AJK2015-XXXXX A DYNAMICALLY ADAPTIVE LATTICE BOLTZMANN METHOD FOR FLAPPING WING AERODYNAMICS
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1 Poceedings of he ASME-JSME-KSME Join Fluids Engineeing Confeence 015 AJK015-FED July 6-31, 015, SEOUL, KOREA AJK015-XXXXX A DYNAMICALLY ADAPTIVE LATTICE BOLTZMANN METHOD FOR FLAPPING WING AERODYNAMICS Sephen L. Wood Univesiy of Tennessee Knoxville The Bedesen Cene 81 Volunee Blvd., Knoxville, TN 37996, USA Ralf Deieding Geman Aeospace Cene (DLR) Insiue of Aeodynamics and Flow Technology Bunsens. 10, Göingen, Gemany ABSTRACT The essenial componens of a paallel dynamically adapive laice Bolzmann mehod coupled o a 6-degee-offeedom igid body moion solve ae pesened. This Caesian appoach wih auomaic fluid meshing is paiculaly well suied fo simulaing he ineacion of low Reynolds numbe flows and moving sucues wih good accuacy and high compuaional pefomance. The fully coupled fluid-sucue simulaion mehod is demonsaed fo he expeimen of a wosegmen hinged wing wih osion dampe by Toomey & Eldedge, a simplisic model of a flapping wing in ai. A gid convegence sudy assesses he pedicion accuacy of he oveall mehod and equied CPU imes. Ou compuaions show vey good ageemen wih measuemens of he evolving hinge angle by Toomey & Eldedge; foces and momens ae pediced wih an eo magin of geneally <5% wih espec o hei compuaional esuls. NOMENCLATURE C, T Collision and anspo opeao of Laice Bolzmann mehod, Physical speed of sound, laice speed Se of laice velociy vecos, Disibuion and equilibium disibuion funcion, Discee disibuion funcions, Discee disibuions on nex fine and coase level p Dynamic fluid pessue, Fluid velociy veco and ih velociy componen Wall velociy, Δx Poin in space, mesh spacing, Time, discee ime sep Fluid densiy Viscosiy Relaxaion ime Collision fequency Time sep index Refinemen level index ϕ Level se funcion, Foce and oque acing on a body and is cene of mass / Tanslaional and oaional consains a join, x-coodinae and angle of diven componen in expeimen, Tanslaional and oaion shape funcion o pescibe moion Re,Re Reynolds numbe defined on anslaional and oaional velociy of diven componen,, Φ Kinemaic paamees fo diffeen cases Hinge angle /, Mean foces and momen on hinge INTRODUCTION The aeodynamics of flapping wings ae chaaceized by a complex ineacion beween moving sucues and fluid flow. Unsucued finie volume mehods ha use gids following he moion have o deal wih difficul mesh mophing, unangling and possibly gid egeneaion and daa emapping poblems. As an alenaive o he usually employed implici, ypically pessue-coecion based CFD soluion algoihms, we adop in hee he laice Bolzmann mehod (LBM), cf. Aono e al. [1]. The LBM is based on solving he Bolzmann equaion in a specially chosen, discee phase space and fully explici in ime []. The LBM is consuced on unifom Caesian gids and geomeically complex boundaies ae consideed wih an immesed bounday appoach, making he mehod well suied fo consideing moving sucues. Hee, we
