1. Introduction
Recently, pumpjet propulsors (PJPs) have been used in submarines for military purposes. The PJP is renowned for its outstanding cavitation performance. The PJP has improved cavitation performance, with propulsion efficiency comparable to that of a conventional propeller. The PJP consists of a rotor, a stator, and a duct surrounding the two aforementioned components. By reducing the flow velocity in the duct, the pressure of the flow field in the duct is increased. This improves cavitation performance and increases resistance to noise caused by cavitation. In a PJP, the function of the stator is to recover the rotational energy by controlling the tangential velocity of the flow entering the rotor, which is the same function as in an energy-saving device. The stator also improves cavitation performance by correcting the downstream swirl of the rotor. There is a slight difference in the role of the stator depending on whether it is located upstream or downstream of the rotor. The PJP with the stator in front of the rotor is called a stator–rotor PJP (S–R PJP), whereas the PJP with the stator behind the rotor is referred to as the rotor–stator PJP (R–S PJP) [
1]. The S–R and R–S PJP are shown schematically in
Figure 1. Because the stator of the R–S PJP is responsible for approximately 25% of the total thrust, it serves to reduce the load on the rotor. This reduces the noise and cavitation generated by the rotor. However, the stator of the S–R PJP effectively improves the noise performance of the rotor by improving the inflow to the rotor. In addition, the velocity of the stator inflow is lower for the S–R PJP than for the R–S PJP.
Typically, an optimized PJP has a smaller diameter than a conventional propeller. In particular, the ratio of the propeller diameter to the ship’s hull is smaller for a submarine than for a torpedo. This means that the PJP of the submarine is highly dependent on the boundary layer flow. Thurston and Evanbar [
2] derived the energy relationship defining the efficiency of a propeller-inducing boundary layer flow in a body of revolution and published the quantitative interrelationships of the corresponding parameters. They demonstrated that a substantial gain in propulsion efficiency can be achieved by inducing a boundary layer flow as opposed to a free stream flow and claimed that significantly higher propulsion efficiency was possible with the PJP with submerged bodies of revolution. However, the limitations and compromises imposed by the advance ratio, cavitation, and structure were not analyzed in detail in this study. The design of a PJP is challenging due to the difficulty of analyzing the turbulent boundary layer flow and the complexity of the interactions between the components of the PJP.
The PJP was designed using methods based on the potential flow and simplified hydraulic analogies and charts. McCormick and Eisenhuth [
3] described a design procedure for PJPs that takes cavitation and efficiency into account. They mentioned that the duct should be designed first because it regulates the velocity and pressure of the flow entering the rotor and has a large effect on the load at the rotor tip. In this study, the duct was replaced by a ring vortex, and momentum theory was used to determine the velocity and pressure upstream of the rotor section as well as the required pressure increase. The chord line of the duct should be aligned with the velocity determined by the hull and rotor to avoid negative pressure peaks at the leading edge of the duct. The duct section should be determined after all design procedures, taking into account the possibility of flow separation on the outer surface of the duct. Henderson et al. [
4] investigated the design of ducts for PJPs. To design the duct, the flow between the duct and the central body of the propulsor was simplified as a one-dimensional flow. Bruce et al. [
5] published a design procedure for a wake-adapted pumpjet attached to the stern of an axisymmetric hull in terms of energy and resistance and used this procedure to design the PJP of the Akron airship. In this study, the mass flow of the propulsor was chosen to minimize kinetic energy loss through the duct. Furuya and Chiang [
6] developed a quasi-three-dimensional PJP design method combining blade-to-blade flow theory and blade-through flow theory to overcome the limitations of previous approaches. Whang et al. [
7] presented a panel method using a tip-leakage vortex model for analyzing the steady-state hydrodynamic performance of the PJP. The PJP was separated into two independent systems, the rotor-hub and stator-duct, and the interaction between the systems was evaluated based on the induced velocity.
