Cheung Lele JFM 09

J. Fluid Mech. (2009), vol. 625, pp. 321351. c 2009 Cambridge University Pressdoi:10.1017/S0022112008005715 Printed in the United Kingdom

321

Linear and nonlinear processes intwo-dimensional mixing layer dynamics

and sound radiation

LAWRENCE C. CHEUNG1 AND SANJIVA K. LELE1,21Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-3030, USA

2Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305-3030, USA

(Received 23 June 2008 and in revised form 10 December 2008)

In this study, we consider the eects of linear and nonlinear instability waves onthe near-eld dynamics and aeroacoustics of two-dimensional laminar compressiblemixing layers. Through a combination of direct computations, linear and nonlinearstability calculations, we demonstrate the signicant role of nonlinear mechanisms inaccurately describing the behaviour of instability waves. In turn, these processes havea major impact on sound generation mechanisms such as Mach wave radiation andvortex pairing sound. Our simulations show that the mean ow correction, which isrequired in order to accurately describe the dynamics of large-scale vortical structures,is intrinsically tied to the nonlinear modal interactions and accurate prediction ofsaturation amplitudes of instability waves. In addition, nonlinear interactions arelargely responsible for the excitation and development of higher harmonics in theow which contribute to the acoustic radiation. Two ow regimes are considered:In supersonic shear layers, where the far-eld sound is determined by the instabilitywave solution at suciently high Mach numbers, it is shown that these nonlineareects directly impact the Mach wave radiation. In subsonic shear layers, correctlycapturing the near-eld vortical structures and the interactions of the subharmonicand fundamental modes become critical due to the vortex pairing sound generationprocess. In this regime, a method is proposed to combine the instability wave solutionwith the LilleyGoldstein acoustic analogy in order to predict far-eld sound.

1. IntroductionThe major role that discrete instability waves play in determining the acoustic

radiation from mixing layers and jets has prompted a need to understand thefundamental linear and nonlinear processes that aect their behaviour. Several majoradvances in aeroacoustics, such as the work of Tam & Morris (1980), Tam & Burton(1984a , b) and Goldstein & Leib (2005), have been predicated on the ability of linearinstability wave theory to predict sound from supersonic shear layers. However, fewstudies thus far have addressed the relative importance of nonlinear instability waveinteractions and mean ow interactions to the sound generation process. In thispaper, we contrast the sound radiation mechanisms from two-dimensional supersonicand subsonic laminar mixing layers forced by linear and nonlinear instability waves.Using the nonlinear Parabolized Stability Equations (PSE) and direct calculations,

Email address for correspondence: [email protected]

322 L. C. Cheung and S. K. Lele

we demonstrate the importance of nonlinear interactions in accurately predicting thespatial evolution of these mixing layers and, as a result, the far-eld radiated sound.

This work seeks to answer several open questions concerning the behaviour ofinstability waves and their link to acoustic radiation. For instance, the importanceof modal interactions in capturing the overall shear layer dynamics and the resultingacoustic radiation has not been addressed directly in the literature. Also, in cases ofMach wave radiation from mixing layers and jets, questions often arise regarding thenecessity of including nonlinear eects, or whether a simple accounting of the meanow spreading is sucient. To address these questions, we consider the behaviour ofdiscrete instability waves which are at the origin of two sound generation processesin mixing layers: Mach wave radiation and vortex sound. Through careful analysiswe can isolate the various eects due to nonlinear modal interactions, mean owcorrections and inlet conditions and also determine their relative inuence on theradiated sound levels in each case.

1.1. Prior work

Several previous theoretical studies are directly relevant to the present study and areworth noting. From the early work of Crighton & Gaster (1976), Tam & Morris(1980) and Tam & Burton (1984a , b), to the more recent studies by Avital, Sandham& Luo (1998a , b), Wu (2005) and Goldstein & Leib (2005), these theories have formedthe basis for explaining the dynamics of linear instability waves, and established theirrole in the aeroacoustics of jets and mixing layers. For instance, in the work ofCrighton & Gaster (1976), they developed methods to model the growth of instabilitywaves in a slowly diverging jet ow through the multiple scales method. This wasalso used in Tam & Morris (1980) and Tam & Burton (1984a , b), where they foundthat the linear stability solution could be extended into the far eld to capture theacoustic radiation from high-speed supersonic mixing layers.

At the same time, however, several important studies also indicated that interactionsbetween instability waves were responsible for the generation of large vorticalstructures that appear in mixing layers when harmonically forced. From experimentalevidence gathered by Brown & Roshko (1974) and Winant & Browand (1974), thework of Ho & Huang (1982) and Laufer & Yen (1983) noted that the interactionbetween the subharmonic and fundamental modes leads to the vortex pairing processand plays a signicant role in determining the acoustic source location. In the reviewby Tam (1995), the link between large vortical structures, instability waves and soundemission is further discussed in the context of supersonic jet noise.

More recent work has focused attention on the nonlinear developmentof instabilitywaves and more complex models of instability wave based sound generation in mixinglayers. For instance, wave packet models were employed by Avital et al. (1998a) tostudy Mach wave radiation for time-developing mixing layers. Their analysis indicatedthat the most dominant mode for acoustic radiation was a two-dimensional modewhich evolved from the nonlinear development of the mixing layer. Additional workregarding the nonlinear evolution of supersonic instability waves in Mach waveradiation was carried out by Wu (2005). Using a matched asymptotic expansioncombined with the multiple scales method, he accounted for the nonlinear spatialevolution of the instability wave through nonlinear terms arising from the critical layer.The inuence of nonlinear mechanisms in sound generation has also been examinedby Sandham, Morfey & Hu (2006) in their study of convecting vortex packets. Theirformulation was based on a pair of linearized Euler equations and restricted thenonlinear interactions to the source terms of the LilleyGoldstein acoustic analogy.

Linear and nonlinear processes in mixing layers 323

A comprehensive study has not yet been undertaken to determine the role ofnonlinear instability wave interactions in capturing the dynamics of vortical structures,nor their importance in determining the level and directivity of acoustic radiation. Atnite amplitudes, the growth of the instability waves aects the evolution of the meanow, which can in turn trigger the growth (or dampening) of higher harmonics in theow. This phenomena naturally raises questions of whether the initial amplitudes andinlet conditions for instability waves have a sizable eect on the downstream evolutionof the shear layer and the resulting acoustic radiation. One can also ask whether theeventual saturation of instability waves is mainly due to the eects of mean owspreading, or if the transfer of energy between modes plays a more signicant role.Some questions still remain as to whether the presence of large vortical structuresin shear layer and jet ows can be accounted for independently of the instabilitywave dynamics, or if the development of such structures is intrinsically tied to theinteractions of the instability waves. Previous work by Hultgren (1992) sought toanswer these questions through the use of matched asymptotic analysis on weaklynon-parallel incompressible mixing layers, but a fully nonlinear study has yet to beundertaken.

A natural approach to investigating these questions is to use the NonlinearParabolized Stability Equations (NPSE). Since its development by Herbert &Bertolotti (1987) and Bertolotti, Herbert & Spalart (1992) in the study of boundarylayer transition, the PSE has been used in many relevant studies of aeroacoustic andfundamental mixing layer problems. For instance, Balakumar (1994, 1998) and Yen& Messersmith (1998) used linear PSE to calculate the instability wave behaviour forMach 2.1 jets, and more complex linear and nonlinear PSE calculations were alsocarried out by Malik & Chang (2000). In their study of the structure and stability ofcompressible reacting mixing layers, Day, Mansour & Reynolds (2001) also used thetechnique to examine mixing eectiveness.

NPSE can accommodate the eects of nonlinear mode interactions, non-parallelow and nite amplitude disturbances. It therefore extends beyond linear instabilitywave methods and more accurately models the near-eld evolution and the far-eldaeroacoustics of the mixing layer. Furthermore, the ability of PSE to probe eachof the nonlinear eects in isolation or in a combined manner can yield tremendousphysical insight into the processes underlying the mixing layer dynamics and soundradiation. For instance, one can investigate the eects of multiple mode interactionseither alone, or in conjunction with the mean ow correction.

Although the general shear layer problem includes complex three-dimensional uidmotions arising from a broadband spectrum of forcing, the current study focuses onnonlinear eects between discrete instability waves leading to the creation of largetwo-dimensional roller structures in the ow and their implications on the subsequentsound generation. While many previous DNS investigations (Sandham & Reynolds1991; Rogers & Moser 1992) have shown that the three-dimensional evolution of themixing layer eventually leads to the growth of streamwise rib vortices and spanwisekinking of the rollers, our objective is to gain a fundamental understanding ofthe two-dimensional processes rst, before attacking the complete three-dimensionalproblem. While the formulations assumptions limit the method to slowly evolvingconvectively unstable ows, the capabilities of the PSE are well suited for the scopeof this papers objectives.

