1 Introduction

The growing demand for heavily loaded,highly efficient propellers that generate very low on board noise and vibration levels requires that the marine propeller designs be increasingly based on analyses of the inception and development of cavitation,which is the more important inhibitor of the propulsion system.If not specifically scheduled,as in supercavitating propellers,cavitation can generate a number of problems such as additional noise,vibrations,and erosion as well as variations in the delivered thrust and torque.In modern propeller designs,m id-chord bubble cavitation may also appear and face cavitation is very common,especially if the propeller operates behind a strongly non-uniform hull wake or in inclined shaft conditions.

There is a clear need for accurate and efficient solvers that can address performance problems and predict cavitation in the preliminary design stage and in the more accurate final propeller optimization phase,during which features like tip vortex cavitation or cavity bubble dynamics are considered.In daily practice,in place of the traditional lifting line and lifting surface approaches,boundary element methods(BEMs)are applied with increasing frequency for design purposes,as in Gaggero et al.(2010,2014b),because of their overall higher accuracy in both non-cavitating and cavitating conditions.The increase in hardware resources has led to a shift toward viscous solvers for performing more detailed propeller analyses,especially in loaded and highly skewed geometries.

BEM s,from their original formulation(Morino and Kuo 1974)to more advanced applications,including sheet cavitation and unsteadiness(Fine and Kinnas 1993;Hsin 1990;Vaz and Bosschers 2006),have been extensively applied for design optimization purposes(Bertetta et al.2012b;Gaggero et al.2012)and the analysis(Gaggero et al.2010,2014a,b,2016a,b;Bertetta et al.2012a)of marine propeller performance in open water and behind-wake functioning in noncavitating and cavitating conditions.Convincing results have also been obtained for unconventional geometries like contracted and tip-loaded(CLT)propellers and decelerating ducted propulsors.Computational efficiency and reliable predictions at the design point are key factors in the application of BEMs in multi-objective design approaches.Different functioning conditions have been simultaneously addressed and trade-off geometries between cavitation and efficiency have been derived based on systematic explorations of the design space,which has led to overall better designs thatalso account for side effects such as induced pressure pulses and radiated noise.

On the other hand,the application of Reynolds-averaged Navier-Stokes equation(RANSE)codes has proven to be more reliable and accurate than the abovementioned approximations.RANSE codes are useful in the completion of experimental tests by providing the opportunity to analyze and visualize phenomena beyond the assumptions of inviscid and irrotational flows that are the basis of BEM formulations.Off design operation and unconventional geometries(or when attention is focused on local phenomena,like tip vortex interactions and cavitation)are the conditions for which their greater usefulness is evident.As a correction step in the design phase(Gaggero et al.2012),but especially in the analysis of propeller characteristics,the application of RANSE solvers results in numerical predictions that,despite the necessary approximations of the models(turbulence and multiphase flow modeling),are in good agreement with experimental results.These results have been promising not only when global parameters such as thrust and torque coefficients are considered but also in terms of the predicted cavity extension.The analyses conducted by Li(2011),Sipila et al.(2011),and Morgut and Nobile(2011,2012a,b)and the systematic comparison in the E779A propeller test case by Vaz et al.(2015),for instance,demonstrate that,especially when close to the design conditions,the RANSE solvers reach a level of confidence sufficient for the final design and the detailed analysis of propellers(Bensow and Bark 2010a,b),albeit with a nonnegligible computation cost.

In parallel,interest in open-source approaches has grown rapidly,also for solving Engineering problems,and their application to marine problems may represent a substantial and convincing alternative to commercial software,not only from the academic perspective.Especially during the preliminary design phase,the possibility for accurately and efficiently investigating a rather wide set of alternatives without encountering any expensive licensing issues represents excellent added value and an inevitable step toward the direct application of viscous-based approaches for design by optimization.Thus,an extensive verification and validation of open-source solvers is necessary in more significant cases than the common straight-ahead conditions.

Most of the analyses in the literature have focused on open water conditions in homogeneous inflow.The First Workshop on Cavitation and Propeller Performance(SVA Potsdam 2015,Heinke and Lubke 2011),for instance,was specifically devoted to cavitation in open-water conditions and a number of papers(Gaggero and Villa 2016;Morgut and Nobile 2012a)were dedicated to analyses of that test case.

In correspondence to the design advance coefficient,the calculated values are slightly underestimated,with differences less than 2%for both thrust and torque.The slope of the curves is slightly higher than those of the experimental measurements,resulting in negligibly overestimated values in correspondence with the loaded conditions(+0.9%for KT,+1.4%for 10KQ at J=0.6)and slightly under-predicted values for the higher advance coefficient(-4.5%for KT,-3.7%for 10KQ at J=1.4).

Few authors have reported on the hydrodynamic characteristics of a marine propeller in non-homogeneous flow.In 1990,the ITTC Propulsion committee(Hoshino 1998)promoted a systematic comparison of the numerical tools available for unsteady propeller analyses,which were of course mainly based on potential theories.El Moctar and Bertram(2000)used a RANSE solver to investigate the flow around a four-bladed propeller at oblique angles up to 12°.Krasilnikov et al.(2009)analyzed the unsteady forces acting on a podded propeller operating in oblique inflow.Dubbioso et al.(2013,2014)used RANSE to analyze the non-cavitating performance of the INSEAN E779A propeller subjected to very high shaft angles(up to 50°),with particular attention to loads acting on a single blade.These analyses highlighted how oscillating blades force an increase in their amplitude and frequency with an increase in the shaft inclination.However,the absence of high-quality experiments,especially for extreme configurations such as those analyzed in Dubbioso et al.(2013,2014),has prevented a reliable validation of these calculations.

