ABO Paper02 Momentum

8/11/2019 ABO Paper02 Momentum

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Theor. Comput. Fluid Dyn.DOI 10.1007/s00162-009-0173-y

O R I G I N A L A R T I C LE

Ali B. Olcay Paul S. Krueger

Momentum evolution of ejected and entrained fluid duringlaminar vortex ring formation

Received: 2 July 2008 / Accepted: 24 September 2009 Springer-Verlag 2009

Abstract The evolution of total circulation and entrainment of ambient fluid during laminar vortex ringformation has been addressed in a number of previous investigations. Motivated by applications involvingpropulsion and fluid transport, the present interest is in the momentum evolution of entrained and ejected fluidand momentum exchange among the ejected, entrained fluid and added mass during vortex ring formation.To this end, vortex rings are generated numerically by transient jet ejection for fluid slug length-to-diameter(L/D)ratios of 0.53.0 using three different velocity programs [trapezoidal, triangular negative slope (NS),and positive slope (PS)] at a jet Reynolds number of 1,000. Lagrangian coherent structures (LCS) were uti-lized to identify ejected and entrained fluid boundaries, and a Runge-Kutta fourth order scheme was used foradvecting these boundaries with the numerical velocity data. By monitoring the center of mass of these fluidboundaries, momentum of each component was calculated and related to the total impulse provided by thevortex ring generator. The results demonstrate that ejected fluid exchanges its momentum mostly with addedmass during jet ejection and that the momentum of the entrained fluid at jet termination was


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A. B. Olcay, P. S. Krueger

applications including flow control by synthetic jets[37], fluid mixing and heat transfer [26], and propulsion[3,6,16], respectively. Several studies have revealed aspects of the transport processes associated with vortexring formation and advection.

The roll-up of the vortex sheet and the connection of this process with the circulation of the resulting

ring has been studied experimentally, numerically, and analytically [10,13,24,30]. It is well recognized thata dominant factor controlling the circulation of the vortex ring is the magnitude of the jet velocity, owingto its control over strength of the jet shear layer, giving the so-called slug model for predicting vortex ringcirculation entirely from the jet axial velocity [12,36]. As apparent from the detailed measurements of Didden[10], however, transients during jet initiation lead to a non-uniform jet flow and a disproportionately largecontribution to circulation from the initial jet transient, which is not predicted by the slug model. Krueger [15]proposed a simple model to account for the initial transient based on over-pressure at the nozzle exit planeduring jet initiation for rapidly initiated jets. Approximations made in the model make it inapplicable to slowlyinitiated jets, but based on the model premise one can anticipate that slowly initiated jets will not experiencethe same disproportionate circulation increase at initiation due to lower nozzle exit over-pressure.

The roll-up of the jet shear layer is also responsible for entrainment of ambient fluid into the forming ringby Biot-Savart induction [1,26] and as such, the evolution of vortex ring circulation is directly relevant to masstransport by vortex rings. The dye visualization of Didden [10] allows a qualitative assessment of the evolutionof fluid entrainment during shear layer roll-up, showing a tight spiral initially with little entrained fluid, whichlater opens up and allows a large amount of fluid to be entrained into the ring boundary immediately followingjet termination. Olcay and Krueger [26] used dye visualization and Lagrangian coherent structures (LCS)obtained from digital particle image velocimetry (DPIV) data to identify the vortex ring boundary and quan-titatively assess ambient fluid entrainment during vortex ring formation. Their results show a nearly twofoldincrease in the rate of fluid entrainment into the ring after jet termination as compared to during jet ejection.The variation of the jet velocity with time (jet velocity program) significantly affected the observed trend andthe final fraction of ambient fluid present in the vortex ring bubble. Specifically, the fraction of ambient fluidentrained into the ring could be increased by as much as 30% for velocity programs with a rapid initiation andgradual termination as compared to velocity programs with more gradual initiations and rapid terminations,even though both the slug model predictions and actual DPIV measurements of final ring circulation were thesame for both cases. The entrainment benefit afforded by the rapidly initiated jets is attributable to the thin-ner shear layer (tighter vortex spiral) and increased initial vorticity flux characteristic of these jets. Recently,Shadden et al. [35] also studied transport and stirring features during vortex ring formation using LCS withexperimental and numerical data to monitor vortex ring evolution throughout the formation process. Usingthese results, they were able to measure overall entrainment quantitatively and to investigate stirring charac-teristics of vortex rings and stretching of interfaces between fluid regions as the vortex ring forms. Followingring formation, ambient fluid continues to be entrained into the vortex ring by viscous diffusion of vorticity,which has been quantified and modeled in previous studies[8,20].

