Influence of Ice Size Parameter Variation on Hydrodynamic Performance of Podded Propulsor

During ice-breaking navigation, a massive amount of crushed ice blocks with different sizes is accumulated under the hull of an ice-going ship. This ice slides into the flow field in the forward side of the podded propulsor, affecting the surrounding flow field and aggravating the non-uniformity of the propeller wake. A pulsating load is formed on the propeller, which affects the hydrodynamic performance of the podded propulsor. To study the changes in the propeller hydrodynamic performance during the ice-podded propulsor interaction, the overlapping grid technique is used to simulate the unsteady hydrodynamic performance of the podded propulsor at different propeller rotation angles and different ice block sizes. Hence, the hydrodynamic blade behavior during propeller rotation under the interaction between the ice and podded propulsor is discussed. The unsteady propeller loads and surrounding flow fields obtained for ice blocks with different sizes interacting with the podded propulsor are analyzed in detail. The variation in the hydrodynamic performance during the circular motion of a propeller and the influence of ice size variation on the propeller thrust and torque are determined. The calculation results have certain reference significance for experiment-based research, theoretical calculations and numerical simulation concerning ice-podded propulsor interaction.


Introduction
With the opening of polar routes and corresponding resource development, the number of ice-going ships has gradually increased in recent years. Thus, ice-going ship propulsion systems have also received attention as one of the most important components of this class of ship. Among them, the podded propulsor is widely used in icebreakers and ice-going ships because of its good maneuverability, low vibration, and high reliability. However, when a ship navigates through pack ice channels and ice floe zones, especially under tail icebreaking conditions, the entire podded propulsor lacks the protection of the hull and rudder, and is exposed to the ice−water mixed medium, which increases the probability of interaction between the ice block and podded propulsor. As a result, extreme loads are generated on the propeller, pod, and strut, yielding problems such as performance degradation, severe vibration, and cavitation erosion of the podded propulsor (Walker, 1996). Therefore, research on podded propulsor−ice interaction is of great sig-nificance.
Previous studies on propeller-ice interaction have reported valuable findings. In the 1980s and 1990s, Kannari (1988); Jussila and Soininen (1991) conducted a series of full-scale measurements and gave the load magnitude, the position of the ice−water mixed load on the blade, and the load distribution during ice−propeller interactions. These findings are of great value, but full-scale measurements are expensive and can yield highly uncertain results because of limitations of the mechanical properties of ice, test conditions, and data acquisition system. To negate these issues, some researchers have performed propeller model tests. Among them, Doucet (1996); Walker (1996); Wang (2016); Guo et al. (2018a) performed model test studies of propellers in uniform and blocked flows in a towing tank, circulating water channel, cavitation tunnel, and ice tank. Variations in propeller thrust, torque, and efficiency were measured for different ice−propeller gaps, propeller rotational speeds, feed rates, blockage heights, and cavitation num-bers, as well as for different ice locations and ice shapes. Furthermore, Sampson et al. (2009Sampson et al. ( , 2013 conducted an experimental study of a podded propulsor model under the condition of ice blockage in a cavitation tunnel; the flow field between the ice and the propeller, cavitation evolution, and propeller performance were measured for different ice−propeller gaps, and cavitation numbers. With the development of a potential flow theory, Liu et al. (2007Liu et al. ( , 2010) used a panel method package called PRO-PELLA to simulate the blocked flow conditions for three different ice shapes (a blade milling contour, a wedge, and a sphere), different ice−propeller gaps, and heavy loads. Their numerical results revealed high blade loads during blockage and performance trends similar to the experimental findings in terms of the mean values. Wang andAkinturk (2006, 2009) used the low-order panel method to perform a numerical simulation of the interactions between ice and a podded propulsor. The separable hydrodynamic loads, inseparable hydrodynamic loads, and ice milling loads arising from the ice−propeller interaction were predicted. In addition, Wang (2016) used the panel method to predict propeller hydrodynamic performance for different ice block sizes. The results show that the thrust and torque of a propeller remain unchanged with the increase of ice length. However, they increase with the increase of ice width and thickness. However, the selected ice size range was small, and did not offer a detailed explanation for the variation of propeller hydrodynamic performance.
In recent years, with the rapid development of computer hardware and remarkable improvements in data computing speed, the advantages of computational fluid dynamics (CFD) numerical simulation methods such as accuracy, efficiency, cycling, and data integrity have become increasingly obvious. Employing CFD numerical simulation software, Wang et al. (2017aWang et al. ( , 2019; Guo et al. (2018b) used an overlapping grid technology to perform a numerical simulation of the blockage effect of ice on a propeller and podded propulsor. The blade pressure, pod and strut pressure, exciting force, and flow-field characteristics during propeller rotation were analyzed.