The workshops organized in the framework of the EU-funded project VIRTUE provided further insight into cavitation modeling capabilities in steady and unsteady conditions.The results obtained by different institutions and research groups were collected,for instance,by Salvatore et al.(2009),but similar calculations for the same propeller(again,the well-known INSEAN E779A)were also presented by Vaz et al.(2015),as part of the Cooperative Research Ships SHARCS project,and by Morgut and Nobile(2012a).Calculations in open-water conditions have shown reasonable agreement in both non-cavitating and cavitating conditions,with the predicted sheet cavitation only slightly overestimated in both the radial and chordwise directions.However, nonhomogeneous calculations have produced scattered results in terms of cavity volume and area.The quality of calculations improved in Vaz et al.(2015)with respect to the results generated by the VIRTUE workshops thanks to the use of refined grids and different cavitation models.In any case,most of the discrepancies have been due to the difficulties in reproducing the non-homogeneous inflow wake and the numerical dissipation inherent in RANSE modeling,which significantly weakens the inflow at the propeller plane.Only through large-eddy simulation(LES)calculations,which were extensively proposed for the same test case by Bensow and Bark(2010a,b),was it possible to partially identify phenomena like the interaction of re-entrant jets with the tip vortex and the upstream desinence of the cavity bubble when the blade exits from the region in which most of the velocity defect is concentrated.The computational effort required for these kinds of calculations are obviously significant.Using LES to resolve detailed structures in time and space with a nonexcessively refined mesh of about 4.5 million cells,an equivalent time step of 1/20 of deg.has been necessary.Even more detailed LES calculations have been proposed by the same authors(Lu et al.2014)for two very similar highly skewed propellers with slight alternations in their tip geometries in inclined shaft conditions,which showed the possibility for discriminating small but crucial differences in the flow features of the two propeller designs and the possible interactions of different vortical structures.The role of the leading edge vortex on cavitation inception in highly skewed blades has been reported.However,even the finest mesh arrangement consisting of 14 million cells and solved with an equivalent time step of 1/120 of deg.was only sufficient to predict cavitation on the blade surfaces,without the possibility for sustaining any detailed flow development behind the trailing edge,nor the shedding from the sheet cavity which is of crucial importance for broadband noise propagation.A complete assessment of the role of re-entrant jets,the features related to periodic shedding at cavity closure,and the risk of secondary cavitation has been possible only for simpler geometries like the Delft Twist wing and bi-dimensional hydrofoils(Bensow and Bark 2010b).

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In any case,due to the geometry(the constant pitch E779A propeller)or the operating conditions(high cavitation indexes,as for the propellers analyzed in Lu et al.2014),the cavitating phenomena characterizing these calculations have been relatively typical.Sheet cavitation(even if significant)in almost any of the investigated operating conditions is characteristic in the E779A propeller,whereas relatively thin and only leading edge sheet bubbles are present in the propellers analyzed by Bensow and Bark(2010a).

For these reasons,the Second Workshop on Cavitation and Propeller Performance(Kinnas et al.2015;SVA Potsdam 2015,in the following the “Workshop”)represented a valuable opportunity for making a careful comparison of the accuracy and reliability of many different codes and numerical approaches developed and/or employed by many internationally recognized research institutes,universities,and companies.The experimental datasets provided by SVA Potsdam cover both open-water and cavitation tests in oblique flow,with well-defined cases,operating conditions,and geometries.Pure oblique flow is a more controlled operating condition,especially from the numerical perspective,with respect to spatial non-homogeneous inflow using wake screens or appendages,hardly reproducible in simulations,as in Salvatore et al.(2009).Furthermore,depending on the operating conditions,different types of propeller cavitation have been characterized,including significant leading edge sheet cavitation with a cavitating tip vortex on the suction side,bubble cavitation at midchord,bubbles at the root,and pressure side sheet cavitation.

Of course,the availability of data on propeller function in inclined shafts is not limited to this test case.Recently,as a part of the European Project SONIC,systematic measurements were proposed regarding the performance of a propeller when changing the shaft inclination(Aktas et al.2016a,b).However,attention was mainly focused on the influence of shaft inclination on propeller thrust,efficiency,and inception points,again considering only two types of cavitation(tip vortex and leading edge sheet)with systematic measurements of radiated noise and extrapolation rules for URN based on simplified model-scale measurements.

The amount of data collected for the Potsdam Propeller Test Case(PPTC)geometry and the availability of numerical calculations has made the SMP Workshop the preferred test case for validation purposes.The results collected during the Workshop showed significant variation in the various codes,especially in cavitating conditions.Moreover,the influence of the mesh was not fully resolved.Calculations were performed,in some cases,with meshes of up to 50 million cells,without any significant improvements(except for the detailed prediction of the propeller wake)on the computed propeller performance,regardless of the cavitation or fully wetted analyses performed.Commercial software featuring ease of use strongly encouraged and widened the application of RANSE solvers to the maritime field,as demonstrated in Gaggero et al.(2012,2014a,2016a,b),Bertetta et al.(2012a),Morgut and Nobile(2011,2012a,b),and Li(2011),as well as in-house developed codes oropen-source libraries,which have shown very similar performances and limitations.In addition,the use of LES approaches,as reported by a few authors(Asnaghi et al.2015),has not significantly improved the accuracy o f the calculations.

从图4可以看出：相同振动参数下，耦合振动流化床平均轴向颗粒速度最大，普通流化床轴向颗粒速度也大，水平振动流化床轴向颗粒速度较小，垂直振动流化床轴向颗粒速度很小；垂直振动促进大气泡形成，会产生节涌，水平、耦合振动不产生气泡，颗粒运动较为平稳；二维垂直振动流化床在颗粒轴向速度0.06～0.08 m/s时流化效果最好。

Among the open-source RANSE solvers employed for the PPTC test case,the OpenFOAM package(OpenFOAM Foundation 2016)has been the preferred choice,likely being the most widely adopted open-source code in engineering and academic contexts.On this basis,we applied the OpenFOAM libraries in this study to characterize the unsteady operation of the Potsdam propeller that,for the reasonsstated above,can be considered to be a mandatory test for the verification and validation of this RANSE solver in propeller performance prediction with an inclined shaft with cavitation.As proposed by Gaggero and Villa(2016)for homogeneous inflow conditions at the First Workshop on Cavitation and Propeller Performance(Heinke and Lubke 2011),we focus our attention here,among the available tests(SVA Potsdam 2015;Kinnas et al.2015),only on open-water non-cavitating performance and unsteady cavitation prediction.

After a brief introduction of the PPTC test case(Section 2),in Section 3,we summarize the numerical formulations of the adopted solvers for non-cavitating and cavitating flows.In Section 4,we propose a mesh sensitivity analysis in light of the results collected during the Workshop.To achieve computational efficiency,based on the experience gained by Gaggero et al.(2014a,b)and Gaggero and Villa(2016)and on the heterogeneous conclusions regarding mesh influence in the results discussed in the Workshop,we used a RANSE setup and calculations to achieve the highest possible accuracy with the lowest computational resources to obtain almost daily affordable simulations on common workstations.The limitations of the RANSE approximation with respect to turbulence modeling and of the multiphase mixture approach based on the volume of fluid(VoF)for cavitating flows are well reported in the literature.The PPTC test case,in this sense,represents a crucial application due to nature of the cavitating phenomena to which it is subjected.The aim of our calculations and analyses,in light of the scattered data collected throughout the Workshop,is to establish a baseline for reliable yet lightweight predictions with the necessary simplifications inherent to the assumed computational models for short-term design cycles.We summarize these results in Section 5.Our goal in comparing our calculations with the cavitation tunnel measurements for the whole range of experimentally tested cavitation indexes and with the numerical calculations submitted to the Workshop is to verify the improvements and drawbacks of the RANSE solver of the open-source OpenFOAM library approach with respect to state-of-the-art methodologies with comparable modeling and flow assumptions.Another task of this study was to assess the reliability and limitations of the proposed numerical approach in predicting propeller performance in relatively severe operational conditions,which were not completely addressed during the Workshop.

