Ilyas Ed -Daoui

Abdelkhalak El Hami

Mhamed Itmi

Nabil Hmina

Tomader Mazri

Resilience assessment as a foundation for systems-of-systems safety evaluation: Application to an economic infrastructure

Keywords: Resilience, Risks assessment, Safety, Structural analysis

In this paper, the authors propose two complementary approaches in an attempt to contribute to systems-of-systems (SoS) safety evaluation through resilience assessment. The first approach is a risk monitoring design, it is conceived to monitor, evaluate and analyze risks that represent destabilizations' catalyzers. The second one is a structural analysis that begins with the estimation of criticality and frailty levels which leads to the calculation of failure impact and susceptibility measures of a component system on/to the SoS performance and process continuity. The combination of these approaches helps to assess SoS resilience through building a futurist, quantitative and anticipative perspective to evaluate the potential risks, their influences and impacts on SoS structure. Accordingly, this embraces a step towards safety forecast, evaluation and enhancement. A case study of a real-based economic infrastructure of a geographic area in France approached as a SoS model, is provided to experiment the proposition. The outcome of the presented approach's application shows that: (1) the use of the risk's monitoring dashboard helps to qualitatively illustrate risks striking the SoS or could possibly affect it in the future; (2) the structural analysis evaluates the impact of a component system's failure on the overall performance and efficiency of the SoS embracing it and vice versa; (3) the proposed approach could be used for anticipative and preventive reasons.

Introduction

Systems-of-systems (SoS) have received extensive attention from the scientific community in the past years and numerous definitions were proposed to sire this concept. Some of the potential definitions of SoS are enumerated:

• Jamshidi Mo: "SoS are large-scale integrated systems which are heterogeneous and independently operable on their own, but are networked together for a common goal" [START_REF] Itmi | A deterministic approach for systems-of-systems resilience quantification[END_REF], [START_REF] Jamshidi | System of systems engineering-New challenges for the 21st century[END_REF].

• Maier Mark W.: "SoS are a collection of systems that must have two features: its components must be able to operate independently by the whole system and they do operate independently, being managed at least in part for their own purpose" [START_REF] Maier | Architecting principles for systems-of-systems[END_REF].

• Department of Defense (USA): "A SoS is a large-scale composite system, which can realize specific function" [36].

• Xia Boyuan, Zhao Qingsong, Dou Yajie et al.: "SoS are special systems, they are composed of systems which can run independently and have their own benefits and values. Once the In this work, the authors answer to the demand of safety evaluation through resilience assessment. A risks monitoring dashboard is proposed for risks characterization for monitoring, prevention and anticipation purposes. They understand that it is mandatory to identify, monitor, classify and evaluate risks for a better protection of the system.

A structural approach dedicated to the assessment of SoS architecture is coupled with that. It aims to evaluate component systems failure impact and susceptibility on/to SoS performance and process continuity and vice versa. It represents an extension of the approaches presented in [START_REF] Filippini | A modeling framework for the resilience analysis of networked systems-of-systems based on functional dependencies[END_REF].

An application of the theory is done on a real-based economic infrastructure of a geographical area in France approached as a SoS. The economic infrastructure represents the internal facilities of a country that eases business activity, such as communication, transportation, distribution networks and markets. Component systems symbolize enterprises and geographical locations are differentiated by colors to emphasize the regional competitiveness.

The remaining part of the paper is structured as follows:

• Section 2 introduces the related literature • Section 3 outlines the contribution • Section 4 and section 5 describe the proposed approaches for resilience assessment • Section 6 explains the correlation between resilience and safety • Section 7 presents the application results of the theory on a case study • The last section summarizes the work and draws conclusions

Background

An interdisciplinary discussion has developed concerning how designers can incorporate resilience into the engineering of complex systems in general and especially in SoS.

As authors of [START_REF] Tran | A framework for the quantitative assessment of performance-based system resilience[END_REF] state, that there are multitudinous works and publications, from different domains, that attempt to lead the effort behind shedding more light on resilience: Ecological systems [START_REF] Holling | Resilience and stability of ecological systems[END_REF],

Safety engineering [START_REF] Woods | Engineering organizational resilience to enhance safety: A progress report on the emerging field of resilience engineering[END_REF], critical infrastructures [START_REF] Itmi | A deterministic approach for systems-of-systems resilience quantification[END_REF], [START_REF] Filippini | A modeling framework for the resilience analysis of networked systems-of-systems based on functional dependencies[END_REF], [START_REF] Turnquist | Design for resilience in infrastructure distribution networks[END_REF], [START_REF] Alderson | Operational models of infrastructure resilience[END_REF], communication networks [START_REF] Sterbenz | Evaluation of network resilience, survivability, and disruption tolerance: analysis, topology generation, simulation, and experimentation[END_REF], logistics and transportation networks [START_REF] Ip | Resilience and friability of transportation networks: evaluation, analysis and optimization[END_REF], [START_REF] Zhao | Analyzing the resilience of complex supply network topologies against random and targeted disruptions[END_REF] and organizational resilience [START_REF] Mendonça | Factors underlying organizational resilience: The case of electric power restoration in New York City after 11 September 2001[END_REF], [START_REF] Woods | Engineering organizational resilience to enhance safety: A progress report on the emerging field of resilience engineering[END_REF].

This section gives a brief overview of prominent resilience definitions. Following this overview, relevant frameworks and metrics for assessing resilience are discussed to establish a foundation for the proposed work.

Resilience definitions

Much of the early work focusing on resilience has been about proposing definitions and common properties of resilient systems. They appear within various scientific fields and are often tailored to specific applications of interest [START_REF] Tran | A framework for the quantitative assessment of performance-based system resilience[END_REF].

Therefore, to get a holistic view of resilience, the authors will briefly review insights from various disciplines. Although it is not their intent to provide an in-depth review of such diverse literature, they will refer to some definitions in an attempt to identify those commonalities.

Resilience is defined as a system's ability to continue operations or recover a stable functional state after a major mishap or event and prevent or adapt to changing conditions in order to maintain a system property or properties [START_REF] Leveson | Engineering resilience into safety-critical systems[END_REF]. However, this definition of resilience can hardly be distinguished from robustness which says that the system can maintain its function within a controlled tolerance under disturbances [START_REF] Zang | A review of robust optimal design and its application in dynamics[END_REF], [START_REF] Wang | Measurement of resilience and its application to enterprise information systems[END_REF].

Another definition of resilience has been proposed in [START_REF] Zhang | Is resilience the destiny for safety management paradigm? Presentation at the Northeastern University of China[END_REF], [START_REF] Zhang | Resilience engineering[END_REF], it is seen as a system's property that can still function to the desired level when the system suffers from a partial damage. A more generalized definition has also been proposed in (Merriam-Webster Online Dictionary) [START_REF] Erol | A framework for investigation into extended enterprise resilience[END_REF]. It is defined as the ability to recover from or adjust easily to misfortune or change.

Even in psychology, a definition of resilience has been proposed. It has been characterized as the positive capacity of individuals to cope with stress and catastrophic events and their level of resistance to future negative events [START_REF] Erol | A framework for investigation into extended enterprise resilience[END_REF]. While in computer networks, resilience has been expressed as the ability to provide and maintain an acceptable level of service in the face of faults and challenges to normal operations [START_REF] Hollnagel | Resilience engineering: Concepts and precepts[END_REF].

Considering discussions of resilience from a variety of communities, the common aspect of all these definitions is that resilience is defined as a response to unexpected or unforeseen changes and disturbances and the ability to adapt and respond to such changes [START_REF] Itmi | A deterministic approach for systems-of-systems resilience quantification[END_REF], [START_REF] Erol | A framework for investigation into extended enterprise resilience[END_REF].

