Vortex Shedding and Modal Behavior of a Circular Cylinder Equipped with Flexible Flaps

When a cylinder is subject to a flow, vortices will be shed that can lead to strong tonal noise. The modification of the cylinder with soft, flexible flaps made of silicone rubber has been shown to affect the vortex shedding cycle in a way that the Strouhal number associated with the vortex shedding suddenly jumps to a higher value at a certain Reynolds number. In the present study, the effect of the flexible flaps on the vortex shedding is further examined by subsequently reducing the number of flaps and additionally shortening their length. Acoustic measurements and camera recordings of the flap motion, performed in an aeroacoustic wind tunnel, suggest that the sudden jump of the Reynolds number is caused by the movement of the outer flaps. A comparison with the eigenfrequencies obtained from a numerical modal analysis of the different flap rings revealed that the cause of the Strouhal number jump is most likely a lock-in of the natural vortex shedding cycle with the next higher eigenfrequency of the outer flaps.

In aw ater tunnel study by Kunze and Brücker [15] it has been shown that the presence of flexible flaps at the aft part of ac ircular cylinder strongly affects the vortex shedding behavior.This led to the fact that at acertain spe-cificReynolds number Re (based on cylinder diameter d) the Strouhal number Sr associated with vortexs hedding quite suddenly increased from av alue of about 0.23 to a value of 0.29.The result wasaj ump in the corresponding Reynolds-Strouhal number diagram (when instead of the cylinder diameter the streamwise length of the separation bubble wasused to calculate the Strouhal number,no jump wasvisible).The reason for this effect wasfound to Received16February 2018, accepted 4December 2018.be alock-in effect between the vortexshedding and the oscillation of the flexible flaps.Particle Image Ve locimetry (PIV)m easurements were conducted at Reynolds numbers of 5,000 to 31,000.The results showed that, due to the presence of the flaps, the vortices were not shed in a zig-zag likea rrangement as in the classical vonK ármán vortexs treet, butr ather in-line in ar ow with the cylinder wake axis.Additionally,i tw as found that the size of the recirculation area behind the cylinders with flaps is notably smaller than the size of the recirculation area behind areference cylinder.
As ubsequent investigation [16], which wasp erformed in an aeroacoustic wind tunnel, revealed that the change in vortexshedding also affected the emission of tonal noise, as the aeolian tones generated by acylinder modified with eight flexible flaps were shifted to higher frequencies when above ac ertain Reynolds number.B esides the acoustic measurements, hot-wire anemometry measurements were performed, with the probe positioned one cylinder diameter downstream and one half diameter off center from the cylinders approximately at mid-span.Those measurements provided the power spectral density of the turbulent velocity fluctuations at this position, which also showed ap eak due to the vortexs hedding.Fort he cylinder with flaps, this measurement resulted in the same jump in the Reynolds-Strouhal number diagram, thus confirming the results from the acoustic measurements.It wasconcluded that the jump of the Strouhal number is caused by alock-Vol.105 (2019) in between the vortexs hedding peak and the resonances of the flap structures.Additionally,i tw as concluded that the outer flaps play am ore important role than the inner flaps since theyinteract directly with the shear layer.Additionally,flow visualization experiments were performed.The results confirmed that the flexible flaps lead to amore slender separation bubble in the wake recirculation region compared to the reference cylinder.
It is interesting to note that an early experimental study has been performed by Grimminger [17] on cylinders modified with rigid "guide vanes", which, if properly designed, lead to an oticeable reduction in drag.However, although it is obvious that the cause of the drag reduction has to be an effect of the guide vanes on the flowpattern behind the cylinder,itwas not observed if these guide vanes also affect the Strouhal number in aw ay similar to the observations made for the flexible flaps.
The effect found for the flexible flaps is also different from that observed for as ingle flexible splitter plate attached to ap lain cylinder.I na ne xperimental study by Shukla et al. [18] in aw ater tunnel at Reynolds numbers between 1,800 and 10,000 (based on cylinder diameter), twom odes of periodic splitter plate motion were identified.Both modes featured afrequencyclose to the vortex shedding frequencyo ft he cylinder,a nd thus no Strouhal number jump occurred.However, the Reynolds numbers were belowthe value were the jump wasidentified [15,16] and the length of the flexible splitter plate wasa lways greater than at least three cylinder diameters.In addition, the flexible splitter plates were constructed with spanwise stiffeners, and hence at hree-dimensional deformation of the plates wasprevented on purpose.Another experimental study on aflexible splitter plate in the wake of acylinder wasd one by Teksin and Ya yla [19], who performed PIV measurements in aw ater tunnel at aR eynolds number (based on cylinder diameter)of2,500.Theyvaried the length of the splitter plate (1.25 d,2.25 d and 2.5 d,with d being the cylinder diameter)and found that it has anotable influence on the turbulence statistics in the wake.H owever,e ff ects liket he shedding of vortices in ar ow with the cylinder wake axis or achange of the size of the separation bubble, as observed for the cylinder with flexible flaps, were not visible.
The work presented here is ac ontinuation of the study presented in [16].Thereby,t he focus is to further examine the fluid-structure-interaction and the assumed cause of the Strouhal number jump.To this end, acoustic measurements and flap motion measurements were performed on the original flap cylinder with eight flaps as used in [16] as well as on modified versions where, subsequently,flaps were cut off in order to determine their individual contribution to the observed vortexshedding behavior.

