Assistant Professor, Mauro Gatti

Withdrawing from the EU: Between Sovereign Rights and Union Autonomy

Through Article 50 TEU, the Treaty of Lisbon expressly acknowledged, for the first time, the right of EU member states to withdraw from the Union. On 29 March 2017, the United Kingdom became the first member state to make use of it. This provision, however, does not define every detail of the withdrawal process. 1

Table of contents

1. The Genesis of Article 50 TEU 2. The "Sovereign Right" to Withdraw Unilaterally from the Union 3. The withdrawal Procedure: An Example of the Autonomy of the Union 4. Sovereignty and Autonomy in the Wightman Judgment

Conclusion

This contribution is part of a special issue of the Working Papers of the "Blog droit européen" and is a sequel to the conference organised at the Université Libre de Bruxelles on 28 and 29 March 2019 entitled "The Lisbon Treaty 10 years on: a Union at a crossroads? ». The conference was organized in cooperation with the Institute of European Studies and the Faculty of Law and Criminology of ULB, the Institute of European Studies of the Université Saint-Louis and Alliance Europa, with the support of the Belgian National Fund for Scientific Research.

This question was addressed by the Court of Justice in the Wightman case. 2 The Court, as well as Advocate General Manuel Campos Sánchez-Bordona, held that the withdrawing State has a "sovereign right" to revoke its notification. 3 Although it mentioned some principles of EU constitutional law, the Court based its reasoning primarily on the international principle of sovereignty, thus suggesting that withdrawal represents "a typical international law issue", as affirmed by the Advocate General. 4 Moreover, both the Court and the Advocate General argued that the Vienna Convention on the Law of Treaties (hereinafter: Vienna Convention) was "taken into account" in the preparatory work for the Treaty establishing a Constitution for Europe. 5 In light of Wightman, it might seem that, to fill the gaps in Article 50 TEU, one might simply use international law, particularly the principle of states' sovereignty and the law of treaties.

Such an approach may have practical consequences, since there are various aspects of Article 50 TEU that remain unclear. For instance, one may wonder whether a state should promptly notify its intention to withdraw as soon as it has decided to withdraw, or if it has a "sovereign right" to play for time (despite its duty of loyalty to the Union). 6 More generally, one might wonder to what extent recourse may be had to international law to fill the gaps in EU primary law, or if the concern for the autonomy of EU law should suggest different solutions.

This paper seeks to demonstrate that the design of Article 50 TEU is only partially inspired by international law and, in particular, the principle of sovereignty. The procedural aspects of Article 50 TEU find their roots in the autonomy of the Union. Had the Court adopted this perspective in Wightman, it might have reached a different conclusion. The analysis begins with a presentation of the drafting history of Article 50 TEU (section 2). It subsequently shows that some elements of Article 50 TEU are inspired by international law (the right to withdraw, section 3), whereas others exemplify the EU's autonomy (the withdrawal procedure, section 4). Section 5 suggests that the Wightman judgment acknowledges the specificity of Article 50 TEU in principle but fails to emphasise the EU's autonomy in Special Working Papers Series 3/2020 practice. The concluding remarks explore the apparent inconsistency of Wightman and its consequences (section 6). In the first place, draft Article 46 recognises the Member States' right to withdraw from the European Union. While it is desirable, according to the Praesidium, that an agreement should be concluded between the Union and the withdrawing state on the arrangements for withdrawal, "it was felt that such an agreement should not constitute a condition for withdrawal so as not to void the concept of voluntary withdrawal of its substance". 8 This explains why draft Article 46, like Article 50 TEU, contemplates the possibility of unilateral withdrawal two years after the notification of the intention to withdraw.

The Genesis of

The second element of the Praesidium text is the withdrawal procedure. This procedure is "partly based on the procedure under the Vienna Convention on the Law of Treaties, but at the same time it provides for the Union to conclude an agreement with the member state concerned setting out the arrangements for its withdrawal". 9 The Praesidium text, in the English version, later introduces a slightly different statement: "The procedure laid down in this provision draws on the procedure in the Vienna Convention on the Law of Treaties." 10 The French version is clearer, since it consistently affirms that "la procédure prévue dans cette disposition s'inspire en partie de celle prévue dans la Convention de Vienne sur le droit des traités." 11 The "partial" character of this inspiration is relevant for the interpretation of Article 50 TEU, as shown below, in section 4.

The members of the European Convention proposed numerous amendments to Article 46. Several proposals regarded the right to withdraw. While many Convention members agreed with the existence of this right, others opposed it or proposed the introduction of conditions for its exercise. For example, some speakers suggested that the member states could request withdrawal but could not withdraw until an agreement with the Union was concluded.

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proposed limiting the right of withdrawal to the scenario in which the withdrawing state did not approve an amendment to the Constitution. 12 Several Convention members addressed procedural issues, too. According to two speakers, the withdrawal agreement should have been concluded by the Member States, not by the Union. According to another speaker, the procedure for withdrawal should have been eliminated, and the Council empowered to define it at a later stage. Some speakers made less radical proposals, suggesting, for instance, that the conclusion of the withdrawal agreement should have required unanimity within the Council (and not qualified majority, as proposed by the Praesidium). 13 Notwithstanding the above proposals, the essential elements of draft Article 46 remained unchanged in the final version of Article I-60 of the Constitution. The European Convention introduced only a few procedural innovations, compared to the original Praesidium text. For instance, Article I-60 stipulates that the negotiation of a withdrawal agreement should take place within the framework of "guidelines" adopted by the European Council. Article I-60 specifies that the withdrawal agreement should be negotiated according to Article III-325(3) (now Article 218(3) TFEU) and should take account of the "framework" for the withdrawing state's future relationship with the Union. Moreover, Article I-60 introduces the possibility to extend, through a European Council Decision, the negotiation of the withdrawal agreement, beyond the two years deadline imposed by original Praesidium text. 14 After the failure of the European Constitution, the text of Article I-60 was placed under Article 50 TEU, with no substantial change. Article 50 TEU, like the original Praesidium proposal, contains two elements: the right to withdraw and the withdrawal procedure. While the first is strongly influenced by international law, and particularly the principle of sovereignty (see the next section), the withdrawal procedure appears to be grounded on the EU's autonomy (see below, section 4).

The "Sovereign Right" to Withdraw Unilaterally from the Union

Article 50(1) TEU, like draft Article 46(1) of the European Constitution, enables any member state to "decide to withdraw from the Union in accordance with its own constitutional requirements". The withdrawing state must subsequently notify its intention to the European Council (Article 50(2) TEU). EU Treaties cease to apply to the withdrawing State from the date of entry into force of a withdrawal agreement concluded by the withdrawing State and the Union or, failing that, "two years after the notification" (Article 50(3) TEU). The European Council, in agreement with the withdrawing state, may unanimously decide to extend this period (Article 50(3) TEU).

12 See European Convention Secretariat, Summary sheet of proposals for amendments concerning Union membership: Draft Articles relating to Title X of Part One (Articles 43 to 46), CONV 672/03, 10-12. See further European Convention Secretariat, Summary Report of the Plenary Session, doc. CONV 696/03 10. 13 Ibid. 14 In addition, Art. I-60 of the European Constitution introduced a clarification regarding the future status of the withdrawing state: "If a State which has withdrawn from the Union asks to rejoin, its request shall be subject to the procedure referred to in Article I-58".

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The explicit recognition of a right to withdraw unilaterally constitutes a novelty for the Union, since, before the Lisbon reform, EU primary law did not expressly authorise unilateral withdrawal. The Member States had created a "Community of unlimited duration" (Costa) 15 and had undertaken obligations "irrevocably" (Simmenthal) 16 to ensure an "ever closer union". Therefore, according to some authors, unilateral withdrawal was impossible.17

Others, however, argued that the Member States could withdraw unilaterally from the Community. 18 Under Article 56 of the Vienna Convention on the Law of Treaties, -which might be declaratory of a rule of customary law19 -States may withdraw from a treaty containing no provision regarding its termination under two alternative conditions: (a) if the parties intended to admit the possibility of denunciation or withdrawal and (b) a right of withdrawal is implied by the nature of the treaty. 20 The latter condition could be relevant for the European Community, as it has been argued that the nature of treaties establishing international organisations generally implies the right to unilateral withdrawal. [START_REF] Th | Article 56[END_REF] The existence of a right to unilateral withdrawal from the European Community was apparently confirmed by the practice. Greenland left the European Community in 1985 but this precedent has limited value, as the exit of Greenland was consensual (i.e. approved by the other Member States) and, above all, Greenland was not a Member State, but a territory. [START_REF] Harhoff | Greenland's Withdrawal from the European Communities[END_REF] The precedent of the UK is perhaps more relevant: this country held a referendum on European Community membership in 1975 and, although the "remain" option won, the other Member States never contested the existence of the UK's right to withdraw from the Community. 23 The members of the European Convention had different positions regarding the existence of a right to withdraw. A few speakers argued that the Member States should not have any right to withdraw unilaterally, which is allegedly incompatible with the EU's nature. [START_REF] Van Lancker | Proposition d'amendement à l'article 46[END_REF] Numerous others, however, noted that the right to withdraw unilaterally is implicit in the EU's nature as an international organisation. The Union being the result of the will of sovereign states, "no one questions their right to withdraw from the Union." [START_REF] Lopes | Suggestion for amendment of Article :Article I-59[END_REF] Since "the Member States' right to Special Working Papers Series 3/2020 withdraw follows from the basic principles of international law",26 "paragraph 1 [of draft Article 46] only spells out what already follows from the basic principles of international law".27

