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

Transcription

1 Fourth International Symposium on Marine Propulsors smp 5, Austin, Texas, USA, June 5 Cavitating Flow Calculations for the E779A Propeller in Open Water and Behind Conditions: Code Comparison and Solution Validation Guilherme Vaz, David Hally, Tobias Huuva 3, Norbert Bulten 4, Pol Muller 5, Paolo Becchi 6, Jose L. R. Herrer 7, Stewart Whitworth 8, Romain Macé 9, Andrei Korsström, The Netherlands. Defence Research and Development Canada. 3 Caterpillar, Sweden. 4, The Netherlands. 5 DCNS Research, France. 6 CETENA, Italy. 7 Navantia, Spain. 8 Lloyd s Register, UK. 9 DGA Techniques hydrodynamiques, France. ABB, Finland. ABSTRACT As part of the Cooperative Research Ships SHARCS project, calculations of the E779A propeller in open water and in a cavitation tunnel behind wake generating plates have been performed by ten different institutions using eight different flow codes. Both full RANS and RANS-BEM coupled approaches have been used to predict wetted and cavitating flows. Propeller performance characteristics, pressure distributions, limiting-streamlines and cavitation volumes have been analyzed. Cavitation patterns have been compared with photographs. In addition, pressure fluctuations at the cavitation tunnel walls and at some hydrophones have been computed and compared against available experimental data. For loading and cavitation extents, there is good agreement among the different calculations. Compared with measurements, the predicted cavity extents are good but the propeller thrust for the behind condition was uniformly under-predicted. The overall agreement is an improvement over earlier studies using the same data set. The pressure fluctuations predicted by half the participants were in reasonable agreement with measurements, but the remaining calculations predicted pressure levels a factor of four or five too high. Keywords CFD, BEM, URANS, RANS+RANS, RANS+BEM, Propellers, E779A, Open water, Behind condition, Cavitation, Pressure fluctuations, ANSYS R CFX R, ANSYS Fluent R, Excalibur, FINE TM /Marine, OpenFOAM R, PROCAL, ReFRESCO, Star CCM+ R Introduction At present, viscous flow Reynolds-Averaged Navier-Stokes (RANS) codes for marine propellers, thrusters, complex propulsors, and even complete ships with propulsor arrangements, are available within the maritime industry. During the design of propulsors, potential flow tools such as Boundary Element Methods (BEM) are still the work-horse codes; for analysis, viscous flow tools using RANS, URANS or even hybrid DES/SAS approaches are taking over. This is due to wide availability of commercial and open-source CFD solvers, as well as cheaper and more powerful hardware which, allied with parallelization acceleration techniques, make calculations feasible now that were impossible in the past. However, even though wetted flow CFD calculations for propulsors are becoming straightforward exercises, the same is not true for cavitating flow simulations which are inherently unsteady and more expensive, and are done using cavitation models which are semi-empirical and not yet mature enough. Also, for accurate predictions of pressure fluctuations on ship hulls caused by the dynamic behaviour of the cavity, the interaction between turbulence modelling, cavitation modelling and numerical details needs to be completely controlled. This is not an easy task even when considering only the first harmonic of the propeller bladepassage frequency. In SMP-9 [], results obtained for the well-known E779A benchmark test-case [] with five different CFD codes were published. Both open water flow and the flow behind a prescribed non-uniform velocity field (trying to model the effect of a wake stimulator) were considered. The results were compared against BEM results and experimental data. Bensow and Bark [3] present implicit LES results for this case. In both these works, only performance characteristics and qualitative comparisons of cavity extents were made for a simplified geometrical setup and for few flow conditions. In SMP- a workshop was organized for two common testcases: the twisted Delft foil and the Potsdam Propeller Test Case (PPTC). For the propeller case [4], the emphasis was only on the comparison of open water cavitating flow performance predictions; the PPTC propeller had little cavitation mostly at the root. The ITTC- committee [5] gave a detailed assessment of the status of numerical tools for simulation of cavitation and associated pressure fluctuations. They found that: Promising advances in propulsion simulation by both in-house and commercial CFD software are made during the last three years. It seems necessary to model the true geometry of propeller and make the computational mesh sufficiently fine for both hull and propeller, in order that the propulsion factors are predicted more accurately. Meanwhile, the body-force approach remains to be an alternative that is efficient and easy-to-use for engineering purposes. There is, however, a lack of benchmark data for the validation of propulsion prediction. Further R&D work for numerical propulsion simulations are proposed as follow: ) Study of numerical uncertainties arising from mesh resolution, turbulence modelling and numerical discretization schemes; ) Full scale propulsion prediction; 3) Prediction of cavitation and fluctuating 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, 5.

