Stability characteristics of a periodically unsteady mixing layerMuhammad R. Hajj Citation: Physics of Fluids (1994-present) 9, 392 (1997); doi: 10.1063/1.869239 View online: http://dx.doi.org/10.1063/1.869239 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/9/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Turbulent mixing measurements in the Richtmyer-Meshkov instability Phys. Fluids 24, 074105 (2012); 10.1063/1.4733447 Three-dimensionalization of the stratified mixing layer at high Reynolds number Phys. Fluids 23, 111701 (2011); 10.1063/1.3651269 Bi-stability in turbulent, rotating spherical Couette flow Phys. Fluids 23, 065104 (2011); 10.1063/1.3593465 Anisotropy of turbulence in stably stratified mixing layers Phys. Fluids 12, 1343 (2000); 10.1063/1.870386 Length scales of turbulence in stably stratified mixing layers Phys. Fluids 12, 1327 (2000); 10.1063/1.870385

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Stability characteristics of a periodically unsteady mixing layerMuhammad R. HajjDepartment of Engineering Science and Mechanics, Virginia Tech, Blacksburg, Virginia 24061-0219

~Received 9 April 1996; accepted 16 October 1996!

In nature, in many technological applications and in some laboratory experiments, the basic state ofshear flows can be time-varying. The effects of such variations on the stability characteristics ofthese flows are not well understood. In previous work, Miksadet al. @J. Fluid Mech.123, 1 ~1982!#and Hajjet al. @J. Fluid Mech.256, 385~1992!#, it has been shown that low-frequency components,generated by nonlinear difference interactions, play an important role in the redistribution of energyamong spectral components. In particular, phase modulation was found to be the most effectivemechanism in energy transfer to the sidebands of unstable modes. In this work, the effects ofsmall-amplitude low-frequency mean flow unsteadiness on the stability of a plane mixing layer aredetermined. By extending earlier analytical arguments, it is shown that periodicity in the mean flowcauses modulations of the most unstable modes. The analysis is then verified experimentally bycomparing levels of amplitude and phase modulations in mixing layers with steady and unsteadybasic flows. The results show that small-amplitude low-frequency unsteadiness results in enhancedmodulations of the fundamental mode. These modulations cause variations in the growth rates of theunstable modes and energy redistribution among them. ©1997 American Institute of Physics.@S1070-6631~97!01502-X#

I. INTRODUCTION

In the majority of existing studies on the transition toturbulence of shear flows, the mean flow is considered~orassumed to be! steady. Yet, in nature, in many technologicalapplications, and in some laboratory experiments, the meanflow is inherently time-varying. These variations can resultfrom nonuniformities in the external flow field or from self-induced motions of bodies immersed in such a field. In com-parison with the stability of shear flows with steady basicstates, the stability of unsteady shear flows has received littleattention. This is due to many difficulties. Mathematically,the linearized Navier–Stokes equations have coefficients thatvary with time and the method of normal modes is not di-rectly applicable. Experimentally, generating well-controlledunsteady mean flow to assess its effects is not easy. In gen-eral, the effects of time variations in the basic state on thestability of the flow have been analyzed by applying steadystate results to instantaneous velocity distributions, i.e. re-sorting to the quasi-steady approach. The objective of thiswork is to examine how small-amplitude low-frequencysinusoidal variations in the mean flow of a plane mixinglayer affect its stability.

Shen1 put forward some ideas on the stability of timedependent flows and explained how the quasi-steady ap-proach is only applicable when the time scale of the unsteadybasic flow is very large in comparison to the time scale of theinstability disturbances. Shen also showed that for specialunsteady basic states, where the steady velocity profile ismodified by a temporal function, the solution of the stabilityproblem in the inviscid limit, can be reduced to that of thesteady flow by a transformation of the time variable. Davis2

concluded that for parallel shear flows, and convective andcentrifugal instabilities, the stability limits can be modified ifmean flow unsteadiness is imposed. Soliset al.3 extended thearguments of Shen to show that mean flow unsteadiness

causes amplitude and phase modulations of the instabilityfluctuations. Through these modulations mechanisms, themean flow unsteadiness causes acceleration of the transitionof a plane wake.