Computational fluid dynamics (CFD) analysis has been used in multiple studies to simulate the flow around the PJP and estimate its performance because the potential flow theory has theoretical limitations for propeller problems inside the duct [
1,
8,
9,
10,
11]. Suryanarayana et al. [
9,
10,
11] designed a PJP for a torpedo using CFD, and its performance was validated by experiments. Qin et al. [
12] performed CFD analysis based on an improved delayed detached eddy (IDDE) simulation to study the flow around a PJP. CFD analysis was performed for the R model (rotor only), R-D model (rotor within the duct), and PJP model (rotor and stator within the duct) to investigate the role of each component. Li et al. [
13] analyzed the ambient flow and propulsion performance of a PJP attached to a submarine using the same CFD analysis method as the study by Qin; the results showed that the performance of the PJP was significantly different from the performance in open-water conditions due to the hull-retarded flow.
As mentioned earlier, the PJP consists of a duct, stator, and rotor; consequently, there are many design parameters. Therefore, it is time-consuming and expensive to analyze the effects of the parameters through experiments. In the majority of studies, the effects of the parameters on the PJP have been analyzed numerically. Due to the narrow distance between the inner surface of the duct and the tip of the rotor, the performance of the PJP, including cavitation and efficiency, is sensitive to variations in the distance. Many studies have examined the hydrodynamic characteristics as a function of the gap [
8,
14,
15]. As the tip clearance increases, the propulsion and cavitation performances of the PJP decreases. Lu et al. [
16] compared the results for cavitating and non-cavitating conditions and discovered that at high advance ratios, the difference in efficiency was smaller at small clearances for the non-cavitating condition and was bigger at small clearances for the cavitating condition. In addition to tip clearance, numerous parametric studies have been conducted on other parameters. Yu et al. [
17] investigated the variation of the propulsion performance of a PJP as a function of stator parameters, such as stator pitch angle, chord length, and rotor–stator spacing, using CFD analysis. As the stator pitch angle increased, the circumferential velocity of the rotor inflow and the overall thrust and propulsion efficiency of the PJP increased. However, the noise performance may degrade due to the sharp increase in unsteady pressure fluctuations. The rotor–stator spacing had no significant effect on the propulsion performance. However, if the rotor–stator spacing is appropriately selected, the fluctuation amplitudes of the unsteady force are significantly reduced. The chord length of the stator had little effect on the propulsion performance and maximum fluctuation amplitude of the unsteady force. Whang et al. [
18] and Huang et al. [
19] investigated the effects of duct parameters on the PJP. Whang et al. [
18] used the same analytical model as the previous study to analyze the propulsion performance of a PJP according to the gap clearance, camber, and angle of attack of the duct. Huang et al. [
19] systematically analyzed the effects of duct parameters (length–diameter ratio, incidence angle, shrinkage ratio at the duct inlet, expansion ratio at the duct outlet, and tip clearance) in the PJP using CFD analysis. In this study, the duct parameters were analyzed in terms of open-water performance, thrust fluctuation, pressure field and its fluctuation, velocity field, and vortices.
The effects of duct parameters have also been investigated in studies of typical ducted propellers. Theoretical calculations performed by Oosterveld [
20] demonstrated the effects of a decelerating duct on the length and thickness of the duct. The risk of flow separation at the outer surface of the duct was reduced by increasing the length or decreasing the thickness of the decelerating duct. Huyer and Dropkin [
21] performed a CFD analysis of a two-dimensional axisymmetric duct behind a ship’s hull to determine the effects of duct parameters. This study did not parameterize the geometry of the duct because it focused on the effect of the flow inside the duct. The study presented the velocity distribution of the flow within the duct as well as the pressure distribution on the duct based on the difference between the mean flow velocities within the duct. The results showed that the drag on the hull decreased, while the drag on the duct increased with an increase in the difference between the mean flow velocities. As the difference in mean flow velocities increased, the mass flow rate inside the duct decreased. Bontempo et al. [
22] applied axial momentum theory and a nonlinear semi-analytical actuator disk model to investigate the flow around a decelerating duct with varying length, camber, and thicknesses of the duct. The study showed that the length had the smallest effect on the drag of the duct compared to its camber and thickness. Although the cavitation behavior of the rotor can be improved by increasing the camber and length of duct or decreasing the thickness of duct, increasing the camber causes cavitation in the mid-chord of the duct, and decreasing the thickness causes cavitation at the leading edge of the duct at very high advance ratios. Gaggero et al. [
23] presented an optimization approach for the design of accelerating and decelerating ducts. Multiple control points were used to modify the geometry of the duct, which was represented by a B-spline curve. This method can be performed with an optimization process using diverse data.
