Optimal vortex ring formation at the exit of a shock tube

(c)2002 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A02-13562

AIAA 2002-0161Optimal Vortex Ring Formationat the Exit of a Shock Tube

Kamran MohseniDepartment of Aerospace Engineering SciencesUniversity of Colorado, BoulderCO 80309-0429

40th AIAA Aerospace SciencesMeeting and Exhibit

14-17 January 2002 / Reno, NV

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AIAA 2002-0161

OPTIMAL VORTEX RING FORMATIONAT THE EXIT OF A SHOCK TUBE

Kamran Mohseni

Department of Aerospace Engineering Sciences,University of Colorado at Boulder, CO 80309-0429,

mohseniOcolorado.edu

ABSTRACTFormation of vortex rings at the exit of a shock tubeis studied theoretically. A model for predicting themaximum circulation that a vortex ring can attain atthe exit of a shock tube is offered. To achieve maxi-mum circulation the initial conditions and the designof the driver and driven sections must be such thatthe generated shock wave reaches the shock tube exitat least 4Dn/^2 time units (Dn is the nozzle diam-eter and U2 is the fluid velocity behind the shockwave) before the leading front of the expansion wavesarrives at the shock tube exit. A simple way to sat-isfy this condition is to design a shock tube withlong driver and short driven sections. The model isverified by comparison with available computationaland experimental data.

1 INTRODUCTIONThe first shock tube was invented by Vieille1 in 1899for investigation on the flame propagation problem.Shock tubes are now a common tools for the studyof gas dynamic problems. A shock tube converts thepotential energy stored in the pressurized gas to ac-celerate the fluid inside the shock tube. There aremany variety of shock tubes available now. Whilethe design and the range of applicability of variousshock tubes are different their working principles arevery similar. In this study we will focus on a sim-ple but widely used shock tube, that is a constant-area shock tube with a diaphragm separating thelow- and high-pressure regions. Our analysis can beeasily extended to other designs of shock tubes asdiscussed in §6.

Vortex rings are one of the simplest three dimen-sional coherent structures in fluid flows. They arevery persistent structures and decay only very slowlyat high Reynolds number. The formation and prop-agation of vortex rings attracted many researchers

Copyright © 2001 by the author. Published by the Amer-ican Institute of Aeronautics and Astronautics, Inc. with per-mission.

over the last century, von Karman & Burgers2 stud-ied vortex ring generation by an impulsive pressureacting over a circular area. Another method of gen-erating vortex rings is by impulsive acceleration ofthe air at the open end of a shock tube by the emer-gence of the generated shock wave when the shocktube is fired. The formation of a vortex ring at theexit of a tube by an emerging shock wave was es-tablished by Gawthrop, Shepherd and Perrott3 in1931. They generated the shock wave by detonationof an explosive charge at the opposite end of thetube. Elaborate photographs of such vortex ringswas provided by Sturtevant.4 Usually the resultingvortex rings are turbulent and very energetic. Theadvantage of this method of vortex ring generationis that the flow field in the tube is almost perfectlyuniform and well-defined.

Different initial conditions and designs of thedriver and driven sections of a shock tube result inthe formation of vortex rings with various size, cir-culation, propagation speed, etc. In this paper con-ditions for generating an optimal vortex ring, withmaximal circulation, at the exit of a shock tube isinvestigated. Such vortex rings have also maximalmixing properties.8

Vortex ring formation at the exit of a tube isa complicated process even in the incompressiblecase and no comprehensive theoretical description isavailable. The compressible Hill's spherical vortexwas recently studied by Moore & Pullin.5 The ini-tial stage of the vortex ring formation in the incom-pressible flows (applicable only to thin core vortexrings) is described by the local similarity theory ofthe flow near a sharp edge offered by Pullin,6 whichwas generally confirmed in the water tank experi-mental study of Bidden.7

This paper is organized as follows. In the nextsection we review the relaxational approach in thevortex ring pinch-off process. This will motivate theapplication of the same approach to the vortex ringpinch-off process at the exit of a shock tube. Theoryof shock tubes will be reviewed in section 3. A vortex

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ring pinch-off model is developed in section 4 and acomputational procedure for its implementation willbe discussed in section 5. In section 6 our vortex ringpinch-off criteria is compared with some availableexperimental and computational data. Finally wesummarize our results in section 7.

