S Rahman

Johan-Georg G Granström

Z Salloum

IBN SĪNĀ'S APPROACH TO EQUALITY AND UNITY

Aristotle did not develop the quantification of the predicate, but, as shown in a recent paper by Hasnawi, Ibn Sīnā did. In fact, assuming the Aristotelian subject-predicate structure, Ibn Sīnā qualifies those propositions that carry a quantified predicate as deviating ḥ (mu arrafah ) propositions. A consequence of Ibn Sīnā's approach is that the second quantification is absorbed by the predicate term. The clear differentiation between a quantified subject, that settles the domain of quantification, and a predicative part, that builds a proposition over this domain, corresponds structurally to the distinction, made in constructive type theory, between the type of sets and the type of propositions.

Neither did Aristotle combine his logical analysis of quantification with his ontological theory of relations or equality. But Ibn Sīnā makes use of syllogisms that require a logic of equality, and considered cases where quantification combines via equality with singular terms. Moreover these reflections provide the basis for his theory of numbers that is based on the interplay between the One and the Many. If we combine Ibn Sīnā's metaphysical theory of equality with his work on the quantification of the predicate, a logic of equality comes out naturally. Indeed, the interaction between quantification of the predicate and equality can be applied to Ibn Sīnā's own examples of syllogisms involving these notions. By using the formal instruments provided Martin-Löf's constructive type theory, the present paper establishes links between Ibn Sīnā's metaphysics and his logical work: links that have been discussed in relation to other topics by Thom and Street. Ibn Sīnā did not develop a logic of identity, but he did develop the conceptual means to do so.

Introduction

As pointed out by Sundholm [START_REF] Sundholm | Conservative generalized quantifiers[END_REF] (cf., [START_REF]Semantic and Linguistic Theory Conference[END_REF]), since the work of Frege, quantifiers are intended to range over the universe of all objects. Hence, since all quantifications concern the same domain, there seems to be no practical or theoretical need to include explicit information about the domain of quantification in the quantifier notation. In this setting, the role of the predicate is to pick out, from an all encompassing universe, the subset of objects appropriate for analysis of the sentence at hand. Such a strategy has the side-effect that it liberates the logical form from the subject-predicate structure, explicitly imposed on propositions in the Aristotelian tradition.

However, the Fregean move is utterly unfaithful to the corresponding natural language expressions. It seems natural, on one hand, to attach to every, a noun, such as every student and every philosopher, but, on the other hand, if someone asserts that Some elephants are small, it looks like we pick, from the restricted universe of elephants, those that are small, and not, from the universal domain of objects, those objects that are small and elephants. It may well be that the elephant in question is small (for an elephant), but even small elephants are big creatures in the animal kingdom.

Crubellier, in a new French translation of the Analytica [START_REF] Aristote | Premiers Analytiques[END_REF] proposes, to translate Aristotle's formulation of universal expressions with a paraphrase like To be small applies to some elephants Crubellier's translation strongly suggests that the subject, restricts the predicate (to be small), to the domain of elephants.

Note that every and some always are attached to a noun. In a paper on the notion of number in Plato and Aristotle, Crubellier [START_REF] Crubellier | Platon, les nombres et Aristote[END_REF] points out that, for the ancient Greeks, a number is always attached to a noun: two apples, two flowers, etc. The pure number of the mathematicians is, according to Aristotle, an abstract presentation that achieves generalitye.g. two abbreviates two whatever objects. Perhaps, similarly, we should think of every as an abstraction from every apple, every flower, etc.

A crucial point in the type-theoretic formalization of the Aristotelian forms of propositions is the distinction drawn between two forms of judgement that both formalize natural language sentences of the form a is B, namely:

a : B and B(a) : true, where, in the first case, B is a set, and, in the second case, a : A and B(x) is a proposition under the assumption that x is an element of the set A . This distinction is linked to Lorenz and Mittelstrass' beautiful analysis of Plato's Cratylus [START_REF] Lorenz | On rational philosophy of language: the programme in Plato's Cratylus reconsidered[END_REF]. In this paper, the authors point out two different basic acts of predication, viz., naming (όνοµάζειν) and stating (λέγειν). The first amounts to the act of subsuming one individual under a concept and the latter establishes a true proposition. Naming is about correctness, i.e., one individual reveals the concept it instantiates if the naming is correct. Stating is about the truth of the proposition that results from the second kind of predication act. If the predicate indeed applies to the individual, the associated proposition is true.

In the context of our own reconstruction, naming corresponds to the assertion that an individual is an element of a given set, or, which amounts to the same, that an individual falls under a given concept, that is, it involves judgements of the form a : B; and stating corresponds to asserting the truth of a proposition, i.e., to asserting B(a) : true.

The conventional type-theoretic notation for quantifiers stresses the two-term structure of the Aristotelian forms of propositions: (∃ x:S)P(x) and (∀x : S)P(x).

In the example above, S ≡ {elephant}, and P(x) ≡ ''x is small'', where the variable x stands for an elephant2 . This clearly reveals that S and P have different status.

Aristotle did not apply his logical analysis of quantification either to relations or equality, but Ibn Sīnā considered the latter, as well as propositions and syllogisms where quantification combines via equality with singular terms. These reflections provide the basis for his theory or numbers that is based on the interplay between One and the Many3 . Certainly, as pointed out by Hasnawi 4 , Aristotle discusses the case of relations and unity in many places of his work, such as in the books ∆ and Ι of his Metaphysics. Moreover, Ibn Sīnā's classification of unity, which we discuss in the present paper, is Aristotelian.

However, and this is the main claim of our paper, it is Ibn Sīnā's quantification of the predicate that allows him to introduce these distinctions into the object language and prefigure, perhaps for the first time, a logic with an equality predicate.

Quantification of the predicate

Many logicians of the middle-ages avoided relations, particularly in the context of building syllogisms. Thus, syllogisms were based on monadic predicates and the logic of monadic predicates does not naturally lead to repeated or nested quantification.

However, Ibn Sīnā devotes two chapters of al-`Ibāra to the quantification of the predicate. Al-`Ibāra ( رة )ا is the third book of the logical collection of his philosophical encyclopaedia entitled al-Shifā (the Cure ء .)ا It is in these texts that Ibn Sīnā shows that his approach to quantification is very close to Aristotle's analysis of relations, mentioned above, though Ibn Sīnā's own understanding of quantification allows him both to study double quantification and defend its study from detractors. This has been made apparent in the excellent paper of Hasnawi [START_REF] Hasnawi | Avicenna on the Quantication of the Predicate[END_REF] on Ibn Sīnā's double quantification -a paper that, by the way, contains an English translation of these two chapters, the first in any language.

Indeed, one of the mediaeval objections against the quantification of the predicate is that, because of the possibility of a negation in the scope of the second quantifier, propositions could not anymore said to be either affirmative or negative 5 . For example, does some man is not every animal express an affirmative or a negative proposition? Hasnawi sums up Ibn Sīnā's position very well: ''Avicenna6 upholds here a radical point of view: according to him the predicate in double quantified propositions is constituted by the quantifier plus the initial predicate, which form together a unit. So a proposition which has the form of a normal affirmative proposition will keep this quality even though a negative quantifier has been prefixed to its predicate. The quantifier of the predicate is conceived of as a predicate-forming operator on predicates: it generates new predicates from previous ones by attaching a quantifier to them.'' [12, pp. 304].

It very much looks as if Ibn Sīnā thinks of quantification of the second term as building a new predicate, and this deviates from the normal use of quantification. Let us once more quote Hasnawi, who this time quotes Ibn Sīnā himself: ''The same semantic core, namely that a clause added to a proposition or to a part of it, makes the proposition deviate from its normal functioning, is present in the description ḥ Avicenna gives of propositions with a quantified predicate as mun arifāt ( ت ): ''If you try ḥ then to add a quantifier, the proposition will be deviate (in arafat :)ا the predicate will no longer be a predicate, but rather it will become part of the predicate. The consideration of the truth will thus be transferred to the relation which occurs between this sum and the subject. That is why these propositions were called deviating'' (al-`Ibāra: 64,17-65,1).'' [12, pp. 323].

