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LIBRARY T37” Michigan State , cf _ University This is to certify that the dissertation entitled AN ACOUSTIC INTENSITY-BASED METHOD AND ITS AEROACOUSTIC APPLICATIONS presented by CHAO YU has been accepted towards fulfillment of the requirements for the Ph.D. degree in MECHANICAL ENGINEERING Ema/OW.V\ Major Professor’s Signature .96 NOVEMAA-U‘ wag Date MSU is an Affirmative Action/Equal Opportunity Employer PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 KzlPrq/AcaPresICIRC/Dateoua.indd AN ACOUSTIC INTENSITY-BASED METHOD AND ITS AEROACOUSTIC APPLICATIONS By Chao Yu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Mechanical Engineering 2008 ABSTRACT AN ACOUSTIC INTENSITY-BASED METHOD AND ITS AEROACOUSTIC APPLICATIONS By Chao Yu Aircraft noise prediction and control is one of the most urgent and challenging tasks world- wide. A hybrid approach is usually considered for predicting the aerodynamic noise. The approach separates the field into aerodynamic source and acoustic propagation regions. Conventional CF D solvers are typically used to evaluate the flow field in the source region. Once the sound source is predicted, the linearized Euler Equations (LEE) can be used to extend the near-field CFD solution to the mid-field acoustic radiation. However, the far- field extension is very time consuming and always prohibited by the excessive computer memory requirements. The F W-H method, instead, predicts the far-field radiation using the flow-field quantities on a closed control surface (that encloses the entire aerodynamic source region) if the wave equation is assumed outside. The surface integration, however, has to be carried out for each far-field location. This would be still computationally in— tensive for a practical 3D problem even though the intensity in terms of the CPU time has been much decreased compared with that required by the LEE methods. For an accurate far-field prediction, the other difficulty of using the FW—H method is that the complete control surface may be infeasible to accomplish for most practical applications. Motivated by the need for the accurate and efficient far-field prediction techniques, an Acoustic Intensity-Based Method (AIBM) has been developed based on an acoustic input from an OPEN control surface. The AIBM assumes that the sound propagation is governed by the modified Helmholtz equation on and outside a control surface that encloses all the nonlinear effects and noise sources. The prediction of the acoustic radiation field is carried out by the inverse method with an input of acoustic pressure derivative and its simultaneous, co-located acoustic pressure. The reconstructed acoustic radiation field using the AIBM is unique due to the unique continuation theory of elliptic equations. Hence the AIBM is more stable and the reconstructed acoustic pressure is less dependent on the locations of the input acoustic data. The solution of the modified Helmholtz equation in the frequency domain is approximated by finite linear combination of basis functions. The coefficients associated with the basis functions are obtained by matching the assumed general solution to the given input data over an open control surface. The details on the optimization method, the instability issue and the numerical implementation of the AIBM have been discussed in the dissertation. To verify the AIBM model, several acoustic radiation examples are solved, e.g. multiple sources radiation. The analytical acoustic pressure and its normal derivative on a partial spherical control surface are used as the input for the AIBM. The reconstructed acoustic field is obtained then compared with the analytical acoustic field. Excellent agreement is achieved and demonstrated. Some affecting factors on the AIBM, e.g. input locations and the signal-to-noise ratio, are also investigated. In addition, the potential of AIBM in broad- band noise prediction is examined in vortex/trailing edge interaction problem. Furthermore, a series of real world mode] problems are chosen to demonstrate the capability and potential of AIBM in CAA applications. Two important aircraft noises: turbofan noise and airframe noise, are studied in detail. Both the permeable surface FW—H equation method and the AIBM are used to evaluate the radiated field. The prediction results obtained from the AIBM and the FW—H integral method are compared with the solution from the CPD/CAA method. The accuracy and efficiency of both the AIBM and the F W-H integral method are analyzed. In summary, the “open surface” AIBM makes up the drawbacks of traditional “closed sur- face” approaches. It provides an effective alternative for the far-field acoustic prediction of practical aeroacoustic problems. Acknowledgments I would like to express my gratitude to my advisers Prof. Zhuang, for all good advices and encouragement during the work and Prof. Zhou, for sharing his great mathematics knowledge and leading causes in my dissertation. I am grateful for Prof. Radcliffe, Prof. Wright and Prof. Jaberi, who not only gave me advice, but also kept track of all the details of my research work. I want to acknowledge their help for serving as my dissertation defense committee and their time in reviewing my dissertation. Many thanks to Prof. Thiele from TU-Berlin, Berlin, Germany for providing me the re- search collaboration opportunity with the scientists and specialists worldwide. I would also like to thank Prof. Li of Beihang University (BUAA), Beijing, China for giving me valuable advice and discussions on aeroacoustic inverse problems and Mr. Lin of BUAA for his assistance on CAA solutions of the duct intake problem. Also I want to thank my colleagues in the Computational Fluid Dynamics Research Labo- ratory for creating a stimulating working atmosphere. Finally, I would like to acknowledge the financial support of this work from the Na- tional Science Foundation (N SF-ITR-0325760), TU-Berlin Research Scientist Grant (IP- 10022152) and Graduate Office Fellowship. iv Contents List of Tables .................................. vi List of Figures ................................. vii Nomenclature ................................. x 1 Introduction ................................ 1 1.1 Traditional Techniques for Noise Prediction --------------- 2 1.1.] Numerical Simulation Methods ----------------- 2 L12 Acoustic Analogy ........................ 4 1.1.3 Aeroacoustic Hybrid Prediction Methods ------------- 4 1.1.4 Inverse Acoustic Methods -------------------- 5 1.2 Acoustic Intensity-Based Method (AIBM) ---------------- 7 1.3 Organization of the Dissertation --------------------- 8 2 Aeroacoustic Governing Equations --------------------- 10 2_1 Introduction .............................. 10 22 Wave Equation ............................. 10 2.3 Helmholtz Equation .......................... 14 3 Acoustic Intensity-Based Method for Acoustic F ar-Field Prediction ----- 15 3.] Introduction .............................. 15 3.2 Mathematical and Numerical Formulations --------------- 16 3.2.1 Mathematical Formulation -------------------- 16 3.2.2 Numerical Formulation --------------------- 19 3.2.3 Numerical Implementation ------------------- 20 3.3 Numerical Results and Discussions ------------------- 23 3.3.1 Verification and Advantages of AIBM -------------- 23 3.3.2 Effect of Signal-to-Noise Ratio (SNR) -------------- 27 3.3.3 Application to Multi-Frequency Acoustic Radiation Problem - - - 35 4 2D AIBM with Subsonic Mean Flow -------------------- 44 4,] Introduction .............................. 44 4.2 Mathematical Formulations ----------------------- 45 42.1 A Single Frequency ....................... 45 4.2.2 Multiple Frequencies ---------------------- 5] 4.3 Numerical Examples and Discussions ------------------ 54 4.3.1 Monopole in a Uniform Flow ------------------ 54 4.3.2 Sound Radiation by a Flow Around a NACA Airfoil. ------- 59 4..33 Sound Scattering ........................ 63 5 3D AIBM with Subsonic Mean Flow -------------------- 72 5 . 1 Introduction .............................. 72 5.2 Mathematical Formulations ----------------------- 73 5.2.1 Basic Formulation for AIBM ------------------ 73 5.2.2 Improved Formulation for AIBM ---------------- 74 5.2.3 Simplified 2.5D Formulation for AIBM ------------- 75 5.2.4 Numerical Implementation ------------------- 76 5.3 Results and Discussions ------------------------ 77 5.3.] Multiple Sources in a Uniform Flow --------------- 77 5.3.2 Acoustic Scattering ----------------------- 83 5.3.3 Acoustic Radiation from an Axisymmetric Duct Intake ------ 87 6 Conclusions ................................. 97 6_] Conclusion ............................... 97 6.2 Suggestions for Future Work ---------------------- 99 Appendix A: Associated Legendre Polynomial ----------------- 101 Appendix B: Spherical Hankel Function -------------------- 103 Bibliography .................................. 104 vi List of Tables 3.1 The strengths and distributions of the 2D acoustic sources .......... 26 5.1 The strengths and distributions of the 3D acoustic sources .......... 79 vii List of Figures 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.1 4.2 4.3 4.4 Acoustic radiation field with input acoustic data locations. ......... l8 Schematic diagram of acoustic sources and each input segment’s location. . 25 Comparisons of reconstructed acoustic radiation pressure with analytical solution at r = 50m for different 7 ....................... 28 Effects of SNRs on reconstructed acoustic radiation pressure at r 2 10m. . 32 Comparisons of reconstructed acoustic radiation pressure P (7‘, 6) contours for SNR=100 with analytical solution. .................... 33 Schematic diagram of 2D vortex filament moving around the edge of a semi-infinite plane and input acoustic data points on specified segments. . . 37 Spectrum of acoustic radiation pressure time history of 2D vortex model at r = 75m, 6’ = 7r/4, Mo = 0.01 and d =1m .................. 39 Acoustic radiation pressure and directivity calculated from the input data with frequency range of 0.05 - 0.2Hz. .................... 40 Acoustic radiation pressure and directivity calculated from the input data with frequency range of 0.05 - 0.45Hz ..................... 41 Acoustic radiation pressure and directivity calculated. from the input data with frequency range of 0.05 - 1.45Hz ..................... 42 Acoustic radiation pressure time history (r 2 100m, 6 = 7r / 5). ....... 43 Schematic diagram of sound propagation field and locations of acoustic measurements. ................................ 47 Schematic diagram of a monopole radiation in a uniform flow and locations of acoustic measurements. .......................... 56 Far-field directivity (r 2 100m) comparison of a monopole radiation in a Mo = 0.5 flow. ................................ 57 Pressure contours of a monopole radiation in a Mo = 0.5 flow. ....... 58 viii 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Instantaneous pressure perturbations of the flow around the NACA 0018 airfoil along with the location of the FW—H surface. ............. 60 F ar-field directivity (r = 100m) of the flow around the NACA 0018 airfoil. 61 Pressure contours of the sound propagation generated by the flow around the NACA 0018 airfoil ............................. 62 Schematic diagram of the sound scattering by a cylinder. .......... 64 Frequency spectrum for the sound scattering, r = 6.125. .......... 65 Pressure time history reconstruction with different frequency range at r = 7.25 and 0 = 00 ................................. 66 Pressure time history reconstruction with different frequency range at r = 7.25 and 6 = 900 ................................ 67 Pressure time history reconstruction with different frequency range at r = 7.25 and 6 = 1800. .............................. 68 Instantaneous pressure contours of the sound scattering at t=152 CAA (top), AIBM (bottom). ............................ 69 Instantaneous pressure contours of the sound scattering at t=1 1: CAA (top), AIBM (bottom). ............................ 70 Instantaneous pressure contours of the sound scattering at t=9: CAA (top), AIBM (bottom) ................................. 71 ~ Schematic diagram of multiple sources in a uniform flow ........... 78 Comparisons of the predicted pressure solutions with the analytical solu- tions: along x-axis (top), along y-axis(bottom). ............... 81 3D instantaneous pressure contours for the sound radiation of multiple sources: Analytical (top), AIBM (bottom) ................... 82 Schematic diagram of acoustic scattering by a sphere ............. 84 Instantaneous pressure contours of the sound scattering in the plane y = 0: CAA (top), AIBM (bottom). ......................... 85 Comparisons of the predicted pressure solutions with the CAA solutions: along x-axis (top), along y-axis (bottom). .................. 86 Schematic diagram of acoustic radiation through an axisymmetric duct intake. 89 ix 5.8 A comparison of the directivity along the x—direction with z = 6.5m and y = 0 for the duct mode of m=2 and n=1. .................. 90 5.9 Instantaneous pressure contours in the plane 6 = 0 for the duct mode of m=2 and n=1: CAA (top), AIBM (bottom). ................. 92 5.10 Pressure amplitude contours in the plane (7’ = 0 for the duct mode of m=3 and n=1: CAA (top), AIBM (bottom). .................... 93 5.1] Comparison of the predicted pressure solution with the CAA solution at z = 15m andy = 0for the duct mode ofm = 2andn =1. ........ 94 5.12 Comparison of the predicted directivity with the CAA solution at z = 15m and y = 0 for the duct mode of m = 3 and n = 1 ............... 95 5.13 3D reconstructed pressure contours for the duct mode of m=4 and n=1: CAA (top), AIBM (bottom). ......................... 