This is a repository copy of PIV studies of coherent structures generated at the end of a stack of parallel plates in a standing wave acoustic field.

Transcription

1 This is a repository copy of PIV stuies of coherent structures generate at the en of a stack of parallel plates in a staning wave acoustic fiel. White Rose Research Online URL for this paper: Version: Accepte Version Article: ao, X, Yu, Z, Jaworski, AJ et al. (1 more author) (2008) PIV stuies of coherent structures generate at the en of a stack of parallel plates in a staning wave acoustic fiel. Experiments in Fluis, 45 (5) ISSN Reuse Unless inicate otherwise, fulltext items are protecte by copyright with all rights reserve. The copyright exception in section 29 of the Copyright, Designs an Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private stuy within the limits of fair ealing. The publisher or other rights-holer may allow further reprouction an re-use of this version - refer to the White Rose Research Online recor for this item. Where recors ientify the publisher as the copyright holer, users can verify any specific terms of use on the publisher s website. Takeown If you consier content in White Rose Research Online to be in breach of UK law, please notify us by ing eprints@whiterose.ac.uk incluing the URL of the recor an the reason for the withrawal request. eprints@whiterose.ac.uk

2 PIV stuies of coherent structures generate at the en of a stack of parallel plates in a staning wave acoustic fiel Xiaoan AO, Zhibin YU, Artur J JAWORSKI 1 an Davi ARX School of echanical, Aerospace an Civil Engineering The University of anchester Sackville Street, PO Box 88, anchester 60 1QD, UK Abstract Oscillating flow near the en of a stack of parallel plates place in a staning wave resonator is investigate using Particle Image Velocimetry (PIV). The Reynols number, Re, base on the plate thickness an the velocity amplitue at the entrance to the stack, is controlle by varying the acoustic excitation (so-calle rive ratio) an by using two configurations of the stacks. As the Reynols number changes, a range of istinct flow patterns is reporte for the flui being ejecte from the stack. Symmetrical an asymmetrical vortex sheing phenomena are shown an two istinct moes of generating vortex streets are ientifie. 1. Introuction Flow structures generate by steay flows past bluff boies have been a subject of many theoretical an experimental stuies, the classic example being formation of the von Karman vortex street behin circular cyliners (Kovasznay, 1949). This class of phenomena is important in many inustrial problems incluing: aerospace flows, civil an marine engineering, esign of heat exchangers or the behaviour of overhea power cables. Somewhat more complex situation arises when steay flows are replace by oscillatory flows (with or without the steay component), the funamental ifference being that vortices she in one half of the cycle impinge on the bluff boy when the flow reverses an may interact with vortices she uring the other half of the cycle. This may lea to interesting lock-on effects resulting in an interaction between the flow an the structural components within (Chung an Kang, 2003, Barbi et al, 1986). Within the class of purely oscillatory flows, by far the most stuie geometrical configurations were flows past circular cyliners (Obasaju et al., 1988; Iliais an Anagnostopoulos, 1998) although other geometries have been consiere incluing a square cross-section (incline at various angles to the flow) or a flat plate perpenicular to the flow (Bearman et al., 1985, Okajima et al., 1997) an triangular an T-shape geometries (Al-Asmi an Castro, 1992). Other stuies investigate the effects of the proximity of the external bounaries on the flow (Sumer et al. 1991). As well as external flows, internal oscillatory flows have been investigate. These inclue oscillatory flows in pipes with wavy walls (Ralph, 1986), internally place orifices (De Bernarinis et al., 1981) or internally baffle channels (Roberts an ackley, 1996). It is wiely accepte that the morphology of the flow structures present within oscillatory flows is governe by three similarity numbers: the Reynols number (Re), the Keulegan-Carpenter number (KC) an the Stokes number ( ), although only two 1 Corresponing author; Tel: +44 (0) ; Fax: +44 (0) ; a.jaworski@manchester.ac.uk 1

3 out of these are really inepenent, as Stokes number can be expresse as the ratio of Reynols number to KC number. Tatsuno an Bearman (1990) stuie the morphology of the flows generate from the oscillatory cyliner as a function of KC an while similar stuies were performe by Okajima et al. (1997) for square cyliners. These have shown a range of flow regimes ranging from fully attache symmetrical pair of vortices through to symmetrical an alternating vortex sheing. In all of the experimental stuies mentione above, the typical setup inclues either the bluff boy being oscillate through the stationary flui, using some form of mechanical rive, or an oscillating incompressible flui within U-tube type of water tunnel, with the bluff boy being hel stationary. However, it shoul be note that similar flow problems incluing vortex sheing phenomena also arise in acoustic systems when the level of acoustic excitation is relatively high. These inclue systems such as pulse tube refrigerators, staning or travelling wave thermoacoustic evices or their components such as jet pumps, Stirling engines an refrigerators an others, where high intensity acoustic wave (or oscillatory flows in general) encounter suen iscontinuities in the cross-section of an acoustic uct. The initial motivation for the current paper came from the nee to unerstan the behaviour of the flow in a staning wave thermoacoustic evice in the vicinity of the so calle thermoacoustic core. This typically comprises of a stack of parallel plates (thermoacoustic stack) sanwiche between two heat exchangers (often also constructe as a set of shorter but thicker parallel plates with a somewhat larger pitch). The role of the thermoacoustic core is to either prouce acoustic power ue to the temperature graient impose by the heat exchangers or to consume externally supplie acoustic power in orer to facilitate heat pumping from col to hot heat exchanger by virtue of the so-calle thermoacoustic effect (Swift, 2002). In the high-intensity acoustic fiel, the flow structures at the en of the stack, or the heat exchanger (or in the region in between) are very complex ue to the iscontinuities of the cross section an the oscillatory nature of the flow. Clearly, the energy transfer taking place within the thermoacoustic core will be affecte by entrance effects, vortex sheing an generation (or suppression) of turbulence over ifferent parts of the acoustic cycle. The existing moels to calculate the performance of the thermoacoustic systems are base on the linear acoustic moels (for example DeltaE, as escribe by War an Swift, 2001) with only some corrections being mae to account for non-linear acoustics effects such as turbulence. The evelopment of such coes is hinere by the lack of unerstaning of the funamental thermalflui processes. Despite being roote in the area of thermoacoustics, the current investigation shoul be seen on a more general level, namely the funamental flui ynamical processes of interest to a wier auience. This is the reason why the current research covers somewhat larger parameter space than woul be expecte from the point of view of thermoacoustic stacks alone an covers the range characteristic for finne heat exchangers an possibly beyon. The experimental stuies of the above phenomena in the context of thermoacoustics are very limite. Gopinath an Harer (2000) stuie the heat transfer effects from a single circular cyliner place in an acoustic resonator (with a possible application to thermoacoustic heat exchangers). They ientifie two flow regimes: the laminar attache flow regime an the less unerstoo regime where the vortex sheing is prevalent with much higher heat transfer coefficients. Blanc-Benon et al. (2003) use PIV measurements to investigate the flow fiel aroun an vortex sheing from a stack of parallel plates for relatively low rive ratios % an 2