In this work, an experimental investigation is conductedto determine the effects of small-amplitude low-frequencymean flow unsteadiness on the stability characteristics of theplane mixing layer. Interpretation of experimental measure-ments can be made much easier when they are tied to ana-lytical arguments. In the next section, the analyses presentedby Shen and by Soliset al. are detailed to show how small-amplitude, low-frequency mean flow unsteadiness causesmodulations of the instability modes in the initial linearstage. Time series of velocity fluctuations in a mixing layerwith steady and unsteady basic states are then analyzed tosubstantiate the analytical arguments. In particular, ampli-tude and phase modulations of instability fluctuations aremeasured by applying complex demodulation techniques totime traces of velocity fluctuations. The effects of mean flowunsteadiness are isolated by comparing measurements of lev-els of amplitude and phase modulations and power spectraunder the steady and unsteady basic states of the mean flow.

II. STABILITY CHARACTERISTICS OF TIMEDEPENDENT FLOWS

Monkewitz and Huerre4 solved the eigenvalue problemof the linear stability theory that corresponds to differentmixing layer velocity profiles. Several experimental studieshave shown that linear stability theory is adequate in predict-ing the frequencies and growth rates of the most unstablemodes in steady, incompressible inviscid mixing layers~Hoand Huerre5!. The application of these results to predict thestability characteristics of mixing layers with an unsteadybasic state is not direct. When the flow is unsteady, the co-efficients of the governing equations are time dependent and

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the method of normal modes is not directly applicable. Shen1

considered the effect of mean flow unsteadiness on thegrowth of the instability fluctuations for the case where thebasic flow is represented as a basic spatial profile,U(y),modified by a time function,T(t/tm), i.e.,

U~y,t !5T~ t/tm!U~y!, ~1!

where tm is the time scale associated with the basic flowunsteadiness. For the case whereT(t/tm) is given by

T~ t/tm!511eeivmt, ~2!

the flow would correspond to a mean flow that is subject toan unsteadiness of small amplitude,e, and low frequency,vm . By stretching the time variable, a new time scalet canbe defined as

t51

ivmE0

t

~11eeivmt!d~ ivmt !5t1e

ivmeivmt2

e

ivm.

~3!

Based on this definition, the time derivative]/]t can be re-placed byT]/]t. The inviscid linearized differential equationgoverning two-dimensional streamfunction disturbances thensatisfies

]

]t¹2c1U~y!

]

]x¹2c2U9~y!

]c

]x50. ~4!

Equation ~4! is identical to the one obtained under steadybasic flow, only with a different time scale, i.e.,t instead oft. Consequently, Eq.~4! can admit solutions of the form

c5f~y!eia~x2ct!, ~5!

wherea is the wavenumber andc is the wave velocity and istaken complex (cr1 ic i). The real part,cr , is the phasespeed and the imaginary part results in the growth rate,aci ,when multiplied bya. Using equation~3! and substitutingfor t, the following expression forc as a function oft isobtained:

c5f~y!e~aci t1A cos~vmt1u!!ei ~a~x2cr t !1A sin~vmt1u!!

3e2A cosue2 iA sin u, ~6!

whereA is equal to~ae/vm! (ci21cr

2)1/2 and u is equal totan21 ci /cr . The first exponential term governs the amplitudebehavior of the instability mode. The periodic portion of thatterm,A cos~wmt1u!, implies a modulation of the amplitudeof this mode. The second exponential term expresses thewavelike nature of the instability mode and its argument rep-resents the phase. The periodic term in the phase,A sin~wmt1u!, represents a modulation of the phase of theinstability mode. The last two terms represent a constantphase shift that may be absorbed in the phase off(y). Theexperimental work presented next is directed to substantiatethe implications of equation~6! that small-amplitude low-frequency variations cause modulations of the unstablemodes.

