________.....,--- d

J. Fluid Meek. (1971), vol. 48, part 3, pp. 507-527

Printed in Great Britain

On the phenomenon of vortex street breakdown

By WILLIAM W. DURGINtAND STURE K. F. KARLSSON

Division of Engineering, Brown University, Providence, R.I. 02912

(Received 6 October 1970)

507

A von Karman vortex street generated in the usual way was subjected to adeceleration, thereby changing the ratio of longitudinal to lateral spacing be­tween the vortices. Distortion of the individual vortices followed which resultedin annihilation of concentrated vortex regions and creation of a stationary wakeflow. This wake flow was itself dynamically unstable and developed into anew vortex street of a different frequency from the initial one. The breakdownof the initial vortex street is qualitatively explained by considering the convec­tion of a concentrated vortex region due to the motion imposed by all the othervortices.

IntroductionFor nearly a century the von Karman vortex street has been the subject of

many experimental and theoretical investigations. Seemingly basic to the studyof time-dependent flows because of its simple periodicity, the vortex streethas, nevertheless, evaded all but the most rudimentary of mathematical de­scriptions. Being relatively unguided by theory, experimentalists have, for themost part, always made the same type of measurements, namely of velocity andgeometry in a plane through the vortex filaments.

Classically the vortex street has been divided into three regions (cf. Schaefer &Eskinazi 1959): the formation, a stable, and an unstable region. The forma­tion region has been synthesized utilizing numerical integrations on a digitalcomputer (Payne 1958). The stable region was originally described by vonKarman (von Karman & Rubach 1912) and more recently by Schaefer &Eskinazi. The unstable region was thought to precede turbulence or accomplishthe viscous annihilation of the vortex street.

In 1959 Taneda found an alternative development for the vortex street. Hediscovered that far downstream the original vortex street disappeared and anew one oflarger scale appeared. That he was the first to observe this phenomenonis not surprising since his apparatus was unique in beingable to preserve thevortextrail for long distances. Credibility was lent to Taneda's observations by Zdrav­kovich (1968) who photographed the vortex trails of three cylinders in closeproximity. For certain spacings of the cylinders the centre four vortex rows

t Present address: Department of Engineering Science and Mechanics, University ofFlorida, Ga,inesville, Florida 32601.

I

508 W. W. Durgin and S. K. F. Karlsson

annihilated each other leaving the two outside rows which quickly changed tothe lower frequency, larger scale vortex street observed by Taneda. Still anothersituation which produces essentially the same phenomenon was observed byKarlsson (unpublished report). In this case the vortices shed from a circularcylinder are convected into the region of deceleration in front of a larger circularcylinder placed perpendicular to the first cylinder and the flow. Somewherein the region between the cylinders a similar change of scale is observed. Thepresent study utilizes this geometry.

Experimental apparatus and instrumentationTwo different low-speed wind tunnels were used in the course of the experi·

mental work, both having rectangular cross-section test sections measuring10 in. by 15 in. and 22 in. by 32 in. respectively. These tunnels have recently beendescribed by Sadeh, Sutera & Maeder (1968) and Wood (1969).

FIGURE 1. Geometrical arrangement in test section.

In the smaller of the two wind tunnels we studied a vortex street approachingthe forward stagnation point of a 1·5 in. diameter circular cylinder, and the corre­sponding large cylinder in the large wind tunnel was 4·5 in. in diameter. Figure 1illustrates the experimental arrangement.

All velocity measurements were made using the anemometer systems de­scribed in Sadeh et al. and Wood operated in the constant-temperature mode.Preliminary experiments showed that a probe with long thin support needlesoriented across-stream was the only type which did not affect the flow field ofinterest significantly and was used exclusively in the final measurements.

On the phenomenol

One hot wire was supported on atest section, which allowed continu(co-ordinate axes. The hot wire wacylinder. A two-dimensional lathe coon the opposite side of the test sect

The flow fieldThe flow in the wake of the sheddi:

regions, each of which will be discuwere made in the plane of symmetr~

Within a range of free-stream VI

shedding cylinder exhibited a perio€single frequency sinusoidal wave w:velocity fluctuations due to convectprobe was moved in towards the I

fluctuations had a dominant secondforms were characteristic of the VOll

wake flow behind a cylinder appro:and 300 and were extensively studieHowever, as the probe was moved (finally becoming undetectable. We.calm region. Here no velocity fluehowever, showed a large deficit aerodownstream was characteristically aat least.

The end of the calm region was !

instability and the formation of anwavelength than the original. FigurEas a function oftime in this wake. Forfluctuations in the original vortex stdownstream, but outside the bounteristically 0·03in. thick), this secorlow frequency fluctuations.

A wave analyzer was used to detersignal, and the fundamental was aCIfrequency standard. Second and tJespecially prominant immediately bE

A comparison between the two simin figure 3 in which the non-dimensioiagainst the non-dimensional secondshedding cylinder diameter and v thesome of the data is from the small "\\turbulence level than the large tunm

One hot wire was supported on a traversing device, mounted external to thetest section, which allowed continuous displacement along each one of the threeco-ordinate axes. The hot wire was always oriented parallel to the sheddingcylillder. A two-dimensional lathe compound (oriented horizontally) was mountedon the opposite side of the test section to hold the reference hot-wire probe.

The flow fieldThe flow in the wake ofthe shedding cylinder can be divided into three distinct

regions, each of which will be discussed separately. The following observationswere made in the plane of symmetry of the flow around the large cylinder.

Within a range of free-stream velocities the wake immediately behilld theshedding cylillder exhibited a periodic structure. Near the outside ofthe wake asingle frequency sinusoidal wave was observed on the oscilloscope representingvelocity fluctuations due to convected vortices on that side of the wake. As theprobe was moved in towards the centre of the wake the measured velocityfluctuations had a dominant second harmonic component. The observed wave­forms were characteristic of the von Karman vortex street which exists ill freewake flow behind a cylinder approximately between Reynolds numbers of 50and 300 and were extensively studied by Kovasznay (1949) and Roshko (1953).However, as the probe was moved downstream, the fluctuations grew smaller,finally becomillg undetectable. We have called this region of the flow field thecalm region. Here no velocity fluctuations were visible. The mean velocity,however, showed a large deficit across the wake. The length of this calm regiondownstream was characteristically an inch at most and practically non-existentat least.

The end of the calm region was signalled by the development of a growinginstability and the formation of a new vortex street of much larger width andwavelength than the original. Figure 2 (plate 1) shows the velocity fluctuationsas a function oftime in this wake. For reference the upper trace shows the velocityfluctuations in the original vortex street close to the shedding cylinder. Furtherdownstream, but outside the boundary layer of the large cylinder (charac­teristically 0·03 in. thick), this second vortex street disappeared, leavillg somelow frequency fluctuations.

