LI Qi, and LIU Pengfei

1) Australian Maritime College, University of Tasmania, Launceston, Tasmania 7250, Australia

2) Marine, Offshore and Subsea Technology, School of Engineering, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

Abstract A systematic method was developed for ice-class propeller modeling, performance estimation, strength and integrity evaluation and optimization. To estimate the impact of sea ice on the propeller structure, URI3 rules, established by the International Association of Classification Societies in 2007, were applied for ice loading calculations. An R-class propeller (a type of ice-class propeller) was utilized for subsequent investigations. The propeller modeling was simplified based on a conventional method, which expedited the model building process. The propeller performance was simulated using the computational fluid dynamics (CFD)method. The simulation results were validated by comparison with experimental data. Furthermore, the hydrodynamic pressure was transferred into a finite element analysis (FEA) module for strength assessment of ice-class propellers. According to URI3 rules, the ice loading was estimated based on different polar classes and working cases. Then, the FEA method was utilized to evaluate the propeller strength. The validation showed that the simulation results accorded with recent research results. Finally, an improved optimization method was developed to save the propeller constituent materials. The optimized propeller example had a minimum safety factor of 1.55, satisfying the safety factor requirement of ≥1.5, and reduced the design volume to 88.2% of the original.

Key words polar class propeller; URI3 rules; propeller blade strength and integrity design and optimization; ice-class propeller hydrodynamics-strength coupled

1 Introduction

The polar region navigation has strategic importance in scientific research, resource development, and tourism.However, navigation in such severe environments threatens shipping safety. In particular, the impact of sea ice on the propeller may cause structural deformation and failure;it also has the potential to result in costly damages or loss of life. For example, DNV conducted several experiments on propellers and found that the reverse operation under ice condition will bend the blade (Norhamoet al., 2009).Therefore, the strength evaluation of propellers navigated on polar regions (polar propeller or ice-class propeller) is fundamental and necessary.

However, the interaction between the propeller and sea ice is complex. Many researchers have attempted to establish appropriate ice-propeller interaction models. For instance, in one study, a load model was built to evaluate ice load levels, and a model of ice failure process was also developed based on experiments (Soininen, 1998).Different nations and classification societies created their own ice load standards before 2008 (Liuet al., 2015). In 2007, the International Association of Classification Society (IACS) established the Unified Rules I3 (URI3),which came into effect in 2008, to identify ice class, the ice load magnitude, and the propeller loading area (IACS,2011). Although many class and ice load standards have been established and unified, further developments and research on these models are ongoing. For example, Vroegrijk and Carlton (2014) proposed a new material model that can describe the orthotropic rate-dependent properties of sea ice.

Since after 2007, many researchers have applied the URI3 rules to evaluate ice-class propeller strength. In 2007, Lee applied URI3 relatively early and evaluated the strength of a sample propeller based on finite element analysis (FEA)and summarized some general assessment results. In 2008,Lee investigated a Baltic ice-class controllable pitch propeller using URI3. Liuet al.(2015) evaluated an R-class propeller based on an advanced 3D unsteady panel method coupled with a structural model and URI3 rules for blade ice loading calculations. In 2018, a strength assessment method was developed based on IACS URI3 and FEM(Yeet al., 2019). In conclusion, IACS URI3 has been widely used since 2007. Moreover, owing to its usefulness in investigating the ice-class propeller strength, FEA has become a crucial and widely accepted tool. This research will apply FEA and URI3 rules for the propeller strength evaluation.

To meet the safety criteria of the polar area navigation regarding the propeller design, the propeller structure may be overbuilt, resulting in manufacturing waste; moreover,there are still weak sections along the blade span, resulting in a low safety factor (SF) of the blade structure. Thus,optimizing polar propellers in terms of both strength and material consumption is required. In 2015, an advanced 3D unsteady panel method was developed for the optimization of both hydrodynamics and structural strength to reduce the amount of propeller constituent material (Liuet al., 2015). The current research aims to optimize polar propellers by modifying the blade geometry based on requiredSFsviaa joint computational fluid dynamics (CFD)and FEA model.

