Barbara Livieri
email: barbara.livieri@unisalento.it

Marco Zappatore
email: marcozappatore@unisalento.it

Mario Bochicchio
email: mario.bochicchio@unisalento.it

Forms of Collaboration in Collaborative Enterprises: An Ontological Model

Keywords: Collaborative Enterprises, Ontologies, Enterprise Modeling, Business Rule

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

Introduction

The transition from intra-organizational to inter-organizational relationships characterized the last twenty years of the evolution of enterprises. More recently the increasing importance of Collaborative Enterprises (CEs) and the growing impact of technologies on businesses drove the transition towards a new phase of trans-organizational relationship, characterized by an increased speed to create value [START_REF]FInES Research Roadmap Task Force: Future Internet Enterprise Systems (FInES)-Research Roadmap[END_REF][START_REF] Bititci | Performance Measurement: Challenges for Tomorrow[END_REF]. Indeed, CEs, defined as "a number of autonomous organizations working together […] for mutual benefit" [START_REF] Bititci | Creating and managing value in collaborative networks[END_REF], are often used as catalysts of competitive advantages. However, it is known that globally 50%-70% of CEs fails, often due to the lack of a comprehensive analysis that combine strategic goals and KPIs [START_REF] Kaplan | Managing Alliances with the Balanced Scorecard[END_REF][START_REF] Bititci | Collaboration: a key competence for competing in the 21st century[END_REF] with a possible negative impact on component firms. Indeed, CEs are heterogeneous clusters of partnerships among enterprises. CEs can be of different types (e.g., horizontal or vertical) and have different goals; therefore, they need for different KPIs [START_REF] Parung | A conceptual metric for managing collaborative networks[END_REF]. In other words, firms and CEs have to understand which KPIs are relevant and what a KPI means in a given CE with defined set of goals. However, this kind of understanding is not immediate, especially in several SMEs, which lack of the know-how needed to perform this kind of analysis and often choose the more "known" KPI, instead of the more relevant, with possible negative effects on the CE equilibrium. In order to measure performance and to create governance tools for inter-organizational environments, it is essential to understand which KPIs are useful for which types of CE [START_REF] Popova | Modeling organizational performance indicators[END_REF].

In this scenario, ontologies and reasoning techniques can help in the understanding of complex entities and in the development of specific Enterprise Information Systems for intra-and inter-organizational settings. By committing to the same ontological specification, different applications share a common vocabulary with a formal language and clear semantics. Also, by representing knowledge with a wellestablished formalism [START_REF] Guarino | Understanding, Building and Using Ontologies: a Commentary to Using Explicit Ontologies in KBS Development[END_REF], internal consistency and compliance checking can be performed in order to determine content adequacy. The main objective of this work is to develop an ontology of CEs, starting from the taxonomy of Collaborative Networks (CNs) developed by [START_REF]FInES Task Force on Collaborative Networks and SOCOLNET: Taxonomy of Collaborative Networks Forms[END_REF], and to infer knowledge from a domain body. To this aim, we take into account the type of a CE, its maturity (i.e., the lifecycle phase), size and other characteristics. The work is structured as follows. In Section II we analyze related works on enterprise modeling. Section III is for the modeling methods and Section IV for the representation of the ontology and the analysis of the case study. Finally, in Section VI we discuss the ontology and draw conclusions.

Related Works

Ontologies can be very effective to represent shared conceptualizations of specific domains [START_REF] Bertolazzi | An approach to the definition of a core enterprise ontology: CEO[END_REF]. To this aim, ontologies can be useful, since they can be seen as repositories of concepts, intended as complex information structures tightly interconnected with each other, on which reasoning functionalities can be applied. An ontology for CNs has been proposed in [START_REF] Plisson | An ontology for virtual organization breeding environments[END_REF], where the organizational structure and the domain specific knowledge of Virtual Breeding Environments (VBEs) are represented. In particular, each VBE has some assets and have some participants.

Each VBE participant has a VBE Role and to each role some tasks are associated. Also, a VBE has some business opportunities related to the development and commercialization of products and services. VBEs are defined as organizations, to which are connected competencies and processes. In turn, each Process uses some resources and produce or use as inputs other products and services. Although this ontology provides a general representation of CNOs and VBEs, it is possible to analyze the roles of participants but not to understand CEs types, since the ontology is focused on VBE. Therefore, the ontology is unable to provide a more comprehensive classification of CESs. The same considerations apply to the ontology for VBE capabilities and profiles [START_REF] Ermilova | Modeling and management of profiles and competencies in VBEs[END_REF]. Moreover, in [START_REF]FInES Task Force on Collaborative Networks and SOCOLNET: Taxonomy of Collaborative Networks Forms[END_REF], a taxonomy of Collaborative Networks (CNs) is provided. The authors start from a definition of CNs in order to classify 26 types of CNs, among which digital ecosystems. Although the study is quite interesting and takes into account criteria such as the time perspective, the analysis should be broadened. Also, the lack of a formal language doesn't allow the semi-automatic classification of instances (CEs). In [START_REF] Madni | IDEONTM: An extensible ontology for designing, integrating, and managing collaborative distributed enterprises[END_REF] the IDEON ontology is proposed in order to support the design and the management of collaborative and distributed enterprises. To this aim, the authors take into account four views of the collaboration, namely, a) Enterprise Context View; b) Enterprise Organizational View; c) Process View; and d) Resource/Product View. Finally, in [START_REF] Ye | An ontology-based architecture for implementing semantic integration of supply chain management[END_REF] a model for supply chain is presented with the aim of enabling the semantic integration of Information Systems. In order to do so, some basic concepts, such as the supply chain structure (SC_Structure), the participants (Party), their roles (Role), the purpose of the alliance (Purpose), the Activity, the Resource, the Performance and the Performance_Metric. However, not even in these two cases there is a classification of the types of CEs, although the basic concepts used in these ontology can be borrowed

Motivation and Requirements

This works is part of a wider project regarding performance measurement in CEs. This is a non-trivial task, which has not yet been addressed by current literature [START_REF] Livieri | Information Systems and Performance Management for Collaborative Enterprises: a proposal[END_REF][START_REF] Livieri | Performance Modeling for Collaborative Enterprises: Review and Discussion[END_REF]. In this context of high-complexity, enterprise modeling techniques can be helpful [START_REF] Dietz | Enterprise Ontology: Theory and Methodology[END_REF]. In more detail, in our proposal, the performance measurement is based on the definition of domain-specific KPIs with the aim of supporting the management of CEs. Domain-specific KPIs are derived starting from the analysis of the CEs and of the participants' goals and form the type of CE. Indeed, there are several kinds of CEs, which need different KPIs [START_REF] Bititci | Creating and managing value in collaborative networks[END_REF]. Therefore, it is particularly relevant to classify CEs in order to identify the peculiarities of each type and, consequently, to offer specific support for the measurement of performance. Using this scenario as a starting point, a set of competencies questions, subdivided in simple (SCQ) and complex (CCQ), were elicited. As an example, we provide the following:

[SCQ.

Modeling Methods

When dealing with ontologies, the first trade-off to be taken into account is between their level of semantic description and their level of decidability (i.e., the capability of specific software applications, the reasoners, to produce new knowledge by inferencing over the explicitly defined semantic relationships). Different ontology languages and flavors are available so that the modeler can tune its knowledge base properly.

For such reasons we selected the OWL 2 DL language [START_REF]OWL 2 Web Ontology Language Primer (2nd Edition)[END_REF] since it is fully decidable.

In addition, OWL DL can be combined with SWRL, a language for the Semantic Web used to express rules as well as logic. Moreover, knowledge resources can be represented as classes or instances depending on specific modeling requirements. We adopted a class-based approach to model the CE macro types and their characteristics, while we used business rules based on SWRL in order to infer the CE macro-types from the association of an instance of the class CE with some characteristics. In this way, end-users can easily populate the ontology by adding instances to CE classes, which inherit the semantic structure associated to the CE class they belong to. The ontology library named "CE Ontology" was developed with a modular architecture, as modularity is advantageous in terms of portability and reuse, domain decomposition and content categorization [START_REF] Van Heijst | Using Explicit Ontologies for KBS Development[END_REF]. We modeled a core ontology (ontoPM) and a set of domain-dependent modules, namely ontoKPI for Key Performance Indicators, ontoFirm and ontoCE, which gather concepts describing firms and CEs respectively. The adoption of core and domain ontologies is a well-established custom in ontological engineering [START_REF] Breuker | Developing Content for LKIF: Ontologies and Frameworks for Legal Reasoning[END_REF][START_REF] Pedrinaci | A Core Ontology for Business Process Analysis[END_REF]. CE ontology was developed by using the Protégé Desktop editor (v.4.3) [START_REF]Protégé Ontology[END_REF], the de-facto freeware open-source ontology modeling tool. Ontology consistency, class expression satisfiability and finite reasoning time complexity were checked by using Pellet reasoner 2.2.0 [START_REF] Sirin | Pellet: A Practical OWL-DL Reasoner[END_REF]. The DL expressivity of the ontology is SHROIQ(D).

Collaborative Enterprise Ontology

This section provides some ontology fragments exemplifying our modeling approach.

The first excerpt illustrates CollaborativeEnterprise, the class describing a CE, its subclasses and associations with other classes. As depicted in Fig. 1, each CE has a CELifecyclePhase, an OrganizationalStructure and a Size (small, medium, large). This is codified by using existential (owl:someValuesFrom) or universal property restrictions (owl:allValuesFrom). Also, each CE is composed by at least 2 firms, which is codified with owl:ObjectMinCardinality and it has Relation and Alliance as subclasses. In turn, Alliance has an object property based semantic characterization, involving integrationLevel (low, medium, high), timeHorizon (shortTerm, mediumTerm, longTerm) and integration-Type (horizontalIntergration or verticalIntergration. Finally, Alliance have as subclasses StrategicAlliance and TacticAlliance.