Resilience engineering and assessment

Resilience Engineering is an emerging discipline [START_REF] Hollnagel | Resilience engineering: Concepts and precepts[END_REF] which aims to enhance an organization's ability to target safety investments proactively in the face of ongoing production and economic pressures [START_REF] Woods | Engineering organizational resilience to enhance safety: A progress report on the emerging field of resilience engineering[END_REF]. Methods and metrics for quantitatively assessing resilience are also proposed to enable rigorous and traceable comparisons between potential system designs [START_REF] Cutter | Disaster resilience: A national imperative[END_REF]. Several quantitative assessment methods have been proposed in the literature [START_REF] Tran | A framework for the quantitative assessment of performance-based system resilience[END_REF]. Authors propose discussions of resilience assessment from related literature.

In [START_REF] Reed | Methodology for assessing the resilience of networked infrastructure[END_REF], a method to characterize the behavior of networked infrastructure for natural hazard events and improve infrastructures resilience is proposed. It includes resilience and interdependency measures. Authors focused their study on the contribution of power delivery systems to post-event infrastructure recovery. The model is a component of a scheme that develops design strategies in order to increase the resilience of infrastructures for extreme natural hazard scenarios.

The goal is to capture the recovery aspects to identify the trends in interconnections in order to assist others who are developing the intricate models and databases required for regional planning and evaluation.

A framework for resilience engineering is proposed in [START_REF] Madni | Towards a conceptual framework for resilience engineering[END_REF]. Authors define resilience from different perspectives and provide a conceptual framework dedicated to analyzing disruptions and present principles for the creation of resilient systems. It includes disruptions, system attributes, methods and metrics. The idea behind such classification is to allow systems engineers to focus on what are the impacted attributes whenever resilience is needed and what methods are appropriate to achieve resilience.

They began by emphasizing that there is a reflex of misattributing systems failure and mishaps occurrence to human error. They also proposed a clarification to the difference between safety, reliability, survivability and resilience. Accordingly, they have emphasized that resilience engineering does not see failure as a breakdown, but, it is viewed as an inability of the system to either absorb perturbations or adapt to changes in real-world conditions.

In [START_REF] Filippini | A modeling framework for the resilience analysis of networked systems-of-systems based on functional dependencies[END_REF], an infrastructure resilience-oriented modeling language (IRML) is proposed to facilitate the analysis of operational interdependencies of infrastructure's components, resilience, the ability to withstand risks and recover.

The IRML comes with a set of analysis tools and procedures that investigate structural properties and resilience. Its analysis leads to a screening of structural and dynamic properties that are related to the resilient behavior of a SoS, in order to provide additional insights about possible misbehaviors at a large-scale.

In [START_REF] Zhang | On the principle of design of resilient systems-application to enterprise information systems[END_REF], authors define some principles to enhance enterprise information systems' resilience. They propose an architecture of what they call "resilient enterprise information systems". It is elaborated on a particular identity of resilience which is related to human as it is implicated in its safety and health. Authors see that resilience has roots in biological and ecological systems which leads to derive the proposed five design principles for resilient systems. These design principles are well applicable to enterprise information systems in order to be resilient. Table 1 summarizes the contribution and situates this work with regards to the current literature. As illustrated there is a need for further development in some aspects such as risks management, structural analysis, monitoring, resilience quantification and their influence on SoS safety. The contribution is an answer to this demand.

Contribution outline

This section is introductive. Explanations and details regarding each approach are all presented further in the paper. However, the authors outline the essential features of the proposition in order to put the reader in the context.

This work aims to address the assessment of resilience in SoS through two complementary approaches: one dedicated to risks management and the other to structural analysis, as illustrated in Figure 1.

Risks management approach is based on two important steps:

-Risks classification -Risks monitoring As a result, the monitoring (aspect) is explicitly included in the risk analysis. While structural analysis approach is based on:

-Dependency network -Criticality and frailty analysis -Failure impact and susceptibility calculations

Figure 1. Overview of the contribution

In fact, failure impact and susceptibility metrics are both structural metrics dedicated to the evaluation of a SoS dependence on each one of its component systems and vice versa (in addition to the capability of the SoS to face disturbances). This implies the use of a structural analysis in order to assess and measure resilience.

This paper tackles safety in SoS through risk and resilience analysis with respect to structural models in addition to operational and functional processes amid the SoS. The combination of the approaches (forming the contribution) quantitatively anticipates SoS resilience measurements in the architecting phase. This implicitly embraces a step towards safety evaluation and enhancement as the more the SoS is resilient and capable of performing as expected without failing, the safer it is.

Risks management

Risk model

As SoS have a special architecture with special properties as distribution, heterogeneity, complexity, etc. it is crucial to inspect the potential sources of risks that could disturb the operational and functional return of SoS for resilience assessment purposes.

Figure 2. Risks' classification

In this work, the authors consider any barrier that could continuously or in an intermittent manner discommode, interrupt or put an end to an interaction between two (or more) enterprises as a risk.

Enterprises are represented as component systems amid the SoS. In Figure 2, SoS risks are classified based on their natures and sources. Here are the main risks classes:

• Vulnerabilities • Obstacles • Emergences
Vulnerabilities represent the weaknesses of the system that can be the subject of possible exploitation and consequently put the system at risk. They also can be classified into two categories:

• Physical vulnerabilities: linked to the physical architecture of the system, i.e. the entities, the used links, machines and server rooms. An unauthorized access of a malicious person to the infrastructure may lead to titanic problems. • Logical vulnerabilities: related to the software, applications, protocols or procedures that can be exploited by a malicious activity may put the SoS at huge risks.

While obstacles represent the barriers that could possibly disturb, interrupt or intercept the interdependency between interacting systems. A taxonomy of obstacles is proposed, it will be adopted in the proposed approach. Here are the four classes and their definitions:

• Organizational obstacles: they concern human, legislative, decisional, financial obstacles, commercial approaches and enterprises' cultures that can discommode the interactions between enterprises.

• Functional obstacles: they are related to the incompatibility of procedures, norms and standards to present and communicate information, as well as the methods of work and technical incompatibilities that may perturb the interactions between communicating enterprises.

• Technical obstacles: they are related to the technical support of interactions. The authors classify them into two levels: logical and physical. Logical is about the obstacles related to exploited software, programs, etc. and physical is related to the physical architecture supporting the logical solutions.

• Geographical obstacles: they represent anything that blocks the pathway between two systems, this can be any natural feature such as mountains or even natural disasters that prevent the interaction from being successful.

Finally, emergence represents a principle in classical systems theory that generally suggests that system properties (patterns, capabilities, structure and behaviors) may be developed from the interaction of system elements [START_REF] Hitchins | Advanced systems thinking, engineering, and management[END_REF]. Emergences may represent prominent risks to the SoS if they affect its performance.

Other definitions are proposed to sire the concept of emergence. In [START_REF]Personal Communication[END_REF], emergent behavior is defined as what cannot be expected through analysis. While in [START_REF] Kuras | Complex-system engineering[END_REF], emergent behaviors refer to the properties arising from cumulative interactions between systems inside the SoS.

In complex systems, this notion generally includes the following commonly held points [START_REF] Jamshidi | System of systems engineering-New challenges for the 21st century[END_REF]:

• Emergent properties exist only at the system level.

• Emergent properties are not held by any of the isolated elements.

• Emergent properties are irreducible. They simply cannot be understood, explained, or inferred from the structure or behavior of constituent elements or their local properties.

• Understanding cause-effect relationships can only be established through retrospective interpretation. This renders traditional reduction-based analytic techniques are incapable of useful predictions of emergent system-level behavior Figure 2 summarizes the detailed classification of risks. To effectively deal with them, an appreciation of the philosophical, methodological and axiomatic underpinnings is required. The non-governance of the disorder at the very beginning can complicate the restoring of systems' performance after an incident.

