Numerical analysis of accelerating and decelerating ducts modified by camber on the unsteady flow field of pump-jet

Utilizing the Reynolds-averaged Navier-Stokes equations method in conjunction with Detached Eddy Simulation model, this paper investigates effects of duct camber (f) on transient hydrodynamic characteristics, particularly propulsion performance and flow field behavior. The research begins by validating the numerical methodology through experimental and numerical assessments of propeller VP1304 and a pump-jet operating under mooring conditions. The exploration commences with examining propulsion characteristics, followed by analyzing time-domain and frequency-domain data, wherein thrust fluctuations and pulsating pressures are scrutinized via fast Fourier transform (FFT). The pressure distribution and velocity field are subsequently presented to unveil the mechanisms triggered by variations in f. Comparative findings highlight that in cases of pump-jets with lower camber, the outlet velocity exceeds the inlet velocity, a trend contrary to scenarios involving higher f values. Additionally, by analyzing vorticity magnitude distribution and vortices, this study attains comparative insights into the effects of accelerating and decelerating ducts on rotor and stator trailing vortices, offering a window into the flow instability mechanism amidst diverse duct configurations. The results showcased to demonstrate the substantial impact of f variation on hydrodynamic properties. This comprehensive investigation yields practical guidance for the optimal design of pump-jets, potentially informing future design endeavours.


Introduction
Comprising a rotor, stator, and duct, the pump-jet stands out as a highly promising alternative propulsion system.Noteworthy advantages set it apart from other propulsion mechanisms, including its elevated critical speed, remarkable propulsion efficiency, excellent resistance to cavitation, and minimal radiated noise during operation.The intricate duct structure serves multiple functions: safeguarding internal blades, providing acoustic shielding, enhancing hydrodynamic performance (in the case of accelerating ducts), or mitigating cavitation generation (in the case of decelerating ducts).Nevertheless, the coexistence of multiple components within the pump-jet, combined with the non-uniform wake flow, leads to intricate hydrodynamic interactions.These interactions can precipitate phenomena like cavitation erosion, induced pressure fluctuations, and structural vibrations, underscoring the complex nature of the system's operational dynamics.
Back in 1963, a seminal exploration into pump-jet design and characteristics was initiated by McCormick et al. [1], who also pioneered the identification of the optimal duct configuration.Concurrently, the inception of numerical investigations applied to pump-jets for torpedoes was attributed to Ivanell [2].Given the presence of the duct within the pump-jet assembly, the flow dynamics are bifurcated into the inner fluid region and the exterior fluid surrounding the duct.This dichotomy introduces a discernible pressure differential between the inner and outer surfaces of the duct's profile section, thereby engendering axial component forces within the duct (John Carlton, 2010).Employing a combined approach of numerical simulation and experimental exploration proves reliable in conducting an inquiry into the selection and impact of critical parameters.These parameters relating to the duct can yield diverse influences on the pump-jet, along with intricate interactions.Researchers such as Huang et al. [4] and Wang et al. [5] have conducted comprehensive investigations into the hydrodynamic performance of pump-jets with varying duct parameters.The findings from these endeavors underscore the intricate and multifaceted influence stemming from alterations in duct parameters, encompassing aspects such as camber, tip-clearance size, duct attack angle, length-diameter ratio, and the pump-jet outlet's expansion ratio.These variations intricately contribute to the substantial and complex ramifications for pump-jet hydrodynamic performance.
To predict the performance and flow dynamics of pump-jets, two primary methodologies come into play: numerical simulations and experimental techniques.Numerical simulations hold distinct advantages due to their cost-effectiveness and ability to offer clearer insights into flow behaviors, yielding richer and more comprehensive datasets when compared to experimental studies.The works of Suryanarayana et al. [6], [7] stand as notable examples where both numerical and experimental investigations were conducted on a pump-jet propulsor for an axisymmetric body within a wind tunnel.The outcomes emphasized the efficacy and reasonable accuracy of Computational Fluid Dynamics (CFD) in rapidly predicting pump-jet performance at a relatively low cost.In response to the need for precise flow simulation within pump-jet propulsors that incorporate novel stator designs, Li et al. [8] harnessed a combination of Reynolds-Averaged Navier-Stokes (RANS) and hybrid RANS/Large Eddy Simulation (LES) techniques.This approach enabled the simulation of unsteady hydrodynamic performance, transient flow patterns, and comparative evaluations of wake vortices.Notably, their findings underscored the enhanced capability of the Delayed Detached Eddy Simulation (DES) method in capturing comprehensive turbulence structures and accurately modeling turbulent properties in contrast to conventional RANS solvers.
The present design challenges in pump-jet ducts lie in their intricate integration with other components and achieving optimal compatibility.The complex interplay between the instable flow and the mutual interactions of the duct with other components remains insufficiently understood, necessitating further investigation.This includes a comprehensive exploration of the effects of accelerating and decelerating ducts, which are characterized by variations in camber (f), on the instantaneous properties within the pump-jet flow field.
With the aim of addressing the aforementioned challenge, this study undertakes instantaneous simulations to examine the impact of accelerating and decelerating ducts with distinct cambers (f) and angles of attack (α).This is achieved by employing a combination of the Reynolds-averaged Navier-Stokes equations (RANSE) method and Detached Eddy Simulation (DES) model.The subsequent sections delineate the geometry configurations, wherein models featuring accelerating and decelerating ducts modified by different values of f and α are established.These models serve to isolate and differentiate the roles played by the various types of ducts.The approach involves ensuring the accuracy of the numerical simulations by structurally meshing the entire computational domain, as elaborated in Section 4. The methodology's reliability is further confirmed through a comparison of experimental and simulated non-cavitating cases involving propeller VP1304.Additionally, numerical simulations of the pump-jet under mooring conditions with varying rotor speeds are conducted to validate the accuracy of the numerical simulation approach.Section 5 presents the analysis and discussion of the obtained results, focusing on flow characteristics.This includes a comparison of the open water performance and transient flow field attributes of pump-jets equipped with accelerating and decelerating ducts altered by different f values.Finally, the study's findings are summarized and implications are drawn in Section 6.