In addition to circulation and mass transport, the total impulse supplied by the jet to the flow (IT)is trans-ported downstream by the vortex ring and can be related to the fluid momentum associated with the vortexring as

IT = (mEJ + mE + mA)UR, (1)

wheremEJ, mE, andmAare ejected, entrained, and added mass of the vortex ring, respectively, andURis thevelocity of the vortex ring [9,16]. For starting jets, ITcan also be expressed as

IT = IU + IP, (2)

where

IU =

tp0

A

u2J(r, t)dAdt (3)

is the impulse supplied by the jet momentum and

IP =

tp0

A

(p(r, t) p)dAdt (4)


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Momentum evolution of ejected and entrained fluid

is the impulse due to nozzle exit over-pressure[16]. In Eqs.3and4,uJ(r, t)andp(r, t)are the jet velocity andpressure at the nozzle exit plane, respectively, and tpis the jet duration.

1 The pressure impulse, IP, providesadditional impulse over what would be expected from the jet momentum alone, which leads to thrust augmen-tation in pulsed jets [6,17] and improved propulsive efficiency in pulsed jet propulsion utilizing short jet pulses

that yield isolated vortex rings with no trailing jets [3,4,23]. Relating Eqs.1and2,Krueger and Gharib [16]reasoned thatIPis related to the acceleration of entrained andadded mass during vortex ring formation, but thissort of before-and-after comparison cannot resolve the relative contributions of the entrained and added massor the phase of the ring formation process most responsible for their contribution. Nevertheless, understandingthe evolution of the added massand entrained fluid momentum may help optimize propulsive benefitsobtainedthroughIP. Moreover, for applications such as mixing one may be interested in knowing the impulse requiredto move m Edownstream. Is a significant increase in total impulse required to entrain mEor does m Estealsome of the momentum of the ejected fluid? For applications where advection of the entrained fluid, mE, is thegoal (e.g., convective cooling), entrainingmEwith minimum additional input from the vortex ring generatorwould be desirable.

Several studies have observed the relationship between the momentum gained by fluid entrained aftervortex ring formation is complete and the motion of the vortex ring. During this phase of motion the ring slowsas additional mass is entrained due to conservation of momentum [2,21,31]. These studies do not address thering formation process or quantify the momentum of the entrained fluid. The vortex ring entrainment resultsof Olcay and Krueger[26] provide a quantitative perspective of entrainment during the formation process andsuggest that the entrained fluid obtains most of its momentum following jet termination because the largestquantity of ambient fluid is entrained during this period. This is certainly not conclusive, however, since thering velocity is highest just after jet termination (suggesting the fluid already entrained during jet ejection hashigh momentum) and the fluid entrained after jet termination was already set in motion by the ring forma-tion. Clearly, Lagrangian tracking of the ejected and entrained fluid domains during vortex ring formation isrequired to fully assess the evolution of their momenta. A few studies[2729] have considered the momentumassociated with vortices in animal wakes for analysis of locomotion. While effective and reasonably general,these methods consider the vortex as a whole and do not allow segregation of momentum associated withejected and entrained fluid as observed in starting jet flows.

