Numerical Study of the Hydrodynamic Characteristics Comparison Between a Ducted Propeller and a Rim-Driven Thruster

applied sciences

Article Numerical Study of the Hydrodynamic Characteristics Comparison between a Ducted Propeller and a Rim-Driven Thruster

Bao Liu and Maarten Vanierschot *

Department of Mechanical Engineering, Group T Leuven Campus, KU Leuven, B-3000 Leuven, Belgium; [email protected] * Correspondence: [email protected]

Abstract: The Rim-Driven Thruster (RDT) is an extraordinary innovation in marine propulsion applications. The structure of an RDT resembles a Ducted Propeller (DP), as both contain several propeller blades and a duct shroud. However, unlike the DP, there is no tip clearance in the RDT as the propeller is directly connected to the rim. Instead, a gap clearance exists in the RDT between the rim and the duct. The distinctive difference in structure between the DP and the RDT causes significant discrepancy in the performance and flow features. The present work compares the hydrodynamic performance of a DP and an RDT by means of Computational Fluid Dynamics (CFD). Reynolds-Averaged Navier–Stokes (RANS) equations are solved in combination with an SST k-ω turbulence model. Validation and verification of the CFD model is conducted to ensure the numerical accuracy. Steady-state simulations are carried out for a wide range of advance coefficients with the Moving Reference Frame (MRF) approach. The results show that the gap flow in the RDT plays an important role in affecting the performance. Compared to the DP, the RDT produces less thrust on the propeller and duct, and, because of the existence of the rim, the overall efficiency of the RDT is Citation: Liu, B.; Vanierschot, M. Numerical Study of the significantly lower than the one of the ducted propeller. Hydrodynamic Characteristics Comparison between a Ducted Keywords: Rim-Driven Thruster (RDT); Ducted Propeller (DP); Computational Fluid Dynamics Propeller and a Rim-Driven Thruster. (CFD); hydrodynamic performance; RANS Appl. Sci. 2021, 11, 4919. https:// doi.org/10.3390/app11114919

Academic Editor: Hoyas Calvo Sergio 1. Introduction Propulsion systems play an indispensable role in marine vessels. There are various Received: 29 April 2021 designs applied to present ship propulsors, such as the open propeller, the ducted propeller, Accepted: 25 May 2021 podded and azimuthing propulsors, etc. Each kind of propulsor has its advantages and Published: 27 May 2021 disadvantages. Regardless of different types of design, they share the same propulsion technology, i.e., a shaft-driven propulsion, in which the shaft is imperative because the Publisher’s Note: MDPI stays neutral propeller must be connected to it in order to rotate. Furthermore, the other side of the shaft with regard to jurisdictional claims in is mounted on the primary engine, which lays within the ship’s hull, making the whole published maps and institutional afﬁl- structure stretch through the cabin space. In addition, substantial friction power loss and iations. high-level noise and vibrations can occur when they operate. In order to overcome the main drawbacks in conventional shaft-driven propulsion systems as mentioned above, a novel propulsion concept has been brought forward, namely the Rim-Driven Thruster (RDT), also known as the Integrated Motor Propulsor (IMP) [1]. Copyright: © 2021 by the authors. As the name implies, this kind of thruster is rim driven instead of being shaft driven. In an Licensee MDPI, Basel, Switzerland. RDT device, the driver motor is integrated into the structure, with the stator embedded in This article is an open access article the shroud and the rotor distributed on or in the rim. As the thruster works under water, distributed under the terms and both the stator and rotor are coated with protection, such as epoxy resin, to prevent water conditions of the Creative Commons erosion. The propeller blades are directly connected to the rim and rotate with the rotor Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ when the motor works. Figure1 gives the inner layout of an RDT, in which some key 4.0/). components are shown: the driven motor, the propeller blades, the bearings, and the duct.