Geometry configuration
Figure 1 provides an overview of the computational geometry configuration for the prototype pump-jet system.The physical arrangement comprises key components, including a pre-stator equipped with 3 blades, a rotor with 7 blades, and a stator featuring 9 blades.The central hub, serving as the underwater vehicle's stern, is positioned coaxially and successively aligned with the pre-stator, rotor, and stator elements.It's worth noting that the pre-stator, with its 3-blade design, assumes a dual role as a pre-swirl guide and a structural support framework.Furthermore, the rotor exhibits rotation about the x-axis, achieving a rotational speed of 2400 revolutions per minute (r/min), while the flow direction coincides with the x-axis orientation.The vertical upward direction, designated as the y-axis, is crucial for regulating tip clearance adjustments.To ensure a broader and more generalizable scope of conclusions, this research employs the NACA5510 airfoil profile as the basis for the pump-jet duct.The initial profile shape conforms to a camber parameter of f=0.5t,where "t" represents the airfoil thickness.
To establish distinct differentiations between accelerating and decelerating ducts based on their structural characteristics, our primary focus is directed towards variations in camber.A significant point of contrast becomes evident when examining the inlet and outlet areas of these ducts: the accelerating duct showcases a larger inlet area relative to its outlet, while the decelerating duct exhibits the converse structural arrangement.Notably, considering the non-parallel orientation of the pump-jet's hull surface and inner hub in relation to the horizontal plane, the hull surface serves as the reference horizontal plane for camber adjustments.As a result, in the process of discerning ducts by means of camber variation and aligning them with the pump-jet's hub, the airfoil sections of the decelerating and accelerating ducts are positioned at chord angles α of 4° and 11.5°, respectively.
To thoroughly investigate and draw comparisons regarding the effects of accelerating and decelerating ducts on pump-jet propulsion performance, we subject the physical model to examination, employing diverse camber sizes (f=0.5t,0.25t, 0, -0.25t, -0.5t).The objective is to discern the influence of distinct camber settings within the same plane, as depicted in figure 3. Notably, it's worth highlighting that the duct profile is rotated around the leading edge's vertex to accommodate the study's requirements.Furthermore, a consistent tip clearance size of 3 mm is maintained by carefully adjusting internal pumpjet components and subsequently modifying the duct profile along the y-axis direction.This meticulous measure ensures the insulation of results from potential external influences.In summary, figure 1 offers a comprehensive visualization of the computational geometry configuration inherent to the prototype pump-jet.This depiction encapsulates the array of components involved, elucidating their roles and dynamic interplays.The choice of the airfoil profile for the pump-jet duct is delineated, centering on the structural differentiations discernible between accelerating and decelerating ducts.This distinction is achieved through diligent considerations of camber.The study's overarching methodology is further elaborated, incorporating angle adjustments and meticulous tip clearance management to safeguard the integrity of the results obtained.

Performance parameters of pump-jet
In this investigation, a dimensionless approach is employed to analyze the relevant physical performance parameters.This approach enables a comprehensive evaluation and comparison of the hydrodynamic performance of pump-jet propulsion.The results of the numerical simulation are succinctly summarized and visually presented in Table 1.The information presented in the aforementioned table reveals the categorization of components through subscripts: "r" pertains to the rotating domain, encompassing the rotor, while "s" represents the static system comprising the pre-stator, stator, and duct.The parameters T and Q are significant metrics, representing thrust and torque, respectively.It is evident from the equation that altering the advance coefficient J is attainable by manipulating both the inflow velocity U and the rotor's rotational speed n.In the context of depicting pressure distribution, the pressure coefficient is denoted as Cp, embodying its dimensionless nature.
where ρl denotes the fluid density, and p∞ signifies the far-field pressure.Notably, the value of p∞ is considered as the outlet pressure within the calculation domain for the purposes of this study.

