Suppression of vortex-induced forces on a two-dimensional circular cylinder by a short and thin splitter plate interference

Transcription

1 Suppresson of vortex-nduced forces on a two-dmensonal crcular cylnder by a short and thn spltter plate nterference Andre S. Chan & Antony Jameson Stanford Unversty Aerospace Computng Laboratory Report ACL December 2007

2 Suppresson of vortex-nduced forces on a two-dmensonal crcular cylnder by a short and thn spltter plate nterference Andre S. Chan 1 Htach GST & Stanford Unversty Antony Jameson 2 Stanford Unversty Ths paper brefly revews bluff body aerodynamcs and focuses manly on the use of short and thn spltter plates to nterfere wth the vortex wakes and suppress vortex-nduced forces (VIF). The present nvestgaton prmarly examnes the suppresson of unsteady, two-dmensonal wake nstabltes over a low Reynolds number range of less than 250. A crcular cylnder s used as the bluff body n ths study. Wake nstabltes of both bounded and unbounded flows are nvestgated. It s found that channel walls have a stablzng effect on the sheddng, and, n ths case, VIF can be elmnated entrely by a correctly placed spltter plate. The nvestgaton s carred out by way of numercal smulaton usng a commercal ncompressble, vscous computatonal flud dynamcs (CFD) solver. A smple experment s also performed to confrm the effectveness of a spltter plate n suppressng flow nduced vbraton. Nomenclature 1UDS = 1 st order upwnd dfference scheme 2UDS = 2 nd order upwnd dfference scheme A = matrx coeffcents u l C D = drag coeffcent CDS = central dfference scheme CFD = computatonal flud dynamcs C D = drag coeffcent, based on U 0, C D * = drag coeffcent, based on U*, C D * C D D = 1 ρ U 2 D = 1 ρ U d *2 d 1 Senor Engneer, Htach Global Storage Technologes, Inc., 5600 Cottle Road, San Jose, CA 95193, AIAA Senor Member. 2 Thomas V. Jones Professor, Dept. of Aeronautcs and Astronautcs, Stanford Unversty, Stanford, CA 94305, AIAA Fellow. 2

3 L C L = lft coeffcent, based on U 0, C L = 1 ρ 2 U 0 d 2 * C L = lft coeffcent, based on U * * L, C L = 1 ρ *2 U d 2 p p C p = pressure coeffcent, C p = 1 2 ρ U 0 2 C pb = mean base pressure coeffcent at 180 from the front stagnaton pont d = dameter of crcular cylnder D = drag force f = sheddng frequency h = channel heght l = spltter plate length l s = dstance of spltter plate separaton from obstructng body l sc = dstance of spltter plate from center of crcular cylnder L = lft force p = pressure Re d = ρ U 0 d Reynolds number based on d and U 0, Re d = µ Re d * = effectve Reynolds number based on d and U *, S φ = sources or snks of a scalar quantty φ f d St = Strouhal number, St = U 0 Re * d * ρ U d = µ St * = * f d Modfed Strouhal number, St = * U t = tme th = thckness of spltter plate U = velocty U 0 = ncomng streamwse velocty U * = mean cross-sectonal velocty along the vertcal lne at cylnder center u, u, v, w = velocty vector u ~ = velocty correcton term VIF = vortex nduced forces x, x, y, z = Cartesan coordnates t = mplct real tme step β = d blockng factor, β = h φ = scalar quantty Γ = dffusvty Λ s = length of spltter plate 3

4 µ = vscosty ω z = z-component of vortcty ρ = densty σ j = vscous stress tensor τ = nondmensonal tme, τ = U l t 0 I. Introducton A vast majorty of past research papers on bluff body aerodynamcs has been devoted to the flow past a crcular cylnder. The research hstory can be traced all the way back to d Alembert wth hs famous paradox. As noted by Roshko [14], the crcular cylnder s, by far, the quntessental bluff body. The popularty of the crcular cylnder comes from ts smplcty and ts practcal mportance n real engneerng applcatons such as offshore ppelnes, brdge towers, pers, etc. Yet, the flow past the two-dmensonal crcular cylnder can be profoundly complex and beautful across the varous ranges of Reynolds numbers. For expermental studes, t s convenent that crcular-shaped tubngs and rods are readly avalable n most places. Ths current work presents a study of vortex sheddng off crcular cylnders n both unbounded flows and bounded channel flows. The goal of ths research s to study the control of vortex sheddng by spltter plate nterference at a relatvely low Reynolds number where three-dmensonalty does not play a role. Structural vbraton caused by vortex nduced forces (VIF) s an mportant specal case of flow nduced vbraton (FIV). Vortex suppresson technques have always been of great nterest to both structural and flud dynamc engneers across many engneerng dscplnes rangng from applcatons n aerospace, cvl, mechancal and even boengneerng. We know from many lfe examples that structural vbraton nduced by wake nstabltes can have a catastrophc mpact on the structure tself. One of the most notorous examples of FIV s the Tacoma Narrows 4

5 Brdge that was destroyed n Another less damagng example s vbraton of telescopes caused by wnd buffetng. Many other FIV examples and countermeasures have been llustrated and analyzed such as those mentoned n Blevns [9]. In addton, an overvew of vortex dynamcs and VIF n the wake of a cylndrcal body and a lst of references has been gven by Wllamson can be found n [2] and [3]. The use of spltter plates to control vortex formaton n cylnder wakes s not a new concept. Roshko, n hs 1954 NACA report [1], had already expermented wth vortex suppresson by placng a thn spltter plate at varous separaton dstances along the mean zero streamlne downstream of the crcular cylnder base n order to nterfere wth ts natural sheddng at Re d = 14,500. The wake nterference caused the alteraton of the sucton base coeffcent of pressure, C pb, and the Strouhal number, St, dependng on the separaton dstance between the spltter plate and the crcular body. Roshko reported less varaton on the pressure dstrbuton over the cylnder crcumference n the presence of a spltter plate, whch would mean less lft and drag fluctuaton or VIF. Roshko also noted that the mean C D was reduced from 1.15 to 0.72 n one of the spltter plate confguratons. Thus, we can further postulate from ths report that spltter plate deployment, n addton to nhbtng vortex sheddng nstablty, produces a streamlnng effect on the flow past a bluff body. Followng the work of Roshko, Apelt and West [15] nvestgated flows past crcular cylnders wth spltter plates attached to a crcular cylnder n the Re d range of 10 4 to 5x10 4. They found that at any rato of the spltter plate length, l, to the cylnder dameter, d, or l/d, between 0 and 7, the flow characterstcs past the cylnder can be altered, as s evdent from the varaton seen on the Strouhal numbers, C D and C pb. They concluded that the effectveness of a spltter plate depends sgnfcantly on the length the plate. At l/d > 5, they observed that the drag component 5

