An analysis of booster tone noise using a time-linearized Navier-Stokes solver

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2 An analysis of booster tone noise using a time-linearized Navier-Stokes solver A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science in the Department of Aerospace Engineering and Engineering Mechanics of the College of Engineering and Applied Science by Nathan A. Wukie B.S., University of Cincinnati (2012) March 2016 Author Nathan A. Wukie Department of Aerospace Engineering March 21, 2016 Committee Chair Paul D. Orkwis, Ph.D. Department Head, Bradley Jones Professor Department of Aerospace Engineering Thesis Supervisor

3 Abstract This thesis details a computational investigation of tone noise generated from a booster(lowpressure compressor) in a fan test rig. The computational study consisted of sets of timelinearized Navier-Stokes simulations in the booster region to investigate the blade-wake interactions that act as the primary noise-generating mechanism for the booster blade-passing frequency and harmonics. An acoustic test database existed with data at several operating points for the fan test rig that was used to compare against the predicted noise data from the computational study. It is shown that the computational methodology is able to capture trends in sound power for the 1 st and 2 nd booster tones along the operating line for the rig. It is also shown that the computational study underpredicts one of the tones at low power and is not able to capture a peak in the data at the Cutback condition. Further investigation of this type is warranted to quantify the source of discrepancies between the computational and experimental data as the reflected transmisison of sound off the fan through the bypass duct was not accounted for in this study. ii

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5 Acknowledgments I would like to thank my advisor Dr. Paul Orkwis for his support both in my undergraduate years starting in the field of computational fluid dynamics as well as through the completion of this thesis. He has always made a tremendous effort to be available when needed to discuss research and provide feedback. He has also allowed me to have a great amount of flexibility in my studies and research path, which I value very much. I also thank Dr. Mark Turner, who has generously provided his time and insight on fluid dynamics, numerical methods, and turbomachinery on numerous occasions over the past several years. I consistently find myself being challenged in our discussions and I have benefited greatly from his breadth of technical knowledge. My two additional committee members, Dr. Shaaban Abdallah and Dr. Ephraim Gutmark, also deserve special thanks for their time and thoughtful feedback on this thesis. Dr. Abdallah has also been the instigator of many lively discussions over the past several years that are simultaneously entertaining and thought-provoking. I appreciate the creativity and enthusiasm that he contributes to any topic. I owe a great deal of thanks to Rob Ogden for his support in keeping computers running, licenses available, and hard drives spinning. Not only has Rob provided consistent technical support, but he has also been a great friend to sit and chat with about music, gardening, and languages. He has pushed me in these areas and I enjoy the challenge. Additionally, I would like to thank Chris Porter for his thoughtful discussions on unsteady turbomachinery and also for his flexibility and accomodation while I was writing this thesis since we were sharing a computer. This required a good bit of coordination and Chris was always positive and happy to be flexible. The background information and technical expertise contributed by Dr. John Wojno, iv

6 Trevor Goerig, and Greg Szczepkowski were key to the completion of this work. Without their combined knowledge and willingness to advise and provide guidance this work would not have been possible, so I thank them greatly for their combined time and effort invested in me. I would also like to thank my family for their unwaivering support for me as I have gone through my studies, traveled about from place to place, and generally maintained an unpredictable schedule that comes along with graduate studies. From an early age, I was always encouraged and enabled to take part in new activities and do my best. I am very grateful for those opportunities and the support of my family and I thank them all very much. v

7 Contents Abstract ii Acknowledgments iv Contents vi List of Figures ix List of Tables xii Nomenclature xiii 1 Introduction Motivation Numerical methods for predicting tone noise in turbomachinery Analytical techniques Linearized computational methods Nonlinear computational methods Duct modes Modes in an annular cylindrical duct with uniform mean flow Tone noise physics Model-scale fan test rig Geometry diagram Experimental data vi

8 2 Numerical Method Nonlinear, steady, Reynolds-Averaged Navier-Stokes Time-linearized, unsteady, Navier-Stokes D Nonreflecting boundary conditions Tone noise prediction methodology Mean flow solution Noise source decomposition Interaction propagation Selection of relevant modes Results Convergence metrics Grid resolution study Wake resolution Wave resolution Initial generated noise Mean flow calculations Linearized-unsteady analysis st BPF modes nd BPF modes Total Power Levels Effect of R2 potential field interaction Potential sources of error Conclusions and future work Conclusions Future work A Tone noise extraction process 76 B Supporting mathematical expressions 79 B.1 Mean flow flux Jacobians vii

9 B.2 Boundary condition matrices C Grid convergence study for wave propagation 82 viii

10 List of Figures 1-1 Effective percieved noise levels from a high bypass-ratio turbofan engine that was tested as a part of the Advanced Subsonic Technology Progrm. Plots reformatted from [1] Progress in aircraft noise reduction.[2] Diagram of frequency-domain methods Radial mode profiles for a m = 10 mode assuming an annular cylindrical duct and uniform mean flow First three radial modes of a Circumferential mode (m = 10) assuming an annular cylindrical duct and a uniform mean flow Duct mode transmission properties Diagram of blade wake interactions in a compressor stage Wake velocity defect and modes from fourier decomposition Hypothetical velocity defect profile and the first three modes of a corresponding Fourier decomposition Fan test rig diagram (not to scale) Fan rig test configurations (not to scale) Fan rig test data (Cutback) - highlighting booster 1 st /2 nd BPF tones Experimental, Aft-radiating 1 st /2 nd booster tones along speed line Acoustic source diagram Noise prediction methodology Mode scattering characteristics across Rotor and Stator blade rows Analysis of modes generated from the interaction of the S1 wake with R ix

11 2-5 Analysis of modes generated from the interaction of the R2 wake with S Nonlinear steady RANS convergence metrics Linearized unsteady convergence metrics Diagram of noise-generating, blade-wake interactions Blade-wake profile convergence S1 grid resolution levels at the trailing edge for wake resolution study R2 grid resolution levels at the trailing edge for wake resolution study Comparison of S1 wake harmonics across grid resolutions Comparison of R2 wake harmonics across grid resolutions SPL at the R2 exit plane for modes generated from the S1 1 st wake harmonic interacting with R SPL at the R2 exit plane for modes generated from the S1 2 nd wake harmonic interacting with R SPL at the S2 exit plane for modes generated from the R2 1 st wake harmonic interacting with S SPL at the S2 exit plane for modes generated from the R2 2 nd wake harmonic interacting with S Operating points computed from mean flow solution Comparison of S1 wake harmonics along the rig operating line Comparison of R2 wake harmonics along the rig operating line S1/R2 modes - 1 st BPF R2/S2 modes - 1 st BPF st BPF response from the S1/R2 and R2/S2 interactions S1/R2 modes - 2 nd BPF R2/S2 modes - 2 nd BPF nd BPF response from the S1/R2 and R2/S2 interactions Trends in experimental and predicted booster tones Diagram of rotor-locked potential field interaction Aft SPL at S1 exit plane of modes generated from R2 potential field interaction. 64 x

12 3-25 R2 potential field interaction propagation Potential sources of error Most significant predicted modes contributing to the 1BPF response at the Sideline condition Scattered modes from R2/S2 interaction contributing to 1BPF tone Predicted propagation properties for mode (-22, 0) Potentially significant effects unaccounted for Spanwise SPL of upstream and downstream-traveling modes generated from the interaction of the 1 st S1 wake harmonic with R2 at PT A Spanwise SPL of upstream and downstream-traveling modes generated from the interaction of the 1 st and 2 nd R2 wake harmonics with S2 at PT A Integrated SPL of upstream and downstream-traveling modes generated from initial blade-wake interactions at operating point PT A A-1 Diagrams for tone noise extraction process C-1 S1/R2 Interaction. R2 blade row. Response from S1 1st harmonic C-2 S1/R2 Interaction. R2 blade row. Response from S1 2nd harmonic C-3 R2/S2 Interaction. S2 blade row. Response from R2 1st harmonic C-4 R2/S2 Interaction. S2 blade row. Response from R2 2nd harmonic C-5 R2/S2 Interaction. Response out of Strut blade row from R2 1 st harmonic coming from S2 (k = 1) C-6 R2/S2 Interaction. Response out of Strut blade row from R2 2 nd harmonic coming from S2 (k = 1) xi

13 List of Tables 1.1 Booster blade counts S1 wake convergence grids R2 wake convergence grids Wave convergence grids Predicted axial wavenumber for mode ( 22, 0) at S2 passage exit C.1 R2 wave convergence grids C.2 S2 wave convergence grids C.3 Strut wave convergence grids xii

14 Nomenclature α β Blade-row index Shear work F ( Q) Q Ω ω Φ Jacobian matrix of the nonlinear flux Rotational rate (rev/s) Rotational frequency Inter-blade phase angle ψ m,n Radial profile of mode (m,n) ρe ρu ρv ρw ρ τ Total energy x-momentum y-momentum z-momentum Density Shear stress a m,n Amplitude of mode (m,n) F k Vector of conservative fluxes Mode interaction integer xiii

15 m n Circumferential mode order Blade harmonic number N α Number of blades in blade row α N B Number of blades N V Number of vanes Q Array of conservative varibles q Small perturbation of conservative variables r H Hub radius r T Tip radius V r Radial velocity component (ft/s) V t Circumferential velocity component (ft/s) V z Axial velocity component (ft/s) PWL Sound power level SPL Sound pressure level xiv

16 Chapter 1 Introduction 1.1 Motivation Aircraft engine noise has been an important area of research since the first jet flow. However, the introduction and subsequent wide-spread use of jet-powered aircraft for commercial civilian transport in the late 1950 s triggered a response by commercial airports, industry, governments, and academic institutions to actively address the problem of jet engine noise[3]. The reason for this abrupt reponse centered around the fact that jet-powered aircraft were significantly noisier than their propeller-driven predecessors. Moreover, several commercial airports at the time were already engaged in lawsuits filed by surrounding communities regarding the noise pollution from propeller-driven aircraft[4]. Initial measurements of noise levels for the introduction of the Boeing 707(one of the first commercially successful jetliners) indicated its percieved noise level was 15dB higher than a Lockheed Super-Constellation; a large propeller driven transport[5]. This prompted the addition of mufflers, flight path alterations, and take-off/landing restrictions for jet-powered aircraft in order to meet airportimposed noise level requirements[4]. Today, noise requirements for aircraft and airports in the United States are regulated by the Federal Aviation Administration s(faa) Federal Aviation Regulations(FAR), specifically Part 36 and Part 150[6, 7]. Part 36 outlines a staged program for reducing aircraft noise over time, which gradually phases out older, noisier aircraft and also sets standards for the future that allows engine and airframe manufacturers to plan accordingly. 1

17 Noise regulation and industry efforts to reduce noise are largely driven by the adverse effects of noise on people, so it is instructive to identify some of the more significant annoyances that people experience as a result of aircraft and engine noise. Two main groups of people affected by commercial aviation noise are the passengers themselves and people in communities surrounding airports, which experience the integrated effect of aviation noise. The most prominent effects of noise that these groups experience include: Annoyance Sleep Interference Hearing Loss Communication Interference These effects can in turn be correlated to decreased residential property values for communities surrounding airports[8]. The FAA s 2015 Aerospace Forecast report projects a 1.9% average yearly increase in U.S. air passengers through 2035[9]. Likewise, the International Air Transport Association(IATA) projects a 4.0% yearly increase in global commercial air passengers between 2014 and 2034[10]. Given these projections of long-term growth, it is natural for airports and their surrounding communities to be concerned about the potential impact that noise will have on quality of life and property values. Jet noise can even be an occupational hazard for military personnel operating on aircraft carrier flight decks[3]. Jet engine noise may be split into two categories. These are broadband noise and discrete frequency or tonal noise. Broadband noise is defined as noise distributed across the range of frequencies and is a direct result of the random flow fluctuations associated with turbulence. Tonal noise is defined as noise occuring at discrete frequencies and is most often associated with rotor blade-passing frequencies, their harmonics, and sum/difference tones resulting from the interaction between different frequencies. Other sources of tone noise may include cavity resonance, vortex shedding, or multiple pure tones(mpt s) that result from the propagation of the nonlinear pressure field of a transonic fan[11, 12]. The individual noise sources that exist in turbofan engines are generally attributed to the following components: Fan noise Tones, Broadband, MPT s 2

