Alice Martzloff

Nicolas Posocco

Quentin Ferré

A Fair Classifier Embracing Triplet Collapse

Keywords: Équité, apprentissage métrique, fonction de coût par triplet, collapse, classifieur équitable, IA responsable Fairness, deep metric learning, triplet loss, collapse, fair classifier, responsible AI

Dans ce papier, nous étudions les comportements de la fonction de coût par triplet et montrons que ceux-ci peuvent être exploités pour limiter les biais crées et perpétués par les modèles d'apprentissage automatique. Notre classifieur équitable utilise le collapse de la fonction de coût par triplet, qui nécessite une marge supérieure à la distance maximum entre deux points dans l'espace latent et une sélection stochastiques des triplets.

Introduction

Fairness is an important component of responsible AI frameworks and is a growing concern, especially from social and regulatory perspectives. It broadly aims at avoiding the learning of historical biases, such as those based on race or gender. We consider the problem of building a fair representation of data, so that sensitive features (i.e. race, gender) cannot be recovered from the representation. To achieve this, deep metric learning is a potentially interesting approach since it builds an embedding space with desirable characteristics. Since the behaviour of associated triplet learning is not completely understood as of today, we propose new insights on the behaviour of the loss and use this new knowledge to create an embedder into a fair embedding space.

Our main contributions are:

• Theoretical and experimental evidence to produce knowledge on triplet learning

• A novel solution for learning representations that are fair and with high utility using triplet loss collapse

Related Works

Fairness is all about the non-discrimination of populations. This means avoiding the creation or reinforcement of human prejudice and promoting inclusion and diversity. This task is difficult in practice, since datasets with supervision carry the historical annotation bias. The scientific community proposed metrics for the study of various definitions of fairness like Demographic parity [START_REF] Dwork | Fairness through awareness[END_REF], Equality of opportunity [START_REF] Hardt | Equality of opportunity in supervised learning[END_REF] and Counterfactual fairness [START_REF] Kusner | Counterfactual fairness[END_REF]. After observing that some models were unfair, special efforts have been allocated to finding ways to mitigate those biases in the ML pipeline, with pre-processing approaches [START_REF] Barocas | Fairness in machine learning[END_REF][START_REF] Zemel | Learning fair representations[END_REF], which impact the data before it is ingested by the model, post-processing approaches [START_REF] Kamiran | Discrimination aware decision tree learning[END_REF], which straighten out the model's output, and in-processing approaches [START_REF] Bilal Zafar | Fairness constraints: Mechanisms for fair classification[END_REF][START_REF] Sorelle | A comparative study of fairness-enhancing interventions in machine learning[END_REF], which get into the core of the model to make a difference. Collecting more representative data is most of the time either impossible or too costly. As a result, the issue is usually tackled by processing existing data. To that end, pre-processing methods aim at creating a new representation of the data that is fair with respect to one of the many fairness definitions. It was first tried using rough baselines [START_REF] Kamiran | Data preprocessing techniques for classification without discrimination[END_REF][START_REF] Feldman | Certifying and removing disparate impact[END_REF], and, more recently, variational autoencoders [START_REF] Louizos | The variational fair autoencoder[END_REF][START_REF] Locatello | On the fairness of disentangled representations[END_REF][START_REF] Creager | Flexibly fair representation learning by disentanglement[END_REF] and modern generative models [START_REF] Balunović | Fair normalizing flows[END_REF].

In this very last publication, Balunovic et al. explain how most fair representation still allow discrimination based on the sensitive feature, in the learned representation. Metric learning for fairness [START_REF] Ilvento | Metric learning for individual fairness[END_REF][START_REF] Kulis | Metric learning: A survey[END_REF] aims at achieving individual fairness, which means that the embeddings of similar samples (except for the sensitive feature) should be very close. Recent papers like the one of Shen et al. [START_REF] Shen | Contrastive learning for fair representations[END_REF] already looked into the potential of contrastive learning for fairness.

Quite unexplored yet in this context, we focus on triplet learning. First used by [START_REF] Schroff | Facenet: A unified embedding for face recognition and clustering[END_REF] for computer vision, Hermans et al. [START_REF] Hermans | In defense of the triplet loss for person re-identification[END_REF] re-argue its arbitrary capacity as a key component for outperforming other methods. The use of metric learning has increased since its rise in computer vision, but few had the objective of in-depth studying the behaviours related to the use of triplet learning. Some [START_REF] Wang | Normface: L2 hypersphere embedding for face verification[END_REF] try to explain the "tricks" to favour a quality training like adding a softmax activation function on the last layer of the embedder. Others have shed some light on collapse modes. A generalised collapse (all embeddings are projected onto a single point) can occur when using the triplet loss [START_REF] Shadley | Trivial solutions in triplet loss. Reddit shorturl[END_REF] with a hard or semi-hard selection of triplets [START_REF] Schroff | Facenet: A unified embedding for face recognition and clustering[END_REF]. An intra-class collapse would group all the data points of the same class into one point [START_REF] Levi | Reducing class collapse in metric learning with easy positive sampling[END_REF] if the chosen margin allows the optimisation to reach that state [START_REF] Hermans | In defense of the triplet loss for person re-identification[END_REF].

We use supervised triplet learning with several different triplet selection methods to create representations where similar samples have similar representations [START_REF] Ilvento | Metric learning for individual fairness[END_REF][START_REF] Kulis | Metric learning: A survey[END_REF], and from which the target feature can be recovered (downstream task) but not the sensitive feature(s).

To do so, we study and propose an explanation of the following phenomena: Why does the We chose a criterion for measuring fairness that relies on the accuracy of recovering the sensitive feature. We measure how much any adversary can recover sensitive information from the embedding. By doing so, we want to ensure that no sensitive information is available and therefore cannot contribute to the decision. This should cover any type of harm resulting from the use of a dataset. We consider that an embedding is fair if for any embedding z ∈ Z, it is impossible to retrieve the corresponding sensitive feature s, while z remains a good predictor for y. The fairness of the embedding is verified through a supervised classification algorithm that:

• shall achieve a much better performance (e.g. area under the ROC) on the task of predicting s from x 1 , ..., x K than from z ; • shall achieve a similar performance on the task of predicting y from x or from z. In the following, we use a deep metric learning approach to compute such a fair embedder.

Proposed Solution

We propose to train a small multi-layer perceptron using a triplet loss, so that the selection of triplets induces desirable properties in the latent space with regard to fairness.

Given a triplet ((x a , y a ), (x + , y + ), (x -, y -)) ∈ D 3 (anchor, positive, negative), (z a , z + , z -) are the corresponding embeddings. With α ∈ R the triplet margin, and as-suming the use of the Euclidian distance, the optimised loss is:

L = max ∥z a -z + ∥ 2 2 -∥z a -z -∥ 2 2 + α, 0
Benchmarked triplet selection methods. We investigate five different triplet selection methods. The first three are studied as potential bias mitigation techniques. The latter two are control baselines meant to support our hypotheses made on the expected behaviours of the first three methods.

1. classical: x + is selected at random so that y + = y a (the anchor can't be re-selected). x -is selected randomly amongst samples for which y -̸ = y a .