While previous studies have addressed circulation and entrainment evolution during vortex ring forma-tion and subsequent advection, little information is available on the evolution of ejected and entrained fluidmomenta during vortex ring formation or the fraction of ITrelated to entrained fluid. The objective of thisstudy is to investigate the ejected and entrained fluid momentum during vortex ring formation, determine therelative momentum contribution ofmEJcompared tomEandmA, elucidate the momentum exchange process,and analyze how this may be leveraged to optimize IPand minimize the impulse required to entrain fluid byadjusting parameters controlling vortex ring formation such as jet velocity program. To this end, vortex ringformation by transient jet ejection from a tube is studied numerically to obtain velocity data inside and outsidethe tube. LCS results obtained from the CFD data are used to identify the ejected and entrained fluid regions atjet initiation (t = 0). Then, bymonitoring thecenterof mass (CM) of these fluid regions,momentum evolutionof the ejected and entrained fluid is obtained and related to the fully developed vortex rings momentum.

In Sect.2the numerical model and methods are briefly described. In Sect.3the time evolution of ejectedand entrained fluid is presented for three velocity programs and jet slug length-to-diameter (L/D)ratios inthe range between 0.5 and 3.0 are discussed. In Sect.4ejected and entrained fluid momenta are discussed incomparison to the added mass effect.

2 The numerical model

Transient jet ejection from a tube is simulated using the domain geometry shown in Fig.1. The geometry isaxisymmetric, therefore, only the x rplane is shown in Fig.1.The domain dimensions in the x rplanewere chosen as 15Din length and 3.5Din height, where D is the tube diameter, to ensure negligible domainsize influence on the flow evolution. Tube length was arbitrarily set to 5D.

On the outer boundary, ambient pressure conditions and zero normal derivatives of velocity were specifiedfor the left, right, and top boundaries in Fig. 1. These outflow conditions yielded faster solution convergence for

1 Strictly speaking, the upper limit of time integration in Eqs.3and4should be taken tot > tpwheret is sufficiently large

that the vortex ring has convected away from the nozzle exit plane by this point. For the Reynolds numbers and jet slug length todiameter (L/D)ratios of interest in this investigation, however,IPand IUare determined primarily during jet ejection, so Eqs.3and4are utilized here.


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5D15D

0.5D

3.5D

Wall

Outlet

Axis

Nozzle exitplane

x

rUP(t)

Fig. 1Solution domain and boundary conditions

each time step and the results showed negligible difference (


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t / tP

UP

(t)/U

M

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

(a)

t / tP

UP

(t)/UM

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

(b)

t / tP

UP

(t)/U

M

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

(c)

Fig. 2Normalized velocity programs for L/D = 2.0:atrapezoidal,btriangular NS, andctriangular PS velocity programs

The governing equations were solved using a finite volume method with a fixed time step. The solver schemewas second order accurate for both time and space. The SIMPLE algorithm was used for pressurevelocitycoupling and the QUICK scheme was used for discretization of the convective terms.

The domain wasdiscretized into388120 nodes (see Fig. 3) in the axial and radial directions, respectively.A total of 46,560 quadrilateral cells were employed with increased mesh density near the nozzle exit planeand the tube wall region to properly resolve wall gradients during the formation process.


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Fig. 3Sample mesh used during the study. 388120 nodes were used in the axial and radial directions, respectively. Only everyfifth mesh point is shown

Table 1Space and time convergence for the trapezoidal velocity program, R eJ = 1,000, and L/D = 2.0 att = 2.5

ITUMD3

% Difference

Mesh resolution258 80 1.6878 1.834388 120 1.6591 0.103582 180 1.6574 0

Time step0.04 1.6310 1.7590.02 1.6557 0.2710.01 1.6602 0

Table 2Summary of the tested cases

L/D Velocity program

2.0 Triangular NS2.0 Triangular PS0.5, 1.0, 2.0, 3.0 Trapezoidal

ReJ = 1,000 for all cases

Because of the present focus on impulse, the hydrodynamic impulse [18,22] was used to monitor spaceand time convergence since it represents the total impulse required to generate the flow. The hydrodynamicimpulse was computed as

IT =

r

2dxdr, (8)

whereis the azimuthal component of vorticity. The integration domain was chosen as 0 x/D 10 and

0 r/D 3.5.Mesh convergence was tested for the trapezoidal velocity program,L/D = 2.0, and t= tUM/D = 0.02

on meshes with resolutions 258 80, 388 120, and 582 180. ITwas computed from the vorticity fieldat t = 2.5, where t is non-dimensional time defined as t/tP. The results given in Table1show that the388 120 mesh (given in Fig.3) is sufficient for this study since this mesh provided 0.103% difference in thetotal impulse compared to the total impulse for the 582 180 mesh.

