Numerical study on transient four-quadrant hydrodynamic performance of cycloidal propellers

Aiming to investigate the transient four-quadrant hydrodynamics on blades under the circumferential motion of the steering center, an efficient three-dimensional (3D) model with half blades was established in this paper. The 3D model with half blades accounts for the hydrodynamic loss induced by the complicated mechanical structure, predicting the hydrodynamic performance of the high-load cycloidal propeller with high accuracy. On this basis, the open water manoeuvering performance of the cycloidal propellers in fully azimuth angle was simulated by Reynolds-averaged Navier-Stokes (RANS) solver, and the hydrodynamic loads and flow field characteristics were analyzed in detail. The research shows that the variation of azimuth angle φ induces the rapid response of hydrodynamic loads, which directly affects the direction of ship motion. Compared with φ = 0°, the increase in the azimuth angle of the steering center will enhance the negative impact of the inflow velocity on the wake flow field and the internal flow field within the blades. Additionally, the non-uniform flow in the main thrust direction leads to a significant increase in the undesired net lateral force. Our primary findings revealed in present paper should contribute to accurate prediction of hydrodynamic performance for high-load cycloidal propellers during maneuvering process.

Main thrust coefficient of a single blade K S Side thrust coefficient of the propeller K S1 Side thrust coefficient of a single blade K Q Torque coefficient of the propeller K Q1 Torque coefficient of a single blade λ Advance coefficient η Open-water efficiency ρ Fluid density S Span of the blade R Radius of the cycloidal propeller e Eccentricity, e = e 2 1 + e 2 e 1 Eccentricity in 'y' direction from disc center e 2 Eccentricity in 'x' direction from disc center t Time n Number of revolutions per second of the propeller Z Number of blades c Chord length of blade C Steering center P Time-averaged pressure u i Velocity in x i direction u j Velocity in x j direction μ Dynamic viscosity coefficient μ t Turbulent viscosity S i Quality force per unit mass −ρu i u j Reynolds stress term f β * Free-shear modification factor f β Vortex-stretching modification factor k Turbulent kinetic energy ω Specific dissipation rate σ k Turbulent Prandtl numbers for k σ ω Turbulent Prandtl numbers for ω P k Production term of k P ω Production term of ω

Introduction
With the development of large-scale ships in the past few decades, the issue of maneuverability has been a worldwide concern, especially for ships with conventional propellers as the propulsion system, the problem of maneuverability under low-speed conditions is particularly prominent. The conventional screw propeller usually requires an extra rudder to steer the ship, and the lateral steering force generated by the rudder depends on the propeller's revolution speed. As the revolution speed of the propeller is reduced, the rudder becomes less effective in providing lateral force, which leads to a sharp decline in the manoeuvering performance of the ship. To overcome this barrier, a large number of special thrusters with excellent performance represented by cycloidal propellers, waterjet propulsors, and podded propulsors have been developed and applied. The cycloidal propeller is generally composed of four to six parallel straight blades arranged vertically to the bottom of the ship. Due to the special hydrodynamic design, cycloidal propellers can generate 360°vector thrust, without changing the revolution speed. It is utilized for ships that require high maneuverability, including tugs, luxury yachts, ferries, minehunters (Jürgens et al., 2007), etc. Back in the 1920s, Kirsten-Boeing Propeller (KBP) (Sachse, 1926), the first cycloidal propeller, was invented. Soon, Voith also developed their own applicable Voith-Schneider Propeller (VSP). The biggest difference between VSP and KBP is the motion pattern of the blade. In KBP, the steady angular velocity of the blade is half the angular velocity of the propeller disc at all the blade orbital positions (Nandy et al., 2018). However, VSPs can change the unsteady pitch motion of the blades by adjusting the radial position of the steering center, to achieve the purpose of changing the thrust magnitude. Due to the better performance in terms of maneuverability, VSP has been more widely used. With the widening application of cycloidal propellers, relevant theoretical and experimental studies have also been carried out. Taniguchi (1962) proposed a quasi-steady flow theory to solve the induced velocity on the blade, and introduced a correction coefficient based on experimental studies to increase the accuracy of the calculation results. Through detailed experimental data (Dickerson & Dobay, 1970;Nakonechny, 1961), Taniguchi's method was validated for advance coefficients 0.4-0.5 but for other advance coefficients or large eccentricities, the calculated results deviated greatly from the experimental results. Zhu (1981) considered the effect of the curved orbit of the blades and the blade rotation on camber and zero lift angle. The accuracy of the calculation results was quite improved under large advance coefficients and eccentricities. Bose and Lai (1989) carried out extensive experimental studies to investigate the thrust and torque of cycloidal propellers. It is found that methods such as the compound flow tube model are invalidated for high loading conditions, and further proposed a more advanced theoretical method to predict the overall performance of cycloidal propellers. Jürgens and Heinke (2009) conducted an experimental study on the cavitation behavior of high-load VSP under bollard pull conditions. The study shows that VSP's high lift profile highly improved cavitation behavior. Besides, Van Manen (1966), Ficken and Dickerson (1969), Li (1991), andVeitch (1992) have contributed a lot of fruitful work to the experimental research of cycloidal propellers. It is worth mentioning that Ficken and Dickerson's work has provided authoritative references for extensive research (e.g. Hu et al., 2021 andLi et al., 2021).
