Studies of Two-Dimensional Vortex Streetsenstrophy.mae.ufl.edu/publications/MyPapers/AIAA2001... · 2012. 12. 6. · lutions of vortex streets consisting of vortex patches (vortices

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A01-31268

AIAA 2001-2842Studies of Two-Dimensional Vortex Streets

Kamran MohseniDivision of Engineering and Applied ScienceCalifornia Institute of TechnologyPasadena, CA91125

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AIAA 2001-2842

STUDIES OF TWO-DIMENSIONALVORTEX STREETS

Kamran MohseniDivision of Engineering and Applied Sciences,

107-81, California Institute of Technology.mohseniOcds.caltech.edu

ABSTRACTPeriodic vortex streets, often considered as a modelfor the organized structures observed in the wakeof two-dimensional (2D) bluff bodies, are revisited.After the generation of the shear layers the forma-tion of the vortices is mainly an inviscid process. Anintrinsic scaling for the formation of vortex streetsis found based on the invariants of motion for 2Dinviscid flows; namely the kinetic energy, impulse,circulation and the translational velocity of the vor-tex system. We show that in the frame of referenceof the invariants of motion the intrinsic sheddingStrouhal number St defined based on the invariantsof motion and the aspect ratio of the vortex street K(the ratio of the lateral distance to the streamwisedistance between the centers of vortices) are liter-ally the same. The formation of the vortex streetis a manifestation of Kelvin's variational principle.Using the computational results from Saffman andSchatzman1 for the inviscid vortex streets of vortexpatches we estimate values of the nondimensional en-ergy and circulation of the vortex system for a wideclass of vortex patches. A relaxational explanationof the vortex shedding is also offered. In this picturethe bluff body is considered as a source of provid-ing the system with invariants of motion. After theformation of the shear layers the system will relaxtoward its final equilibrium state where the forma-tion of a vortex street is mandated by the invariantsof motion for the two-dimensional Euler equations.Similar to the vortex ring pinch-off process it is spec-ulated that the characteristics of the vortex systemcan be modified by modification in the rate of gen-eration of invariants of motion. The two main meth-ods for modifying a vortex system are the dynamicalchanging of the speed and the lateral spacing of thegenerated shear layers during the formation of cir-culation regions. This is often achieved by periodicstreamwise and lateral oscillations of the bluff body.

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

1 INTRODUCTIONThe wake of the two-dimensional bluff bodies in auniform stream usually shape into a regular pat-tern of two parallel staggered rows of vortices fora large range of Reynolds numbers; see for exampleWilliamson2 and the references in there. Vorticeswhich are formed at the two points of separationon the bluff body, are seen to be shed off regularlyin an alternating fashion, as shown in figure 1, inwhich C/oo is the uniform velocity of the approach-ing stream, and Ut is the translational velocity of thevortex street with respect to the free stream. It is ap-parent that viscosity does not play an essential rolein the formation of the vortex street once the shearlayers have been developed. This is supported by in-vestigations on the inviscid evolution of infinite par-allel vortex layers with small periodic disturbances(see Aref and Sigga3 and the references in there);the Karman vortex street seems to be the genericresult in such inviscid calculations. Furthermore,it has long been experimentally observed that theStrouhal number does not change significantly for awide range of Reynolds numbers (e.g. see Roshko4).

The double trail of vortices formed alternately onboth sides of a cylinder was modeled by von Kar-man5 as a regular pattern of point vortices in invis-cid flows, von Karman not only analyzed the sta-bility of a system of vortex streets, but also estab-lished a theoretical link between the vortex streetconfiguration and the drag coefficient on the body.He studied both the symmetric and antisymmetricconfigurations of a double row of point vortices. Heshowed that the only configuration that does not ex-hibit linear instability is for an antisymmetric con-figuration (see figure 1) with a vortex spacing ofKC = sinh"1 I/TT « 0.28. See Lamb6 and Saffman7

for a review of the stability analysis including theeffect of finite core size and three dimensional in-stability. It should be noted that these analyses donot address the issue of formation of vorticity onthe body. However, they suggest that if such vor-tex streets are formed they should be observed for a

