A PARTICLE-BASED LAGRANGIAN CFD TOOL FOR FREE-SURFACE SIMULATION

Transcription

1 C A N A L D E E X P E R I E N C I A S H I D R O D I N Á M I C A S, E L P A R D O Publcacón núm. 194 A PARTICLE-BASED LAGRANGIAN CFD TOOL FOR FREE-SURFACE SIMULATION POR D. MUÑOZ V. GONZÁLEZ M. BLAIN J. VALLE J. C. DÍAZ-CUADRA Mnsteo de Defensa MADRID MARZO 006

2 A PARTICLE-BASED LAGRANGIAN CFD TOOL FOR FREE-SURFACE SIMULATION POR D. MUÑOZ V. GONZÁLEZ M. BLAIN J. VALLE J. C. DÍAZ-CUADRA Tabao pesentado en la confeenca Wold Matme Technology Confeence, WTMC 006. Londes, Reno Undo, Mazo 006

3 A patcle-based Lagangan CFD tool fo fee-suface smulaton D. Muñoz Next Lmt Technologes V. Gonzalez Next Lmt Technologes M. Blan Next Lmt Technologes J. Valle CEHIPAR J C Díaz-Cuada NAVANTIA SYNOPSIS Computaton of flud moton n the pesence of fee sufaces s essental n many engneeng applcatons. Nowadays, howeve, t s dffcult to fnd technologes capable of obustly handlng ths type of scenao. Tadtonal mesh-based CFD codes often stuggle to povde meanngful solutons to poblems nvolvng lage fee suface defomaton and complex nteacton wth movng boundaes. Ths pape pesents a new tool fo flud dynamcs smulaton that focuses on complex flud moton wth lage fee suface defomaton, sloshng phenomena, beakng waves, floatng bodes and fully coupled flud-stuctue nteacton. INTRODUCTION Fom the pont of vew of ndusty, a tool capable of accuately solvng flud dynamcs poblems whee fee sufaces ae pesent s of geat value. Povdng ths type of tool to the engnee s a must fo detemnng the coect dmensons of engneeng stuctues at the desgn stage. Wth technologes cuently avalable, engnees ae not able to coectly take nto account loads due to fee flud mass moton wthn tanks patally flled and foced by an extenal exctaton. Futhemoe, unde some ccumstances, flud motons wth lage fee-sufaces nsde contanes could lead to mpotant stablty ssues. Wthn the doman of naval achtectue and mane engneeng, the sloshng poblem s of patcula elevance. The sloshng phenomena s a hghly nonlnea esonant effect that ases due to couplng between an extenal exctaton that moves a tank patally flled wth lqud and the ntenal foces geneated by the flud tself. Sloshng ases n all mane stuctues contanng lquds. The effects of sloshng loads ae of geat mpotance when, fo example, desgnng LNG and FPSO tankes. The mpotance of sloshng effects n these knds of shps s elated to the dmensons of the tanks. Ol tankes may have two o thee tanks along the beadth n ode to educe the effects of sloshng loads. Sloshng may also be a poblem encounteed by offshoe cago opeatons. In the case of LNG s the sloshng loads ae an mpotant subect as sloshng effects ntoduce sevee lmtatons on the level to whch tanks can be flled. DISCUSSION Famewok Tadtonal softwae fo flud dynamcs computaton typcally have at the heat the long-establshed fnte element method. Most of these sutes smply cannot smulate cases whee movng boundaes ase. Those who cate fo these knds of poblems, also usually have to make use of ndect appoaches to model the movng boundaes effect (as ntoducng netal foces)