2 In this context, within the Cooperative Research Ships (CRS) consortium ( the SHARCS (Ship Hydrodynamic Advanced RANS Cavitation Simulations) working-group is currently investigating the performance of state-of-the-art CFD tools for predicting pressure fluctuations on ships due to cavitating propellers. Three cases with increasing complexity are being considered: an open water propeller, a propeller in the behind condition and a fully appended and propelled ship. In this paper results from the first two cases are presented. The E779A propeller was again used as a test case because it is one of the only high-quality publicly available test cases which considers both open water and behind conditions with a large amount of available experimental data. Due to the advances in the capabilities of the codes and hardware available, fewer concessions in terms of computational grids, time-steps and geometrical simplifications have been made relative to the SMP-9 study. To date, only model-scale conditions have been considered in CRS SHARCS work. Ten different institutions have actively participated in this work: 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), -Netherlands, Navantia-Spain and -Netherlands. Both potential flow (BEM) and viscous flow (RANS) approaches have been considered, including the coupling of both in a so-called RANS-BEM coupled approach. For the potential flow calculations only one tool (or tool combination) has been used: PROCAL [6] and Excalibur [7]. These are tools developed within CRS and available to all participants. For the viscous flow approach, opensource, open-usage and commercial tools have been used (in alphabetical order): ) ANSYS CFX [8]; ) ANSYS Fluent [9]; 3) FINE/Marine []; 4) OpenFOAM []; 5) ReFRESCO []; 6) STAR-CCM+ [3]. Due to the number of different participants and codes participating, this study is representative of the current ability of RANS and BEM tools to tackle cavitating flow simulations on a propeller and associated pressure fluctuations. This paper presents results for the E779A propeller in open water and in a cavitation tunnel positioned behind wake generating plates. For open water wetted flow, propeller performance characteristics have been quantitatively analyzed. For cavitating flow, pressure distributions, limiting-streamlines, the cavity patterns, cavity volume and sensitivity to the volume fraction chosen to represent the cavity have been studied. For the behind condition, attention was paid to the nominal velocity field behind the wake generating plates. Cavitating flow predictions for the propeller operating behind the stationary plates were then made. The same quantities as for the open water case have been analyzed. In addition, pressure fluctuations at the tunnel walls and at some hydrophones have been computed and compared against available experimental data. The paper is organized as follows. The E779A propeller testcases are described in Section. Section 3 gives details of the numerical approaches, numerical settings and grids used. The results for open water conditions are presented in Section 4 followed by the results for the behind condition in Section 5. Finally, Section 6 presents conclusions made from the study. Test-Case The test-case consists of the flow around the INSEAN E779A propeller inside a cavitation tunnel []: Fig.. A comprehensive series of experimental data addressing the propeller in uniform, as well as non-homogeneous, flow was gathered at INSEAN over the last decade [, 4, 5]. The E779A propeller has also previously been subject of calculations by the EU VIRTUE project [, 6], the EU STREAMLINE project [7] and the CRS PROCAL project [8]. In the current work, the cavitation patterns, cavitation dynamics and associated pressure fluctuations are the major focus. Nevertheless, wetted flow calculations are also considered and whenever relevant compared against experimental data. In order to minimize the differences between the calculations, the geometry, physical and numerical conditions were the same for all participants. The following section describes the set-up in detail. Figure : E779A propeller inside the cavitation tunnel.. Geometries INSEAN E779A is a four-bladed, fixed pitch, right-handed model propeller, originally designed in 959. Its diameter, D, is 7.7 mm. Details of its geometry can be found in [,4,5]. During the VIRTUE project the actual geometry of the propeller blade was measured and an IGES file created. The IGES geometry was cleaned and smoothed and a new solid geometrical description was prepared and used in the current work.. Open Water Calculations The cavitation tunnel of Fig. has a test section with square cross-section of width.6 m and length.6 m. To simplify the computational modelling of the propeller in open water conditions, an idealized tunnel was used having a circular cross section equal in area to the actual tunnel, as was done in [, 6]. Fig. illustrates the prescribed computational domain; the domain diameter was.94 D. A common definition of boundary conditions was used: prescribed velocity and turbulence intensity at the inlet, uniform pressure at the outlet (although not strictly compatible with swirling flow), and slip at the tunnel walls. Noslip conditions were required on shaft, fairing, hub, propeller and cap surfaces. While the propeller, hub and cap rotated with rate n, the shaft and fairing were stationary. We emphasize that this

3 test case corresponds to the cavitation-tunnel set-up and not the towing-tank set-up, also available in []. In the experiments, no roughness was used at the leading edges to stimulate transition to turbulence. The following physical parameters were used: T = C, ρ = 998 kg/m 3, µ =.8 3 Ns/m, ν =. 6 m /s. p vap = P a, ρ vap =.7kg/m 3, µ vap =. 5 Ns/m 3. Uniform velocity field V = V in at the inlet. Inlet and background turbulence intensity of %. Propeller rotation rate n = 36 rps, with the different advance coefficients, J = V in /nd, being obtained by changing the inflow velocity V in. For wetted flow, an open water diagram was calculated. The advance coefficients J =.7 and J =.83 were obligatory conditions since those were also used for the cavitating flow cases. For cavitating flow three conditions were considered: ) (J =.7, σ n =.63); ) (J =.7, σ n =.763); 3) (J =.83, σ n =.9), where σ n = (p p ref )/ ρn D. In order to keep the conditions trim-independent relative to the moving blade, gravity was not included. The reference pressure used in the definition of the cavitation number was the outlet pressure. The Reynolds number based on chord length at r/r =.7 is 5 5 when J =.7. Therefore the flow cannot be considered fully turbulent; rather, it is critical or even subcritical for the lower radii sections where large parts of laminar, transitional and turbulent flow can coexist. of the original geometry were not described in any of the available references [,4,5]: some decisions were made so the flow domain and tunnel were the same for all participants. The tunnel, wake generator and shaft/hub were generated so that the center of the propeller plane is at (,, ). The cross section of the tunnel has a rectangular section with sides of.6 m and corner radius of. m. The length of the tunnel was chosen to be. m. The tunnel was longitudinally positioned so that the distance between the propeller plane and the outlet was 4D, as was done for the open water case. The wake generator was modelled as described in [5]. The length of the spacer where it attaches to the tunnel roof was unknown and therefore was assumed to be.76 m in the longitudinal direction. The dimensions of the shaft were also unknown and therefore simplified and estimated. The vertical plate upstream of the wake generator which attaches the shaft to the tunnel roof, clearly visible in the photo of Fig., was omitted. The larger radius of the shaft upstream of the wake generator was estimated as.8 m and it was extended all the way to the inlet. A common definition of boundary conditions was also required for this case: they were the same as for the open water calculations except that the tunnel walls and wake stimulator were noslip walls. Fig. 3 illustrates the chosen domain and boundary conditions. Figure 3: Domain and boundary conditions for the calculations in the behind condition. Figure : Domain and boundary conditions for the open water calculations..3 Calculations in the Behind Condition For the calculations in the behind condition, the cavitation tunnel and some of its components were modelled. Several details The physical parameters were the same as for the open water test case except that the rotation rate was n = 3.5 rps corresponding to (open water) J =.897. The propeller operated in a non-uniform wake caused by the boundary layers of the plates and other appendages. The interaction between the stationary and rotating parts of the computational domain was modelled according to the CFD codes and/or users best practices. For this case, only one loading condition, J =.897, was computed. For wetted flow conditions, both a nominal-wake and a total-wake case were considered: i.e. a case without the propeller and with the operating propeller, respectively. The nominal velocity field in a plane.6d upstream of the propeller disk was compared among all participants and compared with existing time-averaged experimental data [5]. With the propeller included, the emphasis lies on the propeller performance, cavitation dynamics and pressure fluctuations. Two cavitation numbers were considered, σ =.5 and σ = 5.5, where σ = (p p ref )/ ρv in. For the cavitation dynamics, the calculations and experiments were compared at different propeller blade positions. Four pressure sensors, P to P4, were placed on the tunnel walls directly to right, left, above and below the centre of the propeller disk. Four hydrophones, H to H4, were

