A Computational Aeroacoustics Method Using Large ...

A Computational Aeroacoustics Method Using Large Eddy Simulation And Acoustic Analogy

Zur Erlangung des akademischen Grades eines

Doktors der Ingenieurwissenschaften an der Fakultät für Maschinenbau der Universität Karlsruhe (TH) genehmigte Dissertation

von M.Sc. Esra Sorgüven Karlsruhe

Tag der mündlichen Prüfung: 19.07.2004

Hauptreferent: Prof. Dr.-Ing. M. Gabi Korreferent: Prof. Dr.-Ing. W. Schröder

Preface

This thesis is created during my Ph.D. study from April 2001 until July 2004 at the Department of Fluid Machinery, Faculty of Mechanical Engineering, University of Karlsruhe. My work has been funded by the Karlsruher Universitätsgesellschaft. I would like to express special thanks to Prof. Dr.-Ing. Martin Gabi, who gave me the opportunity to join to his group and work on computational aeroacoustics. I also wish to thank Prof. Dr.-Ing. Wolfgang Schröder for his insightful comments that helped me to improve this dissertation. I am greatly indebted to Dr.-Ing. Franco Magagnato, who guided me through the Ph.D. process with valuable advises and inspiring discussions. My time in the Department of Fluid Machinery would not have been nearly as pleasant without my room-mates, colleagues and friends. I am grateful for their friendship, cooperation, help and interest in my work. I would like to thank them all very much. I especially thank to Kevin, Balazs, Iris, Ion, Jarek, Stefan and Claas. Last but not least, I am grateful to my family for their continuing support, love and encouragement.

Preface

viii

Zusammenfassung

Zusammenfassung Die physische und psychologische Belastung der Schallabstrahlung, insbesondere in Wohn- und Lebensbereich der Menschen, ist eines der größten Probleme der modernen Gesellschaft. Daher gehört die Geräuschverminderung zu den wichtigsten Aufgaben der Konstruktion in vielen technischen Bereichen. Die Optimierung des Geräuschverhaltens wurde traditionell mit empirischen oder analytischen Methoden durchgeführt, die Ungenauigkeiten beinhalteten, da hier detaillierte Strömungsdaten nicht berücksichtigt werden. In den letzten Jahren ersetzten immer öfter numerische Vorhersageverfahren die traditionellen Methoden, nicht zuletzt dank ihrer immer weiter verbesserten Genauigkeit. Diese Verfahren berechnen die Schallgenerierung und –ausbreitung basierend auf Strömungssimulationen. Obwohl die numerischen Verfahren vielversprechend sind, sind sie bisher nicht standardisiert – wie es beispielsweise bei Strömungsrechnungen der Fall ist -, sondern verfolgen je nach Komplexität der zu untersuchenden Geometrie und Strömung unterschiedliche Ansätze. Das Ziel dieser Arbeit ist es, ein ausgereiftes numerisches Verfahren für die Geräuschvorhersage im akustischen Fernfeld zu entwickeln. Das Verfahren basiert auf dem am Institut langjährig entwickelten Strömungslöser SPARC, welcher um ein akustisches Modul erweitert wurde. Die akustische Vorhersage wird in zwei Schritten durchgeführt. Als erstes wird das instationäre Strömungsfeld berechnet. Die Strömungsberechnung wird nach dem aktuellen Stand der Technik mit standardisierten CFD - Verfahren durchgeführt. Im Rahmen dieser Arbeit wurden kompressible Strömungsgleichungen mit instationären Reynolds Averaged Navier-Stokes Simulationen und Large Eddy Simulationen gelöst. Nach der Strömungssimulation extrahiert das akustische Modul die nötigen Daten aus dem Strömungslöser. Das akustische Modul berechnet die akustischen Quellen im Strömungsfeld basierend auf den Daten des Strömungslösers. Anschließend wird das ins Fernfeld ausstrahlte Signal als eine Überlagerung der Schallquellen im Strömungsgebiet berechnet. In jedem Zeitschritt wird dieses Zwei-Schritt-Verfahren wiederholt, so dass ein zeitabhängiges Signal im Fernfeld erreicht wird. Am Ende der Berechnung wird eine Fast Fourier Transformation durchgeführt, um das akustische Signal im Frequenzraum zu analysieren. Die Eignung des Verfahrens wird mit Hilfe dreier Anwendungen demonstriert. Der erste Anwendungsfall ist das aerodynamische Geräusch eines kreisförmigen Zylinders. Die Zylinderumströmung und die dabei entstehenden äolischen Töne gehören zu den meist ix

Zusammenfassung

untersuchten Fällen in der Aerodynamik und –akustik. Daher eignet sich dieser Fall sehr gut für die Validierung und auch um die Effekte der numerischen Methoden über die Geräuschvorhersage zu testen. Wie die Turbulenzmodellierung, Feinstrukturmodellierung, Längenkorrelation in quasi-2D Strömungen und Gitterverfeinerung die numerische Geräuschvorhersage beeinflusst, wurde umfangreich untersucht. Die Strömung hat eine Reynoldszahl von 48,000 und befindet sich damit im unterkritischen Bereich. Viele gut dokumentierte Experimente wurden zum Vergleich mit den numerischen Ergebnissen verwendet. Die Basis des zweiten Anwendungsfalles ist ein aktuelles Experiment, das den numerischen Vorhersageverfahren als Standarttestfall – sog. Benchmark - dienen soll. Ein NACA 0012 Profil im Nachlauf eines Zylinders wurde experimentell durch aerodynamische und akustische Messungen untersucht. Die von Kármán’sche Wirbelstraße, die im Nachlauf des Zylinders gebildet wird, imitiert den periodischen Windstoß bei rotierenden Propellerblättern, der an der Profilvorderkante ankommt. Diese Konfiguration erlaubt, in einem stationären Aufbau das Geräuschverhalten eines Propellerblattes zu untersuchen, ohne die Rotation berücksichtigen zu müssen. Die numerische Untersuchung dieses Falles, der eine Reynoldszahl von 480,000 hat, wurde mit Large Eddy Simulationen in einem numerischen Netz von etwa 7.2 Millionen Kontrollvolumina durchgeführt. Sowohl die aerodynamischen als auch die akustischen Ergebnisse der Simulationen stimmen mit den experimentellen Daten überein. Die letzte Anwendung ist ein praxis-relevantes Beispiel. Im Rahmen eines Projektes der KSB-Stiftung wurde ein generischer Propeller entworfen und dessen Geräuschverhalten experimentell und numerisch untersucht. Der Propeller besitzt zwei Blätter mit einem Anstellwinkel von 45°. Die Machzahl an der Blattspitze beträgt ca. 0.14, die Reynoldszahl ca. 130,000. Die akustischen Daten wurden an Beobachterpunkten im Abstand von 1,8 m vom Propeller aufgezeichnet. Die numerische Simulation wurde mit Large Eddy Simulationen in einem bewegten Rechennetz durchgeführt und das akustische Signal wurde in einem stationären Beobachterpunkt evaluiert. Bedauerlicherweise standen keine aerodynamischen Messungen zur Verfügung. Daher wurden ausschließlich die akustischen Daten von Experiment und Simulation verglichen. Die Übereinstimmung der akustischen Ergebnisse ist zufriedenstellend. Mit Hilfe dieser drei Anwendungsfälle wurde das numerische Geräuschvorhersageverfahren primär validiert und seine Anwendbarkeit demonstriert. Darüber hinaus wurden die numerischen Verfahren der Strömungssimulation auf ihre Eignung für die Geräuschvorhersage überprüft. Die wichtigste Aussage dieser Arbeit ist es, dass die Genauigkeit einer Fernfeld-Geräuschvorhersage von der Genauigkeit der Strömungssimulation bestimmt wird. Daher ist es ratsam, in der numerischen Akustik weitgehend Large Eddy Simulation für die Strömungsberechnung zu verwenden. Falls eine x

Zusammenfassung

Geräuschvorhersage basierend auf Reynolds Averaged Navier-Stokes Gleichungen zu machen ist, dann muss man mit einer Überschätzung der Schalldruckpegel rechnen und idealerweise diese Abweichung abschätzen und korrigieren. Die Feinstrukturmodellierung der Large Eddy Simulation spielt ebenfalls eine große Rolle bei der Genauigkeit der Strömungsdaten und daher auch der Schaldruckpegel. Alle Anwendungsbeispiele zeigen eine Übereinstimmung der experimentellen und numerischen Daten. Das numerische Vorhersageverfahren ist in der Lage das akustische Signal im Fernfeld genau zu berechnen. Darüber hinaus ist das entwickelte Verfahren schnell. Es benötigt nur etwa 2% mehr CPU-Zeit als die reine Strömungssimulation. Das Verfahren eignet sich aufgrund seiner hohen Genauigkeit und minimalen Kosten hinsichtlich Zeit und Rechenkapazität sowohl für die akademische als auch für die industrielle Anwendung.

xi

Table of Contents

Table of Contents Preface ..........................................................................................................................vii Zusammenfassung .........................................................................................................ix Table of Contents..........................................................................................................xii 0 Nomenclature ......................................................................................................xiv 1 Introduction ............................................................................................................1 1.1 Motivation ......................................................................................................1 1.2 Computational Aeroacoustics .........................................................................1 2 State of the Art........................................................................................................3 2.1 Milestones of the Acoustic Research ..............................................................3 2.2 Modern Studies in CAA .................................................................................5 3 Theory of the Sound .............................................................................................12 3.1 Basics............................................................................................................12 3.2 Wave Equation..............................................................................................14 3.3 Lighthill ........................................................................................................16 3.4 Solid Boundaries...........................................................................................18 3.5 Merits and Limits of the Ffowcs Williams and Hawkings Equation.............24 4 Numerical Code....................................................................................................27 4.1 Basics............................................................................................................27 4.2 Special Topics...............................................................................................30 4.2.1 Turbulence Modeling............................................................................31 4.2.2 Large Eddy Simulation .........................................................................32 4.2.3 Artificial Dissipation.............................................................................36 4.2.4 Preconditioning.....................................................................................37 4.2.5 Moving Grid .........................................................................................38 4.3 Acoustic Module...........................................................................................39 5 Test Case 1: Circular Cylinder..............................................................................46 5.1 Flow Properties.............................................................................................46 5.2 Configuration................................................................................................48 5.3 Length Correlation........................................................................................49 5.4 Numerical Details .........................................................................................51 5.5 Results ..........................................................................................................53 5.5.1 Turbulence Modeling: uRANS and LES ..............................................54 5.5.2 Computational Span Length..................................................................60 xii

Table of Contents

6

7

8 9

5.5.3 Grid Dependency ..................................................................................64 5.5.4 Sub Grid Scale Model...........................................................................68 5.5.5 Summary...............................................................................................72 Test Case 2: Airfoil in the Wake of a Circular Cylinder.......................................74 6.1 Flow Properties.............................................................................................74 6.2 Configuration................................................................................................74 6.3 Numerical Details .........................................................................................76 6.4 Results ..........................................................................................................77 Test Case 3: Propeller...........................................................................................87 7.1 Flow Properties.............................................................................................87 7.2 Configuration................................................................................................88 7.3 Numerical Details .........................................................................................92 7.4 Results ..........................................................................................................95 Conclusion and Outlook .....................................................................................102 References ..........................................................................................................104

xiii

Nomenclature

0

Nomenclature

Symbols A c cD cL d F f f h I k k l lc Lc lexp Lrecirc Ls Lt lz L∆ m M M Mr p pij Q R xiv

