ACAT1 Benchmark of RANS-Informed Analytical Methods for Fan Broadband Noise Prediction: Part II—Influence of the Acoustic Models
ACAT1 Benchmark of RANS-Informed Analytical Methods for Fan Broadband Noise Prediction: Part...
Guérin, Sébastien;Kissner, Carolin;Seeler, Pascal;Blázquez, Ricardo;Carrasco Laraña, Pedro;de Laborderie, Hélène;Lewis, Danny;Chaitanya, Paruchuri;Polacsek, Cyril;Thisse, Johan
acoustics Article ACAT1 Benchmark of RANS-Informed Analytical Methods for Fan Broadband Noise Prediction: Part II—Inﬂuence of the Acoustic Models 1, 1 1 2 Sébastien Guérin * , Carolin Kissner , Pascal Seeler , Ricardo Blázquez , 3 4 5 6 Pedro Carrasco Laraña , Hélène de Laborderie , Danny Lewis , Paruchuri Chaitanya , 7 8 Cyril Polacsek and Johan Thisse Department of Engine Acoustics, Institute of Propulsion Technology, German Aerospace Center (DLR), 10623 Berlin, Germany; email@example.com (C.K.); firstname.lastname@example.org (P.S.) Department of Engine Propulsion and Fluid Dynamics, Universidad Politécnica de Madrid (UPM), 28040 Madrid, Spain; email@example.com Aerodynamic Technology Department, ITP Aero, 28108 Alcobendas, Spain; firstname.lastname@example.org Aerodynamics and Acoustics Department, Safran Aircraft Engines, 77550 Moissy-Cramayel, France; email@example.com CNRS, Laboratoire de Mécanique des Fluides et d’Acoustique, INSA Lyon, Univ. Lyon, Université Claude Bernard Lyon I, École Centrale de Lyon, 69134 Écully, France; firstname.lastname@example.org Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK; C.C.Paruchuri@soton.ac.uk Department of Aerodynamics, Aeroelasticity, and Acoustics, ONERA—The French Aerospace Lab, 92322 Châtillon, France; email@example.com Airbus Commercial Aircraft, Acoustics Methods, 31060 Toulouse, France; firstname.lastname@example.org * Correspondence: Sebastien.Guerin@dlr.de; Tel.: +49-30-310006-55 Received: 19 June 2020; Accepted: 13 August 2020; Published: 16 August 2020 Abstract: A benchmark dedicated to RANS-informed analytical methods for the prediction of turbofan rotor–stator interaction broadband noise was organised within the framework of the European project TurboNoiseBB. The second part of this benchmark focuses on the impact of the acoustic models. Twelve different approaches implemented in seven different acoustic solvers are compared. Some of the methods resort to the acoustic analogy, while some use a direct approach bypassing the calculation of a source term. Due to differing application objectives, the studied methods vary in terms of complexity to represent the turbulence, to calculate the acoustic response of the stator and to model the boundary and ﬂow conditions for the generation and propagation of the acoustic waves. This diversity of approaches constitutes the unique quality of this work. The overall agreement of the predicted sound power spectra is satisfactory. While the comparison between the models show signiﬁcant deviations at low frequency, the power levels vary within an interval of 3 dB at mid and high frequencies. The trends predicted by increasing the rotor speed are similar for almost all models. However, most predicted levels are some decibels lower than the experimental results. This comparison is not completely fair—particularly at low frequency—because of the presence of noise sources in the experimental results, which were not considered in the simulations. Keywords: RANS-informed noise prediction; fan broadband noise; ACAT1 fan benchmark 1. Introduction Research and development activities regarding the design of turbomachinery components of commercial aero-engines call for reliable and efﬁcient methods to predict the noise emission. Hybrid RANS-informed analytical methods can help to reach that objective. RANS simulations are Acoustics 2020, 2, 617–649; doi:10.3390/acoustics2030033 www.mdpi.com/journal/acoustics Acoustics 2020, 2 618 indeed powerful methods, which are standardly applied in the ﬁeld of engineering. A dedicated post-processing of the RANS results can be applied to reconstruct the input needed by analytical models of fan noise. Thus, if the method works, an acoustic prediction could be achieved as a by-product of a RANS simulation. However two main questions arise regarding that approach—(i) Is RANS able to properly predict the input for the acoustic models, in particular the crucial turbulence statistics needed for broadband noise prediction? (ii) Are analytical models, which tend to strongly simplify reality, sensitive enough to capture the effects of the sough-after design modiﬁcations. The benchmark organised as part of the European project TurboNoiseBB is a contribution to the assessment of RANS-informed analytical methods applied to rotor–stator interaction (RSI) broadband noise. While a ﬁrst part reported in the companion paper by Kissner et al.  focuses on the effect of the RANS model, the present work focuses on the impact of the acoustic model. For several reasons, the low-pressure compressor of an aero-engine, the so-called fan, is the ideal candidate for