2 UNIVERSALITY IN VORTEX RING PINCH-OFFPROCESS

Generation, formation, evolution, and interactionsof vortex rings have been the subject of numerousstudies (e.g., Shariff & Leonard9 and the referencestherein). However, we focus our attention on a spe-cific characteristic of vortex ring formation; namelythe vortex ring pinch-off process.10 In a labora-tory, incompressible vortex rings can be generatedby the motion of a piston pushing a column of fluidthrough an orifice or nozzle. The boundary layer atthe edge of the orifice or nozzle will separate androll up into a vortex ring. Gharib et a/.10 observedthat for large piston stroke versus diameter ratios(L/D), the generated flow field consists of a leadingvortex ring followed by a trailing jet.10 The vortic-ity field of the formed leading vortex ring is discon-nected from that of the trailing jet at a critical valueof L/D (dubbed the "formation number"), at whichtime the vortex ring attains a maximum circulation.The formation number was in the range 3.6 to 4.5for a variety of exit diameters, exit plane geome-tries, and non-impulsive piston velocities. Mohseni& Gharib11 offered a relaxational model for the vor-tex ring pinch-off process. Numerical simulationsof the Navier-Stokes equations were performed byMohseni et a/.12 where they verified the model-ing assumptions in Mohseni & Gharib.11 An ex-planation for this phenomenon was given based onKelvin's variational principle. It was both experi-mentally10 and analytically11 observed that the lim-iting stroke L/D occurs when the generating appa-ratus is no longer able to deliver energy E, circula-tion F and impulse / at a rate comparable with therequirement that a steadily translating vortex ringhas maximum energy with respect to kinematicallyallowable perturbations.

Recently Mohseni13 argued that the energy ex-tremization in Kelvin's variational principle has aclose connection with the entropy maximization instatistical equilibrium theories. We think that sincethe formation of vortex rings involves strong mixingof the generated shear layer with the ambient fluid(the same applies to the formation of vortices intwo-dimensional flows), the ergodicity requirement

of statistical equilibrium theories has a chance tobe satisfied. Inspired by these observations we of-fered a relaxational (statistical) approach to the vor-tex ring pinch-off process.11'13 Numerical evidencefor a relaxation process in axisymmetric flows to anequilibrium state has already been provided in a di-rect numerical simulation of the vortex ring pinch-off process.12 Mixing entropy miximization offers analternative explanation of the vortex ring pinch-offprocess besides the energy extremization approachin Kelvin's variational principle. From this point ofview, any vortex ring generator can be viewed asa tool for initializing an axisymmetric flow with aparticular rate of generation of invariants of motion.Each vortex ring generator has a specific rate forfeeding the flow with the kinetic energy, impulse,circulation, etc. In this picture, at small strokes(small L/D) one will find that all of the initial vor-ticity density will coalesce into a steadily translatingvortex ring. As the stroke length increases the sizeand strength of the resulting vortex ring increase.This process persists until a critical formation num-ber is reached, when the vortex generator is not ableto provide invariants of motion compatible with asingle translating vortex ring. Equivalently, beyondthe critical formation number a single vortex ringat equilibrium (steadily translating) that maximizesthe mixing entropy for a given energy, impulse andcirculation is not possible. In this case the lead-ing vortex ring will pinches off from the trailing jetand will relax to a steadily translating vortex ringwith the translational velocity Utr dictated in themaximum entropy principle. For very large strokes(greater than twice the critical formation number)successive vortex rings will pinch-off from the thetrailing jet. This scenario was verified in the nu-merical simulations of the vortex ring pinch-off pro-cess.12 The general observation in these simulationswas that the main invariants of motion in the pinch-off process are the kinetic energy, circulation and im-pulse. The higher enstrophy densities did not play asignificant role as long as the Reynolds number wasrelatively high.13