2 uilize a level se disance funcion o epesen embedded objecs. Dynamic mesh adapaion is applied in addiion o incease he local esoluion based on he level se funcion and feaues deeced in he flow field [3]. Disibued memoy paallelizaion is adoped o allow fo lage-scale simulaions. The pape is oganized as follows: We fis ecall he consucion pinciples of he LBM. In he nex secion, he block-based mesh adapaion pocedue and in paicula he incopoaion of he LBM ae pesened. The hid secion explains ou appoach in dealing wih embedded geomeies in he LBM and how we compue igid body dynamics. Finally, validaion esuls of a wo-segmen hinged wing wih osion dampe modeling a simplisic flapping wing as poposed by Toomey & Eldedge [4] ae pesened and discussed. The compuaions confim he benefi of he poposed oveall appoach fo flapping wing dynamics and biologically inspied fluid-sucue ineacion poblems. LATTICE BOLTZMANN METHOD The laice Bolzmann mehod is based on compuing appoximaions of he Bolzmann equaion wih a simplified collision opeao f eq u f f f (1) on a ecangula gid of chaaceisic domain lengh L wih isoopic mesh spacing Δx unde he assumpion of a small Knudsen numbe / 1, whee he mean fee pah lengh is eplaced wih Δx. A cucial idea of he LBM is o appoximae Eq. (1) in a specially chosen discee phase space, in which a paicle disibuion funcion, is associaed o evey discee laice velociy. The oal densiy disibuion is given as,, and he macoscopic momens as,, e i,. A spliing appoach is hen adoped ha fis solves he homogeneous anspo equaion wih he ime-explici updae sep () xe x T: f, f,. Hee, we apply he D3Q19 model fo which he laice velociies ae defined as 0, 0, e 1,0,0 a, 0, 1,0 a, 0,0, 1 a, 1,...,6, 1, 1, 0 a, 1, 0, 1 a, 0, 1, 1 a, 7,...,18, wih /. The physical speed of sound is elaed o a by / 3. The igh-hand of Eq. (1) is inegaed subsequenly by he collision opeao x x eq f x f x C : f, f,,,, (3) (4) FIGURE 1. THE VELOCITIES e α OF THE D3Q19 LATTICE. wih equilibium funcion f eq e u u 3e 9 u 3, u 1, 4 a a a wih 1 3, 1 18 fo 1,,6and 1 36 fo 7,, 18. The vaiaion in hydodynamic pessue fo he equilibium funcion (5) eads. Applying a Chapman-Enskog expansion pocedue, i can be shown [5] ha he skeched LBM conveges o a soluion of he weakly compessible Navie-Sokes equaions (5) u 0, (6) uuu u. (7) p I can be shown fuhe, cf. [], ha he kinemaic viscosiy ν and collision fequency ω ae conneced by he elaion c 1 s c. (8) s While he skeched model can be used diecly o simulae lamina flows, i is mandaoy o apply a ubulence model in addiion in high Reynolds numbe siuaions. In he conex of LBM, i is mos common o adop a lage eddy simulaion appoach, cf. [5]. DYNAMIC MESH ADAPTATION Fo local dynamic mesh adapaion we have adoped he block-sucued adapive mesh efinemen (SAMR) mehod afe Bege & Collela [6]. In ode o fi smoohly ino ou exising, fully paallelized finie volume SAMR sofwae sysem AMROC [3], we have implemened he LBM cellbased, which makes he scheme also consevaive in and. In he SAMR appoach, finie volume cells ae cluseed wih a special algoihm ino non-ovelapping ecangula gids. The gids have a suiable laye of halo cells fo synchonizaion