A wave analyzer was used to determine the frequency content of the hot-wiresignal, and the fundamental was acourately determined by comparison with afrequency standard. Second and third harmonics of the fundamental wereespecially promillant immediately behind the shedding oylinder.

A comparison between the two simultaneously existing vortex streets is shownin figure 3 in which the non-dimensional primary street frequency fd2Jv is plottedagainst the non-dimensional secondary street frequency o-d2Jv where d is thesheddillg cylinder diameter and v the kinematic viscosity. As noted on the figuresome of the data is from the small wind tunnel which has a higher free-streamturbulence level than the large tunnel.

1!. Karlsson

: rows which quickly changed tobserved by Taneda. Still another~ phenomenon was observed by~e vortices shed from a circularration in front of a larger circularnder and the flow. Somewherechange of scale is observed. The

tionused in the course of the experi­-section test sections measuringThese tunnels have recently beennd Wood (1969).

lent in test section.

judied a vortex street approaching~er circular cylinder, and the corre­leI was 4·5in. in diameter. Figure 1

ing the anemometer systems de­l the constant-temperature mode.be with long thin support needleslich did not affect the flow field ofy in the final measurements.

On the phenomenon of vortex street breakdown 509

510 W. W. Durgin and S. K. F. KarlssonOn the phenomenon oJ

'8

·0·9

·9

'8

·8

8

6

7

6

7

4

3

2

I

8

f 1ft/sec

.

II

[211

11

II

II

II·

II·

II·

II·

11·

11·

10·

10·7

10·2

0·10

0'15

0·20

0·25

0·30

0·35

0·40

0·45

0·50

0·55

0·60

0·65

1·15

" 0'90

0·95

1·05 10·7

1·35 8·5

1·25 9·1

0'185 0'15

y firFIGURE 4. Transverse profiles 0

0'70 ]' II·

0·75 ~ II·

0·80 ~ 11·2.......,.§. 0'85 11·

defect lfo - U, where lfo is the mean velociwith downstream distance x for all crOSEthe centre ofthe calm region. In other woin the x direction, 0(00 - U)/ox is negative.positive values of this gradient (Schaefer

Figure 5 shows the mean velocity plo1three cross-wake locations. The upper curwake. The broken curve represents the thestagnation streamline in front of a circulfluid. The lower curve represents the velo(

353025105

d=0'028

,~0-- 0 D=I·5, small tunnel

d=0'029

0-1 b -o D=4·5, large tunnel

/ ~0

~

yP

V"U

pP

5

IS 20

fd 2/v

FIGURE 3. Secondary wake frequency VB. primary wake frequency.

10

15

Velocity profilesThe wake profile measurements were made in the large ~nd tunnel, th~s t~e

large cylinder had a diameter of 4·5in. The shedding cyhnder was 0·029m. III

diameter and the distance between the cylinders was 1·8in. The free-streamvelocity was 15·9 ft/sec, the shedding frequency 700 Hz, and the air temperature75°F.

The mean velocity U and the normalized r.m.s. fluctuations of the velocity,((u2)!/U), were measured in the plane of symmetry of the large cylinder usi~g

standard hot-wire techniques. Velocity measurements were made every 0·001 m.in the cross-wake (y) direction and every 0·05in. in the downstream (x) directionin the primary vortex street, increasing to every 0·1 in. in the calm and secondarystreet regions. The wake was found to be symmetrical, therefore, the measure­ments were limited to one side.

Figure 4shows the mean velocity plotted against y for each downst:-eam station.The heavy curve at x = 0·85 in. marks the disappearance of the primary vortexstreet as indicated by the fluctuation measurements. The second heavy curve atx = 1·25in. marks the appearance of the secondary vortex street. These profilescan be seen to drop from some value outside the wake to a minimum on the w~ke

centreline where the slope must be zero if the wake is symmetrical. The veloCIty

[. F. Karlsson On the phenomenon of vortex street breakdown 511

, VB. primary wake frequency.

de in the large wind tunnel, thus thete shedding cylinder was 0·029in. inylinders was 1·8 in. The free-streammcy 700 Hz, and the air temperature

19ainst y for each downstream station.disappearance of the primary vortexurements. The second heavy curve atlcondary vortex street. These profilese the wake to a minimum on the wakeihe wake is symmetrical. The velocity

o0·050·10

Y [in.]

0·150·185

FIGURE 4. Transverse profiles of mean velocity in the wake.

1·35

\·25 9·1 ~--.....----_+-----+_...1·15 IO·2,--t--__---!_---

f I ftJsec

1·05 10·7 I----+-----t-__'-

0·10 11·8 t====+=====1Cc::c:::::::-T----,0·15 12·0 \--.----+----'-----1---,..--..~.

O' 20 11·9 I-_--r--t----'---.L---t~ ........~__,..__.l ~~-0·25 11·9 \--.----+-------.-1----.........0·30 11·8 !----+-----1----,....--....JO· 35 11·8 1-----+-----'-1"--0---_

0·40 11·8 1----+-------1------...---.....0·45 11·6 r----t--.L-.&-.&----1........--"--':'-o.-

0·50 11·7 1-----+-------1---"'--__0·55 11·6 1----+--r_-_---1~-_~

0·60 11·7 I---.........-+-----.~--- .......0·65 11·41----+-----1---.....

0·70 ~ 11·3 1-----1~----_t_~~-.....5

0·75 i::.. 11·2 1------4----.....--+--~--.

0·80 ~ 11·21----+-----+--............,.-,i 0·85 11·1 1----+-----+--........"0·90 I 0·8 I--_.--+_--.L---:.-.&-t__ ~

0·95 10·7 I-~--+-~---t--_

defect Uo- U, where Uois the mean velooity outside the wake, is seen to increasewith downstream distance x for all cross-wake locations up to, approximately,the centre of the calm region. In other words, the mean deficit velocity gradientin the x direction, o(Uo - U) /ox is negative. Vortex streets in uniform flow exhibitpositive values of this gradient (Schaefer & Eskinazi 1959).

Figure 5 shows the mean velocity plotted against downstream distance forthree cross-wake locations. The upper curve represents the velocity outside thewake. The broken curve represents the theoretical velocity distribution along thestagnation streamline in front of a circular cylinder in an unbounded inviscidfluid. The lower curve represents the velocity along the wake centreline, De. The

353025

d r.m.s. fluctuations of the velocity,ymmetry of the large cylinder usingLsurements were made every 0·001 in.05in. in the downstream (x) directionvery 0·1 in. in the calm and secondary;ymmetrical, therefore, the measure-

20

nnel

rge tunnel

/ ~0

~y'P

where u and vare the mean velocities in the x and y directions respectively. Itis easy to demonstrate from our measurements that Iov!oxI ~ IoU!oyIin the wake.