4 compare the experimental results to CFD simulations. They have shown the presence of symmetrical vortices, which on their two experimental configurations ( thin an thick plates) took an elongate an concentrate form, respectively, but never fully etache from the plates. ao et al. (2005) conucte similar PIV stuies using a somewhat larger geometrical arrangement an higher rive ratios (up to 3%) an showe that at higher rive ratios the symmetrical vortices are replace by alternately she vortices. The current paper is an extension of this early work by a proviing more complete experimental ata an its more etaile iscussion an analysis. 2. Experimental apparatus an proceure The experimental apparatus use in the current stuy (Figure 1) was iscusse in some etail by arx et al. (2006) an therefore only a brief escription is given here. Its main part is a 7.4m long transparent Perspex pipe, with the internal cross-section 134 x 134 mm, an the wall thickness of 8 mm. One en of the pipe (to the left of Fig. 1) is close by an en-cap with a flush mounte pressure transucer. The other en is connecte to a relatively large louspeaker box (600 x 600 x 600 mm) through a 0.3 m long pyramial section to match the change in cross-sectional imensions. The resonator is fille with air at atmospheric pressure an room temperature. The first moe of operation (quarter-wavelength) has a funamental frequency f = 13.1 Hz. The experiments were conucte for two stacks of plates (shown schematically in Fig. 2). The length, l, of both stacks was 200 mm, while their with was 132 mm (some clearance ha to be left between the stack an the internal resonator wall). Stack I comprise of 21 Perspex plates of thickness = 1.1 mm with the plate-to-plate spacing D = 5 mm. Stack II was mae out of eight Perspex plates with = 5 mm an D = 10 mm. Both stacks were place in the resonator 4.1 m from the en as schematically shown in Fig. 1. The rive ratio in the experiments was varie by changing the excitation voltage of the louspeaker an controlle by measuring the amplitue of pressure oscillations recore by the ynamic pressure transucer mounte insie the en-cap of the resonator (Enevco oel 8510B-2). Flow fiel measurements were performe using a PIV system by TSI. The light sheet from a ual N:YAG laser enters the resonator perpenicular to its axis, is reflecte by a small mirror an becomes parallel to the resonator axis an normal to the surface of the stack plates. The flow is seee by particles prouce by a smoke generator using 50-50% mixture of glycerol an water, with typical roplet iameter in the range 1 10 microns. Images are taken by a 4 mega-pixel camera (TSI POWERVIEW) an post processe using commercial software (TSI INSIGHT). Cross correlation is use an the interrogation winow is 16 pixels by 16 pixels. The flow fiel measurements are performe at 20 phases within the acoustic cycle (i.e. every 18º). Therefore an appropriate phase locking mechanism ha to be evelope. In this stuy the pressure oscillation measure by the pressure transucer mounte at the en cap was use as a reference. The output signal from the pressure transucer is connecte to an in-house mae pulse generator, which can generate a pulse sequence of the same frequency as the acoustic wave, but with variable time elay from the trigger point. The output of the pulse generator is connecte to the synchroniser (TSI LASERPULSE) that controls the timing of the laser action an triggers the frame grabbing of camera. In this way, the capture images can be phase locke relative to the pressure oscillation (an by the same token to the velocity oscillation, which coul be verifie inepenently both by PIV an hot wire measurements). Figure 3 illustrates the timing of 20 phases ( 1, 2 20 ) within the 3

5 cycle, relative to both velocity an the pressure graient oscillations. For each of the 20 phases, 100 pairs of images were taken to erive the phase-average velocity fiel. The fiel of view of the PIV images varie from 25mm 25mm to 60mm 60mm, epening on the flow features to be image. This correspons to the resolution of the velocity vector fiel between 0.10 mm an 0.23 mm. 4.1 m Fig. 1 Sketch of the experimental apparatus Fig. 2 Geometry of the stack (only three plates shown) Fig. 3 Phase-average velocity oscillation at point an corresponing phases at which the PIV measurements are taken. p/x is the axial graient of acoustic pressure in an oscillation perio. (Simple-harmonic oscillation is assume so that p/x is 90 egrees out of phase of velocity oscillation.) 4

