ABSTRACT - WIT Press€¦ · 30 cnf capacity. The test cylinder of 20.2 mm diameter is towed at a constant speed of 5.53 mm/s, the resultant Reynolds number being 100 approximately

Experimental measurements on cylinder

wakes and vortices

K. Ohmi

Nakagaito, Daito-shi, Osaka 574, Japan

ABSTRACT

The wake characteristics of a two-dimensional circular cylinder are examined bycombined use of image processing velocimetry and numerical calculation basedon the measured velocity field. The results are mainly presented in the form ofcontours of stream function and vorticity and discussion is centered upon thetemporal variation of wake parameters. Then, some additional calculations areattempted as to circulation of wake vortices and vorticity discharge along theseparating shear layer. Some consideration is made on the possibility of applyingthese experimental data to discrete vortex calculations.

INTRODUCTION

The vortex wake of a circular cylinder in a uniform flow has been one of themost challenging problems in fluid mechanics for long decades. However, evenat low Reynolds numbers (Re =100), where a regular shape of von Karmanvortex street is present, the behavior of vorticity-bearing fluid all the way fromthe cylinder surface to the downstream wake has not been closely examined sofar, although this is obviously related to the mechanism determining the dragforce on the cylinder and the wake frequency.In the present study, the author is concerned with visualization experiments of2-D wakes of a circular cylinder accompanied by the use of image-processingvelocimetry, and with some numerical calculations of vorticity and circulationbased on the velocity field obtained from the image-processing velocimetry. Themain interest of the study is to examine both spatial and temporal variation ofvorticity from the generation phase along the wall surface of the cylinder down

Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

160 Computational Methods and Experimental Measurements

to the convection and dissipation phase giving rise to downstream vortices.On the other hand, it is probable that the major parameters investigated in thepresent experiment bear a certain importance when starting a purely numericalcalculation in the same conditions of flow. Since the discrete vortex calculation,above all, seems to benefit from the results of this investigation, some practicalsuggestions are given as to modeling the viscous wake by a series of potentialvortices.

FLOW VISUALIZATION AND IMAGE PROCESSING

The visualization experiment is conducted in a towing water tank of 250 x 30 x30 cnf capacity. The test cylinder of 20.2 mm diameter is towed at a constantspeed of 5.53 mm/s, the resultant Reynolds number being 100 approximately.The flow visualization is carried out under the water (not on the free surface) byusing very fine aluminum particles in suspension, producing a large number ofparticle streaks of the relevant flow. These streak images are photographed withan adequate exposure with a view to visualizing quasi instantaneous distributionof velocity. In the present experiment, the photographic camera is situated in arelative frame fixed to the towed cylinder.The image processing technique is then introduced to quantify the distribution ofvelocity recorded on the photographic images. Local velocity is defined as lengthand gradient of each particle streak and their effective values are computed fromdivision by the exposure time. All these steps are computerized and processedautomatically, except for the assignment of velocity directions in which somemanual operations are needed [1].The resulting distribution of velocity is useful indeed in itself, but not necessarilysuitable for the subsequent computation of vorticity (differentiation of velocity)because local vectors are located at random intervals in both x and y directions.Therefore, the local vectors are processed by a 2-D interpolation algorithm usinga least squares method and transferred into mesh-pattern distribution of velocity.In consequence, the computation of vorticity is conducted by the use of a usualfinite-difference scheme and the results are presented in the form of contours.Then, the circulation is evaluated through spatial integration of vorticity aroundevery vortex-forming region. Further detail of this numerical process is describedelsewhere by the present author et al.[2].It is often pointed out that vortex circulation is rather difficult to evaluate withaccuracy by means of experiments. If especially, the evaluation is based solelyon distribution of velocity, the circulation of wake vortices cannot be properlycomputed because of ambiguity in integration paths. In their famous experimentin early days, Page and Johansen [3] attempted to estimate vortex circulation inthe wake of a flat plate using a potential theory of vortex street (geometry of thevortex street and rate of their streamwise displacement are the known parametersin the experiment), while they employed another method for computing the rateof vorticity discharge at separation from velocities along the two boundaries of ashear layer. More recently, Timme [4], Schaefer and Eskinazi [5] and Berger [6]