testing RANS-informed analytical approaches for turbomachines. Firstly, it is composed of a single rotor–stator stage unlike high-pressure compressors or turbines, which combine several blade rows interacting in a very complex manner. Secondly, blades have a high-aspect ratio; this minimises the contribution of endwall effects, which are difﬁcult to predict. Thirdly, because the duct contours are slowly varying and the mean ﬂow is predominantly axial, weakly sheared and moderately swirling, the sound propagation can be reasonably approximated by analytical models, whose complexity can be increased for more accurate results . Finally, contrary to turbine blades, the airfoils of transonic fan stages are thin and only slightly cambered, which are favourable conditions when using the ﬂat plate (zero camber and zero thickness) hypothesis. Speciﬁcally, rotor–stator interaction broadband noise occurs in subsonic conditions so that turbulence–shock noise  does not have to be considered. Furthermore, the stochastic nature of turbulence is expected to make RSI broadband noise well suited for analytical modelling. The development of analytical models for fan noise prediction has a long tradition, which is detailed in many papers, for example, by Posson et al.  and Moreau . Therefore, the intent of the subsequent review is not to be exhaustive but rather to highlight references that help to speciﬁcally understand the models included in this benchmark. Note that the references are not introduced in chronological order. Instead, the focus is on the works of Amiet  and Hanson , which are typical of two different modelling approaches, valid for a single isolated airfoil and a cascade of airfoils, respectively. The notations introduced in Figure 1 will be used throughout the paper. tip 𝑺 ,𝑨 hub 𝟐𝟐 𝑺 ,𝑨 𝟐𝟐 Figure 1. Notations as used in this paper for a rectilinear cascade of ﬂat plates. Acoustics 2020, 2 619 (x, y, z) and (x, r, q) refer respectively to a cartesian and a cylindrical coordinate system, where the x-axis corresponds to the duct axis. (1, 2, 3) are indexes referring to the streamwise, upwash and spanwise components of ﬂow. The variable K refers to convective wavenumbers (of the incoming gust). The variable k refers to acoustic wavenumbers (of the radiated pressure waves). In this context, an oblique gust is understood to be a vortical disturbances featuring a spanwise wavenumber component, that is, K 6= 0. On the contrary, K is equal to zero for parallel gusts 3 3 (the wavefront is parallel to the leading edge of the blades). Finally, regardless of the dimension of the turbulence wavenumber spectrum, the same notation F is used to denote the upwash velocity component. The dimension of the function is indicated by the number of dependent variables. The ﬁrst comprehensive theory to predict turbulence–airfoil interaction noise was formulated by Amiet  (1975). The author considered the case of homogeneous, isotropic turbulence impinging onto an isolated ﬂate plate at zero mean-ﬂow incidence. Amiet adhered to the acoustic analogy, speciﬁcally to the ﬁndings of Curle . Consequently, he assumed that the unsteady lift produced by the upwash velocity component was the principal noise source mechanism. Amiet developped a formulation and an understanding of the problem, which is still the foundation for many of today’s models used to predict fan noise. If the turbulence is frozenly convected, Amiet showed that the turbulence representation required for the acoustic models can be simpliﬁed to a two-dimensional wavenumber spectrum obtained by integrating the three-dimensional wavenumber spectrum along its wavenumber component normal to the airfoil. For airfoils with a large aspect ratio, he further showed that the acoustic pressure in the plane at midspan can be calculated by only considering the component of the turbulence oriented parallel to the leading edge. The two-dimensional wavenumber spectrum retaining only parallel gust components is actually equal to the one-dimensional wavenumber spectrum multiplied by the spanwise correlation length of the upwash velocity component and by a factor 1/p. As the one-dimensional wavenumber spectrum is easily measured by hot-wire anemometry, it is a useful turbulence representation to work with. To calculate the unsteady pressure jump, Amiet used a closed-formed expression  based on the Sears function  for low frequencies and a successive approximation solution  for high frequencies. Amiet’s work demonstrated that a prediction of turbulence–airfoil interaction noise is achievable as long as the kinetic energy and integral length scale of the incoming turbulence are known. An important milestone to consider more representative cases of turbomachines was marked by the work of Glegg in 1999 . The author generalised previous analytical works on two-dimensional cascades of airfoils to derive a theory for three-dimensional rectilinear cascade of inﬁnite-span swept blades interacting with three-dimensional harmonic gusts convected in a