Together with viscous calculations,we also propose and compare BEM results.Due to their computational efficiency,BEMs are still widely adopted in design and optimization and a further assessment of their accuracy and application limits in demanding conditions with respect to viscous calculations is valuable.

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2 Test Case:PPTC Propeller

The model propeller is the five-bladed,controllable pitch Potsdam Propeller(SVA Potsdam 2015),with a diameter D of 0.250,a hub ratio of 0.3,and a pitch-to-diameter ratio of 1.635 at 0.7 radial section.The skew at the tip is about 19°.The propeller operates in a pulling configuration with a shaft inclination of 12°.Table 1 shows a summary of the geometrical characteristics of the propeller and Fig.1 shows a rendered view of the propeller.

Table 2 lists the operating conditions of the propeller considered in this work.We performed open-water,noncavitating calculations(case 1)with five different advance coefficients ranging between 0.6 and 1.4,in correspondence with the towing-tank tests performed at the Potsdam SVA basin.

The numerical solution consists of an inner iterative scheme that solves the nonlinearities connected to the Kutta condition and the dynamic and kinematic boundary conditions on the unknown cavity surfaces until the cavity-closure condition has been satisfied.We use an outer iterative cycle to integrate over time to obtain a periodic solution after the virtual numerical transient of the key blade approach(Hsin 1990).Finally,the shear stresses,which are ignored by the potential approximation,can be included by simplified and semi-empirical formulations(standard Frictional lines with corrections to account for thickness and camber),whereas the pressure forces,once the potentials are defined,can be computed by app lying the unsteady formulation of Bernoulli’s theorem.

3 Numerical Formulations

3.1 RANSE Calculations with OpenFOAM

The viscous calculation framework for non-cavitating computations uses continuity and momentum equations for an incompressible fluid,in their simpler forms,by employing RANSE solvers with the tensor of the Reynolds stresses computed in accordance with a turbulence model.When addressing cavitation,a modification of these equations is required.Even if most of the theoretical cavitation models describe the dynamics of bubbles by solving for the vapor-liquid interface(the traditional Rayleigh-Plesset equation is probably the most well-known approach),in the RANSE solver framework,most of the cavitating engineering flows are solved using a homogeneous mixture approach.In the case of separated systems,the so-called Eulerian-Eulerian approach for incompressible phases can be applied,in which the flow is treated as a single variable density fluid,i.e.,a homogeneous two phase(vapor and liquid)incompressible mixture without explicit phase interfaces,whose dynamics are governed by tailored continuity and momentum equations.In contrast to compressible homogeneous approaches,wherein a barotropic constitutive relation is used to correlate pressure with density in the mixture and,in turn,with the void(vapor)fraction,in the homogeneous incompressible mixture case,a convection equation for the void fraction is introduced to explicitly account for the phase transition and mass transfer through the vapor/liquid interface.The VoF method can be used to resolve this additional advection equation corrected for the mass transfer.We define the vapor volume fractionαas the percentage of the volume domain(and for each discrete computational cell)occupied by the vapor phase,as follows:

The physical proprieties of the mixture can be defined as a weighted mean of the physical proprieties of the pure liquid and vapor phases:

Assuming an incompressible mixture,we obtain the following continuity and momentum equations:

where T Re is the tensor of the Reynolds stresses,which is modeled through the Boussinesq eddy viscosity assumption in accordance with the turbulent closure equations.S represents the additional momentum sources and ρliq/ρmix∙dα/dt is the net interphase mass flow rate per unit volume.This system of equations is completed by an additional convective equation that solves for the transport of the vapor volume fraction by which the interaction between the vapor and the liquid phases is modeled.The OpenFOAM libraries use a modified version,Eq.(5)(Rusche 2002),of this transport equation,based on the“compression” of the interface with an artificial velocity ur,as follows:

which is oriented orthogonally to the interface of the normal n int and is scaled by a user-defined factoris the velocity magnitude on the cell faces,which is obtained through the volumetric flux φ.

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The role of“artificial compression” is to “push” the volume fraction toward the interface,which allows for the widespread use of monotonic discretization schemes.The α(1-α)term ensures that the resulting artificial sharpening takes place only in proximity to the interface.

Of the available mass transfer models(Kunz et al.2000;Merkle et al.1998;Sauer and Schnerr 2000)for calculating m˙,we used the Schnerr and Sauer approach for all the calculations.

3.2 BEM Calculations

The BEM models the flow field using a scalar function,the perturbation potential φ(x,t),whose spatial derivatives represent the perturbation velocity vector component.Irrotationality,incompressibility,and absence of viscosity are the hypotheses required to express the continuity equation as a Laplace equation for the perturbation potential,as follows:For the more general problem of cavitating flow,using Green’s third identity allow s us to solve the three dimensional differential problem as a simpler integral equation regarding the surfaces(the fully wetted surface SB,the cavitating surface SCB,and the trailing wake surface SW,i.e.,the zero thickness layer,which depart at the trailing edge of the lifting surfaces,where vorticity is shed into the downstream flow)that bound the domain.The solution is obtained as the intensity of a series of mathematical singularities(sources∂φ(x,t)/∂n and dipoles-φ(x,t))whose superposition models the inviscid cavitating flow on and around the surfaces(Morino and Kuo 1974):

where n is the unit normal,r is the distance between points~x and x,and Δ denotes the potential jump across the wake surface.Ignoring the supercavitating case(for which computation is stopped when the cavity bubble reaches the blade trailing edge)and assuming that the cavity bubble thickness is small with respect to the profile chord,singularities that model the cavity bubble can be placed on the blade surface instead of on the real cavity surface(Fine and Kinnas 1993).This approach can be considered to be partially nonlinear,taking into account the weak nonlinearity of the boundary conditions(the dynamic boundary condition on the cavitating part of the blade and the closure condition at its trailing edge)without the need to collocate the singularities on the effective cavity surface.The typical set of required boundary conditions for steady problems is as follows:

·A cavity-closure condition on the cavity bubble trailing edge to drive the convergence of the iterative solution of the cavity bubble extension on the surfaces at each time step

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·A Kutta condition on the blade trailing edge,forced iteratively by requiring a zero-pressure jump between the pressure and suction side at the blade trailing edge

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·A dynamic boundary condition on the cavitating surfaces

·A kinematic boundary condition on the cavitating surfaces to define the cavity thickness by setting the strength of the sources on the cavitating panels

·A kinematic boundary condition on the wetted solid boundaries

We can obtain the arbitrary detachment line on the back and/or on the face of the blade,iteratively,by applying a criterion equivalent in two dimensions to the Villat-Brillouin cavity detachment condition.We can solve the unsteadiness of the problems due to non-homogeneous inflow as proposed by Hsin(1990),by applying the Kelvin theorem.

