text:
Julien Marty
email: julien.marty@onera.fr

Hélène Gaible
email: helene.gaible@onera.fr

Hervé Bézard
email: herve.bezard@onera.fr

H B Ézard

ASSESSMENT OF SCALE ADAPTIVE SIMULATION OF A ROTOR OF HIGH PRESSURE COMPRESSOR

Keywords:

come

Assessment of scale adaptive simulation of a rotor of high pressure compressor

INTRODUCTION

The high pressure compressor design is mainly driven by the search of the highest efficiency and the largest stable operating range for a given stagnation pressure ratio. These design objectives are directly and strongly impacted by the secondary flows such as the corner separation and the tip leakage vortex. As shown by Herzig et al. [START_REF] Herzig | A visualization study of secondary flows in cascades[END_REF], the interaction of the tip flow with the main flow, the boundary layer developing at the shroud and the incoming wakes are responsible for penalizing losses. According to Bindon [START_REF] Bindon | The Measurement and Formation of Tip Clearance Loss[END_REF], the two main sources of losses relative to tip clearance flow are the mixing and the gap shear, not the interaction with endwall. He also conceptualized the flow within the gap by hypothesizing that there are different kinds of channels with and without separation bubbles within the gap. The axial position of these different channels depends on the pressure distribution along the chord, and thus the loading. With a stationnary endwall Kang and Hirsch [START_REF] Kang | Experimental Study of the Three-Dimensionnal Flow within a Compressor Cascade with Tip Clearance : Part II -The Tip Leakage Vortex[END_REF] highligths a tip flow pattern involving a tip leakage vortex, a secondary vortex and a tip separation vortex. Mailach et al. [START_REF] Mailach | Periodical Unsteady Flow within a Rotor Blade Row of an Axial Compressor -Part II : Wake -Tip Clearance Vortex Interaction[END_REF] studied the interaction between incoming wake and tip leakage flow. They highlighted the periodic variation of orientation and extension of tip clearance vortices. Suder and Celestina [START_REF] Suder | Experimental and Computational Investigation of the Tip Clearance Flow in a Transonic Axial Compressor Rotor[END_REF] investigated the tip clearance flow at different rotational speed highlighting the interaction with shock wave at nominal speed and the appearance of a second vortex only at partial speed.

The designers use more and more computational fluid dynamics (CFD) to design compressor as CFD offers a flexibility to evaluate numerous design parameters with low cost. Nonetheless, CFD results depend significantly on the Navier-Stokes closure i.e. on the used turbulence model and the modeling turbulence approach (RANS, DES, LES, DNS). Classically, the RANS turbulence model is based on the Boussinesq hypothesis by using a model with one or two transport equations such as the Spalart-Allmaras [START_REF] Spalart | A One-Equation Turbulence Model for Aerodynamic Flows[END_REF] or the Menter SST [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] models. Borello et al. [START_REF] Borello | Computation of tip-leakage flow in a linear compressor cascade with a second-moment turbulence closure[END_REF] show that the turbulence is clearly anisotropic in the vicinity of the tip gap and second moment closure (Differential Reynolds Stress Model) is able to capture this characteristic. Several authors studied the tip leakage flow with high fidelity methods. Using IDDES, Shi et al. [START_REF] Shi | Numerical study of tip leakage flow of a transonic compressor rotor with iddes method[END_REF] investigate the tip flow interacting with shock wave resulting in a curved shock surface and shock oscillations. Zonal Detached Eddy Simulation was also performed by Riéra et al. [START_REF] Riéra | Inlet condition effects on the tip clearance flow with Zonal Detached Eddy Simulation[END_REF][START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF] to study the tip leakage vortex and its interaction with incoming wakes and vortices. You et al. [START_REF] You | Computational methodology for large-eddy simulation of tipclearance flows[END_REF] and Boudet et al. [START_REF] Boudet | Large-eddy simulation of a single airfoil tip-clearance flow[END_REF] performed Large Eddy Simulation (LES) of a clearance flow. Nonetheless, the cost of LES is still expensive.

The Scale Adaptive Simulation (SAS) approach is a way to enhance traditional RANS model with low implementation effort. It can improve dramatically URANS computations especially when massive flow separation occurs, even for non refined grids (obstacle in cross flow, airfoil beyond stall). A well-known example is the jet in crossflow [START_REF] Duda | Numerical investigations on a hot jet in cross flow using scale-resolving simulations[END_REF]. The classical turbulence models (even Reynolds stress models) fail reproducing the diffusion of the jet downstream, whereas the same models coupled with the SAS approach reproduce perfectly the experiments, using classical RANS grids. The SAS approach was also used in previous studies in turbomachinery framework. Wang and Yuan [START_REF] Wang | Unsteady mechanisms of compressor corner separation over a range of incidences based on hybrid les/rans[END_REF] investigated the unsteady behavior of corner separation. Lucius and Brenner [START_REF] Lucius | Unsteady cfd simulations of a pump in part load conditions using scale-adaptive simulation[END_REF] performed SAS computations on a pump in partial load conditions. Both URANS and SAS approaches are able to capture the rotating stall. Ravelli and Barigozzi [START_REF] Ravelli | Application of unsteady computational fluid dynamics methods to trailing edge cutback film cooling[END_REF] showed that the SAS approach improves the prediction of vortex shedding at the trailing edge and the film cooling effectiveness in the framework of high pressure turbine nozzle guide vane with trailing edge cutback film cooling.

The literature review shows that the combination of SAS approach with DRSM turbulence model is promising and has never been assessed in the turmachinery framework, especially for the tip leakage flow prediction. The present study aims at validating this combination by comparison to experimental data, especially to laser measurements in the tip. The second objective of this paper is to highlight and discuss the improvement obtained with this combination SAS-DRSM by comparison to turbulence approaches classically used in CFD and industry: URANS simulations and standard SAS simulation i.e. combined with SST turbulence model. After a short description of the experimental facility CREATE (high pressure compressor), the numerical test bench is detailed and the turbulence models and approaches are presented. Then the numerical results are validated at several levels (from integral performances to spectral analysis according to Sagaut and Deck [START_REF] Sagaut | Large eddy simulation for aerodynamics: status and perspectives[END_REF]) by comparison to experimental data.

CREATE COMPRESSOR Experimental facility

The studied configuration is the experimental compressor CREATE [START_REF] Ottavy | Experimental and Computational Methods for Flow Investigation in High-Speed Multistage Compressor[END_REF] which is located at the Laboratory of Fluid Mechanics and Acoustics (LMFA) in Lyon, France. This axial compressor from SAFRAN AIRCRAFT ENGINES comprises three and a half stages presented in figure 1 and is representative of the median stages of modern high pressure compressors. The number of stages was chosen to have a magnitude of the secondary flow effects similar to a real compressor, and to be within the rig torque power limitation. SAFRAN AIRCRAFT ENGINES and the research team of LMFA have taken into account technological constraints coming from the experimental part of the project, very early in the compressor design. To have traversal probes between blade and vane rows, the axial gap was slightly increased compared to current compressors, and an outer-case movingrings technology was implemented to perform probe measurements in the circumferential direction at constant radius location. The circumferential periodicity of the whole machine (obviously 2π in the general case with primary blade numbers to avoid generating resonant waves) has been reduced to 2π 16 on the compressor CREATE, choosing the number of blades of each rotor and stator [inlet guide vane (IGV) included] as a multiple of 16. The blade number for each row is detailed in table 1. Thus the analysis of rotor-stator interaction does not require measurements over a full annulus sector as a sector of only 22.5 • ( 2π 16 ) should contain all the spatial information useful to rotor-stator interaction analysis, at least for stabilized operating point. Inlet Mach number at tip of R1 0.92 Some characteristics are summarized in table 2. Measurements are carried out with both pneumatic and unsteady pressure probes [START_REF] Ottavy | Experimental and Computational Methods for Flow Investigation in High-Speed Multistage Compressor[END_REF], which are located between the axial sections 25A and 26A, visible in figure 1. The measurement of the overall compressor performance was performed at several speeds, including the stall line position. At several operating points (near maximum efficiency, near choke and near surge), conventional pressure, temperature, and angle probe traverses were performed downstream of each blade and vane. The four-holes probes had dual pressure and temperature sensors, together with two extra static pressure sensors (mounted on each side of the probe) providing the flow angle when balanced. Thus the Mach number and also the three velocity components can be computed. The quantities measured with pneumatic probes are then averaged circumferentially to obtain the radial profile and averaged circumferentially and radially to obtain the performances (pressure ratio, efficiency). The same method is used in CFD.

Moreover, two-dimensional velocity measurements are available through Laser-Doppler-Anemometry. Two pairs of beams (l = 488 nm and l = 514.5 nm) are used for simultaneously measuring two velocity components which lead to the de-termination of the axial and tangential velocity components. The focal length of the front lens is 250 mm. A diameter of 76 m and a length of around 0.9 mm characterize the measurement volume.

