On the relationship between the vortex formation process and cylinder wake vortex patterns

Transcription

1 J. Fluid Mech. (4), vol. 59, pp. 68. c 4 Cambridge Universit Press DOI:.7/S48 Printed in the United Kingdom 6 On the relationship between the vorte formation process and clinder wake vorte patterns B DAVID JEON AND MORTEZA GHARIB Graduate Aeronautical Laborator, California Institute of Technolog, Pasadena, CA 95, USA (Received Jul 4 and in revised form Jul 4) The idea of vorte formation time, originall developed for vorte ring formation, is etended to bluff-bod flows. Effects related to characteristic vorte formation time are shown for both the clinder starting from rest and the clinder undergoing forced oscillations in a stead flow. B looking at how wake vortices are formed when the clinder is accelerated from rest, it is found that similarities eist between the formation process for wakes and for vorte rings. This formation process is then observed for forced oscillating clinders, where the characteristic formation time interacts with the oscillation period. Frequentl observed bluff-bod phenomena will be recast in light of the vorte formation process.. Introduction Considerable progress has recentl been achieved in the understanding of the nature and characteristics of the wakes of clinders. Williamson & Govardhan (4) give a comprehensive review of wake pattern formation for oscillating and stationar clinders. Gu, Chu & Rockwell (994) shed light on the formation of vortices in the near wake of oscillating clinders. Eplanations of wake behaviour are generall based on the interaction of full developed vortices rather looking at where those vortices lie in terms of their formation process. Although studies of clinders starting from rest do look at the initial formation stage of the wake vortices, these studies generall stop before the formation time becomes sufficientl large to have interesting effects. And for studies of clinders in a stead free stream, the more tpical approach is to stud the interaction of vortices in the wake of the clinder, rather than how those vortices formed. Following the methodolog of Gharib, Rambod & Shariff (998), an analsis of the vorte formation behind a clinder was undertaken. The work is split into two parts: first, an eamination of the initial formation process when a clinder is accelerated from rest, followed b a stud of how this formation process manifests itself in the case of the oscillating clinder. The flow behind a clinder starting from rest is a canonical problem in the stud of wakes. In addition, it is also a favourite for computational studies, perhaps because the initial conditions are clearl defined. Consider, for eample, Honji & Taneda (969), Bouard & Coutanceau (98), or Koumoutsakos & Leonard (995). However, like the vorte ring, most studies onl looked at the earl stages of development of this flow. When the clinder begins to move, the wake is initiall smmetrical and dominated b a pair of counter-rotating vortices. Usuall, the studies focus solel on this regime of development. In an analogous manner, vorte ring studies are generall limited to rings of short formation time, when the vorte is compact. As shown in Gharib et al. (998), interesting features of the formation process can be found b

2 6 D. Jeon and M. Gharib etending the stud to longer formation times. In this stud, the evolution of the flow is studied well past the regime where the flow is nominall smmetrical the equivalent of a clinder wake of long formation time. Much as a vorte ring of indefinitel large formation times transitions into a jet, the wake of a starting clinder eventuall transitions into the classical von Kármán vorte street tpe configuration. In the first part of this paper, this transition and its relation to the vorte formation process will be discussed. In contrast to the two flows mentioned above, the oscillating clinder is a problem of more practical concern. The coupling of a bluff-bod wake and its structure has been demonstrated to occur in a wide variet of manners. Of particular interest here is the situation where the vorte shedding process itself occurs at a frequenc near to a resonant frequenc of the structure (Parkinson 989). A phenomenon known as lock-in can occur in this situation, where the wake is able to adjust itself to the structural frequenc (Sarpkaa 979). Such coupling can lead to large-amplitude vibrations of the structure, which can in turn lead to premature fatigue failure. This problem is tpicall understood in terms of modes of vorte shedding. One of the classical observations (Bishop & Hassan 964) is that the lift force on the clinder undergoes a nearl 8 phase shift at a frequenc ver near to the nominal shedding frequenc of the stationar clinder. Ongoren & Rockwell (988) showed that the vorte shedding process also appears to flip in phase as well. Williamson & Roshko (988) demonstrated that the wake patterns fell into distinct modes. In particular, on crossing the phase transition line, the wake appeared to switch from shedding two single vortices (known as the S mode) to shedding two counter-rotating pairs of vortices (P mode). More recentl, works b Gu et al. (994) and Carberr, Sheridan & Rockwell () have confirmed that the qualitativel observed behaviour persist when using quantitative vorticit measurements of the near wake. In the second part of this paper, the vorte formation process will be shown to link these various observations together. The interaction of the formation process and the motion of the clinder will be shown to lead to a straightforward mechanism b which these transitions occur. Therefore, we intend to show how these disparate phenomena can be viewed as the product of one process, rather than having to resort to multiple mode and phase transitions to eplain them.. Background To summarize Gharib et al. (998), the vorte formation process can be viewed as having three stages: initial vorte growth, saturation, growth of a trailing shear-laerlike structure. At earl times, the separating boundar laer rolls up into a vorte, in a manner similar to the Kaden spiral (Saffman 99; Pullin 979). However, this process reaches saturation at some point, as the vorte cannot be of infinite size. Afterwards an additional vorticit in the separated boundar laer ends up in a shear-laer-like structure connecting the vorte back to the generator. If continued indefinitel, this trailing structure becomes the potential core of a jet. In this paper, the etension of this formation process to wake-tpe flows is proposed. The qualitative change in the vorte should be appreciated first. This morphological change from compact, circular vortices to elongated vortical structures with trailing shear laers is ke to understanding the formation process. In particular, as vortices are fed circulation b their generators, the do not grow indefinitel large. Rather the process saturates, after which the additional circulation does not enter the vorte itself but trails behind. The vorte then undergoes pinch-off, where it separates from the