Amplitude of a sound wave Speed of sound Drag coefficient Lift coefficient Diameter Impulse source Frequency Aerodynamic force on a surface Local grid size Sound intensity Wave number Turbulent kinetic energy Length Chord length Correlation length Aspect ratio in the experimental set-up Recirculation length Spatial filter size Temporal filter size Computational span length Filter size Molecular weight Mach number Mach vector Magnitude of the Mach vector M in the direction of r Pressure Compressive stress tensor Acoustic source term Distance of the acoustic observer from the solid body

[Pa] [m/s]

[m] [kg/(m2s2)] [Hz] [N/m2] [m] [Watt/m2] [1/m] [m2/s2] [m] [m] [m] [m] [m] [m] [m] [m] [m] [kg]

[Pa] [N/m2] [m]

Nomenclature

r R12 Re S SPL St t T Tij tobs. u, v, w V V2 vs x x, y, z y y+

Distance vector between the source and the observer Correlation coefficient Reynolds number Mass source Sound pressure level Strouhal number Time Absolute temperature Lighthill’s tensor Advanced time Streamwise, transverse, spanwise convection velocity Volume of the acoustic sources Volume of the solid body Velocity of the solid body Position vector of the acoustic observer Streamwise, transverse, spanwise coordinates Position vector of the acoustic source Dimensionless wall distance

[m] [kg/(m3s)] [dB] [s] [K] [kg/(m·s2)] [s] [m/s] [m3] [m3] [m/s] [m] [m] [m]

Greek Symbols ρ ∆ δij φ η ηa λ µ θsep τ τ τe τij ω |ω|

Density Step size Dirac Delta function Arbitrary flow variable Coordinate system of the acoustic source Acoustic efficiency Wave length Viscosity Separation angle Turbulent time scale Emission time variable Emission time Shear stress Angular frequency Magnitude of the vorticity

[kg/m3]

[m] [m] [m2/s] [°] [s] [s] [s] [N/m2] [Hz] [1/s]

xv

Subscripts ac. aerodyn. D i,j,k mean obs. Q ref rms 0

Acoustic Aerodynamic Dipole Spatial directions in a Cartesian coordinate system Mean value Observer Quadrupole Reference value Root mean square Mean flow

Superscripts ´

Fluctuation

Other Symbols

( ∂t )

(?)

Time derivation ∂

(¯)

Mean value

xvi

Introduction

1

1.1

Introduction

Motivation

Noise reduction is among the most important design criteria in a variety of technical fields and a challenging task in mechanical engineering. The increasing awareness on the effects of noise on physiological and psychological health and the resulting strictness of the governmental regulations concerning the noise emission, enforce designers to focus on noise reduction more than ever. Especially the recent governmental regulations enforce noise reduction in aerospace engineering, climatization and fluid machinery. An adequate noise prediction is necessary in order to reduce noise emission satisfactorily and to prevent expensive after-design treatments. The goal of this study is to build a reliable and efficient tool to predict the far field noise of fluid machinery. Primary application area of this tool will be industrial problems. Hence, the noise prediction tool has to fulfil certain requirements. First of all, it has to provide a detailed insight of the noise generation mechanisms. A prerequisite of noise reduction is the knowledge on the characteristics of noise generation. Based on this knowledge, main noise sources can be determined and eliminated. Furthermore, the tool has to be precise and cheap in terms of the human and computing resources. Empirical or semi-empirical methods of noise reduction do not suffice for this purpose, since they fail to provide detailed information. Experiments are too costly in terms of financial and human resources. Furthermore they are time-consuming. Semi-empirical or analytical methods are mostly like black boxes, which fail to provide insight and involve numerous assumptions. Computational methods of noise prediction are valuable alternatives to these methods.

1.2

Computational Aeroacoustics

In recent years, computational aeroacoustics (CAA) is proven to be an efficient tool for noise prediction. Acoustic research made use of the fast development in computer 1

Computational Aeroacoustics

technology in one hand and of the advances in computational fluid dynamics on the other hand. During the last decade, CAA attracted attention from acoustic community both for the academic researches and for the industrial applications. CAA is, temporarily, mostly employed in academic research, in order to investigate the noise generation and propagation mechanisms in detail. The practical use in industrial applications is still rare, although especially in the far field acoustics, where reflection and refraction of acoustic waves are negligible, CAA provides good noise predictions of fluid machinery. CAA methods compute the sound field directly from the unsteady flow data. Hence, they provide insight to the relation between the aerodynamics of the flow and the corresponding sound emission. Thus, the sound field can be analyzed and dominating sound sources can be detected with the help of CAA methods. Another advantage of CAA is that an accurate solution can be obtained in a reasonable CPU-time, thanks to the advances in computational technology and in methods of computational fluid dynamics. CAA is a promising noise prediction method and fulfills the requirements for the use in industrial applications. Thus, a CAA-tool is developed for the far field noise prediction. The in-house developed CFD-code SPARC, which is enhanced with an acoustic module, is employed for far field noise prediction. The acoustic computation is performed in two steps. First, the unsteady flow field is computed. The unsteady flow simulation can be performed with the unsteady Reynolds Averaged Navier-Stokes equations, Large Eddy Simulation or Direct Numerical Simulation. The choice of the simulation model depends on the flow characteristics (like Re, Ma, etc.) and the available computing capacity. For CAA of a fluid machinery, where an appropriate accuracy within an affordable CPU-time is a must, LES is the optimum method. In the second step, the acoustic module extracts necessary information out of the unsteady CFD computation, performs Ffowcs Williams and Hawkings (FWH) integration and computes time dependent density and pressure fluctuations as well as the corresponding frequency spectra. In this thesis, first an overview of the acoustic research is given. Historical development of the acoustic research and contemporary methods in computational aeroacoustics are explained briefly. Then a description of the numerical code is given. The state of the art methods of CFD are only mentioned, whereas the advanced methods, which affect the accuracy of the aerodynamic or acoustic prediction, are explained in detail. Three chapters are devoted to the applications of the CAA-tool, where the acoustic predictions of three test cases are demonstrated. A circular cylinder, an airfoil in the wake of a cylinder and a generic propeller are simulated, and the far field noise predictions are performed. The numerical results are compared with recent experimental data. The numerical code is validated. Furthermore, the effect of numerical methods on the aerodynamic and acoustic predictions is investigated. The thesis ends with a discussion of the achieved results and an outlook of the future work. 2

State of the Art

2

2.1

State of the Art

Milestones of the Acoustic Research

Until 20th century, acoustics was considered rather as art than science. Acoustic research depended on common sense or trial and error. During 19th century, valuable contributions to acoustic research were made among others by Rayleigh, Stokes, Kundt, Helmholtz and Kirchhoff [1-3]. In early 20th century, architectural acoustics became an important subject. In 1920s, as radio broadcasting began, research in the field of communication acoustics made advances. Later, ultrasonic methods were evolved in medicine. With the begin of the Second World War, acoustic research gained importance in Europe and United States of America. Great laboratories were formed in order to research phenomena like acoustical detection of submerged submarines or speech communication in noisy environments such as aircraft and armored vehicles. Furthermore, psychological aspects of noise became a research subject. After the war, emphasis of acoustic research was shifted to architectural and industrial applications. During this period scientists like Beranek, Stephens, Lighthill, Goldstein and Witham made great contributions to acoustic research [48]. In 1950s, aircraft with jet engines, which were developed during the war, entered commercial service. Because of their effectiveness in transport, they replaced the traditional propeller-driven aircraft. In the early 1960s fleets of jet-engined airplanes were used for long distance travels. As a consequence, noise exposure near the airports became a great problem, due to the high noise level of jet engines. In the late 1960s, governments finally introduced regulations and noise certificates to prevent noise pollution. Since then, governmental regulations get stricter and more widely applied. Today, noise regulations are not limited to the aircraft industry, but also concern other industrial fields like automotive industry, fluid machinery, etc. During the three decades between 1940 and 1970, jet noise was the most important noise source because of the modern aircraft with jet engines. Aerodynamically generated noise attracted interest of many researchers [9-14]. Lighthill proposed a methodology for the 3

Milestones of the Acoustic Research

prediction of jet noise. He has suggested that in the neighborhood of jet engines, space can be divided into two regions. Noise sources are generated in the near vicinity of the jet, which has a relatively extent small in space, and then they are propagated linearly through the undisturbed atmosphere. Lighthill introduced this space splitting into the conversation equations of the fluid dynamics and derived the so-called Lighthill analogy. The major significance of Lighthill’s analogy is that the sound generated aerodynamically is computed based on aerodynamical flow data. Provided that the turbulent flow field is known, the corresponding acoustic field can be computed accurately. Since Lighthill’s analogy is based on the conservation laws and no simplifications are made during the derivation, the analogy is valid for all flows without limitation. Lighthill’s analogy depends on the idea, that aerodynamical noise can be computed from the basic conservation equations of fluid dynamics. Lighthill has rearranged the governing equations and eliminated a particular term by adding the equations, in order to achieve the inhomogeneous wave equation for the fluctuating flow density ρ'. Lighthill’s methodology can easily be modified by changing the eliminated term and the depending variable of the resulting wave equation. Phillips [15] applied the same methodology as Lighthill, but chose the depended flow variable of the inhomogeneous wave equation as ln(p). Similar derivations were performed for different flow variables by Lilly [16,17], Legendre [18] and Howe [14]. Möhring [19,20] has modified Howe’s model in order to remove the restriction to the lowest order in Mach number. This generalized self-adjoint form is valid for arbitrary non-uniform flows. These alternative equations reduce to the homogenous wave equation outside the flow, like Lighthill’s analogy. Although all formulations depend on the same philosophy, their success and advantages and disadvantages differ from each other [7]. Lighthill’s analogy served as a starting point for numerous scientific studies. His papers are among the most cited articles on aerodynamic noise. Depending on his analogy jet noise is predicted, analyzed and reduced. In early 70’s, with the introduction of new propulsion concepts, which use advanced swept multi-blade propellers; propeller noise became a dominating noise source [19]. Since Lighthill’s analogy was derived for jet noise, where the effect of solid boundaries is negligible, it is not applicable for propeller noise. In propellers, the effects of solid boundaries, i.e. thickness and loading noises are important and dominate the purely aerodynamic noise sources. The thickness and loading noises are monopole and dipole sources, respectively, and have a much more efficient propagation than the turbulent noise, which is quadrupole in nature. Curle introduced noise sources due to stationary solid boundaries into Lighthill’s equation [22]. Curle has enhanced the conversation equation of momentum with the surface sources applied by the fluid on the solid surface, and performed the same derivation as Lighthill did. The additional surface force became visible in the resulting inhomogeneous 4

State of the Art

wave equation as an additional dipole term, i.e. the loading noise. Later Ffowcs Williams and Hawkings [23] enhanced Curle’s equation for moving boundaries. Due to the volume displacement caused by the motion of the solid body, an additional monopole term appeared in the right hand side of the equation, the so-called thickness noise. The Ffowcs Williams and Hawkings equation is a general method for noise prediction, since all possible noise sources are present in the equation. Ffowcs Williams and Hawkings equation is extensively applied to noise prediction in rotating blades [24-26]. The original formulation of FWH equation, although valid for all flows, is not easily computable in supersonic flows. In order to avoid the singularities appearing at high Mach number flows, the equation was reformulated [27-29]. Far field noise can also be computed with the help of different formulations. One of the popular far field prediction methods is Kirchhoff’s formulation, where a surface integration is carried out instead of a volume integration [30,31]. As computer efficiency is increased rapidly in the last decades, computational methods become more feasible and accurate. Academic and industrial interests in computational methods grow both in fluid dynamics and acoustics. Acoustic community, which considers Lighthill’s analogy as the beginning of the first golden era of acoustic research, considers that the second golden era has jet begun thanks to the advanced computer technology. Modern acoustic research makes use of experimental and numerical methods, whereas the numerical methods of noise prediction attract increasing interest, due to their high efficiency.