It is important to note that the formation numberof 4 is only achieved if the rates of generation of theintegrals of motion are constant during the forma-tion process. One can change the formation numberby varying the rate that the invariants of motionare delivered to the system. This is verified numer-ically in high Reynolds number flows in Mohseni eta/.12 In summary there are two ways to modify theformation number and other properties of the lead-



ing vortex rings:11'12'14 1. Time varying speed ofthe shear layer during the vortex formation 2. Timevarying diameter of the shear layer (diameter of theexit). Decelerating the out going fluid slug, whichis equivalent to decelerating the exiting shear layer,results in a smaller vortex ring and a smaller for-mation number. On the contrary accelerating theexiting shear layer during the vortex formation re-sults in a larger vortex ring and a larger formationnumber.

In the next sections we utilize the same approachto study optimal vortex ring formations at the exitof a shock tube. We view such vortex ring forma-tions as a relaxational process where the generatedshear layers relaxes into coherent vortical structureswhile maximizing a mixing entropy or equivalentlyextremizing the energy of the system13 (Kelvin'svariational principle).* One also expects that simi-lar optimal vortex ring pinch-off process happens invortex ring formation at the exit of a shock tube.In fact a close study of the available literature onvortex ring formation at the exit of a shock tubereveals strong indirect evidence for such universal-ity. However, to our knowledge the only systematicexperimental investigation on this topic is by Das eta/.15 Figure 4 in their article clearly shows a time se-quence visualization of such a process. We compareour theoretical predictions with such experimentaland numerical studies in section 6.

3 THEORY OF SHOCK TUBESIn a shock tube shock waves are usually generatedby the sudden rupture of a diaphragm under a com-pressed gas load. In the analysis of a shock tubepresented in this section the gasses will be assumedto be perfect with constant specific heats, and wewill ignore viscous effects. While theses assumptionsare accurate for low-temperature conditions their fi-delity deteriorates at high-temperatures. This anal-ysis, while simple and crude, will be useful for show-ing the important parameters necessary for the pro-duction of strong vortex rings and the conditions forthe vortex ring pinch-off at the exit of a shock tube.We assume that at time t = 0 the diaphragm sepa-rating the low- and high-pressure regions is removedinstantaneously. An advantage of vortex ring gen-eration by a shock tube is that the flow field in thetube is well defined and almost perfectly uniform.

The velocity history at the exit of a shock tube isideally a top-hat profile until the leading front ofthe expansion wave reaches the exit of the shocktube (see figure 2). The potential energy stored inthe high pressure gas is utilized in a shock tube toaccelerates and later decelerates the fluid almost im-pulsively, rather than to use the force applied to apiston to do work on the fluid, as with the moreconventional vortex-ring generators. Consequently,much higher fluid velocities and accelerations can beachieved.

The shock tube described in figure 1 consists oftwo sections of low- and high-pressure regions sep-arated by a thin diaphragm. By rupturing this di-aphragm a traveling shock wave penetrates the lowpressure region, while an expansion wave propagatesinto the high pressure area. The shock wave trav-els at a relatively constant Mach number Ms towardthe exit of the shock tube, and accelerates the fluidbehind it. When the shock wave exit the shock tubean slug of fluid follows the shock and rolls up intoa vortex ring. On the other hand the leading frontof the expansion waves travels at the speed of soundtoward the closed end of the shock tube where theyreflect back and proceed toward the exit of the shocktube, while leaving the fluid behind it stationary.The end of the slug of fluid is marked when theexpansion waves exit the shock tube and leave thefluid behind them in a relatively stationary state (seeSturtevant16'17). There are more complicated inter-actions in a shock tube that are not considered here.For simplicity we also assume that the shock wavesexit the shock tube before the expansion waves reachthe exit of the shock tube.

