three-dimensional effects in turbulent bluff body wakes

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Three-Dimensional Effects in Turbulent Bluff Body Wakes

Anil Prasad Charles H. K. Williamson Department of Mechanical and Aerospace Engineering, Comell University, Ithaca, New York

• Recent investigations have shown that it is possible to control three-dimensional patterns in a cylinder wake at low Reynolds numbers (where the vortex shedding is laminar) by altering the end boundary conditions. However, very little work has been done to understand three-dimensional phenomena at higher Reynolds numbers. In the present study, we demonstrate the effect of end conditions on the cylinder wake at moderately high Reynolds numbers (200 < Re < 10,000). By suitably manipulating the end conditions, it is possible to induce oblique and parallel vortex shedding patterns across large spans (80 cylinder diameters) over a large Re range. Measured parameters in the wake display marked differences between oblique and parallel shedding. The practical significance of such a study is that the total spanwise-integrated unsteady fluid forces on the body can be dramatically reduced to a value close to zero, by inducing oblique vortex shedding or indeed other three-dimensional phenomena.

We have found that the instability of the separated shear layer is also affected by the end conditions: with parallel shedding, the instability first manifests itself at Re = 1200; but, with oblique shedding, the instability is inhibited until a significantly higher Reynolds number of about 2600. We show that the variation of normalized shear-layer frequency with Reynolds number is not accurately represented by a Re °5 power law, which has hitherto been used extensively in the literature. A power law that closely models not only our data, but all of the data from earlier studies, is of the form,

fsL - - = 0.0235 x Re °'67. fK

Physical reasons why one should naturally expect an exponent larger than 0.5 are included. © Elsevier Science Inc., 1997

Keywords: wake, shear-layer instability, shedding-pattern control

I N T R O D U C T I O N

A number of investigations have recently been concerned with the general problem of the development of three-dimensional structure in turbulent shear flows and with the corresponding implications for mixing in such flows. In the past eight years, there has been a surge of activity in wake flows from analytical, computational, and experimental approaches. These developments are described compre- hensively in the review by Williamson [1]. It is perhaps not surprising that the bulk of the recent experimental developments has been studied at low Reynolds numbers, where smaller scales have not yet developed. A number of new flow phenomena have been discovered, many of which are influenced by the spanwise end boundary conditions. It is

indeed surprising that almost no studies have been under- taken to investigate the presence of the new phenomena at moderate and high Reynolds numbers and to study the possible control of wake patterns through boundary-condition control. It is to these questions that we address our work here.

In the case of low Reynolds numbers, a number of three-dimensional phenomena have been discovered, such as various oblique and parallel shedding wake patterns [2-4], transient three-dimensional phase patterns such as "phase shocks" [5, 6], and "phase expansions" [7, 8], the existence of cellular shedding near the ends of a body [9] and in the central part of the span [2, 5, 10, 11], and "vortex dislocations." Such "vortex dislocations," whereby vortices move in and out of phase with neighboring vor-

Address correspondence to Prof. C. H. K. Williamson, 252 Upson Hall, Cornell University, Ithaca, NY 14853.

Experimental Thermal and Fluid Science 1997; 14:9-16 © Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010

0894-1777/97/$17.00 PII S0894-1777(96)00107-0

10 A. Prasad and C. H. K. Williamson

tices along the cylinder span, are found to be fundamental to three-dimensional wake transition [12].

In the course of the development of a new understanding of these low Reynolds number phenomena, it has been found that the end boundary conditions on the cylinder can control many of these spanwise three-dimensional structures. However, there has been apparently no work to show what extent these low-Reynolds-number phenomena occur at higher Reynolds numbers. Although the work of Stager and Eckelmann [13] showed that an end cell (of lower frequency) forms near the end condition (end plate in their study) even at Re above the "laminar" shedding regime, there has been no work to investigate the control of three-dimensional phenomena over long cylinder spans. In the present work, we demonstrate that the flow can indeed be controlled over long span lengths, at moderate Reynolds numbers (200 < Re < 10,000), in a manner not unlike that employed at low Reynolds numbers. The physical motivation for such a study lies in its importance to the total spanwise-integrated unsteady forces on a body, which depends on the shedding phase distribution and correlation along the span.

