Investigation on the Excitation Force and Cavitation Evolution of an Ice-Class Propeller in Ice Blockage ^†

Abstract

When an ice-class propeller is operating in an ice-covered environment, as some ice blocks slide along the ship hull in front of the propeller blades, the inflow ahead of the propeller will become non-uniform. Consequently, the excitation force applied to the blades will increase and massive cavitation bubbles will be generated. In this paper, a hybrid Reynolds-Averaged Navier–Stokes/Large Eddy Simulation method and Schnerr–Sauer cavitation model are used to investigate the hydrodynamics, excitation force, cavitation evolution and flow field characteristics of the propeller in ice blockage conditions. The results show that the numerical method adopted has a relatively high accuracy and the hydrodynamic error is controlled within 3.0%. At low cavitation numbers, although the blockage distance decreases, the cavitation phenomenon is still severe and the hydrodynamic coefficients hardly increase accordingly. Ice blockage causes a sharp increase in cavitation. When the distance is 0.15 times the diameter, the cavitation area amounts to 20% of the propeller blades. As the advance coefficient grows, the total cavitation area diminishes, while the cavitation area of the blade behind ice does not decrease, resulting in an increment in excitation force. Ice blockage also causes backflow in the wake. At this time, the largest backflow appears at the tip of the blade behind the ice. The higher the advance coefficient, the more significant the high-pressure area of the pressure side and the greater the pressure difference, causing the excitation force to rise sharply. This work offers a positive theoretical basis for the anti-cavitation design and excitation force suppression of propellers operating in icy regions.

1. Introduction

The opening and utilization of Arctic Sea routes have shortened the voyage of merchant ships and promoted the development and utilization of Arctic resources. Whether a vessel is an icebreaker for opening a route or an ice-strengthened ship in the route, in an ice-covered environment, it will be affected by ice resistance [1]. However, the proportion of ice resistance is often more than half of the total resistance [2]. It is necessary for an ice-class propeller to raise its rotational speed, aiming to obtain greater thrust and maintain its ship’s navigating efficiency [3]. However, since the rotational speed rises and the navigating speed maintains, the ice-class propeller will undergo heavy loads. Meanwhile, the pressure on the propeller blade drops to the saturated vapor pressure, leading to the appearance of cavitation. In particular, when an ice-blockage phenomenon occurs, the ice-blockage will intensify the cavitation phenomenon [4,5].

Due to the complexity and importance of the hydrodynamic and cavitation evolution of propellers in ice blockage and cavitation environments, some scholars carry out research in terms of experiments, theories and numerical simulations. Sampson et al. carried out model tests in ice blockage and milling environments in the Emerson cavitation tunnel, and it was confirmed that as ice and propeller collided, accompanied by an increment in cavitation bubbles, it not only hindered propulsion but also caused structure damage [6,7]. In an environment without cavitation bubbles, the ice block was close to the propeller, causing its thrust to grow by 40%. However, in an environment with serious cavitation, the change in propulsion performance was not significant when the ice–propeller distance was reduced [8]. Since the ice block moved closer to the suction surface of the blade, an induced conjoined vortex was observed, along with vortex cavitation [9,10]. Simultaneously, this led to a significant increase in the excitation force, especially as the blade behind the blockage was subjected to a heavy load and the low cavitation number brought a drastic growth in the amplitude of the pressure fluctuation in some high-order frequencies [10]. We measured the propulsion performance of an ice-class propeller, considering the effect of different ice–propeller distances, advance coefficients and cavitation numbers in the cavitation tunnel, which indicated that ice blockage intensified the cavitation phenomenon and severe cavitation will reduce the growth in thrust caused by the ice block [5].