uniform cross-ﬂow. Glegg solved the problem with the Wiener–Hopf technique. In that approach, the acoustic pressure is obtained directly without resorting to the acoustic analogy. To calculate sound power, an integration of the acoustic intensity  over the faces of the cascade was performed. For a given gust mode, Glegg identiﬁed an effective frequency, below which the generated pressure waves are evanescent. The cutoff–cuton transition is delayed when the spanwise component of the incoming gust or the blade sweep increase. Both effects are similar and are potentially beneﬁcial for noise reduction. This was also shown in previous works by Graham  for an inﬁnite, isolated plate. Graham proved that an oblique gust sweeping the airfoil’s leading edge at supersonic speed emits sound waves efﬁciently, while acoustic waves are evanescent if the trace speed is subsonic. Based on Glegg’s cascade model, Hanson  (2001) developed a comprehensive theory to predict the broadband noise radiated by a cascade of blades with lean and sweep. As a ﬁrst step, Hanson extended Glegg’s cascade theory to turbulent gusts. Hanson introduced several coordinate transformations to convert a cascade of blades as arranged in an annular duct into a cascade of rectilinear blades. Besides isotropic turbulence, he also considered the case of axisymmetric turbulence using the model proposed by Kerschen and Gliebe . He further extended the method to integrate the case of inhomegeneous turbulence distribution featuring a higher level of turbulence in the Acoustics 2020, 2 620 rotor wakes. Finally, he generalised the solution to rotating blades. Hanson proposed to apply a strip-based approach, which consists in dividing the stator into radial slices, in order to take into account the radial variation of the turbulence characteristics and of the geometry. The sound power is obtained by summing up the contribution of all strips. Hanson performed comparisons of his calculations (made for a typical radial position) to the experimental data of several fans covering a large range of parameters. Because of a lack of experimental data, turbulence intensity and integrale length scales were chosen to minimise the offset between predicted and experimental results. As an alternative to this method of reverse engineering, Hanson identiﬁed the possibility of using RANS inputs for future investigations. Before Glegg’s model was introduced, methods considering the cascade effect or blade-to-blade interaction had relied on two-dimensional solutions in the plane (e , e ). This option had been pursued 1 2 by Ventres et al.  (1982). In their approach, the acoustic analogy was applied using the ﬂuctuating load on the blades as source mechanism. As a beneﬁt, the chosen approach enabled to consider the duct acoustic effect by using the Green’s function for an inﬁnite annular duct expressed as an inﬁnite series of normal modes . The strip theory approximation was applied. Thus, the rest of the problem had been reduced to the calculation of the pressure jump on each strip as though it were a linear cascade of two-dimensional, thin, ﬂat plates. To obtain the blade pressure distribution, Ventres and co-authors applied a numerical method solving an integral equation relating the source strength of dipoles distributed on the plates to the velocity disturbance. The angle of the plates was assumed to match the incoming mean ﬂow angle. The turbulent velocity ﬁeld was modelled by the product of three Gaussian functions representing the spatial correlation of the turbulence in the three directions. Thus, the strips were cross-correlated in the radial direction (e ) via the radial correlation of the turbulence velocity, while the blade response remained two-dimensional. In 2005, Nallasamy and Envia  ﬁrst published a study dedicated to the prediction of fan broadband noise based on a RANS-informed analytical approach. The Source Diagnostic Test (SDT) fan rig equipped with three different designs of stator was used as a test case for the validation. The comparisons were done at three operating points relevant for the acoustic certiﬁcation: Approach, Cutback and Sideline. The turbulence kinetic energy and the turbulence length scale at the stator leading edge position were extracted from RANS k e simulations. The integral length scales were deﬁned based on standard hypotheses for homogeneous isotropic turbulence. The used acoustic model corresponded to an extension of the Ventres’ method, which cannot account for the effect of oblique gust as mentioned before. Nallasamy and Envia were able to reproduce the general trends observed experimentally although the slope at high frequency was overpredicted due to the use of Gaussian functions to model the turbulence rather than the more physically realistic Liepmann or von Kármán turbulence models. Based on a similar approach to Nallasamy and Envia and still using the SDT case for validation, Grace and co-authors published a series of papers, in which they investigated the sensitivity of the