As indicated in the aforementioned studies regarding research on the ice-podded propulsor interference problem, the interference characteristics of ice and the propeller are analyzed only macroscopically. The changes of hydrodynamic performance, pressure, and flow field characteristics of propeller blades in blocked and non-blocked areas have not been described in detail. In addition, in the study of the interference problem of differently sized ice blocks, theoretical studies, numerical predictions, and experimental investigations of the interactions between the ice and podded propulsors have only been performed for ice blocks of a fixed size or with small size variations. Thus, the effects of ice blocks of variable lengths, widths, and thicknesses on the hydrodynamic performance of a podded propulsor have not been discussed in depth. ω In this study, the hydrodynamic loads of all blades and key blade, blade pressures, and flow-field characteristics for one revolution of the propeller, during ice-podded propulsor interaction, are analyzed. The analysis is based on the Reynolds-averaged Navier-Stokes (RANS) method. It uses the overlapping grid technology and is combined with the shear stress transport (SST) k-turbulence model. The law of variation of propeller blades in the blocked and nonblocked areas behind the ice is revealed. Meanwhile, the hydrodynamic performance of a podded propulsor encountering ice with different block sizes under blockage conditions is simulated numerically. The effects of varying ice block sizes on the hydrodynamic performances of propellers are summarized. The main causes of the differences in hydrodynamic performances of propellers are explored. The results of the numerical simulation provide a reference for the selection of ice size for theoretical research, for the numerical simulation and experiment-based research of ice-podded propulsor interaction, for the setup of the hydrodynamic performance of a podded propulsor in a coupled research, and for the design of an ice-class podded propulsor.

Overset grid method
The blockage of a podded propulsor due to ice is essentially a multi-body unsteady disturbance problem with relative motion. Numerical simulation of this problem has been a challenging area for CFD research. In particular, for objects with complex geometric shapes and for a large relative motion amplitude between the objects, generation and division of the mesh near the object surfaces and dynamic adjustment of the mesh during the object's motion greatly increase the computational grid processing complexity. Previously, Steger et al. (1983) proposed the overlapping grid technology, which is saliently characterized by dividing the complex flow-field area into multiple simple sub-regions as needed. The grid overlap in each sub-region of each generated grid is then used for data exchange to transfer the flowfield information. As a result, this approach has a strong advantage with regard to the unsteady disturbance problems caused by large-amplitude relative motions among multiple bodies in a flow field. Overlapping grid generation mainly involves two aspects. First, the background domain grids are "excavated"; that is, the inner area of the hole surface is shielded and the grid inside the hole surface is marked and discarded from the CFD calculation. Second, the information transfer between the boundaries of the hole are numerically calculated through interpolation; this is, the "pointsearching" problem. An overlapping grid diagram is shown in Fig. 1 (Wang et al., 2017b). The specific algorithm and application of the overlapping grid can be found in Koblitz et al. (2017).

Geometric model
The podded propulsor model used in the simulation conducted in this study was based on the ASEA Brown Boveri (ABB) CO1250 prototype pod with a certain scale ratio and local correction. The propeller was that of the puller-type podded propulsor. The main dimensions and parameters of the podded propulsor are listed in Table 1.

Computational domain and grid partition
In the numerical simulation process employed in this study, the area of the outward flow field assumed to be infinitely large was established. Its boundary was considered to have no influence on the podded propulsor. This area was referred to as the "outward flow domain" for which a cylindrical flow domain was adopted. The cylindrical flow domain was coaxial with the propeller and contained two subdomains for calculation: an ice-flow domain in which the ice was wrapped in a hexahedral area, and a rotation domain in which the propeller was wrapped in a cylindrical area. The ice-flow domain partially overlapps the rotation domain. A local coordinate system was established in the rotation domain, with the propeller rotating in the positive direction of the x-axis at a speed of 600 rpm. The computational domain and boundary conditions were established as shown in Fig. 2.
In this paper, the computational domain was meshed using the Trimmer meshing method, which has good robustness and high efficiency, along with good adaptability to complex geometric models (Baek et al., 2015). In the grid generation process, local encryption of locations with large curvature variations, such as the leading edge, trailing edge, and blade root, required attention. In addition, a ten-layer prism grid was generated on the pod unit surface. The dimensionless distance, y+, between the first layer and the wall was 30, and the total boundary-layer thickness was 1.4 mm. The meshing form is shown in Fig. 3.

Calculation condition setting
To reflect the ice size parameter variation, the ice length, width, and thickness are represented as L, B, and H, respectively. The distance between the propeller and ice surface is the axial distance between the propeller disk and ice surface near the side of the blade, and is represented as X. As for the axial position of the ice, according to Guo et al. (2018b), the axial position of the ice was set to X = R/2. The radial position of the ice's upper surface was h = 1.1R. The geometric size of the ice and its location relative to the propeller are shown in Fig. 4. Fig. 4 also shows the inertial coordinate system employed in the numerical calculation. Note that, when L varied, the position of the ice surface near the side of the propeller blade remained unchanged; instead, the position of the ice surface on the far side of the blade changed. When H varied, the position of the ice upper surface remained unchanged, but the underside of the ice changed along the z-axis. The ice size parameter variations employed in the numerical simulation process are detailed in Table 2.
