Leila Amgoud

The outcomes of logic-based argumentation systems under preferred semantics

Logic-based argumentation systems are developed for reasoning with inconsistent information. They consist of a set of arguments, attacks among them and a semantics for the evaluation of arguments. Preferred semantics is favored in the literature since it ensures the existence of extensions (i.e., acceptable sets of arguments), and it guarantees a kind of maximality, accepting thus arguments whenever possible. This paper proposes the first study on the outcomes under preferred semantics of logic-based argumentation systems that satisfy basic rationality postulates. It focuses on systems that are grounded on Tarskian logics, and delimits the number of preferred extensions they may have. It also characterizes both their extensions and their sets of conclusions that are drawn from knowledge bases. The results are disappointing since they show that in the best case, the preferred extensions of a system are computed from the maximal consistent subbases of the knowledge base under study. In this case, the system is coherent, that is preferred extensions are stable ones. Moreover, we show that both semantics are useless in thic case since they ensure exactly the same result as naive semantics. Apart from this case, the outcomes of argumentation systems are counter-intuitive.

Introduction

An important problem in the management of knowledge-based systems is the handling of inconsistency. Inconsistency may be present because the knowledge base includes default rules (e.g. [START_REF] Pearl | System z: A natural ordering of defaults with tractable applications to default reasoning[END_REF]) or because the knowledge comes from several sources of information (e.g. [START_REF] Cholvy | Automated reasoning with merged contradictory information whose reliability depends on topics[END_REF]).

Argumentation theory is an alternative approach for reasoning with inconsistent information. It is based on the key notion of argument which explains why a conclusion may be drawn from a given knowledge base. In fact, an argumentation system is a set of arguments, an attack relation and a semantics for evaluating the arguments (see [START_REF] Bondarenko | An abstract, argumentation-theoretic approach to default reasoning[END_REF][START_REF] García | Defeasible logic programming: an argumentative approach[END_REF][START_REF] Gorogiannis | Instantiating abstract argumentation with classical logic arguments: Postulates and properties[END_REF][START_REF] Pollock | How to reason defeasibly[END_REF] for some examples of such systems). Surprisingly enough, in most existing systems, there is no characterization of the kind of outputs that are drawn from a knowledge base. To say it differently, the properties of those outputs are unknown. These properties should broadly depend on the chosen semantics. It is worth mentioning that in all existing systems, Dung's semantics [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] or variants of them are used. The so-called Preferred semantics is the most favored one. It enjoys a kind of maximality which leads to the acceptance of arguments whenever possible. This semantics was mainly proposed as an alternative for stable semantics which does not guarantee the existence of stable extensions. While we may find some works that investigate the outcomes of particular systems under stable semantics [START_REF] Cayrol | On the relation between argumentation and non-monotonic coherence-based entailment[END_REF], there is no such work under preferred semantics. Thus, the outcomes under this semantics are still completely mysterious and unexplored. This paper investigates for the first time the outcomes under preferred semantics of argumentation systems that are grounded on Tarskian logics [START_REF] Tarski | Logic, Semantics[END_REF] and that satisfy the basic rationality postulates proposed in [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF]. We identify for the first time the maximum number of preferred extensions those systems may have, and characterize both their extensions and their sets of conclusions that are drawn from knowledge bases. The study completely abstracts from the logic and the attack relation. The results are disappointing. They show that in the best case, the preferred extensions of a system are computed from all the maximal consistent subbases of the knowledge base under study. In this case, the argumentation system is coherent, i.e., its preferred extensions coincide with its stable ones. In a companion paper [START_REF] Amgoud | Stable semantics in logic-based argumentation[END_REF], we have shown that stable semantics does not play any role in this case since the output of a system under this semantics is exactly what is returned by the same system under naive semantics (i.e., the maximal conflictfree sets of arguments). Consequently, preferred semantics is also useless in this case. In all the remaining cases we identified, the argumentation systems return counter-intuitive results. To sum up, preferred semantics is not commended for instantiations of Dung's framework with Tarskian logics.

The paper is organized as follows: we start by defining the logic-based argumentation systems we are interested in and by recalling the three basic postulates that such systems should obey. In a subsequent section, we investigate the properties of the preferred extensions of those systems. Next, we study the inferences that are drawn from a knowledge base by argumentation systems under preferred semantics. The last section is devoted to some concluding remarks.

Logic-based argumentation systems and rationality postulates

In this paper, we consider abstract logic-based argumentation systems; that is systems that are grounded on any Tarskian logic [START_REF] Tarski | Logic, Semantics[END_REF] and that use any attack relation. Such abstraction makes our study very general. According to Alfred Tarski, an abstract logic is a pair (L, CN) where L is a set of wellformed formulas. Note that there is no particular requirement on the kind of connectors that may be used. CN is a consequence operator that returns the set of formulas that are logical consequences of another set of formulas according to the logic in question. It should satisfy the following basic properties:

1. X ⊆ CN(X) (Expansion) 2. CN(CN(X)) = CN(X) (Idempotence) 3. CN(X) = Y ⊆ f X CN(Y ) 1 (Finiteness) 4. CN({x}) = L for some x ∈ L (Absurdity) 5. CN(∅) = L (Coherence)
The associated notion of consistency is defined as follows: A set X ⊆ L is consistent wrt a logic (L, CN) iff CN(X) = L. It is inconsistent otherwise. Besides, arguments are built from a knowledge base Σ ⊆ L as follows:

Definition 1 (Argument) Let Σ be a knowledge base. An argument is a pair (X, x) s.t. X ⊆ Σ, X is consistent, and x ∈ CN(X)2 . An argument (X, x) is a sub-argument of another argument (X , x ) iff X ⊆ X .

Notations: Supp and Conc denote respectively the support X and the conclusion x of an argument (X, x). For all S ⊆ Σ, Arg(S) denotes the set of all arguments that can be built from S by means of Definition 1. Sub is a function that returns all the subarguments of a given argument. For all E ⊆ Arg(Σ), Concs(E) = {Conc(a) | a ∈ E} and Base(E) = a∈E Supp(a). Max(Σ) is the set of all maximal (for set inclusion) consistent subbases of Σ. Free(Σ) = S i where S i ∈ Max(Σ), and Inc(Σ) = Σ \ Free(Σ). Finally, C Σ denote the set of all minimal conflicts3 of Σ.

An argumentation system for reasoning over a knowledge base Σ is defined as follows.

Definition 2 (Argumentation system) An argumentation system (AS) over a knowledge base Σ is a pair

T = (Arg(Σ), R) such that R ⊆ Arg(Σ) × Arg(Σ) is an attack relation. For a, b ∈ Arg(Σ), (a, b) ∈ R (or aRb) means that a attacks b.
Throughout the paper, the attack relation is left unspecified.

Arguments are evaluated using preferred semantics [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. For the purpose of this paper, we also need to recall the definition of stable semantics. Preferred semantics is based on two requirements: conflict-freeness and defence. Recall that a set E of arguments is conflict-free iff a, b ∈ E such that aRb. It defends an argument a iff ∀b ∈ Arg(Σ), if bRa, then ∃c ∈ E such that cRb.

Definition 3 Let T = (Arg(Σ), R) be an AS and E ⊆ Arg(Σ).

-E is an admissible extension iff E is conflict-free and defends all its elements. -E is a preferred extension iff it is a maximal (for set inclusion) admissible extension.

-E is a stable extension iff E is conflict-free and attacks any argument in Arg(Σ) \ E.

It is worth recalling that each stable extension is a preferred one but the converse is not true. Let Ext x (T ) denote the set of all extensions of T under semantics x where p and s stand respectively for preferred and stable. When we do not need to specify the semantics, we use the notation Ext(T ) for short.

The set of conclusions drawn from a knowledge base Σ using an argumentation system T = (Arg(Σ), R) contains only the common conclusions of the extensions. Definition 4 (Output) Let T = (Arg(Σ), R) be an AS over a knowledge base Σ.

Output(T ) = {x ∈ L | ∀E ∈ Ext(T ), ∃a ∈ E s.t. Conc(a) = x}.
In [START_REF] Caminada | An axiomatic account of formal argumentation[END_REF], it was shown that not any instantiation of Dung's abstract argumentation framework is acceptable. Some instantiations like [START_REF] Governatori | Argumentation semantics for defeasible logic[END_REF][START_REF] Prakken | Argument-based extended logic programming with defeasible priorities[END_REF] may lead in some cases to undesirable outputs. Consequently, some rationality postulates that any system should obey were proposed. Postulates are desirable properties that any reasoning system should enjoy. In [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF], those postulates were revisited and extended to any Tarskian logic. The first postulate concerns the closure of the system's output under the consequence operator CN. The idea is that the formalism should not forget conclusions.

Postulate 1 (Closure under CN) Let T = (Arg(Σ), R) be an AS over a knowledge base. T satisfies closure iff for all E ∈ Ext(T ), Concs(E) = CN(Concs(E)).

The second rationality postulate ensures that the acceptance of an argument implies also the acceptance of all its sub-parts.

Postulate 2 (Closure under sub-arguments) Let T = (Args(Σ), R) be an AS. T is closed under sub-arguments iff for all E ∈ Ext(T ), if a ∈ E, then Sub(a) ⊆ E.

The third postulate ensures that the set of conclusions supported by each extension is consistent.

Postulate 3 (Consistency) Let T = (Arg(Σ), R) be an AS over a knowledge base Σ. T satisfies consistency iff for all E ∈ Ext(T ), Concs(E) is consistent.

The following interesting result is shown in [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF] under any acceptability semantics.

Proposition 1 [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF] Let T = (Arg(Σ), R) be an AS over a knowledge base Σ. If T satisfies consistency and closure under sub-arguments, then for all E ∈ Ext(T ), Base(E) is consistent.

It was shown in [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF] that in order to satisfy these postulates, the attack relation should capture inconsistency. This is an obvious requirement especially for reasoning about inconsistent information. Note also that all existing attack relations verify this property (see [START_REF] Gorogiannis | Instantiating abstract argumentation with classical logic arguments: Postulates and properties[END_REF] for an overview of those relations defined under propositional logic).

Definition 5 (Conflict-dependent) An attack relation R is conflict-dependent iff ∀a, b ∈ Arg(Σ), if aRb then Supp(a) ∪ Supp(b) is inconsistent.

Properties of preferred extensions

Throughout the paper, we assume argumentation systems T = (Arg(Σ), R) that are built over a knowledge base Σ. These systems are assumed to be sound in the sense that they enjoy the three rationality postulates described in the previous section. It is worth recalling that the attack relation is a crucial parameter in a system since the satisfaction of the postulates depends broadly on it. For instance, it was shown in [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF] that argumentation systems that use symmetric relations may violate the consistency postulate. This is particularly the case when the knowledge base contains a ternary or a n-ary minimal conflict (with n > 2). Thus, such symmetric systems [START_REF] Coste | Symmetric argumentation frameworks[END_REF] should be avoided and are not concerned by our study.

Our aim in this section is to investigate the properties of preferred extensions of sound argumentation systems. We will answer the following interesting questions.

1. What is the number of preferred extensions an AS may have? 2. What is the link between each preferred extension and the knowledge base Σ? 3. Is the set of formulas underlying a preferred extension consistent? 4. What is the real added value of preferred semantics compared to stable semantics?

To put it differently, does preferred semantics solve any problem encountered by stable one?

We start by showing that the argumentation systems that satisfy consistency and closure under sub-arguments, satisfy also the strong version of consistency. Indeed, the union of the supports of all arguments of each preferred extension is a consistent subbase of Σ. This result is interesting since it is in accordance with the idea that an extension represents a coherent position/point of view.