3 and applying ine-level and physical bounday condiions. Refinemen levels ae inegaed ecusively. The spaial mesh widh and he ime sep ae efined by he same faco,whee we assume fo 0and 1. Noe ha in an adapive LBM he collision fequency is no a consan bu needs o be adjused accoding o Eq. (8) fo he updae on each level. In addiion o his, he ineface egion equies a specialized eamen. Disinguishing beween he anspo and collision opeaos, T and C, cf. Eqs. () and (4), he seps of ou mehod fo a efinemen faco of ae: 1. Complee updae on coase gid: f C,n1 : CT f C,n C,n. Use coase gid disibuions f,in ha popagae ino he fine gid, cf. Fig. (a), o consuc iniial fine gid halo f,n values f,in, cf. Fig. (b). 3. Complee anspo on whole fine mesh. Collision cells (yellow in Fig. (b)). is applied only in he ineio 4. Repea 3. o obain and f f,n1 f,n1/ : Cf. 5. Aveage ougoing disibuions fom fine gid halos f,n1/ (Fig. (c)), ha is f,ou in he inne halo laye and (oue halo laye) o obain. 6. Reve anspo fo aveaged ougoing disibuions,, and ovewie hose in he pevious coase gid ime sep, cf. Fig. (d). 7. Paallel synchonizaion of on enie level. 8. Repea complee updae on coase gid cells nex o coasefine bounday only: This algoihm is compuaionally equivalen o he mehod by Chen e al. [7] bu ailoed o he SAMR ecusion ha updaes coase gids in hei eniey befoe fine gids ae compued. Because of he nonlineaiy of he collision opeao C i becomes necessay unde his paadigm o epea he LBM updae fo hose coase gid cells ha shae a face o cone wih a fine gid. EMBEDDED STRUCTURE HANDLING We epesen non-caesian boundaies implicily on he adapive Caesian gid by uilizing a scala level se funcion ϕ ha soes he disance o he bounday suface. The bounday suface i locaed exacly a ϕ=0 and he bounday oue nomal in evey mesh poin can be evaluaed as n= ϕ/ ϕ [8]. We ea a fluid cell as an embedded ghos cell if is midpoin saisfies ϕ<0. In ode o implemen non-caesian bounday condiions wih he LBM, we have chosen o pusue a 1s ode accuae ghos fluid appoach ha was aleady available in AMROC [4]. In ou echnique, he densiy disibuions in embedded ghos cells ae adjused o model he bounday condiions of a non- Caesian eflecive wall moving wih velociy w befoe applying he unaleed LBM. The las sep involves inepolaion and mioing of, u acoss he bounday o and ū and modificaion of he maco velociy in he immesed bounday cells o, cf. [3]. Fom he newly consuced macoscopic values he densiy disibuions in he embedded ghos cells ae simply se o,. Real-wold geomeies ae consideed in AMROC as iangula suface meshes. The compuaion of he level se disance infomaion in evey Caesian mesh poin could FIGURE. DISTRIBUTIONS INVOLVED IN NECESSARY DATA EXCHANGE AT A COARSE-FINE BOUNDARY. (A) COARSE DISTRIBUTIONS GOING INTO FINE GRID; (B) INGOING INTERPOLATED FINE DISTRIBUTIONS IN HALOS (TOP), OUTGOING DISTRIBUTIONS IN HALOS AFTER TWO FINE-LEVEL TRANSPORT STEPS (BOTTOM); (C) AVERAGED DISTRIBUTIONS REPLACING COARSE VALUES BEFORE UPDATE IS REPEATED IN CELLS NEXT TO BOUNDARY.
4 pincipally be accomplished by simply ieaing ove he enie suface mesh; ye, his would lead o deimenal pefomance fo inceasing mesh size. The poblem is equivalen o deemining fo evey Caesian cell he closes face on he suface mesh. Fo his pupose, we employ a specially developed algoihm based on chaaceisic econsucion and scan convesion developed by Mauch [9] ha is used o compue he disance exacly only in a small band aound he embedded sucue. The dynamics of muli-body sysems undegoing ineacion wih he fluid ae modeled as ses of iangulaed suface meshes configued in kineic chains. The dynamics of hese mechanisms ae solved by a ecusive Newon-Eule mehod a each ime sep [10]. Consideing an abiay link wih a coodinae fame locaed a poin P ha is no