Fourth International Symposium on Marine Propulsorssmp’15, Austin, Texas, USA, June 2015

Cavitating Flow Calculations for the E779A Propeller in Open Water and BehindConditions: Code Comparison and Solution Validation

Guilherme Vaz1, David Hally2, Tobias Huuva3, Norbert Bulten4, Pol Muller5, Paolo Becchi6, Jose L. R. Herrer7,Stewart Whitworth8, Romain Mace9, Andrei Korsstrom10

1 MARIN, The Netherlands. 2 Defence Research and Development Canada. 3 Caterpillar, Sweden.4 Wartsila, The Netherlands. 5 DCNS Research, France. 6 CETENA, Italy. 7 Navantia, Spain.

8 Lloyd’s Register, UK. 9 DGA Techniques hydrodynamiques, France. 10 ABB, Finland.

ABSTRACT

As part of the Cooperative Research Ships SHARCS project, cal-culations of the E779A propeller in open water and in a cavitationtunnel behind wake generating plates have been performed by tendifferent institutions using eight different flow codes. Both fullRANS and RANS-BEM coupled approaches have been used topredict wetted and cavitating flows. Propeller performance char-acteristics, pressure distributions, limiting-streamlines and cav-itation volumes have been analyzed. Cavitation patterns havebeen compared with photographs. In addition, pressure fluctu-ations at the cavitation tunnel walls and at some hydrophoneshave been computed and compared against available experimen-tal data. For loading and cavitation extents, there is good agree-ment among the different calculations. Compared with measure-ments, the predicted cavity extents are good but the propellerthrust for the behind condition was uniformly under-predicted.The overall agreement is an improvement over earlier studies us-ing the same data set. The pressure fluctuations predicted byhalf the participants were in reasonable agreement with measure-ments, but the remaining calculations predicted pressure levels afactor of four or five too high.

Keywords

CFD, BEM, URANS, RANS+RANS, RANS+BEM, Propellers,E779A, Open water, Behind condition, Cavitation, Pressurefluctuations, ANSYS R© CFX R©, ANSYS Fluent R©, Excalibur,FINETM/Marine, OpenFOAM R©, PROCAL, ReFRESCO,Star CCM+ R©

1 Introduction

At present, viscous flow Reynolds-Averaged Navier-Stokes(RANS) codes for marine propellers, thrusters, complex propul-sors, and even complete ships with propulsor arrangements, areavailable within the maritime industry. During the design ofpropulsors, potential flow tools such as Boundary Element Meth-ods (BEM) are still the work-horse codes; for analysis, viscousflow tools using RANS, URANS or even hybrid DES/SAS ap-proaches are taking over. This is due to wide availability of com-mercial and open-source CFD solvers, as well as cheaper andmore powerful hardware which, allied with parallelization ac-celeration techniques, make calculations feasible now that wereimpossible in the past.

However, even though wetted flow CFD calculations for propul-sors are becoming straightforward exercises, the same is not truefor cavitating flow simulations which are inherently unsteady andmore expensive, and are done using cavitation models which aresemi-empirical and not yet mature enough. Also, for accuratepredictions of pressure fluctuations on ship hulls caused by thedynamic behaviour of the cavity, the interaction between tur-bulence modelling, cavitation modelling and numerical detailsneeds to be completely controlled. This is not an easy task evenwhen considering only the first harmonic of the propeller blade-passage frequency.

In SMP-2009 [1], results obtained for the well-known E779Abenchmark test-case [2] with five different CFD codes werepublished. Both open water flow and the flow behind a pre-scribed non-uniform velocity field (trying to model the effectof a wake stimulator) were considered. The results were com-pared against BEM results and experimental data. Bensow andBark [3] present implicit LES results for this case. In both theseworks, only performance characteristics and qualitative compar-isons of cavity extents were made for a simplified geometricalsetup and for few flow conditions.

In SMP-2011 a workshop was organized for two common test-cases: the twisted Delft foil and the Potsdam Propeller Test Case(PPTC). For the propeller case [4], the emphasis was only onthe comparison of open water cavitating flow performance pre-dictions; the PPTC propeller had little cavitation mostly at theroot. The ITTC-2011 committee [5] gave a detailed assessmentof the status of numerical tools for simulation of cavitation andassociated pressure fluctuations. They found that: “Promisingadvances in propulsion simulation by both in-house and com-mercial CFD software are made during the last three years. Itseems necessary to model the true geometry of propeller andmake the computational mesh sufficiently fine for both hull andpropeller, in order that the propulsion factors are predicted moreaccurately. Meanwhile, the body-force approach remains to bean alternative that is efficient and easy-to-use for engineeringpurposes. There is, however, a lack of benchmark data for thevalidation of propulsion prediction. Further R&D work for nu-merical propulsion simulations are proposed as follow: 1) Studyof numerical uncertainties arising from mesh resolution, turbu-lence modelling and numerical discretization schemes; 2) Fullscale propulsion prediction; 3) Prediction of cavitation and fluc-tuating pressure for propeller operating behind the hull.”

c© Cooperative Research Ships and Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2015.

In this context, within the Cooperative Research Ships (CRS)consortium (www.crships.org), the SHARCS (Ship Hydrody-namic Advanced RANS Cavitation Simulations) working-groupis currently investigating the performance of state-of-the-art CFDtools for predicting pressure fluctuations on ships due to cavitat-ing propellers. Three cases with increasing complexity are beingconsidered: an open water propeller, a propeller in the behindcondition and a fully appended and propelled ship. In this paperresults from the first two cases are presented. The E779A pro-peller was again used as a test case because it is one of the onlyhigh-quality publicly available test cases which considers bothopen water and behind conditions with a large amount of avail-able experimental data. Due to the advances in the capabilities ofthe codes and hardware available, fewer concessions in terms ofcomputational grids, time-steps and geometrical simplificationshave been made relative to the SMP-2009 study. To date, onlymodel-scale conditions have been considered in CRS SHARCSwork.

Ten different institutions have actively participated in thiswork: ABB-Finland, Caterpillar-Sweden, CETENA-Italy,DCNS Research-France, DGA Techniques hydrodynamiques-France (DGAH), Defence Research and Development Canada(DRDC), Lloyd’s Register-UK (LR), MARIN-Netherlands,Navantia-Spain and Wartsila-Netherlands. Both potential flow(BEM) and viscous flow (RANS) approaches have been consid-ered, including the coupling of both in a so-called RANS-BEMcoupled approach. For the potential flow calculations only onetool (or tool combination) has been used: PROCAL [6] and Ex-calibur [7]. These are tools developed within CRS and avail-able to all participants. For the viscous flow approach, open-source, open-usage and commercial tools have been used (in al-phabetical order): 1) ANSYS CFX [8]; 2) ANSYS Fluent [9];3) FINE/Marine [10]; 4) OpenFOAM [11]; 5) ReFRESCO [12];6) STAR-CCM+ [13]. Due to the number of different partici-pants and codes participating, this study is representative of thecurrent ability of RANS and BEM tools to tackle cavitating flowsimulations on a propeller and associated pressure fluctuations.

