An acoustic model of a Helmholtz resonator under a grazing turbulent boundary layer

Acta Mechanica Preprint of V olume 230, Issue 6, pp 2013-2029 https://doi.org/10.1007/s00707-018-2354-5 An Acoustic Model of a Helmholtz Resonator under a Grazing T urb ulent Boundary Layer Lewin Stein · Jörn Sesterhenn Receiv ed: 11 July 2018 / Accepted: 11 November 2018 Abstract Acoustic models of resonant duct systems with turbulent flo w depend on fitted constants based on expensi ve e xperimental test series. W e introduce a new model of a resonant cavity , flush-mounted in a duct or flat plate, under grazing turb ulent flow . Based on pre vious work by Goody , Howe and, Golliard, we present a more univ ersal model where the constants are replaced by physically significant parameters. This enables the user to understand and to trace back how a modification of design parameters (geometry , fluid condition) will affect acoustic properties. The deri vation of the model is supported by a detailed three dimensional Direct Numer- ical Simulation as well as an experimental test series. W e show that the model is valid for low Mach number flows ( M = 0 . 01-0 . 14) and for low frequencies (below higher transverse cavity modes). Hence, within this range, no expensi ve simulation or experiment is needed any longer to predict the sound spectrum. In principle the model is applicable to arbitrary geometries: Just the provided definitions need to be applied to update the significant param- eters. Utilizing the lumped element method, the model consists of exchangeable elements and guarantees a flexible use. Even though the model is linear , resonance conditions between acoustic cavity modes and fluid dynamic unstable modes are correctly predicted. Keyw ords Acoustic Model · Impedance Model · Helmholtz Resonator · T urbulent Boundary Layer · Kelvin-Helmholtz Instability · Direct Numerical Simulation P A CS 43.50.Gf · 43.50.Cb · 43.28.Py · 43.55.Ka · 43.20.Hq · 47.27.ek · 47.27.N- · 47.20.Ft Lewin Stein Institut für Strömungsmechanik und T echnische Akustik, Müller-Breslau-Str . 15 10623 Berlin E-mail: Lewin.Stein@tu-berlin.de ORCID: 0000-0002-4298-2001 Jörn Sesterhenn Institut für Strömungsmechanik und T echnische Akustik, Müller-Breslau-Str . 15 10623 Berlin 2 Lewin Stein, Jörn Sesterhenn Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 Motiv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Research Goal: Acoustic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Reference Case: DNS of a Helmholtz Resonator under Grazing T urbulent Flo w . . . . . . . . . . 4 3 The New Helmholtz Resonator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1 Basic Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Model Source T erm of a Turb ulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . 6 3.3 Flow-Acoustic Interaction of an Opening with a V ortex Sheet Redefined . . . . . . . . . . . 8 4 Results: Interpretation, V alidation, and V alidity Range of the Model . . . . . . . . . . . . . . . . 15 4.1 V alidation of the Flow-Acoustic Interaction Impedance Z f l ow ( u + , β ) . . . . . . . . . . . . . 15 4.2 A New V iew on the Model Parameters u + and β . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 V alidation of the Helmholtz Resonator Model . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 V alidity Range of the Helmholtz Resonator Model . . . . . . . . . . . . . . . . . . . . . . . 18 5 Model Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1 Introduction 1.1 Motiv ation T ypically , when gases (compressible fluids) stream along a hollow space, acoustic and turb u- lent flo w strongly couple. Many e xamples can be listed in which this coupling is of greatest importance. Noise silencers consisting of cavity arrays are installed in most duct systems, in which tonal noise (due to a constant operating frequency) needs to be reduced, among others: Air conditioning systems, ventilation plants, combustion engines. Beside silencing properties, cavities may give rise to desirable tones of wind instruments like transverse flutes and or gans or undesired ‘windo w buf feting’ of mo ving vehicles. Acoustic cavity resonances may ev en cause se vere material damage for instance in pipeline intersections. 1.2 Research Goal: Acoustic Model Cavities under grazing turbulent flow are commonly surve yed [ 1 – 3 ], mostly for industrial applications. In practice, series of many different cavity configurations are experimentally tested for a certain design goal, e.g. noise cancellation in [ 4 ]. An understanding of the under- lying physical processes, especially of the acoustic turbulence interaction is missing. Such, there is a lack of easy applicable but realistic models, which are not dependent on expensi ve parameter studies. In this work, we derive a new model of a Helmholtz r esonator , which simplifies the de- sign pr ocess and which is widely applicable due to a modular principle (Lumped-Element Method) . By Helmholtz resonator , we mean an almost closed cavity except for a neck open- ing (s. Sect. 2 ). The Helmholtz resonator is excited by a grazing turbulent boundary layer (TBL) flow: In the present case, the thickness of the TBL is smaller than the streamwise extension of the neck, δ 99 , neck < L x − neck . As a consequence, an unstable shear layer arises and thus a strong turbulence-acoustic interaction at the neck area is expected. Particularly , the ne w model will be specialized in this turbulence-acoustic interaction (s. Sect. 3.3.1 ), in An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 3 contrast to common, purely acoustical descriptions. The novelty of the model is character - ized by its r efined par ameters. The parameter s ar e no longer meaningless fit parameters but r eplaced by clearly defined quantities with a physical meaning. Thereby , we focus espe- cially on the conv ection velocity , which is the key factor to model the acoustic impedance of a neck with a shear layer (Sect. 3.3.2 ). The model is optimized for a low Mach number flow and lo w-frequenc y acoustics, which are typical operating conditions of duct systems. In the following, previously established models are briefly surveyed. Due to missing numerical studies (s. Sect. 2 ), they are predominantly based on experiments. Often, the res- onance frequencies but not the amplitudes are predicted: An acoustic feedback mechanism of Kelvin-Helmholtz wa ves at the neck is described by [ 5 ]; [ 6 ] studied the additional cou- pling of these K elvin-Helmholtz wav es at the neck with the acoustic modes of an open cavity (valid for Mach numbers M > 0 . 2). Models which can predict the sound pressure lev el (SPL), too, usually are either based on se vere theoretical simplifications [ 7 , 8 ] or rely on empirical fits with a limited validity [ 2 ]. This work is organized as follows: Sect. 2 introduces the geometry and conditions of our reference DNS. In Sect. 3 the new Helmholtz resonator model is deri ved in three steps: Sect. 3.1 establishes the fundamental model structure. Sect. 3.2 implements the TBL as an acoustic source term into the model. Based on theoretical works, Sect. 3.3 generally rede- fines a model of the turb ulence-acoustic interaction. The physical meaning and the v alidity of the model components is explained in Sect. 4 . A final model summary is made in Sect. 5 . T o conclude, Sect. 6 discusses the impact of the new model. 