Benefiting from the development of theoretical methods and computer technology, the assessment of the hydrodynamics and cavitation evolution of propellers according to viscous flow theories has achieved a swift development. Under an ice blockage condition with no cavitation bubbles, Wang adopted the panel method, Reynolds-Averaged Navier–Stokes equations (RANS) and model tests to research the load on the propeller surface and its hydrodynamic performance in icy conditions [11]. Moreover, the RANS numerical method and overset grid were also applied to simulate the hydrodynamics as the ice block neared the propeller and the hydrodynamics of a single blade with the circumferential position and the pressure distribution were obtained [12]. Ice blockage induced great oscillations in the thrust and torque by obstructing the incoming flow of the propeller, thereby increasing the loads on the propeller [13]. Particularly when cavitation occurred, its excitation force increased rapidly. For this reason, Sun et al. used the RANS method in order to analysis the excitation force caused by ice–propeller interaction and mainly analyzed the cavitation effect on the propeller’s excitation force and its evolution process when the ice block approached the propeller [14]. The ice–propeller distance affected the propeller’s propulsion and cavitation performance; this numerical method was able to effectively predict the hydrodynamics as the blockage and cavitation appeared simultaneously [15]. We also adopted the RANS method to carry out a numerical simulation study. The hydrodynamic coefficients and the cavitation phenomena were in accordance with the experimental results in the cavitation tunnel. The flow characteristics showed that the ice blockage resulted in the turning of the direction of the propeller wake, cavitation appeared for the low pressure area on the blade’s suction surface, and cavitation decreased the vorticity on the suction surface of the propeller blade. Further, a hybrid method combining Reynolds-Averaged Navier–Stokes equations and Large Eddy Simulation (LES) methods was adopted to obtain detailed flow field information. The findings revealed that the ice blockage gave rise to a significant excitation force and delayed the excitation force occurrence at closer distances [16].

The model tests and theoretical and numerical studies carried out by previous scholars are all of great guiding significance for the improvement of the propulsion performance of ice-class propellers and the design of anti-cavitation performance. However, there are still deficiencies in the research on the evolution process of cavitation in an ice-blockage environment. In this work, the hybrid RANS/LES method combined with the Schnerr–Sauer cavitation model is used to explore the excitation force characteristics and the evolution process of cavitation in an ice blockage. First, the precision of the numerical method is validated through comparing numerical and experimental hydrodynamic coefficients; then, the hydrodynamic, excitation force, cavitation performance and flow field are analyzed; finally, the influencing mechanisms of the ice–propeller distance and advance coefficient on the excitation force and cavitation evolution are summarized, as well as their internal relations.

2. Numerical Theory

2.1. Governing Equations

In fluid dynamics, even in the case of multiphase flow with cavitation, the continuum hypothesis is still employed to simplify the problem, enabling the equations of continuum mechanics to be utilized for describing the fluid behavior. Under the continuum hypothesis, the physical properties of the fluid are considered as quantities that vary continuously in space, which allows for the use of partial differential equations to depict the flow motion. The Navier–Stokes equations are the fundamental sets of equations for describing fluid motion and the governing equations are presented as follows:

$${\displaystyle \frac{𝜕{\rho }_m}{𝜕t}}+{\displaystyle \frac{𝜕\left({\rho }_m{u}_j\right)}{𝜕x_j}}=0$$

(1)

$${\displaystyle \frac{𝜕\left({\rho }_m{u}_i\right)}{𝜕t}}+{\displaystyle \frac{𝜕\left({\rho }_m{u}_i{u}_j\right)}{𝜕x_j}}=-{\displaystyle \frac{𝜕p}{𝜕x_i}}+{\displaystyle \frac{𝜕}{𝜕x_j}}\left[({\mu }_m+{\mu }_t)\left({\displaystyle \frac{𝜕u_i}{𝜕x_j}}+{\displaystyle \frac{𝜕u_j}{𝜕x_i}}-{\displaystyle \frac{2}{3}}{\displaystyle \frac{𝜕u_k}{𝜕x_k}}{\delta }_{ij}\right)\right]$$

(2)

where u_i and u_j represent velocity vectors, p stands for static pressure, μ_t denotes turbulent viscosity and δ_ij means the Kronecker function. ρ_m and μ_m represent the density and viscosity coefficients and ρ_m = ρ_lα_l + ρ_vα_v, μ_m = μ_lα_l + μ_vα_v and α_l + α_v = 1. m, l and v stand for the mixture flow, water and cavitation, respectively.

