CFD Simulation of Cloud and Tip Vortex Cavitation on Hydrofoils

Transcription

1 ICMF 007, Lepzg, Germany, Juy 9 3, 007 CFD Smuaton of Coud and Tp Vortex Cavtaton on Hydrofos Th. Fran, C. Lfante, S. Jebauer, M. Kuntz and K. Rec ) ANSYS Germany, CFX Deveopment Staudenfedweg, D-8364 Otterfng, Germany Thomas.Fran@ansys.com, Conxta.Lfante@ansys.com ) Schffsbau-Versuchsanstat Potsdam GmbH (SVA) Marquardter Chaussee 00, D-4469 Potsdam, Germany Rec@SVA-Potsdam.de Abstract Keywords: cavtaton, CFD, vadaton The onset of cavtaton around propeers, hydrofos, shps, etc represents an mportant ssue n terms of reduced performance, eroson and passenger/crew comfort due to cavtaton nduced vbratons and nose among other drawbacs. Consequenty cavtaton has been studed by many researchers, but up to now most of the nvestgatons are st experments. Snce expermenta nvestgatons for marne appcatons are expensve, CFD smuatons represent a powerfu too n order to nvestgate the phenomenon and consequenty to mprove the desgn of such components. A mode to dea wth cavtaton and the pressure fuctuatons ntroduced by t has been deveoped n ANSYS CFX, and a vadaton of t has been carred out. Two test cases have been chosen for ths purpose. The frst one s a D case contanng a pano-convex profe, where coud cavtaton can be observed. The second one conssts of a 3D case, where the fud fows around a NACA hydrofo. A tp vortex s generated wth hgh rada veocty gradents orgnatng cavtaton. In both cases the smuatons have been carred out on refned grds and the numerca resuts have been compared to those n terature, showng good agreement wth them n most of the cases.. Introducton Cavtaton Modeng for Marne Appcatons Cavtaton n marne appcatons e fows around hydrofos, shps and propeers s a phenomenon, whch can ead to serous performance deteroraton of propeers, to cavtaton eroson damages to propeer bades and to the oss of passenger comfort due to cavtaton nduced pressure fuctuatons nteractng wth the shp hu. Therefore arge expermenta and smuaton efforts are spent nto the nvestgaton of cavtaton ncepton and accurate predcton of cavtaton for exstng and new marne technoogy desgns. Due to hgh operatona costs of expermenta nvestgatons n scaed cavtaton tunnes and uncertantes n the upscang from the expermenta data to rea scae desgns t s hghy desrabe to be abe to study cavtaton wth reabe and accurate CFD smuaton technques. The am of the presented wor s the further deveopment of a cavtaton mode n ANSYS CFX and vadaton aganst avaabe expermenta data. The cavtaton mode n ANSYS CFX s based on the homogeneous mutphase fow framewor of the CFD sover tang nto account the dynamcs of the cavtaton bubbes by sovng a smpfed Rayegh-Pesset equaton for the cavtaton bubbe radus. The cavtaton mode can be combned wth any of the turbuence modes, whch are avaabe n ANSYS CFX, where the SST turbuence mode of Menter [6] has been chosen as the bass for the present study. Turbuent pressure fuctuatons and ther nfuence on the cavtaton process can be taen nto account ether by resovng them n the numerca smuaton, by appyng a arge scae resovng turbuence mode (e.g. LES/DES/SAS) or by reatng them to the turbuent netc energy and tang nto account an addtona pressure fuctuaton term n the Rayegh-Pesset equaton. The vadaton of the ANSYS CFX cavtaton mode (Jebauer, [4]) was based on avaabe terature data from Le et a. [5], Franc [3] and Arndt & Dugue []. In the frst test case a pano-convex profe wth cavtaton couds on upper and ower sde has been studed n a two-dmensona confguraton. Cavtaton ncepton, cavtaton bubbe szes and hydrofo ft have been examned for dfferent anges of attac and for dfferent cavtaton numbers. Transent averaged pressure profes on the hydrofo upper and ower sde have been compared aganst expermenta data for dfferent fow regmes. The second test case was the fow around a NACA hydrofo wth eptca panform, where the formaton of a tp vortex can be observed. Hgh rada veocty gradents ead to ow pressure beow saturaton pressure n the vortex core nducng cavtaton. The expected cavtaton ncepton n the vortex core has been nvestgated and compared to a correaton vs. ft coeffcent and Reynods number as obtaned by Arndt et a. [] from a arge number of experments. In order to assess the mnmum pressure n the trang vortex core wth strong swrng moton and hgh veocty gradents a curvature correcton term n the SST turbuence mode was apped, eadng to a substanta mprovement n the accuracy of predcted fud veoctes. Further enhanced resuts were obtaned by changng the turbuence mode to a Reynods Stress Mode [4] [5]. Numerca smuatons usng ANSYS CFX have been performed on herarchcay refned meshes appyng the est Practce Gudenes by Menter [7]. Comparson of ntegra data (as hydrofo ft), transent averaged voume fracton feds and pressure profes on hydrofo surface as we as rada veocty profes n the tp vortex showed good agreement wth expermenta data. Aso for the three-dmensona test case of Arndt & Dugue [] t was found, that the mesh resouton of the fnest mesh was st too coarse n order to fuy resove the very sharp veocty gradents n crcumferenta fud veoctes n the tp vortex cose to the hydrofo, when usng the SST turbuence mode.