Governing equations
Within this paper, the fluid is assumed to be both incompressible and single-phase, with the added assumption of homogeneity.The continuity equation and momentum conservation equation, averaged over time and excluding effects from body forces and gravitational acceleration, are formulated as ∂ ∂t where ρ is the fluid medium density, μ denotes the fluid dynamic viscosity,   ′   ′ represents the Reynolds stresses, p represents the pressure and u is the time averaged velocity.Moreover, x is the spatial coordinate, S is the source term, while the subscript i and j represent the coordinate component.

Turbulence model
The DES model (Detached Eddy Simulation) is considered as a typical RANS model sensitive to the grid spacing.It is a hybrid approach specifically designed for instantaneous flow analysis, where it selectively employs URANS or LES in adequate areas of the domain [10].In regions where the grid is not sufficiently fine, DES resorts to the non-steady RANS model on which it is based.However, in high Reynolds number regions far from the walls, DES operates in the LES mode.Therefore, the DES model is relatively appropriate and precise for numerical simulation when addressing various boundary layers of pump-jet.Hence, the turbulence model adopted in this investigation is DES model based on SST kω to simulate the flow field for the pump-jet properties, while the k equation and ω equation for the SST k-ω model are as follows, where the dimensionless eddy viscosity factor µt is Otherwise, F1 represents the weighting function, F2 represents the mixing function, Pω and Pk are the dissipation source and turbulence generation terms caused by viscous force, S represents the curl amplitude.
The turbulence scale parameter lRANS in the dissipation term of the k equation is defined as follow For the DES method, the scale parameters of turbulence lRANS is replaced by the DES scale parameters lDES. is the maximum value of grid scale in the three directions, which is for a heterogeneous grid.The constant CDES is the model coefficient achieved through the mixed function, with the value of 0.65.
In the near-wall boundary layer, lDES = lRANS and the dissipation term for the turbulent kinetic energy k is the same as for the conventional RANS, namely the DES model switches to SST k- turbulence model.In the boundary away from the wall, lDES = lLES and the dissipation term of the transport equation is (k 3/2 )/(CDES), the model uses the Subgrid Reynolds Stress Model in the large eddy simulation method.

Computational domains and boundary conditions
To ensure an accurate simulation of the pump-jet's underwater performance, a comprehensive computational domain has been devised, encompassing the entire pump-jet structure.This encompassing domain is further subdivided into four distinct regions: the pre-stator domain, the rotor domain, the stator domain, and the external flow field domain.In this context, the symbol "D" denotes the maximum diameter of the pump-jet's rotor.As illustrated in figure 4, the outlet of the external flow field domain is positioned 8D away from the pump-jet outlet, while the inlet is strategically placed approximately 2D upstream from the pump-jet inlet.The overall diameter of the external flow field domain measures 6D.
The rotor domain operates as a rotating system, whereas the pre-stator domain, stator domain, and external flow field domain are considered stationary systems.Given the horizontal inflow conditions, the centerlines of these subdomains coincide.Furthermore, for improved simulation accuracy, local grid refinement is employed in internal regions and the flow field surrounding the duct.This refinement is particularly crucial due to the intricate flow patterns that are concentrated in close proximity to the pump-jet.The establishment of boundary conditions plays a pivotal role in ensuring the completeness and accuracy of numerical simulations.In this study, specific boundary conditions have been defined to govern the behavior of the flow field.The inlet of the external flow field is configured as a speed inlet with 5% turbulence intensity, while the outlet of the pump-jet is designated as a static pressure outlet.Additionally, the cylindrical surface employs a free slip wall boundary condition, while other surfaces like the duct and blades utilize a non-slip wall boundary condition.To address interactions and facilitate the transmission of flow field information, the frozen rotor approach is employed.The frozen rotor serves as a mixing model between the pre-stator domain and rotor domain, as well as between the rotor domain and stator domain.The solution time is set at 1000 iterations, and the convergence criterion is established with a residual target of 10^-5.The rotational speed of the rotor domain remains constant across all configurations, maintaining a value of n = 2400 r/min.Furthermore, for unsteady numerical simulations, a time step size of Δt = 1.3889×10^-4s is implemented.This value corresponds to a 2° rotation of the blade within a single circular period.These defined parameters collectively guide the numerical simulations, facilitating the investigation into the transient behavior of the pump-jet.

