Ida Reyt

Hélène Bailliet

Solène Moreau

Jean-Christophe Valière

Diana Baltean-Carlès

Catherine Weisman

Virginie Daru

Fast acoustic streaming in standing waves: Generation of an additional outer streaming cell

Keywords: numbers: 43, 25, Nm,43, 25, Gf

Rayleigh streaming in a cylindrical acoustic standing waveguide is studied both experimentally and numerically, for nonlinear Reynolds numbers from 1 to 30 (Re N L = U 0 c 0 2 R δν 2 with U 0 the acoustic velocity amplitude at the velocity antinode, c 0 the speed of sound, R the tube radius and δ ν the acoustic boundary layer thickness). Streaming velocity is measured by means of Laser Doppler Velocimetry (LDV) in a cylindrical resonator filled with air at atmospheric pressure at high intensity sound levels. The compressible Navier-Stokes equations are solved numerically with high resolution finite difference schemes. The resonator is excited by shaking it along the axis at imposed frequency.

Results of measurements and of numerical calculation are compared with results given in the literature and with each other. As expected, the axial streaming velocity measured and calculated agrees reasonably well with the slow streaming theory for small Re N L but deviates significantly from such predictions for fast streaming (Re N L > 1). Both experimental and numerical results show that when Re N L is increased, the center of the outer streaming cells are pushed toward the acoustic velocity nodes until counter-rotating additional vortices are generated near the acoustic velocity antinodes.

I. INTRODUCTION

Rayleigh streaming refers to the second order mean velocity that is generated by viscous effects related with the interaction of a solid surface and an acoustic wave in a fluid. In thermoacoustic devices, acoustic streaming results in convected heat flow that can reduce the efficiency of the devices. Because the power density of a thermoacoustic device is roughly proportional to the square of the acoustic pressure amplitude, the study of high acoustic amplitude phenomena is important to the field of thermoacoustics. A better understanding and characterization of the corresponding so-called "fast" Rayleigh streaming is therefore useful to assess its impact on the efficiency of thermoacoustic devices.

The reference solution derived by Rayleigh 1 concerns streaming generated by a standing wave between two infinite, widely separated plates. Together with further studies by Nyborg 2 and Westervelt [START_REF] Westervelt | The theory of steady rotational flow generated by a sound field[END_REF] , this work yielded the so-called RNW streaming theory. For a better application to thermoacoustics, this theoretical description has evolved in order to refine thermal effect description and to remove the restrictions to wide channels and to slow streaming. Concerning the first two points, during the last decades, thermoacoustic applications have motivated several theoretical works to take into account thermal effects [START_REF] Olson | Acoustic streaming in pulse tube refrigerators: tapered pulse tubes[END_REF][START_REF] Hamilton | Thermal effects on acoustic streaming in standing waves[END_REF] and/or to remove the restriction to large channels [START_REF] Bailliet | Acoustic streaming in closed thermoacoustic devices[END_REF][START_REF] Hamilton | Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width[END_REF] . However, all these theoretical developments were restricted to slow streaming where fluid inertia is neglected. The dimensionless number characterizing the inertial effects with respect to viscous effects is the nonlinear Reynolds number Re N L defined by Menguy and Gilbert [START_REF] Menguy | Nonlinear effect of the inertia of the fluid on acoustic streaming in cylindrical guides[END_REF] . For slow streaming the nonlinear Reynolds number is such that Re N L << 1. The influence of fluid inertia was studied only by Menguy and Gilbert and for Re N L up to about 4 only; using a perturbation method with asymptotic expansions followed by time-averaging, they developed boundary layer streaming equations, matched with nonlinear streaming equations in the main acoustic field, thus demonstrating a) Current address:Laboratoire Roberval CNRS 6253, Université de Technologie de Compiègne BP 20529, 60205 Compiègne Cedex, France b) Also at LIMSI-CNRS UPR 3251, 91403 Orsay, France a noticeable distortion of the streaming velocity field.

Previous simulations of acoustic streaming in the linear regime were conducted by solving the flow dynamics between two plates inside a thermoacoustic refrigerator [START_REF] Marx | Computation of the mean velocity field above a stack plate in a thermoacoustic refrigerator[END_REF] . More recently, Aktas and Farouk 10 performed numerical simulations in the linear and nonlinear streaming regimes, solving the full compressible Navier-Stokes equations in the plane case with a FCT scheme. The acoustic excitation was provided by a moving wall. Their results show a distortion of the streaming field and the existence of multiple streaming cells for small values of Re N L , even for values less than 1. In their analysis, the authors correlate the existence of the nonlinear streaming regime with the values of the aspect ratio of the waveguide and of a different streaming Reynolds number (Re s = U 2 0 /(νω), ω being the angular frequency).

When their results are analyzed as a function of Re N L , the tendencies depicted are in contradiction with experiments [START_REF] Moreau | Measurements of inner and outer streaming vortices in a standing waveguide using laser doppler velocimetry[END_REF]12 . One of the goals of the present paper is to present direct numerical simulations of the Navier-Stokes equations in the cylindrical case using a recently developed high resolution numerical scheme 13 tested successfully in several cases (including acoustics and streaming 14,15 ). In the present simulations, the acoustic wave is excited by shaking the entire resonator and there are no restrictions on the value of the

Re N L .
With respect to experimental studies, thanks to improving techniques in fluid dynamics, measurements of the streaming in a standing wave guide have been performed recently [START_REF] Moreau | Measurements of inner and outer streaming vortices in a standing waveguide using laser doppler velocimetry[END_REF][16][17][18][19] .

Experimental works by Sharpe, Greated and collaborators 16 In the present study, numerical and experimental results are successfully compared with available theoretical expectations in the linear regime. For this purpose, a slow streaming analytical model that has been developed previously [START_REF] Bailliet | Acoustic streaming in closed thermoacoustic devices[END_REF] for the calculation of Rayleigh streaming in a cylindrical waveguide is used. Then the analysis of the nonlinear regime is conducted for the first time with a coupled numerical/experimental approach on a cylindrical geometry, and the results are confronted. Both simulations and experiments show that streaming at high Re N L is associated with additional outer streaming cells appearing at a critical value of this number of about Re N L = 30.

Section II presents the different methods for the calculation of streaming that are used or referred to in this paper (Direct numerical method, work by Bailliet et al. [START_REF] Bailliet | Acoustic streaming in closed thermoacoustic devices[END_REF] , and work by Menguy and Gilbert 8 ). The dimensionless numbers used to describe the problem of Rayleigh streaming in a standing wave guide are presented as it is necessary to consider the respective values of these numbers in order to validate the comparison between numerical and experimental results. In Section III, experimental set up and procedure are presented.

Section IV presents the numerical method used to calculate streaming velocity. Finally in Section V, these numerical and experimental results are compared with each other and with the available literature.

II. FUNDAMENTAL EQUATIONS AND NON-DIMENSIONAL ANALYSIS

All the methods used to calculate the streaming flow in this paper are based on the fundamental equations that describe an acoustic wave propagation in the laminar regime along the z axis, in a cylindrical tube of radius R filled with air at atmospheric pressure.

The ratio of the wavelength to the transverse dimension is supposed to be large so that the wave is plane. Air is considered as a Newtonian compressible fluid and there is no mean flow apart from acoustic streaming. Gravity is neglected. Neglecting the bulk viscosity in comparison with the shear viscosity, these fundamental equations (Navier-Stokes equations) that describe the fluid motion are written as

                   ∂ρ ∂t + ∇ • (ρu) = 0 ∂(ρu) ∂t + ∇ • (ρu ⊗ u) + ∇p = ∇ • ( τ ) ∂(ρE) ∂t + ∇ • (ρEu + pu) = ∇ • (k∇T ) + ∇ • ( τ u) (1)
where ρ is the density, p is the pressure, u is the flow velocity vector, E = e+ 1 2 u•u is the total energy, with e = p (γ-1)ρ the internal energy, γ the specific heat ratio, τ = -2 3 µ(∇ • u) Ī + 2µ D the viscous stress tensor, with Ī the identity tensor, D = 1 2 (∇u + ∇u T ) the strain tensor, µ the dynamic viscosity, k the thermal conductivity. The gas is considered as a perfect gas obeying the state law p = r g ρT , where T is the temperature and r g is the constant of the perfect gas (air in the present case).

A. Non-dimensional analysis

In the previous equations, there are 6 unknown physical variables, ρ, u (3 components), p and T (the internal energy e can be replaced by (c p /γ)T , c p being the specific heat capacity at constant pressure) and 6 equations. The problem is written in a dimensionless form, using standard fluid mechanics scalings:

∇ = ∇ L 0 , t = t t 0 , û = u U 0 , p = p p 0 , ρ = ρ ρ 0 , T = T T 0 , μ = µ µ 0 , k = k k 0 , (2)
where L 0 , t 0 , U 0 are reference scales for length, time and velocity (based on the acoustic propagation), and p 0 , ρ 0 , T 0 , µ 0 , k 0 are scales for pressure, density, temperature, dynamic viscosity and conductivity (based on fluid properties at rest). In the context of acoustics, the adiabatic speed of sound is defined as c 0 = ∂p ∂ρ s,0

, also written as c 0 = γ p 0 ρ 0 using the first-order Taylor development of pressure. The characteristic acoustic time and length scales are given by t 0 = 1 ω , L 0 = c 0 ω , where ω is the fundamental acoustic angular frequency. U 0 is a reference value of the acoustic velocity, for example the maximum amplitude of the fundamental component of the wave. With these reference scales, 3 independent dimensionless numbers are defined by rewriting the set of equations 1 : the acoustic Mach number M = U 0 c 0 , the acoustic Reynolds number Re = ρ 0 c 2 0 µ 0 ω and the Prandtl number P r = µ 0 c p k 0 .

