A study for cavitating flow analysis using DES model

Ocean Engineering 160 (2018) 397–411

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

A study for cavitating flow analysis using DES model Onur Usta a, b, *, Emin Korkut a a b

Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, 34469, Maslak-Istanbul, Turkey Faculty of Engineering, Bartin University, Turkey

A R T I C L E I N F O

A B S T R A C T

Keywords: Cavitation modelling CFD Delft twisted hydrofoil PPTC propeller VP1304 DES turbulence model Mesh refinement

This paper presents the ongoing research on cavitation modelling so far which include validation studies to simulate the hydrodynamic characteristics of the Delft twisted hydrofoil and a controllable pitch propeller VP1304 (PPTC propeller) with zero shaft and inclined shaft cavitating flow conditions. Cavitating flow characteristics are modelled by performing Computational Fluid Dynamics (CFD) simulations using Detached Eddy Simulation (DES) method with SST (Menter) k-ω turbulence model. Multiphase viscous flow, water and air, are modelled using Eulerian multiphase approach and motion of the fluid volume throughout the computational domain is modelled with Volume of Fluid (VOF) approach capturing the interface. Cavitation is modelled by Schnerr-Sauer cavitation model with Reboud correction. The predicted three dimensional cavity structures around the twisted hydrofoil agree fairly well with the experimental observations given in open literature. Lift forces predicted using DES calculation are very close to that of the calculated ones, using a recommended quadratic function. The performance of the PPTC propeller for different conditions are conventionally represented in terms of non-dimensional coefficients, i.e., thrust coefficient ðKT Þ, torque coefficient ðKQ Þ and efficiency ðηÞ and cavity patterns on the propeller blades compared with the literature show good agreement with the experimental data.

1. Introduction Cavitation typically occurs when the fluid pressure is lower than the vapour pressure at a local thermodynamic state and the occurrence of cavitation is often inevitable for hydro machinery. Cavitation is a complex fluid mechanics phenomenon for nozzles, pumps, injectors, turbines, propellers and a variety of other fluid machinery components. It causes undesired effects such as noise, vibration, power loss and erosion. Cavitating flow consists of a combination of fluid dynamics and bubble mechanics, and therefore, it is unsteady, unstable and stochastic. It involves irregular phase-change dynamics, large density ratio between phases, and multiple time scale (Sauer and Schnerr, 2000; Usta and Korkut, 2015). Computational Fluid Dynamics (CFD) simulations of turbulent cavitating flows have been improved with the advent of powerful supercomputers and more understanding of cavitation physics. Fluid flows are being simulated using various methodologies depending on the nature of the flow problem and the availability of the computational resources. However, cavitation has still challenges because of the complex and unsteady interaction associated with turbulence and cavitation dynamics

(Bensow and Bark, 2010; Campana et al., 2011; Shin, 2010). Major difficulties in numerical modelling of the turbulent cavitating flows are to represent rapid changes in fluid density, pressure fluctuations and phase change. Because of these complexities, there is not a single sophisticated mathematical cavitation model that can take care of all these factors (Hejranfar et al., 2015). In the numerical modelling of the cavitating flows, the selection of turbulence models to be used plays a crucial role in the prediction of the complex, multiscale and multiphase behaviour of cavitating flows (Ji et al., 2013). Commonly used turbulence models from the simplest to the most complex are Reynolds Averaged Navier–Stokes (RANS), Detached Eddy Simulation (DES), Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). While the RANS simulations use equations which give a good estimations of the turbulent flow physics, LES and DNS resolve the governing equations and thus give more insight into the flow details (Bensow and Bark, 2010; Maasch et al., 2015; Usta et al., 2017). In other words, although computationally cheap, RANS fails to predict the fluid behaviour accurately in complex flow problems where the underlying physics is dominated by unsteady complex physical phenomena (Mahesh et al., 2015).

* Corresponding author. Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, 34469, Maslak-Istanbul, Turkey. E-mail address: [email protected] (O. Usta). https://doi.org/10.1016/j.oceaneng.2018.04.064 Received 31 July 2017; Received in revised form 9 February 2018; Accepted 18 April 2018 0029-8018/© 2018 Elsevier Ltd. All rights reserved.

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Ocean Engineering 160 (2018) 397–411

for this purpose whenever possible. However, it was also remarked that the solution time for the LES simulations were quite long for practical applications. Wu et al. (2016), investigated the effect of mesh resolution on the LES of cloud cavitating flow around the Delft twisted hydrofoil. In the study, two sets of computational domains with 2 million and 10 million meshes were generated to investigate the influence of mesh resolution on the results and the parameters affecting the mesh resolution. It was observed that the small size shedding vortex can only be captured by the fine mesh and the smaller grid spacing in the span wise direction is needed to capture the details of re-entry jet. Many researchers have also reported on their predictions for the hydrodynamic behaviour of the Potsdam Propeller Test Case (PPTC) VP1304 model propeller with zero shaft inclination (smp’11) and 120 inclined shaft (oblique flow) (smp’15) in cavitating flow conditions used in this paper. Salvatore et al. (2011) presented the numerical results of the workshop on cavitation and propeller performance organised by the smp’11 by employing the INSEAN-PFC propeller flow code based on an inviscid-flow BEM model. They remarked some weakness in the capability to correctly describe the complex flow structures, which was alleviated by the very limited computational effort typical of BEM calculations. Sipila et al. (2011) conducted RANS predictions of the PPTC propeller of SVA Potsdam with FINFLO, a general purpose CFD code. They considered that the CFD predictions of the wake field are highly dependent on the mesh resolutions in the regions of high velocity gradients, i.e. at the tip vortex and blade wake location. The other conclusion from the study was the turbulence models have also an effect on the strength of the tip vortices and blade wakes, although the influence was significantly lower than that of the mesh resolution. In 2015, Lloyd et al. (2015) reported the results of the PPTC oblique flow cases with various mesh densities (course, medium and fine) for both open water and cavitating conditions in terms of pressure pulses and cavitation pattern using CFD code ReFRESCO. They reported that the reduction in the thrust and torque due to cavitation was underpredicted compared to the experimental observations. Guilmineau et al. (2015) studied the PPTC propeller with the inclined shaft in cavitating and non-cavitating conditions with k-ω SST model of ISIS-CFD flow solver. Pressure distribution and cavitation pattern on blade surfaces were evaluated as well as propeller performance characteristics. Yao (2015) presented a numerical study on hydrodynamic characteristics of PPTC propeller in oblique flow. The study was performed by unsteady RANS simulations on an open source platform – OpenFOAM with a sliding mesh approach. It was indicated that the computed results showed generally good agreement with experimental data under non-cavitating and cavitating flow conditions. Budich et al. (2016) conducted a study using a fully compressible numerical approach around the model propeller VP1304 in cavitating flow conditions given in Heinke (2011a). They modelled turbulent flow using implicit LES (ILES) with a barotropic, homogeneous mixture approach. It was stated that under the cavitating conditions, the ILES correctly yields a cavitating vortex core in the wake of the propeller. Also the cavitation-induced thrust-breakdown was well matched with the ILES. However, a full resolution of the near-wall region with their model was too demanding, in terms of the computing resources and time. Huuva and T€ ornros (2016) have studied the same case using an in-house modified RANS code, based on the finite volume of Open-FOAM. They modelled the cavitation by applying the Kunz mass transfer model (Kunz et al., 2000). Based on their study, an improvement of the cavitating results could be achieved by using a polyhedral mesh rather than the currently applied hybrid mesh for the PPTC propeller. Hanimann et al. (2016) reviewed the existing models for cavitation analysis and examined PPTC 2011 cavitating flow conditions for steady state using the Kunz cavitation model. They conducted the analysis by an in-house modified version of Open-FOAM code. They concluded that it would require more investigation in the turbulence modelling and mesh requirements for better prediction of the pressure in the regions where high-pressure gradients occur.