4 The pncpal poblem wth the fnte element appoach s that t s necessay to compute a mesh that pattons the smulaton space. Typcally, fo smple steady cases the mesh geneaton step occupes ove eghty pecent of the tme of a flud dynamcs smulaton un. Moeove, mesh qualty s a ctcal facto detemnng the accuacy of the soluton. In addton, when teatng poblems whee fee sufaces ae pesent, the use of addtonal technques fo nteface tackng s essental. Fnte elements ae defntely not natually suted fo fee suface smulaton. X Thee ae two possble pespectves fo handlng mass tanspot poblems n contnuum mechancs: Eulean s the most common and most ntutve appoach n whch sensos that measue medum popetes ae placed at fxed locatons thoughout the smulaton space. By popely combnng these measues estmatons of seveal mass tanspot elated popetes can be obtaned. On the othe hand n a Lagangan appoach sensos follow the flud mass and consde the vaatons whch happen whle tackng the flow. Mesh-based fomulatons of the ncompessble Nave-Stokes equatons usually need atfcal numecal stablzaton due to the way n whch ncompessblty s enfoced (often a Posson equaton fo the pessue). In addton, fomulatons that take the Eulean appoach also encounte poblems due the pesence of a convectve tem n the equatons that whch descbes the tanspot of mass. Even fo small defomatons the dstoton ntoduced n the mesh leads to naccuaces and nstabltes that ae usually esolved by expensve emeshng pocedues. Futhemoe, attemptng to captue compessblty effects wth mesh-based technques aggavates of these poblems. Recently, geat stdes have been taken n the development of mesh-less technques able to cope wth mass tanspot equatons. Lagangan fomulatons do not eque the dscetzaton of a convectve tem as evoluton laws fo flud magntudes ae expessed n mateal devatves unlke mesh-based methods. The am of mesh-fee methods s to povde accuate and stable numecal solutons fo ntegal and patal dffeental equatons elyng solely upon a set of abtaly dstbuted nodes wthout any knd of mesh famewok. Ths new geneaton of computatonal mesh-fee methods pomses to be supeo to the conventonal gd-based fnte dffeences and fnte elements. In ode to mpove upon conventonal mesh-based methods effots have been pncpally focused on tyng to solve poblems fo whch these methods pefom pooly o ae dffcult to apply such as poblems wth fee sufaces, defomable boundaes, movng ntefaces and lage defomatons that would eque an extemely complex mesh geneaton pocedue. Smoothed Patcle Hydodynamcs (SPH) s one of the ealest mesh-fee methods. It was pesented as an appoxmaton of the Monte Calo method fo the esoluton of gas dynamcs poblems. It was fst appled by Lucy (1977) fo modellng astophyscal phenomena, and late wdely extended fo applcaton to poblems n contnuum sold and flud mechancs. The SPH method and ts vaants have been ncopoated nto many commecal codes. Smoothed Patcle Hydodynamcs uses an ntegal epesentaton fo feld functon appoxmaton and effcently epoduces the behavou of compessble and quas-ncompessble flow. The use of an ntegal epesentaton of feld functons tansfes dffeentaton opeatons on the feld functon to the smoothng (weght) functon. Ths elaxes equements egadng the ode of contnuty of the appoxmated feld functon. As a esult, SPH has been poved to be vey stable fo many poblems wth extemely lage defomatons. In ode to solve flud dynamcs poblems, the SPH fomalsm can easly be appled to the Nave-Stokes equatons. The pue Lagangan natue of SPH method handles flud dynamcs poblems wth extemely lage fee suface defomaton effotlessly snce patcles mplctly delneate the fee suface poston. In contast to fnte element methods thee s no need to explctly captue and tack the fee suface wth SPH. The classc SPH fomulaton povdes vey good consevaton popetes and has been poved vey stable unde hghly demandng ccumstances. On the othe hand a smple analyss of the epoducng condtons of the SPH fomulaton eveals a lack of consstency when the nodes ae not evenly dstbuted. The Nave-Stokes equatons ae second ode patal dffeental equatons and theefoe lnea consstency s a suffcent condton to ensue that n the lmt the numecal model conveges to the tue equatons. Lnea consstency s always a desable condton although t s not stctly necessay. Classc SPH wth andomly dstbuted nodes does not consttute even a patton of unty and thus SPH cannot even guaantee zeo-ode consstency. It s known that the lack of consstency wth SPH s a souce of eo and can lead to apd deteoaton n the soluton accuacy. Recently, movng least squaes coectons (MLS) have been appled to SPH n ode to ensue n-ode consstency. - -

