Computational and Empirical Investigation of Propeller Tip Vortex Cavitation Noise

In this study, non-cavitating and cavitating flow around the benchmark DTMB 4119 model propeller are solved using both viscous and potential based solvers. Cavitating and non-cavitating propeller radiated noises are then predicted by using a hybrid method in which RANS (Reynolds-averaged Navier-Stokes) and FWH (Ffowcs Williams Hawkings) equations are solved together in open water conditions. Sheet cavitation on the propeller blades is modelled by using a VOF (Volume of Fiuld) method equipped with Schnerr-Sauer cavitation model. Nevertheless, tip vortex cavitation noise is estimated by using two different semi-empirical techniques, namely Tip Vortex Index (TVI, based on potential flow theory) and Tip Vortex Contribution (TVC). As the reference distance between noise source and receiver is not defined in open water case for TVI technique, one of the outputs of this study is to propose a reference distance for TVI technique by coupling two semi-empirical techniques and ITTC distance normalization. At the defined distance, the starting point of the tip vortex cavitation is determined for different advance ratios and cavitation numbers using potential flow solver. Also, it is examined that whether the hybrid method and potential flow solver give the same noise results at the inception point of tip vortex cavitation. Results show that TVI method based on potential flow theory is reliable and can practically be used to replace the hybrid method (RANS with FWH approach) when tip vortex cavitation starts.


Introduction
Propeller radiated noise is a significant topic that should be investigated during the design stage of ships and underwater vehicles. This topic is important for civil vehicles (in terms of passenger comfort and marine environment) as well as for military purposes (underwater signature). Propeller noise can be mainly divided into two sub-categories: cavitation noise and non-cavitation noise. Besides, cavitation on the propeller blades can also be divided into two categories: cavitation attached to blade surfaces and tip vortex cavitation (Oshima, 1994). Interest towards tip vortex (cavitation) problem is increasing and avoiding tip vortex noise on cavitating propellers can be crucial for cruise liners, passenger ferries, and naval ships. Noise signatures of marine propellers are generally predicted by applying noise measurements in the cavitation tunnel or numerical calculations using viscous and potential based flow solvers. As computer technologies develop, acoustic predictions with computational fluid dynamics (CFD) tools are becoming more interesting.
In the past, numerous researchers studied and predicted non-cavitating and blade sheet cavitation noise of marine propellers through numerical tools and experiments. Seol et al. (2002) investigated non-cavitating noise of a single and ducted propeller using a potential based flow solver. Authors examined the effects of the duct on the propeller hydroacoustic performance. Results showed that the effects of the duct geometry on the propeller hydroacoustic performance is small in the far-field under non-cavitating conditions since the noise signatures and directivities of two propellers are nearly the same. Kowalczyk and Felicjancik (2016) predicted the cavitating propeller radiated noise by numerical and experimental methods under non-uniform flow conditions. Hydroacoustic features of the propeller were investigated under different loading conditions. Numerical results were experimentally validated for sound pressure level, cavity patterns and hydrodynamic characteristics. Bagheri et al. (2017) investigated radiated noise under non-cavitating and cavitating conditions for two different model propellers using RANS (Reynolds-averaged Navier−Stokes) with FWH (Ffowcs Williams Hawkings) approach. Cavitation patterns and hydrodynamic characteristics of the model propellers were experimentally compared. Inception and development of sheet cavitation were examined to understand the effects of rotational speed on propeller radiated noise.
On the other hand, some studies were conducted to predict the tip vortex cavitation noise using different numerical tools and semi-empirical approaches. As is known, tip vortex cavitation generally occurs as the first type of cavitation which can be usually observed in most practical cases. Therefore, it is very important to predict the propeller noise related to tip vortex cavitation. Raestad (1996) investigated the propeller tip vortex cavitation noise and developed a semi-empirical formula (TVI) with full-scale experiments. In his study, TVI technique based on potential flow theory was developed to predict the inboard noise at a certain location for twin screw passenger vessels. However, it was stated that the proposed formulation can be safely used for single screw ships, different propeller geometries for any ship types. Furthermore, results of TVI technique which was stated in the final report of the specialist committee on hydrodynamic noise in 27th ITTC (2014) are in line with measured inboard noise data. Lafeber et al. (2015) investigated propeller radiated noise due to propeller cavitation for three different propellers. Sheet cavitation noise was modelled using the proposed method by Matusiak (1992) and Brown (1999) while ETV (Empirical Tip Vortex) model was used to predict the tip vortex cavitation noise. Numerical results showed that ETV model predicts the tip vortex cavitation noise caused by cavitating vortex accurately compared with experimental results while ETV model underpredicts the noise level if there is significant sheet cavitation on the propeller blades. Kim et al. (2016) proposed a semi-empirical formulation based on experiments to predict the tip vortex cavitation noise. Authors modified the formulation developed by Brooks and Marcolini (1986) for aeroacoustic researchers with performing the model propeller experiments in a medium-sized water tunnel. Taskar et al. (2017) investigated wave effects on cavitation and pressure pulses of a tanker. In their study, TVI technique was used to observe the difference in noise level both for waves and calm water. Bosschers (2017) improved a semiempirical formula to predict the hull pressure fluctuations and underwater radiated noise due to the tip vortex cavitation. In his formulation, principal parameter is vortex cavity size which can be calculated using a potential based flow solver. Results indicated that the presented formulation can easily be used for propeller noise estimation in design process.
All studies discussed above made a certain significant contribution to the prediction of noise due to non-cavitating and cavitating propeller. In this study, first, the flow around a model propeller was solved by using both potential based k − ω and viscous flow solvers to compare the results. Numerical results were also validated with available experimental data. DTMB 4119 benchmark model propeller operating in uniform flow (open water condition) was used for calculations. Commercial CFD software Star CCM+ was used to discretize RANS equations with SST (Shear Stress Transport) turbulence model. After this step, GCI (grid convergence index) method was implemented for verification of RANS solver. Non-cavitating and cavitating propeller radiated noise was then predicted by using a hybrid method RANS with FWH equations. Owing to the lack of experimental noise data for the selected propeller, numerical hydroacoustic results were compared with other numerical studies in the literature. Moreover, acoustic pressure signatures provided by a hybrid method which is RANS solver/FWH equations were also compared with direct RANS solver to analyse the consistency of the numerical solution. Second, tip vortex cavitation noise was predicted by using two different semi-empirical techniques (namely, TVI and TVC). A reference distance between noise source and receiver was proposed for TVI technique in open water conditions for this model propeller as distance in this technique was poorly defined. To do this, TVC and ITTC distance normalization procedures were utilized for different cavitation numbers and design advance coefficient. Reference distance was proposed for design advance coefficient. When tip vortex cavitation and associated noise start, TVI technique based on potential flow theory instead of FWH approach was examined at this suggested distance.

