Hugues Bonnin
email: hugues.bonnin@continental-corporation.com

The Certification Challenges of Connected and Autonomous Vehicles

Keywords: autonomous and connected vehicle, safety critical software, certification, big-data, data-centric, cloud computing, Artificial Intelligence/Machine Learning

In the context of connected cars, some technologies or approaches are unavoidable: data-centric and big data; AI/MLbased systems; cloud computing and agility. These points are not limited to connected cars, but are trends in all transportation fields at least. In this paper, we analyze the interest of these technologies. Then, we analyze that these techniques are not completely in phase with traditional ways of developing safety critical software because the standards used for this purpose do not rely fundamentally on the same approaches to develop software.

I. INTRODUCTION

In the beginning of 2017, Continental Automotive founded a new entity: Continental Digital Services France (CDSF). This new entity will propose services around one basic principle: generating added value data from automotive on-board data.

Modern vehicles produce a huge quantity of data through their numerous sensors; as they are connected to the internet, this data can be uploaded on off-board infrastructures. Data coming from a large number of vehicles can be mixed together to constitute a very large data-lake. From this data-lake a quasiinfinite number of computations can be imagined to produce added value services based on data: weather, road information, traffic, car maintenance, etc. The data produced by these services may be used in different ways: either it can return to the vehicles, to be used by on-board computers; or it can be used by third party services on the internet.

II. TECHNOLOGICAL TRENDS AND THEIR CHALLENGES

There are some fundamental technologies or approaches that underlie these services.

A. Data Centricity

First of all, these services are data centric, and the volume, the diversity and the velocity of data is why we call it big data. One particular and typical example of data is maps, with their multiple information layers. In the case of connected cars, a lot of use cases produce, enhance or use maps: local weather, road and traffic signs recognition, etc. But maps are only one example of big data: other use cases are data collected for car maintenance or data collected for driver monitoring. Note that these cases are not specific to the automotive field domain: maps, maintenance data, driver or pilot data are also available for aircraft or rail transportation, for example.

B. Artificial Intelligence and Machine Learning

A lot of these use cases are well supported by machine learning approaches, because these techniques allow us to enhance automation, to address the complexity of the relationship between data, and to address the high volume that they often represent. So, even if the technologies behind artificial intelligence and machine learning (AI/ML) are not so recent, their usage has been boosted by the emergence of these data collections. In the context of the automotive industry, this approach is absolutely essential to paving the way to autonomy: analyzing the environment, which is very rich and complex. Visual, radar or sonar image analyses are the main examples. To build and use a correct and sufficiently precise representation of their environment, the sensors of the Autonomous vehicles (AV) have to analyze a very large collection of data. AI/ML approaches are thus fundamental in achieving this, because the intrinsic complexity of this information makes traditional approaches inapplicable.

C. Cloud Computing / Infrastructure

The "large horizon" perception, which is the information based not only on one vehicle, but on all the vehicles present in the same area, in a time frame that ranges from a few seconds to a few days, is the natural extension of the vehicle's environment. Then AI/ML approaches occur not only in embedded systems, but in off-board systems as well. The collection from multiple vehicles is off-board, on some type of cloud infrastructure. Whatever the nature of this cloud infrastructure and associated services (private, public, mix, SaaS, IaaS, etc.), the big data nature of the collections makes them unavoidable. These solutions are not only useful to address the high volume of collected data, but they are essential in dealing with scalability. In fact, scalability is an essential dimension of these new services around collected data. The launch of such services is in no case a "one shot" operation. The scalability is multidimensional: the number of vehicles is one dimension; but the type of data may vary in time, too; the accuracy of data could also vary.

D. Agility

This variability implies adopting a global approach that is also agile. Indeed, agility becomes a fundamental pillar of these services, because all is not predefined at the beginning; the developments have to start before all the solutions are found. Development is launched before all aspects of the design and even the specifications are defined. And we anticipate that it will continue to evolve during operation, making a continuous and permanent life-cycle from the first inception to operation and continuous evolution. The necessity of relying on Agility and DevOps concepts is (at least) twofold: first, the real needs of users are only known when they use their product; second, the speed at which technology is evolving would prevent traditional (V cycle) approaches from integrating them.

We have seen that the services around connected vehicles are closely linked to four fundamental pillars: big data and data centricity, AI/ML based systems, cloud computing, and agile approaches (embracing DevOps). In fact, these considerations are global trends not only in the automotive industry, but also in other fields like aeronautics or railway.

The question then is how to include these services in safety critical functions. Indeed, these services will be part of critical functions, especially in the context of AV, because all the information necessary to build the environment of the vehicle is fused, wherever it comes from. More than that, the more information the vehicle has, the safer it is: then inclusion of this information concurs to the overall safety. It means that we must be confident in technologies and approaches.

III. CURRENT SOFTWARE CONFIDENCE MEANS

Traditional ways to be confident in software aspects of critical systems are based on conformance of the software development processes to a standard. The standards are written by a community of experts who capture the state of the art of software development. The best practices, which is the fruit of experience feedback, are those which instill confidence in the software built following these practices. Therefore the standards are written based on these best practices, and give the communities that follow these standards a way to build safety critical software with an acceptable level of malfunction risk. But even if these standards are built to be as independent as possible from technologies, some technological breakthrough may show some limits in regards to existing standards, because these breakthroughs deeply stir the state of the art.

In this article, we mainly reference two standards: DO178 [START_REF]/ED-12C, Software Considerations in Airborne Systems and Equipment Certification[END_REF] and ISO26262 [START_REF]Road Vehicles -Functional Safety[END_REF]. The first version of DO178 was issued in 1982, the second, DO178A, in 1985, and the DO178B in 1992. The last revision, DO178C, was released in 2012. Without minimizing the effort that has been done to issue the last version, there was no extensive revisit of the way to develop software. ISO26262 was issued in 2012; the second version will be issued in 2018. Part 6 of this standard addresses software development. This part wasn't modified between the two versions. The fundamental principles for software are roughly the same in DO178 and ISO26262. It is to be noted that one part of the revision of ISO26262 has been separated in a new document, and is addressing the Safety Of The Intended Functions (SOTIF) of systems ( [START_REF]Road vehicles -Safety of the Intended Functionality (SOTIF)[END_REF]); it is still under development. In aeronautics, DO200B [START_REF]Standards for processing aeronautical data[END_REF] defines the way to certify databases used in aeronautical embedded systems. Other standards could have been referenced like IEC61508 (generic for industry), EN50128 (railway) or ED153 (European CNS/ATM), or ISO12207; but for the purpose of this article, which is not a detailed study, [START_REF]/ED-12C, Software Considerations in Airborne Systems and Equipment Certification[END_REF] and [START_REF]Road Vehicles -Functional Safety[END_REF] are considered sufficiently representative.

IV. CERTIFICATIONS CHALLENGES BROUGHT BY THE FOUR PILLARS

We think that the technologies exposed in section II constitute a breakthrough point regarding the existing standards. Indeed, we can observe how big data, AI/ML, cloud computing and agile approaches are taken into account in the existing standards.

A. Data Centricity

The first concept is "data centricity." Traditional approaches are more "algorithm-oriented" than "dataoriented". The practices described in standards rely on how to be confident in writing code that manipulates data, not in describing data itself. In a data-centric approach, we would like to evaluate how we can rely confidently on data, without knowing how data is built. Indeed, the way the data is established can be very complex. For example, look at the way a map is established: if it comes from satellites images, the huge amount of transformations from the first signals in the payloads of the satellites, to the pixels displayed in a map on a screen is so complex, that it is unfeasible to trace and make them conform to a standard. The DO200A [START_REF]Standards for processing aeronautical data[END_REF] attempts to formalize these transformations, but is only applicable to a small part of these transformations: the last part of the long chain. We have to find some ways to make robust data, for example including or using already existing intrinsic properties, like redundancy, internal dependencies, diversity, quality factors, etc. We don't know if standardizing a general way to be confident in data is possible, or if only domain and specialized perspectives are real possibilities. Nevertheless, the data centricity has to be fully taken into account for the future development of critical systems.

B. Artificial Intelligence and Machine Learning

The second technology, AI/ML technologies, relies on the first one (data), but adds one supplementary layer in the engineering, which is not taken into account in existing standards: the learning phase. We know learning processes for humans, but we don't know them for machines. Traditional approaches rely on best practices to write software, not on machine training practices during a learning phase. So we have to invent a way to be confident in the systems which will be built through a learning phase: the choice of data sets, the learning process roles, the characteristics of trained models, model verification, etc. Note that this reflection has to be conducted even in the cases where the learning phase is offline (i.e. before operation of the system).

C. Cloud Computing / Infrastructure

The third dimension is the infrastructure: cloud computing. In fact, cloud computing embeds a lot of incompatibilities with traditional approaches: it is made by a lot of COTS (commercial/component off the shelf), when traditional approaches have difficulties dealing with COTS, because their development processes are black boxes, and the confidence is entirely based on the way the development is done. So the only accepted COTS, in the context of a standard, are those in which development has followed this standard. This is a real hard point to build trust in black boxes. The fact that the workgroup writing the latest version of the DO178 standard (DO178C) can't reach an accepted text is proof of the immense challenge.

Moreover, in the case of cloud computing, more than simple software component COTS, these technologies are totally black box services based on software and hardware packages provided by third party specialized companies. The hardware and software infrastructures are highly complex, while traditional approaches tend towards simplicity and transparency. These infrastructures are adapting permanently and changing continuously, contrary to the stability of traditional approaches. Then the general idea to build confidence despite all these "non-conventional" features would be to consider that these black boxes can only provoke a limited number of types of faults. Indeed, traditional approaches are binary: either the confidence is total in the case where the standards are followed, or there is no confidence at all in the other cases. We think that for some systems, we could have confidence that certain classes of fault can't arise. The elements for that are twofold: first, the system can't provoke everything (by its architecture, and its functionality), and there is only a certain range of results that are possible; second, there are some visible characteristics that give confidence: experience made by the service, or renowned experience of the provider, open source access, etc.