2 ICMF 007, Lepzg, Germany, Juy 9 3, 007 Nomencature r α Phase voume fracton u Veocty component (m s - ) ɺ Phase mass transfer rate (Kg m -3 s - ) S α g Gravty component (m s - ) P Pressure (N m - ) ubbe radus (m) R u Average veocty component (m s - ) u Fuctuatng veocty component (m s - ) Knetc energy (m s - ) Gree etters ε Turbuence dsspaton rate (m s -3 ) ω Turbuence frequency (s - ) ρ α Phase densty (Kg m -3 ) τ Stress tensor component (Kg m s - ) σ Surface tenson coeffcent (m 3 s - ) Subsrpts m Mxture v Vapour Lqud ubbe mn mnmum sat saturaton n net. The ANSYS CFX Cavtaton Mode The cavtaton mode deveoped by ANSYS CFX s based on the Rayegh-Pesset equaton, whch descrbes the growth of a vapour bubbe n a qud. Ths effect s taen nto account by addng a speca source term nto the contnuty equaton. A homogeneous approxmaton to the vapour-water fow s adopted, consderng the same veocty fed for a phases by assumng that the vapour bubbes are movng wth the contnuous phase wthout sp veocty. The governng equatons for the two-phase fow then read: Contnuty equaton for each phase ( rα ρα ) ( rα ραu ) + = S t Momentum conservaton equaton ɺ ( ) ( ρ ) ( ) ( ) mu ρmuu P τ + = + ρmrα g + t where r α, u, ρ α, S α ɺ, g, α ( ) τ and P, are the phase voume fracton, the cartesan veocty components, the phase densty, the phase mass generaton rate, the acceeraton components due to gravty, the pressure and the stress tensor, respectvey. Subscrpt m refers to mxture propertes. Snce the sum of a phases must occupy the whoe doman voume, the foowng constrant must be satsfed: N r α α = = ( 3 ) where N = s the number of phases. In addton, assumng that the mass sources are due to the nterphase mass transfer, t becomes that: N S α α = ɺ = 0 ( 4 ) When ony two phases are nvoved, as occurs wth cavtaton (vapour and qud) the mass transfer rates are reated by: Sɺ Sɺ Sɺ. ( 5 ) = = v v The expresson to evauate ths source term can be derved from the Rayegh-Pesset equaton, whch n ts fu verson can be wrtten as: where d R 3 dr σ Pv R + dt + =, ( 6 ) dt R ρ R represents the bubbe radus, σ s the surface tenson coeffcent and P v s the pressure n the bubbe, whch s assumed to be the vapour pressure. Negectng the second order terms and the surface tenson, the equaton reduces to dr dt Pv 3 ρ = ( 7 ) The rate of change of bubbe mass s then predcted as: dm dt dv Pv = ρv = ρv 4π R ( 8 ) dt 3 ρ Assumng that there are N bubbes per unt voume, the vapour voume fracton may be expressed as: 4 3 rv = V N = π R N ( 9 ) 3 And therefore the tota nterphase mass transfer due to cavtaton per unt voume becomes: S v 3rv ρv Pv = R 3 ρ ɺ ( 0 ) Ths expresson has been derved assumng bubbe growth (evaporaton). It can be generased to ncude condensaton by ncudng an emprca factor (F) n the foowng manner 3rv ρ P v v Sv = F sgn( Pv P) R 3 ρ ɺ ( ) whch may dffer for condensaton and vaporsaton, and t s desgned to tae nto account the fact that both processes occur at dfferent rates, snce the condensaton process s usuay much sower than evaporaton. Despte the fact that the mode has been generased for evaporaton and condensaton, t requres further modfcaton n the case of evaporaton. Evaporaton s ntated at nuceaton stes. As the vapour voume fracton ncreases, the nuceaton ste densty must decrease accordngy, snce there

3 ICMF 007, Lepzg, Germany, Juy 9 3, 007 s ess qud. For evaporaton v The fna form of the cavtaton mode s: S v r s repaced by r ( r ) nuc 3 rnuc ( rv ) ρv Pv Fvap f P < Pv R 3 ρ = 3rv ρv Pv Fcond f P > Pv R 3 ρ. ɺ ( ) And the foowng mode parameters have been apped: R = m r = F = F = , nuc 5 0, vap 50, cond Interacton of Cavtaton and Turbuence Modeng Most of the fows that can be observed n nature or engneerng processes are turbuent. It s due to the fact that they are three dmensona fows, unsteady and may contan many dfferent ength scaes, orgnatng a compex process. The Naver-Stoes equatons are st vad for turbuent fows. However, turbuent fows span the range of ength and tme scaes nvovng scaes much smaer than the smaest fnte voume sze. The computng power requred for the Drect Numerca Smuaton (DNS) of ths nd of fows s further beyond the avaabe one, partcuary n cases of ndustra nterest. Maor effort has been carred out by the scentfc communty n order to tae nto account the turbuent effects on the fow. Dfferent approaches can be apped such as resovng the arge-scae turbuent fuctuatons contanng the maor part of the turbuent netc energy (LES, DES, SAS) or modeng the phenomena entrey. When attemptng to mode the turbuence, turbuence vscosty modes can be apped. The turbuence or eddy vscosty modes are statstca modes and consder that the man varabes are compound by an average component and an addtona tme-varyng fuctuatng one, e u = u + u ( 3 ) Introducng ths decomposton nto the Naver-Stoes equatons (-) and tme-averagng them, the so-caed Reynods Averaged Naver-Stoes (RANS) equatons are obtaned ( ρmu ) ( ρmuu ) P + = + ρmrα g + t + u u + S ( τ ρm ) M v ( 4 ) Smuaton of RANS