditions" [START_REF] Ruthenbeck | Virtual reality for medical training: the state-of-the-art[END_REF]. Virtual Reality allows us to create exercises representing a wide range of conditions, with "customized learning, challenging environment, and multi-sensory effects" [START_REF] Holopainen | Exploring the Learning Outcomes with Various Technologies -Proposing Design Principles for Virtual Reality Learning Environments[END_REF]. The immersion and involvement of users make the exercises more realistic and impactful. Besides, it is possible to create a consistent scenario that can be repeated as often as needed [START_REF] Dukes | Development of a scenario builder tool for scaffolded virtual patients[END_REF], [START_REF] Izard | Virtual Reality as an Educational and Training Tool for Medicine[END_REF] with no further costs. If it is not possible to burn down a building to practice fire management, "for training and education purposes, Virtual Reality allows trainees to practice cognitive, and sometimes physical, skills in simulation as risks to safety and material costs can be much lower in Virtual Reality than in physical training" [START_REF] Kang | Advancing Ethical Decision Making in Virtual Reality[END_REF]. This reduction of risks also allows learning by trial and error: "Players are encouraged to learn by trying different strategies, succeeding and failing as they would in an entertainment game" [START_REF] Saunders | Validating Virtual Reality as an Effective Training Medium in the Security Domain[END_REF]. This can be used to teach processes. Immersive Factory© 1 proposes Environments, Health, and Safety workshops in Virtual Reality where trainees must perform some tasks such as replacing a bulb in height or using a forklift truck. Mistakes and missed steps in the process cause injuries to the avatar, and the end of the training session. However, the user remains safe, and can thus learn from his mistakes and try new solutions without risks. Mimbus©2 proposes Virtual-Indus, a tool dedicated to teaching industrial procedures such as electrical intervention or maintenance procedures. Virtual-Indus offers a realistic environment in which it is possible to implement a scenario that would be too dangerous or too expensive in real life.

Several studies have proven that Virtual Reality has great potential for learning and practising. [START_REF] Heinrichs | Simulation for Team Training and Assessment: Case Studies of Online Training with Virtual Worlds[END_REF] findings prove that "virtual environments fulfil their promise of providing repeated practice opportunities in disperse locations with uncommon, lifethreatening trauma cases in a safe, reproducible, flexible setting" while being "more cost-effective than many other methods". Studies involving 16 surgical residents show that those training with Virtual Reality can perform operations 29% faster than those trained traditionally [START_REF] Seymour | Virtual Reality Training Improves Operating Room Performance[END_REF].

In crisis management, exercises, though mandatory, present several drawbacks. In the next section, we are going to see how those drawbacks can be addressed by Virtual Reality.

Virtual Reality for crisis management

In [START_REF] Heinrichs | Simulation for Team Training and Assessment: Case Studies of Online Training with Virtual Worlds[END_REF], Heinrichs sums up the advantages of virtual worlds training:

the context (layout, resources) can be exactly replicated, -trainees can participate in the exercises from different locations, -exercises can be conducted at any time without having to wait for specific conditions since multiple scenario and conditions can be recreated in the virtual environment, -scenarios can be repeated as many times as needed in a safe environment, allowing trainees to learn from their mistakes, -recording an exercise is possible, which enables analyses of trainees' performances.

Those advantages can be applied to crisis management, to improve the responders' preparation. In a crisis, decisions and actions have to be precise, quick, and effective in a stressful and urging context. This means that the crisis response's actors need to master skills perfectly, to be able to use them during the crisis. Thus they need to train in realistic environments "to ensure that these skills can be used in emergencies and while under stress" [START_REF] Meesters | Information Management in Largescale Disaster Exercises: An Integrated Perspective[END_REF]. Virtual Reality gives the possibility to train in the digital twin of a site, under specific chosen conditions, and to repeat a crisis scenario as many times as needed to master skills and be able to rely on them in a real crisis. Because of the digital aspect of Virtual Reality, it is also possible to record the training and thus to watch it and analyse it to improve.

Crisis management is an area in which there are risks to people, civil casualties as well as crisis responders. Simulation-based training offers the advantage to practice in realistic and safe conditions, where mistakes can be made without endangering people or materials. "The use of simulators can help a lot in communicating emergency procedures to a wide range of people, by challenging them in a variety of foreseeable crises without running the risk of exposing them to real risks before they are fully aware and trained, with the further benefit of achieving, through an immersive situation, better retention of information received" [START_REF] Gallerati | The Use of Virtual Reality and Simulators for Emergency Management Training Increases Employees' Role Awareness, Commitment and Information Retention, While Helping Companies to Stay Compliant[END_REF].

When a crisis occurs, several actors are involved in the response: firemen, policemen, medical doctors, sometimes some other professionals. Each of them has its way of communicating, its professional language, and it can be difficult for them to communicate and to work together in the most effective way, which means that the collaborative aspect in the four learning steps of Figure 1 is crucial in crisis management.

Based on the advantages of Virtual Reality as a training and learning tool, we want to propose a virtual environment dedicated to the training of crisis responders. In the next section, we are going to present our proposal, and how it will improve crisis management preparation by helping trainees gain knowledge about theory, practice, processes, and collaboration.

APPROACH

Introduction

To make this training in Virtual Reality we start to create one use case defined with basics intentions. The simulation had to include collaboration and dedicated exercise according to each trade. The first use case has been developed and proposed here [START_REF] Conges | On the usage of Virtual Reality for Crisis Management exercises in Critical Industrial Sites[END_REF]. It allowed people to choose between the roles of doctor, firefighter, and police officer, each with their mission. The environment was a metro where the fire did not spread as it went along, and injured people had to be found.

This simulation in a fictitious location was based on the principle of declaring a process for 3 users. Each had their dedicated tasks according to their roles and from start to finish had the same mission. They each had their tools from the beginning and therefore had only one interaction available apart from moving around.

For this use case, we use Unity©3 and like the previous, we made this choice by comparing several technologies. One of the most important points is to have a quick and easy Virtual Reality preview of the chosen decor. Compared to Unreal Engine©4 , Godot© 5 or Babylon© 6 it turns out that for building research prototypes Unity© is easier to access for the developers in our team. Its ergonomics and robustness will allow it to quickly go further than others and be supported by a wide variety of our machines. Working with Unreal Engine was considered but this would have resulted in too much learning when the foundations are already built with Unity. And finally, the graphical rendering achieved with Unreal Engine is compensated by the new rendering brought by the latest Unity advances. The C# is a pure object-oriented modern language. C# programming is often be used to create mobile and console applications and this is helping for our training who want to develop a small project on their machine.

When using Virtual Reality, several technologies are available and robust enough to be used for research. For this case, the most appropriate being HTC Vive© 7 we have multiplied them to have a total of five people connected. The latter being wireless they were able to be free to move freely. Depending on the script we arrange the equipment in wireless or wired to obtain or better capture. We have kept the wireless for the majority of the time because in order to free ourselves from the KatVR if necessary and to be able to have someone try it out who could visit or participate in an event as a spectator, we have kept the wireless for the majority of the time. Three of them were placed on KatVR©8 , which allowed us to optimize the demonstration space. For the moment they are only used to allow easy restriction of the playing areas. It is planned to use them more extensively.

The integration is made by the Software Development Kit: SteamVR© 9 . It is the common one and the most accessible to use. For its use, only footprints have been added instead of the teleport cursor. These footprints rotate according to where the gaze turns. Moreover, we have modified the teleportation to be a linear movement so that it can be understood and not disturbing by the players who could interact with it.

For multiplayer, the simulation uses the asset PUN210 on the asset store which has the advantage of being free and above all accessible very quickly. Their service offers voice and text chat in addition to their more classic service. A way to extend the link between players in real-time. This communication extends the possibilities of collaboration. Other technologies exist, such as the one Unity has set up with UNet, but it has been put on hold until a better service can be offered. Now it is possible to demonstrate more advanced agility in our use case. To respond to a major crisis that evolves in real-time in an existing site.

Scenario

We would like to have a simulation that can be used for training on the sites that are selected. The scenario has been written with the help of a professional group composed of firefighters and a person from the security services of the site. The advantage of this result is that it can be shown to all the personnel of the chosen location.

And to have the potential to apply and exploit it in different ways.