2. counterfactual: the positive is a copy of the anchor but with the sensitive feature flipped, thus

x + = [x a 1 , ..., x a K , s].
x -is selected randomly amongst samples for which y -̸ = y a .

target-agnostic counterfactual

: x + = [x a 1 , ..., x a K , s].
x -is selected randomly among all samples except the anchor.

4. random: Both x + and x -are selected randomly among all samples but the anchor.

identical positive:

The positive is a copy of the anchor, thus x + = x a . x -is selected at random amongst samples for which y -̸ = y a .

Note that we don't use hard or semi-hard selection of triplets which has O(n) time complexity, but a stochastic selection of triplets which has time complexity of O(1). Benchmarked latent activations/normalisations. We also study the influence of the use of different activations/normalisations of the output layer of the MLP embedder, used to produce the embedding. Inspired by the existence of a "softmax trick" mentioned in many papers [START_REF] Wang | Normface: L2 hypersphere embedding for face verification[END_REF], we look into the following activations/normalisations: no activation, batch normalisation, L1 normalisation, L2 normalisation, sigmoïd, softmax and hyperbolic tangent, and study their performance qualitatively and quantitatively.

Understanding Triplet Learning

In this section, we discuss the expected behaviour of the triplet loss based on a theoretical analysis.

The activation shapes the embedding space. We apply an activation function on the embedding defined by our embedder. This operator determines the shape of the embedding space, as well as the maximum distance between two samples.

For the softmax activation, for instance, the embedding space is a d-dimensional simplex, thus the maximum distance between two points is √ 2. A sigmoid activation restricts the embedding space to a d-dimensional hypercube, thus the maximum distance is Using the same idea and the same d, one can prove that for hyperbolic tangent, L1 normalisation and L2 normalisation, the maximum euclidean distances are respectively 2 √ 3, 2 and 2.

√ 3 if d = 3.
A margin greater than the maximum distance prevents convergence. If we note:

Λ(z a , z + , z -) = 1 if α ≥ ∥z a -z -∥ 2 2 -∥z a -z + ∥ 2 2 0 otherwise,
then the partial derivatives of the loss with respect to each embedding component is:

∂L ∂z a = D i=1 2(z - i -z + i ) if Λ(z a , z + , z -) = 1 0 otherwise ∂L ∂z + = D i=1 -2(z a i -z + i ) if Λ(z a , z + , z -) = 1 0 otherwise ∂L ∂z -= D i=1 2(z a i -z - i ) if Λ(z a , z + , z -) = 1 0 otherwise
Conceptually, the impact of the loss optimisation on the embeddings can be decomposed into two forces, whose magnitude depends on the loss gradients:

-an attractive force applied to the pair of embeddings for the anchor and the positive sample -a repulsive force applied to the pair of embeddings for the anchor and the negative sample. These forces are null if and only if for one of these two pairs: either the two embeddings are distant by more than the margin, or they are exactly equal.

If the margin is zero, the optimisation goes on until positive samples are at least as close as corresponding negative samples to the anchor:

α = 0 ⇒ Λ = 1 iff. ∥z a -z + ∥ 2 2 ≥ ∥z a -z -∥ 2 2
If the margin is positive, the optimisation instead goes on until the negative samples are farther from the anchor by α compared to the positive samples:

α > 0 ⇒ Λ = 1 iff. ∥z a -z + ∥ 2 2 ≥ ∥z a -z -∥ 2 2 -α
If the margin is positive and above a maximal distance (which we call this an outrageous margin in the following), there will always be non-null gradients:

α > max z,z ′ (∥z -z ′ ∥ 2 2 ) ⇒ Λ = 1
A collapse happens for triplet selection with small attractive force. A collapsed embedding space consists of samples distributed in a small number of clusters (which we define as the intra-cluster distance being less than 10 -5 ).

We hypothesise that a collapse happens due to an imbalance between attractive and repulsive forces. As we have shown, the derivatives of the loss are proportional to the difference between embeddings components. If the embeddings of the anchor and the positive sample are close, we expect the attractive force (i.e. the result of the gradient w.r.t. to anchor and positive) to be very small. In this case, if we assume there is some distance between anchors and negatives because of the selection method, then the repulsive force will drive the optimisation step, and we hypothesize this will result in the embeddings clustering at the corners of the embedding space, far away from the center of mass.

Figure 2: The farther apart the embeddings in a pair are, the greater the force that is applied to them.

A random triplet selection with an outrageous margin scrambles the features. We hypothesise that in the case of a random triplet selection with an outrageous margin, all features will be scrambled by the collapse of the space, since no feature is targeted by the triplet loss. We define a scrambled feature as a feature that cannot be recovered from the embedding i.e. its recovery AUC is as close to 0.5 as possible.

6 Experiments

Experimental Setup

In this section, we empirically verify previously presented theoretical insights, and show that these can be used to build fair representations in practice. We train the embedder -a simple multilayer perceptron -using a triplet approach, with previously mentioned triplet selection methods. It projects the input feature vector x into a representation with half the number of input components, and returns the embedding z of output dimension 3.

Benchmark Datasets

Experiments are done on the following benchmark data:

Adult Income Dataset. 1 is a tabular dataset widely used in the modern fairness literature. It contains 48 842 samples and 15 features including the sensitive features race and sex to predict whether an individual makes more or less than $50 000 a year. It contains one third of women, of whom only 10 % are labelled as earning more than $50 000 a year.

Law School Dataset. 2 comes from the Law School Admission Council (LSAC) Research Report [START_REF] Linda F Wightman | Lsac national longitudinal bar passage study[END_REF]. It contains 20 798 samples and 12 features including the sensitive features race and sex to predict whether an individual will pass the exam to be admitted to law school. It contains 16 % of non-white, of whom 72 % are labelled as passing the bar exam compared to 92 % for white individuals.

We consider one single sensitive feature for each dataset to produce the following results that are respectively sex and race. The ROC AUC on the task of predicting s and y from the Adult raw dataset is respectively 0.942 and 0.926. For the Law School raw, it is 0.816 and 0.907.

Collapse & Triplet Selection Method

In this section, we present the observations deduced from our current preliminary experiments. The full experimental results are not included, but we present our most salient findings in Table 1, namely the results for the best performing activations and triplet selection methods, and in the appendices.

As expected, we can observe clusters in the corners of the embedding space. These clusters are not reduced to singular points and maintain some diversity.

When using the identical positive triplet selection , the embedding space collapses, because of the single repulsive force induced by such a triplet selection. Experiments with a counterfactual triplet selection result in a collapse, likely due to the fact that the flip of the sensitive feature s is not enough to change the embedding significantly. As such, it results in a behaviour similar to that observed with the identical positive triplet selection. Classical triplet selection did not collapse even after 1,000 epochs on the Adult dataset (see Figure 8 in the Appendix). However, the histogram of all distances between embeddings shows that it is in the process of separating into two distinct clusters, and the embeddings are not spread evenly between them. This supports the idea that a classical triplet selection induces an intra-class collapse [START_REF] Levi | Reducing class collapse in metric learning with easy positive sampling[END_REF] in slow motion. The space may has been better structured because of the remaining positive force.

More research is needed to ensure the space has time to acquire a desired structure before the collapse. Indeed, since the collapse is not finely controlled yet, we observe some variations in the results depending on the initialization of the optimisation.