Usually the computation cost of the simulations suffers when LES is used over unsteady RANS due to the increased mesh density required. However, with VOF multiphase approach high mesh resolution is necessary irrespective of the turbulence modelling technique. A hybrid turbulence modelling technique, DES, which attempts to treat near-wall regions in a RANS-like manner, and treat the rest of the flow in a LES-like manner, thus can be offered as a good alternative for turbulence modelling. DES and other hybrid RANS-LES approaches has emerged as a potential compromise between RANS based turbulence models and LES (Kunz et al., 2013). The DES turbulence model is a RANS sub grid model with the characteristic length scale reinterpreted in terms of the local grid scale. When the local mesh is fine relative to the turbulence mixing length, the DES model becomes a LES with a Smagorinsky-like sub grid closure (Spalart et al., 1997). In these resolved regions, modelling error is minimized. The objective of using DNS is to solve the time-dependent N-S equations resolving all the scales for a sufficient time interval, so that the fluid properties reach a statistical equilibrium. However, the computational cost limits the use of DNS even in simple geometries (Mahesh et al., 2015). Therefore DNS is not a suitable turbulence modelling technique to investigate complex flow problems such, as cavitating flows using CFD with current computing sources due to the computer power and time requirements. For cavitation studies the flow around a twisted hydrofoil, namely the Delft twisted hydrofoil test case has been extensively studied and experimental results were documented by Foeth (2008a); and the simulation results was published by Huuva (2008). In addition to that, many researchers, e.g., Schnerr et al. (2008), Koop (2008), Li et al. (2010), Whitworth (2011) and Koukouvinis et al. (2016) have investigated the cavitating flows around the Delft twisted hydrofoil and it was utilized as the benchmark data in two workshops, ‘The Virtual Tank Utility in Europe’ VIRTURE WP4 (Salvatore et al., 2009) and Second International Symposium of Marine Propulsors (smp’11) (Hoekstra et al., 2011). Li et al. (2009) revealed that the modified SST model together with Singhal's cavitation model improves the RANS solver's capability to predict the essential features like re-entrant jets and shedding of macro scale cloud structures observed in the experiment. Lu et al. (2010) investigated unsteady cavitation prediction on the Delft twisted foil, using LES in combination with a volume of fluid implementation to capture the liquid-vapour interface and Kunz's model for the mass transfer between the phases (Kunz et al., 2000). Non-cavitating and cavitating simulations of the Delft twisted hydrofoil were performed with the RANS solver FreSCo þ by Maquil et al. (2011). The results showed that three-dimensional cavitating flows can be simulated with URANS approach combined with VOF-based Euler-Euler methods. However, the method uses empirical coefficients which have a substantial influence on the attainable predictive accuracy (Maquil et al., 2011). Whitworth (2011) predicted the cavitating flow around the Delft twisted hydrofoil with RANS turbulence model using the commercial solver, Star-CCMþ. It was indicated that RANS solver is inadequate to model such a problem and DES model is better to predict highly dynamic cavitation pattern and extreme pressures on the surface of the foil. Ji et al. (2013) simulated Delft twisted hydrofoil cases using Partially-Averaged Navier–Stokes (PANS) method and a mass transfer cavitation model with the maximum density ratio effect between the liquid and the vapour. The numerical results of the study indicated that the cavity volume fluctuates dramatically as the cavitating flow develops with cavity growth, destabilization, and collapse. The time-averaged lift coefficient predicted by the PANS calculations was about 12% lower than the experimental value. Lidtke et al. (2014) investigated the same cases using three different turbulence models and observed clear differences in the predictions of the all approaches. It was mentioned that judging from the substantial difference between the RANS and LES approaches in both the lift and cavity volume variations, one may conclude that the latter should be used 398

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Ocean Engineering 160 (2018) 397–411

In the light of the findings and suggestions mentioned in the examined studies so far, the present work is carried out by using the unsteady DES solver with SST Menter k-ω turbulence model. The above studies also indicated that the mesh resolution is one of the crucial parameter to model cavitation using CFD flow solvers whatever the turbulence model is. In order to generate suitable mesh for the cavitating flow simulations and reduce the numerical uncertainty related to the mesh, the sliding mesh approach is adopted to compute the rotating motions of the propeller, and a mesh refinement study where regions have high pressure fluctuations and volume of cavitation is conducted. Details of the mesh generation and physical model used in this study is given in Section 2. The main purpose of this study is to predict the hydrofoil and propeller's performances in different cavitating flow conditions generating a computational model that includes the mesh modelling and physical modelling using DES turbulence model. This paper also focuses on the investigation of the interaction between the regions where high pressure fluctuations and cavitation occur and developing a mesh strategy to simulate various cavitation types such as tip vortex, sheet and root cavitations for the propellers. The mesh refinement study around the propeller blades are conducted analysing the velocity and pressure gradients and vapour volumes on and around the surface of the blades. In order to achieve the above objectives, Section 2 of the paper describes theoretical background and numerical modelling study applied to Delft twisted hydrofoil and PPTC propeller (VP1304) with zero shaft and inclined shaft cavitating flow conditions. In Section 3, the results of the seven different cases are presented and compared with the experimental data given in the literature and discussions are made. Finally, Section 4 presents the conclusions obtained from the study.