In the scenario, six people are connected in the same virtual environment. Here are the roles for users: the Game Master, two civils: an Assistant and a Technician and also three firefighters: a Chief Firefighter and two Firefighters.

Around them is a replica representing our lab building. The Game Master authorized to trigger events will launch a first fire start. This will cause an alert to the technical service player who will confirm the fire. The first firefighter extinguishes the fire at the end of the corridor to free the person behind. The second firefighter has the task of extinguishing the first fire that has spread throughout the office and spread to the adjoining offices. The last firefighter head supervises the relationship between the firefighters and the civilians outside as well as with the people in charge of the establishment.

At the same time, the Game Master can add new fires to make it harder for the fire brigade.

Outside the firefighter collects new information: a gas bottle was stored in an office and screams were heard behind a door. The firefighter informs the crisis unit which modifies the response process via the decision support platform as the context is no longer the same. As this process is different, the fire brigades on-site are notified of the changes and must operate differently. One must force a door and evacuate a civilian while the other must retrieve the gas cylinder.

This fire team had come because they were already close to the school, but they did not have the necessary equipment. So, they operated while waiting for the arrival of better-equipped backup.

The spread of the fire does not allow the firefighters to stay too long because the smoke becomes too present and they, unfortunately, must evacuate the premises now that there is more danger. The scenario ends when the firefighters who are better equipped for the situation approach (Fig 4).

ARCHITECTURE

The design of this simulation led to a more in-depth project. For this use case, the environment is a digital replica of our lab. To enhance user collaboration, the simulation includes a major new element compared to the previous use case: voice communication.

Depending on their distance, the players will be able to talk to each other to hear each other and exchange information. The most challenging and the most interesting part of this demo is the architecture between the different services (Fig 6). Indeed, assembling the non-playable characters with the immersion players while having created a dynamic environment. The latter evolves according to an external person called Game Master who writes the scenario in real-time from a computer or a Windows tablet. The demo of the previous Use Case did not allow for the evolution of the process deduced at launch. Here the scenario can evolve and the actions that the players will carry out there is a renewal of the terms in the process. This will influence a different deduction. Compared to our previous simulation where the players had their tools directly embedded (e.g. the firefighter carrying an extinguisher), here the tools are integrated into the environment and this involves the players more in their roles. They must follow the process to not make mistakes in the virtual world. For example, fire extinguishers are placed in the building but have been used by the civilians, so they will have to use those near their emergency vehicle and therefore take them from the start to avoid going back and forth. The addition of different elements integrates more complexity during this immersion and can more easily modulate the scenario editor.

Scenario Editor

We have therefore set up a new central dynamic that we call Game Master, this person oversees writing the scenario in real-time and has the power to react according to the actions of the players.

For a question of accessibility, this new player has been designed to carry out these tasks outside of a Virtual Reality immersion. He is therefore located in front of a screen from a tablet or a computer with a top view of the place and can see each person in the environment and declare fires. The latter can be set and triggered in most of the school's rooms. The fire will spread in each room as the simulation unfolds. But the editing can be made in immersion with a lot of interactions.

When the player clicks or touch on one of the locations in the environment, a meeting room for example. An interface is displayed indicating the possibility to create a fire. The parameters are quite simple:

type of fire size of the source object when to start the fire (with a delay or not)

Depending on what is started the fire behaves almost naturally by a limited propagation system to be easily adjusted. Several sensors located in the building are connected to our decision support platform and will be able to send information according to the situation and the places where smoke is present.

The Game master can use tactile support or not but it is possible to continue this tactile extension by providing support such as a tablet or a phone that could use Augmented Reality but also in exploring an immerged view. These points are exactly what we are going to work on next. This will lead to a more accessible immersion while being more agile.

Virtual environment

In addition to being able to see teammates, the addition of voice communication functionality is also an important point as it allows players to talk to each other without going through a system external to the simulation. During the game, the players will be able to choose different channels to have a parallel with an equivalent of a radio where the firefighters can communicate without any distance problem.

Specifically, we have 1 Game Master, 3 firefighters and 2 civilians.

Each with a different course with appropriate tasks. The different objectives can change depending on the situation. This effect makes the scenario more dynamic.

Players will be surrounded by a host of non-player characters with civilian roles modelled and animated to fit into the set. They will have several animations including phone calls, typing or chatting. They are integrated to amplify the immersion of the players. The look is somewhere between a low-polished style and a realistic style. The characters are easily animated on a generic basis while keeping a modular design.

One of the difficulties of this use case has been to reproduce an environment like the one that exists. It took us several months to recreate the environment so that it could be used in several simulations. Using 2D plans but also 3D scans with drones, the school could be reconstructed to form a building close to the existing one. In this one, the model is simplified to guarantee stability in terms of performance. The various tools and other useful objects are anchored in the environment and can be unhooked by the player for use. For example, an extinguisher will be at the same place that in the reallocation. The place of the simulation therefore includes dynamic objects programmed to be used or not by the players. It is therefore up to them to find these elements and use them.

With increased agility, the simulation offers a multitude of interventions. The environment is directly integrated into an existing place. It allows having real time-sensitive sensors present on the real site. This setting is amplified by the presence of characters with realistic behaviour. And having made the tools useful in their dedicated spaces and not fixed to their hands from the start adds a significant difficulty and coherence. Here you can see an example of an office with a fire in it (Fig 8).

The smoke is close to the ceiling and will be more and more intense if the fire is not extinguished.

Decision support tool

This simulation needed to provide a certain amount of data to be able to deduce a coherent process when orchestrating the tasks to be carried out. Several smoke sensors are present in the establishment and we have reproduced them in Unity© as well as on the decision support platform R-IO Suite [START_REF] Benaben | A conceptual framework and a suite of tools to support crisis management[END_REF]. They are connected and will be able to exchange information depending on the situation. Each sensor has an alert threshold on Unity© when fire and smoke are close to a sensor with a certain intensity it sends to the decision support platform the information that the sensor detects smoke.

The decision platform will send each player a set of tasks to perform. The purpose here is not to explain their handling but rather to indicate the directions as well as the fruitfulness of the collaboration between the different units. Thanks to a tablet attached to their arm, they can see the tasks they have received.

Their tasks may have a description with coordinates but also details of the dangerousness. For example: Securing the building will have a description: Evacuate the gas bottle in office 0A02. Once completed, each player will be able to update the process by clicking on Send Update on their tablet (Fig 9). To install the simulation, all you need to do is to connect each pc by making public information such as their IP address. This will allow our decision support tool to exchange with each player individually. They will communicate with basic web requests.

The connection with our decision support platform is more dynamic than in the previous use case, which was fixed from start to finish. This scenario performs two different processes in a short period with a significant change in tasks and their priorities.

Depending on the situation, the sensors will modify the process. And this will modify the tasks to be performed by the players by adding more information if possible.

OUTCOMES

Results

The result is the sum of the work carried out by the feedback from the experts who witnessed our experiences but also from the users who were able to share their feelings and indicate valuable corrections. To find out if our solution works, we have carried out a battery of tests, some of which were filmed. The aim is to see how well the application offers a wide range of scenarios. The anchored points will establish a structure for our results. People have freedom outside these points that are relevant to observe. With several tests, we have been able to test the efficiency of the simulation. It is interesting to observe the different dynamic factors, which offer the necessary modularity to obtain an agile scenario.

A capture was carried out to effectively show our results. It can be found as "Unique immersion experience inside the IOMEGA VR platform" on the YouTube platform, here at https://youtu.be/F-dO_TSOA4o. This video takes place in one single shot and focuses on the different essential elements of our simulation.

Evaluation

Through this development, we have been able to discover difficulties, particularly through having people who rarely use Virtual Reality and therefore had to become familiar with the controllers. The HTC Vive© controllers are not known for their ergonomics and the user experience can be very difficult when the person is an amateur, especially if he or she is not a video game player. The majority of users admit that they have never had experiences using virtual reality and have been disturbed by their integration in an environment they are familiar with. They have been able to learn and navigate in their environment and master interactive objects. All this to fulfil their roles.

Simulation is highly dependent on manipulations in the virtual world. If one of the VR players makes a mistake, forgets, misunderstands, or interacts badly with an element it can completely disrupt the simulation. Beyond the user experience, there are still hardware and development problems. It can happen that a loss of a tracking break or a fire does not synchronize in multiplayer for all users. These points need to be more robust in the future. As our HTC Vive Pro© devices are wireless, we have had several difficulties with tracking that could interrupt. It is possible that for the next use case we will test on two methods a wired and a wireless one to confirm this bias.