Appl. Sci. 2021, 11, 4919. https://doi.org/10.3390/app11114919 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, x FOR PEER REVIEW 2 of 17

some key components are shown: the driven motor, the propeller blades, the bearings, and the duct. Several outstanding improvements are realized due to the integrated design of RDTs. The component arrangement of RDTs is much more compact compared to conventional propulsion systems and therefore installation and management are more flexible. The elimination of intermediary equipment, such as a motor cooling system, reduction gear box, and bearing lubricating system, can save a lot of cabin space and reduce ship building costs. The motor efficiency is higher when the RDT is powered by a Permanent Magnet (PM) radial field motor, and power does not have to be applied to the field and thus field power losses are eliminated. In addition, the application of water-lubricated bearings ensures a low risk of water pollution [2]. Moreover, the application of the RDT is not limited to ordinary ships, as underwater Remotely Operated Vehicles (ROVs) and Autonomous Appl. Sci. 2021, 11, 4919 2 of 17 Underwater Vehicles (AUVs) can also be equipped with lightweight and highly compact RDTs to operate with a high performance [3,4].

FigureFigure 1.1.Schematic Schematic layoutlayout ofof anan RDT.RDT.

SeveralDespite outstanding many distinguished improvements merits are of realizedRDTs, there due toare the still integrated many challenges design of in RDTs. their Thehydrodynamic component design arrangement and performance of RDTs is much analysis more compared compact to compared traditional to conventionalpropellers. In propulsionorder to understand systems and the thereforeperformance installation of RDTs, and Sharkh management et al. [5] designed are more and flexible. manufac- The eliminationtured a demonstrator of intermediary RDTequipment, with a 250 suchmm diameter. as a motor Experimental cooling system, tests reduction were conducted gear box, andto investigate bearing lubricating the influence system, of the can duct save and a lot propeller of cabin space profiles and on reduce the overall ship building performance costs. Theof the motor thruster. efficiency It was is found higher that when at bollard the RDT conditions is powered the byduct a Permanentfairings could Magnet have (PM)nega- radialtive effects field motor,on the andthrust power production does not due have to tothe be resistance applied toof thethe fieldfairings and to thus the field flow. power How- lossesever, at are higher eliminated. advance In ratios, addition, the existence the application of the fairings of water-lubricated improved the bearings performance. ensures As afor low the risk propellers, of water symmetric pollution and [2]. asymmetric Moreover, theones application were compared of the under RDT the is notsame limited work- toing ordinary conditions. ships, The as asymmetric underwater propeller Remotely produced Operated a Vehicles higher thrust (ROVs) in andthe forward Autonomous direc- Underwatertion compared Vehicles to the (AUVs) symmetric can alsopropeller. be equipped This was with expected lightweight because and highlythe asymmetric compact RDTsprofile to increased operate with the apitch high ratio, performance resulting [ 3in,4 a]. higher thrust at the same speed. The study Despite many distinguished merits of RDTs, there are still many challenges in their also concluded that the smoothness of the surface of the duct and propeller could have an hydrodynamic design and performance analysis compared to traditional propellers. In influence on the performance of the thruster. order to understand the performance of RDTs, Sharkh et al. [5] designed and manufactured For the hydrodynamic performance evaluation of an RDT, experimental methods are a demonstrator RDT with a 250 mm diameter. Experimental tests were conducted to widely employed because of their high reliability and accuracy. However, it is often quite investigate the influence of the duct and propeller profiles on the overall performance of costly to build an experimental setup and sometimes even impractical. Therefore, Com- the thruster. It was found that at bollard conditions the duct fairings could have negative effects on the thrust production due to the resistance of the fairings to the flow. However, at higher advance ratios, the existence of the fairings improved the performance. As for the propellers, symmetric and asymmetric ones were compared under the same working conditions. The asymmetric propeller produced a higher thrust in the forward direction compared to the symmetric propeller. This was expected because the asymmetric profile increased the pitch ratio, resulting in a higher thrust at the same speed. The study also concluded that the smoothness of the surface of the duct and propeller could have an influence on the performance of the thruster. For the hydrodynamic performance evaluation of an RDT, experimental methods are widely employed because of their high reliability and accuracy. However, it is often quite costly to build an experimental setup and sometimes even impractical. Therefore, Computational Fluid Dynamics (CFD) has become a popular choice in dealing with such problems due to its extraordinary convenience and low cost. Among various numerical methods, the RANS methods stand out and can be found in extensive research studies. Cai et al. [6] investigated the performance of an RDT with a commercial RANS solver after validating the numerical model by comparing the simulated results of a Ka 4-55 Appl. Sci. 2021, 11, 4919 3 of 17