This paper presents results for the E779A propeller in open wa-ter and in a cavitation tunnel positioned behind wake generatingplates. For open water wetted flow, propeller performance char-acteristics have been quantitatively analyzed. For cavitating flow,pressure distributions, limiting-streamlines, the cavity patterns,cavity volume and sensitivity to the volume fraction chosen torepresent the cavity have been studied. For the behind condition,attention was paid to the nominal velocity field behind the wakegenerating plates. Cavitating flow predictions for the propelleroperating behind the stationary plates were then made. The samequantities as for the open water case have been analyzed. In ad-dition, pressure fluctuations at the tunnel walls and at some hy-drophones have been computed and compared against availableexperimental data.

The paper is organized as follows. The E779A propeller test-cases are described in Section 2. Section 3 gives details of thenumerical approaches, numerical settings and grids used. Theresults for open water conditions are presented in Section 4 fol-lowed by the results for the behind condition in Section 5. Fi-nally, Section 6 presents conclusions made from the study.

2 Test-Case

The test-case consists of the flow around the INSEAN E779Apropeller inside a cavitation tunnel [2]: Fig. 1. A compre-hensive series of experimental data addressing the propeller inuniform, as well as non-homogeneous, flow was gathered atINSEAN over the last decade [2, 14, 15]. The E779A pro-peller has also previously been subject of calculations by theEU VIRTUE project [1,16], the EU STREAMLINE project [17]and the CRS PROCAL project [18]. In the current work, thecavitation patterns, cavitation dynamics and associated pressurefluctuations are the major focus. Nevertheless, wetted flow cal-culations are also considered and whenever relevant comparedagainst experimental data. In order to minimize the differencesbetween the calculations, the geometry, physical and numericalconditions were the same for all participants. The following sec-tion describes the set-up in detail.

Figure 1: E779A propeller inside the cavitation tunnel.

2.1 Geometries

INSEAN E779A is a four-bladed, fixed pitch, right-handedmodel propeller, originally designed in 1959. Its diameter, D,is 227.27 mm. Details of its geometry can be found in [2,14,15].During the VIRTUE project the actual geometry of the propellerblade was measured and an IGES file created. The IGES ge-ometry was cleaned and smoothed and a new solid geometricaldescription was prepared and used in the current work.

2.2 Open Water Calculations

The cavitation tunnel of Fig. 1 has a test section with squarecross-section of width 0.6m and length 2.6m. To simplify thecomputational modelling of the propeller in open water condi-tions, an idealized tunnel was used having a circular cross sectionequal in area to the actual tunnel, as was done in [1, 16].

Fig. 2 illustrates the prescribed computational domain; the do-main diameter was 2.942D. A common definition of boundaryconditions was used: prescribed velocity and turbulence intensityat the inlet, uniform pressure at the outlet (although not strictlycompatible with swirling flow), and slip at the tunnel walls. No-slip conditions were required on shaft, fairing, hub, propeller andcap surfaces. While the propeller, hub and cap rotated with raten, the shaft and fairing were stationary. We emphasize that this

test case corresponds to the cavitation-tunnel set-up and not thetowing-tank set-up, also available in [2]. In the experiments, noroughness was used at the leading edges to stimulate transitionto turbulence. The following physical parameters were used:

• T = 20◦C, ρ = 998 kg/m3, µ = 1.008 × 10−3Ns/m2,ν = 1.01 × 10−6m2/s.

• pvap = 2.337 × 103Pa, ρvap = 0.017kg/m3,µvap = 1.02 × 10−5Ns/m3.

• Uniform velocity field V = Vin at the inlet.• Inlet and background turbulence intensity of 2%.• Propeller rotation rate n = 36 rps, with the different advance

coefficients, J = Vin/nD, being obtained by changing theinflow velocity Vin.

For wetted flow, an open water diagram was calculated. Theadvance coefficients J = 0.71 and J = 0.83 were obligatoryconditions since those were also used for the cavitating flowcases. For cavitating flow three conditions were consid-ered: 1) (J = 0.71, σn = 0.63); 2) (J = 0.71, σn = 1.763); 3)(J = 0.83, σn = 1.029), where σn = (p− pref)/

12ρn

2D2. Inorder to keep the conditions trim-independent relative to themoving blade, gravity was not included. The reference pres-sure used in the definition of the cavitation number was the out-let pressure. The Reynolds number based on chord length atr/R = 0.7 is 5× 105 when J = 0.71. Therefore the flow cannotbe considered fully turbulent; rather, it is critical or even sub-critical for the lower radii sections where large parts of laminar,transitional and turbulent flow can coexist.

Figure 2: Domain and boundary conditions for the open watercalculations.

2.3 Calculations in the Behind Condition

For the calculations in the behind condition, the cavitation tun-nel and some of its components were modelled. Several details

of the original geometry were not described in any of the avail-able references [2,14,15]: some decisions were made so the flowdomain and tunnel were the same for all participants. The tunnel,wake generator and shaft/hub were generated so that the centerof the propeller plane is at (0, 0, 0). The cross section of thetunnel has a rectangular section with sides of 0.6m and cornerradius of 0.1m. The length of the tunnel was chosen to be 2.2m.The tunnel was longitudinally positioned so that the distance be-tween the propeller plane and the outlet was 4D, as was donefor the open water case. The wake generator was modelled asdescribed in [15]. The length of the spacer where it attaches tothe tunnel roof was unknown and therefore was assumed to be0.76m in the longitudinal direction. The dimensions of the shaftwere also unknown and therefore simplified and estimated. Thevertical plate upstream of the wake generator which attaches theshaft to the tunnel roof, clearly visible in the photo of Fig. 1, wasomitted. The larger radius of the shaft upstream of the wake gen-erator was estimated as 0.08m and it was extended all the wayto the inlet.

A common definition of boundary conditions was also requiredfor this case: they were the same as for the open water calcula-tions except that the tunnel walls and wake stimulator were no-slip walls. Fig. 3 illustrates the chosen domain and boundaryconditions.