As show in Fig 2, the CE classification based on size, time horizon, type (strategic or tactic, horizontal or vertical) and organizational structure is performed with reasoning support [START_REF] Eiter | Reasoning with Rules and Ontologies[END_REF], which enables to infer new knowledge from the original domain body. To do so, we used functional dependencies, represented by means of business rules, coded with Semantic Web Rule Language (SWRL) [START_REF] Horrocks | SWRL: A semantic web rule language combining OWL and RuleML[END_REF]. In particular, in order to determine the size of a CE, we use the datatype property hasNumberOfFirms and we define the following rule: CollaborativeEnterprise(?x), hasNumberOf-Firms(?x, ?integer), greaterThan(?integer, 4), lessThanOrEqual(?integer, 10) -> hasSize(?x, Medium). In this way, we assert that if a CE has a number of firms greater than 4 but less or equal to 10, then it has a medium size. Similar rules have been used in order to define the small and the large size. The timeHorizon has been determined by means of the datatype property hasTimePerspective and of the following rule: Alliance(?x), hasTimePerspective(?x, ?integer), greaterThan(?integer, 3) -> hasTimeHorizon(?x, long-Term). Through this rule we assert that if an alliance has a time perspective greater than 3 years, then it has a long term horizon. Moreover, the alliance type is defined by means of a business rule that takes into account the integration type and the time horizon. The last one is useful in order to distinguish between strategic alliances (medium and long term) and tactic alliances (short term), while the integration type helps in order to divide horizontal alliances from vertical ones. In more detail, with the rule "Alliance(?x), hasIntegra-tionType(?x, horizontalIntegration), hasTimeHorizon(?x, long-Term) -> HorizontalStrategicAlliance(?x)" we assert that if an alliance has a horizontal integration as well as a long-term time horizon, then it can be considered a horizontal strategic alliance. Also, the classification based on the organiza-tionalStructure is based on the number of firms, the participation of firms to the CE, the existence of a focal firm and the presence of other forms of organizational structure. In more detail, the following rule "CollaborativeEnterprise(?x), hasNumberOfFirms(?x, 2) -> hasOrganizationalStructure(?x, dyad-icStructure)" asserts that if a CE is composed by exactly two firms, then it has a dyadic structure. Moreover, the rule "CollaborativeEnterprise(?x), hasFocalFirm(?x, ?y), hasPart(?x, ?y), hasNumberOfFirms(?x, ?integer), greaterThan(?integer, 2) -> hasOrganizationalStructure(?x, organizationWithFocalFirm)" asserts that if CE is composed by more than two firms, has a focal firm and this focal firm is part of the CE, then the CE has an organizationalStructureWithFocalFirm. Finally, the rule "Collabo-rativeEnterprise(?x), hasNumberOfFirms(?x, ?integer), greater-Than(?integer, 2), DifferentFrom (?y, organizationWith-FocalFirm) -> hasOrganizationalStructure(?x, organizationWith-outFocalFirm)" asserts that if a CE is composed by more than two firms and it can't be associated with the class organizationalStructureWithFocalFirm, then it has an organizationalStructureWithoutFocalFirm. The clause Dif-ferentFrom was used in association with an owl:allDifferents axiom declaration to assess that the individual organizationalStructureWithFocalFirm is different from organizationalStructureWithoutFocalFirm. That clause was chosen because SWRL does not support negation as a failure, being a monotonic language.

Validation

In order to assess the quality of the proposed ontology, a way to evaluate how accurate is the encoded knowledge is needed. As a consequence, competency questions (CQs) were introduced in Section 3 in order to model our knowledge base accordingly. The ontology should contain enough information to be able to answer to CQs and these aspects have to be checked during validation. The importance of CQs during the entire ontology development and validation process was highlighted by Presutti et al. in [START_REF] Presutti | eXtreme design with content ontology design patterns[END_REF]. According to them, CQs must be formalized into queries that have to be run against the ontology, so that query outcomes must be compared with the expected results. However, translating CQs into formal queries is not a straightforward task [START_REF] Damljanovic | Natural language interfaces to ontologies: Combining syntactic analysis and ontology-based lookup through the user interaction[END_REF], since many aspects should be taken into account: presence of rules, necessity of considering inferred knowledge, etc. Some alternatives are available and the user should select the most suitable one according to its application scenario. SPARQL (Simple Protocol and RDF Query Language) [START_REF]SPARQL 1.1 Query Language. W3C Recomm[END_REF] has high expressivity and allows to query and manipulate any RDF graph but considering inferences can be troublesome (unless SPARQL 1.1 entailment regimes are used). DL Queries [29] are more suitable to OWL class expressions and allow to capture inferences but cannot manage SWRL rules. SPARQL-DL [START_REF] Sirin | SPARQL-DL: SPARQL Query for OWL-DL[END_REF] combines the best features of SPARQL and DL Queries but it is not a W3C standard and its future is still not very clear. Finally, SQWRL [START_REF] O'connor | SQWRL: A Query Language for OWL[END_REF] is a SWRL-based language capable of querying OWL ontologies thanks to an ad hoc library of SWRL built-ins. Therefore, we used SPARQL to check SCQs and SQWRL to ascertain that CCQs can be answered as well (see Section 3). For the sake of brevity, an example from each of the two approaches is reported. The SCQ.2 was associated with the features of a CE: it can be translated into the following SPARQL query:

Fig. 1 .Fig. 2 .

 12 Fig. 1. Representation of a fragment of the ontology

  1] which types of alliances are possible? [SCQ.2] what characteristics should a CE satisfy in order to be a strategic horizontal alliance? [SCQ.3] which organizational structures are possible? [CCQ.1] how many firms should a CE be composed of, to have a dyadic structure? [CCQ.2] what is the size of a collaborative enterprise with more than 10 firms? [CCQ.3] what type of CE is the one that has a time perspective of more than 3 years and a vertical integration?

MINUS {ontoCE:CollaborativeEnterprise a ?o } } that searches for both predicates and objects of any triple having the CE class as subject domain. In the same way, CCQ.3 aimed at classifying a CE having a time perspective of more than 3 years and a vertical integration. It can be rendered in SQWRL as in the following:

CollaborativeEnterprise(?x), hasIntegrationType(?x, verticallIntegration), hasTimeHorizon(?x, ?integer), greaterThan(?integer, 3) -> sqwrl:select(?x)

Where all the CEs satisfying the conditions mentioned above are enlisted. By adopting this methodology, we managed to verify that all the CQs can be answered by selecting a proper querying approach

Discussion and Conclusions

In this work, we introduce a semantic model for a large class of Collaborative Networks (CNs) defined as Collaborative Enterprises (CEs), in order to enable their understanding and analysis. The emphasis of the paper is on structuring the characteristics of CEs with respect to size, integration type and level, time horizon and so on. The classification of the instances is achieved by means of business rules. The development of the Collaborative Enterprise ontology, supported with business rules, enable the semi-automatic classification of CEs, which, if associated with a KPI Ontology brings to the development of an interpretative framework able to support both CEs and firm managers to understand which KPIs to use and how to integrate them in a performance framework. Future works will include the online publication of the ontology in order to make the collection generally accessible to the community of researchers and practitioners interested in performance indicator modeling. For the purpose of our research, the ontology will be completed with a set of suitable software components with the aim of supporting users to effectively build, analyze and manage CEs.


Dongping Chen

Xuening Chu
email: xnchu@sjtu.edu.cn

Yupeng Li

Applying Platform Design to Improve Product-Service Systems Collaborative Development

Keywords: Product-Service Systems platform, platform design, collaborative development, modular technology

The promise of Product-Service Systems (PSS) is that it might revolutionize the consumer experience, increase the manufacture's profits and reduce environmental impacts by providing comprehensive solutions instead of pure physical products or services. However, most of the existing researches and applications on PSS are focus on the new PSS development (NPD) which could increase the customer satisfaction but could not enhance the profits of an enterprise effectively. Therefore, the platform design theory is adopted to support collaborative development of PSS. The customer requirements are forecasted by Kano model. Instead of completely innovation design, existing products and services are analyzed by function decomposition methods and the modular technology to support the PSS development. Finally, a case study of the crane machine PSS portfolios shows the effectiveness of the proposed approach.

Introduction

Product-Service System (PSS) is emerging as an important method for the manufactures to realize servitization. However, most manufacturing enterprises suffered drops in profits when increasing investment in services. One of the important reasons is that the manufacturing company's core competitiveness still stays on product design and manufacturing, and cannot expand its capabilities to all the services in the product life cycle to keep superior competitiveness. Cooperation with companies who have complementary core competitiveness has become mandatory for the PSS offers to obtain competitive advantage. For example in the communications industry, smartphone manufactures, operating-system developers, application developers and mobile operators often collaborate together to provide wireless service. Long-term collaborative relationships result in fast project development times, lower development and production costs, increased cooperators originated innovation and better product quality. At the same time, it also challenges the cooperators to rearrange their business models in terms of developments, manufacturing, sales, and so on to adapt to the collaborative environments.

PSS conceptual design plays an important role in PSS development, because most life cycle cost and critical performance of a product or service is determined in this stage, and it is very difficult to correct the fundamental shortcomings in the later embodiment and detail design phases [START_REF] Pahl | Engineering design. A Systematic Approach[END_REF]. In spite of the pivotal role of conceptual design, companies find it difficult to design a PSS, especially in the collaborative environments where different organizations take charge different components of the PSS. Companies tend to design their components separately, and then an authoritative company or an third party takes on the role of system integrators and integrates the sub-systems into a whole PSS. When the sub-systems are not compatible, redesign is inevitable. Besides, many innovative parts exist in the new developed PSS and the customer requirement changes quickly. Thus, the design cycle of PSS is very long. Although there are some researches dealing with PSS conceptual design approaches [START_REF] Morelli | Developing new product service systems (PSS): methodologies and operational tools[END_REF][START_REF] Shimomura | A unified representation scheme for effective PSS development[END_REF][START_REF] Aurich | Configuration of product-service systems[END_REF], little considered this problem in the collaborative environments. This article adopts the platform design methods and the modular technology to solve the collaborative development of PSS.

In product design area, platform strategies provide sufficient derivative products for the market while maintaining economies of scale and scope within their manufacturing processes. Platform-based product development offers a multitude of benefits including reduced development time and system complexity, reduced development and production costs, and improved ability to upgrade products. Platforms also promote better learning across products and can reduce testing and certification of complex products [START_REF] Simpson | Product platform design and customization: Status and promise[END_REF]. Therefore, platform strategies are adopted to assist the system integrators with developing PSSs in the collaborative environments. Evans et al. [START_REF] Evans | Industrialization as a key element of sustainable product-service solutions[END_REF] argues that platform strategies can also be a significant enabler to multi-actor PSS. By considering the product-service system as being made of multiple elements, potentially delivered by different actors and integrated through a platform architecture, it may be feasible to create high-performing PSSs. However, no studies have shown clearly how to develop a PSS platform. This paper is dedicated to answering these questions:

(1) What is a PSS platform? What constitutes a PSS platform?