D. Agility

The last point is about organization. Agility principles are not those which have been taken as fundamentals in the existing standards that rely on a strong planning and predictability. Agility requires performing activities before they are completely defined, preferring feedback loops to intangible ways to proceed. Indeed DO178C [START_REF]/ED-12C, Software Considerations in Airborne Systems and Equipment Certification[END_REF] or ISO26262 [START_REF]Road Vehicles -Functional Safety[END_REF] have a large part devoted to planning, and even if modification can be made to the plans, the general idea is to follow them during development, because they have been accepted by all stakeholders (including authorities in the case of aeronautics). In fact, agile is "change-centric", and traditional approaches are more "plan-centric"; in the first approach, the normal way is continuous change and even uncertainty and unknown; in the second approach, the change is taken into account, but more as an exception than a normality. For example, in DO178C [START_REF]/ED-12C, Software Considerations in Airborne Systems and Equipment Certification[END_REF], change and problem management are considered in the same chapter.

Agile and existing standards are opposed on another point: oral and written exchange. The first one promotes oral exchanges between the stakeholders, because it is easier, more efficient, and, in the end, less ambiguous. The standards, on the contrary, have the culture of written record of all activities, and leave no room for activity with no written record. A typical activity is coding. In agile approaches significant confidence and autonomy are given to the coder, because it is considered that doing so will be more efficient, even in terms of bug rates. In standards like DO178C [START_REF]/ED-12C, Software Considerations in Airborne Systems and Equipment Certification[END_REF] or ISO26262 [START_REF]Road Vehicles -Functional Safety[END_REF], coding is considered to be a very defect-prone activity, and is therefore strongly limited and "fenced" by strict detailed design and strict coding rules.

So we can really ask ourselves if agile approaches do not have something new to bring to traditional approaches. Some experience (see [START_REF] Chenu | Agile & Lean development for avionic software[END_REF]) have shown that agility is compatible with DO178 [START_REF]/ED-12C, Software Considerations in Airborne Systems and Equipment Certification[END_REF]. But, by definition, agile is compatible with everything; in [START_REF] Chenu | Agile & Lean development for avionic software[END_REF], the general agile principles are applied to the standard, making the standard's activities an objective of the agile production. But doing so doesn't explore the principles and the reasons of its efficiency. It's like doing sport with a Citroën 2CV: you can do it, by adding a motor and reinforcing the chassis; but fundamentally, you don't have the best candidate for sport if you don't completely revisit the design. So making approaches compatible doesn't mean that you use them in an optimal way. The idea here would be more to "rebase" the standards on agile principles.

V. CONCLUSION

We've seen that some technologies or approaches in the context of connected cars are unavoidable: data-centric and big data; AI/ML-based systems; cloud computing and agility. These points are not limited to connected cars, but are trends in all transportation fields at least. Then we analyzed that these techniques are not completely in phase with traditional ways of developing safety critical software because the standards used for this purpose do not rely fundamentally on the same approaches to develop software. The reason is that these standards rely on principles which have not been fundamentally revisited for more than 20 years.

The idea of this paper is neither to provide a magic solution nor to declare that all the standards content is not adequate, but simply to ask the question: are we sure that the lessons learned during the last 20 years of software development have all been explored, regarding the development of safety critical software, to build them more adequately, more efficiently, and, why not, safer? It could be that [START_REF]Road vehicles -Safety of the Intended Functionality (SOTIF)[END_REF] will address some of these lessons learned; perhaps the principles of Overarching Properties developed by FAA and RESSAC project ( [START_REF]An alternative approach to DO178B, Duncan Brown, Rolls Royce[END_REF]) could cover some of these innovative approaches. This paper focuses on software aspects of confidence in safety critical digital systems. A lot of aspects are linked to this point, but are not developed here: safety aspects could be linked to cybersecurity, and to privacy; the connection between cloud and embedded software is significant (bandwidth, latency, integrity, etc.). Otherwise, the safety process of a digital system is not limited to software aspects: system view or architecture are for example fundamental dimensions to consider; only the global view makes the actual safety of the system. It could be that the technological trends treated in this paper have an impact on all of these related aspects; therefore, the task is far larger than described in the paper…


Sylvain Joubaud

Nicolas B Garnier

Sergio Ciliberto

Fluctuations of the total entropy production in stochastic systems

Keywords: 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion 05.70.-a -Thermodynamics

Thermodynamics of systems at equilibrium has been developed by defining the internal energy, the injected work, and the dissipated heat. The first law of Thermodynamics describes energy conservation and gives a relation between those three energies. The second law of Thermodynamics imposes that the entropy variation is positive for a closed system. Statistical Physics further gives a microscopic definition of entropy, which allows analytical results on entropy production. The extension of Thermodynamics to systems in non-equilibrium steady states is an active field of research. Within this context, the first law has been extended for stochastic systems described by a Langevin equation [1][2][3]. It has been noted that the second law is not verified at all times but only in average, i.e. over macroscopic times : entropy production can have instantaneously negative values. The probabilities of getting positive and negative entropy production are quantitatively related in non-equilibrium systems by the Fluctuation Theorem (FT). This theorem has been first demonstrated in deterministic systems [4][5][6] and secondly extended to stochastic dynamics [7][8][9][10]. FTs for work and heat fluctuations have been theoretically and experimentally studied in Brownian systems described by the Langevin equation [2,3,[11][12][13][14][15][16][17][18]. For these systems in non-equilibrium steady state, Fluctuation Theorems hold only in the limit of infinite time:

Φ(a) ≡ ln p(X τ = +a) p(X τ = -a) → a k B T for τ → ∞ (1)
where k B is the Boltzmann constant, T the temperature of the heat bath and p(X τ ) is the probability density function (PDF) of X τ . Φ is called symmetry function and X τ stands for either injected work or dissipated heat, averaged over a time lag τ . For the injected work, eq. 1 is valid for all fluctuation magnitudes a ; for the dissipated heat, eq. 1 is satisfied for values lower than the mean value [13].

We are interested here in the total entropy production in a non-equilibrium steady state (NESS), introduced in ref. [19][20][21] and directly related to previous work on housekeeping heat [22,23]. Entropy production has been studied both theoretically and experimentally in several systems [24][25][26][27][28][29][30]. In [19], Seifert and Speck have shown that the total entropy production for a stochastic system described by a first order Langevin equation in a NESS satisfies a Detailed Fluctuation Theorem (DFT) :

Φ(a) = ln p(X τ = +a) p(X τ = -a) = a k B T ∀τ ∀a (2)
The relation ( 2) is valid for all integration time τ and all fluctuation magnitudes of the total entropy production. This relation is closely related to the Jarzynski and Crooks non-equilibrium work relations [31,32] which can be exploited to measure equilibrium free energy in experiments [33][34][35][36][37] and it has been extended to Markov processes [38].

In this Letter, we measure the total entropy production in two out-of-equilibrium systems and show that this quantity verifies eq. ( 2). In the first part of the letter, we recall the general definition of dissipated heat, "trajectorydependent" entropy and total entropy production. In the second part, we detail our two experimental systems. The first one is an electric circuit maintained in a NESS by an injected mean current. The second system is a torsion pendulum driven in a NESS by forcing it with a periodic torque.

The heat dissipated by the system is the heat given to the thermostat during a time τ ; we note it Q τ . It is related to the work W τ , given to the system during the time τ , and to the variation of internal energy ∆ τ U during this period, thanks to the first law of Thermodynamics:

Q τ = W τ -∆ τ U . (3)
Expressions of W τ and ∆ τ U for our two experimental setups are given below. Following notations of ref [19], we define the entropy variation in the system during a time τ as :

∆s m,τ = 1 T Q τ ( 4
)
where T is the temperature of the heat bath. For thermostated systems, entropy change in medium behaves like the dissipated heat. The non-equilibrium Gibbs entropy is :

S(t) = -k B d xp( x(t), t, λ t ) ln p( x(t), t, λ t ) = s(t) (5)
where λ t denotes the set of control parameters at time t and p( x(t), t, λ t ) is the probability density function to find the particle at a position x(t) at time t, for the state corresponding to λ t . This expression allows the definition of a "trajectory-dependent" entropy :

s(t) ≡ -k B ln p( x(t), t, λ t ) (6)
The variation ∆s tot,τ of the total entropy s tot during a time τ is the sum of the entropy change in the system during τ and the variation of the "trajectory-dependent" entropy in a time τ , ∆s τ ≡ s(t + τ ) -s(t) : ∆s tot,τ ≡ s tot (t + τ ) -s tot (t) = ∆s m,τ + ∆s τ (7) In this letter, we study fluctuations of ∆s tot,τ computed using (4) and (6). We will show that ∆s tot,τ satisfies a DFT (eq. ( 2)). In ref. [38], the relevance of boundary terms like ∆s τ has already been pointed out. Our first experimental system is an electric circuit composed of a resistance R = 9.22 MΩ in parallel with a capacitance C = 278 pF [15]. The time constant of the circuit is τ 0 ≡ RC = 2.56 ms. The voltage V across the dipole fluctuates due to Johnson-Nyquist thermal noise. We drive the system out of equilibrium by injecting a constant controlled current I = 1.06 • 10 -13 A. After some τ 0 , the system is in a NESS. Electric laws give in this setup [15] :

τ 0 dV dt + V = RI + ξ , ( 8
)
where ξ is the Gaussian thermal noise, delta-correlated in time of variance 2k B T R. Multiplying (8) by V and integrating it between t and t+τ , we define the work W τ , injected into the system, and the dissipated heat Q τ , together with the internal energy U :

W τ ≡ t+τ t V (t ′ )Idt ′ (9) Q τ ≡ t+τ t V (t ′ )i R (t ′ )dt ′ (10)
∆ τ U ≡ 1 2 C V (t + τ ) 2 -V (t) 2 = W τ -Q τ (11)
where i R is the current flowing in the resistance : i R = I -C dU dt . This system has only one degree of freedom, so the trajectory in phase space is defined by the voltage V (t) alone, and the only external parameter is the constant current I. So the variation of the "trajectory-dependent" entropy during a time τ is :

∆s τ = -k B ln p(V (t + τ )) p(V (t)) (12)
Fluctuation relations for the injected work and the dissipated heat (or entropy change in medium) in this system have been reported in [15]. Let us recall the experimental results for the dissipated heat T.∆s m,τ , for several values of the integration time. Its average value T.∆s m,τ is equal to the average of injected work and linear in τ . The PDFs of T.∆s m,τ are plotted in figure 1a) for four values of τ /τ 0 . They are not Gaussian for small times τ and extreme events have an exponential distribution. We now describe new results in this system. The PDF of the "trajectory-dependent" entropy ∆s τ is plotted in figure 1b) ; we have superposed to it the PDF of ∆s m,τ at equilibrium (I = 0 A). These two PDFs are independent of the integration time. The two curves match perfectly within experimental errors. Therefore the "trajectorydependent" entropy is equal in this case to the fluctuations of the entropy exchanged with the thermal bath at equilibrium. The average value of this "trajectorydependent" entropy is zero within experimental errors ; so the total entropy production has the same average value than the entropy ∆s m,τ . The PDFs of the total entropy ∆s tot,τ / ∆s tot,τ , computed by adding ∆s τ to ∆s m,τ , are plotted in figure 1c) for different values of τ ; they are all Gaussian.