Even those who, within the European Convention, opposed the introduction of draft Article 46, often did so for political, rather than legal, reasons: while "no State can be forced to remain in the Union", the "explicit inclusion" of a withdrawal clause is allegedly incompatible with the idea of solidarity of citizens and states. [START_REF] Meyer | Suggestion for amendment of: Article 46, Part I[END_REF] With the explicit acknowledgement of the right to unilaterally withdraw, "le compromis pour l'édification d'une Union chaque fois plus étroite, ne deviendra plus qu'une hypothèse, une simple possibilité." 29 The introduction of an explicit reference to the Member States' right to unilaterally withdraw from the Union in draft Article 46(1) probably was not intended as a legal innovation, but as a mere confirmation of an existing international customary rule, grounded on the principle of sovereignty. As noted by the German Bundesverfassungsgericht, Article 50 TEU "makes explicit" for the first time in primary law "the existing right" of each member state to withdraw from the European Union. 30 The right to withdraw thus "underlines the Member States' sovereignty". 31 Sovereignty was also central to the Spanish Tribunal constitucional's reasoning on Article I-60 of the Constitution (now Article 50 TEU): this provisions allegedly allows the Member States to recuperate, "by virtue of their sovereignty" ("soberanamente"), the competences they have conferred, "through the sovereign will of the state", to the Union. 32 In other words, the explicit mention of the right to withdraw unilaterally in Article 50 TEU is not meant to address an outstanding legal issue: "le véritable objet de la clause de retrait est celui de la souveraineté des Etats". [START_REF] Grosclaude | La clause de retrait du Traité établissant une Constitution pour l'Europe : réflexions sur un possible marché de dupes[END_REF] Given the centrality of an international law principle -sovereignty -to the right to unilateral withdrawal, it appears reasonable that draft Article 46 (now Article 50 TEU) should be partially based, as noted by members of the European Convention, on international law, particularly the Vienna Convention on the Law of Treaties. [START_REF]Intervention du Représentant Suppléant du Gouvernement Portugais Manuel Lobo Antunes[END_REF] Under Article 65 of the Vienna

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Convention, if a treaty permits withdrawal, 35 a party may notify the other parties of its intention to withdraw; if, after the expiry of a period of at least three months no party has raised any objection, the party making the notification may unilaterally carry out the withdrawal. The Court of Justice held in Racke that the specific requirements laid down by Article 65 of the Vienna Convention "do not form part of customary international law". 36 Nonetheless, they reflect a widespread practice, at least up to a point. 37 The founding treaties of several international organisations enable member states to notify their intention to withdraw and effectively withdraw after a certain period, ranging from a few months (e.g. World Trade Organization) to two years (e.g. International Labour Organization). 38 Therefore, it would seem that the right to withdraw unilaterally enshrined in Article 50 TEU is modelled after a pre-existing international practice, partially codified in the Vienna Convention on the Law of Treaties, and grounded on the principle of sovereignty.

The Withdrawal Procedure: An Example of the Autonomy of the Union

While the right to unilaterally withdraw from the Union, enshrined in Article 50 TEU, is certainly rooted in international law, the foundations of the withdrawal procedure are more complex.

From a procedural perspective, Article 50 TEU has two elements in common with Article 65 of the Vienna Convention on the Law of Treaties (and with the founding treaties of numerous other international organisations): (a) the withdrawal procedure begins with a notification of the withdrawing state to the international organisation (the Union); (b) the withdrawal ends some months or years after the withdrawal.

However, the process that leads from (a) to (b) is very different in the cases of Article 50 TEU and other treaties. The foundational treaties of other international organisations generally provide for a simple withdrawal procedure, composed of the notification and the subsequent withdrawal. The Vienna Convention on the Law of Treaties is more articulate, as it expressly addresses the possibility that, after the withdrawing state's notification, another party raises an objection. In this case, the parties must seek a solution through peaceful means, e.g. negotiation. 39 Article 50 TEU is different, as it prescribes that "the Union shall negotiate and conclude an agreement" with the withdrawing state (paragraph 3). The European Council may approve the

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withdrawal agreement "by a qualified majority", after obtaining the consent of the European Parliament (paragraph 2). In other words, as noted by the European Convention Praesidium, while the withdrawal procedure is "partly" based on the Vienna Convention on the Law of Treaties, "at the same time it provides for the Union to conclude an agreement with the Member State concerned setting out the arrangements for its withdrawal". 40 Two innovative aspects of Article 50 TEU are particularly important. In the first place, Article 50 TEU imposes an obligation to negotiate a withdrawal agreement. The obligation is directly binding on the Union and indirectly binding on the withdrawing State, which must "facilitate the achievement of the Union's tasks" (Article 4(3) TEU). 41 It has been argued that Article 50 TEU does not significantly innovate with respect to international law, as it merely prescribes an obligation of means (the negotiation) and not an obligation of result (the conclusion of a withdrawal agreement). 42 The existence of such an obligation of means is nonetheless relevant: in the EU, negotiations are the necessary consequence of the notification of the intention to withdraw; in other organisations, they are the possible product of an objection raised by a Member State. Although EU Treaties do not ensure an orderly withdrawal, they facilitate, at least, a negotiated solution. The reason why EU Treaties seek to facilitate negotiation is evident: unilateral withdrawal from the EU would greatly affect the economy and citizens of both the EU and the withdrawing state, as opposed to the relatively more limited practical consequences of unilateral withdrawal from most international organisations. 43 Secondly, Article 50 TEU introduces an obligation to negotiate with the Union, not the other member states. The significance of this innovation was not lost on the members of the European Convention. Two speakers argued that the withdrawal agreement should have been negotiated by "the other Member states" and not by the Union, as "the membership or nonmembership of the Union is not conferred competence". 44 This proposal was rejected, thus suggesting that the European Convention intended to introduce an innovative solution, aimed at facilitating withdrawal negotiations. One can imagine that 27 parallel negotiations between the UK and each of the other member states could have been chaotic. As argued by Lenaerts and Gerard, Article 50 TEU recognises the Union "as an autonomous entity entitled to negotiate on an equal footing with its Member states the respect of rights and obligations 40

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towards it". [START_REF] Lenaerts | The structure of the Union according to the Constitution for Europe: the emperor is getting dressed[END_REF] Therefore, the Article 50 TEU procedure may be described as an "example of the autonomy of the Union, of the [Treaties] and of the constitutional regime" they establish. 46 The autonomous procedure of Article 50 TEU seems capable of promoting an orderly withdrawal from the Union, as shown by the practice. Despite the vicissitudes of the Brexit negotiations -motivated by domestic political issues in the UK -the withdrawal process followed the path defined by Article 50 TEU. The UK negotiated with the EU and did not manage to divide its representation by conducting separate negotiations with individual member states. [START_REF] Mctague | How the UK lost the Brexit battle[END_REF] The UK did not "control the process of withdrawal to [its] own benefit", as some expected. [START_REF] Friel | Providing a Constitutional Framework for Withdrawal from the EU: Article 59 of the European Constitution[END_REF] On the contrary, the UK government accepted several compromises, regarding for instance the possibility of customs checks on goods traded between Northern Ireland and the rest of the UK. 49 After Wightman, however, the effectiveness of the Article 50 TEU withdrawal procedure might be put into question.

Sovereignty and Autonomy in the Wightman Judgment

Although the Article 50 TEU procedure exemplifies the autonomy of EU law, the Wightman judgment arguably pays lip service to this essential characteristic of the EU legal order.

The Wightman case originated from the question of a British judge, who asked the Court of Justice whether a Member State may unilaterally revoke the notification of its intention to withdraw from the Union (and, if so, subject to what conditions and with what effects) 50 -a matter that is not regulated expressly by Article 50 TEU. Advocate General Campos Sánchez-Bordona and the Court of Justice adopted partially different readings of Article 50 TEU in Wightman, but eventually reached similar solutions.

The Advocate General postulated that Article 50 TEU is "not a self-contained provision which exhaustively governs each and every detail of that withdrawal process": 51 to fill the lacunae in Article 50 TEU, there is nothing to preclude recourse being had to Article 68 of the Vienna Convention on the Law of Treaties, even if the EU is not a party to the Vienna Special Working Papers Series 3/2020 Convention (nor are several of its member states) and Article 68 may not reflect rules of customary international law. 52 Pursuant to this provision, a notification of the intention to withdraw from a treaty may be revoked "at any time before it takes effect". According to the Advocate General, this means that a notification can be revoked "at any time before the withdrawal takes effect". [START_REF]para.108 (emphasis added). Alternatively, it may be argued that revocation can be performed only "until the other party has started undertaking[END_REF] Therefore, there is "no reason not to apply, by analogy, the same rule in the framework of the procedure for withdrawing from the European Union." [START_REF][END_REF] I am unconvinced by this argument. If the procedure for withdrawal from the Union is autonomous from international law, the lacunae in Article 50 TEU should be filled by having recourse primarily to the norms and principles of the EU's constitutional order -not to international law. Recourse should be had, in particular, to the objectives of Article 50 TEU and the essential characteristics of the EU's legal order, including the EU's autonomy. [START_REF] Ch | The Irrational Brexit and the Revocability of the Withdrawal Notification of the United Kingdom to the European Council[END_REF] The Court of Justice apparently stressed the autonomy of EU law in Wightman, at least at first sight. The Court started its analysis by recalling that EU Treaties are "unlike ordinary international treaties", as they set up a new legal order, possessing its own institutions, for the benefit of which the members thereof have limited their sovereign rights, and the subjects of which comprise not only those states but also their nationals. 56 The essential characteristics of the European Union justify the "autonomy of EU law" with respect both to the law of the member states and to international law. 57 Furthermore, and unlike the Advocate General, the Court did not fill the gaps in Article 50 TEU by using the Vienna Convention on the Law of Treaties -at least not directly. This Convention is mentioned in the Wightman judgment as part of a subsidiary argument, based on a historical interpretation of Article 50 TEU. According to the Court, the Vienna Convention was "taken into account in the preparatory work for the Treaty establishing a Constitution for Europe". 58 Therefore, the Court used the Vienna Convention "simplement comme un élément lié aux travaux préparatoires de l'article 50 TUE, qui 'corrobore' une solution fondée sur une interprétation autonome de cet article", as noted by judge Rossi. [START_REF] Rossi | Droits fondamentaux, primauté et autonomie: la mise en balance entre les principes "constitutionnels" de l'Union europénne[END_REF] However, one might wonder whether the Court's interpretation of Article 50 TEU is truly "autonomous" from international law. An historical interpretation of Article 50 TEU should acknowledge that the European Convention intended to introduce a new -autonomouswithdrawal procedure, specific to the European Union. It is true that Article 50 TEU is "partly" based on the Vienna Convention on the Law of Treaties. However, the documents of Special Working Papers Series 3/2020 the European Convention (see above, section 2) show that Article 50 TEU was intended to introduce an original procedure and has original elements, notably the withdrawing States' obligation to negotiate with the Union. Using the Vienna Convention to fill the gaps in the Article 50 TEU procedure -either directly or via a reference to the works of the European Convention -is arguably inappropriate, because it denies the autonomous design of the EU withdrawal procedure, as well as the intention of its drafters. [START_REF] Manzini | Brexit: Does Notification Mean Forever[END_REF] More generally, the Court's purportedly "autonomous" interpretation of Article 50 TEU appears problematic, as it is based mainly, not on EU law, but on an international principle: sovereignty. [START_REF] Cf | L'arrêt Wightman du 10 décembre 2018 : la réversibilité du retrait au service de l'irréversibilité de l'intégration ?[END_REF] The Court notes that Article 50 TEU seeks to enshrine "the sovereign right of a Member State to withdraw from the European Union" 62 and repeatedly stresses the central role of sovereignty in this area. The "sovereign nature" of the right of withdrawal allegedly supports the conclusion that the withdrawing state "has a right to revoke the notification of its intention to withdraw from the European Union". 63 Indeed, the revocation of the intention to withdraw "reflects a sovereign decision" by the withdrawing state to retain its status as a member state of the European Union. 64 In other words, the revocation of the notification is a "unilateral sovereign right". 65 The Court subsequently performs a contextual interpretation of Article 50 TEU in light of the EU's objectives and principles (paras 61-67), but this analysis remains subordinated to the main argument, based on sovereignty. The Court notes that EU Treaties are based on the values of liberty and democracy and aim at creating an ever closer union among the peoples of Europe. 66 It would be inconsistent with these values and objectives to prevent a Member State from remaining a member of the Union, despite its wish -as expressed through its democratic process in accordance with its constitutional requirements -to reverse its decision to withdraw. 67 In any event, the arguments based on EU principles and objectives are introduced in a succinct manner and seem merely subsidiary to the main argument. The first and principal part of the Court's reasoning (paras 50-59) focuses on the sovereignty of the withdrawing state; remarkably, sovereignty is mentioned again ad abundantiam towards the end of the judgement (para 72).