Following a more generally accepted notation, theinitial conditions in the low- and high-pressure re-gions are marked by indices 1 and 4, respectively.As the shock wave propagates into the gas initiallyat rest (region 1), the state of the gas after the shockwave (region 2) is given by the normal shock waverelationships with u\ — 0 (see, e.g., Liepmann &Roshko18)

* We would like to point out that similar phenomena is ob-served in vortex shedding behind bluff bodies where vorticesare shed off regularly in an alternating fashion (see Mohseni14

for details).

Pi Oil

^ + cT2 _ Pz pi(2)

Pi



Figure 1: Wave propagations in a shock tube.

P2

Pi

1 + ai —_____Pi_.

+ — 'Pi

Pi

1/2 '

(4)

where QI = (71 + I)/(71 - 1). For high tempera-ture conditions produced by the strong shock waves,these equations must be modified to include the realgas effects. The region behind the contact surfaceis marked as region 3. Regions 2 and 3 are usuallyseparated by an entropy discontinuity at the contactsurface which travels with the velocity of the gas inthe region 2. The velocities and pressures are thesame on both sides of the contact surface

P2 =(5)(6)

while the density and temperature could be different.Simultaneously with the rupture of the diaphragm

a centered expansion wave is propagating into thehigh pressure region (region 4). From the un-

steady isentropic expansion, the characteristic quan-(3) tity P = -^a+u is constant through this expansion

fan connecting regions 3 and 4. Therefore, the isen-tropic relation

P4

P3

274

(7)

is valid between regions 3 and 4. The expansionwave traveling toward the end wall is of the Q-typewhile the reflected waves are of the P-type. Now,since the value of P (Riemann invariant) is constanton a P-wave we can write the Riemann invariant

2 274-1 74-1

a4; (8)

where -u4 — 0, and assuming 73 =74. One can showthat

PI27l

a4 71 + 1(9)

This equation relates the shock Mach number tothe pressure ratio and sound velocity across the di-



aphragm for a constant-area shock tube. To pro-duce strong shock waves (resulting in stronger vor-tex rings) 0,4/0,1 must be as large as possible. Onecould show that

£iVQ>iJ

747i

T4 (10)

where m is the gas molecular weight. Therefore thesound velocity in the driver section of the shock tubecan be increased by using a light gas at high tem-perature. Another method to increase the shockstrength is by using larger tube diameter in thedriver section. The shock tube with contraction(variable-area shock tube) can be analyzed similarlyand are not presented here.

and (4) are the initial states of the low- and high-pressure sections of the shock tube, respectively. Arelatively steady flow of fluids at velocity u^ will fol-low the exiting shock and will persist until the re-flected expansion waves from the end wall reachesthe exit of the shock tube and signal the flow that thefluid behind them is stationary. Therefore, the totalduration for the exiting slug of fluid at velocity u2 isthe time between the exit of the shock wave and theexit of the expansion waves (see figure 2). This qual-itative description was suggested by Sturtevant.16'17

Here we quantitatively calculate it based on the ini-tial conditions of regions 1 and 4. The expansionwaves have a leading front and a tailing front, non-simple effects due to the interaction of the expansionwaves with the end wall.

4 A MODEL FOR VORTEX RING PINCH-OFFPROCESS AT THE EXIT OF A SHOCK TUBE

Our objective in this section is to calculate thelength of the fired slug of fluids from a shock tubebased on the initial states of the gases in the driverand driven sections of the shock tube. It is believedthat there is a similarity between the vortex ringgeneration at the exit of a shock tube and the vor-tex ring formation in a cylinder piston mechanism.11

In both cases an impulsive movement of a slug offluid generates a cylindrical shear layer that rolesup into a vortex ring. The length of the ejectedslug of fluid determines the size of the resulting vor-tex ring. Therefore, to generate energetic thin-coredvortex rings it is important that a relatively shortslug of fluid be ejected at high velocity. We willderive equations for the equivalent slug of fluid forany initial conditions in the low- and high-pressuresections of a shock tube.