Figure 7 shows - (jj computed ilfor the same downstream 10cationEthe area under the curves is seen tcbeginning of the calm region wherldiminishes in the secondary streetnet r.m.s. velocity falls in the prinobserved in figure 5 are associated

On the phenomenl

Typically Iv!ul ~ 0,03, I(ov!ox)!(o~that

Geometrical structure of the v~

The curvature, or longitudinal slby correlating the hot-wire signal vlocated 1in. above the stagnation Ihot-wire probe was moved downstrlwhen the signals were in phase, am±90

0

phase angle. Thus the zero COlthe vortex positions.

Figure 8 shows the downstreamy = 0·04 in. at various distances froIconnecting these points represent v

It is evident that any given vort,cylinder but rather begins somewhloff' the cylinder towards the plane 0

the large cylinder and get closer t,secondary street region, correlationmetry of the large cylinder (z = 0).as they travel downstream.

In the primary vortex street regiolocation, x, ofthe vortices at a givenbetween successive locations was ta:downstream location given bythe mevalues are plotted in figure 9. Vallsubsequent calculations were tak81decrease in downstream spacing is (and the value just as the primary."noted, this behqviour deviates consuniform flow which exhibits a relati(Schaefer & Eskinazi 1959).

Early measurements of the distaspacing) were made using photogr2street. Any point about which the flucentre and spacing measured from

33

(I)

1-4

-I .. 250Hz~

1·21·0

(jj = ov!ox-oU!oy,

700 Hz ~~I..__ Calmregion

0-4

W. W. Durgin and S. K. F. Karlsson

0·2

,..

0·6 0·8x [in.]

FIGURE 5. Longitudinal profiles of mean velocity in the wake. 0, y = 0;/':,., y = 0·25; D, y = 0·075; ---, potential flow.

2

o

512

velocity defect is very apparent here. The remaining curve shows similar varia­tion at an intermediate position. A recovery in deficit velocity can be seen afterthe calm region. The foremost part of the large cylinder lies at x = 1·8 in.

12r~~~~~~~=-=-r--I-I--r-I

41----+-----+----+---+---=o..-t-=---_+----t-----;

10 I-----j---'''--I...''''cl--=..........--t---+----'''--+----'''''"''-:-+---f------j

'0! 61----+-----+----+-~-+----j---_+----t--_=__1~

Figure 6 shows (u2)t plotted against y for sequential downstream locations.The slope of these curves is zero at the centreline, as it was for the mean velocitycurves, because ofthe wake symmetry. The value is a minimum at the centrelineincreasing at first with y until a maximum is achieved whereupon it begins tofall steadily to the value prevailing outside the wake. The maximum value aswell as the area enclosed by these curves is seen to fall steadily with distancedownstream until the formation of the new street at x = 1·25in. where (u2)!increases sharply. In this secondary street region there is at first a maximum onthe centreline. Further downstream a minimum forms on the centreline with themaximum moving off-centre as in the primary street region.

The centreline minimum and off-centre maximum are characteristic (Schaefer& Eskinazi 1959, Kovasznay 1949, Roshko 1953) of vortex streets in uniformflow as is the decay in the maximum value and area enclosed by the curves. Themean vorticity in the wake also is strongly affected. Although not directlymeasured we can obtain a fairly good estimate for it from our velocity measure­ments. The mean vorticity is given by

rL

On the phenomenon of vortex street breakdown 513

Typically IV/ul :::::: 0,03, \(ov!ox)!(oUjoy)! :::::: 0·01. Hence it is justified to assumethat

· K. F. Karlsson

remaining curve shows similar varia­~y in deficit velocity can be seen afterlarge cylinder lies at x = 1·8 in. W -:- - ou/oy. (2)

m velocity in the wake. 0, y = 0;---, potential flow.

the x and y directions respectively. It.entsthat loVjox\ ~ \ou!oyl in the wake.

for sequential downstream locations.treline, as it was for the mean velocitye value is a minimum at the centrelinem is achieved whereupon it begins tode the wake. The maximum value asi is seen to fall steadily with distanceleW street at x = 1·25in. where (u2)!''region there is at first a maximum onimum forms on the centreline with thenary street region.maximum are characteristic (Schaeferko 1953) of vortex streets in uniforme and area enclosed by the curves. Thengly affected. Although not directlymate for it from our velocity measure-

Figure 7 shows - w computed in this fashion from figure 4 plotted against yfor the same downstream locations as used in figure 6. The amplitude as well asthe area under the curves is seen to increase with downstream distance until thebeginning of the calm region where the amplitude levels out somewhat and thendiminishes in the secondary street region. The net mean vorticity grows as thenet r.m.s. velocity falls in the primary wake region. The large velocity deficitsobserved in figure 5 are associated with maximum values of mean vorticity.

Geometrical structure of the vortex motionThe curvature, or longitudinal shape, of the vortex filaments was determined

by correlating the hot-wire signal with a reference hot-wire signal from a probelocated 1in. above the stagnation plane at x = 0·1 in. and y = - 0·08 in. As thehot-wire probe was moved downstream the correlation was periodic; a maximumwhen the signals were in phase, a minimum when 1800 out of phase and zero at±900 phase angle. Thus the zero correlation points provide a convenient map ofthe vortex positions.

Figure 8 shows the downstream locations of zero correlations in the planey = 0·04 in. at various distances from the plane of symmetry (z = 0). The curvesconnecting these points represent vortex filaments.It is evident that any given vortex is not shed all at once from the shedding

cylinder but rather begins somewhere outside the stagnation region and 'peelsoff' the cylinder towards the plane of symmetry. The filaments then bend around

1the large cylinder and get closer together as they travel downstream. In the~I

secondary street region, correlations were only measured in the plane of sym-metry of the large cylinder (z = 0). Here also the vortices grow closer togetheras they travel downstream.

In the primary vortex street region, in the plane y = 0·055 in., the downstreamlocation, x, ofthe vortices at a given instant of time was measured. The differencebetween successive locations was taken to be the downstream spacing, a, at thedownstream location given bythe mean ofthe successive in-phase locations. Thesevalues are plotted in figure 9. Values of the downstream spacing, a, used insubsequent calculations were taken from this figure. Approximately a 60 %decrease in downstream spacing is observed between the initial value of 0·129and the value just as the primary vortex street disappears. As was previouslynoted, thi:;? behaviour deviates considerably from that of a vortex street in auniform flow which exhibits a relatively constant value in downstream spacing(Schaefer & Eskinazi 1959).

Early measurements of the distance between the rows of vortices (lateralspacing) were made using photographs of particle or dye trails in the vortexstreet. Any point about which the fluid appeared to rotate was taken as a vortexcentre and spacing measured from it. Hooker (1936) pointed out that these

33 FLM 48

(1)

1-4

"I. 250 Hz -..\

1·2

-ou!oy,

1·0

Calm+--- region

~ff'

_1!I!!III'!Il'_->~C-; ~-.