Computational Methods and Experimental Measurements 161

Figure 1 One cycle of vortex shedding in a circular cylinder wake (Re=100).

tried to estimate vortex circulation in a circular cylinder wake from experimentaldistribution of velocity but they all used a vortex street model consisting ofviscous vortices. By contrast, Cantwell and Coles [7] and Tanaka and Murata [8]evaluated circulation without vortex model through spatial integration ofvorticity. This method has the advantage of determining the area of integration inaccordance with vorticity concentration. This is the main reason why the presentexperiment adopted this last method for computing vortex circulation.

RESULTS AND DISCUSSIONS

(1) STREAMLINE PATTERN

Fig.l shows photographic images of the visualization in a 2-D wake of a circular



Figure 2 Contours of stream function corresponding to visualizations in Fig.l.

cylinder at Re-100. The six frames of this figure are recorded with an intervalequal to 1/6 of the vortex shedding period. The velocity filed represented by alarge number of particle paths is then quantified with the aid of image-processingvelocimetry and arranged as mesh-pattern distribution of velocity (35x19 meshpoints, where lateral and transverse intervals are both equal to 1/4 of the cylinderdiameter). On the basis of this form of velocity distribution, stream function andvorticity are computed through a finite-difference approximation. The contoursof the two computed parameters are traced in Figs.2 and 3, in the latter of whichthe contours of zero vorticity are excluded.It is well known that if the flow is viewed from a frame fixed to the cylinder, thefamiliar streamline pattern of a K arm an vortex street cannot be necessarilyrecognized. This fact is confirmed in both Figs.l and 2 of the present work. Inthis frame of a stationary observer, the vortex developed on either side of thecylinder is seen to spread over the near wake and penetrate into the opposite sidearea across the center axis of the wake ((a)-(b) in Fig.2). This vortex is thenswept away gradually by the increasing effect of the S-shaped transverse flow(referred to as "alleyway") between the cylinder and the vortex itself ((c)-(e) inFig.2) and finally detached in full from the body ((f) in Fig.2). In the meanwhile,



Figure 3 Contours of vorticity corresponding to visualizations in Fig.l.

the streamlines around the separating vortex present a slant "U" or "V" patternand, as they travel downstream, change into a sinuous wave pattern. If, on thecontrary, the flow is viewed from a relative frame moving with the free streamvelocity,' the classical streamline pattern of a Karman vortex street is observed.However, this is not included in the present work because the streamline pattern

is not a main subject of the study.

(2) VORTICITY AND CIRCULATION

The vorticity contours are shown in Fig.3, in which the vortex-forming fluidmotions in a cylinder wake are clearly depicted with more physical background.In (a) and (b) of Fig.3, the vorticity accumulated on the two sides of the cylinderis prone to concentration on the upper side and to dissipation on the lower side.As the vortex region of the concentration side is spread downstream, the vorticitybearing fluid penetrates gradually into the opposite dissipation side and gives nseto a constricted part in the latter vortex region. This constricted part on thedissipation side grows furthermore and finally leads to division of the vortexregion itself ((c) in Fig.3). This surely is a vorticity phenomenon correspondingto vortex shedding. After this stage, the leeward portion of the vortex region is



1 2 3 4 5 6 7

Figure 4 Stream wise variation of vorticity peak.