Consider the simple shock tube in Fig. 1. Thelow- and high-pressure sections are initially sepa-rated by a diaphragm. The tube diameter is D,length of the driver section (high-pressure region) isIh, and the length of the low pressure section (low-pressure region) is li. The rupture of the diaphragmgenerates a shock wave that travels into the low pres-sure section. Immediately after the rupture of thediaphragm the high pressure gas is expanded intothe low pressure section of the shock tube. Thisgenerates an expansion wave that travels into thehigh pressure section of the shock tube from the ini-tial position of the diaphragm at the local speed ofsound.

The operation of a shock tube can be easily ex-plained in the x — t diagram in Fig. 1. Regions (1)

Figure 2: Typical exit velocity profile for shock tubeswith relatively long driver section.

After the rupture of the diaphragm the nondimen-sional time^ for the shock wave to reach the exit is

D D

Substituting Ms from equation (1) andequation (3) one can write

*:== 1 +

(H)

from

(12)

Similar quantity for the tailing front of the expansionwave to reach the exit of the shock tube is approxi-

1"Nondimensionalized by the diameter D and the exit ve-locity 7X2. The intrinsic scalings in this case are the transla-tional velocity Utr and the diameter Dr of the leading vortexring. However Dr/D and Utr/uz are usually constant afterthe vortex formation. Therefore, one can equivalently use themore accessible quantities D and U2 for nondimensionaliza-tion.



mately

D D \o3 -

as -+ it (13)

where /* = /* + /*. The leading front of the expan-sion wave travels into the high-pressure region atthe sound speed 04. The reflected rarefaction waveat the end wall proceeds toward the shock tube exitwith the speed of 0,3 + 1*3. The propagation speed ofthis wave will change to 02 + u<z (note that u<2 = u$)if the leading front of the expansion wave catches upwith the contact surface before it reaches the exit ofthe tube. Therefore, one needs to check if the lead-ing front of the expansion wave catches up with thecontact surface moving at the speed u<2 — u% insidethe low-pressure section of the shock tube. At suchan instance, tinc — X/u<2, both the contact surfaceand the leading front of the expansion waves will beat location X. The origin of the coordinate systemis placed at the initial location of the diaphragm. Xis calculated by equating the elapsed time to reachthis location for both the contact surface and theleading front of the expansion wave

XU2

lh04

.03 -f

Hence

Y* - - X - (-\ 4- U2 \ I "h +A . — — _ M -}- — 777- +D d3 0£ ^ O^

,^2 ^2

Therefore, the nondimensional time for the leadingfront of the expansion wave to reach the exit of thetube is

. =el ' ~ D D o4 o3

"~ "04" "f"

^2

u3 ' if X > / z ; (14)

X 7/2

I? = ~D+1)

= X*- X* if X (15)

The approximated elapsed time te for the expan-sion waves to reach the exit of the shock tube will

characterize the total duration of the slug by te-ts.The nondimensional length of the resulting slug offluid is then given by

D D_ _

~ ' ( }

This relation gives the length of the exiting slug offluid as a function of the initial conditions in the low-and high-pressure regions. By knowing T\ , T^ , pi , p± ,the fluid parameters, and the geometrical character-istics of the driver and driven sections of the shocktube one can calculate the duration (an consequentlythe length of the slug) for the exiting fluid the shocktube.

5 COMPUTATIONAL PROCEDUREOur objective in this section is to calculate the ini-tial condition for the vortex ring pinch-off at the exitof a shock tube. Practically we look for the limitingconditions that separate the pinched off cases fromthe non pinched ones. In doing so we seek the initialconditions that result in the total length of the ex-iting slug of fluid to be 4 times the exit diameter aspredicted by the model in Mohseni et al.w~12 Giventhe initial conditions in the low- and high-pressureregions, equation (9) can be used to calculate theshock Mach number M8. This equation needs to besolved iteratively. Knowing Ms, pz/pi can now becalculated using equation (1).