Risks monitoring

The use of the dashboard is an attempt to illustrate, preferably in real-time, qualitative indicators related to risks that are striking the SoS at a given time and in a geographic location or could possibly affect it in the future. The dashboard could be used for both anticipative and preventive reasons. For an optimal exploitation of the dashboard and effective anticipation, it is more advisable to apply it, similarly, on every single dependency and try to anticipate as many scenarios as possible.

Figure 3. Dashboard for risks' supervision

It is worth noting that the elements included in the dashboard, shown in Figure 3, are not exhaustive. The authors prefer to call them control points, as they are used to determine different risk characteristics and implicitly the state of the SoS at a given time.

Figure 4. Example of a dashboard for risks' monitoring

The examined control points may change according to the studied SoS. The idea behind the proposition of the dashboard is not to propose a standard for SoS monitoring but to emphasize the importance of monitoring in such context and suggest an outline of essential features.

Let us examine the key elements included in the dashboard in order to understand their use:

• The origin of risk: in order to correctly address a risk, it is crucial to know its origin which also reflects is nature. Besides, knowing where a risk came from helps to understand the risk and elaborate pertinent countermeasures. In fact, there are numerous sources of risks, it could be environmental, human, technical, etc. Accordingly, a risk may be intentional i.e. it could be organized, managed and targeting a vulnerability in the SoS, in this case, the origin may be internal (e.g. coming from component systems users) or external (e.g. as a consequence of an environmental disaster). Or it could be unintentional e.g. as in the case of an environmental risk or a human intervention that led accidentally to a problem.

• The severity of the risk: it is very important to know how much the SoS performance has degraded. For this reason, the authors propose a classification of degrees of nuisance according to the degree of the SoS disturbance:

o It is called 1st degree if it is quick and does not disturb the performance of the SoS. o It is called 2nd degree if it remains weak but affects slightly the performance of the SoS for a short period of time before the system returns to its initial state. o It is called 3rd degree if it is able to significantly disrupt the performance of the SoS. o It is referred to as a 4th degree if it may provoke an interruption to the SoS performance and it becomes difficult for it to return to its initial state.

o It is called 5th degree if it can cause a breakdown of the system which makes it impossible for him to regain its initial state • The duration of the risk: represents the duration that took (or may take) a system to resist the risk. As the risk may be instant or slow, the resistance duration also changes according to the risks duration. Authors insist on the fact that this has no relation to the degree of severity of the risk.

• The duration of the disturbed state: represents the period where the system leaves its initial state (this depends on the degree of the risk and its duration). In some cases, it may be significantly greater than the duration of the risk, and this may be due to several factors including the degree of risk and the criticality of the systems amid the SoS undergoing this risk. The notion of criticality will be discussed further in this paper.

• The failure rate: represents the rate of component systems that failed to return to their initial states after the occurrence of the risk.

FR % = Number of failed systems Number of component systems

× 100 1

• Risk's type: refers to the class of the risk according to the risk model in the third section.

A major reason why risks may occur and may have predominant consequences is the existence of vulnerabilities. They have existed since the system was implemented. Some of them can be planned from the design stage to be corrected before the system is built, others can be unpredictable and become identifiable only after the SoS has been set up. This triggers the need for frequent maintenance of the system's infrastructure, entities, links, programs and software in order to fix them.

But, why do we need to monitor risks?

First, there are preventive reasons as it is important for engineers and management authorities to have an anticipative and futurist perspective to the SoS behavior, interdependencies' states and overall performance. This helps them to be prepared for eventual risks.

The second reason behind monitoring is a real-time supervision and protection of the SoS. The proposed approach helps to get the real-time state of the performance of the system. In case of a problem, the supervision authority is notified right away. Therefore, some countermeasures to be considered.

The general idea behind the use of a risks monitor is to reduce the response time of the SoS to face a risk as the more the problem is identified earlier the more it is handled efficiently and its consequences are limited.

Structural analysis

In this section, an approach based on structural analysis is presented. It aims to evaluate the impact of a component system's failure on overall performance and efficiency of the SoS embracing it and vice versa.

The structural analysis is based on the assessment of functional dependencies between systems. This leads to the evaluation of the criticality and frailty levels of each component system on/to the group it belongs to. Finally, the failure impact and susceptibility of a component system on/to the performance of the whole SoS are deducted.

This gives us an idea about the dependability of the global system on each component system. In addition, this process should be applied, similarly, on every single component system based on the SoS structural architecture in order to be able to locate impactful and frail component systems.

In this section, the structural analysis is detailed via four important steps:

-First is the dependency network elaboration -Second is the frailty and criticality assessment -Third one is the failure impact metric -The last one is the susceptibility metric

Dependency network

The idea behind dependency analysis is to focus on workflow pathways and directions, as it is illustrated by black arrows in Figure 5. The analysis of dependencies' set emphasizes the functional dependencies relevance. In addition, it identifies clearly the process sequencing by representing functional services to be acquired by systems and dependencies between the systems or between the capabilities by links.

A SoS can be given a topology that accounts for the static representation of its components and the manner they interact and cooperate [START_REF] Itmi | A deterministic approach for systems-of-systems resilience quantification[END_REF], [START_REF] Mazri | Security Enhancement Architectural Model for IMS based Networks[END_REF], [START_REF] Ed-Daoui | Towards Reliable IMS-based Networks[END_REF], [START_REF] Filippini | A modeling framework for the resilience analysis of networked systems-of-systems based on functional dependencies[END_REF]. The idea is to focus on the component's interface, where data, services and quantities are exchanged through functional relationships, i.e. functional dependencies.

It is important to evaluate the effect of topology and possible systems' performance degradation on the SoS as it helps us implicitly evaluate its resilience and capability to face partial failures and component systems' loss of operability.

Frailty and criticality analysis

In [START_REF] Itmi | A deterministic approach for systems-of-systems resilience quantification[END_REF][START_REF] Filippini | A modeling framework for the resilience analysis of networked systems-of-systems based on functional dependencies[END_REF], frailty (or vulnerability, as it is called in the cited reference. The word vulnerability is not used here as it is exploited to express a class of risks) and criticality sets are presented as structural properties that can be analyzed in the dependency network. A system is affected by the systems on which it depends on and it is critical to the systems depending on it. The dependency is related to the workflow pathway between component systems.

Figure 5. Frailty and criticality positions towards workflow pathway

Figure 5 represents a simple example illustrating three systems. The idea is to locate frailty and criticality sets for system '2' with regards to the workflow pathway. System '2' is critical to system '3' and frail to system '1' at the same time. This depicts the difference between frailty and criticality and their positions towards the SoS workflow pathway. The criticality represents how much the process continuity of the SoS relies on the systems and groups. Practically, it is calculated by using the formula (2). While the frailty, in SoS context and according to [START_REF] Filippini | A modeling framework for the resilience analysis of networked systems-of-systems based on functional dependencies[END_REF], represents how the component system relies on the process continuity. It is calculated by using the formula (3). At this stage, the SoS' groups are supposed to be represented by the set 2# 9 , # : , … 5. Moreover, frailty metric values range goes from 0 for not frail at all to 1 for extremely frail. The frailty value may be multiplied by 100 in order to get the criticality rate.

Failure impact calculation

Failure impact is a structural metric conceived to measure each system's failure impact on the rest of component systems and SoS viability with consideration to the repartition of the SoS into groups. The failure impact value of a system is obtained by multiplying its criticality value (with correspondence to its position towards the process inside the containing group) by the same group's criticality value (corresponding to the process inside the SoS). As it is shown in formula (4). ∀ . ! , ; 0 / ∈ # × 7: 7<. !0 / = %&'(')*+'(, =>?@AB ! × %&'(')*+'(, CDEFG .; 0 / 4 With:

' ∈ 1, 2, … , %*&-# 6 ∈ 1, 2, … , %*&-7
! : represents a system inside the group ; 0 .

; 0 : represents a group inside the SoS.

%*&-I 0 : represents the total number of systems forming the group.