In figure 1d), we have plotted the symmetry functions of the dissipated heat (T ∆s m,τ ) together with those of the total entropy. Φ(∆s m,τ ) is a non linear function of ∆s m,τ . The linearity is recovered for ∆s m,τ < ∆s m,τ . The slope tends to 1 for large integration time. Entropy change in medium satisfies relation (1) [15]. On the contrary, the symmetry functions for the total entropy are linear with ∆s tot,τ for all integration times τ and for all values of ∆s tot,τ : Φ(∆s tot,τ ) = Σ(τ )∆s tot,τ . The slope Σ(τ ) is equal to 1 for all integration times within experimental errors. Measurements can be done for other values of injected current and we find the same results. Thus total entropy satisfies relation (2). This experimental result can be explained using a first order Langevin equation and noting that fluctuations of the voltage δV (t) = V (t) -V (t) , when a current is applied, are identical to those at equilibrium [15]. Thus the voltage V (t) has a Gaussian distribution with mean V (t) = R.I and variance (k B T /C), whereas its autocorrelation function is the same out of equilibrium and at equilibrium:

δV (t + τ )δV (t) = k B T C e (-τ /τ0) . ( 13
)
So we can compute the expression of the "trajectorydependent" entropy from eq. ( 12) and we find:

T.∆s τ = 1 2 CδV (t + τ ) 2 - 1 2 CδV (t) 2 (14)
Fluctuations of the voltage are identical to those at equilibrium, thus "trajectory-dependent" entropy is equal to the variation of internal energy divided by T at equilibrium and to the opposite of the entropy variation in the system at equilibrium. Using eq. ( 7), ( 10) and ( 14), the expression of the total entropy is :

T.∆s tot,τ = I t+τ t V (t ′ ) -Iτ 0 (δV (t + τ ) -δV (t)) dt ′ (15)
We experimentally see that the PDF of ∆s tot,τ is Gaussian, so fully characterized by its mean value and its variance. Its average value is linear in τ ( ∆s tot,τ = RI 2 τ /T ) and equals the mean injected work divided by the temperature T . Its variance is computed using eq. ( 13) and we obtain σ 2 ∆stot,τ = 2RI 2 k B τ /T . For a Gaussian distribution, the symmetry function of ∆s tot,τ is linear with ∆s tot,τ , that is Φ(∆s tot,τ ) = Σ∆s tot,τ . The slope Σ is related to the mean and the variance of the total entropy : Σ = 2 ∆s tot,τ /σ 2 ∆stot,τ = 1/k B . Experimentally, we have normalized the total entropy by k B , and the slope is equal to 1 for any τ as shown in fig. 1d).

We now consider the second experimental system. It is a torsion pendulum in a viscous fluid which acts as a thermal bath at temperature T . The system is driven out of equilibrium by an external deterministic time dependent torque M . All the details of the experimental setup can be found in [3]. The torsional stiffness of the oscillator is C = 4.65 • 10 -4 N.m.rad -1 , its viscous damping ν, its total moment of inertia I eff , its resonant frequency f 0 = ω 0 /(2π) = C/I/(2π) = 217 Hz and its relaxation time τ α = ν/(2I eff ) = 9.5 ms. The angular motion of the oscillator obeys a second order Langevin equation :

d 2 θ dt 2 + 2 τ α dθ dt + ω 2 0 θ = M + η C , ( 16
)
where η is the thermal noise, delta-correlated in time of variance 2k B T ν. The work injected into the system between t i and t i + τ is :

W τ = ti+τ ti M (t ′ ) dθ dt (t ′ )dt ′ . ( 17
)
The dissipated heat is computed according to eq. ( 3) where, in this case, the internal energy U (t) is :

U (t) = 1 2 Cθ(t) 2 + 1 2 I eff θ(t) 2 (18)
We investigate a periodic forcing : M (t) = M 0 sin(ω d t) ( M 0 = 0.78 pN.m and ω d /(2π) = 64 Hz). The integration time τ is chosen to be a multiple of the period of the forcing : τ n = 2nπ/ω d . The average responses of the system < θ(t) > and < θ(t) > are periodic function of the pulsation ω d and the system is in a steady state. Results for injected work and dissipated heat in this case are reported in [3,17].

The "trajectory-dependent" entropy is not as simple as in the case of the resistance. The system has two independent degrees of freedom (θ and θ) and its expression is :

∆s τn = -k B ln p(θ(t i + τ n ), ϕ).p( θ(t i + τ n , ϕ)) p(θ(t i + τ n ), ϕ).p( θ(t i + τ n , ϕ) (19)
The DFT (eq. ( 2)) is valid for each fixed starting phase ϕ = t i ω d [21]. For computing correctly the total entropy, we have to calculate the PDFs of the angular position and the angular velocity for each initial phase ϕ. Then we compute the "trajectory-dependent" entropy. As fluctuations of θ and θ are independent of ϕ [3]. These distributions correspond to the equilibrium fluctuations of θ and θ around the mean trajectory defined by θ(t) and θ(t) .

As a consequence, we can average ∆s τn over ϕ which improves a lot the statistical accuracy. We stress that it is not equivalent to calculate first the PDFs over all values of ϕ -which would correspond here to the convolution of the PDF of the fluctuations with the PDF of a periodic signal -and then compute the trajectory dependent entropy.

In figure 2a), we recall the main results for the dissipated heat T ∆s m,τn . Its average value T.∆s m,τn is linear in τ n and equal to the injected work. The PDFs of T.∆s m,τn are not Gaussian and extreme events have an exponential distribution. Let us now describe new results in this system. The PDF of the "trajectory-dependent" entropy is plotted in fig. 2b); it is independent of n. We superpose to it the PDF of the variation of internal energy divided by T at equilibrium : the two curves match perfectly within experimental errors, so the "trajectory-dependent" entropy can again be considered as the entropy exchanged with the thermostat if the system is at equilibrium. The average value of ∆s τn is zero, so the average value of the total entropy is equal to the average of injected power divided by T . In fig. 2c), we plot the PDFs of the normalized total entropy for four typical values of integration time. We find that the PDFs are Gaussian for any time.

The symmetry functions of the dissipated heat Φ(∆s m,τn ) and the total entropy Φ(∆s tot,τn ) are plotted in fig. 2d). For the dissipated heat, the symmetry function is a non linear function of ∆S m,τ and we observe a linear behavior for ∆s m,τn < ∆s m,τn with a slope that tends to 1 for large time. For the normalized total entropy, the symmetry functions are linear with ∆s tot,τn for all values of ∆s tot,τn and the slope is equal to 1 for all values of τ n . Note that it is not exactly the case for the first values of τ n because these are the times over which the statistical errors are the largest and the error in the slope is large.

In ref. [3], we have already shown that, when a torque is applied, the fluctuations around θ(t) have the same statistics and the same dynamics as fluctuations at equilibrium. Using the expression of the distribution of the angular position and the angular velocity, we can compute the "trajectory-dependent" entropy from eq. ( 19):

T.∆s τn = 1 2 C δθ(t i + τ n ) 2 -δθ(t i ) 2 + 1 2 I eff δ θ(t i + τ n ) 2 -δ θ(t i ) 2 (20)
where δθ and δ θ are the fluctuations around the mean trajectory : θ(t) = θ(t) + δθ(t) and θ(t) = θ(t) + δ θ(t).

The "trajectory-dependent" entropy is equivalent to the variation of internal energy for the system at equilibrium divided by the temperature of the bath. The total entropy is :

T.∆s tot,τn = W τn - 1 2 C θ(t i )(δθ(t i + τ n ) -δθ(t i ) - 1 2 I eff θ(t i )(δ θ(t i + τ n ) -δ θ(t i )) (21)
Experimentally, we observe that the PDF of the total entropy is Gaussian. We can calculate the mean value and the variance of the total entropy for any τ n . We find that T.∆s tot,τn is equal to the injected work. Using the first law of thermodynamics and the equation of motion, we write the dissipated heat as the difference between the viscous dissipation and the work of the thermal noise :

T.∆s m,τn = ti+τn ti ν θ(t ′ ) 2 dt ′ - ti+τn ti θ(t ′ )η(t ′ )dt ′ (22)
This expression holds at equilibrium as well as out of equilibrium. The only difference is that when out of equilibrium θ(t) = 0 and θ(t) = 0. After some algebra the total entropy is : + 2ν

T.∆s tot,τn = ν ti+τn ti θ(t ′ ) 2 dt ′
ti+τn ti θ(t ′ ) δ θ(t ′ )dt ′ - ti+τn ti θ(t ′ ) η(t ′ )dt ′ (23)
The average value of the total entropy is ν ti+τn ti θ(t ′ ) 2 dt ′ and its variance is :

σ 2 ∆stot,τ n = 2k B ∆s tot,τn + B/T 2 B = 4ν dudv θ(u) θ(v) ψ(u, v) (24) ψ(u, v) = ν δ θ(u)δ θ(v) -δ θ(u)η(v) (25)
The first term in the function ψ is the autocorrelation function of the angular velocity. This function is symmetric around u = v. The second term is the correlation function of the angular velocity with the noise. Due to the causality principle, this term vanishes if u < v. Changing variables to r = (u + v)/2 and s = u -v, eq. ( 24) can be rewritten :

B = ti+τn ti dr τn 0 θ(r + s/2) θ(r -s/2) ψ(r, s) (26)
ψ(r, s) = 2ν δ θ(s)δ θ(0) -δ θ(s)η(0) (27) After some algebra we can show that the correlation function δ θ(s)η(0) is two times the autocorrelation function δ θ(s)δ θ(0) , therefore ψ = 0. Thus we obtain that the variance of the total entropy is equal to 2k B ∆s tot,τn . The Fluctuation Relation for the total entropy is : Φ(∆s tot,n ) = 1/k B ∆s tot,n for all times τ , for all values of ∆s tot,n and for any kind of stationary forcing.