6 Due to its nature, the PIV technique is not sufficiently time-resolve to permit measurements of the fluctuating components of velocity. Therefore the investigations of the vortex sheing frequencies, to be use in estimating typical Strouhal numbers, ha to be conucte using stanar hot wire methos. Velocity fluctuations were measure with SN type hot-wire probe (DANTEC) operate in the constanttemperature moe using a TSI IFA300 system. The probe is place normal to the plate, while the sensor is normal to the axis of the resonator. The position of the probe is schematically shown in Fig. 2. For Stack I, the sensor is place 2 from the ege of the plate, while for Stack II this istance is 4 because the large iameter of the probe support prevents the sensor to be closer to the plate en (the istance from the plate is enote in Fig. 2 as h w ). A high-pass filter set at 30Hz is use to remove the signal component relate to the funamental frequency of 13.1Hz. Both the filtere signal an the unfiltere signal are recore with a sampling frequency of 5000 Hz ata points are acquire in a typical experimental conition. 3. Experimental results 3.1 Overview of the experimental parameters an conitions Table 1 summarises the stack imensions an the basic acoustic excitation an flow conitions. In aition to stack geometry, the table contains porosity,, efine as the ratio of the total cross-sectional area of the channels within the stack over the cross-sectional area of the resonator. For a stack of evenly space plates this can be approximate by D/(D+), assuming that the gaps between the stack an the walls can be neglecte. Drive ratio D r is the ratio of the acoustic pressure amplitue at the close en of the resonator, p 1, a, to the mean pressure in the resonator p m. The values of the amplitue of the axial velocity, u, at point (Fig. 2) were extracte from the phase-average velocity fiel for the 20 phases, by using the least square fitting metho for a sinusoial function. is the isplacement amplitue of the oscillating gas parcel. It can be calculate from u as = u /. The Reynols number use in the current analysis, Re, is base on the axial velocity amplitue, u, an the plate thickness,, as u Re (1) where is the kinematic viscosity of air at ambient conitions. However, as will be shown later, for convenience of comparisons between ifferent phases within the cycle it seems useful to introuce a Reynols number base on the instantaneous value of velocity, u, at point (later referre to as phase Reynols number ): u Re. (2) u sin u t, but coul be zero, if one chooses to count the time Of course, from the moment when u 0. Similarly, although strictly Re can be either positive or negative (epening on the irection of u ), it is easy to specify the flow as out of the stack an into the stack to avoi confusion. Of course, the relationship between the two Reynols numbers is Re Re sin. (3) t 5

7 Table 1: Summary of stack imensions an experimental conitions (mm) D (mm) l (mm) Stack I Stack II D r u (mm) Re (%) (m/s) It is worth making two comments at this point: Firstly, introucing phase Reynols numbers is not an entirely new iea. Yellin (1966) consiere flow transition in pulsatile flows within bloo vessels an use an analogue of Re efine here. Seconly, as will become clearer in later sections (especially 4.2 an 4.3) Reynols number cannot be a unique similarity number efining the oscillatory flow behaviour in general. This shoul be remembere when looking at labelling of the experimental ata which is mae using Reynols numbers efine by equations (1) an (2). 3.2 Oscillatory flow aroun the en of the stack Figure 4 represents a typical example of the flow visualisation obtaine in the oscillatory flow near the en of the stack, shown here to illustrate its main features. Here the flow past Stack I was investigate at the rive ratio of 1.0%, giving the Reynols number Re 207 (other relevant parameters can be foun in Table 1). The flow features are visualise by using the phase-average vorticity fiel. For brevity, only nine most characteristic phases are selecte out of the 20 phases capture within the cycle. These are shown as a time sequence to illustrate the evolution of the flow structures. The phase Reynols number, Re, is inclue in the graphs; grey areas inicate the presence of the plates within the imaging omain. As can be seen from Fig. 4, the flow can be ivie into two main stages: the ejection stage, when the velocity at point is towar x>0 (i.e. for phases 1 10 as illustrate in Fig. 3) an the suction phase, when the velocity at point is towar x<0 (i.e. for phases as illustrate in Fig. 3). As can be seen the generation of coherent structures takes place in the ejection stage iscusse in Section Ejection stage In general the ejection stage can be further subivie into two ifferent situations: acceleration stage (phases 1 5) an eceleration stage (phases 6 10). Several characteristic flow phenomena/patterns can be observe: (A) Formation of a pair of attache symmetric vortices; (B) Elongate vortex structures; (C) Breakup of elongate vortices into an asymmetric vortex street ; (D) Alternate vortex 6

8 sheing from the en of the plate. The etails of these characteristic flow patterns are escribe with reference to Fig. 4 (a-e). (A) Formation of a pair of attache symmetric vortices: The flow in phase 1 (Fig. 4a) has a positive velocity an starts to accelerate. Base on the instantaneous velocity at point, the corresponing phase Reynols number, Re, is about 47. A pair of vortex structures is forme at the en the plate. They are symmetrical relative to the centre-line of the plate in the x-y plane. Similarly as in the classic case of the wake behin a plate with a square trailing ege (as foun in Bachelor, 2000), a recirculation region is forme at the en of the plate, where the pair of vortices remains attache. In the inner region of the channel forme by two neighbouring plates, there is a pair of shear layers with their vorticity in an opposite irection to the vorticity within the plate bounary layer. (a) Re =46 (b) Re =205 (c) Re =177 () Re =81 (e) Re =18 (f) Re =46 (g) Re =106 (h) Re =205 (i) Re =18 Figure 4 Vorticity contour map for the flow aroun the en of Stack I uring one acoustic cycle. (a) 1, (b) 5, (c) 7, () 9, (e) 10, (f) 11, (g) 12, (h) 15, (i) 20 (Re = 207, Dr = 1.0%). The unit of vorticity legen is s -1 7