To facilitate subsequent analyses and comparisons, the following dimensionless coefficients are defined: where V is the inflow velocity; n is the propeller rotational speed; D is the propeller diameter; T and Q are the propeller thrust and torque, respectively; is the propeller efficiency; K t and K q are the thrust and torque coefficients, respectively; and is the density of water.

Mesh independence verification
In the CFD numerical simulations, uncertainty is usually divided into three main sources: iterative (U I ), grid (U G ), and time-step (U T ) uncertainties. Among them, grid uncertainty is the most important source, being one order of magnitude larger than the other two (Wilson et al., 2004). Therefore, this paper chooses grid uncertainty for the analysis. The open-water performance of the podded propulsor was selected for the uncertainty analysis because experimental studies on the hydrodynamic performance of podded propulsor encountering different ice block sizes under blockage conditions have not been performed yet. During the calculation process, three different grids, i.e., coarse, medium, and fine grids, were established for the podded propulsor. K t and 10K q of the propeller in the podded propulsor were analyzed after the calculation, and the results of this analysis are listed in Table 3. From these results, it is obvious that the two coefficients are insensitive to grid density.
To verify the grid convergence of the podded propulsor, a relevant process discussed in Wang et al. (2015); Wang et al. (2017b) was adopted. In this approach, the grid converge ratio, R G , is defined as follows: where S i (i = 1, 2, 3) indicates the calculation results for the coarse, medium, and fine meshes, respectively. Before the mesh convergence verification was performed, it was necessary to define the value of the refinement ratio, r G . As the cut-cell method was used to calculate the mesh partition of the computational domain, r G was defined as follows: where N is the total grid number and d indicates the dimension. The calculation of the hydrodynamic performance of the podded propulsor performed in this study was a three-dimensional problem; thus, d = 3. Furthermore, for the three different grids, r G ≈1.15. Table 4 gives the calculation results for the grid convergence verification for J = 0.3 and 0.7, where P G is the estimated order of accuracy, U G is the mesh uncertainty (Wang et al., 2015), and E is the experimentally obtained value (Zhao et al., 2017). This table reveals that the convergence rate of the grid obtained from the calculation results for the propeller K t and 10K q was smaller than 1, which indicates that the grid was monotonically convergent. The mesh uncertainties of K t and 10K q were 8.6%E and 7.8%E, respectively, when J = 0.3. For J = 0.7, the mesh uncertainties of K t and 10K q were 6.1%E and 3.4%E, respectively. Therefore, it can be concluded that a good mesh convergence was obtained.
In addition to the grid convergence study, prior to the numerical simulations of the ice−podded propulsor interaction, the hydrodynamic performance of the podded propulsor under open-water working conditions was calculated using the overlapping grid technique. Here, N was 8.22 million. The CFD calculation results for J = 0.2−0.9 were compared with the experiment results to verify the correctness and validity of the calculation method employed in this study. Fig. 5 presents the computed and experimentally measured hydrodynamic performance curves of the propeller in the podded propulsor obtained from open-water working conditions (Zhao et al., 2017). Obviously from Fig. 5, the CFD calculation results are in good agreement with the experiment data; the calculation error is within 5%.
In the numerical simulations, the ice block size was 5R/3 × 5R/3 × R/3, and J = 0.3 and 0.7. The grid partition and boundary condition settings were implemented on the basis of the podded propulsor performance calculations under open-water working conditions, and were appropriately encrypted. Note that the propeller K t and 10K q were calculated on the basis of the three grids presented in Table 5. From the table, it is obvious that the variations in the propeller K t and 10K q were minor as N increased to 9.38 million during the numerical simulation of the ice−podded propulsor behavior. Therefore, to improve the calculation accuracy and avoid prolonged computation time, based on N = 9.38 million, the hydrodynamic performance variations during one revolution of the propeller under ice−podded propulsor interaction were studied. In addition, based on N = 8.22 million, open-water working conditions for the podded propulsor, and appropriate encryption, unsteady propeller loads were investigated under different ice−podded propulsor interaction conditions.

Analysis of propeller blade load during one revolution
The hydrodynamic analysis setup for the blade section in the blocked and non-blocked areas used to analyze the forces acting on a single blade during the rotation process is shown in Fig. 6. θ As shown in Fig. 6, the blade that was about to enter the blocked area was set as the key blade, having a circumferential position of = 0°; this position was sequentially increased in the counterclockwise direction. During the numerical simulation of the ice−podded propulsor behavior, the blade section inflow mainly included the axial and circumferential velocity components (Wang et al., 2017b), which are defined as follows:

Ω
where V is the axial inflow velocity, v tan is the circumferential velocity, and is the propeller rotational angular velocity. In the case of ice having a blockage effect on the podded propulsor, V can be divided into blocked (V B ) and uniform (V U ) flow velocities.