We need to calculate a%/u<2 from the initial states.Using equations (8) and (7), one can write

^3 74 104

However, using equation (8), u% — ^3, and ^2 —one can write

03 74-1 1

Pi Pi- 1

where p^/pi is given by equation (9) and Pi/p2 isgiven by equation (1). Now that 03/1/2 is calculatedone can easily compute 04/1^2 from equation (8) as

04 03 74 — 1

02/^2 is required in calculating X, and can be cal-culated by noting that

02 02 Oi



and using equation (3) and

P2

PiAt this point all the terms on the right hand side

of equation (16) are described in terms of knownquantities. Therefore, the stroke ratio of the equiva-lent slug model in the shock tube mechanism is givenby equation (16) in terms of the known initial con-ditions in the low- and high-pressure regions of theshock tube.

Mohseni & Gharib11 offered a model for the vor-tex ring pinch-off process by equating the invariantsof motion of the system initially and after the vor-tex ring formation. The initial state was approx-imated by a fluid slug moving at a fixed velocityand the final state was approximated by a vortexin the Norbury family of vortices.20 They predictedan average formation number around 4. The sametechnique can be used in the case considered in thisstudy. We calculated the length of the slug of fluid ina compressible process. Therefore, by substitutingthe formation number by 4 in equation (16) we ob-tain a relation to calculate the pressure ratio p^/pifor any given temperature ratio T^/T\. It is clearthat such a relation depends on how the arrival ofthe end of the slug of fluid is marked in defining £*in equation (16). The optimal vortex ring with max-imal circulation will be generated for the formationnumber of 4 and by marking the end of the slug offluid by the emergence of the leading front of theexpansion wave from the shock tube £* = t*el.

6 COMPARISON WITH EXPERIMENTAL ANDCOMPUTATIONAL DATA

In this section the pinch-off criteria predicted in theprevious sections will be contrasted with availableexperimental and numerical data. All of the casesconsidered here are summarized in Table 1.

Recently Das et a/.15 carried out experiments onthe vortex ring formation at the exit of a shock tubewhere they observed vortex ring pinch-off process.Knowing the length of the slug of fluid emergingfrom the shock tube, for any given initial condi-tions, one can use the vortex ring pinch-off crite-ria proposed by Mohseni & Gharib11 to predict thepinch-off at the exit of a shock tube. This is done infigure 3 for the initial conditions of case Al in Table1. Similar results for the cases considered in Elder& de Haas21 are presented in figure 4.

t =t.

2 3T4/T,

Figure 3: Vortex ring pinch-off criteria for experi-mental setup in case Al in Table 1 from Das et a/.15

/* = 12.7/5.2 = 2.44 and /f = 105/5.2 = 20.19.

The lines marked by £* = t*t in figures 3 and 4are calculated from equation (16) by marking theend of the fluid slug by the initial emergence of theleading front of the expansion wave from the shocktube. In the region above this line one is guaranteedto have a vortex ring pinch-off with the formationnumber 4, uniform fluid slug velocity of u<2 duringthe pinch-off, and with the nondimensional energy,End, and circulation Fnc/, of the leading vortex ringas predicted in Mohseni & Gharib.11 That is End —E/tF3/2/1/2) * 0.3 and Tnd = F/t/1/3!/2/3) w 2.Note that all the quantities in these relations needto be calculated in the frame of reference of invari-ants of motion that is moving with the speed of theleading vortex ring. For all the initial conditionsbelow the line marked by t* — t*t we expect thatEnd > 0.3 and Tnd < 2.

The lines marked by £* = £*t are calculated fromequation (16) by marking the end of the fluid slugby the time that the tailing front of the expansionwaves arrives at the shock tube exit. The region be-low this line characterizes the vortex ring formationwithout a pinch-off process; We predict no trailingjets following the leading vortex ring in this region.In the region between the line t* — t*t and t* — t*tthere is a possibility of vortex ring pinch-off at asmaller formation number.

Note that in the region below the lines markedby t* = t*el there is a deceleration in the exit ve-locity of the slug of fluid due to the expansion fan.



