A Review on Hydrodynamic Performance and Design of Pump-Jet: Advances, Challenges and Prospects

: A pump-jet, which is generally and widely adopted on underwater vehicles for applications from deep sea exploration to mine clearing, consists of a rotor, stator, and duct, with the properties of high critical speed, high propulsion efficiency, great anti-cavitation performance, and low radiated noise. The complex interaction of the flow field between the various components and the high degree of coupling with the appendage result in the requirements of in-depth research on the hydrodynamic performance and flow field for application and design. Due to the initial application on the military field and complicated structure, there is scant literature in the evaluation of pump-jet performance and optimal design. This paper, in a comprehensive and specialized way, summarizes the pump-jet hydrodynamic performance, noise performance, and flow field characteristics involving cavitation erosion and vortices properties of tip-clearance, the interaction between the rotor and the stator and the wake field, as well as the optimal design of the pump-jet. The merits and applications range of numerical and experimental methods are overviewed as well as the design method. It also concludes the main challenges faced in practical applications and proposes a vision for future research. It was found that the compact structure and complex internal and external flow field make the pump-jet significantly different, also leading to higher performance. As the focus of cavitation research, vortices interact with the complex structure of the pump-jet, leading to instabilities of the flow field, such as vibration, radiated noise, and cavitation erosion. The effective approaches are adopted to reduce radiated pump-jet with minimal influence on the hydrodynamic performance, such as eliminating the tip clearance and installing the saw-tooth duct. Advanced optimal technology can achieve high performance, cavitation performance, and acoustic performance, possessing good prospects. Further developments in investigation and the application of pump-jets in the multidisciplinary integration of fluid dynamics, acoustics, materials, chemistry, and bionics should be the main focus in future research.


Introduction
The present development in propulsion systems has enabled numerous advances across underwater vehicles, including the dramatic increment in the technology and its scope of application.With the variation of underwater tasks and the development of ocean detection technology, underwater vehicles face serious challenges in maneuverability and crypticity.However, one of the limiting factors with the current technology is the duration of missions, to which the significant contributing factor is the efficiency of the propulsion system.The propulsion system is adopted to provide power for the automatic navigation of the underwater vehicle so that the underwater vehicle has a certain speed and range [1,2].Meanwhile, the research direction about how to reduce the propulsion noise and improve hydrodynamic performance has been brought to the fore.As such, the pump-jet, which was first equipped on a Trafalgar-class submarine in the 1980s [3], is one of the most suitable choices for the high-speed underwater vehicle.The low-noise feature and high-efficiency performance, respectively, improve the overall acoustic concealment of underwater vehicles and guarantee the maneuverability of underwater vehicles.Compared with traditional propellers or ducted propellers, pump-jets are capable of meeting the requirements of underwater vehicles, such as high speed, high efficiency, no cavitation, and low noise [4,5].Contra-rotating propellers, another propulsion widely used in underwater vehicles, can also better meet the relevant requirements, but the shafting is complex, and the sealing requirements are high [6].By contrast, the pump-jet still has room for improvement, while its complex structure makes the design and performance evaluation more difficult, which leads to the more complex flow characteristics and noise mechanisms and higher hull-matching requirements.Therefore, the exploration of pump-jet design methods and performance is critical [7].
Pump-jet propulsion, which consists of a rotor, stator, and an axisymmetric duct, has primarily been applied in the military field, while on account of excellent performance, pump-jets have been gradually applied to the civil field.The compact structure and complex internal and external flow field make the pump-jet significantly different from the ducted propeller with stator.The rotor is commonly located in the symmetrical duct, which slows down the inflow through the rotor blades to improve cavitation performance and underwater critical speed.The stator is a set of stationary blade cascades positioned at an incidence angle to the direction of the inflow, either in front of or behind the rotor in the axial direction.According to the axial positions of the stator relative to the rotor, the pump-jet can be divided into post-swirl pump-jet and pre-swirl pump-jet, as shown in Figure 1.The pump-jet-installed rear stator, which can contribute by the order of 25% to the total thrust and reduce the circumferential rotating wake of the rotor, focuses on high efficiency and moment balance performance, commonly used in small high-speed vehicles such as torpedoes [8].On the other hand, pump-jet with front stator, which filters out the wakes from the appendages prior to reaching the rotor and provides pre-swirl resulting in quieter propulsion, is focused on cavitation performance and radiated noise reduction [8].Pump-jets are generally installed with large but unequal numbers of blades, which avoid impacts arising from issues such as blade passing frequency owing to the key influence on the pump-jet performance by the number of blades.The number of blades is supposed to be a prime number, which can avoid harmonics, while it is important to avoid the same number of blades in more than one row.Typically, blade numbers of the rotor and stator are often between 12 and 20.In order to meet the required requirements, the corresponding design is more complicated.The approach to make full use of the tail section line and wake of the underwater vehicle, which can weaken the wake effect, reduce the radiated noise, improve the hydrodynamic performance, and match the various components to each other, is supposed to be considered in the research of pump-jets for underwater vehicles.Therefore, the design of any component needs to meet the requirements and match the vehicle; otherwise, it will have a detrimental impact on the pump-jet flow field, which in turn influences hydrodynamic performance and acoustic properties.There is still controversy about the pump-jet design principle, which is roughly divided into two ideas.One is to regard it as a more complex ducted propeller with a stator; in other words, the design of pump-jet is based on the propeller circumfluence theory [9].The other one is to regard it as turbomachinery, considering that the flow field of the pump-jet is closer to the axial-flow pump, while the design theory of the axial-flow turbomachinery needs to be applied for the design [10].Therefore, the design and analysis of pump-jet propulsion are able to refer to related underwater propulsion for research, such as propellers and water-jets.In recent years, the design and performance analysis of marine propulsion has attracted more and more attention, but the reviews related to the key technology of marine propulsion are still rare.Yan et al. [11] reviewed the research on theory of the design, performance analysis, and control of the rim-driven thrusters.In this review paper, the typical application and future research directions are given with the unsophisticated part of the technology.However, the vast majority of this review focuses on the control of the rim-driven thrusters, such as motor performance and motor control systems, while the description referring to design and performance is limited (in total 15 pages, with less than 3 pages describing the propulsor hydrodynamic design and performance).W. Lam et al. [12] carried out a comprehensive review of equations that are used to predict the velocity distribution of propeller, including the time-averaged axial, rotational, and radial velocity in the flow establishment zone and the established flow of propeller zone.Nevertheless, most of space focuses on the semi-empirical equations for propulsors, with just a small part of the propulsor principle and flow field characteristics explained.In addition, Liu et al. [13] paid attention to tip clearance of marine equipment, including the propeller, pump, and turbine, with the reveal of vortex types, trajectory, evolution, and cavitation behaviors induced by the tip-clearance flow.To a large extent, the review merely provides limited insight into the tip-clearance flow properties and their relative application to engineering design.
Due to the particularity of pump-jets and theirs main application in the military field, the amount of published literature on pump-jets is not large.The research started relatively late, and the earliest research literature began with the AD (ASTIA Document) report published in 1963.In the report, McCokmick et al. [9] briefly described the design theory of the propeller and pump-jet and elaborated on the design method of various parts of the pump-jet, including the shroud, rotor, and stator.Meanwhile, they carried out the experiment to conclude that the pump-jet has obvious advantages over the propeller in terms of efficiency, noise, and anti-cavitation performance.At present, the key technologies are still being explored, and the theory on pump-jet design and analysis is still controversial and immature.Therefore, up to now, the reviews of pump-jet performance are rare, which are almost always about structural and working principles as well as application status [14].Especially, the specialized reviews of pump-jet performance analysis and design technology with optimization are not published yet.With the growing demand for ocean exploration and performance requirement for propulsion, more and more attention has been paid to solving the difficulties of pump-jet design and flow-field characteristics.Recently, in the last five years, there has been an obvious increase in the publication of numerical simulations and experiments of pump-jets.Hence, it is necessary to publish a specialized review with recent advances in pump-jet development technology referring to design and performance analysis.The aims of the review are to state the basic logic of underlying operating theory and design principle on pump-jets while discussing design guidelines in a pump-jet to achieve stable propulsion stabilization and sufficiently good performance on propulsion, cavitation, and acoustics.Therefore, the review mainly carries out the analysis of the latest research advances in the analysis of hydrodynamic performance and flow characteristics and the design theory of pump-jets.It presents an overview of the development of investigation technology and prediction of pump-jet flow fields, such as cavitation vortex and acoustic performance.The merits and drawbacks of numerical and experimental methods are overviewed as well as the design method.The future development tendency and outlook are also discussed.The review is comprehensive and systematic.It is the first review on the technology of performance analysis and design so far and will provide profound guidance for the development and research of underwater vehicle propulsion.It also serves as inspiration to identify and synthesize the design and optimization methods according to the flow characteristics, pump-jet properties, and requirements of the pump-jet.