2.2

Modern Studies in CAA

Modern CAA techniques can be separated into two steps; the determination of the unsteady flow data and the computation of the acoustic signal. Unsteady flow field can be predicted by a number of methods. Experiments, for example, can be employed for the determination of flow data, but they are limited in temporal and spatial resolution and indeed expensive. The huge costs of the experimental equipments and measurements on one hand and enhancements in computational technology on the other hand, made computational methods a valuable tool in noise prediction. Modern computers enable us to simulate unsteady turbulent flows with high accuracy within reasonable CPU-times. Hence today, the unsteady flow data, which is necessary for the computation of the source term in wave equations, is generally computed with the help of CFD techniques. The developments in computer technology have affected the scientific research in fluid dynamics extensively. Over the last two decades great accomplishments were made in computational fluid dynamics. As a result of these accomplishments, the numerical methods 5

Modern Studies in CAA

in CFD became well investigated and well developed. Modern CFD methods provide reliable flow simulations, and the errors made in modeling, discretizing and convergence are predictable. The most popular CFD models are Reynolds Averaged Navier-Stokes (RANS), unsteady RANS (uRANS), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). The most accurate model is DNS, since it solves the Navier-Stokes equations without any modeling approximations, and aims to resolve the whole range of time and length scales; from integral scales to Kolmogorov scales. Hence, the acoustic scales are directly resolved by a DNS. An additional computation of acoustic sources and wave propagation is redundant. DNS

LES

RANS

′ . + φac ′ φ = φ + φturb

′ . φ = φ + φturb

φ =φ Empiric methods ′ . φ = φ + φturb

Wave Propagation Acoustic Near Field ′ . = f (φ + φturb ′ .) ρ ac (e.g. LEE, APE, IEF...) Linear Wave Propagation Acoustic Far Field ′ . = f (φ + φturb ′ .) ρ ac (e.g. Lighthill’s analogy, FWH, Kirchhoff…)

Fig. 2.1 Overview of the modern CAA methods 6

State of the Art

LES is the second accurate method, which resolves the large and energetic scales of turbulence and models the small and dissipative scales. URANS equations are derived from the RANS equations by reducing the time limit of the Reynolds averaging, so that the time dependent flow data can be predicted. However, the time-dependent solutions of uRANS still involve the same deterministic modeling errors as in RANS solutions. Therefore, the unsteady flow field computed by uRANS is less accurate than a LES solution. But, since both LES and uRANS provide unsteady flow data, the solutions of both methods can be used to compute acoustic sources in the same manner. Since the acoustic scales are not resolved in these models, the acoustic sources have to be computed from the unsteady turbulent flow data. The last CFD method, RANS is based on averaging the flow variables over a large time range, so that only a steady state solution is achieved. Since, unsteady flow fluctuations cause acoustic sources, a time-dependent solution of the flow field is a must to compute the sound emission. Some empirical methods are evolved in order to estimate the turbulent fluctuations depending on the averaged flow field [32-34]. Accordingly, a RANS solution can be combined with a stochastic method in order to estimate unsteady turbulent fluctuations, which can than be used to compute acoustic sources of the flow field. Formally, all of the above methods can be employed in CAA. However, the accuracy of acoustic prediction depends on the accuracy of the aerodynamical prediction. DNS is the most accurate modeling, however it is only applied in academic low Reynolds number flows, because of the practically unaffordable discretization requirements (numerical effort of DNS ~ Re4). RANS method is the fastest approach of flow prediction, however the stochastic methods for the prediction of unsteady turbulent fluctuations involve empirical assumptions and hence inaccuracies. URANS fails in accuracy mainly due to the deterministic approach of its formulation. LES represents a compromise between the accuracy level and the computational effort. Because of its high accuracy and acceptable computational requirements, LES is the most appropriate method in CAA. Ones the unsteady turbulent flow data is available, acoustic sources can be computed. This is handled differently in acoustic near field and in acoustic far field. Acoustic far field is defined as a region in the sound field where the distance to the source is much larger than the linear dimensions of the source itself. At distances much larger than the wavelength of the acoustic signal, the relation between the sound pressure, velocity and sound intensity become simple ones. Here, the fluid velocity is in phase with the pressure, so that all the energy density is radiant, moving outward with the speed of sound. In far field the acoustic signals propagate directly to the acoustic observer with the speed of sound. The acoustic signals do not affect each other’s amplitude or direction; there is no shielding effect of outer sources to the inner sources. Wave propagation is linear. The amplitude of the emitted signal 7

Modern Studies in CAA

is not modified due to the propagation; hence the signal has the same amplitude as it achieves the observer. The far field assumption simplifies computation of wave propagation extensively. The sound signal at an observer point in the far field can be computed as a superposition of emitted acoustic sources in the turbulent flow region, where the acoustic sources in the flow region can be computed by an acoustic analogy. In contrast, in the acoustic near field the distance to the acoustic source is small compared to the wavelength. Usually within about two wavelengths from a sound source is considered as near field. In the near field, there exists a large velocity component out of phase with the pressure. Relationships between the acoustic variables are not as simple as in far field. In the acoustic near field, wave propagation has to be solved in addition to wave generation. In the literature, a great number of models exist to solve the wave propagation. Most of these models are based on linearizing the governing fluid dynamics equations. The basic conservation equations of fluid dynamics (Euler or Navier-Stokes equations) describe also the wave propagation. Usually Euler equations are preferred in the computation of wave propagation, since the effect of viscosity in wave propagation is negligibly small. The governing equations are linearized by decomposing the actual flow variables φ into a mean part φ0, and a fluctuating part φ’. (2.1) φ = φ0 + φ ′ There are numerous models for the decomposition. Some important ones are listed below. • Fluid – Acoustic Decomposition: Mean flow and acoustic fluctuations are decomposed [35-38]. • Acoustic Perturbation Equations (APE): Decomposition is based on a filtering of the non-linear and viscous terms of the Navier-Stokes and continuity equations, respectively, in the Fourier/Laplace domain [39,40]. • Acoustic – Viscous Splitting: Compressible flow data is decomposed into an incompressible part and a compressible fluctuations part [41,42]. • Expansion about Incompressible Formulation (EIF): A modification of the acousticviscous splitting method [43,44]. • Galbrun’s equation: The mean flow motion and the molecular motion are decomposed in a Langrangian approach [45]. The mean part of the flow φ0 is known from the CFD computation, and the fluctuating part is to be computed with the linearized equations. After the linearization, the equations are rearranged in order to collect the known terms in the right hand side. If the Euler equations are employed, the Linearized Euler Equations (LEE) of the following form are achieved. 8

State of the Art

∂u ′j ∂ρ ′ ∂ρ ′ + u j0 + ρ0 =S ∂t ∂x j ∂x j  ∂u ′ ∂u ′  ∂p′ ρ0  i + u j0 i  + = Fi  ∂t ∂x j  ∂x j 

(2.2)

The source terms S and Fi on the right hand side of the linearized equations are computed from the CFD data. The term S represents any sources of mass that may be present in the computational domain, such as vibrating solid surfaces. The forcing term Fi contains the viscous forces, which usually have a negligible effect on sound propagation. The models differ from each other in the formulation of these source terms. The advantages and disadvantages of the models, i.e. of the formulations of source terms, are a subject of current research. An important problem arises by the numerical computation of the linearized equations. Numerical methods used in computational acoustics depend on CFD methods, which were developed in the last 20 years. However, the resolved scales in fluid dynamics differ from the acoustic scales. Acoustic waves inherit less amplitudes and higher frequencies than the pressure waves, which are usually resolved in fluid dynamics (Fig. 2.2). Hence, the usual discretization methods of CFD are too dissipative for the solution of acoustic waves. Principally different numerical algorithms are required for computational acoustics. The numerical errors causing dissipation and dispersion have to be negligible in the computation of wave propagation.

p'/pmean

10

0

10

-1

10

-2

10-3 10

-4

10

-5

Computational Fluid Dynamics

Computational Aeroacoustics

Classical Theoretical Aeroacoustics

-6

10 -1 0 1 2 3 4 5 10 10 10 10 10 10 10 f [Hz]

Fig. 2.2 Levels and frequencies in computational aeroacoustics [46] 9

Modern Studies in CAA

In order to overcome this problem, the numerical computation in the near field is performed in two steps. First, the turbulent flow field is simulated with common CFD methods. Computational grids and discretization methods, which are optimized for the selected CFD-model, are employed to achieve the necessary accuracy in the aerodynamical field. In the second step, the linearized equations for the wave propagation are solved in a different grid and with different discretization methods, which are optimized for the solution of the wave equation. The computational grid for the acoustic computation is prepared to resolve the acoustic wavelengths of concern. Hence, it is usually coarser than the CFD-grid, and grid size is the same in each spatial direction, due to the nature of wave propagation. An orthogonal grid is preferred to reduce the numerical discretization errors. In order to compute wave propagation accurately, special discretization algorithms are developed. These algorithms have to prevent any possible degradation of physical characteristics of the wave through numerical dissipation or dispersion after long distances and long times. However they have to model dissipation near shocks. Tam’s DispersionRelation- Preserving Scheme is one of the mostly employed methods in this area [47]. Another important point in the computation of wave propagation is boundary conditions, which have to be strictly non-reflecting [48,49]. Linearized Euler Equations are today the mostly employed method in near field acoustics. However, there are alternative methods, which are investigated and developed by numerous research groups. Some examples of these alternative methods are Lattice Gas Method, Boundary Element Method and Galerkin Method. Computation of the wave propagation is costly in terms of computer power and CPUtime. Therefore generally, propagation is resolved only in the near field, where it is important, and then a far field acoustic analogy is used. For example, an appropriate way to predict the sound field of an airfoil is to employ a near field acoustic method (e.g. Linearized Euler Equations) in the near vicinity of the airfoil, where noise generation and propagation are resolved; and to employ a far field acoustic analogy (e.g. FWH) in the acoustic far field, where the relation between the acoustic variables is simple. In this work, we define the combination of acoustic methods in order to increase accuracy and speed of computation, as hybrid methods.