96 Images in this dissertation are presented in color Nomenclature A! [If a (m, n) Acoustic source strength Speed of sound n-th order generalized Hankel function of the second kind n-th order Hankel function of the second kind \/——_1 Unit vector in 1‘ direction Wave number, w /c Number of measurements (inputs) Mach number Dust acoustic mode Unit normal vector, (no. ny, n2) Terms of approximation in the asymptotic formulations Acoustic pressure in the time domain Acoustic pressure from a monopole sound source Acoustic pressure in the frequency domain Exact acoustic pressure in the frequency domain Approximated acoustic pressure in the frequency domain Associated Legendre polynomial Polar coordinates Spherical coordinates Modified polar coordinates Modified spherical coordinates xi TC Radius of the control surface (minimum sphere) RN Random noise, or random pressure perturbation 3 Entropy u Unsteady velocity vector U Free-stream velocity vector, (U, V, W) Um Maximum speed of the vortex motion x Cartesian coordinates associated with observation point, (x, y, z) i Modified Cartesian coordinates, (it, 3), :2) t Time associated with the arrival of sound wave at observation point Greek a Shifted angle a t/ 1 — Ma? 7 Angle between the two measurement segments F Spherical boundary containing all the acoustic sources F1 Part of I‘ (D Velocity potential function p Density w Angular frequency Mathematics Dot product or scalar product V2 Laplacian operator Superscripts ’ Fluctuating quantities Subscripts 0 Value of quantity in steady free stream xii Chapter 1 Introduction As one of the most important pollutions, aircraft noise draws intensive attention worldwide. As early as the 19705, the United States has issued the Federal Aviation Regulation (FAR) Part 36 Noise Standards [1] to limit the transportation aircrafi noise. From then on, noise level has been one of the crucial guidelines on navigability. In the 1990s, the International Civil Aviation Organization (ICAO) announced more stringent Noise Standards for Aircraft Type Certification (N SATC) [2], which further spurred the “green aircraft” development initiated by the National Aeronautics and Space Administration (NASA) and the aircraft manufacturers. In the 19705, NASA commenced the Aircraft Noise Prediction Program (A‘NOPP) [3] and conducted more than ten years of study on new technology and comprehensive prediction methods for aircraft noise. In 1992, NASA, in partnership with Federal Aviation Admin- istration (FAA) and the US. industry giants Boeing, Pratt & Whitney and GE, developed the Advanced Subsonic Technology (AST) program [4] on both airframe and engine noise reduction. The initial goal was to achieve 10 dB reductions in each flight phase: take- off, sideline, and approach in 20 years, relative to 1992’s technology. Soon after, in 1997, NASA further raised the three pillar noise stretch goals, which are reducing the perceived noise level of future aircraft by 10 dB in ten years and 25 dB in 20 years relative to 1997. In Europe, similar research is also expanded around aircraft noise supported by European Union and European industry companies (Airbus, Roll-Royce etc.), e.g. TurboNoiseCF D program aimed at jet engine noise, and the RAIN program on airframe noise [5]. Despite significant progress having been made in reducing aircraft noise over the past decade, further improvements are required because of increasing community noise expo- sure caused by the growth in aircraft fleet. To develop low noise aircraft and control the noise at the design stage, developing accurate and efficient noise prediction tools is un- doubtedly important. In this chapter, the traditional computational techniques for aerody- namic noise prediction and underlying theories are briefly reviewed. Then, a new advanced method, Acoustic Intensity-Based Method (AIBM), is introduced and afforded particular attention. Finally, the structure of the dissertation is outlined. 1.1 Traditional Techniques for Noise Prediction Half a century ago, Lighthill [6] first proposed the “aerodynamic noise” concept in his re- search on quiet jet engines. His Lighthill acoustic analogy also became the foundation of aeroacoustics. Nowadays varieties of techniques have been developed to predict aerody- namic noise either theoretically or numerically. In this section, four important methodolo- gies are presented and reviewed. These are numerical simulation techniques, i.e. Computa- tional Fluid Dynamics (CF D) and Computational Aeroacoustics (CAA), acoustic analogy methods, aeroacoustic hybrid techniques, which combines the advantages of numerical methods and acoustic analogy methods, and inverse acoustic methods. 1.1.] Numerical Simulation Methods In this section, three most commonly used CFD and CAA techniques for aerodynamic noise simulation have been reviewed. Direct numerical simulation (DNS) is a simulation in CFD in which the Navier-Stokes equations are numerically solved without any turbulence model [7]. DNS resolves the whole range of spacial and temporal flow scales from the smallest dissipative scales to the largest integral scale. Therefore, the computational cost of DNS is very high, even at low Reynolds numbers. For the Reynolds numbers encountered in most industrial applications, the computational resources required by a DNS would exceed the capacity of the most powerful computers currently available. However, DNS is a useful tool in fundamental re- search in turbulence. It provides a standard tool for evaluating different acoustic prediction model [8]. Unsteady Reynolds-averaged Navier—Stokes (RANS) equations are a set of time-averaged equations, which are derived from Navier-Stokes equations, dealing with turbulent flows. These equations can be used to provide approximate averaged solutions to the Navier- Stokes equations. In aerodynamic noise simulation, MNS can simulate the noise of the largest flow features. However RANS suppresses the acoustic field and under predicts dynamic loads [9]. Another alternative computational technique, Large-eddy simulation (LES), resolves only the dynamically important flow scales and models the effects of small scales using a sub- grid scale (SGS) model. LES requires less computational effort than DNS but more effort than RANS. The main advantage of LES over computationally cheaper RANS approaches is the increased level of detail it can deliver. While RANS methods provide ”averaged” results, LES is able to predict instantaneous flow characteristics and resolve turbulent flow structures. LES is a suitable compromise of DNS and RANS in accuracy and cost. Generally these numerical simulation techniques, DNS, RANS and LES methods offer at- tractive alternatives. But they are not always affordable even with today’s high-performance computers and parallel computing technology. Hybrid approaches are usually considered for predicting aerodynamic noise, in which the field is divided into aerodynamic source and acoustic propagation regions. For example, LES/FW-H and RANS/FW-H hybrid methods. 1.1.2 Acoustic Analogy In 1952, the famous Lighthill theory [6] of sound generation by turbulence was brought forward and soon became the birthstone of aeroacoustics. Lighthill’s equation was accu- rately derived from compressible Navier—Stokes equations without any assumptions. With the “analogy” idea of representing a complicated fluid mechanical process that acts as an acoustic source by an acoustically equivalent source term [10], the Navier-Stokes equations have been rearranged into the form of an inhomogeneous wave equation. Within this equa- tion, all the noise effects on the right hand were described as noise source terms including pressure and velocity fluctuation as well as stress tensor and force terms. Lighthill’s equation is limited to radiation in free space when it was first developed. In 1955, Curle [1 l] expanded a more general equation based on Lighthill’s analogy using Kirchhoff method. The Curle equation takes still solid boundary effects into consideration. In 1969, F fowcs Williams and Hawkings [12] further expanded Curle equation introducing general fiinctions and developed FW-H equation. F W-H equation is valid for aeroacoustic sources in a relative motion with respect to a solid surface. Nowadays, FW—H equation is known as an effective acoustic far-field prediction method, in which Farassat’s Formulation 1A [13—15] is commonly used. 1.1.3 Aeroacoustic Hybrid Prediction Methods The hybrid prediction technique based on CFD and F W—H with penetrable surface has ad- vanced considerably. And much experience has been gained in its use. The technique separates the field into aerodynamic source and acoustic propagation regions. Conven— tional CFD solvers are typically used to evaluate the flow-field solution in the near field to provide the aerodynamic sound source. Once the sound source is predicted, the linearized Euler Equations (LEE) or the integral methods based on Lighthill’s analogy [6] are used for the prediction of the acoustic wave propagation. The LEE methods assume the flow field to be a time-averaged mean flow and a time-dependent small disturbance. The extension of the near field CF D solution to the mid-field acoustic radiation can be achieved using the LEE methods. However, the evaluation of the far-field radiation is prohibited by excessive computer memory requirements. The integral methods (i.e. Kirchhoff method [16] or per- meable surface FW-H equation method [12]), instead, predict the far-field radiating sound using the flow field quantities on a closed control surface (that encloses the entire aero- dynamic source region) if the wave equation is assumed outside. The surface integration, however, has to be carried out for each far-field location. This would be still computation- ally intensive for a practical three-dimensional problem even though the intensity in terms of the CPU time has been much decreased compared to that required by the LEE methods. For an accurate far—field prediction, the other difficulty of using these integral methods for some aeroacoustic problems is that the control surface must completely enclose the aerody- namic source region. This may be infeasible or impossible to accomplish for some practical cases. 1.1.4 Inverse Acoustic Methods In order to reduce aeroacoustic noise effectively, it is crucial to understand the character- istics of the aerodynamic noise sources. These characteristics, however, are not known analytically for aeroacoustic applications. With an inverse approach, the acoustic measure- ments in the radiated field have been used for the characterization and identification of the unsteady aerodynamic sources. In the past decade, the inverse aeroacoustic problems have been investigated with various objectives. Grace and Atassi [17, 18] first introduced the in- verse method into unsteady aerodynamics and aeroacoustics. They developed an inversion model of gust/plate interaction based on solving the Helmholtz equation. The unsteady pressure on a zero-thickness plate was successfully reconstructed using the acoustic mea- surement in the radiated field. Li and Zhou [19] then proposed an inversion model for the reconstruction of steady pressure distribution on a propeller surface based on the three- dimensional F W-H equation. In the work of Luo and Li [20] and Li et al. [21] on the gust cascade interaction and rotor wake/ stator interaction, it has been shown that the accuracy of the reconstruction of the unsteady pressure distribution on the cascade and stator surfaces from the far-field acoustic measurement is excellent when the signal-to-noise ratio is not very low. Recently Gerrard et al. [22] have developed an inverse aeroacoustic model of subsonic axial flow fans, which can determine the circumferential blade loading variations from far-field acoustic measurements. Holland and Nelson [23] applied the inverse meth- ods to study the distributed acoustic sources by exploiting known correlation structures among the sources. Later on, Nelson and Yoon [24] used the inverse methods to deduce the acoustic source strength from radiated field measurements. Near—field Acoustic Holograph (NAH) is one important family of inverse acoustic meth- ods. Several prominent inverse acoustic methods have been developed for NAH [25—34]. These methods can be classified into three categories: ( l) Fourier acoustics, (2) the inverse boundary element methods (IBEM), and (3) the Helmholtz equation least squares (HELS) method. Among these traditional inverse methods, the acoustic pressure measurement by itself is considered as the input of the inverse methods. The solution of these inverse meth- ods, therefore, is not unique unless the input acoustic pressure is provided over a surface enclosing all the acoustic sources, i.e., a closed surface [35—37]. As a result, the effective- ness of these methods weakens when the input acoustic pressure is only available over a portion of a closed surface (i.e., open surface). As a general rule, the more complete the input around a sound source, the more accurate the solution of the inverse problem. However, the far-field acoustic measurement over a surface enclosing the sound sources under consideration is often infeasible or impossible, in particular for the far-field measurement. In addition, very large numbers of the field measurements are also inconceivable. Therefore, there is a need to improve the accuracy and consistency of the inverse methods especially in the case where the input acoustic data is only available over an open surface. 1.2 Acoustic Intensity-Based Method (AIBM) Recently, with the advent of new signal processing techniques and the advances in trans- ducer technology, acoustic intensity measurement devices have been improved to make them more reliable, accurate, and compact [45]. Using these devices, the acoustic inten- sity as well as simultaneous, co-located acoustic pressure can be evaluated. The acoustic pressure derivative can thus be derived along a given axis, e.g., the axis of a microphone pair. Motivated by the need for an accurate and efficient prediction of the far-field acoustic radia- tion, an Acoustic Intensity-Based Method (AIBM) has been developed based on an acoustic input from an open control surface in a two-dimensional and three-dimensional configura- tions [38—43]. The AIBM assumes that the sound propagation is governed by the modified Helmholtz equation on and outside a control surface that encloses all the nonlinear effects and noise sources. The prediction of the acoustic radiation field is carried out by the inverse method with an input of the acoustic pressure derivative and its simultaneous, co-located acoustic pressure over a portion of the control surface. The reconstructed acoustic radiation field using the AIBM is unique due to the unique continuation theory of elliptic equations. Hence the AIBM is more stable and the reconstructed acoustic pressure is less dependent on the locations of the input acoustic data. The solution of the modified Helmholtz equation in the frequency domain is approximated by a finite linear combination of basis functions. The coeflicients associated with the basis fiinctions are obtained by matching the assumed general solution to the given input data over an open control surface. The details on the optimization method, the instability issue and the numerical implementation of the AIBM will be discussed in the following chapters. The “open surface” AIBM makes up the drawbacks of traditional “closed surface” ap- proaches. It provides an effective alternative for the far-field acoustic prediction of practical aeroacoustic problems. 