The formation and evolution of synthetic jets
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PHYSICS OF FLUIDS VOLUME 10, NUMBER 9 SEPTEMBER 1998
The formation and evolution of synthetic jets
Barton L. Smitha) and Ari Glezer
Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0405
~Received 18 December 1997; accepted 6 May 1998!
A nominally plane turbulent jet is synthesized by the interactions of a train of counter-rotating
vortex pairs that are formed at the edge of an orifice by the time-periodic motion of a flexible
diaphragm in a sealed cavity. Even though the jet is formed without net mass injection, the
hydrodynamic impulse of the ejected fluid and thus the momentum of the ensuing jet are nonzero.
Successive vortex pairs are not subjected to pairing or other subharmonic interactions. Each vortex
of the pair develops a spanwise instability and ultimately undergoes transition to turbulence, slows
down, loses its coherence and becomes indistinguishable from the mean jet flow. The trajectories of
vortex pairs at a given formation frequency scale with the length of the ejected fluid slug regardless
of the magnitude of the formation impulse and, near the jet exit plane, their celerity decreases
monotonically with streamwise distance while the local mean velocity of the ensuing jet increases.
In the far field, the synthetic jet is similar to conventional 2D jets in that cross-stream distributions
of the time-averaged velocity and the corresponding rms fluctuations appear to collapse when
plotted in the usual similarity coordinates. However, compared to conventional 2D jets, the
streamwise decrease of the mean centerline velocity of the synthetic jet is somewhat higher
(;x 20.58!, and the streamwise increase of its width and volume flow rate is lower ~;x 0.88 and
;x 0.33, respectively!. This departure from conventional self-similarity is consistent with the
streamwise decrease in the jet’s momentum flux as a result of an adverse streamwise pressure
gradient near its orifice. © 1998 American Institute of Physics. @S1070-6631~98!00909-X#
I. INTRODUCTION It has been known for some time that streaming motions
The concept of synthesizing a turbulent shear flow by in fluids can be induced without mass addition by the trans-
controlled coalescence of its rudimentary coherent vortical mission of sound ~often referred to as acoustic streaming! or
structures ~e.g., turbulent spots in a transitional boundary by oscillating the boundary of a quiescent medium. In a re-
layer or vortex rings in a round jet! was proposed by Coles in view of streaming motions induced by acoustic waves
the early seventies and was later tested in a flat plate bound- Lighthill6 noted that acoustic streaming results from the dis-
ary layer experiment ~Savas and Coles1!. While in the sipation of acoustic energy or the attenuation of the transmit-
boundary layer experiments of Savas and Coles, turbulent ted sound. Such attenuation can occur either within the body
spots were triggered by hairpin vortices induced by the peri- of the fluid ~i.e., away from solid surfaces! at very high fre-
odic protrusion of a spanwise array of small pins into the quencies ~e.g., Meissner7!, or due to viscous effects near a
flow, in the present work, synthetic jets are engendered by solid boundary ~Andres and Ingard8!. Streaming motions as-
the interaction of discrete vortical structures which are sociated with oscillating solid boundaries have been the sub-
formed by time-periodic ejection of fluid out of an orifice at ject of a number of investigations, most notably time-
the flow boundary. Unlike conventional continuous jets ~e.g., harmonic oscillations of a cylinder normal to its axis ~e.g.,
Gutmark and Wygnanski,2 2D jet! or pulsed jets ~e.g., Brem- Stuart,9 Davidson and Riley,10 Riley and Wibrow11! leading
horst and Hollis,3 axisymmetric jet! a unique feature of syn- to streaming velocities on the order of 1 cm/s in water at a
thetic jets is that they are formed from the working fluid of nominal frequency of 45 Hz.
the flow system in which they are deployed, and thus transfer Jet flows without net mass addition can be produced by
linear momentum to the flow system without net mass injec- an oscillatory flow having a zero ~time-averaged! mean ve-
tion across the system boundary. Thus, the interaction of locity through an orifice, provided that the amplitude of os-
synthetic jets with an external flow near the flow boundary cillations is large enough to induce flow separation at the
can lead to the formation of closed recirculation flow regions orifice and the time-periodic rollup of a train of vortices.
and consequently to an apparent modification of the flow Ingard and Labate12 used standing waves in an acoustically
boundary ~Smith and Glezer,4 Amitay, Honohan, Trautman, driven circular tube to induce an oscillating velocity field in
and Glezer5!. This attribute enables synthetic jets to effect the vicinity of an orifice plate placed near a pressure node
significant global modifications of the base flow on scales and observed the formation of jets from trains of vortex rings
that are one to two orders of magnitude larger than the char- on both sides of the orifice with no net mass flux. More
acteristic length scales of the jets themselves. recently, Lebedeva13 created a round jet with velocities of up
to 10 m/s, by transmitting high amplitude sound waves ~150
a!
Author to whom correspondence should be addressed. dB! through an orifice placed at the end of a tube. In a related
1070-6631/98/10(9)/2281/17/$15.00 2281 © 1998 American Institute of Physics
Downloaded 01 Mar 2006 to 129.123.120.190. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp2282 Phys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer
investigation, Mednikov and Novitskii14 reported the forma-
tion of a jet without net mass flux and average streaming
velocities of up to 17 m/s by inducing a low frequency ~10–
100 Hz! oscillatory velocity field with a mechanical piston.
The evolution of a submerged synthetic round turbulent
water jet that is formed without an orifice by an oscillating
diaphragm flush-mounted on a flat plate was recently inves-
tigated by James, Jacobs, and Glezer.15 The jet which was
produced without net mass injection normal to and at the
center of the diaphragm, was comprised entirely of radially
entrained fluid, and was formed only when a small cluster of
cavitation bubbles appeared near the center of the diaphragm
during each oscillation cycle. The authors conjectured that
the time-periodic formation of these bubbles displaces vor-
ticity from the actuator’s boundary layer, and leads to the
formation of vortical puffs ~in the parlance of Kovasznay,
Fujita, and Lee16! that coalesce to synthesize a turbulent jet.
Laser Doppler velocity measurements showed that the time
averaged jet is similar to a conventional turbulent round jet
in that both its nominal diameter and the inverse of its cen-
terline velocity increase linearly with the distance from the
actuator.