Numerical models
In this section, hydrodynamic models which are viscous (RANS) and potential based solvers (LSM) are briefly explained to solve the flow around the model propeller. Following this, propeller radiated noise due to sheet and tip vortex cavitation is predicted by using hydroacoustic model (FWH) and semi-empirical formulations (TVI and TVC).

Hydrodynamic models
In this section, numerical models for hydrodynamic analyses are given. Cavitating and non-cavitating flow around DTMB 4119 model propeller were solved by using potential based (LSM) and viscous (RANS) flow solvers. Since there is no sufficient available experimental data, especially for the cavitating case, a potential based code was used to compare hydrodynamic results.

Lifting Surface Method (LSM)
A lifting surface method was applied to calculate the propulsive performance, similar to that given in Bal (2011aBal ( , 2011b. In this method, three-dimensional unsteady cavitating flows around a propeller were modelled by representing the blade and wake surfaces as a finite number of discretized vortices and sources. These discretized singularities were located on the mean camber line of sections of blade and wake surfaces of each blade. Sources were put to represent the blade thickness and these sources were selected as line sources along the span-wise direction. Strengths of the line sources were computed by taking the derivatives of thickness geometry along the chord-wise direction and these computations are time independent. On the other hand, unknown cavity sources and bound vortices on the blade surface can be found with implementing kinematic and boundary conditions. Kinematic boundary condition must be satisfied at the control points where the mean camber surface is located. Cavity thicknesses were assumed to vary linearly through panels along the direction of chord-wise and were selected as piecewise fixed through panels in the span-wise direction. The effects of flow on span-wise were not found in the cavity closure condition. By implementing a uniform frictional drag coefficient, the viscous force was computed. Details of this method can be found in Bal and Güner (2009) and Kerwin (2001).

