Direct Numerical Simulation of a low Reynolds number subsonic jet and the associated sound field
Direct Numerical Simulation of a low Reynolds number subsonic jet and the associated sound field
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 5 oktober 2009 om 15.00 uur
door
Peter David MOORE
Bachelor of Science (Advanced), Honours Class I, University of Sydney, Australia. Geboren te Sydney, Australia.
Samenstelling promotiecommissie: Rector Magnificus, voorzitter
Prof. dr. ir. B.J. Boersma, Technische Universiteit Delft, promotor Prof. dr. A.E.P. Veldman, RU Groningen
Prof. dr. J.J.M. Slot, Technische Universiteit Eindhoven Prof. dr. ir. A. Hirschberg, Universiteit Twente
Prof. dr. ir. B.J. Geurts, Universiteit Twente
Prof. dr. ir. C. Vuik, Technische Universiteit Delft Prof. dr. ir. G. Ooms, Technische Universiteit Delft
This project was financially supported by the Dutch Technology Foundation STW under grant number DSF:6181. We also thank the foundation NCF of the Netherlands Foundation for Scientific Research (NWO) for the use of supercomputing facilities.
Copyright c 2009 by P.D. Moore All rights reserved.
ISBN 978-90-9024705-2
Summary ix
Samenvatting xi
1. Introduction 1
1.1. Background and motivation . . . 1
1.1.1. Sound and noise . . . 1
1.1.2. Managing noise . . . 1
1.1.3. Jet noise in aircraft . . . 2
1.1.4. Aeroacoustics . . . 2
1.2. Objectives of the research . . . 3
1.3. Outline of the thesis . . . 4
2. Theoretical Background 7 2.1. Abstract . . . 7
2.2. Basics of acoustics . . . 7
2.2.1. Human acoustical perception or hearing. . . 7
2.2.2. Acoustic disturbances . . . 8
2.2.3. Monopoles, dipoles and quadrupoles . . . 9
2.3. Background . . . 9
2.3.1. Some historical developments in jet aeroacoustic research . . . 10
2.3.2. Developments in jet computational aeroacoustics (CAA) . . . 11
2.4. Jet description . . . 13
2.5. Compressible jet flow . . . 15
2.5.1. State variables . . . 15
2.5.2. Governing equations . . . 15
2.5.3. Non-dimensional parameters. . . 16
2.5.4. Static vs stagnation temperature . . . 17
2.6. Numerical scheme to simulate jet flow . . . 17
2.6.1. Direct Numerical Simulation . . . 17
2.6.2. Staggered grid formulation . . . 18
2.6.3. Spatial discretization . . . 19
2.6.4. Calculation of derivatives . . . 19 v
2.6.5. Temporal discretization . . . 20
2.6.6. Computational cost of a direct numerical simulation . . . 20
2.6.7. Parallelization and MPI . . . 21
2.6.8. Boundary Conditions . . . 23
2.6.9. Data storage . . . 26
2.7. Jet aeroacoustics . . . 27
2.7.1. Lighthills acoustic analogy . . . 27
2.7.2. Porous Ffowcs Williams and Hawkings method . . . 32
2.7.3. Some general numerical details of acoustic analogy implementations 32 Appendix 2.A. Alternative techniques used to investigate compressible jet aeroa-coustics . . . 34
2.A.1. U-RANS . . . 34
2.A.2. Large eddy simulation . . . 36
2.A.3. Linearized Euler Equations . . . 36
2.A.4. Kirchoff formulation . . . 36
2.A.5. Alternative acoustic analogies . . . 36
3. Simulation and measurement of flow generated noise 37 3.1. Abstract . . . 37
3.2. Introduction . . . 37
3.3. Geometry, governing equations and numerical procedure . . . 40
3.3.1. Boundary conditions . . . 42
3.4. Acoustic field continuation . . . 43
3.4.1. Experimental setup . . . 45 3.4.2. Error Analysis . . . 45 3.4.3. Numerical setup . . . 48 3.5. Results . . . 49 3.5.1. Flow field . . . 49 3.5.2. Acoustic field . . . 53 3.6. Conclusion . . . 54
4. On the application of acoustic integral techniques and some acoustic properties of a compressible subsonic jet by direct numerical simulation 57 4.1. abstract . . . 57
4.2. Introduction . . . 58
4.3. Simulation details . . . 59
4.3.1. Data storage . . . 59
4.4. Porous Ffowcs Williams and Hawkings method . . . 59
4.4.1. Implementation details . . . 60
4.4.2. Contour selection . . . 60
4.4.3. Quadrature selection . . . 62
4.5. Lighthill’s acoustic analogy . . . 62
4.5.2. Quadrature selection . . . 65
4.5.3. Volume selection . . . 68
4.5.4. Windowing methods . . . 68
4.5.5. Calculation Cost . . . 69
4.5.6. Acoustic properties of the jet flow simulation using Lighthill’s acous-tic analogy . . . 69
4.6. Conclusions . . . 71
Appendix 4.A. Runge effect on a simple test case . . . 73
Appendix 4.B. Derivative in time . . . 75
Appendix 4.C. Filter . . . 76
Appendix 4.D. Integration algorithms . . . 76
4.D.1. Integration of a time series at a single observer location . . . 76
4.D.2. Integration of a 2-D field . . . 76
Appendix 4.E. Newton Coates formulas . . . 77
5. Some properties of the aeroacoustic emissions of hot and cold jets 79 5.1. abstract . . . 79
5.2. Introduction . . . 79
5.3. Details of the simulation . . . 80
5.4. Results . . . 84
5.4.1. Flow and acoustic visualizations . . . 84
5.4.2. Comparison of J-ISO with existing DNS, LES and experiment . . . 86
5.4.3. Flow properties of three jets . . . 87
5.4.4. Aeroacoustic properties of three jets . . . 88
5.4.5. Aeroacoustic sources . . . 90
5.5. Conclusion . . . 92
6. A Direct Numerical Simulation of a low Reynolds number compressible jet 95 6.1. Abstract . . . 95
6.2. Introduction . . . 95
6.3. Numerical details . . . 96
6.4. Results . . . 97
6.4.1. Problem with flow development . . . 97
6.4.2. Jet development . . . 99
6.4.3. Visualization of axial instabilities . . . 99
6.4.4. Visualization of development of instabilities in time . . . 102
6.4.5. Mean results . . . 102
6.4.6. Aeroacoustic results . . . 106
7. Some conclusions 109 7.1. On this thesis . . . 109 7.2. On the role of direct numerical simulation in aeroacoustics research . . . . 110 7.3. References . . . 112
List of publications 119
Acknowledgments 121
Aeroacoustics of compressible subsonic jets - P. D. Moore
Jet noise is an extensively studied phenomenon since the deployment of the first civil jet aircraft more than 50 years ago. Jet noise makes up a considerable portion of the total noise of jet aircraft, and the expansion of the numbers of airplanes and airports has only been possible by keeping the noise of airplanes under control. However, while large efforts have been made to measure jet noise and to predict the features of the noise theoretically, the mechanisms by which turbulent jet flows produce noise are still not fully understood. The continued development of low noise jet engines will certainly be aided by improved understanding of these mechanisms. In the past, directly testing acoustic theory has been difficult, because experiments only provided limited information about the jet mechanisms. Recent developments in computational aeroacoustics have shown a promising avenue to provide the required information. The main aim of this thesis is to develop methods to simulate high speed jet flows and to study the results of completed simulations for special cases. These methods should provide an accurate solution of the equations of motion with only limited introduction of simplifying assumptions about the nature of the flow. Then simulation databases could be obtained for interesting cases to investigate the flow and acoustic features of these jet flows.