10

State of the Art

Acoustic Sources (e.g. DNS) Acoustic Near Field (e.g. LEE) Acoustic Observer

Acoustic Far Field (e.g. far field approximation of FWH)

Fig. 2.3 Hybrid methods

11

Basics

3

3.1

Theory of the Sound

Basics

Sound is due to mechanical vibrations in fluids. Rapid, small-scale pressure fluctuations overlying the atmospheric pressure are called sound pressure, p(t). These pressure fluctuations can be caused due to various mechanisms. Vibration of a solid body or turbulence in a flow can act as a sound source. After the emission, sound propagates in a waveform through the medium. Sound waves can be reflected, partially absorbed or attenuated. The sound waves, which are parallel to the direction of propagation, are called longitudinal waves. Whereas, the perpendicular ones are called lateral waves. Sound waves in fluids are waves of longitudinal type. A sound wave can have an arbitrary shape. However, any wave function can be defined as a superposition of harmonic signals with different amplitudes and frequencies. A sound wave with a distinct amplitude A and a distinct frequency f, is called a pure tone and represented as follows:

p (t ) = A ⋅ cos(2πf ⋅ t )

(3.1)

Human ear can hear sound waves within a frequency range of about 16-16,000 Hz. Pressure fluctuations with lower frequencies are called infrasound, and with higher frequencies ultrasound. Sound due to normal speech has frequencies in the vicinity of 1,000 Hz. Sensibility of human ear to acoustic signal around that frequency is the highest. A sound wave moves through a medium with the sound speed c, which is defined as: λ c= (3.2) T The wavelength λ is the distance between points of corresponding phase of two 12

Theory of the Sound

consecutive cycles of a wave and the period T is the elapsing time between two consecutive waves. Some common quantities used in acoustics are the angular frequency ω and the wave number k: ω = 2πf (3.3) k = 2π

c=

λ

λ ω = fλ = T k

(3.4) (3.5)

Normally, the propagation of sound waves in fluids is independent of the type or the origin of the sound signal. Sound speed is a property of the fluid and depends on the compressibility, K, and density, ρ, of the medium. Accordingly, the speed of sound in the dry air at 20°C is c.a. 343 m/s. Human ear does not respond linearly to the amplitude of the sound pressure. The correspondence is rather logarithmic. Therefore, a logarithmic value, the sound pressure level SPL is defined:  p2  SPL = 10 ⋅ log10  2  (3.6) p   ref  whereas the reference pressure pref is mostly set as 2·10-5 Pa in air. The unit of the SPL is dB. SPL of the threshold of hearing is 0 dB and SPL of the threshold of pain is 140 dB. A loud conversation has a SPL of about 60 dB at the ear position. Note that SPL is a property of the field position. It does not only depend on the sound power of the source but also on the distance from the source, the directivity of the propagation and the properties of the medium. Time signal of a sound pressure is not sufficient to characterize the source of the sound. Since the amplitude and the frequency of the signal are appointing the characteristics of the sound, the representation of the signal in the frequency domain is more insightful. Frequency analysis helps to distinguish the tonal components and the broadband swishing of the sound signal. Fourier Transformation is employed to transform the acoustic signal in the time domain into a function in the frequency domain. Modern acoustic measurement instruments include frequency analyzers, which involve electronic filters to determine the SPL of the acoustic signal within given frequency bands. The frequency analyzers can be classified as constant bandwidth analyzers and octave analyzers. In the first group, each frequency band has the same width, ∆f. Narrow frequency bands allow to determine the shape of the signal in a great detail. Harmonic components of the sound can be distinguished. The second group, octave analyzers are frequently employed in practice. Here, the ratio of the upper and lower frequencies, which bound the band, is fixed. The ratio 13

Wave Equation

is defined as n 2 for an 1 n -octave band. Some of the common octave bands are the 1/1octave, 1/3-octave and 1/12-octave bands. The octave spectra are preliminary used in the presentation of broadband signals with no prevalent frequencies. Frequency spectra with higher resolutions can be transformed into a spectrum with a lower resolution. However, the reverse is not possible, since the information content is not sufficient. For the transformation, all data points, SPLi, which lie within a frequency band of the coarser resolved spectrum, are added up.

 n SPL i  SPLsum = 10 ⋅ log10  ∑ 10 10    i =1

(3.7)

The value SPLsum is the sound pressure level of the corresponding frequency band in the coarse spectrum, and n is the total number of the sound pressure levels within the coarse frequency band. Note that, this transformation leads to an increase in the SPL. An SPLcurve in the 1/3-octave spectrum has higher values in dB than the SPL-curve in the 1/12octave spectrum of the same acoustic signal. The difference is minimal at the pure tones, but remarkable in the broadband distribution.

3.2

Wave Equation

The wave equation has many physical applications from sound waves in air to magnetic waves in the Sun’s atmosphere. Because of the various application fields, there exist different approaches for the derivation of the wave equation, for example by considering a stretched elastic string. Here, since the sound waves in fluids are of concern, the basic conservation laws of the fluid dynamics are used as the starting point. ∂ρ ∂ + (ρui ) = 0 ∂t ∂xi

(

)

∂ (ρu i ) + ∂ ρui u j − τ ij + ∂p = Fi ∂t ∂xi ∂x j

Fi represents the external forces and the shear stress τij is defined as:

14

(3.8)

(3.9)

Theory of the Sound

 ∂u ∂u j 2 ∂u k  − δ τ ij = µ  i +  ∂x j ∂xi 3 ij ∂x k   

(3.10)

The momentum equation can be simplified by assuming an inviscid fluid ( µ = 0 ), a uniform flow ( ∂ ρu i u j ∂xi = 0 ) and the absence of the external forces ( Fi = 0 ).

(

)

∂ (ρui ) + ∂p = 0 ∂t ∂x j

(3.11)

By subtracting the divergence of the impulse equation (3.9) from the time derivative of the continuity equation (3.8), the term ∂ 2 (ρu i ) ∂t∂xi is eliminated. ∂2ρ

−

∂t 2

∂2 p =0 ∂xi ∂x j

(3.12)

The variables of an infinite, homogenous fluid can be linearized as ρ = ρ 0 + ρ ′ and p = p 0 + p ′ . Here, the subscript 0 labels the properties of the steady background flow, and ´ labels the acoustic disturbances, which overlie the background flow. Substituting this linearization and the thermodynamic relationship ∂p = c 02 for isentropic ∂ρ flows, the homogenous wave equation is achieved. ∂2ρ′ ∂t

2

− c02

∂2ρ′ ∂xi2

=0

(3.13)

Propagation of a sound wave in a homogenous, motionless fluid is described with the homogenous wave equation (3.13). Note that, the homogenous wave equation describes only the propagation of the sound waves; sound sources are not present in this equation. In the presence of a sound source, a source term Q is added in the right hand side of the equation (3.14). The inhomogeneous wave equation with a source term of order 2n is written as follows. ∂2ρ′ ∂t 2

− c02

∂2ρ′ ∂xi2

=

∂ n Qij ... ∂xi ∂x j ...

(3.14)

15

Lighthill

Green’s formula is an appropriate method to solve this type of equations [7]. The solution of the generalized wave equation with the Green’s formula is: r 4πc02 ρ ′(x , t ) =

r ∂n dy ∞ Q ... ij ∫ ∂xi ∂x j ... − ∞ r

(3.15)

where, r is the distance from the source. A sound source emerges when kinetic energy of the fluid is converted into acoustic energy. Such a conversion can take place by fluctuations of mass, momentum or strength. These are the elementary solutions of the wave equation, with sources of order 20, 21 and 22. Mass fluctuations in the fluid generate sound sources of 1st order, i.e. the monopole sources. Examples for this type of sound sources are a pulsating sphere, a diaphragm of a loudspeaker or volume displacement by a propeller blade. The 2nd order source term is called a dipole, and is generated due to a fluctuating momentum. This type of sound sources appear on the solid boundaries, due to vibration of solid bodies or varying aerodynamical force fields on the surfaces. The last elementary solution of the wave equation is the 4th order source term, the quadrupole. Quadrupoles are strictly aerodynamical sound sources, generated due to the fluctuations in the flow field, e.g. due to turbulence. One important fact is that each type of the sound sources is less efficient than the former one. The quadrupole source is the least efficient type of the sources. This statement gains more importance at low frequencies, or in other words at high wavelengths (Stokes effect [1]).

3.3

Lighthill

Lighthill’s study on sound generated aerodynamically is a milestone in the aeroacoustics. He has introduced a methodology to compute the radiated sound depending on aerodynamical quantities. His work considered primarily the jet noise, without the effect of solid bodies effecting the generation or radiation of sound. Analog to the derivation of the wave equation, the starting point is the conservation laws of the fluid dynamics. The conservation of mass (3.8) and momentum (3.9) equations are used here in their original forms, i.e. without the assumptions of negligible viscosity and uniform flow. So, the ∂ neglected terms ρu i u j − τ ij in the derivation of the homogenous wave equation are ∂xi taken back into the account. Since the flow is unbounded, no external forces are present, i.e. Fi=0. By introducing the identity:

(

16

)

Theory of the Sound

c02∇ 2 ρ = c02

( )

∂ 2 ρδ ij

(3.16)

∂xi ∂x j

the inhomogeneous wave equation becomes: ∂2ρ′ ∂t 2

− c02

∂2ρ′ ∂xi2

=

∂ 2Tij

(3.17)

∂xi ∂x j

where, Tij is defined as:

(

)

Tij = ρu i u j + p ′ − c02 ρ ′ δ ij − τ ij

(3.18)

The equation (3.17) is known as the Lighthill’s equation and Tij as the Lighthill’s tensor [9,10]. Note that, Lighthill’s equation is the direct result of the conservation laws. No simplification is made during the derivation of the Lighthill’s tensor, hence the equations (3.17) and (3.18) are valid for any arbitrary flow without external forces. With the help of the Lighthill’s equation, the sound field is deduced from the aerodynamic flow field. A back-reaction of the sound on the flow field is precluded. Fortunately, both theory and experiments show that the sound produced is so weak compared to the motions producing it, that no significant back-reaction can be expected, unless there is a resonator present to amplify the sound. Lighthill’s equation does not only cover the sound sources, but also the propagation of sound waves due to convection with the flow ρu i u j , the propagation due to conduction of

(

heat (

p ′ − c02 ρ ′

)

( )

) and the dissipation due to viscous effects τ ij .

The dissipation of the acoustic energy due to viscosity is negligibly small in nearly all flows in practice. The term ( p ′ − c02 ρ ′ ) represents the deviation of the flow from the isentropic condition. This term gains importance where fast chemical reactions occur, such as combustion. Otherwise it is small and becomes zero for isentropic flows. Hence, in most of the practical applications, the convection due to the flow is the dominating part in Lighthill’s tensor and Tij can be approximated to ρuiuj. This approximation involves an error in the order of M2, since the pressure variations are proportional to the density fluctuations times the square of the sound speed ( p ′ ≈ ρ ′ ⋅ co2 ≈ M 2 ). Especially for low Mach number flows, Lighthill’s tensor is in the order of the square of the mean flow velocity (Tij ~ u2). The radiation field of the aerodynamical sound sources is however, proportional to the multiplication of the strength of the Lighthill’s tensor and the square of the frequency of the acoustic signal. Theory and experiments show that the frequency of aerodynamic sound is proportional to the velocity of the mean flow. Hence, the radiation of purely aerodynamic 17

Solid Boundaries

sound waves is in the order of the 8th power of the mean flow, u8 [10].

3.4

Solid Boundaries

Lighthill’s acoustic analogy neglects the influence of the solid boundaries. Solid boundaries affect the aerodynamical sound field in two ways: • The quadrupole sources generated aerodynamically are reflected or diffracted due to the solid boundaries. • Additional sound sources are generated on the solid boundaries. These dipole sources are either due to the vibrations of the solid bodies or due to aerodynamical forces applied from the flow onto the boundaries. Curle has enhanced Lighthill’s equation for stationary solid boundaries [22]. Later, Ffowcs Williams and Hawkings have enhanced the formula further for moving solid boundaries [23]. In order to understand the effect of the moving solid boundaries on the acoustic field, let us consider an impermeable solid body positioned in a fluid flow enclosed by the surface Σ with the volume V (Fig. 3.1). r Σ(x, t ) f >0

r r vS ( x , t )

r VS (x , t )

r S (x , t )

r V (x , t )

Acoustic Observer

Fig. 3.1 Acoustic analogy with moving solid bodies

18

Theory of the Sound

r

The solid body has the volume Vs and its surface S is defined with a function f (x , t ) . r The surface is smooth, without discontinuities. It can move arbitrarily with the velocity v s , and change its shape and orientation. < 0 in the solid body r  f (x , t )= 0 on the boundary > 0 outside the solid body 

(3.19)

Since the solid boundary is impermeable and the fluid is viscose, the fluid velocity on r r r r the boundary is equal to the velocity of the boundary, i.e. if f = 0 then u (x , t ) = v s (x , t ) r r r r r r and u (x , t ) ⋅ n (x , t ) = u n (x , t ) = 0 , where n is the normal vector of the solid boundary directing outside. For this case, continuity and momentum equations are written as follows. ∂ρ ∂ + (ρu i ) = ρ 0 v si δ ( f ) ∂f ∂t ∂xi ∂xi

(

(3.20)

)

∂ (ρui ) + ∂ ρui u j − τ ij + ∂p = pij δ ( f ) ∂f ∂t ∂xi ∂x j ∂xi

(3.21)

where the compressive stress tensor is defined as pij = p ′δ ij − τ ij and dirac delta function is δ . Applying the same procedure as in Lighthill’s analogy, Ffowcs Williams and Hawkings equation is achieved. ∂2ρ′ ∂t

2

− c 02

∂2ρ′ ∂xi2

=

∂ 2Tij ∂xi ∂x j

−

∂ ∂xi

  p δ ( f ) ∂f  ij ∂x j 

 ∂   +  ρ v δ ( f ) ∂f  ∂t  0 i ∂xi  

   

(3.22)

In this equation two new terms appear in addition to the Lighthill’s equation, which are due to the presence and motion of solid bodies. The second term in the right hand side is called the loading noise. It represents the effects of the aerodynamical forces acting on the solid surface. Hence, it is a dipole source. The last term is called the thickness noise. The motion of the solid body in the fluid is the origin of this term. The volume displacement acts as a monopole sound source.

