Léo Girier
email: leo.girier@ec-lyon.fr.

Michel Roger

Anthony Lafitte

Hélène A Posson

Two-Dimensional Mode

A Two-Dimensional Mode-Matching Technique for Wake-Interaction Tonal Noise Including Rotor-Stator Coupling

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A Two-Dimensional Mode-Matching Technique for Wake-Interaction Tonal Noise Including Rotor-Stator Coupling

Noise generated by the impingement of rotor wakes on guide vanes is still widely studied in the context of turbofan noise. However, analytical models are often limited to an isolated stator made of flat-plate vanes, neglecting the presence of the rotor. The present work proposes an iterative procedure based on successive applications of a mode-matching technique on the stator and the rotor to model the acoustically coupled rotor-stator system, including vane camber. This allows one to study the "shielding" effect of the rotor and the occurrence of inter-stage trapped modes, in order to better predict the relative balance between upstream and downstream sound powers, as well as modal and frequency contents. The influence of interstage distance and rotor rotational speed on sound generation and propagation is investigated in this paper. It appears that the additional scattering by the rotor can have a significant impact on the emerging tones by redistributing sound through frequencies. The ratio of downstream to upstream acoustic power tends to increase compared to the single stator computation, as previously reported in the literature, even if an artificial swirl is imposed upstream of the compressor stage in the present study. However, the total sound power with and without the rotor scattering remains the same in the performed test cases.

I. Nomenclature

II. Introduction

Despite the drop in air traffic since the Covid-19 outbreak and a growing societal concern regarding aircraft pollution, air traffic in Europe is recovering and might already return to its 2019 value by 2024 * . Aircraft noise pollution therefore remains an important societal problem. In this context, it is crucial to reduce engine noise, which remains one of the major sources of overall aircraft noise, particularly during take-off. A study made in the early century, on typical long-range four-engine aircraft [START_REF] Huber | Jet noise assessment and sensitivity at aircraft level[END_REF], pointed out that the noise generated by the fan stage has become as important as jet noise during take-off and is also crucial in approach condition. Reducing fan-related noise then appears as one of the main lever to reduce the overall aircraft noise emission. The present work focuses on the modeling of fan-related noise, and more specifically on rotor-stator wake-interaction tonal noise.

Most of the analytical studies on wake-interaction tonal noise have been devoted to the improvement of the disturbance and cascade-response models. One major flaw of these models is that they are often limited to a single cascade of vanes interacting with prescribed wakes coming from upstream, neglecting the effect of sound transmission and reflection by the rotor. Typically, trapped modes can develop between the rotor and the stator, and the modal and frequency scattering of stator noise by the rotor redistributes noise to other harmonics of the Blade Passing Frequency (BPF). Topol et al. [START_REF] Topol | A reflection mechanism for aft fan tone noise from turbofan engines[END_REF] demonstrated a significant impact of noise reflection from the rotor in full-scale engine measurements. In particular, data from a high bypass ratio engine showed evidence of mode trapping and frequency scattering. A surge at 3 × BPF was observed and attributed to the tone 1 × BPF that turned cut-on in the inter-stage while remaining cut-off outside, allowing the rotor to scatter its energy to other frequencies. Topol et al. [START_REF] Topol | A reflection mechanism for aft fan tone noise from turbofan engines[END_REF] were among the first to propose a correction for the aft noise, including rotor reflection and frequency scattering in a one-way coupling. Kaji and Okazaki did pioneering works on the rotor-stator interaction in 1970, but their model was restricted to a single frequency and did not include the swirl effect in the inter-stage. In order to account for the full rotor-stator coupling with swirl, Hanson [START_REF] Hanson | Mode trapping in coupled 2D cascades -Aerodynamic and acoustic results[END_REF][START_REF] Hanson | Coupled 2-Dimensional Cascade Theory for Noise and Unsteady Aerodynamics of Blade Row Interaction in Turbofans[END_REF] developed a fully coupled methodology and applied it to a two-dimensional rotor-stator stage. The scattering coefficients of the rotor and of the stator were computed from a cascade-response theory developed by Smith [START_REF] Smith | Discrete frequency sound generation in axial flow turbomachines[END_REF], based on the Wiener-Hopf technique, and gathered in a global scattering matrix to be inverted. Hanson showed that trapped modes are more likely to occur in the inter-stage because of the impact of swirl on the inter-stage cut-on condition. Hanson [START_REF] Hanson | Acoustic reflection and transmission of rotors and stators including mode and frequency scattering[END_REF] focused on the effects of mode and frequency scattering by a rotor and a stator separately, to break down some of the reflection and transmission phenomena occurring in a rotor-stator stage. One of the most noticeable results was that modes reflected by the rotor could have more energy than the incident mode that generated them. This phenomenon was attributed to the rotor motion, increasing the energy of the modes scattered at higher frequencies. Modes scattered without frequency shift did not show power increase but the analysis was limited to incident modes at 1 × BPF.

The purpose of the present work is to address the interaction with a mode-matching technique, presented as an alternative to other approaches, able to include realistic design parameters. The mode-matching technique, introduced some decades ago, has recently seen a revival of interest. It already showed promising capabilities to model sound generation and propagation in isolated stator stages. For a comprehensive review, the reader can refer, for instance, to the work by Bouley and co-authors [START_REF] Bouley | On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes[END_REF][START_REF] Bouley | Modélisations analytiques du bruit tonal d'interaction rotor / stator par la technique de raccordement modal[END_REF]. An extension to cascades of thick-walled channels is also possible, as shown by Oulmi et al. [START_REF] Oulmi | Analytical Modelling of Sound Transmission Through a Row of Thick-walled Channels[END_REF], or to cambered vanes, ensuring a continuous deviation of the flow, with some increase of the mathematical efforts, as discussed by Girier et al. [START_REF] Girier | A Two-Dimensional Model of Sound Transmission Through Curved and Staggered OGV: Effect of Inter-Vane Channel Mode Transitions[END_REF][START_REF] Girier | Mode-matching techniques for the modeling of rotor-stator wake-interaction noise, with emphasis on the effects of vane camber[END_REF]. More recently, a mode-matching based model has been developed to study sound generation and propagation in a coupled system consisting of two stationary periodic rows of cooling channels in a railway ventilation system [START_REF] Oulmi | Sound Generation and Propagation in a System Consisting of Two Periodic Rows of Channels[END_REF]. The model has also been extended to rotating rows [START_REF] Oulmi | Analytical Modelling of the Aeroacoustic Performances of Self-ventilated Motors Used in Railway Applications[END_REF]. The present work is similar in essence, combining the multiple-scattering formulation with the cambered-vane extension. In Hanson's works [START_REF] Hanson | Mode trapping in coupled 2D cascades -Aerodynamic and acoustic results[END_REF][START_REF] Hanson | Coupled 2-Dimensional Cascade Theory for Noise and Unsteady Aerodynamics of Blade Row Interaction in Turbofans[END_REF][START_REF] Hanson | Acoustic reflection and transmission of rotors and stators including mode and frequency scattering[END_REF], alternatively, the actuator disk theory was used to model jumps in the flow conditions at the rotor leading-edge and stator trailing-edge interfaces. Due to the importance of the flow deviation in the occurrence of trapped modes, it seems interesting to develop a model with continuous flow evolution through the entire rotor-stator stage for more reliable predictions.

In summary, wake-interaction tonal noise in a coupled rotor-stator system has only been modeled analytically with flat blades/vanes and discontinuous flow conditions in the past, motivating extensions. A fast analytical prediction tool could be of great value to get a better understanding of fan noise in early design steps in an engineering context. For that purpose, a two dimensional mode-matching based model for fully coupled rotor-stator wake-interaction tonal noise is proposed, in the continuity of the aforementioned recent works.

The outline of the paper is as follows. The mathematical formulation and the mode-matching technique background are presented in section III. In section IV, the validity of the method is first demonstrated by analyzing the acoustic power balance with an incident acoustic wave scattered by a stator-stator system, and a rotor-alone configuration. Computations of sound generation and propagation in a rotor-stator stage are conducted in section V alongside parametric studies on various flow and geometrical parameters. Finally, some limitations and possible improvements of the current model version are discussed in section VI and concluding remarks are drawn.