2 Methodology

This study mainly utilized the CFD and FEA methods to estimate the propeller performance and assess the propeller strength. The detailed optimization steps are described in Section 6.1. The entire process of evaluation and optimization is recurring. Thus, any mistake or error will be amplified in the steps (Fig.1). The propeller performance estimation aims to obtain the necessary hydrodynamic pressure for strength evaluation. This propeller performance estimation method was also used for performance comparison between the original propeller and the optimized propeller to investigate the influence of optimization on the propeller performance.

Fig.1 Polar class propeller strength and integrity evaluation and optimization process.

3 Propeller Modeling

Propeller modeling is an essential process for propeller design, test, and production. Notably, generating the propeller blade is the most critical part of the propeller 3D modeling. In this study, the investigated propeller model was modified repeatedly to ensure the research correctness and achieve the corresponding objective. Therefore,a simple and efficient propeller 3D modeling method was established. To investigate the strength and integrity of iceclass propellers, the geometry of an R-class propeller was utilized, which has been employed for three Canadian Coast Guard ship (CCGS) icebreakers in Canada (Liuet al.,2015). Table 1 displays the basic parameters of the researched propeller.

The primary propeller geometry data were output from Propella (Liu, 2019) in meshing format. Propella is a panel method code that can efficiently provide basic propeller geometer information for six different propeller types. The conventional propeller model generation process involves many steps, such as the point, generation based on propeller drawings or an offset sheet, the generation of blade section outline, the lofting of the blade surface, and the combination of surfaces to generate the solid model. Applying Propella facilitates the basic geometry of six widely used types of propellers. In this research, propeller modeling was conducted in Rhinoceros 5.0 (Rhino). Themodeling details are shown in Fig.2 and can be concluded as follows: the meshing model is imported into Rhino; the propeller section outline is generated by duplicating borders; the section outline is rebuilt into smooth and highorder curves; the section outline is divided into two segments by using the chord line; the suction face is generated,the pressure face and tip face are lofted individually; all surfaces are combined, and a hub is generated by executing the instruction ‘join two naked edges’; the instruction ‘create solid’ is used to enforce the solid model generation.

Table 1 CCGS R-class icebreaker parameters(Liu et al., 2015)

Compared with the conventional propeller generation process, the presented method reduced the modeling time in obtaining the propeller geometry. The blade surface generation could also be quantitatively controlled. Moreover,the scale effect could be neglected for full-scale propeller modeling. However, some errors were generated during the modeling. First, the rebuilding of the blade section curve and the lofting of the blade surface produced geometry differences. However, a proper operation and setting will generate smooth section curves, which will benefit the blade surface modeling and the meshing in the following CFD and FEA simulation. Second, the selection of section curves influenced the propeller geometry, but the appropriate selection can reduce workloads. Moreover,Rhino, the surface modeling software, cannot generate a real solid model; therefore, the solid model file needs to be imported into the workbench, which may cause some problems. To avoid such problems, the Rhino model should be imported into other solid model design programs, such as Inventor, to enforce the real solid model generation.

Fig.2 Propeller modeling process and solid model.

4 Propeller Performance Estimation

For investigating the propeller performance, the towing tank test is widely used as an effective experiment method.However, the towing tank test requires experiment facilities, propeller models, and skilled laboratory personnel.These requirements are not only expensive but also timeconsuming. The CFD method has been used for propeller performance estimation for many years. For example, the hydrodynamic performance of a system consisting of a propeller and a rudder was predicted in 2007 (Ghassemi and Ghadimi, 2008). Also, Reynolds-averaged Navier-Stokes simulations based on OpenFOAM have been used to investigate the hydrodynamic performance of a propeller in oblique flow (Yao, 2015). Therefore, applying CFD for propeller performance estimation will help to validate the model and to obtain hydrodynamic load in the following propeller strength assessment. In the current study, ANSYS Fluent (Fluent) was applied as the CFD solver.