ΩR α
By neglecting the induced velocity of the propeller itself, the circumferential velocity, , of the propeller blade is constant for a given radial position during a one-revolution rotation in the blocked area. However, V is not uniformly distributed, which is the main cause of the unsteady forces generated by the propeller. As a result of the blockage effect of the ice on the propeller, V changes constantly during one revolution, causing the angle of attack of the blade section, , to also change continuously: Θ α where is the blade section pitch angle. It is obvious from Eq. (5) that the smaller the value of V, the larger the value of . α α Fig. 7 illustrates the behavior of as the blade rotates under both uniform and blocked flows. Hence, it can be concluded that V decreases when the blade enters the blocked flow, increases, and the propeller thrust increases. θ To analyze the periodic variation of K t and 10K q of the key blade, and of all blades, under blocked flow in terms of , a numerical simulation of the ice−podded propulsor interaction was performed for an ice block size of 5R/3 × 5R/3 × R/3 and J = 0.3 and 0.7. The pulsation curves of K t and 10K q obtained during one revolution of the key blade, and of all blades, are shown in Figs. 8 and 9, respectively. θ As shown in Fig. 8, the key blade K t and 10K q exhibit a small range of fluctuations during rotation, when the podded propulsor operates under open-water working conditions. This is because of the mutual interference among the propeller, pod, and strut. When the ice generates a blockage effect on the podded propulsor, the fluctuation ranges of K t and 10K q of the key blade are larger for the one-cycle rotation of the key blade. From Figs. 6 and 8, it can be concluded that, when the key blade is located at = 0°, the leading edge moves to the vicinity of the blocked area behind the ice. This causes the axial induced velocity to change from positive to negative at the blade surface with the pressure difference at the propeller disk being reduced.   Fig. 8. Pulsation curves of key blade thrust and torque coefficients K t and 10K q , respectively. GUO Chun-yu et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 30-45 θ θ θ Hence, the propeller K t and 10K q are smaller than those obtained under open-water working conditions. This phenomenon has the same effect on the propeller K t and 10K q when the ice moves to different radial positions as reported by Chang et al. (2016). As the propeller rotation angle increases such that the key blade rotates to near = 45°, the key blade begins to enter the blocked area and V in the forward side of the key blade decreases. This is equivalent to the low advance operation condition. Thus, K t and 10K q of the key blade are increased. When the key blade rotates to near = 80°, the area on the key blade blocked by the ice is at its maximum causing the propeller K t and 10K q to reach their maximum values. Then, the key blade gradually exits and moves away from the blocked area behind the ice. The value of V increases gradually, and the key blade K t and 10K q decrease gradually. When the key blade rotates to near = 270°, it is farther from the blocked area, and the blocked flow has no effect on its behavior. The propeller K t and 10K q are equal to those of the podded propulsor under openwater working conditions. α From Fig. 9, it can be concluded that the total K t and 10K q of the propeller under blockage conditions are higher than those under open-water conditions. The main reason for this difference is that the blockage effect of ice on the propeller reduces the inflow velocity on the suction side of the propeller and increases of the blade section (as shown in Fig. 7), therefore, the propeller K t and 10K q are increased. In addition, as obvious from Fig. 9, when the ice has a blockage effect on the podded propulsor's inflow, four periodic distributions are obtained for the total K t and 10K q of the propeller during one revolution. This result was mainly obtained because the model propeller considered in this study was a four-bladed propeller. Note that, during the propeller rotation, the blockage effect of the ice is similar to that of a single-peak wake. When any blade passes through the high-wake region, it causes a significant increase in the propeller K t and 10K q . From Figs. 8 and 9, it is clear that the maximum and minimum points of the key blade K t and 10K q appear at the same angles as some of the maximum and minimum points of the total propeller K t and 10K q under ice−podded propulsor interaction conditions. This implies that the main components of the propeller's total thrust and torque ripple extremes are generated by the blades entering the blocked area. This finding is consistent with conclusions drawn by Wang et al. (2017a). θ In this study, the key blade was taken as the research object and the blade pressure variation was analyzed for the period between the key blade entry to and exit from the blocked area. The key blade rotation angle was = 0°-180°, and the pressure distributions on the blade face and blade back of the key blade for J = 0.3 and 0.7 are shown in Fig. 10 and Fig. 11, respectively. θ θ θ From Figs. 10a and 11a, the high-pressure area on the blade surface is mainly distributed near the leading edge, and the pressure distributions of the four blades under openwater working conditions are the same. However, when the ice exerts a blockage effect on the propeller, differences appear in the pressure distributions of the four blades. When the key blade is at = 0°, the range of the high-pressure area on the key blade surface decreases relative to that obtained under open-water conditions. The main reason for this decrease is that the axial induced velocity of the ice on the blade changes from positive to negative at the key blade surface, and the high-pressure area on the blade surface decreases. As the blade rotates such that the