Cylinder Models
The flap cylinders consist of ac ore cylinder with as panwise length l of 0.28 mand adiameter of 20 mm, on which 22 flap rings made of the silicone rubber Elastosil RT 601 were threaded.Thus, the flaps are not extended overt he whole cylinder length, butdivided into finite spanwise segments, which wasdone in order to disturb spanwise coherences.The thickness of the flap rings, and hence the width h of the flaps, was12mm.This value waschosen in order to obtain the same ratio of flap width to cylinder diameter h/d = 0.4 as in the original study by Kunze and Brücker [15].
The flap rings were cast using acasting mold consisting of metal plates with small gaps that will form the flaps.The original flap rings, as used in [16], contained eight flaps with athickness of 0.3 mm.Aschematic of this flap ring is shown in Figure 1.In the present study,m easurements were also performed on modified flap rings, where anumber of flaps wascut off from the original flap rings, resulting in cylinders with six, four and twofl aps instead of eight.Figure 2shows CAD models of the different flap rings used for this study.I nmost cases, the flaps had the original streamwise length s of 9m m( Figure 2a through  2d).In the final experiments, the tworemaining outer flaps were shortened to alength of s = 4.5 mm (Figure 2e).
Additionally,ap lain cylinder wasu sed as ar eference in the experiments.All cylinder models had an outer diameter d of 30 mm, resulting in an aspect ratio l/d of 9.3.Ap hotograph of the cylinder with six flaps is shown in Figure 3.

Modal Analysis
In order to enable ab etter understanding of the observed lock-in effect, the eigenmodes of the flap rings shown in Figure 2were obtained numerically for frequencies up to 250 Hz using the block Lanczos algorithm implemented in Ansys (Academic Ve rsion R15.0).The flap rings were modelled as elastic, isotropic materials with aY oung's modulus of 1.2 MPa, adensity of 1080 kg/m 3 and aPoisson ratio of 0.495.The meshes for the Finite Element  Method (FEM)calculation consisted of 3-D 20-node solid tetrahedral elements with amaximum side length of 1mm.Forexample, the mesh of the original cylinder with 8flaps (see Figure 4) consisted of 6,840 elements.The influence of the mesh resolution on the resulting eigenfrequencies wastested for one of the models by using elements with a smaller maximum side length of 0.5 mm and 0.2 mm, thus increasing the element number from 6,540 to 49,008 and 730,500, respectively.S ince the difference in the resulting eigenfrequencies wasn ot significant (less than 2% ), ar efinement of the mesh wasn ot found necessary.A sa boundary condition for the modal analysis it wasd efined that the inner surface of the flap ring (where the flap ring is in contact with the rigid core cylinder)w ill not be displaced.
It can be expected that the dimensions of the physical models of the flap rings may differ slightly from the CAD models due to the manufacturing process.This is especially true for the thickness of the flaps.To takeacertain deviation of the nominal thickness of the flaps of 0.3 mm into account, eigenmodes and eigenfrequencies were additionally obtained for the case that the overall thickness of the flaps wasi ncreased by 5%a nd 10 %t ov alues of 0.315 mm and 0.33 mm as well as for the cases of a5% and 10 %decreased thickness of 0.285 mm and 0.27 mm.Of course, this rather simple procedure does not account for local deviations of the thickness of asingle flap.