19

Solid Boundaries

ηk

ηk ηi

ηj dη

r v (η , t )

ηi

ηj dη r r (η , t )

r y (η , t )

r r (η , t + ∆t )

r y (η , t + ∆t )

yk r r x (η , t ) = x (η , t + ∆t )

yj

r v (η , t + ∆t )

Acoustic observer (stationary with respect to the main frame of reference)

yi

Fig. 3.2 Coordinate systems for the acoustic source and the observer Until here, we have used a frame of reference, which moves with the sound sources. However, in numerous practical cases, the sound sources move with respect to the acoustic observer, like a moving aircraft or a propeller blade. Therefore, it is more convenient in practice to choose a frame of reference, which is stationary with respect to the acoustic observer. Let us define a coordinate system η, which is stationary with respect to the sound source and moving with respect to the observer with the velocity of the sound source, i.e. the r velocity of the solid boundary v s . r r The sound travels from the sound source at y to the observer at x with the speed of r r x−y sound c0 in a time of . Within this time the origin of the coordinate system η travels c0 r r r r r r x−y a distance of v ⋅ = M x − y . If we assume that both coordinate systems coincide at a c0

20

Theory of the Sound

time t, then

r r r r r η = y+M x−y

(3.23)

∂η = ∂y + (1 − M r )

(3.24)

r r r v where, Mr is the magnitude of the component of M in the direction of r = x − y . The acoustic signal, which is emitted at the emission time τe is received by the acoustic observer at tobs.. r r x−y τ e = tobs. − (3.25) c0

If the new coordinate system is introduced into the Ffowcs Williams and Hawkings equation and Green’s approach is employed to solve it, the following equation is achieved. Here, the brackets imply that the value inside belongs to the emission time τe. r 4πc02 ρ ′(x , t ) = −

∂ ∂xi

∂ − ∂xi −



∂2 ∂xi ∂x j

∫ S

∫  r 1 − M Tij

V

 pij n j   r 1 − M r

 ρ0v&si   r 1− Mr VS 

∫

∂2 ∂xi ∂x j

r

  dη 

  dS (η )    dη 

(3.26)

 ρ0vsi vsj   dη r   VS

∫  r 1 − M

Due to the far field approximation and rearrangements of the equation, the thickness noise in the equation (3.22) is split here into two terms, one containing the acceleration of the solid body, the other containing the surface velocity. Since in practical applications, the acoustic signal at a large distance from the sound source is of interest, the far field approximation of the equation (3.26) gains importance. In order to carry out the far field approximation, let us first consider the spatial derivations of r an arbitrary function φ (x , τ ) , where τ represents the emission time variable: 21

Solid Boundaries

∂φ  ∂φ  ∂φ  ∂τ    +    =   ∂xi  ∂xi τ  ∂τ  x  ∂xi  e

(3.27)

 ∂φ   ∂φ  ri =   −  ∂ − Mr ) ∂ x c r ( τ 1   i 0   x r If φ (x , τ ) =

ϕ (τ ) , then r 1− M r

 ϕr ri ∂φ ∂ ϕ = − 3i − ∂xi c 0 r (1 − M r ) ∂τ r 1 − M r  r

  

(3.28)

 ri ∂ ϕ  −2 = −  +O r 2 1 ∂ − M τ ( 1 ) − c r M r  r   0

( )

accordingly, second order spatial derivations can be evaluated as follows.  ri r j 1 ϕ  ∂ 2φ ∂ ∂ −2 = 2 3 +O r ( ) 1 1 τ τ xi ∂x j ∂ − M ∂ − M r r    c0 r (1 − M r )

( )

(3.29)

If the terms falling off by 1/r2 are neglected for large values of the distance r, and the function ϕ is replaced with the terms in the equation (3.26) ( ϕ = Tij , ϕ = f i , ϕ = ρ 0 v&i and ϕ = ρ 0 vi v j ), then the far field approximation of the Ffowcs Williams and Hawkings Equation is achieved.

22

Theory of the Sound

r ρ ′(x , t ) = +

1 4πc04 1 4πc03

1 + 4πc03 +

1 4πc04



∂

∂

∫  r (1 − M ) ∂τ (1 − M ) ∂τ 1 − M ri r j

1

Tij

3

r

r

V

∫ S

 ∂ r fi  2 i ∂ − Mr τ 1 ( ) − r M 1 r 

  dS (η ) 

 rj ∂ ρ0v&si  2  r (1 − M r ) ∂τ 1 − M r VS 

∫

r

  dη 

  dη 

(3.30)

 ri rj ∂ ∂ ρ0vsi vsj  1  3  dη  r (1 − M r ) ∂τ (1 − M r ) ∂τ 1 − M r  VS

∫

The first term of the Ffowcs Williams and Hawkings equation involves Lighthill’s tensor and represents the purely aerodynamical sound sources, which are mainly due to the turbulent fluctuations in the flow. This term is integrated over the whole fluid region. The second term in the right hand side represents the effect of the dipole sources at the solid surface, which are due to fluctuating aerodynamical forces on the boundary. The fluctuating aerodynamical force fi consists of the fluctuations of pressure and viscose shear stress, f i = pij n j = pδ ij − τ ij . Lighthill has shown, that the sound intensity of a quadrupole source is in the order of the 8th power of the mean velocity. Similar reasoning shows that the intensity of a dipole source is proportional to the 6th power of the mean flow velocity. These considerations result in the following proportionality of the sound intensities of the quadrupole and dipole sources, IQ and ID, respectively. IQ ID

2

u  ~   = M 2  c0 

(3.31)

At low Mach numbers the contribution of the dipole sources into the far field acoustic signal is larger than the contribution of the quadrupole sources. The radiation of the quadrupole sources is diminishing with respect to the radiation of the dipole sources. The acoustic efficiency ηa is defined as the ratio of the acoustic power output to the total energy supplied to the flow. Accordingly, the acoustic efficiency of the dipole sources is ηaD ~ M3, whereas the efficiency of the quadrupole sources is ηaQ ~ M5. The last two terms represent the effect of the volume displacement. The moving solid body initiates dipole sources due to acceleration, ρ 0 v&si , and quadrupole sources due to 23

Merits and Limits of the Ffowcs Williams and Hawkings Equation

surface velocity, ρ 0 v si v sj . These sources are integrated over the volume of the solid body, which is equal to the volume of the displaced fluid due to the motion of the solid object. If the solid body is at rest, i.e. vs=0, then the FWH equation is reduced to Curle’s equation, which is an enhancement to Lighthill’s equation for stationary surfaces.

r ρ ′(x , t ) =

1



ri r j

∂

1

 ri fi ∂ + 3 ∫ 2 4πc0 S  r (1 − M r ) ∂τ 1 − M r 1

3.5

∂

∫  r 3 (1 − M ) ∂τ (1 − M ) ∂τ  r r

4πc04 V

  dη 1 − M r  Tij

  dS (η ) 

(3.32)

Merits and Limits of the Ffowcs Williams and Hawkings Equation

In the frame of this work the far field approximation of the FWH equation (3.30) is employed for the aerodynamic noise prediction. In the following important characteristics of this method, its applicability, merits and limits will be discussed. The accuracy of the predicted sound pressure level with FWH equation depends on the accuracy of the aerodynamical data, which are used in the computation of the acoustic sources. The sound sources in FWH equation (3.26) are a direct result of the governing conservation equations. Since no simplification is made in the derivation, a perfectly accurate and detailed knowledge about the unsteady flow data would result in an exact computation of acoustic sources. Unfortunately, such a detailed knowledge on the flow field is nearly impossible to achieve neither numerically nor experimentally. The numerical errors in a flow simulation, which arise due to modeling, discretization and convergence, decrease the accuracy of the far field noise prediction. An important property of the far field FWH equation is that a direct propagation of the sound waves from the sound source to the acoustic observer is assumed. The sound sources in the FWH equation are computed from the unsteady flow field. The computed sound sources are assumed to propagate directly to the acoustic observer along the distance vector r r with the sound velocity c0. This approximation is a logical consequence of Lighthill’s considerations, where the sound waves are supposed to be generated in a turbulent flow and propagate undisturbed through a steady flow. For FWH equation is similar to Lighthill’s equation in this aspect, its applicability is limited in the same manner. The far field FWH 24

Theory of the Sound

equation (Eqn. (3.30)) can only be applied to cases, where no obstacles exist between the sound sources and the observer. An obstacle would reflect, diffract or absorb the sound waves, hence change the path of the sound wave. However, according to the equation (3.30), sound waves have to travel through the shortest way to the acoustic observer, i.e. through r r. The term 1 − M r , also called the Doppler factor, is present in the denominator of each term in the FWH equation. For subsonic flows this term is in the range of (0;1]. For supersonic flows, it tends to zero when Mr approaches to 1 and becomes zero where θ = cos −1 1 . Small values of the Doppler factor increase the importance of the terms in M the FWH equation extensively, and if the Doppler factor becomes zero, then singularities occur. Under such circumstances the FWH equation does not provide a proper description to the sound field, although it is still valid. In order to avoid difficulties at transonic or supersonic flows, a better description of the sound field has to be employed. Since in the frame of this work only low Mach number flows (M = 0.2) are handled, the FWH equation in the form represented in (3.30) is used. Aeroacoustic methods for supersonic flows will not be explained here; the works of Brentner, Ianniello, and Farassat [27-29] provide insight on this subject.