EXPERIMENTAL DETAILS

The experiments were performed in an open-circuit suction wind tunnel. The free-stream turbulence level was less than 0.08% and flow uniformity better than 0.3% in the 30-cm by 30-cm test section. Cylinders of diameter 0.318 and 0.635 cm were mounted near the upstream end of the test section. The Reynolds number is defined as Re = U~D/v, where Uo~ is the free-stream velocity and D is the cylinder diameter. End plates (of diameter = 16D) fitted on the cylinder produced aspect ratios of 40-80. To produce parallel vortex shedding across the entire span, each end plate was positioned with its leading edge in- clined inward about 12 ° . By inclining each end plate in the same direction, oblique vortex shedding was induced. In the present paper, these end-plate configurations are referred to as the "parallel shedding end conditions" and "oblique shedding end conditions," respectively; the latter produces a mean shedding angle of about 13-15 °.

The origin of the wake coordinate system is fixed on the axis of the cylinder. The x-axis is directed downstream, the y-axis is perpendicular (defined as transverse) to the flow direction and the cylinder axis, and the z-axis lies along the axis of the cylinder (defined as spanwise).

Wake velocity measurements were made by using a miniature hot-wire probe in conjunction with a two-chan- nel anemometer system. The probe was mounted on a three-axis traversing mechanism that allowed it to be placed at any point in the wake. Oblique shedding angles were measured by using two hot wires that were positioned at a known spanwise distance apart and offset in the streamwise direction such that each hot wire would experience the same phase of a passing oblique vortex, as indicated by Lissajous figures. A Stanford Research Sys- tems SR760 spectrum analyzer was used for the spectral measurements. Long-time-averaged velocity spectra were produced by averaging measured spectra for time dura- tions in excess of 20,000 vortex-shedding cycles. The total fluctuation intensity was measured on a Hewlett-Packard 3400A true-RMS meter. The experimental uncertainty in the measurement of velocity was determined to be less

than 1%. Velocity spectra were measured with a constant analyzer bandwidth of 0.122 Hz over a span of 48.8 Hz centered on the shedding frequency peak. The error in estimating the peak frequency is less than 2%. The hot- wire probe was positioned at any point in the wake with an inaccuracy of less than 1%.

Flow visualization was conducted by using a vertical smoke-wire system, as originally described by Corke et al. [14]. A GenRad 1540 Strobolume provided the intense illumination required to capture photographic images on ISO 400/27 ° film with a Nikon F3 camera.

T H E CONTROL OF VORTEX SHEDDING PATTERNS

In our present study, we have indeed shown that one can induce oblique and parallel vortex shedding through manipulating the end boundary conditions--in this case, end plates. A clear example of such control is shown in Fig. 1, where we demonstrate the existence of oblique or parallel turbulent shedding at Re = 2000, a result that typifies a range of Re up to 10,000 by simply altering the end conditions. Smoke-wire visualization is employed in this figure and the flow is upward. Shedding angles in the range of 0 ° to 15 ° can be induced by suitably angling the end plates. As in the laminar shedding regime (Re < 190), it has been found that the Strouhal number decreases as shedding angle (0) increases. There is, however, some

(a)

(b)

Figure 1. Control on the shedding pattern at Re = 2000. By suitably angling the end plates one can induce (a) oblique shedding and (b) parallel shedding.

Three-Dimensional Effects in Bluff Body Wakes l l

deviation from the "cos 0" formula [2], which accurately represents this variation at low Re. Because one can control the vortex shedding angle, it is of practical significance that one can influence the fluid-induced unsteady forces on a bluff body.