Collapses & Features Scrambling

If the scrambling can be controlled so that s can't be recovered from the embedding while y still can, this would result in increased fairness (see next section). Here, we validate that a controlled collapse leads to the scrambling of features.

We use a the ROC-AUC of a Random Forest (RF) classifiers with 200 trees of depth 8 averaged on three different random seeds to evaluate how much a feature can be recovered from the embedding. The RF tries to predict s or y based on the embedding z. It is important to note that these models are used for evaluation only and have absolutely no impact on the training procedure.

For the Adult dataset, the highest ROC-AUC observed (i.e. the least scrambled) for both the target and the sensitive feature are of respectively 0.925 and 0.933, as we can see in Table 1. This serves as our baseline for what happens with this dataset if nothing is done to hide s.

A target-preserving triplet selection induces collapse where some target information is kept whereas a target-agnostic triplet selection may induce a total scrambling. In the Adult dataset, the classical and the counterfactual triplet selection methods have similar results in terms of AUC. They induce some drop on the target feature predictability, but most importantly a far more significant one for the sensitive feature. In the Law School dataset, the same behaviour is observed for the classical triplet selection, but not for the counterfactual. This is likely due to the problem being too hard on this dataset: it is here desirable to use a triplet selection that induces a collapse but tries to preserve some information on the target. This hypothesis is reinforced by the observation that on the Adult dataset that with the classical triplet selection, the sensitive feature is already scrambled at 100 epochs, which suggests that the space acquires a structure quickly for this simpler dataset.

With a lower margin, the learning stops before the collapse. We indeed observe that the AUC on the sensitive feature drops steadily while the AUC on the target feature remains similar when we progressively increase the margin until we reach an outrageous margin.

Collapse is needed for the total scrambling to take place.

We observe for both datasets that the sensitive and the target feature are scrambled and get poor AUC with the targetagnostic counterfactual triplet selection. It collapses in two or more singularities. The random triplet selection seems to shift the shape of the embedding space as previously mentioned, yet does not scramble the features in terms of AUC.

Discussion on our Solution for Learning Fair Representations

A collapse into multiple singularities in the embedding space would result in a fair classifier, if it produces a representation from which the sensitive feature cannot be recovered, while the target feature still can. Based on the insights from the previous section, such an outcome is possible if the following conditions are met:

• Using an activation providing corners in the embedding space

• Setting an outrageous margin

• Properly selecting the triplets One central result of this paper is that a classical triplet selection seems to scramble the sensitive feature by collapsing into multiple clusters (usually, but not always, one per class), despite the sensitive feature being completely ignored in the triplet selection process. This may be contrasted with the general scrambling of all features occurring in the target-agnostic counterfactual triplet selection. Collapses tend to scramble all features. However, when using a triplet selection method that separates embeddings based on a target feature, this target feature will be less scrambled.

As such, one can build a fair classifier by engineering a collapse that protects the target while scrambling the rest of the features (like with counterfactual or classical triplet selections) instead of scrambling them all (like with the targetagnostic counterfactual triplet selection). Compared to already existing approaches, our setup has the advantage of not being adversarial: the RFs are not used in training. Thus, our AUC can be considered unbiased unlike what was highlighted in [START_REF] Balunović | Fair normalizing flows[END_REF]. The many AUCs around 0.7 we observe on the sensitive feature are higher but not much higher than the one obtained by Balunovic et al. [START_REF] Balunović | Fair normalizing flows[END_REF], whose best AUC was 0.61. In any case, it is lower than the best AUC of 0.8 reached out for the adversarial methods (after correction) mentioned in the same paper. Our approach also has the advantage of being far more lightweight than Normalizing Flows-based methods.

A fair classifier emerges from a balance between collapse and target-preservation. Our best result regarding fairness is with the classical triplet selection. We hypothesise it reduces the AUC on the sensitive feature even if it is completely ignored by the selection because the sensitive feature is scrambled. This scrambling occurs because the space is given structure too quickly: far away elements exert a greater force on each other, so that a sample won't have the opportunity to leave a cluster to join another once it has entered it. This space structure acquired early cannot be corrected. One reason for the aforementioned phenomenon is that we randomly select valid samples for the triplets instead of using a semi-hard or hard selection. Our method ensures that all samples may be selected in a triplet and hence moved by the gradient during the entirety of the training. Secondly, the embedding has a high ROC-AUC in the first epochs of the training for both the sensitive and the target feature regardless of the triplet selection method. Thus, it was not obvious that a target-agnostic counterfactual selection collapse would scramble those features: this information could have been preserved by splitting into clusters. Finally, the ROC-AUC on the target is not increased in our approaches, suggesting that the learning does not simply converge to the trivial solution "embedding equals the target". It is indeed scrambling.

Essentially, the collapse induced by the outrageous margin and the repulsive force will outpace the target-preservation induced by the triplet selection. The next research question to be answered is how to efficiently find a balance point between them.

Conclusion

We proposed triplet learning collapse as a bias-mitigation technique to solve fairness issues in machine learning, after highlighting new insights on the associated optimisation. By controlling the behaviour of the associated triplet loss function, we are able to learn fair representations, in the sense that a predictor cannot recover the sensitive attribute, while the embedding retains enough information about the target attribute to solve the downstream classification task. Such a representation is computationally cheaper than approaches based on adversarial models or normalizing flows.

Future Work. Our highest priority is to conduct further experiments to provide convincing evidence for the insights presented in this paper. Then, our current analysis does not rely on any hyper parameter tuning, and thus performance can be further improved. Future experiments will focus on progressively increasing the optimisation strength, to wait until the space is well structured before forcing the collapse. One way to optimise would be to change the way the triplets are selected while the training is in progress. Additional efforts are also needed to validate that we can preserve any needed feature with the triplet selection. Experimenting with a control variable different than the target feature and doing a triplet selection in such a way that more than one feature is retained will strengthen our solution.

That is, this work could be extended to several sensitive features if we manage to retain several control variables, despite the scrambling.

Figure 1 :

 1 Figure 1: State of the embedding manifold at initialization, for L2 normalisation (A), with no activation (B), for L1 normalisation (C), for softmax activation (D). Bi-coloration helps distinguish between different values for the sensitive feature.

  With a target-agnostic counterfactual triplet selection , we observe what we call a "rushed" collapse, meaning the collapse occurs from the very first epochs of the training. For a random triplet selection , the embedding spaces at different epochs are isomorphic to each other: they vary in boundaries, but maintain somewhat constant relative positions and avert collapse. This can be explained by the average resulting force exerted on each point.

1 https://archive-beta.ics.uci.edu/ml/datasets/adult/adult.data 2 http://www.seaphe.org/databases/FOIA/lawschs1_1.dta

Table 1 :

 1 AUC comparison across triplet selection methods for Adult and Law School datasets. The identical positive triplet selection does not appear in this table because we do not expect a scrambling. Instead, the goal of this control experiment is to study the repartition of the samples in the embedding space.

