Experimental investigation of flow past open and partially covered cylindrical cavities

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DOTTORATO DI RICERCA IN INGEGNERIA MECCANICA E

INDUSTRIALE

CICLO XXIV

Experimental investigation of flow past open and partially covered

cylindrical cavities

Francisco Rodriguez Verdugo

Tutor:

Prof. Roberto Camussi

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Abstract

Flows past wall-mounted cavities are found in a wide variety of applications, including side-branches, organ pipes, automobile sunroofs, inter-car gaps in trains and aircraft bays. Under certain conditions, flow excited cavities can generate large pressure fluctuations, undesirable noise and significant structural loads. To date, most of the studies have been focused on rectangular cavities while little attention has been given to cylindrical cavities despite their widespread use.

Two different types of cylindrical cavities were experimentally investigated in low speed wind tunnels: an open mouth cavity and a deep cavity with a small rectangular opening. The measurements included hot wire anemometry, particle image velocimetry (PIV) and unsteady surface pressure measurements. Additionally, numerical analysis of the test section/cavity systems were carried out with the finite element program COMSOL Multiphysics and with a wave expansion method (WEM) code developed by the Trinity College Dublin.

Important flow features are described by evaluating the pressure measurements conducted in several positions over the walls of an open mouth cavity, the PIV measurements performed over horizontal planes inside the cavity and the hot-wire measurements on the shear layer and on the wake of the cavity.

Pressure Fourier spectra evidence the presence of the first three shear layer hydrodynamic modes at frequencies well predicted by classical formulation for rectangular cavities (Rossiter). When the cavity is open, the acoustic modes of the test section are found to be excited by the flow but when the cavity is partially covered, the shear layer hydrodynamic modes are more likely to lock on the natural frequencies of the cavity. The position of the opening has an influence on the lock-on acoustic modes.

The acoustic energy generated by the shear layer is calculated by applying the vortex sound theory of Howe: the flow velocity and the vorticity are extracted from the PIV data and the acoustic particle velocity field from the WEM calculation. The acoustic sources are localised in space and quantified over an acoustic period providing insight into the sound production of flow-excited partially covered cylindrical cavities.

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Acknowledgements

Vorrei ringraziare innanzitutto il prof. Roberto Camussi, per avermi accolto nel suo gruppo di ricerca. Questi quattro anni passati a Roma Tre, prima per la tesi magistrale e dopo per il dottorato, sono stati molto formativi, non solo dal punto di vista professionale ma anche dal punto di vista personale.

Un particolare ringraziamento va ai miei cari colleghi della sezione di termofluidodinamica e aerodinamica per i loro aiuti, i loro consigli, e per tutti i momenti passati insieme all’interno del dipartimento e al di fuori. In ordine alfabetico ringrazio: Giovanni Aloisio, Alessandro Di Marco, Dajana Giulieti, Emanuele Giulietti, Daniele Grassucci, Silvano Grizzi, Riccardo Moscatelli, Tiziano Pagliaroli, Alessandra Parlato e Francesco Tomassi.

Ringrazio a tutti gli stagisti e i tesisti che mi hanno circondato durante la mia attivit`a di ricerca. In particolare ringrazio in ordine cronologico a Chistophe Perge, Stefano Valerio, Alessandro Guerriero, Federico Gargano, Gabriele Baiocco e Ludovica Penten`e per il loro supporto tecnico in galleria del vento e ad Andrea Serrani per le sue simulazioni acustiche.

Non mancherei di ringraziare i dottorandi del GRACO per le pause pranzo, per i loro aiuti con LA_TEX,

per le cene, per i loro consigli durante la mia ricerca di lavoro, per i TrovaRoma, per l’Internazionale, per le giornate in montagna, per il calcetto e tant’altro. Loro sono: Alessandro Anobile, Paolo Gradassi, Eugenio Lombardi, Simone Menicucci, Emanuele Piccione.

This research project has been supported by a Marie Curie Early Stage Research Training Fellowship of the European Community’s Sixth Framework Programme under Contract number MEST CT 2005 020301. This financial support was greatly appreciated. I thank Prof. Aldo Rona for leading the AeroTraNet project, for bringing us a nice cavity model back in 2007 and also for the useful acoustic measurement equipment that he lent us. I would like to express my thanks to all the AeroTraNet fellows for the interesting discussions that we had during the meetings and the conferences. Many thanks to Marco Grottadaurea for the fructose collaboration.

Parmi les participants de l’AeroTraNet, je tiens également à citer Antoine Guitton et Julien Grilliat avec qui j’ai partagé de magnifiques moments à Roma Tre. Je souhaite leur exprimer mes sincères remerciements pour leurs précieux conseils et l’appui qu’ils ont pu me fournir.

I would like to thank my second advisor, Dr. Gareth Bennett for giving me the opportunity to spend thirteen month in the Trinity College. Dublin was, without doubts, a wonderful experience. Moreover I

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this work has been invaluable.

I also thank Dr. Craig Meskell and Prof. John Fitzpatrick for the interesting discussions about the physical phenomena studied. Special thanks go to Shane Finnegan whose experimental methodology has stimulated much of my research. I also want also to acknowledge my Fluids Lab’s colleagues Miguel Pedroche, Ian Davis and John Mahon. Furthermore I want to give a special thanks to Eoin King, Donal Lynch, Dorota Skupi˜nska, Emer Walsh and Rory Stoney with who I spent many late evenings in the ‘dungeon’. I will not forget to thank my lunch-break mates from the Mechanical and Manufacturing Engineering Department: Paul Ervine, Paul Harris, Karl Brown, Peadar Golden, Emma Brazel, Stuart Murphy, Kevin Kerrigan and Robert Smyth.

Many thanks to Prof. Marc Jacob for reading this thesis. I have greatly appreciated his meticulous corrections and suggestions.

Desde luego, mi más amplio y sincero agradecimiento a mi familia, cuyo apoyo incondicional hizo posible la culminación mi formación universitaria en Europa.

Ringrazio Sandro e Ivana che mi hanno generosamente accolto nella loro famiglia facendomi sentire a casa.

Vorrei infine ringraziare Ambra, per la sua pazienza e il suo amore e per essermi stata vicina in tutti questi anni.

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Introduction

Motivation for the research

Flows past wall-mounted cavities are found in a wide variety of applications, including side-branches, organ pipes, automobile sunroofs, inter-car gaps in trains and aircraft bays. Under certain conditions, flow excited cavities can generate large pressure fluctuations, undesirable noise and significant structural loads.

To date, most of the studies have been focused on rectangular cavities while little attention has been given to cylindrical cavities despite their widespread use. In the aerospace sector, cylindrical cavities are present as pressure relief valve of the fuel vents (figure 1), circular anti-icing vent holes or cylindrical landing gear wheel wells, just to cite some examples.

The purpose of this thesis was to extend the existing knowledge in this area through an experimental investigation. Two different cases were studied: an open mouth cavity and a partially covered cavity. The first cavity has an aspect ratio of 1.357 which was dictated by the need of reproducing typical geometries present on commercial aircraft. The second cavity (mouth partially covered) was designed in such a way that different acoustic resonances can be excited.

Frameworks

Part of the PhD work presented here was performed in the framework of the AeroTraNet project, an Early Stage research Training (EST) program funded by Marie-Curie Actions. The AeroTraNet project was launched in 2006 in order to study the unsteady flow in selected airframe cavities of a wide-body civil transport aircraft. This four-year European initiative brought together the University of Leicester, the Uni-versit`a degli Studi Roma Tre, the Politecnico di Torino and the Institut de M´ecanique des Fluides de Toulouse around a common research topic. This successful program brought to an extensive list of publications which can be found in the AeroTraNet official webpage: http://aerotranet.imft.fr.