For our two experimental systems, we have obtained that the "trajectory-dependent" entropy can be considered as the entropy variation in the system in a time τ that one would have if the system was at equilibrium. Therefore the total entropy is the additional entropy due to the presence of the external forcing : this is the part of entropy which is created due to the non-equilibrium stationary process. The derivation we gave for a second order Langevin equation can be extended to a first order Langevin equation : total entropy (or excess entropy) satisfies the Fluctuation Theorem for all times and for all kind of stationary intensity injected in the circuit or all kind of stationary external torque. The ratio between the average value of the total entropy at τ = τ α (relaxation time) and the variance σ 2 eq of fluctuations the entropy variation at equilibrium characterizes the distance from equilibrium d where n d is the number of degrees of freedom. For the second equality, we use the Gaussianity of ∆s tot,τ , i.e. σ 2 ∆stot,τ = 2k B ∆s tot,τ and the variance of of the fluctuation of entropy variation at equilibrium in terms of k B , i.e. σ 2 eq = 3/2n d k 2 B . It turns out that this expression for d is the same that was defined in [3] with a completely different approach. Eq. ( 28) indicates that, when the system is driven far from equilibrium, σ eq becomes negligible.

As a consequence the fluctuations of the total entropy become equal to the fluctuations of the entropy variation in the system in a time τ . In other words, the PDFs of the dissipated heat far from equilibrium are Gaussian.

In conclusion, we have studied the total entropy when the system is driven in a non-equilibrium steady state. We have shown that the Fluctuation Theorem for the total entropy is valid not only in the limit of large times, as it is the case for injected work and dissipated heat, but also for all integration times and all fluctuation amplitudes. In the two examples we have discussed, the total entropy corresponds to the difference between the entropy change in medium out of equilibrium and at equilibrium.

Fig. 1 :

 1 Fig.1: Resistance. a) PDFs of the normalized entropy variation in the system ∆sm,τ / ∆sm,τ , with τ /τ0 = 2.4 (•), τ /τ0 = 4.8 ( ), τ /τ0 = 14.5 (⋄) and τ /τ0 = 29 (×). b) PDF of the variation of "trajectory-dependent" entropy ∆sτ for τ /τ0 = 4.8. The distribution is independent on τ /τ0. Continuous line is experimental PDF of the entropy variation at equilibrium (∆sm,τ,eq at I = 0 A). c) PDFs of the total entropy production ∆stot,τ / ∆stot,τ , with τ /τ0 = 2.4 (•), τ /τ0 = 4.8 ( ), τ /τ0 = 14.5 (⋄) and τ /τ0 = 29 (×). d) Symmetry functions Φ for ∆sm,τ / ∆sm,τ (small symbols in light colors) and ∆stot,τ / ∆stot,τ (large symbols in dark colors) for the same values of τ /τ0.

Fig. 2 :

 2 Fig. 2: Torsion pendulum. a) PDFs of the normalized entropy variation ∆sm,τ n / ∆sm,τ n integrated over n periods of forcing, with n = 7 (•), n = 15 ( ), n = 25 (⋄) and n = 50 (×). b) PDFs of ∆sτ n , the distribution is independent of n and here n = 7.Continuous line is the theoretical prediction for equilibrium entropy exchanged with thermal bath ∆sm,τ n,eq . c) PDFs of the normalized total entropy ∆stot,τ n / ∆stot,τ n , with n = 7 (•), n = 15 ( ), n = 25 (⋄) and n = 50 (×). d) Symmetry functions for the normalized entropy variation in the system (small symbols in light colors) and for the normalized total entropy (large symbols in dark colors) for the same values of n.

:

We thank U. Seifert and K. Gawedzki for useful discussions. This work has been partially supported by ANR-05-BLAN-0105-01.


Amira Barhoumi
email: amira.barhoumi.etu@univ-lemans.fr

Yannick Estève
email: yannick.esteve@univ-lemans.fr

Chafik Aloulou
email: chafik.aloulou@fsegs.rnu.tn

Lamia Hadrich Belguith
email: l.belguith@fsegs.rnu.tn

Document embeddings for Arabic Sentiment Analysis

Keywords: Sentiment analysis, document embedding, paragraph vector, arabic language

Research and industry are more and more focusing in finding automatically the polarity of an opinion regarding a specific subject or entity. Paragraph vector has been recently proposed to learn embeddings which are leveraged for English sentiment analysis. This paper focuses on Arabic sentiment analysis and investigates the use of paragraph vector within a machine learning techniques to determine the polarity of a given text. We tested some preprocessing method, and we show that light stemming enhance the performance of classification.

Introduction

With the widespread of Internet and the revolution of social networks, any person could follow his opinion and express his feelings and emotions regarding various topics, products, ideas, persons, etc. Many academic and industrial efforts are focusing on analyzing opinions and sentiments by investigating automatic techniques to extract convenient information.

Sentiment Analysis (SA) involves building systems that recognize the opinion expressed in a textual unit. It aims mainly to identify the subjectivity and polarity of a given text. Generally, the polarity consists of positive, negative or neutral, with or without their strength. SA and its applications have spread to many languages and most of the works deal with Indo-European ones. Indeed, several researches have been carried out for the English language. However, few works have been done for Arabic. In this work, we are interested in Arabic language.

In this paper, we present an application of sentiment analysis to the Arabic language. The main contributions of this work are as follows: [START_REF] Pang | Opinion Mining and Sentiment Analysis[END_REF] we measure the efficiency of distributed representation for Arabic sentiment analysis ASA, [START_REF] Korayem | Subjectivity and sentiment analysis for arabic: A survey[END_REF] we evaluate the performance of neuronal techniques for sentiment classification.

The rest of the paper is structured as follows. Section 2 discusses some related works dedicated mainly to Arabic. In section 3, we present our methodology for ASA. We report, in section 4, our experimental framework and discuss the obtained results. Finally, we conclude in section 5 and give some outlooks to future work.

Related work

Nowadays, sentiment analysis is becoming very interesting1 due to the explosion of the number of internet users and the proliferation of social networks.

The largest amount of SA researches has been carried out for the English language. There are few works have been done for other languages. Recently, there has been a considerable effort to develop SA systems for the Arabic language. In this section, we focus on works dedicated for ASA. Most of the existing methods in sentiment analysis can be divided into three categories: knowledge based approach, machine learning based approach and hybrid approach.

The knowledge based approach uses lexicon or patterns. [START_REF] Almas | A note on extracting sentiments in financial news in english, arabic & ardu[END_REF] proposed an approach based on a local grammar which contains patterns that extract sentiment from a given document. [START_REF] Farra | Sentence-level and document-level sentiment mining for arabic texts[END_REF] followed the same approach based on patterns. For works based on lexicon, we quote the work of [START_REF] Abdulla | Towards Improving he Lexicon-Based Approach for Arabic Sentiment Analysis[END_REF]. They manually construct a lexicon that contains 4815 words (1942 positive words and 2873 negatives ones). Their system compute the number of positive and negative words in a text in order to generate the overall polarity. Another work is that of [START_REF] Al-Kabi | Opinion Mining and Analysis for Arabic Language[END_REF] who implemented a tool which determine the subjectivity, the polarity and the strength of an opinion. They used two general lexicons and 16 specific lexicons (8 for positive polarity and 8 for negative polarity). For the strength computation, they manually added a score between 1 and 10 to each term in the lexicon. Another work has been done in [START_REF] Bayoudhi | entiment classification at discourse segment level: Experiments on multi-domain arabic corpus[END_REF] where the authors presented a lexicon based approach for MSA. First, a lexicon has been built by applying a semi automatic method. Then, the lexicon entries were used to detect opinion words and assign to each one a sentiment class. [START_REF] Al-Ayyoub | Lexicon-based sentiment analysis of arabic tweets[END_REF] built a sentiment lexicon of about 120,000 Arabic words and created a SA system on top of it. They reported a 86.89% of classification accuracy.

Machine learning based approach views SA as a classification task. Annotated data sets are used to train classifiers. [START_REF] Abdulla | An extended analytical study of arabic sentiments[END_REF] proposed a system that performs subjectivity and sentiment analysis for social media using morphological features. [START_REF] Bayoudhi | Sentiment Classification of Arabic Documents: Experiments with multi-type features and ensemble algorithm[END_REF] compared Support Vector Machines SVM, Naive Bayes NB classifiers and neural networks which are trained on Opinion Corpus for Arabic OCA [START_REF] Rushdi-Saleh | Bilingual Experiments with an Arabic-English Corpus for Opining Mining[END_REF] and ACOM corpus [START_REF] Mountassir | Sentiment classification on Arabic corpora: A preliminary cross-study[END_REF] with different combinations. Another machine learning approach was used in [START_REF] Rushdi-Saleh | Bilingual Experiments with an Arabic-English Corpus for Opining Mining[END_REF] where they build the corpus OCA which consists of movie reviews written in Arabic. They also created an English version translated from Arabic and called EVOCA. SVM and NB classifiers are then used to create SA systems for both languages. For instance, SVM gives 90% F-measure on OCA compared to 86.9% on EVOCA. In multi-way sentiment analysis, [START_REF] Shboul | Multi-Way Sentiment Classification of Arabic Reviews[END_REF] performed a multi-class classification, using a scale from 1 to 5 to measure polarity. They tested SVM, decision tree C4.5, decision table J48, KNearest Neighbors KNN, NB, MultiNaive Bayes MNB and voting (a combination of KNN, decision tree and NB). They concluded that MNB is more efficient. The authors did a flat classification, i.e there is only one level in the hierarchy. However, [START_REF] Al-Ayyoub | Hierarchical Classifiers for Multi-Way Sentiment Analysis of Arabic Reviews[END_REF] shows that a hierarchical classification of the multi-way sentiments is better than an ordinary flat classification. They have implemented two hierarchical structures: one with two levels and the other with four levels. They tested SVM, NB, KNN and decision tree techniques. They concluded that KNN is more efficient.