Before formalizing Ibn Sīnā's quantified predicates, recall that the subject term S in the Aristotelian forms of propositions some/every S is P can be taken in two different ways when formalized in type theory : either as a set, as above, or as a subset, of a larger domain of quantification, the universe of discourse D, i.e., as a propositional function, or a predicate, in the logical sense of the word, on the set D. For example, the proposition some white thing is round can be formalized using D ≡ {physical thing} universe of discourse, S(x) ≡ x is white subject, P(x) ≡ x is round predicate.

Clearly, some S is P has to be formalized as (∃x:D) S(x) & P(x) in this case. We introduce the following compact notation for the case when the subject term is taken as a subset of a larger universe of discourse: some S is P ≡ ∃S(x)) P(x) ≡ (∃x:D)S(x)& P(x), every S is P ≡ (∀S(x)) P(x) ≡ (∀x : D)S(x) ⊃ P(x), where both S(x) and P(x) are propositions under the assumption that x is an element of the universe of discourse D. The symbol ⊃ stands for logical implication. This interpretation of quantification over a restricted domain is standard in modern predicate logic (cf., e.g., [START_REF] Van Dalen | Logic and Structure[END_REF]).

Using this notation, we can now make sense of Ibn Sīnā's cryptic remark that ''the predicate will no longer be a predicate, but rather it will become part of the predicate''. For example, consider the sentence every S is some P, or, which amounts to the same, every S is equal to some P, where P is a propositional function on S: we get the formalization (∀x:S)( ∃P(y)) (x = y) ≡ (∀x:S)(∃y:S) P(y)& (x = y).

By substitution, P(x) is logically equivalent to (∃P(y))(x=y) , whence ''every S is some P'' is logically equivalent to ''every S is P'' 7 .

The discussions on the One and the Many are ubiquitous in the work of Ibn Sīnā, and they share the features of many Neo-Platonists who, quite often, attempt to combine the Platonic and Aristotelian Metaphysics. In fact these two notions set the basis for Ibn Sīnā's theory of numbers. A salient sense of the notion of One is, according to our author, identity developed in the first chapter on Metaphysics in the Book of Science [5, pp. 121-125] and in the second chapter of third book of the Metaphysics of the Shifā [6, pp.160-164] 8 . In this context, Ibn Sīnā distinguishes between

• One in nature: no aspect of multiplicity is involved, such as, God and the geometrical point.

• One in an aspect: this is said of specific things either -according to essence, or -per accidens.

The first sense of One, i.e., One in nature, is related to the unity of an object -even with the very concept of existence of an object. The second sense of One (by essence or per accidens) is the one that builds Ibn Sīnā's theory of equality and that combines with his quantification of the predicate. It is crucial that the interaction between quantification of the predicate and equality can be applied to Ibn Sīnā's own examples of syllogisms involving these notions. The investigation of this second sense of One will occupy the rest of this paper. We will interpret One in essence as dealing with equality, interpreted in type theory either as definitional equality, a ≡ b : A , or as propositional equality, (a=b) : true.

In his Autobiography our author claims that he has reconstructed all the inference steps of the Euclid's geometry. This certainly requires a profuse use of the logic of equality. On the topic of equality, Ibn Sīnā discusses transitivity of equality in Qiyās i.6: ''Thus when you say C is equal to B and B is equal to D, so C is equal to D .'' 9There are quite a number of syllogisms in Ibn Sīnā's work on logic involving equality and equivalence: they have not all been compiled yet, but let us analyze a few examples.

At Qiyās 472.15f we find:

Zayd is this person sitting down, and this person sitting down is white So, Zayd is white.

Here Ibn Sīnā makes use of the substitution rule

true b P true a P true b a : ) ( : ) ( : ) ( =
where a and b are elements of a set A and P(x) is a predicate defined on the set A.

The syllogism discussed at Qiyās 488.10 could be read with equality too: Pleasure is B. B is the good. Therefore pleasure is the good. However, it is perhaps more natural to interpret the word is in these sentences as logical equivalence.

One per accidens

Ibn Sīnā distinguished six cases of unity per accidens10 : unity by species, unity by genus, unity by accidental predicate, unity by relationship, unity in subject, and unity in number. In general, sentences expressing this form of unity have the form a and b are one by/in A.

Our analysis of this form of sentences will associate a (possibly higher order) schema S with the aspect A, and the interpretation of the sentence will be that the predicate applies equally to both a and b, i.e., the interpretation will be that S(a) and S(b) are both true.

This analysis holds good for the first four cases of unity per accidens, but, as we shall see, it breaks down for unity in subject and unity in number.

Let us first consider unity by species (wāhid bilnw .ع / .ا10 .)ا An example due to Ibn Sīnā is given by

Zayd and Amr are one by humanity

In this case, the schema is simply the predicate SH(x) ≡ x is human, where x ranges over some suitable domain, such as living being. That is, Zayd and Amr are one in the sense that both propositions SH(Zayd) and SH(Amr) are true. Put differently, Zayd and Amr are one by humanity is interpreted as Zayd and Amr are both human.

Next, let us consider unity by genus (wāhid biljns 2 3 / .ا10 .)ا An example due to Ibn Sīnā is given by The human and the horse are one by animality.

Again, we pick living being as the universe of discourse L, and we interpret Human(x) and Horse(x) as predicates on L. In this case, the schema is of the second order as it takes a predicate X on L as argument:

SA(X) ≡ (∀x : L )X(x) ⊃ Animal (x).
The sentence is now interpreted in type theory as SA(Human) and SA(Horse) both being true. ḥ ḍ Unity by a ccide nta l pre dica te (a l-wāid bi -a l-ara ض / .ا10 .)ا An example due to Ibn Sīnā is given by Snow and camphor are one in being white.

Here we take the universe of discourse to be substance, abbreviated S, and the schema is SW (X) ≡ (∀x : S)X(x) ⊃ White(x), where X is a predicate on S. If we view Snow(x) and Camphor(x) as predicates on S, it is clear that both propositions SW(Snow) and SW(Camphor) are true. Note that, in the formalization of unity by genus, the universe of discourse cannot be taken to be animal, as this would make the schema SA(X) vacuously true for any predicate X Interestingly, the type-theoretic formalization is the same as for unity by genus. This is because the Aristotelian distinction between essential and accidental predicates pertains to semantics, or meaning, rather than to logical form. In type theory, a formal distinction is made between the universe of discourse (which always is a set) and predicates on this universe, but a distinction between essential and accidental predicates can only be done we have access to the definitions of the predicates involved: a predicate on a genus is essential if it is a mark11 , of the definition of the genus, or a combination of such marks. In summary, the Aristotelian distinction between essential and accidental predicate holds good in type theory, but it is not visible in the logical form of the identity sentence -however it can be made visible in the so-called formation rules for the semantic elements of those sentences.

A particularly interesting kind of unity is unity by relationship (wāhid bilmunāsaba .ا10 ا 8 9 /). An example due to Ibn Sīnā is given by The relation of the sovereign to the city and the relation of the soul to the body are one.

We read this example as unity with respect to the relation is governed by. Let us introduce two universes of discourse, one for things governed, governees, abbreviated GE, e.g., cities and bodies, and one for governors, abbreviated GN, e.g., sovereigns and souls. We view City(x) and Body(x) as predicates on the set GE. In addition, we view Sovereign(x, y) as a dyadic predicate on a governee x and a governor y, so that Sovereign (x, y) means y is a sovereign of x. Similarly, Soul(x, y) means that y is the/a soul of x. Other formalizations are also possible, such as taking Sovereign and Soul to be functions, or restricting the first argument x to be an element of the subset of cities or bodies respectively. However, the formalization we have chosen is probably the simplest. The schema related to unity with respect to governance is given by SG(X, Y) ≡ (∀x : GE)(∀Y : GN) X(x)& Y(x,y) ⊃ Governs(x, y), where X ranges over predicates on GE and Y ranges over dyadic predicates on GE and GN. The propositions SG(City, Sovereign) and SG(Body, Soul) come out as true under this interpretation.

Next we have unity in subject (wāhid bilmwdu 9.?.ع / .ا10 .)ا An example due to Ibn Sīnā is given by Whiteness and softness are one, in sugar.

