the formation of vortex rings in a strongly forced round jet

8/6/2019 The Formation of Vortex Rings in a Strongly Forced Round Jet

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RESEARCH ARTICLE

The formation of vortex rings in a strongly forced round jet

E. Aydemir

N. A. Worth

J. R. Dawson

Received: 12 October 2010 / Revised: 8 March 2011 / Accepted: 25 April 2011 Springer-Verlag 2011

Abstract The periodic formation of vortex rings in thedeveloping region of a round jet subjected to high-ampli-tude acoustic forcing is investigated with High-SpeedParticle Image Velocimetry. Harmonic velocity oscillationsranging from 20 to 120% of the mean exit velocity of the jet was achieved at several forcing frequencies determinedby the acoustic response of the system. The time-resolvedhistory of the formation process and circulation of thevortex rings are evaluated as a function of the forcingconditions. Overall, high-amplitude forcing causes theshear layers of the jet to breakup into a train of large-scalevortex rings, which share many of the features of starting jets. Features of the jet breakup such as the roll-up locationand vortex size were found to be both amplitude and fre-quency dependent. A limiting time-scale of t / T & 0.33based on the normalized forcing period was found torestrict the growth of a vortex ring in terms of its circula-tion for any given arrangement of jet forcing conditions. Insinusoidally forced jets, this time-scale corresponds to akinematic constraint where the translational velocity of thevortex ring exceeds the shear layer velocity that imposespinch-off. This kinematic constraint results from thechange in sign in the jet acceleration between t = 0 andt = 0.33 T . However, some vortex rings were observed topinch-off before t = 0.33 T suggesting that they hadacquired their maximum circulation. By invoking the slugmodel approximations and dening the slug parametersbased on the experimentally obtained time- and length-scales, an analytical model based on the slug and ringenergies revealed that the formation number for a

sinusoidally forced jet is L / D & 4 in agreement with theresults of Gharib et al. (J Fluid Mech 360:121140, 1998 ).

1 Introduction

Although free round jets are self-similar in the far eld, it iswell known that the initial development region exhibitslarge-scale orderly structures. The classical experiment of Crow and Champagne ( 1971 ) observed the formation of large-scale vortical puffs in the near eld of an excitedround jet. The phenomena of vortex pairing in an excited jet was investigated by Zaman and Hussain ( 1980 ). Theseand numerous other experiments over the years have shownthat small levels of forcing applied to the base ow affectshow the ow eld develops by amplifying instabilitygrowth rates, enhancing the transition to turbulence,affecting sound generation, and mixing characteristics(Becker and Massaro 1968 ; Gutmark and Ho 1983 ; Leeand Reynolds 1985 ; Raman et al. 1989 ). This paperinvestigates the effect high-amplitude forcing has on thestructure of the near eld of a jet, particularly the onset of large-scale unsteadiness characterized by the periodic for-mation of vortex rings. The term high amplitude in thispaper refers to uctuations in the exit velocity being ordersof magnitude greater than the turbulent uctuations u0 x; t :Vortex rings formed by starting and unsteady jet owsplay a key role in a variety of emerging technologies. Forexample, unsteady propulsion devices such as pulse deto-nation engines and pulse combustors generate periodicvortex rings whose formation time-scales correlate withimproved thrust augmentation (Opalski et al. 2005 ; Krue-ger and Gharib 2005 ; Mason and Miller 2006 ). They arealso an integral feature of unsteady aerodynamics found inbio-inspired propulsion such as underwater autonomous

E. Aydemir N. A. Worth J. R. Dawson ( & )Department of Engineering, University of Cambridge,Trumpington Street, Cambridge CB2 1PZ, UK e-mail: [email protected]

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vehicles (UAVs) (Krieg and Mohesni 2008 ) or where thecontrolled apping of wings causes the ow to separate androll-up into vortices (Rival et al. 2009 ). In combustioninstability, vortices are a source of ame sheet kinematicsresponsible for nonlinearity that leads to limit-cycleoscillations and is a signicant development issue in leanburn gas turbine combustion (Kulsheimer and Buchner2002 ; Balachandran et al. 2005 ).

Fundamental investigations into vortex rings formed bystarting jets discharging into a quiescent uid have elu-cidated many aspects of the formation process togetherwith models to predict a variety of ring parameters of interest such as trajectory, circulation, and vorticity dis-tribution in the core (Pullin 1979 ; Didden 1979 ). Using apiston-cylinder apparatus to generate vortex rings fromstarting jets, Gharib et al. ( 1998 ) showed that forincreasing aspect ratio of the uid slug ( L / D) the circu-lation of the vortex ring increased until L / D & 4. Furtherincreases in L / D did not increase the ring size and cir-culation with the excess vorticity ux forming a trailing jet in the wake of the vortex. This transitional statebetween a single ring without a trailing jet and a ring witha trailing jet is commonly referred to as the formationnumber, which is L / D & 4. Linden and Turner ( 2001 )analytically demonstrated that the limiting processdescribed by the formation number corresponds to anoptimal vortex ring as it possesses the maximum impulse,circulation, and volume for a given energy input. Thepresence of counter and coow on the formation numberwere also investigated by Dabiri and Gharib ( 2004 ) andKrueger et al. ( 2006 ), respectively. Johari ( 2006 ) inves-tigated the vortex formation in a fully pulsed jet incrossow. However, no studies have investigated whetheror not such a limiting process exists in forced round jetswith a mean ow.

