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Modelling and Characterization of Perforates in Lined Ducts and Mufflers

Tamer Elnady

Stockholm 2004

Doctoral Thesis The Marcus Wallenberg Laboratory for Sound and Vibration Research

Department of Aeronautical and Vehicle Engineering The Royal Institute of Technology (KTH)

Akademiska avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framläggs till offentlig granskning för avläggande av teknologie doktorexamen den 14 september 2004, kl 10:00 i Kollegiesalen, Valhallavägen 79, KTH, Stockholm. Fakultetsopponent är Dr. Yves Aurégan, Laboratoire d’Acoustique de l’Université du Maine, Le Mans, France. Avhandlingen försvaras på engelska.

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Abstract Increased national and international travel over the last decades has caused an increase in the global number of passengers using different means of transportation. Great effort is being directed to improving the noisy environment in the residential community. This is to face the growing strict noise requirements which are implemented by international noise regulatory authorities, governments, and local airports. There is also a strong competition between different manufacturers to make their products quieter. The propulsion system in an aircraft is the major source of noise during relevant flight conditions. The engine noise in a vehicle dominates the total radiated noise at low speeds especially inside cities. Many recent studies on noise reduction involve the use of perforated plates in the air and gas flow ducting connected to the engine. This thesis deals with the modelling of perforates as an absorbent.

There are many difficulties in using liners in these applications. The most important is that there is no large surface area to which the linings may be applied. Equally, the environment in which linings have to survive is hostile. Therefore, liners have to be carefully tailored in order to achieve the most efficient attenuation. The full-scale simulation testing, which is usually necessary to define the noise attenuation produced by a liner installation, is both time-consuming and expensive. Therefore, a need for accurate models is a must. This thesis fills some gaps in the impedance modelling of perforated liners. It also concentrates on those complicated situations of sound propagation in ducts that were solved earlier using Finite Element Methods. Alternate analytical solutions to these problems are developed here, which gives more physical insight into the results.

The key design parameter of perforates is the acoustic impedance. The impedance is what determines their efficiency to absorb sound waves. A semi empirical impedance model was developed to be capable of accurately predicting the liner impedance as a function of its physical properties and the surrounding conditions. It was compared to all previous models in the literature. Nothing in the literature has been reported on the effect of temperature on the perforate impedance, therefore a complete study was performed. A new inverse analytical impedance measurement technique was proposed. It is based on educing the impedance value based on the measurement of the attenuation across a lined duct section. Two applications were further considered: The effect of hard strips in lined ducts on there attenuation properties; and the modelling of perforations in a complicated automotive muffler system.

Keywords: Perforates – Liners – Acoustic impedance – Hot stream liners – Hard splices – Mufflers – Lined ducts – Collocation – Flow duct.

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This thesis consists of the following papers:

Paper I Application of the Point Matching Method to Model Hard Strips in Lined Ducts.

Paper II Investigation of Mode Scattering by Hard Strips in Lined Ducts Using Collocation.

Paper III On the Modelling of the Acoustic Impedance of Perforates with Flow.

Paper IV An Inverse Analytical Method for Extracting Liner Impedance from Pressure Measurements.

Paper V Impedance of SDOF Perforated Liners at High Temperatures.

Paper VI On Acoustic Network Models for Perforated Tube Mufflers and the Effect of Different Coupling Conditions.

Division of work between authors:

The theoretical and experimental work presented in this thesis was done by Tamer Elnady under the supervision of Hans Bodén. Paper VI was supervised by Mats Åbom. In Paper V, Timo Kontio, an M.Sc. student at the department, participated in the measurements with the help of Tamer Elnady and supervision of Hans Bodén.

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The content of this thesis has been presented in the following conferences:

1. 7th AIAA/CEAS Aeroacoustics Conference, Maastricht, The Netherlands, 28-30 May, 2001. AIAA 2001-2202: “Application of the Point Matching Method to Model Circumferentially Segmented Non-Locally Reacting Liners”.

2. 8th AIAA/CEAS Aeroacoustics Conference, Breckenridge, CO, USA, 17-19 May, 2002. AIAA 2002-2444: “Hard Strips in Lined Ducts”.

3. 9th AIAA/CEAS Aeroacoustics Conference, Hilton Head, SC, USA, 12-14 May, 2003. AIAA 2003-3304: “On Semi-Empirical Liner Impedance Modelling with Grazing Flow”.

4. 10th AIAA/CEAS Aeroacoustics Conference, Manchester, UK, 10-12 May, 2004. AIAA 2004-2836: “An Inverse Analytical Method for Extracting Liner Impedance from Pressure Measurements”. And AIAA 2004-2842: “Impedance of SDOF Perforated Liners at High Temperatures”.

5. Sixth International Conference on Production Engineering and Design for Development (PEDD-6), Cairo, Egypt, 12-14 Feb., 2002. “Impedance Modelling of Acoustic Liners”.

6. 10th International Congress on Sound and Vibration (ICSV10), Stockholm, Sweden, 7-10 July, 2003. “Mode Scattering by Hard Strips in Lined ducts”.

The work in this thesis was performed within the following Projects:

1. DUCAT: funded by EC (European Commission), contract number BRPR-CT-97-0479. (Paper I)

2. Aeroacoustics: funded by NFFP (National Flight Research Program), project number P460. (Papers II to V)

3. ARTEMIS: funded by EC (European Commission), contract number G3RD-CT-2001-0511. (Paper IV)

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Acknowledgements If I want to write a detailed acknowledgement, I will have a long list to thank. But, the top of this list is always booked to the two persons who were the reason that I am writing these lines now, and those to whom I relate any success I achieve, my Parents. They say that I behave like my father and I hope that I can achieve half of what he did. My mother is always there for me as a friend more than a mother. Now, two other people have jumped to the top of my priorities, my wife Eman and my son Yusuf. Eman has smoothly joined my life giving it strength and encouragement to always be the best. Yusuf added joy and happiness to the life of all the family with his charming smile. I dedicate this thesis and all my future work to both of them.

In historical order, I must thank all my teachers at school and at Ain Shams University in Cairo. On the other hand, it is a must to mention Dr. Ahmed Hussein and Dr. Mansour Elbardisi. The first contributed a lot to my engineering sense and the second introduced me to the field of Acoustics.

The person who invited me to Sweden and welcomed me at the airport is my supervisor, Hans Bodén. Since then, he has been helping me in almost everything. He is always there for me whenever I need him and it usually takes no more than ten minutes of discussion with him until I find the answer to my question or the solution to my problem. For me, he is the kind of supervisor that anyone would like to work with. I would also like to thank Mats Åbom, the head of MWL, for helping me throughout the thesis especially with Paper VI.

I really enjoyed working in the friendly atmosphere of the Department of Vehicle Engineering and I wish to thank all its members for that, especially Anders Nilsson. I learned from him many things I will need when I become a Head of a Department. I wish I could name all of them, but I just say:

Tack, Thanks, Merci, Gracias, Grazie, Danke, Kiitus, ًشكرا

Financial support from NFFP as well as European funded projects DUCAT and ARTEMIS is gratefully acknowledged. Last but not least, I must thank my Egyptian friends in Stockholm, Hussein, Magdy, and Sabry who reduced the feeling of homesickness.

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Table of Contents I. Introduction .......................................................................................................................9 II. Acoustic Propagation in Ducts.......................................................................................12 III. Perforated Liner Impedance Modelling ........................................................................14 IV. Experimental Techniques for Determining Liner Impedance.......................................17 V. Perforated Tube Muffler Modelling...............................................................................20 VI. Summary of the Papers and Main Results ....................................................................22

A. Paper I: Application of the Point Matching Method to Model Hard Strips in Lined Ducts..........................................................................................................................23

B. Paper II: Investigation of Mode Scattering by Hard Strips in Lined Ducts using Collocation ................................................................................................................24

C. Paper III: On the Modelling of the Acoustic Impedance of Perforates with Flow ....26 D. Paper IV: An Inverse Analytical Method for Extracting Liner Impedance from

Pressure Measurements .............................................................................................29 E. Paper V: Impedance of SDOF Perforated liners at High Temperatures ....................31 F. Paper VI: On acoustic network models for perforated tube mufflers and the effect of

different coupling conditions.....................................................................................33 VII. Recommendations for Future Research ......................................................................35 References ..........................................................................................................................35

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I. Introduction The attenuation of sound propagating in fluids contained in ducts can be due to a variety of causes. The least significant of these for gases, normally, is the “volume absorption” due to irreversible conversion of acoustic energy into heat in the fluid itself, through viscous and thermal diffusion processes. More prominent is the absorption at the wall. This can be due either to motion of the wall and the resulting transmission of sound through it, or to viscous and thermal diffusion processes at the wall. For relatively rigid walls of fairly high thermal conductivity relative to that of the fluid, these processes are more effective at the wall, in producing acoustic attenuation, than in the fluid itself. In general, the attenuation is dependent on frequency and on the degree of complexity of the individual amplitude patterns of the duct modes by which the sound is being transmitted. Presence of a mean flow modifies the attenuation both by convection effects and by shear layer effects. It can also change the acoustic properties of the wall.