require that the time scales of the turbulence be disparate from the meanflow unsteadiness. Furthermore, the limitations of the turbulence model may preclude good unsteady results. DES turbulence model is set up so that boundary layers and irrotational flow regions are solved using a base RANS closure model. However, the turbulence model is intrinsically modified so that, if the grid is fine enough, it will emulate a basic LES subgrid scale model in detached flow regions (STAR-CCM þ Documentation, 2016). For the Shear Stress Transport (SST) Menter k-ω Detached Eddy turbulence model the turbulent length scales are modelled using IDDES turbulent length scale approach. Three different mesh densities are investigated considering verification and validation methodology and procedures given in Stern et al. (2006) and Celik et al. (2008). Two phases, water and air, are modelled using Eulerian multiphase approach and the motion of the fluid volume throughout the computational domain is modelled with Volume of Fluid (VOF). The VOF model assumes that the viscous fluid is homogenous so that both phases share the same properties, such as velocity and pressure. To simulate cavitating flows, the two phases, namely, liquid and vapour, need to be represented in the problem, as well as the phase transition mechanism between the two (Bensow and Bark, 2010). Cavitation is modelled by Schnerr-Sauer cavitation model with Reboud correction (Reboud et al., 1998). Schnerr-Sauer cavitation model implements a reduced Rayleigh-Plesset equation, which neglects the influence of bubble growth acceleration, viscous effects, and surface tension effects. According to this model, the bubbles are considered as spherical and all seeds have the same radius at the beginning of the simulations. Since each bubble cannot be modelled individually, the cavitation is modelled using a number of bubbles in a control volume. Mathematical background of the present model was reported by Schnerr and Sauer (2001).

2. Numerical modelling 2.1. Theoretical background and numerical models

2.2. Cavitating flow analysis of the Delft twisted hydrofoil The predictions are carried out using Star-CCM þ flow solver to simulate the cavitating flow conditions. The solver uses a finite volume method which discretises the governing equations. A second order convection scheme is used for the momentum equations and a first order temporal discretization is used. The flow equations are solved in a segregated manner. The mass conservation, continuity and momentum equations for incompressible flows are given in tensor notation and Cartesian coordinates by the following (Ferziger and Peric, 2002):

∂ρ þ divðρUÞ ¼ 0; ∂t

(1)

∂ρ ∂ðρui Þ þ ¼ 0; ∂xi ∂t

(2)






The Delft twisted hydrofoil has been tested in the cavitation tunnel of Delft University (Foeth, 2008a). The hydrofoil was a wing having a rectangular platform of a NACA0009 foil section with varying twist angle along the span, yielding an angle of attack of 2 at the wall of the cavitation tunnel up to 9 at the centerline. The chord length of the foil was c ¼ 0.15 m and the span length was 0.3 m. The angle of attack of the entire hydrofoil was 2 . Details of the study can be found in (Foeth, 2008a). Two different cases are simulated representing experiments performed in the cavitation tunnel at Delft University of Technology. In the first case a fixed velocity of 6.97 ms1 is given from the inlet and the simulation is carried out at the cavitation number of σ ¼ 1.07 which is achieved by using a fixed value of pressure at the outlet of 29 kPa. In the second case, to speed-up the formation and shedding of the cavity sheet in the numerical simulations, the calculations are performed at a freestream velocity of U ¼ 50 ms1 at the cavitation number of σ ¼ 1.1 and the pressure at the outlet of 1375 kPa as in the experiments of Foeth (2008a). The flow conditions simulated for Case 1 and Case 2 are summarized in Table 1.




∂ρ τij ∂ðρui Þ ∂ρ ui uj ∂P þ ¼  þ ρgi ; ∂t ∂xj ∂x j ∂xi

(3)

Where ρ is the fluid density, t is time, U is velocity vector in three dimensions, ρðui uj Þ is the Reynolds stress and P is the pressure. τij are the viscous stress tensor components as given below: 

τij ¼ μ



∂ui ∂uj þ ; ∂xj ∂xi

2.2.1. Computational domain and boundary conditions In all CFD problems, the initial and boundary conditions must be defined depending on the features of the problem to be solved (Tezdogan et al., 2015). Fig. 1 illustrates that the flow is given from velocity inlet and the end of the computational domain is defined as pressure outlet boundary condition. The top and bottom boundaries and the hydrofoil are defined as walls. In order to reduce computational complexity and solution time, only half of the hydrofoil is modelled. This is acquired by using symmetry plane boundary condition. The symmetry plane feature enables to accurately simulate the other half of the computational domain (see Fig. 2). The computational domain dimensions are taken as the same as in the

(4)

where, μ is the dynamic viscosity. An unsteady Detached Eddy Simulation (DES) method is used to solve the governing equations in this study. DES is a hybrid modelling approach that combines RANS features of the flow near of the boundary layer and LES in the unsteady separated regions in the computational domain. For modelling the complex flows, such as cavitating, transient simulations often yield better results than attempting to use a steadystate approach. However, successful unsteady RANS simulations

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Table 1 Flow conditions for Delft twisted hydrofoil simulations.