4 placed in a vertical line on the centreplane one radius aft of the propeller disk and 8 mm, mm, mm and mm below its centre, respectively. Fig. 4 shows their locations in the tunnel and relative to the propeller. Pressure fluctuations were calculated, analyzed and compared against experiments for these eight locations. Figure 4: Locations of pressure taps and hydrophones. 3 Numerical Formulation 3. BEM Approach As mentioned above, both BEM potential flow and CFD viscous flow tools, and their combination, have been used in the current work, the emphasis of this paper being however on the CFD tools. BEM open water calculations, whether for wetted or for cavitating flow, are usually performed in an infinite domain (no tunnel walls) assuming that the flow is steady. The propeller then operates on an undisturbed uniform velocity field V = V in. These calculations are straightforward and computationally very efficient, with computational times of the order of minutes, maximum hours, on a typical workstation. Vaz and Bosschers [9] provide details of the set-up of a BEM calculation using the same cases that are studied here. For the behind condition, a potentialflow-only approach implies that the propeller is operating in the effective wake (see Carlton [] for more details), which takes into account the velocity deficit due to the plates wake and has to be determined a priori using some other method. The calculations are then unsteady and, even though still computationally efficient, the computation times are usually of the order of many hours, maximum a day, also on a workstation. In the current work, for open water conditions the same approach is used. However, for the behind condition the effective wake is not calculated a priori but during the computational process, using RANS-BEM coupling [8,,, 3]. The procedure used here performs a (cheap) steady RANS calculation for the domain containing the tunnel, plates and shaft, together with an (also cheap) unsteady BEM calculation (wetted and cavitating flow) for the propeller blades. The coupling is done iteratively, where in the RANS domain the propeller is modelled via body-forces, and in the BEM domain the interaction is done via the effectivewake. The detailed procedure employed here by DRDC is explained by Hally [4]. In order to compute the pressure fluctuations caused by the cavitating rotating propeller, use of the acoustic potential flow BEM tool Excalibur is also made. In this case, the hydrodynamic propeller noise/pressure sources computed by the hydrodynamic BEM are passed to the acoustic BEM code which solves the scattering effect of the solid boundaries on the incident pressures (one has therefore a RANS-BEM-BEM coupled approach). Van Wijngaarden [7] provides details of Excalibur and hydro-acoustic coupling. 3. CFD Approach 3.. Open water calculations: The propeller rotation was modelled in several ways: ) using a body-fixed reference system, on which calculations can be steady; ) using an earthfixed reference system where the complete domain is rotating and calculations are unsteady; 3) using an earth-fixed reference system where only a sub-domain involving the blades is rotating, and sliding-grids or interfaces are needed; 4) using a body or earth-fixed reference system and considering only a π/z angular sector of the computational domain shown in Fig. together with cyclic/periodic boundary conditions. While for the wetted flow case steady RANS calculations could be performed, for the cavitating flow case all calculations were unsteady except those of CETENA and two of the three performed by Navantia. All cavitating flow calculations were restarted from the wetted flow converged solution. To obtain the correct cavitation number σ, some participants decreased σ from a high value to the lower desired value in steps in order to prevent large sudden variations of the cavitation pattern and possibly divergence of the calculation. In addition, some participants modified the reference pressure p ref to obtain the desired cavitation numbers while others modify the vapour pressure, p vap. Four different cavitation models were used: the Kunz model [5], the Zwart model [6], the Singhal model [7] and the Sauer model [8]. There are several differences in the numerical and physical behaviour of these models but, in general, and based on previous experience of the authors, one can state that the Kunz model has the numerical advantage (and physical disadvantage) of having the condensation source/sink term independent of the pressure p and the Sauer model of being the most realistic, and therefore the most widely used. Huuva [9] provides a detailed description and a comparison of these models as applied to propellers. All partners used eddy-viscosity turbulence models, variants of the k ɛ or k ω models, commonly used for non-cavitating flow simulations. Caterpillar however used the Reboud cavitating flow correction [3], a damping of the eddy-viscosity in the mixture region. All partners except used wall-functions for near-wall turbulence model boundary conditions and therefore had high y + values. 3.. Calculations in the Behind Condition: All calculations in the behind condition were unsteady and used slidinggrids/interfaces in an earth-fixed reference frame. In some cases, the wetted flow calculations were initially done using a quasisteady frozen-rotor approach, followed by the fully-unsteady wetted flow sliding-interfaces approach. and LR solved the turbulence-model equations up to the wall instead of using a wall-function approximation. For the rest of the set-up, most numerical choices made for the open water calculations were maintained here. For the calculation of the pressure fluctuations, no compressibility effects were considered in the CFD approach and