The equations (1) in dimensionless form are indeed

                               ∂ ρ ∂ t + M ∇ • (ρû) = 0 ∂(ρû) ∂ t + M ∇ • (ρû ⊗ û) + 1 γM ∇p = 1 Re ∇ • ( τ ) 1 γ ∂(ρ T ) ∂ t + M 2 (γ -1) ∂(ρ 1 2 |û| 2 ) ∂ t + M γ ∇ • (ρ T û) + M 3 (γ -1) ∇ • (ρû 1 2 |û| 2 )+ γ -1 γ M ∇ • (pû) = 1 ReP r ∇ • ( k ∇ T ) + 1 Re M 2 (γ -1) ∇ • ( τ û) (3)
The working fluid being air, the Prandtl number is fixed, of order 1. This leaves out two non-dimensional numbers to monitor wave propagation in the tube (M and Re). In order to characterize the streaming flow associated with a standing wave, the ∇ operator should be separated into an axial and a radial part, that is a separate reference length scale for the radial direction has to be introduced, inducing that a supplementary non-dimensional number is needed. Menguy and Gilbert 8 showed that this additional required number is the Shear number, defined as Sh = δ ν R , δ ν = 2ν ω being the acoustic boundary layer thickness in the near wall region (with ν = µ/ρ 0 the kinematic viscosity).

Here, the acoustic wave amplitude ranges from linear (M ≪ 1) to weakly nonlinear.

The Reynolds number compares inertial effects and viscous effects over the wavelength and is large with respect to 1 for the present range of audible frequencies. Then, the third dimensionless number compares the boundary layer thickness to the tube radius. The hypothesis Sh ≪ 1 corresponds to large tubes, which is the case in all the studies we deal with in this paper. Finally when working with air, the same flow behavior is expected for studies where M ≪ 1, Re ≫ 1 and Sh ≪ 1. This asymptotic similarity will be considered as valid for the numerical and experimental flows under study in the following.

In order to be able to compare our experimental and numerical results to a reference, in the following section, two available models for streaming flow are briefly presented: a linear model developed by Bailliet et al. [START_REF] Bailliet | Acoustic streaming in closed thermoacoustic devices[END_REF] (used for quantitative comparison with direct numerical simulations and experiments in the slow streaming regime) and a nonlinear model developed by Menguy and Gilbert 8 (used for the description of the evolution of the streaming flow for moderate streaming regime). The latter shows that when dealing with streaming flow, an additional dimensionless number has to be considered, as discussed in the following.

B. Slow streaming

In the literature, there are several analytical models for slow streaming [START_REF] Nyborg | Acoustic streaming[END_REF][START_REF] Westervelt | The theory of steady rotational flow generated by a sound field[END_REF][START_REF] Hamilton | Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width[END_REF] . In this paper, the model developed by Bailliet et al. [START_REF] Bailliet | Acoustic streaming in closed thermoacoustic devices[END_REF] is used. This model is based on the fundamental equations presented above. Additional assumptions allow to obtain an expression of the second order time averaged axial velocity as function of the first order acoustic quantities.

The streaming is supposed to be slow so that effect of inertia on the streaming motion is neglected by comparison with viscous effects. The solid is supposed to have an infinite heat capacity. Under these assumptions, the method of successive approximation is applied, the first order quantities having a e -iωt dependence. Through time averaging a combination of the continuity equation and of the axial component of the Navier-Stokes equations written to the second order, the following equation is obtained:

1 R 2 µ 0 ∇ 2 η u 2 = ∂ z p 2 + ∂ z (ρ 0 u 2 1z ) + 1 R ∇ η (ρ 0 u 1z u 1η ) - 1 R 2 ∇ η ( µ 1 ∇ η u 1z ) , (4)
where -denotes the time average and η = r R , with r the radial coordinate. An expression for the second order pressure gradient ∂ z p 2 is obtained by writing that, in a closed resonator, there is no net mass flow across the section, i.e. the second order time average mass flow rate across the cross section must be zero. Successive integration of the expression obtained yields the second order pressure gradient as a function of the first order acoustic quantities.

In turn, successive integration of eq. ( 4) yields the axial streaming velocity (see formula (16) from Bailliet et al. [START_REF] Bailliet | Acoustic streaming in closed thermoacoustic devices[END_REF] ). In the case of widely spaced parallel plates, this velocity is equal to the one obtained by RNW theory, but the calculation is valid also for narrow guides (that is in the near wall region of any guide) and it takes into account the dependence of viscosity on temperature. This calculation is also valid in the presence of a temperature gradient (not set in the present study). This analytical study will be used as a reference for our experimental and numerical results. However, high acoustic amplitudes will be considered here and therefore the slow streaming assumption is expected to be restrictive, and the effect of streaming inertia should be analyzed.

C. Effect of inertia on streaming

The only available theoretical study that is not restricted to slow streaming is the one by Menguy and Gilbert 8 . By writing the fundamental equations up to the third order in the main acoustic field followed by time averaging over the acoustic period, Menguy and Gilbert derived the governing equations for the streaming field in the boundary layer and in the main central part of the tube. Under the above presented assumptions (M ≪ 1, Re ≫ 1, Sh ≪ 1), an asymptotic analysis can be performed, allowing to derive the fluid behavior at the first and second order. A time averaging over the acoustic period yields equations for streaming flow both inside and outside the acoustic viscous/thermal boundary layer.

Under the previous hypotheses on M, Re and Sh numbers, streaming equations inside the boundary layer are always linear. Outside the boundary layer, streaming equations can be linear or nonlinear depending on the ratio between inertia and viscous terms. Menguy and Gilbert [START_REF] Menguy | Nonlinear effect of the inertia of the fluid on acoustic streaming in cylindrical guides[END_REF] showed that in the central part of the tube, the equation that governs the streaming velocity ûs has a unique nonlinear term

Sh 2 M 2 1 r ∂ ∂r r ∂ ûs ∂r .
This clearly shows that the outer streaming nonlinearity depends only upon one dimensionless number :

Re N L = M 2 Sh 2
called the nonlinear Reynolds number, that compares inertia and viscosity. For

Re N L = O(1)
the classical streaming models are no longer valid and the nonlinear effects of inertia must be taken into account. The latter cause distortion of the streaming velocity field in the tube and Menguy and Gilbert 8 discuss the distortion of the toroidal vortex patterns. When

Re N L ≪ 1, inertia terms are negligible with respect to viscous terms and the streaming equation reduces to the linear model of streaming (see Section II.B).

Finally, four dimensionless numbers are to be considered when dealing with Rayleigh streaming. At fixed P r, for asymptotic similarity when M ≪ 1, Sh ≪ 1 and Re ≫ 1, the nonlinear Reynolds number is expected to be the driving parameter for acoustic streaming.

The behavior of the streaming flow will then be analyzed and discussed upon the values of the nonlinear Reynolds number.

III. EXPERIMENTAL APPARATUS AND PROCEDURE

The setup used to observe the phenomenon of acoustic streaming is shown in Fig. 1 and consists in a cylindrical tube connected at each end to a loudspeaker via connecting tubes designed to avoid separation effect related to the singularities in change of section. The cylindrical part of the waveguide has inner radius R = 19.5 mm and the waveguide is filled with atmospheric air. The total length of the wave guide is L exp = 2.13 m and the system is tuned on two distinct frequencies corresponding to two different resonant modes: the first mode corresponding to L exp = λ/2 and the third mode corresponding to L exp = 3λ/2, λ being the wavelength of the fundamental mode.

In the following, results will be presented for acoustic streaming across one or two cells and whatever the working frequency, the position along the guide axis will be given by the axial coordinate z, with z = ±λ/4 corresponding to two adjacent streaming cells.

The acoustic wave is generated by two loudspeakers driven by a wave generator whose frequency and amplitude are controlled. Wood smoke is used as seeding particles in order to perform LDV measurements. The single component LDV system is a Dantec Dynamic Model performed more than 26 min after the acoustic field is switched on and are stable after this time. Also, following this latter reference, in order to reach convergence in the measured streaming velocity, we choose to acquire either 70000 points or to stop acquisition after 10 seconds. This gives a good compromise between a sufficient number of measured tracer particles per point for a precise streaming velocity measurement and a small measuring time resulting from a correct seeding.

In order to get as close as possible to the isothermal boundary conditions used in the numerical calculation and because it was found in the literature that temperature gradients along the guide walls greatly influence Rayleigh streaming 19 , the resonator wall is forced to correspond to an isothermal boundary condition with the help of fans used to insure that the wall temperature is equal to the room temperature. The temperature is measured along one streaming cell using 7 thermocouples with a resolution of about 0.

IV. NUMERICAL APPROACH A. Numerical method

In order to initiate an acoustic standing wave in the cylindrical tube of length L, it is shaken in the longitudinal direction z, so that an harmonic velocity law is imposed,

V (t) = (0, V (t))
T , with V (t) = z p ω cos(ωt), z p being the amplitude of the tube displacement.

The flow can be modeled by the compressible Navier-Stokes equations (1), expressed in the moving frame attached to the tube, so that a forcing source term h s is added to the system of equations (1). The model can be written in vector form :

∂w ∂t + 1 r ∂ ∂r [r(f -f v )] + ∂ ∂z (g -g v ) = h + h s ( 5
)
where w is the vector of conservative variables (ρ, ρu, ρv, ρE) T , f and g are the inviscid fluxes f = (ρu, ρu 2 + p, ρuv, ρEu + pu) T and g = (ρv, ρuv, ρv 2 + p, ρEv + pv) T , f v and g v being the viscous fluxes f v = (0, τ rr , τ rz , k ∂T ∂r + uτ rr + vτ rz ) T , g v = (0, τ rz , τ zz , k ∂T ∂z + uτ rz + vτ zz ) T . The source terms read h = (0, -τ θθ r + p r , 0, 0), h s = (0, 0, -ρ dV dt , -ρu dV dt ) T . The model is numerically solved by using high order finite difference schemes, developed by Daru and Tenaud 13 . An upwind scheme, third order accurate in time and space, is used for convective terms, and a centered scheme, second order, is used for diffusion terms.

Traveling shock waves are present in the flow for high acoustic levels, that generate numerical oscillations. However they are of weak intensity and produce very small oscillations that do not spoil the solution. Thus it was not necessary to add a costly shock capturing procedure.