Figure 3: Domain and boundary conditions for the calculationsin the behind condition.

The physical parameters were the same as for the open watertest case except that the rotation rate was n = 30.5 rps corre-sponding to (open water) J = 0.897. The propeller operated ina non-uniform wake caused by the boundary layers of the platesand other appendages. The interaction between the stationaryand rotating parts of the computational domain was modelled ac-cording to the CFD codes’ and/or users’ best practices.

For this case, only one loading condition, J = 0.897, wascomputed. For wetted flow conditions, both a nominal-wakeand a total-wake case were considered: i.e. a case without thepropeller and with the operating propeller, respectively. Thenominal velocity field in a plane 0.26D upstream of the pro-peller disk was compared among all participants and comparedwith existing time-averaged experimental data [15]. With thepropeller included, the emphasis lies on the propeller perfor-mance, cavitation dynamics and pressure fluctuations. Two cav-itation numbers were considered, σ = 2.5 and σ = 5.5, whereσ = (p − pref)/

12ρV

2in. For the cavitation dynamics, the cal-

culations and experiments were compared at different propellerblade positions. Four pressure sensors, P1 to P4, were placedon the tunnel walls directly to right, left, above and below thecentre of the propeller disk. Four hydrophones, H1 to H4, were

placed in a vertical line on the centreplane one radius aft of thepropeller disk and 80mm, 100mm, 120mm and 200mm be-low its centre, respectively. Fig. 4 shows their locations in thetunnel and relative to the propeller. Pressure fluctuations werecalculated, analyzed and compared against experiments for theseeight locations.

Figure 4: Locations of pressure taps and hydrophones.

3 Numerical Formulation

3.1 BEM Approach

As mentioned above, both BEM potential flow and CFD vis-cous flow tools, and their combination, have been used in thecurrent work, the emphasis of this paper being however on theCFD tools.

BEM open water calculations, whether for wetted or for cavitat-ing flow, are usually performed in an infinite domain (no tunnelwalls) assuming that the flow is steady. The propeller then op-erates on an undisturbed uniform velocity field V = Vin. Thesecalculations are straightforward and computationally very effi-cient, with computational times of the order of minutes, max-imum hours, on a typical workstation. Vaz and Bosschers [19]provide details of the set-up of a BEM calculation using the samecases that are studied here. For the behind condition, a potential-flow-only approach implies that the propeller is operating in theeffective wake (see Carlton [20] for more details), which takesinto account the velocity deficit due to the plates’ wake and hasto be determined a priori using some other method. The calcu-lations are then unsteady and, even though still computationallyefficient, the computation times are usually of the order of manyhours, maximum a day, also on a workstation.

In the current work, for open water conditions the same approachis used. However, for the behind condition the effective wakeis not calculated a priori but during the computational process,using RANS-BEM coupling [18, 21, 22, 23]. The procedure usedhere performs a (cheap) steady RANS calculation for the domaincontaining the tunnel, plates and shaft, together with an (alsocheap) unsteady BEM calculation (wetted and cavitating flow)for the propeller blades. The coupling is done iteratively, wherein the RANS domain the propeller is modelled via body-forces,and in the BEM domain the interaction is done via the effective-wake. The detailed procedure employed here by DRDC is ex-plained by Hally [24]. In order to compute the pressure fluctua-tions caused by the cavitating rotating propeller, use of the acous-tic potential flow BEM tool Excalibur is also made. In this case,the hydrodynamic propeller noise/pressure sources computed bythe hydrodynamic BEM are passed to the acoustic BEM codewhich solves the scattering effect of the solid boundaries on the

incident pressures (one has therefore a RANS-BEM-BEM cou-pled approach). Van Wijngaarden [7] provides details of Excal-ibur and hydro-acoustic coupling.

3.2 CFD Approach

3.2.1 Open water calculations: The propeller rotationwas modelled in several ways: 1) using a body-fixed referencesystem, on which calculations can be steady; 2) using an earth-fixed reference system where the complete domain is rotatingand calculations are unsteady; 3) using an earth-fixed referencesystem where only a sub-domain involving the blades is rotating,and sliding-grids or interfaces are needed; 4) using a body orearth-fixed reference system and considering only a 2π/Z angu-lar sector of the computational domain shown in Fig. 2 togetherwith cyclic/periodic boundary conditions. While for the wettedflow case steady RANS calculations could be performed, for thecavitating flow case all calculations were unsteady except thoseof CETENA and two of the three performed by Navantia.

All cavitating flow calculations were restarted from the wettedflow converged solution. To obtain the correct cavitation numberσ, some participants decreased σ from a high value to the lowerdesired value in steps in order to prevent large sudden variationsof the cavitation pattern and possibly divergence of the calcula-tion. In addition, some participants modified the reference pres-sure pref to obtain the desired cavitation numbers while othersmodify the vapour pressure, pvap.

Four different cavitation models were used: the Kunz model [25],the Zwart model [26], the Singhal model [27] and the Sauermodel [28]. There are several differences in the numerical andphysical behaviour of these models but, in general, and based onprevious experience of the authors, one can state that the Kunzmodel has the numerical advantage (and physical disadvantage)of having the condensation source/sink term independent of thepressure p and the Sauer model of being the most realistic, andtherefore the most widely used. Huuva [29] provides a detaileddescription and a comparison of these models as applied to pro-pellers.

All partners used eddy-viscosity turbulence models, variants ofthe k − ε or k − ω models, commonly used for non-cavitatingflow simulations. Caterpillar however used the Reboud cavitat-ing flow correction [30], a damping of the eddy-viscosity in themixture region. All partners except MARIN used wall-functionsfor near-wall turbulence model boundary conditions and there-fore had high y+ values.

3.2.2 Calculations in the Behind Condition: All calcu-lations in the behind condition were unsteady and used sliding-grids/interfaces in an earth-fixed reference frame. In some cases,the wetted flow calculations were initially done using a quasi-steady frozen-rotor approach, followed by the fully-unsteadywetted flow sliding-interfaces approach. MARIN and LR solvedthe turbulence-model equations up to the wall instead of using awall-function approximation. For the rest of the set-up, most nu-merical choices made for the open water calculations were main-tained here. For the calculation of the pressure fluctuations, nocompressibility effects were considered in the CFD approach and

the pressure values coming directly from the URANS calcula-tions were monitored in time at the required points without anyadditional post-processing.

3.3 Codes, Numerical Settings and Grids

Tabs. 1 and 2 summarize the most relevant numerical choices andsettings used for all cavitating flow calculations performed. Formore details on the tools, the reader may consult the respectiveweb sites cited. One may observe that the tools, grid generators,approaches, numerical discretization choices and resolution, arevery heterogenous and representative of the current state-of-the-art for application of CFD to hydrodynamic problems.