(2) What is the different among the PSS platform, the product platform and the service platform?

(3) What are the procedures to develop a PSS platform? Are there any methods that can be used to support the development?

(4) How can the platform methods assist the collaborative development of PSS?

PSS Platform

Definition of PSS platform

Synthesizing the definitions of product platform [START_REF] Meyer | The Power of Product Platforms: Building Value and Cost Leadership[END_REF], service platform [START_REF] Meyer | Product development for services[END_REF] and PSS [START_REF] Mont | Clarifying the concept of product-service system[END_REF], we define PSS platform as: A set of common assets such as components, modules, parts, processes, knowledge, people and relationships from which a stream of derivative product-service systems can be efficiently developed and launched. The PSS platform is designed, manufactured and operated in a value created network involved many stakeholders. The PSS provider is the system integrator who develops platform plans, selects cooperative partners, integrates subsystems or modules and offers a whole solution to customers. The customers take a more important and active role in the PSS development. Elaborate requirements and timely feedback help the provider to adjust planning and other partners, ensure the quality of the final solution. Local providers adjust the service contents or frequency to customers' personalized needs. For example, rust-proof is more important in humid areas than arid areas. Local service providers, especially the maintenance centers, usually need components from component manufactures.

Comparison among PSS platform, product platform and service platform

The main differences among PSS platform, product platform and service platform are the component and interface (see Table 1). Product module, service module, interface, support system between the product platform and the service platform

The Process of PSS Platform Development

In order to assist PSS providers to develop a PSS platform in a collaborative environment, an integrated approach is proposed (Shown in Fig. 1).

Fig. 1.

The process of PSS platform development.

Phase 1. Determine PSS family functions

A PSS family is a group of individual PSSs that share common subsystems and yet possess specific functional features to satisfy a variety of market niches. A platform may support one or more PSS families. In this paper, we constructed a PSS platform from one product family and relative services. One of the most important objectives of PSS is to maximize the customer value. The customer values must be analyzed before determining PSS family functions. Customer value is a customer's perceived preference for and evaluation of those product attributes, attribute performances, and consequences arising from use that facilitate (or block) achieving the customer's goals and purposes in use situations.

Step 1. Define customer value elements The means-end type of model is used to capture the customer value. Firstly, customers' goals and purposes are described. Then, considering use situations customers' desired consequences are analyzed. Finally, customer expected value elements are obtained.

Customer value often changes based on time, technology and market niches. These changes will influence the platform's component. Kano model is adopted to classify the customer value elements into basic value elements, expectable elements and adjunctive elements.

Step 2. Analyze the functions of existing products Axiomatic design has been widely used in new product design, and it can also assist the product redesign [START_REF] Dong | Parsing of functional decomposition methods for methods for mechanical-electronical product creative conceptual design[END_REF]. The function decomposition process is a zig-zagging process. Functional requirements (FR) are mapped to design parameters (DP). A product's total FR is first specified, and then the corresponding DP satisfying this FR is found. This DP leads to the analysis of lower lever of FRs. Thus, the product's function is decomposed into a hierarchy structure.

Step 3. Analyze the functions of existing services Existing services are analyzed by service blueprint. Service is in nature a process thus its function is constructed in the form of a process structure [START_REF] Geng | A new PSS conceptual design approach driven by user task model[END_REF].Every action performed by the customers and activity performed by the staff are analyzed to obtain the function.

Step 4. Determine PSS family functions [START_REF] Pahl | Engineering design. A Systematic Approach[END_REF]. Construct the mapping matrix between PSS family required function and existing product and service's function (as shown in Fig. 2). The columns are PSS family required functions obtained in Section 3.1.1, and they are classified into basic functions, expectable functions and adjunctive functions. The rows are functions of existing products and services. If a FR can deliver a customer value element, the corresponding cell is filled with 1.

Fig. 2. The mapping matrix between customer value and existing product and service's function (modified from [START_REF] Salhieh | A methodology to redesign heterogeneous product portfolios as homogeneous product families[END_REF]) [START_REF] Morelli | Developing new product service systems (PSS): methodologies and operational tools[END_REF]. Check the rows to find the replicated functions, which may be between two products, two services or a product and a service. The first two kinds of replication are caused by the lack of commonality between different products/services in the sense that different components/activities are used to deliver similar functions [START_REF] Salhieh | A methodology to redesign heterogeneous product portfolios as homogeneous product families[END_REF]. These replicated functions should be redesigned to be delivered by uniform components/activities. The replicated functions between products and services should be considered in a different way. In PSS, product and service may support or replace each other, in which situation there is no need to abandon one of them. For example, although both the automatic teller machine and the bank staff can provide accepting deposits service, a bank may retain both of them to serve the customer. Besides, some FRs may have no relation with any customer value elements (see FR5 in Fig. 2). These functions may no longer be needed by customers, and should be abandoned in the new developed PSS family.

(3). Check the columns to find the value elements which are delivered by no functions, such as EV2 and AVna. For basic value elements and expectable elements, new functions should be added to achieve these value elements for all customers, while for adjunctive elements, new functions could be added as optional functions for those customers who are willing to pay for that.

(4). Arrange the remaining functions in step2 and the new functions added in step 3. The functions of new developed PSS family are obtained. Classify the functions into basic functions, expectable functions and adjunctive functions based on their corresponding value elements.

Phase 2. Modularize PSS

Step 1. Partition PSS module [START_REF] Pahl | Engineering design. A Systematic Approach[END_REF]. Product module partition. There are many mature approaches for product partition, such as heuristic methods and clustering methods. This paper will not designate one approach, and the reader can choose any based on the concrete case.

(2). Service module partition. Analyze the correlation coefficient between the service activities from the viewpoint of function correlation, class correlation and process correlation [START_REF] Li | Module partition process model and method of integrated service product[END_REF]. The function correlation describes the closeness of two activities in achieving the same function. The correlation coefficient has five levels (1.0, 0.7, 0.5, 0.3, 0). The definition of class correlation is that some service activities have the same characteristics, and these activities can be put into the same service groups and form a module, which will facilitate the service management and provide more similar services [START_REF] Li | Module partition process model and method of integrated service product[END_REF]. The correlation coefficient has four levels (1.0, 0.8, 0.4, 0). The process correlation describe that whether the service activities are continuous or not.

Establish correlation integration analysis matrix, M [START_REF] Li | Module partition process model and method of integrated service product[END_REF].

M={mij} ( 1)

mij=∑rijFf+∑rijFc+∑rijFp ( 2
)
where rij is the correlation between the service activities respectively. According to the correlation integration analysis matrix, we can calculate the correlation between service activities and get the service module partition scheme at different levels [START_REF] Li | Module partition process model and method of integrated service product[END_REF].

(3). Determine the basic module, the indispensable modules and optional modules. The modules that deliver the basic functions are basic modules, the modules that deliver the expectable functions are indispensable modules, and the modules that deliver adjunctive functions are optional modules.

Step 2. Design PSS interface Design the interface between different modules. In a PSS platform, there are three kinds of interface, the interface between product modules, the interface between service modules, and the interface between product and service modules. The interface between product modules has been fully discussed by many scholars [START_REF] Chen | Interface strategies in modular product innovation[END_REF][START_REF] Hamraz | Change prediction using interface data[END_REF]. This article concentrates on the interface between service modules and the interface between product and service modules. These interfaces are classified into two categories: one is the customer interface, which has direct interaction with customers, and the other is the module interface. Design the customer interface could be performed in the following dimensions: purpose, duration and time delay; breadth and depth of options, nature of contact and media employed [START_REF] Bitran | A structured product development perspective for service operations[END_REF]. The service blueprint is a good tool to find interactions. The module interface should be designed considering its purpose. For example, designing the interface between product modules and its corresponding maintenance service modules, equipment failure information is important.

Phase 3. Assign and Integrate modules

After module partition and interface design, the modules detailed design tasks are assigned to cooperators. Lead times, prices, quality, credibility and other factors should be considered when selecting cooperators. The system integrators configure these modules according to customer's requirements.

Case Study

This case (crane machine PSS concept evaluation) comes from an engineering machine manufacturer (company H) in Shanghai, which provides overall solution (including product and service) for customers. Company H has five industrial parks in China and four R&D and manufacturing bases in America, Germany, India and Brazil. The main products consist of concrete machinery, excavator, crane machinery, pile driving machinery, road construction machinery, port machinery, and wind turbine. In order to respond rapidly to changing demands in today's competitive markets, company H launched PSS platform plan. The crane machinery PSS with well structured product modularization and extensive service experience is selected as a pilot project. Fig. 3 shows the hierarchical structure of a crane machine. Company H defines the crane machinery characteristics and select appropriate internal and external suppliers for corresponding product modules. The crane machine family has 6 kinds of lifting capacity, 450,500,600,650,900 and 1400 tons, and 28 types of crane machine.

Determine the functions of crane machine PSS

Customer value are collected through survey questionnaire and semi-structured interviews. The value elements are shown in Fig. 4.

Existing product (28 types of crane machine) and services (45 kinds of service) are analyzed to obtain the existing functions. According to the matrix in Fig. 2, some functions are abandoned. For example, the air conditioner is not needed in Indian, and headlight is not needed if the operating time is very shot. Some functions are added. For example, 24-hour rescue service, oil analysis service, and lending service are added to serve the customer. The family functions are as follows: core capability, module reliability, control technology, safety guard, operating mode, environmental protection and energy saving, cramped construction, operating temperature, construction guarantee, maintenance technology, maintenance cost, service professionalism, service timeliness, 24-hour rescue service, oil analysis service, and lending service.

Modularize the crane machine PSS

The crane machine has been modularized. Thus we only need to divide service modules. Construct the correlation integration analysis matrix between the 45 services (see Fig. 5), such as hydraulic motor maintenance, chassis frame maintenance, operation training service et al. Every element in line i and column j in Fig. 10 represents the correlation between activity i and activity j. For example, the number 0.76 in line 2 and column 3 means that the correlation coefficient is 0.76. This number is the weighted sum of the function correlation, class correlation and process correlation coefficient. This matrix provides a basis for module partition. Activities with bigger correlation coefficient are more likely to be clustered into a module. After module partition, 8 service modules are obtained: maintenance module, training module, lending module, operating condition analysis module, installation module, transportation module, oil analysis module and pre-sales consulting module. Each module has more detailed modules.