For heavy loading conditions, rapid spinning of the vertical blade leads to strong downstream vortical flow. In some cases, the potential flow method is less effective due to the neglection of fluid viscosity, and thus Computational Fluid Dynamics (CFD)  simulation is a preferred option for evaluating cycloidal propellers. Jürgens and Grabert (2003) studied the optimization of ships equipped with cycloidal propellers by numerical means. Palm et al. (2011) investigated the hydrodynamics of a cycloidal propeller with ventilation effect and the subsequent thrust loss by using CFD simulation. Complicated blade motion leads to a massive need for computational sources. Thus, in some studies, the cycloidal propeller is assumed two-dimensional (2D) to facilitate numerical simulation. Hu et al. (2020) used a simplified 2D model in the hydrodynamic study of cycloidal propellers and performed an extensive comparative analysis of the effect of geometric particulars and kinematic motion on the hydrodynamic performance. To effectively improve the hydrodynamic efficiency and eliminate the adverse effect of lateral thrust, Sun et al. (2021) established a 2D model to study the influence of flap on the hydrodynamic performance of cycloidal propellers. The results show that the flap rotation according to the anti-virtual camber law is beneficial to offset the lateral thrust, and the rotation according to the sine law is helpful to improve the efficiency of the propeller at a high advance coefficient. Yang (2010) used the CFD method to numerically simulate the cycloidal rotor. Studies revealed that both the 2D and three-dimensional (3D) models can accurately capture the vertical forces, but the 2D model cannot capture the vortex shedding at the tip of the blade. Ghassemi (2019, 2020) stretched a chordlength distance of the 2D model in the blade-span direction to ensure the necessary 3D flow. It is proposed as a 2.5D numerical model and applied to the related research of cycloidal propellers. Hu et al. (2018) used different numerical models such as 2D, 2.5D, 3D models with half blades, and 3D models with full blades to simulate aerial cycloidal rotors. It is found that the 2.5D model cannot achieve more accurate results than the 2D model. The results of the 3D model with half blades and 3D model with full blades are consistent, but the 3D model with half blades has higher computational efficiency.
From the literature review, it can be discovered that the hydrodynamic performance of the cycloidal propeller as a thruster is the major concern in most previous studies. However, it is also important to investigate the hydrodynamic performance of the cycloidal propeller as a steering device by altering the location of the steering center. In this aspect, the present study on the four-quadrant hydrodynamics of cycloidal propellers is of great interest and significance. Nevertheless, the hydrodynamic discrepancies caused by the complex mechanical structure of cycloidal propellers represent a challenge for accurate CFD prediction. Prior to performing our study, it is a problem that must to addressed. In our study, it is regarded that the prediction accuracy at low advance coefficients can be significantly improved by neglecting the hydrodynamic increment from the rotating disc to offset the hydrodynamic loss induced by the mechanical structure. As a result, the contribution to the computational accuracy obtained by the model without the disc in predicting high-load cycloidal propellers was investigated. According to that, the 3D model with half blades was finally adopted in this research to evaluate high-load cycloidal propellers to simultaneously gain considerable computational savings. On this basis, the variation patterns of the transient four-quadrant hydrodynamic forces under different eccentric azimuths were further analyzed, and the unequal effects of the wake flow field on the hydrodynamic performance of the cycloidal propeller under different eccentric azimuths were studied. And in the last part of the article, the research results are briefly summarized.