propeller with existing experimental data. From their results, rim-driven thruster propellers could produce a larger thrust by absorbing a higher torque from the motor compared to conventional shaft-driven propellers, but this resulted in a lower efficiency. However, this situation can be alleviated by applying thinner propeller blades. Moreover, a long rim can slightly increase friction torque and affect the performance of the propeller blades as a circumferential velocity will be introduced to the local flow, causing a reduced tangential velocity and angle of attack for the propeller. Due to the fact that no design guidelines were available for propeller design in hub-less RDTs, Zhang et al. [7] designed a propeller based on current Ka-series propeller geometry. Three configurations of hub-less RDTs were modeled and calculated numerically, i.e., 5 blades, 7 blades, and 5-4 contra-rotating blades. They concluded that the hydrodynamic performance of RDTs is similar to that of ducted propellers, and, because there is no hub in the hub-less RDT, no hub vortex exists and therefore cavitation is diminished. The rim gap is a characteristic structure in RDTs and, even though the dimensions of the gap are usually small, its influence on the performance of the thruster can hardly be ignored. Cao et al. [8,9] studied the effects of the gap flow with and without axial pressure difference using RANS simulations. When there was no pressure gradient in the axial direction, the angular momentum current analyzing method was introduced to analyze the gap flow. The developed method was validated by experiments in the large cavitation tunnel of CSSRC. Compared to the calculations with empirical formulas, the predicted results by the numerical method were improved significantly. However, when a pressure difference existed between the rim ends due to the rotational movement of the propeller, the flow regimes in the gap flow changed significantly, resulting in a higher torque friction. The thrusts produced by the surfaces surrounding the gap, including the duct inner surface, rim outer surface, and rim end surface, were balanced out. The propeller loading distributions and steady wake flow fields were also investigated by studying four cases of different propellers. The propeller blade radial circulation and chord- wise circulation density distributions were analyzed. It was found that the maximum radial circulation occurs at the blade tip, which is different from conventional shaft-driven propellers, and the right adjustment of the blade loading distribution could restrain root and tip region vortices. As an innovative invention, the RDT is quite similar to a conventional DP in its structural design. However, a detailed comparison study of hydrodynamic performance between them is difficult to find. Therefore, the present work aims to find out the difference in hydrodynamic performance between the DP and the RDT with the aid of numerical simulations. A verification and validation study is carried out for the CFD model to ensure the numerical accuracy. The CFD model is then applied to the performance prediction of the DP and the RDT. Based on the results, the local and global hydrodynamic characteristics of the DP and the RDT are compared and analyzed, including the open water performance and several important flow features. The model setup and validation is given in Section2. A detailed discussion for the DP and RDT is given in Section3, and finally the conclusions are summarized in Section4.

2. CFD Model Setup and Validation A typical four-blade RDT is used for this study. The geometric details are provided in Table1. The propeller under consideration is the Ka 4-70 [ 10] propeller with a uniform pitch ratio of 1.0 and area ratio of 0.7. The hub has a total length of 0.52D with 0.28D before the propeller and 0.24D after the propeller, in which D represents the diameter of the propeller, i.e., 250 mm in this case. The hub shroud is round shaped with a diameter of 0.167D. The duct is modiﬁed from the most commonly encountered duct, i.e., the No. 19A duct, with a length of 0.65D. Considering the accommodation of the driven motor, the duct length along the chord-wise direction is extended instead of using the classical length of 0.5D. A gap with a width of 2 mm and a length of 90 mm represents the air gap of the driven motor, as also found in the study of Cao [8]. The same structural and dimensional Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 17

The duct is modified from the most commonly encountered duct, i.e., the No. 19A duct, with a length of 0.65D. Considering the accommodation of the driven motor, the duct Appl. Sci. 2021, 11, 4919 length along the chord-wise direction is extended instead of using the classical length4 of of 17 0.5D. A gap with a width of 2 mm and a length of 90 mm represents the air gap of the driven motor, as also found in the study of Cao [8]. The same structural and dimensional configurationsconﬁgurations are are kept kept for for the the DP, DP, except th thatat the tip clearance between the the propeller propeller tip tip andand the the duct duct is is designed designed as as 1.5 1.5 mm mm (Wang (Wang [11]). [11]). A A schematic schematic view view of of the the DP DP and and the the RDT RDT isis shown shown in in Figure Figure 22,, withwith thethe MRFMRF shownshown inin yellow.yellow.

