Recent Reseches in Mechnics Investigtions of Boundy Tetments in Incompessile Smoothed Pticle Hydodynmics fo Fluid-Stuctul Intections Fnfn Sun, Mingyi Tn, nd Jing T Xing Astct Two oundy tetment methods wee developed fo incompessile flow simultions nd fluid-stuctul intection polems using Smoothed Pticle Hydodynmics (SPH): ) To pply epulsive foce on the oundy pticles while keeping the sme pticle spcing fo inne fluid pticles nd wll oundy pticles; ) To use dense wll pticles without ny dditionl foce. The dm-eking polem nd nothe testing exmple e used to demonstte the pefomnce of this method. Results otined fom the pesent ppoch show esonle geement with expeimentl dt. The fluid pessue vlues otined with SPH method is investigted. Bsed on the esult of the study, it cn e concluded tht the pesent ppoch is elile to simulte incompessile fluid nd the pessue vlue otined cn e used to solve fluid-stuctul intection polems. Keywods Boundy condition tetments, incompessile SPH, pessue pediction. I. INTRODUCTION The Smoothed Pticle Hydodynmics (SPH) method is fully Lgngin mesh-fee method used widely in lge defomtion polems such s fluid motions whee the continuum hydodynmic equtions e solved with set of intecting fluid pticles [], []. The oiginl equtions tht e discetised e those fo compessile viscous fluid. When SPH is pplied to simulte incompessile flows, thee e genelly two wys to impose incompessiility: one is to un the simultions in the qusi-incompessile limit y ssuming smll Mch nume to ensue density fluctutions within % [3]-[5], which is known s Wekly Compessile Smoothed Pticle Hydodynmics (WCSPH); the othe one is clled tuly Incompessile SPH (ISPH) in which incompessiility is enfoced y solving Poisson eqution t evey time step. The velocity divegence is set to zeo s condition to ensue the incompessiility in this method [6]- [8]. It is noted tht incompessile condition lso mens tht the volume of ech fluid pticle should not chnge. Hence, the incompessiility cn e enfoced y setting the volume of Mnuscipt eceived My, 0. F. Sun is with the Fluid Stuctul Intection esech goup of Univesity of Southmpton, SO7BJ UK (e-mil: ffsg09@soton.c.uk). M. Tn is with the Fluid Stuctul Intection esech goup of Univesity of Southmpton, SO7BJ UK (e-mil: mingyi@soton.c.uk). J. T. Xing is with the Fluid Stuctul Intection esech goup of Univesity of Southmpton, SO7BJ UK, (e-mil: txing@soton.c.uk). ech fluid pticle s constnt in the simultion using Lgnge multiplies [9]. Anothe wy to enfoce incompessile fluid is to set the density vition nd velocity divegence to e zeo. This method is used fo multi-phse fluid simultions to enfoce the incompessiility [0]. All those tetment methods, eithe setting density vition to e zeo o foce the velocity divegence to e zeo, equie dditionl considetion on the fluid density vition. In fct the density of the fluid cn e simply set to e constnt fo the incompessiility, nd the zeo velocity divegence cn e stisfied utomticlly []. This method povides stightfowd ppoch to the incompessile fluid polem nd it is dopted in this ppe. It is well known tht SPH does not hve zeoth ode consistency in oundy e. On the oundies, the filue of SPH modelling is chcteized y wll penettion of fluid pticles. Genelly, thee e thee wys to pevent this fom hppening: ) mio pticles [0], ) epulsive foces [3] o 3) dummy pticles [6], []. Usully, epulsive foces e used in WCSPH whees mio pticles need specil considetion on cones o cuved sufces. Hence, dummy pticles o ghost pticles e pefeed in ISPH method [3]. This ppe focuses on investigtion of oundy tetment methods in ode to impove the iciency of SPH model fo incompessile flow simultions. Ghost pticles e useful to keep the symmety configution of the pticles ne the wll. Theefoe, the kenel domin of the pticles cn emin complete nd the physicl popeties such s density cn e clculted coectly. Howeve, when deling with polems with complex solid oundies the ghost oundy tetment ecomes difficult. Tking comptment flooding s n exmple, wte cn fill oth inside nd outside of the stuctue nd t lest two lyes of ghost pticles need to e plced on the inside wll nd outside wll espectively. These ghost pticles sometimes ovelp the tue fluid pticles, which cuses inccute neighouing pticles counting nd esults in wong pedictions. It is lso difficult to tet the ngled oundies y using ghost pticles. Specil considetion is equied to clculte the exct position of the ghost pticle fo the ngled points since the ghost pticle position is impotnt to ensue ISBN: 978--6804-00-6 9