	Dataset	Triplet Selection Method	Activation Margin ROC-AUC on Y ROC-AUC on S Epoch Collapsed?
	Adult	classical	batchnorm	0	0.925	0.933	1000	no
	Adult	classical	softmax	0	0.891	0.888	1000	no
	Adult	classical	softmax	1	0.918	0.792	1000	no
	Adult	classical	softmax	3	0.861	0.679	1000	no
	Adult	classical	softmax	100	0.861	0.679	1000	no
	Adult	counterfactual	softmax	100	0.792	0.738	1000	yes
	Adult	counterfactual	sigmoid	100	0.815	0.716	1000	yes
	Adult	counterfactual	tanh	100	0.834	0.735	1000	yes
	Adult	target-agnostic counterfactual	softmax	100	0.677	0.630	1000	yes
	Adult	random	softmax	100	0.870	0.862	100	no
	Law School target-agnostic counterfactual	softmax	100	0.533	0.553	1000	yes
	Law School counterfactual	softmax	100	0.6237	0.614	1000	yes
	Law School classical	none	100	0.953	0.963	100	no
	Law School classical	sigmoid	100	0.953	0.768	100	no
	Law School random	softmax	100	0.806	0.950	100	no

Acknowledgement

This work was supported by the French National Research Agency in the framework of the TAUDoS project (ANR-20-CE23-0020).

Appendix


Vincent Clair
email: vincent.clair@ec-lyon.fr

Edouard Salze

Pascal Souchotte

Emmanuel Jondeau

Turbulence interaction noise from a rectilinear cascade of airfoils and effects of porous material inclusions

Turbulence interaction noise from a rectilinear cascade of airfoils and effects of porous material inclusions

I. Introduction

Aircraft noise is a major source of nuisance for the communities living in the vicinity of airports. As the noise radiated from the engines is one of the main contributors to aircraft noise during the approach and take-off phases, its reduction is a major concern for engine manufacturers in order to comply with future regulations and guideline, such as the ACARE flight path 2050. Modern turbofan engines architectures known as Ultra-High Bypass Ratio (UHBR) engines aim at improving the engine efficiency by increasing the bypass ratio, using a larger fan with a lower rotational velocity. For such architectures, the fan-Outlet Guide Vanes (OGV) stage is one of the main source of engine noise, especially at approach conditions [START_REF] Envia | Fan Noise Reduction: An Overview[END_REF][START_REF] Peake | Modern Challenges Facing Turbomachinery Aeroacoustics[END_REF]. Typical fan-OGV noise consists in a tonal component at multiples of the Blade Passing Frequency (BPF) and a broadband component. The tonal component is mostly associated with the highly coherent wakes developing downstream of the fan and impinging on the OGV. This contribution has been widely studied in the literature and a number of solutions have been proposed and applied over the past decades [START_REF] Envia | Fan Noise Reduction: An Overview[END_REF]. For example, the number of fan blades and outlet guide vanes can be chosen to take advantage of duct propagation effects and ensure that the BPF is cut-off [START_REF] Tyler | Axial Flow Compressor Noise Studies[END_REF]. Acoustic liners are now commonly used in the casing of engines to damp tonal components as well [START_REF] Astley | Computational aero-acoustics for fan duct propagation and radiation. Current status and application to turbofan liner optimisation[END_REF]. The broadband component of fan-OGV noise is a more recent concern that emerged with the reduction of the tonal components and the move towards larger fan diameters. Broadband noise is associated with the finer and less coherent turbulent structures present in the boundary layers and in the wakes of the fan. The main broadband noise generation mechanism is known as turbulence interaction noise and is due to the turbulent structures of the fan's wake impinging on the OGV. Because of its wide frequency range and very rich modal content, some of the noise reduction solutions designed for tonal noise are much less efficient on broadband noise.

In the past decade, passive solutions were imagined and investigated to reduce this broadband component of the turbulence interaction noise. A first set of solutions, often referred to as serrations, consists in modifying the shape of the leading edge by introducing periodic patterns in the spanwise direction. The acoustic performances of serrations have been widely investigated, mostly on single airfoil configurations. Sinusoidal serrations are the most commonly studied [START_REF] Clair | Experimental and Numerical Investigation of Turbulence-Airfoil Noise Reduction Using Wavy Edges[END_REF][START_REF] Roger | Reduction of Airfoil Turbulence-Impingement Noise by Means of Leading-Edge Serrations and/or Porous Material[END_REF][START_REF] Chaitanya | Performance and mechanism of sinusoidal leading edge serrations for the reduction of turbulence-aerofoil interaction noise[END_REF][START_REF] Bampanis | Airfoil Turbulence-Impingement Noise Reduction by porous cells or wavy leading-edge design[END_REF][START_REF] Bampanis | On a three-dimensional investigation of airfoil turbulence-impingement noise and its reduction by leading-edge tubercles[END_REF], but other shapes such as the superposition of two sine curves [START_REF] Chaitanya | Aerofoil broadband noise reductions through double-wavelength leading edge serrations; a new control concept[END_REF] and a sawtooth pattern with or without slits [START_REF] Ayton | An analytical and experimental investigation of aerofoil-turbulence interaction noise for plates with spanwise-varying leading edges[END_REF] were also considered. These concepts showed promising performances, provided that the parameters defining the leading edge shape (pattern wavelength, serration depth) are carefully chosen with regards to the characteristic length of the incident turbulence.

Another kind of passive solutions that were investigated rely on the use of porous material on the OGV. Some of the work was dedicated to trailing edge noise [START_REF] Sarradj | Noise Generation by Porous Airfoils[END_REF][START_REF] Geyer | Trailing Edge Noise of Partially Porous Airfoils[END_REF] which is not the noise source of interest in the present paper. The use of porous material on airfoils was also considered as a way to reduce turbulence interaction noise and tested on single airfoils at relatively low flow velocities [START_REF] Roger | Reduction of Airfoil Turbulence-Impingement Noise by Means of Leading-Edge Serrations and/or Porous Material[END_REF][START_REF] Bampanis | Airfoil Turbulence-Impingement Noise Reduction by porous cells or wavy leading-edge design[END_REF][START_REF] Bampanis | On the Turbulence-Impingement Noise of a NACA-12 Airfoil with Porous Inclusions[END_REF]. Broadband noise reductions were observed, but the impact of such a concept on the aerodynamic performances of the airfoils remains to be assessed, particularly for loaded airfoils at higher flow velocities.