The Universit`a degli Studi Roma Tre encourages PhD students to spend a period of time in a foreign research center. Part of the experimental results was therefore obtained during a thirteen months stay at the Trinity College of Dublin (TCD). The School of Engineering at Trinity was founded in 1841 and is one of the oldest Engineering Schools in the English-speaking world. The Department of Mechanical

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Figure 1: Photograph of a commercial aeroplane. Detail of circular pressure relief valve on the lower surface of the wing.

and Manufacturing Engineering has been conducting research for many years on modelling and analysis of flow/structure interactions and vibro-acoustic problems. The results achieved by the author was an invited research student at TCD were presented in four different international conferences (refer to list of publication on page 113).

Outline of the thesis

This thesis is organized as follows: the first chapter introduces the state of the art; the two main parts, each of which contains four chapters, treat the open mouth case and the partially closed mouth case; the final chapter summarises the main results. Hereafter the organization of the two main parts of the thesis is described: chapters 2 and 6 detail the experimental facilities and the measurement techniques; in chapters 3 and 7 the acoustic modes are calculated analytically and numerically; the experimental results are given in chapters 4 and 8; chapters 5 and 9 draw the conclusions of each part.

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Chapter 1

Background

This chapter is not intended to be an exhaustive literature survey on cavity flow studies: special emphasis is given to research on deep cylindrical cavities with open or partially covered mouths in low Mach number flows. For general surveys on cavity flow, see for instance Rockwell & Naudascher (1978, 1979). More recently, Cattafesta et al. (2003) and Rowley & Williams (2006) have reviewed the studies on cavity flow with a particular focus on the control of flow-induced resonance.

1.1 Open mouth cavities

1.1.1 Rectangular cavities

Early studies and oscillations classification

Anatol Roshko and Krishnamurty Karamcheti, both from California Institute of Technology, performed the first mayor studies on rectangular cavities in the 1950’s. Roshko (1955) reported the pressure distribution on the walls of cavities tested in a low speed wind tunnel. Karamcheti (1955, 1956) studied the acoustic radiation of a rectangular cavity in a transonic wind tunnel by means of schlieren observation, interferometry and hot-wire anemometry. These pioneer researchers are still largely cited nowadays and have inspired many studies on flow-excited cavities over the years.

Rockwell & Naudascher (1978) divided the shear layer driven cavity oscillations into three categories: fluid-elastic, fluid-dynamic, and fluid-resonant. The primary condition for fluid-elastic interactions to exist is the structural elasticity of the cavity walls. Fluid-dynamic oscillations are triggered by the interaction between the upstream edge of the cavity and the pressure wave generated on the cavity’s downstream edge by the impingement of the shear layer. A fluid-resonant interaction may occur if the acoustic wavelength is of the same order of magnitude than the dimensions of the cavity. The last two cavity oscillation categories are further analyzed in the next two sections.

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Fluid-dynamic oscillations

The fluid-dynamic oscillations, also called self-sustained cavity oscillations or shear layer hydrodynamic modes are the cavity oscillations in an acoustic-free system. Figure 1.1 shows the essential features involved in the feedback mechanism which can be summarized in the following way: creation of an instability at the upstream edge which is amplified as it is convected downstream until it interacts with the downstream edge of the cavity; the impingement generates an acoustic perturbation which propagates upstream and triggers the generation of another instability at the upstream edge of the cavity.

Many studies have been conducted in order to predict of the frequencies of the shear layer modes over a rectangular cavity. Rossiter (1964) proposed a semi-empirical formula based on the vortex shedding phenomenon:

St = f Lchar

U '

n − α

M + 1_κ (1.1)

where n is the order of the shear layer mode, Lchar is the characteristic length of the shear layer,

α describes the phase delay between the hydrodynamic forcing and the acoustic feedback and κ is ratio between the convection velocity in the shear layer and the free stream velocity. This formula correctly fitted Rossiter (1964)’s experimental data which were obtained with pressure transducers and flow visualisation (shadowgraph) for Mach numbers between 0.3 and 1.2.

Even if many different values for the convection speed coefficient κ can be found in the literature, the original value proposed by Rossiter (κ = 0.57) is often used. Chatellier et al. (2004) and El Hassan et al. (2007) argued that there is no need to consider a phase delay when the convection speed is much lower than the speed of sound and therefore α = 0. Usually the selected characteristic length Lchar is the length of the

cavity in the streamwise direction.

Over the years, equation 1.1 has been subjected to small changes introduced after analytical developments: see for instance Bilanin & Covert (1973), Heller & Bliss (1975), Block (1976) and Howe (1997).

Flow-acoustic coupling

The shear layer hydrodynamic modes can excite different types of acoustic modes. Plumblee et al. (1962) showed that for shallow rectangular cavities the predominant excited acoustic mode is the lengthwise mode whereas for cavities of aspect ratio higher than unity the depth acoustic mode is the one excited. Similar flow-acoustic coupling mechanisms exist in the case of coaxial side branches (Arthus & Ziada (2009); Dequand et al. (2003); Oshkai & Yan (2008); Ziada & Shine (1999)) or in the case of axisymmetric internal cavities mounted on pipes (Aly & Ziada (2010)).

Ziada et al. (2003) noticed that when a shallow rectangular cavity is mounted in a closed test section wind tunnel a flow-acoustic coupling can occur between the shear layer modes and the acoustic modes of the cavity-tunnel. Alvarez & Kerschen (2005) analytically evaluated the influence of the confinement on the acoustic resonances of a two-dimensional cavity. In Kerschen & Cain (2008) a good agreement was found

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1.Background

Figure 1.1: Schematic diagram of cavity flow.

between the predicted and the experimental trapped modes. In order to avoid the confinement effects, Yang et al. (2009) studied a deep rectangular cavity mounted at the outlet of a duct.

1.1.2 Cylindrical cavities

Mean flow

As opposed to rectangular cavities for which the flow can be considered two-dimensional when the spanwise length is large compared to the streamwise length, the flow in a cylindrical cavity is always fully three-dimensional. The organization of the mean flow depends exclusively on the aspect ratio H/D of the cavity, where H and D denote respectively the depth and the diameter. Gaudet & Winter (1973), Hiwada et al. (1983) and Dybenko & Savory (2008) showed that the flow is asymmetric with respect to the central stream-wise plane for aspect ratios between 0.2 and 0.8. Through long pressure measurements, Hiwada et al. (1983) identified two different dynamics: the flow can either flap for H/D = 0.2 − 0.4 or switch orientation for H/D = 0.4 − 0.7. For aspect ratios such as H/D < 0.2 or 0.8 < H/D, the flow inside the cavity was found to be stable and symmetric.

The parametric study of Gaudet & Winter (1973) includes a set of oil-flow visualisations at the walls of 9 different cavities (H/D between 0.04 and 1.34). The employed experimental technique revealed impor-tant information about the flow near the walls. Gaudet & Winter (1973) presented such information with streamline pattern drawing. However, as they mentioned, “a certain amount of imagination has been used in drawing the streamline patterns”. An example of these results is shown in figure 1.3: a strong asymmetric case (H/D = 0.47) as well as a symmetric one (H/D = 1.07) are presented.

