An Empirical Approach for Propeller Tip Vortex Cavitation Noise

2.1. PLL and LSM

PLL (Propeller Lifting Line) theory and a LSM (Lifting Surface Method) have been coupled to design the propeller with optimum circulation distribution. PLL has first been applied to represent the propeller as a set of a number of blades and straight and radial lifting lines. The blades have angular spacing and equal loading. The continuous distribution of vortices along the lifting line is discretized by vortex lattice elements with constant strengths. The velocity induced at the lifting line by this system of vortices can be computed using the very efficient formulas given by Kerwin [15]. They are not repeated here. The PLL uses the Betz-Lerbs condition to obtain the optimum circulation distribution over blades. Detail information for PLL theory can be found in [15] and [16]. LSM has also been applied to calculate the propulsive performance and to check the optimum circulation distribution obtained by PLL, similar to the one given [16], [17]. The lifting surface method models the three-dimensional unsteady cavitating flow around a propeller by representing the blade and wake as a discrete set of vortices and sources, which are conveniently located on the blade mean camber surface and wake surface. In this method, a discretized version of the kinematic boundary condition can be employed as:

where υ Γ the velocity vector is induced by each unit strength vortex element, υQ is the velocity vector induced by each unit strength source element, and n m is the unit vector normal to the mean camber line or trailing wake surface. QB and QC represent the magnitude of the line sources that model the blade thickness and cavity source strengths, respectively. Details about LSM and coupling with PLL are given in [16].

2.2. Unsteady RANS Method

The governing equations are the continuity equation and the RANS equations for the unsteady, three-dimensional, incompressible flow. The continuity can be given as;

Velocity Ui can be decomposed into mean velocity part and fluctuating velocity part, respectively; U i  U i  ui '

The momentum equations are expressed as; ' ' U i  U iU j  1 P    U i U j   uiu j           t x j  xi x j   x j xi   x j

In this study, the analyses have been conducted in steady state manner under non-cavitating conditions while transient analyses have been carried out in order to simulate the cavitation on propeller blades. In momentum equations, U i (m/s) states the mean velocity while ui' states the fluctuation velocity components in the direction of the Cartesian coordinate xi (m). P expresses the mean pressure (Pa),  (kg/m3) the density and  (m2/s) the kinematic viscosity of the fluid. The k-ε turbulence model is applied to simulate the turbulent flow around the model propellers

under both non-cavitating and cavitating conditions. Detailed explanations for the k-ε turbulence model can be found in the literature [18].

2.3. Uncertainty Assessment

In this study, uncertainty assessment has been carried out by Grid Convergence Index (GCI) as recommended in the ITTC procedure for CFD verification [19]. This method firstly was proposed by Roache [20] and then improved with other studies. The procedure implemented in this study has briefly been explained below [21]: Let h1, h2 and h3 are grid lengths and h1 < h2 < h3. The refinement factors are as follows:

Refinement factors should be greater than 1.3 in accordance with the experiments [21]. Grid lengths’ refinement has been selected as 2 as in the studies of Tezdogan et al. [22] Luca et al. [23] and Ozdemir et al. [24]. 1/ 3

Differences between generated grid numbers can be calculated as below:

At this point, the convergence condition R can be examined as,

The solution has an oscillating convergence when -1<R<0 while the solution is monotonically convergent if 0<R<1 [25].

2.4. TVI Technique

TVI (Tip Vortex Index) technique has been implemented to predict the tip vortex cavitation noise of marine propellers. This technique has been developed in accordance with the experimental studies for different type of vessels such as cruise liners, passenger vessels [6]. In this method, the propeller has been considered as the acoustic source of transmitting the noise. In the report of the 27th ITTC conference [7], it has also been mentioned that TVI technique gives satisfactorily good results with experimental measurements. TVI technique has been widely used for prediction of tip vortex cavitation noise at a reference distance (three decks above the propeller and aft perpendicular). The non-dimensional TVI factor is given as;

Here, ktip is the relative tip loading factor. ktbl is the non-dimensional thrust coefficient per blade. Z is the blade number and σ is the cavitation number. Cavitation number is given as;

Cavitation number is calculated by considering that the operation depth of the propeller (h) is 5 meters while the saturation pressure (P v) is taken as 2000.7 Pa (for water at T=17.5 oC). Patm is 1130

the atmospheric pressure level (Pa), D is the propeller diameter (m) and ρ is the fluid density (kg/m3). The propeller number of revolution (n, rps) is calculated from the non-dimensional cavitation number.

