Gabriel Mondin
email: gabriel.mondin@ec-lyon.fr

William Riéra
email: william.riera@safrangroup.com

Pierre Duquesne
email: pierre.duquesne@ec-lyon.fr

Xavier Ottavy
email: xavier.ottavy@ec-lyon.fr

BENEFICIAL EFFECTS OF GUIDE FINS ON CORNER SEPARATION

Keywords: Corner separation control, guide fin, optimization, RANS, Kriging surrogate, experimental validation, physical mechanisms

Corner separation is a major phenomenon limiting the operability of aeronautical compressors. Passive control devices such as guide fins are envisioned to reduce its deleterious effects. In this paper, guide fins are optimized to reduce the total pressure losses downstream of a subsonic compressor cascade (M=0.11, Re=3.2 10 5 , modern stator blades) at design and near-stall incidences. Guide fins shapes are defined using 15 parameters. An optimization strategy relying on the iterative refinement of surrogate model with Reynolds-Averaged Navier-Stokes (RANS) computations is carried out. A database of 1033 guide fins is obtained, a large number of which strongly reduces the near-stall losses. Three guide fins are thoroughly investigated and referred to as Short Fence (short and straight guide fin), Long Fence (long and straight guide fin) and 3D (long guide fin with a strong pyramidal aspect). Their effect on the downstream measurement plane is validated experimentally, as well as their integrated performances. Large reductions of total pressure losses at near-stall incidence are obtained (-2.4 pt to -2.9 pt, or a relative reduction of -50% to -60%) without degrading neither the nominal losses nor the stator deflection. These guide fins are used to highlight three beneficial mechanisms that alter the corner separation development within the blade passage a) Tip Vortex (energizes the passage flow) b) Fence (induces 2 passage vortices) c) Guide (induces a local favourable pressure gradient). The role of these mechanisms in the generation of total pressure losses are analyzed in depth with streamlines visualisations and with a topological analysis. Kriging surrogate models reveal that three parameters drive the guide fins performances : the axial position, height, and thickness value at hub. The performances of the investigated guide fins are robust to significant variations of these parameters, as well as small variation of the inlet boundary layer thickness.

𝑈 ∞

Upstream velocity magnitude ω Total pressure loss coefficient ω 𝐸𝑊

Endwall losses Δω 𝐸𝑊

Endwall losses with a guide fin minus baseline endwall losses at the same incidence Δω 0 /Δω 4.9 Endwall loss variation at i=0° / i=4.9° when adding a guide fin compared to the baseline 𝑉 𝑥 ̃ Axial velocity divided by inlet axial velocity 𝑥/𝑦 Engine axis / pitchwise direction 𝑦 ̃

Pitchwise position in the passage along an isox line. 𝑦 ̃=0: suction side. Related to guide fin hub / tip profile 0D / MID Averaged on downstream plane / at midspan

INTRODUCTION

Corner separation is one of the most limiting flow features in terms of operability in modern aeronautical compressors. It corresponds to a three-dimensional separation occurring at the blade-endwall junctions. Corner separation originates from complex interactions of secondary flow structures [START_REF] Horlock | Wall Stall in Compressor Cascades[END_REF], [START_REF] Kang | Three Dimensional Flow in a Linear Compressor Cascade at Design Conditions[END_REF]. It is particularly enhanced by the blade-to-blade pressure gradient that exists between two consecutive blades, which causes the socalled passage flow in the near-endwall regions. Under the influence of the passage flow, the low momentum fluid of the endwall boundary layer is driven towards the suction side of the blades. This results in an accumulation of low momentum fluid yielding a separation in the corner region, hence the denomination corner separation. Corner separation can be characterized by the topology of skin friction line patterns [START_REF] Délery | Robert Legendre and Henri Werlé: Toward the Elucidation of Three-Dimensional Separation[END_REF]. Its magnitude is greatly influenced by the inflow incidence ( [START_REF] Taylor | Competing Three-Dimensional Mechanisms in Compressor Flows[END_REF], [START_REF] Dawkins | The Unsteady Topology of Corner Separations[END_REF]) and by the geometry of the passage itself. Several passive means of control are investigated in the literature to reduce corner separation. 3D stator blading techniques were notably developed ( [START_REF] Taylor | Competing Three-Dimensional Mechanisms in Compressor Flows[END_REF], [START_REF] Harvey | Some Effects of Non-Axisymmetric End Wall Profiling on Axial Flow Compressor Aerodynamics: Part II-Multi-Stage HPC CFD Study[END_REF]). Fillets and leading edge shapes can also significantly alter the corner separation development ( [START_REF] Goodhand | The Impact of Real Geometries on Three-Dimensional Separations in Compressors[END_REF], [START_REF] Sun | Corner separation control by optimizing blade end slots in a linear compressor cascade[END_REF]).