6 becomes constant and the unsteady vortex sheddng s elmnated from the cylnder body even though they noted that the exstence of a vortex street further downstream of the spltter plate. Unal and Rockwell [4] expermented wth the nserton of a long and thck plate wth a sharp, wedge-lke leadng edge n the wake of a cylnder to control the development of wake nstabltes n the Reynolds number range of 140 < Re d < The plate s relatvely thck (of the order of d) and ts length s much greater than d,.e. l/d = 24. Note that Unal et al used crcular cylnders of varous dameters to cover the range of Reynolds number n ther study. Dependng on the separaton dstance from the cylnder, l s, a spltter plate can break the wake flow characterstcs nto two regmes pre-vortex formaton and post-vortex formaton. It was reported that at a relatvely low Reynolds number, the unsteady von Kármán vortex street can be elmnated wth a thck spltter plate at a separaton dstance of 2.8 d for Re d = 142. We smulated a few selected cases of Unal s experments and present the results n the Appendx as a benchmark study for comparson. Mttal [12] suggested the use of a long detached spltter plate for control of vortex suppresson. He found that n an unbounded flow at Re d = 100, a spltter plate whch has the length of 2 d or greater can completely suppress the fluctuatng lft and drag components when placed at a separaton dstance of l sc = 2.68 d. Some notceable amount of VIF suppresson can also be acheved when the spltter plate length, l, s shorter than 2 d. It was also found that when a spltter plate s placed at a separaton dstance of l sc > 2.68 d, VIF s worse than the case of no spltter plate. The queston arses f t s possble to suppress the unsteadness by a thn (.e. th < 0.1 d) and short spltter plate (.e. l 1 d). In some real engneerng applcatons, t s mpractcal to have a controllng devce such as a spltter plate several tmes longer than the cylnder body tself. 6

7 Furthermore, how effectve would a spltter plate be n a bounded flow? Several of the prevous researches nvolved a spltter plate that s attached to the bluff body. In order to mnmze VIF on the bluff body tself, t s mportant that the spltter plate be physcally separated from the bluff body. Attachng the spltter plate would drectly connect ts aerodynamc forces to the bluff body. By detachng the spltter plate, any fluctuatng forces actng on the spltter plate wll not be drectly coupled nto the bluff body. Ths would allow the maxmum attenuaton of VIF on the bluff body. Hence, the present research work wll focus on the confguraton where the spltter plate s deployed n the downstream wakes but physcally detached from the cylnder body. Our work s partcularly motvated by the need to fnd ways of reducng VIF of the read-wrte arm of a hard dsk drve, whch les between the upper and lower surfaces of two dsks n a multple-platter confguraton. Hence, we wsh to address the ssue of how effectve a detached spltter plate can be n suppressng unsteady vortex sheddng behnd a cylnder n a channel. We examne both the case of a cylnder between movng walls, whch may be regarded as an dealzed model of a dsk drve, and the case of a cylnder n a Poseulle flow wth fxed walls. II. Numercal method and valdaton CFD has been establshed as a valuable tool n the development of aerodynamc desgn, partcularly n the aerospace ndustry. In ths study, a commercal CFD code wll be used as the prmary nvestgatve tool snce t offers benefts n terms of ease of use n geometrcal and mesh modfcaton, reasonable soluton algorthm effcency and good post processng capablty. Most commercal CFD codes solve the ncompressble Naver-Stokes equatons usng the SIMPLE (Sem-Implct Method for Pressure-Lnked Equaton) algorthm, frst developed by Patankar and Spaldng n 1972 [5], or one of ts varants such as SIMPLE-C and SIMPLE-R. The frst step of ths study s to establsh confdence n the numercal results by comparng our smulatons wth 7

8 some well-known numercal and expermental work of prevous nvestgators. Some examples are presented n the Appendx. A typcal commercal CFD code uses the fnte volume approach to solve for an ncompressble, vscous flow. Consder the followng conservaton equatons: Conservaton of Mass ρ + t ( ρu x ) = 0 (1) Conservaton of Momentum ( ρ u j ) ( ρuu j ) + t x = σ x j p x j (2) where ρ s the densty, u s the velocty component, p s the pressure and τ j s the stress tensor. Most commercal codes solve the partal dfferental equatons as an nstance of the generc conservaton equaton. The detaled mplementaton s well documented n varous publcatons and textbooks, namely Ferzger & Perć [16]. In summary, the generalzed transport equaton can be expressed n the followng manner: ( ρφ) ( ρu φ + ) = t x x φ Γ x + S φ (3) The two terms on the left hand sde of the above equaton (3) are the transent term and convecton terms of any scalar quantty φ, respectvely. These terms are balanced by the dffusve transport term and any sources or snks of φ on the rght hand sde, respectvely. The calculaton of pressure generally poses a problem n obtanng a soluton to the Naver- Stokes equatons at a constant densty. Ths s due to the lack of an ndependent equaton for 8

9 pressure. The Naver-Stokes equatons are non-lnear wth unknown velocty components. In addton, the pressure gradent contrbutes as a source term n the three momentum equatons (2). Therefore, solvng the flow feld requres a known pressure feld. In compressble flow, the contnuty equaton, n practce, can be solved to determne the densty of the flud. Pressure can then be determned from an equaton of state. However, n an ncompressble flow or flow at a low Mach number, t s not approprate snce the flud densty s treated as constant and has no drect relaton wth pressure. There are four unknowns u 1, u 2, u 3 and p n four equatons contnuty, x 1, x 2, and x 3 -momentum: = 0 x u (4) ) ( x p x u x x u x u x x u u t u + = + µ µ ρ (5) ) ( x p x u x x u x u x x u u t u + = + µ µ ρ (6) ) ( x p x u x x u x u x x u u t u + = + µ µ ρ (7) Essentally, ths means we cannot solve for velocty components untl we fnd out what the pressure feld s, and vce versa. By an teratve approach, the SIMPLE algorthm couples the pressure and velocty terms after solvng the dscretzed equatons separately. The detals of the SIMPLE algorthm can also be found n Ferzger & Perc [16]. 9

10 In 1980, Patankar proposed a new algorthm called SIMPLE-R to address the problem of dvergence as a result of the omsson of the velocty correcton term,, P u A l l u, l u AP u~ =, from the corrected dscretzed momentum equatons. The acronym SIMPLE-R stands for SIMPLE- Revsed. The algorthm SIMPLE-R essentally requres an ntermedate step of determnng the pressure feld from a Posson equaton based on pseudo-velocty components obtaned from the velocty correcton equaton of the SIMPLE algorthm. Ths ntermedate pressure p m s used n the correcton equatons to agan solve for velocty components and pressure, before movng on to the next teraton untl convergence s acheved. In general, SIMPLE-R wll result n fewer teraton steps but can be computatonally more expensve than SIMPLE, as t wll requre more calculatons n determnng the ntermedate pressure n each teraton. Also commonly mplemented n commercal solvers s SIMPLE-C, whch stands for SIMPLE-Consstent. Ths varant also solves the conservaton equatons by couplng the pressure and velocty terms smlar to the orgnal SIMPLE. However, as ponted out by Patankar n the mplementaton of SIMPLE-R, the problem of gnorng the velocty correcton term can lead to nconsstency when solvng for the pressure correcton term. In 1984, Van Doormaal and Rathby proposed that snce u A u l s of the same order as l u, A l u, P. It therefore mposes a l l less severe constrant on the pressure correcton equaton f one replaces u A P n SIMPLE wth u u u AP + l Al rather than gnorng A l u l, l altogether [6]. Ths can be done wth mnmum change to the orgnal SIMPLE algorthm, but does have a sgnfcant and postve mpact on convergence. It can also be computatonally less expensve than SIMPLE-R. Many of the commercal CFD codes, ncludng CFX, Fluent, and Star-CD, mplement SIMPLE and/or any one of ts varants [7]. The code chosen for ths work s CFD-ACE+ by ESI 10