In the present implementation, plane ~or round! turbulent
jets having finite streamwise momentum are synthesized nor-
mal to an orifice in a flat plate by a train of vortex pairs ~or
vortex rings!. The vortices are formed at the edge of an ac-
tuator orifice without net mass injection by the motion of a
diaphragm in a sealed cavity. Because the characteristic di-
mensions of the jet scale with the characteristic dimension of
the orifice, it is possible to synthesize jets over a broad range
of length scales ~microfabrication of synthetic jets having a
nominal orifice dimensions of 150 mm using standard silicon
micromachining techniques was reported by Coe, Allen,
Trautman, and Glezer17 and by Coe, Allen, Smith, and
Glezer18!. The present work focuses on the evolution of a FIG. 1. Schematic diagrams of synthetic jet: ~a! side view, ~b! top view.
nominally two-dimensional ~aspect ratio 150! synthetic jet.
The jet actuator and other experimental hardware are de-
scribed in Sec. II. The formation and evolution of the two- vortex sheet that rolls into a vortex pair and begins to move
dimensional vortex pairs that synthesize the jet are described away from the orifice under its own self-induced velocity
in Sec. III A, while the far-field structure of the ensuing jet is ~Auerbach19!. When the diaphragm begins to move away
discussed in Sec. III B. from the orifice, the vortex pair is already sufficiently re-
moved and is thus unaffected by the ambient fluid that is
drawn into the cavity. Therefore, during each cycle the net
II. EXPERIMENTAL METHODS AND PROCEDURE
mass flux out of the cavity is zero while the mass and hy-
In the work reported here, the synthetic jet is formed in drodynamic impulse of each vortex pair are nonzero.
air at a rectangular orifice measuring 0.5375 mm flush A schlieren image of the ensuing two-dimensional jet is
mounted in a flat plate measuring 30338 cm as shown sche- shown in Fig. 2. For the purpose of the schlieren visualiza-
matically in Figs. 1~a! and 1~b!. The exit plane of the jet is tion, the air inside the actuator’s cavity is slightly heated
instrumented with a linear array of 17 static pressure ports using a thin-film surface heater that is internally mounted on
equally spaced along z/h50 between y/h56.3 and 39, and one of the cavity walls. The schlieren view is in the x-y plane
connected to a Scannivalve pressure switch. and extends approximately through x570h. The motion is
The jet is synthesized by the time-harmonic formation recorded at standard video rate using a CCD camera having
and subsequent interactions of a train of vortex pairs that are an exposure time of 100 ms. The image shows a vortex pair
formed at the edge of the orifice by the motion of a dia- that is formed near the orifice, and a turbulent jet farther
phragm mounted in a sealed cavity. The circular diaphragm downstream. Although this image does not show the motion
is driven at resonance ~nominally 1140 Hz! by a centrally of the ambient air that is drawn towards the cavity along the
bonded piezoceramic disk. During the forward motion of the surface of the flat plate, such motion is evident in a similar
diaphragm, fluid is ejected from the cavity. The flow sepa- image of the merging of two identical coflowing synthetic
rates at each of the sharp edges of the orifice forming a jets operating side by side ~with their orifices parallel length-
Downloaded 01 Mar 2006 to 129.123.120.190. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jspPhys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer 2283
controlled traversing mechanism and are calibrated in an ad-
jacent conventional laboratory jet. A laboratory computer
system equipped with a 12 bit 100 kHz A/D board is dedi-
cated to experiment control and data acquisition.
III. THE SYNTHETIC JET
The evolution of the synthetic jet can be divided into two
distinct domains which are described in Secs. III A and III B.
Near the jet exit plane ~Sec. III A!, the flow is dominated by
the time-periodic formation and advection of discrete vortex
pairs which ultimately undergo transition to turbulence, slow
down and lose their coherence. The transition process is fol-
lowed by the emergence of a fully-developed turbulent jet
~Sec. III B! which is similar in some respects to a conven-
tional 2D jet.
A. Near-field formation and evolution
FIG. 2. Schlieren image of a rectangular synthetic jet. ReI0518124 (ReU0
The formation of a synthetic jet at ReU05383 and ReI0
5383), h50.5 mm, f51140 Hz. 518,000 ~referred to below as the ‘‘nominal case’’! is shown
in a sequence of digitized video schlieren images ~Fig. 3!
that are each taken phase-locked to the actuator driving sig-
nal at 27 equal time intervals ~33.8 ms apart! during the
wise and 3h apart! as shown in Fig. 11 below. The schlieren forcing period. The sequence begins with the forward motion
image in Fig. 11~a! shows evidence of strong entrainment of the actuator diaphragm (t/T50) which results in the ejec-
along the plate towards the jet orifices which is manifested tion of fluid from the jet cavity. ~The coordinate system is
by the rollup of 2D vortices at the left and right edges of the shown for reference in the image corresponding to t/T
plate ~the edges of the plate are not shown in Fig. 2!. 50.481 which is repeated on the bottom right hand side.! It
As for axisymmetric vortex rings ~e.g., Didden20 and should be noted that while the images in Fig. 3 are phase-
Glezer21!, each vortex pair may be characterized by two pri- locked to the actuator’s driving signal, the video frame rate is
mary dimensionless parameters based on a simple ‘‘slug’’ a submultiple of the forcing frequency, and thus successive
model; ~i! the dimensionless ‘‘stroke’’ length L 0 /h images do not show the same vortex pair.
(L 0 5 * t0 u 0 (t)dt where u 0 (t) is the velocity at the exit plane The front end of the fluid slug that is ejected out of the
of the orifice and t 5T/2 is the time of discharge or half the orifice and leads to the formation of the vortex pair is appar-
period of the diaphragm motion!, and ~ii! a Reynolds number ent on the left at time t/T50.11. Some traces of the previous
based on the impulse per unit width ~i.e., the momentum vortex pair are still discernible near x/h511 and the emerg-
associated with the discharge per unit width! ReI05I0 /mh ing turbulent jet is visible farther downstream. In subsequent
(I 0 5 r h * t0 u 20 (t)dt, r and m are is the fluid density and vis- images (0.15,t/T,0.41), the new vortex pair continues its
cosity, respectively!. When these vortices are generated rollup as it is advected downstream while the previous vortex
time-periodically to synthesize a jet, additional formation pa- pair becomes indistinguishable from the background flow
rameters include the formation frequency and the duty cycle, ~and, as discussed further below, it is no longer phase locked
both of which are fixed in the present experiments. Under to the excitation signal!. The new vortex pair and the remain-
these conditions ~and for a fixed orifice width!, the formation der of the ejected fluid behind it appear to be laminar after
parameters of the jet depend only on the amplitude of the the rollup process is completed and while the vortex core is
diaphragm motion and cannot be varied independently. In advected through x/h58.5 (t/T50.407).