On top of these modifications on the stator geometry, several technological effects were investigated. Axisymmetric deformations of the endwall were studied with the objective of energizing the endwall boundary layer [START_REF] Denton | Loss Mechanisms in Turbomachines[END_REF]. Non-axisymmetric endwall deformations were also used to smoothly deflect the passage flow [START_REF] Reutter | Advanced Endwall Contouring for Loss Reduction and Outflow Homogenization for an Optimized Compressor Cascade[END_REF] or generate a local aerodynamic barrier [START_REF] Hergt | Advanced Non-Axisymmetric Endwall Contouring for Axial Compressors by Generating an Aerodynamic Separator--Part II: Experimental and Numerical Cascade Investigation[END_REF]. However, in these examples, the non-axisymmetric deformation relies on negative height values, which might yield practical issues in real compressor stages. Numerous investigations on the use of vortex generators were also carried out, to energize the near-hub flow ( [START_REF] Chima | Computational Modeling of Vortex Generators for Turbomachinery[END_REF], [START_REF] Hergt | Effects of Vortex Generator Application on the Performance of a Compressor Cascade[END_REF], [START_REF] Ortmanns | Numerical investigation of vortex generators to reduce cross-passage flow phenomena in compressor stator end-walls[END_REF]) or the stator boundary layer [START_REF] Hergt | Experimental Investigation of Flow Control in Compressor Cascades[END_REF]. With vortex generators, the main challenge is to compensate the extra losses added by the tip vortex with an efficient reduction of the corner separation losses. Fences were also investigated to block the passage flow [START_REF] Hergt | A New Approach for Compressor Endwall Contouring[END_REF] in the context of compressors, even though it is mostly used in the context of turbines ( [START_REF] Kawai | Secondary flow control and loss reduction in a turbine cascade using endwall fences[END_REF], [START_REF] Cho | Controlling the secondary flows near endwall boundary layer fences in a 90° turning duct using approximate optimization method[END_REF]).

Despite the large literature available in that topic, no clear design rules have been developed for any of these technological effects. Their design most often relies on a strong empiricism or on complex optimisation procedures. In this paper, an approach based on "guide fins" is implemented. Guide fins are defined as profiled devices of a priori non-conventional shapes fixed to the hub. Contrarily to the technological effect usually investigated, guide fins are not constrained to act through a dedicated physical mechanism. Their design includes basic aerodynamic rules, but is let free to combine several mechanisms to reduce the losses most efficiently. In this work, guide fins are intended to increase the operability of the stator row without degrading its nominal performances. Several guide fins extremely efficient for that purpose are found. They implement a combination of beneficial mechanisms linked to the non-axisymmetric hub deformation, vortex generation and fence techniques. These mechanisms are investigated in depth with several level of fidelity (RANS, kriging surrogate model) and are validated with a comprehensive experimental campaign. With these results, this paper aims at providing examples and guidelines for future designs of passive means of control.

In this paper, guide fins are evaluated in a low Mach number compressor cascade at the Laboratoire de Mécanique des Fluides et Acoustique (LMFA, France). This set up enables detailed measurements, yielding a precise characterization of the corner separation modifications when adding a guide fin. An optimization strategy is implemented to detect numerous efficient guide fins. It relies on the iterative refinement of Kriging surrogate models, which are trained with CFD results. An automated numerical chain is implemented for that purpose. RANS modelling is selected as it enables numerous evaluations (≃ 2000 CFD in this work). Its use in the context of corner separation evaluation is carefully justified in Section 3.2. Three guide fins that particularly reduce the endwall losses at near-stall incidence are then selected. An experimental campaign is carried out, validating their performances as well as the flow structures predicted with RANS. Three physical mechanisms explaining the corner separation reduction are found, and their interactions with the corner separation development is investigated in depth using RANS results. Finally, the sensitivity of these guide fins to some shape variations is evaluated using Kriging surrogate models. A preliminary analysis on the sensitivity of these means of control to the inlet boundary layer thickness is also carried out.

MATERIALS AND METHODS

The configuration of reference (baseline) is first presented, together with the experimental set up. The methodology to generate, evaluate and optimize guide fin shapes is then explained.

Experimental set up

The experiments are performed in a low Mach number compressor cascade presented in Figure 1 and schematized in Figure 2. Its characteristics are summarized in Table 1. An original and specific methodology detailed in [START_REF] Dawkins | The Unsteady Topology of Corner Separations[END_REF] enables to accurately monitor the inlet incidence (i) (uncertainty of ±0.1°) and to accurately impose a systematic, very thin inlet hub boundary layer. The integral characteristics of the inlet boundary layer are measured 0.25 stator chord (c) upstream of the cascade by [START_REF] Dawkins | The Unsteady Topology of Corner Separations[END_REF]: 𝛿 = 1.9%𝑐 ± 5%, 𝛿 2 = 0.20%𝑐 ± 6%, 𝐻 = 1.69 ± 3%. The cascade features a large span-to-chord ratio of 2.32. As a consequence, the flow at midspan it is locally independent of the spanwise position as long as it remains attached. A region of twodimensional flow of reference can thus be defined (midspan flow). The time-averaged performances of the cascade are evaluated at several incidences with five-hole probe measurements in a plane 0.2c downstream of the cascade. In particular, the total pressure loss coefficient (𝜔) is computed as the total pressure difference across the cascade divided by the inlet dynamic pressure measured at the same time (Equation 1). Two consecutive passages are investigated with a highly refined sampling grid (825 and 525 sampling points), yielding a high resolution of flow patterns. The mass averaged loss coefficient on the entire measurement plane (𝜔 0𝐷 ) and at midspan only (𝜔 𝑀𝐼𝐷 ) are computed. The endwall losses (𝜔 𝐸𝑊 , Equation 2) are defined as the difference between 𝜔 0𝐷 and 𝜔 𝑀𝐼𝐷 . 𝜔 𝐸𝑊 therefore quantifies the loss variation induced by the presence of viscous endwalls, which major contribution is corner separation. Indeed, 𝜔 0𝐷 would equal 𝜔 𝑀𝐼𝐷 if the endwalls were inviscid, as the stator blades are designed with the same profile along the span (no stacking law).The settings of this cascade were thoroughly chosen by [START_REF] Dawkins | The Unsteady Topology of Corner Separations[END_REF] to ensure an accurate measurement of the endwall losses (uncertainty of ±0.1 𝑝𝑡) and an excellent periodicity (variation of 𝜔 𝐸𝑊 between two consecutive passages within ±0.1 𝑝𝑡).