5 the pressure values coming directly from the URANS calculations were monitored in time at the required points without any additional post-processing. 3.3 Codes, Numerical Settings and Grids Tabs. and summarize the most relevant numerical choices and settings used for all cavitating flow calculations performed. For more details on the tools, the reader may consult the respective web sites cited. One may observe that the tools, grid generators, approaches, numerical discretization choices and resolution, are very heterogenous and representative of the current state-of-theart 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 = plane through each grid used for the behind condition. For the partners that performed both cases, the blade surface grid of the open water calculation presented in Fig. 5 was kept similar for the calculations in the behind condition (though the grid was finer close the leading-edge). However, in the rotating sub-grid, most partners refined the grids at the propeller slip-stream but not all the way to the tunnel walls or to the locations of the hydrophones (a) ABB+PROCAL. (b) Caterpillar+OpenFOAM. (c) CETENA+ANSYS CFX. (d) DCNS+FINE/Marine. (e) +ReFRESCO. (f) Navantia+STAR-CCM+. Results: Open Water Calculations Wetted Flow All results now presented are based on the previously described numerical settings and best-practice guidelines of each partner and numerical codes. The iterative, spatial and timediscretization convergence of each calculation is therefore considered adequate. Some partners have performed extra studies to study these effects but those results are not shown here. Fig. 7 compares the results of six partners for the thrust, torque and efficiency in an open water diagram. The experimental data [, 4, 5] were measured in a towing-tank at a slightly different Reynolds number and are therefore used here simply for qualitative comparison. The difference between all numerical results is less than 5% for KT and KQ, and % for η. Compared with earlier studies (e.g. the EU VIRTUE project []), the spread between the results is lower in the current work. When compared with the experimental data, the averaged differences are even lower than 5%. The differences in KT and KQ are larger at higher propeller loadings (lower J). The potential flow results for the open water diagram have similar accuracies as the viscous flow ones, although there is a small but consistent over-prediction of KQ. Fig. 8 shows the distribution of the pressure coefficient Cpn on the propeller blades and Fig. 9 shows Cpn versus x/c for two different radial sections: r/r =.7 and.9. The pressure distributions differ mainly for contours between Cpn =. and 3.. There are also point-to-point oscillations in some cases due to interpolation errors by the visualization package at hanging nodes: they are not in the original solution. This is corroborated by Fig. 9 which does not show any oscillations. Also, Fig. 9 shows that all codes (including the potential flow code) agree reasonably well, with some differences at the leading-edge, suction peak and trailing-edge. Close to the propeller tip these differences are larger. These small discrepancies are caused by the (g) Wa rtsila +STAR-CCM+. Figure 5: Grids for the open water calculations. different levels of grid resolution in those zones, different turbulence models and boundary conditions and intrinsic inviscid flow assumptions of the BEM code for the trailing edge. 4. Cavitating Flow In this section we present first several results for the nominal condition with J =.7 and σn =.763, then the comparison between all numerical results and the experimental data for all three conditions. An interesting issue that has been discussed in the past [] is how to define the cavity surface. As an iso-surface of the vapour volume fraction? For which value? And are the cavity extents sensitive to this value? Or based on the iso-surface where Cp = σ? Figs. and show the pressure distribution on the blade and at some radial sections. One can see a clear Cpn = σn contour in Fig. and line in Fig. but with some differences between all partners, especially when close to the cavity detachment and re-attachment. It is also known that there must be a pressure lower than the vapour pressure in order for cavitation to start and