4 Lewin Stein, Jörn Sesterhenn 2 Reference Case: DNS of a Helmholtz Resonator under Grazing T urbulent Flow This section introduces the setup of the reference case, in particular how it was obtained by means of a Direct Numerical Simulation (DNS). More details are published in [ 9 ]. A sketch of the geometry is provided in Fig. 1 . In streamwise x -direction, a zero-adverse-pressure- gradient turb ulent boundary layer streams ov er a rectangular cavity , which is flush-mounted inside the bottom wall. The cavity is connected via a rectangular neck (aka opening) with the flat-plate. The reference geometry is moti v ated by Golliard [ 2 ]. Experimentally Golliard in vestigated a Mach number range from 0.01 to 0.14 and δ 99 , neck / L x − neck ratios (boundary layer thickness to streamwise neck length) from 0.7 to 5.5. This allows comparisons in Sect. 4.4 . W e conducted one DNS at a Mach number of M ∼ = 0 . 11 and at δ 99 , neck / L x − neck ∼ = 0 . 7. T o our knowledge, we conducted for the first time a DNS of this setup without simpli- fication of the compressible Navier -Stokes equations. In case a cavity with a neck is con- sidered (Helmholtz resonator) the inflo wing TBL is often missing [ 10 – 12 ] or not all system scales are resolved, but some form of turb ulence model is assumed [ 13 ]. The geometrically closest match is the simulation of a cylindric cavity by [ 14 ]. An acoustic solver (Lighthill’ s analogy) with source terms giv en by an incompressible fluid flow cannot be used due to the expected non-linear coupling of the TBL and the sound field in the neck re gion. In order to simulate 30 ms of physical time, we inv ested 7 × 10 6 CPU-hours with a grid composed of 1 . 2 × 10 9 grid points. Due to the computational cost, a parameter study is only experimentally feasible. But in contrast to an experiment, our single DNS run supplies a dataset of all flow variables at all times and spaces (20 terabytes were stored.). W e will ev aluate this reference dataset to identify major and minor mechanisms and to legitimate our model assumptions. x 10 . 8 δ 15 . 5 δ 12 . 6 δ 10 . 8 δ y z 0 . 1 δ 1 . 5 δ Fig. 1: Reference case setup of a Helmholtz resonator under grazing turbulent flow . At the top, vorticity isosurfaces ( ± 3000 Hz colored blue and red) of the TBL are visible. Below the flat-plate flow , the rectangular cavity of the resonator is mounted. All units are given in terms of the boundary layer thickness, defined at the neck center δ 99 , neck = 9 . 28 mm. Around the neck, the grid is further refined as indicated by the zoom window . An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 5 3 The New Helmholtz Resonator Model In the following Sect. 3.1 the basic model structure of a Helmholtz resonator with a TBL flow is introduced. The model consists of acoustical elements and flow-related elements. The focus in this paper is to ne wly deriv e all flow-related elements (purely acoustical elements stem from [ 2 ]). The two flow-related elements are the TBL, which serves as a broadband source term for the new model (selected in Sect. 3.2 ), and the most important flo w-acoustic interaction element of the neck (defined in Sect. 3.3 ). 3.1 Basic Model Structure Based on the Lumped-Element method [ 15 ], the underlying basic model structure is  p bot 0  =  cos ( kL y − cavit y ) i sin ( k L y − cavit y ) Y cavit y − i sin ( k L y − cavit y ) / Y cavit y cos ( kL y − cavit y )  | {z } M cavity  1 Z neck 0 1  | {z } M neck  p t o p v t o p  . (1) As illustrated in Fig. 2 , Eq. 1 relates the acoustic pressure p t o p and the acoustic v olume flux v t o p abov e the cavity (“top” position) with the acoustic pressure at the bottom of the ca vity (“bot” position with a hard w all i.e. v bot = 0). The Lumped-Element method assumes linear one-dimensional plane harmonic wa ves with the relation ω = k c of frequency , wa ve v ector , and speed of sound, respectiv ely . The standard transfer matrix M cavit y describes acoustic wa ve propagation in y -direction inside the ca vity with the constant cross-section S cavit y , the characteristic impedance Y cavit y = S cavit y / ρ c and the cavity height L y − cavit y . Emphasis is to be laid on the neck impedance Z neck , which combines all ef fects related to the neck geometry and the flow . Its detailed discussion follo ws. flat plate infinite flange flow interaction area jump cavity M neck M cavit y l rad l f l ow l jum p r rad r f l ow S neck S cavit y y t o p = 0 y bot = − L y − neck − L y − cavit y v bot = 0 y x Fig. 2: Basic model structure of the Helmholtz resonator model using the Lumped-Element Method. T o provide a clear view of the neck elements the figure proportions are distorted (for corrects proportions s. Fig. 1 ). The red horizontal lines mark the start and end surfaces of the elements. The cavity has a height L y − cavit y and a constant cross-section S cavit y . The cavity and the neck ha ve the same spanwise depth L z − cavit y = L z − neck . 6 Lewin Stein, Jörn Sesterhenn T ypical challenges in dealing with acoustically-resonant cavities are either to silence existing tonal noise or to prev ent cavity resonances before they happen (Sect. 1.1 ). In the first case the transfer or damping function of external acoustic wa ves passing by the ca vity is seeked (i.e. Z t o p = p t o p / v t o p of Eq. 1 ). In the second case the penetration of acoustic waves into the cavity resonator is of interest (i.e. p bot / p t o p of Eq. 1 ). As a proof of concept, this work discusses the second case exclusi vely , the prediction of the sound pressure level (SPL) spectrum at the bottom of the cavity: Φ bot ( ω ) = T ( ω ) Φ t o p ( ω ) , (2) being Φ the power spectral density of pressure and T = | p bot / p t o p | 2 the transmission func- tion. Without constraints, the model elements, explicitly deri ved in this paper , can be directly applied in the first case, too. The source term Φ t o p of Eq. 2 will be specified in Sect. 3.2 . In the following, the transmission function T is deriv ed from Eq. 1 . Flow related and purely acoustical effects are separated from each other by different impedance elements: Z neck = Z jum p + Z f l ow + Z rad . Fig. 2 qualitati vely illustrates the horizontal positions of these three neck elements separated by red lines. The separation brings the advantage of differenti- ating and in vestigating physical effects independently . An individual element can be adapted to the specific application. Z jum p and Z rad are purely acoustical elements: Z jum p accounts for the cross-section jump between the ca vity and the neck; Z rad describes the radiation losses of a neck opening mounted in an infinitely extended plate (infinite flange). All interactions of acoustic wa ves with the shearing flo w around the cavity opening are incorporated in Z f l ow . W e split the real and imaginary-valued part of each impedance element according to Z = ( r + ik l ) Y . Altogether (analogous to [ 2 ]) the transmission function for Eq. 2 deriv ed from Eq. 1 is T − 1 = sin 2 ( kL y − cavit y )  Λ 2 l + ∆ 2 r  , being (3) Λ l = cot ( kL y − cavit y ) − S rat io k ( l jum p + l f l ow + l rad ) the effecti v e length, (4) ∆ r = S rat io ( r f l ow + r rad ) the energy transfer , (5) and S rat io = S cavit y / S neck the cross-section surface ratio. Λ l can be related to the total effecti ve length of the cavity including all length corrections. Its zeros are the longitudinal angular y -wa venumbers k = ω / c of the cavity: The first zero corresponds to the Helmholtz base fre- quency; the second zero is the lar gest y -wav elength fitting into the cavity and so on. The total resistance ∆ r can be interpreted as amplitude modulation of acoustic wav es (sign dependent excitation or damping). T o determine the transmission function T of Eq. 3 , its impedance elements Z jum p , Z f l ow , Z rad must be specified first. The purely acoustical elements (no-flow) l jum p = L x − neck ln 2 L x − cavit y π L x − neck , r rad = S neck k 2 2 π , l rad = L x − neck π ln  8 L z − neck eL x − neck  (6) are adapted from Golliard [ 2 ]. The ke y point of this paper is to derive the flow related impedance Z f l ow = ( r f l ow + ikl f l ow ) Y neck fail-safe (s. Sect. 3.3 ), which is missing in the trans- mission function T (Eq. 4 and Eq. 5 ). 