5 Patcle methods usually also encounte dffcultes when handlng bounday condtons. A key aspect n ode to ensue coect teatment of bounday condtons s to be awae of the non delta-konecke popety of SPH and MLS-SPH shape functons. In classc fnte elements, nfomaton s easly ntepolated between nodes, and the value of nodal popetes match the values measued usng the shape functons. Howeve, n SPH o MLS-SPH, nodal popetes do not match the measued feld magntudes appoxmated by the shape functons. Ths pape pesents a new appoach fo soluton of the vscous Nave-stokes equatons that takes nto account ecent advances n mesh-fee patcle methods. Ths new technology explots the poweful capabltes of mesh-less patcle methods by usng a Lagangan fomulaton that handles compessblty effects n lquds, fee and foced floatng bodes, movng boundaes, multphase flow, and complex coupled flud-stuctue nteacton. Ths technology s also hghly applcable to the study of sloshng phenomena. Standad SPH Fomulatons The key dea of SPH s that any feld functon f ( ) can be appoxmated by ( ) f defned as: = ( ) ( ) ( ) f fw d Ω,h (1) whee W, the kenel o smoothng functon, s a scala functon of the dstance between patcles and h s efeed to as the smoothng length. The kenel must satsfy the followng condtons: lm W,h h 0 Ω ( ) = δ( ) ( ) W,hd= 1 () (3) A Gaussan smoothng functon nomalzed fo D computatons would have the followng fom ( ) W,h = The Gaussan s a C functon that fulfls all the kenel equements. Fom a computatonal pont of vew t s moe effcent to defne a kenel wth compact suppot that vanshes fo patcle sepaatons > h 1 h π e h (4) ( ) 4h π( 3 log( 16) ) W,h = + h 0 fo h ( 4) 1 h ( 4) fo 0 h ( ) ( < ) (5) Fg 1: Shape of the kenel fo D smulatons Equaton (1) can be appoxmated by the followng expesson: ( ) ( ) N f m f ( ) = W( -,h) ρ ( ) d f ( ) W( -,h) ρ Ω = 1 ρ (6)

6 Ths can used to measue feld popetes at any locaton wthn the smulaton doman by ust usng the nodal nfomaton stoed wth each patcle. Wtng equaton (1) fo the gadent of a functon f ( ) and ntegatng by pats gves: f = f W -,h d = f W -,h d + f W -,h n ds ( ) ( ) ( ) ( ) ( ) ( ) ( ) Ω Ω Ω (7) Whee Ω s the bounday of the flud doman Ω. The suface ntegal s usually neglected because the functon o the kenel s supposed to vansh at the bounday of the doman. Howeve, cae must be taken n the vcnty of sold boundaes. Standad SPH s geneally used to solve the Eule equaton whch s lmted to nvscd flows snce t s not easy to obtan expessons fo the second devatves n a staghtfowad manne usng SPH fomalsm. Howeve, the SPH fomalsm may be appled to the full Nave-Stokes equatons n ode to smulate geneal-pupose flow smulatons. The contnuty equaton n Lagangan fom s: Smlaly, the momentum equaton s: Dρ = ρ v Dt (8) Dv Dt α αβ 1 σ = + F β ρ x whee the supescpts α, β denote the coodnate dectons, v s the velocty and F s the extenal body foce pe unt mass and σ s the full ntenal stess tenso: σ = pδ + τ αβ αβ αβ Fo Newtonan fluds, the vscous shea stess s popotonal to the shea denoted by the stan tenso ε by a facto known as the dynamc vscosty µ : α (9) (10) whee: ε αβ αβ αβ τ = ε µ β α v v = + α β x x 3 ( v) δ αβ (11) (1) Fnally the mateal devatve fo the ntenal enegy may be expessed as the combnaton of two pats. The fst tem s the sotopc pessue contbuton multpled by the volumetc stan wheeas the second tem epesents the enegy dsspaton due to vscous shea foces. De p µ αβ = v + ε ε Dt ρ ρ αβ (13) Usng the equalty ρ v = ( ρv) + v ρ and applyng SPH fomalsm we can expess the mateal densty devatve fo each patcle as follows: Dρ Dt ( ) ( ) = m v W-,h v (14) Altenatvely, mass consevaton may be exactly enfoced by dectly applyng equaton [3] fo the densty: ρ = m W-,h ( ) (15)