6 Settings/Partner ABB Caterpillar CETENA DCNS Navantia Code PROCAL v. OpenFOAM v. ANSYS CFX v5 FINE/Marine v3.. ReFRESCO v.3. Star CCM+ v8.4 Star CCM+ v7.4 Calc.Type Steady Unsteady+MVG+Cyclic-bc Steady+Cyclic-bc Unsteady+SI Steady/Unsteady+AFM Steady/Unsteady+MVG Unsteady+MVG Grid Default (radial) Hybrid (hexahedrals+prisms) Hybrid (tetrahedrals+prisms) Trimmed Hexahedrals Structured Hexahedrals Polyhedrals Trimmed Hexahedrals Provise [3] ANSA TM [3] ANSYS [33] HEXPRESS TM [34] GridPro TM [35] Star CCM+ [3] Star CCM+ [3] MCells ( 4) panels Time-step (Max/Avg y + ) (7, 75) (5, 7) (6, ) (,.) (5, 6) (33, 5) Discretization Default nd order for mom. nd order for mom. nd order for mom. nd order for mom. nd order for mom. nd order for mom. st order for turb. st order for turb. st order for turb. st order for turb. nd order for turb. nd order for turb. st order cav. st order cav. st order cav. st order cav. st order cav. st order cav. nd order in time st order in time st order in time nd order in time st order in time st order in time Turb.Model RNG k-ɛ + Reboud cor. Std. k-ɛ k-ω SST 994 k-ω SST 994 Real. k-ɛ Std./Real. k-ɛ Cav.Model Default Kunz Model Zwart Model Sauer Model Sauer Model Sauer Model Sauer Model Cc = Fc =.3 n = 8 m 3 n = 8 m 3 n = m 3 n = m 3 Ce = 4 Fe = 3 R = 5 m R = 6 m R = 6 m < 6 all residuals Iterative Conv. tol =. L Other Detach.=LE Only blade Only blade Table : Settings for the open water cavitating flow calculations. Settings/Partner Caterpillar DGA DRDC LR Code OpenFOAM v. ANSYS Fluent v3 ANSYS CFX v5+procal v.7 Star CCM+ v8.6 ReFRESCO v.. Star CCM+ v7.4 +Excalibur v.6..3 Calc.Type Unsteady+SI Unsteady+SI Steady(RANS)+Unsteady(PROCAL) Unsteady+SI Unsteady+SI Unsteady+SI +Freq.(Excalibur) Grid Polyhedral Structured Hexahedrals Structured Hexahedrals + BEM Unstructured Polyhedral Structured + Unstructured Hexahedrals Trimmed Hexahedrals ANSA [3] ICEM-CFD [36] Pointwise [37] + Provise [3] Star CCM+ [3] GridPro [35] + HEXPRESS [34] Star CCM+ [3] MCells ( )panels Time-step (Max/Avg y + ) Propeller: (85, ) Propeller: (33, ) Propeller: Propeller:(3,.6) Propeller:(.4,.5) All: (65, ) Tunnel: (45, 83) Tunnel: (3, 3) Tunnel: (3, ) Tunnel: (85,.7) Tunnel:(37, 6) Discretization nd order for mom. nd order for mom. BEM: Default nd order for mom. nd order for mom. nd order for mom. st order for turb. st order for turb. RANS: nd order for mom. nd order for turb. st order for turb. nd order for turb. nd order cav. st order cav. st order for turb. nd order cav. st order cav. st order cav. st order in time st order in time nd order in time nd order in time st order in time Turb.Model RNG k-ɛ + Reboud cor. RNG k-ɛ k-ω SST 994 DES k-ω SST k-ω SST 3 Real. k-ɛ Cav.Model Kunz Model Singhal Model PROCAL: Default Sauer Model Sauer Model Sauer Model Cc = 4 Non-cond.mass frac. = 7 n = m 3 n = 8 m 3 n = m 3 Ce = 5 R = 6 m R = 5 m R = 6 m iterations per time-step 5 5 all residuals iterations per time-step Iterative Conv. 3 iterations per time-step 5 L < or 6 iterations per time-step Other Initialized with FR PROCAL:Detach.=LE, Prescribed wake Initialized with FR Table : Settings for the in-behind cavitating flow calculations.

7 (a) Caterpillar+OpenFOAM. (b) DGAH+ANSYS Fluent. (c) DRDC+ANSYS CFX (d) LR+STAR-CCM+. (a) ABB. (b) Caterpillar. (c) DCNS. (d). (e) Wa rtsila. Figure 8: Open water wetted flow Cpn distribution. J = r/r =.7 (e) +ReFRESCO. (f) Wa rtsila +STAR-CCM+. Cpn 4 Figure 6: Grids for the calculations in the behind condition. 6.9 KQ Fractional Chord Length 6 η.7.6 Cpn r/r =.9 KT Fractional Chord Length ABB CETENA Navantia... ABB.4.6 J.8 Experiment Caterpillar Navantia. CETENA Figure 7: Open water wetted flow KT, KQ and η. that this under-pressure is very much dependent on the cavitation model. Nevertheless, the low pressures within the cavity seen in the Caterpillar and DCNS distributions in Fig. are considered outliers. Fig. shows the influence of the value of the vapour volume iso-surface on the cavity extents for one calculation; however, it is representative of all the calculations and independent of the cavitation model used. One can see that the lower the value the larger the cavity. However, varying the value between % and Caterpillar DCNS Figure 9: Open water wetted flow Cpn vs x/c on radial sections r/r =.7 and.9. J =.7. 5% causes only small qualitative differences in the cavity extents. Notice that for a potential flow approach, as in PROCAL, this is not an issue since well-defined dynamic and kinematic boundary conditions are used to define the cavity surface (see [6] for more details). Based on these results we chose a vapour volume iso-surface of. (% of vapour in a grid cell) to define the cavity surface. Figs. 3 5 show the cavity extents for all participants, for the three flow conditions, together with the available experimental data. Several interesting trends are worth emphasizing: For the nominal condition with J =.7 and σn =.763, the comparison between all numerical results, and against the experimental data, can be considered good. Nevertheless, some calculations under-predict the extents while others overpredict them. Also, the viscous flow approaches detect cavita-