For time-step convergence, simulations with t = 0.01, 0.02, and 0.04 on the 388 120 mesh wereperformed for the trapezoidal velocity program with L/D = 2.0. The results for ITare shown in Table1. The0.02 time step was chosen since this time step provided 0.271% difference in the total impulse compared tothe total impulse for the 0.01 time step.

As L/Dand the velocity program are key controlling parameters in the vortex ring formation and fluidentrainment process, a wide range of L/Dand velocity programs were studied. A list of cases simulated isshown in Table2.


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2.2 Lagrangian coherent structures (LCSs) implementation

Olcay and Krueger [26] and Shadden et al. [33,34] briefly discuss the use of LCS. In the present study, theEulerian CFD data were used to obtain Lagrangian information about the flow as expressed using the finite

time Lyapunov exponents (FTLE). Details of this approach may be found in Shadden et al.[33]; however, abrief overview following Olcay and Krueger [26] will be given here for completeness.The equation describing the trajectory of a fluid particle at positionx0at timet0may be expressed as

x(t; t0, x0) = V(x(t; t0, x0), t), (9)

wherex(t0; t0, x0) = x0. The right side of Eq. 9 can be obtained from the CFD velocity field data. The solution

to (9) is a flow map (t0+Tt0

(x0))describing the position at timet= t0 + Tof the fluid particle initially at x0at timet0, namely,

t0+Tt0

(x0) = x(t0 + T; t0, x0) (10)

Then the FTLE is defined as

T

t0(x)

1

|T|ln max, (11)

wheremaxis the maximum eigenvalue of

t0+Tt0

(x)

t0+Tt0

(x)

(12)

and() denotes the adjoint operation. It can be shown [33] that the separation of particles advected by the flow

is proportional toeTt0

(x)|T|to leading order. Hence, the FTLE is roughly a measure of the maximum expansion

rate of particle pairs advected by the flow. For example, strong and convex peaks (ridges) in the FTLE fieldcorrespond to the stable (for T> 0)and unstable (for T< 0)manifolds of hyperbolic fixed points in steadyflows (separatrices).

Generalizing to aperiodic and unsteady flows, LCS are defined as the ridges in the FTLE field. Shadden

et al. [33] show that the flux across a LCS scales like 1|T|

and thus, for large |T| can be treated as a material

line, transport barrier, or separatrix in the flow. LCS can be obtained for both forward and backward time inte-gration of the velocity field data (e.g., right side of Eq.9). The resulting LCSs can be regarded as the unsteadygeneralization of the stable (i.e., T>0, also called repelling LCS) and unstable manifolds (i.e., T0, the boundaries of these regions were advected using a Runge-Kutta fourthorder scheme [5] to determine the time evolution of the ejected and entrained fluid.


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Fig. 4Color contour plot of the FTLE field withT= 10s for anL/D = 2.0 trapezoidal velocity program at t = 0

x /D

r/D

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

Nozzle wall

AttractingLCS

RepellingLCS

Fig. 5Ejected and entrained fluid boundaries att= 0 for the trapezoidal velocity program with L/D = 2.0

As thedomain boundaries were stretchedduringadvection, the distance between points forming theejectedor entrained fluid boundaries were monitored and additional points were added where excessive stretchingoccurred. The maximum allowable distance between points, zMAX, was adjusted to provide sufficient res-olution without excessive points. The appropriate zMAXwas chosen based on a convergence study. To testthe convergence, the entrained fluid was advected using zMAX= 0.05D, 0.025D, and 0.01D. Total volumeconservation and time evolution of the axial location of the center of mass (CMx) were monitored during thisprocess. The results are plotted in Fig.6.The center of mass axial location was calculated using

CMx=

x f(x)dx f(x)dx

, (13)

where f(x) represents the dimensionless area obtained by revolving a slice of the fluid domain of interest aboutthe tube centerline. The integration was taken over the fluid domain of interest (e.g., ejected fluid or entrainedfluid). Based on the results shown in Fig.6, zMAX =0.025Dwas chosen for further calculations since itprovided constant volume with < 1% deviation from the 0.01Dcase and CMxproved nearly insensitive tozMAXover the tested range.