Here we cannot fit the formalization into the general schema presented above. Instead we directly formalize this sentence as (∀x : S)Sugar(x) ⊃ (White(x) ⊃ ⊂ Soft(x)), where ⊃ ⊂ stands for bidirectional implication, the universe of discourse is substance, abbreviated S, and Sugar(x), White(x), and Soft(x) are predicates on the universe of discourse. That is, our interpretation is every substance that is sugar is white if and only if it is soft.

Finally, we have unity in number (wāhid bil dad 0د / .ا10 .)ا It is important to recall here that Ibn Sīnā thought, as did Aristotle before him, that numbers are properties of terms -a claim that lost popularity after Frege's incisive objections in his Grundlagen der Arithmetik [START_REF] Frege | Die Grundlagen der Arithmetik[END_REF]. Be this as it may: it is natural to formalize a sentence such as A and B are one in number, as the sets or subsets A and B admitting a bijection between them.

Conclusion

This paper suggests that Ibn Sīnā explored new paths of logic beyond the work of Aristotle, particularly so in relation to equality. As already mentioned in the introduction, Ibn Sīnā's quantification of the predicate, rejected by Aristotle 12 , allows him to combine, perhaps for the first time, a logical analysis of quantification combined with a theory of equality.

A topic that deserves further study is Ibn Sīnā's theory of the relation between equality and existence in the context of his overall philosophical view on mathematics, and particularly in the context of his theory of numbers.

shahid.rahman@univ-lille3.fr (Université Lille3, STL : UMR 8163), georg.granstrom@gmail.com, salloum@ul.edu.lb (Université Lille3, STL : UMR 8163, Université Libanaise, Faculté de Sciences, Section I, Département de Mathématiques, Beyrouth, Liban)

In this paper, we use the sign ≡ for definitional equality: in this case, the definition of S has type `set' and the definition of P(x) has type `prop'.

Rahman/Salloum: The One, the Many and Ibn Sīnā 's Logic of Identity. Paper in preparation.

Personal communication of Ahmed Hasnawi.

In the 19th century, this debate was revived, with, on one side, W. Hamilton defending quantification of the predicate, and, on the other side, A. De Morgan strongly objecting to it[START_REF] Fogelin | Hamilton's Quantification of the Predicate[END_REF].

Recall that Ibn Sīnā's Latinized name is Avicenna.

Cf., Maritain: ``In every affirmative, the predicate is taken particularly''[14, p. 125].

The Schoolmen referred to the distinction between one in nature and one in an aspect as ``duplex est unum'', cf., Aquinas De Potentia, q. 3, a. 16, ad 3.

Transl. W. Hodges.

This classification of the various senses of identity is derived from Aristotle, Metaph., Bk. 5, Ch. 6.

Nota, in the scholastic terminology or Merkmal in the terminology of Frege. .

Acknowledgements

The final form of the paper is due to discussions with Ahmed Hasnawi, who encouraged the first author to the exploration of the fascinating issues underlying the subject of the paper and who shared some wise commentaries on an early draft of the paper, the assistance of my colleague Michel Crubellier and the deep insights of Göran Sundholm during his visit at our university in Lille. We would like to record herewith our fond recollection of the learned interactions with all of them.

Dedication

The first author wishes to dedicate the paper to his colleague and friend Michel Crubellier who just started the exercise of his (administrative) retirement and with whom the first author engaged in many fascinating discussions on Aristotle, Arabic philosophy and beyond.


E Friot
email: friot@lma.cnrs-mrs.fr

Time-domain versus frequency-domain effort weighting in active noise control

Although Active Noise Control aims at reducing the noise at a set of error sensors, it is often designed by minimizing an error index that includes a weighted penalty on the actuator inputs. In this way, the control tends to be more robust and the effort-weighting parameter allows the maximum voltages applied to the control sources to be monitored. Two similar effort-weighting techniques have been widely implemented in active control studies: optimal control can be computed using Tikhonov regularization in frequency-domain simulations, whereas the leaky Filtered-reference least mean squares algorithm can be implemented for real-time feedforward control. This paper makes explicit the relationship between the two effort-weighting parameters which lead, in the case of a single-tone noise, to exactly the same error index in both the time and frequency domains.

The best real-time leakage factor can then be computed from frequency-domain optimization. This paper also discusses numerical simulations of a single-channel set-up, showing that with these two related parameters, the control performances are indeed very similar. One exception occurs in the case of a control filter with a very short impulse response, when the control is more conservative in the timedomain simulations than in those of the frequency-domain.

Introduction

Loudspeakers or shakers can be destroyed when they are driven by a high voltage. To prevent this risk, in situations where the primary noise to be reduced requires high voltages at the secondary actuators, an Active Noise Control set-up is usually designed so as to minimize an error index including both the control signals (the actuator inputs) and the error signals (the residual noise at the error sensors) (cf. [Nelson(1992)], [START_REF] Elliott | [END_REF]], [START_REF] Kuo | [END_REF]]). By adjusting a weighting coefficient in the error index, the trade-off between noise reduction and voltage limitation can be optimized.

From a theoretical point of view, the inclusion of an effort-weighting parameter in the minimization index also allows ill-posed control problems (e.g. in situations where there are more actuators than error sensors) [START_REF] Tikhonov | [END_REF]] to be regularized. It has also been reported [Nelson(1992)] that effort weighting in active control may enlarge the area in which noise is reduced, because it forces noise minimization at a discrete set of sensors to be better matched to an underlying, continuous-space, global control problem. Finally, leakage has the virtue of increasing the robustness of a real-time adaptive algorithm to errors in the secondary path estimation. This is a useful property for on-line computations (see [START_REF] Elliott | [END_REF]], section 3.4.7). Therefore, besides monitoring the control inputs in the case of a loud primary noise, it is generally a good idea to include an effort-weighting parameter in the design of an active control set-up.

Most active control studies address the weighting of the control inputs through the use of two different techniques:

1. at the design stage, the control signals and the noise reduction are usually computed off-line, in frequency domain simulations. In this case the actuator inputs can be limited by minimizing, at each frequency, a quadratic index which weights the norm of the sensor signal vector and the norm of the actuator input vector (see [START_REF] Elliott | [END_REF]], section 4.2.6). Optimal control can then be determined by performing the regularized inversion of the theoretical or measured actuator-tosensor response matrix [START_REF] Tikhonov | [END_REF]].

2. For effective real-time control, the actuator inputs are usually computed on-line in the time domain, using an adaptive algorithm such as the Filtered Reference Least Mean Square algorithm (FxLMS). The FxLMS constantly adapts the coefficients of the transverse Finite Impulse Response (FIR) control filters which generate the control signals. In this case, it is straightforward to take the effort weighting into account, by including the norm of the FIR filter coefficients in the error index. This leads to the introduction of a so-called leakage factor (cf. [START_REF] Elliott | [END_REF]], section 3.4.7) in the FxLMS updating formula.

These approaches are similar. They both include a quadratic vector norm of residual noise in the error index and, in the time domain, the leakage factor monitors the tradeoff between control inputs and performance, as does the regularization parameter in the frequency domain. In the case of a single-tone noise, these approaches are exactly equivalent, because a change in magnitude in the control filter coefficients leads to the same change in the control signals. However, no obvious relationship has been established between the two approaches, even though it would be advantageous if the leakage factor to be implemented in real-time could be deduced from preliminary frequency-domain simulations. Indeed, although a trial-and-error procedure can be used in the latter simulations, in order to determine the best effort-weighting parameter, the leakage factor cannot be optimized in the time domain without running the risk of damaging the actuators.

The present paper provides, in the case of a single-tone noise, the theoretical relationship between the regularization parameter in the frequency domain, and the leakage factor in the time domain, which both lead to the minimization of the same error index.

Using this relationship, the leakage factor to be used on-line can be computed from a regularization parameter selected by means of frequency-domain simulations. This paper also presents numerical simulations of FxLMS control for a simple acoustic set-up, in which the leakage coefficient has been computed from the frequency-domain regularization parameter. The simulations show that the real-time control leads to slightly lower control signals and performance than predicted in the frequency domain.

The gap between the respective control results becomes narrower, as the length of the control FIR is increased.