Case 1 Case 2

Angle of attack (deg)

Outlet pressure (kPa)

Vapour density (kg/ m3)

Inlet velocity (m/s)

Vapour pressure (kPa)

Water density (kg/ m3 )

Cavitation number

2 2

29 1375

0.023 0.023

6.97 50.0

2.970 2.170

998 997.5

1.07 1.10

convection time scale, relates the mesh cell dimension Δx to the mesh flow speed U as given below:   UΔt 1 CFL criteria ¼  Δx 

(5)

The Courant number is typically calculated for each cell and should be less than or equal to 1 for numerical stability. Most of the calculated CFL numbers on the hydrofoil in this paper are smaller than 1, providing the CFL criteria. CFL numbers vary between 0.00012 < CFL< 1 and 1 and 0.0438 < CFL<1 for the Case 1, and Case 2, respectively, as shown in Fig. 3. The discretization of the time directly affects the truncation error present in the simulation and plays a crucial role in the accuracy of a time accurate computation. Time step is determined for the unsteady flow where the properties of the flow varies with time and it defines the time of each iterative solution. In the analysis, the time-step values are systematically lessened based on the refinement ratio ðrT Þ of 2 and 2.5 considering the determined CFL numbers. Time-step value is 2x105 in the normal mesh analysis, reduced to 1x105 and increased to 5x105 in the verification studies. In the analysis, the first grid point from the wall, yþ value is sufficiently small for the turbulence to be captured in the boundary layer. If yþ 5, the solution is in linear sub-region, if 5 < yþ<30, it is in buffer region and if 30  yþ800, it is in log-law region. The yþ values on the hydrofoil surface are kept in linear sub-region. yþ values vary between 0.0230 < yþ< 5 and 0.86 < yþ<5 for the Case 1 and Case 2, respectively, as shown in Fig. 4. Since the velocity of the Case 1 (6.97 m/s) is less than the velocity of the Case 2 (50 m/s), the Case 1 yields less yþ and courant number values compared to those of the Case 2. Based on the mesh refinement ratio, a uniform refinement ratio ðrG Þ is pffiffiffi chosen to be 3 2, which is applied to the computational domain. Based on the mesh generation explained the mesh numbers for the half domain in both cases are listed in Table 2. The experimental lift force (L) is calculated by using a recommended quadratic function given in TU Delft reports (Foeth, 2008b) and Maasch et al. (2015) as:

Fig. 1. Computational domain and boundary conditions for the hydrofoil at 2 angle of attack.

smp’11 workshop given in Hoekstra et al. (2011). The length of the domain is generated as 7 chord lengths (7c), starting 2c ahead of the leading edge and ending 4c behind the trailing edge. The height of the domain is generated as 2c corresponding to the height of the test section. 2.2.2. Mesh generation Mesh generation is performed using Star-CCM þ automated mesh facility which uses the Cartesian cut-cell method (STAR-CCM þ Documentation, 2016). Structured mesh is generated on the computational domain; however, the mesh is composed of unstructured hexahedral cells on the hydrofoil. A trimmed cell mesh model is employed to produce high-quality mesh cells on the hydrofoil. Three different mesh density levels, from coarse to fine mesh, are simulated for each case. To improve the resolution in the region with cavitation, finer mesh is adopted generating volumetric controls and refining the region of the hydrofoil to approximately 5% of the base size. In order to capture turbulence effects in the boundary layer, prismatic layers are generated around the hydrofoil. Prism layer characteristics (number of prism layers, prism layer thickness and layer stretching) are determined to achieve a target wall-y þ value using a specific code. This code is generated considering the flat-plate boundary layer theory of White (2010). 15 prism layers are generated to model boundary layer flow accurately by the prism layer stretching of 1.5, with approximately 1.5 mm total thickness. With the use of these layers around the hydrofoil, the resolution of the mesh near the surface of the hydrofoil can be improved even further in order to capture the re-entrant and side-entrant flows more accurately. Mesh on the hydrofoil is generated based on the Courant-FriedrichsLevy (CFL) Stability Criterion (Courant number condition). The Courant number (CFL) which is the ratio of the physical time step ðΔtÞ to the mesh

L ¼ 10:052 U 2

(6)

where U is the flow speed. The dimensionless lift coefficient CL is defined as:

Fig. 2. (A) Mesh on the computational domain and (b) around the Delft twisted hydrofoil. 400

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Fig. 3. Convective courant number values on the hydrofoil surface for.

Fig. 4. Wall yþ values on the hydrofoil surface for (a) Case 1 and (b) Case 2. Table 2 Mesh numbers for uncertainty analyses of the Delft twisted hydrofoil. Velocity (m/s)

6.97 (Case 1) 50.0 (Case 2)

Table 4 Operating conditions for PPTC 2011 with zero shaft inclination.

Number of cells Coarse

Normal

Fine

5,705,130 6,228,757

9,071,499 10,678,943

17,066,570 18,368,038

L CL ¼ 0:5 ρ U 2 A

Advance coefficient Cavitation number Propeller rate of revolution Water density Kinematic viscosity of water Vapour pressure Inlet velocity

(7)

where A is the projected area, given as A ¼ 2c2 in Maquil et al. (2011). Lift forces are calculated using the above recommended quadratic function given by Foeth (2008b) as the experimental lift values. Then, the lift coefficients are calculated using equation (7). The experimental and predicted lift forces and coefficients are given in Table 3 and Table 7. The lift forces are time-dependent and the predicted values are the mean lift forces after the simulations were converged. For the case 1 the lift force predicted by the DES simulation is about 1.22% lower than the experimental value for the normal mesh density. As for the case 2 the lift force predicted by the DES simulation is about 1.44% lower than the experimental value for the normal mesh density.

Case 3

Case 4

Case 5

J[] σ [] n[1/s]

1.019 2.024 24.987

1.269 1.424 24.986

1.408 2.000 25.014

p[kg/ m3] v [m2/ s] pv [Pa] U[m/s]

997.44

997.44 7

9.337 x 10

9.337 x 10

2818 6.365

2818 8.051

997.37 7

9.272 x 107 2869 8.804

2.3. Cavitating flow analysis of PPTC propeller VP1304 The Potsdam Model Basin SVA provided extensive experimental test data for open water and cavitation characteristics of a controllable pitch propeller, VP1304, which were used as the test cases at the smp’11 and smp’15 workshops. The results are available at https://www.svapotsdam.de/en/potsdam-propeller-test-case-pptc/. The VP1304 model propeller is a five bladed controllable pitch propeller with a diameter of D ¼ 0.25 m. The model propeller VP1304 can be seen in Fig. 5.

Table 3 Comparison of measured (calculated) lift force with respect to mesh density and time-step. Experiment (Calculation)

Case 1 Case 2

488.35 N 25130 N

Time-step

Time-step

Coarse

Mesh density Normal

Fine

5 x 105

1 x 105

448.76 N 24516 N

480.04 N 24772 N

491.85 N 24860 N

472.82 N 24652 N

480.78 24864 N

401

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Operating conditions for zero shaft inclination and 12 inclination cases are given in Tables 4 and 5, respectively.