11 Corporaton. It uses the SIMPLE-C pressure-correcton algorthm along wth varous spatal dscretzaton schemes. Several optonal schemes are avalable n CFD-ACE+ for spatal dscretzaton of the convecton term, namely 1 st order upwnd (1UDS), 2 nd order upwnd (2UDS) and central dfferencng (CDS). Typcally, the soluton from a 1 st order upwnd scheme tends to be more dffusve. It s therefore avoded n ths study as t can adversely affect the sheddng characterstcs of the wake vortces. In the Reynolds number range of our study, a 2 nd order scheme should be suffcent n yeldng a reasonably accurate result. For tme dscretzaton, both the 1st order backward Euler and the 2 nd order Crank-Ncholson (CN) schemes are avalable. The dscretzaton scheme of the dffuson and source terms s fxed and no opton can be set; however, t should produce 2 nd order accuracy. III. Vortex sheddng n channel flows A major goal of ths study s to nvestgate boundary effects on bluff body aerodynamcs. There are plenty of real engneerng examples of a bounded flow movng past an obstructon of an arbtrary shape bluff body or otherwse for example, an arplane n a wnd tunnel, a boat movng through a canal, a tran movng through a tunnel, and of partcular nterest to us, the flow around a read-wrte arm of a hard dsk drve. We examne frst the case of a cylnder n a channel wth movng walls. We smply set the ncomng flow to be unform. In order to numercally nvestgate the response of the flow past a crcular cylnder to the walls, we brng the top and bottom boundares progressvely closer to the body whle adjustng grds accordngly to properly address the smulaton accuracy and numercal stablty. We start our calculaton by keepng a constant U 0 wth top and bottom boundares far apart. We then ncrementally regenerate the grds as we brng the top and bottom boundares closer together to nvestgate the blockage effects. Note that the crcular cylnder s kept perfectly at the center 11

12 between the two boundares. In addton to the same coarse meshng scheme that we employ n the unbounded case, the mesh spacng s vared n a geometrc progresson n order to resolve the boundary layer flows near the top and bottom walls as shown n Fgure 1. The grd spacng adjacent to the cylnder and wall surfaces s set to be approxmately d. Dependng on the stuaton, a tme step n range of 1e-4 t 1e-5 s used throughout ths research. A mnmum of a four-order reducton n the error resdual s sought as a convergence crteron. It s essental that we balance the soluton accuracy wth the computatonal speed to enable systematc numercal expermentaton wth spltter plate length and separaton. After extensve nvestgaton, the current coarse grd arrangement was found satsfactory for our study. Fgure 1: Coarse mesh around the body for bounded flow study The Strouhal number s tradtonally defned as a functon of a gven ncomng unform freestream velocty. Typcally, t s used n an unbounded flow condton where the upper and lower bounds are far enough away so that they have no effect on the flow near the body. In Fgure 2, the Strouhal number s plotted on a dashed lne as a functon of the blockage factor, d β = that s when β = 0, we recover the case of unbounded flow, and when β = 1, we would h 12

13 have a complete blockage. In a practcal sense, we cannot smulate the flow at these two lmts. We would have to make an assumpton so that when β s approachng zero,.e. h >> d, we consder t equvalent to the unbounded case snce the top and bottom boundares should have neglgble effect on the flow characterstcs past the body. In that case, t s found on Fgure 2 that the Strouhal number St, based on ncomng unform velocty, approaches an approxmate asymptotc value of 0.18 at the unbounded lmt (β = 0). The trend lne of St (dashed lne) also suggests that there s a fnte value at the physcal lmt of β = 1, whch cannot be true snce the flow velocty at ths lmt would approach nfnty, resultng n a St value of zero. Calculatng the Strouhal number based on ncomng freestream velocty, U 0, would lead to a non-zero value and therefore a modfcaton to the Strouhal number s needed to address the value at ths lmt. The most logcal modfcaton s to base the Strouhal number on the mean cross-sectonal velocty at the body center. We wll use an astersk to denote ths modfcaton. At the lmt of β = 1 where the mean velocty approaches nfnty, St * wll approach zero. If we calculate Re * d as shown on Fgure 3, usng the mean cross-sectonal flow velocty at the body center, nstead of smply the unform ncomng flow velocty, we can see that, when β > 1/2, Re * d s already well over 300. As observed by dfferent nvestgators, when the Reynolds number surpasses the value of about 200 to 250, vortex wakes start to become unstable n the spanwse drecton for the case of a wake of a cylnder n an unbounded flow. Forcng the flow past the cylnder body through the movng walls would result n sheddng characterstcs that would most lkely cease to be purely two-dmensonal. One should expect a transton of flow from beng purely two-dmensonal to the formaton of three-dmensonal vortex wakes at the approxmate blockage range of β between 0.25 and 0.5 where Re * d s n the range of

14 * Flow past a crcular cylnder St St* 0.6 Strouhal Number Blockage factor, β Fgure 2: Blockage effect on Strouhal and modfed Strouhal numbers of flow past a crcular cylnder at constant U 0 based on unbounded Re d = Flow past a crcular cylnder Red Blockage factor, β Fgure 3: Re d * amplfcaton as a functon of β, based on unbounded Re d =

15 Fgure 4: Instantaneous velocty profles of flow between crcular cylnder body and upper wall Fgure 5: Instantaneous vortcty plot of flow past a crcular cylnder, at Re d = 150, β = 1/20 (.e. unbounded), cells 15

16 Fgure 6: Instantaneous vortcty plot of flow past a crcular cylnder, same U 0 whch produces Re d = 150 on Fgure 5, β = 1/2, cells Because of the nlet condton, as β ncreases, the flow has to speed up and, as a result, the vortex wakes ntensfy. Fgure 4 shows the velocty profles at varous blockages from β = 1/20 and β = 2/3. As the blockages ncrease, the flow becomes more jet-lke forcng a change n sheddng characterstcs. Fgure 5 shows the vortcty plot when the channel s wde (β = 1/20). In ths case, the wall effect can, for all ntended purposes, be neglected and thus the flow can be consdered to unbounded. Fgure 6, on the other hand, shows a much more ntensfed vortex sheddng (plotted on the same scale as Fgure 5) when the channel s narrowed (β = 1/2). Both fgures show von Kármán vortex streets smlar to what would be produced by cylnder of crcular shape. It s apparent from Fgure 6 that, n the flow near the top and bottom boundares, there s enough evdence of a strong nteracton of shear layers between those produced by the obstructng body and those produced by the movng walls. The von Kármán vortex street that sheds from the body s seen to nteract wth a par of counter-rotatng vortex streets on the boundares. Interestngly, the modfed Strouhal number, St *, on Fgure 2 s relatvely constant at an approxmate value of 0.2 for the blockage range of 0 < β < 2/3. Ths s because, as the blockage ncreases, the sheddng frequency of the vortex wakes ncreases, St * s balanced by the ncrease n the average mean flow velocty past the cylnder body. It s useful n a sense that t can be 16