We also computed the influence(and extension)of cavitation on the propeller performance(case2)for three revolution based cavitation indexes measured at the cavitation tunnel of the same basin.

4 Mesh and Sensitivity Analysis

As noted in the introduction,the mesh sensitivity analysis,which is mandatory prior to any numerical prediction,is particularly significant in light of the results collected during the Second Workshop on Cavitation and Propeller Performance(Kinnas et al.2015).Figures 2 and 3 summarize the results of the cavitation tunnel configuration(case 2),for both the noncavitating and cavitating conditions of Table 2.The influence of the mesh was clearly not resolved and the results of participants who had proposed their own mesh sensitivity analyses were quite heterogeneous.In addition,the application of the same code(in particular the commercial codes that are available to more participants)often led to substantially different results.Differences in the turbulence modeling(calculations were performed using simpler two-equation models,with and without wall functions,and implicit LES),in the mesh topology(some used structured or block-structured meshes whereas most computations were performed using unstructured tet-,hex-,and poly-meshes),and,finally,in the experience of the users may partially justify these discrepancies.In any case,however,an increased number of cells that at first sight may represent the most straightforward(and the first)attempt to resolve complex and specific interactions like those of the PPTC test case(tip vortex and root bubbles,for instance)does not seem to be the most convenient.Furthermore,calculations with resolved boundary layers in spite of wall functions did not significantly improve the reliability of the calculations.Most of the calculations were performed with meshes consisting of a total of between 5 and 10 million cells and few were performed with meshes of 30 million cells but the results,especially in the cavitating case,were not as good as expected.

These considerations influenced the analyses we propose in this paper.In addition to the analysis of the PPTC performance with OpenFOAM and in light of the limitations of current models(i.e.,homogeneous multiphase mixture and VoF assumptions,mass transfer models)in the severe cavitating conditions being considered,we believed it to be equally important to take the opportunity to employ the lightest mesh arrangement to make the cavitating calculations affordable on a daily basis.This is necessary to make this kind of calculation truly suitable on low-end infrastructures(i.e.,desktop workstations)and for short-term design cycles.

We conducted mesh sensitivity analyses with the cavitation tunnel configuration(case 2)considered to be more significant with respect to the scope of our analyses than the simpler towing-tank,open-water non-cavitating configuration.We also applied relevant information about the cell densities,prism layer arrangements,and simulation strategies to the less complex towing-tank configuration calculations(case 1)for which the Workshop results shown less dispersed data.

We arranged five different meshes using the main parameters summarized in Table 3.Figure 4 shows a schematic illustrating the coarser grid arrangement.Figure 5 shows closer views of the selected meshes in proximity to the propeller leading edge.To predict the unsteady functioning of the propeller as shown in Fig.6 for example,we achieved the rigid motion of the propeller using sliding meshes and arbitrary mesh interfaces(AM Is)(Farrell and Maddison 2011)to match the innerrotating domain with the outer,fixed region.For the towing-tank configuration(case 1),the only modification consists of a larger outer domain(at least 3 or 4 propeller diameters)with a slip(or pressure)condition rather than a no-slip wall condition on the external surface.Table 4 lists the boundary conditions adopted for these calculations.With respect to the OpenFOAM package,we specified the “zero gradient” condition for the pressure field through the fixed Flux Pressure boundary conditions,whereas we imposed wall functions for the turbulent kinetic energy k and for the specific dissipation ω using,respectively,the kqR Wall Function and the omega Wall Function with a 1%turbulence intensity level and an eddy viscosity ratio of 10.We arranged the first three meshes(coarse,medium,and fine)with a“reference” prism layer,which had a constant firstcell height,to keep the non-dimensional wall distance constant.We used the medium-A and medium-B meshes to further analyze the influence of the wall treatmenton the solution,and both meshes had the same global and local refinements(same minimum surface mesh size at the leading edge,same average surface size on the blades).We arranged medium-A,however,with a thicker prism layer to reasonably encompass most of the vapor bubbles after chordw ise development.

We arranged medium-B,instead,with half the first cell thickness at a constant total prism layer height to gain insight into the role of the mesh with respect to the re-entrant jets and their influence on bubble development.Bearing in mind the limitations of the OpenFOAM native snappy Hex Mesh tool(Gaggero and Villa 2016),especially in dealing with prism layer extrusion and highly curved surfaces(as at the blade leading edge)that are particularly important for the reliable prediction of smooth pressure distributions,we performed the calculations with the polyhedral meshes arranged in StarCCM+(CD-Adapco 2014).

We adopted the two-equation SST k-ω turbulence model(Menter 1994)without any compressibility corrections for the cavitating calculations(Coutier-Delgoshaet al.2003),with scalable wall functions(OpenFOAM Foundation 2016)for an average non-dimensional wall distance,with the reference prism layer between 30 and 40,depending on the advance coefficients.The need for computationally efficient calculations makes the use of wall functions(based on the nut US palding Wall Functionimplementation)unavoidable and even with the finer arrangement(medium-B),we maintained the average non-dimensional wall distance(≈20)within the range of their applicability.

We preferred second-order and blended first-/second-order accurate numerical schemes for spatial derivatives for all the calculations,whereas we used a first-order scheme to integrate over time,as detailed in Table 5.

We computed the gradients using second-order Gaussian quadrature with the cell MD Limited multi-dimensional limiter(OpenFOAM Foundation 2016)to improve stability(setting the limiting coefficient to 1)without causing excessive dissipation.For momentum,we preferred the linear Upwind scheme with a velocity gradient limiter scheme.With respect to the turbulence discretization,we achieved blending between the first-order,upw ind,and second-order linear scheme using the limited Linear scheme.