The focus of this paper is to study the formation of vortex rings that occur when a round jet is subjected tohigh-amplitude acoustic forcing. Emphasis is placed uponthe formation time-scales and the circulation of the vortexrings as a function of forcing conditions. A particular aimis to identify whether or not a formation number exists forforced jets and if so what are the dening nondimensionalparameters. For this purpose, experiments have been per-formed using 2D High-speed particle image velocimetry(HSPIV) for the quantitative examination of the near eldof an acoustically forced round jet.

2 Experimental methods

A diagram of the apparatus and experimental setup isshown in Fig. 1. The apparatus was based on the setup of

Balachandran et al. ( 2005 ) to study forced ames and wasdesigned to provide large-amplitude, sinusoidal velocityoscillations at the nozzle exit by exciting various acousticmodes of the system. This experimental arrangement couldgenerate velocity amplitudes C 100% of the mean exit owfor a discrete set of frequencies. The apparatus consists of a200-mm long-cylindrical plenum chamber with an innerdiameter of ID = 100 mm. Flow comes in through thebottom of the plenum chamber and passes into a 300 mmlong tube with an ID = 35 mm. The converging exitnozzle of the round jet was knife-edged and designed witha matched cubic prole having an exit diameter of D = 10 mm to give a top-hat exit velocity prole. A largerdiameter exit nozzle D = 23 mm was also used for someexperiments but most of the results presented herein wereobtained with the 10 mm nozzle. To generate harmonicvelocity oscillations at the nozzle exit, a TTi 40 MHzwaveform generator was used to provide monochromaticsinusoidal input signals, which were amplied to drive twodiametrically opposed loudspeakers tted to the plenumchamber.

The air mass ow rates were set by an Alicat MCseries mass ow controller and passed through an oilseeder consisting of double Laskin nozzles to produce anoil mist with mean diameters ranging from 0.3 to 0.5 l mfor the PIV measurements. The rst step was to carry outa frequency sweep and measure the amplitude response Aof the round jet by placing a hot-wire anemometer(HWA) at the center of the exit nozzle, where the A u0 f =U : The value of u0 is the magnitude of theFourier transform centered at the forcing frequency, andU is thetime-averaged velocityfrom thehot-wiredata.As willbe discussed shortly, it is important to note that physicallymeaningful denitions of A and other ow variables needto be carefully dened in order to relate the vortex for-mation process with the forcing conditions. Figure 1shows that the largest amplitude response occurs atforcing frequencies of 40, 150, and 260 Hz correspondingto the acoustic modes of the system. The mean exitvelocity proles under forced and unforced conditionswere obtained from both hot-wire anemometry (HWA)and the high-speed PIV measurements (HSPIV) and areshown in Fig. 2. The velocity proles are approximatelytop hat and show good agreement apart from the shearlayer region where the HSPIV results are not as spatiallyresolved as the HWA measurements due the relativelylarge eld of view required to capture the vortex forma-tion process. Figure 2 shows the phase-averaged exitvelocity at the jet centerline obtained from HSPIV can bewell approximated as sinusoidal and scales linearly withforcing amplitude, A. This permits the adoption of amodied slug model to characterize vortex ring formation

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from a sinusoidal velocity programme superimposed upona mean background ow.

Following the usual notation, we can decompose thevelocity into a mean and uctuating component U t u0t U : Here, the prime denotes the large-amplitudesinusoidal disturbances generated by the acoustic forcingnoting that the turbulent uctuations are comparativelysmall. The time-averaged mean velocity of the jet is

U 1T Z

T

0

U t dt : 1Neglecting the effects of phase, the velocity at the exit

of a jet subjected to harmonic forcing is

U t U 1 Asin2p ft 2where the amplitude A u0max =U and u0max is the peak amplitude of the sinusoidal velocity oscillation calculatedfrom the PIV measurements. As hot-wire measurementswere only acquired to calibrate the boundary conditions,

this denition of A serves as a suitable proxy to the fre-quency centered Fourier transform method.

In order to make meaningful comparisons with theliterature on vortex formation, the forcing parametersneed to be expressed in terms compatible with the slugmodel (Lim and Nickels 1995 ; Shariff and Leonard1992 ). This requires suitable denitions for the sluglength ( L ), velocity ( U p), and discharge time ( t ) in termsof the forcing conditions. The slug length is a derivedparameter where L R

t 0 U t dt and in these experiments,

both L and t are functions wavelength k and the inverse of

forcing frequency f , respectively. These parameters areused to describe a complete vortex formation process eachforcing cycle and are treated as quasi-steady independentevents. From physical arguments, we dene the slugdischarge time over half a forcing period from t = 0 tot = T /2, i.e., when the jet velocity is greater than U :Setting the limits of integration to the slug discharge time,the mean slug which we term the effective piston velocitycan be dened as

(a)

0 50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

f ( Hz )

A

(b)

Fig. 1 A schematic of the experimental setup a and b the acoustic response as a function of forcing frequency, f

0.6 0.4 0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

r/D

U /

U

HWunforcedHWforcedPIVunforcedPIVforced

(a)

0 0.2 0.4 0.6 0.8 1

t / T

0

0.5

1

1.5

2

U ( t ) /

U

Increasing A

(b)Fig. 2 The exit velocity prolemeasured by HWA and PIV in afor both forced and unforced

conditions, b shows thevariation in centerline velocityas a function of forcingamplitude, A