The evolutionary presence of the non-player characters fulfils its role perfectly. Despite a rather limited number of animations, their fidelity and their settings improve the feeling of realism for most of the immersed players. During the simulation, the players could notice the absence of sound from the unplayed civilians. It is planned to enhance discussions or shouting to better integrate the non-player characters and their interactions in the environment.

With this simulation, it is easy to reproduce a situation and to observe it. Each test can have a different course from other thanks to the modularity that can be set up. Nevertheless, we have fixed the elements described in the scenario.

Referring to figure 1, the expectations regarding collaboration and interactions correspond with the results. We can see that the business processes are well respected and well replicated.

Knowledge: It is not an in-depth direction that is set on this simulation.

Action:

The development is still too recent to allow a realistic approach to interactions and actions with tools in the field. For the moment, the symbolic approach of a fire extinguisher is sufficient.

The interaction with objects in the environment is more subtle because it is coherent. However, it requires knowledge of the joystick keys to carry out the interactions easily. Once this learning period is over, the gesture is carried out naturally. The aim is not to reproduce gesture by gesture the right movements to use a particular fire extinguisher. The interactions are simplified but remain unexplained which can be frustrating at the beginning. Once understood this is functional and it accentuates the ergonomics of the user experience.

Process: Although the exercise did not have this aim, the experts applied their protocols and positioned themselves in the same situations as in reality. Putting one's hand close to a suspicious door is a consistent technique. Placing a hand on a door to check the temperature of the room for example. This reinforces the effectiveness of the management of our research.

Collaboration: They will coordinate well according to their trade assignment. Important information is exchanged with the voice channels. And their collaboration is more refined thanks to vocal interaction.

CONCLUSION

This experiment was complex to develop because it was intended to be close to reality. It is necessary to consider a plurality of parameters that have been suggested by professionals such as the spread of fire and the appearance of smoke. The requested requirement made it possible to carry out this simulation and to satisfy the participants as well as the professionals.

Increasing the potential of the Game Master in terms of interaction on the simulation but also in terms of visualization is one of the avenues that will have to be advanced. It is easy to continue tactile 2D support as here but it is envisaged to imagine its role in virtual immersion or to have exploitation using augmented reality.

Integrating a system where each player could have his summary of his simulation to observe his course and potentially his mistakes. A dashboard summarising the actions of the game would be saved. The point could be, for example, to emphasize a civil loss due to a loss of time or a bad method follow up. This could be exploited by the Game Master who would have access to all his interventions.

The success of this experiment has confirmed the questioning around the subject and adds new and very interesting issues. It is possible to replicate situations to derive results concerning the behaviour and actions of the players. This creates a safe context where nothing dangerous is involved and everything is virtual. The application can be used in multiplayer and can be played all over the world.

Future simulations will include new interactions, not just creating fires and a parallel of different situations that the Game Master will have to manage with further interaction. Several avenues have already been discussed with partner companies.

The scenarios could have consequences for every other place. The vision may have several scenario managements states. The vision of the scenario writer will not be there to see the whole site or to generate a problem in a particular room.

The aim will be to have several sites, each with its particularities.

It is planned to have an industrial site categorised as CBRN, another historical site.

In future cases, it will be necessary to integrate the opportunity to join as a civilian in addition to the professional trades to establish a civilian opportunity during the simulation. Dealing with Virtual Reality provides real support to immerse different people. This technology offers an answer that positions itself as a solution.

Figure 1 :

 1 Figure 1: Our decomposition of learning levels

Figure 2 :

 2 Figure 2: View of the Game Master Once confirmed, the school alarm goes off and causes the exit of the non-player characters as well as the player from the technical department who has finished his scenario (Fig 3). The player in charge of checking that the offices are empty goes around the school. He notices one person still working because the character was wearing a headset and did not hear the alarm. When the player enters the room, he signals the alert, which makes the person leave. The player continues his way and sees a second fire with a person trapped by the fire as well as shouting behind a locked door. He goes out and warns the firefighters who have just arrived. The firefighters receive their tasks sent by the decision support platform and go to fulfil them. The smoke sensors have sent the decision support platform information about the position of the fires.

Figure 3 :

 3 Figure 3: People exit the school

Figure 4 :

 4 Figure 4: Firetruck near the building

  It has been programmed under Unity© 2019.3.2f1 with the default settings to integrate the Asset SteamVR© and Photon Unity Engine 2©. SteamVR© makes it possible to quickly have a VR situation in our 3D environment and finally have little interaction to develop for the VR. The simulation uses the High Rendering Pipeline technology of the Unity© graphics engine which is a very recent technology that gives a more advanced graphics rendering.

Figure 5 :

 5 Figure 5: Deploy Plan

Figure 6 :

 6 Figure 6: Architecture System

Figure 7 :

 7 Figure 7: Office with one non-player character

Figure 8 :

 8 Figure 8: Fire in an office

Figure 9 :

 9 Figure 9: Tablet connected to the decision support

Figure 10 :

 10 Figure 10: Image from the YouTube video

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Romain Gojon
email: romain.gojon@ec-lyon.fr.

Christophe Bogey
email: christophe.bogey@ec-lyon.fr.

Olivier Marsden
email: olivier.marsden@ec-lyon.fr.

Large-eddy simulation of underexpanded round jets impinging on a flat plate 4 to 9 radii downstream from the nozzle

Supersonic round jets have been computed by compressible Large Eddy Simulation (LES) using low-dispersion and low-dissipation schemes. The jets are underexpanded, and are characterized by a Nozzle Pressure Ratio of N P R = Pr/Pamb = 4.03, where Pr is the stagnation pressure and Pamb is the ambient pressure. They have a fully expanded Mach number of Mj = 1.56, an exit Mach number of Me = 1, and a Reynolds number of Rej = ujD/ν = 5 × 10 4 , where uj and D are the jet fully expanded exit velocity and the nozzle diameter, respectively. A free jet is first considered. Four jets impinging on a flat plate normally are then examined. The distance L between the nozzle lip and the flat plate varies from L = 4.16r0 up to L = 9.32r0 where r0 = D/2, for the impinging jets. The effects of the plate on the aerodynamic and acoustic properties of the jets are thus studied. For the free jet, snapshots of density, pressure and vorticity are presented. Mean velocity fields are displayed, they are in good agreement with experimental data. The near pressure field of the jet is investigated using Fourier decomposition. A screech tone component is found, at a frequency comparing well with experimental data and theoretical models. For the four impinging jets, similarly, flow snapshots and mean flow fields are shown. The results obtained are similar to the corresponding measurements. The convection velocity of large-scale structures in the jet shear layers is then evaluated and an expression giving the average convection velocity between the nozzle lips and the flat plate is proposed. The near pressure fields are then explored, and the main properties of the aeroacoustic feedback mechanism occurring between the nozzle lip and the flat plate are presented. The results are consistent with theoretical models and experimental data.

I. Introduction

Very intense tones were observed experimentally by Powell 1 and Wagner, 2 among others, in the acoustic field of high subsonic and supersonic jets impinging on a flat plate normally. Moreover, as the distance between the nozzle and the flat plate varies, a staging phenomenon of the main tone frequency was obtained. Powell [START_REF] Powell | On edge tones and associated phenomena[END_REF] suggested that such tones are generated by a feedback mechanism between hydrodynamic instabilities propagating downstream from the nozzle to the plate and acoustic waves propagating upstream from the plate to the nozzle.