analytical models regarding some of the assumptions. Thus, in 2012, Grace et al.  showed that the method may create peaks with a high amplitude in the predicted noise spectra, which are not present in the measurements. These peaks are linked to resonances of the two-dimensional cascade model, that are much weaker for non-parallel blades. Grace et al. also started to investigate the validity of their formulation, in which the three-dimensional wavenumber spectrum F (K , K , K ) is replaced 22 1 2 3 by a two-dimensional wavenumber spectrum multiplied by a radial correlation function, denoted by F (K , K )R (r). For one example, they found results that were signiﬁcantly different between 22 1 2 r the two approaches. The computation of the three-dimensional cascade response, required for the exact approach, was a time-consuming process at high frequency, which explains the few frequencies considered in the study. In the same paper, Grace et al. found that modeling the inhomogeneity of the turbulence in terms of energy and length scale across the passage is not important (provided that an appropriate turbulence averaging is applied as recently showed by Kissner et al.  by applying an hybrid numerical method coupling the generation of synthetic turbulence and the linearised Acoustics 2020, 2 621 Euler equations). An analysis of the correlation lengths by Grace et al. indicated that none of the known isotropic models of turbulence could well reproduce the experimental data. They concluded that anisotropy is important. Finally, they showed that whatever the choice of the stagger angle representing the cambered airfoil was—either the metal angle at the leading edge, at the trailing edge or a combination of both values, no good agreement in amplitude and phase between the analytically calculated pressure jump across the blade surface and an accurate numerical solution accounting for the real blade geometry could be observed. One year earlier, Grace et al.  published a sensitivity study of the RANS turbulence model, in which they highlighted the fact that an accurate prediction of the turbulence intensity and turbulent length scale of background turbulence can be of importance to obtain a good match with the measurements. They found that the plate angle have a signiﬁcant impact on the noise prediction. Note that the difference of angle between the leading edge and the trailing edge of a stator depends on the fan loading as explained, for example, by Moreau and Guérin . For the SDT as for the ACAT1 fan, this difference is about 30 to 40 . By choosing the trailing edge rather than the leading edge to ﬁx the angle of the plate, Grace and co-authors found that the broadband noise levels measured at the exhaust position were lowered at low frequency but increased at high frequency. On the upstream side, the impact can be more substantial as shown by Jaron et al. , who reported a global decrease of the noise of several decibels. An explanation for that behaviour, which is linked to the orientation of the dipole sources with respect to the duct cross-section, was proposed by Blandeau et al. . Grace and co-authors were also interested in the deﬁnition of the length scales for RANS simulations. They evidenced its importance for the acoustic results. In 2015, Grace  extended once again Ventres’s solution to three-dimensional gusts. This time, the unsteady response of the cascade to a three-dimensional vortical disturbance was solved by using the integral equation approach of Ventres together with the similarity rules proposed by Graham . Graham’s similarity rules relate a three-dimensional gust to a two-dimensional problem. Grace showed that only considering parallel gusts (i.e., only retaining the contributions for K = 0) led to a strong underestimation of the sound power levels by about 20 dB over the relevant frequency range. Setting the unsteady vane response to the same value as that obtained for K = 0 (8K ) produced a good 3 3 agreement at high frequency but an overprediction at low frequency. A similar result was achieved by using the two-dimensional solution. Adopting an approach similar to Ventres, but using a three-dimensional solution  for the unsteady blade loading derived from an extension of Glegg’s model, Posson et al.  (2011) ﬁrst developed a method to account for three-dimensional gusts in annular ducts. The unsteady loading was used as a distribution of dipole sources in the acoustic analogy together with the strip theory approach. To avoid having some of the drawbacks linked to the rectilinear cascade hypothesis, Posson et al.  proposed some corrections, in particular to minimise the resonance effects related to the presence of parallel, adjacent blades. As the hub-to-tip ratio decreases, the formulation is increasingly less exact and the solution is more prone to resonances. Finally, it should be noted that comparisons between Posson’s model and some of the other models cited above (Ventres, Hanson, Amiet) using the SDT experimental results for validation were presented, for example, by de Laborderie  and Lewis et al. . Their results evidenced discrepancies between the models, especially at low frequency. Accounting for the cascade effect for skewed gusts greatly improved the prediction compared to solutions calculated either with a two-dimensional cascade model or with an isolated-airfoil model. The presence of the duct proved to be important too. As shown by the literature review, the validation of RANS-informed analytical methods has been mostly restricted to the SDT data provided by NASA. The present study uses a new, independent data set obtained in 2018 at AneCom AeroTest during a test campaign organised in the framework of the European project TurboNoiseBB . These data give the opportunity to further assess the RANS-informed analytical method. With the two questions raised at the beginning of the introduction in mind, a benchmark was organised reconsidering the impact of the two main ingredients of Acoustics 2020, 2 622 the method—(1) the RANS calculation (in particular the choice of the turbulence model) and (2) the aeroacoustic models. While the ﬁrst part of the study is addressed in a companion paper by Kissner et al. , this second part deals with the impact of the acoustic models. The benchmark character is unique as more than ten independent European institutions using different CFD and acoustic solvers were involved. This large diversity guarantees that several key aspects of the RANS-informed analytical approach for fan broadband noise will be addressed. The paper is structured as follows. The methodology to prepare the benchmark is described and the common input data for the noise calculation are presented. Then the acoustic prediction models are brieﬂy introduced. Finally, the results are analysed in terms of the prediction of absolute levels and trends. 2. Benchmark Preparation Section 2 provides some relevant information regarding the benchmark, including an overview of the data delivered to the participants. 2.1. ACAT1 Fan Benchmark Data 2.1.1. Tests at AneCom AeroTest A short description of the TurboNoiseBB test campaign, which provided the validation data for the benchmark, was given by Guérin et al. . Speciﬁc details on the instrumentation and data post-processing can be found in several publications [31,32]. The ACAT1 fan is a transonic fan composed of 20 rotor blades and 44 stator vanes. A longitudinal cut of the AneCom test rig is shown in Figure 2. The noise instrumentation is highlighted in red. Two conﬁgurations with different rotor–stator gaps were measured in the project TurboNoiseBB but only the short gap variant was considered for the benchmark. Furthermore, the focus was restricted to the three operating conditions relevant for acoustic certiﬁcation: Approach, Cutback and Sideline. These points were distributed along a single, the so-called “Sea Level Static” working line. Note that the stator geometry was simple, with no lean and nearly no sweep. According to Hanson , sweep has no signiﬁcant effect for low angles as the impact on the peak amplitude PW L is approximately proportional to the cosine of the peak sweep angle f , hence DPW L = 10 log (cos f ). swee p swee p peak Figure 2. UFFA rig of AneCom AeroTest with the acoustic instrumentation as used during the TurboNoiseBB tests (TurboNoiseBB consortium, reprint with permission). 2.1.2. Acoustic Data The acoustic and aerodynamic tests were conducted separately in order to avoid a contamination of the acoustic results by the instrumentation (hot-wire probes and total pressure sensor rakes mounted along the stator leading edge). The noise results for the bypass duct are based on measurement data provided by the line array of condenser microphones AX1 indicated in Figure 2. The microphones were wall-ﬂush mounted in a section of constant radii located far downstream of the stator. The microphone signals were ﬁltered using an axial wavenumber decomposition technique to efﬁciently separate hydrodynamic and acoustic pressure ﬂuctuations . The signals were synchronised with the rotor shaft so that the rotor-locked part of the ﬂuctuations could be removed . The model used to deduce the sound power spectra based on the sound pressure assumes an equal energy density distribution Acoustics 2020, 2 623 between the propagating acoustic modes of the same frequency band [33,35]. Very similar results were obtained by Pereira and Jacob . The results in the forward arc are less prone to uncertainty. They were obtained by processing the sound pressure spectra measured by the 25 far-ﬁeld microphones arranged in a broken semicircle inside the large anechoic chamber (see Figure 2). The bypass and core ﬂows of the UFFA rig are piped outside of the anechoic chamber. Furthermore, the intake of the rig protrudes inside the anechoic chamber from one sidewall. Thus, the far-ﬁeld microphones of the UFFA rig measure only the noise component radiated through the inlet. As the ambient ﬂow velocity in the plenum was very small during the test, it was