3 Relation between vorte formation and wake vorte patterns Strouhal period. Strouhal period.9 Strouhal period Formation time (diameters) /f St Path of non-oscillating clinder One period Time (periods) Figure. Distance travelled (i.e. non-dimensional time) b an oscillating clinder during one shedding ccle with a transverse amplitude of.5 diameters. Within the lock-in region, the wake is phase-locked to the oscillating frequenc. Therefore this is a simplistic measure of the formation time of the wake vortices as well. boundar laer that provided its circulation. The details of the saturation mechanisms ma var (which leads to the vorte no longer accepting additional circulation), but this morphological change is a consistent identifier of this tpe of vorte formation. Equall intriguing, the forming vorte appears to reach saturation at nearl the same non-dimensional time if the initial acceleration or final Renolds numbers are varied. This led to the notion of a characteristic formation time, a natural pacemaker in the sstem dictating when saturation begins. For the vorte ring, the critical time is approimatel four diameters, where time (t ) is measured in terms of the average velocit of the generator and its diameter; this definition happens to be the same as the stroke ratio of the generator (ratio of the distance the generator travels divided b its diameter): t = Ut d = L d. (.) Consequentl, an analogous measure of time is used for the clinder. In this case, time is scaled b the average velocit of the clinder and its diameter. This, in turn, represents the distance the clinder travels, in diameters. For the forced oscillating clinder, the equivalent time can be defined as the distance travelled per ccle, provided that the clinder is oscillating in the lock-in region, where the vorte shedding is phase locked to the clinder motion. This problem is, in some sense, an inverse one. Since the preferred shedding time is known (the inverse of the Strouhal number), the issue becomes what changes occur as a consequence of shedding either before or after this time. As a simple eample, the path distance travelled b the clinder is plotted in figure for the case of a clinder oscillating transversel at an amplitude of.5 diameters. The -ais is time (measured in periods) and the -ais is the path distance travelled (measured in diameters). The solid line

4 64 D. Jeon and M. Gharib. P + S Amplitude/diameter S P. Coalescence of vortices Period/Strouhal period Figure. Transition point based on the simple model of path distance travelled b the oscillating clinder. The triangles represent the locations on the Williamson Roshko wake mode map where the path distance of the oscillating clinder equals that of a non-oscillating clinder. For details of what the various modes look like, refer to the original Williamson & Roshko (988) paper. corresponds to the case of the non-oscillating clinder a straight line with slope /St. For oscillation frequencies in the lock-in range, the wake frequenc follows the oscillation frequenc of the clinder; hence, the shedding time of the oscillating case depends on the frequenc of motion. For frequencies above the Strouhal frequenc, the path distance can actuall be less than for the non-oscillating case. In this simple model, the frequenc at which the path distance is equal to the inverse of the Strouhal number corresponds to saturation in the wake formation process, since that is where the clinder would have travelled the same distance per vorte shedding as the nonoscillating clinder. Neglecting the additional acceleration due to oscillation, it is presumed that an oscillating clinder would therefore shed a vorte when its path distance equals the inverse of the Strouhal number. The frequenc at which the path distance equals the inverse of the Strouhal number is plotted in figure for a range of oscillation amplitudes, overlaid upon the Williamson & Roshko mode map. Although the frequenc does not lie on the SP transition curve, it is close especiall for such a simplistic model. More importantl, it suggests that a formation-time-tpe transition ma occur in this region, one where vorte satuation and vorte formation time pla a role in wake behaviour. Therefore we will see if the observed wake transitions like those seen in Williamson & Roshko (988) and Ongoren & Rockwell (988) can be recast in terms of vorte formation time. As an additional note, it is worth speculating on whether or not the value of the Strouhal number is directl connected to the vorte formation time. If we can observe vorte formation time effects that hinge around a formation time of /St,it would be natural to conclude that the value of the Strouhal number is in fact set b the vorte formation time. Therefore understanding the timing of the vorte formation process can be the wa to understand wh the Strouhal number is so remarkabl constant, even across differentl geometries and wide ranges of Renolds number.