3.2 Model Source T erm of a T urbulent Boundary Layer The examined Helmholtz resonator is dri ven by a TBL. In this section, we select the source term Φ t o p of the new model, that most realistically models the natural broadband excita- tion generated by a TBL. First, we ev aluate the SPL frequency spectrum of the DNS data. Second, we adopt a validated generally usable model to the present case. An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 7 T o set up the source term Φ t o p ( ω ) of Eq. 2 we need to determine the power spectrum of pressure fluctuations below an undisturbed TBL ( y = 0). Because the TBL of the DNS de- scribed in Sect. 2 is disturbed by the presence of the cavity , in this section, we will ev aluate a second DNS without a ca vity but with the same conditions otherwise. Details of this undis- turbed TBL simulation are specified in [ 9 ]. As an intermediate step, the Φ t o p ( k x , ω ) po wer spectra of the DNS are calculated by a Discrete F ourier T ransformation (DFT) in space and time (W iener-Khinchin theorem, for details s. [ 9 ]). This more comprehensiv e k x − ω repre- sentation is needed later in Sect. 3.3 . As the spatial input of the DFT , two dif ferent top-hat windows ( L x − DF T = 1 . 5 δ 99 , neck , 7 . 5 δ 99 , neck ) in the streamwise direction of the TBL are se- lected (at the bottom w all y = 0). The DFT output is av eraged ov er all spanwise z locations. In this section (in contrast to Sect. 3.3 ) the k x − ω − spectra are averaged over all k x , where k x is the streamwise wav e vector . The two resulting narrowband SPL ( ω ) are displayed in Fig. 3 . The ev aluable DNS time window of 20 ms results in a narrowband frequency bin of f bin = 1 / T = 50 Hz. T o set up a generally applicable model source term, which is v alid for different kinds of TBL flows, we compared different spectral models to our DNS spectra. In conclusion, we found that the model of Goody [ 16 ] is the best fit for the present TBL. This is con- sistent with the revie w paper of [ 17 ]. Goody ’ s model features the typical Strouhal number scaling St T BL = ω δ 99 / u 0 , a prefactor ρ 2 w τ w 4 δ 99 / u 0 like the model by Chase [ 18 ], an cor- rect exponential gro wth with ω 2 at low frequencies, an exponential decay with ω − 5 at high frequencies, and a wall friction Reynolds number Re τ dependency: Φ T BL ( ω ) = ρ 2 w τ w 4 δ 99 u 0 C St T BL 2  St T BL 3 / 4 + 0 . 5  3 . 7 + n 1 . 1 St T BL ( Re τ u τ / u 0 ) − 0 . 57 o 7 . (7) The lower index “ w ” denotes quantities defined at the wall. In Fig. 3 , beside the DNS SPL, Eq. 7 is ev aluated and displayed, too. Goody’ s model agrees with the DNS within an accuracy of ± 2 dB for all frequencies above 300 Hz. Only the offset constant C of Eq. 7 was increased from 3 to 25 ( + 9 dB) to fit the DNS data. This is comparable to the increase 10 2 10 3 10 4 70 80 90 f [Hz] SPL [dB] Goody’ s SPL model (Eq. 7 ) DNS SPL with a DFT window of 1 L x − neck = 1 . 5 δ 99 , neck DNS SPL with a DFT window of 5 L x − neck = 7 . 5 δ 99 , neck Fig. 3: Pressure fluctuations of a undisturbed TBL at the wall ( k x -av eraged, y = 0, z - av eraged). Goody’ s Model is contrasted with the DNS results for dif ferent streamwise DFT windows. 8 Lewin Stein, Jörn Sesterhenn needed to match the Chase model in [ 2 ]. Hence, the value of C is likely to be v alid in other similar cases, too. The first four frequency bins are overestimated by the DFT , due to the noise lev els of the short DNS time series av ailable (20 ms, s. Sect. 2 ). Ideally , to calculate a univ ersal TBL spectrum, the streamwise input signal length L x − DF T DFT is as long as possible. Howe ver , by comparing the two different L x − DF T lengths in Fig. 3 , no substantial differences occur . The wav enumber filtering caused by the spatial confinement (i.e. finite neck opening L x − neck ) is ne gligible. Therefore, the TBL spectrum by Goody’ s model can be utilized directly as a source term of the Helmholtz resonator model Φ t o p = Φ T BL , which acts on the localized surface of the neck only . Goody’ s model is generally usable without DNS because it is only dependent on univ ersal TBL parameters ( δ 99 , u 0 , Re τ , . . . ). 3.3 Flow-Acoustic Interaction of an Opening with a V ortex Sheet Redefined This section elaborates on the central impedance element Z f l ow of the new Helmholtz res- onator model: the incorporation of the interaction between the shearing flo w around the cavity opening and the acoustic waves resonating inside the cavity . So far the source term Φ t o p of Eq. 2 describes the outer undisturbed TBL source, only . In the next Sect. 3.3.1 , Z f l ow = ( r f l ow + ikl f l ow ) Y neck of the transmission function T (Eq. 3 ) is linked in a new way to Howe ’ s theory [ 19 ]. Finally , Sect. 3.3.2 completely defines Z f l ow . The nov elty of the new definition lies in a more general description. Later , in Sect. 4.2 , a physical reinterpretation of the ne w definition is giv en. 3.3.1 Con version of Howe’ s Rayleigh Conductivity into an Acoustic Impedance Element T o legitimate the use of Howe’ s extensi ve theoretical work on the Rayleigh conductivity of apertures in turbulent flow we ev aluate the DNS data around the neck opening first. The incoming boundary layer thickness is smaller than the streamwise neck length. As a result, an unstable shear layer with K elvin-Helmholtz waves arises inside the neck zone. In Fig. 4 (b) vorticity isosurfaces with ∇ × u = ± 3000 Hz are depicted. A secondary vorte x sheet cov ering all the opening surface at constant height y = 0 can be visually distinguished from the dominant streaks of the incoming TBL. For a better understanding, this vortex sheet is colored red and blue. This secondary vorte x sheet leads to a reattachment of the TBL on the wall for a small stretch do wnstream of the neck: In Fig. 4 (b) for about one L x − neck down- stream of the neck length no (white) spacing between the lowest TBL whirls and the plate is visible. Inside the cavity , whirls are only present in the immediate vicinity of the opening. The size ratio of the small neck and the large cavity is apparent in Fig. 4 (a). The same is true for a snapshot at another time or for a dif ferent v orticity frequenc y than 3000 Hz. Inside the cavity , except the neck region, we observe nearly zero flow and acoustical phenomena dominate. Based on these observations, the following assumptions for the Helmholtz resonator model are reasonable: Acoustic wav es can propagate inside the cavity without an y fluid related effects. The crucial interaction of the acoustic waves with the turbulent flow is re- stricted to the vorte x sheet of the opening. The vorte x sheet is very thin compared to the acoustic wa velengths of interest (ca vity size and larger). Under these assumptions, Howe ’ s model of acoustic w aves tunneling perpendicular through an opening with an infinitely thin vortex sheet (caused by grazing flow) can be used. Howe models the local y -displacement ζ ( x , z ) of the vorte x sheet dependent on the frequency ω of the tunneling acoustic wav e, the xz -shape of the neck (not dependent on An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 9 (a) x y cavity plate (b) 190 200 230 220 210 10 0 -10 x [mm] y [mm] Fig. 4: DNS Simulation of the reference case (Sect. 2 ): (a) calculation domain in exact proportions with hard walls (black lines) and nonreflecting boundary conditions (red dashed line), (b) snapshot of two vorticity isosurfaces with ± 3000 Hz (blue and red). In order to highlight the vortex sheet at y ≈ 0, everything else is grayed out. (b) shows a section near the neck only . Its trim area is indicated by the dashed blue box of (a). L y − neck ), and two characteristic flow