Fourth Internatonal Symposum on Marne Propulsors smp 15, Austn, Texas, USA, June 2015 Workshop: Propeller Performance Computatonal Analyss of Cavtatng Marne Propeller Performance usng OpenFOAM Abolfazl Asnagh, Andreas Feymark, Rckard E. Bensow Department of Shppng and Marne Technology, Chalmers Unversty of Technology, Gothenburg, Sweden ABSTRACT In ths paper, numercal results, based on mplct LES and transport equaton mass transfer modellng approach, submtted to the Second Internatonal Workshop on Cavtatng Propeller Performance at the Fourth Internatonal Symposum on Marne Propulsors (SMP 15) are presented. The numercal smulatons are performed usng OpenFOAM. The present work s focused on the second task of the workshop, Propeller n oblque flow nsde tunnel n wetted and cavtatng condtons. We summarse results from the three dfferent operatng condtons gven n the task, where the expermental data of one condton s revealed, and the two other condtons are put forward as blnd tests to workshop partcpants. For the condton where expermental data s known, we see good agreement for the forces n wetted condtons that slghtly deterorate n cavtatng condtons. Cavtaton extent s over predcted, where md-span bubbly cavtaton n the experments s predcted as a sheet cavty; cavtaton n the tp regon does however seem reasonable. Ths s also lkely the reason for the larger error n force predcton. A lmted mesh resoluton study has been performed. Keywords Cavtaton, Numercal Smulaton, OpenFOAM. 1 INTRODUCTION A controllable ptch propeller wth extensve expermental data s provded by SVA Potsdam to be consdered for SMP 15 Workshop on Cavtaton and Propeller Performance. The current work covers the performance predcton of the propeller n the cavtatng condtons (Case 2) n three dfferent operatng condtons. The workshop s organzed n a blnd test format where none of the partcpants knows the experment results pror to the workshop. The current numercal smulatons have been conducted usng a modfed nterphasechangedymfoam solver of OpenFOAM (OpenFOAM foundaton). Implct LES approach s adopted to account for turbulence (Bensow and Bark, 2010). The Schnerr-Sauer mass transfer model s employed to mmc the phase change between vapour and lqud (Schnerr and Sauer, 2001). The presented results consst of smulatons of three dfferent operatng condtons for wetted and cavtatng flows, denoted as Case2.1, 2.2, and 2.3, see Table 1. For each of the operatng condtons, the thrust and torque coeffcents are presented. The effects of the mesh resoluton, the nlet locaton and also the tme dscretzaton scheme on the numercal smulaton are also nvestgated n ths work and reported for Case2.1. The cavtaton pattern at dfferent blade postons for sucton sde and pressure sde are presented. For Case2.1 that the expermental data are avalable the numercal results are analysed and compared wth experment. 2 Governng Equatons In the current study, the effectve flow and each phase have been consdered ncompressble and sothermal whch s a common engneerng approach for cavtatng flows. A mxture assumpton s adopted usng Transport Equaton Modellng of volume fracton (TEM) for the phase dstrbuton. 2.1 Mass and momentum equatons The conservaton equatons of mass and momentum for the effectve flud can be wrtten as follow, ( ρ u ) ρ m m + = 0 t x ( ρ u ) ( ρ u u ) m m j τ + = + ρmg (2) t x j x j The stress tensor n Newtonan fluds s conventonally wrtten n the form of summaton of pressure stress and shear stresses as follow. τ = pδ + S 2 u µ 3 x m m δ (1) (3)