To date, few studies have examined the link between the near-eld dynamicsof nonlinear instability waves with the radiated acoustic eld. In Mach waveradiation, the instability wave is directly coupled to the acoustic eld, but in


vortex sound generation, an additional nonlinear mechanism is responsible for theacoustic radiation. The creation of sound sources through the nonlinear interactionof instability waves can be analysed through the acoustic analogy method (Lighthill1952; Phillips 1960; Lilley 1974), and may capture the sound generation process morecompletely than a pure instability wave theory. Recent advances by Goldstein (2001,2003, 2005) have led to the development of a generalized acoustic analogy, and amethod based on ltering in the wavenumber frequency domain was proposed tobetter understand the true sources of sound. The generalized acoustic analogy hasbeen used by Goldstein & Leib (2005) to attack the problem of instability waves ina non-parallel base ow, but remains restricted to linear processes.

1.2. Outline

The primary objective of this study is to address several unanswered questionsregarding the linear and nonlinear behaviour of instability waves and their relationshipto sound generation in mixing layers. Using a combination of direct computation,linear and nonlinear stability methods, the importance of nonlinear interactionsand mean ow corrections to the shear layer dynamics will be illustrated. Threetwo-dimensional mixing layers, one supersonic and two subsonic, are examined indetail, and two dierent mechanisms for sound generation Mach wave radiationand vortex pairing sound are discussed. In each case, the coupling between theinstability waves near-eld hydrodynamics and their radiated far eld sound will beexamined. In instances where instability wave theory is not expected to provide acomplete description of the acoustic radiation, we show how the formulation can beextended using Lilleys acoustic analogy.

The basic problem description and computational details are provided in 2. Theshear layer dynamics and the acoustic radiation characteristics of a typical supersonicand subsonic mixing layers are compared and contrasted in 3. In 4, the mixinglayer results from linear and nonlinear theory are presented, followed by a detaileddiscussion of the results for the subsonic mixing layer. Finally, the combined instabilitywave acoustic analogy technique is shown in 6 before the conclusion.

2. Methodology2.1. Problem description

The primary focus of this investigation is on the aeroacoustics of two-dimensionallaminar compressible mixing layers. An upper speed stream with velocity U1 overliesthe lower speed stream U2 and the resulting shear causes the mixing layer to developdownstream in the x direction, as represented schematically in gure 1. Conceptually,we divide the domain into a near-eld region, where the hydrodynamic motions aredominant, and a far-eld region, where the acoustic behaviour is to be determined.In addition, the mixing layer is articially excited by particular instability waves thatare imposed at the inlet location. These inlet instability waves for all calculationswere obtained by solving the parallel ow linear stability problem dened by theRayleigh equation (A 8). The fundamental frequency of the instability waves 0 ischosen based upon the most unstable frequency at the inlet of the mixing layer. Aninstability wave corresponding to the subharmonic frequency 0/2 was also includedin the subsonic mixing layer case where vortex pairing was to be studied.

This method of articial excitation generally leads to the development of highlyorganized periodic vortex structures as the mixing layer evolves downstream. Althoughthis articial forcing limits a broadband spectrum from developing and precludes


Simulations M1 M2 ReU St0 Uc/a1 Mr,1 Mr,2

M29M1 2.9 1.0 1283 0.0419 1.95 0.762 1.138M25M15 2.5 1.5 3000 0.230 2.00 0.509 0.491M050M025 0.50 0.25 250 0.201 0.375 0.128 0.122

Table 1. Flow conditions for the mixing layer computations. The Reynolds number iscalculated as ReU = 1(U1 U2)0/1.

U1 Far field

Near field

Far field

y

x

U2

Figure 1. Schematic representation of two-dimensional compressible forced mixing layer.

the more complicated turbulent motions of realistic shear ows, it does allow usto investigate the sound generated by specic instability modes and examine theinteractions between dierent modes. Oblique modes were also excluded from theforcing in order to focus on two-dimensional nonlinear interactions and their eects.While the oblique modes tend to be more unstable under higher speed conditions,the inuence of three-dimensional structures on sound generation rst requires adetailed understanding of the mechanisms of sound radiation from two-dimensionalstructures.

In table 1, the relevant parameters for three compressible two-dimensional shearlayers are given. Both subsonic and supersonic shear layers are included, and theycan be characterized based on the convective velocity Uc (Papamoschou & Roshko1988) and relative phase velocity Mr,i , as dened by

Uc =a2U1 + a1U2

a1 + a2, Mr,i =

|/Re {} Ui |ai

,

where i =1, 2, and and are the wavenumber and temporal frequency of theprimary instability mode, respectively. The speed of sound in the upper and lowerstreams is denoted by a1 and a2, respectively, and Re{} denotes the real part of theargument. In this study, mixing layers are classied as subsonic if Mr,i < 1 in bothstreams, and as supersonic otherwise. In all of the tabulated cases, the temperatureratio between the upper and lower streams was set to T2/T1 = 1, the Reynolds numberReU was based on the velocity dierence U =U1 U2, initial vorticity thickness 0and upper stream density 1 and 1. The Strouhal number of the forced fundamentalfrequency was dened by St0 =00/(2U ). The parameter Mr,i indicated the generalacoustic radiation characteristics of the instability wave into the upper (i =1) andlower stream (i =2) of the mixing layer. In cases where Mr,i > 1, the instability waveis capable of radiating directly into the corresponding stream in the form of Machwaves, while cases of Mr,i < 1 suggest that a dierent mechanism of sound radiation


100 200 3000

0.5

1.0

1.5

0.5

1.0

1.5

M050M025

M29M1

x/0

100 200 3000x/0

Mr,i

(a) (b)

M050M025

M29M1

Figure 2. The phase speeds Mr,1 ( ) and Mr,2 ( ) for the forced instability modes.(a) The fundamental mode of M29M1 and subharmonic mode of M050M025. (b) The rstharmonic of M29M1 and fundamental mode of M050M025.

is responsible. In the subsonic mixing layers M25M15 and M050M025, Mr,i statedin the table corresponds to the fundamental mode at the inlet. The variation of Mr,iwith downstream distance for these ows is shown in gure 2. For the supersonicmixing layer M29M1, Mr,i is given for the rst harmonic, which is the mode initiallyradiating at the inlet. Results from these shear layers, and simplied models of theirbehaviour, will be presented in subsequent sections.

2.2. Computational models

The behaviour of the instability waves in two-dimensional compressible laminarmixing layers can be determined through the direct computation of the governingNavierStokes equations (2.1a)(2.1d) for the density , velocities ui , temperature T ,pressure P and dissipation function . Details regarding the simulations, includingthe numerical methodology and boundary conditions used in the code, are discussedin Appendix A 1.

t+

xi(ui) = 0, (2.1a)

(ui

t+ uj

ui

xj

)= P

xi+

1

Re

xjij , (2.1b)

(T

t+ uj

T

xj

)=

RePr

(

xj

T

xj

) P

(uj

xj

)+

1

Re, (2.1c)

P = 1

T . (2.1d)

In conjunction with the direct computations, an instability wave model was alsoused to probe the nonlinear interactions between modes. Using =[ u1 u2 T ]

T asthe vector of ow variables, this model represented the discrete instability waves in

terms of spatially evolving, nite amplitude modes m(x, y) in a slowly developingmean ow. Through the use of the parabolizing approximations (A 5)(A 6), thenonlinear evolution equation (2.2) can be used to track the behaviour of the instabilitywaves given the nonlinear interaction term Fm and amplitude factor Am dened in


40(a)

20

0

20

40

80(b)

604020

20406080

0

50 100 150 200 250 300 50 100 150 200 250

Figure 3. Real part of the far-eld pressure P1(x, y), superimposed on the instantaneousnear-eld spanwise vorticity, for the (a) M29M1 and (b) M050M025 mixing layers. Pressurecontour levels: (a) 3.5 102 to 3.5 102 in steps of 3.5 103, (b) 5 106 to 5 106in steps of 5 107.

Appendix A 2.

Lm{m} = FmAm . (2.2)In some subsonic shear layers, this model was used to compute the source terms inLilleys acoustic analogy (Lilley 1974) in order to capture the sound radiated fromvortex pairing processes. Further discussion of the combined PSE-acoustic analogymethod is found in Appendix A 2, along with a more comprehensive description ofthe PSE implementation and methodology.

3. Direct computationsThe dierences between Mach wave radiation and vortex sound generation can be

best illustrated by examining the direct calculations of the supersonic (M29M1) andsubsonic (M050M025) mixing layers. In the supersonic mixing layer, the instabilitymode forcing at the inlet triggers a direct response in the far-eld pressure, resultingin strong Mach wave radiation downstream (gure 3a). This is evident in situationswhere the phase speed of the instability mode is supersonic relative to the free stream,such as for the fundamental and rst harmonic modes. For the rst harmonic, Mr,2is supersonic at the inlet (gure 2), and thus the mode can radiate into the lowerstream immediately, while the fundamental mode is initially subsonic (Mr,1 = 0.97 andMr,2 = 0.93 at the inlet). However, as the fundamental mode evolves downstream, Mr,2becomes supersonic and Mach wave radiation is also seen at that frequency. A roughestimate of the wavefront geometry in the lower stream of gure 3(a) gives a Machangle of 60, corresponding to the downstream value of Mr,2 = 1.15 in gure 2(a).

