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Aeroacoustic Computation of Low Mach Number Flow
Kristian Skriver Dahl
RECEIVED FFB 1 3 W97 OSTI
MASTER
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Ris0 National Laboratory, Roskilde, Denmark December 1996 DISCLAIMER
Portions of this document may be illegible in electronic image products. Images are produced from the best available original document Aeroacoustic Computation of Ris0-R-947(EN) Low Mach Number Flow
Kristian Skriver Dahl
Ris0 National Laboratory, Roskilde, Denmark December 1996 Note This report is the corrected version of the thesis submitted in August 1996 to the Technical University of Denmark in partial fulfillment of the require ments for the Ph.D. degree. The thesis was defended in November 1996. Examin ers were Dr. J.C. Hardin, NASA Langley Research Center, Dr. C. Westergaard, LM Glasfiber A/S, and Associate Professor P. Grove Thomsen, Department of Mathematical Modelling, Technical University of Denmark. Advisors were Dr. H. Aagaard Madsen, The Test Station for Wind Turbines, Risp National Laboratory and Dr. J.N. Sprensen, Department of Energy Engineering, Technical University of Denmark.
ISBN 87-550-2249-9 ISSN 0106-2840
Grafisk Service • Risp • 1996 Abstract This thesis explores the possibilities of applying a recently devel oped numerical technique to predict aerodynamically generated sound from wind turbines. The technique is a perturbation technique that has the advantage that the underlying flow field and the sound field are computed separately. Solution of the incompressible, time dependent flow field yields a hydrodynamic density correction to the incompressible constant density. The sound field is calculated from a set of equations governing the inviscid perturbations about the corrected flow field. Here, the emphasis is placed on the computation of the sound field. The nonlinear partial differential equations governing the sound field are solved numerically using an explicit MacCormack scheme. Two types of nonreflecting boundary conditions are applied; one based on the asymptotic solution of the gov erning equations and the other based on a characteristic analysis of the governing equations. The former condition is easy to use and it performs slightly better than the characteristic based condition. The technique is applied to the problems of the sound generation of a pulsating sphere, which is a monopole; a co-rotating vortex pair, which is a quadrupole, and the viscous flow over a circular cylinder, which is a dipole. The governing equations are written and solved for spherical, Cartesian, and cylindrical coordinates, respectively, thus, representing three com mon orthogonal coordinate systems. Numerical results agree very well with the analytical solutions for the problems of the pulsating sphere and the co-rotating vortex pair. Numerical results for the viscous flow over a cylinder are presented and evaluated qualitatively. The technique has potential for applications to airfoil flows as they are on a wind turbine blade, as well as for other low Mach number flows.
This report is submitted to the Technical University of Denmark in partial fulfill ment of the requirements for the Ph.D. degree.
Cover figure. Acoustic pressure for flow around a circular cylinder; carpet plot version of Figure 6.9d on page 65. Abstrakt Denne afhandling unders0ger mulighederne for anvendelse af en nyud- viklet teknik til at forudsige aerodynamisk genereret lyd fra vindm0ller. Teknikken er en perturbationsteknik, som bar den fordel, at det underliggende str0mningsfelt og lydfeltet kan beregnes hver for sig. L0sning af det inkompressible, tidsafhaengige strpmningsfelt giver mulighed for beregning af en hydrodynamisk densitetskorrek- tion af den konstante densitet (massefylden af fluiden). Lydfeltet beregnes som inviskpse perturbationer omkring det korrigerede strpmningsfelt. I denne afhan dling er vaegten lagt pa den numeriske beregning af lydfeltet. De ulineaere, par- tielle differentialligninger, som beskriver perturbationerne, loses ved hjaelp af et eksplicit MacCormack-skema. To slags ikke-reflekterende randbetingelser anven- des. Den ene er baseret p& en asymptotisk lpsning af de styrende ligninger, den anden er baseret p& den karakteristiske form af de styrende ligninger. Den fprste betingelse er let at anvende, og den er en simile bedre end betingelsen baseret p& den karakteristiske form. Teknikken er anvendt til bestemmelse af lydgenererin- gen af: en pulserende kugle, som er en monopol; et roterende hvirvelpar, der er en kvadrupol; og den viskpse strpmning omkring en cirkulaer cylinder, som er en dipol. De styrende ligninger er skrevet og Ipst i henholdsvis sfasriske, kartesiske og cylindriske koordinater, s&ledes er tre gaengse ortogonale koordinatsystemer repraesenteret. De numeriske resultater er i meget god overensstemmelse med de analytiske Ipsninger for den pulserende kugle og det roterende hvirvelpar. Resul- taterne for cylinderstrpmningen praesenteres og vurderes kvalitativt. Teknikken bar potentiale til beregning af lydgenereringen af luftstrpmningerne, der finder sted pa vindmpllevinger, samt andre fluidstrpmninger ved lave Mach-tal.
Denne afliandling er et led i erhvervelsen af Ph.D.-graden ved Danmarks Tekniske Universitet. Afliandlingens danske titel er: “Aeroakustisk beregning af fluidstrpm ninger ved lave Mach-tal. ” Contents
Preface and Acknowledgements 7
1 Introduction 9 1.1 Noise from wind turbines 9 1.1.1 Mechanical noise 10 1.1.2 Aerodynamic noise 10 1.2 Computational aeroacoustics 11 1.3 The present study 13
2 Theoretical background 14 2.1 Governing equations for compressible flow 14 2.2 Governing equations for incompressible flow 15 2.3 Hydrodynamic density correction 15 2.4 Governing equations for the perturbations 16 2.5 Solution strategy 17
3 Numerical method and boundary conditions 19 3.1 The MacCormack scheme 19 3.2 Boundary conditions 19 3.2.1 Asymptotic approach 20 3.2.2 Characteristics approach 20
4 Pulsating sphere 23 4.1 Incompressible flow 23 4.2 Hydrodynamic density correction 24 4.3 Governing equations for perturbations 25 4.4 MacCormack scheme 25 4.5 Boundary conditions 26 4.5.1 Asymptotic approach, TWBC 26 4.5.2 Characteristics approach, ThBC 27 4.6 Results 28 4.6.1 Numerical set-up 28 4.6.2 Comparisons of numerical and analytical solutions 29 4.6.3 Comparison of outer boundary conditions 34 4.6.4 Initial conditions 35 4.7 Summary 35
5 Co-rotating vortex pair 37 5.1 Incompressible flow 37 5.2 Hydrodynamic density correction 39 5.3 Governing equations for perturbations 41 5.4 Boundary conditions 4% 5.4.1 Asymptotic approach, TWBC 4% 5.4.2 Characteristics approach, ThBC 4% 5.5 Results 4$ 5.5.1 Numerical set-up ^5 5.5.2 Effect of boundary conditions. 40 5.5.3 Effect of domain size 51 5.5.4 Effect of temporal resolution 53
Ris0-R-947(EN) 5 5.5.5 Numerical solution for M = 0.11 54 5.6 Summary 55
6 Viscous flow over cylinder 56 6.1 Incompressible flow 56 6.1.1 Vorticity-stream function formulation 57 6.1.2 Numerical results for incompressible flow 59 6.2 Hydrodynamic density correction 61 6.3 Governing equations for perturbations 61 6.4 Boundary conditions 62 6.5 Results 62 6.6 Summary 65
7 Conclusion 66
References 68
N omenclature 71
List of Figures 75
List of Tables 76
6 Ris0-R-947(EN) Preface and Acknowledgements
This report is submitted to the Technical University of Denmark (DTU) in partial fulfillment of the requirements for the Ph.D. degree. I did the work at The Test Station for Wind Turbines (PFV), Ris0 National Laboratory and at Christopher Newport University (GNU) and NASA Langley Research Center (LaRC), Virginia, USA during the period from January 1993 to August 1996 . The work was funded by the Danish Research Academy and Risp National Laboratory. Thanks go to my research advisors Dr. Helge Aagaard Madsen, Senior Research Scientist at PFV, and Dr. Jens N0rkaer S0rensen, Associate Professor at the Fluid Mechanics Section, Department of Energy Engineering, DTU. During the study, I spent a year at CNU, working under the supervision of Dr. James E. Martin, Assistant Professor at the Department of Mathematics, CNU and Dr. Jay C. Hardin, Chief Scientist at the Acoustics Division, NASA LaRC. I am indebted to Dr. Martin and Dr. Hardin for making the stay very beneficial and worthwhile for me. I thank D. Stuart Pope at Lockheed Engineering and Sciences Company, Hampton, Virginia, USA, for fruitful discussions and for letting me use his code for the computations of the viscous flow over a cylinder. I acknowledge the kind treatment by Dean George R. Webb and the staff and faculty at the College of Science and Technology at CNU. In the final writing stages of this thesis, Dr. John A. Ekaterinaris, Senior Re search Scientist at PFV has tirelessly reviewed, corrected, and suggested improve ments for my writing. I am grateful to him for that, and it goes without saying that all remaining errors and occurrences of bad English are my own doings. Other colleagues at PFV have contributed with valuable comments in the review process, which I very much appreciate. Finally, I am especially grateful to Monica for having taken care of our seven month old son, Gustav, without much assistance from me lately. Though she has done a very good job, and he has not suffered from the overwhelming female in fluence, I look forward to teach Gustav some of that male stuff involving watching complete Tour de France mountain stages on television and following RBOG’s quest for promotion to Superligaen.
Kristian Skriver Dahl
Roskilde, Denmark August 1996
Ris0-R-947(EN) 7
1 Introduction
The increasing concern for a more pleasant environment to live and work in, has in recent years drawn attention to noise pollution. Noise does not impose a threat to the future of this planet, as opposed to the chemical and radioactive pollution of air and soil. However, particularly, in developed countries, restrictions on noise emission force the industrial manufacturers to consider noise reduction and control as a part of the design process.
1.1 Noise from wind turbines The objective of this work is the application and testing of a numerical method capable of predicting aerodynamic noise from wind turbines. A numerical method, that eventually can supplement the commonly used semi-empirical methods. For wind turbines in operation, the only pollutant is noise. (Even though, some find that the mere sight of wind turbines is polluting, whether operating or not.) In densely populated countries, such as Denmark, the noise is often a serious obstacle to the siting of new wind turbines, and today we always see a noise emission curve alongside the power curve in the wind turbine sales material.
Sound Power Level dB(A) re 10~12W
10000 Rated Electric Power, kW Figure 1.1: Sound power levels (SWL) for different sizes of wind turbines. SWL = 10 log(W z/Tyrcf), where W is the sound power and Wrct = 10-12W for air. [31]
The straight line in Figure 1.1 indicates that a rough estimation of the sound power emission from a wind turbine is obtained by multiplying its rated power with a factor of 10-7 . For example, a 500 kW wind turbine operating 24 hours a day for nine days, produces sound energy just enough to boil an egg. Thus, acoustic energy losses are negligible and do not influence at all the yearly energy production of a wind turbine. However, very little acoustic energy output can cause much nuisance. (Other examples of the sound power emission of every day events are: the human shout emits about 10-5 W, and a large jet transport at take-off emits about 100 kW [10, page 12].) Therefore, public acceptance of wind turbines (good will), in addition to current regulations and mutual competition
Ris0-R-947(EN) 9 in the wind turbine industry, compel the wind turbine manufacturers to take the sound emission seriously. In Germany, for example, subsidies to wind turbines are dependent upon their noise emission [12]. A typical noise control concept for, e.g., jet noise is the division of a large single jet into many smaller jets which enhances the mixing with ambient air [14]. An ex ample of noise control of internal flows is the case of cavitation noise in hydraulics systems, where pressure reductions should happen in several stages rather than a single big reduction [25]. For wind turbines the following characteristics complicate noise control. • The source is elevated, which is the cause for minimal attenuation and shelter at the ground. • The source operates in natural windy conditions, sometimes complicated by complex terrain and wind farms. The classical theory of acoustics, e.g., [26, 39, 44], deals with sound propagation in stationary media. • Wind turbines operate and emit sound continuously, as opposed to, e.g., traffic noise that reaches its peak at rush hours. The noise from wind turbines can be divided into two main categories: mechanical noise and aerodynamic noise.
1.1.1 Mechanical noise The mechanical noise is mainly due to the gearbox. As the noise is tonal and unnatural, it is not masked by the wind to the same extend as is the aerodynamic noise. In general, the mechanical noise is the main contributor to the total noise from smaller wind turbines (< 100 kW). The mechanical noise generation mecha nisms are well understood and can be addressed at the design stage through, e.g., vibration insulation and use of acoustically absorbing material.
1.1.2 Aerodynamic noise The aerodynamic noise from wind turbines is the most important noise source especially for larger wind turbines. Aerodynamic noise can be divided in three main categories. • Discrete frequency noise, originating from tower/blade-interaction and wind shear • Airfoil self-noise, i.e.: turbulent boundary layer interacting with the trailing edge, separation-stall, tip vortex formation, laminar boundary layer causing vortex shedding, and vortex shedding from blunt trailing edge. • Inflow turbulence noise. The process of the aerodynamic sound generation of a wind turbine is not well understood and further research on the subject is necessary to come to a better understanding of the fundamental mechanisms involved, and finally, find means to control it. Towards this end, several prediction methods are available. Semi-empirical methods for the prediction of aerodynamically generated wind turbine noise are given by Grosveld (1985) [16] and Lowson and Fiddes (1994) [32], and by Brooks et al. (1989) [7], for the prediction of airfoil self-noise, in partic ular. The basis for these prediction models is the boundary layer parameters on the airfoil sections. However, capturing of all physical processes involved with the aerodynamic sound generation and propagation requires the use of a method ca pable of predicting these phenomena from first principles. This is the topic of
10 Ris0-R-947(EN) computational aeroacoustics (CAA), a fairly new discipline that emerged in the 1980s. It is the objective of this thesis to explore the possibilities of applying a recently developed technique for CAA and perform computations for model problems. This technique has potential not only for the prediction of airfoil self-noise of wind turbines, but also has applications in aeronautics, automotive and home appliance industries.