We also employed the Multidimensional Universal Limiter with Explicit Solution(MULES)algorithm to avoid the unboundedness of the phase fractions in Eq.5.We adopted limited van Leer schemes to model the convective terms of the VoF transport equation.We also adopted a conservative value for the sharpening factor(kc=1)in light of the results collected for the simpler case of cavitating hydrofoils by Gaggero and Villa(2016).The semi-implicit nature of the MULES algorithm allowed for simulations with a relatively high Courant number.In all our calculations,we adopted a simulation time step equivalent to an angular increment of 0.2°to ensure maximum Courant numbers(in a few deteriorated cells due to the chopping of the prism layers of the unstructured mesh)below 10.We considered this value to be an acceptable threshold in light of previous calculations(Gaggero and Villa 2016)for balancing computational efficiency and numerical stability.

We used the PIMPLE(OpenFOAM Foundation 2016)algorithm with two iterations per time step to couple the pressure and velocity.This choice,which weights the calculations with respect to a pure PISO coupling scheme,is related(and was preferred for the Euler first-order time integration scheme)to the stability concerns for calculations at Courant numbers greater than 1.

Also on the basis of the preliminary sensitivity analyses reported by Gaggero and Villa(2016),we performed all calculations with the nuclei concentration(n0=1013)and nuclei diameter(R=10-6 m)calibrated to ensure parameter independent predictions.In any case,the influence of these Schnerr-Sauer model parameters was demonstrated to be almost negligible and was definitely lower than in the Kunz or Zwart models.

Figures 7 and 8 show the results of the sensitivity analyses,respectively,for the thrust and torque coefficients in the cavitating and non-cavitating functions of case 2.1(loaded condition)and case 2.2(design condition).We repeated the analysis for these two distinct conditions due to the peculiarity of the phenomena under investigation.We found the loaded condition,as in the Workshop findings,to be clearly subject to suction side sheet cavitation whereas the design condition suffered a mix of pressure side sheet and suction side midchord/root bubble cavitation,the prediction of which,especially for the latter,may be unfeasible.Consequently,when selecting the mesh arrangement,both phenomena should be kept in mind.

In non-cavitating conditions,the convergence trend is particularly good and exactly in line with the tendencies observed during the Workshop.In correspondence with the loaded condition,the difference between the coarse and fine meshes is less than 1%and we overestimated the medium mesh results(KT=0.398,10KQ=1.019)by less than 0.5%with respect to the fine mesh calculations for both thrust and torque.With respect to the design advance coefficient,the differences are slightly higher,especially for torque,with the coarse mesh value over-predicted by about 2%versus the fine arrangement.In contrast,the medium mesh results(KT=0.262,10KQ=0.753)are less than 0.75%overestimated with respect to the fine mesh case.The use of a different prism layer arrangement at the walls has a negligible influence on noncavitating performance.Medium-A mesh results are equivalent to those of the medium mesh,both having the same refinements and non-dimensional distance.Also,by halving the first prism cell height,the non-cavitating performance result is almost unchanged,with a negligible overestimation trend observable mainly for the torque coefficient and the loaded functioning.In cavitating conditions,the mesh has a slightly higher influence but is not sufficient to justify weighting the calculations.In correspondence with the loaded case,again,the discrepancy between the different mesh arrangements is only about 1%,with the medium(KT=0.368,10KQ=0.985)and fine very similar(less than 0.5%of difference).At the design point,the coarse configuration show s a nonnegligible difference with respect to the more refined arrangements(about a 3%overestimation for both thrust and torque),but the medium configuration(KT=0.201,10KQ=0.664)limits this difference to an acceptable 1.5%.Similar to the non-cavitating calculations,different prism arrangements slightly affect the predicted propeller performance.Using the finer prisms at the walls(medium-B)yields only a 1%difference in both thrust and torque with respect to the reference medium arrangement in correspondence with the loaded condition.At the design point,the use of thinner prisms is slightly more influential,yielding thrust and torque values about 2%lower than those calculated with the reference configuration(at the cost,however,of a reduction in the simulation time step to maintain a maximum Courant number below 10).

Increasing the total prism thickness(medium-A)causes the reverse result:in the loaded condition,it leads to overestimated values with respect to the medium configuration,whereas at the design point,a 1%underestimation is generated.

In terms of cavity extensions(isosurfaces of vapor fraction equal to 0.5 shown in Figs.9,10,and 11),the influence of the mesh is hardly visible.The medium and fine meshes show a slightly more extended cavity bubble on both the pressure and suction sides with respect to the coarse configuration.Tip vortex cavitation prediction is beyond the capability of any of the mesh arrangements proposed in this analysis and only with the fine mesh can we observe a hint of cavity extension in the propeller wake at the 90°blade position.Also,the midchord and root bubble stake the form of sheetcavitation due to the limitations clearly observed by all the participants at the Workshop,which we discuss in detail in the next section with respect to the mixture modeling of a multiphase flow.

The only appreciable difference we found was in the prediction of the re-entrant jets at the cavity closure.As shown in Fig.12,the use of finer meshes favors the prediction of this feature,which is reflected in the significant portion of the blade surface below the vapor bubble,as evidenced by the isosurfaces of the above figures,partially or even completely wetted.This phenomenon is particularly evident in the medium-B mesh arrangement that,along with the presence of thinner prism layers,facilitates the wetting of the surface(and,sometimes,the unphysical inclusion of spurious portions of water into the vaporization region).A similar behavior was observed in simpler 2D RANSE calculations of the flow around cavitating hydrofoils(Gaggero and Villa 2016).In that case,the combination of the overestimation of turbulent viscosity at the cavity interface and closure,due to the averaging nature of RANSE in a region with multiple spatial and temporal scales,resulted in a re-entrant jet seized by the vapor region,in which the intrinsic unsteadiness of the cavitatingflows(and,in turn,the periodic shedding)were unrealistically damped.Similarly,our calculations suggest this behavior and the first cell at the wall for the medium-B mesh of Fig.12e is filled by water,with the exception of the blade leading edge where the vaporization process is stronger.The current implementation of the SST k-ω turbulence model without any Reboud correction(Coutier-Delgosha et al.2003)further exaggerates these issues.In any case,as shown by Li(2015),not even this correction is sufficient,in the case of complex three dimensional configurations,to account for the complexity of bubble shedding.A lso,scale-resolving simulations using LES,as those proposed by Bensow and Bark(2010a,b),Lu et al.(2014),and Asnaghi et al.(2015),were identically ineffective in accurately resolving shedding and bubble cavitation in complex three-dimensional flows.