2.2. Turbulence Model

In this paper, the hybrid RANS/LES method is employed, aiming to solve the Navier–Stokes equations. This method combines the high precision of the LES method and the high efficiency of the RANS method and realizes the transformation between the LES and RANS methods by controlling the physical quantities. The Improved Delayed DES method (IDDES) on the basis of the hybrid RANS/LES concept is widely used in the calculations of traditional propellers [17], podded propellers [18], pump-jet propulsors [19,20] and shaftless rim-driven propulsors [21]. The IDDES expression is as follows:

$${\displaystyle \frac{𝜕{\rho }_{m}k}{𝜕t}}+\nabla \cdot ({\rho }_m\overrightarrow{U}k)=\nabla \cdot \left[\left(\mu +{\sigma }_k{\mu }_t\right)\nabla k\right]+P_k-{\rho }_m\sqrt{k^3}/l_{IDDES}$$

(3)

$${\displaystyle \frac{𝜕{\rho }_m\omega }{𝜕t}}+\nabla \cdot ({\rho }_m\overrightarrow{U}\omega )=\nabla \cdot \left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right)\nabla \omega \right]+2(1-F_1){\rho }_m{\sigma }_{\omega 2}{\displaystyle \frac{\nabla k\cdot \nabla \omega }{\omega }}+\alpha {\displaystyle \frac{{\rho }_m}{{\mu }_t}}{P}_k-\beta {\rho }_m{\omega }^2$$

(4)

$${\mu }_t={\rho }_m{\displaystyle \frac{a_1\cdot k}{\max (a_1\cdot \omega ,F_2\cdot S)}}$$

(5)

where k represents turbulent kinetic energy, ω stands for turbulent dissipation rate, σ_k and σ_ω donate the turbulent Prandtl numbers, P_k represents the productions of k and ω, F₁ and F₂ stand for the SST blending functions and l_IDDES donates the length scale for the transition from RANS to LES, which is as follows:

$$\begin{array}{l}{l}_{IDDES}={\tilde{f}}_d\cdot (1+f_e)\cdot l_{RANS}+(1-{\tilde{f}}_d)l_{LES}\\ l_{RANS}=\sqrt{k}/(C_{\mu }\omega ),l_{LES}=C_{DES}\Delta \\ C_{DES}=C_{DES1}\cdot F_1+C_{DES2}\cdot (1-F_1)\end{array}$$

(6)

where ${\tilde{f}}_d$ represents the empiric blending function and f_e stands for the elevating function. C_DES1, C_DES2 and C_μ are the constants, which are 0.78, 0.61 and 0.09, respectively.

2.3. Cavitation Model

In this paper, the cavitation phenomenon is described by the Schnerr–Sauer cavitation model. The relational formula between the cavitation mass conversion rate and volume fraction is

$$\dot{m}={\displaystyle \frac{{\rho }_v{\rho }_l}{{\rho }_m}}{\displaystyle \frac{d\alpha }{dt}}, \alpha ={\displaystyle \frac{\frac{4}{3}\pi R_B^3{n}_0}{1+\frac{4}{3}\pi R_B^3}}{n}_0$$

(7)

where R_B represents the radius of the cavitation bubble and n₀ stands for its number. The cavitation mass change rate can be expressed as

$$\dot{m}=sign{\displaystyle \frac{3{\alpha }_v\left(1-{\alpha }_v\right)}{R_B}}{\displaystyle \frac{{\rho }_l{\rho }_v}{{\rho }_m}}\sqrt{{\displaystyle \frac{2}{3}}{\displaystyle \frac{\left|P_v-P\right|}{{\rho }_l}}}$$

(8)

where sign represents the sign function and P_v stands for the saturated vapor pressure.

2.4. Hydrodynamic Coefficients

For hydrodynamic performance tests of model-scale propellers, it is required to meet the similarity criteria of dimensionless coefficients stipulated by the International Towing Tank Conference (ITTC), which include

$$J={\displaystyle \frac{V}{nD}}, K_T={\displaystyle \frac{T}{\rho n^2{D}^4}}, K_Q={\displaystyle \frac{Q}{\rho n^2{D}^5}}, {\eta }_0={\displaystyle \frac{J{K}_T}{2\pi K_Q}}$$

(9)

where V represents inflow velocity, n donates rotational speed, D stands for the diameter, T represents thrust, Q donates torque, J stands for the advance coefficient, K_T represents the thrust coefficient, K_Q stands for the torque coefficient and η₀ donates the open-water efficiency.