8 (a) ABB. (b) Caterpillar. (a) αv =.. (b) αv =.5. (c) αv =.9. Figure : Influence of αv iso-surface on cavitation extent. J =.7, σn =.763. (c) CETENA. (e). tion does not show any cavitation and the viscous flow approaches (all of them!) consistently show a super-cavitating sheet. It should be noted, however, that the potential flow calculation predicted pressure levels below the vapour pressure at the lower radii, but because only sheet cavitation is modelled and because the detachment point was constrained to be at the leading edge, the algorithm did not let cavitation occur. (d) DCNS. (f) Navantia. (g) Wa rtsila. Figure : Open water Cpn distribution for cavitating flow. J =.7, σn = r/r =.7 Cpn Fractional Chord Length 8 6 r/r =.9 Cpn Fractional Chord Length ABB CETENA Navantia Caterpillar DCNS Figure : Open water cavitating flow Cpn vs x/c for radial sections r/r =.7,.9. J =.7, σn =.763. When J =.83 and σn =.9, a very small low pressure region is visible for all calculations. Depending on the cavitation model and its settings, some of the RANS calculations predict a small tongue-shaped mid-chord sheet cavity. The potential flow cavitation model does not permit mid-chord cavitation. In general, the, Navantia and Wa rtsila results look very similar to each other, even though the codes, grids and numerical settings are very different. The common numerical feature between these three approaches is the Sauer cavitation model. The pressure distributions predicted by both potential and viscous flow approaches are considered to be accurate and similar for the seven different calculations. All of them predict regions of pressure lower than vapour pressure that are larger than the cavity extents present in the experiments. But it is also known from 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 is laminar. For that reason, leading edge roughness is commonly used at some model-basin facilities to promote cavitation. The differences between the numerical results and the experiments are thought to be explained by this lack of nuclei, roughness and associated turbulent flow. In general, the results of all viscous flow calculations are very much alike, even when using completely different codes, numerical settings and turbulence/cavitation models. This is reassuring and shows the maturity of CFD for cavitating flows. 5 tion at the blade lower radii which is not present in the experiments nor in the potential flow code predictions. When J =.7 and σn =.63, the difference between the potential flow and viscous flow approaches and the experiments are very large. While in the experiments one can very clearly see bubble cavitation, the potential flow calcula- 5. Results: Behind Condition Nominal Wake Flow Fig. 6 presents the computed axial velocity distribution normalized by the inlet velocity, Vx /Vin, at a plane.6d upstream of the propeller disk. In Fig. 6 all results have been obtained using a steady RANS approach. The differences between them

9 (a) ABB+PROCAL. (b) Caterpillar+OpenFOAM. (c) CETENA+ANSYS CFX. (d) DCNS+FINE/Marine. (e) +ReFRESCO. (f) Navantia+STAR-CCM+. (g) Wa rtsila +STAR-CCM+. (h) Experiments. Figure 3: Open water cavity extents. J =.7, σn =.763. (a) ABB+PROCAL. (b) Caterpillar+OpenFOAM. (c) CETENA+ANSYS CFX. (d) DCNS+FINE/Marine. (e) +ReFRESCO. (f) Navantia+STAR-CCM+. (g) Wa rtsila +STAR-CCM+. (h) Experiments. Figure 4: Open water cavity extents. J =.7, σn =.63. (a) ABB+PROCAL. (b) Caterpillar+OpenFOAM. (c) CETENA+ANSYS CFX. (d) DCNS+FINE/Marine. (e) +ReFRESCO. (f) Navantia+STAR-CCM+. (g) Wa rtsila +STAR-CCM+. (h) Experiments. Figure 5: Open water cavity extents. J =.83, σn =.9.

10 are clear, with better similarity between and LR, and DGAH and. Nevertheless, the wake is very sharp with large gradients in the circumferential direction, much larger than would normally be experienced behind a ship. In some cases, a strong asymmetry is also visible as well as large low-velocity areas which correspond to the wakes of the rods connecting the planes. The large variation in the steady wakes is due to unsteady von Karman vortex shedding from the rods; they cannot be captured correctly with a steady approach. Time-averaged wakes from unsteady calculations performed by some partners compared much better with the experimental data (see Fig. 7) but the experiment still shows a region of velocity deficit below the shaft that is not found in the calculations. 5. Wetted Flow Tab. 3 lists the predicted averaged loads, K t and K q, for the propeller rotating behind the wake generating plates. The measured loads are K t =.75 and K q =.334 [5] so the values are very low for all partners. The spread in computed averaged loads is around % for K t, about twice as large as in the open water case; the reason for the relatively poor agreement is unclear. For K q, the spread is 5%, about the same as for the open water case. Partner-Code Avg. K t Avg. K q Caterpillar+OpenFOAM DGAH+ANSYS Fluent DRDC+ANSYS CFX PROCAL LR+STAR-CCM ReFRESCO.5.3 +STAR-CCM Table 3: Mean wetted flow loads. Fig. 8 shows the variation of thrust and torque for one blade passage; one can see that the minimum thrust/torque is achieved at a blade angle near θ = 3 and the maximum at 8, for all viscous flow calculations. The potential flow results present some oscillations deviating from the normal sinusoidal-like behaviour of thrust and torque temporal distribution. Fig. 9 shows C pn on radial sections r/r =.7 and.9 for one blade angle, θ =. For this angle the pressure distributions at two high-radii sections are very similar for all calculations. The three major visible differences are: ) the oscillations on the potential flow results, which are due to the wake velocity content and the lack of diffusion in a potential flow code; ) different values of the minimum pressure, a quantity very sensitive to the leading-edge grid quality and resolution; 3) a hump in the pressure distribution for section r/r =.9 which is due to a leading-edge vortex, only captured by the calculations not using wall-functions. 5.3 Cavitating Flow Fig. shows the variation with time of the cavity volume. The error bars in this figure represent the standard deviation computed from the several steady-state revolutions (different for all (a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e). (f). Figure 6: Nominal axial velocity at x/d =.6 for the behind condition. J =.897. (a) +ReFRESCO. (b) Experimental data. Figure 7: Axial velocity at x =.6D for the behind condition. (left) Time-averaged unsteady calculations; (right) Timeaveraged experimental LDV data. J = K Q.3 K T Angle (degrees) Angle (degrees) Caterpillar DGAH DRDC LR Figure 8: Wetted flow propeller loads for the behind condition. J =.897. partners and computed when available). Notice that this volume comprises the complete domain and not only one blade or only the propeller; if cavitation appears on the plates or inside the hub vortex this volume considers those locations too. Fig. shows that the cavity volume variation is qualitatively the same for all viscous flow approaches, the results of the potential flow approach being different in terms of amplitude and phase. The results of the viscous flow approach present sharper temporal variations, and the maximum volume does not occur at but closer to 5. Two viscous flow results show high cavity volume outside the range of the wake peak. It is known from