III. Mathematical Formulation

A. Mode-Matching Technique

General Principle

The mode-matching technique is explained for the case of an incident acoustic wave, for which an illustrative scheme is given in Fig. 1, without stagger or camber. Detailed explanations of how to add stagger and camber can be found in Refs. [START_REF] Girier | A Two-Dimensional Model of Sound Transmission Through Curved and Staggered OGV: Effect of Inter-Vane Channel Mode Transitions[END_REF][START_REF] Girier | Mode-matching techniques for the modeling of rotor-stator wake-interaction noise, with emphasis on the effects of vane camber[END_REF]. An incident acoustic mode, described by its velocity potential 𝜙 𝑖 , is scattered at the cascade Leading-Edge (LE) interface, generating reflected modes 𝜙 𝑟 and transmitted modes 𝜙 𝑑 in the channels. The latter are then scattered at the Trailing-Edge (TE) interface, giving rise to reflected channel modes 𝜙 𝑢 and transmitted modes 𝜙 𝑡 . Hard-walled boundary conditions at the walls of the inter-vane channels and periodic boundary conditions in the 𝑦-direction are imposed.

Fig. 1 Scattering of an incident acoustic wave by a linear cascade of flat vanes.

The mode-matching technique can be described in three steps: partitioning, solving, and matching. The partitioning consists in dividing the domain into different sub-domains in which the boundary conditions are uniform and the wave equation separable, allowing a solution on a local modal basis. In Fig. 1, the sub-domains are the upstream medium, each inter-vane channel and the downstream medium. The solving step provides the wavenumbers and modal shapes of the scattered waves in each sub-domain, based on an appropriate Helmholtz equation. For simple geometries such as in Fig. 1, these wavenumbers and eigenfunctions are analytically known. Finally, the only remaining unknowns are the modal coefficients of the waves. They are defined by matching the different modal solutions at both leading-edge and trailing-edge interfaces. To do so, continuity equations specific to the problem are used to build a set of equations on the modal coefficients, which can be solved by matrix inversion. The equations that need to be satisfied through a compressor stage are derived from the classical conservation equations (mass and energy) applied to an adiabatic lossless turbomachine [START_REF] Dixon | Fluid Mechanics and Thermodynamics of Turbomachinery[END_REF]. In this context, the mass conservation is equivalent to mass-flow continuity and the energy conservation is expressed through stagnation enthalpy. The solving and matching steps are detailed in what follows.

Solving Step: Wave Equation in the Sub-Domains

Compressibility is accounted for in the Euler equations and the ideal gas law is used. Furthermore, the fluid is considered inviscid and the flow isentropic. Defining the following quantities: 𝜌 * as the density, u * as the velocity, 𝑐 * as the sound speed and 𝑝 * as the pressure, the flow is then split up into a steady component (time average) and its fluctuations. The fluctuating part varies harmonically in time, with angular frequency 𝜔, and is assumed to be small enough to allow linearization. Thus, the decomposition reads

[u * , 𝜌 * , 𝑝 * , 𝑐 * ] = [U, 𝐷, 𝑃, 𝐶] + [u, 𝜌, 𝑝, 𝑐] e -i𝜔𝑡 . (1)
The fluctuating velocity u can be expressed as the sum of independent acoustic and vortical motions according to the analysis of Chu & Kovásznay [START_REF] Chu | Non-linear interactions in a viscous heat-conducting compressible gas[END_REF] at leading order. The fundamental theorem of vector calculus states that u can be decomposed into the sum of a rotational-free field ∇𝜙 and a divergence-free field u 𝑅 , which are associated, in this case, to the acoustic and vortical motions, respectively. Thus, the fluctuating velocity u reads

u = ∇𝜙 + u 𝑅 . (2)
Defining the pressure field as

𝑝 = -𝐷 (-i𝜔𝜙 + U • ∇𝜙) , (3)
the vortical field u 𝑅 is then uniquely defined by

(-𝑖𝜔 + U • ∇)u 𝑅 = 0, (4)
where the second-order vortical-acoustic and vortical-vortical coupling terms are neglected (see for example Ref. [START_REF] Girier | Mode-matching techniques for the modeling of rotor-stator wake-interaction noise, with emphasis on the effects of vane camber[END_REF] for detailed explanations). The pressure field, on the other hand, is given by the following homogeneous compressible wave equation

𝐷 (-𝑖𝜔 + U • ∇) 1 𝐶 2 (-𝑖𝜔 + U • ∇)𝜙 -∇ • (𝐷∇𝜙) = 0.
(5)

Matching Step: Jump Conditions at the Interfaces

The mass conservation at an interface between a sub-domain 1 and a sub-domain 2, equivalent to the mass-flow conservation through the interface, reads 𝑝 𝑍

𝑀 2 𝑥 + 𝑀 𝑥 𝑢 𝑥 2 1 = 0, (6)
where the brackets denote the difference of the quantity between both sides of the interface, 𝑍 = 𝐷𝐶 and 𝑀 𝑥 = 𝑈 𝑥 /𝐶 (the subscript 𝑥 referring to the projection along the corresponding axis of Fig. 1). The conservation of stagnation enthalpy [START_REF] Dixon | Fluid Mechanics and Thermodynamics of Turbomachinery[END_REF] states that 𝑝 𝑍

+ 𝑀 𝑥 𝑢 𝑥 + 𝑀 𝑦 𝑢 𝑦 2 1 = 0. ( 7
)
Writing 𝑈 as the norm of the mean velocity and Ψ as the angle it makes with the direction perpendicular to the interface, the 𝑥-axis in Fig. 1, allows one to express the mean velocity components as 𝑈 𝑥 = 𝑈 cos Ψ and 𝑈 𝑦 = 𝑈 sin Ψ. Equation ( 7) is then recast as 𝑝 𝑍

+ 𝑀 𝑥 𝑢 𝑥 + 𝑢 𝑦 tan Ψ 2 1 = 0. ( 8
)
Combining Eq. ( 6) and ( 8) allows one to recover the conservation of the fluctuating pressure 𝑝 and fluctuating axial velocity 𝑢 𝑥 , commonly used in mode-matching procedures, but only for the particular case where the mean flow is perfectly perpendicular to the matching interface, i.e. when Ψ = 0. In general, when Ψ ≠ 0, new variables need to be introduced to ensure the conservation of mass-flow and stagnation enthalpy through the interface. These conservative variables are written Γ 1 and Γ 2 in the following, and are gathered into a vector Γ 𝛾 = (Γ 1𝛾 , Γ 2𝛾 ), where the index 𝛾 stands either for the incident (𝑖), reflected annular (𝑟), transmitted annular (𝑡), downstream-propagating channel (𝑑) or upstream-propagating channel (𝑢) field (see Fig. 1). The matching equations then read

Γ 𝑖 + Γ 𝑟 = Γ 𝑑 + Γ 𝑢 at LE, Γ 𝑑 + Γ 𝑢 = Γ 𝑡 at TE. (9) (10)
In case of an incident acoustic wave, the velocity field is purely potential. The first variable is chosen as the velocity potential Γ 1 = 𝜙 and the second one as a combination of axial and transverse velocities Γ 2 = 𝛽 2 𝑥 𝑢 𝑥 -𝑀 𝑥 𝑀 𝑦 𝑢 𝑦 , where

𝛽 𝑥 = √ 1 -𝑀 2 𝑥 .
If a vortical wave is present, as for sound generation by wake impingement and/or when a Kutta condition is imposed, then Γ 1 = 𝛽 2 𝑥 𝑝/𝑍 + 𝑀 𝑦 𝑢 𝑦 , whereas Γ 2 remains unchanged.

B. Rotor-Stator Back-and-Forth Iterative Procedure

Recursive Algorithm

The procedure begins by an iteration of direct noise generation or acoustic scattering on the stator, depending on the vortical or acoustic nature of the incident wave, respectively. This produces the first set of acoustic modes. A recursive algorithm is then called to iterate back-and-forth between the secondary cascade (rotor) and primary cascade (stator) until a convergence on the modal coefficients is reached. To do so, a filter is applied at each step on the scattered acoustic modes to retain only the most significant ones for the next iteration. A mode is filtered out if its amplitude is at least two orders of magnitude lower than a reference amplitude. This reference is chosen as the initial wave in the case of acoustic scattering, or the strongest acoustic mode produced by the initial iteration in the case of wake interaction. The recursive algorithm iterates on each mode independently by virtue of linearity, so a graph tree is constructed gradually. A branch reaches an endpoint when the filter retains no modes. All coefficients and acoustic powers are finally summed up together on mode order and frequency for the annular domains, and also on inter-vane phase shift for the channel modes. The algorithm is implemented in the MATLAB environment and runs in a few seconds on a personal laptop for a given disturbance component.