18 Compressor noise High-frequency tones, Broadband Combustor noise Low-frequency broadband Turbine Noise High-frequency tones, High-frequency broadband Jet Exhaust Noise Low-frequency broadband One can get an idea of their respective contributions to the overall noise level from Figure 1-1. A significant observation from Figure 1-1 is that at the Approach condition, the Inlet(Fan) and Aft Fan sources dominate the engine noise components. This highlights the challenge of jet engine noise, since there are many contributing sources that have varying levels of significance at different points in the flight path. A thorough overview of the many aeroacoustic challenges related to turbomachinery is given by Peake and Parry[13]. Historically, the high-velocity, high-temperature jet exhaust from the engine was the dominant noise source attributed to any jet aircraft. The significant dependence of jet noise on jet velocity can be observed analytically as AcousticP ower = Kρ 0 U 8 a 5 0 l 2 (1.1) which is Lighthill s celebrated U 8 law.[14, 15] The expression in Equation 1.1 assumes constant density and sound speed, but for low Mach number flows with small temperature variations it allows one to compute the acoustic power generated from flow disturbances on the order of length l traveling with velocity U[16]. In the context of jet noise, the lengthscale l of turbulence structures would be on the order of the jet shear layer thickness. The introduction of high bypass-ratio turbofan engines greatly reduced jet noise levels as a result of the cooler, lower-velocity bypass flow. This was largely a byproduct of manufacturers efforts to increase fuel efficiency. Although Equation 1.1 has many inherant assumptions, it does provide context for the large noise reductions that were achieved by increasing bypass ratios and reducing jet velocities. Indeed, modern turbofan engines are on the order of EPNdB quieter than the turbojet and initial turbofan engines of the 1960 s, as shown in Figure 1-2. In general terms, since EPNdB is a logarithmic scale, a 10 EPNdB reduction is percieved as being about half as noisy. It follows then that a 30 EPNdB reduction is percieved 3

19 (a) Approach (b) Sideline Figure 1-1: Effective percieved noise levels from a high bypass-ratio turbofan engine that was tested as a part of the Advanced Subsonic Technology Progrm. Plots reformatted from [1]. as being one-eighth as noisy. With the large reduction in noise attributed to jet exhaust, the relative significance of fan, compressor, and turbine noise has steadily increased. Along with the increased significance of the turbomachinery noise sources has also come increased importance in predicting the noise associated with these components. 1.2 Numerical methods for predicting tone noise in turbomachinery Computational methods for noise prediction range from analytic and semi-empirical methods to fully-unsteady, turbulence resolving, computational fluid dynamics(cfd) simulations. The span of physical phenomena that influence an acoustic field is vast and computational methods offer varying abilities to predict these phenomena. The trade-off between computational expediancy and physical accuracy is one that is constantly being balanced in design and analysis efforts. A range of computational methods are discussed here, highlighting the physical phenomena they are designed to predict along with their relative computational cost. 4

20 Figure 1-2: Progress in aircraft noise reduction.[2] Analytical techniques Analytic methods for noise prediction, while fast, have difficulties accounting for threedimensional(3d) geometry and are generally not adequate for modern detailed design efforts. These methods are sometimes used to solve a simplified or canonical problem that can then be used as a verification data set for more complex models or can provide insight into fundamental physical processes that can guide design efforts. Some of the first analytical investigations concerning turbomachinery-related noise were performed by Kemp and Sears[17, 18], who approached the problem of blade-row interaction noise by considering a stator-rotor stage and represented each blade by a thin-airfoil theory approximation; first considering the potential field interaction and in the later investigation considering the effect of viscous wake interactions. These analyses allow one to investigate some of the fundamental physical phenomena that lead to blade-row interaction noise and provide useful insights that can guide design practices. However, approximations employed by Kemp and Sears limit the applicability of the analysis to incompressible flows and lightly loaded airfoils, where the blades are aligned with the relative flow vector. 5

21 Tyler and Sofrin[19] approached the problem of turbomachinery noise from a different perspective in their seminal paper considering the rotor self-noise and rotor-stator interaction problems in the context of duct modes and the allowable pressure patterns that result from rotor-stator interaction. Their work also details duct mode transmission behavior, which focuses on the properties of a given mode to describe its behavior as being cut-on and cutoff; cut-on indicating that a mode pattern will travel down a duct unattenuated; cut-off indicating that a mode amplitude will diminish as it travels the length of a duct. The concepts introduced by Tyler and Sofrin can be found throughout the literature on tone noise in turbomachinery and are the underpinning of many modern analysis techniques. Their work went a long way towards advancing the understanding of the form of the blade row interaction and mode propagation, but it does not provide a method of predicting the strength of any given interaction or source. Following the work of Kemp and Sears, and taking insights from Tyler and Sofrin, Kaji and Okazaki[20] modeled a pair of rotor and stator blade rows with an array of doublets, taking the effects of compressibility into account and assuming a harmonic form. As noted by Silkowski and Hall[21] however, the blade rows were coupled by only a single mode. Smith[22] developed a method for computing the response of an isolated blade row due to some forcing function, such as incoming acoustic pressure or wake velocity perturbations, allowing one to compute the transmission and reflection coefficients for a given blade row. Hanson[23], using Smith s method to compute reflection and transmission coefficients for modes in each blade-row individually, used modal coupling to generate a system of linear equations describing a rotor-stator system which could be inverted to achieve the coupled system response. Hall and Silkowski[24] used the same approach to investigate multiple stages coupled together in an aeromechanics context, but the same method is applicable for acoustics investigations. The previously discussed methods do not take real geometry effects such as blade thickness and camber into account. Additionally, forcing functions such as wake profiles are generally assumed or correlated, rather than computing them from first principles. Silverstein, Katzoff, and Bullivant[25] give an experimental database of airfoil downwash and wake properties along with correlations to the data, which have at times been used to generate 6

22 viscous wake profiles to assist in the analytical studies Linearized computational methods Time-linearized Navier-Stokes calculations provide the benefit of being able to account for real, 3D geometry while maintaining computational efficiency. An assumption on the form of the solution is made such that it consists of a fixed mean component and a sum of harmonic perturbations. Substituting the assumed form of the solution into the governing set of equations and neglecting products of perturbations yields a conservation expression governing the transport of perturbation variables via the fixed mean flow. The linear assumption in the derivation neglects the influence of the unsteadiness on the time-mean component of the flow and also any potential nonlinear interaction between separate frequencies. As such, multiple frequencies may be accounted for by running separate time-linearized calculations; each with a separate frequency. A time-domain representation of the solution could be reconstructed by combining the mean and computed harmonic components; each instant in time being computed with a phase shifted value of the harmonic perturbation. A significant advantage of the time-linearized method in computing in the frequency-domain is that the method is solving a set of steady-like equations, while providing an unsteady solution. This allows various convergence acceleration techniques to be used. Time-linearized Navier- Stokes calculations are the method used for the present work to predict tone noise associated with turbomachinery blade row interactions. A more thorough discussion of the method is deferred to a section on the methodology of the present investication. Many previous investigations have looked at predicting tone noise using time-linearized approaches[26, 27, 28, 29]. Korte et al.[28] used a time-linearized Euler solver to predict tone noise in a multi-stage turbine. Pinelli et al.[26, 27] used a time-linearized Navier-Stokes approach to analyze a cold flow turbine rig for tone noise emission predictions. The linearized calculations are uncoupled and the rotor-stator interactions are propagated by successively applying the downstream propagating waves to the inlet of the next blade row until all of the interactions have been computed through the last row of the geometry. The previously mentioned investigations by Korte et al.[28] and Pinalli et al.[27] do not account for coupled blade-row interactions such as reflections. Hall et al.[29] developed a methodology to capture 7

23 coupled blade-row effects for time-linearized unsteady computations by running several timelinearized unsteady computations in parallel at different frequencies for each blade row under investigation. The modes at the interface of the two blade rows are then coupled such that as a mode crosses from one row to the next, it is passed to the computation in the next blade row with the correct frequency. However, this method excludes any coupling between frequencies in a local blade row. There have been several previous investigations that have applied the time-linearized unsteady methodology in a GE internal CFD code to different acoustics problems for turbomachinery. Sharma et al.[30, 31] applied this methodology to several canonical tone noise problems in addition to the NASA Source Diagnostics Test(SDT) geometry, which was representative of a fan wake interacting with an outlet guide vane(ogv). Additionally, Sharma et al.[32] used the same methodology with a separate Ffowcs Williams-Hawkings solver to predict tone noise for a counter-rotating open rotor configuration Nonlinear computational methods There are several computational methods in existence that are able to capture nonlinear, unsteady phenomena associated with a given problem. The methods discussed here are the Nonlinear Harmonic, Harmonic Balance, Time-marching unsteady RANS, and Large-eddy Simulation (LES). These can be separated into separate categories based on their treatment of time or the method by which the effect of unsteadiness is included in the calculation, and also by their treatment of turbulence. The following discussion describes these distinguishing features for each method. Nonlinear Harmonic The Nonlinear Harmonic(NLH) method[33, 34, 35, 36, 37, 38, 39] can be interpreted as an extension of a time-linearized unsteady method. Instead of considering a fixed mean flow and solving for a single frequency, the Nonlinear Harmonic method uses a variable mean flow and solves for multiple frequencies simultaneously. The nonlinear interaction between a given frequency and the mean flow occurs via deterministic stresses which are computed 8

24 from the harmonic perturbations and are incorporated into the variable mean flow solution as source terms. This provides a nonlinear influence of the unsteadiness on the mean flow. Since each frequency is solved as a perturbation on top of the mean flow, an individual frequency can feel the effect from the other frequencies via the variable mean flow. This provides a nonlinear interaction between frequencies. Since the form of the unsteadiness is assumed to be periodic in time, a turbomachinery calculation using the Nonlinear Harmonic method requires only one blade passage and the boundary condition in the circumferential direction can be imposed as a phase-shifted periodic boundary condition. This is a significant advantage for computational efficiency compared with a nonlinear time-marching RANS method, where a series of blade rows is required to be spatially periodic, or to use a phase-lagged periodic boundary condition that requires boundary solution data to be stored at every instance in time over a period. Imposing spatial periodicity can result in a significant portion of the annulus being included in the calculation if multiple blade rows are being analyzed; a significant disadvantage for efficient use of computational resources. Imposing a phase-lagged boundary condition can act as a filter on the time-marching calculation, which reduces its benefit of providing the full unsteady solution(in a RANS sense). Harmonic Balance The Harmonic Balance method[40, 41, 42, 43, 44, 45] provides a capability similar to that of the Nonlinear Harmonic method, but the mathematical underpinning of the two approaches are quite different. The Harmonic Balance method makes the assumption that the solution consists of a mean component and a sum of periodic perturbations. Knapke[45] points out that in contrast to the Nonlinear Harmonic method, the perturbations in the Harmonic Balance method are not assumed to be small. Instead of computing explicitly in the frequency domain, the Harmonic Balance method computes a series of time instances, which are then linked via what is termed a pseudo-spectral operator that enforces the requirement that the time instances satisfy the original assumption on the form of the solution. That is, that it be composed of a time-mean component and a sum of periodic perturbations. A result of this formulation is that the frequencies and mean component being computed are explicitly 9

25 (a) Linearized Unsteady (b) Nonlinear Harmonic (c) Harmonic Balance Figure 1-3: Diagram of frequency-domain methods coupled with each other and do not rely on the feedback mechanism of deterministic stress terms as in the Nonlinear Harmonic method. Additionally, as more time instances are added to the calculation, the solution approaches the fully unsteady form. In the context of turbomachinery calculations, only a single blade passage is required by the Harmonic Balance method in order to compute the unsteady solution since the assumption of time-periodicity was made. Figure 1-3 shows diagrams of the Time-Linearized Unsteady, Nonlinear Harmonic, and Harmonic Balance methods in an effort to provide a more intuitive comparison between the techniques. Nonlinear, time-marching RANS Nonlinear, time-marching RANS calculations[46, 47, 48, 49, 50, 51, 52] make no assumptions on the form of the solution and, as the name suggests, march in time until an acceptably converged solution is reached. The benefit of the time-marching method is that all frequencies are captured within the grid and time-step resolutions. Additionally, this method intrinsically captures interactions between frequencies and blade-row coupling. However, the added physics from the direct discretization of the temporal-derivative are balanced by additional dissipation and dispersion errors that arise as a result of the discretization. Low-dispersion, low-dissipation time integration schemes can be used to reduce the these errors[53] The most 10

26 significant disadvantage of the time-marching method is its high computational expense. Instead of just one blade passage per row, the time-marching method often requires a significant portion of the annulus to be included in the computational domain in order to obtain spatial periodicity. Additionally, a small time-step size is often required, either as a constraint of the numerical method, or as required to resolve a particular range of frequencies, which extends the time to convergence. Due to the expense of the time-marching method, it is not necessarily practical in a design or optimization framework for turbomachinery. 1.3 Duct modes The discrete frequency components in a turbomachinery noise spectrum associated with rotating blade-rows and blade-row interactions are often the result of the pressure fields associated with those interactions coupling with various duct modes, which may propagate unattenuated to the inlet or exit of an engine and radiate to the far-field. Since such duct modes are the primary method by which the previously mentioned noise sources in turbomachinery are transmitted, an understanding of their form and behavior is beneficial from a computational perspective for making accurate predictions Modes in an annular cylindrical duct with uniform mean flow A simplified analysis of duct modes can be performed by considering an annular cylindrical duct with a uniform mean flow. The analysis detailed here is not used directly in the numerical study presented later in this work, but serves to provide context and insight on the fundamental physical phenomenon of duct modes and their behavior. This type of analysis and its results have been known for quite some time and are documented in many references. Rienstra and Hirschberg[54] and Moinier and Giles[55] give particularly nice presentations of the duct mode analysis, which are reflected here. Under the assumption of small, linear perturbations and a uniform mean flow, the unsteady pressure can be modeled using the convected wave equation in cylindrical coordinates 11