III. EXPERIMENTAL SET-UP AND DATA ANALYSIS

The experiments were conducted in a low-turbulenceopen return type wind tunnel. The mixing layer was formed

by the merging of two laminar streams. The upper and lowerfree stream velocities were 7.17 m/s and 1.51 m/s, respec-tively. This resulted in a velocity ratioR5(U12U2)/(U11U2)50.65. The frequency of the dominant instabilitymode, f 0, is 215 Hz and its streamwise wavelength,l0, is1.98 cm. More details about the wind tunnel and proceduresof velocity measurements are given by Hajjet al.6,7 Alldownstream measurement locations are normalized withRandl0, in accordance to our previous work, Hajjet al.

6 Thetest section is isolated from the pump noise by a sonic throat.This arrangement provides an extremely quiet test sectionflow. The free-stream turbulence intensity is 0.0004U1 in thehigh-speed stream and 0.0008U1 in the low-speed stream.This background turbulence is distributed over a broad rangeof frequency components. The generation of the sinusoidalmean flow unsteadiness simultaneously in both streams wasaccomplished by varying the cross sectional area of the sonicthroat of the tunnel. A brief description of the sonic throatmodulator used to produce these variations and its effects onthe mean flow are given below.

The gap of the sonic throat has a height,h, equal to 1.5cm and a width,w, equal to 10.2 cm. A schematic of themodulator used in these experiments is shown in Fig. 1. Thekey idea behind the design of this modulator is that changesin the sonic throat flow area have a direct effect on the flowrate in the test section. For a sonic nozzle, the flow rate in thetest section can be shown to be linearly proportional to thecross sectional area of the sonic throat. This area is varied byoffsetting a hinged flat plate of lengthl ~55.1 cm! with asimple cam. The plate, hinged at the upstream end, is posi-tioned flush to the bottom wall of the sonic throat such thatthe downstream end of the plate coincides with the locationin the sonic-throat which has the minimum mean cross sec-tional area.

The cam consists of a circular cylinder of radiusr ~56.35 mm! and an eccentricitye~50.09 mm!. The centerof rotation is set under the modulator plate by a horizontaldistancez~56.35 mm! from endB. The magnitude of throatarea variationuA(t)u is adjusted by varying the throw of thecam. As the cam rotates, the free end of the plate periodicallyconstricts the sonic throat. A tension spring is used to keepthe plate in contact with the cam. The cam in each case wasdriven by a 1/4 HP, DC-motor. A transistorized speed con-troller powered by a power supply was used to set the motorspeed. A small flywheel was attached to the cam shaft tostabilize the rotation rate. For a sonic throat of mean verticalopeningh and mean cross sectional areaA0, it can be shown

FIG. 1. Schematic of a sonic throat modulator.

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that, becausee/ l!1, z/ l!1, andr / l!1, the throat area variessinusoidally. For the dimensions presented above, the throatarea variation,A(t)/A0 is equal toe/h~50.6%!.

Measurements ofDU1rms/U1 vs modulator frequency areshown in Fig. 2. Here,DU1rms is defined as the rms of themean high-speed velocity variations generated by the sonicthroat modulations. The figure shows that these variationsare a strong function of the modulator frequency. At thelowest and highest frequencies,DU1rms/U1 approached 0.5%which is close toe/h. Near the tunnel resonance frequency,i.e., at 35 Hz,DU1rms/U1 surges to a value near 3%. A timerecord of the free stream velocityU1(t) is plotted in Fig.3~a!. The power spectrum of this record is shown in Fig.3~b!. It is obvious that the low-frequency velocity variations,when unsteadiness is imposed, are sinusoidal with an ampli-tude level that is about two orders of magnitude larger thanthe background turbulence observed under steady conditions.Note also that the harmonic distortions are at or below back-ground noise level in the tunnel. It is thus clear that thecharacter of mean flow unsteadiness~in frequency and am-plitude! is different than that of the background turbulence.Measurements of free-stream velocity in the low-speed sideshowed the same characteristics as in the high-speed. Morespecifically, the rms level surged to about 3% at the modu-lation frequency of 35 Hz and showed a distinct sinusoidalcharacter.