Acoustic Power

On top of the recursive algorithm, radiated acoustic powers are computed at each iteration based on Goldstein's formulation [START_REF] Goldstein | Aeroacoustics[END_REF]. This is used as an indicator of energy conservation a posteriori, but the convergence of the algorithm is solely based on the modal coefficients. For rotor-stator sound generation, a global power balance is not accessible for now, because the hydrodynamic power carried by the vortical gust is not estimated. However, for validation purposes, cases of acoustic scattering are presented in section IV.A and IV.B, for which the global sound power balance can be computed.

IV. Preliminary Analyses on Acoustic Power Balance

A. Stator-Stator System with Incident Acoustic Wave

The developed model presents two new additions compared to the aforementioned literature that need to be investigated beforehand: the recursive algorithm coupling the cascades and the rotating motion. The latter has, in fact, already been modeled and studied, essentially by Hanson [START_REF] Hanson | Acoustic reflection and transmission of rotors and stators including mode and frequency scattering[END_REF], but lacks some physical understanding. In a first instance, the validity of the recursive procedure is tested with a system composed of two stators excited by an incident acoustic wave. The algorithm should converge in terms of modal coefficients and the acoustic power balance should remain within 1% error. A schematic view of the problem is depicted in Fig. 2. A primary cascade of 𝑉 1 = 23 vanes and a secondary cascade of 𝑉 2 = 17 vanes are separated by a distance 𝑑/𝑙 2 = 1.5. Considering a radius of 𝑅 = 22.35 cm, the two sets of geometrical parameters are defined by 𝑎 1,2 /𝑅 = 2𝜋/𝑉 1,2 and 𝑙 1,2 = 𝜎 1,2 𝑎 1,2 , where the solidity values are 𝜎 1 = 1.5 and 𝜎 2 = 1.2. The incident acoustic wave is defined by its azimuthal order 𝑚 = -7, 5, 11 and its dimensionless frequency 𝜔𝑑/𝜋𝐶 ranges from 2 to 7 approximately. This range of frequencies and azimuthal orders will give a sufficient basis of validation to build confidence in the results obtained later on for wake-interaction. Fig. 2 Sketch of the stator-stator system with an incident acoustic wave coming from the right. The colors are here to remind that the modes are scattered with the periodicity of both cascades. Fig. 3 shows the acoustic power balance (P 𝑖 + P 𝑟 -P 𝑡 )/P 𝑖 † against the angular frequency for the different incident azimuthal orders. The conservation of the acoustic power is achieved across the whole frequency range, except at some specific frequencies.

Fig. 3 Acoustic power balance against angular frequency for different mode orders 𝒎.

Looking at Fig. 4a, which represents the relative reflected and transmitted powers for 𝑚 = 5, a resonance seems to take place in the missing frequency range. It is known from previous studies [START_REF] Girier | Mode-matching techniques for the modeling of rotor-stator wake-interaction noise, with emphasis on the effects of vane camber[END_REF] that at resonant frequencies the method becomes much more sensitive to modal truncation. In absence of dissipating mechanism, it might be impossible to converge to a power-conserving solution. Adding a mean flow of Mach number 𝑀 = 0.45 in the opposite direction to that of the incident wave is sufficient to attenuate the resonance and allows the solver to converge, even without imposing a Kutta condition in this case (see Fig. 4b). As expected, the resonant frequencies are shifted in the low-frequency range. Furthermore, the second resonance, not observable before, is now clearly evidenced. Note that the total acoustic power is still well conserved. A computation on the primary cascade alone, not presented in this paper, has shown that both resonant frequencies in the case 𝑚 = 5 are due to the primary cascade, neither to the secondary cascade nor to the inter-stage area. The test demonstrates the relevance and the validity of the iterative procedure for a tandem of stationary cascades. The next section is naturally devoted to the case of a rotating cascade alone, before going to the rotor-stator case.

B. Rotor Alone with Incident Acoustic Wave

This time, a rotating cascade is examined alone in order to emphasize on the sound-power imbalance due to the motion of a scattering surface, which will help analyze sound power outputs in the rotor-stator case. The problem is sketched in Fig. 5, where the absolute mean velocity is noted 𝑈, the relative mean velocity 𝑊 and the rotation rate Ω. Note that 𝑊 is always aligned with the blades and 𝑀 = 𝑈/𝐶 is fixed to 0.55, so that |𝑊 | changes when the rotation rate varies as |𝑊 | 2 = |𝑈| 2 -|Ω𝑅| 2 . The cascade is made of 𝐵 = 31 blades with a chord-to-gap ratio 𝜎 2 = 1.2, and the acoustic excitation is defined by 𝑚 = 16 and 𝜔𝑎/𝜋𝐶 = 1.5. The radius of the unwrapped cut is taken as 𝑅 = 22.35 cm. The scattering of this incident mode generates only two cut-on modes: 𝑚 + 𝑠𝐵 = 16 (𝑠 = 0) and 𝑚 + 𝑠𝐵 = -15 (𝑠 = -1). Fig. 6a and 6b display the reflected and transmitted modal contents, respectively, for a rotational Mach number 𝑀 Ω = Ω𝑅/𝐶 = 0.4, while Fig. 6c and 6d do the same for the opposite rotation 𝑀 Ω = -0.4. In the first case, the cascade scatters the incident mode into the opposite direction to that of its rotation (𝑠Ω < 0), which leads to a lower frequency 𝜔 + 𝑠𝐵Ω = 𝜔 -𝐵|Ω|. In the second case, the rotating motion is reversed and the scattered mode 𝑚 + 𝑠𝐵 = -15 is then scattered at a higher frequency: 𝜔 + 𝐵|Ω|.

The relative reflected and transmitted acoustic powers are plotted against the rotational Mach number in Fig. 7a. What is noticeable from this plot is that the transmitted power is greater than the incident power for 𝑀 Ω < 0, and then drops for 𝑀 Ω > 0 without being compensated by the reflected power. A closer look on the sound power balance is given in Fig. 7b. An increased scattered power is found when the mode -15 is scattered at a higher frequency (𝑀 Ω < 0), whereas a decrease is found when this mode is scattered at a lower frequency (𝑀 Ω > 0). This is in remarkable agreement with the theory of Hanson [START_REF] Hanson | Acoustic reflection and transmission of rotors and stators including mode and frequency scattering[END_REF], who made a quick mathematical derivation based on Goldstein's energy flux [START_REF] Goldstein | Aeroacoustics[END_REF] expressed in the relative and absolute reference frames. According to Hanson, this results from the cascade motion, giving or taking energy from the scattered waves ‡ . The rotor does work on the acoustic field, since there is no other way to exchange energy in the model. This is because the conservation of stagnation enthalpy in the rotating frame of reference is equivalent to the conservation of rothalpy. The rothalpy, for rotational enthalpy, is the physical quantity conserved through the rotor in this case, as suggested by thermodynamics [START_REF] Dixon | Fluid Mechanics and Thermodynamics of Turbomachinery[END_REF]. It seems that when the rotor "pulls" the incident mode in the same direction as its rotation (𝑠Ω > 0), it gives part of its kinetic energy to the scattered mode. On the other hand, when it "pushes" the incident mode in the opposite direction to that of its rotation (𝑠Ω < 0), the rotor takes part of the acoustic energy of the scattered mode.