equatons substantay reduces the computatona effort n comparson wth DNS and t s generay adopted for engneerng appcatons. However, the averagng procedure ntroduces addtona unnown terms contanng products of the fuctuatng components, whch act e addtona stresses n the fud. These stresses are dffcut to determne drecty and must be modeed by means of addtona equatons or quanttes n order to cose the set of equatons. Eddy vscosty modes assume that the Reynods stresses can be reated to the mean veocty gradents and turbuent vscosty by the gradent dffuson hypothess n an anaogous manner to Newtonan amnar fow as: ρmuu = ρmδ + µ t uδ 3 3 µ u + u t ( 5 ) where µ t s the eddy vscosty or turbuent vscosty, and needs to be evauated. In ths wor a two-equaton turbuence mode s apped. It represents a good compromse between numerca effort and computatona accuracy. Two extra equatons must be soved (-ε, or -ω), The turbuent vscosty s modeed as the product of a turbuent veocty and turbuent ength scae. The turbuent veocty scae s computed from the turbuent netc energy (), and the turbuent ength scae s estmated from ether the turbuence netc dsspaton rate (ε) or the turbuence frequency (ω). A representatve of the two-equaton modes s the SST (Shear Stress Transport) turbuence mode. The SST mode [6] [9] s based on the combnaton of two underyng two-equaton turbuence modes, the ndustray wde-spread -ε -mode (Jones and Launder, []), and the -ω mode n the formuaton of Wcox [0][]. The hybrd procedure conssts of the -equaton and a speca form of the ω-equaton, whch enabes through changng the vaue of a bend factor F swtchng between a ω-equaton (F =) and a ε-equaton (F =0). The two equatons read as: and ( ρ ) ( m ) m ρ u + = P β ρmω + t ( ρ ) ( ρ u ω) mω γ t µ t + µ + σ x βρ ω m + = P m + υt σ µ t ω ω ω µ + + ( F ) ρ m σ ω ( 6 ) ( 7 ) The vaue P represents the turbuent netc energy producton term u u u u P mn = µ t + + ρmδ,0ε x x 3 whe the bendng functon oos e 4 500υ 4ρmσ ω F = tanh mn max,, β ω y y ω CDω y, ω 0 CDω = max ρmσ ω,0,β =0.0. ω beng 3

4 ICMF 007, Lepzg, Germany, Juy 9 3, 007 Then the turbuent vscosty can be computed as: a t = m, wth s SS max ( aω sf ) µ ρ 500υ F = tanh max, β ω y y ω and =, a = 0.3 In order to become free from effects of the curvature or rotaton of the overa system, correctons to the mode were ntroduced. One of them was suggested by Spaart and Shur [3], based on the vaue s ω (ω s the thcness of the eddy). A factor ntroducng a correcton of the turbuence szes ncuded. For the SST mode apped, the correcton factor f r (Langtry and Menter, 005) for the producton term s computed as f r where = max * ( + cr)r mn * { cr 3 tan ( cr rɶ )} cr,.5, + r 0.0 ] r * = s ɶ ɶ ω ω ; RC ɶ ω rɶ = ω D ɶ ωg RC DS ω = S + ε mns n + ε mns n Ωm D Dt u u ɶ ω = ε mωm ; ( ) 3 3 ɶG = ɶ + ɶ + ɶ ; ω ω ω ω. ; ( 8 ) ε mno s the permutatons symbo, Ω m s the rotaton veocty of the system, D = max( s, c, ω ), c r =.0, c r =.0, c r3 =.0, r 4 c r4 =0.09. When the stress tensor components must be computed more accuratey or the underyng assumpton of sotropc turbuence s voated, Reynods Stress Modes can be empoyed. They are based on transport equatons for a components of the Reynods stress tensor and the dsspaton rate (or the turbuence frequency). Agebrac Reynods Stress modes sove agebrac equatons for each ndvdua component of the tensor, whe dfferenta methods sove a dfferenta transport equaton. In ths case the computatona effort s consequenty ncreased. An ω-based Reynods Stress mode was chosen for the present wor: the so-caed SL Reynods stress mode. In ths case the modeed equatons for the Reynods stresses can be wrtten as foows: ( ρτ ) ( uρτ ) + = ρp + t 3 β ρωδ µ t τ ρ Π + µ + * σ ; ( 9 ) And the correspondng ω-equaton read as: ( ρω) ( u ρω) ω + = α3 P β3ρω t µ t ω ω + µ + + ( F )ρ σ ω3 σ ω ( 0 ) Agan the mode bends from a ω-based mode to an ε-based mode. In the frst case, the foowng parameters are empoyed, * σ =,.0.0 σ =, β = 0.075, α = whe n the second case, they are * σ =, σ =, β = 0.088, α = The bendng s done by means of a smooth near nterpoaton n a smar way as for the SST method [4]. The consttutve pressure-stran correaton s gven by ˆ Π = β Cω τ + δ α P δ 3 3 ˆ β D ˆ δ γ S Sδ 3 3 where the producton tensor P s computed as u u P = τ + τ ; P = P and the tensor D as D u = τ + τ u Fnay the turbuent vscosty can be computed as µ t ρ ω ( ) ( ) ( 3 ) = ( 4 ) The vaues of the coeffcents apped by ANSYS CFX for the computaton of ths mode are: β = 0.09, ˆ α = (8 + C) /, ˆ β = (8C ) /, ˆ γ = (60C 4) / 55, C =.8, C = 0.5 In addton to the turbuence vscosty modes, another famy of methods can be used nown as LES, consstng of fterng the Naver-Stoes equatons and the decomposton of the fow varabes nto a arge scae and a sma scae. However, ths technque s computatonay very expensve when t s apped to ndustra probems. In ths context arses the need of the use of Scae-Adaptve Smuatons (SAS). It s an mproved URANS formuaton, whch aows the resouton of the turbuent spectrum n unstabe fow condtons. The SAS method [6] s based on the Von Karman ength scae. Dependng on t, the mode adusts to a URANS smuaton, wth LES-e behavour n unsteady regons, or to RANS smuaton n stabe fow regons. As t w be shown n next secton, t was found that the use of ether a scheme or another pays an mportant roe n the smuaton. The Reynods Stress Mode apped (SL RSM) eaded to more accurate predctons of the rotatona veocty (whch presents a steep profe) n case of tp vortex cavtaton 4