The numerical procedure is briefly described thereafter. Denoting by w n i,j the numerical solution at time t = nδt and grid point (r, z) = (iδr, jδz), with δt and δr, δz the time and space discretization steps respectively, the following Strang splitting procedure is used to obtain a second order accuracy every two time steps:

w n+2 i,j = L δr L δz L δz L δr w n ij ( 6
)
where

L δr (resp. L δz ) is a discrete approximation of L r (w) = w + δt(-1 r (rf ) r + 1 r (rf v ) r + h + h s ) (resp. L z (w) = w + δt(-g z + g v z )
). The 1D operators being similar in both directions, only the r operator is described. The scheme is implemented as a correction to the second order Mac-Cormack scheme. It consists of three steps, as follows:

w * i,j = w n i,j -δt r i,j δr r i+ 1 2 ,j (f i+1,j -f v i+ 1 2 ,j ) -r i-1 2 ,j (f i,j -f v i-1 2 ,j ) n + δt(h + h s ) n i,j w * * i,j = w * i,j -δt r i,j δr r i+ 1 2 ,j (f i,j -f v i+ 1 2 ,j ) -r i-1 2 ,j (f i-1,j -f v i-1 2 ,j ) * w n+1 i,j = 1 2 (w n i,j + w * * i,j ) + 1 r i,j (r i+ 1 2 ,j C r i+ 1 2 ,j -r i-1 2 ,j C r i-1 2 ,j ) (7)
In (7), the subscripts (i ± 1 2 , j) denote a value at the interface between cells (i, j) and (i±1, j). The viscous fluxes are discretized at each interface using centered second order finite differences. The corrective terms C r i± 1 2 ,j provide the third order accuracy and upwinding for the inviscid terms. Details on the model can be found in Daru et al. 14,15,22 .

The physical boundary conditions employed in the moving frame are : no slip on the wall, symmetry at the ends of the tube, and isothermal walls.

The acoustic streaming is generated by the interaction of the imposed plane standing wave and the tube wall. Resonant conditions are imposed, for which L = λ/2, λ being the wavelength calculated for a dissipative gas. The boundary layer δ ν is of small thickness and must be correctly resolved by the discretization mesh. After several trials, it was found that a value of 5 points per boundary layer thickness is sufficient for reasonable accuracy of the simulations. All results presented below are thus obtained using a cartesian mesh of rectangular cells of constant size δr and δz, composed of 500 points in the axial direction z, and of 5 × R/δ ν points in the radial direction.

The flow being assumed to be axisymmetric (at least in the range of parameters treated), only the region 0 ≤ r ≤ R and 0 ≤ z ≤ L was considered. Also, the scheme being fully explicit, the time step δt is fixed such as to satisfy the stability condition of the scheme. In all cases considered here, the time step limitation is acoustic, it reads δt ≤ 1 2 δr/c 0 . Taking δt = [START_REF] Rayleigh | On the circulation of air observed in Kundt's tubes, and on some allied acoustical problems[END_REF] 2 δr/c 0 and δr = δ ν /5, this results in a number of time steps N T per period of oscillation proportional to

√ L, N T = 1/(f δt) = 10 2πc 0 ν √ L.
Because the LDV measurements yield the Eulerian particle velocity numerical results are based on the Eulerian time-average of fluid velocity. The numerical streaming velocity is obtained by calculating a simple mean value of the particle velocity over the discretized acoustic period, thus a time average made with respect to the fixed measuring position, for each mesh point.

B. Simulation parameters

A cylindrical tube is considered, initially filled with air at standard thermodynamic conditions, p 0 = 101325 Pa, ρ 0 = 1.2 kgm -3 , T 0 = 294.15 K. The thermo-physical properties of air are µ 0 = 1.795 10 -5 kgm -1 s -1 and k 0 = 0.025 Wm -1 K -1 . Viscosity is expressed as a power law of temperature, µ(T ) = µ 0 (T /T 0 ) 0.77 . Also for air, γ = 1.4 and r g = 287.06 Jkg -1 K -1 .

The Prandtl number P r is equal to 0.726. This results in an initial speed of sound c 0 = 343.82 ms -1 .

In the experimental case presented in Section III, the half-wavelength is about 0.7 m long, which would yield a limiting time step corresponding to N T ≈ 25000 √ L, that is 21000 iterations per period. Since transients of several hundreds of periods may be needed in order to reach stabilized streaming flow, several millions of iterations are necessary for each simulation. Considering these numerical constraints, a high-frequency wave is chosen for numerical calculations, with f = 20000 Hz. Taking into account the viscous dissipation, this corresponds to a shorter wave guide with L = 8.396 mm. The resulting boundary layer thickness is δ ν = 1.54 10 -5 m. For computational ease, the Shear number was fixed in the numerical simulations to be 1/40.

The time step δt = 8 10 -9 s is chosen in order to satisfy the numerical stability condition, corresponding to 6250 time iterations per period. The acoustic velocity produced in the wave guide depends on the imposed amplitude of the horizontal displacement z p . It varies approximately linearly with z p , for a given ratio R/δ ν . Therefore it is possible via the only choice of z p to approximately set the value of the desired nonlinear Reynolds number Re N L , so that to approximately fit the ones obtained in experimental configurations.

V. RESULTS AND DISCUSSION

As seen in sections III and IV several differences exist between experimental and simulated configurations. As discussed in section II these differences should not be associated Table I summarizes then the values of these dimensionless parameters for both the experimental and numerical configurations together with the ones for several studies of the literature [START_REF] Menguy | Nonlinear effect of the inertia of the fluid on acoustic streaming in cylindrical guides[END_REF][START_REF] Aktas | Numerical simulation of acoustic streaming generated by finite-amplitude resonant oscillations in an enclosure[END_REF]12,19 that are used for comparison with the present study.

As shown by Table I and as stated in the previous section IV, all numerical calculations were performed for a given frequency in air at atmospheric pressure, corresponding to one acoustic Reynolds number Re. Because the experimental setup could be tuned at two different resonant modes, two different Re numbers could be achieved. Whatever the frequency condition is in numerical and experimental configurations, the acoustic Reynolds number is always much larger than 1. Therefore, we can expect similarity to hold between the different configurations studied here in terms of Re.

When working on the λ/2 experimental mode, the Sh number is 1.2 • 10 -2 (the corresponding experimental configurations are labeled E 1 and E 2 ), whereas when working on the 3λ/2 experimental mode, the Sh number is 7.2 • 10 -3 (labeled E 3 to E 5 ). In the numerical simulations the Sh number is 2.5 • 10 -2 . As will be shown in the following section, such differences in Sh numbers do not yield any significant difference in the acoustic streaming regime.

Finally, by changing the acoustic amplitude, the M and Re N L numbers are modified.

Since the Re N L number is the relevant parameter to study nonlinear acoustic streaming, numerical and experimental configurations were chosen such that the Re N L numbers are as close as possible for comparisons. The corresponding acoustic Mach number ranges from 8.2•10 -3 to 4•10 -2 for experiments and from 1.2•10 -2 to 1.5•10 -1 for numerical simulations, and therefore attention should be paid to the possible effects of such different dynamic ranges.

In the following Sections V.A-C, results for experiments and numerical simulations are presented for different ranges of Re N L , then in Section V.D the effect of other dimensionless numbers are considered and discussed.

A. Streaming velocity at low nonlinear Reynolds number

The results presented in this section refer to situations where the nonlinear Reynolds number Re N L is smaller than 1. The experimental case selected for this comparison at low nonlinear Reynolds number corresponds to Re N L = 0.47 (E 1 ), and the numerical simulation case corresponds to Re N L = 0.25 (N 1 ). in the present coordinate system. Figure 2 shows that it is indeed the case for all results presented in the figure, except for a slight difference between numerical and both analytical and experimental results. The source of this difference is unclear. The acoustic wave is created in a different manner in the three studies (analytical, numerical and experimental). This can create discrepancies between the results. Moreover, it is important to recall that the numerical simulations are long and the streaming flow is of the second order. These are difficult conditions, especially for small values of the nonlinear Reynolds number. Therefore this difference in streaming velocity between numerical and analytical results is very satisfying.

The streamlines of the streaming flow are plotted in Fig. 3. Two outer streaming cells and two inner cells can be observed, corresponding to the classical Rayleigh streaming pattern.

The streaming vortices are present either sides of the central axis and spaced at interval of λ/4. For outer streaming, the flow along the central axis is directed toward the acoustic velocity antinodes and returns in the vicinity of the wall to complete a closed loop. In the near wall region, the inner streaming vortices have directions of rotation opposite to those of the outer cells. In the cylindrical resonator, those vortices form toroidal flows around the central axis. between numerical and experimental results can be observed near the walls, when 0.8R < r < R that can be attributed to the difference between the Shear numbers in the experimental and numerical configurations. To cope with this effect of different Shear numbers, the evolution of the streaming radial dependency u s (r) will no further be discussed, but only the evolution of the axial dependency u s (z) along the axis with Re N L .

Also, given the similarity between the two analytical curves along the center line, it is chosen in the following to plot only the analytical result corresponding to the experimental configuration. it gets significantly lower than Rayleigh expectation. As shown in Fig. 6, the streaming structure keeps its symmetry with respect to the z = 0 position, both in numerical simulations and in experimental measurements. It was checked that this symmetry is kept even for higher amplitude, and therefore, in the rest of the figures, measurements are presented along one cell only (from about z = 0 to z/λ/2 = 0.5).

B. Streaming velocity at moderate nonlinear

The figures clearly show a continuous decrease of the maximum value of the streaming velocity as Re N L increases both in the experimental results and the numerical simulations even so the decrease is not as pronounced in the numerical simulations as in experiments (see Figure 7). Potential sources of this discrepancy will be discussed in Section V.D.

C. Streaming velocity at high nonlinear Reynolds number

As the Reynolds number is further increased, as shown in Fig. 8, the difference between measured and calculated streaming on one hand, and linear theory on the other hand is even more pronounced.

In Figure 8

D. Discussion

As seen in the previous sections, numerical, experimental and analytical results all agree for small values of Re N L . As Re N L increases, the tendencies are similar for numerical simulations and experiments: the levels of dimensionless streaming decrease and new streaming cells appear at a critical value Re N L ≃ 30. In order to identify a physical source for this evolution of streaming with Re N L , several hypotheses are discussed in the following sections on the basis of the present literature review.

Thermal effects

Following Thompson et al. 19 , the influence of thermal effects can be questioned. They performed measurements in a standing wave guide for nonlinear Reynolds numbers up to 22, the three other dimensionless numbers (M , Re and Sh) respecting the above mentioned asymptotic conditions of our study, see Table I. They explored the evolution of the streaming for three different thermal conditions : isothermal, uncontrolled and insulated and for different nonlinear Reynolds numbers. They observed that when the magnitude of the tem-perature gradient was increased by changing the thermal condition for a given high nonlinear Reynolds number, the magnitude of the streaming decreased and the shape of the streaming cell became increasingly distorted. They found that the thermoacoustically induced axial temperature gradient strongly influences the axial component of the acoustic streaming velocity and concluded that the deviation from the theories of Rott (characterizing slow streaming with an axial temperature gradient) is due to the influence of thermal effect more than fluid inertia.