17 used to predct how the sheddng characterstcs of a crcular cylnder would behave later on n a bounded flow stuaton. In addton, both the mean and fluctuatng components of drag and lft are also relatvely constant along the same blockage range as shown on Fgure 7. Plotted are mean C * D and fluctuatng components of C * L and C * D ; all of whch are based on mean crosssectonal flow velocty past the body center. Note that the mean lft s zero. On the other end of the blockage range,.e. 2/3 < β < 1, t s mportant to note that apart from flow threedmensonalty, compressblty effects would become ncreasngly sgnfcant n our numercal solutons. 2 Flow past a crcular cylnder 10 C L * Oscllaton Mean C D * C D * Oscllaton Coeffcents of Lft and Drag Blockage factor, β Fgure 7: Forces on the crcular cylnder body as a functon of blockage factor Movng walls versus statonary walls Another classcal channel flow s the vscous Poseulle flow between two statonary walls. In settng up the computatonal model, the only dfference from the prevous case s to fx the upper 17

18 and lower boundares. In comparson to the unform flow wth movng walls (Fgure 8), the flow s allowed to fully develop nto a parabolc profle upstream of the crcular cylnder (Fgure 9). In these llustratons, the walls are shown to be postoned to be equvalent to the blockage factor of β = 0.5. It s mportant to note that wth ths condton, the cylnder sheds steady and symmetrcal wakes at Re * d = 150 n the statonary wall case. Ths would not be nterestng from the pont of vew of vortex suppresson. Therefore, n the nterest of VIF suppresson, the ncomng velocty wll be set to produce an equvalent of Re * d = 250 at β = 0.5 for the case study of spltter plate n the next secton. Fgure 8: Incomng velocty profle of unform flow wth movng walls Fgure 9: Incomng velocty profle of developed flow between statonary walls A notceable dfference of body forces s observed to be that wth the movng walls, the fluctuatng components of lft and drag are much larger than those generated by flow n between statonary walls. The VIF s much more ntense due to the more energetc shear layer nteracton that happens between the movng walls and the cylndrcal body than n the case of statonary walls. However, the mean drag n the case of statonary walls s much hgher than the case of 18

19 movng walls as the overall flow over the cylnder s beng retarded more. The mean C * D values are for movng walls and for statonary. For the fluctuatng components of force, the movng walls produce C * L and C * D n the range of and respectvely whle the statonary ones gve and Fgure 10 and Fgure 11 show the nstantaneous vortcty contours of the two cases, llustratng the dynamcs of flow n between movng walls versus the flow n between statonary walls. The vortex wakes s enhanced by the movng walls to form nto a much more organzed von Kármán vortex street than the statonary case. Fgure 10: Movng walls, bounded flow, β = 0.5, Re d * = 250 Fgure 11: Statonary walls, bounded flow, β = 0.5, Re d * =

20 IV. Systematc studes of spltter plate deployment As reported by past researchers, a spltter plate n an unbounded flow can effectvely change the sheddng characterstcs and, as a result, VIF. We shall now systematcally nvestgate the mechansm of decouplng the effect of vortex wakes on the VIF of bluff bodes for both unbounded and bounded flows. We expect that the proper postonng and length of the spltter plate to be the crtcal factors n the effectveness of the VIF suppresson. To carry out these numercal experments, we modfy the coarse mesh of the crcular cylnder, whch was used n the verfcaton tests, to study dfferent spltter plate confguratons. Among all the confguratons that mght be devsed, we focus on two man factors the effect due separaton dstance, l s, and the effect due to spltter plate length, l, as shown on Fgure 12. The thckness of the spltter plate, th, used wll be kept constant at 0.1 d throughout the current nvestgatve research. d l s l th Fgure 12: Schematc drawng of cylnder and spltter plate parameters Case 1: Effect of a spltter plate n an unbounded flow, Re * d = 150 In ths case, we vary the spltter plate length and the separaton dstance from the crcular cylnder body n an unbounded flow condton,.e. β 0. The mean lft s zero, and we focus on how spltter plate nterference suppresses the fluctuatng component of lft and drag on the cylnder from ts own vortex wake effects. In studyng the separaton dstance, the computatonal grds are remeshed and adjusted so the physcal representaton s not lost. We also pcked a representatve confguraton and ran a denser mesh calculaton to verfy the results. 20

21 It s clear from the Strouhal plot on Fgure 12 that a spltter plate that s placed downstream of the cylnder body can nterfere wth the development of vortex wakes. The degree of effectveness n vortex and VIF suppresson depends largely on the spltter plate length and the separaton dstance. Sharp transton ponts at separaton dstances between 1.65 d to 1.85 d can be seen for all plate lengths whch range from 0.25 d to 1.0 d. Ths also corresponds to the same pont, where the sudden change n lft fluctuaton (Fgure 14) and mean drag (Fgure 15) are observed. From these fgures, we can determne the optmal separaton dstance for a gven spltter plate length. The same effect on drag fluctuaton can also seen n Fgure 16. In comparson to study by Unal & Rockwell [4] (See also the benchmark study shown n the Appendx), usng ths type of thn spltter plate does not completely suppress the unsteadness n the vortex wakes, gven an unbounded flow stuaton. However, some amount of reducton n vortex nduced force fluctuaton can stll be realzed when a spltter plate s placed at a proper dstance downstream of the crcular cylnder. In agreement wth past researches, one of the most benefcal qualtes of spltter plate nterference n an unbounded flow, however, s the fact that there s a reducton n mean drag all across the varous confguratons. The beneft s greatly enhanced, agan by the proper placement of the spltter plate. A thn spltter plate n an unbounded flow can therefore be a very effectve streamlnng devce. Thus, the relatve placement between the crcular cylnder body and the spltter plate s extremely crtcal. In a real world applcaton, for example, there would be a drastc dfference f the spltter plate were to be placed slghtly behnd the crtcal dstance due to a postonng 21

22 Fgure 13: Strouhal number of unbounded flow past a crcular cylnder wth spltter plate nterference, Re d = 150 Fgure 15: Mean C D of crcular cylnder wth spltter plate nterference, unbounded flow, Re d = 150 Fgure 14: C L oscllaton ampltude of crcular cylnder wth spltter plate nterference, unbounded flow, Re d = 150 Fgure 16: C D oscllaton ampltude of crcular cylnder wth spltter plate nterference, unbounded flow, Re d =

23 a) No spltter plate b) l s = 1.5 d c) l s = 1.8 d d) l s = 1.9 d e) l s = 2.5 d Fgure 17: Instantaneous plots of streamlne (left column) and vortcty (rght column) of flow past a crcular cylnder n an unbounded flow, l = 1.0 d, Re d =