Governing equations for RANS approach
k − ω A commercial computational fluid dynamics code (STAR-CCM+) was used in the analysis as a viscous solver. The discretizing of governing equations was done using a finite volume method. Segregated flow model solves the flow equation in a segregated manner. The connection between the momentum and continuity equation was enabled using a predictor-corrector approach. Besides, a second-order upwind scheme was used for discretizing the convection terms in the equations while a first-order scheme was applied in momentum equations for unsteady terms. SST model was implemented to model the turbulence since such a model enables more accurate results for cavitating flow around model propellers (Zhu, 2014). List of numerical discretization is briefly given in Table 1.
Besides, cavitation on the propeller blades was modelled using VOF (Volume of Fluid) method to represent the water−vapour interaction with Schnerr−Sauer cavitation model which is based on Rayleigh Plesset Equation in Star CCM+ (CD Adapco, 2010).

Hydroacoustic models
In this section, methods to predict propeller radiated noise under non-cavitating and cavitating conditions are presented. A hybrid method RANS with FWH equation was used to predict the non-cavitating and blade sheet cavitation noise for different advance ratios at a constant cavitation number. On the other hand, tip vortex cavitation noise was predicted by using TVI and TVC techniques. These methods are explained briefly in the following subsections.

FWH equationp
FWH equation is an inhomogeneous wave equation derived from the conservation of mass and momentum equations. The noise around the arbitrary bodies such as rotors, propellers and different geometries can be predicted by discretizing FWH equation (Williams and Hawkings, 1969) which is based on Lighthill's acoustic analogy (Lighthill, 1952). Acoustic pressure can be calculated over a specific surface that represents the noise source. There are different ways in solving the surface and volume integrals in FWH equation. Brentner and Farassat (2003) proposed a solution by using Green functions. Total acoustic pressure can be expressed as follows: where, is the thickness noise and is the loading noise terms,respectively. Noise terms can be calculated as follows by solving the surface integrals: Eqs. (2) and (3) indicate the solution of surface integrals for thickness and loading noise terms of FWH equation. Here, r is the distance to the receiver, M r is Mach number in the radiation direction to the receiver. v i is the flow velocity and l i is the surface area in the i direction. c 0 is the sound speed and ρ 0 is the density. Symbol [.] represents derivatives taken with respect to the retarded time.

TVI technique
Tip vortex cavitation is one of the significant noise sources of marine propellers. Noise level and vibration increase when there is tip vortex cavitation. In this study, propeller radiated noise related to tip vortex cavitation was predicted by using TVI technique based on potential flow theory proposed by Raestad (1996). Reference distance between receiver and noise source is stated as approximately three decks above the propeller and at aft perpendicular for inboard noise estimation.
One of the main outputs of this study was to define a reference distance between the receiver and noise source in open water conditions since the exact receiver location is not specified in TVI technique for open water conditions. In this way, TVI technique would be practically used to predict the tip vortex cavitation noise at the desired receiver locations by using ITTC distance normalization procedure. Details are explained in the following section. TVI technique are briefly explained here for the completeness of the paper; where, SPL TVI is the noise level due to the tip vortex cavitation (dB), TVI is the non-dimensional factor describing the pressure field from the propeller tip vortices, n is the propeller rotational speed (rps), and D is the propeller diameter (m). Non-dimensional factor TVI can be calculated as follows: σ where, k tip is the tip loading factor. This factor is nearly constant for a wide range of speed in fixed pitch propellers while it is dependent on the pitch settings for controllable pitch propeller. k tbl is the non-dimensional thrust coefficient per blade. Z is the blade number and is the cavitation number.
Cavitation number can also be calculated for TVI technique: ρ where, the operating depth of the propeller h was taken as 5 m. Saturation pressure P V and atmospheric pressure P atm were taken as 2338 and 101325 Pa, respectively. g is the gravitational acceleration (m/s 2 ) and is the fluid density (kg/m 3 ). On the other hand, tip loading factor is calculated as follows: Determining circulation distribution on propeller blades was significant in calculating k tip . Circulation values which were calculated by potential based flow solver were taken from the propeller blade section r/R=0.997 and 12 o'clock blade position (Taskar et al., 2017). Reference circulation and thrust values per blade were obtained from design advance coefficient (J design =0.833) since this value gives the optimum circulation distribution on the blade (Bal, 2011a). Detailed information about TVI technique can be found in Raestad (1996) and Taskar et al. (2017).