The basic module contains: power system, driver cab, hoisting gear, main platform, chassis, track assembly, main boom, pre-sales consulting module, training module, transportation module, installation module, et al.

The indispensable module contains: hydraulic motor module, auxiliary jib, maintenance module, lending module, operating condition analysis module, oil analysis module et al.

The optional module contains: remote control system, 24-hour rescue service, High pressure warning, the third winding engine, fuel preheating et al. Company H assign these modules to different cooperators, and integrates these modular into a whole solution. After the new developed crane PSS was put on the market, customer feedback was gathered through questionnaires and interviews. The acceptance of the new solution by the users was very high, including comments that they would like the service to continue after the pilot test. Although the cost exceeded the budget, it can be accepted by the company. The practical operating results show that the selected solution is good.

Conclusions

As customer requirements become more complicated and change rapidly, platform design is introduced to improve the responsibility of PSS. An integrated PSS platform development approach is proposed in a collaborative environment. The main contributions are as follows:

(1) Platform design is introduced to develop PSS in a collaborative environment. The system integrator designs the PSS platform with basic module, indispensable module, optional module and uniform interfaces. The modules are designed by cooperators separately and integrated by the system integrator.

(2) An integrated PSS platform development approach is proposed based on existing product and service. Customer value is obtained by means-end type of model and Kano model. Existing product/service function is analyzed by axiomatic design and service blueprint. Module partition approach is used to divide PSS.

Future work will concentrate on PSS module partition. When the system become complex, the matrix that represents the interdependent relationship between components or service activities will be too huge to implement the corresponding clustering algorithms effectively. The complex network will be introduced to solve the PSS module partition problem.

Fig. 3 .

 3 Fig. 3. The hierarchical structure of a crane machine.

Fig. 4 .

 4 Fig. 4. Value elements of crane machine PSS.

Fig. 5 .

 5 Fig. 5. The correlation integration analysis matrix.

Table 1 .

 1 Comparison among PSS platform, product platform and service platform.

		Component	Interface
	Product	Product module, interface	Industry standards determines the
	platform		interfaces and technologies
	Service	Service module, interface	Little	influence	form	industry
	platform		standards
	PSS platform

D. Chen, X. Chu and Y. Li

Acknowledgment

The project was supported by National Natural Science Foundation, China (No.51075261、51205242、71301104), Research Fund for the Doctoral Program of Higher Education of China ( No.20120073110096, 20133120120002 ) Shanghai Science and Technology Innovation Action Plan (No.11DZ1120800 , No. 12dz1125600).


Colin B Leclercq

Jamie R Partridge

Pierre Augier

Stuart R Dalziel

Rich R Kerswell

Jamie L Partridge

Stuart B Dalziel

Using stratification to mitigate end effects in quasi-Keplerian Taylor-Couette flow

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

Introduction

In the past decade or so there has been much controversy about transition to turbulence in laboratory experiments of centrifugally-stable Taylor-Couette flow [START_REF] Balbus | Fluid dynamics: a turbulent matter[END_REF]. The motivation behind these experiments has been to uncover physical mechanisms to explain the origin of turbulence in accretion disks where, in the simplest model of a non-selfgravitating disk, the angular velocity Ω ∼ r -3/2 decreases with radius r while the angular momentum L = r 2 Ω ∼ r 1/2 increases: so-called Keplerian flow. This flow is centrifugallystable according to Rayleigh's criterion [START_REF] Peyret | Spectral methods for incompressible viscous flow[END_REF]. The connection with Taylor-Couette flow [START_REF] Zeldovich | On the friction of fluids between rotating cylinders[END_REF]) lies in the fact that a similar 'Rayleigh-stable' flow can be set up by appropriate motion of the inner and outer cylinders. Specifically, if we define the ratios η := r i /r o and µ := Ω o /Ω i , where the inner (resp. outer) cylinder at r = r i (resp. r = r o ) rotates at Ω i (resp. Ω o ) -see figure 1 -then the angular velocity in the interior is

Ω T C = Ω i 1 -η 2 µ -η 2 + (1 -µ) r 2 i r 2
(1.1) with 'quasi-Keplerian' flows -d|Ω|/dr < 0 and d|L|/dr > 0 -corresponding to the parameter range η 2 < µ < 1.

(1.2) only useful if stratification does not introduce any linear instabilities into the problem. In the absence of axial boundaries, stratified Taylor-Couette flow is known to be prone to the stratorotational instability (SRI) (so named by [START_REF] Dubrulle | A hydrodynamic shear instability in stratified disks[END_REF]) caused by the resonance between boundary-trapped inertia-gravity waves. Originally discovered by [START_REF] Molemaker | Instability and equilibration of centrifugally stable stratified Taylor-Couette flow[END_REF] and [START_REF] Yavneh | Non-axisymmetric instability of centrifugally stable stratified Taylor-Couette flow[END_REF], the instability was theoretically predicted to develop for any value of µ < 1 (cyclonic regime) in the inviscid, small-gap limit. [START_REF] Shalybkov | Stability of density-stratified viscous Taylor-Couette flows[END_REF] and [START_REF] Rüdiger | Stratorotational instability in MHD Taylor-Couette flows[END_REF] later extended these results to the wide gap and viscous case with the help of numerical methods. These authors showed that the instability range was narrower than initially thought, as SRI could only be found for a very limited range of rotation ratios µ beyond η, if at all, depending on the radius ratio and stratification. In parallel, Le [START_REF] Bars | Experimental analysis of the stratorotational instability in a cylindrical Couette flow[END_REF] verified their predictions experimentally, providing the first evidence of the existence of SRI. Given the development of SRI in 'fully' stratified flows, the effect of the stratified layers on the global stability of the axisymmetric base flow is systematically assessed in this work.

The plan of the paper is as follows. In §2, we introduce the governing equations of the problem, and then present the numerical and experimental methods in §3. In §4, the effects of stratification 'strength' and layer thickness are assessed for different rotation ratios. In §5, we discuss the physical mechanisms responsible for generation and suppression of meridional flow, the limits of our numerical model, and ultimately the nonlinear stability of the optimally-controlled flow. Finally, section 6 summarizes our main results.

Governing equations

The dimensional parameters of the problem are defined in figure 1. There are seven independent non-dimensional parameters specifying the problem. The geometry of the container requires two:

η := r i r o and Γ := h d , (2.1)
where h and d := r o -r i are respectively the height of the cylinders and the radial gap between them. In all computations and experiments, the radius ratio and aspect ratio are set to η = 0.417 and Γ = 3 (except for validation, cf. figure 2(b)). The rotation and shear imposed on the fluid are quantified by the Reynolds number and rotation ratio

Re := r i Ω i d ν and µ := Ω o Ω i , (2.2)
where ν is the kinematic viscosity. The quasi-Keplerian range η 2 < µ < 1 is bordered by solid-body rotation (no shear) µ = 1 and the so-called 'Rayleigh line' µ = η 2 where the angular momentum is constant over radius. Describing stratification requires three more non-dimensional numbers. Let ρ tot denote the density field and ρ := ρ tot -ρ 0 be the deviation with respect to the reference value ρ 0 := ρ tot (z = 0, t = 0), such that ρ is initially antisymmetric with respect to z. The initial buoyancy frequency within the layers is N := -(g/ρ 0 )∂ρ/∂z, where g denotes gravity. The strength of the initial stratification is quantified by the local Richardson number

Ri l := N 2 /Ω 2 i . (2.
3)

The ratio of the momentum diffusion coefficient ν to the density diffusion coefficient κ is either the Prandtl number P r for thermal stratification or the Schmidt number Sc for salt stratification P r, Sc := ν κ .

(2.4)

Momentum and density diffusion time scales across the gap d are given by τ ν := d 2 /ν and τ κ := d 2 /κ. Finally, the stratified fraction

0 α := l h 1 (2.5)
is defined as the ratio between the thickness l/2 of each stratified layer and the half height h/2 of the apparatus. With these definitions, the buoyancy frequency can be expressed as N = g∆ρ/(ρ 0 αΓd).

The flow is governed by the incompressible Navier-Stokes equations in the Boussinesq approximation, relying on the assumption that ∆ρ/ρ 0 ≪ 1 and that curvature of the isopycnals due to centrifugal effects can be neglected. In the following, we choose d, r i Ω i and ∆ρ as typical length, velocity and density scales in order to obtain non-dimensional quantities. In cylindrical coordinates (r, θ, z), the dynamical equations for the velocity u = ue r + ve θ + we z and density ρ read:

∂ t u + u∂ r u + v r ∂ θ u + w∂ z u - v 2 r = -∂ r p + 1 Re ∇ 2 u - u r 2 - 2 r 2 ∂ θ v , (2.6a) ∂ t v + u∂ r v + v r ∂ θ v + w∂ z v + uv r = - 1 r ∂ θ p + 1 Re ∇ 2 v - v r 2 + 2 r 2 ∂ θ u , (2.6b) ∂ t w + u∂ r w + v r ∂ θ w + w∂ z w = -∂ z p + 1 Re ∇ 2 w -Ri g ρ, (2.6c) ∂ t ρ + u∂ r ρ + v r ∂ θ ρ + w∂ z ρ = 1 ReSc ∇ 2 ρ, (2.6d )
where

∇ 2 := ∂ 2 rr + 1 r ∂ r + 1 r 2 ∂ 2 θθ + ∂ 2 zz (2.7)
is the scalar Laplacian operator, p := Π/ρ 0 + gz is a potential based on the pressure Π, and finally

Ri g := αΓ (1 -η) 2 η 2 Ri l (2.8)
is a global Richardson number (created in the equations by the choice of reference scales).