7 Afte some manpulaton the SPH appoxmaton of the momentum equaton fo any gven patcle can be wtten n the followng dscetzed fom αβ αβ p + p W µε + µ ε W α β ρρ ρρ α Dv = m + m + F Dt x x (16) Usng equaton (1) SPH appoxmaton fo ε we fnally obtan: ε m W W = v + v δ v W 3 αβ β α αβ α β ρ x x (17) The enegy equaton can be obtaned by followng a smla pocedue to that of the momentum equaton. The dscetzed enegy equaton has the fnal fom: De Dt p 1 p + µ αβ αβ = m v W + ε ε ρρ ρ (18) Patcle postons ae easly updated by usng followng evoluton law: D Dt = v (19) In standad SPH fo compessble flows, the patcle moton s dven by the pessue gadent, whee the patcle pessue s calculated usng the local patcle densty and ntenal enegy n a state equaton. Fo ncompessble flows, howeve, thee s no equaton of state fo pessue. Moeove, the actual equaton of state fo the flud wll lead to pohbtvely small tmesteps. Although t s possble to nclude a constant densty constant nto the SPH fomulaton the esultant equatons ae too cumbesome to be solved. Usually an atfcal compessblty appoach s used. It s based n the fact that evey ncompessble flud s theoetcally compessble and theefoe t s feasble to use a quasncompessble equaton of state to model the ncompessble flow. The pupose of ntoducng the atfcal compessblty s to poduce a tme devatve of pessue. The followng atfcal equaton of state s wdely used γ ρc s ρ P = 1 γ ρ o (0) whee cs the sentopc speed of sound and ρ o s a nomnal efeence densty. Impovng consstency of SPH Although the tadtonal SPH fomulaton pesents vey good consevaton popetes and has been poved stable s wdely known that ths fomulaton loses ts consstency as soon as the nodes ae not unfomly dstbuted. Ths apdly deteoates the soluton qualty. MLS shape functons can be appled n ode to mpose lnea ode consstency. Ths mpoves stablty and accuacy of standad SPH algothms. Fo evenly dstbuted nodes, classc kenel appoxmatons epoduce constant and lnea scala felds povded a suffcent numbe of neghbous ae pesent. As soon as the nodes ae not placed unfomly wthn the doman we have n 1 = ( ) W x (1)

8 Ths means that SPH appoxmatons ae not able to epoduce even a constant scala feld. Ths poblem s magnfed at the boundaes whee the lack of nfomaton always ntoduces sevee eos n kenel appoxmatons even fo unfomly dstbuted nodes. Recent SPH coectons esolve ths stuaton makng use of movng least squaes (fst successfully appled by Dlts [3]). MLS coectons have made t possble to mpose an abtay ode of consstency n shape functon appoxmaton gven suffcent nfomaton (nodes). MLS exactly epoduces an abtay functonal base p = 1,x,y,z,x,y,... { } () The pocedue conssts of modfyng the kenel weght functon n the followng way Mls ( ) ( )( ( ) ( ) ) N x = N x p x β x,x ρ whee β( x ρ ) s a vecto of coeffcents that depends on the spatal dstbuton of Lagangan nodes. When the coeffcents have been computed, feld functons may be appoxmated by: a% a N x = ( ) MLS (3) (4) Fo example, n ode to mpose lnea consstency the followng base must be adopted p = { 1,x,y,z} (5) Thus, the MLS shape functon takes the followng fom ( ) = ( ( ) + ( )( ) + ( )( ) + ( )( )) N x β x β x x x β x y y β x z z N Mls 0 x y z (6) ρ β ρ The coeffcents vecto ( x) must be computed n the followng manne β0( x ) 1 βx( x) 1 0 β ( x) = = A ( x) β y( x) 0 β z( x) 0 Ax ( ) = AN % x ( ) 1 ( x x) ( y y) ( z z) ( x x) ( x x) ( x x)( y y) ( x x)( z z) A% = ( y y) ( x x)( y y) ( y y) ( y y)( z z) ( z z) ( x x)( z z) ( y y)( z z) ( z z) (7) (8) (9) As soon as n-ode consstency has been mposed va MLS coectons the epoducng condtons of MLS kenels ensue that any functon expessed as a lnea combnaton of the base functons wll be exactly epoduced. The fgue below shows the shape functons of standad SPH (Wo) wth Gaussan kenel functon n a 3 + x undmensonal bounded doman attemptng to epoduce a scala feld of the fom f (x) = (gey 4 lne) wth evenly dstbuted nodes. Followng ths the MLS-0 and MLS-1 esults ae shown