key blade rotates to the vicinity of the blocked area behind the ice ( = 45°), the range of the high-pressure area near the key blade leading edge increases. As the key blade enters the blocked area, the high-pressure area of the blade surface has a tendency to increase toward the trailing edge, and a distinct high-pressure area appears. As the blade's entry area increases, the range of the high-pressure area becomes increasingly large, as shown in Figs. 10a and 11a, for which the key blade is located at = 90°. As the blade rotates further, the key blade exits the blocked area. The pressure distribution on the blade surface is then opposite to that obtained when the blade is screwing into the blocked area. The range of the high-pressure area near the blade leading edge gradually de-   Figs. 10b and 11b, it can be concluded that, when the ice has a blockage effect on the propeller, and the key blade is located at = 0°, the range of the low-pressure area at the back of the key blade decreases with respect to that obtained under open water conditions. This behavior is the same as the pressure distribution variation trends on the blade face. As the blade rotates, the key blade rotates into the blocked area behind the ice. The low-pressure area of the key blade back moves gradually from the leading edge to the trailing edge direction. With the increase in the blade's entry area, the range of the low-pressure area increases, and a distinct low-pressure area is generated near the leading edge and blade tip. The formation of this lowpressure area inevitably aggravates the cavitation. This is similar to the phenomenon observed during the experiment (Walker, 1996;Sampson et al., 2013). When the key blade exits the blocked area, the range of the low-pressure area on the blade back decreases gradually. As the blade rotates, the overall variation trend of the pressure on the blade back is the same as that of the pressure on the blade face.
In this paper, unsteady solutions indicating the interference between the ice and podded propulsor are presented. Thus, the variations of axial velocity caused by the interactions between the ice and key blade at different rotation angles were analyzed. For this analysis, the cross-sections were located in front of the propeller at X = 0.36R and at the propeller disk, and J = 0.3 and 0.7. The calculation results are shown in Figs. 12−15. Fig. 12 illustrates that a distinct low-velocity region is  generated in the blocked area when the ice has a blockage effect on the podded propulsor. Further, the induced velocity has a negative value, the velocity contours are gradually denser, and a plurality of velocity closure zones is formed. At a position far from the blocked area, the axial induced-velocity distribution is approximately identical to that obtained for the podded propulsor under open-water conditions. When the key blade rotates from = 0° to = 90°, a low-velocity region is generated near the blade leading edge. The range of the high-velocity region near the forward side of the leading edge increases gradually. When the key blade rotates from = 90° to = 180°, the change in velocity in the vicinity of the leading edge and the blade tip is opposite to that of the key blade when it enters the blocked area. However, V of the key blade at = 135° is obviously higher than that of the key blade at = 45°. The main reason for this difference is that the blade behind the key blade is about to enter the blocked area. Under key blade rotation, the axial induced-velocity distribution is approximately identical to that obtained for the podded propulsor under open-water working conditions. The changes in the axial induced velocity at the cross-section for J = 0.7 are illustrated in Fig. 13. From this figure, it is obvious that, during the process of the key blade entering and exiting the blocked area, the variation trends of the low-velocity region near the leading edge, the high-velocity region near the tip, and the high-velocity region at the front of the leading edge are identical to those obtained at low J. The difference is that larger J corresponds to higher-density velocity contours.
From Fig. 14, the presence of ice has a considerable influence on the axial velocity in the blocked area. On the portion of the blade swept by the blocked area, a distinct lowvelocity region forms near the blade tip, and the induced velocity is negative. The axial induced velocity distribution on the propeller disk is approximately identical to that ob-

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GUO Chun-yu et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 30-45 θ θ θ tained under open-water working conditions at a position far from the ice-blocked area. When the key blade rotates from = 0° to = 90°, the range of the low-velocity region near the blade tip increases gradually, extending on both sides in the circumferential direction. Meanwhile, the high-velocity zone of the blade back has a tendency to gradually decrease. Then, the velocity contour between the key blade and the front and rear blades is gradually denser. As the key blade rotates, the low-velocity region of the blade tip and the high-velocity region of the blade back continue to increase. When the key blade is at = 135°, the ranges of the low-velocity region of the blade tip and the high-velocity region at the blade back reach their maximum values. As the key blade exits the blocked area, the ranges of these regions gradually decrease, and the contour and gradient distribution gradually return to those obtained under open-water working conditions. Under high J, as obvious from Fig. 15, the variation trends of the low-velocity region of the key blade tip and the high-velocity region of the key blade back are identical to those for low J. The only difference is that the range of the low-velocity region at the blade tip in the blocked area is larger and the blockage effect is more obvious.