9 Y (mm) u (m/s) (a) Y (mm) u (m/s) (b) Figure 5 Transverse profile of axial velocity taken at x=5 mm. Stack I, rive ratio 1%. (a) 1, (b) 5. (Y remains the same as the coorinate system use in Figure 4) Such a shear flow pattern exhibite in this phase is irectly relate to the typical velocity profile shown in Fig. 5a (taken at x = 5mm accoring to the coorinate system use in Figure 4). The velocity oscillation of the flow in the central region is elaye in phase, compare with the flow in the bounary layer region. A peak value of velocity appears at some istance from the plate. (B) Elongate vortex structures: As the velocity increases ue to the flow acceleration the relate phase Reynols number increases an the flow pattern changes accoringly. At first (uring phases 2 4) the attache vortex structures remain symmetric but become elongate. However, as the velocity almost reaches its peak aroun phase 5, the very elongate vortex structures in the wake become asymmetric by exhibiting somewhat wavy pattern (as shown in Fig. 4b), which seems to be relate to the loss of stability in the subsequent phases. The phase Reynols number Re is about 206. Within the channel, the axial velocity profile changes significantly between phases 1 5 by becoming flattene (Fig.5b). The flow velocity in the central region catches up with the velocity closer to the channel walls, which weakens the vorticity in this central region. (C) Break-up of elongate vortices into an asymmetric vortex street : As illustrate in Fig. 3, velocity of the flui leaving the channels of the stack reaches its maximum between 5 6 an so from then on the flui enters the eceleration phase. Fig. 4c ( 7) shows that the pair of elongate vortex structures has broken up into a street of iniviual vortices in the plate s wake, very much resembling the classical von Karman street. At phase 7, the corresponing phase Reynols number Re is about 177. (D) Alternate vortex sheing from the en of the plate: Once the elongate vortex structures have broken up into the street of iniviual vortices, a ifferent mechanism seems to take over, namely the vortices seem to be she in an alternating fashion from the en of the plate in a manner resembling the classical bluff boy vortex sheing. Figure 4 shows the resulting flow pattern for phase 9 for the corresponing phase Reynols number Re aroun 81. The flow slows own even further, an Fig. 4e shows an almost stationary suspene vortex pattern just before the flow reverses ( 10, Re 18). 8

10 3.2.2 Suction stage As shown in Fig. 3, the irection of the flow reverses between phase 10 an 11. The flui an the vortex structures generate in the ejection stage are to be sucke back into the channels between the plates. Figure 4f (phase 11) shows the beginning of such a process. The remains of the vortex street impinge on the en of the plate, the iniviual vortices becoming split into a series of vortices of much smaller size. The shear layers close to the channel walls are pushe away (isplace) further into the central region of the channel by the continually eveloping shear layers in an opposite irection see the channels in Fig. 4f. In aition, in the region near the en of the plate, the entry flow generates a strong shear region, which pushes back (eliminates) the shear layer forme in the ejection stage of the cycle. In Fig. 4g ( 12), the isplace shear layer (which was originally forme in the ejection stage) an the scattere remains of the small vortices finally ie out. The newly generate shear layer corresponing to the irection of the suction flow grows an evelops. In Fig. 4h ( 15), the negative flow velocity is at its peak value. The unsteainess of the shear layer still visible in Fig. 4g an the scattere remains of the weak vortices close to the channel centre have isappeare. In general the shear layer next to the surface of the plates is quite similar with the eveloping bounary layer on a flat plate in the steay flow. Following phase 15 the suction flow starts to ecelerate an the negative velocity approaches zero at about 20 (see Fig. 4i). The flow irection is about to change, when the flow enters the ejection stage of the next cycle. 3.3 Comparisons between flows at various peak Reynols numbers, Re To gain further insight into the Reynols number effects on the flow structure, the flow aroun the en of the stack was stuie at four more rive ratios (0.3, 0.6, 1.5 an 2.0%). In this section, the iscussion will focus on the flow structures in the ejecting stages for varying rive ratios. To save space, the PIV results are shown for a single plate, not an array of plates as previously shown in Fig. 4. Figure 6 shows the flow visualisations for five rive ratios stuie (Fig. 6a through to 6e), however instea of using rive ratios, the figures are labelle with the values of Reynols number, Re, which is more meaningful from flui mechanics point of view. In each of the figures, the images for five selecte phases are shown: 2, 4, 6, 7 an 9, each image also being labelle with the value of the phase Reynols number - Re. In the simplest approach, one can compare the evolution of the flow patterns uring the ejecting stage for ifferent rive ratios (ifferent Re ) by looking at each column in Fig. 6 (i.e. columns a e). However, one coul also imagine a flow pattern evolution by looking at a selecte phase in the cycle (for example 4) an comparing the flow patterns for varying rive ratios (i.e. Re ). To o this one woul have to inspect each row of Figure 6 (rather than each column). In this way one coul try to evelop a relationship between the flow patterns an the phase Reynols number, Re. 9

11 2 4 (a) Re = 62 Re = (b) Re = 123 (c) Re = 207 Re = 63 Re = () Re = 317 (e) Re = 417 Re = Re = Fig.6 Vorticity fiel aroun the en of one plate in Stack I, shown for five rive ratios an five selecte phases within the cycle; (a) Dr=0.3%; (b) Dr=0.6%; (c) Dr=1.0%; () Dr=1.5%; (e) Dr=2.0%; colour scale is the same as for Figure 4. 10