American Institute of Aeronautics and Astronautics


ooooooe

h

Figure 1: Vortex street behind a bluff body. Note that for an observer traveling with the vortices the velocityat infinity is E/t, and the cylinder moves away from the vortices at a relative velocity of UQQ — U±.

reasonable distances downstream.Vortex streets are the two-dimensional (2D) coun-

terpart of an array of axisymmetric vortex rings.Our results here is a natural extension of the re-laxational idea developed in Mohseni,8 Mohseni andGharib,9 and Mohseni et al.1Q A brief review of thevortex ring pinch-off process in axisymmetric flowsis presented in section 2.

In this paper an intrinsic scaling for a regular vor-tex street based on the main invariants of motionof the 2D Euler equations is introduced. The rele-vant quantities are then nondimensionalized basedon these scales. This scaling fits well with Kelvin'svariational principle. Saffman and Schatzman1'11

and Schatzman12 studied the steadily translating so-lutions of vortex streets consisting of vortex patches(vortices of finite area and uniform vorticity) as thesimplest extension of von Karman's point vortexmodel. Saffman and Schatzman11 mathematicallyaddressed the question of whether constant vortic-ity structures of finite extent can stabilize a vortexstreet. They found numerical solutions of the Eulerequation for their model and calculated the proper-ties of the street, that we used in the present study.In passing we note that Meiron et al.13 concludedthat the modeling of a vortex street by regions ofconstant vorticity did not fundamentally alter thestability results of Karman's point vortex model; atleast for moderate values of the area of the individualvortices, there is only a single value of K, for whichthe street is stable. This behavior is a consequenceof the Hamiltonian nature of the Euler equationsand its symmetries14 and it should hold for any in-viscid model of the street that retains the back tofore symmetry of the basic flow. Jimenez14 arguesthat the viscous effects might be responsible for the

persistence of a natural vortex street observed in ex-periments.

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 Karmanvortex street. Equivalence of the intrinsic sheddingStrouhal number St in the frame of reference of in-variants of motion and Karman's aspect ratio K wasstudied in section 3. Kelvin's variational results werealso addressed in section 3. Nondimensionalizationof the Karman vortex street based on the invariantsof motion is performed in section 4. In section 5the numerical computations of Saffman and Schatz-man1 were used to estimate the nondimensional en-ergy and impulse of the vortex system for variousaspect ratios K. An explanation for the vortex shed-ding behind bluff bodies based on the relaxationalmodel is discussed in section 6. Finally we summa-rize our results in section 7.

2 UNIVERSALITY IN VORTEX RING PINCH-OFFPROCESS

Our main motivation for the present investigationcomes from the universal formation number of vor-tex ring pinch-off process observed in experimentsby Gharib et al.lb Mohseni & Gharib9 offered a re-laxational model for the vortex ring pinch-off pro-cess. Numerical simulations of the Navier-Stokesequations were performed by Mohseni et allQ wherethey verified the modeling assumptions in Mohseni& Gharib.9 In the laboratory, vortex: rings canbe generated by the motion of a piston pushing acolumn of fluid through an orifice or nozzle. Theboundary layer at the edge of the orifice or nozzlewill separate and roll up into a vortex ring. We