For round subsonic impinging jets, Ho & Nosseir [START_REF] Ho | Dynamics of an impinging jet. Part 1. The feedback phenomenon[END_REF] and Nosseir & Ho [START_REF] Nosseir | Dynamics of an impinging jet. Part 2. The noise generation[END_REF] built a model to predict the tone frequencies of the feedback mechanism. This model is based on cross-correlations between microphones in the near field of impinging jets. Later, Tam & Ahuja [START_REF] Tam | Theoretical model of discrete tone generation by impinging jets[END_REF] proposed that the upstream-propagating waves of the feedback mechanism correspond to neutral acoustic wave modes of the jets. They found an allowable frequency range for each upstream propagating neutral acoustic wave mode of the jet flow using a vortex sheet jet model. They obtained results in line with the experimental data of Wagner 2 for high subsonic round jets. Round supersonic jets impinging on a flat plate normally have been investigated experimentally by Henderson & Powell, [START_REF] Henderson | Experiments concerning tones produced by an axisymmetric choked jet impinging on flat plates[END_REF] Krothapalli et al. [START_REF] Krothapalli | Flow field and noise characteristics of a supersonic impinging jet[END_REF] and Henderson et al. [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] In some cases, a feedback mechanism is observed as in subsonic jets. This is very often the case when the jet is ideally expanded, but only for some nozzle-to-plate distances when the jet is imperfectly expanded. Henderson & Powell [START_REF] Henderson | Experiments concerning tones produced by an axisymmetric choked jet impinging on flat plates[END_REF] suggested that, in the latter case, the feedback establishes only when a Mach disk forms upstream from the plate. Recirculation zones were also observed near the flat plate by Krothapalli et al. [START_REF] Krothapalli | Flow field and noise characteristics of a supersonic impinging jet[END_REF] More recently, for underexpanded impinging jets, Risbord & Soria 9 explored the instability modes of the jets using ultra-high-speed Schlieren and shadowgraphy techniques. Axial and helical modes have been identified and the Mach disk located upstream from the plate has been observed to oscillate. For similar jets, Buchmann et al. [START_REF] Buchmann | High spatial resolution imaging of a supersonic underexpanded jet impinging on a flat plate[END_REF] observed the periodic formation of large-scale structures in the jet shear layers using a high spatial resolution Schlieren imaging. The complete feedback mechanism is visible, and includes large-scale structures in the shear layers propagating downstream from the nozzle to the plate and acoustic waves propagating upstream from the plate to the nozzle. Finally, Mitchell et al. [START_REF] Mitchell | The visualization of the acoustic feedback loop in impinging underexpanded supersonic jet flows using ultra-high frame rate schlieren[END_REF] studied the periodic oscillation of the shear layer of underexpanded impinging jets using time-resolved Schlieren image sequences. Unfortunately, the connections between the different flow features listed above, namely, the shock-wave oscillations, the periodic shear-layer instabilities, the recirculation zones near the plate, and the production of tones, remain unclear. In order to understand these interactions, Kuo & Dowling [START_REF] Kuo | Oscillations of a moderately underexpanded choked jet impinging upon a flat plate[END_REF] considered that there is a flux resonance in the region of impact between the Mach disk and the flat plate, and they developed a 1-D model of the Mach disk motion involving acoustic and entropy waves. With this model, a resonance condition is found, and for a given mean velocity flow, the frequency of resonance of the Mach disk motion can be estimated. The predicted resonance frequencies matched the experimental tone frequencies observed by Powell [START_REF] Powell | The sound-producing oscillations of round underexpanded jets impinging on normal plates[END_REF] only for small plates, although the model was supposed to be adapted to all plate sizes. Moreover, the model is only valid when there is no recirculation zones between the Mach disk and the plate. When the jet is strongly underexpanded and the plate is located in the first cell of the free equivalent jet, a feedback loop involving disturbances in the shear layer between the subsonic region downstream of the Mach disk and the supersonic peripheral flow was proposed by Henderson et al. [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] Numerical simulations were conducted by Dauptain et al. [START_REF] Dauptain | Large eddy simulation of stable supersonic jet impinging on flat plate[END_REF][START_REF] Dauptain | Large Eddy Simulation of Supersonic Impinging Jets[END_REF] for such configurations, and a modified model to predict the tone frequencies of the feedback loop was proposed.

In the present work, the LES of five round jets are carried out in order to investigate the feedback mechanism occurring between the nozzle lip and the flat plate for impinging jets. In particular, the spectral and hydrodynamic properties of the jets are studied and compared with experimental data and theoretical models. The paper is organized as follows. The jets parameters and the numerical methods used for the LES are given in section II. The aeroacoustic properties of the free jet is then presented in section III, and compared with experimental data and theoretical models. The four jets impinging on a flat plate are then described in section IV, and the feedback mechanism is studied by evaluating the convection velocity and describing the near pressure fields of the jets. Concluding remarks are given in section V.

II. Parameters

II.A. Jets parameters

Large-eddy simulations are performed for four jets impinging on a flat plate with nozzle-to-plate distances L/r 0 of 4.16, 5.6, 7.3 and 9.32, and for a free jet, as shown in table 1. The jets are referred to as JetL4, JetL5, JetL7 JetL9, and Jetfree, respectively. The ejection conditions of the jets and the nozzle-to-plate distances are similar to the conditions in the experiments of Henderson et al. [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] The jets have a Nozzle Pressure Ratio of N P R = P r /P amb = 4.03, a fully expanded Mach number of M j = 1.56, a Reynolds number of Re j = u j D/ν = 5 × 10 4 and an exit Mach number of M e = 1. They originate from a pipe nozzle whose lip thickness is equal to 0.1r 0 . At the nozzle exit, a Blasius boundary-layer mean velocity profile is imposed with a boundary-layer thickness of δ BL = 0.15r 0 .

II.B. Numerical parameters

The unsteady compressible Navier-Stokes equations are solved on a cylindrical mesh (r, θ, z) by using an explicit six-stage Runge-Kutta algorithm for time integration, and low-dispersion and low-dissipation explicit eleven-point finite differences for spatial derivation. [START_REF] Bogey | A family of low dispersive and low dissipative explicit schemes for flow and noise computations[END_REF][START_REF] Berland | High-order, low dispersive and low dissipative explicit schemes for multiple-scale and boundary problems[END_REF] At the end of each time step, a relaxation filtering is applied to the flow variables in order to remove grid-to-grid oscillations and to dissipate subgrid-scale turbulent energy. [START_REF] Bogey | Large eddy simulations of transitional round jets: influence of the Reynolds number on flow development and energy dissipation[END_REF][START_REF] Bogey | Turbulence and energy budget in a self-preserving round jet: direct evaluation using large eddy simulation[END_REF] The radiation conditions of Tam & Dong 20 are implemented at the inflow and lateral boundaries of the computational domain. A sponge zone combining grid stretching and Laplacian filtering is also employed to damp the turbulent fluctuations before they reach the lateral boundaries. The present solver has been previously used [START_REF] Bogey | Large-eddy simulation of the flow and acoustic fields of a Reynolds number 10 5 subsonic jet with tripped exit boundary layers[END_REF][START_REF] Bogey | Influence of initial turbulence level on the flow and sound fields of a subsonic jet at a diameter-based Reynolds number of 10 5[END_REF] to simulate 3-D round jets at a Mach number M e = 0.9. Adiabatic conditions are imposed to the nozzle walls and to the flat plate. A shock-capturing filtering is applied in order to avoid Gibbs oscillations near shocks. It consists in applying a conservative second-order filter optimised in Fourier space at a magnitude determined each time step using a shock sensor. [START_REF] Bogey | A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations[END_REF] This method was successfully used by de Cacqueray et al. [START_REF] De Cacqueray | Investigation of a High-Mach-Number Overexpanded Jet Using Large-Eddy Simulation[END_REF] for the LES of an overexpanded jet at Mach number M e = 3.3. Moreover, low-amplitude vortical disturbances, not correlated in the spanwise direction, [START_REF] Bogey | Large-eddy simulation of the flow and acoustic fields of a Reynolds number 10 5 subsonic jet with tripped exit boundary layers[END_REF] are added in the boundary layer in the nozzle, at around z = -r 0 and r = 0.90r 0 , in order to generate velocity fluctuations at the nozzle exit. The strength α of the forcing is fixed to 0.02 to obtain turbulent intensities between 5% and 10% of the fully expanded jet velocity at the nozzle exit. This value is similar to those used in Bogey et al. [START_REF] Bogey | Large-eddy simulation of the flow and acoustic fields of a Reynolds number 10 5 subsonic jet with tripped exit boundary layers[END_REF] The axis singularity is treated with the method proposed by Mohseni & Colonius. [START_REF] Mohseni | Numerical treatment of polar coordinate singularities[END_REF] Notably, the first point close to the axis is located at r = ∆r/2, where ∆r is the radial mesh size. Finally, a reduction of the effective resolution near the origin of the polar coordinates is implemented, [START_REF] Bogey | Finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations[END_REF] The simulations are carried out using an OpenMP-based in-house solver, and a total of 180, 000 iterations are computed in each case for the steady state. The simulation time is thus equal to 1000r 0 /u j . The cylindrical meshes (n r , n θ , n z ) contain between 171 and 400 million points, as reported in table 2. The variations of the radial and axial mesh spacings are represented in figure 1. For the impinging jets, the minimal axial mesh spacing, near the nozzle lip and the flat plate, is equal to ∆z = 0.0075r 0 , and the maximal axial mesh spacing, between the nozzle and the plate, is equal to ∆z = 0.015r 0 for JetL4 and JetL5, and to ∆z = 0.03r 0 for JetL7 and JetL9. The minimal radial spacing is ∆r = 0.0075r 0 at r = r 0 , and the maximal radial spacing is ∆r = 0.06r 0 for 5r 0 ≤ r ≤ 15r 0 . Then, for r ≥ 15r 0 , a sponge zone is implemented. For the free jet, the grid is the same in the radial direction, but differs in the axial direction far from the nozzle with a maximal axial mesh spacing of ∆z = 0.03r 0 for 5r 0 ≤ z ≤ 30r 0 , and a sponge zone for z ≥ 30r 0 . These discretizations allow acoustic waves with Strouhal numbers up to St D = f D/u e = 6.4 to be well propagated, where f is the frequency. Finally, note that grids are stretched at rates lower than 1% outside the sponge zones, in order to preserve numerical accuracy.