The pneumatic probe and LDA measurements are performed downstream of each row, at mid-distance between rotor and stator. For the pneumatic probe, a system with rotating ring allows the scanning at different azimuthal position over a 22.5 • sector and the measurements can be performed from 5% to 95% relative height. For LDA, windows are not only placed downstream each row with a size allowing measurements over the same azimuthal sector but also in front of the rotor tip. The measurements were triggered with the rotation frequency of the machine, in such a way that the flowfield is described either inside a single-blade passage, or within several blade passages covering the circumferential periodicity of the whole machine. More details about LDA measurements are given in [START_REF] Arnaud | Experimental Investigation of the Rotor-Stator Interactions, within a High Speed, Multistage, Axial Compressor. Part 1 -Experimental Facilities and Results[END_REF][START_REF] Arnaud | Experimental Investigation of the Rotor-Stator Interactions, within a High Speed, Multistage, Axial Compressor. Part 2 -Modal Analysis of the Interactions[END_REF][START_REF] Ottavy | Experimental and Computational Methods for Flow Investigation in High-Speed Multistage Compressor[END_REF].

CFD setup

Computational domain In the present study, only the first rotor of the compressor (R1) is investigated at the nominal rotational speed and the nominal operating point. The inlet boundary of the test bench is set to section 25A of the experimental test rig, located between the IGV and the R1. The reason is that measurements are available on the experimental compressor for different operating points at this section. Besides, the possible comparisons with experimental data at section 25A make the numerical setup more reliable for the evaluation of the method. The location of the outlet boundary is based on the hypothesis that the outflow is axisymmetric, therefore, it must be far from the trailing edge of the R1. Furthermore experimental measurements are available at section 26A, so this section has to be inside the computational domain for validation purposes. Nonetheless, the mesh size is a key element in the domain definition. This is why the outlet boundary of the computational domain is defined at two axial chords downstream the R1.

Mesh and boundary conditions

In this study, two meshes are used. All grids are obtained with a multi block approach using an O4H topology. The rotor tip clearance is meshed with an additional O-H block. The normalized wall cell dimension normal to the wall fulfils ∆y + of the order of 1 in every zone. The first grid is composed of one blade passage (1/64 of the compressor) and can be considered as a fine mesh for URANS simulation. This mesh is called "M1" and consists in 4.3 million points. The IGV wakes are not taken into account in the simulation: the inlet boundary conditions are based on the radial profiles of stagnation pressure, stagnation enthalpy, velocity direction and turbulent primitive variables obtained by azimuthal average of the two-dimensional cartography used with the second grid.

The second grid (called "M2") is the same used for Zonal Detached Eddy Simulation by Riéra et al. [START_REF] Riéra | Inlet condition effects on the tip clearance flow with Zonal Detached Eddy Simulation[END_REF][START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF]. The normalized wall cell dimension ∆x + and ∆z + are detailed in [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF]. Two passages are simulated in order to fit the spatial periodicity of the IGV. The whole grid comprises 88 million points in order to model 1/32 of the compressor. The computational domain of mesh M2 is shown in figure 2. For the first mesh, the boundaries are located at the same location except the periodic boundaries by rotation as only one blade passage is simulated. The mesh at hub and blades walls is depicted in figure 3 with one point over two in all mesh directions. The mesh density is clearly high near leading and trailing edge but also in the blade passage in order to capture the tip leakage vortex. In the real configuration, there are two leakage flows at the hub. The first one is upstream of the rotor and the second one is linked to the labyrinth seal under the first stator. These two leakage flows and blade fillets are not included in the computational domain as the massflow are very small and the impact on tip leakage vortex is several order of magnitude smaller than the IGV wakes and vortices. In this configuration, the IGV effects are reproduced with specific inlet boundary conditions based on 2D cartography resulting from a previous IGV-R1 computation [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF]. It consists in a rotating distortion cartography, whose method was used for centrifugal compressor by [START_REF] Tartousi | Numerical Investigation of a Turbocharger Centrifugal Compressor: Volute Influence on the Performance of The Compressor[END_REF], based on a Fourier decomposition with 60 harmonics of the two-dimensional map of the flow. The IGV wakes and vortices are observed at the 25A section as shown in figure 4 which depicts the axial velocity and the entropy variation fields. The convection of this secondary flows originating from the IGV across the rotor passage is shown in figure 5. Only where the reference state is taken at the computational domain inlet. The physical values for stagnation pressure, stagnation enthalpy, primitive turbulence quantities of the turbulence model, and direction of the flow are given at each cell of the map. Therefore, with this rotating boundary condition, the unsteady distortion effects induced by the IGV are modeled in the simulation of R1 flow.

For the outlet boundary condition, the static pressure is specified at the hub and then a simplified radial equilibrium law. All simulations are based on the same static pressure value as in [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF]. Finally, a classic rotation periodicity condition is set at the azimuthal boundaries and a no-slip adiabatic wall condition is applied at all wall surfaces.

Numerical methods

The numerical simulations have been performed using the elsA software [START_REF] Cambier | The Onera elsA CFD Software: Input from Research and Feedback from Industry[END_REF], developed at ON-ERA and co-owned by AIRBUS, SAFRAN and ONERA. This software is based on a cell centered finite volume technique and structured multiblock meshes. The viscous fluxes are computed with a second-order centered scheme. For efficiency, an implicit time integration is employed to deal with the very small grid size encountered near the wall.

As one of the aims of this study is to compare SAS approach coupled with turbulence models to ZDES-SA method, the same numerical parameters as Riéra et al. [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF] are used. The computation is performed in the rotor frame of reference. The casing wall is then moving regarding this relative frame. The spatial discretization scheme for the inviscid fluxes [START_REF] Mary | An algorithm for low mach number unsteady flows[END_REF] is based on the third order accurate Advection Upstream Splitting Method for low Mach numbers (AUSM+P), initially developed by Edwards and Liou [START_REF] Edwards | Low diffusion fluxsplitting methods for flows at all speeds[END_REF]. The numerical dissipation is proportional to the local fluid velocity. A classic second order centred formulation is used for the viscous fluxes. The time discretization scheme is the second order accurate Gear scheme. At each time step, an approximate Newton method based on the LU factorization solves the non-linear problem. The time step is set to 1.6 10 -7 s which leads to a Courant-Friedrich-Levy number lower than 1 except for the boundary layers. It corresponds to 1000 time steps per IGV passing period. The use of 8 sub-iterations per time-step is required to reach a decay superior to 1.5 orders of magnitude for the residuals. This criterion is a compromise between precision and cost of the computation and takes into account the small time step involved [START_REF] Daude | Méthode d'Intégration Temporelle Implicite pour la Simulation des Grandes Échelles Application à la Réduction du Bruit de Cavité[END_REF]. Moreover, as the physical timestep is very small (CFL < 1 outside the boundary layer, CFL ≈ 15 near the wall). Daude et al. [START_REF] Daude | Méthode d'Intégration Temporelle Implicite pour la Simulation des Grandes Échelles Application à la Réduction du Bruit de Cavité[END_REF] use a ratio of CFL/N equal to 2 where N is the sub-iteration number, to perform LES in order to reduce CPU cost in comparison to an explicit time integration scheme while the result quality is identical. In the present case, the maximum CFL is equal to 15. Thus 8 sub-iterations ensure the fully-converged solutions.

TURBULENCE MODELING SST Menter kω model

The transport equations for the turbulent kinetic energy k and for the turbulent specific dissipation rate ω are the following:

∂ ρk ∂t + u j ∂ ρk ∂ x j = P k -β ρkω (2)
+ ∂ ∂ x i (µ + σ k µ t ) ∂ k ∂ x i ∂ ρω ∂t + u j ∂ ω ∂ x j = γ µ t P k -β ρω 2 + ∂ ∂ x i (µ + σ ω µ t ) ∂ ω ∂ x i (3)
+ 2 (1 -F 1 ) σ ω2 ρ ω ∂ k ∂ x j ∂ ω ∂ x j
F 1 = tanh ζ 4 ζ = min max √ k 0, 09ωd ; 500ν ωd 2 ; 4σ ω2 k CD kω d 2 (4)
CD kω = max 2σ ω2 1 ω ∂ k ∂ x j ∂ ω ∂ x j ; 10 -20
The constants of the model are a combination of the inner [kω model (index 1 )] and outer [ kε model (index 2 )] models:

C = F 1 C 1 + (1 -F 1 )C 2 (5)
σ k1 = 0.5 σ ω1 = 0.5 β 1 = 0.075 (6)
σ k2 = 0.5 σ ω2 = 0.856 β 2 = 0.0828 (7)
The γ constant is defined by equation ( 8) which gives γ 1 = 0.555 and γ 2 = 0.664.