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U p 1T Z

t T =2

t 0U t dt : 3

Where this integral is evaluated from the ensembleaveraged exit velocity of the jet obtained from the PIVmeasurements and forms the relevant velocity scale in

terms of vortex formation. Evaluating U p over a dischargetime of half the forcing period also relates the slug length L to the appropriate forcing length-scale k /2 via thefrequency as t = T /2 = 1/2 f giving a stroke length of

L k2

U p2 f

: 4It is important to note that the denominator 2 f is the slug

discharge time expressed in terms of the forcing frequency.As will be discussed later in this paper, identication of theformation number of a given ow hinges on the correctdetermination of L . The above expressions provide a directanalogy with the vortex ring experiments of Gharib et al.(1998 ) and Krueger et al. ( 2006 ). Combining Eqs. ( 2) and(3) and integrating, we obtain

U p U 1 2 Ap : 5

Equation ( 5) shows how the mean slug velocity U p is afunction of both mean ow U and the forcing amplitude A.The forcing conditions can be nondimensionalized usingEq. ( 4) by incorporating both A and f

L D

U p2 fD : 6

This denition of L / D describes both the aspect ratio of the uid slug and the nondimensional formation time forvortex rings in sinusoidally forced jets. In other words, itexpresses the magnitude of the forcing in terms of thelength- and time-scales relevant to vortex formation. Usingthis equation, we can determine the formation number forhigh-amplitude forced jets. U p relates the forcing periodT /2 and wavelength k /2 to the slug length where L is theeffective slugvolume displacedby theswept length of a pistonstroke over the characteristic discharge time t = 1/2 f .

However, later on in this paper, we demonstrate that forsinusoidal forcing, the integration limits that determine thecirculation of the vortex ring are actually from t = 0 tot = T /3, which can only be determined empirically andchanges L : k /3 and equation ( 6) accordingly.

Table 1 lists the forcing conditions investigated with 2DHSPIV. The HSPIV system was comprised of a Pegasus-PIV dual pulse 527 nm Nd:YLF laser with a maximumrepetition rate of 10 kHz, a set of sheet forming optics anda Photron Fastcam SA-1.1 monochrome camera with a

maximum frame rate of 5.1 kHz at 1MP (max) resolution.The camera timing and the synchronization of the laserpulses were achieved using a LaVision high-speed con-

troller. A 1 mm thick laser sheet was passed through the jetcross-section as shown schematically in Fig. 2 with thecamera positioned normal to the imaging plane. The eldsof view (FOV) were adjusted to cover the developingregion of the jet up to x / D * 6, giving the images spatialresolutions between 0.06 and 0.09 mm/pixel. The timedelays between image pairs were varied between 10 and60 l m to ensure in-plane particle translations of 45 pixelsdepending on inlet conditions. In most of the cases, 33image pairs per forcing period T were obtained requiringsampling frequencies of 1,320 and 4,950 Hz for f = 40 and150 Hz, respectively. In order to maintain maximum

camera resolution, 19 image pairs were obtained with asampling rate of 4,940 Hz when the jet forcing frequencywas f = 260 Hz.

The cross-correlations were performed using Davissoftware. Firstly, image preprocessing which includessliding background subtraction and particle intensity nor-malization was applied to the raw images. Second, a fourpass cross-correlation scheme was applied, with a circularweighting function and window sizes of 32 9 32 pixels forthe rst two passes and 16 9 16 pixels for the nal two

Table 1 List of experimental conditions including dimensional andnondimensional forcing parameters

f (Hz) U ms1 U p ms1 A Re40 10.40 15.85 0.82 6630

10.56 16.71 0.91 6735

6.86 10.33 0.80 4373

6.89 11.10 0.96 43967.05 11.86 1.07 4493

6.72 8.46 0.41 4285

150 9.68 15.47 0.94 6176

9.41 15.18 0.96 6003

9.78 16.52 1.08 6237

6.36 9.50 0.78 4055

6.51 10.61 0.99 4150

6.60 11.49 1.17 4206

6.37 7.52 0.28 4061

260 10.28 14.10 0.58 6555

10.02 14.97 0.78 6393

10.01 15.99 0.94 6383

6.37 10.03 0.90 4062

6.34 11.02 1.16 4044

6.45 11.93 1.33 4116

6.30 7.74 0.36 4017

U and U p calculated from the ensemble averaged PIV measurements. Re is based on D and U

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passes. A 50% overlap of adjacent windows was appliedto each image pair providing a ow eld resolution of 0.50.7 mm. A vector postprocessing step including;median lter, smoothing, and lling up of empty spaceswas applied. In Fig. 3, a sample raw PIV image with thecorresponding velocity vector elds show the formingvortex ring. In order to determine the circulation, C of thevortex rings the vorticity elds were obtained using a 2ndorder least-squares method following the recommendationof Raffel et al. ( 1998 ). The vortex ring circulation, Cvr wascalculated using Stokes theorem combined with a suitablethreshold

Cvr I C

udl ZZ S

x dS : 7The circulation of a vortex ring can be quantied by

measuring the circulation around the largest iso-vorticitycontour which denes its boundary. However, dening theboundary of a vortex ring is problematic given that theformation process is a transient event. This has an impacton dening other aspects of vortex ring formation such aspinch-off, separation, and the core size. To this extent, weadopt a similar approach to others (Gharib et al. 1998 ;Krueger et al. 2006 ) by dening a threshold around anappropriate iso-vorticity contour level. A sensitivity anal-ysis showed that this threshold value can signicantlyinuence the value of vortex ring circulation since it affectsthe area and the sum of bounded vorticity. Krueger et al.(2006 ) used a threshold with a normalized vorticity of x * = 0.91, which corresponded to approximately 20% of the peak vorticity in the pinched off ring. In the currentstudy, we found that the smallest value which gave sensibleresults was x * = 1, for a normalization based on the mean jet velocity U ; making the vortex ring boundaries and theseparation events clear and identiable. In the worst cases,this value corresponds to approximately 5% of the peak vorticity of the ring, which is suitably resolved.

