acoustics and dynamics of coaxial interacting vortex rings

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 129.99.135.146This content was downloaded on 21/11/2014 at 21:36

Please note that terms and conditions apply.

Acoustics and dynamics of coaxial interacting vortex rings

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

Fluid Dynamics Research 3 (1988) 337-343North-Holland

7. Vortex and sound

Acoustics and dynamicsof coaxial interacting vortex ringsKarim SHARIFFNASA-Ames Research Center, Moffett Field, CA 94035. USA

Anthony LEONARDCalifornia Institute of Technology., Pasadena, CA 91125, USA

Norman J. ZABUSKYUniversity of Pittsburgh, Pittsburgh, PA 15260, USA

Joel H. FERZIGERStanford University, Stanford, CA 94305, USA

337

Abstract. Using a contour dynamics method for inviscid axisymmetric flow we examine the effects of coredeformation on the dynamics and acoustic signatures of coaxial interacting vortex rings. Both "passage" and"collision" (head-on) interactions are studied for initially identical vortices. Good correspondence withexperiments is obtained. A simple model which retains only the elliptic degree of freedom in the core shape isused to explain some of the calculated features.

1. Methods

We consider inviscid, swirl-free, axisymmetric flow in which the azimuthal vorticity isconfined to regions in which it varies linearly with respect to distance from the axis ofsymmetry. The vorticity of a circular vortex line also increases linearly with its radius as itstretches so that such a distribution is maintained for all time. Hence one needs to follow onlythe motion of the boundary. Its evolution can be formulated as a I-D integro-differentialequation which is solved numerically (Shariff, 1987),The results will be compared with a classical model due to Dyson (1893). This is an

asymptotic limit for the class of rings studied here. It assumes (a) the core size is sufficientlysmall compared to the radius allowing circular cores to be steady in isolation; (b) thatinteracting vortices remain distant so that one vortex influences the other only via the leadingterm in the far-field expression for its induced velocity, for which its vorticity may beconsidered to be concentrated on an infinitesimally thin circle and (c) that deviations of thecores from being circular due to straining by other vortices may be neglected.A simple model was constructed which relaxes assumption (c) above by allowing the cores to

be strained into ellipses. The model was inspired by a similar model for 2-D vortices byMelander, Zabusky and Styczek (1986). There are five ODEs per vortex which govern itsposition and shape. The model incorporates Moore's (1980) solution of an elliptic core vortexring to obtain the self-induced motion and Kida's (1981) solution of the behavior of a 2-Delliptic patch of vorticity in a strain to obtain the core deformation due to the plane-strain part

0169-5983/88/$2.75 1) 1988, The Japan Society of Fluid Mechanics

FLUID DYNAMICS RESEARCH is now available from the IOP Publishing website http://www.iop.org/journals/FDR

338 K. Shariff et al. I Coaxial interacting vortex rings

of the velocity gradient. The model was found useful in understanding some of the coredeformations and accompanying acoustic signals observed with contour dynamics.Acoustic signals were calculated using the theory presented by Kambe and Minota (1983).

To obtain the time behavior of the acoustic pressure requires merely the evaluation of the thirdtime derivative of the centroid of the vorticity as defined by Saffman (1970).The temporal part of the acoustic pressure is denoted as Q'" (t) in the figures. Ur.) is the

translational velocity of the initial rings in isolation and L o is the initial mean toroidal radius.The initial core shapes were chosen from the steadily translating family of solutions computedby Norbury (1973). Each member of the family is characterized by the core thickness parametera, which is the ratio of area-effective core radius to L o.

2. Collision cases

(i') Fig. 1 shows a meridional cross section for a colIision of thin rings (a = 0.2). The cores atthe last instant shown are magnified in fig. 2a and approximate the shape of the 2-D steadilytranslating pair (dotted) first calculated in Sadovskii (1971). The radial speed was found toapproach the translational speed of a 2-D pair that is being stretched along its span. One mighthope that subsequently the cores retain this shape. However, a simple argument as welI as anumerical check showed that this would violate energy conservation. In order to conserveenergy the cores continualIy deposit vorticity in a thin tail while maintaining roughly the shapeof the translating pair as shown in fig. 2b. This process could not be studied precisely becausethe calculation breaks down not long after the last instant shown here.(ii) Fig. 3 shows the pattern of core deformation in a meridional plane for the colIision of

a = 0.5. Fig. 4 shows the corresponding acoustic signature (dotted) compared with the experi-mental result (solid) from Kambe and Minota (1983). Except for the final peak the agreementis very good. Dyson's model (dashed) fails to predict the dip at 'B' where the cores distortstrongly. In Kambe and Minota a viscous model based on a solution for colIiding layers ofopposite vorticity was proposed to account for the dip but the present result suggests thatinviscid core deformation may be sufficient.

ell, ,

(0) (b)Fig. 1. Collision of a 0.2 shown at equal time inter-vals Uot1tlLo = 0.543 except between the last two in-stants for which Lo 0.68. +, vorticity centroid.