Effects of ice blocks with different lengths on hydro-
dynamic performance of podded propulsor By keeping the B and H of the ice unchanged, the hydrodynamic performance of the propeller was simulated for ice blocks of four different L values, which generated a blockage effect on the podded propulsor. In the numerical simulation, B = 5R/3 and H = 2R/3 were adopted for the ice, with L = 4R/3, 5R/3, 2R and 7R/3. The calculation results are shown in Fig. 16.
As shown in Fig. 16, when the ice has a blockage effect on the podded propulsor, K t and 10K q of the propeller increase significantly. For the ice blocks with four different lengths having a blockage effect on the podded propulsor, the changes in L were found to have a relatively small effect on the propeller hydrodynamic performance. The calcu-lation results are consistent with those obtained by Wang (2016) using the panel method. There are two main reasons for the negligible change in the propeller hydrodynamic performance with varying L. On the one hand, there are no changes in the blocked area and the ice position on the propeller, and the variation range of V is small; thus, the propeller K t and 10K q remain almost unchanged. On the other hand, when L exceeds the lengths of the top and bottom recirculation zones of the ice, only the ice wake has an effect on the flow field in the suction side of the propeller (Martinuzzi and Tropea, 1993). Thus, changes in L have little effect on the ice wake and on the flow field in front of the propeller. Therefore, the propeller K t and 10K q remain almost unchanged.
To further analyze the influence of different L values of ice on the podded propulsor hydrodynamic performance, numerical simulations of the interactions between the ice blocks with various L and podded propulsor were carried out for J = 0.3 and 0.7. The L values considered here are listed in Table 2, and the calculation results are shown in Fig. 17.
From Fig. 17, it can be concluded that the changes in L have essentially no influence on the propeller K t and 10K q when J = 0.3, except for ice blocks with L = R/24. When J = 0.7 and for ice with L > R, changes in L also have little influence on the propeller K t and 10K q . For L ≤ R, a smaller L corresponds to a greater influence on the propeller K t and 10K q . The main reason for this trend is that the influence ranges of the top and bottom recirculation zones of different lengths of ice vary for the same J (Mi, 2012), generating differences in the flow field in front of the propeller. Thus, the variation trends of the propeller K t and 10K q are different.
To analyze the influence of different values of L on the hydrodynamic performance of the propeller under different J values, the axial dimensionless velocity contours of the mid-longitudinal section of ice blocks with L = R/24, 2R/3, R and 4R/3 during the interactions between the ice and podded propulsor were obtained, as shown in Fig. 18.   Fig. 16. Influence of four different ice lengths L on propeller hydrodynamic load for different J. GUO Chun-yu et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 30-45 In Fig. 18a, when J = 0.3 and L = R/24, there are obvious low-and high-velocity regions behind the ice. The lengths of the top and bottom ice recirculation zones exceed L. Therefore, the velocity of the flow field between the ice and the propeller is greatly affected. As regard the velocity contours for ice with other L values, the length of the ice recirculation zone is smaller than that of the ice. The wake vortex behind the ice becomes the main factor affecting the propeller inflow, and the effects on the flow field in front of the propeller are basically the same for those different L values. Therefore, these different L values have essentially the same influence on the propeller K t and 10K q . In addition, when J = 0.7, the influence range of the top and bottom ice recirculation zones is larger because of the greater inflow velocity. When L = R/24, 2R/3 and R, the lengths of the top and bottom recirculation zones of the ice exceed L. Furthermore, a smaller L corresponds to a larger range for the lowvelocity region after the ice; that is, the blockage effect on the propeller inflow is more obvious. When L = 4R/3, the lengths of the top and bottom ice recirculation zones gradu-ally become smaller than L. The smaller the influence of the flow field in the recirculation zone on the field in the suction side of the propeller, the smaller the influence on the propeller K t and 10K q .
With interactions between the ice and podded propulsor, the axial velocity distribution is directly related to the magnitude of the hydrodynamic load; therefore, the axial inflow velocities at the X = 0.36R cross-section in front of the propeller were selected as the research objects, and the influence of different L values in the flow field in front of the propeller was analyzed. The analysis results are shown in Fig. 19.
From Fig. 19a, when J = 0.3, and the ice has a blockage effect on the podded propulsor, a distinct low-velocity zone is generated at the leading-edge front end and in the vicinity of the trailing edge in the blocked area at the cross-section. Furthermore, the range of the low-velocity region near the leading-edge front end is relatively large, and the induced velocity value is relatively small. The maximum ranges of the low-and high-velocity regions in the cross-  section are obtained for L = R/24. Because the range of the low-velocity region is larger for this L value, the value of the induced velocity is smaller, and thus the influence on the propeller is greater. For L = 2R/3, R, and 4R/3, the ranges of the low-and high-velocity regions in the cross-section are basically the same, and the effects on the propeller are also the same. The results for J = 0.7 are shown in Fig. 19b. It is obvious that a smaller L corresponds to a larger range of the low-velocity region near the leading edge in the blocked area at the cross-section. Therefore, the influence on the propeller K t and 10K q is also greater for a smaller L.