For the simulation of the jet flows, a staggered high order finite difference scheme was used. The implementation of robust boundary conditions that simulated free jet con-ditions proved to be particularly challenging. The simulations presented use a collection of different techniques which are designed to resemble physical jet conditions in the interior portion of the domain, with a surrounding region used to resemble the effect of an infinite extension of the finite interior.
This work took place as part of a larger project that also involved the creation of an experimental facility for investigating high speed, low Reynolds number jets. Results from this part of the project, compared with data obtained by the simulation data demonstrated that the simulations were able to accurately predict the flow and acoustic features of these jet flows.
Investigation of the acoustic features of the flow took place by means of solutions of Lighthill’s acoustic analogy and the porous Ffowcs Williams and Hawkings method. The implementations of both of these methods required the development of special techniques to optimize the order and number of I/O operations, which was found to be essential for
efficiency. Solution of Lighthill’s acoustic analogy were found to become more spectrally limited as the spatial integration order was increased. No such effect was seen with solutions of the porous Ffowcs Williams and Hawkings method. The high frequency corruption of the signal was linked to Runge’s phenomenon, and sound pressure levels were only accurate when the acoustic signals were appropriately filtered. In the region of resolved frequencies, it was found that the effects of non-linear propagation are contained in both the “Entropy” source term and the “Reynolds Stress” source term of Lighthill’s acoustic analogy for emissions at low angles to the downstream direction of the jet axis, even for essentially isothermal jets.
The effect of inflow temperature was investigated independently of nozzle Reynolds number, by varying the temperature of the inflow for three jet simulations, with the nozzle Reynolds number fixed. The effect of increasing temperature was found to increase the sound pressure level of the acoustic emissions at all angles and to broaden the spectrum of the acoustic emissions. Investigation of the terms of Lighthill’s acoustic analogy at 90◦
reveal that the Entropy term makes a significant contribution to the noise emission at these angles for the heated jets, but very little for the isothermal jet. This gives some evidence for the view that this term is related to an independent emission mechanism in these jets. Finally, a detailed simulation is performed at a larger Reynolds number in the final chapter. This simulation gave emissions that were around 5dB higher than the lower Reynolds number jet. It was found that due to an interplay of the various boundary conditions, a recirculation developed in the flow. It is believed that this is responsible for the increase in noise emissions in this case. In other words, the acoustics of jet flows are greatly impacted, not only by nozzle conditions, but also on the surrounding environment, that can alter the way in which the jet develops.
Aero-akoestiek van compressibele, subsone vrije straal - P. D. Moore
Het geluid, dat door een vrije straal wordt geproduceerd, is een veel bestudeerd verschijnsel sinds de eerste straalvliegtuigen, meer dan 50 jaar geleden, in gebruik kwa-men bij de burgerluchtvaart. Het geluid van de vrije straal maakt een belangrijk deel uit van het totale geluid van een straalvliegtuig en de toename van het aantal vliegtuigen en vliegvelden was alleen mogelijk door de hoeveelheid geproduceerd geluid te reduceren. Of-schoon er veel tijd en moeite is gestoken in het meten van het geluid van een vrije straal en in het theoretisch onderzoek van geluid, worden de mechanismes waardoor een vrije, turbulente straal geluid produceert echter nog niet volledig begrepen. De ontwikkeling van straalmotoren met een laag geluidsniveau zal dan ook gebaat zijn bij een beter begrip van deze mechanismes. In het verleden was het toetsen van akoestische theorie moeilijk, omdat experimenten slechts beperkte informatie verschaften over de mechanismes die een rol spelen in een vrije straal.
Met de recente vooruitgang in de numerieke aero-akoestiek is een veelbelovende weg geopend die kan leiden tot het verkrijgen van de benodigde, ontbrekende informatie. Het hoofddoel van dit proefschrift is om methodes te ontwikkelen om vrije stralen van hoge snelheid te simuleren en om de data, verkregen uit simulaties, van speciale gevallen te bestuderen. Deze methodes zouden een nauwkeurige oplossing van de bewegingsvergelijkin-gen moeten geven met slechts gebruik van een beperkt aantal versimpelende aannames over het karakter van de stroming. Daarmee zouden databases kunnen worden gegenereerd voor interessante gevallen om de stroming van vrije stralen en hun akoestische eigenschappen te bestuderen.
Voor de simulatie van deze vrije stralen is een ’staggered’, hogere orde, eindige differentie code gebruikt. Hierbij bleek met name de implementatie van randvoorwaarden die een vrije straal goed representeren een grote uitdaging. Er zijn bij de simulaties diverse methodes gebruikt om het nabije veld van de jet weer te geven, plus een extra domein eromheen om een oneindig groot buitengebied te representeren.
Dit werk was een deel van een groter project waarin ook een experimentele op-stelling werd gebouwd voor het onderzoek aan vrije stralen met hoge snelheid en een laag Reynoldsgetal. Een vergelijk van de numerieke met de experimentele resultaten van het project lieten duidelijk zien dat de simulaties in staat waren om nauwkeurig de stroming en de akoestische eigenschappen van deze vrije stralen te voorspellen.
Het onderzoek van de akoestische eigenschappen van de stroming werden gedaan met oplossingen van Lighthills akoestische analogie en de poreuze methode van Ffowcs Williams en Hawkings. Voor de implementatie van deze beide methodes was het nodig om speciale technieken te ontwikkelen om de orde van de methode en het aantal I/O operaties te optimaliseren. De oplossing van de analogie van Lighthill liet spectrale convergentie zien bij vergroting van de orde van de ruimtelijke discretisatie. Dit werd niet geconstateerd bij toepassing van de methode van Ffowcs en Williams. De corruptie van het signaal bij hoge frequenties werd in verband gebracht met het Runge-verschijnsel, en de hoogte van de geluidsdruk kon alleen nauwkeurig worden berekend na filteren van de akoestische signalen. In het gebied waar de frequenties goed werden opgelost, vonden we dat de effecten van niet-lineaire voortplanting aanwezig zijn in zowel de ”Entropie” bronterm als in de ”Reynolds stress” bronterm van Lighthills akoestische analogie voor uitstraling van geluid onder lage hoeken ten opzichte van de as van de vrije straal in stroomafwaartse richting, zelfs voor isotherme vrije stralen.
Het effect van verandering van de temperatuur bij de instroommond bij gelijkbli-jvend Reynoldsgetal bij de instroommond is ook onderzocht. Het effect van een verhoging van de temperatuur bleek, voor alle hoeken, een verhoging van het niveau van de gelu-idsdruk van het uitgestraalde geluid te zijn en een verbreding van het spectrum van het uitgestraalde geluid. Onderzoek van de termen van Lighthills akoestische analogie bij 90◦
laat zien dat voor verwarmde vrije stralen, bij deze hoeken, de Entropie term een belangrijke bijdrage levert aan de uitstraling van het geluid terwijl deze bijdrage voor isotherme vrije stralen zeer gering is. Dit levert enig bewijs voor de opvatting dat deze term gerelateerd is aan een onafhankelijk mechanisme in deze vrije stralen.
Afsluitend wordt in het laatste hoofdstuk een simulatie beschreven voor een vrije straal bij een hoger Reynolds getal. Deze straalde een geluid uit dat 5dB sterker was dan voor de vrije straal bij een laag Reynolds getal. Het bleek dat, door een wisselwerking tussen de verschillende randvoorwaarden toegepast op de verschillende randen, een recirculatie in de stroming ontstond. Er wordt van uitgegaan dat deze verantwoordelijk is voor de toename in het uitgestraalde geluid. Dit betekent, dat de akoestische eigenschappen van een vrije straal niet alleen worden be¨ınvloed door de omstandigheden bij de instroommond, maar ook door de omgeving, die mede bepaalt hoe de vrije straal zich ontwikkelt.