4.1 Fluid Field Model

This research applied the multiple reference frame (MRF)method in Fluent to simulate the rotation of a propeller.This method simplifies the fluid field as two zones: a stationary zone and a moving reference frame zone. The machinery rotation is represented by the rotation of the moving reference frame zone, which contains the machinery geometry data (ANSYS, 2013). The motion of the moving reference frame zone not only simulates the machinery motion but also accelerates the fluid field surrounding the machinery in the moving reference frame zone. This method is useful and straightforward for engineering estimation and the preliminary design of propellers.

Before the fluid field model was built, the propeller model was checked in the following perspectives: angle,edge, face, hole, and body. The potential geometry errors will cause severe problems during the fluid field modeling, meshing, and calculation. In this study, the towing tank was generated as a rotating domain and a stationary domain (Fig.3). The rotating domain contained the geometry data of the researched propeller. To reduce the boundary effect and control the simulating calculation, the length and radius of the stationary domain were 10 m and 6 m,respectively, which were seven and five times the rotating domain length and radius, respectively.

Fig.3 Right, geometry with one rotating impeller (ANSYS, 2013); Left, the fluid field model.

4.2 Meshing Model

Meshing is a crucial process in CFD simulation, as it remarkably affects the required calculation resource and simulation results. Especially, the boundary layer is an important factor in this simulation. The boundary layer will result in reduced speed, a steady fluid boundary on the propeller surface, and a change in the propeller geometry during the rotating motion due to as the surface viscosity effect. Hence, the hydrodynamic performance will be influenced. According to standards (ITTC, 2002,2011), the first layer thickness of inflation meshingyand the total boundary layer thicknessyBLcan be calculatedviathe following equations:

A medium-size meshing model with 782634 elements was utilized in this simulation to balance the result correctness and computational resources. The fluid field model applied unstructured meshing elements for the complex propeller structure. The curvature and approximation were captured for more precise meshing. The propeller surface was refined to generate a higher-quality inflation layer to avoid a high aspect ratio, Jacobin ratio, and skewness,which significantly influence the calculation accuracy, as shown in Fig.4. The first layer thickness method was applied for simulating the boundary layer. According to the formulas introduced above, the first layer thickness was 0.000975 m, the growth ratio was 1.2, and the total number of layers was 11 (see Figs.4c-e).

The computing cost,i.e., the CPU time, for each combined hydrodynamic and structural simulation was about 30 min, on an Intel(R) Core i5-6300HQ CPU at 2.30 GHz with a DRAM of 4 GB. The 30 min CPU time included the following: 1) hydrodynamic load calculation for meshing (5 min) and solution based on Fluent (10 min); 2) static analysis for meshing (5 min) and solution based on ANSYS Workbench (10 min).

4.3 CFD Pre-Setup

The flow surrounding the propeller rotation can be assumed as a steady fluid when the fluid field is fully developed. The specific fluid properties are unknown. Thus,this study utilized the fluid properties of freshwater as defined in the Fluent database: the freshwater density was 998.2 kg m-3, and the kinematic viscosity was 0.001003 m2s-1. This research investigated three different widely used turbulence models for analyzing the effect of the turbulence model on the propeller motion performance estimation: thek-ε,k-ω, and shear-stress transport (SST)models. Thek-εmodel is more known than the other two turbulence models (Jones and Launder, 1972). This model has great estimation for zero, and a small mean pressure gradient wall-bounded flow (Wilcox, 1993). It works well when the fluid is far away from the wall. Thek-ωmodel will better agree with experimental data in the logarithmic region for mild adverse pressure gradient flow (Bardinaet al., 1997). Moreover, the model works well when the fluid is close to the wall. The SST model combines elements of thek-εandk-ωmodel so that it generally behaves better than thek-εandk-ωmodels in the overall fluid field.

Fig.4 (a) and (b), high aspect ratio and high-skewness elements; (c) - (e), meshing model.