4.0

3.0

2.0

1.0

0.010

X/d

Figure 5 Streamwise variation of circulation.of the wake vortices.

separated from the near wake rapidly. The remaining portion of the vortex, onthe other hand, retains relatively high intensity of vorticity but is now difficult toextend its domain because this is covered with large extension of the oppositevortex region. At this moment, the vorticity concentration and dissipation areturned over on the two sides of the cylinder And by iterating this switchingprocess, alternate vortex shedding from the cylinder and staggered arrangementof the downstream vortices are establishedThe vortices shed from the cylinder lose their vorticity as they travel downstreamand finally vanishes by viscous dissipation. In order to get some understandingof this vorticity behavior, stream wise variation of the vorticity peak and that ofthe vortex circulation are examined from the vorticity distribution of Fig.3. Therespective results are given in Figs.4 and 5, where both of the parameters arecomputed in non-dimensional unit ((*)*=wd/U, P= T/Ud). The vortex circulation



is calculated through spatial integration of voracity but the integration is limitedto non-dimensional vorticity greater than 0.2 in absolute value.

As shown typically in Fig.4, the voracity peak in the wake is reduced rapidly atfirst, but more and more slowly as it goes downsaeam, the variation being tracedby a damping curve. The peak plots in the vicinity of the cylinder suffer from acertain inaccuracy because of the use of a Cartesian mesh pattern for the presentnumerical calculation. But it is still estimated that the peak values are reduced byhalf at about 6 diameters downstream of the cylinder.The plots of vortex circulation in Fig.5 are not abundant because the computationis limited to the detached vortices. But one tendency recognized in comparisonwith the vorticity peak is that the reduction of circulation is rather linear thandamping This fact leads to an observation that the viscous vortex shed from thecylinder loses more vorticity from the central part at initial stages, but from thecircumferential part at later stages. In fact, many of the original visualizations inthe present experiment seems to give proof of this view of vortex behavior.There exist some other experiments attempting experimental determination ofvorticity in a circular cylinder wake; e.g. Cantwell and Coles [7], Tanaka andMurata [8], Okude and Matsui [9] and Green and Gerrard [10]. The first twogroups are concerned with high Reynolds number flows in a wind tunnel (Re-1 4x105 and 3 7xW respectively), whereas the last two groups with much lowerReynolds numbers (Re= 140 in a wind tunnel and Re= 70 to 120 in a towing

water tank respectively).The experimental results of Cantwell and Coles [7] and Tanaka and Murata [8]are often similar except that the vorticity peaks are 30 to 40 % lower in the firstsroup The streamwise reduction of vorticity peak is fit by a damping curvewhile'that of the vortex circulation is relatively linear. The estimated values ofvortex circulation are very close to each other, though they are 30 to 35 % lowerthan the present experiment. The higher vorticity peaks of Cantwell and Colesare still 35 % lower than the present experiment.By contrast, the results of Okude and Matsui [9] and Green and Gerrard [10]come closer to those of the present experiment. Better agreement is met with thefirst group in estimating the vortex circulation (T = 2.6 at x/d= 6.0) Circulationestimated by Green and Gerrard is 15 to 20 % higher than that of the presentexperiment around Re=100. Unfortunately, both groups do not give the stream-wise variation of vorticity peaks comparable with that of the present author.Other authors concerned with experimental determination of vortex circulation,such as Timme [4], Schaefer and Eskinazi [5] and Berger [6], computed just afew values of circulation using a viscous vortex model and their results can beplotted within about 10 % deviation from the experimental curve of the present

experiment.

(3) VORTICITY DISCHARGE AND DISCRETE VORTEX MODEL

Finally some additional calculations are made for examining temporal variationof total circulation in the near wake of the cylinder. In other words, circulation iscomputed around the upper (or lower) vortex region attached to and spread out



Time

Figure 6 Variation of total circulation of the near wake vortex during one cycleof vortex shedding (T : period of vortex shedding).