velocities u ± abov e and belo w the thin vortex sheet. Howe [ 19 , Sec. 5.3.1] demonstrates, that his Rayleigh conducti vity definition depends on the integrated v ortex sheet displacement K R i ω ρ ≡ v p + − p − = π L x − neck 2 Z S neck ζ ( x , z ) d x d z , (8) being p ± the acoustic pressure abov e and below the opening, and v = v + = v − the total acoustic volume flux through the opening. K R includes no-flow effects (due to the narro wing of the neck), too. In contrast to the present DNS case, Ho we considers an opening inside an infinite e xtended and infinite thin w all. Considering the left part of Eq. 8 as equi v alent to the transfer matrix M neck ( Z neck = ( r neck + ikl neck ) Y neck ) of Eq. 1 , we deriv e the relation e l neck = − ℜ  S neck K R  , and e r neck = k ℑ  S neck K R  . (9) Positiv e reactance e r neck > 0 implies acoustic damping, which is equiv alent to ℑ [ K R ] < 0. The tilde of e l neck and e r neck is needed, to denote Howe’ s different geometry . Since only the impedance Z f l ow of the v ortex sheet tunneling process itself is required to complete the ne w Helmholtz resonator model, we simply subtract the dispensable no-flo w part from both sides of Eq. 9 . In that way , effects of Ho we’ s infinite extended wall are excluded. Z f l ow becomes l f l ow = ℜ h S neck K R 0 − K R i and r f l ow = k ℑ  S neck K R − K R 0  , (10) being K R 0 = lim u 0 → 0 K R the Rayleigh conductivity in the case of an opening without flow nor vorte x sheet. 10 Lewin Stein, Jörn Sesterhenn T o determine the Rayleigh conductivity K R , the vorte x sheet displacement ζ ( x , z ) of Eq. 8 needs to be computed. For arbitrary xz -shapes of the opening, the vortex sheet dis- placement ζ ( x , z ) can be determined by numerically solving the conditional equation Z S ζ ( x , x 0 ) x − x 0 d x 0 + λ 1 ( z ) e i σ 1 x + λ 2 ( z ) e i σ 2 x = 1 (11) for all locations x = ( x , z ) T ∈ S neck of the opening surface [ 20 ]. σ 1 , 2 are Kelvin-Helmholtz wa venumbers dependent on the Strouhal number S t neck of the vorte x sheet: σ 1 , 2 = St neck 2 1 ± i 1 ± i β , St neck = ω L x − neck u + , β = u − u + . (12) β = 1 implies a real-valued wavenumber σ 1 , 2 scaling with St neck . Only in case β 6 = 1 the wa venumber becomes comple x, i.e. the K elvin-Helmholtz wa ves e i σ x are amplified. In case of simple geometries like a rectangular neck with L z − neck  L x − neck as in the present case (cf. Fig. 1 ), ζ ( x , z ) can be analytically determined [ 7 ]. The solution K R = π 2 L z − neck / { F ( σ 1 , σ 2 ) + Ψ } , (13) F ( σ 1 , σ 2 ) = { J 0 ( σ 1 ) f 1 ( σ 2 ) − J 0 ( σ 2 ) f 1 ( σ 1 ) } / { f 0 ( σ 2 ) f 1 ( σ 1 ) − f 0 ( σ 1 ) f 1 ( σ 2 ) } , Ψ = ln ( 2 ) − π 2 ( a − + a + ) will be utilized to ev aluate Z f l ow of the Helmholtz resonator model for the present setup. f 1 ( σ ) = σ { J 0 ( σ ) − 2 f 0 ( σ ) } , f 0 ( σ ) = i σ { J 0 ( σ ) − iJ 1 ( σ ) } are shorthands of combined Bessel functions J 0 , 1 . The spatial dimensions are expressed as a ± = ln ( eL x − neck / 4 L z − neck ) / π . W ithout flo w , the function lim u + → 0 F = lim St neck → ∞ F = − 2 becomes real-valued and there is no energy e xchange of the acoustics with the vorte x sheet: ℑ [ S / K R 0 ] = 0. By now we derived all lumped impedance terms ( r ’ s and l ’ s) of the transmission function T (Eq. 3 ) for the new Helmholtz resonator model (Eq. 2 ). W ith Eq. 10 we formally linked the impedance element Z f l ow = ( r f l ow + ik l f l ow ) Y neck to Howe’ s K R . Howe ver , both the analytical (Eq. 13 ) and the numerical solution (Eq. 11 ) of K R are dependent on two unknown constants: the characteristic flow velocities u ± (s. σ 1 , 2 of Eq. 11 ). Most users fit u ± to match their particular case only . T o solve this outstanding issue of Howe’ s theory , we propose a general usable u ± definition in the next Sect. 3.3.2 . 3.3.2 Con vection V elocities in Howe’ s Theory: A Refined, Unique Definition Below , we deri ve a nov el, unique definition of the parameters u ± of Eq. 12 , which are gov erning Howe’ s Rayleigh conductivity of an opening, i.e. the impedance r f l ow and l f l ow (Eq. 10 ). Though the theory of Ho we is widely-used, u ± are unclearly defined so far . Some- times u ± are referred to as a mean velocity [ 19 ], sometimes interpreted as turbulent con- vection velocity [ 21 ]. This section begins with a definition of the general con vection ve- locity u c (Eq. 14 ). u c varies depending on the spatial position and frequency . Then, the four-dimensional field u c is spectrally and spatially av eraged to deduce the single parameter u + (Eq. 15 ), which best represents the neck vorte x sheet v elocity , within Howe’ s theory . β is set as a constant (Eq. 16 ). In doing so, u + and β become uni versally applicable quantities and gain a physical meaning. An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 11 General Con vection V elocity u u u c c c The turbulent conv ection velocity is an ill-defined property , which describes the speed of vortices in general. Depending on the application, many definitions exist [ 22 , 23 ], both in real and spectral space. The definitions range from broadband or group velocities to narrowband or phase v elocities. Sometimes the definitions depend on spatial (wa velength) and temporal (frequency) length-scales, sometimes a verages are taken into account. A common, robust definition of the narro w band con vection velocity u c is the con vecti v e ridge maximum of the SPL k x − ω − spectrum [ 24 ]: u c ( y , z , ω ) ≡ ω k max ( y , z , ω ) with 0 ≡ ∂ ∂ k x SPL ( k x , y , z , ω )    k x = k max . (14) Below , we will demonstrate how to apply Eq. 14 on the k x − ω − spectrum pro vided by our DNS of the undisturbed TBL (s. previous Sect. 3.2 ). Subsequently , we will calculate the general conv ection velocity in the more complex case of a TBL with a cavity , which is needed to deriv e a ne w definition for u + . In Fig. 5 the full DNS spectrum of the undisturbed TBL is shown. This more detailed k x − ω − representation separates acoustical and fluid-related (so-called conv ectiv e) contri- butions. All acoustic fluctuations are located around the frequency axis between the phase velocities u 0 ± c , while most fluid related pressure fluctuations are centered around the con- vecti ve ridge with a phase velocity below the free stream velocity u 0 . Inside the conv ectiv e ridge, most ener gy is stored, which is also reflected by the highest sound pressure levels. The maxima of the con vecti ve ridge, which define the general con vection velocity u c (Eq. 14 ), u 0 + c u 0 − c u 0 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 f δ 99 , neck / u 0 k x [1/mm] 0 20 40 60 80 100 SPL [dB] Fig. 5: SPL k x − ω − spectrum of a turbulent flat-plate flow at y = 0 ( z -averaged, stream- wise window of 13 δ 99 , neck ). The characteristic phase velocities u 0 and u 0 ± c are marked by dashed black lines. The green and blue bullets denote the conv ecti ve ridge. Only the blue bullets are utilized later to define u + (Eq. 15 ). 12 Lewin Stein, Jörn Sesterhenn 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 0 . 2 0 . 4 0 . 6 0 . 8 f δ 99 / u 0 h u c i z / u 0 Fig. 6: Conv ection velocity of a turbulent flat-plate flow calculated by Eq. 14 ( z -averaged, streamwise window of 13 δ 99 , neck (8 L x − neck ), y = 0, same case as Fig. 5 ). Above the typical maximal curvature at ω δ ∗ / u 0 ≈ 0 . 3 the conv ection velocity is approximately constant. Be- low , u + is defined as av erage from ω δ ∗ / u 0 = 0 . 3 to ω δ ∗ / u 0 = 3 (denoted by blue bullets). are marked by blue and green bullets. In the interest of a clearer presentation, these max- ima are replotted in Fig. 6 . At around ω δ ∗ / u 0 ≈ f δ 99 / u 0 ≈ 0 . 