TVC method
Tip vortex cavitation noise of marine propeller can also be predicted by using another semi-empirical technique (TVC) proposed by Kim et al. (2016). As a result of the experimental studies in the water tunnel for different types of marine propellers, a formulation developed by Brooks and Marcolini (1986) is optimized for marine propellers. In this formulation, the flow noise component due to the turbulent is considered near the propeller tip in the local separating flow. Reference distance between receiver and noise source is defined as 1 m in TVC technique.
TVC technique was summarized as follows: where, Z is the blade number and D is the propeller diameter (m), n is propeller rotational speed (rps), f is the noise frequency (Hz) and St is the Strouhal number. Strouhal number can be calculated using the Reynolds number at blade section r/R=0.7 in the propeller radius direction (White, 2006).
where, l chord is the chord length (m) which is assumed to be the quarter of the propeller diameter, V A is the average incoming flow velocity at the propeller plane (m/s). Detailed information about TVC technique can be found in Kim et al. (2016).

Geometry, boundary conditions and grid structure
This section presents the main dimensions of benchmark model propeller, boundary conditions of the problem and the appropriate grid structure.
3.1 Model propeller geometry DTMB 4119 model propeller was selected as a benchmark propeller for hydrodynamic and hydroacoustic performance prediction in numerical analyses. A 3-D view of the model propeller and main particulars are given in Fig. 1 and Table 2, respectively.

Boundary conditions
Initial and boundary conditions must be defined according to flow characteristics of the problem. A computational domain which represents the open water test condition was generated to predict the hydrodynamic and hydroacoustic Savas SEZEN, Sakir BAL China Ocean Eng., 2020, Vol. 34, No. 2, P. 232-244 235 performance of the model propeller. Origin of coordinate system was defined as the centre of the propeller blades. The computational domain was divided into the rotating region around propeller blades and static region which are related to an interface defined between these two regions. As can be seen in Fig. 2, the right-hand side of the computational domain was defined as velocity inlet while the lefthand side was pressure outlet. Remaining surfaces were also defined as symmetry plane. Propeller blades and shaft were identified as no-slip wall. Thus, the kinematic boundary condition on the propeller blades was satisfied. Besides, the computational domain dimensions were also given in Table 3. 3.3 Grid structure Unstructured grid with hexahedral elements was used to discretize the computational domain by implementing FVM. As seen in Fig. 3, local mesh refinements were applied to refine the grid around the propeller blades since the flow field should be well captured in the near-field region to precisely predict propeller radiated noise in the far field.
The motion of propeller rotation can be simulated by RBM (Rigid Body Motion or Sliding Mesh) or MRF (Moving Reference Frame) methods. In this study, RBM method was used to simulate propeller rotation for propeller hy-droacoustic and cavitating analysis while MRF method was implemented for non-cavitating hydrodynamic analysis. RBM is the rotating mesh with a sliding interface which is considered as the most accurate way to solve unsteady problems. RBM method involves accurate time-dependent simulations where transient features of the flow are most significant. At every time step, unsteady governing equations are solved for each cell and the communication with the interface region is made by calculation of the fluxes at the boundaries of the interface. Both methods (RBM and MRF) give similar results for propeller hydrodynamic characteristics. However, several studies such as Kellett et al. (2013), Noughabi et al. (2017, Moussa (2014) and Hallander et al. (2012) stated that RBM method generates better results for prediction of propeller noise under cavitating and non-cavitating conditions. Additionally, the time step size was set corresponding to 1degree of propeller rotation (Δt = 2.77 × 10 −4 ). Thus, CFL number was calculated as 0<CFL<1.
Furthermore, two-layer wall y+ treatment was used to define the mean flow quantities around the near-wall region of turbulent boundary layers. Increase of cell size was set at fixed growth rate from the boundary layer to outer boundaries of the computational domain. Average wall y+ values were found in the range of 50−60 for different advance coefficients. Grid structures and element numbers implemented in this study were also given in Table 4.