Previous studies have shown that ducts have a significant impact on PJP performance. This is because the duct has a significant effect on flow entering the stator and duct. As mentioned earlier, potential analysis is a limitation in analyzing the closed-tip clearance in the duct–propeller problem. In addition, the majority of previous studies have analyzed the characteristics of the PJP under open-water conditions. However, the flow behind bodies of revolution, such as submarines, is quite different from the flow under open water conditions due to the large inclination angle of the astern body hull form.
In this study, a parametric analysis of the PJP of the SUBOFF submarine was performed. The incidence angle and camber were selected as the duct parameters and the pitch angle was selected as the stator parameter. The hydrodynamic characteristics of the PJP were investigated for each parameter, and the results were summarized and analyzed according to the variations of the area ratio difference between the inlet and outlet stations of the duct and the operating station of the rotor. CFD was used for most of the computations and the initial design of the rotor and stator was performed using the potential analysis. The in-house developed potential code (PASTA) was used for the design [
24], and Star CCM+ v15.04 was used for CFD computations. The CFD analysis was verified by a grid dependence test, and a comparison with experimental results from the David Taylor Research Center (DTRC) and the Korea Research Institute of Ship & Ocean Engineering (KRISO) was performed for validation.
For the proper design of a submarine–PJP, the current study focuses on a parametric study that takes favorable flow conditions and hydrodynamic efficiency into account. Although not specified, favorable flow conditions may also contribute to the improvement of cavitation performance. It is expected that cavitation performance and noise will be studied when cavitation inception speed (CIS) and cavitation conditions are explicitly specified, and future research will investigate optimization with two objective functions of efficiency and cavitation, including CIS. The current study may be the first step toward achieving this objective.
The first section of the current study presents the specifications of the target submarine. It describes the types of design variables of the PJP used in the analysis and their definitions, and the dimensionless coefficients used for the performance analysis. The second section describes the numerical method used in the current study. The numerical methods include governing equations, turbulence model, grids, and boundary conditions. The third section presents the results of the grid dependence tests and comparison with experimental results for validation and describes the results of the parametric study.
5. Conclusions
In this study, a parametric study of a PJP was conducted from a hydrodynamic point of view. The PJP was designed for the SUBOFF submarine, and its propulsion performance was evaluated for the design parameters (duct incidence angle (DIA), duct camber (DC), and stator pitch angle (SPA)). The propulsion performance for each design parameter was also analyzed using the non-dimensional mass flow rate, and area ratio difference (). Star-CCM+ was used as a tool for the parametric study.
The CFD analysis method was validated by a grid dependence test and the experimental results from DTRC and KRISO were compared. For a parametric study of the PJP, each case’s propulsion performance at the self-propulsion point was compared. The propulsion performance was also analyzed using a non-dimensional mass flow rate, and area ratio difference (). DIA is the most dominant influence on the hydrodynamic characteristics and efficiency of the PJP. This is because of the large variation in the non-dimensional mass flow rate and area ratio difference and the dominant effect on flow separation at the leading and trailing edge of the duct. DC has the same effect as DIA, but it is expected to change the rotor advance ratio mainly because it changes the non-dimensional mass flow rate more than the area ratio difference. SPA changes the performance of the rotor itself because it changes the tangential velocity, which causes a change in the angle of attack of the rotor. The highest efficiency was attained when the non-dimensional mass flow rate was approximately 0.77 and the area ratio difference was approximately 0.1. However, a PJP with high non-dimensional flow rate and low area ratio difference are not suitable for a PJP because the duct functions as an accelerating duct. Therefore, to design a PJP that is optimal in terms of efficiency and cavitation, the flow rate and area ratio of the PJP must be properly determined. In addition, the outer contour of the duct must be determined by taking into account the flow separation that occurs at the leading and trailing edge of the duct.
This study is expected to be the initial step toward the optimal design of PJPs for submarines. It is also anticipated that additional parameters, including the diameter, number of stator and rotor blades, and length of the duct, will be incorporated into a broader spectrum of applications in the near future.