3 The evolution of vortex rings as a functionof forcing conditions

The formation of a vortex rings over a normalized forcingperiod t / T are shown by iso-vorticity contour plots inFigs. 4, 5, and 6, which correspond to forcing frequenciesof f = 40, 150 and 260 Hz, respectively. For comparison,

each gure shows the effect of two values of A on theformation process. The vorticity is normalized by the meanvelocity x x D=U for contour ranges of 20 B x

*C

- 20 with increments 1 with normalized time-stamps shownat the top of the gures. For reasons that will becomeapparent later, the vortex ring formation will be describedwith reference to the forcing frequency f and amplitude A rather than in terms of their slug aspect ratio L / D.

In each of the gures, the initial roll-up, growth, pinch-off, and advection of the vortex rings is clearly seen. Ingeneral, an increase in f promotes earlier breakup of the jetinto smaller vortex rings. For a xed f , an increase in Acauses the formation process to move upstream closer tothe jet exit. This effect is nicely illustrated in Fig. 4 bycomparing the vortex formation process when A = 0.41and 0.96. For A = 0.96, the vortex ring rolls up earlier inthe forcing cycle and closer to the jet exit as shown whent = 0.15 T . Consequently, the vortex ring formed for A = 0.96 is much bigger by t / T = 0.45. This A-depen-dence amounts to a phase shift between the formation time-scales of the vortex ring and the forcing period. This effectis also evident when the jet is forced at f = 150 and260 Hz. Figures 5 and 6 show that the number of ringsvisible within the FOV decreases from two rings to one andfrom three rings to two, respectively, as the spacingbetween consecutive rings increases with A. This increasein spacing between consecutive rings occurs becauseduring formation the ring convects with U p which scaleswith both U and A according to Eq. ( 5). These effects arediscussed further in Sect. 5.

2 1.5 1 0.5 0 0.5 1 1.5 20

1

2

3

4(a)10 m/s

2 1 0 1 20

1

2

3

4(b)Fig. 3 Mie-scattering imageshowing the roll-up of a vortexring in a and b thecorresponding vector eld

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For f = 40 and 150 Hz, the existence of a trailing jet isevident, with the former resembling the type of ow eldencountered in the experiments of Gharib et al. ( 1998 ) forstarting jets with long nondimensional stroke ratios ( L / D).In Fig. 4, the separation of the leading vortex ring from the

trailing jet occurs when part of the ring has convected outof the FOV by t & 0.5T . This only occurs when the jet isforced at f = 40 Hz due to the longer wavelength, whereasthe formation process is fully captured in Figs. 5 and 6 andlasts up to t & 0.5 T . Forcing at f = 150 Hz shows some

x / D

t = 0.00 T

0

1

2

3

4

5

6t = 0.15 T t = 0.30 T t = 0.45 T t = 0.61 T t = 0.76 T t = 0.91 T

10

5

0

5

10 *

r/D

x / D

t = 0.00 T

1 0 10

1

2

3

4

5

6

r/D

t = 0.15 T

1 0 1

r/D

t = 0.30 T

1 0 1

r/D

t = 0.45 T

1 0 1

r/D

t = 0.61 T

1 0 1

r/D

t = 0.76 T

1 0 1

r/D

t = 0.91 T

1 0 110

5

0

5

10 *

Fig. 4 Normalized iso-vorticity contours x * illustrating the effect of amplitude A on the formation of a vortex ring over a forcing period, f = 40 Hz, A = 0.41 ( top ) and 0.96 ( bottom )

x / D

t = 0.00 T

0

1

2

3

4

t = 0.15 T t = 0.30 T t = 0.45 T t = 0.61 T t = 0.76 T t = 0.91 T

10

5

0

5

10 *

r/D

x / D

t = 0.00 T

1 0 10

1

2

3

4

r/D

t = 0.15 T

1 0 1

r/D

t = 0.30 T

1 0 1

r/D

t = 0.45 T

1 0 1

r/D

t = 0.61 T

1 0 1

r/D

t = 0.76 T

1 0 1

r/D

t = 0.91 T

1 0 115

10

5

0

5

10

15 *


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interesting phenomena. First of all, the lead ring separatesfrom the trailing jet much later in the forcing period andsecondly, the vorticity left in the wake of the leading ringstarts to roll-up before it is entrained into the oncomingring. For A = 0.9, the entrainment process is completed bythe end of the forcing period as shown at the time-stept = 0.91 T in Fig. 5. It is also interesting to note that themagnitude of the wake vorticity reduces when A isincreased due to the formation of a larger ring. Thisentrainment phenomena is somewhat similar to the leap-frogging effect reported by Gharib et al. ( 1998 ).