From further extensive wake measurements, we have also found a marked effect on the development of the wake velocity fluctuation spectra by increasing the shedding angle, 0. As the angle 0 is increased, the spectrum broadens dramatically (shown in Fig. 2 for Re = 2600), whereas its peak value at the Kfirmfin (primary vortex) shedding frequency, fK, diminishes. Interestingly, our measurements show that the total fluctuation intensity, ( U ' r m s / U ~ ) T o t a l , remains approximately constant as the shedding angle increases, indicating that a redistribution of spectral energy takes place as the shedding angle is increased. It is suggested that the broadening of the spectral peak is associated with two phenomena: it is due

-47

-52

-57

~,~ -62

-67

-72

-77

' ' 1 ' ' 1 ' ' 1 ' ' 1 ' '

I I I I I I I I

0 = 9.4 °

' ' I ; ' I ' ' I ' ' I ' '

O = 1 3 . 1 °

I I , , I , , I ,

805 815 825 835 845 855 f (rIz)

Figure 2. Evolution of wake-velocity spectra with increasing shedding angle, at Re = 2600. The shedding angle, 0, is indicated in each case. The measurements are made at x /D = 10.0, y /D = 1.0.

to (1) a slight wavering of the shedding angle about its mean and (2) the possibility of vortex dislocations being introduced by the end conditions. With regard to the latter point, it has been suggested that vortex dislocations are a fundamental aspect of high Re turbulent wake flows [15], although one may note that these recent deductions are based on short cylinder wakes. Our own study indicates that dislocations do not appear along the span of long-cylinder wakes (although further work is needed to clarify this) for Reynolds numbers above the three-dimensional wake transition regime (190 < Re < 260), over which regime they were found to occur spontaneously along the span [16]. Nonetheless, dislocations definitely occur near the ends of a cylinder, under some end conditions, and therefore for short-cylinder wakes (low aspect ratio), one might expect to find such dislocations dominat- ing the entire spanwise wake pattern, which would be consistent with recent suggestions from other studies. We have found almost no control on the flow in the three-dimensional wake transition regime but find that strong control on the wake shedding pattern exists for Re > 260, as we indicate below.

The spectral broadening and the consequent diminution of the shedding frequency peak, referred to above, are represented in this paper as the nondimensional spectral bandwidth, (AfKD2/v) , and the velocity fluctuation at the Kfirmfin shedding frequency, (U'rmJU~)fK, respectively. As Re is varied over a range (Fig. 3), the Khrmfin fluctuation intensity and the spectral bandwidth follow the character of our preceding example (for Re -- 2600). In Fig. 3a, it is observed that the Kfirmfin intensity for oblique shedding end conditions is consistently lower than that for parallel shedding end conditions over the entire Reynolds number range up to 3000, and further data (not shown here) demonstrate the same trend up to Re > 6000. The variation in spectral bandwidth in Fig. 3b comple- ments that of the Kfirmfin intensity. In this case, the spectral bandwidth with oblique shedding conditions is markedly larger than that with parallel shedding conditions, rising to beyond four times as large near Re = 3000. Although these conclusions are based on measurements made at a single location in the wake, we find from other extensive measurements (not shown here) that the preceding trends persist for streamwise distances of the order of 40 diameters.

T H E INSTABILITY OF T H E SEPARATED SHEAR LAYER

At this point, we would like to address the effect of the development of turbulence in the separating shear layers from the sides of a bluff body. It may be noted that there are a local minimum in the (parallel shedding) Kfirmfin fluctuation intensity at about Re = 1200 in Fig. 3a and a corresponding maximum in the spectral bandwidth in Fig. 3b. We have found that, above this Reynolds number, the shear layers separating from the body are themselves susceptible to a Kelvin-Helmholtz instability. Turbulence in the form of shear-layer vortices (of higher frequency than the Kfirmfin frequency) is found to develop in the near wake, and the effect is to increase the Reynolds shear stress, as Re is increased beyond 1200. The vortex formation is brought closer to the body, the

0.04

0.03

~ 0.02

0 . 0 1

0 . 0 0 0

6 t~

4

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!

* i • • i . . . . I . . . . i . . . . I , . . , i . . . .

5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0

Re

1 0 . . . . , . . . . , . . . . , . . . . , . . . . ,

(a)

Figure 3. Characterization of the spectral peak at the vortex shedding frequency, over a range of Re. Here we show the variation of (a) peak turbulence intensity, (u--s/U=)y~:, and (b) nondimensional spectral bandwidth, ( A f K D 2 / v ) . The solid symbols represent parallel shedding end conditions and the open symbols oblique shedding end conditions; the measurements are made at x / D = 10.0, y / D = 1.0.

0 0

r . . . . i , . . , | . . . . i . . . . i . . . . i . . . .