Haigermoser et al. (2009) investigated a cylindrical cavity of aspect ratio H/D = 0.5 with stereo and tomographic particle image velocimetry achieving the first detailed description of the three-dimensional macroscopic organisation of the flow inside a cylindrical cavity. Chicheportiche & Gloerfelt (2010),

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Desvi-Figure 1.2: Schematic of a cylindrical cavity

gne et al. (2010), Marsden et al. (2010) and Mincu et al. (2009), within a Fondation de Recherche pour l’A´eronautique & l’Espace (FRAE) project called AEROCAV, studied numerically a cylindrical cavity (H = D = 10 cm). Their results were compared to experiments performed in two different wind tunnels: the Ecole Centrale de Lyon’s anechoic wind tunnel and the ONERA’s F2 closed section wind tunnel (experi-mental details can be found in Marsden et al. (2008) and Mincu et al. (2009)). Even if the main objective of the AEROCAV project was the investigation of the flow unsteadiness and the noise generated by cylindrical cavities, some important results on the mean flow were also obtained. Some results are reported in figure 1.4. Grottadaurea (2009) simulated the flow over a deep cylindrical cavity (H/D = 1.4) with a Detached Eddy Simulation (DES). One of the important results of this study, from the aerodynamic point of view, is the analysis of the wake behind the cavity. Specific features are described as for example the counter-rotating convective eddies generated by the cavity downstream edge. Figure 1.5 gives an example of the results obtained.

Shear layer hydrodynamic mode

An important question for the aeroacoustic viewpoint is if Rossiter’s formula can be used to estimate the frequencies of the shear layer modes on cylindrical cavities. Bruggeman et al. (1991) assumed that circular side branches can be treated as rectangular side branches for the prediction of the shear layer modes as long as an effective length Weff is used:

Weff =

πD

4 (1.2)

This effective length represents the streamwise dimension of a rectangular opening with the same surface area and the same spanwise dimension (D) as the original circular opening. If the upstream edge of the side branch is not sharp, an additional term should be added to Weff (see Bruggeman et al. (1991) or Tonon

et al. (2011)).

Czech et al. (2006) recently studied an array of circular vent holes in a wind tunnel. In order to accom-modate Rossiter’s formula, another expression for the effective length, based on an equivalent streamwise

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1.Background

(a) (b)

(c) (d)

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Figure 1.3: Oil-film flow visualisation and streamline patterns of a cylindrical cavity of aspect ratio H/D = 0.47 (a, b) and H/D = 1.07 (c, d), Gaudet & Winter (1973). Legend (e).

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(a) (b) (c)

Figure 1.4: Numerical results: (a) Non-dimensionalized mean streamwise U velocity profiles, in black lines, and vector plot of in-plane velocity (U, W ) as grey arrows in the middle spanwise plane of the cavity; (b) Non-dimensionalized mean cross-stream V velocity profiles and vector of in-plane velocity (V, W ) in the middle streamwise plane of the cavity. Free-stream flow velocities of 70 m/s, Marsden et al. (2010). (c) Flow dynamics using Q criterion applied to the mean flow, iso-surfaces of Q = 0.25(U∞/D)2, Mincu (2010)

Figure 1.5: Grottadaurea numerical simulation of a H/D = 1.375 cylindrical cavity. ReD= 34800 and M∞= 0.235. (a) View of the cavity with streamlines over streamwise planes. The mean velocity contours are also plotted over to the walls. A color-bar range from blue to red is used (0.1 < u/u∞ < 0.5). Image taken from Rodr´ıguez Verdugo et al. (2010)

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1.Background

length for a square vent hole of the same area, has been introduced:

Leff =

√ πD

2 (1.3)

Figure 1.6 shows the equivalent rectangular openings according to Bruggeman et al. (1991) and Czech et al. (2006).

Figure 1.6: Schematic representing the original circular opening and the equivalent rectangular opening according to Bruggeman et al. (1991) and Czech et al. (2006).

Mery et al. (2009) found a better agreement with Block (1976) ’s formula in the prediction of the shear layer modes over a cylindrical cavity:

St = ₁ n

kR + M (1 +

0.514 L/D)

(1.4)

where kR is the real part of the wave number of the disturbance travelling downstream. This formula

includes the effect of the bottom-reflected acoustic wave originated at the downstream edge.

Acoustic modes of an open mouth cylindrical cavity

When studying the acoustic resonances of a cavity without mean flow, it is fundamental to distinguish the open and the closed mouth cases: the boundary conditions influence indeed drastically the acoustic modes. The presence of a mean flow, especially a shear layer over the mouth of the cavity, is believed to change the characteristics (shape and frequency) of the modes. In Rona (2007), an extensive discussion about which boundary condition should be imposed at the opening of a cavity when solving the acoustic eigenvalue problem for low Mach number flows is given. Rona (2007) chose a simple acoustic reflecting boundary condition for the cavity open end while admitting that this represents a “strong hypothesis” in his model. In the next section, important experimental results will show that the acoustic modes of an open mouth cylindrical cavity excited by a low Mach number flow are better estimated when a no reflection condition is imposed at the open end. Therefore, in the following the acoustic modes of a cylindrical cavity with an open end condition are described.

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The first longitudinal mode of a deep cylindrical cavity, sometimes referred as the depth acoustic mode or as the quarter-wavelength, is the lowest tone which can resonate in the cavity. Euler and Lagrange assumed the pressure at the mouth of an open pipe to be zero and therefore the wave-length of the first longitudinal mode to be exactly 4H. Rayleigh (1894) introduced a correction for the open end of a tube with an infinite flange. The general expression for the frequency of the first longitudinal mode of a H-long open-closed pipe is:

fc=

c

4(H + αR) (1.5)

where c is the speed of sound and α is the end correction factor and R = D/2 is the radius of the pipe. The original value calculated by Rayleigh (1894) for an infinite flange tube is α = 0.8242. Nomura et al. (1960) studied the sound radiation from a flanged circular pipe with Weber-Schafheitlin type integrals and Jacobi’s polynomials. They reported how the end correction varies with the frequency of the incident plane wave and found lim

ka→0α = 0.8217 which is close to the coefficient calculated by Rayleigh. Unfortunately,

Nomura et al. (1960) did not propose a simple expression fitting their results for α. Norris & Sheng (1989) studied the same problem with Green’s functions and proposed a rational function to approximate their numerical solution:

α ≈ 0.82159 − 0.49(kR)

2

1 − 0.46(kR)3 (1.6)

where k is the wavenumber of the incident acoustic wave. According to Norris & Sheng (1989), their own results graphically match the Nomura et al. (1960) ones. Another function that approximates correctly the numerical results of Nomura et al. (1960) and that removes the singularity of equation 1.6 is (Dalmont et al. (2001)): α ≈ 0.8216 1 + (0.77kR) 2 1 + 0.77kR −1 (1.7)

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1.Background

The frequencies of higher order longitudinal modes can be predicted by the generalization of equation 1.5.

fc=

c(2q + 1)

4(H + αR) (1.8)

where q ∈ N and 2q +1 is the order of the mode. This expression guarantees a pressure node at a distance of αR from the opening.

The longitudinal modes are not the only resonant acoustic modes of a cylindrical cavity: there are also the azimuthal and the radial modes and all the combinations of these three types of modes. Marsden et al. (2008), while testing a cylindrical cavity in an anechoic wind tunnel, found a peak on the pressure spectra whose frequency was much higher than the frequencies of the quarter wave length: “The peak at 2160 Hz [...] is more strongly visible on the cavity floor, and it exhibits a strong symmetry with respect to the flow direction, almost disappearing for φ = ±π. This peak is not a multiple of one of the two main frequencies present, nor of the cavity resonant frequency. Its origin is at present not clearly identified.”. Mincu et al. (2009) performed a numerical simulation of a cylindrical cavity with exactly the same dimensions. They demonstrated that the 2160 Hz peak reported by Marsden et al. (2008) corresponds in fact to the first azimuthal mode. Furthermore, they described four other modes present in the pressure spectra. The shape of the modes are given in figure 1.8.

Figure 1.8: Shape of the acoustic modes (pressure amplitude) inside a cylindrical cavity at the half depth plane given by Large Eddy Simulation (Mincu (2010)). For the coordinate system see figure 1.4.

The acoustic modes can be calculated analytically from the Helmholtz equation which is an eigenvalue problem: the eigenvectors represent the pressure distribution of the acoustic modes of the cavity and the eigenvalues represent the square of the natural frequencies. In section 3.1, the acoustic modes of an open mouth cylindrical cavity are calculated analytically whereas in section 7.1 the closed case is treated.