M j Re j L/

III. Results for the free jet

The results obtained for the free jet are first described for validation of the numerical set-up and for comparison with the results for the impinging jets.

III.A. Flow snapshots

In order to illustrate the shear-layer development downstream of the nozzle, a snapshot of the vorticity norm obtained in the plane (z, r) is presented in figure 2. The shear layer develops rapidly downstream of the nozzle exit with both small and large turbulent structures, in agreement with the Reynolds number of Re j = 5 × 10 4 . The end of the jet potential core appears to be located around 17.5r 0 . Lau et al. [START_REF] Lau | Measurements in subsonic and supersonic free jets using a laser velocimeter[END_REF] proposed an empirical model for the length of the potential core z p for isothermal jets with Mach number up to 2.5, which writes

z p D j = 4.2 + 1.1M 2 j ( 1
)
where D j and M j are the fully expanded diameter and the Mach number of the jet.

For the present free jet, the parameters M j = 1.56 and D j = 2.2mm give a potential core length of z p = 15r 0 . Tam et al. [START_REF] Tam | A multiple-scales model of the shock-cell structure of imperfectly expanded supersonic jets[END_REF] included in the expression (1) the temperature ratio between the stagnation and the ambient temperatures in order to take into account compressibility effects. This expression gives z p = 17.5r 0 . A good overall agreement is thus found between the present result and the empirical expression derived by Tam et al. In order to visualize both the flow and acoustic fields, a snapshot in the (z, r) plane of the density and the fluctuating pressure is provided in figure 3(a). A shock-cell structure, typical of an underexpanded jet, is obtained with around ten shock cells visible. Because the jet is highly underexpanded, a Mach disk is found in the first cell, at z = 2.35r 0 and shear layers can be observed in figure 2 between the subsonic and the supersonic region downstream of the Mach disk. This is in agreement with the results of Powell, [START_REF] Powell | The sound-producing oscillations of round underexpanded jets impinging on normal plates[END_REF] Henderson, [START_REF] Henderson | The connection between sound production and jet structure of the supersonic impinging jet[END_REF] and Addy, 30 who noted that a Mach disk is generated in underexpanded jets for N P R > 3.8 or 3.9. In the pressure field, acoustic waves and two acoustic contributions can be seen. First, circular wavefronts are visible and seem to originate from the first five cells. They are due to the interactions between the shocks and the turbulence in the jet shear layers. Upstream propagating acoustic waves can also be observed on each side of the jet, in the vicinity of the nozzle. For the comparison, a Schlieren picture of an underexpanded jet obtained by André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] is displayed in figure 3(b). The fully expanded Mach number of the jet is M j = 1.55 and the exit Mach number is M e = 1. A great similarity is noted with the computed jet. In particular, the Mach disk is clearly visible in the first cell as well as the shear layers downstream.

III.B. Mean fields

The mean axial and radial velocity fields are presented in figure 4, where the experimental PIV results of André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] are also displayed. The geometry of the shock-cell structure and the levels obtained in the LES and in the experiment are in good agreement. In figure 4(a), the length of the first shock cell L s of the jet appears to be 3.20r 0 . This result is identical to those of Henderson et al. [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] for a similar free jet. Besides, the length of the first cell can be estimated by using a first-order shock solution based on the pressure ratio p s /p a , where p s is the pressure perturbation of the shock-cell structure and p s + p a is the actual pressure in the jet. This solution was first proposed by Prandtl 32 in 1904. Later, the following approximated solution was given by Tam & Tanna: 33

L s = πD j β µ 1 ≃ 1.306βD j (2)
where µ 1 is the first zero of the zero-order Bessel functions of the first kind and β = M 2 j -1. This expression is called Prandt formula. Expression (2), for the present jet, provides L s = 3.45r 0 , which is slightly larger than the measured and computed values. Later, the expression (2) was improved by Pack 34 by calculating the first forty coefficients of the model of Prandtl, 32 yielding L s = 1.22βD j , hence L s = 3.20r 0 for the present jet, which is similar to the values reported above.

The shock-cell size seems to decrease with the downstream distance in figure 4(a). This evolution is due to the growth of the shear layer and to the dissipation of the shock-cell structure. The decrease of the normalized shock-cell size is presented in figure 5 for the present jet and for the jets of André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] A good agreement is found between the present jet and the experimental results. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] A linear variation of the shock-cell size is also obtained, as observed by Harper-Bourne & Fisher 35 for instance. For the first eight cells of the present jet, the mean velocity field provides a total size of 20.5r 0 . For underexpanded supersonic jets, Harper-Bourne & Fisher [START_REF] Harper-Bourne | The noise from shock in supersonic jets[END_REF] and Seiner & Norum 36 obtained an empirical expression for the size of the first eight cells, which writes L t = 1.11βD. For the present jet, the formula provides L t = 21.5r 0 , which is consistent with the above value.

Finally, the Mach disk located at z M = 2.3r 0 is a normal shock wave. For a normal shock, the Rankine-Hugoniot jump condition leads

u 2 u 1 = (γ -1)M 2 1 + 2 (γ + 1)M 2 1 (3)
where u 1 and u 2 are the velocities upstream and downstream of the normal shock wave respectively, and M 1 is the upstream Mach number. In the present jet, one sets u 1 = 595 m.s -1 and M 1 = 2.75 before the Mach disk and u 2 = 166 m.s -1 after the Mach disk. The latter value compares very well with the value u 2RH = 164 m.s -1 predicted by the Rankine-Hugoniot jump condition. The Mach disk position z M and diameter D M can also be estimated from the mean velocity field, yielding z M = 2.3r 0 and D M = 0.25r 0 . Experimentally, Addy 30 observed a Mach disk for jets with a N P R exceeding 3.9, and obtained the following empirical expressions:

z M D j = 0.65 √ N P R (4)
D M D j = 0.36 √ N P R -3.9 (5)
The Mach disk positions and diameters obtained from the present simulation, the experiments of Addy 30 and expressions (4) and (5), are plotted as a function of the Nozzle Pressure Ratio in figure 6. A good overall agreement is found. It can be noted that expression (4) slightly overestimates the position of the Mach disk for N P R < 6.