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While Wightman overemphasises the protection of member states' sovereignty, it pays little attention to the other objective of Article 50 TEU: establishing a procedure to enable withdrawal to take place "in an orderly fashion". 68 This objective is arguably consistent with the innovative character of the Article 50 TEU procedure: while the withdrawal procedures of other international organisations emphasise the sovereignty of their member states, Article 50 TEU introduces novel elements, with specific purposes: (a) The obligation to negotiate a withdrawal agreement, which is aimed at preventing the legal uncertainty caused by a "no deal" scenario; (b) The EU's direct participation in the negotiation, which has the objective of avoiding parallel negotiations between the withdrawing state and all the other member states.

Ensuring an "orderly" withdrawal may be described as a specific objective of the EU's withdrawal procedure and an example of the EU's autonomy. Although the Court acknowledges the existence of this objective in abstracto, 69 it does not use it to interpret Article 50 TEU. Had it done so, it might have reached the same conclusion as the Council and the Commission: allowing a withdrawing member state to revoke unilaterally its notification paves the way "for abuse by the Member State concerned to the detriment of the European Union and its institutions." 70 The right to revoke unilaterally the notification enables the withdrawing state "to unilaterally stop the withdrawal process and subsequently trigger it again with a fresh notification, thereby clearly circumventing the need for unanimity in the European Council for the prolongation of the two-year period foreseen in Article 50(3) TEU". 71 Thanks to the right to revoke unilaterally its notification, recognised in Wightman, a withdrawing member state might now bypass the autonomous rules of Article 50 TEU, which should have ensured an "orderly" withdrawal from the European Union. The UK did not revoke its notification, but some British politicians supported the use of a "tactical revocation". [START_REF] For | then, you can invoke it again once you've made sufficient process[END_REF] It is perhaps "extremely difficult for tactical revocations to proliferate", as the Advocate General suggests, [START_REF][END_REF] but it is by no means impossible.

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Conclusion

Article 50 TEU is a complex provision, which has its roots in international law and in the EU legal order. This provision underlines the sovereign right to withdrawal of the member states and, in this respect, is not different from similar provisions in the foundational treaties of other international organisations. Article 50 TEU nonetheless provides for a unique withdrawal procedure that embodies the Union's autonomy. The two sides of Article 50 TEU thus mirror the ambivalent nature of the European Union, part international organisation, part federation.

In the Wightman judgment, the Court of Justice acknowledged the Janus-faced nature of Article 50 TEU but underlined the international roots of this provision. 74 This choice might be understandable from a political perspective, as it gave a positive message to the public opinion of the United Kingdom: the EU is not a prison or an empire, as the "leavers" claim, but an organisation based on the sovereignty of the member states. The decision to withdraw from the Union (or not) is for the member states' alone to take. However, one might doubt whether the Court stroke the right balance between national sovereignty and EU autonomy in Wightman. The Lisbon Treaty has introduced an original procedure with a specific purpose: enabling orderly withdrawals from the Union. By reading Article 50 TEU in light of international law, the Court arguably undermined the EU's autonomy -the same autonomy it has buttressed for decades 75 -and paved the way for abuses of the withdrawal procedure. Ten years after the Lisbon Treaty, an orderly withdrawal of the member states can no longer be taken for granted.

  Article 50 TEU To interpret Article 50 TEU, one may consider the preparatory works that generated it, i.e. the works of the Convention on the Future of Europe (European Convention), which operated between early 2002 and July 2003. The Praesidium of the European Convention presented draft Article 46 (later renumbered Article 59 of the draft Constitution, Article I-60 of the European Constitution, and eventually Article 50 TEU) in April 2004. 7 The Praesidium's comments annexed to the proposed text of draft Article 46 suggest that this provision contains two elements.

  Some elements of Article 65 of the Vienna Convention on the Law of Treaties seem to reflect customary See Art. 65(2) and (3) of the Vienna Convention on the Law of Treaties. See also Art. 33 of the UN Charter.

	35 See above, text to n 19.
	36 Judgment of 16 June 1998, C-162/96, Racke, EU:C:1998:293, para. 59; see also AG Opinion Wightman (n
	1), para. 75.

37

rules, see M. BUSCEMI and L. MAROTTI, "Obblighi procedurali e conseguenze del recesso dai trattati: quale rilevanza della Convenzione di Vienna nella prassi recente?" (2019) Rivista di diritto internazionale 939.

38

Agreement Establishing the World Trade Organization, 15 April 1994, Art. XV; Constitution of the International Labour Organisation, 1 April 1919, Art. 1(5); v. amplius Henry G. Schermers, Niels M. Blokker, International Institutional Law (Brill 2011) 99-100. 39

Judgment of 15 July 1964, Case 6-64, Costa, EU:C:1964:66.

Judgment of 9 March 1978, Case 106/77, Simmenthal, EU:C:1978:49, para. 18.

See e.g. F. X. PRIOLLAUD and D. SIRITZKY, La Constitution européenne -texte et commentaires (La documentation française 2005) 157.

See e.g. A. VAHLAS, "Appartenance à l'Union européenne", in V. CONSTANTINESCO, Y. GAUTIER, and V. MICHEL (ed.), Le Traité établissant une Constitution pour l'Europe (Presses universitaires de Strasbourg 2005) 267-270.

M. VILLIGER, Commentary on the 1969 Vienna Convention on the Law of Treaties (Brill

2009) 705.20 In addition, the member states could possibly invoke a "fundamental change of circumstances", which justified their withdrawal, as long as that those circumstances constituted an essential basis of their consent to be bound and the effect of the change was radically to transform the extent of obligations to be performed under the EC Treaty, see Art. 62 of the Vienna Convention on the Law of Treaties.

K. KILJUNEN and M. VANHANEN, Suggestion for amendment of Article 46: (Title X : Union membership, part I of the Constitution).

T. TIILIKAINEN, A. PELTOMÄKI, R. KORHONEN, Suggestion for amendment of Article 46: (Title X : Union membership, part I of the Constitution).

See, to that effect, CASOLARI (n 41) 1012; A. CUYVERS, "Wightman, Brexit, and the sovereign right to remain" (2019) Common Market L Rev 1320-1321.

See e.g. Opinion 1/91 of 14 December 1991, EU:C:1991:490; Opinion 1/17 of 30 April 2019, EU:C:2019:341.


Sareh Mahmoudian Moghaddam

Dr Blandine Quelennec

Dr Nicolas Delpouve
email: nicolas.delpouve1@univ-rouen.fr

Dr Bienvenu Atawa

Prof Laurent A Delbreilh

Allisson Saiter

Elisa Passaglia

Dr Stefano Fiori

Allisson Saiter-Fourcin

Saiter-Fourcin

Prof E Passaglia

Fragility of short-chain poly(lactic acid)s derivatives by combining dielectric spectroscopy and fast scanning calorimetry

Keywords: DSC, cooperativity, glass transition, relaxation, PLA, oligomers

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

Introduction

Oligomeric materials represent a growing area of research, [1] in which polymer chemistry is fed by organic synthesis to enable the design of new material properties. Processing biopolymers such as poly (lactic acid) (PLA) with their oligomeric homologues constitutes an interesting way to tune their macroscopic properties, while keeping their environmental friendly character.

PLA brittleness is among the main drawbacks limiting its industrial use. Although flexibility improvement was demonstrated from the addition of many plasticizers, the research for efficient additives is still intensive to answer other requirements like decent barrier properties and biocompatibility. [2][3] For the latter, lactic acid oligomers (OLAs), which were characterized by Cicogna et al. [4] in terms of structure, thermal features, and biodegradability once mixed with PLA, showed promises. In the current study, the investigation of OLAs molecular mobility at the glass transition is motivated by its connection with the free volume, [5] an essential structural criterion to understand barrier properties.

The calorimetric signature of the glass transition in PLA and OLAs is easily obtained from modulated temperature differential scanning calorimetry (MT-DSC). To characterize the molecular mobility, one can use the concept of fragility index m introduced by Angell: [6] m depicts how abrupt is the increase of relaxation time  when a polymer is cooled down to the glass transition. An abrupt increase of , which leads to high values of m, is characteristic of fragile polymers, by opposition to strong ones.