5 0 0 I 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0

1 2 A. Prasad and C. H. K. Williamson

R e

(b)

base "suction" coefficient ( - C e B ) increases, and the intensity of vortex shedding is increased. Here the base suction coefficient is defined as the negative of the base pressure coefficient.

Figures 3a and 3b show that the variation at about Re = 1200 is distinctly different between parallel and oblique shedding end conditions, which led us to investigate the effect of the end conditions on the shear-layer instability. In addition, there is the question regarding the critical Reynolds number, Re c, for the onset of the shear- layer instability. This question arises because a fairly large range for Re c has been quoted in the literature, as the following examples illustrate: Bloor [17] did not detect the instability for Re < 1300, and Unal and Rockwell [18] estimated that the instability manifested itself for Re > 1900 from their hot-film and flow-visualization studies. Most recently, Wu et al. [19] have stated that Rec can lie anywhere between 1000 and 3000 and have suggested that this discrepancy can be attributed to background disturbance conditions.

Our spectral measurements of velocity fluctuation demonstrate that a peak at the shear-layer frequency fSL, is inhibited with the use of oblique shedding conditions

when compared with the parallel shedding conditions. Because the shear layer is known to be convectively unstable, one can expect that the measurement of disturbance growth would depend on the location of measurement, resulting in some variation in the measurement of Re c. Nevertheless, to obtain a realistic estimate of the critical Reynolds number, it seems reasonable to make the measurements at a point sufficiently downstream of the separation point but before large-scale Kfirmfin vortex roll-up, for which purpose we have selected x / D = 1.0. In Fig. 4, we show that the intensity at the shear-layer frequency, (Utrms/U~)fsL, increases rapidly with Re for parallel shedding but that the increase with oblique shedding is only moderate. Peterka and Richardson [20] also deduced, from limited measurements, that the shear-layer instability intensifies as Re is increased. From Fig. 4, one can also determine that Re c = 1200 with parallel shedding conditions but that it is closer to 2600 with oblique shedding conditions. Such differences in the end conditions may suggest an explanation for the large Re-range quoted in the literature for the onset of the shear-layer instability.

The Re-variation of the shear-layer frequency is also, an outstanding question in the literature. Bloor [17] was the

Three-Dimensional Effects in Bluff Body Wakes 13

0 .006

0 .006

|

"~ 0 .004

"::I

0 .002

0.000 0

' • • • I . . . . I " " " " l . . . . I . . . . I . . . .

perallel */ _ ~ oblique

~ e shodding

. . . . W . _ . , . . . . . . . .

1000 2000 3000 4000 5000 6000

Re

Figure 4. Variation of the intensity at the shear-layer frequency, (U--s/U~)~ , with Re. The intensity with parallel s'~dding is markedly larger than that with oblique shedding. These hot-wire measurements were made at x / D = 1.0, y / D = 0.8.

first person to measure transition waves in the separated shear layer from a cylinder. Based on parameters from laminar separated boundary layers, she suggested that fSL/ fK would scale with Re °5. Since then, most investigators have attempted to fit their data to such a power law. However, Okamoto et al. [21] and Wei and Smith [22] did suggest other values of the exponent.

Our own analysis of the actual data points from various investigators who have measured f s L / f n , shown in Table 1, indicates that the exponent, in each and every case, is significantly greater than 0.5. Moreover, there appears to be variation of the order of about 30% in the value of the exponent, p. In an attempt to resolve this discrepancy, we have made independent measurements of the shear-layer frequency. Noting that, in many of the previous studies, end conditions were not well defined, we have purpose- fully set up our cylinder with parallel shedding end conditions. The shear-layer frequency is determined from long- t ime-averaged velocity spectra. In Fig. 5, we show data from several investigators who have measured the shear- layer frequency, including our own. The least-squares best-fit line through all of this data is

fSL - - = 0.0235 × Re °'67. (1) fK

We would also like to point out that the data from the present study lie quite close to this line. Table 1 includes

the power-law fit from our data and that from all the data concatenated. Should it be construed that the reasonably large number of data points in the present study weights the exponent of the concatenated data close to our own exponent, we would like to point out that, if one deter- mines the average exponent from all of the other data (excluding ours), one still arrives at an exponent close to 0.67. It appears that one can conclude quite definitively that the exponent, p, is significantly higher than 0.5.