注浆可沿隧道方向分段进行，根据类似工程的经验，注浆压力控制在0.2 MPa以内，注浆量为待加固土体体积的10%～20%，则对于粉质黏土、粉细砂地层，单孔单位长度注浆量约为0.2 m3，对于中、粗砂地层，单孔单位长度注浆量为0.5 m3，具体的数值应根据现场情况确定。

These results,when compared with those in Figs.2 and 3,are perfectly in line with the average of the collected data and,in non-cavitating conditions in particular,are very close(1%)to the experimental measurements.The analyzed meshes have a cell count between one fifth and one half the coarsest mesh proposed in the Workshop.Consequently,the choice of the medium-size mesh with about 2.4 M elements seem s reasonable for making reliable comparisons with the available experimental observations and for the aims of this paper.

5 Resu lts

5.1 Non-cavitating Performance in Inclined Shaft

Figure 13 shows the results computed using the RANSE solver of the OpenFOAM package in the open-water,towing-tank configuration(case 1).As noted,open-water calculations were performed using the same numerical setup of the cavitating calculation(Table 5)and with the globally arranged medium mesh of case 2.Only the outer domain was extended to avoid any interaction with the boundaries,so the total number of cells was slightly increased.The analysis also includes an envelope of the minimum and maximum values calculated by the various participants at the Workshop.In addition,the vertical bars represent the standard deviation of the available data with respect to the average of the collected values.Figure 14 presents an identical analysis but is based on potential flow calculations.

Our open-water calculations,even with a shaft inclination of 12°,prove the reliability of the OpenFOAM libraries for predicting unsteady propeller performance.

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We observe a similar trend of the thrust coefficient by considering the calculations of Fig.14,which were performed using the BEM based on reliable surface meshes such as those devised by Gaggero and Villa(2016)and Gaggero et al.(2016a,b).Also,with the simplified inviscid approach,the measurement discrepancies at the design advance coefficient are limited to a 2%of underestimation.In correspondence with the loaded conditions,thrust,in contrast,is slightly underestimated(-2%),whereas with respect to the zerothrust condition,the values predicted using the BEM are slightly higher than the measured values(+4%).

We must highlight the qualitatively different torque coefficient behavior,which is however in accordance with almost all the BEM calculations available in the literature and among the Workshop participants(Kinnas et al.2015).The combination of higher angles of attack(lower advance coefficients)with a reasonably high shaft inclination of 12°stresses the BEM to its inherent limits related to the inviscid approximation.The empirical correction for the frictional forces cannot account,for instance,for the non-negligible interaction between inviscid pressure distributions and the relatively thick boundary layers,or the leading edge suction forces.The

resulting torque is significantly underestimated in off-design conditions.At the advance design coefficient,this difference is reasonable(about 1%)but the envelope of the collected results and a rough estimation of the calculation uncertainties based on standard deviation values suggest an overall lower confidence level for the BEM in dealing with these operating conditions.

5.2 Cavitating Performance in Inclined Shaft

We performed cavitating calculations,initially,only for the three operating conditions of Table 2 in correspondence with the measurements and the numerical calculations available from the various Workshop participants.As a second step,we offer a complete analysis of the cavitating performance of the propeller while varying the cavitation index at selected advance coefficients to completely reproduce the measurements and gain insight into the inception and influence of the various cavitating phenomena.

Tables 6(non-cavitating)and 7(cavitating)list the results computed using OpenFOAM.Similar to the open-water analyses,we have included the average plus the minimum and the maximum values collected in the Workshop,together with the relative standard deviations.Table 8 summarizes the results obtained using the BEM.Due to the absence of tunnel walls,we employed the thrust identity assumption to account for the confined flow and we performed BEM calculations based on selected tuned advance coefficients to achieve the same noncavitating thrust.In addition,due to the small number of participants who employed BEMs,we report no analysis of the collected data here.

In contrast,the computed cavity extensions are closer to those observed.Again,due to modeling limitations,bubble cavitation is predicted as sheet cavitation.In this case(case 2.2),however,the mid-chord detachment of the vapor,also foreseen numerically,leaves no doubt about the bubbly nature of the phenomenon.

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Figures 15,17,and 19 also show the observed cavity extensions compared to calculation results using OpenFOAM and the BEM.As usual,we selected a vapor fraction of 0.5 to visualize the cavity bubble computed by the RANSEsolver.

We found overall acceptable agreement between the calculations and measurements in light of the scattered results of the Workshop.For the loaded conditions(case 2.1),both the noncavitating and the cavitating predicted thrust coefficients are in good agreement with the measurements.The thrust breakdown due to the presence of cavitation is successfully predicted(-8%for both measurements and calculations),despite the RANSE results slightly overestimating(+1.5%)both the noncavitating and cavitating performances.

Figure 15 shows the predicted cavity extensions.From an experimental viewpoint,the sheet cavitation from r/R=0.9 to the tip merges with a strong cavitating tip vortex.Cavitating bubbles characterize the blade rootatany angular position and bubbly cavitation relatively close to the blade leading edge is also present on blades in correspondence with the loaded angular position(90°-200°).The pressure side is free from cavitation.

Numerical calculations can sufficiently foresee the following features:the cavitating tip vortex is partially predicted as a vapor region that entirely covers the tip of the blade and extends into the wake.Tip phenomena,however,are under resolved because of the RANSE assumptions and the consequent under-prediction of the turbulent structures in the wake.Lack of sufficient grid resolution in the downstream region favors further numerical smearing of the solution and only massive mesh refinements could begin to deal with the inception and development of tip vortex cavitation.

A lso,the presence of bubble cavitation at the hub is in agreement with the observations,even if numerically the phenomenon takes the form of an unrealistic sheet cavitation and it is slightly under-predicted,especially at 0°.