In addition, the similarity of the cavitation numbers also needs to be satisfied. Since the method of changing the inflow velocity at a fixed rotational speed is used to adjust the advance coefficient, a rotational-speed cavitation number σ_n is defined and the expression is presented as follows:

$${\sigma }_n={\displaystyle \frac{P-P_v}{\frac{1}{2}\rho {(nD)}^2}}$$

(10)

3. Numerical Strategy

3.1. Models

The research object is an ice-class propeller with 4 blades. In order to study the performance of the propeller with an ice blockage and cavitation, a model test is conducted in the cavitation tunnel of China Ship Scientific Research Center. The scale ratio of the model is 1:28 and the scaled-down model diameter D is 0.25 m. The geometric model of the propeller is depicted in Figure 1 and its main parameters are presented in

Table 1. The main parameters of the propeller.

Blade Number	Scale Ratio	Diameter	Pitch Ratio	Disc Ratio	Hub Diameter Ratio
Z	Λ	D/m	(P/D)0.7R	AE/A0	dh/D
4	1:28	0.25	0.84	0.75	0.21

. To simulate the ice-blockage environment, a block is installed in front of the propeller. The ice block has a length of 1.72D, a width of D and a height of 0.5D and the distance between it and the propeller is L. The propeller model is fixed inside the cavitation tunnel through the central propeller shaft, while the ice block is suspended and fixed through the upper fixing bracket. The change in the distance between the ice block and the propeller can be achieved by adjusting the fixing bracket. The experiment of the hydrodynamic performance is carried out by the method of fixed rotating speed and variable inflow velocity. The rotating speed of the propeller n = 35 rps. The inflow velocity and the pressure are adjusted to the advance coefficient and cavitation number. This ice-class propeller is a right-hand propeller and the definitions of the propeller rotation direction and the coordinate system are presented in Figure 1.

3.2. Computational Domain

Figure 2 illustrates the computational domain and its boundary condition settings. The dimensions of the computational domain are identical to that of the cavitation tunnel, having a length of 12.8D and a diameter of 3.2D. On its left side is set the velocity inlet, while on the right side is set the pressure outlet. The distance from both sides to the center of the propeller is 6.4D and the cylindrical side is a slip wall. The boundary conditions of both the propeller and ice are non-slip walls. Given that the propeller needs to rotate within the computational domain, the computational domain is therefore divided into a stationary domain and a rotating domain and the interface between them is an internal interface. The rotating domain encloses the propeller blades and has a diameter of 1.2D.

The grid generation of the computational domain model is displayed in Figure 3. For the intricate geometry of this propeller, a cut-body grid with high stability is utilized to achieve the spatial discretization of the computational domain. In order to ensure the calculation accuracy, a greater grid concentration is provided for the propeller, the ice blockage and the wake of the propeller. The basic grid size of the propeller blade is 0.001 m and further refinement is carried out at the blade edges. The basic grid size of the ice blockage is 0.002 m and its edges are also refined. Because of the high-speed rotation of the propeller, a high-gradient boundary layer is formed around it. To ensure the accuracy of the flow calculation within the boundary layer, a prismatic layer grid is used on the propeller and the ice blockage. The height of first prismatic layer of the ice-class propeller is 2.8 × 10⁻⁵ m, the extension ratio of the prismatic layer is 1.2 and there are 16 layers in total. The height of the first prismatic layer of the ice blockage is 4.48 × 10⁻⁵ m, its extension ratio is 1.2 and there are 16 layers in total. There are a total of 1.26 × 10⁷ element grids in the computational domain.

3.3. Settings

Based on the previous work on the hydrodynamics and cavitation of the propeller within the ice-blockage environment, we further investigate the excitation force, cavitation evolution and flow field characteristics of the propeller when σ_n = 1.5, J = 0.35, 0.45 and 0.55 and L/D = 0.15 and 0.5. The Reynolds number is 2.46 × 10⁶. The calculation is carried out on a 48 core workstation using STAR-CCM+ software version 2302 and a calculation duration of 120 h. The calculation is divided into two steps. Firstly, the steady-state flow field is solved based on the RANS method. The SST k-ω turbulence model is employed to close the N-S equations and the SIMPLEC numerical algorithm is utilized to solve the discrete difference equations of the velocity and pressure terms. At this moment, the moving reference frame (MRF) method is adopted to simulate the rotation of the propeller. After 1000 iteration steps, a stable flow field in a steady state is obtained. Then, the hybrid RANS/LES method is used to precisely solve the unsteady turbulent flow field, with the IDDES as the turbulence model. The Schnerr–Sauer cavitation model is introduced to simulate the cavitation. In this working condition, the propeller rotates at a constant speed of 35 rps, the rotation period T_n is 1/35 s and the inflow velocity is determined by the advance coefficient. The rotational movement of the rotor is achieved by the sliding grid. The time step is 6.25 × 10⁻⁵ s, which is equivalent to 0.7875° per time step, and the total calculation is 35 T_n.