coinciden wih is associaed body s cene of mass, he foce and oque applied by he peceding link ae m ap m α mω ω c m F m1 c τ P mcicm c c, ω Icm c c ω whee m is he mass of he body, denoes he 44 ideniy maix, a is he acceleaion of link fame wih oigin a P in he peceding link's fame. is he momen of ineia abou he cene of mass, and ae angula velociy and acceleaion, especively, is he locaion of he body's cene of mass expessed in he associaed link's fame, and, denoe skew-symmeic coss poduc maices. Hee, we addiionally define he oal foce and oque acing on a body, FFFSI Fpescibed Cxyz and τ τ FSI τpescibed C especively. Whee C xyz and C ae he anslaional and oaional consains, especively. F FSI and τ FSI ae deemined fo each body by inegaing he fluid hydodynamic pessue and viscous foces on he iangula faces of he especive body s suface mesh. Each suface mesh (lofed fom.xyz cuves o loaded fom.obj,.ply,.sl fomas) is associaed wih a kineic link in a chain ha begins wih a base link in he global coodinae fame. Links ae conneced by joins ha may be independenly consained in six degees of feedom elaive o he peceding link. The evoluion of he iangula suface mesh as well as he velociy w in each node ae communicaed o he LBM fluid solve in dedicaed coupling ime seps. The daa exchange coesponds o he ime sep of an SAMR level bu his does no have o be he fines efinemen level available, cf. [11]. This fomulaion eadily faciliaes he kineic analysis of each link and iangulaed suface in he global coodinae fame o in any of he link coodinae fames. (9) FIGURE 3. MODEL SYSTEM CONSISTING OF TWO RIGID ELLIPTICAL SECTIONS CONNECTED BY A HINGE WITH TORSION SPRING AND DAMPER. RESULTS A canonical poblem of fluid-sucue ineacion and wake pedicion poposed by Toomey & Eldedge [4] is seleced as a validaion es case fo coupled flapping wing aeodynamics. This model, depiced in Fig. 3, uilizes a sysem of wo aiculaed igid bodies conneced by a osion sping and dampe. The kinemaics of he cenoid of he diven wing ae pescibed, while he ailing body esponds passively o he aeodynamic and ineial/elasic foces. The pinciple unknown in his igid body dynamics poblem is he hinge angle,. The paameic kinemaic equaions G A G f G f X,, (9) max maxg C anh anh 8 anh, G anh cos d, anh cos G (11) (1) (13) descibe he moion of he diven body. The paamees uilized in his wok and in [1] o specify he kinemaics hough he anslaional,, and oaional,, shape funcions ae given in Table 1. The sa-up condiione,, is applied o he anslaional kinemaics o avoid an impulsive sa. The anslaional and oaional Reynolds numbes ae based on he peak anslaional, V, and oaional, anh, velociies as shown in Re Vc, Re fc anh. (14) The used osion sping and dampe coefficiens ae kg m /s and kg m /s especively. A no-slip bounday condiion is applied a he wing suface and a he op and boom (y diecion) boundaies. The simulaion domain is a box of exens