Ol....~

~

~

l...;::l

~~

~~~I;:l

;'!...CoCo

~

o

.\"=0·90 in.

......

0·95

......

1·05

----j·15

.........

1·25 ..-----1·35 ./'

--------1·45 ~

-~

1·55---I-"'"-o0·135 0·10 0·05

Y [in.]

(c)

0

I

0

I

0

,-., I

'"d)

~o

~

110

I

0

I

0

o

x=0·50in.-- ----.,0·55

- ----0·60

............ --

0·65

...... ---.......----0·70

0·75

0·80

0·85

oO· 135 0·10 0·05

y [in.]

(b)

0

I

0

1

0

'0d)

-!E-.~

0

110

1

0

1

0

o

x=O·lO in. -~" '--0·15 ~

~ ""0·20

---------- '"0·25

~ --- '-.,0·30

------IK" ".,

0·35

~,...-~

0·40

........-- I-'"""~

0·45~.............,-----o

O'135 0·10 0·05

Y [in.]

(a)

o

o

o

o

o

'01~~O

11 I

o

FIGURE 6. Profilos of r.m.s. velocity fluctuations.

" >;'"i_- 'T1'iTF~·

---......---....."""'............-." _. 4

0

~D O'::f/ ~· ,J ~

I:~

~

I'I-r-~/ I"t:>;;,..

~

~'"~

x=0·50in.

0·55

0·60

0·65

..........

o

o

o

n

'I'0~'I'

x=O·IO in.

~0·15

0·20

~0·25

~

0

I0

1

0'",L,

'"d)

.:!3..,

'I' 0

o0·135 0·10 0·05

y [in.]

(b)

o0·135 0·10 0·05

Y [in.]

(0)

a

~~

~"'l

[~o

o

o

--

0·75

0·80

0·85

o

o

v

0' .I • ...---~ I ...,0·135 0·10 0·05 0

y [in.]

(a)

oI ,• .--- I ..,

oI I' .---- ~ I l

u 'i

FIGURE 6. Profiles ofr.m.s. velocity fluctuations.

Ot.....Ot

a;:l

~

I~~;:l

~

:5"'l

~""~~

i~;:l

o

,

x=0'90 in. / ~

~ ,( --0·95 I '\

~

~ A1·05 .;I.

...-A'l/

1·15 /./""

1·25 /7

1·35 /'V

. 1·45

/", -

1·55--~

o

o

o0·135 0·10 0·05

Y [in.]

(0)

o

o

o

o

x:3I

..I,-,0)

!bO......

o

x=0'50 in.

-~0·55

___V~0·60 ...............

-,V \0·65 ./\

----

V \-0·70 /\

~V !\ '0·75 j ~

~ I-' /\ "0·80 I ~.

II - \~

0·85 f"\.~ \

/

FIGURE 7. Mean vorticity.

o0·135 0·10 0·05

Y [in.]

(b)

o

o

o

o

o

o

x:3I

.. 0,!...,

!io......

o

'.

x=O'1O in.

-~0·15

0·20

~0·25

~

0·30

~0·35

I~0040

~0·45

V~

o

o

o

o

o

o

o

o0·135 0'10 0·05

Y [in.]

(a)

;L,!io......x:3I

~N

points differed from the point of maximum vorticity or actual vortex centrebecause of the diffusion of vorticity. He showed how to correct for this error byestimating the age, r, of the vortices and assuming them to diffuse in the Hamel­Oseen fashion, (cf. equation (3)).

Schaefer & Eskinazi adapted essentially the same method for use with hot·wire anemometry. They assumed the vortices to be of the Hamel-Oseen typewhence the maximum peripheral velocity is at a distance of r* = (5'04vr)l fromthe vortex centre. Using the additional assumption that the maximum in thecross-wake plot ofr .m.s. velocity coincides with the outside maximum peripheral

0·8

~ 0-6<l...::-i*...::

-i 0-4...::

0·2

On the phenomena

velocity, they calculated the laterawhere h* is the distance between thEskinazi obtained good theoreti<justifying their assumptions, the VOl

not circular and therefore the applieable.

A reasonable assumption here ilvortex centres coincides with the 1

approximation W ~.- oU/oy, this celwake mean velocity plots. An exaindeed show that for their vortex smaximum slopes of their mean veloevalues of h.

FIGURE 11

00'-----'---­0·2

1-0 r----r--

Figure 10 is a comparison betwee:and h as functions of x, all normalizeFor comparison, von Karman's valur = r(x) obtained from the measuremto traverse a vortex spacing given by

Whereas the data of Schaefer & E~

figure 10 reveals that, for the presel1across-wake spacing ratio based onsubstantially greater than that caletmaxima and vortex age, hla, indicatvorticity.

The phase of the hot-wire signal reIfrom photographs of oscilloscope tra

0-80-60-2 0·4

x [in.]

FIGURE 9. Downstream spacing.

-.............

"~

oo

W. W. Durgin and S. K. F. Karlsson

Primary Dlml Secondary Vvortex vortex Vstreet street Large

V cylinder/

l Y:\ \ \ 1\ I v:

1\ \ \\ \ V

0-16

0-12

C"""""'

i 0-08Ij

0-04

-0,75

-0·25

o 0·2 0-4 0·6 0·8 1·0 1·2 1·4 1·6 1·8x [in.]

FIGURE 8. Correlation zeros.

-1,0

-0,50

C"""""'

g Flow",-0

516

]1'. Karlsson

vorticity or actual vortex centreld how to correct for this error byLing them to diffuse in the Hamel-

)·0 r---,----,----,.-----,

velocity, they calculated the lateral spacing of the vortex rows as h = h* - 2r*where h* is the distance between the r.m.s. velocity peaks. Although Schaefer &Eskinazi obtained good theoretical and experimental agreement, therebyjustifying their assumptions, the vortices in the present case are (as will be shown)not circular and therefore the applicability here of the above method is question­able.

A reasonable assumption here is that the across-wake location, h"" of thevortex centres coincides with the maximum in the mean vorticity. Using theapproximation W ~ - oujoy, this centre is also the maximum slope of the across­wake mean velocity plots. An examination of Schaefer & Eskinazi's data didindeed show that for their vortex street the across-wake distance between themaximum slopes of their mean velocity plots, h"" was identical to their computedvalues of h.

517On the phenomenon of vortex street breakdown

Largecylinder

aryxt

1-4 1·6 1·8

n zeros.

0·8

~ 0·6:a...::

..s£*...::

..s£ 0-4

...::

0·2 1--~x;...--+_+_=---t--~__1

0·80·60·2 0·4

x [in.]

FIGURE 10. Spacing ratios.

0·6 0·8

m spacing.

ihe same method for use with hot­es to be of the Hamel-Oseen typeat a distance of r* = (5'04vr)i fromImption that the maximum in theth the outside maximum peripheral

Figure 10 is a comparison between the different cross-wake spacings h*, h""and h as functions of x, all normalized by the local downstream spacing ratio a.For comparison, von Karman's value of 0·281 is also shown. Figure II showsT = r(x) obtained from the measurements by noting that a vortex took Ij700secto traverse a vortex spacing given by figure 9.

Whereas the data of Schaefer & Eskinazi show h",ja and hja to be coincident,figure 10 reveals that, for the present case, there is a marked difference. Theacross-wake spacing ratio based on mean vorticity maxima, h",ja, is alwayssubstantially greater than that calculated from the spacing of r.m.s. velocitymaxima and vortex age, hja, indicating a reduction in the lateral diffusion ofvorticity.

The phase of the hot-wire signal relative to the reference signal was measuredfrom photographs of oscilloscope traces representing the velocity fluctuations

518 W. W. Durgin and S. K. F. Karlsson On the phenomenl

and the reference signal. These measurements were made at several across-wakelocations for various downstream locations. Figure 12 shows the phase angleplotted versus y for fixed values of x, the downstream stations. The inside of avortex, i.e. the part nearest y = 0, is seen to lag the outer part for all but onecurve. This lag becomes especially pronounced, almost a full cycle, just priorto the calm region. The missing points ofthe last two frames correspond to weaksignals whose phase could not be reliably determined.