III.C. Acoustic results

The sound pressure level (SPL) obtained near the nozzle exit at z = 0 and r = 2r 0 is displayed in figure 7 as a function of the Strouhal number St D = f D/u e . One tone emerges 15 dB above the broadband noise at a Strouhal number of St D = 0.38. Such a result is typical of a screeching jet, see for instance in Westley & Woolley, [START_REF] Westley | The near field sound pressures of a choked jet during a screech cycle[END_REF] Tam & Tanna, [START_REF] Tam | Shock associated noise of supersonic jets from convergent-divergent nozzles[END_REF] Panda, [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] and André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] The tone at St D = 0.38 can therefore be attributed to the generation of screech noise by the jet. The near pressure fields in the (z, r) plane have been recorded every 50 th time step. The results are arranged in the M × N matrix:

P all =      P 1 1 P 2 1 . P N 1 P 1 2 P 2 2 . P N 2 . . . . P 1 M P 2 M . P N M      (6)
where N is the number of samplings, and M = n z × 2n r is the total number of points in a (z, r) plane. The pressure field obtained at a given time is thus provided by one column of P all . A Fast Fourier Transform is applied to each row of the matrix P all . In this way, for a given frequency, the amplitude and the phase fields can be displayed. For the tone at St D = 0.38, they are given in figure 8. The amplitude field in figure 8(a) exhibits a cell structure different from the shock-cell structure. Such a structure is observed in the presence of an hydrodynamic-acoustic standing wave, as is the case in supersonic screeching jets, see in Panda. [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] The typical cell length in this structure is equal to the wavelength L sw of the standing wave formed between the downstream propagating large-scale structures in the shear layers and the upstream propagating acoustic waves. The screech frequency f s associated with this wavelength is provided by the model of Panda:

38 f s = u c L sw (1 + u c /c 0 ) (7)
where u c is the mean convection velocity of the structures in the shear layers and c 0 is the ambient sound speed. For the present jet, the cell structure in figure 8(a) shows a spatial periodicity of L sw ≃ 2.5r 0 . The mean convection velocity, computed from cross-correlations of axial velocity fluctuations in the shear layers, see in section IV, is u c = 0.65u j , where u j is the fully expanded jet velocity. Expression (7) thus leads to a screech Strouhal number of St D = f s D/u e ≃ 0.38, which is in agreement with the tone frequency of figure 7. In figure 8(b), a 180 degree phase shift with respect to the jet axis is visible. This indicates that the mode at St D = 0.38 is sinuous or helical. More precisely, using a Fourier decomposition of the fluctuating pressure on 32 sensors regularly-spaced in the azimuthal direction, at z = 0 and r = 2r 0 , it turns out to be helical. This mode is compared in figure 9 with the dominant screech tones found in the experiments of Powell et al. [START_REF] Powell | Observations of the oscillation modes of choked circular jets[END_REF] for round supersonic underexpanded jets. These tones are associated with varicose modes (A1 and A2), with sinuous and sometimes helical modes (B), with helical modes (C), and with sinuous modes (D). The screech tone in the present jet falls in the vicinity of tones C, in agreement with the results reported above. Consequently, both the frequency and the nature of the screech mode in Jetfree are consistent with the theoretical predictions of Panda [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] and the measurements of Powell et al. In the next section, jets with the same exit conditions as Jetfree impinging normally on a flat plate for four different nozzle-to-plate distances are presented. The positions of the plate with respect to the shock-cell structure of Jetfree are shown in figure 10. For JetL4, with L = 4.16r 0 , the plate is located in the first half of the second cell of the shock-cell structure, where the velocity increases and the pressure decreases with the axial distance on the jet centerline, whereas for JetL5, with L = 5.6r 0 , it lies in the second half of the second cell, where velocity decreases and pressure increases. For JetL7 and JetL9, with L = 7.3r 0 and L = 9.32r 0 , the plates are located in the first halves of the third and of the fourth cells of the shock-cell structure, respectively. These remarks are of interest, given that, according to Henderson et al., [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] the occurrence of intense tones in imperfectly expanded impinging jets depends on the location of the plate in the shock-cell structure of the corresponding free jet.

IV. Results for the impinging jets

IV.A. Flow snapshots

In order to illustrate the flow fields in the impinging jets, isosurfaces of density are displayed in figure 11 for JetL9. In the jet, a shock-cell structure appears clearly between the nozzle and the plate in blue an red. The boundaries of the mixing layers are also well visible in cyan. In addition, longitudinal structures can be seen on the outer border of the first shock cell. They have been described in different experiments, including those by Arnette et al. [START_REF] Arnette | On streamwise vortices in high Reynolds number supersonic axisymmetric jets[END_REF] They are due to small perturbations at the nozzle exit, which are amplified by Taylor-Goertler-type instabilities, and are specific to underexpanded jets. Snapshots of the vorticity norm obtained in the (z, r) plane for JetL4, JetL5, JetL7 and JetL9, are presented in figure 12. In JetL4, in figure 12(a), large vortical structures of typical size 0.2r 0 are found in the shear layers. They are strongly correlated in the azimuthal direction (not shown here). The distance between the two sliplines visible downstream of the Mach disk at z ≃ 2r 0 , equal to 0.5r 0 , is larger than that in the free jet, equal to 0.23r 0 , which results from a larger diameter of the Mach disk. In JetL5, JetL7 and JetL9, the shear layers develop in the same way as in Jetfree with both small and large turbulent structures. The distance between the sliplines downstream of the Mach disk is also the same as in Jetfree. Snapshots of density and fluctuating pressure in the (z, r) plane are provided in figure 13 for the four jets. In figure 13(a), for JetL4, one shock cell appears in the density field. Compared to the free jet where the Mach disk is located at z M = 2.3r 0 , the cell here is shorter, with z M = 1.85r 0 . The Mach disk is also thicker than in the free jet. An annular oblique shock is visible around the Mach disk, as in Jetfree. The temporal evolution of the jet shows a strong axial motion of the Mach disk, as experimentally observed by Risborg & Soria. [START_REF] Risborg | High-speed optical measurements of an underexpanded supersonic jet impinging on an inclined plate[END_REF] Moreover, the periodic motion of the Mach disk is synchronized with the strong spherical sound waves visible in the pressure field. In figure 13(b), for JetL5, the first cell of the shock-cell structure is similar to that in Jetfree. A Mach disk forms upstream from the plate, at z = 4.25r 0 , in the second cell. In the pressure field, sound waves seem to propagate from the region of the impact. In figure 13(c), for JetL7, the two first cells of the shock-cell structure resemble those in Jetfree. Two acoustic contributions are visible in the pressure field. First, wavefronts centered around the annular oblique shock of the first cell appear. They are due to the interactions between the oblique shock and the turbulence in the shear layers. A second acoustic contribution seems to come from the region of impact. Finally, in figure 13(d), for JetL9, a Mach disk forms upstream from the plate at z = 7.7r 0 , in the third cell. The pressure field looks like the pressure field of JetL7.

IV.B. Mean flows

The mean velocity fields obtained in the (r, z) plane are presented in figure 14. A good agreement is found with the measurements of Henderson et al. [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] For JetL5 and JetL7, however, shock cells are slightly shorter in the simulation than in the experiment. For JetL4, in figure 14(a), the effects of the flat plate on the first shock cell mentioned previously are clearly visible. For JetL5 and JetL9, a second cell and a third cell, respectively, cannot fully form before the plate, and a Mach disk is generated close to the wall. On the contrary, for JetL7, in figure 14(c), the two first cells of the shock-cell structure appear to spread over the entire space between the nozzle and the plate, and no Mach disk is created upstream from the plate. Finally, for JetL4, JetL5 and JetL9, recirculation zones are found on each side of the jet near the plate, in agreement with the experiments of Henderson & Powell, [START_REF] Henderson | Experiments concerning tones produced by an axisymmetric choked jet impinging on flat plates[END_REF] Krothapalli et al. [START_REF] Krothapalli | Flow field and noise characteristics of a supersonic impinging jet[END_REF] and Henderson et al. [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] These results are also in line with the observations of Kuo & Dowling 12 who noted the appearance of a recirculation zone when the distance between the Mach disk and the flat plate exceeds 1.2r 0 . This distance is indeed equal to 2.3r 0 , 1.35r 0 and 1.6r 0 for JetL4, JetL5 and JetL9.