5 TABLE 1. KINEMATIC PARAMETERS (cm) 7.1 Φ 0, 45 (cm) , 370 (cm) , 500 /4 / , 1.885, , 1.885, TABLE. CONVERGENCE STUDY PARAMETERS ne efine fines /.45E-0 1.E-0 6.1E E-03 fines / 3.53E E E-05.1E-05 : 0.5,0.5, :0.5,0.5], z:[-0.31,0.31]. I is peiodic in he x-diecion and has slip walls a he z-boundaies o minimize cone effecs simila o he confguaion used by Aono e al. [1]. A convegence sudy was conduced fo he modeae kinemaic paamees ( 1.85, 1.885,Φ=0) assigned o fo he simulaion paamees given in Table. The eo in pediced hinge deflecion elaive o he expeimens conduced by Toomey & Eldedge fo each of he convegence sudy cases is pesened in Table 3 along wih he wall ime of each simulaion on 4 Inel-Ivybidge CPUs. The eos in mean and peak foces and momens elaive o he values pediced by VVPM [4] ae shown in Tables 4 and 5 especively. The spaial and empoal esoluion a he wing suface in. pedics peak hinge deflecion, foces and momen accuaely a modeae compuaional cos. These paamees whee seleced o simulae he seven kinemaic cases invesigaed by Toomey & Eldedge [4]. Eddies shed by he moving wing in ae clealy depiced in he voiciy field a wo imes in Fig. 4. Regions of mesh efinemen ae shown in Fig. 5 and he domain decomposiion is displayed in Fig. 6 plainly pesening he adapive efinemen and load balancing duing unime wihin AMROC. Figues 7 9 display he hinge deflecion angle fo expeimens and ou simulaions hough hee peiods of moion fo s 1, and 4. The mean and peak fluid loads ae simulaed in his wok ae wihin 5% of hose pediced by he VVPM [4] as shown in Tabs. 6 and 7. Hinge deflecions pesened in Figs. 7 9 ae in good ageemen wih he expeimenal esuls [4]. Compaing s 1,, and 4, whee he anslaional and oaional shape paamees ae inceased simulaneously he expeced inceases in deflecion angle, mean and peak foces and momen ae obseved. Figue 9 clealy depics he expeced lage deflecion opposie he iniial oaion followed by a ecoil. In conas, seady anslaion causes a small af oaion. Compaing s 4 and 6 he expeced decease in hinge deflecion coesponds o he educed oaion ae caused by he oaional shape paamee,. The insensiiviy of hinge deflecion o anslaion ae conolled by is shown in he compaison of s 4 and 7. The deviaion a small hinge angles obseved in 7 (Fig. 9) coesponds o peiods of he flaping cycle when he diven lead wing componen is anslaing and oaing vey slowly and he ailing wing componen is ecoiling. In he expeimens he hinge was obseved o behave nonlinealy a small angles and unde small loads. This nonlineaiy was no accouned fo in his wok o in he simulaions by Aono e al. [1] o hose by Toomey & Eldedge [4, 1] o simila effec. TABLE 3. CONVERGENCE STUDY RELATIVE ERROR OF MEAN AND PEAK HINGE DEFLECTION VS WALL TIME Re 100 Re 500 Mean Peak Wall [s] Mean Peak Wall [s] Ref % -0.95% % -0.93% % -0.81% % -0.79% % -0.03% % -0.03% % -0.01% % -0.01% TABLE 4. CONVERGENCE STUDY RELATIVE ERROR OF NONDIMENSIONAL MEAN FORCE AND MOMENTS Re 100 Re 500 Ref % 3.74% 5.55% 5.35% 6.83% -7.9%..47% 0.74%.55%.35% 3.83% -4.9%.3.% 0.49%.30%.10% 3.58% -4.04%.4.17% 0.44%.5%.05% 3.53% -3.99% TABLE 5. CONVERGENCE STUDY RELATIVE ERROR OF NONDIMENSIONAL PEAK FORCE AND MOMENTS Re 100 Re 500 Ref % 5.77% 5.97% 6.07% 7.68% -5.69%. 4.46%.4%.6%.7% 4.33% -.34%.3 4.1%.08%.8%.38% 3.99% -.00% %.04%.4%.34% 3.95% -1.96%