the present experiments 5.3,L 0 /h,25 and 1400,I 0 / m h The cores of the vortex pairs begin to exhibit small-
,30 000. The corresponding Reynolds number of the syn- scale motions and undergo transition to turbulence around
thetic jet, ReU0 ~based on the orifice width h and the average t/T50.5 which, as shown in Fig. 8 below, is accompanied
orifice velocity U 0 5L 0 /T! varies between 104 and 489. by a reduction in their advection velocity. The transition pro-
Cross stream distributions of the streamwise and cross cess begins with the onset and rapid amplification of a span-
stream velocity components are measured at a number of wise instability of each ~primary! vortex that leads to the
streamwise and spanwise stations ~0,x/h,177 and 280,z/ formation of nominally spanwise-periodic counter-rotating
h,80! using hot wire anemometry with 1 mm long, 5 mm streamwise vortex pairs that are wrapped around the cores of
diameter single- and X-configuration sensors. The single- the primary vortices and ultimately lead to a cellular breakup
sensor probe was used primarily in the near field of the jet of their cores ~as shown in the spanwise view in Fig. 4!. The
and the two sensor probe was used for x/h.10 ~where cross formation of these streamwise vortices and the small-scale
stream distributions of the mean streamwise velocity mea- transition of the primary vortices is shown in a sequence of
sured with both probe types are virtually identical!. The hot phase-locked smoke visualization images taken in the x-z
wire probes are traversed using a three-axis computer- plane y50 using a laser sheet. In order to maintain smoke-
Downloaded 01 Mar 2006 to 129.123.120.190. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp2284 Phys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer FIG. 3. Phase-locked schlieren images of the synthetic jet in the cross stream (x-y) plane taken at 27 equal intervals during the actuator cycle. The forward and backward motions of the diaphragm from the rest position begin at t/T50 and t/T50.5 respectively. ReI0518,124 (ReU05383). concentration that is adequate for spanwise visualization, the The images in Figs. 4~a!–4~d! show a spanwise section of jet frequency was lowered to 360 Hz and, as a result, the the jet that is approximately 30h wide ~about z50! and are advection velocity of the vortex pairs is reduced to approxi- captured at t/T50.5, 0.625, 0.75, and 0.875, respectively. mately one tenth the advection velocity for the nominal case. Figure 4~a! shows a new spanwise vortex on the left and Downloaded 01 Mar 2006 to 129.123.120.190. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
Phys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer 2285
spanwise spacing of 2.5h. As the primary vortex is advected
downstream @Figs. 4~b!, t/T50.625 and 4~c!, t/T50.75#, the
secondary vortical structures intensify and shortly thereafter
appear to lead to a cellular breakdown of the core of the
primary vortex @Fig. 4~d!, t/T50.875#. As is evident from
the image of the downstream primary vortex in Fig. 4~a!, the
cellular segments apparently continue to break down to
smaller and smaller scales until the primary vortex loses its
identity @e.g., on the right hand side of Fig. 4~d!#. Similar
secondary vortex tubes that are wrapped around the core of
an isolated vortex ring appear during the final stages of its
transition following an azimuthal instability of the vortex
core ~Didden 1977,22 Schneider 198023!, and were also ob-
served in a turbulent vortex ring ~Glezer,21 Glezer and Coles
199024!. The appearance of counter-rotating pairs of stream-
wise vortices around the cores of the spanwise ~primary!
vortices in plane shear layers ~e.g., Bernal and Roshko
1986,25 Nygaard and Glezer 199126! and wakes ~e.g., Rob-
erts 1985,27 Williamson 199128! marks the appearance of
small scale motion within the cores of the primary vortices
and the onset of mixing transition.
The schlieren images for t/T.0.444 in Fig. 3 suggest
that similar to a vortex ring ~Glezer21!, the onset of small-
scale transition appears to take place near the front stagna-
tion point of the primary vortex where the strain rates are
high. Based on the schlieren visualization, the transition pro-
cess seems to proceed towards the rear of the vortex, and
ultimately progresses through the fluid stem behind it. In Fig.
3 for 0.67,t/T,1 the entire vortex pair appears to be tur-
bulent and its celerity, or propagation velocity, is diminished
as it merges into the ensuing turbulent jet. An important
feature of this sequence of images is that unlike vortex pairs
that form near the edges of the potential core of conventional
2D jets, consecutive vortex pairs in the present jet do not
coalesce or undergo pairing and @as shown in Figs. 20~a! and
20~b! below# there are no subharmonic components in power
spectra of the streamwise velocity.
Time series of the streamwise velocity component are
measured along the centerline of the jet ~y50! using a single
sensor hot wire probe. The sensor is operated at low overheat
ratio ~1.2! to minimize heat transfer to the jet orifice, and the
measured velocity is corrected for changes in the room tem-
perature. These data are taken phase-locked to the actuator
signal ~1,140 Hz, T50.877 ms! at 88 equal time intervals per
cycle ~i.e., 10 ms apart! for 1200 cycles. Figure 5 shows a
sequence of phase-averaged velocity traces ^ u(t/T;x) & /U 0
measured in the domains 0,x/h,5 ~at five equally-spaced
positions, marked with closed symbols!, and 5,x/h,25 ~at
nine equally-spaced positions, marked with open symbols!
for the nominal case. Near the jet orifice, the velocity traces
are rectified by the hot wire sensor when the velocity re-
verses its direction at mid-cycle. Thus, for x/h2286 Phys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer
well as the rollup and ultimately the advection of the vortex
pair during the discharge period, and the flow toward the
orifice behind the advected vortex pair during the suction
period. At the center of the orifice ~x/h50!, the two halves of
the velocity cycle are virtually identical and the time-
averaged velocity and the net mass flux are indeed zero. The
local velocity extreme at t/T50.08 and 0.58, mark the sym-
metrical rollup and advection of a vortex pair at the upstream
and downstream sides of the orifice during both the ejection
and suction parts of the cycle ~symmetric rollup on both
sides of a circular orifice was also reported by Ingard and
Labate12!.