neglected for the calculation of the sound power. Contrary to the in-duct results, the far-ﬁeld measurements were not rotor-synchronised. Therefore, the tones are (strongly) present in the spectra. The acoustic results for the Sideline conditions are shown in Figure 3. Two curves were drawn per hand on one of the graphs to exemplary suggest the presence of at least another source of broadband noise besides rotor–stator interaction (RSI) broadband noise. The peak frequency of RSI noise is approximately 3 times the blade passing frequency (BPF), which is close to the empirical factor of 2.5 proposed by Heidmann . The Strouhal number St given in x-axis of the graphs in Figure 3 is deﬁned in accordance with Kissner et al. : St = f R/W , with f the frequency, R 4.23 m the radius at the duct casing upstream of the stator leading edge, and W (240 m/s at Sideline) the averaged ﬂow velocity upstream of the stator. The second non-dimensional number kR shown on the top x-axis of the graphs corresponds to the Helmholtz number. This number permits to identify the frequency range within which pressure peaks, produced by acoustic duct modes while they become cut-on, are likely to be visible (typically for kR < 10). Figure 3. Sound power spectra at Sideline used for the comparison with the predictions: (top) (black line) result obtained from the far-ﬁeld microphones located in the forward arc of the test rig as illustrated in Figure 2, (bottom) (black line) result obtained from the line array AX1 located in the bypass duct after the rotor-locked contribution and the hydrodynamic pressure component had been removed; (dashed red lines) hand drawn curves suggesting the presence of an additional, dominant source of noise at low frequency. 2.2. Input for the Analytical Models 2.2.1. RANS Calculations As the ﬂow Mach number is one of the key parameter for fan noise, it was decided to extend the benchmark, which was focused on the condition Approach in Part I, to two more operating conditions, namely Cutback and Sideline. The corresponding values of rotation speed and mass ﬂow are provided in Table 1. The structure of the ﬂow differed signiﬁcantly between the three operating conditions (see Guérin et al. ). At Approach, the ﬂow detached at the leading edge as the rotor was highly Acoustics 2020, 2 624 loaded. This ﬂow separation observed in the RANS calculations could not be evidenced as such by the experimental data. Due to a reduced rotor loading, no ﬂow detachment was found at the highest investigated speed. As shown in Part I of the benchmark, the choice of the turbulence model has a substantial impact on the predicted acoustic levels. All the RANS solutions used in Part II were produced by DLR with the CFD solver TRACE  using the Shear-Stress-Tensor (SST) k w turbulence model from Menter . For Approach, this corresponds to solution RANS #2 presented in the Part I paper . This turbulence model seems to be predominantely used in the community. In fact, the study of Part I has shown that the agreement between CFD and the hot wire measurements is not satisfactory for any of the studied turbulence models. The hot-wire measurements data could not be used either, because of the current uncertainty in the data, in particular at high speed. As a consequence, the goal of the present benchmark was not to ﬁnd out the most suitable acoustic model, but to investigate the relative impact of the model assumptions on the acoustic predictions and to quantify the variations. Table 1. True and corrected operating conditions on the SLS working line as measured during the acoustic tests. Short Gap Approach (AP) Cutback (CB) Sideline (SL) rpm 3856.1 (50%) 6175.1 (80%) 6945.7 (90%) massﬂow (kg/s) 54.85 88.80 101.32 corr. rpm 3797.9 6077.3 6836.5 corr. massﬂow (kg/s) 56.48 91.61 104.53 2.2.2. RANS Data Processing For a given operating point, all the acoustic simulations were based on the same input obtained by analysing the geometry and the ﬂow solution of the RANS calculation. The data analysis—done with the DLR in-house tool C3D_T2P —was conducted at 97 radial positions equally distributed along the whole span (see Figure 4). The streamline positions in the (x, r)-plane were determined after having circumferentially averaged the mean ﬂow. Only the part of the ﬂow going into the bypass duct was considered in the prediction. Thus, the interaction with the Engine Support Stator (ESS) located at the core entry was ignored. For each radial position of the stator, the following values were provided: the axial and tangential speeds up- and downstream of the stator (see Figure 5), the turbulent kinetic energy and the turbulence length scale (see Figure 6). Additionally, averaged values of the speed of sound and of the mean ﬂow density calculated upstream of the leading edge were provided. Only one acoustic code was able to consider the real proﬁle of the airfoil. In the other models, the stator vanes were discretised into ﬂat thin plates, whose stagger angle varied along the span (see Figure 7). Acoustics 