Grid generation and grid independence verification
The quality and convergence of numerical simulation outcomes directly hinge on the precision of the computational grid.While both hexa-structured and hybrid-unstructured grids can yield comparable levels of accuracy during simulations, hexa-structured grids are particularly adept at facilitating detailed analysis of the flow field.This preference is due to the fact that structured grids excel in computing boundary layers and effectively reducing grid volume [11], [12].During the process of mesh generation, segmenting the entire computational domain proves to be advantageous.As a result, this study adopts structured grids generated using ANSYS ICEM CFD for all investigated computational domains.Visual representation of the structured grids encompassing the entire pump-jet propulsion setup is showcased in figure 5.Meanwhile, figure 6 provides a depiction of the grids within the internal pump-jet computational domains, including the pre-stator, rotor, and stator regions.For a more focused examination of the grid distribution surrounding the ducts, Figure 7 presents intricate configurations.This figure presents mesh patterns for pump-jets exhibiting different cambers (e.g., f=0.25t, 0, -0.25t) and angles of attack (e.g., α=-4°, 0°, 4°).Notably, the quality assessment of each computational domain's grid exceeds a threshold of 0.42.This underscores the meticulous approach employed in grid generation across all domains, ensuring optimal simulation accuracy.To ensure precise predictions while efficiently utilizing computational resources and avoiding unnecessary grid complexity, it becomes essential to undertake a rigorous evaluation of grid independence within the computational domain.This validation process involves generating seven distinct schemes, each utilizing different grid densities, as outlined in Table 2. Subsequent to this, a comprehensive comparative study is conducted.It entails evaluating thrust coefficient, torque coefficient, and propulsion efficiency across a range of grid scales for the entire computational domain, under the operational condition of J=0.97.The findings of this comparative analysis are pivotal in identifying the most suitable grid scale.The presented table illustrates a consistent trend whereby performance parameters exhibit an initial increase up to a peak as the number of grids is augmented, followed by a subsequent decline that levels off after further increments.Simultaneously, the disparities in propulsion efficiencies gradually diminish, especially following the second increase in grid density.In light of these observations, with the objective of achieving enhanced accuracy in hydrodynamic performance outcomes and capturing near-wall flow field details, while also being mindful of computational resource optimization, the scenario involving 17.68 million grids is deemed appropriate for the numerical analysis conducted within the scope of this study.

Numerical method validation
This paper establishes the feasibility of numerical simulation through a validation process employing the propeller VP1304.This 5-bladed propeller is renowned for its widespread application and holds a significant position as a benchmark test propeller in cases involving numerical investigations.It stands as a classic test propeller developed by SVA, further enhancing its credibility.Notably, the open water experiments pertaining to propeller VP1304 were executed during the smp'11 workshop on propeller performance focusing on non-cavitating cases at SVA in Hamburg.Delving into specifics, the geometry and fundamental structural parameters of this propeller are depicted in Figure 8 and summarized in Table 3. Visualized in figure 9 are the boundary conditions delineating the VP1304 setup, which encompass both the external flow field and the rotational domain.Aligning with the methodology of this study, the numerical simulation for VP1304 employs the SST k-ω turbulence model.To ascertain the precision of the hydrodynamic performance coefficients derived from the simulation results, a direct comparison is made against experimental data gathered by SVA [13].The comparison, graphically depicted in Figure 10, showcases a minute error of less than 10%, thus affirming the accuracy of the simulation outcomes.In an effort to fortify the reliability of the numerical investigation, the SST k-ω turbulence model is also employed to analyze the original pump-jet model within the scope of this study.Due to constraints within the experimental setup, the pump-jet testing takes place under moored conditions.In this controlled environment, the pump-jet is situated within a water channel devoid of any inflow velocity, and the experimental variables are confined to alterations in the rotational speed.As depicted in figure 11, a comparison between the outcomes of the numerical simulation and the experimental data unveils a marginal discrepancy of under ten percent.This outcome serves as evidence of a robust alignment between the results derived from the numerical approach and the experimental observations for the pump-jet operating under moored conditions.
Collectively assessing these two sets of data, a clear convergence emerges between the outcomes derived through the numerical methodology adopted in this study and the experimental results.This alignment emphasizes the steadfastness and dependability of the numerical approach applied in this research, concurrently corroborating the accuracy and validity of the study.