Turbulence interaction noise received a lot of experimental, numerical and analytical attention on single airfoil setups, especially for the assessment of noise reduction concepts. Analytical techniques have been developed to estimate the acoustic response of airfoil cascades [START_REF] Glegg | The Response of a Swept Blade Row to a Three-Dimensional Gust[END_REF][START_REF] Hanson | Theory for Broadband Noise of Rotor and Stator Cascades With Inhomogeneous Inflow Turbulence Including Effects of Lean and Sweep[END_REF][START_REF] Posson | On a uniformly valid analytical rectilinear cascade response function[END_REF][START_REF] Posson | On the use of a uniformly valid analytical cascade response function for fan broadband noise predictions[END_REF][START_REF] Bouley | On a two-dimensional mode-matching technique for sound generation and transmission in axial-flow outlet guide vanes[END_REF][START_REF] Bouley | Modélisations analytiques du bruit tonal d'interaction rotor/stator par la technique de raccordement modal[END_REF] impinged by turbulent flows, usually considering that the airfoils can be assimilated to flat plates with no thickness. Some work is ongoing to account for the camber of the airfoils [START_REF] Girier | A two-dimensional model of sound transmission through curved and staggered OGV: effect of inter-vane channel mode transitions[END_REF][START_REF] Girier | Mode-matching techniques for the modeling of rotor-stator wake-interaction tonal noise, with emphasis on the effects of vane camber[END_REF]. Experimentally, turbulence interaction noise on airfoil cascades configurations has received limited attention. A very early study took interest in the aerodynamic and acoustic performances of airfoils in a rectilinear cascade configuration, including the effects of leading edge serrations [START_REF] Smith | Cascade Tests of Serrated Leading Edge Blading at High Subsonic Speeds[END_REF]. More recent experimental work has been carried out in anechoic wind tunnels to investigate turbulence interaction noise radiated by a rectilinear cascade of airfoils [START_REF] Sabah | Etude expérimentale du bruit à large bande d'une grille d'aubes. Application au calcul du bruit des soufflantes[END_REF], an annular cascade of OGV [START_REF] Posson | Experimental Validation of a Cascade Response Function for Fan Broadband Noise Predictions[END_REF], and a rectilinear cascade of flat plates with leading edge serrations [START_REF] Mazella | Experimental investigation of the noise control performance of leading edge serrations in a rectilinear cascade[END_REF]. These studies were carried out at low to moderate flow speed.

The present paper is dedicated to an experimental study of the broadband noise radiated by a rectilinear cascade of airfoils impinged by grid generated turbulence, for flow velocities up to a Mach number M = 0.53. This test campaign took place within the framework of the European project InnoSTAT, aiming at developing and assessing broadband noise reduction concepts applied to OGVs in order to increase their Technology Readiness Level. Within this project, a concept of OGV with inclusions of porous materials was developed based on previous work [START_REF] Roger | Reduction of Airfoil Turbulence-Impingement Noise by Means of Leading-Edge Serrations and/or Porous Material[END_REF][START_REF] Bampanis | Airfoil Turbulence-Impingement Noise Reduction by porous cells or wavy leading-edge design[END_REF][START_REF] Bampanis | On the Turbulence-Impingement Noise of a NACA-12 Airfoil with Porous Inclusions[END_REF] and tested. The experimental setup of the test campaign, and the design of the airfoil with porous inclusions is presented in section II of this paper. The aerodynamic and acoustic results are then presented for the cascade nominal frontline angle in section III.

II. Experimental setup A. Test rig

The experimental campaign took place in LMFA's anechoic open jet wind tunnel at Ecole Centrale de Lyon, shown in figure 1. The airflow is generated by a centrifugal fan powered by an 800 kW engine, with a maximum mass flow rate of about 20 kg/s, and passes through a series of acoustic treatments before entering the 720 m 3 anechoic room through a 0.56 × 0.56 m 2 section. The cascade is composed of 7 NACA 7310 airfoils with a chord 𝑐 = 0.12 m, a span 𝐿 = 0.2 m and an inter-vane spacing 𝑠 = 0.085 m. One of the objectives of this experimental campaign is to perform acoustic measurements of the noise radiated by the cascade for flow velocities up to a Mach number of M = 0.53. A dedicated test section has been designed to install the airfoil cascade (see figure 2). A 0.5 m long nozzle adapts the test section from 0.56 × 0.56 m 2 to 0.56 × 0.2 m 2 . A turbulence grid can be inserted just upstream of the nozzle. The leading edge of the central airfoil is located at about 0.5 m downstream of the nozzle. The vertical side plates downstream of the nozzle are guiding the flow down to the trailing edge of the cascade, while the horizontal plates extend downstream of the cascade as semicircles centered on the trailing edge of the central airfoil. Brushes have been attached to the horizontal plates, as visible in figure 2 (right), to limit the presence of additional acoustic sources. The airfoils are mounted inbetween two large disks embedded in the horizontal plates, which allows the rotation of the cascade around the trailing edge of the central airfoil. Thus, the angle 𝛽 𝑐 between the incident flow and the normal to the cascade frontline, as defined in figure 3 (left), can be adjusted. In this experimental campaign, the stagger angle is kept constant at a value 𝜒 = 13 • . Thus, variations in the cascade frontline angle 𝛽 𝑐 correspond to variations of the incidence angle of the airfoils 𝛼 𝑐 . The nominal value of the cascade frontline angle is 𝛽 𝑐 = 34 • and it can vary in the range 𝛽 𝑐 = 34 ± 8 • . The nominal and extreme angular positions of the airfoils are shown in figure 3 (right). Note that, because the flow is deflected by the airfoil cascade, a bend is located upstream of the nozzle (see figure 2) to ensure that the flow downstream of the cascade exits the anechoic room through the dedicated opening and does not impinge on the acoustic treatments of the room. A set of deflector plates is installed in this bend to allow for the flow rotation without separation.

B. Instrumentation

Aerodynamic measurements are realized upstream of the cascade to check the homogeneity of the flow through the test section, and to characterize the turbulent flow generated by the turbulence grid. The locations of these measurements is presented in figure 4 (left). The green line corresponds to a set of 35 static pressure probes, flush mounted on the lower horizontal plate of the test section to assess the homogeneity of the flow in the transverse direction (notably to check that there is no separation because of the bend installed upstream of the test section). The red lines and dots correspond to hot-wire measurements. Velocity profiles are measured in the mid-span plane over lines parallel to the cascade front line located 0.2 m and 0.05 m upstream of the cascade. These profiles are measured both with a single component hot-wire probe and with a two component cross wire probe, shown in figure 4 (bottom, right). Boundary layer velocity measurements are also performed over the lower horizontal plate with a hot-wire probe just downstream of the nozzle, as visible in figure 4 (top, right). Additionally, velocity profiles are also measured in the wake of the airfoil, 0.015 m downstream of the trailing edges, with a single component hot-wire probe. These wake profiles are only measured for the nominal value of the cascade frontline angle 𝛽 𝑐 = 34 • . Additionally, for the reference configuration (rigid airfoils), the central airfoil is equipped with 24 pinholes connected to tubes that allow to perform both static pressure measurements and fluctuating pressure measurements using remote microphones and a dedicated calibration procedure. The positions of the pinholes on the surface of the airfoil is shown in figure 5 (right).

C. Airfoil with porous inclusions design

The concept tested in this experimental campaign relies on previous work by Bampanis [START_REF] Bampanis | Airfoil Turbulence-Impingement Noise Reduction by porous cells or wavy leading-edge design[END_REF] and Bampanis & Roger [START_REF] Bampanis | On the Turbulence-Impingement Noise of a NACA-12 Airfoil with Porous Inclusions[END_REF] on a single airfoil configuration at low flow velocities. In their work, the design consisted in a rigid centerplate with stiffeners following the shape of the targeted airfoil geometry. The space between the chordwise and spanwise stiffeners was filled with a porous material (melamine foam or metal wool) and a wiremesh was attached to the stiffeners to form the surface of the airfoil.

The noise reduction effects were expected to be associated with two mechanisms. The first one is an attenuation of the impingement of turbulent structures in the leading edge region, which is the source of turbulence interaction noise. The second effect is related to the propagation of the acoustic waves radiated from the leading edge along the airfoil, acting as an acoustic liner.