Complex demodulation techniques have been used to de-termine the levels of amplitude and phase modulations influctuating waveforms. Details concerning the complex de-modulation procedures can be found in Khadra.8 Briefly, amodulated time series of the velocity fluctuations can bewritten in the form

u~ t !5a~ t !cos~2p f ct1p~ t !!, ~7!

wherea(t) and p(t) are the amplitude and phase modula-tions of a carrier wave with frequencyf c . By multiplyingEq. ~7! by 2 exp(i2p f dt), one obtains

a~ t !$exp@2 i2p~ f c2 f d!t2 ip~ t !#

1exp@ i2p~ f c1 f d!t1 ip~ t !#%. ~8!

The first term includes a difference frequency (f c2 f d) andthe second term includes a sum frequency (f c1 f d) compo-nent. The difference frequencyf c2 f d , can be nullified by

letting f d be equal tof c . The sum frequency component canbe eliminated by applying a low-pass filter. The remainingcomponent of the signal is the complex demodulate and isgiven by

c~ t !5a~ t !exp@2 ip~ t !#. ~9!

The amplitude part ofc(t) describes the amplitude modula-tion of the chosen carrier componentf c . The phase part ofc(t) is the phase modulation of that component. From theestimates of the instantaneous amplitude and phase modula-tions,a(t) andp(t), one can determine other parameters thatare related to the physics of the modulated signal. Theseparameters include the rms amplitude and phase modulationindices defined, respectively, as

a5~E@a~ t !2# !1/2

a0~10a!

and

b5~E@p~ t !# !1/2, ~10b!

whereE@•••# denotes a time average anda0 is the mean valueof a(t). Estimates of these indices will be used to quantifythe separate roles of amplitude and phase modulations in themixing layer under steady and unsteady basic states.

FIG. 2. Normalized rms-amplitude of high-speed mean flow unsteadiness vsfrequency atx50.0 cm.

FIG. 3. ~a! Sample free-stream time trace.~b! Power spectrum of time traceshown in a.

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IV. RESULTS AND DISCUSSION

The effects of mean flow unsteadiness on the stability ofthe plane mixing layer are isolated by comparing measure-ments and results taken under steady and unsteady basicstates. In our previous work on the transition of steady mix-ing layer, Hajjet al.,6 we showed that the region limited byRx/l0,1.0 represents the initial stability region. Figure 4shows the characteristic power spectra measured at fourdownstream locations in, and just beyond, the initial instabil-ity region at the cross-stream location of maximumurms8when no unsteadiness is imposed. This location was chosenin order to avoid riding up and down a modal profile shapeand because, as shown by Miksad,9 the growth rates of thefundamental and subharmonic modes along maximumurms8differ only slightly ~less than 10%! from those determined bycross-stream integration of totalurms8 .

At Rx/l050.32, the spectra shows the early emergenceof a continuous band of instability modes with frequenciesup to 250 Hz. AtRx/l051.0, the band becomes centeredaround the most unstable mode with frequency 215 Hz. Thismode is referred to as the fundamental mode and will benoted by f 0. A relatively lower peak is also noticed at themode with frequency equal to half that of the fundamentalmode. This corresponds to the subharmonic component,f 0/2,whose growth at this early stage is a result of a pressurewave feedback from the vortex pairing mechanism, Corke.10

Two other peaks also appear at the first and second har-monic, 2f 0 and 3f 0, respectively. The enhanced growth ofthe low-frequency components~up to 50 Hz! should also benoted. The spectrum, atRx/l051.3, is characterized by theenhanced growth of the band of frequencies aroundf 0, 2 f 0,and 3f 0, and the emergence of a peak at 4f 0. The low-frequency components continue to gain energy. These mea-surements, along with the detailed analysis of Hajjet al.,6

indicate that theRx/l051.3 location marks the beginning ofeffective nonlinear interactions that result in energy transferto harmonic bands and low-frequency modes. In this work,we will concentrate on the effects of mean flow unsteadinessin the linear stability region, i.e.,Rx/l0,1.0.