Propulsion performance
Figure 12 presents a comprehensive analysis of pump-jet propulsion efficiency, encompassing ducts characterized by varying camber values (0.5t, 0.25t, 0, -0.25t, -0.5t) across a range of operational conditions spanning from J=0.24 to J=1.06.Throughout these assessments, a consistent rotational speed of n = 2400 rpm is maintained across all configurations.The curves depicted in figure 12 reveal a distinct trend: as cambers transition from positive (f=0.5t) to negative (f=-0.5t), the overall maximum open water efficiency of the pump-jet demonstrates an initial modest rise followed by a subsequent decline.Notably, decelerating ducts exhibit the ability to sustain significantly high propulsion efficiencies over a wider range of advance coefficients compared to their accelerating counterparts.
The rate of efficiency ascent intensifies with increasing camber until it culminates at its zenith.This phenomenon can be attributed to the decreasing rate of the pump-jet's thrust coefficient Kt and the marginal variance observed in the torque coefficient KQ.Furthermore, the position of the peak propulsion efficiency is influenced by the camber parameter: as the camber of the duct section decreases, the peak efficiency point shifts towards lower advance coefficients.For instance, it transitions from η of 85.40% at J=1.06 (f=0.5t) to η of 89.27% at J=0.97 (f=0.25t), and further to η of 63.22% at J=0.65 (f=-0.5t).
Regarding thrust and efficiency, the values for pump-jet parameters Kt, KQ, and η generally exhibit an upward trend, with an exception observed in the case of the duct with camber f=0.5t at low advance coefficients.Interestingly, while this particular duct with camber f=0.5t records the lowest thrust and efficiency values at lower advance coefficients, it surpasses all the accelerating ducts at higher advance coefficients.This distinct behavior arises from the steeper ascent rate of efficiency and the gradual decline of the thrust coefficient, setting it apart from the other duct configurations.
In this comparison, the torque coefficient curves exhibit marginal variations across the entire range of advance coefficients, whereas the thrust coefficient curves demonstrate greater sensitivity to alterations, particularly in the context of time-based considerations.

Unsteady thrust of rotor and stator
From Figure 13 to Figure 16, the time domain and frequency domain curves of each blade transient thrust coefficient of pump-jet with different f at the design working point J = 0.97 are indicated.The thrust coefficient of the pump-jet rotor and stator is made up of each blade thrust of the rotor and stator, respectively.The KT curves composed of KTr and KTs are periodic fluctuant, which means that per blade works in a periodic flow field.Figure 12 and figure 15 show the general tendency of rotor and stator transient force per blade in the time domain on the pumpjets with different duct cambers in one period.The variations of J = 0.97 for rotor blade curves has a period of 1/3T (where T is the rotation period of the hub: 1/40s = 0.025s), whereas for stator blade curves, a different period of 1/7T is observed.
In figure 16, the comparative results are discussed by the subtitles (a) -(e), corresponding to the transient force per rotor blade of pump-jet with duct cambers 0.5t, 0.25t, 0, -0.25t, -0.5t, respectively.The results indicate that lower f leads to the decrease of values of the thrust peak and trough.Meanwhile, the decline of f not only reduces the average value of unsteady force, but also lowers the differences in thrust coefficient values between the peak and trough of curves.On the other hand, Figure 18 shows the comparative time domain curves of unsteady thrust per stator blade of pump-jet with duct cambers.The time-domain curves demonstrate the similar tendency as described in the transient force per rotor blade.Additionally, the magnitude of the fluctuations shows an upward trend along with the increment of f, except for the situation of pump-jet with f=0.5t and -0.5t.
From the concern of figure 14 and 16, the frequency domain curves of thrust coefficients, which are adopted to study the frequency characteristics of the unsteady force, are transformed from the time domain curves through the method of Fast Fourier transform (FFT) at all the advance coefficients.The frequency domain spectrum of per rotor blade thrust coefficient has the same peaks in nfpre (n=1, 2, 3,…, and fpre=nprefn is the pre-stator blades passing frequency), while for per stator blade thrust coefficient, the frequency domain amplitude exhibits the same peaks in nfBRF (n=1, 2, 3,…, and fBRF=nrotorfn is the prestator blades passing frequency).It is found that per rotor and stator blade amplitude of direct-current (DC) which is at the zero frequency is equal to the periodic averaged values of per rotor and stator blade, respectively.In terms of the loading of rotor blade, the amplitude at the blade passing frequency is about 8.5% of the direct-current component, and about 5.1% at double times fpre.Moreover, the amplitude becomes lower at the high order of the harmonic.Compared to the frequency domain curves of rotor blade, the peak value of the stator blade tends to decay more slowly, that is 16.4% of the direct-current component at the blade passing frequency and 6% at the double times of the blade passing frequency.With the exception of the direct-current components, the main component parts of the rotor and stator blade thrust are made up of the amplitudes at peaks, which is prone to the vibration and fluctuation.
In figure 14, the frequency domain curves are compared by the subtitles (a) -(e), corresponding to the pump-jet with different duct cambers.It is demonstrated that increasing f can achieve the improvement of direct-current component value and the averaged amplitude value at overall harmonic.The amplitudes at peaks contribute the major components of thrust coefficients of the blade, while the increment of curve amplitude stands for the enhancement of thrust and torque as the fluctuation and time-averaged values are improved, simultaneously.Therefore, the value of pump-jet Kt becomes higher with the increment of f, which is consistent with the regularity shown in figure 14.Except for the camber f = 0.5t and -0.5t, the values at curves peaks are decayed more rapid as the camber becomes smaller.Additionally, the minimum difference between different rotor blades is occurred in pump-jet with f = 0.25t, which indicates that the internal flow is the most stable in this situation.Figure 19 shows the comparative frequency domain curves of per stator blade of pump-jets with different duct cambers.With the reduction of f, the values of peaks are decayed slower.Meanwhile, the direct-current component value presents a distinct tendency of descent as the camber decreased, as well as the averaged amplitude value at overall harmonic.