1.2 Computational aeroacoustics The origin of computational aeroacoustics (CAA) is computational fluid dynamics (CFD). CFD has matured during the past 20 years due to advances in computer technology and numerical algorithms, and is now routinely being used by industry for research and even for design purposes. Current methods in CFD are predominantly Eulerian. The flow domain of in terest is discretized, i.e., a mesh is constructed in the computational domain of interest. A numerical scheme is used to solve the discretized governing equations. The scheme can be based on finite differences, finite volume, finite element, or spectral methods. For external flow problems, since computer resources are not unlimited, the computational domain has to be finite, and artificial boundary conditions must be imposed on the edges of the domain. Hardin (1994) [17] defines CAA as “the employment of computational fluid dynamics (CFD) techniques in the direct calculation of all aspects of sound generation and propagation with application to aeronautics in which the sound field, and perhaps the underlying flow field as well, is calculated starting from the fundamental differential or integral equations describing the fluid motion. ” or, following Crighton (1988) [8], as “the study by largely numerical techniques of the distant acoustic fields generated by unsteady fluid flows” What makes CAA a truly new scientific discipline amongst all the disciplines with the prefix computational, is, citing Crighton (1993) [9] “ ... that all numerical procedures are inherently noisy (in the acoustic sense). ” That is, the numerical treatment itself introduces acoustic sources in the flow; the strength of these sources depends on the accuracy of the numerical scheme. There fore, CAA requires development of new numerical schemes with minimal diffusion (dissipation and dispersion). In addition, nonreflective numerical boundary condi tions must be developed to avoid generation of spurious reflections which contam inate the numerical solution. Sound propagation is different in many respects from fluid flow. Straightforward application of CFD methods to aeroacoustic problems is not always possible for the following reasons. In most cases, problems encoun tered in aerodynamics are time independent, whereas, by definition, aeroacoustic problems are time dependent. Another complication is the disparity of scales in aeroacoustics. The magnitude of the acoustic quantities of interest is typically very small compared to the magnitude of the flow variables. For instance, referring to the previously mentioned 500 kW wind turbine example, when dealing with acous tics, the interest is on energy with a magnitude that integrated over nine days is just enough to boil an egg. In aerodynamics, on the other hand, the interest is on energy with a magnitude that integrated over the same period of time is enough to supply 20 families with electricity for a year. Furthermore, the ranges of audible
Ris0-R-947(EN) 11 sound fields are enormous. The dynamic range is from 0 dB at the threshold of hearing to 140 dB at the threshold of pain, i.e., a variation by a factor of 1014. The audible frequency range is from 20 — 20000 Hz, where the ear is most sensitive to the range 2000 — 4000 Hz. Aerodynamicists are mostly interested in the near-field, e.g., lift and drag on an airfoil. This requires highly stretched meshes close to the body surface to resolve the near-wall viscous layers. The viscous mechanisms take place due to eddies of dimension L/Re3/4 (the Kolmogoroff scale) [27]. Here, L is a typical length scale for the flow field and Re is the Reynolds number (a measure of the ratio of the inertia to viscous forces). Aeroacousticians, on the contrary, are mainly interested in the far-field sound level. And as a result, CAA requires a large computational domain with uniform mesh distribution to ensure good resolution of the acoustic waves outgoing from the noise source. The need for large computational domains comes from the fact, that a typical wavelength A for the acoustic field scales like L/M, where M = U/c is the Mach number, defined as the ratio of the convective speed U and the speed of sound c. Hence, at low Mach numbers the computational domain should be several times the size of the typical length scale in the flow field, e.g., the airfoil chord in aerodynamic flow problems. For a wind turbine, Rem 2 x 106 and M m 0.2 in the tip region of the blades. The speed of sound, c = 343 m/s in air at 20°C. A fluid flow can usually be assumed to be incompressible for M < 0.3; a complication of aeroacoustic computation at low Mach numbers is that the flow field is weakly coupled to its acoustic field. Crighton (1988) [8] argues that high Mach number flows are more amenable to numerical computation. Sound sources are usually inefficient in low Mach number flows; acoustic power to flow power scales as M3 for dipole sound, e.g., flow over cylinder, and as M5 for quadrupole sound, e.g., free turbulence away from solid bodies [30]. This fact further complicates things, because it raises the risk that the numerical procedure applied might be noisier than the flow when dealing with the low Mach number flows typical for wind turbines. Two approaches can be followed in CAA. The most obvious, perhaps, is direct numerical simulation (DNS). In DNS the full set of equations governing the time dependent, compressible, viscous flow are solved, yielding the complete flow and sound fields, e.g., [36]. This approach implies that the same numerical scheme is used for both the flow field and the acoustic field. Because of the weak coupling between flow and acoustic field and the acoustic inefficiency at low Mach numbers, this may not be the best approach, since the acoustic field may be corrupted by numerical errors in the underlying flow quantities. In addition, DNS requires a mesh and a discretization procedure which enables adequate resolution of the vis cous region and simultaneously, secures good wavetracking in the far-field. These are contradicting requirements. The Euler equations are capable of describing sound propagation once the noise source is known (the classical wave equation is derived from the linearized Euler equations). This is the basis for the perturbation approaches, where the under lying flow is solved, followed by a solution of perturbations around the flow, as in [19, 48]. This approach allows the use of optimal schemes and meshes for the computation of both the underlying flow field and the acoustic field. By ignoring viscous effects on the perturbations, it furthermore yields a hyperbolic set of equa tions. Thus, simplifying the formulation of boundary conditions, because of the limited domain of dependence of hyperbolic equations as opposed to the time de pendent, compressible Navier-Stokes equations solved in DNS, which are a mixed set of hyperbolic-parabolic equations. The downside of ignoring viscous effects on the perturbations is that transients can persist in the computational domain. In [37], without judging either approach superior, Myers (1993) leans towards
12 Ris0-R-947(EN) the perturbation approach as long as the acoustic field remains continuous, i.e., shocks and other discontinuities do not occur.
1.3 The present study Since this thesis is a step towards an aeroacoustic computation of airfoil flows on wind turbine blades, the perturbation approach appears to be ideal for the Mach number range for which wind turbines as well as automobiles operate in. It also has potential applications for computation of the sound generated by home appliances, such as lawn mower, vacuum cleaners, hair dryers, etc. Furthermore, the perturbation approach allows use of already developed CFD codes for the calculation of the underlying incompressible flow field, so the emphasis can be placed on the perturbation solver. The method is a newly developed technique by Hardin and Pope (1992) [19] for aerodynamic noise calculation. They base the aeroacoustic computation on a solution of the time dependent, incompressible flow equations. Thus, the technique is only valid for low Mach- number flows where the assumption of incompressibility is valid, which is the case for wind turbines. The acoustic field is then found from the solution of the perturbed, time dependent, inviscid, compressible equations governing the pertur bations about the corrected incompressible flow field. The theory is presented in more detail in Section 2. In Section 3, the numerical scheme used to solve the equations governing the acoustic field is described. Two types of numerical boundary conditions are de scribed. One based on the characteristic form of the governing equations; the other based on an asymptotic solution of the governing equations. The sound emission from a pulsating sphere is computed in Section 4. This problem is one-dimensional and can be described using spherical coordinates. In Section 5, the two-dimensional problem of two co-rotating vortices is considered and the governing equations are written in Cartesian coordinates. For both prob lems of Section 4 and 5, analytical solutions exist for both the incompressible flow field and the acoustic field. Section 6 presents results from an aeroacoustic computation of the viscous flow over a circular cylinder. The governing equations are cast in cylindrical coordi nates. The incompressible, viscous flow field is computed using a CFD code based on the vorticity-stream function formulation, and a Poisson equation for pressure. The results are only evaluated qualitatively, since the aeroacoustic solution never stabilized. Given a working aeroacoustic code in cylindrical coordinates, the exten sion to airfoil flows is straightforward using conformal mapping to transform the cylinder to a Joukowski-profile. Thus, allowing the investigation of the different airfoil self-noise mechanisms of wind turbine blades. However, this is not done in the present work. The three problems represent different orthogonal coordinate systems, as well as the three elementary sound sources, i.e., a monopole, a quadrupole, and a dipole, respectively. Conclusions and recommendations for future work conclude the thesis in Sec tion 7.
Ris0-R-947(EN) 13 2 Theoretical background
This section presents a new technique for aerodynamic noise calculation, proposed by Hardin and Pope (1992) [19]. In 1994 [20], the authors named this technique: “An Acoustic/Viscous Splitting Technique for Computational Aeroacoustics. ” The technique belongs to the category of perturbation approaches. The approach consists of the following steps. It begins with computation of the underlying, time dependent, incompressible flow field. A correction to the constant density field is obtained using isentropic relations and a pressure fluctuation which is the difference between the instantaneous incompressible pressure and its time- average. Finally, the perturbations about the corrected flow field are computed solving a set of equations, from now on referred to as Hardin ’s equations, for convenience. To the best of my knowledge, Hardin ’s equations has until now only been applied to the following problems: computation of the acoustic field from a pulsating sphere and an oscillating sphere, Hardin and Pope (1992) [19]; acoustic field of viscous, incompressible flow over cavity, Hardin and Pope (1995) [21]; and finally, Lee and Koo (1995) [28] computed the acoustic field from a co-rotating vortex pair. Assuming that Hardin ’s equations are not as yet widely known, their derivation is presented below, based on [19, 20].
2.1 Governing equations for compressible flow The equations governing the time dependent motion of a compressible, viscous fluid are the Navier-Stokes equations (strictly speaking, the N-S equations only refer to the momentum equations). The vector form of these equations is, continuity equation, % + V' (H = °: (2.1) momentum equations ^(pv) + V- (pvv) = -Vp + V-r, (2.2) energy equation, P°p^ ~ = T®Vy - V* 9, (2.3) supplemented by the equation of state which relates density p and pressure p,
P = p(p, S). (2.4) Here v denotes the vector of the velocity components, r is the viscous stress tensor, which for a Cartesian coordinate system is
(i,j,k = 1,2,3), where p, is the dynamic viscosity, a fluid property dependent on temperature T, and Sij is the Kronecker delta function (= 0 if i ^ j and = 1 if i = j). The substantial derivative is >=!< )+”-v( >■ ct, is the specific heat at constant pressure, (3 is the thermal expansion coefficient, q = —kVT is the heat flux vector, k is the thermal conductivity. The first term on the right-hand side of (2.3)—the inner product of two tensors, Tijdvi/dxj— represents the temperature increase due to viscous dissipation. The second term
14 Ris0-R-947(EN) on the right-hand side of (2.3), is the temperature change as an effect of heat transfer, and S is entropy. A direct numerical simulation (DNS) based on (2.1)-(2.4), yields the viscous as well the acoustic field. Here, however, for the reasons pointed out in the In troduction, the perturbation approach is preferred to the DNS approach. The perturbation approach requires a little more rewriting of the equations, shown in the following.
2.2 Governing equations for incompressible flow The compressible flow equations are simplified, when the fluid or the flow is in compressible, i.e., p = po- Letting capital letters denote the incompressible flow field variables, the complete set of incompressible flow equations is, continuity equation, V- V = 0, (2.5) and the momentum equations, po + V-V V) = - VP + p 0V2V. (2.6) where po denotes the constant dynamic viscosity of the fluid. This is a closed set of equations. Solution of (2.5) and (2.6) yields the time-dependent, incompressible velocity V(x, t) and pressure field P(x, t) which, subsequently, will be used to find the hydrodynamic density correction to the incompressible solution.
2.3 Hydro dynamic density correction The assumption of incompressibility is only an approximation, and hydrodynamic density fluctuations are always present in the flow. For a free stream Mach number below 0.3, the maximum density variations are less than 5 percent of the farfleld or ambient density [11, p.40]. These fluctuations can safely be ignored for the com putation of the flow field, but they are important for the acoustic field. Consider, for example, a plane harmonic sound wave in air, with a sound pressure level of 140 dB, corresponding to the threshold of pain. In this wave, the density variations are less than 0.4 percent of the ambient density. Thus, the hydrodynamic density fluctuations become important. The pressure changes from the ambient pressure po in an incompressible flow are dp = P(x,t) -p Q. Equation (2.4) implies that
where c is the speed of sound defined by
Now, Hardin and Pope assume that the flow is stationary or time periodic so that time averages are meaningful, and obtain a time averaged pressure distribution as, 1 P{ x) = lim (2.8) T-hx ) T
Ris0-R-947(EN) 15 where T is averaging time. For time periodic flow, such as the flow over a cylinder, (2.8) can be simplified to _ i rto+T p P(x) = 7jT P{x,t)dt, (2.9) 1p Jt0 where Tp is the fundamental period of the flow oscillations, and to is the point in time, where the flow has reached time periodicity, i.e., transients have died out. Following Batchelor (1967) [4, pages 167-168], who states that the effects of viscosity and heat conduction normally control the pressure distribution rather than the pressure fluctuations, Hardin and Pope regard the time averaged pressure to be a result of the dissipative mechanisms and the pressure fluctuation pi (x, t) = P(x, t) — P(x) to be isentropic, i.e., from (2.7) Pi =Pi/c2, (2.10) where pi is an estimate of the hydrodynamic density fluctuations taking place in an incompressible flow. Division by po, the ambient undisturbed fluid density, on both sides of (2.10), and use of the Mach number definition, M = U/c, gives
Po PoU 2 The ratio pi/(poU 2) on the right-hand side of this expression is of the order of one. Therefore, pi is negligible for low Mach number fluid flow. Substitution of pi(x, t) = P(x,t) — P(x) in (2.10) yields the correction of the constant density field P(x,t) - P(x) pi(x,t) (2.11)
2.4 Governing equations for the perturbations To derive the set of equations governing the perturbations, (referred to as Hardin ’s equations), Hardin and Pope let
P = Po + Pi + p\ v = V + v', (2.12) p = P+p ‘, where p ' and the vector v1 are the fluctuations about the incompressible pressure and velocity fields, respectively, p ' is the density fluctuation about the corrected density field po + Pi- The perturbations, p 1, v1, and p ‘ constitute the acoustic variables in the farfield where the underlying incompressible flow field has reached the ambient state. Substitution of (2.12) into (2.1) and (2.2), ignoring viscous effects on the per turbations, yields with the use of (2.5) and (2.6) the following equations for the perturbations, ^ + v-Vp' + PV-v' = -^-v-VPl, (2.13) and, ^+v.Vv- + Yil Pi + p 1 dV (pi + p 1 -V + V') w. (2.14) at p p dt By definition p ‘ = p(p, S) - P{po, S), and dp 1 _ (dp\ dp \dp) s
16 Ris0-R-947(EN) Since, p' = p'(p) dp 1 _ dp' dp 2dp = c and Vp' = c2Vp, dt dp dt dt ’ where,
- W + 1%' and VA>-Vp! + Vp'.