A sound absorbing material is placed along a flow passage to attenuate noise generated by some upstream device, i.e. a method for passive noise control. It is usually attached to a hard surface to attenuate noise by converting sound energy into heat through viscous and thermal diffusion processes at the walls.

Liner effectiveness is a function of several parameters; some have to do with the configuration of the liner itself and others with the environment in which it has to operate. The full-scale simulation testing, which is usually necessary to define the noise attenuation produced by a liner installation, is both expensive and time consuming. Therefore, it is desirable to produce mathematical models which are capable of predicting liner performance inside the system. The key design parameter is the acoustic impedance of the treatment panel. The impedance is used as input for propagation models to predict overall attenuation inside the engine duct.

A typical liner structure consists of a perforated face-sheet covering a honeycomb or porous material, and having a solid back-plate (Fig. 1). Acoustic liners are of two types namely locally and non-locally reacting. These are defined below.

Locally reacting liners do not allow sound propagation inside the liner, parallel to the liner surface. The face-sheet is backed by a regular partitioned single-layer cellular structure (such as metal honeycomb (Fig. 2)) with solid walls perpendicular to the face-sheet plate. Such a design is called a single degree-of-freedom liner (SDOF). The locally reacting liner can be considered as an array of Helmholtz resonators, whose theory of operation is analogous to that of the mechanical mass-spring system. The mass of the gas oscillating in the aperture corresponds to a mechanical mass sliding over a resistive surface and the compressibility of the gas in the resonator cavity acts as the spring in the mechanical system. Non-locally reacting liners allow the sound waves to propagate within the liner material (Fig. 1). They are often called bulk absorbers. They usually have a single-layer construction in which fibrous material fills the panel between the porous face-sheet and the back-plate. A typical example of this material is the glass wool that is used, for instance, inside anechoic rooms.

The SDOF liner is effective over the narrowest range of frequencies (one octave) and must be tuned to the frequency band containing the single tone of greatest concern. A multiple-degree-of-freedom (MDOF) liner has a wider bandwidth that can cover the main tone and its next two harmonics (about two octaves), using various internal depths and several

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layers with different resistance, but it is more difficult to manufacture. Bulk absorbers have the widest bandwidth, extending over three octaves if the panel is made sufficiently deep to be effective at the lowest frequency.

Fig. 1 Types of acoustic liners.

Fig. 2 Honeycomb structure of SDOF locally reacting liner

Perforates are used to attenuate sound in ducts in many applications. The applications considered in this thesis are related to the reduction of transportation noise, either due to aircraft or vehicles. Increased national and international travel over the last decades has caused an increase in the global number of passengers using different means of transportation. Great effort is being directed to improving the noisy environment in the residential community. There has been a significant progress during the last forty years concerning the overall vehicle and aircraft noise reduction; but the noise level and annoyance for people living in residential areas and in the vicinity of airports is constantly increasing. The appearance of low-cost airlines helped increasing air traffic in small airports that compete with low airport taxes and more relaxed regulations. Residents around airports and close to highways blame aircraft and vehicle noise for the appearance of sleeping disorders and other noise related diseases such as hypertension, fatigue, and loss of concentration.

Stricter noise requirements are being implemented by international noise regulatory authorities and governments. In order to achieve long-term improvements, a need for quieter means of transportation has evolved. The propulsion system in an aircraft is the major source of noise during relevant flight conditions. The engine noise in a vehicle dominates the total radiated noise at low speeds especially inside cities. Many recent studies on noise reduction involve the use of perforated plates in the air and gas flow ducting connected to the engine.

One of the main applications of using liners is inside aircraft jet engines. Forty years ago, the engines were of the turbojet type where jet noise dominated. Recently, turbofans, having large bypass ratios, are commonly used in modern aircrafts. In a turbofan engine, a large-diameter fan spins at a high speed and fan noise dominates all other noise sources in

Perforated face-sheet Partitions (e.g. honeycomb)

Porous material Solid back-plate

Locally reacting liner Non-locally reacting liner

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the engine. Fig. 3 shows the sources of noise and the share of each source in the total radiated noise. The engine is wrapped in a nacelle and fan noise has two paths to propagate; the inlet and bypass ducts. The ducts have acoustic treatment to damp and absorb noise generated by the fan. Sound, which is not absorbed, propagates out the front and back of the nacelle and can be heard at the ground. In addition, liners are used inside the exhaust duct to minimize turbine and combustor noise.

(a) Turbojet (low bypass ratio engine). (b) Turbofan (high bypass ratio engine).

Fig. 3 The percentage of each noise source in different types of engines 1.

The liners in service inside aircraft engines are of the locally reacting type. Bulk absorber treatment has not been used in aircraft engines in commercial service because of structural design difficulties. A liner with the fibrous form will be faced with a high-porosity screen to hold the fibres in place; however, because of its high porosity, it allows the material it contains to absorb and retain the fluid. Bulk absorbers cannot be used either in the cold ducts (where freeze/thaw action on retained water is the problem) or in the hot ducts (where combustible fluid retention is a fire hazard). Aircraft manufacturers are interested to use porous ceramic materials or metallic foams in liners to obtain better performance. This will be the main trend for future liner research.

Fig. 4 The fan of an aircraft engine (Courtesy of Airbus).

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Other applications of perforates are inside car mufflers to reduce the engine noise, inside combustion chambers of rockets or gas turbines to control combustion instabilities, and inside ventilation ducts to attenuate flow induced noise.

II. Acoustic Propagation in Ducts Although much success has been achieved in understanding and optimizing the attenuation of sound waves in acoustically lined ducts, there still remain various areas that require further investigation. In particular the effects of a circumferential variation in wall admittance on the propagation and radiation of sound from ducts has received relatively little attention. Although the lined wall may have been designed to be uniform, a variation in admittance may result from the mounting techniques, which can make the existence of hard strips inevitable, to hold the liners in place. On the other hand, it may be possible that by a deliberate choice of a non-uniform liner, an increase in attenuation can be achieved.

Fig. 5 The inlet nacelle being mounted in front of the fan (Courtesy of Boeing).

The segmented liner problem (circumferential or axial variation of liner impedance) was first addressed in the seventies. A mathematical model was developed for rectangular ducts with uniform mean flow 2. The aim was to define methods for obtaining improved multi-segment lining performance by taking advantage of relative placement of each lining segment. Properly phased liner segments reflected and spatially redistributed the incident acoustic energy and thus provided additional attenuation. Optimal configurations were presented to improve the attenuation rate inside infinite ducts.

Watson 3 developed Finite Element Methods to analyze sound attenuation in ducts with a peripherally variable liner. For a finite duct with no flow, he showed that the attenuation rate near the frequency of maximum attenuation drops significantly if a small portion of a peripheral liner is removed.

From the in-flight circumferential modal spectra of the Rolls-Royce Tay 650 engine mounted on the Fokker 100 aircraft, Sarin and Rademaker 4 found that the sound field propagating upstream in the inlet is strongly modulated by hard-walled strips in the lined area, the non-cylindrical geometry of the duct and the non-axisymmetric flow velocity distribution. In order to study the effect of the modulation of the acoustic field by the hard-walled strips separately, an experimental program in the NLR spinning mode synthesizer

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was carried out 5. The m-mode spectra of the transmitted field clearly showed modulation effects caused by the hard-walled strips and the modulation increased with increasing mode number. Nevertheless, it was found that the total in-duct transmitted acoustic energy flux is not much influenced by the scattering of the incident modes.

It is believed that this modal modulation is caused by scattering which occurs at the hard-wall strips. As an incident mode passes over these hard-walled strips, its acoustic energy tends to be scattered into different lower-order modes, thus altering the modal character of the radiated field. The major concern is that scattering into lower-order modes has an adverse effect on the liner performance, since energy is transferred into modes which are less attenuated by the liner. In addition, it is possible that cut-off modes may scatter into cut-on modes with a resulting increase in radiated energy.