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Acoustic-hydrodynamic coupling

Similarly to rectangular cavities, the hydrodynamic modes of a shear layer spanning a cylindrical cavity can be amplified when their frequencies match the acoustic modes of the cavity. Parthasarathy et al. (1985) performed a series of experiments on deep cylindrical cavities at low Mach numbers (M = 0.12 - 0.24). They reported a strong whistle when the quarter wave length matches the first shear layer mode. Marsden et al. (2008) found a strong coupling mechanism between the first two shear layer modes and the quarter wave length of a cylindrical cavity tested in an anechoic wind tunnel. Dybenko & Savory (2008) did not find a fluid acoustic coupling for any of their cylindrical cavities (H/D = 0.20, 0.47 and 0.70) tested at 27 m/s because, as it is mentioned in the paper, the first shear layer mode was expected at 145.5 Hz and the quarter wave length at 1156 Hz for the deepest cavity studied.

1.2 Partially covered cavities

Cavities whose openings are partially covered, commonly called Helmholtz resonators, are found not only in the transportation industry (window buffeting in cars/trains, aircraft landing gear wheel well) but also in musical instruments (jug bands) or in duct applications (side branches).

When these systems are exposed to a grazing flow, periodic pressure fluctuations can be generated inside the cavity. However, as noticed by Rayleigh (1894) in §310, Panton & Miller (1975a) and De Metz et al. (1977), two different oscillatory mechanisms can be excited: either periodic compressions of the volume of fluid inside the cavity, commonly called the Helmholtz resonance, or the excitation of a standing acoustic wave in the resonator. Both mechanisms fall into the Rockwell & Naudascher (1978) category of fluid-resonant oscillations.

1.2.1 Helmholtz resonance

The ‘gravest mode of vibration’ of a cavity communicating with the exterior space though an orifice is the Helmholtz resonance. This acoustic mode can be excited either by acoustic pressure fluctuations or by fluid-dynamic pressure fluctuations. The response of a Helmholtz resonator acoustically excited is presented first, followed by the examination of the flow excited case.

The classical Helmholtz theory

The easiest way to model the Helmholtz resonance is with the mechanical analogy of a spring-mass system. The air in the opening acts as lumped mass and the air volume inside the cavity acts as spring. If an external perturbation (incident acoustic waves for example) displace the slug of air in the neck, the volume inside the cavity undergoes an adiabatic transformation (compression or dilatation). By applying Newton’s second law, one can find the second order differential equation governing the displacement of the slug of air in the neck. The natural frequency of vibration of the system, Helmholtz resonance, takes the flowing expression:

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1.Background

(a) (b)

Figure 1.9: Two different types of Helmholtz resonator: (a) thin opening cavity and (b) long neck resonator

fHR= c 2π s S V leq (1.9)

where c is the sound speed, V the volume of the resonator, S is the cross-section area of the orifice, leq

is the equivalent neck length. For the extended development, refer any textbook of acoustics, for example Rienstra & Hirschberg (2011). This generic formula can be applied to different types of Helmholtz resonators by correctly estimating the equivalent neck length which is the height of the slug of air. Figure 1.9 gives a schematic drawing of two types of cavities: the slug of air is represented with a dashed line.

Conceptually, leq is the actual length of neck plus an interior and an exterior end correction (∆lint and

∆lext):

leq = l + ∆lint+ ∆lext (1.10)

Different expressions for the end corrections have been developed depending on the shape of the aperture, on the flange and on the dimensions of cavity. For circular openings, the Rayleigh end correction formerly introduced in equation 1.5 is often used for the exterior part (∆lext = 0.8242a). In order to apply the

Rayleigh end correction for non-circular openings, an effective radius aeff = 1.06 S3/4U−1/2, where U is the

perimeter of the opening, has proven to give reasonable agreement with experiments (Crighton et al. (1994)). Ingard (1953) proposed different expressions for the interior end correction, based on the shape of both the cavity and the opening and on the position of the opening. Interior and exterior end corrections are often grouped in a single parameter 2∆l.

Influence of the geometry

Some researchers have shown that the classical Helmholtz theory presented so far does not correctly predict the frequency of the Helmholtz resonance because it does not take into account the geometrical characteristics

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of the cavity. Selamet et al. (1995), for example, investigated the effect on the resonance frequency of changing the aspect ratio of a cylindrical cavity while maintaining its volume. Their computational and experimental results showed that the resonance frequency depends on the cavity’s aspect ratio: the deeper the cavity is, the lower the frequency is.

Panton & Miller (1975b) proposed a transcendental equation for the resonant wavenumbers k of a cylin-drical Helmholtz resonator based on one-dimensional wave propagation inside the cavity:

leqA

H S kH = cot(kH) (1.11)

where A is the cavity cross section area. In Panton & Miller (1975b)’s development, the slug of air in the neck was treated as a lumped mass. A more sophisticated analysis was performed by Tang & Sirignano (1973) who assumed one-dimensional wave propagation not only inside the cavity but also on its neck. The first-order approximation gives:

tan(kl) tan(kH) = S/A (1.12)

This expression yields to equation 1.11 when the cavity has a short neck (kl 1) because in this case tan(kl) ≈ kl. Furthermore, when the wavelength λ is much larger than the dimensions of the cavity (kH 1), equation 1.11 reduces to the classical formula for the Helmholtz resonance (equation 1.9) because cot(kH) ≈ 1/kH. It is interesting to note that Selamet et al. (1995), through a different conceptual develop-ment, found equation 1.12 as well. The trend of the Helmholtz resonance frequency observed experimentally is correctly predicted by equation 1.12 even if some quantitative discrepancies have been noticed for small aspect ratios H/D (Selamet et al. (1995)). Tang & Sirignano (1973) recommended the addition of the end corrections to the length of the orifice and the inclusion of the flow contraction effects to the model for small aspect ratio cavities in order to improve the prediction of the resonant modes.

The model of Panton & Miller (1975b) and Tang & Sirignano (1973) does not only predict the frequency of the Helmholtz resonance but also the longitudinal modes of the cavity. In 1.10 a graphical resolution of equations 1.11 and 1.12 is given.

Panton & Miller (1975b) also proposed an explicit expression for the Helmholtz resonator that improves the classical Helmholtz resonance formula (equation 1.9) by retaining two terms in equation 1.11:

fHR= c 2π s S V leq+ H2S/3 (1.13)

This formula has been used by different authors to predict the Helmholtz resonance (Panton (1990) or Mechel (2002)).

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1.Background

(a) (b)

Figure 1.10: Graphical resolution of two different transcendental equations for the resonant wavenumbers. (a) Panton & Miller (1975b) : blue line cot(kH) and red line leqA

H S kH. (b) Tang & Sirignano (1973): black line cot(lkH/H) S/A and magenta line tan(kH)

Flow effect on the resonance frequency

Several researchers have reported an increase of the Helmholtz resonance frequency when the cavity is excited by a flow. Anderson (1977), while testing side branch Helmholtz resonators of different sizes in a circular duct with fully developed turbulent flow, reported an increase of the Helmholtz resonance frequency for flow velocities higher than 30 m/s. Their 53.1 mm long cavity underwent a 100 Hz shift of the resonance frequency from 250 Hz at 0 m/s to 350 Hz at a flow velocity of 80.7 m/s. For flow velocities lower than 30 m/s the resonant frequency did not change. Phillips (1968) reported an increase of fHRbetween 20 m/s and

80 m/s for a partially covered cavity tested in a wind tunnel. The experiments of Panton & Miller (1975a) on the fuselage of a glider with a free-stream speed of 30 m/s also indicated an increase in the frequency of the fundamental acoustic mode of different Helmholtz resonators. Their hot wire measurements assessed a fully developed boundary layer in the orifice region.