1.3 Organization of the Dissertation The dissertation is organized centering at the development and application of AIBM ap— proach. In Chapter 2, the acoustic governing equations are derived starting at the fluid mechanics conservation laws. In Chapter 3 through Chapter 5, the mathematical model, verification examples and application problems are discussed in detail. In Chapter 3, AIBM has been proposed and modeled without considering the effect of mean flow. The mathematical and numerical formulations are first developed in a 2D con- figuration. Three numerical examples of acoustic radiations from either single or multi- frequency acoustic sources are presented for the verification. The advantages of the AIBM over a traditional inverse method, HELS, are demonstrated. Furthermore, the sensitivity of the AIBM to random noises with various signal-to-noise ratios (SNR) is examined. In Chapter 4, AIBM is extended to problems with subsonic uniform flows in the 2D con- figuration. Firstly, The mathematical formulation for sound propagations in uniform flows is described. In addition, examples are given for the verification and demonstration of the AIBM’s capability and potential in aeroacoustic applications. The results of the AIBM are also compared with that from the F W—H integral method. In Chapter 5, 3D AIBM with subsonic uniform flows is developed. The model is verified by examples of the propagation of multiple acoustic sources in a uniform flow and the acoustic scattering of a time dependent source by a sphere. The effectiveness of AIBM in aeroacoustic applications is demonstrated by the accuracy and efliciency of the predicted acoustic radiations from an axisymmetric duct intake by a hybrid CAA/AIBM approach. The AIBM is much more eflicient than the other methods for the far-field acoustic predic- tion and can use the input acoustic data from an open surface instead of a closed FW-H surface. In the last chapter, the concluding remarks are drawn through the study in this dissertation. And the potential future work is also suggested. Chapter 2 Aeroacoustic Governing Equations 2.1 Introduction For acoustic propagation problems, the viscosity and thermal conductivity have little ef- fects on the sound wave propagation. Hence the motion of the propagation media is al- ways determined by solving Euler’s equations instead of Navier-Stokes (N-S) equations. In this chapter, the linearized aerodynamic equations for an inviscid flow are firstly de- rived. Then the wave equation is derived with small disturbances assumption. Furthermore the Helmholtz equation, the governing equation in the frequency domain, has been obtained by performing Fourier transformation from the wave equation. As regards of the symbols, the boldface type (u) denotes a vector and the same letter (u) in italic type is used to denote its components. The detailed physical meaning of the symbols and operators used in this chapter are listed in the Nomenclature. 2.2 Wave Equation From the conservation laws in fluid mechanics, we can obtain the fundamental equations describing the fluid motion. These equations are called N-S equations for viscous flow or Euler’s equations for inviscid flow. The wave equation for a uniform mean flow can be derived based on the fundamental fluid mechanics equations. Assuming a sound wave propagating in an inviscid flow and there is no external force or quantity source, the governing equations (Euler’s equations) are expressed as below 10 [10, 44]. Mass conservation: 0p 52+V-(pu)=0 or Momentum conservation: p =—Vp or Energy conservation (isentropic flow): g+uov.s=0 or Equation of State: 0;) 5p _ —— I " I‘ = 0 0t + 01,- (”l”) 01/; + 0a,- 8]) '_—— '1' ' ——"' = ““7— ” at '1 (911- 6.1:,- p = p(p, .9), (1]) = ('2(1[) + (98) ds ()3 p where the speed of sound (3 is defined as 02 = ( 3%) 9. Introducing the acoustic perturbations, P=I)0+P’a p=p0+p’, U=UO+ll,, 8:80-l-SI, (f 2 = (:02 + 6,2 (2.1) (2.2) (2.3) (2.4) (2.5) where p0, p0, uu, so, Co and p’, p’, u’, s’. c' are for the uniform and perturbation variables, respectively. And they satisfy ll I pl pl 'I (.12 — <<1, — <<1, — <<1, — <<1, —2 <<1 ' P0 P0 50 ('0 (2.6) For steady flow, Polio ' Vuo = -VP0~ V ' P0111) = 0» u0 - V80 = 0, no ° VPO = €200 - VPO (27) Substituting above relations into Equation (2.1) - Equation (2.4), we can get [10,44] a/__’_ I I _ 57 + V (pou + I) no) — 0 (2.8) Bu’ / I ()t + uo Vu’ + u .Vuo + p u0 - Vuu = —Vp (2.9) ()s’ W‘FUO' V9, +u" VSO— - (210) I . I I I 6p 2 (9/) P P V = —— ———— . :0 dt +u0 Vp +u’- VPo (aim +00 VP +u VP0+ 0(p0 p0 u0 PO (2.11) These are the Linearized Aerodynamic Equations. They govern the propagation of small disturbances through a steady flow. If the disturbances and their gradients are not small, we can no longer apply these equations to simulate the propagation. For a uniform fluid, u0 = const, p0 = const, p0 = const, and so = const (2.12) Taking the time derivative on Equation (2.8) and the divergence on Equation (2.9), we get a? ’ au ' a' 02—2 +p0V aT +u0 Va—’:= 0 (2.13) a I [)0 (V 7‘; +V- (u()-V)u’> = —V2p’ (2.14) 12 Then, subtracting Equation (2.13) from Equation (2.14), it yields 2/ a! —%t—/2)—+pOV-(u0-Vu'-d)—u0 V7— p _V2p I From Equation (2.8), we know, d_/_)’_ (It— Combining above two equations, we can get I %——2/:+ (uo- V)‘2 p +2u0 Viz—l}: V21)I Considering the isentropic state equation, I 2 I P = (‘0 P Finally, the wave equation with a uniform mean flow is obtained. 2 2; 1 a 2; (9 22/ Vin—C—2 0dt+u0 V p20 or dt+u0 V —con=0 -V- (poll, + ping) = —p0V ° u’ — “()Vp, (2.15) (2.16) (2.17) (2.18) (2.19) All of the small disturbance quantities describing the acoustic field satisfy the wave equa- tion. For the notational convenience, we will use p (in the time domain) or P (in the frequency domain) to represent acoustic pressure instead of p’ in the rest of this disserta- tion. 13 2.3 Helmholtz Equation In the frequency domain, the acoustic field is governed by the Helmholtz equation which can be obtained by Fourier transformation from the wave equation. Assuming the mean flow Mach number is zero, the Helmholtz equation can be written as V219 + k2P = 0, (2.20) where P is the acoustic pressure in the frequency domain. When p is a simple harmonic function of time, p and P can be related as p =2 P 6“”. k = (12/60 is the wave number, where w is the angular frequency and co is the speed of sound. If the standard polar coordinates are used in a two-dimensional configuration, Equation (2.20) can be written as 82F 16F 182P __+ _______ k2pzo, 2.21 (9r2 + 7' 07' + 7'2 662 + ( ) In three dimension spherical coordinates, Equation (2.20) is in the following form. . 2 i2 7.2.63 +_¥__‘?_ Sin995+ 1 8P+k2P=0 (2.22) .. 723111606 7‘2 sin2 6 89")2 If a mean flow is considered, that is Mo # 0, modifications need to be made to the above equations. The detailed derivation for the solution in this case is given in Chapter 4 when needed. 14 Chapter 3 Acoustic Intensity-Based Method for Acoustic F ar-Field Prediction 3.1 Introduction Recently, the new signal processing techniques in transducer technology has been advanced dramatically. Using these devices, the acoustic intensity as well as simultaneous, co- located acoustic pressure can be evaluated. The acoustic pressure derivative can thus be derived along a given axis of a microphone pair. Motivated by the advances in acoustic measurement technology and the Helmholtz Equation Least Squares (HELS) method [32], an Acoustic Intensity-Based Method (AIBM) is proposed for the accurate reconstruction of the acoustic radiation pressure in the far field. The method uses the acoustic pressure derivative and its simultaneous pressure as the input acoustic data for the computations. Because of the addition of the acoustic pressure gradient in the input acoustic data, the solu- tion of the AIBM becomes unique with the input given over an open surface [35]. However, the solution is not stable since it is a Cauchy problem for Helmholtz equation [46]. Mathe- matically, this instability comes from the highly oscillatory modes. By carefully removing these modes, the AIBM, compared with the traditional inverse methods, could provide a measurable improvement in terms of accuracy and consistency of the reconstructed acous- tic pressure. The objectives of the study in this chapter are two-fold (1) developing, nu- merically implementing and verifying the AIBM and (2) demonstrating the advantages, effectiveness and potential of the AIBM for engineering applications. 15 The chapter is organized as follows. The mathematical and numerical formulations of the AIBM are discussed in Section 3.2. The exact mathematical solution of the Helmholtz equation is approximated by finite linear combinations of basis functions. The numerical formulation and implementation are developed to effectively determine the coeflicients of these basis functions by the method with an input of acoustic pressure and its derivative of a given direction. In Section 3.3, three numerical examples of acoustic radiations from either single or multi-frequency acoustic source are presented and verified. The advantages of the AIBM over a traditional inverse method are demonstrated. Furthermore, the sensitivity of the AIBM to random noises with various signal-to-noise ratios in the input acoustic data is examined and analyzed. The concluding remarks are drawn in Section 3.4. 3.2 Mathematical and Numerical Formulations 3.2.1 Mathematical Formulation The acoustic pressure field P in the frequency domain is governed by the Helmholtz equa- tion V2P + 1.21) = 0, in :2 = R2\Sl,:,,, (3.1) where (In, is a bounded domain in R2 containing all acoustic sources, I: = w/c is the wave number with the angular frequency w and the speed of sound 0. If the standard polar coordinates are used in a two-dimensional configuration, Equation (3.1) can be written as 62F 16F 18210 .2 _ - = 2 .- 67‘2 + 7‘ 07 + T2 {)02 +A P — 0’ m Q R \Szm- (3.2) With the Sommerfeld radiation condition rill-.90 7'1/2[0,‘P(r. 6) — ikP(r, 6)] = 0, the solu- l6 tion of Equation (3.2) can be written as [47,48] 00 P(r. 6) = 2 (an cos I26 + bu sin 726)H,,,(A:r), (3.3) 71:0 where H n is the nth-order Hankel function of the second kind. In order to obtain a solution from Equation (3.3), it is necessary to determine the coefficients an and bn. These coeffi- cients are determined by matching the assumed form of the solution to the input acoustic pressure in the HELS method [32]. Strictly speaking the solution is unique only if the in- put acoustic pressure is made on a boundary I‘ that encloses all the acoustic sources in the domain 52. The acoustic pressure measurement on the boundary F is usually impractical or infeasible for engineering applications. In practice, the input acoustic pressure is only available on a number of segments of the boundary, I‘t (an open surface, see Figure 3.1) . Although the AIBM could be implemented in various inverse acoustic methods, it is for- mulated here similar to that of the HELS [32]. It will be demonstrated later that when the input acoustic data is given over an open surface the AIBM strengthens the HELS method, and improves its mathematical well-posedness and practical applicability. In the AIBM, both the acoustic pressure and its co-Iocated derivative (normal to F1) on the boundary P1 are considered as the input acoustic data for the reconstruction of acoustic radiation pressure in the domain $2. With the pressure derivative boundary condition as an additional input, the uniqueness of the reconstructed acoustic pressure solution is guaran- teed from the unique continuation theory of elliptic equations [35-37]. In the AIBM, the partial boundary value problem for acoustic radiation pressure is defined as V2P + 1:213 = 0. in 12 = 122w)", (3.4) Plrl = P, anPlr1 = Pn. 713590 r1/2[a,.P(r, 6) — ikP(-r, 0)] = 0, l7 _'—-—.—-_—_ ~— ‘ ’-' -‘~__-— .— \ Sound Sources ,4 Figure 3.1: Acoustic radiation field with input acoustic data locations. where n is the outward normal to the boundary F1, P and Pu are the input acoustic pres- sure on F1 and the pressure derivative normal to F1, respectively. The solution can then be expressed as Equation (3.3) on and outside the control surface, which encloses all the acoustic sources under consideration. It is worth mentioning that the boundary F1 where the input acoustic data is provided needs to be outside of the control surface. Although this partial boundary value problem is unique, it is not stable. Small variations in the input data may lead to large differences in the solution P('r, 6). In the following section, the numerical techniques and schemes are described for solving this partial boundary value problem. The stability conditions and other restrictions of the numerical formulation and implementation are discussed in details. 18 3.2.2 Numerical Formulation In the current study, the numerical solution of Equation (3.4) is being sought by the follow- ing three steps: Step 1: Instead of using the infinite summation in Equation (3.3), the exact solution of Equation (3.4) is approximated by a finite summation, i.e., N P(r, 6) ~ (1.0H0(kr) + 2 (an cos 71.6 + bn sin n6)Hn(k7'), (3.5) 1121 where N is a suitable integer. The choice for N will be discussed later in the section. In the AIBM, the solution P is achieved by obtaining the coeflicients orn and bn. The coeflicients are numerically determined by matching the assumed form of the solution to the input acoustic data over the boundary I‘t. One obvious restriction of N is that the number of coefficients (2N + 1) to be determined must be less than the number of the input data over r1. Step 2: In order to determine the coeflicients an and bu numerically, an accurate and efficient method must be developed to evaluate each H 7,. When kr is relatively large, the first Q terms in the asymptotic expansion of H 0( kr) [47] are used in the current numerical calculation H0(A:T)~H0,Q(kr)=i/%o—i( W— mi gin—1) q(1/!2)2c:§f:/)2lq], lav—>00, (3.6) where (1/2)q = 1/2(1+1/2)...(q — 1+ 1/2). It can be shown that the accuracy established . . . . g . 