Reference

Das et a/.15

Das et a/.15

Elder & de Haas21

Elder & de Haas21

Minota et a/.22

Sturtevant16

Sturtevant16

Sturtevant16

Sturtevant16

Table 1: Parameters for experimental cases considered in this study.

Case

AlA2BlB2ClDlD2D3D4

li105 cm105 cm176 in176 in

1250 mm160 cm160 cm160 cm160 cm

lh12.7 cm5.1 cm24 in24 in

100 mm15 cm10 cm5 cm

2.5 cm

D5.2 cm5.2 cm

3.375 in3.375 in40 mm7.2 cm7.2 cm7.2 cm7.2 cm

Dn

5.2 cm5.2 cm

3.375 in3.375 in6 mm7.2 cm7.2 cm7.2 cm7.2 cm

P4

Pi6.24.13.721.68

52,362.362.362.36

±!T\111111111

This is equivalent to decelerating an exiting shearlayer suggested in Mohseni & Gharib11 and studiednumerically in Mohseni et al12 As suggested andverified in those studies one expects the observedformation number for a pinched off vortex ring tobe smaller that 4. This decrease in the formationnumber was observed by Das et a/.,15 and it is onlya consequence of the geometrical characteristics oftheir shock tube and the chosen initial conditions.In their case the length of driver section is relativelyshort and the length of the driven section is relativelylong. Consequently, the leading front of the expan-sion wave will quickly reflects back from the wall atthe end of the driver section and follows the shockwave. Usually the leading expansion wave travels ata lager velocity of as + u^ than the shock wave ve-locity. Therefore, the longer the length of the drivensection the shorter the length of the uniform slug offluid with velocity u<2- In fact for shock tubes witha very long driven section the leading front of theexpansion wave might even catch up with the shockwave. As it is clear from figure 3 the cases consideredby Das et a/.15 do not have a uniform exit velocityu^ during the entire vortex ring formation process.The last stage of the formation process in the casesconsidered by them is significantly affected by theexit of the expansion wave and its consequent de-celeration in the exit velocity u^. We predict thatin their cases End > 0.3 and Tnd < 2. To guaranteethe formation of optimal vortex rings with formationnumber 4 one requires to be above the line markedby £* = t*i, where the exit velocity during the vortexformation is uniformly u^. In fact one can generatean even stronger vortex ring if the exit velocity isaccelerated at the final stage of the vortex ring for-mation.12

Elder & de Haas21 used a relatively long high pres-

sure chamber. See cases Bl and B2 in Table 1. Con-sequently, they had a better chance for generatingoptimal vortex rings at lower pressure ratios p±/p\.This is clear in figure 4 where the pinch-off curveshave smaller slopes than the corresponding curves infigure 3 generated for the data from Das et a/.15 Thephotographs of the vortex rings generated in experi-ments by Elder & de Haas,21 however, were reportedonly for the initial phase of the vortex propagationwhere the vortices were very close to the exit of theshock tube or just focusing on the leading vortexring and not its possible trailing jets. As a result,Elder & de Haas21 did not provide enough experi-mental information to verify the pinch-off process intheir experiments.

Case C in Table 1 represents a case where theinner diameter of the shock tube is significantly dif-ferent than the diameter of the the exit nozzle. Sucha change could result in a new formation number de-fined based on the nozzle diameter and with longerfluid slug with uniform velocity. This case was in-vestigated by Minota et al.22 in a study of shockformation by compressible vortex rings impinging ona wall. There are evidence of a vortex ring pinch-offprocess in figures 4 and 6 of Minota et al22 Theyplaced a glass wall at a distance of 2,7 times the noz-zle diameter away from the nozzle. Therefore, Theexiting shock wave from the shock tube will quicklyreflects back from the glass wall and interacts withthe exiting shear layer from the shock tube. Thismakes it difficult to make a conclusion about thepinch-off process in their experiments and computa-tions.