27 as ( t + M ) 2 ( p 2 z z r r r + 1 ) 2 p = 0 (1.2) r 2 θ 2 The geometric configuration considered here is an annular, hard-walled duct of infinite length. As such, no end reflections or acoustic liner effects are considered in this analysis. Such effects can certainly be important and are analyzed several places in the literature[56, 57, 58]. However, their inclusion is not required to understand fundamental duct mode phenomena. Assuming a harmonic form for the pressure perturbation in Equation (1.2) with each Fourier harmonic composed of a series of radial modes as p (t, r, θ, z) = p mn (r)e i(mθ+kmnz+ωt) (1.3) m= n=0 and substiting this form into Equation (1.2), one finds that each mode in the expansion must satisfy the following equation 2 p mn (r) r r p mn (r) r + [α 2mn m2 r 2 ] p mn (r) = 0 (1.4) which is known as the Bessel equation, the solution of which is a combination of Bessel functions as p mn (r) = AJ m (α mn r) + BY m (α mn r) (1.5) where J m and Y m are the first and second Bessel functions and α mn is defined as α 2 mn = (ω k mn M) 2 k 2 mn (1.6) Subjecting Equation (1.5) to boundary conditions at the inner and outer radius locations of the annulus, such that the radial derivative of pressure is zero as 12

28 p mn (h) = 0 r p mn (1) = 0 r (1.7) produces the system J m(α mn ) Y m(α mn ) A = 0 (1.8) J m(α mn h) Y m(α mn h) B 0 Meaningful values for α mn can be found by ensuring the determinant of the matrix in Equation (1.8) is zero as J m(α mn )Y m(α mn h) Y m(α mn )J m(α mn h) = 0 (1.9) Nontrivial values of α mn must then be found that satisfy the above equation. With the eigenvalues in hand, the allowable radial mode profiles( p(r)) can be constructed. As an example, the first four radial mode profiles for a m = 10 mode are plotted in Figure 1-4. Their corresponding two-dimensional form for a given axial location in a duct are shown in Figure 1-5. The axial propagation characteristics of a given mode are determined by the axial wavenumber k mn. In general, duct mode transmission is categorized as cut-on or cut-off. A cut-on mode travels down a duct unattenuated. A cut-off mode travels down a duct and attenuates a defined amount per unit length traveled. These two transmission properties are diagramed in Figure Tone noise physics Turbomachinery tone noise resulting from rotor-stator interactions has three primary sources, as discussed in the seminal paper by Tyler and Sofrin[19]. These sources include rotor blades cutting the wakes of upstream stators, rotor wakes impinging on downstream stators, and the interaction of the rotor-locked potential pressure field with adjacent blade rows. In the case of blade-wake interactions, noise is generated when a vortical wake interacts with the leading 13

29 Figure 1-4: Radial mode profiles for a m = 10 mode assuming an annular cylindrical duct and uniform mean flow. (a) (10, 1) (b) (10, 2) (c) (10, 3) Figure 1-5: First three radial modes of a Circumferential mode (m = 10) assuming an annular cylindrical duct and a uniform mean flow. 14

30 (a) Cut-off duct mode (b) Cut-on duct mode Figure 1-6: Duct mode transmission properties Figure 1-7: Diagram of blade wake interactions in a compressor stage edge of a downstream blade row. The wake defect passing over the leading edge creates an oscillating pressure field, which manifests itself as noise. Figure 1-7 shows a conceptual diagram of blade-wake interactions. Figure 1-8 demonstrates the concept of decomposing a wake into its modal components. Figure 1-9 demonstrates how a hypothetical wake profile of a complete turbomachinery row may be decomposed into a series of harmonic waves. The deterministic pressure field generated by a rotor-stator interaction consists of a infinite number of spinning modes at harmonics of the blade passing frequency(bpf). For a given harmonic, n BP F, there exist an infinite number of spinning modes, each of which is governed by the following equation: m = nn B + kn V (1.10) where m is the mode order, which can be thought of as the number of lobes about the circumference in the spinning mode; n is the blade passage harmonic number; and k is 15

31 Figure 1-8: Wake velocity defect and modes from fourier decomposition (a) Wake profile (b) 1 st Harmonic (c) 2 nd Harmonic (d) 3 rd Harmonic Figure 1-9: Hypothetical velocity defect profile and the first three modes of a corresponding Fourier decomposition 16

32 any positive or negative integer. Positive values of m correspond to modes spinning in the direction of rotor rotation, whereas negative values spin in the direction opposite of rotor rotation. For a given temporal frequency(ω), the corresponding spinning modes must all rotate at different angular rates(ω) in order to achieve the same frequency. Note that when considering modes propagating across multiple blade rows in a computational framework, it is necessary to adjust the frequency to the reference frame of the calculation. For example, consider several spinning modes of different circumferential order(m) generated in a rotor reference frame. If they are to be propagated to a downstream stator row, each of the spinning modes at the same frequency in the rotor reference frame will appear at different frequencies in the stator reference frame. The frequency and inter-blade phase angle of a mode m from row α 1 being applied to a downstream row α can be calculated as follows: ω α = ω α 1 + m α 1 (Ω α 1 Ω α ) (1.11) Φ = 2πm α 1 N α (1.12) Additionally, spinning modes are further separated into a series of radial modes for each circumferential mode, and can be classified as propagating(cut-on) or non-propagating(cutoff or evanescent), depending on whether the associated wave number is real or imaginary. Cut-on modes travel unattenuated down the length of a duct, whereas cut-off modes decay exponentially down the length of the duct[19]. For a given frequency, lower circumferential mode orders tend to be cut-on since they are spinning the fastest. orders spin slower and therefore tend to be cut-off. Likewise, high mode 1.5 Model-scale fan test rig The work performed for this thesis evaluates a linearized Navier-Stokes computational method for predicting booster tone noise. The geometry chosen for this investigation was a modelscale fan test rig for which an acoustic test database already existed. The rig geometry 17

33 OGV! R1! S1! R2! S2! Strut 1! Strut 2! Figure 1-10: Fan test rig diagram (not to scale) resembled an engine fan system design, including scaled blade geometry and an integrated vane frame OGV in the bypass duct. The flow splitter and primary flow is modeled using a multi-stage, single rotor booster to establish the desired fan bypass ratio for the test. The fan is representative of a commercial jet engine fan, whereas the booster was designed strictly to set the bypass ratio for the fan rig and is not representative of any actual engine. This investigation is focused specifically on tone noise generated in the booster section. Although the rig booster is not a scale model for an actual engine, the booster interaction noise physics are representative of low-pressure boosters in general. Additionally, the acoustic test database of the rig provided an opportunity to evaluate the ability of the computational approach to predict booster tone noise in general Geometry diagram Figure 1-10 shows a notional diagram of the fan test rig geometry that was used for this investigation. Table 1.1 includes the blade counts for the booster region of the test rig. As mentioned before, tone noise due to rotor-stator interaction is either a product of a rotor cutting a stator wake, a rotor wake impinging on a stator blade, or the potential pressure field interacting with an upstream/downstream blade row. For the booster in the test rig, there are two such rotor-stator interactions that would produce tones at multiples of the booster blade-passing frequency: the interaction of S1 with R2(S1/R2); as well as the interaction of R2 with S2(R2/S2). Each of these interactions will be investigated in the computational study. 18

34 Table 1.1: Booster blade counts Blade row S1 R2 S2 Strut1 Blade count Microphone traverse! Microphone traverse! Acoustic barrier! (a) Baseline (b) With acoustic barrier Figure 1-11: Fan rig test configurations (not to scale) Experimental data The first configuration of interest from the fan rig test was a baseline setup as shown in Figure 1-11(a). The second configuration included an acoustic barrier wall which was used to isolate the forward radiated noise from the test, as shown in Figure 1-11(b). Acoustic measurements were taken using a sideline traversing microphone. A representative operating point was selected from the test data and the total sound power levels for both configurations were plotted against each other, shown in Figure In the baseline configuration with no acoustic barrier, it can be seen that the 1 st and 2 nd booster BPF tones were particularly offending. In the configuration including the acoustic barrier, these tones are drastically reduced. In this way, it was confirmed that the booster tones were primarily emanating from the aft region of the test rig and this information helped guide the computational investigation to focus on the first two, downstream-propagating harmonics of the booster blade-passing frequency. The 1 st and 2 nd aft-radiating booster harmonics were then extracted along the speed line of the test rig to identify measured trends in the data. This was done by removing the forward-radiating tones(identified in the configuration with the acoustic barrier) from the total component(identified in the baseline configuration without the acoustic barrier). These measured trends are shown in Figure Surprisingly, 19

35 the 2 nd BPF tone was stronger than the 1 st BPF at operating conditions from low to mid power settings. At the highest powers, the 1 st BPF tone overtakes the 2 nd BPF in relative intensity. It was noted later in this investigation that the drive turbine for the rig had the potential to generate a tone equivalent to the 2 nd booster harmonic. This was identified as a potential source of error in the efforts of the team to identify pure booster tones. The rig was equipped with a drive turbine muffler designed to attenuate noise using a bulk absorber. However, at the time of this test, the muffler had been used for multiple test campaigns and the efficiency may have been degraded due to absorption of lubricating oil in the bulkabsorber. With the data set available, there was no possibility to deterministically verify the 2 nd BPF contribution from each of these potential sources. It is expected that the booster remains the primary source of tone noise at its harmonics due to its location relative to the microphone traverse and also since the possible source from the drive turbine is a scattered interaction tone and expected to be lower in intensity. The present study seeks to predict the levels of the 1 st and 2 nd booster blade-passing frequencies over a range of operating points using a time-linearized Navier-Stokes solver. The ability of the method to predict the trends in the acoustic data is of interest in addition to predicting the relative significance of the 1 st and 2 nd blade-passing frequencies with respect to each other. 20

36 Total Sound Power Levels! No acoustic barrier! With acoustic barrier! 1 st BPF! 2 nd BPF! PWL! 10 db! Frequency! Figure 1-12: Fan rig test data (Cutback) - highlighting booster 1 st /2 nd BPF tones Figure 1-13: Experimental, Aft-radiating 1 st /2 nd booster tones along speed line 21

37 Chapter 2 Numerical Method This chapter describes the numerical methods used for this investigation. This includes the governing equations, underlying assumptions, and modeling choices. The methodology used for tone noise prediction is also detailed here. The solver used was a GE internal CFD code for solving the Navier-Stokes equations. This is documented in Refs. [59, 60]. Nonlinear steady and time-linearized(frequency-domain) unsteady methods were applied in the present investigation. These numerical techniques along with the methodology for tone noise prediction are discussed in the following sections. 2.1 Nonlinear, steady, Reynolds-Averaged Navier-Stokes The nonlinear, steady, Reynolds-Averaged Navier-Stokes solver in the GE code was used in this investigation to obtain a steady mean flow solution that is utilized later in the linear, acoustic response calculations. Multi-blade row turbomachinery calculations were facilitated with mixing-plane boundary conditions between blade rows. The k-ω turbulence model was used with wall functions and a Durbin-style limiter[61]. Time advancement was achieved with a multi-stage Runge-Kutta algorithm. Grid sequencing was utilized to accelerate convergence. The grid sequencing technique operates on the original grid by coarsening a specified number of levels. The solver then operates on the coarsest grid for a specified number of iterations; moving to the next finest grid upon completion. This process is continued until the solver is operating on the original grid and has achieved a specified error tolerance. 22

38 This improves convergence by allowing for a larger time-step to be used when advancing the solution and it also acts as a filter on small-scale transient flow features due to the inability of the larger computational cells to resolve the smaller features. Steady, 2-D, nonreflecting boundary conditions based on the formulation by Giles were used[62]. Inlet boundary conditions were specified with total pressure and total temperature along with flow direction and inlet turbulence level. Exit boundary conditions were imposed by specifying a radial pressure profile. The profile was then adjusted to achieve a specified mass flow rate. Since this investigation is comparing against an experimental data set, the mass flow rates and associated rotation-rate of the rig were known a-priori. 2.2 Time-linearized, unsteady, Navier-Stokes The discussion here is based on the documentation of the method as exists in the relevant literature. The derivation of the time-linearized method is discussed in many different places in the literature[63, 64, 65, 66, 67, 68]. A derivation of the time-linearized method will be discussed here in the context of the Navier-Stokes equations. The unsteady Navier-Stokes equations which govern viscid and inviscid compressible flows are written here in conservation form as Q t + F (Q, Q) = S(Q, Q) (2.1) where Q is the array of conservative variables, F is the vector composed of conservative inviscid and viscous fluxes, and S is an array of source term components. These are defined as ρ ρu Q = ρv ρw ρe F (Q, Q) = F (Q) inv + F (Q, Q) vis (2.2) 23