( )

In the light of these considerations, some examples, where the far field FWH equation can be employed for noise prediction are: • Aeolian sound of wires or cylinders • Airframe noise of an aircraft or automobile • Noise of propellers, fans or helicopter blades (in subsonic regime) On the other hand the use of far field FWH equation is inappropriate for the noise prediction of internal flows, like in a duct, in a pump or inside an automobile. In such cases, the employed noise prediction method has to resolve the propagation of the sound waves as well as their generation. Near field acoustic methods, like Linearized Euler Equations are appropriate to employ in such cases. One of the most important advantages of FWH equation is that all terms in the equation are physically meaningful. The quadrupole sources account for the nonlinear effects, such as turbulence in the flow. Loading noise represents the acoustic contribution of the force acting on the fluid in the presence of a solid surface. Thickness noise appears due to the motion of the solid body and can be determined completely by the geometry and kinematics of the body. Hence, each noise source appearing in the FWH equation is related to an aerodynamical process. Each term is independent and can be computed separately. This fact allows comparing the contribution of each noise source to the total acoustic signal detected by an observer in the far field. FWH formulation provides a detailed insight on the relation between the aerodynamics of a flow and its acoustic response. Two uses can be made from 25

Merits and Limits of the Ffowcs Williams and Hawkings Equation

this knowledge. Firstly, the dominant noise sources can be identified by comparing the values of the terms and this knowledge can be used for the purpose of noise reduction. Secondly, some of the terms, which are known to be diminishing in a particular application for example quadrupole sources in a low Mach number flow - may be neglected, in order to reduce the required computational power and CPU-time. The first term of the far field FWH equation is the volume integral of the turbulent acoustic sources. This integral can easily be rewritten as a surface integral. If the integration surface is chosen as the boundaries of the computational domain, both integrations are equivalent. In numerous implementations of FWH equation, the surface integral formulation is preferred in order to spare computational power. Furthermore, the integration surface is usually chosen nearer to the solid boundaries instead of the boundaries of the computational domain. This application is justified with the fact that the quadrupole sources are the least important and the least effective sources. Accordingly, only the quadrupole sources inside the integration surface are taken into account, and the rest is ignored. Provided that the ignored part of the turbulent noise sources have really a negligible contribution to the acoustic signal at the observer, then the position of the integration surface does not falsify the acoustic computation. Note that in other acoustic methods, like Kirchhoff’s analogy, the integration surface has to be positioned carefully, in order to prevent numerical instabilities. Fortunately, the mathematical formulation of FWH equation is stable and such instabilities do not occur [50]. However, recent researches [51] indicate that the effect of the turbulent quadrupole sources on the total acoustic signal may be important, especially if a high order of accuracy in the acoustic prediction is required. Errors in the noise prediction, which are due to the limits of the integration domain, can be eliminated by computing the acoustic contribution of the whole computational domain. In the simulations carried out here, the volume integral of the quadrupole sources through the whole computational domain is computed, in order to prohibit errors due to a priori assumptions. Another advantage of the FWH formulation is the robustness of the formulation. The computation of FWH equation is numerically robust and straightforward. Furthermore, its requirements of computational power and CPU-time are relatively low.

26

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4

4.1

Numerical Code

Basics

The numerical simulations carried out in the frame of this thesis are performed with the flow solver SPARC, which is developed in the Department of the Fluid Machinery of the University of Karlsruhe [52]. In this chapter an overview of the numerical code is given. Important features of the code, which effect the quality of the aerodynamical solution and hence the accuracy of the acoustical prediction, are discussed intensively. The starting point in Computational Fluid Dynamics (CFD) is the mathematical model, which describes the physical phenomena. The mathematical model may deviate from the real fluid dynamics because of assumptions like inviscid fluids or incompressible flows. Such assumptions are well suited for some practical applications and simplify the mathematical model, hence the simulation procedure, extensively. The main scope of SPARC is the computation of compressible, turbulent flows. Therefore it is optimized for the computation of the compressible, unsteady Navier-Stokes equations based on a densitybased algorithm. However, special features of the code allow the user to choose between the Euler or Navier-Stokes equations (i.e. viscid or inviscid fluids), and between compressible or incompressible flows. SPARC can compute turbulent flows with Direct Numerical Simulation (DNS), Large Eddy Simulation (LES) or Reynolds Averaged Navier Stokes (RANS) equations. Turbulent flows involve a wide range of length and time scales. DNS is based on the idea to solve the unsteady Navier Stokes equations without any simplification in the mathematical model. So, whole kinetic energy dissipation is captured by resolving the smallest structures with Kolmogorov scale, as well as the large structures with the integral scale of turbulence. For such a resolution, computational domain must be resolved with about Re9/4 control volumes. Since the time step is proportional to the grid size, the cost of a DNS becomes Re3. This is an unaffordable cost for most of the practical applications. LES decreases the computational cost, by resolving only the large, energetic scales of the flow and modeling the small, dissipative structures. Structures with larger length scales than the grid size are resolved. Hence, the required number of control volumes depends on the size of the minimum length 27

Basics

scale, which has to be resolved. Even though LES is cheaper in terms of computational power and CPU-time with respect to DNS, it is still expensive for most of the industrial applications, where a coarse prediction of mean flow variables is satisfactory. RANS is appropriate, whenever a quick estimation of the flow is desired and a detailed information about the fast processes is unnecessary. In this model the governing equations are averaged in time, and the terms containing the turbulent fluctuations (Reynolds stress) are modeled. The whole range of time and length scales are modeled. Due to this modeling, information about turbulent fluctuations gets lost. If the averaging is performed over a restricted time, unsteady RANS (uRANS) equations are achieved instead of the steady state RANS equations. URANS allows temporal fluctuations in the flow, however still some information about the turbulence gets lost, due to modeling of the Reynolds stress terms. The second step in CFD is the discretization of the mathematical model, in other words approximation of the differential equations by a system of algebraic equations, which are functions of flow variables at some set of discrete locations in space and time. The CFD code SPARC is based on the Finite Volume Method (FVM). The computational grid is curvilinear and block structured. In SPARC, spatial discretization can be carried out with 2nd order or 4th order accurate methods. In the frame of this work, second order central differencing scheme is employed for the spatial discretisation. Time discretization can be carried out with explicit or implicit discretization schemes. In SPARC both methods are implemented, since each are optimal for different type of problems. As the explicit scheme, the 4th order Runge Kutta method and as the implicit scheme, the 2nd order Dual Time Stepping method are implemented. In explicit schemes, the unknown variables of the next time step are described as functions of known variables from previous time steps. Therefore, explicit time discretization schemes have the advantage of computing the variables of the next time step with only one binary operation. On the other hand, implicit schemes describe the variables of next time step as a function of variables belonging to previous and future time steps. Hence, an iterative procedure is required for the solution of an implicit scheme. The so-called inner iterations are carried out at each time step. Therefore, for the same size of the time step, implicit methods require more operations, i.e. more CPU-time, than explicit ones. On the other hand, the disadvantage of explicit schemes is their stability limit. The condition under which explicit schemes are stable is firstly noticed in 1920s by Courant [53]. The stability condition is named as CFL-limit after the scientists discovered it Courant, Friedrichs and Lewy [54]. CFL =

u ( x , t ) ⋅ ∆x ∆t

(4.1)

where, u(x,t) is the local velocity, ∆x the grid size and ∆t is the time step. The CFLlimit depends on the stability analysis of the employed numerical algorithm. The explicit 28

Numerical Code

time step has to be set according to this stability condition in order to achieve a stable numerical computation. Implicit methods are on the contrary, unconditionally stable. If the time step is set too large, then they result in a steady state solution. Thus, implicit schemes provide a high degree of freedom in the decision of the size of the time step. It is suggested to make use of this freedom whenever the smallest interested time scale is about two times larger than the temporal step size allowed for the explicit scheme. Under circumstances, implicit time discretization may reduce the required CPU-time for a computation remarkably, without decreasing the accuracy of the unsteady solution. A quantitative example of this is demonstrated in Chapter 5. Note that for both temporal and spatial discretizations, the order of accuracy determines only the rate at which the discretization error vanishes as step size goes to zero. This is only valid if the step size is small enough, where “small enough” is problem dependent and cannot be determined a priori [55]. For each numerical simulation the time and length scales of the problem have to be analyzed carefully and the smallest scales of interest have to be determined, before setting the step size for spatial and temporal discretizations. Furthermore a grid dependency analysis is required to check the spatial refinement, although it may be too costly in LES (or DNS) computations. Third and last part of a CFD algorithm is the solver. Numerous iterative algorithms are available in the literature [55,56]. These are developed over the last decades; hence there are numerous mature algorithms, which provide fast convergence and high accuracy. The effectiveness of a solver is described by the ratio of the number of binary operations per iteration to the convergence rate. The number of operations per iteration is determined by the algorithm of solver. Convergence rate of the solution algorithm depends on the eigenvalues of the iteration matrix. The eigenvalue with the largest magnitude determines how rapidly the converged solution is reached. The corresponding eigenvector determines the spatial distribution of the convergence error. A possibility to increase the rate of convergence is the multigrid method. Multigrid method is based on the fact that the wavelength of discretization error depends on the grid size. Hence, on coarse grid levels the discretization error of large wavelengths can be reduced faster than on fine grid levels. Here, multigrid method is employed with the dual time stepping algorithm. If in an inner iteration loop, computations on more than one grid level are performed, discretization error of a larger range of wavelengths will be smoothed. In multigrid method, the solution on the fine grid is restricted, i.e. smoothed, to obtain the solution on a coarser grid. After some iterations on the coarse level the solution is prolonged, i.e. interpolated, to the fine grid and iterations are carried on there. The full multigrid method is implemented in SPARC, which uses the solution of a coarse grid as the initial flow field of a next finer grid level. If the computational mesh has more than four grid levels, then only the finest four grid levels are used, since more than four levels decrease the efficiency of the method. The number of iterations for prolongation and restriction at each 29

Special Topics

grid level is determined, depending on CFD experience with the chosen solver. SPARC is designed for the simulation of practice relevant problems, where usually high Reynolds number flows in complex geometries are of concern. Under such circumstances flow is usually unsteady and the required number of control volumes is very high. Proportional to the number of control volumes, the required memory space and CPUtime increase. In practice, one processor does not suffice. SPARC is parallelized to overcome this problem with the Message Passing Interface (MPI). Blocks are distributed among the available processors. Load distribution is a criterion to describe the efficiency of parallelization, which is defined as the ratio of the maximum computational load of a processor to the minimum load of a processor. A load distribution of 1.0 is optimum, which means that all processors have the same amount of computations to handle. Low load distributions result in inefficient parallel computations. In order to achieve an efficient load distribution block splitting may be carried out. Block splitting option of SPARC determines the largest blocks and splits them into two blocks in the direction of the maximum control volumes. Splitting may be repeated, if necessary. Due to splitting, a more equivalent distribution of the control volumes among the blocks is achieved, and hence, a more equivalent distribution of blocks among the processors is possible. For the simulations carried out in the frame of this work mainly two platforms are employed; the Local Area Network (LAN) of the Department of Fluid Machinery and the super computer IBM RS/6000 SP of the Computational Department of University Karlsruhe. The computational details of every simulation; i.e. chosen platform, number of processors, load distribution and CPU-time requirements are mentioned in the following chapters. SPARC involves a graphical user interface (GUI) to enable the user an easy usage of the program. GUI is programmed with tcl/tk, and attention is paid during implementation, that the GUI is easy to handle and self-explaining.

4.2

Special Topics

In this section special aspects of the numerical code, which affect the accuracy of the computational method extensively, are described in detail. In the frame of this work, two turbulence models – uRANS and LES – are performed. The differences in their mathematical formulation and its effects on the numerical results are discussed. The filtering method and the sub grid scale models in LES are explained, in order to provide insight to the merits and limits of the performed LES simulations. Details of the employed artificial dissipation and the preconditioning methods are presented. Finally, the moving grid method is described, which is employed in the simulation of the generic propeller (Chapter 7). 30

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4.2.1

Turbulence Modeling

The choice of the turbulence model is directly related to the goals of the simulation. For our purpose is to compute the acoustic signal of technically relevant flows, either uRANS or LES was chosen as the turbulence model. In the derivation of uRANS equation, the exact flow variables, (for example the exact flow velocity u (x, t ) ) are split into a time average ( u T (x, t ) ) and a turbulent fluctuation term ( u ′(x, t ) ). The first term represents the coherent structure of the flow and it is solved directly from the uRANS equations. The second one represents the turbulent part of the turbulence and it is modeled. Since this is an unsteady approach, it contains more information than RANS, but it still precludes a deterministic description of a particular event. The decomposition of the energy spectrum with uRANS approach can symbolically be presented as follows.