5 Relation between vorte formation and wake vorte patterns Eperimental set-up These eperiments were performed in two separate facilities. The starting flow eperiments were conducted in the GALCIT X-Y Towing Tank facilit. This consists of a large glass tank, approimatel 4.5 m long and m wide, with optical access from all sides. The tank was filled to a depth of approimatel 5 cm of water, resulting in a tpical aspect ratio of. To minimize three-dimensional effects, the free end of the clinder was held around.5 diameters from the floor, supported from above in a cantilever fashion. Atop this tank is mounted a two-ais traversing sstem capable of towing the clinder down the length of the tank. The traverse is computer controlled, allowing us to precisel specif the acceleration profiles used in the starting the clinder flow. The oscillating clinder eperiments were conducted in a low-speed, free-surface water tunnel facilit. This tunnel has a glass test section, about cm in crosssection. A two-ais, computer-controlled traversing sstem was used to move the clinder in arbitrar two-degree-of-freedom patterns in a stead free stream. Again the clinder was supported in cantilever fashion from above; in this case, the free end was placed around.5 diameters from the floor, to match the boundar laer thickness on the bottom surface of the facilit. Aspect ratios varied between around and 5 for these cases. In both cases, the primar diagnostic technique was digital particle image velocimetr (DPIV). For these eperiments the images were taken using a video camera and interrogated using cross-correlation. See Willert & Gharib (99) for more details on DPIV. For the starting flow case, a continuous argon-ion laser was mounted beneath the tank and the beam was formed into a sheet in a plane parallel to the floor. A mechanical chopper, snchronized to the video signal, was used to produce pulses of light at known intervals. The camera was attached to the traverse moving the clinder and hence moved with the clinder. For the oscillating clinder case, a pair of Nd:YAG pulsed lasers were used. Again the light sheet was formed in the plane parallel to the floor. The camera viewed the flow from beneath the test section through a mirror arrangement. In both cases, the data were processed with a window size of 3 3 piels with a step size of 8 8 piels. We then used a discrete-window-shifting algorithm with spatial filtering to reduce noise and effectivel decrease the window size to 6 6 piels. The window size is between.3 and.7 diameters. Error in PIV measurements is affected b the nature of the flow itself (for eample, high gradients skew the results). However, in laborator testing, our PIV processing software delivers around 5% error in the velocit data, and about 5% error in the vorticit. 4. Starting clinder The vorte formation process for the starting clinder wake begins when the initiall attached boundar laer on the clinder separates. A clinder, initiall at rest, is quickl accelerated up to a constant velocit, tpicall at constant acceleration. The separated boundar laers rolls up into vortices and begins to move into the wake. As the vortices are fed vorticit from the boundar laer, the continue to grow roughl smmetricall. At a later time, the wake undergoes a transition and starts to become strongl asmmetrical. This flow is loosel analogous to the vorte ring, ecept that a momentum sink is turned on at some time instead of a momentum source. (Here momentum source and sink are analoguous to their use in potential flow.) Consequentl, the details of the formation process differ, but not the general