Proposition 2 Let T = (Arg(Σ), R) be an AS s.t. R is conflict-dependent and T satisfies consistency and closure under sub-arguments. For all E ∈ Ext p (T ), Base(E) is consistent.

Proof. Let T = (Arg(Σ), R) be an AS s.t. R is conflict-dependent and T satisfies consistency and closure under sub-arguments. From Proposition 1, it follows immediately that for all E ∈ Ext p (T ), Base(E) is consistent.

In addition to the fact that the subbase computed from a preferred extension is consistent, we show next that it is unique. Indeed, the subbase computed from one extension can never be a subset of a subbase computed from another extension. Thus, the preferred extensions of an argumentation system return completely different subbases of Σ.

Proposition 3 Let T = (Arg(Σ), R) be an AS s.t. R is conflict-dependent and T satisfies Postulates 2 and 3. For all

E i , E j ∈ Ext p (T ), if Base(E i ) ⊆ Base(E j ) then E i = E j .
Proof. Let T = (Arg(Σ), R) be an AS s.t. R is conflict-dependent and T satisfies consistency and closure under sub-arguments. Assume that

E i , E j ∈ Ext p (T ) and Base(E i ) ⊆ Base(E j ). We first show that ∀a ∈ Arg(Base(E i )), E j ∪ {a} is conflict- free. Let a ∈ Arg(Base(E i )). Assume that E j ∪ {a} is not conflict-free. Thus, ∃b ∈ E j such that aRb or bRa. Since R is conflict-dependent, then Supp(a) ∪ Supp(b) is incon- sistent. Besides, Supp(a) ⊆ Base(E j ). Thus, Base(E j ) is inconsistent. This contradicts Proposition 2. Let E = E j ∪ (E i \ E j ).
From above, it follows that E is conflict-free. Moreover, E defends any element in E j (since E j ∈ Ext p (T )) and any element in E i \ E j (since E i ∈ Ext p (T )). Thus, E is an admissible set. This contradicts the fact that E j ∈ Ext p (T ).

In [START_REF] Amgoud | Stable semantics in logic-based argumentation[END_REF], we have shown that the subbases computed from the stable extensions of any argumentation system that satisfies the postulates are maximal (for set inclusion) consistent subbases of Σ. In what follows, we show that this is not necessarily the case for preferred extensions. Note that this does not mean that a preferred extension can never return a maximal consistent subbase. The previous result guarantees that the maximal consistent subbases containing a non-maximal subbase computed from a given extension will never be returned by any other extension of the same system.

Proposition 4 Let T = (Arg(Σ), R) be an AS s.t. R is conflict-dependent and T satisfies consistency and closure under sub-arguments. The non-maximality of the subbases that are computed from preferred extensions is due to the existence of undecided arguments under preferred semantics. Indeed, in [START_REF] Caminada | On the issue of reinstatement in argumentation[END_REF] another way of defining Dung's semantics was provided. It consists of labeling the nodes of the graph corresponding to the argumentation system with three possibles values: {in, out, undec}. The value in means that the argument is accepted, the value out means that the argument is attacked by an accepted arguments, and finally the value undec means that the argument is neither accepted nor attacked by an accepted argument. It is thus possible that some formulas appear only in undecided arguments.

Let E ∈ Ext p (T ). If Base(E) / ∈ Max(Σ), then ∀S ∈ Max(Σ) s.t. Base(E) ⊂ S, E ∈ Ext p (T ) s.t. Base(E ) = S. Proof. Let T = (Arg(Σ), R) be an AS s.t.
Another particular property of preferred extensions is the fact that they may not be closed in terms of arguments. Indeed, they may not contain all the arguments that may be built from their bases. Indeed, it is possible that E is a preferred extension of a system and E = Arg(Base(E)). Surprisingly enough, the supports and conclusions of the missed arguments are conclusions of the extensions. Thus, even if an argument of Arg(Base(E)) does not belong to the extension E, all the formulas of its supports are conclusions of arguments in the extension, and the same holds for its conclusion. Proof. Let T = (Arg(Σ), R) be an AS s.t. T is closed under CN and under subarguments. Let E ∈ Ext p (T ) and a ∈ Arg(Base(E)). Thus, Supp(a) ⊆ Base(E). Since T is closed under sub-arguments, then Base(E) ⊆ Concs(E) (proved in [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF]). Thus, Supp(a) ⊆ Concs(E). Besides, by monotonicity of CN, CN(Supp(a)) ⊆ CN(Base(E)). Since T is also closed under CN, then Concs(E) = CN(Base(E)) (proved in [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF]). Thus, CN(Supp(a)) ⊆ Concs(E) and Conc(a) ∈ Concs(E).

We show next that the free part of Σ (i.e., the formulas that are not involved in any conflict) is inferred by any argumentation system under preferred semantics. The reason is that the set of arguments built from Free(Σ) is an admissible extension of any argumentation systems whose attack relations are conflict-dependent. Thus, this is true even for systems that do not satisfy the postulates.

Proposition 6 Let T = (Arg(Σ), R) be s.t. R is conflict-dependent. The set Arg(Free(Σ)) is an admissible extension of T . Proof. Let (Arg(Σ), R) be s.t. R is conflict-dependent. Let a ∈ Arg(Free(Σ)). As- sume that ∃b ∈ Arg(Σ) s.t. aRb or bRa. Since R is conflict-dependent, then ∃C ∈ C Σ such that C ⊆ Supp(a) ∪ Supp(b)
. By definition of an argument, both Supp(a) and Supp(b) are consistent. Then, C ∩ Supp(a) = ∅. This contradicts the fact that Supp(a) ⊆ Free(Σ). Thus, Arg(Free(Σ)) is conflict-free and can never be attacked.

We show next that the set Arg(Free(Σ)) is contained in every preferred extension. This is true for any argumentation system that uses a conflict-dependent attack relation. That is, it is always true.

Proposition 7 Let T = (Arg(Σ), R) be s.t. R is conflict-dependent. For all E ∈ Ext p (T ), Arg(Free(Σ)) ⊆ E. Proof. Let (Arg(Σ), R) be s.t. R is conflict-dependent. Assume that ∃E ∈ Ext p (T )
such that Arg(Free(Σ)) ⊆ E. Thus, either E ∪ Arg(Free(Σ)) is conflicting or E does not defend elements of Arg(Free(Σ)). Both cases are impossible since arguments of Arg(Free(Σ)) neither attack nor are attacked by any argument.

The next result shows that formulas of Free(Σ) are always drawn from Σ under preferred semantics.

Proposition 8 Let T = (Arg(Σ), R) be s.t. R is conflict-dependent. It holds that Free(Σ) ⊆ Output(T ).

Proof. Let T = (Arg(Σ), R) be s.t. R is conflict-dependent. From Proposition 7, ∀E ∈ Ext p (T ), Arg(Free(Σ)) ⊆ E. Besides, ∀x ∈ Free(Σ), ({x}, x) ∈ Arg(Free(Σ)), thus ({x}, x) ∈ E. Consequently, ∀E ∈ Ext p (T ), Free(Σ) ⊆ Concs(E). It follows that Free(Σ) ⊆ Concs(E) where E ∈ Ext p (T ).

In [START_REF] Amgoud | Stable semantics in logic-based argumentation[END_REF], we have shown that formulas of the set Free(Σ) may be missed by argumentation systems under stable semantics. This is particularly the case when the systems do not have stable extensions. We have also seen that this problem is due to the use of skewed attack relations. Even if those relations ensure the rationality postulates, the corresponding systems do not return satisfactory results since they may miss intuitive conclusions like Free(Σ). The previous results show that since preferred semantics guarantees the existence of preferred extensions, then it guarantees also the inference of elements of Free(Σ).

The previous results make it possible to delimit the maximum number of preferred extensions a system may have. It is the number of consistent subbases of Σ that contain the free part of Σ and which are pairwise different. Note that this number is less than the number of consistent subbases of Σ. Proof. From Proposition 2, each preferred extension returns a consistent subbase of Σ. From Proposition 3, it is not possible to have the same subbase several times. Finally, from Proposition 7, each preferred extension contains Arg(Free(Σ)).

Until now, we showed that preferred extensions reflect coherent points of view since they rely on consistent subbases of Σ. We also showed that when Σ is finite, each argumentation system that enjoy the rationality postulates has a finite number of preferred extensions. Proposition 9 provides the maximum number of such extensions. We thus answered all our questions.

Inferences under preferred semantics

In this section, we investigate the characteristics of the set Output(T ) of any argumentation system T that satisfies the postulates. Indeed, we study the kind of inferences that are made by an argumentation system under preferred semantics. From the results of the previous section, this set is defined as follows.

Proposition 10 Let T = (Arg(Σ), R) be an AS s.t. R is conflict-dependent and T satisfies the three postulates. It holds that Output(T ) = CN(S i ) s.t. S i ∈ Cons(Σ) and S i = Base(E i ) where E i ∈ Ext p (T ).

Proof. Let T = (Arg(Σ), R) be an AS s.t. R is conflict-dependent and T satisfies the three postulates. Let E ∈ Ext p (T ). Since T is closed both under CN and under subarguments, then Concs(E) = CN(Base(E)) (result shown in [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF]). From Proposition 7, Free(Σ) ⊆ Base(E). Moreover, from Proposition 2, Base(E) is consistent. Thus,

Base(E) ∈ Cons(Σ). From Definition 4, Output(T ) = Concs(E i ), E i ∈ Ext p (T ). Thus, Output(T ) = Base(E i ), E i ∈ Ext p (T ).
It is worth noticing that preferred semantics is more powerful than stable semantics only in case stable extensions do not exist and Free(Σ) = ∅. Indeed, in this case the output set of any argumentation system is empty ((Output(T )) = ∅) under stable semantics. Thus, the free formulas of Σ will not be inferred while they are guaranteed under preferred semantics. However, this does not mean that outputs under preferred semantics are "complete" and "intuitive". We show that some argumentation systems may miss in some cases some interesting conclusions. Worse yet, they may even return counter-intuitive ones. Let us illustrate our ideas on the following example.

Example 1 Let us consider the following propositional knowledge base Σ = {x, ¬x ∧ y}. The two formulas are equally preferred. From Proposition 10, it follows that any reasonable argumentation that may be built over Σ will have one of the three following outputs:

-Output 1 (T ) = ∅. This is the case of systems that have a unique and empty extension, or those which have two extensions E 1 and E 2 where Base(E 1 ) = {x} and Base(E 2 ) = {¬x ∧ y}. -Output 2 (T ) = CN({x}). This is the case of systems that have E 1 as their unique extension. -Output 3 (T ) = CN({¬x ∧ y}). This is the case of systems that have E 2 as their unique extension.

Let us analyze the three cases. In the first one, the result is not satisfactory. Indeed, one may expect to have y as a conclusion since it is not part of the conflict in Σ. Assume that x stands for "sunny day" and y for "My dog is sick". It is clear that the two information x and y are independent. This shows that argumentation systems are syntax-dependent.

The two other outputs (Output 2 (T ) and Output 3 (T )) are not satisfactory neither. The reason in these cases is different. For instance, in Output 2 (T ), the formula x is inferred from Σ while ¬x is not deduced. This discrimination between the two formulas is not justified since the two formulas of Σ are assumed to be equally preferred.

Let us now analyze in detail all the possible situations that may occur. Throughout this sub-section, p denotes the set of all attack relations that ensure the three postulates for any argumentation system, that is for any Σ. Indeed, p = {R ⊆ Arg(Σ)×Arg(Σ) | R is conflict-dependent and (Arg(Σ), R) satisfies Postulates 1, 2 and 3 under preferred semantics} for all Σ. We distinguish two categories of attack relations: those that always lead to a unique extension ( u ) and those that may lead to multiple extensions ( m ), where p = u ∪ m .