%*&-7 : the total number of groups forming the SoS.

%&'(')*+'(, =>?@AB values range goes from 0 for not critical at all to 1 for extremely critical. %&'(')*+'(, CDEFG is equal to 1 in case there is no dependency between groups.

Furthermore, groups criticality values are calculated following the same tactic that has been adopted to calculate each component system's criticality on the rest of component systems within the same group, with consideration of itself. This means that in addition to the groups following the same workflow pathway, the group in question joins to the group's criticality set.

The failure impact metric takes into account all variables taking part of the system's forming. If a system has a high failure impact that means that an important part of the SoS could be affected in case of its deficiency. This means that the infrastructure is not resilient and robust enough to overcome its failure.

Susceptibility calculation

Contrarily to the failure impact metric, susceptibility is a metric that evaluates component systems fragility to the process continuity, with consideration to the repartition of the SoS in question into groups.

The susceptibility of a component system inside a SoS is obtained by the multiplication of its frailty (with correspondence to its position towards the process inside the containing group) by the frailty value of the same group (corresponding to the process inside the SoS). As it is shown in formula [START_REF] Bukowski | System of systems dependability-Theoretical models and applications examples[END_REF]. ∀ . ! , ; 0 / ∈ # × 7: J. !0 / = 7&*'+(, =>?@AB ! × 7&*'+(, CDEFG .; 0 / 5 With:

' ∈ 1, 2, … , %*&-# 6 ∈ 1, 2, … , %*&-7
! : represents a system inside the group ; 0 .

; 0 : represents a group inside the SoS.

%*&-I 0 : represents the total number of systems forming the group.

%*&-7 : the total number of groups forming the SoS.

7&*'+(, =>?@AB values range goes from 0 for not frail at all, which means that the component system is independent inside its group and does not receive any workflow from any component system, to 1 for extremely frail, which means that the component system receives flaw from all systems inside the same group. 7&*'+(, CDEFG is equal to 1 in case there is no dependency between groups.

Correspondingly, the calculation of the frailty of each group on the rest of groups within the SoS is done following the same tactic that has been adopted to calculate the criticality of each group on the rest of groups within the same SoS, with consideration of itself. This means that in addition to the groups following the same workflow pathway, the group in question joins the group's frailty set.

Failure impact and susceptibility metrics are both structural metrics dedicated to the evaluation of a SoS dependence on each one of its component systems and vice versa. This implies the evaluation of SoS resilience and capability to overcome disturbances.

The correlation between resilience and safety

One the founding principles of safety science is the need to take a systems approach to understand how an organization or a composition of components succeeds and sometimes fails in managing increasingly complex systems in more highly pressured contexts [START_REF] Reason | Managing the risks of organizational accidents[END_REF]. A systems approach to safety in complex systems requires a shift in how to study, model and measure operational processes [START_REF] Woods | Engineering organizational resilience to enhance safety: A progress report on the emerging field of resilience engineering[END_REF], [START_REF] Hollnagel | Resilience engineering: Concepts and precepts[END_REF], [START_REF] Zhang | Restoration resource allocation model for enhancing resilience of interdependent infrastructure systems[END_REF], [START_REF] Tran | A framework for the quantitative assessment of performance-based system resilience[END_REF], [START_REF] Cedergren | Challenges to critical infrastructure resilience in an institutionally fragmented setting[END_REF], [START_REF] Patriarca | Resilience engineering: Current status of the research and future challenges[END_REF], [START_REF] Stroeve | Agent-based modelling and mental simulation for resilience engineering in air transport[END_REF], [START_REF] Harvey | Safety in System-of-Systems: Ten key challenges[END_REF], [START_REF] Alexander | Supporting systems of systems hazard analysis using multiagent simulation[END_REF].

In safety science literature, resilience represents the ability of a system to "adjust its functioning prior to, during, or following changes and disturbances, so that it can continue to perform as required after a disruption a major mishap, and in the presence of continuous stresses" [START_REF] Hollnagel | The four cornerstones of resilience engineering[END_REF], [START_REF] Cedergren | Challenges to critical infrastructure resilience in an institutionally fragmented setting[END_REF], [START_REF] Cedergren | Challenges to critical infrastructure resilience in an institutionally fragmented setting[END_REF], [START_REF] Patriarca | Resilience engineering: Current status of the research and future challenges[END_REF]. The introduction of the concept of resilience in safety science has contributed to shifting focus from including strength to tackle functional recovery and survivability [START_REF] Hollnagel | Resilience engineering: Concepts and precepts[END_REF], [START_REF] Cedergren | Challenges to critical infrastructure resilience in an institutionally fragmented setting[END_REF]. The same goes for the relationship between reliability and resilience, as one of the recent definitions of reliability is related to resilience [START_REF] Avizienis | Basic concepts and taxonomy of dependable and secure computing[END_REF], [START_REF] Itmi | A deterministic approach for systems-of-systems resilience quantification[END_REF], [START_REF] Birolini | Reliability engineering: theory and practice[END_REF].

In this study, the authors tackle safety in SoS through risks and resilience analysis with respect to structural models in addition to operational and functional processes of the SoS. Risks analysis addresses menaces that could possibly affect component systems. While the structural analysis anticipates the impact of these menaces on the architecture of the SoS and implicitly on the process continuity. These approaches are effective manners to anticipate risks, their influence and impact on the SoS. The combination of both approaches quantitatively anticipates resilience measurements of SoS in the architecting phase. This implicitly embraces a step towards safety evaluation and enhancement.

Accordingly, the authors insinuate by process continuity the resumption of the performance of systems, groups and the SoS after the occurrence of a disturbance. The correlation between the concept of process continuity and the proposed metrics is that the anticipation of the impact of a risk, based on a structural analysis, can help to foresee its impact on the performance on SoS and the process continuity after recovery.

Safety and resilience concepts are two strongly related notions. This study aims to emphasize the mutual correspondence between the two concepts. Resilience evaluation and assessment implies the implicit evaluation and assessment of the safety. See Figure 1 that represents an overview of the proposed model which also explains the correspondence between resilience and safety.

With this in mind, when a reorganization of a SoS architecture is triggered by an application of the proposed combination of approaches, it assesses the SoS structure and evaluates its capability to survive and keep performing as expected during and after the disturbance(s). Therefore, this implicitly concerns the safety enhancement as the more it is resilient and capable of performing as expected without failing, the safer it is. Another theory is that resilience assessment through this proposition contributes to the quantitative anticipation of the degree of safety of the SoS.

Case study

In this section, a projection of the analysis and approach is done on a case study (see Figure 6). An economic infrastructure of an area in France is represented by the global SoS. Geographical locations are differentiated by colors in order to emphasize territorial region and regional competitiveness. They form three groups of fourteen enterprises presented as component systems. Dependencies' arrows represent both production lines and relations between enterprises.

Figure 6. Dependency network of the studied SoS

In fact, economic infrastructure refers to the internal facilities of a country that ease business activity, such as communication, transportation, distribution networks and markets.

A SoS can be given a topology that accounts for the static representation of its components and the manner they can interact and cooperate [START_REF] Itmi | A deterministic approach for systems-of-systems resilience quantification[END_REF], [START_REF] Filippini | A modeling framework for the resilience analysis of networked systems-of-systems based on functional dependencies[END_REF]. The idea is to focus on the component's interface, where services and quantities are exchanged through functional relationships, i.e. functional dependencies.

Risk's monitoring application

In practical terms, each aspect of organizational, functional, technical and geographical barriers is evaluated by virtue of the taxonomy presented in section 4. Therefore, if an interdependency has a barrier or an obstacle that prevents the interaction from being successful it should be mentioned on the dashboard, represented by Table 2, as the latter embraces all information about the obstacles hindering the SoS viability in order to effectively address them.