In strongly rotating flows, 'centrifugal buoyancy' can be taken into account by including the nonlinear term (∆ρ/ρ 0 )ρ(u • ∇)u in the convective derivative [START_REF] Lopez | The Boussinesq approximation in rapidly rotating flows[END_REF]). This term is O(∆ρ/ρ 0 ) with our choice of reference scales, whereas the usual buoyancy term related to gravity is O(Ri g ). In the present study, Ri g ≫ ∆ρ/ρ 0 (see parameter values in table 1), which justifies the use of the traditional Boussinesq approximation. The incompressibility constraint can be written as

∂ r u + u r + 1 r ∂ θ v + ∂ z w = 0.
(2.9)

The meridional velocity u ⊥ := (u, w) vanishes at the boundaries whereas the zonal component satisfies

v =      1 if r = r ⋆ i , |z| Γ/2, µ/η if r = r ⋆ o , |z| Γ/2, µ(r/r ⋆ i ) if r ⋆ i < r r ⋆ o , |z| = Γ/2, (2.10) where r ⋆ i = η/(1 -η) and r ⋆ o = 1/(1 -η)
denote respectively the nondimensional inner and outer radii. Finally, we impose the piecewise-linear density profile

ρ =            - 1 2α 2z Γ -(1 -α) if 1 -α 2z/Γ 1, - 1 2α 2z Γ + (1 -α) if 1 -α -2z/Γ 1, 0 otherwise.
(2.11) either initially in the volume in the case of salt stratification, or at the boundaries at all times in the case of temperature stratification. Indeed, in the former case, the density satisfies no-flux (Neumann) boundary conditions at the walls ∂ n ρ = 0 (where ∂ n denotes the wall-normal derivative), whereas in the latter case, the temperature on the walls can be controlled such that (2.11) holds for all time there (Dirichlet boundary conditions).

Methods

Numerical methods

In the following, the flow variables are jointly written as q := (u, p, ρ). Base flow quantities are denoted with capital letters or subscript 'B' for Greek characters, hence Q = (U, P, ρ B ). Perturbations with respect to the base flow are denoted q ′ := q -Q.

Temperature stratification: steady base flows

In the case of temperature stratification, the base flow is steady, axisymmetric and has the z-symmetry as a consequence of the Boussinesq approximation and the imposed boundary conditions. Each field was discretized using a double expansion of Chebyshev polynomials,

Z : (U, V, W, P, ρ B )(r, z) → (U, V, -W, P, -ρ B )(r, -z) (3.1)
Q(r, z) = I,J i=0,j=0 Qij T i (2(r -r ⋆ i ) -1) Φ j (2z/Γ) (3.2)
where T i (x) := cos(i cos -1 x) and Φ j (x) := T 2j (x) or T 2j+1 (x), depending on whether the field Q is symmetric or antisymmetric in z. Because of the absence of boundary conditions on P , the corresponding expansion (3.2) was truncated at order (I -2, J -1) instead of (I, J) in this case (see [START_REF] Peyret | Spectral methods for incompressible viscous flow[END_REF]). The governing equations were solved directly in the physical space using a collocation method. Discontinuities in (2.10) (at r = r ⋆ i for |2z/Γ| = 1) and (2.11) (at |2z/Γ| = 1 -α) were smoothed in order to preserve spectral accuracy. This was achieved by introducing a small length scale δ, following [START_REF] Lopez | An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries: I. axisymmetric cases[END_REF], and defining new boundary conditions for the azimuthal velocity

V = r r ⋆ i µ + (1 -µ) exp - r -r ⋆ i δ , (3.3)
and density ρ B , by integration of

∂ z ρ B = - 1 2αΓ 2 + tanh z -Γ/2(1 -α) δ -tanh z + Γ/2(1 -α) δ (3.4)
with respect to z. A value of δ = 0.006 was taken for the computations, following [START_REF] Avila | Global endwall effects on centrifugally stable flows[END_REF]. Solutions were computed by Newton-Raphson iteration until the residuals

Q n+1 -Q n ∞ / Q n ∞ converged below 10 -8 for each component Q of Q.
All base flow computations were performed with (I, J) = (101, 150), ensuring that the trailing coefficients were always between 4 and 5 orders of magnitude lower than the maximum coefficient. Figure 2(a) shows a typical 'spectral' convergence of the coefficients of (3.2). The solver was also validated against results from [START_REF] Avila | Global endwall effects on centrifugally stable flows[END_REF] (see figure 2(b)) and [START_REF] Abshagen | Multiple localized states in centrifugally stable rotating flow[END_REF].

Linear stability

We investigated the linear stability of the base flows computed using temperature boundary conditions. As Q is steady and axisymmetric, infinitesimal perturbations can be expressed as a superposition of normal modes of the form q ′ = q′ (r, z) exp[i(mθ -ωt)],

with azimuthal mode number m and complex frequency ω. These modes either share the z-symmetry (3.1) of the base flow, so that Zq ′ = q ′ , or are antisymmetric under the action of Z: Zq ′ = -q ′ . Substituting the normal mode form into the Navier-Stokes equations (2.6)-(2.11) linearized about Q leads to a generalized eigenvalue problem in ω and q ′ . Following [START_REF] Leclercq | Temporal stability of eccentric Taylor-Couette-Poiseuille flow[END_REF], this problem was reduced to a much smaller standard eigenvalue problem by eliminating the pressure, boundary points, and one velocity component ( w′ if m = 0 and ṽ′ otherwise) and then solved using standard LAPACK and ARPACK++ routines. The solver was validated against figure 6 of [START_REF] Avila | Global endwall effects on centrifugally stable flows[END_REF] with perfect agreement between our computed values of critical Re and values digitized from the plot for Γ = 3, 4, 6 and the various m shown.

Salt stratification: quasi-steady base flows

In the case of salt stratification, the no-flux boundary conditions lead to a very slow relaxation of the density field to an eventually unstratified state. The time scale of density diffusion τ κ is larger than 5 months in our apparatus, however, so that the laboratory experiments are quasi-steady after a few momentum diffusion times (τ ν ∼ 5h30). The resulting time-dependent base flow is also axisymmetric and invariant under the action of Z. In order to compute these solutions, a time-stepper was developed with no-flux boundary conditions on the density ∂ n ρ B = 0, as in the experiments. The flow was taken initially at rest with a stratification profile given by (3.4) at any radial position. The cylinders and end-plates were then accelerated to their final rotation speed over a few periods τ = 0.05τ ν , by multiplying the boundary condition (3.3) by the factor (1 -exp[-t/τ ]). A streamfunction-vorticity formulation was used and spectral expansions of the form given in (3.2) were again used to discretize the problem (see Peyret (2002), pp. 188-195). Nonlinear terms were treated explicitly with a second-order extrapolation (Adams-Bashforth) whereas the diffusion terms were treated implicitly (backward differentiation). Incompressibility was enforced with an influence matrix. Helmholtz and Poisson problems were solved using a full diagonalization method. The code was validated by applying Dirichlet boundary conditions (3.4) on the density and checking consistency with solutions given by the Newton-Raphson solver.

Experimental methods

Experiments were conducted in a Taylor-Couette tank of height 50 cm with (r i , r o ) = (10, 24) cm resulting in a radius ratio η = 0.417. The tank was filled to a depth h = 42 ± 2 cm, giving an aspect ratio Γ = 3 ± 0.14. The top and bottom boundaries both rotated with the outer cylinder. A transparent perspex lid, placed in contact with the fluid in the annulus, was fixed to the outer cylinder after filling and formed the top endplate, while the bottom end-plate was securely attached to the rotating table forming the base of the tank. The inner cylinder was driven independently from above by a mounted motor unit, at its maximum rotation rate Ω i = 1 ± 0.03 rad s -1 for all experiments. The rotation rate Ω o of the rotating table was varied to achieve the desired value of µ.

The experiments began by filling the tank to the desired stratification using two computer-controlled peristaltic pumps. The two pumps were connected to different reservoirs of fluid, one of relatively dense NaCl solution and the other of fresh water, both of which had been de-aerated using vacuum cylinders to reduce unwanted air coming out of the solution during the course of an experiment and hindering the visualisation. The reservoirs also sat out for approximately one day to equilibrate with the ambient temperature conditions of the laboratory. The density of the two reservoirs were measured just prior to filling using an Anton Paar DMA5000 density meter with a resolution of 10 -6 g cm -3 . After filling, a conductivity probe was then traversed through the depth of the tank to measure the resulting stratification. Due to the long filling time (∼ 1 -2 hours), the flow rates in the pumps could deviate during the filling procedure resulting in a small error of the desired fill height (±5%). We do not expect this small change in aspect ratio to alter the overall flow patterns observed.

To reduce the impact of any initial transient, the cylinders were slowly ramped up over 8 hours (∼ 1.5 τ ν ), with the rotation ratio µ held constant during the course of the ramp. After the initial ramp period, the cylinders' rotation rates were held constant for another momentum diffusion time before acquiring any raw images for PIV. After the raw images had been acquired, the cylinders were both ramped down to rest over a momentum diffusion time to minimise unwanted shear before taking a final density profile using the conductivity probe.

To monitor the behaviour of the stratified layers, shadowgraph images were automatically recorded at 30 minute intervals during the experiment. To enable simultaneous capture of PIV and shadowgraph images, the inner cylinder was painted white except for a black band that spanned z = -9 cm to z = 10 cm. A slide projector positioned approximately 3 m from the tank illuminated the flow and projected refractive index variations onto the white regions of the inner cylinder, which were then recorded using a Uniq Vision UP1830-12B 1 MPixel resolution camera.

Raw images for PIV were captured using a 4 MPixel Dalsa Falcon 2 camera at 160 fps for 30 seconds (∼ 5 rotation periods of the inner cylinder). The camera was mounted directly above the apparatus, viewing down through the top transparent end-plate. The flow was seeded using 20 µ m diameter polyamid particles and illuminated using a 700 W arc lamp. The resulting light sheet was approximately 3 mm thick and was horizontally orientated so all velocity data are in the (r, θ) plane. For all the data reported here, the light sheet was positioned at z = 0 ± 1 cm so data are acquired in the centre of the unstratified region. The region illuminated and captured by the camera was 14 cm × 3 cm which corresponded to 1800 pixels × 400 pixels. Particle image velocimetry was analysed using DigiFlow [START_REF] Dalziel | Simultaneous synthetic Schlieren and PIV measurements for internal solitary waves[END_REF]) with an interrogation window size 32 pixels × 32 pixels and an overlap of 50% that results in a spatial resolution of ∼ 1.24 mm. The error in particle displacements was ∼ 0.02 of a pixel which gives an error in velocity of ∼ 0.25 mm s -1 (∼ 0.025% of r i Ω i ).

Mitigating end-effects with stratification

Table 1 summarizes the values of the different control parameters taken in our experiments and computations. Two rotation ratios were considered in both: a 'high-shear' (HS) value where µ = η 3/2 and a 'low-shear' (LS) one where µ = η 1/2 . We recall that the quasi-Keplerian range is η 2 < µ < 1 with the upper limit µ = 1 corresponding to uniform rotation and therefore no shear. Here we use the terms 'low' and 'high' only to distinguish the two cases by comparison with one another, but note that arbitrarily large shear rates can be achieved even in the LS case, by setting Re to a large enough value.