The propeller performance is highly relevant to the advance ratio. For generating different work conditions in different advance ratios, the propeller rotational speed was fixed at 300 r min-1in the simulation. Therefore, the advanced speed of the propeller, which is represented by the fluid speed of inlet flow in the stationary domain, was calculated according to Eq. (5). The result is presented in Table 2. To transfer the calculation information between two domains, an interface is defined between the two domains. The simulation based on the SST model is more challenging to converge, compared with those based on the other two turbulence models. The residual value of the monitoring parameter was set as 1×10-6, and the calculation iteration step was set as 2000 so that the calculation can converge or complete the 2000 iterations.

Table 2 Zone condition in different working cases

4.4 Result and Analysis

The open-water performance of a propeller is mainly describedviathree parameters: thrust coefficientKT, torque coefficientKQ, and propeller efficiencyη(ITTC, 2002),which are defined by the following equations:

The propeller performance estimation and experiment data are shown in Fig.5. The figure demonstrates that the estimations of the open-water performance all agreed with the experiment data. For the thrust coefficient estimation,there were slight differences between the simulation results of the three turbulence models. However, the differences between the CFD simulation and experiment were more significant for the torque coefficient estimation. The multiplier 10 is one influencing factor. Fig.5 shows that three estimations based on CFD are more significant than the experiment data, fromJ= 0.1 toJ= 0.4. The results of thek-ωmodel showed the best conformance with experiment data. For the efficiency estimation, the result of thek-εmodel had the ideal estimation. However, the estimation based on the SST model was the most coincident with laboratory results, fromJ= 0.1 toJ= 0.5. In conclusion, thek-εmodel is the best turbulence model in this simulation for good estimation conformance and less calculation. Therefore, in the following strength evaluation through FEA, the results based on thek-εmodel were used as the resource of hydrodynamic loads.

Fig.5 Open-water curve and validation.

The errors of this simulation can be concluded to have emerged from four aspects: propeller model generation,simulation method, meshing model, and CFD simulation setup. First, the solid model propeller generation process was based on the meshing model of Propella. In the rebuilding process, errors were generated during the blade section curve modification and blade surface generation by lofting. The surface simplification of the propeller reduced the structure lines and caused geometry differences.Second, the Fluent MRF method was used to simulate the propeller rotation. The rotation of the moving reference frame zone was used to represent the propeller motion.However, the fluid in this zone was also accelerated with the rotation of the complete rotating zone, which does not agree with the real towing tank test. Third, the meshing model significantly influences the simulation result. The meshing quality can be described by different parameters,such as skewness, aspect ratio, and Jacobin ratio. These parameters show the difficulties to solve the controlling equations. If these equations cannot satisfy the corresponding requirements, the low-quality meshing will reduce the result accuracy and even cause solver problems.Finally, the CFD simulation setup generates errors from many perspectives, such as fluid properties, turbulence model, calculation iteration steps, and residual values.The essence of the simulation setup is the simplifications and assumptions for the mathematical problems based on the fluid dynamics theory. Therefore, only suitable simplifications and assumptions can generate realizable and reasonable simulation results.

5 Strength Evaluation

Propeller strength evaluation is a safety assurance for polar navigation. To estimate the propeller blade strength behavior under the influence of ice loading, ice forces and hydrodynamic loads were considered during the simulation. First, ice forces were estimated by applying URI3 rules, and hydrodynamic loads were transferred from the CFD module directly into ANSYS Workbench (Workbench). Then, the Workbench as the FEA solver was utilized for static strength assessment. In this paper, the pretreatment settings, especially the meshing process, are displayed and discussed. Finally, the simulation results were validated and analyzed.

5.1 Ice Loading Calculation

For quantifying ice loading, URI3 rules were applied for calculations (IACS, 2011). The navigation environment is described using seven polar classes with three different factors in URI3 rules, as presented in Table 3. Polar class 1 (abbreviated as PC1) means the most severe sea ice situation, and PC7 means the mildest environment. The URI3 rules specify the interaction between ice and propeller as two main loads: maximum blade backward forceFband maximum blade forward forceFf, which are calculated by following Table 3 and Eqs. (9) to (12).