Zoccola (2000), reported in his PhD thesis a decrease of the resonance’s frequency of flow excited Helmholtz resonators. He tested three different cavities and measured the frequency fHRfor a purely

acous-tic excitation (0 m/s) and for a 6.9 m/s flow: 346 Hz (333 Hz), 636 (554 Hz) and 436 Hz (416 Hz). Zoccola (2000)’s experiments were done at a flow speed inferior than the velocity range analysed by Anderson (1977), Phillips (1968) and Panton & Miller (1975a). Unfortunately Zoccola (2000) did not give the characteristics of his boundary layer.

1.2.2 Shear layer over the mouth of a Helmholtz resonator

Of particular interest is the understanding of the shear layer organization when the cavity is subject to a fluid-resonant oscillations. Elder (1978) experimentally investigated a deep cylindrical cavity with a rectangular opening. He characterized the shear layer with a hot wire probe sampled simultaneously with a microphone

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at the bottom wall the cavity: after phase-averaging the velocity, shear layer profiles at different phases during an acoustic cycle were presented. Elder (1978) also introduced the concept of the “interface wave” surface which is the locus of the inflection points of the shear layer profiles along the opening. He proposed a model to predict interface wave surface based on an initial displacement wave generated by the traverse acoustic particle motion. Good agreement was found between the experiments and the proposed model.

Nelson et al. (1981) used two-component Laser Doppler Anemometry (LDA) and a flow visualisation technique (stroboscopic light) to characterize the shear layer spanning the rectangular opening of a cuboidal Helmholtz resonator. The free stream velocity chosen for the experiments was 22 m/s which corresponds to the first shear layer mode strongly coupled with the Helmholtz resonance of the cavity. At the upstream lip they observed the shedding of a single vortex per acoustic cycle.

Ma et al. (2009) studied the shear layer over a rectangular Helmholtz resonator with PIV. They analysed the same resonant case as Nelson et al. (1981): the first shear layer mode coupled with the Helmholtz resonance. However Ma et al. (2009) reported phase-averaged contours of the spanwise vorticity for three different velocities corresponding to one strongly resonant case and two weakly resonant cases. They found that the shear layer has a sheet-like character in the region closer to upstream edge and tends to roll up into a single discrete vortex in the downstream portion of the opening (figure 1.11).

Figure 1.11: Contours of phase averaged vorticity for three different flow speeds. In all the cases the predominant acoustic mode is the Helmholtz resonance and the amplified shear layer instability is the first hydrodynamic mode (Ma et al. (2009)).

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1.Background

1.3 Noise source characterisation

Aeroacoustic source characterization has been facilitated by the development of full field velocity mea-surement techniques, as indicated by the number of publications in the last decade: Geveci et al. (2003), Haigermoser (2009), Velikorodny et al. (2010) and Finnegan et al. (2010) just to cite a few. Morris (2010) has recently provided a review on the use of PIV to examine how shear layer instabilities and turbulence lead to radiated sound. The common strategy is to apply an acoustic analogy to the experimental data, either the Curle (1955) acoustic analogy or the vortex sound theory introduced by Powell (1964) and extended by Howe (1975), in order to identify the spatial distribution of the sound sources. Both analogies were applied by Koschatzky et al. (2010) to estimate the cavity sound emission: the overall sound pressure level was correctly predicted by the two methods even if the vortex sound theory appears to predict better the amplitude of the tonal component.

Howe (1975, 1980) estimated that with the low Mach number, inviscid, constant entropy approximation, the total flow velocity can be decomposed into an incompressible vorticity-bearing velocity and an irrotational acoustic velocity. The generation of acoustic power by the vortical field can be calculated by the following formula:

Π = −ρ0

Z

V

( ~w × ~v) · ~uacoustdV (1.14)

where ρ0 is the fluid density, ~v and ~w are the fluid velocity and vorticity, and ~uacoust is the acoustic

particle velocity. The sign of Π determines if vorticity acts as a source or sink for the acoustics. For the details on the derivation of equation 1.14 see Appendix A.

Interesting results have been found by applying equation 1.14 to strongly resonant cases: Velikorodny et al. (2010), for example, estimated the distribution of the acoustic sources and sinks on a duct with coaxial side branches (figure 1.12).

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Figure 1.12: Patterns of time-averaged acoustic power on a duct with coaxial side branches corresponding to the second hydrodynamic oscillation mode (Velikorodny et al. (2010)).

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Part I

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Chapter 2

Experimental set-up

2.1 Wind tunnel

The experimental investigation was conducted in a closed circuit low speed wind tunnel designed by the Mechanical and Industrial Engineering Department (DIMI) of Roma Tre University. The facility is located in the Italian National Agency for New Technology Energy and Environment (ENEA) research center of Casaccia, 28 km from Rome. The closed test section is 2.49 m long (Lx) and has a 0.89 × 1.16 m2 _cross

section (Ly × Lz). Other important dimensions of the wind tunnel are given in figure 2.1. The fan is able to generate a flow ranging from 0 to 90 m/s in the centreline of the test section with a relative turbulence level of 0.1% at a velocity of 40 m/s. Further details about the wind tunnel properties can be found in Camussi et al. (2006a).

2.2 Test model

The test model was designed within the framework of the AeroTraNet project and manufactured by the Uni-versity of Leicester. A Perspex cylindrical pipe with an interior diameter (D) of 210 mm was mounted flush to the bottom wall of the test section, 1780 mm downstream the end of the convergent section (figure 2.2). A flat Perspex disk sealed the cylinder from underneath, creating a 285 mm deep (H) cavity (figure 2.3). These dimensions lead to a cavity aspect ratio of 1.357 . According to the above mentioned literature studies (section 1.1.2), the selected aspect ratio should guarantee the symmetry of the flow with respect to the central streamwise wall-normal plane.

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Figure 2.1: Schematic drawing of the ENEA Casaccia wind tunnel (top view).

(a) (b)

Figure 2.2: Schematic drawing of the test section with the cylindrical cavity studied. Dimensions of the test section are: Lx = 2.49 m, Ly = 0.89 m, Lz = 1.16 m. An azimuthal angle ϕ has been introduced to follow the pressure at the walls of the cavity.

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2.Experimental set-up

Figure 2.3: Photography of the cavity before the preliminary characterization campaign. During the experiments, one microphone was mounted in one of the orifices. See figure 2.5 for further details on the set-up of the microphones.

2.3 Instrumentation

2.3.1 Pitot-static tube

The flow velocity in the test section was monitored with a Pitot tube connected to a Kavlico pressure transducer model P592. The probe was introduced in the wind tunnel through the “breather”, a small slot around the perimeter at the downstream end of the test section. The purpose of this vent is to keep the test section close to atmospheric pressure and to isolate it from vibration that could otherwise be transmitted through the diffuser. The calibration of the pressure transducer was done with a pressure pump and a U-tube manometer.

2.3.2 Hot wire anemometry

Velocity in the shear layer and in the wake of the cavity was measured with a 55P11 single component Dantec probe connected to a constant temperature hot-wire anemometer (A.A. Lab System AN-1003). To reach the desired positions on a given yz-plane, the probe was mounted on a two axis traverse system equipped with stepping motors (Rexroth Compact Module CKK). The calibration of the probe was done with the Pitot tube.

Because the hot wire anemometry is an intrusive technique, a verification that the presence of the hot-wire probe does not affect the flow significantly was performed. A microphone in the downstream cavity

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wall was used as controller. A hot-wire probe was introduced in the test section and placed at 16 different locations above the cavity. Figure 2.4 presents two pressure spectra corresponding to two different cases: the hot-wire probe in the shear layer (y/H = 0) and outside the shear layer, far above the cavity (y/H = 0.35). Significant differences are not observed, leading to the conclusion that the presence of the hot-wire probe in the shear layer does not strongly influence the flow.