think that since the formation of vortex rings in-volves strong mixing of the generated shear layerwith the ambient fluid (the same applies to the for-mation of vortices in two-dimensional flows), the er-godicity requirement of statistical equilibrium theo-ries has a chance to be satisfied. The experimentsof Gharib et a/.15 have shown that for large pis-ton stroke versus diameter ratios (L/D), the gener-ated flow field consists of a leading vortex ring fol-lowed by a trailing jet. The vorticity field of theformed leading vortex ring is disconnected from thatof the trailing jet at a critical value of L/D (dubbedthe "formation number"), at which time the vortexring attains a maximum circulation. The formationnumber was in the range 3.6 to 4.5 for a varietyof exit diameters, exit plane geometries, and non-impulsive piston velocities. An explanation for thisphenomenon was given based on Kelvin's variationalprinciple. It was both experimentally15 and analyti-cally9 observed that the limiting stroke L/D occurswhen the generating apparatus is no longer able todeliver energy, circulation and impulse at a rate com-parable with the requirement that a steadily trans-lating vortex ring has maximum energy with respectto kinematically allowable perturbations. RecentlyMohseni8 argued that the energy extremization inKelvin's variational principle has a close connectionwith the entropy maximization in statistical equilib-rium theories. Numerical evidence for a relaxationprocess in axisymmetric flows to an equilibrium statehas already been provided by Mohseni et a/.10 in adirect numerical simulation of the vortex ring pinch-off process.

Inspired by these observations we offered a re-laxational (statistical) approach to the vortex ringpinch-off process.9'10 This is an alternative expla-nation of the vortex ring pinch-off process, basedon a mixing entropy maximization, besides the en-ergy extremization approach in Kelvin's variationalprinciple. From this point of view, any vortex ringgenerator can be viewed as a tool for initializing anaxisymmetric flow with a particular rate of the gen-eration of invariants of motion. Each vortex ringgenerator has a specific rate for feeding the flow withthe kinetic energy, impulse, circulation, etc. In thispicture, at small strokes (small L/D) one will findthat all of the initial vorticity density will coalesceinto a steadily translating vortex ring. As the strokelength increases the size, strength, and the transla-tional velocity of the resulting vortex ring increase.This process persists until a critical formation num-ber is reached, when the vortex generator is not able

to 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 given energy, impulse andcirculation is not possible. In this case the lead-ing vortex ring will pinch-off from the trailing jetand will relax to a translating vortex ring with thetranslational velocity Ut dictated in the maximumentropy principle. For very large strokes (greaterthan twice the critical formation number) successivevortex rings will pinch-off from the the trailing jet.This scenario was verified in the numerical simula-tions of the vortex ring pinch-off process in Mohseniet a/.10 The general observation in these simula-tions was that the main invariants of motion in thepinch-off process are the kinetic energy, circulationand impulse. The higher enstrophy densities did notplay a significant role as long as the Reynolds num-ber was relatively high.

In the next sections we utilize the same approachto study the Karman vortex street behind bluff bod-ies. We view the vortex street formation as a re-laxational process where the generated shear layersrelaxes into coherent vortical structures while maxi-mizing a mixing entropy or equivalently extremizingthe energy of the system (Kelvin's variational prin-ciple). Statistical equilibrium of a vorticity distribu-tion in 2-D flows was studied by Miller,16 Miller eta/.,17 and Robert and Sommeria.18

3 FRAME OF REFERENCE OF INVARIANTS OFMOTION

Apart from the vorticity generation process, the for-mation of vortex streets is mainly an inviscid pro-cess. The invariants of motion for 2-D Euler equa-tions include the energy JE, circulation F, impulse I,and the generalized enstrophies. However, on anyfinite resolution the only invariants of motion thatsurvive the mixing process are the linear functionalof vorticity,8'17 namely #,F, and /. The transla-tional velocity Ut of the center of the vorticity fieldis also an invariant of motion. Note that Ut appearsas the Lagrange multiplier for the impulse in thevariational formulation underlying the Euler equa-tions. Therefore, the main kinematical invariants ofmotion in the formation of vortex streets are: J5, F, /and Ut. Apart from these kinematical invariants thegeometrical parameters h and I (see figure 1) are alsoinvariants of motion.