After calculation of TVI, the sound pressure level due to the tip vortex cavitation is calculated as follow,

Detailed information about TVI technique can be found in [6], [8].

3.1. Optimum Blade Design

In this study, DTMB 4119 model propeller has been chosen for the investigation of the tip vortex cavitation noise. The original DTMB 4119 propeller has three blades. The number of blades of the original DTMB 4119 propeller has later been changed to 2 and 4, respectively. All other geometrical properties (such as blade sections, pitch ratios, chord lengths etc.) are same as in the original three-bladed propeller. The geometrical properties of the model propellers have been given in Table 1. Detailed information about the propeller geometry can be found in [26]. 3D views of the DTMB 4119 model propellers have also been presented Figure 2.

Table 1. Main particulars of DTMB 4119 propellers DTMB 4119 Model Propeller D (m) 0.3048 Z 2-3-4 Skew (0) 0 Rake (0) 0 Blade section NACA66 a=0.8 Rotation direction Right

Figure 2. 3-D views of DTMB 4119 model propellers 1.2 1.1 1 0.9 0.8

Figure 3. Comparison of circulation distribution with PLL and LSM for non cavitating DTMB 4119 propeller (Z=3) (Bal, 2011a) The PLL (propeller lifting line) method and LSM (lifting surface method) have been implemented as seen in Figure 3 for design J=0.833, respectively. Here the radial optimum circulation distribution which is non-dimensionalized by 2 RVR (here VR is the resulting velocity at 0.7 R, VR 

V   0.7 nD  ). The obtained radial circulation distributions of PLL 2

and LSM have been compared in Figure 3. The differences of radial circulation distribution of the two methods are quite small. By this way, it can be specified that the blade geometry of the DTMB 4119 model propeller (Z=3) has optimum radial circulation distribution under the design conditions. The non-dimensional thrust and torque values are also given in Figure 3 [16]. For tip vortex cavitation noise calculation, the reference circulation value has been taken at the r/R=0.997

3.2. Verification and Validation for Propeller Hydrodynamic

The grid convergence index (GCI) method has been applied for uncertainty assessment using RANS method. It is implemented for original DTMB 4119 model propeller (Z=3) for thrust value. The optimum grid number has also been used for other blade numbers (Z=2 and Z=4). As seen in Table 2, three different grid sizes have been used for open water propeller analyses to model the computational domain. The significant part of the uncertainty of the numerical results is the selection of grid [27]. The uncertainty study has been done at design J=0.833. Table 2. Grid numbers for uncertainty analysis Open water condition (OWC) #

As a result of the uncertainty study at J=0.833, the convergence condition (R) is calculated as 0.26 which means that the solution is converged monotonically. As a result of the GCI method, the uncertainty value has been found as 0.032 %. Fine mesh algorithm has then been selected for open water propeller analyses. Validation study of RANS method has also been done with specifying the optimum grid number using GCI method. The numerical results have been compared with the experiments under non-cavitating conditions (Z=3). The relative absolute difference has been found as 0.68 % at J=0.833 with the experiments. After verification and validation study, the rest of the analyses have been done using optimum grid number.