V. Wake-Interaction Noise Results

A. Rotor-Stator vs. Stator Alone

Now that energy conservation of the recursive algorithm has been assessed in the test case of section IV.A, and that the power imbalance due to the rotor scattering has been explained in section IV.B, the complete and fully coupled rotor-stator system is studied in this section, and results are compared with the stator-alone case. Analyses of instantaneous pressure maps, modal contents and sound power outputs are performed in what follows. A sketch of the system is represented in Fig. 8. The stator is made of 𝑉 = 31 cambered vanes with a camber angle of Ψ = 20 • and a stagger angle of Ψ/2 = 10 • . The incoming mean flow Mach number is 𝑀 = 𝑈/𝐶 = 0.4, with artificial swirl upstream of the stage. This is due to the use of flat rotor blades, because blade camber is not currently available in the code, and the need for a continuous mean flow evolution. The rotor is made of 𝐵 = 16 zero-stagger flat blades rotating in the positive direction at a rate 𝑀 Ω = Ω𝑅/𝐶 = 𝑀 sin Ψ 0.137, and the relative mean flow Mach number is 𝑀 𝑟 = 𝑊/𝐶 = 𝑀 cos Ψ 0.376. Compared to section IV, a Kutta condition is now applied to the trailing edges of both cascades, enforcing a zero pressure jump and adding an infinitely-thin vortical sheet downstream, the velocity field of which is noted 𝑢 𝑅 𝐾 . Sound generation is computed for the 5 th BPF wake harmonic 𝑛 = 5 and the amplitude of the vortical disturbance is fixed to unity in all test cases: |𝑢 𝑅 𝑖 | = 1. No wake evolution is modeled from the rotor TE to the stator LE. The generated pressure field is displayed in Fig. 9a, and without the rotor in Fig. 9b. A strong interaction is visible in the inter-stage area in Fig. 9a, as well as a disturbed wave-front upstream that indicates a multimodal propagation. In the stator-alone scenario (Fig. 9b), a single mode propagates upstream. By looking at the modal contents in Fig. 10, it is found that only the mode 𝑛𝐵 + 𝑠 1 𝑉 = -13 (𝑠 1 = -3) is cut-on upstream of the stator among the modes generated by the wake impingement, thus no sound is radiated downstream. The rotor is able to scatter this mode into 𝑛𝐵 + 𝑠 1 𝑉 + 𝑠 2 𝐵 = 3 (𝑠 2 = 1), which is now cut-on in the entire stage and propagates noise downstream through the stator. With a single stator, noise is only produced at the 𝑛 th BPF noise harmonic, in a one-to-one relationship with the incoming BPF wake harmonic. The scattering by the rotor redistributes noise upstream between the 5 th BPF noise harmonic, carried by the mode -13 (𝑠 2 = 0), and the 6 th BPF noise harmonic, carried by the mode 3 (𝑠 2 = 1). It also produces noise downstream at 6 × BPF via the same mode 3. Figure 11 shows the distribution of Sound Power Level (SWL) per BPF noise harmonic in both cases. The SWL is computed as SWL = 10 log 10 P/P 𝑟 𝑒 𝑓 , with P 𝑟 𝑒 𝑓 = 1 pW. In this case, the stator alone produces 125 dB upstream and nothing downstream. The rotor spreads an amount of the energy at the 5 th BPF to the 6 th BPF, resulting in 122 dB and 114 dB upstream, respectively, and 106 dB downstream. If the inter-stage distance is increased too much, the rotor and stator can no longer interact because of the exponentially decaying nature of the mode -14 produced by the stator. The Sound Power Levels (SWL) evolution per BPF noise harmonic is plotted against the inter-stage distance in Fig. 15b. The exponential decay of the noise radiating upstream and downstream is evidenced in this case. Notice that the downstream part is only visible at very small distances because of (i) the decrease in the reflection coefficient of the rotor and (ii) the threshold filtering of the algorithm. The previous case (𝑛 = 5 and 𝑀 Ω 0.137) is also depicted in Fig. 15a. It is seen that both 5 th and 6 th BPF noise harmonics remain constant with 𝑑, demonstrating that the noise levels do not depend upon the inter-stage distance if the interaction is due to cut-on modes with a fixed perturbation amplitude (independent to the distance 𝑑). In real applications, the velocity deficit is expected to decrease with the distance, thus the SWL in this case would decrease proportionally to the decaying prescribed velocity deficit. In conclusion, depending on the compressor design, cut-off modes could play a significant role on the radiated noise by energizing new tones. They should be accounted for in rotor-stator noise prediction tools.

Rotor Rotational Mach Number

For this parametric study, the inter-stage distance is fixed to 𝑑/𝑙 2 = 1 and now the rotor rotational speed varies. This increases the BPF, allowing more modes and tones to be produced. The first example is made with the 2 nd BPF wake harmonic and a rotational Mach number ranging from 0.0171 to 0.222. Figure 16 shows the evolution of upstream and downstream SWL for the rotor-stator case and the stator alone. The radiated noise is dominated by the downstream field up to 𝑀 Ω 0.15 in both cases, and is balanced for larger rotational speeds. The only difference when accounting for the rotor scattering is a decrease of 2 dB of the upstream power at the lowest rotational speed. This is attributed to the reflection of the stator noise by the rotor. When looking at higher wake harmonics, the pattern becomes more complex. Tones produced by the BPF wake harmonic 𝑛 = 5 are displayed in Fig. 17. The stator alone (Fig. 17b) generates equivalent upstream and downstream noises at 5 × BPF above 𝑀 Ω = 0.14, and is silent below this value because the generated modes cannot propagate at theses frequencies. The rotor-stator case (Fig. 17a) displays evidences of frequency scattering by the rotor. The tone 6 × BPF is generated at rotational speeds starting from 0.1, when the mode order 3 becomes cut-on, and the tone 7 × BPF appears around 0.17 to 0.19, when the mode order 19 is able to propagate. The emerging 7 th BPF noise harmonic has a significant impact on the noise produced at 5 × BPF upstream of the compressor stage. Starting from 𝑀 Ω 0.19, the tone 7 × BPF completely replaces the tone 5 × BPF when comparing SWL with the stator alone scenario (Fig. 17b). This results in a sharp drop of 10 dB of the noise radiated upstream at 5 × BPF. This shift of tonality is explained by the mode order 19 (𝑠 1 = -3 and 𝑠 2 = 2) becoming cut-on between 0.171 ≤ 𝑀 Ω ≤ 0.188 and the energy distribution switching from the mode order -13 to 19, as shown in Fig. 18. Finally, the pressure maps at both rotational speeds are presented in Fig. 19, in which the shift from the mode -13 to 19 is clearly observable in the upstream region.

To conclude on the effect of the rotor rotational speed on sound generation, the total noise produced by the first six BPF wake harmonics is studied. The acoustic powers carried by all modes, at all frequencies, generated by the first 6 wake harmonics are summed up together by virtue of linearity and plotted in dB in Fig. 20a. On the other hand, Fig. 20b displays the difference in dB between the upstream and downstream sound power levels as ΔSWL = SWL up -SWL down . It is found that, in this case, adding the rotor increases the ratio downstream/upstream because of the reflection at the rotor trailing edge (so called "shielding" effect), which is stronger around 𝑀 Ω = 0.04 and 𝑀 Ω = 0.15 from Fig. 20b. However, the total power does not vary significantly between both cases in Fig. 20a. It seems that the work done by the rotor on the scattered waves demonstrated in section IV.B is either negligible in this case, or compensates itself by giving energy to some modes and taking to others. Section IV.B has shown power imbalances lower than 10% for this range of rotational speeds, which might lead to the conclusion that this effect is negligible here. Yet keep in mind that Fig. 20a is indeed for the total noise summed over all BPF noise harmonics. When looking at individual noise harmonics, significant discrepancies occur between the rotor-stator case and the stator alone.

VI. Conclusion

A two-dimensional mode-matching technique for wake-interaction tonal noise has been developed, including rotor-stator coupling and the most recent achievements on mode matching in this context. The model consists of a primary sound generation computation by rotor wakes impingement on the stator, and then successive acoustic scattering iterations on the rotor and stator are performed until a convergence on the modal coefficients is reached. Thanks to previous works by the authors [START_REF] Girier | A Two-Dimensional Model of Sound Transmission Through Curved and Staggered OGV: Effect of Inter-Vane Channel Mode Transitions[END_REF][START_REF] Girier | Mode-matching techniques for the modeling of rotor-stator wake-interaction noise, with emphasis on the effects of vane camber[END_REF], a continuous evolution of the mean-flow conditions is considered through the entire rotor-stator stage, and advanced geometrical design parameters such as stagger and camber are accounted for. The current version of the code runs in a few seconds on a personal computer for a given Fourier component of the wake mean-velocity deficit.

The consistency of the iterative procedure has been demonstrated on a stator-stator scenario with incident acoustic waves, which give access to a power balance. When an incident vortical wave is considered, only the "local" power balance at each iteration of acoustic scattering can be tracked. An estimation of the hydrodynamic power carried by the vortical wave could be implemented in a future work, based on Maierhofer & Peake's formulation [START_REF] Maierhofer | Acoustic and hydrodynamic power of wave scattering by an infinite cascade of plates in mean flow[END_REF] for example, to give further validations. It would also give guidance regarding the number of modes to account for in the modal truncation. Indeed, the radiated sound powers can be sensitive and simple convergence hard to obtain when camber is modeled, depending on the truncation [START_REF] Girier | Mode-matching techniques for the modeling of rotor-stator wake-interaction noise, with emphasis on the effects of vane camber[END_REF]. Further comparisons with, for example, Hanson's results [START_REF] Hanson | Coupled 2-Dimensional Cascade Theory for Noise and Unsteady Aerodynamics of Blade Row Interaction in Turbofans[END_REF] and eventually experimental or numerical results with a hub-to-tip ratio close to unity is also something to aim for, in order to assess the physical consistency of the proposed model.