1n the above approx1matlon 13 of order 0 ((3597762). It 15 also noted that the error generated by the approximation becomes unacceptable when Q is much larger than 2131*. Since the accuracy of the method depends on a good approximation of the Hankel functions, the restriction of Q g 1.51m: is used. For the approximation of other H n, the recurrence relation 19 for Hn is employed. Because of the initial error of 0(1/rQ), N has to be restricted to prevent the propagation of the errors for H 7,. From a careful analysis, N should be less than 2137* to ensure that the error for Hn is also in the order of 0(1/rQ). In addition, since the partial boundary value problem, Equation (3.4), is unstable due to the presence of functions for large index n in Equation (3.5), a proper limit on the upper bound of n is needed to control the solution instability. When k7: is relatively small, one can use the expansion of H 0(k7') for r ——> 0, or a higher order numerical integration method to evaluate H0(kr). Other H n, can be obtained again by the recurrence relation. In the current numerical calculation, a relatively large kr (i.e., kr = 6) is considered and the above asymptotic expansion formulation is used. Thus, the definition of the control surface in terms of the numerical solution of P needs to be extended beyond enclosing all the acoustic sources. For a given wave number k, the radius of the control surface, rc, is given by, e.g., r0 2 6 / k. The numerical solution is valid on and outside the control surface. Step 3: After the evaluation of the Hankel fiinctions, a suitable optimization method is used to determine the coeflicients an and (In in Equation (3.5) for the partial boundary value problem (Equation (3.4)). The simplest method is the least squares technique. Because of the underlying partial differential equation and the particular basis fiinctions used in this calculation, some numerical techniques are introduced to improve the condition numbers of the numerical schemes. 3.2.3 Numerical Implementation If the input acoustic data, P and Pn, are known at M discrete points (7'1, 61), (TM, 6 M) over the boundary Ft, the linear system for the coefficients an, and bn is given by the 20 following 2M equations N (1,0H0.Q(krj) + 2: ((1,, cos 7163- + bu sin Tlij)Hn.Q(Ii'I'j) = P('I’j, 6]), (3.7) 11:] N aoanH0,Q(krj) + 2 an [(an cos'n6j + 1),, sin n6j)Hn,Q(k-rj)] = Pn(7'j,6J-), (3.8) 7221 where j = 1. M. The above system can be expressed in a matrix form as A3: = B, where .r = [(10.---,(11\r.b1,---,bN]T (3.9) 7 B = [PINK/1).. - -°,130211911).Pn(7'1,f)1), - --,Pn(7'.1[,011)lT, (3-10) and A = (A1, A2)T, A1 corresponds to P and .42 corresponds to Pn. The vector coefli- cients a: can be determined by minimizing “Ar: - B|| directly for a choice of norm || - ll. However, when 7"] = 7'2 = = TM, i.e, the input acoustic data is prescribed on a circular segment, it is well known that 8nH,,,Q(I-cr) = 0,-H,,,Q(lcr) = ian,Q(kr) + O(1/(kr)) and therefore the matrix A2 ~ ikAt for a large T. In many cases, this will result a large condition number for the system Ar = B, and more importantly, this pr0portionality be- tween A2 and A1 will not allow us to take the full advantage of the additional input acoustic data Pn. To overcome this difliculty, the original system is first modified as follows cAikr(Ag —-ich1):r = CPkr(Pn — ikP). (3.11) Then, to further improve the condition number for the linear system, the coefl'lcients vector x is replaced by the vector y defined as y = (aoHoeaiHL -~aal\I'H..\'~. b1H1~ bNHN)T. (3.12) 21 The new linear system for the vector y has a much simpler matrix form CD10?!) 2 cPk:I'(Pn — iAfP), (3.13) where K 1 cos 61 cos N61 sin 61 sin N61 \ 1 cos 6; cos N 62 sin 62 sin N62 Dl :- (3.14) K 1 cos 6M cos N611 I sin 6n I sin N6M } ( E0 E1 cos 61 EN cos N61 E1 sin 61 EN sin N61 \ D E0 E1 cos62 EN cos N62 E1 sin62 EN sin N62 k 2 = 7 \ E0 E1 COS 0.)! EN (“USA-r63” E1 SIIl (In! EN sin N701]! / (3.15) HI-(kr) . . . . and Ej = T6075 - 2k, E J- can be obtamed easrly from a recurrence relation [47]. The least squares method that minimizes the standard L2 norm H Dy — B” is used to deter- mine the vector y, which in turn gives values of the coefficients an and bn. In the current numerical study, it is observed that although some regularization methods may be needed to improve the stability of the system for large N, they are not necessary for relatively small N. Based on the analysis given in section 2.2 (b) (N S 2kr), N is initialized within a given range of 2 - 12. The reconstructed solutions of acoustic radiation pressure for various N are compared with the input acoustic data given on the boundary F1. The total error at all the input points is then computed for each N. The optimum N within the given range is determined from the minimum overall error. The reconstruction of the acoustic field is 22 then carried out using the optimum N. It should be pointed out that without the addition of the pressure gradient to the input acous- tic data the reconstructed acoustic pressure could not be unique if only PIP1 is specified. For example, let 12,-” be the unit disk centered at the origin and all the acoustic sources are enclosed by the unit circle, the acoustic pressure should then satisfy the following equation (101-10M?) + Z (an cos 116 + I)” sin II6)Hn(/r) = P(1,6), 6 E [0, 7r] (3.16) 71.21 if the upper half boundary of the unit circle ( 0 S 6 S It) is considered as the boundary F1. It is well known that an and bn are not unique from the Fourier analysis. Especially Pn has always a cosine expansion for 6 E [0, 7r], i.e., bi = 02 = = 0. It is therefore impossible to predict solutions for 6 E (7r, 27r). 3.3 Numerical Results and Discussions Numerical examples are considered in this section for the purposes of: (1) verifying the AIBM and indicating the advantages of the method, (2) discussing the effects of the signal- to-noise ratio, and (3) demonstrating the potential of using the AIBM to reconstruct the acoustic radiation field for engineering applications. 3.3.1 Verification and Advantages of AIBM The example used here is formulated by the acoustic radiation from a combination of one monopole, one dipole, and two quadrupoles in a two-dimensional domain. The acoustic pressure generated by a dipole and quadrupole source can be expressed as a summation J of the pressure generated by a monopole, i.e., P('r, 6) = Z (Ao)j H0(kRj), where A0 is 3:1 23 the strength of the monopole and R is the distance between a field point and the monopole source. A dipole (J = 2) consists of two monopole sources separated by a small distance compared with the wavelength of acoustic radiation. In the frequency domain the strengths of these two monopoles are of equal amplitude but opposite sign. A quadrupole (J = 4) is made of two opposite dipole. The strengths and locations of these acoustic sources used in this example are given in Table 3.1. The acoustic sources are enclosed in the circle of radius r =1 m. The wavenumber of the sources is considered as k = 2m’1 and the control surface is then chosen as the circle of radius rc=3 m (see discussions on the size of the control surface in section 3.2). The label P(7‘, 6) used in the figures of this chapter denotes the real part of the acoustic pressure in the frequency domain. The units used for r and 6 are the meter and the radian, respectively. The reconstruction of acoustic radiation pressure is carried out on and outside the control surface. The input acoustic data is given at two circular (or straight line) segments. The schematic diagram of the acoustic sources and each input segment’s location is given in Figure 3.2. The starting and ending point coordinates of the two segments are chosen as (6m, 637r/128), (6m, 657r/128) and (6m, 637r/128 + 7), (6m, 657r/128 + 7) in the polar coordinates. It is noted that the input data is given over an open surface and the circular arc length of each segment is only 1/ 128 of the circumference of the circle. Each segment is then discretized uniformly into ten grid points where the analytical acoustic pressure and its derivative normal to the segments are given as the input acoustic data. The angle '7 shown in Figure 3.2 is a measure of the dimensionless distance between the two segments of the input, with 7 = It being the farthest, i.e., the segments are at the opposite sides of the control surface. Different values of 'y are considered to examine the sensitivity of the reconstructed acoustic radiation pressure against various input locations of an open surface. In order to determine if the addition of the pressure gradient in the input acoustic data would improve the accuracy and consistency of the inverse method, two sets of the input acoustic data are considered. The first set of the input consists of the analytical acoustic 24 yA Circular arc segments .9 + . . .92 (Ir 1 Y) Strarght l1nc segments“ ) .\ o \o (r. (12+)) °--- ‘ (r. 91) C ”D . oQo MO “Y I . D (3 .' x D C83 Sound sources . Control surface . Figure 3.2: Schematic diagram of acoustic sources and each input segment’s location. 25 Table 3.1: The strengths and distributions of the 2D acoustic sources Ago/Wm?) x yon) Quadrupole I 1.00 0.30 0.60 -l .00 -0. 10 0.60 1.00 -0.10 0.20 -l .00 0.30 0.20 Quadrupole II - l .20 0.49 -0. 12 1.20 0.20 -0.20 -1 .20 0.28 -0.49 1.20 0.57 -0.41 Dipole -0.80 -0.54 -0.16 0.80 -0.78 -0.45 Monopole 0.90 -0.58 -0.58 pressure and its normal derivative, i.e., the AIBM, and the second set of the input consists of only the analytical acoustic pressure, which is referred as the AIBM_without. Since in this study both the AIBM and AIBM_without are implemented using the least squares method, it allows us to examine the sole effect of the addition of the pressure derivative in the input acoustic data. The reconstructed acoustic radiation pressure at 7‘25011‘1 is compared with the analytical solution for four different values of ’7 in Figure 3.3. Since the reconstructed pressure cal- culated using the input acoustic data from the two straight line segments are identical to that using the input from the two circular segments, only the results based on the input from the circular segments are shown in Figure 3.3. It can be seen clearly that the recon- structed acoustic pressure obtained from the AlBM.without deteriorates in some regions as 'y deviates from 7r. On the other hand, the reconstructed acoustic pressure from the AIBM 26 agrees reasonably well with the analytical solution for 7 = 0.87r and 0.67r. The results demonstrate that the reconstructed acoustic pressure using the AIBM is less dependable on the input location of the acoustic data. The AIBM gives more consistent, reliable and accurate reconstruction of acoustic radiation pressure when the input acoustic data is avail- able over an open surface. However, for the case of 7 = 0.47r, the solution of the AIBM starts to deviate from the analytical solution. As 7 decreases further, the effectiveness of the AIBM becomes hindered. As a general rule, the more complete the input around an acoustic source, the more accurate the reconstructed acoustic pressure solution. If the input is given on a single segment over an open surface, a considerable extent of the segment is needed in order to achieve acoustic pressure reconstructions with acceptable accuracy. If the input is available on multiple segments over an open surface around an acoustic source, then a better accuracy of the reconstructed solution will be obtained when the segments are on the opposite sides of the control surface, although the exact location of the acoustic source may not be known. It is also worth mentioning that for an input given over a closed surface, the reconstructed acoustic pressure solution is unique without the addition of the pressure gradient to the input acoustic data. The advantage of the AIBM over the AIBM_without diminishes. The formulation used for the AIBM _without is identical to that of the HELS [32]. Therefore, by including the pressure gradient in the input of the HELS, the mathematical well-posedness of the HELS will be improved and the method will be enhanced. 3.3.2 Effect of Signal-to-Noise Ratio (SN R) Until now the analytical solution has been used as the input acoustic data for the calcula- tion. In practice, however, the input data is usually experimentally measured. The acoustic measurements are unfortunately prone to random errors. It is therefore desirable to have the reconstructed acoustic radiation pressure less sensitive to random noises. In the following, 27 0.4 0 Analytical solution AIBM ------- AlBM_wlthout 0.3 O N ITrijrrrrlrrrrTirrll itfltjrrnIIIITli - 4111l11111111ll1imL111111111LLm 1 2 3 4 5 6 0 o rTir (a) '7' 21.07r 0.4 0 Analytical solution AIBM ------- AlBILwlthout 0.3 0.2 'TI'TIIT ITIj’ IIII III] ,1 1 1 T 1 IIIWIIII'IITI1IIIII _05IlIJIIIJIILHJLLIIIIIII111111114 '0 I 2 3 4 5 6 0 (b) 1., = 0.8a Figure 3.3: Comparisons of reconstructed acoustic radiation pressure with analytical solu- tion at r = 50m for different 7. 0.4 0.3 0.2 0.1 IIIIIIIIII'IIIIIIIIHI .5 & .6 01 O rrrrerTT11rTT'rrirl ‘ 0.4 Figure 3.3 continued L111111L1J111111111IL111l1L1114 1 2 3 4 5 6 0 (c) ’7' = 0.67r E 0 Analytical solution 0.3 : -— AIBM : ------- AlBM_w1thout 0.2 :— o.1 :— 3 o : kq b O \r o 0‘ '0 .1? I" \‘00000 " \ : .’ \, ~02 j" s‘ :3; “ -o.3 r"- x ‘ : “ I : ~ ’ '0.4 '_— “‘ ”l’ _0._IJJLIIIIJIJIIIIIILIIIIIIIIIIIJL 50 1 2 3 4 5 6 0 (d) ’y = 0.47r 29 the effect of the SNR is investigated in a two-dimensional configuration using an acoustic source similar to a monopole but with an amplitude proportional to a sine fimction. Assuming the sound source is located at (10.310), i.e., 1‘0 2 0.8m, yo = 0.6m, the exact solution is constructed in the following form Po(r, 6) = sin (1H1 (ktR), (3.17) where R = \/(‘T — $0)2 + (y — y0)2. a is the shifted angle and is given by tana = (.2: - :170)/ (y — yo). Like the previous example, the acoustic source is enclosed in the circle of r = 1m. The wavenumber and the radius of the control surface are given as k = 2m"1 and rc = 3m, respectively. The input acoustic data consists of the exact acoustic pressure solution with an added random noise RN (r, 6). The location of the input is chosen as the upper half circle of radius r =2 6111 (an open surface). The SNR is defined as M M SNR: Z lPe(r,6,n)!2/ Z |RN(1~,6m)|2 (3.19) 112.21 771.21 where m represents each uniformly distributed grid point (M = 200) over the upper half circle, and the random noise has zero mean and the identity covariance matrix. The recon- structed acoustic radiation pressure from the AIBM and AIBM.without is compared with the analytical solution in Figure 3.4 for different SNRs. The results demonstrate that the accuracy of the reconstructed acoustic pressure with the SNR up to 10 is overall accept- able for the AIBM. However, without the addition of the acoustic pressure gradient in the input acoustic data, the reconstructed acoustic pressure from the AIBM.without is unac- 30 ceptable even for the SNR as large as 100. In order to have an overview of the reconstructed acoustic radiation field, the acoustic pressure contours of the analytical solution and the re- constructed solutions from the two inverse methods are given in Figure 3.5. Since the input acoustic data is given over the upper half circle, as expected the reconstructed acoustic pressure for the upper half domain is very accurate using both inverse methods. The con- tour plots also show that the reconstructed pressure contours in the lower half domain of the AIBM-without are completely different from the analytical solution even with SNR=100. On the other hand, the accuracy of the reconstructed acoustic pressure in the lower half domain of the AIBM is very good for the same SNR. The results indicate that by including the pressure gradient in the input acoustic data, the AIBM can handle random noises much more effectively. As it is shown in Figure 3.4, with up to 10% random noises (SNR-=10), the reconstructed acoustic radiation pressure using the AIBM agrees with the analytical solution reasonably well. 31 0.5 -1 Figure 3.4: Effects of SNRs on reconstructed acoustic radiation pressure at r = 10m. I I 177 I 0 Analytical solution - - — - SNR=10 ........... SNR=40 SNR=100 Llll‘llIiJJILIIIIJIIIIIIIIIIJLL 1 2 3 4 5 6 0 (a) AIBM £- 0 Analyticalsolutlon - ---- SNR=10 .. ........... SNR=40 r SNR=100 )— .. \ I — \ I l- \/ _LlllllllLlIllIIIIIIIIIJIIIJJIJL 0 1 2 3 4 5 6 0 (b) AIBM.without 32 (21) Analytical Solution (b) AIBM Figure 3.5: Comparisons of reconstructed acoustic radiation pressure P(r, 6) contours for SNR=100 with analytical solution. 