Transient flow field 5.2.1 Pressure field
As shown in the following part of this section, the comparative effect of decelerating and accelerating ducts with different f on pressure contours are discussed at the J = 0.97 (n=2400 r/min, U=6 m/s).As illustrated in figure 17 and figure 18, a prominent intensity of pressure fluctuation on the leading edge of rotor blade and stator blade.There are two areas on the suction side with apparent low-pressure, mainly distributed in the middle area of the suction surface, respectively near the leading edge and trailing edge.Increasing the f expand the low-pressure region near the leading edge and reduce the pressure value of the region, while the lowest-pressure region near the trailing edge of pump-jet with accelerating duct obtained by changing f is reduced relative to that with the accelerating ducts.On the pressure side of rotor blade, a significant high-pressure area occurs at the side of trailing edge, while the region of pump-jet with accelerating duct is smaller by comparing to that with the decelerating duct.
Though by decreasing f, the pressure on rotor suction side is reclined, the weakened interaction causes the reduction of pressure on rotor pressure side, thereby causing the loss of rotor thrust owing to that the pressure side has larger reduction of pressure.Meanwhile, increasing f causes the global improvement of pressure difference between suction side and pressure side of rotor blade, which is the main reason for causing the trend of the pump-jet thrust.It shows that the accelerating duct obtained by changing f increases the velocity of the fluid in the duct, thereby reducing the relative advance velocity of the rotor, resulting in a decrease in the thrust generated by the rotor, while the decelerating duct obtained in this way makes the flow field in the duct emerges the opposite trend.Increasing f results in a significant variation of stator blade length in the radial direction and thus changing the high-pressure area on pressure side.Increasing f causes narrowing low-pressure region on the suction side of stator blade, which represents the cavitation of suction side is more likely to happen by the reduction of f.It well corresponds with the pressure distribution on rotor blade, indicated in figure 17.The larger camber f is friendly with the increasing of duct thrust and intenser fluctuation at the leading edge of stator, as well as the pressure of stator pressure side going up.Moreover, larger f strengthens the interaction between the rotor wake and the leading edge of stator, but exacerbates the interaction between the tip of leading edge of rotor blade and the stator wake.Combined with figure 17 and figure 18, it is illustrated that the larger f is better for the stator cavitation, but the leading edge is under the easier cavitation condition.

Velocity field
On the basis of the open water performance of unsteady results above, the impacts of accelerating and decelerating ducts on the pump-jet performance can be further concluded from the velocity field.With advance ratio J=0.97 (n=2400 RPM, U=6 m/s).The axial velocity magnitude contours of the internal flow field at different radial places of transient simulations of pump-jets with accelerating and decelerating ducts distinguished by f is presented in figure 18 and 19.In figure 18, the axial velocity magnitude distribution of internal flow filed in the pump-jet with different f, from left to right presented as f = 0.5t, 0.25t, 0, -0.25t and -0.5t, while from top to bottom are represented as different radial locations from hub to tip region, correspond to span = 0.02, 0.5, 0.9 and 0.98, respectively.Overall, from hub to tip region, the averaged axial velocity distribution shows a downward trend after the increment at the middle of blade.It is found that the axial velocity of pump-jet with decelerating duct becomes smaller as the inflow passes through the pre-stator, while the higher-velocity region appears on the internal flow field of pump-jet with accelerating ducts as the inflow passes through the pre-stator.Increasing f leads to the improvement of the velocity distribution at the inlet area and the variation amplitude between the inlet and outlet of pump-jet, corresponding to the characteristics of accelerating and decelerating ducts.The low-velocity region occurs at the downstream area of rotor which is caused by the tip-clearance flow.Meanwhile, it becomes higher as the camber improves, indicating that increasing f results in the intenser unsteady tip-clearance flow.Additionally, the difference of axial velocity magnitude between different f becomes more apparent with the increment of radial places from span = 0.02 to span = 0.98.).In figure 20, the velocity contours of pump-jet with different f on the axial velocity, the circumference velocity and radial velocity components in x-y are shown.In the whole, the axial component of velocity distribution is the major velocity constituent in the wake of pump-jet, which is generated by the rotor acceleration and the recycling effect by the stator.Meanwhile, the circumference and radial components of velocity are mainly produced by the effect of rotation and acceleration by rotor.The lowvelocity region of the axial velocity component is occurred at the tip region in the rotor downstream, resulting from generation of the tip-clearance flow.Moreover, the low-velocity region of the axial velocity component, including the hub wake and tip region, shows the high velocity value in the circumference and radial components of velocity distribution.
As shown in figure 20, the rising f give rise to the increment of the pump-jet inlet velocity, owing to the movement of the locations of maximum curvature.In the axial component of velocity distribution, the low-velocity region generated by the tip-clearance flow shares lower value as the camber increases, correspondingly presenting the higher velocity distribution in the circumferential distribution.In the radial components of velocity, the velocity distribution is slightly influenced by the variation of f, owing to the slight change in the shrinking extent of the internal flow field generated by the angle between the hub and duct.With the reduction of f, the outlet velocity in axial direction gradually drops off, caused by the higher energy loss.The radial velocity distribution at the leading edge of duct shares the opposite pattern between accelerating and decelerating ducts.

