The ducted propeller, given its multiple excellent characteristics, has been used increasingly on a variety of vessels in recent years. For a shallow-draft ship, the duct can protect the propeller and prevent stranding. For a tugboat, more thrust is generated by using ducted propellers with accelerating nozzles, so that it can pull large-sized vessels. For some high-speed propellers, the ducted propellers with decelerating nozzles can increase the static pressure inside the duct and reduce propeller cavitation. Therefore, this study aims to adopt the CFD to conduct the self-propulsion simulation, and discuss the fluid acceleration quality on the propulsive performance.

Sung et al. [1] applied the CFD to simulate the performance of underwater vehicles. Kim et al. [2] recommended three turbulence models, namely, the standard k-ε model (SKE), the RNG-based k-ε model (RNG), and the realizable k-ε model (RKE), for calculating the nominal wake of surface vessels. Among the three models, the RKE model could identify the strong vortex position most accurately. Huuva [3] discussed the two-phase flow problem that resulted from the Reynolds-averaged-Navier-Stokes (RANS), Detached-Eddy Simulation (DES), and Large-Eddy simulations (LES) equations. Yang et al. [4] employed the commercial software Fluent with the k-ω turbulent model to conduct the self-propulsion simulation of large oil tankers. They found that the axial wakes calculated at the circumferential positions, except for the 180° position (six o’clock position), are close to the measured values. Toxopeus et al. [5] analyzed the wake field of the DARPA SUBOFF submarine by the DES turbulent. They used RANS for calculating the boundary layer and the DES for predicting the flow field. Although the difference between the computed results and the experimental values was low, they suggested that the mesh should be optimized, different inflow angles should be considered, and the results of different turbulent models should be compared. Vaz et al. [6] applied the MARIN ReFRESCO program proposed in Vaz et al. [7] and AcuSolve to predict the flow field of the DARPA SUBOFF submarine. They analyzed the maneuvering quality, and found that the difference between the numerical simulations and the measured values was within the acceptable range. Seo et al. [8] adopted flexible meshing techniques in the CFD for the self-propulsion simulation of ships, and the obtained stern wake was relatively small. Korkmaz [9] employed the RANS with explicit algebraic stress model and the potential theory for the self-propulsion simulation of a bulk carrier. The predicted results were compared with the experimental values, and the comparisons indicated that the bare hull resistances were quite close. Kao et al. [10] used the CFD for the self-propulsion simulation of the DARPA SUBOFF submarine. They compared the simulation results with the measured values of Vaz et al. [6], and found that the maximum error was only 3.5%. This study refers to the calculation settings and methods mentioned by Kao et al. [10], and compares the calculation results on the propeller performance and resistance estimation with the experimental values to determine the correctness of the self-propulsion simulation.

Yu et al. [11] combined the panel method panMARE with the RANS code ANSYS-CFX to calculate the open-water performance of a Ka series propeller with a 19A duct. Their analysis on the flow regime of the fluid at the tip clearance suggested that the SST turbulence model was more suitable for the calculation of a ducted propeller system than the k-epsilon turbulence model. Focused on a ducted propeller, Yongle et al. [12] discussed the influence of its tip clearance on its tip clearance loss at different advance ratios. Gao et al. [13] integrated the CFD technique with the finite element and boundary element methods to estimate the vibration and noise generated by a ducted propeller. Gaggero et al. [14] adopted the OpenFOAM blockMesh library and the RANS solver simpleFOAM for the geometric design of accelerating and decelerating nozzles, in order to increase the thrust and delay the cavitation phenomenon. Gong et al. [15] applied the DES model to calculate the wake vortex evolution of a ducted propeller, and confirmed that the duct can delay the wake contraction and result in different trajectory behaviors of the tip vortices. Andersson et al. [16] utilized the commercial CFD software STAR-CCM + v10.06 to conduct the self-propulsion simulation on the ducted propeller. They also discussed the main engine horsepower loss that resulted from the viscous losses. Gong et al. [17] employed the CFD to calculate the flow field of a ducted propeller with an inclined shaft angle, and measured the tip vortex and blade surface pressure. Kinnas et al. [18] introduced a perturbation potential based panel method with full wake alignment to calculate the ducted propellers. It was shown that the fully aligned blade wake gets more accurate results than a simplified wake alignment model. In Bosschers et al. [19], an iterative procedure to analyze ducted propellers was carried out by a RANS solver and a boundary element method (BEM). The RANS solver analyzed the viscous flow by body forces obtained from the BEM. However, the absence of the tip leakage vortex which leads to a different flow behavior in the tip region should be further investigated. Bontempo et al. [20] applied the axial momentum theory and the nonlinear semi-analytical actuator disk model in analyzing the ducted propeller with a decelerating nozzle. It was indicated that some duct geometrical parameters such as the profile camber, the chord, and the maximum thickness affect the appearance of the cavitation. Bontempo et al. [21] investigated the accelerating duct by the axial momentum theory and the nonlinear semi-analytical actuator disk model. According to Bontempo et al. [21], the propulsive efficiency can be improved by increasing the thickness, chord, camber, or decreasing the incidence angle. Du et al. [22] provided the 3D VII method (the viscous/inviscid interaction method) to solve the flow around three-dimensional wings and propellers. It was shown that the 3D VII method can improve the results significantly as compared with the full-blown RANS simulations. Villa et al. [23] used the viscous-based CFD code to analyze the flow features around the ducted propellers, and got a good agreement with LDV measurements in terms of predicted wake shape and vortex positions. According to Villa et al. [23], the effects of the shaft configuration were non-negligible, while the shape of the boss cap had practically no effects on the computed forces. Amiri Delouei et al. [24], Karimnejad et al. [25], and Afra et al. [26] applied the Immersed boundary method (IBM) for dealing with moving objects in fluids. The IBM has the behavior of good stability. There is no arbitrary parameter used in IBM. The IBM along with diffuse interface scheme is often employed to take into account the forces arising from the particle and fluid interactions. By using of the fixed, non-body-conformal grids in the IBM, time-consuming re-meshing process for simulation of moving bodies in fluid flows is unnecessary.