5 ICMF 007, Lepzg, Germany, Juy 9 3, 007 than SST computatons... The Turbuent Pressure Fuctuaton Mode As dscussed before, the nfuence of the turbuence on the cavtaton process has been wdey observed n mutpe expermenta nvestgatons. A dfferent approach to account for enhancement of cavtaton due to turbuent pressure fuctuatons conssts of reatng them to the turbuence netc energy. In ths case, the threshod pressure has been changed from saturaton pressure to where P = P + P, ( 5 ) v sat turb P turb 0.39ρ = ( 6 ) Thus, the Rayegh-Pesset equaton (0) apped for the computaton of the cavtaton bubbe growth becomes: dr dt Psat + Pturb 3 ρ = ( 7 ) Ths strategy was found to be not competey physcay reastc and therefore a further sght modfcaton was done by the authors n order to mae t more rgorous. Snce the netc energy s reated to the turbuence of the qud phase, water n the current stuaton, t seemed more approprate to appy the pressure turbuence term ony n ts presence. Therefore, the expresson (6) was changed to: turb P = 0.39( r ) ρ, ( 8 ) whch vanshes when the voume s fed up ony wth vapour. v Menter [7]. 3.. The Le et a. Test Case Sheet Cavtaton on Pano-Convex Hydrofo Profe 3... Test case defnton A schematc of the expermenta setup [3] of Le s gven n Fgure. For the orgna experment the hydrofo was at a submerson depth of 0 cm under a free surface. Its upper sde s pane and ts ower sde crcuar (radus 6 cm) wth a maxmum thcness of 0mm. The eadng edge s rounded wth a radus of mm, so that the chord (c 0 ) s about 96 mm (Fgure ). Experments nvovng dfferent anges of attac (from -8 to 8 ), dfferent cavtaton numbers and dfferent Reynods numbers (from 0 6 to x0 6, whch correspond to net veoctes from 5 m/s to 0 m/s) were performed as reported n the orgna pubcaton of Le [3]. The cavtaton number (σ) many charactersng the fow pattern s defned as: σ P 0.5ρv = ( 9 ) v n Confguratons wthn the range of vaues descrbed by Le were chosen to run the numerca computatons, and vadate the mode n ANSYS CFX. 3. Mode vadaton Two dfferent test cases were chosen to vadate the ANSYS CFX cavtaton mode. The frst one conssts of a two-dmensona test wth pano-convex profe, where sheet or coud cavtaton taes pace. Man fow characterstcs as cavtaton ncepton, cavtaton ength or hydrofo ft have been anayzed for dfferent anges of attac and dfferent Reynods numbers. Resuts have been compared aganst expermenta data n terature [3][5]. The second vadaton case studed was a three-dmensona fow around a NACA hydrofo wth eptca panform. A tp vortex can be observed wth appearng cavtaton n ts core, due to the hgh rada veocty gradents and the ow pressure (beow saturaton pressure) at the vortex core ocaton. Cavtaton ncepton at the vortex has been anayzed and compared to correaton vs. ft by Arndt et a []. The vortex core shows a strong swrng moton wth hgh veocty gradents. In order to estmate the mnmum pressure vaue at dfferent cross sectons behnd the tp of the hydrofo, dfferent turbuence technques have been apped. Incudng a curvature correcton term n the SST mode or empoyng a Reynods Stress Mode was found to mprove sgnfcanty the accuracy of predcted fud veoctes. Integra data as ft, transent averaged voume fracton feds or pressure profes have been compared to those by Arndt []. For both test cases the numerca computatons were carred out usng ANSYS CFX on a herarchy of three consecutvey refned meshes appyng the est Practce Gudenes by Fgure : Schematc representaton of the fow around a pano-convex hydrofo Mesh herarchy and CFD setup The confguraton chosen to run the CFD smuatons s presented n Fgure. In dfference to the orgna expermenta setup the hydrofo was submerged n a wa bounded channe, thereby avodng the predcton of the free surface. Symmetry Pane Wa Fgure : Representaton of the setup used for the CFD computatons 5