𝑚

= 𝑑𝑙𝑜𝑔 10 𝜏 𝑑 𝑇/𝑇 𝑔 𝑇=𝑇 𝑔 (1)
Often, dielectric relaxation spectroscopy (DRS) analyses are performed to plot the temperature dependence of , so to obtain m. Moreover, it is possible to extend the window in which the molecular mobility can be probed by combining DRS with calorimetric techniques, such as fast scanning calorimetry (FSC), and using the concept of fictive temperature Tf introduced by

Tool. [7] Tf depicts how far a polymer in the glassy state is from equilibrium. When the polymer is non-aged, Tf is renamed limiting fictive temperature Tf' and directly depends on the cooling rate |c|. This dependence can be adjusted, at least on a limited window, [8][9] by Vogel-Fulcher-Tammann (VFT) equation ( 2), similarly to the temperature dependence of . To improve m estimation accuracy, data issued from FSC can therefore be superimposed to the DRS results, using the same VFT parameters: pre-exponential factor 0, strength parameter D and Vogel temperature TV.

𝜏 = 𝜏 0 𝑒𝑥𝑝 𝐷𝑇 𝑉 𝑇 -𝑇 𝑉 (2)
The plasticizing efficiency of oligomers increases with the decrease of their chain length, presumably because the free volume increases. Thus, the idea of this study is to combine FSC and DRS to provide a very accurate description of the molecular mobility at the glass transition for amorphous PLA and OLAs, and then to estimate the size of the cooperative rearranging regions related to the free volume, quantity directly linked to the barrier properties. [5,10] The characteristic cooperativity volume  3 T at the dynamic glass transition temperature T has been calculated from MT-DSC data by applying Donth's equation: [11]

𝜉 𝑇 𝛼 3 = 1 𝐶𝑝 𝑔𝑙𝑎𝑠𝑠 𝑇𝛼 - 1 𝐶𝑝 𝑙𝑖𝑞𝑢𝑖𝑑 𝑇𝛼 𝑘 𝐵 𝑇 𝛼 2 𝜌 𝛿𝑇 2 (3)
Where T is the mean temperature fluctuation related to the dynamic glass transition of one CRR, kB is the Boltzmann constant,  is the density and Cp is the heat capacity at constant pressure.

Experimental

Oligomers of D-/L-lactic acid (Glyplast® OLAs, code OLAMw, where Mw is the weight average molecular mass in g/mol, being equal to 1400, 1900, or 2700) were synthesized and supplied by Condensia Química SA®. [4] Mw values were obtained by size exclusion chromatography (Agilent Technologies 1200 Series instrument equipped with a degasser, an isocratic HPLC pump, a refractive index (RI) detector, and two PLgel 5 μm MiniMIX-D columns), [4] and did not change after thermal treatment. PLA pellets were provided by Natureworks® (grade 4042D having 4.3% D-, and 95.7% L-lactic acid, named PDLLA in the rest of the manuscript), with Mw = 188 kg/mol. Films were obtained by applying 2 bars pressure on melted pellets, initially dried at 50 °C for several hours, under steel plates of Specac® manual hydraulic press. Amorphous samples were obtained by quenching from the melt, in liquid nitrogen for OLAs, in cold water for PDLLA.

MT-DSC experiments were performed at 2 K/min under 50 mL/min nitrogen flow gas in a TA® DSC Q2000 apparatus equipped with a refrigerating cooling system. The calibration was performed using indium for temperature and energy, and using sapphire for heat capacity. The modulation parameters were 60s period and ± 0.318K amplitude. In line with previous works, the deconvolution procedure, [START_REF] Lacey | Modulated Temperature Differential Scanning Calorimetry[END_REF] and phase lag correction, [START_REF] Weyer | [END_REF] were done. DRS analyses were performed using an Alpha Analyzer from Novocontrol® technologies. Spectra were collected from -150 up to 100 °C in a frequency range of 10 -1 -2*10 6 Hz. PDLLA film was placed between 30 mm gold parallel electrodes. OLAs were placed on interdigitated electrodes (20 mm sensor diameter and 0.001 loss factor accuracy). DRS responses were analyzed using the Havriliak-Negami complex function.

FSC experiments were performed in Mettler Toledo® Flash DSC1 apparatus equipped with a Huber® TC100 intracooler. Each MultiSTAR UFS 1 MEMS® chip sensor was conditioned before the low thickness sample was positioned on it. The static thermal lag was corrected from the response of a gallium standard. The dynamic thermal lag was corrected as described elsewhere. [8] The sample mass was estimated from the amplitude of the glass transition, by assuming that the heat capacity step of the amorphous system was the same than the one determined from MT-DSC. [8] The atmosphere was continuously flushed with nitrogen (20 mL/min).

Results and discussion

Figure 1 shows the MT-DSC responses of amorphous PDLLA and OLAs recorded in the glass transition range. From MT-DSC, the observation of the glass transition is facilitated. One can separate the kinetic events which sometimes occur in its temperature domain from its calorimetric signature itself, i.e., the heat capacity increment, which manifests in the reversing heat capacity signal. The results are presented in Table 1. As expected, the glass transition Tg shifts to lower temperatures with the decrease of the chain length. The heat capacity step Cp at Tg slightly increases from 0.51 (PDLLA) up to 0.60 (OLA1400) J/(g.K). In Figure 2 the dielectric response of the imaginary part of complex permittivity noted '' is presented for all the samples in the glass transition domain, at various temperatures, as a function of frequency. For each temperature, the slope in the low frequency domain is due to the conductivity, whereas the signature of the glass transition is recorded at high frequency. As expected, this signature shifts to higher frequencies with increasing temperature. Finally, one can observe a shouldering close to the conductivity slope, likely due to a slower relaxation mode. Figure 3 shows the analytical fitting procedure of OLA2700 dielectric response '' at 299 K.

MT-DSC FSC + DRS

Tg (K) Cp (J/(g.K)) log C log (0) (s -1 )] D TV (K) Tg ( = 100s) (K)
The relaxation time  was extracted using a conductivity  term and two Havriliak-Negami contributions. The low frequency contribution (in red) might be the signature of the normalmode, as reported for PLA, [14][15] related to the fluctuation and orientation of the end-to-end chain, [16] but this is not ascertained and goes beyond the scope of the study. The high frequency contribution (in blue), that is the contribution of interest in this study, from which  is obtained, is associated with the main relaxation of polymer chains, or -relaxation. The general expression for the Havriliak-Negami complex function is given below.

𝜀 * 𝜔 = -𝑖 𝜎 𝜔𝜀 0 𝑛 + ∆𝜀 𝑘 1 + 𝑖𝜔𝜏 𝑘 𝛼 𝑘 𝛽 𝑘 + 𝜀 ∞ 𝑘 (4)
Where  is the angular pulsation, k is the number of contributions for the fitting procedure (equal to 2 in the present study), n is a fitting parameter related to the conductivity slope (black dashed line),  is the dielectric strength,  and  are the symmetric and asymmetric broadenings of the complex dielectric function respectively, ∞ is the permittivity at high frequency of the relaxation process, 0 is the vacuum permittivity. The procedure is extended to the measurements performed at different temperatures in the glass transition range to obtain the temperature dependence of the -relaxation time. The results are presented in Figure 4 showing the temperature dependence for all the relaxation processes detected from DRS measurements. The DRS signatures of local relaxations are given in Supporting Information (Figure SI1). One can notice first that PDLLA and OLAs exhibit different local relaxation processes. In PDLLA, two relaxations are recorded in the same temperature range. On the other hand, OLAs undergo a common fast local process at temperatures situated farther below. Besides, the detection of a slow local process is more difficult. It has only been recorded for OLA2700 and OLA1900, still on a very narrow temperature range. Furthermore, the temperature dependences of both -relaxation and low frequency process are obviously steeper for PDLLA than those of oligomers.

Figure 5 shows FSC curves of PDLLA and OLAs. The analyses were performed at constant heating rate h =1500 K/s for various |c|. The glass transition appears as an endothermic step with a superimposed recovery peak used to calculate the dependence of Tf' with |c|. By decreasing the cooling rate, the peak increases in amplitude and shifts towards higher temperatures. The area matching method, [17] which allows calculating the limiting fictive temperature Tf' is presented in Supporting Information (Figure SI2). Data from FSC and DRS are plotted in Figure 6 and superimposed by using Frenkel-Kobeko-Reiner (FKR) equation ( 5), using same D, 0 and TV fit parameters, and applying a simple constant logarithmic shift log C (Fitting parameters given in Table 1). Tg ( = 100s) is the glass transition temperature arbitrary defined as the temperature for which  and the experimental time scale are of the same order, i.e., about 100s. (ranging from 0.01 K/s up to 1500 K/s) at constant heating rate h =1500 K/s. The sample masses have been estimating by comparing the Cp step values obtained by MT-DSC and FSC. [8] MT-DSC results are also added. It is worth mentioning that the validity of this approach can be questioned since a decoupling between vitrification and molecular mobility has recently been evidenced. Nevertheless, as can be seen in Figure 6, dielectric and calorimetric data superimpose well in the present study. In consistence with previous studies, [5] PDLLA exhibits a fragile behavior (m = 155). On the other hand, oligomers exhibit lower m, which decreases from 99 (OLA2700) down to 68 (OLA1400) with the decrease of Mw. The dependence of the segmental dynamics on the chain length has been the focus of several studies. [20][21][22][23][24] A correlation between the fragility index and the chain length is frequently observed. On the other hand, some systems show no dependence of the fragility index on the molecular weight. [20] According to Santangelo and Roland, [21] only polymers which exhibit a high cooperativity length at the glass transition, that is the case of PDLLA, [25] show a significant decrease of m with the decrease of Mw. weight. This result might be linked to their chemical structure. They carry an end-chain group that increases their plasticizing efficiency. [4] Determining the influence of this group on the segmental dynamics will require further investigations. .