Wind Tunnel
The experimental part of the study included acoustic measurements and measurements of the flap motion in the small aeroacoustic wind tunnel at BrandenburgU niversity of Technology Cottbus -S enftenberg [ 20].The test section used for the experiments has ar ectangular crosssection of 0.28 mheight ×0.23 mwidth.The top and bottom of this test section are made from acrylic glass, while the twos ide windows are covered with tensioned Kevlar, thus providing atwo-dimensional flow, while at the same time enabling the use of an acoustic measurement technique positioned outside of the flow.Surrounding the test section is ac abin with absorbing side walls that lead to an early anechoic environment for frequencies above approximately 100 Hz.
In the experiments, the velocity wasadjusted by setting the pressure in the wind tunnel settling chamber.The corresponding velocities were determined at the exit of the test section (with the reference cylinder in place)inaseparate measurement, using av ane anemometer with an ac-curacyof±0.2 m/s.The blockage due to the cylinders was corrected using the simple approximative blockage correction method proposed by Barlow et al. [21].
Measurements were conducted at 13 subsonic flow speeds between 7m /s and 17 m/s, leading to Reynolds numbers (based on outer cylinder diameter d)r anging from 14,600 to 34,000.In this range of flows peeds, the vortexshedding noise of the cylinders will be at frequencies below200 Hz.In order to examine the reproducibility and to obtain ab etter statistical significance, each measurement wasperformed twice in individual runs.

Acoustic Measurements
The acoustic measurements were performed using twosingle one fourth inch free-field microphones positioned in a distance of 0.6 mo ne ach side of the cylinder models.In the vertical direction, the microphones were pointed approximately at the mid-span location of the cylinders.The data were recorded with as ampling frequencyo f 51.2 kHz overalong time period of 90 s, using a24Bit National Instruments digital dynamic signal acquisition module (NI-USB 4431).Following the approach used in [16], the time signals from both microphones were added with a phase difference of 180 • ,assuming atheoretical dipole behavior of the cylinder generated noise.The data were then transformed in the frequencydomain according to Welch's method [22] using aF ast Fourier Transformation (FFT) with aH anning windowo n5 0%o verlapping blocks of 131,072 samples each and converted to sound pressure levels re 20 µPa.T he resulting frequencyr esolution is only 0.39 Hz.Finally,6d Bw ere subtracted to correct for the increased amplitude due to the summation of both time signals.

Flap Motion Measurements
The movement of the flexible flaps of afl ap ring, positioned at mid-span, wasm easured using ah igh speed camera (Phantom V12.1-8 G-M, Vision Research)w ith a 35 mm Nikon lens, which wasp ositioned belowt he test section (see Figure 6).T he frame rate wass et to 500 Hz with atotal measurement duration of approximately 14 s.The exposure time per frame was5 00 µs.Using the procedure described in [16], the time-series of the movement of the centroid point of each single flap of ac hosen flap ring wasderivedfrom the camera recordings.These results were then converted to corresponding power spectral densities of the flap movement using aBurgalgorithm [23].
Due to the fact that neighboring flaps were found to collide at high flows peeds as ac onsequence of the increased amplitude of the flap oscillations, the flap motion measurements were only performed up to Reynolds numbers of 26,000 to 28,500.Forthe case of the cylinder with six flaps, however, those measurements could only be conducted up to aReynolds number of 20,900.

Modal Analysis
The numerical modal analysis revealed that, for all cases where the flaps were not shortened (Figure 2a through 2d), the eigenmodes and eigenfrequencies in the examined range up to 250 Hz are identical.Furthermore, since the shape and material of each flap on the flap ring is identical, the eigenmodes are also the same for each flap.
The first eigenmode at 22 Hz corresponds to the first bending mode of the flap.The second mode (the first torsion mode)can be observed at afrequencyof39Hz, while the third mode, which appears to be the second torsion mode, is visible at 97 Hz.Another bending mode, but in ad irection perpendicular to the first bending mode at 22 Hz, occurs at 137 Hz.Due to the fact that the flaps are firmly attached to the cylinder body,the resulting shape of that particular mode resembles ac ambered surface.Further eigenfrequencies of 156 Hz, 236 Hz and 243 Hz were found, the corresponding modes, however, takemore complexs hapes.Of course, it cannot be expected that each eigenmode will affect the vortexs hedding cycle in the same way, it seems rather more likely that only those eigenmodes that lead to as trong deformation of the flap in the direction perpendicular to the shear layer will have an influence.As an example, Figure 7s hows the shapes of three selected eigenmodes of one flap from the original flap ring with eight flaps.Fort he case where the remaining twoo uter flaps were shortened (see Figure 2e), the modal analysis revealed completely different eigenmodes and eigenfrequencies.Fore xample, the frequencyo ft he first bending mode increases to av alue of 86 Hz, that of the first torsion mode to 107 Hz and that of the second torsion mode to 171 Hz.Table Ilists the eigenfrequencies of the different flap rings.
When the thickness of the flaps is increased or decreased by 5%c ompared to their nominal thickness of 0.3 mm, the modal analysis revealed that the corresponding eigenfrequencies increase/decrease by about 3to5% as well.If the thickness increases/decreases by 10 %, the eigenfrequencies subsequently showanincrease/decrease of about 8to10%.This agrees with basic mechanical theory,w here an increase of the thickness of ar ectangular plate will result in an increase of the corresponding eigenfrequencies (see for example [24]).