Performance Evaluation
The basic principle of underwater propulsion is that the propulsor directly converts energy power into propulsion power, that is, doing work on the fluid to drive the movement of marine propulsion.Pump-jets are different from traditional propellers or waterjets with structure and working principle, such as scale effects and the radiated noise mechanism.Therefore, the performance prediction methods are also distinct so that researchers are currently try to find a more suitable method for pump-jets.Underwater vehicles adopt pump-jets as their main propulsion since they generally demand a propulsion system to meet certain requirements, such as reaching high efficiency and suppression of cavitation and noise.Therefore, the research on cavitation performance and the mechanism of cavitation thereby aims to suppress cavitation and improve the critical speed of pump-jets, while minimizing noise emissions, which is the most significant performance target of pump-jets, requires obtaining the characteristics and applicability of each research method as well as the commonly applied research model of the pump-jet sound field on the basis of flow-field evaluation.
Performance analysis, as a basis and indicator for design and optimization, is applicable f through two main methods: numerical simulation and experiment.Among them, numerical simulation methods are mainly divided into potential-flow method and viscous-flow method.Moreover, the experimental method is constituted by open-water performance test, the cavitation test, and the self-propulsion test, which are, respectively, applied to measure the pump-jet open-water performance, predict the cavitation performance, and verify whether the pump-jet matches the hull.In order to visually present the results of hydrodynamic performance and to better analyze the pump-jet, dimensionless processing can be conducted on the relevant physical parameters, as shown in Table 1.
Table 1.Dimensionless expression of related physical parameters.

Investigation Techniques
Investigation of pump-jet hydrodynamic performance is capable of aiding researchers to comprehend the operational properties and evaluate the procedure of design as well.As stated above, the experimental method and numerical simulation method are relatively commonly carried out for hydrodynamic performance in the published literature.As regards the experimental types of research in pump-jets, the vast majority only carried out open-water performance experiments and verified the feasibility of numerical simulation by comparing with simulation results [15][16][17][18][19][20][21][22][23][24][25][26] without the comparatively complete experiment procedure, such as a cavitation study in a cavitation tunnel, which was concluded by very few authors [27][28][29][30].However, the experiment procedures described are limited by the reason that most experiments are influenced by the Reynolds number and scale effects, and the cost of the pump-jet experiment is too high and relatively complex.In order to compensate for the lack of experimental investigation, most of the researches chose numerical simulation methods for pump-jet performance, which are composed of potential-flow theory and viscous-flow theory.The controversy about the pump-jet design principle is roughly divided into two ideas that are similar to axial-flow pumps or marine propellers.Meanwhile, both the potential-flow theory and viscous-flow theory can be utilized for performance prediction as the basis of design [9,10].Thus far, the potential-flow method, which is widely applied to design and performance forecasting of propulsion, includes the lifting-line method, lifting-surface method, and panel method.Moreover, CFD (computational fluid dynamics) method is the most commonly used and burgeoning viscous-flow method, which differs from the potential-flow method in fundamental logic.Among them, fast calculation speed and effective calculation results can be achieved by the potential-flow method.However, it has a negative impact on the accuracy of the numerical simulation to a certain extent, which considers the viscous effect through an empirical formula.On the other hand, viscous flow has strong applicability, and its accuracy of calculation results are also recognized, yet the calculation speed is limited by the computer, with a certain CFD uncertainty as well.Meanwhile, the calculation results of the CFD method are capable of being applied as the verification for potential-flow calculation results.In addition, other methods have also been applied to the pump-jet performance prediction, such as that by Dong et al. [31].They combined the water-jet propulsion theory with the existing experimental data of the propulsor pump, which makes the correction coefficient, to obtain the pump-jet performance prediction method.However, the potential-flow theory and the viscous-flow theory are still the major prediction methods on account of the more accurate results and convenience.

Propulsion Performance
The pump-jet propulsion performance is generally expressed by performance curves, which mainly include thrust coefficient, torque coefficient, and efficiency.Compared with propellers and ducted propellers, the range of the advance coefficient of the pump-jet can be adjusted in a wider range according to the design requirements.Hence, in terms of propulsion properties, the high-efficiency zone of pump-jet is wider whether it is applied to high-speed sailing or low-speed low-noise propulsion.As the most intuitive means to obtain the pumpjet performance, there is not much published experimental research on pump-jets, so the review of the experimental methods can also refer to the ducted propeller.By now, the propulsion performances of different types of underwater vehicles equipped with pump-jets or ducted propellers have been studied experimentally by scholars [17,18,27,[32][33][34][35][36].The early experiment procedure was introduced by McCokmick et al. [9] using a pump-jet designed and tested at the Garfield Thomas Water Tunnel, whose experimental data agreed satisfactorily with the predicted performance from the point of view of efficiency.Then, Suryanarayana et al. [27,28] carried out experimental evaluation of a full-scale pump-jet with rear stator equipped on the torpedo to obtain the performance properties in a wind tunnel and cavitation tunnel, while Shirazi et al. [34] planned several experimental tests in a towing tank, such as bare-hull resistance test, bollard pull-condition test, and SPP (self-propulsion point) test.Both of these researches investigated the wake analysis and flow characteristics of their own designed pump-jet, proving that their design makes the pump-jet operate consistently and that the components are well-matched.As shown in Figure 2, the propulsion performance test of the pump-jet with the tests of forces acting on both the rotor and the stator as well as the shroud was carried out in the cavitation tunnel at Shanghai Jiao Tong University to verify feasibility of the numerical simulation [17].With the same water tunnel, Shi et al. [30] carried out further experiments on the pump-jet characteristics of acoustic radiation and flow-induced vibration.They established and verified the numerical methodology based on refined CFD, coupled FEM, and BEM to achieve the flow-induced vibration and acoustic radiation by comparing with experimental results in the water tunnel.Najafi et al. [37] carried out an investigation on the effects of three different pre-swirl ducts compared to the case with no pre-swirl duct on the propulsion performance via self-propulsion experimental tests.It was concluded that a higher propulsive efficiency at a high Froude number can be achieved by using the preswirl ducts.As far as numerical simulation of pump-jets is concerned, both the potential-flow method and viscous-flow method have been widely utilized for investigation.A comparison of results was shown by Wang et al. [38][39][40], who predicted the hydrodynamic performance of a pre-swirl pump-jet through both the CFD method and the potential-flow method.Meanwhile, a tip-leakage vortex model was established by the potential-flow method for the gap-leakage vortex simulation of the pump-jet, which successfully improved the prediction accuracy of the hydrodynamic performance of the pump-jet.Then, a gap-flow model was proposed based on the existing leakage vortex model, which concentrates on the impact of the tip-leakage vortex model on the viscosity of the gap flow.The comparison of the simulation results is illustrated in Figure 3; it was found that the CFD method has higher accuracy and reliability than the potential-flow method in the pump-jet prediction, while the introduction of the tip vortex model and the gap-flow model improves the prediction accuracy of the hydrodynamic performance.The results are also consistent with the conclusion obtained by the CFD method based on the viscousflow method.
Compared with the propeller, the range of the advance coefficients can be adjusted to a wide range of design requirements of underwater vehicles whether referring to a preswirl pump-jet or post-swirl pump-jet.Figure 4 [19] concludes the hydrodynamic performance curves of the pre-swirl pump-jet and the post-swirl pump-jet.The hydrodynamic performance curves of both types of pump-jets are generally consistent, indicating that the thrust coefficient of the rotor system is significantly greater than that of the stator system, while the thrust of both the rotor and the stator system have a relatively good linear relationship with the advance coefficient J. From the overall viewpoint of the pump-jet performance curves, the increasing advance coefficient causes the reduction of propulsion efficiency after a significant increment in the company of the linear decline in the thrust coefficient and torque coefficient of the rotor and stator system.The decreasing trend of the stator thrust coefficient is less pronounced than that of the rotor.The value of the stator thrust coefficient is negative at a certain advance coefficient, while the negative values indicate that resistance is developed rather than thrust.In addition, there are significant differences between the pre-swirl pump-jet and post-swirl pump-jet that show, compared with the pre-swirl pump-jet, that the post-swirl pump-jet has better maximum efficiency, while the efficient zone is more distributed in high J working conditions.The rotor of the pre-swirl pump-jet contributes nearly overall to the thrust of the propulsion in the calculated J range.The more rapid variation occurs in the performance curves of the post-swirl pump-jet, which is also the most obvious difference between the hydrodynamic performance of the two types of pump-jet [19].In addition, the post-swirl pump-jet requires a higher balance of torque and efficiency, with the best torque balance performance among traditional propulsions.The regularities are also the reasons that the post-swirl pump-jet is mainly adopted as the propulsion of small and high-speed underwater vehicles, while the low-speed and heavy-duty underwater vehicles are generally equipped with the pre-swirl pump-jet.
When the investigation is carried out on the pump-jet equipped on the large-scale underwater vehicle, there is a significant hydrodynamic scale impact caused by the variety of flow fields.Li Han et al. [18], Jun Yang et al. [41], and Mingyu Sun et al. [42], respectively, carried out numerical simulations of the open-water performance of the pumpjet in model scale and full scale and evaluated the scale effect of each component with hydrodynamic differences.The results compared the performance diversities of each component between the full scale (λ = 1) and model scale (λ = 20), as shown in Figures 5 and 6.It is indicated from the figures that compared with model scale, KTd, KTs, and KQ decline in full scale with the increase of KTr, while the variation in full scale is insusceptible to the change in advance coefficients.From Figure 5, in terms of the rotor system, the more prominent scale effects occur in high advance coefficients, and the larger relative value of the thrust coefficient KTr is shown in full scale at all operating condition.There is a significant impact on the torque coefficient KQr of the rotor by the loading condition of the rotor.The scale effects on its KQr are influenced by the advance coefficient.For the duct and stator, the decrement is exhibited in thrust coefficients (KTd and KTs) in full scale as well as presented in the torque coefficient of the stator.Furthermore, it is found in Figure 6 that compared with the pump-jet in model scale, the high-pressure region on the pressure side of the rotor in the full scale narrows, with a larger low-pressure zone on the suction side of the rotor.The velocity of flow through the stator is higher due to the higher suction of the rotor in full scale, which leads to a significant decrease in pressure distribution on the stator suction side and, consequently, to the reduction of the stator thrust.For the fullscale pump-jet, the non-uniformity of the stator wake falls obviously, leading to a lower fluctuation of rotor thrust and a weaker flow separation on the outside of the pump-jet outlet.Additionally, the tip-clearance leakage flow in the advance direction of the pumpjet is remarkably restrained through the higher-velocity mainstream and the shrinkage of the duct.