Table 1. Geometric details for the ducted propeller and the RDT used in this study. Table 1. Geometric details for the ducted propeller and the RDT used in this study. Parameter Unit Value Parameter Unit Value Number of blades - 4 Number of blades - 4 Blade diameter m 0.25 Blade diameter m 0.25 BladeBlade area area ratio ratio - - 0.7 0.7 BladeBlade pitch pitch ratio ratio - - 1.0 1.0 DuctDuct type type - - 19A 19A PropellerPropeller type type - - Ka Ka

(a) (b)

FigureFigure 2. 2. GeometricalGeometrical models models of of ( (aa)) the the ducted ducted propeller propeller and and ( (bb)) the the RDT. RDT.

2.1.2.1. Numerical Numerical Model Model TheThe Reynolds-Averaged Reynolds-Averaged Navier–Stokes Navier–Stokes (RANS) (RANS) equations equations are are employed employed to to resolve thethe flow flow field field around around the thrusters. For For inco incompressiblempressible Newtonian Newtonian fluids, fluids, the the continuity andand momentum momentum equations equations are are given given as as

∂=0ui and (1) = 0 and (1) ∂xi ! " !# ∂ui+ ∂ui=− ∂p+ ∂ ∂+ui ∂uj + −∂ , 0 0 (2) ρ + uj = − + µ + + −ρuiuj , (2) ∂t ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj where is the fluid density, (i = 1, 2, 3, representing the component in the x, y, z di- where ρ is the fluid density, u (i= 1, 2, 3, representing the component in the x, y, z direction) rection) is the mean velocity icomponent, t is the flow time, p is the pressure, is the dy- is the mean velocity component, t is the flow time, p is the pressure, µ is the dynamic namic viscosity, and − is the Reynolds stress term. viscosity, and −ρu0u0 is the Reynolds stress term. The direct modelingi j of the Reynolds stress is based on the Boussinesq hypothesis, The direct modeling of the Reynolds stress is based on the Boussinesq hypothesis, which assumes the Reynolds stress to be a linear function of the mean velocity gradients which assumes the Reynolds stress to be a linear function of the mean velocity gradients [12]. [12]. For incompressible Newtonian flows, the Reynolds stress is calculated as For incompressible Newtonian flows, the Reynolds stress is calculated as ! ∂u ∂uj 2 − 0 0 = i + − ρuiuj µt ρkδij (3) ∂xj ∂xi 3 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 17

2 − = + − (3) 3

where is the turbulent eddy viscosity and is the Kronecker symbol. To complete the set of equations, the SST k-ω turbulence model developed by Menter [13] is adopted, as the SST k-ω model takes advantage of both the k-ω model in a robust near-wall treatment due to the simple low Reynolds number formulation and the ability to compute flows with moderate adverse pressure gradients, and the k-ε model in a better performance near the boundary layer edge and away from the walls because of its insen- sitivity to the free stream values. In this model, two additional transport equations are solved for the turbulent kinetic energy, k, and the specific turbulent dissipation rate, ω.

Appl. Sci. 2021, 11, 4919 These equations are solved together with Equations (1) and (2) using the commercial 5soft- of 17 ware ANSYS Fluent.