γ = β β * - σ ω κ 2 β * κ = 0.41 β * = 0.09 (8)

Wall condition on ω

The ω-scale behaves theoretically as 1 d 2 when approaching a wall, which is hard to deal with from the numerical point of view. Different wall conditions on ω can be specified at a wall cell interface. Menter suggests to prescribe the ω wall value as

60ν d 2 1
where d 1 is the wall distance to the first cell point. It is a kind of quadratic extrapolation and can lead to great ω wall values. A more robust way implemented in elsA code is to extrapolate linearly the first cell ω value. Computations performed on different cases showed no influence of these conditions on the solution (friction coefficient, velocity profiles). Wilcox [START_REF] Wilcox | Formulation of the kω turbulence model revisited[END_REF] introduces the notion of slightly-rough surface and prescribes the ω wall value as 40,000ν ks 2 where the sand-grain roughness height ks should be taken to ensure a reduced wall value ks + below than 5. However this condition produces a small increase of the flat plate friction coefficient of about 5%. Both conditions (linear extrapolation and pseudo-roughness) were tested in the present simulations.

DRSM model

The Differential Reynolds Stress Models transport seven equations: six equations for the turbulent stresses and one equation for a turbulent scale (ε, ω, ...). On the contrary to standard two-equation models, the DRSM features many physical behaviors of the turbulence: anisotropy, rotation, curvature, history ef-fects, strong shear. The model formally reads:

DR i j Dt = P i j -ε i j + Φ i j + D i j +C i j (9)
where P i j is the production (no modelization), ε i j the dissipation, Φ i j the pressure strain, D i j the diffusion and C i j the rotation term in case of a rotating relative frame (no modelization). Many different expressions exist for ε i j , Φ i j and D i j and are not detailed here. In this paper the LRR/SSG-ω DRSM of Eisfeld et al.is used (see [START_REF] Eisfeld | Verification and validation of a second-moment-closure model[END_REF] for more details). It is based on a blending of the LRR and SSG redistribution terms, a classical isotropic dissipation term, a simple or generalized gradient diffusion term and the BSL Menter ω-scale transport equation (without SST correction on the eddy viscosity).

SAS model Standard model

The Scale Adaptive Simulation (SAS) approach of Menter [START_REF] Menter | A scale-adaptive simulation model using two-equation models[END_REF][START_REF] Menter | Revisiting the turbulent scale equation[END_REF][START_REF] Menter | The scale-adaptive simulation method for unsteady turbulent flow predictions. part 1: Theory and model description[END_REF] can be considered of an improved URANS model which mimics the behavior of hybrid RANS/LES methods: when the flow is attached, the standard RANS solution is obtained; when the flow is separated, an extra term in the scale equation becomes active and reduces the eddy viscosity, which allows capturing large turbulent eddies and resolve part of the turbulence spectrum.

The standard SAS approach of Menter associated to the SST kω model is here presented. The k transport equation is the same as before. The ω transport equation is slightly modified with the addition of a Q SAS term on the right side (source term). This term is responsible for the decrease of the eddy viscosity in case of occurrence of flow instabilities. It has the following expression:

∂ ρω ∂t + u j ∂ ω ∂ x j = γ µ t P k + Q SAS -β ρω 2 + ∂ ∂ x i (µ + σ ω µ t ) ∂ ω ∂ x i (10)
+ 2 (1 -F 1 ) σ ω2 ρ ω ∂ k ∂ x j ∂ ω ∂ x j
The Q SAS term, which is responsible for the decrease of eddy viscosity in case of flow instabilities, has the following expression:

Q SAS = max ρζ 2 κS 2 L L vK 2 (11) -C 2ρk σ Φ max 1 k 2 ∂ k ∂ x j ∂ k ∂ x j , 1 ω 2 ∂ ω ∂ x j ∂ ω ∂ x j , 0
with the following constants: ζ 2 = 3.51, σ Φ = 2 3 and C = 2. In the Q SAS expression, L is the length scale deduced from the turbulent scales transported by the model

L = k 3 2 ε = √ k C 1 4 µ ω
.

L vK is a 3D generalization of the von Kármán length scale expression coming from the boundary layer and defined by L vK = κS ||∆U|| where S is the second invariant of the strain rate tensor: S = 2S i j S i j and the denominator is the norm of the Laplacian of the velocity vector:

||∆U|| = ∂ 2 U i ∂ x j 2 ∂ 2 U i ∂ x j 2 .
Small scales limiter To limit the classical phenomenon of energy accumulation at small turbulent scales, Menter et al. introduce lower bounds on the von Kármán length scale, which are function of the cell size ∆:

L vK new = max L vK , L min vK (12)
There can be of two types: ¯Smagorinsky limiter with C S = 0.11 and ∆ = 3 √ Ω CV where Ω CV is the cell volumne.

L min vK = C S ζ 2 κ β C µ -γ ∆ (13)
¯WALE-LES limiter of Nicoud and Ducros [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF], with

C W = 0.4, S W = 1 √ 2 4S d i j S d i j 3 2 S 5 + 4S d i j S d i j 5 4
and S d i j = S ik S k j +Ω ik Ω k j -1 3 δ i j (S mn S mn -Ω mn Ω mn ).

L min vK = C W ζ 2 κ β C µ -γ - S W S ∆ (14)
αL correction It can be shown that in the original SAS expression of Menter, the von Kármán length scale goes to infinity in the center of a 2D mixing layer. This implies that the Q SAS term is zero in a region close to the mixing layer center. This unwanted feature prevents or causes delay to the development of mixing layer instabilities (Kelvin-Helmholtz). A correction was introduced by Benyoucef et al. [START_REF] Benyoucef | SAS-SST model assessment and improvement[END_REF] to limit the upper bound of the von Kármán length scale:

L vK = min (L vK , F 1 L vK + [1 -F 1 ] αL) ( 15
)
where L is the large turbulent length scale, α is equal to 2.5, F 1 is the Menter's kω model blending function. The blending is present here to prevent this new expression to be active in the inner boundary layer and ensure a RANS behavior in the attached boundary layer.

Extension to DRSM models

As mentioned before, the SAS approach can be generalized to any RANS turbulence model. It was thus extended to the DRSM-ω models already implemented in elsA code. The implementation is thus straightforward as only the Q SAS term is added to the ω-scale transport equation. Using DRSM-SAS approach ensures that both the modelized and the resolved part of the turbulence are well captured, on the contrary to SAS-SST approach where the modelized part of turbulence is isotropic. Moreover the DRSM are more sensible to flow instabilities than standard eddy viscosity models, which ensures a better development of the scale resolving mode when coupled to the SAS approach. The DRSM-SAS was successfully used on a heated square cylinder in a crossflow as shown by Didorally et al. [START_REF] Didorally | Numerical simulation of the unsteady flow past a heated square cylinder[END_REF]. It exhibits very close behavior to ZDES simulations in terms of turbulence spectrum, or mean and rms velocity and temperature profiles.

Turbulence model summary

The present study aims at evaluating the SAS approach coupled with two different RANS turbulence models (SST and DRSM). For the SST model, two meshes are used and for the finest grid, the IGV wakes can be taken into account through the inlet boundary condition (the rotating cartography). Thus, the number of simulations is significant. The name of simulation is normalized according to the different options used. The table 3 summaries the simulations performed in this study.

RESULTS

Unsteady flow in blade passage

The instantaneous entropy variation field at four axial locations (22.3% x/c, 31% x/c, 46% x/c and 96% x/c) for all simulations is depicted in figure 6. In figure 6(n), there is not only the tip leakage vortex but also the IGV tip vortex which can be observed at 22.3%, 31% and 46% x/c planes. The analysis of the entropy variation field shows that the switch from URANS to SAS approach with both SST and DRSM turbulence model, is responsible for a decrease of losses near the casing. The vortices seem to be more numerous but also more coherent, especially with the DRSM model. The effect of SAS is not visible at the leading edge as flow topology is very similar between URANS and SAS. However, significant discrepancies are observed from 46% x/c. This is partly due the tip leakage vortex dissipation across the weak shock as shown by Riéra et al. [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF]. It should be noticed that the rebound effect linked to the induced tip vortex is better captured with SAS approach, especially with DRSM turbulence model. Moreover, secondary vortices can be observed with these approaches, in particular in the vicinity of the trailing edge while all URANS-SST simulations are unable to capture them. The only discrepancy between M1-URANS-SST-wlin and M1-URANS-DRSM-wlin on one hand, between M1-SAS-SSTwlin and M1-SAS-DRSM-wlin on the other hand, is the use of the DRSM turbulence model instead of the SST one. It should be noticed that the suffix "-wlin" refers to the linear extrapolation for the ω value at the wall. Without this suffix, the pseudo roughness condition is used. Since the generation of the tip leakage vortex due to pressure difference between pressure and suction sides leads to a shear layer at the blade tip, there is a discrepency between DRSM and SST models. The roll-up is quite different and less stable with DRSM. At 31% x/c, the tip leakage vortex is less spread. After the weak shock located just downstream the last mentioned plane, the dissipation is very different between the two RANS turbulence models. The tip leakage vortex is still coherent with DRSM. With SAS approach, the differences are more significant as the shear layer is more unstable leading to a more fluctuating vortex. The dissipation reduction and the fluctuating tip vortex lead to decreased losses (both area and levels). The comparison between M1-URANS-SST and M1-URANS-SST-wlin shows that the ω value at the wall based on linear extrapolation leads to a small reduction of entropy i.e. of losses in the tip region. The high loss area linked to the tip leakage vortex has both smaller level and extent. With SAS approach, this is also observed, except in the last plane where the losses are higher with linear extrapolation. This is due to a combination of boundary condition and turbulent viscosity destruction with SAS. The comparison of mesh M1 and M2 shows that, for URANS, the mesh density has quite no significant effect on entropy generation. For SAS, some discrepancies appear from the 46% x/c plane, especially in the tip shear layer. The amplitudes of the ω value effect and of mesh size are at least one order of magnitude smaller than the other modeling parameters. The comparison between all DRSM results obtained on both M1 and M2 meshes shows that the mesh has quite no effect on URANS predictions while the entropy field predicted by SAS approach is reduced with the finest mesh, especially in the tip leakage vortex core and in the vicinity of the casing. The combination of the DRSM model with the SAS approach allows the generation of a consequent content of fluctuations near the tip without any strong dissipation of the tip leakage vortex. Far from the IGV wakes, the tip leakage vortex is very similar for URANS and SAS approaches at plane 22.3% x/c and 31% x/c even if the rebound effect is slightly differently predicted. At plane 91% x/c, the SAS simulation highlights a second vortex at mid-passage which is not observed as well without IGV wakes and vortices. Moreover, in this plane, the entropy level is smaller meaning that the losses are smaller with incoming wakes and vortices. Figures 6(i) and 6(j) depict the instant where there is no significant interaction between the tip leakage vortex and the IGV wakes and vortices. An another instant is depicted in figures 6(m) and 6(n) in order to highlight this interaction, especially at the 46% x/c plane. This significantly impacts the tip leakage vortex topology, especially with SAS approach. As previously observed by Riéra et al. [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF], this interaction is responsible for the tip leakage vortex flutter leading to the breakdown of the tip leakage vortex into small coherent structures. This interaction modifies also the loss area (radial and azimuthal extents, loss level). Losses are reduced thanks to the interaction with IGV wakes and vortices.

