Computers & Fluids 38 (2009) 1011–1017

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Computers & Fluids

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Induced co-flow and laminar-to-turbulent transition with synthetic jets

John Abraham *, AnnMarie ThomasUniversity of St. Thomas, School of Engineering, 2115 Summit Avenue, St. Paul, MN 55105-1079, United States

a r t i c l e i n f o

Article history:Received 18 August 2007Received in revised form 14 December 2007Accepted 20 January 2008Available online 14 April 2008

0045-7930/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.compfluid.2008.01.028

* Corresponding author. Tel.: +1 651 962 5766; faxE-mail address: jpabraham@stthomas.edu (J. Abra

a b s t r a c t

A detailed numerical solution of the fluid flow patterns engendered by a synthetic jet has been carriedout. The synthetic jet is caused by a reciprocating piston assembly which is attached to a large, stationarycavity. It was found that a significant momentum efflux is produced by the synthetic jet assembly. Also,fluid entrainment by the synthetic jet causes a coincident flow around the exterior of the cavity (self-induced co-flow). Numerical solutions allow the investigation of the effect of reciprocation stroke lengthand piston speed on the resulting flow patterns and momentum flows. For all investigated cases, the con-tribution made by the co-flow to the momentum flowrate is found to be small. In order to account for thesimultaneous existence of both laminar and turbulent regions, two numerical approaches were taken.One approach used the Shear Stress Transport (SST) turbulence model while the other used a newlydevised transitional turbulence model. Of particular interest was a comparison between the predictedlocations of the laminar-to-turbulent transition based on the two independent models. The excellentagreement between the two models reinforces the use of the SST model throughout the domain.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Synthetic jets can be used to provide a means of control forvehicles or for causing a directed flow within a larger fluid region.The jet is created by a vibrating wall or piston which is attached toa cavity. One wall of the cavity is an orifice plate through whichfluid is allowed to flow during the reciprocating cycles. Inspirationfor the use of synthetic jets for this purpose is taken from observa-tion of living animals which use similar jets for their locomotion[1–5].

Alternating compression and expansion strokes by the wall orpiston engender the fluid motion required for the formation ofthe pulsating jet. Throughout an entire reciprocating cycle, thereis no net mass flow out of the cavity. Despite this fact, the jet cre-ates a net flow of momentum in the direction of motion of theunderwater body.

Pioneering work on synthetic jets has provided experimentalvisualization of the flow patterns [6–9]. This body of work includessynthetic jets created by vibrating motion and by acoustic excita-tion. In [10], Schlieren and smoke tracing showed the vortexdynamics of the synthetic jet.

More recently [11,12], simulations of synthetic jets have beencompared to experiments. In [11] for instance, the solution domaindid not extend into the cavity proper. Rather, the presence of thecavity was simulated by an imposed velocity which was appliedto a replica of the orifice. In addition, the lateral extent of the cavity

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: +1 651 962 6419.ham).

was infinite so that entrained fluid was drawn in only in the lateraldirection.

A number of works have been exclusively numerical. Includedhere are calculations on a shallow, wide cavity [13] with a planargeometry and an orifice plate of infinite size. In [14], calculationsshow the effect of cavity size on the jet. In addition, interactionsbetween the synthetic jet and a crossflow were presented. Thegeometry of [14] was planar and again the orifice plate was infinitein extent.

To the best knowledge of the authors, the numerical researchreported in the literature has focused primarily on orifice plateswhose lateral dimension is infinite. The infinite extent of the orificeprohibits flow from passing around the body of the cavity andthereby eliminates the possibility of the development of co-flow.An exception to this is the work done on the effects of a syntheticjet on a bump. Synthetic jets which are created by wall bumps arepartially wall-bounded flows and exhibit entirely different flowcharacteristics than synthetic jets created by orifice cavities [15].

The present investigation makes use of information from anexperimental investigation carried out with water as the workingfluid [16]. There, a cavity was connected to a piston-cylinder de-vice, as shown in Fig. 1. The experimental results provided conclu-sive evidence of the existence of a co-flow around the exterior ofthe device. The experiments were performed with a single opera-tional protocol which is one case among many considered here.