9 Fg : SPH and MLS kenels fo evenly dstbuted nodes The fgue clealy llustates that, fo evenly dstbuted nodes, the classc SPH appoxmaton epoduces the lnea functon eveywhee except at the boundaes due to lack of nfomaton. The fgue also shows that MLS-1 exactly epoduces the lnea scala functon acoss the whole doman. The fgue below shows the shape functons of standad SPH (Wo), MLS-0 and MLS-1 econstuctng the same feld wth nodes at andomly dstbuted postons. Fg 3: SPH and MLS kenels fo andomly dstbuted nodes It s clealy evdent that the classc SPH appoxmaton leads to sevee eos whle the MLS-1 shape functons exactly epoduce the lnea scala feld acoss the ente doman

10 Implementaton The new wok caed out by the authos mpoves the exstng mesh-less methods solvng most of the outstandng ssues elated to the teatment of bounday condtons. Ths technology also explots the mpoved consstency fully Lagangan fomulaton povded by the use of MLS-lke shape functons. New appoach takes specal cae n bounday condtons teatment. Bounday condtons play a decsve ole on the soluton of a flud dynamcs smulaton. In fnte elements, mposng these condtons s not vey dffcult. In most commecal pe-pocesso, uses can easly specfy these condtons ethe to the geometc enttes o dectly to the mesh. Howeve, smulate accuately bounday condtons wth actual engneeng systems eques expeence, knowledge, tme and accuate engneeng decson. Although all the pocedues developed n fnte elements ae applcable wth some modfcatons to mesh-less methods, bounday condtons mplementaton usually eques specal cae. In classc SPH fomulatons, thee s no dect bounday condton. Nodes nea the sold boundaes lack of nfomaton as thee s no patcles outsde the flud doman. Indect appoaches as bounday patcles whch exet cental foces on flud patcles ae the common choce fo modellng the sold bounday. But the defct of nfomaton nea the walls usually leads to ncoect solutons. Fnally, even usng MLS appoxmatons fo constuctng shape functons, specal technques ae equed to mpose essental bounday condtons, because the shape functons ceated do not satsfy the koneke delta condtons. The technology developed makes use of n-house specal technques that physcally mposes bounday condtons takng nto account these subtle ssues. Ths has been mplemented n both and 3 dmensons and solves the full vscous Nave-Stokes equatons wth compessblty effects. It povdes a pomsng soluton to easly cope wth complex poblems whch nclude fee sufaces, multple phases, abtay movng sold boundaes and coupled flud-stuctue nteacton. The authos have decded to call ths method XPH (Extended Patcle Hydodynamcs) due to the smltude wth SPH and elated wok. XPH has been tested n seveal scenaos elated to the sloshng phenomena. In the followng fgue, the smulaton (bght wate) has been supemposed ove the eal vdeo footage of the tank (dak wate). The fgue shows how the smulated esults match almost pefectly the pofle of the eal wate. Fg 4: smulaton wth XPH - 8 -

11 VALIDATION The new method has been extensvely compaed aganst both theoetcal and expemental data n seveal scenaos. Ou fst tests focus on the capabltes of the method to smulate sloshng phenomena. Sloshng tests In ode to test the method s capabltes fo solvng the sloshng poblem we stat wth a smple paallelepped tank. Ths tank s subected to seveal ollng motons of vaous ampltudes and fequences whlst flled to vaous levels. The goal s to captue fo each wate level the magntude and fequency of the esonant sloshng effects. In ths pape the tank dmensons ae (length x wdth x heght) 800mm x 400mm x 50mm. The followng sequence shows some snapshots fom a smulaton wth fllng level of 50mm subected to an exctaton of ampltude 10mm wth fequency 5 ad s -1. The numecal smulaton has been supemposed ove the expemental vdeo data to llustate the emakable coespondence between the numecal and expemental fee-sufaces. Fg 5: The expemental tank vdeo and the ovelad smulaton vsualzaton show emakable coespondence. In ode to poduce the foced moton of the tank a movable platfom has been constucted. Ths platfom estcts all degees of feedom except the angula otaton about an axs nomal to the longest sde of the tank. A compute contolled lnea actuato geneates the foced moton and also measues the angle of oscllaton and the magntude of the oveall ollng momentum on the tank due to the moton of the fee flud mass. Below we pesent the numecal and expemental esults fo a complete ange of fequences fo ampltude 10º and fllng level 100mm. The numecal smulatons ae fully 3 dmensonal. The fgues show the ampltude of the ollng momentum due to wate moton and the phase offset between exctaton and esponse of the ollng momentum. The esults fo ollng momentum confm that the numecal model closely matches the expemental data and that the esonant fequency s captued both qualtatvely and quanttatvely