Figure 2.4: Auto-spectra of the pressure signal with a hot-wire probe in the shear layer (black line) and outside the shear layer (grey line). The hot-wire probe was located in the center plane at 0.05D of the downstream edge. U∞= 40 m/s.

2.3.3 Microphones

Two 1/4-inch Br¨uel&Kjær free-field microphones (type: 4939 and 4135) were used to measure the fluctuating pressure at the cavity walls. The microphones were connected to pre-amplifiers B&K 2670 and to a signal conditioner Br¨uel&Kjær NEXUS 2692. Even if several holes for the microphones are available on the walls of the cavity, a toothed graduated ring mechanism (precision of 1◦) was designed in order to reach any azimuthal angle by rotating the cavity (figure 2.3).

The microphones were connected to the interior of the cavity through 1 mm diameter pinholes. The geometry of the pinholes is the same as that adopted in Camussi et al. (2006a,b, 2008). By using pinholes the spatial averaging effects are minimized. The main drawback of this layout is the presence of a non negligible volume between the microphone diaphragm and the pinhole. This volume can act as a Helmholtz acoustic resonator. In the present case the corresponding cut-off frequency has been accurately calculated to 3347 Hz with equation 1.9. The expression leq = l + 1.64a was taken for the equivalent neck length

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(a) (b)

Figure 2.5: Detail on the microphone set-up. The volume of air between the microphone diaphragm and the interior surface of the cavity represents a Helmholtz resonator which dimensions are: length and cross surface area of the pinhole l = 1.3 mm and S = π0.52mm2 respectively and volume of the resonator V = 99 mm3.

(Blake (1986)), where a represents the pinhole’s radius, which is a simplification of a double Rayleigh end correction. The maximum frequency of interest (shear layer modes) is expected to be around 300 Hz for the velocity range explored, thus the Helmholtz resonator should not affect the frequency range of interest.

2.3.4 Particle image velocimetry

To investigate the flow inside the cavity, Particle Image Velocimetry (PIV) was used. The planes of interest were illuminated by a BigSky Twins Ultra/CFR 200 Nd:YAG laser with the maximum energy of 50 mJ per pulse. The images were taken with a PCO Pixelfly VGA (1280 × 1024 pixels) camera equipped with a Docter Optics Tevidon 1.8/16 lens. Synchronization between the laser and the camera was achieved with a BNC digital pulse/delay generator 575 controlled with an in-house Labview program.

Velocity fields over horizontal streamwise planes (Oxz) were explored by introducing the laser light sheet horizontally through the cavity sidewall and placing the camera under the cavity. The measurements were performed at different depths by moving the laser along the y-axis. A picture and a schematic of the experiment are given in figure 2.6.

The PIV processing was done with PIVDEF, a software developed by the Istituto Nazionale per Studi ed Esperienze di Architettura Navale (INSEAN) also known as the “Italian Ship Model Basin”. This software is based on a standard PIV cross-correlation algorithm (Cotroni et al. (2000)), a recursive window offset (Westerweel et al. (1997)) and a multiplication between adjacent correlation tables (Hart (1998)). A window deformation techniques similar to the one developed by Lecordier et al. (1999) is also used. Further details on the processing algorithm are given in Di Florio et al. (2002).

For each plane of interest and for each flow velocity investigated, 600 couples of images were recorded. The output velocity fields obtained with PIVDEF were exported to Matlab for further processing.

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(a) (b)

Figure 2.6: (a) Photograph taken from the exterior of the test setion showing, from left to right, the laser beam, a cylindrical and a spherical lens, the generated laser sheet, the cavity and the camera underneaths. (b) Schematic of the experimental rig showing the position of the PIV apparatus.

2.3.5 Data acquisition card and processing of the pressure signals

Signals from all the instruments were acquired using a National Instrument SCXI-1600 Data Acquisition Module. The signal from the hot wire was sampled at a frequency of 40 kHz, acquired for 5 seconds and low-pass filtered at a cut-off frequency of 10 kHz to avoid aliasing. The pressure signals were acquired for 4 seconds using a sample rate of 25 kHz and were band-pass filtered (between 20 Hz and 10 kHz) by the signal conditioner.

The pressure signals were processed in Matlab using a Fast Fourier Transform (FFT) technique. The number of points per segment was 25000 and therefore the frequency resolution was 1 Hz. The 4 data sets were then averaged.

2.4 Test conditions

2.4.1 Flow conditions

The wind tunnel velocity was set manually by rotating a knob which controls the speed of the fan: the RPM can be then read on a digital display located in the control room. Before every measurement, enough time is waited in order to obtain stable flow conditions. Upstream of the wind tunnel’s contraction, the temperature was kept at 21◦C ±1◦C by a heat exchanger. For the aeroacoustic characterization of the cavity, the flow speed was increased from 5 to 55 m/s (M = 0.015 - 0.161) by steps of 1 m/s.

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2.4.2 Incoming boundary layer

The boundary layer over the test section floor was characterized by hot-wire anemometry. The properties of the incoming boundary layer at 46 mm from the upstream corner of the cavity are summarized in table 2.1 for three velocities (20, 30 and 40 m/s). The length scale of the boundary layer (momentum thickness) represents less than 2 % of the diameter of the cavity thus this parameter should not influence the aerodynamics and acoustics of the cavity. Velocity profiles are given in figure 2.7 jointly with the 1/7 power profiles. The velocity measurements match the theoretical turbulent boundary layer trend.

U∞ (m/s) δ (mm) δ∗ (mm) θ (mm)

19.9 35.2 4.9 3.8

30.4 34.4 4.7 3.6

40.2 31.7 4.3 3.3

Table 2.1: Boundary layer properties for 3 free stream velocities (20, 30 and 40 m/s). Boundary layer thickness δ (99 %), displacement thickness δ∗and momentum thickness θ.

Figure 2.7: Velocity profiles of the boundary layer measured at 0.22D from the upstream corner of the cavity at 3 different flow velocities: U∞= 20, 30 and 40 m/s.

2.4.3 Background pressure fluctuations

The background pressure fluctuations in the test section were explored in order to check if their intensity is lower than the flow-excited cavity unsteady pressure level. Camussi et al. (2000) have already clarified most

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of the background unsteady pressure features of the ENEA Casaccia wind tunnel. Therefore the results from this previous study were used to identify some of the frequency peaks in the pressure spectra.

The measurements were done by means of an in-flow microphone fitted with a nose cone and located 150 mm (y) above the cavity. The mouth of the cavity was covered with a wooden plate. The pressure spectra obtained for flow velocities between 5 and 55 m/s are given in figure 2.8 with the theoretical blade passing frequency and its harmonics superimposed. The pure tone at 300 Hz, present for all the flow velocities, has been ascribed by Camussi et al. (2000) to the constant speed fan installed for cooling the wind tunnel blower.

Figure 2.8: Background pressure fluctuations in the test section without the cavity. The white dots represents the blade passing frequency and its first two harmonics.

2.5 Measurement matrix

The fluctuating pressure was measured at the side and bottom walls of the cavity. A total of 325 different positions were surveyed over 6 different depths (y), 4 different radii (r) and 36 different azimuthal angles (ϕ). Figure 2.9 shows the positions adopted by the microphone. In this schematic drawing, the lateral wall has been unrolled and the circular bottom reproduced underneath. With this plane representation, the reader can have a panoramic view of the cavity walls. To achieve this measurement matrix, a single microphone was moved from one pinhole to another and the cavity rotated by steps of 10 degrees. Results can be found in section 4.4.