Once CMxwasobtained, theaxial velocity of CMxwascalculatedusing a secondorder centraldifferencingmethod [5] for a constant-time step. The computed velocity of CMxhad second order truncation error.


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t / tP

VE

/V

ME

0 0.5 1 1.5 2 2.5 30.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

0.01D

0.025D

0.05D

(a)

t / tP

CM

X/D

0 0.5 1 1.5 2 2.5 3

0.5

1

1.5

2

2.5

0.01D

0.025D

0.05D

(b)

Fig. 6Fluid domain boundary convergence test forzMAX=0.05D, 0.025D, and 0.01D.aVolume change of entrained fluid

with thetime; b time evolutionof CMxforthe entrainedfluid. Theresults arefor thetrapezoidal velocity program withL/D = 2.0.VMEis the entrained fluid volume at t = 2.5

3 Results

3.1 Time evolution of ejected and entrained fluid

Figure7illustrates the evolution of ejected and entrained fluid for L/D =2.0 and the trapezoidal velocityprogram. The ejected fluids shape initially looks like a cylinder as one would expect; however, the back edgeis not straight like the front edge. This actually indicates that fluid near r= D/2 is not pulled into the ringtoward the end of the jet pulse. As the flow evolves, the entrained fluid domain is swept over the developingvortex ring spiral by Biot-Savart induction and enters the spiral through the rear such that the union of the twofluid domains gives the expected ellipsoidal shape to the vortex ring boundary in Figs. 7e, 8. The initial location

and time evolution of the ejected and entrained fluid domains agrees qualitatively with Peng and Dabiri [27]although Peng and Dabiri do not investigate the two domains separately.

The precise overlapping of the domain boundaries illustrated in Fig.8confirms the accuracy of the advec-tion scheme. As further confirmation of the results, comparing the final volume of the vortex ring with theexperimental results ofOlcay andKrueger [26] (forReJ = 1,000,L/D = 2.0 andtrapezoidal velocity program)gives0), which is a consequence of the


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x /D

r/D

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

Nozzle wall

Entrained fluiddomain

Ejected fluiddomain

Nozzle exitplane

(a)

x /D

r/D

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

Nozzle wall

Entrained fluiddomain

Ejected fluiddomain

(b)

x /D

r/D

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

(c)

x /D

r/D

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

(d)

x /D

r/D

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.5

1

1.5

(e)

Fig. 7Evolution of ejected and entrained fluid. Panelsaeshow the ejected and entrained fluid at t = 0.0, 0.5, 1.0, 1.5, and2.0, respectively, for the trapezoidal velocity program with L/D = 2.0. Thered solid lineandblue dash-dot linerepresent ejectedfluid and entrained fluid domains, respectively


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x /D

r/D

0.5 1 1.5 2 2.5

0

0.2

0.4

0.6

0.8

1

Fig. 8Enlarged view of ejected and entrained fluid regions att = 2.0 for the trapezoidal velocity program with L/D = 2.0

x /D

r/D

-0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

Trapezoidal

Triangular NS

TriangularPS

Fig. 9 Shape of the entrained fluid regions at t = 0 for the trapezoidal, triangularNS, and PS velocity programs withL/D = 2.0

stronger overall Biot-Savart induction provided by the strong initial shear layer associated with the rapid jetinitiation for these cases.