Introduction
1.1
Background and motivation
1.1.1
Sound and noise
Sound is a concept that most of us are familiar with. It is a part of our everyday experience, from the alarm clock in the morning to the snoring of bed partners as we try to fall asleep, we are constantly cataloging sounds to enrich our sense and understanding of the world around us. Humans have developed the ability of speech to convey rapidly large amounts of information for the purposes of communication. Humans have also developed devices that create sound for specific purposes, such as music instruments, sonar detection systems and ultrasound imaging.
However, sound is all-to-often an unwanted by-product of the tools and vehicles we have developed, especially those mechanical in nature such as vacuum cleaners, au-tomobiles, computers, industrial machinery and astronautic vehicles. The word “noise” is reserved to designate these kinds of unwanted sounds. Such sounds constitute envi-ronmental noise, which is a significant form of pollution in the man-made environment. Noise can have severely detrimental effects on human health, ranging from annoyance to sleep deprivation, hypertension, increased stress levels, increased blood pressure, tinnitus and hearing impairment. Efforts to reduce these various noises are an important part of scientific research, and an important issue for civilian governing bodies.
1.1.2
Managing noise
In some countries, local community councils regulate the hours at which typical noisy activities (such as using vacuum cleaning systems, using washing machines in apartment buildings, or recycling glass in street containers) make take place. On the level of city and state planning, motorways and airport runways are placed strategically giving due consideration to the effect of noise emissions on the surrounding populace. Motorways are often built with barriers designed to mitigate the noise effect, while houses near to airports are sometimes built or supplied with acoustically insulating materials by the
evant authority. Low frequency vibrational noise from wind turbine rotation as well as the much higher frequency broadband and discrete frequency noise associated with the unsteady aerodynamic forces on the blades, often in the blade tips can be an issue and new designs such as the SWIFT Wind Turbine focus largely on solving these issues. Noise reduction is also an issue in the design of virtually every type of transportation vehicle -cars, trains, ships, helicopters and aeroplanes, both for reasons of civilian annoyance as well as military stealth. QuietCar, an automobile paint coating, is advertised as “a new high-tech polymer that absorbs twice as much noise & vibration as old techniques such as asphalt, rubber and elastomer coatings”, while “the new Toyota Crown Hybrid will come with three microphones that will work with three always-on active phase speakers that will make the interior of the car much quieter”. Helicopter pilots can select operating modes which limits the engine torque and other parameters to reduce noise, while the new Airbus A380 for example is said to be so quiet, that pilots have trouble sleeping because the thrum of the engine is too low to drown out the noise of their passengers (Morrison December 3, 2008).
1.1.3
Jet noise in aircraft
The jet engine in aircraft are responsible for a significant amount of the total noise emis-sions of airplanes, with the rest of the noise from the interaction of the airframe with the surrounding air. Approaching jet aircraft are associated with a high pitched noise as fans suck air into the jet engine, while retreating jet aircraft are associated with a low pitched rumble from exhaust leaving the jet engines. The reduction of this type of noise has had some success. The use of chevron type nozzles for example allow the air from the engine to mix more thoroughly with the external air, decreasing turbulence and noise. Large fan designs lead to a slower exhaust speed and a dramatic reduction in noise. However, the ongoing development of these techniques requires better theoretical understanding of jet noise mechanisms.
1.1.4
Aeroacoustics
Aeroacoustics deals specifically with noise created by flowing gases and liquids and the interaction of flows with solid surfaces. The field of aeroacoustics is regarded to have begun with the defining work of Lighthill (1952). It was no coincidence that this work came at the time that the modern aviation industry was beginning. With the growth in civilian jet aircraft, noise pollution was becoming of particular concern, and aircraft companies were keen to understand the problem. Lighthill states the problem as
... given a fluctuating fluid flow, to estimate the sound radiated from it.
What Lighthill did, was to mathematically demonstrate the link between fluid flow, and the radiation of acoustic waves. His equation is referred to now as Lighthill’s acoustic analogy and is used for many types of aeroacoustic flows.
At the time of Lighthill, most aeroacoustic investigation was performed experi-mentally, since this was the main means of investigating flow behavior. Today, while ex-periments remain an important component of aeroacoustic research, an increasingly large component is done numerically. With modern, powerful computers, the equations govern-ing the flow are solved directly, or with as minimal degree of modelgovern-ing as is necessary. Then, with detailed knowledge of the flow, equations such as Lighthill’s acoustic analogy can be solved directly. While the techniques for aeroacoustic investigation have changed in recent times, the goal has remained the same - to improve theoretical models of noise generation mechanisms.
And as for jet aeroacoustics? The original example Lighthill used to demonstrate his theory was that of the turbulent jet. The turbulent jet case has remained as an impor-tant topic since this time and today, is just as imporimpor-tant as it ever was.
1.2
Objectives of the research
This study was carried out as part of a project entitled “Measurement and computation of sound generated by turbulent flows”. The objectives for this research were twofold, with a computational aspect and an experimental aspect.
• Computational: Development of computational methods for the determination of aeroacoustical sound sources in low speed flows and methods to use these sources to compute the far field sound in all directions.
• Experimental: Development of an experimental setup for a jet flow at low speed with flow observation techniques that allow the validation of the computations of the aeroacoustical sound sources and with acoustical measurements that allow the verification of the radiated sound.
This thesis concerns the work related to the computational aspect of the project, while the experimental aspect was carried out largely by another Promovendi, Harmen Slot. However relevant material from the experimental aspect is used and cited accordingly.
It is the objective of this work to develop and utilize a numerical method for the simulation of an aerodynamic jet and its associated sound field, to complete fully resolved simulations of jet flows under a variety of conditions and to analyze the results of these simulations to investigate the physical mechanisms that link the aerodynamic aspects of the flow with the acoustic aspects of the flow. The development of the method will require attention paid to robustness under a variety of conditions, including a variety of jet Mach numbers, Reynolds numbers and temperature variations. Further, the development of robust boundary conditions is an area of particular importance, which should allow entrainment of flow into the domain, while at the same time, not generating spurious sound, or reflecting internally generated sound back into the domain.
The accuracy of the methods will first be demonstrated by comparing completed simulations with published experimental data such as that of Stromberg, Mclaughlin, and
Troutt (1980) and Slot, Moore, and Boersma (2007), the latter being specifically undertaken for the verification of these methods. This comparison should attempt to verify that acoustic features as well as the unsteady and steady aerodynamic flow and are in agreement between the simulation and experiment. Further, the mechanisms of noise creation should be investigated for these cases and conclusions drawn from this investigation compared with theory and hypotheses in existing literature. This comparison will also give us the confidence to examine jet flows for which detailed experimental knowledge does not yet exist.
It is also intended to perform a limited parametric study of jet noise, by modifying the temperature and Reynolds numbers of the jets. While such a study will be necessarily limited by the demanding nature of these calculations to a handful of cases it is hoped that insight will be gained by investigating the difference in the mechanisms of each case. In particular it is hoped that some progress will be made in answering the following questions of ‘what is the effect of increasing the inflow temperature of a jet?’, ‘how is this effect related to Reynolds number changes?’, and ‘what are the underlying physical mechanisms of the noise in these cases for both the small and large scales?’ While studies have been performed before, direct numerical simulations of compressible jet flows are almost non-existent.
1.3
Outline of the thesis
In Chapter 2 of this thesis, we give an introduction of the topic and introduce the techniques that are used for the simulations.
Chapter 3 gives the main details of a scheme for simulating a low Reynolds number jet flow and extension of the acoustic portion of the flow to arbitrary locations in the far field. A direct numerical simulation is carried out and the results are compared with results from a similar jet created with our experimental facility.