Cavitation
The distinct structure of the pump-jet, the components of which interact with each other, results in the complexity of the flow field.Although the pump-jet takes a dominant position in high performance and low noise, the development of the underwater vehicle in the direction of large scale and high speed contributes to the upward probability of pump-jet cavitation erosion.Cavitation is an abnormality of the fluid flow leading to appreciable effects that not only include the reduction of propulsion performance and the vibration of the underwater vehicle but also the production of cavitation noise, which may expose itself [43].In particular, the interaction of the cavitation bubbles with the tip vortex will be more complicated, which may deteriorate the consistency of the flow field.Hence, it is significant to understand the mechanism of production of pump-jet cavitation and the suppression of cavitation.
At present, the investigation of cavitating flow can be carried out by the experimental and numerical method.Experimental investigation has superiority in the accurate reflection of the flow-field features, while it requires a copious cost of manpower and financial resource, with unworkable experiments under complex conditions.Therefore, very limited published literature covering experimental investigations of pump-jets is available for reference, as it is mainly combined with the aspect of propeller and hydraulic machinery.Suryanarayana et al. [27][28][29] carried out the experiments of pump-jet equipped on an axi-symmetric underwater body in the wind tunnel and the cavitation tunnel, in which propulsion properties and cavitation performance were studied.The results indicate that the cavitation inception of the rotor appears on the tip surface and develops towards the suction side of the rotor with the growth of rotation speed.Furthermore, the cavitation expands fully on the suction side of the rotor blade when the pump-jet operates at a very low cavitation number.Ebrahimi et al. [44] experimented on a five-bladed B-series in the Sharif University of Technology cavitation tunnel, which can be a reference for the investigation of cavitation experiments.Comprehensive cavitation tests were conducted to investigate inception points and the development of cavitation during possible advance ratios with the tunnel limitations in rpm and inlet velocity, as shown in Figure 7.Meanwhile, the impacts of the cavitation number and ambient pressure on the sound-pressure level of the propeller were ascertained.The results can also be utilized to find the flow parameters' effects on the hydrodynamic and acoustic characteristics.From another point of view, numerical simulation plays an increasingly significant role in the prediction of cavitation, with the advantages of unlimited conditions and lower cost.Shi et al. [45] investigated the influence of rate of rotation, cavitation number, and inflow velocity on the pump-jet cavitation performance with the mixture homogeneousflow cavitation mode and found that the cavitation phenomenon becomes more obvious with the increment of the rate of rotation under the same cavitation number or the decline of the cavitation number under the same rotation velocity.Pan et al. [46] carried out cavitation performance research on pump-jets with the Schnerr-Sauer cavitation model, which was improved by introducing non-condensable gas (NCG).The results indicate that cavitation appears in the leading edge of the rotor blade tip and gradually extends to the exit edge and suction side of the stator blade with the decrease of the advance ratio.
The existence of tip-clearance cavitation results in the potential negative consequences of noise and vibration.In addition, navigation safety is the increasingly prominent issue, so some investigations focused on the cavitation performance of the pump-jet under special operation conditions, such as the oblique flow.Sun et al. [47] compared the cavitation performance of pre-swirl and post-swirl pump-jets in the oblique flow, while Qiu et al. [48] investigated the cavitation performance of a pump-jet with different tip clearances under the oblique condition.Until now, inadequate attention has been paid to the numerical simulation of the pumpjet cavitation performance, which focused on the vortex characteristics or wake instabilities of the pump-jet.Therefore, this section will take propulsion, which is similar to pump-jets, as a reference, such as in marine propellers, while other literature about the vortex properties and wake instability will be reviewed in the next subsection [45,46].Experiments and numerical investigations of two low-noise propellers, which were highly skewed with the modification of tip geometry, were carried out by Abolfazl et al. [49].It was found that the predictions of the cavitation inception were matched successfully with the trend of experimental data by different types of cavitation patterns in detail, including the wetted flow analysis, Eulerian cavitation simulation, and bubble dynamics model, with emphasis on the contributing vortical structures.On the other hand, Zhao et al. [50] proposed a novel approach to study the cavitation performance of a propeller under uniform and non-uniform flows, which was the adoption of interPhaseChangeDyMFoam in the OpenFOAM with Schnerr-Sauer cavitation model.The function input utilized in the research simulated the unsteady cavitation of a propeller with fewer calculations and higher calculation efficiency.In addition, Yaw-Huei et al. [51] investigated the impact of the tunable parameters of cavitation models on the prediction of cavitation performance.The comparison results of experimental cavitation occurrence probability (COP) and numerical vapor volume fraction (VVF) were found by simulating different cavitation models at default values, including the full cavitation model (FCM), Zwart-Gerber-Belamri (Z-G-B), and Schnerr-Sauer (S-S) models.As shown in Figure 8, it is indicated that the most accurate cavitation predictions were yielded from the FCM model at default setting, which is beneficial to the further investigation of cavitation performance.For pump-jet cavitation, the tip-clearance cavitation plays an important role, which has been the focus of more and more research groups, so the review mainly analyzes the cavitation and its effects on tip-clearance vortices in the next subsection.