𝑙𝑜𝑔( 𝛽 𝑐 ) = 𝑙𝑜𝑔( 1 𝜏 ) + 𝑙𝑜𝑔 𝐶 = 𝑙𝑜𝑔 1 𝜏 0 - 𝐷𝑇 𝑉 ln 10(𝑇 -𝑇 𝑉 ) + 𝑙𝑜𝑔 𝐶 (5)

Conclusion

The tracking of the segmental relaxation dynamics by combining calorimetric and dielectric investigations has provided a relaxation map from which the fragility has been estimated without much uncertainty. The global agreement between cooperativity and fragility variations is sufficiently convincing to interpret the difference of segmental dynamics between the lactic acid oligomers and poly (lactic acid) by a fall of cooperativity with the reducing of the chain length. This assumption is not fully ascertained when it comes to explaining the variations of fragility among oligomers only, since the cooperativity volume is about the same. Free volume measurements would therefore be of interest to complement the results of this study.

Sareh Mahmoudian

Figure 1 :

 1 Figure 1: MT-DSC reversing heat capacity signals recorded in the glass transition range for amorphous

Figure 2 :

 2 Figure 2: Frequency dependence of the imaginary part of complex permittivity '' at several

Figure 3 :

 3 Figure 3: Fitting procedure used to investigate the segmental relaxation region with a conductivity

Figure 4 :

 4 Figure 4: Relaxation map including all the relaxation processes recorded from DRS measurements for

Figure 5 :

 5 Figure 5: FSC normalized heat flow of amorphous PDLLA and OLAs for various cooling rate |c|

Figure 6 :In Figure 7 ,

 67 Figure 6: Temperature dependence of the relaxation rate (1/max) (left coordinate) and cooling rate |c|

Figure 7 :

 7 Figure 7: Fragility index as a function of the cooperativity volume for PDLLA and OLAs. Additional

  Moghaddam, Blandine Quelennec, Nicolas Delpouve * , Bienvenu Atawa, Elisa Passaglia, Stefano Fiori, Allisson Saiter-Fourcin, Laurent Delbreilh Fragility of Short-Chain Poly(lactic acid)s Derivatives by Combining Dielectric Spectroscopy and Fast Scanning Calorimetry Table of contents: Segmental relaxation dynamics of lactic acid oligomers depend on the molecular weight, as evidenced from the spectacular variations regarding the fragility index that has been calculated by combining the temperature dependence of the relaxation time (dielectric measurements) with the cooling rate dependence of the limiting fictive temperature (calorimetric measurements). The investigated systems are also characterized by a low cooperativity at the glass transition. As shown in Figure SI.1, PDLLA exhibits two local relaxation processes in close temperature domains. OLAs exhibit a different behaviour. All share the same fast local relaxation which occurs between 128 and 158K, farther below local relaxations of PDLLA. A slower process can be clearly observed only for OLA1900 and OLA2700. This one is recorded between 233 and 248K.

Figure SI. 1 :

 1 Figure SI.1: DRS signatures of local relaxation processes recorded for PDLLA and OLAs.

Figure SI3 :

 SI3 Figure SI3: MT-DSC reversing heat capacity signal recorded in the glass transition range for

Table 1 :

 1 Glass transition characteristic parameters for PDLLA and OLAs from calorimetric and dielectric measurements.

((Supporting Information can be included here using this template))

Supporting Information

Fragility of Short-Chain Poly(lactic acid)s Derivatives by Combining Dielectric

In Figure SI.2 the area matching method [1] is presented, which allows calculating the limiting fictive temperature Tf' from the following equation:

Where Cp,g and Cp,l are the heat capacity of the glass and the liquid respectively, Cp is the heat capacity of the material, T1 and T2 are two arbitrary temperatures, chosen in the glass and the liquid like states respectively, in such a way that Cp = Cp,g at T1 and Cp = Cp,l at T2. In this study, the signal used to apply this method was the FSC heat flow, having been normalized to the heating rate h and the sample mass, to be expressed in unit equivalent to the heat capacity one. The procedure is extended to the measurements performed at different cooling rates |c| to obtain the cooling rate dependence of the limiting fictive temperature.


Emmanuel Motheau
email: emmanuel.motheau@cerfacs.fr

Franck Nicoud
email: franck.nicoud@univ-montp2.fr

Thierry Poinsot
email: thierry.poinsot@cerfacs.fr

USING BOUNDARY CONDITIONS TO ACCOUNT FOR MEAN FLOW EFFECTS IN A ZERO MACH NUMBER ACOUSTIC SOLVER

The present study is devoted to the modeling of mean flow effects while computing thermoacoustic modes under the zero Mach number assumption. It is first recalled that the acoustic impedance modelling a compressor or a turbine must be prescribed under an energetical form instead of the classical acoustic variables. Then we demonstrate the feasibility to take into account the coupling between acoustic and entropy waves in a zero Mach number framework to capture a family of low frequency entropic modes. The proposed approach relies on a new Delayed Entropy Coupled Boundary Condition (DECBC) and proves able to capture a family of Low frequency entropic mode even though no mean flow term is included into the fluctuating pressure equation.

NOMENCLATURE

A + , A -Amplitudes of forward and backward acoustic waves Cp Heat capacity per mass unit at fixed pressure Cv Heat capacity per mass unit at fixed volume J Total enthalpy M Mach number R Reflection coefficient Z Complex impedance c Sound celerity k Wave number m Mass flow rate p Static pressure

Introduction

Lean premixed combustor systems in aero-engines are promising devices to meet the future NOx emission reduction requirements. However, they are also more prone to combustion instabilities compared to classical combustors [START_REF] Lieuwen | Combustion instabilities in gas turbine engines. operational experience, fundamental mechanisms and modeling[END_REF]. These oscillations of the flow may lead to many undesirable effects, such as large-amplitude structural vibrations, flame flashback or blowoff, or an abnormally high temperature of the wall of the combustor. In the best case, the consequences would be a simple loss of performance or a premature fatigue of materials. In the worst case, a spectacular destruction of the system can occur. Consequently, there is a need to better understand combustion instabilities and to be able to predict them at the design level.

Several methods have been developed to study these instabilities, such as low-order methods [START_REF] Dowling | The calculation of thermoacoustic oscillations[END_REF][START_REF] Polifke | Reconstruction of acoustic transfer matrices by instationnary computational fluid dynamics[END_REF] which solve only the unidimensional acoustic problems, or at the opposite the Large Eddy Simulation (LES) which solves numerically the complete flow thanks to the reactive Navier-Stokes equations. As an intermediate, the problem complexity can be reduced by the derivation of the Linearized Euler Equations (LEE) which neglect viscous effects, or by the derivation of the Helmholtz equation when neglecting the mean flow motion. Although this last method may be a crude simplification of the problem, this approach combined with LES proved useful to better understand the structure and nature of the instabilities observed in academic or industrial burners [START_REF] Selle | Joint use of compressible large-eddy simulation and helmoltz solvers for the analysis of rotating modes in an industrial swirled burner[END_REF].

Although it is tempting to restrict the study of the behaviour of thermo-acoustic instabilities to only the combustion chamber, it is however important to take into account the proper acoustic environment of the gas turbine, as for example the presence of a compressor or a high pressure distributor. Since it is difficult to simulate the whole configuration, a possibility is to reduce the computational domain to only the flame tube where the mean Mach number is approximatively zero (M = 0) while the compressor and the turbine are modelled by prescribing appropriate complex valued impedances at the boundaries.

At least two difficulties arise when using a Helmholtz solver (thus assuming M = 0) for the combustion chamber and boundary impedances to represent the upstream/downstream environment:

• In order to match the zero Mach number assumption used to formulate the thermo-acoustic problem, the impedance should be defined in theory in a section of the combustion chamber where the Mach number is very small. In practice, the location of the outlet section is often imposed by geometrical considerations and the effective Mach number at the boundary is not necessarily very small, • Three modes of fluctuations can perturb a steady baseline flow [START_REF] Lieuwen | Modeling premixed combustionacoustic wave interactions: A review[END_REF]: acoustic perturbations propagates at the speed of sound augmented by the local mean velocity while vortical and entropy perturbations are simply convected by the mean flow. As a consequence, the zero Mach number assumption necessarily neglect the convection of entropy or vorticity spots to the downstream. As shown by many authors [START_REF] Morfey | Amplification of aerodynamic noise by convected flow inhomogeneities[END_REF][START_REF] Williams | The generation of sound by density inhomogeneities in low mach number nozzle flows[END_REF][START_REF] Howe | Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute[END_REF][START_REF] Candel | Acoustic conservation principles, application to plane and modal propagation in nozzles and diffusers[END_REF], entropy inhomogeneities generate acoustic waves when accelerated in a mean flow pressure gradient, which is the case when the combustor opens onto a high pressure distributor. Acoustic waves transmitted through the distributor generate indirect combustion noise, while acoustic waves travelling back to the flame may generate a low frequency resonant mode called rumble [START_REF] Polifke | Constructive and Destructive Interference of Acoustic and Entropy Waves in a Premixed Combustor with a Choked Exit[END_REF][START_REF] Bloxsidge | Active control of reheat buzz[END_REF].

It has been demonstrated in a previous study [START_REF] Nicoud | About the zero mach number assumption in the calculation of thermoacoustic instabilitie[END_REF] that the zero Mach number assumption for the mean flow can lead to significant errors both in the prediction of the frequency and the growth rate of the thermo-acoustic mode, suggesting that the two points described above are of major importance. Since accounting for the non zero Mach number terms in the equations leads to a drastic increase of the problem complexity [START_REF] Rao | Use of finite element methods in frequency domain aeroacoustics[END_REF], the aim of this paper is to investigate an alternative approach where the thermo-acoustic problem in the domain is still solved under the zero Mach number assumption and the coupling with the convec-tive quantities is modelled through an appropriate downstream boundary condition.

The basic equations are recalled in section 1 while in section 2 the proper way to couple a Helmholtz solver with an analytical and/or numerical tool for the computation of the upstream/downstream boundary impedance is discussed. In section 3 we demonstrate the feasibility to take into account the coupling between acoustic and entropy waves in a zero Mach number framework and to capture a family of low frequency entropic modes. The analysis focuses on the configuration of a simple tube containing a flame and where a shocked nozzle is located downstream.

Mathematical formalism

This study is conducted in the frequency domain, i.e. g ′ (x,t) = ℜ ĝ(x)e -jωt for any fluctuating quantity; it also focuses on unidimensional configurations so that only purely longitudinal acoustic and entropy convection effects are present. Consequently, vorticity perturbations and their interactions with the acoustics are neglected in the remainder of the whole study.