6 66 D. Jeon and M. Gharib pattern. After an overview of the flow at earl time, the details of the formation process will be eamined. The growth of asmmetr, coming from an instabilit of the smmetrical flow, will be shown as a likel mechanism controlling the formation and saturation process. In the net section, we will use the ideas of short and long formation time shown here to eplain the dnamics of the oscillating clinder wake. A snapshot of the clinder flow at earl time is presented in figure 3. For this discussion, time is measured in terms of the number of diameters the clinder has travelled from its initial position; this time scale tends to minimize the effects of different initial acceleration profiles. In this particular case, the final clinder Renolds number is and the final velocit is reached in.5 diameters from rest. (In real units, this case represents a.54 cm clinder accelerating to 7.87 cm s in s; the fluid is industrial lab water.) At ver earl times (less than four diameters), the wake remains ver smmetrical, consisting primaril of two large vortices in the near wake. At later times, the wake becomes asmmetrical and eventuall sheds a vorte. Thereafter, the wake graduall enters a nearl periodic shedding state. Around a time of four diameters, a transition occurs between the two states, from the initiall smmetrical to the asmmetrical. Note however that for sufficientl low initial acceleration, this pattern does not manifest itself. (The eact critical acceleration depends greatl upon Renolds number and background noise. For eample, in a well-resolved DNS computation, the wake ma never become asmmetrical before numerical errors accumulate.) Rather, the smmetrical state appears to graduall fade into Kármán vorte shedding instead of going through a dramaticall asmmetrical state first. This can be seen in figure 4; this is presented simpl to point out how the process changes at ver low acceleration, and will not be discussed further. Using a methodolog similar to Gharib et al. (998), the circulation of the growing vortices is measured as a function of time. In ever data frame, the total circulation was measured b taking the area integral of the vorticit field where the vorte was visibl located. Care was taken to ensure that onl the vorticit in the vorte was counted as the circulation. The circulation is scaled b the circulation of the first shed vorte, immediatel after it is shed. (The first shed vorte is the first vorte that separates from the clinder and moves independentl downstream. This vorte ma come from either side of the clinder. It is believed that this vorte contains the critical amount of circulation, which is wh it is shed in this manner. In the vorte ring stud, the equivalent of the first shed vorte is the vorte ring itself.) The results are shown in figure 5. This is a composite plot of four runs, all at Renolds number of, but with initial acceleration ramps ranging from.5 s to. s (.3 to.5 diameters). The vorte formation time is defined as the time when the circulation of the wake has reached the level of that which ends up in the first shed vorte; in other words, when the non-dimensional circulation reaches. B coincidence, the clinder wake seems to have the same formation time as the vorte ring four diameters. It would appear that a similar saturation event occurs in the clinder wake. It is worth noting that this number is not close to 4.8, which would be the Strouhal period. We believe this arises from the etra vorticit flu coming from the clinder acceleration, which is not present for stead clinder motion. Not onl does the circulation growth share the vorte ring s pattern, but also the vorte morpholog. Recall that the vorte ring begins life as a compact vorte with a ver thin shear laer connecting it back to its generator. After saturation, the vorte begins to move awa and the remaining circulation ends up in a trailing

7 Relation between vorte formation and wake vorte patterns 67 t * = 3 4 t * = t * = Figure 3. Initial wake behind a clinder starting from rest. Time is non-dimensionalized b the diameter and the average clinder velocit equivalentl, the distance travelled in diameters. Contours are non-dimensional vorticit levels of 7, On this and similar figures negative levels are shown as dashed lines, zero vorticit is white and deeper gre denotes larger magnitude.

8 68 D. Jeon and M. Gharib t * = t * = t * = Figure 4. Comparison of wake development at two different initial accelerations. On the left is a case above the critical acceleration, on the right, below. Note how the lower acceleration case retains the initial smmetr to long times. Contours are non-dimensional vorticit levels of 8.5, jet-like structure. In the case of the clinder, the same pattern eists. The earl time (t = ) compact smmetrical state gives wa to an intermediate (t = 4) elongated vorte. This structure splits and the wake sheds a vorte whose circulation is equal the circulation of the whole vorte around time four diameters. This morphological transition is important when we come to consider oscillating clinder flows in 5. To better stud this transition, the asmmetr of the wake was measured. In principle, this can be done b decomposing the wake into orthogonal modes and measuring the power in the asmmetrical modes. R. Henderson (995, personal

9 Relation between vorte formation and wake vorte patterns 69. Circulation/circulation of first shed vorte.5..5 First shed vorte Non-dimensional time (diameters) Figure 5. Circulation in the wake of the starting clinder, normalized b the circulation of the first shed vorte, as a function of non-dimensional time (i.e. distance travelled). Different smbols refer to different acceleration profiles. communication) suggested, as a simple alternative, that the wake could be divided into two parts, one of which was mirror smmetrical about the wake centreline and the other being everthing else. For eample, the smmetrical part is that which has a streamwise component of velocit of the same sign above and below the ais of smmetr and opposite sign for the transverse component. Since the wake at ver earl times is nearl mirror smmetric about the centreline, growth of the other mode should be an indicator of wake asmmetr. The norm of the asmmetrical part, computed b taking the vector norm of the velocit magnitudes, is considered to be the measure of asmmetr of the flow field. The resulting decomposition is shown in figure 6, where the contours represent the local magnitude of the respective modes. At time two diameters (t = ), virtuall all of the flow is in the smmetrical state, as epected. Around time four diameters (t = 4), a significant region of asmmetr has formed where the two main vortices touch. B time eight diameters (t =8), the asmmetrical mode has nearl the same magnitude as the smmetrical state in the near-wake region. In figure 7 the norm of the asmmetrical mode is plotted on semi-log aes. (Norms are computed in the usual sense. For eample the norm is sum of the absolute values of the mode.) At earl times, the measurement is probabl noise limited, but shows a low level of asmmetr. Around time four diameters, the asmmetrical mode eperiences nearl eponential growth before saturating at late time as the flow transitions towards shedding. Readers might note that the peak in the asmmetr curve is nearl coincident with the peak in the circulation and wonder if there might be a connection. In general, while the behaviour at t =4 is quite consistent, this is not the case near where asmmetr and circulation peak. Presumabl this is because the wake is most nonlinear around that transition time and small details matter more. Regardless, it would appear that an instabilit in the flow triggers the growth of asmmetr in the wake, which in turn switches the wake from the initial smmetrical to the Kármán shedding state.