In our numerical simulations, isothermal boundary conditions are imposed and thermal effects are accounted for. Nevertheless, the axial streaming velocity profile is deformed with increasing Re N L . The same tendency is observed in the experimental results presented in Fig. 5678, even if the thermal condition maintained for our set up is only quasi-isothermal (as a 0.3 • C temperature difference between acoustic velocity node and acoustic velocity antinode is measured). For Re N L = 31.8 the streaming flow already presents additional vortices compared to Rayleigh expectation, as shown by the change of sign of the axial streaming velocity in Fig. 8.

Therefore, if we agree that thermal effects are very important in understanding acoustic streaming (the difference between experimental and numerical thermal conditions could be at the origin of the difference in results) it appears from our study that the presence of a temperature gradient is not the cause of the divergence from slow streaming expectation at high Re N L observed both numerically and experimentally.

Irregular streaming

Another possible cause of streaming distortion at high amplitudes exposed in the numerical and experimental literature is the change of regime from "regular" (or classical)

to "irregular" streaming. In the literature the term "regular" is used when streaming flow exhibits the well-known patterns (toroidal outer and inner cells), while the term "irregular" is used to characterize any streaming flow that deviates from these patterns.

Aktas and Farouk 10 studied acoustic streaming in a 2D rectangular channel numerically.

They found that there is a limit in the value of the enclosure height to wavelength ratio above which the streaming flow structures are always irregular and complex. Since their working frequency is constant, it is equivalent to consider a limit value of the inverse of the Shear number. Based on their numerical results, they estimate this limit value of 1/Sh Finally, we choose not to use the terms "regular" or "irregular" to describe the streaming flow regimes observed both numerically and experimentally. Instead, we would rather describe them as "regular" streaming flow regimes with either two (Re N L < 30) or three (Re N L > 30) toroidal structures, with a smooth transition as Re N L increases. Our analysis has been done under the assumption of low Mach number. But as the nonlinear Reynolds number increases, so does the Mach number, creating situations where nonlinear acoustic propagation should be of importance, thus the generation of higher harmonics has to be considered. This brings up the question of higher harmonics influence on the streaming in competition with inertial effects associated with high values of nonlinear Reynolds number.

Figure 11 shows the harmonic distribution of the acoustic field, obtained by projecting the total velocity signals on a sine basis, in both numerical (Fig. 11.a) and experimental (Fig. 11.b) results for Re N L ≈ 30. It shows that the nonlinear propagation induces the existence of shockwaves in the simulations whereas the magnitude of higher harmonics never reaches more than 4% of the magnitude of the fundamental in the experiments, due to the difference in geometrical configurations between experimental and numerical studies. Now for a first approach a superposition principle is applied in order to identify the acoustic streaming due to multiple frequency components. But this crude assumption cannot cope with nonlinear effects. So, as stated in section II.C, because the boundary layer can be considered as a region where the streaming is linear, we choose to superimpose the streaming fields resulting from higher harmonics in the vicinity of this region. We choose the particular radial position corresponding to the limit between inner and outer streaming cell, that is at (R -r) ≈ 3δ ν , at which the streaming is expected to be close to linear streaming. Figure 12 shows the streaming velocity obtained from the numerical simulation at Re N L = 38.6 (N 5 ), the analytical calculation resulting from the fundamental component of the numerical signal, and the analytical calculation resulting from the linear superposition of all the harmonic components of the numerical signal. First, the distortion of the streaming close to the boundary layer (Fig. 12) is found to be much weaker than in the core of the resonator (Fig. 8 and Fig. 10). The effect of the harmonics superposition at (R -r) ≈ 3δ ν results in a shift of the streaming velocity maxima towards the acoustic velocity nodes associated with a small change in the velocity magnitude. The same type of distortion can be observed on the numerical profile (Fig. 12), but with a stronger change in the velocity magnitude (the maximum value and the slope of the curve at z = 0 are much smaller). Moreover, as the distortion in acoustic streaming is qualitatively the same in both experimental and numerical study along the centerline of the guide, we conclude that while the harmonic content of experimental and numerical signals are very different, the influence of higher harmonics is not the main cause of the divergence of the streaming field from Rayleigh's expectation.

Finally we are left with only one hypothesis to explain this divergence: the inertial effects cannot be neglected anymore when the streaming is fast.

VI. CONCLUSION

In this paper a coupled experimental and numerical study of Rayleigh streaming from low to high acoustic amplitudes was conducted. As expected, slow streaming (both numerically and experimentally obtained) agrees with analytical slow streaming expectations for small

Re N L . The divergence from these analytical results that occurs when the Re N L is increased is very similar for numerical and experimental results. As Re N L increases, the cell centers are pushed towards the acoustic velocity nodes and the maximum axial streaming velocity decreases with respect to the analytical Rayleigh reference amplitude. It was also shown that for Re N L larger than about 30, additional outer cells are formed through a smooth transition between coherent streaming flow patterns. This is why the streaming flow regimes can be called "regular" throughout the transition between two (Re N L < 30) and three (Re

  and by Thompson and Atchley 19 first allowed quantitative study of streaming velocity in the core region on the basis of Particle Image Velocimetry (PIV) or Laser Doppler Velocimetry (LDV) measurements. Moreau et al. 11 first performed LDV measurements of streaming in both the core and the near wall region to characterize inner streaming vortices. Their measurements were performed for Re N L up to 140 but they mainly focused on inner streaming cells evolution. In their experimental study, Thompson and Atchley 19 considered streaming for Re N L up to 20 and found that the streaming fields is distorted for high nonlinear Reynolds number in correlation with the increasing temperature gradient along a streaming cell. Nabavi et al. 12 analyzed stream-ing in a square channel on the basis of PIV measurements; they found that for Re N L up to 25, streaming cells are regular and correspond to "classical" streaming flow, whereas when Re N L exceeds 25, streaming deforms to irregular and complex shaped patterns. Another goal of the present work is to present results of measurements of the streaming velocity field generated by an acoustic standing wave in a cylindrical resonator by means of LDV from low to high acoustic amplitudes (up to Re N L greater than 25).

FIG. 1 .

 1 FIG. 1. (color online) Photo and diagram of the experimental apparatus.

  05 • C. These thermocouples are placed at the outer wall of the resonator and are insulated from the external environment by a polystyrene covering. Using fans to force heat transfer from the guide to the environment yields a residual temperature difference of 0.3 • C between the acoustic velocity node and antinode at Re N L = 31.8. This temperature difference would be equal to 3.1 • C if no fans were used.

2 2 is

 22 with different behaviors of acoustic streaming if the nonlinear Reynolds number Re N L = M Sh the same, and if the other relevant dimensionless numbers Sh = δ ν R , of the same order of magnitude. The acoustic amplitude of the fundamental mode at the velocity antinode z = 0 is chosen as the velocity reference U 0 .

Figure 2 8 U 2 0c 0 1 FIG. 2 . 8 U 2 0

 28201282 Figure 2 displays results from measurements and from numerical simulations for the axial streaming velocity component u s along the scaled central axis z/(λ/2) together with two different results from the analytical analysis presented in Section II corresponding to the numerical and experimental cases respectively. In all figures, the velocities are normalized by U Rayleigh = 3 8

1 FIG. 3 .

 13 FIG. 3. Streamlines of the streaming flow, from numerical simulation at Re N L = 0.25.

Figure 4 1 FIG. 4 .

 414 Figure4shows the variation of the axial streaming velocity component along the radial axis on a cross-section positioned at z = -λ/8, for the same cases as in Fig.2. There is good agreement between all results in the core flow, away from the walls. Small discrepancies

1 FIG. 5 . 1 FIG. 6 .

 1516 FIG. 5. (color online) Variation of the normalized axial streaming velocity component along the central axis of the guide. (--) Analytical results using the conditions of the experiment ; (•) E 2 , experimental measurements at Re N L = 4.8 ; (-) N 2 , numerical simulation Re N L = 4.91.

1 FIG. 7 .

 17 FIG. 7. (color online) Variation of the normalized axial streaming velocity component along the central axis of the guide. (--) Analytical results using the conditions of the experiment ; (•) E 4 , experimental measurements at Re N L = 14.6 ; (-) N 4 , numerical simulation at Re N L = 15.2.

1 FIG. 8 . 1 FIG. 9 .FIG. 10

 181910 FIG. 8. (color online) Variation of the normalized axial streaming velocity component along the central axis of the guide. (--) Analytical results using the conditions of the experiment ; (•) E 5 , experimental measurements at Re N L = 31.8 ; (-) N 5 , numerical simulation at Re N L = 38.6.

Fig. 3 .

 3 Fig.3. In this figure, the streaming velocity field is clearly regular. Although our results are obtained for axisymmetrical configurations, whereas those of Aktas and Farouk 10 concern a rectangular enclosure, previous work 14 showed similar behavior in planar geometry compared to the present observations in the axisymmetrical case. This is indeed coherent with the asymptotic approach.

FIG. 11

 11 FIG. 11. (color online) Distribution of the harmonics along the guide. (a) Numerical simulation N 5 ; (b) Experimental study E 5 .

2 )

 2 FIG. 12. (color online) Streaming velocity profile along the axis of the guide at (R -r) ≈ 3δ ν , for Re N L = 38.6. (-) Analytical solution ; (-•-) Streaming velocity from analytical superposition of frequency components of the numerically calculated acoustic velocity; (--) Numerical simulation.

• 10 6 4 8 U 2 0

 6482 FIG. 1 (color online) Photo and diagram of the experimental apparatus. . . . . . . 12

  Logically the Re N L is the driving parameter for the acoustic streaming in the regime considered here. acoustic streaming patterns in a rectangular enclosure", Wave Motion 46, 312 -322. 13 Daru, V. and Tenaud, C. (2004). "High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations", J. Comp. Phys. 193, 563 -594. Moreau, S., Bailliet, H., Valière, J.-C., Boucheron, R., and Poignand, G. (2009). "Development of laser techniques for acoustic boundary layer measurements. part ii: Comparison of ldv and piv measurements to analytical calculation", Acta Acustica united with Acustica 95, 805-813. 21 Valière, J.-C., Moreau, S., and Bailliet, H. (2009). "Development of laser techniques for acoustic boundary layer measurements. part i: LDV signal processing for high acoustic displacements", Acta Acustica united with Acustica 95, 585-594. 22 Daru, V. and Tenaud, C. (2001). "Evaluation of tvd high resolution schemes for un-steady viscous shocked flows", Computers and Fluids 30, 89 -113.TABLE I. Non-dimensional numbers used for the present study together with those of several studies in the literature.