Results from Acoustic Measurements
The sound pressure levels pectra measured for the cylinders from Figure 2aswell as for the reference cylinder are shown in Figure 8asafunction of Strouhal number based on the outer diameter.Inaddition, the spectra obtained for the empty wind tunnel are included.It can be seen that in the region around the vortexshedding peak the signals obtained for the different cylinders exceed the background noise by more than 40 dB.
Basically,t he above discussed effect that the vortex shedding peak obtained for the flap cylinders suddenly jumps towards ahigher Strouhal number at Reynolds numbers between 23,300 and 26,000 is visible, while the peak of the reference cylinder stays at ac onstant value just above 0.2.Thereby,the spectra obtained for the cases with unshortened flaps all showe ssentially the same behavior,while the peak obtained for the cylinder with the two shortened flaps (Figure 2e)j umps to an oticeably lower Strouhal number.T his implies that, in air,t he observed jump is not caused by an oscillation of the flap system as awhole (consisting of eight equally-spaced flaps with a fluid volume in between), butessentially by the movement of the outer flaps.In contrast, in the water tunnel experiments in [15], the lock-in wasfound to occur between the vortexshedding and atraveling wave running through the bundle of flexible flaps in adirection perpendicular to the flowa nd the cylinder axis.This difference is presumably caused by the different properties of the twofl uids.Due to the fact that the bulk modulus of water is much higher than that of air,the oscillating system consisting of the silicone flaps and the fluid-filled spaces between those flaps is much stiffer in water than in air,leading to astronger coupling between the single flaps.Additionally,inthe case of water the difference in density between flap and fluid is much smaller than in air.
To further illustrate the differences between the cases with the long flaps and the shortened flaps,    ber,which is in good agreement with the theoretical value of 0.21 [25] known for this flowr egime (the subcritical range, characterized by al aminar near-wakew ith vortex street instability).In addition, Figure 10 shows howthe maximum sound pressure levelofthe vortexshedding peak of the different cylinders changes with Reynolds number Re and Mach number M.F or the baseline cylinder,t he peak leveli ncreases with increasing Re.A tR eynolds numbers up to approximately 26,000, the peak levelfollows the theoretical scaling of adipole sound source with the sixth power of the flows peed.At higher Reynolds numbers, the peak levelsomewhat decreases, leading to aslightly reduced velocity dependence.This trend wasa lso observed in [14] and can be attributed to the complexvelocity dependence of cylinder flownoise in general and to achange in boundary layer thickness on the Kevlar windows of the test section, which results in adecrease in measured peak amplitude.It is clearly visible from Figure 10 that the presence of the flexible flaps leads to astrong reduction of the vortexshedding peak level.The cause of this noise reduction is assumed to be the fact that ap art of the energy contained in the boundary layer is needed to move and deform the flaps, and hence less energy is converted to noise.This agrees with the work of Kunze and Brücker [15] who found that the flaps lead to ad ecrease of the rms-values of the velocity components within the wake region.Teksin and Ya yla [19] also observed that the presence of aflexible splitter plate decreases the turbulent kinetic energy and the rms-values of the streamwise and transverse velocities of the flowfield.
However, while the peak levelo ft he cylinder models with long flaps showas imilar increase with Reynolds number as the reference cylinder without flaps, the peak levelo btained for the cylinder with the shortened flaps shows ad i ff erent behavior.A tR eynolds numbers up to 20,900, the levelr emains approximately constant at around 62 dB.With further increasing Reynolds number,t he peak leveld ecreases considerably.Acomparison with Figure 8r eveals that the vortexs hedding peaks appear much broader in this range, suggesting that the decrease in peak leveld oes not correspond to ad ecrease in sound power or sound intensity of the peak, butmainly to a change of the spectral shape of the peak.Such achange is not visible in the spectra obtained for the other flap cylinders.
Since it wasshown in [16] that the aeolian tones generated by the flap cylinder are strongly coupled to the movement of the flaps, the following section will provide results from the flap motion measurements.