Vortex
Vortices, which are typical flow structures in the wake, interact with the complex structure of the pump-jet, leading to instabilities of the flow field, such as vibration, radiated noise, and cavitation erosion.As a sophisticated propulsion with multiple components that create internal and external flow fields, the pump-jet has both great commonalities and differences with other marine propulsors.For the different types of pump-jets, there are partial variations that are attributed to the discrepancies.As for the post-swirl pump-jet, it directly works on the inflow without the disturbance of the wake developed by the stator.Meanwhile, the rotor tip flow field and the wake of the rotor interact with the duct and stator, and the impact on the flow field in the area of the duct guide edge is caused by the strong suction of the rotor.For the pre-swirl pump-jet, the rotor tip of it is under the influence of the boundary layer on the inner wall of the duct, operating in the stator wake.Despite the fact that the wake of the rotor does not suffer from significant interference by the stator, the interaction with the duct substantially contributes to the rotor forces and the development of tip-clearance vortex in the wake.The main vortex system of pump-jet propulsion is concentrated in the rotor system, in the interaction between rotor and the stator, as well as the wake of the pump-jet.In particular, the vortex system of the rotor is the most complicated of the internal flow field of the pump-jet regardless of whether the object is the pre-swirl pump-jet or post-swirl pump-jet [16].In addition, a strong vortex emanates in the wake of the pump-jet.These vortices interact with each other, bringing about the flow instabilities, while there is not yet a proper description, which is under investigation by many researchers.
The vortex system of the rotor is mainly divided into the tip vortices, the hub vortex, the trailing edge shedding vortex, and the blade root horseshoe vortex [16], of which the tip-clearance flow is the most elaborate in the whole flow field.The tip vortices are generated by the difference of pressure between the suction side and the pressure side of the rotor blade, with the inflow passing through the clearance between the blade tip and duct.The tip vortices are mainly composed of the tip-leakage vortex (TLV), the tip-separation vortex (TSV), and the induced vortices (IVs), etc. [43,[52][53][54].Among them, the most common forms are TLV and TSV in pump-jet propulsion, as displayed in Figure 9.When cavitation occurs in the pump-jet, the TLV intensity in the downstream non-cavitation and the TLV range will grow with the decrease of the TSV [55], which not only results in the degradation of pump-jet performance but also noticeably intensifies vibration, radiated noise, and cavitation erosion as well as seriously compromise the safety of the propulsion systems and the underwater vehicle [56][57][58][59][60].For the investigation of pump-jet vortical structure and the complex vortices' evolution under special conditions, Li et al. [61] found that a complicated vortical system occurs in the pre-swirl pump-jet.In oblique flow, the duct separating vortices makes the interaction between the duct shed vortices and tipclearance leakage vortices more intense.The stator vortices play a significant role in the downstream interaction with wake vortices of the rotor and hub.Owing to the existence of the considerable separating vortices, the interaction with the wake vortices of the rotor and stator domain enhances with the destabilization process in oblique flow.
Moreover, there is insufficient literature on the characteristics of pump-jet tip-clearance flow and vortices, which are not adequately understood, with little available research, which limits the further improvement of pump-jet performances.Li et al. [62] concentrated on the transient flow field and tip-clearance effect, and they found a remarkable amplification of the notable difference in stator and duct hydrodynamics with the increase of tip clearance.Meanwhile, the leaking flow brings about two typical vortices in the tipclearance region and its downstream.These vortices are composed of tip-sweeping vortex and tip-leakage vortex, which, respectively, exist in the tip region and the stator domain.
Through the cutting effect of the stator, the tip-leakage vortex extends to the wake of pump-jet with discrete segments, while the vortices are enhanced through the amplifying of tip clearance [38].For attenuation and control of the tip-clearance flow, Ji et al. [63] and Zhang et al. [64], respectively, carried out study by thickening and raking the tips of rotor blades and by applying groove structure.Through raising the rotor section thickness and raking the blades from 95% tip radius to the tip, the TLV decreased, which resulted from the improvement of the negative pressure peaks.The reasons are that the TLV of the proposed tip-modification model takes form later and develops a higher core pressure than the unmodified pump-jet, with reduction of the main amplitudes of TLV core pressure fluctuations.However, as indicated in Figure 10, the TSV downstream of the leading edge is strengthened with the modified rotor, which can be alleviated by further improvement of the rake distribution near the tip [62].By means of groove structure, the weakening of the rotor tip vortex becomes more pronounced, while tip vortex weakening amplitude first increases and then the change trend of decreases as the distance from the trailing edge of the rotor increases [64][65][66][67].Unlike other researchers, Chao Wang et al. [38] took advantage of the effect of tip vortex for a supplement of the panel method to predict the hydrodynamic performance accurately.When considering the impact of tip vortex, the tip load and the overall annular distribution of the rotor blade is more exact, which is considerably closer to the measured data.The tip-clearance flow field described above reflects the interaction between the rotor and the duct, but the tip-clearance flow field also has the implication of the interaction between the rotor and the stator.For the pre-swirl pump-jet, the rotor operates in the wake of the stator, whose structure has a negative impact on the surface flow of the rotor.From the research of Li et al. [15], it was found that the stator blade generates the stator-trailing edge vortices and the stator surface-separating vortices, which move into the rotor region before interacting with the rotor-trailing edge vortices, as shown in Figure 11.Moreover, the stator blade root vortices participate in significant interactions with the rotor blade root vortices, which are considerably affected by upstream vortices from the stator, as shown in Figure 12.The vorticity exchange depends on the relative intensity between the two root vortices.The destabilization of the hub vortices is significantly affected by upstream vortices, particularly the rotor blade root vortices.Hence, in the case of low advance coefficients, the hub vortices become unstable as soon as they are generated [15].On the other hand, in terms of the post-swirl pump-jet, Han Li et al. [62] investigated the post-swirl pump-jet for an unsteady flow field, the tip-clearance flow effect, and the evolution of tip vortices.As shown in Figure 11, they concluded that the stator blade load is significantly influenced by the rotor blade wake and tip-clearance flow owing to the periodic rotor blade wake and tip-clearance flow propagating to the stator system caused by rotor revolution.Meanwhile, the stator has a slight effect on tip-clearance flow while obtaining better recycling of rotor blade wake [62].The vortices within the wake and their interaction with the surrounding components determine the propulsion performance, cavitation erosion, and noise performance to a large extent [65][66][67][68][69]. Therefore, understanding the wake flow and wake instability mechanisms is significant for pump-jets, while the wake instabilities of pump-jets or propellers have been investigated through theoretical [15,16,19,[70][71][72][73][74][75] and experimental approaches.The results of the investigation of the post-swirl pump-jet carried out by Han Li et al. [15,62] indicated that the stator recycles the wake and tip-clearance flow from rotor, which may result in fluctuations in the flow field.When the vortex propagates through the stator domain, the intensity of the vortex changes significantly under the influence of low-velocity flow and duct shear layer.It is implicated from the vortex evolution and the distributions of pressure and velocity in Figure 13 that the change of wake energy and hydrodynamic efficiency is attributed to the decrease of the capability of doing work on the flow by the rotor as well as the weaker effect of the stator ability to recycle the tip-clearance flow.In addition, as shown in Figure 14, the tip-clearance vortices become unstable during their evolution passing through the stator, with significant short-wave instability.When the flow is out of the pump-jet, the tip-leakage vortex breaks up after the interaction with the shedding vortex of duct, generating secondary vortical structures, which further accelerate the break-up of the mainstream vortices [15].Denghui et al. [19] compared the wake vortices between the pre-swirl pump-jet and post-swirl pump-jet, and they found that the wake instability of both pump-jets is caused by the generation of concentrated rotor-trailing vortices by the roll-up of the rotor-trailing edge wake and interaction with rotor-tip vortices.Then, the stable wake region of the tip and hub vortices of the post-swirl pump-jet is smaller than that of the pre-swirl pump-jet, while the instability of the vortices of the post-swirl pump-jet is more likely to occur.For further research on the evolution mechanism of wake vortex structure, they also carried out the investigation of the effects of each component of the pre-swirl pump-jet on the wake dynamics.It was concluded that the shedding vortices from the outer and the inner surface of the duct, which compose the duct-induced vortex, are attracted and merged by the adjacent tip vortices after falling from the trailing edge of the duct.The strength of the duct-induced vortex also has a significant influence on the conversion of the tip vortex from stable to unstable.In addition, the short-wave instability mode of tip vortices is produced by a small, periodic fluctuation of the rotor profile resulting from the existence of the pre-swirl stator blade [69].