In section 2 of this paper, the two conventionally used techniques for effort weighting in active noise control are briefly recalled. Notations are introduced, enabling the subsequent derivation of the correspondence between these two techniques. The relationship between the regularization parameter and the leakage factor is derived in section 3. Section 4 presents numerical simulations of a simple control set-up, showing that when the corresponding minimization indexes are used, control with a leaky FxLMS algorithm is slightly more conservative in terms of control input limitations than the direct frequency domain optimization.

Weighted error indexes and optimal control

Fig. 1 is a drawing of a typical feedforward active noise control set-up. The reference signal, x, which is correlated with the noise to be reduced by the active control, drives a set of linear filters with vector w(ω) of Frequency Response Functions (FRF) at the angular frequency ω. u(ω) is the vector corresponding to the control filter outputs, which are the actuator inputs, and e is the vector describing the noise at the set of error sensors. e p is the so-called primary noise at the sensors, i.e. the noise without control.

Matrix H(ω) denotes the so-called secondary path matrix of the FRFs between the actuator inputs and the error sensors, at the angular frequency ω. Similarly, the vector f (ω) denotes the primary path between the reference signal x and the primary noise e p .

Optimal control in the frequency domain

Active control at the angular frequency ω can easily take effort weighting into account by using the minimization index (see [START_REF] Elliott | [END_REF]], section 4.2):

J ω = e(ω) 2 + γ 2 u(ω)| 2
(1) where • is the usual vector 2-norm and γ 2 is the effort-weighting parameter. Index 1 can be expanded as:

J ω = H(ω)u(ω) + e p (ω) 2 + γ 2 u(ω) 2 (2)
The input signal vector which minimizes this index is then given by (cf. [START_REF] Tikhonov | [END_REF]]):

û(ω) = H † e p (ω) (3)
where H † is the so-called Tikhonov-regularized pseudo-inverse matrix of H(ω). H † can be computed by inverting H(ω) through Singular Value Decomposition, and by replacing every singular value 1 σ i of the inverse matrix H by σ i σ 2 i +γ 2 . General purpose scientific computing packages include routines for regularized matrix inversion, such as function pinv in Matlab R .

Optimal control in the time-domain

According to [START_REF] Elliott | [END_REF]] (section 3.4.7), effort weighting can be taken into account in real-time FxLMS control by minimizing the error index:

J n = e(n) 2 n + ν w(n) 2 n ( 4
)
where

• e(n) is the vector of the instantaneous error signals at discrete time n and e(n) t is its mean square value over time,

• w(n) is the stacked vector of all the control FIR at time n,

• ν is an arbitrary effort-weighting coefficient.

The filter responses which minimize index J t can be computed using the adaptive form:

w(n + 1) = (1 -νβ)w(n) -βR(n)e(n) (5)
where

• β is a proper convergence coefficient • R(n)
is the so-called filtered reference, providing filtering of the reference signal

x(n) through FIR estimates of the secondary path.

The updating formula 5 is constitutive of the so-called leaky FxLMS algorithm.

Linking the effort-weighting parameters

In this section, the error indexes 1 and 4 are manipulated in order to determine the values of the weighting parameters γ and ν which lead to the same minimization problem.

At first, as a consequence of Parseval's theorem, the error signal contributions have the same form in both indexes, as soon as the FxLMS has converged to the steady-state optimal control. The time-domain minimization index can thus be rewritten:

J n = e(ω) 2 + ν w(n) 2 n = e(ω) 2 + ν w(n) 2 n u(ω) 2 u(ω) 2 (6)
By comparing Eq. ( 4) and Eq. ( 6), it can be seen that both minimization indexes will lead to the same solution if, and only if:

ν = γ 2 û(ω) 2 ŵ 2 (7)
where ŵ is the stacked vector of the optimal control filter FIRs.

Eq. ( 7) formally connects the two effort-weighting factors but vector ŵ in 7 is not a direct result computation of optimal control in the frequency domain. However, in the frequency domain, the FRFs ŵ(ω) of the optimal filters are easy to compute, because the optimal control inputs û are the filtering of the reference signal:

ŵ(omega) = û(ω) (2) x(n) n (8)
The factor (2) used in this formula arises from the definition of x(n) n , as the quadratic mean square value of the reference signal x(n).

The FIRs of the optimal control filters can now be determined from the FRFs by using the fact that, among all the FIR filters with frequency responses ŵ(ω), the optimal control FIR vector has a minimum vector 2-norm because it minimizes index 4.

In the case of a single actuator, the optimal FRF ŵ(ω) is a single complex number and ŵ is a single FIR [ ŵ0 ŵ1 . . . & ŵN-1 ] t , where N is the FIR length, of the form: ŵ(ω) = ŵ0 + ŵ1 e -iω/fs + . . . + ŵN-1 e -i(N -1)ω/fs = 1 e -iω/fe . . . e -i(N -1)ω/fs t ŵ (9)

In the above expression, f s is the sampling frequency of the real-time control. The vector ŵ is theferore the least squares solution of the linear equation:

    ( ŵ(ω)) ( ŵ(ω))     =     1 . . . cos((N -1)ω/f s ) 0 ldots sin((N -1)ω/f s )     t ŵ (10)
This can, for example, be solved in Matlab code by a matrix division using the symbol \.

It should be noted that the FIRs, ŵ, and therefore the computed leakage factor, ν, depend strongly, but not in a straightforward manner, on the length N of the control filter FIRs. Real-time changes in the control filter thus lead to the need for changes in the leakage factor, if the effort weighting is to be maintained at the same level.

In the multiple-actuator case, the stacked vector ŵ of optimal FIRs is obtained in the same way, by stacking the optimal control FRFs on the left-hand side of Eq. (10).

To summarize, the leakage factor corresponding to a given regularization parameter γ can be computed using the following procedure:

• compute the optimal control inputs û(ω) by means of a pseudo-inversion, as discussed in section 2.1

• compute the FRFs of the optimal filters ŵ(ω) using Eq. ( 9)

• compute the FIRs of the optimal filters ŵ using a least-square inversion (Eq. ( 10))

• compute the leakage factor from Eq. ( 7):

ν = γ 2 û(ω) 2 ŵ 2 (11)
The above procedure was implemented for the numerical simulation of a very simple, single-channel control case.

The primary noise was a 300Hz pure-tone e p (t) = π sin(2π300t). The reference signal was taken as x = 3 √ 2 sin(2π300t), which was perfectly correlated with the primary noise. FxLMS was simulated at the sampling frequency f s = 1000Hz. The secondary path was chosen as a pure two-sample delay with FIR h = [0 0 1].

For the results given below, the FxLMS convergence coefficient (β in Eq. ( 5)), normalized by the control filter FRF length and the mean square value of the reference (cf.

[ [START_REF] Elliott | [END_REF]], section 3.4.4), was assigned to 0.4. The mean-square values of the error and control signals were computed over a 2 second period, after running the FxLMS for 8 seconds, which was much longer than the apparent control convergence time.

Table 1 shows the resulting control performances for a set of regularization parameters γ and control FIR lengths N . u ω and u n are the mean-square values of the control signal in the frequency and time domains, respectively, and A ω and A n are the corresponding values of noise attenuation.

Firstly, table 1 shows that, as expected, the regularization monitors the control signal;

the greater the effort weighting, the smaller the control signal and noise attenuation.

Secondly, table 1 shows that the regularization parameter γ and the leakage factor ν, computed as discussed in section 3, lead to a control signal magnitude and a noise attenuation that are very close. This result confirms, if necessary, that the procedure discussed in section 3 gives effort-weighting parameters leading to the same minimiza- tion problem. However, the control in the time-domain is slightly more conservative when the control filter FIR is very short in length. This is probably due to the lack of accuracy of the FxLMS, when the selected length of the control filter FIR is too short.

Concluding remarks

In the case of a single-tone primary noise, Tykhonov regularization in the frequency domain and leaky-FxLMS active control are exactly equivalent. A simple numerical procedure has been introduced in this paper to compute the leakage factor correspond-ing to a given regularization parameter; simple numerical simulations have confirmed that the two effort-weighting approaches lead to almost the same control results.