The rollup of the vortex pair proceeds as it is advected
downstream and the velocity peak induced by its passage at
a given streamwise position increases in magnitude, while
the magnitude of the velocity minimum associated with the
suction decreases. These changes are accompanied by an in-
crease in the mean ~time-averaged! velocity. It appears that
at x/h54, the vortex pair is fully formed and the suction
cycle no longer affects the phase-averaged velocity. Similar
to the streamwise velocity measured along the axis of a vor-
tex ring,24 the centerline velocity reflects the passage of the
cores of a vortex pair where the peak corresponds to the
center of the cores. As demonstrated in Fig. 5 (3.9,x/h
,9.8), the magnitude of the induced velocity peak on the
centerline decreases monotonically as the vortex is advected
downstream, ostensibly as a result of the transition to turbu-
lence ~cf. Fig. 3! and loss of vorticity to the wake which are
also accompanied by reduction in phase coherence. Figure 5
(x/h.3.0) also shows that in addition to the time-dependent
velocity induced by the passage of the vortex pair, the mean
velocity at a given streamwise position includes a time-
invariant offset component u os(x)5min(^u(t/T;x)&) ~as
marked in Fig. 5! which increases with downstream distance.
The evolution of u os is discussed further in connection with
Fig. 9 below.
As the magnitude of the velocity that is induced by the
passage of the vortex pair diminishes farther downstream, it
becomes evident that the centerline velocity of the emerging
synthetic jet has a low-level time-periodic component at the
frequency of the actuator and its higher harmonics. Figures
6~a!–6~c! show phase averaged time traces ~with the local
time-averaged velocity subtracted! at x/h515.7, 17.7, and
19.7. While at x/h515.7 @Fig. 6~a!#, the velocity increase
associated with the passage of the vortex pair is still detect-
able during the first half of the cycle, at x/h519.7 @Fig.
6~c!#, the velocity distributions during each of the two halves
of the cycle are virtually identical. As shown in Fig. 21 be-
low, although the magnitude of the spectral component at the
actuator frequency decreases with downstream distance, it is
nevertheless detectable throughout the present domain of
measurements (x/h,180). That the phase of this spectral
component relative to the actuator motion does not change
appreciably with downstream distance suggests that it is in-
duced by the oscillating pressure field which is associated
with the pumping of the jet fluid in and out of the cavity.
The time t p corresponding to the passage of the ~phase-
averaged! velocity peak on the jet centerline during the pas-
sage of the vortex pair at a given measurement station allows
Downloaded 01 Mar 2006 to 129.123.120.190. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jspPhys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer 2287
FIG. 6. Phase-averaged centerline velocity during one cycle of the actuator:
x/h515.7 ~a, s!, 17.7 ~b, h!, 19.7 ~c, L!. ReI0518,124 (ReU05383).
for the determination of the streamwise position of its core.
Figure 7~a! shows the mean trajectories of a family of vortex
pairs that are produced with increasing total impulse ~i.e.,
1,400,I 0 / m h,30,000! at a fixed actuator frequency. ~Time
is measured from the beginning of the forward motion of the
actuator or the beginning of the ejection period.! It is striking
that regardless of the magnitude of the impulse, the resulting
trajectories are quite similar and are comprised of three dis-
tinct domains that are characterized by changes in slope ~or
celerity!. In Fig. 7~b!, the same data are plotted in dimen-
sionless form where the streamwise trajectory of each pair is
normalized by the corresponding ‘‘stroke’’ length L 0 , and
the reasonable collapse of the data suggests that the trajecto-
ries of the vortex pairs indeed scale with L 0 . In the domain
x/L 0 ,40, U c 'U 0 and, as can be shown from these data, it FIG. 7. Vortex pair trajectories ~a! global normalization, ~b! individual nor-
increases approximately like (I 0 ) 1/3. The vortex pair begins malization: ReI051,396 ~s!, 3,171 ~h!, 4,967 ~L!, 9,072 ~n!, 12,552 ~,!,
18,124 ~d!, 20,761 ~j!, 22,282 ~l!, 27,025 ~m!, 29,654 ~.!.
to slow down at t p /T'0.6 and x/L 0 '45 which is where it
begins to undergo transition to turbulence as suggested by
flow visualization ~cf. Fig. 3!. Finally, at t p /T'1, the vortex
pair begins to move faster until it loses its phase coherence mately at the beginning of the suction cycle of the actuator
and becomes part of the ensuing jet. Note also, that regard- and thus may be triggered by the reversal of the streamwise
less of the total impulse, no vortex pairs are still phase- velocity near the exit plane. Following transition (0.5,t/T
locked to the actuator signal much beyond t/T.1.3. ,0.8), the celerity decreases like @ (t/T) 22 # which is faster
The celerity U c (x,t) of the vortex pair is determined by than for the laminar vortex pair and considerably faster than
taking the time derivative of its trajectory. The variation of for an isolated turbulent vortex pair ~for which U c }t 20.5!.
the celerity ~normalized by the characteristic ejection veloc- The celerity reaches a minimum at t/T'0.8 and then in-
ity U 0 ! with t/T for the family of vortex pairs in Figs. 7~a! creases again like (t/T) 2 until the vortex pair becomes indis-
and 7~b! is shown in Fig. 8. For 0.25,t/T,0.5, the vortex tinguishable from the jet flow and its fluid effectively moves
pairs are nominally laminar and the celerity decreases like with the mean flow of the jet.
(t/T) 20.5 ~a straight line segment m520.5 is shown for ref- The offset velocity u os ~cf. Fig. 5! may be thought of as
erence!. As noted in connection with Figs. 3 and 4 above, the the time-invariant velocity of fluid that is entrained into the
transition to turbulence of the vortex pair starts approxi- jet in part as a result of the suction at the orifice and its
Downloaded 01 Mar 2006 to 129.123.120.190. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp2288 Phys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer
when the vortex pair undergoes transition to turbulence
around x/h57, the streamwise rate of decay of the centerline
velocity and the celerity increases substantially. The celerity
and the offset velocity change again at x/h'10 and ulti-
mately merge with the mean velocity at x/h.20.
