-
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.
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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
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324 L. C. Cheung and S. K. Lele
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
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Linear and nonlinear processes in mixing layers 325
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
-
326 L. C. Cheung and S. K. Lele
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
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Linear and nonlinear processes in mixing layers 327
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
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328 L. C. Cheung and S. K. Lele
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.
-
Linear and nonlinear processes in mixing layers 329
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
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330 L. C. Cheung and S. K. Lele
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
-
Linear and nonlinear processes in mixing layers 331
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).
-
332 L. C. Cheung and S. K. Lele
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
-
Linear and nonlinear processes in mixing layers 333
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.
-
334 L. C. Cheung and S. K. Lele
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
-
Linear and nonlinear processes in mixing layers 335
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
-
336 L. C. Cheung and S. K. Lele
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
-
Linear and nonlinear processes in mixing layers 337
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
-
338 L. C. Cheung and S. K. Lele
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
-
Linear and nonlinear processes in mixing layers 339
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.
-
340 L. C. Cheung and S. K. Lele
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
-
Linear and nonlinear processes in mixing layers 341
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
-
342 L. C. Cheung and S. K. Lele
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
-
Linear and nonlinear processes in mixing layers 343
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
-
344 L. C. Cheung and S. K. Lele
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)
-
Linear and nonlinear processes in mixing layers 345
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
-
346 L. C. Cheung and S. K. Lele
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
-
Linear and nonlinear processes in mixing layers 347
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
-
348 L. C. Cheung and S. K. Lele
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 ,
-
Linear and nonlinear processes in mixing layers 349
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).
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