An insertion system is developed by [START_REF] Mondin | Experimental validation of a corner stall control methodology using parametrised guide fins[END_REF] to rapidly equip the cascade with guide fins in 6 consecutive passages. In each passage, one guide fin is inserted on each endwall to respect the passage symmetry in the spanwise direction. This system can mimic the baseline configuration (no guide fins) with a negligible impact on the endwall losses (below ±0.1 𝑝𝑡) and no effect on the flow periodicity.

In this configuration, the endwall losses rise from 1.1 pt at i=0° (incidence of minimal losses) to 4.8 pt at i=5.4°. Above 5.4°, the midspan flow starts separating. In order to investigate the effect of guide fins on the isolated phenomenon of corner separation, i=4.9° is selected as the high incidence value of reference. The efficiency of guide fins is evaluated with the endwall losses they induce minus the baseline endwall losses. Guide fins are intended to reduce the corner separation magnitude at high incidences without degrading the nominal performance. Their efficiency is therefore monitored at nominal incidence (Δ𝜔 0 , Equation 3) and high incidence (Δ𝜔 4.9 , Equation 4). A negative value of these quantities therefore indicates a beneficial effect of the guide fin (loss reduction). In Equations 3 and 4, "GF" refers to Guide Fin and "BSL" to Baseline.

𝜔 = 𝑃 𝑡,∞ -𝑃 𝑡 𝑃 𝑡,∞ -𝑃 𝑠,∞ (1)
𝜔 𝐸𝑊 = 𝜔 0𝐷 -𝜔 𝑀𝐼𝐷 (2)
𝛥𝜔 0 = 𝜔 𝐸𝑊,𝑖=0°( 𝐺𝐹) -𝜔 𝐸𝑊,𝑖=0°( 𝐵𝑆𝐿)

𝛥𝜔 4.9 = 𝜔 𝐸𝑊,𝑖=4.9°( 𝐺𝐹) -𝜔 𝐸𝑊,𝑖=4.9°( 𝐵𝑆𝐿) (4)

Cascade parameters Value

Profile

Numerical methodology RANS evaluation

A numerical chain is implemented to generate various guide fin shapes and evaluate their performances with RANS. This chain is precisely detailed in [START_REF] Mondin | Experimental validation of a corner stall control methodology using parametrised guide fins[END_REF] and is organised in three steps: 1. Parametrisation of guide fin shape: A stacking methodology is used. It relies on two parametrised profiles (at hub and at guide fin tip, 7 parameters each) and on a height value. These 7 parameters are: the position of maximum of camber, the value of the angles at the camber line tips, the thickness value (from a thickness law of reference), the chord value and the streamwise and pitchwise position (constrained to fit in the authorised green zone in Figure 4). Guide fins are therefore defined with 15 parameters (7 for each profile + height). This parametrisation correspond to the one thoroughly described in [START_REF] Mondin | Experimental validation of a corner stall control methodology using parametrised guide fins[END_REF] without considering fillets. Their formulation natively includes positioning and aerodynamic constraints. 2. Meshing: Only half of the passage is meshed, from the hub to midspan. A hybrid meshing strategy is implemented, using both structured and unstructured blocks. This strategy is illustrated in Figure 3. Each guide fin is meshed within an unstructured block of fixed dimension, obtained using ANSA (Beta). This enables a robust discretization of various shapes, uncoupled from the stator part. The rest of the domain, and notably the O-grid around the stator blade is meshed with a fixed structured mesh obtained using AutoGridV5 (Numeca). This ensures a systematic discretization of most of the domain, a high resolution of the stator boundary layer region, and faster computations than with a fully unstructured approach. The resulting meshes typically contain 1.1 × 10 6 to 1.6 × 10 6 cells and are characterised by a typical maximum y+ value of 4. This is up to modern standards for one half-passage of a subsonic compressor cascade. The sensitivity to the unstructured mesh resolution is evaluated with one guide fin. With that guide fin, the endwall losses vary by less than 0.02 pt when the number of point is increased by 40%, thus validating the initial mesh resolution.

3. CFD evaluation: RANS is selected to enable numerous computations. The choice of the turbulent modelling has a large impact on the prediction of corner separation. In [START_REF] Mondin | Experimental validation of a corner stall control methodology using parametrised guide fins[END_REF], the turbulence models of 𝑘 -𝜔 Menter BSL, 𝑘 -𝑙 of Smith and Spalart-Allmaras are evaluated with the CFD solver elsA (ONERA) on the same baseline as in the present work.