The surviving invariants of motion may be usedto find an intrinsic scaling for vortex streets. This is



done in the next section. In this section we show thatin the frame of reference of the invariants of motionthe intrinsic Strouhal number St and the Karmanaspect ratio K are the same. We define the intrinsicStrouhal number based on the separation distance /i,shedding period T = I//, in which / is the sheddingfrequency measured by an observer moving with thefree stream and measuring a translational velocityUt for the vortices. Therefore

Ut TUt(1)

Note that All the parameters in defining this intrin-sic Strouhal number are invariants of motion. Theshedding time is clearly the same as the time re-quired for a vortex in the vortex street to travel oneperiod 1. Hence, T = l/Ut, and we obtain

(2)

Therefore, for an observer in the frame of referenceof invariants of motion the intrinsic Strouhal numberSt defined in (1) and the aspect ratio K = h/l (remi-niscence of the von Karman aspect ratio used in thestability analysis of Karman's point vortex street)are the same. Note that h is twice the distance ofthe center of a vortex from the wake centerline.

Apart from the measurement of the shedding fre-quency in the frame of reference of invariants of mo-tion the Strouhal number defined in equation (1) alsodeviates from that introduced by Roshko19 in thelength scale parameter used, and is different fromthat introduced by Bearman20 in the velocity pa-rameter used. Instead of the vortex street width /i,Roshko based it on the distance of the two free shearlayers separated from the tripping cylinder. For ve-locity scale, we use the translational speed of the vor-tex street, while the velocity at the separation pointon the cylinder surface was used by Bearman.20 Al-though our definition of the Strouhal number is dif-ferent than the available definitions in the literature,their values are relatively close. Consequently thismight explain why for a fixed stationary bluff bodythe Strouhal number for vortex shedding behind abluff body is very close to the von Karman's stabilityaspect ratio, 0.28.

Prom the point of view of the theory developed inthis study the departure of the values of the Strouhalnumber and the Karman aspect ratio for a stablytranslating vortex street stems from the fact that thequantities that are used in the literature are mostlynot an invariant of motion for 2-D Euler flows. In

general the invariants of motion (e.g. Ut and h) arenot known a priori, while the dimensions of the bluffbody D and the free stream velocity UQQ are oftengiven. This fact makes the definition of the Strouhalnumber based on easily available scales more desir-able and practical.

To connect the intrinsic Strouhal number (1) toany other Strouhal number defined in the literaturewe assume a general Strouhal number Stfj definedfor a general frame of reference with a distinctivevelocity and length scales, V and H respectively.Therefore,

(3)V

where H is the characteristic length of the bluff bodyand fn is the shedding frequency measured in thisframe of reference. One can easily show that

StH H ( UtSt h \ V (4)

where the Doppler effect is invoked to relate the fre-quency in the two frame of references. This relationshows that to connect the vortex street in the waketo the bluff body one requires to measure two param-eters experimentally, namely H/h and Ut/V. Thisis consistent with the previous observations that anyconnection between the wake and the vortex shed-ding of a bluff body requires two free parametersthat needs to be measured or calculated indepen-dently.

If we assume that in the frame of reference of in-variants of motion for the Euler equations St = Kis universally invariant for a fixed stationary body(as argued in the next section with a value com-parable with Karman's aspect ratio for the stabil-ity of Karman's point vortex streets) one realizesthat the vortex shedding for various bluff bodies orfree stream flow parameters are characterized by theratios H/h and Ut/V. Therefore the main experi-mental relations needed to relate a general Strouhalnumber Sin to the intrinsic Strouhal number St arerelations for H/h and Ut/V. For a fixed station-ary bluff body, where the relative rates of generationof the main invariants of motion are approximatelyconstant during the formation process, we speculate(see next sections) that the resulting Karman as-pect ratio would be close to 0.28. However if byany mechanism* the relative rates of generation of

"This can be achieved by forced lateral or streamwise mo-tion of the cylinder or application of a synthetic jet to thenear wake of the bluff body



the main invariants of motion are changed duringthe formation of each vortex region one expects astrong modification in the characteristics of the re-sulting vortex streets, including strong deviation ofthe Karman aspect ratio from 0.28. See next sec-tions for more discussion on this issue.