Fig. 5 illustrates the grids used for the open water calculations.Fig. 6 shows a slice by the y = 0 plane through each grid used forthe behind condition. For the partners that performed both cases,the blade surface grid of the open water calculation presented inFig. 5 was kept similar for the calculations in the behind condi-tion (though the MARIN grid was finer close the leading-edge).However, in the rotating sub-grid, most partners refined the gridsat the propeller slip-stream but not all the way to the tunnel wallsor to the locations of the hydrophones.

4 Results: Open Water Calculations

4.1 Wetted Flow

All results now presented are based on the previously de-scribed numerical settings and best-practice guidelines of eachpartner and numerical codes. The iterative, spatial and time-discretization convergence of each calculation is therefore con-sidered adequate. Some partners have performed extra studies tostudy these effects but those results are not shown here.

Fig. 7 compares the results of six partners for the thrust, torqueand efficiency in an open water diagram. The experimental data[2, 14, 15] were measured in a towing-tank at a slightly differentReynolds number and are therefore used here simply for qualita-tive comparison. The difference between all numerical resultsis less than 5% for KT and KQ, and 10% for η. Comparedwith earlier studies (e.g. the EU VIRTUE project [1]), the spreadbetween the results is lower in the current work. When com-pared with the experimental data, the averaged differences areeven lower than 5%. The differences in KT and KQ are largerat higher propeller loadings (lower J). The potential flow resultsfor the open water diagram have similar accuracies as the viscousflow ones, although there is a small but consistent over-predictionof KQ.

Fig. 8 shows the distribution of the pressure coefficient Cpn onthe propeller blades and Fig. 9 shows Cpn versus x/c for twodifferent radial sections: r/R = 0.7 and 0.9. The pressuredistributions differ mainly for contours between Cpn = 2.0 and3.0. There are also point-to-point oscillations in some cases dueto interpolation errors by the visualization package at hangingnodes: they are not in the original solution. This is corroboratedby Fig. 9 which does not show any oscillations. Also, Fig. 9shows that all codes (including the potential flow code) agreereasonably well, with some differences at the leading-edge, suc-tion peak and trailing-edge. Close to the propeller tip these dif-ferences are larger. These small discrepancies are caused by the

(a) ABB+PROCAL. (b) Caterpillar+OpenFOAM.

(c) CETENA+ANSYS CFX. (d) DCNS+FINE/Marine.

(e) MARIN+ReFRESCO. (f) Navantia+STAR-CCM+.

(g) Wartsila+STAR-CCM+.

Figure 5: Grids for the open water calculations.

different levels of grid resolution in those zones, different turbu-lence models and boundary conditions and intrinsic inviscid flowassumptions of the BEM code for the trailing edge.

4.2 Cavitating Flow

In this section we present first several results for the nominalcondition with J = 0.71 and σn = 1.763, then the comparisonbetween all numerical results and the experimental data for allthree conditions.

An interesting issue that has been discussed in the past [1] is howto define the cavity surface. As an iso-surface of the vapour vol-ume fraction? For which value? And are the cavity extents sensi-tive to this value? Or based on the iso-surface where −Cp = σ?Figs. 10 and 11 show the pressure distribution on the blade andat some radial sections. One can see a clear −Cpn = σn contourin Fig. 10 and line in Fig. 11 but with some differences betweenall partners, especially when close to the cavity detachment andre-attachment. It is also known that there must be a pressurelower than the vapour pressure in order for cavitation to start and

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(c) DRDC+ANSYS CFX (d) LR+STAR-CCM+.

(e) MARIN+ReFRESCO. (f) Wartsila+STAR-CCM+.

Figure 6: Grids for the calculations in the behind condition.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2

J

KT

10KQ η

Experiment

ABB Caterpillar CETENA

MARIN Navantia Wärtsilä

Figure 7: Open water wetted flow KT , KQ and η.

that this under-pressure is very much dependent on the cavitationmodel. Nevertheless, the low pressures within the cavity seen inthe Caterpillar and DCNS distributions in Fig. 11 are consideredoutliers.

Fig. 12 shows the influence of the value of the vapour volumeiso-surface on the cavity extents for one calculation; however, itis representative of all the calculations and independent of thecavitation model used. One can see that the lower the value thelarger the cavity. However, varying the value between 10% and

(a) ABB. (b) Caterpillar.

(c) DCNS. (d) MARIN. (e) Wartsila.

Figure 8: Open water wetted flow Cpn distribution. J = 0.71.

−16

−12

−8

−4

0

4 0 0.2 0.4 0.6 0.8 1

Cp

n

Fractional Chord Length

r/R = 0.9

ABB Caterpillar

CETENA DCNS

MARIN Navantia Wärtsilä

−6

−4

−2

0

2

4

6 0 0.2 0.4 0.6 0.8 1

Cp

n

Fractional Chord Length

r/R = 0.7

Figure 9: Open water wetted flow Cpn vs x/c on radial sectionsr/R = 0.7 and 0.9. J = 0.71.

50% causes only small qualitative differences in the cavity ex-tents. Notice that for a potential flow approach, as in PROCAL,this is not an issue since well-defined dynamic and kinematicboundary conditions are used to define the cavity surface (see [6]for more details). Based on these results we chose a vapour vol-ume iso-surface of 0.10 (10% of vapour in a grid cell) to definethe cavity surface.

Figs. 13–15 show the cavity extents for all participants, for thethree flow conditions, together with the available experimentaldata. Several interesting trends are worth emphasizing:

• For the nominal condition with J = 0.71 and σn = 1.763,the comparison between all numerical results, and against theexperimental data, can be considered good. Nevertheless,some calculations under-predict the extents while others over-predict them. Also, the viscous flow approaches detect cavita-

(a) ABB. (b) Caterpillar.

(c) CETENA. (d) DCNS.

(e) MARIN. (f) Navantia. (g) Wartsila.

Figure 10: Open water Cpn distribution for cavitating flow.J = 0.71, σn = 1.763.

−8

−6

−4

−2

0

2

0 0.2 0.4 0.6 0.8 1

Cp

n

Fractional Chord Length

r/R = 0.9

ABB Caterpillar

CETENA DCNS

MARIN Navantia Wärtsilä

−4

−2

0

2

4

0 0.2 0.4 0.6 0.8 1

Cp

n

Fractional Chord Length

r/R = 0.7

Figure 11: Open water cavitating flow Cpn vs x/c for radialsections r/R = 0.7, 0.9. J = 0.71, σn = 1.763.

tion at the blade lower radii which is not present in the experi-ments nor in the potential flow code predictions.