Fig. 2. (a) Last instant in fig. 1 shown magnified; ... ,2-D translating pair. (b) Shows the shape a short timeinterval VotlLa 0.05 after (a).


K. Shariff et al. / Coaxial interacting cortex rings 339

Fig. 3. Collision of a = 0.5. Values ofVot / La at the successive instants are: O.1.33, 2.67, 4.00, and 5.49. The pointsmarked A and B on the trajectory of thevorticity centroid correspond to the pointsmarked likewise on the acoustic signaturein fig. 4. The arrow points to the sameparticle on the boundary to provide asense of core rotation. Only between thelast two instants is this rotation largerthan 2'll.

It should be mentioned that the experimental curve represents the average of severalrealizations. The individual realizations show oscillations which are not reproducible betweentrials. This case (with rather thick rings) provides the best overall agreement with the average.Calculations with thinner cores showed that large oscillations are produced as a result ofamplification of core unsteadiness present in the initial condition by the increasing strain rateas the rings collide.(iii) Fig. 5 shows the motion for very thick cores (a = 1.0). At (b) the cores have flattened to

an aspect ratio greater than the value (z 3 : 1) for the 2-D pair. Then as a means of formingsuch a pair the cores "fill-out" and at (c) a head which has very nearly the shape of thetranslating pair has formed trailing a long thin tail. The flow-visualization photograph (d) fromOshima (1978) shows a similar head-tail structure. The head region is shown magnified in fig. 6and compared with the shape of the 2-D pair (dotted).

3. Passage cases

For these interactions, four outcomes were observed in contrast with the "leapfrogging" that1S predicted by the classical model. The determining parameters are a and the initial

f0.5

...0

::J0

--.JI::: 0

<b -0.5I

-10

85 6 Fig. 4. Acoustics of a = 0.5 collision.... ,

contour dynamics; --, experiment;------, Dyson's model.


340 K. Sharif] et al. I Coaxial interacting vortex rings

0 0 ill0 0 rn

(0) (b)

11 (c) (d)

Fig. 5. Collision of a = 1.0. (a) UatlLa 0; (b) UotlLo5.92; (c) UatlLo 8.89; (d) Photograph from Oshima

(1978).

Fig. 6. Comparison of the head region from fig. 5c withthe 2-D transla ting pair ( ).

separation scaled with La. An exhaustive study of the parameter space was not conducted; wemerely suggest trends and sketch the relevant physical effects. The value of d is unity unlessotherwise specified.(i) The classical picture holds for only the thinnest of cores; this is represented by the first

case (0: = 0.1). Core deformations are weak, with a maximum aspect ratio of 1.14 andquasi-periodic with no permanent deformations. However, fig. 7 shows that they introduceacoustic oscillations (solid) with large frequency and amplitude superimposed on the circularcore result (chain-dashed). These oscillations enhance the total radiated power by a factor of6.8. To explain and predict this behavior with the simplest possible arguments, use was made ofa model which retains only the elliptic degree of freedom in the shape of the core. Thisrepresents leading order corrections to Dyson's model for the effects of core deformation. Theprediction of the model is excellent as shown by the dashed curve in fig. 7.Although Dyson's model fails to predict the acoustic signal it is still a good approximation

for the velocity field in this case since the deviation from circularity is small. Using Dyson'smodel, motion of passive particles was investigated by calculating the unstable manifold. It

10

,,--..... 5..0

=:J0

0---ll:::,3-

-5bI -10

-150 0.5 1 1.5 2

Ua t / La2.5 Fig. 7. Acoustics of a = 0.1 passage--, contour dynamics, - - -,Dyson's model; - - - - - -, elliptic model.


K. Shariff et al. / Coaxial interacting rortex rings 341

bFig. 8. (a) Smoke visualization from Yamada and Matsui (1978). (b) Unstable manifold for passive particle motions forthe passage of a = 0.1 using Dyson's model.

emanates from the periodic "stagnation" point on the axis of symmetry. The unstable manifoldis an evolving curve that always contains the same fluid particles and at least near its origin hasthe property that particles tend to converge to it from both sides. Points on the manifold areshown in fig. 8 for one phase in the motion together with the smoke visualization from Yamadaand Matsui (1978); the agreement is excellent at other phases also. Why smoke should tend toaccumulate near the manifold is a question that is yet to be addressed. It is misleading tointerpret the experimental photograph as depicting the distortion and wrapping around of thepassing vortex. The present result suggests that the observed pattern may be due merely tocomplex motion of tracer with vorticity behaving in a simple and approximately classicalmanner. This interpretation is appropriate only for this particular experiment in which a smokewire was stretched across the entire diameter of the orifice causing smoke to be introduced notonly in the emitted shear layer but also into the volume of fluid initially transported with eachvortex.(ii) For the previous case (0: = 0.1) the elliptic model showed that as the strain rate to which