	Reference	Name	Re N L	M	Sh	Re
	14 Daru, V., Baltean-Carlès, D., Weisman, C., Debesse, P., and Gandikota, G. (2013). E 1 0.47 8.2 • 10 -3 1.2 • 10 -2 1.4 • 10 7 E 2 4.8 2.6 • 10 -2 "Two-dimensional numerical simulations of nonlinear acoustic streaming in standing Experiments E 3 7.7 2 • 10 -2 waves", Wave Motion 50(5), 955 -963. 7.2 • 10 -3 5.2 • 10 6 (present study) E 4 14.6 2.7 • 10 -2 15 Daru, V. and Gloerfelt, X. (2007). "Aeroacoustic computations using a high order shock-E 5 31.8 4 • 10 -2 capturing scheme", AIAA J. 45, 2474-2486. N 1 0.25 1.2 • 10 -2
		N 2	4.91	5.5 • 10 -2
	Numerical	N 3	7.93	7.0 • 10 -2	2.5 • 10 -2	6.3 • 10 4
	(present study)	N 4	15.2	9.7 • 10 -2
		N 5	38.6	15 • 10 -2
	Menguy & Gilbert 8		[0.5 -2]	<< 1	<< 1	>> 1
	Thompson et al. 19

N L > 30) toroidal structures. After having considered all potential sources for the observed divergence from slow streaming flow, we have concluded that inertial effects on acoustic streaming are the main source.

16 Sharpe, J., Greated, C., Gray, C., and Campbell, D. M.

(1989)

. "The measurements of acoustic streaming using particle image velocimetry", Acustica 68, 168 -172.

17 Campbell, M., Cosgrove, J., Greated, C., Jack, S., and Rockliff, D.

(2000)

. "Review of lda and piv applied to the measurement of sound and acoustic streaming", Optics and Laser Technology 32, 629 -639, optical methods in heat and fluid flow. 18 Thompson, M. and Atchley, A. (2005). "Simultaneous measurement of acoustic and streaming velocities in a standing wave using laser doppler anemometry", J. Acoust. Soc. Am. 117, 1828-1838. 19 Thompson, M., Atchley, A., and Maccarone, M. (2005). "Influences of a temperature gradient and fluid inertia on acoustic streaming in a standing wave", J. Acoust. Soc. Am. 117, 1839-1849.

20

Acknowledgments

All the experiments were conducted at the Pprime Institute. The authors wish to aknowledge the technical support of Pascal Biais, Laurent Philippon and Philippe Szeger.


S Harrat

K Meftouh
email: karima.meftouh@univ-annaba.org

M Abbas
email: m_abbas04@yahoo.fr

K Smaili
email: smaili@loria.fr

GRAPHEME TO PHONEME CONVERSION AN ARABIC DIALECT CASE

Keywords: Modern Standard Arabic (MSA), Algiers Dialect, Grapheme-to-phoneme conversion, Statistical Machine Translation

We aim to develop a speech translation system between Modern Standard Arabic and Algiers dialect. Such a system must include a Text-to-Speech module which itself must include a grapheme-phoneme converter. Algiers dialect is an Arabic dialect concerned by the most problems of Modern Standard Arabic in NLP area. Furthermore, it could be considered as an under-resourced language because it is a vernacular language for which no substantial corpus exists. In this paper we present a grapheme-to-phoneme converter for this language. We used a rule based approach and a statistical approach, we got an accuracy of 92% VS 85% despite the lack of resource for this language.

INTRODUCTION

Grapheme-to-Phoneme (G2P) conversion or phonetic transcription is the process which converts a written form of a word to its pronunciation form. G2P conversion is not a simple deal, especially for non-transparent languages like English where a phoneme may be represented by a letter or a group of letters and vice-versa [START_REF] Mininni | Applied Psycholinguistics, Positive effect a d ethical per pective[END_REF].

Unlike English, Arabic is considered as a transparent language; in fact, the relationship between grapheme and phoneme is one to one, but note that this feature is conditioned by the presence of diacritics. The lack of vocalization generates ambiguity at the phonetic level and consequently at the lexcial, syntactic and semantic levels. For exapmle the word "كتب" /ktb/, can have different phonetic transcriptions 1 such as /kataba/, /kutiba/, /kutubun/, /kutubi/, /katbin/... Algiers dialect obeys to the same rule: without diacritics, G2P conversion will be a difficult issue to 1 We use SAMPA Speech Assessment Methods Phonetic Alphabet for phoeneme representation. resolve. That is why we first dedicated our efforts to develop an automatic diacritizer system for this language.

Most of works on G2P conversion have used two approaches: the first one is a dictionary-based approach, where a phonetized dictionary contains for each word of the language its correct pronunciation. The G2P conversion is reduced to a lookup of this dictionary. The second approach is rule-based [START_REF] Alghamdi | Arabic Phonological Rules[END_REF][3] [START_REF] Zeki | Development of An Arabic Text-To-Speech System[END_REF], in which the conversion is done by applying phonetic rules, which are deduced from phonological and phonetic studies of the considered language or learned on a phonetized corpus using a statistical approach based on significant quantities of data [START_REF] Taylor | Hidden Markov model for grapheme to phoneme co ver io[END_REF], [START_REF] Ogbureke | Hidden Markov Models with Context-Sensitive Observations for Grapheme-to-Phoneme Conversion[END_REF].

For Algiers dialect which is a non-resourced language, a dictionary based solution for a G2P converter is not feasible since a phonetized dictionary with a large amount of data is not available. The first intuitive approach (regards to the lack of resource) is a rule-based one, but the specificity of Algiers dialect had led us to a statistical approach in order to consider all features related to this language.

ALGIERS DIALECT

The Algiers dialect is one of the most important dialects of Algeria; it represents the dialectal Arabic spoken in Algiers and its periphery. Algiers dialect simplifies the morphological and syntactic rules of the written Arabic. It uses the Arabic alphabet which includes 28 letters (14 solar consonants which assimilate the ل of a preceding definite article ال and 14 lunar consonants which do not assimilate it) and three non Arabic letters ڤ /G/ , ڢ /V/ and پ /P/. It uses all Arabic diacritics except the Tanween doubled case endings see [START_REF] Harrat | Diacritic Restoration for Arabic Dialect Texts[END_REF] for more details.

Writing System

Algiers dialect is a spoken language without conventional writing rules. No writing system is adopted for it by scientific community of the country. In fact, the absence of writing system for Algiers dialect is due to the lack of resources such as corpora for this language. Working on this issue (lack of resources) began with a corpus construction which has taken us to adopt a writing system. The main idea for elaborating writing rules is based on the fact that Algiers language is an under-resourced Arabic language, so we adopted the Arabic writing system when it is possible. That means when writing a word in Algiers dialect we look if there is an Arabic word close to this dialect word: if it exists, we adopt the Arabic writing for the dialect word; otherwise, the word is written as it is spoken. It should be noted that Algiers dialect is written from right to left (like Arabic language).

Issues of G2P Conversion for Algiers Dialect

Algiers dialect G2P conversion obeys to the same rules as MSA. Indeed, Algiers dialect could be considered as a transparent language since alignment between grapheme and phoneme is one to one when the input text is vocalized. This transparency conducts us to adopt a rule-based approach to build a G2P converter. Furthermore, Algiers dialect contains borrowed words from foreign languages (mainly French). Its vocabulary contains many French words used in everyday conversation. French borrowed words could be divided into two categories: the first one includes French words phonologically altered such as the word َة" ي َ َل َام ف َ َ َ َ َ َ" (famille, family) and the second one includes words which are pronounced as in French like the word "ََ َور س َ َ َ َ " (sûr, sure). This last category constitutes a serious deal for G2P conversion since these words do not obey to Arabic pronunciation rules.

Dialect word

Dialect phonetic transcription French word Meaning

َ َ ك َ َ َ َ وز َ َ َ َ َ ين َ ََ ة /ku:zina/ Cuisine Kitchen َ َ ط َ َ َ َ اب َ ََ َ َ ل َ َ ة / tʻa:bla/ Table Table َ َ ك َ َ َ َ ون َ ََ َ َ ك َ َ َ َ س َ َ َ َ ي َ َ ون / Connexion Connection َو د َ َ َ َ ڢ َ َ ـيز dəviz Devise Currency Table 1.

Examples of French words used in Algiers dialect

In the examples of Table1, although the first two words are French, they are phonetized as Arabic words. The french word "table" i pho ologically altered a d writte i Algier dialect (with Arabic cript) ة" َ َل اب َ ط َ َ ََ َ ". On the other side, the last two words are phonetized as French words since they are pronounced as in French by Algiers dialect speakers. In order to take account of this word category, the French phonemes like / , a d ə mu t be i cluded i Algiers dialect phonemes. Note that, depending on the speaker, we can found different words related to the same meaning falling into both of the two categories. An exemple of this, the word َة" ي َ َل َام ف َ َ َ َ َ َ" (family), which is a french word (famille) phonologically altered is widely used in Algiers dialect, but even the word " َ َ ف َ َ َ َ ام َ َ َ ََ ي َ َ " (famille)َ with French pronounciation is used.

RULE BASED APPRAOCH

As mentioned above the rule-based approach for G2P conversion applied to Algiers dialect requires a diacritized text. Hence, we implemented ADAD, the Automatic Diacritizer of Algerian Dialect texts which then can be converted into their phonetic forms by applying the established rules mentioned in Section 3.2.

