Unsteady Flow and Its Reduced-Order Modeling of A Finite-Aspect-Ratio Flapping Foil

Transcription

1 Usteady Flow ad Its Reduced-Order Modelg of A Fte-Aspect-Rato Flappg Fol Zogxa Lag 1, Hu Wa 2, ad Habo Dog 3 Departmet of Mechacal & Materals Egeerg, Wrght State Uversty, Dayto, OH Phlp Bera 4 Ar Force Research Laboratory, AFRL/RBSD, BLDG 146, 2210 EIGHTH ST, WPAFB, OH Drect umercal smulato (DNS) ad proper orthogoal decomposto(pod)- Galerk projecto are coducted for a 3-D fol udergog ptchg ad plugg moto. POD aalyss shows that most turbulet ketc eergy (96%) s cotaed by the frst sx POD modes, whereas the frst four modes are suffcet to recostruct the vortces formato the far wake zoe. Recostructo coeffcets of the frst two POD modes could be accurately predcted by usg a pressure correcto based Galerk projecto method. Complex topologcal structures are foud exstg so-surfaces of velocty ad Q-crtero of POD modes. u u p Nomeclature = POD base elemet of velocty = POD base elemet of pressure gradet = Velocty sapshot = Arthmetc average of velocty feld of sapshots = Arthmetc average of pressure gradet of sapshots I. Itroducto T s terestg to vestgate the usteady flow caused by a three-dmesoal flappg fol sce t ot oly helps I to better uderstad the trsc ature of a physcal pheomeo but also has practcal usages, e.g. desgg flappg wg devces utlzg the flow features, or furthermore, cotrollg the flow to ehace performace. Reduced-order models (ROMs), whch are based o the data acqured from drect umercal smulatos (DNS) or expermets, ca provde accurate recostructos of orgal flow felds at a rather low cost feld aalyss ad data storage. It s also deal for real-tme computato-based cotrol desg uder the codto of curret mcroelectrocs techology. Amog all kds of ROMs, proper orthogoal decomposto (POD) 1 method costructs flow felds based o a eergetc sese. Usg POD modes, ordary dfferetal equatos obtaed from Galerk projecto of the Naver-Stokes equatos ca be used for flow cotrol 2. May POD applcatos flud dyamcs focus o flow felds whch boudares do ot move or dsplace. Few studes are o flappg wgs because of complexty of boudary treatmets for POD-Galerk method. Several approaches have bee used to solve ths problem. Oe of them s usg coordate trasformato techques to tur a movg arfol to a statoary oe 3. The umercal smulatos ad POD-Galerk projectos are performed o modfed mometum equatos whch are reformulatos of the Naver-Stokes equatos a o-ertal referece 1 Ph.D Studet, AIAA studet member, lag.5@wrght.edu. 2 Research Assocate, hu.wa@wrght.edu. 3 Assocate Professor, AIAA Assocate Fellow, habo.dog@wrght.edu. 4 Prcple Aerospace Research Egeer, AIAA Assocate Fellow, Phlp.Bera@wpafb.af.ml. 1 Amerca Isttute of Aeroautcs ad Astroautcs