2020, 2 625 rotor stator (for the rotor) mean stagger (𝜒 ) Figure 4. Extraction of geometry and ﬂow parameters from RANS simulations by means of the post-processing method implemented in C3D_T2P (adapted from Jaron ). Figure 5. Spanwise distribution of the axial (M ), tangential (M ) and absolute (M) Mach numbers obtained by circumferential averaging at positions A and B, respectively located at one quarter chord length upstream of the stator leading edge and one quarter chord length downstream of the trailing edge (see positions A and B approximately indicated in Figure 4). θrel θrel θ Acoustics 2020, 2 626 ¯ ¯ Figure 6. Turbulent kinetic energy k (top) and turbulence length scale L (bottom) RANS RANS reconstructed at the stator leading edge for the three investigated operationg points: Approach (AP), Cutback (CB) and Sideline (SL). Figure 7. Spanwise distribution of (left) the ﬂow (b) and stator (c) angles, and of (right) the stator solidity (chord-to-pitch ratio). Background and wake turbulence contributions were averaged and modelled by one single contribution having equivalent TKE and TLS values. Acoustics 2020, 2 627 The turbulence kinetic energy k was circumferentially area-averaged and the turbulence RANS length scale L was weighted by the local value of TKE as proposed by Jaron et al. : RANS Dq k (r) = k (r, q)dq, (1) RANS RANS Dq Dq L (r) = k (r, q)L (r, q)dq. (2) RANS RANS RANS Dqk (r) RANS From RANS, only a single integral length scale can be calculated locally. It corresponds to the average size of the largest energy containing eddy. The local value of the turbulence length scale L RANS was calculated based on the local values of k and w , where w is the speciﬁc turbulence dissipation rate . The following relationship was applied: k (r, q) RANS L (r, q) = C , (3) RANS L Cm w (r, q) RANS with the two constants C = 0.09 and C 0.4. As a consequence of this averaging technique, background and wake turbulence are mixed. Further ways to determine the turbulence integral length scale are discussed in the companion paper . Note that the turbulence characteristics were extrapolated downstream of the mixing plane up to the stator leading edge position by using the reconstruction method based on a semi-empirical model proposed by Jaron . This extrapolation aimed at improving the comparison to the experimental data. Indeed as the turbulence in the rotor wakes is convected towards the stator, its intensity tends to decrease, while its length scale tends to increase. These two effects shift the peak frequency to a lower value and produce a slight increase of the peak amplitude as shown in Part I. 3. Acoustic Models Some general features of the acoustic models of the benchmark are presented in preamble to a more detailed analysis of the models. 3.1. Preamble All methods of the benchmark are formulated in the frequency domain. They target a representation of broadband noise in the form of a frequency spectrum but not as a time signal. It was assumed that broadband noise was generated by the interaction of the incoming turbulence with the blades. Other sources of broadband noise like rotor self-noise, stator self-noise and rotor–ESS interaction noise were ignored for the benchmark. The turbulence was assumed to be homogeneous, isotropic turbulence at each radial position/for each strip. In all calculations, the turbulence was imposed as if it were a background turbulence but of course using the equivalent TKE and TLS values of the benchmark, which include the wake and background contributions. Either the von Kármán or the Liepmann model was used to describe the turbulence. The difference between the two models is rather small. In fact, the differences are smaller than 1 dB for the one-dimenional wavenumber spectrum. As observed by Grace , the agreement with experiments is better using the Liepmann model than using the Gaussian model . Most of the methods are mathematical expressions containing integrals and summations. A few methods resort to a very complex modelling of RSI noise, which has a direct impact on computation time. The latter can potentially exceed one day as reported by Grace . The solution labelled BB1 is partly numerical as it used a CAA solver to calculate the acoustic response of the stator. That method was the only one able to account for the real blade proﬁle including the effects from the mean ﬂow. Acoustics 2020, 2 628 All other methods replaced the stator vanes by ﬂat plates as isolated airfoils or arranged in a cascade. The “ﬂat plate” hypothesis implies that the most representative stagger angle is used. All the methods relied on the angle at the leading edge except for results TA1 and TA2, which considered the inﬂow angle. As the ﬂow incidence is small at the stator leading edge, no strong effect is expected from that choice, even though the stagger angle is known to be a sensitive modelling parameter. 