Unique extension Let us focus on argumentation systems T = (Arg(Σ), R) that satisfy the three postulates and that use attack relations of the set u . Thus, Ext p (T ) = {E}. Three possible situations may occur:

Ext p (T ) = {∅}: In this case, the output set is empty, and consequently, there is no free formula, i.e. all the formulas of Σ are involved in at least one conflict.

Property 1 Let T = (Arg(Σ), R) be an AS s.t. R ∈ u . If Ext p (T ) = {∅}, then Output(T ) = ∅ and Free(Σ) = ∅.
Proof. Since Ext p (T ) = {∅}, then from Definition 4 Output(T ) = ∅. From Proposition 8, it follows that Free(Σ) = ∅.

Note that the fact that Free(Σ) = ∅ does not imply that Ext p (T ) = {∅}. At a first glance, the previous result may seem reasonable since all the formulas in Σ are conflicting and are all equally preferred. However, Example 1 shows that this is not the case since there are some interesting formulas that may be missed.

Ext p (T ) = {Arg(Free(Σ))}: Argumentation systems that have the unique preferred extension Arg(Free(Σ)) return as conclusions all the formulas that follow under CN from Free(Σ).

Property 2 Let T = (Arg(Σ), R) be an AS s.t. R ∈ u . If Ext p (T ) = {Arg(Free(Σ))}, then Output(T ) = CN(Free(Σ)).
Proof. Let T = (Arg(Σ), R) be an AS s.t. R ∈ u . Assume that Ext p (T ) = {Arg(Free(Σ))}.

Let E = Arg(Free(Σ)). Since T is closed both under sub-arguments and CN, then

Concs(E) = CN(Base(E)) ([1]). Thus, Concs(E) = CN(Free(Σ)). Besides, Output(T ) = Concs(E), thus Output(T ) = CN(Free(Σ)).
It is worth mentioning that such outputs correspond exactly to the so-called free consequences developed in [START_REF] Benferhat | How to infer from inconsisent beliefs without revising?[END_REF] for handling inconsistency in propositional knowledge bases. The authors in [START_REF] Benferhat | How to infer from inconsisent beliefs without revising?[END_REF] argue that this approach is very conservative. Indeed, if Free(Σ) is empty, then nothing can be drawn from Σ. This may lead to miss intuitive formulas as shown in the next example.

Example 2 Let us consider the following propositional knowledge base Σ = {x, ¬x ∧ y, z}. It can be checked that Free(Σ) = {z}. Thus, any reasonable argumentation system that may be built over Σ and that uses an attack relation of category u will have the set CN({z}) as output. However, y should also be inferred from Σ.

Ext p (T ) = {E} where Arg(Free(Σ)) ⊂ E: In this case, there is at least one argument in the extension E whose support contains at least one formula which is involved in at least one conflict in Σ. However, since Base(E) is consistent, then there are some formulas involved in the same conflict which are not considered. Then, there is a discrimination between elements of Σ which leads to ad hoc results as shown by the following result. x / ∈ Output(T ).

Proof. Let T = (Arg(Σ), R) be an AS s.t. R ∈ u . Assume that Ext p (T ) = {E} and Arg(Free(Σ)) ⊂ E. Thus, ∃a ∈ E and a / ∈ Arg(Free(Σ)). Consequently, Supp(a) ⊆ Free(Σ). Thus, ∃x ∈ Supp(a) and x / ∈ Free(Σ). Thus, x ∈ Inc(Σ). Moreover, since T is closed under sub-arguments, then from [START_REF] Amgoud | Postulates for logic-based argumentation systems[END_REF], Base(E) ⊆ Output(T ), then x ∈ Output(T ). Besides, {x} is consistent since Supp(a) is consistent. Thus, ∃C ∈ C Σ such that |C| > 1 and x ∈ C. Since T satisfies consistency, then C ⊆ Output(T ). Thus, ∃x ∈ C such that x / ∈ Output(T ).

Let us illustrate this result on the following critical example.

Example 2 (Cont): Assume again the propositional knowledge base Σ = {x, ¬x ∧ y, z}. Argumentation systems of the previous category may return either E 1 or E 2 (not both) such that Base(E 1 ) = {x, z} and Base(E 2 ) = {¬x ∧ y, z}. In the first case, Output(T ) = CN({x, z}). Thus, x ∈ Output(T ) while ¬x / ∈ Output(T ). In the second case, Output(T ) = CN({¬x∧y, z}), thus ¬x ∈ Output(T ) while x / ∈ Output(T ). Both cases are undesirable since there is no reason to privilege x over ¬x and vice versa. Remember the case where x stands for "sunny day" and y for "my dog is sick".

Multiple extensions

Let us now tackle the second category of attack relations: the ones that may lead to multiple preferred semantics. Let T = (Arg(Σ), R) be an AS s.t. R ∈ m and let Ext p (T ) = {E 1 , . . ., E n } such that n ≥ 1. We have seen previously that each preferred extension gives birth to a consistent subbase of Σ. This subbase may be either maximal (for set inclusion) or not. Moreover, the subbases of some extensions of the same system may be maximal while those of the remaining extensions not. In what follows, we study three possible cases.

Case 1: In this case, an argumentation system has at least one preferred extension whose corresponding base is not maximal (i.e. ∃E ∈ Ext p (T ) such that Base(E) / ∈ Max(Σ)). The output set may be counter-intuitive since some priority will be given to some formula. Let us consider the following example.

Example 3 Let us consider the following propositional knowledge base that contains four equally preferred formulas: Σ = {x, x → y, z, z → ¬y}. It can be checked that Free(Σ) = ∅. An argumentation system that fits in Case 1 would have for instance, two preferred extensions E 1 and E 2 such that Base(E 1 ) = {x, x → y, z} and Base(E 2 ) = {z, z → ¬y}. Note that Base(E 1 ) ∈ Max(Σ) while Base(E 2 ) / ∈ Max(Σ). The output of this system is Output(T ) = CN({z}). Thus, z ∈ Output(T ) while the three other formulas of Σ are not elements of Output(T ). This result is unjustified since all the formulas of Σ are involved in the conflict and are equally preferred.

Case 2:

The bases of all the preferred extensions of an argumentation system are maximal (for set inclusion). However, not all the maximal consistent subbases of Σ have a corresponding preferred extension (i.e. ∀E ∈ Ext p (T ), Base(E) ∈ Max(Σ) and |Ext p (T )| < |Max(Σ)|). The same problem described in Case 1 is encountered here. Indeed, some formulas are privileged over others in an ad hoc way. Let us consider the following example.

Example 3 (Cont): Let us consider again the knowledge base of Example 3. An argumentation system that fits in Case 2 would have for instance, two preferred extensions E 1 and E 2 such that Base(E 1 ) = {x, x → y, z} and Base(E 2 ) = {x, x → y, z → ¬y}.

Note that both subbases are maximal. It is easy to check that x, x → y ∈ Output(T ) while z, z → ¬y / ∈ Output(T ). This result is again unjustified.

Case 3:

The bases of all the preferred extensions of an argumentation system are maximal (for set inclusion). Moreover, any maximal consistent subbases of Σ has a corresponding preferred extension in the argumentation system (i. Proof. This follows from the definition of Output(T ) and the fact that for each preferred extension E, Concs(E) = CN(Base(E)).

It is worth noticing that this output corresponds to the universal consequences developed in [START_REF] Rescher | On inference from inconsistent premises[END_REF] for handling inconsistency in propositional knowledge bases. Thus, argumentation systems of this category generalize the coherence-based approach to any logic. Consequently, they inherit its problems, namely the one described in Example 1 (missing intuitive conclusions). It is also worth recalling that there exist attack relations that lead to this result. Assumption attack developed in [START_REF] Elvang-Gøransson | Acceptability of arguments as 'logical uncertainty[END_REF] is one of them. Indeed, any argumentation system that use this relation will have the output described in Property 3. Finally, we have shown in another paper that the stable extensions of any argumentation system (that satisfies consistency and closure under sub-arguments) return maximal consistent subbases of Σ. We show next that when all the maximal subbases of Σ have a corresponding stable extension in a system, then this latter is certainly coherent, i.e., its preferred extensions are stable ones.

Proposition 12 Let T = (Arg(Σ), R) be an AS over a knowledge base Σ s.t. R ∈ p . If |Ext s (T )| = |Max(Σ)|, then T is coherent.
Proof. Let T = (Arg(Σ), R) be an AS over a knowledge base Σ such that R ∈ p . By definition of stable semantics, Ext s (T ) ⊆ Ext p (T ). Assume now that E ∈ Ext p (T ). From Proposition 2, Base(E) is consistent. Thus, ∃S ∈ Max(Σ) such that Base(E) ⊆ S. From Property 4 in [START_REF] Amgoud | Bridging the gap between abstract argumentation systems and logic[END_REF], it holds that Arg(Base(E)) ⊆ Arg(S). However, E ⊆ Arg(Base(E)) and Arg(S) ∈ Ext s (T ). Thus, E ⊆ Arg(S) where Arg(S) ∈ Ext p (T ). This contradicts the fact that E is a preferred extension, thus maximal.

The results of this section show that reasoning under preferred semantics is not recommended. Figure 1 summaries the different outputs that may be encountered under this semantics.

Conclusion

In this paper, we characterized for the first time the outcomes of argumentation systems under preferred semantics. To the best of our knowledge there is no work that tackled this issue. In [START_REF] Cayrol | On the relation between argumentation and non-monotonic coherence-based entailment[END_REF], the author studied the outcomes of a very particular system under stable semantics. In [START_REF] Gorogiannis | Instantiating abstract argumentation with classical logic arguments: Postulates and properties[END_REF], the authors focused on argumentation systems that are grounded on propositional logic and studied the properties of various systems using specified attack relations. The focus was mainly on the satisfaction of rationality postulates. In this paper, we assume abstract logic-based argumentations in which neither the underlying logic nor the attack relations are specified. This abstraction makes our results more general and powerful. Moreover, among all the possible instantiations of this setting, we considered those that satisfy some basic rationality postulates. Indeed, systems that violate those postulates should be avoided as they certainly lead to undesirable results.

A first important result consists of delimiting the maximum number of preferred extensions a system may have. We have shown that if the knowledge base under study is finite, then any argumentation system buit over it has a finite number of preferred extensions. Then, we have shown that from each preferred extension, a (maybe maximal) consistent subbase of the knowledge base is computed. This subbase contains all the formulas that are not involved in any conflict. We have then shown that in the best case, reasoning under preferred semantics may lead to missing some interesting conclusions. This is mainly due to the fact that argumentation systems are syntax-dependent. Moreover, they coincide with the coherence-based approach [START_REF] Benferhat | How to infer from inconsisent beliefs without revising?[END_REF][START_REF] Rescher | On inference from inconsistent premises[END_REF], thus inherit all its weaknesses. In the worst case, preferred semantics will lead to undesirable conclusions.

The main problem here is that the attack relation defines some "artificial" priorities between the formulas of the knowledge base leading to ad hoc outputs. The only good news is that preferred semantics performs better than stable one in the sense that it guarantees the inference of the free formulas (i.e., the formulas that are not involved in any conflict).

To sum up, we have shown (for the large class of argumentation systems we discussed) that preferred semantics should be avoided.

Proposition 5

 5 Let T = (Arg(Σ), R) be an AS s.t. T is closed under CN and under sub-arguments. Let E ∈ Ext p (T ). For all a ∈ Arg(Base(E)), Supp(a) ⊆ Concs(E) and Conc(a) ∈ Concs(E).

Proposition 9

 9 Let T = (Arg(Σ), R) be an AS s.t. R is conflict-dependent and T satisfies consistency and closure under sub-arguments. It holds that 1 ≤ |Ext p (T )| ≤ |Cons(Σ)| where Cons(Σ) = {S | S ⊆ Σ, S is consistent and Free(Σ) ⊆ S}.