Recent Reseches in Mechnics tht thee is no fluid pticle penettion. Pcticlly, s long s the density cn e kept s constnt, peventing pticles fom penetting the wlls is the mo concen of these oundies. Theefoe, epulsive foce cn e pplied on the wll pticles insted of using sevel lines of dummy pticles which not only inceses computtion time ut lso complictes the model especilly in fluid stuctul intection polems. Anothe oundy tetment using dense wll pticles is lso investigted. With epulsive foce, ll the pticles cn e mintined in unifom ngement ut the dditionl foce my ffect the pessue vlues otined. This polem cn e ovecome y using dense wll pticles, sy hlf spcing wll pticles, ut it equies slightly moe computtion time in this cse. These two oundy tetments cn e chosen ccoding to diffeent situtions. Both the pesent oundy tetments llow icient simultions with complex solid oundies nd even povide new coupling ppoch fo fluid stuctul intections in the futue.. SPH fomultion II. NUMERICAL MODEL The SPH fomultion is sed on the theoy of integl inteplnt tht uses kenel function to ppoximte delt function. A physicl popety is otined y the intepoltion etween set of points inside cetin e. These points known s pticles cy ll the popeties the fluid hs, such s mss nd velocity. The sic ide of this method is to ppoximte function A() s [4] A A( ) = m W (, h) () A model of SPH fomulted gdient tem in the N-S eqution is employed to peseve line nd ngul momentum [3] ( whee W P P) = m + x x = W. Govening equtions P W The govening equtions fo incompessile continuum including the consevtion of mss nd momentum e pesented in the following equtions, espectively. D + v = 0 Dt Dv = g + τ P Dt () (3) (4) (5) Whee t is the time, is density, g is the gvittionl cceletion, P is pessue, v is the velocity, τ is viscous stess tenso nd D/Dt efes to the mteil deivtive. The momentum equtions include thee diving foce tems, i.e., ody foce, foces due to divegence of stess tenso nd the pessue gdient..3 Incompessile SPH nd numeicl fomultion Fo incompessile flow the mss density is constnt. Accoding to (4) the velocity divegence will e zeo []. v = 0 (6) Splitting the momentum eqution into two pts, one is with the ect of ody foce nd viscosity intoducing n intemedite velocity, v / v δt n n = g + τ Anothe is fom pessue influence nd the new velocity cn e updted sed on the intemedite one otined fom pevious step v / v = P δt Tking the divegence of (8) nd sustitute (6) so the pessue Poisson eqution cn e deived / P = v δt The Poisson eqution is fomulted with SPH method m P n n = + η δt m u / Hence, the pessue cn e updted y (0) implicitly. The viscous foce is computed in SPH fom s τ = τ τ + W m n (7) (8) (9) (0) () Whee the stess tenso τ is elted to the stin tenso. The suffix nd epesent diffeent pticles. u u i τ + i = τ i = µ () x xi The full deivtive etween two pticles is fist otined using finite diffeence efoe decomposing it into x nd y diections. Thus ISBN: 978--6804-00-6 93
Recent Reseches in Mechnics ui x ui = x i u u = i x x (3) Fo Newtonin fluids such s wte, the viscosity coicient µ (ective viscosity) hs constnt vlue µ. Hence, the SPH fomultion of viscosity tem cn e witten s [6]: µ 4mµ W ( u u ) u = (4) + η The quntities e updted following the steps u = + 4mµ u δt g + W u / ( ) + η δt m u / = P m P n + η n n u n n u / t + + + + = m + u (5) (6) δ P P (7) f ( ) p p 0 0 D = (9) It is zeo when > 0 so tht the foce is puely epulsive. Mostly, p 4 nd p, D=5gH ccoding to [3]. The = = length scle 0 is tken to e the initil spcing etween the pticles. In incompessile SPH method, ghost pticles which mio the physicl popeties of inne fluid pticles e the usul tetment of wll oundy conditions. These ghost pticles mke smoothing domin complete fo the ne wll fluid pticles so tht the consistence of SPH simultion ne wll oundies is ensued. The epulsive foce oundy tetment s shown in Fig. hs not een used in incompessile SPH method. Nomlly ghost pticles e consideed to e necessy to void unphysiclly lge density vition fo the ne oundy pticles. Howeve, when the density of ll the pticles is set to e constnt, ghost pticles will not e necessy ny moe. Wll pticles e involved in the Poisson eqution, using dense pticles on the wll oundy cn poduce pessue to keep the inne pticles wy fom the oundy. A hlfed spcing is set on the wll pticles comped with the inne