Cavitation Simulation on Conventional and Highly-Skewed Propellers in the Behind-Hull Condition

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1 Second Internationa Symposium on Marine Propusors smp 11, Hamburg, Germany, June 2011 Caitation Simuation on Conentiona and Highy-Skewed Propeers in the Behind-Hu Condition Keun Woo Shin 1, Pou Andersen 2, Robert Mikkesen 2 1 MAN Diese & Turbo, Frederikshan, Denmark 2 Department of Mechanica Engineering (MEK), Technica Uniersity of Denmark (DTU), Kgs.Lyngby, Denmark ABSTRACT The caitating fows around conentiona and highyskewed propeers in the behind-hu condition are simuated by an in-house RANS soer, EipSys (Sørensen 2003), with the caitation mode, based on the homogeneous equiibrium modeing (HEM) approach and a apor transport equation. The aidation of the caitation mode in EipSys has been conducted for the caitating fows on 2D/3D hydrofois (Shin 2010). Prior to the caitation simuation, the open-water characteristics of the propeers from the computation are compared with those from the propusion test for the fuy-wetted fows (Li & Lundström 2002, Linde 2005). The caitation simuation is performed for the fow condition corresponding to that in the caitation tunne test for the ship mode equipped with the propeer (Johannsen 2004, Linde 2005). Instead of modeing the hu for the behind-hu condition, the measured wake fied in the propeer pane is appied by using a non-homogenousy oaded actuator disk (Mikkesen et a 2007) paced in a pane upstream of the propeer. The ariation of the computed caitation profie with respect to the bade ange is compared with that from the caitation tunne test. The present work describes the study of impementing a HEM caitation mode for computing unsteady caitation patterns in behind-hu condition with respect to bade anges and caity extent on the compicated geometry of a conentiona/highy-skewed propeer. In the computations, the efficiency of the non-homogeneousy oaded actuator disk as behindhu wake fied for the propeer infow is demonstrated successfuy. The computed unsteady caitation patterns in behind-hu condition and with respect to the bade ange has quaitatiey acceptabe accuracy, but with respect to the caity extent, there are quantitatie discrepancies. Keywords propeer caitation, RANS, hu wake, actuator disk, highy-skewed propeer 1 INTRODUCTION The CFD anaysis by using a turbuent iscous fow soer is common for practica industria appications in many discipines nowadays. Caitation modes for the CFD soers hae been deeoped in the ast decade. The popuar type of caitation modes is the HEM with a apor transport equation. Two-phase mixture is handed as a singe-phase fuid with ariabe fuid properties corresponding to the composition of two phases, and phase changes are goerned by a transport equation for either apor oume fraction or apor mass fraction. Such modes hae shown the potentia for the simuation of propeer caitation, but the considered cases are imited to the conentiona propeer with a moderate skew. In the present work, a caitation mode anaogous to the existing modes is impemented in EipSys. The unsteady caitation on the conentiona and highy-skewed propeers in the behind-hu condition is simuated by EipSys with the impemented caitation mode. The measurement of the behind-hu wake fied is appied by the actuator disk, instead of using the inet boundary condition, so that the we-presered wake fied may reach the region of the propeer fow. Firsty, the mathematica formuation and numerica schemes for the impementation are summarized. Next, the meshed modes are presented and the preiminary aidation of the computationa modes is made for the open-water fuy-wetted fow on the propeers. Lasty, the caitation simuations in the open-water and behind-hu conditions are presented. Before the behind-hu caitation simuation, the wake fied generated by the actuator disk is erified against the intended wake-fied measurement ony with an axia fow in a rectanguar grid. 2 FORMULATION AND IMPLEMENTATION The constant density and iscosity in the RANS equations for incompressibe fows are substituted with ariabe mixture properties for caitating fows. The RANS equations are written in Einstein notation and Cartesian coordinates as:

2 ( ( u u ) ( u ) 0 (1) t x u ui ) i ui p (2) ( t ) t x x x x i xi By adopting the HEM, the mixture properties are approximated on a oume fraction basis as: ( 1 ), (1 ) (3) The apor oume fraction α is obtained by soing the continuity equation for the apor phase: ( u ) ( ) m (4) t x By the equations (3) and (4), the mass transfer rate m between two phases can be reated to the materia deriatie Dα /Dt. By assuming that apor is distributed as a constant number density of spherica microbubbes with a consistent radius R, m is rewritten with R and the time deriatie R as: D 3 R m (5) Dt R By integrating the Rayeigh-Pesset equation with ignoring the effects of surface tension, iscosity and non-condensabe gas, R is expressed as a function of the oca pressure p and the initia apor oume fraction α 0 as: p p R (6) 3 Eq.(6) is appied to Eq.(5) with the assumption that bubbes grow or coapse rapidy, i.e. R R max, α 1 or R R min, α 4πR 3 min /3. By coecting constants into C e for eaporation and C c for condensation, m is rewritten as: 2 p p Ce for p p (7) m 2 p p for p p Cc 0 3 Most of the caitation modes (Kunz et a 2000, Singha et a 2002, Zwart et a 2004, Kim & Brewton 2008), based on the HEM are formuated anaogousy to Eq.(4) and Eq.(7). It is common that the mass transfer depends firsthand on the oca pressure, the amount of iquid for eaporation or apor for condensation and numericay-determined coefficients. The integra forms of the partia differentia equations (2) and (4) are soed by the coocated finite oume method. By appying Eq.(5) to Eq.(1), the continuity equation is couped with m as: u 1 1 m x (8) The integra form of Eq.(8) is soed by the SIMPLE method with the Rhie-Chow interpoation. After the momentum conseration equation (2) is soed for the fow fied, the pressure fied is corrected by Eq.(8) with from the preious time-step. The eddy iscosity is updated by the k-ω SST turbuence mode. The apor transport equation (4) is soed with the corrected pressure fied and from the preious time-step and the mixture properties are updated by Eq.(3). The expressions in Cartesian coordinates are transformed into those in cyindrica coordinates with a rotating reference frame for the propeer fow. Body force is added to the integra form of the momentum conseration equations to generate a wake fied. Based on the Rankine-Froude momentum theory, the oca body force F=(F r, F θ, F z ) on the actuator disk corresponds to the intended oca wake w=(w r, w θ, w z ) for the uniform infow eocity V aong the axia direction, as: F A( V w ) w F z r A( V w ) w F 0.5 A( V z z 2 r 2 z w ) (9) Where ΔA = oca area eement perpendicuar to the axia direction. The hydrostatic pressure effects on the caitation are incuded by subtracting the reatie hydrostatic pressure from the apor pressure p. Figure 1: Surface mesh on the conentiona (top) and highy-skewed (bottom) propeers

3 EipSys uses curiinear coordinates and parae computation with the muti-bock topoogy and the MPI. 3 COMPUTATIONAL MODEL We consider the conentiona and highy-skewed propeers, for which the caitation tunne tests hae been conducted in the open-water and behind-hu conditions on the EU research proect Leading Edge. A the reports from Leading Edge are aaiabe for the pubic. The conentiona propeer has a mode-scae diameter of D = m and a pitch ratio of P 0.7R /D = The highy-skewed propeer has D = m and P 0.7R /D = Figure 1 shows the structured rectanguar mesh on the propeer surface. The grid size is decreased around the bade edge to resoe the curature. The surface mesh is twisted chordwisey to aoid an excessiey skewed oume mesh. A cyinder extending through the entire domain with a sip boundary condition substitutes for the propeer hub. The oume mesh consists of an O-O grid in the near fied around the propeer surface and a H-C grid in the far fied. The fuid domain extends about 5D in a directions from the center. The ce number is and for the conentiona and highy-skewed propeers, respectiey. The first-ce height is m resuting in y CAVITATION SIMULATION 4.1 Open-water Caitating Fow Steady-state computations are made for open-water caitating fows with J = 0.447, σ N = 1.60 on the conentiona propeer and J = 0.603, σ N = on the highy-skewed propeer. The caitation number is defined by: p p N (10) N D where p = ambient static pressure. The aerage pressure on the inet boundary is taken as p in computation. Figure 3: Vapor oume as a function of iteration number Figure 2: K T and K Q from the experiment (dotted ine) and computation (soid ine) in the open-water condition Steady-state computations are made for the fuywetted open-water condition with arying V according to the adance ratio J. The comparison with the experimenta resut (Li & Lundström 2002, Linde 2005) in Figure 2 shows underestimation for reatiey ow aues of J and oerestimation for higher J in both K T and K Q, which may be reated to insufficient resoution of high gradients of fow ariabes for high oadings and increase of turbuent fow error in high oca Reynods number for high J. Figure 4: Snapshot from the experiment (Li & Lundström 2002) (top) and iso-contour of α = 0.1 from the computation (bottom) for the conentiona propeer with J = 0.447, σ N = 1.60