Regan & Eaton developed a FE model 6, 7 and used it to analyze a finite duct lined with locally reacting liner with different number of hard strips of different widths. They demonstrated that the transmitted modal spectrum can be significantly modulated by the presence of hard strips, but for the frequency range considered (ka < 10), the overall transmitted power is not significantly affected.

McAlpine et al. 8 presented an assessment of the effect of scattering by the liner hard strips for both finite element and boundary element calculations. The results demonstrated that the scattering depends on the fan speed and the number and widths of strips. They showed that the level of the scattered tones can be significantly reduced by having thinner splices. But at high fan speeds, the rotor-alone field is well cut-on and propagates close to the duct axis. It was anticipated that more novel noise control methods will be required to significantly improve the sound attenuation at these speeds.

Fig. 6 The intake of an aircraft engine showing the position of hard strips (Courtesy of Rolls-Royce).

The solution of the eigenvalue problem inside a circular duct is simple as long as the duct is axi-symmetric, because the variables are separable. Reference 9 presents a complete review of the available methods to analyze lined ducts having basic shapes (circular or rectangular cross-section) with uniformly distributed boundary conditions. When the boundary condition varies around the circumference, the property of the field being separable breaks down. Therefore, a more complicated technique has to be used. The most well known method is probably FEM 6, 10. FEM may, especially for the analysis of higher order modes, be somewhat cumbersome and is thus not ideally suited in an iterative design process. However, there are a number of other methods such as null-field 11, Rayleigh-

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Ritz 12 and collocation 13. The collocation technique has been already tested to handle hard strips in lined ducts and showed good efficiency 14. The null field approach, not only gives better physical insight but also faster convergence and more accurate results. This is to the cost of a geometrical restriction upon the cross sectional shape regarding the eigenmodes. The Rayleigh-Ritz formulation appears to be an approach which combines the flexibility of FEM, regarding cross sectional shape and in-homogeneities of the absorbent, with the speed of the null field approach. It is however best suited for analysis of the lowest eigenmodes and it does not, at least not directly, give so much insight into the overall behaviour of the system as does for example the null field approach.

Fuller 15 developed a model which considers the acoustic propagation in an infinite circular duct with circumferentially varying wall admittance with no flow. An exact solution was obtained and was used to investigate the characteristics of wave dispersion and mode shapes. He specified the admittance distribution as a Fourier series.

Bi et al. 16 used the Multimodal Method to solve the problem of sound transmission through circular cylindrical ducts lined with a non-uniform impedance in the absence of flow. The liner impedance was assumed to be piecewise constant along the axis of the duct, and can arbitrary vary along the circumference. First, the Euler and continuity equations are projected over the eigenfunctions of a rigid uniform duct. Mode coupling effects are then explicitly expressed by the inverse Fourier transformation of liner admittance. Second, a scattering matrix is used to express the reflection and transmission coefficients of each axial uniform segment from which a global scattering matrix can be constructed.

Another well-known analytical technique which is useful to analyze acoustic propagation through ducts with finite lengths is mode matching. It is used to determine how energy is transferred and scattered between modes at an interface where there is a discontinuity in either duct dimensions or transversal boundary conditions. Cummings & Chung 17 used mode matching to calculate the sound attenuation of a circular dissipative silencer with flow. Glav 18 used it to calculate the transfer matrix for a dissipative silencer of arbitrary cross section but with no mean flow. Peat 19 calculated the transfer matrix for an absorption circular silencer element with flow. The aforementioned studies were done for silencers in automotive applications; therefore, the used liner was non-locally reacting. There was sound propagation inside the liner but the mean flow is neglected compared to that in the main duct. Peat 20 also used mode matching to determine equivalent acoustic impedance at the junction of an extended inlet or outlet inside a circular silencer to be able to calculate its transfer matrix. Lansing & Zorumski 21 used the technique to evaluate the effect of axially changing the wall admittance of a circular lined duct with no flow in order to evaluate the effect on the transmitted power.

In paper I of this thesis, the point matching method is presented in details applied to lined ducts having hard strips. It was used to obtain the wavenumbers and mode shapes for an infinite lined duct. Both locally reacting and non-locally reacting liners were studied. In paper II of this thesis, the effect of hard strips on the transmitted energy across a lined duct of finite length is studied. The model was constructed using mode matching.

III. Perforated Liner Impedance Modelling The key design parameter of perforates is the acoustic impedance. The impedance is what determines their efficiency to absorb sound waves. Several studies have been conducted to

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develop impedance models, including the effect of different configurations and surrounding conditions. In spite of the large number of published research, a single verified global model does not exist. The type of liners considered in this thesis is the single-degree-of-freedom (SDOF) liner. It consists of a sandwich structure: a solid back-plate and a perforated face-sheet enclose a honeycomb cellular separator (Fig. 7). The locally reacting liner can be considered as an array of Helmholtz resonators, analogous to mechanical mass-spring systems. The mass of the gas oscillating in the aperture corresponds to a mechanical mass sliding over a resistive surface and the compressibility of the gas in the resonator cavity acts as the spring in the mechanical system.

In the early seventies, researchers began to investigate the impedance of SDOF liners with a perforated face-sheet. In this paper, the theoretical approach is based on Crandall’s theory of acoustic propagation in perforates 22. Melling 23 and Guess 24 used this approach, together with some measurement results from Ingård 25 and Sivian 26. In order to derive the expression of the viscous impedance due to lumped face-sheet resistance and mass reactance of an array of orifices, Crandall first assumed the perforate hole to be an infinite duct, then end corrections are added to account for the finite length of the perforate holes.

Later on, other approaches have been used to develop other models. Motsinger & Kraft 27 assumed an acoustic resistance equivalent to a DC flow resistance inside the hole. Therefore, there was no frequency dependence for the resistive part in their impedance model. This works well for low frequencies. Hersh, Walker & Celano 28 derived a simple, one-dimensional, fluid mechanical-lumped element model applying the conservation of mass and momentum across a control volume just outside the orifice. They assumed the flow inside the orifice to be consisting of a uniform part within an inviscid core, and a laminar boundary layer part. They determined some parameters in the model empirically.

Fig. 7 Location of acoustic liners in an aircraft engine.

Acoustic liners can be subject to one of two types of flow, or both. These are grazing flow and bias through flow. Grazing flow is the flow of air outside the liner and parallel to the face-sheet (Fig. 7). It normally exists as a mean flow inside the duct in which the liner is installed, e.g., inside a turbofan engine. Bias flow is the introduction of airflow, blowing or suction, perpendicular to the perforate of the acoustic liner, and through its holes. It is usually used to cool the face-sheet when the temperature of the air inside the duct is very

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high. Through flow configurations can be also found in many advanced car mufflers. It has been known for several decades that flow strongly affects the acoustic behaviour of perforated structures, whether the flow is purely grazing, purely normal (through the holes), or both.

Many researches were dedicated to investigate the physical reasons behind the change in impedance with grazing flow. Flow visualization techniques, and hot-film and laser anemometry were used by a number of researchers to observe the flow patterns close to the orifice. Investigations concentrate on how sound energy is dissipated in the near field of the orifice. Ronneberger 29 used a thin filament of coloured dye in water to visualize the forced oscillation of the interface between the fluid at rest inside a cavity and the grazing mean flow on the outside of the cavity. He compared the experimental results to his previously developed model 30. The values of impedance predicted by the model were larger than the measured values by a factor of 3, but the general dependence of the impedance on the mean flow velocity was correctly predicted. The model assumes that distortions of the mean flow emerge at the leading edge of the orifice and travel downstream. The interaction of these distortions with the trailing edge of the orifice affects the generation of the new distortions constituting some kind of feedback loop.

In the context of the self excited oscillations of a resonator in grazing flow, Nelson, Halliwell & Doak 31, 32 reported an extensive experimental and theoretical investigation of an orifice flow field, with emphasis on the production and absorption of acoustic energy. They showed that the fluctuating Coriolis force brought about by the combination of potential and vortical motion leads to an energy interaction between mean and fluctuating flows; thus acoustic energy is extracted from the mean flow.

Worraker & Halliwell 33 performed detailed experiments to understand the detailed flow-acoustic interactions taking place at the liner surface. They discussed their results in the light of existing theoretical models concerning flow induced impedance changes of cavities. They found that generally, these models are not appropriate for explaining phenomena in the real jet engine liner. In particular, they found that the approach presented by Nelson et al. 31, 32 is the most matching model to their experiments.