Vorticity distribution
In order to further investigate the vorticity distribution of pump-jet with accelerating and decelerating duct varied by f, the instantaneous iso-surfaces of vorticity magnitude are demonstrated in Figure 21.The variation in the intensity of vortices is mainly caused by the tip-clearance flow change induced by the f and α.The tip vortices are apparently generated in the tip region between the rotor and duct, which starts at the leading edge of the rotor blade tip and extends backward in a slender strip with the direction of rotor rotation.Meanwhile, the obvious vorticity distribution occurs at the locations on the leading edge, trailing edge, and suction side.It is found that the stators in all of the pump-jets shares the relative well effects of recovering tip-clearance flow, benefiting to the fluctuation and stability of flow field.In figure 19, the angle between the vortex line and the rotor blade gradually improves along with the decline of f, as well as the increment of the intensity of tip vortices.The instability of tip vortices becomes intenser in the pump-jet with accelerating duct, even with bifurcation.Meanwhile, the area of the vortices occupied on the suction side of the stator narrows, whereas the length of the tip vortices becomes longer, caused by the stronger effect of radial velocity.

Conclusion
This study embarks on an exhaustive analysis to explore the intricate effects of varying camber (f) and angle of attack (α) parameters on transient flow fields within accelerating and decelerating ducts.To accomplish this, a numerical framework is employed, incorporating the Reynolds-averaged Navier-Stokes equations (RANSE) method, Detached Eddy Simulation (DES) model, and sliding mesh technology.This amalgamation of techniques aims to unravel the complex interplay of flow characteristics and interactions inherent in pump-jets operating under diverse duct profile conditions.The robustness of the numerical methodology is meticulously validated against experimental data obtained from both the pump-jet under mooring conditions and the propeller VP1304, spanning various rotational speeds.The investigation extends its purview to pump-jets encompassing different duct profiles, specifically focusing on five camber sizes of f (f=0.5t,0.25t, 0, -0.25t, -0.5t).The heart of the analysis lies in the scrutiny of pump-jets featuring accelerating and decelerating ducts, each modified with varying f values.This systematic inquiry delves into two primary dimensions: firstly, a comprehensive comparison of open water performance, and secondly, an intricate exploration of transient flow characteristics displayed by pump-jets incorporating two distinct duct types, each characterized by distinct f values.The paramount discoveries and deductions gleaned from this extensive analysis can be summarized as follows.a) Concerning the overall pump-jet performance curves, augmenting the parameter f results in a sharper ascent of the propulsion efficiency until it reaches its apex.Consequently, the resulting trends become more straightforward and pronounced.When it comes to the parameter f, while an increase does improve propulsion performance, surpassing a certain threshold can lead to significant performance degradation, especially at lower J values.Remarkably, even with the decrease in performance at lower J values, a higher level of performance is maintained at higher J values.In a comprehensive assessment, it is evident that the decelerating duct is adept at sustaining high hydrodynamic efficiency over a wider range of J values in comparison to the accelerating ducts.This observation highlights the beneficial impact of utilizing decelerating duct configurations in terms of maintaining propulsion efficiency across different operational conditions.b) The mutual interference among the rotor, duct and stator results in the exciting force of the rotor and stator blades.Time domain curves showed periodic fluctuations, related to the blade number and the rotational speed of rotor.Decreasing f not only reduces the average value of unsteady rotor thrust, but also lowers the differences in rotor thrust coefficient values between the peak and trough of curves.The fluctuation magnitude of stator shows an upward trend with the increment of f, except for f = 0.5t and -0.5t.Increasing f results in the enhancement of thrust and torque as the fluctuation and time-averaged values are improved, while the values of peaks are decayed more rapidly.Additionally, the minimum difference between different rotor blades occurs in f = 0.25t.c) The f significantly influences the pressure distribution on the whole components of pump-jet.