2.2. Grid Generation and Validation Study whereIn µthist is study, the turbulent a cylindrical eddy viscositycomputational and δij domainis the Kronecker sized 14D symbol. in length and 5D in radius isTo adopted, complete in the which set of D equations, represents the the SST diamk-ω eterturbulence of the propeller, model developed as depicted by Menter in Figure [13] 3a.is adopted,The inlet asis thelocated SST 4Dk-ω upstreammodel takes of th advantagee propeller of center both theand kthe-ω modeloutlet 10D in a down- robust stream.near-wall The treatment mesh is generated due to the in simple a hybrid low Reynoldsform in which number the formulationstationary domain and the is ability filled withto compute a structured flows mesh with and moderate the rotating adverse doma pressurein with an gradients, unstructured and themesh.k-ε Prismmodel layers in a arebetter placed performance on the propeller, near the hub, boundary and duct layer wi edgeth smooth and away transition from thenormal walls to because the walls, of itsas showninsensitivity in Figure to the 4. freeThe streamfirst layer values. height In thisis so model, small that two the additional maximum transport y+ value equations on the k propellerare solved is forless the than turbulent 15. As it kinetic is difficult energy, to maintain, and the a specificuniform turbulent grid size dissipationnear the wall, rate, a ω. These equations are solved together with Equations (1) and (2) using the commercial scalable wall function is employed, which resolves the boundary layer with varied near- software ANSYS Fluent. wall treatment based on local y+ values. To resolve the high velocity gradients in the clear- ance2.2. Gridand Generationgap, 15 prism and Validationlayers are Study placed between the propeller tip and the duct inner surface for the DP, while 20 prism layers are placed between the rim and the duct inner In this study, a cylindrical computational domain sized 14D in length and 5D in radius surface for the RDT. is adopted, in which D represents the diameter of the propeller, as depicted in Figure3a. The computational domain is divided into stationary and rotating fluid zones by con- The inlet is located 4D upstream of the propeller center and the outlet 10D downstream. formal interfaces, and the propeller, hub, and duct are contained in the rotating zone. The The mesh is generated in a hybrid form in which the stationary domain is filled with a Moving Reference Frame (MRF) approach is utilized to handle the relative motion of the structured mesh and the rotating domain with an unstructured mesh. Prism layers are propeller. In the MRF, the governing equations for the flow in the selected rotating zone placed on the propeller, hub, and duct with smooth transition normal to the walls, as shown are solved in a relative rotating frame. At the interfaces, appropriate transformations of in Figure4. The first layer height is so small that the maximum y+ value on the propeller is the velocity vector and velocity gradients are performed to compute fluxes of mass, mo- less than 15. As it is difficult to maintain a uniform grid size near the wall, a scalable wall mentum,function isand employed, other scalars. which As resolves the MRF the is boundarya steady-state layer approximation with varied near-wall of the actual treatment flow, thebased interactions on local y+ between values. the To stationary resolve the and high ro velocitytating components gradients in are the ignored. clearance Therefore, and gap, it15 requires prism layers much are less placed computational between theresources propeller compared tip and theto other duct approaches, inner surface such for theas the DP, slidingwhile 20 mesh prism model. layers are placed between the rim and the duct inner surface for the RDT.

(a) (b)

Figure 3. Computational domain: (a) overall mesh; (b) boundary conditions.

The computational domain is divided into stationary and rotating fluid zones by conformal interfaces, and the propeller, hub, and duct are contained in the rotating zone. The Moving Reference Frame (MRF) approach is utilized to handle the relative motion of the propeller. In the MRF, the governing equations for the flow in the selected rotating zone are solved in a relative rotating frame. At the interfaces, appropriate transformations of the velocity vector and velocity gradients are performed to compute fluxes of mass, momentum, and other scalars. As the MRF is a steady-state approximation of the actual flow, the interactions between the stationary and rotating components are ignored. Therefore, it requires much less computational resources compared to other approaches, such as the sliding mesh model. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 17

Appl. Sci. 2021, 11, 4919 6 of 17 Figure 3. Computational domain: (a) overall mesh; (b) boundary conditions.

(a) (b)

Figure Figure4. Mesh 4. Meshdistribution: distribution: (a) surface (a) surface mesh mesh on onduct, duct, propeller, propeller, and and hub; hub; ((bb)) volumevolume mesh. mesh.