33 Figure 3.5 continued (c) AIBM_without 34 3.3.3 Application to Multi-Frequency Acoustic Radiation Problem The acoustic radiation fiom 2D vortex filament moving around the edge of a semi-infinite plane is a simplified model for the radiation of flow—airfoil interaction problem in relation to the airframe noise study. Ffowcs William & Hall [49] initially developed the general theory of the scattering of aerodynamic noise by flow-surface interaction. Shortly after, Crighton [50] developed the simplified 2D vortex model and derived the analytical acoustic solution in the form of the potential function using the singular perturbation method. Since then, the model has been used [51] to verify the effectiveness of the numerical methods of solving the F fowcs Williams and Hawkings (F W-H) integral equation. In the current study, the vortex model is used to show the capability and potential of apply- ing the AIBM for multi-frequency acoustic radiation problems in general. A schematic diagram of a 2D vortex moving around a semi-infinite half plane is shown in Figure 3.6. The origin of the polar (or Cartesian) coordinates is located at the edge of. the semi-infinite plane. Based on the analysis given by Crighton [50], the first term in the asymptotic expansion of the analytical potential function as U m —+ 0 is given by 4 i Um (12 sin is 1 1 ’ [1113(7‘ — 156)2 + 4]? 7‘? (-r. 6, t) = 6 6 (—7r, 7r), (3.20) where c is the speed of sound, d is the shortest distance between the vortex and the edge (see Figure 3.6), Um is the maximum speed of vortex motion, and the Mach number M a is defined as Mo = Um/c. These variables are considered as c = 344m/s, d = 1m and Mo = 0.01 in the current numerical investigation. It can be easily shown that the acoustic pressure derived from the above potential fianction satisfies the wave equation and the solid wall boundary condition, Zip/59y = 0 at the top and bottom surfaces of the plane. Since the acoustic radiation of the vortex model problem is not in a free space, the general 35 approximated solution (Equation (3.5)) for the Helmholtz equation needs to be modified to satisfy the solid wall boundary condition and can be written as TV ‘ 1 P(r, 6) = Z on cos n6Hn(kr) + 1),, sin (n. + 2 >011 1 (1.4)] . (3.21) ”:0 n+2 As it can be seen from Equation (3.20), the acoustic radiation of the vortex model problem is not a single frequency problem. The FFT of the acoustic pressure time history at a field point with the polar coordinates ( 75m, 7r / 4) is shown in Figure 3.7. The continuum spectrum demonstrates the multi-frequency nature of the acoustic radiation. It can also be shown mathematically that the pressure time history of any given point in the field has the same frequency spectrum although the amplitude may vary. Since the AIBM is a frequency domain method, the reconstructed acoustic pressure field for each frequency component needs to be calculated. A superposition of the contribution from each frequency gives the total acoustic pressure field in the frequency domain. The reconstructed acoustic pressure field in the time domain can then be obtained by an inverse FFT. In order to accurately reconstruct the acoustic pressure, all the dominant frequency components need to be included in the calculation. The input acoustic data for the AIBM consists of the analytical acoustic pressure at four uniformly distributed points on each of the two circular segments (7' = 50m) as shown in Figure 3.6. The reconstructed acoustic pressure time history at a field point with the polar coordinates (100m, 7r / 5) and the directivity pattern at the radius of r = 100m are calculated by the AIBM and compared with the respective analytical solutions in Figures 3.8 - 3.10 for three selected ranges of the frequency components. The poor agreement between the analytical and the reconstructed acoustic pressure shown in Figure 3.8 is expected since the input acoustic data includes only a small portion (0.05 to 0.2Hz) of the dominant frequency com- ponents. As the frequency range widens to include more dominant frequency components (0.05 to 0.45Hz), the accuracy of the reconstructed acoustic solutions using the AIBM im- 36 segment I Vortex path 01s= 3-13rad Half Plane ete= - 3.13rad ._ Figure 3.6: Schematic diagram of 2D vortex filament moving around the edge of a semi- infinite plane and input acoustic data points on specified segments. proves significantly (see Figure 3.9). Finally, an excellent agreement is achieved between the analytical and the AIBM solutions if the frequency range widens further to include all frequency components from 0.05 to 1.45Hz in the input acoustic data (see Figure 3.10). After the verification of the AIBM for the reconstruction of acoustic radiation pressure generated by the multi-frequency model problem, it is important to examine the sensitivity of the reconstructed acoustic pressure to the SNR to ensure the potential of the AIBM for engineering applications. Since the acoustic radiation of the vortex model problem involves multi-frequency components, a random noise is added to the input acoustic data in the time 37 domain. For a given set of the input acoustic data, the SNR is defined as SNR = (3.22) I 1 ' 1 2 2.2. 2);. 1.41,. x.) / 111- N-a where M and N are the numbers of the input locations (points) and the time discretization over a period of measurement time (or data collection time), a is the variance of the random noise. The mean of the random noise is zero. It is noted from Equation (3.22) that the SNR is defined based on the average of the input acoustic data in the time domain. Since the magnitude of the input data varies, the SNR could be larger or smaller than defined for an input at a specific location. Using the input locations shown in Figure 3.6, the effect of the SNR on the acoustic radia- tion pressure time history at a given point in the field is shown in Figure 3.11 for different SNRs. As it is shown, a very good agreement between the reconstructed solution and the analytical solution is obtained for the case of SNR=10. As the SNR decreases to 5, the error of the reconstructed acoustic pressure increases as expected, particularly in the region where the amplitude of the analytical pressure is small as expected. Nevertheless, the re- sults demonstrate that the AIBM can effectively handle the input acoustic data with up to 20% random noise. 38 015 OJ O IIIIIIIIIIIIII'IIIIIIIII'IITI' -01 _ 4 1 1 1 l 1 1 1 1 I 0'15-10 5 10 (3) Acoustic pressure time spectrum 0.03 l- 0.02 - Q‘ . “ 7 I . I I (b) Frequency spectrum Figure 3.7: Spectrum of acoustic radiation pressure time history of 2D vortex model at r = 75m, 6 = 7r/4, Illa = 0.01 and d=1m. 39 T Analytical solution r - -O- - AIBM solution 0.1 s 0.05 s .\ _ N .. \., _ a, o -0.05 fiIIIIrTIII‘, 1| 1141 .. .— .— .— — )— (a) Acoustic pressure time history (r=100m, 6 = 7r / 5) 0-1 — —— Analyflcalsolutlon - - -o— - AIBM solution 0.05 P pm Sine -0.05 - l I J -o.1 cos 0 0.05 pm COS0 (b) Directivity (r=100m) Figure 3.8: Acoustic radiation pressure and directivity calculated from the input data with frequency range of 0.05 - 0.2Hz. 40 0.1 -0.05 frrrrllrrnjrirrl —-— Analytical solution - -O— - AIBM solution -0.1 l L L 1 l J. I L l l J l l L L L l I l I -10 -5 0 5 10 t (3) Acoustic pressure time history (r=100m, 6 = 7r / 5) 0-1 r —— Analytical solution # — -o- - AlBMsolutlon L 0.05 — G 1- .5 L W L E o — Q. s -0.05 — t 1 1 1 I (b) Directivity (r=100m) Figure 3.9: Acoustic radiation pressure and directivity calculated from the input data with frequency range of 0.05 - 0.45Hz. 41 0'15 _— Analyticalsolutlon ~ - -o- - AIBM solution 0.1 - o.05 — ’\ r "' r \" +- Q 0'-122;3-_ -0.05 IIfIITTIIII‘ is O I 01 No— 0" d O (a) Acoustic pressure time history ('r=100m, 6 = 7r/5) 0-1 V —— Analyticalsolutlon ~ - -o— - AIBM solution 0.05 - Pm sine -0.05 — I A J I 1 J I 1 1 I -o.1 43.05 0 0.05 prms case (b) Directivity (7'=100m) Figure 3.10: Acoustic radiation pressure and directivity calculated from the input data with frequency range of 0.05 - 1.45Hz. 42 0'1 ’ Analytical solution - -O- - AIBM solution r 0.05 19(1) -0.05 (a) SNR=10 0'1 '— Analytical solution '- -o- - AIBM solution 0.05 - ’\ O °¢O ? N \, 6 O a 3 -0.05 l l l l I l I J—J -0.1 5 10 (b) SNR=5 Figure 3.11: Acoustic radiation pressure time history (1" 2 100m, 6 = 7r / 5). 43 Chapter 4 2D AIBM with Subsonic Mean Flow 4.1 Introduction In Chapter 3, an acoustic intensity-based method (AIBM) has been proposed without con- sidering the effects of mean flow. The objectives of this chapter are to extend the AIBM to include sound propagations in uniform flows and demonstrate the capability of the method when coupling with the near-field CF D/CAA methods for the prediction of far-field sound radiations in a two-dimensional configuration. First of all, the mathematical formulation and numerical implementation of the AIBM for sound propagations in uniform flows is described in Section 4.2. In Section 4.3, The method is then coupled with the near-field CFD/CAA methods for the prediction of sound radia- tions in the far field. An example of sound radiation from monopole in a uniform flow is solved for the verification. Two aeroacoustic problems, sound radiation by a flow around a NACA airfoil and sound scattering by a cylinder, are solved in the following sections to demonstrate the effectiveness of the CF D/AIBM and CAA/AIBM hybrid techniques. At the same time, the sound radiation field obtained from the AIBM and the FW—H integral method are compared. The effects of input locations and the signal-to-noise ratio (SNR) on the accuracy of the AIBM solution are examined. Conclusions are drawn in Section 4.4. 44 4.2 Mathematical Formulations In this section the formulation of the AIBM [3 8] is extended to include sound propagations in uniform flows. Without loss of generality, we assume the uniform mean velocity is in .r-direction u0 = uui. Let 12,-” be a bounded domain in R2 containing all acoustic sources in this mean flow (see Figure 4.1), and c be the speed of sound, it is well-known that the acoustic pressure p(;r, y, 1‘) is governed by the following homogeneous wave equation 2 1 2 Vp—Z§(at+u0-V)p=0. (4.1) Two cases will be considered in this paper. One is the problem with a single frequency, the other is with multiple frequencies. 4.2.1 A Single Frequency Assume that p = e'MPCr. y) with angular frequency w, P(.r, 3)) satisfies V217 — 11,211,, — ant/1,1214 k2P = 0, (4.2) where k = w/c is the wave number. Mo = uO/c represents the Mach number, which is assumed to be less than 1 in the current study. In order to get the solution of Equation (4.2), we want to convert this equation to the standard Helmholtz equation. Set W(;f, 3)) = P(.r, y) with (in g) = (:r/J. y) and 0’ = t/I — 21102, the governing equation of P, Equation (4.2), can be rewritten as Qikfila V211" — 113,. + k2W = 0. (4.3) 45 To eliminate the first order term W 5., we define the function 3(5, 3)) as S(:i:, g) = exp[—‘ik1Ma;it/,U]I'V(§:, 3}). (4.4) The equation for S is therefore expressed as V25 + +3 = 0. (4.5) In terms of the polar coordinates for ig—plane, :i: = 7“ cos 6, g) = 7" sin 6 and f = 1/ £2 + {12. the general solution for S on or outside a control surface, which encloses all the sound sources under consideration, is given by (x: A. A S(.i:, ) = 2: (an cos 716 + bn Sin n6)Hn (kf/b’) , (4.6) 7120 where H n is the nth—order Hankel fimction of the second kind. Combining these equations yields m A A P(;r, y) = exp('z'kA/I:r,{5_2) 2 (an cos 726 + bn sin n6)Hn (kr/U) . (4.7) 7120 Note that .1: is scaled by the factor (3 == 1/ 1 — 1110?, I2 is not the usual 6.122 + y2, and 6 are also different from the usual angle 6 in the polar coordinates for cry-plane. In order to obtain the solution of Equation (4.7), it is necessary to determine the coefficients an and bn. These coefficients are determined by matching the assumed form of the solution to the measured acoustic pressure and its normal derivative over the input surface segments. Once these coeflicients are determined, the solution can be quickly evaluated at any field point on or outside the control surface. In the AIBM, both the acoustic pressures and its simultaneous, co-located derivative (in out normal direction) on the boundary [‘1 are given as the input for the reconstruction of the acoustic field in the domain 52 (see Figure 4.1). With the pressure derivative boundary condition as an additional input, the uniqueness of 46 the reconstructed solution is guaranteed from the unique continuation theory of elliptical equations [35—37]. The method also yields a consistent and accurate solution on and out- side of the control surface. When using the AIBM, it is assumed that the control surface is known although the exact locations of sound sources may not be available. ‘ _-——-—_ - “_ ._—' '- Mean Flow ‘. , Figure 4.1: Schematic diagram of sound propagation field and locations of acoustic mea- SUI’CITICIIIS. With the consideration of sound propagations in uniform flows, the partial boundary value problem is defined as V21) — 01(12PJrr ‘ QIICI‘IIQPJ: + [”213 = 0 in Q = R2\Qm, (4 8) Plri = R anplri = Pn» where n is the outward normal to F1. Similar to the procedures given in our earlier work 47 [38], the following three steps are used to solve Equation (4.8) in the solution form given by Equation (4.7). Step 1: The infinite summation in Equation (4.7) is replaced by a finite summation, i.e., P(.r, 3)) ~ exp(i/c]\l.f/d) (10Ho(kf/t )+ i)(§: (an cos 726 + bn sin n6) HAM/,6) ”2 1 (4.9) where N is a suitable integer. The choice for N will be discussed later in the section. Obviously, one restriction is that the number of coefficients (2N + 1) to be determined must be less than the number of the measurement or input points. Step 2: An accurate and efficient method must be developed to evaluate each Hn. When kf/ 6 is relatively large, the first Q terms in the asymptotic expansion of H n(7), namely Hn(7‘) ~ 72,620), are used in the current numerical study. For the completeness, we derive the expression for Hn,Q here. Even though the asymptotic expansion for Bessel functions are well documented, it seems that the expansion for H n cannot be found easily. It is known that H n has the following explicit integral expression [52] H. - = /_ c"‘t(1—t "*1/212. 