The difficulties associated with the theoretical modelling of the acoustic behaviour of perforates subjected to grazing flow have resulted in a reliance on empirical models based on experimental data. Early attempts to relate the orifice impedance to the mean flow parameters tended to rely on a mean flow Mach number as an appropriate measure of flow speed, or merely on a mean flow velocity 34. It was later realized that the physics of the problem was not so simple as to permit such elementary universal correlations 35, 36. It was found that it is more convenient to include the properties of the turbulent layer in the form of the friction velocity which is defined as the square root of the ratio between the wall shear stress and the density of the fluid at the wall. Three main semi-empirical models were developed by Kooi & Sarin 37 , Cummings 38 and Kirby & Cummings 39. Each model covers a certain range of perforate parameters and generally they do not agree with each other for the aeronautic liners.

In paper III of this thesis, a model based on Crandall’s theory 22 was developed. Some parameters were included from later models which were not in the original model. Other parameters were determined from experiments and semi-empirical formulae were derived. In paper V, investigated the effect of high temperature and temperature gradients on the perforate impedance was investigated without flow.

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IV. Experimental Techniques for Determining Liner Impedance The normal incidence acoustic impedance of passive linear materials can be experimentally determined in several ways. The first known method is the standing wave tube method which has been a commonly used procedure for impedance measurements for over 80 years 40. This method is based on the unique relationship between the standing wave parameters and the test specimen impedance. It is implemented by means of a tube that guides plane waves such that normal incidence is achieved on the test specimen surface. In its original version, a continuous trace of sound pressure level versus distance was measured, and the standing wave ratio and the null location were determined from this analogue trace. This procedure was time consuming and involved inaccuracy in the measurement of distances.

The development of Fast Fourier Algorithms and their implementation in laboratory analyzers provided new possibilities. The standing wave method evolved to an easier and more accurate method using fixed microphone positions, the two-microphone technique. Seybert & Ross 41 were able to separate the incident and reflected wave spectra from measurements of auto- and cross-spectral densities between the microphones. Chung & Blaser 42 used the frequency response function between the two microphones to determine the acoustic reflection coefficient perpendicular to the duct axis. The method uses an impedance tube with a sound source connected to one end and the test sample mounted in the tube at the other end. This method has been used extensively and has become an ISO standard method 43. A modified version of this method is used in this paper to decompose the acoustic fields in the hard ducts before and after the lined section, in order to determine the reflection coefficient perpendicular to the duct axis. Åbom & Bodén 44 discussed and analyzed the influence of all types of measurement errors on the two microphone methods in ducts with and without flow. They drew conclusions concerning the best configuration and useful frequency range to avoid errors in the frequency response function estimate.

Another very widely used method is the in-situ technique. Dean 1 introduced this method in 1974. Since then, it has been extensively used to measure the impedance of locally reacting acoustic liners with and without flow. It is based on the fact that for a resonant cavity, under certain conditions, a simple relationship exists between cavity acoustic pressure and particle velocity. It is further assumed that the particle velocities on either side of the facing sheet are identical. The conditions of application are that the liner is locally reacting and the wavelengths are long compared to the cavity cross dimensions. The only quantity which is required to be measured is the frequency response function between the acoustic pressure at the surface and the acoustic pressure at any point inside the cavity. The microphones are mounted flush to the back-plate inside the cavity and to the facing sheet outside the liner.

The in-situ technique has some advantages over the two-microphone technique. The impedance is measured at the actual liner surface, whereas in the two-microphone technique, knowledge of the sound propagation inside an impedance tube is required. The in-situ method does not have the same frequency limitations as the two microphone method. In practice, there are some problems associated with the measurements when higher order modes exist, because of the distance between the two microphones. The theory assumes that the two pressures are measured at the same location, one inside the cavity and the other at the liner surface. There are many references, for example 28, which describe the experience gained using this method. Measurements can be carried out in

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laboratories with and without flow, and even in a real engine during flight 46. It can also be extended to measure the impedance of two 47 and three 48 degrees of freedom liners.

Several drawbacks and problems have been reported with the use of this method. There are usually strong near field effects which influence the reading of the microphone flush to the surface, which is subject to the incident sound wave. These effects are even stronger with grazing flow. A correction scheme is proposed in paper IV of this thesis for the no flow impedance measurements inside an impedance tube. Moreover, the in-situ method does not work when there is, for example, porous material inside the liner so that the sound propagation inside the cavity can not be predicted.

Malmary et al. 49 presented a comparison between the in-situ method and another method using a moving microphone probe. They found that the in-situ method is easier to use and gives better results by comparison with theoretical models from literature. However, they recommended that the moving probe method should not be given up because it gives more information because the pressure field is measured at numerous points across the duct.

Fig. 8 First development of indirect measurement techniques. (Courtesy of NASA Langley).

A continuing measurement problem in acoustic treatment technology is the accurate determination of normal incidence impedance of acoustic material in grazing flow environments. The problems discussed above motivated the evolution of indirect approaches, which depend only on the measurement of the acoustic pressure at selected locations outside the liner. These indirect methods have an extra advantage of not destroying the sample by drilling holes for the microphones. Usually, it requires extremely precise instrumentation of the liner sample. Inverse techniques based on propagation models for the lined duct are becoming popular because of their convenience and advantages.

The classical approach, the so-called infinite-wave-guide method, involves measuring the sound attenuation properties of an assumed single, unidirectional propagating mode, in a wave-guide lined with the acoustic material over a sufficient length to be effectively

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infinite. This data is then used with the solution to the wave equation in an infinite wave-guide to establish the impedance of the material at the boundary. The evolution of wave-guide models for this purpose began over thirty years ago with a uniform mean flow model 50. An analytical expression for the impedance of the material was derived using known functions and the measured axial wavenumber. Infinite wave-guide models are applicable, in a very straightforward manner, to situations for which a non-reflecting single mode propagates within the wave-guide containing the liner whose impedance is unknown. However, many conventional liners generate more complex acoustic fields. Thus, measured data must now be interpreted as the superposition of many propagating modes.

Watson et al. 51 developed this approach further and presented a Finite-Element-based numerical method for educing the impedance of an acoustic material placed at the wall of a two dimensional duct that conveys a multi-modal sound field. Their proposed method depends on the measurement of acoustic pressure at selected locations at the upper wall of a rectangular duct to determine the normal incidence impedance of the material placed at the lower wall. These measurements are then used as input to a Finite Element propagation model combined with a coarse/fine grid impedance plane search technique to extract the impedance of the material. The procedure consists of repeatedly cycling through the solution to the boundary value problem, and obtaining a set of upper wall pressures for each impedance function. As each new set of wall pressures is computed, it is compared to the known (measured) values until convergence is achieved. Convergence of the procedure is guaranteed, since the boundary value problem is well-posed. They first tested the method with input data computed from modal theory and then contaminated by random error. Then they validated the impedance extraction method using measured data and a finite-length liner 52. Later 52, they used the Davidon-Fletcher-Powell optimization algorithm to educe the impedance that minimizes the difference between the measured and the numerically computed upper wall pressure. Results showed that this more robust method reduces the time required to make impedance determinations by a factor of 30 as compared with the contour determination method. After developing and validating the method for the no flow case, they could reproduce the measured normal incidence impedance in a uniform flow 54. They later showed that the liner resistance educed in the presence of shear flow can differ from that educed when only uniform flow is modelled 55. The effect of shear flow on the reactance of the liner was found to be not significant. They developed their code further to be able to handle segmented liners in the circumferential and axial directions (checkerboard liners) 56.

Aurégan et al. 57 proposed another method based on the measurement of the scattering matrix of the lined section using the two-source method. The transmission and reflection coefficients before and after the lined section can be calculated from the transfer matrix. They proposed a new theoretical model called “Scattering Matrix by Multimodal Method (S3M)” to compute the scattering matrix of a lined duct with flow. The basis of this method is to project the propagation equations on a complete set of basis functions. They used uniform mean flow reasoning that the use of shear flow does not affect the transfer matrix coefficients. On their first attempt, they used a minimization criteria based on one of the transmission coefficient. There are still some unsolved problems with the technique; for instance, different coefficients resulted in different values for the liner admittance.

In paper IV of this thesis, a similar, yet simpler indirect method is proposed to educe liner impedance from the measurement of the pressure field using an analytical propagation model constructed using mode matching.