Another distinguishing feature between the forcing fre-quencies is the extent of the shear layer disruption. Whenthe jet is forced at f = 40 Hz with A = 0.41, the vorticityeld clearly uctuates in response to the velocity oscilla-tions, however, for the most part x * -contours remainattached to the jet exit extending throughout the near eld.As A is increased, the shear layer is weakened as morevorticity is rolled-up into the vortex ring, yet there is evi-dence that the shear layer remains attached to the jet exit.Forcing at wavelengths longer than the near eld of the jet,i.e., x / D & 5, did not promote complete breakdown of the jet shear layer. The other extreme is found when the jet isforced at f = 260 Hz where the entire near eld is domi-nated by the formation of a train of vortices each formingwithin 1 D of the jet exit with no discernible structureassociated with the mean ow remaining. The relativephase between the time- and length-scales between vortexformation and the forcing conditions in the near eld of jetows have important consequences on a variety of tech-nical applications, such nonlinear ame dynamics in lean

premixed ames that result from thermo-acoustic instabil-ity, and thrust augmentation techniques discussed inBalachandran et al. ( 2005 ), Mason and Miller ( 2006 ),respectively.

To further elucidate the effects of forcing amplitude, thevorticity elds from A = 0 to the maximum value areplotted at the same normalized time-step of t / T = 0.2 for f = 150 Hz ( top ) and f = 40 Hz ( bottom ) in Fig. 7. Theunforced jet shows the natural development of the KelvinHelmholtz instability along the jet shear layer, which islocked into the forcing frequency as A is increased. Thisgure provides a nice illustration of how increasing A causes the vortex roll-up progressively closer to the jetexit. The value of A at which this occurs is clearly f -dependent, with higher values needed as f is decreased.Figure 7 shows that for f = 150 Hz, the critical value of A lies between 0.4 and 0.5. whereas for f = 40 Hz, thevalue of A [ 1. The increase in spacing between consec-utive rings is also captured for the f = 150 Hz case.

These results suggest that in the case of forced jets other L / D-dependent regimes might be considered in addition tothe formation number. For example, the minimum L / Drequired to form a coherent vortex. This was investigatedby Kulsheimer and Buchner ( 2002 ) in a swirling jet withand without combustion excited by a siren. These authorsshowed that the minimum amplitude required to form acoherent vortex decreased hyperbolically with increasingnondimensional frequency (Strouhal number). A hint of this trend can be seen in Fig. 7, however, lower values of Awere not investigated in this study. Moreover, their de-nitions of A and St were based on r.m.s. and mean

x / D

t = 0.00 T

0

1

2

3

4

t = 0.16 T t = 0.32 T t = 0.47 T t = 0.63 T t = 0.79 T t = 0.95 T

15

10

5

0

5

10

15 *

r/D

x / D

t = 0.00 T

1 0 10

1

2

3

4

r/D

t = 0.16 T

1 0 1r/D

t = 0.32 T

1 0 1r/D

t = 0.47 T

1 0 1r/D

t = 0.63 T

1 0 1r/D

t = 0.79 T

1 0 1r/D

t = 0.95 T

1 0 120

10

0

10

20 *


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quantities and are distinctly different from the derived slugparameters in Eqs. ( 5) and ( 6) making exact comparisonsdifcult.

The effect of forcing conditions on the length- and time-scales associated with the formation process can also beexamined in terms of the spatial location where lead ringseparates from the jet and moves off at its own inducedvelocity. Separation is dened as the moment when thelowest iso-vorticity contour of the lead ring separates fromthe trailing jet. As will be discussed in Sect. 4, the differ-ence between separation and pinch-off of a vortex ring issignicant and care should be taken not to confuse the two.The vorticity elds for all the experiments were examinedto identify when and where separation occurred during theforcing cycle. The separation time t s and location xs / Dobtained from all the experiments are plotted in Fig. 8. The x-axis is the nondimensional time dened as t t sU p= D:The location of separation as a function of nondimensionaltime follows an approximately linear trend whose slopecorresponds to a velocity of & U p /2. These results showgood agreement with those of Didden ( 1979 ) and Mohseniand Gharib ( 1998 ).

4 The circulation of vortex rings as a functionof forcing conditions

This section of the paper discusses how the forcing con-ditions relate to the size of the vortex rings in terms of their

circulation, Cvr . Figure 9 shows the total and vortex ringcirculations as a function of normalized period for a rangeof forcing conditions. As before, each graph compares theeffect of different A for forcing frequencies of f = 150 in

Fig. 9a, b and f = 260 Hz in Fig. 9c, d. The circulationvalues are normalized by CC=UD : The total circula-tion C is represented by the solid lines, whereas the cir-culation of the vortex rings Cvr are denoted by the verticaldashed lines. Normalizing the circulation with U opposed

to U p2t =2; which would collapse the data, was done to

illustrate the effect of different forcing conditions. Forconsistency, when calculating Cvr in Fig. 9a, b, the trailingvorticity left over from the previous cycle as shown in

x / D

A = 0.00

0

1

2

3

4

A = 0.28 A = 0.53 A = 0.78 A = 0.99 A = 1.17

20

10

0

10

20 *

r/D

x / D

A = 0.00

1 0 10

1

2

3

4

r/D

A = 0.41

1 0 1

r/D

A = 0.59

1 0 1

r/D

A = 0.80

1 0 1

r/D

A = 0.96

1 0 1

r/D

A = 1.07

1 0 1

10

5

0

5

10 *

Fig. 7 The amplitude dependence on the roll-up location for f = 160 Hz ( top ) and f = 40 Hz ( bottom ) at the same normalized time-step

0 2 4 6 8 100

1

2

3

4

5

t

x s /

DRe = 4,250Re = 6,750Re = 10,500Re = 7,500Re = 7,000Re = 15,000

Re = 22,500

decreasingfrequency

Fig. 8 Normalized downstream position xs / D of all vortex rings atthe rst observed separation from the trailing jet in formation timet * = Ut / D

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Fig. 5 was excluded as the entrainment process continuesafter the ring has left the FOV for some values of A.