3.2. Classiﬁcation of the Methods The methods used for the benchmark were classiﬁed into two different categories. The result of that classiﬁcation is shown in Figure 8. One group contains methods that explicitly refer to the acoustic analogy, while the other group contains methods that rely on a direct calculation of the pressure cascade response: Methods based on the acoustic analogy were assembled in Group A. The models use a source term (the unsteady lift produced by the turbulence on the blade surface) in combination with a Green’s function to calculate the acoustic pressure. They either make the assumption of a single, isolated airfoil or consider a cascade of airfoils. The methods of Group B follow a different approach. They rely on a direct calculation of the acoustic pressure response of the cascade of blades without requiring a source term. Therefore there is one step less in the workﬂow represented in Figure 8. All of the studied methods account for the cascade by considering separate radial strips. These strips are then unwrapped to match the theoretical case. Tables A1 and A2 in Appendix A summarise some important characteristics of the models and refer to the publications where more details can be found. Group A Group B Acoustic analogy Direct acoustic calculation (PN1, PN2, TA1, TA2, OB1, OB3) (OB2, BB1, LN1, LN2, LN3, OP1) Skewed gust Parallel gust Skewed gust Parallel gust (LN1, LN2, BB1, OP1, (PN1, PN2, TA1) (TA2, OB1, OB3) (LN3) OB2) gust gust description description velocity ﬁltering low. freq. - high. freq. - Amiet rectilinear cascade - Sears/Amiet (TA1,TA2, OB1) Posson (OB3) (PN1, PN2) airfoil single airfoil 3D cascade response Smith + Smith LNS Glegg Graham (LN3, LN1, OP1) (BB1) (OB2) (LN2) unsteady pressure 3D cascade 2D cascade jump acoustic response annular duct free ﬁeld (PN2, OB1) (PN1, TA1, TA2, OB3) acoust. ﬁeld Green's (LN3, LN1, function OP1, BB1, LN2, OB2) modal modal pressure pressure amplitude amplitude free ﬁeld annular duct free ﬁeld annular duct (PN2, OB1) (PN1, TA1, TA2, OB3) (LN3,LN1, BB1, OB2, LN2) (OP1) sound sound power power Figure 8. Classiﬁcation of the models used in Part II of the TurboNoiseBB benchmark. Acoustics 2020, 2 629 3.3. Methods Based on the Acoustic Analogy (Group A) Approaches of Group A all rely on the acoustic analogy as mentioned before. To get an acoustic pressure, two main steps are necessary—(1) the calculation of the unsteady pressure distribution on the plates created by the vortical disturbance and (2) the integration along the stator radius of the source term multiplied by the appropriate Green’s function. The result is a distribution of the pressure at any point in space, which can then be used to calculate the sound power. Two options are possible regarding the calculation of the pressure jump on the surface of the plates. Five of the six methods considered that the blades are isolated but used different unsteady lift response models. The sixth method considered the cascade effect, that is, accounted for the blade-to-blade interactions. Regarding turbulence, three solutions assumed that the gusts impinged parallel to the leading edge, while the other three methods accounted for the oblique component. Concerning the choice of the Green’s function, two options were tested in the benchmark. Two solutions considered the Green’s function in free-ﬁeld with a uniform axial mean ﬂow as if there were no hub and no casing surrounding the stator. Four solutions resorted to the Green’s function for an inﬁnitely long annular duct with a uniform axial mean ﬂow. This last Green’s function is expanded in normal modes [2,13]. The methods of Group A are now described, starting with the one implemented in the solver PropNoise. The latter is presented in more detail than the other methods as it was used to perform all the noise calculations presented in Part I of the benchmark . Even though all methods in Group A are different, they must follow the same key steps. 3.3.1. Solutions PN1 and PN2 The methods implemented in the DLR in-house code PropNoise  is among the simpler methods used in the benchmark. It is subsequently presented using the formalism of Moreau and Guérin . The objective is to enumerate the various steps necessary in order to obtain a noise prediction based on RANS data and to illustrate the ambiguity and complexity of some choices. As mentioned before, the acoustic prediction relies on the acoustic analogy. For the prediction of RSI noise, only the source term related to the vane unsteady loading mechanism is considered. The other contributions for example, due to the turbulence–potential-ﬁeld interaction are not modelled. The unsteady loading is calculated by assuming that the vanes are isolated, which means that blade-to-blade interactions are not considered. For a harmonic gust, Moreau and Guérin  showed that the pressure complex amplitude A mn of the in-duct acoustic mode with azimuthal and radial orders m and n, at angular frequency w = 2p f , can be written as A (w) = iV g ˆ (w, r) exp( ik x (r) imq (r))s (w, r)dr, (4) mn x S,A S,A mn mn h R with the superscript indicating the direction of progagation of the waves (“