FIGURE 13. Wave analy

FIGURE 12. Relative phasl[·00·80·60·4

x [in.]

FIGURE 11. Vortex age.

0·2

1,./

JI

J

//-

VV

2

oo

4

8

10

12

Figure 13 shows wave analyses of the hot-wire signal at various downstreampositions along the wake centreline. The band pass of the wave analyzer was1octave and the vertical axis represents (u2)!. The data was taken in the smallwind tunnel using a cylinder separation of 1·2 in. and a primary wake frequencyof 670Hz. The curve at x = 0·24in. shows a peak at the primary frequency andlesser peaks at the first and second harmonics. The curve at x = 0·4 shows peaksat the fundamental and first harmonic only. These locations were in the primaryvortex street region. At x = 0·6 and 0·7 in. an entirely different spectrum ispresent; this is the secondary vortex street region. The fundamental of 240Hzpresents the largest peak and first and second harmonics are visible. At anystation, if the probe was moved away from the centreline, the harmonics wouldquickly disappear and the fundamental gradually die in amplitude.

jill? ----------- l..... _

Karlsson

'e made at several across-wakeIre 12 shows the phase angleream stations. The inside of athe outer part for all but onealmost a full cycle, just priorwo frames correspond to weakned.

j

!

On the phenomenon of vortex street breakdown

J$~

:~f 0'20t.=ffi

0·25

::1 0'35~0·30

~:l 0-40~0·45

~til

::fb 0'50~illtil 0·55

..c:p.

:~t ~::~~

::1 ~·75

::1 ~0·85V) V) V) V) 0

0 '" <') :::0 06 6 6 6

Y [in.]

519

0·8 '·0 FIGURE 12. Relative phase angle of velocity fluctuations.

Frequency [Hz]

FIGURE 13. Wave analysis along the wake centreline.

~e.

e signal at various downstreampass of the wave analyzer was~he data was taken in the small. and a primary wake frequencyk at the primary frequency and'he curve at x = 0·4 shows peaksse locations were in the primaryL entirely different spectrum is.on. The fundamental of 240 Hzharmonics are visible. At any

Jentreline, the harmonics wouldly die in amplitude.

200 300 1000.

J

Other relevant experimentsTwo experimental investigations into vortex street breakdown by Taneda

(1959) and Zdravkovich (1968) are of particular interest in connexion with thepresent work.

Using a glass-sided towing tank Taneda was able to observe the wakes behindtowed cylinders and plates for very long distances via the aluminium dust, sideslit lighting technique. His improved experimental techniques allowed muchmore extended observations of the vortex street development than had beenpossible in earlier wind-tunnel investigations where typically the free-streamvelocity fluctuations due to background turbulence were comparable in magni­tude to fluctuations in the vortex street at 30 diameters downstream from theshedding cylinder, effectively terminating the study of further vortex streetdevelopment.

For Reynolds numbers less than 150, Taneda found the primary vortex streetlasted for about 50 cylinder diameters before breaking down. The secondaryvortex street formed slowly, reaching full development about 103 diametersdownstream. He found the secondary vortex street wavelength to be between1·8 and 3·3 times the primary wavelength. For Reynolds numbers greater than150 he found that the vortex street breaks down to turbulence and then formsa secondary vortex street with about 10 times larger wavelength than theprimary one. In a more recent investigation Taneda (1965) has found that theturbulent wake regime behind a flat plate also shows a strong tendency to becomeunstable and form a vortex street.

Taneda also found that the secondary vortex street either broke down forminga third vortex street or died out through viscous diffusion.

Zdravkovich qualitatively studied the wake of a group of three cylindersusing smoke to make the wake visible in a vertical wind tunnel. By carefulpositioning of the three cylinders he was able to generate single vortex streetswith charaoteristically large spacing ratios (e.g. 0·67 for one experiment). Fromhis photographs it can be seen that the vortices of this vortex street continuallydeform; the outside of a vortex appears to advance and the inside slows downrelative to the 'vortex centre'. The vortices thus become somewhat oval inshape finally aligning themselves with the mean flow direotion and become in­distinguishable. Sinusoidal oscillations are then seen to arise and develop into

520 W. W. Durgin and S. K. F. Karlsson On the phenome

a new vortex street of larger waVIis usually about 2 to 1.

The relatively large spacing ravortex street behind three cylinapparently governs in some way 1oircular shape to one conducive 1of the secondary vortex street.

DiscussionThere are essentially three diffl

the vortex street breakdown in tlthe vortex street behind a oylin~

filaments ourve about the large cyof the mean flow around the larglThirdly, the vortices grow closer ttravel downstream because the mlNeither Taneda's nor Zdravkovicor stretching of the vortex filamlbehaviour similar to that observexperiment was, however, the largforced to have because of the meexperiments, bending and stretchprimary causes of vortex street bbreakdown prooess sought, whiclstrongly on the spacing ratio of thl

Taking the same point of view:and letting the flow in the vortexposition, we see that anyone vorteall the others. It is thus approprialof one selected vortex by all the ot

Shortly after formation close to 1vortex street is a highly concentratcalled the vortex centre. If, initiainfinitesimal filament of circulatiobounded, the ensuing diffusion of V(

x in phase (in.)

0·1230·2520·3810·5090·6270·7300·816

a spacing (in.)

0·1290·1290·1280·1l80·1030·086

TABLE 1

x location of a (in.)

0·1l80·3160·4450·5680·6780·773

]w=-

471

where r2 = x 2 + y2 and the co-ordinlIn this solution the vorticity diffmcreating a 'diffuse' vortex or patch

At distances r ~ (VT)! from the ceto a potential vortex. In fact, a cjradius of maximum induced peripl

'. Karlsson On the phenomenon of vortex street breakdown 521

[ street breakdown by Taneda: interest in connexion with the

ble to observe the wakes behindles via the aluminium dust, sideental techniques allowed much,et development than had beenvhere typically the free-streammce were comparable in magni­liameters downstream from thestudy of further vortex street

found the primary vortex streetbreaking down. The secondary{elopment about 103 diameterstreet wavelength to be betweenReynolds numbers greater than'n to turbulence and then formsles larger wavelength than thetneda (1965) has found that theows a strong tendency to become

street either broke down forminglS diffusion.B of a group of three cylindersertical wind tunnel. By carefulto generate single vortex streets,0,67 for one experiment). From, of this vortex street continuallyrance and the inside slows downthus become somewhat oval inm flow direction and become in­n seen to arise and develop into