IV.C. Convection velocity

The convection velocity u c of the large-scale structures in the jet shear layers has been estimated. It has been calculated at the center of the shear layer, whose position is given by the maximum of the root mean square of velocity fluctuations. The resulting path for JetL9 is for example displayed in figure 15. Along this path, axial velocity cross-correlations are computed to obtain the convection velocity. The results are shown in figure 16(a) for the free jet and the impinging jets. The convection velocity in the shear layers is not constant but varies according to the shock-cell structure. In the first cell, the convection velocity increases between the nozzle lips and the position of the Mach disk, as the velocity inside the jet grows, and then decreases down to the end of the cell, following the velocity reduction in the jet downstream of a Mach disk or an oblique shock. Similar variations can be seen for all cells. They are consistent with the experimental results obtained by Westley & Woolley [START_REF] Westley | The near field sound pressures of a choked jet during a screech cycle[END_REF] and by André et al. [START_REF] André | Investigation of the mixing layer of underexpanded supersonic jets by particle image velocimetry[END_REF] for imperfectly expanded round jets. The modification of the first shock cell in JetL4 described earlier is also visible in figure 16, with a maximum of the convection velocity reached around z M = 1.85r 0 for JetL4 but around z M = 2.3r 0 for the other jets. Moreover, for Jetfree, the convection velocity tends to a value around u c = 0.65u j in the downstream direction, which is in good agreement with the experimental observations of Harper-Bourne & Fisher, [START_REF] Harper-Bourne | The noise from shock in supersonic jets[END_REF] who found u c = 0.70u j using a crossed-beam Schlieren technique.

Given the variations of the convection velocity in figure 16, it does not appear relevant to use the same convection velocity between the lip and the plate for all nozzle-to-plate distances, as was the case in the model of Ho & Nosseir 3 and Nosseir & Ho [START_REF] Nosseir | Dynamics of an impinging jet. Part 2. The noise generation[END_REF] to predict the frequencies of the feedback mechanism. Indeed, the average convection velocity between the nozzle and the flat plate for the four impinging jets, displayed in figure 16(b), varies from 0.54u j for JetL4 up to 0.59u j for JetL9. It can be approximated by the average convection velocity

< u c > (L) = 0.65u j -(0.65u j -0.5u e ) 1 1 + L/D j (8)
which is represented in figure figure 16(b) for a nozzle-to-plate distance ranging from L = 2r 0 to L = 10r 0 . Expression (8) tends to 0.65u j for large distances L, in agreement with the result of Jetfree, and tends to 0.5u e near the nozzle, as expected.

IV.D. Pressure spectra

The sound pressure levels obtained at r = 2r 0 and z = 0 are plotted in figure 17 as a function of the Strouhal numbers St D = f D/u e . The Strouhal number of the tones whose levels are 5 dB higher than the broadband noise are given in table 3. For JetL4, JetL7 and JetL9, a Mach disk forms just upstream of the flat plate, and strong tones are obtained. On the contrary, for JetL7, the emerging tones are weak. These results are consistent with the experimental results of Henderson & Powell 6 who noted that the feedback mechanism establishes when there is a Mach disk upstream from the plate and ceases when there is a conical shock wave instead.

IV.E. Tone frequencies

In order to explain the tone frequencies measured for supersonic impinging jets, Powell 1 suggested that a feedback mechanism establishes between the nozzle lips and the flat plate. This mechanism consists of two steps. First, in the jet shear layers, a coherent structure is convected downstream from the nozzle to the plate. The structure impinges on the plate, and generates an acoustic wave propagating upstream towards the nozzle. This wave is then reflected by the nozzle lip, which excites the shear layer, and leads to the formation of a new coherent structure. The fundamental period T 0 of this feedback loop corresponds to the sum of the time necessary for a coherent structure to travel downstream from the nozzle to the plate and the time of propagation of an acoustic wave from the plate to the nozzle, yielding

T 0 = L 0 1 u c (z) dz + L c 0 = N + p f (9)
where u c (z) is the convection velocity in the shear layers, c 0 is the speed of sound, p is a phase lag at the nozzle exit, and the mode number N indicates the number of time the feedback mechanism occurs during the fundamental period T 0 . Powell 1 argued that the phase lag p is not necessarily zero because the reflection of the acoustic wave on the nozzle lips and the creation of a coherent structure in the shear layer are not simultaneous phenomena. In experiments, Krothapalli et al. [START_REF] Krothapalli | Flow field and noise characteristics of a supersonic impinging jet[END_REF] found for instance p = 0 for subsonic round jets and p = -0.4 for supersonic round jets. Later, Ho & Nosseir 3 and Nosseir & Ho 4 investigated the near field pressure of jets impinging on a flat plate normally, and proposed an empirical model for predicting the frequencies of the feedback mechanism, this model writes

N f = L < u c > + L c 0 (10)
where < u c > is the mean convection velocity of the large-scale structures in the shear layers between the nozzle and the plate. Here, the phase lag p is considered null. The Strouhal numbers of the dominant tone frequencies collected in table 3 for the present jets are shown in figure 18 as a function of the nozzle-to-plate distance L/r 0 . The experimental tone frequencies obtained by Henderson [START_REF] Henderson | The connection between sound production and jet structure of the supersonic impinging jet[END_REF] for underexpanded supersonic round jets characterized by Nozzle Pressure Ratios of 3.80, 4.03, 4.15 and 4.50 as well as the tone frequencies predicted by the expression (10) are also displayed. In the latter case, the mean convection velocity < u c > (L) is given by the proposed expression (8). A good agreement is found between the simulation and the experimental data. Moreover, the tone frequencies seem to be well predicted by the model of Ho & Nosseir 3 and Nosseir & Ho 4 using expression (8). The dominant tone frequencies in JetL4 are associated with the second and third modes of the model. As the nozzle-to-plate increases, a switch from the second mode to the third mode is obtained for JetL5, and a second switch from the third mode to the fourth mode happens in JetL7 and in JetL9. The dominant tone frequencies thus remain between St = 0.43 and St D = 0.645 for the present jets. Such a staging behaviour of the dominant tone frequency as the nozzle-to-plate distance varies, is typical of feedback mechanisms, see the experiments of Wagner 2 and Henderson, 29 among others.

IV.F. Fourier decomposition of the pressure field

The near pressure fields of the impinging jets have been recorded every 50 th time step and decomposed using a Fast Fourier Transform the same way as the pressure field of Jetfree. The amplitude fields thus obtained for the dominant tone frequencies are shown in the top views of figure 19, and the phase fields are displayed in the bottom views.

As there is no phase shift on both sides of the jet axis in figure 19(e), the fundamental tone frequency of JetL4 is associated with a varicose mode. This result is consistent with the axially pulsing motion of the Mach disk reported previously and with the experiments of Henderson et al. [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] who obtained symmetrical jet oscillations for a similar impinging jet. On the contrary, a 180 degree phase shift is visible with respect to the jet axis in the phase fields of the dominant tone frequencies of JetL5, JetL7 and JetL9, indicating a sinuous or helical mode. More precisely, based on a Fourier decomposition of the fluctuating pressure on 32 sensors regularly spaced in the azimuthal direction at z = 0 and r = 2r 0 , the mode is helical for the three cases.

Information on the sound sources are also provided by the amplitude and phase fields of figure 19. For a better description, let α be the angle at the impingement region between the upstream direction and the waves propagating from the flat plate, as illustrated in figure 19(a). For the dominant tone frequency of JetL4, in figures 19(a) and 19(e), two acoustic contributions are revealed. Indeed, two regions of constructive interferences appear in the amplitude field. The first acoustic contribution can be seen in the shear layers of the jet and in the near field on each side of the nozzle, for 0 ≤ α ≤ 20 degrees. This contribution corresponds to the upstream propagating acoustic waves which close the feedback mechanism. The second contribution is visible for 40 ≤ α ≤ 70 degrees and seems to be generated from a point located on the plate at r ≃ 3r 0 . This position coincides with an intense spot in the amplitude field. Destructive interferences can be seen for α ≃ 30 degrees between the two acoustic contributions in figure 19(e) with a strong reduction of the amplitude in this direction. The far-field noise in the sideline directions is thus mainly due to the second contribution and the localisation of the intense spot in the amplitude field on the plate at r ≃ 3r 0 in figure 19(a) corresponds to the acoustic source found experimentally by Henderson et al., [START_REF] Henderson | An experimental study of the oscillatory flow structure of tone-producing supersonic impinging jets[END_REF] for a similar impinging jet, on an annular region on the plate, at r = 2.6r 0 . For the dominant tone frequencies of JetL5 and JetL9, two acoustic contributions can similarly be noted. The second contribution seems to come from a point located on the plate at r ≃ 4r 0 . Finally, for the dominant tone frequency of JetL7, there is no clear destructive interferences. Only one acoustic contribution appears to propagate from the region of impact.