The numerical work to be presented will provide a detailedanalysis of the fluid flow associated with a synthetic jet. The calcu-lations will investigate the effect of operating parameters on thekey features of the flow. Consideration of the transitional natureof the flow will be made and comparisons of results obtained from

Nomenclature

A areadc channel widthdo orifice widthE destruction termF1 turbulence model blending functiong gravitational accelerationhc cavity heightk turbulent kinetic energy_M momentum flowrate

p pressureP production termr radial coordinateReht transition onset Reynolds numberS turbulence model constantt timeT periodt velocity

wc cavity widthxi tensor coordinatexpiston position of pistonz axial coordinate

Greek symbolsa turbulence model constantb turbulence model constantc turbulence intermittencyq Densityr Prandtl numberx specific turbulent dissipation rate

Subscriptst turbulentc turbulence intermittencyht transition onset

1012 J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017

a purely turbulent simulation model and a newly developed tran-sitional model will be given. An assessment of the effect of the co-flow on the propulsive ability of the synthetic jet will also be made.The calculations will span a fourfold variation in both the strokelength and the reciprocation period. The effect of these variationson the fluid flow and momentum transfer will be assessed.

2. Details of the numerical solution

A generic description of the expected flow patterns is depictedin Fig. 2. Arrows show the patterns of co-flow around the cavitybody and the jet flow emanating from the orifice. Also shown are

Fig. 1. Schematic diagram of the synthetic jet apparatus and relevant operatingnomenclature.

symbolic representations of detached vortices which are createdas flow passes into and out of the cavity. The annotations ofFig. 2 show some of the key boundaries of the fluid domain whichare relevant to the computational work. As mentioned in the pre-ceding section, the channel is bounded at the top by a moving platewhich is allowed to reciprocate vertically. The other boundingwalls of the cavity and channel are nonmoving and the no-slip con-dition is enforced there. At the upper right- and left-hand sides ofthe domain, fluid is allowed to enter into the zone while at the bot-

Fig. 2. Expected flow patterns and relevant components of the solution domain.

Table 1Values of geometric parameters

Parameter Symbol Value (cm)

Cavity width wc 7.0Cavity height hc 9.9Channel width dc 2.5Orifice width do 1.9

Table 2Values of the stroke and reciprocation period

Parameter Case number1 2 3 4 5

Stroke (cm) 5.1 5.1 5.1 2.5 1.3Period (s) 0.83 1.66 3.32 0.83 0.83

J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017 1013

tom of the diagram, the fluid exit is marked where the flow leavesthe region. The right and left vertical edges are non-moving wallsand the no-slip condition is enforced there.

A listing of values of dimensional and operational parametersare set forth in Tables 1 and 2, respectively. It can be seen from Ta-ble 2 that the investigation spans a four-fold range for both thereciprocation period and the stroke length. The effect of varyingthese parameters on the flow patterns and momentum flowrateswill be assessed in Section 3 of this report.

2.1. The governing equations

The complicated flow patterns and regions of free shear virtu-ally guarantee that portions of the fluid will be turbulent. This pre-diction is reinforced by the flow separation which will occur nearthe orifice. Furthermore, the pulsating motion of the jet createdby the piston reciprocation requires the solution be carried out inan unsteady fashion. In order to accommodate these features, theunsteady form of the Shear Stress Transport (SST) model [17]was utilized. This model combines features of the popular k–emodel of [18] with the k–x taken from [19]. The combination ofthese approaches is performed in such a manner that the k–xequations dominate in the near-wall region [20] while k–e holdsaway from the wall. In this way, the advantage of the near-wall cal-culations of k–x are realized yet its sensitivity to freestream valuesof the turbulence level is mitigated [21]. It has been shown that theSST approach provides superior results for near-wall and separatedflow calculations [22–27].

The unsteady RANS equations of motion are written in tensorform as

oui

oxi¼ 0; ð1Þ

q uiouj

oxi

� �¼ � op

oxiþ o

oxiðlþ ltÞ

ouj

oxi

� �þ qgi j ¼ 1; 2; 3; ð2Þ

where lt is the turbulent viscosity. Additional transport equationsare presented for the turbulent kinetic energy, k, and the specificdissipation rate, x. Those transport equations, written in tensorform are

oðqkÞotþ oðqtikÞ

oxi¼ Pk � bqkxþ o

oxilþ lt

rk

� �okoxi

� �ð3Þ

and

oðqxÞotþ oðqtixÞ

oxi¼ aqS2 � bqx2 þ o

oxilþ lt

rx

� �oxoxi

� �

þ 2ð1� F1Þqrx21x

okoxi

oxoxi

: ð4Þ

In these expressions, q is the fluid density, r represents the Prandtlnumber associated with the transport of k or x. The terms a, S, and bare model constants, and Pk is a production term for turbulent ki-netic energy. F1 is a blending function that takes on a value of zeroaway from solid surfaces and one near surfaces. The effect of blend-ing is that the SST model behaves as k–e away from the wall and k–x within the boundary layer.