Table 3 Ice-class factors (IACS, 2011)

According to the URI3 rules, the interaction between the polar propeller and sea ice can be divided into five cases (Fig.6 and Table 4). In Fig.6, the force-applied area is highlighted with a different color. The forcesFbandFf(Table 4) were applied as uniform pressure distributed on the propeller surfaces. The calculation results for the seven polar classes and the five working cases are listed in Table 5.

Table 4 Loading cases for open propeller (IACS, 2011)

Table 5 Ice loading calculation results (Unit: kn)

Fig.6 Propeller ice loading implementation (Liu et al., 2015).

5.2 FEA Setup

The FEA module should be correctly set up before calculation. There are three different ice loading application situations (with the same force application section), according to Table 4: cases 1 and 3; cases 2 and 4; and case 5. Thus, the calculation cases are divided into three groups in this simulation. Three different models were built for the different groups (Fig.7). There was no root fillet in the original propeller model; the root fillet was imported from Propella. However, the actual manufactured propeller has a root fillet (see Fig.7d). Therefore, a fillet with a radius of 20 mm (blade length is 1430 mm) was added in each FEA model to transform the load without changing the propeller geometry. The influence of the root fillet is discussed in Section 5.3. In this simulation, the properties of nickel aluminum bronze as used by Liuet al.(2015)were taken as the blade material properties (Table 6). The propeller was assumed to behave linearly and elastically.To simplify the simulation, only one propeller blade was analyzed for strength evaluation. Then, the intersection surface between the blade and hub was assumed as fixed.Ice loads in the seven polar classes were applied to the corresponding blade area according to the different cases.

Fig.7 FEA model and real propeller blade model (in Australian Maritime College).

Table 6 Blade material properties (Nickel aluminum bronze) (Liu et al., 2015)

The hydrodynamic pressure was transferred from the CFD module into the FEA module for static strength analysis. Given that the ice loading will dominate the propeller strength behavior, the hydrodynamic load should be adjusted to avoid the ice loading reduction. For cases 1, 2,and 5, the ice force and hydrodynamic pressure were applied in opposite directions. Therefore, an advanced ratio of 0.9713 was selected to ensure that the total hydrodynamic force was zero (Liuet al., 2015). Also, the imported hydrodynamic pressure will vary with the meshing precision, as shown in Fig.8, which will influence the simulation results.

The meshing model will dramatically affect the FEA simulation results. First, different sections of blades after slicing were formed as a new ‘part’ for better meshing.Thereafter, a similar meshing method was applied for the three FEA solid models. An unstructured meshing element was utilized for the uniform blade section shape after slicing. Moreover, the intersection surface between hub and blade was refined, so that it would capture more stress information around the root. Third, a convergence study was conducted to improve the balance between the calculation resource requirement and simulation result correctness, and the results are shown in Fig.9. The maximum stress for the three groups will converge at 400000 blade elements.Therefore, the meshing model was applied with the element size of 20 mm for investigation. As shown in Fig.9,among the three meshing models, only the meshing models of cases 1 and 3 were similar.

Fig.8 Hydrodynamic pressure variation with meshing fineness.

Fig.9 (a) Convergence study; (b) meshing model (for cases 1 and 3).

5.3 Results and Analysis

In this section, the influence of the root fillet on propeller strength behavior is first discussed. Then, the detailed strength evaluation of the polar class propeller blade is discussed from four perspectives: comparison with Liu’s research (Liuet al., 2015), comparison among different cases, the high-stress area distribution, and error analysis.

The maximum stresses of the blade for the seven polar classes and five cases are shown in Fig.10. The figure shows that the estimated stress for cases 1 and 5 will drop significantly when the propeller blade is modeled with the root fillet. The estimation of other cases also shows a reduction after the root fillet was applied, which is mainly ascribed to the stress concentration at the root of the blade.Therefore, the root fillet can effectively reduce the stress concentration problem. These models with the root fillet are considered in the following discussion and optimization.