Table 5 Operating conditions for 120 inclined shaft conditions.

Advance coefficient Cavitation number Propeller rate of rotation Water density Kinematic viscosity of water Vapour pressure Inlet velocity

J[]

σ [] n[1/s] p[kg/m3] v [m2/s] pv [Pa] U[m/s]

Case 6

Case 7

1.019 2.024 20 997.78 9.567 x 107 2643 5.095

1.408 2.000 20 997.41 9.229 x 107 2904 7.04

2.3.1. Computational domain and boundary conditions In both cases, the flow is given with a velocity inlet boundary condition and set in the positive x direction. The negative x direction is modelled as a pressure outlet. Top and bottom of the computational domain and propeller are given as the wall. The computational domain dimensions are taken as same as those in the smp’11 workshop given in (Heinke, 2011a). The total length is generated as 14 propeller diameter (14D), starting 3D ahead of the propeller and 10D behind the hub of the propeller. The height and width of the domain are generated as 3D. Boundary conditions for the cavitating flow around VP1304 propeller in zero shaft conditions are shown in Fig. 6. For the 12 inclined shaft condition, the computational domain dimensions were generated as 5D for the height and width, 14D for the length, starting 4D ahead the hub of the propeller and ending 10D behind the propeller. Boundary conditions for the cavitating flow around VP1304 propeller in the inclined shaft conditions were demonstrated in Fig. 7.

Table 6 Number of cells for the numerical uncertainty analysis of the VP1304 propeller. Number of cells

Zero shaft inclination (case 3,4,5) 120 shaft inclination (Case 6,7)

Rotating Stationary Total Rotating Stationary Total

Coarse

Normal

Fine

7,312,868 2,048,747 9,361,615 6,394,021 2,278,173 8,672,194

11,792,818 3,953,193 15,746,011 10,556,379 2,281,655 12,838,034

19,634,077 6,731,305 26,365,382 16,983,038 4,092,744 21,075,782

2.3.2. Mesh generation The mesh generation procedure, which was given in Section 2.2.2 for the Delft twisted hydrofoil, is applied to the VP1304 propeller. Three different meshes are generated for each cases; coarse, normal and fine meshes. A mesh adaptation study is applied to the VP1304 propeller. To increase the accuracy of the solution the mesh can be refined based on a solution quantity (such as pressure or volume fraction of vapour). In order to do this, a field function of the quantity is created and then this function is converted into a table to apply to the mesh. So, a mesh adaptation study is carried out based on velocity, pressure and volume fraction of vapour on the propeller blades and tip vortex region. After the mesh adaptation study, the mesh generation is repeated by considering the change of the velocity and pressure gradient and the cavitation volume around the propeller. Thus, in the regions where high pressure fluctuations and cavitation occur, have better mesh resolution. In order to capture turbulence effects and model cavitation in the boundary layer, the prismatic layers are generated around the propeller. Boundary layer characteristics, such as number of prism layers, prism layer thickness and layer stretching are determined by the help of a special code based on the White's boundary layer theory (White, 2010). Prism layer around the VP1304 propeller blades with zero shaft inclination condition is generated using 13 prism layers by the prism layer stretching of 1.5 for approximately 1.5 mm total thickness. Prism layer in the 12 inclined shaft condition is generated with 15 prism layers by the prism layer stretching of 1.5, for approximately 2 mm total thickness. With the use of prism layers around the propeller, resolution of the mesh near the boundary layer is improved and cavitation dynamics are captured more accurately.

Table 7 Comparison of measured (calculated) lift coefficient with respect to mesh density and time-step. Experiment (Calculation)

CL (case 1) CL .(case 2)

Mesh density

Timestep

Timestep

Coarse

Normal

Fine

5 x 105

1 x 105

0.448

0.411

0.440

0.450

0.433

0.440

0.448

0.438

0.441

0.443

0.439

0.443

The Potsdam Propeller Test Case (PPTC) studies include the investigations of the open-water performance given in Barkmann et al. (2011), LDV velocity measurements in the propeller wake given in Mach (2011), and test results given in Heinke (2011a). Additionally, the predictions of propeller were studied at the Workshop on Propeller Performance within the 2nd Symposium on Marine Propulsors (smp’11), (see Abdel-Maksoud, 2011a; b; c). These studies provide an extensive database for the validation. The cases 3, 4, and 5 in this paper are the validation of the cavitation tests given in smp’11. The cases 6 and 7 are the validation of the cavitation tests given in smp’15, which were conducted in the cavitation tunnel of the SVA Potsdam. The prediction of CFD analysis is validated based on the thrust identity, which were given in smp’15 (Lübke, 2015). During the tests, the propeller was positioned with a 12 inclination towards the inflow direction as shown in Fig. 5.

Fig. 5. PPTC propeller (VP1304) with no inclination and inclined shaft in open water condition. 402

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Fig. 6. A general view of the computational domain with boundary conditions for the cavitating flow around VP1304 propeller with zero shaft inclination condition.

Fig. 7. A general view of the computational domain with boundary conditions for the cavitating flow around VP1304 propeller with 12 inclined shaft condition.

For the grid convergence study, a uniform refinement ratio ðrG Þ is pffiffiffi chosen to be 3 2, which is applied to the computational domain. The time-step values are systematically reduced based on the refinement ratio ðrT Þ of 2 and 2.5. The time-step values used in this study are 5x105 , 2x105 and 1x105 . Number of cells generated for the numerical uncertainty analysis are given in Table 6.