CFD verification study
Uncertainty assessment for RANS approach was applied by using GCI (Grid Convergence Method) based on     Richardson (1911) extrapolation. This method was first proposed by Roache (1998) and has been implemented in numerous studies in the literature. Verification of grid resolution was implemented with the methodology designated by Celik et al. (2008). Refinement factor is defined larger than 1.3 in line with experiments. Therefore, the refinement factor r was selected as as in Tezdogan et al. (2015), Sezen et al. (2018) and Cakici et al. (2018). Uncertainty study was conducted for the thrust coefficient at J=0.5. To apply this procedure, three solutions were selected. ε The difference of the solution scalars should be determined by Eq. (10) where, , and are fine, medium and coarse mesh grid solutions, respectively. The ratio of solution scalars is used to calculate the convergence condition by Eq. (11) Convergence condition R defines the solution type. Thus, possible R values and solution types are listed below (Stern et al., 2006): (1) −1 < R < 0, oscillatory convergence; (2) 0 < R < 1, monotonic convergence; (3) R < −1, oscillatory divergence; (4) R > 1, monotonic divergence. If convergence condition R is found as in Case (2), the procedure can be directly employed.
Approximate relative error and extrapolated relative error are: Finally, GCI index is calculated by: Verification results are given in Table 5.
As seen from Table 5, the convergence condition R is between 0 and 1 (monotonic convergence). As a result of the uncertainty study, the fine mesh was selected for both hydrodynamic and hydroacoustic analyses. On the other hand, the difference between numerical and experimental study for thrust coefficient K T was found approximately 1.33% at J=0.5.

Results and discussions
This section presents the numerical results and discussions on hydrodynamic and hydroacoustic performance prediction of DTMB 4119 model propeller in (uniform flow) open water conditions.

Hydrodynamic results
Hydrodynamic characteristics of the marine propeller are determined with open water tests. In open water tests, propeller thrust, torque, and efficiency values are calculated for different advance ratios. Propeller hydrodynamics equations are briefly explained below: ρ where T is the thrust (N), Q is the torque (N·m), D is the diameter (m), n is the propeller rotational speed per second (rps) and is the density of water (kg/m 3 ). V A is the incoming averaged flow velocity at the propeller plane (m/s).

Non-cavitating hydrodynamic analysis
Hydrodynamic characteristics of model propeller were predicted at different advance ratios using potential based (LSM) and viscous (RANS) solvers. The rotational speed of model propeller was set at 10 rps. Incoming average velocity (V A ) was changed to achieve different advance coefficients. Propeller rotation was simulated by MRF method in a steady manner. As seen in Fig. 4, numerical results were validated with experimental results (Jessup, 1989). All results show good fit.

Cavitating hydrodynamic analysis
Sheet cavitation on the propeller blades was modelled by using viscous and potential based solvers to compare the results since there are no sufficient available experimental cavitation data in the literature for the selected propeller. For cavitation simulations in the viscous solver (Star Fig. 4. Comparison of open water characteristics with RANS, LSM and experiment (Jessup, 1989) under non-cavitating conditions.
ρ where, p sat is the saturation pressure (Pa), p ∞ is the pressure of the liquid (Pa) and is the liquid density (kg/m 3 ). Moreover, cavitation number can also be calculated by Eq.
ρ where P 0 is the static pressure (Pa), is the fluid density (kg/m 3 ), D is the propeller diameter (m) and n is the propeller rotational speed per second (rps). Detailed information about the cavitation model can be found in Schnerr and Sauer (2001). σ Here the cavitation simulations related to sheet cavitation were conducted at J=0.5 and =2.5 by using both solvers in open water conditions. Propeller rotational speed (n) was kept at 10 rps and static pressure was changed to obtain the desired cavitation number. Table 6 shows non-dimensional thrust, torque and efficiency values computed by both methods. Viscous solver predicts higher hydrodynamic characteristics with respect to the potential based code. Based on the obtained results, compliance between the two methods is acceptable.
In addition, sheet cavitation pattern on the propeller blades for unsteady RANS analysis was generated by using the VOF method in Star CCM+. Cavity pattern was also computed by potential based flow solver. Fig. 5 shows the cavity patterns due to sheet cavitation by using both methods. As seen in Fig. 5, the locations of sheet cavitation covers on the propeller blades predicted by both methods are highly close.