Governing equation
In this paper, the unsteady RANS equations are used in the solving process, and the average mass and momentum equations are as follows: At this point, SSTk − ω turbulence model (Menter, 1994) that can produce superior simulation results for strong inverse pressure gradients and separation phenomena is introduced to close the above equations (Song et al., 2021). Based on the Boussinesq hypothesis (Boussinesq, 1877), Equation (3) representing the relationship between Reynolds stress and average velocity gradient is established. The turbulent viscosity can be expressed by Equation (4) as follows: The turbulent kinetic energy k and specific dissipation rateω are calculated from transport Equations (6)-(7).

Kinematics of cycloidal propellers
The kinematics of the cycloidal propeller is shown in Figure 1. The origin O of the coordinate system established in this paper is located at the center of the propeller disc. The positive direction of the x-axis coincides with the forward direction, and the positive direction of the y-axis points to the starboard direction. The coupled motion of the blade can be decomposed into the steady revolution motion relative to the disc center and unsteady pitch motion relative to its chord center. When the blade is performing coupled motion, the connecting line between the chord center of the blade and steering center C is always perpendicular to the chord line of the blade. The distance between the steering center C and origin O denotes the eccentric distance, and the ratio of eccentric distance to the orbit radius R is defined as the eccentricity e given by Equation (10).
The angle θ indicates the angle of the blade revolution around the disc center. The derivative of θ with respect to time yields the angular velocity of the disc in Equation The angle β is the angle between the chord line and the tangent to the trajectory circle of the blade. According to geometric relations in Figure 1(a), Equation (12) that describes the β can be got. Furthermore, the pitch motion equation (Equation (13)) of the blade is obtained by calculating the derivative of β with respect to time.
The steering center can be treated as the controlling center of the hydrodynamics of cycloidal propellers. The angle ϕ indicates the azimuth angle of the steering center. As depicted in Figure 1(b), the angle ϕ is defined as 0°when e 1 < 0 and e 2 = 0. Therefore, the pitch angle β and pitch angular velocity at any ϕ satisfy the following Equations (14)- (15).
(15) Figure 2 shows the variation of the pitch angle β and pitch angular velocity with the revolution angle θ under different azimuth angles. It can be observed that the discrepancy presented in the curves of the β under different ϕ is embodied in significant phase differences. Similarly, this variation rule is also perfectly applicable to the curves of the under different ϕ. According to the relation of the  velocity vectors in Figure 1(a), the resultant velocity can be described by Equation (16), Therefore, the forces acting on the blades are clarified, namely the lift force L perpendicular to the resultant velocity and drag force D parallel to the resultant velocity. By decomposing the lift L and drag D acting on the blade, the thrust of a single blade can be obtained. Moreover, the hydrodynamic thrusts and torque in Equations (17)-(19) of cycloidal propellers are obtained by superposition of the instantaneous values on each blade in an operating cycle.
The above hydrodynamic thrusts and torque are further transformed into dimensionless forms to obtain the main coefficients that appraise the hydrodynamic performance of cycloidal propellers. The hydrodynamic coefficients can be described by Equations (20)-(24).

The 3D model with half blades
As mentioned above, the 3D model with half blades was initially applied to the aerodynamic performance investigation for aerial cycloidal rotors in the literature (Hu et al., 2018 ). Nevertheless, the structure of aerial cycloidal rotors differs from that of marine cycloidal propellers in that their blade ends possess identical mechanical properties. One side of cycloidal propellers (e.g. VSP) blade is installed on a rotating disc, resulting in the structure of cycloidal propellers in the span direction lack symmetry. Actually, the influence of the rotating disc is ignored in the present study on the high-load cycloidal propellers, which will be discussed in a later section. Based on this, a 3D model with half blades is built, evaluating the high-load cycloidal propeller to achieve computational savings. Considering the slight variation of hydrodynamics in the blade span direction, the blade   thickness and chord length are consistent in the span direction. 3D full-blade with the disc is segmented on the midspan section of the blade span, and the new section of one half without rotating disc is set as symmetric planes, resulting in a 3D model with half blades (visible in Figure 3) obtained.