The most obvious difference with respect to cavitation tunnel observations is the overestimated leading edge sheet cavitation,which is clearly predicted by both the RANSE andBEM solvers in the angular range between 0°and 180°in place of bubbly cavitation.The bubbly cavitation evidenced by the cavitation tunnel observations close to the leading edge isnumerically predicted as a sheet cavity that covers almost all of the suction side of the blade starting from the leading edge and which is attached to the near-tip sheet cavity at r/R=0.9.The multiphase mixture approach cannot accommodate the grow th of individual nuclei and all the Workshop participants who employed VoF and mass transfer models achieved similar results(i.e.,observed bubbly cavitation modeled as sheet cavitation).The reasons for the overestimation may also be found in the assumption of the cavitation model and in the fully turbulent nature of the simulations.As reported by Kuiper(1981)and Franc and Michel(2004),for instance,cavitation may not occur,even if the pressure falls below the vapor tension,as a consequence of an insufficient content of gas nuclei in water or when laminar flow occurs.The latter condition,in principle,could be representative of the flow nature at the leading edge of the blade and at lower radii when operating at lower inflow speed(even if the chord Reynolds number seems sufficiently high to prevent laminar flow).A fully turbulent simulation,consequently,cannot resemble these operating conditions,which results in an overestimated cavitation stimulated by the turbulence itself.Such a condition could partially explain the overestimated sheet cavitation on the blade suction side at the 0°blade position of Fig.15.Moreover,the Schnerr-Sauer cavitation model could affect the development of cavitation,since the vaporization and condensation processes are directly and equally correlated with the pressure differences.The numerical tendency to overestimate sheet cavitation at the blade leading edge has already been evident for this propeller using different mass transfer approximations(Li 2011;Sipila et al.2011;Morgut and Nobile 2011,2012a).Only a well-calibrated full cavitation model has been able,in the simpler case of steady uniform inflow(Morgut and Nobile 2012a),to partially reduce the amount of predicted sheet bubble at the blade leading edge.The mesh resolution,as evidenced in our sensitivity analysis,has almost no influence on the nature of the predicted cavitation that,due to modeling discrepancies,takes the form of a sheet cavity even with(relatively)finer cell densities.

Figure 16 shows details of the predicted cavity bubble and dynamics.From the color of the semi-transparent vapor isosurface,we can deduce the presence of re-entrant jets,which bring water between the cavity and the blade surface:bright red characterizes vapor bubbles that are detached and“distant” from the wetted blade surface(i.e.,over a re-entrant jet)whereas red-brown identifies the portion of the blade with vapor directly on the surface.

Among the phenomena that characterize the cavity dynamics(shedding,secondary cavitation,desinence),the interaction between the re-entrant jets and the development of the tip vortex is the only observable phenomenon with this choice of mesh and simulation modeling.As evidenced by Bensow and Bark(2010a,b)who used LES analyses,the presence of re-entrant jets changes the dynamic of the cavitating tip vortex.Rather than detaching from the blade tip as a consequence of the cross flow from the pressure to the suction side of the blade(or being purely a leading edge vortex,which by nature is closer to the edge of the blade due to the geometric skew),the cavitating vortex develops around the edge of the cavity(Fig.16b,c)thanks to re-entrant jets which favor the roll up process of the trailing part of the bubble.

Qualitatively,beyond the modeling assumptions of panel methods,our BEM calculations are in agreement with the RANSE results in terms of the predicted cavity extension,with the exception o f the tip cavitating vortex.An overestimated leading edge sheet cavitation bubble is predicted in the same angular range of the RANSE calculations,showing,as already evidenced by the non-cavitating RANSE results,a suction peak at the leading edge that is reasonably lower than the vapor pressure,which is not in agreement with the bubbly nature of the cavitating phenomena observed experimentally.Root bubbles are qualitatively evidenced but,as in the case of RANSE calculations,are modeled with sheet cavitation,which is the only model available in the BEM.The predicted performances are affected by the limitations of the BEMs,as has been highlighted in similar calculations(Gaggero and Villa 2016).Due to an excessive bubble pressure recovery in correspondence with a sensibly overestimated cavity extension,the predicted propeller thrust in cavitating conditions is overestimated and even higher for the thrust delivered in non-cavitating operations.

At the design and with an unloaded advance coefficient(case 2.2,Fig.17,and case 2.3,Fig.19),the trendsare similar,with both RANSEand BEM underestimating the thrustreduction due to cavitation.The RANSE calculations predict a thrust breakdown of about 25%for both the loading conditions versus the measured values,which are about 35%lower with respect to the non-cavitating functioning.BEM shows an appreciable thrust reduction(-13%)only in correspondence with the unloaded case,which is far from the measured values,however.

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The cavitating tip vortex,on both the suction and pressure sides,is reasonably underestimated due to the unloading of the blade at a higher advance,which naturally reduces the strength of the vortex,which makes its numerical prediction even more challenging.

Also,the cavitating tip vortex from the leading edge pressure side sheet cavitation at the 270°(cases2.2 and 2.3)and 0°(case 2.3)blade positions is beyond the adopted mesh resolution(compare,for instance,the fine mesh arrangements proposed for case 2.2),but the relevant phenomena on the blade surface are reasonably foreseen.

The presence of jets in the trailing part of the bubble is clear in Figs.18 and 19 and,similar to what we observed for case 2.1,the interaction of the re-entrant jets and the vapor bubble is substantial.The hint of a cavitating vortex,which is obviously under-resolved due to the relatively coarse mesh for sustaining any cavity development in the wake,is well detached from the blade due to the rollup process fostered at the closure of the bubble more than by the natural crossflow between high-and low-p ressure regions by the jets themselves.

Root cavitation is fairly predicted,with both RANSE and BEM slightly overestimating its extension on the suction side for case 2.2.Similarly,the pressure side sheet bubble is overestimated for both loading conditions(and especially for the BEM at the 140°and 210°blade positions),with the exception of the RANSE in case 2.3.These discrepancies may partially justify the underestimation of the thrust reduction in cavitating conditions.

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Figure 20 shows a summary of a more detailed analysis of the cavitating performance of the PPTC propeller,with the predicted cavity extensions by RANSE and BEM shown in Figs.24,25,and 26.At the three advance coefficients of case 2,we repeated the calculations while varying the cavitation index from non-cavitating(σN=8.0)to cavitating(σN=1.0 or 1.5,depending on the case)conditions to compare predicted values with the whole thrust breakdown measurements.

Similarly,in the absence of dedicated observations,we collected and compared the cavity extension predictions for the considered operational points to gain insight into the inception of the various phenomena.We also collected the unsteady performances of a reference blade,as shown in Figs.21,22,and 23,to correlate the cavity extensions with the delivered propeller forces.

Based on our findings,we confirmed our discussion and general observations of cases 2.1-2.3 at the reference cavitation indexes.In the loaded condition,we verified the good agreement between the measurements and RANSE calculations with the OpenFOAM libraries at almost any cavitation index.Our calculations resemble the thrust reduction measurements,even in severe conditions,with a sheet cavity that completely covers the blade suction side atthe 90°-180°blade position and determines the thrust breakdown mainly observable from 0°to 180°in Fig.21.The overestimation of the pressure recovery by the BEM is made even clearer:the thrust breakdown is reasonably postponed and only at very low cavitation indexes is it possible to appreciate a reduction in the propeller performance due to cavitation.