Automatic Diacritic Restoration

ADAD [START_REF] Harrat | Diacritic Restoration for Arabic Dialect Texts[END_REF] is based on a statistical approach; we used available tools for machine translation where source and target were respectively undiacritized and diacitized texts. We considered diacritization as a Machine Translation problem; we built a Statistical Machine Translation system based on parallel corpora of diacritized and undiacritized texts. We used a trigram language model trained on diacritized texts, a phrase table with undiacritized and diacritized entries. This choice is due to the fact that this approach does not require any linguistic knowledge or resources except vocalized corpora. First, we experimented our solution on MSA diacritized corpora Tashkeela2 and LDC Arabic Treebank (Part3, V1.0) [START_REF] Maamouri | Arabic Treebank: Part 3 v 1.0[END_REF], we got motivating results: a DER (Diacritization Error Rate) of respectively 4.1% and 5.7%, and precision rate of 93.1% and 96%. For dialectal side, we vocalized by hand a part of Algiers dialect corpus and we used it for training our system, the results achieved a DER of 12.8% and precision rate of 98% despite the very little size of this corpus. Considering this interesting precision rate, we used an iterative process to vocalize the rest of the dialectal corpus. We begin by automatic vocalization of a small amount of data, then we proceed to the correction of errors (by hand). This task is not time consuming regarding the high precision rate of the system. The vocalized text obtained constitutes the input to the training corpus then we restart training process, and proceed to automatic vocalization of another portion of the corpus, and so on (see figure 1).

Rule Based G2P Conversion

After the diacritization step described above, the text is converted into phonemes by applying a set of rules detailed in the following. It should be noted that most of these rules are those adopted also for Arabic [START_REF] Alghamdi | Arabic Phonological Rules[END_REF] [3] and are applicable only for Arabic words and foreign words phonologically altered in our corpus.

Let us consider: BS is a mark of the beginning of a sentence, ES is a mark of the end of a sentence, BL is a blank character, C is a consonant, V is a vowel, LC is a lunar consonant, SC is a solar consonant, and LV is a long vowel. A conversion rule can be written as follows:

LFT + GR + RGT  /PH/
The rule is read as follows: a grapheme GR having respectively as left and right context LFT and RGT is converted to the phoneme PH. Left and right contexts could be a grapheme, a word separator, the beginning or the end of a sentence or empty.

,ذ ظ and ث rules

Definite article rule

When the definite article "ال" is followed by a lunar consonant (which does not assimilate the ("ل" the "ا" is not pronounced.

{ LC}+{+الBL,BS}  /l/+/LC/ Example: ََ ر َ م َ الق َ َ َ َََ (the moon)  /laqmar/
This rule is the same as in MSA with the difference that in MSA the "ا" is pronounced if the definite article is in the beginning of the sentence.

When the definite article "ال" is followed by a solar consonant theَ "ل" is not pronounced and the consonant following it is doubled (gemination). It should be noted that the hamza in the middle of the word is replaced by the long vowels ا or ي in Algiers dialect. For example the Arabic words " َ بئر " (a well) and "فأس" (a poleax) correspond respectively to /bi:r/ and /fa:s/.

Alif Maqsura َ

ىَ rule Alif Maqsuraa َ ىَ (which is always preceded by a fatha /a/) at the end of a word is realized as the short vowel /a/.

STATISTICAL APPRAOCH

The rule based approach that we adopted in our research didn't take account of French words used in Algiers dialect and pronounced the same way as in French language. Statistical approaches are the best solution to tackle this kind of issues. Again, we considered this situation as a machine translation problem in which the source language is the text (a set of graphemes) and the target language is its phonetic representation (a set of phonemes). This system uses Moses package [START_REF] Koehn | Moses: Open Source Toolkit for Statistical Machine Translation[END_REF], Giza++ [START_REF] Och | A Systematic Comparison of Various Statistical Alignment Models[END_REF] for alignment and SRILM [START_REF] Stolcke | SRILM An Extensible Language Modeling Toolkit on Spoken Language Processing[END_REF] for language model training.

The main motivation of using a statistical approach is that we can include French phonemes in the training data.

To build this system, the first component is a parallel corpus including a text and its phonetic representation. Actually, this resource is not available, so we created it by using the rule-based converter described above.

This system operates at a grapheme and phoneme level, we split the parallel corpus into individual graphemes and phonemes including a special character as word separator in order to restore the word after conversion process (see example below).

ــ

َ ــ ََ ََ ت ـ ــ َ ـ ََ َ ك ــ َ ـــ ََ َََ س ــ َ ــ ََ ََ ن Null /t/ /u/ /k/ Null /s/ /a/ /n/

EXPERIMENTS

Experimental Material

We used an Algiers dialect corpus constructed by hand. It consists of more than 6K sentences including 10.7k different words. This corpus is a collection of movies and TV shows and some everyday recorded conversations which we transcribed into text (in Arabic script) by hand 8by adopting writing rules. Initially, this corpus was not vocalized, we proceeded to vocalize it as explained in section 3.1.

Our corpus is consisting of three categories of words: 1. Arabic words.

2. French words phonologically altered and their pronunciation is realized with Arabic phonemes. 3. French words for which the pronunciation is realized with French phonemes.

Results for Rule Based Approach

We applied phonetization rules seen below on a corpus of Algiers dialect words with Arabic words(8993 words) and French words phonologically altered (801 words), the system accuracy was 100%. In addition to Arabic words, French words in the second category are correctly phonetized because their phonetic realization is close to Algiers dialect. For example the word َة" ين َ وز َ ك َ ََ َ َ َ "َ (kitchen, original French word is "cui i e") which i a borrowed French word phonologically altered is correctly converted as /ku:zina/, while a word in the third category as " َ َون َي س َ َك ون َ ك َ َ َ َ َ ََ َ " (connection, original French word connexion) is incorrectly converted to /ku:niksju:n/ since it is realized / / with French phonemes. Considering these words(906 words), the system accuracy decreases to 92%. The issue of these words is that we cannot introduce rules for French words written in Arabic script, since the relationship between Arabic graphemes and French phonemes is not a one-to-one. For example, the graphemes " ـ َ ــ َ ََ و " in a French wordَwritten in Arabic script could correspond to the French phonemes /y/, /u/,/ / or /O/ (see some examples in Table 2).

Dialect word

French phonetic transcription

French word

Meaning

َ َ س َ َ و ر syʀ Sûre Sure ُ ﭙ ُ ـ و ر p ʀ Port Port َ َ س َ َ و َ َ د َ َ و ر sudoeʀ Soudeur Welder

Results for Statistical approach

In order to evaluate the statistical approach, we needed a parallel corpus (text and its phonetic representation) which was not available. To build this resource we proceeded as follows: we used the rule-based system described below to convert Arabic words and French words phonologically altered (category 1 and 2) into Arabic phonemes. Whereas French words (of category 3) were identified and transliterated into Latin script by hand and then converted to French phonemes by using a free French converter. For example the word َون" َي س َ َك ون َ ك َ َ َ َ َ ََ َ " i tra literated to "co exio " then converted to / /.

For test, we split randomly the parallel corpus into three datasets: training data (80%, 8560 words), tuning data (10%, 1070 words) and testing data (10%, 1070 words). Note that the test set includes 913 words of category 1, 77 words of category 2 and 80 words of category 3.

First we tested the statistical approach on a test set containing only Arabic words and French words phonologically altered (category 1 and 2). We got an accuracy of 93%. Then we proceeded to a test on a corpus including the three words categories, system accuracy decreases to 85%. This result is due to the increase of hypothesis number of each grapheme because of introducing French phonemes in the training data. The grapheme " ـو َ ــ َ َ ََ " for example in some Arabic words (category 1) are phonetized as the French phonemes /y/ or / / instead of the Arabic long vowel /u:/, the phoneme / / instead of /u:n/. Whereas, some words in category 3 are phonetized with Arabic phonemes by substituting for example the phonemes /y/, /u/,/ / or /O/ the /u: /, and / / by /a:/.

Discussion

In Table3 we summarize the achieved results. At first glance, and regarding the accuracy rate, we could deduce that rule-based approach is more efficient than statistical approach.

Statistical approach

Rule-based approach Rule-based approach does not take account of French words of category 3; it achieves efficient results only for Arabic words and French phonologically altered words (category 1 and 2). The results of statistical approach must be analyzed regarding the small amount of training data. Accuracy rate of 85% could be easily improved by increasing the size of training data.

On another side, a hybrid approach could be adopted: instead of using one corpus including all categories of words for training the statistical G2P converter, we can use two corpora: the first one including words of categories 1 and 2, could be processed by rule-based approach. The second corpus is a parallel corpus including words of category 3 with their French phonetization used for training the statistical G2P converter. Unfortunately, we have not sufficient data to test such a converter, since our corpus includes only about 1k words of category 3.

CONCLUSION

We presented two approaches to build a G2P converter for Algiers dialect which is an under-resourced language. The first approach is rule based; it gives perfect results for Arabic words and French words phonologically altered. Unfortunately, Algiers dialect includes a big proportion of French words pronounced as in the original language. This category of words could not be phonetized with a rule-based approach since the relationship between Arabic grapheme and French phoneme is not a one-to-one process. We used the statistical approach to overcome this problem. Regarding the small amount of training data, we can consider that the results are acceptable.

To improve the statistical G2P converter, French words pronounced as in the original language must be processed in a separate corpus. That is why we have to develop a module for identifying these words. We can use the rule-based approach for Arabic and French words phonologically altered. This hybrid approach could carry out better results.

In terms of resources, this work allowed us to build a phonetized dictionary for Algiers dialect; at our knowledge no such resource is available at this time.

Figure 1 .

 1 Figure 1. Iterative Diacritization Process

  In Algiers dialect, the letters ذ /D/, ظ /Dʻ/ and ث /T/ are not used, they are in most cases pronounced respectively as the graphemes د /d/, ض /dʻ/ and ت /t/. {C, V} ذَ+ + {C,V}  /d/ {C, V} ظَ+ + {C,V}  /dʻ/ {C, V}+َث + {C,V}  /T/

	3.2.2. Foreign letters rule
	Algiers dialect alphabet corresponds to Arabic alphabet
	extended to three foreign letters g, v and p.
	{C, V} + ڤ َ + {C,V}  /g/
	{C, V} + ڢ َ + {C,V}ََ/v/
	{C, V} + پ َ + {C,V}  /p/

Table 2 . Examples of mappings between Arabic graphemes " ـ ُ ـ ُ ُ و " and French phonemes

 2

Table 3 . Results Summary

 3

http://sourceforge.net/project/tashkeela


Julien Heulot
email: jheulot@insa-rennes.frweb

Maxime Pelcat
email: mpelcat@insa-rennes.frweb

Karol Desnos
email: kdesnos@insa-rennes.frweb

Jean-Franc ¸ois Nezan
email: jnezan@insa-rennes.frweb

Slaheddine Aridhi
email: saridhi@ti.com

SPIDER: A SYNCHRONOUS PARAMETERIZED AND INTERFACED DATAFLOW-BASED RTOS FOR MULTICORE DSPS

This paper introduces a novel Real-Time Operating System (RTOS) based on a parameterized dataflow Model of Computation (MoC). This RTOS, called Synchronous Parameterized and Interfaced Dataflow Embedded Runtime (SPiDER), aims at efficiently scheduling Parameterized and Interfaced Synchronous Dataflow (PiSDF) graphs on multicore architectures. It exploits features of PiSDF to locate locally static regions that exhibit predictable application behavior. This paper uses a multicore signal processing benchmark to demonstrate that the SPiDER runtime can exploit more parallelism than a conventional multicore task scheduler. By comparing experimental results of the SPiDER runtime on an 8-core Texas Instruments Keystone I Digital Signal Processor (DSP) with those obtained from the OpenMP framework, latency improvements of up to 26% are demonstrated.