The particulars of the 3D model with half blades established in this paper are listed in Table 1. Figure 4 illustrates the hydrodynamic force comparison between the half-blade model and full-blade model without the disc. After comparison, it is found that the hydrodynamic curves of the two models are completely consistent. Figure 5 depicts the blade surface pressure distributions of the two models at the same solving time when e = 0.6, λ = 0.127. The pressure difference between both sides of the blades reaches its maximum at the guide edge. It can be observed that the pressure distribution on the blade surface of the full-blade model without the disc is strictly symmetrical about the mid-span section, and the pressure distributions on the blade surface of the half-blade model are completely consistent with that of one half of the full-blade model.

Boundary conditions and meshing
To reduce the influence of boundary conditions on the calculation results, the size of the computational domain should be large enough. The calculation domain is divided into the fluid region and rotating region. As depicted in Figure 6, the fluid region extends 30R in the x and y-axis directions respectively, and 5R in the z-axis direction. The rotating region inside the fluid region is a 4R-high, 6R-diameter cylinder centered on the propeller shaft. The propeller shaft is 13R from the inlet, 17R from the outlet, and 15R from the left and right boundaries. The blade boundary is set to a non-slip wall, the inlet boundary condition is set to velocity inlet, the outlet boundary condition is set to pressure outlet, and the other boundary conditions are set to symmetric planes.  In the present study, the structured grid method is used to discretize the computational domain. The sliding mesh technology is adopted to transmit data and information between the rotating region and fluid region. To ensure the accuracy of the calculation results, the same mesh dimension is set on both sides of the sliding mesh interface. Usually, the grids require gradual refinement from the far-field of the fluid region to the rotating region with large flow field disturbance, to ensure a superior transition between different dimension grids and appropriately reduce the total number of grids. As depicted in Figure 7, considering the direction change of the wake flow field under different ϕ, mesh refinement regions centered on the propeller are set up, and gradually refined to the rotating region in the form of cylinders with different diameters. Additionally, a rectangular mesh refinement region is added on the backside of the first cylinder outside the rotating region. The above measures, including the arrangement of the fluid region around the propeller in the x-y plane as a square, are to ensure that there is enough space around the propeller and sufficiently fine grids to capture the trailing vortex shedding and evolution in different directions.

Convergence analysis
Owing to ensure the accuracy of the numerical simulation results, the convergence analyses of the grid and time step are carried out below to determine the appropriate grid dimension and time step. Main parameter settings of numerical model in the convergence analysis: λ = 0.127, e = 0.6, ϕ = 0°and n = 1rps.
In the assessment of the meshing scheme, the uncertainty analysis method recommended by Roache (1997) is adopted in the study. Before grid convergence analysis, three grid schemes are determined according to the  Table 3. Results of time step convergence analysis. above meshing method. Of these, the grid basic size of the fine grid scheme is set to 0.260 m, the growth rate of the grid basic size among the three schemes is √ 2, and the number of grids in the three schemes is 3.6, 2.2, and 1.2 million in turn. The thickness and layer number of prism layers of the three schemes are 0.01 m and 15 layers, respectively. The related parameters of the prism layer are determined by the y+ wall function, which details in Guo et al. (2021). Table 2 exhibits the grid convergence analysis results. Compared with the results of the fine mesh, the discrepancies of the coarse grid are larger than that of the medium grid but it is still within an acceptable range. Nevertheless, at the peaks of the curves in Figure 8, the results for the coarse grid present more pronounced discrepancies with the results for the other two grids. Particularly at the peak of θ = 200°∼ 220°, the coarse grid underestimates the K T1 value by 17.4% compared to the fine grid, which would be extremely misleading for the hydrodynamic analysis. To this end, considering the requirements of calculation efficiency and calculation accuracy, the medium grid scheme is adopted in the subsequent calculations.