On the other hand, the values of Mr,i remain rmly subsonic for the entire domainfor the M050M025 mixing layer, and the acoustic radiation mechanism is distinctlydierent. Rather than strongly emitting sound at a particular angle downstream, themajority of the sound in the subsonic mixing layer originates from a single pointinside the shear layer. The inclusion of both the fundamental and the subharmonicfrequency in the subsonic mixing layer leads to vortex roll up, and consequently, vortexpairing occurring near the apparent acoustic source origin. This correlation was notedpreviously by Ho & Huang (1982) and in the work of Colonius, Lele & Moin (1997).In addition, the location of vortex pairing is linked to the point of saturation forthe subharmonic and fundamental modes. From gure 4(b), one can determine thatthe maximum modal energy, as determined by (A 2), for the subharmonic (E1) and


100 200 300

1010

108

106

104

102

105

100(a) (b)

E1E1

E2E2

x/0 x/00 100 200 300

Figure 4. (a) Modal energy for the fundamental mode (E1(x)) and rst harmonic (E2(x)) ofthe supersonic mixing layer M29M1. (b) The modal energy for the subharmonic (E1(x)) andfundamental (E2(x)) mode of the subsonic mixing layer M050M025.

the fundamental (E2) also occurs near x 100, and around this location the largestnonlinear interactions between the two modes can be expected to occur.

In contrast, the nonlinear interactions of simulation M29M1 alter the characteristicsof the hydrodynamic near eld to a lesser degree, but still inuence the resulting soundeld. The near-eld vorticity in gure 3(a) shows a lack of vortex pairing due to theabsence of the subharmonic frequency, and less well-dened vortex roll up comparedto the subsonic M050M025 mixing layer. However, nonlinear interactions play asignicant role in the excitation of the rst harmonic, which can also radiate stronglyto the far eld. Unlike the fundamental mode, the rst harmonic is not forced at highamplitudes at the inlet, but grows to an appreciable amplitude due to interactionswith other modes (see gure 4a). The radiation patterns of the rst harmonic aresimilar to the Mach wave radiation of the fundamental, as discussed in 4.1.

Lastly, the direct calculations of the supersonic and subsonic mixing layers alsoillustrate dierences in the behaviour of the pressure eigenfunction, depending onthe phase speed of the instability wave relative to the free stream. In the subsonicM050M025 mixing layer, the phase speed of the instability wave is subsonic relativeto both streams, leading to two distinct regions of behaviour. Close to the shearedregion of the ow, the pressure eigenfunction exhibits purely exponential decay inthe y-direction, which is the expected behaviour from classical instability wave theory(gure 5b). For |y|> 15, however, the pressure eigenfunction transitions to a sloweralgebraic decay, which would signal a shift to acoustic wave propagation. In thecorresponding gure for the supersonic mixing layer (gure 5a), this transition inthe pressure P1(y) only occurs on the upper stream side, where Mr,1 = 0.855 at thelocation x =275. In the lower stream, where the relative phase speed Mr,2 = 1.04 atthe same location, the instability wave is directly coupled to acoustic disturbancespropagating in the free stream and a very slow exponential decay persists, becomingvisible as Mach wave radiation.

4. Linear and nonlinear stability calculationsThe extent to which linear and nonlinear stability calculations can capture the

ow and acoustic behaviours observed in 3 is examined next using a series of PSEsimulations. The results for the supersonic mixing layer are discussed rst, followedby the computations for the two subsonic cases.


20 0 20

101(a) (b)

102

103

102

104

106104

y/0y/0

40 20 0 20 40

Figure 5. Pressure cross-sections |P1(y)| for (a) M29M1 mixing layer at x =275 and(b) M050M025 mixing layer at x =250.

40(a) (b)

20

0

20

4050 100 150 200 250 300

40

20

0

20

40 50 100 150 200 250 300

Figure 6. (a) Pressure eld Re{P1} of the fundamental mode, as calculated by nonlinear PSE,plotted using same contours as gure 3(a). (b) Pressure eld Re{P2} of the rst harmonic,plotted using contours from 2.75 103 to 2.75 103 in steps of 2.75 104.

4.1. Supersonic mixing layer

4.1.1. Acoustic and near-eld predictions

The ability of nonlinear PSE to capture the acoustic and hydrodynamic behaviourof the supersonic mixing layer M29M15 is shown qualitatively in gure 6. StrongMach wave radiation is encountered at both the fundamental and rst harmonicfrequencies, and matches the acoustic behaviour of the direct calculation shown ingure 3(a). The angle of the Mach wave radiation to the upstream axis matches thevalues determined from the results of 3. At the fundamental frequency, the Machangle in the lower stream agrees with the previously estimated 60, and measurementsof the Mach wave give a value of Mr,2 1.05 for the rst harmonic mode, which isin close agreement with the value obtained from the direct calculation.

Contours of the spanwise vorticity plotted in gure 7 show that large-scale featuresof the ow are well represented by the current instability wave model. The streamwiseevolution of the instability waves responsible for creating vortical structures in theow is illustrated in gure 8(a). The growth of the modal energy E1(x) at thefundamental frequency and E2(x) at the rst harmonic closely follows the resultsfrom the direct calculations. Similarly, the behaviour of the acoustic pressure Pm inboth the streamwise and cross-stream directions is consistent with the Mach waveradiation previously observed. In gure 8(b), we can see that the strength of the


5

(a) (b)

0

5

5

0

5

100 150 200 250

Direct calculation Nonlinear PSE300 100 150 200 250 300

Figure 7. Near-eld vorticity contours for the M29M1 mixing layer, computed by (a) directcalculation and (b) nonlinear PSE. Contours range from 1.5 to 1.5 in steps of 0.1.

100 200 300

100

105

1010

100

102

104

106

E1

E2

x/0

y/0

100 200 300x/0

(a)

102

103

101

104

104

106

102

108

(c)(d)

(b)

y = 0

y = 30

20 0 20

x = 60

x = 275

2040 0 20 40y/0

Figure 8. (a) Integrated modal energy E1(x) for the supersonic mixing layer M29M1. (b)

streamwise pressure behaviour |P1(x)| at y =0 and y =30. (c) The pressure |P1(y)| at x =60and x =275 at the fundamental frequency. (d) The pressure |P2(y)| at x =290 at the rstharmonic. Direct calculation: solid line ( ), nonlinear PSE: dashed line ( ), linear PSE:dash-dot line ( ).

radiated pressure P1(x, y =30) is directly linked to the growth of the instabilitywave itself, and the dierence in downstream eigenfunction behaviour, depending onwhether Mr,i is supersonic or subsonic (gure 8c).

4.1.2. Nonlinear eects

The eects of nonlinearity can be easily shown by comparing the nonlinearcalculations with results from linear PSE and direct calculations. At the fundamentalfrequency, the evolution of the instability wave is well captured by the linearcalculation (gure 8a), and the radiated Mach waves show good agreement with


0 100 200 3001.0

1.5

2.0

2.5

3.0

x/00 100 200 300

x/0

w

(a) (b)

1

2

3

4

5

6

Figure 9. Comparison of the mean vorticity thickness for (a) M25M15 and (b) M050M025subsonic mixing layer. Direct calculation: solid line ( ), nonlinear PSE: dashed-dot line( ), linear PSE: dashed line ( ).

previous calculations. On the other hand, linear predictions for the developmentof the rst harmonic diverges with results from both nonlinear PSE and directcalculations. The initial amplitude of the rst harmonic in the linear PSE simulationis identical to the nonlinear case, but the growth rates remain constantly lower.As a consequence, the far-eld acoustic radiation is severely underestimated at thisfrequency (gure 8d ). This suggests that in order to accurately account for the Machwave emission at all of the radiating frequencies, nonlinear interactions betweeninstability modes must be included.

4.2. Nonlinear eects in subsonic calculations

The presence of nonlinear eects become more visible in the two subsonic mixinglayers M25M15 and M050M025. As instability waves cause vortex roll up and pairingto occur in the subsonic mixing layers, nonlinear mechanisms such as the mean owcorrection and the transfer of energy between modes drastically alter the downstreamevolution of the overall mixing layer. The precise inuence of these nonlinear eectscan be determined using a series of three PSE computations of the subsonic mixinglayers.

In the rst series, a purely linear stability theory (linear PSE) is used to calculatethe behaviour of the instability waves without the presence of a mean ow correctionand excludes the possibility of nonlinear interactions between modes. The secondset of calculations uses a partially linear theory, where linear PSE is used with apreviously computed mean ow correction but still excluding nonlinear interactions.The third set of calculations is fully nonlinear, using nonlinear PSE with the meanow correction and including all interactions between modes.

For both subsonic mixing layers, the correction to the mean ow was quite sizable,and was seen to have a large eect on the evolution, and hence, acoustic behaviourof the instability modes. At its peak location near the point of saturation, thecorrected mean ow vorticity thickness was two to three times the uncorrected value(gure 9). Nevertheless, the mean ow correction procedure in the full nonlinear PSEcalculations accurately predicted the growth of the mean ow.

The sudden expansion of the mean ow due to vortex pairing led to signicantchanges in instability wave growth. In the linear and nonlinear cases where themean ow correction was included, the location of maximum vorticity thickness alsocorresponded to the saturation locations of the rst two instability modes (gure 10).