Acoustic power imbalances in presence of rotating scattering surfaces have been highlighted, which are in agreement with Hanson's theory [START_REF] Hanson | Acoustic reflection and transmission of rotors and stators including mode and frequency scattering[END_REF]: an energy increase is found for modes scattered at higher frequencies and an energy decrease for modes scattered at lower frequencies. An exchange of energy between the acoustic field and the rotor is made possible by the conservation of the stagnation enthalpy in the rotating frame of reference, which is equivalent to the conservation of the rothalpy. Regarding the rotor-stator system, comparisons of radiated sound power levels between the rotor-stator system and a stator alone have shown expected outcomes: the rotor reflection generally increases the downstream to upstream power ratio and also redistributes power to a larger number of BPF noise harmonics due to frequency scattering, giving rise to tones that would not be present in a stator alone computation.

Concerning possible improvements of the current model version, the rotor blades geometry could benefit from the improvements made for the stator. Staggered and cambered blades would enable different flow conditions between the inter-stage and upstream areas, thus being more realistic and favoring the occurrence of trapped modes, which was not accessible in the current study.

The methodology could be extended to three dimensions without difficulty, only at the cost of a higher computation time. Hanson [START_REF] Hanson | Broadband Noise of Fans -With Unsteady Coupling Theory to Account for Rotor and Stator Reflection/Transmission Effects[END_REF][START_REF] Hanson | Broadband Theory for Coupled Fan Stages Including Blade Row Reflection/Transmission Effects[END_REF][START_REF] Hanson | Broadband Noise Source Studies for a Fan with a Coupled Rotor/Stator[END_REF] treated turbulence impingement noise on the stator with full unsteady rotor-stator coupling. He showed that accounting for the rotor-stator coupling can lead to a substantial increase of the noise radiated downstream and a decrease of the noise radiated upstream of the compressor stage. Accounting for the fully coupled system instead of an isolated stator seems mandatory for reliable noise predictions and the mode-matching model presented herein could be extended to broadband noise in that purpose. Hanson also explained how scattered modes spread over limited mode subsets, thus reducing the possibilities of coupling. This principle of independent mode subsets, as well as considerations in frequency truncation explained in Ref. [START_REF] Hanson | Broadband Noise Source Studies for a Fan with a Coupled Rotor/Stator[END_REF], could help to build an efficient broadband noise prediction strategy.

Scattering index 1 𝜔

 1 𝑈 = Mean velocity in the stationary reference frame 𝑊 = Mean velocity in the rotating reference frame Ω = Rotation rate 𝑀 = Mach number 𝐷 = Mean Density 𝑃 = Mean Pressure 𝐶 = Mean sound speed 𝑅 = Radius of the unwrapped cut 𝑙 = Chord length 𝑎 = Inter-vane channel height / angular spacing 𝑑 = Inter-stage distance 𝐵 = Rotor blades number 𝑉 = Stator vanes number 𝑛 = BPF wake harmonic 𝑚 = Acoustic incident azimuthal order 𝑠 =

Fig. 4

 4 Fig. 4 Relative reflected and transmitted sound powers without flow (a) and with flow of Mach number 𝑴 = 0.45 in the opposite direction to that of the incident wave (b).

Fig. 5 Fig. 6

 56 Fig. 5 Sketch of the model rotor cascade with an incident acoustic wave, featuring the velocity triangle.

Fig. 7

 7 Fig. 7 Relative reflected and transmitted acoustic powers (a) and power acoustic balance (b) against rotational Mach number: the co-and counter-rotating indications refer to the scattered mode -15.

Fig. 8

 8 Fig. 8 Sketch of the rotor-stator system with an incident vortical wave and a Kutta condition imposed at the trailing edges of both cascades, featuring the velocity triangles.

Fig. 9

 9 Fig. 9 Instantaneous pressure maps for the rotor-stator case (a) and the stator alone (b), with 𝒏 = 5 and 𝑴 𝛀 0.137.

Fig. 10 Fig. 11 Fig. 14

 101114 Fig. 10 Upstream and downstream modal contents for the rotor-stator case (a-c), and stator alone (b-d), with 𝒏 = 5 and 𝑴 𝛀 0.137. Blue bars indicate cut-on modes, red bars are cut-off.

  BPF harmonic with 𝒏 = 5 and 𝑴 𝛀 0.137 (a), and 𝒏 = 3 and 𝑴 𝛀 0.188 (b).

  Fig. 16 Upstream ( ) and downstream ( ) SWL per BPF harmonic with 𝒏 = 2 for the rotor-stator case (a) and the stator alone (b).

  Fig. 17 Upstream ( ) and downstream ( ) SWL per BPF harmonic with 𝒏 = 5 for the rotor-stator case (a) and the stator alone (b).

Fig. 18

 18 Fig. 18 Upstream modal contents at 𝑴 𝛀 0.171 and 𝑴 𝛀 0.188, respectively, with 𝒏 = 5 for the rotor-stator case (a-c), and stator alone (b-d)

Fig. 19 Fig. 20

 1920 Fig. 19 Instantaneous pressure maps at 𝑴 𝛀 0.171 and 𝑴 𝛀 0.188, respectively, with 𝒏 = 5 for the rotorstator case (a-c) and the stator alone (b-d).

  Léo Girier * and Michel Roger † Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon I, CNRS, Laboratoire de Mécanique des Fluides et d'Acoustique, UMR 5509, 36 Avenue Guy de Collongue, F-69134, Écully, France Anthony Lafitte ‡ and Hélène Posson § Safran Aircraft Engines, Rond Point René Ravaud -Réau, 77550, Moissy-Cramayel, France

† The acoustic power is defined as the integral over the duct height of the energy flux projected on the horizontal axis (see Ref.[START_REF] Goldstein | Aeroacoustics[END_REF]), and thus is negative for upward-traveling waves and positive for downward-traveling waves.

‡ Notice that computations performed with no frequency shift (no scattered mode 𝑠 ≠ 0), not presented here, have shown no power imbalance. Hanson[START_REF] Hanson | Acoustic reflection and transmission of rotors and stators including mode and frequency scattering[END_REF] also mentioned the same outcome.

Acknowledgments

This work was performed within the framework of the industrial chair ARENA (ANR-18-CHIN-0004-01) cofinanced by Safran Aircraft Engines and the French National Research Agency (ANR), with the help of the French governmental plan "France Relance", and is also supported by the Labex CeLyA of the Université de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ANR-16-IDEX-0005).

Note that, due to the lack of mean flow deviation through the rotor in the current model, the flow conditions are the same in the inter-stage and upstream areas. This allows the mode -13 to propagate upstream, whereas it would be trapped with a purely axial upstream mean flow. This could be an interesting test case when adding stagger and camber to the rotor blades in the future.

Parametric studies are performed in the following to assess the impact of the rotor scattering with different inter-stage distances and rotor rotational speeds.

B. Parametric Studies

Inter-Stage Distance

The same test case is used but the inter-stage distance is increased from 𝑑/𝑙 2 = 1 to 𝑑/𝑙 2 = 3. The vortical amplitude |𝑢 𝑅 𝑖 | at the stator leading-edge location is kept the same in both cases to emphasize on the effect of acoustic coupling, thus not accounting for the wake decay from the rotor TE to the stator LE. The real part of the resulting pressure map is plotted in Fig. 12 and the inter-stage modal content in Fig. 13. The strong interaction due to the mode -13 and its reflection is still visible. The incident disturbance is now changed to the third BPF wake harmonic 𝑛 = 3 and the rotational Mach number is increased to 𝑀 Ω 0.188. The wake-interaction in this case generates only cut-off modes as depicted in Fig. 14b. Adding the rotor at a close distance 𝑑/𝑙 2 = 1 is able to scatter the cut-off mode 𝑛𝐵 + 𝑠 1 𝑉 = -14 (𝑠 1 = -2) before it


Chieh-Chun Chen
email: chieh-chun.chen@eurecom.fr

Mikel Irazabal
email: mikel.irazabal@openairinterface.org

Chia-Yu Chang
email: chia-yu.chang@nokia-bell-labs.com

Alireza Mohammadi
email: alireza.mohammadi@eurecom.fr

Navid Nikaein
email: navid.nikaein@eurecom.fr

FlexApp: Flexible and Low-Latency xApp Framework for RAN Intelligent Controller

RAN openness is a vision for 5G and beyond to avoid unnecessary lock-in effects. In this regard, the O-RAN alliance provides a new architecture with RAN intelligent controller (RIC) for both non-real-time and near-real-time cases, together with running applications, i.e., rApps and xApps. However, two key challenges remain in the current xApp framework: platform lock-in and xApp reusability. Therefore, we introduce a novel FlexApp framework to address both issues, as well as a new E2* interface that consumes less latency and CPU utilization. This new interface speeds up the xApp development process. Our performance evaluation of the FlexApp prototype shows that it can realize scalability and ultra-low latency operations (<10 ms), as well as the capability of two-level abstraction for xApp development.