While for the orifice geometry presented here, the vortex
pairs are advected along the centerline of the jet, this trajec-
tory and consequently the direction of the ensuing jet can be
easily altered. Lee and Reynolds29 demonstrated that small
changes in the azimuthal formation of successive vortex
rings in a circular jet can lead to changes in their trajectories
and consequently to substantial changes in the far field struc-
ture of the jet ~which the authors refer to as ‘‘bifurcation’’ or
‘‘blooming’’!. In order to demonstrate the role of the vortex
dynamics in the formation of the synthetic jet, the formation
process is modified by placing two synthetic jets in close
proximity. Each of the jets is essentially similar to the single
jet described above and they are placed side by side in a flat
plate so that they are parallel along the long dimension of
their orifices and 3h apart. The resultant jet can be effectively
manipulated by modifying the formation and evolution of the
vortex pairs of each jet by varying the amplitudes or the
relative phase of the driving waveforms. In particular, phase
variation between the driving signals effectively changes the
relative timing of the rollup of the adjacent vortex pairs and
thus leads to strong vortex interactions that alter the trajec-
tories of the vortex pairs and the direction of the ensuing jet.
Figure 11 shows a schlieren image of the jets and demon-
strates the effect of phase variation between two driving sig-
nals having the same frequency and amplitude. When the
two jets are in phase @Fig. 11~a!#, the inner vortices of each
vortex pair cancel each other, resulting in a single, larger
synthetic jet. As mentioned in Sec. II above, Fig. 11~a! also
shows evidence of the strong entrainment flow along the
plate towards the jet orifices which is manifested by the rol-
lup of 2D vortices at the left and right edges of the plate.
When one of the jets is leading in phase, the interaction
between the adjacent vortex pairs ~which is also affected by
the suction flow! alters their ultimate trajectories and the
merged jet is vectored towards the leading jet. When the jet
FIG. 8. Variation of vortex pair celerity U c (x,t) with time ~symbols as in on the right is leading in phase by 60°, the merged jet is
Fig. 7!. vectored to the right. When the phase angle is 150° @Fig.
11~c!#, the merged jet becomes almost attached to the exit
plane.
streamwise dependence is shown in Fig. 9~a! for different
impulse levels. It is remarkable that ~except for the lowest
B. The mean flow
I 0 ! regardless of the impulse, u os of all vortex pairs initially
increases along the same curve before it reaches a maximum Cross stream distributions of the time-averaged stream-
value which depends on and increases with the initial im- wise ~U! and cross stream ~V! velocity components along
pulse. Past the maximum, for a given impulse level, u os be- with the corresponding rms velocity fluctuations u8,v8, and
gins to decrease with streamwise distance ostensibly as a u8v8 of the nominal case are plotted in Figs. 12~a!–12~e! in
result of the cross stream spreading of the jet. Finally, for the usual similarity coordinates of conventional 2D jets ~the
x/h.20, u os is equal to U cl , which, as is shown in Fig. 13, cross stream coordinate is normalized with the local jet width
decreases like x 20.58. The same data are plotted in dimen- b(x) based on U cl/2!. These data are measured at a 11
sionless form in Fig. 9~b! which shows a reasonable collapse streamwise stations between x/h59.8 and 78.7 and, at least
with the possible exception of the vortex pair that is formed within this streamwise domain, collapse reasonably well de-
at the lowest impulse level. The streamwise dependence of spite the fact that the jet is formed by time-harmonic motion.
the celerity, offset velocity and the mean velocity for the The mean cross stream velocity component @Fig. 12~b!# is
nominal case is shown in Fig. 10. It is interesting to note that nominally antisymmetric about the jet centerline and its nor-
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FIG. 9. Offset velocity u os . ~a! Dimensional variables, ~b! normalized variables: ReI054,967 ~s!, 9,072 ~h!, 12,552 ~L!, 18,124 ~n!, 20,761 ~,!, 22,282
~d!, 27,025 ~j!, 29,654 ~l!.
malized magnitude is similar to the cross section velocity of 9.8,x/h,177 are shown in Fig. 13. For x/h,80, U cl de-
corresponding conventional jets.30 The normalized distribu- creases like x 20.58 and u8 decreases like x 20.5, while for
tions of the rms velocity fluctuations u8 @Fig. 12~c!#, v 8 @Fig. conventional 2D fully developed turbulent jets, both U cl and
12~d!# and of u8v 8 @Fig. 12~d!# are also very similar to and u8 decrease like x 20.5. Note also, that for x/h.80, the rate of
have approximately the same magnitudes as corresponding
streamwise decay of the centerline velocity diminishes to
distributions in conventional jets.2,30,31,32
x 20.25 ostensibly due to three-dimensional effects associated
The cross stream distribution of u8 @Fig. 12~c!# exhibits
with the streamwise decrease in the aspect ratio of the jet
two distinct peaks on both sides of the centerline ~where u 8
cross section in the y-z plane ~cf. Fig. 19 below!. It is inter-
'0.25U cl! which coincide with the peaks of the cross stream
velocity components. In conventional jets, u8 and v 8 typi- esting to note that u8 appears to be unaffected by these
cally increase rapidly downstream of the potential core and changes and continues to decrease like x 20.5. Figure 13 also
their cross stream peaks are normally between 0.2U cl and shows that for 10,x/h,80, the jet width b(x) ~based on
0.3U cl where the flow becomes fully developed2,30,32 and in- U cl/2! in the cross stream plane z50 increases like x 0.88
crease with decreasing Reynolds number ~based on the jet while in conventional 2D jets, b}x. The streamwise rate of
height!.32 In contrast to conventional plane jets which be- increase of the jet width at x/h530 is 0.194 and is almost
come fully developed at x/h.40 ~e.g., Gutmark and twice the corresponding streamwise increase in the width of
Wygnanski2!, the mean flow of the synthetic jet appears to conventional 2D jets at Reynolds numbers on the order of
become fully developed considerably closer to the jet exit 104 ~which varies between 0.09 to 0.12!.2,30,33 Note also that
plane (x/h.10). the linear fit b 1.136}x yields a virtual origin for the nominal
The streamwise variation of the mean velocity and case of x 0 '24h which is comparable to what was mea-
rms velocity fluctuations along the jet centerline for sured by Gutmark and Wygnanski2 (22.5h) and Krothapalli
Downloaded 01 Mar 2006 to 129.123.120.190. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp2290 Phys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer
FIG. 10. Mean centerline velocity ~h!, celerity ~s! and offset velocity ~L! FIG. 11. Schlieren images of the interaction between two adjacent synthetic
for the nominal case ReI0518,124 (ReU05383). jets: DF50° ~a!, 60° ~b!, and 150° ~c!.