Thus, ~it = c2~£t +c2%~’ and Vp ' = c2Vpi +c2Vp'.
Now, (2.13) can be written in terms of p' yielding an equation for the pressure perturbations
+ u-Vp' + pc 2V- v' = 0.
The equations governing the perturbations, or Hardin ’s equations, are
-7^- + v-Vp' + pV* v' = —7^- — u-Vpi, (2.15)
^+„.v„- + ^ = - (ajv r+„.).w, (2.16 ) dt 0 0 dt \ 0 /
+ u-Vp' + pc 2V-1/ = 0. (2.17)
In the original paper [19], Hardin and Pope present the equations in a slightly different Cartesian tensor notation,
(2.18)
[pvi' + p'v^j + A (p(Viv/ + VjVi' + Vi'v/) + p'PilA + p'5y
= -^M-i{,^(piVi), (2.19)
2.5 Solution strategy Now, having derived the equations governing the incompressible flow, the hydrody namic density correction, and the perturbations, the steps in the acoustic/viscous splitting technique of Hardin and Pope are given in Table 2.5, in summary.
Ris0-R-947(EN) 17 Table 2.1: Solution strategy for the acoustic/viscous splitting technique.
Step 1. Solution of (2.5) and (2.6), in principle, using whatever mesh and numerical scheme is optimal for the problem of interest. The solution is the incompressible flow field, po, V(x,t) and P(x,t).
Step 2. Correction of constant density by estimation of p\(x,t) using (2.8) or (2.9) and (2.11).
Step 3. Solution of Hardin ’s equations, i.e., (2.15)-(2.17) or (2.18)- (2.20) which yield the perturbations, p'{x,t), v'{x,t) and p'(x,t), which are the acoustic variables in the farfield.
Note, that the discretization of the domain of interest and the numerical scheme does not have to be the same in Step 1 and Step 3. Entirely different meshes and schemes can be used. An example of the use of different meshes is in the case of the flow over a cylinder in Section 6, Figure 6.7. Even the size of the domain can be different. Often, a smaller domain can be applied for the flow calculation (Step 1), since the incompressible flow falls off rapidly to the ambient state in the farfield. This was done by Hardin and Pope for their cavity calculations [21].
18 Ris0-R-947(EN) 3 Numerical method and boundary conditions
To solve Hardin ’s equations, a numerical scheme must be applied, and numerical boundary conditions must be imposed at the boundaries of the computational domain. This section presents the MacCormack scheme along with two types of numerical boundary conditions.
3.1 The MacCormack scheme The MacCormack scheme [33] is a variation of the Lax-Wendroff scheme. It is especially well suited for the solution of nonlinear partial differential equations [2], and it has been applied successfully to Hardin ’s equations [19, 21, 28], and other aeroacoustics problems [34, 47]. The explicit MacCormack scheme exists in various forms: the original 2-2 scheme [33] (second order accurate in time and space), the 2-4 scheme [15] (second order accurate in time and fourth order accurate in space), the 2-6 scheme [5] (second order accurate in time and sixth order accurate in space), and a 4-4 scheme [47] (fourth order accurate in time and space). Here, we use the original 2-2 MacCormack scheme. Consider the general form of the following two-dimensional hyperbolic partial differential equation, dU OF dG „ (3.1) where F = F(U), G = G(U), and S = S(U). A finite difference approximation of (3.1) is
u#1 - uh . - Ki. Gh +1 - + + = S At Ax Ay ij- Rearranging this, yields the Predictor step At At "u1 = vh - W+i,j - F&] - + AlSJ„ (3.2) Ay where Un+1 is the the predicted value of U at time (n+l)At. To find the corrected value of U at time (n + 1) At, the Corrector step ’nr = \ {uh +vft - £ Kf - F2ui
-^[Glf-Gal.l+AtSjf}, (3.3) is applied. To eliminate any bias due to the one-sided differencing, expressed by the terms in the square brackets of (3.2) and (3.3), the differencing sequence in Table 3.1 is applied. This sequence alternates forward and backward differencing between predictor and corrector step and x- and ^-derivatives.
3.2 Boundary conditions Since the computational domain has to be finite, numerical boundary conditions must be imposed on the edges of the domain. Hixon et al. (1995) [24] classify the different types of artificial boundary conditions in three categories: quasi-one- dimensional characteristics [45, 46], decomposition of the solution into Fourier
Ris0-R-947(EN) 19 Table 3.1: Differencing sequence for the MacCormack scheme *
Predictor Corrector Step z-derivative {/-derivative z-derivative {/-derivative 1 F F B B 2 B F F B 3 F B B F 4 B B F F 5 F F B B
• • . • •
* P, forward difference; B, backward difference. From [2, Table 9-1]. modes [13], and asymptotic solutions of the governing equations [43]. To these three types can be added absorbing boundary conditions, where a buffer zone around the computational domain prevents reflections [42]. Here, two different types of boundary conditions are applied, one belonging to the category of asymp totic solutions of the governing equations, the other belonging to the quasi-one- dimensional characteristics approach. Physical boundary conditions, such as con ditions on solid bodies are described where used in, e.g., Section 4 and Section 6.
3.2.1 Asymptotic approach In [43], Tam and Webb (1993) constructed a set of nonreflective boundary con ditions (TWBC) from the asymptotic solutions of the linearized, Fourier-Laplace transformed Euler equations. These boundary conditions have been extensively used by the CAA community ever since, and they are widely considered to be the best choice of boundary conditions for a variety of problems. Tam and Webb found, that at boundaries where there are only outgoing acoustic waves the solution of the two-dimensional Euler equations has the following form,
f[r-{Vr + c)t J 4>{i\ t) =------, (3.4) where df> / r 0 dr rl/-2 y.l/2 2r Eliminating /', yields the Tam and Webb ’s nonreflective boundary condition (TWBC) for the acoustic variable 3.2.2 Characteristics approach In [45], Thompson (1987) bases his nonreflective boundary conditions on a char acteristic analysis of the governing equations and determines which waves are 20 Ris0-R-947(EN) incoming to and which are outgoing from the computational domain. The out going waves are completely determined from variables inside the computational domain, i.e., one-sided differences can be used at the boundaries. To achieve a nonreflecting boundary condition, the amplitudes of the incoming waves are set to zero. One-dimensional case For the one dimensional case Thompson uses the following system of equations w+Alr+c'™0 (3.6) where U is a vector of dependent, primitive variables, A(Z7) is a square matrix and the vector C(U) contains source terms which are not derivatives of U. Let A,- be the eigenvalues of A and let h be the corresponding left eigenvectors of A, i.e., ZjA ” AjZj, then the characteristic form of (3.6) is liiir+Xilii&+liC-0- (3.7) To avoid incoming waves at the boundaries of the computational domain, referring to Hedstrom (1979) [22], Thompson lets, li^+liC~0, (3.8) for the waves with characteristic velocities, i.e., eigenvalues, directed inwards. Now, we have two sets of equations to be used at the boundaries: one for outgoing waves (3.7), and another for incoming waves (3.8). The general, and more compact, form of (3.7) and (3.8) is li -q£ + + hC — 0, (3.9) where 8U £ _ I Xili-g— for outgoing waves, I 0 for incoming waves. Equation (3.9) is the boundary equation to be used in Thompson ’s nonreflective boundary condition (ThBC). Note, that for outgoing waves the solution at the boundary is completely determined from interior. Therefore, in effect, no bound ary condition is needed and the equations that govern the interior, can be applied at the boundary using one-sided differences. So, boundary conditions in the true sense are only imposed for incoming waves. Two-dimensional case For the two dimensional case Thompson performs a quasi-one-dimensional char acteristics analysis. Transverse terms are considered to be source terms and only normal terms are analyzed, since it is not possible to diagonalize both the co efficient matrices simultaneously. For example, at an %-boundary, ^-derivatives are the normal terms and the ^-derivatives are regarded as source terms. The equations become quasi-one-dimensional, and it is possible to proceed in a man ner similar to the one-dimensional case. Thompson uses the following system of conservative equations dU OF f-' = 0, (3.10) Ris0-R-947(EN) 21 or, on non-conservation form, (3.11) where au_ _ au_ d£_ du_ dG _ du_ dt ~ dt ’ 8x ~ ^ dx ’ Sy dy p _dU dF dG P~dU’ Q~dU’ R~du’ A = P_1Q, B = P-1R, C = P-1C'. First let us write an equation for the x-boundaries. Since SAS-1 = A, (3.10) becomes ^ + p(s-'AS^+^ + C' = 0, (3.12) where the rows of the matrix S are the left eigenvectors Z; of A, and A is the diago nal matrix of the eigenvalues A; of A. Abbreviating the quantity in the parentheses as —8U/dt x, we get an equation for the ^-boundaries: (3.13) Since by definition S 1ASdU/dx = —dU/dtx , we find a characteristics equation analogous to (3.7), (3.14) Equivalent to the one-dimensional case, we write: ZiE+A-0, (3.15) where dU £ _ 1 AjZi—— for outgoing waves, I 0 for incoming waves. This definition of £j along with (3.14) and (3.15) are used at the ^-boundaries. Similarly at y-boundaries, (3.16) rrii^+Mi = 0, (3.17) Oty where du ^ fiiTTii-^- for outgoing waves, 0 for incoming waves, where m; are the left eigenvectors of B, and m are the corresponding eigenvalues of B. At a corner point all terms are considered normal, and du du dtx + dty (3.18) Note the complexity of ThBC compared to TWBC. 22 Ris0-R-947(EN) 4 Pulsating sphere This section presents a solution of Hardin ’s equations for the pulsating sphere problem. The problem is one-dimensional, and since it has analytical solutions for both the incompressible flow field and the acoustic field, it is well-suited as a test case for the numerical scheme and validation of the computer code. The pulsating sphere is a monopole sound source, i.e., it has omni-directional directivity (radiates sound of equal intensity in all directions). The monopole is the most efficient type of the elementary sound sources, it corresponds to injection and removal of fluid. Hardin and Pope used the pulsating sphere as a test case in their original paper [19] and tested the effect of temporal and spatial resolution. Here, the problem will be used to test the Thompson conditions (ThBC) and Tam and Webb ’s boundary conditions (TWBC). First, the analytical incompressible flow solution is given. The derivation of the analytical hydrodynamic density correction follows. Next, we write Hardin ’s equations in spherical coordinates and discretize using the MacCormack scheme. Physical and artificial boundary conditions are derived based on Hardin ’s equa tions in spherical coordinates. Finally, the numerical results and comparisons with the analytical solution for the acoustic field are presented. 4.1 Incompressible flow Let a sphere with radius a vibrate radially with the surface velocity Uq sin(wf), Figure 4.1, where Uo is the velocity amplitude of the sphere surface, w = 2tt/ is the angular frequency of the pulsation, / is the frequency, and t is time. Due to Figure 4-1: Flow configuration. Pulsating sphere. conservation of mass, the radial velocity field is then, sin(cvf) U(r,t) = a2Uo (4.1) where r is the radial distance to the sphere center. Since the flow is irrotational, the unsteady Bernoulli equation is valid, p 00 PoU 2 p = po ~po di~ 2 ' Using the definition of the velocity potential Ris0-R-947(EN) 23 Figure 4-2: Incompressible flow solution for pulsating sphere. shows how pressure (solid line) falls off to its ambient value po with increasing radial distance, and how the velocity amplitude (dashed line) falls off to zero with increasing radial distance. The derivatives of (4.1) and (4.2) are dU 2 cos(wt) _ = va’Vo-^-, dr r3 8P sin(wt) 4tt2 sin(wt) cos (cut) ~aT — — pou> 2a2Uo—-—- — po or rL r° Equations (4.1) and (4.2) constitute the incompressible flow solution, and Step 1 in the solution strategy, Table 2.5 on page 18, is accomplished. 4.2 Hydro dynamic density correction Now, to find the hydrodynamic density correction pi, the average pressure is computed using (2.9). P^ = Y jQ P(r’t)dt=P°-\p° U where Tp = 1// is the period of the pulsation. Using (2.11), the hydrodynamic density correction is, P{r, t) - P(r) Pi (r, t) cos(cvt) cos(2 tot) = pQioa 2 Mq + poaf Mq (4.3) rco 4 r4 ’ where Mq = Uq/cq is the Mach number. The derivatives of the density correction (4.3) with respect to time and radial distance are, dpi 2 sin(wt) 4;,r2 sin(2wf) —pow~a Mq — powa Mq ■ dt rc0 2 r4 dpi cos (wt) 2 cos(2 cot) —poioa~Mo —f—- — poaf Mq dr 7- Co With (4.3) and the derivatives, Step 2 in the solution procedure is completed. 24 Ris0-R-947(EN) 4.3 Governing equations for perturbations Hardin ’s equations, (2.15) and (2.16) in spherical coordinates and only radial dependency become, dj_ dP' , du' u' dpi dpi ~fr+27 (4.4) dt dt U dr' (t5) where p = po + pi + p', p = P + p 1, and u — U + u'. The flow is isentropic, i.e., p/po = (p/po) 7 i where 7 is the ratio of specific heats (7 = 1.4 for air). Hence, the algebraic equation, P'=P°(£)7-P, (4-6) can be used instead of the partial differential equation (2.17) for the pressure perturbation. Equations (4.4)-(4.6) are solved with the following numerical scheme. 4.4 MacCormack scheme Using the MacCormack scheme described in Section 3, equations (4.4)-(4.6) are discretized as follows, Predictor step P"i+1 ,n At = Pi~A^ ui (p'i+1 - pT) + p ”(u'”+i - UT) + At 5", (4.7) At u7 +1 = V" - u?(u'7+i ~u'7) + (p'IV i - p'D/p? + a tr/*, (4.8) A?- where pf = Po + Pi” + p'\ and uf = U-1 + u'i, and and P”+1 p'i +l = Po -FT. (4.9) Po Equations (4.7), (4.8), and (4.9) are the discretized equivalents of Hardin ’s equa tions, (4.4), (4.5), and (4.6), respectively, and they constitute the predictor step in the MacCormack scheme. Corrector step or ‘ = 5 />'?