I. INTRODUCTION

The roll-out of 5G elevates user experience among distinct aspects and brings new opportunities to reshape mobile networks. Specifically, some key features, such as flexibility, openness, and intelligence, are expected to evolve monolithic infrastructures in legacy Radio Access Network (RAN). In this regard, standardization development organizations and industry fora, such as O-RAN Alliance and Telecom Infra Project, are formed to not only concretize this vision into requirements, but also standardize open network interfaces among disaggregated RAN entities (e.g., Distributed Unit [DU] and Centralized Unit [CU]). Therefore, "Open RAN" is made possible with adequate programmability and extensibility.

Moreover, RAN Intelligent Controllers (RICs) are highlighted in the O-RAN architecture in Fig. 1 to offer the Software-Defined RAN (SD-RAN) capability for both Near-Real-Time (NearRT-RIC) and Non-Real-Time (NonRT-RIC) cases. The NonRT-RIC [START_REF] Nikaein | Near-Real-time RAN Intelligent Controller, Near-RT RIC Architecture[END_REF] is within service management and orchestration framework, and it performs a control loop larger than 1 sec. It relies on data transmitted over the A1 interface [1, ch. 7], like policy guidance (e.g., slice priority) or enrichment information (e.g., application-level metrics). While the NearRT-RIC [START_REF] Group | Non-RT RIC Architecture[END_REF] handles control loops between 10 ms and 1 sec over RAN entities (CU and DU) via the E2-Node, which logically terminates the E2 interface [START_REF]Near-Real-time RAN Intelligent Controller Architecture E2 General Aspects and Principles[END_REF]. The O1 interface [1, ch. 7] serves the purposes of op- erations, administration, and management and is beyond our scope.

To go one more step, two types of applications are utilized: rApps [2, ch. 4] running on NonRT-RIC and xApps [2, ch. 6] running on NearRT-RIC. The rApps provide value-added services related to RAN operation and optimization (e.g., energy saving) using the REpresentational State Transfer (REST)-based protocol over the R1 interface [1, ch. 7] to the NonRT-RIC. While the xApps utilize the E2 interface based on the Service Models (SMs) [START_REF]Near-Real-time RAN Intelligent Controller, E2 Service Model (E2SM)[END_REF] to interact with RAN Functions (RFs), which provide controllable RAN functionalities in an E2-Node. For example, Key Performance Measurement (KPM) SM provides RAN statistics and performance metrics.

Beyond the above recap, two key challenges remain. First is the interoperability of xApps across different NearRT-RIC platforms. Due to the lack of a standardized interface between NearRT-RIC and xApps, the current xApp uses a platform-specific NearRT-RIC Application Programmable Interface (API) [2, ch. 7] to communicate with NearRT-RIC (i.e., light blue arrows in Fig. 1), which clearly violates Open RAN vision. Second, the scalability of xApp deployments is an issue due to the latency and overhead incurred by the NearRT-RIC API.

To make matters worse, all messages sent to the xApp must first go through NearRT-RIC and then be exposed via the NearRT-RIC API to the embedded xApp.

To address these challenges, we propose the FlexApp framework, which will not only provide interoperability for any native or third-party xApps, but also enable ultralow latency operations (<10 ms). This framework relies on our newly-designed E2* interface between NearRT-RIC and xApps (i.e., dark red lines in Fig. 1), which can be viewed as a natural extension of the E2 interface and is considered as a potential candidate for standardization. Further, this framework enables the virtualization of RFs recursively to produce high-level abstraction and trigger future intent-based networking.

In the following, we first compare the FlexApp framework with other open-source works in Section II. An overview of the proposed framework is presented in Section III, and its design details are given in Section IV. Our FlexApp prototype in Section V demonstrates its low-latency in a scaled deployment and two-level abstraction ability for recursive xApp deployment.

II. RELATED WORKS

To avoid lock-in with a closed and/or proprietary xApp framework, we compare the FlexApp framework to two others in the open-source community: one developed by the O-RAN Software Community (OSC) and the other by the Open Networking Foundation (ONF), both on their respective NearRT-RIC platforms.

The OSC presents its xApp framework [START_REF]O-RAN Software Community[END_REF] (see Fig. 2a) on its own NearRT-RIC platform (OSC RIC [START_REF]RIC platform source code[END_REF]) and introduces the RIC Message Router (RMR) to communicate with the xApps through its Software Development Kit (SDK). The RMR is a library that abstracts the message transport mechanism (e.g., Nanomsg, Nanomsg next generation, or Socket Interface-95 [SI95] [START_REF]RMR Developer's Guide[END_REF]) to be used by xApps to send/receive messages to/from an E2-Node. The message routing and endpoint selection in RMR are based on a pair of message type and subscription ID, which need to be translated into the endpoint via a table entry. A non-negligible overhead and latency are observable when this framework is employed due to the process of selecting an endpoint for each message [8].

Moreover, the ONF introduces their RIC SDKs (see Fig. 2b) to develop xApps on their own NearRT-RIC platform (µONOS RIC) [START_REF]µONOS RIC architecture[END_REF]. These xApps communicate with the µONOS RIC using gRPC-based APIs, which provide HTTP request/response communication. Such an SDK encapsulates some of the complexities of dealing with individual gRPC services (e.g., threading and session management) to offer routing capabilities, and it requires xApps to provide the endpoint address (e.g., DNS server hostname, IP address, and port number) [START_REF]µONOS RIC architecture[END_REF]. Despite claims that gRPC is faster than other RESTlike transaction implementations, when compared to the RMR on top of SI95 (RMR/SI95) [START_REF] Community | Suitability of gRPC for RMR Communications[END_REF], gRPC cannot achieve the same throughput and with only acceptable latency when the message rate is less than 10,000 per second.

In contrast, the FlexApp framework (see Fig. 2) extends the standardized E2AP protocol as an E2* interface. This new interface can onboard the xApps into the NearRT-RIC and supports bidirectional communication between xApps and NearRT-RIC while inheriting the existing mechanism in the E2, i.e., Stream Control Transmission Protocol (SCTP) socket. One key aspect of this framework is to avoid platform lock-in; hence, the NearRT-RIC platform in Fig. 2 can be any O-RAN compliant RIC [START_REF] Group | Non-RT RIC Architecture[END_REF] (e.g., OSC RIC in Fig. 2a, µONOS RIC in Fig. 2b, or FlexRIC [START_REF] Schmidt | FlexRIC: An SDK for next-generation SD-RANs[END_REF]). This framework also supports xApp development in a variety of high-level programming languages (for example, Python, GoLang, and Java).

Another advantage of this framework is that it enables xApps to specialize the desired set of SMs, effectively decoupling the SMs from the NearRT-RIC. Currently, different formats for a given SM are standardized while letting xApps manage SMs' internal data model (by subscribing to events and actions); therefore, an xApp composes the desired SMs from different E2-Nodes and aggregates the data accordingly. In contrast, the FlexApp framework provides composability of RFs and SMs at the intermediate level by using the virtualized RF (vRF) to consolidate the data for xApp.

III. FLEXAPP OVERVIEW

As mentioned in Section I, the FlexApp framework is proposed to tackle both platform lock-in and xApp reusability challenges. A key approach adopted in FlexApp is to offer a novel set of abstractions for the virtualization of RFs to provide a high level of SM compatibility and composability, as depicted in Fig. 3. In fact, this approach can distinguish the xApp development model by relying on SMs' logic from that relying on business logic [2, ch. 7]. To provide more insight, three cases are presented in the same figure.