et al.30 (22h) in conventional 2D jets. Gutmark and Ho34 first domain x/h,2, the jet centerline velocity increases rap-
suggested that the disparity in streamwise spreading rates of idly to a level which scales with the average orifice velocity
2D jets in earlier investigations could be attributed to the ~which depends on the formation amplitude!. In the second
spontaneous emergence of different instability modes of the domain, the streamwise rate of increase of the centerline ve-
jet shear layers. Thus, because the synthetic jet is formed by locity is much smaller ~although not zero!. Farther down-
a train of 2D vortex pairs which do not interact ~or pair! it is stream ~nominally x/h'10! the centerline velocity begins to
expected that at least near the exit plane, the cross stream decay with streamwise distance ~within the third domain!. As
spreading of the synthetic jet would be limited. noted in Sec. III A, the streamwise decay of the centerline
As noted in Sec. II, when the motion of the diaphram is velocity begins at t/T50.5, and thus the corresponding
time-harmonic ~and for a fixed orifice width!, the formation streamwise locations increase linearly with the formation
parameters of the jet depend only on the amplitude of the amplitude ~or the slug length L 0 !. Note that all the data
actuator signal, and cannot be varied independently. Figures within the third domain ultimately collapse onto a single
8 and 9~b! demonstrate that the celerity and offset velocity of curve given by x 0.58 ~cf. Fig. 13! which is also shown for
the vortex pairs, respectively, scale with the average orifice reference.
velocity and thus with the amplitude of the actuator signal. The streamwise variation of integral quantities such as
The effect of the amplitude on the global properties of the the jet volume flow rate and its streamwise momentum flux
ensuing jet is demonstrated by considering the dependence are assessed using a least squares fit of the hyperbolic cosine
of the centerline velocity ~normalized by U 0 , Fig. 14! on function U f 5U cl cosh22(hy) ~where h is a parameter of the
x/h. These data show that the existence of three distinct fit! to cross stream distributions of the streamwise velocity.
streamwise domains corresponding to the formation of the The quality of the fit at x/h520 is demonstrated in Fig. 15.
vortex pairs, their laminar advection and transition to turbu- Similarity arguments for conventional 2D turbulent jets
lence, and finally the emergence of the turbulent jet. In the suggest that the volume flow rate per unit width i.e.,
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2292 Phys. Fluids, Vol. 10, No. 9, September 1998 B. L. Smith and A. Glezer
FIG. 13. Streamwise variation of: U ~d!, u 8 ~s!, and b ~l!. ReU05383.
FIG. 15. Least-squares fit of a cross stream distribution of the mean stream-
plane is lower than the ambient pressure. This is evident in wise velocity at x/h520 to a hyperbolic cosine function.
measurements of the static pressure on the exit plane along
z50 at different Reynolds numbers. The static pressure ports
are equally spaced (2.3h apart!, and unfortunately, owing to pressure gradient near the jet orifice and consequently a
structural constraints, it is not possible to achieve better reso- streamwise decrease in the momentum flux of the synthetic
lution near the jet orifice. The resulting pressure coefficient jet. The streamwise variation of the momentum flux per unit
~normalized by r U 20 ! in Fig. 17 shows that the mean static width normalized by the average momentum flux of the
pressure near the jet orifice is lower than the ambient pres- ejected fluid is shown in Fig. 18. The closed symbols corre-
sure and is consistent with the steady suction of ambient spond to integration of the ~fitted! hyperbolic cosine profiles
fluid towards the jet orifice as is evident in Fig. 11 above. ~which near the jet exit plane yields a value near unity!,
Figure 17 also shows for reference a line segment which while the open symbols are based on integral limits of half
represents the radial decrease of the static pressure in the the centerline velocity ~i.e., 2b,y,b!. The data set repre-
flow field of a 2D potential sink ~i.e., p}r 22 !. These mea- sented by open symbols is effectively based on measured
surements suggest the existence of an adverse streamwise velocity ~rather than the fitted curve! and is included for
FIG. 14. Variation of centerline velocity U cl(x,t) with axial distance ~sym- FIG. 16. Streamwise variation of the volume flow rate. The straight line
bols as in Fig. 7!. segment denotes Q}x 0.5 ~for self-similar 2D jet!.
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FIG. 19. Contour maps of the streamwise velocity in the y-z planes x/h
519.7 ~a!, 39.4 ~b!, and 78.7 ~c!. The first contour is 1 m/s and contour
FIG. 17. Distributions of the pressure coefficient at the exit plane: increment is 0.5 m/s. ReU05383.
ReU05270 ~d!, 383 ~j!, 424 ~l!, and 489 ~m!.
As noted by Kotsovinos and Angelidis,35 the streamwise
reference. For a conventional self-similar flow, both curves variation of the time-averaged momentum flux in plane ~or
should be streamwise invariant. However, the momentum axisymmetric! jets depends critically on the pressure field,
flux of the synthetic jet decreases monotonically with and on the geometry of the jet. Based on data published by
streamwise distance. The decrease is further complicated by other investigators since 1957, these authors assert that in
the spanwise nonuniformities in the jet cross section and the conventional jets emanating normal to a plane surface the
streamwise decrease in its aspect ratio as shown in Fig. 19. A momentum flux decreases with downstream distance. Varia-
single measurement taken at x/h578.7, suggests that far tion in the streamwise rate of decrease among the different
enough downstream (x/h.100), the momentum flux as- data sets results in momentum flux levels at x/h580 that are
ymptotes to a constant value around 0.55. between 75% and 85% of the level at the exit plane ~at
x/h50!.
The streamwise variation of the jet cross section in the
y-z plane can be assessed from contours of the mean stream-
wise velocity at x/h519.7, 39.4, and 78.7 shown in Figs.