In Chapter 4 we discuss details of the implementation and solution of both the porous Ffowcs-Williams and Hawkings method and Lighthill’s acoustic analogy. The meth-ods are verified using a new simulation database that was similar to the one completed in Chapter 3 (with a larger domain). This chapter also provides a discussion on the role of these techniques in aeroacoustic studies and gives additional acoustic properties of the jet flow in Chapter 3.
In Chapter 5 we investigate temperature effects through three fully-resolved jet simulations. For these simulations, the Reynolds number and inflow velocity are held constant, while the ratio of the static inflow temperature to ambient temperature is varied. Specific ratios of the inflow to ambient temperature were chosen for this study, Ts/To = 1.0,
Ts/To = 1.8 and Ts/To = 2.7 representing an isothermal jet, and two heated jets.
In Chapter 6 we discuss details of the direct numerical simulation of a jet at a Reynolds number of 3,960 using approximately 113 million grid points. This is to date, one of the larger direct numerical flow simulations completed for compressible jet flows, compared with what can be found in the literature. The simulation is compared with a similar jet from Chapter 5 (with lower Reynolds number) and with previously published
results.
In Chapter 7 are some conclusions for the thesis as a whole, and some suggestions where such research could be carried in the future.
Theoretical Background
2.1
Abstract
We introduce material relevant to the work presented in this thesis. Firstly, the basics of sound and aeroacoustics are discussed. Next, a historical overview of jet aeroacoustic research is given, and the context of the present work is made clear. A detailed description of the problem under investigation is given, along with details of the problem geometry.
Following this, the governing equations and numerical implementation details for the compressible jet problem are presented. Computational aspects related to the flow geometry are discussed, and the differences of the geometry used for numerical simulation with an idealized geometry are emphasized. A high order scheme is introduced, along with a rationale for the scheme, and details of the boundary treatment are given. Moreover, details of the implementation are provided, including some description of various pitfalls and a high level of the technical details that may be useful for someone about to embark upon this kind of research, but not always found in scientific journal publications (or the later chapters of this thesis).
Finally we continue in the same vein presenting the governing equations and nu-merical implementation details for the aeroacoustic portion of the flow. While sound waves are necessarily captured by the compressible jet simulations, we present the techniques for extending these sound waves to arbitrarily distant locations and for attempting to connect the created sound waves to particular structures in the jet flow. This part is mainly con-cerned with Lighthill’s Acoustic analogy and the porous Ffowcs Williams and Hawkings method.
2.2
Basics of acoustics
2.2.1
Human acoustical perception or hearing.
Humans perceive sound through the vibration of air molecules acting on the tympanic membrane of our ears (eardrum). It has been found empirically, that there is a minimum
amplitude of vibrations to which our ears are sensitive. This is known as the threshold of sound. This threshold varies slightly from person to person, and also with the frequency of the vibration of the air molecules. Nevertheless, it is usual to choose a common reference value when speaking about sound pressure levels. This common reference value in air is taken as
pref = 2 × 10−5Pa. (2.1)
This contrasts with the standard atmospheric pressure (the typical pressure of the air around us) of 105 Pa. In other words, humans can perceive pressure variations which are 1
part in 5 billion of the background pressure. The perceived loudness of a sound is not linear with the amplitude of the pressure. In fact, the perceive loudness changes logarithmically with the root mean square of the pressure variation. In light of this, a scale has been invented to describe sound, not in terms of Pascals, but in terms of decibels (dB). The quantity is then referred to as the sound pressure level. The definition of this is
SPL = 20 log10 prms pref . (2.2)
On this scale, an increase of 20 dB (or a ten-fold increase in pressure) corresponds roughly to a doubling in the perceived loudness of sound, although this is fairly arbitrary and differs from individual to individual, and also with the frequency of the sound. In order to make this scale comprehensible, some typical sounds along with their sound pressure level in decibels are given in table 2.1.
Pressure (Pa) SPL (dB)
Pneumatic drill 6 × 101 130
Car horn (close by) 2 × 101 120
Airport 6 × 100 110
Inside a metro 2 × 100 100
Inside a bus 6 × 10−1 90
Next to a main road 2 × 10−1 80
Human speech 6 × 10−2 70
Bedroom 2 × 10−3 40
Recording studio 6 × 10−4 30
Threshold of hearing 2 × 10−5 0
Table 2.1: Noise associated with some common sounds.
2.2.2
Acoustic disturbances
In a given turbulent flow, different disturbances can be identified, such as instability waves, vortical disturbances, entropic disturbances and acoustic disturbances. These disturbances each have a different character. Vortical disturbances are associated with the turbulence
of the flow, and entropic disturbances are associated with temperature gradients. Both vortical and entropic disturbances convect with the flow, and have little effect away from the flow region. Acoustic disturbances can be a by-product of the vortical or entropic disturbances, or created by some other mechanism. They are a much weaker form of disturbance in general in the flow region, however, acoustic disturbances can propagate efficiently over long distances. Moreover, humans have a very fine sensitivity to even weak acoustic disturbances. It is these properties that make acoustic disturbances important to tackle, while it is the weakness of the disturbances in the flow region which makes them difficult to handle.
2.2.3
Monopoles, dipoles and quadrupoles
In aeroacoustics, the terms monopole, dipole and quadrupole are used quite frequently. To explain the concepts of monopoles, dipoles and quadrupoles we make reference here to similar phenomenon in electromagnetic theory. If we consider charged particles, then the electric field of a single charged particle takes a monopole form, which is spherically uniform in shape. The electric field associated with a two charged particles of opposite sign in close proximity takes on the much different, dipole character. While the net charge in this case is zero, since the two particles are displaced relative to each other, the electric field at locations closer to one of the particular charges has a small net value of that charge. The total electric field of the dipole system is bilobal in shape. Similarly, a quadrupole results from the close proximity of 4 charges, two of one sign and two of another. The associated electric field typically has 2 or 4 lobes, depending on the orientation of the charges.
In fluid dynamics, the positive and negative charges are replaced by mass source and sinks. Monopole noise is associated with fluctuating mass inflow, dipole noise with turbulent eddies deforming near aerodynamic surfaces and quadrupole noise with a pair of turbulent eddies in close proximity deforming each other.
2.3
Background
Jet aeroacoustics is a well established field. Several excellent older and more recent reviews exist covering various aspects of this topic. Less recently, Tam (1998) gives an account of developments from 1952 until 1988 on jet noise research, while Jordan and Gervais (2008) gives a modern, easy to read account of the development of our understanding of the acoustic source in turbulent jets. While supersonic jets was covered by Tam (1995), the largest developments in our understanding of them had already occurred. Computational aspects of jet flow are covered specifically by Bailly and Bogey (2004), while a more general review of computational aeroacoustics is provided by Wang, Freund, and Lele (2006). Several reviews exist on computational boundary treatment, including Colonius (2004) and Hixon (2004). A discussion of the relevance of Direct Numerical Simulation as a research tool is given by Moin and Mahesh (1998). Moreover, several dedicated books exist on aeroacoustics including those of Goldstein (1976) and Howe (2003). This list is
merely a selection from a large number of publications and items not included may also be worth reading. Here we give a discussion incorporating elements of these, pertaining to the work in this thesis.
2.3.1
Some historical developments in jet aeroacoustic research
The field of aeroacoustics in general, is usually regarded to have begun with the defining work of Sir James Lighthill on jet noise (Lighthill 1952). Lighthill’s work was to provide the first theory that gave some mathematical unification to the before separate fields of acoustics and aerodynamics. Lighthill developed this theory in the context of attempting to predict the intensity of noise produced by turbulent jets. Lighthill reformulated the equations of fluid motion as a wave equation, and demonstrates that in regions where only acoustic flow modes are present, then the wave reformulation is an appropriate tool to describe the acoustic waves. The equation is now referred to as Lighthill’s acoustic analogy. The mathematical character of the source term in this equation is similar to that of a quadrapole, leading to a model of the source of noise in a turbulent jet as being due to the quadrupolar interaction of mutual deformation of vortices in close proximity.