3 the conv ection velocity ov er the frequency exhibits a typical maximal curvature in accordance with V iazzo et al. [ 25 ], Gloerfelt and Berland [ 26 ] and Hu et al. [ 27 ], where δ ∗ is the displacement thickness. Abov e this maximal curvature, the conv ection v elocity remains approximately constant. In the follo wing subsection, we will use this characteristic z -av eraged shape of u c to define u + . T o accurately determine the maxima the calculation of a narro w band k x − ω − spectrum requires a resolution which fully resolves the location of the conv ectiv e ridge in time and space (like our DNS [ 9 ] or the experiment by Arguillat et al. [ 28 ]). Measurements which are based on only a few microphones are incorrect, due to the known frequency and scale dependencies (i.e. streamwise probe separation) of the con vection v elocity [ 26 , 29 ]. Integral definitions of the con vection velocity lik e in Alamo and Jiménez [ 23 ] are not recommended in the present case of a distorted TBL since the con vecti ve ridge does not decay rapidly at high | k x | or ω . This implies that the integral definition depends on the integration limits (of k x or ω ), which in turn depend on the data sampling rate. In case a cavity is present below the general conv ection velocity u c can be equally calcu- lated. Ho we ver , the acoustic range (defined between u 0 + c and u 0 − c ) should be excluded from the determination of the maximum k max (Eq. 14 ), because localized SPL peaks of cav- ity modes can exceed the SPL of the con vectiv e ridge. u u u + Definition Based on the characteristic frequency dependence of the general con- vection velocity h u c i z , the first step to spectrally define u + is to av erage h u c i Sr , z between ω δ ∗ / u 0 = 0 . 3 and ω δ ∗ / u 0 = 3 (denoted by blue bullets). Just the upper bound is arbi- trary . At least a shift of the upper bound has a marginal influence on the con vection veloc- ity because h u c i z is roughly constant around S r = 3 (s. Fig. 6 ) and decays only slowly at high frequencies far beyond S r = 3. Beside this spectral determination of the con vection An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 13 velocity , h u c i Sr will be also spatially confined to define u + . T ypically , larger TBL streaks occur further away from the no-slip wall, in faster moving fluid regions. Consequently , the con vection velocity increases with the streamwise length of the input signal L x − DF T . An asymptotic value of the frequency and z -averaged con vection velocity h u c i Sr , z / u 0 ≈ 0 . 7 is reached as soon as the largest streaks of the TBL are captured by the streamwise signal win- dow . For the definition of u + , only the structures which fit into the neck are of interest. In the case of a TBL with a flush-mounted cavity , the natural window length to calculate the SPL k x − ω − spectrum is the neck dimension. Only inside the neck surface S neck the vortex sheet is modeled by Howe (Eq. 11 ). Hence, the streamwise DFT window is set equal to the streamwise neck length: L x − DF T = L x − neck . In the following, the notation h u c i implies an av erage ov er ω δ ∗ / u 0 ∈ [ 0 . 3 , 3 . 0 ] and over S neck . So far , the con vection v elocity h u c i , which is a veraged o ver x , z , and ω , still depends on y . In the following, this y dependency of h u c i is examined, before the characteristic height y + is selected, which ultimately defines u + = h u c i ( y = y + ) . For this purpose, the DNS data of our reference case (s. Sect. 2 ) is further ev aluated. In Fig. 7 h u c i ( y ) is contrasted with the mean velocity h u x i ( y ) . For comparison, the same quantities of a plain TBL without a cavity are charted, too. The mean velocity h u x i of the shear layer at the opening (cf. Fig. 4 ) is greater than the plain TBL profile near the wall. Only for y > 0 . 1 δ 99 , neck both cases, with and without mounted cavity , coincide (consistent with the analytical V an-Driest profile [ 9 ]). The conv ection v elocity h u c i increases with increasing wall distance, as noted before. Remarkably , h u c i ev en exceeds h u x i near the wall in the viscous sublayer . A possible explanation is that u c is a phase velocity (s. Eq. 14 ) and not a group velocity . In the study of [ 23 ], the same beha vior appears. By comparing the case with an opening to the undisturbed TBL, we find an ev en lower h u c i , despite a higher h u x i . W e − 0 . 2 − 0 . 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 0 . 2 0 . 4 0 . 6 0 . 8 y / δ 99 V elocity / u 0 h u c i / u 0 TBL with a cavity underneath h u c i / u 0 TBL of a flat-plate h u x i / u 0 TBL with a cavity underneath h u x i / u 0 TBL of a flat-plate h u x i Difference Fig. 7: V elocity profiles plotted over y of a plane TBL and a TBL with a cavity underneath. The notation h • i denotes an average ov er ω δ ∗ / u 0 ∈ [ 0 . 3 , 3 . 0 ] and ov er x , z ∈ S neck . The neck location is indicated by the v ertical black lines between the ca vity ceiling at y = − 1mm and the flat-plate surface at y = 0mm. 14 Lewin Stein, Jörn Sesterhenn − 0 . 16 − 0 . 12 − 0 . 08 − 0 . 04 0 0 . 04 0 . 08 0 . 12 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 y / δ 99 h u x i / u 0 Fig. 8: Indication of the constancy of the inflection point of the mean shear layer profile for different x positions, starting with the leading edge of the neck at 197mm (black line) up to 209mm (yello w line). All profiles are z -a veraged. Again the neck is aligned y -wise between y = − 1mm and y = 0mm. The first inflection point (black bullets) is located at the wall y = 0 (vertical black lines). The second inflection point (blue b ullets) is located at y + ≈ 6. can assume that the ne wly formed eddies of the upstream edge hav e to be accelerated first so that the av erage (eddy) con vection v elocity drops. In order to identify a characteristic height y + , we sho w in Fig. 8 a whole series of bound- ary layer profiles ( z and time-a veraged) at dif ferent streamwise neck positions, starting with the upstream edge. The colormap varies from black to yellow with increasing x . Beautifully , the first two inflection points of the profile have a constant height y for all streamwise x positions within the accurac y of the DNS mesh with d x + = 1 . 5. An upper " + " indicates δ ν normalization. A third top inflection point scattering around y / δ 99 = 0 . 08 is not marked to keep the Fig. 8 clear . The curvature of the two outer inflection points (first and third) meets the Fjørtoft’ s criterion necessary for a shear layer instability [ 30 ] in contrast to the second inner inflection point. T o conclude the determination of u + , we select the second central inflection point (located between the two others), as the characteristic location of the neck shear layer y + = y central inflection pt. : u + ≡ h u c i Sr , S neck ( y central inflection pt. ) , (15) being u c defined by Eq. 14 and a veraged over S neck and ω δ ∗ / u 0 ∈ [ 0 . 3 , 3 . 0 ] . In the present case the point of inflection is y + + = 6 ± 1 . 5 ( y + = 0 . 31 ± 0 . 08mm), which results according to Fig. 7 in u + / u 0 = 0 . 39 ± 0 . 03. u + directly scales with u 0 here. This u 0 dependency is in agreement with experiments carried out at various v elocities u 0 ∈ [ 5m / s , 47m / s ] (see the following Fig. 9 , Fig. 11 and Fig. 12 in Sect. 4 ). Thus, we set up a rob ust u + definition. β β β Definition As the most obvious approach, corresponding to the common original interpretation of u ± , we set β = u − / u + to be proportional to the mean v elocities h u x i below An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 15 and abov e the opening. Though, we will clarify in the following, that this choice is incapable to describe a realistic opening with a vortex sheet. Subsequently , an alternative definition is suggested. In the present case of one-sided flow , the obvious (but discarded) approach results in the hypothetical proportional relation β = h u x i ( y = δ ∗ − ) / h u x i ( y = δ ∗ + ) ∝ 1 / u 0 , where δ ∗ ± represent a measure of the shear layer thickness abov e and below the opening. Since δ ∗ ± typically grows linearly in streamwise x direction [ 31 , Sect. 5.4.2], β is roughly x independent. δ ∗ + is approximated as the location, where the dif ference between the shear layer profile and the plane TBL profile falls until 0 . 