• When J = 0.71 and σn = 0.630, the difference betweenthe potential flow and viscous flow approaches and the ex-periments are very large. While in the experiments one canvery clearly see bubble cavitation, the potential flow calcula-

(a) αv = 0.1. (b) αv = 0.5. (c) αv = 0.9.

Figure 12: Influence of αv iso-surface on cavitation extent.J = 0.71, σn = 1.763.

tion does not show any cavitation and the viscous flow ap-proaches (all of them!) consistently show a super-cavitatingsheet. It should be noted, however, that the potential flow cal-culation predicted pressure levels below the vapour pressure atthe lower radii, but because only sheet cavitation is modelledand because the detachment point was constrained to be at theleading edge, the algorithm did not let cavitation occur.

• When J = 0.83 and σn = 1.029, a very small low pressure re-gion is visible for all calculations. Depending on the cavitationmodel and its settings, some of the RANS calculations predicta small tongue-shaped mid-chord sheet cavity. The potentialflow cavitation model does not permit mid-chord cavitation.

• In general, the MARIN, Navantia and Wartsila results lookvery similar to each other, even though the codes, grids andnumerical settings are very different. The common numericalfeature between these three approaches is the Sauer cavitationmodel.

• The pressure distributions predicted by both potential and vis-cous flow approaches are considered to be accurate and similarfor the seven different calculations. All of them predict regionsof pressure lower than vapour pressure that are larger than thecavity extents present in the experiments. But it is also knownfrom the literature (for example see Kuiper [38] or Franc [39])that cavitation does not occur in water with low nuclei content,even with high under-pressures, and also not when the flow islaminar. For that reason, leading edge roughness is commonlyused at some model-basin facilities to promote cavitation. Thedifferences between the numerical results and the experimentsare thought to be explained by this lack of nuclei, roughnessand associated turbulent flow.

• In general, the results of all viscous flow calculations are verymuch alike, even when using completely different codes, nu-merical settings and turbulence/cavitation models. This is re-assuring and shows the maturity of CFD for cavitating flows.

5 Results: Behind Condition

5.1 Nominal Wake Flow

Fig. 16 presents the computed axial velocity distribution normal-ized by the inlet velocity, Vx/Vin, at a plane 0.26D upstreamof the propeller disk. In Fig. 16 all results have been obtainedusing a steady RANS approach. The differences between them

(a) ABB+PROCAL. (b) Caterpillar+OpenFOAM. (c) CETENA+ANSYS CFX. (d) DCNS+FINE/Marine.

(e) MARIN+ReFRESCO. (f) Navantia+STAR-CCM+. (g) Wartsila+STAR-CCM+. (h) Experiments.

Figure 13: Open water cavity extents. J = 0.71, σn = 1.763.

(a) ABB+PROCAL. (b) Caterpillar+OpenFOAM. (c) CETENA+ANSYS CFX. (d) DCNS+FINE/Marine.

(e) MARIN+ReFRESCO. (f) Navantia+STAR-CCM+. (g) Wartsila+STAR-CCM+. (h) Experiments.

Figure 14: Open water cavity extents. J = 0.71, σn = 0.630.

(a) ABB+PROCAL. (b) Caterpillar+OpenFOAM. (c) CETENA+ANSYS CFX. (d) DCNS+FINE/Marine.

(e) MARIN+ReFRESCO. (f) Navantia+STAR-CCM+. (g) Wartsila+STAR-CCM+. (h) Experiments.

Figure 15: Open water cavity extents. J = 0.83, σn = 1.029.

are clear, with better similarity between MARIN and LR, andDGAH and Wartsila. Nevertheless, the wake is very sharp withlarge gradients in the circumferential direction, much larger thanwould normally be experienced behind a ship. In some cases,a strong asymmetry is also visible as well as large low-velocityareas which correspond to the wakes of the rods connecting theplanes.

The large variation in the steady wakes is due to unsteady vonKarman vortex shedding from the rods; they cannot be capturedcorrectly with a steady approach. Time-averaged wakes from un-steady calculations performed by some partners compared muchbetter with the experimental data (see Fig. 17) but the experimentstill shows a region of velocity deficit below the shaft that is notfound in the calculations.

5.2 Wetted Flow

Tab. 3 lists the predicted averaged loads, Kt and Kq , for the pro-peller rotating behind the wake generating plates. The measuredloads are Kt = 0.175 and 10Kq = 0.334 [15] so the values arevery low for all partners. The spread in computed averaged loadsis around 11% for Kt, about twice as large as in the open watercase; the reason for the relatively poor agreement is unclear. ForKq , the spread is 5%, about the same as for the open water case.

Partner-Code Avg. Kt Avg. 10Kq

Caterpillar+OpenFOAM 0.154 0.303DGAH+ANSYS Fluent 0.147 0.298DRDC+ANSYS CFX 0.145 0.291

+PROCALLR+STAR-CCM+ 0.140 0.289

MARIN+ReFRESCO 0.151 0.300Wartsila+STAR-CCM+ 0.139 0.302

Table 3: Mean wetted flow loads.

Fig. 18 shows the variation of thrust and torque for one blade pas-sage; one can see that the minimum thrust/torque is achieved ata blade angle near θ = 30◦ and the maximum at 80◦, for all vis-cous flow calculations. The potential flow results present someoscillations deviating from the normal sinusoidal-like behaviourof thrust and torque temporal distribution.

Fig. 19 shows Cpn on radial sections r/R = 0.7 and 0.9 forone blade angle, θ = 0◦. For this angle the pressure distribu-tions at two high-radii sections are very similar for all calcula-tions. The three major visible differences are: 1) the oscillationson the potential flow results, which are due to the wake velocitycontent and the lack of diffusion in a potential flow code; 2) dif-ferent values of the minimum pressure, a quantity very sensitiveto the leading-edge grid quality and resolution; 3) a hump in thepressure distribution for section r/R = 0.90 which is due to aleading-edge vortex, only captured by the calculations not usingwall-functions.

5.3 Cavitating Flow

Fig. 20 shows the variation with time of the cavity volume. Theerror bars in this figure represent the standard deviation com-puted from the several steady-state revolutions (different for all

(a) Caterpillar. (b) DGAH. (c) DRDC.

(d) LR. (e) MARIN. (f) Wartsila.

Figure 16: Nominal axial velocity at x/D = −0.26 for the be-hind condition. J = 0.897.