10 7 D. Jeon and M. Gharib Smmetrical t * = Asmmetrical t * = t * = Figure 6. Smmetric and asmmetric components of the wake of a starting clinder. Contours denote magnitude. Contours are.,....5 on the smmetrical side, and.,.5... on the asmmetrical side. 5. Oscillating clinder Like the starting clinder, the oscillating clinder flow has been studied for a long time. Of particular interest here is the phenomenon of vorte-induced vibration. Researchers have etensivel mapped the response of simple geometries (like clinders and prisms) to changing flow conditions. The results have traditionall been eplained in terms of mode changes in the wake. (Consider Brika & Laneville 993, especiall, figure 9.) Our goal is to offer an alternative view on wh these wake changes occur, one based of the concept of vorte formation time. We will now attempt to portra

11 Relation between vorte formation and wake vorte patterns 7. Level of asmmetr 5. 5 norm norm inf. norm Non-dimensional time (diameters) Figure 7. Growth of the norm of the asmmetrical mode of the wake with time. On this semi-log plot, the nearl linear growth between times 4 and 8 impl an eponential growth of asmmetr. the behaviour of the oscillating clinder s wake in terms of concepts learned from the stud of the starting clinder case. We shall see that looking at the wake behaviour in terms of short and long formation time effects will clarif a range of previousl noted effects under a simple unifing model. Although the base flows are quite different, the underling mechanisms that affect the formation of vortices remains the same for the two flows. Seeing how wake vortices form under the controlled conditions of the starting clinder flow will make it easier to understand how the same processes are present in the oscillating case. To illustrate the behaviour of the wake of a forced oscillating clinder, two sets of vorticit plots are shown in figures 8 and 9. In both cases, non-dimensional vorticit fields (scaled b the free-stream velocit and clinder diameter) are phase-averaged and plotted for si phases in the upper half of the clinder motion. In our phase time, the clinder is moving downward during phases 4/ to 9/, and moving upward between phases / to 3/. Phase time is linked onl to the transverse motion, and an streamwise motion is alwas relative in phase to the transverse. In these figures, the clinder was forced in sinusoidal motion in both the streamwise and transverse directions simultaneousl. The streamwise motion has twice the frequenc of the transverse and no phase difference, so that the motion would draw a figure-eight pattern if one were moving with the clinder. In figure 8, the clinder is oscillating at.84 Strouhal periods (hence has a formation time less than that of the nonoscillating clinder), with a transverse amplitude of.5 diameters. The streamwise motion is about. diameters, with zero phase difference between the two directions. This produces a wake similar to regular von Kármán shedding, albeit with a ver short wake. In contrast, figure 9 is for a similar motion, ecept that the transverse motion was at. Strouhal periods (therefore with a formation time longer than that of the non-oscillating clinder). The streamwise motion is not important at this time;

12 7 D. Jeon and M. Gharib T = 4/ T = 3/ 3 3 T = 5/ T = / 3 3 T = 6/ T = / Figure 8. Wake of an oscillating clinder for the upper half of motion. T refers to the phase time (from / to /). Renolds number is, transverse period is.84 the nominal Strouhal period, transverse amplitude is.5 diameters, streamwise amplitude is. diameters, and the relative phase between the two is. Contours are non-dimensional vorticit levels of 8.5, we are using these figures because the are representative of the changes that the wake undergoes without the additional factor of mode change involved. (However, in a later section, we will illustrate how streamwise motion affects the appearance of the wake.) Although the wake is nominall shedding one vorte per side per ccle, it is qualitativel different from the wake at higher oscillation frequenc. Note especiall the much longer wake with the elongated shear laer: this feature is also present

13 Relation between vorte formation and wake vorte patterns 73 T = 4/ T = 3/ 3 3 T = 5/ T = / 3 3 T = 6/ T = / Figure 9. As figure 8 but Renolds number is 5, transverse period is. the nominal Strouhal period. in the P mode. Other researchers have documented similar changes in the wake (albeit without the streamwise motion). Carberr et al. () and Govardhan & Williamson () offer similar plots of oscillating clinder wakes; the change from compact vorte to vorte with elongated shear laer is apparent in their plots as well. In figure, the three classes of flow are presented side-b-side. Notice the characteristic change in vorte morpholog as formation time increases. In ever case, there is a change from a compact vorte to a vorte attached b a long shear