Linearized Euler Equations for the eigenvalue problem

For an homogeneous reacting mixture with constant heat capacities C p and C v , the linearised harmonic form of conservation equations for mass, momentum and energy in a quasi-1D domain of cross section area S(x) read respectively:

û ∂ ρ 0 ∂ x + ρ 0 ∂ û ∂ x + u 0 ∂ ρ ∂ x + ρ ∂ u 0 ∂ x + ρ 0 û + ρu 0 S ∂ S ∂ x = jω ρ (1) 1 ρ 0 ∂ p ∂ x + û ∂ u 0 ∂ x + u 0 ∂ û ∂ x + u 0 ρ ρ 0 ∂ u 0 ∂ x = jω û (2) û ∂ s 0 ∂ x + u 0 ∂ ŝ ∂ x + rq 0 p 2 0 p - r p 0 q = jω ŝ (3)
with the linearized state equation and entropy expression:

p p 0 - ρ ρ 0 - T T 0 = 0 and ŝ C v = p p 0 -γ ρ ρ 0 (4)
Assuming that the unsteady heat release amplitude q is modeled as a linear operator, i.e. q = q ρ ρ + q û û + q ŝ ŝ, Eqs. 1-3 define an eigenvalue problem M W = jωW where M is :

    ∂ u 0 ∂ x + u 0 ∂ ∂ x + u 0 S ∂ S ∂ x ∂ ρ 0 ∂ x + ρ 0 ∂ ∂ x + ρ 0 S ∂ S ∂ x 0 1 ρ 0 ∂ c 2 0 ∂ x + u 0 ρ 0 ∂ u 0 ∂ x + c 2 0 ρ 0 ∂ ∂ x ∂ u 0 ∂ x + u 0 ∂ ∂ x T ( 1 p 0 ∂ p 0 ∂ x + ∂ ∂ x ) γrq 0 ρ 0 p 0 -r p 0 q ρ ∂ s 0 ∂ x -r p 0 q û u 0 ∂ ∂ x + T T 0 q 0 p 0 -r p 0 q ŝ     (5)
and (ω, W ) the eigenpair, the eigenvector being W = ( ρ, û, ŝ) T , S the varying area section and T = (γ -1)T 0 .

Plane waves in an homogeneous media

In the case of a 1D domain with homogeneous baseline flow, the following expressions can be obtained [START_REF] Davies | Practical flow duct acoustics[END_REF] for the complex amplitudes:

p(x) = A + e jk + x + A -e -jk -x (6) û(x) = 1 ρ 0 c 0 [A + e jk + x -A -e -jk -x ] (7) ŝ(x) = σ e jk s x (8)
with

k + = ω c 0 + u 0 = k 1 + M k -= ω c 0 -u 0 = k 1 -M (9)
k s = k M (10)
where A + and A -are the amplitudes of the forward and backward propagating acoustic waves, respectively, σ the amplitude of the forward propagating entropic wave, k = ω/c 0 is the acoustic wave number and M the Mach number.

It should be noted that Eq. 8 represents an idealistic situation where the spatial diffusion of entropy spots is neglected. Although this situation is not supported by previous studies [START_REF] Sattelmayer | Influence of the combustor aerodynamics on combustion instabilities from equivalence ratio fluctuations[END_REF], it is judged appropriate for this proof of concept study.

Injecting the above expression of entropy into Eq. 4 leads to

ρ = p c 2 0 - ρ 0 σ
C p e jk s x and T

T 0 = γ -1 ρ 0 c 2 0 p + σ C p e jk s x (11)
Eqs. 6, 7 and 11 describe the harmonic small amplitude perturbations in the domain. Relations between complex valued amplitudes A + , A -and σ are prescribed by boundary conditions and jump relations through the interfaces.

Influence of the boundary impedance on thermoacoustic eigenfrequencies

When studying the acoustic behaviour of a combustor by computing the eigenmodes thanks to a Helmholtz solver, appropriate boundary conditions must be used in order to represent the acoustic environment. Upstream/downstream impedances can be deduced from transfer functions describing the response of acoustic elements to acoustic or entropic perturbations, either analytically under the compact hypothesis [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF] or numerically by solving the LEEs [START_REF] Lamarque | Boundary conditions for acoustic eigenmodes computation in gas turbine combustion chambers[END_REF]. The impedance should be defined in a section of the combustion chamber where the Mach number is very small, consistently with the zero Mach number assumption which is valid when M ≪ L f /L a , where L f is the thickness of the reaction zone and L a the typical acoustic wavelength. Since this is not always the case in practice, there is a need to quantify the errors inherent to this practice when the outlet/inlet Mach number is not very small.

The configuration considered is shown in Fig. 1.a. It consists in two connected tubes of section S 1 and S 2 . The subscripts 1 and 2 refer to parameters in the left and right tube, respectively. Only isentropic fluctuations are considered (the inlet boundary condition is s ′ = 0 and there is no entropy source within the flow domain).

The eigenmodes of this simple configuration are computed by four different methods to assess the errors made when the boundary Mach number increases and deduce the proper way to prescribe boundary impedance.

Method A0: Numerical results from the LEE's

The LEEs presented in section 1.1 are solved numerically so as to provide a reference solution accounting for mean flow effects and the complete geometry [START_REF] Nicoud | About the zero mach number assumption in the calculation of thermoacoustic instabilitie[END_REF].

Method A1: Global acoustic model

The forward and backward waves travelling in both tubes lead to four unknown A + 1 , A - 1 , A + 2 and A - 2 . These waves are solutions of an homogeneous linear system of equations obtained by requiring that the boundary conditions and jump relations are fulfilled. The two boundary conditions form a first set of relations between waves amplitudes. A constant mass-flow rate m ′ 1 = 0 is imposed at the intlet x = 0 of the domain, while the outlet at x = L is modelled by an acoustic open-end, so that p ′ 2 = 0. Introducing the plane wave decomposition of pressure and velocity fluctuations leads to:

A + 1 (1 + M 1 ) -A - 1 (1 -M 1 ) = 0 at x = 0 (12) A + 2 e jk + 2 L + A - 2 e -jk - 2 L = 0 at x = L (13)
As explained in [START_REF] Davies | Practical flow duct acoustics[END_REF] the jump relations for the section change located at x = l can be expressed by integrating over an infinitesimal control volume the conservation of mass (Eq. 1) where the time averaged mass flux term has been substracted, and energy (Eq. 3) as long as the specific stagnation enthalpy within the volume remains invariant. The following set of equations is then obtained :

(1 + M 1 )A + 1 e jk + 1 l -(1 -M 1 )A - 1 e -jk - 1 l S 1 c 1 = [(1 + M 2 )A + 2 e jk + 2 l -(1 -M 2 )A - 2 e -jk - 2 l S 2 c 2 (14)
(1 + M 1 )A + 1 e jk + 1 l + (1 -M 1 )A - 1 e -jk - 1 l 1 ρ 1 = [(1 + M 2 )A + 2 e jk + 2 l + (1 -M 2 )A - 2 e -jk - 2 l 1 ρ 2 (15)
One obtains the algebraic system M W = 0 where M is defined as :

     M + 1 -M - 1 0 0 0 0 e jk + 2 L e -jk - 2 L S 1 c 1 M + 1 e jk + 1 l -S 1 c 1 M - 1 e -jk - 1 l -S 2 c 2 M + 2 e jk + 2 l S 2 c 2 M - 2 e -jk - 2 l M + 1 ρ 2 ρ 1 e jk + 1 l M - 1 ρ 2 ρ 1 e -jk - 1 l -M + 2 e jk + 2 l -M - 2 e -jk - 2 l      (16)
where

M + 1 = (1 + M 1 ), M - 1 = (1 -M 1 ), M + 2 = (1 + M 2 ), M - 2 = (1 -M 2 ) and W = (A + 1 , A - 1 , A + 2 , A - 2 )
T . The dispersion relation is then obtained by requiring the matrix M to be singular, producing the solution of the acoustical problem.

Method A2: modeling of the section change by an impedance

As shown in Fig. 1.b the computational domain is reduced to the region between the inlet x = 0 where the boundary condition is still m ′ 1 = 0 and x = l z < l where the impedance Z ac is imposed. The algebraic system described at Eq. 16 reduces to:

(1 + M 1 ) -(1 -M 1 ) (1 -Z ac )e jk + 1 l Z (1 + Z ac )e -jk - 1 l Z A + 1 A - 1 = 0 (17)
The impedance Z ac is defined by

Z ac = 1 + R ac 1 -R ac (18)
where R ac = A - 1 /A + 1 is the coefficient of reflection defined as the ratio between the reflected and the incident wave. The transfer function of the modeled part of the domain (l z < x < L) can be found by setting A + 1 to unity so as to express A - 1 . After cumbersome but straightforward algebra, Eq. 16 leads to:

R ac = e 2 jk 1 (l-l Z ) 1-M 2 1 1 + M 1 1 -M 1 A + B A -B (19)
with

A = S 1 c 1 (1 -M 2 )e 2 jk 2 (L-l Z ) 1-M 2 2 -(1 + M 2 )e 2 jk 2 (l-l Z ) 1-M 2 2 (20) B = S 2 c 2 ρ 2 ρ 1 (1 -M 2 )e 2 jk 2 (L-l Z ) 1-M 2 2 + (1 + M 2 )e 2 jk 2 (l-l Z ) 1-M 2 2 (21)

Method A3: zero Mach number domain

Method A3 is similar to method A2 as described above. The only difference is the use in the computational domain of a wave number k derived under the zero Mach number assumption instead of k ± and M 1 is set to zero in Eq. 17. The relation A +

1 -A - 1 = 0 is now imposed at x = 0. Moreover, the expression of the impedance Z ac and the reflection coefficient R ac remains unchanged. Thus, Eq. 17 becomes :

-1.377i 1 -1 (1 -Z ac )e jk 1 l Z (1 + Z ac )e -jk 1 l Z A + 1 A - 1 = 0 (22)

Results

Thermodynamic and geometrical parameters used for the computation of eigenmodes are presented in Table 1. The temperature T 1 and pressure P 1 are imposed to the computation, while these parameters with the subscript 2 are deduced from the steady mean flow equations.