0 100 200

1010

100

105

100

105

(a) (b)

x/0

0 100 200x/0

E1

E2

E1E2

Figure 10. Comparison of energies for the linear and nonlinear PSE simulations. (a) M25M15,(b) M050M025 subsonic mixing layer. Fully nonlinear PSE: solid line ( ), fully linear PSE:thin dash-dot line ( ), linear PSE with mean ow correction: thick dashed line ( ).

In the case of the purely linear PSE simulations, the absence of any mean owcorrection resulted in continued exponential growth for the rst two modes. While allcalculations yielded similar growth rates in the linear regions of the ow, using a purelylinear theory without compensating for the mean ow would lead to unreasonablylarge amplitudes for these modes. These observations are consistent with the ndingsof Hultgren (1992).

Nonlinear interactions between modes also played a signicant role indetermining the proper nal amplitudes for instability modes. Using a linear PSEapproach with mean ow correction captured the eventual saturation of thefundamental mode (E2) of mixing layer M050M025, but a fully nonlineartheory is required to predict the correct saturation amplitude. The inclusion ofnonlinear interactions is also required to capture the growth of higher harmonics.The fully or partly linear PSE calculations predict that the rst harmonic mode (E2) ofM25M15 remains relatively neutral, and stays at a relatively small amplitude beforeeventually decaying downstream (gure 10a). However, under the fully nonlineartheory, the rst harmonic is correctly shown to be initially unstable, and eventuallyreach a point of saturation.

The physical dierences in the ow between the linear and nonlinear PSEcalculations are illustrated in gures 11 and 12. When compared to the equivalentvorticity contours from the direct calculations, we can conclude that the linear PSEcalculations fail to completely capture the vortex roll up and pairing mechanisms. Inthe uncorrected linear PSE simulations, the perturbations in the fundamental modelead to alternating patterns of vorticity, but the roll up phenomena is not present.The unbounded growth of the mode also causes increasingly high concentrationsof vorticity which eventually exceed the contours of the plot. The inclusion of thecorrected mean ow helps to alleviate this problem, but the vortex roll up pattern isstill seen to be incomplete. In gure 12, the mean ow corrected linear PSE simulationfor M050M025 eventually results in discrete vortices, but the pairing process nearx =100 is still very dierent from the phenomena shown by the correspondingnonlinear PSE or direct calculation plots.

Comparisons of the fully nonlinear PSE simulations with the equivalent directcalculations show the importance of including the mean ow correction and modalinteractions. Once both eects are accounted for, the large-scale vortical structures


5

0

550 100 150

DNS

200

5

0

550 100 150

Linear PSE

200

5

0

550 100 150

Linear PSE with mean flow correction

200

5

0

550 100 150

Nonlinear PSE

200

Figure 11. Spanwise vorticity contours z for the subsonic mixing layer M25M15, ascomputed by direct calculation (upper left), nonlinear PSE (upper right), linear PSE (lowerleft) and linear PSE with corrected mean ow (lower right). Contour levels range from 1.0to 0.3 in steps of 0.025.

10

5

0

10

5

0 50 100 150

DNS

200

10

5

0

10

5

0 50 100 150 200

10

5

0

10

5

0 50 100 150 200

10

5

0

10

5

0 50 100 150 200

Linear PSE Linear PSE with mean flow correction

Nonlinear PSE

Figure 12. Spanwise vorticity contours z for the subsonic mixing layer M050M025, ascomputed by direct calculation (upper left), nonlinear PSE (upper right), linear PSE (lowerleft) and linear PSE with corrected mean ow (lower right). Contour levels range from 0.20to 0.05 in steps of 0.01.

computed from instability theory closely match those from direct computations.These results are in good agreement with the ndings of Day et al. (2001), whoalso compared linear and nonlinear PSE simulations of reacting mixing layers. Theimplications of these nonlinear eects for sound radiation are stressed in the presentpaper.


500 100 150 200 250

100 100

104

106

108

102

104

106

x/0 x/0

1/101/10

1

21

21

Modal Energy E1

0 50 100 150 200 250

1

y = 15

(a) (b)

Figure 13. (a) Integrated modal kinetic energy E1 for the fundamental mode with initial

amplitudes 21, 1 (thicker lines) and 1/10 (thickest lines). (b) Pressure magnitudes |P1(x)| aty/0 = 15 of the fundamental mode. Nonlinear PSE: solid lines ( ). Linear PSE: dashedlines ( ).

4.3. Initial amplitudes and saturation location

From the results shown in 4.2, one could surmise that if the correct mean owwere obtained, linear theory would be sucient in predicting the evolution for themost unstable instability waves. However, the initial amplitudes of the instabilitymodes provided at the inlet can also play a large role in determining the downstreamdynamics of the mode, and ultimately the emitted sound from the mixing layer. Inthe following section, we show the necessity of using both the corrected mean owand the corresponding initial mode amplitudes in order to obtain correct predictionsfrom linear theory.

For the subsonic M25M15 mixing layer, three pairs of linear and nonlinearPSE simulations were computed, with each pair of simulations using a dierentinitial amplitude (21, 1 or 1/10) for the fundamental mode. In the nonlinear PSEsimulations, the higher harmonics were fully coupled and the mean ow correctionappropriate to the set initial amplitude was accounted for. In all of the complementarylinear PSE simulations, only the corrected mean ow corresponding to the 1 casewas used, but the simulations did not include any coupling to the instability modes.Hence, for the 21 and 1/10 linear PSE simulations, there exists a mismatch betweenthe corrected mean ow and the corresponding initial amplitude of the instabilitywave.

Comparisons of the modal kinetic energy growths are shown in gure 13(a). Asexpected, the fundamental mode in the linear PSE simulations behave identically,except for a scaling factor determined from the initial amplitude. Following aninitial exponential growth region, all of the linear PSE modes saturate at the samedownstream position, and the nal amplitudes vary according to the scaling factor.On the other hand, the fundamental modes in the nonlinear PSE simulations saturateat dierent downstream positions depending on their initial amplitude, while thesaturation amplitudes remain approximately constant. Similar behaviour for thepressure amplitudes away from the centreline is shown in gure 13(b). The far-eldpressures from the nonlinear PSE simulations all converge to approximately the same


amplitude downstream, while the pressure amplitudes from the linear PSE simulationsdepend on their initial amplitudes.

From these comparisons, we can conclude that for a given instability mode (closeto the most unstable mode), the saturation amplitudes and saturation locationare correctly predicted by linear theory only if both the correct mean ow andcorresponding initial amplitudes are provided. Because some of this informationcannot be found through linear theory alone, it appears that some elements ofnonlinearity must be incorporated to obtain an accurate picture of the most unstableinstability wave evolution. As noted earlier, nonlinear eects are much stronger forother instability modes which grow via modemode interactions.

5. Subsonic mixing layers acoustic radiationThe previous section demonstrated the inuence of nonlinear interactions and

mean ow corrections in predicting the evolution of instability waves inside the shearlayer. In this section, we now consider the far-eld behaviour for the fully nonlinearsubsonic simulations. Of particular interest is the transition of the instability wavefrom hydrodynamic behaviour in the near-eld region to acoustic behaviour inthe far eld. Results from the nonlinear PSE simulation of M050M025 are shownalong with the complementary results from the direct calculations mentioned in 3.Because the phase speed of the fundamental and subharmonic mode of M050M025are subsonic relative to both streams of the mixing layer, the instability waves arenot directly coupled to the far eld, and the sound generation mechanism in this caseis due to vortex pairing.

The rst comparisons shown in gure 14(a) depict the streamwise evolution of thepressure P1 at the subharmonic frequency at two vertical locations: inside the shearregions of the ow (at y =0) and in the far eld (y =30). In the near-eld region of theow the pressure uctuations grow exponentially during the initial linear instabilityphase, before eventually saturating near the vortex pairing location. Not surprisingly,the growth rate and the saturation amplitudes for P1(x, y =0) along the centrelineare well predicted by the nonlinear PSE computation. However, in the acoustic eld,the predictions from nonlinear PSE begin to diverge from the direct calculation. Aty =30, the PSE calculations generally underpredict the pressure radiated to the fareld by an order of magnitude or greater, except near the vortex pairing location,where the pressures are comparable. In contrast to the radiation patterns of thesupersonic mixing layer, sound is radiated in both the downstream and upstreamdirections of the vortex pairing. From this near-eld data it is not clear whether theacoustic radiation is superdirective or a simpler composition of multipole elds.

This behaviour is further explored by considering the cross-sections of the pressurein the y-domain at dierent streamwise locations. In gure 14(b), the subharmonicpressure P1(x =50, y) is considered at a point upstream of the vortex pairing location.The direct calculation and nonlinear PSE calculation both agree in the near eldregion (|y| 20), but the pressure calculated by PSE decays to much lower levelsthan its directly computed counterpart. The directly calculated pressure shifts to theslower algebraic decay at an earlier point than PSE, leading to a larger amplitudepredictions in the far eld.