14 74 D. Jeon and M. Gharib S mode (short period) P mode (long period) 3 3 t * = diameters t * = 6 diameters Figure. Comparison of vorte morpholog for three different forming vortices. Lines are drawn for comparison between the equivalent structures seen in the three cases. laer. In the vorte ring, as the formation time is increased past four diameters, the etra circulation pumped into the fluid remains behind in a trailing jet-like shear laer. In the starting clinder, after the clinder moves more than four diameters from rest, the wake destabilizes, becomes asmmetrical, and one vorte elongates. This vortical structure splits off a vorte, with the remaining circulation trailing back to the clinder. In the forced oscillating clinder, as the clinder oscillation period increases somewhat beond the Strouhal period, the etra circulation remains behind in a trailing-shear-laer structure connecting the main vorte with the clinder. Measurements of the vorte circulation in the oscillating cases showed that there is in fact little difference in the circulation of the shed vorte at short or long oscillation period. The etra circulation is ending up in the trailing shear laer. Consequentl, it is believed that the vorte formation process also affects the oscillating clinder case, causing a straightforward transition in the wake from compact vortices to elongated vorte and trailing-shear-laer structures. Other consequences of formation will be discussed below.

15 Relation between vorte formation and wake vorte patterns 75 S S + S Transverse.5. P + 45 P + P 3 4 Streamwise Figure. Phases of motion when a vorte is attached to the clinder. Plotted are the times when a vorte is attached to the upper side of the clinder. (The sinusoid represents the motion of the clinder through space.) The dashed lines represent the higher frequenc case (.84 Strouhal periods) and the solid lines, the lower (. Strouhal periods). In all cases, the vorte begins forming on the falling stroke (leeward side). The wake appears to flip at lower frequenc because the vorte sheds on the rising stroke. 5.. Effect of the formation process on the apparent phase of the wake An interesting consequence of the change in vorte formation as the oscillation period crosses the Strouhal period is its effect on the phase of the wake (relative to the motion of the clinder). As mentioned in the introduction, an oft-noted observation is that the lift forces on the clinder flip in phase when the forced oscillation period varies across the nominal Strouhal period. Researchers have noticed that the wake, as well, appears to have flipped phase across this transition (i.e. the wake appears to be upside down at the same part of the motion ccle) (Ongoren & Rockwell 988). However, b plotting the phase time that the vorte remains attached the clinder, the role of formation in this phase change is evident. Since the vorte is onl fed b the clinder, its formation time is intimatel related to the time it is attached to the clinder. Figure plots the attached times for a variet of cases. The lines indicate phase of the motion when a vorte is attached to the upper side of the clinder (in our case, vortices of negative circulation). Let us first concentrate on the upper dashed lines these are for the higher frequenc/lower formation time cases. Notice that the lines start and end on the falling stroke, indicating that a vorte has formed on the leeward side and shed before reaching the transverse minima of motion. However, for the lower solid lines (lower frequenc/longer formation time cases), the behaviour has changed. Although the vorte has begun to form on the leeward side, as before, the increased formation time means that the vorte ends up shedding on the opposite stroke. In all cases, the vorte begins to form on the leeward side. But at longer formation times, the vorte does not shed until after the clinder changes direction. Hence the wake appears to have switched phase.

16 76 D. Jeon and M. Gharib That the flipping of the phase of the wake is enough to cause the phase transition is illustrated below. We start with the impulse formulation, but ignoring the added mass force, F = d ( ) ρ ω dv. (5.) dt Taking the component in the lift direction, and considering onl the most dominant vorte, this becomes lift d dt (ργ )= ρ(γ ẋ + Γ ) (5.) where Γ refers to the circulation of that vorte and its position downstream from the clinder. Since both and the circulation are increasing during the formation process, the sign of the lift force is determined b the sign of Γ relative to the motion of the clinder. To obtain positive coupling (between the motion of the clinder and the lift force), the dominant vorte in the wake should be negative on the rising stroke. In other words the lift force and the transverse velocit must have the same sign. Eceeding the critical formation time causes the wake to move into the oppositestroke shedding state which results in a positive coupling between the wake forces and the clinder motion. It is worthwhile to emphasize that these formation time effects are not the same as phase shifting. The magnitude of perturbation involved in our eperiment is quite large. Whereas for small perturbations one might think of this problem as a forced linear oscillator (with the clinder motion forcing the wake), that kind of model does not work here. For instance, Loft & Rockwell (993) show a good eample of a linear perturbation on the wake. Their model of the phase clock is quite reasonable for their kind of perturbation. (The used a flapping trailing edge to modif the oscillating wake of a bluff bod.) However, to borrow their clock model, our eperiment would be one where the clock is so perturbed that it does not make hour ccles anmore. At shorter formation times, we are interrupting the normal formation time; it is as if the clock onl runs from to 8 before resetting. Likewise at longer formation times, the formation is prolonged as if the clock ran past to 4 before resetting. Our work shows that the wake appears to flip phase because the reset point of the clock appears to be on opposite sides of the clock, even though the starting phase time might be the same. And this results simpl from the relationship between formation time and the forcing period. In a sense, it is a string of coincidences that leads to the strong coupling of the structure to its wake. Since a passive structure cannot send net energ to the wake, the structure must necessaril oscillate at frequenc less than or equal to the frequenc of the wake. Therefore a passive structure selects lower frequencies, which are necessaril at longer formation time. This leads to a change in the morpholog of the wake which allows the vortices to remain attached long enough to switch the phase of shedding and hence the forcing. If it were not for the changing morpholog as the critical formation time is eceeded, the strong fluid structure coupling might not materialize since the fluid forcing and the structure motion would have the wrong phase relationship. 5.. From whence comes the pairing mode? The question still remains what role does the transition to the P state have to do with the vorte formation process? The P mode is where two pairs of vortices are formed and shed as dipole pairs per side per ccle. (See Williamson & Roshko