Lighthill’s acoustic analogy was valid only for the acoustics of isolated regions of free turbulence, such as found by a jet in an open location. A short time later, Curle (1955) extended the acoustic analogy to include the effect that turbulence interacting with a solid body has on the creation of aeroacoustics. Curle’s equation was found to be appropriate for problems of acoustics from wall-bounded jets and flows over surfaces such as wings. Curle’s equation is usually referred to as Curle’s analogy. Curle’s analogy was generalized by Ffowcs Williams and Hawkings (1969) to allow for arbitrary motion of solid surfaces near a flow, particularly useful for problems involving moving blades and propellers.
The role of flow-acoustic interaction was highlighted experimentally by Atvars, Schubert, and Ribner (1965) and alternative acoustic analogies were developed such as Phillip’s (Phillips 1960) and Lilley’s (Lilley 1974) to account for this interaction. The idea behind these was that since Lighthill’s acoustic analogy does not include the effect of refraction due to non-uniform flow as a propagation term, then these effects are present in the analogy as a source term. By modifying the wave operator suitably, a more suitable source term could be chosen. Other acoustic analogies attempted to emphasize the roll of vorticity, notably those of Powell (Powell 1964) and of Howe (Howe 1975a;b).
These acoustic analogies formed the basis for investigation in the first period of research into the acoustics of jet flow. However, a fundamental shift began to occur with the discovery of large scale turbulent structure by Crow and Champagne (1971), Brown and Roshko (1974) and Winant and Broward (1974). It was begun to be recognized, that, the mathematical nature of Lighthill’s analogy being that of a quadrupole, actually reveals little about the physical mechanisms involved in noise generation, and indeed, may obscure the essential questions about the physics of jet noise generation. For an account of this see Tam (2002).
By the late 1970’s, attention was focused on the role of instability waves in the shear layer. Moore (1977) investigated the relationship between instability waves and jet
noise experimentally. In his view, the primary effect of instability waves in a subsonic jet was to modify the downstream turbulence, but not to be a direct source of noise itself. This idea was pursued, it was hoped that accurate prediction of downstream turbulence could lead to an accurate model of jet noise. Tam and Chen (1979) developed a model to attempt to predict turbulent statistics in a shear flow by using linear stability theory to determine the modes of the large scale structure. The continued development of these models by Morris, Giridharan, and Lilley (1990), Plaschko (1981), Viswanathan and Morris (1992) and others has lead to the U-RANS approaches suggested by Bailly, Lafon, and Candel (1996; 1997) and Khavaran and Bridges (2004; 2005).
As it became clear that instability waves influence the mixing process of jets, and indirectly the noise produced, some focus has been placed on influencing jet development, either through conditions at the nozzle, or by influencing the instability waves in the shear layers. The idea of acoustic excitation of the modes in a shear layer is not recent (Crighton 1981, Heavens 1980) and Miyagi, Hodoya, Fujita, Shoji, and Kimura (2006) and Miyagi, Hodoya, Fujita, Shoji, and Kimura (1999) have attempted to exploit such techniques to reduce acoustic emission intensities. Nozzle conditions were have also been investigated. Boersma, Brethouwer, and Nieuwstadt (1998) show that altering nozzle conditions not only influences the mixing region, but can even influence turbulent properties well downstream of the initial mixing region, and such results are behind the motivation of alternative nozzle types such as the Chevron. Still, to date, linear stability theory has not had a dramatic impact on developing a physical model for the noise generation process in subsonic jets, and the essential production mechanisms are under discussion.
However for supersonic jet flow, the approach has proven directly successful. For supersonic jet flow, it was already believed by Moore (1977) and others that Mach wave radiation was related to the instability waves of the flow. The physical mechanism was developed, first by the use of linear stability theory (Tam and Morris 1980) and later by the use of matched asymptotic expansions (Tam and Burton 1984, Tam and Chen 1994). The properties and character of Mach wave radiation has now been well established both numerically (Freund, Lele, and Moin 2000) and experimentally (Seiner, Ponton, Jansen, and Lagen 1992, Troutt and McLauglin 1982) and the matched asymptotic expansion model in particular has been demonstrably successful in predicting the features of Mach radiation. At this point, the view of Tam (2002) and others is worth emphasizing. That is, while acoustic analogies have been employed quite successfully for acoustic predictions, they do not of themselves ellicit the underlying physics of the noise generation mechanism. The development of a generalized framework (Goldstein 2003) for acoustic analogies is useful, as it is likely that specific acoustic analogies are required that target specific turbulent processes, if the link between them is to be fully understood.
2.3.2
Developments in jet computational aeroacoustics (CAA)
The modern period in aeroacoustic research began in the 1990’s, in parallel with devel-oping computer power. Prior to this, jet noise prediction was largely based on empirical methods (Bailly and Bogey 2004). Three types of simulation techniques have been
devel-oped - Direct Numerical Simulation (DNS), Large Eddy Simulation (LES) and unsteady RANS simulations. These techniques either obtain acoustic information directly by solv-ing the compressible flow equations, or obtain acoustic information in a secondary step in what is known as a hybrid technique. For compressible simulations, various methods have been developed which extend acoustic waves beyond the domain of the simulation. Among these are the porous FW-H analogy (Di Francescantonio 1997, Ffowcs Williams and Hawkings 1969), and the Kirchoff formulation (Lyrintzis 1994). Alternatively, one of the various acoustic analogies (Howe 1975a;b, Lighthill 1952, Lilley 1974, Moehring 1978, Phillips 1960, Powell 1964, Ribner 1962) can be applied to both data from an incompressible or compressible simulation. Typical approaches are illustrated in figure 2.1.
prediction Acoustic Compressible? Simulation type LES DNS U−RANS
Yes Yes Yes Yes or no
other analogy LEE
Porous FWH
Kirchoff / Lighthill /
Figure 2.1: Typical CAA noise prediction approaches
The DNS approach is typically extremely expensive and limited to low Reynolds numbers. Freund (2001) simulated a Reynolds number 3,600 jet and showed very good agreement with the experimental results of Stromberg et al. (1980). Additionally, Fre-und et al. (2000) demonstrated DNS capability for supersonic flows. Moore, Slot, and Boersma (2007) also demonstrated convincing results with their DNS. However few other 3-D compressible jet DNS studies exist, probably due to the prohibitive expense of such calculations.
Undoubtedly, the current trend in CAA is the use of Large Eddy Simulation (LES). In this approach, a filtered version of the flow equations is solved, and the effect of subgrid scales is modeled. Various LES have been performed, including Andersson, Eriksson, and Davidson (2004), Bodony and Lele (2004; 2005), Boersma and Lele (1999), Bogey, Bailly, and Juv´e (2003a), Constantinescu and Lele (2001), Lew, Blaisdell, and Lyrintzis (2005),
Seror and Sagaut (2000), Sinha, Chidambaram, Dash, and Seiner (1998), Uzun, Blaisdell, and Lyrintzis (2003; 2004). However LES as a design tool is still someway off. Typical air-craft jets can have Reynolds numbers exceeding 107, whilst the highest Reynolds numbers
LES have achieved are of the order 105 (Bogey and Bailly 2007, Uzun et al. 2004), and a
considerable amount of modeling and computational resources must be used to achieve such Reynolds numbers. In this gap, mixed LES-RANS approaches (Arunajatesan, Kannepalli, and Dash 2002, Tristanto 2004) and unsteady-RANS simulations (Wright, Blaisdell, and Lyrintzis 2004, Yan, Tawackolian, Michel, and Thiele 2007) can be found.