Time-averaged flow in blade passage

Figure 7 and 9 compare the experimental data to numerical results in terms of axial and circumferential velocities and standard deviations of these quantities in two axial locations. The first one is located at 21.7% chord and 98% of relative height (h/H) while the second one matches the 26A plane where compressor performances are measured. The radial location of the second plane is chosen in order to highlight the presence of tip leakage vortex. In each figure, the different numerical results highlight the influence of grid density (meshes "M1" vs. "M2"), of ω value at the wall (ω wall , 'roughness' vs. 'linear extrapolation'), of RANS turbulence model ('SST' vs. 'DRSM'), of SAS model ('URANS' vs. 'SAS') and of IGV wake and tip vortices.

In the first plane (21.7% chord), the mesh size, the value of ω wall and RANS turbulence model have no significant effect as no velocity fluctuations are observed. These fluctuations can be seen as root mean square value of velocity components. It should be noticed that the SAS modeling generates small fluctuations. This is still better than URANS modeling. Nonetheless, this Thus, this seems that some axial perturbations are not taken into account in the simulations. A part of the underprediction can be due to the absence of the downstream stator row.

In terms of time-average of velocity components, the agreement with experimental data is quite good, especially for the circumferential velocity. A deficit of axial velocity is not well captured. This discrepancy can be explained by the average used to compute the time-averaged data. Numerically, as the simulation is performed in the rotor frame, a simple time average is performed. Experimentally, in order to be in the same relative position to the rotor while the LDA measurements are performed in absolute frame, the phase average is also used. This phase average implies that the IGV wakes and vortices are not azimuthally averaged. On the contrary, as the time-average of numerical results is performed on 20 IGV periods, the IGV wakes and vortices can not be directly observed. Only the influence on the velocity field around the rotor blade can be captured.

Time-averaged flow downstream of the rotor

The axial and circumferential velocities and standard deviations of these quantities in plane 26A (performance measurements plane) are plotted in figure 9.

The profile of the circumferential velocity is well predicted by all simulations in terms of location and width of the rotor wake. A small discrepancy is observed on the amplitude of the peak within the wake which is over-predicted by CFD. There are two exceptions. The first one concerns the URANS-DRSM, for both M1 and M2 meshes, which does not predict the good shape of the velocity profile as the peak is not well located and gradients are not similar to the experimental profile. Indeed, experimentally, the gradient is stronger at the smallest azimuthal edge of the peak (around 5 • -5.5 • ) than at the other edge (around 5.5 • -7 • ). The opposite behavior is observed for URANS-DRSM simulations, despite the discrepancy is reduced with the finest mesh M2. The SAS-DRSM with mesh M1 is the second exception. Despite the fact that the shape of the velocity profile is well captured, the peak level is under-predicted by this simulation. On the mesh M2, the circumferential velocity profile is quite well captured by SAS-DRSM as the wake is only a little bit thinner than the experimental profile.

The experimental profile of axial velocity is similar to the circumferential velocity except that there is a deficit of axial ve-locity within the wake instead of an increase of azimuthal velocity. As for the other velocity component, the shape of the axial velocity profile is well predicted outside and within the wake. The velocity deficit is well located by simulations. Nonetheless, the width of the peak relative to the velocity deficit is underpredicted by CFD. As for the the circumferential velocity, for both meshes, the URANS-DRSM fails correctly predicting the shape of the axial velocity profile (location of the peak and gradients) while SAS-DRSM underestimates the velocity deficit. It can be noticed that the finest mesh M2 leads to a thinner wake, especially with the SAS approach. The comparison between the resultats obtained with both meshes (URANS-SST and SAS-SST) shows that there is a very small impact on the velocity profile. On the contrary, the velocity shape is significantly sensitive to the presence of IGV wake and vortices in the simulation as the width is reduced with these rotating secondary flows at the inlet.

The rms value of circumferential velocity represented by the standard deviation of this quantity is not well predicted in terms of level by any simulation. With the first mesh "M1", some fluctuations are captured only if the SAS model is used, especially with the DRSM turbulence model. For these simulations, the location of the peak is well reproduced. A similar behavior is obtained with SAS-SST or SAS-DRSM and the second mesh "M2". Nonetheless, the width of the peak is too small. The inclusion of IGV wakes and vortices allows the prediction of a wider peak. All URANS simulations fail predicting the shape of standard deviation profile. With the second mesh "M2" and IGV vortices, the URANS-SST-wake is able to capture some fluctuations. However, the shape of the profile is far from being well reproduced.

The standard deviation of axial velocity is shown in figure 9(c). The experimental profile is characterized by a double peak in the rotor wake at 5.25 • and 6.25 • . This double peak shape is well captured by all SAS simulations (SST or DRSM models). It should be noticed that the simulation M2-SAS-DRSM predicts a second peak too close to the first one. This is linked to the thinner wake as observed on axial velocity. Despite the good agreement in double peak locations, the fluctuation level and the double peak width are not well predicted. This can be due to the absence of the interaction of the first rotor and the first stator located downstream as the computational domain includes only the rotor and IGV wakes are taken into account through the inlet boundary condition. Except with the mesh "M2" and the IGV vortices, all URANS simulations fail correctly predicting the axial velocity fluctuations.

Time-averaged performances

The overall performances of the first rotor, computed between the experimental planes 25A (upstream of the rotor as shown in figure 1) and 26A (downstream) are plotted in figure 10. These figures depict the evolution of stagnation pressure ratio (figure 10(a)) and isentropic efficiency (figure 10(b)) against the massflow. In the present study, only the nominal point is computed. Three nominal isospeeds predicted by RANS simulations are also plotted in order to show the trend of the evolution of stagnation pressure ratio and isentropic efficiency with respect to massflow. The main discrepancy of stagnation pressure ratio between experimental data and numerical results is due to the choice of the operating point. The static pressure specified at the hub was set in order to have the best agreement with the experimental axial momentum, especially in the tip region as shown in figure 8. This outlet boundary condition allows a consistent comparison to experimental data near the tip, especially for LDA measurements. Considering the evolution ot the stagnation pressure ratio against massflow, the definition of the operating point seems to be responsible for an error about 0.75% for the stagnation pressure ratio and 1% for the massflow.

Except the SAS-DRSM simulation, all computations pre-dict the stagnation pressure with an error less than 1%. With another operating point and the trend given by RANS-SST isospeed, this discrepancy could be reduced up to 0.5% for both stagnation pressure ratio and massflow. The mesh density has a small impact on performances as stagnation pressure ratio and massflow obtained by M1-URANS-SST and M2-URANS-SST are very similar. The mesh effect is quite identical for SAS-SST model. For DRSM computations, the mesh has a small influence with URANS approach, mainly on massflow (reduction with the finest mesh). The combination with SAS is much more sensitive to the mesh density. With the finest mesh, the stagnation pressure ratio is significantly reduced and a discrepancy close to 2.5% is observed on this ratio (2% for the first mesh M1). The value of ω at the wall has a significant impact on performances as stagnation pressure ratio is higher with the linear extrapolation, for both URANS and SAS approaches. On the finest mesh "M2", the influence of the IGV wakes and vortices is not similar between URANS and SAS. For URANS, the stagnation pressure is decreased while there is an increase for SAS model. It should be noticed that the trend observed with SAS is similar to those of Zonal Detached Eddy Simulation as shown by Riéra et al. [START_REF] Riéra | Inlet condition effects on the tip clearance flow with Zonal Detached Eddy Simulation[END_REF].