17 Relation between vorte formation and wake vorte patterns ). To answer this question, it is worth a quick aside on the effect of streamwise acceleration on the formation process. B superimposing streamwise acceleration on top of the regular transverse motion, the rate of formation can be altered. If the clinder were to accelerate forward during formation, there would be less time for the vorte to form before it sheds. Since the P mode appears on the long formation side, we believe there is a connection between long formation and the appearance of the P mode. Upon closer eamination, it appeared to us that the etra vorte formed in the P mode forms on the long shear laer that is characteristic of long formation time wakes. In the framework of formation time, our supposition is that the second vorte in the pair forms from a secondar roll-up of the connecting shear laer in the sense of Kelvin Helmholtz instabilities, and hence is just a b-product of long formation time. For eample, in figure 9, if the clinder motion did not include the streamwise component, the wake would be the classic P mode. But with the small streamwise forward acceleration, the pairing mode disappears. The forward motion during figure-eight motions cuts short the formation process before the second vorte can appear. However, if this were true, then the P state should reappear if the figure-eight motion were run backwards. In this case (streamwise motion at 8 phase difference to the transverse), the clinder would run back towards the vorte during the formation process, prolonging the formation process. If our supposition is correct, then this reverse motion during formation should bring about the return of the P mode. As shown in figure, the actual result is more interesting. Rather than simpl forming two vortices per side, the prolonged formation time allows a third vorte to form during the same ccle. (The first two correspond to the pair of vortices formed in the P state.) It would indeed appear that the pairing state is most likel a result of etended formation (longer than normal shedding period) rather than a mode change in and of itself. Alternativel, the appearance of etra vortices is a quick wa of seeing that etended formation is occurring, although not uniquel so. These three tpes of vorte groupings are summarized in figure 3, where the forward and backward accelerations are the same magnitude, differing onl in sign. Note the general similarit of the vorte patterns (aside from number) and the same curvature of the long shear laer. In a straightforward wa, the formation time is linked to the formation of etra vortices during shedding, which makes it appear that a mode transition has occurred. So while the mode shift to P is a useful diagnostic in looking at formation time, we believe the formation time effect is more fundamental to processes in the near wake. It is also worth noting that the effect is the reverse of what one might epect if the single vorte of the S mode is split into two to form the pair in the P mode. The amount of strain on the vorte should decrease in the 8 phase case (where the clinder goes downstream during formation), et the number of vortices formed increases. B using the idea of vorte formation time, an alternative perspective on the mode transition in the wake of the oscillating clinder can be presented. 6. Conclusions As important as vortices are in bluff-bod flows, how those vortices form (and how long it takes to form them) is equall important. The vorte formation described in Gharib et al. (998) has been found to have bluff-bod analogues, and has been shown to have a significant effect on the development and nature of the flow. Much like the formation process defines a characteristic time for vorte ring flows, so there are also characteristic formation times in bluff-bod flows.