The aerodynamic campaign consisted in a three-dimensional grid of 2520 points distributed over 8 dif-ferent streamwise (x), 7 difdif-ferent vertical (y) and 45 difdif-ferent spanwise (z) positions. A schematic drawing indicating the hot wire probe positions adopted for the velocity measurements is given in figure 2.10: a top

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and a lateral view of the experimental rig are represented. In section 4.1, the results from this campaign are discussed.

For the PIV measurements, three different depths inside the cavity were explored: y/H = -0.25, -0.50 and -0.75. The obtained mean velocity fields are reported in section 4.2.

Figure 2.9: Schematic drawing of the wall pressure measurements grid, see figure 2.2 for the reference frame adopted. The lateral wall has been unrolled and the circular bottom reproduced underneath.

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(a)

(b)

Figure 2.10: Schematic drawing of the velocity measurements grid, see figure 2.2 for the reference frame adopted. Positions taken by the hot wire probe for the aerodynamic campaign: top view (a) and lateral view (b)

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Chapter 3

Acoustic mode calculation

3.1 The acoustic modes of an open-closed cylindrical cavity

The aim of this section is to find an analytical expression for the natural acoustic modes of a cylindrical cavity with an open mouth. The starting point is the three-dimensional wave equation that is the second-order linear partial differential equation governing the acoustic pressure p:

∇2p = 1 c2

∂2

∂t2p (3.1)

where c is the speed of sound. The pressure can be decomposed into a sum of orthogonal Fourier components pheiωhtwhich individually satisfy the wave equation at every moment t. Therefore equation 3.1

becomes the Helmholtz equation:

∇2_p

h+ k2hph= 0 (3.2)

where kh= ωh/c the wave number of the corresponding Fourier component.

In cylindrical coordinates (r, θ and y, see figure 3.1) the Helmholtz equation becomes:

∂2_p h ∂r2 + 1 r ∂ph ∂r + 1 r2 ∂2_p h ∂θ2 + ∂2_p h ∂y2 + k 2 hph= 0 (3.3)

This equation can be solved analytically by the separation of variables technique (details are not given here for sake of brevity and because it can be found in many acoustic textbooks). The general solution of equation 3.3 takes the following form:

ph(r, θ, y) = Jm(krr) eimθ(C1eikyy+ C2e−ikyy) (3.4)

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Figure 3.1: Definition of the cylindrical coordinates for the analytical resolution of the Helmholtz equation.

In order to find the acoustic eigenfrequencies of the cavity, the boundary conditions have to be specified: the wall-normal derivative of p has to vanish at the walls (bottom and side) and an assumption has to be made for the mouth of the cavity (y = H). As mentioned in section 1.1.2, physically the pressure node is not exactly at the opening of the cavity. A simple way to proceed is to take an end correction: the additional boundary condition adopted here is that the pressure has to vanish at y = H + αR. The general expression for an eigenfrequency is:

fm,n,q= ω 2π = c 2π _j0 m,n R 2 + (2q 0_{+ 1)π} 2(H + αR 2!1/2 (3.5)

where j0_m,n is (n + 1)th _{positive zero of J}0

m and m, n and q are the azimuthal, radial and longitudinal

order of the mode and q = 2q0 + 1 where (m, n, q0_{) ∈ N}3_{. It is interesting to notice that if m = n = 0,}

equation 3.5 reduces to equation 1.8.

With the dimensions of the present cavity (H and D) and the temperature and the ambient pressure during the tests (for the calculation of c), the frequencies of the acoustic modes of the cavity can be calculated. The end correction coefficient α that depends on ky can be evaluated with equation 1.7 (figure 3.2). The

predicted frequencies are given in table 3.1 and the shapes of the first eight modes are reported in figure 3.3.

Mode λ/4 3λ/4 AZ1(λ/4) AZ1(3λ/4) 5λ/4 AZ1(5λ/4) AZ1R1(λ/4) AZ1R1(3λ/4)

(m, n, q) (0,0,1) (0,0,3) (1,0,1) (1,0,3) (0,0,5) (1,0,5) (1,1,1) (1,1,3)

Frequency 236 760 994 1246 1333 1663 2792 2904

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3.Acoustic mode calculation

Figure 3.2: Correction coefficient predicted by equation 1.7. The red cross represent the quarter wavelength (α = 0.7536), the black dot the mode 3λ/4 (α = 0.5153) and the green star the mode 5λ/4 (α = 0.3562).

3.2 The acoustic modes in a wind tunnel

When performing aeroacoustic experiments in a closed test section, it is important to study the acoustic behaviour of the facility in addition to its aerodynamic performance. If the walls of the test section are not acoustically treated, the pressure measurements may be affected by the acoustic resonant modes of the wind tunnel. Furthermore the presence of a model in the test section can modify the acoustic resonances of the wind tunnel and, more important, excite strongly localized resonant modes with zero radiation loss called trapped modes.

In order to predict the frequency and the shape of the resonant acoustic modes of an experimental rig, finite element analysis has become popular in the past decade: Aly & Ziada (2010) used the commercial software ABAQUS to study the trapped modes of a ducted axisymmetric internal cavity, Finnegan et al. (2010) calculated the acoustic field around a bluff body with ANSYS and Oshkai et al. (2008) predicted the acoustic modes of a coaxial side branch resonator with COMSOL. Some advantages of the use of a commercial software for the calculation of the acoustic modes are the promptness and the reliability of the results when the boundary conditions are properly set.

3.3 The computational model

The acoustic modes of the test rig were studied using the commercial software COMSOL Multiphysics 4.0. This package includes a finite element solver able to evaluate partial differential equations of different disciplines. Even if several physical models can be solved jointly, a no-flow case was studied. The default eigenfrequency formulation available within this software was used for the calculation:

∇ −1 ρ∇p +λ 2_p ρc2 = 0 (3.6)

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(a) (0,0,1) (b) (0,0,3)

(c) (1,0,1) (d) (1,0,3)

(e) (0,0,5) (f) (1,0,5)

(g) (1,1,1) (h) (1,1,3)

Figure 3.3: Shape of the first height acoustic modes of a cylindrical cavity with a closed bottom wall and an open top surface. The infinite flange has not been represented. The color bar gives the normalized real pressure values.

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(a) (b)

Figure 3.4: Overview (a) and detail (b) of the mesh. The spatial domain chosen for the calculation includes the contraction and the test section of the wind tunnel.

where ρ is the air density, c the speed of sound, p the pressure and λ an eigenvalue. From this equation, it is clear that the fluid is modeled as a lossless medium. Once the equation has been solved, the eigenfrequency can be evaluated through:

λ = i2πf (3.7)

3.4 The geometry

A preliminary investigation brought into evidence that the simulation of the test section without the con-vergent nozzle does not predict correctly the frequencies measured experimentally. Thereby a simplified contraction was included into the calculation leading to a five-meter-long geometry.

As already mentioned in the previous chapter, the frequency range of interest is [0 - 300 Hz]. In order to resolve correctly the highest frequency of this range, every mesh element does not have to exceed 0.2 meters. This is dictated by the five-mesh-elements-per-wavelength rule. The wind tunnel section was meshed with approximately 53 000 tetrahedrons and the cavity with 6 000 elements. Figure 3.4 gives an idea of the meshed geometry.

All the walls (contraction, test section and cavity) were modelled with rigid boundaries (called ‘sound-hard’ in the COMSOL Multiphysics) meaning that the normal derivative of the pressure is set to zero. At the end of the contraction and at the end of the test section ‘soft boundary’ conditions were imposed: the acoustic pressure vanishes.