39 ρu ρv ρw ρu 2 + p ρvu ρwu F inv = ρuv, ρv 2 + p, ρwv ρuw ρvw ρw 2 + p ρuh ρvh ρwh τ xx τ xy τ xz F vis = τ xy, τ yy, τ yz τ xz τ zy τ zz β x β y β z (2.3) The total enthalpy and static pressure terms are given as H = ρe+p ρ and p = (γ 1)(ρE 1 ρ V 2 ). In the base form of the Navier-Stokes equations, the source term S is zero. However, 2 turbulence models can introduce additional source terms that would be included in S. The viscous stress and work terms in F vis are defined as [ ( )] τ xx = µ 2 u x 2 u 3 x + v y + w z [ ( )] τ yy = µ 2 v y 2 u 3 x + v y + w z [ ( )] τ zz = µ 2 w z 2 u 3 x + v y + w z β x = uτ xx + vτ xy + wτ xz + µc p P r β y = uτ xy + vτ yy + wτ yz + µc p P r β z = uτ xz + vτ zy + wτ zz + µc p P r T x T y T z ( ) u τ xy = µ y + v x ( ) u τ xz = µ z + w x ( ) v τ yz = µ z + w y (2.4) The Reynolds-averaging process applied to the Navier-Stokes equations results in the addition of the Reynolds stress tensor, which is composed of products of turbulence-induced fluctuating components. This results in an undetermined system with more unknowns than equations since no new equations were added describing the Reynolds stress terms. The Boussinesq approximation is often invoked to relate the Reynolds stress terms to the timeaveraged components, closing the system of equations. This introduces the concept of tur- 24

40 bulent eddy viscosity(ν t ) and turbulent kinetic energy(k). A turbulence model is required to describe the behavior of these or other related turbulence properties. This usually takes the form of an additional partial differential equation or set of partial differential equations added to the system above in addition to turbulent components added to the momentum and energy equations, coupling the turbulence model with the original equation set. A specific implementaion is not discussed here, as the approximations and models become increasingly application dependent and an implementation effort was not apart of this work. Implementations of the Spalart-Allmaras turbulence model in a time-linearized framework are detailed by Clark[69], Clark and Hall[67] and Sbardella and Imregun[70]. The present work uses a version of the k-omega turbulence model and a discussion of its implementaion is given by Holmes and Lorence[59]. To linearize the governing equations, the solution variables are assumed to consist of a mean component and a small perturbation component as Q( x, t) = Q( x) + q ( x, t) (2.5) The present study does not include grid motion in the analysis so it is neglected in the derivation here. However, the grid coordinates can also be expanded as consisting of a mean and perturbation component. The solution expansion is substituted into Equation (2.1) and simplified by neglecting products of perturbations. The remaining terms consist of zero-order mean components and first-order perturbation components. The zero-order mean terms turn out to be the steady flux vectors and the coefficients of the first-order perturbations are the Jacobian components of the steady flux vectors. Grouping terms together, the original flux vectors can be represented by a perturbation series as F (Q( x, t)) = F ( Q) + F ( Q) q Q = F inv ( Q) + F vis ( Q) + F inv ( Q) Q q + F vis ( Q, Q) (2.6) Q q 25

41 If the grid coordinates are also expanded as mean and perturbation components, there will be extra terms that appear in Equation (2.6) that account for the effect of grid motion on the solution variables. The mean flux Jacobian matrices for this derivation are included in Appendix B. Considering the linearized solution and flux expressions substituted into the original conservation equation gives Nonlinear mean flow equations {[ }}{ Q ] t + F ( Q) Linear perturbation equations [{}}{ q + t + F ( Q) ] q = 0 (2.7) Q From Equation (2.1) we know that the nonlinear mean flow component is identically zero, which allows the nonlinear and linear equation sets to be separated. The result is a conservation equation for the linear perturbation variables that depends on the mean flow flux Jacobians as q t + F ( Q) Q q = 0 (2.8) Since the mean flux Jacobian terms are time-independent and only affect the perturbation solution by acting as a constant(variable in space), they can be provided by any means available. This could be an analytical expression for the mean flow or a mean flow that was solved numerically using a separate spatial discretization. The important consideration here is that the mean flow provided satisfies the original conservation expression in Equation (2.1). Often, the nonlinear mean flow is obtained by numerically solving the Euler or Navier-Stokes equations using a time-marching scheme and ensuring that the residual of the time derivative is as near to zero as possible. A residual flux imbalance in the nonlinear mean flow would manifest itself as a source in the linear solution which could cause inaccurate results or numerical instability. as A further assumption on the perturbation component is that its form is harmonic in time q ( x, t) = q ( x)e iωt (2.9) where q ( x) and ω specify the complex amplitude and frequency of the perturbation respec- 26

42 tively. This assumed form of the linear solution is then substituted into the conservation expression in Equation (2.8) for the linear perturbations which gives iωq ( x) + F ( Q) Q q ( x) = 0 (2.10) These are the time-linearized Navier-Stokes equations. There is no explicit time-dependance in Equation (2.10), however, a pseudo-time term is often added and the linear perturbations are marched in pseudo-time until convergence is reached, at which point, the pseudo-time derivative is equal or near to zero and the original equation is recovered. This allows one to take advantage of common convergence acceleration techniques used in time-marching schemes such as multigrid and local time-stepping. The pseudo-time form of the timelinearized equations is q ( x, τ) τ + iωq ( x, τ) + F ( Q) Q q ( x, τ) = 0 (2.11) At this point, Equation (2.11) may be discretized in space by any preferred method (Finite Difference, Finite Volume, etc.). A practical note regarding Equation (2.11) is that its numerical implementation requires complex arithmetic and associated data types. This can result in decreased computational performance due to complex arithmetic operations. However, the equation set can be solved using purely real arithmetic by separating out the real and imaginary components. complex amplitude of the perturbation is expressed in polar form as The q ( x) = Ae iθ (2.12) Euler s equation yields the complex perturbation in cartesian form as q ( x) = A[cos(θ) + isin(θ)] = [Re{q } + Im{q }i] (2.13) 27

43 where the Real and Imaginary components are Re{q } = Acos(θ) Im{q } = Asin(θ) (2.14) Substituting the cartesian form of the complex perturbation into Equation (2.10) gives iω[re{q } + Im{q }i] + F ( Q) Q [Re{q } + Im{q }i] = [0 + 0i] (2.15) The Real and Imaginary components can then be split into separate equations, which yields ω[im{q }] + F ( Q) Q [Re{q }] = [0] ω[re{q }] + F ( Q) Q [Im{q }] = [0] (2.16) Equation (2.16) can then be solved as a coupled set of equations using real arithmetic D Nonreflecting boundary conditions The time-linearized Navier-Stokes solver applied in the present work uses a 3D nonreflecting boundary condition based off of a formulation by Verdon[71] that is documented in Refs.[30, 31, 32]. As laid out in the references, the boundary condition is formulated beginning with the time-dependent, linearized Euler equations in cylindrical coordinates and in primitive variable form as U t + 1 r r (rāu ) + 1 U U B + C r θ z DU = 0 (2.17) 28

44 where U is the vector of perturbed variables defined as U = ρ v r v θ v z p (2.18) The matrices Ā, B, C, and D are listed in Appendix B for reference. Similar to the derivation of the time-linearized method itself, the form of the solution is assumed to be harmonic in time and further, that it is composed as a sum over circumferential(m) and associated radial(n) modes as U (r, θ, z, t) = Û mn(r)e i(ωt+mθ+kmnz) (2.19) m= n=0 Substituting this assumed form of the solution into Equation (2.17) gives iωû mn + 1 r r (rāû mn) + i m r BÛ mn + ik mn CÛ mn DÛ mn = 0 (2.20) which governs the set of radial modes(n) for a given circumferential mode order(m). The partial derivative term is approximated by a numerical differential operator Γ r, allowing the following eigensystem to be formed where Γ r is defined as [ iωi + Γ r + i(m/r) B + ik mn C D] Û mn = 0 (2.21) Γ r Û mn = 1 r r (rāû mn) (2.22) The practical description of the problem posed by Equation (2.21) is that for a given cir- 29

45 cumferential mode(m), the profiles of its corresponding radial modes(n) must be found. These are the eigenvectors of the above problem. The axial wave number(k z ) for each mode must also be found, which is the associated eigenvalue. These are computed numerically; a requirement due to the numerical representation of the differential operator. Spurious, convective, and acoustic modes are generated during the solution of the eigensystem and only the acoustic modes(designated here as R mn (r)) are used by the boundary condition. With the radial mode profiles in-hand, the form of the solution at the boundary is completely defined as a sum of these patterns and takes the following form U (r, θ, z, t) = a mn R mn (r)e i( ωt+mθ+kmnz) (2.23) m= n=0 The remaining task is to find the amplitudes a mn of each mode defining the boundary solution. This is accomplished by computing the inner product of the perturbation field solution with the individual mode profiles at the boundary. The modes can then be categorized as incoming or outgoing modes. This is based on the sign of the imaginary component of the axial wave number Im{k z } or the group velocity depending on if the mode is evanescent or propagating. Incoming modes from this decomposition can be interpreted as a reflection and are not used to update the boundary values, effectively eliminating the reflected waves. User imposed incoming perturbations along with the outgoing modes from the boundary decomposition are used to update the boundary values via their characteristics. A noteworthy observation here is that the boundary condition is based on the Euler equations and does not make any attempt to include viscous effects in the boundary update calculation. While this did not result in stability issues for the investigation, it is not known how significant the error is in the Euler approximation and may result in unwanted reflections. 2.3 Tone noise prediction methodology The discussion here details the methodology used in the present work to predict booster tone noise for the fan test rig, which was presented in Section 1.5. The tone noise prediction methodology makes use of both the nonlinear-steady and linearized-unsteady methods. As 30

46 discussed in Chapter 1, tone noise generation associated with turbomachinery blade-row interactions results primarily from three physical mechanisms. These are: Interaction of a rotor-locked potential pressure field with an upstream blade row Impingement of viscous wakes on a downstream blade row Cutting of an upstream wake by the current blade row As the generation and propagation of tone noise are inherantly unsteady processes, the linearized-unsteady method is used to facilitate their prediction. As discussed in Section 2.2, the linearized-unsteady method requires a base flow of the form Q( x) upon which the perturbed flow variables are solved. This base flow is provided by a converged nonlinear-steady, multi-blade row simulation using mixing-plane boundary conditions at blade row interfaces, as detailed in Section 2.1. It is important that a sufficiently converged nonlinear-steady simulation is achieved if it is to be used as the base flow for a linearized-unsteady analysis. An unconverged nonlinear solution implies a residual flux imbalance and a corresponding temporal variation. This is inconsistent with the assumption made in the derivation of the linearized-unsteady method that the base flow has only spatial dependence. Failure to meet this criteria will adversely affect the convergence of the linearized-unsteady analysis. The remaining constraints for the linearized unsteady analysis are the boundary conditions for the solver. More specifically, an incoming perturbation is required that provides a forcing for the acoustic response calculation, and the form of this perturbation must be determined. As mentioned before, we are interested in two physical cases; noise generation and noise propagation. The noise generation process here consists of imposing a known noise-generating disturbance at the boundary of a linearized-unsteady simulation and observing the response. This is termed a forced-response problem in the literature. The nonlinear, steady calculation serves a second purpose here by providing the form of these disturbances from the viscous wakes and potential pressure field. In order to obtain the harmonic form of these disturbances, an axial slice of the nonlinear solution is extracted just upstream of a mixing plane interface. This provides the spatial form of the disturbance in that frame of reference. Recognizing that this will appear as a temporal disturbance in the downstream blade 31

47 row, a circumferential Fourier transform of the extracted data will provide the harmonic amplitudes that can be imposed as the boundary condition to the linearized solver. The noise propagation process takes the modes from a noise generation calculation at the exit plane and applies them as incoming perturbations for a linearized-unsteady calculation on the inlet plane for the downstream blade-row. These processes are detailed more clearly in the following discussion Mean flow solution The first step in the computational process was to run a nonlinear, steady Reynolds-Averaged Navier-Stokes(RANS) calculation. The geometry in this study was run as a multi-stage calculation, including the Fan, S1, R2, S2, and Strut blade rows. The Fan was included in the investigation in order to set the appropriate flow condition ahead of the S1 blade row. This nonlinear mean flow solution was used to capture both viscous wakes and also the potential pressure field so they could be decomposed into Fourier harmonics. The nonlinear steady solution also provided a mean flow for the linearized-unsteady calculations Noise source decomposition An initial perturbation for a noise generation calculation was found by taking a circumferential Fourier transform of the primitive variables at the exit plane of an upstream blade row; decomposing the vortical wake profile into its individual frequency components. As pointed out by Sharma et al.[31], a circumferential Fourier transform in the disturbance frame of reference is equivalent to a temporal Fourier transform in the frame of reference of the downstream blade-row. Each harmonic may then be applied in separate calculations as a perturbation to the downstream row Interaction propagation This investigation was interested in predicting booster tone noise, which in this case involved two rotor-stator interactions(s1/r2 and R2/S2) and multiple blade-rows across which the corresponding interaction noise had to be propagated. Each initial wake harmonic applied 32