24 tolerance. Fgure 17c llustrates the optmal separaton dstance of 1.8 d, whch allows the 1.0 d long spltter plate to break down ncomng vortces whle Fgure 17b also shows a very smlar vortcty contour plot even when the same spltter plate s moved slghtly closer to a separaton dstance of 1.5 d. Ths s seen as an effectve mechansm of decouplng the nducton of force oscllatons to the man body. Fgure 17d, on the other hand, shows that by allowng a slght ncrease n separaton dstance,.e. 1.9 d separaton dstance, an ndependent vortex lobe appears n the wakes. It s mportant to note also that when the spltter plate s placed further downstream of the optmal dstance, t would ncrease the ampltude of lft oscllaton to a degree hgher than the baselne wthout a spltter plate (Fgure 17a). The reason s partally due the upstream wake effects of the spltter plate that adversely nterferes wth the crcular cylnder wakes. Such a placement of spltter plate allows the frst ndependent vortex lobe to develop and n the meantme mpedes ts natural downstream moton. As the spltter plate s moved further away, the separaton dstance provdes an area where vortex sheddng can be locally ntensfed and the development of the frst ndependent vortex lobe can even be enhanced by mergng wth the front vortex that rolls off from the leadng edge of the spltter plate (Fgure 17e). As a result, t exacerbates the vortex-nduced forces on the crcular cylnder body. As the spltter plate gets shorter, the mechansm of breakng down an ndependent vortex lobe becomes less effectve as the lobe tself s allowed to form by rollng around the spltter plate. Whle Fgure 17c shows an effectve lobe break down usng a 1.0 d long spltter plate at 1.8 d separaton dstance, Fgure 18 llustrates what happens to the vortex moton when the spltter plate length s shortened. In ths example, we show the vortcty contour of the flow past crcular cylnder wth a 0.25 d long spltter plate whch s the shortest length we use n our study. The rollng moton allows an alternatng vortex lobe to spread out n the cross plane drecton, but each lobe has the chance of 24

25 nteractng wth ts counter-rotatng par effectvely n a wder area. The phenomenon thus seems to further worsen the pressure nstablty surroundng the body and ultmately ncrease the lft oscllaton ampltude. Wth the excepton of the oscllaton components of lft and drag, we can see that as the spltter plate length becomes very short, St and mean C D also approach the value obtaned from the baselne of flow wthout spltter plate. Fgure 18: Vortcty contour, l = 0.25 d, l s = 1.8 d, unbounded flow, Re d = 150 Our numercal experments establsh that the effectveness n controllng VIF depends largely on the combnaton of spltter plate length and separaton dstance. The optmal separaton dstance that we obtaned s the dstance whch yelds the lowest fluctuaton ampltudes, e.g. 1.8 d for 1.0 d long spltter plate. In comparson to the study by Unal et al., we have shown that t s possble to use a relatvely short and thn spltter plate for VIF reducton although not to the extent of achevng the complete suppresson of VIF as shown n the case of l s = 2.3 d on top of Fgure A10 n the appendx. However, t can be postulated from our data that f a thn and short spltter plate s placed such that the separaton dstance s larger than 0 but less than the optmal dstance, some notceable amount of suppresson can stll be realzed. On the other hand, havng a spltter plate of an arbtrary length at a separaton dstance that s more than the optmal seems to exacerbate both lft and drag oscllatons. 25

26 Case 2: Effect of a spltter plate n a bounded unform flow wth movng walls Plug flow at Re * d = 150 As dscussed n Secton III, the presence of movng walls on the upper and lower bounds at U 0 has a sgnfcant effect on the vortex wakes of the cylnder. We wll now examne how the presence of a spltter plate effects the vortex sheddng and VIF n a bounded flow. Usng β = 0.5 as the test case, Re * d s set to 150, whch would correspond to the same unform ncomng velocty, U 0, used n an unbounded flow of Re d = 75. Note that the confned Strouhal number for wthout a spltter plate s approxmately As shown on Fgures Fgure 19 to Fgure 22, t appears that n the bounded flow case, the prmary factor that s responsble for suppressng the VIF oscllaton s the separaton dstance alone. The effect from the length of a spltter plate appears to be mnmal. The most notable observaton s that there appears to be a crtcal separaton dstance of approxmately d. Insde ths crtcal separaton dstance, the spltter plate n the presence of movng upper and lower walls can completely elmnate or partally reduce the unsteady vortex wakes and, as a result, VIF oscllaton. The complete suppresson of flow nstablty appears to be mnmally dependent of the spltter plate length, of whch we have expermented wth 0.25 d, 0.5 d, 0.75 d and 1.0 d, at the separaton dstance between d to d. However, f the spltter plate s too short, 0.25 d for example, t s less effectve and a complete suppresson cannot be realzed. The suppresson mechansm appears to depend on an exstence of a body, n ths case the spltter plate, to break down the frst ndependent vortex lobe wth the full or partal physcal presence of the spltter plate n the regon. The prmary factor s seemngly the separaton Ths s an approxmate number based on the average dstance between the largest effectve l s ( d) and the smallest neffectve l s ( d) used n our smulaton. Smlarly, ths s an approxmate number based on the average dstance between the largest sem-effectve l s (0.75 d) and the smallest effectve l s (1.0 d). 26

27 dstance whch decdes whether there would be an attenuaton of force oscllaton. The spltter plate length, on the other hand, plays a role n the degree of effectveness n attenuaton,.e. a complete or partal suppresson of force oscllaton. In bounded flow, the movng upper and lower walls also help stablze the flow by dampng out the vortex wakes. The crtcal separaton for maxmzng the suppresson of VIF oscllaton can vary accordng to the Reynolds number but can be found teratvely. In summary, we can classfy the effectveness of suppresson nto 3 categores: 1) Sem-effectve: 0 < l s < d 2) Effectve: d l s d 3) Ineffectve: l s > d In contrast to the unbounded flow, the poston of the spltter plate of varous lengths n the flow at ths Re * d does not seem to alter the value of St * sgnfcantly as shown n Fgure 19. Usng a spltter plate of length 1 d as an example as shown n Fgure 23, the spltter plate effect n the bounded flow confguraton yelds essentally two modes of vortex wakes one that s steady and symmetrcal (c & d) and the other that s unsteady and asymmetrcal wth a constant sheddng frequency (b, e, & f). Note that the Strouhal number of the unsteady wake mode at ths Reynolds number approxmately concdes wth the baselne value of St * = 0.21 when spltter plate s absent. When a 1 d long thn spltter plate s deployed at a slghtly hgher Reynolds number, Re * d = 250, a smlarty of the spltter plate effect can be seen such that there appears to be a regon of separaton dstance where the unsteadness of the vortex wakes can be completely elmnated. However, the total suppresson regon s shortened at a hgher Reynolds number. As shown n Fgure 24 to Fgure 27, ths regon for Re * d = 250 s reduced to 1.125d l s d. Fgure 28 27

28 llustrates some examples of streamlnes and vortcty contour of the three regons of effectveness sem-effectve (b), effectve (c & d) and neffectve (e & f). In the neffectve range of separaton dstance the vortex wakes can actually be ntensfed and all the fluctuatng forces are correspondngly exacerbated. For example, n the effectve regon of the movng wall case, the peak-to-peak value of C * L of crcular cylnder wthout spltter plate presence at Re * d = 250 s If the spltter plate s mproperly placed at say, l s = 2 d, the peakto-peak value of C * L ncreases to as much as Smlar behavor s observed n the neffectve regon of the statonary wall case where a spltter plate s mproperly placed at, l s = 2 d, the peakto-peak value of C * L get exacerbated to a value of 0.34 as compared to 0.21 when the spltter plate s absent. Whle the spltter plate can completely suppress drag and lft oscllatons n a bounded flow, the reducton of mean drag s less than that obtaned n an unbounded flow. In fact, n the neffectve range of separaton dstance, the mean drag actually ncreases slghtly. Ths can be attrbuted to the fact that the shear layers between the crcular cylnder and the walls do not vary much n all of the smulated confguratons. Ths can easly be seen n Fgure 23 and Fgure 28 that n the mmedate area adjacent to the cylnder body for the most part, the vortcty contour reveals a remarkably smlar profle n all of the confguratons baselne and wth spltter plate of any length and separaton. 28