4 NONDIMENSIONALIZATION

We would like to consider vortex shedding behind abluff body as a relaxation problem similar to therelaxation of axisymmetric flows to a regular ar-ray of vortex rings discussed in section 2 and inMohseni.8 As discussed in the previous sections themain parameters for the resulting Karman street areE, F, /, Ut, /i, and I, in which h and / are the lateraland streamwise distance between the center of vor-tex patches, respectively. Note that the period ofvortex shedding in the frame of reference of invari-ants of motion is related to Ut and I through I = TUt-

It was pointed out by Kelvin21 (see also Arnold22)that for, given vorticity and momentum, steadystates of 2D Euler flows correspond to stationarypoints of the kinetic energy with respect to kinemat-ically allowable isovortical perturbations. When thestrength of the vorticity is uniform, the requirementof kinematically allowable perturbations in Kelvin'svariational method corresponds to keeping the areaconstant. Kelvin also remarked that the configura-tion would be stable if the stationary value were amaximum or minimum, but unstable if it were a min-imax. In this case the total area A of the uniformvorticity is also an invariant of motion. Thereforeone can expect the functionality

(5)

for the aspect ratio K. Note that E and / are calcu-lated per unit depth and unit streamwise period. Astraight forward dimensional analysis results in thefollowing three nondimensional numbers, namely

Tnd =

EIUtr

(6)

(7)

(8)

Since in the frame of reference of invariants of motionSt = /c we can cast the relation (5) as

In the next section we explore the dependence of Endand Tnd on the values of St = « and And for a wideclass of inviscid vortex streets.

5 ESTIMATIONS FOR End AND Tnd

To estimate the behavior of End and Tnd for vari-ous K we use the numerical simulations of the Eulerequations performed in Professor Saffman's group atCaltech in the 80's. Saffman & Schatzman1'11'12'23

studied the Karman street of the two dimensional ar-rays of vortices of finite area and uniform vorticity.This arrangement is believed to be a better approxi-mation of the vortex streets formed behind cylindersthan the point vortices of von Karman. Saffman '&Schatzman consider a double staggered array of vor-tices with their centers h apart and with the wave-length of the periodic vortex array being I.

Here we use the calculations by Saffman & Schatz-man1'23 to estimate values of End and Tnd for var-ious aspect ratios ft. We denote any quantity ex-tracted from Saffman & Schatzman's data by '"'.Note that in Saffman & Schatzman1'23 all quantitiesare nondimensionalized by the circulation T and thewavelength /. Their energy is calculated per unitlength, while in our calculations energy is calculatedper unit period /. Saffman & Schatzman1'23 curve-fitted the results of their numerical calculations forvalues of K between 0.1 and 0.8 which are used inthe present study.

Relations for the energy and the translational ve-locity of the curve-fitted results are given by Saffman& Schatzman.1 The area of a vortex is clearly re-lated to the strength, i.e. circulation, of each vortex.The area parameter in Saffman & Schatzman1'23 isa = All2. Therefore And = a//c2. The relationshipbetween the area of the vortices and K, required inthe calculation of E and Ut, can be obtained fromSchatzman.12

Now since Ut is only a function of K and a, we canwrite

= -7T- (10)

Therefore both Tnd and And are at most functionsof K and a. Consequently, the nondimensional re-lation (9) may be translated to a relation for thenondimensional energy End as a function of K anda. Hence

End = End (11)St = « = (9)

Using Saffman & Schatzman's result1 the nondimen-sional energy End(/£,a) is calculated and shown in



1.5

0.5

00 0.1 0.2 0.3K

(a)

0.4 0.5

10 r

0 0.1 0.2 0.3 0.4 0.5K

(b)

Figure 2: Nondimensional energy (a) and circulation(b) versus ft for various a.

figure 2(a). The corresponding nondimensional cir-culation is shown in figure 2(b). For some class ofvortex streets there is a preferred value of K = Staround 0.26. In analogy with the vortex ring pinch-off process we speculate that these vortex streets aregenerated by a fixed stationary bluff body where therelative rate of generation of invariants of motion arerelatively constant during the formation of each vor-tex. However if these rates are changed dynamicallyduring the formation of each vortex a different equi-librium state with possibly different value of spacingratio ft will be formed. This idea is explored in thenext section.