2 frame. The use of coordate trasformato s twofold: o the oe had, t avods movg boudares, o the other had, t complcates Galerk projectos ad s hardly applcable to multple wgs ad/or flexble wg systems. Thus, Lag et al. 4, 5 proposed to used membrae plates wth fte small thckess. The effect of movg boudares s cosdered by coductg POD of pressure gradet 5. The addtoal formato of the POD of pressure gradet makes Galerk projectos ot oly more accurate but also more flexble to varous types of boudary codtos. The preset work s a applcato of the method by Lag et al. 5 to a fte-aspect-rato flappg fol case whch has bee studed by Dog et al. 7 Vortex formatos geerated by a fte-aspect-rato flappg fol s examed frst. The POD egevalue spectrum aalyss s coducted. Ad recostructos based o dfferet umber of modes are performed to evaluate the ecessary umber of POD modes for the vortex formato recostructo the wake zoe. A pressure-corrected Galerk projecto method s appled ad the results are used to evaluate the applcablty of the projecto method three-dmesoal flow feld. 3-D topologcal structures of POD modes are show last. II. Methodology A. Proper Orthogoal Decomposto (POD) Two kds of methods, the drect method 8 ad the method of sapshots 9, are avalable to calculate the POD modes. Egevalues are solved from a characterstc matrx, whose sze s proportoal to the umber of grd pots for the drect method, or the umber of sapshots for the method of sapshots, respectvely. The method used our study s the method of sapshots, whch s sutable to process the datasets from CFD smulatos, especally for three-dmesoal smulatos, whch the umber of sapshots s far less tha the umber of grd pots. Accordg to prevous studes 5, POD s ot oly appled to the velocty feld, but also to the pressure gradet feld: u x t u x N t x 1, N p x t p x t x 1, where superscrpt meas the th sapshot ad subscrpt s the th mode, orthogoal modes, Each egemode 2 Amerca Isttute of Aeroautcs ad Astroautcs (1a) (1b) ad are the ormalzed ad are the recostructo coeffcets for velocty ad pressure gradet, respectvely. represets a certa type of steady flow. The lear combato of wth coeffcets recostructs the stataeous flow felds. Note that the pressure gradet feld has to be carefully calculated ear the boudares sce the smulatg case s related to mmersed movg boudares. B. Galerk Projecto The modes, ad, are used to derve a reduced-order modelg for flow cotrol through a Galerk projecto of the compressble Naver-Stokes equatos: U 1 2, U U, p, U (2) t Re t ca be formed as: Substtutg Eq. 1 to Eq. 2, a set of evoluto equatos for the mode ampltude d dt N N N a bj dj cjk j k j1 j1 k1 (3)

3 where s parameter. Provded by tal values, Eq.3 ca be solved. Detals of POD-Galerk projecto method used curret study ca be foud Ref. 5. III. Results A membrae plate ecouters uform comg flow wth velocty U whle the plate s udergog a ptchad-heave moto three-dmesoal space. Drchlet velocty boudary codto s appled except at the dowstream where t s of Neuma B.C. The pressure s of homogeeous Neuma B.C. o all the boudares. The flow feld s computed by umercally solvg the compressble Naver-Stokes equatos o a Cartesa mesh. The solver employs a cetral dfferece scheme ad fractoal-step method to acheve secod-order accuracy space ad tme. The pressure-posso equato s solved usg geometrc mult-grd method tegrated wth the mmersedboudary methodology, whch provdes great flexblty hadlg complex geometres. Detals of valdatos of the code ca be foud Ref.10. The membrae plate s ellptc ad ts geometry s defed by two major dameters: the chord legth c ad the spa b. The spawse axs also serves as a pvot axs, whose posto s defed by: y( t) H cos(2 f t) (4) where H s the heave ampltude, f s the frequecy. The ptchg agle of the plate s prescrbed by ( t) As(2 f t) (5) where A s the ampltude of the susodal ptch agle varato. The actual parameters used DNS are tabulated Table 1. The grd umber all three drectos s 128. Table 1. Flow parameters. U H A() f c b/c Re = U c/ν St=2H f /U The dataset s obtaed whe perodc vortex sheddg has reached. Fgure 1 shows the vortex topology at the ed of a heavg cycle. The plate s at the lowest posto ad s ready to move upward. The vortces are detfed by Q-crtero, whch s defed as: 1 Q S where Ω s the vortcty tesor ad S s the tesor of rate of stra. A rego wth Q>0 s domated by hgh vortcal flow. The vortex structures Fgure 1 ca be categorzed to two groups accordg to ther movemets crossflow drecto. I the process of covectg to dowstream, two groups of vortces twst ad the shape of each vortex structure s dstorted uder the effect of duced flow caused by the other vortces. A two-dmesoal case whch employs the same kematcs setup geerates verse Karma vortex streets wth two rows of early parallel vortces 5. Ulke to that, the wake of curret case s gradually expadg heavg drecto (sde vew, Fgure 1b) ad shrkg spawse drecto (top vew, Fgure 1c), showg strog three-dmesoal effects of elogato ad cotracto of vortces. Detaled dscussos about the vortex formato ca be foud Ref. 7, despte that the plate used curret study s a membrae of zero thckess stead of a plate wth fte thckess. (6) 3 Amerca Isttute of Aeroautcs ad Astroautcs