(a) MARIN+ReFRESCO. (b) Experimental data.

Figure 17: Axial velocity at x = −0.26D for the behind condi-tion. (left) Time-averaged unsteady calculations; (right) Time-averaged experimental LDV data. J = 0.897.

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0 30 60 90

Angle (degrees)

KT0.26

0.28

0.30

0.32

0.34

0 30 60 90

Angle (degrees)

10KQ

Caterpillar DGAH DRDCLR MARIN Wärtsilä

Figure 18: Wetted flow propeller loads for the behind condition.J = 0.897.

partners and computed when available). Notice that this volumecomprises the complete domain and not only one blade or onlythe propeller; if cavitation appears on the plates or inside the hubvortex this volume considers those locations too.

Fig. 20 shows that the cavity volume variation is qualitativelythe same for all viscous flow approaches, the results of the po-tential flow approach being different in terms of amplitude andphase. The results of the viscous flow approach present sharpertemporal variations, and the maximum volume does not occur at0◦ but closer to 15◦. Two viscous flow results show high cavityvolume outside the range of the wake peak. It is known from

−10

−8

−6

−4

−2

0

2

4

6 0 0.2 0.4 0.6 0.8 1

Cp

n

Fractional Chord Length

r/R = 0.7

−25

−20

−15

−10

−5

0

0 0.2 0.4 0.6 0.8 1

Cp

n

Fractional Chord Length

r/R = 0.9

Caterpillar DGAH DRDC

LR MARIN Wärtsilä

Figure 19: Cpn vs x/c on radial sections r/R = 0.7 and 0.9 forwetted flow in the behind condition. θ = 0◦, J = 0.897.

0

1

2

3

4

5

6

−40 −20 0 20 40

Ca

vity V

olu

me

(cm

3)

Angle (degrees)

DGAHDRDC

LRMARIN

Wärtsilä

Figure 20: Cavity volume for the behind condition. J = 0.897,σ = 2.5.

the classical theory of cavitation and propellers [39,20], that thesecond time-derivative of the cavity volume is the main contri-bution to the cavitating flow pressure fluctuations and thereforethe results presented in Fig. 20 are important for understandingand explaining the results obtained for the pressure fluctuationspresented below.

Fig. 21 shows the variation of KT and KQ over one blade pas-sage when σ = 2.5. The differences between the numerical re-sults have increased relative to the open water flow. Non-physicalhigh-frequency oscillations appear in the Wartsila calculations.

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0 30 60 90

Angle (degrees)

KT

0.22

0.24

0.26

0.28

0.30

0.32

0.34

0 30 60 90

Angle (degrees)

10KQ

Caterpillar DGAH DRDCLR MARIN Wärtsilä

Figure 21: Cavitating flow propeller loads for the behind condi-tion. J = 0.897, σ = 2.5.

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0 30 60 90

Angle (degrees)

Wettedσ = 5.5σ = 2.5

(a) LR+STAR-CCM+.

0.12

0.13

0.14

0.15

0.16

0.17

0.18

0 30 60 90

Angle (degrees)

Wettedσ = 5.5σ = 2.5

(b) MARIN+ReFRESCO.

Figure 22: Kt for wetted and cavitating flow in the behind con-dition. J = 0.897.

The effect of cavitation on the loads can be seen in Fig. 22 whichcompares the loads calculated by MARIN and LR for wettedflow and for σ = 2.5 and σ = 5.5; the results of DGAH andWartsila were qualitatively similar. For the σ = 2.5 conditionthe effect is considerable between 10◦ and 40◦, with an increasein loading occurring during the collapse of the cavity. A smalldecrease of performance is seen while the cavity is growing, i.e.between 70◦ and 10◦ (or −20 and 10◦ in Fig. 20).

Figs. 23 and 24 show the calculated pressure distributions forσ = 2.5 when the blade is at θ = 0◦. The results from all calcu-lations are similar, with the differences between viscous and po-tential flow approaches being visible especially close to the cav-ity re-attachment area. Differences at the beginning of the cavityare also visible with some results showing pressures lower thanvapour pressure. While for a potential flow approach p = pvap

is an imposed dynamic boundary condition, for a viscous flowcavitation modelling approach an under-pressure is needed tofeed the evaporation process. However, the amount of the under-pressure and the size of the area where it occurs depend on thecavitation model, its constants and the grid resolution.

(a) Caterpillar. (b) DGAH. (c) DRDC.

(d) LR. (e) MARIN. (f) Wartsila.

Figure 23: Cpn distribution for cavitating flow in the behind con-dition. θ = 0◦, J = 0.897, σ = 2.5.

Figs. 25–27 and Figs. 28–30 show the results obtained for thecavity extents for σ = 2.5 and σ = 5.5, respectively. Onlyangles θ = −20◦, 0◦ and +20◦ were chosen for illustrative pur-poses. For the viscous flow results, the cavity is represented byan iso-surface of vapour volume equal to 10%. The followingobservations can be made:

−6

−4

−2

0

2

4

6 0 0.2 0.4 0.6 0.8 1

Cp

n

Fractional Chord Length

r/R = 0.7

−4

−3

−2

−1

0

1

2

3

4 0 0.2 0.4 0.6 0.8 1

Cp

n

Fractional Chord Length

r/R = 0.9

Caterpillar DGAH DRDC

LR MARIN Wärtsilä

Figure 24: Cpn vs x/c on radial sections r/R = 0.7 and 0.9 forcavitating flow in the behind condition. θ = 0◦, J = 0.897, σ =2.5.

• In general, the agreement between the viscous flow results andthe experiments is good, for both cavitation numbers, and ismuch better than was found for open water conditions. The ex-perimental results now show a cavity that extends to low radialsections in a smooth distribution likely due to the non-uniformwake that disturbs the flow enough to promote cavitation in-ception and growth (see Kuiper [38]).

• For θ = −20◦, the potential flow results show an isolatedpatch of cavitation at the lower radii similar to the open wa-ter potential flow results at J = 0.71 and σn = 0.63 (seeFig. 14a). This deserves further investigation (Vaz and Boss-chers [19] showed cavitation at lower radial sections thanshown here). This is probably due to the modelling usedfor detachment-point, but RANS-BEM coupling for cavitatingflow conditions could also play a role.

• Apart from one calculation, all the results seem to under-predict the cavity extents. This is explained by the under-prediction of the propeller loads previously discussed.