For hybrid approach, it is a combination of the two previous ones: it uses both lexicons and machine learning algorithms. The earliest work is of [START_REF] El-Halees | Arabic opinion mining using combined models[END_REF] how presents a combined classification hierarchy by applying sequentially multiple classifiers. Moreover, [START_REF] Ibrahim | Sentiment analysis for modern standard Arabic and colloquial[END_REF] use a lexicon of 5244 adjectives, a lexicon of 3296 idioms to improve the classification of sentences made with SVM. [START_REF] Refaee | iLab-Edinburgh at SemEval-2016 Task 7: A Hybrid Approach for Determining Sentiment Intensity of Arabic Twitter Phrases[END_REF] apply a hybrid approach to predicting the sentiment strength of an Arabic tweet. In fact, they used a set of linear regression models for predicting initial scores for sentences, then they adjusted these scores by applying a set of rules extracted from existent sentiment lexicon.

Works on Arabic SA are fewer than those on English. The mainly reasons behind that are the followings:

-Limited number of resources developed for ASA: there are few corpora and lexicon freely available [START_REF] Al-Kabi | A Prototype for a Standard Arabic Sentiment Analysis Corpus[END_REF]. For more details about previous works on ASA, we refer the reader to the extensive surveys presented in [START_REF] Biltawi | Sentiment classification techniques for arabic language: A survey[END_REF]. [START_REF] Mdhaffar | Sentiment Analysis of Tunisian Dialect: Linguistic Resources and Experiments[END_REF] summarizes the list of all freely available SA corpora for MSA and its dialects. -The MSA is a semitic language with rich morphology.

-The diacritization problem of MSA (Table 1 shows the meaning change of the word " " /jml/ while changing its diacritics). -The way of negation detection: the existence of a negation term reverses the polarity. -The structure of the statement (structered, semi-structered or non structered) has an impact on polarity prediction [START_REF] Mohey | A survey on sentiment analysis challenges[END_REF].

-The problem of figurative language: irony, sarcasm, etc.

-The use of foreign words (English, French, Italian, etc) in Internet user's content makes the ASA more difficult.

Methodology

This work falls within the framework of the machine learning based approach. In fact, many machine learning algorithms require, as input, vector representations. The most common representation used in NLP is the bag of words (BOW) representation. Despite its popularity, the BOW has two major drawbacks: the lost of the order words and the semantic ignorance of words. Distributed representations resolve these problems. We distinguish mainly two types of embeddings:

word embeddings: word2vec [START_REF] Mikolov | Efficient estimation of word representations in vector space[END_REF] and Glove [START_REF] Pennington | Glove: Global vectors for words representation[END_REF], etc.

document embeddings: paragraph vector [START_REF] Le | Distributed representations of sentences and documents[END_REF] for variable length texts, sentence vector [START_REF] Socher | Recursive deep models for semantic compositionality over a sentiment treebank[END_REF], etc.

Paragraph vector algorithm allows obtaining distributed representations (Doc2vec) for any length sequence, ranging from phrases to documents. It efficiently computes document vector representations in a dimensional vector space. Word vectors are located in the vector space where words that have similar semantic and share common contexts are mapped nearby each other in the space. The Doc2vec representations were used for English sentiment analysis by [START_REF] Le | Distributed representations of sentences and documents[END_REF]. The authors, Le and Mikolov, achieved the best performance with paragraph vector compared to other approaches on IMDB [START_REF] Maas | Learning word vectors for sentiment analysis[END_REF] dataset which contains 100000 film reviews. Motivated by their work, we propose using Doc2vec embeddings for Arabic sentiment analysis. The main question asked in this work consists on measuring the efficiency of Le and Mikolov's SA method for Arabic language. We built a system composed with two parts: the first one applies some linguistic preprocessing on the input text, and the second uses a classifier in order to predict the polarity of the input. We trained two classifiers: a logistic regression LR and a multilayer perceptron MLP2 . The input vector of the classifier is the embeddings obtained by learning paragraph vector. This vector is a concatenation of the two learned vectors, one from distributed memory version DM and one from distributed bag of words version DBOW, each have 400 dimensions. So that, 800 is the dimension of the classifier's input. In fact, we kept the same neural architecture and the same hyperparameters of paragraph vector model used by Le and Mikolov [START_REF] Le | Distributed representations of sentences and documents[END_REF].

Experiments and results

In this section, we perform experiments for two tasks: binary sentiment polarity classification and five-class classification. We test two classifiers: MLP and logistic regression.

Training data and feature extraction

The learning of Doc2vec representations needs a big corpus. According to our knowledge, LABR dataset [START_REF] Mahmoud | LABR 2.0: Large Scale Arabic Sentiment Analysis Benchmark[END_REF] is the biggest arabic dataset for SA that is freely available 3 . We used the corpus LABR for ASA . This corpus consists of 63257 book reviews written in MSA and colloquial Arabic, each with a rating 1 to 5 stars. Table 2 describes the distribution of the reviews on different classes.

Data preprocessing

We use LABR dataset that contains book reviews. The plain reviews without any preprocessing consists the baseline of our experiments. In other words, each token in the review is considered as a normal word. For sentiment analysis, some special characters such as ! ? carry sentiments. Moreover, some combinations of these special characters, for example :) :(, are smileys which are significant for our task. So it is important to consider them as words. Following an analysis of our corpus, we found that many punctuations are agglutinated to words. For this reason, the first preprocess applied over LABR consists on separating punctuations from words and considers them as normal words.

An other experimentation consists on applying stemming for LARB. In fact, the stemming (either light or not) reduces the size of vocabulary. The stemming is the process of eliminating the affixes of words and reducing them to their roots. However, the light stemming removes only prefixes and/or suffixes, without manipulation of the infixes of the word. For example, the two words et (

Arabic SA experiments

In this work, two types of classification are performed: binary classification and multiclass one. Binary classification considers only two classes: positive and negative. However, in multi-class classification, there are five classes: very positive, positive, neutral, negative and very negative. The same method and hyperparameters are used for both classification tasks: binary sentiment classification and five-classes classification.

To evaluate the performance of SA on the LABR dataset, we carried out several experiments using various configuration. All the experiments were conducted in Python using Theano 5 for classification and gensim 6 for learning vector representation. For machine learning methods, we investigate two classifiers: logistic regression LR and multi-layer perceptron MLP. The input of the each sentiment classifier is a set of features vectors obtained with paragraph vector algorithm. In fact, we tested three different types of Doc2vec vectors: (1) vectors obtained with DM version of paragrath vector algorithm, (2) vectors obtained with DBOW version, and (3) concatenation of the vectors obtained separately with DM and DBOW.

Results and discussion

In binary classification framework, the results of the different classifiers with different experimental prepocessing are presented in Table 4. The empty set symbol ∅ means that there is no preprocessing step: we used the review as it stands, without any modification. It represents the baseline of the experiments conducted. The MLP classifier is more efficient than the logistic regression. However, this is not the case when applying preprocessing: the regression classifier becomes more efficient. We notice that there is a little difference in the performances of two classifiers. The lower error rate is 23.31% and it is obtained with logistic regression by applying light stemming. There is a 2% gain after light stemming and special character preprocessing. We think that this low value of gain obtained by applaying light stemming comes from the quantity of MSA words that exist in the corpus. In fact, The reviews of LABR dataset are written in MSA and dialectal Arabic. We conclude that Arabic language, as opposed to English, requires a specific processing process in order to enhance the performance of SA. It's well known that paragraph vector can capture the semantic similarity between words. Or, among the objectives of this paper is to measure the effeciency of document embeddings for ASA. For example, the words "good" and "excellent" are close to each other. To ensure the effectiveness of Doc2vec algorithm for arabic language, we look to the most similar words to some sentiment words. Here, we report the 10 top words similar to the word "good" which are in the following order: beautiful, fabulous, enjoyable, useful, interesting, boring, light, excellent, nice , very. Among these words, seven words are semantically similar to "good". We notice some similarity errors:

-The word "boring" is close to "good", which is not true.

-The word "boring" is closer to "good" than "excellent", which is false.

We think that this similarity error is strongly linked to the way of Doc2vec learning. In fact, paragraph vector algorithm extract representations that covers syntactic and semantic information based on the context. This means that words with similar context are very near in the vector space, even antonyms. To circumvent this problem, the representations should be constructed by predicting the context and the polarity at a time. For multi-class classification, we tested also MLP and LR classifiers. The performances obtained in multi-class classification framework are reported in table 5. In this framework, the input of each classifier is LABR dataset after application of light stemming and special character preprocessing. Table 5 shows that logistic regression is more efficient that MLP. In fact, the error rate with regression is lower than with MLP. Moreover, the error rate in the binary classification framework is lower than in multi-class framework. Indeed and under the same dataset preprocessing and classifier hyperparameters, the error rate obtained with regression is 23.31% with binary classification and 67.62% in multi-class classification which is obviously much harder to handle. In fact, having more classes is not the only challenge imposed by multi-class classification. The other difficulty comes from the relation between some classes, i.e the relation between positive and very positive polarities and relation between negative and very negative polarities.

works with LABR Accuracy Our work 32.38% [START_REF] Shboul | Multi-Way Sentiment Classification of Arabic Reviews[END_REF] 45% [START_REF] Al-Ayyoub | Hierarchical Classifiers for Multi-Way Sentiment Analysis of Arabic Reviews[END_REF] 45.7% Table 6. Flat hierarchy for multi-class classification

In this work, we adopted a flat classification (table 6) and we obtained an accuracy equal to 32.38% by using regression classifier over Doc2vec representations. However, [START_REF] Shboul | Multi-Way Sentiment Classification of Arabic Reviews[END_REF] used muti-Naive Bayes over BOW vectors. They obtained 45% as accuracy. All works mentioned in tables 6 and 7 use LABR dataset for their experiments.

Works with LABR

#Levels Accuracy [START_REF] Al-Ayyoub | Hierarchical Classifiers for Multi-Way Sentiment Analysis of Arabic Reviews[END_REF] 2 46.2% 4 57.8% Table 7. multi-level hierarchy for multi-class classification [START_REF] Al-Ayyoub | Hierarchical Classifiers for Multi-Way Sentiment Analysis of Arabic Reviews[END_REF] prove that multi-level hierarchy enhance the performance of multi-class framework (table 7). They used KNN classifier and they obtained an accuracy equal to 46.2% with 2-level hierarchy. But, they obtained 57.8% as accuracy with a 4-level hierarchy.

Table 1 .