During the rst half of the forcing cycle, the total cir-culation grows steadily before levelling off around mid-cycle after which the circulation of the vortex ring can bedetermined. The results show that both C and Cvr increasewith A suggesting that the vortex rings in these cases arenot optimal in terms of their circulation, in other words,they are below the formation number following the ter-minology of Gharib et al. ( 1998 ). However, a closer look at

Fig. 9 reveals a relationship between C and Cvr : In each of the gures, the vortex ring circulation is approximatelyequal to the total circulation discharged within a narrowrange of time-scales from t / T = 0.28 to 0.35 with theaverage being t / T & 0.33 or a third of the cycle. Physically,this means that the vorticity ux issuing from the jet exit upto this time determines the circulation budget of theforming vortex ring. For these cases, it also corresponds tothe time when pinch-off occurs. As mentioned by Dabiri(2009 ), the difference between pinch-off and separation

requires succinct denition. Pinch-off refers to the momentwhen vorticity from a discharging slug stops owing intothe developing ring. By denition, the dynamics of thisprocess cannot be resolved experimentally as there is noviscous length-scale associated with a vortex sheet; how-ever, the experiments of Gharib et al. ( 1998 ) showed thatthe moment of pinch-off can be inferred from time-scalesobtained in the C plots, which for a wide range of forcingconditions was found to be t / T & 0.33. Separation on theother hand refers to when the lowest iso-vorticity contour

around the vortex ring disconnects from the trailing jet owas shown at t / T = 0.76 in Fig. 5(bottom ) and is sensitive tothe value selected for thresholding. This time-scale wasfound in all the experiments listed in Table 1 with theexception of some cases where pinch-off happens earlierdue to the formation of optimal vortex rings, which isdiscussed shortly.

Further insight can be gained by comparing the ringvelocity W r with the shear layer velocity. Figure 10 plotsthe time-dependent core location in the x-direction, which

0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

t /

vr

at A=1.17

at A=1.17

vr at A=0.99

at A=0.99

vrat A=0.78

at A=0.78

(a)

0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

7

8

t /

vr

at A=1.08

at A=1.08

vrat A=0.94

at A=0.94

(b)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

t /

vr

at A=1.33

at A=1.33

vrat A=1.16

at A=1.16

vrat A=0.90

at A=0.90

(c)

0 0.2 0.4 0.6 0.80

1

2

3

4

t /

vr

at A=0.94

at A=0.94

vrat A=0.78

at A=0.78

vrat A=0.58

at A=0.58

(d)

Fig. 9 Vortex ring circulationas a function of normalizedperiod for different forcingamplitdue A according toTable 1. Top a , b f = 150 Hz, Bottom c, d f = 260 Hz

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is used to calculate W r and compare with the shear layervelocity in Fig. 10. The core location was found bytracking the peak vorticity, which in these data was foundto be more reliable than tracking the centroid, which wouldinclude vorticity left in the wake of the previous ring. Thering velocity was obtained from the core position using acentral differencing scheme at each time-step to comparewith the shear layer velocity U t =2: As shown in Fig. 10,W r exceeds the shear layer velocity at t / T & 0.35, which isconsistent with the range of time-scales centered aroundt / T & 0.33. It is interesting to note that this time-scale is

the same as the pinch-off time-scale of the leading edgevortex (LEV) formed by a harmonically plunging aerofoilin the experiments of Rival et al. ( 2009 ).

There are several interesting aspects to this time-scale.Recall from Sect. 2 that integration limits of t = 0 tot = T /2 were selected in order to dene the discharge timeof the uid slug hence the mean slug velocity U p: In all thecases, shown in Fig. 9, the vortex ring pinches off at theearlier time-scale of t & 0.33 T . However, we could haveobtained the same value for U p by setting the integrationlimits from t = 0 to t = T /4 due to symmetry in the forcingconditions. This would require pinch-off to occur at

t = T /4, which is an inexion point i.e., is when dU / dt goes from positive values at t = 0 to zero at t = T /4.The wide range of experimental results suggest that it isthe change in sign of dU / dt that initiates pinch-off att / T & 0.33, which means that the value of U p is slightlyunderestimated when calculated over half of the forcingperiod. Thenext section in this paper discusshow t / T = 0.33represents an empirically derived kinematic constraint for jets subjected to harmonic forcing and cannot be analyticallydetermined from energy arguments. Moreover, it enables the

effective slug length L to be determined and hence the for-mation number.

So, if a formation number exists in forced jets, theforcing conditions need to be arranged to deliver the max-imum impulse before or at t = 0.33 T . Figure 11 plots thechange in C and Cvr when the jet is forced at f = 40 Hz for arange of A. The vortex ring circulation intersects the totalcirculation curve at t / T & 0.17 in Fig. 11 and t / T & 0.12 inFig. 11 . The fact that the time when C %Cvr occurs beforet = 0.33 T suggests the formation of optimal rings. In thesecases, the mechanism of pinch-off is due to the energy

constraint described by Gharib et al. ( 1998 ) and Shusserand Gharib ( 2000 ) rather than the kinematic limit describedpreviously supporting the supposition of optimality. Com-paring the pinch-off times in Fig. 11 with the time when W r equals or exceeds the shear layer velocity in Fig. 12 furtherillustrates the change in pinch-off mechanism and optimalring circulation. According to Fig. 11 ,W r intersects theshear layer velocity at t / T = 0.31, however by this timepinch-off has already happened. By contrast, separation of the lead ring from the trailing jet occurs several time-stepslater as shown in the bottom of Fig. 4.