Near the vortex pairing location the far eld behaviour of the pressure is morecomparable between the two methods of calculation. At the streamwise location ofx =125.0 (gure 14c), both calculation methods predict a similar transition pointto the slower algebraic decay of the pressure eigenfunction. Near this location the


100 200 300

105

102

104

106

108

1010

100

x/0

y/0 y/0

x/0

y = 0

y = 30

y/0 = 0, y/0 = 30

x/0 = 125.0

x/0 = 50.0

x/0 = 250.0

40 20 0 20 40

40 20 0 20 40

102

104

106

102

104

106

40 20 0 20 40

(a) (b)

(c) (d)

Figure 14. The streamwise pressure |P1(x)| for the subsonic M050M025 mixing layer (a) alongy =0 and y =30. Also shown are the cross-stream pressure |P1(y)| at downstream positions(b) x =50 (c) x =125 (d) x =250. Direct calculation: solid lines ( ). NPSE: dashed-dotlines ( ).

far-eld pressures calculated by the PSE simulation are generally on the same orderof magnitude. For the M050M025 mixing layer this region of agreement is limited toa small region around the vortex pairing location, near 120 x 130.

Further downstream of the vortex pairing location the PSE predictions and directcalculations of the far eld again diverge. At position x =250.0 (gure 14d ), thepressure calculated by nonlinear PSE is approximately two orders of magnitudesmaller than the direct calculations. Inside the near-eld region, however, the PSEmethod correctly predicts the hydrodynamic behaviour for both the subsonic shearlayers.

6. Acoustic analogyIn previous sections, we noted that nonlinear instability wave theories could

correctly predict the near-eld hydrodynamic behaviour of subsonic mixing layers, butunderpredicted the far-eld acoustic radiation emanating from the vortex roll up andpairing triggered by the instability. In this section we show it is possible to combine theinstability wave approach with an acoustic analogy, and properly capture the acoustic


20(a)

(b)

0

0 50 100 150

PSE momentum source term

DNS momentum source term

200 250 300 350

0 50 100 150 200 250 300 350

20

20

0

20(c)

(d)

0

0 50 100 150

DNS thermodynamuc source term

200 250 300 350

0 50 100 150 200 250 300 350

20

20

20

020

PSE thermodynamuc source term

Figure 15. Plot of the source term magnitudes |m|and |t | for the subsonic mixing layerM050M025. Contours for (a) and (b) range from 106 to 105 in steps of 106. Contours for(c) and (d) range from 107 to 106 in steps of 106.

wave propagation. We demonstrate both the accuracy of the source terms computedusing nonlinear PSE information and the predictions obtained from applying Lilleysacoustic analogy. Lastly, we discuss the importance of using nonlinear theory incomputing the source terms versus using a purely linear theory.

6.1. Source terms

We rst consider the overall structure of the source terms computed by the nonlinearPSE and direct computations for the M050M025 mixing layer. As described by(A 13) and (A 14), the source term at the subharmonic frequency 1(x, y) can bedivided into momentum (m) and thermodynamic (t ) components in order to assesstheir relative contributions to the predicted far-eld sound. The linearized versionof the inhomogeneous LilleyGoldstein equation also requires a parallel mean ow(y), which we take near the point of subharmonic saturation, at x =100, for theM050M025 mixing layer. The sensitivity of the acoustic predictions to the choice ofthe mean ow location is discussed in 6.4. Also, in the computation of the nonlinearPSE, the inlet amplitudes 1 were not altered and consistent with the mean ow usedin the LilleyGoldstein equation.

A qualitative assessment of the source terms m and t is provided by the contourplots in gure 15. When the results from the nonlinear PSE calculations are comparedagainst their directly computed counterparts, we can see that the magnitude and extentof the source terms have been relatively well predicted. The peak source terms occurnear the vortex pairing location and are relatively limited in the transverse direction.The thermodynamic source terms in gure 15(c, d ) are also an order of magnitudesmaller than the momentum source terms, as would be expected for an isothermalmixing layer.

Cross-sections of the source terms for the isothermal subsonic shear layer ingure 16 provide a more quantitative examination of the source term structure. Ingure 16(a), the transverse structure of the momentum and thermodynamic sourceterms is shown at the location of vortex pairing (x =100) and the compact natureof the source terms in the y-direction is visible. Both nonlinear PSE and directcalculations predict that the source terms decay exponentially rapidly outside therange 10 y 10 and are essentially negligible in the free stream. From gure16(b), the streamwise structure of the momentum source term m(x, y =0) involvesan initial development region from about 0 x 60, also peaks near the vortexpairing location. Downstream of the vortex pairing location the source terms decayby approximately an order of magnitude by x =200. As noted previously, thethermodynamic source terms for the M050M025 mixing layer are small compared tothe momentum source terms. Although the nonlinear PSE version predicts a slightly


30 20 10 0 10 20 30

1010

108

106

104

(a) (b)

|m| |m|

|t | |t |

0 100 200 300

104

105

106

107

108

109

1010

Figure 16. Source terms for the isothermal subsonic shear layer. (a) Cross-section of source

terms |m(y)| and |t (y)| at x =100. (b) Cross-section of source terms |m(x)| and |t (x)|at the centreline y =0. Direct calculation: solid lines ( ). NPSE: thick dashed-dot lines( ).

40 20 0 20 40

102

104

106

108

1010

102

104

106

108

1010

yx = 75

yx = 175

40 20 0 20 40

(a) (b)

Figure 17. Far-eld acoustic predictions from DNS, PSE and hybrid PSE-acoustic analogy

methods for the isothermal subsonic shear layer. (a) |P1(y)| at x =75 and (b) |P1(y)| at x =175.Direct calculation: solid line ( ), PSE: dashed line ( ), Lilleys equation: dashed-dotline ( ).

higher peak for t (x, y =0) than the direct computation, it is still negligible comparedto the momentum source term.

6.2. Lilleys far-eld predictions

Given the source terms calculated in 6.1, we now make use of the acoustic analogyto recover the acoustic eld that was missing from the nonlinear PSE solution.By employing (A 16) to calculate the far eld pressures, we can compare thenew predictions to values previously obtained from the nonlinear PSE and directcalculations.

In gure 17 we re-examine the near-eld and far-eld pressure behaviours ofthe subsonic M050M025 mixing layer. Upstream of the vortex pairing location atx =75, the calculations from Lilleys equation correctly predict the far-eld behaviourin the upper and lower free stream, while the pressure eigenfunction in the PSE


0 100 200 300

102

104

106

108

1010

Figure 18. Far-eld acoustic predictions, |P1(x)| at y =30, from DNS, PSE and hybridPSE-acoustic analogy methods for the isothermal subsonic shear layer. Direct calculation:solid line ( ), PSE: dashed line ( ), Lilleys equation: dashed-dot line ( ).

50

0

50

50 100 150 200 250

Figure 19. Far-eld acoustic predictions, Re{P1(x, y)} using Lilleys acoustic analogy on thesubsonic M050M025 mixing layer. Pressure contours are the same as in gure 3(b).

method continues to decay exponentially. Downstream of the vortex pairing location,at x =175, the predictions from Lilleys acoustic analogy also correctly predict theamplitude of sound radiated away, while the PSE method alone underestimates thepressure by approximately an order of magnitude.

The streamwise behaviour of the far-eld pressure is also improved using predictionsfrom the combined nonlinear PSE-acoustic analogy approach. Whereas much of theupstream and downstream sound is absent from nonlinear PSE calculations, thepressure amplitudes |P (x, y =30)| given by the acoustic analogy predictions arecomparable to the directly calculated pressures (gure 18). Finally, the radiationpatterns shown in gure 19 illustrate the acoustic eld resulting from vortex soundgeneration, and agrees qualitatively with gure 3(b) from the directly computedmixing layer.


50 100 150 200 250 300

102100

105

1010

(a) (b)

104

106

108

1010

1012

1014

x/d0 x/d0y = 14 x = 190

20 0 20

Figure 20. Comparisons of the pressure |P1(x, y)| for the subsonic mixing layer M25M15,comparing results from the direct calculations (solid line, ), Lilleys acoustic analogy withnonlinear source terms (dash-dot line, ) and Lilleys acoustic analogy with linear sourceterms (dashed line, ).

6.3. Linear versus nonlinear source terms

In the next example of the combined PSE-acoustic analogy approach, we consideracoustic predictions for the subsonic M25M15 mixing layer and compare the eectsof source terms calculated with and without nonlinearity. Two series of calculationswere carried out, one using data from the nonlinear PSE, and the other using linearPSE.

In the rst series, nonlinear PSE was used to calculate the source terms for theLilleyGoldstein equation, and the predictions were compared against the directcalculations described in 3. The comparisons for the pressure P1 at the fundamentalfrequency, shown in gure 20, resemble the predictions for the M050M025 subsonicmixing layer from 6. The results from Lilleys acoustic analogy generally capturethe levels of acoustic radiation in the far eld, although the predictions for thelower stream are generally more accurate than in the upper stream. In the initialregion (0


150 100 50 0 50 100 1500

1

2

3

4

5

6

7

8

9 106

Figure 21. Sensitivity of Lilleys acoustic analogy predictions |P1( )| to mean ow prolesfor M050M025 mixing layer. Mean ow proles (y) chosen at the locations x =50 (thickdashed line, ), x =100 (thick solid, ), x =125 (dash-dot ), x =150, (thin dashed,

), x =200 (thin solid, ).

rst harmonic 2. However, because the linear PSE calculations failed to properlyexcite the rst harmonic ( 4), the resulting source term at the fundamental is alsoinaccurate. Not surprisingly, when the source terms from linear PSE are examinedin this particular case, we nd that their typical amplitudes are several orders ofmagnitude below those computed from nonlinear PSE.