6 ICMF 007, Lepzg, Germany, Juy 9 3, 007 The dscretzaton of the doman has been performed by usng ICEM CFD Hexa as a grd generator. The bocng structure shown n Fgure 3 has been desgned to generate the grds. In ths manner a smooth and hgh quaty mesh can be obtaned (n terms of grd nes ange and aspect rato). In order to appy the est Practce Gudenes, the smuatons were computed on h-refned grds. Three eves of refnement are performed obtanng fner meshes, snce the quaty of the mesh can determne sgnfcanty the accuracy of the smuaton executed on t. The refnement factor s n each coordnate drecton, whe the mnmum grd ange vaue s around 40 for a three cases. An mportant attrbute of the mesh to tae nto account s the dstance of the frst node of the grd to the wa, partcuary when turbuence modes are apped. For a three meshes ths vaue s sma enough to expect a satsfactory resouton of the turbuent boundary ayer near the wa. It can be computed as 3/4 + y = L y ( 30 ) 80 Re L The grd has been changed not ony by refnement but aso by rotatng the ange of attac of the fow aganst the hydrofo n order to dea wth dfferent confguratons. In ths case, the same bocng structure can be empoyed, and by rotatng the bocs adacent to the hydrofo, the grds can be updated to the current ange. Fgure 4: Coarse mesh Once the meshes were generated, steady state smuatons were carred out. However, some confguratons appeared to be transent, specfcay those wth ower cavtaton number or arger ange of attac. In these cases the cavtaton bubbes become oscatory or are even partay removed from the hydrofo surface by the ncdent fud fow. Thus, transent smuatons had to be carred out for these confguratons. The ANSYS CFX setup then must be updated ntroducng an arthmetca averagng procedure to be apped to the man fow varabes, whch orgnates an average pressure, average veocty and average voume fracton fed to be compared to the expermenta data. The descrbed oscatory fow behavor can be observed n Fgure 5, where a whoe transent cyce s shown for a confguraton of α=4, and σ=0.5. Top Inet Out Fgure 3: ocng structure ott The man characterstcs of the grds created for the numerca smuatons are summarzed n Tabe, and a representaton of one of the coarse meshes nvoved n the cacuatons s shown n Fgure 4. Grd Coarse Medum Fne # nodes 56,45 4,64 893,986 # eements 7,840, ,440 Mnmum grd ange Frst ayer dstance y [µm] Average y + 4 Tabe : Grd characterstcs Fgure 5; Transent cyce of an oscatng cavtaton regon on upper sde of the hydrofo for α=4, and σ= Cavtaton cavty ength In order to examne cavtaton for the dfferent confguratons the ength of the cavtaton zone attached to the upper sde of the hydrofo s measured. An nvestgaton of the nfuence of both the cavtaton number and the ange of attac was performed. It s observed n Fgure 6 that the arger the cavtaton number s, the ower cavtaton ength s obtaned. In addton, the mpact of the ange of attac can be seen. The arger the ange of attac s, the arger becomes the cavtaton zone and ts ength. 6

7 ICMF 007, Lepzg, Germany, Juy 9 3, 007 vapour voume fracton fed but aso by anayzng the pressure vaues and comparng to drect pressure measurements at specfc ocatons on the hydrofo surface. A pressure coeffcent can be defned as Fgure 6: Cavtaton ength vs. cavtaton number for dfferent anges of attac Two representatve resuts of the computed seres out of test case condtons are shown n Fgure 7 and Fgure 8, correspondng to the vapour voume fracton for an ange of attac of α=0 at a cavtaton number of σ=0.4 and α=4 wth σ=0.5, respectvey. For the frst case, sma cavtatng areas appear on both upper and ower sde, whe for the second case ony one arger cavtaton bubbe appears to be attached to the upper sde of the hydrofo. c p p stat ρu = ( 3 ) In Fgure 9 to Fgure, the pressure coeffcent obtaned wth medum grd smuatons s potted aganst the expermenta resuts. They correspond to dfferent anges of attac (α=.5, 3.5, 4. and 5. respectvey), whe the cavtaton number s 0.55 for the frst two cases and 0.8 for Fgures and. At the zone where the pressure coeffcent s ower than the cavtaton number, evaporaton s occurrng. It can be notced by comparng Fgure 9 and Fgure 0 that the ength of the vapour bubbe attached to the upper sde of the hydrofo s arger for the case of α=3.5 as expected. The same effect can be seen n Fgure and Fgure. Nevertheess both predcted cavtaton bubbe engths are shorter snce the cavtaton number s arger. Comparng the dfferent curves to the expermenta vaues reasonabe agreement n shape s observed, specfcay for the frst three confguratons whe for the arger ange of attac at α=5. dscrepances appear. Fgure 9: Pressure coeffcent, α=.5, σ=0.55 Fgure 7: Vapor voume fracton. α=0, σ=0.4. Fgure 8: Vapor voume fracton. α=0.4, σ=0.5. Fgure 0: Pressure coeffcent, α=3.5, σ= Pressure coeffcent data The cavtaton arses when the pressure drops beow the saturaton pressure. Ths can be detected not ony by the 7