The first legacy case is xApp-I in Fig. 3, in which this xApp needs to handle different RFs (i.e., RF-1 and RF-2) through the corresponding SMs, data, and control actions and form a virtualized base station to process the data, even in a vendor-specific case. In contrast, the FlexApp framework provides the notion of vRFs to enable flexible SM composition based on the xApp requirements. Therefore, vRF-I in Fig. 3 is formed, which reveals a single SM abstracting data from the 2 RFs from different vendors to the xApp-II. Further, a higher level of abstraction can be built via recursive vRFs referencing, e.g., vRF-2 in Fig. 3 can handle the data from both RF-3 and vRF-1 among different Radio Access Technologies (RATs) to serve the purpose of xApp-III. Such virtualization allows the construction of new vRFs from existing RFs/vRFs as well as the recomposition of SMs, while still complying with E2 interface requirements.

As a concrete example, the Quality of Service (QoS) vRF can be composed by virtualizing other RFs that han-dle the data rate (e.g., by scheduling resource blocks) and latency (e.g., by managing queues). Furthermore, this vRF can also virtualize another vRF that handles servicespecific metrics, such as RAN availability, according to the business logic in xApp.

To conclude, our proposed FlexApp framework can benefit both RAN providers and xApp developers. On one hand, it simplifies the development of RFs and lowers the overhead on the E2-Nodes, since more advanced vRFs become composable via multi-level abstractions. On the other hand, it abstracts the heterogeneous RAN deployment and allows the newly composed vRF to be tailored to the desired xApp objectives, thereby simplifying the development of xApp. Furthermore, unlike the relationship between NearRT-RIC and xApps using platform-specific NearRT-RIC APIs (cf. Fig. 2a and Fig. 2b), xApps under the FlexApp framework are independent components. They have independent lifecycles and are treated as "first-class citizen applications" with a dedicated E2* interface to transfer the E2-based context for compatibility.

IV. SYSTEM DESIGN Generally, the FlexApp framework follows the zerooverhead principle to achieve low latency, and it includes four main elements, as shown in Fig. 4a: (1) E2* interface, (2) server library, (3) virtualization layer, and (4) agent library. From the viewpoint of the xApp, a specialized service controller is formed by utilizing these elements to realize its control logic. We can see a depiction of such a controller in Fig. 4b, and the design details of these elements are given as follows:

A. E2* interface

The E2* interface supports elementary procedures (e.g., E2* setup request/response) and services (e.g., report and control) similar to the E2AP protocol [START_REF]Near-Real-time RAN Intelligent Controller Architecture E2 General Aspects and Principles[END_REF]. First, like the E2 interface, the E2* setup request message is initiated by the agent towards the NearRT-RIC, and, once successfully received, the NearRT-RIC assigns a particular xApp ID and responds with the information of connected E2-Nodes. However, unlike E2, the E2* subscription request message is generated by the agent to the NearRT-RIC, including the E2-Node(s) that this xApp would like to subscribe to. This message is bookkept (e.g., update the subscription request ID) and then forwarded to the corresponding E2-Node by the NearRT-RIC. A similar process is followed by a control request message. In total, four new elementary procedures are added to the E2 ASN.1 specification: E2* Setup, RIC Subscription, RIC Subscription Delete, and RIC control. Note that other encoding and decoding schemes (compared to the standard ASN.1) are supported in the E2* interface using C11 Generics, which can alleviate existing bottlenecks at the CPU.

B. Server Library

The server library is essential for extending the NearRT-RIC with a virtualization layer that can aggregate vRFs from RFs/vRFs, as mentioned in Section III. Moreover, it manages the connection towards the agents and multiplexes messages between the virtualization layer and the agents by using the E2* termination, E2* management, and E2* message handler modules in Fig. 4b. Also, this library is designed as an eventdriven/callback-driven system, adhering to the aforementioned ultra-lean design principle to impose minimal overhead. Therefore, it activates the virtualization layer only when there are new messages to be handled.

C. Virtualization Layer

The virtualization layer can implement new SMs on top of the newly composed vRF and expose specific information to xApps. From the xApp viewpoint, this layer is the key in the specialized service controller, and there are four modules in this layer (cf. Fig. 4b). First, the E2SM abstraction handles the vRFs and the respective SMs to expose the prepossessed information to the northbound xApp via the E2* interface. Moreover, the E2SM management module handles the information of the supported SMs in the NearRT-RIC. Then, the subscription management module is responsible for the subscription procedure and tracks the existing RF subscription statuses for each xApp. Finally, the service management module takes care of the interaction between xApps and (v)RFs by processing the messages related to the basic services specified in E2SM, including report, insert, control, and policy.more than one xApps sends conflicting messages to the same (v)RFs.

D. Agent Library and xApp development

From the perspective of xApp, the agent library is the cornerstone of realizing its logic because it is built on top of this library in the southbound. Furthermore, as shown in Fig. 4a, an xApp can recursively expand the interface at its northbound by reusing the server library (cf. Section IV-B). In detail, two building blocks exist in this library as a part of the SDKs: C SDK and SWIG1 as shown in Fig. 4b. The C SDK refactors all common functionalities of an xApp (e.g., communication, message handling, encoding/decoding, and data layer) under a single C library to be exposed to xApps for the development of C/C++ based xApp. In contrast, as mentioned in Section II, to enable the xApp development using other high-level programming languages, the SWIG compiler is apoted to create the wrapper codes for the C-based library of E2/E2* interfaces.

To develop an xApp, the northbound SDK2 is provided in the agent library functions including:

V. PERFORMANCE EVALUATION

In this section, the performance of FlexApp framework is validated in three aspects, based on a prototype implemented on top of the O-RAN compliant FlexRIC [START_REF] Schmidt | FlexRIC: An SDK for next-generation SD-RANs[END_REF]. First, we measure the End-to-End (E2E) latency from the E2-Node to the xApp under different numbers of E2-Nodes, which represents the capability to realize ultra-low latency operations even in a scaled deployment. Second, benchmarks are made between the proposed E2* interface 3 and two other transport mechanisms adopted by the OSC and the ONF, i.e., gRPC and RMR/SI95 (cf. Fig. 2). Finally, the efficiency and effectiveness of the xApp control logic are verified after applying the virtualization layer for multi-level abstraction in the FlexApp framework.

Furthermore, as shown in TABLE I, the experiments are evaluated in two setup environments: containerized (cloud-native) and non-containerized (bare metal). To represent the RAN functionalities and perform the report service, we use the E2-emulator provided by the FlexRIC. Within each E2-emulator, the Medium Access Control (MAC) RF is enabled, and the corresponding data is encapsulated into the MAC SM. Then, our selfdeveloped xApp subscribes to the MAC SM with a periodicity, and the composed E2SM will be periodically updated by the E2-emulator using E2AP and sent to the xApp using the E2* interface.

A. Latency and Scalability

In this evaluation, three Precision Timing Protocol (PTP) synchronized machines are used to deploy the E2emulator, FlexRIC, and xApp. We cover both containerized and non-containerized environments. The xApp subscribes to the MAC SM and obtains the monitoring data from the E2-emulator(s) via the E2* interface for every Transmission Time Interval (TTI) in a 12-byte message format (including E2AP & E2SM protocols). Two relationships between xApps and E2-emulator are evaluated: 1-to-N and N-to-1.

As shown in Fig. 5a, both the average and the maximum values of E2E latency are presented for 10 ms TTI in a containerized environment. Therefore, within 1 sec, there are 100 messages to be processed for each connected E2-emulator or xApp. The average 3 In the FlexRIC implementation, the E2* is labelled as E42 interface. E2E latency remains constant when the number of E2emulators or xApps is low (i.e., 8); however, its values increase when there are either 32 E2-emulators 4 or 32 xApps, but remain below 1 ms for ultra-low latency operations. A similar trend is observed in the maximum E2E latency, where the values are all lower than 10 ms. This guarantees that the FlexApp approach will not produce an internal bottleneck under a 10 ms TTI.

Based on the results shown in Fig. 5a, we can conclude that the FlexApp framework can support 32 xApps using its northbound to monitor SM from an E2-Node, and an xApp can simultaneously monitor 32 E2-Nodes connected to the NearRT-RIC. In parallel, we perform a stress test on the FlexApp framework, which shows that up to 350 xApps could be supported per machine, and the average E2E latency is around 6 ms. This limitation is due to the thread mode implemented in our prototype and the way messages are enqueued. We plan to make improvements in the future.