Proposition 11

 11 Let T = (Arg(Σ), R) be an AS s.t. R ∈ u . If Ext p (T ) = {E} and Arg(Free(Σ)) ⊂ E, then ∃x ∈ Inc(Σ) s.t. x ∈ Output(T ) and ∃x ∈ Inc(Σ) s.t.

Property 3

 3 e. ∀E ∈ Ext p (T ), Base(E) ∈ Max(Σ) and |Ext p (T )| = |Max(Σ)|). The outputs of such systems are exactly the common conclusions that are drawn under CN from the maximal consistent subbases of Σ. Let T = (Arg(Σ), R) be an AS s.t. R ∈ m . If ∀E ∈ Ext p (T ), Base(E) ∈ Max(Σ) and |Ext p (T )| = |Max(Σ)|, then Output(T ) = S i where S i ∈ Max(Σ).

Fig. 1 .

 1 Fig. 1. Outcomes under preferred semantics (M stands for missing conclusions and U for undesirable ones)

  R is conflict-dependent and T satisfies consistency and closure under sub-arguments. Let E ∈ Ext

p (T ). Assume that Base(E) / ∈ Max(Σ) and that ∃S ∈ Max(Σ) s.t. Base(E) ⊂ S and ∃E ∈ Ext p (T ) s.t. Base(E ) = S. Thus, ∃x ∈ S \ Base(E ). Moreover, ∃a ∈ E such that x ∈ Supp(a) and a / ∈ E. Besides, from Proposition 2, it holds that E = E . Contradiction.

Y ⊆ f X means that Y is a finite subset of X.

Generally, the support X is minimal (for set inclusion). In this paper, we do not need to make this assumption.

A set C ⊆ Σ is a minimal conflict iff C is inconsistent and ∀x ∈ C, C\{x} is consistent.


Florent Kravtsoff
email: florent.kravtsoff@ec-lyon.fr

Michaël Schvallinger

Xavier Ottavy
email: xavier.ottavy@ec-lyon.fr

A CFD-BASED THROUGHFLOW SOLVER USING CASCADE POTENTIAL THEORY FOR AXIAL COMPRESSOR FLOW MODELING

Keywords:

This paper describes the use of the Hess & Smith cascade potential method as an additional feature for throughflow modeling. This method is incorporated inside ASTEC, a CFD-based throughflow solver developed by Safran Tech providing local blade loading information for blade force and loss modeling. The test cases are all subsonic, composed of two compressors cascades and a low speed compressor. The two profile families for the cascades are a classical NACA 65 009 and a controlled diffusion cascade designed by the Whittle Laboratory. The compressor is the well studied CME2, a research single stage designed by Safran Aircraft Engines. From the potential calculation, boundary layers and wakes are calculated on the blades providing deviation angles and profile losses. The tip leakage losses are also estimated from blade loading at the tip of a rotor. The results are compared to MISES code predictions, RANS simulations, experimental data and classic Lieblein empirical correlations. This work aims to model compressors flow with geometry-based classic analytical methodologies for a minimal use of empiricism. The predictions are good on bi-dimensional cascades cases compared to several other methodologies and at mid-span for the multiple incidences considered compared to experimental cascade results. For the compressor case, the blade loading near mid-span of the rotor blades show good agreement compared to RANS results leading to an overall good capture of the performance characteristic trends near design point. There is still room for improvements near the end-walls, mostly due to the lack of hub boundary layer model in the actual methodology. These results still build a step on which the enrichment of the flow model can be pursued.

Transpose operator

INTRODUCTION

The use of throughflow methods in aeronautical compressors design is historically strongly established. However, the past few decades demonstrated a growing interest for high fidelity methods as the computational costs decreased. This led to two phenomena: on one hand, the research interest for throughflow modeling declined but on the other hand understanding of 3D complex features of compressor flows has strongly increased. In compressor design actual trends, a reduced number of stages is observed. This involves increased blade loading as already observed by Cumpsty [START_REF] Cumpsty | Compressor Aerodynamics. Longman Scientific & Technical[END_REF] and the generation of secondary flows of greater importance. To control these undesirable flows, 3D complex shapes of blades have emerged thanks to better understanding of compressor flow topologies. Although throughflow methodologies were lacking popularity compared to more advanced high fidelity simulations they are still an essential design tool in the process of making a new engine [START_REF] Horlock | A Review of Some Early Design Practice Using Computational Fluid Dynamics and a Current Perspective[END_REF]. In this context, a recent interest in CFD-based throughflow solvers is observed [START_REF] Simon | Contribution to Throughflow Modelling for Axial Flow Turbomachines[END_REF][START_REF] Pacciani | A CFD-based Throughflow Method with an Explicit Body Force Model and an Adaptive Formulation for the S2 Streamsurface[END_REF][START_REF] Ricci | Computational Fluid Dynamics-Based Throughflow Analysis of Transonic Flows in Steam Turbines[END_REF]. Thanks to increased CFD power, the computational time required for this type of simulations is now of the same order of the well-known streamline curvature method (SCM) during its advent [START_REF] Novak | Streamline Curvature Computing Procedures for Fluid-Flow Problems[END_REF]. CFDbased method does not suffer from some drawbacks imputed to SCM method such as the handling of recirculations or compressible effects (although the quality of shock capturing properties of CFD-based throughflow must be treated with caution [START_REF] Baralon | Validation of a Throughflow Time-Marching Finite-Volume Solver for Transonic Compressors[END_REF]). Also, it is easier to consider a multi-fidelity conception approach when the throughflow code and the higher fidelity methods share the same CFD solver. Finally, keeping the throughflow models upto-date is easier in CFD-based methodologies, especially if the throughflow routines are external to the CFD solver. If the recent better understanding of 3D flow features is admitted, the reduction of this knowledge to reliable reduced order model adaptable to throughflow simulations is not yet systemically observed as noticed by Banjac et al. [START_REF] Banjac | Secondary Flows, Endwall Effects, and Stall Detection in Axial Compressor Design[END_REF]. In addition to this, the legacy models used in the past show limited accuracy for modern designs [START_REF] Manfredi | Transonic Axial Compressors Loss Correlations: Part I -Analysis and Update of Loss Models[END_REF]. This is why models must continuously be developed to improve new generation of throughflow codes suitable for modern compressor design methodologies.

To address this objective, this paper present the use of literature models showing a low level of empiricism and using the local blade geometry. They are incorporated in ASTEC (Aerodynamic Source Term for Efficient Computation), a CFD-based throughflow solver relying on elsA CFD solver. In details, the use of the well-known Hess & Smith cascade potential method [START_REF] Hess | Calculation of Potential Flow about Arbitrary Bodies[END_REF] allows estimating the local blade pressure coefficients which are used to compute profile boundary layers using an integral boundary layer method. This leads to the estimation of profile drag losses. The deviation angles resulting from the development of the profile boundary layers are estimated using a simple coupling between the potential method and the boundary layer method by the modification of the profile geometry. In addition, wakes are also estimated as an axial evolution of the state of the boundary layer at the trailing edge. Finally, in the case of a rotor, tip leakage flow losses are computed from the pressure distribution at the blade tip. The results are confronted to MISES, RANS, experimental and empirical correlation data.

ASTEC: A CFD-BASED THROUGHFLOW SOLVER

ASTEC refers to a newly developed1 throughflow code which solves the circumferentially averaged Navier-Stokes equations. The circumferentially averaged flow is calculated with the stateof-the-art finite volumes CFD solver elsA [START_REF] Cambier | The Onera elsA CFD Software: Input from Research and Feedback from Industry[END_REF] developed by ONERA and the volumic source terms arising from the average process in blades rows are calculated with ASTEC external modules. This non-invasive coupling between elsA and the external source terms occurs at each elsA iterations. Similar coupling exists in the literature such as the work done by Thollet et al. [START_REF] Thollet | Body-Force Modeling for Aerodynamic Analysis of Air Intake -Fan Interactions[END_REF] to study air intake distortions. The resulting solver is fully parallelized. It uses one processor per blade zone and one processor per inter-blade zone making the calculation response time independent of the stage number. For commodity and simplicity ASTEC will refer to the ensemble elsA + source terms in the following.

General governing equations

CFD-based throughflow solvers intend to resolve the circumferentially-average flow. The rigorous establishment of the circumferentially averaged 3D-Navier-Stokes equations is well described by Simon [START_REF] Simon | Contribution to Throughflow Modelling for Axial Flow Turbomachines[END_REF]. It consists in the successive application of Reynolds ensemble-average, a time-average, the averagepassage and finally the circumferential-average as detailed by Adamczyk [START_REF] Adamczyk | Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design[END_REF]. Each successive average process will create additional terms. Those terms, Reynolds, unsteady, aperiodic and circumferential stresses, added to the stationary axisymmetric flow field, allow the reconstruction of the original 3D-unsteady turbulent flow. The general governing throughflow equations written in conservative form following these average process read:

𝜕𝑏𝜌 𝜕𝑡 + div 𝑏𝜌V = 0 (1)
𝜕𝑏𝜌V 𝜕𝑡

+ div 𝑏(𝜌VV 𝑡 -𝝉 + 𝑝I) = 𝑏S 𝜌𝑉 (2)
𝜕𝑏𝜌𝑒 𝑡 𝜕𝑡 + div 𝑏(𝜌ℎ 𝑡 V -𝝉 • V + q) = 𝑏𝑆 𝑒 𝑡 (3)
In those equations one can notice the blockage factor 𝑏 appearing in the average process. Its value is less than one and equals one in the perfect case of no existent aerodynamic blockage outside a blade row. In the left hand-side, one can recognize the Navier-Stokes conservation equations omitting 𝑏. In the right hand-side two composite source terms appears, S 𝜌𝑉 and 𝑆 𝑒 𝑡 . They are composed of stresses arising from the average process and, in presence of blades, also blade forces, contributing in momentum conservation Eq. ( 2) and total energy conservation Eq. ( 3). For the sake of readability, although nothing denote the consecutive averaging operations in the above system, the variables must be understood as the multiple-average resultant version of themselves obtained at the end of the whole averaging process. 2 The left-hand side can be solved with a CFD solver and the right-hand side must be estimated. In the special case of no further simplifications, the estimation of the right-hand side can be tedious since it implies the precise estimation of all the stresses and blade terms obtained in the average process. Such a task has been partly addressed by Thomas et al. [START_REF] Thomas | Towards a High Order Throughflow: Part I-Investigating the Effectiveness of a Harmonic Reconstruction for 3D Flows[END_REF] with the use of the circumferential stresses from 3D Navier-Stokes simulations, showing gain in results precision compared to the case without the extracted terms. This is relevant to high order throughflow modeling topic. For the work described in this paper, a particular importance has been placed in the time-efficiency of the simulation. For the authors, what made throughflow simulation so popular in the history of turbomachinery conception tools is their robustness and fast-response abilities in multistage cases. In order to respect this guiding principle, hypothesis are made to simplify the system of equations solved as explained in the following section.

ASTEC governing equations

In this work, the equations solved with ASTEC are obtained after four main hypotheses:

• The Euler form of Navier-Stokes equations are used;

• Heat fluxes with the exterior of the domain are neglected;

• All stresses obtained in the averaging process part of S 𝜌𝑉 and 𝑆 𝑒 𝑡 are neglected;

• The blade force f b are taken into account through two volumic source terms f bi and f bv that are the inviscid and the viscous part respectively. 2 For more details see Simon [START_REF] Simon | Contribution to Throughflow Modelling for Axial Flow Turbomachines[END_REF] Appendix B. p.173.