According to URI3 rules, the minimumSFof propeller stress should not be less than 1.5 (IACS, 2011). TheSFis defined in Eq. (14), whereσrefis the reference stress, and is equal to 0.7 times the ultimate stress in this research.Therefore, the minimumSFs for the seven polar classes and the five cases based on the propeller blade model with the root fillet could be calculated and are listed in Table 7. The results show that the minimumSFfor all the situations satisfied the minimumSFrequirement.

Fig.10 Maximum stress for seven polar classes and five cases. (a), propeller blade without root fillet; (b), propeller blade with root fillet.

The researched CCGS icebreaker is rated as Arctic class 3, which is equal to polar class 3 (Liuet al., 2015). Therefore, the investigated polar propeller only needs to satisfy the PC3 safety requirement. The following section discusses the strength behavior analysis of PC3 and the propeller blade optimization. The comparison between Table 7 and Table 8 shows that the estimation based on the FEA simulation has the same stress behavior trend with the results based on Liuet al.’s (2015) method. This result suggests that stress for the cases decreases as follows:case 1 ＞ case 5 ＞ case 2 ＞ case 3 ＞ case 4. In addition, the FEA simulation results of case 1, case 3, and case 5 are greater than those of Liuet al.’s (2015) method. The main cause of the difference between these two results is the difference in the methods themselves.

To investigate the high-stress area for each case, the stress contours of PC3 were obtained (Fig.11). The two groups of cases 1 and 3 (Figs.11a and c) and cases 2 and 4(Figs.11b and d) had a similar stress distribution for the same loading area (suction surface for cases 1 and 2 and pressure surface for cases 3 and 4). The highest stress area was located at the section close to the root, except for case 5 (Fig.11e). This was mainly because the highest torque was at the longest linear distance of the root, that is, from the force location to the blade root. For cases 1 and 3, the second-highest stress area was distributed at around 0.3r/Rsection; this was due to a low moment of inertia, which cannot reach the increased speed of the moment of ice forces. This also explains why the secondhighest stress areas of cases 2 and 4 were located at around 0.6r/Rsection. For case 5, the highest stress was located at the leading edge, close to the root section. This was due to a low moment of inertia for the low thickness.The result shows that there were some apparent separations of the stress distribution along the trailing edge of case 5. An imperfection of the blade model was generated during the modeling (Fig.11f) by automatically adjusting the curvature of the section line for the curve cohesion in Rhino. Therefore, the propeller blade thickness will decrease at the trailing edge.

The discrepancies between the FEA simulation and the experiment-derived stress can be ascribed to modeling,unconformity of material properties, strength model simplification, and meshing model fineness. Especially, the blade stress was assumed to behave linearly and elastically, indicating a straight stress-strain curve. Other reasons are stated in the conclusion of Section 4.4.

Table 7 Safety factor based on stress for seven polar classes and five cases

Table 8 Safety factor based on stress for seven polar classes and five cases (Liu et al., 2015)

Fig.11 (a-e), Stress contour for five working cases; (f), one imperfection during the modeling.

6 Optimization

6.1 Method

In this research, the propeller blade optimization was mainly investigated based on safety criteria. The optimization aims to adjust the propeller blade geometry to decrease manufacturing cost. The geometry will only be tuned by changing the blade thickness, which avoids decreasing the propeller performance and influences the moment of inertia of the propeller section. This can significantly overcome the ice loads. At the same time, the optimized propeller blade should satisfy the minimumSFof 1.5. To ensure the propeller blade integrity, the optimization also needs to guarantee surface uniformity across the blade span.