Here we present a discussion that suggests why one should actually expect the exponent to be greater than 0.5. On a dimensional basis, one can suggest that the shear- layer frequency would scale on a characteristic velocity and length in the form

U~ep ~ , (2) fSL 0S L

where Us~p is the velocity outside the boundary layer at the separation point and 0SL is the momentum thickness of the separated shear layer. Bloor [17] suggested that the momentum thickness of the separated shear layer (normalized with the cylinder diameter, D) scales with the normalized boundary-layer thickness just before separation, which itself scales as Re -°'5, such that,

0SL ~BL Re- o.s. (3) D D

Table 1. Comparison of Power-Law Variations from Various Studies

A p Investigation Coefficient Exponent Range of Re

Bloor [17] 0.0277 0.6509 1300-25,000 Okamoto et al. [21] 0.0170 0.6925 1700-5600 Wei and Smith [22] 0.0078 0.7997 2500-11,000 Kourta et al. [23] 0.0507 0.5811 2600-15,000 Norberg [24] 0.0405 0.6276 4700-45,000 Present study 0.0269 0.6587 1200-6000 All of the above data 0.0235 0.6742 1200-45,000

The data for normalized shear-layer frequency, from each investigation, has been fitted to an equation of the form fSL//fK = A x Re p.

14 A. Prasad and C. H. K. Williamson

Figure 5. Variation of normalized shear-layer frequency with Re. The data from several investigations are included, and the solid line is the least-squares best-fit line through all of this data. The present hot-wire measurements are made at x / D = 1.0, y / D = 0.8.

10 z

" ~ 1 0 s

100 10 a

. . . . . . . ! . . . . . . .

/s____~_z __- 0.0235 x Re °'m

~ ~ t study Norl~rI[ (1M7) moor (1964)

• Okamoto eL al. (1981) * l ( o u r ~ e l . al. (11145'7) • Wel & Smith (1986)

, . . . . . . . . ! . . . . . . . 10 4 10 5

Re

This scaling, when inserted into Eq. (2), in conjunction with the definition of the Strouhal number, S = f K D / U ~ , gives

/SL ( sop t Re°5 fK ~ U~ ) S (4)

If one then assumes that Usep/U~ is approximately constant and because from experiment S = 0.2, then the following variation is easily deduced:

/sL - - ~ Re °5. (5) /K

However, such assumptions do not appear to be justi- fied over a large range of Re. The base suction coefficient ( - C e B ) increases with Reynolds number, for Re > 1200, as shown by the meticulous work of Norberg [25]. We expect the ratio Use_/U~ to increase over a large range of • P Re, because it depends on -Cp~ as follows:

Usep Us = ~ - CeB " (6)

Consequently, from Eq. (4), one could expect a value of p > 0.5. Moreover, the position of transition in the shear layers moves upstream as Re increases, as shown by Schiller and Linke [26], implying that this transition point would move into a region where the shear-layer thickness is smaller. Thus the most unstable frequency would be larger than if the analysis were made at a fixed point in the shear layer. Such a trend would further increase the value of the exponent.

PRACTICAL SIGNIFICANCE

The flow past a bluff body has innumerable applications in a wide variety of engineering situations. The ability to control the vortex shedding pattern from a bluff body has practical importance because the total spanwise-integrated fluid forces on a body depend on the phase of shedding and its correlation along the span. By purpose- fully decorrelating the spanwise shedding pattern through the use of end manipulation, one can significantly reduce these forces, thereby diminishing structural vibrations,

acoustic noise, or resonance, which may lead to failure• An investigation concerning the development of instability in the separated shear layers assists in an understanding of the near-wake vortex dynamics that contribute significantly to the process by which the transition to turbulence Occurs•

CONCLUSIONS

We have demonstrated that one can control patterns of turbulent vortex shedding behind a bluff body by manipulation of end boundary conditions• It appears distinctly surprising that this straightforward result for moderate and high Reynolds numbers has not been shown before, because the discovery of such control of oblique and parallel shedding wake patterns was made several years ago [2-4].