11 C pn C pn r/r = Fractional Chord Length r/r = Fractional Chord Length Caterpillar DGAH DRDC LR Figure 9: C pn vs x/c on radial sections r/r =.7 and.9 for wetted flow in the behind condition. θ =, J =.897. Cavity Volume (cm 3 ) DGAH DRDC LR 4 4 Angle (degrees) Figure : Cavity volume for the behind condition. J =.897, σ =.5. the classical theory of cavitation and propellers [39,], that the second time-derivative of the cavity volume is the main contribution to the cavitating flow pressure fluctuations and therefore the results presented in Fig. are important for understanding and explaining the results obtained for the pressure fluctuations presented below. Fig. shows the variation of K T and K Q over one blade passage when σ =.5. The differences between the numerical results have increased relative to the open water flow. Non-physical high-frequency oscillations appear in the calculations K T K Q Angle (degrees) Angle (degrees) Caterpillar DGAH DRDC LR Figure : Cavitating flow propeller loads for the behind condition. J =.897, σ = Wetted σ = 5.5 σ = Angle (degrees) (a) LR+STAR-CCM Wetted σ = 5.5 σ = Angle (degrees) (b) +ReFRESCO. Figure : K t for wetted and cavitating flow in the behind condition. J =.897. The effect of cavitation on the loads can be seen in Fig. which compares the loads calculated by and LR for wetted flow and for σ =.5 and σ = 5.5; the results of DGAH and were qualitatively similar. For the σ =.5 condition the effect is considerable between and 4, with an increase in loading occurring during the collapse of the cavity. A small decrease of performance is seen while the cavity is growing, i.e. between 7 and (or and in Fig. ). Figs. 3 and 4 show the calculated pressure distributions for σ =.5 when the blade is at θ =. The results from all calculations are similar, with the differences between viscous and potential flow approaches being visible especially close to the cavity re-attachment area. Differences at the beginning of the cavity are also visible with some results showing pressures lower than vapour pressure. While for a potential flow approach p = p vap is an imposed dynamic boundary condition, for a viscous flow cavitation modelling approach an under-pressure is needed to feed the evaporation process. However, the amount of the underpressure and the size of the area where it occurs depend on the cavitation model, its constants and the grid resolution. (a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e). (f). Figure 3: C pn distribution for cavitating flow in the behind condition. θ =, J =.897, σ =.5. Figs. 5 7 and Figs. 8 3 show the results obtained for the cavity extents for σ =.5 and σ = 5.5, respectively. Only angles θ =, and + were chosen for illustrative purposes. For the viscous flow results, the cavity is represented by an iso-surface of vapour volume equal to %. The following observations can be made:

12 C pn C pn r/r = Fractional Chord Length r/r = Fractional Chord Length Caterpillar DGAH DRDC LR Figure 4: C pn vs x/c on radial sections r/r =.7 and.9 for cavitating flow in the behind condition. θ =, J =.897, σ =.5. In general, the agreement between the viscous flow results and the experiments is good, for both cavitation numbers, and is much better than was found for open water conditions. The experimental results now show a cavity that extends to low radial sections in a smooth distribution likely due to the non-uniform wake that disturbs the flow enough to promote cavitation inception and growth (see Kuiper [38]). For θ =, the potential flow results show an isolated patch of cavitation at the lower radii similar to the open water potential flow results at J =.7 and σ n =.63 (see Fig. 4a). This deserves further investigation (Vaz and Bosschers [9] showed cavitation at lower radial sections than shown here). This is probably due to the modelling used for detachment-point, but RANS-BEM coupling for cavitating flow conditions could also play a role. Apart from one calculation, all the results seem to underpredict the cavity extents. This is explained by the underprediction of the propeller loads previously discussed. LR predicts a cavitating hub vortex; these were the calculations using the finest grid of all partners. LR, and, all using the Sauer cavitation model, predict very similar cavity extents. A small cavitating tip-vortex is seen in some of the calculations. At, only Caterpillar predicts the mid-chord cavitation seen in the experiments. 5.4 Pressure Fluctuations In this section only the results for the lowest cavitation number σ =.5 will be presented. Figs. 3 and 3 show histograms of the first four harmonics of blade passage frequency at the sensors H, inside the propeller wake, and P, on the wall to the left of the propeller. The results from the other sensors are qualitatively similar. Figs. 33 and 34 show the time histories of the pressure for the same sensors. For these plots, the time histories of the experimental data were obtained from the 48 harmonics of shaft rate frequency provided in the experimental data set [5]. Similarly, the DRDC time history at P was obtained from four harmonics of blade rate frequency calculated using Excalibur; DRDC is not included in the results at H as Excalibur cannot provide realistic predictions of the pressure in the wake. Since neither the experimental nor the DRDC data sets include an absolute pressure reference, the mean pressure was subtracted from all data sets to aid in comparison. It should be noted that, for the experiments, the rotation angle of the blades at time t = is not known. Therefore, the experimental curve may be shifted right or left by an arbitrary amount. The pressure fluctuations predicted by Caterpillar, LR and are four to five times higher than the experimental values and, as can be seen by comparing the upper portions of Figs. 33 and 34, the time histories of the pressure are insensitive to the sensor location, the results for all pressure taps and hydrophones being almost the same. The LR and predictions are very similar; each used STAR-CCM+ with the Sauer cavitation model. The Caterpillar prediction is similar except that it lacks the pronounced peaks near 3,, and 8 msecs; they used OpenFOAM with the Kunz cavitation model. DGAH, DRDC and predict more realistic pressure fluctuations when compared with the reconstructed experimental signal. For the P sensors, the amplitude tends to be under-predicted, perhaps due to the lower loading of the propeller relative to the experiment. At some of the P sensors, DGAH and predict the complete signal quite well, as is the case for in Fig. 34. At the H sensors, there was generally worse agreement between the numerical and experimental values which is understandable given the rapid coarsening of the grids in the axial direction. Nevertheless, the values obtained by and DGAH were of the same order of magnitude as the experimental values. With respect to potential versus viscous flow method results, the results by DRDC are comparable with the more expensive URANS approaches; however, Excalibur is not capable of predicting the pressure inside or very close to the propeller wake, so the pressures at H, H and H3 were not correctly computed. Even though the RANS-BEM coupling approach does not suffer from numerical diffusion, the pressure fluctuations predicted by this method also under-predicted the experimental results. All organizations predicted P to have the highest levels for the first harmonic among the P sensors; the experiment shows that P3 has the highest level with P second. All organizations except Caterpillar correctly predicted that P3 would have higher levels than P (these are the sensors on the left and right walls of the tunnel). All organizations except DRDC correctly predicted that P4 would have the lowest amplitude for the first harmonic among the P sensors, though in the DRDC results the P and P4 levels only differ by P a. In the experiments, the amplitudes of the third and fourth harmonics at P exceed the first and second harmonics. This behaviour was not predicted by any of the organizations. Two major poorly-understood weaknesses of these calculations are the high-frequency oscillations seen in the results of some partners even for non-cavitating loads, and the very high pressure fluctuations predicted by some partners. To try to shed some