19~a!–19~c!, respectively ~contours start at 1 m/s and the
contour increment is 0.5 m/s!. These plots indicate that the
aspect ratio of the jet cross section ~based on contour level of
1 m/s! decreases from approximately 6 at x/h519.7 to 3 at
x/h578.7. While near the exit plane @x/h519.6, Fig. 19~a!#
the jet appears to be reasonably spanwise-uniform, farther
downstream, @x/h539.4, Fig. 19~b!# the cross-stream width
of the jet near its spanwise edges is larger than at the mid
span. At z/h5655, the streamwise velocity has local span
wise maximas, and the normalized momentum flux in these
x-y planes is 1.16 compared to 0.58 at z/h50 which may be
associated with the streamwise decrease in the jet aspect ra-
tio. At x/h578.7, the centerline velocity is relatively low
~2.8 m/s! and the cross section of the jet appears to be
slightly rotated about its centerline. Similar saddle-like dis-
tributions of the streamwise velocity was also observed in
conventional high aspect ratio rectangular jets.30,36
Additional insight into the evolution of the synthetic jet
FIG. 18. Streamwise variation of the jet momentum flux based on fitted cos
may be gained from spectra of the streamwise velocity.
hyperbolic distribution, and on the velocity data for 2b,y,b. Power spectra of the jet centerline velocity measured at
ReU05383. x/h55.9, 9.8, 19.7, 98.4, and 177.2, are shown in Figs.
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20~a!–20~e!, respectively @each of the curves in Figs. 20~b!–
20~e! are successively displaced by seven decades, and the
power spectrum at x/h55.9 is replotted for reference using a
shaded curve#. Near the jet exit plane @Fig. 20~a!#, the spec-
trum is dominated by the formation frequency of the vortex
pairs and its higher harmonics ~although hot-wire rectifica-
tion of velocity traces within this domain clearly contributes
to the spectral contents at the higher harmonics!, while the
spectral distribution below the fundamental frequency is vir-
tually featureless. The harmonics of the formation frequency
are rapidly attenuated with downstream distance and by
x/h59.8, only four harmonics are present. Concomitantly,
there is also a significant increase in the magnitude of the
spectral band below the formation frequency which is indica-
tive of the decay of the vortex pairs and the development of
the jet flow. However, with the exception of a weak band of
spectral components centered around 10 Hz, which disap-
pears by x/h598, the spectral band below the formation
frequency remains featureless throughout the present domain
of measurements and shows no evidence of subharmonics of
the formation frequency.
A striking feature of the velocity spectra in Figs. 20~b!–
20~e! is the rapid streamwise attenuation of virtually all spec-
tral components indicating strong dissipation within the syn-
thetic jet and a reduction in the total turbulent kinetic energy.
The spectral decay is initially more prominent at frequencies
that are above the formation frequency of the jet @Figs. 20~a!
and 20~b!# while, as noted above, there is a concomitant
increase in the magnitude of the spectral band below the
formation frequency. Thus, it is conjectured that following
the time-harmonic formation of discrete vortex pairs, energy
is transferred from these primary ~‘‘large scale’’! eddies,
which coalesce to form the jet, to the mean flow and also
cascades down to smaller scales at which dissipation ulti-
mately takes place. Farther downstream @Figs. 20~c!–20~e!#,
the low frequency components of the jet are continuously
attenuated and by x/h5177 @Fig. 20~e!#, the nominal mag-
nitude of the band f,100 Hz is comparable to the corre-
sponding band near the jet exit plane suggesting energy
transfer to the smaller scale. At the same time, the ‘‘roll-
over’’ frequency ~at which the low-frequency end of the
spectrum begins to undergo a change in slope!, moves to-
wards lower frequencies @in Fig. 20~c!, the roll-over fre-
quency is below the formation frequency#. The spectral dis-
tributions in Figs. 20~c!–20~e! also include a relatively
narrow frequency band having a slope of approximately
2 35 suggesting the existence of an inertial subrange which is
limited by the low Reynolds number of the flow. It is note-
worthy that because the characteristic local ~centerline! ve-
locity decreases with downstream distance, the spectral peak
at the formation frequency actually shifts towards higher
wave numbers where the dissipation ultimately takes place
@e.g., Fig. 20~e!#. FIG. 20. Power spectra of the centerline velocity ~each curve is successively
displaced 7 decades!: x/h55.9 ~a!, 9.8 ~b!, 19.7 ~c!, 98.4 ~d!, 177.2 ~e!.
As mentioned in Sec. III A above, a notable feature of
ReU05383.
the synthetic jet is the absence of pairing interactions be-
tween the vortex pairs that form the jet and consequently the pairs undergo transition and breakdown to smaller eddies
absence of subharmonic frequencies in spectra of the stream- and that the jet is ultimately formed by the coalescence of
wise velocity component in Fig. 20. The phase-locked clusters of such smaller eddies. The breakdown of the span-
schlieren images in Fig. 3 indicate that the primary vortex wise vortex pair is alluded to by an abrupt and rapid decrease
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FIG. 21. Streamwise variation of the magnitude of the spectral component
at the forcing frequency. ReU05383.
~around x/h510! in the magnitude of the spectral compo-
nent at the forcing frequency, a f o ~Fig. 21!. This change is
also apparent in the gray-scale raster plot of the auto corre-
lation function r ( t ,x) of the centerline velocity shown in
Fig. 22. For ~a fixed! large x, r ( t ).0 ( r (0,x)51) and de-
cays monotonically to zero for large t as in other fully de-
veloped turbulent flows. However, as a result of the coherent
vortex motion near the exit plane, the auto correlation is
FIG. 23. Contour map of the spanwise correlation function of the stream-
wise velocity component R 11(x,Dz). The lowest and highest contour levels
are 0.8 and 20.15, respectively, and the contour increment is 0.05.
ReU05383.
nominally time-harmonic with a zero cycle average ~negative
grayscale values are marked with contours!. As x increases, r
becomes gradually non-negative and although fluctuations at
the forcing frequency are still apparent, their amplitude is
considerably diminished indicating loss of coherence of the
vortical structures. The streamwise domain where r becomes
non-negative, coincides with the abrupt decrease in the am-
plitude of the spectral peak of the formation frequency ~Fig.
21!.