As shown in Figs. 8 and 9, the mesh on the propeller is formed primarily of unstructured hexahedral cells with trimmed cells adjacent to the surface. The mesh generation is progressively refined around the rotating part to ensure that the complex flow features like cavitation are appropriately captured. The refined mesh density in these zones is achieved using volumetric controls applied to these regions. The number of cells for the rotating and stationary parts are given in Table 6. In this study the sliding mesh approach is used to model the rotating motion of the propeller. This technique is quite useful for simulations involving dynamic parts e.g. rotating propeller. With the sliding mesh approach the computational domain is divided into two parts as an outer static part and an inner dynamic (rotational) part. The inner part is cylinder-shaped and could rotate relatively to the outer part. The two parts are connected by an interface by which flow information is transformed between relative sliding interfaces (Yao, 2015). The CFL number, time-step and yþ values are determined, based on the procedure given in Section 2.2.2. CFL numbers for the zero shaft inclination condition vary between 0:00014  CFL  1 and yþ values vary between 0:028  y þ  5 for the Case 3, as shown in Fig. 10. For the case 6, CFL numbers vary between 0:0012  CFL  1, while yþ values vary between 0:045  y þ  10 as shown in Fig. 11. The courant numbers and yþ values, for the all cases are very similar. Then the figures for the Case 3 and Case 6 are only given as the verification study is also performed for these cases.

3. Presentation and discussion of the results 3.1. Delft twisted hydrofoil 3.1.1. Verification of the Delft twisted hydrofoil analysis The numerical error of a CFD prediction has three components, such as the round-off error, the iterative error and the discretization error (Eca and Hoekstra, 2014). The source of the round-off error is a finite precision of the computers and its importance tends to increase with grid refinement. The iterative error is unavoidable due to the non-linearity of the mathematical equations, even if direct (non-iterative) methods can be used for the linear equations. The iterative error is usually quantified in terms of a residual or a residual norm. The discretization error is a consequence of the approximations made, e.g. finite-differences, finite-volume, finite elements, to transform the partial differential equations of the continuum formulation into a system of algebraic equations. Unlike

Fig. 8. Mesh refinement (a) around the computational domain and (b) on the PPTC propeller blades in zero shaft inclination condition. 403

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Fig. 9. Mesh refinement (a) around the computational domain (b) on the PPTC propeller blades in 120 inclined shaft condition.

Fig. 10. (A) Convective courant number and (b) Wall yþ values on the propeller surface for Case 3.

Fig. 11. (A) Convective courant number and (b) Wall yþ values on the propeller surface for Case 6.

the other two error sources, the relative importance of the discretization error decreases with the grid refinement (Eca and Hoekstra, 2014). A verification study is undertaken to assess the numerical uncertainty, USN and numerical errors, δSN for the cavitating flow simulations of Delft twisted hydrofoil. In the present study, it is assumed that the numerical error is composed of iterative convergence error (δI), grid-spacing convergence error (δG) and time-step convergence error (δT), which

gives the following expressions for the simulation numerical error and uncertainty (Stern et al., 2001; Tezdogan et al., 2015): 2 ¼ UI2 þ UG2 þ UT2 δSN ¼ δI þ δG þ δT USN

(8)

The verification study is carried out for two cases. The recommended method for discretization error estimation is the Richardson extrapolation method (Richardson, 1911). It is the basis for the existing 404

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3.2. PPTC VP1304 propeller

quantitative numerical error/uncertainty estimates for time-step convergence and grid-spacing. With this method, the error is expanded in a power series with integer powers of grid-spacing or time-step taken as a finite sum (Celik et al., 2008; Tezdogan et al., 2015). Grid convergence index (GCI), proposed by Roache (1994, 1998) is useful for estimating uncertainties arising from grid-spacing and time-step errors. Roache's GCI was recommended for use by both the American Society of Mechanical Engineers (ASME) (Celik et al., 2008) and the American Institute of Aeronautics and Astronautics (AIAA) (Cosner et al., 2006). Grid-spacing and time-step convergence studies are carried out the following correlation factor (CF) and GCI methods of Stern et al. (2006). The convergence studies are performed with triple solutions using systematically refined grid-spacing or time-steps. For example, the grid convergence study is conducted using three calculations in which the grid size is systematically coarsened in all directions whilst keeping all other input parameters (such as time-step) constant. The time-step convergence study conducted using three calculations in which the time-step is systematically reduced and all the other parameters kept constant. The mathematical background of the verification study conducted in this paper is taken from Roache (1998), Stern et al. (2006); Celik et al. (2008) and Tezdogan et al. (2015). For the grid convergence study, a uniform refinement ratio ðrG Þ is pffiffiffi chosen to be 3 2, which is applied to the computational domain. The time-step convergence study is conducted with triple solutions, using systematically reduced time-steps based on the refinement ratio ðrT Þ of 2 and 2.5. The time-step values used in this study are 5x105 , 2x105 and 1x105 . Numerical uncertainty analysis is conducted using CL values of the performed mesh density and time-step configurations. Considering the procedure for estimation of discretization error given above, the numerical uncertainty, USN for the case 1 is calculated as USN ¼ 1:838%; while the total numerical uncertainty for case 2 is calculated as USN ¼ 1:623%.

3.2.1. Verification of the VP1304 propeller analysis 3.2.1.1. Zero shaft condition. A verification study is undertaken to assess the numerical uncertainty, USN and numerical error, δSN for cavitating flow simulations of the VP1304 Propeller in zero shaft conditions following the procedure given in Section 3.1.1. The verification study is carried out for the thrust coefficient only in the case 3 and assumed to be the same in the cases 4 and 5. The numerical uncertainty is predicted for the thrust coefficients with respect to mesh density and time-step given in Table 8. Considering the procedure for estimation of discretization error given above, the numerical uncertainty, USN for the case 3 is calculated as USN ¼ 3:323%. 3.2.1.2. 12 inclined shaft condition. A verification study is also performed to assess the numerical uncertainty, USN and numerical errors, δSN for cavitating flow simulations of the VP1304 propeller in the inclined shaft conditions. The verification study is carried out for the thrust coefficient only in the case 6 and assumed to be the same in the case 7. The results are given in Table 9. Considering the procedure for the estimation of discretization error given above, the numerical uncertainty, USN for the case 6 is calculated as USN ¼ 2:129%. 3.2.2. Cavitation predictions for the VP1304 propeller in zero shaft condition case 3, 4 and 5) Comparison of CFD predictions of the propeller characteristics with the experiments is given in Table 10. As far as the propeller characteristics are concerned, the differences in the thrust coefficient compared to experimental work are about 0.35%, 2.77% and 4.98% for the case 3, 4 and 5, respectively. For the torque coefficient the differences are about 1.40%, 2.03% and 4.17% for the case 3, 4 and 5, respectively. The differences for the open water efficiency are about 1.03%, 0.76% and 0.77% for the case 3, 4 and 5, respectively. As the cavitation developed more on the blades, the predictions deviate more from the experiment results. Therefore, the results of the CFD study are very similar to experimental work of Heinke (2011a) for all the cases.