Miksadet al.,11 in experimental measurements of a tran-sitioning plane wake, showed that amplitude and phasemodulations play an important role in the redistribution ofenergy to the spectral modes and valleys fill-up. More spe-cifically, those results showed that amplitude and phasemodulations provide a very efficient mechanism to achievespectral energy transfer. Table I shows the amplitude andphase modulation indices, as defined in Eqs.~10a! and~10b!,in the natural steady mixing layer for which the spectra ispresented above. In this initial stage, up toRx/l051.3, thefundamental mode is the most organized and is the one thatis drawing most of the energy from the mean flow. Conse-quently, it was chosen as the carrier mode. The results showthat, in the region up toRx/l051.0, before any strong non-linear development, the index of amplitude modulation in-creases significantly while the index of phase modulationremains constant. This indicates the organization of the flowrepresented by the exponential amplification of the instabilitymodes and especially of the carrier frequency atf 05215 Hz.

At Rx/l051.3, the index of phase modulation increases dra-matically while that of the amplitude modulation decreases.This indicates the growing importance of the phase modula-tion beyond the initial linear instability stage. As seen in thespectrum this location is characterized by enhanced energytransfer to the bands around the most unstable mode and itsharmonics. Therefore, the increase in the level of phase

FIG. 4. Power spectra of velocity fluctuations in the steady mixing layeralong maximumurms8 .

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modulation is related to the nonlinear activity associatedwith the energy transfer to the valleys between the spectralpeaks.

The effects of mean flow unsteadiness are brought outby comparing measurements taken under steady mean flowconditions, as documented above, to those taken under un-steady mean flow conditions. In both cases, no acoustic forc-ing is used. Such forcing would have caused enhancement ofthe excited frequencies as well as other components. Forinstance, exciting the flow at two modes with a small differ-ence in their frequencies can enhance low-frequency compo-nents via difference interactions. Exciting the flow at a fre-quency near that of the most unstable mode would have asimilar effect. Consequently, the natural excitation condition,as considered here, allows the study of effects due strictly tomean flow unsteadiness and comparison with the analyticalargument of Sec. II. We should note that while the steadymixing layer is excited by small amplitude and random back-ground fluctuations, the free stream of the unsteady mixinglayer shows a clean sinusoidal character as seen in Figs. 3~a!and 3~b!. Moreover, because of the experimental setup, bothstreams undergo the same acceleration and deceleration at alltimes. Consequently, it is expected that distortions to themean velocity profile, if there are any, are a part of the ef-fects of the imposed unsteadiness. The effects of mean flowunsteadiness are determined by comparing the indices of am-plitude and phase modulations under steady and unsteadymean flow conditions at the same downstream and crossstream locations. Table II shows the values ofa andb whenunsteadiness is present. The results show that when low-frequency variations are introduced in the mean flow, theamplitude modulation index atRx/l050.064, has a value of0.717 in comparison to a value of 0.489 measured understeady conditions~Table I!. The phase modulation index hasa value of 0.579 compared to 0.121 measured under thesteady conditions. The enhanced amplitude and phase modu-lations atRx/l050.064 and 0.32 are in qualitative agreementwith the analysis put forward by Shen1 and Soliset al.3 and

detailed in Sec. II. Table II also shows that further down-stream, atRx/l050.64 and 1.0, the dynamics and effects ofamplitude modulations when the unsteadiness is introducedare different. As discussed above, the amplitude modulationincreases and the phase modulation index remains constantover this region, when the basic state is steady. In contrast,when mean flow unsteadiness was introduced, the initiallyhigh amplitude modulation decreases while the phase modu-lation increases. The physical implication is that while theamplitude modulation continues to play its role in transfer-ring energy between the low-frequency and fundamentalmodes, the enhanced phase modulation allows more meanflow energy to be funneled through the fundamental mode tothe sidebands when the mean flow is unsteady. This indicatesthat the mean flow unsteadiness increases the redistributionof energy among spectral modes.