Hydroacoustic results
In this section, non-cavitating and cavitating noise results related to sheet and tip vortex are presented by using a hybrid approach (RANS with the acoustic analogy (FWH)) and the semi-empirical techniques. In the hybrid approach, time-dependent acoustic pressures were calculated during analyses and these data were transferred from time domain to frequency domain by using fast Fourier transform (FFT). Thus, sound pressure level (SPL) was calculated as follows: where, p is the acoustic pressure (Pa), p ref is the reference acoustic pressure (for water p ref =10 −6 Pa). On the other hand, overall sound pressure level (OASPL) can be calculated as follows: where p rss is the acoustic pressure (Pa). Decibel equivalent of the root sum square (RSS) of pressure was used in OAS-PL calculation as in NASA (1996). In addition, sound pressure levels can be corrected to a standard measuring distance of 1 m by using Eq. (20) (ITTC, 2014), SPL where, SPL p is the reference noise level (dB) and SPL s is the normalized noise level (dB). d ref is taken as 1 m. For noise prediction simulations in Star CCM+, the propeller rotation was simulated by using RBM (Rigid Body Motion) as explained in Section 3.3. Grid structure determined by uncertainty study was implemented in this analysis. Acoustic pressures were collected at receiver locations as shown in Fig. 6.
where, n is the propeller rotational speed (rps), Z is the blade number, and k is the harmonic number.

Non-cavitating noise results
Non-cavitating propeller radiated noise was predicted at different loading conditions. To show the consistency of numerical calculations, pressures in the near field were calculated by solving direct hydrodynamic solver and FWH acoustic analogy and results of this method as well as other numerical studies in the literature were compared. It should be noted that there were no sufficiently available experimental noise data in the literature for the benchmark propeller.
As the first step, pressures derived by a hybrid method RANS+FWH and directly solving the hydrodynamic solver (RANS) were compared at J=0.5 for different receiver locations. Generally, in numerical simulations, noise prediction dρ = 0 c 0 2 = dp/dρ can be made by direct approach or hybrid methods. Direct numerical simulations require compressible flow solution because sound speed is finite and sound is defined as a pressure fluctuating propagating in a medium. On the other hand, in marine applications, hydroacoustic field based on incompressible flow theory requires incompressibility assumption ( ). Accordingly, sound speed is infinite based on the isentropic flow hypothesis ( ). Therefore, the direct approach cannot be used to predict the noise level instead of a hybrid method which includes acoustic analogy with hydrodynamic solver in marine problems. Besides, acoustic pressures received from other distances where such signals may come from all possible noise sources in the fluid domain and instantaneous signals overlap at a specified distance. Hence, random signals generate a time shift called compressibility delay at receiver positions (Ianniello et al., 2013;Lloyd et al., 2015). The propeller rotation speed is relatively smaller than sound speed (c 0 =1500 m/s in water). Therefore, time shift from noise sources might be negligible. In this way, incompressible RANS pressure and FWH acoustic pressure in the near field can be compared if the propeller rotational speed and loading on the blades are low. Comparison study enables us to analyse the reliability of two numerical solutions, especially if experimental noise data are insufficient. Fig. 7 presents a comparison of pressure fluctuations obtained from both hydrodynamic solver and acoustic analogy for one propeller rotation at near field at J=0.5 under non-cavitating conditions. Here, the signal is characterized by blade passage frequency (three peak values mean three blades). Results showed that pressure values were compatible with Receivers 1 and 2. However, the difference between pressures starts to increase with an increase in receiver distance, i.e. from near field to far-field as shown in Fig. 8. It should be noted that with an increase in distance between noise source and receiver, numerical diffusion probably affects RANS pressure. However, FWH pressures are not changed as a result of numerical diffusion since these pressures only depend on blade shape and hydro-  Savas SEZEN, Sakir BAL China Ocean Eng., 2020, Vol. 34, No. 2, P. 232-244 239 dynamic load on propeller blades. Note that the near field noise can be predicted by using RANS pressure, while the hydroacoustic solver is necessary to predict far-field noise. Second, non-cavitating noise results were also compared with that of other two numerical studies in the literature. Since there are no experimental data on the noise of the selected propeller in the current literature, the results of the present method were compared with that of other numerical studies. Non-cavitating noise spectrum was compared with Seol et al. (2005). In their method, potential based panel method was coupled with FWH analogy at Receiver 6 (z=1.5 m, d=10R), V A =1.6 m/s, n=120 rpm. As shown in Fig. 9, distributions of the noise spectrum in frequency domain were compliant. Also note that although Seol's data exclude BPF values that define the characteristics of the propeller noise level, BPF value was well captured in the current study, especially in the first BPF.
SPL values for non-cavitating noise by present method was also compared with that of Bagheri et al. (2017) at Receiver 6 (z=1.5 m, d=10R), n=960 rpm, V A =2.6 m/s, and J=0.54. As shown in Fig. 10, SPL distribution by the present method was found slightly higher than that of the study of Bagheri et al. (2017). This difference might be caused by grid structure and/or signal to process. It should be noted that there was no comparison of RANS and FWH approach for the near field in Bagheri et al. (2017). Validation study of RANS and FWH in near-field also makes it possible to find the suitable fine grid structure for acoustic analyses. In addition, BPF values at f=48, 96 and 144 Hz were as expected in the present study but this values were not presented in Bagheri et al. (2017). BPF values dominate the noise spectrum, especially in the low-frequency region.