4.10 ”(7) 271 1F( (+12 +1/2) fi 1+m 2) ( ( ) By the change of variable for integration, 7 = —1 + 2'7), we have 26.—(72 1/2) 7ri/2,I.72 726 —z'rr /: "2 H = Vi); e"'(n"’" 7' V221. 4.11 Furthermore by letting 7' = 77), the asymptotic expansion of H n, for 'r —» 00 is derived as Q s) ,- . _ 2 —I r—II‘IT 2__7r 4()’l]*1/2)j11'(77+]+1/2) - Hn(7“) — fie ( / / )[1-1— 32:1 ([312JTJF(JII. + 1/2) 1‘ 72,62 48 It is easy to see that the order of error established in the above approximation is Q E(n, Q)(-) 7*)3' H( (n—j+1/2) (72+j —1/2) (4.13) jzl which tends to 00 when N —> 00 or Q —+ 30. Hence the asymptotic expansion is not convergent, it is necessary to choose N and Q carefully. For fixed 7 and 72, the error E (n, Q)(7) is roughly (28):? the restriction of Q S 27 should be used to control the approximation errors. Furthermore, for fixed 7 and Q, N (the maximum of 72) must also be controlled from E (N, Q)(7) ~ (21%? which requires that N cannot be larger than 27' as well. In this study, we will restrict our choice so that Q g 1.519776 and N g 1.5197713. Step 3: Suitable optimization method is used to determine the coefficients an and bn. For the partial boundary value problem, Equation (4.8), the simplest method is the least squares technique because the resulting equations for an and bn are linear. If 91 and g2 are known at M discrete points, (71, 61), ..., (7 A 1, 6 M), the linear system for the coefficients an and bn is given by the following 2M equations ' . '2 . IV A A erlIan/J (HOHOQ + 2 (an cos 726J~ + 1),, sin 726j)Hn‘Q) = P(:1:j, yj), (4.14) 7221 ' .2 '. , ‘ IVY A A an [e’UI‘IJJ/‘j (aOHOQ + 2 (an. cos 7261' + bn si11n6j)Hn,Q) = Pn(.rj, yj), (4.15) 72:1 where j = 1,. .,.M It 18 noted that in order to solve the above linear system, the terms H n, Bf/an and 86/6n need to be evaluated first. The linear system can be expressed in a matrix form as AX = B, where _ T X — [(1.0,---,a‘,v,b1,-~,bN] , (4.16) B = [P(é171~y1)a ' ° nP(f111»y111)apn(131~.y1)a ' ' ',Pn(JF11Ia3/111)]Ta (4-17) 49 and A = (A1, A2)T, A1 corresponds to P and A2 corresponds to normal derivative Pn. One could find X by minimizing ||AX — B||2, where H.“ is the standard L2 norm. Dur- ing the numerical study, it is observed that although some regularization methods may be needed to improve the stability of the system for large N, they are not necessary for rela- tively small N. In the current work, N is initialized within a given range of 0 — 20. The reconstructed solutions for various N are compared with the input acoustic solutions of the inverse problem. The total error at all the input points is then computed for each N. The optimum N within the given range is determined from the minimum overall error. Finally, the reconstruction of the entire acoustic field is carried out using the optimum N. It is worth a mention that without the inclusion of the pressure gradient to the input of the inverse methods the reconstructed solution cannot be unique if only PIP1 is spec- ified. For example, if 52,-" is the unit disk centered at the origin, we assume that all the acoustic sources are enclosed by the unit circle, and the boundary F1 is defined as F1 = {(123, y) | 2:2 + 92 2 f2, 2) > 0}. That is, F1 is the upper halfboundary ofan ellipse, 7‘ is chosen large enough so that the ellipse will include all the acoustic sources. The acoustic pressure for 6 E [0, 7r] should satisfy Equation (4.7). It is well known that an and bn are not unique from Fourier analysis. Especially P always has Fourier cosine expansion for 6 E [0, 7r], namely one can set b1 2 b2 2 = 0. In any case, it is impossible to predict solutions for 6 E (71,27r). Furthermore, it should be pointed out that even though our formula are derived for the uni- form flow u0 = uoi = (220, 0), it is very easy to modify it for a general case no = (210,120). In that case, by rotating the coordinates for an angle a, where a in the range of (0, 271) is given by (110,120) = 621.8 + 228(cos a, sin a), the pressure P(.r. y) in the frequency domain with angular frequency 122 can then be calculated by CX: A P(.'27, y) = exp(2'k]\[;ifJ—2) 2 (an cos n6 + I)" cos 72,6)Hn(kf‘/1’3), (4.l8) 72:0 50 where Ma ’ 18 + l’8/C, i‘ = (:r cosa + ysin (2)/,6 = fcos 6, (4.19) 9 = —;27 sina + ycosa = 7‘sin6, A (.9 .2 r = \/.’I“" + y . 4.2.2 Multiple Frequencies When the scattering problem involves multiple frequencies, the F FT can be used to decom- pose the problem into the linear combinations of several dominant frequencies. Since the AIBM is a frequency domain method, the reconstruction of the sound pressure for each frequency component needs to be individually calculated. A superposition of the contri- bution from each frequency gives the total sound pressure in the frequency domain. The reconstructed sound pressure field in the time domain is obtained by an inverse FFT. The details of the solution of the AIBM are explained in the following steps: Step 1: Input Acoustic Data. The input acoustic data are obtained from measurements, or from numerical solutions of CAA calculation on a circle with radius 70 for 0 g t g to, where to is large enough so that the solution asymptotically decays at large t. Assuming the dimensionless speed of sound c = 1, then the wave number for each frequency kj = 1123-. The fast Fourier transform is then used to decompose the solution in terms of its frequencies wj, i.e., J Mir, 1% t) = Z €XP('iw‘jt)Pj(-r, y) (420) i=1 with PJ-(r. y) satisfies the Helmholtz equation v21),- + 11,210, = 0. .r e 12 = R2\B(0,7'1) (4.21) 51 where B (O, 71) is a circle of radius 71, centered at the origin. The boundary condition P]- on the circle of 7 is given as the input acoustic data for the AIBM. If the polar coordinates are used in a two-dimensional configuration, the solution of Equation (4.21), which also satisfies the radiation condition, can be written as DC [DJ-(37,31) = E(aJ-m cos 726 + (2%,, sin 726)H.n(wj7‘) (4.22) [=0 where H" is the nth-order Hankel function of the second kind. Step 2: Determination of the coefficients 0),, and b )n The numerical solution of Equation (4.22) is obtained by replacing infinite summation in Equation (4.22) by a finite summation, i.e., [VJ P(:2:, y) % anOHofiujr) + Z (cry-fl cos 726 + by, sin n6)Hn(wJ-7) (4.23) 7221 where N]- is a suitable integer for each frequency component. Because of the broadband nature of the problem, it is necessary to accurately determine the coefficients a j.” and bjm. for each frequency wj. As in the case for single frequency, these coefficients are determined by matching the assumed form of the solution to the input acoustic data in the frequency domain, i.e., P(;2:, y) and P“, on the circle of radius 70. The choice for N J- must be carefully chosen to ensure both the accuracy and stability as discussed in [38]. Apparently, the accuracy of the solution naturally requrres a large value of Nj. However, a large N j can also result in computational instability. For any given 1123-, a mathematical analysis yields 272 n Hn(wjr)=( ) , n—mo (4.24) GUJJ'T When wjr is small, Hn(wjr) is very large when n 2: (egg r/ 2) . A small perturbation in the input acoustic data may result in large variations in (13-, ,1 and 1))". Hence the value for 52 N J- has to be restricted for the stability of the numerical solution. In the current study, the restriction is given as N j S [wjro] + 1, where [z] is the greatest integer less than or equal to 2:. In the current work, the value of N j is chosen as N) = rnin{Nj S [adj-7'0] + 1,30} (4.25) with considerations of numerical stability and computational efficiency. For any given wj and N j, the coefficients am and bjm are determined for each frequency component of 1 g j 3 J and 0 S 72 g N J- based on the acoustic input provided on the circle of 7. Step 3: Prediction of sound radiation p(7, 6, t). For any given a: E B(0, 7‘2) with polar coordinate (7, 6), and t E [0, to], with 72 not far away from 71, N (x) , (21,-, cos [6 + 11,-) sin (6)6thH,(wj)z|), (4.26) ‘ _-0 . J p(7‘162t) N Z j=1 [- where the upper limit N j (.23) is chosen as NJ-(gr) = n1in{[wJ-|.T|] +1,[wj7‘0]+ 1,30}. (4.27) It is noticed that Nj (.2) can’t be larger than ([wj 70] + 1, 30) due to the value of N j (27) used in determining the coefficients am and bjm- The additional restriction [wjr] + 1 is added for a similar reason mentioned earlier in the discussion of Equation (4.24). Without the additional restriction on N j, the solution p(-r, 6, 2‘) may become unstable. 53 4.3 Numerical Examples and Discussions In this section, the AIBM is first verified for sound source propagations in a uniform flow. The method’s effectiveness and capability for aeroacoustic applications are then demon- strated by airfoil noise and sound scattering problems. 4.3.1 Monopole in a Uniform Flow The AIBM is first applied to a stationary monopole source placed in a uniform flow. A schematic diagram of the source in a uniform flow of Ma = 0.5 in the +2: direction is shown in Figure 4.2. The two circular arc segments are the locations of the acoustic input and the angle 7 is a measure of the dimensionless distance between the two segments, with 7 = 7r being the farthest, i.e., the segments are at the opposite sides of the control surface. The units used for 7 and 6 are the meter and the radian, respectively. The monopole is placed at (1m, 0.27r) in the polar coordinates. The wavenumber and intensity of the monopole is given as k = 2m‘1 and A0 = 0.001m2/s. The control surface is considered here as the circle of radius 7 = 2m. The analytical complex potential for the monopole is given by Dowling and Ffowcs Williams [53] as 2' , k 52:2 (;2:, y, 2) = 1401561))th + Mam/152))H0 B‘ I 13—, + y2 (4.28) The analytical acoustic pressure and its normal derivative, over the two circular arc seg- ments, can be derived from Equation (4.28) and are used as the input for the inverse cal- culation. The polar coordinates of the starting and ending points of the two segments are (10m, 0), (10m, 0.171) and (10m, 7), (10m, 0.17r + ’7’), respectively. Ten uniformly spaced grid points are used on each of the segments. With {3 = 7r, the far-field directivity from 54 the AIBM is calculated and compared with those from the analytical solution and the F W- H integral equation in Figure 4.3. The results show excellent agreement among the three methods. It is important to point out that the arc length of the each input segment for the AIBM is only about 1/20 of the circumference of the circle (7 = 10111), that is used as the FW-H surface. The reconstructed acoustic pressure contours from the AIBM and the analytical solution are shown in Figure 4.4. As it is indicated, an accurate reconstruction of the radiated field is achieved on and outside the control surface using the AIBM. As it has been shown in the work of Yu et (21 . [39,40], though an accurate reconstruction can be obtained from the input given over an open surface, the AIBM becomes less effective when the input segments become clustered. As a general rule, the more scattered the input segments around a sound source, the more accurate the reconstructed acoustic solution. Since the choices of the input segments are limited by the accessibility and practicality of the acoustic measurement in the radiated field, some regularization techniques may be needed to improve the effectiveness of the AIBM when the input locations are not scattered far enough around the sound source. 55 yA Circular arc segments ' \\\ (r, 02) Sound sourcc'-_ r Mcan flow d ’ ("2 91+Y) 7/ (r, 91) 4 z; 5 x (2'. 02+7) Control surface Figure 4.2: Schematic diagram of a monopole radiation in a uniform flow and locations of acoustic measurements. 56 1-2 ‘ 0 Analytical solution ~ —— AIBIM _ .................. FW-H 0.6 — CD _ E % ._ E l- K 0 r- 2 -O.6 — _ I l I I I I l -1.2 -O.6 2 o 0.6 r|P| cosO Figure 4.3: Far-field directivity (r 2 100m) comparison of a monopole radiation in a 57 -20 -1O 0 10 20 x (a) Analytical solution -20 -10 0 1O 20 x (b) AIBM Figure 4.4: Pressure contours of a monopole radiation in a Ma = 0.5 flow. 58 4.3.2 Sound Radiation by a Flow Around a NACA Airfoil After the above verification of the AIBM for the prediction of sound radiation in a uniform flow, the method is used to obtain the sound radiation from a uniform flow around a NACA airfoil. Based on the near field CFD solution, the aerodynamic sound generated in the far-field is calculated using the AIBM. The predicted sound radiation of the AIBM is then compared with that from the F W—H integral equation. The far-field acoustic solution is commonly obtained by solving the FW—H integral equation based on the unsteady CFD solutions on a F W-H surface that encloses the airfoil. The FW- H equation is a rearrangement of the exact continuity and momentum equations to a wave equation with source contributions from the monopole, dipole and quadrupole terms. The contribution of the quadrupole term, the Lighthill stress tensor, is neglected since the FW- H surface, as indicated in Figure 5 for the current study, is placed outside of all regions where the stress tensor is significant. The quadrupole contribution is in fact included by the surface sources. The contours shown in Figure 4.5 are the instantaneous sound pressure contours of the flow around NACA0018 airfoil of a chord length 0.3m. As also shown in Figure 4.5, the free-stream Mach number, Ma, is 0.2 and the angle of the attack is 20°. The details of the CF D solutions were given by Greschner et al. [54]. The AIBM and the FW—H integral method are used for the prediction of the far-field sound radiation by the flow around the airfoil. The AIBM is carried out by using the unsteady pressure solution over an open surface, formed by the curved segments I and II or segments III and IV of the FW—H surface (see Figure 4.5), as the acoustic input. The two-dimensional formulation of the FW—H equation in the frequency-domain [55] is used with the input of the unsteady pressure and velocity solutions over the entire FW-H surface. The comparison of the far-field directivity obtained from the AIBM and the FW—H integral method is shown in Figure 4.6. As can be seen in Figure 4.6, the results of the AIBM based 59 Figure 4.5: Instantaneous pressure perturbations of the flow around the NACA 0018 airfoil along with the location of the FW-H surface. on the inputs of two chosen open surfaces agree reasonably well with that of the FW-H integral method. In order to have an overview of the sound propagation in the far-field, the sound pressure contour plots from the AIBM and the FW-H method are shown in Figure 4.7. It is noted that the radius of the control surface used for this problem is r = 3m, which is ten time of the chord of the airfoil. It should also be pointed out that the FW-H surface used in the study, though not a circular shape, is enclosed in the control surface. The close agreement among the three contour plots indicates that the AIBM is capable to effectively obtain the radiated sound field based on the acoustic input from an open surface. The method, therefore, has a potential application for the far-field sound reconstruction for problems where a closed FW-H surface is not possible. 