ThereThere are aresome some hydrodynamic-related hydrodynamic-related coefficientscoefficients that that are are used used in this in study, this study, and they and are defined as follows: they are defined as follows: Va J= , (4) = , nD (4) T Kt = , (5) ρn2D4 = , (5) Q Kq = , (6) = ρn2, D5 (6) J Kt η = , (7) = 2π, Kq (7) 2 p − p∞ Cp−= , (8) 1 2 = 2 ρ,v 1 (8) −1 −1 where Va is the inflow velocity in m·s , n2is the rotation rate of the propeller in s , ρ is the −3 · D −1 J K T K Q −1 η wherefluid density is the inflow in kg m velocity, is thein m·s diameter, isof the the propellerrotation rate in m, of and the ,propellert, , q, in, ands , is are the advance coefficient,−3 total thrust coefficient, total thrust, total torque coefficient, total the fluid density in kg·m , is the diameter of the propeller in m, and , , , , , torque, and efficiency, respectively; C is the pressure coefficient, p is the local pressure in and are the advance coefficient, totalp thrust coefficient, total thrust, total torque coeffi- Pa, p is the free stream pressure in Pa, and v is the representative velocity of the propeller, ∞ cient,which total torque, in this study and isefficiency,πnD. respectively; is the pressure coefficient, is the local pressureAs relatedin Pa, experimental is the free data stream for RDTs pressure are scarce, in Pa, a and validation is the study representative is carried out ve- locityto of verify the propeller, and validate which the numerical in this study model. is The physical. model used for validation with Asexperimental related experimental data was obtained data for from RDTs the studyare sc ofarce, Wang a validation [11], in which study open is carried water tests out to verifywere and performed validate forthe anumerical Ka47 ducted model. propeller The under physical a wide model range ofused advance for validation coefficients. with experimentalThree data sets ofwas meshes obtained have beenfrom generated the study to of perform Wang a [11], mesh in sensitivity which open analysis, water from tests were aperformed coarse to a for fine a meshKa47 withducted cell propeller numbers under increasing a wide by arange certain of growth advance ratio coefficients. in each direction. Detailed mesh parameters are given in Table2. As the flow in the rotational zone Three sets of meshes have been generated to perform a mesh sensitivity analysis, changes much more rapidly than in the stationary zone, more grid cells are placed in the from formera coarse to to resolve a fine themesh complex with cell flow numbers field around increasing the propeller. by a certain The grid growth growth ratio ratio in iseach direction.kept almostDetailed constant mesh to parameters ensure a better are evaluation given in ofTable the discretization2. As the flow error. in the rotational zone changesThe Gridmuch Convergence more rapidly Index than (GCI) in describedthe stationary in Celik zone, [14] more is introduced grid cells to evaluateare placed in thethe former discretization to resolve error the for complex the thrust flow and field torque around of the the propeller propeller. under The two grid representative growth ratio is keptloading almost conditions: constant to J = ensure 0.1 and a Jbetter = 0.5. evaluation GCI is a discretization of the discretization error estimation error. method Thebased Grid on RichardsonConvergence extrapolation Index (GCI) and described is calculated in in Celik the following [14] is introduced way: to evaluate the discretization error for the thrust and torque of the propeller under two representative loading conditions: J = 0.1 and J = 0.5. GCI is a discretization error estimation method based on Richardson extrapolation and is calculated in the following way:

Appl. Sci. 2021, 11, 4919 7 of 17

Table 2. Discretization error estimation using the GCI.

J = 0.1 J = 0.5 Mesh Total Cells Reﬁnement GCI (%) GCI (%) Density (Million) Ratio r Ktp/Kq Ktp/Kq Ktp/Kq Ktp/Kq Coarse 5.02 - 0.257/0.0456 - 0.170/0.0334 - Medium 9.79 1.25 0.262/0.0465 4.32/4.39 0.174/0.0340 5.23/3.19 Fine 19.6 1.26 0.263/0.0461 0.86/1.95 0.175/0.0337 1.22/1.88