Many of the physical mechanisms underlying jet noise creation in both high and low Reynolds number jets are believed to be similar, so that understanding of the physics gained by studying low Reynolds number jets should lead to valuable insight in deriving models for the high Reynolds number flows. It is in this context of a desire for more detailed flow simulations in which the flows are solved directly and the impact of modeling is minimized, that this thesis takes place.
2.4
Jet description
A jet is a coherent stream of fluid that is projected into a surrounding medium, usually from some kind of nozzle. The stream of fluid may be initially laminar or turbulent, while the surrounding medium may be stationary with respect to the nozzle, or moving, in which case it is said to be a co-flow. From the flow on an initially laminar jet to the surrounding medium, there exists a layer called the shear layer across which the properties of the flow change sharply from the jet to the surrounding medium. This shear layer is usually unstable, and wave-like disturbances may form and grow outwards from the nozzle which disrupt the regular laminar behavior. Commonly these disturbances take a Kelvin-Helmholtz form which are characterized by an initially linear growth. Downstream, the disturbances lead to a rapid breakdown of the laminar jet flow to turbulent flow, and the individual disturbances modes couple in a nonlinear fashion. The initial laminar region of the jet is distinctly different from the downstream turbulent region of the jet and is often referred to as the jet potential core or the laminar jet core. The downstream region of the jet is characterized by chaotic turbulent behavior. In this region, as the jet incorporates surrounding fluid in a process known as entrainment, the jet mass and radius increases. If the Reynolds number of the jet is sufficiently high, then after a certain downstream distance, this region will self-assemble in such a manner that many of the averaged jet properties, when scaled appropriately, collapse to a single value. This region is referred to as the similarity region. Typically, the interactions of turbulent eddies with each other, and with the shear layer, may produce acoustic waves. The process by which this occurs is an important topic for this thesis. The features of the jet described in this section are illustrated in figure 2.2.
Note that by a jet, we do not mean the entire jet engine, which usually consists of several several component stages, such as a fan, low-pressure compressor, shaft region, combustion chamber, various turbines and a nozzle. In this work, we are only interested
Nozzle Entrainment Potential core Acoustic waves Kelvin-Helmholtz Instability Shear layer Similarity Region
Figure 2.2: Sketch of main features in the turbulent jet. For clarity, entrainment and acoustic emissions are only illustrated on one side of the jet, but in practice are symmetric on both sides of the jet.
in the features of jet flow, which is emitted from the exit nozzle, and not the internal mechanisms.
2.5
Compressible jet flow
Jet flow, like other forms of gas and liquid flows, is governed by well known equations that predict the evolution of the properties of the flow in time, from an initially known state. Both Lagrangian and Eulerian formulations exist (equations of motion following the flow and equations of motion at fixed points in space) and for this problem the Eulerian formulation is chosen as the most appropriate. For the purposes here, we are considering jet flows of a gas (such as air, or hydrogen) from a nozzle, into an initially quiescent volume of the same kind of gas.
2.5.1
State variables
For Newtonian fluids the complete state of the flow system at a given time is governed by the density ρ, the velocity u and the temperature T of the gas. Each of these variables is a function of the spatial location x where compression, expansion and heating are local effects related to specific flow conditions. Other flow properties are either universal, or can be determined from the knowledge of the above variables alone, so that ρ(x), u(x) and T (x) form a set of state variables. This choice of state variables is not unique and in this study the internal energy E is used in place of T .
2.5.2
Governing equations
The equations governing the flow of liquids and gases are well known (see for instance Batchelor (1967)). The equations themselves have a simple origin from a physical point of view. The Navier-Stokes equations are derived by application of Newton’s law of con-servation of momentum and forces to a fluid in each of the spatial dimensions. These are complemented by an equation derived by consideration of conservation of mass, and an equation derived by consideration of conservation of energy. These five equations are then able to completely determine the evolution of the five state variables. The equation for conservation of mass is ∂ρ ∂t + ∂ ∂xi ρui = 0, (2.3)
the Navier-Stokes equations in vector form is ∂ρui ∂t + ∂ ∂xj [ρuiuj+ pδij] = ∂ ∂xj τij, (2.4)
and the equation for conservation of energy is ∂E ∂t + ∂ ∂xj uj(p + E) = ∂ ∂xi κ∂T ∂xi + ∂ ∂xj uiτij (2.5)
where τij is the viscous stress tensor given by τij = µ ∂ui ∂xj + ∂uj ∂xi − 2 3δij ∂uk ∂xk , (2.6)
in which p is the pressure, ui is the velocity, µ the dynamic viscosity of the fluid and the
total energy E is the sum of the internal energy ρCvT and the kinetic energy ρuiui/2 where
κ is the thermal diffusion coefficient and δij is the Kronecker delta given by
δij =
(
1, if i = j
0, if i 6= j (2.7)
The thermodynamic quantities p, ρ and T are related to each other by the equation of state for an ideal gas
p = ρRT (2.8)
where R is the gas constant. The speed of sound is defined as c2 = ∂ ∂ρ s , (2.9)
where the subscript s denotes evaluation at fixed entropy.
For an ideal gas it follows that c =√γRT where γ is the specific heat ratio. For an ideal gas the speed of sound is thus only a function of the temperature and of the composition of the gas and independent of the density and pressure. The viscosity of the flow varies with temperature. The ambient value µ∞ occurs when the temperature is at
its ambient value T∞. Following Freund, Moin, and Lele (1997), this dependence is given
by Sutherland’s Law µ µ∞ = T T∞ 3 2 1.4T ∞ T + 0.4T∞ . (2.10)
2.5.3
Non-dimensional parameters.
All the variables in these equations are made non-dimensional using the ambient speed of sound c∞ as the reference velocity, the ambient density ρ∞ as the reference density, ρ∞c2∞
as the reference pressure, c2
∞/Cp as the reference temperature and the nozzle diameter Dj
as the reference length. The non-dimensionalization using sound speed is preferred due to the presence of sound waves in compressible flows which must also be captured by any compressible numerical scheme. These non-dimensionalizations are
x′ = x/D j (2.11) u′i = ui/c∞ (2.12) ρ′ = ρ/ρ∞ (2.13) T′ = c2∞/Cp (2.14) t′ = tDj/c∞. (2.15) p′ = p/(ρ∞c2∞). (2.16)
A second velocity scale is also important in this flow, that of the jet exit velocity Uj.
For analysis and description we use the Reynolds number based on the jet exit velocity, density, viscosity and nozzle diameter, and the Mach number as the ratio between the nozzle velocity and the nozzle sound speed:
Re = ρjUjDj µj , (2.17) Ma = Uj cj . (2.18)
2.5.4
Static vs stagnation temperature
The static temperature Ts is simply the temperature at a given point, while the stagnation
temperature Tt is the temperature that point would have if the flow at that point was
brought to rest, and the kinetic energy has been converted to internal energy and is added to the local static enthalpy. The ratio of these quantities is given by
Tt/Ts = 1 + γ − 1
2 M
2 (2.19)
where M is the local Mach number. The stagnation temperature is never smaller than the static temperature.
2.6
Numerical scheme to simulate jet flow
This section describes a discretization scheme for the aerodynamic simulation of compress-ible jet flows. In section 2.2, we introduced the concept of aeroacoustic phenomenon as a pressure variation that can be as little as 1 part in 5 billion of the background phenomenon, at the limit of hearing. Even loud acoustic waves are pressure variations that are between a thousand and a million times smaller than the ambient pressure. These acoustic waves represent only a small fraction of the total energy in the system and require very accurate numerical schemes to resolve properly. Hence we have developed a high order scheme, and perform simulations at double precision, in order to resolve the acoustic waves and the acoustic production process. The scheme is designed for performing Direct Numerical Sim-ulations (DNS) at low Reynolds numbers and this is discussed further. While the scheme is capable of treating general flow phenomenon, it has been designed with the case of jet flow in mind, and suitable boundary conditions for a jet flow are presented.