The spatial scheme of Roe of order 2 is used with no limiter for variable reconstruction at interfaces. 𝑘 -𝜔 Menter BSL is selected as it best retrieves the endwall losses evolution with the incidence and the pitchwise losses evolution at midspan. In the present work, the Quadrative Constitutive Relation (QCR) correction is added, as suggested by [START_REF] Spalart | Strategies for turbulence modelling and simulations[END_REF] to improve corner flow prediction. By comparison with RANS and experimental results on the baseline case, the author verified that the QCR correction leaves the midspan region unchanged while improving the prediction of the loss gradients in the hub region.

Optimization process

An optimization strategy is implemented to build a large database of efficient guide fins. In this work, two objective functions are considered (Δ𝜔 0 and Δ𝜔 4.9 ), with a relatively large number of parameters [START_REF] Hergt | Experimental Investigation of Flow Control in Compressor Cascades[END_REF] and expensive evaluations (≃ 100 CPU.Hours each). In this context, a strategy relying on the iterative refinement of Kriging surrogate models is selected. This strategy is used to explore two input spaces described hereafter (15p and 6p). This strategy relies on a first exploration of the input space (Design Of Experiment or DOE). A first Kriging surrogate model is trained with the RANS results of the DOE. Such a model provides predictions of the two objective functions on the entire input space, as well as a measure of the prediction uncertainty. The Expected Hypervolume Improvement (EHI) criterion, introduced by [START_REF] Picheny | Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction[END_REF], is computed using these two characteristics. It provides a set of guide fins (refinement set) to be evaluated in order to increase the surrogate precision in the regions of low objective functions values. The refinement set is evaluated in RANS, which enables to a) populate the RANS database b) generate a refined Kriging surrogate model. The Hyper Volume (HV) metric [START_REF] Zitzler | Multiobjective Optimization Using Evolutionary Algorithms -A Comparative Case Study[END_REF] is used to monitor the convergence of the Pareto front at each iteration. This criterion increases when the Pareto front is improved (in the Pareto sense). In both input spaces 1) the RANS HV (computed with the RANS results) increases 2) the surrogate HV (computed with the surrogate predictions) decreases towards the RANS HV values. Several refinement steps are performed iteratively (10 for the 15p and 16 for the 6p input space). In each case, the surrogate models are close to convergence, but the maximum budget (≃2000 RANS computations) is reached before reaching total convergence. Their predictability in the Pareto region is still greatly improved with respect to the DOE, and their predictions will be used to perform a sensitivity analysis in section 3.5.

Two inputs spaces are explored with this methodology, based on 15 parameters (15p) and 6 parameters (6p):

• 15p: The 15 parameters vary on large ranges to explore a large variety of shapes. As a preliminary step, a large sample of 600 000 CAD geometries taken at random in this space is generated. Only 15% (90 000) of these geometries are positioned within the authorised zone (in green in Figure 4) and respect manufacturing criteria. The exploration is therefore performed within this discrete database made of the 90 000 viable geometries. The DOE is built as a selection of 150 guide fins within this database using the Kernel Herding Algorithm as proposed by [START_REF] Chen | Super-Samples from Kernel Herding[END_REF]. The refinement sets are selected within this database. • 6p: Only 6 parameters vary. Their ranges are adapted using the results of the 15p exploration. The objective is to explore a smaller space containing efficient guide fins. In the 6p input space, the viability rate is excellent (≃ 75%) and the input space is continuously explored. The DOE contains 50 guide fins selected with a Latin Hypercube Sampling (LHS) [START_REF] Mckay | Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code[END_REF] coupled to the maxProj algorithm [START_REF] Joseph | Maximum projection designs for computer experiments[END_REF]. Typical geometries illustrating these two input spaces are shown in Figure 4.

RESULTS AND DISCUSSION

The numerical methodology is now used to evaluate numerous guide fins with RANS. Three guide fins are then selected and thoroughly investigated. Their performances and effects on the flow are validated experimentally. RANS results are used to reveal beneficial mechanisms they implement. In the last section, Kriging surrogate predictions are used to assess their sensitivity to three geometrical parameters.

RANS database

For each input space, about 500 guide fins are evaluated at i=0° and i=4.9°. In total, 1033 guide fins are evaluated, corresponding to 2066 RANS computations. Their performances are gathered in Figure 5. They appear able to efficiently lower the losses at i=4.9° (𝛥𝜔 4.9 down to -1.14 pt), but do not provide significant gains at i=0° (𝛥𝜔 0 down to -0.02 pt). The best guide fins (in the Pareto sense) are highlighted with red diamonds. This Pareto set reveals that large variations of 𝛥𝜔 4.9 can be obtained with little impact on 𝛥𝜔 0 . Guide fins can therefore be used to increase the operability without strongly degrading the nominal efficiency. For that purpose, two regions of interesting compromise appear (blue circles): (a) little over loss at nominal incidence and large gains at high incidence (b) no over loss at nominal incidence and little gains at high incidence. The investigation of the RANS database reveals that two parameters monitor the position of the guide fins along the Pareto set: the height and the axial position. To go from compromise (a) to compromise (b), the height must increase and the guide fins must move from downstream to upstream in the passage.

Three guide fins of different shapes are selected among compromise (a) for physical investigations, and are referred to as guide fins SF (Short Fence), LF (Long Fence) and 3D. Their shapes and main characteristics are gathered in Figure 6 and Table 2. Guide fins LF and 3D belong to the Pareto set.