8 ICMF 007, Lepzg, Germany, Juy 9 3, Lft coeffcent Fgure : Pressure coeffcent, α=4., σ=0.8 Goba vaues for the dfferent confguratons were aso nvestgated and compared to data. Ths s the case for the ft coeffcent, defned as: c L = ρ F L u Abade ( 3 ) where F L s the ft force, A bade s the area of the hydrofo and u s the veocty far downstream the hydrofo. Fgure 4 shows the vaue of the ft coeffcent for dfferent anges of attac as we as for dfferent cavtaton numbers. Under non-cavtatng condtons the reatonshp between ft and ange of attac s amost near, however ths behavor s dramatcay modfed when the cavtaton number s decreased and cavtaton appears. 0,04 0,0 ft coeffcent ,0-0,04-0,06-0,08 cav 0.3 cav 0.6 nocav -0, -0, Fgure : Pressure coeffcent, α=5., σ=0.8 Pressure coeffcent can be further used to evauate the nfuence of turbuent pressure fuctuatons on cavtaton n accordance wth equaton (7) and (8). In Fgure 3 the c p curves for three dfferent modeng approaches can be compared. The dagram shows resuts from a smuaton usng the orgna Rayegh Pesset equaton, a smuaton usng the modfcaton to the Rayegh-Pesset equaton descrbed n equaton (6), and fnay a smuaton usng the modfcaton to the Rayegh-Pesset equaton descrbed n equaton (8). As mentoned n secton., the ast expresson eads to more reastc resuts, aso observabe dfferences are not very pronounced for ths partcuar test case. -0,4 ange [deg] Fgure 4: Lft coeffcent vs. ange of attac for dfferent cavtaton number. 3.. The Arndt et a. Test Case Tp Vortex Induced Cavtaton 3... Testcase defnton p ɶ = 0.0 pɶ = 0.39ρ pɶ = ( r )0.39ρ v Fgure 5: Schematc representaton of the NACA cavtaton channe setup Fgure 3; Pressure coeffcent n dependency on the modeng approach for the turbuent pressure fuctuaton term. α=3.5, σ=0.55 In addton to the pano-convex cavtaton test, a three dmensona case consstng of a fow around a NACA hydrofo wth eptca panform was nvestgated. In ths case tp-vortex cavtaton taes pace due to the hgh rada veocty gradents n the vortex tube, whch s reeased from the tp of the hydrofo. Hghy swrng fow generates 8

9 ICMF 007, Lepzg, Germany, Juy 9 3, 007 pressure drop beow saturaton pressure eadng to cavtaton on the tp of the hydrofo and n the vortex core of the tp-vortex. The test body used n the orgna facty [][8] conssts of an eptca panform hydrofo wth a chord ength of 8mm, a semspan of 95mm and a mean ne of 0.8. Fgure 5 shows the representaton of the expermenta fow geometry whch was exacty used for the CFD smuatons as we, whe n Fgure 6 and Fgure 7 the detas of the panform geometry of the hydrofo are ponted out. z/c0 NACA x/c + α 0 Fgure 6: Eptca profe of the NACA Fgure 8: ocng structure around the hydrofo The resutng bocng structure apped near the hydrofo s shown n Fgure 8, whe the coarser mesh obtaned wth ths boc structure s presented n Fgure 9. The desgned grd boc structure guarantees a mnmum grd ange arger then 0 ndependent from the grd refnement eve. As for the prevous case an h-refnement study has been carred out, empoyng three dfferent grds, whch are refned by a factor of 3 4 n each coordnate drecton. The same parameters were taen nto account to evauate the quaty of the mesh: mnmum ange formed by the grd nes, aspect ratos and the near wa dstance of the frst mesh eement (computed as n equaton (30)). The man nformaton reated to the grd propertes and grd quaty on varous mesh eves of refnement used to run the CFD smuatons s summarzed n Tabe. Fgure 7: Eptca profe of the NACA As for the prevous case dfferent confguratons were anayzed by changng the ange of attac, the Reynods number charactersng the fow and appyng dfferent turbuence modeng approaches (SST, SST wth curvature correcton term, SL RSM). In accordance wth the orgna pubcaton of Arndt an α = α α, effectve ange of attac has been defned as eff 0 where α 0 corresponds to the zero ft ange, whch after a parametrc study was chosen as α 0 = Mesh herarchy and CFD setup The ICEM CFD Hexa grd generator has been used to dscretze the doman. A boc structure aowng to refne the grd near the bade surface as we as to perform a smooth transton between coarsey resoved areas n the far fed and fney resoved areas around the hydrofo was desgned. Fgure 9: Representaton of the meshes empoyed. Grd Coarse Medum Fne # nodes # eements Mnmum grd ange Frst ayer dstance y [µm] Tabe : Grd characterstcs Tp vortex traectory Average y Frst the shape of the tp vortex traectory has been nvestgated. It coud be shown that the traectory does not strongy depend ether on the ange of attac, the Reynods number vaue or the cavtaton number. Ths effect can be observed n Fgure 0, 9

10 ICMF 007, Lepzg, Germany, Juy 9 3, 007 where the tp vortex traectory obtaned for an ange of attac equa to 8. and Reynods number of 9.x0 5 s potted as we as for the case of α=.6 and Re=5.x0 5 aganst the expermenta vaues of Arndt. severe dscrepances to the expermenta resuts arose, especay on measurement cross secton further downstream the hydrofo where the meshes are coarsenng due to axa expanson. Whe the strong veocty gradents can be predcted for the cross secton cose to the hydrofo at x/c o =0.06, the potted veocty profes are much smoother then the expermenta data obtaned from the expermenta facty for x/c o =.0 (see Fgure 3). Fgure 0: Tp vortex traectores n the x-y coordnate pane Resouton of crcumferenta veoctes n the tp vortex In order to evauate the quaty of the obtaned numerca resuts, the rada veocty profe at dfferent ocatons has been evauated. These postons are ocated near the tp of the hydrofo and a steep veocty gradent can be observed. Further downstream dsspaton of the tp vortex, a reducton n crcumferenta veocty amptude as we as n veocty gradent can be observed as the