Sheet cavitation noise results
σ σ Hydroacoustic performance of the model propeller was numerically simulated to investigate the noise increase due to sheet cavitation ( =2.5) for a fixed advance coefficient (J=0.5) at z=1 m (Receiver 5). In cavitating hydroacoustic simulations, the propeller rotational speed a as 10 rps and only static pressure was changed to obtain the cavitation number of =2.5 (P V =2338 Pa). Cavity pattern is indicated in Section 5.1.2. Results indicated that sheet cavitation caused an increase in the noise level of approximately 8 dB as shown in Fig. 11.

Tip vortex cavity noise by TVI and TVC techniques
Tip vortex cavitation noise was investigated by using two semi-empirical methods (namely, TVI and TVC techniques) under different loading conditions. In TVI technique, calculation of k tip (tip loading factor) which depends on the circulation and thrust values per blade is crucial for predicting the noise precisely. Circulation values were taken from the blade section of r/R=0.997 at 12 o'clock blade position as also noted by Taskar et al. (2017). Reference circulation and thrust values per blade (Eq. (7)) were obtained from optimum circulation distribution (design J=0.833) (Bal, 2011a). Different cavitation numbers for observed tip vortex cavitation were determined by potential based solver at the design advance coefficient. Propeller rotational speeds required for TVC technique were obtained from computed cavitation numbers. So, tip vortex cavitation noise can be predicted by TVI and TVC techniques by using Eqs. (4) and (8), respectively. In addition, overall sound pressure levels (OASPL) can be calculated by using the recommended formulation [Eq. (18)] by NASA (1996).
Overall tip vortex cavitation noise versus different cavitation numbers are given in Table 7 for both techniques. It is obvious that the tip vortex cavitation noise decreased with an increase in cavitation number at J=0.833. Reference distance was given as 1 m (in the propeller rotation axis) for TVC technique while the distance between the noise source and receiver was not defined clearly for TVI technique in open water conditions.
Later on a reference distance between the noise source σ and receiver was proposed for TVI technique since it was not specified clearly for open water conditions by Raestad (1996). This distance was selected as three decks above at aft perpendicular for inboard noise estimation. TVC technique has been used to do this at design advance coefficient (J=0.833) by utilizing ITTC distance normalization formula (Eq. (19)). Distance proposal was made at cavitation number ( =0.75) where higher tip vortex cavitation occurs. Reference distance was found approximately as z=8.3 m (z≈27D) in the propeller rotation axis for TVI technique (as shown in Fig. 12). Fig. 13 and Table 8 show the overall sound pressure σ Fig. 11. Propeller radiated noise levels at Receiver 5 (z=1 m) under noncavitating and sheet cavitation conditions (J=0.5, n=10 rps, =2.5).   levels at the proposed distance (z=8.3 m) by both techniques. As seen in Table 8 and Fig. 13, OASPL differences between the two techniques were small enough at the proposed distance. This means that the proposed distance (z=27D) can be reliably used for tip vortex cavitation noise estimation by TVI technique under open water conditions.