4. Validation

Grid uncertainty validation is the evaluation and quantification of errors caused by grid division and numerical methods. This paper uses three sets of grids with different sizes; the total grid quantities are 6.8 million, 12.6 million and 22.5 million, respectively. The grid convergence index (GCI) is introduced to analyze grid errors, as shown in

Table 2. Grid convergence validation (σ_n = 1.5, J = 0.35 and L/D = 0.15).

ϕ	φ1	φ2	φ3	${\mathit{e}}_{\mathit{a}}^{21}$	${\mathit{e}}_{\mathit{e}\mathit{x}\mathit{t}}^{21}$	$\mathit{G}\mathit{C}{\mathit{I}}_{\mathit{fine}}^{21}$
KT	0.2755	0.2751	0.2746	0.145%	0.009%	0.011%
10KQ	0.3732	0.3729	0.3725	0.080%	0.601%	0.756%
η0	0.4112	0.4109	0.4106	0.073%	0.028%	0.036%

. ϕ represents the hydrodynamic parameter and φ₁, φ₂ and φ₃ represent the hydrodynamic values under the grid of 22.5 million, 12.6 million and 6.8 million, respectively. $e_a^{21}$, $e_{ext}^{21}$ and $GC{I}_{fine}^{21}$ stand for the relative error, extrapolation error and grid convergence indicator, respectively. Table 2 reveals that the values of $e_a^{21}$, $e_{ext}^{21}$ and $GC{I}_{fine}^{21}$ are less than 0.8%, indicating that the grid is converged and can be used for numerical calculations.

Aiming to validate the accuracy of the simulation results, a validation is made between the computational fluid dynamics (CFD) and experimental fluid dynamics (EFD) when σ_n = 1.5 and J = 0.35, as illustrated in Figure 4. It can be observed that the trends of the hydrodynamic coefficients K_T, K_Q and η₀ in CFD and EFD are in agreement and the errors are within 3.0%. As L/D rises from 0.15 to 0.5, K_T drops from 0.2744 to 0.2726 and 10K_Q declines from 0.3723 to 0.3654, while η₀ ascends from 0.4108 to 0.4173. The variation of the hydrodynamic coefficients does not exceed 2.0% and is relatively steady. When σ_n = 4.0, K_T and K_Q decrease rapidly as L/D increases from 0.15 to 0.25; when L/D is within the range of 0.25~0.5, K_T and K_Q gradually decrease but are always larger than the hydrodynamic coefficients as σ_n = 1.5 [5]. This indicates that a decrease in the ice–propeller distance under a higher cavitation number will lead to an increase in K_T and K_Q, while a low cavitation number will raise the cavitation on the suction surface, thereby leading to a reduction in the hydrodynamic coefficients. Particularly when L/D = 0.15, the hydrodynamic coefficients are more markedly affected by the ice blockage and cavitation performance.

5. Results and Analyses

5.1. Hydrodynamics

Figure 5 presents the comparison of the hydrodynamic coefficients K_T, K_Q and η₀ of the ice-class propeller under the ice-blockage condition at σ_n = 1.5. It can be observed that the CFD results are in line with the EFD results, having the same trends and keeping the error within 3.0%. As J rises from 0.35 to 0.55, both K_T and K_Q for L/D = 0.15 and 0.50 decrease accordingly, while η₀ increases. The rotational speed of the propeller is fixed. The increase in the advance coefficient causes the inflow velocity to increase. The increase in the inflow velocity adds the angle of attack between the propeller blade and the inflow, which in turn results in the decrease of K_T and K_Q. For L/D = 0.15, K_T drops from 0.2751 to 0.2076 and 10K_Q drops from 0.3729 to 0.3004; for L/D = 0.50, K_T drops from 0.2736 to 0.2078 and 10K_Q drops from 0.3654 to 0.2948. Because of the severe cavitation when σ_n = 1.5, the differences in K_T, K_Q and η₀ for L/D = 0.15 and 0.50 are very small and the hydrodynamic coefficients scarcely increase as the blockage distance decreases.