each vortex is subjected varies during the interaction, the cores pulsate about the equilibriumshape corresponding to the instantaneous strain rate, e.g. as the rear vortex completes itspassage and the strain rate to which it is subjected decreases, its aspect ratio also decreases toits initial value. It is in this sense that we referred to the deformations as being quasi-periodic.However, in the second case (0: = 0.14) the elliptic model shows that the strain rate changessufficiently rapidly that each successive passage excites a permanent deformation in the passingvortex. Fig. 9 shows the aspect ratio predicted by the elliptic model (dashed) for one completepassage. Note the sudden departure in the level of oscillations from the equilibrium (chain-

1.4.QeuQ)0...(/) 1.2«

1o 0.5 2.5 3

Fig. 9. Aspect ratio of the passingvortex for ------, el-liptic model; - - -, stable equi-librium for the elliptic model atthe instantaneous value of thestrain-rate; -- contour dy-namics.


342 K. Shariff et al. / Coaxial interacting vortex rings

G G (£)@

JJG GQ)

'.(0) 8 (b) (c)

@"\"

.. .: (fl.-:< (d) (e)

Fig. 10. Passage of IX = 0.2. --, contour dynamics; - - - - - -, elliptic model. Values of Vot / Lo at the successiveinstants are: (a) 0, (b) 0.81, (c) 1.34, (d) 1.62, (e) 1.89, (f) 2.16.

dashed) to a larger value without a return to the initial shape, The contour dynamicscalculation (solid) exhibits the same feature,

(iii) The core shapes (solid) for the third case (a = 0.2) are shown in fig, 10 compared withthe prediction of the elliptic model (dashed), The rear vortex elongates and 60% of it iscaptured by the leading one with only a thin sheet-like region able to pass through. To allowthe calculation to proceed beyond UollLa = 1.5 required the ad hoc removal of thinned regions.This resulted in a 15% loss in the circulation of the vortex. In the future, techniques should bedeveloped which retain these filamentary regions by perhaps an asymptotic treatment thattakes advantage of their thinness.The elliptic model represents the initial elongation process well and from it we learn that

even though the maximum strain rate is small (roughly half that required for tearing under an

Fig, 11. Passage of IX = 0.4. Photographs are from Oshima, Kambe and Asaka (1975).


K. Sharif] ct al. / Coaxial interacting vortex rings 343

axisymmetric analog of the criterion established by Moore and Saffman (1975) based onstability of the equilibrium), it varies sufficiently rapidly that the core overshoots the stableequilibrium and unabated elongation is observed at the very first passage.

(iv) The last case (a = 0.4, J = 2) is shown in fig. 11. The critical value of the strain rate isexceeded according to the elliptic model, th'e rear vortex is considerably elongated along thesymmetry axis, and in the contour dynamics case a thin wisp of it rolls up around the leadingvortex showing good agreement with the photographs from Oshima, Kambe and Asaka (1975).In this experiment, electrolyte grains produced at the edge of the orifice were injected into theshear layer for visualization. Hence, here the tracer is not prone to the effects of passiveadvection discussed above. However, the Schmidt number of tracer is typically very large sothat the vorticity may be considerably more diffused than tracer.

Acknowledgement

We are grateful to Prof. Y. Oshima for sending us photographs and allowing theirreproduction.

References

Dyson, F.W. (1893) Phil. Trans. Roy. Soc London AI84, 1041-1106.Kambe, T. and Minota, T. (1983) Pmc. Roy. Soc. London A386. 277-308.Kida S. (1981) J. Phys. Soc. Japan 50,3517-3520.Melander, M.V., Zabusky, N.J. and Styczek. A.S. (1986) J. Fluid Mech. 167,95-115.Moore, D.W. (1980) Pmc. Roy. Soc. London A370. 407-415.Moore, D.W. and Saffrnan, P.G. (1975) J. Fluid Mech. 69,465-473.Norbury, J. (1973) J. Fluid Mech 57, 417-431.Oshima, Y. (1978) J. Phys. Soc. Japan 44, 328-331.Oshima, Y., Kambe. T. and Asaka, S. (1975) J. Phys. Soc. Japan 38,1159-1166.Saffrnan, P.G. (1970) Stud. Appl. Math. 49 371-380.Sadovskii, V.S. (1970) Prikladnaia matematika i mekhanika 35, 729-735.Shariff, K. (1987) Dynamics of a Class of Vortex Rings Ph.D. Thesis in preparation, Dept. of Mech. Engr., Stanford

University.Yamada, H. and Matsui, T. (1978) Phys. Fluids 21, 292-294.


acoustics and dynamics of coaxial interacting vortex rings

Documents