poston s departng from the tp. It can be ceary observed n Fgure, where the veocty profe at the poston ayng haf chord ength behnd the hydrofo tp s substantay steeper than the profes ocated at a chord ength dstance or twce chord ength dstance. Fgure : Rada veocty profes for dfferent grds cose to the tp of the hydrofo. Veocty w/u _ Experment x/c0=.0 Coarse() x/c0=.0 Medum() x/c0=.0 Fne(3) x/c0=.0 Fgure 3: Rada veocty profes for dfferent grds at a chord ength dstance from the tp. Fgure : Rada veocty profe at three dfferent ocatons after the tp vortex for x/c o =0.5, and. The grd refnement aows to anayze the spata dscretzaton error of the numerca method and to evauate f an asymptotca souton ndependent of the grd resouton can be fnay obtaned. For ths purpose, the rada veocty profe was evauated usng the three refned grds n dfferent ocatons (Fgure and Fgure ). Sma dfferences between the resuts can be observed even on the hghest eve of mesh refnement, ndcatng that a mesh ndependent souton coud not yet be obtaned. However, even more A reason for ths behavour s the strong swr of the veocty fed near the tp of the hydrofo. In order to dea wth ths effect, dfferent strateges have been consdered. The frst one conssted of the use of a Hgh Resouton Scheme to sove the turbuence equatons, whch are soved by defaut usng an upwnd advecton scheme, whch s of cause more dffusve. ut the nfuence of the chosen advecton scheme, shown n Fgure 4, was found to be not sgnfcant. In a second step a curvature correcton term n the SST turbuence mode had been apped (see secton.), n order to account for the strong curvature of streamnes n the tp-vortex fow. The veocty profes obtaned wth ths curvature correcton s aso compared n Fgure 4, showng an mportant mprovement to approxmate the strong veocty gradent. 0

11 ICMF 007, Lepzg, Germany, Juy 9 3, 007 a) Fgure 4: Rada veocty profe wth dfferent numerca schemes for sovng/modeng the fud fow turbuence. b) A further step was done n order to enhance the evauaton of the veocty gradent near the tp vortex by rasng the mtaton of assumed sotropc turbuence, whch mght be not satsfed n the strong swrng fow of the tp vortex behnd the hydrofo. Therefore the turbuence mode was changed from a two-equaton mode (secton.) to the SL Reynods Stress Mode (secton.), where not two turbuence mode equatons but one equaton for each Reynods tensor component s soved. In ths case, the computer and memory resources requred has been ncreased, but anayzng Fgure 5, t can be notced that even for coarser meshes the enhancement s sgnfcant approachng n a more satsfactory comparson of the steep veocty profe to measurement data. SST SST (Fne) SST Hgh Res (Fne) SST Hgh Res CC (Fne) RSM (Medum) RSM (Coarse) Experment c) Veocty w/u Fgure 5: Rada veocty profe for dfferent turbuence modeng The nfuence of the turbuence mode can aso be observed by oong nto the vapour voume fracton obtaned n an ANSYS CFX mutphase fow smuaton appyng the cavtaton mode n combnaton wth SST and SL RSM turbuence modes. A arger tp vortex cavtaton zone appears when the SL Reynods Stress Mode s apped. Sheet cavtaton s coverng the most of the bade surface for both confguratons (Fgure 6) RSM Fgure 6: Vapour voume fracton n cavtatng fow near the tp. Re=5.x0 5. σ= (a) expermenta observaton α eff =9.5, (b) SST turbuence mode α eff =. (c) SL Reynods Stress Mode α eff =

12 ICMF 007, Lepzg, Germany, Juy 9 3, Lft coeffcent In addton to the tp vortex traectory and the veocty profes the vaue of the ft coeffcent (Equaton 3) has been nvestgated. Fgure 7 shows the nfuence of the ange of attac on the ft coeffcent. It has been computed for dfferent Reynods numbers and by usng dfferent grds, however a the computatona resuts are fnay arrangng between the two expermenta resuts at Obernach [7] and SAFL [8]. Fgure 7: Lft coeffcent vs ange of attac The reatonshp between the cavtaton ncepton, the Reynods number and the ft coeffcent has been consdered as we. A correaton can be found n terature for the dependency of these three parameters, whch s σ = c Re ( 33 ) 0.4 Resuts obtaned wth the three refned grds are compared to the expermenta ones, and regressons of the numerca soutons obtaned are computed (to compare ts sope to the one n equaton 33). Fgure 8 shows that the sope of the regresson curves obtaned are ower than the expermenta resuts for the coarse grd, whe t ncreases for the medum grd resuts. Fnay the ony resut whch coud be obtaned on the fnest grd eve due to the nvoved hgh computatona effort s n very good agreement to the expermenta resuts. Concusons A cavtaton mode n ANSYS CFX has been deveoped. It s based on a homogeneous mutphase fow approach and on modeng of the bubbe dynamcs sovng the Rayegh-Pesset equaton for cavtaton bubbe radus. The mode has been combned wth dfferent turbuence modes for the contnuous fud phase. Turbuent pressure fuctuatons and ther nfuence on the cavtaton phenomena were taen nto account by reatng them to the turbuent netc energy of the