FWH versus TVI technique
This section investigated whether TVI technique based on potential theory could be used instead of FWH method when the tip vortex cavitation starts. First, for the advance coefficients, cavitation numbers were analysed by LSM when the tip vortex cavitation starts as given in Table 9.
The viscous solver was implemented to predict the overall sound pressure level related to tip vortex cavitation by using FWH acoustic analogy. Simulations in Star CCM+ were conducted at specified cavitation numbers and advance coefficients which are given in Table 9. Cavity patterns where the tip vortex cavitation starts were obtained σ both from potential based code (LSM) and viscous solver and these patterns are shown in Fig. 14 for J=0.53 and =5.0. As shown in Fig. 14, the tip vortex cavitation has just started to occur at the specified condition in both solvers. In the scope of this study, the main aim is to determine the tip vortex cavitation inception points rather than observation of tip vortex cavitation in the slipstream. Therefore, special grid techniques such as adaptive mesh refinement were not included to observe the tip vortex cavitation in the slipstream since it increases the total element count of the computational domain considerably (Lloyd et al., 2017).
Overall tip vortex noise was then predicted for specified three different cavitation numbers and advance coefficients (Table 9) both using TVI, TVC techniques and FWH acoustic analogy at the proposed distance z=8.3 m. It should be noted that the hydroacoustic simulations in Star CCM+ enable to locate the receivers outside the computational domain (CD Adapco, 2010). Results showed that TVI technique gives promising results compared with FWH acoustic analogy at the proposed distance as shown in Table 10. On the other hand, TVC method predicts the noise level slightly lower than TVI and FWH methods. The relative difference between values of TVI and FWH with respect to FWH acoustic analogy is also added to the table and it can be noted that the results are compliant.

Conclusions
In this paper, flow around the model propeller was solved in uniform flow (open water conditions) using viscous and potential based solvers. Numerical results were also validated by available experimental data for thrust, torque coefficients and efficiency values under non-cavitating conditions. Verification study was implemented by GCI method which is recommended in ITTC procedure for CFD verification.
After validation and verification studies, numerical simulations were conducted to determine the propeller hydroacoustic performance by using a hybrid method, RANS with FWH equation in uniform flow. Numerical hydroacoustic results were compared with other numerical studies in literature since there is no available experimental noise data for the model propeller. It is found that numerical results are compliant with that of other numerical studies. Moreover, the first BPF value, which dominates the noise σ Fig. 14. Comparison of cavity patterns for potential based code (right) and viscous solver (left) at J=0.53 and =5.0.  spectrum in narrowband, was well captured in accordance with the present study. Another comparison study was conducted for acoustic pressures which were directly computed by the hydrodynamic solver and acoustic analogy (FWH) to analyse the reliability of both numerical solutions. Results showed that hydrodynamic solver can be directly used to predict the hydroacoustic performance of the model propeller in the near-field. On the other hand, for the far field, acoustic analogy must be used to precisely predict the noise level.
It is also found that non-cavitating and sheet cavitation noises were compared well at a constant advance coefficient by taking the same propeller rotational speed n i.e. only the changing static pressure P 0 for the desired cavitation number. Numerical results showed that the propeller noise increased approximately 8 dB when there was sheet cavitation on the propeller blades. Furthermore, tip vortex cavitation noise was predicted by using two semi-empirical approaches namely TVI and TVC for different loading conditions. It should be noted that reference distance between the noise source and receiver for TVI was proposed by using TVC and ITTC distance normalization procedure for the model propeller in open water conditions. The distance was proposed at the design advance coefficient and the lowest cavitation number which corresponded to higher tip vortex cavitation under uniform flow conditions. It is found that TVI method based on the potential-based solver can be used to replace acoustic analogy as tip vortex cavitation starts. Results show that the overall sound pressure levels by TVI and FWH acoustic analogy were relatively close to each other at a proposed distance. Results of TVC was found at an acceptable level for FWH and TVI methods under different loading conditions. Note that TVI method is a highly cost-effective and simple approach compared with acoustic analogy. It should also be noted that the proposed distance for TVI is only valid under uniform flow conditions. Future study will be conducted for the non-uniform flow conditions to check out the proposed reference distance under uniform flow conditions and the reliability of TVI technique with respect to FWH approach when the tip vortex cavitation starts.