S = 2µ (4) D In these equatons, p s the statc pressure, μ s the effectve vscosty and S s the vscous stress tensor and D s the deformaton rate tensor (symmetrc part of the velocty stran), defned as D = ( u x + u x ) 2. j j 2.2 Turbulence modellng For turbulence modellng, mplct Large Eddy Smulaton approach, ILES, s used. Ths turbulence model has been used and proved effectve prevously by the authors for smulaton of the cavtatng flows (Bensow and Bark, 2010), as well as for wetted flows. Usng the low pass flterng approach, the momentum equaton n LES model can be wrtten as Eq. (5). ( ρm ) ( ρm j ) u u u p + = + ( S B ) + ρmg t x x x j j The over bar denotes the low pass flterng quanttes. In ths equaton, the subgrd stress tensor s B =ρ u u u u. ( ) j j In ILES approach no explct model s appled for B, nstead the numercal dsspaton s consdered enough to mmc the acton of B (Bensow and Bark, 2010). Therefore, for momentum convecton term, a relatvely dsspatve scheme should be used to provde approprate numercal dffuson n the soluton procedure. 2.3 Multphase modellng TEM, Transport Equaton Model, s used n ths study to model the transport of phases. In the TEM approach the spatal dstrbuton of each phase s specfed usng a volume fracton functon. Usng the volume fracton, t s possble to calculate the mxture densty and mxture vscosty based on the homogenous flow assumpton. ρ = α ρ + (1 α ) ρ, µ = α µ + (1 α ) µ (6) m l l l v m l l l v α t ( α u ) l l + = x m& ρ l In Eq. (7), whch represents the transport equaton of lqud volume fracton, the source term s the rate of phase change between vapour and lqud phases. Based on the flud propertes and also the local flow propertes, dfferent models have been proposed to predct the cavtaton phase change rate, m& n the Eq. (7). In the current study, the mass transfer model proposed by Schnerr and Sauer s employed, Eq. (8), where average nucleus per lqud volume s consdered constant and n ths study equal 8 to n 0 = 10, and the ntal nucle radus s d = 10 4 m. n 2 2 Psat P 0 & = sgn( Psat P) 4πR 4 (8) 3 3ρ 1+ n πr l 0 m 3 Nuc (5) (7) Usng the saturaton pressure as the pressure threshold for phase change n the cavtatng flows s based upon the lqud rupturng at the statc or quas-statc condtons. In these condtons, the statc pressure n the major part of the lqud s much larger than the vscous shear stresses. Although ths estmaton, usng the saturaton pressure as the pressure threshold, has been used wdely n numercal smulaton of cavtaton, t does not take nto account the effects of the shear stresses n the lqud rupturng and ntatng phase change. In order to consder the vscous stresses, the egenvalue of the stress tensor should be consdered as the crtera on whether the flud wthstands rupturng or phase change. Here the modfcaton proposed n Asnagh et al. (2014) has been employed. p = µ γ& + (9) threshold p saturaton γ& = 2D D (10) The added term s mportant f ether shear stran rate or effectve vscosty s large enough, and comparable wth the statc pressure value. For the flow around the fols, ths s the case near the leadng edge or durng the collapse when the velocty varaton s very hgh, and for the flow around the propellers ths s the case both at the tp and leadng edge regons. 2.4 Non-dmensonal parameters The defnton of advance coeffcent (J), cavtaton number (σ n ), thrust coeffcent (K T ) and torque coeffcent (K Q ) are as follow, A ( n D p ) 2 2 ( p psat ) ( 0.5 n Dp ) 2 4 Tx ( n Dp ) 2 5 Q ( n ) J = V (11) σ = ρ (12) K K n Tx = ρ (13) = ρ (14) Q D p where n these equatons, V A s the advance velocty (.e. n ths case the nlet velocty), n s the rotatonal speed of the propeller, p s the tunnel outlet pressure, T s the propeller thrust, and Q s the propeller torque. 3 Soluton Procedure and Dscretzaton In order to solve the governng equatons, OpenFOAM- 2.3.x whch s an open source CFD software package developed by OpenCFD Ltd at ESI Group and dstrbuted by the OpenFOAM Foundaton s used. In ths software, the spatal dscretzaton s performed usng a cell centered colocated fnte volume (FV) method for unstructured meshes wth arbtrary cell-shapes, and a mult-step scheme s used for the tme dervatves. The nterphasechangedymfoam solver s employed to smulate the cavtaton. The mplct LES model s mplemented nto the orgnal code and n order to reduce the mesh resoluton requrement for LES smulatons near the walls, the Spaldng wall model s employed to correct the turbulent vscosty at the frst cell.