These results suggest that if the formation number can

be applied to harmonically forced jets, it must account forthe kinematic constraint at t / T & 0.33. In other words, theformation number L / D & 4 occurs when the forcing con-ditions are arranged to provide maximum circulation att = 0.33 T .

5 The formation number of sinusoidally forced jets

By investigating the process of vortex formation in high-amplitude forced jets, we are asking what effect does the

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

2.5

3

3.5

t / T

X V R

/ D

(a)

0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

0.8

1

1.2

U ( t ) / U

VR Core VelocityShear Layer Velocity

(b)

t / T

Fig. 10 The position of the vortex ring core for the case in Fig. 9 as a function of normalized period a , b the corresponding shear layer and ringcore velocity

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presence of a mean ow U have on the formation num-ber? In this section of the paper, we will show that thelimiting time-scale t = 0.33 T serves as both a kinematicconstraint and the maximum slug discharge time-scale

associated with the formation number of forced jets.Figures 9, 10, 11 , and 12 demonstrated that two differentpinch-off mechanisms were at play depending on theforcing conditions. For a wide range of experiments,pinch-off occurred at t = 0.33 T , whereas for small rangeof experiments pinch-off happened earlier. In the lattercase, the mechanism of pinch-off is not kinematic butrelated to the relative energy of the forced jet and thevortex ring. For vortex rings formed with long strokeratios, i.e., L / D C 4, Shusser and Gharib ( 2000 ) proposed

that pinch-off occurs when the jet is no longer able todeliver energy at a rate compatible with the energy of asteadily translating vortex ring. In other words, the ringwill continue to accumulate vorticity as long as the

energy of the jet exceeds the energy of the ring, E j[

E r .However, in the forced jet case, there is a constraint onthe maximum time available for the ring to acquire itsmaximum circulation, which is t = 0.33 T and means thatthe condition E r C E j must be met at or within this timeconstraint or else the ring will pinch-off. By taking thisinto account, we can analytically determine the value of L / D when E j = E r at t = 0.33 T .

Employing the slug model approximations for the cir-culation C; impulse I , and the energy E , we obtain

0 0.1 0.2 0.3 0.40

2

4

6

8

10

12

14

16

vr

at A=1.07

at A=1.07

vrat A=0.96

at A=0.96

vrat A=0.80

at A=0.80

(a)

0 0.1 0.2 0.3 0.40

2

4

6

8

10

12

14

16

vr

at A=0.91

at A=0.91

vrat A=0.82

at A=0.82

(b)

t / T t / T

Fig. 11 Vortex ring circulation as a function of normalized period for different f = 40 Hz for a range of velocity amplitudes A

0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

X V R

/ D

(a)

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

U ( t ) /

U

VR Core VelocityShear Layer Velocity

(b)

t / T t / T

Fig. 12 The position of the vortex ring core for a case in Fig. 11 as a function of normalized period a , b the corresponding shear layer and ringcore velocity

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C 12

LU p; 8 I

14

p D2q LU p; 9E

18

p D2q LU p2: 10

where U p is dened according to Eq. 5 and L is the sluglength. Generalizing these expressions in terms of theforcing parameters where L U pt where the dischargetime t is a function of f and the normalized time^

t t =T ft , we obtainC j

^

t 2 f

U p2 ; 11

I j ^

t 4 f

p D2qU p2 ; 12

E

^

t

8 f p D2qU p

3:

13

The energy of a steadily translating vortex ring E vr can

be calculated according to Shusser and Gharib ( 2000 ) by

E vr a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiq I C3 q 14where the limiting value for the dimensionless coefcient awas experimentally determined by Gharib et al. ( 1998 ) asa & 0.33. It has to be mentioned that this denition of E vr makes some assumptions about various properties of thevortex ring. For example, it is assumed that both the vor-ticity distribution in the core and the radius of the ring is

xed. However, the most important assumption is that theformation of the vortex ring is a quasi-steady process. Thequasi-steady assumption is implicit in the denitions of U pand the other slug parameters which treats the formation of each vortex as an independent quasi-steady event occurringonce per cycle.

During the time where E j [ E vr , the jet transfers energyto the ring, which is manifested by an increase in circula-tion. At the moment when E j = E vr , pinch-off occursrestricting any further growth of the ring by cutting-off thevorticity ow. In order to plot the growth in the jet andring energies as a function of the normalized forcing period(t / T ), we use the dimensionless kinetic energy

E nd 4E

p D3qU p2 15

to cast the normalized jet and ring energies in Eqs. 13 and14 into their nondimensional forms where

E j;nd 3

^

t 2

L D 16

E vr ;nd ^

t 2L

D 2 9a

ffiffiffiffiffiffi2pp : 17

When E j,nd = E vr ,nd Eqs. 16 and 17 can be combined tosolve for the value of L / D at that instant to obtain theformation number for harmonically forced jets. Recall thatU p was evaluated from Eq. 5 over T /2 and not T /3.

Consequently, the values of U p in Table 1 are all slightlyunderestimated. Figure 13 shows that the % change of U pas a function of A between the two different integrationlimits is only a few % up to A = 1, which is withinexperimental accuracy. However, the change to the slugdischarge time, which is expressed in terms of frequency issignicant, i.e., t = 1/2 f to 1/3 f and needs to be accountedfor if the effective slug length L and hence formationnumber is to be correctly determined.