+o'r +1 At ,71+1 (P?+1 -p'^)4-pr(»'r — u z“+l\ + A tSf+1 (4.10) Av i-1 ) Ris0-R-947(EN) 25 At + i\lTf“|, (4.11) »r'(»'"+i -<«)+(pT ‘-p'tttm* 1 A r where p" +1 = po + Pi”+1 + p' ”+1 and u"+1 = C7 ”+1 + u'-‘+1, and 71+1 S',+1 =^ ^ ^ ri dt /, Or/, 71+1 Tn+T=_PuZ+Jll(9U'] ,7i+i \ dt ) i ' 71+1 . 771+1 ---- n+l ~r~ P i rfTi+1 , ;7i+l and / n+l\7 =w(V) -P‘+1’ (4.12) where p" +1 = po + Pi"+1 + p' ”+1- Equations (4.10), (4.11), and (4.12) constitute the corrector step in the MacCormack scheme. Note that these discretized equations are only used in the interior, i.e., they are not used at the sphere surface and the outer most mesh point. Here, boundary conditions must be imposed. 4.5 Boundary conditions The inner and outer most mesh points of the computational domain need special treatment. On the surface of the sphere a physical boundary condition is that there is no flow through the surface, i.e., u‘ = 0. The computations show that no condition for p' is needed. Equation (4.4) is simply solved on the surface of the sphere, using one-sided differences. This approach is mentioned in [23, page 380]. For the outer boundary conditions, the form of both TWBC and ThBC is given below. 4.5.1 Asymptotic approach, TWBC The quantities exiting the boundary in the computation of the perturbations are the differences between the total field variables and the uncorrected flow field variables, i.e., p' + pi = p — po, and u 1 =u — U. Now, since this problem is actually three-dimensional, the asymptotic solution has the form fir ~ (Vr+c)t J 4>{i\ t) =------, (4.13) where 26 Ris0-R-947(EN) the speed of sound. Note, that instead of the two-dimensional 1 /r1/2-dependence, we now have a 1/r-dependence, i.e., the amplitudes of three-dimensional, spherical waves fall off quicker than the amplitudes of two-dimensional, cylindrical waves. Taking the time and r derivatives of (4.13), yields d du' (4.16) dt which are the TWBC at the outer boundary point. They are discretized using the MacCormack scheme in the same way, we treat the equations for the interior points. The r-derivatives are calculated using backward differences. 4.5.2 Characteristics approach, ThBC To derive the Thompson boundary conditions (ThBC) for spherical coordinates, (4.4) and (4.5) are written in a matrix form, + c = 0, (4.17) where u p U = A = c2 ip u ’ 2~ + T5l+"Tr r at v or c = Pi+p'dU *±l u + AW + c;|i 1 dP * . p dt v p J dr p dr p dr j We have used that, W=t?(dpi d£\-ML dr \ dr dr J dr ’ from (4.6). The left eigenvectors of the matrix A are and the corresponding eigenvalues of A are A! = — c, A2 = u + c, respectively. Repeated below, for convenience, the general form of the one-dimensional ThBC is (3.9). + IjC = 0, (j = 1, 2), Ris0-R-947(EN) 27 where for outgoing waves, Cj = o for incoming waves. Inserting the eigenvalues and eigenvectors in these expressions yields the boundary equations. For j = 1, 1 du' c + C2 + £i=0, (4.18) 2 dt 2P where dU c dp' Ai l u — c> 0, £1 = dr p dr u — c < 0, and, for j = 2, 1 du' + C2 + £2 = 0, (4.19) 2 dt where , , dU 1. , (du' , cdp' u 4- c > 0, £2 +c) 0 u + c < 0. Note that outgoing waves in this case have positive characteristic velocities (eigen values). Subtracting (4.18) from (4.19) gives an equation for p', ^ ^ (£1 - £2) - Ci. (4.20) Adding (4.18) and (4.19) gives an equation for u', du* -q£ — —£1 — £2 — C2. (4.21) Equation (4.20) and (4.21) constitute the ThBC, and they are used at the outer boundary point. The r-derivatives, which are only needed for outgoing waves, are calculated using backward differences. Equations (4.20) and (4.21) are discretized using the MacCormack scheme, in a way analogous to the treatment of the equa tions for the interior points. 4.6 Results Results for the sphere are obtained for a Mach number range 0.05-0.5 and pulsa tion frequency (or, Strouhal number) of fa/co = 0.2 corresponding to a pulsation period of Tco/a = 5. Since Xf = cq , the fundamental wavelength is A/a = 5. The fluid is a diatomic gas such as air, i.e., 7 = 1.4. 4.6.1 Numerical set-up The numerical set-up is equivalent to that used by Hardin and Pope in [19], except here we use Hardin ’s equations in the form of (2.15) and (2.16) rather than the original form (2.18) and (2.19). Furthermore, Hardin and Pope used the predictor step values of the MacCormack scheme as outer boundary conditions whereas we use ThBC and TWBC. The discretized Hardin ’s equations are solved in the domain 1 < r/a < 26 on a mesh consisting of 501 mesh points with equal radial spacing, i.e., A r/a = 28 Ris0-R-947(EN) 0.05. Thus, the computational domain includes 5 fundamental wavelengths, and provides a spatial resolution of 100 mesh points per wavelength (PPW). This is a rather high resolution of the fundamental wavelength, since about 25 PPW is adequate for the MacCormack scheme [38]. For higher harmonics with shorter wavelengths, the resolution of course decreases. The computations are carried out for 100 periods, i.e., tstop co/a — 500. The time step is found from the fact that the Courant-Friedrichs-Lewy (CFL) number y3 = (cq + Uo)At/Ar should be close to unity for minimal numerical diffusion. Ideally, the time step should be chosen such that the distance the wave propagates in one time step corresponds to the mesh spacing. Here, we use ft = 1. Hence, the temporal resolution for the Mach numbers 0.05, 0.1, 0.2, 0.3, 0.4, and 0.5, is 105, 110, 120, 130, 140, and 150 time steps per period, respectively. Initial conditions are: u'(r, 0) = 0, and p'(r, 0) = ~Pi(r, 0). 4.6.2 Comparisons of numerical and analytical solutions From classical, linear acoustics [26, page 163], the analytical pressure variation for a pulsating sphere is p-po = PoCqUo^ cos (0„) sin (u>t - k(r - a) + Ris0-R-947(EN) 29 Numerical Analytical M = 0.05 Numerical Analytical M = 0.1 Numerical Analytical M = 0.2 Figure 4-3: Numerical, nonlinear solution and anahjtical, linear solution for the following Mach numbers: M = 0.05, M = 0.1, and M = 0.2. For TWBC and tco/a = 500. 30 Ris0-R-947(EN) Numerical Analytical P-Po M = 0.3 Numerical Analytical P-Po M = 0.4 Numerical Analytical 0.1 r P-Pp M = 0.5 Figure 4-4: Numerical, nonlinear solution and analytical, linear solution for the following Mach nujnbers: M = 0.3, M = 0.4, and M = 0.5. For TWBC and tco/a = 500. Ris0-R-947(EN) 31 0.003- 0.002- M = 0.05 0.001-' -0.001- -0.002- 0.00&-r 0.004- M = 0.1 0.002- -0.002- -0.004- 0.014- M = 0.2 0.007- -0.007- 0.024- M = 0.3 -0.012- 0.036- M = 0.4 0.018- -0.01& i i = 0.5 0.02- tco/a Figure 4.5: The last five periods of the time history of (p —po) /(poCq) r/a = 26. 32 Ris0-R-947(EN) 0.00010 M = 0.05 0.00008- 0.00000- 0.00050- M = 0.1 0.00025- 0.00000 M = 0.2 0.001- 0.00250 M = 0.3 0.00125- 0.00000 0.004- M = 0.4 0.002- 0.000 0.004- M = 0.5 0.002- 0.000 fa/co Figure 4.0: Power spectral density of (p — Po)/(pqC q) at r/a = 26. Ris0-R-947(EN) 33 Numerical experiments show that fewer points per wavelength tend to smooth out the higher harmonics, because of inadequate spatial resolution of the shorter wavelengths. 4.6.3 Comparison of outer boundary conditions The performance of the Thompson condition (ThBC) and the Tam and Webb condition (TWBC) is compared in Figure 4.7. This comparison shows that ThBC eventually begins to diverge whereas TWBC remains stable. Numerical exper iments show that ThBC begins to diverge sooner and more severely at higher Mach numbers. In contrast, the TWBC is stable for the entire range of Mach numbers used here. Figure 4.7 also shows that the initial disturbance generated at the sphere surface at r/a = 1 reaches the outer boundary, located at r/a = 26, after the course of nondimensional time tco/a ss 25. This is to be expected, since the disturbance covers a distance equivalent to the fundamental wavelength (A/a = 5) in the time equivalent to the fundamental period {Tco/a = 5). 0.012- Thompson 0.008- 0.004- -0.004- -0.00& P-Po 0.012- Tam and Webb 0.008- 0.004- -0.004- -0.008- tco/a Figure f.7: Time history of (p — Po)Z(PoCo) o.t r/a = 26 for different boundary conditions at M = 0.2. The effect of the location of the outer boundary on the performance of TWBC is displayed in Figure 4.8. Here, results for the original computational domain (1 < r/a < 26) for M = 0.5 are compared with computational domains with extent 1 < r/a < 13.5 and 1 < r/a < 7.25. Spatial and temporal resolution are held constant, i.e., 100 PPW and 150 time steps per period. Figure 4.8 shows that TWBC performs well even when the boundary conditions are imposed at just above one wavelength away from the sphere surface at r&/o = 7.25. 34 Ris0-R-947(EN) rb/a = 26.00 rb/a = 13.50 Tb/a. — 7.25 0.05 V-Vo -0.1 - M = 0.5 -0.15 - Figure 4-S: Effect of distance to outer boundary on the boundary conditions of Tam and Webb, M = 0.5 and tco/a = 500. 4.6.4 Initial conditions In [19], Hardin and Pope used p' = 0 as initial condition. Figure 4.9 shows that the initial condition used here, i.e., p' — -p\ yields a faster convergence to time periodic response. 0.014- 0.007- 0— — -0.007- P-Po 0.014- 0.007- -0.007- tco/a Figure Jt.9: Time history of (p — Po)/(Po co) at r/a = 26 for different initial conditions at M = 0.2. 4.7 Summary Hardin ’s equations have been applied to the problem of the sound generation of a pulsating sphere. The numerical results agree very well with the analytical solution at low Mach numbers where nonlinear effects are negligible. In addition, the solution was found capable of predicting the development of nonlinear waves. Outer boundary conditions have been compared, and it is found that TWBC performs better than ThBC, especially at higher Mach numbers where the latter Ris0-R-947(EN) 35 fails. It is also found that TWBC performs well even for a computational domain with a size of the order of one wavelength. 36 Ris0-R-947(EN) 5 Co-rotating vortex pair This section presents a solution of Hardin ’s equations in two dimensions. The acoustic field generated by a co-rotating vortex is computed. This problem is a good test case for the algorithm because an analytic closed form solution exists for both the incompressible flow field and the acoustic field. In contrast, other more interesting practical problems, such as flow over an airfoil or a bluff body require an accurate incompressible flow solution and detailed measurement of the acoustic field before they can be used for algorithm validation. Therefore, the co-rotating vortex pair problem is chosen for algorithm validation in two dimensions. The co-rotating vortex pair is a lateral quadrupole sound source, i.e., it has four- lobed directivity, Figure 5.1. Quadrupoles are less efficient than monopoles and Figure 5.1: Directivity pattern for lateral quadrupole. dipoles, since the acoustic field cancels in certain parts of space. A quadrupole can be constructed from four monopoles with phase relations as sketched in Figure 5.1. In [28], Lee and Koo (1995) used a modified version of Thompson ’s boundary conditions to solve this problem. Here, we will test both the modified version of ThBC and TWBC. First, the incompressible flow solution and the hydrodynamic density correction are given. Next, we write Hardin ’s equations in Cartesian coordinates, and the ThBC and the TWBC are derived on the basis of the Cartesian form of Hardin ’s equations. Finally, the numerical and analytical solutions are presented. 5.1 Incompressible flow The co-rotating vortex pair is obtained when two point vortices, each with a strength of k = F/(27t) are placed with a distance of 2ro apart. Then each point vortex induces a velocity q on the other, which causes the vortices to rotate around the midpoint distance between them, as the schematic of Figure 5.2 indicates. The induced velocity, e.g., [35, page 74] is given by 9=4^’ where F is the vortex circulation. The rotation period is T = 2nr0Jq = S^rg/T, and the angular speed is w = 2n/T = r/(47rrg). The rotating Mach number is M = q/co = Y/iA-nraCo). Ris0-R-947(EN) 37 Figure 5.2: Flow configuration. The complex potential function for two co-rotating point vortices is the sum of each vortex ’ complex potential, i.e., r r w(z, t) = — ln(z - b) + — ln(z + b), = ^ ln(z 2 _ A=), where z = x + iy = re'0, b = r0e,o,t, and i — \f—l. The complex potential function can be written as w = (p + iip, where ip = 5{w} = — In |z2 - b 21 = ln(% 2 + Y2), 2tt 4tt where 5R{} and Q{} denotes real-part-of and imaginary-part-of, respectively, and X = 5R{z2 — b 2} = x2 — y2 — rg cos 2u>t, Y = 5{z2 — b 2} = 2xy — r$ sin 2wt. The Cartesian velocity components U and V can be found using the fact that dw/dz = d(p/dx + idip/dx (or, equivalently, dw/dz = dip/dy — idcp/dy). Since, U - - d 38 Ris0-R-947(EN) Derivatives of the velocity components U and V in (5.1) are rr du ,ev 2wr zb2 % - —(,2 - vy ’ (5.2) rr au .dv d2w r z2 + b 2 Ux l]x~ dx ldx dz2 TTi (z2 - b 2)2 ’ (5.3) ^ ^ (5.4) i.e., Uy — Vx and Vy — bJ x. From the unsteady Bernoulli ’s equation, the incompressible pressure can be found P — Po — po To find the derivatives of P, let a denote x,y, or t. The derivative of P with respect to the generic variable a is then Pa = dP/da = ~Po((pta + QQa), where QQa = UUa + VVa. Since (f>ix —- fixt — bJt^ (j>ty — 4>yt — and ~ dt* ~ 7T ^ 1 (Z2- 62)2 J ’ the derivatives of the incompressible pressure P (5.5) are Pt — —Poi^tt + UUt + VVt), (5.6) Px = -po(Ut + UUx + VVx), (5.7) Py = ~P0(Vt + UUy + VVy). (5.8) The incompressible flow variables and their derivatives included in the source term of Hardin ’s equations are given by (5.1)-(5.8). Hence, Step 1 in the solution strategy, Table 2.5 on page 18, is accomplished. A snapshot of the incompressible flow solution can be seen in Figure 5.3. The small circle with center in x = y = 0 depicts the path followed by the vortices. 