29 Fgure 19: St * of plug flow past crcular cylnder wth spltter plate, Re d * = 150 Fgure 21: Mean C D * of crcular cylnder, bounded, plug flow, Re d * = 150 Fgure 20: C L * oscllaton ampltude of crcular cylnder, plug flow, Re d * = 150 Fgure 22: C D * oscllaton ampltude of crcular cylnder, plug flow, Re d * =

30 a) Baselne: No spltter plate b) l s = 0.5 d c) l s = 2.0 d d) l s = d e) l s = d f) l s = 3.0 d Fgure 23: Isocontour plots of nstantaneous streamlne (left column) and vortcty (rght column) of flow past a crcular cylnder n a bounded flow wth movng walls (plug flow), top and bottom walls movng at U 0 to the rght, l = 1.0 d, Re d * =

31 Fgure 24: St * of plug flow flow past crcular cylnder wth spltter plate, l=1 d, Re d * = 250 Fgure 26: Mean C D * of plug flow and Poseulle flow past crcular cylnder wth spltter plate, l=1 d, Re d * = 250 Fgure 25: C L * oscllaton of plug flow past crcular cylnder wth spltter plate, l=1 d, Re d * = 250 Fgure 27: C D * of plug flow and Poseulle flow past crcular cylnder wth spltter plate, l=1 d, Re d * =

32 a) Baselne: No spltter plate b) l s = 0.5 d c) l s = 1.5 d d) l s = d e) l s = d f) l s = 2.5 d Fgure 28: Isocontour plots of nstantaneous streamlne (left column) and vortcty (rght column) of flow past a crcular cylnder n a bounded flow wth movng walls (plug flow), top and bottom walls movng at U 0 to the rght, l = 1.0 d, Re d * =

33 Case 3: The effect of spltter plate n a bounded Poseulle flow (statonary walls), Re * d = 250, l = 1 d In the statonary wall case (Poseulle flow), the crtcal separaton dstance s approxmately d. In contrast to the plug flow condton, the effectve suppresson dstance of the spltter plate n extends all the way nwards to the cylnder body. There appears to be only 2 categores of suppresson (Fgure 29 to Fgure 32): 1) Effectve: 0 < l s d 2) Ineffectve: l s > d Fgure 33 llustrates some examples of streamlnes and vortcty contour of these two regons of effectveness effectve (b, c & d) and neffectve (e & f). Apart from the absence of a sem-effectve range of separaton dstance, the statonary wall case shows smlartes n the behavor and effectveness of spltter plate to the flow and VIF, although the magntudes of the both mean and fluctuatng components are qute dfferent. In the effectve range, both cases exhbt the ablty of a spltter plate to completely suppress the unsteadness of the flow, whle n the neffectve range the unsteady forces can actually be exacerbated. 33

34 Fgure 29: St * of Poseulle flow past crcular cylnder wth spltter plate, l=1 d, Re d * = 250 Fgure 31: Mean C D * of Poseulle flow past crcular cylnder wth spltter plate, l=1 d, Re d * = 250 Fgure 30: C L * oscllaton of Poseulle flow past crcular cylnder wth spltter plate, l=1 d, Re d * = 250 Fgure 32: C D * of Poseulle flow past crcular cylnder wth spltter plate, l=1 d, Re d * =

35 a) Baselne: No spltter plate b) l s = 0.5 d c) l s = 1.5 d d) l s = 1.84 d e) l s = 1.85 d f) l s = 3.0 d Fgure 33: Isocontour plots of nstantaneous streamlne (left column) and vortcty (rght column) of flow past a crcular cylnder n a bounded flow wth statonary top and bottom walls (Poseulle flow), l = 1.0 d, Re d * =

36 V. Expermental verfcaton An experment to verfy our numercal results of vortex nduced vbraton and ts suppresson usng thn spltter plate has been carred out n a very low speed wnd tunnel wth an achevable ar speed range of 0.5 to 5.0 m/s. The correspondng Reynolds number ranges from approxmately 70 to 1000 dependng on the blockage factor and the dameter of the test cylnder. The wnd tunnel s made of entrely out of Plexglass clear acrylc materal whch would allow for the use of a laser Doppler velocmetry (LDV) to detect the test specmen moton. The wnd tunnel length s approxmately 3.5 feet long and s smply set up to operate n a sucton mode where the drvng fan s postoned at the dvergng end of the tunnel as shown on Fgure 34. The test secton has a cross sectonal area of 4.5 nches by 1.5 nches. A crcular 416 stanless steel rod wth a dameter of 1/16 nch s used as a test specmen. The 416 stanless steel s chosen because t s magnetc and the cylnder can thus be levtated horzontally as shown n a couple of arrangements on Fgure 35 and Fgure 36. Note that the cylnder has ponted ends one has a drect contact wth one of the magnets and the other s floated but held horzontally by the frnge feld of the other magnet. When the flow s ntroduced at the low Reynolds number range, the vortex sheddng from the test cylnder nduces a forcng functon whch causes a snusodal moton. The oscllatory moton s easly detectable near the floatng end by processng the sgnal of an LDV beam va a dynamc sgnal analyzer. The cylnder does not move n a purely vertcal moton. Rather t pvots around the ponted end that s physcally connected to one of the magnets. The unbounded flow set up can be realzed as shown on Fgure 35 snce the rato of the test cylnder to the test secton heght,.e. blockage factor, s much less than 0.1. When the test cylnder s placed n a secondary enclosure, as shown on Fgure 36, the setup s equvalent to the 36

37 case of a bounded flow wth statonary walls. In both cases, the test cylnder s placed n the developed flow regon whch s determned by theoretcal calculaton, and verfed by usng a ptot tube and a Baratron pressure sensor at a few locatons along the length of the test secton. In order to determne the effectveness of a spltter plate at varable separaton dstances, t s mounted on a moveable, low profle stand, whch can then be systematc placed drectly downstream of the cylnder. It s mportant to note that the spltter plate needs to be made out of a non-magnetc materal. For our experment, we smply chose a pece of plastc shm stock wth a thckness of 0.15 mm. The reason for such a choce s to prevent any nteracton due to magnetc couplng force that can transmt back nto the crcular cylnder. Fgure 34: Wnd tunnel setup LDV beam Clear Plexglass wnd tunnel Magnet Test cylnder Magnet Fgure 35: Test secton, unbounded flow 37

38 LDV beam Clear Plexglass wnd tunnel Magnet Test cylnder Clear Plexglass Secondary enclosure Magnet Fgure 36: Test secton, bounded flow The frequency of the cylnder moton due to vortex sheddng s dentfed by processng the LDV sgnal va fast Fourer transform usng a dynamc sgnal analyzer. Because of the presence of nose n the expermental system, the dentfcaton of ths frequency requres careful examnaton of the data. In the two-dmensonal sheddng regme, ths frequency wll be dstnct but depend solely on changes n the ar speed. However, t s possble to falsely dentfy one of the harmoncs of the fan modes, for example, to be the sheddng frequency. One of the keys n solatng the vbraton mode due to sheddng s to normalze the sgnal measured at the outboard locaton of the cylnder, as shown n Fgure 37, to the sgnal measured at the root secton of the cylnder. Normalzng the two sgnals yelds a dstnctve mode of moton due to vortex sheddng alone as shown n Fgure 38. The frequency obtaned from ths vbraton mode s used to calculate the Strouhal number. It wll become clear later when a spltter plate s placed at a proper poston that the vbraton mode can be completely elmnated, whch confrms the valdty of ths dentfcaton technque. Outboard Root Test cylnder Fgure 37: Test setup for dentfyng vbraton mode due to vortex sheddng 38