3.1.2. Cavitation predictions on Delft twisted hydrofoil Comparisons of the predicted cavitation pattern with the experiment and some reviewed numerical studies from the literature are given in Fig. 12, Fig. 13 and Fig. 14 for the Case 1 and Case 2. The unsteady cavitation patterns and their evolution around the Delft twisted hydrofoil with dramatic cavity volume fluctuations are captured by the present method. The predicted three dimensional cavity structures around the hydrofoil agreed fairly well with experimental observations for both cases.

3.2.2.1. Comparison of the cavitation patterns on the propeller blades for case 3. In smp’11, CFD predictions from 11 different groups were evaluated for the cavitating flow analysis, employing 12 different solvers

Fig. 12. Cavitation pattern on the hydrofoil for the case 1. (a) Experimental result (Whitworth, 2011), (b) Numerical result of Ji et al. (2013) and (c) Numerical prediction of this study (vapour volume fraction 50%). 405

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Fig. 13. Full view of Cavitation pattern on the hydrofoil for the case 1: (a) Numerical result of Ji et al. (2013) and (b) Numerical prediction of this study (vapour volume fraction 50%).

Fig. 14. Cavitation pattern on the hydrofoil for the case 2: (a) Experimental result (Foeth, 2008a); (b) Numerical result of Koop (2008) and (c) Numerical prediction of this study (vapour volume fraction 50%).

The comparisons of the predicted cavitation patterns with the experiment and the numerical results obtained from smp’11 are given in Fig. 15 and Fig. 16 for the Case 3. The experiments showed that tip vortex and suction side sheet cavitation developed in the radius range of r/R > 0.90. A root cavitation was also observed. In addition a strong hub vortex cavitation structures developed (Heinke, 2011b). Therefore, the results of the CFD study fairly predict the tip and hub vortex types; and sheet and root cavitation types.

Table 8 Comparison of measured thrust coefficient in zero shaft condition with respect to mesh density and time-step. Experiment

Case 3

0.3725

Mesh density

Time-step

Coarse

Normal

Fine

5 x 105

1 x 105

0.3845

0.3712

0.3715

0.3789

0.3763

Table 9 Comparison of measured thrust coefficient in 12 inclined shaft condition with respect to mesh density and time-step. Experiment

Case 6

0.363

Mesh density

3.2.2.2. Comparison of the cavitation patterns on the propeller blades for case 4. The comparisons of the predicted cavitation pattern with the experiment and the numerical results obtained from smp’11 are given in Fig. 17 and Fig. 18 for the case 4. During the experiments, tip vortex cavitation and strong root type of cavitation on the suction side of the blade were observed, as shown in Fig. 18. Intermittent foam cavitation also developed on the suction side (Heinke, 2011a). The predicted results show good agreement with the experiments.

Time-step

Coarse

Normal

Fine

5 x 105

1 x 105

0.345

0.356

0.361

0.351

0.359

(Heinke, 2011b). The cavitation pattern predictions of this study are compared with the experimental figures (Heinke, 2011b) besides CFD figures obtained from the smp’11 predictions (Heinke, 2011b). The CFD figures, which are denoted by the letter (b), shows the best result given in smp’11.

3.2.2.3. Comparison of the cavitation patterns on the propeller blades for case 5. The comparisons of the predicted cavitation patterns with the experiment and the numerical results obtained from smp’11 are given in Fig. 19 and Fig. 20 for the case 5.

Table 10 Comparison of CFD predictions of the propeller with zero shaft inclination characteristics to experiments. Experiment (Heinke, 2011a)

KT 10KQ

η0

CFD

Difference (%)

Case 3

Case 4

Case 5

Case 3

Case 4

Case 5

Case 3

Case 4

Case 5

0.3725 0.9698 0.6246

0.2064 0.6312 0.6604

0.1362 0.4890 0.6241

0.3712 0.9564 0.6293

0.2123 0.6443 0.6654

0.1297 0.4694 0.6193

0.35 1.40 1.03

2.77 2.03 0.76

4.98 4.17 0.77

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Fig. 15. Comparison of cavitation pattern on the propeller blade for the Case 3: (a) Experimental result (Heinke, 2011b); (b) CFD result of UniTriest-CFX (Kunz) in smp’11 (Heinke, 2011b) and (c) Numerical prediction of this study (vapour volume fraction 20%).

Fig. 16. (A) Experimental result (Heinke, 2011b) and (b) Numerical prediction of this study for the cavitation development on the propeller in the case 3.

Fig. 17. Comparison of cavitation pattern on the propeller blade for the case 4: (a) Experimental result (Heinke, 2011b); (b) CFD result of SSPA-Fluent in smp’11 (Heinke, 2011b) and (c) Numerical prediction of this study (vapour volume fraction 20%).

The results of the CFD study given in Table 11 is for the fine mesh case. The results indicate that the differences between the CFD and experiment for the thrust coefficient are about 0.55% and 1.65% for the cases 6 and 7, respectively that are given in Table 11. Therefore, the results of the CFD study can be accepted as similar to those measured in the experiment.

For the case 5 the experiments showed that pressure side sheet cavitation was observed on the leading edge between the radius of 0.4 < r/R < 0.95. Root type of cavitation developed on both pressure and suction sides, as shown in Fig. 20. The predictions of the CFD study agree well with the experiments for the developed cavitation types. 3.2.3. Cavitation predictions for the VP1304 propeller in 12 inclined shaft condition (case 6 and 7) In smp’15, CFD predictions from 12 different groups were evaluated for the cavitating flow analysis, employing 19 different solvers (Lübke, 2015). The cavitation pattern predictions of this study are compared with the experimental figures (Lübke, 2015) besides the figures obtained from the smp’15 predictions (Lübke, 2015). The CFD figures, which are denoted by the letter (b), shows the best result given in smp’15. The comparison of the CFD predictions for the thrust coefficients in 12 inclined condition with the experiments is given in Table 11.