5.2 Hydro dynamic density correction To find the hydrodynamic density correction pi, the average pressure is computed using (2.9), - i rT P(x,y) = ? J 0 - Po ~ ^r[0]o “ 2p0 T J0 ®2dt' = Po~ PoQ/2, Ris0-R-947(EN) 39 ■10 (a) Mesh (b) Stream function X X \ \ \ \ \ \ \ \ ill \ \ \ \ ' III I i _ III, 1 1 t \ / / z ' ' \ \ \ (c) Velocity vectors (d) Pressure Figure 5.3: Incompressible flow solution where n= 1 [T Qidt - r2 x2+ V2 - 7-2 r Vo ^ - rg M - rg' For x = y — 0: _ wr P(x,y) = P0 - p 0—. 7 T The derivatives of the average pressure are, Px - PoQx/2, Py — PoQy/2, where the derivatives of Q are, 7) _ 2r2 %(r4 + rg) ' 7f2 (r" - rg): ' 3 _ 2r2 y{r4 + rg) a-2 (r< - rg):" 40 Ris0-R-947(EN) The hydrodynamic density correction is by definition P-P (5.9) cl ’ and its derivatives are, i , II (5.10) Px-Px Plx — 2 * (5.11) L0 />.»= (5.12) c0 With (5.9)-(5.12) Step 2 in the solution strategy, Table 2.5, is completed. 5.3 Governing equations for perturbations The two-dimensional Cartesian form of Hardin ’s equations, (2.18) and (2.19) is (5.13) where U is the vector of the conservative variables, i.e., density and generalized momentum in x and y, respectively, U = < (p 0 + pi+ p')v! + p'U { (po + Pi + p')v' + p'V The nonlinear convective flux vectors F and G are given by {po + Pi + p')u' + p'U F — ^ (po + pi + p')(2Uu' + u'~) + p'U 2 + p' (po + Pi + p')(Vu' + Uv' + u'v') + p'UV and (po + Pi+ p')v' + p'V G = { (po + pi + p')(Vu' + Uv' + u'v') + p'UV (po + Pi + p')(2Vv' + v'2) + p'V 2 + p' respectively. C' is the source term vector, composed by the substantial derivatives of the hydrodynamic density correction pi and its fluxes piU and p\V . dt dx 8y ^(piV) + y_(^y) p = po + pi + p' , p = P + p' , u = U + u', and v = V + v'. The flow is isentropic, i.e., p/po = (p/po)1 . Hence, the algebraic equation, can be used in place of the partial differential equation (2.20) for the pressure perturbation. Equations (5.13) and (5.14) are solved using the MacCormack scheme described in Section 3.1. Ris0-R-947(EN) 41 5.4 Boundary conditions In contrast to the pulsating sphere problem, here, there is no need for inner bound ary conditions. Only outer boundary conditions are needed and below we derive the TWBC and the ThBC for Cartesian coordinates. 5.4.1 Asymptotic approach, TWBC From (3.4) we have, P1 + Pi u' = 0. (5.15) \Vr + cdt dr 2 r v' The radial distance r is measured from x = y = 0, and r = \Jx2 + y2. Using the chain-rule d _ dx d dy d dr dr dx dr dy' and the fact that x = rcos# and y — rsin#, where the angle # = arctan {y/x), (5.15) can be written in Cartesian coordinates f—- + cos6——1- sin#——K — v! > = 0, (5.16) \Vr + cdt dx dy 2 r where Vr = u cos 6 + v sin #. 5.4.2 Characteristics approach, ThBC To derive the ThBC following the approach outlined in Section 3.2.2, the vectors U, F, and G in (5.13) are linearized as follows U Hn Poll' + p'U > , p 0v' + p'V ) p 0v! + p'U ) F li„ 2p0 Uu' + p'{U 2+cl) \ , p 0(Vu' + Uv') + p'UV ) p 0v' + p'V G||n p0 (Vu' + Uv')+p'UV 2p 0Vv' +p'(V 2 + c20) Since the vector of primitive variables is U = {p' , u', v'}T then o o " 1 0 0' dUi in P = U Po 0 , P-1 = —U/po 1/po 0 dU V 0 po. . — V/ po 0 1/po. o § ' U po 0" TP to 9F ]jn o Q = U2 + cl 2poU 0 , A = P-1Q = du UV p 0V PoU_ 0 0 u _ o o s s dGlin UV p0v p0U , B = P-^R = 0 V 0 T? dU o V2 + 4 0 2poV 42 Ris0-R-947(EN) Now, the eigenvalues of A and B are, Ai = Z7-Co, A2 = U, A3 = [7 + co, and, Mi = V - Co, M2 = y, M3 = V + co, respectively. The corresponding left eigenvectors of A and B are, / Co 1 ii = {0,0,l}, Is = {^.5.0}, I 2#'2' and, f co „ 1 m2 = {0,1,0}, m3 = |^> °> 2} ’ m,=rw°'2 respectively. Mi > 0 Mi < 0 Figure 5.4: Computational domain with wave directions. The equations constituting ThBC at re-boundaries, i.e., where x is constant {Bw and Be in Figure 5.4), are: 9U , dGUn . „,_n dU at + ay (5.17) where the right-hand side is found from liWx+Ci~Q' (i = 1, 2, 3), (5.18) and fxidu , for A, < 0 at Bw, and Aj > 0 at Be 0 , for A, > 0 at Bw, and A, < 0 at Be- Ris0-R-947(EN) 43 Writing out (5.18) yields Kfe_Sfe)+£i=o’ (5.19) lk + Cl =0' (5.20) KE+SE)+z:,=o ' (5.21) Now, subtracting (5.19) from (5.21) yields §r>-£s)' (5.22) where for f/ — cq < 0 at Bw, and U — co > 0 at Be A dx po dx 0 , for t/ — co > 0 at Bw , and U — co < 0 at Bg. from (5.20) dv' aT - ”£=' (5.23) where TTdv U-z— , for y < 0 at Bw, and U > 0 at Be A = OX 0 , for y > 0 at Bw, and U < 0 at Be- adding (5.19) and (5.21) yields = —(A + A), (5.24) for y + cq < 0 at Bw, and U + co > 0 at Be , for y + cq > 0 at Bw, and U + co < 0 at Be- Analogous to TliBC for .-E-boundaries, the boundary equations for ^-boundaries (Bs and Bn in Figure 5.4), are: dUnn , 9Ain , r, _ p dU (5.25) dt dx dty’ where the right-hand side is found from + = 0, (z = 1, 2, 3), and du P-iTTti for p-i < 0 at Bs, and pi > 0 at Bn Mi = dy 0 for pi > 0 at Bs, and pi < 0 at Bn- Thus, = —{Mi - M3), (5.26) Oty CQ where 1/T, . (dv co dp . . i — (F — Co) I -5------%— ) , for V - co < 0 at Bs, and V — co > 0 at Bn Mi = < 2 \dy po dy ' 0 , for V — co > 0 at Bs, and V — co < 0 at Bn- 44 Ris0-R-947(EN) and (5.27) where V-r— , for V < 0 at Bs, and V > 0 at Syv 0 , for V > 0 at Bs, and V < 0 at Bp/. and Qyt — = -(Mi + Ms), (5.28) g(y+co) ^ dv' cq dp' ) , for V + co < 0 at Bs, and V + co > 0 at B^ Ms = dy + pa dy 0 , for V + co > 0 at Bs, and V + cq < 0 at Bpf. To summarize ThBC, (5.17) is used at boundaries where x is constant and (5.25) is used at boundaries where y is constant. The right-hand sides of (5.17) and (5.25) are found using (5.22)-(5.24) and (5.26)-(5.28), respectively, using one-sided finite differences for outgoing waves. 5.5 Results Results for the co-rotating vortex pair are obtained for rotating Mach numbers of M — 0.08 and M = 0.11 corresponding to nondimensional rotation periods of Tcq/tq = 2tt/M = 78.54 and Tcq/tq = 2tt/M = 57.12 and rotation frequencies of fro/co = 0.0127 and fro/co = 0.0175, respectively. Note, that these periods and frequencies correspond to a full rotation; due to symmetry the solution actually repeats itself after half a rotation. The wavelengths based on a full rotation are X/ro = 78.54 and X/r0 = 57.12. The fluid is a diatomic gas with 7 = 1.4. 5.5.1 Numerical set-up The discretized Hardin ’s equations are solved in the domain —100 < x/ro < 100 and —100 < y/ro < 100 covered by a uniform mesh of 81 x 81 points yielding a mesh spacing of Ax/ro = Ay/ro = 200/80 = 2.5. There is a mesh point at x/ro — y/ra — 0, thus the vortices which represent singularities and follow a circular path with center in x/ro = y/r0 = 0 and a radius of 2ro, will never coincide with a mesh point, see Figure 5.3a. Hence, there is no need for special treatment of the case of a vortex located at a mesh point. In fact, numerical experiments (Ax/7 ’0 = Ay/7'0 = 2.0) showed that the mesh spacing can not be much smaller than Ax/7 ’0 = Ay/7 ’0 = 2.5 since the mesh points get too close to the vortex path resulting in very high amplitude source terms. In [28], Lee and Koo (1995) used vortex core models with good results to be able to apply finer meshes. On the other hand, the mesh spacing can not be much bigger, since the MacCormack scheme needs a spatial resolution of about 25 points per wavelength (PPW). Based on the fundamental wavelengths for M = 0.08 and M = 0.11, the mesh provides a resolution of 78.54/2.5 = 31 PPW and 57.12/2.5 = 23 PPW, respectively. And the spatial resolution is actually halved when considering the wavelengths based on half a rotation which perhaps is the correct thing to do. For comparisons of results the computations were carried out for about 19 rota tion periods. To test the stability of the boundary conditions some computations were carried out for 60 rotation periods. The time step is found from [2, page 484] Ris0-R-947(EN) 45 where a « 0.9 is the recommended safety factor. Here, we use a = 0.9 as the basis, but a — 0.45 is also used to test effect of temporal resolution. (A safety factor of one did not lead to a converged solution.) \U\ and |V| are the maximum numeric values of U and V, respectively, at the mesh points next to the center. Lee and Koo (1995) found in [28], that the application of ThBC for all the acoustic variables finally led to divergence, which numerical experiments confirm. Lee and Koo suggest using ThBC only for p'. And, assuming that the waves at the boundaries are locally planar, the acoustic pressure can be found from p' — c^p' and the radial particle velocity from u'r = p' /(pqCq), where the Cartesian particle velocity components are v! = u'r cos 9 and v' = u'r sin 9. This Lee and Koo version of ThBC (LKBC) is used in the following instead of the original ThBC. Initial conditions are: u'(x, y, 0) = 0, v'(x, y, 0) = 0, and p'{x, y, 0) = -pi (x, y, 0). For evaluation of numerical result, the leading term of the far-held acoustic pressure can be obtained using the method of matched asymptotic expansions, e.g., [28, 36] = Q4°3r4c2 (Ji(kr) Sin (2(cot - 9)) - Y2(kr) cos (2(ut - #))) where k = 2w/co and H^(lcr) = J2(&r) — iY2(fcr) is the Hankel function of second order and second kind. J2(kr) and Y2(fcr) are second-order Bessel functions of the first and second kind, respectively. 5.5.2 Effect of boundary conditions. Iso-contours and carpet plots of p'/{ poCq) for the numerical and analytical solu tions are given in Figure 5.5 for M = 0.08. They show a double spiral pattern with nearly cylindrical waves in the far held. Note the four-lobed directivity pat tern especially visible in the analytical solution, Figure 5.5c and compare with the sketch in Figure 5.1. There is good agreement in the far held between numerical and analytical so lutions. It appears that the farheld contour lines for the numerical solution with LKBC, Figure 5.5a, are bulkier than is the case for the numerical solution with TWBC, Figure 5.5b. The TWBC solution has almost circular farheld contour lines much like the analytical solution in Figure 5.5c. In the nearheld the numerical and analytical solutions do not and should not compare well, because of the nonlinear effects in the numerical solution are very strong. 46 Ris0-R-947(EN) (b) Numerical solution with Tam and Webb boundary condition (TWBC). (c) Analytical solution. Figure 5.5: Acoustic pressure p'K poCq) for tco/ro = 1542.20, M = 0.08, and a — 0.9. Acoustic pressure profiles along a diagonal line from the upper-left corner to the lower-left corner of the computational domain are displayed in Figure 5.6a and Figure 5.6b for LKBC and TWBC, respectively. Note, that Figure 5.6 gives an impression of the spatial resolution of the waves. There are no irregularities in the waveform at the outer most mesh points, that could indicate spurious reflections for any of the two boundary conditions. But, TWBC performs slightly better in the Ris0-R-947(EN) 47 far field compared to LKBC. This is further confirmed by the time series for two 0.0003 Numerical 0.0002 Analytical 0.0001 -0.0001 -0.0002 -0.0003 -100,100 -50,50 0,0 50,-50 100,-100 2/7-0, y/r0 (a) Lee and Koo boundary condition (LKBC) 0.0003 Numerical — 0.0002 Analytical 0.0001 -0.0001 -0.0002 -0.0003 -100,100 -50,50 0,0 50,-50 100,-100 z/t-o , y/ro (b) Tam and Webb boundary condition (TWBC) Figure 5.6: Acoustic pressure on diagonal from upper left comer to lower right comer of computational domain, tco/ro = 1542.20, M = 0.08, and a = 0.9. farfield monitor points, Figure 5.7. There is almost no difference in performance between the two types of boundary conditions. Consequently, the following results are obtained only with the marginally better TWBC. 48 Ris0-R-947(EN) 0.00015 Tam and Webb ------0.0001 Lee and Koo------Analytical 5e-05 -5e-05 -0.0001 tco/ro (a) x/ro = — 100 and y/ro = —100 0.0002 0.00015 Tam and Webb ------Lee and Koo------Analytical ...... 0.0001 -5e-05 -0.0001 -0.00015 tco/r 0 (b) x/ro = —100 and y/ro = 0 Figure 5.7: Time series of acoustic pressure atfarfield monitor points, M = 0.08, and a = 0.9. In Figure 5.8, the acoustic pressure profiles along the horizontal line from the midpoint of the left boundary to the midpoint of the right boundary of the domain is given, Figure 5.8 should be compared to Figure 5.6b which is for a diagonal line. It could be expected that numerical results would compare better with analytical results on the horizontal line, since it coincides with a coordinate line and has bet ter spatial resolution. Considering Figure 5.8 and Figure 5.6, it is not necessarily so. Ris0-R-947(EN) 49 0.0003 Numerical — 0.0002 Analytical 0.0001 -0.0001 -0.0002 -0.0003 -100,0 -50,0 Figure 5.8: Acoustic pressure on horizontal line from midpoint of left boundary to midpoint of right boundary of computational domain, tco/ro = 1542.20, M = 0.08, and a = 0.9. For TWBC. In Figure 5.9, particle velocity vectors for the numerical solution based on TWBC indicate that there are no spurious reflections of importance. TTTTTTTTTT-rI I I I I I t t t t t t t t f t 1 t t 1 f f t t f t f f t 1 t t / i i i t t i f 1 t t t f t f f / t t I t t t t f f f t / / MM/ M M M M / t M M M M M ' ' itMM//////'' '////////' t f t f / / ////////// \ f f t t t t t t ********* t t f t t f t t ff ********* .iiittttfff/// itiittff/////*f t t t t / / / / / / / MM1'*' \ * I t t i * ' ' ' ' HW#:;;{///////«" Figure 5.9: Particle velocity in first quadrant of computational domain, tco/ra — 1542.20, M = 0.08, and a = 0.9. For TWBC. 50 Ris0-R-947(EN) 5.5.3 Effect of domain size The effect of the location of the outer boundary on the performance of TWBC is displayed in Figure 5.10 and Figure 5.11. Here, results for the original computa tional domain, —100 < x/tq < 100 and —100 < y/ro < 100 are compared with results for a smaller domain, -50 < x/tq < 50 and -50 < y/ro < 50. Spatial arid temporal resolutions are held constant. The performance of TWBC is obviously not much influenced by the smaller domain size. There are small differences in the (a) Small domain, 100ro x 100ro (b) Big domain, 200ro x 200ro Figure 5.10: Acoustic pressure p'/(poc q ) for tco/ro = 1542.20, M = 0.08, and a = 0.9. peak values, but the phase does not change in the numerical results going from the larger domain to the smaller domain. Ris0-R-947(EN) 51 0.0003 0.0002 0.0001 -0.0001 Small domain -0.0002 / Big domain ' Analytical -0.0003 Figure 5.11: Acoustic pressure on diagonal from upper left comer to lower right comer of computational domain, tco/ro = 1542.20, M = 0.08, and a = 0.9. 