20

V. Perforated Tube Muffler Modelling One of the biggest challenges facing modern car and truck manufacturers is to reduce the overall vehicle noise. Noise regulations are now considered one of the main parameters to evaluate the quality of the vehicle. One of the main sources is the engine where its noise is usually damped by means of acoustic mufflers. Perforated tubes are commonly found inside automotive mufflers. They are used to confine the mean flow in order to reduce the back-pressure to the engine and the flow generated noise inside the muffler. Ideally, perforates are acoustically transparent and permit acoustic coupling to an outer cavity acting as a muffler. Being able to theoretically model these mufflers enables car manufacturers to optimize their performance and increase their efficiency in attenuating engine noise. Therefore, there has been a lot of interest to model the acoustics of two ducts coupled through a perforated plate or tube. Generally, the modelling techniques can be divided into two main groups, the distributed parameter wall impedance approach, and the discrete or segmentation approach.

Fig. 9 A complete exhaust system (Courtesy of Faurecia)

In the distributed approach, the perforated tube is seen as a continuous object, and the local pressure difference over the tube is related to the normal particle velocity via surface averaged wall impedance. The main challenge facing this approach is the decoupling of the equations on each side of the perforate. Using this approach, however, results in closed form expressions for the acoustic transmission and therefore the calculations are very fast. Sullivan & Crocker 58 presented the first analysis of this approach. They only considered through flow concentric resonators with the flow confined in the main duct. They did not have the decoupling problem because the flow inside the cavity was assumed to be zero. Therefore, their model cannot be applied to situations with cross flow. Moreover, it does not work for non-rigid boundary conditions at the side plates of the muffler; e.g. extended inlet and outlet configurations. Later, Jayaraman & Yam 59 introduced a decoupling approach based on a mathematical assumption which requires the mean flow Mach number to be equal on both sides of the perforate. This is considered a major disadvantage because this case is hardly found in practice.

Rao & Munjal 60 presented a method to overcome this problem with a generalized decoupling analysis which allows for different flow Mach numbers in the two pipes. They used the same equations as Sullivan & Crocker 58. They also extended the method to be able to handle any boundary conditions at the muffler end plates. Peat 61 pointed out that their decoupling conditions are only satisfied for the two simple cases of M1 = M2 = 0 or M1 = M2. He was unable to find analytical expressions which fully satisfy the generalized

21

approach and hence resorted to a numerical decoupling solution. Munjal et al. 62 presented a first attempt to use the numerical decoupling technique but they reported problems of numerical instabilities close to the peaks of the transmission loss.

Peat 61 derived more general equations than all previous models by allowing the Mach number and impedance to vary along the length of the perforate. This generalization contains some of the properties and advantages of the segmentation approach. He only made calculations using constant parameters. Numerical decoupling overcomes the modelling deficiencies of the analytical techniques, introducing an additional cost in terms of computing time as it requires solving an eigenvalue problem.

Dokumaci 63 developed a new approach for numerical implementation of the distributed parameter method based on the so called Matrizant theory. This approach is comparable to Peat’s numerical decoupling method in that an eigenvalue problem has to be solved; however, it is able to correctly handle a mean flow velocity gradient which was earlier a shortcoming of the distributed parameter methods. Dokumaci criticized Peat’s method pointing out that a gradient term appears to be missing in the governing equations, and that he neglected the mean flow variation in some terms which might be of the same order as the retained terms. There was a published correspondence on this issue 64.

The discrete or segmentation approach was first developed by Sullivan 65, 66. In this approach, the coupling of the perforate is divided into several discrete coupling points with straight hard pipes in-between. Each segment consists of two straight hard pipes and a coupling branch. The total 4x4 transmission matrix of the perforated element is found from successive multiplication of the transmission matrices of each segment. Kergomard et al. 67 used this concept and presented, for the case of two waveguides communicating via single holes, a model for waves transmission in a periodic system. Dokumaci 68 presented another discrete approach based on the scattering matrix formulation. He discussed several possibilities for modelling the connecting branch. He proposed a continuous visco-thermal pipe model, a continuous isentropic pipe model with end corrections, and a lumped impedance model (as in Sullivan). The conclusion was that the lumped element model is the most appropriate for this problem. He sometimes added a correction to the segment length so that his results match Sullivan experiments. It was unclear how he determined this end correction and on what basis he includes it or not.

The distributed approach is mainly convenient for relatively simple perforated mufflers. There are however many complicated muffler configurations in which perforations are used in non-standard ways. The discrete approach is more convenient to analyse advanced muffler systems because of numerical simplicity and flexibility. As will be demonstrated in this paper, it cannot only handle arbitrary complex perforated systems, but it is also straightforward to model gradients in mean flow and temperature.

One main difference between the distributed and discrete approaches is the definition of the coupling conditions over the perforate. In the distributed approach, continuity of acoustic momentum is usually imposed (which means that there is no transfer of axial momentum through the perforations), whereas in the discrete approach, continuity of acoustic energy is usually assumed (which means that there is no dissipation inside the pipe). Recently, Aurégan & Leroux 69 presented an experimental investigation which proved the failure of the discrete models with flow. There was no previous careful experimental evidence of the accuracy of these models with flow, although they have been always used with flow. Aurégan & Leroux suggested that the coupling condition should be

22

something in-between conservation of energy and conservation of momentum. In paper VI of this thesis, the effect of different coupling conditions is further investigated. A generalized segmentation model was developed to be able to incorporate any coupling condition. In addition, another version of the segmentation approach is also presented.

VI. Summary of the Papers and Main Results The objective of Paper III is to review all available models in the literature and include all effects in a single global model. A model based on Crandall’s theory 22 was developed. Some parameters were included from later models which were not in the original model. Other parameters were determined from experiments and semi-empirical formulae were derived. Measurements were performed with and without flow to validate the model. The effect of nonlinearities at high SPL was also investigated without flow. No measurements were performed with through flow but the paper reviews the literature on this aspect. The developed model is an effective tool for the designer to predict the impedance of acoustic liners in order to have an accurate estimation of the boundary condition in propagation models. Paper V investigated the effect of high temperature and temperature gradients on the perforate impedance without flow. Impedance measurements were performed at high temperatures and the results were compared to the theoretical model developed in Paper III.

The rest of the thesis has a common theme, to find alternative analytical solutions rather than numerical methods for some problems. Numerical methods include Finite Element Methods (FEM), Boundary Element Methods (BEM) or combined Computational Fluid Dynamics and Computational Aeroacoustics (CFD/CAA). They can handle complicated geometries and complex problems as found in practice, and give more information about flow distributions. Analytical solutions provide a deep physical understanding of the nature of the results, whereas numerical methods give the solution directly. Moreover, analytical methods are easier to use for setting up the model and to code without the need of complicated and expensive software. They require less time to implement and run.

Paper IV presents an analytical inverse method to extract liner impedance from pressure measurements. It is based on the mode-matching technique instead of the previously developed Finite Element Methods. The new method was validated with measurements of perforate impedance. Two applications of using perforates to reduce engine noise were considered. The first is the investigation of the effect of the hard strips which exist in the lined ducts of aircraft jet engines. A new analytical method based on collocation or point-matching technique is presented in Paper I for the infinite cylindrical duct. The new method is extended in Paper II to include scattering at the finite ends of the lined duct when it is connected to a hard duct at each end.

The second application is the modelling of perforated tubes that can be found in automotive silencers. This is another perspective to duct propagation modelling where only low frequencies are considered (plane wave propagation). A generalisation to the classical Sullivan model 65 was developed in Paper VI incorporating different coupling conditions between the acoustic fields on both sides of the perforate. An example of the analysis of a complicated muffler system is presented using both the generalized segmentation model and a new approach which is based on the two-port formulation.

The papers in the thesis are numbered and arranged in chronological order.

23

A. Paper I: Application of the Point Matching Method to Model Hard Strips in Lined Ducts In order to increase the modelling capabilities of the acoustic propagation in lined ducts, an alternative technique is presented in this paper; the point matching method. This method is based on the well-known numerical concept of collocation 13. The used point-matching formulation is valid for any cross-section with a circular central passage. The given solution fulfils both the governing equations and the boundary conditions at the collocation points. If the number of points is large enough, the eigenvalues and eigenmodes can be obtained with an acceptable accuracy. This technique has been previously applied to lined ducts for the exhaust systems silencers in automobile applications 70.

The above analytical approach is presented in details in this paper. The first part of the paper gives an introduction to acoustic duct propagation. Two cases are presented, the locally and non-locally reacting liners. For the non-locally reacting lining, the propagation inside the liner must be considered; therefore, two regions are included in the formulation. The first is inside the central passage and the other is inside the liner itself. There is continuity of acoustic pressure and normal particle displacement at the surface between the two regions. The envelope surface is assumed to be completely rigid. The normal velocity at this boundary is equal to zero.

(a) The collocation points used with the bulk absorber with 0,1,2 and 3 hard strips.