Noise Characteristics
The noise of the propulsor, as one of the main kinds of underwater radiated noise, is an important indicator of underwater vehicle performance.In the process of navigation, pump-jet noise consists of cavitation noise, low-frequency discrete noise, low-frequency continuous spectrum noise, and high-frequency continuous spectrum noise, of which low-frequency noise is the main contributor to the hydrodynamic noise of pump-jets [72].Accurate description of the sound source in flow is the primary problem in the simulation of flow-induced noise, and the main approaches of the extraction and calculation of sound sources are mainly composed of the Lighthill acoustic analogy [73], Kirchhoff method [74], and the Powell vortex sound theory [75].Numerical prediction of pump-jet flow noise faces many difficulties [76].The investigation requires consideration of how to unify linear and non-linear problems, as radiated noise and flow-field prediction are non-linear, non-stationary processes, whereas the process of noise propagation ignores the consideration of non-linearity.Moreover, the step of the unsteady simulation determines the resolution of the sound-field frequencies, which poses a challenge for simulation resources that require sufficiently small steps if broadband noise needs to be calculated [77].A simple fact currently is that due to the difficulties of prediction and the close relationship between low-noise performance and concealment of underwater vehicles as well as the military application field, there is little published research about the prediction of pumpjet radiated noise.In terms of the investigation of noise mechanisms, Huang et al. [78] demonstrated the significant impact of the characteristics of distributed unsteady hydrodynamic forces on the vibro-acoustic responses of a pump-jet.Xiaoxu et al. [79] carried out the acoustic noise simulation for the sound source distribution around the pump-jet at different speeds on the basis of finite element/infinite element technology and based on the Lighthill acoustic analogy theory.The results show that surrounding the pump-jet, the sound-pressure level is directivity fixed, where the sound-field leaf frequency directivity of the circumferential polar field is "o"-distributed, and the axial leaf frequency soundpressure level is "8"-distributed, as shown in Figure 15.In addition, as the monitoring distance increases, the sound-pressure level shows a decaying characteristic, with consistent decay in the inlet, outlet, and radial directions of the propeller.The distribution of the axial and circumferential leaf frequency sound-pressure level is consistent at different rotational speeds.In terms of the effects of the components on pump-jet noise, Su et al. [80] investigated the impacts of the duct on the sound radiation of the coupled pump-jetshaft-submarine system.They found that the main source of acoustic radiation at the range investigated was the hull except for near the shaft frequency.The duct plays an important role at blade-passing frequency in the acoustic radiation.Meanwhile, acoustic signatures in the hull domain are caused by vibration at the duct modes, which transforms from the stator to the hull.Moreover, the greatest contribution to the coupled system is by the pulsating forces generated by the rotor.With regard to noise-reduction technology, Huang et al. [81] eliminated the tip clearance for the reduction of underwater radiated noise of pump-jets through the SUBOFF model.The results were compared with a pump-jet with 2 mm tip clearance to conclude that this approach effectually suppresses the sound radiation power level.Especially in the lower frequency range, the sound radiation power shows a lower level owing to the great suppression of rotor contribution.In addition, Yu et al. [82] and Denghui et al. [83] installed the sawtooth duct on the post-swirl pump-jet, with the geometry of the pumpjet with a sawtooth duct as shown in Figure 16.The results indicate that with the set of the serrated trailing edge, the vortex intensity is greatly reduced, with a directional noisereduction effect and a reduced vortex intensity, with circumferential continuity weakening, which can reduce the radial force fluctuation and noise radiation of the duct.Meanwhile, the axial sound-pressure level of the low-frequency noise is reduced significantly.Therefore, the sawtooth duct, which has excellent noise reduction at low frequencies, is an effective approach to reduce radiated noise of pump-jets with minimal influence on the hydrodynamic performance.

Design Principle of Pump-Jet
The basic principle of underwater propulsion is the direct conversion of energy power into power of propulsion, which is on the basis of the momentum theorem.The procedure consists of the thrust developed by the momentum of the flow caused by the propulsion; namely, the axial thrust generated by the propulsion is considered to be equal to the momentum difference of the mass flow of the fluid, leading to the following parameters: where vs (m/s) represents the undisturbed navigation speed of propulsion, vout (m/s) is the outlet velocity of the pump-jet, Q (m 3 /s) is flow volume, PT (N*m) is the effective thrust power, and PE (N*m) is the denotes the effective power.When operating in an actual working environment, the pump-jet is retrofitted to the rear or side of the underwater vehicle, which is affected by the wake of the appendage prior to the boundary.Since the inflow velocity in the boundary layer is lower than the flow velocity outside the duct, this leads to a smaller velocity in the duct than the navigating velocity of the underwater vehicle, vs [84,85].Therefore, a correction for each influence factor in the boundary layer is required to be more accurate, considering the wake impact during navigation [86][87][88][89][90][91].It is assumed that the coefficient of the momentum effect of the boundary layer on the flow into the pump-jet is represented by α, while β is the kinetic energy impact coefficient with the jet velocity ratio μ = vout/vs, so by considering this fact, Equations ( 1)-( 3) can be modified to where Vr (m/s) indicates the velocity of the cross-section at different radial positions of the inlet, and Ai (m 2 ) is the area of the inlet cross-section.The larger the jet velocity ratio, the greater the derived momentum difference and with greater thrust.Meanwhile, for ensuring the overall propulsion efficiency of the pump-jet, the velocity ratio is supposed to be taken as an appropriate value, generally 1.2~1.5.Low radiated noise and high critical speed are distinctive performance properties of the pump-jet as well as a measurement of the pump-jet design level.As mentioned above, the pump-jet, a combined propulsion, is more difficult to design than traditional propulsion.The duct, rotor, and stator are closely associated with each other, with the complicated mechanism of interaction between the flow and sound fields of these three components [9].The difference from the traditional propeller is that the rotor and stator are located within the duct, which operate in the boundary layer flow of the prior appendages.Within limits, it is possible to control the velocity and static pressure at the rotor by suitable shaping of the shroud [9].Compared with propellers, the diameter of the pump-jet rotor would be designed larger to develop the same thrust at the same flow rate.The increase in diameter also increases the drag of the pump-jet, so the advantage of pump-jet cavitation performance is obtained at the expense of propulsor efficiency, with less wake kinetic energy and energy loss than with a propeller [3,86,87], as demonstrated in Figure 17, according to momentum theorem.The momentum theorem for pump-jets and propellers is described in that the axial thrust is equal to the difference in momentum of the mass flow, as in Equation (1).In terms of pump-jet structure, the rotor is typically located in the duct, which decelerates the flow through the rotor blades to delay cavitation erosion, improving pump-jet cavitation performance and critical speed.The stator, which has a commutating effect on the flow, is a group of stationary cascades at an angle to the inflow direction.It plays different roles in the pre-swirl pump-jet and post-swirl pump-jet.In the case of the former type, the stator provides pre-whirl for the rotor system to improve rotor inflow conditions.On the other hand, for the post-swirl pump-jet, the stator is installed to adjust the circumferential rotational wake of the rotor, which is beneficial to the hydrodynamic performance [88][89][90][91][92][93][94].Therefore, the parameters of the stator have a significant influence on the hydrodynamic performance and flow field whether for the pre-swirl pump-jet or postswirl pump-jet.Li et al. [22] and Negin et al. [89], respectively, investigated the influence of various stator parameters on pump-jet performance, and the results show that stator structure, stator angles, and rotor-stator spacing as well as the thickness and camber size of the stator cross-section have significant effects on the pump-jet performance.In addition, Denghui Qin et al. [69] carried out a study on the effect of the pre-swirl stator on the wake dynamics of a pre-swirl pump-jet, indicating that the impact of the pre-swirl stator on the rotor blade is strong due to the rotor-stator interference, which may develop undesirable and unstable vortices.
The existence of the duct in a pump-jet causes the flow through the whole assembly to be divided into the inner fluid and fluid outside the duct.With the presence of a certain pressure difference between the inner and outer surfaces of the duct profile, it generates the axial component forces.Proper design of the duct profile is very crucial in that an incorrect design can waste a considerable amount of global pump-jet thrust [86].The control of the airfoil shape and inlet angle allows the duct to be regulated to produce resistance or thrust or to have zero force so that the duct can be sorted into the accelerating duct and decelerating duct [3,90,91], the schematics for which can be referenced in Figure 18.The accelerating duct, which is originally equipped in cases where the propeller is heavily loaded, extensively develops a positive thrust, thus offering the opportunity to increase the propulsion efficiency.Nevertheless, owing to the low pressure at the internal face of the duct, the weakening of anti-cavitation capability occurs [92].On the other hand, the other duct type, the decelerating duct, can be adopted as a retardation of the cavitation and a reduction of its side effects such as noise and vibrations.Decreasing the flow rate swallowed by the pump-jet gives rise to an increase in the static pressure at the propulsion inlet, resulting in the improvement of the pump-jet cavitation.However, in order to maintain the global thrust, the rotor load must be increased, and the fluid will exert an axial force on the duct, which is opposite to the propulsion [93].A comparison of the partly hydrodynamic performance of the accelerating duct and decelerating duct is shown in Figure 19.Moreover, Qiaogao Huang et al. [94] and Chao et al. [40] investigated the impact of duct parameters on pump-jet hydrodynamic performance, obtaining the results that variations of tip-clearance size, camber, and attack angle of duct as well as the length-diameter ratio and the expansion ratio of the duct outlet result in a significant impact on the pump-jet hydrodynamic performance.It is necessary to adopt suitable duct geometry parameters for a significant increase of the propulsion efficiency, expansion of the efficient operating region, and improvement in blade loading.Xinguo et al. [95] carried out investigations to analyze the influence of the selection of the duct with an NACA66-0.8profile installed at the SUBOFF stern, and the results implicated that as the ratio of the section inclination angle increased, the thrust-reduction fraction and momentum impact factor showed negative influence.Moreover, the increase of duct camber plays a different role on the accelerating duct and decelerating duct.The thrust-reduction fraction and momentum impact factor of a pump-jet with an accelerating duct are reduced with the reduction of blade loading, while the pattern is the opposite for the decelerating duct.The above descriptions are design principles and design guidelines for each important component, followed by an overview of the main design methods for the pumpjet blade, which is mainly composed of the direct method and the inverse design method, with both of the design procedures shown in Figure 20.The process shown in Figure 20a, the direct method, which is one of the traditional blade design methods, uses the known flow conditions to obtain the initial blades.The optimization for the final model on the basis of the hydrodynamic performance parameters is calculated by CFD method or experiment, combined with the design requirements for the blades.On the other hand, the inverse design method is primarily segmented into flow-field calculation and blade shape calculation.The results of flow-field calculation lead to the iterative improvement of the blade shape and the determination of the optimal impeller [96][97][98].In this way, the design procedure is capable of getting rid of the cumbersome process depending on traditional empirical parameters, repeating verification, and impeller optimization.The requirements of the pump-jet are to produce the highest possible thrust with the lowest possible power consumption, namely to achieve efficiency as high as possible while delaying cavitation generation and reducing radiated noise.When designing the pump-jet, it is necessary to take into account the internal three-dimensional flow characteristics, particularly the interaction between the thruster and the hull, that is, the thrust deduction and the wake of the prior appendage.Whether adopting propeller-based design methods or pump-based design methods, ultimately, the three-dimensional geometry of the pump-jet blade must be derived to meet the propulsion performance, cavitation performance, and acoustic performance.Among them, blade pressure loading is a necessary issue.Therefore, the challenge of the procedure of the rotating machinery blade design and inverse design method is the determination of the rotor and stator blade shape and the precision of the blade pressure load.