a new vortex street oflarger wavelength than the original. The wavelength ratiois usually about 2 to 1.

The relatively large spacing ratio is the only obvious difference between thevortex street behind three cylinders and that behind a single cylinder. Thisapparently governs in some way the deformation of each vortex from its usuallycircular shape to one conducive to the conditions which precede the formationof the secondary vortex street.

DiscussionThere are essentially three differences between the flow field associated with

the vortex street breakdown in the present case and the flow field surroundingthe vortex street behind a cylinder in an unbounded fluid. First, the vortexfilaments curve about the large cylinder as shown in figure 8. Secondly, by virtueof the mean flow around the large cylinder, the vortex filaments are stretched.Thirdly, the vortices grow closer together (in the downstream direction) as theytravel downstream because the mean flow is slowing as shown in figures 8 and 9.Neither Taneda's nor Zdravkovich's experiments apparently involved bendingor stretching of the vortex filaments on any appreciable scale yet they showbehaviour similar to that observed here. The unique feature of the latter'sexperiment was, however, the large spacing ratio the primary vortex street wasforced to have because of the method of its generation. On the basis of theseexperiments, bending and stretching will be tentatively ruled out as possibleprimary causes of vortex street breakdown, and a mechanism to describe thebreakdown process sought, which is two-dimensional in nature and dependsstrongly on the spacing ratio of the primary vortex street.

Taking the same point of view as von Karman (and, of course, others later)and letting the flow in the vortex street be given by the vortices in type andposition, we see that anyone vortex can be deformed by the combined action ofall the others. It is thus appropriate to examine, in some detail, the convectionof one selected vortex by all the other vortices.

Shortly after formation close to the shedding cylinder a vortex in the primaryvortex street is a highly concentrated distribution of vorticity about some pointcalled the vortex centre. If, initially, the vorticity is all concentrated into aninfinitesimal filament of circulation r (potential vortex), and the field is un­bounded, the ensuing diffusion of vorticity is given by the Hamel-Oseen solution

where r2 = x 2 + y2 and the co-ordinate system is fixed in the centre ofthe vortex.In this solution the vorticity diffuses radially outward from the vortex centrecreating a 'diffuse' vortex or patch of vorticity.

At distances r ~ (vT)i from the centre of a Hamel-Oseen vortex, it is identicalto a potential vortex. In fact, a circle with the radius r = r* = (5'04vT)i, theradius of maximum induced peripheral velocity, includes approximately 70 %

x location of a (in.)

0·1180·3160·4450·5680·6780·773

r (r2)w = 41TVT exp - 4VT ' (3)

8

~

"'\4

2

On the phenome

and the ellipse bounding the enreplacing R. The principal axes 0

o

6

C=1

The area of the ellipse, 7Tcb, is couniform distribution of vorticity

A uniform elliptical vorticity divelocity (Lamb 1945, p. 232)

n,

Thus the elliptical vorticity distlin the same sense as its circulatdeformation increases.

xy

-4-__ x

(b) a<O

xy

)(--11--__ X

W. W. Durgin and S. K. F. Karlsson

(a) co 0

C(x, y, t) = (xjR)2 cosh (2cd) - 2(xyjR2) sinh (2at) + (yjR)2 cosh (2at) = 1. (5)

FIGURE 14. Modes of vortex straining.

C(x, y, t = 0) = (xjR)2 + (yjR)2 = 1,

vortex region is in general convected into an elliptic shape. More precisely, acircle about (0,0) given by

is, to second order of approximation, deformed into

522

of the vorticity associated with the vortex. Providing none of the vortices'overlap', that is (!a)2+h2 > 4r~ and a > 2r* where h is used for the distancebetween rows since there is no ambiguity here, we may say that to a good approxi­mation all the vortices look like potential vortices to the one under consideration.Further assuming a constant downstream spacing, a, and across-wake spacing,h, it can be shown (Durgin 1970) that the initially circular cross-section of the

FIGURE 15. Tb

-2o 0·10

As soon as a vorticity distributiorthe other vortices induce, and conSEthe problem of finding the motion I

and the foregoing analysis is not sclear, however, that the same proce,qualitatively some insight may bediscrete.

Comparison with experimentThe phenomenon observed in the

than 0·366 and thus in the a < 0 regthe part ofa vortex outside of the v(stream direction. This mode is in agJvariations (figure 12).w = rj7Tr~

If a = 0 the curve C remains circular. If a > 0 the original circle deforms withtime as shown in figure 14 (a). The major axis ofthe ellipse lies on the X axis whilethe minor lies on the Y axis. If a < 0 the reverse situation is true as shown infigure 14(b) with the major axis ofthe ellipse on the Y axis and the minor axison the X axis.

The parameter a is a function of the vortex spacing ratio which has been Iiplotted in figure 15 in terms of 27Ta2ajr VS. hja. It can be seen that a > 0 ifhja < 0·366 and a < 0 if hja > 0'366.1 Thus the fluid near the centre of anyvortex can be strained into two possible modes depending on the values of thespacing ratio.

As soon as a vortex deviates from its idealized circular shape its shape can befurther changed by its own induced velocities. For the purpose of finding theself-induced motion ofthe elliptically deformed vortex it is convenient to assumethat all the vorticity associated with the vortex is initially uniformly distributedwithin a specified radius, say r*. This uniform vorticity is given by

This is the equation of an ellipse which upon rotation of the co-ordinatesystem by 45° to the principal axes becomes

;ex straining.

n elliptic shape. More precisely, a

523

0·500·400·10 0·20 0·30hja

FIGURE 15. The function 27Ta2ajr V8. hja.

~

l'",'\

~1\

~\0

f'"ex; \0

~N M6 6

2

4

On the phenomenon of vortex street breakdown

o

-2 o

6

8

As soonas a vorticity distribution is strained by a finite amount by the velocitiesthe other vortices induce, and consequently rotated by its own induced velocities,the problem of finding the motion of the deformed vortex is no longer separableand the foregoing analysis is not strictly valid for the subsequent motion. It isclear, however, that the same processes ofdeformation and rotation continue andqualitatively some insight may be gained by imagining that the processes arediscrete.

and the ellipse bounding the ensuing strained vortex by equation (6) with r*replacing R. The principal axes of the ellipse have lengths

c = r*eext, b = r*e-ext. (7)