(b) The collocation points used with the locally reacting liner with 0,1,2 and 3 hard strips.

Fig. 10 The configurations investigated in this paper (The black points represent hard surfaces).

The problem is simpler when the non-locally reacting liner is replaced by a locally reacting liner. There is only one region for sound propagation which is analogous to the inner region in the previous case, because of the nature of locally reacting liners which do not allow sound propagation inside the liner parallel to the liner surface, as non-locally reacting liners do. The effect of the liner can be modelled, due to its locally reacting nature, as an impedance boundary condition at the boundary of the flow duct. The ratio between the acoustic pressure and the particle normal velocity at the duct wall is equal to the liner impedance, which is a physical property of the liner. The basic idea behind point-matching or collocation is to assume the each mode inside the duct to be approximated by finite expansions in the polar eigenfunctions. Each mode is then forced to fulfil the boundary conditions, corresponding to each case, at the collocation points which are predefined at the boundaries. The dependence of the formulation on points allows the variation of the duct shape or boundary condition between different points.

24

A test duct, 1 m in diameter was considered. Locally and non-locally reacting liners were added with 0, 1, 2 and 3 hard strips of an included angle of 10o. For the case of 1 strip, the included angle was varied to test the effect of varying the width of the strip. 5, 10, 15 and 20 degrees were considered. Although the formulations presented in the paper can handle flow in the main duct, the results shown are only for cases without flow. The choice of points must be adapted to the geometry of the duct. To obtain accurate results, two points are allocated very close to each discontinuity, one on the hard part and the other on the lined part. Fig. 10 shows the distribution of the points for different cases.

It was found that non-locally reacting liners exhibit some advantages over the locally reacting ones. Their frequency range of effective attenuation is wider and they offer more attenuation. Moreover, they are more robust in behaviour when hard strips are present inside the lined duct. Few changes in their behaviour can occur and attenuated modes may propagate and vice versa. On the other hand, when hard strips exist in ducts lined with locally reacting liners, the modes are greatly affected by the number and widths of the strips. The main reason for this is that during material stepping, modes move in a complicated pattern and exchange their order. There is a high possibility that a mode, which is attenuated inside a uniform liner duct, starts to propagate. Thus, the total attenuation inside the duct will be affected by the energy distribution among the modes.

B. Paper II: Investigation of Mode Scattering by Hard Strips in Lined Ducts using Collocation The collocation technique was used in Paper I to calculate the wavenumbers of the modes in an infinite duct of an arbitrary cross section or boundary conditions. The problem is taken a step forward in this paper by considering a duct of a finite length. The field in the lined duct must be linked to the field in the hard ducts before and after the lined section. Mode matching was used to determine how energy is transferred and scattered between modes across the lined section when hard strips exist. The formulation presented here is valid for any cross section with flow, as long as the fields in the three ducts are pre-determined, provided that that the liner in the intermediate duct is locally reacting. The boundary conditions at the interface between any two adjacent ducts implies the continuity of acoustic pressure and axial velocity at z = 0 and z = L

Fig. 11 Incident, reflected and transmitted modes in the calculation domain.

The considered configuration is shown in Fig. 11. A single mode, of any order m and n, is incident in the inlet duct emanating from a sound source, e.g. a rotating fan. The generated wave, Ψ1i, travels towards the lined section and is scattered at z = 0 to a reflected, Ψ1r, and transmitted Ψ2i wave. The travelling wave in the lined duct is further scattered at z = L into

Ψ1i Ψ2i

Ψ3i

Ψ2r Ψ1r

Lined section Outlet Hard Duct Inlet Hard Duct

M

Fan

z=0 z=L

z

25

reflected, Ψ2r, and transmitted waves, Ψ3i. The hard ducts are assumed to be semi-infinite so that no waves are reflected from the other ends. The numbering of the fields is as follows: 1 for the inlet hard duct, 2 for the lined duct and 3 for the outlet hard duct. The letter i refers to the incident wave and the letter r refers to the reflected one. The boundary condition in the lined duct can vary around the circumference to allow the definition of the longitudinal hard strips.

The combined collocation and mode matching technique was coded and a series of simple duct tests were examined to validate the formulation and the computer code. All the test cases were compared with reference 7 to verify the application of the new technique. Three fundamental cylindrical duct test configurations were used for validation before including hard strips. In the first case, the intermediate duct was completely hard as the inlet and outlet ducts. The second test case was a semi-infinite hard walled duct with uniform flow and one end closed by an acoustically hard termination. This configuration is physically unrealistic, but it provides a useful test case to verify the correct prediction of the reflected modal amplitudes at the modal input plane. The third test case was more appropriate where a liner of finite admittance is placed at the walls of the intermediate duct. In Fig. 12, the acoustic field is plotted as contours of the amplitude of the computed complex pressure. It can be seen that there is close agreement between the mode matching and Finite Element calculations.

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

z/a

r/a

(a) Mode matching.

(b) FEM from reference 7.

Fig. 12 Calculations of amplitude pressure contours for incident mode (0, 2), impedance (2 , -1) in the intermediate duct, M = -0.3 and ka = 7.4, L/a = 2.

For the uniform liner case, only the incident mode and other modes having the same circumferential order m appear in the transmitted spectrum. This is because there is no mechanism to generate modes of other circumferential orders. Finally, the combined collocation mode matching technique was applied to a configuration where the lined section has two diametrically opposite hard splices. Splices scattering effects cause modes of other circumferential orders to be excited. By analogy to the rotor-stator interaction theory 71, the circumferential order of the modes excited by scattering can be specified by

26

where mi is the circumferential order of the incident mode and N is the number of splices. This is illustrated in Fig. 13, where the scattered spectrum of the collocation-mode matching technique is compared to that calculated by finite elements in reference 7 and the multimodal method in reference 16. Although there are some discrepancies between the three methods in the exact values of the amplitudes of the scattered waves, the general shape looks similar for the three methods. The result of the collocation mode matching technique is similar to the finite element calculations in that the amplitude of the scattered mode (1, 1) is higher than that of the mode (1, 0). On the other hand, the collocation mode matching technique is similar to the multimodal method in the decreasing amplitudes of the scattered higher order modes of zero radial order with increasing m (or decreasing –m), whereas in the Finite Element calculations, the amplitude of the scattered mode (-1, 0) is smaller than that of the mode (-3, 0).

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00

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Circumferential mode number (m). (a) Collocation and mode matching. (b) FEM from reference 7. (c) Multimodal Method from

reference 16.

Fig. 13 Comparison of the scattering spectra calculated by the collocation mode matching technique, FEM from reference 7, and the multimodal method from reference 16. M = 0, ka = 6.8, 2

strips (.75 rad), and incident mode (1, 0).

C. Paper III: On the Modelling of the Acoustic Impedance of Perforates with Flow In spite of the large number of published research papers, a single verified global model does not exist for the acoustic impedance of perforates with grazing flow. Some models give results close to each other but others give significantly different results. There are still some points of debate. First, there is still an argument about defining the end correction that is added due to the finite length of the orifices. Second, the interaction between neighbouring orifices is still under investigation. Third, there is no grazing flow model that can be applied for all liner configurations. Fourth, there is not much published information on the effect of temperature and temperature gradients inside the liner. The temperature issue will be dealt with in Paper V in this thesis. The objective of this paper is to attempt to find an answer to the above questions by performing experiments and developing a semi-empirical model. It will be an effective tool for the designer to predict the impedance of acoustic liners. In order to have a complete model for the impedance, different terms are considered separately. First, the viscous term caused by the dissipation at the orifice walls is considered. Then allowances are made for the nonlinearities caused by high incident particle velocities and airflow over and through the surface.

The value of the discharge coefficient plays an important role in impedance prediction; therefore, it has to be determined accurately. Measurements were made on different

Nmm i ⋅±= β β = 0, 1, 2, … (1)

Radial mode order (m)

|c+(q)|

|a+(2)|

27

orifices to determine the discharge coefficient as a function of the thickness and diameter. The discharge coefficient was determined by measurement of the DC flow resistance of the hole, then a least squares solution for the discharge coefficient was found that best fits a theoretical model based on fluid mechanics principles. The result is a contour plot that gives the discharge coefficient for any orifice of thickness up to 2.5 mm and diameter 5 mm.