The PIMPLE algorthm s used to solve the couplng between the velocty and pressure. The resdual of solvng pressure and velocty n each teraton s set equal to 1e-6 for wetted flow and 1e-12 for cavtaton smulaton. A second order mplct tme scheme (backward scheme) s used for tme dscretzaton. For one condton, the effects of usng frst order Euler scheme, often suggested suffcent when usng small tme steps, s also nvestgated and dscussed. A blendng scheme of frst order upwnd and second order central dfference schemes s used for the convectve term. The constant of ths blendng s set equal to 0.2. All of the s have been corrected to consder non-orthogonalty effects of neghbourng cells. For the volume fracton transport equaton, frst order upwnd scheme s utlzed. In order to handle the rotaton of the propeller, the computatonal doman s decomposed n two regons, the rotatng regon close to the propeller and the outer regon, coupled va the standard sldng mesh mplementaton n OpenFOAM. The data across the regons are nterpolated through the AMI boundares n OpenFOAM. Fgure 1: Propeller geometry Flow drecton 4 Test Condtons The propeller geometry and three dfferent operatng condtons are provded by SMP 15 workshop organsers. The propeller s a model scale, fve bladed propeller wth a dameter equal to 250 mm, Fg. 1. The cavtaton tests were conducted n the cavtaton tunnel K 15 A of the SVA Potsdam. Durng testng the propeller was postoned accordng to the Fg. 2 wth a 12 nclnaton of the propeller towards the nflow drecton. In Table 1, the operatng condtons are brefly presented. Table 1: Operatng condtons Case J σ n V nlet (m/s) n (rev/sec) 2.1 1.019 2.024 5.095 20 2.2 1.269 1.424 6.345 20 2.3 1.408 2.0 7.04 20 5 RESULTS AND DISCUSSION The numercal results consst of propeller performance (.e. thrust and torque coeffcents) predctons n the wetted and cavtatng flows for three dfferent operatng condtons, descrbed n Table 1. For the cavtatng flows, the cavtaton pattern at dfferent blade postons are also plotted and nvestgated. The angular postons of the blades are descrbed accordng to the rght-handed rotaton of the propeller wth zero degree beng equvalent to the 12 o clock poston. Fgure 2: Test secton Snce the expermental data are provded just for Case2.1, the numercal results are compared wth the expermental data just for ths case. In order to elaborate the study, for the operatng condton Case2.1, effects of usng frst order Euler tme scheme, dstance of the nlet boundary locaton relatve to the propeller, and also mesh resoluton are nvestgated and results are presented and compared wth expermental data. 5.1 Boundary condtons A summary of the numercal boundary setup s presented n Table 2. In order to reduce the requrement of mesh resoluton near the tunnel wall, slp boundary condton s appled for the tunnel wall. The unform nlet velocty and unform outlet pressure are adopted to adjust the flow advance rato and cavtaton number. 5.2 Mesh specfcatons The blades surface mesh conssts of quad surfaces, whch then have been extruded n the wall normal drecton (y + =10) to create prsm cells (hexahedrals) n order to better capture the boundary layer over the blades. The rest of the doman s flled wth unstructured tetrahedral cells. Snce the flow has hgher s near the leadng and tralng edges and also near the tp regon of the blades, the mesh has fner resoluton at these areas. In order to lmt the mesh sze n a reasonable range, the mesh gets coarser by ncreasng dstance from the blades.
Table 2: Numercal boundary setup Boundary Velocty Pressure nu Sgs Vapour (α) Inlet Outlet Propeller surfaces Tunnel wall Fxed No-slp Slp Fxed Wall model Fxed In order to handle the rotaton of the propeller, the computatonal doman s decomposed n two regons, the rotatng regon close to the propeller, and the statonary regon where the total sze of the mesh s around 4.7 M cells, called MeshI n ths paper. For ths mesh, the doman sze has been kept the same as the geometry provded by the workshop commttee. In the provded geometry of the tunnel, the nlet s located almost n 2D upstream of the propeller. Snce the nlet s relatvely close to the propeller, t s possble that usng unform nflow as nlet velocty boundary condton affects the flow around the propeller (e.g. pressure dstrbuton and cavtaton pattern). Therefore, another mesh s also created where the nlet s moved 4D further upstream, MeshII n Fg. 3. In order to nvestgate the effects of the mesh resoluton on the results, MeshIII s created from MeshII where the prsm cells around one blade are refned usng refnemesh command n OpenFOAM. Ths command splts a hex cell nto 2 cells n each drecton. Therefore, the fnal cells are 8 tmes smaller than the orgnal one. The fnal total cell sze for ths mesh s around 8.5 M cells. The blades surface mesh s presented n Fg. 4. MeshII Fgure 3: The nlet locatons for MeshI and MeshII MeshI Fgure 4: The blade surface mesh for MeshIII 5.3 Wetted flow results In Table 3, the thrust and torque coeffcents for the three dfferent operatng condtons are presented. Comparson between the expermental data and numercal results for Case2.1 shows that the obtaned results have a good agreement wth the experment. Table 3: Thrust and torque coeffcents for wetted flow smulatons Operatng condtons Case2.1 Refned blade Mesh Tme scheme Base blade Method K Tx 10K Q ------ Exp 0.397 1.02 MeshI backward ILES 0.405 1.01 MeshI Euler ILES 0.408 1.01 MeshII backward ILES 0.404 1.00 MeshIII backward ILES 0.406 1.01 Case2.2 MeshI backward ILES 0.262 0.72 Case2.3 MeshI backward ILES 0.181 0.55 5.4 Cavtatng flow results In Table 4, the thrust and torque coeffcents for the three dfferent operatng condtons of cavtatng flow are presented. For Case2.1, where the expermental data are avalable, comparson between numercal results and expermental data reveals that the comparson error s around 8% for K Q and 4% for K Tx predcton usng backward scheme. However, the results related to Euler scheme show a severe over predcton of K Q by 35%. 5.4.1 Case2.1 In Fg. 5, cavtaton pattern for two so-surfaces of alpha (40% and 60%) are presented for sucton and pressure sdes of the propeller. These results are related to the MeshI wth backward tme scheme. Note that we do not see any pressure sde cavtaton, but the mage only reveals the extended sheet of the sucton sde.