2.6.1
Direct Numerical Simulation
Direct Numerical Simulations are simulations in which the full Navier-Stokes equations are solved for the investigated problem, with all the scales of motion fully resolved without any turbulence model. Such a simulation will typically determine all the flow variables at dis-crete locations and times with sufficient accuracy to accurately describe the continous state
of the variables through interpolation. Thus an almost arbitrarily complete description of the simulated phenomena can be made available. This is especially useful in turbulence research where the time and space variation of flow variables can be difficult to measure accurately.
However, even if the flow is completely resolved, to simulate a flow in the real world requires matching initial conditions and boundary conditions. This is difficult for two reasons: one is that such data is often not completely available, and two is that a DNS by nature is attempting to simulate a flow in a finite domain, while real world flows occur in an effectively infinite domain, and it is usually necessary to introduce boundary and initial conditions which attempt to approximate the effect of an infinite domain on a flow of finite domain.
2.6.2
Staggered grid formulation
A staggered formulation of the flow variables is used, where the scalar quantities are defined at the cell centers and the velocity components are defined at the cell faces. More commonly used collocated schemes place all the flow quantities at a single grid location, and Lele (Lele 1992) has given generic formulae to derive such schemes. The situation for a staggered scheme is illustrated for two dimensions in figure 2.3. How does this work in practice?
Ui−1/2,j Ui+1/2, j i,j+1/2 V i,j−1/2 V i,j E , pi,j
Figure 2.3: An illustration of the staggered formulation.
Consider the calculation of ρ∂U∂x at the cell center. For a collocated second order scheme with a uniform grid, this would be evaluated as
ρ∂U ∂x i,j,k = ρi,j,k Ui+1,j,k− Ui−1,j,k 2dx . (2.20)
For the staggered formulation, instead this would be evaluated as ρ∂U ∂x i,j,k = ρi,j,k Ui+1/2,j,k − Ui−1/2,j,k dx . (2.21)
However, the calculation of, say, U∂V∂y at the cell face with a staggered scheme would require that U is first interpolated by
Ui,j,k = 0.5(Ui+1/2,j,k+ Ui−1/2,j,k). (2.22)
Then, U∂V ∂y i,j,k = Ui,j,k Vi,j+1/2,k− Vi,j−1/2,k dy . (2.23)
2.6.3
Spatial discretization
The numerical scheme is similar to that of Boersma (2005) and Moore et al. (2007), utilizing a staggered grid formulation. All the derivatives are calculated with the following compact finite difference formulation
a(fi+1′ + fi−1′ ) + fi′ = b
∆X(fi+1/2− fi−1/2) + c ∆X(fi+3/2 − fi−3/2) + d ∆X(fi+5/2− fi−5/2) + e ∆X(fi+7/2 − fi−7/2) (2.24) In which f′
i is derivative of f with respect to X in point i and ∆X is the grid spacing. For
the interpolation between various grid locations we use the following formula fi+ a(fi+1+ fi−1) =b(fi+1/2+ fi−1/2) + c(fi+3/2+ fi−3/2)
+d(fi+5/2+ fi−5/2) + e(fi+7/2+ fi−7/2) (2.25)
The values of the coefficients a, b, c, d, e can be found in Chapter 3. Close to the boundaries of the domain the order of the scheme has to be reduced. The exact procedure for this is given in Boersma (2005).
Note that this is a scheme for a uniform mesh, while the simulations performed use in fact a non-uniform mesh. To do this, the coordinates of the grids used in each simulation are given analytically by polynomials of order up to degree three. These are then mapped to a uniform grid where the computations take place.
2.6.4
Calculation of derivatives
The derivatives in equation (2.24) and the interpolations in equation (2.25) can not be calculated independently of each other with the scheme as described, because each equation contains 3 unknowns. Rather, the unknown required values of f′
i+1, fi−1′ , fi′ for derivatives
and fi, fi+1, fi−1for interpolations, each form a tridiagonal matrix. Various algorithms exist
for solving tridiagonal matrices. Two methods were evaluated for use in our simulations. The first algorithm implements straight forward Gaussian elimination, while the second adds partial pivotting to avoid the possibility of the calculation becoming ill-conditioned. It was found that for calculations performed in double precision, no significant difference was observed in the accuracy of the two methods, however, the addition of partial pivotting
added approximately 10% to the computational cost of the entire simulation. For this reason, the first algorithm is used in the present simulations. Still, it should be noted that in single precision, there is a loss of accuracy in the last 3 significant digits, so calculations performed in single precision (which can be faster on certain computer architectures and use less memory) can benefit from the use of the more sophisticated algorithm.
2.6.5
Temporal discretization
We give here a description of how solutions are updated in time. The updating is performed with a standard fourth order explicit Runge-Kutta method with a fixed time step. Suppose at the time step tn, the state of the system is described by yn. Suppose also that y is
governed by the equation
dy
dt = f (t, y) (2.26)
then if the solution at a time interval dt later is yn+1, it can be approximated numerically
by yn+1 = yn+ dt 6(K1+ 2K2+ 2K3+ K4), (2.27) where K1 =f (tn, yn) (2.28) K2 =f tn+1/2, yn+ dt 2 K1 (2.29) K3 =f tn+1/2, yn+ dt 2 K2 (2.30) K4 =f (tn+1, yn+ dt K3). (2.31)
and dt is determined by the CFL condition.
2.6.6
Computational cost of a direct numerical simulation
The main drawback of DNS is the computational cost of performing a calculation. The computational cost of a calculation typically scales as Ng × Nt, the total number of grid
points times the number of computational timesteps. The number of grid points in each dimension scales as Nx = L/dx, the domain length divided by the grid spacing, the number
of time steps scales as Nt ∝ L/Ma/dt and the domain length by L ∝ 1/Ma. The CFL
condition gives dt ∝ dx/Ma, while resolution of Komogorov length scales requires L/dx ∝ Re3/4. Hence for a three dimensional flow,
Cost ∝ Ng× Nt
∝ Nx3× L/Ma/dt
∝Re3/43/ (Ma)2× Re3/4/ (Ma)2 ∝ Re3/Ma4
A review of this is given by Bailly and Bogey (2004).
On current trends, computing power doubles roughly every two years. A simulation completed as part of this thesis took longer than a month, using part of a powerful, modern super computer. Supposing that 20 years from now, a PhD student was given access to equivalent resources, then that student could hope to achieve a Reynolds number of around 45, 000. So in the near future, DNS is unlikely to become a practical tool for flows of this type at industrial scales (where Re > 106), but it is possible to be used now for low
Reynolds number investigations, although even this is still quite costly.
2.6.7
Parallelization and MPI
The heavy requirements of performing these simulations means that powerful computer systems are required to solve them in a reasonable time frame. The code has been designed to make use of the multiple CPU cores that are present in supercomputer facilities and modern personal computers. This has been done by using the Message Passing Interface (MPI). MPI implementation is used for distributed memory systems, where each processor core has access to a unique section of the system memory, independently from each other processor core. If data in the memory of 1 computer core is required by another core, then communication must take place between the cores, via an MPI routine.
Implementation details
Parallelization is implemented for the simulations presented in this thesis by a method of domain decomposition. Initially, each process contains the 5 state variables, with a com-plete domain for the x and z axes, but divided across the y axis. Calculations are carried out independently by each process core for the data in the domain it is currently stor-ing. Quantities such as x spatial derivatives and z spatial derivatives can proceed directly. However, derivatives along y axis can not be carried out directly because information is required across several process boundaries. The MPI all to all command is first used to redistribute the data, so that domain distribution is divided along the z axis as illustrated in figure 2.4. Calculation of the derivative in the y then proceeds, and the result is redis-tributed again along the y axis with a further call to the MPI all to all command. Such an approach is required (as opposed to local communication between just a few nodes), because the calculation of derivatives along each axis requires the solution of a tridiagonal system involving all of the grid nodes along the axis.