The authors aim to evaluate the interdependencies between enterprises that are considered as component systems within the studied SoS through risks' classification. They note that this case study is based on a real case and the information within Table 2 is based on real information.

Frailty and criticality calculations

Figure 7. Frailty and criticality values distribution of all component systems

Accordingly, 'A' and 'G' are the most critical component systems in their groups by reaching around 0.8 in criticality values, which means that more than 80 % of each single group is affected by their failures.

Figure 8. Groups criticality and frailty values distribution

The calculation of the criticality and frailty levels of each group on the rest of groups within the SoS is done as presented previously. In Figure 8, second and third groups are the most critical one among the three. While, the first group is the frailest group among the others. This seems logical because if we return to Figure 6 illustrating the functional dependencies of the SoS, we intuitively deduct that the first group represents the end of the workflow pathway between groups within the studied SoS, hence its frailty. In addition, since both second and third groups represent simultaneously the start of the workflow pathway, they are equally critical to the first group and implicitly to the global system. Figure 10 illustrates the contrast between criticality, frailty, failure impact and susceptibility metrics. It demonstrates through the studied SoS that a highly critical component system does not necessarily have a high failure impact on the SoS and global process continuity. The same thing goes for frailty and susceptibility.

Failure impact and susceptibility calculations

And by highly critical, the authors mean that the element in question, whether it is a system or a group, is impactful more than the other elements.

Towards safer economic infrastructure

Practically, there are numerous manners to address safety issues through the results generated from the combination of both risks management and structural analysis approaches. In this case study, it is supposed that the SoS is already established. Thus, a structural enhancement is triggered in order to assure SoS safety. Accordingly, the authors propose some measures in order to enhance the SoS safety, therefore, the economic infrastructure's safety.

Some measures can be triggered by the results generated from the risk's monitoring dashboard, which evaluates the interdependencies between enterprises (approached as component systems). For example, we can address risks by their severity order (from the most influential to the least).

An alternative to address SoS safety is proposed by failure impact results, which locate and classify impactful component systems on SoS safety and process continuity. For example, we can address dependencies linking component systems by respecting their failure impact order (from the most influential to the least).

Another alternative is proposed by susceptibility calculations which locate frail component systems.

For example, we can address issues related to dependencies linking component systems by respecting their susceptibility order (from the most influenced to the least).

Moreover, a more balanced option lays in equilibrating between approaches results and avoiding prioritization between them instead of merely depending on one approach. Other aspects could be considered for pertinent safety enhancement, for instance: (political, strategic, natural, etc.) environment embracing the economic infrastructure, expectations from it, the process amid the SoS, etc.

The authors believe that the proposed approaches can be useful at the conceptual level (SoS architects could use these approaches in order to conceive a well-balanced and safe SoS) and after the construction of the SoS (as engineers could use the presented approaches in order to enhance structural SoS safety).

Epilogue

In this work, the authors responded to the concerns related to SoS safety through resilience assessment by managing risks and analyzing the structural architecture of SoS. They proposed an approach to anticipate risks, their influences and impacts, which contributes to the quantitative anticipation of SoS resilience and safety. This implicitly embraces a step towards safety evaluation and enhancement. It is authors' belief that safety and resilience concepts are two strongly related notions. This study aims to emphasize the mutual correspondence between the two concepts.

An application of the theory is effectuated on a real-based economic infrastructure of a geographic area in France approached as SoS approach. The economic infrastructure represents the internal facilities of a country that eases business activities, such as communication, transportation, distribution networks and markets.

As a perspective, the authors believe that this work can be extended to cover this limitation. Furthermore, a proactive approach for SoS resilience assessment while integrating new systems and removing existing ones could also be a judicious perspective; since it is common for SoS to be heterogeneous and to support systems integration and segregation while maintaining the performance.

In addition, the automatization of the reorganization process is also an important perspective as the current proposal requires the existence of a management authority for the evaluation and the rearrangement of the SoS architecture.

  -.# 0 /: the number of component systems forming the group embracing the component system ! .%*&-.% ! /: represents the number of component systems that are directly or indirectly affected by the failure of the system ! . The component systems should be in the same group as the system ! .

  ∀ ! ∈ # : 7&*'+(, ! = %*&-.7 ! / %*&-# 0 3 With: ' ∈ 21, 2, … 5 and 6 ∈ 21, 2, … 5 %*&-.# 0 /: the number of component systems forming the group embracing the component system ! . %*&-.7 ! /: represents the number of component systems that affect directly or indirectly the system ! by their failures.

Figure 7

 7 Figure 7 illustrates frailty and criticality values distribution across the studied SoS. It is evident that component systems 'D', 'E', 'F' and 'K' are "weaker" than the others as their frailty values are around 0.8 which means that they are affected by the failure of more than 80 % of their groups.

Figure 9 .Figure 9

 99 Figure 9. The distribution of failure impact and susceptibility values of all component systems

Table 1 . Literature positioning towards different aspects siring the concept of resilience Risk analysis Structural analysis Monitoring Resilience quantification and measurement Safety Reliability Recovery Reed et al. [21] x x x Filippini and Silva [10] x x Zhang and Lin [31] x x Tran et al. [40] x x Liu et al. [33] x x Wang et al. [32] x Ed-daoui et al.

 1

	[7]		x		x	x
	This work	x	x	x	x	x

Table 2 . Illustration of the risks' monitoring dashboard for the studied case Interdependence Class of risk The origin of the risk

 2

	The severity of the risk	The duration of the risk	The duration of the disturbed state	The failure rate

Acknowledgements

This research is supported by the European Union (EU) with the European Regional Development Fund (ERDF) and the Regional Council of Normandie.


S Baleriola

A Leroy

S Loyer

Philippe Devinant

Sandrine Aubrun-Sanches

S Aubrun

M Forte

J Jolibois

J Pons

Scaling investigation of plasma-induced flows over curved and flat surfaces: Comparison to the wall jet

Keywords: DBD, curvature effect, plasma wall jet

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

Introduction

In the field of active flow control, plasma-based actuators are widely used to achieve flow manipulation and enhance aerodynamic performances over a large range of applications (e.g., aerospace, wind energy, aeronautics, turbomachinery). However, these kinds of actuators are not yet mature for industrial applications as reported in [START_REF] Kriegseis | Towards in-flight appli-1097 cations? a review on dielectric barrier discharge-based boundary-1098 layer control[END_REF]. Nevertheless, surface plasma actuators based on a Dielectric Barrier Discharge (DBD) are frequently studied because of their easy implementation, short time-response and versatility. In the domain of flow control over airfoils, many studies have analysed the aerodynamic effects of this actuation which can be used to successfully achieve boundary layer separation control [START_REF] Post | Separation control on high angle of attack 1145 airfoil using plasma actuators[END_REF]), circulation control by acting near a rounded trailing-edge [START_REF] Zhang | Airfoil flow control using plasma actuation and coanda effect[END_REF], [START_REF] Meijerink | Plasma actuators for ac-1116 tive flow control on wind turbine blades[END_REF], [START_REF] Pereira | Effect of external flow veloc-1136 ity on momentum transfer of dielectric barrier discharge plasma 1137 actuators[END_REF] and [START_REF] Baleriola | Cir-1000 culation control on a rounded trailing-edge wind turbine airfoil 1001 using plasma actuators[END_REF] or by the implementation of Gurney flaps with plasma induced jets (Feng et al. (2015)), and boundary layer transition control [START_REF] Schuele | Control of stationary cross-1164 flow modes in a mach 3.5 boundary layer using patterned passive 1165 and active roughness[END_REF], [START_REF] Lovig | Passive laminar flow con-1104 trol at low turbulence levels[END_REF] and [START_REF] Yadala | Experimental control of swept-wing 1192 transition through base-flow modification by plasma actuators[END_REF]) among others.