The main discrepancy between experimental data and numerical results is obtained with SAS-DRSM approach for the mesh "M2". The stagnation pressure ratio is significantly underpredicted for both meshes with SAS-DRSM. This can be due to an overestimation of separation and blockage, especially near the hub corner. The isentropic efficiency is depicted in figure 10(b). As only the first rotor of the 3.5 stage compressor is simulated, there is no experimental data. Thus this is not possible to highlight the wrong prediction of an approach. Nevertheless, the comparison between all numerical results shows that there is a significant sensitivity of the isentropic efficiency to the following parameters: ¯turbulence modeling as SAS approaches predict smaller isentropic efficiency than URANS up to 0.75 point for SST model (M2-URANS-SST-wake vs. M2-SAS-SST-wake) and 1.5 points for DRSM model (M2-URANS-DRSM-wlin vs. M2-SAS-DRSM-wlin). Moreover the RANS turbulence model (DRSM vs. SST) influences the prediction of isentropic efficiency with a difference about 0.5 point for both URANS and SAS approaches on mesh M1. ¯ω value at wall as the linear extrapolation from the first cell center leads to an isentropic efficiency increased by 0.3 point. This effect is compensated by the effect of RANS model between results on mesh M2 with URANS approach (M2-URANS-SST vs M2-URANS-DRSM-wlin). On contrary, with SAS approach, the compensation is not sufficient and the combination with DRSM model leads to a decrease of 1 point of efficiency (M2-SAS-SST vs M2-SAS-DRSMwlin). ¯the IGV wakes and vortices, especially for SAS approach coupled with SST model, with a difference of 0.5 point (M2-SAS-SST vs. M2-SAS-SST-wake). The URANS approach is less sensitive to the incoming wakes and vortices as the difference is only of 0.2 point. ¯mesh density at least for URANS simulations with a discrepancy of 0.2 point. The mesh effect is amplified with DRSM model and SAS approach (0.75 point).

In the experimental plane 26A, the time-averaged numerical data are azimuthally averaged in order to obtain the radial profile of some quantities. In figure 11 are depicted the radial profiles of absolute deviation angle, of absolute stagnation pressure and of absolute stagnation temperature.

The comparison between experimental and numerical data shows that there is a good agreement for the absolute deviation angle, except near the hub and near the casing. The discrepancy between all numerical results is about 1 • which is the order of magnitude of experimental uncertainty. Near the hub, similar levels are obtained by all simulations. The discrepancy with experimental data can be due to both (i) a misprediction of the hub corner flow as a small hub corner separation is predicted leading to a decrease of the absolute deviation angle and (ii) the leakage flow generated by the gap between fixed and rotating wall at the hub [START_REF] Marty | Interaction of shrouded stator flow and main flow and its influence on performances of a three-stage high pressure compressor[END_REF]. At the casing, the discrepancy of prediction of tip leakage flow, especially the vortices, leads to a significant difference in terms of level and radial gradient. The highest dis-crepancy concerns the M1-SAS-DRSM-wlin simulation with an underestimation about 4 • . The analysis of the different modeling parameters (mesh density, ω value at the wall, RANS modeling, SAS modeling and IGV wakes) shows that: ¯the value of ω at the wall has no influence on absolute deviation angle. ¯the mesh refinement does not modify significantly the radial profile of this angle. ¯the IGV wakes and vortices impact significantly the prediction of the tip leakage flow. It should be noticed that the gradient near the casing is well captured with URANS approach. ¯the use of DRSM model instead of SST modifies strongly the radial profile, especially when combined with the SAS approach especially in the vicinity of the tip leakage flow.

The radial profile of absolute stagnation pressure in plane 26A is shown in figure 11(b). As the absolute deviation angle, the agreement with experimental data is globally good for all computations except the M1-SAS-DRSM-wlin and M2-SAS-DRSMwlin ones. With this turbulence modeling, the absolute stagnation pressure is significantly underpredicted along the whole span, with the highest discrepancy between 15% and 60% of the vane height (h/H). Thus the underestimation of stagnation pressure ratio (figure 10(a)) is not only due to an amplified hub corner separation but also to an amplified stagnation pressure loss outside the influence area of endwall. This underprediction is increased with the finest mesh M2. The comparison between all other simulations shows that the main discrepancy is due to the choice between SAS and URANS approaches. The effects of mesh density and ω value at the wall are smaller than the switch from URANS to SAS approach i.e. two families of nu-merical results are obtained: SAS on one hand, URANS one the other hand. For each family, the discrepancy between the predictions is smaller than the difference between the two experimental data sets (pneumatic probes vs. time-averaged unsteady probes). With the DRSM model, the stagnation pressure within the tip leakage flow is significantly overpredicted (close to the 85% relative height), especially with the mesh M2. It should be noticed that the underprediction of the stagnation pressure near the hub is due to the absence of the leakage flow itself due to the axial gap between rotating and fixed endwall. The analysis of the radial profile between 80% and 90% h/H shows that the IGV wakes impact significantly the prediction of stagnation pressure as the gradient prediction is better when these wakes are taken into account through the inlet boundary condition.

Figure 11(c) depicts the radial profile of absolute stagnation temperature. The agreement with experimental data is good as the maximum discrepancy is about 0.9%. The comparison between all numerical results shows that the SAS approach modifies strongly the profile near the casing, with a discrepancy of 0.75%. The modification of the radial profile shape is one of the explanation of the difference in terms of isentropic efficiency. The other variants of flow modeling (mesh density, turbulence modeling) have a very small impact on stagnation temperature prediction. For these secondary influence parameters, the highest effect is obtained with DRSM model which predicts a smaller stagnation temperature.

Spectral analysis

In the following, all power spectral densities (PSD) are computed with the Welch method [START_REF] Welch | The Use of Fast Fourier Transform for the Estimation of Power Spectra : A Method Based on Time Averaging Over Short, Modified Periodograms[END_REF] using an overlap of 50% and at least 10 Hann windows with a linear mean for each. The PSD of the static pressure signal recorded at the casing in plane 26A is plotted in figure 12. All simulations are depicted and the two signals obtained by Riéra et al. [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF] are also added as references ("M2-URANS-SA-wake" and "M2-ZDES-SA-wake", the last one being obtained with Zonal Detached Eddy Simulation based on Spalart-Allmaras turbulence model (noted SA) [START_REF] Spalart | A One-Equation Turbulence Model for Aerodynamic Flows[END_REF]). First of all, the frequency resolution is 2000 Hz for the numerical data set arising from a temporal range of 20 IGV periods (even if the IGV is not taken into account in the simulation). The experimental data (frequency of 2 Hz) is resampled in order to have the same resolution frequency as numerical results and to perform consistent comparison as explained in [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF].

For the first mesh, as there is no rotor-stator interaction (no IGV wake and absence of first stator in computational domain), no specific frequency is observed. The only exceptions are M1-URANS-SST and M1-URANS-SST-wlin for which there is quite no fluctuations as the PSD is very close to zero and a specific frequency and its harmonics arises from the PSD. This frequency is clearly articifial and has no physical meaning. Switching to DRSM turbulence model, the fluctuations of static pressure increases leading to a higher PSD on the whole frequency range. For both SST and DRSM turbulence models, the ω value at the wall has no significant impact on PSD, especially with SAS approach. Switching from URANS to SAS approach, for both SST and DRSM turbulence models, the fluctuations amplitude are increased up to 2.5 orders of magnitude (for URANS approach). Nonetheless, the PSD is still far from the experimental data.

Considering all PSD obtained with the second mesh "M2", three main effects are highlighted: ¯the IGV wakes. Without any IGV wakes and vortices, the blade passing frequency of IGV (BPF) and its harmonics cannot be captured. There is only a broadband spectral content with a discrepancy of 5 orders of magntitude. With the IGV wakes and vortices specified at the inlet boundary condition, the fluctuations are higher leading to a higher PSD, especially at the lowest frequencies. Moreover, the BPF and harmonics can be observed. Despite the good agreement for the first three harmonics of the PBF, the cutoff frequency arises too early. It should be noticed that SST and SA turbulence models lead to similar PSDs (URANS approach). In terms of physical mechanisms, the IGV tip vortex is responsible for the flutter of the rotor tip leakage vortex. Due to the fact that the rotor blade count is twice the IGV one, the flutter appears at a specific frequency linked to BPF. In compressor with prime numbers, this frequency is not directly linked to the compressor rotational speed but to rotor-stator interaction (Tyler-Sofrin mode). ¯the URANS vs. SAS approach. Switching from URANS to SAS approach, the PSD level without IGV is increased and the broadband spectral content is closer to experimental data. With IGV wakes, the broadband spectral content is also amplified and the BPF and some of its harmonics are captured. It should be noticed that the switch from URANS to SAS approach delays the frequency where the numerical results become far from experimental data and the PSD gradient is not captured. ¯the switch from SST to DRSM turbulence model increases the broadband spectral content on the whole frequency range, by three orders of magnitude. With SAS approach, the PSD is close to the result of the reference PSD obtained by Riéra et al. [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF]. Nevertheless the BPF and its harmonics are not captured by M2-SAS-DRSM-wlin simulation as the IGV wakes are not taken into account in the simulation.