12 Figure 6a shows the results at rive ratio 0.3% ( Re 62). It can be seen that a pair of vortex structures remains attache to the en of the plate in each phase an remains symmetric relative to the centre line in all phases. The size an strength of the vortices clearly increases as the phase Reynols number increases throughout the accelerating stage (see 2 an 4) an increases further in the ecelerating flow up until phases 6 or 7. As the vorticity fe from the plate bounary layer ecreases, so oes the strength of the vortices (see 9). Figure 6b shows the results at rive ratio 0.6% ( Re 123). Compare with the vortex structures in Fig. 6a, the vortices become more an more elongate in the accelerating stage. Eventually (aroun phase 7), the previously symmetrical structures become wavy an this instability amplifies in the eceleration phase (see the asymmetric wavy structure in phase 9). Figure 6c shows the results at rive ratio 1.0% ( Re 207). They correspon to the results alreay shown in Fig. 4. From both figures, one can fin that the vortex structure exhibits asymmetry in phase 5 an 6, that is at an earlier phase than in Fig. 6b. The break-up of this wavy pattern into a vortex street an subsequent sheing from the plate occurs between phase 7 an 9. Quite similar results can be seen in Fig. 6, for rive ratio 1.5% ( Re 317 ). The vortex structures become unstable even earlier ( 4). Further increase of rive ratio, as illustrate in Fig. 6e (2.0%, Re 417), leas to even earlier break-up of the initial symmetrical structures - by phase 4 a vortex street is alreay in place). One coul attempt a qualitative analysis by comparing images in Fig. 6 row by row ; that is by looking at increasing rive ratios for a fixe phase. For example, the first row of figures shows phase 2. From Fig. 6a to Fig.6e the phase Reynols number increases from 31 to 214. One can fin that, the attache pair of vortices elongates, but remains symmetric as the phase Reynols number increases. The secon row of figures shows the flow behaviour for phase 4, as the phase Reynols number increases from 56 to 383. One can clearly see an evolution of the flow patterns similar to phase-by-phase evolution: a pair of symmetric vortices (6a/ 4), elongate symmetric vortices (6b/ 4 an 6c/ 4) unstable/wavy elongate structure (6/ 4) an alternate vortex sheing (6e/ 4). The thir, fourth an fifth rows show the flow pattern evolution for phases 6, 7 an 9, respectively. Similar trens in the evelopment of the flow patterns can be observe. However, clearly, the vortex sheing pattern is present earlier when comparing rive ratios left to right. 3.4 Comparisons between flows aroun Stack I an Stack II Similar experiments to those escribe with reference to Stack I were carrie out for the secon configuration: Stack II (see Table 1 for etails). The main reason for selecting this configuration was a further increase in the Reynols number ( Re ) an a change in the porosity of the stack. The same rive ratios were teste which, given a ifferent porosity, resulte in somewhat moifie velocities an isplacements. The Reynols numbers were increase roughly about five-fol. Figure 7 shows the vorticity fiel aroun the en of one plate in Stack II (keeping the convention of Fig. 6, for Stack I). Results at five phases of the ejection stage are presente for the stack at the five rive ratios teste, corresponing to the Reynols number Re of 317, 650, 1080, 1680 an

13 By comparing the flow aroun the en of plate in Figures 6 an 7a, one can see that the value of Re in both cases is the same: 317. However, instea of an elongate vortex structure which breaks-up into a vortex street which was the main feature for Stack 1, Stack II exhibits a ifferent flow structure, namely a fully attache an symmetrical pair of vortices, somewhat similar to what was observe for rive ratio 0.3% on Stack I. However, for Stack II, the pair of vortices is much less elongate in the stream-wise irection (only about 1 in terms of plate thickness). 2 (a) Re = (b) Re = 650 (c) Re = 1080 Re = 335 Re = () Re = 1680 (e) Re = 2300 Re = 865 Re = Fig.7 Vorticity fiel for the flow aroun the en of Stack II (a) Dr = 0.3%; (b) Dr = 0.6%; (c) Dr=1.0% () Dr =1.5% (e) Dr = 2.0% Increasing Re to 650 leas to the vortex structures becoming asymmetric (see phase 9 in Fig. 7b). This feature is more pronounce for Re see phases 6 to 9 in Fig. 7c. easurements at even higher rive ratios (Figs. 7 an 7e), show that the higher Re, the earlier in the cycle the asymmetric vortex sheing takes place. It is interesting to note that for Stack II, the two flow patterns at the ejection stage escribe uner (B) an (C) in section (elongate vortex structures an their break-up) are absent. 12