12 Fg 6: ampltude of wate momentum (expemental vesus numecal) Fg 5: phase offset (expemental vesus numecal) Fg 7: expemental tank vesus smulaton

13 Wave geneaton Lnea wave geneaton has been tested and compaed wth analytcal models. Good ageement wth lnea wave theoy has been obseved. A b-dmensonal numecal wave tank 36 metes long flled wth patcles to a depth of 1.5 metes has been used fo wave geneaton smulatons. The waves ae geneated by usng a movng wall. A numecal beach dsspates the waves as appoach the channel end. Lnea wave theoy states that the elaton between wave numbe, channel depth and wave-make fequency s as follows: ( ) Tanh h k s ω =- kg s (30) Fg 8: wave geneaton test The above mage shows a wave geneated by a pston wth a 0.m stoke and a peod of 1.5 seconds. Equaton (30) pedcts a wavelength of.4 m that closely matches the smulated wavelength of.6m. Fg 9: wave geneaton test In ths case the peod s.13 seconds and the wave-make stoke s 0.4 m. The wavelength pedcted by lnea wave theoy s 6.1 m whle the smulated wave length s 6m

14 Shp sunk The method can easly deal wth the behavou of fee floatng bodes as they nteact wth fee sufaces. Below s a vsualzaton of a b-dmensonal smulaton of a 150 metes long tanke wth a beach ust aft of the bow. The smulaton contans moe that 800,000 patcles and s un at full-scale usng the tue speed of sound fo wate. The mage focuses on a subset of the numecal doman whch s thee tmes longe than the shp. Fg 10: snk smulaton The sequence shows below seveal snapshots fom the smulaton. Pessue waves can be obseved adatng fom the shp. Fg 11: snk smulaton sequence - 1 -

15 Multphase tests Ths sequence shows a coss-secton fom a full 3 dmensonal computaton of the evoluton of a bubble fully mmesed n a dense medum. A hgh densty ato was used wthout suface tenson effects. Fg 1: bubble smulaton Below we show anothe coss-secton, ths tme of a 3D Raylegh-Taylo nstablty smulaton. Once agan the model neglects suface tenson effects. Fg 13: Raylegh-Taylo nstablty

16 ACKNOWLEDGEMENTS The new methodology pesented n ths pape has been developed by the authos as pat of an offcal Spansh eseach and development pogam suppoted by the Mnsty of Industy, the Mnsty of Scence and Technology and the Mnsty of Defence. Next Lmt Technologes has povded the expetse fo the development of the smulaton and vsualzaton tools. Commecal applcatons of ths technology wll be suppoted by Next Lmt Technologes. The expemental valdaton of the sloshng cases has been pefomed n collaboaton wth CEHIPAR (El Pado Model Basn). CEHIPAR s an autonomous oganzaton belongng to the Spansh Mnsty of Defence. Its actvtes nclude shp and popelle desgn and model tests and eseach and development n hydodynamcs and Computatonal Flud Dynamcs (CFD) technologes all focused on obtanng of optmal shp hydodynamc behavou. NAVANTIA s the leadng Spansh company n the mltay shpbuldng secto. Fom the pespectve of sze and technologcal capablty t occupes a leadng poston n Euopean and Woldwde mltay shpbuldng. NAVANTIA s engnees have povded nfomaton esouces to the eseach and development team ncludng specfc techncal dscussons egadng applcatons n the mane ndusty. REFERENCES 1. J.J. Monaghan, Smulatng Fee Suface Flows wth SPH. Annu. Rev. Aston. Astophys., 30, ; 199. P.W. Randles, L.D. Lbesky, SPH: Some ecent mpovements and applcatons. Comput., Methods n Appl. Mech. And Engtg., 139, ; G.A. Dlts, Movng-Least-Squaes-Patcle Hydodynamcs. Int. J. Nume. Meth. Engng. Pat I 44, (1999), Pat II 48, 1503 (000)