60 _L l _ AlBlMsolution _ - - - - FWH solution 0.5— CD 0“- .= r a; _ E _ “—0.5— {2 _1-5_Llll 111111111144 -1 0.5 1 , 1.5 r|P|2 0039 Figure 4.6: Far-field directivity (7 :2 100m) of the flow around the NACA 0018 airfoil. 61 (a) FW-H 20 -20 -10 0 10 20 JC (b) AIBM Figure 4.7: Pressure contours of the sound propagation generated by the flow around the NACA 0018 airfoil. 4.3.3 Sound Scattering This example is an ideal model of the physical problem of predicting the sound field gen- erated by a propeller scattered off by the fuselage of a moving aircraft. In the model, the fuselage is considered as a circular cylinder and the noise source (propeller) as a line source such that the computational problem is two-dimensional. A polar coordinate system and the Cartesian coordinate system centered at the center of the circular cylinder of the di- mensionless radius 0.5 are shown in Figure 8. The mean flow is given as Mach number of zero. The governing equations for this problem are the linearized Euler equations (LEE). The equations are discretized using the optimized upwind dispersion-relation-preserving scheme (DRP) of Zhuang and Chen [56]. The detailed implementations of the boundary and initial conditions are given by Chen and Zhuang [57], in which the numerical solution (CAA solution) was also verified by the analytical solution. In the current study, the CAA solution is used as the acoustic input for the AIBM. The input is given at forty uniformly distributed points on each of the two circular segments (see Figure 4.8) with the radius of 6.125. The initial pressure pulse located at (4, 0) is given as (x ---4)'2 +y2 0 22 (4.29) p(:2:, y) 2: exp -— 1112 and the perturbation velocity components in ar- and y-directions are considered as zero, u=v=0. The results of predicted sound pressure history for different frequency ranges are also com- pared with the corresponding CAA solutions at three locations are shown in Figures 4.10- 4.12, respectively. As it is indicated that the accuracy of the predicted solution improves significantly as all the dominant frequencies are included in the AIBM calculation. In terms of peak pressure values and locations of the incident and reflected waves, excellent agree- ments between the two methods, AIBM and CAA, are demonstrated. The oscillations at 63 y ‘1 ,x / \‘\.CAA boundary / \. /' \, / \. o/ . \‘ o/ . . . n . . I ' . \| / '0" =6 12 s‘ \ / ’0' r =05 r . ‘3 \ I .0 “ \I i -' . " 1 ; .' norse source: ; _ l . l V " '1‘ cylinder 4—4—4 '2' j X . " l ' \‘ \ .0 I \‘ ‘s‘ '5' ./ \‘ 'Q‘ ' . ..I '/ \.\AIBM surface I. ' ' ' - -- ---""AIBM surface 11/ ‘\. o/o \ ,/ \ '/ \ ~ ' / \ ' x . I / - - T . .,. Figure 4.8: Schematic diagram of the sound scattering by a cylinder. the lower amplitudes are due to a relatively large value of N. As it is discussed in Sec- tion 11, the number of summations, N (Equation (4.8)), needs to be reduced as the radius 7' decreases to provide a converged solution. The pressure contour plots from the CAA calculation and the AIBM are compared in Figures 4.13-4.15 for various times. The results of these contour plots demonstrate that the AIBM can effectively predict the propagations of both the incident and reflected sound waves. 64 9=3rtl4 0:0 --------- 9:11'14 ———- 0:11: _._._._ 9: 30 25 Figure 4.9: Frequency spectrum for the sound scattering, 7* = 6.125. 65 OJ (105 pm 4105 01 005 pm {105 ——CAA -—--AIBM ‘f 7' 0 (a)w=0.307——3.07 _ ---- C‘UK ~ ----AIBM 1. _ I 11 1 r 1 r 1 I 1 r 1 111 11 1 I 5 10 15 20 (b) w = 0.307 — 24.56 Figure 4.10: Pressure time history reconstruction with different frequency range at 7 = 7.25 and 6 = 00. 66 006 f ——CAA _ ----AIBM 004"- "\0D2" ~ .— \_, ._ p Q‘ 7 I h I 0 .. \ -002'- — 111111 111 111 1%] 0 10 15 20 t (a)w=0.307-3.07 0.06p' CM . —-—-AIBII 004- : 1 «0.02— ' ‘“ u l V — 1 Q1 1 l 0% ____ - 1.1 v _ '1’ -002F- P 14 1 a 141 11111111] 0 10 15 20 (b) w 2 0.307 — 24.56 Figure 4.1 1 : Pressure time history reconstruction with different frequency range at 7 = 7.25 and 6 = 900. 67 0.05 : -—-——CM : ----AIBM 0.04} 0.03:— «0.02:- N II- \.+ 1- Q... L 0.01_ ’\ : Ill _ - a. ‘ I ‘ 0 I a 1- I : v -0.01:' _ h 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 J 0.020 5 10 15 20 t (a)w:0.307—3.07 0.05? CAA : ----AIBM 0.04:— 0.03} \0027 ~ b \, _ “0.015- 1 0M ____ 0.01:— _711111111111111111111 0020 5 10 15 20 t (b) w = 0.307 — 24.56 Figure 4.12: Pressure time history reconstruction with different frequency range at 7 = 7.25 and 6 = 1800. 68 Figure 4.13: Instantaneous pressure contours of the sound scattering at t=15: CAA (top), AIBM (bottom). 69 Figure 4.14: Instantaneous pressure contours of the sound scattering at t=1 l: CAA (top), AIBM (bottom). 70 0.0186856 410216143 43.028543 Figure 4.15: Instantaneous pressure contours of the sound scattering at t=9: CAA (top), AIBM (bottom). 71 Chapter 5 3D AIBM with Subsonic Mean Flow 5.1 Introduction Motivated by the need for an accurate and efficient prediction of the far-field acoustic radi- ation, an Acoustic Intensity-Based Method (AIBM) has been developed based on acoustic input from an open control surface in a two-dimensional configuration [38—40]. In this chapter, a three-dimensional AIBM for the reconstruction and prediction of radiated acous- tic fields is developed [41—43]. The method is verified by examples of the propagation of multiple acoustic sources in a uniform flow and the acoustic scattering of a time dependent source by a sphere. The effectiveness of AIBM in aeroacoustic applications is demon- strated by the accurate and efficient prediction of acoustic radiations from an axisymmetric duct intake using a hybrid CAA/AIBM approach. The results of the radiated acoustic field from the AIBM agree well with the solutions of CAA and the FW-H integral equation. The AIBM is much more eflicient than other methods for the far-field acoustic prediction and can use the input acoustic data from an open surface instead of a closed F W—H surface. The organization of this chapter is outlined. First of all, the details on the extension of the AIBM to a three-dimensional configuration with a subsonic uniform flow are derived. Furthermore, the accuracy of the AIBM for the far-field acoustic prediction is shown by numerical examples and the advantages of the method with respect to both efficiency and choice of locations for the acoustic input are demonstrated. 72 5.2 Mathematical Formulations Let F be the boundary of a sphere containing all acoustic sources, and F1 be a (usually very small) part of F. In the AIBM, both the acoustic pressure P and its collocated normal derivative BP/On on F1 are considered as the input acoustic data. In practice, P and 6P/6n are given at a finite number ofpoints (:rj, yj, 2]) E F1,j = 1, M. 5.2.1 Basic Formulation for AIBM Assuming that the mean flow is in z direction and employing the standard separation of variables in terms of spherical coordinates, the general solution for the Helmholtz equation can be approximated by a linear combination of basis functions 00 72 . A P(7‘, 6, 02) = exp('ik]l~[af cos 0213—2) 2 2 ((111111 cos 7726 + bum sin 722.6)P72nG71,(k7‘,U—2), 7220 77220 (5.1) where Ma is the free-stream Mach number, ,6 = (/1 — Mg, 11' is the wave number, Pi,” = P,’,"(cos 03) is the associated Legendre polynomial and G n represents the generalized Han- kel function or spherical Hankel function. The coordinates (7, 6, 6)) are defined as the modified spherical coordinates from Cartesian coordinates (:r, y, z) in the physical domain. Let (2:, g). :3) = (6.7:, fly. 2:), then i=75in6cos6, Qz'fsincgsin6, 2=7°cosq:), 7‘=\/:i?2+3}2+:32. (5.2) Differentiating both sides of Equation (5. 1) with respect to the unit normal 11 = (72 x, 72y, 77.3) and using the chain rule, we have the formula for normal derivative of P, 0P 012231: 010019 0130;?) 0n(t, :13, y, z) = fl exp[i(wt — 117(7 — (”amid—2)] (5.16) where ,13 = (/1 — Mg, 7 = \/;2:2 + ,d2(y2 + 22) and A0, w are the strength and the angular frequency of the monopole, respectively. In the current study, the multiple sources, consisting of two quadrupoles, two dipoles (formed by superposing monopoles) and one monopole, are distributed inside a control 77 Figure 5.1: Schematic diagram of multiple sources in a uniform flow. surface of 7 = 1m. The coordinates and strengths of the sources are given in Table 5.1. The acoustic pressure and its normal derivative on the two planar surfaces are considered as the input for AIBM (see Figure 5.1) and obtained by using the FFT of the analytical solution of Eq. (5.14). In the current implementation, the distance of each planar surface to the origin is given as 10m and the size of the each surface is 4m by 4m. The angle 7 between the surfaces is 120°. Each surface was discretized into 10 by 10 uniform gn'd lines. The radiated acoustic field for the range of 4m 3 r 3 40m is then predicted by AIBM. Quantitative comparisons of the predicted pressure with the analytical pressure are shown 78 Table 5.1: The strengths and distributions of the 3D acoustic sources A0(W/m2) x 21m) Quadrupole I 1.0 0.0 -0.5 0.5 -l .0 0.0 0.5 0.5 1.0 0.0 0.5 -0.5 -l .0 0.0 -O.5 -0.5 Quadrupole II 0.8 -0.6 0.0 0.6 -O.8 0.6 0.0 0.6 0.8 0.6 0.0 -0.6 -0.8 -0.6 0.0 -O.6 Dipole l 1.1 0.4 0.3 0.1 -1.1 0.7 —0.1 0.5 Dipole II 0.9 0.3 0.2 -0.2 -0.9 -0.2 0.5 -0.3 Monopole 1.0 -0.5 —O.5 -0.1 79 in Figure 5.2 along the x—axis and the y-axis. Excellent agreement of the radiated pressure solutions from the near-field to the far-field is demonstrated in the figure. Contours of the instantaneous pressure are compared with the respective analytical solutions in Figures 5.3. These pressure contour results substantiate that AIBM accurately predicts the radiated acoustic field in a uniform flow of a three-dimensional configuration. 80 I -——Analytlcal 4:— - ------ AIBM 3E- 2:— n *\ i {11: Q E llllllllllll -2 M 11111111L111111111J -40 -20 0 20 40 x 3r _ Analyflcal ; - ------ AIBM 2:- A 1; v\ 0- © _ °~ : -1: C 1. .2- -3; 1 ‘41 1 1 1 11 1 111 Ll 11 1L44 -4O -20 0 20 40 Figure 5.2: Comparisons of the predicted pressure solutions with the analytical solutions: along x-axis (top), along y-axis(bottom). 81 Figure 5.3: 3D instantaneous pressure contours for the sound radiation of multiple sources: Analytical (top), AIBM (bottom). 82 5.3.2 Acoustic Scattering This example is an ideal model of the physical problem of predicting the sound field gen- erated by a propeller scattered by the fuselage of a moving aircraft. In the model, the fuselage is considered as a sphere and the noise source (propeller) as a time-dependent, single frequency acoustic source. The mean flow is given as Mach number of zero. The dimensionless radius of the sphere is given as l. The Cartesian coordinates centered at the center of the sphere are shown in Figure 5.4. The governing equations for this problem are the linearized Euler equations. The acoustic source is located on the :1:—axis at :27 = 1.5 and expressed as S = 0.01exp[—161n2((x — 1.5)2 + y2 + 2.2)] cos(27rt) (5.17) The CAA solution [56] verified by the analytical solution of the scattering by a sphere [58] was used as the acoustic input for AIBM calculations. The input surface of 7 = (£132 + y2 + 22 = 4 is also depicted in Figure 5.4. It is worth mentioning that the input surface for AIBM can be an open surface as indicated in the previous example. Since the CPU time is not at all an issue here, a spherical surface was used for the sake of simplicity. 83 . . ~.-....—.’ \ \ Input surface / \ \ / \ I \ ’ ‘1" Figure 5.4: Schematic diagram of acoustic scattering by a sphere. 84 Figure 5.5: Instantaneous pressure contours of the sound scattering in the plane y = 0: CAA (top), AIBM (bottom). 85 0.0006 ———CAA ----AIBM n 0.0004 0.0002 19 (x) lllllllITTrIjll1 -0.0002 U 111111111111111111111111111111LL -6 -4 -2 0 2 4 6 x _- .— ~— _— l— .— b 1— l -0.0004 0.0002 - L ——CAA — ———-AIBM 1 1 [9 (y) - i U U U U P. 1i111111111111114111111111111411 -6 -4 -2 0 2 4 6 y Figure 5.6: Comparisons of the predicted pressure solutions with the CAA solutions: along x-axis (top), along y-axis (bottom). 86 The predicted instantaneous pressure contours by AIBM are compared with the CAA so- lution in Figure 5.5 for the plane of y = 0. As it is noted from the figure, the predicted radiated acoustic field agrees very well with the CAA solutions outside a minimum sphere (7 = 2) that encloses the acoustic sources. The proper size of a minimum sphere was dis- cussed in our previous work [38]. Quantitative comparisons of the perturbation pressure along the as—axis and the y—axis are shown in Figure 5.6. Again an excellent agreement is achieved. 