2.2. Transitional flow

Eddy-viscosity turbulence models such as the k–e, k–x, or theirderivative (SST) have been developed and evaluated for situationsin which the freestream flow was either fully laminar or turbulent.Their ability to model flow that is undergoing laminar-to-turbulenttransition has been a source of doubt for some time. This inabilityhas spawned a very extensive body of research, particularly in re-cent years which has resulted in a new approach to modeling flowthat expresses laminar and turbulent regions within a single fluiddomain.

Pioneering work such as that carried out by [28–31] has dem-onstrated various modes of laminar–turbulent transition includ-ing natural transition, bypass transition, separation-inducedtransition, and wake-induced transition. Efforts to predict thelocation at which laminar boundary-layer flow begins to breakdown has been based on empirical correlations which relate thefreestream turbulence levels to the transition Reynolds numberbased on the momentum thickness [32]. These correlations, cou-pled with the concept of flow intermittency first set forth in[33,34] have enabled a new approach to be developed in whicha new transport equation is proposed for intermittency, c [35–41]. Of the various models, the most viable for implementationin a modern CFD environment with an unstructured mesh andparallel processing, is that of [41]. That model requires the solu-tion of additional transport equations for the intermittency, c, andfor the transition onset Reynolds number, Reht. The new equationsare

oðqcÞotþ oðqticÞ

oxi¼ Pc � Ec þ

o

oxilþ lt

rc

� �ocoxi

� �ð5Þ

and

oðqRehtÞot

þ oðqtiRehtÞoxi

¼ Pht þo

oxirhtðlþ ltÞ

oReht

oxi

� �: ð6Þ

The terms Pc, Pht, and Ec, are production and destruction terms,respectively. Details for their evaluation can be obtained from [41].

For the calculations which are provided in the present report, acomparison between results obtained from the SST and the transi-tional models will be made. The comparison will demonstratewhether a traditional two-equation turbulence model such as theSST is suitable for capturing a flow that exhibits both laminarand turbulent characteristics.

2.3. Initial and boundary conditions

In order to complete the description of the fluid flow problem,specification of the flow at all boundaries is required. At all walls,the no-slip condition was enforced so that the fluid was eitherstationary or moved with the velocity of the wall. The reciprocatingmotion of the piston was sinusoidal so that

xpiston ¼Stroke

2sin

2pT

t� �

; ð7Þ

where T is the period of the motion. At the rightmost and leftmostboundaries, a wall was employed so that fluid there would be

1014 J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017

motionless. At the lower boundary, an exit condition was givenwith weak closure conditions enforced on the second derivativesin the flow direction.

Initially, at t = 0, the entire fluid region was at rest. The simula-tion spanned ten full reciprocating cycles of the piston to ensurethat timewise periodic flow was achieved.

2.4. The numerical method

The circular geometry of the cavity, channel, and the sur-rounding water was initially constructed using more than80,000 elements to resolve the spatial domain. These elementswere preferentially deployed in regions where large variationsin the transported variables were expected. In addition, thin pris-matic elements were deployed along interfaces between the fluidand solid surfaces to ensure proper resolution of the boundarylayers.

The unsteady computations were performed with 100 time-steps for each reciprocating cycle. All timesteps contained 10 iter-

Fig. 3. SST velocity magnitude contours obtained for as the piston is (a) moving downwa

ations, each of which was performed using a two-step, multi-gridcomputational algorithm. The time stepping was carried out usingan Euler backward scheme of second-order accuracy.

Coupling of the velocity–pressure equations was achieved on anon-staggered, collocated grid using the techniques developed byRhie and Chow [42] and Majumdar [43]. The inclusion of pres-sure-smoothing terms in the mass conservation equation sup-presses oscillations which can occur when both the velocity andpressure are evaluated at coincident locations.

The advection term in the momentum equations was evaluatedby using the upwind values of the momentum flux, supplementedwith an advection-correction term. The correction term reducesthe occurrence of numerical diffusion and is of second-order accu-racy. Further details of the advection treatment can be found in[44].