Table 4: Thrust and torque coeffcents for cavtatng flow smulatons Operatng condtons Case2.1 Mesh Tme scheme Method K Tx 10K Q Exp 0.36 0.94 MeshI backward ILES 0.373 1.07 MeshI Euler ILES 0.351 1.34 MeshII backward ILES 0.374 1.05 MeshIII backward ILES 0.375 1.04 Case2.2 MeshI backward ILES 0.196 0.73 Case2.3 MeshI backward ILES 0.157 0.53 (a) Pressure sde, alpha vapour 0.6 (b) Pressure sde, alpha vapour 0.4 The lnes on the surface of the blade represent the radus rato, r/r, where R s the propeller radus and r s the dstance from the centre of the propeller n the cylndrcal coordnate system algned wth the propeller shaft, Fg. 6. In Fg.7, the cavtaton predcton for dfferent settngs and mesh resolutons are presented for Case2.1 where the vapour so-surface s 60%. For MeshIII, the pcture s modfed n a way that each blade poston s replaced wth the correspondng results of the blade havng the refned mesh. Therefore, the pcture somehow represents the results for an magnary fully refned propeller. Comparson between results of Fg. 7-c and Fg. 7-d wll reveal the effects of mesh resoluton on the cavtaton predcton. From the results t can be deduced that the fner mesh s more capable of capturng and preservng the vortex rolled up nto the blade tp regon; note that the only the regon around the blade s refned and not when the vortex has left the blade. From blade postons zero degree, t can be seen that fner mesh resoluton was able to preserve the tp vortex cavtaton longer, tll the end of blade tp whle n the coarser mesh the tp vortex cavtaton s ended before reachng the blade tp. From the blade poston 72 degree, t can be seen that n the fner mesh the vortex s rolled up earler nto blade tp regon, and also from the blade poston 216 degree, t can be seen that the preserved cavty s bgger than n the coarser mesh. These three man effects are hghlghted by yellow ovals n the fgures. We remark that the mesh refnement does not affect the over predcted md rad sheet. (c) Sucton sde, alpha vapour 0.6 (d) Sucton sde, alpha vapour 0.4 Fgure 5: Case2.1, vew along x-axs 0.8 0.9 0.95 (a) MeshI, backward scheme (b) MeshI, Euler scheme 07 0.6 0.5 0.4 (c) MeshII, backward scheme (d) MeshIII, backward scheme Fgure 6: Descrpton of radus rato over the blade surface, vew along x-axs Fgure 7: Case2.1, vew along x-axs, Sucton sde, vapour sosurface 0.6
(a) Blade poston zero degree (a) Blade poston zero degree (b) Blade poston 90 degree (b) Blade poston 90 degree (c) Blade poston 180 degree (c) Blade poston 180 degree (d) Blade poston 270 degree Fgure 8: Comparson between numercal results and expermental sketches for cavtaton n Case2.1, vew along x-axs, sucton sde, numercal results: MeshIII, vapour so-surface: 0.6 In Fgs. 8 and 9, the cavtaton predctons are compared wth the expermental sketches for Case2.1 for the sucton sde and pressure sde at dfferent blade postons. As t s shown n Fg. 8, the general trend of the cavtaton has been predcted reasonably well. The man dfference between numercal results and the expermental data s related to the regon wth the bubbly cavtaton pattern. In Fg.8-a, the bubbly root cavtaton s predcted as sheet cavty, and n Fg.8-b the bubbly cavtaton near the leadng edge s predcted wth the sheet leadng edge cavtaton. Ths sheet cavty then s attached to the near tp sheet cavty (radus 0.9) and covers almost all of the sucton sde of the blade. The type of bubble cavtaton n the experments ndcates a blade pressure close to, or even below, vapour pressure. The modellng used here can not accommodate the growth of ndvdual nucle to ths type of bubble cavtaton, nstead leadng to ths formaton of a sheet over the leadng half of the blade. The pressure sde of the blade experences root cavtaton at blade postons of zero and 270 degrees durng the experment. The numercal smulaton under predcts root cavtaton at zero degree poston, and 270 degree blade poston. (d) Blade poston 270 degree Fgure 9: Comparson between numercal results and expermental sketches for cavtaton n Case2.1, vew along x-axs, pressure sde, numercal results: MeshIII, Vapour Iso-surface: 0.6 In Fg. 10, the pressure coeffcent of the wetted flow and also the vapour so-surface 60% are presented for Case2.1. The pressure coeffcent values, Fg. 10-a, are adjusted to show the values below C p = -2, whch represent regons wth pressure lower than the saturaton pressure. As t s dscussed before, the man dscrepancy between numercal predcton of cavtaton extent and the expermental observatons s related to the predcton of leadng edge sheet cavtaton, e.g. at the blade postons 72 and 144 degrees n Fg. 10. In the leadng edge regons where the numercal predcton show pressure lower than the saturaton pressure, the computatonal model wll start to produce vapour. In the experments, the formaton of a sheet cavty depends as well on the nucle content and nucle resdence tme n the low pressure regon. Ths s a modellng dscrepancy between the numercal and expermental procedures. Bubble cavtaton s observed n the experment to ncept from the leadng edge at these postons whch suggests a blade pressure close to, or possbly even below, vapour pressure whle the numercally predcted pressure at the leadng edge s far lower than the saturaton pressure n a consderable regon. Wthout further expermental data,
clarfyng the actual blade pressure, t s dffcult to assess whether the dfference n predcton s related to an error n the flow modellng, or f there are, e.g., geometrcal dfferences between the tested and modelled propeller causng ths devaton. However, t s also known that a lamnar boundary layer can supress the cavtaton ncepton even though pressure s far below the saturaton pressure. 5.4.3 Case2.3 In Fg. 12, cavtaton predcton of Case2.3 s presented. The root cavtaton s predcted for both sucton and pressure sdes of the blade at dfferent postons. The leadng edge cavtaton s predcted for just the pressure sde of the blade. At poston 135 degree, tp cavtaton s predcted for both sdes of the blade. (a) Sucton sde, wetted flow, pressure coeffcent (b) Sucton sde, cavtatng flow, alpha vapour 0.6 (a) Pressure sde, alpha vapour 0.6 (b) Pressure sde, alpha vapour 0.4 Fgure 10: Case2.1, MeshI, backward scheme, vew along x-axs 5.4.2 Case2.2 Cavtaton predcton of Case2.2, presented n Fg. 11, shows cavtaton appearances n both pressure sde and sucton sde of the blade. It should be noted that the mesh s constructed n a way that has fner resoluton n the sucton sde of the blades. As a result the cavtaton s less resolved on the leadng edge of the pressure sde comparng to the sucton sde. The most pronounced feature s the leadng edge cavtaton whch seems to start from the md-chord of the blade on the sucton sde and then cavty extends tll the tralng edge. (a) Pressure sde, alpha vapour 0.6 (b) Pressure sde, alpha vapour 0.4 (c) Sucton sde, alpha vapour 0.6 (d) Sucton sde, alpha vapour 0.4 Fgure 12: Case2.3, vew along x-axs 6 CONCLUSION Numercal smulatons of cavtaton of the Potsdam propeller test case (Case2) at three operatng condtons are presented n ths paper. For Case2.1 that the expermental data are avalable, results ndcate that the employed numercal tool can predct the thrust and torque coeffcents n the wetted and cavtatng flows reasonably well. The cavtaton smulaton shows over predcton of the cavty sze especally at the regon that the bubbly cavtaton s observed durng the experment. ACKNOWLEDGMENTS Fnancal support of ths work has been provded by Rolls-Royce Marne through the Unversty Technology Centre n Computatonal Hydrodynamcs hosted at the Department of Shppng and Marne Technology, Chalmers. Computatonal resources have been provded by Chalmers Centre for Computatonal Scence and Engneerng, C3SE. (c) Sucton sde, alpha vapour 0.6 (d) Sucton sde, alpha vapour 0.4 Fgure 11: Case2.2, vew along x-axs
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