In the case of a primary noise involving several tones, the convergence of the FxLMS algorithm can be considered independently at each frequency. A single leakage factor ν(ω k ) could thus be computed from the regularization parameters γ(ω k ), for each frequency. However, since Eq. ( 10) is frequency dependent, a single global leakage factor cannot be computed, even if the frequency-domain parameter is the same at all frequencies. In the case of broadband noise, control cannot be computed independently at each frequency, as a consequence of the causality constraint (cf. [START_REF] Elliott | [END_REF]]); in this case, as discussed in section 2.1, even optimal frequency-domain control cannot be easily computed in the multi-channel case.

Finally, although this paper has focused on active noise control, real-time effortweighting is of interest for the solution of other problems in acoustics. For example, so-called Adaptive Wave-Field Synthesis, which was introduced for the purposes of 3D sound field generation, involves the on-line minimization of an index including a quadratic penalty [Gauthier(2008)]. In this case, the time-domain regularization parameter can also be computed from the frequency-domain parameter, as discussed in section 2, for the case of single-tone synthesis.

Figure 1 :

 1 Figure 1: A feedforward active noise control set-up

Table 1 :

 1 Optimal control indexes related to regularization parameter γ and control

	γ	N u f req	u temp A ω (dB) A n (dB)
	0	2 0.7405 0.7407	+∞	223
	0.05 2 0.7386 0.7099	52.1	27.2
	0.05 10 0.7386 0.7385	52.1	50.7
	0.05 50 0.7386 0.7385	52.1	52.1
	0.2 2 0.7120 0.4357	28.3	7.20
	0.2 10 0.712 0.7076	28.3	27.0
	0.5 2 0.5924 0.1818	14.0	1.1
	0.5 10 0.5924 0.5666	14.0	12.6
	1	10 0.3702 0.2920	6.0	4.3
	FIR length N


Colette Nicoli

Bruno Denet

Pierre Haldenwang

M2p2

Lean flame dynamics through a 2D lattice of alkane droplets in air

Keywords: spray-flame, two-phase combustion, heterogeneous combustion, vaporizationdiffusion front, droplet combustion Nomenclature

Flame propagation along a 1-D array or through a 2D-lattice of fuel droplets has long been suggested to schematize spray-flames spreading in a two-phase premixture. The present numerical work considers the fresh aerosol as a system of individual alkane droplets initially located at the nodes of a face-centred 2D-lattice, surrounded by a variable mixture of alkane and air, in which the droplets can move. The main parameters of the study are s , the lattice path and L ϕ , the liquid loading, which are both varied, whereas T ϕ , the overall equivalence ratio, is maintained lean

). Main results are: a) for large lattice path (or when the droplets are large enough), spreading occurs as resulting from two stages : a short time of combustion followed by a long time lag of vaporization; a classical triple flame (with a very short rich wing) spreads around the droplets; b) spray-flame speed decreases as liquid loading increases; c) an elementary model invoking both propagation stages allows us to interpret flame speed as a function of the sole parameter L s ϕ × ; d) when the lattice path shortens, the spray-flame exhibits a pattern that continuously goes from this situation to the plane flame front;

Introduction

Combustion spreading through a spray is an important concern in a large number of applications, such as diesel engines or gas turbines. On the one hand, the state of the art has clearly identified different regimes of propagation: group combustion of droplets, vaporization controlled propagation, pulsating spray-flames [START_REF] Umemura | Percolation theory for flame propagation in non-or less-volatile fuel spray: A conceptual analysis to group combustion excitation mechanism[END_REF][START_REF] Mikami | Microgravity experiments on flame spread along fuel-droplet arrays at high temperatures[END_REF][START_REF] Suard | Different Spreading Regimes of Spray Flames[END_REF]. Recent microgravity experiments have brought improvements of our understanding of the different mechanisms involved [START_REF] Pichard | Average Droplet Vaporization Rates in Partially Prevaporised turbulent Spray Flames[END_REF][START_REF] Nunome | Flame Propagation of n-Decane Spray in Microgravity[END_REF][START_REF] Nomura | Microgravity Experiments of Flame Propagation in Ethanol Droplet-Vapor_Air Mixture[END_REF][START_REF] Nomura | Effects of pressure on flame propagation in a premixture containing fine fuel droplets[END_REF]. Those well-controlled contributions have been particularly useful in order to provide information on droplet size influence and liquid loading effects on spray vaporization and spray-flame speed promotion. On the other hand, the development of monodisperse droplet generator has led to a number of recent works on flame propagation in an array of droplets under different configurations: 1D array [START_REF] Kikuchi | Numerical study on flame spread of an n-decane droplet array in different temperature environment under microgravity[END_REF], [START_REF] Mikami | Microgravity experiments on flame spread along fuel-droplet arrays using a new droplet-generation technique[END_REF], Chen and Lin (2012), [START_REF] Nomura | Droplet ignition behavior in the vicinity of the leading edge of a flame spreading along a fuel droplet array in fuel-vapor/air mixture[END_REF]) regular 2D array [START_REF] Wu | Transient Convective Burning of Interactive Fuel Droplets in Double-Layer Arrays[END_REF] or random configurations [START_REF] Oyagi | Flame-spread probability and local interactive effects in randomly arranged fuel-droplet arrays in microgravity[END_REF].

In the paper, we present a set of numerical studies of combustion spreading through a facecentred 2D-lattice of droplets. The simulations are carried out using a simplified chemistry with a global exothermic reaction, and we are particularly interested in the combustion of a globally lean spray, a regime which has not much benefited from numerous studies, although this corresponds to a general trend in the applications. Combustion spreading through this system is here aimed at providing new insights into the problem of spray-flame propagation through a mist composed of droplets, the vaporization time of which is not negligible in comparison with the chemical times. Note furthermore that the lattice of droplets is an initial condition only: when the flame propagates, the droplets are allowed to move. They are driven by two cooperative features: droplet vaporization and thermal expansion of the mixture.

Although the overall spray composition is supposed lean (the overall equivalence ratio is set to 85 . 0 = T ϕ

), the local equivalence ratio (ϕ ) can be found rich close to droplets. To study the combustion in such a heterogeneous medium, we have implemented two different chemical schemes, which are both of one-step reaction type. The first scheme is the classical one-step irreversible reaction model, which is known as acceptable in lean pre-mixture. Borrowed from the recent literature [START_REF] Garrido-Lopez | Effect of imperfect premixing coupled with hydrodynamic instability on flame propagation[END_REF], the second one introduces a progress variable that allows us to adapt heat release to fresh composition, in order to get satisfactory results on the rich side, too. Their comparison however shows that the spreading features reported here do not markedly depend on those chemical models. Non-dimensioning is carried out thanks to time and length scales of the stoichiometric premixed flame. Four lattice paths have been investigated:

Modelling the spray flames

At low pressure, flame thickness often appears large in comparison with droplet interspacing. In the recent years, this allowed us to resort to homogenization for developing an appropriate numerical modelling. In such an approach which also neglects droplet inertia, liquid fuel appears as an additional species subjected to enter into the chemical scheme after a vaporization step. Several spreading regimes have been predicted [START_REF] Suard | Different Spreading Regimes of Spray Flames[END_REF], in particular an intrinsic oscillatory regime [START_REF] Hanai | Pulsating flame propagation of PMMA particle cloud in microgravity[END_REF], Atzler et al. 2001] occurring as a Hopf bifurcation. The existence of this regime does not require the implication of differential diffusivity effects [START_REF] Nicoli | Analysis of pulsating spray flames propagating in lean twophase mixtures with unity Lewis number[END_REF][START_REF] Nicoli | Effects of substituting fuel spray for fuel gas on flame stability in lean premixtures[END_REF].

At moderate and high pressure, spray-flame thickness can no longer be large enough -in comparison with droplet interspacing-to allow any process of homogenization. In such a system, spray-flame tends to be controlled by vaporization, the chemical heat release permitting the vaporization of the droplets one after another. Therefore, spray-flame propagates within a heterogeneous mixture with large droplets, as it is the case of the n-decane experiment by [START_REF] Nunome | Flame Propagation of n-Decane Spray in Microgravity[END_REF]. This is also the configuration of the experiments by [START_REF] Nomura | Microgravity Experiments of Flame Propagation in Ethanol Droplet-Vapor_Air Mixture[END_REF], which was concerned with globally lean mixtures of ethanol and air (with an equivalence ratio about 0.8): with droplet interspacing typically of the same size as spray-flame thickness. The aim of the present work is hence to provide a numerical analysis of such lean configurations where the flame "feels" heterogeneities at the droplet length scale. We fix the overall equivalence ratio to 0.85 for the sprays we study. The mist structure is schematized by a face-centred 2D-lattice of alkane droplets in a leaner pre-mixture alkane-air.