A comparison of the power spectra under steady andunsteady mean flow conditions atRx/l050.064, 0.32, 0.64,and 1.0 are shown in Fig. 5. Both spectra show peaks at thefrequencies of the fundamental~f 05215 Hz! and subhar-monic modes~f 0/25107.5 Hz!. Yet, there are several differ-ences. First, the unsteady spectra at all locations show a peakat the frequency of the unsteady component~f m535 Hz!.Second, while the fundamental mode maintains the same am-plitude under both steady and unsteady conditions, the sub-harmonic mode is enhanced significantly by the presence ofmean flow unsteadiness. This amplification starts at the firstmeasurement location,Rx/l050.064 which corresponds tox50.2 cm. Third, the sideband structures surrounding thefundamental and subharmonic modes are broader under un-steady mean flow than under the steady flow counterpart.Moreover, the spectral valleys, i.e., those regions betweenthe spectral peaks, fill in more rapidly at the successivedownstream locations when the mean flow unsteadiness ispresent. The enhancement of the sideband structure and en-ergy transfer to the spectral valleys is consistent with theenhanced modulations, in particular, phase modulation, bythe mean flow unsteadiness as discussed above.

The stability characteristics are best understood by mea-surements of growth rates in the initial stage of the transition.Figure 6 compares the growth rates of the unstable modesunder steady and unsteady mean flow conditions. Althoughat first sight, both growth rates may appear similar, severalsignificant differences exist. First, the growth rates of themodes with frequencies of 35 Hz~which corresponds tof m!and 107.5 Hz~which corresponds tof 0/2! are enhanced sig-nificantly by the presence of mean flow unsteadiness. In con-junction, Fig. 6 also shows that the modes with frequenciesnear 70 and 140 Hz have higher growth rates due to thepresence of mean flow unsteadiness. The enhanced growthrates of all of these modes is accompanied by reducedgrowth rates of their immediate neighboring modes. Theseresults suggest that the mean flow unsteadiness componentcan interact with the unstable subharmonic mode and causethe enhanced growth of their sum and difference modes. Thisinteraction mechanism can be explained in terms of ampli-tude and/or phase modulations of the subharmonic mode bythe mean flow unsteadiness, which is consistent with the dis-cussion of Sec. II.

TABLE I. Amplitude and phase modulation indices in the steady mixinglayer.

Rx/l0 a b

0.064 0.489 0.1210.32 0.504 0.1030.64 0.751 0.1511.0 0.9 0.1641.3 0.738 0.413

TABLE II. Amplitude and phase modulation indices in the unsteady mixinglayer.

Rx/l0 a b

0.064 0.717 0.5790.32 0.665 0.7330.64 0.531 0.9891.0 0.415 0.9941.3 0.254 0.964

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Another feature of the growth rate characteristics of theunstable mode under steady and unsteady mean flow condi-tions is the enhanced growth of the upper sideband and somemodes in the lower sideband of the fundamental mode.These results are consistent with the earlier results in Table IIwhich show enhanced amplitude and phase modulation ofthe fundamental mode under unsteady mean flow conditions.

It is important to note here that the spacing of the sidebandsaway from the fundamental are not at integer multiples off mas in the case of the subharmonic. This is due to the fact thatin contrast with the subharmonic, the band of modes aboutthe fundamental mode are highly unstable also. Within thisband, these modes can interact to enhance low-frequencydifference modes which in turn cause modulation of the fun-damental. Thus, the mean flow unsteadiness acts as an addi-tional modulating frequency of the fundamental and resultsin significant variations in the growth rates.