An objective in the modeling of vortex sheddingin the wake of bluff bodies could be the modeling ofH/h and Ut/V. Note that since Ut is provided in thecalculations by Saffman &; Schatzman one can havea graph for Ut/V. Now if the preferred intrinsicStrouhal number is approximately 0.28 for a vortexstreet generated by a fixed stationary bluff body onecan relate it to the more conventional Strouhal num-ber defined based on the characteristic length scaleof the bluff body D (e.g. cylinder diameter) and thefree stream velocity UQQ. As an example considerthe Strouhal number St& defined as

StD = fpD (12)

where /# is the shedding frequency measured by anobserver attached to the cylinder. An approximatevalue of h/D w 1.2, and t/$/t/oo ~ 0-16 was reportedby Tyler24 for a fixed stationary bluff body in a flowwith the free stream velocity U^. Therefore

StD 1.2 (1 - 0.16) = 0.75t = 0.19 (13)

which is very close to the value reported in the lit-erature.

6 AN ALTERNATIVE EXPLANATION FOR THEVORTEX SHEDDING BEHIND BLUFF BODIES

A descriptive explanation of the nearfield wake vor-tex formation was offered by Gerrard25 in the 60's.He suggested that a forming vortex draws the shearlayer of the opposite sign from the other side ofthe wake across the wake centerline, eventually cut-ting off the supply of vorticity to the growing vor-tex. Gerrard's qualitative description of the vortexshedding process considers vortex formation solelyin terms of the interaction of the main shear layerformed on the upper and lower part of the bluff body.Perry et al.26 described the same process from the



topology of the instantaneous-streamline patterns.The formation of a new vortex (circulating region)was characterized by the formation of a saddle pointin the streamline topology.

In this section we speculate another qualitativedescription of the vortex shedding in the wake ofbluff bodies. Instead of looking at the local dy-namics of the shear layers (governed by the Eulerequations) and trying to relate it to the final con-figuration of the resulting vortex street we take arelaxational approach (governed by e.g. statisticalequilibrium theory. We look at the bluff body as asource for providing the flow with the invariants ofmotion for the Euler equations, namely the energy,impulse and circulation. Prom this point of view inhigh Reynolds number flows after the generation ofthe shear layers the formation of vortices is mainlyan inviscid process. This assumption can be easilyverified for high Reynolds number flows by compar-ing the viscous time scale with the relaxational timescale of the system. When this condition is satisfiedthe system relaxes to an almost equilibrium state,before the viscous effects alter the invariants of mo-tion. Modification in the rate of generation of in-variants of motion results in a different equilibriumstate; in other words a different vortex system.

It has been experimentally observed that the as-pect ratio K in the vortex street gradually change asthe viscous effects diffuse the vortices in the vortexstreet. In the far field of bluff bodies viscous effectscause changes in the invariants of motion (as de-scribed by the Euler equations). For example whilethe impulse is still conserved in a viscous fluid thekinetic energy and circulation decay. Changes inthese parameters derives the system toward a differ-ent equilibrium state which itself changes continu-ously. Consequently, the spacing of vortices vary asthe viscosity modifies the invariants of motion.

For a fixed stationary cylinder (or any other bluffbody) there is a specific rate for the generation of theinvariants of motion. This will result in the forma-tion of a vortex street that is usually characterizedby an aspect ratio of K, = 0.28. However any mod-ification in the rate of generation of the invariantsof motion during the formation of each circulationregion might result in a different equilibrium statethat might be characterized by a different spacingratio of the vortex system. Similar to the vortexring formation9'10 (2D axisymmetric counterpart tothe vortex shedding in 2D plane flows) it is expectedthat two main methods for modifying the character-istics of the resulting vortex system to be:

• changing the local speed of the shear layers

• varying the lateral spacing of the shear layers

during the formation of vortex regions. These meth-ods substantially alter the rate of generation of theinvariants of motion.