Mesh and time-step values were sufficiently small to ensure asolution that was independent of their values. The selected valuesresulted from a thorough independence study during which boththe element sizes and time steps were halved and results were

rds, (b) at bottom-dead center, (c) is moving upwards, and (d) is at top-dead center.

J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017 1015

compared. When the sequential reductions failed to yield notice-able changes in the results, it was determined that the settingswere sufficiently refined.

3. Results and discussion

3.1. Comparison of SST and transitional results

Fig. 3a–d show a series of global velocity magnitude contourdiagrams. The figures correspond to results obtained using theSST model for turbulent simulation. In Fig. 3a, the piston is passingdownward through its center point, and a mass of fluid is seen tobe ejected from the cavity at relatively high speed as indicatedby the contour. In Fig. 3b, the fast-moving fluid is seen to be de-tached from the orifice and is moving downwards, away fromthe cavity. In Fig. 3c the piston is traveling upwards, drawing fluidinto the cavity from the surrounding region. A remnant jet of fluidis seen to be continuing its downwards motion, away from the cav-

Fig. 4. Transition velocity magnitude contours obtained for as the piston is (a) moving dcenter.

ity. Finally, when the piston is at a top-dead-center position inFig. 3d, a fast-moving mass of fluid is observed being pulled intothe cavity. An overall view of the velocity patterns from Fig. 3 de-pict the oscillating inflow and outflow during an entire reciprocat-ing cycle. The contour is scaled by the legend at the left-hand sideof the figures.

Fig. 4a–d show the counterpart results based on the transi-tional model. An examination of Figs. 3 and 4 reveals a close sim-ilarity between the results for the SST and transitionalapproaches. This similarity includes both the flow patterns andvelocity magnitudes inside and outside of the cavity space. Theseresults strongly support the congruence of the SST and transi-tional models. Similar agreement was seen with the other casesindicated in Table 2. As a consequence, the remaining discussionwill solely make use of results obtained from the SST model. Thatdiscussion will focus on the mass and momentum flowratesengendered by the synthetic jet throughout an entire reciprocat-ing cycle.

ownwards, (b) at bottom-dead center, (c) is moving upwards, and (d) is at top-dead

Table 5Momentum flowrate imparted to the co-flow during one cycle

Case Momentum flowrate per cycle(N)

Average momentum flowrate per second(N/s)

1 1.31 � 10�4 1.58 � 10�4

2 6.14 � 10�5 3.70 � 10�5

3 4.94 � 10�5 1.49 � 10�5

4 1.28 � 10�5 1.54 � 10�5

5 1.34 � 10�6 1.61 � 10�6

1016 J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017

3.2. Mass and momentum flowrates through the orifice

While conservation of mass guarantees that there can be no netmass efflux from the cavity, the synthetic jet is capable of generat-ing a net momentum flow in the streamwise direction. The netmomentum flow is a critical measure of the propulsive capabilityof the synthetic jet device.

The total momentum imparted to the fluid during a cycle wascalculated by timewise integration of the local momentum varia-tions at the orifice. The integral, expressed in both its integral formand its numerical analog is

_Morifice ¼Z T

0

ZOrifice Area

qt2 dA� �

dt ¼Xt¼T

t¼0

XOrifice Area

qt2i DAi

" #Dti; ð8Þ

where q is the fluid density, t is the streamwise component of thefluid velocity, and DA is the horizontal projection of the area ofthe elements which span the orifice.

A tabulation of resulting momentum flowrates is presented inTable 3. The results are keyed to the five cases set forth in Table2. In all cases, it was found that there was a net momentum flowin the streamwise direction. This finding is verified from experi-mental measurements made in [16]. The table presents two setsof results. The first results list the total momentum flowrate pass-ing through the orifice throughout an entire reciprocation of thepiston. The second set shows the time-averaged rate of momentumtransfer to the fluid. From the tabulated results, it is seen that thevariation of momentum flowrate very nearly follows the behavior

_Morifice �stroke length2

Period2 : ð9Þ

The dependence of momentum flowrate to the operating parame-ters indicated in Eq. (9) agrees qualitatively with results of experi-mentally observed in [16].