E(k)

E(k)

E(k)

=

+

k Total energy spectrum

k Resolved part

k Modeled part

Fig. 4.1 Kinetic energy decomposition with uRANS formalismus in the presence of a predominant frequency LES formulation depends on the idea of splitting the exact turbulent motion into large and small structures. The structures with large length scales, i.e. large eddies, are resolved with the LES and the rest is modeled.

31

Special Topics

E(k)

E(k) k

E(k)

k Total energy spectrum

k Resolved part

Modeled part

k +

Fig. 4.2 Kinetic energy decomposition with LES formalismus in the presence of a predominant frequency [57] Although it is well known and clearly seen in the energy spectra decomposition, that the formulation of LES is more accurate than the formulation of the Reynolds Averaged Navier Stokes equations, still uRANS is preferred in many applications. A major advantage of uRANS with respect to LES is that it can predict the flow data with some degree of accuracy in relatively coarser grids, thus the computational resource and CPU-time requirements are much less than in LES. Today, uRANS is affordable with the state of the art personal computers, and therefore it is suitable for many industrial applications. But for applications where more accuracy is required, it is not satisfactory. Especially for acoustics, where the small scales of turbulence determine the characteristics of sound, the deterministic approach of uRANS is misleading. Experience shows that RANS overestimates the sound pressure level, as well as LES with insufficient grid resolution. LES with an adequate grid resolution is a must for reliable noise prediction.

4.2.2

Large Eddy Simulation

LES makes use of the knowledge that the large structures of turbulence contain most part of the turbulent energy of the flow. Turbulent energy is transferred from large structures to smaller ones, and it is finally dissipated through the smallest structures. Large structures are coherent, inhomogeneous and energetic, whereas small structures are isotropic, homogeneous and dissipative. The effect of the small structures is comparable to the viscosity. Hence, instead of resolving the whole range of structures, to resolve only the 32

Numerical Code

energetic ones and model the dissipative part, is a good approximation. Two questions arise in this methodology; the first question is how to differentiate between the small and large structures and the second one is how to model the small structures. In order to split the length scales of the flow filtering methods are used, and to model the dissipative scales the Sub Grid Scale (SGS) models are employed. A large number of filters are defined in the literature [57]. The three classical filters of LES are: • Box Filter or Top-Hat Filter: Structures having a lesser size than the filter size are considered as dissipative. • Gaussian Filter: This type differs from the box filter by weighting the filtered scales with a Gaussian function. • Spectral Cutoff Filter: All fluctuations belonging to wave numbers above a cutoff wave number are filtered. This filter is employed in frequency domain. Box filter is the mostly preferred filter in CFD codes. The filter size L∆, splits the length scales of the problem. Eddies smaller than L∆ are modeled. Filter size is usually a function of the local grid size h.

h = (∆x ⋅ ∆y ⋅ ∆z )

1

3

(4.2)

where ∆x, ∆y and ∆z are the length of the local control volume in the three space directions. Assuming that the temporal resolution is finer than the spatial resolution, then it is appropriate to set the filter size proportional to the grid size (L∆∼h). As h goes to zero, LES approaches to DNS. As h increases, the deviation from DNS enlarges and the effect of the SGS model increases. There is a modeling error proportional to the filter size. If the filter size is defined as a function of the grid size, then the modeling error becomes proportional to the grid size. Then, both this modeling error in LES and the discretization error due to spatial resolution become proportional to the grid size. Changes in the grid resolution affect both the discretization and the filtering errors in the same manner. With this definition of the filter size, there is no possibility to study the modeling error due to filtering separately. An interesting research subject is to decouple these errors by changing the filter description and to study the modeling error due to filtering alone. To define the filter size independent of the grid size is called pre-filtering. With this method, grid size can be decreased without changing the filter size. However it is usually not preferred, due to the huge computational cost of this method. Filter size is also a function of the temporal resolution besides the spatial resolution. The appropriate time scale for temporal filtering is proportional to the time step of the unsteady simulation ∆t and local velocity u(x,t). 33

Special Topics

Lt = u (x, t ) ⋅ ∆t

(4.3)

In simulations with explicit time discretization schemes, this temporal filter is usually equal or smaller than the spatial filter, because of the CFL criterion. Implicit time discretization on the other hand, allows a larger range of time resolution. Hence, time filter may be larger than the spatial one. The filter size used in SPARC is a combination of both filters.  Ls  L∆ = c∆ max   (4.4)  Lt  whereas the spatial filter size is defined in SPARC as Ls = 2 ⋅ h and the constant c∆=0.9 is determined from numerical experiments. Filtered scales are modeled with SGS models. A SGS model has to describe the energy transfer between the resolved and unresolved structures and the effect of the small structures on the large ones. A large number of SGS models exist in the literature, which can be classified as follows: • Spectral models: Turbulence simulations in the Fourier space employ spectral models. • Algebraic models: These are eddy viscosity models, which are based on the Boussinesq approximation. The sub grid eddy viscosity is defined as a function of a constant model coefficient, which is determined empirically. This type of SGS model is the most prevalent one, because of its easy application, low CPU-time requirement and relatively high accuracy, whenever the modeled part of the turbulence is totally isotropic. • One equation models: Transport equation of one turbulent variable is solved additionally. • Two equation models: Transport equations for two turbulent variables, e.g. for turbulent energy k and turbulent time scale τ, are solved. Since the transport of the turbulence field is taken into account, this model is more precise than algebraic or one equation models. This class of SGS models can be applied to a wider range of flows and exhibit better dynamic properties. However the two additional transport equations cost more CPU-time than the former models.

34

Numerical Code

• Dynamical models: This class makes use of the fact that the length scales of the smallest resolved structures are similar to the scales of the largest modeled structures. Depending on this similarity the model parameters are computed from the resolved solution dynamically. Formally, a second filtering is carried out and the Germano identity is calculated. The major advantage of this methodology is that the coefficients are not empirical and fixed, but computed from the results of the resolved flow field. • Pseudo DNS: In this class no explicit SGS model is employed. It is assumed that the effect of the numerical dissipation due to the spatial discretization scheme accounts as a SGS model. Since both a SGS model and numerical dissipation result in dissipation of turbulent energy, a controlled numerical dissipation is used instead of a SGS model.

In this work two SGS models are used in the simulations and compared to each other. First one is the algebraic model of Smagorinsky and Lilly [58,59]. The second SGS model is a two equation model proposed by Magagnato [60], the adaptive k-τ model. “Adaptive” means here that the turbulence model approaches to DNS as temporal and spatial step sizes go to zero, and it approaches to RANS as step sizes increase extensively, so that no turbulent fluctuations are resolved. In between these two extremes the model adapts itself to the resolved turbulent fluctuations. The model is derived by splitting the resolved and unresolved parts of the turbulent variables, the turbulent kinetic energy k, and the turbulent time scale τ. k = k + k′ τ =τ +τ′

(4.5) (4.6)

The resolved part of the turbulent kinetic energy k is a part of the solution, achieved directly from the computation of the large scales. The unresolved parts of the turbulent kinetic energy k′ and the time scale τ ′ are modeled with the transport equations. The resolved part of the turbulent time scale τ is computed from the unresolved kinetic energy by a relation for isotropic high Reynolds number flows. τ =

L∆ k′

(4.7)

Any transport equation for k and τ can be chosen for the modeling of k′ and τ ′ . Here 35

Special Topics

a modified form of the two equation model of Craft, Launder and Suga [61] is used. This model is chosen because of its applicability in a wide range of flow fields. The major advantage of this method is the computation of the transport equations for turbulence variables. The importance of this advantage increases as the unresolved part of the turbulence is increased. For low Reynolds number flows, a good resolution of turbulence, in other words a small filter size is affordable. Then, the unresolved part is relatively small. The unresolved structures are homogenous, isotropic and dissipative. Thus, the assumptions of eddy viscosity models are valid and these become fair descriptions of the real flow. However at high Reynolds number flows, the affordable spatial and temporal resolution is much larger than the required filter size. A greater part of the flow has to be modeled. The homogeneity and isotropy of the unresolved structures are questionable. Therefore, better modeling is necessary for the unresolved part. Another advantage of this model is that information about the turbulence is given through the boundary conditions. Especially if the surrounding of the flow is highly turbulent, then this additional information may make a difference in LES. The disadvantage of the model is the additional requirements in computational memory and CPU-time. Due to two additional variables and equations, more computational power is necessary. Whether this additional cost is worth, is a question of the fluid dynamics of the specific problem, the required accuracy level and the available computational power.

4.2.3

Artificial Dissipation

Due to the mathematical formulation of central differencing schemes (CDS) in spatial discretization, odd-even decoupling occurs in the equation system [56]. Especially discontinuities in the flow, like shock waves or phase boundaries, cannot be captured cleanly with CDS. Upwind differencing schemes (UDS), on the other hand can handle discontinuities without oscillations. The difference between these two schemes arises from the difference in handling the propagation direction of certain waves. Per definition, UDS take the direction of the wave propagation into account and can handle sudden changes of the propagation direction. However CDS formulation lacks this information and causes nonphysical oscillations around discontinuities. These uncontrolled oscillations can lead to serious inaccuracies and numerical instability. In order to overcome this problematic in CDS formulation, Von Neumann and Richtmyer [62] introduced the concept of artificial dissipation or artificial viscosity. UDS can be represented as a combination of CDS and an additional term, which is similar to artificial dissipation term. So, the artificial dissipation term may be regarded as an upwind 36

Numerical Code

correction term, which removes nonphysical effects of CDS in wave propagation. An artificial dissipation term has to damp nonphysical oscillations where discontinuities occur and be negligible where the solution is smooth. Here “negligible” means being in the order of truncation error. Numerous methods are developed for artificial dissipation. Local extremum diminishing (LED), total variation diminishing (TVD), symmetric limited positive scheme (SLIP) are among these methods. In SPARC the artificial dissipation term is computed according to the work of Jameson et al. [63]. This artificial dissipation term is a blending of second difference and fourth difference terms. The second difference terms prevent the oscillations at discontinuities, the fourth difference terms are important for stability and convergence to a steady state. Several modifications are carried out to optimize the method for Navier-Stokes equations.

4.2.4

Preconditioning

Theoretically, compressible Navier-Stokes equations are valid for all flows, without a limitation of Mach number. However, at low Mach numbers, compressible equations result in a large numerical stiffness, so that they are not applicable to nearly incompressible flows. The stiffness is caused due to the large disparity of mean flow speed u and acoustic wave speed u+c0. In many practical applications, part of the flow field is nearly incompressible with low Mach numbers, whereas the Mach number is high in the rest of the flow field, like in low speed turbines or pumps. In order to handle such cases, compressible equations are multiplied with a preconditioning matrix. Preconditioning matrix provides an efficient solution for both constant and variable density flows at all speeds. There are numerous preconditioning methods in literature. Turkel provided a helpful review of preconditioning methods [64]. The preconditioning method implemented in SPARC is based on the work of Weiss and Smith [65], who proposed a time-derivative preconditioning of Navier-Stokes equations. This preconditioning includes transformation of the dependent variables. Instead of density, pressure is chosen as a primitive variable in order to single out the acoustic wave propagation and to increase the numerical accuracy. Eigenvalues of the equation system, which are in the order of (u, u, u, u+c0, u-c0) before the preconditioning, become in the order of (u, u, u, u, u) after the treatment. The ratio of the eigenvalue with the highest magnitude to the one with the lowest magnitude, in other words the condition number of the governing system of equations, is reduced dramatically. Hence stiffness is removed from the equation system. Preconditioning affects convergence rate and artificial dissipation, hence the quality of computational results, positively. Time-derivative preconditioning destroys the time accuracy of the system; hence the 37

Special Topics

application is limited. In the frame of this work, dual time stepping is employed as the time discretization algorithm. Since this algorithm involves an inner iteration loop at each time step, the solution at each time step can be considered as a steady state solution at that pseudo-time. Efficiency of the inner iteration loop is increased by the preconditioning, whereas the physical time stepping in the outer loop is remained unchanged and physically meaningful. The combination of the time-derivative preconditioning method and dual time stepping results in time accurate solutions for variable or constant density flows over a wide range of speeds.