14 Another interesting flow feature is the appearance of a pair of counter-rotating vortices, relative to the sense of rotation of the main vortices she from the plate. The initial formation of these vortices is clearly seen for phase 2 in Figs. 7a 7e. These counter-rotating vortices seem to be convecte far away from the plate, preceing the ominating von Karman-type of vortices subsequently she over the ejection stage of the cycle. By phase 9, the vorticity of this pair of counter rotating vortices is practically issipate. These small vortices seem to originate from the remains of the shear layer forme uring the suction cycle. Similar feature is inee present for Stack I, but has a much more elongate shape an is far less concentrate in terms of vorticity levels (see for example phase 4 in Fig. 6). In summary, it seems that the main ifference in the flow behaviour on two stacks investigate (I an II) is the moe of vortex sheing. While Stack I is ominate by elongate shear layers, which then suenly break up into a street of iniviual vortices, Stack II exhibits alternate (bluff-boy type) sheing very early on within the cycle (as long as the Reynols number is large enough). This may well be responsible for the ifferences in Strouhal numbers iscusse in Section Frequency of vortex sheing As illustrate in the previous sections, for both stacks, the vortices will start to she in an alternate fashion for a sufficiently large Reynols number, Re. This behaviour is similar to vortex sheing from bluff boies in steay flows. This part of the stuy looke at the epenence of vortex sheing frequency (an thus the Strouhal number) on the Reynols number, Re, an the geometry of the stack. The measurements were performe using stanar hot-wire anemometry methos (Section 2) in orer to collect the fluctuating velocity signal behin the stack of plates. Typical signal traces of the hot-wire anemometer output are shown in Figs. 8a an 8c for Stack I an Stack II, respectively. The top signal trace represents the unfiltere signal an the bottom one represents the filtere signal. Each fluctuation burst in the bottom signal trace represents an event of vortex sheing in the ejection stage of the acoustic cycle. The frequency spectrum of the filtere signal trace is analyze using Fast Fourier Transform (FFT). The signal trace is ivie into ata blocks, which contain 256 ata points, starting at the same phase of an acoustic cycle, an incluing the whole fluctuation burst event. For each block of ata, a frequency spectrum is obtaine. Subsequently all such frequency spectra, obtaine for a given experimental run, are ensemble-average an a mean frequency spectrum is obtaine. This enables extracting a characteristic peak frequency. Figs. 8b an 8 show the mean frequency spectra of the corresponing signal traces in the left column of Fig. 8. As can be seen from the plots, the curve covers a narrow ban of frequencies, as oppose to a sharp spike which woul normally be obtaine in steay flows past bluff boies. This is most likely ue to variations in the sheing frequency as the instantaneous velocity u changes over the ejection stage, but it may also reflect the fact that vortex sheing is quasi-perioic with the pattern changing slightly from cycle to cycle. The peak frequency is subsequently use to calculate the values of Strouhal number, St efine here as f SH / u. Symbol f SH is use to enote the sheing frequency (such as epicte in Fig. 8.) an to istinguish it from the rig s operating frequency f=13.1 Hz. u is also use to calculate the corresponing values of Re. As shown in Fig.9, for the cases investigate, the Strouhal number is between

15 an 0.20 for Stack II, an between 0.08 an 0.10 for Stack I. As alreay mentione this ifference between the values of the Strouhal number coul be relate to ifferent phenomena that seem to govern the vortex structure formation: a loss of stability of elongate shear layers for Stack I an bluff-boy type of sheing for Stack II. (a) (c) Power ensity [(m/s)^2/hz] Power ensity [(m/s)^2/hz] Frequency [Hz] (b) Frequency [Hz] () Fig.8 Signal trace of hot-wire sensor output close to the stack en, an the frequency spectrum of the signal trace; (a, b) Stack I (Dr = 1.0%; u = 2.7m/s; Re = 194); (c, ) Stack II (Dr = 2.0%; u = 6.5m/s; Re = 2165) 0.25 Strouhal number Stack I Stack II 0 1.E+02 1.E+03 1.E+04 Reynols number Fig.9 Strouhal number, St, versus Reynols number, Re, base on the plate thickness 4. Discussion of results an irection of further stuies This iscussion section highlights some interesting problems in three areas that the current investigation touche upon: the issue of the interaction of the vortex structures originating from ajacent channels with each other, the somewhat 14

16 paraoxical issue of the influence of the Reynols number on the flow patterns for Stack I, an finally the effects of stack geometry. These are iscusse in the three subsections below. 4.1 irror vs. translational symmetries between ajacent vortex streets When inspecting the flow patterns generate by the stack of plates such as those shown in Figs. 4 an 4e, it is clear that the resulting vortex streets may exhibit either mirror or translational symmetries in relation to the channels centrelines. For example, looking at Fig. 4, an counting the channels from the bottom of the graph, it is clear that for the first an secon channel, the vortex streets have mirror symmetries, relative to the channel centre-line. However, for the thir channel the symmetry is translational (or there is an anti-symmetry ) in that the vortex pattern she from the thir plate coul be overlappe with the vortex pattern she from the fourth plate if the image was simply shifte upwars. It shoul be note that similar problems are encountere in the steay flows past an array of plates (e.g. Guillaume an LaRue, 2002). The future stuies shoul investigate how the spacing between the plates within the stack influences this kin of symmetrical or anti-symmetrical alignment of vortices, an whether there is some kin of a lock-on effect when the plates are sufficiently close to one another. This relates to the question of porosity values aresse in section Reynols number paraox When Fig. 6 is stuie row-by-row, that is for ifferent rive ratios, but the same phase, one can see that as the phase Reynols number increases the flow pattern evolves through the stages that are escribe as (A) (D) in section The latter pattern (D) occurs at higher phase Reynols number than patterns (A) (C). However, when the flow patterns are stuie column-by-column, that is for ifferent phase Reynols numbers within the same rive ratio, the relationship between the flow pattern an the phase Reynols number is somewhat ifferent. For example in Fig. 6c, the flow becomes unstable at 6 ( Re =200), an the vortex sheing is observe at 7 ( Re =176) an 9 ( Re =80). This inicates that the flow pattern that usually correspons to a high Reynols at an earlier phase can take place at a lower Reynols number in the ecelerating stage. One can fin more examples of this phenomenon in Fig. 6. In Fig. 6e Re is 214 at 2, an the flow has a pair of elongate symmetric vortex structure. On the other han, in Fig.6b, Re equals 48 at 9, an the flow shows a pair of elongate wavy vortices. This clearly illustrates that the transition between ifferent flow patterns cannot be efine by Reynols numbers alone. There are at least two explanations for this flow behaviour (referre to here for brevity as Reynols number paraox ). The first factor coul be the oscillating pressure graient. It is well known that a favourable pressure graient (responsible for the flow acceleration) tens to suppress flow instabilities, while the averse pressure graient (responsible for the flow eceleration) tens to amplify them (e.g. Lee an Buwig, 1991). This may explain why the vortex structures represente in Fig. 6c break up into a vortex street after phase 6. However, an alternative explanation coul be foun for the flows with higher Reynols numbers (e.g. Figs. 6 an 6e, where the break-up occurs in the accelerating phase): namely that the vorticity generate on the surface of the plates is too strong to be convecte ownstream in the 15