The area of the ellipse, 7Tcb, is conserved so that within this approximation theuniform distribution of vorticity is maintained.

A uniform elliptical vorticity distribution in an ideal fluid rotates with angularvelocity (Lamb 1945, p. 232)

n = [cbj(c+b)2]w. (8)

Thus the elliptical vorticity distribution rotates, because of its self-induction,in the same sense as its circulation with the rotation rate decreasing as thedeformation increases.

(4)

xy

--4-__ x

(b) 0:<0

(

rtex spacing ratio which has beenI. hla. It can be seen that IX > 0 ifs the fluid near the centre of anydes depending on the values of the

upon rotation of the co-ordinate

[YjRe-a-t)2 = 1. (6)

> 0 the original circle deforms withofthe ellipse lies on the X axis while)Verse situation is true as shown ine on the Y axis and the minor axis

r(yjR)2 = 1,

~d into

lh(2at) + (yjR)2 cosh (2IXt) = 1. (5)

F. Karlsson

Providing none of the vorticeswhere h is used for the distance~emay say that to a good approxi­3es to the one under consideration.,cing, a, and across-wake spacing,tially circular cross-section of the

lized circular shape its shape can beiies. For the purpose of finding theled vortex it is convenient to assumetex is initially uniformly distributed~m vorticity is given by

,2

*

Comparison with experimentThe phenomenon observed in the experiments occurred at spacing ratios larger

than 0·366 and thus in the IX < 0 regime in which, according to the simple model,the part of a vortex outside of the vortex street leads the inside part in the down­stream direction. This mode is in agreement with the observed across-wake phasevariations (figure 12).

Neglecting u(ov/ox) and u'(ov'lox) in comparison with the other terms the circula­tion becomes

At x = 0·65 (just preceding serious phase shifts) the downstream spacinga == 0·107in. and the spacing ratio h,ja = 0·4 which will be taken to be h/a forthe purpose of this comparison. The circulation around any vortex is given by

On the phenomenori

Stability of the calm regionAn examination of the mean veloc

profiles, figure 7, reveals that the calvorticity of opposite signs. These prformed behind a flat plate in a unijflow to sinusoidal perturbations in x(1961). Although their analysis wasinteresting, nevertheless, for the pUIresults to the present case, overlookiJFigure 16 shows the mean velocity a1compared to their velocity distributil([fo - Uc)/Uo = 0'609 for the present ca

T!

(9)

(10)

(ll)

r = fA wda

_ _ ov __ ou ,ov' ,ou'uw = u- -u-+u --u -.ox oy ox oy

W. W. Durgin and S. K. F. Karlsson524

The integrand may be expanded to read

providing the vortices do not overlap and the integration is only carried out overthe vorticity belonging to the vortex under consideration. Neglecting diffusionin the downstream direction the mean vorticity flux per unit depth through aplane, x = const. from y = - th to 00 (here - th is the wake centreline) is thenjrwherefis the frequency at which vortices pass a fixed x. But the mean vorticityflux may also be written

(12)

Substituting the appropriate values for x = 0·65in. from figures 4 and 6 andf = 700Hz the circulation around a vortex at x = 0·65in. is r = 9·9in.2/sec.From figure 11 the vortex age is found, whence r* = 0·03 in. From figure 1527Ta2ajr = 0'56, when a = 76. Using t = 1/700 sec in equation (7)

c/r* = 1,12, b/r* = 0·89

in one cycle. From equation (8) the corresponding rotation rate becomesn = 861 rad/sec or in terms of rotation, e, the ellipse would rotate e = 70° inone cycle. Evidently, this estimate is too large, because rotation ofthis magnitudedoes not occur in the experiment. Clearly, simple superposition of the twomechanisms at work here is not valid for that large deformations.

Limitation on the extent of rotation in the present model can be explainedby noting that as a strained vortex rotates its extremes enter regions of largevelocities induced by its neighbours of the same row which are undergoingsimilar processes. These velocities counteract the self-induced rotation of thedeformed vortex. As the vortices rotate, they assume some alignment in thedownstream direction whereupon the character of the convection has changeddrastically. This alignment causes the vortices to become indistinguishable fromone another and, in essence, a shear layer is formed in each row. These shearlayers would correspond to what has previously been called the calm region.

2

y

FIGURE 16. Non-dimensional velcO,x=:O·

Computing the frequency of the nSato & Kuriki, it is found that (T = ,

x = 0·95. These values are somewhat Itally. It should be noted that Sato &experimental values for Reynoldsnuml500. The same Reynolds number forrelatively low value of the Reynoldseffects are not completely negligible. Ununknown, is the influence of the decelThe analysis does, however, predict a700 Hz primary wake frequency.

Conclusions

Throughout this study of vortex streof view that a certain distribution of vthe wake. This idea was originated by .of velocities induced by potential vorticdown, we have found that it is necess

11'0-.---------__1 _

________________J

Stability of the calm regionAn examination of the mean velocity profiles, figure 4, and the mean vorticity

profiles, figure 7, reveals that the calm region consists of two adjacent layers ofvorticity of opposite signs. These profiles are reminiscent of the laminar wakeformed behind a flat plate in a uniform stream. The stability of this type offlow to sinusoidal perturbations in x and t has been studied by Sato & Kuriki(1961). Although their analysis was developed for uniform external flow it isinteresting, nevertheless, for the purpose of a rough comparison, to apply theresults to the present case, overlooking for the moment this obvious difference.Figure 16 shows the mean velocity at x = 0·85 and 0·95 in. for the present casecompared to their velocity distribution. The agreement is reasonably good but(UO - Uc)/Uo = 0·609 for the present case as compared to 0·692 for Sato & Kuriki.

F. Karlsson

shifts) the downstream spacingwhich will be taken to be h/a forn around any vortex is given by

(9)

ntegration is only carried out overonsideration. Neglecting diffusionity flux per unit depth through a1h is the wake centreline) is then2

ss a fixed x. But the mean vorticity

On the phenomenon of vortex street breakdown 525

)ll with the other terms the circula-

ly.

11 ,ou'--U-.IX oy

(10)

(11)

,...--,..-----r----::::o-o-~--_.---__r-__, 1·0

i:J Ii:JcI--+-----=:;!-----t-----'k:-----t--j 0·5 I I

tftf

~=:-.L.._ .L.._ .L.._ .L.- J....::=_O

(12)2 -1 -2

0'65in. from figures 4 and 6 andat x = 0·65in. is r = 9·9in.2/sec.lence r* = 0·03in. From figure 15Osee in equation (7)

= 0·89

esponding rotation rate becomeshe ellipse would rotate () = 70° inbecause rotation ofthis magnitude