This paper is mainly to observe the effect of the fluid acceleration due to the ducted propulsion system. The fluid acceleration is related to the propulsive efficiency and cavitation control. A real workboat case is treated as the computing sample. According to the propulsive theory, the propeller pitch ratio, propeller camber ratio, and the duct length are modified to avoid the problems due to the fluids acceleration. The most effective way to solve the problems caused by the flow acceleration is recommended in this paper. The pressure distribution induced by the fluid acceleration on the hull is also discussed in order to investigate the influence on the hull resistance. This study provides four computing samples with different propellers and ducts. The calculated resistances include the viscous resistance and wave-making resistance. In the self-propulsion simulation, the effective wake, thrust deduction factor, hull efficiency, and relative rotative efficiency of an actual ship are obtained. Besides, in order to correctly simulate the self-propulsion test, the calculated propeller performance curve (K-J chart), and the resistance are compared with the measured values.

In this study, the CFD is employed to compute the viscous resistance of the hull, the wave-making resistance, the propeller performance curve (i.e., K–J chart), and simulate the self-propulsion test. The turbulent model adopted in this paper is the SST k-ω model. The viscous flow is solved in the Reynolds-averaged Navier-Stokes (RANS) equations, and the SIMPLE algorithm is applied. The spatial discretization scheme is second-order accurate. The SIMPLE method, as deduced by Patankar et al. [27], is commonly used to calculate the incompressible flow. The turbulent model, computing settings and mesh generation used in this study are referred to Kao et al. [10].

The self-propulsion simulation is conducted in this study to discuss the performance of a ducted propulsion system with the accelerating nozzle. When the total propeller thrust is equal to the total hull resistance (including the resistance generated by the propeller suction effect (R_S), the viscous resistance (R_V), the waving-making resistance (R_W), and the sea-margin resistance (R_SM)), the balance point of the self-propulsion is reached. The balance point is as indicated in Eq. (1). (1)∑T=RS+RV+RW+RSM

Here T is the single propeller thrust.

Then, the thrust coefficient, KT, is obtained by Eq. (2). (2)KT=Tρn2D4

Here ρ is the seawater density, n is the propeller speed, and D is the propeller diameter. According to the thrust identity, the advanced ratio (JA=VAnD) and torque coefficient (KQo=Q0ρn2D5) can be interpolated by the propeller K–J chart calculated by CFD. Here V_A is the propeller inflow velocity, Q₀ is the propeller torque in the open-water condition. Thus, at the balance point, the ship effective wake (1 − w), thrust deduction (1 − t), hull efficiency (η_H), and relative rotative efficiency (η_R) can be obtained according to following equations: (3)(1−w)=VAVS=JA×n×DVS (4)(1−t)=RTT (5)ηR=Q0Q (6)ηH=1−t1−w

Here VS is the ship speed, R_T is the total bare hull resistance (R_T = R_V + R_W + R_SM), and Q is the propeller torque in the behind condition. The above computing process is also shown in Fig. 1. Fig. 1 indicates the procedure of the self-propulsion simulation. According to Fig. 1, the bare hull resistances (R_V and R_W) at the specified ship speed, and the propeller performance (i.e., K–J chart) are calculated in advance by the CFD. The propeller speed is unknown, and set to an initial value. Then the total propeller thrust (∑T), and the self-propulsion resistance (R) without the free surface are computed by the self-propulsion simulation. Determine whether the results of the self-propulsion simulation meet Eq. (7), it not, the propeller speed will be modified, and the process will be executed once again. As the results of the self-propulsion simulation meet Eq. (7), the propulsive factors can be obtained by interpolating K_T value in the K–J chart.

The operating propeller and ship hull are both considered in the self-propulsion simulation. It is noteworthy that, in order to save computational time, the free surface is disregarded in the self-propulsion simulation (i.e., disregarding the wave-making resistance (R_W)). The value of wave-making resistance (R_W) is computed by using the bare hull. In other words, the ship resistance (R) obtained by the self-propulsion simulation only includes the resistance caused by the propeller suction effect (R_S) and viscous resistance (R_V). Therefore, when the self-propulsion reaches the balance point, the total propeller thrust is equal to the summation of the ship resistance obtained by the self-propulsion simulation (R), the wave-making resistance (R_W) and the sea-margin resistance (R_SM), as expressed in Eq. (7). (7)∑T=R+RW+RSM=RS+RV+RW+RSM

The dimensional formulation and governing equations in this section are referenced in HYKAT HSVA [28].

The turbulent model, computing settings and mesh generation used in this study are referred to Kao et al. [10]. Table 1 shows that the predicated results by Kao et al. [10] are in good agreement with the measured values. In other words, the turbulent model, computing settings and mesh generation used in this study are verified.

In this paper, an actual two-screw workboat is used as a computing sample. Table 2 shows the related data of the hull geometry.

This study employed CFD to conduct the self-propulsion simulation of a ducted propeller with four combinations of different propeller geometries and duct lengths. The conclusions are as follows:

In order to compute the wave-making resistance correctly, the mesh needs to be refined on the free surface. The result is consistent with the Kevin wave system. The maximum pressure occurs at the stagnation point of the bow. Consequently, the waves begin to form, and a wave crest is generated. The results are consistent with the physical phenomenon.