3.3. Mass and momentum flowrates through the co-flow region

The numerical simulation clearly shows a directed streamwiseflow in the co-flow region throughout the entire period of recipro-cation. This finding is in contradistinction to the observationsmade at the orifice where the direction of flow reverses with thedirection of the piston. As a consequence, it is relevant to calculatethe mass flowrate through the co-flow region throughout thereciprocation. The resulting mass flowrates are shown in Table 4.The table displays two sets of results. First, the total mass passingthrough the co-flow region for one cycle is presented. Next, the

Table 3Momentum flowrate of fluid passing through the orifice throughout one cycle

Case Momentum flowrate per cycle (N) Momentum flowrate per second (N/s)

1 4.1 � 10�3 5.0 � 10�3

2 2.0 � 10�3 1.2 � 10�3

3 9.5 � 10�4 2.9 � 10�4

4 9.5 � 10�4 1.1 � 10�3

5 2.0 � 10�4 2.4 � 10�4

Table 4Mass passing through the co-flow region

Case Mass flow per cycle (kg) Mass flow per second (kg/s)

1 0.0932 0.1122 0.0914 0.05503 0.137 0.04144 0.0185 0.02225 0.000509 0.000614

respective masses are divided by the corresponding time periodsto obtain the average mass flowrate for the individual cases. Itcan be seen from the results that the effect of an extended timeperiod (Cases 2 and 3) is to reduce the average mass flowrate. Like-wise, a shorter stroke length gives rise to lesser mass flowrates. It isalso seen that the mass flowrate is substantially more sensitive tostroke length than to reciprocation period.

Next, attention is turned to the contribution of momentum bythe entrained fluid in the co-flow region. The calculation of theco-flow momentum was performed by the numerical integrationpresented in Eq. (10). There, the timewise variation of the momen-tum flow was integrated spatially and over time to give the resultsshown in Table 5. Also shown in the table are the average momen-tum flow rates.

_Mco-flow ¼Z T

0

ZCo-flow Area

qt2 dA� �

dt

¼Xt¼T

t¼0

XCo-flow Area

qt2i DAi

" #Dti: ð10Þ

Inspection of the table reveals key results regarding the impor-tance of the co-flow region in overall momentum transfer. First,a comparison of Tables 3 and 5 show that for all cases, the contri-bution to the momentum transfer by the co-flow is seen to bemuch smaller than the jet flow. This finding suggests that themomentum flowrate in the co-flow region can be neglected withrespect to its propulsive contribution. However, it raises the possi-bility that modifications to the cavity shape may enable anenhancement of the contribution of the co-flow to the overallmomentum transfer.

In addition, the results show the effect of reciprocation periodand stroke length on momentum transfer. It is seen that whenthe time period increases (Cases 2 and 3), the momentum flowratedecreases due to the lower piston velocity. Also, when the strokelength decreases (Cases 4 and 5), the momentum flow decreases.The sensitivity of the momentum flowrate to stroke length is sub-stantially greater than the sensitivity due to period. In fact, the var-iation of co-flow momentum with respect to operating parametersdiffers from that of the orifice momentum.

4. Concluding remarks

A comprehensive numerical simulation was performed toinvestigate the fluid flow patterns associated with a synthetic jet.A reciprocating piston-cylinder assembly attached to a cavity cre-ates a pulsating synthetic jet which passes through an orifice plate.The calculations set forth here shows evidence of the existence of aco-flow in the region surrounding the cavity. The co-flow, which iscreated by entrainment of fluid into the emerging jet was investi-gated in terms of its overall mass flowrate and its contribution tothe momentum imparted to the fluid by the jet.

The simulations showed the agreement between fully turbulentcalculations with calculations made using a new two-equationlaminar–turbulent transition model. The assessment was basedon global and local velocity patterns.

J. Abraham, A. Thomas / Computers & Fluids 38 (2009) 1011–1017 1017

The complete calculations were carried out using a range ofoperating parameters. The effect of operating parameters was clas-sified by reciprocation period and piston stroke length. It was seenthat both at the orifice and in the co-flow region, the momentumtransfer depended strongly on these parameters. In both regions,the momentum flow decreased when the period increased or whenthe stroke length of the piston decreased. In general, the momen-tum flow was more sensitive to changes in the stroke length thanto changes in the time period.

Acknowledgement

Support of H. Birali Runesha and the Supercomputing Institutefor Digital Simulation & Advanced Computation at the Universityof Minnesota is gratefully acknowledged.

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