Figure10illustrates the initial shape of the entrained fluid regions for the trapezoidal velocity programwith L/Dof 0.5, 1.0, 2.0, and 3.0. As in Fig. 9,most of the entrained fluid initially resides in front of the

nozzle exit plane. In fact, as the volume of entrained fluid increases with L/D, the fraction of it residing infront of the nozzle increases until forL /D = 3 the largest portion of entrained fluid comes from very near tothe nozzle centerline. This trend may seem at odds with the fact that entrainment into the ring occurs from therear of the forming spiral and closer to the nozzle lip, but it results from the increased effect of Biot-Savartinduction sweeping the entrained fluid over the vortex spiral for larger L /Ddue to the larger ring circulationand longer formation period as L /Dis increased.

3.3 Impulse evolution of ejected and entrained fluid

Figure11shows the ejected and entrained fluid momentum evolution of a vortex ring from jet initiation to thesteadily translating state for the trapezoidal velocity program with L/D =2.0. Axial ejected and entrainedmomenta (IEJand IE) were calculated using


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x /D

r/D

-0.2 0 0.2 0.4 0.60

0.2

0.4

0.6

0.8L /D= 0.5L /D= 1.0L /D= 2.0L /D= 3.0

Fig. 10Shape of the entrained fluid regions att = 0 for the trapezoidal velocity program with L/D = 0.5, 1.0, 2.0, and 3.0

t / tP

I(t)/IT

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.2

0.4

0.6

0.8

1

1.2

IE

IEJ

IR

= IE

+IEJ

Added mass momentum

Fig. 11Time variation of ejected and entrained fluid momenta for the trapezoidal velocity program withL/D = 2.0

IEJ = mEJUEJ(14)

IE = mEUE,

whereUEJand UErepresent the axial velocities of the center of masses of the ejected and entrained fluids,respectively. The results are normalized using ITcomputed att

=2.5, which is the point where ITreachesits final, steady value (i.e., the ring is fully developed and boundary effects are minimal). Figure11indicates

that the bulk of the ejected fluid impulse is obtained during the initial acceleration similar to an isolated rigidparticle. Subsequently, IEJdecreases as its momentum is shared with the entrained mass and external fluid(i.e., fluid not in either the ejected or entrained fluid domain). The latter represents what eventually is the addedmass component, and is clearly more significant than the entrained mass momentum because IEJdecreasesfaster than IEincreases for t


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t / tP

IE/IT

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.05

0.1

0.15

0.2

0.25TrapezoidalTriangular NSTriangularPS

Fig. 12Time variation of entrained fluid momentum for trapezoidal, triangular PS, and NS velocity programs withL/D = 2.0

t / tP

IE/IT

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3L /D= 0.5L /D= 1.0L /D= 2.0L /D= 3.0

Fig. 13Time variation of entrained fluid momentum forL/D = 0.5, 1.0, 2.0, and 3.0 for the trapezoidal velocity program

that entrainment contributes very little to any impulse benefit provided by pulsation. On the other hand, thisalso indicates that the momentum cost to transport entrained fluid with the ring is small.

Figure12illustrates the time variation of IEfor the three studied velocity programs. Note that IEfor thetriangular PS case is negligible until t =0.75. On the other hand, for the triangular NS case IEincreasesthroughout the pulse duration. Thus, a rapid initial acceleration (e.g., trapezoidal and triangular NS velocityprogram) immediately impartsmomentum to the entrainedfluid, anda gradual acceleration delays thisprocess.

Somewhat counter intuitively, Fig.12also shows that IErises rapidly when UP(t)begins decelerating. Thisobservation is related to the more rapid fluid entrainment into the ring following jet termination[26].

The entrainedfluid momentum variationfordifferent stroke ratios is given in Fig. 13. Clearly the maximumIEis a larger fraction of the total impulse as L/Ddecreases, except for L/D =0.5. Figure13also showsthat entrained fluid momentum for all of the L/Ds except for L/D =3.0 makes a dip during jet ejection.This indicates that the entrained fluid for these cases is slowed (or even moves backward as for L /D =0.5,i.e., the entrained fluid gets pushed back during jet start up) by its interaction with the forming ring. During0.9 t 1.0, a kink appears in entrained fluid momentum plot (Fig.13). This is due to the jet terminationwhich pulls the fluid around the nozzle lip. As in Fig.12, IEincreases significantly for all L/Ds following jetdeceleration.