The airfoil considered was symmetric (NACA 0012), and it was tested at a 0 • angle of attack, thus with no mean loading. The possibility to leave a gap between the leading edge and the rigid centerplate was investigated and showed to improve the acoustic performances of the airfoil. A drawback of this idea is that, for a loaded airfoil, some flow would go through the airfoil in the gap and deteriorate the aerodynamic performances of the airfoil. The main difficulty encountered in this previous work is associated with the wiremesh being attached only to the stiffeners to form the surface of the airfoil. Even at low flow flow velocities, it was observed that the wiremesh would deform, and thus not preserve the airfoil shape. This is expected to be even more problematic for higher flow velocities and for loaded airfoils.

In the present experiments, the flow velocity is up to M = 0.53 and the airfoils of the cascade are highly loaded. Thus, the solution previously investigated by Bampanis & Roger had to be adapted to provide more rigidity to the surface of the airfoil and prevent deformation. The design still relies on a rigid frame, consisting in a centerplate that follow the camber line of the airfoil and stiffeners in the streamwise and spanwise direction. In the trailing edge region, because of the limited thickness available, the airfoil is fully rigid. A view of this rigid frame is visible in figure 6 (left). The spaces between the stiffeners is filled with melamine foam. Note that, because of the overall thickness of the airfoil, the maximum thickness of melamine foam on each side of the airfoil is about 5 mm. In order to provide a more rigid layer at the surface of the airfoil, a perforated plate with square perforations is bent to the shape of the airfoil and welded on top of the rigid frame. This grid also provides for more surface to glue the wiremesh on top of it. The final assembly can be seen in figure 6 (right), where the outline of the grid is visible under the wiremesh.

In terms of the noise reduction effects described above, the addition of the grid reduces the porosity of the surface and might thus lead to a reduced efficiency. However, as the airfoils will be installed in a cascade configuration, the absorption of acoustic waves propagating between the airfoils might be improved compared to a single airfoil configuration.

III. Results at the nominal cascade frontline angle

In this section, the results are presented for the nominal value of the cascade frontline angle 𝛽 𝑐 = 34 • . The trends observed at other values of 𝛽 𝑐 considered in the test campaign do not differ significantly.

A. Upstream flow characteristics

The flow is first investigated upstream of the cascade to assess its homogeneity through the test section. The velocity spectrum of the turbulence generated by the turbulence grid located upstream of the nozzle is also characterized, as it is the main parameter of the turbulence interaction noise mechanism, which is the source of interest in this work.

Figure 7 shows the static pressure profiles measured in the transverse direction along the lower horizontal plate, upstream of the airfoil cascade (but downstream of the nozzle). These results show a uniform distribution of static pressure across the section which is indicative that the flow does not display any region of separation due to the presence of the bend upstream of the nozzle. For the two flow velocities presented, only a ±1% deviation from the averaged value is visible. These profiles were measured without the turbulence grid and it was observed that the distance between the velocity deficits corresponds to the distance between the deflector plates located in the bend upstream of the nozzle.

The turbulent velocity spectra calculated from single hot-wire measurements at the position circled in blue in figure 4 (left) are presented in figure 9 in black lines for two flow velocities. These experimental spectra can be used to estimate an integral length scale Λ and the turbulent intensity 𝑇 𝐼 of the turbulence impinging on the cascade.

The turbulent intensity is defined as 𝑇 𝐼 = √ 𝑢 ′2 /𝑈 2 0 , where 𝑈 0 is the mean value of the flow velocity and 𝑢 ′2 is the mean square value of the velocity fluctuation component measured by the hot-wire probe. The latter can be calculated from the experimental one-sided frequency spectra showed in figure 9:

𝑢 ′2 = ∫ +∞ 0 𝑆 𝑢𝑢 ( 𝑓 ) d 𝑓 (1)
where 𝑆 𝑢𝑢 is the velocity spectrum. The integral length scale can be calculated as:

Λ = 𝑈 0 𝑆 𝑢𝑢 ( 𝑓 = 0) 4𝑢 ′2 (2)
In practice, an average of the experimental spectra over the first 10 points is used instead of 𝑆 𝑢𝑢 ( 𝑓 = 0). These values of 𝑢 ′2 and Λ can then be used in a standard homogeneous and isotropic turbulent velocity spectrum model such as the Liepmann model to assess its adequacy with the experimental results. It is also possible to assume a turbulence model (here the Liepmann spectrum) and perform a least square fit of the model to the experimental data to deduce values for 𝑢 ′2 and Λ.

The expression the one-sided frequency spectrum of the axial velocity fluctuation for the Liepmann model is:

𝑆 𝑢𝑢 ( 𝑓 ) = 4𝑢 ′2 Λ 𝑈 0 1 1 + 𝑘 2 1 Λ 2 (3)
with 𝑘 1 = 2𝜋 𝑓 /𝑈 0 .

In figure 9, the Liepmann spectra calculated using the theoretical values of 𝑢 ′2 and Λ as defined in equations ( 1) and ( 2) are shown in blue lines, and the Liepmann spectra deduced from a least square fit are shown in red lines. It can be seen that both the theoretical spectrum and least square fit match very well with the experimental data. Only a slight vertical shift can be observed between the two Liepmann spectra at 𝑈 0 = 43 m/s. The values of 𝑇 𝐼 and Λ deduced from equations ( 1) and (2), as well as from the least square fit of a Liepmann model are presented in table 1. The two set of values are very close to one another. It can be noted that the values of 𝑇 𝐼 and Λ are evolving slightly between the two flow velocities presented.

B. Wake profiles

Velocity measurements have been performed in the wake of the cascade using a single component hot-wire probe, 15 mm downstream of the trailing edges along a line represented in red in figure 4 (left). The velocity profiles (mean and RMS values) are normalized by the mean velocity measured in the middle of a cascade channel (between two airfoils).

The mean velocity wake profiles are presented in figure 10 for two flow velocities. For the porous configuration, the maximum velocity deficit is significantely larger than for the reference, and the wake is much wider. The widening of 1 Values of the turbulence intensity and the integral length scale deduced from the experimental velocity spectra.

the wake appears even more important for the highest velocity. The position of the wake is shifted towards the suction side for the porous configuration. Overall, these wakes may be indicative of a flow separation on the suction side, or at least an important thickening of the boundary layers, and lead to a deterioration of the aerodynamic performances for the porous configuration.

The profiles of the RMS value of the velocity fluctuations measured in the wake are shown in figure 11. The amplitude and width of the velocity fluctuation are increased for the porous case, particularly at the highest velocity. The trace of the boundary layers forming on both sides of the airfoil are visible on these profiles. It appears that the thickness of the boundary layer on the suction side has increased a lot for the porous airfoil, which is consistent with the hypothesis of a flow separation.

It is unclear if this supposed separation is associated with the behavior of the flow along the airfoil, as the fluid may flow through the porous surface and interact with the stiffeners, or if it associated to a difference in the geometry of the airfoil compared to the reference because of its complex assembly.

C. Acoustic results

In this section, the acoustic results will be presented for two upstream flow velocities corresponding to Mach numbers M = 0.34 and M = 0.53.