The previous system immediately become:

𝜕𝑏𝜌 𝜕𝑡 + div 𝑏𝜌V = 0 (4) 𝜕𝑏𝜌V 𝜕𝑡 + div 𝑏(𝜌VV 𝑡 + 𝑝I) = 𝑏f b (5) 𝜕𝑏𝜌𝑒 𝑡 𝜕𝑡 + div 𝑏𝜌ℎ 𝑡 V = 𝑏f b • 𝛀𝑟 (6)
With 𝑟 the local radius and 𝛀 the blade rotation velocity vector around the shaft axis. The final non-invasive ASTEC coupling system is obtained using property of the divergence operator and the non-dependence in time of the blockage factor 𝑏:

𝜕 𝜌 𝜕𝑡

+ div 𝜌V = - 1 𝑏 (𝜌V • ∇𝑏) (7)
𝜕 𝜌V 𝜕𝑡 + div (𝜌VV 𝑡 + 𝑝I) = f bi + f bv - 1 𝑏 (𝜌V • ∇𝑏)V (8)
𝜕 𝜌𝑒 𝑡 𝜕𝑡 + div 𝜌ℎ 𝑡 V = (f bi + f bv ) • 𝛀𝑟 - 1 𝑏 (𝜌ℎ 𝑡 V • ∇𝑏) (9)
In this system one can identify the Euler equations integrated by elsA at the left hand-side and an additional blockage source term, identified by terms with 𝑏 in their expression calculated by ASTEC jointly with f bi and f bv .

Baseline source term models

Blockage model. The blockage factor involve blade circumferential thickness 𝜀 visible in Fig. 1 and blade number 𝑁.

𝑏 = 1 - 𝑁 𝜀 2𝜋𝑟 ( 10
)
This formulation is consistent with the circumferential averaging process. However, for thick or highly staggered blades the blockage factor may reach values that prevent the correct estimation of compressor choke massflow rate for high Mach number cases. In these cases the use of a normal to flow direction blockage factor may allow better predictions as studied by Baralon et al. [START_REF] Baralon | Validation of a Throughflow Time-Marching Finite-Volume Solver for Transonic Compressors[END_REF]. The blockage factor may also be increased with aerodynamic blockage as it will be seen in Sect. 5.1. For this work, the formulation in Eq. ( 10) is used as the test cases presented in the results sections belong to subsonic flow domain.

Blade force model. As illustrated in Fig. 1 the blade force is decomposed into the two terms f bi and f bv to account for the two effects of blades on the flow: flow turning and viscous loss.

The inviscid term f bi must not create entropy since it is the part of f bv . Thus the flow turning model must deliver a force that is normal to the mean relative flow field, summarized by the orthogonality condition:

f bi • W = 0 (11)
In practice, the flow turning procedure is an iterative process and the orthogonality condition is satisfied when the calculation is converged. More precisely the procedure assume that the flow deflection is known a priori, for example it is possible to assume that the flow follows the camber line direction so an orthogonality condition between the mean relative flow and the normal vectors n to the camber line is resolved to obtain the inviscid blade force modulus:

|df bi | 𝑘 = -K(𝑊 𝑘 𝑥 𝑛 𝑥 + 𝑊 𝑘 𝑟 𝑛 𝑟 + 𝑊 𝑘 𝜃 𝑛 𝜃 ) (12)
Where |df bi | 𝑘 is the contribution obtained at iteration 𝑘 to total inviscid blade force modulus |f bi | = ∑ 𝑘 |df bi | 𝑘 during ASTEC convergence process. The minus sign is due to the outward orientation of the normal vectors and the constant K must be interpreted as a mean to adjust the rate of response to a violation of orthogonality condition as stated by Simon [START_REF] Simon | Contribution to Throughflow Modelling for Axial Flow Turbomachines[END_REF]. The modulus of the force is then projected on the normal vectors n and added to the right-hand side of ASTEC conservative equations.

The estimation of f bv is still an active research topic for throughflow simulation accuracy improvement. The overall precision of a throughflow model rely on the accurate estimation of the most relevant sources of loss and deviation and their correct distribution in the (𝑥, 𝑟) meridional plane. As a pure dissipative force f bv is oriented in the opposite direction of the relative mean flow and may be estimated from entropy generation sources as suggested by Denton [START_REF] Denton | Loss Mechanisms in Turbomachines[END_REF] or by estimation of a drag force from boundary layer momentum thickness calculations.

Concerning the deviation angles arising from viscous flows such as profile boundary layers, or secondary flows, they are taken into account through a rotation of the normal vectors, used in f bi calculation, of the corresponding predicted deviation angle from viscous models. The calculation then converge to an updated prescribed flow angle taking into account viscous effects and so during the complete convergence procedure.

Comments on the described methodology

About the first ASTEC hypothesis described in Sect. 2.2, the use of Euler equations in place of their viscous RANS version is motivated by two complementary considerations:

• Blade rows are fictitious in the simulation, so only axisymmetric end-wall boundary layers would be predicted by the turbulence model of a RANS solver;

• It is known that the turbulent boundary layers (TBL) show 3D features at the end-wall region of the blades. Moreover, the TBL at the hub of a multistage compressor does not behave as a regular TBL at end walls [START_REF] Smith | Casing Boundary Layers in Multistage Axial-Flow Compressors[END_REF].

Running ASTEC with a viscous set of equations would require a refined mesh at end walls, to satisfy turbulence model usage with still questionable end-wall predictions once the flow passes blades rows. The gain in precision of the results by running viscous seems not worth the time invested in heavier simulation, that is why Euler mode is preferred by the authors as it also encourages the use/development of accurate secondary flow models. For the record, the use of viscous conservatives equations in ASTEC is described in the work of Budo et al. [START_REF] Budo | Application of a Viscous Through-Flow Model to a Modern Axial Low-Pressure Compressor[END_REF] applied to a realistic modern low pressure compressor.

Recent designs of axial compressors blade showed diversified profile shapes, increased solidity due to larger chords and reduced blade aspect ratios compared to the precedent generations. This brings difficulties to estimate the correct loss levels with classical semi-empirical models mostly based on cascade test cases such as the well-known Lieblein correlations [START_REF] Manfredi | Transonic Axial Compressors Loss Correlations: Part I -Analysis and Update of Loss Models[END_REF][START_REF] Lieblein | Incidence and Deviation-Angle Correlations for Compressor Cascades[END_REF]. This fact has motivated the testing of a more locally driven approach based on local geometry as it will be described in the next section of this paper.

To conclude this section, in the following paper, ASTEC is used in analysis mode. Design mode is theoretically possible, but it has not been addressed in this work.

POTENTIAL FLOW APPROACH FOR INVISCID BLADE FORCE

As a first step toward detailed geometry driven throughflow modeling, the estimation of a realistic inviscid blade loading is incorporated in ASTEC. This is currently done by the use of the well-known singularity method also known as panel method in the literature [START_REF] Katz | Low-Speed Aerodynamics[END_REF].

Singularity method

In this work, a 2D singularity method is used on slices of the real 3D blade geometry along the blade span as illustrated in Fig. 2. It consists in the estimation of a steady, incompressible, irrotational non-viscous potential flow by the use of singularities, particular solution of Laplace equation, placed on the blade profile. If needed, blades slices are flattened in the well-known (𝑚 ′ -𝜃) stream surface coordinates before any potential calculation as described in [START_REF] Mcfarland | A Rapid Blade-to-Blade Solution for Use in Turbomachinery Design[END_REF]:

𝑚 ′ = ∫ 𝑑𝑚 𝑟 = ∫ √ 𝑑𝑥 2 + 𝑑𝑟 2 𝑟 (13)
In this conformal transformation, profiles angles are preserved and 𝜃 is unchanged. Any calculation is then performed on the flattened profile and the results are transposed to the original slice. Laplace equation for the incompressible flow potential 𝜙 reads:

Δ𝜙 = 0 (14)
Under the small perturbation hypothesis, the flow potential can be separated into the contribution of the onset flow potential 𝜙 ∞ and a perturbation potential ˜ 𝜙 induced by the presence of the profile:

𝜙 = 𝜙 ∞ + ˜ 𝜙 (15)
The Laplace equation being linear, ˜ 𝜙 still satisfies it. It is then possible to use particular solutions of Laplace equation to find the perturbation potential. Those solutions are called the singularities. The commonly used singularities are sources, vortices or doublets. In addition, the potential flow is linked to the flow speed by the relation V = ∇𝜙 so in terms of speed Eq. ( 15) is written:

V = V ∞ + ˜ V ( 16
)
Singularities definition. The presence of the profile is materialized by an ensemble of singularities, sources and vortices distributed on the profile. The profile is thus discretized in 𝑁 elem , the so-called panels. Each element is given lineic source and vortex densities. For great simplification of relations, the complex formalism is used in the following.

α i dz i x jy n p,i z i+1 z i

FIGURE 3: Profile element definition

The complex conjugate velocity induced at a point 𝑀 (𝑧) of 2D complex space by a point source of intensity Q located in 𝑧 0 reads:

𝑣 * 𝜎 (𝑧) = Q 2𝜋(𝑧 -𝑧 0 ) (17)
Thus, the complex conjugate velocity induced by an element 𝑑𝑧 𝑖 = |𝑑𝑧 𝑖 |𝑒 𝑗 𝛼 𝑖 of the profile possessing a lineic source density is obtained by integration on the length of the element |𝑑𝑧 𝑖 | = 𝑑𝑧 𝑖 𝑒 -𝑗 𝛼 𝑖 (see Fig. 3):

𝑣 * 𝜎 𝑖 (𝑧) = ∫ 𝑧 𝑖+1 𝑧 𝑖 𝜎 𝑖 𝑒 -𝑗 𝛼 𝑖 2𝜋(𝑧 -𝑧 𝑖 ) 𝑑𝑧 𝑖 ( 18
)
Where 𝛼 𝑖 if the orientation angle of the element in the complex plane, 𝑒 the complex exponential and 𝑗 is the complex number defined as 𝑗 2 = -1.

The complex conjugate velocity induced by the lineic vortex density 𝛾 𝑖 on 𝑑𝑧 𝑖 is obtained by an indirect rotation of 𝜋/2 of the complex conjugate velocity induced by a lineic source density.

𝑣 * 𝛾 𝑖 (𝑧) = ∫ 𝑧 𝑖+1 𝑧 𝑖 𝛾 𝑖 𝑒 -𝑗 ( 𝛼 𝑖 + 𝜋/2) 2𝜋(𝑧 -𝑧 𝑖 ) 𝑑𝑧 𝑖 (19)
For the whole set of singularities the total complex conjugate perturbation speed at 𝑀 (𝑧) is the added contributions of all singularities on the profile so that:

Ṽ * (𝑧) = ∑ 𝑁 elem 𝑣 * 𝜎 𝑖 (𝑧) + 𝑣 * 𝛾 𝑖 (𝑧) (20)

Neumann boundary condition.

A non penetration condition is then applied on the surface of the profile. In practice the condition is applied at the centers 𝑐 𝑗 of the elements 𝑑𝑧 𝑗 , 1 ≤ 𝑗 ≤ 𝑁 𝑒𝑙𝑒𝑚 using real velocity vectors, giving 𝑁 𝑒𝑙𝑒𝑚 equations:

∀ 𝑗, V(𝑐 𝑗 ) • n 𝑝, 𝑗 = V ∞ • n 𝑝, 𝑗 + ˜ V(𝑐 𝑗 ) • n 𝑝, 𝑗 = 0 (21) ⇔ 𝑁 elem ∑ 𝑖=1 (v 𝜎 𝑖 (𝑐 𝑗 ) + v 𝛾 𝑖 (𝑐 𝑗 )) • n 𝑝, 𝑗 = -V ∞ • n 𝑝, 𝑗 (22)
Where n 𝑝, 𝑗 is the normal at the center of element 𝑗 located at 𝑧 𝑗 .