Three main factors were considered for the propeller optimization:SF, thickness factor (TF), and blade section thicknesst. TheSFis the most vital factor to evaluate the optimization results, and it should be greater than 1.5 according to URI3 rules. TheTF, defined in Eq. (14), was used for the propeller blade model adjustment. From Eq.(14), theTFof each section of the original blade (before adjustment) was 1.0. This factor quantifies the thickness for the propeller blade section after optimization. The blade section thickness had the maximum thickness of the blade section outline, which facilitates the evaluation and ensures the blade surface integrity.

The optimization involved the following steps:

First, the relationship between the propellerTFandSFof the whole blade was investigated. The original propeller was adjusted by changing the thickness of each section used for lofting (as shown in Fig.2c) according to the threeTFs (0.7, 0.8, and 0.9). Then, the maximum stress of each optimized propeller was collected, as shown for the five cases in Fig.12. Based on four points, the mathematical relationship between the thickness and safety two factors could be generated using the trending line linear equation.The trending line can be matchedviadifferent forms of the equation. However, functions with variable monotony are not recommended. For example, the power function is ideal for fitting points, but it will have inaccurate estimation in other sections, as shown in Fig.12b, which leads to errors in the following estimation. Moreover, the trending line was translated to ensure it passes through all the points so that the estimation results will more easily satisfy the safety requirement. Therefore, the exponential function,log function, and linear function are recommended in this method. The linear equation was applied in this research to simplify the calculation. Therefore, the relationship between theTFandSFfor the whole blade in the five working cases can be described by Eqs. (15) - (19):

Fig.12 Estimation of the relationship between TF and SF for case 1. (a), linear function; (b), quadratic function.

Second, the weakest section (section with the highest stress) of the propeller blade at different working cases was found; the optimized blade could satisfy the minimumSFof 1.5. The highest stress and lowestSFwere then obtained.

Third, based on the lowestSFof sections and the relationship between theSFand theTFdefined by Eqs. (15) -(19), the available optimized minimum thickness factor(T.F.opt_blade) could be estimated. However, thisTFis for the whole blade. Eq. (20) can be applied to estimate the actual available optimized minimumTF(T.F.opt_sec) for each section. The key assumption in this optimization method is that the relationship between theTFandSFon the whole propeller blade is the same as that on each section. Thus, the relationship between blade sectionTFand localSFcan be simply estimated. According to the lowestSFin Table 9 for each section, the corresponding original propeller blade section thickness factor (T.F.orig) could also be obtained.

Fourth, Fig.13 illustrates the optimization results considering three factors: structure safety, material reduction,and integrity. To ensure the structure safety, the optimized blade outlines (solid lines) should all be located at the right side of the available optimized minimum thicknesstoptim. At the same time, the optimized blade outline should be located leftward as much as possible to save more material. The solid red line shows the original propeller blade outline, which is the upper limit of optimization. The propeller blade integrity is maintained by keeping the blade surface uniform across the blade span, which is controlled by the actual optimized propeller section thickness. Therefore, the optimized blade outline should be uniform and smooth. In this study, the trending curve of the available optimized minimum thicknesstoptimwas generated using the parabolic equation (dashed line). Subsequently, the final optimization result was obtained by translating the hull curve (solid line) to satisfy the structure safety requirement.

Table 9 Stress evaluation for blade sections

6.2 Results and Analysis

The optimization range is show in Table 10. Two optimization results are shown in Table 11. The optimized propeller blade was evaluated by the FEA method mentioned in Section 4. Table 11 shows that the first optimized blade did not satisfy theSFof 1.5 for case 5. Thereafter, suitable adjustment was implemented to generate a second optimized blade (Fig.13). When applied, the second optimized blade could decrease the amount of utilized blade material by 11.8% (the decrease will vary depending on propeller type). As shown in Fig.14, the optimized propeller exhibited the same efficiency trend as the original propeller at the same advance ratio. The optimized design showed a decrease in propeller thrust. Moreover, for the multiplier 10, the optimized design showed a significant decrease in torque coefficient compared with the original design.