V. Conclusion

In this paper the flow and near pressure fields of a free jet and four impinging underexpanded jets have been presented. They have a fully expanded Mach number of M j = 1.56, a Reynolds number of Re j = 5×10 4 and an exit Mach number of M e = 1. The free jet is first investigated. Flow snapshots of vorticity, density and pressure as well as mean velocity fields are shown. The results, including the shock-cell structure, are consistent with experimental data, empirical and theoretical models. A tone frequency associated with screech noise is obtained in the near pressure field of the jet. Its frequency and helical nature agree well with the model of Panda [START_REF] Panda | Underexpanded screeching jets from circular, rectangular and elliptic nozzles[END_REF] describing the formation of an hydrodynamic-acoustic standing wave and with the measurements of Powell et al. [START_REF] Powell | Observations of the oscillation modes of choked circular jets[END_REF] The four impinging jets are then studied. Flow snapshots and the mean fields of the jets are presented and compared with experimental data and with the results of the free jet. The convection velocity in the shear layers of the jets is evaluated. An expression yielding the mean convection velocity between the nozzle and the flat plate, for nozzle-to-plate distances L ranging from 2r 0 to 10r 0 , is then proposed. The near pressure fields reveal several tone frequencies. A staging phenomenon of the main tone frequency with the nozzle-to-plate distance is found, which is typical of an aeroacoustic feedback mechanism occurring between the nozzle lip and the flat plate. The different models existing for the mechanism are presented and the model of Ho & Nosseir 3 and Nosseir & Ho 4 is applied using the expression (8) giving the average convection velocity < u c > (L). A very good agreement is found for the tone frequencies obtained. Finally, the nature of the dominant modes is identified. The results are consistent with experimental and numerical results. Notably, the strength of the feedback mechanism appears to be correlated with the formation of a Mach disk upstream of the plate, as noted by Henderson & Powell. 6 In further work, the underlying physics of the feedback mechanism will be explored. This will be done by extracting the most energetic modes by proper orthogonal decomposition and by using dynamic mode decomposition. The effect of the Mach disk position and motion on the feedback mechanism will then be studied.

Figure 1 .

 1 Figure 1. Representation of: (a) radial mesh spacings, and (b) axial mesh spacings: JetL4, . JetL5, JetL7, JetL9 and + + + Jetfree.

28

 28

Figure 2 .

 2 Figure 2. Snapshot in the (z, r) plane of the vorticity norm |ω| for Jetfree. The colour scale ranges up to the level of 10ue/D.

Figure 3 .

 3 Figure 3. (a) Snapshot in the (z, r) plane of the density in the jet and close to the flat plate, and of the fluctuating pressure for Jetfree. The colour scale ranges from 1 to 3 kg.m -3 for the density and from -2000 to 2000 Pa for the pressure; (b) Schlieren picture of an underexpanded jet obtained by André et al. 31 for a Mj = 1.55 and Me = 1 jet.

Figure 4 .

 4 Figure 4. Mean fields in the (z, r) plane of (a) axial and (b) radial velocities for Jetfree. The colour scale ranges from 0 to 600 m.s -1 for the axial velocity and from -150 to 150 m.s -1 for the radial velocity; the PIV results of André et al. 31 for a Mj = 1.55 and Me = 1 jet are displayed in the black rectangles.

Figure 5 .

 5 Figure 5. Normalized lengths of the first ten shock cells in -•the present jet, and the experimental jets of André et al. 31 with -• -Mj = 1.50, -• -Mj = 1.35, -• -Mj = 1.15 and -• -Mj = 1.10.

Figure 6 .

 6 Figure 6. (a) Position and (d) diameter of the Mach disk at the end of the first shock cell obtained: • for Jetfree, × in the experiment of Addy 30 and using expressions (4) and (5).

Figure 7 .

 7 Figure 7. Sound pressure level (SPL) at r = 2r0 and z = 0 as a function of the Strouhal number for Jetfree.

Figure 8 .

 8 Figure 8. Amplitude (a) and phase (b) fields in the (z, r) plane obtained for the dominant tone frequency of Jetfree at StD = 0.38.

39

 39

Figure 9 .

 9 Figure 9. • Dominant screech tone frequencies obtained in the experiments of Powell et al. 39 as a function of the fully expanded Mach number Mj ; × dominant screech tone frequency in Jetfree.

Figure 10 .

 10 Figure 10. Mean density field in the (z, r) plane for Jetfree. The colour scale ranges from 1 to 3 kg.m -3 . The black lines correspond to the position of the plates for the impinging jets.

Figure 11 .

 11 Figure 11. Isosurfaces of density for JetL9. The blue, cyan and red isosurfaces correspond to the values of 0.8, 1.25 and 2.5 kg.m -3 , respectively. The nozzle and the flat plate are represented in grey.

Figure 12 .

 12 Figure 12. Snapshots in the (z, r) plane of the vorticity norm |ω| for for (a) JetL4, (b) JetL5, (c) JetL7 and (d) JetL9. The colour scale ranges up to the level of 10ue/D.

Figure 13 .

 13 Figure 13. Snapshots in the (z, r) plane of the density in the jet and close to the flat plate, and of the fluctuating pressure for (a) JetL4, (b) JetL5, (c) JetL7 and (d) JetL9. The colour scale ranges from 1 to 3 kg.m -3 for the density and from -2000 to 2000 Pa for the pressure.

Figure 14 .

 14 Figure 14. Mean velocity fields in the (z, r) plane for (a) JetL4, (b) JetL5, (c) JetL7 and (d) JetL9. The colour scale ranges from 0 to 500 m.s -1 . The experimental results of Henderson et al. 8 are represented in the black rectangles.

Figure 15 .

 15 Figure 15. Root mean square values of velocity fluctuations for JetL9. The color scale ranges from 0 to 100 m.s -1 . The black line shows the position of the maximum values.

Figure 16 .

 16 Figure 16. (a) Convection velocity of the large scale structures in the shear layers for JetL4, . JetL5, JetL7, JetL9 and + + + Jetfree. The dashed vertical grey lines correspond to the end of the cells in the shock-cell structure of the free jet. (b) Mean convection velocity between the nozzle and the plate for: × the four impinging jets and the expression 8.

Figure 17 .

 17 Figure 17. Sound pressure levels (SPL) at r = 2r0 and z = 0 as a function of the Strouhal number for (a) JetL4, (b) JetL5, (c) JetL7 and (d) JetL9.

Figure 18 .

 18 Figure 18. Strouhal number of the tone frequencies obtained: • for the present impinging jets, experimentally by Henderson et al. 8 for N P R = 4.03 and × experimentally by Henderson 29 for N P R = 3.80, 4.15 and 4.50. The grey lines show the values predicted by the expression (10).

Figure 19 .

 19 Figure 19. Amplitude (top) and phase (bottom) fields obtained for the dominant tone frequency in (a) and (e) JetL4 (StD = 0.645), (b) and (f ) JetL5 (StD = 0.525) (c) and (g) JetL7 (StD = 0.44) and (d) and (h) JetL9 (StD = 0.43).

Table 1 .

 1 Jet parameters: fully expanded Mach number Mj , Reynolds number Rej , and nozzle-to-plate distance L/r0.

	r 0
	JetL4 1.56 5 × 10 4 4.16 5.6 JetL5 1.56 5 × 10 4 7.3 JetL7 1.56 5 × 10 4 JetL9 1.56 5 × 10 4 9.32 Jetfree 1.56 5 × 10 4 -

Table 2 .

 2 in order to increase the admissible time step of the simulation. Mesh parameters: number of points in the radial, azimuthal and axial direction, and total number of points.

	n r	n θ	n z	number of points
	JetL4 500 512 668 JetL5 500 512 764 JetL7 500 512 780 JetL9 500 512 847 Jetfree 500 512 1561	171 × 10 6 195 × 10 6 200 × 10 6 217 × 10 6 400 × 10 6

Table 3 .

 3 Strouhal numbers emerging in the spectra of figure17. The Strouhal number of the dominant tone in each case appears in bold.

Acknowledgments

This work was performed using HPC resources of P2CHPD (Pôle de Calcul Hautes Performances Dédiés) and IDRIS (Institut du Développement et des Ressources en Informatique Scientifique) under the allocation 2014-2a0204 made by GENCI (Grand Equipement National de Calcul Intensif). This work was performed within the framework of the Labex CeLyA of Université de Lyon, within the program "Investissements d'Avenir" (ANR-10-LABX-0060/ ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).