6 The influence of oaional phase is obseved by compaing s and 3, as well as, 4 and 5. In boh compaisons he mean y foce is slighly inceased and hinge deflecion is only changed by a phase shif. These simulaions show ha he ae of oaion of he diven body is he majo cause of hinge deflecion as was found in he expeimens conduced by Toomey & Eldedge [4, 1]. CONCLUSIONS The fis pooype of a dynamically adapive, heedimensional laice-bolzmann mehod fo simulaion of flapping wing dynamics has been developed. Fis validaion has been achieved fo a canonical FSI poblem fom [4]. We have confimed ha ou appoach is able o simulae he popagaion of wake fields ceaed by he anslaion and oaion of simplified hinged wing geomey, including he ineacion wih peviously shed voices, wih appaen good qualiy and compaably modeae compuaional coss. Immediae fuue wok will concenae on incopoaing he dynamic elasic esponse of he componens ino simplified biologically elevan models and validaing he appoach fo available benchmaks. ACKNOWLEDGMENTS Sephen L. Wood was suppoed by he TN-SCORE Enegy Schola pogam funded by NSF EPS duing his wok. REFERENCES [1] H. Aono, A. Gupa, D. Qi, Wei Shyy. The Laice Bolzmann Mehod fo Flapping Wing Aeodynamics. AIAA , 40h Fluid Dynamics Confeence and Exhibi, Chicago, Illinois, June 8-1, 010. [] D. Hähnel. Molekulae Gasdynamik. Spinge, 004. [3] R. Deieding. Block-sucued adapive mesh efinemen - heoy, implemenaion and applicaion. Euopean Seies in Applied and Indusial Mahemaics: Poceedings, 34:97 150, 011. [4] J. Toomey and J. D. Eldedge. Numeical and expeimenal sudy of he fluid dynamics of a flapping wing wih low ode flexibiliy. Physics of Fluids (1994- pesen), 0(7):, 008. [5] S. Hou, J. Seling, S. Chen, and G. D. Doolen. A laice Bolzmann subgid model fo high Reynolds numbe flows. In A. T. Lawniczak and R. Kapal, edios, Paen fomaion and laice gas auomaa, volume 6, pages Fields Ins Comm, [6] M. Bege and P. Colella. Local adapive mesh efinemen fo shock hydodynamics. J. Compu. Phys., 8:64 84, [7] H. Chen, O. Filippova, J. Hoch, K. Molvig, R. Shock, C. Teixeia, and R. Zhang. Gid efinemen in laice Bolzmann mehods based on volumeic fomulaion. Physica A, 36: , 006. [8] R. Deieding. A paallel adapive mehod fo simulaing shock-induced combusion wih deailed chemical kineics in complex domains. Compues & Sucues, 87: , 009. [9] S. P. Mauch. Efficien Algoihms fo Solving Saic Hamilon-Jacobi Equaions. PhD hesis, Califonia Insiue of Technology, 003. [10] L. Tsai. Robo Analysis: The Mechanics of Seial and Paallel Manipulaos. Wiley, 1999 [11] R. Deieding and S. L. Wood. Paallel adapive fluidsucue ineacion simulaions of explosions impacing building sucues. Compues & Fluids, 88:719 79, 013. [1] J. D. Eldedge, J. Toomey, and A. Medina. On he oles of chod-wise flexibiliy in a flapping wing wih hoveing kinemaics. Jounal of Fluid Mechanics, 659:94 115, TABLE 6. RELATIVE ERROR (%) OF NONDIMENSIONAL MEAN FORCE AND MOMENTS Re =100 Re = TABLE 7. RELATIVE ERROR (%) OF NONDIMENSIONAL PEAK FORCE AND MOMENTS Re =100 Re =
7 FIGURE 5. CASE. 1.85, Φ=0 100 : 3 REFINEMENT LEVELS (INDICATED BY COLOR) AT / 1.05LEFT, 1.58(RIGHT). FIGURE 6. CASE. 1.85, Φ=0 100 : DOMAIN DISTRIBUTIONS TO 4 PROCESSORS (INDICATED BY COLOR) AT / 1.05LEFT, 1.58(RIGHT). FIGURE 4. CASE. 1.85, 1.885, Φ=0 100 : VORTICITY AT / 1.05LEFT, 1.58(RIGHT). FIGURE 8. CASE 1.85, 1.885, Φ=0 : HINGE DEFLECTION ANGLE OVER TIME. EXPERIMENTAL RESULTS ( ); CURRENT (- -). FIGURE 7. CASE , 0.68, Φ=0 : HINGE DEFLECTION ANGLE OVER TIME. EXPERIMENTAL RESULTS ( ); CURRENT (- -). FIGURE 9. CASE , 3.770, Φ=0 : HINGE DEFLECTION ANGLE OVER TIME. EXPERIMENTAL RESULTS ( ); CURRENT (- -).
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