INTRODUCTION

The current limitation of the processing power of individual Processing Element (PE) due to power consumption considerations is fostering the integration of more and more PEs into Multiprocessor System-on-Chip (MPSoC) devices such as Texas Instruments' Keystones and other multicore devices. This trend is even more marked because of the rising complexity of applications in the signal processing systems domain.

Concurrently, signal processing applications are becoming increasingly dynamic which leads to more complex hardware resource requirements. Data dependencies between different subsections of an algorithm can also change over time. This fact is due to the growing number of conditional operations in algorithms so as to achieve better performance in terms of latency and reliability. For example, the variation of the number of User Equipments (UEs) scheduled to be transmitted by a base station implementing the 3rd Generation Partnership Project (3GPP) Long Term Evolution (LTE) needs to be taken into account to process each UEs data reliably and efficiently. The modification in the number of scheduled UEs between iterations (in this case with a period of 1ms) leads to high variations in the complexity of uplink and downlink data stream processing [START_REF] Pelcat | Adaptive multicore scheduling for the LTE uplink[END_REF].

When designing and implementing multicore signal processing systems, one of the main challenges is to dispatch computational tasks efficiently onto the available PEs while also considering dynamic changes in application functionalities and resource requirements. The process of assigning, ordering and timing actors on PEs in this context is referred to as multicore scheduling.

In multicore scheduling, inefficient use of the PEs leads to additional costs ranging from longer processing times to higher energy consumption. Data dependencies and dynamic signal processing algorithms make multicore scheduling an interesting and difficult challenge [START_REF] Marwedel | Mapping of applications to MPSoCs[END_REF].

This paper describes a new RTOS called SPiDER runtime to address these challenges. The SPiDER runtime takes scheduling This work is supported by the ANR COMPA project.

decisions at runtime as soon as it gets information on the current algorithm topology. In relation to the scheduling taxonomy defined by Lee and Ha [START_REF] Lee | Scheduling strategies for multiprocessor realtime DSP[END_REF], this RTOS implements a fully dynamic scheduling strategy.

Instead of Tasks and Threads as in other RTOS, applications managed by SPiDER runtime are described using the PiSDF dataflow MoC, which is a specialization of the general Dataflow Process Network (DPN) MoC. The DPN MoC decomposes algorithms into pieces of computation, called actors, that exchange data, decomposed into tokens (atomic data element), through First In, First Out data queues (FIFOs). Actors and FIFOs compose a directed graph called dataflow graph.

The PiSDF model is obtained through the use of Parameterized and Interfaced dataflow Meta-Model (PiMM) over a Synchronous DataFlow (SDF) graph. PiMM integrates two main features: an acyclic graph of parameters to transmit control values between actors, and an interfaced hierarchical scheme ensuring schedulability over hierarchy exploration [START_REF] Desnos | Pimm: Parameterized and interfaced dataflow meta-model for mpsocs runtime reconfiguration[END_REF]. The SPiDER runtime exploits these features of the PiSDF model to efficiently extract parallelism and reduce the overall latency of a dataflow graph execution. The SPiDER runtime is an open source project 1 .

This paper is organized as follows. After Section 2 which presents related works, Section 3 describes the SPiDER runtime components and Section 4 details its implementation on the TMS320C6678 platform. Finally, Section 5 presents our experimental results.

RELATED WORKS

To address the multicore scheduling challenge, various frameworks based on OpenMP [START_REF] Dagum | Openmp: an industry standard api for shared-memory programming[END_REF] and OpenCL [START_REF] Stone | Opencl: A parallel programming standard for heterogeneous computing systems[END_REF] language extensions are currently proposed. However, these extensions are based on imperative languages (e.g., C, C++, Fortran) that do not provide mechanisms to specify signal algorithms with complex task parallelism.

The Open Event Machine (OpenEM)2 is a multicore runtime for Texas Instruments Keystone platforms. It allows dispatching tasks on several working cores following prioritized queues of tasks called events. It can be seen as hardware abstraction layer for the SPiDER runtime as it mainly proposes tools for multicore programming but does not provide complex scheduling strategies. The OpenMP framework implementation for Keystone platforms is implemented over OpenEM.

Dataflow MoCS are widely used for specification and implementation of data-driven signal algorithms in many application areas [START_REF] Lucarz | Dataflow/actor-oriented language for the design of complex signal processing systems[END_REF], telecommunication [START_REF] Pelcat | Adaptive multicore scheduling for the LTE uplink[END_REF], and computer vision [START_REF] Sen | Dataflow-based mapping of computer vision algorithms onto fpgas[END_REF]. The popularity of dataflow MoCS in the design and implementation of signal processing systems is largely due to their analyzability, their predictability and their natural expressivity of task parallelism in signal processing algorithms.

In [START_REF] Desnos | Pimm: Parameterized and interfaced dataflow meta-model for mpsocs runtime reconfiguration[END_REF], Desnos and al. define a meta model called PiMM that can be applied to an SDF MoC to obtain PiSDF. This meta model brings multiple features such as hierarchy interfaces and parameterization. Hierarchy interfaces make possible to explore one hierarchy level entirely without any information on the inner parameterization of the subgraphs. Parameterization introduced by PiMM can mix data flow and parameter flow making possible complex parameterization of the application. The meta model also introduces configuration actors having a dedicated firing rule and able to set parameter values dynamically for the current subgraph.

The SPiDER runtime is an evolution of the work in [START_REF] Pelcat | Adaptive multicore scheduling for the LTE uplink[END_REF]. In [START_REF] Oliva | Building a rtos for mpsoc dataflow programming[END_REF], Oliva and al. propose an RTOS based on this previous work. It uses the µC/OS II Operating System (OS) and a Master/Slave pattern for the runtime architecture. This work takes as input dataflow MoC a Parameterized Cyclo-Static Directed Acyclic Graph (PCSDAG) that does not consider application hierarchy and feedback loops. This work was also focused only on a specific application, the 3GPP LTE Uplink Physical Layer data processing (PUSCH) algorithm. The method presented in this paper is based on another dataflow MoC called PiSDF. This MoC allows more flexible parameterization schemes, not only as preprocessing. This MoC also handles hierarchical programming and feedback loops.

Nollet and al. define a taxonomy of MPSoC runtime architectures in [START_REF] Nollet | A quick safari through the mpsoc runtime management jungle[END_REF]. They define terms such as Manager, and Runtime Library. The SPiDER runtime introduces a Quality Manager called Global Runtime based on PiSDF MoC and a Runtime Library called Local Runtime. These both components are platform independent.

In [START_REF] Neuendorffer | Hierarchical reconfiguration of dataflow models[END_REF], Neuendorffer, et al. define quiescent points as points where parameters influencing an execution are allowed to change. Between two quiescent points, the application can be considered static. In this paper, decisions taken by the SPiDER runtime on actor ordering and mapping are taken after the quiescent points are reached.

The Jade (Just-in-time adaptive decoder engine) scheduler from Gorin, et al. [START_REF] Gorin | Just-in-time adaptive decoder engine: a universal video decoder based on mpeg rvc[END_REF] is a scheduler based on dataflow methods. The difference between the Jade scheduling method and the SPiDER runtime is that Jade is based on the CAL language implementing the dynamic dataflow MoC. This model gives knowledge on quiescent points only after graph execution, providing a posteriori informations rather than predictability information.

In [START_REF] Singh | Mapping on multi/many-core systems: survey of current and emerging trends[END_REF], Singh presents a survey on multi/manycore mapping methodologies. The SPiDER runtime can be classified as "On-thefly" mapping, targeting heterogeneous platform with a centralized resource management strategy. This survey does not reference any "On-the-fly" mapping method based on dataflow.

SPIDER RUNTIME

DataFlow MoC

The SPiDER runtime is based on a Dataflow MoC. The MoC chosen for this RTOS is PiSDF for its parameterization and hierarchy properties. The predictability of this MoC allows taking relevant scheduling decisions a priori without a strong penalty on its expressivity.

An example of a PiSDF graph can be seen in Figure 1. The parameters are variables that can influence production and consumption of data tokens by actors in the graph. They can be static such as NbS which is constant for the whole graph execution or they can be set dynamically; e.g. N and M are set by the config and setM actors respectively. These actors are called configure actors because they have special firing rules related to their use in updating parameter values. Configuration actors (the ones marked a small white circle in Figure 1) are fired only once at the beginning of the execution of the hierarchical graph containing them. The FIR Chan actor is hierarchical and contains, as a nested subgraph, the graph shown under it. More details about PiSDF can be found in [START_REF] Desnos | Pimm: Parameterized and interfaced dataflow meta-model for mpsocs runtime reconfiguration[END_REF].

In order to use most of the parallelism of a PiSDF representation, the method developed in this paper transforms the PiSDF graph into an single rate Synchronous DataFlow (srSDF) graph at runtime as soon as parameters are resolved. An srSDF graph is an SDF graph where token production and consumption rates are equal on each edge. This is achieved by replicating actors that need to be fired more than once, thus meaning that each actor of srSDF graph is executed only once in the graph execution. Doing so, the scheduler has a global view of actors dependencies for the whole graph iteration.

Since parameters are resolved after all configuration actors have been executed, the PiSDF to srSDF transformation cannot be completed at once. Each hierarchy level has to be configured sequentially as the lower hierarchy level depends on parameters and/or data coming from the upper one. To perform this transformation, multiple steps of scheduling need to be completed, each step configuring a hierarchy level and then revealing a part of the srSDF graph.