4 The ratio between the iquid density and the apor density is set to ρ /ρ = The coefficients in the equation for are set to C e = 75, C c = 30. The propeer reoution is set to N = 14 rps and 30 rps for the conentiona and highy-skewed propeers, respectiey. The same reoution of N = 30 rps has been appied to the experiment for the highy-skewed propeer, but the appied reoution is not reported in the experiment for the conentiona propeer. The soutions are conerged with normaized residuas beow Figure 3 shows that the apor oume grows with reducing σ N to an intended aue and afterwards it is conerged. σ N is graduay decreased from σ N = 5 to an intended aue between iteration numbers of 500 and In Figure 4 and 5, the iso-contour of α = 0.1 from the computation is compared to the experimenta snapshot. The distribution of the sheet caity on the suction side has a good agreement with that from the experiment for both propeers. The sheet caitation continues to be in a form of ortex caitation, but it is not extended away from the bade surface probaby due to a reatiey ow grid resoution outside the boundary ayer. axia component of the wake fied is appied to the actuator disk. Figure 6: Structured rectanguar grid for the wake-fied test without propeer fow Figure 7: Wake fied behind a tanker from the measurement (Kuiper 2004) (eft) and the computation (right) Figure 5: Snapshot from the experiment (Lydorf 2005) (eft) and iso-contour of α = 0.1 from the computation (right) for the highy-skewed propeer with J = 0.603, σ N = Wake Fied Modeing Before we appy a behind-hu wake fied to the caitation simuation, the wake fied, generated by the actuator disk without propeer fow, is erified by the comparison with the intended wake-fied measurement (Kuiper 2004). We perform a steady-state computation on the structured rectanguar grid with a propeer diameter coering 24 ces in the fine-grid region and an extent of 10D. As shown in Figure 6, the actuator disk is appied to the fine-grid region and the eocity distribution on the cross-section 1D downstream from the actuator disk is taken for the comparison. Ony the Figure 8: Wake fied behind a ferry from the measurement (Kuiper 2004) (top) and the computation (bottom) The conentiona and highy-skewed propeers are designed for a singe-screw tanker and a twin-screw ferry, respectiey. The wake fied from the tanker is

5 amost symmetric and the wake at inner radii is higher in the ower haf, but the wake peak at outer radii exists at the upright ange. The wake fied from the ferry is on the port side propeer pane and the eft side corresponds to the port side. There is no wake in the ower haf and the wake peak appears in the tip region of a bade ange φ = 200, where φ = 0 indicates the 6 o cock position. The circe in Figure 7 and 8 indicates the propeer disk area. The propeer disk area for the tanker wake fied is roughy approximated because imited information for the test of the tanker wake-fied measurement is aaiabe. The normaized axia eocity component (V-w z )/V is dispayed in Figure 7 and 8. The comparison in Figure 7 and 8 shows that the wake fied from the actuator disk agrees we with the measurement in magnitude and distribution. minimize the diffusion of the wake fied without a numerica confict between the propeer fow and the actuator disk. After the wake fied is deeoped, the caitation number is graduay decreased to the intended aue. We consider a case for K T = 0.164, σ N = 2.2 on the conentiona propeer. Since the aue of J is not reported in the experiment, we find J resuting in a oading simiar to K T from the experiment by appying seera different aues of J to the computation. In the computation, we use J = 0.58 resuting in K T = We consider a case for J = 0.915, σ N = 1.49 on the highy-skewed propeer. K T = is reported in the caitation tunne test (Johannsen 2004) and the aue of J corresponding to K T is found from the propusion test (Mrugowski 2003). In the computation, J = resuts in K T = Figure 9 shows that the ariations of K T and apor oume on each of two opposite bades with respect to time are periodic with the bade rotating frequency. As the wake fied is deeoped, the ariation ampitude is increased to a constant. Since the oera magnitude of the wake fied from the tanker is higher than that from the ferry, the increase of K T in the behind-hu condition is aso arger. The initia aue of K T is from the open-water computation. Figure 9: K T and apor oume on each of two opposite bades as functions of time for the conentiona (top) and highy-skewed (bottom) propeers in the behindhu condition 4.3 Behind-hu Caitating Fow We start an unsteady-state computation from the conerged soution of the steady-state computation with a time step corresponding to 0.5 rotation of the propeer. First, the actuator disk is appied to a pane 0.5D upstream from the propeer pane. The actuator disk is paced cosey to the propeer pane in so far as it is outside the upstream propeer fow in order to Figure 10: K T and apor oume on a bade as functions of bade ange in a singe cyce for the conentiona (top) and highy-skewed (bottom) propeers in the behind-hu condition