17 form of an elongate vortex structure an breaks-up so that the flow may assume a more efficient form (vortex street) to convect the vorticity ownstream. It is likely that it is the combination of these two factors that ecies on the exact nature of the transition between the flow patterns. These aspects of the flow coul be stuie by means of a rigorous flow stability analysis, which however is somewhat beyon the scope of the current experimental stuies. 4.3 Similarity issues geometry an flow parameters The geometry of a stack of parallel plates (in the 2D cross-sectional sense) can be escribe by three parameters: the plate thickness,, the spacing between plates, D an the length of plates, l. In all tests presente here, the isplacement amplitue of the oscillating gas parcel,, is much smaller than the length of the plates l. Therefore, the effect of the stack length on the flow aroun the en of the stack can be neglecte (Swift, 1988). Furthermore, the stack porosity can be efine as D/(D+), when the plates of the stack are place evenly. In this case any two of the three parameters, i.e., D an, can uniquely efine the stack geometry. It is also obvious from the current experimental results that changing the Reynols number, Re, can rastically change the observe flow patterns. However, a tria of two geometrical parameters an a velocity relate parameter (Reynols number or velocity itself) are not sufficient to efine the problem entirely. Inee a large boy of literature relate to the oscillatory flows past cyliners (Bearman et al. 1985, Bar et al. 1995, Lin et al. 1996, Iliais an Anagnostopoulos 1998) suggests that frequency of the forcing flow must appear in the problem escription in one form or another. A wiely accepte parameter of this kin is Keulegan-Carpenter number (KC), usually efine as (velocity)/(frequency x imension). It is easy to show that this can be expresse here as the ratio of the isplacement amplitue to the thickness of the plate, /. Of course can be calculate from u as = u /, where = 2 f is the angular frequency of the acoustic oscillation. Unfortunately, the ata available in the open literature, relate to the geometry iscusse in this paper (parallel plates), is rather scarce an thus insufficient to perform meaningful similarity stuies. The alreay mentione paper by Blanc-Benon et al. (2003) allows extracting two experimental cases (which are only for relatively small rive ratios). The current work provies a few more cases, but there is no inepenent ata to verify any similarities for larger rive ratios. It is therefore hope that the current work will motivate other researchers to carry out similar stuies in flow rigs of ifferent esigns. Table 2 Parameters of stack geometry an oscillating flow D l D r Re (mm) (mm) (mm) (%) (m/s) (mm) KC Stack I Stack II Configuration A Configuration B Data aopte from Blanc-Benon et al. (2003) u 16

18 Table 2 attempts to compare the experimental parameters between the current stuy an that of Blanc-Benon et al. (2003) for the situations where the flow visualisations in both stuies are relatively similar. The vorticity fiel for Stack I an Stack II at the conitions liste in the table is shown in Fig.10 (with superimpose black arrows inicating the velocity vector). Blanc-Benon et al (2003) use slightly ifferent measurement protocol: only 16 phases were measure in an acoustic cycle an the phases were counte an labelle in a somewhat ifferent way. Figures 10a an 10b show phases 7 an 9, which seem to be closest to phases t 0 +7T/16 an t 0 +9T/16 in Fig. 3 of the paper by Blanc-Benon et al. (2003), relate to the flow patterns for their Configuration B. In both situations one can see elongate symmetrical vortex structures. 7 9 (a) (b) 9 11 (c) () Fig.10 Vorticity fiel (contour) an velocity fiel (vector) of the flow at the stack en. a, b, Stack I (Dr=0.3%); c,, Stack II (Dr=0.3%). Figures 10c an 10 show phases 9 an 11, which seem to be closest to phases t 0 +T/4 an t 0 +3T/8 in Fig. 2 of the paper by Blanc-Benon et al. (2003), relate to the flow patterns for their Configuration A. In both situations one can see more concentrate forms of vortices with the length comparable to the plate thickness. As can be seen from Table 2, KC number has a value of 9.3 for Stack I when the rive ratio, Dr is 0.3% an 14 for Configure B. For these two configurations, the vortex structures behin the plates are of a similar elongate form. When the vortex structures are concentrate at the en of plates, such as for Stack II an Configuration A, KC number has a relatively small value of 2.3 for Stack II an 1.4 for Configuration A. Therefore, in aition to the usual choice of the Reynols number, porosity an KC number seem to be a promising group of non-imensional numbers that coul be use to escribe the effect of the stack geometry on the flow 17