13 (a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e). (f). (g) Experiments. Figure 5: Cavity extents for the behind condition. θ =. J =.897, σ =.5. (a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e). (f). (g) Experiments. Figure 6: Cavity extents for the behind condition. θ =, J =.897, σ =.5. (a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e). (f). (g) Experiments. Figure 7: Cavity extents for the behind condition. θ = +, J =.897, σ =.5. (a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e). (f). (g) Experiments. Figure 8: Cavity extents for the behind condition. θ =, J =.897, σ = 5.5. (a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e). (f). (g) Experiments. Figure 9: Cavity extents for the behind condition. θ =, J =.897, σ = 5.5. (a) Caterpillar. (b) DGAH. (c) DRDC. (d) LR. (e). (f). (g) Experiments. Figure 3: Cavity extents for the behind condition. θ = +, J =.897, σ = 5.5.

14 Amplitude (kpa) Harmonic Experiment Caterpillar DGAH LR Figure 3: Amplitudes of the first four harmonics of pressure at sensor H. J =.897, σ =.5. For the two partners using STAR-CCM+, Fig. 35 shows C pn close to the sliding-interfaces at one blade angle. Clearly some pressure perturbations exist at those areas. The results of DGAH, and Caterpillar (not presented here) showed no abnormal pressure jumps between the interfaces, and obviously the results by DRDC do not use interfaces. Therefore, these anomalies could explain some high-frequency peaks but not the very high pressure fluctuations. Amplitude (kpa) Harmonic Experiment Caterpillar DGAH DRDC LR Figure 3: Amplitudes of the first four harmonics of pressure at sensor P. J =.897, σ =.5. Pressure (kpa) Pressure (kpa) Time (msecs) Time (msecs) Caterpillar LR Experiment DGAH Experiment Figure 33: Pressure time history at sensor H. J =.897, σ =.5. Pressure (kpa) Pressure (kpa) Time (msecs) Time (msecs) Caterpillar LR Experiment DGAH DRDC Experiment Figure 34: Pressure time history at sensor P. J =.897, σ =.5. light on these issues, two extra studies were performed: ) a detailed analysis of the pressure distribution close to the interfaces between the non-rotating and the rotating grids; ) a comparison between all numerical predictions for the pressure fluctuations for the wetted flow case. Figure 35: Pressure perturbations close to/at the sliding interfaces in cavitating flow: (top) LR; (bottom). θ =, J =.897, σ =.5. Fig. 36 shows the results obtained for the wetted flow case at the P sensor (LR did not perform these calculations). The predicted pressures are now of the same order of magnitude but highfrequency oscillations are present in the Caterpillar and results. Even for the smoother DGAH and results there are some minor numerical (non-physical) oscillations. However, notice that the scale of the ordinate is much smaller with amplitudes of the order of to Pa. The equivalent variation in C pn is C pn <.. It is possible that the iterative convergence per time-step is not low enough to permit such a level of accuracy. Pressure (kpa) Time (msecs) Caterpillar DGAH DRDC Figure 36: Pressure time history at sensor P for wetted flow. J =.897. We conclude that: ) the origin of the outlying pressure fluctuations by some of the partners is likely due to the cavitation modelling (not only the cavitation model itself but its implementation and inter-connection with the other equations solved); ) the higher-frequency perturbations are probably due to numerical errors at the interfaces and/or lack of iterative convergence,