In addition, we measure the E2E latency in a noncontainerized environment to plot the viable regions for different latency restrictions, as shown in Fig. 5b, in terms of various combinations of TTI values and the number of E2-emulators. Take the example case with 1 ms latency restriction, its viable region covers both the light orange and the light red parts in Fig. 5b, since these two parts can support latency below 1 ms. We can see a tradeoff between how frequently the data can be updated and how many E2-Nodes can be monitored simultaneously, e.g., shorter TTI values must come with fewer monitored E2-Nodes. Also, the red star in the same figure represents the setting used in Fig. 5a, i.e., 10 ms TTI and 32 E2-Nodes, and this setting is confirmed to be functional for 1 ms latency restriction.

B. Benchmarking on transport mechanism

Afterwards, different transport mechanisms between NearRT-RIC and xApp are compared. Since our focus is on the transport mechanism, a simple 1-to-1 relationship between xApp and FlexRIC is applied (similar to [START_REF] Community | Suitability of gRPC for RMR Communications[END_REF]), but with two different TTI values, i.e., 250 µs and 10 ms. Specifically, an xApp is used to measure the latency by calculating the time difference from when the indication message is forwarded by the NearRT-RIC to when it arrives at xApp callback function. In Table II and Fig. 6a, both E2* interface and gRPC are compared at different message rates, and the E2* interface outperforms at every percentile value.

Furthermore, in Fig. 6b, we compare the CPU utilization of the xApp to receive the transported messages among the E2* interface, the gRPC, and the RMR/SI95. We can see that the E2* interface consumes considerably less CPU resources (i.e., 101.5% compared with 260% and 235%) than the other two.

In summary, by design, the E2* interface can reach a lower latency than the gRPC because it performs fewer work/abstractions, e.g., a raw SCTP connection in the E2* interface versus an HTTP-based approach in the gRPC. Moreover, the E2* interface consumes less CPU resource at the xApp owing to the simpler FlexApp ar-

C. Two-level abstraction and Control loop RTT

To validate the multi-level abstraction approach described in Section III, we build a virtualization layer (see Fig. 4b) in the NearRT-RIC platform and test its ability to interact with two separate slicing control xApps. Based on the respective service requests from these xApps, radio resources from the E2-emulator, i.e., 162 Resource Blocks (RBs) in a 60 MHz bandwidth with numerology 1 can achieve up to 200 Mbps, will be assigned accordingly.

Moreover, to compare different perspectives after and before the two-level abstractions (see xApp-II in Fig. 3), two respective figures are presented in Fig. 7a and Fig. 7b. The former shows the virtual RBs assigned to the xApp by the virtualization layer, whereas the latter represents the physical RBs seen from the NearRT-RIC perspective. Initially, only xApp1 is present, and it needs 80 Mbps for its service. Therefore, only 40% of the physical RBs are requested by the virtualization layer from the E2-emulator (see vRIC-s1-pRBs in Fig. 7b). But from xApp1's point of view, it assumes it receives 100% of the virtual RBs because of the second-level ab-straction (see xApp1-vRBs in Fig. 7a). At the 7 th second, xApp2 is deployed and requests 120 Mbps instead. In this regard, the virtualization layer requests 60% of the physical RBs from the E2-emulator (see vRIC-s2-pRBs in Fig. 7b), but xApp2 receives 100% of the virtual RBs (see xApp2-vRBs in Fig. 7a), similar to xApp1. Then, at the 14 th second, xApp2 lowers its request to 40 Mbps, and thus the corresponding decreases by a factor of three are made in both Fig. 7a and Fig. 7b. Finally, xApp2 releases its request at the 21 st second but has no effect on xApp1.

Furthermore, we measure the control loop Round-Trip-Time (RTT) between the E2-emulator and the xApp after considering the virtualization layer as the secondlevel abstraction. In specific, the measured RTT as shown in Fig. 8 represents the time difference from when the xApp sends a control message until it receives the control acknowledgement. Compared with the measurements conducted in Section V-A, these results are more informative in terms of considering the second-level abstraction as well as the RTT statistics for the control loop. We can see that the RTT after considering the secondlevel abstraction is approximately three times higher than that of the first-level abstraction case. However, it still fulfills the requirements for control loops in NearRT-RIC (between 10 ms and 1 sec), implying that extralevel abstractions (e.g., third-level abstraction xApp-III in Fig. 3) are potentially feasible.

D. Summary

In this section, three aspects of the performance evaluation are conducted for our FlexApp prototype. These results prove that the FlexApp framework is not only suitable for inclusion in the current NearRT-RIC platform, but also outperforms other xApp frameworks in terms of better performance (i.e., lower latency, less CPU usage) and newer functionalities (i.e., two-level abstraction). In addition, we provide informative results (e.g., feasible region in Fig. 5b and control loop RTT in Fig. 8), which can be used for further analysis of the different xApp deployment possibilities. In summary, the FlexApp framework enables an xApp to perform ultralow latency operations thanks to the new E2* interface and is built on the virtualization layer, abstracting the raw data from the SM logic to the business logic to simplify the xApp development.

VI. CONCLUSION AND FUTURE DIRECTIONS

In this paper, we presented FlexApp, an xApp framework that supports latency-sensitive use cases and enables multi-level abstractions, as well as a future standardized E2* interface, to communicate between xApp and NearRT-RIC. Additionally, by design, it is compatible with other NearRT-RIC platforms. The performance evaluation of our FlexApp prototype was thoroughly conducted, and the results indicated its benefits include low latency, high scalability, slim overhead, and multilevel abstraction functionality.

In the future, we plan to extend this work in three directions. First, related to the FlexApp framework, we plan to enhance its scalability in its northbound and exploit its virtualization layer to abstract a heterogeneous and multi-X (e.g., multi-tier, multi-RAT) RAN. Then, as for the xApp, we will extend its northbound interface to expand its flexible and recursive relationship toward the NearRT-RIC and explore new xApps (e.g., interference management) for RAN optimization and coordination. Finally, we consider porting existing xApps (e.g., traffic steering) to enrich FlexApp compatibility and interoperability between different NearRT-RIC platforms.

Fig. 1 :

 1 Fig. 1: O-RAN architecture with the proposed FlexApp framework and E2* interface.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Comparison of xApp framework from OSC, ONF and the proposed FlexApp

Fig. 4 :

 4 Fig. 4: (a) FlexApp framework design; (b) A specialized service controller based on the FlexApp framework .

•

  init_xapp_api initializes the xApp by sending E2* setup request to the NearRT-RIC, • e2_nodes_xapp_api can get the information of the connected E2-Nodes from the NearRT-RIC, • report_sm_xapp_api enables the report service by subscribing to the RF with arguments, and • control_sm_xapp_api can control the targeting RFs by specifying the input arguments E. Summary In short, the FlexApp framework provides several key takeaways: (a) Forward compatibility, such as dedicated controllers and xApps, (b) Ultra-lean design by extending E2AP without extra overhead, (c) A unified SDK exposing generic APIs for xApp development, and (d) Multilingual xApp development kit.

  Fig. 5: Latency and scalability evaluation.

Fig. 6 :

 6 Fig. 6: Benchmarking on transport mechanisms.

Fig. 7 :

 7 Fig. 7: Two-level abstraction.

Fig. 8 :

 8 Fig. 8: Control loop round-trip-time.

TABLE I :

 I Experimental environment.

		Cloud-Native	Bare Metal
	Ubuntu	20.04.5	20.04.1
	CPU(s)	4.8GHz x 20	4.7GHz x 12
	Memory	64 GiB	32 GiB
	Kubernetes	v1.24.5	-
	Containerd	v1.6.8	-

TABLE II :

 II Latency comparison in milliseconds.

		Min	Max 50% 95% 99% Total	Rate
							Sent	(msg/s)
	gRPC 0.07	>1.0 0.15	0.18	0.20	100K 4K
	E2*	0.02	0.39	0.11	0.11	0.11	100K 4K
	gRPC 0.20	0.54	0.31	0.37	0.39	25K	0.1K
	E2*	0.03	0.21	0.06	0.06	0.06	25K	0.1K
				4K(msg/s)
					gRP C
					E2*
	(a) Latency comparison: 100K mes-
	sages sent at 4K rate (msg/s)

https://www.swig.org/

https://gitlab.eurecom.fr/mosaic5g/flexric/-/tree/master/examples/ xApp/c/

The maximum number of supported E2-emulators is currently limited by FlexRIC; hence, we only evaluate up to 32 E2-emulators.

ACKNOWLEDGMENTS

This work is funded by the European Commission Horizon 2020 5G-PPP Program under Grant Agreement Number H2020-ICT-2020-2/101017226 "6G BRAINS: Bringing Reinforcement learning Into Radio Light Network for Massive Connections".