Increasing f not only raises the pressure globally on the whole surfaces of duct, rotor and stator, but also causes the expansion of high-pressure area.It results in the relatively higher hydrodynamic loading and decreases the likelihood of cavitation at the leading edge.Moreover, larger f strengthens the mutual effects between the rotor wake and the leading edge of stator, benefiting to the cavitation resistance of rotor and stator.However, it also intensifies the mutual impacts between the tip of leading edge of rotor blade and the stator wake, leading to the higher efficiency.d) The analysis of the velocity field provides insights into the impact of accelerating and decelerating ducts on pump-jet performance.Contour plots of axial velocity magnitude at different radial locations reveal that pump-jets with decelerating ducts exhibit reduced axial velocity as the flow passes through the pre-stator, while pump-jets with accelerating ducts show higher velocity regions after the pre-stator.Increasing f improves the velocity distribution at the inlet and increases the variation between the inlet and outlet, reflecting the characteristics of accelerating and decelerating ducts.The tip-clearance flow causes a low-velocity region downstream of the rotor, which becomes more pronounced with higher camber values.Decreasing f results in reduced outlet velocity due to higher energy loss.e) The investigation of complicated vortex system in flow field is carried out by the analysis of vortices distribution inside the pump-jet.The tip region and the central area after the pump-jet hub exhibit high turbulence levels due to rotor tip vortices and hub vortices instability.Increasing distance inside the pump-jet leads to a larger angular displacement between the tip vortex and trailing vortex, caused by trailing edge vortices rolling up.Decreasing f leads to the higher intensity of vortices, especially, the tip-clearance vortices are stronger in the pump-jet with decelerating duct, while the outlet of pump-jet with accelerating duct shares the stronger rolling up process.The rotor trailing wake flows through the stator blades and becomes more uniform with the decline of intensity of duct-induced vortex, especially pronounced at f = 0.25t.By varying the parameter f, ducts can be categorized into accelerating and decelerating configurations.Specifically, ducts with f values less than 0 are considered accelerating ducts, while those with f values greater than 0 act as decelerating ducts.An exception to this classification arises for the duct with camber f = 0 or α = 0°, which closely resembles a zero-accelerating duct.It's noteworthy that within a certain range, an increase in f leads to performance enhancement, but surpassing this range can result in performance deterioration.In the context of this study, the pump-jet equipped with a duct featuring f = 0.25t exhibits the most favorable performance, boasting the highest propulsion efficiency (η) and thrust, along with a relatively more consistent transient flow field.Generally, the decelerating duct showcases superior propulsive efficiency, a reduced susceptibility to cavitation erosion, and presents more prominent fluctuations, accompanied by smoother tip-clearance flow patterns.This configuration excels at high values of advance coefficient (J), although its performance relative to accelerating ducts diminishes at lower J values.On the other hand, the accelerating duct generates higher outlet velocities and maintains more stable performance at low J values.However, it also experiences intensified instability in tip vortices and encounters greater energy loss at high J values.Consequently, decelerating ducts prove better suited for high-speed underwater vehicles, while accelerating ducts demonstrate enhanced performance characteristics for low-speed applications.

Figure 2 .
Figure 2. The diagrammatic sketch of the parameters which distinguish accelerating and decelerating ducts.

Figure 3 .
Figure 3. Schematic diagram of duct profile with different cambers.

Figure 4 .
Figure 4.The computational domain and boundary conditions of pump-jet.

Figure 5 .
Figure 5. Structured grids of the entire computational domain with the internal domain of pump-jet.

Figure 6 .
Figure 6.Structured grids of each internal region of pump-jet.

Figure 7 .
Figure 7.The close-up views of the mesh around the accelerating and decelerating ducts.The mesh for the

Figure 11 .
Figure 11.Validation of CFD results with experimental data of pump-jet under moored condition.

Figure 19 .
Figure 19.The velocity magnitude distribution at different radial places of pump-jets with different f (from left to right: f = 0.5t, f = 0.25t, f = 0, f = -0.25tand f = -0.5t.from top to bottom: span = 0.98, span = 0.9, span = 0.5, span = 0.02.span represents the relative location to the maximum distance between duct inner surface and hub.).In figure20, the velocity contours of pump-jet with different f on the axial velocity, the circumference velocity and radial velocity components in x-y are shown.In the whole, the axial component of velocity distribution is the major velocity constituent in the wake of pump-jet, which is generated by the rotor acceleration and the recycling effect by the stator.Meanwhile, the circumference and radial components of velocity are mainly produced by the effect of rotation and acceleration by rotor.The lowvelocity region of the axial velocity component is occurred at the tip region in the rotor downstream,

Table 1 .
Dimensionless presentation of pertinent physical performance parameters.

Table 2 .
Comparison of different quantity of grids for grid independence analysis.