Céline De Looze
email: celine.delooze@lpl-aix.fr

Stéphane Rauzy
email: stephane.rauzy@lpl-aix.fr

Automatic Detection and Prediction of Topic Changes Through Automatic Detection of Register variations and Pause Duration

Keywords: register variations, pause duration, topic changes, automatic detection and prediction

In this article a clustering algorithm, allowing the automatic detection of speakers' register changes, is presented. Together with automatic detection of pause duration, it has shown to be efficient for the automatic detection and prediction of topic changes. The need to take into account other parameters such as tempo and intensity, in the framework of Linear Discriminant Analysis, is proposed in order to improve the identification of the topic structure of discourse.

Introduction

In this article, a clustering algorithm is presented, which allows the automatic detection of a speaker's changes of register. Developed to be implemented in the automatic coding of intonation models, it has been used, together with automatic detection of pause duration, to detect and predict topic changes automatically. We assume that two types of fundamental frequency patterns need to be distinguished: on the one hand, local pitch characteristics corresponding to changes in the phonological representation of intonation and, on the other hand, more global pitch changes determined by variations in register as defined by key (or level) and span (or range). Assuming that a speaker's register may vary, especially when analyzing spontaneous speech, and that these variations may convey linguistic, extra-linguistic as well as para-linguistic functions, will certainly improve the (automatic) description of intonation patterns. More particularly, register is reported to throw light on the informational organisation of discourse structure, the information weight carried by the discourse element compared to its preceding or following neighbour as well as the hierarchical dimension and relational organisation of linguistic units. Discourse units are more or less glued together depending on their semantic/pragmatic relations so as to form a coherent whole. It has been shown, for example, in many languages, that the first sentences of paragraphs in reading tasks are uttered with higher register than sentences within the paragraph, a pitch reset mostly explained by the introduction of a new topic ([1], [START_REF] Brazil | Discourse Intonation and Language Teaching[END_REF], [START_REF] Menn | Fundamental frequency and discourse structure[END_REF], [START_REF] Pierrehumbert | The Meaning of Intonational Contours in the Interpretation of Discourse[END_REF], [START_REF] Arons | Pitch-based Emphasis Detection for Segmenting Speech Recordings[END_REF], [START_REF] Kong | The role of pitch range variation in the discourse structure and intonation structure of Korean[END_REF], [START_REF] Mayer | Pitch range and pause duration as markers of discourse hierarchy: perception experiments[END_REF], [START_REF] Chiu-Yu | Investigating F0 reset and Range in relation to Fluent Speech Prosody Technical Acoustics[END_REF]). Authors also report a declination or downtrend throughout the paragraph, the last sentences being realized in a low and compressed register ( [START_REF] Menn | Fundamental frequency and discourse structure[END_REF], [START_REF] Thorsen | Text and intonation. A case study[END_REF], [START_REF] Hirschberg | Intonational structuring of discourse[END_REF], [START_REF] Nakajima | Prosody as a cue for discourse structure[END_REF], [START_REF] Sluijter | Beyond Sentence Prosody: Paragraph Intonation In Dutch[END_REF], [START_REF] Nicolas | Symbolic coding of higher level characteristics of fundamental frequency curves[END_REF]). It appears in fact that the higher an element is connected in the text or in the discourse structure, the higher the register ( [START_REF] Fon | A cross-linguistic study and discourse boundary cues in spontaneous speech[END_REF], [START_REF] Den Ouden | Prosodic realizations of global and local structure and rhetorical relations in read aloud news reports[END_REF]). Register expansion is therefore associated with elements carrying new (or relevant) information, indicating topic change and usually positioned at the beginning of the structure. Register lowering is used for topic continuity, where elements within the lowering convey the same information.

Register compression is found at the end of a topic discussed, hence found in final parts of the discourse. Without expectation then, sub-topics and parenthetical comments are reported to be associated with a compressed register ( [START_REF] Kutik | Declination of fundamental frequency in speakers' production of parenthetical and main clauses[END_REF]). Consequently, according to what has been found in the literature, we expect the clustering algorithm to detect a register reset at the beginning of a discourse structure or at the beginning of a new discussed issue, separating these units from the preceding ones in the clustering structure. We also suppose that the algorithm may detect a declination trend throughout the discussed issue, specifically at the end of the discourse structure, grouping together the units that are semantically linked. This last point which is currently under investigation, it will not be presented in this article. Such a function-oriented approach confronted with acoustic data may indeed allow the automatic extraction and prediction of functional information, hence contributing to the automatic mapping of prosodic form and function for speech synthesis.

Corpora

Four corpora were used in this study: PAC (Phonologie de l'Anglais Contemporain, [START_REF] Carr | La Phonologie de l'Anglais Contemporain : usages, variétés et structure : The Phonology of Contemporary English: usages, varieties and structure[END_REF]) -A total of 30 minutes of newspaper article-like readings were selected from the PAC (5 female and 3 male speakers from Northern England; Lancashire, Greater Manchester and West Yorkshire).

AIX-MARSEC ([17]

) -A total of 54 minutes of recording (15 female and 39 male speakers of standard British English) were selected from the AIX-MARSEC corpus. Mainly prepared monologues, the recordings correspond to commentaries, new broadcasts, lectures, religious broadcasts, magazine-style reporting, fiction, poetry, dialogues and propaganda. PFC (Phonologie du Français Contemporain, [START_REF] Delais-Roussarie | Corpus et variation en phonologie du français: méthodes et analyses[END_REF]) -A total of 80 minutes of recording (6 female and 4 male speakers of regional French -Marseille) were selected from the PFC corpus. The recordings, being of three different speaking styles consist of newspaper article-like readings as well as guided and spontaneous conversations. CID (Corpus of Interactional Data, [START_REF] Bertrand | Le CID-Corpus of Interactional Data-: protocoles, conventions, annotations[END_REF]) -A total of 30 minutes of dialogue recording in a sound-proof room (3 female and 3 male speakers of regional French -Marseille) were selected from the CID corpus.

Measuring speakers' register

Detecting variations in speakers' register implies at first the efficient detection of their global shape, i.e. their global key and span. Many studies have used median or mean f0 to express register key and the difference between the extreme values or the standard deviation to express its span ( [START_REF] Huttar | Relations between prosodic variables and emotions in normal American English utterances[END_REF], [START_REF] Jassem | Pitch and compass of the speaking voice[END_REF], [START_REF] Van Bezooijen | Sociocultural aspects of pitch differences between Japanese and Dutch women[END_REF], [START_REF] Clark | Using prosodic structure to improve pitch range variation in text to speech synthesis[END_REF], [START_REF] Mozziconacci | Speech variability and emotion: production and perception[END_REF], [START_REF] Lennes | Prosodic features associated with the distribution of turns in Finnish informal dialogues[END_REF], [START_REF] Rietveld | Cues for perceived pitch register[END_REF]). However, since fo detection is very sensitive to microprosodic effects and octave errors, hence resulting in very error-prone results, many authors have chosen, instead of the extrema, to limit their measure to quantiles, the difference between the 90 th and 10 th quantiles or the 95 th and the 5 th quantiles for example ( [START_REF] Patterson | Pitch Range Modelling: Linguistic dimensions of variation[END_REF], [START_REF] Patterson | A linguistic approach to pitch range modeling[END_REF]). Therefore, an experiment was conducted ( [START_REF] De Looze | In progress. Analyse et interprétation de l'empan temporel des variations prosodiques en français et en anglais contemporain[END_REF]) to test which quantiles give the best estimate of a speaker's register. First, manual annotation of maximum and minimum pitch of the AIX-MARSEC and PFC speakers were made from the editor window of the objects Sound and Pitch (the scale being adapted for each speaker's voice), thus dismissing microprosodic effects and octave errors in the analysis; then, a comparative study of different quantiles (from q05 to q95) was carried out to estimate which were best correlated with manual estimate of pitch extrema. In [START_REF] De Looze | Detecting changes in key and range for the automatic modelling and coding of intonation[END_REF], the formulae 1.75*q75 and 0.75*q25 were taken to give best estimations of fo extrema while implemented as ceiling and floor (with +/-10 Hz) in the MOMEL-INTSINT algorithm ( [START_REF] Hirst | A Praat Plugin for MOMEL and INTSINT with improved algorithms for modelling and coding of intonation[END_REF], [START_REF] Boersma | Praat: doing phonetics by computer[END_REF]). New investigation showed that q65*1.90 and q35*0.72 slightly improve the estimation of these extrema. Comparing the detection of the manually annotated extrema (MMIN, MMAX; serving as reference) to the extrema values as obtained with the defined formulae (MIN35; MAX65) and to the extrema values as obtained in Praat (MINPRAAT, MAXPRAAT; i.e. by using the default floor of 75 and ceiling of 600 and then the functions Get minimum... and Get maximum...), it appears that the algorithm greatly improves the detection of speakers' register key and span (Figures1&2).

Automatic detection of variations in register key and span

In [START_REF] De Looze | Automatic detection of register changes for the analysis of discourse structure[END_REF], an improved version of our clustering algorithm, allowing the automatic detection of register variations, was proposed. First, the algorithm (1) recursively reduces the Euclidian distance between two consecutive tonal units in a space defined by key and span parameters (normalized values), and (2) calculates the difference between two consecutive units for key and span separately. Key is calculated in hertz then normalized with a logarithmic transformation. Span is calculated in semitones, thus normalized too. The formulae used in the calculation of the differences in key and span (DIFFKEY, DIFFSPAN) are given in (f1) and (f2). (f1) sqrt(log2(median_unit)log2(median_prevUnit))^2 (f2) sqrt(log2(max/min_unit)-log2(max/min_prevUnit))2 Then, after obtaining consecutive differences between units, the clustering algorithm groups the units together according to their difference in key and span. The smaller the difference between two units, the sooner these units are branched together. The output generated by the algorithm is a binary tree structure in the form of a layered icicle diagram (Figure 3). This representation allows the definition of the hierarchical structure and relational organisation of tonal units as reflected by register changes. Groups of units are therefore distinguished and an analysis of the distance between the leaf nodes according to key and span parameters (NDKEY & NDSPAN) allows boundary strength measurements between them. The larger the distance, the stronger the boundary between two groups. On the contrary, a short distance suggests that two consecutive units belong to the same group of units. The benefit of such an algorithm is that different units may be tested and that smaller units may be grouped together so as to indicate which unit may be under investigation, allowing "theory-neutral" analyses.

Figure 3: Extract of a layered icicle diagram representation as obtained with the algorithm. The representation suggests that units "Christmas" to "evangelist" belong to the same group and are separated from the group of units starting with "if television". In fact, the distance between leaf nodes "evangelist" and "if television" indicates the presence of a strong boundary. The colour scale used indicates register key for each unit. The darker the colour, the higher the key.

Detection and prediction of topic changes

Manual annotation of topic changes

To test the relevance of the algorithm, the topic structure of discourse was annotated in terms of topic change. The annotations were made by two labellers (the first author and a colleague) on 30 minutes of read speech from 8 English native speakers (PAC), 30 minutes of read speech from 10 French native speakers (PFC), 30 minutes of dialogue from 9 Englishnative speakers (AIX-MARSEC), and 30 minutes of dialogue from 6 French-native speakers (CID). Specifically, three levels of disjuncture between adjacent units were defined, a simplified version of Grosz & Sidner [START_REF] Grosz | Attention, intentions and the structure of discourse[END_REF] as used in Fon [START_REF] Fon | A cross-linguistic study and discourse boundary cues in spontaneous speech[END_REF] and Kong [START_REF] Kong | The role of Pitch Range Variation in the Discourse Structure and Intonation Structure of Korean[END_REF]. When a DSP0 is indicated between two units, it means that these units share the same topic and the same relation to the units that dominate them, hence DSP0 indicates no discourse unit boundary. DSP1 stands for a hierarchically superior relation between two units, meaning that the units share related purposes, indicating, for example, a causeconsequence relation or again a clarifying relation. Finally, when two discourse units with no related discourse purposes or topics side together, a DSP2 boundary is inserted to indicate the introduction of a new issue. However, we already assume that further work is needed to refine this manual annotation.

Discourse structure and variations in register and pause duration

ANOVA analyses, carried out for the four corpora, showed a significant correlation between discourse structure and register changes. The DIFFKEY and NDKEY between two consecutive units, for DSP1 and DSP2 levels, significantly increases while approaching higher levels of disjuncture (pval < 2.2e-16) (Figure4), except for the CID data where only DSP2 is reported significant. DIFFSPAN is significatively correlated with DSP1 and DSP2 levels for the AM corpus (F(2,3447)=23.98, pval=4.549e-11); it is reported less significant than DIFFKEY for the PFC and the CID corpora (pval= 0.001995 & 0.02847 respectively) and not significant for the PAC Corpus (F(2,3003)=0.14, pval=0.8634).

NDSPAN is highly correlated with DSP2 level (pval< 2e-16) while it is not with DSP1 level for the four corpora (pval>0.441). Thus, key appears as a stable parameter to indicate topic changes while span may be optional. Variations in span may be rather seen as marking a speaker's involvement while telling his/her story. It may be concluded that key and span parameters may convey different functions and have to be studied separately. Finally, ANOVAs analyses showed that pause duration is correlated with topic changes (pval < 2.2e-16), longer pauses being inserted before the introduction of a new issue.

Predicting topic changes

Having shown that discourse structure is significantly correlated with register key changes and pause duration, we investigate in this section the possibility of using these two parameters to predict the temporal location of discourse structure markers. We consider two classes, the first corresponding to units labelled DSP0 (class 1) and the second regrouping the DSP1 and DSP2 markers (class 2). The training of the binary classifiers is performed on the sample, for which the distribution proportion between the two classes is highly asymmetric (e.g. for the PFC, 93% of the units belong to class 1 and 7% to class 2). The first classifier takes as input parameter feature the pause duration, immediately 1: Scores of F-Measure for the classifiers based on the pause duration feature (column 2), on the node distance feature (column 3) and on the combination of both features (column 4).

The second classifier is based on the register key feature. The relevant quantity considered is herein the weighted node distance (i.e. the node distance weighted by difference in register key, NDKEY) between units obtained from the clustering tree structure. This classifier is less efficient than the one based on duration pause information as shown in Table 1, column 3. Because of the low correlation previously obtained between the discourse structure and variations in span, register span feature has been excluded from classifying analyses. Combining both pause and NDKEY features (Table1, column 4) reveals that adding register key information does not improve the prediction power of the classifier. It appears that pause information only is sufficient to predict topic changes, specifically for read speech in French.

Discussion

Classifiers have shown that pause duration is the best predictor for topic changes. However, it has to be noticed that the magnitude of the prediction power is corpus-dependant (from 0.32 to 0.76). Speakers may use different strategies to indicate a topic change. In reading task for example, while pause variations are widely used by French speakers, the case is not as straightforward for English ones. Moreover and against all expectations, the addition of key feature has been reported as not sufficient enough to improve the prediction power. This may be explained by both the asymmetry of the class distribution and the supremacy of pause feature during the merging process. Again, speakers may vary their register key differently or use other prosodic cues to signal a topic change such as variations in speaking rate and intensity. It may be therefore interesting to merge these other prosodic features in the existing classifiers to improve their prediction power. Furthermore, the objective detection of register variations as obtained with the clustering algorithm is correlated with manual annotation of topic changes. It may be assumed that the results obtained are dependant on the subjective annotation. The choice of three levels of structure boundary (DSP0, DSP1 & DSP2) may be questioned and considered not sufficient enough to understand and capture topic changes. Finally, as the detection is objective, the algorithm may detect variations in register key and span that convey other functions than topic changes (such as focus for example), functions which are not indicated in our annotation and therefore annotated with a DSP0 boundary if not linked to topic changes. This may have lowered the prediction power of the classifier.

Conclusions

If pause duration has been shown to be sufficient enough to predict topic changes, key feature has been reported as not bringing additional information as was expected by statistical analyses. However, the method consisting in using a clustering algorithm may be regarded as very promising, notably to better understand the hierarchical and organisational structure of the discourse, ie. how units are embedded in it and therefore improve the (automatic) representation of intonation patterns. For example, as already mentioned, the algorithm may be used for the analysis of declination phenomenon. Indeed, it may reveal the way declination is captured in phonological units such as the Accentual phrase or the Intonational Phrase.

Figure 1 :

 1 Figure 1: Graphic representation of MMIN (given in Hertz) as compared to MINPRAAT and MIN35 as obtained from the PFC corpus (27 files selected).

Figure 2 :

 2 Figure 2: Graphic representation of MMAX (given in Hertz) as compared to MAXPRAAT and MAX65 as obtained from the PFC corpus (27 files selected).

Figure 4 :

 4 Figure 4: Boxplots of difference in key depending on the three levels of discourse boundary, i.e. DSP0, DSP1 and DSP2 as obtained from the PFC data.

  preceding the unit under consideration. Training the classifier allows the definition of a pause duration threshold (from 0.49s to 0.79s, depending on the corpus) delimiting the two classes. The unit will be classified in class 1 if its feature value is lower than the threshold and in class 2 otherwise. The evaluation of the classifier is obtained on the sample by computing the confusion matrix (i.e. the counts of observed versus predicted classes on the sample). The scores of F-Measure associated with the prediction of discourse markers (class 2) are given for the four corpora in column 2, Table1.

	Corpus	Pause	NDKEY	Both Features
	PAC	0.321	0.181	0.325
	PFC	0.766	0.298	0.754
	AM	0.564	0.305	0.566
	CID	0.472	0.232	0.468
	Table


Alexis Fernandez

Paul Malumba

Olivier Roiseux

Gilles Olive

Christophe Blecker

Does pearling process concentrate β-Glucans in barley grain ?

come

INTRODUCTION

Dietary ibres are increasingly used in the Western diet due to for example their lower carbohydrate intake or their hypocholesterolemic action and preventive efects against colon's infections and diseases.

-glucans, a soluble dietary ibre found in diferent cereals (mainly in oat and barley), consist of glucose units linked by 1-3 bonds and 1-4. In contrary to starch, linked by 1-4 and 1-6 bonds, -glucans are less degraded during gastic and ileal phases of digestion, inducing less increase of glycaemic index during the postprandial phase of digestion. As they are not hydrolysed by digestive enzymes, -glucans are fermentable substrates for the microbiome implanted in the colon of monogastrics and contribute to the production of short-chain fatty acids, useful for the regeneration of cells lining the walls of the colon. The aim of this study was to develop the barley grain fractionation process, in order to isolate fractions, with higher amounts in -glucans, by isolation of grain subcomponents where these polysaccharides are preferentially stored; this fraction willing to be used as a health ingredient according to the EFSA advice that allow to mention an hypocholesterol efect for barley's beta-glucans.

Corresponding author : alexisfernandez2803@gmail.com CONCLUSION This experimental process allows to demonstrate that -glucans are heterogeneously distributed through the diferent tissues of the barley grain. Indeed, it was found that the lowest concentration in -glucans is present irst in the husk and then in the bran. In view of this results, we can deduce that pearled grains have an higher level in -glucans than the whole barley grains. Pearled barley could thus be part of dietary meal to ight too high rate of bad cholesterol in blood.


Miguel Fosas De Pando

Peter J Schmid

Denis Sipp

On the receptivity of aerofoil tonal noise: an adjoint analysis

For moderate-to-high Reynolds numbers, aerofoils are known to produce substantial levels of acoustic radiation, known as tonal noise, which arises from a complex interplay between laminar boundary-layer instabilities, trailing-edge acoustic scattering and upstream receptivity of the boundary layers on both aerofoil surfaces. The resulting acoustic spectrum is commonly characterised by distinct equally-spaced peaks encompassing the frequency range of convectively-amplified instability waves in the pressuresurface boundary-layer. In this work, we assess the receptivity and sensitivity of the flow by means of global stability theory and adjoint methods which are discussed in light of the spatial structure of the adjoint global modes as well as the wave-maker region. It is found that for the frequency range corresponding to acoustic tones the direct global modes capture the growth of instability waves on the suction surface and the near wake together with acoustic radiation into the far field. Conversely, it is shown that the corresponding adjoint global modes, which capture the most receptive region in the flow to external perturbations, have compact spatial support in the pressure surface boundarylayer, upstream of the separated flow region. Furthermore, we find that the relative spatial amplitude of the adjoint modes is higher for those modes whose real frequencies correspond to the acoustic peaks. Finally, the analysis of the wave-maker region points at the pressure surface near 30% of chord as the preferred zone for the placement of actuators for flow control of tonal noise.

Introduction

Fluid systems that are governed by multiple physical processes often exhibit instabilities that are composed of a complex interplay of many or all prevailing amplification and propagation mechanisms. Tonal noise about an aerofoil falls into this category, as convective hydrodynamic instabilities in both the pressure-and suction-boundary layers interact with acoustic waves radiating upstream from the trailing edge, which in turn retrigger the boundary layer instabilities, and in doing so sustain distinct pressure sound levels and select discrete frequencies via subtle feedback loops. Due to their very nature, the analysis of such instabilities requires a global (rather than local) point of view, since neither component in isolation -hydrodynamic or acoustic -can fully capture the tonal noise mechanism. An analysis of this type has been reported in Fosas de [START_REF] Fosas De Pando | A global analysis of tonal noise in flow around aerofoils[END_REF] where the full global spectrum has been presented. The results of this study, including the global spectrum of the mean-flow-based linear operator, has shown that only a rather small number of global modes has a significant acoustic footprint to explain experimentally observed sound pressure levels and frequencies. The same study could identify and classify the shape and structure of flow patterns responsible for noise generation.

Equally important for a full understanding of tonal aerofoil noise is the quest for the origin of the acoustically active instabilities; in other words, we seek flow field structures (in type and location) that most efficiently trigger and sustain the noise-generating global modes. Adjoint methodology [START_REF] Hill | Adjoint systems and their role in the receptivity problem for boundary layers[END_REF][START_REF] Luchini | Adjoint equations in stability analysis[END_REF] will be used to furnish sensitivity information for these modes, and a pointwise superposition of local sensitivity measures (contained in the adjoint structure) and local amplification rates (contained in the direct/modal structure) will point towards positions in the flow field that act as so-called wavemakers. These positions optimally exploit both the responsiveness and the growth of global structures, and hence can be thought of as the source or origin of a global instability. Analyses of this type, also referred to as structural stability (see, e.g. [START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF][START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF], have previously been applied to flows with simpler transport processes, where respective regions of amplification and sensitivity are convectively linked [START_REF] Bottaro | The effect of base flow variation on flow stability[END_REF][START_REF] Chomaz | Global instabilities in spatially developing flows: Non-normality and nonlinearity[END_REF]. Here, a more complex situation has to be addressed since flow information can be transported hydrodynamically and acoustically, and the full feedback loop consists of both components.

The tonal noise phenomenon on aerofoils, the focus of this study, has been investigated both experimentally and numerically over the past decades, starting with the experimental work of [START_REF] Paterson | Vortex noise of isolated airfoils[END_REF] that led to an empirical expression for the dominant tonal frequency as a function of the Reynolds number. This was followed by an identification of multiple frequency peaks (see, e.g. [START_REF] Arbey | Noise generated by airfoil profiles placed in a uniform laminar flow[END_REF]) and a resulting ladder structure as the free-stream velocity is varied. The multi-peak structure has been confirmed in further experiments as well as in numerical simulations by, e.g., [START_REF] Desquesnes | Numerical investigation of the tone noise mechanism over laminar airfoils[END_REF]; [START_REF] Jones | Numerical analysis of tonal airfoil self-noise and acoustic feedback-loops[END_REF]; Fosas de [START_REF] Fosas De Pando | A global analysis of tonal noise in flow around aerofoils[END_REF] and has been linked to feedback loops consisting of boundary layer instabilities and acoustic radiation. Simplified theoretical models -relying on a phase condition between local spatial growth, acoustic scattering on the trailing edge and a receptivity process at a prescribed location -have been proposed by [START_REF] Kingan | Laminar boundary layer instability noise produced by an aerofoil[END_REF], which reproduced well the observed frequencies for the tonal noise problem. A recent experimental study [START_REF] Pröbsting | Experimental investigation of aerofoil tonal noise generation[END_REF] has presented detailed time-resolved particle-image-velocimetry (PIV) measurements of acoustic sources near the trailing edge of the aerofoil, reporting that vortical structures, emanating from boundary layer instabilities, pass the trailing edge at the frequency of the dominant tone. Even though these vortical structures have been identified as one component of the tonal-noise generation mechanism, a full closed-loop analysis was beyond the experimental measurements. In the same study, a crucial involvement of the pressure side has also been suggested. Already early experimental work [START_REF] Paterson | Vortex noise of isolated airfoils[END_REF] has similarly highlighted a characteristic sensitivity of acoustic tones on the details of aerofoil's pressure-surface boundary layer.

The goal of the current study is then to use numerical simulations and adjoint techniques to extract, isolate and quantitatively describe the regions and structures of the flow that are principally responsible for the production of tonal noise. It aims at providing evidence and support for previous models, empiricisms and measurements. While the identification of noise-generating structures near the trailing edge has been the subject of various previous investigations, the current study shall not focus on this issue, but rather concentrate on the full (global) feedback loop consisting of multiple hydrodynamic instabilities, noise generation and propagation, and boundary layer receptivities to acoustic forcing. The article is organised as follows. After a description of the flow configuration and the numerical setup ( § 2), which also contains a brief introduction to the adjoint analysis and the wavemaker argument, the global spectrum will be dissected (in § 3) into acoustically active modes (responsible for the acoustic tones) and the rather quiet high-frequency and low-frequency modes. Each type will be analyzed as to its degree of sensitivity and its wavemaker location, and appropriate means for the passive or active manipulation of these modes will suggested. The conclusions in § 4 give a summary and a complete structural description of tonal noise generation in flow about an aerofoil.

Flow configuration and numerical setup

We take a numerical approach for the receptivity and sensitivity analysis of aerofoil tonal noise, consisting of a modal decomposition of the linearised governing equation, together with an accompanying analysis of the adjoint operator. A brief introduction to the principles of this type of analysis is given below.

Compressible Navier-Stokes numerical solver

We base our analysis on numerical simulations of the compressible Navier-Stokes equations on cuvilinear structured grids using a pseudo-characteristics formulation [START_REF] Sesterhenn | A characteristic-type formulation of the Navier-Stokes equations for high order upwind schemes[END_REF]. After proper non-dimensionalisation, the governing equations are discretised in space using high-order compact schemes: a fifth-order compact-upwind lowdissipative scheme [START_REF] Adams | A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems[END_REF] for the advective terms and a third-order centred scheme [START_REF] Lele | Compact finite difference schemes with spectral-like resolution[END_REF] for the diffusive terms. Characteristics-based boundary conditions [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF][START_REF] Lodato | Three-dimensional boundary conditions for direct and large-eddy simulation of compressible viscous flows[END_REF] are implemented, and non-reflecting inlet and outlet conditions at the computational domain are augmented by sponge layers to further attenuate spurious reflections [START_REF] Bodony | Analysis of sponge zones for computational fluid mechanics[END_REF]. At this point, the governing equations consist of a system of ordinary differential equations of the form

dv dt = F(v) (2.1)
which has to be integrated in time. In the above expression, F(v) denotes the nonlinear function representing the spatially-discretised governing equations, and v stands for a composite vector containing the dependent flow variables (pressure p, entropy s and velocity vector u) at every grid point. For further details on the numerical solver, the reader is referred to Fosas de Pando et al. (2014).

Evaluation of the linearised operators

Our study focuses on the dynamics of infinitesimally small perturbations superimposed on a reference flow field. To this end, we set v(t) = v + v (t) in equation (2.1) and, upon linearisation, arrive at

dv (t) dt = A(v)v (t), with A(v) = ∂F ∂v v , (2.2)
where A(v) is referred to as the direct operator. In what follows, we will concentrate on the perturbation dynamics and omit the primes. While the analysis of the direct operator provides pertinent information on the flow dynamics and instability mechanisms, even more valuable information can be obtained by accessing the associated adjoint operator, denoted by A * . This operator is defined by the duality relation w, Av = A * w, v , where the inner product reads w, v = w H Mv with M as a Hermitian, positive definite weight matrix. Throughout this study, we choose the inner product w, v based on the small-perturbation energy norm derived in [START_REF] Chu | On the energy transfer to small disturbances in fluid flow (Part I)[END_REF] and [START_REF] Hanifi | Transient growth in compressible boundary layer flow[END_REF] for compressible flows.

We stress that we consider the discrete adjoint, i.e., the adjoint of the discretised equations; for this reason, the above duality relation is satisfied up to machine precision. Based on this fundamental relationship and our specific choice of inner product, the operator A * can be easily expressed in terms of the transconjugate of the direct operator A H given by A * = M -1 A H M. The numerical implementation of both linear operators A and A * can conveniently be performed in matrix-free form using the methodology described in Fosas de [START_REF] Fosas De Pando | Efficient evaluation of the direct and adjoint linearized dynamics from compressible flow solvers[END_REF]; in this reference, algorithms are described for the evaluation of the matrix-vector products A(v)v and A * (v)w, directly from the implementation of the right-hand side in equation (2.1). This form of access to the linear operators is preferred over explicit storage of the direct operator A, since, for typical numbers of degrees of freedom and our choice of spatial discretisation, storage requirements well exceed typical memory resources.

Direct and adjoint global modes and computational details

We next turn our attention to the eigenvalue spectra of the direct operator A and its adjoint A * . Starting from a modal decomposition for the perturbations in the form v k exp (-iω k t), we arrive at the eigenvalue problem given by

-iω k v k = Av k (2.3)
with v k as the direct global mode and ω k = ω r,k + iω i,k as the corresponding complex global frequency. Analogously, for each direct global mode, the associated adjoint global mode w k is given by the eigenvalue problem

iω * k w k = A * w k , (2.4)
with ω * k = ω r,kiω i,k , i.e., the complex conjugate eigenvalue of the direct problem (2.3). From the duality relationship, it is readily deduced that direct modes v k and adjoint modes w l are mutually bi-orthogonal unless k = l. The direct and adjoint global modes are defined up to a multiplicative constant; hereinafter, all direct modes are normalised according to v k 2 = 1, and the associated adjoint global modes are rescaled such that w k , v k = 1. Numerically, the large-scale eigenvalue problems (2.3) and (2.4) are solved using the open-source software SLEPc [START_REF] Hernández | SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems[END_REF] interfaced to our compressible simulation code. More specifically, a Krylov-Schur technique has been applied to the evolution operators exp(∆tA) and exp(∆tA * ), which represent the time advancement over one time step ∆t by the direct or adjoint operator, respectively. The Krylov subspace is spanned by 2048 vectors, and the time step has been chosen as ∆t = 0.015. Once a number of direct-adjoint eigenpairs has been computed, the degree of convergence is measured using the relative residuals given by Av

k +iω k v k 2 / ω k v k 2 and A * w k -iω * k w k 2 / ω * k w k 2 .
In contrast to Fosas de Pando et al. (2014), we do not employ a harmonic extraction technique, which yields lower relative residuals.

Wavemaker analysis

The direct global modes and the corresponding adjoint modes determine the structural sensitivity of the linearised operator A [START_REF] Hill | A theoretical approach for analyzing the restabilization of wakes[END_REF][START_REF] Bottaro | The effect of base flow variation on flow stability[END_REF][START_REF] Chomaz | Global instabilities in spatially developing flows: Non-normality and nonlinearity[END_REF][START_REF] Giannetti | Structural sensitivity of the first instability of the cylinder wake[END_REF][START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF]. The sensitivity of the eigenvalues to perturbations in the operator A can be assessed by considering the perturbed eigenvalue problem (A+δA)(v k +δv k ) = -i(ω k +δω k )(v k +δv k ) and retaining only first-order terms. By computing the inner product of either side of the equation with the adjoint mode, we obtain -iδω k = w k , δAv k . This latter expression indicates that the effect of a unitary proportional feedback between flow variables at different locations on any eigenvalue can be quantified by the scalar product between the adjoint at the output location and the direct global mode at the input location. We note that in general the feedback affects many eigenvalues of the matrix A. For the case of a diagonal operator perturbation δA, we restrict ourselves to a local feedback between each individual component at its given location. This approach leads to the concept of a wavemaker: the structural sensitivity of the flow to local (in-place) feedback quantified by the entrywise (Hadamard) product of the direct global mode and the corresponding adjoint global mode at a given location.

Flow configuration and parameters

We reconsider the flow case analysed in Fosas de Pando et al. (2014) but, for completeness' sake, provide a brief summary of the flow configuration and selected parameters. We compute the compressible flow about a NACA 0012 aerofoil section, at a two-degree angle of attack, with a sharp trailing edge. The chord-based Reynolds number Re c is taken as 2 • 10 5 and the Mach number M is 0.4. The specific heat ratio γ and the Prandtl number Pr have been chosen as 1.4 and 0.71, respectively. Two-dimensional nonlinear simulations reveal that the flow reaches a quasi-periodic state characterised by vortices shedding into the wake and by a pressure spectrum in the far-field that is dominated by distinct tones at equally-spaced frequencies. The corresponding non-dimensional values for the dominant angular frequencies are ω ≈ 37.02, ω ≈ 39.47, ω ≈ 41.93 and ω ≈ 44.38. Proceeding with our analysis, the compressible Navier-Stokes equations are then linearised around the time-averaged flow field, depicted in figure 1 by the mean streamwise velocity component. This mean flow presents spatially-growing boundary layers on both aerofoil surfaces, with flow separation on both sides: the boundary layer detaches on the suction surface between x s ≈ 0.405 and x r ≈ 0.686 with a maximum reverse flow of 7%, while on the pressure surface the mean flow detaches between x s ≈ 0.728 and x r ≈ 0.977 with a maximum reverse flow of 10%.

Receptivity and sensitivity analysis

A study of the receptivity and sensitivity of tonal noise generation around an aerofoil must commence with the analysis of the global spectrum. Specifically, we will classify the modal structures according to their frequencies and the dominance of their physical components, and particular attention shall be directed towards the pressure component (or their dilatation field) as it indicates the acoustic footprint of the associated modal structure. In a second step then, the corresponding adjoint modes are computed which will furnish valuable information about the region of strongest sensitivity of the flow to a particular modal structure and the region of strongest localised (in-place) feedback; this latter analysis will point towards the spatial origin of a particular instability mechanism and guide possible (active or passive) control efforts.

Features of the global spectrum

We present in figure 2 the global spectrum of the linearised compressible Navier-Stokes operator A, where each direct global mode v is represented in the complex plane by its associated complex frequency or eigenfrequency ω = ω r + iω i , where ω r is the angular frequency and ω i is the temporal growth-rate. Following the outline of the previous section, the associated adjoint global mode is denoted by w and its eigenfrequency is -ω * = -ω r + iω i . Since the linearised Navier-Stokes operator A is real, the spectrum is symmetric with respect to the vertical axis (ω r = 0). It thus suffices to focus on the features of the global modes in the right half-plane, for positive angular frequencies.

The eigenvalues are colored according to their relative residual value, following the iterative computation of the spectrum by a Krylov-Schur subspace technique. Only eigenvalues above an angular frequency of ω r ≈ 10 could be identified with an acceptable relative residual value; eigenvalues below this frequency will be neglected in our analysis.

A first inspection of figure 2 confirms that the operator A is stable, since all eigenvalues are contained in the lower half of the complex plane ω i < 0. No isolated branches of global modes can be observed in the spectrum. This suggests that the features of the direct and adjoint global modes vary, at least in principle, in a continuous manner. The shape of the spectrum suggests a division of the modes into three subgroups, each representing a different physical perturbation mechanism. The main group (labelled by M in figure 2) contains the least stable eigenvalues and thus captures the most dominant features of the flow. Lower-frequency modes (labelled L) and higher-frequency modes (labelled H) emanate from this main group and describe distinct perturbation dynamics (see more details below).

Leading modes: the coupled dynamics of the separation bubbles

In figure 3 we provide a more detailed view of the spectrum for the subdomain M comprising the leading, least stable, global modes. The leading global modes consist of multiple local maxima of growth rate for varying frequency, centred around the maximum growthrate peak at ω r ≈ 43.91 and ω i ≈ -0.29. The frequency difference between consecutive maxima is nearly constant and equal to ∆ω r ≈ 2.44, and the temporal growth rate for the local maxima is approximately ω i ≈ -0.29. Between the peaks, the frequency difference of neighbouring eigenvalues is ∆ω r ≈ 0.34, and their minimum growth-rate is approximately ω i ≈ -0.60. These spectral features (also reported in Fosas de Pando et al. (2014)) have been found in good agreement with numerical simulations and experiments.

The corresponding least stable mode, associated with M 1 in figure 3, is shown in figure 4(a). It is visualised by the pressure and reveals a complex spatial structure that emphasizes a distinct disturbance wavepacket downstream of both separation bubbles, a vortex street shedding into the wake and a pronounced acoustic wave radiating from the aerofoil's trailing edge. The inset, showing the streamwise velocity component, displays the modal features near the trailing edge, where a wavepacket structure on the suction side links, via the trailing edge, to a more compact disturbance stemming from the pressure side. As we will demonstrate below, this link between the pressure-side and suction-side dynamics is a quintessential characteristic of the tonal-noise phenomenon. It is equally important to note that the acoustic wave reaches both the pressure-and suction-side boundary layer where it retriggers a respective boundary-layer instability; this feature has been illustrated in Fosas de [START_REF] Fosas De Pando | A global analysis of tonal noise in flow around aerofoils[END_REF] by following an impulse response. The global mode M 1 thus represents in one structure the various components of the double feedback loop: boundary-layer instability → acoustic wave → reseeding mechanism.

The description of the perturbation dynamics only in terms of global modes is incomplete if no further thoughts are given to the processes by which an initial perturbation (or, equivalently, a given forcing) projects onto the various direct global modes. In more physical terms, such a projection is linked to receptivity mechanisms. As was argued earlier, the adjoint global modes represent a powerful theoretical concept to assess receptivity mechanisms of the flow: within the limitation of linear disturbance analysis, an arbitrary perturbation can be expressed as a linear combination of global modes u = j a j v j . In order to obtain the coefficients a i , we take the inner product on both sides with the corresponding adjoint global mode w i . By invoking the bi-orthogonality condition, and noting that we have chosen the adjoint global modes such that w i , v i = 1, we arrive at a i = w i , u . From an optimisation point of view, the adjoint global mode corresponds to the perturbation that produces the largest projection onto the corresponding direct mode, and its norm w i constitutes a useful measure to evaluate overall receptivity. More physically, the adjoint mode corresponding to M 1 will pinpoint the location, structure and flow components where the double feedback loop (via the pressure and suction side) that sustains M 1 is most easily excited or otherwise manipulated.

Figure 4(b) displays the adjoint mode that corresponds to the direct mode M 1 , visualised by the adjoint streamwise velocity u * . Within the domain depicted in this figure, the adjoint mode is spatially localised in the boundary layer on the pressure side between about 20 to 50 percent of chord (see inset). The amplitude of the adjoint mode, with the normalisation convention introduced above, confirms a significant amount of receptivity of the structure displayed in figure 4(a) to localised perturbations in the pressure-side boundary layer. The receptivity anywhere else, including the suction-side boundary layer, is orders of magnitude less. These results suggest that in order to break the feedback loop that maintains the least stable mode (and its associated acoustic foot print) a minimal effort is required if control is applied at the location where the corresponding adjoint is maximal, i.e. at 30 percent of chord on the pressure side.

A clearer picture of the least stable M-modes emerges when considering the direct and adjoint global modes corresponding to M 1 -M 4 in figure 3. In figures 5(a) and 5(b), we depict the streamwise velocity peak along the aerofoil chord on the suction-and pressure-surface of the aerofoil, respectively, for the four modes M 1 -M 4 . All four modes display a similar behavior: on the suction side, up to 60 percent of chord, we observe a significant rise, after which the peak streamwise velocity steadily decreases towards the trailing edge; on the pressure side, we have a continual rise in the peak velocity, which becomes even more pronouced as the trailing edge is approached. The corresponding adjoint modes are displayed in figures 5(c) and 5(d), visualised by the adjoint streamwise velocity magnitude |u * | in the suction-and pressure-side boundary layers, respectively. Recalling the normalisation convention of the adjoint modes, the magnitude of u * is a meaningful measure to compare the relative receptivity of different direct modes. As was the case for the direct modes, the spatial distribution of the four adjoint global modes for M 1 -M 4 is very similar: the support of the adjoint mode is highly localised on the pressure-surface boundary layer, between the leading edge and upstream of the separation point, i.e., 0.2 < x < 0.5, with the maximum value at approximately 10 9 . Among the four modes considered, M 1 and M 4 show nearly the same receptivity, while M 2 and M 3 display an order of magnitude less. Some receptivity is also observed in the free-stream (not shown) in the region where the base flow convects perturbations to the pressure-surface boundary layer, i.e., near the stagnation line. We note that the levels of receptivity on the suction surface are nearly 10 3 times smaller than the corresponding receptivity measures on the pressure surface.

The difference in spatial support between the direct global modes and their adjoint equivalents is striking and has implications on the understanding of the underlying physical phenomena. It is well-known that in convectively unstable shear flows, a common manifestation of the non-normality of the Navier-Stokes operator is the spatial separation between the direct and adjoint global modes in the downstream and upstream direction [START_REF] Chomaz | Global instabilities in spatially developing flows: Non-normality and nonlinearity[END_REF], respectively. The direct global modes typically display exponential growth in the convectively unstable region, and if the flow is stable downstream, their amplitudes reach nearly a constant value in that region. Conversely, from a global point of view, the adjoint modes show increased spatial support in the region where a localised perturbation yields a large projection onto the direct global mode. From a local point of view, the adjoint modes show the highest value where a localised pointwise perturbation undergoes maximum amplification by means of the instability mechanisms present in the flow. In our case, however, the direct global mode is concentrated on the suction surface and in the near wake, whereas the adjoint global mode is predominantly located on the pressure surface. Paradoxically, the perturbation that causes the largest growth of a global mode with spatial support on the suction surface is not convectively linked to the location on the pressure surface that causes maximum receptivity. As a consequence, the spatial structure of the adjoint mode suggests a different route for the excitation of the instabilities in the flow via (i) the growth of instabilities by convective mechanisms on the pressure surface, (ii) the scattering of an acoustic wave at the trailing edge, (iii) the receptivity at the leading edge, where the suction-surface boundary layer displays convective instabilities. This receptivity mechanism prevails over the direct growth of instabilities on the suction surface as the total spatial growth along the pressure-surface is larger than the analogous growth on the suction-surface. This observation is supported by previous experiments [START_REF] Nash | Boundary-layer instability noise on aerofoils[END_REF] and numerical calculations [START_REF] Desquesnes | Numerical investigation of the tone noise mechanism over laminar airfoils[END_REF]: the frequency of the dominant tone in the spectrum coincides with the most amplified frequency along the pressure surface based on local stability theory.

Besides, not all modes appear to be equally receptive. We depict the magnitude of the adjoint mode on the suction-surface, figure 5(c), and on the pressure surface, figure 5(d). It can be observed that the receptivity on the pressure surface to streamwise velocity perturbations is higher for the modes at the peaks in growth rate, M 1 and M 4 , than for the modes M 2 and M 3 . As discussed before, the frequencies of the modes M 1 and M 4 are in excellent agreement with the frequencies that appear in the acoustic spectrum of the nonlinear numerical simulation.

The occurrence of multiple peaks in the sound spectrum cannot solely be explained by convective instability properties of the boundary-layer: the range of frequencies where the boundary layer displays convective growth is larger than the frequency difference between the peaks. In this respect, it is suggested that global effects further discretise the components that are most amplified.

Wavemaker analysis and structural sensitivity

In figures 5(e) and 5(f) we display the peak amplitude of the wavemaker of the leading mode M 1 (together with the respective contribution of the direct and adjoint modes) along the suction-and pressure-side boundary layer, respectively. We observe that the spatial support of the wavemaker is closer to the adjoint than to the direct mode, since the compactness of the adjoint is more pronounced than that of the direct mode.

This fact suggests that the individual leading eigenfrequency is more sensitive to operator perturbations A on the pressure surface of the aerofoil (and, in particular, upstream of the separation point) than in the rest of the domain. A first indication of the wavemaker location on the pressure surface has been reported in the previous numerical study of [START_REF] Desquesnes | Numerical investigation of the tone noise mechanism over laminar airfoils[END_REF], where they state: "The computation of the amplification ratio for different chordwise stations and the same x 0 [distance from the leading edge] shows that the most amplified frequency is constant and equal to 845Hz from x/c = 0.7 to x/c = 0.96. The tone noise frequency is therefore selected before the boundary-layer separation." In this respect, the wavemaker describes the localised region in the flow that imposes its dynamics on the remaining flow. The results presented here are therefore in good agreement with both numerical and experimental studies; we can quote [START_REF] Paterson | Vortex noise of isolated airfoils[END_REF] who reports: "A boundary-layer trip wire was found to have no effect on the tone when placed at various chord-wise positions on the suction surface of this airfoil but caused the tone to disappear when placed forward of 80% chord on the pressure surface."

The above observations, describing the sensitivity of the flow to changes in the base flow, are in excellent agreement with the spatial location of the wavemaker. However, it is less clear whether a link between the triggering of the pressure-surface boundary layer and the small operator perturbations (considered above) can be established and whether this link can be made responsible for the vanishing of tonal-noise effects.

Another implication is related to the choice of base flow. Although the instantaneous flow and the mean flow are not steady solutions of the Navier-Stokes equations, the time residual at the pressure-surface boundary layer upstream of the separation bubble is very small. This indicates that the modes of interest exhibit sufficient robustness, provided that the pressure-surface boundary layer remains unaltered.

High-frequency modes: suction-surface shear-layer instabilities

We next consider in figure 6 the range of high frequencies of the global spectrum (with respect to the leading eigenfrequencies). A rather continuous branch is observed, and for our analysis four representative global modes are selected and identified, to assess the parameterisation along the high-frequency branch.

The direct mode H 4 , depicted in figure 7(a) in terms of the pressure, is singled out and shown to display features that are concentrated in the detached area of the suctionside boundary layer, together with its effect on the shear layer past the reattachment point. A wave-packet pattern with a distinct spatial wavenumber is observed, with its maximum near 60 percent of chord and a steadily diminishing amplitude as it sheds towards the trailing edge and into the aerofoil's wake. An involvement of instability features from the pressure-side boundary layer is not observed. We thus conclude that high-frequency modes, such as H 4 describe the Kelvin-Helmholtz-like instabilities induced by the separated suction-side boundary layer.

Only a very small amount of acoustic radiation is present in this mode, far less than the acoustic footprint of any of the leading modes: the high-frequency vortex-shedding dynamics of the suction-side separation bubble is a very minor contributor to the tonalnoise sound pressure levels.

Again, the corresponding adjoint mode provides information about the receptivity of this instability and its origin. The adjoint H 4 -mode is displayed in figure 7(b), visualised by the adjoint streamwise velocity. The insets clearly show non-zero adjoint amplitude at both the pressure and suction side of the aerofoil; the suction side amplitudes are, however, orders of magnitude larger, indicating a preferred receptivity location near the upper leading edge (10 percent chord) of the aerofoil. On the pressure side, the region of maximum receptivity is at about 50 percent of chord. A more complete analysis of the high-frequency modes and their structural connection to the leading modes of the previous section consists of a comparison of several (direct and adjoint) modes along the high-frequency branch in terms of their amplitudes and localisation along the pressure and suction side of the aerofoil. To this end, we investigate four selected high-frequency modes, labelled H 1 to H 4 in figure 6, and display their streamwise velocity amplitudes in the suction-and pressure-side boundary layer. On the suction side, see figure 8(a), the modal structure remains very similar as we move from H 1 to H 4 : a strong exponential rise in amplitude towards a peak near 60 percent of chord, followed by an exponential decay towards the trailing edge. On the pressure side, see figure 8(b), however, we observe markedly larger differences. While the highfrequency modes H 1 and H 2 (which are closest in frequency to the leading modes) are characterised by a steady and pronounced rise in amplitude towards the trailing edge, this rise is absent in the higher-frequency modes H 3 and H 4 . In effect, towards the lower edge of the high-frequency branch, the direct modes still resemble the leading modes in shape, i.e., a dominant localised structures on the suction side, coupled to an even more localised counterpart near the pressure-side's trailing edge. As we progress towards higher frequencies, only the suction-side structures remains; the pressure side amplitudes are considerably smaller along the entire chord. This observation is in accordance with the structure of H 4 , figure 7(a), where only features near the suction-side separation bubble and its wake have been detected. The analysis of the adjoint modes confirms the above findings. On the suction side (see figure 8c), we observe an amplitude evolution for H 1 reminiscent of the leading modes, compare to figure 5(c). As higher frequencies are approached, however, the maximum amplitudes decay two orders of magnitude, while preserving the approximate dependence on the chordwise coordinate direction. More importantly, this decay in amplitude for higher-frequency modes is more pronounced on the pressure side (see figure 8d), where H 1 still displays a receptivity of 10 9 , comparable to the value for the leading modes, while H 4 shows a receptivity measure of more than four orders of magnitude smaller. In other words, higher-frequency modes increasingly become unreceptive to perturbations on the pressure side of the aerofoil. It should be stressed again that the choice of normalisation allows direct comparison of the direct and adjoint modal structures.

The wavemaker analysis of the highest-frequency mode H 4 is depicted in figures 8(e,f) for the suction and pressure side, respectively. Few distinguishing features can be found in these amplitude curves, indicating the absence of a strong selection principle for localisation of receptivity measure in the case of high-frequency modes.

Low-frequency modes: separation-bubble flapping and reattached flow dynamics

The global modes at lower frequencies (compared to the leading modes) capture the dynamics of the unsteady separation bubbles together with the induced flow dynamics in the reattachment zone. The low-frequency part of the spectrum, displayed in figure 9, shows a continuous branch reaching up to the quasi-steady (ω r ≈ 0) regime. Along this branch, we select four representative modes, labelled L 1 through L 4 , which we will analyse in more detail to determine the qualitative changes in the modal structures and their receptivity as we approach the lower-frequency limit.

The shape of mode L 3 , visualised by pressure, is displayed in figure 10a. It shows flow features concentrated on the separated region of the suction-side boundary layer. Due to the low frequency, large-scale spatial structures are encountered. They describe the "breathing" unsteadiness of the separation bubble, coupled to the remainder of the complex flow dynamics which even includes a weak but non-negligible acoustic component. The inset shows more details, such as the link via the trailing edge to the separated flow region on the pressure surface: a vortical element near the very end of the pressure surface -an essential part of the pressure-surface dynamics -is clearly visible.

The corresponding adjoint mode (figure 10b) shows nonzero features on both the pressure and the suction side of the aerofoil. Either structure is characterised by lowwavenumber elements due to the low temporal frequency. While the suction-side structure is concentrated near 20 percent of chord, the pressure-side structure is localised near half-chord. In terms of amplitude, the two adjoint structures are nearly equal (about 10 4 using our standard normalisation), suggesting that triggering this particular mode L 3 is equally effective by perturbations in either of the two general locations identified by the adjoint mode.

As before, more insight can be gained by evaluating the direct and adjoint streamwise velocity along the pressure and suction side for each of the four selected modes L 1,...,4 .

Figure 11a

,b presents the results for the direct modes, where subfigure a shows localisation of the modal structure on the suction surface in the second half of the aerofoil, nearly independent of the frequency, except for the quasi-steady case L 4 . On the pressure surface (subfigure b), similar tendencies are observed: modes L 1,...,3 exhibit nearly identical behavior, reflecting the presence of a strong separation bubble near the trailing edge. Again, the quasi-steady mode L 4 shows a markedly different shape.

Figures 11c,d display the adjoint modes L 1,...,4 evaluated along the suction and pressure surface of the aerofoil, respectively. The suction surface depicts a dominant receptivity measure (quantified by the adjoint streamwise velocity) near the upstream half of the aerofoil, which is rather pronounced for L 1 , closest in frequency to the leading modes, and gradually diminishes in amplitude (without too strong a change in shape) as the frequency is lowered. In this respect, the curve for adjoint mode L 1 should be compared to the one for adjoint mode M 1 (from figure 5c): similarities are clearly present. The same can be said about the analogous results for the pressure surface (figure 11d). For L 1 we nearly recover the receptivity curves of the leading modes (see 5d), even though at lower maximum amplitude. Moreover, the low-frequency modes L 1,2 inherit from the leading modes their preference of the pressure surface (over the suction surface) regarding the most effective manner of triggering of the associated global direct modes. This preference is more conspicuous for the leading modes (with more than two orders of magnitudes in receptivity measure max x |u * | between the two surfaces); nonetheless, it is also present (with one order of magnitude between the two surfaces) for the low-frequency modes L 1,2 . As the frequency is further lowered, such as for mode L 3 , the partiality in receptivity of the pressure surface vanishes, and mode L 3 is as easily triggered (and manipulated) from the pressure as from the suction surface. This result can be inferred from figure 11c,d.

Finally, the wavemaker analysis (see figure 11e,f) for the low-frequency mode L 2 shows that both surfaces of the aerofoil are nearly equally structurally stable. In addition, similar to the high-frequency modes, there is little that distinguishes specific regions on either airfoil surface as far as structural stability is concerned.

Summary and conclusions

A complete and global analysis of the compressible flow about a NACA-0012 aerofoil at a 2 o angle of attack, a chord-based Reynolds number of 2 • 10 5 and a Mach number of 0.4 has been presented, with emphasis on extracting the dominant physical mechanisms underlying the generation of tonal noise. This flow configuration was chosen to match The modal structures span a rather wide frequency range, but only a few modes (in the leading M-category) contain a significant acoustic component to account for the observed sound pressure levels as well as the observed frequencies. An analysis of the adjoint modal structures gives insight into receptivity and sensitivity measures for the associated direct counterparts, and a pointwise superposition of direct and adjoint modes identifies the wavemaker region of maximal structural instability. It should be stressed that the details of noise generation near the trailing edge -the subject of a large body of literatureare of less importance in our study, since they only represent one component of a global feedback loop; rather, it is the details of the complete feedback loop, and the role of its components, that we are interested in.

In the case of the acoustically active M-modes this type of analysis revealed that tonal M. Fosas de Pando, P.J. Schmid and D. Sipp noise is most easily triggered, or manipulated, at the pressure side of the aerofoil, more specifically near 35% of chord. This finding corroborates previous observations, including the early investigations of [START_REF] Paterson | Vortex noise of isolated airfoils[END_REF] as well as the numerical simulations of [START_REF] Desquesnes | Numerical investigation of the tone noise mechanism over laminar airfoils[END_REF] and Fosas [START_REF] Fosas De Pando | A global analysis of tonal noise in flow around aerofoils[END_REF]. It identifies the most sensitive part of the hydrodynamic-acoustic feedback loop and suggests the placement of actuators or surface elements on the aerofoil's upstream pressure side for an active or passive strategy to weaken or suppress acoustic tones.

With the direct M-modes appearing predominantly on the suction side of the aerofoil and the associated adjoint modes showing compact support on the pressure side, it is worth pointing out that these two structures are not convectively connected. This is in contrast to previous studies where the upstream sensitivity and the downstream amplification are solely joined by an advective process via the base or mean flow. In our present case, a link between the input and output structures for the M-modes is established by the acoustic waves, which form an integral part of the feedback loop. Ascribing the generation of tonal noise solely to boundary layer instabilities would amount to an incomplete description of the overall mechanism.

For completeness, modes with lower and higher frequency (compared to the M-branch) have also been included in our analysis. Either type shows rather negligible acoustic radiation, and, according to their adjoint counterparts, they can be most easily manipulated from the suction side (in case of high-frequency modes) or equally from the suction and pressure side (for the low-frequency modes). As the frequency range of the leading Mmodes is approached, however, the aerofoil's pressure side increasingly dominates the receptivity of either modes.

Lastly, the present analysis of the tonal noise problem using adjoint techniques showcases a tool which provides valuable insight into instability mechanisms that entail a complex interplay of multiple physical processes. The same analysis would, for example, also be appropriate for thermo-acoustic or combustion systems where hydrodynamic, acoustic and reactive instabilities co-exist and interact in a non-trivial manner. Similar to the present study, a global stability and receptivity analysis would capture pertinent feedback loops and suggest means of disrupting them most efficiently.

Figure 1 .

 1 Figure 1. Mean flow considered in the present study, visualised by the streamwise velocity component. The boundary layer on the suction surface detaches between xs ≈ 0.405 and xr ≈ 0.686, while the pressure-surface boundary layer detaches between xs ≈ 0.728 and xr ≈ 0.977. The reader is referred to Fosas de Pando et al. (2014) for further details on the mean-flow characteristics.

Figure 2 .

 2 Figure 2. Global spectrum of the tonal noise problem in the complex frequency plane. The eigenvalues are colored according to the size of the relative residual with respect to the operator A. The spectrum is divided into least stable modes (labelled M), low-frequency modes (labelled L) and high-frequency modes (labelled H).

Figure 3 .

 3 Figure 3. Zoomed view of the global spectrum showing the least stable (or leading) global modes. The modes M1-M4 are considered in more detail, and the corresponding eigenvalues are ωM1 ≈ 43.91 -0.29i, ωM2 ≈ 44.63 -0.61i, ωM3 ≈ 45.59 -0.59i and ωM4 ≈ 46.25 -0.34i. The relative residuals for the displayed modes fall below 10 -5 .

Figure 4 .

 4 Figure 4. (a) Spatial structure of the global mode labelled M1, visualised by the real part of the associated near-field pressure levels, and the real part of the streamwise velocity levels in the vicinity of the aerofoil surface (inset). The mode has been normalised by the maximum value of the velocity field in the near wake 1 < x < 1.2. (b) Spatial structure of the associated adjoint global mode (labelled M1 in figure 3), visualised by the magnitude of the streamwise velocity levels, and the real part in the insets.

Figure 5 .

 5 Figure 5. Evolution of the streamwise velocity peak along the aerofoil chord (a) on the suctionand (b) on the pressure surface of the aerofoil for the leading modes labelled M1-M4 in figure 3. (c,d) Same for the adjoint streamwise velocity peak. (e,f) Same for the direct mode, adjoint mode and wavemaker corresponding to M1.

Figure 6 .

 6 Figure6. Global spectrum displaying high-frequencies ω > 45; the modes labelled H1-H4 will be analysed further. Their associated eigenfqrequencies are: ωH1 = 60.06 -0.85i, ωH2 = 69.90 -0.92i, ωH3 = 80.17 -1.00i, ωH4 = 89.97 -1.05i. The residual associated with the iterative eigenvalue solver is smaller than 5 • 10 -3 .

Figure 7 .

 7 Figure 7. Direct and adjoint global mode corresponding to H4 indicated in figure 6, visualised by the real part of the pressure field (a) and absolute value of the adjoint streamwise velocity (b). The insets, showing the real part of the streamwise velocity, give more details of the respective flow fields.

Figure 8 .

 8 Figure 8. Evolution of the streamwise velocity peak along the aerofoil chord (a) on the suctionand (b) on the pressure surface of the aerofoil for the high-frequency modes labelled H1-H4 in figure 6. (c,d) Same for the adjoint streamwise peak. (e,f) Same for the direct mode, adjoint mode and wavemaker corresponding to H4.

Figure 9 .

 9 Figure9. Low-frequency modes in the spectrum. The modes labelled L1-L4 are depicted in figure10, and the corresponding eigenvalues are ωL1 = 29.98 -0.63i, ωL2 = 20.15 -0.70i, ωL3 = 10.12 -0.65i ωL4 = 0.001 -0.19i.

Figure 10 .

 10 Figure 10. Direct and adjoint global modes corresponding to L3 indicated in figure 9, showing streamwise velocity (a) and adjoint streamwise velocity (b). The insets give more details of the respective flow fields.

Figure 11 .

 11 Figure 11. Evolution of the streamwise velocity peak along the aerofoil chord (a) on the suction-and (b) on the pressure surface of the aerofoil for the low-frequency modes labelled L1-L4 in figure 9. (c,d) Same for the adjoint streamwise velocity peak. (e,f) Same for the direct mode, adjoint mode and wavemaker corresponding to L3.

† Email address for correspondence: miguel.fosas@uca.es