 1 Different meaning for the word " "

	Word Possible Diacritics Transliteration Translation	Polarity
	/joumalun/	sentenses	neutral
	/jamalun/	camel	neutral
	/jammala/	beautify	positive

Table 2 .

 2 LABR corpus: the reviews distribution on different classes

		Very negative Negative Neutral Positive Very positive Total
	Training	2331	4195	9762	15189	19129	50606
	Test	608	1090	2439	3865	1649	12651

table 3

 3

	Stem	Light stem	Transliteration	Translation	Polarity
			/mrwE/	terrible	negative
	( , , )
			/r|}E/	fabulous	positive
	( , , )

) have the same stem [rgb]0.24,0.7,0.44or root ( , , ) but, they don't have the same polarity. So, applying light stemming

4

is relevant for Arabic SA.

Table 3 .

 3 Light stemming and polarity

Table 4 .

 4 Error rate of various experiments over LABR dataset

		Regression	MLP
	∅	25.60%	24.61%
	Special character	25.32%	25.46%
	Light stemming	23.31%	23.35%

Table 5 .

 5 Error rate in a multi-class classification framework

		Regression	MLP
	Error rate	67.62%	69.42%

https://trends.google.com/trends/explore?q=sentiment%20analysis#qusentiment%20analysis

The MLP contains one hidden layer with 50 units in order to predict the sentiment.

LARB dataset is available on http://www.mohamedaly.info/datasets/labr

In this work, we use the light stemmer https://github.com/motazsaad/arabic-light-stemmer

http://deeplearning.net/software/theano/

https://radimrehurek.com/gensim/

Conclusion and future works

In this paper, we have made an Arabic sentiment analysis which uses embedding. The aim of this study is to measure the utility of Doc2vec embeddings in Arabic SA framework. The results reported in this paper match the difficulty of Arabic with respect to English. Arabic is morphological rich language. So dealing with Arabic requires preprocessing step. With the purpose to study the potential of preprocessing, we had principally tested the contribution of light stemming in improving performance.

As future work, we think that using tokenization in preprocessing could enhance the performance [START_REF] Duwairi | Arabic Sentiment Analysis using Supervised Classification[END_REF]. Moreover, Adding a stop word list consists an other way of preprocessing. We would like also to test the common BOW representation: the input of our classifiers becomes BOW vectors, not Doc2vec embeddings. So that we could compare the two different representations.


J Boudet

J Caro

L Shao

E Lévêque

Numerical studies towards practical large-eddy simulation

Keywords: LES, adaptable methods, validation

Large-eddy simulation developments and validations are presented for an improved simulation of turbulent internal flows. Numerical methods are proposed according to two competing criteria: numerical qualities (precision and spectral characteristics), and adaptability to complex configurations. First, methods are tested on academic test-cases, in order to abridge with fundamental studies. Consistent results are obtained using adaptable finite volume method, with higher order advection fluxes, implicit grid filtering and "low-cost" shear-improved Smagorinsky model. This analysis particularly focuses on mean flow, fluctuations, two-point correlations and spectra. Moreover, it is shown that exponential averaging is a promising tool for LES implementation in complex geometry with deterministic unsteadiness. Finally, adaptability of the method is demonstrated by application to a configuration representative of blade-tip clearance flow in a turbomachine.

Nomenclature

) u c large-scale characteristic velocity < . > average u i velocity component in the direction i

Introduction

Improvement of computational fluid dynamics for the prediction of internal flows needs progress in different but complementary directions. These are mainly:

-The enlargement of the computational extent (in space, time, towards other physics…), in order to include more physical phenomena. Schlüter et al. [START_REF] Schlüter | Large-scale integrated LES-RANS simulations of a gas turbine engine[END_REF] investigate such effects in the aerodynamic computation of a complete gas-turbine.

-A better representation of turbulence, particularly for 3D interaction with complex mean field. The present paper mainly addresses the second point. A preference is given to direct simulation of turbulence, compared to modeling of averaged effects. Indeed, increasing the complexity of turbulence modeling is of limited efficiency: only average data are computed, and validity is not guaranteed for complex flows that are not comparable to calibration cases. In comparison, direct simulation delivers more physical information (spectra, correlations...) and is more adaptable. Large-eddy simulation (LES) [START_REF] Sagaut | Large eddy simulation for incompressible flows[END_REF] allows direct simulation of turbulence with a reduced number of points: only the largest turbulent eddies are represented, out of a spectral width that grows with Reynolds number. LES already represents a major tool for academic study of turbulent flows, from the transition in boundary layer [START_REF] Huai | Large-Eddy Simulation of Transition to Turbulence in Boundary Layers[END_REF] to separated flows on bluff bodies [START_REF] Rodi | Large-Eddy Simulations of the Flow past Bluff Bodies: State-of-the Art[END_REF]. Application to practical flows is developing, and a particularly interesting example is given by Raverdy et al. [START_REF] Raverdy | High-Resolution Large-Eddy Simulation of Flow Around Low-Pressure Turbine Blade[END_REF] on a turbine blade. These developments also benefits from theoretical and numerical studies, such as the analysis of compressible aspects by Erlebacher et al. [START_REF] Erlebacher | Toward the large-eddy simulation of compressible turbulent flows[END_REF].

It can be noted here that direct representation of turbulence can also improve description of other physical aspects. For example, it provides spectra and two-point correlations used for broadband noise simulation, of major importance to study nuisance from turbo-engines. Such computations are presented on airfoils by Terracol et al. [START_REF] Terracol | Airfoil noise prediction using large-eddy simulation, linearized Euler equations and Kirchhoff integral[END_REF] or Boudet et al. [START_REF] Boudet | Wake-airfoil interaction as broadband noise source: a large-eddy simulation study[END_REF].

The present paper intends to analyze adaptable methods for LES, from academic test-cases where turbulent physics is precisely isolated, to complex configurations and associated numerical constraints.

Next section presents numerical methods, chosen for their numerical characteristics (precision, transfer function…) and their adaptability. Afterwards, the numerical methods are analyzed on two academic test-cases: isotropic turbulence and channel flow, in order to abridge with fundamental developments on LES. Finally, a section presents application to a complex configuration, representative of turbomachine blade-tip clearance flow.

Adaptable numerical methods

General equations

The Turb'Flow solver uses the filtered equations of continuity, momentum and energy:

0 ~= ∂ ∂ + ∂ ∂ j j x u t ρ ρ j ij j ij i j j i i x x x P x u u t u ∂ Π ∂ + ∂ ∂ + ∂ ∂ - = ∂ ∂ + ∂ ∂ τ ρ ρ ~ ( ) [ ] ( ) [ ] j j j p j ij ij i j j t j t x x T C x u x u P e x t e ∂ ∂ +         ∂ ∂ ∂ ∂ + Π + ∂ ∂ = + ∂ ∂ + ∂ ∂ θ µ τ ρ ρ Pr ~
The commutation error of the filter operator with the partial derivatives is neglected. The subgrid-scale stress tensor is:

j i j i ij u u u u ρ ρ - = Π
~, and the subgrid-scale heat flux:

( ) ( ) ( ) ij i ij ij i j t j t j u u u P e u P e τ τ ρ ρ θ + Π + - + - + = ~
The fluid is air, considered Newtonian:

        ∂ ∂ - ∂ ∂ + ∂ ∂ = ij k k i j j i ij x u x u x u δ µ τ 3 2 ~
with constant dynamic viscosity. Finally, the ideal gas law is used:

T r P ρ = (with r=287 J/kg-K)
The equations are solved to obtain the filtered quantities. The possible use of a discrete filter is presented below, but the approach relying only on the natural filter of the grid is shown more reliable.

Discretisation

Discretisation uses finite volume approach, with explicit time marching and 5-step Runge-Kutta. For convection fluxes, value of conservative quantity q is given at index j-1/2 by: 12 7 7

1 1 2 2 / 1 + - - - - + + - = j j j j j
q q q q q corresponding to 4 th order when computing the spatial derivative at index j with: ( )

x q q j j ∆ -- + / 2 / 1 2 / 1
, on uniform grid. The Fourier analysis of the scheme is briefly presented here, to evaluate the capabilities regarding the representation of the turbulent spectrum. Following Lele [START_REF] Lele | Compact Finite-Difference Schemes with Spectral-like Resolution[END_REF], the Fourier component ( )

x x i x q j j ∆ = / exp ) ( ω , is considered for [ ] π ω , 0 ∈
. The analytical derivative is: ( ) ( )

x x i x i x q j j ∆ ∆ = / exp / ) ( ' ω ω
, while the discrete derivative is:

) ( ' ) ( 1 j x q G × ω
, with:

ω ω ω ω 6 ) 2 sin( ) sin( 8 ) ( 1 - = G
This numerical damping coefficient is presented in Fig. 1. Sometimes,

ω ω × ) ( 1 G
is presented as a modified wavenumber, but this can be misleading because the actual wavenumber (in the exponential) is not altered by the linear discrete scheme. In Fig. 1, it can be noted that a wide band of the spectrum is correctly described ( Diffusion fluxes are calculated with a second order centered scheme.

Subgrid-scale models

Models are required for ij Π and j θ in order to close the system. These quantities represent the influence of the subgrid-scales on the resolved field, regarding momentum and heat flux respectively. Following the theory of Kolmogorov (cf. Pope [START_REF] Pope | Turbulent Flows[END_REF]), the smallest scales are considered as mainly dissipative, at the end of the energy cascade coming from the largest eddies. This leads to the following models:

        ∂ ∂ - ∂ ∂ + ∂ ∂ = Π ij k k i j j i sgs ij x u x u x u δ µ j sgs p sgs j x T C ∂ ∂ = Pr µ θ
In the present study, two models for sgs µ are investigated: the standard model of Smagorinsky [START_REF] Smagorinsky | General circulation experiments with the primitive equations: I. the basic experiment[END_REF],

and a shear-improved version. In both cases, 9 . 0 Pr = sgs and filtering is implicit (by the grid). A third approach will be presented, using an additional explicit filter to avoid accumulation of the smallest resolved scales, and 0 = sgs µ .

• Smagorinsky model

In the well-known Smagorinsky model [START_REF] Smagorinsky | General circulation experiments with the primitive equations: I. the basic experiment[END_REF], subgrid-scale viscosity is computed locally:

S x C s sgs 2 2 ∆ = ρ µ with: ij ij S S S 2 ~=
, and

        ∂ ∂ + ∂ ∂ = i j j i ij x u x u S 2 1 ~
where x ∆ is calculated as the cubic root of the cell volume.