4.2.5

Moving Grid

There are two ways to compute the acoustic signal of a moving sound source. Either the source domain is considered to be at rest, and the acoustic observer will be moved with respect to the source domain, or the observer is considered to be at rest and the source domain is in motion. The second approach is chosen in SPARC. It also corresponds to the real physical system. The unsteady flow is simulated with the moving grid method. In other words, the computational grid is allowed to change its position by transversal or rotational motions. The motion is simulated by a fixed frame of reference, i.e. the position of the control volumes is computed at every time step with respect to that fixed frame of reference. Correspondingly, the governing conservation equations are modified with the grid velocity, vg. The conservation equation of an arbitrary variable φ is written as follows.

r ∂ r r  r ∫∫∫ ρφdV + ∫∫ ρφ  v − v g  ⋅ ndS = ∫∫ Γgradφ ⋅ n dS + ∫∫∫ qφ dV   ∂t V S S V

(4.8)

Due to the grid motion, the computational volume is no longer unconditionally conserved. Control volumes may deform, and their volume may be modified. Therefore an additional equation for the conservation of the cell volumes, W, has to be solved. The socalled space conservation law (SCL) is defined as follows. d dt

38

∫ dΩ − ∫ v ⋅ ndS = 0 g

Ω

S

(4.9)

Numerical Code

4.3

Acoustic Module

In this chapter the implementation of the far field FWH equation in SPARC is described. The acoustic computations are carried out in a separate module. The acoustic module extracts the flow data, which is necessary for the acoustics, at each time step out of the main program. The module computes sound sources in the flow field and the resulting acoustic signal in the far field depending on the aerodynamical data at each time step. At the end of the computation, the far field acoustic signal is analyzed in the frequency domain. This module is also parallelized like the rest of the CFD code in order to increase the efficiency of the procedure. Each processor computes acoustic contribution of a particular part of the computational domain. For an efficient simulation, the amount of the data transfer between processors is decreased by forcing the processors work separately as long as possible. Far field FWH equation includes four integrals; first for the turbulent sources, second for the surface forces, third and fourth for the volume displacement. ρ ′ = I1 + I 2 + I 3 + I 4

I1 =

I4 =

1



∂

ri r j

1

(4.10) ∂

∫  r 3 (1 − M ) ∂τ (1 − M ) ∂τ  r r

4πc04 V I2 =

1 4πc03

I3 =

1 4πc03

1 4πc04

∫

VS



  dη 1 − M r  Tij

  dS (η )   rj ∂ ρ0v&si   2  dη r (1 − M r ) ∂τ 1 − M r    VS ri

∂

∫  r 2 (1 − M ) ∂τ S r 

fi 1− Mr

∫

 ri rj ∂ ∂ ρ0vsi vsj  1  3  dη  r (1 − M r ) ∂τ (1 − M r ) ∂τ 1 − M r 

(4.11)

(4.12)

(4.13)

(4.14)

Each integral includes time derivations. In order to implement this differential equation easily, the time derivations are performed analytically. Here, only the time derivation for the first term will be presented. Similar procedures are applied to the rest of the terms. A more detailed explanation of this procedure can be found in [66]. 39

Acoustic Module

Il =

i1 =

1 4πc04

∫ il dη

(4.15)

ri r j Tij ∂ 1 ∂ r 3 (1 − M r ) ∂τ (1 − M r ) ∂τ 1 − M r

(4.16)

For the sake of simplicity in the presentation, let us define two new variables F and F* r and use the common notation for vectors, i.e. r = r . 1 1 1 r r= = 1 − M r 1 − M ⋅ r 1 − mr 1 1 1 F* = = r r = 1− Mr 1 − mr 1− M ⋅ r F=

(4.17) (4.18)

So, the integrated term i1 can be rewritten as follows. i1 =

ri r j r

F

3

∂ ∂ F Tij F * ∂τ ∂τ

(4.19)

Analytical transformation of the time derivatives result in the following equation, where the time derivatives are presented as ∂ F = F& . ∂τ i1 =

ri r j r3

(T (FF&F& ij

*

)

(

)

+ F 2 F&& + T&ij FF&F * + T&&ij F 2 F *

)

(4.20)

The equation can be further simplified as: i1 =

1 r3

((3F

*5

)

(

)

A 2 + FF *3 B ri r jTij + 3FF *3 A ri r jT&ij + F *3 ri r jT&&ij

)

(4.21)

where the parameters A and B are defined as: A=

40

& (rm )(rr& ) r&m rm + − r r r3

(4.22)

Numerical Code

B=

(

(

)) (

1 (&r&m + 2r&m& + rm& ) + 13 2rr& (r&m + rm& ) + rm r 2 + r&r& + 3 rm(rr& )2 r 5 r

)

(4.23)

After performing this transformation for all terms, the integrated terms are presented as functions of r, m and their derivatives. Time derivations are performed with a second order accurate method in order to assure sufficient accuracy. In order to achieve this accuracy, data from the last four time steps are necessary for the computation of the acoustic signal. Hence, the first acoustic signal is computed at the fifth time step after the initialization of the acoustic computation. Before the initiation of the acoustic computation, a statistically converged flow solution has to be achieved. An unsteady simulation begins with an initialization of the flow field. Usually the flow field is initialized with constant values for the flow parameters. A better approximation is yielded in a full multigrid code, where the initial solution is taken from the converged solution of the coarser grid level. In both cases, CFD simulation needs some time to achieve a statistically converged state. At the beginning of the simulation transient fluctuations occur, due to the deviation of the initial solution from the real flow. Any evaluation of the CFD-data, e.g. averaging, post processing or acoustic evaluation during this transient phase is misleading. The duration of the transient phase depends on the characteristics of the investigated flow, the CFD-code and the initial solution. When the statistically converged flow solution is achieved, the acoustic computations may be initialized. The first step is the computation of the acoustic source in each control volume (CV). The acoustic source at a control volume is calculated according to the equation (4.10). If the control volume is located inside the flow field, then only the ′ = I1 . Acoustic sources quadrupole terms differ from zero and the equation reduces to ρ CV in a control volume are considered as compact. Inside every control volume in the flow field the distribution of quadrupole sources are assumed to be fully correlated inside the control volume, and fully uncorrelated outside. So, the total quadrupole sound sources inside one control volume can be considered as a point source at the center of the control volume, i.e. as a compact source (Fig. 4.3).

41

Acoustic Module

Correlation 1 0 ∆xi Fig. 4.3 Correlation of the Tij in a control volume

The acoustic sources are evaluated at the aerodynamic time of the CFD simulation taerodyn., which is equal to the emission time τe. Additionally, the time at which this signal will achieve to the observer, i.e. the advanced time t’obs., is computed according to the ′ . = τ e + r . The distance between the control volume and the observer r is equation t obs c0 computed at each time step based on their up-to-date positions. Hence, the acoustic contribution of the motion of both the computational domain and the observer are taken into account. The advanced time of each control volume is different, since their distances to the observer is different. The acoustic signal of the control volume nearest to the observer, will reach the observer before the rest of the signals, t’obs.,min. The signal from the farthest one, will achieve the observer as the last one, t’obs.,max. Between the time t’obs.,min and t’obs.,max, observer will detect the acoustic signals of only a part of the computational domain. The acoustic evaluation begins after that the acoustic responds of the whole domain is reached to the observer, i.e. after the time is equal to t’obs.,max. In the first acoustic computation, the maximum advanced time t’obs.,max is determined. Since the acoustic time step is given as a model parameter a priori, it is easy to calculate the discrete time point at which the acoustic evaluation has to be carried out. The acoustic source in each control volume and the corresponding advanced time are saved in SPARC in a matrix format (4.24). The parameters in the matrix are the number of control volumes - a function of the block number block# and the coordinate indices i,j,k - the advanced time and the acoustic signal of the corresponding control volume.

42

Numerical Code

′ . (τ e ) ρ CV ′ (τ e ) = ρ CV ′ (tobs ′ . ) # CV = f (block # , i, j, k ) tobs   M M M  

(4.24)

Since the time at which the acoustic evaluation at the observer has to be performed, is also known from the former considerations, the acoustic signal at the observer can be calculated. The matrix (4.25) represents this solution at the observer. If there are more than one observer, than the size of this matrix increases proportional to the number of observers.

tobs.   M

′ . ρ obs  M 

(4.25)

The acoustic signal at the observer is the superposition of the acoustic signals emitted from the whole computational domain. ′ . (tobs. ) = ρ obs

CVmax

′ (tobs. ) ∑ ρ CV

(4.26)

CV =1

If the advanced time of the acoustic signal from a control volume does not match the ′ . ≠ t obs. , then the value at the evaluation time ρ CV ′ (t obs. ) is evaluation time, i.e. t obs

′ (t obs ′ .) . interpolated from the known values ρ CV

In most of the applications, the additional CPU-time requirement for the acoustical module is negligibly small. Since the acoustic module involves straightforward computations, the computational cost is much less than the flow solver, where iterative schemes are employed. Additional computer memory requirement of the acoustic module depends on the number of variables, which have to be saved for the computation. Data from the last four acoustic time steps has to be saved for restarting the computations. This means 9 variables for each control volume and time step, i.e. 4·9·(total number of control volumes). The saved 9 variables are 6 Lighthill tensors Tij and 3 surface forces fi. This additional memory space is not important in moderate grid sizes. However, if a fine grid is employed, it may be limiting. One way to overcome this problematic is to ignore the acoustic sources due to turbulence. It is claimed that, the effect of the turbulent acoustic sources is negligible for low Mach number flows. The acoustic module in SPARC is enhanced with an option, so that the user may choose to ignore the Lighthill tensor. If this option is chosen, then the first term I1 is not evaluated, and the computation reduces itself basically to a surface integral and the volume integrals over the solid body. Hence, only information of the control volumes on the solid surface has to be saved. The amount of the saved data reduces to 4·3·(total number of the control volumes on the solid surface). 43

Acoustic Module

The acoustic time step ∆tac. is the time difference between two acoustic evaluations. The size of this time step defines the range of the resolved frequency domain. The smallest acoustic fluctuation resolved has the size of the acoustic time step; hence the highest frequency resolved is 1/∆tac.. The frequency resolution is determined by the duration of the acoustic evaluation. The longer the acoustic signal at the observer is computed, the smaller is the bandwidth of the frequency. The relation between the bandwidth ∆f and the duration of the acoustic evaluation T is ∆f=1/T. Before the acoustic computation the required range and resolution of the frequency analysis of the acoustic signal has to be determined. The acoustic time step and the duration of the acoustic computations are estimated based on these criteria. If implicit time stepping is employed in CFD computation, then usually the aerodynamical and the acoustical time scales are similar and hence the aerodynamical and the acoustical time steps are set equal, ∆taerodyn.=∆tac.. Under these circumstances, acoustical module will be executed after each aerodynamical time step. If explicit time stepping is employed for the same flow problem, then the aerodynamical time step will probably be much smaller than the necessary acoustic time step, ∆taerodyn.