The comparison to the reference PSD obtained by Riéra et al. [START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF] shows that the SAS approach improves certainly the prediction of static pressure PSD by comparison to URANS method, especially at low and medium frequencies where the SAS is very close to ZDES. With SST turbulence model and with the IGV wakes, a good agreement is obtained up to the fifth harmonics of IGV BPF (third hamonics for URANS approach). This can be explained by the cutoff effect of modeled eddy viscosity on high frequencies. With the DRSM turbulence model, the agreement is even better at high frequency range. A better agreement should be expected with the IGV wakes at the compu- tational domain inlet. The comparison of the two meshes shows that with the first mesh, the SAS-DRSM technique is able to capture the same PSD as the second mesh up to the ninth harmonics (on the same mesh, the discrepancy between ZDES and SAS-DRSM arises at the 20 th harmonics) while the first mesh is ten times smaller (4.3 million points vs 44 million points per blade passage). This means also that the PSD prediction could be im-proved by using a more elaborated RANS model (DRSM) in the ZDES framework [START_REF] Uribe | Zonal Detached Eddy Simulation extension to kω models[END_REF][START_REF] Uribe | Evaluation of Zonal Detached Eddy Simulation kω and Scale Adaptive Simulation methods on a realistic rotor of a highpressure compressor[END_REF].

CONCLUSION

The prediction of secondary flows (corner separation and tip leakage flow) and of the compressor performances (compression ratio, efficiency, operating range) is influenced by the turbulence modeling. The aims of the present study are (i) the validation of the combination of the SAS approach with the DRSM turbulence model by comparison to experimental data, especially to laser measurements in the tip of a rotor of a high pressure compressor and (ii) the discussion of the flow prediction improvements with respect to turbulence approaches classically used in CFD and industry: URANS simulations and standard SAS simulation i.e. combined with SST turbulence model. On one side, the SAS approach can be viewed as an improved URANS one as the turbulent viscosity is reduced allowing the appearance of instability, especially in shear layer. For this reason, one mesh adapted to RANS simulation is used. On another side, this approach acts with the same idea of Detached Eddy Simulation: the reduction of eddy viscosity in detached flow region. Thus a second mesh adapted to Zonal Detached Eddy Simulation was used, allowing to highlight the mesh effect on URANS and SAS approaches. The RANS turbulence model plays also a significant role in URANS prediction. Thus two turbulence models are assessed both with URANS and SAS simulations: kω SST model and a DRSM-SSG/LRR-ω model.

The analysis of entropy field and the comparison to experimental data (LDA measurements, performances) show that the tip leakage vortex predicted by SAS and/or DRSM is significantly different from URANS-SST simulation. The vortex dissipation through the weak shock is strongly reduced. The tip leakage vortex is still observed in the vicinity of the trailing edge. Only the simulations with IGV wakes predict the velocity fluctuations near tip gap, from the leading edge. Except the combination SAS with DRSM model and with the first mesh, all simulations are able to capture the time-average velocity in the vicinity of the tip. The prediction of SAS-DRSM is clearly improved by using a fine mesh. Concerning the time-averaged performances, the stagnation pressure losses are slightly overestimated by SAS. This is due to an amplification of the hub corner separation. The combination with DRSM model increases the pressure stagnation losses. Moreover, the isentropic efficiency is very sensitive to the SAS approach and to the turbulence model. The spectral analysis shows that the prediction of the amplitude and frequencies of the power spectral density of static pressure is improved using the SAS approach instead of URANS one, especially with the DRSM turbulence model. The SAS approach leads to PSD similar to ZDES, especially with the DRSM model. Thus, the SAS-DRSM is able to well predict the tip leakage flow with the fine mesh. Nevertheless, this approach amplifies the hub corner separation leading to a strong underestimation of overall performances. Future work will concern the simulation of SAS-DRSM with incoming wakes and further analysis at design point, especially in the flow field and at near surge point in order to better understand the effect of turbulence modeling. Moreover, the comparison between SAS and ZDES will be pursued. The present configuration is quite simple with one rotor subjected to incoming wakes and vortices. The next step is to extend the present study to real rotor-stator interactions by including the downstream rows in the computational domain. Then, technological effects such as axial gap (and seal teeth), fillets will be added to the computational domain.

ACKNOWLEDGMENT

All simulations presented in this article have been funded in the framework of the elsA three-party agreement between AIR-BUS, SAFRAN and ONERA which are co-owners of this software. The support of the different partners for the research compressor CREATE program is fully acknowledged. The authors thank gratefully X. Ottavy (LMFA) for providing the experimental data.

FIGURE 1 .

 1 FIGURE 1. CREATE COMPRESSOR MERIDIAN VIEW

FIGURE 2 .

 2 FIGURE 2. Computational domain

FIGURE 5 .

 5 FIGURE 5. Instantaneous entropy variation field at several axial sections, from 25A (foreground) to 26A sections (background) for the simulation M2-SAS-DRSM-wlin

FIGURE 3 .

 3 FIGURE 3. Computational domain and mesh over hub and blade walls (one point over two in all mesh directions)

FIGURE 4 .

 4 FIGURE 4. IGV wakes and vortices at section 25A (simulation M2-SAS-DRSM-wlin)

FIGURE 6 .

 6 FIGURE 6. INSTANTANEOUS ENTROPY VARIATION FIELD AT 22.3% X/C, 31% X/C, 46% X/C AND 96% X/C PLANES (FROM TOP TO BOTTOM).

FIGURE 8 .FIGURE 7 .

 87 FIGURE 8. TIME-AVERAGED AXIAL MOMENTUM UP-STREAM OF THE COMPRESSOR ROTOR.

FIGURE 9 .

 9 FIGURE 9. TIME-AVERAGED VELOCITY FIELD NEAR THE CASING AT 26A PLANE.

  (a) Stagnation pressure ratio (b) Isentropic efficiency

FIGURE 10 .

 10 FIGURE 10. TIME-AVERAGED OVERALL PERFORMANCES OF THE COMPRESSOR ROTOR.

  (a) Absolute deviation angle (b) Absolute stagnation pressure (c) Absolute stagnation temperature

FIGURE 11 .

 11 FIGURE 11. TIME-AVERAGED RADIAL PROFILES IN 26A PLANE.

  (a) All meshes, without IGV wakes (b) Only mesh M2, wake effect and comparison to ZDES-SA[START_REF] Riéra | Zonal detached-eddy simulation applied to the tipclearance flow in an axial compressor[END_REF]

FIGURE 12 .

 12 FIGURE 12. POWER SPECTRAL DENSITY OF STATIC PRESSURE IN PLANE 26A AND AT CASING.

TABLE 1 .

 1 BLADE NUMBER FOR THE ROWS

	Row	IGV R1 S1 R2 S2 R3 S3
	Blade number	32	64 96 80 112 80 128

TABLE 2 .

 2 CHARACTERISTICS AT DESIGN OPERATING POINT

	Outer casing diameter	0.52 m
	Rotating speed	11543 rpm
	Mass flow	12.7 kg.s -1

TABLE 3 .

 3 SUMMARY OF SIMULATIONS

	Name	Mesh SST vs. DRSM URANS vs. SAS	ω wall	IGV wakes
	M1-URANS-SST	M1	SST	URANS	pseudo roughness
	M1-SAS-SST	M1	SST	SAS	pseudo roughness
	M1-URANS-SST-wlin	M1	SST	URANS	linear extrapolation
	M1-SAS-SST-wlin	M1	SST	SAS	linear extrapolation
	M1-URANS-DRSM-wlin	M1	DRSM	URANS	linear extrapolation
	M1-SAS-DRSM-wlin	M1	DRSM	SAS	linear extrapolation
	M2-URANS-SST	M2	SST	URANS	pseudo roughness
	M2-SAS-SST	M2	SST	SAS	pseudo roughness
	M2-URANS-SST-wake	M2	SST	URANS	pseudo roughness
	M2-SAS-SST-wake	M2	SST	SAS	pseudo roughness
	M2-URANS-DRSM-wlin	M2	DRSM	URANS	linear extrapolation
	M2-SAS-DRSM-wlin	M2	DRSM	SAS	linear extrapolation


domain: spi phys

text:
C Stephan

P Lubrina
email: pascal.lubrina@onera.fr

J Sinske

Y Govers
email: yves.govers@dlr.de

N Lastère

AIRBUS BELUGA XL STATE-OF-THE-ART TECHNIQUES TO PERFORM A GROUND VIBRATION TEST CAMPAIGN OF A LARGE AIRCRAFT

Keywords: Ground Vibration Testing, structural nonlinearities, modal identification, Phase Separation Method

After the completion of the Airbus A350 XWB 900 Ground Vibration Test (GVT) in 2013 and the A320 NEO GVT in July 2014, the ONERA-DLR joined team performed the GVT of the new BelugaXL aircraft in May-June 2018. The test was completed in 8 measurement days for 2 mass configurations, thanks to efficient methods and tools. The originality of this structure comes from its very large fuselage and its rear section with two large additional vertical planes, which made the modal identification particularly challenging. This paper describes the process followed, the methods used, and the interactions with engineers in charge of aeroelastic computations during the test, and how they allowed the success of this campaign.