First, the Power Spectral Density (PSD) of the pressure fluctuation measured by the microphone located at 𝜃 = -60 • on the downstream circular array are presented in figure 12. The reference pressure used for the dB scale is 𝑝 𝑟 𝑒 𝑓 = 2 × 10 -5 Pa. For the two flow velocities considered, it appears that the use of porous airfoils leads to an increase of the noise up to 2-2.5 kHz. This increase is particularly visible from 1 kHz to 2.5 kHz. At higher frequencies, a reduction of the noise is observed consistently up to at least 10 kHz. Since the most energetic part of these spectra is located below 2 to 3 kHz, this would indicate that the overall performances of the porous airfoils may not be positive in terms of noise reduction. The Overall Sound Pressure Levels (OASPL), integrated between 200 Hz and 10 kHz, measured over the downstream circular array are shown in figure 13. These confirm that, over this whole frequency range, the porous airfoils do not seem to perform very well. A small but consistent increase in OASPL of about 1 dB can be seen at M = 0.34. For M = 0.53, the OASPL of the reference and porous configurations show very similar results, with a small but almost insignificant noise reduction at some angles for the porous configuration.

Using the microphone measurements over the downstream circular array, the acoustic power radiated by the cascade in the downstream direction can be estimated, assuming that the radiation from the cascade behaves as spherical waves in the far-field. Thus, the acoustic power radiated through a surface 𝑆 can be calculated by:

𝑊 ( 𝑓 ) = ∫ 𝑆 𝑆 𝑝 𝑝 ( 𝑓 ) 𝜌 0 𝑐 0 d𝑆 (4)
where 𝑆 𝑝 𝑝 is the power spectral density of the pressure fluctuation, 𝜌 0 the mean density and 𝑐 0 the speed of sound at the microphone positions. Here, since the microphones of the circular array have an angular spacing Δ𝜃 = 10 • , the surface element used for each microphone to approximate the integral in equation ( 4) corresponds to a sphere surface element with 𝑅 = 1.95 m and Δ𝜃 = Δ𝜑 = 10 • .

The acoustic power in dB is then defined as:

𝑃𝑊 𝐿 ( 𝑓 ) = 10 log 10 𝑊 ( 𝑓 ) 𝑊 𝑟 𝑒 𝑓 (5)
with 𝑊 𝑟 𝑒 𝑓 = 1 × 10 -12 W. The downstream acoustic power spectra are presented in figure 14. They show trends that are quite similar to the PSD shown in figure 12, with an increase of the noise radiated below 2-2.5 kHz (especially from 1 kHz to 2.5 kHz), and a noise reduction at higher frequencies. Upstream of the cascade, in the duct, the pressure signals recorded by a flush mounted microphone alone would not be indicative of the noise radiated by the airfoil cascade, as these signals also contain the wall pressure fluctuations of the turbulent boundary layer developing along the plate. Thus, as mentioned in section II.B, the duct section upstream of the cascade is equipped with a linear array of microphones flush mounted on the lower horizontal plate and aligned in the axial direction. This array can be used to separate the acoustic waves propagating upstream (radiated by the airfoil cascade) from the perturbations traveling in the downstream direction, such as the wall pressure fluctuations of the turbulent boundary layer, as depicted in figure 15.

x ξ ij k k x M 26 M 1 M i M j
where 𝜉 𝑖 𝑗 = 𝑥 𝑗 -𝑥 𝑖 is the separation between the microphones 𝑖 and 𝑗. The elements of 𝑆 𝑖 𝑗 are then ranked by increasing value of 𝜉 𝑖 𝑗 and, when several pairs of microphones have the same value of 𝜉 𝑖 𝑗 , only one pair is kept. This results in an array 𝑆(𝜉, 𝜔) containing a cross-spectrum for each value of the separation 𝜉 that can be obtained with the microphone array.

The wavenumber-frequency spectrum is then defined by:

Φ 𝑝 𝑝 (𝑘 𝑥 , 𝜔) = 1 2𝜋 ∫ ∞ -∞ 𝑆(𝜉, 𝜔)e -i𝑘 𝑥 𝜉 d𝜉 (7)
A schematic of the different regions of the wavenumber-frequency spectrum Φ 𝑝 𝑝 corresponding to upstream or downstream propagating perturbations of an acoustic or turbulent nature is presented in figure 16.

Finally, the power spectral density of the pressure fluctuations propagating in the upstream direction 𝑆 (-) 𝑝 𝑝 can be retrieved by integrating Φ 𝑝 𝑝 over 𝑘 𝑥 < 0: This is demonstrated in figure 18 where the raw pressure spectrum or one of the microphones from the linear array is compared to the filtered pressure spectrum 𝑆 (-) 𝑝 𝑝 containing only the upstream propagating fluctuations, calculated using equation [START_REF] Bampanis | Airfoil Turbulence-Impingement Noise Reduction by porous cells or wavy leading-edge design[END_REF]. The levels of the filtered spectrum are well below the levels of the raw spectrum over most of the frequency range, but especially at low frequencies, below 600 Hz, and at high frequencies, above 2 kHz. The capability to separate the upstream propagating components of the pressure fluctuations is thus necessary to compare the acoustic performances of the porous airfoil concept to the reference. The same filtering procedure is applied to the measurements performed with the porous airfoil concept, and the upstream propagating pressure spectra are compared to the reference airfoil cascade in figure 19 for the two Mach numbers M = 0.34 and M = 0.53, and with a cascade frontline angle 𝛽 𝑐 = 34 • . The observations are slightly different than for the downstream PWL presented in figure 14. The upstream pressure spectra for the porous airfoil concept show lower levels of about 2 to 3 dB for frequencies below 600 Hz. The significant noise increase that was observed downstream around 1 kHz to 2.5 kHz does not appear upstream. Finally, there appear to be some noise reduction above 2.5 kHz, similar to what was observed downstream. Overall, the porous airfoil concept developed in this work appear to show some noise reduction at high frequencies, that are unfortunately counterbalanced by an increase of the noise at lower frequencies downstream of the cascade, in the 1 kHz to 2.5 kHz range, where most of the acoustic energy is located. This noise increase is mostly visible downstream of the cascade, but not upstream. The origin of this noise increase is unclear, but may be associated with a possible flow separation for the porous airfoil concept and requires further investigations.

𝑆 (-) 𝑝 𝑝 (𝜔) = ∫ 0 -∞ Φ 𝑝 𝑝 (𝑘 𝑥 , 𝜔) d𝑘 𝑥 (8) k x f k c k 0 =(1 + M ) -k 0 =(1 -M )

IV. Conclusion

In this paper, the broadband noise resulting from the interaction of a turbulent flow with a rectilinear cascade of airfoils is investigated experimentally in an anechoic wind tunnel. This noise generation mechanism is one of the major source of turbofan engine noise, when the wakes from the fan interact with the OGVs. An airfoil concept consisting in inclusions of porous material in the airfoil and a wiremesh at the surface of the airfoil as been designed and tested in the rectilinear cascade configuration. As the airfoils are loaded and the flow velocities under consideration are up to a Mach number of 0.53, the design of the porous airfoil had to be adapted from previous work for these conditions. Upstream of the cascade, static pressure and hot-wire anemometry measurements were performed and showed the flow to be homogeneous through the test section. The turbulence spectra are consistent with a classical isotropic turbulence model. Hot-wire measurements performed in the wake of the airfoils, near the trailing edge, showed that the wakes for the porous airfoil concept are much wider and have a larger velocity deficit than those of the reference airfoil. The velocity fluctuations in the wake are also much larger for the porous concept. This suggests a possible flow separation for the porous concept, even though the origin of this separation remains unclear at the moment. Acoustic measurements were performed both downstream of the cascade in the far-field, and upstream of the cascade in the duct. The results downstream show a noise reduction with the porous concept above 2.5 kHz, which is counterbalanced by an increase of the noise between 1 kHz and 2.5 kHz, where most of the energy is located in the spectra. This results in an slight overall noise increase at M = 0.34 at all observation angles, and a minimal noise reduction at M = 0.53. Upstream of the cascade, measurements performed with a linear array of microphones aligned in the axial direction are used to separate the noise radiated from the cascade from the turbulent boundary layer pressure fluctuations. The acoustic spectra display a noise reduction above 2.5 kHz similar to the one observed downstream. At lower frequency, no noise increase is visible and a noise reduction is even visible below 600 Hz, contrary to the noise increase observed downstream. Overall, the porous airfoil concept shows a net noise reduction upstream of the cascade, but the noise increase at mid-frequencies downstream of the cascade balances the gains visible at higher frequencies. As the porous airfoil was shown to have much larger and more turbulent wakes, the hypothesis that this noise increase is related to a possible flow separation needs to be investigated further.

Fig. 1

 1 Fig. 1 Anechoic wind tunnel at Ecole Centrale de Lyon.

Fig. 2

 2 Fig. 2 Test section designed for the experimental campaign.

Fig. 3

 3 Fig. 3 Definitions of the stagger angle 𝜒, the incidence angle 𝛼 𝑐 and the cascade front angle 𝛽 𝑐 (left). Positions of the airfoils for values of the 𝛽 𝑐 angle ±8 • around its nominal value.

Fig. 4

 4 Fig. 4 Positions of the static pressure probes in green and of the hot-wire measurements in red (left). Picture of the hot-wire probes used for the boundary layer measurements (right, top). Picture of the X wire probe used for the 2 components velocity measurements (right, bottom).

Fig. 5

 5 Fig. 5 Positions of the microphones in the test section and on the far-field array (left). Positions of the pinholes used for static pressure and fluctuating pressure measurements on the surface of the central vane for the reference geometry (right).

Fig. 6

 6 Fig. 6 Airfoil with porous inclusions. CAD view of the rigid centerplate and stiffeners (left), and picture of an assembled airfoil (right).

Fig. 7

 7 Fig. 7 Transverse profiles of the static pressure upstream of the cascade for the upstream flow velocities: 45 m/s (black), 75 m/s (red) and 100 m/s (blue).

Figure 8

 8 Figure 8 presents the mean velocity profiles measured with a single component hot-wire probe along a line parallel to the cascade frontline, located 0.2 m upstream of the airfoil leading edges. The position of this profile is visible in figure 4 (left) in red. The mean velocity profiles are normalized by the average value calculated over the whole profile.For the two flow velocities presented, only a ±1% deviation from the averaged value is visible. These profiles were measured without the turbulence grid and it was observed that the distance between the velocity deficits corresponds to the distance between the deflector plates located in the bend upstream of the nozzle.The turbulent velocity spectra calculated from single hot-wire measurements at the position circled in blue in figure 4 (left) are presented in figure9in black lines for two flow velocities. These experimental spectra can be used to estimate an integral length scale Λ and the turbulent intensity 𝑇 𝐼 of the turbulence impinging on the cascade.

Fig. 8

 8 Fig. 8 Normalized mean velocity profiles 0.2 m upstream of the cascade, measured without the turbulence grid, for the upstream flow velocities: 45 m/s (blue) and 100 m/s (red).

Fig. 9

 9 Fig. 9 Turbulent velocity spectrum of the axial velocity component measured 0.2 m upstream of the cascade.

Fig. 10

 10 Fig. 10 Mean velocity profiles measured downstream of the airfoils, normalized by the mean velocity in the middle of a cascade channel. The cascade frontline angle is 𝛽 𝑐 = 34 • and the upstream flow has a velocity 𝑈 0 = 46 m/s (left) and 𝑈 0 = 100 m/s (right).

Fig. 11 RMS

 11 Fig. 11 RMS velocity profiles measured downstream of the airfoils, normalized by the mean velocity in the middle of a cascade channel. The cascade frontline angle is 𝛽 𝑐 = 34 • and the upstream flow has a velocity 𝑈 0 = 46 m/s (left) and 𝑈 0 = 100 m/s (right).

Fig. 12 Power

 12 Fig. 12 Power Spectral Density of the pressure fluctuation measured on the downstream circular array at 𝜃 = -60 • (microphone 33). The cascade frontline angle is 𝛽 𝑐 = 34 • and the upstream flow has a Mach number M = 0.34 (left) and M = 0.53 (right).

Fig. 13 Overall

 13 Fig. 13 Overall Sound Pressure Level, integrated between 200 Hz and 10 kHz, measured on the downstream circular array. The cascade frontline angle is 𝛽 𝑐 = 34 • and the upstream flow has a Mach number M = 0.34 (left) and M = 0.53 (right).

Fig. 14

 14 Fig. 14 Acoustic power spectra integrated over the downstream circular array. The cascade frontline angle is 𝛽 𝑐 = 34 • and the upstream flow has a Mach number M = 0.34 (left) and M = 0.53 (right).

Fig. 15

 15 Fig. 15 Schematic of the upstream linear microphone array, and of the upstream and downstream propagating pressure perturbations.

Fig. 16

 16 Fig. 16 Regions of the wavenumber-frequency spectrum 𝜙 𝑝 𝑝 (𝑘 𝑥 , 𝜔) corresponding to upstream or downstream pressure perturbations associated with acoustic waves or the turbulent boundary layer.

Fig. 17

 17 Fig. 17 Wavenumber-frequency spectrum Φ 𝑝 𝑝 for the reference case with a flow Mach number M = 0.34 and a cascade frontline angle 𝛽 𝑐 = 34 • .

Fig. 18

 18 Fig. 18 Comparison between the raw Power Spectral Density of the pressure fluctuations measured for a microphone of the array (gray) and the filtered spectrum accounting only for the upstream propagating waves (black). Reference case with a flow Mach number M = 0.34 and a cascade frontline angle 𝛽 𝑐 = 34 • .

Fig. 19 Power

 19 Fig. 19 Power Spectral Density of the pressure fluctuation radiated upstream, measured by the linear microphone array. The cascade frontline angle is 𝛽 𝑐 = 34 • and the upstream flow has a Mach number M = 0.34 (left) and M = 0.53 (right).

Acknowledgments

This work was supported by the Labex CeLyA of the Université de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ ANR-16-IDEX-0005).

This project has received funding from the Clean Sky 2 Joint Undertaking (JU) under agreement n°865007. The JU receives support from the European Union's Horizon 2020 research and innovation program and the Clean Sky 2 JU members other than the Union. This publication reflects only the author's view and the JU is not responsible for any use that may be of the information it contains.