Once this srSDF graph has been generated, the scheduler can dispatch actors onto the MPSoC. To do so, the process is separated into: task ordering and mapping. The task ordering consists of sorting all non-executed tasks of the srSDF graph into one list. This list is then used to dispatch tasks onto each PE of the MPSoC. This task is called mapping. Both the tasks of ordering and mapping can be optimized to improve different metrics such as: latency, throughput, memory utilization or energy efficiency.

RTOS Topology

One of the uses of the SPiDER runtime is on heterogeneous platforms. In this usage, a local decision on actor firing may lead to a bad global decision, as there may be another PE available for this actor which could have permitted earlier completion of the task. In order to ensure efficient global decisions, a Master/Slave execution scheme is thus preferred for heterogeneous platforms.

The SPiDER runtime uses a PiSDF graph as an input algorithm graph. This MoC defines parameters as integer values that influence algorithm execution. Since configuration actors can be executed on any PE of the MPSoC, it is important to send parameters back to the Master. It will then take scheduling decisions based on these parameters.

The runtime (Figure 2) requires the following elements:

• Local RunTime (LRT): low footprint OS that processes actors. It can be implemented over multiple types of PE: General-Purpose Processor (GPP), DSP, accelerator, etc... • Global RunTime (GRT): This is the master of the system, and knows the algorithm topology and takes multicore scheduling decisions. It is usually implemented over a GPP core but can also be a DSP core. The GRT can also process actors. • Data tokens: atomic data exchanged by actors. A data token has a predefined size: it can be one bit, one byte, a data structure, etc... A data FIFO can be implemented over any data medium (e.g. a shared memory or a network on chip). • Jobs: a job embeds all data required to execute one instance of an actor. In particular, it includes information on actor code location, which FIFO receives input data and which FIFO sends output data. Each LRT has a job queue. • Parameters: A parameter influences the algorithm graph topology or the execution timing of actors. When it is set by a configuration actor processed by a LRT, its value is sent to the GRT via a dedicated queue. • Timings: To have timing feedback on the previous and the current execution, LRTs send back actor start and end times through low priority timing FIFOs. All timing information is based on the same timing reference.

IMPLEMENTATION ON C6678

The experimental platform used is the Texas Instruments Keystone I architecture (EVM TMS320C6678) composed of 8 c66x DSP cores. They are interconnected by a Network-on-Chip (NoC) called TeraNet which accesses an internal shared memory called MSMC. The Keystone platform also provides an external memory access to DDR memory. Control queues for parameters, timings and jobs are implemented using the hardware queues present in the Keystone Multicore Navigator [START_REF]Texas Instruments: KeyStone Architecture Multicore Navigator[END_REF]. It embeds 8192 hardware queues for multicore synchronization. These queues exchange data through descriptors stored in memory regions. We allocate one memory region in cached MSMC for the descriptors of these control queues. Since control data sent through these queues are relatively small, descriptors are configured monolythic, i.e. all data is present in the descriptor.

For data token communication, data is stored in MSMC or DDR (depending on the memory allocation). Synchronizations between cores is performed using one hardware queue for each data transfer. A descriptor present in a hardware queue means that the corresponding data is available in shared memory. This makes access to shared memory predictable and suitable for enabling caches.

For timing information, the keystone architecture provides 16 shared timers. One of these shared timers is used for global timestamp information. This ensures relevant (global) timing information on current and previous executions.

Figure 3 represents the implementation of the SPiDER runtime on a c6678 keystone architecture. The hardware independent GRT and LRT need services from hardware which are provided by an abstraction layer called Platform Library.

EXPERIMENTS

This paper describes an RTOS used to distribute efficiently at runtime signal processing applications. In this context, experiments will focus on comparison with another widely used framework called OpenMP. Results have been acquired by studying single and multi-iteration latencies of a benchmark application on the Texas Instruments c6678 multi-core DSP platform. To compare our approach and the OpenMP frameworks, we chose a generic benchmark of signal processing. This benchmark is an extension of the MP-sched benchmark [START_REF] Zaki | Integration of dataflow-based heterogeneous multiprocessor scheduling techniques in gnu radio[END_REF]. The MP-sched benchmark can be viewed as a two-dimensional grid involving N channels, where each channel consists of M cascaded Finite Impulse Response (FIR) filters of NbS samples. Here, we extend the MP-sched benchmark by allowing the M parameter to vary across different channels. We refer to this extended version of the MP-sched benchmark as heterogeneous-chain-length MP-sched (HCLM-sched).

The OpenMP framework cannot implement the HCLM-sched as a double nested loop since FIRs are pipelined on each channels, this data dependencies is not suitable for an OpenMP "parallel for". However, OpenMP framework is used to parallelize channels making them monolithic tasks.

In PiSDF, the HCLM-sched description can be found in Figure 1. To handle the versatility of the application, two parameters, called N and M, are used. The N parameter corresponds to the number of channels of FIRs. The M parameter corresponds to the number of FIRs in each channel. Following the PiSDF semantic, as M parameter is inside the hierarchical actor FIR Chan, it can be different for each channel of FIR. To represent the HCLM-sched application, many control actors have been added: This implementation of the HCLM-sched algorithm exploits the Round Buffer (RB) behavior of interfaces in PiSDF MoC. As explained in [START_REF] Desnos | Pimm: Parameterized and interfaced dataflow meta-model for mpsocs runtime reconfiguration[END_REF], to maintain schedulability of the upper graph without regarding a lower graph, input and output interfaces behave like RBs. If too much data is sent into an output interface, only the last one will be returned to the upper graph. If not enough data is produced by an input interface, data will be duplicated. However, repetition number in one graph iteration for each actor is computed in such a way that all data from an input interface is consumed at least once and at least enough data are produced to output interfaces. In the HCLM-sched graph, the RB behavior of the output interface of the lower graph is used to keep only the result of the last execution of the FIR actor.

For these experiments, and to be compliant with the default implementation of the OpenMP framework for this platform, both cached and uncached shared L2 memory have been used by the SPiDER runtime to allocate data FIFOs.

Results

We have proceeded with two experiments in our comparison between SPiDER runtime and OpenMP.

For the first experiment (Exp. 1), we fixed M = 12 for all stages. 512-tap FIR filters of NbS = 4000 samples have been implemented using the dsplib library of Texas Instruments. Latencies for each iteration have been measured for N varying from 6 to 17 and displayed in Figure 4. The OpenMP implementation latency curve displays a step shape when increasing N. This is due to channels distribution on the platform. Since each stage is implemented as a monolithic block with OpenMP, as soon as 9 channels are reached, 2 channels have to be completed on one PE making the overall latency double. The execution Gantt chart for N = 9 can be found at Figure 5. In the Gantt charts, FIR of same level on all channels have the same color. With the SPiDER runtime, the graph transformation and scheduling phases introduce a visible overhead. This overhead can be seen in Figure 6 where GRT first red tasks represent scheduling overhead. However, the transformation to srSDF extracts more parallelism than OpenMP from the subdivision of channels into multiple FIRs. These choices make SPiDER runtime suitable for applications that do not fit to the architecture topology. In the HCLM-sched benchmark with 9 channels, the overall latency is reduced of up to 26%. The second experiment (Exp. 2) is based on multiple iterations of the HCLM-sched benchmark. As the first experiment, 512-tap FIRs of 4000 samples have been used. We fix the number of channels N = 8 and the number of FIRs pipelined in each channel varies with : M = 8 -Chan Id . Then, multiple iterations of the application are launched with a fixed period of 700 kcycles. As we can see in Figure 7, if the latency of the OpenMP implementation is superior to the period, the latency is growing at each new iteration. This is due to the global synchronization at the end of each OpenMP parallel blocks which occurs on the Gantt chart at Figure 8 For the SPiDER runtime, the latency remains constant over iterations. By having prior knowledge on how the application will behave, the GRT can start an execution on LRTs which have already finished the previous execution. It can then start the following iteration as soon as the next period tick occurs, see Figure 9. With a knowledge of the application execution, the SPiDER Runtime can pipeline iterations.

Figure 1 :

 1 Figure 1: PiSDF representation of HCLM-sched benchmark

Figure 2 :

 2 Figure 2: Runtime execution scheme

Figure 3 :

 3 Figure 3: Hardware arch. of SPiDER Runtime on c6678

•

  config: Configuration actor defining the parameters of the whole graph. The N parameter is set and a list of corresponding M values is sent to MFilter . • MFilter : This actor is used to filter the M values to only N values. This will lead to the N executions of the FIR Chan hierarchical actor. • Src and Snk : These actors are used respectively to retrieve raw data and to send results. • setM : A basic configure actor used to set the M value of the current chain of FIRs. • Switch: This actor is used to select the input data of the FIR actor. Depending of the select input, it chooses data from interface or feedbacked data. • Init: This actor sets select values. It will make the Switch actor choose input data from the interface on the first iteration and feedbacked data on other iterations. • Broadcast(Br ): This actor duplicates data for the output interface and for the feedback edge.

Figure 4 :

 4 Figure 4: Exp. 1: Latency vs N

7 Figure 5 :

 75 Figure 5: OpenMP Gantt chart for Exp. 1 (N = 9)

7 Figure 6 :

 76 Figure 6: SPiDER Gantt chart for Exp. 1 (N = 9)

Figure 7 :

 7 Figure 7: Exp. 2: Latency vs iterations

7 Figure 8 :

 78 Figure 8: OpenMP Gantt chart for Exp. 2

  This paper presents a novel multicore RTOS called SPiDER runtime. SPiDER runtime exploits parallelism from a PiSDF dataflow graph for a multicore execution. It enables efficient assignment and ordering of actors into PEs with a better knowledge of actor interactions. Experiments conducted on an 8-core Texas Instruments DSP demonstrate on a benchmark that the SPiDER runtime provides more parallelism to the execution than the OpenMP framework. Results have shown that the SPiDER Runtime reduces the execution latency by up to 26% and allows handling multiple executions.

	PE 7
	PE 6
	PE 5
	PE 4
	PE 3
	PE 2
	PE 1
	PE 0
	0	200 400 600 800 1000 1200 1400 1600 kcycles
		Figure 9: SPiDER Gantt chart for Exp. 2
	6. CONCLUSION AND FUTURE WORKS

http://github.com/COMPA-Runtime/COMPA-Runtime

http://sourceforge.net/projects/eventmachine/