Oblique and parallel shedding patterns can be induced at moderate Re, not unlike control that is possible at low Re. With an increase in the controlled shedding angle, there are a decrease in the Kfirm~in fluctuation intensity and a dramatic increase in the spectral bandwidth. It is suggested that there are two reasons associated with the spectral broadening: (1) a slight wavering of the shedding angle about its mean and (2) the possibility of vortex dislocation being introduced by the end conditions.

Our measurements indicate that the spontaneous gen- eration of vortex dislocations along the span for Reynolds numbers within the three-dimensional wake transition regime (190 < Re < 260) precludes the possibility of control through the use of end manipulation• Furthermore, it appears that dislocations do not manifest themselves along the span of long-cylinder wakes, for Re > 260, although additional work is required to clarify this point. The strong control on the shedding pattern, found for Re > 260, is characterized by substantial differences in wake parameters between parallel and oblique shedding end conditions over a large Re range, which has strong implications on the magnitude of integrated unsteady fluid forces on the body.

Three-Dimensional Effects in Bluff Body Wakes 15

The effect of the end conditions on the instability of the separated shear layer has also been investigated. The critical Reynolds number for the inception of this instability depends on the end conditions: with parallel shedding, Re c = 1200, but it is higher (near 2600) with oblique shedding end conditions. The difference in end conditions may suggest an explanation for the large range quoted in the literature for values of the critical Reynolds number. Contrary to previous studies, from which a variation of the form

fSL = constant × Re °5

fK

has been suggested, we have found that the normalized shear-layer frequency varies as

fSL - - = 0.0235 × Re °'67. f~

A complete reanalysis shows that data from all previous investigations are remarkably well represented by such a power-law variation. Indeed, the fact that the power-law variation is significantly larger than Re °5 is to be ex- pected, based on the variation of characteristic velocity and length scales in the near wake, which influence this frequency.

This work has been supported by O.N.R. Contract No. N-00014-95-1- 0332.

- - C p B

D

fK fSL

P

(Utrms/U~)fK

(drms/U~o)fsL

( U'rms/ U~)TotaL U+

Usep

NOMENCLATURE

base "suction" coefficient, dimensionless cylinder diameter, m Kfirmfin vortex-shedding frequency, Hz shear-layer frequency, Hz Reynolds number exponent in Eq. (1), dimensionless

Re Reynolds number based on cylinder diameter, Re = U ~ D / v , dimensionless

Re c critical Reynolds number for shear-layer instability, dimensionless

S Strouhal number, S = f K D / U ~ , dimensionless intensity of fluctuation at vortex-shedding frequency, dimensionless intensity of fluctuation at shear-layer frequency, dimensionless total fluctuation intensity, dimensionless free-stream velocity, m / s velocity outside the boundary layer at the separation point, m / s

Greek Symbols 0 vortex shedding angle, degrees

AfK spectral bandwidth, Hz OSL momentum thickness of separated shear

layer, m 6BL boundary-layer thickness just before

separation, m v kinematic viscosity, mZ/s

REFERENCES

1. Williamson, C. H. K., Vortex Dynamics in the Cylinder Wake. Annu. Rev. Fluid Mech. 28, 477-539, 1996.

2. Williamson, C. H. K., Defining a Universal and Continuous Strouhal-Reynolds Number Relationship for the Laminar Vortex Shedding of a Circular Cylinder. Phys. Fluids 31, 2742-2744, 1988.

3. Eisenlohr, H., and Eckelmann, H., Vortex Splitting and Its Con- sequences in the Vortex Street Wake of Cylinders at Low Reynolds Number. Phys. Fluids A 1, 189-192, 1989.

4. Hammache, M., and Gharib, M., A Novel Method to Promote Parallel Vortex Shedding in the Wake of Circular Cylinders. Phys. Fluids A 1, 1611-1614, 1989.

5. Williamson, C. H. K., Oblique and Parallel Modes of Vortex Shedding in the Wake of a Cylinder at Low Reynolds Numbers. J. Fluid Mech. 206, 579-627, 1989.

6. Albar~de, P., and Monkewitz, P. A., A Model for the Formation of Oblique Shedding and "Chevron" Patterns in Cylinder Wakes. Phys. Fluids A 4, 744-756, 1992.