Effects of ice blocks with different widths on hydrodynamic performance of podded propulsor
Keeping the ice's L and H unchanged, the effects of four different B values for the ice on the propeller K t and 10K q were simulated. In the numerical simulation, L = 5R/3 and H = 2R/3, whereas B = 2R/3, 4R/3, 2R and 7R/3. The calculation results are shown in Fig. 20.
As shown in Fig. 20, when B varies between 2R/3 and 2R, the influence of ice on the propeller K t and 10K q in-creases with increasing B. The main reason for this greater influence is that the blocked area in the suction side increases the reduction range of V. The more obvious situation is where the propeller appears a low J. The calculation results are consistent with those obtained by Wang (2016) using the panel method. When B varies between 2R and 7R/3, the increasing trends of the propeller K t and 10K q with increased B are not obvious. Therefore, to study the influence of B on the hydrodynamic performance of the podded propulsor, the propeller area blocked by the ice was divided into two parts: a) the area within the propeller disk and b) the area beyond the region swept by the propeller disk. To further analyze the influence of B on the hydrodynamic performance of the propeller for variation of B in these areas, the effects of multiple B values on the propeller K t and 10K q were analyzed for J = 0.3 and 0.7.The B values considered here are detailed in Table 2, and the calculation results are shown in Fig. 21.
In Fig. 21, this plot reveals that the propeller K t and 10K q increase linearly at first with increased B, and then slowly increase to a maximum before stabilizing. During the GUO Chun-yu et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 30-45 linear increase, B ≤ 5R/3, and the whole area of the propeller blocked by the ice is within the propeller disk. Note that the blocked area increases linearly with the increasing B. During the slow increase of the propeller K t and 10K q , 5R/3 ≤ B ≤ 2.0R. With the increasing B, the area blocked by the ice increases linearly. However, the area of the propeller blocked by the ice consists of two parts, and the part located within the propeller disk has a greater impact on the propeller than that located beyond the region swept by the propeller disk. Thus, the propeller K t and 10K q increase slowly. When B ≥ 2R, the propeller K t and 10K q no longer increase with the increasing B, and tend to gradually stabilize. The main reason for this behavior is that the size of the blocked area in front of the propeller increases with the increasing B, when B is larger than the propeller diameter. However, the increased area is located farther from the region swept by the propeller disk, thus having a relatively small influence on the flow field in front of the propeller. Consequently, it has less influence on the propeller K t and 10K q than the other factors.
To further explore the differences in the propeller hydrodynamic performance for different B, the axial velocity contours at the X = 0.36R cross-section in front of the propeller was determined for the interactions between the ice blocks of four different B values and podded propulsor. Here, B = R/3, R, 2R and 7R/3, and the J values of the propeller were 0.3 and 0.7. The analysis results are shown in Fig. 22.
As shown in Fig. 22a, when B increases within the region swept by the propeller disk, a larger B corresponds to a larger area of the propeller blocked by the ice, along with a larger range for the low-velocity region in the blocked area. When B is increased to 2R, the range of the blocked area continues to increase, and the range of the low-velocity region in the blocked area also increases. However, the range of the low-velocity region has a tendency to increase radially away from the propeller disk. When B is increased to 7R/3, the effect of the ice on the axial velocity within the propeller disk is almost equal to that for B = 2R. The influence on the region outside the area swept by the propeller disk shows an increasing trend in the low-velocity region. However, the range of the low-velocity region is far from the zone of the propeller disk; thus, the increase of the range of the low-velocity region has a relatively small influence on the hydrodynamic performance of the propeller. As shown in Fig. 22b, when J = 0.7, the influence trend of the variation in B on the axial velocity in the blocked area is equal to that for low J.
5.4 Effects of ice blocks with different thicknesses on hydrodynamic performance of podded propulsor Keeping the ice's L and B unchanged, the effects of four different H values for the ice on the propeller K t and 10K q were simulated. In the numerical simulations, L = 5R/3 and B = 5R/3, while H = R/3, 2R/3, 4R/3 and 2R. The calculation results are shown in Fig. 23.
It can be concluded from Fig. 23 that the propeller K t and 10K q increase with increasing H under the same J. The main reason for this change is that the area of propeller blocked by the ice increases with increasing H. Furthermore, the decrease in inflow velocity in the suction side of the propeller increases with increasing H, and more obvious the case where the propeller appears a low J. Hence, the propeller thrust and torque coefficient increase. The results are the same as those obtained by Wang (2016) and Walker (1996). For H ≤ 4R/3, the propeller K t and 10K q decrease linearly with J. For H = 2R, the propeller K t and 10K q increase with increased J. The main reason for this increase is that, when H = 2R, all the four blades of the propeller are wholly located in the blocked area behind the ice. With increasing inflow velocity, fewer amounts of fluid enter the space between the ice and the propeller, and the range of the low-velocity region in the blocked area increases. Therefore, the propeller K t and 10K q increase the area of the cross-section along with a steeper decrease in V and denser velocity contours. In addition, the range of the low-velocity region at the front of the blade leading edge is larger.