V. CONCLUSIONS

In previous work, Miksadet al.11 and Hajjet al.,6 it hasbeen shown that low-frequency components generated bydifference interactions among the unstable modes play a spe-cial role in the breakdown to turbulence. Through modula-tions mechanisms, specifically phase modulation, low-frequency components enhance the energy transfer to thesidebands. In this work, the effects of the presence of small-amplitude low-frequency variations in the mean flow on thestability of a plane mixing layer have been examined. Byextending the analytical arguments of Shen1 and Soliset al.,3

it has been shown that such variations cause enhanced modu-lations of the instability modes. The experimental resultssupport the analytical arguments and show that small-amplitude low-frequency unsteadiness can act as a modula-tion mechanism. The growth rates of the sum and differencefrequency components of the unsteady and subharmoniccomponents increase significantly. Moreover, the enhancedamplitude modulation of the fundamental mode play an im-portant role in passing energy from the low-frequency tospectral modes. On the other hand, the enhanced phasemodulation redistributes the energy to the sidebands of thefundamental. In particular, the upper sidebands of the funda-mental exhibit higher growth rates than the rates measuredunder the steady basic state. These results indicate that thebreakdown to turbulence may be enhanced by the presenceof mean flow unsteadiness.

1S. F. Shen, ‘‘Some considerations on the laminar stability of time-dependent basic flows,’’ Aerosp. Sci.28, 397 ~1961!.

2S. H. Davis, ‘‘The stability of time-periodic flows,’’ Annu. Rev. FluidMech.8, 57 ~1976!.

FIG. 5. Comparison of power spectra of velocity fluctuations under steadyand unsteady basic states along maximumurms8 .

FIG. 6. Normalized growth rates of unstable spectral modes in the mixinglayer under steady and unsteady basic states.

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3R. S. Solis, R. W. Miksad, and E. J. Powers, ‘‘Experiments on the influ-ence of mean flow unsteadiness on the laminar–turbulent transition of awake,’’ 10th Symposium on Turbulence, edited by X. B. Reed, ContinuingEducation~The University of Missouri, Rolla, 1985!.

4P. A. Monkewitz and P. Huerre, ‘‘The influence of the velocity ratio onthe spatial instability of mixing layers,’’ Phys. Fluids25, 1137~1982!.

5C. M. Ho and P. Huerre, ‘‘Perturbed free shear layers,’’ Annu. Rev. FluidMech.16, 365 ~1984!.

6M. R. Hajj, R. W. Miksad, and E. J. Powers, ‘‘Subharmonic growth byparametric resonance,’’ J. Fluid Mech.236, 385 ~1992!.

7M. R. Hajj, R. W. Miksad, and E. J. Powers, ‘‘Fundamental–subharmonicinteraction: Effect of phase relation,’’ J. Fluid Mech.256, 403 ~1992!.

8L. Khadra, ‘‘Digital complex demodulation of nonlinear fluctuation data,’’Ph.D. dissertation, The University of Texas at Austin, 1981.

9R. W. Miksad, ‘‘Experiments on the non-linear stages of free shear layertransition,’’ J. Fluid Mech.56, 695 ~1972!.

10T. C. Corke, ‘‘Measurements of resonant phase locking in unstable axi-symmetric jets and boundary layers,’’ Nonlinear Wave Interactions in Flu-ids, The Winter Annual Meeting of ASME, Boston, MA, edited by R. W.Miksad, T. R. Akylas, and T. Herbert, AMD 87, December 1987, pp.37–65.

11R. W. Miksad, F. L. Jones, E. J. Powers, Y. C. Kim, and L. Khadra,‘‘Experiments on the role of amplitude and phase modulations during thetransition to turbulence,’’ J. Fluid Mech.123, 1 ~1982!.

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