As in all flow control studies, the analysis in quiescent air conditions of the flow induced by DBD actuators needs model designed with a specific rounded trailing-edge (TE) as reported in [START_REF] Baleriola | Cir-1000 culation control on a rounded trailing-edge wind turbine airfoil 1001 using plasma actuators[END_REF]. Due to the dimensions of the model designed for wind-tunnel tests, the curvature radius of the rounded trailing-edge is of an order of magnitude equal to a few millimeters. More conventional circulation control strategies are traditionally implemented by means of concepts including shape change, TE flaps or air blowing and often rely on the Coanda effect that keeps a tangential jet attached over a curved surface.

Whereas blowing techniques usually show high momentum coefficients (Joslin and Jones ( 2006)), it is not possible to achieve similar performances so far with DBD actuators due to their inherent limitation to produce a high level of momentum, in spite of the promising outcomes of the numerical approach proposed by [START_REF] Zhang | Numerical simulation on plasma circulation control airfoil[END_REF]. According to Joslin and Jones (2006) and demonstrated by [START_REF] Pereira | Effect of external flow veloc-1136 ity on momentum transfer of dielectric barrier discharge plasma 1137 actuators[END_REF] and [START_REF] Baleriola | Cir-1000 culation control on a rounded trailing-edge wind turbine airfoil 1001 using plasma actuators[END_REF], lift increase achieved thanks to circulation control is mainly due to a mechanism of boundary layer separation manipulation around the trailing-edge which modifies the mixing of the near wake without totally suppressing it. Nevertheless, by considering specific arrangements of electrodes, it is possible to produce a tangential jet over a curved wall with a small radius of curvature that will stay more or less attached to the surface. Several DBD actuators were used for the intended flow control application. Their induced flows were characterised in quiescent air conditions and constitute the working basis of this study.

Generally, DBD induced jets are compared to fluidic wall jets even if some differences remain between them.

In the case of a DBD induced jet, the electrically charged particles close to the wall are accelerated from the active electrode and along the grounded one, following the electrical field. This particle motion entrains the ambient air in the same direction along the wall, creating an induced jet. There is no nozzle or jet exit and the flow accelerates continuously until roughly the end of the last grounded electrode. The velocity magnitude induced by a single DBD is weak (max. ≈ 8 m/s as in [START_REF] Moreau | Airflow control by non-thermal plasma actuators[END_REF]) compared to most fluidic jets. In the case of a DBD induced jet, there is no mass addition by the actuator but only mass entrainment and momentum transfer.

The present study focuses on the experimental characterisation by PIV (Particle Image Velocimetry) of the plasma induced jets evolving along a specific airfoil with a rounded trailing-edge as a function of the electrode arrangement and without an external flow. In order to highlight the effects of possible centrifugal forces on curved induced jets, a DBD induced jet in a flat configuration equivalent to the curved ones was also studied. Induced jets main features were analysed by using dimensional and normalised variables commonly used for fluidic wall jets.

This approach made it possible to assess the transition of such plasma induced jets from a laminar to a turbulent regime and to evaluate the evolution of DBD induced flows in terms of self-similarity of the mean flow and of its turbulent components as well as in terms of rate of spread and maximum velocity decay evolution. Furthermore, this 109 analysis enabled us to determine whether or not plasma in- (Reynolds number between 100 and 600) and found that Kelvin-Helmholtz instability of the free shear layer plays an important role in the WJ transition. [START_REF] Schneider | Laser doppler measurement of 1161 turbulence parameters in a two dimensional plane wall jet[END_REF] carried out a detailed analysis on the turbulent WJ evolution with three different measurement techniques (pressure probe, Laser Doppler Anemometry and Hot Wire Anemometry). [START_REF] Bradshaw | Turbulent Wall Jets with and without 1028 an External Stream[END_REF] carried out experiments on a WJ developing over a circular cylinder. The growth of the curved WJ thickness proved to be linear and equal to 1.5 times that of the plane WJ. According to the authors, this thickness increase may be caused by a negative radial gradient of angular momentum that causes a centrifugal instability on the turbulent flow. The authors defined an angle θ * from the nozzle to the azimuthal position of the velocity profile. In their experiment, the curved WJ remained attached to the wall until an angle θ * of 150 • . An in-depth study on the plane WJ was carried out by [START_REF] Wygnanski | On the applicability of var-1187 ious scaling laws to the turbulent wall jet[END_REF] and pursued by [START_REF] Neuendorf | On a turbulent wall jet flowing over 1133 a circular cylinder[END_REF] who also examined the WJ development around a circular cylinder. The rate of spread and the velocity decay of the curved WJ appeared to be greater than those of the plane WJ, as were the levels of turbulence and Reynolds stresses.

Wall jets induced by DBD actuators

Wall jets induced by DBD actuators (DBD-WJs) have been studied only on plane configurations. Table 1 shows a summary of DBD-WJ characteristics found in the literature. Only publications showing velocity profile investigations relevant to this study are cited. The table specifies the type of actuator, the method used for the measurement of the velocity profiles, the maximum jet velocity U j and its corresponding thickness (maximum velocity halfwidth) b and jet Reynolds number Re j . A precise definition of these variables is given below in section 3.1. For thick dielectrics (thicker than 2 mm) the jet velocities are between 3 and 5 m/s and jet thicknesses between 2 mm and 5 mm at the end of the last grounded electrode (beginning of jet diffusion). This leads to Reynolds numbers from 400 to 900 suggesting that the plane DBD induced jets found in the literature are laminar or in a transitional regime. DBD-WJs seem to be similar to laminar WJs as pointed out by [START_REF] Roth | Flow field measurements of paraelectric, peristaltic, and combined 1155 plasma actuators based on the one atmosphere uniform glow dis-1156 charge plasma[END_REF] and Jukes et al. (2006). As for fluidic WJs, jet velocity decay seems to evolve as x -1 2 and jet thickness as x As mentioned in [START_REF] Prasad | Particle Image Velocimetry[END_REF], PIV errors can be decomposed in: (1) random errors, (2) bias errors, (3) gradient errors, (4) tracking errors and (5) acceleration errors. With the exception of random and bias errors, these errors are not deterministic and it is therefore difficult to give a quantitative estimation of the overall systematic errors on the measured velocity fields. However, experiments were carried out carefully in order to minimise these errors. Wall reflections were reduced with the use of non-reflective coatings and the ensemble average of minimum intensity was subtracted from the raw images during the pre-processing as mentioned in Kotsonis (2015). Also, particle diameter was equal to about 2 px as suggested in [START_REF] Prasad | Particle Image Velocimetry[END_REF]. The frame time interval was chosen such as particles are displaced of 1 4 of the final interrogation window as mentioned in Keane and Adrian (1990). Furthermore, during the post-processing, two passes were used with decreasing interrogation areas and a 50% overlap as also advised in [START_REF] Prasad | Particle Image Velocimetry[END_REF]. Vectors were accepted if the ratio between the 1st and the 2nd correlation peaks was above 1.2 and a median filter was applied to the resulting 351 velocity field.

352

To quantify random errors, statistical uncertainty of the 353 mean velocity can be computed using the central limit

359

The statistical error u r is defined as:

360 u r = 1 √ N s z α 2 U RM S U (1)
where N s is the number of samples, z α 2 is equal to 1.96 for

Plane and curved wall jet variables

To analyse the wall jet development over a plane surface some quantities are usually defined as shown in figure 5. A new coordinate system related to the WJ and different from the one presented in the section above is defined here. It is defined in such a way that the abscissa x is in where b is the nozzle width and ν is the kinematic viscosity of the flow. In the case of a plane DBD-WJ (see figure 6), as there is no physical nozzle, the origin of the x coordinate (x = 0) is set at the beginning of the first active electrode and the abscissa of the virtual origin x 0 , corresponding to the maximum jet velocity, is also defined as

U max (x 0 ) = U j .
The virtual slot width b is defined as the jet thickness at the virtual origin b = z 1 2 (x 0 ). Assuming that the surface discharge has a negligible effect on the gas properties as mentioned in [START_REF] Moreau | Airflow control by non-thermal plasma actuators[END_REF], the kinematic viscosity of the induced jet is assumed to be equal to the kinematic viscosity of air. Then, the jet Reynolds number

Re j can be defined by analogy with the fluidic WJ defined above.