19 characteristics when an oscillating flow aroun the en of a stack of parallel plates is consiere. Nevertheless further work woul be require (especially in experimenting with ifferent frequencies an characteristic stack imensions) to exten this kin of comparisons to other flow patterns, especially the alternating sheing that occurs for larger isplacement amplitues. 5. Conclusion In this paper, the flow structures aroun the en of the stack of parallel plates in the oscillatory flow generate by an acoustic staning wave were investigate using PIV. The flow aroun two stack configurations was measure for a series of acoustic excitation levels (an thus isplacement amplitues). The resulting flow patterns have been ocumente an escribe in some etail. The main finings in this respect are as follows: 1. For the relatively small rive ratios the flow structures alreay ientifie by Blanc- Benon et al. (2003) are present within the flow. These inclue symmetrical an attache pairs of vortices which coul be either elongate or concentrate. However when the rive ratios are increase, other flow patterns exist which lea to alternate type of vortex sheing (similar to von Karman vortex streets characteristic for flows past bluff boies) 2. Two moes of the above mentione alternate sheing were ientifie on the two stacks consiere. The first moe seems to be relate to an instability of the elongate shear layers, which leas to their break-up an fragmentation into a vortex street pattern. The secon moe seems to be relate to the classical von Karman vortex street typically foun in bluff boy vortex sheing in steay flows. Interestingly, the two ifferent moes seem to lea to two ifferent values of Strouhal number. Furthermore, the problem of flow similarities was aresse, which was iscusse in some etail in the context of the mirror an translational symmetries an the Reynols number paraox, the latter relate to the appearance of seemingly similar flow patterns at ifferent phases of the cycle for ifferent rive ratios on Stack I. It is thought that a rigorous stability analysis woul be require to explain this flow behaviour. Finally, the results available in the open literature were compare to some of the results of the current work. The comparison showe that, in orer to escribe the kin of oscillations investigate here, other non-imensional parameters shoul be consiere, besies the Reynols number. In particular, KC number has been pointe out as one possible similarity number. The conclusion that a pair of similarity numbers: Re an KC woul be require for characterising oscillatory flows is not surprising given that it is known to appear for oscillatory flows past bluff boies (e.g. the alreay mentione works le by Bearman an Graham). However, the current work essentially eals with perioic structures present in the oscillatory flows. Unfortunately, investigations of such cases are few an far between. Research into the flows past multiple (perioic) bluff boies are nevertheless available for steay (oneirectional) cases (e.g. Auger & Coutanceau 1978, Hayashi et al. 1986, oretti 1993 an Le Gal et al. 1996). An aitional parameter, to just using a Reynols number, use in such stuies is the ratio of pitch-to-iameter. In the present work a similar approach is aopte by introucing porosity,, as an inepenent similarity number (notation D/(D+) is in effect an inverse of pitch-to-iameter). Therefore, it seems reasonable to suggest that the tria of similarity numbers: Re, KC an coul be 18

20 use as a starting point for characterisation of oscillatory flows past perioic structures. Further work to answer some of these important questions is planne. An interesting point raise by one of the reviewers was relate to fining the right scaling parameter for the Strouhal number analyse in section 3.5. In the current work it is base on the plate thickness (), by analogy to bluff boy sheing, where typically St is base on the transverse imension of the boy. The reviewer pointe out that there coul be an analogy to jet flows mae (the jet iameter being the spacing between the plates, D). Simple experiments consisting of measuring sheing frequencies for stacks which ha every other plate remove i not support the reviewer s suggestion. Sheing frequency i not scale with D in the configurations stuie. Nevertheless, it is possible to imagine that for certain configurations (e.g. very thick plates separate by narrow gaps) plate spacing may be more appropriate for calculating Strouhal numbers, because there woul be very little interaction between ajacent jets. Selection of the scaling length is somewhat arbitrary (see for example comments by oretti 1993 regaring flows past arrays of tubes ), an often counterintuitive. For example Bunerson an Smith (2005) investigate two planar parallel jets, but erive their Strouhal numbers base on the with of the centreboy separating the two jets, not on the with of the jets. Clearly, in the case of oscillatory flows past the series of plates such as escribe in the current paper, a much larger boy of ata woul be require before one coul make a more informe choice regaring the best length scales for non-imensional stuies. Acknowlegment The authors woul like to acknowlege the support receive from the Engineering an Physical Sciences Research Council (EPSRC), UK an Universities UK. References Al-Asmi K an Castro IP (1992) Vortex sheing in oscillatory flow: geometrical effects. Flow easurement an Instrumentation, vol. 3, pp Auger JL an Coutanceau J (1978) On the complex structure of the ownstream flow of cylinrical tube rows at various spacings, echanics Research Communications, vol. 5, pp Bar H, Dennis SCR, Kocabiyik S an Nguyen P (1995) Viscous oscillatory flow about a circular cyliner at small to moerate Strouhal number, Journal of Flui echanics, vol. 303, pp Barbi C, Favier DP, aresca CA an Telionis DP (1986) Vortex sheing an lock-on of a circular cyliner in oscillatory flow, Journal of Flui echanics, vol. 170, pp Bachelor GK (2000) An introuction to flui ynamics. Cambrige, Cambrige University Press Bearman PW, Downie J, Graham JR an Obasaju ED (1985) Forces on cyliners in viscous oscillatory flow at low Keulegan-Carpenter numbers, Journal of Flui echanics, vol. 154, pp Blanc-Benon Ph, Besnoin E an Knio O (2003) Experimental an computational visualization of the flow fiel in a thermoacoustic stack. C.R. ecanique, vol. 331, pp Bunerson NE an Smith BL (2005) Passive mixing control of plane parallel jets, Experiments in Fluis, vol. 39, pp Chung YJ an Kang SH (2003) A stuy on the vortex sheing an lock-on behin a square cyliner in an oscillatory incoming flow, JSE International Journal, Series B, vol. 46,pp De Bernarinis B, Graham JR an Parker KH (1981) Oscillatory flow aroun isks an through orifices, Journal of Flui echanics, vol. 102, pp