Direct Design Method
In terms of the direct design method, it is necessary to provide the relevant design indicators before the design of the pump-jet, including the essential performance parameters, such as inflow velocity of pump-jet, thrust, and so on.When the pump-jet is in operation, the inlet of the pump-jet is required to correspond to the resistance of prior appendage for the purpose of developing suitable thrust with a high efficiency of the pumpjet.Additionally, the other parameters are supposed to be matched with the performance result in reducing or delaying cavitation as well as confirming heavy thrust and high efficiency.The direct design method was first applied in research about ducted propellers with stator blades.The review provides an overview of the direct design approach, with two separate idea principles: one is the lifting design method based on the theory of axialflow pump design, and the other is circulation theory, which is based on the theory of propeller design.The former approach is on account of flow features around the airfoil, which combines the flow features with experimental data, leading to the competition of a reasonable correction method.The method belongs to a half theoretical and half empirical method, with a relatively simple calculation formula and a shorter calculation cycle [99][100][101][102][103][104].The circulation theory takes full account of a variety of factors such as the wakes from the prior appendages, cavitation erosion, and fluctuation, but the structure of propulsion designed by circulation theory is commonly more complicated and has been paid more attention, especially in the design of propulsions for ships, torpedoes, etc., where there are special requirements for noise and vibration.For the circumfluence theory design of propulsion blades, the main methods commonly include the lifting-line method, lifting-surface method, and the panel method.At present, as far as the theoretical design of propellers is concerned, the optimal circulation distribution on the blade surface is generally solved by the lifting-line method, and then, other methods are adopted to combine the requirements of the optimal circulation distribution for the profile design [105].

Lifting Design Method
The fluid movement in the rotor is a complex three-dimensional flow.For the movement, the fluid enters the duct along the axial position, then fully develops in the internal channel of the pump-jet, and at last, it is rectified by the stator and ejected through the tail of the duct.The thrust produced by the pump-jet propeller can be explained by Newton's third discipline.The lift method based on one-and two-element design theories was originally applied to the design of axial-flow pump blades, which is still widely used in the design of axial-flow pumps.The design of the rotor by the lift method is completely based on the assumption of the independence of the cylindrical layer.
Therefore, when designing a pump-jet propeller using this semi-empirical and semitheoretical method, the following assumptions need to be made: 1.
The flow in the rotor is regarded as potential flow, and there is no radial velocity component; 2.
The distribution of velocity loop is constant along the radius;

3.
There is no induced velocity in the axial direction.
The complex movement of fluid in the rotor domain mainly includes the movement relative to the rotor blades, circular movement accompanying the rotation of the rotor, and absolute movement relative to the duct, which are expressed as absolute velocity v, relative velocity W, and circumferential velocity u, respectively.The inlet and outlet velocity triangle is shown in Figure 21, where β is the inlet angle of single airfoil profile, and ∆α is the attack angle of single airfoil profile.Among them, the subscript 1 represents inlet rotor parameters, while the subscript 2 represents outlet rotor parameters.For instance, vu1 and vu2 are the circumferential components of the absolute velocity at the inlet and outlet of the rotor; u1 and u2 are circumferential velocity of the rotor inlet and outlet.Based on the assumption of the independence of the cylindrical layer, the fluid at the inlet on the same cylindrical section has no circumferential pre-swirl; that is, vu1 = 0.Then, the velocity circulation of the blade is: where t is the cascade pitch calculated by = ; Z is the number of blades.It can be seen that any fluid particle in the rotor domain can be analyzed by Figures 21 and 22.Then, the basic equation can be derived from the momentum theorem as follows: where the theoretical head of the pump is Ht (m), the fluid gravity acceleration is g (m/s 2 ), and ω (rad/s) is the angular velocity of rotor rotation.The schematic diagram of the force of the rotor airfoil section in the viscous fluid is shown in Figure 22.In the figure, F (N) is the resultant force, Fl (N) is the lift, and Fd (N) is the resistance, which are given by where W∞ (m/s) is the average of the relative velocities, Cy is the lift coefficient, CD is the drag coefficient, A is maximum projected area.Then, the circumferential component of the resultant force is given by where λ (°) is the angle between the lift and the resultant force, namely the airfoil lift angle.(°) represents the inflow angle of the incoming flow at infinity.Then, the lift coefficient can be introduced for definition by The input power of the blade airfoil profile on the cylindrical layer at the radius r is , and the output power is ; then, the hydraulic efficiency is given by = where H (m) is the true head, vm (m/s) is the axial velocity, and hydraulic efficiency can also be expressed as ƞh = H/Ht.Thus, it can be obtained from Formulas ( 14) and ( 12) as Then, from Formulas ( 12), (13), and (15), we can obtain Formula ( 16) is the basic equation for the lifting design method, where l represents the chord length of the airfoil, and ⁄ represents the cascade density, which reflects the effective cascade channel area.The basic equation of the lifting design method expresses the relationship between the geometry, size, and flow characteristics of the designed blade section.In the calculation, it is assumed lift angle = 1 °, and we select ⁄ to calculate the lift coefficient Cy and attack angle, and the result can be finally achieved by iteratively calculating.The related design process for the lifting method is shown in Figure 23.

Lifting-Line Method
With reference to the lifting-line theory used in propeller design, the pump-jet rotor is designed as a wing with a finite span.The rotor blade is regarded as a lifting line, namely an attached vortex line.The circulation of rotor changes in the radial direction, and the lift is also subsequently replaced by the concentrated vortex, and the radial and chordal distributed load can be calculated later.Then, the hydrodynamic performances of the two-dimensional airfoil in each section are superimposed to predict the performance of the propeller rotor.This theory is based on the following assumptions: (1) The fluid is regarded as ideally incompressible; (2) The inflow is treated as steady and axisymmetric; (3) The rotor wake is regarded as non-contracting, and its influence on the shape of the vortex is not considered; (4) The radial induced speed is not considered; (5) It is assumed that the circulation at the hub diameter is 0, but for the rotor with a larger hub, a later correction is required.
According to the principle, the lifting-line method is to use lifting lines to replace rotor blades.At the same time, the circulation and lift coefficient correspond to each other at the airfoil profile positions of different cylindrical radii.When using this theory, the rotor blades are regarded as having no axial deformation.Figure 24 shows the force triangles and velocity triangles on the control points on the lift line.In the figure, there are axial and circumferential induced velocities Va and Vt; the axial and circumferential components of the inlet flow at the rotor are Va and Vt, respectively; the resultant force F is divided into lift and drag; the pitch angle of the airfoil profile is the placement angle of the profile with * (°) means; the angle of attack is α (°); β (°) and βi (°) are the airfoil inlet angle and induced velocity angle, respectively.Assuming that the circulation and the resultant speed v* are known, the rotor thrust and torque can be expressed in the following integral form: where variable c (m) is the chord length of the airfoil profile; CD is used to express the drag coefficient of the airfoil profile.In the lifting-line theory, the axial and circumferential components of the induced velocity can be obtained by integration.Because the vortex line replaces the blade, and the free vortex is released by the blade, the induced factor can also be used to represent the vortex line.At is the dimensionless radius x of the lifting line, and the entire free vortex is obtained as follows: where Vin (m/s) represents the rotor inlet flow velocity, and G is the dimensionless circulation; ia and it are the axial and tangential induction factors, respectively.Moreover, x and x0 are the dimensionless radial coordinates, and xh is the dimensionless radius at the hub of the rotor.The dimensionless circulation is calculated as follows: The integral form of the equation for solving the ring quantity G can be written as The approximate solution equation of the discretized circulation G can be written as The dimensionless circulation G can be rewritten in the series form as follows: The approximate solution equation of the discretized dimensionless circulation G can be written as ℎ ( ) + ℎ ( ) = (1 − ) * (25) According to the above equations, the lift-length product coefficient (the product of the lift coefficient and the chord length) of the pump-jet rotor in a uniform flow field can be calculated according to the following basic equations:

Inverse Design Method
The inverse design method, which is different from the direct design method, is required to determine the initial geometry of the blade and part of the flow parameter distribution and to solve the final blade according to the circulation theory by means of inverse design algorithm.Compared with direct design method, the design method is more theoretical without relying on traditional design experience, while it predominates at aspects of shorter design cycles and clearer objectives, as shown in Figure 25.Generally, the inverse design method is comprised of two categories: one is the concept of single inverse design, and the other idea is the mutual iterative calculation of the direct and inverse approach.However, multiple limitations still exit in the first type, whose research is insufficient.The concept of the other category is to take the blades that are designed by the direct method as the known condition, with the utilization of the evaluation of simulation results as the basis of the inverse design segment [96].Zhang Mingyu [99,100] carried out the pump-jet blade design by adaptation of the second category of three-dimensional inverse design method, and while conducting the performance comparison between the pumpjet propeller and the propeller, the study found predomination at aspects of high-efficiency and high-speed adaptability.Jin et al. [101] took the pump-jet installed on the torpedo as the object; in the investigation, they explored the theory of three-dimensional inverse method with overall selection and shape of pump-jet propulsion by combining theoretical evaluation with experimental and numerical results, which lead to the verification of the accuracy of the pump-jet design method.Blade shape calculation and flow-field calculation roughly make up the inverse design method, while it is essential to make corresponding restrictions and assumptions to characterize the complicated internal flow field in the pump-jet before designing.In the design procedure of inverse design method, It is assumed that the flow fluid in the pump-jet operation environment is steady and incompressible and without viscous flow as well as that the inflow, which is in the rotor, is non-rotating.
When designing pump-jet propeller blades, the thickness of the blades is ignored, and the center surface of the blade is replaced by a vortex so that the effect of the blade on the water is replaced by the vortex on the center surface.The strength of the vortex is controlled by the circumferential average circulation 2πrVθ [102]: where Z represents the number of blades; Vθ (m/s) represents the circumferential speed; r (m) represents the radius or radial distance.
In the above process, the average flow field and periodic flow field of the blade area and the non-blade area are established.Based on the previous assumptions, on the blade surface where the thickness is ignored, the relative flow velocity Wb is tangent to the blade surface and orthogonal to the blade normal vector ∇ ; namely, • ∇ = 0. Expanding the relative speed, we can express this relationship as where f (°) represents the blade wrap angle.Then, the relative velocity direction of the suction surface and pressure surface of the blade is perpendicular to their normal direction and perpendicular to the direction of curl, so the relative velocity increment can be expressed as Formula ( 28) can be simplified by applying the Bernoulli equation along the streamline, and then: where = .Formulas ( 28)-( 30) can control the geometry of the blade.Finally, by combining all the equations in this part, the solution equations for the threedimensional inverse design method can be obtained.After obtaining the conditions of circulation, blade thickness, number of blades, and boundary conditions, a complete blade design can be realized.The load of the rotor and stator blades is shown in Figure 26, where m represents the position of blade airfoil chord length, indicating that when m is 0, it represents the blade inlet edge, and when m is 1, it represents the blade outlet edge.

Optimization
In realistic design procedure, due to the existence of empirical parameters in the design process, numerical simulations need to be carried out after the preliminary design of the pump-jet.Then, the selected structural parameters are accordingly optimized for the final pump-jet structure.However, there is scant literature available on the optimization of pump-jets, while the optimal design of similar equipment is capable of providing the corresponding theoretical basis.With the development of various modelling and calculation techniques, the utilization of intelligent optimization method has gradually become an important step in structural design in recent years as well as a main technique for improving performance [104].For the optimization of research objects, the variables optimized are generally carried out by using the main structural parameters, with the selection of the efficiency and thrust as the often-chosen optimization objectives.Research in related fields is mainly focused on the optimization of equipment such as conventional propellers and axial-flow pumps.In terms of the pump-jet, Section 3.1 above mentioned structural parameters of the stator and duct that affect the performance of a pump-jet.Apart from this, the structural parameters of the rotor blades of the pump-jet and matching between the components have a more significant impact on the hydrodynamic performance of the pump-jet.Moreover, Chunxiao et al. [105] adopted the SOBOL (Sobol quasirandom sequence) algorithm and the NSGA-II algorithm combined with the geometry reconstruction technology and numerical technique to realize the automated design process of pump-jets.They aimed at presenting the geometric properties of pump-jets more simply by establishing the fully parameterized mode, resulting in the establishment of the relationship between the geometric parameters and hydrodynamic performance.Nowadays, the main approaches of optimization that are applied to the pump-jet can refer to marine propulsion and axial-flow pumps, such as ant colony optimization, genetic algorithm, PSO (particle swarm optimization) and neural network, as well as machine learning and so on.For the optimal design of marine propellers, the automated design optimization has been adopted for propellers, which have evolved by an intermediate step integrating human-computer interaction.In order to achieve the optimization objectives, an interactive optimization methodology for propellers, based on interactive genetic algorithms (IGAs), was systematically developed [106].Nowrouz et al. [107] investigated the optimization of marine contra-rotating propellers by adopting the Kriging algorithm coupled with the genetic optimization tool, with which the added points reduced the number of simulations with the increase of the convergence speed.In this process, namely the optimization of a contra-rotating propeller set through finding the best radial distributions of pitch and camber ratios for both propellers and adopting random distributed initial data, the solution converged after fewer iterations, as shown in Figure 27.For the optimization of axial-flow pumps, the investigation carried out by Mohamed et al. [108][109][110] showed the application of a multi-objective cuckoo search method coupled with the inverse method for the axial-flow pump, where maximization of the total efficiency and minimization of the required net positive suction head of the pump are considered as the two objective functions, while a comparative analysis was carried out with the results obtained from a reference machine and genetic algorithm to verify the validation of the approach.