52 Ris0-R-947(EN) 5.5.4 Effect of temporal resolution The effect of higher temporal resolution, i.e. smaller time step, is better prediction of the amplitudes, but worse prediction of the phase, Figure 5.12. 0.00015 a = 0.90 0.0001 a = 0.45 Analytical -5e-05 -0.0001 tco/ro (a) x/ro = —100 and y/ro — —100 0.0002 0.00015 -5e-05 -0.0001 -0.00015 tco/r 0 (b) x/ro = —100 and y/ro = 0 Figure 5.12: Time series of acoustic pressure at farfield monitor points, M = 0.08, and a = 0.9. For TWBC. Ris0-R-947(EN) 53 5.5.5 Numerical solution for M = 0.11 Iso-contours and carpet plots of p'/{poCo) for the numerical and analytical solu tions are given in Figure 5.13 for M = 0.11. Note, that a higher Mach number means shorter wavelengths with lower spatial resolution of the waves as a result of unchanged mesh spacing. Judging from the contour lines in Figure 5.13, the agreement between numerical and analytical solutions is worse than for the case of M = 0.08, Figure 5.5. Figure 5.14a shows that both amplitude and phase is pre dicted well in the diagonal direction. Figure 5.14b, however indicates that in the horizontal direction, the amplitude is under-predicted and the phase is off, despite the better spatial resolution in this direction. As is also the case for M = 0.08, there are no irregularities in the waveform at the outer most mesh points, that could indicate spurious reflections from the application of TWBC. .'A,« ■wo -so so wo (a) Numerical solution with Tam and Webb boundary condition (TWBC). (b) Analytical solution Figure 5.13: Acoustic pressure p'/(pqCq) for tco/ro = 1127.60, M = 0.11, and a = 0.9. It is not possible to go to higher Mach numbers with the present mesh config uration. Higher harmonics will be present in the numerical solution and a finer mesh is needed to resolve the shorter wavelengths. For the reasons pointed out earlier, this necessitates the application of a vortex core model. 54 Ris0-R-947(EN) Numerical Analytical 0.0005 -0.0005 -0.001 -100,100 -50,50 0,0 50,-50 100,-100 z/ro, y/r0 (a) Diagonal from upper left corner to lower right corner of computational domain. Numerical Analytical 0.0005 - -0.0005 -0.001 (b) Horizontal line from midpoint of left boundary to midpoint of right bound ary of computational domain. Figure 5.14: Acoustic pressure, tco/ro = 1127.60, M = 0.11, and a = 0.9. 5.6 Summary Hardin ’s equations have been applied to the two-dimensional problem of the sound generation of a co-rotating vortex pair. The numerical results agree well with the analytical solution at low Mach numbers where nonlinear effects are negligible. It is not possible to predict the higher harmonics without the use of a vortex core model. The performance of TWBC is very slightly better than the Lee and Koo version of ThBC (LKBC), where only p' is found using the original ThBC and the remaining variables are found assuming that the waves at the boundaries are locally planar. The TWBC performs well even with the boundaries located at a distance to the center below one fundamental wavelength. Ris0-R-947(EN) 55 6 Viscous flow over cylinder This section presents prediction of aeroacoustic noise for a more practical problem, the sound from the viscous flow over a cylinder. This two-dimensional problem can be considered as a representative case of real fluid flow as opposed to the two former potential flow test cases. When a stationary cylinder is exposed to a cross flow, it exerts a fluctuating force on the surrounding fluid which leads to the generation of Aeolian tone noise, e.g., the singing of telephone wires. The noise peaks at angles normal to the flow direction, and it has the radiation pattern of a dipole, depicted in Figure 6.1. A Figure 6.1: Directivity pattern for dipole. dipole is a less efficient sound source than a monopole but more efficient than a quadrupole. First, the numerical method used to compute the incompressible, viscous flow is briefly described and the incompressible flow solution is presented. Next, the hydrodynamic density correction is given, followed by Hardin ’s equations in cylin drical coordinates and boundary conditions for the numerical solution. Finally, the numerical solution to Hardin ’s equations is presented. 6.1 Incompressible flow The flow configuration is shown schematically in Figure 6.2. The fluid flows from left to right with the free stream velocity Uq over a cylinder of radius vq. Since an Figure 6.2: Circular cylinder in uniform cross flow. 56 Ris0-R-947(EN) analytical solution to the problem of viscous flow over a cylinder does not exist, the incompressible flow solution is obtained numerically. To this end, we use a CFD code developed by Hardin and Pope 1989 [18]. The code is based on the vorticity-stream function formulation of the governing fluid equations, i.e., (2.5) and (2.6). The vorticity-stream function formulation is presented in the following. 6.1.1 Vorticity-stream function formulation In three dimensions vorticity fi is a vector defined as the curl of the velocity vector V, i.e., fl = VxV. Using this definition and taking the curl on both sides of the momentum equa tion (2.6) yields a transport equation for vorticity ^ - (Cl-V)V + (V-V)fi = z/0V2fi, where i>o = po/po is the kinematic viscosity. In two dimensions the vorticity fZ has only one component fl which is perpendicular to the components of the velocity field. So, the transport equation for vorticity becomes — + V-Vfi = z/0V2fl (6.1) Letting Vr and Vo denote the components of the incompressible velocity vector V in the radial and tangential directions, respectively, the vorticity Q, in cylindrical coordinates is dVr 8Vo Vo 1 8Vr (6 .2) n = -r 89 d7 + V"rW Now, using the definition of stream function 3 v 103 TZ 03 V' = rW V‘ = -W (6.3) the vorticity transport equation (6.1) becomes rd 153 8 103 8 r d2 18 1 d2in Ldt + r~89d~r ~ rdFaer- 1/0Id^ + rd~r + (6.4) Combining (6.2) and (6.3), the equation for the stream function is 18 1 52-i 3 = —fl. (6.5) dr* + rWr + ^W- J Equation (6.4) and (6.5) constitute the vorticity-stream function formulation in cylindrical coordinates of the governing fluid equations. For more detail on the vorticity-stream function formulation, e.g., see [3, pages 54-58]. In the original CFD code by Hardin and Pope 1989 [18], the pressure field is not solved for, so to make the code applicable to the present method, a Poisson equation for pressure is implemented. We find an equation for pressure P by taking the divergence of the momentum equation (2.6) which upon rearranging yields V2P = ~p 0 V- (V-VV). Using the definition of stream function, the pressure equation in cylindrical coor dinates becomes r 8 2 1 8 1 02 1d _o [S23 / 1 523 153\ . dr- v dr r2 89- J dr- \ r- 89' 2 r dr ) n 8 23 _ J_53\2" x r dr89 r2 89 ) (6 .6 ) Ris0-R-947(EN) 57 To facilitate mesh-stretching (concentration of mesh points close to the cylinder surface) and discretization on a rectangular, uniform mesh, the following trans formation is introduced, r = ett<’, 6 = at], where Q and 77 are the coordinates in transformed space. Letting the range of ( be 0 < ( < 1, the constant a can be determined from the outer boundary radius 7- max (where C = 1) a, — ln(r max ). The range for 77 is 0 < 77 < 2tt/o . Letting E = ae0<>, the transformed versions of (6.4)-(6.6) for Q, T, and P are L dt dr] d( (6.7) (6.8) idQ2 dr] 02P d2P 2/90 \/d2^ d'&\ /d2'& d<£\ 5C2 + drf- ~ E2 [U<2 a~dcJ\'W + a z a2* UcSt? "677V J' (6.9) respectively. Equation (6.7) is solved using a second-order upwind differencing scheme de scribed in [18]. The Poisson equations for the stream function (6.8) and the pres sure (6.9) are solved using a direct solver for elliptic partial differential equations, FISHPAK [1, 41]. For details on boundary conditions, refer to [18]. The incom pressible velocity components Vr and Vo are found from central difference approx imations of (6.3). The temporal and spatial derivatives of the incompressible flow field, i.e., VT and Vo, and P are found using backward differences and central differences, respectively. Thus, Step 1 in the solution strategy is accomplished. 58 Ris0-R-947(EN) 6.1.2 Numerical results for incompressible flow The numerical solution of the incompressible flow field for a Reynolds number of Re = UqDq/uq = 200 (Do = 2ro is the diameter of the cylinder) is computed on the mesh shown in Figure 6.3. The mesh has 189 x 161 mesh points in the radial and tangential direction, respectively and it extends 40 radii away from the cylinder center. The spacing between a mesh point on the cylinder surface and the next radial mesh point is 0.04ro . When the solution reaches time pe- Figure 6.3: Viscous mesh with 189 and 161 mesh points in the radial and tan gential direction, respectively. (Every other mesh point is shown.) riodicity, the well known von Karman vortex street (Figure 6.4c) has formed with the fundamental vortex shedding frequency f0 yielding a Strouhal number of St = foDo/Uo = 0.195. This Strouhal number agrees well with experiments, e.g., [6, Figure 4-10]. Figure 6.4c, where bright colors denote high negative vorticity and dark colors denote high positive vorticity, gives an impression of the alter nating shedding of vortices which form the von Karman vortex street where blobs of vorticity are convected downstream. Figure 6.4d, where bright colors denote high pressure and dark colors denote lower pressure, shows a high pressure re gion around the stagnation point at the front of the cylinder and regions with low pressure in the wake of the cylinder. The low pressure regions correspond to the vorticity blobs. This agrees well with simple vortex theory, saying that in the center of a vortex there is low pressure. Ris0-R-947(EN) 59 (a) Zoom of viscous mesh (b) Stream function, # -5 0 5 JO 15 20 25 30 35 40 x/r„ (c) Vorticity field, fl (d) Pressure field, P Figure 6.4: Snapshot of the incompressible solution. 60 Ris0-R-947(EN) 6.2 Hydro dynamic density correction The hydrodynamic density correction pi is found by averaging over one funda mental period To = l//o (or an integer multiple of thereof) when the solution has reached its time periodic state. P-P P(r,6) = ±j\r,8,t) dL Pi y/r„ 0 Figure 6.5: Hydrodynamic density correction, p\. A snapshot of pi corresponding to Figure 6.4 is given in Figure 6.5 where bright colors denote positive corrections and dark colors denote negative corrections. The corrections also form a street in the wake of the cylinder. For a Mach number of 0.5, the order of magnitude of pi is ±6% of po- 6.3 Governing equations for perturbations Hardin ’s equations, (2.15)-(2.17) in two dimensions and in cylindrical coordinates are ° W (dK + <+ldvj, ve dp' 8t P\ dr ^ r r 89 r dd dpi _ dpi _ ve dpi dt Vr dr r dO ’ (6 .10) dv' dv'r ve dv'r 1 dp' =_ei±pLSK_(p i±lVr+v,\w1 (6 .11) p dt \ p J dr 1 Pi + P' ) 9V r i 1 pi + p' V$ +Vg W + 7~T~ r P dv'g dv'g Vg dv'g 1 dp' -dF+Vr~d7 + - ~dd+7rle pi + p' dVg (pi + p\ r , ^ dVg l dVg i Vg + Vg VrVg — -(Vrv'g + vgv'r), r de r p (6.13) \ dr r r d9 ) Ris0-R-947(EN) 61 where p = Po+Pi+p', vr = VT+v'r, vg = Vg+v'g, and p = P+p'. These equations are solved using the MacCormack scheme, analogous to the former two problems of the pulsating sphere and the co-rotating vortex pair. 6.4 Boundary conditions Boundary conditions on the cylinder are dt/ v' = 0 and — 0. or The first condition is exact. The second condition is a linearization of (6.11), which simplifies to dp' dr on the cylinder surface, since here Vr = Vg = v'T — 0. For p' and v'e, (6.10) and (6.12) are solved, respectively, using one-sided differences for the r-derivatives. For outer boundary conditions, both the ThBC and TWBC are used. Their derivation is analogous to the derivation presented in the previous section, dealing with the co-rotating vortex pair. 6.5 Results For this problem, time periodic solutions for the perturbations were not obtained. The amplitudes keep increasing until the solution blows up after the course just above seven fundamental period To- Several different temporal and spatial resolu tions have been applied to no avail and there was no noticeable difference between the performance of ThBC and TWBC. However, an example of the numerical results is presented, since they show some interesting features. The results are for a Mach number of 0.5. This was chosen to have at least a couple of fundamental wavelengths in the computational domain. A smaller Mach number means larger wavelength, hence fewer wavelengths in the computational domain. Of course a Mach number of 0.5 violates the assumption of incompressibility in the flow calcu lations. But, since the computations of the perturbations diverged even for smaller Mach numbers, we chose the Mach number that produces the most instructive (and prettiest!) pictures of the acoustic pressure. Note, that the Mach number does not influence the flow field calculations, but merely the size of the hydrodynamic den sity correction. The mesh used for the computations of the perturbations is shown in Figure 6.6. The distribution of radial mesh points is uniform all the way out to the boundary as opposed to the stretched mesh distribution in the viscous mesh, Figure 6.3. Figure 6.7 shows a close-up of the two meshes at the cylinder surface. For ev ery mesh point in the acoustic mesh there is not a corresponding mesh point in the viscous mesh with the exact same coordinates. Therefore, the viscous mesh solution is interpolated on to the acoustic mesh. Ideally, interpolation should be avoided, by using a viscous mesh where for every acoustic mesh points there is a viscous mesh point with the same location. The CFD code, used for the com putation of the flow field, does not allow such a flexibility for meshing the two solutions simultaneously. It is possible to align the constant 6 mesh lines such that no interpolation is needed. In the r-direction, however, interpolation must be done because of the exponential mesh points distribution in the viscous mesh. We use bilinear interpolation [40, page 96]. A table, containing information on each acoustic mesh point ’s location in the viscous mesh, provides the input necessary for the interpolation scheme. 