INTRODUCTION

The joint team from the French aerospace laboratory ONERA and the German Aerospace Center (DLR) performed the Ground Vibration Test of the new Airbus Beluga XL large transport aircraft in May-June 2018. As for previous tests like on the Airbus A350 XWB 900 in 2013 and the Airbus A320 NEO with Pratt & Whitney engines in 2014 [START_REF] Lubrina | Airbus A350 XWB GVT : state-of-art techniques to perform a faster and better GVT campaign[END_REF] [START_REF] Stephan | Airbus A320 NEO GVT and FEM updating: state-of-the art techniques to perform an industrial vibration test campaign and a rapid process to update renewed FEM clearance of first flight test[END_REF], the specifications of these tests were particularly oriented to the FEM (Finite Element Model) updating for aeroelastic computations. A main challenge of this BelugaXL aircraft GVT (Figure 1) was the excitation and the modal identification of the very large fuselage and the rear section with the two large additional vertical planes, called Auxiliary Fins, on both sides of the horizontal tail plane. To succeed in the completion of the ambitious technical demands, new modal exciters with very large coil strokes and seismic exciters were used. Recent improvements on developed excitation methods and post-processing tools were applied. This paper describes the test equipment, the processes followed and the methods used in this particularly hard context and how those contributed to the successful achievement of this challenging test campaign.

BELUGA XL DESCRIPTION

The BelugaXL program was launched in November 2014 to address Airbus' transport capacity requirements in view of the A350 XWB ramp-up and Single-Aisle production rate increases. Six aircraft will be built between 2019 and 2023 to gradually replace the five BelugaST. The aircraft will operate from 11 destinations as Airbus' method of transporting large aircraft components.

Based on an A330-200 Freighter, the BelugaXL is highly modified with the lowered cockpit, the cargo bay structure and the rear-end and tail, giving the aircraft its distinctive look (Figure 3). This new Airbus oversize transport aircraft is tailored to carry large airframe components such as twin wings of A350 or simultaneous fuselage sections of A350 with sections of Single-Aisle family. The BelugaXL cross section (more than 8m) is much bigger than those of all existing cargo aircraft. The BelugaXL is powered by Rolls Royce Trent 700 engines.

The first aircraft specimen performed its maiden mission from Toulouse (France) on the 19th of July, 2018. Both configurations were empty of fuel. The control surfaces were set in neutral position (direct law with no feedback).

IFASD-2019-041

For this campaign, the objectives were to obtain the modal characteristics of the complete aircraft and its control surfaces, in detail:  Resonance frequencies,  Generalized masses and damping ratios,  FRFs (Frequency Response Functions),  Structural nonlinear behavior. Experimental modal data are necessary for FEM updating, aeroelastic computations and finally certification. As tests were on the critical path of the program, the impact on planning was minimized thanks to an optimized workflow.

The frequency band was defined to be able to identify all modes of interest, from all rigid body modes to the first torsional modes of each lifting and control surface. The 6 rigid body modes of the aircraft were identified, to update the stiffness and the damping coefficient values of the soft pneumatic suspensions.

As the control surfaces and their associated actuators come from the well-known A330 Airbus aircraft, no measurement dedicated to actuator failure configurations was performed.

Equipment

More than 600 ICP accelerometers have been installed over the entire structure of the aircraft in order to be able to identify the global structural behavior together with local modes from control surfaces, the heavy dummy payload structures (HPJ), and also to the dimensions of the fuselage related breathing modes.

The sensors were connected with approximately 7000m cables and combined via special patch panels, then bundled in a multi-wire cable and routed to the measuring system. Roughly 300 accelerometers from a new development [START_REF] Govers | Latest Design Trends in Modal Accelerometers for Aircraft Ground Vibration Testing[END_REF] have been used during this test. These accelerometers can be rotated by 360° in their electrically isolated housings. This type of sensor was used at positions where surface normals differ significantly from global coordinate directions. In order to install and connect such a large number of sensors without mistakes in a short time, it requires a very good installation plan and a well-trained team. 33 exciter locations were used during the test campaign to properly excite all modes of interest. Furthermore, a well-structured measurement plan was needed to quickly switch from one point of excitation to the next, so that only short breaks between the individual measurement runs arise and the short test phase was used as efficiently as possible. In total 19 shakers were used during the test, of 5 different types (Figure 6).

For excitation on the engines, 1000N Prodera exciters were used; on most other structural parts (such as wings, control surfaces, etc.) 500N Prodera exciters were used. Besides, seismic exciters were used to excite the HPJs inside the fuselage.

A particular challenge of this GVT was the excitation on the top of the two big auxiliary-fins of the BelugaXL, which had to be simultaneously excited from two cranes in parallel. For this purpose, special shakers (modified APS 420) with a very large coil stroke of up to 15cm peak-peak were used for the first time during a GVT for Airbus. These shakers were mounted in two so called pendulum devices, which were brought into position with the cranes.

IFASD-2019-041

With the use of these shakers, significantly larger vibration amplitudes could be introduced into the structure and there were, despite the higher amplitudes, fewer aborts because of exceeding the maximum stroke.

The excitation forces were permanently measured by 2 means:

-force cells installed on each active driving point, -the coil currents at the active power amplifiers. The whole data acquisition system was based on ONERA's and DLR's combined LMS Scadas III frontends controlled by the Test.Lab software. Distributed data acquisition has been realized by placing 8 LMS Scadas III frontends around the aircraft. The frontends were optimally distributed under the aircraft so that the analogue signal routing was kept as short as possible in order to achieve a better signal to noise ratio. These frontends were connected by optical fiber cables to allow for data flow in a ring-shaped data bus. The V12-L acquisition modules inside the LMS Scadas III frontends were used due to their very low cut-off frequency of 0.05 Hz of the analogic high-pass filters. This very low high pass filter makes it possible to identify even the low rigid body modes of an aircraft on pneumatic suspensions, so that the boundary conditions can be accurately determined.

Teams

The work was organized in 2 shifts with 7 people per shift. A single team consisted of several positions:

-team manager -technicians for shaker handling -technicians and engineers for excitation control, data acquisition and checks -engineers for modal identification and model correlation The work rate was 6 days a week, from 7 a.m. to 12 p.m. during 8 days of measurement.

METHODS

For modal identification of aircraft structures during a GVT, there are two available methods, (a) the Phase Resonance Method (PRM) and (b) the Phase Separation Method (PSM). Since the test time of GVT has been reduced significantly in the past, the time consuming PRM has been skipped completely from GVT. To achieve similar quality from PSM where a large number of modes are excited in a single excitation run that are separated mathematically afterwards, extended PSM methods have been developed by DLR/ONERA. These methods include user-defined excitation signals based on swept-sine excitation, different levels of excitation to detect nonlinear behavior [START_REF] Lubrina | Airbus A350 XWB GVT : state-of-art techniques to perform a faster and better GVT campaign[END_REF][2], dedicated signal processing algorithms [START_REF] Gloth | Ground Vibration Testing of the AIRBUS A380-800[END_REF][5] and a database for modal model determination [START_REF] Stephan | Airbus A320 NEO GVT and FEM updating: state-of-the art techniques to perform an industrial vibration test campaign and a rapid process to update renewed FEM clearance of first flight test[END_REF].

Phase Separation Method

The excitation of the structure is performed for different exciter configurations at different locations and directions of the structure. The exciter signal can be varied [START_REF] Lubrina | Airbus A350 XWB GVT : state-of-art techniques to perform a faster and better GVT campaign[END_REF] in type, which means to modify the type of signal itself, typically random or swept-sine excitation and (2) force level, which means to raise the amplification factor. Generally random signals are used to get a quick overview about the structural dynamic characteristics of the structure. The energy level at each frequency line for random excitation is very low and therefore the displacement amplitude levels of the responding structure are not realistic for operating conditions. Based on the information from the quick overview, optimized excitation signals for swept-sine runs are determined to achieve much higher displacement levels depending on the force amplitude. Additionally limitations are respected in terms of response or force levels. To identify nonlinear behavior the input force level is increased to analyze the modal parameters over force and displacement level.

A typical data workflow is presented in Figure 8 from time domain data to frequency response function calculation, and then to modal analysis and modal model correlation. These steps are conducted with commercial software Test.Lab and in-house tools to address specific aspects in experimental modal analysis that are related to GVT.