Kutta-Joukowski condition. A particular flow escaping condition is applied at the trailing edge to reflect the real flow behavior. It consists in preventing the flow to go from pressure side to suction side or vice-versa at the trailing edge and so, allows lift generation by the profile. In a pressure point of view it is equivalent to say that the pressure at the suction side equals the pressure at the pressure side at the trailing edge. Using Bernoulli theorem, this condition reads in terms of velocities:

V(𝑧 𝑡𝑒, 𝑝𝑠 ) • t 𝑡𝑒, 𝑝𝑠 = V(𝑧 𝑡𝑒,𝑠𝑠 ) • t 𝑡𝑒,𝑠𝑠 (23)
⇔ (V ∞ + ˜ V(𝑧 𝑡𝑒, 𝑝𝑠 )) • t 𝑡𝑒, 𝑝𝑠 = (V ∞ + ˜ V(𝑧 𝑡𝑒,𝑠𝑠 )) • t 𝑡𝑒,𝑠𝑠 (24)
Where t 𝑡𝑒, 𝑝𝑠 and t 𝑡𝑒,𝑠𝑠 are the tangent vectors at the trailing edge of the profile at pressure side, respectively suction side, oriented from leading edge to trailing edge. In practice this condition is applied at the center of the two trailing edge elements. The application of Kutta-Joukowski condition therefore require a sharp trailing edge and an adapted refinement process at the profile trailing edge. In practice, if the analyzed profile presents round trailing edge, it is then cut and closed conserving profile chord. For the refinement process, distribution laws based on cosine function showed good results in the literature [START_REF] Katz | Low-Speed Aerodynamics[END_REF]. Also, leading edge refinement is important for a good accuracy of singularity method, so the use of symmetric cosine-based points distribution on suction side and pressure side fulfill both requirements and is used in this work.

Solved linear system.

In the singularity method, the unknown of the problem are the singularities intensities 𝜎 𝑖 and 𝛾 𝑖 that simulate the profile influence on the onset potential flow. Equations ( 22) and ( 24) total 𝑁 elem + 1 equations, but there is a total of 2𝑁 elem unknown singularities. To solve this system an assumption is made:

∀𝑖, 𝛾 𝑖 = 𝛾 (25)
Physically this means that the circulation around the profile is homogeneously distributed. The number of unknown then passes from 2𝑁 elem to 𝑁 elem + 1. The unknown vector reads:

X 𝑡 = [ 𝜎 1 𝜎 2 • • • 𝜎 𝑁 𝑒𝑙𝑒𝑚 𝛾 ] (26)
The system solved is of the form:

𝐴X = B (27)
With the 𝑁 𝑒𝑙𝑒𝑚 first lines of 𝐴 and B filled with Eq. ( 22) and their respective last line filled with the Kutta-Joukowski relation Eq. ( 24) using Eqs. ( 18), ( 19) and (25). This system is easily solved with linear algebra routines. Once X is known it is possible to calculate the flow velocity everywhere around the blade and on the blade profile using Eq. ( 16).

Cascade extension: Hess & Smith method

The previous detailing of the singularity method allows to easily understand its extension to cascade geometries done by Hess & Smith [START_REF] Hess | Calculation of Potential Flow about Arbitrary Bodies[END_REF]. The principle of the extension is to consider a cascade of as an infinite repetition of same geometry spaced apart by a distance equals to the pitch 𝑠 in cascade both directions as pictured in Fig. 4.

Singularities expression modification.

In the considered cascade the influence of repeated profiles accumulate and the complex conjugate velocity induced by the entire cascade of lineic sources on corresponding elements on a point 𝑀 (𝑧) can be written using Eq. ( 18):

𝑣 * 𝜎 𝑖 (𝑧) = ∫ 𝑧 𝑖+1 𝑧 𝑖 [ 𝜎 𝑖 𝑒 -𝑗 𝛼 𝑖 2𝜋(𝑧 -𝑧 𝑖 ) + 𝜎 𝑖 𝑒 -𝑗 𝛼 𝑖 2𝜋(𝑧 -𝑧 𝑖 + 𝑗 𝑠) + 𝜎 𝑖 𝑒 -𝑗 𝛼 𝑖 2𝜋(𝑧 -𝑧 𝑖 -𝑗 𝑠) + 𝜎 𝑖 𝑒 -𝑗 𝛼 𝑖 2𝜋(𝑧 -𝑧 𝑖 + 2 𝑗 𝑠) + 𝜎 𝑖 𝑒 -𝑗 𝛼 𝑖 2𝜋(𝑧 -𝑧 𝑖 -2 𝑗 𝑠) + • • • ] 𝑑𝑧 𝑖 (28)
The expression in the brackets regroups to an analytic expression as an alternate series decomposition of coth(𝑧) on C. With the same remark made to obtain Eq. ( 19), expressions of Eqs. ( 18) and ( 19) become in the case of a cascade problem:

𝑣 * 𝜎 𝑖 (𝑧) = ∫ 𝑧 𝑖+1 𝑧 𝑖 𝜎 𝑖 𝑒 -𝑗 𝛼 𝑖 2𝑠 coth(𝜋 𝑧 -𝑧 𝑖 𝑠 )𝑑𝑧 𝑖 ( 29
)
𝑣 * 𝛾 𝑖 (𝑧) = ∫ 𝑧 𝑖+1 𝑧 𝑖 𝛾 𝑖 𝑒 -𝑗 ( 𝛼 𝑖 + 𝜋/2) 2𝑠 coth(𝜋 𝑧 -𝑧 𝑖 𝑠 )𝑑𝑧 𝑖 ( 30
)
The effect on the resolved system Eq. ( 27) is to modify the 𝐴 matrix influence coefficients.

Implicit boundary condition for cascades.

As stated by Lewis [START_REF] Lewis | Vortex Element Methods for Fluid Dynamic Analysis of Engineering Systems[END_REF], singularity method developed in Sect. 3.1 is adapted for isolated profile with inlet flow angle 𝛼 ∞ set from profile chord reference. In a cascade or turbomachinery application the relative inlet velocity vector W 1 is known and W ∞ required for the singularity method can be deduced from blades circulation.

W ∞ = W 1 + W 2 2 ( 31
)
W ∞ = W 1 - Γ 2𝑠 y ( 32
)
Where Γ is the circulation generated by the cascade, counted positively clockwise and y the direct unit vector in the direction of the cascade as illustrated in Fig. 4. Γ is linked to the lineic vortex distribution on profiles by integrating on the profile length (Γ and the 𝛾 𝑖 having opposite sign convention):

Γ = - ∫ C 𝛾𝑑𝑙 (33)
Numerically, Eq. (33) using Eq. ( 25) become:

Γ = - ∑ 𝑁 elem 𝛾 𝑖 |𝑑𝑧 𝑖 | (34) Γ = -𝛾 ∑ 𝑁 elem |𝑑𝑧 𝑖 | (35)
Then W ∞ is substituted to V ∞ using Eq. ( 32) and Eq. (35). The system Eq. ( 27) is thus modified accordingly: the last column of 𝐴 associated with 𝛾 unknown are updated as well as the last line of 𝐴 associated with the Kutta-Joukowksi condition.

Γ FIGURE 4: Cascade definition

Prediction of inviscid blade force

The application of the above procedure make possible the estimation of potential flow relative velocity on the profile and in the ensemble of stream surfaces defined by (𝑥, 𝑟) slices. Thus, incompressible pressure coefficients 𝐶 𝑝 = 1 -W 2 /W 2 1 on the blades are available and also, relative mean flow angles in blade passage since it is possible to analyze the potential velocity everywhere. In particular mean flow angles are used to set the rotation of blade normal vectors n in Eq. ( 12) for invisicd blade force source term f bi and pressure gradients on the profiles can be used for development of viscous effects for f bv source term as detailed in the next section.

VISCOUS FLOW MODELING

Using previously introduced potential methodology, particular identified loss sources are modeled in ASTEC: profile loss and deviation, wake losses and tip leakage flow losses.

Profile loss and deviation

For boundary layers growing on the surface of the blade, an integral boundary layer method for turbulent boundary layers on flat plate with pressure gradient is used. It consists in the space marching integration on pressure side and suction side of the Von Kármán boundary layer integral equation using Head turbulent entrainment equation [START_REF] Head | Entrainment in the Turbulent Boundary Layer[END_REF]:

Von Kármán: 𝑑𝛿 2 𝑑𝑥 = 𝐶 𝑓 2 -(𝐻 12 + 2) 𝛿 2 𝑈 𝑒 𝑑𝑈 𝑒 𝑑𝑥 (36)
Head:

𝑑 (𝛿 -𝛿 1 ) 𝑑𝑥 = 𝐶 𝐸 - 𝛿 -𝛿 1 𝑈 𝑒 𝑑𝑈 𝑒 𝑑𝑥 (37)
To close this system the empirical friction and entrainment coefficients of Ludwieg-Tillman [START_REF] Ludweig | Investigations of The Wall-Shearing Stress in Turbulent Boundary Layers[END_REF] and Head [START_REF] Head | Entrainment in the Turbulent Boundary Layer[END_REF] respectively, are provided: Stall. The previous introduced boundary layer method does not allow stalled regime which appears when 𝐻 12 ≳ 2.4. A simple procedure is adopted to avoid the method to fail in this case. In a physical point of view, when stall appears on the suction side of a cascade profile for example, the profile is no longer able to provide deflection to the flow. The following assumption is thus made: in case of stall, the flow on the suction side will follow the streamline given by the profile enlarged with displacement thickness, which is then extrapolated keeping the same direction from the point of detected stall to the trailing edge. A stalled enlarged profile is thus obtained and 𝛿 1 in stalled region can be deduced geometrically (see Fig. 5). Similarly, 𝐻 12 is linearly extrapolated on the blade surface from the point of detected stall until the profile trailing edge allowing to deduce 𝛿 2 in stalled region. In practice, the stall threshold value of 𝐻 12 is fixed to 2.0 in order to perform extrapolation before the boundary layer predicted by the integral method enters exponential growth regime.

𝐶 𝑓 = 0.24610 -0.678𝐻
Profile loss. The profile losses are then calculated from the well-known Von Kármán relation between momentum thickness and profile drag at a station x:

𝐷 (𝑥) = (𝜌𝑈 2 𝑒 𝛿 2 ) (𝑥) (41)
Equation ( 41) is then discretized to locally update ASTEC f bv source term for each blade slices. Deviation. The deviation induced by the growth of profile boundary layers is obtained by profile geometry modification in the potential calculation. From the estimation of profile displacement thicknesses on pressure and suction sides, an enlarged profile is built from the original one. A new camber line of the profile is then deduced from the mean line of the enlarged profile as shown in Fig. 5. The original profile thickness law is then applied to this new camber line and the potential calculation take into account the new geometry for the next ASTEC iterations until the geometry is modified again. The calculation of the mean angle given by the potential solution, obtained by averaging the velocity in the inter blade channel, is then used to rotate the blade normal vectors n in Eq. ( 12) for f bi source term. By doing so, a blade force including deviation from profile boundary layers is then calculated for each blade slices.