The model of Smagorinsky is constructed from the theory of isotropic turbulence, where the constant can be calibrated to

18 . 0 ≈ s C
. Near walls, mean gradients artificially increase the value of sgs µ . For a plane channel flow, Deardorff [START_REF] Deardorff | A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers[END_REF] reduced s C to 0.1 to compensate this effect. However, it is necessary to consider more general procedures to tackle walls in complex geometries. Two recent approaches are considered here, chosen for their simplicity and their adaptability to various configurations: a shear-improved Smagorinsky model, and explicit filtering.

• Shear-improved Smagorinsky model Lévêque et al. [START_REF] Lévêque | Shear-Improved Smagorinsky Model for Large-Eddy Simulation of Wall-Bounded Turbulent Flows[END_REF] propose the following improvement to the Smagorinsky model:

( ) > < - ∆ = S S x C s sgs 2 2 ρ µ
where the influence of the mean gradients is removed in a way that respects the energy budget for both isotropic turbulence and shear turbulence (cf. Toschi et al. [START_REF] Toschi | Shear Effects in Nonhomogeneous Turbulence[END_REF]). In the present implementation, sgs µ is forced to zero if the formula gives a negative value.

The only difficulty lies in the averaging. Spatial averaging is not adapted to complex geometries, and deterministic unsteadiness must not be removed by temporal averaging. In many practical applications (e.g. turbo-machines), an intermediate time scale τ can be defined between the turbulent time scale and the deterministic time scale. Temporal averaging will be done over this period. The implementation of a running arithmetical average over τ would require the storage of all the flow fields over this period, in order to remove the oldest samples as computation progresses. To counter this problem, exponential averaging is tested here. For a quantity n q , where superscript n represents temporal index:

( )

n n n q c q c q exp 1 exp 1 + > < - = > < - Only storage of the average is required in this case. [ ] 1 , 0 exp ∈ c
is chosen according to the period τ the average has to cover. Practically, the combined influence of the data older than τ will be lower than 5% if ( )

05 . 0 1 / exp = - ∆t c τ , or equivalently: τ / exp 05 . 0 1 t c ∆ - = (1)
• Explicit filtering Bogey and Bailly [START_REF] Bogey | Computation of a high Reynolds number jet and its radiated noise using large eddy simulation based on explicit filtering[END_REF] propose to use an additional discrete filter to continuously remove the smallest computed scales that are fed by the turbulent cascade. The filter plays the role of the subgrid-scales, and there is no other model to implement:

0 = Π ij and 0 = j θ
. Mathew et al. [START_REF] Mathew | An explicit filtering method for large eddy simulation of compressible flows[END_REF] discuss the physical behavior reproduced by the filter as regularization. The first advantage of this approach lies in its simplicity and its generality. There is no complex physical quantity to evaluate, and no calibration. The choice of the filter parameters will be presented below. Moreover, Bogey and Bailly [START_REF] Bogey | Decrease of the effective Reynolds number with eddy-viscosity subgridscale modeling[END_REF] showed that subgrid-scale viscosity can modify the effective Reynolds number of jets, and this effect is suppressed by explicit filtering.

In order to maintain the adaptability of the solver, the stencil of the filter is here limited to 5 points, using a 4th order compact filter of Lele [START_REF] Lele | Compact Finite-Difference Schemes with Spectral-like Resolution[END_REF]. For a conservative variable q, the filtered variable c q is:

( ) ( ) 2 2 1 1 1 1 16 1 2 4 1 2 8 5 6 + - + - + - + - + + + + + = + + j j j j j c j c j c j q q q q q q q q α α α α α
where α is a shape parameter. Solution of the resulting tridiagonal system is obtained by Thomas algorithm. Following again the analysis of Lele [START_REF] Lele | Compact Finite-Difference Schemes with Spectral-like Resolution[END_REF], the transfer function of the filter is:

)) cos( 2 1 ( 8 ) 2 cos( ) 1 2 ( ) cos( ) 1 2 ( 4 ) 5 6 ( ) ( : ] , 0 [ 2 ω α ω α ω α α ω π ω + - + + + + = ∈ ∀ G
and it is plotted in Fig. 1 for α=0.48. This value of α concentrates the damping on the highest wavenumbers, while keeping:

1 ) ( 0 : ] , 0 [ 2 ≤ ≤ ∈ ∀ ω π ω G .
In the solver, the compact filter is applied to the conservative variables q, in the three directions of the grid sequentially, giving c q at each time step. The actual filtered variables q are then obtained with a

relaxation coefficient ] 1 , 0 [ ∈ β : c q q q β β + - = ) 1 (
The overall transfer function is:

) ( ) 1 ( ) ( : ] , 0 [ 2 3 ω β β ω π ω G G + - = ∈ ∀
where β allows controlling the repeated application of the filter without excessive damping at low frequencies ( ) ( 2 ω

G

is not exactly 1 at low ω). The value of β has to be set according to the energy to dissipate. The idea is to filter over a band comparable to the damping band of the spatial scheme (here (cf.

Pope [START_REF] Pope | Turbulent Flows[END_REF]), where l c and u c are characteristic length and velocity for large-scale turbulence. Over K τ , resolved turbulence can be considered frozen and the global transfer function of the filter can be approximated by:

( ) t K G ∆ → / 3 ) ( τ ω ω
. Consequently, β is chosen to obtain: ( ) (2)

For illustration, the transfer function

( ) t K G ∆ → / 3 ) ( τ ω ω
is plotted in Fig. 1 for β=0.138 and 66 / = ∆t K τ (values corresponding to the isotropic turbulence test-case below).

Evaluation on academic test-cases

The adaptable approaches will be first evaluated on classical academic test-cases: isotropic turbulence and bi-periodic channel flow. The present solver is dedicated to complex configurations, but these tests will abridge with more fundamental LES studies, and isolate the major physical mechanisms.

Isotropic turbulence

Study first focuses on the free decay of isotropic turbulence, which was particularly addressed by Comte-Bellot and Corrsin [START_REF] Comte-Bellot | Simple Eulerian time correlation of full-and narrow-band velocity signals in grid-generated, 'isotropic' turbulence[END_REF]. Experimental data from Kang et al. [START_REF] Kang | Decaying turbulence in an active-grid-generated flow and comparisons with large-eddy simulation[END_REF] are used here, because they are obtained at higher Reynolds number, and are consequently more representative of LES applications. In the experiment, turbulence is generated by a grid with step M=0.152m, in a flow with mean velocity U 0 =11.25m/s. Initial measurement position is 20M after the grid, and final position is 48M, corresponding to free decay during time 28M/U 0 according to Taylor hypothesis. 3D computational mesh is made of 128 points in each direction, with step ∆x=0.04 m. Time step is ∆t=10 -4 s. Boundary faces of the cube are treated as periodic. Mean field is still air, at normal atmospheric conditions. Fluctuating velocities are initialized in the corresponding wavenumber domain, with amplitudes respecting the initial energy spectrum of the experiment, and random phases. The flow field is then transposed in the physical domain using 3D Fast Fourier Transform. Following and further investigations must be carried out to study the failure of the present computation. Such complementary study should consider how energy is transferred through the cut-off, taking into account that the smallest scales are removed. Indeed, in the present computation, the spectrum seems to indicate an accumulation of energy in the bump before the cut-off. The choice of the filter and the position of the cut-off must have an influence that could explain the present failure. As a partial conclusion, isotropic turbulence test case shows the good spectral behavior of the present adaptable methods, with implicit filtering by the grid. Regarding the subgrid-scale modeling, explicit filtering appears to distort the spectrum, which needs to be further investigated. The Smagorinsky model, despite its good performance on isotropic turbulence, is not adapted to next configurations with walls. Consequently, only the shear-improved Smagorinsky model will be retained. [START_REF] Moser | Direct numerical simulation of turbulent channel flow up to Reτ=590[END_REF] with Direct Numerical Simulation (DNS). An illustration is presented in Fig. 3: mean flow is oriented on x, walls bound the domain on y direction, and periodicity is imposed on x and z directions. Grid extends over:

Bi-periodic channel flow

πδ δ πδ × × 2 2
with δ=0.01m. Point numbers are: 49x89x41, with tanh repartition in y-direction. In wall units, grid spacing is given by: ∆x + =47, ∆y + =0.5 at wall (11 points below ∆y + =10), ∆y + =22 at centerline, and ∆z + =28. Computation is initialized with Poiseuille's parabolic velocity profile for air at standard conditions (except pressure), plus 2% random velocity perturbation. Pressure is reduced to obtain a centerline Mach number of 0.2, in order to optimize convergence of the compressible solver. x-momentum is maintained by a source term that is adjusted dynamically to match the objective centerline velocity. Shear-improved Smagorinsky model uses exponential average with c exp =8.837 10 -5 , given τ=δ/u w =1.69 10 -2 s as characteristic time (based on objective u w value) and ∆t=5 10 -7 s. Post-treatment uses spatial average in x and z directions, plus "mirror"average in y direction, and temporal average during 0.22s (90 instants). Computation naturally transitions from perturbed Poiseuille flow to turbulent flow. This proves the present numerical methods are not over-dissipative, and the subgrid-scale model allows transition. Indeed, the shear-improved Smagorinsky is designed to yield µ sgs =0 in laminar flow. Resulting turbulent fluctuations are investigated below. Fig. 4 presents the mean axial velocity, compared to DNS results of Moser et al. [START_REF] Moser | Direct numerical simulation of turbulent channel flow up to Reτ=590[END_REF]. A fairly good agreement is obtained over the different layers of the flow. The overestimate of <u 1 >/u w at centerline comes from the underestimate of friction:

53 . 0 / 1 = ∂ > < ∂ = y u u w υ
instead of 0.59 (objective value). However, this 10% error lies within expected levels of accuracy given by Sagaut et al. [START_REF] Sagaut | Filtered subgrid-scale models[END_REF] for quality subgrid-scale models. Thin lines aside the LES curve represent the standard deviation of velocity, taken between the exponential-averaged field at the final instant and the post-processing space-time average (<u 1 >,0,0). A good estimate of the mean field is obtained from the exponential average, at limited constraints. Indeed, it is computed over a very limited period (τ=δ/u w =1.69 10 -2 s), without spatial averaging. This appears as an interesting tool for LES in complex configurations, where no spatial homogeneity or stationarity can be used. Fig. 5 presents the fluctuating velocity components. A good agreement is obtained with DNS, concerning the levels and the evolutions. Wall turbulence anisotropy is captured with present adaptable methods. Fig. 6 presents the two-point spanwise velocity correlations:

> ∆ + =< ∆ ) , , ( ). , , ( ) ( z z y x u z y x u z R i i ii
at y + =10 and y + =60. Post-processing average <…> covers x and z directions, mirror average in ydirection, and time. LES accurately reproduces the two-point correlations, considering the levels, the distribution between the three components of velocity, and the spanwise evolution. Comparing y + =60 to y + =10, the increase of the correlation length is captured, which appears through the spreading of the curves. Also, the increase of R 22 and R 33 shows some reduction of anisotropy, in agreement with DNS. This accurate representation of two-point correlations is one of the advantages of LES for practical applications. Averaged methods do not provide such information, which is particularly important for turbulence phenomena and acoustics. This prediction is achieved with a rather coarse discretisation (∆z + =28), which results in the slope discontinuities of the LES curves in Fig. 6. This is beneficial for the computational time. Finally, Fig. 7 presents the spanwise spectrum E 11 (energy spectrum for u 1 ) at y + =10. Post-processing average covers x-direction, mirror average in y-direction, and time. This figure illustrates one of the main advantages of LES: the spectral description of the flow, for an accurate representation of turbulence eddies and acoustics. Moreover, it allows evaluating the discretisation characteristics. In comparison with reference DNS, present LES accurately represents the most energetic structures, characterized by a flat spectrum at the lower wavenumbers. Present discretisation captures the spectrum up to the decline zone, and one decade of decline is directly represented. This kind of discretisation is consistent with LES approach, which does not represent the small and poorly energetic structures. The LES curve progressively diverges from DNS at the upper end of the spectrum, probably due to the damping associated with the numerical scheme at the high wavenumbers (cf. Fig. 1). The present level of physical discretisation, characterized by nondimensional spatial steps ∆x + , ∆y + , and ∆z + can be used as reference for future applications.

Application: tip clearance flow

This section presents a computation on a complex configuration, in order to demonstrate the adaptability of the numerical methods. The geometry is composed of a NACA5510 airfoil (chord c=0.2m, span: 0.2m), set perpendicularly above a flat plate, with a clearance between the plate and the airfoil tip (gap: 10 -2 m). Incoming air velocity is U 0 =70m/s. Lift on the airfoil generates a large clearance vortex, shown in Fig. 8 ) require the use of adaptable numerical methods, as presented above. This section only demonstrates the applicability of the methods, and no quantitative data are considered. Detailed numerical analysis of this configuration will be presented in a following paper. The grid is composed of about 3 million points, with near wall density For the exponential averaging, characteristic time is τ=c/3U 0 , corresponding to convection over one third of the chord length. LES must be particularly effective in capturing this complex vortical structure through the gap, and the associate broadband spectrum. In summary, present adaptable numerical methods allow turbulent simulation on a complex configuration, at limited computational cost (limited discretisation stencils and number of operations). Thanks to exponential averaging, spatial homogeneity or steadiness are not required. Storage is limited to the instantaneous flow field and exponential-averaged field. Finally, the shear-improved Smagorinsky model allows the development of turbulent fluctuations.

Conclusion

The present paper focused on the development of LES for improved turbulent simulation of practical internal flows. Numerical methods have been chosen according to two competing criteria: their performance (precision, spectral characteristics) and their adaptability to complex configurations. Also, implementation matters are discussed, such as the relationship between the filtering frequency and the spectral characteristics of the discretisation. These methods have been tested on academic configurations, in order to abridge with previous fundamental studies on LES. Finite volume discretisation with higher order advection fluxes, implicit filtering by the grid and a "low-cost" shear-improved Smagorinsky model showed good performances on isotropic turbulence and channel flow. Mean flow, fluctuations, two-point correlations and spectra are particularly well captured. Furthermore, exponential averaging presents good performance, and appears as an interesting tool in complex configurations with no spatial homogeneity and large deterministic unsteadiness. Finally, the adaptability of the method is confirmed by application to a complex configuration, representative of blade-tip clearance flow in turbomachine.

  logarithmic scale for physical reason. The damping becomes significant ( scales smaller than 4 points per wavelength.

Fig. 1

 1 Fig. 1 Transfer functions of the numerical schemes.

Fig. 3

 3 Fig. 3 Channel: contours of u 1 at the final instant. The second test-case is a bi-periodic plane channel flow at 395 / Re = = ν δ w w u (based on objective u w value), as studied by Moser et al.[START_REF] Moser | Direct numerical simulation of turbulent channel flow up to Reτ=590[END_REF] with Direct Numerical Simulation (DNS). An illustration is presented in Fig.3: mean flow is oriented on x, walls bound the domain on y direction, and periodicity is imposed on x and z directions. Grid extends over: πδ δ πδ

Fig. 4

 4 Fig. 4 Channel: mean axial velocity (thin lines represent standard deviation of running exponential average compared to post-processing space-time average).

Fig. 5

 5 Fig. 5 Channel: fluctuating velocity components (rms).

Fig. 6

 6 Fig. 6 Channel: two-point spanwise correlations (y + =10, and y + =60).

Fig. 7

 7 Fig. 7 Channel: spanwise spectrum E 11 at y + =10.Academic test cases have been used to focus study on spectral dynamics and wall-turbulence. Good performances are achieved on mean flow, fluctuations, two-point correlations and spectra. Moreover, discretisation aspects have been discussed, as well as performance of the exponential averaging.

  -upper. Corresponding aerodynamic and acoustic measurements are presented by Grilliat et al.[START_REF] Grilliat | Tip Leakage Experiment -Part One: Aerodynamic And Acoustic Measurements[END_REF]. This configuration is particularly representative of clearance effects in turbomachines. The objective of LES is a combined description of aerodynamics and broadband acoustics, including the turbulent physics of the detached vortical flow. The geometric complexity and the high Reynolds number (

  initialized with converged RANS results on the same grid, and the clearance vortex provides natural forcing for the development of turbulent fluctuations. The shear-improved Smagorinsky model is used.

Fig. 8

 8 Fig. 8 Tip clearance flow: streamlines through the gap (LE: leading edge, TE: trailing edge).

Fig. 8

 8 Fig.8presents results at an instant during the development of turbulent fluctuations. In the top view, streamlines show the development of the main clearance vortex along the suction side, and its deviation by the main flow downstream of the trailing edge. In the bottom view, a secondary vortex is observed on the blade tip. It is generated by the separation of the clearance flow on the pressure side edge. LES must be particularly effective in capturing this complex vortical structure through the gap, and the associate broadband spectrum. In summary, present adaptable numerical methods allow turbulent simulation on a complex configuration, at limited computational cost (limited discretisation stencils and number of operations). Thanks to exponential averaging, spatial homogeneity or steadiness are not required. Storage is limited to the instantaneous flow field and exponential-averaged field. Finally, the shear-improved Smagorinsky model allows the development of turbulent fluctuations.

~

Acknowledgement

This work is carried-out in the frame of Ecole Centrale de Lyon's BQR grant "Contrôle de l'écoulement de jeu en turbomachines". Dr Marc Jacob and EU consortium PROBAND are acknowledged for providing the tip-clearance test-case. Dr Lionel Gamet is acknowledged for numerical support and Dr Jean-Pierre Bertoglio for theoretical discussions on LES.

Kang et al. [START_REF] Kang | Decaying turbulence in an active-grid-generated flow and comparisons with large-eddy simulation[END_REF], computation is run for time 10M/U 0 , in order to initialize the turbulent cascade. Amplitude of the fluctuating velocity spectrum is then re-scaled on the experiment initial spectrum (cf. Fig. 2). Finally, computation is run during 28M/U 0 . Results obtained with the different subgrid-scale models are presented in Fig. 2. The models are tested in their general form, without simplification for the present academic case. Characteristic velocity is

, where k=2.38 m 2 /s 2 is initial turbulent kinetic energy, and characteristic length is l c =128∆x. For the shear-improved Smagorinsky model, the time used for the averaging is the characteristic time of the large scales: . Using Eq.( 1), the coefficient of the exponential average is: c exp =7.37 10 -5 . The initialization of the exponential average field is done with still air, at normal atmospheric conditions. For explicit filtering, Eq.( 2) gives the relaxation coefficient: β=0.138 (with τ K =6.57 10 -3 s). This configuration allows isolating the spectral characteristics of the methods, without walls. It is particularly demanding for adaptable methods because it is fully unsteady (statistical decay) and there is no production of turbulence. Consequently, numerical effects are accumulated, and there is no regeneration of the turbulent characteristics. Fig. 2 shows that LES is correctly initialized over the experimental spectrum. After decay, the following observations can be made.

• When no viscosity is used, energy accumulates at the highest wavenumbers, and spectrum presents an exponential growth, as expected from theory. This clearly shows the need for a subgrid-scale model. Moreover, this is the proof that present methods are not over-dissipative.

• The model of Smagorinsky allows good agreement with the experiment. A slight overprediction is observed at low wavenumbers, but this is probably the influence of the periodic boundary conditions. There is also some accumulation of energy at the highest wavenumbers (weak: logarithmic scale), but this can be explained by the uncertainty on C s . This result shows the good performance of present adaptable methods, when using a model particularly designed for this case.

• Very similar results are obtained with the shear-improved Smagorinsky model. In this case, with perfect averaging (i.e.

~= > < S

), this model actually reduces to the model of Smagorinsky.

Consequently, the present result shows the good behavior of the exponential averaging. At the end of the computation, in the exponential-averaged flow field, the standard deviation of the velocity components is

, where k is the initial turbulent kinetic energy.

• Regarding the LES with explicit filtering, the damping band clearly appears at the upper end of the spectrum. The smallest resolved scales are removed, but a bump appears at the intermediate wavenumbers. Previous authors obtained very good results using such explicit filtering [START_REF] Bogey | Computation of a high Reynolds number jet and its radiated noise using large eddy simulation based on explicit filtering[END_REF][16] [START_REF] Bogey | Decrease of the effective Reynolds number with eddy-viscosity subgridscale modeling[END_REF],