7. Miller, G. D., and Williamson, C. H. K., Control of Three-Di- mensional Phase Dynamics in a Cylinder Wake. Exp. Fluids 18, 26-35, 1994.

8. Monkewitz, P. A., Williamson, C. H. K., and Miller, G. D., Phase Dynamics of K~rmfin Vortices in Cylinder Wakes. Phys. Fluids 8, 91-96, 1995.

9. Gerich, D., and Eckelmann, H., Influence of End Plates and Free Ends on the Shedding Frequency of Circular Cylinders. J. Fluid Mech. 122, 109-121, 1982.

10. K6nig, M., Eisenlohr, H., and Eckelmann, H., The Fine Structure in the Strouhal-Reynolds Number Relationship of the Laminar Wake of a Circular Cylinder. Phys, Fluids A 2, 1607-1614, 1990.

11. K6nig, M., Eisenlohr, H., and Eckelmann, H., Visualization of the Spanwise Cellular Structure of the Laminar Wake of Wall- Bounded Circular Cylinders. Phys. Fluids A 4, 869-872, 1992.

12. Williamson, C. H. K., The Natural and Forced Formation of Spot-like "Vortex Dislocations" in the Transition of a Wake. J. Fluid Mech. 243, 393-441, 1992.

13. Stager, R., and Eckelmann, H., The Effect of Endplates on the Shedding Frequency of Circular Cylinders in the Irregular Range. Phys. Fluids A 3, 2116-2121, 1991.

14. Corke, T., Koga, D., Drubka, R., and Nagib, H., A New Tech- nique for Introducing Controlled Sheets of Streaklines in Wind Tunnels. IEEE Publication No. 77-CH 1251-8 AES, 1977.

15. Szepessy, S., On the Spanwise Correlation of Vortex Shedding from a Circular Cylinder at High Subcritical Reynolds Numbers. Phys. Fluids 6, 2406-2416, 1994.

16. Williamson, C. H. K., The Existence of Two Stages in the Transition to Three-Dimensionality of a Cylinder Wake. Phys. Fluids 31, 3165-3167, 1988.

17. Bloor, M. S., The Transition to Turbulence in the Wake of a Circular Cylinder. J. Fluid Mech. 19, 290-304, 1964.

18. Unal, M. F., and Rockwell, D., On Vortex Shedding from a Cylinder, Part 1: The Initial Instability. J. Fluid Mech. 190, 491-512, 1988.

19. Wu, J., Sheridan, J., Hourigan, K., and Soria, J., Shear Layer Vortices and Longitudinal Vortices in the Wake of a Circular Circular Cylinder. Exp. Thermal Fluid Sci. 12, 169-174, 1996.

20. Peterka, J. A., and Richardson, P. D., Effects of Sound on Separated Flows. J. Fluid Mech. 37, 265-287, 1969.

21. Okamoto, S., Hirose, T., and Adachi, T., The Effect of Sound on the Vortex-Shedding from a Circular Cylinder. Bull. JSME. 24, 45-53, 1981.

16 A. Prasad and C. H. K. Wil l iamson

22. Wei, T., and Smith, C. R., Secondary Vortices in the Wake of Circular Cylinders. J. Fluid Mech. 169, 513-533, 1986.

23. Kourta, A., Boisson, H. C., Chassaing, P., and Ha Minh, H., Nonlinear Interaction and the Transition to Turbulence in the Wake of a Circular Cylinder. J. Fluid Mech. 181, 141-161, 1987.

24. Norberg, C., Effect of Reynolds Number and a Low-Intensity Freestream Turbulence on the Flow around a Circular Cylinder. Publication No. 87/2, Dept. Appl. Thermodynamics and Fluid Mech., Chalmers University o f Technology, Goteborg, Sweden, 1987.

25. Norberg, C., An Experimental Investigation of the Flow around a Circular Cylinder: Influence of Aspect Ratio. J. Fluid Mech. 258, 287-316, 1994.

26. Schiller, L., and Linke, W., Pressure and Frictional Resistance of a Cylinder at Reynolds Numbers 5000 to 40,000. NACA TM 715, 1933.

Received February 23, 1996; revised June 10, 1996

three-dimensional effects in turbulent bluff body wakes

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