On the one hand, the present numerical modeling considers the usual set of conservation laws: mass, momenta, energy and species. Since the accurate chemical schemes for alkane are too complex for efficient numerical simulations, the standard approach searches after a simplified chemical kinetics. As this work is devoted to spray with droplet inter-distance not small in comparison with the characteristic combustion scales (at least, of the same order), our spray-flames will propagate through a medium with varying chemical composition. The simplest manner consists of choosing an irreversible 1-step reaction, the parameters of which are adjusted to mimic the flame dynamics. It is known that the classical one-step Arrhenius law largely overestimates the adiabatic flame temperature on the rich side. Although the calculations presented here concern lean sprayflames, we use two different manners of computing the heat release: either the classical (constant) heat of reaction, or a heat of reaction as a linear function of the fresh gases equivalence ratio [START_REF] Nicoli | Analysis of one-step chemistry models for flame propagation in various equivalence ratio premixtures of high alkane-air[END_REF]. The second model correctly mimics the characteristics of the premixed single-phase flame (adiabatic flame temperature and flame speed). The use of this model in a heterogeneous mixture is carried out by introducing the mixture fraction as proposed in [START_REF] Garrido-Lopez | Effect of imperfect premixing coupled with hydrodynamic instability on flame propagation[END_REF][START_REF] Garrido-Lopez | Effect of imperfect premixing coupled with hydrodynamic instability on flame propagation[END_REF][START_REF] Fernandez-Tarrazo | A simple one step chemistry model for partially premixed hydrocarbon[END_REF] to suitably adapt heat release to the local composition of the unburnt spray. (see below). The latter point seems, a priori, to be important, since the resulting chemical scheme is able to take account of the heterogeneities in the mixture composition (it must be recalled that flame dynamics depends on the manner the mixture is performed, in the close vicinity of the flame especially).

Non-dimensioning

Non-dimensional form of the conservation laws is performed with the use of the theoretical data related the stoichiometric (gaseous) premixed flame, as derived in the theoretical papers by [START_REF] Joulin | Linear stability analysis of two-reactant flames[END_REF], or [START_REF] Garcia-Ybarra | Soret and Dilution Effects on Premixed Flames[END_REF]. We define the stoichiometric flame temperature, as * b T , given by ( )

f f p u u b M C Q Y T T ν * f * + = (1)
Temperature and species mass fractions are handled under the reduced forms ( )

u b u T T T T - - = * ) ( θ ; * ,u i i i Y Y = ψ (2)
(i=f for the considered alkane and i=o for oxygen) As for the time and length scales, we define them with the use of * ,b th D , the thermal diffusivity coefficient of the burnt gases, and * L U , the stoichiometric (single-phase) flame speed, given by

( ) [ ] ) / exp( 4 ) ( * * 2 * * 2 * * 3 2 * , b A b f f b p b b u b L T T B Y M C Ze U u o -         = ν ρ ρ λ ρ ρ (3)
This allows us to establish the scalar conservation laws as follows

) , , ( ) F( ) ( 1 . θ ψ ρ ϕ θ λ ρ θ ∂ ∂θ i p W div C V t + ∇ = ∇ + , ( 4
) ) , , ( ) ( 1 . T W M D div V t i i i i i i i ψ ρ ν ψ ρ ρ ψ ∂ ∂ψ - ∇ = ∇ + , ( 5
)
where the reaction rate is now defined by

      - + -               = ) 1 ( 1 1 exp 4 ) , , W( 3 * 2 * * 3 i θ γ θ ρ ρ ψ ψ ρ ρ θ ψ ρ Ze Ze b o f b u (6) with ( ) 2 * * ) ( b u b A T T T T Ze - = and * * ) ( b u b T T T - = γ .
The overall equivalence ratio of the spray is given by the ratio of the total amount of fuel to the total amount of oxygen in the whole lattice:

∫ ∫ = ice fresh latt o ce fesh latti f f f o o T dv dv M M ρ ρ ν ν ϕ , ( 7
)
while the quantity L ϕ [resp. G ϕ ] only takes account of the fuel density under liquid phase [resp. gas phase]. We obviously have

G L T ϕ ϕ ϕ + = .

Combustion models in a heterogeneous medium

As mentioned before, the first model -denoted by " " 0 Q -supposes that the heat of reaction is independent of the local composition. This assumption is only satisfying for the lean side of the composition. On the other hand, we searched after an improvement useful for the rich side, because rich combustion can affect the close vicinity of the droplets. As well-known, combustion on the rich side is characterized by the production of metastable species, the enthalpy of which reduces the flame temperature and therefore the flame speed. A simple analysis of the enthalpy budget at equilibrium and the resulting flame speed has been carried out for several alkanes in [START_REF] Nicoli | Analysis of one-step chemistry models for flame propagation in various equivalence ratio premixtures of high alkane-air[END_REF]. This approach allowed us to adapt heat of reaction to fresh gas composition. It turned out that the procedure led us to a good agreement with the experimental data on gaseous premixed flame speed. The result took the form of ) F(ϕ , a multiplying factor affecting the heat production term in the energy conservation law.

) F(ϕ depends on the equivalence ratio of the fresh gases, linearly as [ ]

) ( ) F( 1 1 - - = ϕ ϕ (8)
where α is a coefficient depending on the considered fuel [refer to a forthcoming paper by the same authors].

Next, we have to leave the concept of homogeneous fresh mixture for considering a medium of variable composition. We now follow the flamelet spirit developed in Garrido-Lopez and Sarkar (2005) (see also [START_REF] Fernandez-Tarrazo | A simple one step chemistry model for partially premixed hydrocarbon[END_REF]). It is well known that mixture fraction is a quantity preserved when crossing a premixed flame, as long as Lewis number is close to unity. As for the opposite case of a diffusion flame, the mixture fraction only results from the transport-diffusive processes and allows us to find the flame positioning. For the triple flame (the intermediate situation), it permits to predict the correct mixing in the fresh gases just in front of the flame. In other words, mixture fraction allows us to assess the mixture composition that the flame burns. More precisely, we compute at any point of the domain ). In other words, the locus where an alkane spray-flame can propagate occupies a limited area (in Z) during mixing. To sum up, the method considers Z as a continuous marker of the equivalence ratio in the fresh mixture right before the flame location.

( ) ( ) r / r Z o f + + - = ν ψ νψ
Furthermore, for the diffusion flame, a classical result leads to linearly link mass fraction of reactants with mixture fraction as follows :

( )

Z injection f f ψ ψ = and ( ) ) 1 ( Z injection o o - = ψ ψ (10)
Note these hypotheses are consistent with the definition of Z . Consequently, in this context, the equivalence ratio that results from a diffusion problem now reads [START_REF] Garrido-Lopez | Effect of imperfect premixing coupled with hydrodynamic instability on flame propagation[END_REF]]

( ) ( ) ( ) Z r Z Z inj o inj f o f - = - = = 1 ) 1 ( ν ψ ψ ν ψ νψ ϕ (11)
Hence, the previous adjustable heat of reaction introduced in the equation of energy now reads [ ]