3.3. Non-cavitating Open Water Propeller Analyses

The flow around the DTMB 4119 model propellers have been solved using RANS method and LSM, respectively. The reference axis has been chosen as moving reference frame (MRF). As seen in Figure 4, the computational domain is divided into two parts; static region and rotating region. The right and left side of the computational domain has been defined as velocity inlet and pressure outlet respectively. The rest of the surfaces have been considered as symmetry planes in order to capture the normal component of the velocity as zero. The propeller blades and shaft surfaces have been also defined as non-slip wall for no-slip condition [28]. Discretization of the computational domain has been done with the finite volume method. Unstructured mesh algorithm has been used to solve the velocity and pressure fields precisely; mesh refinements have been done on the propeller blade surfaces (Figure 5).

Figure 4. Computational domain and boundary conditions for DTMB 4119 propellers

Figure 5. Unstructured mesh on propeller blade surfaces for propellers with different number of blades Later, the flow around the model propellers has also been solved by the lifting surface method under non-cavitating condition. The panel distributions on the propeller blades and wakes have been given in Figure 6.

Figure 6. Panel arrangement of DTMB 4119 propellers and their wakes As can be seen in Figure 7, the results of LSM and RANS method have been validated with the experimental data of three-bladed model propeller under non-cavitating condition. RANS method gives better results for open water propeller efficiency when compared with LSM. Figure 7 also shows the numerical results of two and four-bladed model propellers.

Figure 7. Comparison of thrust, torque and efficiency of DTMB 4119 propeller for Z=2, 3, 4, respectively.

3.4. Cavitating Open Water Propeller Analyses

The cavitating flow around the model propellers have been solved using unsteady RANS method with same boundary conditions as in Chapter 3.2. To capture the cavitation pattern on the propeller blades, volume of fluid method (VOF) has been used in the analyses. The thrust and torque values and the cavitation patterns by LSM and RANS methods have been compared at J=0.53 and σ=2.2. As can be seen in Table 3, the non-dimensional thrust and torque values computed by LSM and unsteady RANS solver are in good agreement with each other. The absolute relative differences between URANS and LSM are also given in Table 3. Table 3. Comparison of hydrodynamic results of URANS method and LSM under cavitating conditions Z

3.5. TVI Technique for Tip Vortex Cavity Noise

Tip vortex cavitation has a major effect on propeller radiated noise. The higher blade loading on the tip triggers the noise level inevitably. TVI technique has been coupled with lifting surface method based on the potential theory [6]. In this technique, tip loading factor has been calculated by using the circulation (loading) value on the blade section r/R=0.997 at 12 O’clock blade position which is calculated by LSM [8]. The reference circulation value has been obtained from the optimum distribution of circulation for J=0.833 and three-bladed propeller under noncavitation condition. The reference circulation for three-bladed propeller value has then been assumed to be same as for two and four-bladed propellers. The noise related to tip vortex cavitation has been calculated at a fixed distance for all three propellers. Table 4 shows the parameters used in TVI calculation at a constant advance ratio (J=0.53) and cavitation number (σ=2.2) for all blade numbers. Table 4. Numerical values for TVI calculation at a constant advance ratio and cavitation number Z 2 3 4

The noise level due to the tip vortex cavitation has been calculated for different cavitation numbers at a constant advance ratio (J=0.53).The results have been given in Figure 9 for different blade numbers. As expected, the sound pressure level is decreasing with an increase in blade number. Cavitation volume and lengths increase with a decrease in the cavitation number. Therefore, the sound pressure level due to the tip vortex cavitation is increasing while the cavitation number decreases for all three propellers.

Figure 9. Sound pressure level of tip vortex cavitation for different cavitation numbers at a fixed advance ratio (J= 0.53) for different blade numbers. On the other hand, the effect of advance ratios on tip vortex cavitation noise at constant cavitation number (σ=2.2) has been investigated as shown in Figure 10. As can be seen in Figure 10, the noise level due to the tip vortex cavitation is decreasing with an increase in advance ratio. It is also decreasing slightly with an increase in blade number since the loading on the blades are decreasing with an increase in blade numbers.

Figure 10. Sound pressure level related to the tip vortex cavitation versus advance ratio for different blade numbers at σ=2.2