Because of our underlying objective of investigating the existence of turbulence in quasi-Keplerian flow, we indeed chose to maximise the value of the Reynolds number in the experiments, leading to Re = 14000 with our set-up. However, accurate computations could not be achieved at this large value of Re in a reasonable time frame with our serial codes, so an upper limit of Re = 5000 was taken for these.

Salt was used to generate stratification in the experiments, so the density field satisfied no-flux boundary conditions at the walls and the Schmidt number was Sc ≈ 700. However, for reasons which will be discussed further in §5.3, we chose to model the experiment with Dirichlet boundary conditions at the walls, instead of Neumann conditions. Table 1. Comparison between numerical and experimental control parameters. For the stratified experiment with α = 0.5, Ri l = 2, and the larger value of Re = 14000, Rig ≈ 6 and ∆ρ/ρ0 ≈ 0.04 so the condition Rig ≫ ∆ρ/ρ0 is well-satisfied. 'Centrifugal buoyancy' is therefore expected to be negligible compared to 'gravitational buoyancy', justifying the use of the Boussinesq approximation.

In practice, this would correspond to imposing a temperature profile at the walls, which would lead to much quicker density diffusion, as heat diffuses more rapidly than salt in fresh water (P r ≈ 7 instead of Sc ≈ 700). This other discrepancy will be discussed in §5.2. Put simply, these modelling choices for stratification made our equations much more amenable to numerical computation than if actual experimental settings were taken.

The computed meridional and zonal base flows are presented in figures 3 and 4 for the two shear cases and three choices of stratification: no stratification (α = 0), full stratification (α = 1) and partial stratification (0 < α < 1). Each case is discussed in more detail in the following subsections. In the absence of stratification, the meridional circulation for the HS case penetrates deep into the flow and causes a strong distortion of the azimuthal flow field, as can be seen in figure 3(a). The driving mechanism is a broken balance between centrifugal force and pressure gradient at the end-plates caused by the no-slip boundary condition [START_REF] Czarny | Interaction between Ekman pumping and the centrifugal instability in Taylor-Couette flow[END_REF]. The inward radial flow at the endcaps is then redirected to the equatorial region through Stewartson layers once it reaches the inner cylinder. This leads to the formation of two meridional recirculation cells that merge at the equator into a strong outward radial jet [START_REF] Avila | Global endwall effects on centrifugally stable flows[END_REF]. The jet redistributes angular momentum along r, leading to a locally centrifugally unstable angular momentum profile, i.e. d|L|/dr < 0 for 0 r -r ⋆ i 0.63 and z = 0. The structure of the resulting global mode linear instability is represented in figure 5(a). In the experiment, this instability develops into turbulent motion, characterised by high levels of velocity fluctuations in the equatorial plane: δu 2 + δv 2 /v up to 5 -6% near the inner cylinder ((.) denotes temporal average and δu is the fluctuation with respect to the mean: δu := uu), as shown by the dashed line in figure 6. Turbulence tends to homogenise angular momentum in centrifugallyunstable regions, explaining the deviation between the computed base flow (dashed line) and the time-averaged azimuthal velocity (solid line with markers) for 0 r -r ⋆ i 0.7 in figure 7(a). Although the end-plates generate large scale meridional circulation in the LS case, the symmetric cells in figure 4(1a) do not appear to reach the equator. As a result, the axial homogeneity of the zonal flow V is preserved across z = 0 and the flow is numerically found to be linearly stable. Correlatively, velocity fluctuations in the experiments remain small with δu 2 + δv 2 /v ∼ 1% at the equator (solid line in figure 6). In fact, apart from rapid variations in boundary layers near the end-plates, the azimuthal velocity seems virtually independent of z across the entire container. However, a closer look at the radial profile of v in figure 7(b) shows significant differences between the experiments (solid line with markers) and the ideal Taylor-Couette solution (thick solid line). This mismatch can be interpreted as a manifestation of the Taylor-Proudman theorem in rapidly rotating flows, as explained in [START_REF] Hollerbach | End-effects in rapidly rotating cylindrical Taylor-Couette flow[END_REF]. Indeed, in the LS case, the Rossby number (defined as in [START_REF] Hollerbach | End-effects in rapidly rotating cylindrical Taylor-Couette flow[END_REF] and [START_REF] Paoletti | Angular momentum transport in turbulent flow between independently rotating cylinders[END_REF]) the flow is not very far from geostrophic balance. It is therefore nearly invariant along the axial direction and controlled by Ekman pumping at the end-plates. Thus, despite weak meridional circulation and stability, experiments in the LS case do not yield the desired Taylor-Couette solution (1.1).

Ro := (Ω i -Ω o )/Ω o = (1 -µ)/µ is small (Ro ≈ 0.55 for η = 0.417) and

Fully stratified case: α = 1

In this section, we investigate the effect of stratification across the entire container. The choice of Ri l = 2 in figures 3(c), 4(c) and 5(c) will be justified later in §4.3.

4.2.1. 'High-shear case' µ = η 3/2
The addition of stratification in our axisymmetric computations results in a dramatic reduction in meridional circulation and subsequent zonal flow distortion for the HS case as can be seen in figure 3(1c). Apart from boundary layers at the end-plates and a small 'eddy' trapped in the inner corners, the magnitude of U ⊥ has dropped to less than 0.1% of the inner cylinder velocity everywhere in the flow. As a result, the azimuthal flow now weakly depends on z in the equatorial region and solution (1.1) (thick solid red line) is almost perfectly recovered at z = 0, as can be seen in figure 7(a) (dashed-dotted blue line). However, despite the fully stratified solutions being significantly closer to the base flow, the computed axisymmetric solution cannot actually be observed in an experiment as the flow is, again, subject to a linear instability. The structure of the most unstable mode, shown for Re = 1500 in figure 5(c)), is clearly reminiscent of the SRI as described in [START_REF] Shalybkov | Stability of density-stratified viscous Taylor-Couette flows[END_REF] and [START_REF] Rüdiger | Stratorotational instability in MHD Taylor-Couette flows[END_REF]: the mode is nonaxisymmetric and almost axially periodic, with an axial wavelength smaller than the gap (for infinite cylinders, [START_REF] Shalybkov | Stability of density-stratified viscous Taylor-Couette flows[END_REF] found the scaling λ ∼ Ri -1/2 l for the critical axial wavelength λ). However, the production of total disturbance energy is localised near the inner cylinder, which suggests that the instability mechanism for our wide-gap configuration η = 0.417 is closer to the radiative mechanism of Le Dizès & ). Solid black line with markers: time-averaged profiles in the unstratified case, from PIV measurements. There is a small discrepancy in the measured velocity profile due to experimental error (<5%) in setting the rotation rate of the cylinders.

Riedinger (2010) for η → 0 than to a resonance between boundary-trapped inertia-gravity waves as discussed by [START_REF] Molemaker | Instability and equilibration of centrifugally stable stratified Taylor-Couette flow[END_REF] in the limit η → 1.

'Low-shear case

' µ = η 1/2
In the LS case, the effect of stratification on the base flow is similarly beneficial: the meridional flow is negligible except in boundary layers, and the azimuthal velocity is not only almost axially invariant, but also very close to the theoretical solution (1.1) (see figure 7(b); dashed-dotted blue line for the base flow and thick solid red line for the Taylor-Couette solution).

However, a crucial difference with the HS case is that the base flow now remains stable. This is consistent with the predictions of [START_REF] Shalybkov | Stability of density-stratified viscous Taylor-Couette flows[END_REF] and [START_REF] Rüdiger | Stratorotational instability in MHD Taylor-Couette flows[END_REF], who showed that stratified Taylor-Couette flow between infinite cylinders is generally stable for µ < η. For the HS case, the presence of end-plates does not seem to impact the structure of the mode far from the end-plates (see 5(c)). Therefore, it may not be surprising to obtain good agreement with theoretical results obtained for an axially invariant base flow in the LS case too. 4.3. Partially-stratified case: 0 < α < 1

The promising numerical results of section §4.2 indicate that stratification is quite powerful at suppressing large meridional circulation cells, which in turn leads to azimuthal velocity profiles close to (1.1). But the end goal of our investigation is to probe the possibility of purely hydrodynamic turbulence, so the central portion of the apparatus needs to be left unstratified and avoid contamination by the stratorotational instability.

Our control strategy hence relies on two parameters: the stratified fraction α and the local Richardson number Ri l . The effect of these two parameters on the departure of computed solutions with respect to (1.1) is shown in figure 8, in the LS case (results are qualitatively similar for HS). The relative difference between computed and ideal solutions is averaged over a volume spanning half of the unstratified region and centered about the equator. More specifically, we introduce zonal and meridional deviations, where

∆ V := (V -V T C )/V T C V and ∆ ⊥ := U ⊥ /V T C V (4.1)
V T C := rΩ T C , (.) V := 1 V V (.) 2 dV 1/2 (4.2)
and

V := (r, θ, z) ∈ [r ⋆ i , r ⋆ o ] × [0, 2π[× - Γ(1 -α) 4 , Γ(1 -α) 4 . (4.
3)

It appears that both quantities decrease rapidly as Ri l increases in the range 0 < Ri l 5. The meridional deviation ∆ ⊥ even drops by nearly one order of magnitude within the range 0 < Ri l < 1. Increasing Ri l further has virtually no effect on U ⊥ but keeps improving V . Similarly, the thickness of the layers does not seem critical to the suppression of the meridional flow, as long as Ri l 1. However, thick layers are more efficient at correcting the azimuthal velocity profile, as long as the unstratified zone does not become shallow. Indeed, when α 5/6, the depth of the unstratified zone becomes lower than half the gap width and the flow in no longer axially homogeneous in this region. This has a negative impact on the deviations, which start increasing again with α and Ri l .

In this section, we choose to study the case (α, Ri l ) = (0.5, 2), which appears to be a good compromise for various reasons: deviations ∆ V and ∆ ⊥ are both small, the aspect ratio of the unstratified zone Γ(1 -α) = 1.5 is larger than 1, and finally this combination is easily achieved experimentally since ∆ρ/ρ 0 ≈ 4% for Re = 14000. This justifies a posteriori the choice of Ri l = 2 to illustrate the fully stratified case in §4.2. The density field corresponding to this parameter combination is shown in figure 9 for the HS and LS cases. In both, the isopycnals are almost horizontal, indicating that stratification is 'strong enough' to avoid overturning by the meridional circulation, except at the inner corners (more details will be provided in §5.1).