Accordingly, before the time step convergence analysis, three-time step schemes are set in advance, namely t = 0.001s, t = 0.002s, and t = 0.004s. Table 3 shows the results of time step convergence analysis. It can be observed that the discrepancies of t = 0.002s and t = 0.004s are all limited to a small range. A more considerable time step tends to understate the peaks and overestimate the troughs of the curve. As illustrated in Figure 9, other than slight differences at the peaks, the curves of t = 0.001s and t = 0.002s almost overlap each other in one operating cycle. Whereas, the results obtained with t = 0.004s performed unsatisfactorily at the extremums of the curves. In terms of the K T1 curves of θ = 20°∼ 40°in Figure 9(a), the discrepancies of the peak and trough values gained by t = 0.004s are -9.8% and 12.9%, respectively, compared with the results of t = 0.001s. These significant differences are masked in the mean values of hydrodynamic coefficients. Based on the above analysis results, the t = 0.002s is finally selected for all the following calculations.

Verification of numerical models and methods
Referring to the literature (Ficken & Dickerson, 1969), the relatively comprehensive experimental data of cycloidal propellers can be obtained. Table 4 lists the main parameters of the cycloidal propeller in the literature. To verify the accuracy of the CFD models and methods proposed in this paper, the 3D full-blade models with and without disc are established based on the geometric parameter in the literature. Moreover, the turbulence model, meshing and boundary conditions are in strict agreement with those of the 3D model with half blades in this paper.
To this end, the hydrodynamics of the cycloidal propeller with different advance coefficients λ (0 ∼ 0.318) is predicted using both numerical models. In this paper, advance coefficients are obtained by varying the inlet velocity. Figure 10 demonstrates the blade pressure distribution obtained with both models at λ = 0.127. Undoubtedly, the disc will change the flow state and thus affect the hydrodynamic performance. As depicted in Figure 10(b), the disc varies the pressure distribution on the upper half of the blade, which leads to an increase in the pressure difference on the upper half of the blade. Figure 11 shows the validation of the calculated values with the experimental values. As can be observed from Figure 11, the results obtained with the model with the disc are significantly higher than the results for the model without the disc and the experiment. The rotational disc at the top of cycloidal propellers combines propulsion, steering, and roll stabilization, resulting in this part being the most complex mechanical structure of cycloidal propellers. We assume that the extremely complicated mechanical structure of cycloidal propellers contributes to visible hydrodynamic losses. Although the model without the disc fails to consider the hydrodynamic increment induced by the disc effect, its calculations are better fitted to the experimental values thanks to the mechanical losses. It is also worth noting that the discrepancies in the results of the model without the disc will increase considerably with larger advance coefficients. For larger advance coefficients, the contribution from the disc to the hydrodynamic forces will be enhanced,   leading to a severe underestimation of hydrodynamic coefficients by the model without the disc. Table 5 lists the average errors of both numerical models for λ = 0 ∼ 0.318. It can be observed that the model without the disc yields more satisfactory results, especially in the prediction of thrust and torque. As shown in Figure 11, when λ = 0.127, the discrepancies of thrust and torque obtained with the model without the disc are only 1.27% and 1.26%, respectively, while the corresponding discrepancies for the model with the disc are 9.21% and 8.89%. Consequently, in the assessment of the hydrodynamics of the high-load cycloidal propeller in this paper, the influence of the rotating disc is excluded to expect offsetting the hydrodynamic loss caused by the mechanical structure. Additionally, in recent experimental studies of cycloidal propellers (e.g. Fasse et al., 2022), the blades are hooked up to each shaft instead of the rotating disc. In this case, the model without the disc is more consistent with the experimental model.