5.2. Excitation Force

Figure 6 displays the time history curves of four blades’ excitation force superposition within the last 7 T_n. The area below the line represents the excitation force of each blade. It can be found that the K_T exhibits periodic excitation. In each rotation period, a single propeller blade undergoes one excitation behind the ice blockage. Before entering the ice blockage’s wake, the blade’s excitation force rises abruptly, reaches its maximum behind the ice blockage and then decreases gradually. Owing to the effect of the distance between the ice blockage and propeller, the excitation force with L/D = 0.15 occurs later than that with L/D = 0.50 and its excitation force is more prominent. During this, the advance coefficient climbs from 0.35 to 0.55 and the average value of the K_T for a single blade decreases, while the peak value of the excitation force increases. When J = 0.35 and L/D = 0.15 and 0.5, the maximum excitation force is around 0.3. As J increases, the mean K_T decreases but its excitation force increases. When J = 0.55 and L/D = 0.15, the excitation force even reaches 0.43, as shown in Figure 6e. The excitation force at J = 0.55 and L/D = 0.50 does not exceed 0.25, as presented in Figure 6f. The propeller blade generates excitation behind the ice block and there is a phase difference in the occurrence of the excitation forces among different blades. The superposition of the excitation forces of the four blades leads to four excitations of the total excitation force within one rotation period. The greater the excitation force exerted by a single propeller blade, the larger the total excitation force.

5.3. Cavitation Evolution

Figure 7 presents the evolution of cavitation shapes, indicating that the cavitation on blade “1” is the most prominent. Moreover, the cavitation on blade “4”, blade “3” and blade “2” decreases successively and gradually collapses. The ice blockage causes a growth in the cavitation on the blade. The smaller the L/D, the larger the cavitation coverage area will be. The evolution of the cavitation pattern in one rotation period accounts for the change in excitation force. Differing from the cavitation patterns at other propeller blades, multiple protrusions appear at the lower edge of the cavitation on blade “1”, which indicates that cavitation occurs in the vortex, as showed in black circle. Owing to the blockage effect, an induced vortex is formed in its wake. When the induced vortex approaches the blade surface, the pressure in the vortex tube decreases and cavitation occurs when it is lower than the saturated vapor pressure. As the J increases, the cavitation patterns on blade “1” vary from each other, but its volume change is minimal. Meanwhile, the cavitation on blade “4”, blade “3” and blade “2” decreases significantly, which reveals the mechanism of the more significant excitation forces caused by the growth of J. As the L/D increases, the blockage effect decreases, resulting in a reduction in cavitation. The decrease in cavitation at blade “1” is more evident and the reduction in cavitation volume brings about a decline in the excitation forces.

To explore the trend of the cavitation coverage area on the propeller blades, the fraction C_s is defined as C_s = S_c/S, where S_c stands for the cavitation coverage area and S is the total area of the blades. The time-history curves of the cavitation coverage areas superposition is presented in Figure 8. The area below the line represents the C_s of each blade. It can be discovered that the trends of the cavitation coverage area on the propeller blades are nearly identical to that of the excitation force. Both reach their maximum values behind the ice blockage, but the peak value of the cavitation coverage area is relatively stable. When L/D = 0.15 and J = 0.35, the C_s peak value of a single blade reaches 8.0% and the total C_s is approximately 20%. While the advance coefficient rises, the total C_s decreases. However, the C_s peak value of the blade behind the ice block remains almost the same and the change amount of the cavitation coverage area increases, which leads to a growth in the excitation force. When the distance between the ice block and propeller rises, the change in the amount of the cavitation coverage area decreases and the excitation force decreases to a certain extent. When L/D = 0.50 and J = 0.35, the total cavitation coverage area fraction fluctuates around 20.0%. When the advance coefficient grows, the larger the distance between the ice block and propeller is, the smaller the total cavitation coverage area will be, yet the peak of the cavitation coverage area of a single propeller blade is always around 7.5%.

5.4. Flow Field Characteristics

Figure 9 illustrates the axial velocity V_x/nD in the flow field surrounding the ice-class propeller. It is obvious that the suction effect of the propeller makes the velocity of the surrounding flow increase. However, the ice blockage hinders the incoming flow, reduces the wake velocity of the ice block and gives rise to backflow, which enhances the chaos of the incoming flow to the propeller blades. When L/D = 0.15, the maximum backflow takes place at the blade tip behind the ice blockage and has an impact on the velocity field behind the propeller. The greater the advance coefficient is, the stronger the backflow will be and the greater the influence on the velocity field is, thereby leading to more significant excitation forces. As the distance of the ice blockage increases, the obstruction effect of the ice block on the propeller diminishes, the backflow behind the blockage weakens and the effect of the backflow on the velocity field of the ice-class propeller reduces.