6 Figure 10 shows that K T is increased when the bade tip is in a high wake region, whie the ower haf of the propeer disk area has a high wake in the inner radii for the conentiona propeer, K T is ow at φ = This impies that the wake fied in the outer radii at the bade tip is more crucia for K T than that in the inner radii and the effectie wake fied on the propeer pane is contracted due to the upstream propeer fow. The distribution of K T with respect to the bade ange for the conentiona propeer is not as symmetric aong the ertica axis as the appied wake fied. The highest peak of K T is at φ = 190 for the highy-skewed propeer. The bade ange is φ = 0 for generator ine on the 6 o cock position and the generator ine is about 20 ahead of the mid-chord ocus for both propeers. The increase of the apor oume appears ater than that of K T, which may impy that the formation and coapsing of caitation bubbes take time. The highest peak of the apor oume is at φ = 300 and 245 for the conentiona and highy-skewed propeers, respectiey. whereas the caitation exists continuousy around the whoe reoution in the computation. The computed caitation profie corresponds to the iso-contour of α = 0.1. Figure 12: Caitation profie on the conentiona propeer at φ = with 30 interas (from top-eft corner to bottom-right corner) from the computation for σ N = 2.2 in the behind-hu condition Figure 11: Caitation on the conentiona propeer at φ = 180 (top) and 240 (bottom) from the experiment (Kuiper 2004) (eft coumn) and the computation (right coumn) for σ N = 2.2 in the behind-hu condition Figure 11 shows that the caitation profie on the conentiona propeer changes sowy in the computation rather than in the experiment. The caitation competey disappears at φ = and appears again at φ = in the experiment, Figure 13: Caitation on the highy-skewed propeer at φ = 180 (top) and 210 (bottom) from the experiment (Johannsen 2004) (eft coumn) and the computation (right coumn) for σ N = 1.49 in the behind-hu condition

7 At φ = 180, the caitation profie from the computation is simiar to that from the experiment in the starting point at the eading edge of r 0.75R and the transforming pattern from the sheet caitation to the ortex caitation, but the chordwise extent at r 0.75R 0.9R differs. Whie ony the tip ortex caitation is eft at φ = 240 in the experiment, the sheet caitation is sti on the increase in the computation. The caitation profies at φ = 180 and 240 in the experiment are coser to those at φ = and 0-30, respectiey, in the computation. Since the propeer disk area in the appied wake fied is roughy chosen, as mentioned aboe, it is difficut to reate the oera difference to the rates for eaporation and condensation in the caitation mode. Figure 13 shows that the computed caitation profie on the highy-skewed propeer at φ = 180 and 210 start earier at the eading edge and the extent aong the bade tip is ess than that from the experiment. No caitation appears at the outer radii at φ = 90 and 120 in both the computation and the experiment. Root caitation appears around the whoe reoution in the computation differenty from the experiment because the bade foot is not incuded in the computationa mode. Whie the experiment shows unstabe caitation in Figure 14, the computation shows no fuctuating caitation with a higher frequency than the bade rotating rate. Unstabe and stabe caitations are marked by singe and doube diagona ines in the sketches of the experimenta resut. Stabe caitation starts to appear at φ = in both the computation and the experiment. The argest extent of the computed caitation profie appears ater than in the experiment. The sheet caitation disappears at φ = The computed caitation profies at φ = are ess extended aong the chordwise and radia directions than those from the experiment, whereas the ariation patterns with respect to the bade ange hae simiarity. 5 CONCLUSION The open-water caitation simuations on the conentiona and highy-skewed propeers show an acceptabe degree of quantitatie accuracy for steady sheet caitation. The wake fied generated by using the non-homogeneousy oaded actuator disk shows a high degree of accuracy in a simpe rectanguar grid without a propeer fow. The ariation pattern with respect to the bade ange of the computed unsteady caitation in the behind-hu condition has quaitatiey acceptabe accuracy, but the caitation extent has quantitatie discrepancies, which may be reated to the eaporation/condensation rate in the caitation mode and the interaction of the wake fied and the propeer fow. If the wake fied measured upstream from the propeer pane is appied to the computation, it may reduce the infuence of the propeer fow interaction. For a further diagnosis, the cases with simper wake fied and stronger caitation need to be considered. Figure 14: Caitation profie at the different bade anges on the highy-skewed propeer from the experiment (eft coumn) and the computation (right coumn) REFERENCES Johannsen, C. (2004). Leading edge- Part2: Caitation tests for a ferry equipped with skew propeers. HSVA Report K51(03), Hamburg, Germany. Kim, S. E. & Brewton, S. (2008). A mutiphase approach to turbuent caitating fows.

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