Analysis of transient hydrodynamic loads
In this section, an azimuth angle is selected at each interval of 30°for numerical simulation to systematically analyze the maneuvering performance of the cycloidal propeller. Main parameter settings of numerical model: λ = 0.127, e = 0.6 and n = 1rps. Figure 12 illustrates the variation in the x-axial thrust component of the propeller. The thrust value in the x-axis for Figure 12(a) and (b) is opposite of each other, which indicates that the azimuth angle causes significant variation in the direction of the main thrust. To further study, firstly, the mechanism of the main thrust direction varying with the azimuth angle should be investigated clearly. Figure 13 depicts the direction of the main thrust under different ϕ. The main thrust direction is opposite to the angular velocity direction of the disc and is perpendicular to the eccentric distance (Chen, 2013). As shown in Figure 13, the azimuth angle ϕ can range from 0°to 360°. However, the eccentricity e is restricted to be a fixed value. At the present, it is not allowed to control the thrust magnitude of the propeller by changing the eccentricity. While the direction of the thrust can be controlled by varying the azimuth angle (Prabhu et al., 2017). The velocity vector analysis diagram of the cycloidal propeller is shown in Figure 14. It can be observed that the pitch angle of the blade at the same orbit position is significantly different in both cases, which should be attributed to a change in the position of the steering center. Compared with ϕ = 0°, the attack of angle of the blade is significantly increased when ϕ = 180°, resulting in the lift force and drag force acting on the blade being also significantly increased. In both cases, the drag direction is the same, but the lift direction becomes opposite of each other, which is the immediate cause why the main thrust of the cycloidal propeller deflects 180°. It is worth noting that in the symmetrical position of the propeller disc, the side thrust of the blade can offset each other, which effectively eliminates the influence of the undesirable side thrust. Figure 15 presents the variation in the hydrodynamic coefficient when e 1 < 0. It can be observed that the variation in the hydrodynamic coefficient of the propeller over one operating cycle includes the oscillation of five peaks and troughs, and the number of small cycles is related to the number of blades of cycloidal propellers. Figure 15(a) illustrates the variation of K T that occurs over one operating cycle, the phase difference between the peaks when ϕ = 0°and ϕ = 270°is 19°. For the amplitude, the peak values in both cases are about 0.677 and 0.816 respectively, while the trough values are about 0.347 and 0.653 respectively. Compared with the hydrodynamic coefficient when ϕ = 0°, the hydrodynamic coefficient will gradually increase as the steering center approaches the x-axis. Figure 16 depicts variation in the hydrodynamic coefficient when e 1 > 0. As shown in Figure 16, The rule of the propeller hydrodynamic coefficient varying with the azimuth angle demonstrates more obscure. Of particular note is that for the propeller hydrodynamic coefficient over one operating cycle, the convergence between the peaks caused by small period oscillation presents quite unsatisfying results. The negative effect of inflow velocity on the wake flow field may be responsible for this phenomenon, which will be further discussed  in the later part. Figure 17 plots the variation of average hydrodynamic coefficient of various azimuth angles. It should be pointed out that a certain range of thrust can also be obtained with the variation in azimuth angle, which indicates that azimuth angle also affects the thrust magnitude, but the influence is fairly tiny compared with eccentricity e.

Influence of freestream on wake field
The azimuth angle determines the orientation of the vortex shedding and wake flow field of cycloidal propellers. Considering the impact of inflow velocity on the wake flow field the vorticity distribution near the blades also appears to be more complex. Especially, when ϕ = 180°, the hydrodynamic performance of the propeller is severely limited. To study and clarify the role of those influences in the hydrodynamics of cycloidal propellers, the flow field characteristics are analyzed. Figure 18(a) shows the wake vorticity field of the propeller when ϕ = 0°, where the blue arrows indicate the trajectory of the trailing vortices. The interaction between the blade and fluid causes the formation of attached vortices at the blade wall. With the coupled motion of the blade, the vortices continuously fall off from the blade wall and move to the flow field far away from the propeller. As depicted in Figure 18(a), the wake flow field is in the direction of the inflow velocity, and the freestream will strengthen the original motion trend of the wake vortices. Since the motion of the shedding vortex to the far-field is not inhibited, a large number of eddy gathering areas will not be formed near and inside the propeller. Figure 