Changing the local speed of the shear layer isequivalent to changing the effective free stream ve-locity felt by the bluff body; consequently chang-ing the preferred translational velocity of the result-ing vortex system formed at such rate of generationof invariants of motion. Varying the lateral spac-ing of the shear layers could dynamically changethe effective length scale of the body. Similar effectwas observed in the vortex ring pinch-off process9'10

where the dynamical variation of the piston velocity(changing the speed of the shear layers) and the exitdiameter of the nozzle (changing the lateral spacingof the shear layers) resulted in the modification ofthe size and the shedding frequency (formation num-ber) of the pinched-off vortex rings. See Mohseni etal.10 for numerical simulation of such cases.

By appropriately forcing a periodic motion of thecylinder one can significantly modify the vortex pat-tern, spacing ratio of the resulting vortices, the totalcirculation of each vortex, and the resulting induceddrag. This has been repeatedly observed over thelast few decades. See the recent computations byBlackburn & Henderson27 and the experiments byWilliamson & Roshko28 and the references in there.The preceding relaxational ideas can be used to qual-itatively describe such situations. Lateral oscillationof the cylinder would dynamically modify the effec-tive length scale of the body by changing the lateralspacing of the shear layers. Pushing the shear layeraway from the wake centerline at an appropriate fre-quency and amplitude could result in the formationof a stronger vortex, while driving the shear layertoward the wake centerline would cut th_e supply ofvorticity to the vortex and results in a smaller vor-tex. On the other hand, streamwise oscillations ofthe cylinder would effectively change the local speedof the shear layers. A faster shear layer can followand feed the same sign vortex patch for a longer timeand would generate a larger vortex. The appropri-ate amplitude and frequency of the the oscillationsfor the cylinder is determined by the translationalvelocity of the desired vortex street. Verification ofthese ideas and quantitative descriptions of such sit-uations based on the relaxational ideas is the topicof a future study.



7 CONCLUSIONSThe Strouhal number St and the aspect ratio ft (theratio of the lateral to streamwise distances in a vor-tex street) were studied from the frame of referenceof the invariants of motion. For an observer in thisframe of reference the intrinsic Strouhal number St(defined based on only the invariants of motion for2D plane flows) and the aspect ratio AC are the samequantities.

The main invariants of motion are the energy E,impulse /, circulation F, and the translational veloc-ity Ut. These invariants of motion supplemented bythe geometrical invariants of motion h and / definesan intrinsic scaling for the vortex street.

The model of Saffman & Schatzman1'23 for theKarman vortex street was used to estimate thenondimensional energy and circulation of a periodicvortex street. They consider an inviscid, incompress-ible, two-dimensional system consisting of vorticesof finite area and uniform vorticity. For a class ofvortex streets the nondimensional energy of the vor-tex system has a minimum around St = K, w 0.26,which is very close to Karman's stability criteria forpoint vortices. It was speculated that these casescorresponds to the vortex shedding from a fixed sta-tionary bluff body where the rate of generation of in-variants of motion are relatively constant during theformation of each vortex. By changing the relativerate of generation of invariants of motion during theformation of each circulation region one can mod-ify the available invariants of motion in the relax-ational process, resulting in a different equilibriumstate with possibly different aspect ratio K. It wassuggested that relaxational point of view to the vor-tex shedding behind bluff bodies could provide anexplanation for the effect of forced motion of bluffbodies on the characteristics of the resulting vortexstreet.

ACKNOWLEDGMENTSThe author would like to thank Professor A. Roshkofor useful discussions and comments.

REFERENCES[1] P.G. Saffman and J.C. Schatzman. An invis-

cid model for the vortex-street wake. J. FluidMech., 122:467-486, 1982.

[2] C.H.K. Williamson. Vortex dynamics in thecylinder wake. Ann. Rev. Fluid Mech., 28:477-539, 1996.

[3] H. Aref and E.D. Sigga. Evolution and break-down of a vortex street in two dimensions. J.Fluid Mech., 109:435-?, 1981.

[4] A. Roshko. Experiments on the flow past acir-cular cylinder at very high Reynolds number.J. Fluid Mech., 10:345-356, 1961.