Results from Flap Motion Measurements
Figure 11 shows spectra obtained from the flap motion measurements performed with the high-speed camera.As an example, spectra are shown for one outer flap of the original flap cylinder with eight flaps (Figure 2a)and that with the twoshortened flaps (Figure 2e).Each figure additionally contains the theoretical eigenfrequencies derived from the modal analysis as giveninT able I. Also included are the resulting eigenfrequencyr anges derivedn umerically when ad eviation of the flap thickness of ±10 %i s assumed.The spectra obtained for the remaining cylinders were basically identical to those obtained for the cylinder with 8flaps and thus are not shown here.
The spectra from the flap motion measurements reveal that the flaps do not only perform just one single motion related to the vortexs hedding at the cylinder,b ut rather theyd op erform am ultitude of different oscillating motions, most of which are linked to the eigenfrequencies of the flaps.In case of the cylinder with 8flaps (Figure 11a), the first peak seems to be linked to the first bending mode at 22 Hz or the first torsion mode at 39 Hz.The second peak, at frequencies above 50 Hz, can be associated with vortexs hedding, as its frequencyi ncreases with increasing Reynolds number.The third peak is at aconstant frequencyo fa pproximately 170 Hz.When the differences between aC AD model and the actual flap cylinders are taken into account, this third peak may be related to the eigenmode observed at 156 Hz.Interestingly,atthe highest Reynolds number of 26,000 the shape of the flap motion spectrum differs from that obtained at lower Reynolds numbers.Most notably,t his includes as hift of the first and strongest spectral peak towards higher frequencies, butalso agenerally increased amplitude of the flap movement.The measurement at the highest Reynolds number of 26,000 belongs to ac ase where the Strouhal number obtained from the acoustic measurements jumped to anotably higher value of around 0.3.This corresponds to a frequencyo fa bout 131 Hz.Around this frequency, the flap motion spectrum shows as trong peak, which can be assumed to be related to the eigenmode of the flap rings at 137 Hz, as shown in Figure 7c.As described in Section 3.1, this eigenmode is related to abending motion of the flaps, although in adirection perpendicular to the first bending mode at 22 Hz.The result is afl ap that repeatedly cambers and straightens with afrequencyof137 Hz, ap rocess that is very likely to interact with the adjacent flowa nd the regular vortexs hedding.The same shift occurred in the spectra obtained for the cylinder with only the twoo uter flaps left (not shown here).It can therefore be concluded that the jump in the Re-Sr-plot is caused by alock-in of the vortexshedding cycle with an eigenmode of the outer flaps.
The spectra obtained for the cylinder with twos hortened flaps (Figure 11b)showconsiderable differences.At lowR eynolds numbers, the first discernible peak is the one related to the vortexs hedding (atf requencies above 50 Hz).At the twoh ighest Reynolds numbers of 26,000 and 28,500 (which is above the jump), this vortexs hedding peak is already merged with another peak at afi xed frequencyo fa bout 95 Hz, which can be assumed to be related to the first bending mode of the short flaps at afre-quencyof86Hz.Thus, in case of the shortened flaps, the sudden shift of the Strouhal number is caused by alock-in with the first bending mode of the flaps.As econd, much smaller local maximum is visible in the flap motion spectra around 140 Hz.This peak does not seem to be related to anyeigenfrequencyofthe flap ring, as the modal analysis predicted the next eigenmode at afrequencyof170 Hz.The cause of this small peak is currently not clear.
Overall, the present analysis of the flap motion reveals that the observed jump of the Strouhal number related to vortexshedding is caused by alock-in with an eigenmode of the flaps.In case of the unshortened flaps, this eigenmode belongs to ab ending motion of the flaps, buti na direction perpendicular to the first bending mode.Forthe case with the twoshortened flaps, the lock-in happens with the frequencyo ft he first bending mode.It appears, however,that the lock-in always happens with the next "available" mode, and hence the mode with an eigenfrequency just above the natural vortexshedding frequency.It is not fully clear whether the exact shape of this mode is important, as in the present case the corresponding modes were twod i ff erent bending modes.Basically,t his behavior indicates that such effects may also occur at even higher frequencies, when the vortexshedding cycle locks-in with the next higher eigenfrequency.It wasnot possible to test this hypothesis with the present setup, since with further increasing flows peed neighboring flaps started to collide, rendering the analysis of the flap motion impossible.