Taking these ndings into consideration, we can make the following observationsregarding the aeroacoustic behaviour of mixing layers. For instability waves withsupersonic phase speed relative to one of the free streams, they are associated witha direct Mach wave radiation. This Mach wave radiation can be predicted using alinear stability wave theory, but only after appropriate accounting of the nonlineareects. The instability wave growth rate and its saturation need to be accuratelycaptured. However, for instability waves with a subsonic phase speed relative to thefree stream, the sound radiation is due to a dierent physical mechanism vortexpairing noise. Accurate predictions of the far-eld sound in this case heavily dependon using the correct mean ow and accounting for nonlinear modal interactions.In addition, although nonlinear stability wave theory can capture the growth of theinstability wave, it must be supplemented with an additional method such as anacoustic analogy to capture the acoustic radiation.

6.4. Sensitivity to mean ow location

As mentioned in 6.1, a preliminary study regarding the sensitivity of the acousticeld was conducted to determine the most appropriate choice of the mean owprole (y) to use in Lilleys acoustic analogy. Our results indicated that the choiceof the mean ow prole location can aect the directivity of Lilleys acoustic analogypredictions. This is shown in gure 21, where the pressure predictions |P1()| areplotted for mean ows selected at various locations, using data sampled at a distanceR=50 away from the apparent source location, and where the angle is chosen tobe 0 pointing in the downstream direction. From these results we can make twogeneral observations.

First, for mean ow proles located downstream of the instability wave and sourceterm peak location, e.g. the proles taken at x =125, 150 and 200 in gure 21, the


directivity of acoustic radiation shifts towards higher angles. For these proles thehigher angle lobe at 75 dominates over the lower angle lobe at 4045, andradiation towards lower angles is suppressed. This contradicts earlier evidence fromthe direct calculations and previous studies of mixing layer aeroacoustics (Coloniuset al. 1997), which have suggested that the peak acoustic radiation from vortex pairingshould occur at angles less than 60.

Secondly, for mean ow proles located upstream of the sound source peak, theacoustic radiation peak at lower angles is recovered. However, using proles locatedtoo close to the inlet or upstream of the vortex roll up location generally causedover-estimations of the far-eld pressure, as in the case of the prole selected fromx =50.

Given these observations, the choice of the mean ow prole near the saturationlocation and sound source peak at x =100 seems to avoid problems associated withthe directivity and amplitude of the acoustic predictions, and this prole was usedin all computations in 6.16.3. However, from our preliminary investigations, theneed for a comprehensive study on the sensitivity of Lilleys acoustic analogy to themean ow prole is apparent. By using the non-parallel acoustic analogy formulationfrom Goldstein (2003), or the adjoint Greens function approach of Tam & Auriault(1998), better insight may be gained into the refraction of sound from a spreadingmean ow.

Lastly, we also provide some comments regarding the eects of three dimensionalityon the dynamics and aeroacoustics of the mixing layers discussed in this work.Although oblique modes were not considered as a part of this work, some previousstudies have provided hints as to how three dimensionality may aect our currentresults. For supersonic mixing layers, the higher growth rate of oblique modes (Jackson& Grosch 1989) should lead to stronger acoustic radiation, but the mechanism ofsound generation (Mach wave radiation) is expected to remain the same as inthe two-dimensional case. For the subsonic case, the work of Rogers & Moser(1992) in incompressible mixing layers highlighted the collapse of rib vortices and thedistortion of the vortex core in the formation of kinks along the rollers. The nonlinearmechanisms involved in these processes may also serve to generate or modify soundsources, similar to the role of the nonlinear vortex pairing process in the creation ofsound sources in the two-dimensional case.

7. ConclusionsIn this paper, we examined the linear and nonlinear behaviour of instability

waves in two-dimensional mixing layers, and highlighted the inuence of nonlinearityto the dynamics and aeroacoustics of the shear layers. The eects of nonlinearinteractions, mean ow correction and inlet amplitude conditions were examinedusing a combination of direct calculation, linear and nonlinear stability methods.Both supersonic and subsonic compressible mixing layers were considered.

From the results of our computations, several general observations appearedregarding the linear and nonlinear processes in mixing layers can be made. First,nonlinear interactions are largely responsible for the excitation and development ofthe higher harmonic instability waves. Even in cases where the primary instabilitymode behaves linearly (over a specic region of the ow), nonlinear couplingwill trigger the growth of higher harmonics which can contribute to the acousticradiation that is missing from the purely linear analysis. Secondly, the mean owcorrection generated by nite amplitude instability modes was observed to be critical


in determining the appropriate saturation levels of instability modes. Using a lineartheory without any mean ow modication resulted in non-physical instability waveamplitudes downstream, and a fully nonlinear treatment with mean ow correctionwas required to predict the development of instability waves past the saturation point.In addition, both the nonlinear interactions and mean ow correction were observedto be necessary in capturing the dynamics of large-scale vortical structures in the neareld.

Similarly, the results of several computations illustrated the need for a fullynonlinear theory in order to predict the eects of the inlet conditions on the dynamicsand acoustics of the mixing layer. While linear theory predicts a similar saturationlocation for all instability waves regardless of initial inlet amplitude, nonlinear eectswere vital in determining position of the vortical structures, and consequently, thenoise source locations. Based on these observations, the need to account for theinteraction between the mean ow and the instability waves, and among the instabilitywaves themselves, becomes apparent.

The eects of nonlinearity could also be seen in the aeroacoustic behaviour of themixing layers as well. For supersonic mixing layers, where Mr,i > 1 in at least oneuid stream, the dominant mechanism of sound generation was Mach wave radiation.In these cases the far-eld sound was directly coupled to the behaviour and growthof instability wave. Thus, the nonlinear interactions discussed above directly impactsthe amplitude and structure of the Mach wave radiation. For subsonic mixing layers,where Mr,i < 1, the primary mechanism of sound generation is due to vortex pairinginitiated by the interactions of the fundamental and subharmonic instability modes.In this instance, capturing the dynamics of the near-eld vortical structures becomescritical and neither the nonlinear interactions nor the mean ow correction can beneglected.

Although nonlinear PSE provided an accurate description of the near-eldhydrodynamic motions, its solution underestimated the acoustic pressure eld forsubsonically convected waves. However, by using source terms calculated fromnonlinear PSE in the LilleyGoldstein equation, the resulting far-eld soundwas found to be in good agreement with direct calculations. Lastly, this paperdemonstrated the need to account for nonlinear interactions in computing the acousticsource terms. Using linear PSE to calculate the source terms in Lilleys acousticanalogy led to large underpredictions of the acoustic far eld.

This work was sponsored by a National Defense Science and Engineering Graduate(NDSEG) Fellowship, and a National Science Foundation (NSF) Graduate ResearchFellowship. Computational resources were provided in part by the Air Force Oceof Science Research (AFOSR), contract number FA9550-04-1-0031.

Appendix A. Mathematical detailsA.1. Direct calculation

Further details regarding the direct numerical simulation (DNS) of a single phasetwo-dimensional compressible mixing layer are presented in this section.

The variables in (2.1a)(2.1d) have been non-dimensionalized using the initialvorticity thickness 0 and values from the upper stream, including the speed of sounda1, density 1, temperature T1 and viscosity 1. The value for the ratio of specicheats in the upper stream = cp,1/cv,1 is taken to be 1.4. Finally, the non-dimensional


Reynolds number and Prandtl number are set to be Re= 1a10/1, Pr=1cp,1/k1,respectively, where k1 is the thermal conductivity of the uid in the upper stream.

The solution to (2.1a)(2.1d) were found through highly resolved numericalcomputations developed specically for aeroacoustics applications. The basicnumerical code was provided by Lui (2003), and used a sixth order compactnite dierence scheme with spectral-like resolution (Lele 1992) in the transverse(y) and streamwise (x) directions. For time advancement, a two-step fourth-orderLow Dissipation and Dispersion RungeKutta (LDDRK) scheme was employed.Sponge regions were placed in all outow boundaries to suppress reections andcoordinate stretching in the y-direction allowed for an ecient allocation of gridpoints in the near- and far-eld regions.

In order to obtain statistics in the frequency domain, data from the directcalculations were Fourier transformed in time, and instability wave energies werecomputed accordingly. The basic discrete transform

m(x, y) =1

N

N1j=0

(x, y, tj )eimtj (A 1)

was used where is any uid variable and N samples were gathered over a periodT =2/m at the frequency of interest. The values of

m were computed by direct

summation with either N = 32 or 64 samples obtained from the direct calculation.One major statistic used in comparing the eigenmode behaviour between the direct

calculation and PSE simulations is the integrated modal energy of the eigenmode.While the energy of the mode used in Day et al. (2001) involved only the kineticenergy components, the denition used here stems from the work of Chu (1965), andincludes the energy due to temperature and density uctuations.