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Without cavity With cavity

Frequency Mode number Mode number Frequency Difference

Hz Hz % 42.64 1 1 42.63 0.02 66.72 2 2 66.67 0.07 99.95 3 3 99.72 0.23 102.42 4 4 102.42 0.00 118.07 5 5 118.07 0.00 130.86 6 6 130.62 0.18 136.11 7 7 136.11 0.00 152.05 8 8 152.06 -0.01 154.98 9 9 154.81 0.11 160.25 10 10 160.28 -0.02 165.67 11 11 165.47 0.12 167.84 12 12 167.84 0.00 181.13 13 13 181.19 -0.03 183.76 14 14 183.76 0.00 186.95 15 15 186.87 0.04 191.16 16 16 191.16 0.00 196.16 17 17 196.16 0.00 198.32 18 19 198.66 -0.17 199.83 19 18 197.78 1.03 204.28 20 20 204.3 -0.01 212.2 21 21 211.96 0.11 215.95 22 22 212.78 1.47 217.59 23 23 217.61 -0.01 223.86 24 24 223.86 0.00 225.32 25 26 225.32 0.00 226.96 26 25 224.6 1.04 232.04 27 27 232.06 -0.01 232.52 28 28 232.53 0.00 241.29 29 29 236.77 1.87 244.71 30 30 244.72 0.00 do not exist 31 246.76 249.47 31 32 249.47 0.00 250.08 32 33 250.13 -0.02 251.21 33 34 251.29 -0.03 253.97 34 35 256.64 -1.05 258.69 35 36 258.69 0.00 260.52 36 37 260.52 0.00 262.13 37 38 263.08 -0.36

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3.5 Results

3.5.1 Acoustic modes without the cavity

In the range of frequency [0 - 263 Hz], 37 different eigenmodes were found and their frequencies, sorted from smallest to largest, are reported in table 3.2. Six modes were identified with a zero isobar central horizontal plane but without a zero isobar central vertical plane. The frequencies of these modes are highlighted in the table 3.2 and their pressure distributions over the walls of the test section are plotted in figure 3.5. The shapes of six modes not having the characteristics previously described are given in figure 3.6.

3.5.2 Acoustic modes with the cavity

When the cavity is considered, 38 different eigenmodes are found in the same frequency range. The eigenfre-quencies are reported in table 3.2 jointly with the corresponding eigenfreeigenfre-quencies without the cavity. Two minor changes can be noticed: small frequency shifts and two order permutations (modes 18-19 and 25-26). Furthermore, the addition of the cavity into the geometry brings one mayor change: another eigenfrequency appears. The discussion of this extra mode will be detailed later.

As in the previous case, the modes can be divided in two categories according to the pressure pattern on the horizontal and vertical planes of the test section. The second group (31 eigenmodes) is weekly influenced by the introduction of the cavity: the biggest frequency shift is 0.23 %. When comparing figure 3.6 to figure 3.8 it can be seen that the shapes of the modes have not changed significantly. Another observation is that the pressure inside the cavity is constant and equal to zero. It is important to point out that most of the modes of this group have a central vertical plane where the pressure is equal to zero (pressure node). The shape of these modes would has probably changed if the cavity had been added in a plane other than the central one.

Figures 3.7 gives the shapes of the 6 acoustic modes for which the central horizontal plane (0xz) was a pressure node and the central vertical plane was not a zero isobar before the introduction of the cavity. The frequency shifts are reported in table 3.2: the maximum difference is of 1.87 % which is a much higher percentage than for the former category. It can be seen, in figures 3.7, that the pressure amplitude in the cavity is not constant. All the modes of this class have a pressure pattern similar to the cavity quarter-wave length: the bottom of the cavity is a pressure anti-node and the fluctuations at the orifice are minimal. This common shape can be illustrated by the mode 35 (figure 3.10). At the cavity orifice, the pressure pattern is not uniform as it should be for an unconfined case: this mode, as the 6 others, are indeed a combination of a cross-ducted mode with the quarter-wave length of the cavity.

The only eigenfrequency added by the cavity is 246.76 Hz (pressure reported in figure 3.9). This value is close to 326 Hz, the frequency of the quarter-wave length of the cavity analytically calculated (table 3.1). This result is relevant for the present analysis since the signature of this mode will be evidenced by the wall pressure measurements presented in section 4.4 of the next chapter.

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(a) Mode 19 (b) Mode 22

(c) Mode 26 (d) Mode 29

(e) Mode 34 (f) Mode 37

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Figure 3.9: Extra mode: acoustic mode 31

Figure 3.10: Isosurface plot of the acoustic pressure distribution (Pa) inside the cavity at 256Hz. This mode has a quarter-wave length shape

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Chapter 4

Experimental results

4.1 Overall aerodynamics

A single component hot-wire probe was used to measure the velocity above and downstream of the cavity in order to obtain a general understanding of the shear layer and of the wake topology. The incoming free stream velocity was fixed to U∞ = 40 m/s. The flow velocity at a given position (x, y, z) in the test section

is ~U = ~Ux+ ~Uy+ ~Uz. As the wire was orientated along the z direction, the velocity measured by the hot-wire

is Uhw = || ~Ux+ ~Uy||. Profiles of the non-dimensional mean velocity (Uhw/U∞) and of the non-dimensional

standard deviation of the measured velocity (σhw/U∞) are given in figure 4.1 and figure 4.2 respectively.

According to the literature, because H/D > 0.8 both quantities are expected to be symmetric with respect to the central streamwise plane (Oxy) within the range of experimental uncertainties. The shear layer topology is first presented followed by the wake description.

4.1.1 Shear layer topology

Three different streamwise planes are reported in figure 4.1: x/D = 0, 0.25 and 0.50. For each one of these planes, different heights were explored. In the region close to the vertical symmetry plane (z/D = 0) a mean velocity defect is clearly observed as an effect of the shear layer generated over the cavity opening. Because the geometry is cylindrical, the greatest shear layer effects are observed in the central region. The turbulent velocity fluctuations (σhw) confirm this trend since an increase of the turbulent kinetic energy is observed in

the region close to the plane z/D = 0. When moving downstream, the shear layer grows: the mean velocity decreases and the velocity fluctuations increase. This effect is shown both for the cases at x/D = 0.25 and 0.5.

It is interesting to note that when a given longitudinal position x/D is considered, the perturbation induced by the presence of the cavity is restricted to the wall proximity region. Indeed, in the region close to the downstream side of the cavity and for around y/D = 1/10, Uhwand σhw profiles are almost flat and

Experimental investigation of flow past open and partially covered cylindrical cavities

DOTTORATO DI RICERCA IN INGEGNERIA MECCANICA E

INDUSTRIALE

CICLO XXIV

Experimental investigation of flow past open and partially covered

cylindrical cavities

Francisco Rodriguez Verdugo

Tutor:

Prof. Roberto Camussi

Abstract

Acknowledgements

Table of contents

I

Cylindrical cavity with open mouth

19

II

Cylindrical cavity with partially closed mouth

61

Introduction

Motivation for the research

Frameworks

Outline of the thesis

Chapter 1

Background

1.1

Open mouth cavities

1.1.1

Rectangular cavities

1.1.2

Cylindrical cavities

1.2

Partially covered cavities

1.2.1

Helmholtz resonance

1.2.2

Shear layer over the mouth of a Helmholtz resonator

1.3

Noise source characterisation

Part I

Chapter 2

Experimental set-up

2.1

Wind tunnel

2.2

Test model

2.3

Instrumentation

2.3.1

Pitot-static tube

2.3.2

Hot wire anemometry

2.3.3

Microphones

2.3.4

Particle image velocimetry

2.3.5

Data acquisition card and processing of the pressure signals

2.4

Test conditions

2.4.1

Flow conditions

2.4.2

Incoming boundary layer

2.4.3

Background pressure fluctuations

2.5

Measurement matrix

Chapter 3

Acoustic mode calculation

3.1

The acoustic modes of an open-closed cylindrical cavity

3.2

The acoustic modes in a wind tunnel

3.3

The computational model

3.4

The geometry

3.5

Results

3.5.1