4 a) Perspectve vew b) Sde vew c) Top vew Fgure 1. The vortces formato dfferet vew agles. Due to the large sze of data acqured from the three-dmesoal smulato, oly 48 frames over oe cycle are stored ad processed for the followg POD aalyss. Accordg to the prevous study 5, less umber of sapshots helps to reduce the global projecto error because the correlato betwee two sapshots decreases. Therefore 48 frames s eough to recostruct the orgal flow feld despte that the smulato s three-dmesoal. Fgure 2 shows cotours of the mea streamwse velocty whch s computed from the 48 frames of oe cycle. From both cotours, t ca tell that there s a jet flow caused by the ptch-ad-heave moto of flappg fol geeratg thrust the wake. Fgure 2a presets cotour of the mea streamwse velocty alog the crossflow symmetry plae, whch there s a fork-lke structure wth two teeth, surrouded by a coverget evelop. Whle Fgure 2b the jet s smaller. a) Cotours of mea streamwse velocty alog the b) Cotours of mea streamwse velocty alog the crossflow symmetry plae. spawse symmetry plae. Fgure 2. Cotours of mea flow dfferet symmetry plaes, computed from 48 frames. 4 Amerca Isttute of Aeroautcs ad Astroautcs

5 After subtractg the mea flow, the stataeous velocty felds are used to geerate thrty-two POD modes. Fgure 3 shows the egevalue spectrum ad the covergece of the accumulated egevalues wth respect to the mode umber. The cotrbuto of egevalues to the turbulet ketc eergy s lsted Table 2. It s clear to see that the frst two egemodes cota 41% ad 35% of total turbulet ketc eergy, respectvely. The thrd ad fourth egemodes accout lower level eergy at about 8%. Ad up to the 6th mode the cumulatve eergy from correspodg Table 2. Cotrbuto of the frst sx egevalues to the turbulet ketc eergy Mode Cotrbuto(100%) Accumulated(100%) modes s over 95%, whch meas that the frst sx modes capture a majorty of large flow features, e.g. the vortces the vcty of the flappg wg ad the vortces the far wake zoe. The plot for ormalzed egevalues Fgure 3 also dsplays a patter of star-step descedet tred. Every two egevalues form a par wth approxmate values. It dcates that there are two major structures the flow feld led by the frst two modes, respectvely. The structures of hgher modes could be topologcally smlar to those of the frst two modes, or be depedet structures wth smaller scales, whchever s case depedet. We recostruct the global flow felds wth the frst two, the frst four, the frst sx, ad all the thrty two velocty modes, respectvely. These seres of recostructos are to check the effect of umber of modes o vortex formato. The 95% cumulatve eergy s a good threshold value wth statstcal meag. But t does t mea 95% eergy s a ecessary codto to recostruct the flow feld because there s spatal o-uformty the eergy dstrbuto. Hgher eergy modes may cota more eergy the wake zoe ad lttle ear the flappg fol, whle lower eergy modes are a cotrary maer. Thus t s possble that fewer modes are eough to observe the flow pheomea we are terested. The so-surface of vortex structures at the ed of a heavg cycle s show Fgure 4. The recostructo Fgure 3. Egevalue spectrum. from the frst two modes devates pretty much from the DNS result. The vortces ear the flappg plate are rregular, ad the far wake vortces are ot coected to form a whole structure sce some vortces, especally those ear the spawse symmetrc plae, dsappear. Although the frst two modes cota 76% of fluctuatg eergy, they mss some key vortcal features whose eergy s low. Wth supplemet of modes 3 ad 4, the recostructo shows a lot of mprovemets: major vortces the far wake all emerge. The spawse vortex behd the plate also appears to be a straght pole. Addg modes 5 ad 6, the captured eergy s creased from 92% to 96%, wth some chages the shapes of some trval structures ear the flappg plate. The recostructo wth all modes deed s almost same as the DNS result. I geeral, f we focus o the vortces the far wake zoe, the the frst four modes are eough to capture essetal formato of a ptch-ad-heavg moto, eve three-dmesoal space. 5 Amerca Isttute of Aeroautcs ad Astroautcs