fluid pticles s shown in Fig.. Altentively, the simple epulsive foce tetment cn e pplied to sve computtionl time. 3. Fee sufce III. BOUNDARY CONDITIONS Fee sufce pticles e tcked down to set thei pessue to zeo to simplify the dynmic sufce oundy conditions [5]. The following quntity is clculted to identify the fee sufce pticles m = w ' e (8) This vlue equls to in -D pplictions o 3 in 3-D cses when the smoothing domin is not tuncted ut it is f elow these vlues fo sufce pticles, citeion used in this ppe is.6 in -D cses. 3. Wll oundy The solid wlls e simulted y pticles which pevent the inne pticles fom penetting the wll. Conventionlly, the wll oundy conditions e modelled y fixed pticles exeting epulsive foce on inne fluid pticles in wekly compessile SPH method. Fig. : Boundy tetment: using hlfed spcing on wll pticles Fig. : Boundy tetment: using epulsive foce IV. COURANT NUMBER CONDITION Since this incompessile SPH method clcultes pessue implicitly nd othe popeties explicitly, the size of time step must e contolled in ode to hve stle nd ccute esults. The following Count condition must e stisfied [6] h t 0. (0) v mx whee h is the initil pticle spcing nd v mx is the mximum pticle velocity in the computtion. The fcto 0. ensues tht the pticle moves only fction (in this cse 0.) of the pticle spcing pe time step. Anothe constint is ISBN: 978--6804-00-6 94
Recent Reseches in Mechnics sed on the viscous tems [0] h t 0.5 () µ / whee µ is the ective viscosity. The llowle time-step should stisfy oth of the ove citei. V. TESTING EXAMPLES Without ghost pticles, model with complex wll oundies cn e simulted iciently. An exmple is used to test the two oundy tetments in simulting wte flooding into comptments s shown elow in Fig. 3. Fig. 3: wte flooding comptment The wte column is set to e m high nd m wide. It is kept in sttic stte in the eginning, ut suddenly hole is unlocked on the ight wll nd the wte stts to flow out. Simultion of the wte flooding ws ecoded t time 0.4s, 0.6s nd.8s s shown elow in Fig. 4. Fig. 5: testing cse using epulsive foce on wll pticles when t=0.4s, 0.6s,.8s Only smll diffeences cn e oseved fom the esults otined sed on these two diffeent oundy tetments. VI. PRSSURE INVERSITGATION The investigtion of these two oundy tetments e cied out with -D dm ek simultion. The model is set up s shown in Fig. 6. The spcing of fluid pticles is set to e 0.0m, the ovell height of the wte column (H) is 0.6m nd its width () is.m. The size of the solid contine is 3.m long. The wte column is kept in hydosttic stte in the eginning. The flow stts when the ight side gte is suddenly opened so the wte column collpsed. The pessue vlues (P) t point 0.6m on the ight wll e tced. The esults e otined using ghost pticles, hlf spcing wll oundy pticles nd epulsive foce oundy tetments nd the esults e comped with expeimentl dt povided y [6] Fig. 6: Dm ek Fig. 4: testing cse with hlfed spcing fo wll pticle when t=0.4s, 0.6s,.8s Wte flows violently fte the unlocking of the hole. The font of wte impcts the ight side wll nd comes ck impcting the second uilding. Some i ules e geneted duing the pocess, fluid motion chnges quickly. If ghost pticles e used to model the wll oundy, sevel lyes of these ghost pticles must e plced on ech side of the "smll stuctues". This will ffect the fluid pticles when flow fills oth sides of the stuctue. Some of the ghost pticle my ovelp with the tue fluid pticle, which mkes the counting of the neighouing pticles inccute. Results otined using epulsive foce pplied on the oundy pticles e lso displyed in Fig. 5 to mke compison with the dense wll pticle tetment. Wte configution with time is shown in Fig. 7 Fig. 7: Wte configution t time 0s,0.5s,0.75s,.5s Results otined using diffeent oundy tetments in SPH method e comped with expeimentl dt [6] s shown in Fig. 8. ISBN: 978--6804-00-6 95