Parallel efficiency
In figure 2.5, the efficiency of the calculation is given for various architectures, for a test simulation run with 64 ×64×64 grid points. The code was compiled with the Intel Fortran compiler and MPICH 2 for MPI for both the Core 2 Quad and the AMD Opteron, while for the IBM Power6 array, the IBM XL Fortran compiler was used with MPICH 2.
Y Y Px 1 Px 2 Px 3 Px 4 Px 1 Px 2 Px 3 Px 4 Z X X Z
Figure 2.4: MPI parallelization.
0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 9 10 Number threads
CPU time (sec) X n cores per iteration
Intel Core 2 Quad CPU Q6600 (4 X 2.4 Ghz) AMD Dual Core Opteron 875 (8 X 2.2 Ghz) IBM Power6 (32 X 4.7Ghz, SMT)
The IBM Power6 array (representing a single node of a super computer) scales particularly well almost uniform performance regardless of the number of cores used (and even a slight improvement with 64 vs 32 threads on 32 cores, utilizing SMT). The AMD Opteron well for computations using up to 4 cores, with 8 core calculations less efficient, but still worthwhile. The Core 2 Quad is about 50% less efficient when all 4 cores are used instead of 1 and 2. The efficiency of the parallelization is thus highly dependendent on the computer architecture and / or compiler used, but the gain for each case certainly justifies the effort for a parallel implementation.
2.6.8
Boundary Conditions
Devising accurate boundary conditions for aeroacoustic applications can be a complicated task (Colonius 2004, Freund et al. 1997). Common turbulence boundary conditions such as slip walls, no slip walls and periodic flow are inadequate due to the efficiency of acoustic wave propagation and reflection. Moreover, different types of boundary conditions may be required in a single simulation, such as an inflow condition, or an outflow condition. The case of jet flow in particular is certainly not simple. Ideally, to simulate a jet flow we would like to give the values of the state variables at the nozzle at each time step as input to the flow simulation. Boundary conditions would then automatically allow flow from outside the domain to enter, as required, to meet entrainment conditions. Downstream, the jet flow would leave the flow domain without returning, while aeroacoustic waves would propagate freely out of the domain without reflection.
The boundary conditions used for these simulations were developed over the course of the PhD study. As the simulations presented were also performed at different times over the study, they each use slightly different boundary conditions. We describe here the various approaches used. For each simulation, these are a combination of various techniques described by Bogey and Bailly (2002), Colonius, Lele, and Moin (1993), Freund (2001), Tam and Dong (1996). These boundary conditions are ad-hoc, intending to approximate physical conditions, as opposed to being themselves a physical part of the solution. As such, they employ a significant number of parameters that require some tuning for each specific simulation performed.
Flow averaging
For some of the boundary conditions, an estimate of the mean flow at or near the boundary is required. The procedure used in these simulations was to average both in time and tangentially. In order to average in time, two running averages were taken, A1 consisting of the last n time steps, and A2 consisiting of the last n + no time steps, which is used for
the boundary condition. Every n = no time steps, A2 is replaced by A1 and A1 is reset. In
this way, an average is available for the boundary condition which is formed from between the last no and 2no time steps. For spatial averaging in the axisymmetric direction, the
flow quantities are transformed to a polar mesh using a cubic spline fit. This was important for simulations presented in Chapters 4, 5, and 6.
Inflow boundary
The presence of a nozzle is not implicitly included in the simulations. Rather, an inflow profile for the velocity is employed, for the mean jet flow in the potential core region after the nozzle is. Two typical choices for the velocity profile, for example, are those suggested by Michalke (1984), which match the mean velocity flow of real jets just downstream of the nozzle exit.
Profile I: U = 0.5 1 + tanh Rj θ 1 − r Rj (2.32) Profile II: U = 0.5 1 + tanh 0.25Rj θ Rj r − r Rj (2.33) where Rj is the radius of the nozzle and δθ is the incompressible momentum thickness
defined by δθ = Z ∞ 0 U U0 1 − UU 0 dr. (2.34)
For the simulations presented here, Profile II was used exclusively, although in some pre-vious work we have also used Profile I. The temperature variation at the inflow is included by modifying the density and internal energy of the flow. The inflow profiles for density, energy and the other velocity components are given in equations (2.35:a-e).
ui = Mao 2 1 − tanh B r r0 − r0 r (2.35a) vi = 0 (2.35b) wi = 0 (2.35c) ρi = 1 + To Tj − 1 ui Mao (2.35d) Ei = 1 γ(γ − 1) + 1 2ρiu 2 i. (2.35e)
For the inflow boundary, additional terms are added to the equations of motion to force the flow to the inflow state of the jet in a very small region. The purpose of this forcing is to dampen the acoustic waves in that region of the flow and for numerical stability. This region of the jet is not considered to be a physical region in any way. However, downstream of this region the flow closely resembles that of a real jet, from where it will develop according to the conservation equations of jet flow.
Instability at the inflow
A common technique in jet simulations is to add small perturbations to the velocities in order to facilitate the growth of the unstable modes of the jet. However, in these com-pressible simulations, many methods of adding flow perturbations lead to noise generated
directly at the inflow. While noise generated at the nozzle may be a significant acoustic source in experimental facilities (Viswanathan 2004), in these simulations we are interested in isolating the noise generated downstream in the turbulent mixing region of the jet. The method we use to seed instability is similar to that of Freund (2001). Essentially, we add to the parameter controlling the thickness of the shear layer, B, azimuthal modes that are randomly updated in time, so that
B = B(θ, t) = B0 + X m X n Anmcos (fnmt + φnm) cos (mθ + ψnm) (2.36)
where the values of Anm, fnm, φnm, ψnm are updated randomly over time.
Outflow boundary
The outflow condition is the simplest of all the conditions. Flow is artificially convected to a supersonic speed as it approaches the outlow boundary. With a supersonic outflow, the properties of the boundary are well-posed without requiring that any of the flow variables are specified. In addition, the flow is progressively filtered in this region, to remove large fluctuations at the outflow, which were found to sometimes lead to an unstable numerical solution. For each state variable Q, the filtered quantity bQ is obtained by applying along each of the coordinate axes a spatial filter. After the flow quantity has been integrated in time, the value is updated according to
Qnew(x, y, z) = Qold(x, y, z)+0.5(1−tanh[af il(xmax−x−xf il)])( bQold(x, y, z)−Qold(x, y, z)).
(2.37) The values of af il and xf il are chosen independently for each simulation, and the values
used are listed in the appropriate section.
Sideline condition 1: forcing to specified flow state
At the sideline boundaries, we add damping terms to the equations of motion in localized layers in a similar manner as at the inflow. These damping terms are chosen to force the flow to the specified boundary state. For the calculations presented in this thesis, in Chapter 3, the sideline forcing used the ambient state as reference flow, in Chapters 4 and 5, the flow is forced to a local flow average, while in Chapter 6, no forcing of the flow is used (described in the next section).
Sideline condition 2: Radiation condition.
Several methods exist that allow acoustic waves to freely propagate outside numerical do-mains by providing an estimation of the acoustic waves at the boundary. Any inaccuracies in the estimate generate incoming acoustic waves. The most commonly used in aeroacous-tics are probably the characteristic boundary conditions due to Thompson (1987; 1990). However, these boundary conditions are only efficient at absorbing acoustic waves with normal incidence, and can be poor at absorbing acoustic waves with an acute incidence.