1 u 0 . Furthermore, we set δ ∗ − = δ ∗ + . In case of the DNS data with u 0 = 38 . 5 m / s we get β = 8 / u 0 = 0 . 21. Howe ver , this u 0 dependency is not reflected by experiments. As an alternativ e, the best fit of Z f l ow (see the following Sect. 4.1 ) and the complete Helmholtz resonator model (Sect. 4.4 ) with Golliard’ s experiments is achie ved with a constant β ≡ 0 . 21 (16) relation, which is valid for dif ferent u 0 ∈ [ 5m / s , 47m / s ] (s. Fig. 11 and Fig. 12 ). This def- inition of β remains to be further inv estigated (more comments follow in Sect. 4.1 and Sect. 4.2 ). 4 Results: Interpretation, V alidation, and V alidity Range of the Model In this section, the Helmholtz resonator model with its ne w parameters is v alidated by com- parison with an experiment and our DNS: First, the comparison is started with the most crucial model element Z f l ow ( u + , β ) , separately (Sect. 4.1 ). Second, a ne w ph ysical meaning of the governing model parameters u + and β is revealed (Sect. 4.2 ). Third, the complete Helmholtz resonator model is compared with our DNS results (Sect. 4.3 ). F ourth, the v alid- ity range of the new model is demonstrated (Sect. 4.4 ). 4.1 V alidation of the Flow-Acoustic Interaction Impedance Z f l ow ( u + , β ) Fig. 9 shows the impedance Z f l ow / Y neck = r f l ow + ik l f l ow modeled by Eq. 10 and measure- ments by [ 2 ]. Thereby the experimental curves are a veraged over different conditions such as M ∈ [ 0 . 11 , 0 . 12 ] and ratios of δ 99 , neck / L x − neck ∈ [ 0 . 7 , 5 ] . Even though Ho we ’ s theoretical model assumes an infinite thin v ortex sheet, the e xperimental results are in accordance with the model predictions: especially the frequency scaling i.e. the extrema positions and the zero-crossings match. At the zero-flo w limit ( S t neck at infinity) the impedance becomes zero, as expected: lim St neck → ∞ Z f l ow = 0. The sign of r f l ow determines the direction of the energy transfer between the K elvin-Helmholtz wa ves of the neck and the passing sound waves. The acoustic damping at lo w Strouhal numbers S t neck < 3 . 3, as well as the acoustic e xcitation be- tween St neck ∈ [ 3 . 3 , 5 . 9 ] , is typical for all kind of neck geometries like circles, rectangles or triangles [ 20 ]. The coupling of Kelvin-Helmholtz wa ves and acoustic wa ves can be vie wed similarly to Rossiter ’ s feedback mechanism. The crucial difference is that Rossiter’ s self- generated sound waves propagate upstream in the vorte x alley , while in the present model (Sect. 3.3.1 ) external plane wa ves transv ersally pass through the vorte x sheet. The impact of u + and β on Z f l ow is as follows: u + determines the main frequency scaling, which relates the Strouhal number St neck and the frequency f (s. Eq. 12 ). If u + is modified, the entire Fig. 9 remains unchanged with the sole exception of the upper frequency axis. Such Z f l ow plotted over St neck (as in Fig. 9 ) allows a uni versal representation. β tunes 16 Lewin Stein, Jörn Sesterhenn 0 2 4 6 8 10 12 14 16 − 2 − 1 0 1 2 Con vecti ve Strouhal number St neck ( u + ) Scaled resistance & reactance Theoretical r f l ow / M (Eq. 10 ) Experimental r f l ow / M [ 2 ] Theoretical l f l ow / L x − neck (Eq. 10 ) Experimental l f l ow / L x − neck [ 2 ] 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2 2 . 2 2 . 4 2 . 6 2 . 8 3 f [kHz] Fig. 9: V alidation of Ho we’ s impedance model Eq. 10 vs. e xperiment for u + / u 0 = 0 . 39 and β = 0 . 21 (s. Sect. 3.3.2 ). The resistance r f l ow and the reactance l f l ow are rescaled with the Mach number M and the streamwise neck length L x − neck , respectiv ely . the amplitude of the complex-v alued impedance Z f l ow depending on the frequency . If β is increased from zero to one, constantly new positiv e and negativ e bulges of r f l ow and l f l ow arise at higher and higher frequencies [cf. 19 , Fig. 5.3.8]. In doing so, the new bulges have a slowly , shrinking amplitude with higher frequency , while the amplitude and position of the low-frequenc y bulges remain almost the same. Here, our definition of β = 0 . 21 by Eq. 16 ov erestimates low-frequency and underestimates high-frequency amplitudes, as shown in Fig. 9 . This discrepanc y is an indication that the single parameter β of Ho we’ s theory is not sufficient to accurately describe lo w and high-frequenc y amplitudes simultaneously . 4.2 A New V ie w on the Model Parameters u + and β In conclusion of Sect. 3.3 and Sect. 4.1 , a reinterpretation of u + and β suggests itself. W e propose to consider u + as the central vorte x sheet v elocity , i.e. the con vection v elocity at the central inflection point of the neck shear layer . Hence u + is a central quantity of the vorte x sheet, rather than a velocity located abov e the vorte x sheet as suggested by the lower “ + ” index. In the follo wing, the plus sign is kept to remind of its origin within Ho we’ s theory . Instead of understanding β only as a velocity ratio u − / u + (Eq. 12 ), β should be more univ ersally regarded as a measure of the shear layer growth within the opening. Other than the mean velocity difference below and above the vorte x sheet h u x i ± (original interpreta- tion), the gro wth of the shear layer thickness is likely to be affected by the opening thickness L y − neck and the relativ e, incoming boundary layer thickness δ 99 / L x − neck . The motiv ation to regard β as a general measure of the shear layer growth is as follows: In the limit β = 1, Z f l ow (s. Fig. 9 ) has contributions (bulges) over the whole frequency range (and a purely An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 17 imaginary-valued K elvin-Helmholtz wave exponent of Eq. 11 ). If high frequencies have a larger influence, small structures are more important, which indicates a thin vorte x sheet and a minimal shear layer gro wth. By contrast, in the limit β = 0, Z f l ow has only lo w-frequency components, lar ge structures are dominant, which requires a rapid shear layer gro wth within the streamwise opening length. Howe [ 19 , Sect. 5.3.6] himself argues that the conducti vity of an opening in two-sided flo w , in practice, is actually similar to the case of one-sided flow , because of the finite L y − neck thickness (not infinitely thin as theoretically assumed). In other words, the recirculation area (caused by thicker plate) at the upstream side of the opening enhances the shear layer gro wth and therefore can be represented by an effecti vely smaller β . T o sum up, β is (at least) a function of h u x i ± , L y − neck , and δ 99 / L x − neck . It remains to prov e and explicitly quantify these predictions about the qualitati ve beha vior of β . 4.3 V alidation of the Helmholtz Resonator Model The application of the complete Helmholtz resonator model (Eq. 2 with Eq. 3 and Eq. 7 ) is presented in Fig. 10 . Here the SPL narrowband spectrum at the bottom of the cavity is shown. At the bottom of a cavity unsteady , acoustical pressure fluctuations dominate (cf. Fig. 4 ). This has the advantage that without a Helmholtz decomposition the po wer spectral density Φ bot of the pressure is directly related to the SPL. As a comparison, the xz a veraged SPL of the DNS and the experimental SPL of Golliard [ 2 , cavity C1, TBL B, L x − neck = 14mm, δ 99 = 10 . 