7 Conclusions

In this paper, a systematic method of ice-class propellermodeling, performance estimation, strength and integrity evaluation, and optimization is discussed; the method can be also applied for other types of propellers and standards.The URI3 rules were applied for the ice loading estimation. The summary and conclusions are given below:

Table 10 Optimization range estimation

Table 11 Optimization results

Fig.14 Propeller performance comparison. (a), thrust coefficient and torque coefficient; (b), propeller efficiency.

1) One R-class propeller was utilized for the whole research. The propeller modeling method was modified based on the application of Propella to reduce the model generation time.

2) The investigated propeller performance was estimated based on the CFD method, and Fluent was used as a solver. The simulation results agreed with the experiment data and were used to provide precise hydrodynamic loading for the strength evaluation. This method provides reference values for the comparison of the propeller performance between the original design and the optimized propeller.

3) The R-class propeller strength evaluation adopted a modified propeller model, equipped with a root fillet to reduce the stress concentration. Two main loads were applied on the propeller interacting with sea ice: hydrodynamic pressure and ice loads. The hydrodynamic pressure was from the CFD module, and the URI3 rules were utilized for the ice force estimation. TheSFs for seven polar classes and five working cases were validated, which demonstrated the same stress behavior trend. The stress for the cased decreased as follows: case 1 ＞ case 5 ＞ case 2＞ case 3 ＞ case 4.

4) Because the investigated R-class propeller was graded as polar class 3, the high-stress area of the propeller blade at different working cases was displayed by stress contours of polar class 3. From case 1 to case 4, the highest stress area was located at the center area of blade root for the high torque. For case 5, the highest stress area appeared at the trailing-edge area close to the root thickness for the thin blade section.

5) A comprehensive optimization method was developed to ensure that theSFof the propeller blade was higher than 1.5 for all working cases. At the same time, the propeller blade section thickness was adjusted to satisfy the integrity requirement, which requires that the blade surface is uniform across the blade span. The final optimized propeller had the minimumSFof 1.55 and decreased the design volume to 88.2% of the original. The comparison of open-water curves showed that the optimized propeller would lose some thrust and torque. However, there is no significant difference in propeller efficiency.

In this paper, the optimization process is developed, and a new optimization method is described. Some strength evaluation results for the ice-class propeller design are also given.

Acknowledgement

The author would like to thank University of Tasmania and Newcastle University for their support.

Nomenclature

C0.7R: Blade section chord at 0.7R

Cf: Frictional coefficient

D: Propeller diameter

Dlimit: Diameter limit

EAR: Propeller expanded area ratio

Fb: Maximum backward force

Ff: Maximum forward force

hD: Propeller hub diameter

Hice: Ice thickness

J: Advance ratio

KQ: Torque coefficient

KT: Thrust coefficient

Lpp: Length of perpendiculars, which is the propeller blade length in this research

Sice: Ice strength index for blade ice force

Sqice: Ice strength index for blade ice torque

T: Thrust

TF: Thickness factor

T.F.i: Thickness factor for casei

T.F.opt_blade: Available optimized minimum thickness factor for whole blade

T.F.opt_sec: Available optimized minimum thickness factor for blade section

T.F.orig: Section thickness factor of original propeller based on the optimization curve

toptim: The blade section thickness of optimized propeller

torig: The blade section thickness of original propeller

Troot: Root thickness

Va: Ship advance speed, which is flow speed in this research

Va-max: Propeller maximum advanced speed

Nmax: Propeller maximum rotational speed

PC:Polar class

Pdroot: Blade pitch diameter ratio at rootPdtip: Blade pitch diameter ratio at tip

Q: Torque

Re: Reynolds number, which isVL/ν

SF: Safety factor

SFi: Safety factor for casei

X: Characteristic length, which is the propeller radius in this research

Y: First inflation layer thickness for boundary layer simulation

y+: Non-dimensional wall thickness

yBL: Total inflation layer thickness for boundary layer simulation

Z: Number of blades

Η: Propeller efficiency

Ν: Fluid dynamic viscosity

ρ: Fluid density