Sturtevant16 (cases D in Table 1) clearly noticedthe significance of the length of the pressurizedchamber in defining the length of the fluid slug exit-ing from the shock tube. However, Sturtevant used



15

10

case B2

T4/T,

Figure 4: Vortex ring pinch-off criteria for experi-mental setup in cases Bl and B2 in Table 1 fromElder & Haas.21 l*h = 24/3.375 = 7.11 and If =176/3.375 - 52.15.

Helium in the pressurized chamber. Helium has alower molecular weight, 7714 — 4, than the air inthe low pressure region, mi = 28.97. Consequently,the speed of sound in the high-pressure section wasmuch higher, resulting in shorter time for the expan-sion waves to reach the exit of the tube. This willbe evident from the speed of sound

a4 =

where R is the universal gas constant. Therefore thelength of the resulting fluid slugs in Sturtevant's ex-periments are relatively small; They have small for-mation numbers and his test cases resides below thecritical (p^/Pi) — (T^/Ti) curve for a pinch-off. Thesignificant difference between the molecular weightof air and helium would also brings more complica-tions due to mixing of a heavy and light gases.

7 CONCLUSIONSThe optimal formation of vortex rings at the exitof a shock tube was studied theoretically. A modelfor predicting the maximum circulation that a vor-tex ring can attain at the exit of a shock tubewas offered. The formation of vortex rings at highReynolds numbers is mainly an inviscid problemgoverned by the integrals of motion. It is predicted

that if the rate of generation of the integrals of mo-tion at the exit of the shock tube is constant duringthe vortex ring formation (the shaded region in fig-ure 2), and if the shear layer is sufficiently thin (andthe Reynolds number sufficiently high) the leadingvortex ring pinches off with normalized energy Endand circulation Ynd of about 0.3 and 2.0, respec-tively. The main invariants of motion are the en-ergy E, impulse /, circulation F, and the transla-tional velocity Utr. These invariants of motion de-fine an intrinsic scaling for vortex ring formations.An observer in the intrinsic frame of reference is at-tached to the vorticity center of the leading vortexring and is moving with the translational velocity Utrof the leading vortex ring. Since the ratios Utr/u^and DTing/Dn (Dn is the nozzle diameter and w2 isthe fluid velocity behind the shock wave) are usuallyconstant it is easier to nondimensionalize based onu^ and Dn.

To achieve maximum circulation the initial condi-tions and the design of the driver and driven sectionsmust be such that the generated shock wave reachesthe shock tube exit at least 4J9n/u2 time units be-fore the leading front of the expansion waves arrivesat the shock tube exit. A simple way to satisfy thiscondition is to design a shock tube with long driverand short driven sections. At high Reynolds numberand with thin shear layers, the rate of generation ofthe integrals of motion needs to be modified as afunction of time in order to change the formationnumber of the leading vortex ring. This statementis based on the observation that the leading vortexring will pinch off when the generating mechanismis not capable of providing energy and circulationat a rate compatible with the normalized circulationand energy corresponding to a steadily translatingvortex ring. A dynamic interpretation of the pinchoff process suggests that the deceleration of the slugof fluid due to the arrival of the expansion wavesat the shock tube exit results in an early pinch offand smaller formation number. However, in a shocktube the maximum circulation of the leading vortexring is achieved when the length of the exiting anduniform slug of fluid (before the emergence of theleading front of the expansion wave from the shocktube) is at least four times the exit diameter. Fi-nally, the model is verified by comparison with theavailable computational and experimental data.

ACKNOWLEDGMENTSThe author would like to acknowledge his conversa-tions with Professor B. Sturtevant in 1999 when the



author, motivated by figures 5b and 6b in Minota eta/.,22 attempted to model vortex ring pinch-off pro-cess at the exit of a shock tube. Professor Sturtevantgenerously provided copies of his AFOSR annualcontract reports.16'17 Sturtevant's ideas in those re-ports were instrumental in the development of themodel developed in this study.

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10American Institute of Aeronautics and Astronautics

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Optimal vortex ring formation at the exit of a shock tube

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