Em =

{ (|um|2 + |vm|2) + 1

| m|22

+ |T m|22T

}dy (A 2)

A.2. PSE formulation

A simpler more insightful model for the behaviour of the mixing layer can be utilizedin cases where the inlet forcing is harmonic and the development of the shear layeris slow compared to the wavelength of the instability wave. Under these restrictions,the NavierStokes equations (2.1a)(2.1c) can be reduced to a parabolized systemgoverning the evolution of the instability waves and their nonlinear interactions.While the PSE approach has been largely standardized since its development byBertolotti et al., we provide a brief account of this method as it pertains to our work.Additional implementation details can be easily found in Malik & Chang (2000) andDay et al. (2001).

The model begins by separating the vector of ow variables =[ u v T]T into

mean and disturbance components via the assumption = + . The mean owquantities are found through a similarity solution of the compressible boundarylayer equations, as in Schlichting et al. (2004). The disturbance quantities are

assumed to be expressible as an expansion over eigenmodes m:

(x, y, t) =

Mm=M

m(x, y)Am(x)eimt , (A 3)


where the amplitude portion Am(x) is written as

Am(x) = m exp{i

x0

m(x) dx

}. (A 4)

In (A 3) and (A 4), an initial amplitude m is provided for each eigenmode m, withan associated wavenumber m and temporal frequency m. The temporal frequencyof each mode is set to be an integer multiple of a base frequency , so that m =m.

In contrast to parallel linear stability theory, both m and m are allowed to evolvedownstream in x through a set of slowly varying assumptions. If the mean owquantities evolve over a much longer length scale than the spatial wavelength of

the instability wave, then the rst derivatives of m and m can be retained, and aparabolization of the governing equations is possible.

The slow rate of evolution in the mean ow usually translates into the assumptions

1

Re

x O(1) and

2

x2 O(1), (A 5)

and these terms can then be ignored in the derivation of the PSE. Similarly, the PSErestricts the downstream changes in the eigenfunction and growth rate by assumingthat the terms

2m

x2 O(1) and

2m

x2 O(1), (A 6)

and can be neglected as well. The nal restriction on the disturbances admissible inPSE is that the instability waves are convectively unstable, and not absolutely unstable(Huerre & Monkewitz 1990). This ensures that for any given region, any oscillationswill remain bounded in time.

Using (A 3) and the slowly varying assumptions (A 5) and (A 6) in the governingequations (2.1a)(2.1c) yields a set of nonlinear disturbances equations (2.2) whichform the basis of the nonlinear PSE method. The linear operator Lm can be brokendown in terms of

Lm = imG+ A(im +

x

)+ B

y+ C

2

y2+ D+

m

xN, (A 7)

where the matrices A, B, C, D, G and N are functions of the mean ow variables only. The right-hand side of (2.2) contains the nonlinear forcing function Fm due tothe higher order products of disturbances .

By removing all streamwise derivatives, and viscous terms, and setting v=0, theRayleigh operator

LR {m} = {imG + A (im) + B y + C 2y2 + D} m (A 8)can be recovered from (A7). Solutions to (A 8) are used as initial conditions to boththe PSE calculations and the direct numerical calculations.

The linear PSE formulation is obtained from (2.2) by setting Fm =0. BecauseLm contains only a single derivative in the x-direction, (2.2) can be eciently solvedthrough a streamwise marching procedure. A rst-order backwards Euler method withvariable step size was used to speed calculation and to avoid problems with numericalstability Andersson, Henningson & Hani (1998). In general, PSE computationsrequired an order of magnitude fewer computational resources than the equivalentdirect computations, usually about O(102) versus O(103) CPU-hours on an Intel Xeon


cluster. Further details regarding the form of (2.2), the slowly varying assumptionsand the mean ow correction, are given in Appendix A2.1 and Cheung (2007).

A.2.1. Normalization conditions

An additional constraint must be placed on (2.2) in order to make the systemsolvable. The need for this additional normalization constraint arises from the extradegree of freedom inherent in the representation (A 3) streamwise variations of

(x, y) can be absorbed into either (x) or (x, y). This ambiguity in x canbe eliminated by placing an additional normalization constraint on either theeigenfunction, the eigenvalue or both. Many normalization conditions have beenproposed that limit rapid changes in the eigenfunctions (Bertolotti et al. 1992). Forthis work, the usual integral norm is generalized to include all components of the

eigenfunction

[W1

m

m

x+ W2

(um

um

x+ vm

vm

x

)+ W3T

m

Tm

x

]dy = 0, (A 9)

where the weights W1, W2, W3 correspond to dierent normalization conditions. Forthe kinetic energy based norm, we set the weights W1 =W3 = 0 and W2 = 1. A secondnorm with weights W1 = ( 1)T / , W2 = and W3 = ( 1)/( 2T P ) was alsoused to calculate the total disturbance energy (Chu 1965). In practice, the choice ofweights had negligible impact on the overall results (Cheung 2007).

A.2.2. Mean ow correction

When the amplitude of the eigenfunctions in nonlinear PSE calculations reaches anite size and modal interactions start to play an important role in the ow dynamics,

the nonlinear terms will inevitably redistribute some amount of energy to the mode 0at zero frequency. This redistribution of energy indicates that the disturbances havegrown to the extent that they are capable of modifying the mean ow, and a nonlinearcorrection to the mean ow is in order. This correction can be accommodated byadding components of the zero frequency mode to the original laminar mean owcomponents

= + 0(x, y)A0(x). (A 10)Although the zero-frequency mode is solved in a similar manner as all of theother nite-frequency modes, the mode is not present at the initial step of the PSEcalculation. When the maximum of nonlinear forcing term F0 exceeds a set threshold,the zero frequency mode is initialized by solving (2.2) with no /x terms, and 0 = 0,0 = 0 initially. The growth of the mode is controlled by changes in 0,i , and theboundary conditions are identical to the conditions imposed on other modes. Theinuence and importance of the mean ow correction on shear layer ows is evidentin some of the simulations, especially in cases where vortex pairing causes rapidchanges to the thickness of the shear layer.

A.3. Lilleys equation

As mentioned previously, in some situations the instability wave model will fail toprovide a complete description of the far-eld acoustics for subsonic mixing layers.For these subsonic mixing layers, it becomes necessary to extend the formulation toinclude a model of the source terms and a method to propagate the solution to the fareld. In the present case, we describe how the acoustic solution can be found through


an appropriate acoustic analogy using near-eld source terms calculated from PSEand Lilleys wave equation.

Based on the work of Goldstein (2001), we express the acoustic analogy in terms ofa parallel sheared ow U (x2) with a source term composed of a velocity quadrupolarcomponent and a uctuating temperature dipole component

L0 =D0Dt

fi

xi 2 U

xj

fj

x1, (A 11a)

where

fi = xj

(1 + ) uj ui c2 xi

(A 11b)

and c2 = ( 1)T . The third-order linear operator is given by

L0 D0Dt

(D20Dt2

xi

c2

xi

)+ 2

U

xi

x1c2

xi

where the material derivative is D0/Dt = /t + U/x1 and the pressure variable is

=

(P

P0

)1/ 1. (A 11c)

The source terms on the right-hand side of (A 11a) are evaluated using informationfrom the PSE calculations. As shown in A3.1, the far-eld pressure is solvedby converting (A 11a) to a single ordinary dierential equation through Fouriertransforms in x and t .

In general, the solution to the parallel ow LilleyGoldsteins equation will consistof a particular solution plus homogeneous solutions. Goldstein & Leib (2005) pointout that these homogeneous contributions correspond to spatially growing instabilitywaves which have the potential to grow unbounded far downstream in the ow.Their solution to this problem was to use a vector Greens function approach on aslightly non-parallel ow. The diverging non-parallel base ow would then ensure thatthe instability waves contributions grow and eventually decay. However, in this work,the mean ow U (x2) is generally chosen at, or slightly after, the point of saturationof the eigenmode solution. After the point of saturation, the PSE solution for theinstability waves were found to be either neutral or decaying, which would suggest thatthey would remain bounded downstream. On the other hand, if the base mean ow wasset near the inlet position, where the fundamental and subharmonic instability waveswere initially unstable, then the Goldstein & Leib concerns about the unboundedgrowth of the homogeneous solutions would apply.

A.3.1. Solution to Lilleys equation

The problem posed by the LilleyGoldstein equation[D0Dt

(D20Dt2

xi

c2

xi

)+ 2

U

xi

x1c2

xi

] = (A 12)

in A.3 can be converted into a single ordinary dierential equation inthe transverse


direction y and solved numerically. Following Goldstein (2005), the source term can be split into the momentum m and thermodynamic t components

m =

[D0Dt

xi 2U

xi

x1

](

xj(1 + ) uj ui

), (A 13)

t =

[D0Dt

xi 2U

xi

x1

](c2

xi

). (A 14)

For a given source term , we can nd the particular solution using a Fourierrepresentation of and in the streamwise coordinate x and in time t

(x, y, t) = eit (x, y) = eit

(k, y)eikx dk. (A 15)

Applying the transformation to (A 12), we obtain the inhomogeneous Rayleighequation

L = i (k, y)2(U 1)c2 , (A 16)

where the left-hand side operator is

L = d2

dy2(

2

U dU

dy 1

c2

dc2

dy

)d

dy+ 2

[(U 1)2

c2 2

],

= k/ and the Fourier transformed source term is

(k, y) = eit

(x, y)eikx dx.