6 a) Mea+1+2 b) Mea+ (1~4) c) Mea+ (1~6) d) Mea+ (1~32) Fgure 4. Perspectve vews of recostructed flow feld We exted the pressure-corrected Galerk projecto algorthm, whch has bee successfully appled to may two-dmesoal cases, to ths three-dmesoal case. It s terestg to see that the predcto curves of the frst two recostructo coeffcets (Fgure 5) agree well wth the projecto curves, but ot for modes three ad four, where the predcto curves are gradually dverget as tme creases. These results partally proved the applcablty of pressure-corrected projecto algorthm three-dmesoal study. Further vestgatos o the Galerk projecto algorthm are eeded. a) The frst ad secod modes b) The thrd ad fourth modes Fgure 5. Tme hstory of ampltude of bass fuctos May POD modes for flow past a statoary obstacle 2, 6, 11, 12 or a flappg fol 4, 5 have bee show the prevous studes. Costraed by the smulato or expermetal codtos, may of those studes provde lmted aspects of POD modes by showg ther two-dmesoal slces. Naturally, whe we are dealg wth three dmesoal flow felds, merely showg slces of POD modes s ot eough because the flow s ot a moto a plae aymore, ad the effect of the thrd dmeso eeds to be cosdered. To get a better uderstadg to the 3- D POD modes, we observe the POD modes by dfferet kds of so-surfaces. Frst we look at the so-surface of streamwse velocty. Fgure 6a shows the so-surface of streamwse compoet of velocty mode oe. It s a composte structure wth multply strpes. Near the leadg edge posto, there are two spawse strpes, oe upo the other, each followed by fve shelf strpes whose two eds bed to the streamwse drecto. Smlar to the wake of vortces formato, the group of strpes gradually elogates crossflow drecto ad shrks spawse drecto. The so-surface of streamwse compoet of velocty mode two has a smlar patter as mode oe but wth a phase shftg the streamwse drecto. The szes of the frst two pars of strpes mode two ear the leadg edge are comparatve smaller tha those of mode oe, whereas the other strpes the wake zoe are larger. Note that the so-surface value s the absolute value of ormalzed streamwse velocty. I fact, the structure of mode oe 6 Amerca Isttute of Aeroautcs ad Astroautcs

7 ad two Fgure 6 s symmetrc spawse drecto ad atsymmetrc crossflow drecto wth respect to ther respectve ceter plaes. a) Mode 1 b) Mode 2 Fgure 6. Perspectve vew of so-surface of streamwse velocty of POD modes To further explore the structure of POD mode three-dmesoal space, we draw the so-surface of Q-crtero of POD modes oe to four as show from Fgure 7 to Fgure 10. The frst mode s a bad loop followed by a delcate, complex spatal structure wth spral strads. It looks lke four double helx structures (as show Fgure 7a). However, whe looked at the sde vew, the helx structures are dscotuous, especally the wake zoe of modes three ad four (Fgure 9b, Fgure 10b). We have checked the drecto of vortcty for the spral strads from mode oe to mode four. It s foud that the streamwse vortcty of modes oe ad two are spatally atsymmetrc the spawse drecto ad symmetrc the crossflow drecto. However, the streamwse vortcty of modes three ad four s atsymmetrc both spawse ad crossflow drecto. Further vestgato o POD modes s stll udergog. a) Perspectve vew b) Sde vew c) Top vew Fgure 7. Iso-surface of Q-crtero of POD mode 1 7 Amerca Isttute of Aeroautcs ad Astroautcs