• LR predicts a cavitating hub vortex; these were the calcula-tions using the finest grid of all partners. LR, MARIN andWartsila, all using the Sauer cavitation model, predict verysimilar cavity extents. A small cavitating tip-vortex is seen insome of the calculations. At 20◦, only Caterpillar predicts themid-chord cavitation seen in the experiments.

5.4 Pressure Fluctuations

In this section only the results for the lowest cavitation numberσ = 2.5 will be presented. Figs. 31 and 32 show histograms ofthe first four harmonics of blade passage frequency at the sen-sors H1, inside the propeller wake, and P2, on the wall to theleft of the propeller. The results from the other sensors are qual-itatively similar. Figs. 33 and 34 show the time histories of thepressure for the same sensors. For these plots, the time historiesof the experimental data were obtained from the 48 harmonics of

shaft rate frequency provided in the experimental data set [15].Similarly, the DRDC time history at P1 was obtained from fourharmonics of blade rate frequency calculated using Excalibur;DRDC is not included in the results at H1 as Excalibur cannotprovide realistic predictions of the pressure in the wake. Sinceneither the experimental nor the DRDC data sets include an ab-solute pressure reference, the mean pressure was subtracted fromall data sets to aid in comparison. It should be noted that, for theexperiments, the rotation angle of the blades at time t = 0 is notknown. Therefore, the experimental curve may be shifted rightor left by an arbitrary amount.

The pressure fluctuations predicted by Caterpillar, LR andWartsila are four to five times higher than the experimental val-ues and, as can be seen by comparing the upper portions of Figs.33 and 34, the time histories of the pressure are insensitive to thesensor location, the results for all pressure taps and hydrophonesbeing almost the same. The LR and Wartsila predictions arevery similar; each used STAR-CCM+ with the Sauer cavitationmodel. The Caterpillar prediction is similar except that it lacksthe pronounced peaks near 3, 11, 20 and 28msecs; they usedOpenFOAM with the Kunz cavitation model.

DGAH, DRDC and MARIN predict more realistic pressure fluc-tuations when compared with the reconstructed experimental sig-nal. For the P sensors, the amplitude tends to be under-predicted,perhaps due to the lower loading of the propeller relative to theexperiment. At some of the P sensors, DGAH and MARIN pre-dict the complete signal quite well, as is the case for MARINin Fig. 34. At the H sensors, there was generally worse agree-ment between the numerical and experimental values which isunderstandable given the rapid coarsening of the grids in the ax-ial direction. Nevertheless, the values obtained by MARIN andDGAH were of the same order of magnitude as the experimentalvalues. With respect to potential versus viscous flow method re-sults, the results by DRDC are comparable with the more expen-sive URANS approaches; however, Excalibur is not capable ofpredicting the pressure inside or very close to the propeller wake,so the pressures at H1, H2 and H3 were not correctly computed.Even though the RANS-BEM coupling approach does not sufferfrom numerical diffusion, the pressure fluctuations predicted bythis method also under-predicted the experimental results.

All organizations predicted P2 to have the highest levels for thefirst harmonic among the P sensors; the experiment shows thatP3 has the highest level with P2 second. All organizations ex-cept Caterpillar correctly predicted that P3 would have higherlevels than P1 (these are the sensors on the left and right walls ofthe tunnel). All organizations except DRDC correctly predictedthat P4 would have the lowest amplitude for the first harmonicamong the P sensors, though in the DRDC results the P1 and P4levels only differ by 2Pa. In the experiments, the amplitudesof the third and fourth harmonics at P2 exceed the first and sec-ond harmonics. This behaviour was not predicted by any of theorganizations.

Two major poorly-understood weaknesses of these calculationsare the high-frequency oscillations seen in the results of somepartners even for non-cavitating loads, and the very high pressurefluctuations predicted by some partners. To try to shed some

(a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e) MARIN. (f) Wartsila. (g) Experiments.Figure 25: Cavity extents for the behind condition. θ = −20◦. J = 0.897, σ = 2.5.

(a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e) MARIN. (f) Wartsila. (g) Experiments.Figure 26: Cavity extents for the behind condition. θ = 0◦, J = 0.897, σ = 2.5.

(a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e) MARIN. (f) Wartsila. (g) Experiments.Figure 27: Cavity extents for the behind condition. θ = +20◦, J = 0.897, σ = 2.5.

(a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e) MARIN. (f) Wartsila. (g) Experiments.Figure 28: Cavity extents for the behind condition. θ = −20◦, J = 0.897, σ = 5.5.

(a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e) MARIN. (f) Wartsila. (g) Experiments.Figure 29: Cavity extents for the behind condition. θ = 0◦, J = 0.897, σ = 5.5.

(a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e) MARIN. (f) Wartsila. (g) Experiments.Figure 30: Cavity extents for the behind condition. θ = +20◦, J = 0.897, σ = 5.5.

0

0.5

1

1.5

2

2.5

3

1 2 3 4

Am

plit

ud

e (

kP

a)

Harmonic

ExperimentCaterpillar

DGAHLR

MARINWärtsilä

Figure 31: Amplitudes of the first four harmonics of pressure atsensor H1. J = 0.897, σ = 2.5.

0

0.5

1

1.5

2

2.5

3

3.5

1 2 3 4

Am

plit

ud

e (

kP

a)

Harmonic

ExperimentCaterpillar

DGAHDRDC

LRMARIN

Wärtsilä

Figure 32: Amplitudes of the first four harmonics of pressure atsensor P1. J = 0.897, σ = 2.5.

−5

0

5

10

15

0 5 10 15 20 25 30

Pre

ssu

re (

kP

a)

Time (msecs)

CaterpillarLR

WärtsiläExperiment

−1.5−1

−0.5 0

0.5 1

1.5 2

0 5 10 15 20 25 30

Pre

ssu

re (

kP

a)

Time (msecs)

DGAHMARIN

Experiment

Figure 33: Pressure time history at sensor H1. J = 0.897,σ = 2.5.

−5

0

5

10

15

0 5 10 15 20 25 30

Pre

ssu

re (

kP

a)

Time (msecs)

CaterpillarLR

WärtsiläExperiment

−1

0

1

2

0 5 10 15 20 25 30

Pre

ssu

re (

kP

a)

Time (msecs)

DGAHDRDC

MARINExperiment

Figure 34: Pressure time history at sensor P1. J = 0.897,σ = 2.5.

light on these issues, two extra studies were performed: 1) a de-tailed analysis of the pressure distribution close to the interfacesbetween the non-rotating and the rotating grids; 2) a comparisonbetween all numerical predictions for the pressure fluctuationsfor the wetted flow case.