39 12 Rato of vertcal moton at outboard to root 10 amplfcaton freq, Hz Fgure 38: Normalzed power spectrum of outboard to root moton Comparson between numercal and expermental results of crcular cylnder wth and wthout spltter plate n a bounded flow between statonary walls at Re * d = 250 A comparson between the earler numercal predcton and expermental result of flow nduced vbraton as measured by LDV technque can be made for a crcular cylnder n a bounded flow at Re * d = 250. It s necessary to frst establsh a baselne where a crcular cylnder s placed n the bounded flow wthout the presence of a spltter plate. Once the baselne data s establshed, smlar measurement can be made wth the deployment of a spltter plate. 39

40 2 x Moton due to lft fluctuaton, Re * d = 250 No spltter l=1d, l s =1d magntude, m/s freq, Hz Fgure 39: Power spectrum of the crcular cylnder moton wthout and wth (l=1 d, l s =1 d) spltter plate Plotted n Fgure 39 s the power spectrum showng the flow-nduced vbraton mode (black lne) of crcular cylnder at 335 Hz due to vortex sheddng and the complete suppresson (gray lne) when a spltter plate of length 1 d s placed downstream of the crcular cylnder at 1 d separaton dstance. Ths s n good agreement wth the numercal results that were obtaned and shown n case III of secton IV. Note that the spkes between 200 to 250 Hz can be attrbuted to both mechancal and electrcal noses n the expermental system. It can also be shown that once the spltter plate s placed slghtly beyond the crtcal dstance, the oscllatng moton can get worse than the case where the spltter plate s not present. Fgure 40 llustrates a slght amplfcaton of the magntude of the cylnder oscllatng moton as a result of placng the spltter plate at 2 d separaton dstance, whch s n good agreement wth the earler numercal result that predcts the crtcal dstance to be at d. 40

41 2 x Moton due to lft fluctuaton, Re d* = 250 No spltter l=1d, l s =2d magntude, m/s freq, Hz Fgure 40: Power spectrum of the crcular cylnder moton wthout and wth (l=1 d, l s =1 d) spltter plate By ncrementally varyng the poston of a spltter plate downstream of the crcular cylnder, we can obtan the Strouhal numbers as a functon of separaton dstance smlar to the numercal results shown earler. Fgure 41 shows a reasonably good correlaton between expermental result and numercal predcton of the flow nduced vbraton on crcular cylnder wth the presence of a spltter plate. A slght dscrepancy can be attrbuted to the dfferences n constranng the crcular cylnder n the bounded flow between the numercal and expermental results. Whle the aerodynamc forces are obtaned from the 2-dmensonal numercal analyss wth the cylnder beng rgdly fxed n space, the experment setup allows the cylnder to move so that the LDV measurement can be made. 41

42 Numercal Experment Vscous flow over a bluff body St* Separaton ( d) Fgure 41: Numercal and expermental comparson of Strouhal number as a functon of spltter plate separaton dstance, Re d * = 250 It s also possble to compare the forces on the crcular cylnder as a functon of separaton dstance of cylnder. The snusodal acceleraton of the cylnder can be obtaned drectly by dfferentatng the velocty sgnal from LDV as follows: u = X& sn( ωt) a = ωx& cos( ωt) The normalzed quantty of the coeffcent of lft s then drectly proportonal to the normalzed acceleraton of the cylnder between the baselne measurement and the measurement wth spltter plate. ( CL ) ( C ) splt L no _ splt a a splt no _ splt = ( ωx& ) splt fno splt X& _ splt = ( ωx& ) no _ splt fsplt X& no _ splt Fgure 42 shows a smlar trend between numercal and expermental results. In the separaton range of less than d, the spltter plate completely suppresses the vortex nduced vbraton 42

43 mode at 335 Hz. On the other hand, we detect a decreasng trend of amplfcaton as the spltter plate s moved further away from the crtcal dstance. The dscrepancy of between numercal and expermental CL* can agan be attrbuted to the dfference n the appled boundary condtons of the cylnder. Whle the crcular cylnder s rgdly fxed n space n our numercal study, such s not the case n our experment. * Normalzed CL as a functon of separaton dstance Normalzed CL * Numercal Experment Separaton ( d) Fgure 42: Numercal and expermental comparson of normalzed CL* as a functon of spltter plate separaton dstance, Red* =

44 VI. Concluson Both our numercal smulatons and our wnd tunnel experments confrm that a thn spltter plate can be effectvely used to control unsteady vortex sheddng, vortex nduced forces (VIF) and thus vortex nduced vbraton when t s properly placed downstream of a bluff body. Focusng on flows at low Reynolds numbers Re * d 250, we have found that the effectveness s relatvely nsenstve to varatons of the spltter plate length n the range 0.25 d to 1.0 d. However, there s a crtcal separaton dstance where a jump s observed n the fluctuatng components of lft and drag. In an unbounded flow the short spltter plate alters the sheddng characterstcs but does not completely elmnate the oscllaton of VIF. However, the mean drag on the body s generally reduced. In a bounded flow, a spltter plate placed at a proper separaton dstance can completely elmnate the vortex nstablty and thus the oscllaton of VIF on a two-dmensonal crcular cylnder. In future work we ntend to study the effect of spltter plates on flow past other bluff bodes, such as square or rectangular cylnders. It s also mportant to recognze that n realty, the flow past bluff bodes can be complex and three-dmensonal. It wll ultmately be mportant to see how a spltter plate effects the flow n a hgher Reynolds number range. We would lke to thank Professor John Eaton of Stanford Unversty and Dr. Ferdnand Hendrks of Htach Global Storage Technologes for many frutful dscussons and helpful suggestons on ths research topc. The frst author would lke to acknowledge the fnancal support from Htach Global Storage Technologes for hs Ph.D. research. The second author has benefted greatly from the long term and contnung support of the AFOSR Computatonal Mathematcs Program, drected by Dr. Farba Fahroo. 44

45 Appendx Valdaton of prevous studes: In order to establsh confdence n the results obtaned from our smulaton from the commercal CFD code, we compare them to some known solutons. There s a vast amount of nformaton on the wakes of crcular cylnders from the work of varous nvestgators, namely by Wllamson [2] and Belov et al. [8], [13]. Comparson wth Belov s numercal result: We wll frst focus on an unbounded flow past a crcular cylnder at Re d = 150. We have chosen the numercal result of Belov as an establshed baselne for comparson. Instead of usng an O-mesh as Belov dd, t would be more sutable to use a modfed O-mesh ( Fgure A1: Modfed O-Mesh around Crcular Cylnder and Fgure A2) on an x-y Cartesan coordnate system so that we can experment wth spltter nterference by buldng upon ths model. Fgure A1: Modfed O-Mesh around Crcular Cylnder 45