The particular solution to the ordinary dierential equation (A 16) can be found interms of the Greens function G(y, ys), which is dened by

LG(y, ys) = (y ys). (A 17)Equation (A 17) can be solved numerically as a three-point boundary value problem,using adaptive quadrature to integrate a function G(y, ys) from y = to adesignated matching point ys , and a function G

+(y, ys) from y =+ to ys . In theGreens function problem, we replace the Dirac delta function on the right-hand sidewith the jump conditions at the match point

G+|ys G|ys = 0,dG+

dy

ys

dG

dy

ys

= 1 (A18)

and instead solve the homogeneous ODE. The boundary conditions are found byexamining (A 17) in the free stream, where the derivatives dU/dy and dc2/dy vanish,resulting in [

d2

dy2+ 2

((U 1)2

c2 1

)]G(y, ys) = 0.

This yields far-eld solutions in terms of decaying exponentials, where

G(y, ys) exp {iqy} , as y ,


the variable q is dened as q =(

(U 1)2/c2 1), and the sign on the square is

always chosen to yield a decaying function for G as y .Although the numerical solution to (A 17) can be calculated rather quickly for a

single matching point ys , the complete solution for in (A 16) requires G(y, ys) tobe found for many matching points ys . Fortunately, a reciprocity relation exists forthe Greens function at hand

G(y, ys)

(U (y)k )2 =G(ys, y)

(U (ys)k )2 , (A 19)and can be used to calculate G(ys, y) at any ys once G(y, ys) is calculated over the ydomain (Ray 2006). To obtain the nal solution, the convolution integral is used

(k, y) =

i (k, )

(U ( )k ) c2G(y, ) d (A 20)

to nd (k, y) at a particular wavenumber k. To nd at a specic frequency ,(k, y) is calculated at a number of points k, and (A 15) is used to transform into(x, y) space.

An additional complication arises in computing the solution to (A 16) and theintegral (A 20) when considering the critical layer singularity. For solutions wherethe mean ow prole U (y) and frequency cause the term U (y) to vanish, thecontour of integration C must be deformed o of the real y-axis to pass over thecritical point yc. Details regarding this procedure are given in Tam & Morris (1980)and the appropriate branch cut for supersonic disturbances of the Rayleigh equationis described in Lin (1955). In addition, the analytic continuation of the mean owprole U (y) must be accounted for, and we use the procedures detailed in Mitchell,Lele & Moin (1999). In a similar fashion, around the critical point yc, the source term (x, y) must be extended into the complex plane to allow the contour deformationto occur in integral (A 20).

REFERENCES

Andersson, P., Henningson, D. S. & Hanifi, A. 1998 On a stabilization procedure for the parabolicstability equations. J. Engng Math 33, 311332.

Avital, E. J., Sandham, N. D. & Luo, K. H. 1998a Mach wave radiations by mixing layers. Part I:Analysis of the sound eld. Theor. Comput. Fluid Dyn. 12 (2), 7390.

Avital, E. J., Sandham, N. D. & Luo, K. H. 1998b Mach wave radiations by mixing layers. Part II:Analysis of the source eld. Theor. Comput. Fluid Dyn. 12 (2), 91108.

Balakumar, P. 1994 Supersonic jet noise prediction and control. Contractor Report to NASALewis Research Center, Contract No. NAS3-27205.

Balakumar, P. 1998 Prediction of supersonic jet noise. AIAA Paper 98-1057.

Bertolotti, F. P., Herbert, Th. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasiusboundary layer. J. Fluid Mech. 242, 441474.

Brown, G. L. & Roshko, A. 1974 On density eects and large structure in turbulent mixing layers.J. Fluid Mech. 64, 775816.

Cheung, L. C. 2007 Aeroacoustic noise prediction and the dynamics of shear layers and jets usingthe nonlinear PSE. Ph.D. dissertation, Stanford University.

Chu, B. T. 1965 On the energy transfer to small disturbances in uid ow (Part I). Acta Mechanica1, 215234.

Colonius, T., Lele, S. K. & Moin, P. 1997 Sound generation in a mixing layer. J. Fluid Mech. 330,375409.

Crighton, D. G. & Gaster, M. 1976 Stability of slowly divergent jet ows. J. Fluid Mech. 77,397413.


Day, M. J., Mansour, N. N. & Reynolds, W. C. 2001 Nonlinear stability and structure ofcompressible reacting mixing layers. J. Fluid Mech. 446, 375408.

Goldstein, M. E. 2001 An exact form of Lilleys equation with a velocity quadrupole/temperaturedipole source term. J. Fluid Mech. 443, 231236.

Goldstein, M. E. 2003 A generalized acoustic analogy. J. Fluid Mech. 488, 315333.

Goldstein, M. E. 2005 On identifying the true sources of aerodynamic sound. J. Fluid Mech. 526,337347.

Goldstein, M. E. & Leib, S. J. 2005 The role of instability waves in predicting jet noise. J. FluidMech. 525, 3772.

Herbert, T. & Bertolotti, F. P. 1987 Stability analysis of nonparallel boundary layers. Bull. Am.Phys. Soc. 32, 2079.

Ho, C. M. & Huang, L. S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech.119, 443473.

Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing ows.Annu. Rev. Fluid Mech. 22, 473537.

Hultgren, L. 1992 Nonlinear spatial equilibration of an externally excited instability wave in afree shear layer. J. Fluid Mech. 236, 635634.

Jackson, T. L. & Grosch, C. E. 1989 Inviscid spatial stability of a compressible mixing layer.J. Fluid Mech. 208, 609637.

Laufer, J. & Yen, T. C. 1983 Noise generation by a low Mach-number jet. J. Fluid Mech. 134, 131.

Lele, S. K. 1992 Compact nite dierence schemes with spectral-like resolution. J. Comput. Phys.103, 1642.

Lighthill, M. J. 1952 On sound generated aerodynamically I. General theory. Proc. Roy. Soc.Lond. A 222, 132.

Lilley, G. M. 1974 On the noise from jets. AGARD CP-131. 13.1-13.12.

Lin, C. C. 1955 The Theory of Hydrodynamic Stability . Cambridge University Press.

Lui, C. C. M. 2003 A numerical investigation of shock-associated noise. Ph.D. dissertation, StanfordUniversity.

Malik, M. R. & Chang, C. L. 2000 Nonparallel and nonlinear stability of supersonic jet ow.Comput. Fluids 29 (3), 327365.

Mitchell, B., Lele, S. K. & Moin, P. 1999 Direct computation of the sound generated by vortexpairing in an axisymmetric jet. J. Fluid Mech. 383, 113142.

Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layer: an experimentalstudy. J. Fluid Mech. 197, 453477.

Phillips, O. M. 1960 On the generation of sound by supersonic turbulent shear layers. J. FluidMech. 9, 128.

Ray, P. K. 2006 Sound generated by instability wave/shock-cell interaction in supersonic jets. Ph.D.dissertation, Stanford University.

Rogers, M. M. & Moser, R. D. 1992 The three-dimensional evolution of a plane mixing layer: theKelvinHelmholtz rollup. J. Fluid Mech. 243, 183226.

Sandham, N. D., Morfey, C. L. & Hu, Z. W. 2006 Nonlinear mechanisms of sound generation ina perturbed parallel ow. J. Fluid Mech. 565, 123.

Sandham, N. D. & Reynolds, W. C. 1991 Three-dimensional simulations of large eddies in thecompressible mixing layer. J. Fluid Mech. 224, 133158.

Sandham, N. D., Salgado, A. M. & Agarwal, A. 2008 Jet noise from instability mode interactions.AIAA Paper 2008-2987.

Schlichting, H., Gersten, K., Krause, E., Oertel, H. Jr. & Mayes, C. 2004 Boundary LayerTheory , 8th edn. Springer.

Tam, C. K. W. 1995 Supersonic jet noise. Annu. Rev. Fluid Mech. 27, 1743.

Tam, C. K. W. & Auriault, L. 1998 Mean ow refraction eects on sound radiated from localizedsources in a jet. J. Fluid Mech. 370, 149174.

Tam, C. K. W. & Burton, D. 1984a Sound generation by instability waves of supersonic ows.Part 1. Two dimensional mixing layers. J. Fluid Mech. 138, 249271.

Tam, C. K. W. & Burton, D. 1984b Sound generation by instability waves of supersonic ows. Part 2.Axisymmetric jets. J. Fluid Mech. 138, 273295.


Tam, C. K. W. & Morris, P. J. 1980 The radiation of sound by the instability waves of a compressibleplane turbulent shear layer. J. Fluid Mech. 98, 349381.

Winant, C. D. & Browand, F. K. 1974 Vortex pairing. J. Fluid Mech. 63, 237255.

Wu, X. S. 2005 Mach wave radiation of nonlinear evolving supersonic instability modes in shearlayers. J. Fluid Mech. 523, 121159.

Yen, C. C. & Messersmith, N. L. 1998 Application of the parabolized stability equations to theprediction of jet instabilities. AIAA J. 36, 15411544.

Documents

Cheung Lele JFM 09