ν 2 J R Velocity scale U j J ν J R Jet Re Re j = bUj ν - Re N = √ JR ν = 1 2 U 2 j b * R ν 2 Local Re Re x = Umax(x)z 1 2 (x) ν - Re N (x * ) = √ J(x * )R ν
Table 2: Summary of the three employed scalings

WJs (scaling 3) involving the radius of curvature R as the length scale and J R as the velocity scale. For both scalings 2 and 3 the kinematic momentum flux J(x) (or J(x * ))

is defined as:

J(x) = ∞ z=0 U 2 (x, z)dz = z 1 2 (x)U 2 max (x) ∞ z z 1 2 =0 U (x, z) U max (x) 2 d z z 1 2 K(x)≈constant = z 1 2 (x)U 2 max (x)K(x) (2)
which gives at the nozzle or virtual origin x 0 (or x * 0 ):

J 0 = J(x 0 ) (3) = z 1 2 (x 0 )U 2 max (x 0 ) ∞ z z 1 2 =0 U (x 0 , z) U max (x 0 ) 2 d z z 1 2 K(x0)=K0 (4) = bU 2 j K 0 (5)
It should be noted that J 0 is a constant for a given in- 2.9 2.9 -600 - et al., 2008b), [START_REF] Murphy | Scaling of maximum veloc-1126 ity, body force, and power consumption of dielectric barrier dis-1127 charge plasma actuators via particle image velocimetry[END_REF] and [START_REF] Maden | Experimental and computational study of 1108 the flow induced by a plasma actuator[END_REF] studied the evolution of plane DBD induced jets and also found velocity profiles in between laminar and turbulent ones. The velocity gradient in the inner part seems closer to the laminar WJ, but the outer part tends to follow the turbulent velocity distribution. In the case of the curved DBD configuration (figure 12b), the profiles of the higher abscissas x * (lighter colors) fit well the turbulent profile obtained by [START_REF] Schneider | Laser doppler measurement of 1161 turbulence parameters in a two dimensional plane wall jet[END_REF] but the first velocity profiles beyond the virtual origin (darker colors) do not really fit either laminar or turbu- , z * max and Umax) turbulence intensity profiles is not obtained with this scaling and will not be obtained because of the noticeable decline in the turbulent components. This result was indeed expected because self-similarity of the mean flow is not completely obtained. However, the analysed velocity profiles are very close to the virtual origin (5 virtual slot widths for the curved DBD-WJ and 20 for the plane DBD-WJ) and these distances are not enough to properly characterise a WJ. [START_REF] Wygnanski | On the applicability of var-1187 ious scaling laws to the turbulent wall jet[END_REF] found selfsimilarity for the mean velocity profiles of their fluidic WJ but did not find it for the turbulent fluctuations with scaling 1. The authors conducted the experiments up to 120 slot widths downstream of the nozzle and admitted that this distance was not sufficient to attain self-similarity in their range of Reynolds numbers. Wilson and Goldstein (1976) in their study on the WJ around a cylinder did not find self-preservation of the turbulent quantities either, even though the mean flow appeared to be quasi-similar.

V AC [kV] b or b * [mm] U j [m s -1 ] β [ • ]

Transition to turbulence

The transition to turbulence of the induced jets is examined below regarding the DBD set-up and the possible flow instabilities. This evolution is in reasonable agreement with the results obtained by [START_REF] Bradshaw | Turbulent Wall Jets with and without 1028 an External Stream[END_REF] who found that the spread of a fluidic WJ over a cylinder is 1.5 times higher than over a flat plate.

Local maximum velocity U max (x * ) in the streamwise direction x * is plotted in figure 17a. By definition, the velocity decay appears for all the actuators just beyond the virtual origin x * 0 . The maximum velocity is of the same order of magnitude for the three actuators plotted but the velocity decay is slower for the plane DBD-WJ. For this jet, the multi-DBD actuation is clearly visible as the veloc-ity increases in two stages: a first stage up to 1.8 m/s and 5. After the virtual origin x * 0 , the evolution of

U j U max 2 = A u x * -x * 0 b * (6) z * 1 2 b * = A z x * -x * 0 b * (
U j U max 2
is linear for both curved and plane DBD-WJs confirming the theoretical x -1 2 evolution of the velocity decay. In the same way, the rate of spread of DBD induced jets behaves as a classical fluidic WJ and evolves linearly in the diffusion zone. DBD curved WJs exhibit a much higher velocity decay than the plane DBD and a higher rate of spread as well. The effect of the jet Reynolds number is significant in the present study. The higher the Re j , the higher the slopes A u and A z . For instance, the slopes of , whereas the streamwise direction evolves as

(x * -x * 0 )J(x * ) ν 2
. Figures 19a and19b show the

110

  duced jets show similar properties as canonical fluidic wall 111 jets but also to determine which are the most relevant vari-112 ables to describe the evolution of the jets along a curved 113 wall. The causes leading to the jet transition to turbulence 114 were also investigated. 115 First, a short review of previous investigations on fluidic 116 wall jets and on DBD induced jets is presented to intro-117 duce the comparison between these two flows. Then, the 118 mean topology of curved DBD flows is analysed for three 119 actuator positions around the airfoil trailing-edge. The 120 third part establishes the basis for the fourth section that 121 studies the streamwise evolution of the induced jets in the 122 jet-related coordinate system. 123 1.1. Fluidic wall jets 124 Förthmann (1936) studied the wall jet (WJ) for the first 125 time and theoretically compared it to the free open jet. 126 Assuming x as the WJ streaming direction, the author 127 proved that the jet maximum velocity decreases as x -1 2 128 and that the jet spread, defined as the maximum velocity 129 half-width, linearly evolves along the x-direction. In addi-130 tion, velocity profiles from different distances to the nozzle 131 proved to be coincident in one curve showing a similarity 132 law. 133 A wall jet can be divided into two distinct layers on both 134 sides of the velocity peak as introduced by Glauert (1956). 135 On the one hand, the inner layer is located between the 136 wall and the maximum velocity ordinate. Due to the in-137 fluence of the wall, Glauert (1956) predicted that this zone 138 would behave as a turbulent boundary layer, whereas the 139 outer layer would be similar to the shear layer of a free 140 jet. As the WJ develops, the inner layer becomes wider 141 and the maximum velocity decreases. However, jet veloc-142 ity decreases faster than spread increases. This implies 143 that the local Reynolds number of the jet, based on the 144 maximum jet velocity and its related velocity half-width, 145 slowly decreases with the jet development. 146 The wall jet development can also be described in the 147 streamwise direction and be characterised by three dis-148 tinct zones: the potential core, where the nozzle exhaust 149 velocity is preserved; a transitional region where the flow 150 shifts progressively from a laminar to a turbulent regime, 151 and a fully developed zone where the flow is completely 152 turbulent. One of the main difficulties is to determine 153 the critical jet Reynolds number Re jc corresponding to 154 the transitional regime. Kurtz and Crandall (1962) stud-155 ied theoretically the natural convection on a heated ver-156 tical plate on air and found a Re jc of 161. Chun and 157 Schwarz (1967) carried out a theoretical study on the sta-158 bility of an incompressible, viscous plane WJ subjected to 159 small disturbances and Re jc was determined to be about 160 57. An experimental stability investigation of a lami-161 nar two-dimensional plane WJ was carried out by Ba-162 jura and Szewczyk (1970). The authors concluded that 163 2 the critical value of Re jc lies between 500 and 2000. Bajura and Catalano (1971) studied a transitional plane WJ

3 4 (Figure 1 :Figure 2 (

 412 Figure 1: Side view of the model

  Figure 2: Geometry and top-view of the electrodes

  employed is a laboratory-made supply 289 composed of a Crown amplifier (XS1200 series, 22 Hz -290 22 kHz, 2.3 kW) and of a single-phase transformer made by 291 Trabo (90 V -20 kV, 2 kW, 20 Hz -20 kHz). The maximum 292 voltage output is about 20 kV and 2 kW. The sinusoidal 293 function generator is a TTi (TGA1241, 40 MHz, 10 V). 294 The applied high voltage is measured with a high voltage 295 Tektronix probe (P6015A series, 75 MHz, 3 pF).