Experimental validation Performances

Guide fins SF, LF and 3D are evaluated in RANS on a large range of incidence. In addition, they are manufactured and tested experimentally. The endwall losses obtained are compared to the baseline case in Figure 7 (RANS and experimental results).

Experimental results show that every guide fin efficiently reduce the endwall losses on a large range of incidence. Their influence on the losses at i=0° is negligible (±0.1 𝑝𝑡) and the net gain increases as the incidence is raised (see Table 3). At i=5.4°, the highest incidence before midspan separation starts, a relative endwall losses reduction of 50% (-2.4 pt) to 60% (-2.9 pt) is notably obtained in comparison with the baseline case.

In RANS computations, the endwall losses appear to be systematically underpredicted, both in the baseline case and with any guide fin. However, the experimental gains are larger than predicted in RANS, notably at i=0° and i=4.9°. Indeed, for every guide fin, RANS predictions yield Δ𝜔 0 ≃ +0.1 𝑝𝑡 and Δ𝜔 4.9 ∈ [-0.9 𝑝𝑡, -1.1 𝑝𝑡], while experimentally Δ𝜔 0 ∈ [-0.1 𝑝𝑡, +0.1 𝑝𝑡] and Δ𝜔 4.9 ∈ [-1.0 𝑝𝑡, -1.6 𝑝𝑡]. Moreover, it should be noticed that the guide fin best reducing the endwall losses (guide fin 3D) features an experimental trend very well captured by RANS. This illustrates the fact that RANS predictions are more accurate when separated regions are downsized. This participates to justifying the use of RANS in the optimization process. In addition to these results on endwall losses modifications, the effect of guide fins on the stator deflection is assessed. The integrated downstream pitchwise angle (𝛽 0𝐷 ) is used to quantify the deflection. Figure 8 : evolution of the averaged downstream pitchwise angle with the incidence. Experiment.Figure 8 shows that the stator deflection is little affected at nominal incidence (between +0.3° and -0.2° compared to the baseline case) and improved at high incidence (up to -0.9° at i=5.4° compared to the baseline case).

As a conclusion, adding any of these guide fins increases the operability of the stator without deteriorating neither the losses at nominal condition nor the stator deflection. The operability is then no longer limited by the corner separation but by the midspan separation. This validates the potential of guide fins for increasing the operability of stators, as well as the value of the implemented numerical methodology.

Flow structure

The effect of guide fins SF, LF and 3D on the flow structure downstream of the cascade is now investigated. Figure 9 shows cartographies of losses comparing the configurations with and without guide fins. The incidences i=0° and i=4.9° are presented. For each one RANS results are compared to experimental results. At i=0°, RANS predictions are in good agreement with the experiments. The size, shape and magnitude of the corner separations are well retrieved in every configuration. In addition, specific loss patterns are well detected, such as the hub boundary layer (baseline case) and guide fin wakes. At i=4.9°, RANS predictions are acceptable. They tend to underpredict the corner separation size and magnitude, and underpredict the diffusion of the large loss regions. This is consistent with the underprediction of the endwall losses noticed in Figure 9. However, the shapes and flow patterns specific to each guide fin are retrieved. Notably, a region of low losses near the hub is noticed with the guide fin SF. More importantly, the corner separation and the wake of the guide fin 3D are very well captured.

As a conclusion, RANS computations enable to capture the guide fin characteristics downstream of the cascade. RANS is therefore used in the next section to investigate the beneficial mechanisms implemented by the guide fins within the cascade.

Analysis of beneficial mechanisms

The investigation of the RANS flowfield reveals three beneficial mechanisms for reducing corner separation: Tip Vortex, Fence, and Guide. Each guide fin is used to illustrate a different mechanism.

Effect of TIP VORTEX

The SF guide fin features a pressure difference across its tip, which generates a tip vortex (see Figure 10, black streamlines). This vortex is generated much above the inlet boundary layer thickness value (𝐻 𝐺𝐹 ≃ 3.5 𝛿). This is unlike conventional vortex generators whose heights are usually of the order of 𝛿. The descending component of the vortex brings high velocity fluid from the bulk flow towards the near hub region (blue streamlines), and towards the stator suction side thanks to the vortex rotation direction. This high velocity fluid explains the origin of the low loss region near the hub that was noticed in Figure 9.

Effect of FENCE

The LF guide fin is particularly long and thin, thus creating two passages in the near hub region. As the blade-to-blade pressure gradient is still imposed by the stator blades, two passage flows and two passage vortices (PV) are created, one in each passage. These passage vortices are revealed with secondary streamlines in Figure 11 (PV1 in blue and PV2 in red), and coloured with the dimensionless axial velocity. Secondary streamlines are built using the velocity field minus the midspan velocity field at the corresponding pitchwise position. They enable to visualise any flow motion that differs from midspan. In order to analyse the effect of PV1 and PV2 on the near hub flow, the edges of the hub boundary layer and of the corner separations are indicated with a solid black line (iso-contour 𝜔 = 0.3). PV1 appears to drive the hub boundary layer towards the guide fin suction side, resulting in a corner separation on the guide fin suction side. PV2 forms in a high loss region that contains high axial velocity fluid. As it is convected downstream, its descending component incorporates high velocity fluid along the guide fin pressure side, thus making the hub boundary layer thinner. On the last downstream position, the high velocity region exactly corresponds to the low loss region.