In contrast,the cavitation bubble extensions by the two approaches are in reasonable agreement.Both codes foresee a leading edge,suction side cavitation for angular positions between 0°and 180°,root bubbles,and a non-cavitating pressure side.At any cavitation index,the suction side bubble appears as leading edge cavitation,which confirms the modeling limitations(or possible discrepancies in the predicted leading edge pressure)of both codes in dealing with bubbly cavitation.At the design and for the unloaded functioning condition,the numerical predictions confirm the reduction trend but the more the blade is unloaded,the more the thrust breakdown is postponed with respect to the measurements,and in particular(as already observed in similar cases by Gaggero and Villa 2016)for the inviscid flow calculations.These results contrast with the calculations of Morgut and Nobile(2012a),whose numerical predictions for the E779A propeller using calibrated mass transfer modelsshowed earlier performance degradation due to cavitation than in tunnel experiments.The prevalent sheet nature of the cavity bubble on that propeller together with a certain overestimation of the cavity extension(Salvatore et al.2009;Vaz et al.2015)may partially explain this behavior.

Again,the bubbly nature of the suction side cavitation is not predicted but can be deduced from the mid-chord detachment of the unrealistic sheet bubble identified by both approaches.

The presence of leading edge pressure side cavitation is similarly computed,with the BEM calculations slightly overestimating its extension at the root and a thigher cavitation indexes.Both suction and pressure side cavitation contribute to the reduction of blade forces,as illustrated in Figs.22,23.The degradation of blade performance in the range 0°-180°for the design condition is significant and ascribable to the presence of the bubbleover almost all the blade suction side as shown in Figs.24,25 and 26 also for very off design conditions.Pressure side cavitation,instead,seems to be less important in reducing the delivered thrust.In correspondence with the unloaded condition,the reduction in the propeller performance,in contrast,is equally distributed between the suction and pressure sides.

6 Conclusions

In this article,we applied the RANSE solver of the OpenFOAM libraries to predict the non-cavitating and cavitating performances in inclined shaft conditions of the well-known model-scale PPTC test case.We also made a comparison,further than has been done experimentally,with the numerical results obtained by participants in the Second Workshop on Cavitation and Propeller Performance.In addition,we proposed boundary element method calculations with a proprietary code.Because of the broadly heterogeneous results of the Workshop,we conducted a preliminary mesh sensitivity analysis to verify the reliability and accuracy of the open-source tool to define the lightest numerical setup for conducting daily affordable calculations.

The analysis of the propeller performance,and our comparison with measurements and calculations,confirmed the reliability of the OpenFOAM approach for predicting the open-water characteristics of marine propellers even in relatively high inclined shaft,off-design conditions.Our noncavitating calculations were in very good agreement with the measurements and perfectly in line with the calculations proposed in the Workshop.

In cavitating conditions,the agreement was slightly worse but comparable with similar calculations carried out by various institutions attending the Workshop.We observed no significant differences in our comparison of the present viscous approach(with the current computationally efficient mesh setup)with,for instance,OpenFOAM calculations performed using finer meshes and more accurate turbulence modeling.Both calculations showed almost identical cavitation phenomena,with very similar predicted root bubbles and cavitating tip vortexes,limited only by the spatial resolution and the order of accuracy of the adopted computational setup.The agreement with the experimental results was good,in particular for the loaded condition,albeit with a clear overestimation of a leading edge sheet bubble in place of sporadic bubbly cavitation.This phenomenon,which is common to all the calculations proposed in the Workshop,was also closely predicted by the BEM calculations,which basically suggests an under-pressure region from which the cavitation originates,regardless of the cavitation model.This difference,which we can attribute to the partially laminar nature of the model scale flow at the blade leading edge,and which cannot be accounted for in fully turbulent simulations nor by BEMs,indicates the necessity of further calibrations of the masstransfer model.In particular,the influence of the vaporization coefficient,which can moderate the production of vapor even when the pressure is below saturation,is worthy of analysis to enhance the model-scale leading edge predictions together with the use of the increasingly available laminar-turbulent boundary-layer transition models.Moreover,even small geometrical differences in curvature between the model propeller and the CAD model may potentially justify the difference in the pressure distributions(and,in turn,in cavitation predictions)observed in our calculations as well as in the Workshop results with respect to the blade leading edge.

In design and unloaded conditions,the differences between our measurements and the calculations are more significant.The thrust reduction is underestimated as a consequence(which is well evidenced by the analysis of the thrust cavitation diagrams)of a global numerical postponement of the thrust breakdown.Also,cavitation that is qualitatively in agreement with observations of the extension and position on the blade has the form of a sheet cavity rather than bubble cavitation due to the same limitations of the computational model evidenced in correspondence with the loaded condition.Mesh refinements,adopted for instance in the preliminary sensitivity analysis,were ineffective in dealing with this phenomenon and,asconfirmed by the results collected during the Workshop,even calculations with several million of cells and the same homogeneous mixture approach,irrespective of the mass transfer model,were unable to deal with bubble cavitation.In addition to laminar-turbulent turbulence closure relations,partial improvements of the proposed calculations would consist of the application of turbulence models that are able to account for local compressibility effects like those that characterize cavitation bubble closure.RANSE assumptions are subject to a sensible overprediction of the turbulent viscosity in the mixture region,which dampens the bubbles’unsteadiness and partially prevents the shearing effect due to the re-entrant jets,which are partially appreciable in the interaction with the cavitating tip vortex.Such models,combined with finer meshes suitable for the prediction of accurate reentrant jets,may improve the accuracy of the analyses by facilitating more reliable prediction of unsteady cavitation phenomena that are particularly severe at the root or,in the case of bubble-like cavitation,can affect pressure distributions and,in turn,forces.

Ultimately,we found the OpenFOAM calculations to provide a reasonable prediction of the PPTC performance,which was even better depending on the computational resources employed.We obtained all the results with a mesh having half the number of cells of the smallest mesh adopted in the Workshop.This configuration definitely allows daily affordable calculations and partially opens the way to the application of these kinds of tools for design(rather than only analysis)problems that have traditionally used inviscid theories such as lifting line,lifting surface,or optimization by BEM.The exact boundary element method calculations showed acceptable accuracy in open-water conditions,in particular for the thrust prediction and close to the design point,as well as the capability for predicting qualitative trends(occurrence of cavitation rather than reliable prediction of cavitation influence on performance)when considering cavitating conditions.We found significant limitations of the boundary element method in very off-design conditions.However,and comparatively speaking,boundary element methods can still be successfully and routinely employed as computationally efficient and accurate tools for preliminary analysis and design purposes.

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