Conclusions and Challenge
This review presents an overview of the latest advances in the pump-jet involving mechanism, performance, and design approaches to the pump-jet.The specific performance analysis of pump-jets and details of complex flow fields and performance were also discussed.This paper, in a comprehensive and specialized way, reviewed the pumpjet hydrodynamic performance, noise performance, and flow-field characteristics involving cavitation erosion and vortices properties of tip-clearance; the interaction between the rotor and the stator and the wake field; as well as the optimal design of pump-jets.The main conclusions and challenges are as follows: Performance evaluation, as a basis for design, can be achieved through two main approaches: the numerical and experimental methods.The experimental method is constituted by the open-water performance test, the cavitation test, and the self-propulsion test, which are, respectively, applied to measure the pump-jet open-water performance, predict the cavitation performance, and verify whether the pump-jet matches the hull.Potential-flow method and viscous-flow method are the most used for numerical prediction of performance.The CFD method based on the viscous-flow method has higher accuracy and reliability than the potential-flow method in pump-jet prediction.Experiments and simulations have been adopted by many researchers to verify that the pump-jet achieves better results in hydrodynamic performance.In terms of the application of pump-jets, the post-swirl pump-jet is mainly adopted in the propulsion of small and highspeed underwater vehicles, while the low-speed and heavy-duty underwater vehicles are generally equipped with the pre-swirl pump-jet.Moreover, scale effects are significant in precise prediction of pump-jet performance, which has different influences on the different components.The variation by scale effects is complicated for many reasons, such as the change of boundary layer thickness on the blade surface and change in the duct loading.The specific reasons for this are subject to further study.Furthermore, the pump-jet has a larger axial dimension and stronger coupling with the prior appendage, leading to the a significant influence on a pump-jet operating under a complex environment, such as the oblique flow, compared with other propulsions.
Cavitation is an abnormality of the flow leading to appreciable effects that not only result in the reduction of propulsion performance and the vibration of underwater vehicle but also the production of cavitation noise.The existence of tip-clearance cavitation results have potential negative consequences such as noise and vibration.Meanwhile, the interaction of the cavitation bubbles with the tip vortex are more complicated, which may deteriorate the consistently of flow field and is influenced by many factors.As the focus of cavitation research, vortices interact with the complex structure of the pump-jet, leading to the instabilities of flow field, such as vibration, radiated noise, and cavitation erosion.In addition, a strong vortex emanates in the wake of the pump-jet.These vortices interact with each other, bringing about flow instabilities.Under special conditions such as oblique flow, the interaction with the wake vortices of the rotor and stator is enhanced with the destabilization process, owing to the existence of the considerable separating vortices.For the pre-swirl pump-jet, the rotor operates in the wake of the stator, whose structure has a negative impact on the surface flow of the rotor.For the post-swirl pump-jet, the stator has a slight effect on tip-clearance flow while achieving better recycling of the rotor wake.The within the wake and their interaction with the surrounding components determine the propulsion performance, cavitation erosion, and noise performance.
Additionally, the high-accuracy prediction of the tip-clearance flow is a significant challenge for the viscous-flow method in terms of not only mesh generation for complex structures but also the interaction between the mesh and accuracy of turbulence simulation, particularly in cavitation and noise calculations.In a realistic operation environment, the wake of a pump-jet can propagate over very large distances, while the investigation of the mid-and far-field of the pump-jet wake relies on the mesh accuracy and turbulence dissipation in CFD technology, so the existing discussion of the wake field is almost in the near-medium field, where the flow field is more obviously characterized.
In the process of navigation, pump-jet noise consists of cavitation noise, low-frequency discrete noise, low-frequency continuous spectrum noise, and high-frequency continuous spectrum noise, of which low-frequency noise is the main contributor to the hydrodynamic noise.An accurate description of the sound source in flow is the primary problem in the simulation of flow-induced noise, and the main approaches of extraction and calculation of sound sources mainly include Lighthill acoustic analogy, Kirchhoff method, and the Powell vortex sound theory.Effective approaches are adopted to reduce the radiated noise of pump-jets with minimal influence on the hydrodynamic performance, such as eliminating the tip clearance and installing a sawtooth duct.Moreover, the investigation of noise performance requires consideration of how to unify linear and nonlinear problems, as radiated noise and flow-field prediction are non-linear, non-stationary processes, whereas the process of noise propagation ignores consideration of non-linearity.Lastly, the step of the unsteady simulation determines the resolution of the soundfield frequencies, which poses a challenge for simulation resources that require sufficiently small steps if broadband noise needs to be calculated.
The pump-jet design principle is roughly divided into two ideas: one is based on the propeller circumfluence theory, and the other one is achieved by the design theory of the axial-flow turbomachinery.The duct, rotor, and stator are closely associated with each other, with complicated mechanisms of interaction between each other.For the selection of design method, the inverse design method is adopted if the thrust and high efficiency are emphasized.Otherwise, if the stability and anti-cavitation performance are emphasized, the direct design method should be preferred.During an optimal design procedure, it is complicated to take the interaction between each component into consideration, while optimization is mainly carried out independently for one separate component, without taking into account the synergy between the components even under the wake of the appendage, so improvement of present design methods is a challenge.

Emerging Possibilities and Prospects
In this contribution, perspectives on the potential of pump-jets to advance performance and performance prediction have been provided, focusing on optimal design, matching technique, noise prediction, control of flow and vortices, and so on.Despite the fact that the investigation of pump-jets mainly focuses on the traditional hydrodynamics field, current research on conventional underwater propulsion has extended to the interdisciplinary integration of fluid dynamics, acoustics, materials, chemistry, and bionics.It is significant to aim for low noise and high efficiency, continuously improving the mobility and concealment of underwater vehicles.The further investigation and application of pump-jets will benefit from leading-edge achievements in multidisciplinary fields as well as break through the limitations of special propulsions to a wide range of applications in the marine and military field.
During the procedure of optimal design, the optimization is carried out on the individual and independent components without taking account of the synergy between the components even under the wake of the appendage.In addition, the optimal design of the pump-jet is generally to select the optimized algorithm and optimal objectives to optimize the initial model designed by the common design method, based on the flow-field and the performance parameters.Therefore, there are several emerging areas of intelligent approaches that are promising for optimal design as well as flow-field and performance prediction.For example, machine learning is the family of approaches that is becoming widely adopted for flow-field calculation and optimal design.Since machine learning has an excellent capacity for non-linear expression and generalization, it provides an improvement in prediction and optimization capability as well as the development of fluid turbulence theory in hydrodynamics.Qiu et al. [110] established a deep learning framework by convolutional neural networks (CNN) to predict the pump-jet velocity field by discrete pressure points obtained through a pressure sensor.It was verified that it maintains accuracy and rapidity in actual engineering or in experiments.Then, they established the new framework for a variational Bayesian convolutional neural network (VB-CNN) to rapidly predict the pump-jet wake velocity distribution.Compared with CNN method, the VB-CNN method has superior prediction accuracy and performance for the velocity distribution, such as velocity gradient and field contour.
Under the existing comprehensive background of a green-energy-saving environment, the application field of pump-jet propulsion technology has expanded to the field of civil underwater propulsion.Therefore, shaftless pump-jet propulsion technology will become one of the key technologies of future submarine propulsion in the military field.The essence of the shaftless pump-jet is the complete integration of the motor and pumpjet with the elimination of the propulsion shaft system.It distributes the motor in the duct, thus enabling the maximum elimination of the thruster-shaft-stern excitation noise source.Meanwhile, the locomotion of the motor significantly improves the space arrangement in the hull compartment owing to the application of an induction motor or a permanent magnet motor.From the view point of results of the practical research in the field of permanent magnet motors, it is important that permanent magnet motors will be the preference for shaftless pumping by the reason that the high efficiency and small size of the permanent magnet motor coincides with the external requirements.In spite of the key issues of high-efficiency, low-noise propulsion design and permanent magnet motor design, it is necessary to address the adaptation of the motor to the operation environment, i.e., pressure resistance, watertightness, corrosion resistance, and foreign body resistance.

Figure 2 .
Figure 2. Experiment research on pump-jets by the propeller dynamometer in a cavitation tunnel [17].

Figure 5 .
Figure 5.Comparison performance diversities between the full scale and the model scale.

Figure 6 .
Figure 6.Stress nephogram and vortex diversities between the full scale and the model scale.(a) The pressure distribution on each component of the pump-jet; (b) the velocity distribution in x = 0 plane.

Figure 8 .
Figure 8. Distribution contours of propeller blades: (a) experimental COP and numerical VVFs simulated with cavitation models of (b) FCM, (c) Z-G-B, and (d) S-S by default values for tunable parameters.

Figure 11 .
Figure 11.Coherent structures of vortices in the wakes of the stator, rotor, and hub [15].

Figure 12 .
Figure 12.Interaction between the blade root vortices of the stator and rotor at a series of instants.

Figure 14 .
Figure 14.Vortices in the bound of the mainstream [15].

Figure 15 .
Figure 15.Sound-field leaf frequency directivity of the circumferential and axial polar field.

Figure 19 .
Figure 19.Flow-field pressure contour of the accelerating duct (a) and decelerating duct (b).

Figure 20 .
Figure 20.The flowchart of pump-jet design methods: (a) direct method and (b) inverse method.

Figure 21 .
Figure 21.The velocity triangle of the rotor.

Figure 22 .
Figure 22.Velocity triangle of the rotor blade profile.

Figure 23 .
Figure 23.The flowchart of design by lifting design method.

Figure 24 .
Figure 24.The force and velocity triangle at the control point on the lifting line.

Figure 25 .
Figure 25.The flowchart of inverse design method.

Figure 26 .
Figure 26.The load of the rotor and stator blades (a) Load of the rotor, (b) Load of the stator blades [103].

Author
Contributions: Conceptualization, Y.Z. and J.Y.; investigation, Y.Z. and Y.F.; resources, Y.Z., J.Y. and G.P.; writing-original draft preparation, Y.Z. and Y.F.; writing-review and editing, Y.Z. and G.P.; visualization, Y.Z.; project administration, J.Y.; funding acquisition, J.Y.All authors have read and agreed to the published version of the manuscript.Funding: China Scholarship Council: 202108690015.Institutional Review Board Statement: Not applicable.Informed Consent Statement: Not applicable.