As such, the effect of Fence features two beneficial subeffects corresponding to each passage vortex:

1) A portion of the hub boundary layer is blocked and ejected radially (ascending component of PV1).

2) The other portion of the near hub flow is accelerated by incorporation of high velocity flow from the bulk flow along the guide fin pressure side (descending component of PV2).

Effect of GUIDE

The 3D guide fin features an unconventional design, close to a non-axisymmetric profiled hub. It features a thick hub profile (𝑡 𝑀,𝐻 = 15%𝑐 𝐺𝐹,𝐻 ) and a clear pyramidal aspect (𝑡 𝑀,𝑇 /𝑡 𝑀,𝐻 = 14%, 𝑐 𝐺𝐹,𝑇 /𝑐 𝐺𝐹,𝐻 = 25%). These characteristics induce a large rounded leading edge and a smooth evolution of the curvature along the guide fin sides. This guide fin splits the near hub flow in three regions (see Figure 12). Region 1 (yellow streamlines) is unaffected by the guide fin and feeds the stator corner separation (Stator CS). The guiding effect itself appears in the region 2 (black streamlines), where the incoming near hub flow is guided without separation thanks to the smooth guide fin curvature. The guiding effect induces a local pressure drop on the guide fin side facing the stator suction side (white contour). This pressure drop induces favourable gradients in both the engine axis direction and the blade-to-blade direction, thus facilitating the re-orientation of the passage flow towards downstream. In region 3 (blue streamlines), the passage flow is blocked and a small corner separation is triggered on the guide fin side (Guide fin CS). Downstream of the guide fin, the region of guided fluid persists and prevents any interaction between the guide fin corner separation and the stator corner separation.

Topological analysis of the near-hub flow

Skin friction lines (streamlines infinitely close to the hub) are now used to further investigate the near hub flow. Their analysis helps characterizing the near hub flow orientation and the nature of the separated regions [START_REF] Duquesne | Topology Rule-Based Methodology for Flow Separation Analysis in Turbomachinery[END_REF]. Locations where the friction field vanishes are symptomatic of separated regions. These locations are referred to as critical points, and are classified according to the skin friction line patterns in their vicinity. Figure 13 compares the configuration with and without guide fins.

A Saddle/Focus pair of critical points appears in the baseline case (S/F). This is symptomatic of a corner separation located on both the stator suction side and on the hub. Such a separation is said to be double sided as proposed in [START_REF] Dawkins | The Unsteady Topology of Corner Separations[END_REF]. This pair disappears when adding any of the three guide fins, showing that they all remove the separation on the hub. In this case, the corner separation is said to be single sided, as proposed in [START_REF] Dawkins | The Unsteady Topology of Corner Separations[END_REF].

Saddle points (in red) induce separating lines (in white, dotted line) that reveal flow regions that are not accessible to one another. In this context, separating lines are used to highlight the origin of the near hub flow reaching the stator suction side. The portion of the stator colored in yellow is reached by the flow coming from upstream; in blue, by the flow coming from the FIGURE 13 : NEAR-HUB FLOW TOPOLOGY. stator pressure side; in green, by the flow coming from the guide fin. The blue zone is reached by the flow that developed in the longest distance within the passage. The extent of this zone is reduced when adding the SF guide fin, much reduced with the 3D one and disappears with the LF guide fin. It is replaced or partly replaced by green zones, that are symptomatic of a fluid that developed on a shorter distance and therefore contains less low momentum fluid. Interestingly, the SF and LF acts rather upstream in the passage, at the axial position corresponding to the Saddle/Focus pair in the baseline case, and leaves the yellow zone hardly unaffected. On the other hand, the 3D guide fin acts downstream in the passage, and suppresses the Saddle/Focus pair without creating new critical points thanks to its smooth curvature.

Skin friction lines also reveal the near hub flow orientation, highlighted with thick white arrows. The flow appears better oriented when adding guide fins, especially towards green zones. This is attributed to the high axial velocity regions (yellow contours) that helps orienting the flow towards downstream. These high velocity regions are symptomatic of the beneficial effects of Fence and Tip Vortex (SF), Fence (LF) and Guide by smooth flow deflection (3D). Downstream of the 3D guide fin, a pitchwise angle (𝛽) close to the averaged midspan value (𝛽 𝑀𝐼𝐷 ) is observed, consistently with the guiding effect (thin white arrow).

This topological analysis reveals that guide fins alter the near hub-flow reaching the stator. This flow is re-oriented and contains less low momentum fluid, which explains the resulting corner separation reduction. It is worth noticing that these effects can a priori be implemented simultaneously, as with the SF guide (Tip Vortex and Fence effects).

Sensitivity analysis Design parameters

In the previous sections, guide fin performances were assessed and related to geometrical characteristics and flow mechanisms. The sensitivity of these performances to geometrical modifications is now investigated with the Kriging surrogate models output by the optimization process. Three parameters exhibit a significant influence on Δ𝜔 0 and Δ𝜔 4.9 : the axial position of guide fin hub profile leading edge (𝑥 𝐿𝐸 ), the guide fin height (𝐻 𝐺𝐹 ) and the guide fin hub thickness (𝑡 𝑀,𝐻 ). Figure 14 shows the surrogate model predictions of Δ𝜔 0 and Δ𝜔 4.9 when each parameter is varied on a large range around each guide fin configuration.