Figure14gives summary of IE/ITvalues versus IU/(UMD3)for all the studied cases when the jet is

terminated (i.e., t= 1.0). The value of IEat this instant is most relevant for assessing the input requiredfrom the jet to entrain mEsince the flow evolves according to conservation of momentum fort

>1. The jet


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IU

/(UMD

3)

IE/IT

0 0.25 0 .5 0 .75 1 1.25 1 .5 1 .75 2 2.250

0.05

0.1

0.15

0.2

0.25

0.3

0.35Trapezoidal andL/D= 2.0

TriangularPS andL/D = 2.0

Triangular NS andL/D= 2.0

Trapezoidal L/D= 0.5

Trapezoidal andL/D= 1.0


Fig. 14 IE/ITversusIU/(UMD3)for the trapezoidal velocity program with L/D = 0.5, 1.0, 2.0, 3.0 and triangular NS and PS

with L/D = 2.0 att = 1.0

Fig. 15 FTLE fields illustrating entrained fluid in front of the nozzle for the trapezoidal velocity program with L/D =2.0 att = 0.0

momentumIUis chosen as the abscissa in Fig.14since it is an indicator of the bulk effect of the jet for a givenstroke ratio and velocity program combination. Despite variations in the maximum IEobtained (see Fig.13),Fig. 14 shows thatIE/ITat t

= 1.0 is nearly insensitive to the generatingconditions for nearly all cases testedexcept for the triangular NS case. Triangular NS case produces an almost two times larger IE/ITvalue than

other cases due to the increased rate of entrainment occurring near the beginning of jet initiation for this case.In all cases, however, IE/ITis small, indicating the momentum cost of entraining fluid is generally small.

Returning to the overall flow evolution, Fig.11also shows the momentum of the fluid inside the ellipsoidalvortex ring bubble (see Fig.7e), namely, IR IEJ + IE. (The evolution of IRand IEJfor the other casessimulated can be found in Olcay [25].) This ring momentum excludes the momentum associated with addedmass. Figure11shows that IRdecreases fort

2.5. This is due to additional ambient fluid entrainment byvorticity diffusion as documented by Maxworthy[20,21], Dabiri and Gharib [8] and Olcay and Krueger[26].The fluid entrained for t 2.5 initially resides in front of the nozzle near the jet centerline as illustrated bythe FTLE field in Fig. 15 (note the ridge near r/D = 0.2 for 0.0 x/D 4.0). As the ring translates forward,this fluid is swept over the top of the ring by Biot-Savart induction and entrained into the back of the ring. Asthis entrainment follows the mechanism described by Maxworthly [20] and Dabiri and Gharib [8] and occursprimarily after the ring is completely formed, we refer to this fluid as fluid entrained by viscous diffusion.Although this additional fluid steals some of the ejected fluids momentum, the transfer occurs primarily after


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IU

/(UMD

3)

IA/IT

0 0.25 0 .5 0 .75 1 1.25 1 .5 1 .75 2 2.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7


TriangularPS andL/D= 2.0


TrapezoidalL/D= 0.5



Fig. 17 IA/ITversus IU/(UMD3)for the trapezoidal velocity program with L /D =0.5, 1.0, 2.0, 3.0 and triangular NS and

PS with L/D = 2.0 att = 2.5

IU

/(UMD

3)

0 0.25 0 .5 0.75 1 1.25 1 .5 1 .75 2 2.250

0.25

0.5

0.75

1

1.25

1.5

1.75


TriangularPS andL/D= 2.0


Trapezoidal L/D= 0.5


Trapezoidal andL/D

= 3.0

Fig. 18 att = 2.5 versus IU/(UMD3)for all cases considered

In order to quantitatively assess the influence of the entrained and added mass components on IPand theassociated influence of the ring generating conditions (L/Dand velocity program), it is observed that

IP = IE + IA + IEJ IU = (IE + IA) (IU IEJ) (17)

(see Eqs.1and2). Physically, IE + IAis the momentum gained by the ambient fluid and is the resultingoutput of the jet action related to IP. Conversely, IU IEJis the jet momentum contribution, or input, to

the ambient fluid momentum gain. Hence, a useful measure of output to input for the ambient fluid momentumcontribution to IPis