62 Ris0-R-947(EN) Figure 6.6: Acoustic mesh with 151 and 161 mesh points in the radial and tan gential direction, respectively. (Every other mesh point is shown.) (a) Acoustic mesh (b) Viscous mesh Figure 6.7: Close-up of meshes. A series of snapshots of the perturbation pressure, Figure 6.8, shows the devel opment of the wave field. At time t = 1.96To, waves start to form, and as expected the waves come from the upper and lower parts of the cylinder. The wave pattern is very clear upstream of the cylinder whereas the acoustic field is more complex behind the cylinder. Somewhere between t = 7To and t = 8To, the numerical so lution blows up. The reason can be that the computational domain is too small, Ris0-R-947(EN) 63 so that the wake (which, in effect, constitutes an array of sound sources) has not weakened adequately before reaching the outer boundary. Hence, sound sources are actually present at the boundary itself. Careful observation of the computed results, however, indicates no problems at the boundary in the wake region. f = 0.07 T0 t = 0.70 T0 t = 1.33 T0 t = 1.96 T0 t = 2.59 T0 * = 3.22 T0 t = 3.85 To t = 4.48 T0 t = 5.11 T0 t = 5.74 T0 t = 6.37 T0 t = 7.00 T0 Figure 6.8: Snapshots of perturbation pressure, p' . Figure 6.9 illustrates a characteristic difference between a flow field and its corresponding acoustic field. The extent of the interesting parts of the flow field is a relatively limited region close to the cylinder and its wake, whereas the acoustic field fills the entire computational domain both upstream and downstream the cylinder. 64 Ris0-R-947(EN) (c) Hydrodynamic density correction, (d) Perturbation pressure, p'. —0.056 < pi/po < 0.053 Figure 6.9: Extent of flow field and acoustic field, t = 7Tq. The relatively limited extent of the flow field suggests (not surprisingly) that the viscous mesh, Figure 6.3 is not optimal in that there are just as many mesh points upstream as there are downstream the cylinder. A more optimal solution is to decrease the upstream extent of the computational domain and increase the extent downstream. However, the present CFD code does not allow this approach. 6.6 Summary Hardin ’s equations have been applied to the two-dimensional problem of the sound generation from viscous flow over a circular cylinder. It has not been possible to perform long time integration for this problem. However, the solution did produce waves, and it should be possible to solve this problem perhaps with the use of codes based on generalized coordinates which makes numerical experiments with different mesh configurations possible. Ris0-R-947(EN) 65 7 Conclusion The motivation for the present thesis was the application and testing of a nu merical method suitable for prediction of generation and propagation of noise from wind turbines; a full numerical method that can supplement the commonly used semi-empirical methods. To this end, a recently developed technique by Hardin and Pope [19], which belongs to the category of perturbation approaches, was chosen. Application of this technique is explored as an alternative to the com putationally expensive direct numerical simulation, where flow and sound field are computed at the same time. For low Mach numbers, typical for wind turbines flows, the perturbation approach appears to be the best choice. The technique initially requires a solution (numerical or analytical) of the incom pressible time dependent flow field. An estimation of the hydrodynamic density correction to the constant incompressible density is performed using the differ ence between the instantaneous incompressible pressure and its time average. The correction is necessary, since the hydrodynamic density variation in the flow field can be quite large compared to the density variation in a sound wave (5 percent and 0.4 percent of the ambient density, respectively). Finally, the sound field is the solution to the nonlinear partial differential equations (PDF), herein called Hardin ’s equations, which govern the inviscid perturbations about the corrected flow field. The emphasis in this thesis was on the computation of the sound field. We solved Hardin ’s equations numerically using the explicit MacCormack scheme [33], which is second order accurate in time and space. The MacCormack scheme is especially well suited for the solution of nonlinear PDFs. Among four categories of existing nonreflecting boundary conditions: quasi- one-dimensional characteristics, decomposition of the solution into Fourier modes, asymptotic solutions of the governing equations, and absorbing boundary condi tions, where a buffer zone around the computational domain prevents reflections, we have implemented and tested the characteristic and asymptotic boundary con ditions in the present work. The characteristic approach proposed by Thomp son (1987) [45] finds which waves are outgoing from and which are incoming to the computational domain on the basis of the characteristic form of the governing equations. In the asymptotic approach, Tam and Webb [43] establish the func tional form of the solution to the linearized Euler equations in the far field, where only outgoing waves are present. Hardin ’s equations were applied to three problems of flow generated sound. Firstly, the problem of sound generation from a pulsating sphere is examined, as an example of a monopole sound source. The problem is one-dimensional and Hardin ’s equations are solved in spherical coordinates. Secondly, we solved the problem of the sound generated by a co-rotating vortex pair. This is an example of a quadrupole sound source. The problem is two-dimensional and Hardin ’s equations are written in Cartesian coordinates. Finally, the problem of the dipole sound generation of viscous flow over a circular cylinder is examined. The governing equations are solved in cylindrical coordinates. For the first two problems, analytical solutions to the incompressible flow exist. For the cylinder flow, the incompressible flow solution is computed numerically using a CFD code by Hardin and Pope [18] based on a vorticity-stream function formulation of the governing flow equations. This code was extended with a Poisson equation for the computation of the incompressible pressure. Numerical results agree very well with the analytical solutions for the problems of the pulsating sphere and the co-rotating vortex pair. As for the two types of boundary conditions, the asymptotic based condition is more straightforward to 66 Ris0-R-947(EN) implement, and also performs slightly better than the characteristic based condi tion. The latter condition is furthermore disadvantaged by its lengthy derivation. The results for the viscous flow over a cylinder show promising features, but, work still remains to be done on this problem before any definite conclusions can be drawn. However, if both the flow field solver and the perturbation solver are based on generalized coordinates, the cylinder problem probably can be solved using Hardin ’s equations. Solvers based on generalized coordinates leave more room for numerical experimentation with different meshes and eliminate the need for interpolation of the flow field on to the acoustic mesh. Furthermore, this approach allows application of the same computer codes to different test cases; something that eases debugging of the codes. Originally, it was chosen not to go with the generalized coordinates approach to avoid the numerical errors associated with the numerically computed metrics. Instead of one general code, we have written three different perturbation solvers, based on spherical, Cartesian, and cylindrical coordinates, respectively; one for each of the three test cases. So, in summary, future plans are to write a perturbation solver (simultaneous, perhaps, with the writing of a flow field solver to secure optimal communication be tween the two codes) based on generalized coordinates, which, expectively, makes solution of the cylinder problem possible. In turn, the more interesting problem of sound generation of flow over an airfoil can be addressed. We have only dealt with one- and two-dimensional problems here, but an ex tension to three dimensions is possible given the adequate computer resources. Three-dimensional, time dependent problems demand long computing time. How ever, the rapid development in computer technology continuously diminishes this limitation. Ris0-R-947(EN) 67 References [1] Adams , J., Swarztrauber , P., and Sweet, R. FISHPAK. A package of Fortran Subprograms for the Solution of Separable Elliptic Partial Differential Equations, 3 ed. The National Center for Atmospheric Research, June 1979. [2] Anderson , D. A., Tannehill , J. C., and Fletcher , R. H. Computa tional Fluid Mechanics and Heat Transfer. Hemisphere Publishing Corpora tion, 1984. [3] Arpaci , V. S., and Larsen , P. S. Convection Heat Transfer. Prentice-Hall, Inc., 1984. [4] BATCHELOR, G. K. An Introduction to Fluid Dynamics. Cambridge Univer sity Press, Cambridge, 1967. [5] Bayliss , A., Maestrello , L., Parikh , P., and Turkel , E. A Fourth- Order Scheme for the Unsteady Compressible Navier-Stokes Equations. AIAA-85-1694 (1985). [6] Blake , W. K. Mechanics of Flow-Induced Sound and Vibration, vol. I. Academic Press, 1986. [7] Brooks , T. F., Pope, D. S., and Marcolini , M. A. Airfoil Self-Noise and Prediction. NASA Reference Publication 1218, NASA, 1989. [8] C richton , D. G. Goals for Computational Aeroacoustics. In Computational Acoustics: Algorithms and Applications, D. Lee, R. L. Sternberg, and M. H. Schultz, Eds. Elsevier Science Publ. B. V., 1988. [9] CRICHTON, D. G. Computational Aeroacoustics for Low Mach Number Flows. In Computational Aeroacoustics, J. C. Hardin and M. Y. Hussaini, Eds. Springer-Verlag, 1993. [10] D owling, A. P., and Ffowcs Williams , J. E. Sound and Sources of Sound. Ellis Horwood Limited, 1983. [11] Fox, R. W., AND Mc D onald , A. T. Introduction to Fluid Mechanics, 3rd ed. John Wiley h Sons, 1985. [12] Fuglsang , P., and Madsen , H. A. Implementation and Verification of an Aeroacoustic Noise Prediction Model for Wind Turbines. Ris0-R-867(EN) , Ris0 National Laboratory, 1996. [13] Giles, M. B. Nonreflecting Boundary Conditions for Euler Equation Cal culations. AIAA Journal 28, 12 (1990), 2050-2058. [14] Gliebe, P. R., BRAUSCH, J. F., Majjigi , R. K., and Lee, R. Jet Noise Suppression. In Aeroacoustics of Flight Vehicles: Theory and Practice. Vol ume 2: Noise Control, H. H. Hubbard, Ed., NASA Reference Publication 1258, Vol.2. NASA, 1991. [15] Gottlieb, D., and Turkel , E. Dissipative Two-Four Methods for Time- Dependent Problems. Mathematics of Computation 30, 136 (1976), 703-723. [16] Grosveld , F. W. Prediction of Broadband Noise from Horizontal Axis Wind Turbines. AIAA Journal Propulsion & Power 1, 4 (1985), 292-299. [17] Hardin , J. C. Overview of Computational Aeroacoustics in Aerodynamics. In Short Course. Presented In The AIAA Professional Studies Series (Reno, Nevada, January 8-9 1994). 68 Ris0-R-947(EN) [18] HARDIN, J. C., AND Pope, D. S. Active Flow Control for Aeolian Tone Noise Reduction. AIAA 89-1112 (1989). [19] HARDIN, J. C., and Pope, D. S. A New Technique for Aerodynamic Noise Calculation. DGLR/AIAA 92-02-076 (1992), 448-456. [20] Hardin, J. C., AND Pope, D. S. An Acoustic/Viscous Splitting Technique for Computational Aeroacoustics. Theoretical and Computational Fluid Dy namics 6 (1994), 323-340. [21] Hardin, J. C., AND Pope, D. S. Sound Generation by Flow over a Two- Dimensional Cavity. AIAA Journal 33, 3 (1995), 407-412. [22] HEDSTROM, G. W. Nonreflecting Boundary Conditions for Nonlinear Hy perbolic Systems. Journal of Computational Physics 30 (1979), 222-237. [23] HlRSCH, C. Numerical Computation of Internal and External Flows, vol. 2. John Wiley & Sons, 1992. [24] Hixon, R., Shih, S.