18(b) illustrates the wake vorticity field of the propeller when ϕ = 180°, the wake flow field is in the opposite direction of the inflow velocity. Due to the centrifugal force, the shedding vortices on the blades move towards the wake flow field direction of the propeller (the direction of the blue arrows in Figure 18(b)). When subjected to the strong reaction force from the inflow velocity (in the direction of the black arrow in Figure 18(b)), some vortices will be forced to return to the vicinity of the propeller and even to the internal flow field of the propeller. Owing to the vortex movement to the far rear of the propeller being suppressed, the vortices will begin to gather around or even inside the propeller, and the hydrodynamic performance of the propeller will be seriously affected when it reaches a certain level. When e 1 > 0, the unsteady structures in the flow field exacerbated by freestream further undermine the periodicity of hydrodynamic variation. Figure 19 illustrates the wake velocity (x-axial component) field of the propeller. As shown in Figure 19(a), benefiting from favorable freestream conditions when ϕ = 0°, the velocity field behind the propeller has been fully developed. Furthermore, a banded high-velocity region penetrating the flow field appears in the wake velocity field of the propeller, which is correlated to the high-speed rotation of the fluid induced by the vortex shedding. In Figure 19(b), the wake velocity field is compelled to extend laterally after being squeezed by the facing freestream in the x-axis direction. Compared with ϕ = 0°, the shorter banded high-velocity region when ϕ = 180°shows a more noticeable velocity gradient in the length direction, and the high velocity is mainly concentrated in the flow field near the propeller. Fundamentally, when ϕ = 180°, an overwhelming majority of the vortices with a large amount of kinetic energy are confined near the cycloidal propeller, which causes the flow field structure in this area to exhibit highly turbulent pulsations. As presented in Figure 19(c) and (d), the velocity fields of ϕ = 90°and ϕ = 270°are mainly extended along the y-axis. Although, the wake is tilted in the positive x-axis direction due to the influence of the inflow velocity. Although without exhibiting a strong effect comparable to that at ϕ = 180°, the impact of the freestream on hydrodynamic loads and flow fields cannot be discounted.
The consequence of the disordered flow field structure will be mirrored in hydrodynamic performance of the propeller, which is inevitable. However, to what extent this will exert an impact in the hydrodynamics is our priority concern. The pitch and heave of the blade will produce periodic thrust varying with time. The five-blade model of the cycloidal propeller is studied in this paper. Therefore, the periodic curve of the total thrust of the propeller with time includes five peaks. It will take 1s for the propeller blades to complete the disc orbit. Figures 20 and 21 plot the variation of the side thrust with time when ϕ = 0°and ϕ = 180°, respectively. As shown in Figure 20, the side thrust of the propeller will tend to be stable after one operating cycle, which is more obvious for a single blade side thrust. As depicted in Figure 21, it takes more time for the side thrust to reach its stable state, and the curve tends to be stable after about 5s. If the advance coefficient is increased, the time will be longer, and even the stable state is indistinct. The hydrodynamic performance of the cycloidal propeller is closely related to the characteristics of the flow field. Thus, with the change of azimuth angle, the interaction between inflow velocity and wake flow field is inevitable and far-reaching. Figure 22 demonstrates the vorticity distribution of the propeller. The arrow direction indicates the direction of the eddy current near the steering center. As shown in Figure 22, when e 2 > 0, the vorticity gathered in the flow field inside the propeller is pronounced as compared to that when e 2 < 0. When the blade approaches the steering center, the linear increase in the pitch angular velocity will bring stronger oscillation to blade. In this study, the minimum distance between the blade rotation center and steering center is 0.42 m. At this point, the chord length is orthogonal to the eccentric distance, and the pitch angular velocity of the blade reaches the extreme value. Strong interaction between the blade and flow field accelerates the vortex formation, evolution, and shedding, and the eddy gathering area will be formed near the steering center. When e 2 > 0, the propeller center is located on the downstream side of the steering center, and the vortices in the eddy gathering area will flow into the internal area of the propeller, which is most obvious at ϕ = 270°. This will further complicate the flow field inside the propeller. For e 2 < 0, the propeller center is located upstream of the eddy gathering area near the steering center. Thus, the diffusion motion of the vortices has a marginal impact on the internal flow field of the propeller.