IFASD-2019-041

A specialty of aircraft structures is the symmetry. If swept-sine signals are used on a symmetric structure, symmetric and antisymmetric modes can be excited separately. Therefore the modal density can be reduced in the output spectra and the excitation force levels can be increased in every symmetrical and antisymmetrical mode. To realize such excitation configurations, often two or even more shakers are used. In the following you have to deal with correlated input signals (symmetric or antisymmetric swept sine signals). Correlated input signals do not allow a direct estimation of the frequency response functions. The method called Single Virtual Driving Point (SVDP) described in [START_REF] Gloth | Ground Vibration Testing of the AIRBUS A380-800[END_REF] and [START_REF] Fuellekrug | Measurement of FRFs and Modal Identification in Case of Correlated Multi-Point Excitation[END_REF] allows for direct transformation referencing on a "virtual" driving point. This estimation method is commercially not available, but implemented in ONERA and DLR in-house tools. It was used to determine the FRFs from all multi-shaker swept-sine excitation runs.

Modal correlation

After modal analysis has been performed on a specific dataset obtained from an excitation run, the corresponding modal data and the corresponding FRFs are transferred into a dedicated SQL database. Due to the fact that specific modes can be identified multiple times from different excitation runs with slightly different properties, a need of systematic correlation among individual test data sets arises together with a need of the assessment of the accuracy of modal analysis results e.g. based on evaluation of specific quality criteria. Currently there is no software on the market to collect, filter, classify and post-process modal parameters. The DLR correlation tool has been developed to cope with this type of problem; see Figure 9 and Figure 10 from an example test case. means, it has been labelled to be a member of a family of modes that belong to a single mode in the modal model. There are several means to check if modes from a new dataset are already represented in the final modal model or if modes of a new type are contained in the current dataset. The Modal Assurance Criterion (MAC) gives a fast indication if two modes are linearly independent or collinear. In addition, a spike plot and an animated 3d plot are available to support the operator in the interpretation and verification of modal analysis results. This process has to be repeated for each modal dataset from each excitation run and for each mode contained in these datasets to ensure highest quality of the final modal model that is an equivalent mathematical model of the tested object.

Finally the DLR Correlation Tool provides a modal model containing a linear independent set of modes. Each mode in the modal model is the best candidate mode of a family of modes identified from multiple excitation runs and found to be similar in terms of the MAC criterion.

The family or a subset of the family can be used to analyze nonlinearity of modes or to determine the test uncertainty in terms of the scatter among the mode family members.

In the previous paragraph it is mentioned that the correlation tool provides the best mode of each mode family. This mode is the main mode in the final modal model and therefore delivered to the customer. The DLR/ONERA philosophy of establishing modal models in the Correlation Tool is to deliver modes preferably on high excitation levels which are indicated by high values of the parameter generalized force. Proceeding in this way, the delivered results are more realistic since operating conditions of the tested structures usually cause high amplitudes of deformation [START_REF] Lubrina | Airbus A350 XWB GVT : state-of-art techniques to perform a faster and better GVT campaign[END_REF][2].

Nevertheless, it may occur that sometimes modes that have been excited on a much lower level of energy show a better phase purity and reliability. The reason can be found in the linear modal analysis process to which the assumption of linearity and time-invariance of the test object apply. If these assumptions are not fulfilled e.g. with increasing excitation force level or increasing response amplitude level, the accuracy of the corresponding modal parameters will degrade and in such cases the selection of modes from lower excitation force levels can be meaningful.

Test data access and progress overview

In the present Airbus context, a GVT can only be efficient if test data are analyzed on site as fast as possible. In case of BelugaXL GVT, 3 people have been working on data processing described in Figure 8. Final modal data have been stored in a multi-user access database which is accessible for Airbus from DLR correlation tool to view and to compare modes with each other and with modes from the finite element model. Data can be exported in different formats e.g. universal file format (UFF) or Excel tables. Nonlinear modes can be discussed and actions can be taken to place additional excitations if necessary. For example, as the excitation of Auxiliary Fins was particularly important for the model updating, the decision from Airbus of adding runs with increased level of excitation was made by a constant monitoring of vibratory accelerations, in near-live time after each run performed.

The online access to the final data is as important as the information about the current and next excitation run which is performed. Since more than 300 excitation runs have been acquired, such information is shown via a tool called the ONERA test progress manager. Test progress transparency is a key enabler for interactive test management.

RESULTS

Modal model

A total number of 314 runs were made for modal identification of structural configurations C1 and C2. For both mass configurations, in total 2245 modes were identified. From this set of modes, 1017 were selected as reliable and were combined for providing the non-linearity plots. The final modal model consists of 100 master modes (one per family mode) for the 1 st structural configuration and 107 master modes for the 2 nd structural configuration.

During the BelugaXL test, all rigid body modes have been identified below 1 Hz for both configurations, and excited by random excitation. For each mode of a modal model, the following data are provided:  the mode shape (see Figure 12)  the eigenfrequency, damping ratio and modal mass at the highest level of excitation,  the nonlinearity plot (evolution of eigenfrequency and damping over generalized force).

IFASD-2019-041 In general, frequency decreases as a function of force, while damping increases, like the example on Figure 13. Observing this kind of asymptotic trend is auspicious for the manufacturer, because it means that a sufficient level of force was applied during the test. But of course, each mode can show a pattern of its own, because these evolutions are very casedependent. Obtaining these trends still a main challenge of GVT, and absolutely impossible to compute with an FE model at the present time. This explains why GVT is and will remain essential for certification. The auto-MACs of structural configurations C1 and C2 are depicted on Figure 14. The matrices are mainly diagonal, with few noticeable extra-diagonal terms. They are due to modes whose shapes are very similar; in general the movement of a sub-component enables the analyst to distinguish them.

Special case of HTP behavior

One of the main originalities of this aircraft comes from the presence of two large auxiliary fins on the HTP (Figure 15). Due to their sizes and masses, they have a deep influence on the dynamic behavior of the HTP, and hence on the whole aircraft dynamics. Furthermore, strong structural nonlinearities were expected for dedicated modes with dominant motion at the horizontal tail plane and the auxiliary fins.

IFASD-2019-041

Figure 15 : HTP view with the two auxiliary fins Therefore the structure was excited with extra-long coil stroke shakers at the auxiliary fins in order to highlight them as much as possible, with restrictions in acceleration and force limits. The shakers were positioned with pendulum devices from two cranes (Figure 16). This excitation was the most challenging one, because the symmetric and antisymmetric excitations from two shakers supported by pendulum devices were never performed before. It turned out that the whole system was very stable and all excitation runs could be performed without any problems. The maximum force at the excitation was about 500N at each shaker while the maximum coil stroke was at 80mm peak to peak.

Especially the HTP yaw mode shows strong nonlinear behavior. While crossing the eigenfrequency of the 1 st in-plane bending of the HTP, the mode shape changes. In fact the fin amplitudes change direction. This special behavior is clearly visible in Figure 17 where modes 1 to 3 are above the eigenfrequency of the 1 st in-plane bending mode, mode 4 is at the border and modes 5-7 are below that eigenfrequency.

IFASD-2019-041

Figure 1 :

 1 Figure 1 : Airbus BelugaXL during Ground Vibration Test Figure 2 : Right auxiliary fin

Figure 3 :

 3 Figure 3: BelugaXL aircraft

Figure 4 :

 4 Figure 4 : Main landing gear air spring suspension devices Figure 5: HPJ (orange structure) put in the fuselage

Figure 6 :

 6 Figure 6 : View of typical exciters used Left : APS exciter on top of LHS auxiliary fin, middle : Prodera exciter on right engine, right : ONERA seismic exciter for the HPJ

Figure 7 :

 7 Figure 7 : Inside the GVT command room

Figure 8 :Figure 9 :

 89 Figure 8 : Schematic data flow between TestLab and DLR/ONERA software

Figure 11 :

 11 Figure 11 : Airbus on-site access to database and correlation tool

Figure 12 :

 12 Figure 12 : Shape of the 3 nodes wing bending of structural configuration 2

Figure 13 :

 13 Figure 13 : Example of nonlinearity plot

Figure 14 :

 14 Figure 14 : Auto-MAC of structural configurations C1 (left part) and C2 (right part)

Figure 16 :

 16 Figure 16 : Auxiliary fins excitation

Figure 17 :

 17 Figure 17: HTP yaw mode (left) with Auto-MAC matrix (right)

The BelugaXL GVT was fulfilled with respect to all specifications and with all expected results delivered in required quality. Performing such a test relies on the maturity of methods, optimized workflow, and last but not the least, on the high experience of involved teams. Besides, for this GVT the collaboration between ONERA-DLR test providers and Airbus engineers was particularly fruitful; this explains much of the success of this campaign.

COPYRIGHT STATEMENT

The authors confirm that they, and/or their company or organization, hold copyright on all of the original material included in this paper. The authors also confirm that they have obtained permission, from the copyright holder of any third party material included in this paper, to publish it as part of their paper. The authors confirm that they give permission, or have obtained permission from the copyright holder of this paper, for the publication and distribution of this paper as part of the IFASD-2019 proceedings or as individual off-prints from the proceedings.