'High-shear case' µ = η 3/2

There is barely any noticeable difference between the computed meridional flow fields in the fully and partially stratified cases in figure 3 (1b & 1c). Apart from boundary layers in the unstratified region for α = 0.5, the two fields are almost indistinguishable, confirming that partial stratification efficiently suppresses U ⊥ even when α = 1. The azimuthal velocity is also remarkably homogeneous along the axial direction in the unstratified zone, but unfortunately, the computed base flow is linearly unstable. The production of energy of the most unstable mode, displayed in figure 5(b) for Re = 1500, is precisely localized within the stratified layers. Moreover, the distribution of this quantity is very similar to that of the fully stratified flow in the same region. Therefore, we conclude that the mode is subject to a stratorotational instability mode localized inside the layers.

Experimentally, this instability leads to the braid pattern visible in the spatiotemporal shadowgraph of figure 10(a). This pattern corresponds to the nonlinear interaction of two helical modes of opposite 'handedness', which are destabilized simultaneously because of symmetry. In the long-term, this secondary flow generates mixing and destroys the initial density profile, as can be seen in figure 10(b).

4.3.2. 'Low-shear case' µ = η 1/2
The effect of partial stratification on the base flow in the LS case is similar to that observed for the HS case. The meridional flow is suppressed in the unstratified zone,

0 0.25 0.75 r -r ⋆ i V, v 0.5 0.8 0.9 1 1 1.1 1.2 1.3 1.4 1.5 1.6
Figure 11. Comparison between azimuthal velocity profile at the equator with and without control in the LS case. Thick solid red line -ideal Taylor-Couette solution (1.1). Solid black line with markers -time-averages from experiments (circles -unstratified; squares -partially stratified (α, Ri l ) = (0.5, 2)). Blue dashed/dashed-dotted lines -computed base flows for unstratified case/partially-stratified case (α, Ri l ) = (0.5, 2). Additional measurements at z ≈ Γ/6(1 -α) in the partially-stratified case yielded a mean velocity profile nearly indistinguishable from that at z ≈ 0, hence very close to ideality too. This confirms the effectiveness of the control strategy across a large portion of the volume bounded by the stratified layers.

including within the boundary layer running along the inner cylinder, as |z| → 0. The azimuthal velocity is virtually axially homogeneous in the unstratified region, but contrary to the unstratified case, the radial profile now closely matches the ideal solution (1.1), as illustrated in figure 11.

For LS, no instability is found (numerically tested up to Re = 5000 and experimentally up to 14000) in both the fully and the partially stratified cases. This result is consistent with the analysis of [START_REF] Rüdiger | Stratorotational instability in MHD Taylor-Couette flows[END_REF] who always found linear stability at µ = η 1/2 (these authors investigated the range 0.3 η 0.78 and 0.2 Ri l 4).

In conclusion, it appears that the partially-stratified LS case (α, Ri l ) = (0.5, 2) corresponds to a 'sweet-spot' where stratification efficiently mitigates end-effects without introducing any additional instability mechanism. It is also a realistic set of parameters for an experiment since it only requires ∆ρ/ρ 0 ≈ 4% for Re = 14000.

Discussion

Physical mechanism

To understand why end-plates create meridional flow, consider the transport equation for the azimuthal vorticity ζ := ∂ z u -∂ r w. For the axisymmetric and steady base flow, this equation can be written as

U ∂ r + W ∂ z - U r - 1 Re ∇ 2 - 1 r 2 ζ B = ∂ z V 2 r + Ri g ∂ r ρ B (5.1)
In the absence of end-plates and stratification, the right-hand side of (5.1) is exactly zero, and the basic flow solution satisfies ζ B = 0 which implies that there is no meridional flow. In this case, the cyclostrophic balance V 2 /r = dP/dr is satisfied everywhere in the flow. When end-plates are added, the boundary condition on the azimuthal velocity disrupts this equilibrium [START_REF] Czarny | Interaction between Ekman pumping and the centrifugal instability in Taylor-Couette flow[END_REF] and a now non-vanishing torque term ∂ z V 2 /r = 0 (due to an inhomogeneous centrifugal force along z) generates meridional vorticity. When stratification is added, the baroclinic torque Ri g ∂ r ρ B counteracts this effect, leaving the flow virtually free of meridional vorticity when the right-hand side becomes negligible. If we estimate that r ≈ 1/2(r

⋆ i + r ⋆ o ) = 1/2(1 + η)/(1 -η), V ≈ 1 and ∂ z = O (d/(l/2)) = O (2/(αΓ)), this occurs for Ri l ≈ 4η 2 (αΓ) 2 (1 -η 2 )∂ r ρ B . (5.2)
This expression confirms that thick layers, i.e. large αΓ, are more efficient at suppressing meridional motion, as they require a lower value of Ri l . There are indeed two reasons to try and minimize the required value for Ri l in experiments. Firstly, Ri l ∝ Ω -2 i decreases rapidly with increasing Re ∝ Ω i at fixed buoyancy frequency N . Secondly, the maximum buoyancy frequency N is set by saturation of salt into water at ∆ρ/ρ 0 ≈ 20%, limiting in turn the maximum Ri l ∝ N 2 ∝ ∆ρ/ρ 0 . Therefore, thick layers are helpful when trying to maximise Re in an experiment while satisfying (5.2). As discussed in §4.3, this only remains true as long as the aspect ratio of the unstratified zone Γ(1 -α) does not become small.

By estimating an upper bound for ∂ r ρ B , it is also possible to evaluate a lower bound for Ri l . Looking at the density field in figure 9, we observe that the isopycnals are approximately straight lines in the meridional plane, except at the inner corners. In dimensional units, the radial density gradient must therefore be lower than ∆ρ/(2d), since, by definition, the initial density jump across one layer is equal to ∆ρ/2. This leads to the nondimensional estimate ∂ r ρ < 1/2. For α = 0.5, η = 0.417 and Γ = 3, we obtain the upper bound Ri l > 0.75, which is consistent with the observation that ∆ ⊥ is nearly constant beyond that value in figure 8.

Finally, we note that in the case of Dirichlet boundary conditions on the density, the baroclinic term is infinitesimally small in the vicinity of the end-plates, making it impossible to fully cancel the centrifugal torque term. This centrifugal torque is large in the vicinity of the inner corners because of the sharp vertical variation in V , explaining the locally higher levels of U ⊥ .

Effect of Prandtl number

Figure 12 shows the impact of the Prandtl number on the deviations ∆ V and ∆ ⊥ defined in (4.1). It appears that larger P r leads to a significant reduction in both quantities, following approximate power laws ∆ V ∝ P r -0.47 and ∆ ⊥ ∝ P r -0.7 in the range P r ∈ [1,46] for Re = 5000. This behaviour suggests that stratification with salt (Sc ≈ 700) ought to be more efficient than stratification with temperature (P r ≈ 7), ignoring the differing boundary conditions. This observation suggests that our numerical model overestimates the departure of the experiment from (1.1), as can also be seen in figure 11. This results reinforces our confidence in the effectiveness of our method.

Effect of no-flux boundary conditions on density

Aside from the difference in Reynolds numbers, the only remaining discrepancy between the model and the experiment lies in the boundary conditions for the density. In our model, we assumed that stratification was permanently forced by applying a temperature profile at the walls, whereas in the experiments, the density is only controlled at t = 0. In the experiment, a piecewise linear density profile of the form (2.11) is set initially, but it is subject to no-flux boundary conditions ∂ n ρ B at all times. Because of mass diffusion, the only possible steady state solution is unfortunately unstratified. Yet, given the large value of the Schmidt number Sc = ν/κ ≈ 700 in the experiment, there is a separation of time scales between momentum diffusion τ ν = d 2 /ν and mass diffusion τ κ = d 2 /κ which translates into a quasi-steady evolution of the flow after a few units of τ ν . But simulating such large Sc number flows proves computationally expensive as the corresponding separation of length scales is also large. Indeed, if we estimate the smallest inhomogeneity scale in the flow to be given by the [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid. part 1. general discussion and the case of small conductivity[END_REF] scale (formally defined for a stratified turbulent flow) L B ∼ Sc -1/2 (or L B ∼ P r -1/2 for temperature), we expect it to be one order of magnitude smaller for Sc = 700 than for P r = 7. The resulting increase in computational cost did not allow us to perform the calculations in a reasonable timeframe with the present codes. However, it is in principle possible to satisfy the separation of time scales τ κ /τ ν = Sc ≫ 1 without taking Sc as large as 700. One might hope that a lower value of Sc, for instance Sc = 30, would suffice. Figure 13 shows that this is not the case. Indeed, the discontinuity of V in the corners leads to local overturning of the density field, hence strongly curved isopycnals (see figure 9). As a result, the mass diffusion term in equation (2.6d) is not negligible in these corners, despite the small prefactor (Re Sc) -1 ≈ 7 × 10 -6 . Therefore, the rate of expansion of the meridional circulation cannot be approximated by τ -1 κ , as the inhomogeneity length scale in the corners is much smaller than d. We conclude that it is not possible to compute quasi-steady solutions even for Schmidt numbers in the range 1 ≪ Sc ≪ 700. This prohibits the use of linear stability analysis and motivates the use of Dirichlet (temperature) boundary conditions in our model. Despite an inadequate separation of time scales, it is worth noting that the aspect of the solution at the inner corners in the initial stages of our simulation with Neumann (salt) boundary conditions (see figure 13(a) for t = 0.5τ ν ) is quite similar to the equivalent steady flow in figure 3 (1b). This observation strengthens our confidence in the ability of the model with Dirichlet boundary conditions to capture the dynamics of the experiment.

Seeking subcritical transition to turbulence

After demonstrating that the parameter combination (ln µ/ ln η, α, Ri l ) = (0.5, 0.5, 2) satisfies all the constraints set forth in the introduction, we probed the possibility of sustained turbulence in the unstratified zone by carrying out a series of experiments.