contnuous phase. A vadaton of the mode has been performed anayzng two dfferent test cases avaabe from terature and comparng resuts of the CFD smuatons obtaned wth ANSYS CFX to expermenta data. The frst test case s based on the experments made by [3]. In ths test case the fow passes around a pano-convex hydrofo, and cavtaton couds on both sdes can be observed. Three refned grds have been used for the smuaton, ensurng comparabe mesh quaty on a grd eves. The cavtaton engths, pressure coeffcents and ft vaues have been nvestgated and compared aganst the terature vaues. The numerca resuts agree reasonaby we to the experments, even the necessty to use even fner grds coud be shown from the present vadaton study. The second test case s based on the experments by Arndt []. Speca attenton has been pad to the tp vortex, snce ths s the zone of the fow where arger veocty gradents appear as we as arger pressure drop occur, orgnatng the ncepton of the tp-vortex cavtaton. The traectory of the tp vortex and the resouton of the rada veoctes n the tp vortex have been nvestgated and compared to data. The veocty gradents were found to be dffcut to compute and dfferent strateges have been nvestgated. The basc smuatons were run appyng the standard SST turbuence mode wthout any modfcatons, and t has been observed that the use of hgh order resouton schemes and the use of a curvature correcton term mproved the resouton of the steep veocty gradent near the tp of the hydrofo. In addton, a Reynods Stress Mode has been apped showng a more satsfactory agreement to the numerca resuts even on coarser grds by tang nto account the unsotropy of the contnuous phase turbuence n the strong swrng fow n the tp vortex behnd the tp of the hydrofo.. Acnowedgements Cavtaton number σ Experment Coarse () Medum () Fne (3) Regresson, Coarse Regresson Medum Fgure 8: Cavtaton ncepton vs. ft coeffcent. Presented nvestgatons have been supported by the German Mnstry of Educaton and Research (MF) under grant number 03SX0A. References [] ANSYS Inc., ANSYS CFX.0 Users Manua (006). [] Arndt, R.E.A., Dugue, C., Recent Advances n Tp Vortex Cavtaton Research, Proc.The Internatona Symposum on Propusors and Cavtaton, Hamburg, Deutschand,.-5. Jun, 99. [3] Franc J.P., Parta Cavty Instabtes and Re-Entrant Jet,

13 ICMF 007, Lepzg, Germany, Juy 9 3, 007 Keynote Lecture 00, Proc. 4th Internatona Symposum on Cavtaton, Pasadena, Kafornen, U.S.A., June 00. [4] Jebauer S., Numersche Smuaton avterender Strömungen, Dpoma Thess, TU Dresden, pp. -79 (006). [5] Le Q., Franc J.P. und Mche J.M., Parta Cavtes: Goba ehavor and Mean Pressure Dstrbuton, Journa of Fuds Engneerng, Vo. 5-, S [6] Menter F., Two-Equaton Eddy-Vscosty Turbuence Modes for Engneerng Appcatons, AIAA Journa, Vo. 3, No. 8, pp (994). [7] Menter F., CFD est Practce Gudenes for CFD Code Vadaton for Reactor Safety Appcatons, ECORA Proect, pp. -47 (00). [8] Arndt, R.E.A. and Araer, V.H., Hguch, H., 99, Some observatons of tp-vortex cavtaton, J. Fud Mechancs, Vo. 9, pp [9] Manes,.H. and Arndt, R.E.A., 997, Tp Vortex Formaton and Cavtaton, J. Fuds Eng., Vo. 9-, pp [9] Menter, F.R., Rumsey, C.L., Assesment of Two-Equaton Turbuence Modes for Transonc Fows, AIAA , Proc. 5th Fud Dynamcs Conference, Coorado Sprngs, Coorado, U.S.A., Jun 994 [0] Wcox, D.C., 988, Reassesment of the Scae-Determnng Equaton for Advanced Turbuence Modes, AIAA J., Vo. 6, S [] Wcox, D.C., 000, Turbuence Modeng for CFD, DCW Industres [] Jones, W. P.; Launder,. E. The Cacuaton of Low-Reynods- Number Phenomena wth a Two-Equaton Mode of Turbuence. Internatona Journa for Heat and Mass Transfer 973,6, 9. [3] Spaart, P.R., Shur, M.L., 997, On the senstzaton of turbuence modes to rotaton and curvature, Aerospace Scence and Technoogy, Vo. -5, S [0] Abbot, I.H. and Doenhoff, A.E. von, 959, Theory of Wng Sectons, Dover [] Manes,.H., 995, Tp Vortex Formaton and Cavtaton, Dssertaton, Unversty of Mnnesota, U.S.A. []., Taacs, T., Wemsen, S., 00, CFD est Practce Gudenes for CFD Code Vadaton for Reactor-Safety Appcatons, European Commson, ECORA [3] Casey, M., Wntergerste, T., 000, est Practce Gudenes, ERCOFTAC Speca Interest Group on Quaty and Trust n Industra CFD, Fud Dynamcs Laboratory Suzer Innotec, 94 p. [4] Wcox, D.C., Mutscae mode for turbuent fows, In AIAA 4th Aerospace Scences Meetng. Amercan Insttute of Aeronautcs and Astronautcs, 986. [5] Menter, F.R., Mutscae mode for turbuent fows, In 4th Fud Dynamcs Conference. Amercan Insttute of Aeronautcs and Astronautcs, 993. [6] Menter, F.R., and Egorov, Y. A Scae-Adaptve Smuaton Mode usng Two-Equaton Modes, AIAA paper , Reno/NV, 005. [7] Kedsen, M. Arndt, R.E.A., Effertzt, M. Spectra Characterstcs of Sheet/Coud Cavtaton. Journa of Fuds Engneerng. Vo pp [8] Menter, F., Hemstrom,., Henrsson, M., Karsson, R., Latrobe, A., Martn, A., Muhbauer, P., Scheuerer, M., Smth,., Taacs, T., Wemsen, S., 00, CFD est Practce Gudenes for CFD Code Vadaton for Reactor-Safety Appcatons, European Comsson. ECORA, 3