Figure 3 :

 3 Figure 3: Drawing of electrode position around the trailing-edge of the airfoil

354theorem(

  [START_REF] Coleman | Experimentation, Validation, and 1038 Uncertainty Analysis for Engineers[END_REF]). This method was 355 recently used by Van Hooff et al. (2012) and Kaffel et al. 356 (2015) for the evaluation of the repeatability error of PIV 357 measurements on a plane WJ in a confined space and a 358 plane WJ subjected to an external lateral flow respectively.

361aFigure 4

 4 Figure4shows time-averaged velocity magnitude fields

  Figure 4: Time-averaged velocity fields for the three electrode positions in quiescent air conditions (V AC = 18 kV and f AC = 1 kHz)

  the longitudinal direction of the flow and the ordinate z in the normal direction. The origin of the x-coordinate is the exit of the nozzle and the z-coordinate origin is the surface of the wall. Each velocity profile along the x-coordinate has its local maximum velocity named U max (x) located at a distance z max (x) from the wall. The variable z 1 2 (x) is defined as the distance from the wall to the location at which the mean velocity decreases to one half of its local maximum value in the outer flow. The jet maximum velocity U j is the jet velocity slightly downstream of the exit of the nozzle, at the x 0 abscissa defined as U max (x 0 ) = U j . The Reynolds number of the jet is defined as Re j = Uj b ν

Figure 5 :Figure 6 :

 56 Figure 5: Scheme of a fluidic plane WJ over a plane surface and its specific variables

  duced jet whereas J(x) is a local variable that evolves with the WJ development. For scaling 3, the authors redefined the Reynolds number as Re N = the DBD induced jet characteristics for the actuators studied. The Reynolds numbers of the jets are shown according to the two definitions specified in table 2. In the case of Pos. A powered at 18 kV, the virtual origin x * 0 is located at an angle β with respect to the TE extremity equal to -10 • whereas for Pos. B and Pos. C it is located at 60 • and 70 • respectively. The position of the virtual origin is directly related to the WJ topology and therefore to the position of the electrodes around the TE: 518 the more the DBD-WJ attaches to the model curvature, 519 the greater the β angle. While the jet velocity magnitude 520 U j is equivalent between plane and curved DBD-WJs, the 521 jet thickness b (or b * ) remains smaller for the plane cases 522 leading to lower Reynolds numbers. The Reynolds number 523 Re j for curved DBD-WJs varies from 1100 to 1900 whereas 524 the one according to Neuendorf's definition Re N ranges 525 from 900 to 1200 and is more uniform for all the actuators 526 and voltages tested. This is due to the magnitude of the 527 length scales b and √ bR used in the Reynolds numbers Re j 528 and Re N . In our case b ≈ 7 mm and √ bR ≈ 6.6 mm which 529 leads to a Re N slightly smaller than Re j . This is not the 530 case in Neuendorf and Wygnanski (1999) where Re N is 531 almost five times higher than Re j because b ≈ 2.3 mm and 532 √ bR ≈ 15 mm. Whatever the Reynolds number definition 533 used, all the DBD induced jets are in a transitional regime 534 or not very far from it according to the critical Reynolds 535 number suggested by Murphy et al. (2013) which is Re jc ≈ 536 100 for DBD induced jets.

  this section is to characterise the curved 540 and plane induced jets in terms of velocity profiles and 541 self-similarity. Both plane and curved DBD-WJs are pow-542 ered at V AC = 18 kV but have distinct Reynolds numbers: 543 600 for the plane DBD and 1300 for the curved DBD in 544 Pos. C. The latter curved case behaves the closest to the 545 plane WJ of all the curved actuators studied, especially 546 in the diffusion zone. Velocity profiles normal to the wall 547 are computed in the jet-related coordinates (x * , z * ). The 548 equation of the lines perpendicular to the surface and their 549 angle with the horizontal direction are found. Then, the 550 coordinates of the velocity profiles are computed in the 551 curvilinear framework. Finally, the velocity values are lin-552 early interpolated along the profile coordinates. 553 The analysed velocity profiles located in the diffusion 554 zone of each actuator are shown in figures 8a and 8b for 555 the plane and curved DBD-WJ respectively. Markers and 556 line colors shown in the figures correspond to the ones 557 employed all along the section at each given abscissa: the 558 lighter the gray-scale color and the farther the profiles are 559 in the jet streamwise direction.

  560

Figure 8 :J.Figure 9 :

 89 Figure 8: Location of the analysed velocity profiles and their respective markers

Figure 10 :Figure 11 : 2

 10112 Figure 10: Velocity U and centrifugal force fcent profiles at the virtual origin x 0 (or x * 0 ) for different actuators

Figure 13 :

 13 Figure 13: Normalised velocity profiles at different x * positions (scaling based on z * 1 2

Figure 14 :Figure 15 :

 1415 Figure 14: Turbulence intensities and Reynolds stresses at different x positions based on scaling 1 (plane DBD -diffusion zone)

2 .

 2 Figure 16: Maximum velocity ordinate z * max and ordinate z * 1 2evolu-

813ab

  Figure 17: Maximum velocity Umax and local Reynolds number Re x * evolution in the flow direction x * The evolution of the local Reynolds number Re x * with

Figure 18 :

 18 Figure 18: Maximum velocity decay and rate of spread in the flow direction based on scaling 1 (diffusion zone)

892Figure 19 :

 19 Figure 19: Maximum velocity decay and rate of spread in the flow direction based on scaling 2 (diffusion zone)

Figure 20 :

 20 Figure 20: Maximum velocity decay and rate of spread in the flow direction based on scaling 3 (diffusion zone)

984

  This project was funded by the French national project 985 SMARTEOLE (ANR-14-CE05-0034). The authors want 986 to thank the reviewers for their comments which helped 987 to improve the quality of the paper.

  988

Table 1 :

 1 Overview table of plane DBD-WJ characteristics from previous studies (SDBD: Single DBD, S2DBD: Single two-electrode DBD, ExDBD: Extended DBD)

	264

  Re j Re N

	Pos. A	18	4.8	2.8	-10	860	680
	Pos. B	15 18	7.2 7.7	3.4 3.8	50 60	1600 1000 1900 1200
	Pos. C	15 18	5.7 6.2	3.0 3.3	70 70	1100 800 1300 900
	Plane DBD	15 18	2.5	2.8	-	450	-

Table 3 :

 3 DBD-WJ characteristics

Table 4 :

 4 Ordinates of turbulence intensity peaks centrifugal forces acting on the jet that generates large ed-

	676
	677	dies entraining the external air and inducing a broadening
	678	of the DBD-WJ. The ordinates of the maximum inten-
	679	sities are also shown in table 4. This ordinate increases
		progressively and stabilizes at a value of 0.65 for the three

  7)

		Re j	A u	A z
	Curved DBD Pos. B	1900 0.80 0.87
	Curved DBD Pos. C	1300 0.19 0.22
	Plane DBD	600 0.04 0.06
	Wygnanski et al. (1992) 3400 0.15 0.09

Table 5 :

 5 Table summarizing slope values Au and Az of the power laws in equations 6 and 7