The growth in the nondimensional jet and ring energiesas a function of normalized period is shown in Fig. 14 for aselection of forcing conditions. For a given arrangement of U p; U ; f ; and A, a set of curves is produced illustratinghow the relative growth rates of the jet and ring vary over anormalized forcing cycle. Figure 14 shows the growth of the jet and ring energies when the jet is forced at f = 150 Hz with A = 1.17. For this case, the jet energyexceeds the vortex ring energy up to t / T = 0.5. This meansthat purely from an energy argument the energy of the jetcan feed the ring until this time, however Fig. 9 shows thatthe growth of the ring is halted due to the kinematic con-

0 0.5 1 1.5 20

1

2

3

4

5

A

% e r r o r i n U

p

Fig. 13 Percentage change in the mean effective piston velocity U pwhen calculated from Eq. 3 with empirically determined limits fromt = 0 to t = 0.33 T

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straint occurring at t / T = 0.33. The nondimensional energyof the ring is not equal to the jet at this time, and therefore,the ring can be considered suboptimal. Based on the valuesfrom Table 1, the forcing conditions correspond to L = D U p=3 fD 2:55, which is below the formation number.Figure 14 also plots the moment when E j = E vr , which iscoincident with the time-scale t = 0.33 T . This condition ismet when L / D = 3.80, which shows remarkable agreementwith the formation number in starting jets originally pro-posed by Gharib et al. ( 1998 ).

Figure 14 on the other hand shows two forcing caseswhere the nondimensional energy of the ring exceeds the jet prior to t / T = 0.33. In these cases, the jet was forced at f = 40 Hz for values of A = 0.8 and 0.82. By comparing

with the experimental results in Fig. 11 , we see that theanalytical model correctly predicts the pinch-off times. Thefact that pinch-off occurs so early in the forcing cycle isstriking and is due to the fact that they correspond tonondimensional stroke ratios of L / D = 8.6 and 13.2. Thishighlights the importance of recovering the correct slugparameters needed to correctly dene L , particularly thedischarge time as a function of the forcing conditions,which needs to be found empirically. Incorporating thekinematic time-scale into Eq. 6, the formation number forharmonically forced round jets can be expressed as

L

D %U p3 fD %4: 18

where t = 1/3 f corresponds to the discharge time of the uidslug which occurs over one third of the forcing period. It isworth pointing out that there is some evidence that thiskinematic restriction is more universal in character and mayapply to any time-dependent velocity programme, i.e.,velocities programmes where the acceleration changes sign.The kinematic time-scale will shift according to when theacceleration changes sign in the velocity programme, Rival

et al. ( 2009 ) reports such a shift. This kinematic restrictionwas also observed but not commented upon in the experi-ments of Gharib et al. ( 1998 ) for vortex rings formed with aslow-ramp velocity programme shown in their Fig. 2 (whichis a good approximation of the positive half of a sine wave).By cross-referencing their Fig. 2 with the correspondingcirculation plot in their Fig. 12 shows that the actual pinch-off time occurs just after the peak velocity which agrees withour experiments (note: the value of L / D in the captions of their Figs. 12 and 13 should be swapped).

6 Conclusion

The periodic formation of vortex rings in the developmentregion of a round jet subjected to high levels of acousticforcing has been performed for a wide range of experi-mental conditions. When subjected to harmonic velocityoscillations, the near eld of the jet breaks up into vortexrings of various size and circulation depending on theforcing conditions. The spacing between consecutive ringsand the initial roll-up location are both amplitude andfrequency dependent. By considering the total and vortexring circulation over the forcing period revealed that alimiting time-scale of t / T = 0.33 kinematically restricts thegrowth of a vortex ring for any given arrangement of jetforcing conditions. This kinematic constraint is imposed bythe translational velocity of the vortex ring exceeding theshear layer velocity which emanates from the change insign in the jet acceleration. This time-scale was alsoreported by Rival et al. ( 2009 ) for the separation of avortex ring from the leading edge of an harmonicallyplunging aerofoil, which lends credence to the notion thatthis time-scale may have a more universal character forvortex rings formed in other congurations which experi-ence a sinusoidal velocity history.

0 0.2 0.4 0.6 0.80

0.5

1

1.5

2

2.5

3

t / T

E n d

E j L/D=2.55

Evr L/D=2.55

E j L/D=3.80

Evr L/D=3.80

KinematicRestriction

(a)

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

3

E n d

E j L/D=8.61

Evr L/D=8.61

E j L/D=13.21

Evr L/D=13.21

(b)

t / T

Fig. 14 The nondimensionvortex ring and jet energy as afunction of normalized forcingperiod for a variety of experimental conditions and theanalytically determined value of L / D at the kinematic time-scalet = 0.33 T

Exp Fluids

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By considering the energy of the jet and a steadytranslating vortex ring in conjunction with the kinematictime-scale of t / T = 0.33, the formation number for sinu-soidally forced jets was analytically determined to be L / D = 3.80, which agrees well with the results of Gharibet al. ( 1998 ). This illustrates that care needs to be takenwhen determining the slug discharge time in order torecover the effective stroke length L in a forced jet and willmost likely need to be experimentally determined for agiven velocity programme.

Acknowledgments Dr James Dawson is indebted to Dr TimNickels for his many suggestions and helpful discussions on vortexrings related problems, he will be sorely missed. Dr James Dawson issupported by EPSRC under the Advanced Research Fellowshipscheme.

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the formation of vortex rings in a strongly forced round jet

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