48 as a noise-generating perturbation in a linearized unsteady calcuation for the downstream blade row produces upstream and downstream traveling spinning modes, scattered by the blade count of the downstream row. Inspection of the experimental data sets determined that the offending tone noise was primarily originating from the aft section of the rig and so only downstream traveling modes were accounted for in this investigation. The relevant, downstream-traveling, outgoing modes were then applied to each subsequent blade row as an incoming perturbation after adjusting their frequency and inter-blade phase angle to the appropriate reference frame and geometrical configuration using Eqns. (1.11) and (1.12). This process was repeated until each interaction had been propagated through Strut1, as shown in Figure 2-1, where the acoustic response from each interaction was combined to generate predicted total sound power levels(pwl) for the first two booster blade-passing frequencies. Note, that the contribution of fan wakes interacting with the booster rotor(fanbooster interaction) were not accounted for in this study. Previous studies that have used the GE internal CFD solver for predicting tone noise were focused on a single rotor-stator or rotor-rotor interaction. As such, only the cut-on modes were considered in those studies, since cut-off modes would have decayed to trivial amplitudes down the length of the duct. In this study, however, multiple blade rows had to be accounted for that were in close proximity with each other. This would potentially allow a cut-off mode from one blade row to scatter into both radiating and evanescent modes in an adjacent blade row. Therefore, all duct modes of sufficient intensity were taken into account through the entire booster region. Finally, once each interaction had been propagated through Strut1, the cut-on acoustic modes at the exit plane of Strut1 were used to compute the total predicted tone PWL. Figure 2-1 points out the two noise sources of interest in addition to the location at which their respective in-duct PWL levels are computed after being propagated through Strut1. For this investigation, the S1/R2 and R2/S2 interactions were computed separately, where the S1 wake harmonics are imposed on R2 and those interactions are propagated downstream through the Strut. In separate calculations, the R2 wake harmonics are imposed on S2 and the interaction modes are propagated down through the Strut as well. In this way, the noise generated from the individual S1/R2 and R2/S2 interactions can be isolated to compare 33

49 OGV! R1! R2/S2 Interaction S1! R2! S2! Strut 1! In-duct PWL evaluation S1/R2 Interaction Figure 2-1: Acoustic source diagram their relative significance. Figure 2-2 demonstrates this diagramatically. The 3D, nonreflecting boundary condition used in the linearized-unsteady calculations provides the downstream traveling waves from the solution in the form of an expansion in eigenmodes computed from the solution of the eigenvalue system for the boundary condition. With the solution expansion in-hand, the Sound Pressure and Sound Power Levels (SPL, PWL) associated with a particular mode may be computed by summing the integrated acoustic modes for the appropriate quantities over the boundary. The particular expressions used to compute the total SPL and PWL levels downstream of the Strut are given as SP L = 10 log 10 ( 1 r 2 T r2 H ) rt a m,n 2 ψ m,n 2 r dr 10 log 10 (Pref) 2 (2.24) m,n r H { rt P W L = 10 log 10 (πre a m,n 2 m,n r H ψ (a) m,n(ψ (b) m,n)r dr }) 10 log 10 (P W L ref ) (2.25) where ψ (a) m,n and ψ (b) m,n are the eigenmodes corresponding to the quantities ρ 1 (p + ρ v z v z) 34

50 S1/R2 Interaction R2/S2 Interaction Sum propagating PWL levels to obtain overall propagating acoustic response Figure 2-2: Noise prediction methodology. and ρv z + ρ v z respectively. The reference acoustic pressure and power levels are 20µP a and W respectively. Intermediate SPL and PWL values(between blade rows) may include a combination of cut-on and cut-off acoustic modes since cut-off modes have the potential to be scattered to cut-on modes across blade rows. Total SPL/PWL values presented in the results will only include cut-on acoustic modes from the exit plane of Strut1 as shown in Figure 2-1. For the purposes of this study, it is assumed that the downstream duct is of sufficient length to allow cut-off modes to decay. 2.4 Selection of relevant modes The interaction of a frequency from a noise-generating mechanism(such as a blade wake or potential pressure field) with a neighboring blade row produces an infinite number of interaction modes governed by m generated = m incoming + kn v k =,..., 1, 0, 1,..., (2.26) This expression conveys the same information as Eq except here the discussion is limited to a single incoming mode. With the generated interaction modes in hand, the challenge 35

51 then becomes identifying which of the generated modes will be relevant and need to be propagated through the booster. There are several criteria that can assist in this selection process. First, for the case of a stator wake interacting with a downstream rotor, the spinning modes generated from that interaction will exist at different frequencies in the absolute frame that are multiples of the rotor blade passing frequency. This is confirmed by considering an incoming mode of order m in spinning with a frequency ω in sta in the stationary frame of reference, and how this mode interacts with a rotor of N b blades spinning at a frequency ω R and rotational rate Ω R. The rotational rate and frequency of the incoming mode as percieved in the rotational frame of reference by the rotor are computed as Ω in rot = Ω in sta Ω R ω in rot = m in Ω in rot (2.27) The modes generated from this interaction follow the relation given in Equation (2.26) and thus give m gen = m in + kn b (2.28) These interaction modes are generated at the frequency of the incoming disturbance as seen in the rotor frame of reference ω in rot. However, since each interaction mode has a different circumferential order, they rotate with different rotational rates in the rotating frame as Ω gen rot = ω in rot m gen (2.29) An interaction mode is transformed back into the stationary frame via Ω gen sta = Ω gen rot + Ω R (2.30) The frequency of a given mode generated by the k-th interaction may then be computed as 36

52 ω gen sta = m gen Ω gen sta = (m in + kn b )(Ω gen rot + Ω R ) [ ] min Ω in rot = (m in + kn b ) (m in + kn b ) + Ω R (2.31) = m in (Ω in sta Ω R ) + (m in + kn b )Ω R = m in Ω in sta + kn b Ω R = ω in + kω R This result shows that the frequencies generated by the interaction of an incoming mode with a rotor blade row are equal to the frequency of the incoming mode plus an integer multiple of the rotor blade-passing frequency. In the literature, these are often referred to as sum and difference tones. Cumpsty[72] provides a kinematic analysis of rotor stator interaction that describes this phenomenon. In the case of a stator wake impinging on a downstream rotor, the incoming frequency ω in sta is zero, so the frequencies generated by this interaction are simply integer multiples of the rotor blade-passing frequency as ω gen = kω R (2.32) Since we are often only concerned about a subset of these frequencies, the modes generated at frequencies outside of this subset can be discarded. In the present work, only the first two rotor blade-passing frequencies in the booster were of interest. As such, only the k = [ 2, 1, 1, 2] interaction modes of the stator wake being cut by the booster rotor were considered. A spinning mode interacting with a downstream stator will scatter into modes that spin with the same frequency as the incoming mode, but with mode numbers modified by integer multiples of the stator blade count as defined in Eq The characteristics of a mode scattering across rotor and stator blade rows is diagramed in Figure 2-3. Taking these considerations into account, an analysis of the allowable modes generated from the S1/R2 and R2/S2 blade-wake interactions was performed. Figure 2.4 shows the 37

53 (a) Mode scattering across a Rotor (b) Mode scattering across a Stator Figure 2-3: Mode scattering characteristics across Rotor and Stator blade rows. 38

54 analysis of the downstream traveling modes generated from the interaction of the S1 wake with the R2 blade row. This accounts for only the downstream traveling modes and does not account for any reflections between blade rows. As mentioned before, only the k = [ 2, 1, 1, 2] interactions of the S1 wake with R2 were considered, since these included the two frequencies of interest. The generated modes from the initial interaction of the S1 wake with R2 were considered to pass downstream through the S2 blade row; each scattering into an infinite number of modes. The modes scattered across S2 were considered for the scatter indices k = [ 3, 2, 1, 0, 1, 2, 3]. Here, an additional restriction was placed on the modes traveling downstream from the S2 exit into the Strut. Since there is a significant amount of separation between S2 and Strut1, this would allow cut-off modes to decay significantly. As such, only scattered modes satisfying m < 45 were propagated from S2 downstream into the Strut1 row. These modes are highlighted in green in Figure 2.4. The m < 45 selection criteria was based on preliminary analysis of the blade-passage flows and the ranges of modes that could be allowed to propagate, allowing for sufficient margin for variability in the flow conditions. The criteria was also based on the observation that the Strut1 blade count is low relative to these modes, so even modes scattered by the downstream row would likely remain cut-off. Once the green-highlighted modes were passed down through the Strut1 row, there was no other interaction that was accounted for in this work and all PWL levels for the S1/R2 interaction are computed from the propagation of the green-highlighted modes across the Strut1 row at the exit plane. Figure 2.4 shows the analysis of the downstream traveling modes generated from the interaction of the R2 wake with the S2 blade row. Since a given harmonic of the R2 wake scatters into modes that exist at the same frequency, the k = [ 2, 1, 0, 1, 2] modes from the interaction with S2 were investigated. The propagation of those modes downstream through Strut1 was then considered along with the allowable scattered mode orders. In order to select which modes would be considered in the computational study, the k = [ 3, 2, 1, 0, 1, 2, 3] scattered modes were considered in the Strut1 row for each incoming mode. For the modes existing at the first blade-passing frequency in the Strut1 row, mode orders satisfying m < 30 were considered as potentially cut-on along the rig operating line and were highlighted. Similarly, for the second blade-passing frequency, the modes in Strut1 satisfying m < 45 39

55 Figure 2-4: Analysis of modes generated from the interaction of the S1 wake with R2. were considered as potentially cut-on. Based on this investigation, only the k = [0, 1] modes from the interaction of the first R2 wake harmonic with S2 were considered for the computational study. Correspondingly, only the k = [ 1, 2] modes from the interaction of the second R2 wake harmonic with S2 were considered. 40

56 Figure 2-5: Analysis of modes generated from the interaction of the R2 wake with S2. 41

57 Chapter 3 Results Results for the booster tone noise analysis include mean flow solutions at several operating points in addition to the linearized unsteady calculations that provided predictions of induct PWL levels for the 1 st and 2 nd booster blade passing frequencies at the Strut1 exit plane. Note that the intent of this investigation was not to make exact comparisons to experimental data. This is because the experimental PWL values were calculated from traversing microphone data outside the test rig whereas the predicted in-duct PWL values are evaluated at the end of Strut1. While only propagating acoustic modes are used in the in-duct PWL calculation, the effects of downstream geometry such as duct variations and Strut2, as shown in Figure 1-10, were not taken into account. Additionally, as noted below, the 2 nd booster harmonic was potentially contaminated from tones emanating from the drive turbine, as noted previously. Considering these limitations, the present work was focused on the ability of the approach to correctly capture trends in the measured data along the operating line of the experimental rig. PWL was chosen to compare the experimental and computational data because it is a good indicator of source strength and does not vary with distance from the source. Since the computational effort accounts for the most significant acoustic sources(s1/r2 and R2/S2) and their propagation through Strut1, the predicted and experimental trends in the data should be comparable if the computational methodology is successful. The particular characteristic of interest was the ability of the code to predict the trend of booster BPF s along the operating line of the experimental rig. 42

58 The experimental PWL s for the booster harmonics presented in the results section represent the aft-radiating tone components. These were calculated by taking data from the Baseline configuration and removing the corresponding tone levels measured in the Acoustic Barrier configuration. This will allow a more consistent comparison to be made between the experimental and predicted PWL s since the predicted values only account for aft-propagating modes. The broadband noise component was also removed from the experimental data, since it is beyond the scope of this study. The process used to isolate the aft-radiating tones is presented in Appendix A 3.1 Convergence metrics The metrics used to evaluate convergence for the Steady and Linearized Unsteady RANS solvers are presented here. Mass flow rate, averaged exit static pressure, and the residual error in each blade row were monitored to determine that a Nonlinear Steady RANS calculation was sufficiently converged. Figure 3.1 shows these metrics plotted as an example of acceptable convergence. For the Linearized Unsteady calculations, the residual error in the calculation is tracked. Figure 3.1 shows an acceptable convergence history for a Linearized Unsteady calculation. In general, calculations were considered converged once the maximum error had been reduced by four orders of magnitude. 3.2 Grid resolution study A grid resolution study was performed for this work to quantify the effect of grid resolution on the computational results and to ensure that the relevant phenomena were being adequately resolved. Two physical phenomena were of interest for this study. The first of these was the resolution of blade wakes in the Nonlinear Steady calculations, which are decomposed into their Fourier harmonics and used as noise-generating inputs for the Linearized Unsteady calculations. A computational grid with insufficient resolution to retain the structure of the blade wake in the Nonlinear Steady calculations would cause the associated Fourier harmonics to be diminished, changing the input to the Linearized Unsteady calculation and 43

59 (a) Mass flow rate convergence (b) Averaged exit static pressure (c) Equation residual errors Figure 3-1: Nonlinear steady RANS convergence metrics. (a) Equation residual errors Figure 3-2: Linearized unsteady convergence metrics. 44