Firstly, it must be underlined that all guide fins exhibit an optimal value on each parameter and on each objective function. However, the objective functions are moderately sensitive to variations around these optima. Each parameter value can indeed vary by ±15% of the total investigated range around the optimal value and still provide efficient guide fins. With such variations, Δ𝜔 0 remains within [+0 𝑝𝑡, +0.2 𝑝𝑡] and Δ𝜔 4.9 within [-0.8 𝑝𝑡, -1.2 𝑝𝑡].

More specifically, the optimal values and predicted trends can be related to the beneficial effects implemented:

• Axial position of hub profile leading edge (𝒙 𝑳𝑬 ): 𝑥 𝐿𝐸 is the parameter to which Δ𝜔 4.9 is the most sensitive. The optimal position of the SF and LF corresponds to the location where the flow separates at hub in the baseline case. This is explained by the effect of fence that prevents the low momentum flow to reach this sensitive zone. As such, the position of the Saddle point in the baseline case reveals the most efficient axial position for the SF and LF guide fins. • Guide fin height (𝑯 𝑮𝑭 ): no optimal value clearly appears for Δ𝜔 0 , which is degraded as 𝐻 𝐺𝐹 increases. However, optimal values exist on Δ𝜔 4.9 . Interestingly, these values are all much larger than the inlet boundary layer thickness (𝛿). This indicates a little sensitivity of the beneficial effects to the inlet boundary layer thickness. This is particularly remarkable with the SF guide fin, as Δ𝜔 4.9 is insensitive to height values in a large range (𝐻 𝐺𝐹 ∈[7%c,12%c]).

• Guide fin hub thickness (𝒕 𝑴,𝑯 ): As for the height, the thickness tends to only induce additional losses at nominal incidence. However, at high incidence the guide fins behave differently. The SF and LF guide fins should be thin: their beneficial mechanisms do not rely on their thickness, consistently with the previous analysis. However, the 3D guide fin should be thick enough: this is consistent with the pyramidal aspect and the large rounded leading edge that explains the guiding effect.

As a summary, the sensitivity observed on the three efficient guide fins are related to the beneficial mechanisms they implement. In addition, the losses at both incidences are little sensitive to the precise value of each parameter. From a practical point of view, this indicates that these guide fins can be adapted in case of additional constraints (ex: mechanical constraints), or that their performance will not be affected by the manufacturing uncertainties. This is particularly interesting for the SF and LF guide fins, which feature very thin leading and trailing edges. These results indicate that their manufacturing could be eased by thickening their minimum thickness with little impact on their performances.

Inlet boundary layer thickness

The inlet boundary layer thickness (𝛿) plays a major role in the development of the corner separation. In order to assess its influence on the efficiency of the SF, LF and 3D guide fins, RANS computations are run with a doubled value of the inlet boundary layer thickness (2𝛿). Figure 15 shows that the endwall losses (𝜔 𝐸𝑊 ) are increased for all configurations at all incidences in the 2𝛿 case. At i=0°, the endwall losses increase by 0.4 pt for all configurations. At i=4.9°, the endwall losses increase by 1.2 pt in the baseline configuration, and only by 0.8 pt when adding the SF guide fin, by 0.7 pt with the LF and by 0.6 pt with the 3D guide fin. The 3 guide fins therefore still provide an important endwall losses reduction. Interestingly, this gain is even larger than with the initial boundary layer thickness. As such, the efficiency of these guide fins is robust to a small variation of the inlet boundary layer thickness. This is encouraging for the future potential use of guide fins in more realistic flow conditions. However, for much thicker boundary layers, the efficiency of the guide fins might be altered. A dedicated study on the relation between the inlet boundary layer thickness and the guide fin design should be carried out. Such a study constitutes an interesting perspective of this work towards the applicability of guide fins in realistic configurations.

CONCLUSION

The potential of guide fins for reducing corner separation in a low Mach number compressor cascade of reference was assessed experimentally. A RANS based optimization process relying on the iterative refinement of surrogate models enabled to obtain a set of efficient guide fins (Pareto set). Two regions of this set were discussed, corresponding to two different design compromises. Three guide fins were then selected for physical investigation (SF, LF, 3D). Their performances are assessed experimentally, with a loss reduction up to -2.9 pt (-60%) at high incidence, without degrading neither the losses at nominal incidence nor the stator deflection on the whole operating range. With these guide fins, the operability is no longer limited by the corner separation, but by the midspan separation. A detailed RANS investigation enabled to reveal and characterize three beneficial mechanisms that participate to energizing and reorienting the passage flow: Tip Vortex, Fence and Guide.

The tip vortex effect brings high momentum fluid from the bulk flow towards the hub and the stator suction side. The effect of Fence divides the passage flow into two passage flows, which 1) prevent a portion of the near hub flow to reach the stator suction side and 2) energize the near hub flow reaching the stator suction side. The guide effect induces a local favourable pressure gradient within the passage that helps evacuating the near-hub low momentum fluid towards downstream, without reaching the stator suction side. The guide effect is obtained thanks to the smooth rounded shape of the guide fin 3D. Finally, surrogate predictions reveal a significant robustness of guide fins performance to variations of their shapes. The three most influencing parameters were varied: the guide fin axial position, height and thickness at hub. Moreover, a preliminary sensitivity analysis on the effect of the boundary layer thickness is carried out and reveals encouraging results on the robustness to that parameter.