-H., and Mankbadi, R. R. Evaluation of Boundary Conditions for Computational Aeroacoustics. AIAA Journal 33, 11 (1995), 2006-2012. [25] J0RGENSEN, J. U. St0jbekcempelse: Principper og praksis. /Noise Control: Principles and Practice]. Arbejdsmiljpfondet, 1980. In Danish. [26] Kinsler , L. E., Frey , A. R., C oppens, A. B., and Sanders , J. V. Fundamentals of Acoustics, 3rd ed. John Wiley & Sons, 1982. [27] Landahl, M. T. CFD and Turbulence. In The 17tli ICAS Congress: The Daniel & Florence Guggenheim Memorial Lecture (1990). [28] Lee, D. J., AND Koo, S. O. Numerical Study of Sound Generation due to a Spinning Vortex Pair. AIAA Journal 33, 1 (1995), 20-26. [29] Lighthill, J. Waves in Fluids. Cambridge University Press, 1978. [30] Lighthill, J. A General Introduction to Aeroacoustics and Atmospheric Sound. In Computational Aeroacoustics, J. C. Hardin and M. Y. Hussaini, Eds. Springer-Verlag, 1993. [31] Lowson, M. V. Applications of Aero-Acoustic Analysis to Wind Turbine Noise Control. Wind Engineering 16, 3 (1992), 126-140. [32] Lowson, M. V., AND Fiddes, S. P. Design Prediction Model for Wind Turbine Noise. ETSU W/13/00317/REP (1994). [33] Mac C ormacic , R. W. The Effect of Viscosity in Hypervelocity Impact Cratering. AIAA Paper No. 69-354 (1969). [34] Martin, J. E. Aeroacoustic Computation of a Gust-Plate Interacting with a Flat Plate. In IGASE/LaRC Workshop on Benchmark Problems in Com putational Aeroacoustics (CAA) (1995), J. C. Hardin, J. R. Ristorcelli, and C. K. W. Tam, Eds., NASA Conference Publication 3300. [35] MlLNE-THOMSON, L. M. Theoretical Aerodynamics, Dover ed. Dover Pub lications, 1973. [36] Mitchell , B. E., Lele, S. K., and Parviz , M. Direct computation of the sound from a compressible co-rotating vortex pair. Journal of Fluid Mechanics 285 (1995), 181-202. [37] MYERS, M. K. Direct Simulation: Review and Comments. In Computational Aeroacoustics, J. C. Hardin and M. Y. Hussaini, Eds. Springer-Verlag, 1993. Ris0-R-947(EN) 69 [38] Nark, D. M. A Computational Aeroacoustics Approach to Sound Gen eration by Baffled Pistons. Master’s thesis, George Washington University, 1992. [39] Pierce, A. D. Acoustics, An Introduction to its Physical Principles and Applications. Acoustical Society of America, 1989. [40] Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. Numerical Recipes. Cambridge University Press, 1986. [41] Swarztrauber, P., and Sweet, R. Efficient FORTRAN Subprograms for the Solution of Elliptic Partial Differential Equations. Tech, rep., NCAR- TN/IA-109, 1975. [42] Ta ’asan , S., and Nark , D. M. An Absorbing Buffer Zone Technique for Acoustic Wave Propagation. AIAA 95-0164 (1995). [43] Tam, C. K. W., and Webb, J. C. Dispersion-Relation-Preserving Finite Difference Schemes for Computational Acoustics. Journal of Computational Physics 107, 2 (1993), 262-281. [44] TEMKIN, S. Elements of Acoustics. John Wiley & Sons, 1981. [45] Thompson, K. W. Time Dependent Boundary Conditions for Hyperbolic Systems. Journal of Computational Physics 68 (1987), 1-24. [46] THOMPSON, K. W. Time Dependent Boundary Conditions for Hyperbolic Systems, II. Journal of Computational Physics 89 (1990), 439-461. [47] Viswanathan, K., AND Snakar, L. N. A Comparative Study of Upwind and MacCormack Schemes for CAA Benchmark Problems. In ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA) (1995), J. C. Hardin, J. R. Ristorcelli, and C. K. W. Tam, Eds., NASA Con ference Publication 3300. [48] Watson, W. R., and Myers, M. K. Inflow-Outflow Boundary Conditions for Two-Dimensional Acoustic Waves in Channels with Flow. AIAA Journal 29, 9 (1991), 1383-1389. 70 Ris0-R-947(EN) Nomenclature Latin letter symbols A square coefficient matrix a sphere radius or grid stretching parameter B square coefficient matrix Be east boundary Bn north boundary Bs south boundary B\v west boundary b complex position coordinate for vortex C source term vector a source term vector component of vector C c sound speed Co ambient sound speed Cp specific heat at constant pressure E grid stretching parameter e natural base of logarithms F vector f frequency or arbitrary function f derivative of arbitrary function G vector Hankel function of second order and second kind ( J2-iY2) i V=T j2 Bessel function of the first kind and second order k wave number or thermal conductivity L typical length scale for flow field A, £2, £«) £j used in the compact form of ThBC ^1) ^2) lj left eigenvectors of the matrix A M, Mo Mach number Mi, M2, Mi used in the compact form of ThBC mi, m2, m-i left eigenvectors of the matrix B P incompressible pressure P square Jacobian matrix P _ time averaged incompressible pressure -Py derivative of P with respect to x, y Pt, Px, Py derivative of P with respect to t, x, y P total pressure P' pressure perturbation Po ambient pressure Pi hydrodynamic pressure fluctuation Ris0-R-947(EN) 71 Q speed Q square Jacobian matrix Q time average of the speed Q Qx, Qy derivative of Q with respect to x, y <7 induced velocity Q heat flux vector R radius (= \/X2 + Y2) R square Jacobian matrix Re Reynolds number r radius n boundary radius S entropy S source term vector S square matrix with rows of eigenvectors U T temperature or averaging time T square matrix with rows of eigenvectors m; Ty fundamental period of flow oscillation t time to start time for averaging of pressure U convective velocity or incompressible velocity component U solution vector of nonconserved variables U solution vector of conserved variables U0 velocity amplitude of sphere surface Ut, UX, Uy derivative of U with respect to t, x, y u total velocity u' velocity perturbation V incompressible velocity vector incompressible velocity components Vr radial velocity derivative of V with respect to t, x, y Vo tangential velocity V total velocity vector v' perturbation velocity vector Vi, Vj, vk total velocity components Vi', Vj' perturbation velocity components IV sound power Wre( reference sound power (= 10-12W in air) IV complex potential function X position type coordinate X position coordinate X position vector Xi, Xj, xk components of position vector Y position type coordinate Y 2 Bessel function of the second kind and second order y position coordinate complex number = x + hy 72 Ris0-R-947(EN) Greek letter symbols P thermal expansion coefficient or CFL number r circulation 7 ratio of specific heats A r grid spacing in radial direction At time step Ax, Ay grid spacing in Cartesian coordinates 8 ij Kronecker delta function (= 0 if i ^ j and = 1 if i = j) C grid stretching coordinate V grid stretching coordinate K, strength of vortex A diagonal matrix of eigenvalues A, A wave length Ai, Ag, A{, Aj eigenvalues of the matrix A M dynamic viscosity Mo ambient dynamic viscosity Mu M2, Mi eigenvalues of the matrix B fo ambient kinematic viscosity P total fluid density P' density perturbation Po ambient density of fluid pi hydrodynamic density correction Pit, Plx, Ply derivative of pi with respect to t, x, y a safety factor r, Tij viscous stress tensor or time complex potential function (/> acoustic variable or velocity potential 4>a phase angle between complex pressure and particle speed at sphere surface Super- and subscript o ambient state, farfield state lin linearized i, J, grid coordinates or counters n time level ref reference Units °c degree Celsius dB decibel Hz Hertz kW kilowatt m meter s second W Watt Ris0-R-947(EN) 73 Mathematics tensor product arg( ) argument of complex number S{} imaginary part of a complex number %{} real part of a complex number ln( ) logarithm to the base e log( ) logarithm to the base 10 d/dt, d/dx, .. . partial derivatives with respect tot, %, ... D/Dt( ) substantial derivative (= d/dt( ) + v-V( )) v-( ) divergence of a vector V( ) gradient of a scalar Abbreviations CAA computational aeroacoustics CFD computational fluid dynamics CFL Courant, Friedrichs, and Lewy GNU Christopher Newport University DNS direct numerical simulation DTU Technical University of Denmark LKBC Thompson ’s boundary conditions modified by Lee and Koo PDF partial differential equation PFV The Test Station for Wind Turbines Ph.D. Philosophiae Doctor (Doctor of Philosophy) PPW points per wavelength PSD power spectral density RB06 currently, the best soccer team in Roskilde SWL sound power level ThBC Thompson ’s nonreflecting boundary condition TWBC Tam and Webb ’s nonreflecting boundary condition 74 Ris0-R-947(EN) List of Figures 1.1 Sound power levels for different sizes of wind turbines. 9 4.1 Flow configuration. Pulsating sphere. 23 4.2 Incompressible flow solution for pulsating sphere. 24 4.3 Numerical and analytical solution for M = 0.05, M = 0.1, and M 0.2. 30 4.4 Numerical and analytical solution for M = 0.3, M = 0.4, and M 0.5. 31 4.5 Time history of (p — Po)/(poCo) at r/a = 26. 32 4.6 Power spectral density of (p - Po)/(poC q) at r/a = 26. 33 4.7 Time history for different boundary conditions. 34 4.8 Effect of distance to outer boundary on TWBC. 35 4.9 Time history for different initial conditions. 35 5.1 Directivity pattern for lateral quadrupole. 37 5.2 Flow configuration. 38 5.3 Incompressible flow solution 40 5.4 Computational domain with wave directions. 43 5.5 Acoustic pressure p'/(poCo) for tco/ro = 1542.20, M = 0.08. ^7 5.6 Acoustic pressure on diagonal of computational domain. 43 5.7 Time series of acoustic pressure at farfield monitor points. 40 5.8 Acoustic pressure on horizontal line of computational domain. 50 5.9 Particle velocity in first quadrant of computational domain. 50 5.10 Acoustic pressure p'/(poCq) for tco/ro = 1542.20, M = 0.08. 51 5.11 Acoustic pressure on diagonal of computational domain. 52 5.12 Time series of acoustic pressure at farfield monitor points. 53 5.13 Acoustic pressure p'/(pocl) for tco/ro = 1127.60, M = 0.11. 54 5.14 Acoustic pressure, tco/r 0 = 1127.60, M = 0.11, and a = 0.9. 55 6.1 Directivity pattern for dipole. 56 6.2 Circular cylinder in uniform cross flow. 56 6.3 Viscous mesh, 189 x 161. 59 6.4 Snapshot of the incompressible solution. 60 6.5 Hydrodynamic density correction, pi . 61 6.6 Acoustic mesh, 151 x 161. 63 6.7 Close-up of meshes. 63 6.8 Snapshots of perturbation pressure, p'. 64 6.9 Extent of flow field and acoustic field, t = 7Tq. 65 Ris0-R-947(EN) List of Tables 2.1 Solution strategy for the acoustic/viscous splitting technique. 18 3.1 Differencing sequence for the MacCormack scheme 20 76 Ris0-R-947(EN) Bibliographic Data Sheet Ris0—R—947 (EN) Title and author(s) Aeroacoustic Computation of Low Mach Number Flow Kristian Skriver Dahl ISBN ISSN 87-550-2249-9 0106-2840 Dept, or group Date Meteorology and Wind Energy December 1996 Groups own reg. number(s) Project/contract No. Pages Tables Illustrations References 76 2 33 48 Abstract (Max. 2000 char.) This thesis explores the possibilities of applying a recently developed numerical technique to predict aerodynamically generated sound from wind turbines. The technique is a perturbation technique that has the advantage that the underlying flow field and the sound field are computed separately. Solution of the incom pressible, time dependent flow field yields a hydrodynamic density correction to the incompressible constant density. The sound field is calculated from a set of equations governing the inviscid perturbations about the corrected flow field. Here, the emphasis is placed on the computation of the sound field. The nonlinear par tial differential equations governing the sound field are solved numerically using an explicit MacCormack scheme. Two types of nonreflecting boundary conditions are applied; one based on the asymptotic solution of the governing equations and the other based on a characteristic analysis of the governing equations. The for mer condition is easy to use and it performs slightly better than the characteristic based condition. The technique is applied to the problems of the sound generation of a pulsating sphere, which is a monopole; a co-rotating vortex pair, which is a quadrupole, and the viscous flow over a circular cylinder, which is a dipole. The governing equations are written and solved for spherical, Cartesian, and cylindrical coordinates, respectively, thus, representing three common orthogonal coordinate systems. Numerical results agree very well with the analytical solutions for the problems of the pulsating sphere and the co-rotating vortex pair. Numerical re sults for the viscous flow over a cylinder are presented and evaluated qualitatively. The technique has potential for applications to airfoil flows as they are on a wind turbine blade, as well as for other low Mach number flows. Descriptors INIS/EDB ACOUSTICS; AERODYNAMICS; BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; INCOMPRESSIBLE FLOW; MACH NUMBER; NUMERICAL SOLUTION; SUBSONIC FLOW; WIND TURBINES Available on request from: Information Service Department, Rise National Laboratory (Afdelingen for Informationsservice, Forskningscenter Riso) P.O. Box 49, DK-4000 Roskilde, Denmark Phone (+45) 46 77 46 77, ext. 4004/4005 • Fax (+45) 46 75 56 27 • Telex 43 116 TCDTT Objective Jml Ris0’s objective is to provide society and industry with new opportunities for development in three main areas: • Energy technology and energy planning • Environmental aspects of energy, industrial and agricultural production • Materials and measuring techniques for industry In addition, Ris0 advises the authorities on nuclear issues. Research profile Ris0’s research is strategic, which means that it is long-term and directed toward areas which technological solutions are called for, whether in Denmark or globally. The research takes place within 11 programme areas: • Wind energy • Energy materials and energy technologies for thefiiture • Energy planning • Environmental impact of atmospheric processes • Processes and cycling of matter in ecosystems • Industrial safety • Environmental aspects of agricultural production • Nuclear safety and radiation protection • Structural materials • Materials with special physical and chemical properties • Optical measurement techniques and information processing Transfer of Knowledge Risp ’s research results are transferred to industry and authorities through: • Co-operation on research • Co-operation in R&D consortia • R&D clubs and exchange of researchers • Centre for Advanced Technology • Patenting and licencing activities And to the world of science through: • Publication activities Ris0-R-947(EN) • Network co-operation ISBN 87-550-2249-9 • PhD education and post docs ISSN 0106-2840 Available on request from: Information Service Department Key Figures Ris0 National Laboratory Ris0 has a staff of more than 900, including more than 300 BO. Box 49, DK-4000 Roskilde, Denmark researchers and 100 PhD students and post docs. Risp ’s Phone +45 46 77 46 77, ext. 4004/4005 1996 budget totals DKK 471 m, of which 45 % come from Telex 43116, Fax +45 46 75 56 27 http://www.risoe.dk research programmes and commercial contracts, while the e-mail: [email protected] remainder is covered by government appropriations.