ϕ1 − ϕ2 e21 = (9) ϕ1

1.25e21 GCI21 = p (10) r21 − 1

where e21 is the relative error between the result from the coarse and the medium mesh, ϕ1 and ϕ2 are the results from the coarse and the medium mesh, respectively, 1.25 is the safety factor that is adopted when three or more meshes are used, r21 is the refinement ratio between the coarse and the fine mesh, and p is the order of numerical accuracy, which is 2 in this study. For the GCI between the medium and the fine mesh, the same procedures can be followed. The results show that for both thrust and torque the GCI values of the fine mesh are lower than that of the medium mesh, which demonstrates that the numerical uncertainty becomes smaller when the mesh is refined. The highest numerical uncertainty is reported within 5.23% and 1.95% for the medium mesh and fine mesh, respectively. Based on the mesh sensitivity study, the fine mesh is used as the final mesh for the present study. After the grid independence study, the numerical results of the ducted propeller are also compared with experimental data by Wang [11] to validate the CFD model. Figure5 provides an overview of the comparison between the numerical results from the fine mesh and experimental measurements under different advance coefficients. The former is presented in lines and the latter in dots. The curves show that the thrusts on the propeller and duct are predicted well by the simulations. However, the propeller torque is over predicted, causing a discrepancy with the experimental data. The difference in efficiency is within 5% for all the advance coefficients and a similar discrepancy is found in the research of Pawar [15]. The discrepancy can be attributed to experimental errors during measurements, physical modeling errors as the geometry is generated from the coordinates provided in Wang [11], and the errors caused by the turbulence model. Since the SST k-ω model assumes a fully turbulent boundary layer, which may not be the case on certain parts of the propeller surface as the rotational speed is rather small, this may be the reason for the exaggerated torque prediction. Appl. Sci. 2021, 11, x 4919 FOR PEER REVIEW 8 8of of 17

Figure 5. 5. ValidationValidation of ofthe the numerical numerical model model in this in thisstudy study by the by experimental the experimental results results of Wang of Wanget al. [11] et al.for [ 11the] forducted the propeller.ducted propeller.

3.3. Results Results and and Discussion Discussion InIn this this section, the local and global hydrodynamic characteristics are presented and comparedcompared for for the the DP DP and the RDT. For all the calculations, the rotational speed of the propeller is kept constant at 10 ss−−11 andand the the change change of of advance advance coefficient coefﬁcient J J is is achieved by varying the inflow inﬂow velocity. velocity. The Reynolds number, ranging from 6.15 × 10105 5toto 6.42 6.42 × ×10105, is5, determinedis determined based based on onthe the chord chord leng lengthth of ofthe the propeller propeller at at0.75R. 0.75R.