18 78 D. Jeon and M. Gharib T = 4/ T = 3/ 3 3 T = 5/ T = / 3 3 T = 6/ T = / Figure. Wake of an oscillating clinder for the upper half of motion. T refers to the phase time (from / to /). Renolds number is 99, transverse period is. the nominal Strouhal period, transverse amplitude is.5 diameters, streamwise amplitude is. diameters, and the relative phase between the two is 8. Contours are non-dimensional vorticit levels of 8.5, For the starting clinder, this time seems to signal the onset of asmmetr in the wake. It is this smmetr breaking that eventuall leads to the von Kármán vorte street pattern. Otherwise the wake would remain smmetrical and vorticit would be diffused from the wake, as opposed to being shed. At earl times, the wake is smmetrical, and the vortices do not interact strongl. Around a time of four

19 Relation between vorte formation and wake vorte patterns 79 Forward acceleration No acceleration Backward acceleration Figure 3. Comparison of the wake of an oscillating clinder. All three cases have the same transverse oscillation frequenc. Forward acceleration means that the clinder is accelerating into the free stream at the etrema of motion. Note that the general morpholog is preserved, but the change in formation time b the streamwise motion changes the number of vortices formed per ccle. Contours are non-dimensional vorticit levels of 5.5,

20 8 D. Jeon and M. Gharib diameters, the wake appears to sharpl increase the degree to which it is asmmetrical. Also at time four, the wake vortices have the same circulation as the vorte that is first shed, leading us to believe that a critical circulation has been reached. The asmmetr in the wake continues to grow from there until shedding occurs. For the clinder undergoing forced oscillations, the characteristic formation time marks the switch in the phase of the wake, relative to the motion of the clinder. At higher oscillation frequencies, the vortices are formed in less than the critical time and hence form and shed on the same stroke of the clinder motion. At lower oscillation frequencies, the vorte formation stretches past the critical time, leading to formation and shedding occurring on opposite strokes. This accounts for the observed phase change in the wake. In addition, the phase change allows the wake and the clinder motion to positivel couple. Formation time effects were also shown to underlie the P mode of vorte shedding. It is believed that prolonging formation allows time for the trailing-shear-laer structure to have a secondar (or even tertiar) roll-up, leading to the formation of additional vortices per ccle. That said, the detailed mechanisms b which these vortices reach saturation remains to be discovered. For the vorte ring, the critical time corresponds to a minimum in the energ of the vorte, after which it becomes favourable for the vorte to move downstream. Whether a similar mechanism underlies vorte shedding has not been determined. Nevertheless, the wake of the circular clinder seems to be paced b a vorte formation time. And bearing that in mind appears to clarif some of the behaviour of those wakes. The authors would like to thank the Office of Naval Research (grant numbers N and ONR-N ) for their generous support, and Professors A. Roshko and A. Leonard for helpful discussions and useful advice. REFERENCES Bishop, R. E. D. & Hassan, A. Y. 964 The lift and drag forces on a circular clinder oscillating in a flowing fluid. Proc. R. Soc. Lond. A 77, Bouard, R. & Coutanceau, M. 98 The earl stage of development of the wake behind an impulsivel started clinder from 4 < Re < 4. J. Fluid Mech., Brika, D. & Laneville, A. 993 Vorte-induced vibrations of a long fleible circular clinder. J. Fluid Mech. 5, Carberr, J., Sheridan, J. & Rockwell, D.. Forces and wake modes of an oscillating clinder. J. Fluids Struct. 5, Gharib, M., Rambod, E. & Shariff, K. 998 A universal time scale for vorte ring formation. J. Fluid Mech. 36, 4. Govardhan, R. & Williamson, C. Mean and fluctuating velocit fields in the wake of a freel oscillating clinder. J. Fluids Struct. 5, Gu, W., Chu, C. & Rockwell, D. 994 Timing of vorte formation from an oscillating clinder. Phs. Fluids. 6, Honji, H. & Taneda, S. 969 Unstead flow past a circular clinder. J. Phs. Soc. Japan 7, Koumoutsakos, P. & Leonard, A. 995 High-resolution simulations of the flow around an impulsivel started clinder using vorte methods. J. Fluid Mech. 96, 38. Loft, A. & Rockwell, D. 993 The near-wake of an oscillating trailing edge: mechanisms of periodic and aperiodic response. J. Fluid Mech. 5, 73. Ongoren, A. & Rockwell, D. 988 Flow structure from an oscillating clinder. J. Fluid Mech. 9, Parkinson, G. 989 Phenomena and modelling of flow-induced vibrations of bluff bodies. Prog. Aerospace Sci. 6, 694.

21 Relation between vorte formation and wake vorte patterns 8 Pullin, D. 979 Vorte ring formation at tube and orifice openings. Phs. Fluids, Saffman, P. 99 Vorte Dnamics, pp Cambridge Universit Press. Sarpkaa, T. 979 Vorte-induced oscillations: a selective review. J. Appl. Mech. 46, 458. Willert, C. E. & Gharib, M. 99 Digital particle image velocimetr. Eps. Fluids, 893. Williamson, C. H. K. & Govardhan, R. 4 Vorte-induced vibrations. Annu. Rev. Fluid Mech. 36, Williamson, C. H. K. & Roshko, A. 988 Vorte formation in the wake of an oscillating clinder. J. Fluids Struct.,