[5] T. von Karman. Z7ber den Mechanusmus desMiderstandes, den ein bewegter Korper in einerFl-ussigkeit erfdhrt. Nachr. der K. Gesell derWiss. zu Gottingen, 13:547-556, 1912.

[6] H. Lamb. Hydrodynamics. Dover, New York,1945.

[7] P.G. Saffman. Vortex Dynamics. CambridgeUniversity Press, Cambridge, 1992.

[8] K. Mohseni. Statistical equilibrium theory ofaxisymmetric flows: Kelvin's variational prin-ciple and an explanation for the vortex ringpinch-off process. Phys. Fluids, 13(7):1924-1931, 2001.

[9] K. Mohseni and M. Gharib. A model for uni-versal time scale of vortex ring formation. Phys.Fluids, 10(10):2436-2438, 1998.

[10] K. Mohseni, H. Ran, and T. Colonhis. Numer-ical experiments on vortex ring formation. J.Fluid Mech, 430:267-282, 2001.

[11] P.G. Saffman and J.C. Schatzman. Propertiesof a vortex street of finite vortices. SI AM J.Sci. Stat. Comp., 2:285-295, 1981.

[12] J.C. Schatzman. A model for von Karman vor-tex street. PhD thesis, California Institute ofTechnology, 1981.

[13] D.I. Meiron, P.G. Saffman, and J.C. Schatz-man. The linear two-dimensional stability ofinviscid vortex streets of finite-cored vortices.J. Fluid Mech., 147:187-212, 1984.

[14] J. Jimenez. On the linear stability of the in-viscid karman vortex street. J. Fluid Mech,178:177-194, 1987.

[15] M. Gharib, E. Rambod, and K. Shariff. A uni-versal time scale for vortex ring formation. J.Fluid Mech, 360:121-140, 1998.

[16] J. Miller. Statistical mechanics of Euler equa-tions in two dimensions. Phys. Rev. Lett,65(17):2137-2140, 1990.


(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)1 Sponsoring Organization.

[17] J. Miller, RB. Weichman, and M.C. Cross.Statistical mechanics, Euler's equation, andJupiter's red spot. Physical Rev. A, 45(4):2328-2359, 1992.

[18] R. Robert and J. Sommeria. Statistical equilib-rium states for two-dimensional flows. J. FluidMech, 229:291-310, 1991.

[19] A. Roshko. On the development of turbulentwakes from vortex streets. Technical Note 3169,NACA, 1954.

[20] P. W. Bearman. On vortex street wakes. J. FluidMech, 28(4).-625-641, 1967.

[21] Lord Kelvin. Mathematical and Physical Pa-pers, volume IV. Cambridge University Press,Cambridge, UK, 1910.

[22] V.I. Arnold. Mathematical Methods of ClassicalMechanics. Springer-Verlag, New York, secondedition, 1989.

[23] P.G. Saffman and J.C. Schatzman. Stability ofa vortex street of finite vortices. J. Fluid Mech.,117:171-185, 1982.

[24] J. Tyler. A hot wire method for measurnmentof the distribution of vortices behind obstacles.Philosophical Magazine, 7th series, 11:489-490,1930.

[25] J.H. Gerrard. The mechanics of the vortex for-mation region of vortices behind bluff bodies.J. Fluid Mech, 25:401-413, 1966.

[26] A.E. Perry, M.S. Chong, and T.T. Lim.The vortex shedding process behind two-dimensional bluff bodies. J. Fluid Mech,116:77-90, 1982.

[27] H.M. Blackburn and R.D. Henderson. A studyof two-dimensional flow past an oscillatingcylinder. J. Fluid Mech, 385:255-286, 1999.

[28] C.H.K. Williamson and A. Roshko. Vortex for-mation in the wake of an oscillating cylinder. J.Fluid Struct., 2:355-381, 1988.

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Studies of Two-Dimensional Vortex Streetsenstrophy.mae.ufl.edu/publications/MyPapers/AIAA2001... · 2012. 12. 6. · lutions of vortex streets consisting of vortex patches (vortices