from the cylinder, at every time stage during one cycle of vortex shedding. Theresult is shown in Fig.6, where the six plots stand for the computed values ofcirculation. As shown clearly in this figure, the total circulation increases duringone cycle with accumulation of vorticity and, at the final stage, where the vortexshedding takes place, this returns to the initial value within the extent of errors.Needless to say, the difference in circulation between the initial and final stagescorresponds to the initial circulation of the detached vortex (2.6 in this case).Judging from the circulation curve in Fig.6, it is thought that the rate of vorticityproduction is decreased with time during one cycle and reaches a minimum justbefore the vortex shedding. This is however related to the fact that the increase intotal circulation is apparently ceased because the vortex region is spread over alarger area of the downstream wake and met with more dissipation or damping ofvorticity. If the rate of vorticity production is constant during one full cycle inthe vicinity of the cylinder, this rate must be given by the tangent to the curve atzero relative time. In the case of Fig.6, the best-fit parabolic curve for circulationis traced through a least squares calculation and the tangent is also determinedfrom the same calculation. As a result, it is estimated that, if the rate of vorticityproduction were kept at the initial level, the total circulation gained during onecycle of vortex shedding would be 4.6.From this tangent extrapolation, it is possible to estimate the loss of circulationexperienced by the separating vortex until the moment of vortex shedding, whichis as follows:

Loss = (4.6-2.6) /4.6 = 0.43

This loss of circulation is also estimated in some of other experiments, but directcomparison is not possible because the definition is slightly different. In usualcases, the loss is computed from the initial circulation of a detached vortex andthe total circulation generated in a boundary layer at separation during one cycle



of vortex shedding. In the present case, however, the second quantity is definedas total circulation gained by one of the attached vortex regions during one cycle.If neglecting this difference of definition, it can be said that the circulation lossin the present experiment is nearly 10 % greater than that of Green and Gerrard[10] at about the same Reynolds number. This is not understandable because thecirculation gained by the attached vortex region does not count in the inevitableloss across the stationary basic boundary. Consequently, this last circulation issmaller than that of the boundary layer at separation and, therefore, the loss ofcirculation should be estimated smaller in the present experiment. The reason forthis inconsistency remains unexplained as yet.On the other hand, at higher Reynolds numbers of the order 10* to 10 \ Cantwelland Coles [7] and Tanaka and Murata [8] report that this loss be 57 to 61 %. Thislevel of circulation loss seems plausible because, as pointed out in the precedingsection, the circulation of the detached vortices is much smaller in this highReynolds number range. Anyway it is certain, in both high and low Reynoldsnumber ranges, that more data are needed for discussing the vorticity behaviorwith a certain accuracy.The advantage of the present method of determining circulation is that estimationis unique whether it is in the boundary layer or in the wake vortices. This is animportant point when examining the vorticity convection and dissipation fromthe near wake to downstream regions. Other estimation methods using differenttypes of equations or formulae simultaneously are not assured of this consistency

of accuracy.Another advantage is that vorticity data may be more conveniently applicable todiscrete vortex calculations. The experimental values of the rate of vorticitydischarge along the boundary layer at separation are indeed direct indices fordetermining intensity of nascent vortices introduced. But this is a rather inviscidmodel of separation. As the viscous effect of the tlow is increased (or the Rey-nolds number is decreased), the vorticity generated on the cylinder surface is notalways convected downstream straightforward, but accumulated once in the nearwake region and then swept away in some part as demonstrated clearly in Fig.3.Therefore a more realistic model might be that which introduces nascent vorticesnot at the point of separation but in the near wake near the two shoulders of therear part of the cylinder. And in this case, the intensity of nascent vortices isestimated from the rate of vorticity discharge indicated by the tangent in Fig.6.The present author is now working on this type of discrete vortex model.

REFERENCES

1. Imaichi, K. and Ohmi, K., "Quantitative flow analysis aided by imageprocessing of flow visualization pictures", Flow Visualization III, pp. 365-369,Hemisphere Publishing Co., Washington, 19842. Imaichi, K. and Ohmi, K., "Numerical processing of flow-visualizationpictures — Measurement of two-dimensional vortex flow",/. FluidMech. 129,

pp. 283-311, 1983


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ABSTRACT - WIT Press€¦ · 30 cnf capacity. The test cylinder of 20.2 mm diameter is towed at a constant speed of 5.53 mm/s, the resultant Reynolds number being 100 approximately