In addition, considering that H is larger than the propeller diameter under tail icebreaking conditions (Kinnunen et al., 2013), H continues to increase beyond the region swept by the propeller disk. Therefore, taking J = 0.3 and 0.7 as examples, the effects of different H values on the propeller K t and 10K q were analyzed. The H values considered here are presented in Table 2, and the corresponding calculation results are shown in Fig. 24.
As obvious from Fig. 24, the propeller K t and 10K q gradually increase with increasing H. When H increases within the propeller disk, the propeller K t and 10K q exhibit a linearly increasing trend. Furthermore, when H increases beyond the region swept by the propeller disk, the propeller K t and 10K q increase slowly. The main reason for this increase is that the propeller blades are all located in the blocked area behind the ice, and the axial inflow in front of  GUO Chun-yu et al. China Ocean Eng., 2020, Vol. 34, No. 1, P. 30-45 43 the propeller blades is completely blocked. However, the axial inflow is still the major contributor to the main inflow effect (corresponding to suction) during the propeller rotation process. Therefore, when H increases beyond the zone swept by the propeller disk, V continues to be blocked. Therefore, the propeller K t and 10K q increase.
To further analyze the effect of H on the propeller K t and 10K q , the axial velocity contours at the X = 0.36R crosssection in front of the propeller was obtained for interactions between the ice blocks with four different H values and podded propulsor. Here, H = R/3, R, 2R, and 7R/3, respectively. The analysis results of the same are shown in Fig. 25.
From Fig. 25a, it is obvious that increasing H corresponds to a larger range of the blocked area in the cross-section, along with a steeper decrease in V and denser velocity contours. In addition, the range of the low-velocity region at the front of the blade leading edge is larger. Comparison of the effects of ice blocks with H = 2R and H = 7R/3 on the axial velocity in the cross-section reveals that, when H increases beyond the region swept by the propeller disk, V at the bottom of the propeller disk continues to decrease, and the induced velocity exhibits a negative value. The range of the low-velocity region near the blade leading edge also exhibits an increasing trend. However, at a high J as shown in Fig. 25b, the variation trend of the axial inflow velocity in the blocked area of the propeller disk in response to a changing H is the same as that at a low J. The difference is that the blocked area has a larger range of influence at high J.

Conclusions
In this study, changes in propeller hydrodynamic performance in response to the changes in blade circumferential angle and ice size parameters during ice−podded propulsor interaction were studied. The approach employed in this study was based on the RANS method and incorporated the overlapping grid technique. The main conclusions of this study are as follows.
(1) The numerical simulation results obtained via the viscous flow calculation method based on overlapping grids are in good agreement with the test results for a podded propulsor under open-water working conditions; the error margin of calculation is within 5%. This approach has good applicability to the ice−podded propulsor interference problem with good reliability.
(2) When the blade enters the blocked area behind the ice during the ice−podded propulsor interaction, distinct high-and low-pressure areas are generated on the blade face and blade back. The ranges of these areas increase with the increase of the blade's entry area. Hence, the blade K t and 10K q are greatly increased.
(3) The presence of ice greatly influences the axial velocity in the blocked area behind the ice. A distinct low-velocity region is generated near the blade tip in the blocked area, and the induced velocity is negative. In addition, the induced velocity at the blade back decreases when it enters the blocked area and increases when it exits the blocked area.
(4) For a fixed ice position and ice blocks of different lengths, L, interact with the podded propulsor, the effect of ice on the propeller thrust and torque coefficients is related to both L and the inflow velocity, V. For L ≥ R, changes in L have almost no influence on the propeller thrust and torque coefficients. For L < R, the effects of changes in L on the propeller thrust and torque coefficients are related to V.
(5) When ice blocks with different widths, B, interact with the podded propulsor, the propeller thrust and torque coefficients increase linearly with an increasing B that changes within the propeller disk. However, when B in- creases gradually from within the propeller disk to beyond the region swept by the propeller disk, the propeller thrust and torque coefficients gradually stabilize.
(6) When ice blocks with different thicknesses, H, interact with the podded propulsor, the propeller thrust and torque coefficients increase linearly as H increases within the propeller disk. When H increases beyond the region swept by the propeller disk, the propeller thrust and torque coefficients continue to increase.
In future works, interactions between the ice blocks of different sizes and podded propulsors under different operating conditions should be systematically investigated both theoretically and experimentally. Hence, the interference characteristics for the interactions between the ice and podded propulsors should be determined in a more comprehensive manner.