) 1 ) 1 ( / ( 1 ) F( - - - = Z r Z ν α ϕ if ( ) [ ] 2 , 5 . 0 1 ∈ -Z r Z ν or 0 ) ( = ϕ W if ( ) [ ] 2 , 5 . 0 1 ∉ -Z r Z ν . ( 12
)
For the sake of illustrating the efficiency of the adjustment, the homogeneous laminar flame speed of octane-air premixture obtained for 33 . 0 = α is plotted in Fig. 1. (Ze being set to 8) and compared with experimental data. Lastly, as in the conservation laws appears the velocity field V , the reaction-diffusion system is coupled with the Navier-Stokes equations. The overall numerical scheme considers periodic boundary conditions in the (y-) direction transverse to flame propagation, open boundary conditions in the downstream (x-) direction, and closed in the x-direction upstream from the lattice. The numerical approach of low Mach number type has previously been described in [START_REF] Denet | A numerical study of premixed flames Darrieus-Landau instability[END_REF]. In contrast with this previous paper which used constant diffusion coefficients, the present numerical approach uses diffusion coefficients varying with temperature. At low temperature the diffusion coefficient is very small, at high temperature we use the diffusion coefficient of an ideal gas, and the formula used interpolates between these two regimes. The result is that diffusion is effectively frozen at low temperature. Let us also note that we use a high Lewis number for the fuel (1.9), which also tends to reduce droplet diffusion. The resulting model is denoted by

)" ( " z Q

Numerical experiments

The fuel droplets are positioned at the nodes of the face-centered lattice for a given liquid loading. . This definition can sometimes be misleading because f

x is not -by definitiona monotonic function of time: when droplet vaporization is long (in comparison with reaction time), the temperature profile can flatten and f

x can seemingly admit a regressing position. To summarize the conditions of the numerical experiments, in Figure 2 we provide the initial conditions given to the different scalar fields. Figure 2 a recalls the pattern of the initial fuel supply. Figure 2.b illustrates the temperature profile which starts to ignite the lattice, whereas Figure 2.c displays the oxygen initial mass fraction field.

Two-stage spreading

The plot of f x vs. time presents various shapes as shown in Figure 3, where the flame location is drawn versus time. Combustion spreading develops from the right to the left in the x-direction. This is why the overall slope of the flame position vs. time, i.e. the flame speed, is negative. On the one hand, for large liquid loading, or equivalently for large droplets, the flame front does not propagate monotonously. There are in fact two different stages during the propagation from a large droplet to another: the first one corresponds to vaporization; it is characterized by an apparent front regression, due to the fact that the temperature profile becomes less sharp during the preheating of air and fuel vaporization; therefore, the locus where 5 . 0

) ( = > < f y x T
can appear as regressing. This effect is increased by the fact that gas phase is produced and heated, and the flow that results from the gas expansion pushes this locus back. The second stage is much shorter in time and corresponds to the reaction stage, which is itself decomposed into two sub-steps: a very rapid propagation of a triple flame-like front which goes around the droplet, followed by the propagation of a slower flame that burns the mixture between the current droplet and the next one. This two-stage propagation remains noticeable as far as the droplet radius is larger than 0.15 (i.e. a small fraction of the stoichiometric gaseous flame thickness). Indeed, this two-stage propagation also disappears when the initial pre-mixture increases (i.e for vanishing L ϕ ), as illustrated by the lowest two curves in Fig. 3. Those curves correspond to a flammable pre-mixture (their slope is always negative), for which during the vaporization time simultaneously occurs the propagation of a very lean (and therefore slow) flame.

On the other hand, when the droplet radius diminishes (simultaneously with liquid loading), vaporization time becomes shorter, leading us to a flame propagation through a nearly homogeneous medium; two cooperative arguments are advanced for explaining this behaviour: as liquid loading decreases, the pre-mixture increases (since 85 . 0 = T ϕ is given), whereas the vaporisation time becomes negligible for smaller and smaller droplets.

For the sake of illustrating the propagation through a large droplet system, we have plotted heat release at various times after ignition in Figure 4. The succession of events can be told as follows. Figure 4.a) provides us with the first instant after ignition. The flame is nearly flat with a very weak heat production, except in the close vicinity of the droplet where the reaction is developing. Figure 4.b) illustrates that the flame skirts round the droplet that vaporizes in the same time. Note the flame is not a strict triple flame, since the rich wing is hardly visible (likely due to high gradients on the rich side). Figure 4.c) shows the diffusion flame that remains behind the front. Combustion spread is now very slow since a weak front of reaction is sustained before the burning droplet. In Figure 4.d) this reaction front becomes even weaker. In the meantime, the next droplet vaporizes. Note additionally that the diffusion flame occupies a large "volume". This is due to the large stoichiometric coefficients of heavy alkanes. The above sequence of pictures -with their corresponding instants-corroborates the previous interpretation of Figure 3 about the different stages that characterize the combustion spreading through the lattice. Moreover, the monotonic combustion spreading from the right to the left confirms that there is no front regression (as could have been a possible wrong interpretation of Figure 3).

When the lattice path becomes smaller, the vaporizing fuel pocket (as schematized in Figure 4.b) still remains visible. But, since the flame thickness becomes of the same size as the lattice path, the vaporization of the next droplet is quickly activated. In other words, combustion spread appears rather as a continuous process, close to the propagation in a premixed gaseous medium. The transitional process is investigated in the last paragraph (see Fig. 8).

Note also that the droplets can move during the calculation. As an illustration, the comparison of the heat release presented in Fig. 4

Effect of liquid loading

For fixed overall equivalence ratio ( 85 . 0 = T ϕ

) and lattice path s , increasing the liquid loading corresponds to an enhancement of the droplet radius. Consequently, the vaporization characteristic time being a priori scaled by the square of the radius, the time lag devoted to vaporization is expected to increase. This is illustrated in Figure 5 where the front history is plotted for a large lattice path and various liquid loading (or various corresponding vapour press ), the flame has to utterly create the mixture permitting its propagation. This is noticeable that the necessary time becomes very large. The different time lags devoted either for vaporization or for reaction are analysed in Table 1, where we report on the characteristic times as indicated in Figure 5. Let us define as v τ , the characteristic time for vaporization. As mentioned in Figure 5, its value reads ) appear less markedly. Accordingly with Table 1, the characteristic time of the combustion stage only depends on the lattice path (see the fifth column). The combustion step occurs after a vaporization step which also depends on the actual droplet size. Of course, the scaling of the latter time is more or less found in direct ratio to the square of the radius, or equivalently proportional to the product L s ϕ 2 .

To analyse the part played by liquid loading, it must be recalled that combustion must stand far from the droplet, since we are concerned with heavy alkanes. Vaporization must therefore be carried out enough to fill the sphere surrounded by the flame. In other words for globally lean spray, combustion develops intensively only when vaporization is rather complete. This is corroborated by the fact that c τ , the combustion time, does not depend on liquid loading (see the last column of Table 1).

On the other hand, for a given lattice path the liquid loading modifies the droplet radius and the initial surrounding gas composition. Therefore, v τ , the vaporization time depends on liquid loading.

Accordingly with the penultimate column of Table 1, we found v τ proportional to liquid loading.

Although we also possess a large set of results for the cases 3 = s and 5 . 1 = s , let us observe that Table .1 does not contain any data from those cases. The reason is the following: as the droplet radius decreases, the determination of the two stages scaled by v τ and c τ more and more becomes arbitrary. So that, to derive the scalings of v τ and c τ , we have decided to exploit the data from large droplets only.

Spray-flame speed

We have observed that combustion spreads through the lattice with an overall velocity given by the mean slope of the curves drawn in Figures 3 &5. In what follows, this slope is called "spray-flame speed". As mentioned in Introduction, we performed the numerical simulations with the use of two different chemical models. Hereafter, the model that uses Z , the mixture fraction, is called "Zcorrected 1-step" (and labelled by ) (z Q ), while the non-corrected (classical) model is called "Standard 1-step" (and labelled by 0 Q ),. In Figure 6, we plot the numerical spray-flame speed, normalized by the single-phase premixed flame at 85

. 0 = = T G ϕ ϕ
, with respect to droplet radius. In this figure , both chemical models are compared for each case of liquid loading. For a given symbol, the dashed line expresses the results for the standard 1-step model, while the plain line represents the "Zcorrected 1-step model". In the cases drawn here, we observe that the discrepancy remains weak.