60 affecting the acoustic response. The second phenomenon of interest for the grid resolution study was wave resolution. Given some acoustic response to a blade-wake interaction or a propagating spinning mode, it is necessary to make sure that the waves can be adequately represented by the computational grid. An often-used metric for this is the number of grid points-per-wavelength.[73, 74]. Single blade passage grids were generated for each blade row using a combination of in-house GE grid generation tools and the commercial grid generator AutoGrid5 TM, developed by NUMECA International. The wake resolution and wave resolution studies were carried out as separate efforts and their results are presented in the following sections Wake resolution The two noise-generating wakes of interest are those associated with the S1 and R2 blade rows. These wakes are diagramed in Figure 3-3. A series of single blade-passage grids were generated with increasing levels of resolution to evaluate the amount of resolution required to adequately capture the spectral content of the wakes. That is, since only the Fourier harmonics of the wake profiles are used as the noise-generating input for the linearized unsteady calculations, we are concerned with the convergence of the harmonic profiles with grid resolution, as opposed to ensuring that the wake profile itself has completely converged. An example of the wake velocity defect profile at a given spanwise location is shown in Figure 3-4 General characteristics for the wake resolution grids are listed in Tables 3.1 and 3.2. Additionally, the resolution of the wake regions in particular can be seen in Figures 3-5 and 3-6. For each level of refinement, the number of grid points along the blade chord were increased along with the number of grid points in the wake region and also in the spanwise direction. Though there were multiple operating points targeted for this investigation, the Sideline condition was used for the wake convergence study since the higher Reynolds number flows should result in thinner wakes, which represents the more difficult condition for the computational grids in terms of wake resolution. Nonlinear steady calculations were then run in a multi-blade row configuration including the R1 - Strut rows. A calculation was run for 45

61 Figure 3-3: Diagram of noise-generating, blade-wake interactions Figure 3-4: Blade-wake profile convergence. Table 3.1: S1 wake convergence grids Grid Level Chord points Spanwise points Total Points Coarse Million Medium Million Fine Million 46

62 (a) Coarse (b) Medium (c) Fine Figure 3-5: S1 grid resolution levels at the trailing edge for wake resolution study. 47

63 (a) Coarse (b) Medium (c) Fine Figure 3-6: R2 grid resolution levels at the trailing edge for wake resolution study. 48

64 Table 3.2: R2 wake convergence grids Grid Level Chord points Spanwise points Total Points Coarse Million Medium Million Fine Million each of the grid refinement levels and the corresponding wake profiles were extracted. These profiles were decomposed into their Fourier components, as described in Section 2.3 on the prediction methodology. The wake velocity defect is the structure of interest here, so the 1 st and 2 nd harmonic components of the wake defect are plotted in Figures 3-7 and 3-8 for the S1 and R2 rows. The metric for determining convergence was the percent difference in the L2 norm of a given wake harmonic profile from one grid level to the next, which was less than one percent between the Medium and Fine grid levels for all wake harmonics. As such, the results here show that the Medium level grid for both the S1 and R2 rows is sufficient to resolve the frequencies of interest. The gust profiles from the Medium grid are used as the inputs for the linearized unsteady calculations. A note regarding the S1 wake harmonic profiles shown in Figure 3-7 is that one would expect the velocity perturbation to go to zero at the hub and tip locations due to the no-slip condition at the solid boundaries. The S1 wake profiles do not go to zero at the hub location because part of the downstream rotor hub was included in the S1 domain so that the absolute fluid velocity at the hub in the S1 exit plane was non-zero. As such, the shape of the S1 wake profiles near the hub is expected. The amplitudes of the harmonic profiles in Figures 3-7 and 3-8 were computed as 20log 10 ( V ) = 10log 10 (Re{V z} 2 +Im{V z} 2 +Re{V t } 2 +Im{V t } 2 +Re{V r } 2 +Im{V r } 2 ) (3.1) Wave resolution In order to assess the ability of the computational grid to adequately resolve the wave phenomena consisting of noise generation and noise propagation, several computational grids, 49

65 (a) 1 st Harmonic (b) 2 nd Harmonic Figure 3-7: Comparison of S1 wake harmonics across grid resolutions. (a) 1 st Harmonic (b) 2 nd Harmonic Figure 3-8: Comparison of R2 wake harmonics across grid resolutions. 50

66 Table 3.3: Wave convergence grids Grid Level R2 S2 Strut Coarse 0.24 M 0.35 M 0.28 M Medium 0.73 M 1.2 M 1.4 M Fine 2.5 M 5.2 M 4.5 M Extra Fine 7.9 M 7.3 M 8.8 M each with increasing grid density, were generated for the R2, S2, and Strut blade rows. These were then studied individually, starting with R2, by applying the relevant disturbances to each grid refinement level and comparing the downstream traveling modes. Once an acceptable grid resolution was identified for R2, the relevant outgoing modes were applied to S2 in the same manner on the different grid levels to evaluate the S2 resolution requirements. Finally, the same process is repeated with the Strut row. Since the generation and propagation of noise are the phenomena of interest, the profiles of the pressure perturbation amplitude at the exit plane of each blade row were used as the metric for evaluating grid convergence in addition to integrated SPL values. Four different levels of grid resolution were tested to evaluate the resolution requirement for wave propagation in this setting. The characteristics of the wave propagation grids are presented in Table The results of the wave convergence study are presented in Appendix C. It was found that the Fine wave convergence grid level was sufficient to resolve the waves of interest Initial generated noise In order to further identify which modes generated by the S1/R2 and R2/S2 blade-wake interactions contain significant acoustic energy, the Sound Pressure Level (SPL) profile of each circumferential mode of interest was plotted, along with their integrated SPL values at the exit planes of the R2 and S2 blade rows for the Cutback operating point. The mode profiles were computed by taking a circumferential Fourier transform of the perturbation solution at the exit plane of a given blade row. Figure 3-9 shows the modes generated from the 1 st S1 wake harmonic impinging on R2. The (k = 1) and (k = 2) modes contain the 51

67 majority of the acoustic energy for this interaction, with their integrated SPL values nearly 20dB above the (k = 1) and (k = 2) modes. Figure 3-10 shows the modes generated from the 2 nd S1 wake harmonic impinging on R2. The integrated SPL values of all modes generated from this interaction are all at least 10dB below the modes generated from the interaction of the 1 st S1 wake harmonic since these modes are strongly cut-off. Considering now the 1 st R2 wake harmonic impinging on S2, the modes generated from this interaction are shown in Figure The (k = 1) mode contains the only significant acoustic energy from this interaction, existing at over 20dB above the (k = 0) mode. Lastly, the modes generated from the 2 nd R2 wake harmonic impinging on S2 are presented in Figure The integrated SPL for the (k = 1) mode here was greater than 10dB above the (k = 2) mode. The benefit of identifying the modes that contain significant amounts of acoustic energy and those that do not, is that one can filter out the insignificant modes from the propagation process. This can significantly reduce the number of linearized calculations that have to be run in order to obtain the acoustic response downstream of the Strut blade row. However, even if the pressure response from a given interaction is insignificant, one must confirm that the velocity perturbations for the same mode are also negligible as it is possible for the modulated wake profile to generate pressure fluctuations in the downstream row. For the present work, all modes were retained and passed downstream to avoid inadvertently discarding some form of noise generating disturbance. 3.3 Mean flow calculations Five points along the operating line for the test rig were identified for analysis in this investigation. Three of these generally correspond to Approach, Cutback, and Sideline conditions for an aircraft community noise study. Two additional points(pt A and PT B) were added to provide better resolution of the operating line. Experimental freestream flow variables were provided to a GE code that computes the boundary condition variables at the appropriate inlet plane for the steady RANS calculation. The mass flow was then varied in the RANS calculation to match experimental values for pressure ratio across the fan. A more 52

68 (a) k = 2 (b) k = 1 (c) k = 1 (d) k = 2 (e) Integrated SPL Figure 3-9: SPL at the R2 exit plane for modes generated from the S1 1 st wake harmonic interacting with R2. (a) k = 2 (b) k = 1 (c) k = 1 (d) k = 2 (e) Integrated SPL Figure 3-10: SPL at the R2 exit plane for modes generated from the S1 2 nd wake harmonic interacting with R2. 53

69 (b) k = 1 (c) k = 0 (d) Integrated SPL Figure 3-11: SPL at the S2 exit plane for modes generated from the R2 1 st wake harmonic interacting with S2. (b) k = 2 (c) k = 1 (d) Integrated SPL Figure 3-12: SPL at the S2 exit plane for modes generated from the R2 2 nd wake harmonic interacting with S2. 54

70 Figure 3-13: Operating points computed from mean flow solution appropriate measure for matching the operating condition would have been the booster pressure ratio, however the fan pressure ratio was the only data available for this investigation. The mean flow solution results can be seen in Figure The 1 st and 2 nd harmonic profiles of the S1 and R2 wakes were also plotted at the Approach, Cutback, and Sideline operating points. These are shown in Figures 3-14 and Since the blade wake interactions are a primary source of noise generation, the amplitudes of the wake harmonics are indicators for how noise levels may vary for different operating points. 3.4 Linearized-unsteady analysis A linearized unsteady analysis was performed for each of the prescribed operating points. The propagating outgoing acoustic modes were used to evaluate PWL at the exit plane of Strut1. The analysis of the computational results was split into several sections. First, the individual modes from the S1/R2 and R2/S2 interactions were investigated along the different operating points. Then, the total sound power levels for the 1 st and 2 nd BPF s were compared against the trends in the experimental data. 55

71 (a) 1 st Harmonic (b) 2 nd Harmonic Figure 3-14: Comparison of S1 wake harmonics along the rig operating line. (a) 1 st Harmonic (b) 2 nd Harmonic Figure 3-15: Comparison of R2 wake harmonics along the rig operating line. 56

72 st BPF modes Figures 3-16 and 3-17 show the cut-on modes for the primary booster BPF from the S1/R2 and R2/S2 interactions, respectively. Observation of the data in both figures shows that as the operating speed is increased, higher mode orders transition from cut-off to cut-on. This makes sense physically because the 1 st and 2 nd BPF tones are at successively higher frequencies, resulting in circumferential modes spinning at faster angular rates. Also, each mode generally increases in intensity from Approach through Sideline, which is consistent with expected behavior, since the wake harmonic amplitudes increased from the Approach through Sideline points. The additional mode scattering that occurs in the S1/R2 interaction due to the extra blade row is evident when comparing Figures 3-16 and One unexpected trait in these predictions is seen in Figure 3-17, where the total PWL at Approach from the R2/S2 interaction is nearly 30dB lower than the total PWL at Cutback. While one would expect a decrease in PWL at the lower speed, such a large decrease was surprising. Investigating the corresponding circumferential modes provided some insight into the discrepancy. For the Cutback and Sideline conditions, the -6, 2, and 10 modes are well cut-on. For both of these operating points the -6, and 10 modes are 8-10dB higher than the 2 mode. At low-speed this behavior does not hold. The -6, 2, and 10 modes are all relatively equal. Further investigation showed that the parent mode (m = 22), which was scattered by Strut1 to produce the -6, 2, and 10 modes, was in transition from cut-off to cut-on at Approach through PT B in the numerical simulation. A significant amount of duct length between S2 and Strut1 allowed that mode to decay prior to its interaction with Strut1, causing a much weaker interaction and subsequently lower PWL values for the -6, 2, and 10 modes at low-speed. The most significant modes from the S1/R2 interaction were well cut-on through the booster, reducing the potential error due to cut-off modes between blade rows. In particular, the m = 2 mode is responsible for the majority of the noise from the S1/R2 interaction contributing to the 1 st blade-passing frequency. This results from the S1 wake being modulated by R2 and impinging on the S2 blade row. Figure 3-18 shows the total predicted PWL from the individual S1/R2 and R2/S2 interactions along the operating line for the 1 st booster blade-passing frequency. This prediction indicates that at 57

73 OGV! R1! S1! R2! S2! Strut 1! Strut 2! Figure 3-16: S1/R2 modes - 1 st BPF low speed for the rig, the S1/R2 interaction is the most significant contributor to the 1 st booster blade-passing frequency. At high speeds, the R2/S2 interaction is predicted as being most dominant. This is due to the m = 22 mode transitioning from cut-off to cut-on at higher speeds nd BPF modes Similarly, Figures 3-19 and 3-20 show the cut-on modes from the S1/R2 and R2/S2 interactions respectively for the 2 nd booster BPF. As expected, more modes become cut-on as the speed of the rig increases, and the intensity of the modes increase with speed as well. For the S1/R2 interaction, the most significant modes were well cut-on throughout the booster region. In particular, the m = 6 mode is the most significant contributor of the S1/R2 interaction to the 2 nd blade-passing frequency. This is generated by the (k=2) interaction of the S1 wake with the R2 row. Investigating the individual modes for the R2/S2 interaction contributing to the 2 nd blade-passing frequency, the 26 mode was the most significant contributor, but was cut-off at Approach. This is generated from the (k = 1) interaction of the 2 nd rotor wake harmonic with S2. Figure 3-21 shows the total PWL contributions from the S1/R2 and R2/S2 blade-wake interactions. This shows that the S1/R2 interaction 58

74 OGV! R1! S1! R2! S2! Strut 1! Strut 2! Figure 3-17: R2/S2 modes - 1 st BPF Figure 3-18: 1 st BPF response from the S1/R2 and R2/S2 interactions. 59