Conclusions
In arecent wind tunnel study on the vortexshedding noise generated by acylinder equipped with eight flexible flaps made of silicone rubber it wasf ound that the flaps alter the frequencyofthe vortexshedding.This led to asudden jump in the corresponding plot of Strouhal number versus Reynolds number.T he aim of the present study is to further investigate the effect of the flexible flaps on the vortex shedding and the possible cause of the Strouhal number jump.To this end, the original cylinder with eight flaps wasmodified by subsequently cutting off flaps, until only the twoo utermost flaps remained.This wasd one in order to verify whether all flaps contribute to the Strouhal number jump or if only the outer flaps are necessary.F inally,t he remaining outer flaps were additionally shortened in order to investigate the effect of the flap size on the Strouhal number.F or each resulting cylinder model, acoustic measurements were performed in an aeroacoustic wind tunnel at lowR eynolds numbers, using twom icrophones on opposite sides of the wind tunnel test section.A high speed camera wasu sed to capture the motion of the flaps of one flap ring approximately at mid span.In addition to the wind tunnel experiments, am odal analysis of the different configurations wasperformed numerically.
The measurement results revealed that the spectra of the cylinders with flaps of equal length showe ssentially the same behavior,meaning the Strouhal number jumps to the same value within the same Reynolds number range between 23,300 and 26,000.This leads to the conclusion that the observed lock-in effect between the vortexs hedding cycle and the flap motion does not seem to be caused by an oscillation of awhole system of flaps, butrather by the movement of the twoouter flaps.
Additionally,the analysis of the flap motion spectra and asubsequent comparison with the eigenmodes of the single flaps showed that the observed Strouhal number jump is caused by al ock-in of the natural vortexs hedding cycle with the next higher flap eigenfrequency.This means that flexible flaps (which can basically be understood as a flat plate of which one side is clamped and three sides are free)c ould be specifically designed in order to obtain desired eigenfrequencies, thus controlling the Strouhal number associated with vortexshedding.

Figure 1 .
Figure 1.Schematic of the original flap ring with 8flaps, as used in [16], showing outer diameter d as well as streamwise length s and width h of the flaps.

Figure 2 .
Figure 2. CAD models of the different flap ring versions used for the study.

Figure 3 .
Figure 3. Photograph of the cylinder with six flaps (the flap rings at mid span are painted black to give abetter contrast in the flap motion measurements).

Figure 4 .
Figure 4. Mesh used for the modal analysis of the original flap ring with eight flaps.
Figure 5shows aschematic and aphotograph of the acoustic measurement setup.

Figure 5 .
Figure 5. Setup used for the acoustic measurements.Top: Schematic (top view, not to scale), bottom: Photograph.

Figure 6 .Figure 7 .
Figure 6.Schematic (side view) of the setup used for the flap motion measurements (not to scale) Figure 9shows the peak Strouhal number as af unction of the Reynolds number based on cylinder diameter.T hereby,t his peak Strouhal number represents the arithmetic mean of the peak Strouhal numbers from both measurements.While the resulting Strouhal number for the original cylinder with eight flaps suddenly jumps from values around 0.25 at Re ≤ 23,300 to av alue of about 0.3 at Re = 26,000, the Strouhal number for the cylinder with shortened flaps

Figure 9 .
Figure 9. Dependence of the peak Strouhal number Sr obtained from the acoustic measurements on the Reynolds number Re based on cylinder diameter,the lines represent linear approximations of the measured data ( baseline cylinder, • 8flaps, H 2flaps, shortened).

Figure 10 .
Figure 10.Dependence of the peak sound pressure level L p,max obtained from the acoustic measurements on the Reynolds number Re based on cylinder diameter ( baseline cylinder, • 8flaps, J 6flaps, I 4flaps, N 2flaps, H 2flaps, shortened.Dashed line: theoretic dipole).

Table I .
Numerically obtained eigenfrequencies of the different flap rings shown in Figure2.
changes from about 0.24 at Re = 23,300 to approximately 0.26 at Re = 26,000.Forthe reference cylinder,the Strouhal number takes values between 0.22 at the lowest Reynolds number and 0.21 at the highest Reynolds num-