In this paper the vortex generator effect is revealed to have a large beneficial effect even when generated high above the inlet boundary layer thickness (𝐻 𝐺𝐹 ≃ 3.5𝛿). With the Fence effect, the importance of the passage vortex created near the stator suction side (PV2) is highlighted. The effect of Guide highlights the beneficial outcomes on the corner separation development of a local favourable pressure gradient within the channel. Finally, this paper showed an example of three geometries whose efficiency mostly relies on three parameters with an interesting robustness on design parameters and on the inlet boundary layer thickness. As a passive, robust, and efficient control device, guide fins have a high potential of applicability. They should be assessed in more realistic configurations that notably include compressible flows and thick, skewed, realistic inlet boundary layers.

FIGURE 1 :

 1 FIGURE 1 : SIDE VIEW OF THE RIG

FIGURE 2 :

 2 FIGURE 2 : DETAILS ON PROBE LOCATIONS

FIGURE 3 :

 3 FIGURE 3 : EXAMPLE OF HYBRID MESH USED WITH A GUIDE FIN. WHITE ELEMENTS: STRUCTURED BLOCKS. RED AND GREEN ELEMENTS: UNSTRUCTURED BLOCKS. ADAPTED FROM [19].

FIGURE 4 :

 4 FIGURE 4 : ILLUSTRATION OF THE INPUT SPACES EXPLORED. THE COORDINATE SYSTEM USED TO LOCATE THE HUB PROFILE LEADING EDGE IS ADDED.

FIGURE 5 :FIGURE 6 :

 56 FIGURE 5 : PERFORMANCES OF THE RANS DATABASE

FIGURE 7 :

 7 FIGURE 7 : EVOLUTION OF THE ENDWALL LOSSES WITH THE INCIDENCE. RANS VS EXPERIMENT.

FIGURE 8 :

 8 FIGURE 8 : EVOLUTION OF THE AVERAGED DOWNSTREAM PITCHWISE ANGLE WITH THE INCIDENCE. EXPERIMENT.

FIGURE 9 :

 9 FIGURE 9 : TOTAL PRESSURE LOSSES IN THE MEASUREMENT PLANE. EXPERIMENT VS RANS PREDICTIONS. CS=CORNER SEPARATION. BL = HUB BOUNDARY LAYER. ZOOM IN THE MOST REFINED PASSAGE AND IN THE FIRST 25% OF SPAN.

FIGURE 10 :

 10 FIGURE 10 : EFFECT OF TIP VORTEX (I=4.9°). AXIAL SLICES COLOURED WITH DIMENSIONLESS AXIAL VELOCITY. CS = CORNER SEPARATION.

FIGURE 11 :

 11 FIGURE 11 : EFFECT OF FENCE (I=4.9°). SECONDARY STREAMLINES COLOURED WITH DIMENSIONLESS AXIAL VELOCITY. IN BLACK: HIGH LOSS REGIONS. CS = CORNER SEPARATION. PV=PASSAGE VORTEX.

FIGURE 12 :

 12 FIGURE 12 : EFFECT OF GUIDE (I=4.9°). CS = CORNER SEPARATION. NEAR HUB FLOW: 1) UNAFFECTED 2) GUIDED 3) BLOCKED. THE SURFACES ARE COLORED WITH STATIC PRESSURE VALUES. BLUE: MINIMUM, RED : MAXIMUM.

FIGURE 14 :

 14 FIGURE 14 : PREDICTIONS OF SURROGATE MODEL NEAR GUIDE FINS SF, LF AND 3D. 𝒙 𝑭𝒐𝒄𝒖𝒔 : POSITION OF HUB FOCUS IN THE BASELINE CASE. 𝜹: INLET BOUNDARY LAYER THICKNESS.

FIGURE 15 :

 15 FIGURE 15 : EVOLUTION OF THE ENDWALL LOSSES WHEN DOUBLING THE INLET BOUNDARY LAYER THICKNESS. RANS RESULTS. BSL : BASELINE.

TABLE 1 :

 1 CASCADE CHARACTERISTICS

TABLE 3 :

 3 EXPERIMENTAL

RESULTS: BASELINE ENDWALL LOSSES (±𝟎. 𝟏𝟎 𝒑𝒕) AND ENDWALL LOSSES VARIATION 𝜟𝝎 𝑬𝑾 WHEN ADDING GUIDE FINS (±𝟎. 𝟏𝟓 𝒑𝒕).

ACKNOWLEDGEMENTS

The authors would like to thank the CIRT (Consortium Industrie-Recherche en Turbomachine) and Safran Tech for supporting this work. In particular, thanks go to Sebastien Da Veiga, Emma Croner and Etienne Tang for their time and support. At LMFA, the authors would also like to thank Sebastien Goguey, Pierre Laucher, Benoit Paoletti, Gilbert Halter and Lionel Pierrard for their advices on the experimental set up. Finally, special thanks to Ivo Dawkins, for his many discussions and for sharing his experience on the baseline configuration.