Wake evolution

Since the boundary layer characteristics are known until the trailing edge, it is possible to use a wake decay model. To do so the decay correlations of Lakshminarayana et al. [START_REF] Lakshminarayana | Mean Velocity and Decay Characteristics of the Guidevane and Stator Blade Wake of an Axial Flow Compressor[END_REF] are used. They consist in the estimation of the decay of the velocity defect at the center of the wake after the trailing edge of the profile: Near Wake:

𝑣 𝑐 𝑈 𝑒

√

𝐶 𝐷 = 0.168 (𝑑 + 0.12) 0.22 + 0.353 (𝑑 + 0.12) 1.22 (42) Far Wake:

𝑣 𝑐 𝑈 𝑒 √ 𝐶 𝐷 = 0.9792 𝑑 + 0.688 (43)
Where 𝑈 𝑒 is the free stream flow velocity, 𝑣 𝑐 is the velocity deficit compared to free stream velocity at the center of the wake, 𝑑 is the chord normalized distance in the direction of the wake and 𝐶 𝐷 is the profile drag coefficient based on cascade mean velocity (𝑊 1 + 𝑊 2 )/2 as in Fig. 4. The frontier between near and far wake is located at 𝑑 = 0.4. The paper indicate that the correlations are not valid at the vicinity of the trailing edge for 0 < 𝑑 < 0.06. In this application the correlations are simply extrapolated for this domain.

In their work, Lakshminarayana et al. [START_REF] Lakshminarayana | Mean Velocity and Decay Characteristics of the Guidevane and Stator Blade Wake of an Axial Flow Compressor[END_REF] correlated the velocity profile in the wake as a Gaussian function. The correlation use two length scale 𝐿 𝑠𝑠 and 𝐿 𝑝𝑠 , the distances at suction side and pressure side, respectively, where the speed deficit 𝑣 = 𝑣 𝑐 /2. The expression of speed deficit as a function of the normalized distance normal to wake center line 𝜂 = 𝑙/𝐿 is:

𝑣 𝑣 𝑐 = 𝑒 -ln2𝜂 2 (44)
Assuming a total wake width as 2𝐿 𝑠𝑠 + 2𝐿 𝑝𝑠 the paper also defines the pitch normalized width of the wake 𝑤 = 2(𝐿 𝑠𝑠 + 𝐿 𝑝𝑠 )/𝑠:

Near Wake:

𝑤 √ 𝐶 𝐷 = 1
.39(𝑑 + 0.55) 1.22 (45)

Far Wake:

𝑤 √ 𝐶 𝐷 = 1.18(𝑑 + 0.688) ( 46
)
By building an equivalent symmetric wake of characteristic length scale 𝐿 𝑒𝑞 =

𝐿 𝑝𝑠 +𝐿 𝑠𝑠 2
it is possible to integrate Eq. ( 44) on wake width using Eqs. ( 45) and ( 46) and Eqs. ( 42) and ( 43) to obtain wake displacement thickness 𝛿 1, 𝑤 . As shown in [START_REF] Raj | Characteristics of the Wake behind a Cascade of Airfoils[END_REF], 𝑈 𝑒 decreases when the flow leaves the trailing edge of a blade row. To model this, the evolution of 𝑈 𝑒, 𝑤 in the wake is obtained in a small iterative procedure involving basic mass conservation equation 3 . The procedure converge to find 𝛿 1, 𝑤 and 𝑈 𝑒, 𝑤 satisfying Eq. ( 47) in the entire wake. As the wake displacement thickness rapidly decrease (i.e., the aerodynamic blockage induced by the wake), 𝑈 𝑒 rapidly decrease causing increase of wake momentum thickness 𝛿 2, 𝑤 according to Eq. ( 36) applied to wake with 𝐶 𝑓 = 0.

𝑈 𝑒, 𝑡𝑒 (𝑠 -𝛿 1, 𝑡𝑒 ) = 𝑈 𝑒, 𝑤 (𝑠 -𝛿 1, 𝑤 ) = const. ( 47
)
To compute loss arising from the wake decay, the correlation for the shape factor of [START_REF] Raj | Characteristics of the Wake behind a Cascade of Airfoils[END_REF] is used:

1 - 1 𝐻 12, 𝑤 = ( 1 - 1 𝐻 12, 𝑡𝑒 ) ( 40
𝑑 𝑐 + 1 ) -0.5 (
Since 𝐻 12, 𝑤 , 𝛿 1, 𝑤 and 𝑈 𝑒, 𝑤 evolutions are known, 𝛿 2, 𝑤 can be estimated using Eq. (36) applied to wake and loss can be calculated in wakes using Eq. (41), allowing to set ASTEC f bv source term in wake regions for each blade slices.

Tip leakage flow

The tip leakage losses are incorporated in ASTEC with the well-known model of Denton [START_REF] Denton | Loss Mechanisms in Turbomachines[END_REF] since the velocity on the blade tip profile is known from potential calculation. It consists first in the estimation of the leakage massflow rate crossing over the blade tip and then the estimation of entropy created by the mixing of leakage flow arriving at the other side of the blade. In case of a compressor with a leakage flow blowing from pressure side to suction side the specific entropy created Δ𝑠 for the whole rotor row is modeled with:

𝑇Δ𝑠 = 1 𝑚 𝑚 ∫ 𝑐 𝑉 2 𝑠𝑠 ( 1 - 𝑉 𝑝𝑠 𝑉 𝑠𝑠
) 𝑑𝑚 (50) 3 Neglecting changes in hub or casing radius.

Where 𝑑𝑚 is the local leakage massflow rate estimated by the pressure difference across the blade and 𝑚 𝑚 is the massflow rate of the entire blade channel. The integration of the local leakage massflow rate of Eq. ( 51) take place on the camber-line of the blade tip profile in the proposed implementation.

𝑑𝑚 = 𝐶 𝑑 𝑔 √ 𝜌2Δ𝑃𝑑𝑐 𝑙 (51)
Where 𝐶 𝑑 is the vena contracta discharge coefficient. In this work the theoretical value of Moore & Tilton [START_REF] Moore | Tip Leakage Flow in a Linear Turbine Cascade[END_REF] is used (𝐶 𝑑 ≈ 0.84).

With an incompressible assumption using Bernoulli equation:

Δ𝑃 = 0.5𝜌(𝑉 2 𝑠𝑠 -𝑉 2 𝑝𝑠
). The volumic viscous force, in opposite flow direction, obtained from an increase of specific entropy Δ𝑠 on a streamline reads [START_REF] Denton | Loss Mechanisms in Turbomachines[END_REF]:

𝑓 𝑣 = 𝜌𝑇 Δ𝑠 Δ𝑙 ( 52
)
Where 𝑙 is the distance along the streamline. Equation ( 50) is then discretized to locally update f bv source term for rotor using Eqs. ( 51) and (52).

AERODYNAMIC BLOCKAGE COUPLING

As introduced in Sect. 4.1 the effect of the boundary layer on the blade to blade passage averaged flow angle is taken into account by modification of the shape of the profile in the potential method. Still, the influence of the profile boundary layer on the profile potential velocity distribution is not taken into account by this mean. The purpose of the following section is to model that effect. In the literature accurate techniques exist for viscid-inviscid interactions implying a strong coupling between the boundary layer and the potential solver [START_REF] Veldman | Quasi-Simultaneous Viscous-Inviscid Interaction for Transonic Airfoil Flow[END_REF]. However, applying this kind of techniques on multiple slices for multistage application seems not realistic in terms of computation time for a throughflow solver. A simple procedure is described to simulate the potential-boundary layer interaction in the following.

Aerodynamic blockage

The boundary layers that develop on the pressure side and suction side of the profiles create additional blockage in the bladed region as well as secondary flows. This aerodynamic blockage is directly linked to the displacement thickness induced by viscous effects in the blade channel. For now, in ASTEC, aerodynamic blockage is produced by the profile boundary layer and wakes displacement thicknesses. The blockage factor 𝑏 in Eq. ( 10) is updated as follows:

𝑏 = 1 - 𝑁 (𝜀 + 𝜀 1 ) 2𝜋𝑟 (53)
Where 𝜀 1 is the additional thickness in the circumferential direction due to viscous effects displacement thicknesses.

Viscid-inviscid coupling

The consequence in ASTEC of the use of Eq. ( 53) is an acceleration of the computed mean flow in presence of aerodynamic blockage compared to the inviscid case. In order to adjust potential velocities on profile according to the development of viscous effect, a comparison is made between the mean meridional velocity in bladed region given by ASTEC 𝑉 𝑚 and the mean potential meridional velocity obtained in the blade to blade plane 𝑉 𝑝 𝑚 from each stream surfaces. A ratio coefficient, 𝑉 𝑚 /𝑉 𝑝 𝑚 , is obtained for each slice. This ratio is thus ≥ 1. Then the hypothesis of a homogeneous increase of speed in the blade to blade plane thanks to aerodynamic blockage is made to modify the velocity on suction sides respectively pressure side in potential calculations. This coupling update velocities obtained by the potential procedure as a pre-treatment according to Eqs. ( 54) and (55) before the calculation of the profile boundary layer. As an example of interaction, starting with an already thick boundary layer, the coupling will reduce the potential pressure gradients for the next boundary layer calculation. Then, the following calculated boundary layer become thinner, reducing aerodynamic blockage and increasing pressure gradient for next boundary layer calculation and so goes on.

𝑉 updated 𝑠𝑠 = 𝑉 𝑚 𝑉 𝑝 𝑚 𝑉 old 𝑠𝑠 ( 54
)
𝑉 updated 𝑝𝑠 = 𝑉 𝑚 𝑉 𝑝 𝑚 𝑉 old 𝑝𝑠 (55)
The procedure appears to converge in few iterations and the coupling is done simultaneously with the convergence of ASTEC.

RESULTS AND DISCUSSION 6.1 Linear cascades

The first validations where made on two cases of linear cascades: a NACA 65 009 and a controlled diffusion arc blade (CDA), both tested at LMFA by Zambonini [START_REF] Zambonini | Corner Separation Dynamics in a Linear Compressor Cascade[END_REF] and Dawkins [START_REF] Dawkins | The Unsteady Topology of Corner Separations[END_REF] respectively. They have been mounted on the same test rig delivering an inlet Mach number of 0.12. The shape of the blades are visible in Fig. 6 and more details on the test cases are available in the previously mentioned papers. It is important to note that the NACA cascade is equipped with trips at the leading edge allowing the development of a fully turbulent boundary layer whereas the other cascade has no trip. Thus, a boundary layer transition appear on the latter at ≈ 30% of blade chord as shown in [START_REF] Dawkins | The Unsteady Topology of Corner Separations[END_REF].

NACA 65 009 CDA FIGURE 6: Blade passages of validation cascades

Bi-dimensional results. First, results from ASTEC are compared to the well-known turbomachinery design tool MISES developed by M. Drela and H. Youngren. 4 The latter uses an Euler solver coupled with viscous boundary layer calculation allowing boundary layer transition. The simulated case is a profile from the NACA 65 009 cascade at inlet Mach number of 0.12. The Reynolds number is around 385 × 10 3 . Here, to be comparable with the fully turbulent hypothesis of the boundary layer method of ASTEC, the transition is forced at the leading edge in MISES simulations.

The blade pressure coefficients 𝐶 𝑝 for two incidences, 𝑖 = 0 • and 𝑖 = 4 • are compared on Fig. 7. The agreement between the two methods is excellent. At the trailing edge small differences still exists, due to the sharp trailing edge required by the Hess & Smith method. The results validate the implemented potential method and the viscous coupling with the integral boundary layer method of Sect. 5.2. The blockage factor evolution in the blade and wake regions is also showed in Fig. 8. It includes aerodynamic blockage from the boundary layer calculation in both methodologies. In the bladed region, the agreement is matching, which means that both boundary layers calculated from ASTEC and MISES show very close displacement thickness evolution. However, in the wake region, the blockage in ASTEC shows a steeper decrease close to the trailing edge, revealing differences between used wake models of Lakshminarayana and MISES wake handling methods. Since the aerodynamic blockage decays more quickly in ASTEC to a lower level than in MISES, the loss level in the wake can be expected to be slightly higher in ASTEC. The total pressure loss coefficient 𝜔 and the exit flow angle 𝛽 2 for this bi-dimensional case are shown in Table 1, again for 𝑖 = 0 • and 𝑖 = 4 • . Data from ASTEC, MISES and bi-dimensional blade-to-blade steady RANS simulations are compared together with estimations from empirical Lieblein correlations, fitted by Aungier [START_REF] Aungier | Axial-Flow Compressors[END_REF]. The RANS simulations are carried out with elsA using 2-equations 𝑘-𝑙 turbulence model of Smith [START_REF] Smith | A near Wall Model for the k -l Two Equation Turbulence Model[END_REF] with a Reynolds number around 385 × 10 3 on a 𝑦 + ≈ 1 mesh. The results exposed show an overall good agreement between ASTEC and other methods on a bi-dimensional case for both incidences.