Influence of eccentric position on flow field inside the propeller
To further verify the influence of the eddy gathering area on hydrodynamic performance, the pressure distribution on the blade surface is also analyzed. Figure 23 is the pressure distribution on the blade surface at the same simulation time corresponding to Figure 22. From blade 1 to blade 5, the pressure difference on the blade surface first increases and then drops, which is caused by the pitch angular velocity. As the pitch angular velocity reaches its peak, the pressure difference on the surface of blade 3 increases significantly compared with other blades. Given the consistency of hydrodynamics between blades, blades 1-5 can be regarded as different orbital positions of a blade in an operating cycle. As illustrated in Figure 23, when the steering center is located in the symmetrical position of the left and right discs, the pressure difference on the blade surface of e 2 > 0 is significantly greater than that on the blade surface of e 2 < 0. This phenomenon is mainly caused by the change of attack of angle that cannot be denied. In addition, the asymmetric effect of the vortex diffusion near the steering center on the internal flow field of the propeller should not be ignored. Nevertheless, the discrepancy in pressure difference on blade surface is most pronounced between ϕ = 270°and ϕ = 90°, suggesting that the effect of the latter is a critical factor. Figure 24 shows the comparison of K T variation under symmetric azimuth angle. The main thrust difference is maximized for ϕ = 90°and ϕ = 270°, which is absolutely in accordance with the pressure contour on the blade surface.

Influence of non-uniform freestream on the side thrust
Additionally, this paper also focuses on the valuable problem of non-uniform flow relative to the main thrust direction caused by the variation in position of steering center. Figure 25 shows the velocity vector diagram of the cycloidal propeller at ϕ = 60°. It can be found that the angle of attack and effective resultant velocity at the symmetrical position of the propeller disc are significantly different along the main thrust direction. In this case, the uniform flow problem is transformed into a non-uniform flow problem in the main thrust direction, which leads to the asymmetrical distribution of hydrodynamics in the main thrust direction. Consequently, the side thrust will no longer offset each other, and the net side thrust of the propeller will increase significantly, which will make it more difficult for the ship to keep over a straight course. Figure 26 depicts the variation of the side thrust at different azimuth angles. Compared with ϕ = 300°and ϕ = 60°, the peak value and trough value (absolute value) of the side thrust when ϕ = 0°are closer, which indicates that its net side thrust is smaller in an operating cycle.
As shown in Figure 27, compared with ϕ = 0°, the average of K S and K S1 when ϕ = 300°increase by 61.9% and 47.8% respectively. Nevertheless, when ϕ = 60°, the corresponding increments amount to a staggering 249.8% and 253.3% respectively. To this end, cycloidal propellers are usually installed in pairs on the starboard and port of the ship. The disc of each unit rotates in the opposite direction to cancel out unwanted moments and forces on the ship (Nandy et al., 2018).

Conclusions
To accurately predict the transient four-quadrant hydrodynamic performance of cycloidal propellers, a 3D model with half blades was established in this paper. On this basis, the hydrodynamic performance of the propeller in full eccentric azimuths was numerically simulated. The numerical simulation results show: (1) In this investigation, the hydrodynamic losses caused by the complex structure of the cycloidal propeller during the experiment were considered. Additionally, it was evaluated that the feasibility of eliminating the hydrodynamic discrepancies resulting from the mechanical structure of the high-load cycloidal propeller by excluding the rotating disc influence. For lower advance coefficients, the 3D model with half blades achieves visible improvements in accuracy and efficiency. Nevertheless, at larger advance coefficients, this will cause considerable underestimation of the hydrodynamic  performance due to the increased influence of the rotating disc.
(2) By changing the azimuth angle of the steering center, the thrust in an arbitrary direction can be obtained quickly, which highlights the excellent maneuverability of cycloidal propellers. When the steering center is located in the symmetrical position of the upper and lower discs, the hydrodynamic coefficients of e 1 > 0 are larger than that of e 1 < 0, which cannot be simplistically attributed to the larger angle of attack, and the strong interference effect of inflow velocity on the wake flow field should also be considered. Meanwhile, the influence of vortex diffusion motion in the vortex gathering area near the steering center on the internal flow field of the propeller cannot be ignored. Hence, the influence of the wake flow field on the hydrodynamic performance of cycloidal propellers should be paid enough attention to in manoeuvering. Furthermore, the nonuniform flow in the main thrust direction will induce a sharp increase in undesired net lateral force, which requires special attention. The work of this paper contributes reference value to solving the problem of ship maneuverability at low vessel speeds.
The investigation into the transient four-quadrant hydrodynamic performance of cycloidal propellers during maneuvering is a complicated problem that involves the mechanism underlying the influence of the vortex structure of the unsteady flow field on the hydrodynamics. For further research, the relevant model experiments would be worthier work.