Figure 10 presents the pressure C_p distribution in the ice-class propeller flow field, with C_p = (P − P₀)/0.5ρn²D². It can be found that the suction effect of the ice-class propeller on the flow creates a pressure difference on the propeller blades and then generates thrust. In comparison with the pressure of the unblocked propeller blade, the obstruction effect of the ice block on the incoming flow leads to an increase in both the low-pressure area and the high-pressure area on the suction surface behind the ice blockage. When L/D = 0.15, the low-pressure area of suction surface lies within the low-pressure area behind the blockage, which makes the low-pressure even lower and the coverage area larger; the high-pressure area on the pressure surface is influenced by the dynamic pressure of the backflow, which causes the pressure to increase further, and the larger the advance coefficient is, the greater the increase in pressure will be. As shown in Figure 10c, the pressure difference on the propeller blades is the largest, which causes the excitation force to rise sharply.

Figure 11 presents the C_p on the suction and pressure sides of the blade, which discloses the mechanism of cavitation evolution on the blade. The suction effect of the blade results in a low-pressure area on its suction side and a high-pressure area on its pressure side, and thrust is generated under the pressure difference. When the pressure on the suction side is lower than the saturation vapor pressure, water vaporizes to form cavitation. Under the influence of ice blockage, the low-pressure area on the suction side of blade “1” becomes even lower and the distribution range of this low-pressure area is broader. At the same time, the leading edge pressure on the pressure side of blade “1” rises and a low-pressure area emerges at the top of the trailing edge due to the impact of the suction side. When J increases, the low-pressure ranges on the suction sides of blades “2”, “3” and “4” decrease, thereby reducing the cavitation area. However, the low-pressure range on the suction side of blade 1 scarcely changes, leading to a rapid development of cavitation here. The increase in J makes the blade subject to a spatially non-uniformly distributed pressure load, which gives rise to severe cavitation evolution and thus generates an excitation force. Moreover, the increase in J also causes a sharp rise in the leading-edge pressure on the pressure side of blade “1”, as shown in Figure 11e. Additionally, when L/D = 0.15, the low-pressure area on the suction side also extends to the pressure side, making the blade load more complicated. As L/D increases, the influence of the ice blockage decreases accordingly.

6. Conclusions

In this work, the hybrid RANS/LES method combined with the Schnerr–Sauer cavitation model is adopted to study the hydrodynamics, excitation force, cavitation evolution and the flow field characteristics in an ice blockage environment with a low cavitation number. The conclusions are listed as follows:

(1)

The hybrid RANS/LES method and Schnerr–Sauer cavitation model possess good numerical accuracy, with the error between the numerical value and the experimental result being within 3.0%. When the advance coefficient rises, the angle of attack between the propeller blade and the incoming flow also increases, which leads to a reduction in thrust and torque. In the case of a low cavitation number, severe cavitation makes the hydrodynamic coefficient scarcely increase as the distance between the ice and propeller decreases.

(2)

The obstruction effect of the ice block on the incoming flow leads to a great increase in cavitation on the blade behind it. Especially when L/D = 0.15, the total cavitation coverage area reaches 20% and the cavitation-covered area of a single blade reaches 8.0%. As the advance coefficient increases, the total cavitation coverage area decreases, but as the blade locates behind the ice blockage its cavitation coverage area hardly reduces, causing rapid cavitation evolution and an increase in the excitation force. Especially when J = 0.55, the excitation force is twice its average value.

(3)

The ice block gives rise to a backflow behind it. When L/D = 0.15, the maximum backflow takes place at the blade tip behind the ice blockage, which results in an increase in the low-pressure zone on the suction surface and the high-pressure zone on the pressure surface. The greater the advance coefficient is, the more the high pressure rises and the larger the pressure difference is, thereby causing the excitation force to increase sharply. The increase in J makes the blade subject to a spatially non-uniformly distributed pressure load, which gives rise to severe cavitation evolution and thus generates an excitation force.