In our first experiment, the flow was perturbed impulsively by reducing the inner cylinder rotation rate by 75% in about 1s. A first PIV measurement was taken then (defining t = 0), before Ω i was ramped back up to its final value over the course of 1 minute (approximately 10 inner cylinder rotation periods). A second measurement was made 30 minutes later, at t ≈ τ ν /10. Figure 14 shows the deviation from the unperturbed state q ∞ (time-averaged azimuthal profile v ∞ shown in figure 11), measured by ∆ ∞ :=

u 2 + (v -v ∞ ) 2 /v ∞ ,
at both instants. At t = 0 (dashed line), the perturbation extends radially over a distance d ′ ≈ d/10, corresponding to an effective diffusion time τ ′ ν = d ′2 /ν ≈ τ ν /100. At t ≈ τ ν /10 ≈ 10τ ′ ν (solid line), the perturbation has completely diffused and the flow has relaxed to its unperturbed laminar state, as indicated by the low value of ∆ ∞ ≈ 1% over the entire radial range.

In addition to that, we also tried to trigger subcritical transition to turbulence using two types of non-axisymmetric perturbations. After the spin-up procedure, a turbulent radial jet was introduced at the equator using a syringe and a needle of diameter 1 mm. The needle was introduced into the flow through a hole in the top end-plate before passing through the upper stratified region to reach the unstratified core. No discernible mixing was introduced in the stratified region throughout this procedure. The perturbation velocity in the jet was approximately 7% fo the inner cylinder velocity, but sustained turbulence was not triggered, with the flow relaminarising within 30 minutes. The second non-axisymmetric perturbation involved the same needle set-up but this time fluid was removed from the unstratified core region to induce vortex stretching. Once again, no sustainable turbulence was generated using this method.

All these observations show the robustness of the laminar flow with respect to finiteamplitude perturbations, and indicate its nonlinear stability at Re = O(10 4 ) and µ = η 1/2 . This conclusion is consistent with recent numerical and experimental results by [START_REF] Ostilla-Mónico | Turbulence decay towards the linearly stable regime of Taylor-Couette flow[END_REF] and [START_REF] Edlund | Nonlinear stability of laboratory quasi-Keplerian flows[END_REF], at even higher Reynolds number and larger rotation ratios. The first authors used axially periodic direct numerical simulation (with no stratification), and could not find any evidence of self-sustaining process up to Re = O(10 5 ) with µ = η 3/2 . Similarly, the latter authors, from the Princeton group, found robust decay of various types of initial perturbations in their enhanced split-ring apparatus (HTX device) at Re even greater than 10 6 , with µ = η 1.8 .

Conclusions

In this work we have confirmed both experimentally and numerically the severe effect end-plates have on the interior of Taylor-Couette flow in the quasi-Keplerian regime. In the 'high-shear case' µ = η 3/2 , two symmetric recirculation cells are generated in the meridional plane; these merge at the equator to form a strong radial jet. This jet redistributes angular momentum, causing the flow to become turbulent through a linear instability. In the 'low-shear case' µ = η 1/2 , the meridional flow is weaker, and there is no radial jet at the equator for the same value of Re. Despite remaining approximately invariant along the axial direction, the azimuthal flow is however significantly disturbed by the presence of end-plates and deviates from the 'ideal' Taylor-Couette solution corresponding to an axially-unbounded system.

We have then examined how stable stratification can be judiciously used to mitigate these end-effects in order to recover 'ideal' Taylor-Couette flow. When stratification is added across the full height, axisymmetric meridional flow is only able to develop in boundary layers near the end-plates and recirculation becomes essentially confined to the inner corners, where the azimuthal velocity is discontinuous. Everywhere else in the flow, the deviation with respect to the ideal velocity profile drops significantly. The physical mechanism behind this passive control method is the competition between the 'centrifugal' torque term ∂ z (v 2 /r) responsible for the generation of azimuthal vorticity, and the baroclinic torque Ri g ∂ r ρ acting against it. This mechanism is robust enough that similar enhancement of the base flow in the equatorial region can be achieved even when stratification is restricted to layers adjacent to the top and bottom end-plates. As long as the stable stratification is strong enough, typically 1 < Ri l < 5, and that neither the stratified nor the unstratified layers are too shallow (typically α ≈ 0.5 for Γ = 3), the control of the base flow is effective.

One obvious advantage of the approach is that it is much easier to implement than complex designs such as split-endcaps [START_REF] Ji | Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks[END_REF][START_REF] Edlund | Nonlinear stability of laboratory quasi-Keplerian flows[END_REF] or split inner cylinder [START_REF] Paoletti | Angular momentum transport in turbulent flow between independently rotating cylinders[END_REF]. By simply adding salt in varying concentration to the working fluid, we managed to significantly mitigate end-effects and recover the Couette flow solution around the equator z = 0.

However, like any other control method, this approach has its limits. Firstly, the method is only applicable to rotation ratios in the range η µ < 1, corresponding to the least-sheared configurations in the quasi-Keplerian range η 2 < µ < 1. When µ < η, the stratorotational instability develops in the stratified layers, despite the presence of the unstratified region and end-plates. This instability destroys the initial density profile, rendering the method ineffective. Secondly, applying the method at very high Reynolds number, of the order of 10 6 or more, is challenging due to the eventual saturation of water by salt. For example, taking Ri l = 2, ∆ρ/ρ 0 ≈ 20% (saturation threshold for salt in water) and a layer height of 0.75 gap (corresponding to αΓ = 1.5 as in §4.3), means that reaching Re = 10 6 would require a huge tank of outer radius r o ≈ 2 m or more.

This aside, we have successfully demonstrated the applicability of the method at Re = O(10 4 ) for the rotation ratio µ = η 1/2 , and found turbulence decay following several types of large-amplitude disturbances. This result is consistent with the most recent numerical (Ostilla-Mónico et al. 2014) and experimental [START_REF] Edlund | Nonlinear stability of laboratory quasi-Keplerian flows[END_REF] investigations at respectively Re = O(10 5 ) and Re = O(10 6 ), hence further strengthens the view that extra physics beyond that of a simple rotating shear flow is needed to explain the inferred turbulence in weakly ionized accretion disks (see [START_REF] Turner | Transport and accretion in planet-forming disks[END_REF] for a recent review of other possible instability mechanisms).

Figure 1 .

 1 Figure1. Partially stratified Taylor-Couette setup: the flow is linearly stratified in the axial direction z near the end-plates (colour scheme is indicative of temperature). In this paper, the end-plates are attached to the outer cylinder. Rather than the total density, ρ denotes the deviation with respect to a reference value ρ0.

Figure 2 .

 2 Figure 2. (a) Convergence of spectral coefficients Ûij of the radial velocity component, for µ = η 1/2 , α = 0.5 and Ri l = 2. (b) Maximum radial velocity at the equator for η = 0.5, Γ = 6, µ = 1, fixed end-plates and no stratification. Solid line: data digitized from Avila et al. (2008), dots: present code.

4. 1 .

 1 Unstratified case: α = 0 4.1.1. 'High-shear case' µ = η 3/2

Figure 3 .

 3 Figure 3. Base flow for the HS case µ = η 3/2 (1) Meridional flow U ⊥ = (U, W ): streamlines (∆ψ = 10 -3 ) and colourmap of the norm U ⊥ . (2) Azimuthal velocity V : colourmap and contours of V (∆V = 0.05). Dotted lines in (b) indicate the limit of the stratified layers. In this figure and hereafter, the left (resp. right) boundary corresponds to the inner (resp. outer) cylinder.

Figure 4 .

 4 Figure 4. Base flow in the LS case µ = η 1/2 . Same caption as figure 3, except that here ∆ψ = 2.5 × 10 -4 .

Figure 5 .Figure 6 .

 56 Figure5. Structure of the most unstable linear mode for µ = η 3/2 , Re = 1500 and (a) α = 0 (m = 2, symmetric under Z), (b) α = 0.5 (m = 1, antisymmetric under Z), (c) α = 1 (m = 1, symmetric under Z). The production of perturbation energy (kinetic + potential) over the volume V of the container can be expressed as PV = -V u ′ • ∇U • u ′ dV. We introduce the density P(r, z) by writing PV as 4πe i2ω i t P(r, z)rdrdz. The quantity r P(r, z) indicates the distribution of positive production of perturbation energy in the meridional plane, averaged over θ. Contour lines: ∆ log 10 r P(r, z) = 0.25. Dotted lines in (b) mark the limits of the stratified layers. The resolution here is (I, J) = (67, 100).

Figure 7 .

 7 Figure 7. Azimuthal velocity profiles at the equator for the (a) HS and (b) LS cases. Solid red lines: axially-invariant Taylor-Couette solution (1.1), dashed blue lines: computed values in the unstratified case, dashed-dotted blue lines: computed base flows in the fully stratified case (α, Ri l ) = (1, 2). Solid black line with markers: time-averaged profiles in the unstratified case, from PIV measurements. There is a small discrepancy in the measured velocity profile due to experimental error (<5%) in setting the rotation rate of the cylinders.

Figure 8 .

 8 Figure 8. (a) Relative azimuthal deviation ∆V and (b) relative meridional deviation ∆ ⊥ as functions of Ri l , for various values of α, in the LS case.

Figure 9 .Figure 10 .

 910 Figure 9. Base flow density field ρB in (a) HS case (unstable) and (b) LS case (stable); (α, Ri l ) = (0.5, 2). Dotted lines mark the limits of the stratified layers. Density step between solid lines: 0.05; between dashed lines: 5 × 10 -3 .

Figure 12 .

 12 Figure 12. Effect of the Prandtl number on the deviations ∆V and ∆ ⊥ for the HS case µ = η 3/2 , α = 0.5 and Ri l = 2.

Figure 13 .

 13 Figure13. Meridional flow in the case of no-flux boundary conditions on density; HS case, Re = 1000, α = 0.5, Ri l = 2 and Sc = 30. The unit of time here is the density diffusion time τν = d 2 /ν. Despite a large Schmidt number, there is no separation of time scales between diffusion of momentum and diffusion of density, leading to a rapid expansion of the meridional circulation. Contours and colourmap: same as figure3(1) (here ∆ψ = 10 -3 ). Dotted lines indicate the limit of the stratified layers at t = 0.

∆∞Figure 14 .

 14 Figure14. Deviation ∆∞ from the unperturbed state at the equator when searching for subcritical transition to turbulence. The dashed curve shows a measure of ∆∞ right after applying the pertubation and the solid line corresponds to the deviation 30 min later. For this forcing at Re = O(10 4 ), turbulence is not triggered in the unstratified zone as is evident from the low values of ∆∞ ≈ 1% rapidly reached.

† Email address for correspondence: c.leclercq@bristol.ac.uk

Acknowledgements

We gratefully acknowledge the support of EPSRC under grant EP/K034529/1.