Recent Reseches in Mechnics Fig. 8: Results otined fom thee diffeent oundy tetments comped with expeimentl dt nd nothe numeicl method All these thee oundy tetments find the fist pessue pek ound the ight time comped with expeimentl dt. The fist pek vlues otined fom numeicl method e slightly lge thn oseved in the expeiment. Boundy tetments with epulsive foce nd ghost pticles gve simil esults, dense wll pticle oundy tetment give slightly highe pek vlue thn the othe two tetments. Except the second pek vlue, the ovell cuves gee with expeimentl dt well. Thee is no ovious second pek pessue in the simultions. This is pehps ecuse of entined i ects which e not pedicted duing the simultion. But comped with othe numeicl method such s Nvie-Stokes Solve povided y [7], SPH gives close vlues to the expeimentl ones. Fig. 9: Investigtion of epulsive foce oundy tetment with diffeent time stepping size The Fig. 9 shows lge vitions of the fist pek vlue when using diffeent time stepping sizes. The est of the cuves e lmost the sme. When time stepping size is 0.000s, the esults e fily close to the expeiment, which mens tht time stepping size etween 0.0005s nd 0.000s is sufficiently ccute fo the simultion. This time stepping size vlue cn e otined fom Count nume condition. Fig. 0: Investigtion of hlfed wll pticle spcing tetment with diffeent time stepping size Fom Fig. 0, it cn e seen tht simil to epulsive foce tetment, using time stepping size of 0.000s povides ette esults thn using 0.0005s. Howeve, the cuves do not chnge s much s the pevious oundy tetment cse. Futhe time stepping size decesing does not chnge the esults noticely. And second pek is pedicted in this cse when using time stepping size of 0.00005s, which indictes tht using dense wll pticles is ette oundy tetment to otin ccute esults comped with epulsive foce tetment. VII. CONCLUSION Simple oundy tetments cn e used insted of conventionl ghost pticles fo incompessile SPH with constnt fluid density. Simultions with complex solid oundies cn now e modelled without difficulty. Two testing exmples hve een used to demonstte the ppliction of these two oundy tetments. Model set-up cn e done moe iciently using one of these two oundy tetments. They not only offe simple configution fo the model ut lso poduce ette simultions. It is oseved fom the pessue esults nlysis tht incompessile SPH cn povide moe ccute pessue vlue. REFERENCES [] R. A. Gingold nd J. J. Monghn, Smoothed Pticle Hydodynmics: Theoy nd Appliction to Non-Spheicl sts, Monthly Notice of the Royl Astonomicl Society, vol. 8, pp. 375-389, 977. [] L. B. Lucy, Numeicl ppoch to testing the fission hypothesis, Astonomicl Jounl, vol. 8, pp.03-04, 977. [3] J. J. Monghn, Simulting fee sufce flows with SPH, Jounl of Computtionl Physics, vol. 0(), pp. 399-406, 994. [4] J. P. Mois, P. J. Fox, nd Y. Zhou, Modelling low Reynolds nume incompessile flows using SPH, Jounl of Computtionl Physics. vol. 36, pp. 4-6, 997 [5] X. Y. Hu nd N. A. Adms, A multi-phse SPH method fo mcoscopic nd mesoscopic flows, Jounl of Computtionl Physics, vol. 3(), pp. 844-86, 006 [6] S. Sho nd E. Y. M. Lo, Incompessile SPH method fo simulting Newtonin nd non-newtonin flows with fee sufce, Advnces in Wte Resouces, vol. 6(7), pp. 787-800, 003 [7] J. Pozoski nd A. Wwenczuk, SPH computtion of incompessile viscous flows, Jounl of Theoeticl Applied Mechnics, vol. 40, pp. 97, 00 [8] S. M. Hosseini, M. T. Mnzi, nd S. K. Hnnni, A fully explicit thee-step SPH lgoithm fo simultion of non-newtonin fluid flow, ISBN: 978--6804-00-6 96
Recent Reseches in Mechnics Intentionl Jounl fo Numeicl Methods fo Het & Fluid Flow, vol. 7, pp. 75-735, 007 [9] M. Elleo, M. Seno, nd P. Espñol, Incompessile smoothed pticle hydodynmics, Jounl of Computtionl Physics, vol. 6(), pp. 73-75, 007 [0] S. J. Cummins nd M. Rudmn, An SPH poection method, Jounl of Computtionl Physics, vol. 5(), pp. 584-607, 999 [] E. S. Lee, D. Violeu, nd R. Iss, Appliction of wekly compessile nd tuly incompessile SPH to 3-d wte collpse in wtewoks, Jounl of Hydulic Resech, vol. 48, pp. 50-60, 00 [] A. J. C. Cespo, M. Gomez-Gestei, nd R. A. Dlymple, Boundy conditions geneted y dynmic pticles in SPH methods, Computes, Mteils & Continu, vol. 5, pp. 73-84, 007 [3] E. S. Lee, D. Violeu, nd R. Iss, Compisons of wekly compessile nd tuly incompessile lgoithm fo the SPH meshfee pticle method, Jounl of Computtionl Physics, vol. 7(8), pp. 847-8436, 008 [4] G. R. Liu, Meshfee methods: Moving eyond the finite element method, CRC Pess, 7 pges, 00 [5] J. J. Monghn, On the polem of penettion in pticle methods, Jounl of Computtionl Physics, vol. 8, pp. -5, 989 [6] Z. Q. Zhou, J.O. DeKt, nd B. Bunchne, A nonline 3-d ppoch to simulte geen wte dynmics on deck, Poceedings of the 7 th intentionl confeence on numeicl ship hydodynmics, Nntes, July 999 [7] K. Adolmleki, K. P. Thign, nd M. Mois-Thoms, Simultion of the dm ek polem nd impct flows using nviestokes solve, 5 th Austlsin Fluid Mechnics Confeence, 004 ISBN: 978--6804-00-6 97