Results are presented in Table 2. As expected, comparisons between method A0 and the global acoustical model (method A1) give very satisfying results. Real and imaginary parts of the frequency of the first eigenmode are both very close when computed at different inlet Mach number M 1 . The small but existing difference can be explained by the fact that the length of the section change between S 1 and S 2 is not zero in the LEE solver but close to x = 0.5 mm.

Comparisons between methods A1 and A2 prove that truncating the computational domain while modeling the unresolved part through an impedance gives exactly the same results. Based on this observation, the influence of the zero Mach number assumption can be isolated. Computations with method A3 are chosen so as to represent the extreme case where 98% of the domain is solved with a zero mean flow. It is obvious that in such configuration the real part of the frequency f M computed with Mach number effects will be very close to the one ( f 0 ) computed at very low Mach number. As shown in [START_REF] Nicoud | About the zero mach number assumption in the calculation of thermoacoustic instabilitie[END_REF] a relation between frequencies can be found, so that f M ≈ (1 -M 2 ) f 0 . Verification of this relation with results of Table 2 is rather straightforward. The imaginary parts computed from method A3 suggest that energy is entering into the domain while the Mach number increases. This observation is in contradiction with results from the reference methods which show that the first mode is always damped ( f i < 0). The solid line with star symbols on Fig. 2 shows the modulus evolution of the reflection coefficient for different inlet Mach numbers M 1 (note that |R ac | does not depend on the frequency according to Eq. 19). It is obvious that non-physical energy growth is generated because the ratio between the reflected wave and the incoming wave is superior to unity. As explained in [START_REF] Peters | Damping and reflection coefficient measurements for an open pipe at low mach and low helmholtz numbers[END_REF], |R ac | can exceed unity at a critical Strouhal number where vortical energy is generated downstream of an abrupt change in the geometry of a pipe. As vorticity is neglected in the present study, another explanation must be provided.

Boundary conditions: from a non stationary flow formalism to a zero Mach number formulation

As shown by many authors [START_REF] Morfey | Amplification of aerodynamic noise by convected flow inhomogeneities[END_REF][START_REF] Cantrell | Interaction between sound and flow in acoustic cavities: Mass, momentum, and energy considerations[END_REF], the classical derivation of acoustic disturbances must be reformulated to take into account the convection of mechanical work by the mean flow. The starting point is to find an appropriate expression of the energy flux through a boundary [START_REF] Candel | Acoustic conservation principles, application to plane and modal propagation in nozzles and diffusers[END_REF], using fluctuating stagnation enthalpy J ′ and mass flow rate m ′ as state variables. A mathematical proof of the accuracy of such a choice for generalized acoustic field may be found in [START_REF] Doak | Fluctuating total enthalpy as the basic generalized acoustic field[END_REF].

Defining first the total enthalpy J = C p T + 1 2 u 2 and the mass flow rate m = ρu, the linearized isentropic fluctuating part of these variables reads :

Ĵ = p ρ 0 + u 0 û (23) m = ρ 0 û + u 0 ρ with ρ = p c 2 0 ( 24
)
Reformulation of Eqs. 23-24 with the plane wave expansion of the pressure (Eq. 6) and velocity (Eq. 7) leads to:

Ĵ = J + + J -= A + (1 + M)e jk + x ρ 0 + A -(1 -M)e -jk -x ρ 0 ( 25
)
m = m + + m -= A + (1 + M)e jk + x c 0 - A -(1 -M)e -jk -x c 0 ( 26
)
The methodology to express in a general manner a boundary condition in the zero-Mach number framework is described as follow:

1. The boundary condition must be prescribed first with energetic variables without neglecting the mean flow, defining then a functional f ( Ĵ, m, ŝ) = 0 linking the fluctuating variables Ĵ (total enthalpy), m (mass) and ŝ (entropy). Expanded each fluctuating quantity into waves (see Eq. 25, Eq. 26 and Eq. 8), the boundary condition is written under the form:

g(J + , J -, m + , m -, σ ) = 0 (27) 2.
When the computational domain is represented under the zero Mach number assumption, the functional f ( Ĵ, m, ŝ) = 0 describing the boundary condition remains unchanged. However, the energetic variables Ĵ, m and ŝ must be then expressed at the limit when M → 0. It comes:

Ĵ = p ρ 0 m = ρ 0 û ŝ = 0 (28)
As far as the waves are concerned, Eqs. 25 and 26 become when M → 0 :

J + = A + e jkx ρ 0 , J -= A -e -jkx ρ 0 m + = A + e jkx c 0 , m -= - A -e -jkx c 0 (29)
Thus, the functional relationship representing the boundary condition under the zero Mach number assumption becomes:

g A + e jkx ρ 0 , A -e -jkx ρ 0 , A + e jkx c 0 , - A -e -jkx c 0 , 0 = 0 (30)
As an illustration, this methodology is now applied to the boundary conditions used in the present study:

Boundary condition m = 0 For that case, the functional f ( Ĵ, m, ŝ) = 0 is m = 0. Reformulating this condition using waves J + , J -, m + , m -and introducing the limiting behavior when M → 0 leads to :

m + + m -= 0 (31) A + c 0 - A - c 0 = 0 (32)
At the boundary where m = 0, the input energetical reflection coefficient R e defined as the ratio between the reflected wave A + and the incident wave A -becomes R e = 1, which may be viewed as R e = R ac where R ac is the classical acoustical reflection coefficient, which is +1 for û = 0. This result explains in a proper way the validity of û1 = 0 as a choice for the input boundary condition in method A3. Boundary condition p = 0 Combining Eqs. 23 and 24 with p = 0 leads to the functional f ( Ĵ, m, ŝ) = 0:

Ĵ - u 0 ρ 0 m = 0 (33)
Reformulating with waves and using the limiting behaviors Eq. 29 leads to :

J + + J -= u 0 ρ 0 m + + m - (34) A + (1 -M) = -A -(1 + M) (35)
At the boundary where p = 0, the energetical reflection coefficient R e defined as the ratio between the reflected wave A -and the incident wave A + becomes R e = -1-M 1+M which may be viewed as

R e = 1 -M 1 + M R ac ( 36
)
where R ac is the classical acoustical reflection coefficient, which is -1 for p = 0. This latter result can be directly applied to Eq. 19. Injection of R e into 18 leads to an energetical impedance which takes the form of a ratio between stagnation enthalpy and mass fluctuations :

Z e = ρ 1 c 1 Ĵ m ( 37
)
The dashed line with square symbols on Fig. 2 shows the evolution of |R e | as a function of inlet Mach number M 1 . As discussed in [START_REF] Peters | Damping and reflection coefficient measurements for an open pipe at low mach and low helmholtz numbers[END_REF], the reflection coefficient R e defined with the acoustical energy formulation is always equal or inferior to unity, suggesting that the continuity of the energy at the control surface is respected.

Computation of eigenmodes are now performed with R e instead of R ac . Results are presented in Table 2 under the column A3*. Eigenfrequencies computed with a reflection coefficient based on the acoustic energy description are very close to the ones computed with the reference methods (A0 and A1). As the real part of the frequency remains unchanged whatever the choice of the formulation of R, it suggests that the absorption of acoustical energy is included only through the formulation of the impedance. The small discrepancies between the reference solution and method A3 with R ac are related to the length of the domain where the Mach number is neglected. Actually, it can be shown that the frequency tends to the reference solution when l Z approaches x = 0. Such results demonstrate that the impedance must be prescribed under an energetical form in order to obtain an appropriate representation of the whole configuration in the case where the domain is computed under the zero Mach number assumption.

Accounting for entropy convection in a zero Mach mean flow

The acoustic waves generation when entropy inhomogeneities are accelerated in a non-uniform flow is a well known phenomenon that has been extensively studied over the past decades. Early analytical investigations deal with the development of the jet noise theory, extending the work of Lighthill to non-uniform density flows [START_REF] Morfey | Amplification of aerodynamic noise by convected flow inhomogeneities[END_REF][START_REF] Williams | The generation of sound by density inhomogeneities in low mach number nozzle flows[END_REF][START_REF] Howe | Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute[END_REF]. However these analytical solutions were limited to low Mach number flows and focused on the derivation of a formulation for the far-field sound radiation into free space by inhomogeneities swept out of nozzle orifice. In a different way, Marble and Candel [START_REF] Marble | Acoustic disturbances from gas nonuniformities convected through a nozzle[END_REF] proposed a one-dimensional theory based on the compact assumption (the nozzle dimensions are small in comparison with the shortest wave length that appears in the flow field). The nozzle may be viewed as a duct discontinuity and simple relations between upstream/downstream acoustic and entropy waves can be written. Validation of such a theory has been provided recently [START_REF] Leyko | Numerical and analytical modelling of entropy noise in a supersonic nozzle with a shock[END_REF] by comparisons with experiments.

The effect on thermoacoustic instabilities of the presence of a nozzle at the combustor exit is not obvious. As reviewed by [START_REF] Hield | Thermoacoustic limit cycles in a premixed laboratory combustor with open and choked exits[END_REF] some authors found no difference on the thermoacoustic modes of their combustor whatever the type of exit used, while other authors [START_REF] Macquisten | Lowfrequency combustion oscillations in a model afterburner[END_REF] reported that a strong low-frequency instability occured when the open exit was replaced by a chocked nozzle. As explained by [START_REF] Hield | Thermoacoustic limit cycles in a premixed laboratory combustor with open and choked exits[END_REF], the behaviour of such an instability depends to the first order on the geometry. The spatial dispersion of entropy fluctuations by the combustor aerodynamics [START_REF] Sattelmayer | Influence of the combustor aerodynamics on combustion instabilities from equivalence ratio fluctuations[END_REF] or the constructive/destructive phase dependency [START_REF] Polifke | Constructive and Destructive Interference of Acoustic and Entropy Waves in a Premixed Combustor with a Choked Exit[END_REF] play an important role for the establishment of a coupling between acoustic and entropy modes.

The configuration depicted in Fig. 3 has been chosen in order to exhibit the presence of an entropy mode among acoustic modes. A flame is located at x = x f , which constitutes an interface between cold gas and hot gas. A chocked nozzle is located at x = L while the sonic throat where the Mach number reaches unity is located at x = x th . A similar methodology to the previous section is employed by comparing three methods.