3.2.3.1. Comparison of the cavitation patterns on the propeller blades for case 6. The comparisons of the predicted cavitation pattern with the experiment and the numerical results obtained from smp’15 are given in Fig. 21 for the case 6. Cavitation developed on both sides of the propeller blades with 12 inclination angle for the case 6, as shown in Fig. 21(a). On the suction side tip vortex, sheet cavitation from 0.8r position and some bubble cavitation were observed on the blades depending on the blade position. Root type of cavitation developed on the pressure side of the blades. The

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Fig. 18. (A) Experimental result (Heinke, 2011a) and (b) Numerical prediction of this study for the cavitation development on the propeller in the case 4.

Fig. 19. Comparison of cavitation pattern on the propeller blade for the case 5: (a) Experimental result (Heinke, 2011b); (b) CFD result of UniGenua-Panel in smp’11 (Heinke, 2011b) and (c) Numerical prediction of this study (vapour volume fraction 20%).

Fig. 20. (A) Experimental result (Heinke, 2011a) and (b) Numerical prediction of this study for the cavitation development on the propeller in the case 5.

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7.

Table 11 Comparison of CFD predictions of the thrust coefficients in 12 inclination condition with experiments.

For the case 7 cavitation also developed on both sides of the propeller blades with 12 inclination angle, as shown in Fig. 22(a). On both sides of the propeller streak type sheet cavitation, cloud cavitation and root type of cavitation were observed. The results of the current CFD study predict well the cavitation pattern for the sheet cavitation, however fails to predict cloud and streak characteristics of the sheet cavitation observed in the experiments given in Lübke (2015).

Thrust coefficient (KT )

Experiment CFD Difference (%)

Case 6

Case 7

0.363 0.361 0.55

0.123 0.121 1.65

4. Conclusions results of the current CFD study predict the tip vortex and sheet cavitation, except the bubble types observed in the experiments given in Lübke (2015). However, there are clear differences in the cavitation patterns.

A numerical study is carried out to predict cavitation patterns on Delft twisted hydrofoil and PPTC VP1304 propeller with zero shaft and inclined shaft cavitating flow conditions by using a Detached Eddy Simulation (DES) method with SST (Menter) k-ω turbulence model. Based on the foregoing analysis some conclusions drawn from the study are as follows:

3.2.3.2. Comparison of the cavitation patterns on the propeller blades for case 7. The comparisons of the predicted cavitation patterns with the experiment results obtained from smp’15 are given in Fig. 22 for the case

Fig. 21. Comparison of cavitation patterns on the suction (upper figures) and pressure (lower figures) sides of propeller blade for the case 6: (a) Experimental result (Lübke, 2015); (b) CFD result of ACCUSIM CFX-FCM in smp’15 (Lübke, 2015) and (c) Numerical prediction of this study (vapour volume fraction 40%).

Fig. 22. (A) Experimental result (Lübke, 2015) and (b) Numerical prediction of this study for the cavitation development on the propeller for the case 7 (vapour volume fraction 40%). 409

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 The method used, accurately predicts the lift coefficient for the Delft twisted hydrofoil at 2 angle of attack for multiphase cavitating flow conditions and the open water characteristics of the PPTC VP1304 propeller with different flow conditions.  For cavitation predictions of the Delft twisted hydrofoil analysis, the shape of the sheet cavity and the outline of the closure region as predicted by the present numerical method are comparable quite well with the experimental results of Foeth (2008a and 2008b).  Mesh refinement study is carried out using the initial simulation results to determine which part of the propeller regions have high pressure fluctuations and volume of cavitation. After determining the regions the mesh is refined in these regions using volumetric controls. By the way of mesh refinement study, CFD predictions of the propeller characteristics and cavity patterns on and around the propeller are more consistent with the experimental results.  The numerical methodology used in this study, including a mass transfer based on the Schnerr and Sauer cavitation model and the simplified Rayleigh–Plesset equation, the k–ω SST turbulence model, and a sliding mesh accurately simulate the propeller cavitation. The predicted cavitation pattern around and on the propeller is assumed to be well correlate with the experimental data.  The turbulence model used in this study is DES SST (Menter) k-ω Detached Eddy turbulence model with IDDES turbulent length scale model, to keep the computational cost low. The DES model does not

require very powerful computing power as LES does, without neglecting the physical details unlike RANS. This means that although the physics of the cavitating flow is modelled in detail as much as the LES, it requires much less computing resources compared to LES applications.  DES simulations are a powerful and reliable tool for understanding the physical mechanisms of the cavitating flow. Even though it is not yet possible to numerically predict all the dynamics of the tip vortex cavitation completely, compared results showed good agreement with the experimental observations. Acknowledgement The current work is part of Ph.D. study of the first author and supported by TUBITAK 2214-A International Doctoral Research Fellowship Programme, (Grant No. B.14.2.TBT.0.06.01-21514107-020-155998) and this Ph.D. is supported by Faculty of Engineering, Bartin University under OYP Programme and those supports are gratefully acknowledged. The simulations were performed at the University of Strathclyde Glasgow, using High-Performance Computing for the West of Scotland (ARCHIE-WeSt) and at the School of Marine Science and Technology of the University of Newcastle. The authors thank Prof. Osman Turan of the University of Strathclyde Glasgow for his invaluable support and guidance, during the first author's visit in Glasgow.

Nomenclature A AE =A0 CL c D

ρ μ σ

ϑ k

ε P/D

τij n J P Pv T Q U KT KQ rG rT δG δT δI USN Z

projected area propeller blade expansion ratio lift coefficient chord length propeller diameter fluid density dynamic viscosity cavitation number kinematic viscosity of water turbulent kinetic energy turbulent dissipation rate propeller pitch ratio viscous stress tensor components propeller rotation speed advance ratio pressure vapour pressure propeller thrust force propeller torque velocity non-dimensional thrust coefficient non-dimensional torque coefficient uniform mesh refinement ratio time-step refinement ratio grid-spacing convergence error time-step convergence error iterative convergence error total numerical uncertainty number of blades

Evaluation case 2.2 LDV measurements with the model propeller VP1304. In: AbdelMaksoud, M. (Ed.), 2011b. Proceedings of the Workshop onCavitation and Propeller Performance, Second International Symposium on Marine Propulsors (Hamburg). Evaluation case 2.3 cavitation tests with the model propeller VP1304. In: AbdelMaksoud, M. (Ed.), 2011c. Proceedings of the Workshop on Cavitation and Propeller Performance, Second International Symposium on Marine Propulsors (Hamburg).

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