75 OGV! R1! S1! R2! S2! Strut 1! Strut 2! Figure 3-19: S1/R2 modes - 2 nd BPF is the primary noise source contributing to the 2 nd booster blade passing frequency at lowspeed. At the Cutback and Sideline operating points, the S1/R2 and R2/S2 contributions are relatively equal Total Power Levels The aft-propagating PWL levels computed from modes generated in the S1/R2 and R2/S2 interactions were combined at the exit plane of the Strut to produce the total predicted 1 st and 2 nd booster tone levels for each operating point. Figure 3-22 shows the predicted total Aft PWL levels for the first two booster harmonics compared against the Aft PWL levels that were extracted from the experimental data set. In general, the predicted values for PWL follow the trend of the experimental data along the operating line. However, there was a large deficit in the predicted 1 st BPF at Approach, PT A, and PT B compared to the experimental value. This also shows that the computational method captures the relative strength of the 2 nd BPF at low power, and the transition of the 1 st BPF to dominance at high power. The spike in the experimental 2 nd BPF around cutback was not predicted by the computational methodology. 60

76 OGV! R1! S1! R2! S2! Strut 1! Strut 2! Figure 3-20: R2/S2 modes - 2 nd BPF Figure 3-21: 2 nd BPF response from the S1/R2 and R2/S2 interactions. 61

77 Figure 3-22: Trends in experimental and predicted booster tones Effect of R2 potential field interaction The work presented up to this point has focused exclusively on blade-wake interactions. However, as discussed in the Introduction, the rotor-locked potential pressure field is also a source of noise as it interacts with upstream blade-rows. Here, the significance of the potential field interaction will be quantified relative to the blade-wake interactions. In order to evaluate the interaction, a circumferential Fourier transform of the converged Nonlinear- Steady calculation was computed at the inlet plane of the R2 blade row. This is diagramed in Figure The first and second blade-passing frequencies were taken from the Fourier decomposition of the potential field and imposed as incoming perturbations at the exit plane of the upstream blade-row(s1) in separate linearized unsteady calculations. The in-duct SPL values of the downstream-propagating acoustic modes generated from the interactions were computed at the S1 exit plane and are shown in Figure 3-24 to identify the most significant generated modes. The modes generated from this interaction will be scattered across the R2 blade-row while being propagated downstream and so could generate frequencies at both the 1 st and 2 nd R2 blade passing frequencies. The (k = 1) interaction 62

78 Figure 3-23: Diagram of rotor-locked potential field interaction from both the 1 st and 2 nd R2 harmonics scatter to cut-on modes across the R2 blade row so their effects will be computed through the booster at the PT A and Cutback points to compare against the blade-wake interaction SPL levels. The (k = 0) and (k = 2) interactions form the 1 st and 2 nd harmonics respectively scatter to cut-off modes across R2 so their propagation will not be computed. They are also of lower magnitude than the (k = 1) interactions. Figures 3-25(a) and 3-25(b) show the comparison of in-duct SPL for the R2 potential field interaction modes (R2/S1) against the S1/R2 blade-wake interaction modes at the PT A and Cutback operating points. These plots show that the R2 potential field interaction decays significantly downstream through the booster and the S1 wake source accounts for the majority of the noise at low power through Cutback. This demonstrates that the downstream potential field interaction is less significant than the blade-wake interaction sources for this study and would not affect the results with significance. As such, the potential field interactions were not included in the total PWL results and were not investigated further Potential sources of error The data presented previously regarding blade-wake interactions detailed the predicted booster sound power levels along with a more detailed breakdown of the individual modes 63

79 (a) 1 st R2 Harmonic (b) 2 nd R2 Harmonic Figure 3-24: Aft SPL at S1 exit plane of modes generated from R2 potential field interaction. (a) PT A (b) Cutback Figure 3-25: R2 potential field interaction propagation. 64

80 responsible for the majority of the noise. A significant observation was that the method underpredicted the 1 st blade passing frequency at low speed. The discussion here offers some insight on this issue. There are two possible cases that could be made to explain the discrepancy between the predicted and measured 1BPF tone at low speed. The first possibility is that all of the relevant tonal sources have been captured, only their particular propagation behavior is not predicted correctly. The second possibility is that the computational analysis accurately predicts the sources included in the investigation, but some other effect or source was present in the experiment that was not accounted for computationally. Several items fall under these categories and are presented in a high-level diagram in Figure Effect of real geometry Real geometry effects represent the difference between the rig as exists in the experiment compared to the geometry that exists in a CFD environment, the latter of which is often idealized. In the computational study, hub cavities between blade rows were not accounted for and instrumented blades were not represented. There can also be differences between a manufactured part and its representation in a CAD environment. These potential differences were not accounted for in the study, since even greater assumptions are made in the linearized unsteady approximation. In the present work, duct variations downstream of Strut 1 were not accounted for. As noted previously, the primary goal was to capture the trends in the noise generated from rotor-stator interactions in the booster and these trends should be captured by studying the cut-on modes in the booster region. Operating point Since the experimental data for the pressure ratio across the booster was not available for this investigation, the only metric available for matching the operating point of the calculation with the experiment was the mass flow of the booster and bypass regions and the pressure ratio of the fan. A better option would have been to match the pressure ratio of the booster itself and also to match blade loadings at the different operating points. Potential inaccuracies in the prediction of the mean flow could impact the noise source via incorrect 65

81 wake profiles and incorrect mean flow solutions upon which the perturbed variables are computed. Neglected phenomena There are also other effects that have been explicitly neglected in the present work that could have been significant in the experiment. In the computational study, only the downstream traveling noise was computed in the booster region. There is a potential for upstream traveling modes to interact with the fan and reflect back through the bypass duct. Additionally, the fan wake itself interacting with the booster would produce some level of fan-booster interaction, which was not accounted for in this study although many of the interaction tones exist at frequencies independent of the booster blade-passing frequencies. Regarding coupled blade-row interactions, the nature of the linearized calculations used here explicitly neglects this effect since each calculation only includes a single blade row with an incoming fixed perturbation. Additionally, nonlinear interactions between frequencies are not accounted for. In turbomachinery cases with large distances between blade rows(fan-ogv interaction for example), coupled blade row and nonlinear interactions are expected to be less significant. This is confirmed by the widespread use of linearized analyses for predicting tone noise associated with Fan-OGV interactions. In a compressor region however, the coupled and nonlinear interactions are expected to be more significant. Figure 3-30 diagrams some of these neglected sources. Evanescent and radiating modes One potential source of error in the computational investigation was identified by first focusing on the 1 st blade-passing frequency at the Sideline operating point, where the predicted and experimental tone levels were in good agreement. At this operating point, the most relavant predicted modes were identified and isolated for further investigation. These are shown in Figure 3-27, demonstrating that the R2/S2 interaction accounts for the majority of the sound power contributing to the overall 1 BPF response at the Sideline operating point. Within the set of propagating R2/S2 interaction modes, the 22 mode is most significant. In addition, the next nearest significant modes in the R2/S2 interaction, (m = 14, 6, 2, 10), 66

82 (a) Potential causes of inaccurate predictions (b) Uncaptured noise sources Figure 3-26: Potential sources of error. 67

83 Table 3.4: Predicted axial wavenumber for mode ( 22, 0) at S2 passage exit. Operating Point Re{k z } Im{k z } Approach 7.31e e 0 PT A 8.52e e 0 PT B 1.24e e 0 Cutback 1.78e e 3 Sideline 1.69e e 3 are scattered modes produced from the interaction of the 22 mode with the Strut, diagramed in Figure This identifies the 22 mode as the most important mode contributing to the 1BPF tone. This mode was then investigated at the different computed operating points to characterize its numerical behavior. In particular, the real and imaginary components of the axial wavenumber(k z ) for the first radial mode ( 22, 0) were recorded at the exit of the S2 blade passage. These are shown in Table 3.4. The imaginary component of k z for mode ( 22, 0), was plotted to demonstrate the propagation behavior of the mode. This is shown in Figure 3-29, demonstrating that at lower speed the ( 22, 0) mode is cut-off. Unlike the close spacing of the S1 and R2 blade rows, there is significant separation between the S2 blade row and the downstream Strut. This becomes significant in the case that mode ( 22, 0) is cut-off since it allows this mode to decay before being scattered by the Strut into propagating modes. This would cause the overall acoustic response to be lower. Some of the previously mentioned phenomena could have caused this mode to become cut-on at a lower operating point or to scatter to cut-on modes. For example, a coupled blade-row interaction could have caused the ( 22, 0) mode to scatter to lower mode orders before traveling down through the Strut. Unfortunately, without in-duct acoustic data from the experiment, it is difficult to confirm the physical reality occuring for this interaction. However, some higherfidelity computational studies may provide additional clarity and would be warranted for future work. Upstream-traveling modes and reflections One method to evaluate the potential for coupled blade-row interactions to play a significant role in the noise generation and propagation processes is to investigate the SPL levels of 68

84 (a) S1/R2 Modes (b) R2/S2 Modes (c) Total PWL Figure 3-27: Most significant predicted modes contributing to the 1BPF response at the Sideline condition. Figure 3-28: Scattered modes from R2/S2 interaction contributing to 1BPF tone. Figure 3-29: Predicted propagation properties for mode (-22, 0). 69

85 (a) Reflection of forward radiating waves off of the fan through the bypass duct (b) Coupled blade row interactions (c) Nonlinear interaction between frequencies Figure 3-30: Potentially significant effects unaccounted for. upstream traveling modes generated from the initial blade-wake interactions. A comparison of the most significant modes from the interaction of the 1 st S1 wake harmonic with R2 at the PT A operating point are shown in Figure Here, the red, blue, and green contours represent the 1 st, 2 nd, and 3 rd blade-passing frequencies respectively. A similar display for the interaction of the 1 st and 2 nd R2 wake harmonics with S2 is shown in Figure These show that there are significant levels of acoustic pressure generated from the blade-wake interactions that travel upstream. Since the blade-rows in a compressor are so closely spaced together, these modes would be reflected and scattered by upstream blade-rows. Integrated SPL values for these modes are presented in Figure Indeed, in every case except for the k = 2 interaction of the 1 st S1 wake harmonic, the upstream-traveling modes have higher SPL values than their downstream-traveling counterparts. It is worth noting here that the S2 exit plane in the computational grid is significantly downstream of the S2 blade itself. As such, cut-off modes are able to decay in this distance. However, since the upstream-traveling modes would be scattered in a reflection from an upstream row, they could easily be scattered 70

86 Figure 3-31: Spanwise SPL of upstream and downstream-traveling modes generated from the interaction of the 1 st S1 wake harmonic with R2 at PT A. to cut-on modes that would then travel downstream unattenuated. These could reasonably represent a larger noise source than the original downstream traveling modes generated from the blade-wake interactions. This could explain the error in some of the predicted Total PWL values for this investigation, since only the original downstream-traveling modes were accounted for in the downstream propagation process. 71

87 Figure 3-32: Spanwise SPL of upstream and downstream-traveling modes generated from the interaction of the 1 st and 2 nd R2 wake harmonics with S2 at PT A. (a) S1 1 st wake harmonic interactions (b) R2 1 st and 2 nd wake harmonic interactions Figure 3-33: Integrated SPL of upstream and downstream-traveling modes generated from initial blade-wake interactions at operating point PT A. 72

88 Chapter 4 Conclusions and future work 4.1 Conclusions A multi-stage methodology for the linearized-unsteady Navier-Stokes equations in a GE internal CFD code was outlined for predicting booster tone noise. A fan test rig was modeled that featured a booster with multiple blade rows. An existing experimental acoustic data set over a range of operating points was leveraged as reference data against which the predictions were compared. The predicted booster tones were compared against the extracted tones from the experimental data. The multi-stage methodology outlined in the investigation was applied to the booster region of the fan test rig and in-duct sound power levels were computed downstream of the acoustic source. The predicted trends for the 1 st and 2 nd booster harmonics followed the trends in experimental data along the operating line. A key finding was that the computational methodology was able to predict the 2 nd booster harmonic as louder than the 1 st at the low power conditions. However, an under-prediction of the R2/S2 interaction may have yielded an unexpectedly low predicted sound power level for the 1 st booster harmonic at low speed. The interaction of the R2 rotor-locked potential field with the upstream blade row(s1) was investigated and was determined to be much less significant than the blade-wake interactions. One possible cause of the underpredicted 1 st booster harmonic was identified as a key mode contributing to the 1 st harmonic being predicted as cut-off at low speed. In the experiment, this mode could have been more easily scattered to cut-on modes due to coupled blade row interactions that are not captured 73