5.3.3 Acoustic Radiation from an Axisymmetric Duct Intake Both CAA methods [19, 20, 59] and the FW—H integral method coupled with CAA meth~ ods [21] were used in the past for the prediction of acoustic propagation and radiation from an engine inlet. In the current study, an axisymmetric geometric model of a duct intake is considered with a uniform subsonic flow. Small acoustic perturbations are propagating upstream through the axisymmetric duct flow and radiating to the far field. A schematic of the duct intake configuration along with the Cartesian and the spherical coordinates and the domain of acoustic propagation and radiation is shown in Figure 5.7. The far field flow pressure, density, Mach number and the speed of sound are given as P00 2 101.325kPa, ,000 = 1.249kg/m3, MaOO = 0.19 (in negative z-direction) and coo = 337m/s, respec- tively. A single frequency sound source with the non-dimensional angular frequency of 14.2567 is considered. The azimuthal and radial modes used in the CAA calculations are (772, 72) = (2,1),(3, 1) and (4,1). The FW—H integral method is the most commonly used method for the far-field acoustic prediction. The method is based on the N-S equations and therefore is valid in both the near field and the far field. The AIBM, on the other hand, is based on the linearized Euler’s equations and developed for the effective prediction of the far-field acoustic radiation. To show the capability and effectiveness of AIBM in a far-field prediction, the method is 87 coupled with CAA methods for the prediction of the acoustic radiation from the duct intake. The results of AIBM are compared to CAA solutions as well as the solution from the FW- H integral method. Since the acoustic wave is radiating in a 3D configuration, Farassat’s 3D Formulation 1A [15] with the quadruple term neglected was used in the F W-H integral method. The F W-H surface is depicted in Figure 5.7, and the input surface used for AIBM is 1 / 9 of the F W—H surface, with 0 from 00 to 400. The input acoustic data is provided by CAA solutions, and the number of grid points used for the input is 57,960 for the FW-H method and 6,440 for the AIBM method. The input data consists of the unsteady pressure and its normal derivative to the input surface for AIBM, and the unsteady pressure and velocity for F W-H, respectively. Since the mean flow and the duct intake geometry are both axisymmetric, the simplified 2.5D formulation for AIBM is also used. The input data for 2.5D AIBM is given on a number of line segments in the plane of 0 = 0 (see Figure 5.7). The acoustic pressure is predicted by AIBM, FW—H and 2.5D AIBM at 136 grid points in the range of 0 g a: S 6.5m with z = 6.5m and y = 0. The results from these methods are compared to the CAA solution in Figure 5.8. It is shown that the results from both AIBM and F W-H agree reasonably well with the CAA solution. The computation was performed on a personal computer of Pentium (R) 4CPU at 3.ZGHz and memory of lGB. In general, the computational time for AIBM is directly related to the number of input points and the number of coefficients, anm and bum. The CPU time will quadruple as the number of input points doubles and be increased by eight times if the number of coefficients doubles. For this single frequency radiation problem, the CPU times used for 3D AIBM and 2.5D AIBM are 5 minutes and 4 seconds, respectively. The total CPU time for FW-H is 16 hours. However, the time-domain formulation of FW—H is used and 64 time increments were considered in the period. The equivalent CPU time for F W-H is 15 minutes if the frequency-domain formulation of FW—H would be considered. Furthermore, in AIBM the acoustic pressure in a radiated field is calculated analytically after the coefiicients of the 88 Figure 5.7: Schematic diagram of acoustic radiation through an axisymmetric duct intake. basis functions are determined. In FW—H, on the other hand, the surface integration has to be carried out for each far—field location. Therefore, AIBM is much more efficient than FW-H and can lead to a significant CPU time reduction if the entire acoustic far field shown in Figure 7 is to be determined. Furthermore, it is noted from Figure 5.8 that the solution by 2.5D AIBM agrees very well with the CAA solution. The better agreement can be attributed to a relatively more complete input surface. Since the problem considered here is axisymmetric, in the following studies, the radiated acoustic field is calculated based on the 2.5D AIBM formulation. The instantaneous pres- sure contours and the pressure amplitude contours obtained by 2.5D AIBM are compared to the CAA solutions in Figures 5.9 and 5.10 for the duct modes of (2, 1) and (3, 1), re- 89 20— * —— CAA 7 — — — - 2.50 AIBM . - ------ 30711131111 L —--—- 3DFW—H 15—- F‘ 1 -. \ F '1 ,\ ~ \ R 1- \w 310»— 9~ : . 1' _ 1 5— _’ ' _ , \ _ I ‘ 1" \... :7“- ! ‘ ..\. 0"11111L111111LLL1111114_111L111111 0 1 2 3 4 5 6 )C Figure 5.8: A comparison of the directivity along the x-direction with z = 6.5m and y = 0 for the duct mode of m=2 and n=l. 90 spectively. In addition, the quantitative comparisons of the acoustic pressure for the (2, 1) mode and the directivity for the (3, 1) mode are given in Figures 5.1] and 5.12, respec- tively. The results shown in these figures clearly demonstrate the effectiveness of AIBM for the prediction of far-field acoustic radiation. It is noted that the solution inside and in the vicinity of intake although not valid was kept in the contour plots for convenience. In addition, the input acoustic data for AIBM should be given away from aerodynamic noise sources in order to guarantee the accuracy of the predicted acoustic solution in radiated fields. The reconstructed 3D instantaneous pressure contours by AIBM are compared to the CAA solution in Figure 5.13 for the duct mode of (4, 1). The capability and the overall effectiveness of AIBM for aeroacoustic applications are demonstrated by the example. 91 Figure 5.9: Instantaneous pressure contours in the plane 0 = 0 for the duct mode of m=2 and n=l: CAA (top), AIBM (bottom). Figure 5.10: Pressure amplitude contours in the plane 0 = 0 for the duct mode of m=3 and n=l: CAA (top), AIBM (bottom). (D —- CAA ------ 2.50 AIBM #- O) N P(x) 0 o111111—IITTIIIIIIIIFIIIIIFTIIIITFTIIT—II—I 1 N .L l O) I (I) Figure 5.11: Comparison of the predicted pressure solution with the CAA solution at z = 15m and y = 0 for the duct mode ofm = 2 and n =1. 94 — —-—— CAA — ------ 2.50MB" 41— C L. Q 3— \, _ h 7— § _ 1— h 1 1 1 1 I 1 1 J 1 1 1 1 1 1 1 00 5 10 15 Figure 5.12: Comparison of the predicted directivity with the CAA solution at 2: = 15m and y = 0 for the duct mode ofm = 3 and n =1. 95 Figure 5.13: 3D reconstructed pressure contours for the duct mode of m=4 and n=l: CAA (t0p), AIBM (bottom). 96 Chapter 6 Conclusions 6.1 Conclusion An advanced computational methodology, Acoustic Intensity-Based Method (AIBM), has been developed for the acoustic far-field prediction in both 2D and 3D configurations. This method assumes that the sound propagation is governed by the Helmholtz equation on and outside of a control surface that encloses all the nonlinear effects and noise sources. By employing the standard separation of variables, the general solution for the Helmholtz equation can be approximated by finite linear combinations of basis functions. The coef- ficients in the general solution are numerically determined by matching the assumed form of the solution to the input acoustic pressure and its co-located pressure derivative at mea- surements. With the addition of pressure derivative input, the solution of the Helmholtz equation is unique and more stable. Its advantages over traditional methods can be summa- rized as less input data over an “open surface” and computational efficiency. The AIBM method is initially developed in 2D configuration without considering the mean flow effect. Several acoustic radiation problems have been studied to showcase the reli- ability and accuracy of the AIBM. In each example, both the qualitative and quantitative comparisons have been conducted between HELS and AIBM. When the input acoustic data is only provided over an open surface, the AIBM improves the accuracy, reliability, and consistency of reconstructed acoustic radiation pressure. The improvement, however, becomes less significant when the input segments are clustered. The AIBM is especially effective when random noises are added in the input acoustic data. The results indicate that the reconstruction of HELS method, i.e, without the addition of the pressure gradient to the 97 input acoustic data, starts to deteriorate even with 1% of random noise in the input from an open surface. The AIBM, on the other hand, can give reasonably accurate reconstructed acoustic pressure from the input acoustic data with up to 20% random noise. In addition, the capability and efficiency of the AIBM for multi-frequency (broadband) acoustic radia- tion have been demonstrated through a vortex/trailing edge interaction noise problem. Accurate and efficient noise prediction plays a very important role in the aircraft design. The current far-field acoustic prediction technique, FW-H equation method, needs the input acoustic data on a closed surface, which encloses all the noise sources. Clearly, it is not pos- sible for most aerospace application cases. Another important drawback of FW-H method is that the surface integration has to be carried out for each far-field location. Hence, it is very time consuming for F W-H method in a 3D acoustic simulation. Motivated by the strong need for an efficient open surface method in aircraft noise prediction, the AIBM is successfully extended for predicting sound propagations in a uniform flow. By coupling with the CF D/CAA numerical techniques, the extended AIBM has been verified in various acoustic propagation and radiation problems, such as flows around the airfoil, the scatter- ing of a time-dependent acoustic source and the radiation of duct acoustic modes from an axisymmetric duct intake. The predicted acoustic fields by AIBM for all the cases agree very well with the respective CAA and the F W-H solutions. On the other hand, the AIBM only requires partial input data of F W-H and costs less computation time than F W—H. The overall effectiveness of the AIBM indicates that the method has the capability for the pre- diction of acoustic radiations encountered in engineering applications. In addition, relying on the advantage of computation efficiency, the AIBM has the potential to become a part of an integrated computational or design optimization procedure for evaluation of far-field acoustic propagations. Furthermore, the accurate prediction of the sound propagation and reflection of the scattering problem demonstrates that the AIBM can be used for both the near-field and far-field acoustic reconstruction and prediction. However the accuracy of the near—field reconstruction has been sacrificed to increase the reconstruction availability. 98 As pointed out by F fowcs Williams (1993), the nature of aeroacoustics fields “permits many different, but equally exact, computational procedures for evaluation both the sound and its source field.” The results in the current study demonstrate that the AIBM could be used for the far-field sound prediction for aeroacoustic problems when a closed FW—H control surface is not possible. In general, the AIBM provides an effective alternative for the far-field acoustic prediction of practical aeroacoustic problems. 6.2 Suggestions for Future Work AIBM method is a newly developed technique extended from HELS. We first published it in the AIAA/CEAS Aeroacoustics Conference in 2007, later in the Journal of the Acoustical Society of America in 2008. There could be several interesting possibilities in methodology extension and applications for the future study. First, in dealing with practical problems, usually lots of measurements are needed as input data for AIBM. The inverse problem models are usually ill-posed with huge condition numbers. Also the Cauchy problem for the Helmholtz equation is unstable theoretically. The regularization technique could be considered and introduced to effectively solve the ill-posed problems with our AIBM method. Second, the AIBM is initially defined as a far-field acoustic prediction method. The near- field (source region) prediction is also of interest by aircraft and automobile manufacturers. The AIBM has a potential for accurate near-field reconstruction with the additional basis functions. Finally, there is currently no effective way to predict jet noise, which is one of the most challenging tasks in computational aeroacoustics. As our AIBM method was derived based on uniform flow assumption, a new model for dealing with a non-uniform flow is desired 99 for jet noise assumption. We hope to solve this real-world puzzle with further developments of the AIBM. We believe the future of AIBM is promising with more exposure and examination of real industrial projects. 100 Appendix A: Associated Legendre Polynomial The associated Legendre polynomial is derived in this section following the derivation of Numerical Recipes [63]. With 1 g .2“ S 1, the associated Legendre polynomials are defined in terms of ordinary Legendre polynomials by , g . (1m P77172(:E) : (_1)772(1_ 1‘2)m/2WP72(I) (A-l) where 1 (171(332 _ 1)” 12.11:) = 13911:) = 2,”, (1,, (A-2) Introducing the stable recurrence on 72 presented in [63], (n. — 772)P,27" = 7(272 — 1) {[11 — (72 +772 —— 1) 77,712 (A-3) For the starting value, there is a closed-form expression, 17,7 = (~1)”"(2m — 1)!!(1-— 11,-?)"1/2 (A-4) The notation 72!! denotes the product of all odd integers less than or equal to 72. Using Equation A-3 with n = m. + 1, and setting 57,14 = 0, we can obtain ,2?“ = 7(27'72 + 1)P,7,’,l (A-5) Equation A-4 and A-5 provide the two starting values required for Equation A-3 for general 101 n. The derivative of P77," with 772 < 72 can be derived from its definition equation. 6P5” = —772J‘ ———Pm' _ 1 017 1 — :72 " P772+1 v1 — 22"2 72 When 772 = 72, the derivative is (937,” _ —772.7 m (9.27 — 1 —- .72 72 Let PHI (1.) (1777, 311(1) : (1 _nx2)772/2 = (— )m C15,3772 P7101.) Then the recurrence for Q’"( ) is 1n—m) ’7: 12n—1>Q::’;1—1n+m— 1162;." iii = (—1)m(2m —1)!! an+1 - (-1)m(2m +1)11;r The derivative for Qm(.r ) when 772 < 72 is aQii? _ 772+l 8T _ - TL When m = 72, the derivative is a 772 —Q—" =0 8.7 102 (A-6) (A-7) (A-8) (A-9) (A-lO) (A-ll) (A-12) (A-l3) Appendix B: Spherical Hankel Function The 0-th order and first order Spherical Hankel functions are [63] 00(7) = 71/36”) [—2' (7 — g» (B-l) 0111-) = $732212 1—1‘12 — 7)] (1— E) (8-2) Employing the recurrence formulation, n-th order can be obtained based on the O—th and first order formulations. 272—1 07217“) = Gn—l — Gn—Z (3‘3) The corresponding derivative formulations for O-th order is just simply derived from G 0 (7). And the higher orders are obtained by recurrence relations. 05(7) = —Go (1+2) (B-4) 072.”) = ’1 , G72 '1' Gn—l (B-5) 103 Bibliography [1] “Federal Aviation Regulations Part 36 Noise Standards: Aircraft Type and Airwor- thiness Certification,” see http://www.airweb.faa.gov/. 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