Method B0: Numerical reference method

Method B0 is strictly the same as method A0 presented above in section 1.1 [START_REF] Nicoud | About the zero mach number assumption in the calculation of thermoacoustic instabilitie[END_REF]. Both the combustor and the nozzle are computed. The thickness of the flame is 0.005 m.

Method B1: Global thermoacoustic model

Five equations are required to express the global thermoacoustic model. The flame is considered infinitely small, allowing the integration of conservation equations Eqs. 1-3 over the discontinuity. At a first approximation and in the case where the two connected ducts share the same cross section S, this leads to the following set of jump relations [START_REF] Dowling | The calculation of thermoacoustic oscillations[END_REF]:

ρ 1 û1 + u 1 ρ1 = ρ 2 û2 + u 2 ρ2 (38) p1 + 2ρ 1 u 1 û1 + u 2 1 ρ1 = p2 + 2ρ 2 u 2 û2 + u 2 2 ρ2 (39) C p T 01 (ρ 1 û1 + ρ1 u 1 ) + ρ 1 u 1 C p T1 + u 1 û1 = C p T 02 (ρ 2 û2 + ρ2 u 2 ) + ρ 2 u 2 C p T2 + u 2 û2 (40)
where Eq. 3 has been reformulated to express the total energy under a conservative form and the total unsteady heat release has been neglected. The mean stagnation temperature is T 0i = T i + 1 2 u 2 0i /C p where the subscript i = 1, 2 refers to cold and hot gas, respectively.

The inlet and outlet boundary conditions form the last two equations that are closing the problem. At x = 0, the boundary condition imposed is m1 = 0. As recalled by [START_REF] Leyko | Numerical and analytical modelling of entropy noise in a supersonic nozzle with a shock[END_REF], the compact relation that models the supersonic chocked nozzle at x = L reads :

û2 (L) c 2 - γ -1 2 M 2 p2 (L) γP 2 - 1 2 M 2 ŝ2 (L) C p = 0 (41)
Using the usual plane wave expansions and Eqs. 38-41, the eigenfrequencies of the problem are computed by solving det(M ) = 0, where M is the matrix :

       M + 1 e jk + 1 x f -M - 1 e -jk - 1 x f -M + TABLE 3. THERMODYNAMIC PARAMETERS FOR CONFIGURATION B. T 1 P 1 T 2 γ r L x f x th 300 K 101325 Pa 1200 K 1.4 287 SI 1 m 0.5 m 1.087 m with β = 1 2 C p (T 2 -T 1 )(γ -1) c 1 c 2 e jωτ s .
As in the previous section, the proper inlet boundary condition leads to A + 1 -A - 1 = 0. Finally, the problem defined by the matrix of Eq. 42 reduces to Eq. 48 :

      e jk 1 x f ρ 1 c 1 -e -jk 1 x f ρ 1 c 1 -e jk 2 x f ρ 2 c 2 e -jk 2 x f ρ 2 c 2 e jk 1 x f -e -jk 1 x f -e jk 2 x f -e -jk 2 x f 1 -1 0 0 β e jk 1 x f -β e -jk 1 x f (1-γ-1 2 M 2 )e jk 2 L 1+M 2 - (1+ γ-1 2 M 2 )e -jk 2 L 1-M 2       (48)
Consistently with the no flow approximation, Mach numbers M 1 and M 2 do not appear in the first two rows which describe the wave propagation within the cold and burnt gases.

Results

Thermodynamic parameters used in the computation of eigenmodes of the configuration B are presented in Table 3.

Results are presented in Table 5 and Table 4. The Mach number M 2 downstream of the flame is a free parameter driving the system, which is computed directly by the equations with methods B0 and B1, and prescribed in the boundary condition formulation with method B2. The behaviour of the real part of the first eigenfrequency computed with methods B0, B1 and B2 shows a strong dependency on the mean flow Mach number. The real part of the frequency is proportional to the convective time delay of the flow, suggesting that this eigenmode is of entropic nature. The second eigenmode computed does not present the same behaviour and the real part of the frequency is approximately the same whichever the method used or the Mach number prescribed. As shown in section 2, the influence of increasing the mean flow velocity is to damp the imaginary part, suggesting the presence of an acoustic eigenmode.

A good agreement is found between analytical methods B1 and B2 and the numerical reference method B0, excepting that this last one mispredicts the imaginary part of the eigenmodes and underpredicts the real part of acoustic eigenfrequencies. Such discrepancies can be explained by the fact that method B0 solves the whole domain (the combustor plus the nozzle), while methods B1 and B2 are limited by the low-frequency compact model which does not take into account the length of the nozzle nor the phase shift between incoming and reflected waves [START_REF] Goh | Phase prediction of the response of choked nozzles to entropy and acoustic disturbances[END_REF]. However, such a simplified model shows at the first order that the use of a DECBC is able to capture both entropic and acoustic eigenmodes and to predict them correctly, despite the fact that the mean flow is neglected during the computation of the domain. In order to illustrate the effect of the coupling with entropy, methods B1 * and B2 * present the computation of eigenmodes with a purely acoustic boundary condition, i.e. the coupling with entropy waves has been neglected in Eq. 41 (this means using Eq. 46 instead of Eq. 47). Obviously the entropic mode does not appear in this case. The growth rate of the acoustic mode is significantly mispredicted, suggesting that the coupling between entropy and acoustic fluctuations exhibit a strong unstable thermoacoustic mode, while purely acoustic analysis shows a stable eigenmode.

Conclusions

The present study is devoted to the modelling of mean flow effect while computing thermoacoustic eigenmodes under the zero Mach number assumption. It has been shown that:

• when the computational domain is represented under the no flow assumption, the acoustic impedance modelling a compressor or a turbine must be prescribed under an energetical form rather than from the classical acoustic variables. A systematic methodology is proposed to generate the proper reflection coefficient to impose given any physical boundary condition written without the zero Mach number assumption.

• the coupling between entropy and acoustic waves at the entrance of a nozzle generates a low frequency eigenmode and a shift of higher frequency modes that does not appear when the mean flow convective effects are neglected. Such phenomena can be taken into account thanks to the proposed methodology where a zero Mach number formulation is used together with a Delayed Entropy Coupled Boundary Condition.

FIGURE 1 .

 1 FIGURE 1. CONFIGURATION A INVESTIGATED IN SECTION 2. FIGURE (a): THE COMPLETE GEOMETRY IS COMPUTED. FIGURE (b): ONLY A PART OF THE TUBE 1 IS COMPUTED WHILE THE REST OF THE DOMAIN IS MODELLED THROUGH DOWNSTREAM IMPEDANCE.

FIGURE 2 .

 2 FIGURE 2. MODULUS OF THE REFLECTION COEFFICIENT AS A FUNCTION OF INLET MACH NUMBER M 1 . STAR SYM-BOL: ACOUSTICAL FORMULATION. SQUARE SYMBOL: ENER-GETICAL FORMULATION.

FIGURE 3 .

 3 FIGURE 3. CONFIGURATION B INVESTIGATED IN SECTION 3.

TABLE 1 .

 1 THERMODYNAMIC AND GEOMETRICAL PARAMETERS USED IN THE CONFIGURATION.

	S 1	S 2	T 1	P 1	γ	r	L	l	l Z
	0.05 m 2	0.1 m 2	300 K	101325 Pa	1.4	287 SI	1 m	0.99 m	0.98 m

TABLE 2 .

 2 RESULTS FOR THE FIRST EIGENMODE WITH METHODS A0, A1, A2, A3 WITH R ac AND A3* WITH R e , FOR DIFFERENT INLET MACH NUMBER M 1 .

	M 1	0.0005	0.1
	A0	87.154 -0.007i	86.37 -1.360i
	A1	87.233 -0.007i	86.367 -1.364i
	A2	87.233 -0.007i	86.367 -1.364i
	A3 (R ac )	87.233 + 0.0208i	87.226 + 4.194i
	A3* (R e )	87.233 -0.0007i	87.226

TABLE 4 .

 4 ENTROPIC EIGENFREQUENCIES COMPUTED WITH METHODS B0, B1 AND B2.

	M 2	0.0125	0.1
	B0	8.92 -0.89i	67.30 -5.99i
	B1	9.36 -0.80i	73.13 -4.34i
	B2	8.97 -0.80i	73.65 -4.11i

TABLE 5 .

 5 ACOUSTIC EIGENFREQUENCIES COMPUTED WITH METHODS B0, B1 AND B2. THE * SYMBOL REFERS TO RESPECTIVE METHODS WHERE THE COUPLING WITH EN-TROPY IS NEGLECTED AT THE EXIT BOUNDARY CONDITION.

	M 2	0.0125	0.1
	B0	206.21 + 25i	178.04 + 10.57i
	B1	211.67 + 5.04i	189.02 + 13.7i
	B1 *	211.14 -2.28i	210.76 -18.51i
	B2	211.67 + 5.34i	192.27 + 18.63
	B2 *	211.14 -0.82i	210.66 -6.6i

3.3 Method B2: Delayed Entropy Coupled Boundary Condition approach. Combining Eq. 11, Eq. 38 and Eq. 40, and taking the limit u → 0 while the product u 2 ŝ remains finite [START_REF] Dowling | The calculation of thermoacoustic oscillations[END_REF], one obtains the following expression for the entropy produced by the flame:

Entropy fluctuations at the exit x = L may be analytically expressed by the addition of a time delay that mimics the mean flow convective effect:

The methodology explained in the previous section is applied to Eq. 41, by reformulating this boundary condition thanks to an energetical and isentropic functional f ( Ĵ, m, ŝ) = 0. It comes :

Reformulating this equation by using waves and injecting the limiting behaviors Eq. 29 leads to :

This condition handles properly the acoustic reflection that arises when a nozzle is connected to a domain computed under the zero Mach number assumption. However it does not represent the entropy/acoustic coupling since σ was set to zero in agreement with Eq. 29. This is the objective of the DECBC approach to model the effect of the accelerated entropy fluctuations on the acoustics within the zero Mach number region. To this purpose, the modeled entropy at the domain exit, Eq. 44, is injected into Eq. 46 while keeping the energetical form given by the functional at Eq. 45. Expressing û1 (x f ) in terms of waves A + 1 and A - 1 allows us to write the following condition at x = L: