An advanced hybrid method for the acoustic prediction

Advances in Engineering Software 88 (2015) 30–52

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An advanced hybrid method for the acoustic prediction S. Redonnet ⇑, G. Cunha ONERA, Aeroacoustics Dept., BP 72, 29 av. Division Leclerc, Châtillon 92322, France

a r t i c l e

i n f o

Article history: Received 19 December 2014 Received in revised form 16 April 2015 Accepted 24 May 2015

Keywords: Acoustics Prediction Computational acoustics Hybrid method Weak coupling Optimization

a b s t r a c t The present article proposes an advanced methodology for numerically simulating complex noise problems. More precisely, we consider the so-called multi-stage acoustic hybrid approach, which principle is to couple sound generation and acoustic propagation stages. Under that approach, we propose an advanced hybrid method which acoustic propagation stage relies on Computational AeroAcoustics (CAA) techniques. To this end, first, an innovative weak-coupling technique is developed, which allows an implicit forcing of the CAA stage with a given source signal coming from an a priori evaluation, whether the latter evaluation is of analytical or computational nature. Then, thanks to additional innovative solutions, the resulting CAA-based hybrid approach is optimized so that it can be applied to realistic and complex acoustic problems in an easier and safer way. All these innovative features are then validated on the basis of an academic test case, before the resulting advanced CAA-based hybrid methodology is applied to two problems of flow-induced noise radiation. This demonstrates the ability of the here proposed method to address realistic problems, by offering to handle at the same time both acoustic generation and propagation phenomena, despite their intrinsic multiscale character. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Sound is a key component of the human activity. As a consequence, acoustics constitutes a major field of investigation when it comes to improve the human condition. In particular, mastering acoustic phenomena is mandatory in many areas of the industry and technology worlds, for instance in order to improve manufactured products by mitigating their noise issues, to develop innovative sound-based technologies, and so on. On the other hand, since acoustics is a complex discipline, all concerned stakeholders are often bound to make intensive use of numerical simulation, which constitutes a powerful tool, when combined with experimentation. This, however, requires a continuous development and a proper application of advanced modeling and solving techniques, which are mandatory for simulating the sound generation and/or propagation phenomena occurring in practical situations of interest. 2. Context and problem 2.1. Acoustic prediction Nowadays, considerable research is focused on the modeling of acoustic phenomena, which finds many applications within ⇑ Corresponding author at: Aeroacoustics Department, ONERA (French Aerospace Lab), BP 72, Avenue de la Division Leclerc, Châtillon 92322, France. E-mail address: [email protected] (S. Redonnet). http://dx.doi.org/10.1016/j.advengsoft.2015.05.006 0965-9978/Ó 2015 Elsevier Ltd. All rights reserved.

numerous fields. Indeed, adequately modeling and predicting acoustic phenomena can help to improve existing technologies, such as reducing the noise annoyances by industrial products (aircraft, trains or cars, wind turbines, computers, etc.). It can also help optimizing specific sound-based devices, such as those used for non destructive testing, non intrusive medical imaging, seismic monitoring, etc. Finally, it can facilitate the emergence of new technologies, such as alternative and innovative sound-based devices (sensorless tactile screens, ultra-sound medical curative techniques, etc.). On another hand, since acoustics is a complex discipline, researchers are often required to make extensive use of numerical simulation, which constitutes a powerful means of investigation, when used to complement experimentation. This, however, requires the continuous development of advanced modeling and solving techniques, which are needed to simulate the sound generation and/or propagation phenomena occurring in realistic situations. Indeed, sound finds its origin in numerous source mechanisms such as structural vibrations, fluidic motions, flow interactions with structures, gas combustions or explosions, and so on. Once they have been generated by these sources, acoustic waves propagate within the surrounding environment, which is generally comprised of one or several media of various complexity (e.g. including solid bodies and/or medium heterogeneities). During such propagation, acoustic waves may exhibit numerous and important alterations in terms of amplitude, phase or frequency, which all result from mechanisms as diverse as reflection and diffraction effects by solid structures, convection by fluidic

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motions, refraction by the medium heterogeneities (e.g. velocity or temperature gradients), diffusion by the medium turbulence, absorption by the medium viscosity, etc.

weak coupling technique to be used both relies on valid physical principles and offers sufficient numerical robustness, especially in regard to an application within a realistic context.

2.2. Hybrid methods for the acoustic prediction

2.4. Acoustic propagation stage and computational acoustics

Many of the sound generation processes and most of the acoustic propagation mechanisms are relevant to the physics of fluid dynamics, and can thus be simulated by numerically solving the compressible Navier–Stokes equations. At the present time, however, and despite of the continuous development of computational tools and resources, it is still extremely challenging to solve acoustic problems following a direct manner, that is to say, via a single calculation. Indeed, except in particular situations (e.g. academic configurations), it is nearly impossible to simulate altogether the acoustic generation and propagation phenomena, because their underlying mechanisms greatly differ in their intrinsic characteristics (e.g. energy, length scales). As an example, numerous sound emissions by industrial products come from the interaction of their solid structures with a flow, whether the latter occurs in air (e.g. wind turbines, aircraft, trains, cars) or in water (e.g. ships and submarine). In particular, most of the noise annoyances due to modern transportation (aircraft, trains, cars, etc.) or energy (wind turbines) systems come from the so-called aerodynamic noise, which results from either the interaction of the airflow with the structure itself (e.g. airframe noise), or from its ingestion by the engines (e.g. fan and/or turbine noises). On another hand, the aerodynamic noise physics is made of complex phenomena covering a broad range of spatio-temporal scales, with acoustic generation processes that are driven by turbulent structures of high amplitude and small space–time correlations, whereas propagation ones are associated with sound waves of low amplitude and large space–time correlations. Thus, and although both phenomena are ruled by the same compressible Navier– Stokes equations, they cannot be easily predicted via a single calculation because the computational resources required to resolve all of the relevant scales would be far too high. Therefore, to make the numerical approach tractable in a practical context, the overall acoustic problem is usually decomposed into a set of coupled sub-problems that focus on individual sub-regions of the overall spatial domain. Each sub-problem has a specific range of amplitudes and physical scales that can be addressed using a numerical method that is customized to the dominant physics occurring at this stage. Methods involving a mix of techniques in this manner are classified as hybrid approaches for the acoustic prediction.

Regarding more specifically the acoustic propagation stage of an acoustic hybrid method, its role is to propagate within the surrounding environment all the acoustic information that could have been made available from any prior sound generation modeling and/or simulation. As said above, due to the variety and complexity of all physical phenomena involved, numerically simulating such a propagation phase is generally not trivial. In particular, the computational techniques used must simulate accurately the propagation of acoustic waves over relatively large distances across possibly heterogeneous media, while accounting for the possible presence of solid obstacles. This may typically be accomplished with high fidelity acoustic propagation approaches, such as a Computational AeroAcoustics (CAA) method relying on the Perturbed Euler Equations (PEE), or a linearized version thereof (Linearized Euler Equations, LEE). Indeed, one can here recall that only a CAA method can simultaneously account for both the reflection/diffraction effects by solid obstacles and the refraction effects by the medium heterogeneities, in contrast to other techniques that can only model the former (such as the Boundary Element Method, BEM) or even none of them (such as an Integration Method, IM, whether it is based on a Kirchhoff [1] extrapolation or an Acoustic Analogy [2,3]), because of their underlying hypothesis of a free and/or homogeneous propagation medium. As an illustration of typical problems more accurately solved via an acoustic hybrid method relying on a CAA-based propagation stage, consider an acoustic internal propagation problem, such as those occurring in HVAC (heating, ventilation, and air conditioning) systems of buildings, vehicles, computers, etc. Indeed, here, once their generation has been properly modeled by analytical means (e.g. duct mode theories [4]) or simulated via a numerical method (e.g. Computational Fluid Dynamics, CFD), acoustic waves may then be transferred to a CAA solver, for the latter to propagate them through the duct, while accounting for all internal effects to be possibly induced by the presence of flow heterogeneities, solid devices or any other disturbing elements (such as noise absorbing panels, etc.). Another good example of situations where an acoustic hybrid approach based on a CAA propagation stage can be advantageously applied is all external acoustic problems where the propagation phase occurs within a complex environment, such as, for instance, those associated with airframe noise emissions by particular components of modern transportation vehicles (e.g. landing gear of an aircraft, pantograph system of a high-speed train, side-mirror of a car). Indeed, here again, once their generation has been properly simulated (usually via an unsteady compressible CFD method), acoustic waves may then be transferred to a CAA solver, for the latter to propagate them up to the far-field, while accounting from all the installation effects induced by either the vehicle elements (e.g. reflection/diffraction) or the airflow that surrounds the latter (e.g. convection/refraction).

2.3. Multi-stage coupling for hybrid methods in acoustics A critical aspect of developing acoustic hybrid methodologies corresponds to the coupling, i.e. the information exchange to occur between the various stages respectively associated with the individual sub-problems. The nature of this coupling is problem dependent, because of significant variations in the interdependencies between the various stages from one problem to another. However, except in problems involving acoustic feedback (e.g. screech tones, in jet aeroacoustics), the coupling between these stages is weak, i.e. primarily unidirectional. Under this scenario, feedback from a given level to the previous one can be neglected, and the successive stages of an acoustic hybrid calculation can be coupled in a weak sense, all possible retro actions from a given step to the previous one being then neglected. Such a weak coupling process to occur between two successive stages of a computational acoustics hybrid approach is constituted with a data transfer, whose role is to transmit to the next level all the acoustic information gathered at each step. Needless to say, such an operation must be properly achieved, so that it does not degrade the sound signal to be transferred. This requires that the

2.5. Weak coupling techniques for computational acoustics As was said, properly weak coupling a preliminary sound generation stage with a CAA-based propagation one implies that the acoustic information is transferred from the former to the latter in a conservative fashion, that is, without any loss nor duplication. This requires that the weak coupling technique relies on valid physical principles, as well as it offers sufficient numerical robustness to be applicable to realistic configurations. In particular, the technique must possess and maintain certain critical attributes;

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it must first offer enough generality with respect to the physics, being in particular free of any too stringent assumption that would restrict its application to only a certain type of situation (e.g. homogeneous medium, low Mach number flow). It must then guarantee enough consistency with respect to the modeling, being in particular free of any hidden bias coming from the mismatch that might arise between the respective continuous formulations underlying the sound generation and the CAA-based propagation stages (e.g. viscous effects, nonlinear effects). It must also offer sufficient stability with respect to the numerical solution, being in particular free of any issues that might arise from the mismatch between the respective discrete formulations associated with these two latter stages (e.g. sampling and/or interpolation effects). Last but not least, to be applicable to many situations of interest, the coupling technique must offer a perfect numerical anechoicity (or transparency); that is, it must allow proper handling of the acoustic backscatter effects that may occur within the acoustic propagation regions, because of the surrounding environment (see Section 2.1). However, simultaneous fulfillment of the above stated requirements is a quite challenging objective. And, after more than a decade of extensive research, the question of achieving a weak coupling technique that would be accurate and robust enough for a CAA-based hybrid method is applicable to any acoustic problem still remains an open question; Indeed, weak coupling a sound generation and a CAA-based acoustic propagation stages was first attempted following the so-called volumetric approach, which was intensively investigated by various researchers [5–14]; inspired originally from the works by Sir James Lighthill [2], such approach is based on the idea of locally mimicking the sound generation scenario within the CAA-based acoustic propagation stage, this being achieved via an equivalent acoustic source. The latter is to be synthesized from the outputs delivered by the prior sound generation stage, before it is incorporated to the CAA simulation with the help of a so-called source term (i.e., acting on the right hand side of the equations system). Once the sound generation events are mimicked, all the induced acoustic occurrences can then be propagated to the far-field by the CAA solver. Needless to say, such a coupling approach depends entirely on the proper modeling of the sound generation mechanisms themselves – which still eludes us and constitutes an open issue. For instance, regarding aerodynamic noise problems, our lack of understanding into how energy from unsteady flows is converted into acoustics makes it very challenging to isolate the aerodynamically-induced noise generation mechanisms from the acoustically-driven propagation ones [10,15,16]. First, regarding the consistency requirement listed above, this prevents the CAA continuous formulation from being allotted an equivalent source term that would have a true physical meaning,1 preventing advancement much further than Lighthill’s famous Acoustic Analogy.2 Then, regarding this time the stability requirement enunciated previously, this complicates significantly the way the CAA discrete formulation can be numerically handled in practice, especially for what concerns the proper CAA-solving of the source signal (which, in addition to acoustic ones, may also gather unsteady aerodynamics occurrences of small length scale and large amplitude). 2.6. Recent efforts for the development of an advanced CAA-based hybrid method Over the last years, ONERA devoted consequent research means to the development of a robust and accurate CAA-based hybrid 1

Except under particular conditions, e.g. vortex noise theory for low speed flow regimes. 2 Which resulting IM-based hybrid method is however too restrictive, for the reasons enunciated in Section 2.4.

method. Within this framework, several dedicated studies were achieved, which all served the same purpose, whilst focusing each on a particular aspect of the problem; besides previous works that led to the development of an optimal CAA method/solver [17,18], a specific research action [22–24] consisted in developing a weak coupling approach that could allow a CAA-based propagation stage to be forced with any given acoustic signal whilst offering to handle acoustic backscattering phenomena, for the overall hybrid method is compatible with many situations of interest (cf. the anechoicity requirement evocated above). Then, follow-on research actions [25–30] consisted in optimizing such overall hybrid method, for it can cope with all stringent constraints that are dictated by real-life applications without being jeopardized by some of their unavoidable side effects (e.g. signal degradation – see detail below). Thanks to the various methodological outcomes they led to, these studies made it possible to develop an advanced CAA-based hybrid approach, which integrates and connects altogether the diverse aspects of the problem, whilst accounting for their interactions. Such approach was validated through an application to various acoustic problems of increasing complexity [23,24,31] (some of which are presented hereafter). All these works constitute the matter of the present article, which is organized as follows; Section 3 summarizes the various research actions conducted, and their associated outcomes. Section 4 presents the numerical method which resulted from all these outcomes, and that now constitutes the advanced CAA-based hybrid approach. The key features of the latter are then illustrated in Section 5, on the basis of a unique academic test case coming from the BANC (Benchmark for the Airframe Noise Computations) international initiative.3 After that, in Section 6, such advanced CAA-based hybrid method is applied to two problems of aerodynamic noise, the latter of which comes from a real-life application (aeroacoustics of a landing gear, which is also excerpted from BANC initiative). Finally, Section 7 draws the main conclusions about this advanced CAA-based hybrid method, summarizing its main features and discussing how it can effectively help solving practical acoustic problems. 3. Development of an advanced CAA-based hybrid method The present section summarizes the various works that were achieved for developing the CAA-based hybrid method introduced above, and the diverse innovative solutions such works led to. Please note that the latter were thoroughly validated on the basis of various test cases of increasing complexity (only a fraction of which is provided in Sections 5 and 6), which addressed both academic and realistic configurations, and concerned either internal noise problems (e.g. in-duct propagation) or external ones (e.g. aerodynamic noise). On the same way, all the related outcomes were extensively documented through previous publications, either as journal articles or proceedings of international conferences [22–31]. The reason for summarizing these various works and associated outcomes here was that, although focusing each on a particular aspect (and deserving each an entire publication in itself), they all targeted the same common objective, which is the development of the advanced CAA-based hybrid approach that the present article is about. For the latter approach can be shared with the community, it was needed to unify within a common formalism all the innovative solutions (technical developments, methodological guidelines, etc.) these works and outcomes had 3 Managed by NASA Langley Research Center (LaRC) and sponsored by the American Institute of Aeronautics and Astronautics (AIAA), the Benchmark for the Airframe Noise Computations (BANC) is an international joint initiative that seeks at improving the capabilities of numerical simulation techniques to solve airframe noise issues.

S. Redonnet, G. Cunha / Advances in Engineering Software 88 (2015) 30–52

led to (which is made at Section 4), as well as to present some of the typical problems that can be handled through such advanced CAA-based hybrid approach (which is made at Sections 5 and 6). 3.1. Derivation of an innovative weak coupling approach With the view of allowing the proper transfer of any given acoustic signal coming from an a priori sound generation stage to within a CAA-based propagation one, an innovative weak coupling approach was recently derived by the present first author [22–24]. Such approach relies on the idea of letting the sound generation stage (e.g. unsteady CFD calculation) to account not only for the acoustic generation mechanisms, but also for the early propagation processes that are to occur within a given near field encompassing all source regions. From the edge of such a near field, and provided that the latter is defined with enough care4 [19,22], all these early propagated waves can then be constituted as a source signal with which forcing the CAA-based acoustic propagation stage, for the latter to propagate them up to the far field. Compared to the volumetric one, such surface coupling approach offers thus the important advantage of allowing the propagation stage to be forced with real (rather than equivalent) noise sources. This not only spares the effort of tentatively modeling the generation mechanisms, but also avoids the potential risks of wrongly mimicking them. This weak coupling approach relies on the so-called Non Reflective Interface (NRI) technique [22–24], which constitutes an improved version of the original (reflective) interfacing technique that had been initially developed by the present first author a decade ago [17,18]. The latter technique consisted in forcing explicitly the CAA computed field with the acoustic signal to be transferred, such explicit forcing being applied at each time step and over a few rows of ghost cells (see details in Refs. [17,18]). As is, such an interface technique was not only straightforward to implement and use, but it was also general enough to be applicable to many situations of interest. As an illustration, regarding internal noise problems, such interface technique had been successfully applied to various cases (e.g. the aft fan noise emission by turbojet engines [32,33]), for which the acoustic signals to be CAA-forced had been derived via analytical means (based on the modal theory [4]). On another hand, regarding now external noise problems, the technique had been also applied with success to several airframe noise applications (e.g. the acoustic emission by either an in-flight NACA0012 airfoil with a blunted trailing edge [19,20] or a thick plate embedded within a flow [21]), the acoustic signals to be CAA-forced having been, this time, obtained via a compressible unsteady CFD calculation. However, in its original version, and because of the explicit forcing it relied on, the interface technique was of reflective type; that is, it was unable to handle properly backscattered acoustic waves that could possibly occur within the propagation regions. Although such a reflective character of the interface did not really constitute an issue for the various application cases recalled above, it could question its application to more complex configurations. For instance, in internal propagation problems, sound waves that propagate within a duct can be backscattered because of various artifacts such as those constituted by either the termination or any additional devices (bifurcations, splices, etc.) of the duct, as well as by the lining of the latter with sound absorbing materials. Furthermore, in external propagation problems involving installed configurations, any acoustic source region is likely to be 4 For the resulting source signal gathers unsteady perturbations that (i) are accurately predicted by the acoustic generation stage (e.g. are properly solved through CFD), and (ii) are likely to be accurately solved by the acoustic propagation one (e.g. are sufficiently discretized onto the CAA stage) – see [19,22] for more detail on this point.

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surrounded by solid bodies that may produce backscattered waves (consider for instance the situation sketched in Fig. 1, where the interactions between an air flow and a wing with a deployed landing gear makes the latter radiate acoustic waves, some of which being then backscattered towards the landing gear region because of the wing’s presence). For the overall hybrid calculation to behave correctly in these situations, not only must the weak coupling technique accurately transfer the acoustic information from the prior sound generation stage to the CAA-based propagation one, but it must also allow backscattered acoustic waves to cross back into the coupling area in a non-reflective (or anechoic) manner, i.e., without being bounced back or without creating spurious noise. Therefore, with the view of removing its intrinsic limitations, recently, the interface technique was entirely reformulated, so that it exhibits a non-reflective character. Such improved version of the technique, which is called the Non Reflective Interface (NRI), was thoroughly validated and/or applied on the basis of various test cases of increasing complexity [22–24], among which the ones provided in Sections 5 and 6. The NRI technique relies on a dual splitting of the flow field into a mean and a perturbed components, which associated fields can then be defined following a composite manner (see detail in Section 4.1.2 below). Thanks to their composite nature, these mean/perturbed fields can be allotted a given discontinuity that, occurring along the weak coupling interface, corresponds specifically to the source signal to be transferred from the sound generation regions to the acoustic propagation ones. As a result, the CAA stage is implicitly forced with the source signal, which passes from the sound production zones (over which it is left CAA-untouched) to the propagation areas (over which it becomes free of CAA-evolving). More importantly, the CAA perturbed field being not submitted to any other treatment than such an implicit forcing by the source signal, any other disturbance can travel within the whole domain, independently of where the NRI weak coupling interface is located. Therefore, the CAA perturbed field remains free to evolve as it has to do, and in particular to handle backscattered occurrences that may arise within the acoustic propagation regions – a thing that, again, is mandatory for the resulting CAA-based hybrid approach is applicable to a wide range of problems. 3.2. Optimization of the CAA-based hybrid approach In a second time, the resulting CAA-based hybrid approach was further improved, so that it can handle more challenging problems (e.g. those involving complex acoustic signal and/or requiring heavy computations). More precisely, works consisted in optimizing the overall hybrid procedure [25–30], for the latter can (i) cope with all stringent constraints that are dictated by real-life applications (ii) without being jeopardized by some of their unavoidable side effects (such as the signal degradation to which unsteady CFD data are subjected, when manipulated for being acoustically exploited [25]). To this end, and besides specific optimizations of the underlying CAA approach by the present first author (e.g. non linear formulation [17,18], LOC Non Reflecting Boundary Condition [17,18,21]), several research actions were conducted [25–30], being primarily achieved by the present second author in the framework of his PhD thesis [30]. These works consisted in (i) characterizing [25,27,28] and (ii) possibly minimizing [26,29] the various impacts that its data manipulation may have onto a given acoustic signal, when the latter is to be transferred from one to another stage of an acoustic hybrid method. In addition to deliver fundamental insights that may help optimizing any kind of acoustic hybrid method (e.g. IM- or CAA-based one), these actions led to the development of several methodological innovations (SPC criterion [25], IBP interpolator [26], IOFD schemes

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Fig. 1. Aeroacoustic of an aircraft undercarriage system: sketch of the aerodynamic noise emission by a landing gear installed under a high-lift wing, to be numerically simulated via a multi-stage acoustic hybrid approach relying on a hybrid calculation (CFD-CAA-IM).

[29] – see below), which made it possible to relax some of the constraints preventing the present CAA-based hybrid approach to be applied to challenging problems (such as the one presented in Section 6.2). Here again, all these outcomes were thoroughly validated and/or applied through various test cases of increasing complexity, among which the ones provided in Sections 5 and 6.

3.2.1. Minimization of the signal degradation induced by the CFD data manipulation One of these improvements consisted in the accurate characterization of the signal degradation that unsteady data may be subjected to, when sampled and/or interpolated in space and/or time [25,27]. Indeed, hybrid calculations generally require that the acoustic signal to be transferred from one to another stage is submitted to a certain number of manipulations; first, since hybrid calculations are usually not conducted simultaneously, the unsteady data that are computed at a given stage must be stored before they are transferred to the next hybrid step. Such operation is often conducted with a space and/or time sampling applied, so that the storage time and memory requirements are minimized. Second, because of the mismatches that may occur between the various discrete formulations ruling each one of the stages to be weakly-coupled, when transferred from one to another stage, these data must then be interpolated in time and/or space. The point is that both the sampling and the interpolation operations are subjected to side effects that may lead to a complete degradation of the acoustic signal to be transferred, which suffices to question the accuracy and robustness of the resulting overall hybrid procedure. For instance, the theoretical investigations conducted in the present framework demonstrated how far the acoustic information delivered by a given CFD calculation can be dramatically and irremediably degraded, depending on the way the CFD-generated data are effectively stored and/or processed for being acoustically exploited [25,27]. Among other things, it was shown how its sole sampling (in space and/or time) may degrade a CFD-generated acoustic signal as much – and sometimes even more – than its interpolation. More importantly, these theoretical investigations helped in better understanding and modeling the two intrinsic mechanisms that are responsible for the signal degradation, which are the aliasing and the spuriousing phenomena [25]. This led to derive a dedicated formalism [25,27] that allows predicting a priori (i) when and how a given unsteady signal may be degraded when sampled and/or interpolated, as well as (ii) how far such degradation may then impact the acoustic propagation stage (whether the latter is of CAA or IM type). With the view of

minimizing such impact, innovative solutions were then developed; among other things, one can mention the so-called Interpolation By Parts (IBP) technique [26], which allows interpolating accurately a given signal, while minimizing its possible degradation. On the same way, one can mention the specific Signal Preservation Criterion (SPC) [25], which can serve as general guidelines for adjusting the various elements (CFD storage, CAA grid, etc.) of a given hybrid scenario, so that the acoustic signal to be transferred from one stage to the other is preserved at best. These innovative solutions (IBP, SPC) were validated on the basis of various academic test cases [25,26,30] among which the one of Section 5, before they were applied to the aerodynamic noise problems of Section 6. Here, one can underline that the fundamental insights and methodological outcomes that were derived within this particular framework are not restricted to the sole case of an acoustic hybrid method relying on a CAA sound propagation stage. Indeed, most of the theoretical results and/or innovative techniques developed here could easily be extended to simpler acoustic hybrid approaches (e.g. those relying on an IM sound radiation stage), which are also subjected to the same kind of side effects (e.g. signal degradation). On the same way, beyond their application to acoustic hybrid methods, many of the previous outcomes could also be advantageously applied to research areas involving interpolation techniques (e.g. multi-size-mesh multi-time-step problems, immersed boundary methods, overlapping grid techniques).

3.2.2. Relaxation of the constraints weighting on both the CFD data transfer and the CAA stage Last improvements consisted in optimizing both the NRI technique and the CAA stage itself, so that the overall CAA-based hybrid approach can cope with the constraints (e.g. SPC) imposed by the necessary preservation of the acoustic signal to be CAA-forced and further propagated; first, the NRI technique was optimized so that its minimal storage requirements can be relaxed, in order to lower the sampling (and, therefore, possible degradation) of the CFD datasets to be stored. This was achieved by adapting the NRI process so that it can be handled by space operators of reduced stencil sizes, compared with those used for the propagation stage. Once validated on the basis of the test case provided in Section 5.1, this optimized version of the NRI was applied to the one of Section 6.2. Then, with the view of optimizing the CAA stage with respect to the signal preservation constraints, a new class of finite difference (FD) propagation schemes was developed [29]; as detailed in Ref. [29], these so-called Intrinsically Optimized

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FD (IOFD) schemes are of as-like optimal accuracy, thanks to an optimization process that is based on a minimization of the scheme’s leading-order truncation error (rather than on an optimization of the scheme’ spectral properties, such as usually done [43–45]). Thanks to their optimal accuracy, these IOFD schemes allow lightening significantly the CAA stage, which grid density requirements can be relaxed at maximum. Such a benefit makes it possible to either extend the frequency range to be properly solved by the CAA stage, or to reduce the computational cost to be paid for its consumption. As a corollary, regarding the present CAA-based hybrid approach and weakly-coupled CFD-CAA calculation matters, such a relaxation of the constraints weighting on the CAA stage can be directly turned into a beneficial increase of the CFD storage density – which, at the present date, constitutes the best way for minimizing the signal degradation to be possibly induced by the manipulation of CFD data [43]. These IOFD schemes were thoroughly validated on the basis of various test cases [29– 31] (among which the one of Section 5.3), before they were applied to various problems [31,52,53] (among which the aerodynamic problems provided in Section 6). 4. Numerical approach The present section details the numerical approach that underlies the advanced CAA-based hybrid method. Such approach integrates the specific innovations introduced above (NRI, SPC, IBP, IOFD), whose features will be illustrated in next Sections 5 and 6. 4.1. Continuous formulation The present subsection summarizes the continuous formulation underlying the CAA propagation stage, as well as its possible forcing via NRI. 4.1.1. CAA governing equations We start with the compressible Navier–Stokes equations written in conservative form, for a Newtonian fluid and a perfect gas (such as air). We then adopt a perturbation approach, according to which any unsteady phenomena can be seen as composed of a perturbed state oscillating around a mean one. Following this, we split each one of the usual physical quantities into both a perturbed (subscript .p) and a mean (or background, subscript .o) components, which respective definitions are still to be given:

q ¼ qo þ qp

ð1Þ

In the above equation, each q. quantity is composed of its associated density, q., velocity, v., and pressure, p., such that q. = (q., v., p.). We then invoke the classical small perturbation hypothesis, which implies that the background field, qo = (qo, vo, po), can be considered as insensitive to the evolution of its perturbed counterpart, qp = (qp, vp, pp), as long as the latter remains sufficiently small in terms of amplitude. This allows us deriving an equation governing solely the perturbed field, qp, whilst accounting for its background counterpart, qo:

@ t up ðqo ; qp Þ þ r  Fp ðqo ; qp Þ ¼ 0

ð2Þ

with

Fp ðqo ; qp Þ ¼ Fp ðqo ; qp Þ  Fmp ðqo ; qp Þ

ð3Þ

As is, Eq. (2) represents the Navier–Stokes equations in conservative and perturbed form [17,18]. In the above equations, up term stands for the perturbed flow-field vector, whereas Fp and Fpm ones stand for its associated convective and diffusive flux matrixes, respectively. For conciseness, these terms, which complete

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derivation can be found in Refs. [17,18,14], are not detailed here, being however provided in Appendix A. At this stage, one can notice that, whenever the perturbed field evolution can be considered as insensitive to viscous effects (such as what happens to acoustic phenomena, when propagating time and distance are not too large), one can neglect the diffusive terms Fpm. In which case Eq. (2) reduces to the Full Euler equations in conservative and perturbed form [17,18]. Whether it is taken in its Navier–Stokes or Full Euler version, Eq. (2) governs the perturbed field qp, whose evolution incorporates all phenomena to be induced by its background counterpart qo (such as the convection and refraction effects by the medium onto acoustic occurrences, for instance). On another hand, because of the small perturbation hypothesis made here, such perturbed field qp cannot retro-act5 in any way onto the mean one qo. Constituting a data entry for the evolution problem associated with its perturbed counterpart, qp, the background field, qo, must be determined a priori, i.e. it is to be either analytically derived or numerically simulated. This implies that such background field is given a clear definition; the most popular – and certainly most convenient – one is the steady mean (i.e. stationary in time) field, that is qo = hqi, where h.i denotes the time average operator6 such that def

hqiðxÞ ¼

Z

t final

t init

qðx;tÞ dt

ð4Þ

Thanks to its trivial definition, such a steady mean field can generally be obtained in a straightforward manner, whether it is supplied by a proper analytical derivation (e.g. for academic problems) or from the post-processing of a (steady or unsteady) CFD calculation (e.g. for more complex configurations). Because of its definition however (and, in particular, of its time independent character), such a background field can only be composed of time invariant phenomena (e.g. steady aerodynamics). On the contrary, the associated perturbed field may include unsteady features (among which acoustic ones), to be excited or not depending on the problem definition (source, initial conditions, etc.). Finally, it is important to notice that, although such a particular definition of the background field is the one generally used, it is not the sole possibility. Indeed, in some cases, one can advantageously make use of an alternative definition for the background field, such as done in the NRI technique (see below). 4.1.2. Non Reflective Interface (NRI) Let us now consider that one wishes to force the CAA-based acoustic propagation stage with a given source signal, qs = (qs, vs, ps). With that view, let us distinguish within the CAA domain those source regions Xs where such signal is to be made available from a prior sound generation stage, from those propagation areas Xs over which it is to be further propagated, once forced. For doing so, and following NRI approach [22–24], the source signal (qs) is incorporated into the background field (qo) over all source regions (Xs ), while it is incorporated to the perturbed field (qp) over all propagation areas (Xs ).This can be achieved in a straightforward manner by defining a composite representation of the background field qo (and, by extension, of its perturbed counterpart qp = q  qo), to be composed of a different association of hqi and qs, depending on the region considered (Xs or Xs );

qoj

def Xs

¼ hqi

ð5Þ

5 As was said, however, this restriction is only weakly limiting since aero acoustic feedback effects only occur under special circumstances. 6 To be associated with the longest time interval [tinitial, tfinal] possible.

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and def

qojX ¼ hqi þ qs

ð6Þ

s

As a result, although their sum (i.e. the total field, q) remains perfectly continuous within the whole CAA calculation domain, both the background and the perturbed fields (q. = o, p) are now discontinuous, being defined differently over Xs and Xs (see Fig. 2, where Xs is highlighted in gray). Such discontinuity, which solely occurs along the Xs =Xs interface, corresponds to the source signal qs itself;

(

qoj qpj

Xs

¼ qojX  qs

Xs

¼ qpjX þ qs

s

ð7Þ

s

As a consequence of such a composite representation of the background and perturbed fields, the conservative perturbed quantities (Q p ¼ up ; Fp ) are also discontinuous along the Xs =Xs interface:

Q p ðqo ; qp Þj

Xs

¼ Q p ðqo ; qp Þj

Xs

þ Q p ðhqi; qs Þ

ð8Þ

Being defined by the quantity Q p ðhqi; qs Þ, such discontinuity is explicit, and depends only on the steady flow (hqi) and the source signal (qs) components – which are known. It is in such a composite representation of the mean/perturbed fields (qo, qp) and associated conservative perturbed quantities (Qp) that lies the Non Reflective Interface (NRI) technique, which allows all of the source signal to be transferred implicitly from the source areas’ mean field, qojX , to the propagation regions’ pers

turbed one, qpj . In doing so, nothing else is required other than to Xs

solve Eq. (2) in accordance with the composite representation given by Eq. (8), such as detailed for instance in Section 4.1.7 for the particular case of a structured approach relying on finite differences schemes. Since the perturbed field qp may still evolve freely, any perturbations beyond the source signal can still be handled properly by the CAA method, as in the absence of any forcing. This is precisely what makes the NRI technique non-reflective, which constitutes its major strength compared to other forcing techniques.7 4.1.3. Some remarks The present continuous formulation, which was proposed and promoted by the present first author through the years [17,18,22,24], is very general in that it solely relies on a small perturbation hypothesis. In particular, no additional hypothesis (isentropy, incompressibility, linearity, etc.) is made here. Therefore, one can expect the present formulation to offer enough physical generality to allow the modeling of complex acoustic problems, that is, without implying restrictions8 that usually limit other prediction approaches.9 Regarding numerics, one should also emphasize that the present formulation is of conservative nature, which ensures more numerical robustness (compared to those formulations written in a non-conservative form). Finally, this formulation can be solved via various numerical techniques, to either rely on an unstructured approach operating on finite elements (e.g. the so-called Discontinuous Galerkin Method), or to be based on a structured approach operating on multi-block structured grids. The latter structured approach can be handled either through finite volumes or with 7 Such as characteristic [34–35], asymptotic approximation [36] or buffer zone [37,38] approaches, whose intrinsic limitations and/or underlying assumptions prevent from being applicable to many non–trivial configurations [36,37,39]. 8 Such as homogeneous or irrotational mean flows, high frequency sources, etc. 9 Such as any of the Integral Methods (IM), the Boundary Element Method (BEM), the Parabolic Equation (PE), the Equivalent Source Method (ESM), etc.

Fig. 2. Dual splitting of total primitive variables (q) into composite perturbed (qp) and mean (qo) fields, via a dual representation of qo into a source (qs) and/or a time averaged (hqi) components. In the above sketch, the signal associated with the total field is depicted in red, whereas its source and time averaged components are plotted in blue and black respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the help of high-order finite difference schemes – such as is detailed hereafter. 4.2. Discrete approximation of the continuous formulation The present subsection summarizes how the previous continuous formulation is approximated from a discrete point of view, for being solved numerically with the help of finite-difference schemes. Indeed, unless it possesses an analytical solution, no physical problem can be exactly solved through numerical methods since, although computer resources are rapidly and constantly increasing, they are not (and will never be) limitless. Due to such limits in terms of data volume that can be effectively processed by computers, any physical (i.e. continuous and infinite) problem has to be first (i) discretized (or sampled) and (ii) bounded within a finite domain, before it can effectively be numerically solved in time and space. On that stage, one can recall that literature abounds in studies that focused on which discrete schemes to better privilege for approximating at best the various operations (time evolution, spatial derivation, etc.) of the continuous problem. All of these studies clearly showed that only those schemes whose order of accuracy is high enough can ensure sufficiently low numerical dissipation and dispersion for the sound propagation features are rendered properly, at a reasonable CPU cost. These good propagative features of high-order schemes can be further increased through dedicated optimization procedures, as was done here and is detailed hereafter. 4.2.1. Time discretization The discrete approximation of the continuous problem, Eq. (2), is first achieved in time. This is done by approximating the exact temporal evolution operator that acts on the perturbed field, sfg, ~fg ; for a given with the help of a discrete time marching scheme, s step Dt along the temporal axis et, one then gets;

s~ðDtet Þ fup g ffi sðDtet Þ fup g

ð9Þ

In the present case, the time marching scheme corresponds to a multi-stage Runge–Kutta (RK) scheme of M levels:

s~ðDtet Þ fup g ¼ fup gM

ð10Þ

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with, for each stage m = 1, . . ., M;

fup gm ¼ fup gm1 þ cm fwp gm

ð11Þ

Table 1 Discrete approximation of the time advance operator; weighting coefficients associated with the optimized 3-stage Runge Kutta scheme.

and

cm

dm

fwp gm ¼ dm fwp gm1 þ Dtf@ t up gm1

c1 = 0.5 c2 = 0.9106836025229591 c3 = 0.3660254037844387

d1 = 0 d2 = 0.6830127018922193 d3 = 1.333333333333

ð12Þ

knowing that

fwp g0 ¼ 0

ð13Þ

and

fup g0 ¼ up

ð14Þ

as well as, according to Eq. (2),

@ t up ¼ r  Fp

ð15Þ

In the above equations, cm and dm are the RK scheme’s coefficients, which are to be judiciously determined so that the approximation is accurate enough. Here, one can recall that a fair compromise between the performances and the CPU requirements is met by (i) the standard 4-stage (3rd order) RK time marching scheme [17,18], which became quite popular among the CAA community. In the present case, however, choice is made of using a 3-stage (M = 3) RK scheme, which was optimized so that its CFL (Courant– Friedrichs–Lewy) constraints are lower [42]. Table 1 lists the set of weighting coefficients associated with such optimized 3-stage Runge–Kutta scheme. Please note that, regarding its use within a CAA context, this particular optimized scheme was validated on the basis of numerous test cases, before it was applied to various problems [20,32,33,40,41]. 4.2.2. Space discretization In a second time, the divergence of perturbed flux in Eq. (15) is approximated;

e  F ffi r  F r p p

ð16Þ

This is achieved through a discrete approximation of the exact divergence operator, r  fg;

e  fg ¼ r

X ð@~i fgÞ  ei

ð17Þ

Wavelength, PPW) is reduced further. Such optimized schemes are characterized by alternative sets of coefficients aj, to be derived following an optimization process (whether the latter optimization is achieved in the spectral space,12 or not). As was said above, in the present case, an innovative optimization process based on a minimization of the scheme’s leading-order truncation error is employed. The resulting and so-called Intrinsically Optimized FD (IOFD) schemes are of higher accuracy [29], compared to either standard or spectral-like optimized schemes [43–45]; for instance, with no more than 4 (resp. 8) Points Per Wavelength (PPW), an IOFD scheme of 15 (resp. 11) points guarantees that the error made on the group velocity (on which depends the acoustic energy transport and, thus, the overall accuracy of the CAA stage) is less than 0.14%. Compared to a standard 7-point/6th order standard FD scheme (which corresponding minimal PPW is 12), this represents a gain of 3 (resp. 3/2) per direction, i.e. a factor 27 (resp. 3.375) in 3D. Such a benefit can allow either an extent of the frequency range to be properly solved by the CAA stage, or a reduction of the computational cost to be paid for its consumption (with memory requirements that go decreasing by the same factor – while CPU times can be reduced even more, e.g. down to a factor 54, depending on the CFL constraints). Table 2 provides the set of weighting coefficients associated with either the 11-point (6th order) or 15-point (8th order) IOFD scheme. As was said, these two IOFD schemes were validated on the basis of various academic test cases [29] (among which the one of Section 5.4), being then applied to various problems [31,52,53] (among which the aerodynamic problems provided in Sections 6.1 and 6.2, respectively).

i

where @~i fg denotes the approximate derivative operator acting in the ith direction of unitary vector ei. In the present case of a structured approach, the approximate derivative operator is based on finite-difference (FD) schemes. As usually done, the latter are here taken under their explicit10 and centered11 version, which can be expressed as; N 1 X @~i fg ¼ aj sðjDxi ei Þ fg Dxi j¼N

ð18Þ

As one can see, the latter expression is composed with a linear combination of translation operators s that, operating in the ith spatial direction, are weighted by coefficients aj. Here again, a judicious choice of these coefficients is to be made so that the resulting approximation is of enough accuracy. At this stage, one can remind that a fair compromise between the performances and the CPU requirements is met by the standard 7-point (6th order) FD spatial derivative scheme, which became very popular among the CAA community. However, one can also make use of alternative FD schemes, to be optimized such that their computational cost (in terms of minimal Points Per 10 For explicit FD schemes allow an easier handling of spatial boundary conditions, compared with their implicit (or so-called compact) counterparts. 11 Since centered FD schemes are known to be more stable than their non-centered counterparts, which are likely to amplify high frequency contents.

4.3. Discrete approximation of the source signal As for the perturbed physical quantities (up ; Fp ) and/or mathematical operators (@ t fg; r  fg) constituting the continuous formulation, the source signal qs must be approximated so that it becomes compatible with the CAA discrete formulation to be numerically solved. As summarized below, such an approximation consists in one or several operations (namely, sampling rate reduction, interpolation, discretization). As was shown in [25,27], because of their side effects (namely, aliasing and spuriousing phenomena), these operations are likely to impact the signal, degrading its original contents in a non negligible manner. With the view of possibly remedying this issue, as was explained in Section 3.2, several fundamental insights and innovative tools were derived within the present framework. Here, it is worth mentioning that all these outcomes apply to any kind of acoustic hybrid approach, whether its propagation stage relies on CAA, IM or BEM techniques. Here, however, they were implemented solely within the present CAA-based acoustic hybrid method, as described below. They were then validated on the basis of various academic test cases [25] (among which the ones of Section 5), and applied to several problems [30,31] (among which the ones of Section 6). 12 e.g. the so-called spectral-like optimized schemes proposed by Lele [43], by Tam and Webb [44], and, later on, by Bogey and Bailly [45].

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Table 2 Discrete approximation of the space derivation operator via FD schemes; weighting coefficients associated with Intrinsically Optimized Finite Difference (IOFD) schemes of 11 (N = 5) and 15 (N = 7) points, respectively.

a0 a1 = a1 a2 = a2 a3 = a3 a4 = a4 a5 = a5 a6 = a6 a7 = a7

N=5

N=7

0 0.865816409136225 0.275690008254363 0.0811126692593992 0.0165005999177029 0.00164559985302308

0 0.899944730463117 0.326219480460150 0.124540388374523 0.0404962578862881 0.00985168108601892 0.00152738609091035 0.000109143999162341

4.3.1. Discrete projection (or discretization) of the source signal onto the acoustic propagation stage If one assumes that the source signal, qs, is continuous, then, its approximation solely consists in a discretization (or discrete projection) onto the acoustic propagation grid, which is here constituted with the spatio-temporal CAA mesh. Such discretization can be assimilated to multiple samplings, to be applied successively and independently along the various space/time directions (v = xi, t) with, for each, a sampling frequency corresponding to 1/D, where D indicates the CAA grid interval (in space or time) along the concerned direction;

~ sðvÞ ¼ q

þ1 X

qsðjDÞs dðvjDÞ

ð19Þ

j¼1

This can be formally expressed by a convolution product between the signal itself, qs ; and that particular distribution, dD , which corresponds to a Dirac stencil of period D; v

~ s ¼ qs  dD q

ð20Þ

with

dD ¼

þ1 X

sðjDÞ fdg

ð21Þ

j¼1

where d denotes the Dirac distribution at origin. As demonstrated in [25], when subjected to such a discretization operation, the acoustic source signal sees its higher frequency contents to be aliased back onto the lower frequency range. Such so-called aliasing phenomena13 spares only those components which frequency, a, is sufficiently low such that

i p ph a2  ;

D D

ð22Þ

The above equation translates the Specific Preservation Criterion (SPC), as applied to the discrete projection of the (continuous) source signal onto the CAA stage. It ensures that, when discretized onto the CAA grid, such source signal will see its components preserved from aliasing, provided that their frequency (or wavenumber,14 in space), a, is included in the ]p/D, p/D [ interval (that is, verifies the Nqist criterion). On another hand, it predicts that all other components will be aliased back onto this particular frequency range. Therefore, such SPC criterion constitutes a key means for monitoring the preservation of the acoustic source signal, when used along with a proper analysis of the latter (through time and/or space Fourier transforms); indeed, based on the latter criterion, one can first (i) adjust the temporal and/or spatial CAA grid so that 13 Which consists in both a (i) destruction of correct information and (ii) a creation of incorrect information, through the abusive translation of those signal components which are of higher frequency (in terms of wavenumber) into spurious contents, to be spread out all over the spectrum. 14 One can here recall that, for any component of the noise source signal, the wavenumber is related to the frequency via the dispersion relation.

all components of interest fall into the preserved frequency range of ]p/D, p/D[. Then, one can (ii) clean up the signal from all other components (i.e. those that do not verify Eq. (22)), for them not to corrupt the preserved contents, if aliased. Such cleaning of the source signal may be conducted via the application of a selective filtering procedure [43,49,51], which – ideally – should be calibrated in such a way that any occurrence whose scale would fall under a given characteristic length corresponding to SPC minimal requirements is suppressed. On that stage, however, one can recall that filtering procedures may not be so easy to handle nor safe to apply, especially for what concerns spatial aspects. Therefore, whenever possible, one shall better extend the interval of preserved frequencies (see Eq. (22)), through a densification of the CAA grid (in space and/or time)15 [43]. Here, it is worth mentioning that the SPC criterion provided above constitutes a necessary condition for the CAA stage to preserve the acoustic source signal, but it is not a sufficient one. Indeed, once projected onto the CAA grid (in space and time), such signal will then be propagated through numerical schemes. And, as was said above (see Section 4.2), if the latter schemes are not accurate enough, they may pollute the signal by corrupting it with specific spurious artefacts (as illustrated in Section 5.3). 4.3.2. Reconstruction (or interpolation) into a continuous source signal of the discrete solution derived by the sound generation stage As was said, except for those trivial problems where it can be derived analytically (e.g. test cases of Section 5), the sound generation stage must generally be achieved (i.e. discretized and, then, solved) through numerical means, such as an unsteady CFD computation (e.g. problems of Section 6). In this case, the acoustic source signal is not delivered as a continuous function, but as a discrete one – which results from the discretization (or discrete approximation) of the sound generation problem itself. The latter’s discretization has however no reason to fit with the one characterizing the acoustic propagation stage and associated numerical method (e.g. CAA). As a result, in this case, the source signal must first be submitted to a continuous reconstruction (or interpolation) in space and in time, before it can be discretized in accordance with the acoustic propagation stage requirements, following SPC guidelines provided above (see Section 4.3.1). Such spatio-temporal continuous reconstruction of the source signal is generally conducted through multiple mono-dimensional interpolations, to be applied successively and independently along each one of the space and time directions (v = xi, t). As shown in [25], such an interpolation process can be formally expressed as a convolution product between the discrete signal itself, qd , and a given interpolator, cD0 ;

qsðvÞ ¼

þ1 X

qdðjD0 Þ cD0 ðvjD0 Þ

ð23Þ

j¼1

that is v

qs ¼ qd  cD0

ð24Þ

15 Doing so may however be difficult, in practice, for some applications. Consider for instance a situation where the acoustic signal is melted with non acoustical occurrences whose characteristic lengthscale (resp. energetic level) is much smaller (resp. higher). In this case, none of the solutions provided above may be conclusive enough, either because of the CPU price to be paid (e.g. local densification of the CAA grid) or because of a non negligible residual corruption by the unwanted occurrences (e.g. selective filtering of the latter). To date, the sole viable alternative consists in considering only that part of the source signal which effectively meets the SPC criterion, for instance by designing (i.e. shaping) the NRI interface accordingly. This technique proved to be very efficient when applied to several aerodynamic noise problems, such as for instance the trailing edge noise emission by an in-flight NACA0012 (for which the CAA-forcing interface was designed so that the source signal does not gather any of the wake vortices, which were likely to corrupt the CAA stage) [15].

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where D0 now indicates the grid step of the sound generation stage associated with the considered spatio-temporal direction (v = xi, t). The interpolator, cD0 , is specifically designed with the help of an interpolation basis Sj;

cD0 ðvÞ ¼

J max X j¼Jmin

L=7

1ð v0 þjÞ Sjð v0 þjÞ D

ð25Þ

D

where 1 denotes the identity function. For instance, and although other possibilities exist, such interpolation basis can rely on centered Lagrange polynomials, in which case Sj reduces to Pj;



JY max

PjðvÞ ¼



vp

p¼J min ;p–j

jp

ð26Þ

As recalled above and demonstrated in [25], whatever its underlying basis and resulting operator are, the interpolation process is likely to impact the source signal, by degrading its original content because of the so-called spuriousing16 phenomena. With the view of possibly characterizing such impact in a straightforward and explicit manner, in the present framework, a specific formalism was derived; such formalism, which relies on a Fourier analysis of the cD0 interpolator, allows predicting a priori the degradation to which a given signal may be subjected when interpolated (see detail in [25]). In a second time, with the view of partly remedying such signal degradation by the interpolation process, an innovative optimization of the latter was proposed. Such optimization, which is referred to as the Interpolation By Parts (IBP) technique [26], consists in virtually combining the chosen interpolator with a filtering procedure. This can be achieved in a straightforward manner by deriving a linear combination of the latter interpolator, to be weighted with specific coefficients, fj, the resulting weighted combination constituting its optimized version;

cþD0 ¼

L X j¼L

Table 3 Interpolation (or continuous reconstruction) of the discrete source signal, through the use of an Interpolation By Parts (IBP) approach; weighting coefficients associated with the IBP operator relying on 4th order Lagrange polynomials, as optimized through a virtual filtering scheme of 15 (L = 7) points.

f j sðjD0 Þ fcD0 g 2

ð27Þ

The coefficients, fj, which drive the selective accuracy of the virtual filtering procedure, are to be chosen judiciously, based on the degradation features characterizing cD0 (which can be accessed through its Fourier analysis, see above). For instance, in the present case where the interpolator relies on 4th order accurate Lagrange polynomials, the virtual filtering procedure [26] is that of a 15-point stencil (L = 7), which associated coefficients are detailed in Table 3 [26]. As shown in [26], when optimized following the IBP approach, any interpolation scheme sees its features improved, with a generation of spurious modes that is significantly reduced. Regarding the present concerns, this allows interpolating more accurately and thus preserving better the (discrete) source signal that is delivered by the acoustic generation stage. 4.3.3. Sampling of the discrete solution approximated through the acoustic generation When the sound generation stage is handled through numerical means (e.g. CFD), the resulting discrete solution is generally stored, in order to be analyzed and/or exploited in a second step. Such storage is usually conducted along with a sampling reduction applied in time and/or space. The associated sampling rate, which 16 Which consists in both a (i) destruction of correct information and (ii) a creation of incorrect information, through the alteration (i.e. either dissipation or amplification with phase-shifting, depending on the interpolator scheme used) of those signal components that fall within the mid- to high-frequency range (in terms of wavenumber), as well as the generation of spurious contents, to be spread out over the higher part of the spectrum only.

f0 f1 = f1 f2 = f2 f3 = f3 f4 = f4 f5 = f5 f6 = f6 f7 = f7

1/3 0.26598093 0.12936060 0 0.04602726 0.03212998 0 0.01614906

is generally determined a priori, results from a balance between several aspects; first, the storage volume must be minimized, which goes in the sense of a higher sampling rate. Second, the stored data must gather all of the physical information needed for a suitable exploitation (maximal frequency targeted, etc.), which generally goes in the sense of a lower sampling rate. Besides these aspects, however, and for the reasons enunciated previously (see Section 4.3.1, above), one can expect such sampling operation to degrade the discrete solution to be stored – and this, before the latter solution is submitted to an interpolation, and then further discretized onto the acoustic propagation stage. Therefore, whenever the source signal coming from a sound generation stage is to be stored with a sampling applied, one shall also consider the SPC criterion, but – this time – with respect to the storage grid; according to it, the aliasing phenomena induced by the sampling operation will spare only those components which frequency (or wavenumber, in space) fall into the   p=D0 ; p=D0 ½ interval, where D0 indicates the storage grid interval along the concerned space and/or time direction (v = xi, t). On another hand, all other components will be aliased back onto this particular frequency range. As before (see Section 4.3.1), when used along with a proper analysis of its components (through time and/or space Fourier transforms), such SPC criterion constitutes a key means for monitoring the preservation of the source signal, during the storage operation. And again, such preservation of the signal to be stored can be enhanced through either the (i) proper adjustment (i.e. densification) of the CFD storage grid (in time and/or space) or (ii) the selective filtering of the signal itself, to be conducted before any sampling is achieved. 4.4. Discrete approximation of the mean flow field Not only the perturbed quantities (up ; Fp ) and the source signal (qs ) constituting the continuous problem must be approximated so that they can be properly CAA-solved, but the mean flow field (qo ) must also be rendered compatible with a CAA exploitation. As for what concerns the source signal, two scenarios may then happen, depending on if the mean flow is made available from analytical means, or through a preliminary CFD calculation; 4.4.1. Discrete projection (or discretization) of the mean flow onto the acoustic propagation stage In the case the mean flow field is analytically defined (e.g. homogeneous steady medium, parallel shear layers), it is straightforward to project it onto the CAA grid – such projection (or discretization) being generally conducted in space only, considering the stationary nature of the mean flow. In some cases, however, the mean flow may have to be submitted to additional manipulations before it can be safely exploited in

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a CAA sense; for instance, it may be required to regularize mean flow quantities (generally by smoothing their gradients through space filtering, as was done in Ref. [23]), so that they are cleaned up from any pattern which characteristic length would not meet locally the CAA space discretization level. In particular, such regularization of the mean flow may imply that some of its near-wall thin boundary layers are suppressed (as was also done in Ref. [23]). The reason for regularizing that way all mean flow quantities is to avoid the possible emergence of numerical instabilities in the CAA simulation, due to the presence of gradients that would be poorly CAA-resolved.

4.4.2. Reconstruction (or interpolation/extrapolation) and projection of the mean flow onto the acoustic propagation stage In most realistic situations, the mean flow cannot be made available via analytical means, and must therefore be obtained through numerical ways (e.g. CFD). In this case, it is common to first derive mean flow quantities from the ‘sound generation stage’ results, whether it is by using directly the outputs of any preliminary steady CFD calculation, or by post-processing (through time-averages, see Eq. (4)) the unsteady flow realizations coming from an unsteady CFD calculation. Then, these mean flow quantities must be projected from the sound generation grid towards the acoustic propagation one, which generally requires that dedicated interpolations are performed, because of the non coincidence in the space discretization of both stages. Depending on the case, the latter interpolations may be conducted by using multi-linear interpolator (as was done in Ref. [23]), or via more advanced techniques (e.g. high-order Lagrange interpolator, as was done for the aerodynamic noise problem provided in Section 6.1, below [30]). Please note that, in some case, the CAA CFD interpolation in space of the mean flow may have to be completed with an as-like extrapolation operation, since it is not rare that the CAA computational domain extends beyond the CFD grid peripheral boundaries. In this case, the mean flow must be artificially extended up to the limit of the CAA domain, i.e., its missing information must be re-constructed for every CAA point located outside the limits of the CFD domain. This operation can be handled in marge of the mean flow’s near/mid-field CAA CFD space interpolation, via a complementary mid- to far-field interpolation relying on both the mean flow mid-field values (as provided by the peripheral CFD solution) and its far-field ones (as prescribed in the CFD calculation, in terms of flow stream at infinite). Please, note that, ideally, it should be verified 1 a posteriori that mean flow ratio jv o j=jv 1 o j and jpo  po j=jko j (where ko indicates the kinetic energy) are conserved along each streamline. As an illustration of the above, for the aerodynamic noise problem provided at Section 6.1, the mean flow was first derived by time averaging all the flow realizations of the unsteady CFD devoted to the noise generation stage. Such mean flow was then interpolated onto the near/mid-field of the CAA grid via 4th-order Lagrange interpolators, before it was extended up to the CAA far-field though a cubic spline interpolation which outer peripheral values were based on the flow stream at infinity. Finally, at this stage, it is important to stress out that, contrarily to what happens for the source signal (see Section 4.3 above), the possible errors inherited from the approximation (projection, interpolation, etc.) of the mean flow have a rather negligible impact onto the CAA solution, at least as long as the latter solution remains linear.17 17 For the unsteady solution of a hyperbolic linear system is independent of time-independent (source) terms.

4.5. Boundary conditions 4.5.1. Reflecting boundary conditions To simulate efficiently the presence of a rigid Boundary Condition (BC), a classical image source procedure is used [17,18]. The latter procedure simply consists in creating inside the rigid obstacle a mirror image of the physical field which is located within the near-wall fluidic region. For doing so, the computational domain is first extended beyond the wall surface with several lines of ghost points18 (x0 ), over which each (perturbed, mean and/or source) component of the field recorded at the physical (off-wall) points (x) is then duplicated. This is done with a symmetry condition applied, that is, with the velocity vectors’ normal component inverted such that

(

qjx0 ¼ qjx pjx0 ¼ pjx

vjx

0

¼ v jx  2ðv jx  nÞ

ð28Þ

ð29Þ

where n indicates the outgoing normal vector to the wall, and subscript  informs about the nature of the field ( = p, o, s). Please note such wall treatment is applied every iteration to the perturbed field (along with the source one, if applicable), whereas it is enforced to the mean flow only once, before the simulation is initiated. 4.5.2. Non reflecting boundary conditions Regarding the requirement of a finite space–time computational domain, there are several ways to approach the original infinite problem, that is, to mimic unbounded space–time physical field. First, concerning temporal aspects, the causality principle naturally introduces a lower bound to the calculation time duration, which upper bound simply corresponds to the end of the simulation. Regarding now spatial aspects, only the use of specific boundary conditions can allow truncating the computational domain in a safe manner. Indeed, as said above, artificially simulating an infinite free field beyond a calculation grid of finite dimensions implies that specific and so-called Non Reflecting Boundary Conditions (NRBC) are used, such that the free-field radiation of acoustic waves through peripheral boundaries is mimicked at best. In other word, such NRBCs must allow the propagating waves that leave the calculation domain to exit it in a clean way, that is, without generating spurious reflections when crossing the domain outer boundaries. On that stage, one can remind that literature abounds in various NRBC techniques, such as the so-called characteristics [34,35,46], the radiation conditions [36], the Perfectly Matched Layer (PML) [47], and so on. At the light of many comparative studies, however, it appears that none of these NRBC techniques can be blindly applied to all possible cases, because of a lack of generality, efficiency and/or robustness19 [48]. Therefore, here, use is made of an original NRBC technique; originally proposed by the present first author, this so-called LOC (Low Order Centered) technique [17,18,21] consists in the progressive decrease of the spatial derivative/filter order of accuracy, which is obtained via a reduction of the stencil width characterizing the peripheral FD schemes (which are kept centered, see Fig. 3). When coupled with a rapid grid stretching (over a few peripheral rows of ghost points), this technique allows damping rapidly, safely, and blindly all perturbations that reach the periphery of the computational domain. In addition, since it 18 In order avoid using non-centered FD schemes, which are known to be less stable than their centered counterparts. 19 Whether it is because of the too restrictive assumptions they rely on, or because of their possible ill-posed nature.

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41

Fig. 3. Sketch of the Non Reflecting Boundary Condition (NRBC) based on the Low Order Centered (LOC) approach (i.e. progressive decrease of the space derivation and filtering schemes’ stencils, at the domain periphery), as illustrated here for a 7-point/6th order scheme.

solely relies on a local adjustment of FD schemes coefficients, such LOC technique is trivial to implement, as well as it does not induce any extra CPU cost. 4.6. Handling of curvilinear grids Because realistic applications generally involve complicated geometries, the Cartesian-based discrete formulation detailed above must be extended, so that it can handle properly curvilinear grids. In the present case, this is achieved in a standard manner by making use of metrics [49]. The latter metrics allow the curvilinear grid that maps the physical space (of basis x) to be transformed into a virtual Cartesian mesh, to be associated with a computational space (of basis n) over which solving the problem in the exact same manner of what was described in the above sections (see Fig. 4). Such a virtual transformation x ? n can be achieved by using a Jacobian matrix J,20 thanks to which the approximate divergence of the perturbed flux can easily be expressed in an alternative way;

e  Fp ¼ jJjrn  Fn r p

ð30Þ

with a curvilinear flux tensor and a Jacobian matrices that are respectively defined by

Fnp ¼

Fp  J jJj

ð31Þ

and

J ¼ rn

ð32Þ

In the above expressions, rn and r denote the divergence operator associated with the curvilinear (n) and the Cartesian (x) basis, respectively.

could then jeopardize the entire calculation. Such a cleaning of the computed solution must however be conducted in such a way that the latter is preserved at best, that is, its key information is kept. Whereas this can be achieved through other techniques,22 in the present case, the computed solution is regularized by the means of a selective high-order low-pass filter operator [43,49], Fi, to be applied at every time step along each ith spatial direction;

p ¼ u

X F i fup g

ð33Þ

i

Such a filter is defined by a linear combination of translation operators, s, that act along the ith spatial direction, being weighted with specific coefficients fj. The explicit and centered version of such a filtering scheme takes the following form;

F i fg ¼

L X

f j sðjDxi ei Þ fg

ð34Þ

j¼L

Here again, the weighting coefficients fj are to be chosen so that the resulting filter exhibit a sufficient selective accuracy (i.e. fidelity over the low to mid frequency range). In the present case, and as for what concerned the space derivation schemes, the FD coefficients were derived following the innovative intrinsic optimization process, which allowed the resulting IOFD filter to be more accurate, compared to standard or spectral-like optimized [43–45] ones. Table 4 lists the weighting coefficients fj associated with IOFD filtering schemes of either 11-point (6th order) or 21-point (12th order). Please, note that the two latter schemes were thoroughly validated on the basis of various test cases [30], before they were applied to various problems [31,52,53] for which they were used along with IOFD derivation schemes, offering thus to maintain the accuracy level reached by the latter. 4.8. Adaptation of the space derivation and selective filtering to NRI

4.7. Regularization of the discrete solution When solving a discrete problem with the help of high-order FD schemes, it is required to clean up the computed solution every time step, so that all the (high frequency) spurious contents that are inherent to high order schemes21 do not accumulate, which 20 Please, note that, for consistency reasons, the Jacobian matrix shall better be derived numerically, using the same FD schemes as those employed for solving the overall problem [49]. 21 Such as grid-to-grid oscillations, but not only.

Regarding NRI forcing technique, the composite representation of the mean/perturbed fields (q. = o, p) implies that special attention must be paid to the way the latter are numerically managed, depending on the CAA method used. In particular, when the CAA approach relies on Finite Difference (FD) schemes (as is the case here), the discontinuity of the perturbed conservative quantities 22 For instance, by enhancing the discrete formulation to be solved with an unphysical term whose role is to selectively damp all high frequency contents emerging from the calculation [50,51].

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Fig. 4. Numerical handling of curvilinear meshes, via metrics associated with a Jacobian transformation of the curvilinear grid (physical space, left) into a virtual Cartesian mesh (computational space, right).

Table 4 Regularization of the solution via selective filtering FD schemes; weighting coefficients associated with Intrinsically Optimized Finite Difference (IOFD) schemes of 11 (L = 5) and 21 (L = 10) points, respectively.

f0 f1 = f1 f2 = f2 f3 = f3 f4 = f4 f5 = f5 f6 = f6 f7 = f7 f8 = f8 f9 = f9 f10 = f10

L=5

L = 10

0.776262811746421 0.193715328923949 0.123364214552874 0.0544437616555214 0.0147671913203363 0.00184090942052943

0.833898839361257 0.152881679650199 0.118982123129569 0.0777216723835284 0.0418911421581011 0.0180498174958730 0.00590024045861430 0.00134040731824761 0.000176789293756521 6.42315215221764E06 8.75359411794226E07

(Qp) must be correctly accounted for, whenever a spatial (derivative or filtering) operation is conducted nearby the Xs /Xs interface [22–24]. Indeed, as was shown above, differencing or filtering a given quantity with the help of FD schemes consists in weighted-averaging its neighboring values. The point is that, in the present case, for a given point xi located close enough to the Xs =Xs interface, there will always going to be some neighboring points xj obeying an alternative splitting definition (with, for instance, xi 2 Xs whereas xj 2 Xs ). However, consistency imposes that, for correct weight-averaging at point xi, a given perturbed Qp quantity must be evaluated at the xj location in accordance with the splitting definition ruling point xi. This can be achieved in a straightforward manner by employing an alternative weight-averaging procedure [22–24], to be based on the following generic expression

sj fQ p gjx ¼ Q pjx þ i

In the above,



ci  cj 2

j

 Q p ðhqi; qs Þjx

ð35Þ

j

sj fgjx indicates the xi ? xj translation operator; for i

any perturbed quantity Qp to be weight-averaged at point xi, Eq. (35) provides the evaluation that must be made at point xj according to splitting definitions ruling either xi (i.e. sj fQ p gj term) or xj xi

(i.e. Q pjx term) points. Execution only requires setting the coeffij

cient ci (resp. cj) to a proper specific value (1 or 1), depending on if point xi (resp. xj) is located within Xs or Xs , respectively. Indeed, for two points (xi, xj) located within the same region (i.e. sharing the same splitting definition), the second right hand side term of Eq. (35) automatically vanishes (ci = cj). In contrast, for two points located within two distinct (splitting definition) regions, such a term ensures that Eq. (8) is recovered. Thanks to such generic expression of Eq. (35), one can easily derive an alternative weight-averaging procedure that, when

applied to the whole computational domain, will handle automatically the proper evaluation of weight-averaged (i.e. differentiated or filtered) Qp quantities along the Xs /Xs interface. In practice, such an alternative procedure must be conducted in place of the usual one, being applied to Fp (resp. up) during the spatial derivative (resp. filtering) process, i.e. at each time step of the CAA calculation. 4.9. Some remarks As for the continuous one introduced in Section 3, the discrete formulation described above was developed by the present first author through the years [17,18,22,24], and is the one that is now implemented in ONERA’s time domain CAA solver sAbrinA23 [17,18,32,33,40,41]. In particular, the latter solver integrates some of the innovative features presented here (NRI, IOFD), the other ones (SPC, IBP) having being implemented in a dedicated code, named cAmilA24 [30]. Both solvers were intensively used for validating these various features, before the resulting advanced hybrid method was applied to diverse configurations. Sections 5 and 6 summarize some of these validation and application works. 5. Illustration via an application to a canonical problem As was said, the advanced CAA-based hybrid method detailed above was thoroughly validated via an application to various problems of increasing complexity. In a first time, validation efforts focused on a rather simple hybrid scenario, which consisted in weak coupling both an analytical acoustic generation stage and a CAA-based propagation one. For doing so, several academic test cases were considered [22–31,52], some of which concerned the acoustic scattering by a 3D rigid sphere of a quadrupole source within an infinite quiescent medium. This canonical problem, which is well representative of the typical situations one may encounter when addressing external noise configurations, was directly inspired by the validation exercise proposed in the BANC (Benchmark on Airframe Noise Computations) initiative. The latter exercise was jointly designed by NASA, JAXA and ONERA to serve as a generic test case for assessing the propagation stage of acoustic hybrid methods, whether the latter stage relies on a CAA, a BEM, or an IM method. Indeed, for this problem, there exists an analytical solution which can be used for forcing any acoustic propagation tool (in the exact same manner as what would be done with unsteady perturbations coming from a prior CFD calculation, for less trivial situations). Based on this particular test case, several validation exercises were achieved; whereas each exercise addressed one of the various innovative solutions developed in the present framework, they all 23 24

‘‘Solver for Aeroacousic BRoadband INteractions with Aerodynamics’’. ‘‘Clustered Advanced Methods for Interpolation in Aeroacoustics’’.

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relied on a hybrid calculation that consisted in forcing the CAA stage with the analytical solution of the scattering problem. All CAA computations were performed with the help of ONERA’s sAbrinA solver [17,18,32,33,40,41], each being conducted over one core of a laptop (64 bits, Pentium 2.1 GHz, 4 Gb Ram) and requiring approximately 30 min of wall clock time. Next subsection summarizes these validation exercises (which detail can be found in [22– 30,52]), so as to illustrate the main features composing the advanced CAA-based hybrid acoustic method described above.

5.1. CAA-forcing of the source signal via NRI The present subsection illustrates how the Non Reflective Interface (NRI) allows forcing a CAA acoustic propagation stage with a given source signal, including when backscatter effects are present. With that view, we consider a sphere of radius R located at the origin x = 0 of a quiescent medium. Such a sphere acts as a scattering agent to a quadrupole acoustic source that is located 1.5R away from the sphere’s center (at x = (1.5R; 0; 0)) and that pulsates with a frequency k = 2p. Following guidelines of BANC intitiative, the source region is chosen as the area encompassing both the acoustic quadrupole and the sphere (Xs = [2R; 1.5R]  [1.5R; 1.5R]  [1.5R; 1.5R]). From this source region Xs , the unsteady signal resulting from both the acoustic emission by the quadrupole and its scattering by the sphere is NRI-forced within a wider domain (Xs = [5R; 5R]3\Xs ) over which it is CAA-propagated. Such NRI forcing of the CAA perturbed field by the analytical source signal is achieved as described in Section 4, that is, it is prescribed every iteration along an immaterial interface given by the cubic envelop corresponding to the frontiers of the source region Xs (blue dashes in Fig. 5). Please note that, thanks to its analytical nature, the acoustic source signal can here be defined exactly where and when needed by the CAA stage, that is, at the exact space–time locations required by the NRI-forcing. Such signal escapes thus the need to be submitted to any space and/or time interpolation/sampling operations (which could ever degrade its accuracy due to spuriousing and aliasing phenomena, as will be shown in the next subsection). The CAA mesh used here is of homogenous Cartesian type (and, thus, it does not conform to the sphere), the reason for doing so being to minimize possible side-effects (such as metric errors) that could bias the accuracy of the results. The spatial discretization of the Cartesian CAA grid corresponds to Dx = Dy = Dz = R/10, leading to a computational domain comprised of 1013 (i.e. approximately 1 million) points. Such a grid ensures a nominal value of 11 PPW (Points Per Wavelength), guaranteeing a maximal accuracy error of a bit more than 0.14% in terms of acoustic group velocity (all this, with respect to the 7 point/6th-order-finite difference schemes used in the present case by the CAA solver). The simulation is conducted for a simulation time corresponding to 6 source periods, with 20 iterations per period (DtCAA = T2p/20), leading to a CFL (Courant–Friedrichs–Lewy) number of 1/2. Fig. 5 displays the instantaneous perturbed pressure field obtained at the end of the 6th source cycle, either via the NRI-forced CAA calculation, or via the analytical evaluation; as one can see, both results collapse all over the computational box, included close to the NRI interface (depicted in blue dashes). This proves that the source signal is accurately NRI-forced and then CAA-propagated within the calculation domain, which illustrates the efficiency of the Non Reflective Interface. To illustrate the non-reflective character of the NRI technique, the previous test case is now slightly modified, the infinite space being replaced by a semi-infinite one. This is achieved by prescribing a rigid plane over a lateral side of the computational box (as

43

depicted in Fig. 6, where the rigid plane appears in red color). Fig. 6 displays the instantaneous pressure field obtained after 15 periods of source emission, a time necessary for the acoustic field to reach a stationary state all over the domain. By comparing left sides of Figs. 5 and 6, one can clearly observe the acoustic interaction patterns (such as standing waves, etc.), which identify the backscattering effects by the rigid wall.25 These effects are more visible in Fig. 6-c, which isolates the sole scattered field. Please, note that the latter was obtained by subtracting from the total perturbed field computed here the incident field obtained from the previous ‘infinite medium’ calculation (see Fig. 5). When looking at this scattered field, one can clearly distinguish how the acoustic waves are reflected by the rigid plane back into the computational domain. On that stage, one can appreciate how such backscattered field is perfectly continuous over the whole domain, and more especially over the NRI interface (depicted in blue dashes). This confirms that the NRI interface does not influence the propagation of the backscattered waves, which illustrates the non-reflective nature of the NRI technique. 5.2. Degradation of the source signal, and preservation through IBP The present subsection illustrates (i) how far acoustic signals may be degraded when manipulated (i.e. sampled and/or interpolated), and (ii) how the innovative Interpolation By Parts (IBP) process may help in partly remedying such issue. With that end, the baseline test case of previous section is now modified following an alternative manner; the configuration is kept the same, whereas the acoustic source signal originally defined (k = 2p) is now combined with an extra noise of same amplitude, but of much higher dimensionless frequency (k = 9p). Once cumulated, both components result in a single signal (k = 2p + 9p), which is discretized in time, as if the signal had been stored every one tenth of the primary emission source period (DtStorage = T2p/10). This discretized signal is then time-interpolated for fitting a CAA calculation time step that is two third times smaller than the storage one (DtCAA = 2/3DtStorage)26; thanks to the dedicated formalism developed in the present framework [25], one can predict that such manipulations (i.e. time sampling + interpolation) of the signal shall lead to the generation of various spurious modes, which respective wavenumbers correspond to multiples of p (kspurious = mp, m 2 Z). The interpolation stage is carried out either via a standard approach (centered 4th-order Lagrange interpolator, L4) or through the Interpolation By Parts technique (IBP4, based on L4 interpolator – see [26,30] for more details). Each one of the two interpolated signal is then NRI-forced and CAA-propagated, its resulting time signature being recorded at four different CAA points surrounding the coupling interface before it is spectrally decomposed by the means of a Fourier analysis; Fig. 7 plots the spectra delivered by each signal, comparing it to the analytical solution associated with the sole primary signal (i.e. k = 2p). As one can see, whatever the nature of the interpolation to be applied was, the cumulated signal (k = 2p + 9p) saw its lower frequency mode of interest (k = 2p) being well preserved by the data manipulation, NRI-forcing and CAA-propagation. On another hand, for both signals, the spectra exhibit additional tones of lower and higher frequencies, which 25 By the rigid plane alone, since the sphere is still not CAA-accounted for; the acoustic interactions highlighted here concern thus only the primary backscattering effect by the solid surface, all secondary reflections being ignored. Since the latter are expected to be weaker than the former, one can however expect the present result to be quite close to the exact one (which analytical solution was not available). 26 One can notice here that such a configuration (which differs from the one addressed in Section 5.1, for which one had implicitly DtCAA = DtStorage = T2p/20), is quite typical of what can be encountered in real-life applications, for which the CAA time step is generally smaller than the CFD-storage one.

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Fig. 5. Scattering by a 3D rigid sphere of a harmonic (k = 2p) quadrupole within an infinite medium at rest. Instantaneous perturbed pressure field obtained at t = 6T within either the xy (a and b) or the yz (a and c) median planes. Comparison of the NRI-forced CAA calculation results (in flood) against analytical ones (drawn in flood over the cubic envelope on a-side, plotted in lines on b- and c-sides). Sphere is not included in the CAA calculation, and the NRI interface appears in blue dashes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Scattering by a 3D rigid sphere of a harmonic (k = 2p) quadrupole within a semi-infinite domain with a medium at rest (rigid plane drawn in red). Instantaneous perturbed pressure field obtained at t = 15T within xy (a) and yz (a and b) planes. The scattered field induced by the rigid plane alone is drawn in (c). The sphere is not included in the CAA grid, NRI interface appears in blue dashes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

correspond exactly to the spurious modes that were expected to occur, because of the signal manipulation. Such spurious contents, however, were more importantly filtered out by the interpolation stage when the latter relied on an IBP procedure than when it was based on its standard Lagrange interpolator counterpart (see the respective levels of blue and red27 components, in Fig. 7). In particular, one can notice how IBP succeeded in dissipating almost entirely the first spurious mode (kspurious = p) whilst preserving the primary signal component (k = 2p), despite the fact the latter was of higher frequency (and, thus, more likely to be degraded by the selective filtering procedures). This illustrates how the Interpolation By Parts technique can help in remedying the signal degradation to be possibly induced by the manipulation of those unsteady data which are transferred from one to another stage of an aeroacoustic hybrid method (whether the latter is of CFD-IM or CFD-CAA nature). 5.3. Relaxation of the CAA propagation and CFD storage constraints via IOFD schemes The present subsection illustrates how IOFD schemes allow enhancing the accuracy of the CAA stage (and, thus, the preservation by the latter of acoustic signals it is forced with), to the direct benefit of the overall hybrid calculation. With that view, the baseline calculation case of Section 5.1 is again repeated as is, to the exception that the acoustic source signal originally defined (k = 2p) is now replaced with a signal of same amplitude, but of higher dimensionless frequency (k = 6p). With 27

For interpretation of color in Fig. 7, the reader is referred to the web version of this article.

respect to the present computational set-up (which CAA grid is kept the same), such signal is thus discretized with no more than 3.3 PPW (Points Per Wavelength). Thanks to the dedicated formalism developed in the present framework [28], one can predict that, even without being manipulated (i.e. sampled and/or interpolated in time and/or space), such signal shall generate spurious numerical noises from the moment it will be CAA-handled with derivative operators of insufficient accuracy. With the view of highlighting this point, four calculations are thus conducted, each one being performed with a particular derivative operator, to be based on standard finite differences (SFD) schemes of either 7-point (resp. 6th order) or 11-point stencil (resp. 10th order), as well as on Intrinsically Optimized Finite Difference (IOFD) schemes of either 11-point (resp. 6th order) or 15-point stencil (resp. 8th order). For all calculations, the filtering stage is handled with a very high fidelity filter (SFD of 21-point stencil28/20th order), that is, ensuring a negligible bias on the propagation results. Fig. 8 depicts the four calculation results, displaying for each one the instantaneous perturbed pressure field obtained at the end of the computation (out of the source zone), and comparing it to the analytical results (depicted all over the computational box). As one can see, both standard schemes (SFD) failed in propagating properly such a high frequency signal, generating spurious modes that corrupted the acoustic field all over the domain. On the contrary, however, both intrinsically optimized schemes (IOFD) behaved much better; indeed, as one can see, the 11-point stencil/6th order IOFD scheme delivered an acoustic field 28 Consequently, regarding the CAA-forcing, the NRI procedure is here achieved with a modified interface (i.e. of wider extent and thicker envelopes, compared to the baseline computational set-up of previous cases).

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Fig. 7. Scattering of a two-tone (k = 2p + 9p) quadrupolar emission by a 3D rigid sphere within an infinite medium at rest, via analytical-CAA hybrid calculations. Power Spectral Density of both the manipulated (time sampled + interpolated) and the NRI-forced/CAA-propagated signal, as recorded at four locations surrounding the coupling interface. The black circle indicates the (analytical) spectra associated with the sole low frequency tone (k = 2p).

Fig. 8. Scattering of a high frequency (k = 6p) acoustic quadrupole by a 3D sphere within a quiescent medium at rest, via analytical-CAA hybrid calculations. From left to right: standard FD (SFD) schemes of 7- and 11-point stencils, intrinsically optimized (IOFD) schemes of 11- and 15-point stencils. Comparison of numerical outputs (in dashes, out of the source zone/red box) against analytical data (in line, over the entire domain), as obtained within either the xy (top) or the yz (bottom) plane. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

that is almost free of spurious modes, whereas its counterpart of 15-point stencil/8th order counterpart provided a result matching exactly the analytical one. This illustrates how far IOFD schemes effectively offer an advantageous alternative to standard ones, allowing to enhance the accuracy of the CAA stage, and thus, to preserve at best the aeroacoustic signal such stage is forced with. Here, one can notice that the present exercise made it possible to check that, compared to the 7-point/6th order SFD (which is the most popular FD scheme

used among the CAA community), an IOFD scheme of 15-point stencil/8th order (resp. 11-point/6th order) requires no more than 50% (resp. 25%) extra CPU time per operation, while it allows the CAA grid to be coarsen by a factor of 3 (resp. 3/2) per direction,29 offering thus to lower the mesh density by a factor 27 (resp. 29 Considering a maximum error on the group velocity of 0.14%, a level for which the 7-point stencil standard FD, the 11-point stencil IOFD, and the 15-point stencil IOFD schemes require a minimum of 12 PPW, 8 PPW and 4 PPW, respectively.

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3.375) in 3D. Again, because of CFL aspects, these numbers may be multiplied by up to 2, when it comes to consider the CPU time reduction offered by such a grid density decrease. 6. Further illustration via an application to two aerodynamic noise problems With the view of demonstrating further the potentialities offered by the advanced CAA-based acoustic hybrid approach regarding an application to real-life problems, validation works then focused on a more complex hybrid scenario, that is, relying on a sound generation stage based on a compressible unsteady CFD. To that end, several aerodynamic noise problems of increasing complexity were considered [22–31], among which the two ones that are summarized hereafter. Each problem was solved following a hybrid process relying on weakly coupled CFD and CAA calculations; for each one of these CFD-CAA hybrid calculations, a preliminary unsteady compressible CFD calculation was first acquired, from which a proper acoustic source signal was then derived, for the subsequent CAA calculation is then forced with it. Making such CFD-originated acoustic source signal compatible with the CAA consumption led to conduct several preliminary (though sometimes intensive) tasks,30 such as (i) the proper specification of the CFD storage to be conducted in space and/or time, (ii) the analysis of the stored CFD data, (iii) the derivation of a suitable CAA set-up (in terms of space/time grids, steady mean flow, etc.), and (iv) the translation of the stored CFD-generated dataset into a proper acoustic source signal to be CAA-forced. All these operations were achieved with special attention paid to the degradation of the signal due to possible aliasing and/or spurious phenomena, which were here minimized through the use of the dedicated formalism and/or innovative technique (e.g. SPC criterion, IBP interpolation, IOFD schemes) developed in the present framework.31 As with the previous academic test cases, all CAA calculations were performed with the help of sAbrinA solver. However, due to their disparate requirements in terms of computational resources (grid size, time duration, etc.), these calculations were run on various machines; more precisely, for the first problem (see Section 6.1, below), the CAA calculation was run using one core of a laptop (64 bit, Intel Pentium 2.1 GHz, 4 GB RAM), which required approximately 5 min of wall clock time. On the contrary, the CAA calculations associated with the second problem (see Section 6.2, below) were run over 480 cores of ONERA super computer,32 requiring approximately 45 CPU hours each (for a total of 12,000 iterations and 66 millions of grid points computed). 6.1. Acoustic emission by a 2D cylinder in a low Reynolds/Mach number flow First, the problem of the acoustic emission by a 2D cylinder in a cross-flow of low Mach and Reynolds numbers (M1 = 0.2, ReD = 100, with D the cylinder’s diameter) is considered. On this stage, one can recall that such a configuration is particularly well 30 Contrarily to the previous academic test cases, for which the noise source signal could had been derived directly according to the CAA set-up, thanks to its analytical nature. 31 In addition, here, the resulting noise source signals were cleaned up from those unsteady perturbations which were expected to corrupt the CAA stage, due to their insufficient solving by the propagation grid and schemes (see Ref. [19] for more details). The latter perturbations mainly consisted in those aerodynamic (or so-called hydrodynamic) occurrences of high-amplitude/small space–time correlation that were convected by the wake occurring downstream the solid obstacles considered – which was thus faded out numerically. 32 160 Westmere bi-processors, with each processor composed of 6-cores X5675 of 3.07 GHz/4 Go.

suited for investigating aerodynamic noise phenomena, since it includes all the basic mechanisms the latter involve (loading noise, self noise, etc.) whilst remaining quite tractable from a computational effort point of view. This explains why such problem was often numerically addressed by the aeroacoustic research community, and is now well documented [54,55]. On another hand, and from a methodological point of view, this particular configuration involves aeroacoustic phenomena covering a wide range of scales (both in terms of time/space characteristic and amplitude), which makes this problem being well representative of the usual needs and constraints that acoustic hybrid methods need to face – constituting for instance a good intermediate step towards the solving of the real-life aircraft noise case given by the landing gear configuration addressed in next section (see Fig. 9). As was said, the hybrid acoustic calculation of such configuration was achieved in two-steps; the acoustic generation stage was handled with the help of an unsteady CFD computation that was performed by NASA Langley Research Center (LaRC), thanks to the second order compressible CFD solver from NASA/LaRC’s named CFL3D. The calculation was run in laminar mode, and it involved a O-mesh of 289  289 points that extended away in the radial direction up to a distance of approximately 30 times the cylinder’s radius (R = 0.0285 m). Special attention was paid on the correct simulation of the boundary layers.33 As can be seen in Fig. 10a where the instantaneous perturbed field is plotted, the calculation captured all the mechanisms characterizing such a cylinder/flow interaction problem, with – among other things – a strong dipolar acoustic emission (oriented orthogonally to the flow direction, and of a frequency corresponding to a Strouhal number of St = 0.1666). This result is fully consistent with observations of other researchers [54,55], and can be explained by the dynamics of the unsteady loads that wake’s vortices induce onto the cylinder. Based on the unsteady data delivered by this CFD computation, the CAA-based propagation stage of such hybrid calculation was then achieved [30]. This was done in the exact same way as what had been done for the previous academic test cases, to the exception that, here, the advanced hybrid acoustic approach was employed with most of its innovative features (NRI, SPC, IOFD, etc.) used altogether. For instance, the CFD-generated acoustic signal was first analyzed with care, using ONERA’s module cAmilA [30], which incorporates all the signal analysis and interpolation tools evocated at Section 3.2. Based on that, and thanks to SPC insights, a homogeneous Cartesian CAA grid (of 833  833 points34) was then designed such that all of the CFD-generated acoustic occurrences (2 main modes and their 13 harmonics) are preserved at best, once space/time35 interpolated and further propagated on it. Such a space/time interpolation of the CFD dataset onto the CAA mesh was achieved thanks to cAmilA, a special care being paid to the preservation of the CFD-generated acoustic signal. Finally, the CAA computation was run for a total of 3600 iterations (corresponding to 6 cycles of the vortex shedding), the CFD-generated acoustic signal being forced via NRI, and the CAA consumption being handled with the help of 11-point IOFD derivation schemes. Fig. 10c displays the results delivered by such weakly coupled CFD-CAA hybrid calculation after 2000 iterations, i.e. at approximately 3.5 cycles of the vortex shedding; as one can see, the near-field patterns match exactly the CFD ones (to the exception of the wake, which was faded out within the CAA stage, for the CAA grid was not made dense enough for properly handling the unsteady vortices convected downstream [22,56,19]). This confirms

33 With a near-wall cells’ maximum size set to Dr = 1.037  103, that is, approximately R/28. 34 With DxCAA = 4.8  103. 35 Please, note that in the present case, CFL criteria imposed that the CAA time step is set to half the one used in the CFD.

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47

Fig. 9. Towards an application of the advanced CAA-based hybrid method to real-life aircraft noise problem. Aeroacoustics of both a 2D cylinder (left) and a 3D landing gear (right) within a cross flow of low Mach number, and of either a low (left) or a high (right) Reynolds number. Mean density field associated with the flow.

Fig. 10. Acoustic emission by a 2D cylinder in a cross flow of low Mach and Reynolds numbers (M = 0.2, ReD = 100). Near-field acoustic generation and early propagation predicted via CFD (a), CFD body-fitted and CAA Cartesian grids (b), acoustic near- to far-field propagation simulated via the CFD-CAA hybrid calculation (c), where CFD and CAA results appear in lines and flood, respectively. The acoustic signal coming from the CFD calculation is NRI-forced within the CAA domain via the immaterial interface (see the white dashes, on c-side).

the accuracy of the coupling procedure, when applied to unsteady perturbations coming from an a priori CFD calculation. In addition, the relatively large domain used for the CAA consumption (see Fig. 10b) allowed the near-field CFD solution to be extended up to far-field distances of more than 90 times the cylinder’s radius, which illustrates the potential offered by the present advanced CAA-based hybrid approach when it comes to solve acoustic problems that are beyond the limits of more traditional methods. 6.2. Acoustic emission by a 3D landing gear in an approach phase flight With the view of addressing still more realistic situations, we now consider a typical aircraft noise problem, that is, the acoustic emission by a landing gear in an approach flight phase. Being also excerpted from the BANC initiative, such test case was derived from the so-called LAGOON project (LAnding Gear nOise database for CAA validatiON), which was supported by Airbus and conducted by several partners (ONERA, DLR, Southampton University, etc.). The objective of the project was to acquire an extensive experimental database associated with elementary landing gear configurations, so that computational methods dedicated to landing gear noise predictions can be accurately and thoroughly validated. Within this framework, combined experimental and computational campaigns were thus carried out, focusing on both the

aerodynamics and the acoustics of a simplified landing gear geometry,36 which was considered in an isolated configuration in either a take-off or an approach flight condition. The aero-acoustic dual experiments [57,58] were achieved in ONERA’s aerodynamic (F2) and anechoic (CEPRA 19) facilities, respectively (see Fig. 11). The computational counterpart of such aero-acoustic experimental campaign was conducted at ONERA, following an IM-based acoustic hybrid approach; first, aerodynamics computations [59] relied on 3D unsteady compressible CFD calculations (based on the so-called zDES approach), which were all conducted using ONERA’s solver named elsA [60] (see Fig. 12a). As one can see in Fig. 12b, those calculations were favorably compared with the aerodynamic measurements through direct comparison of near-field results (this, to the exception of mismatches over the low and high frequency ranges, which can be reasonably attributed to the high pass filtering induced by the experimental acquisition and the numerical simulation techniques, respectively). In particular, both experimental and numerical outputs exhibited tonal features (at approximately 1 kHz and 1.5 kHz) which, radiating in the sideline directions, were inferred to be induced by resonances coming from the wheels’ inner cavities. These unsteady CFD 36 Nose gear of an Airbus A320 aircraft, with a scale factor of 1:2.5 applied, and with only the main elements (leg, wheels, etc.) kept.

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Fig. 11. Noise emission by a simplified nose landing gear (NLG) in approach flight conditions. NLG model (seen from behind), as installed in both the ONERA’s F2 aerodynamic facility (left side) and CEPRA19 open jet anechoic wind tunnel (right side).

Fig. 12. Aeroacoustics of the NLG in isolated configuration (i.e. within a free-field), as predicted by a CFD-FWH hybrid calculation (M = 0.18, ReD = 1.2  106) . Top: near-field aerodynamics computed by the unsteady CFD (zDES, structured grid) calculation, as plotted in terms of Q-criterion iso-surfaces colored by the stream wise velocity component and instantaneous pressure fluctuation field (a) and validated via direct comparison of the Power Spectral Density (PSD) computed by CFD (in blue) against the one recorded in the experiments (in red and green) for a probe on the right wheel (b). Bottom; validation of the far-field acoustic results, via direct comparison of the PSD predicted by CFD-IM (black and blue) and measured in the experiments (red), for two microphones located at 90° from the model in the flyover (c) and side line (d) directions. Reproduced from [59] and [61] with permission. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

results were then acoustically extrapolated to the far-field [61] via an IM method relying on a FWH technique [3], which was handled thanks to ONERA’s code KIM [62]. As one can see in right side of Fig. 12, these CFD-IM hybrid calculation results were also favorably compared to the experimental measurements recorded in the far-field (to the exception of mismatches coming from errors in the FWH calculation, depending on the integration surface used37).

37 Improper handling of hydrodynamic features convected across permeable (or porous) FWH surfaces in the landing gear’s wake.

Despite the good match between numerical and experimental outputs, it seemed important to possibly improve the fidelity of the acoustic propagation stage by (i) accounting for the noise emission that had been effectively predicted by the CFD stage (rather than to model it via equivalent sources, as implicitly done in the FWH approach), as well as by (ii) considering the realistic jet flow characterizing the experiment (rather than to model it via a simplistic uniform mean flow, as also done in the FWH approach). As was said above, these two requirements could be fulfilled by an acoustic hybrid method relying on a CAA propagation stage. Therefore, two CFD-CAA weakly-coupled calculations of the

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present NLG configuration were achieved with the help of the present advanced CAA-based hybrid approach [31]. As for the problem addressed in the previous section, the latter approach was applied with most of its innovative features used altogether (NRI, SPC, IOFD, etc.); indeed, for both hybrid calculations, the CAA stage was NRI-forced with an identical CFD dataset (coming from that particular LAGOON unsteady CFD computation associated with a flow stream of Mach and Reynolds number M = 0.18 and ReD = 1.2  106, respectively [61]). Here too, such CFD-originated dataset was first submitted to a certain number of preliminary manipulations (e.g. sampling and/or interpolation in time and space, removal of the NLG wake’s unsteady aerodynamic occurrences), such that a proper acoustic source signal can be derived and then NRI-forced within the CAA stage. The latter CAA stage, which was designed following guidelines dictated by SPC insights, relied on the use of IOFD derivation and filtering schemes (of 15and 21 points, respectively). Such use of IOFD schemes made it possible to not only (i) reduce drastically the overall CPU/MEM price to be paid for the CAA stage,38 but also to (ii) minimize the signal degradation of CFD data.39 The first CFD-CAA hybrid calculation corresponded to an isolated NLG, with a computational set up similar to the CFD-IM computation (incorporating in particular a steady mean flow corresponding to a homogeneous free field). This calculation allowed validating the present CFD-CAA coupling through direct comparisons against both the numerical (CFD) and the experimental results, which had been acquired and/or processed under the same assumption of a homogeneous medium.40 For the second simulation, the CFD-CAA calculation considered the NLG as installed in the anechoic facility, with a mean flow matching the heterogeneous and sheared steady jet occurring in the facility (see Fig. 13c). This alternative calculation made it possible to enhance further the fidelity of the prediction by CAA-accounting for the facility jet flow, allowing then to assess the refraction effects such jet flow may have had on the experimental measurements. The reader is referred to Ref. [31] for more details about these two simulations, which only a few excerpts are provided below; Fig. 13 provides results coming from the two CFD-CAA hybrid calculations with, first, a snapshot of the instantaneous perturbed pressure field obtained at the end of the facility-installed NLG calculation (see Fig. 13a). As one can see, once NRI-forced within the CAA domain via the weak coupling interface (small cubic box,41 drawn here in purple), the CFD source signal is properly CAA-propagated up to the far-field. One can notice that the resulting acoustic emission is somehow irregular, translating the intermittent character exhibited by the CFD source signal itself. Regarding now the isolated NLG configuration (i.e. associated with a free-field jet flow), Fig. 13d compares the spectra of the CFD-CAA signals against the ones that were experimentally recorded, this being achieved for 5 probes located in the far-field flyover direction, at approximately 40R away from the model (R standing for the NLG wheel radius). This far-field validation exercise is satisfactory, if one consider the rather good match of spectra delivered by both the experimental measurements (in black lines) and the CFD-CAA coupled calculation

38 With, in particular, a 27 times lighter CAA grid volume and, thus, calculation CPU time, compared to the ones that would have been required if a regular 7-point standard FD scheme had been used instead. 39 By increasing the accuracy of their manipulation (space sampling and interpolation), thanks to the proper densification of the CFD-CAA weak coupling interface. 40 The experimental dataset had been corrected from the refraction effects by the open jet shear layers. 41 Some parts of which were however removed so that all the unsteady aerodynamics occurrences (vortices, etc.) convected by the landing gear’s wake are not NRI-forced into the CAA domain, which grid density was insufficient for handling them accurately enough (see Ref. [31] for more details).

49

(in red lines).42 For indicative purpose, Fig. 13d also depicts (in black dashes) the spectra coming from the CFD-IM hybrid calculation43 that had been previously obtained for the same isolated NLG configuration. Please note that the variability between these CFD-IM and CFD-CAA outputs may be attributed to the way the present CFD calculation was IM- or CAA-exploited, with intrinsic differences whose nature was both practical (e.g. coupling interface shape/density, acquisition time) and fundamental (e.g. equivalent vs. exact acoustic source signal, linear vs. non linear propagation kernel). Finally, the effects of the facility-installed jet flow onto the NLG acoustic emission are highlighted in Fig. 13c, which compares the RMS (Root Mean Square) perturbed pressure field associated with the second tonal emission (f = 1.5 kHz), as delivered at the end of the calculations associated with either the isolated or the facility-installed NLG configurations. Indeed, contrarily to what happens when the medium is taken as homogeneous, when the realistic jet flow is accounted for, the acoustic waves see their patterns modified as they cross the jet shear layers, and then propagate within a region where the medium is at rest. As one can see, these refraction effects impact the acoustic signature in a non negligible manner, leading to differences in terms of directivity, as well as in terms of radiated power.44 Here, it is worth mentioning that a standard CFD-IM hybrid approach could not be used to perform such an assessment of the installation effects by the facility environment because of the underlying hypothesis (e.g. homogeneous propagation medium) the IM stage relies on. This emphasizes again the importance of employing advanced techniques such as the present CAA-based hybrid method for enhancing the fidelity of acoustic predictions. 7. Summary and concluding remarks The present article concerns an advanced method for the numerical simulation of acoustic problems. More precisely, it proposes a robust and accurate multi-stage acoustic hybrid methodology, whose sound propagation step relies on Computational AeroAcoustics (CAA) techniques. To this end, after a CAA method was first derived, an innovative weak coupling technique (NRI) was developed, allowing then to implicitly force the CAA stage with a given source signal coming from an a priori evaluation, whether the latter evaluation is of analytical or computational (e.g. CFD) nature. Then, the resulting CAA-based hybrid approach was optimized so that it can cope with all stringent constraints that are dictated by real-life problems, without being jeopardized by some of their unavoidable side effects (such as the signal degradation to which source signals are subjected to, when manipulated and further exploited through CAA). As a result, the CAA-based hybrid approach was enhanced with several innovative solutions, such as (i) specific guidelines and associated criterion (SPC), (ii) a novel interpolation method (IBP) and (iii) a new class of finite difference derivative schemes (IOFD). Once each of its constitutive ingredients (NRI, SPC, IBP, 42 It is worth mentioning that, compared to the signal acquisition time used in the experiment (20 s), the CFD-CAA simulation time (0.06 s) was much shorter. In addition, because of the transient time needed for the first wave front to reach the probes location, the effective length of the useful signal that was numerically recorded was even shorter (0.05 s for the far-field microphones). As a consequence, one could expect the spectra analysis of the CFD-CAA outputs to be much less accurate than the one applied to the experimental data. More precisely, while the latter had been Fourier Transform (FT) processed with many averaging blocks and a frequency bin width (that is, accuracy) of approx. 10 Hz, the former where FT-processed with only a few averages, and a frequency resolution greater than 100 Hz. 43 Relying on the so-called ‘porous surface’ FWH integration technique. 44 Especially for this particular tonal component, which acoustic energy is more likely to be confined within the jet region because of multiple backscattering of the radiating waves by the shear layer interfaces [53].

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Fig. 13. Noise emission of the Nose Landing Gear (NLG) installed within ONERA’s CEPRA19 anechoic facility, as simulated via CFD-CAA hybrid computations. From top/left, counter clockwise: (a) instantaneous perturbed field obtained at the end of the ‘installed NLG’ calculation, (b) associated confined/sheared steady jet flow of the facility (axial velocity), (c) effects induced by the latter jet flow onto the NLG noise tonal component (f = 1.5 kHz) as highlighted through a comparison of the RMS perturbed pressure field recorded within the xy plane for either the installed (top) and isolated (bottom) configurations, and (d) validation of the ‘isolated NLG’ calculation by cross-comparison of CFD-CAA (red lines), experiments (black lines) and CFD-IM (black dashes) far-field noise results, in terms of spectra recorded for 5 probes located in the flyover direction – see red dots on (b) image. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

IOFD) was validated on the basis of a same academic test case, the advanced CAA-based hybrid approach was applied to two aerodynamic noise problems, which demonstrated its potentialities as an efficient acoustic prediction tool. Such potentialities will be further assessed in the very near future within the framework of two European projects (namely, NINHA and JERONIMO), which are devoted to CROR (Counter Rotating Open Rotor) and HBPR (High By Pass Ratio) powered aircraft, respectively. Indeed, with the view of numerically predicting the noise emission and acoustic installation effects associated with the latter propulsive configurations, CFD-CAA hybrid calculations are currently achieved using the advanced CAA-based hybrid method developed in the present framework. Apart from that, the latter approach will soon be applied within a collaborative project between ONERA and the French Space Agency (CNES), which focuses on the prediction of acoustic levels around launch pads; relying on the present advanced CAA-based hybrid method, CFD-CAA calculations will be achieved for predicting the noise emissions by launchers in harsh environments (involving multi-species and multi scales phenomena, with very high temperatures, Mach numbers, and noise levels included). This shall further demonstrate how the present approach and its underlying innovative techniques make it effectively possible to simulate

realistic noise problems coming from the aerospace industry, despite the stringent constraints such problems place on the method. On that stage, one can precise again that the mitigation of noise by aerospace industry products is not the sole context that could benefit from the here proposed methodology, which could advantageously be applied to any other research domain involving numerical simulations of non trivial acoustic problems (e.g. air/earth/sea transportation, defense, energy, environment, medical, new technologies). Acknowledgements Although mostly supported by ONERA, the present study was conducted within several frameworks and benefited from various key contributions by other Research Establishments. Part of the works achieved was conducted by the present two authors, as part of their respective PhD thesis. Other works were performed by the first author, as part of the International Agreement between NASA and ONERA on ‘‘Understanding and Predicting the Source of Nose Landing Gear Noise’’. Most of validation tasks took direct benefit from diverse joint efforts conducted by various teams coming from different organizations (NASA, ONERA, JAXA).

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In that regard, the first author acknowledges NASA Langley Research Center (LaRC) for having welcomed him as a Visiting Researcher, and provided him with all logistical means needed, including computational resources and data. Then, the authors thank Dr. Tomoaki Ikeda (JAXA) for having derived the analytical solution of the academic test case given by the 3D sphere scattering problem (see Section 5), as a by-product of the international initiative devoted to the Benchmark on Airframe Noise Computations (BANC). Regarding the problem of the 2D Cylinder in a cross flow (see Section 6.1), the authors are grateful to Dr. David P. Lockard (NASA/LaRC) for having provided them with a reliable unsteady CFD dataset, as part of the International Agreement between NASA and ONERA evocated above. Regarding now the realistic test case given by the nose landing gear problem (see Section 6.2), the authors acknowledge Dr. Saloua Ben Khelil for having granted them access to the unsteady CFD data coming from the LAGOON project, which was supported and led by Airbus. Similarly, Dr. Bastien Caruelle (Airbus), Dr. Eric Manoha and Mr. Laurent Sanders (ONERA) are also acknowledged for their help regarding the acquisition and exploitation of LAGOON experimental or numerical data. Finally, authors acknowledge Dr. Mehdi Khorrami (NASA/LaRC), Dr. P. Spalart (Boeing), and Dr. A. Sengissen (Airbus) whose research works and/or informal exchanges helped in guiding the present study. Appendix A As detailed in [17,18], any of the perturbed quantities (Qp = up, Fp) can be expressed as a sum of its linear (Ql) and non linear (Qnl) components (Qp = Ql + Qnl). Regarding for instance the perturbed flow-field vector up, the latter components are respectively given by

8 > <

9 > =

qp q v ul ¼ o p þ qp v o > > : ; 1 qo vo  v p þ 12 qp v o  v o þ c1 pp

unl ¼

> :1 2

qp v p

ðA:2Þ

>

qo vp  v p þ qp vo  v p þ 12 qp v p  v p ;

6 6 Fl ¼ 6 6 4

3

ðqp v o þ qo v p ÞT

qo v o v p þ qo vp v o þ qp vo vo þ pp I

1 2







c c qo v o  vo þ ðc1Þ po v Tp þ qo v o  v p þ 12 qp v o  v o þ ðc1Þ pp v To

7 7 7 7 5

ðA:3Þ and 2 6 Fnl ¼ 6 4

qp v Tp qo vp v p þ qp v o v p þ qp vp vo þ qp vp v p c ðqo v o  v p þ 12 qp v o  v o þ ðc1Þ pp Þv Tp þ



1 2

3 

qo v p  vp þ qp vo  vp þ 12 qp v p  v p ðvo þ vp ÞT

7 7 5

ðA:4Þ

Finally, regarding this time the perturbed diffusive flux, its linear and non linear components are respectively given by

2 16 6 Fl ¼ 6 Re 4 m

3

0T

lo rp þ lp ro T



16 6 6 Re 4

lo ðro  v p þ rp  v o þ CPrp rT p Þ þ lp ro  v o þ CPrp rT o

7 7 7 T 5 ðA:5Þ

3

0T 

lp r p

lo ðrp  v p ÞT þ lp ro  vp þ rp  vo þ rp  vp þ CPrp rT p

7 7 7 T 5

ðA:6Þ The above Eqs. (A.5) and (A.6) exhibit two dimensionless number, namely, the Reynolds (Re) and the Prandtl (Pr) numbers, respectively; the Reynolds number, which expresses the ratio of inertial forces to viscous forces, comes from the non-dimensionalization of the Navier–Stokes equations (more precisely, Re = qrefxref vref/lref, where qref, xref, vref and lref refer to the reference values respectively chosen for non-dimensionalizing the density, the lengthscale, the velocity and the viscosity). The Prandlt number, which denotes the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity, takes a value of 0.72 for ambient air. In the same Eqs. (A.5) and (A.6), ro and rp refer to the mean and the perturbed stress tensors, respectively. Each one of the latter quantities take the following form:

2 3

r ¼ ðrv  þ t rv  Þ  ðr  v  ÞI with  ¼ o; p

ðA:7Þ

where v* = o, p stands for the (mean or perturbed) velocity. These Eqs. (A.5) and (A.6) make also appear both a mean and a perturbed temperature items (To and Tp, respectively). After some manipulations, the latter can be expressed under the following forms:

To ¼

  1 c po C p c  1 qo

ðA:8Þ

1 c ðqo pp  qp po Þ C p c  1 qo ðqo þ qp Þ

ðA:9Þ

and

In the two previous Eqs. (A.8) and (A.9), c stands for the specific heat ratio, whose value is c = 1.4 for air considered in normal conditions of pressure and temperature. Along with the characteristic constant of perfect gases (R = 287.06 J kg1 K1), the latter defines the specific heat ration coefficient (Cp) which is given by

Cp ¼ R

On the other hand, those associated with the perturbed convective flux (Fp) take the following form, respectively;

2

Fmnl ¼

ðA:1Þ

9 > =

0

2

Tp ¼

and

8 > <

and

c c1

ðA:10Þ

Finally, the perturbed viscous flux expression makes appear the mean (lo) and perturbed (lp) dynamic viscosities, which expressions can be approximated by developing at first order the Sutherland’s law. This manipulation, which is here legitimated by the small perturbation hypothesis made from the beginning, leads to

lo ffi lSuth



To T Suth

32

ðT Suth þ CteÞ ðT o þ CteÞ

ðA:11Þ

and

lp ffi lo



 T o þ 3Cte Tp 2T o ðT o þ CteÞ

ðA:12Þ

with, for air in normal conditions of pressure and temperature,

lSuth = 1.711  105 kg m1 s1, TSuth = 273.16 K and Cte = 110.4. References [1] Kirchhoff GR. Zur Theorie der Lichtstrahlen. Ann Phys Chem 1883;18:663–95. [2] Lighthill MJ. On sound generated aerodynamically. I. General theory. Turbulence as a source of sound. Proc R Soc London 1952;A 211; Lighthill MJ. On sound generated aerodynamically. II. General theory. Turbulence as a source of sound. Proc R Soc London 1954;A 222.

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