46 Fgure A2: Modfed O-Mesh for Flow Past Crcular Cylnder showng boundares We have chosen to keep the number of computatonal cells the same as Belov s, that s 256x256. Whle the crcumferental spacng s unform, the normal spacng s ncreased geometrcally wth the grd layer next to cylnder surface beng d, whch s the same as what s used n Belov s. It should also be noted that Belov used CDS for the convecton term and a 3 rd order scheme based on Farmer s [11] as an artfcal dsspaton term. For tme dscretzaton, Belov mplemented a second order accurate mplct backward scheme, whch s based on a three-tme level scheme,.e. n dφ 3φ dt + 1 n n 4φ + φ 2 t 1. 46

47 In summary, usng CN as the temporal dfferencng scheme, we found that both 2UDS and CDS produce very smlar results. However, when a 1 st order backward Euler scheme s used for tme dscretzaton, the Strouhal number as well as the ampltudes of the drag and lft coeffcents are sgnfcantly dfferent. Fgure A3 and Fgure A4 llustrate the dfference between the solutons obtaned from the Euler scheme as compared to that from CN. Note that the plots correspond to flow solutons n whch the steady asymmetrc sheddng characterstcs are establshed. One of the technques used to accelerate the sheddng s to perturb the ntal condton by usng very large t (e.g. 0.1) n the frst few steps. Once the mean flow s establshed throughout the doman, t s then throttled back to the desred target (e.g sec). 1.6 Tme Response, Vscous Flow Over Crcular Cylnder 1.6 Tme Response, Vscous Flow Over Crcular Cylnder Lft and Drag Coeffcents, CL and CD CD CL Lft and Drag Coeffcents, CL and CD CD CL Nondmensonal Tme, τ Nondmensonal Tme, τ Fgure A3: C L & C D of flow past a crcular cylnder, unbounded, 2UDS, Euler Fgure A4: C L & C D of flow past a crcular cylnder, unbounded, 2UDS, CN The shortcomng of usng a 1 st order temporal scheme becomes even more apparent n solutons on coarser grds. Coarser grds have been nvestgated n the nterest of savng computatonal tme to enable the study of numerous varatons of spltter plate deployment. Table 1 shows the comparson of flow past crcular cylnder at Re d = 150 usng these dfferent 47

48 schemes. It s obvous that the accuracy of St and C pb suffers markedly when the backward Euler scheme s used. Thus, the 2UDS and the CN scheme are chosen for our study of wake flow behnd a cylnder wth and wthout spltter plate nterference. In the nterest of reducng computatonal tme, t s noted that a coarse mesh produces reasonable results when the proper temporal and spatal dfferencng schemes are used. Source Total Cells C L C D St C pb Belov [8], [13] ± ± UDS, Euler ± ± CDS, CN ± ± UDS, CN ± ± UDS, Euler (Coarse) ± ± CDS, CN (Coarse) ± ± UDS, CN (Coarse) ± ± Wllamson [2], exp Table 1: Comparson of propertes for flow past a crcular cylnder at Re d = 150 Qualtatvely, the nstantaneous plot of the z-component of vortcty, shown on Fgure A5, reveals the typcal von Kármán vortex street as one would expect from flow past a crcular cylnder at Re d = 150. It shows the formaton of two rows of alternatng vortces n the wake of the cylnder. Ths asymmetrcal flow pattern produces an oscllatng pressure dstrbuton ( Fgure A6) on the cylnder, and leads to fluctuatng C L and C D. Note that C D oscllates twce as fast as C L. The sheddng pattern s regular but s subjected to vscous dsspaton as each vortex moves further downstream away from the obstructng body. The rate of decay may be exaggerated n 48

49 the smulatons due to numercal dsspaton. However, no spurous reflecton s observed downstream, confrmng the correct pressure settng at the far rght boundary. Fgure A5: Instantaneous Plot of vortcty of Flow past a crcular cylnder, 2UDS, CN, Re d = 150 Fgure A6: Instantaneous Cp plot showng sobar of flow past a crcular cylnder, 2UDS, CN, Re d =

50 Fgure A7: Streamlne plot of Flow past a crcular cylnder, 2UDS, CN, Re d = 150 It s well known that the CN scheme can sometmes lead to nonphyscal, oscllatory solutons when t s set too large. Agan, n the nterest of savng computatonal tme, CFD-ACE allows a basng scheme where CN can be modfed such that the code wll solve the soluton wth more or less mplct nformaton. For example, when the blendng factor s set to 0.5, t produces the tradtonal CN scheme where the varables at tme step n and n+1 have equal weghtng. But when t s set to 1.0, t essentally recovers the backward Euler method. It has been found that the default settng of the blendng or basng factor at 0.6 s suffcent to nhbt oscllaton n the soluton. Fgure A8 shows a dverged soluton when the pure CN scheme (blendng factor = 0.5) s used wth t = , whle Fgure A9 shows a stable soluton wth no sgnfcant devaton n C D, C L, or St from the references when the blendng factor s set at

51 1.6 Tme Response, Vscous Flow Over Crcular Cylnder 1.6 Tme Response, Vscous Flow Over Crcular Cylnder Lft and Drag Coeffcents, CL and CD CD CL Lft and Drag Coeffcents, CL and CD CD CL Nondmensonal Tme, τ Nondmensonal Tme, τ 150 Fgure A8: Flow soluton from the tradtonal Crank-Ncholson Scheme Fgure A9: Flow soluton from Crank- Ncholson wth blendng factor of 0.6 Comparson wth thck spltter plate experment by Unal & Rockwell: In establshng the confdence of our smulaton, we have also chosen a benchmark case to compare wth the expermental results obtaned by Unal & Rockwell [4]. As prevously mentoned, Unal & Rockwell reported a total suppresson of unsteady vortex wakes n the von Kármán vortex sheddng regme at Re d = 142 by deployng a thck and long spltter plate downstream of the crcular cylnder. The cylnder body and the spltter plate were placed far enough from the wall to be consdered an unbounded flow condton. We assume the spltter plate to have thckness of 1 d and the leadng edge a length of 2 d for our smulaton purpose. We have found that our numercal smulaton results are n good agreement wth Unal and Rockwell. As shown on Fgure A10, the vortex sheddng from crcular cylnder body wth the spltter plate at the separaton dstance of l sc = 2.8 d s shown to be steady and symmetrc along the mean center lne and s thus classfed as pre-vortex formaton regme. At l sc 3.2 d, the vortex wakes become unsteady and the spltter plate s no longer effectve n controllng the l sc s the locaton of the spltter plate from the center of the crcular cylnder as defned by Unal & Rockwell. As such, l sc = 2.8 d, 3.2 d, 5.2 d and 6.5 d correspond to our conventon of l s = 2.3 d, 2.7 d, 4.7 d and 6.0 d respectvely. 51

52 l s = 2.3 d (l sc =2.8 d) l s = 2.7 d (l sc =3.2 d) l s = 4.7 d (l sc =5.2 d) l s = 6.0 d (l sc =6.5 d) Fgure A10: Instantaneous plots of streamlne (left column) and vortcty (rght column) of flow past a crcular cylnder, 2UDS, CN, Re d =