89 in the linearized analysis. Additionally, the upstream traveling modes generated from the blade-wake interactions were investigated and their SPL levels were greater than or similar in magnitude to the downstream propagating modes that were initially generated. This indicated that coupled blade-row interactions would likely be a significant contributor to the overall acoustic response since the upstream traveling modes would interact and reflect off of upstream blade rows. Other potential sources of error were identified as nonlinear frequency interactions, effects of real geometry, and inaccurate prediction of the wake profiles which are the noise generating sources for the blade-wake interactions. An additional source of error for the analysis exists in using the Euler equations to govern the 3D nonreflecting boundary conditions, as reflections due to viscous effects may be present in the calculations although this has not been quantified. This work was successful in showing that the linearized-unsteady Navier-Stokes method is able to predict some significant noise generating sources and their propagation in a boostertype geometry characterized by high blade counts and relatively close blade spacing in comparison to previous investigations focused primarily on Fan-OGV interactions, characterized by low blade counts and long lengths of duct separating the blade rows. The tone noise predictions using the linearized-unsteady method also followed trends in the experimental data, indicating that the method is well-suited for predicting interactions that occur over a wide range of operating points and blade-loading conditions. This work also highlighted the danger of relying solely on a linearized-unsteady analysis for a complete tone noise investigation for these configurations. The concern here is in the underprediction of the 1 st booster blade-passing frequency. This result indicates that there is some fundamental part of the problem that is being missed in the prediction. This is also not unexpected. Given the close blade-spacings that exist in a compressor stage, the coupled blade-row interactions are expected to produce some significant effects that will not be present in the linearized analysis of an isolated blade-row. 74

90 4.2 Future work There are a number of questions that one could pose as the beginning of a follow-on investigation from this work. How significant are the coupled blade-row effects for acoustics problems? How much noise is reflected off the fan and down the bypass duct? How good is the Euler approximation in the development of 3D nonreflecting boundary conditions for acoustic calculations? Particularly, in configurations with closely spaced blade rows. A valuable calculation to begin investigating these effects would be a nonlinear, timemarching Navier-Stokes calculation including multiple coupled blade rows. The comparison of the acoustic fields from the nonlinear time-marching calculation to corresponding results from a set of linearized calculations would help quantify the differences and potential deficiencies of the linearized method. The goal here would not necessarily be to say whether the linearized calculations are good or bad, but rather to gather quantitative results that can guide future investigations on where the linearized methodology is most appropriate for acoustic analyses and where the method falls short in its capabilities. Another interesting investigation would be to compare results from a set of linearized calculations with some of the nonlinear frequency-domain methods that are well-suited to turbomachinery such as the Nonlinear Harmonic method and the Harmonic Balance method. These methods have the benefits of requiring only a single blade-passage per row(this is in comparison to a time-domain, nonlinear unsteady calculation, which may require many blades passages per row to ensure periodicity), accounting for multiple frequencies and also nonlinear interactions between those frequencies and their effect on the mean flow solution. 75

91 Appendix A Tone noise extraction process The acoustic data provided for the experimental rig was composed of both the tonal and broadband components of noise. It was necessary to isolate the tonal noise from the broadband noise for the frequencies of interest in order to make a valid comparison between the computational predictions and the measured data. The process of extracting the tonal components of noise from the frequency spectrum is detailed here. For this investigation, the 1 st and 2 nd blade-passing frequencies were of interest. The method used here to extract the tonal component of noise from the spectrum was to sample the frequency spectrum on both sides of a given tone to obtain the sound power level of the broadband noise. These values were averaged to obtain a single sound power level that was representative of the broadband noise around a given tone. This was then subtracted pointwise from the original spectrum on a logarithmic scale to leave just the tonal component of noise as T otal = T one + Broadband T one = T otal Broadband (A.1) T one(db) = 10log 10 (10 T otal/10 10 BBavg/10 ) (A.2) This process is presented in diagram form in Figure A-1. With the tone noise components isolated from the original spectrum are then integrated to produce a single power level for that particular blade-passing frequency as 76

92 (a) Noise components (b) Average broadband level (c) Extracted tone component Figure A-1: Diagrams for tone noise extraction process 77

93 T one T otal (db) = 10log 10 (10 T onea/ T one b/ ) (A.3) The tones from two experimental configurations were extracted for the present work. These were a baseline configuration and a configuration including an acoustic barrier that was used to block noise emanating from the aft portion of the fan test rig. The setup using the acoustic barrier effectively provides acoustic measurements for noise from the front portion of the fan test rig. In all the comparisons between predicted and experimental results in the present work, the experimentally measured tones presented are the aft radiating tones. The aft radiating tones were computed by subtracting the forward radiating tones in the acoustic barrier case from the total tones in the baseline configuration as T one total = T one forward + T one aft T one aft = T one total T one forward (A.4) T one aft (db) = 10log 10 (10 (T one total/10) 10 (T one forward/10) ) (A.5) 78

94 Appendix B Supporting mathematical expressions B.1 Mean flow flux Jacobians The expressions for the inviscid mean flow flux jacobian matrices are given by F x (Q) inv Q F y (Q) inv Q F z (Q) inv Q u 2 + p ρ 2u + p ρu p ρv p ρw p ρe = uv v u 0 0 uw w 0 u 0 u(p ρ H) H + up ρu up ρv up ρw u(1 + p ρe ) uv v u 0 0 = v 2 + p ρ p ρu 2v + p ρv p ρw p ρe vw 0 w v 0 v(p ρ H) vp ρu H + vp ρv vp ρw v(1 + p ρe ) uw w 0 u 0 = vw 0 w v 0 w + p ρ p ρu p ρv 2w + p ρw p ρe w(p ρ H) wp ρu wp ρv H + wp ρw w(1 + p ρe ) 79 (B.1)

95 where the Jacobians of pressure (p ρ, p ρu, p ρv, p ρw, p ρe ) are p Q = p ρ p ρu p ρv p ρw p ρe = 2 (u2 + v 2 + w 2 ) (γ 1)u (γ 1)v (γ 1)w (γ 1) γ 1 (B.2) The expressions for the viscous mean flow flux jacobian matrices are given by F x (Q) vis Q F y (Q) vis Q F z (Q) vis Q = = = τxx ρ τxy ρ τxz ρ βx ρ τxx ρu τxy ρu τxz ρu βx ρu τxx ρv τxy ρv τxz ρv βx ρv τxx ρw τxy ρw τxz ρw βx ρw τxx ρe τxy ρe τxz ρe βx ρe τyx ρ τyy ρ τyz ρ βy ρ τyx ρu τyy ρu τyz ρu βy ρu τyx ρv τyy ρv τyz ρv βy ρv τyx ρw τyy ρw τyz ρw βy ρw τyx ρe τyy ρe τyz ρe βy ρe τzx ρ τzy ρ τzz ρ βz ρ τzx ρu τzy ρu τzz ρu βz ρu τzx ρv τzy ρv τzz ρv βz ρv τzx ρw τzy ρw τzz ρw βz ρw τzx ρe τzy ρe τzz ρe βz ρe (B.3) A derivation of the exact form of the viscous flux jacobian terms in (B.3) is given by Galbraith[75]. Note also that the introduction of turbulence models into the governing equations will expand the matrices above to include jacobian terms with respect to the turbulence working variables. 80

96 B.2 Boundary condition matrices v r ρ v θ 0 ρ 0 0 v z 0 0 ρ v r v ρ θ v z Ā = 0 0 v r 0 0 B = v θ 0 C = 0 0 v ρ z v r v θ v z ρ 0 γ p 0 0 v r 0 0 γ p 0 v θ γ p v z ( v θ +Ωr) 2 2( v 0 θ +Ωr) 1 0 ρr r ρr D = vr( v θ+2ωr) ( v θ+2ωr) vr 0 0 ρr r r (γ 1) ρ( v 0 θ +Ωr) 2 (γ 1) ρ vr( v θ+2ωr) (γ 1) vr 0 r r r (B.4) 81

97 Appendix C Grid convergence study for wave propagation Two grid convergence studies were performed for this investigation. First, a wake convergence study was performed to ensure the blade wake profiles that were used as noise source drivers were being predicted independent of the grid resolution. Important metrics for wake convergence are number of grid points along the blade chord for capturing the evolution of the boundary layer and also the grid density in the wake region directly downstream of a blade. Second, a wave convergence study was performed to ensure that an adequate grid density was being used in each blade passage to sufficiently resolve the propagation of the relevant acoustic waves through each passage. The wave convergence study is more concerned with the overall grid density of a given passage than targeting particular geometry metrics. The results of the wave convergence study for this investigation are presented here. The grid resolution characteristics from this study for the R2, S2, and Strut blade rows are presented in Tables C.1, C.2, and C.3 for the Coarse, Medium, Fine, and Extra Fine grid levels that were tested in each row. All wave resolution calculations presented here were performed at the Cutback operating condition. This study was performed in serial fashion, starting with the R2 blade row and continuing with S2 and then the Strut blade rows once the study on the upstream row was completed. For a fixed perturbation at the inlet of a given blade row, a linearized Navier-Stokes calculation was run on four different computational grids, each defining the same geometry but increasing in grid density. The 82

98 results from each calculation were then compared to see how the acoustic response changed with increasing grid density. In order to evaluate a grid for its ability to adequately resolve the waves of interest, the real component of pressure is plotted at the exit plane for the individual modes generated from the interaction of the incoming disturbance with the blade geometry. This provides visual confirmation that a particular mode is sufficiently unaffected by the grid resolution. Additionally, integrated sound pressure levels for the given mode are plotted for each grid level as a more compact and quantitative comparison. The results for a given mode from each grid level are plotted on the same axis to demonstrate their convergence. The most important modes from the S1/R2 interaction were identified to be the (k = 1) and (k = 2) modes from the interaction of the 1 st S1 wake harmonic with the R2 blade row. Figures C-1(c) and C-1(d) show the results of these interactions with varying grid resolution. The R2-Medium grid level is seen to resolve these interactions very well, as both the mode profiles and their integrated SPL values change very little at the Fine and Extra Fine levels. The (k = 2) and (k = 1) interactions are also presented in Figures C-1(a) and C-1(b) but are much weaker, due to their high circumferential mode order. The integrated SPL values can be seen to be changing up to the highest grid resolution, but these are still db lower than the (k = 1) and (k = 2) interactions and increasing grid resolution is not expected to cause that level of variation in the results. The same reasoning was applied to the results for the interaction of the 2 nd S1 harmonic. Some integrated SPL values are seen to be changing 1-2 db from the Fine to Extra Fine grid levels, but these interactions were not found to be significant drivers of overall noise. Based on these results, the Fine grid level for R2 was used for the primary investigation since it captured the most important sources very well. The results showing the convergence of the R2/S2 interaction in the S2 blade row from the R2 1 st and 2 nd wake harmonics are shown in Figures C-3 and C-4 respectively. For all of the modes considered for these interactions, the wake profiles and SPL values are all well converged at the Fine grid level. The S2 blade row is also used to propagate the S1/R2 interaction downstream. The mode orders for the S1/R2 propagation are similar to those of the R2/S2 interaction, so the grid convergence results here are taken to be applicable for 83

99 Table C.1: R2 wave convergence grids Grid Level Circumferential points Chord points Spanwise points Total Points Coarse Million Medium Million Fine Million Extra Fine Million Table C.2: S2 wave convergence grids Grid Level Circumferential points Chord points Spanwise points Total Points Coarse Million Medium Million Fine Million Extra Fine Million the S1/R2 propagation as well. As such, the Fine grid level for S2 was used for the primary investigation. Finally, the Strut blade row was investigated by imposing the (1 R2)-(-1 S2) and (2 R2)- (-1 S2) modes as the incoming perturbations. The results of these calculations are shown in Figures C-5 and C-6. The Fine grid level is shown to provide adequate resolution for these propagations. One noticable difference can be seen in Figure C-6(d) where the mode profile changes from the Fine to Extra Fine levels. Looking at the SPL values for this mode, very little change can be seen between the Fine and Extra Fine levels so the difference in mode profiles is likely just a phase shift. Since the Strut is the last blade row in the investigation, the magnitude of the perturbation is most important and the phase shift should not impact the results. The Fine grid level for the Strut row was used for the primary investigation. Table C.3: Strut wave convergence grids Grid Level Circumferential points Chord points Spanwise points Total Points Coarse Million Medium Million Fine Million Extra Fine Million 84

100 (m = -186) (m = -138) (a) k = 2 (b) k = 1 (m = -42) (m = 6) (c) k = 1 (d) k = 2 Figure C-1: S1/R2 Interaction. R2 blade row. Response from S1 1st harmonic. 85

101 (m = -276) (m = -228) (a) k = 2 (b) k = 1 (m = -132) (m = -84) (c) k = 1 (d) k = 2 Figure C-2: S1/R2 Interaction. R2 blade row. Response from S1 2nd harmonic. 86

102 (m = -22) (m = 48) (a) k = 1 (b) k = 0 Figure C-3: R2/S2 Interaction. S2 blade row. Response from R2 1st harmonic. (m = -44) (m = 26) (a) k = 2 (b) k = 1 Figure C-4: R2/S2 Interaction. S2 blade row. Response from R2 2nd harmonic. 87

103 (m = -14) (m = -6) (a) k = 1 (b) k = 2 (m = 2) (m = 10) (c) k = 3 Figure C-5: R2/S2 Interaction. coming from S2 (k = 1). (d) k = 4 Response out of Strut blade row from R2 1 st harmonic 88

104 (m = 2) (m = 10) (a) k = 3 (b) k = 2 (m = 18) (m = 26) (c) k = 1 Figure C-6: R2/S2 Interaction. coming from S2 (k = 1). (d) k = 0 Response out of Strut blade row from R2 2 nd harmonic 89