In details, the loss level and exit angle predicted by ASTEC agree particularly well with RANS and MISES results. Regarding the results of the Lieblein correlations, they tend to predict lower deviation angles and lower loss levels than other methods. These results validate the loss and deviation models, previously introduced in ASTEC, based on boundary layer integral method, wake decay and the coupling with the potential calculation. Cp comparison with experiment at mid-span. End-wall secondary flows are not yet modeled in this work, but it is necessary to take into account their induced blockage effect to validate against experimental data the presented methodology at mid-span. The choice is made to include the experimental endwall aerodynamic blockage of the cascades in ASTEC. To do so, a span-wise distribution of pitch-wise displacement thickness is calculated from experimental data just after the trailing edge of the cascades. The resulting end-wall aerodynamic blockage contribution is linearly distributed in the axial direction from trailing edge to leading edge, with leading edge value set to 0, and then added to the local aerodynamic blockage due to the profile boundary layers. In order to determine axial distribution of 𝜀 1 of Eq. ( 53) for each blade slices, the axial distribution of end-wall blockage is added to the boundary layer contribution to aerodynamic blockage estimated in the methodology. with versus without exp. end-wall blockage Figure 9 compares the pressure coefficient at mid-span estimated with ASTEC on NACA 65 009 case for 4 • degrees of incidence with and without experimental end-wall aerodynamic blockage. Regarding those results, once the end-wall blockage is introduced, a good agreement is observed. Figure 10 summarize the pressure coefficients for the CDA cascade at mid-span using experimental end wall blockage as well for multiple incidences also showing an excellent agreement. This again validates the simple inviscid/viscid coupling introduced in Sect. 5.2. 3 for • of incidence where 𝛽 2 is the mean cascade exit angle and 𝜔 the total pressure loss coefficient, both measured at mid-span. The comparison with experimental measurements are done in the wake at 𝑥 = 0.363 𝑐 𝑎𝑥 after the trailing edge. About the CDA cascade, the blade mean exit angle and total pressure loss coefficient are plotted against inlet flow incidence in Fig. 11. The plane of measurement is situated 0.2𝑐 after trailing edge in axial direction. The agreement between the experiment and ASTEC on performance data is good on the NACA 65 009 test case. Although the pressure coefficients showed good matching in Fig. 10 for the second cascade case, the models predicted over deviation and over loss. This can be explained by the nature of the boundary layer developing on the CDA cascade. A transition exists on the suction side of the blade at ≈ 30% of chord and no transition modeling is implemented in ASTEC at the moment. Consequently, levels discrepancies on loss and deviation obviously appear. But the trends are good despite the observed offsets caused by the ignored laminar fraction of the boundary layer. However, in the most aeronautical axial compressor applications, Reynolds numbers are high enough to make profile boundary layers transitioning close to the leading edge allowing the authors to say that the presented methodology is adapted for future applications.

CME2 low speed compressor

The third test case is the research low speed compressor CME2. It is a single stage axial compressor with relative inlet mach number about 0.6 at blade tip at design point. More characteristics are summarized in [START_REF] Gourdain | Prediction of the Unsteady Turbulent Flow in an Axial Compressor Stage. Part 1: Comparison of Unsteady RANS and LES with Experiments[END_REF]. For comparison purpose, single passage RANS steady simulations of the compressor are carried out with elsA using 2-equations 𝑘-𝑙 turbulence model of Smith [START_REF] Smith | A near Wall Model for the k -l Two Equation Turbulence Model[END_REF] on a 𝑦 + ≈ 1 mesh. About simulations, the mesh used is visible in Fig. 14. Each blade have approximately 60 points in the axial direction.

CME2 characteristics.

The compressor total to total pressure coefficient and isotropic efficiency are plotted against standardized massflow rate for nominal speed in Fig. 12. The data are also compared to experimental results from Gourdain [START_REF] Gourdain | Prediction of the Unsteady Turbulent Flow in an Axial Compressor Stage. Part 1: Comparison of Unsteady RANS and LES with Experiments[END_REF]. The design point is evidenced by the only available experimental point in the efficiency plot. However since the efficiency is difficult to measure in a low speed compressor test-rig, extra care must be taken for the interpretation of its position.

In terms of pressure ratio, the agreement is good near design point but deteriorates as the compressor is throttled. This can be the evidence of several phenomena occurring and not taken into account in the actual methodology yet: as the massflow rate decreases, the blade loading increases and the secondary flows intensify leading to high deviation angles at hub and casing and so reduced useful work. For secondary flows, only tip leakage loss are today taken into account. The differences can also be explained by the lack of additional losses coming from hub (and casing for the stator) boundary layers in the current version of the code. As Denton remarked [START_REF] Denton | Loss Mechanisms in Turbomachines[END_REF], end-wall secondary flows remains the most difficult flow mechanism to precisely model. Nonetheless, the integral boundary layer method coupled with the potential cascade calculation allow capturing a correct level of flow turning near the design point in the present methodology. In terms of efficiency, the relative position of ASTEC and RANS results is easily explained by the same lack of end-wall boundary layer losses, but the trend is quite well predicted.

Rotor pressure coefficients. Rotor blade pressure coefficients from ASTEC compared to RANS results are visible on Fig. 13. They derive from the operation points circled on Fig. 12 for several reduced blade heights ℎ/𝐻. This time, there is no additional aerodynamic blockage coming from experimental values. Still, for 0.3 ≲ ℎ/𝐻 ≲ 0.7 a very good agreement is observed. Below ℎ/𝐻 ≈ 0.3 the influence of hub end-wall boundary layer is observed in the RANS results. Above ℎ/𝐻 ≈ 0.7 the influence of casing end-wall boundary layer and increasing compressible effects cause the observed discrepancies.

[pa] A very good agreement on the absolute value of total leakage massflow rate with an error less than 1% is observed and the overall trend along the blade tip is well captured by the model compared to the RANS simulation.

Then the evolution of tip leakage loss predicted by the model while throttling the compressor is shown on Fig. 16. The loss and the channel massflow rate 𝑚 𝑚 are normalized by reference values extracted at the ASTEC circled operation point on Fig. 12. During the compressor throttle, the loss produced by the leakage flow shows an intense growth, up to close to tree time the loss produced at the reference point. The main objective for the future of ASTEC is to pursue the modeling of secondary flows, especially the compressor hub boundary layers. This would bring radial distribution of losses and blockage, improved results on the axial compressor case and make the code ready for future realistic multistage applications. The current methodology developed involves subsonic potential flow theory, so a second objective is to extend this methodology to be suitable for higher Mach numbers. On the long term, ASTEC would be used in an analysis optimization process for design purposes.

FIGURE 1 :

 1 FIGURE 1: Blade thickness and blade force components

FIGURE 2 :

 2 FIGURE 2: Blade slice definition

original pro�ile enlarged with δ 1 ,FIGURE 5 :

 15 FIGURE 5: View of trailing edge region of a profile showing enlarged profile with 𝛿 1 and new camber line definition for deviation in a stalled case

FIGURE 7 :

 7 FIGURE 7: ASTEC vs MISES Cp on NACA 65 009 profile for i = 0 • and i = 4 •

FIGURE 8 :

 8 FIGURE 8: ASTEC vs MISES blockage factor Eq. (10) on NACA 65 009 profile for i = 0 • and i = 4 •

FIGURE 9 :

 9 FIGURE 9: Cp comparison on NACA 65 009 at mid-span at i = 4 •

FIGURE 10 :

 10 FIGURE 10: Cp comparison on CDA cascade at mid-span for multiple incidencesPerformances comparison at mid-span. The performances of the NACA 65 009 at mid-span are summarized in Table3for • of incidence where 𝛽 2 is the mean cascade exit angle and 𝜔 the total pressure loss coefficient, both measured at mid-span. The comparison with experimental measurements are done in the wake at 𝑥 = 0.363 𝑐 𝑎𝑥 after the trailing edge. About the CDA cascade, the blade mean exit angle and total pressure loss coefficient are plotted against inlet flow incidence in Fig.11. The plane of measurement is situated 0.2𝑐 after trailing edge in axial direction.

FIGURE 11 :

 11 FIGURE 11: CDA cascade mid-span performances against incidence measured 0.2c after trailing edge

FIGURE 13 :

 13 FIGURE 13: CP comparison on rotor blade at 100Nn near design point for multiple reduced blade heights

FIGURE 14 :

 14 FIGURE 14: CME2 ASTEC mesh colored with static pressure at 100NN near design point, Approx. 6000 cells Tip leakage model evaluation. First a comparison is made between RANS and ASTEC at the operation points circled on Fig. 12 presenting same compressor massflow rate. Figure 15 show the distribution of tip leakage massflow rate 𝑑𝑚 of Eq. (50) normalized by the total leakage massflow rate against the chordnormalized axial position.

FIGURE 15 :

 15 FIGURE 15: Tip leakage massflow distribution at 100NN near design point

FIGURE 16 :

 16 FIGURE 16: Tip leakage loss evolution with throttle at 100NN from the model

  𝐻 12, 𝑠𝑠 𝛿 2, 𝑠𝑠 + 𝐻 12, 𝑝𝑠 𝛿 2, 𝑝𝑠 𝛿 2, 𝑠𝑠 + 𝛿 2, 𝑝𝑠

				48)
	where:	𝐻 12,𝑡𝑒 =	| | | |	𝑡𝑒

TABLE 1 : NACA 65 009 profile performances comparison 0.5c ax after trailing edge for

 1 i = 0 • and i = 4 • 𝛽 2 37.8 • 38.2 • 38.9 • 39.8 • 38.7 • 39.6 • 38.6 • 39.5 •

		Lieblein	ASTEC	MISES	RANS
	𝑖	0 •	4 •	0 •	4 •	0 •	4 •	0 •	4 •
	𝜔 1.5% 2.1% 2.1% 2.4% 2.0% 2.3% 2.1% 2.4%

TABLE 2 : NACA 65 009 mid-span performances at i = 4 • 0.363c ax after trailing edge

 2

		ASTEC exp.
	𝛽 2	38.9 •	38.5 •
	𝜔	2.2%	2.3%

TABLE 3 : NACA 65 009 mid-span performances at i = 4 • 0.363c ax after trailing edge

 3

		exp. ASTEC w/ EW blockage ASTEC w/o EW blockage
	𝛽 2 38.5 •	38.9 •	39.8 •
	𝜔	2.3%	2.2%	2.4%

ASTEC was recently developed by Safran Tech the Research & Technology center of Safran Group.

See A User's Guide to MISES 2.63, http://web.mit.edu/drela/Public/web/mises/

ACKNOWLEDGMENTS

The authors would like to thank Safran Aircraft Engines and the French Ministry of Higher Education and Research for the funding of this work. They are also thankful to Safran Tech for accepting to share ASTEC source code to conduct this research.

For the purpose of Open Access, a CC-BY public copyright licence has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.