9mm] are plotted. Also, the model predictions of an alternati ve empirical model are taken from Golliard [ 2 , Sect. 5.3.1]. The predicted extrema frequencies of the Helmholtz resonator model match with the experimental and the DNS results. Zeros of the 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 70 80 90 100 110 120 f [kHz] SPL [dB] Direct Numerical Simulation [ 9 ] Golliard’ s experiment [ 2 ] Golliard’ s empirical model [ 2 ] New Helmholtz resonator model (Eq. 2 ) Zeros of the ener gy transfer term ∆ r (Eq. 5 ) Zeros of the length term Λ l (Eq. 4 ) Fig. 10: Narro w-band SPL spectrum ( f bin = 50 Hz) centered at the bottom of the Helmholtz cavity (setup s. Fig. 1 ). For comparison, the measured SPL and the empirical model of [ 2 ] are sho wn. x - and z -cavity modes, which are not part of the model, begin at 1700 Hz. This explains the de viation starting near this frequency . 18 Lewin Stein, Jörn Sesterhenn length term Λ l (Eq. 4 ) are denoted by violet circles: The first SPL peak of Fig. 10 at 300 Hz is the Helmholtz base frequency . The second peak at 1200 Hz is the first vertical y -cavity mode. Below 1500 Hz the SPL of the new model, the experiment, and the DNS de viates by ± 7dB. x - or z -cavity modes begin at 1700 Hz ( c / 2 L x − cavit y , where c is the speed of sound). Hence, the negati ve deviation of the model at 1700 Hz is expected. By virtue of simplicity , these higher transverse x - or z -modes are not included in the model because they are be yond the typical range of operation. Zeros of the ener gy transfer term ∆ r (Eq. 5 ) are denoted by red stars. Here the interchange between acoustical and fluid energy is balanced ov er the time av erage. Since the ratio S rat io = S neck / S cavit y = 0 . 14 of Eq. 3 is small, they hardly contribute to the overall SPL at the cavity bottom. Only in the DNS results, small local increases are visible at zero resistance (red stars in Fig. 10 ). Ho wev er , this might be up to the noise due to the limited time series av ailable of 20ms (restriction of DNS resources). Golliard’ s empirical model is only ev aluated within its frequencies validity range. Thereby , the frequency prediction of the first vertical y -cavity mode is in agreement with the other curves, ho we ver , a constant ne gativ e of fset of about − 10dB appears. 4.4 V alidity Range of the Helmholtz Resonator Model During the deriv ation of the model, the focus is on universal definitions of the go verning pa- rameters. Ho wev er , the reference data is provided by one single DNS run with u 0 = 38 . 5m / s (Sect. 2 ). T o demonstrate its uni versal validity , the model is ev aluated for the whole veloc- ity range from 5m / s to 47m / s ( M ∈ [ 0 . 01 , 0 . 14 ] ) measured by Golliard [ 2 ]. In Fig. 11 and Fig. 12 the SPL model spectra can be compared with the experimental measurements, re- spectiv ely . T o a great extent, both figures are in agreement at all velocities. The zeros of the effective length Λ l (Eq. 4 , dotted black mainly vertical lines) are pre- dominantly geometry dependent so that the position of the Helmholtz base frequency (at 304 Hz) and the first vertical y -ca vity mode (at 1154 Hz) are almost velocity independent. In contrast, the zeros of the ener gy tr ansfer ∆ r (Eq. 5 , dashed white diagonal lines), which cor- respond to a balanced energy interchange between the acoustical and the K elvin-Helmholtz wa ves, scale as expected with the free stream velocity u 0 . The overlap of both resonance conditions leads to a strong nonlinear interaction of the vortex-sheet with the ca vity modes. A qualitativ e similar behavior of these resonance conditions is also observed for example by Y ang’ s [ 32 ] experimental study of deep, open ca vities (without a neck). The ne w Helmholtz resonator model successfully predicts where the occurring physical phenomena are coupled: the Helmholtz base frequency strongly couples with the Kelvin-Helmholtz wa ves of the neck between 5 m / s and 23 m / s, resulting in a SPL beyond 120dB (s. Fig. 11 ). It is known that such high sound pressure levels cause a scattering to higher harmonics [ 33 ]. Experimentally these higher harmonics are visible in Fig. 12 at 330, 660, 990 and 1320 Hz. The model predicts the initial resonance condition at ≈ 300 Hz and u 0 ∈ [ 5 m / s , 23 m / s ] , but no scat- tered higher harmonics (nonlinear effect). The right prediction of the resonance conditions by Λ δ ( u + , β ) and ∆ r ( u + , β ) for the frequencies f ∈ [ 0 , 1500 Hz ] and for the Mach numbers M ∈ [ 0 . 01 , 0 . 14 ] verifies the new u + (Eq. 15 ) and β (Eq. 16 ) definitions, within these ranges. An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 19 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 10 20 30 40 50 f [kHz] u 0 [m / s] 50 60 70 80 90 100 SPL [dB] Fig. 11: SPL spectrum ( f bin = 4 . 3 Hz) predicted by the new Helmholtz resonator model Eq. 2 for the cavity setup of Fig. 1 . The dotted black vertical lines mark the resonance conditions i.e. the length term zeros Λ l = 0 (Eq. 4 ). The white lines mark r f l ow + r rad = 0 and the dashed white lines mark | r f l ow + r rad | = 0 . 07. Consequently , both white lines are related to the energy transfer term Eq. 5 . The black horizontal double line corresponds to the DNS case with u 0 = 38 . 5m / s (s. Fig. 10 ). 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 10 20 30 40 50 f [kHz] u 0 [m / s] 50 60 70 80 90 100 SPL [dB] Fig. 12: Measured SPL spectrum extracted from [ 2 ] for the same conditions as in Fig. 11 (effecti v e f bin = 4 . 3 Hz, original f bin , ex p = 1 Hz). The dotted black and the white lines are a copy from Fig. 11 . Lines which are only present in Fig. 12 are artifacts caused by the data extraction. 20 Lewin Stein, Jörn Sesterhenn 5 Model Summary The Helmholtz resonator model as gi ven by Eq. 3 and Eq. 7 with its generalized parameters u + and β is ready for use. An industrial user can directly apply it, by evaluating Φ bot = T ( Z neck , S neck , S cavit y , k L y − cavit y ) Φ T BL ( ω , δ 99 , neck , u 0 , u τ , ρ w , ν ) . Thereby , the neck impedance is given in detail by Z neck = Z jum p ( S neck , S cavit y ) + Z f l ow ( L x − neck , St neck , β ) + Z rad ( L x − neck , L z − neck ) , where the individual impedance elements originate from Eq. 6 and Eq. 10 . The new u + is determined as follows: First, the k x − ω − spectrum is calculated exactly abov e the neck surface at a constant height of the central inflection point of the shear layer . Second, the con- vection velocity u c is deriv ed as the phase v elocity ω / k x of the pressure k x − ω − spectrum at the con vectiv e ridge (maximum). Third, u + is the mean of u c ( f ) for f δ 99 , neck / u 0 ∈ [ 0 . 3 , 3 . 0 ] and for all spanwise neck positions (Eq. 15 ). In the present case of one-sided flow , β is defined as constant 0 . 21 (Eq. 16 ). The model is validated for low frequencies belo w the cutof f of higher transverse cavity modes ( f ∈ [ 0 , 1500 Hz ] ) and for lo w Mach numbers from 0 . 01 to 0 . 14 (Sect. 4.4 ). These ranges comply with typical operating conditions of duct systems. The model is based on a modular design (Eq. 1 ), which guarantees a flexible use and e xpandability . 6 Conclusion Altogether , by combining and carrying on works by Goody , Howe, and Golliard we estab- lished a ne w model of a Helmholtz resonator in grazing flow . The deri v ation of the model is based on DNS data. Goody’ s model is selected as the most realistic acoustic source term of a TBL and adapted to our model (Sect. 3.2 ). The physical k ey point of this paper is the correct description of the turbulence-acoustic interaction. The understanding of the related energy interchange is vital to predicting acous- tical damping or excitation. T o this end, Howe’ s extensi ve theory of this interaction is put on a new ph ysical footing and incorporated in the new model. The main contrib ution is the novel definition and reinterpretation of the parameters u + and β , which are governing Howe’ s theory . So far , these parameters are empirically fitted depending on the particular application. Now u + and β are uniquely defined and directly related to physically significant quantities: u + is the central, temporal and spatial specified con vection velocity of the neck shear layer . 