For the two partners using STAR-CCM+, Fig. 35 shows Cpnclose to the sliding-interfaces at one blade angle. Clearly somepressure perturbations exist at those areas. The results of DGAH,MARIN and Caterpillar (not presented here) showed no abnor-mal pressure jumps between the interfaces, and obviously the re-sults by DRDC do not use interfaces. Therefore, these anomaliescould explain some high-frequency peaks but not the very highpressure fluctuations.

Figure 35: Pressure perturbations close to/at the sliding inter-faces in cavitating flow: (top) LR; (bottom) Wartsila. θ = 0◦,J = 0.897, σ = 2.5.

Fig. 36 shows the results obtained for the wetted flow case at theP1 sensor (LR did not perform these calculations). The predictedpressures are now of the same order of magnitude but high-frequency oscillations are present in the Caterpillar and Wartsilaresults. Even for the smoother DGAH and MARIN results thereare some minor numerical (non-physical) oscillations. However,notice that the scale of the ordinate is much smaller with ampli-tudes of the order of 100 to 200 Pa. The equivalent variation inCpn is ∆Cpn < 0.01. It is possible that the iterative conver-gence per time-step is not low enough to permit such a level ofaccuracy.

−0.3 −0.2 −0.1

0 0.1 0.2 0.3

0 5 10 15 20 25 30

Pre

ssu

re (

kP

a)

Time (msecs)

CaterpillarDGAHDRDC

MARINWärtsilä

Figure 36: Pressure time history at sensor P1 for wetted flow.J = 0.897.

We conclude that: 1) the origin of the outlying pressure fluc-tuations by some of the partners is likely due to the cavitationmodelling (not only the cavitation model itself but its implemen-tation and inter-connection with the other equations solved); 2)the higher-frequency perturbations are probably due to numeri-cal errors at the interfaces and/or lack of iterative convergence,

with or without the sliding-interfaces. This clearly deserves fur-ther investigation, both by the users and by the developers of theviscous flow tools used.

6 Conclusions

In this paper, calculations for the E779A propeller in open waterand in a cavitation tunnel behind wake generating plates havebeen presented. Ten different institutions performed calcula-tions and eight different codes were used including full RANSand RANS-BEM coupled approaches. The results are thereforerepresentative of the state-of-the-art for cavitating simulationson propellers and associated pressure fluctuations using incom-pressible flow (potential and viscous) numerical approaches.

From the open water results one can draw the following majorconclusions:

• Very different grids were used, both structured and unstruc-tured, although unstructured grids were more common. Allexcept two partners used large y+ values and wall-functionboundary conditions. Both viscous flow RANS and potentialflow BEM approaches were employed.

• Despite the variety of methods used, very similar loads wereobtained. Differences between all results are about 5% forKT and KQ. Difference in flow fields are more visible, butnot large.

• There were large differences in cavitation extent which seemdue to the different cavitation models and associated constantsrather than to any other numerical settings. The results ob-tained by different codes using the same cavitation model arevery similar. Differences in cavity size may be due to the lackof turbulence, seeding/nuclei issues and the interaction withcavitation detachment (a known problem with E779A pro-peller).

From the results for the behind condition one can state:

• CPU times varied between the calculations as is to be ex-pected given the different methods employed. Full RANS cal-culations with sliding-interfaces typically take weeks, whileRANS-BEM calculations typically take one or two days.

• For wetted flow, the nominal wake predicted using steadyRANS differed among the partners. However the unsteadytime-averaged nominal wake was close to the (time-averaged)experimental LDV data. Nevertheless, the average thrust valuenumerically predicted was far from the experiments for allcodes/partners. Given the good agreement with the LDV dataand the good prediction of open water performance character-istics, this is difficult to explain and deserves further attention.

• For cavitating flow, the loads and pressure distributions on theblade and their variation with time look qualitatively similar.Once again, predictions by different partners using the samecavitation model are very similar. In general, the cavity ex-tents agreed well with the experimental data though all codesunder-predicted them slightly and the potential flow code haddifficulty modelling the cavitation at lower radii.

• Predictions of pressure fluctuations fell into two groups: onewith high-frequency oscillations and levels over-predicted byfour to five times; the other with a generally fair agreement for

the first harmonic of the blade-passage frequency. The pres-sures at the sensors at the wall were better predicted than atthe hydrophones in the flow. The RANS-BEM approach waslocated in the group with fair agreement with the experiments.

• Although further verification is needed, it is conjectured thatthe high levels of the pressure fluctuations for the cavitatingflow case are due to the cavitation model implementation, andthat the high-frequency oscillations are due to numerical ac-curacy at the interfaces and/or enough iterative convergence toachieve accurate results for very low pressure levels.

Compared with previous comparative studies, the current resultspresent both a better agreement between all viscous and poten-tial flow results and with the experimental data. When comparingthe viscous versus the potential flow approaches, one has to bal-ance the pros and cons of both approaches (accuracy versus cost).From the results obtained here, and ones available in the litera-ture, a RANS-BEM approach seems the best option to use whenfirst tackling a new cavitating flow problem in the behind con-dition; a full RANS calculation can be used once more detailedresults are needed. We must emphasize however that, while afull RANS approach is inherently expensive, a RANS-BEM ap-proach is very sensitive to numerical settings and is not as robustnor as accurate as a full RANS approach. Future work in the CRSSHARCS working group will deal with similar comparative nu-merical studies for a real appended ship plus cavitating propellerconfiguration, associated pressure fluctuations, together with val-idation against available experimental data.

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DISCUSSION

Question from Jacques Andre Astolfi

Did you trigger the turbulence at the leading edge of the pro-pelle model in the experiments? This can have a strong influenceon the cavitation development and could explain differences be-tween experiment and numerical results.

Authors’ closure

As described in Section 2.2, no roughness was used at the leadingedges of the model propeller to stimulate transition to turbulence.As discussed in Section 4.2, we agree that our failure to modelthe effects of transition correctly is probably one of the principalreasons for the relatively poor agreement of the open water cal-culations with the experiments. In future test cases we intend toaddress this deficiency by including transition models.

Question from Sverre Steen

Did you in any way include the effect of laminar flow and transi-tion in the numerical calculations?

Authors’ closure

All the RANS calculations were performed using only turbu-lence models; no transition models have been employed. Thus

laminar-turbulent flow transition is captured too early in terms ofReynolds number, and not accurately enough, as is widely knownfrom the literature on the subject. In this particular open-waterpropeller case, the flow is fully turbulent at the leading edge ofthe propeller. As mentioned in the answer to the previous ques-tion, better modelling of the effects of transition is somethingthat we will be pursuing in the future.