Next, we want to use the previous analysis in terms of characteristic vaporization and combustion time scales, in order to interpret the data shown in Figure 6. If we consider a combustion spread through a distance corresponding to the lattice path s . As the lattice is face-centred, we meet 2 droplets per lattice period. Therefore, according to Figure 5, we need to spend two vaporization times and two combustion times. Thus, the spray-flame speed reads: Now, we turn towards Table 1, where the scales for vaporisation and combustion are analysed. Taking the mean value of the concerned column, we can set Next, we compare the results of this elementary theory with the data obtained from the numerical simulation. This is carried out in Figure 7, where we plot the same velocity data as those of In Figure 7, we perform the plot for the data resulting from both kinetics models. We again observe that the discrepancy between both models remains feeble. Therefore, we conclude that the rough theoretical expression ( 14) is valid for both chemical models. Note that the cases s=1.

) . /( /U U L L SF ϕ 5 0 1 1 + = (15)
which corresponds to the dashed line in Figure 7, while the plain line is related to equation ( 14). Note additionally that -for a given liquid loading-droplet size are in direct ratio to lattice path. In other words, for large droplets, both equations ( 14) or (15) retrieve the classical result characterizing the spray combustion regime controlled by vaporization, which behaves as

Effect of diminishing the lattice path

As the lattice path shortens, the characteristic time for vaporization (i.e. ), total droplet vaporization and some degree of oxygen diffusion (inside the vapour puff resulting from the droplet vaporization) are carried out before flame spreading. Therefore, when a critical value of s is reached (from above), a rich premixed flame starts to slowly cross the vapour fuel pocket. Then, if the lattice path still decreases (and oxygen diffusion increases), this premixed flame crosses the vapour pocket faster,. Behind the rich premixed flame, a diffusion flame takes place around the puff of fuel vapour. This process for small s is described in Figure 8, where heat release and the corresponding fuel mass fraction are presented at three consecutive times (with a time interval of 5 . 0 = ∆t

). In Figure 8, the puff of vaporized fuel (added with oxygen diffused from its vicinity) is swept by a rich flame, leaving behind a diffusion flame surrounding a hot gas pocket, which still contains an amount of fuel. In the present situation, no clear triple flame occurs since the flame thickness is large enough to vaporize the droplet in the pre-heating zone and to allow diffusion of oxygen towards the fuel puff. The transition to classical (single-phase) premixed flame occurs when this fuel pocket appears negligible for smaller lattice path.

Conclusion

The present numerical study on lean spray combustion uses a lattice to schematize the droplet deposition. Our results concern the influence of the spray characteristic parameters ("lattice path and liquid loading" or equivalently "droplet radius and liquid loading") on flame dynamics. A particular attention has been paid to focus on large lattice path in comparison with the flame thickness. Additionally, we investigate the transition to premixed flame when the lattice path becomes small.

Two simple chemical schemes with a 1-step exothermic reaction have been used. The first one is associated with the standard Arrhenius law with a reaction heat independent of the equivalence ratio of the mixture to burn. The second one adapts the heat of reaction to the equivalence ratio deduced from the field of mixture fraction, with the introduction of a correction applied particularly to rich mixtures.

For both cases of chemical models, we have obtained similar qualitative results in flame structure, as well as quantitative values of spray-flame speed. This indicates that corrections of the triple flame structure on the rich side are probably not very important for flame propagating in a globally lean spray.

Our simulations include a prevaporized fraction of the fuel, which allows the flame propagation from one droplet to the next one through a variable lean pre-mixture. Even for ), no flame extinction has been observed (although combustion spreading was found very slow). We have shown that the combustion spreading for relatively large droplets (compared to the premixed flame thickness) is characterized with two stages: a slow phase corresponding to the partial vaporization of the next droplet until a flammable mixture is carried out, and a faster phase corresponding to the propagation of a triple flame around the droplet.

On the other hand, for smaller droplets, the flame propagates in a regime much closer to a premixed flame, but a premixed flame with a heterogeneous equivalence ratio inside the flame thickness. This regime should deserve a further study because we did not obtain (as visible in Figures 6 and7) any maximum of the flame speed at non-zero droplet radius. This maximum has, however, been observed in experiments on lean spray-flames in microgravity by [START_REF] Nomura | Microgravity Experiments of Flame Propagation in Ethanol Droplet-Vapor_Air Mixture[END_REF].

Further studies will turn our attention towards other situations of overall equivalence ratio. Concerning even leaner sprays, we envisage the effects of a random distribution of droplets to explain unexpected flammability of spray, as observed in the experiments of [START_REF] Nunome | Flame Propagation of n-Decane Spray in Microgravity[END_REF]. With respect to rich sprays, the experimental contributions are more numerous. The corresponding amount of data should help us to further check the different reduced chemical schemes we have presently used. .

ϕ

  is the equivalence ratio linked to the fuel initially under liquid phase. For each set of these parameters, the droplet size is assessed to keep the overall equivalence ratio fixed

  oxidizer-alkane mass ratio for a heavy alkane and r , the oxidizer proportion in air with . The dotted line, in figure1, represents the mixture fraction for any fresh mixture equivalence ratio. It can be observed that the whole flammability domain corresponds to very small mixture fractions:

Figure 1 :

 1 Figure 1: Octane-air flame speed vs. equivalence ratio: comparison between numerical premixed gaseous flame speeds computed with 33 . 0 = α

ϕ

  radius will only depend on lattice path s and liquid loading L ϕ Four lattice paths have been investigated: the stoichiometric premixed flame thickness. Four different liquid loadings have been investigatedis the equivalence ratio linked to the amount of fuel initially under liquid phase. At the open (downstream) end, the temperature field is initiated with the profile of a premixed flame, that allows us to ignite the first droplet of the lattice. To follow the combustion spread, we compute) (x T y > <, the mean temperature averaged in the periodic (y-)direction. Then, we decide to consider as the flame front, fx ,

Figure 2 :

 2 Figure 2: initial fields; a) fuel mass fraction, b) temperature, c) oxygen mass fraction.

Figure 3 :

 3 Figure 3: "flame location" vs. time for lattice path s= 6 ( 85 . 0 = T ϕ

  Figure 4.e) illustrates a new stage of intense heat release that corresponds to the combustion of the next droplet, similarly with Figure 4.b). Lastly, Figure 4.f) displays the last instants of life of the diffusion flame, that surrounds the current burning droplet.

  Figure 4: Snapshots of heat release at various times after ignition ( 6 = s ,

Figure

  Figure 5: "Front" location vs. time; for 12 = s and various vapour pressures.

Figure 6 :

 6 Figure 6: Spray-flame speed (i.e. the combustion spread through the lattice) versus droplet radius for various liquid loading, accordingly with both chemical models

  This expression being in terms of the stoichiometric gaseous premixed flame, we are interested in its normalization with respect to the gaseous flame of the same equivalence ratio, denoted by )

  Figure 6 (again normalized with 0.153, the single-phase flame speed at 85

Figure 7 :

 7 Figure 7: Spray-flame speed (reduced by the single-phase flame speed in a gas mixture of the same overall equivalence ratio) versus the factor s L ϕ .

  5 and s=3 correspond to the smallest values of the factor s L ϕ in figure 7. It is therefore not surprising that the agreement of the numerical data with our simple model [i.e. formula (14)] is weak in the domain of the small factors s L ϕ . Of course, a better agreement would be obtained with the following fit curve

  s

Figure 8 .

 8 Figure 8. Spreading through a small lattice (s= 3 ; φ L = 0.25): heat release (top) and fuel mass fraction (bottom) at three times separated by 5 . 0 = ∆t .

Table 1 : Characteristic times of vaporization and combustion, and their corresponding scalings, for various liquid loadings and lattice paths (with the "Z-corrected 1-step" model).

 1

		85	0.705 164	34	330	1.34	2.8
	12	0.65	0.615 118	28	311	1.26	2.3
	12	0.45	0.51 77	28	298	1.19	2.3
	12	0.25	0.37 51	22	373	1.42	1.8
	6	0.85	0.351	52	11	422	1.70	1.83
	6	0.65	0.3	41.5	11.8	461	1.77	1.97
	6	0.45	0.255	31.3	10.6	481	1.93	1.77
	6	0.25	0.185	19.8	11	578	2.20	1.83

-R (as observed by Ballal & Lefebvre 1981 or predicted in Suard et al

. 2001).

Acknowledgements

The present work has received the support of the Research Program "Micropesanteur Fondamentale et Appliquée" GDR n°2799 CNRS/CNES under the contract CNES/11-5349.