1 / β is a measure of the shear layer growth (Sect. 4.2 ). Because the turbulence-acoustic interaction is based on Howe’ s simplified impedance model (s. Sect. 3.3.1 ), some restriction apply: Ef fects caused by a modified w all thicknesses L y − neck or induced by an altered boundary layer thickness δ 99 require to set up an extended database and to re-e valuate u + , β by using their ne w uni v ersal definitions derived here (con- tinuation option of this work). By applying the new u + , β definitions to the DNS data and by comparison with ex- periments by other researchers it is sho wn that changes in the frequency or the free stream velocity (Sect. 4.4 ) have an appropriate impact on the final model predictions. Different free stream velocities u 0 , frequencies f and most geometry modifications 1 can be studied 1 Eq. 11 can be solved for any S neck ( x , z ) shape. An Acoustic Model of a Helmholtz Resonator under a Grazing T urbulent Boundary Layer 21 within the ranges f ∈ [ 0 , 1500 Hz ] , M ∈ [ 0 . 01 , 0 . 14 ] . Hence, the change of outer parameters is now traceably linked to governing parameters, which in turn directly update the model predictions. For instance, a lar ger free stream velocity u 0 increases the conv ection velocity u + . The updated Strouhal number St neck ( u + ) then newly determines at which frequencies acoustic energy is damped or excited (sign of ∆ r , Eq. 5 ) and restates the resonance condi- tion of the instability wa ves of the opening and the cavity modes depending on the frequency and the velocity u 0 (cf. Fig. 11 ). A priori, rather than by e xpensiv e tests, the sound spectrum can be directly tuned for frequencies of interest. Acknowledgements W e gratefully ackno wledge the financial support of the Deutsche Forschungsgemein- schaft (SE824/29-1) and the pro vision of computational resources by the High-Performance Computing Cen- ter Stuttgart (A CID11700). References 1. Kook, H., Mongeau, L.: Analysis of the Periodic Pressure Fluctuations Induced By Flow Ov er a Cavity. J. Sound V ib. 251 (5), 823–846 (2002) 2. Golliard, J.: Noise of Helmholtz-resonator like cavities excited by a low Mach-number turbulent flo w . Ph.D. thesis, University of Poitiers (2002) 3. Hémon, P ., Santi, F ., Amandolèse, X.: On the pressure oscillations inside a deep cavity excited by a grazing airflo w . Eur. J. Mech. - B/Fluids 23 (4), 617–632 (2004) 4. Dequand, S., Luo, X., W illems, J., Hirschberg, A.: Helmholtz-Like Resonator Self- Sustained Oscillations, Part 1: Acoustical Measurements and Analytical Models. AIAA J. 41 (3), 408–415 (2003) 5. Rossiter, J.: W ind-T unnel Experiments on the Flow ov er Rectangular Cavities at Sub- sonic and T ransonic Speeds. Aeronaut. Res. Counc. Reports Memo. 3438 (1964) 6. T am, C., Block, P .: On the tones and pressure oscillations induced by flow over rectan- gular cavities. J. Fluid Mech. 89 (02), 373 (1978) 7. Howe, M.S.: The influence of mean shear on unsteady aperture flow , with application to acoustical dif fraction and self-sustained cavity oscillations. J. Fluid Mech. 109 (-1), 125 (1981) 8. Howe, M.S.: On the Theory of Unsteady Shearing Flow ov er a Slot. Philos. T ransactions Royal Soc. A: Math. Phys. Eng. Sci. 303 (1475), 151–180 (1981) 9. Stein, L., Reiss, J., Sesterhenn, J.: Numerical Simulation of a Resonant Ca vity: Acous- tical Response Under Grazing T urbulent Flow. New Results Numer . Exp. Fluid Mech. XI 136 , 671–681 (2018) 10. T am, C., Ju, H., Jones, M., W atson, W ., Parrott, T .: A computational and experimental study of slit resonators. J. Sound V ib. 284 , 947–984 (2005) 11. T am, C.K., Ju, H., Jones, M., W atson, W ., Parrott, T .: A computational and e xperimental study of resonators in three dimensions. J. Sound V ib. 329 (24), 5164–5193 (2010) 12. Roche, J.M., Leylekian, L., Delattre, G., V uillot, F .: Aircraft Fan Noise Absorption: DNS of the Acoustic Dissipation of Resonant Liners. In: 30th AIAA Aeroacoustics Conference (2009) 13. Eldredge, J., Bodony , D., Shoe ybi, M.: Numerical In vestigation of the Acoustic Behav- ior of a Multi-Perforated Liner. In: 28th AIAA Aeroacoustics Conference. American Institute of Aeronautics and Astronautics (2007) 14. Roche, J., V uillot, F ., Leylekian, L.: Numerical and Experimental Study of Resonant Liners Aeroacoustic Absorption under Grazing Flow. AIAA Pap. 16 , 1–18 (2010) 22 Lewin Stein, Jörn Sesterhenn 15. Munjal, M.L.: Acoustics of Ducts and Mufflers with Application to Exhaust and V enti- lation System Design. W iley (1987) 16. Goody , M.: Empirical Spectral Model of Surf ace Pressure Fluctuations. AIAA J. 42 (9), 1788–1794 (2004) 17. Hwang, Y ., Bonness, W .K., Hambric, S.A.: Comparison of semi-empirical models for turbulent boundary layer wall pressure spectra. J. Sound V ib. 319 (1-2), 199–217 (2009) 18. Chase, D.: The character of the turbulent wall pressure spectrum at subcon vecti ve wa venumbers and a suggested comprehensiv e model. J. Sound V ib. 112 (1), 125–147 (1987) 19. Howe, M.S.: Acoustics of Fluid-Structure Interactions. Cambridge monographs on me- chanics (1998) 20. Grace, S.M., Horan, K.P ., Howe, M.S.: The Influence of Shape on the Rayleigh Con- ductivity of a W all Aperture in the Presence of Grazing Flow. J. Fluids Struct. 12 (3), 335–351 (1998) 21. Peat, K.S., Ih, J.G., Lee, S.H.: The acoustic impedance of a circular orifice in graz- ing mean flo w: Comparison with theory . The J. Acoust. Soc. Am. 114 (6), 3076–3086 (2003) 22. Blake, W .K.: T urb ulent boundary-layer wall-pressure fluctuations on smooth and rough walls. J. Fluid Mech. 44 (04), 637 (1970) 23. Alamo, J.C., Jiménez, J.: Estimation of turb ulent con v ection v elocities and corrections to T aylor’ s approximation. J. Fluid Mech. 640 , 5 (2009) 24. Goldschmidt, V .W ., Y oung, M.F ., Ott, E.S.: T urbulent con vectiv e velocities (broadband and wa venumber dependent) in a plane jet. J. Fluid Mech. 105 (-1), 327 (1981) 25. V iazzo, S., Dejoan, A., Schiestel, R.: Spectral features of the wall-pressure fluctuations in turbulent wall flows with and without perturbations using LES. Int. J. Heat Fluid Flow 22 (1), 39–52 (2001) 26. Gloerfelt, X., Berland, J.: T urbulent boundary-layer noise: Direct radiation at Mach number 0.5. J. Fluid Mech. 723 , 318–351 (2013) 27. Hu, N., Appel, C., Herr , M., Ewert, R., Reiche, N.: Numerical Study of W all Pressure Fluctuations for Zero and Non-Zero Pressure Gradient T urbulent Boundary Layers. In: 22nd AIAA/CEAS Aeroacoustics Conference (2016) 28. Arguillat, B., Ricot, D., Bailly , C., Robert, G.: Measured w av enumber: Frequency spec- trum associated with acoustic and aerodynamic w all pressure fluctuations. The J. Acoust. Soc. Am. 128 , 1647–1655 (2010) 29. Kim, J., Hussain, F .: Propagation velocity of perturbations in turbulent channel flow . Phys. Fluids A: Fluid Dyn. 5 (3), 695–706 (1993) 30. Schmid, P .J.: Stability and Transition in Shear Flo ws. Springer (2012) 31. Pope, S.: T urbulent Flo ws. Cambridge Univ ersity Press (2000) 32. Y ang, Y ., Rockwell, D., Lai-Fook Cody, K., Pollack, M.: Generation of tones due to flow past a deep cavity: Effect of streamwise length. J. Fluids Struct. 25 (2), 364–388 (2009) 33. Förner, K., T ournadre, J., Martinez-Lera, P ., Polifke, W .: Scattering to Higher Harmon- ics for Quarter W av e and Helmholtz Resonators. AIAA J. 55 (4), 1194–1204 (2017)