, simple superposition of the twot large deformations.Le present model can be explainedits extremes enter regions of large: same row which are undergoingct the self-induced rotation of theley assume some alignment in thelter of the convection has changedlS to become indistinguishable froms formed in each row. These shear[sly been called the calm region.

Computing the frequency of the most amplified scale from the results ofSato & Kuriki, it is found that (T = 465 Hz for x = 0·85 and (T = 426 Hz forx == 0·95. These values are somewhat higher than the 250Hz found experimen­tally. It should be noted that Sato & Kuriki found good agreement with theirexperimental values for Reynolds numbers based on the wake halfwidth, b, above500. The same Reynolds number for the present case was about 100. Thisrelatively low value of the Reynolds number indicates that perhaps viscouseffects are not completely negligible. Undoubtedly more important, but presentlyunknown, is the influence of the decelerating external flow on the frequency.The analysis does, however, predict a considerably lower frequency than the700 Hz primary wake frequency.

ConclusionsThroughout this study of vortex street breakdown we have taken the point

of view that a certain distribution of vortices determines the velocity field ofthe wake. This idea was originated by von Karman in his original summationof velocities induced by potential vortices. In the case of vortex street break­down, we have found that it is necessary to extend the above approach by

526 W. W. Durgin and S. K. F. Karlssonconsidering, in some detail, the convective processes acting on any given vortex.

The phenomenon of vortex street breakdown is characterized by the dis­appearance of a vortex street into a region where no significant velocity fluctua­tions are observed. This region, in turn, gives rise to another vortex street oflarger scale and lower frequency than the original one. The detailed experimentalfindings suggest and are explained by a model in which a local concentration ofvorticity, or vortex, is strained or convected by the other vortices of the streetto an elliptical shape. This distorted vortex then rotates in the proper direction,through its self-induced velocities, approaching alignment of its major axis inthe downstream direction. Because these processes take place as the vortices aretravelling downstream, a region is reached where they touch or overlap to theextent they become shear layers on either side of the wake. These shear layersare then unstable and give rise to the secondary vortex street.

The convection of vorticity within a vortex by the other vortices of the streetinitiates the phenomenon. A spacing ratio greater than 0·366 is indicative of thestraining of the vortices in the proper direction for the calm region to be formed.The reverse situation of straining such that the velocity fluctuations outside thestreet lag those inside (hja < 0'366) has, apparently, not been observed by otherinvestigators and deserves further investigation.

It is interesting to speculate that similar processes may be present in turbulentflows. If small vortices making up a turbulent flow are subjected to the properstraining by other vortices a regrouping of vortices to a larger scale and hencelower frequency might occur.

The financial support of the National Science Foundation under NSF GrantGK-3854 is gratefully acknowledged.

REFERENOES

DURGIN, W. W. 1970 An experimental and analytical investigation of the mechanismcausing the phenomenon of vortex street breakdown. Ph.D. thesis, Brown University.

HOOKER, S. G. 1936 On the action of viscosity in increasing the spacing ratio of a vortexstreet. Proc. Roy. Soc. A 154, 67-89.

KARMAN, TH. VON & RUBACH, H. 1912 Uber den Mechanismus des Flussigkeits undLuftwider standes. Phys. Z. 13, Heft 2, 49-59.

KOVASZNAY, L. 1949 Hot-wire investigation of the wake behind cylinders at low Reynoldsnumbers. Proc. Roy. Soc. A 198,175-190.

LAMB, H. 1945 Hydrodynamics, 6th edn. New York: Dover.PAYNE, R. 1958 Calculations of unsteady viscous flow past a circular cylinder. J. Fluid

Meck. 4, 81-86.ROSHKO, A. 1953 On the development of turbulent wakes from vortex streets. NACA

TN 2913.SADEH, W., SUTERA, S. P. & MAEDER, P. F. 1968 An investigation of vorticity amplifica.

tion in stagnation flow. Brown Univ., Div. of Engrg., Tech. Report WT-50.SATO, H. & KURIKI, K. 1961 The mechanism of transition in the wake of a thin flat

plate placed parallel to a uniform flow. J. Fluid Mech. 11, 321-352.SCHAEFER, J. W. & ESKINAZI, S. 1959 An analysis of the vortex street generated ina

viscous fluid. J. Fluid Mech. 6, 241-260.TANEDA, S. 1959 Downstream development of the wakes behind cylinders. J. Phys. SIX.

Japan, 14, 843-848.

On the phenomenon

TANEDA, S. 1965 Experimental investig1714-1721.

WOOD, R. T. 1969 An experimental andlying the influence of free-stream tfrom cylinders in cross flow. Ph.D. t

ZDRAVKOVICH, M. 1968 Smoke observaat low Reynolds number. J. Fluid 11<

r-- ..--------------A- _!f

I

TANEDA, S. 1965 Experimental investigation of vortex streets. J. Phys. Soc. Japan, 20,1714-1721.

WOOD, R. T. 1969 An experimental and analytical investigation of the mechanism under·lying the influence of free-stream turbulence on the stagnation line heat transferfrom cylinders in cross flow. Ph.D. thesis, Brown University.

ZDRAVKOVICH, M. 1968 Smoke observations of the wake of a group of three cylindersat low Reynolds number. J. Fluid Mech. 32, 339-351.

· F. Karlsson)Cesses acting on any given vortex.own is characterized by the dis­lere no significant velocity fluctua­~s rise to another vortex street ofnal one. The detailed experimental:1 in which a local concentration ofby the other vortices of the streetlen rotates in the proper direction,ing alignment of its major axis in:esses take place as the vortices are-here they touch or overlap to theIe of the wake. These shear layerstry vortex street.: by the other vortices of the street:ater than 0·366 is indicative of then for the calm region to be formed.1e velocity fluctuations outside the,rently, not been observed by otherion.'ocesses may be present in turbulentlt flow are subjected to the proper'ortices to a larger scale and hence

,nee Foundation under NSF Grant

!ES

lytical investigation of the me?han~sm~down. Ph.D. thesis, Brown Umverslty.increasing the spacing ratio of a vortex

len Mechanismus des Flussigkeits undl.~ wake behind cylinders at low Reynolds

)rk: Dover.3 flow past a circular cylinder. J. Fluid

lent wake~ from vortex streets. N ACA

An investigation ofvorticity amplifica.Engrg., Tech. Report WT·50.)f transition in the wake of a thin flatuid Mech. 11, 321-352.{sis of the vortex street generated in a

Ie wakes behind cylinders. J. Phys. Soc.

On the phenomenon of vortex street breakdown 527