A model based on Crandall’s theory 22 was developed for the no flow case. Some parameters were included from later models which were not in the original model. Other parameters were determined from experiments and semi-empirical formulae were derived. The normal incidence impedance of 36 orifices was measured at low and high incident sound pressure levels in order to drive the orifice impedance into the nonlinear region. The in-situ method was used to measure the impedance in an impedance tube. The details of the test setup with and without flow are described in more detail in Paper IV in this thesis. New values for the end corrections and the nonlinear factor of the resistive and reactive parts were deduced from the experiments. Comparisons of the experiments and the theoretical model using the new formulae for end corrections and nonlinear factor are shown in Fig. 14 and Fig. 15.

(a) Resistance. (b) Reactance.

Fig. 14 The no-flow impedance of some samples using the new end corrections.

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t=2.5mm, d=1.6mm t=2mm, d=2.5mm

(a) Resistance. (b) Reactance.

Fig. 15 The no-flow impedance (including non-linear effects) of some samples.

28

Generally speaking, the dependence of the hole impedance on the grazing flow can be explained as follows: The flow blows away some of the end effects which result in a decrease of the so-called attached mass, which means that some of the kinetic energy stored in the oscillating medium across the orifice is lost; and this results in an increase in the acoustical resistance of the hole and a shift in the resonator resonance frequency. The orifice resistance increases with mean flow speed while the reactance decreases. Few purely theoretical predictions of the effects of grazing flow on orifice impedance have been attempted, probably because of the complexity of the problem. The models could predict the impedance behaviour with respect to grazing flow only qualitatively not quantitatively. Therefore, models based on experiments have evolved to predict the impedance.

The semi-empirical models can be divided into two main groups; the first depends on the grazing mean flow Mach number and the second depends on the friction velocity at the liner surface. For a fully developed turbulent boundary layer, there is a fixed relation between the grazing mean flow velocity and the friction velocity. Therefore, when no boundary layer effects are to be studied, a simple equation is sufficient for the calculation of the effect of grazing flow on the perforate impedance which only depends on the Mach number. The impedance of fifty samples was measured at different flow speeds. All available models were compared to the measurement results. It was found that the Kooi & Sarin model in reference 37 gives the best fit to the measurements compared to the others. To get the best fit for the simple model to the measurement results, the grazing flow terms added to the resistance and reactance were 0.5M/σ and -0.3M/σ respectively. Fig. 16 shows a comparison between the measured impedance value, the Kooi & Sarin model, and the simple model developed in this paper.

The impedance without flow shown in the figure was calculated using the model developed previously. The impedance with flow shown in the figure is the summation of the no-flow impedance and the grazing flow term, either calculated from Kooi & Sarin, or the simple model.

1200 1400 1600 1800 2000 2200 2400 2600 2800 30000

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(a) Resistance (b) Reactance

Fig. 16 Comparison of the measured resistance with Kooi & Sarin 37 and the simple model.

Through flow in perforated plates is often called bias flow. It can be found in many practical applications, for instance car mufflers to increase losses, and perforates installed inside combustion chambers with the purpose of cooling. In this paper, no bias flow measurements were performed, but a literature review was made and the best available model was included in this paper in order to have a complete impedance model.

29

D. Paper IV: An Inverse Analytical Method for Extracting Liner Impedance from Pressure Measurements A new indirect method for measuring liner impedance is proposed in this paper. The liner sample is placed inside a rectangular duct. The amplitude of the plane wave incident towards the lined section is measured using the two-microphone technique 42. The reflection coefficient at the exit plane is also measured using the same technique. These measured values are fed to an analytical model for sound propagation through the lined section, which is constructed using mode matching and similar to the formulation presented in Paper II of this thesis. A solution of the impedance value is educed in the complex plane to match the calculated sound field to the measured one. Two cost functions were tested and compared. The first is the summation of the differences between the measured and calculated complex pressures at five duct microphones (Fig. 17). These measurement positions are already used for the two-microphone technique. The second cost function is the difference between the measured and calculated complex amplitude of the transmitted plane wave. The minimization algorithm used the Matlab function “fminsearch” which uses the simplex search method.

Fig. 17 Impedance eduction measurement points.

The paper gives an overview of the practical details of performing several kinds of measurements. The first is using the in-duct two-microphone technique for determining plane wave propagation inside flow ducts, the second is the in-situ technique for measuring locally reacting liner impedance, the third is performing acoustic measurements in a flow environment, and finally the measurement of the wall friction velocity using the Preston tube.

The first step to validate the new method was to consider a hard surface in the intermediate duct, because the admittance of a hard surface is exactly known and is equal to zero. The first estimate of the admittance that was fed to the minimization algorithm was 1+1j, a value which is quite far from the correct one to check if the algorithm really converges to the zero admittance. It was shown that the new method could reproduce the admittance of the hard sample at all frequencies at all flow speeds up to 0.3 Mach. The educed admittance is shown in Fig. 18 using the criteria of minimizing the difference in the transmitted wave amplitude c+. Fig. 19 shows the percentage error between the measured and calculated complex pressures at positions A and D in Fig. 17.

The mode matching code with a locally reacting liner placed in the intermediate duct was validated by comparison with published numerical simulations and experimental data 72. The comparison showed good agreement with the experimental data. The educed impedance without flow is shown in Fig. 20 for one sample compared to the first

a+ a-

s

Mic 3 Mic 4

pA pB c+

c-

s

Mic 5 Mic 6

pC pD

Mic 1

Mic 2

30

impedance estimate obtained form the in-situ measurement in the flow duct, and the normal incidence impedance measured using the in-duct two-microphone technique inside an impedance tube. If more microphones are used, higher accuracy would be achieved. The educed impedance in the presence of flow is shown in Fig. 21 for the same sample. The educed impedance agrees quite well with the theoretical model by Kooi & Sarin 37. With flow, the resistance increases and the reactance decreases. The first guess from the in-situ measurement gives impedance values very far from both educed and calculated values but they are not shown in the figure.

1000 1500 2000 2500 3000 3500 4000-0.04

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Fig. 18 The educed admittance at different flow

speeds. Fig. 19 The percentage error in the pressures at

points A and D.

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Fig. 20 The educed impedance without flow,

and comparison with the impedance tube measurement.

Fig. 21 Comparison of the educed impedance at different flow speeds and the theoretical model

by Kooi and Sarin in reference 37.

The power of the method arises from the simplicity in constructing the equations and that all the terms in the system matrix can be evaluated analytically. The use of the minimization algorithm reduces the time needed to find the correct impedance compared to the grid methods. Moreover, the measurement input data to the computer code can be easily collected using a single measurement with fixed microphone positions. It has been shown that the new method could reproduce the admittance of a hard sample and several perforated samples with flow. Apart from the development of the new method, a correction scheme was also developed for the measurement using the in-situ technique, when the liner sample is placed at one end of an impedance tube, to overcome near field disturbances to the measurement results.

31

E. Paper V: Impedance of SDOF Perforated liners at High Temperatures Perforates are often found in a high temperature environment; e.g. car exhaust mufflers, exhaust ducts of aircraft jet engines, and inside combustion chambers of gas turbines. Several studies have been conducted to develop impedance models, including the effect of different configurations and flow conditions. Nevertheless, little has been published on the effect of high temperature and temperature gradients across the liner on the impedance.

The objective of this paper is to investigate the effect of high temperature on the impedance of perforated liners. The aim is to evaluate the assumption that the effect is limited to the change of the fluid properties (density, viscosity, and speed of sound). If this assumption proves to be true, then future hot measurements can be replaced by cold measurements of the same system but injecting another gas which simulates the properties of air at higher temperatures. The impedance of different orifices was measured at different temperatures and incident sound pressure levels.

The in-situ technique 45 was used to measure the liner impedance. It is based on the fact that under certain conditions, a simple relationship between cavity acoustic pressure and particle velocity exists. The idea of the in-situ technique is to measure the pressure inside the cavity at the back-plate. From this pressure, the velocity inside the cavity can be calculated; particularly, at the other end of the cavity just below the top sheet. The top- sheet is then assumed to be thin compared to the wavelength which means that the normal particle velocity at the surface is equal to the particle velocity inside the cavity just below the top-sheet. Knowing the pressure at the surface from another microphone, the impedance can be calculated using the estimated velocity. The particle velocity at the other end of the cavity can be easily calculated if the medium is uniform inside the cavity; i.e., the wavenumber is constant. But when the temperature in the duct at the liner surface is higher than that behind the liner, a temperature gradient exists and the air properties change along the cavity axis.

A segmentation approach based on the transfer matrix formulation was developed to calculate the surface velocity. Several numerical simulations of the cavity with different temperature gradients revealed that the error is small if the cavity is considered to have uniform mean temperature. This requires a careful mapping of the temperature inside the cavity to be able to estimate the mean temperature. The temperature at the orifice should also be known to be used for normalizing the impedance. The temperature gradients inside the cavities were measured and found to be linear which decreased the error even more.