8 a) Perspectve vew b) Sde vew c) Top vew Fgure 8. Iso-surface of Q-crtero of POD mode 2 a) Perspectve vew b) Sde vew c) Top vew Fgure 9. Iso-surface of Q-crtero of POD mode 3 a) Perspectve vew b) Sde vew c) Top vew Fgure 10. Iso-surface of Q-crtero of POD mode 4 IV. Summary POD aalyss s coducted based o the data acqured from the umercal smulato of a fte-aspect-rato flappg plate udergog ptchg ad plugg moto. It s foud that the frst four modes cota 92% turbulet ketc eergy of the flow feld ad are suffcet to recostruct the vortex formato the far feld of wake zoe. A 8 Amerca Isttute of Aeroautcs ad Astroautcs

9 pressure correcto method for the POD-Galerk projecto o the Naver-Stokes equatos s appled to the flow feld. The frst two modes accurately predct the tred of bass fuctos. The POD modes show well orgazed patters wth spatal symmetry spawse ad crossflow drecto based o so-surface vsualzato. Ackowledgmets Ths work s supported by AFOSR FA motored by Dr. Douglas Smth. Refereces 1 Holmes, P., Lumley, J.L., ad Berkooz, G Turbulece, Coheret Structures, Dyamcal Systems Ad Symmetry. 2 Graham, W.R., Perare, J., ad Tag, K.Y Optmal cotrol of vortex sheddg usg low order models. Part I: Ope-loop model developmet, Iteratoal Joural for Numercal Methods Egeerg, Vol. 44, No. 7, pp Lew, G.C. ad Haj-Harr, H Reduced-Order Modelg of a Heavg Arfol, AIAA Joural, Vol. 43, No Lag, Z., Dog, H., Wa, H., Bera, P.S., ad We, M., Wg-Wake Iteracto ad ts Proper Orthogoal Decomposto, AIAA Paper Lag, Z., Dog, H., ad Bera, P.S., POD-Galerk Projecto of Flappg Wgs, AIAA Paper Ma, X. ad Karadaks, G.E A low-dmesoal model for smulatg three-dmesoal cylder flow, Joural of Flud Mechacs, Vol. 458, pp Dog, H., Mttal, R., Najjar, F. M Wake topology ad hydrodyamc performace of low aspect-rato flappg fols, Joural of Flud Mechacs, Vol. 556, pp Smth, T.R., Moehls, J., ad Holmes, P Low-Dmesoal Modellg of Turbulece Usg the Proper Orthogoal Decomposto: A Tutoral, Nolear Dyamcs, Vol. 41, No. 1, pp Srovch, L Turbulece ad the dyamcs of coheret structures. parts -, Quarterly of Appled Mathematcs Vol. 45, No. 3, pp , Mttal, R., Dog, H., et al A versatle sharp terface mmersed boudary method for compressble flows wth complex boudares, Joural of Computatoal Physcs, Vol. 227, No. 10, pp Kostatds, E., Balaba, S., ad Yaesks, M Bmodal vortex sheddg a perturbed cylder wake, Physcs of Fluds, Vol. 19, No. 1, pp Feg, L.-H., Wag, J.-J., ad Pa, C Proper orthogoal decomposto aalyss of vortex dyamcs of a crcular cylder uder sythetc jet cotrol, Physcs of Fluds, Vol. 23, No. 1, pp Amerca Isttute of Aeroautcs ad Astroautcs