Finally, the spanwise correlation function R 11 is mea-
sured along the z axis at a number of streamwise stations
using two single-element hot wire probes Dz apart ~one of
the sensors is located on the jet centerline!. A contour plot of
R 11(x,Dz) ~Fig. 23! shows that for x/h,6, the jet is nearly
spanwise-uniform ~the highest contour level is 0.8! but that
FIG. 22. Gray-scale raster plot of the autocorrelation function of the stream-
wise velocity r (x, t ). Negative levels are marked with contours ~contour as a result of the transition process, R 11 decreases rapidly
increment 0.1!. ReU05383. with streamwise distance and at x/h'12 the spanwise coher-
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ence is almost lost. It is conjectured that similar to thin iso- of three distinct domains that are characterized by changes in
lated vortex rings ~as shown in the experiments of the vortex pair celerity. Following the formation process, the
Sturtevant37!, the instability of the vortex pair cores is vortex pairs are advected at almost constant speed which
quickly amplified because the high length to core diameter scale approximately with (I 0 ) 1/3. After the transition to tur-
ratio ~approximately 50!. However, the spanwise correlation bulence (0.5,t/T,0.8), the celerity decreases as (t/T) 22
does not exhibit spanwise variations at the wavelength of the which is considerably faster than for an isolated turbulent
vortex core instability ~e.g., Fig. 4! which indicates ~as con- vortex pair (U c }t 20.5). The celerity reaches a minimum at
firmed separately by flow visualization! that the instability is t/T'0.8 and then increases again like (t/T) 2 until the vortex
not locked to spanwise disturbances upstream of the jet ori- pair becomes indistinguishable from the jet flow (x/h.12)
fice. The distortion of R 11 near the edge of the jet suggests and effectively moves with the mean flow of the jet. While
that the primary vortices bend in the streamwise direction. the celerity of the vortex pairs decreases monotonically in
This distortion presumably occurs as a result of a local inter- the domain x/h,10, the mean ~centerline! velocity of the jet
action between the vortex pair and the vortex segment that increases until it reaches a local maximum at x/h'7 and
forms along the short side of the jet orifice similar to the then begins to decay monotonically.
streamwise distortion of elliptic vortex rings in the cross The synthetic jet is similar to conventional 2D turbulent
stream view along their minor axes ~e.g., Gutmark and jets in that cross stream distributions of the time-averaged
Ho34!. Such a streamwise distortion is consistent with nega- streamwise and cross stream velocity components and the
tive values of R 11 near the edge of the jet and before the corresponding rms velocity fluctuations u8 and v 8 and the
spanwise breakdown of the primary vortices takes place.
correlation u8v 8 appear to collapse when plotted in the usual
similarity coordinates for conventional 2D jets. However,
IV. SUMMARY AND CONCLUSIONS compared to conventional 2D jets, the streamwise decrease
of the mean centerline velocity of the synthetic jet is some-
A high aspect ratio rectangular air jet is synthesized by
what higher (x 20.58 vs x 20.5 for 2D jets!. Also, the width of
the time-harmonic formation and subsequent interactions of
the synthetic jet b(x) ~based on U cl/2! increases like x 0.88
a train of counter-rotating vortex pairs. Each pair is formed
~for conventional 2D jets, b}x!, and its volume flow rate
at the edge of an orifice of an otherwise sealed cavity by the
Q(x) increases like x 0.33 ~for conventional 2D jets Q}x 0.5!.
motion of a flexible diaphragm that is mounted on one of the
Despite the lower streamwise increase of b(x) and Q(x) of
cavity walls and is driven at resonance. Even though the net
the synthetic jet, db/dx is almost twice the value measured
mass flux out of the cavity during each cycle of the dia-
for conventional 2D jets at much higher Reynolds numbers
phragm motion is zero, the mass and hydrodynamic impulse
~of order 104 !. Furthermore, even though dQ/dx is smaller
of the ejected fluid are nonzero. The flow separates at the
than in conventional jets, the net entrained volume flow rate
sharp edges of the orifice and the resulting vortex sheet rolls
of the synthetic jet within the present domain is nearly 4Q 0
into a vortex pair which is advected away from the orifice
and substantially larger than for conventional 2D jets.
under its own self-induced velocity. When the diaphragm
begins to retract from the cavity, the vortex pair is already This departure from conventional self-similarity is asso-
sufficiently removed and is thus relatively unaffected by the ciated with a streamwise decrease in the jet’s momentum
motion of the ambient fluid that is drawn into the cavity. flux. While for conventional self-similar 2D jets the momen-
The evolution of the synthetic jet near its exit plane is tum flux is presumably an invariant of the motion, the mo-
dominated by the time-periodic formation and advection of mentum flux of synthetic jets decreases with streamwise dis-
these vortex pairs which ultimately undergo transition to tur- tance as a result of an adverse streamwise pressure gradient
bulence, slow down and lose their coherence. Schlieren vi- near the jet orifice that is associated with the suction cycle of
sualization shows that despite the relatively high formation the actuator and an induced mean static pressure which is
frequency of the jet, successive vortices do not pair, and the lower than the ambient.
spectral band below the formation frequency in velocity Finally, a striking feature of the velocity spectra of the
spectra remains relatively featureless throughout the present synthetic jet is the rapid streamwise attenuation of virtually
domain of measurements. The passage of the vortex pairs is all spectral components indicating strong dissipation within
manifested by a strong time-periodic component of the the jet and reduction in the total turbulent kinetic energy.
streamwise velocity which diminishes rapidly with down- Following the time-harmonic formation of the discrete vor-
stream distance until the vortex pairs are no longer phase- tex pairs, energy is transferred from these primary ~‘‘large
locked to the excitation signal and become indistinguishable scale’’! eddies which coalesce to form the jet both to the
from the background flow. Spanwise flow visualization mean flow and to smaller scales at which dissipation ulti-
shows the appearance rib-like secondary vortical structures mately takes place. Ultimately, the spectral components
that are wrapped around the cores of the primary ~spanwise! within the ~low! frequency band below the formation fre-
vortices which lead to the formation of spanwise-periodic quency begin to decay and the energy is transferred primarily
cellular structure within the cores of the vortices and ulti- to the smaller scales. Because the characteristic local ~cen-
mately to their small scale breakdown. terline! velocity decreases with downstream distance, the
The mean trajectories of vortex pairs at a given forma- spectral peak at the formation frequency continuously shifts
tion frequency scale with the ‘‘stroke’’ length L 0 regardless towards higher wave numbers where the dissipation ulti-
of the magnitude of the formation impulse and are comprised mately takes place.
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This work has been supported by AFOSR Grant No. 19
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