The test sample was an orifice backed by a single cavity of diameter 10 mm and depth 17.5 mm. Another adjacent identical cavity was used for temperature measurements. The sample was placed at the end of an impedance tube with a loudspeaker placed at the other end. The tube was fixed inside an industrial oven where cables, cooling tubes, and the temperature transducer are connected through a hole in the door. The tube extends outside the oven from the other side opposite to the door through a hole drilled in the oven wall. Two pressure transducers (KULITE, WCT-312) were used to collect the acoustic data. They were water cooled using tap water at the rate of 0.5 litres per minute.

Four sample configurations were measured. Experiments were made for each sample at four different controlled sound pressure levels at the liner surface (120 to 145 dB). The temperature was nominally varied from ambient temperature up to 300oC with 100oC steps. A stepped sine excitation was used in the frequency range from 400 to 3500 Hz

32

depending on the sample resonance frequency and the temperature. The used step was 50 Hz around the resonance and 100 Hz at other frequencies.

It was noticed that the temperature inside the cavity increases during the acoustic measurement and then decreases again to its initial value after the loudspeaker is switched off. This could be explained as follows. Before the acoustic signal is applied, the temperature has a certain distribution inside the duct and the cavity, and it is obvious that the temperature inside the duct is higher than that of the cavity. When an acoustic pressure wave is applied, it circulates air through the orifice and pushes some hot air from the duct into the cavity; thus, the air temperature inside the cavity increases. When the strength of the pressure wave increases, the air circulation increases; thus the temperature inside the cavity increases. This increase in the temperature was recorded and included in the calculations.

The behaviour of the impedance at high temperature was as follows. When temperature increases, the resonance frequency of the liner increases due to the change of the speed of sound inside the cavity. The resistance and mass reactance increase slightly with increasing temperature. This increase is masked out when the incident sound pressure is high and the orifice becomes very nonlinear. The values of the resistance and the viscous part of the mass reactance are almost unchanged with increasing temperature. The measurements were compared to the theory which was found to be capable of following the aforementioned trends. An example of the comparison is shown in Fig. 22 for one sample (1 mm thickness, 1.5 mm diameter, 2.25% porosity, and 17.5 mm depth of the cavity). The impedance model developed in Paper III was used here.

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, Hz

Nor

mal

ized

Res

ista

nce

20 deg100 deg200 deg300 degTheory

0 500 1000 1500 2000 2500-12

-10

-8

-6

-4

-2

0

2

4

Frequency, Hz

Nor

mal

ized

Rea

ctan

ce

20 deg100 deg200 deg300 degTheory

Resistance Reactance

Fig. 22 Impedance of sample 3 at 120 dB.

It was found that the impedance change at high temperatures and combinations of high temperature and high sound pressure levels can be well predicted by the present theory by simply changing the properties of air (density, viscosity, and speed of sound). The calculated increase in resistance with temperature is slightly larger than the measured increase. This difference can be related to small temperature effects which are not included in the model. For example, the coefficient of viscosity close to a highly conducting wall was assumed to have a constant relation to the adiabatic value of the viscosity. This relation can have a dependence on the temperature difference between the air inside the orifice and the adjacent wall. The overestimation was very small, and even masked out at high sound pressure levels; therefore this effect can be neglected for practical use.

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F. Paper VI: On acoustic network models for perforated tube mufflers and the effect of different coupling conditions A generalized segmentation model was developed to calculate the four-port transmission matrix across two ducts coupled through a perforate with flow. This general model is able to use any specified coupling condition, besides mass conservation, based on a single parameter, α, which was assigned the value of 1 for continuity of energy, the value of 2 for continuity of momentum, or it can be assigned any value in-between. The objective of this paper is to investigate the sensitivity of the model results to the assumption of the coupling condition. The generalized coupling condition has the following form

constant00 =′+′ uup αρ (2)

where p´ and u´ are the fluctuating pressure and velocity respectively, ρ0 is the fluid density and u0 is the mean flow velocity. The sensitivity of the transmission loss results are compared for simple muffler configurations then tested on a complete muffler system. Fig. 23 shows the transmission loss of a through flow muffler and a plug flow muffler with flow using different coupling conditions. It can be seen that with flow, the choice of the coupling condition has a significant effect on the TL results for the through flow muffler, whereas it is of less significance for the plug muffler, except at the peaks.

0 500 1000 1500 2000 2500 3000 35000

5

10

15

20

25

Frequency, Hz

TL

M=0M=0.3, α=1M=0.3, α=2

0 500 1000 1500 2000 2500 3000 35000

10

20

30

40

50

60

Frequency, Hz

TL

M=0M=0.3, α=0M=0.3, α=1M=0.3, α=2

Through flow muffler Plug flow muffler

Fig. 23 Effect of coupling condition on transmission loss of simple muffler configurations.

In previously published articles, only simple muffler configurations were analyzed. These configurations are hardly found in advanced muffler systems where combinations are used to improve the acoustic attenuation of the system. An example of these advanced systems (Fig. 24) was considered in this paper and the detailed analysis is presented. The simulations were compared with measurements. Practical considerations are described to guide the future modelling of similar systems. A new approach is also described to calculate the flow distribution inside the muffler using electric circuit analogy.

Fig. 24 Sketch of the muffler internal scheme.

1

2

3 4 5 6 7

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The difference between the three curves for different coupling conditions in Fig. 25 is not significant as it was for simple configurations. The explanation for this observation is that the acoustic behaviour of the muffler is not only controlled by the perforated tubes and the effect of the choice of the coupling conditions inside the four-ports are masked out by other acoustic elements.

0 100 200 300 400 500 6000

10

20

30

40

50

60

Frequency, Hz

TLMeasurementsSimulations

0 100 200 300 400 500 6000

10

20

30

40

50

60

Frequency, Hz

TL

Measurementsα = 0α = 1α = 2

M = 0 M = 0.15

Fig. 25 Comparison of the modelled and measured transmission loss.

Another version of the segmentation approach was developed. It uses a two-port transfer matrix to model the perforated branches and the intermediate hard pipe segments. A convergence test is presented for the new method, and examples are given for simple perforated tube configurations. This new technique is more flexible as it facilitates the modelling of perforated pipes within any muffler system with an arbitrary configuration. The calculations were done using an in-house computer code for modelling low frequency sound propagation in complex duct networks, called SID 73. The present version of SID can only apply the classical coupling conditions of the continuity of pressure and volume velocity between different elements. The SID simulations are therefore compared to the generalized Sullivan simulations in Fig. 26 for the case when α = 0. There is a small shift in the third peak which is due to the few number of segments considered. Otherwise, the agreement is excellent with and without flow between the generalized segmentation model with 50 segments and the transfer matrix model applied in SID with only two segments.

0 100 200 300 400 500 6000

10

20

30

40

50

60

Frequency, Hz

Tran

smis

sion

loss

, dB

MeasurementsSimulations in section IVSID Simulations

0 100 200 300 400 500 6000

10

20

30

40

50

60

Frequency, Hz

TL

Measurements Simulations in section IVSID Simulations

M = 0 M = 0.15

Fig. 26 Comparison between the conventional and modified segmentation approaches.

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VII. Recommendations for Future Research The recommendations for possible continuation of the research presented in this Thesis can be divided into three categories. The first is related to the newly developed technique in Paper IV. The technique can be further validated by comparisons with other test rigs or by measurements on reference test samples. A desired reference test sample would have constant properties with flow or at high sound pressure levels. The code can be further developed to include shear flow effects. Moreover, the measurement results can be enhanced by using more microphones.

The second category is related to the investigations of different liner top sheets rather than perforated sheets, which have been studied throughout this thesis. Adding resistive material to the face-sheet can give more stable properties with respect to flow and temperature 74. They are some types of metallic foams brazed on a perforated sheet. It is of interest to develop theoretical models for these materials to be included in liner impedance prediction and optimization codes. Another related work is to perform similar impedance measurements to those performed in Paper V at high temperatures with the in-duct two-microphone technique. This will add some complexity to the measurement because the temperature distribution inside the pipe has to be measured. If the in-duct technique is validated, impedance measurement of materials other than the single degree of freedom liners would be possible.

The third category of the recommended future research is related to the muffler considered in Paper VI. The model presented in the paper is a 1-D model and is limited to the plane wave region. It is of interest to include the effect of higher order modes in a 3-D model in order to model its acoustic behaviour at higher frequencies. A possible approach would be to use Finite Elements.

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