= IE + IA

IU IEJ. (18)

Based on Eq.17,it is desirable to maximize the performance metric. In particular, maximizingmaximizesthe pressure impulseper unit of jetmomentum contributedto the acceleration of ambient fluid (IP/(IU IEJ)).Figure18plots evaluated at t =2.5 for all cases considered. From Fig.18it is observed that increasesas L /Dis increased, indicating that higher a stroke ratio provides a stronger relative contribution to pressureimpulse and ambient fluid momentum. The rate of increase slows, however, asL/Dincreases and the forma-tion number [11] is approached, consistent with the results of Krueger and Gharib[16]. Comparing the opensymbols shows that the trapezoidal and triangular PS velocity programs exhibit higher values compared tothe triangular NS velocity program for the same L/D, indicating that rapid decelerationvelocity programs


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provide higher ambient fluid impulse per amount of momentum transferred from ejected fluid. This result issomewhat surprising as a rapid initial acceleration might seem to have the stronger effect as it promotes rapidacceleration of the ambient fluid. Therapid deceleration of the trapezoidal and triangular PS velocity programs,however, prevents as much IUfrom being used to accelerate this fluid and is higher for these cases. In any

case, the effect of velocity program on is small for the cases considered.

4 Conclusions

Ejected and entrained impulse calculations were performed to investigate each terms contribution to theformed ring impulse. CFD combined with LCS were utilized to identify initial fluid boundaries. Then, thesefluid boundaries were advected via fourth order Runge-Kutta integration scheme. The effect of velocity pro-grams and L/Don the evolution and momentum of the ejected and entrained fluid domains were investigated.

The velocity program used to generate a single jet pulse played a key role in the IEand IAresults. Particu-larly, IEwas low while the jet was on, indicating that most of the ejected fluids momentum was transferred tothe added mass during this phase and that the momentum input required from a starting jet to entrain ambientfluid is small. Most of the momentum gained by the entrained fluid occurred after the jet began to decelerate.

This constrained IEat jet termination (t

=1.0)to nearly the same value for all but the triangular NS case,which initiated deceleration very early in the jet pulse. Once the jet was terminated, both IEand IEJoscillatedas the ejected and entrained fluid rotated about the center of vorticity in the ring. On the other hand, it wasobserved that IAcaptured about 50% of total impulse of all the cases studied when the ring was formed. Itwas also determined that lower L/Dsand/or gradual deceleration in the velocity program could increase themomentum of added mass more than 20% indicating that the effect of velocity program and associated strokeratio can be significant in the determination of added mass momentum. Generally, changes in the velocityprogram had a much larger effect than might otherwise be anticipated from a simple analysis based on the slugmodel since the jet impulse and circulation predicted by the slug model were the same for the triangular NSand PS programs.

Since IP = IE + IA (IU IEJ), = (IE+ IA)/(IU IEJ)was used as a performance metric to gagethe effectiveness of different cases in contributing to IPas a function of the vortex formation process. It wasobserved that the rate of increase ofslows as L /Dincreases and the formation number[11] is approached.

It was also observed that the triangular PS velocity program and higherL/Dprovided a larger, indicating alarger contribution to IPfor a given input IU IEJin these cases.An additional benefit of this study is the identification of the initial shape and location of entrained fluid

around the nozzle exit plane. This information can be useful forpromoting efficientmixing of tworeactive sub-stances by indicating where a regent should be concentrated or what formation parameters (velocity programsor L /D)should be adjusted to enhance the process.

While this study only investigates the impulse variation of starting jets, the many of the general trendsin IEand IAcan be applicable to pulsed jets where the time duration between the pulses is long enough toavoid strong vortex interaction between pulses [17]. Often applications involving pulsed jets such as pulsed jetpropulsion[3,4,23], the jet pulses are widely separated so that the flow between pulses is nearly undisturbed.The present results can be expected to be representative of the trends observed in such cases.

Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 0347958.

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