3.1.3.1. Open Open Water Water Performance Performance TheThe open open water water performance performance comparison between the ducted propeller and the RDT is given in Figure6, in which the propeller thrust coefficient, K , duct thrust coefficient, K , is given in Figure 6, in which the propeller thrust coefficient,tp Ktp, duct thrust coefficient,tn propeller torque coefficient, Kqp, and overall coefficients, Kt, Kq and efficiency are shown Ktn, propeller torque coefficient, Kqp, and overall coefficients, Kt, Kq and efficiency are shownbased onbased the hydrodynamicon the hydrodynamic coefficient coefficient definitions definitions as mentioned as mentioned before. before. The green The curves green curvesrepresent represent the hydrodynamic the hydrodynamic coefficients coefficients for the ducted for the propeller ducted propeller and the blue and curves the blue for the RDT. For the DP, the total thrust coefficient includes the propeller thrust coefficient and curves for the RDT. For the DP, the total thrust coefficient includes the propeller thrust duct thrust coefficient, and the torque coefficient includes the propeller torque coefficient, coefficient and duct thrust coefficient, and the torque coefficient includes the propeller while, for the RDT, the total thrust and torque coefficient also include an additional term, torque coefficient, while, for the RDT, the total thrust and torque coefficient also include i.e., the rim coefficient. an additional term, i.e., the rim coefficient. From Figure6a–c, in which the thrust and torque contribution from the rim is excluded From Figure 6a–c, in which the thrust and torque contribution from the rim is ex- for the RDT to make a clear comparison, it can be observed that the propeller thrust cluded for the RDT to make a clear comparison, it can be observed that the propeller thrust coefficient, duct thrust coefficient, and propeller torque coefficient for the RDT are smaller coefficient, duct thrust coefficient, and propeller torque coefficient for the RDT are smaller than those for the DP under all advance coefficients. In Figure6d, it can be seen that with than those for the DP under all advance coefficients. In Figure 6d, it can be seen that with the thrust contribution from the rim, the total thrust coefficient for the RDT is still below the thrust contribution from the rim, the total thrust coefficient for the RDT is still below the DP. On the other hand, the additional torque of the rim in the RDT makes the torque the DP. On the other hand, the additional torque of the rim in the RDT makes the torque coefficient of the RDT larger than the one of the DP. The overall effect is that the efficiency coefficientof the RDT of is the significantly RDT larger lower than comparedthe one of th toe the DP. DP The and overall the discrepancy effect is that increases the efficiency with ofan the increase RDT is in significantly the advance lower coefficient. compared The maximumto the DP and discrepancy the discrepancy is up to increases almost 0.18 with at anJ = increase 0.7, and, in at the the advance design point, coefficient. i.e., at The J = 0.5, ma theximum discrepancy discrepancy is about is up 0.15. to almost The efficiency 0.18 at J =curve 0.7, and, is always at the below design 0.4 point, for the i.e., RDT, at J and= 0.5, in the the discrepancy study of Dubas is about et al. 0.15. [16] similarThe efficiency results curvecan be is observed, always below in which 0.4 for a smaller the RDT, RDT and with in the a diameter study of ofDubas 150 mm et al. was [16] studied similar and results the canmaximum be observed, experimental in which efficiency a smaller wasRDT around with a diameter 0.15, which of 150 may mm suggest was studied that RDTs and with the maximumsmall dimensions experimental tend to efficiency have a poor was performance. around 0.15, which may suggest that RDTs with small dimensions tend to have a poor performance.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 17 Appl. Sci. 2021, 11, 4919 9 of 17

(a) (b)

FigureFigure 6. 6.OpenOpen water water performance performance comparison comparison between between the the ducted ducted propeller propeller (DP) (DP) and and the the RDT: RDT: (a) ( apropeller) propeller thrust thrust coefficient;coefficient; (b) duct (b) thrust duct thrust coefficient; coefficient; (c) propeller (c) propeller torque torque coefficient; coefficient; (d) ( doverall) overall coefficient. coefficient.

TheThe thrust thrust and and torque generatedgenerated on on the the propeller propeller are are caused caused by by pressure pressure differences differences acrossacross the the blades blades and viscous shearshear stresses. stresses. Table Table3 gives 3 gives an an overview overview of theof the contribution contribution ofof pressure pressure and and viscosity toto thethe thrustthrust and and torque torque production production on on the the propeller propeller for for two two extremeextreme off-design off-design points. ItIt cancan bebe seenseen that that pressure pressure is is the the dominant dominant factor factor in generatingin generating bothboth thrust thrust and and torque andand thatthat thethe viscous viscous shear shear is is similar similar for for both both the the ducted ducted propeller propeller andand the the RDT. RDT. The pressure distributionsdistributions on on the the propeller’s propeller’s pressure pressure and and suction suction side side are are providedprovided in Figure7 .7. The The pressure pressure distribution distributi onon theon suctionthe suction side displaysside displays a similar a similar pattern pat- except for a small low-pressure region found near the leading edge at the propeller’s tip tern except for a small low-pressure region found near the leading edge at the propeller’s for the RDT (the propeller is rotating in the anti-clockwise direction in this view). On tip for the RDT (the propeller is rotating in the anti-clockwise direction in this view). On the pressure side, the main difference can also be found near the leading edge where the thelocation pressure and side, area ofthe the main high-pressure difference region can also are slightlybe found different near the (the leading propeller edge is rotating where the locationin the clockwise and area direction of the high-pressure in this view). region are slightly different (the propeller is rotating in the clockwise direction in this view). To quantitatively understand the pressure difference on the propeller surface of the DP and the RDT, the pressure distribution on the blade at various radial cross-sections is presented in Figure 8, in which the pressure coefficient is given as a function of the normalized chord-wise direction z. Coordinate 0 represents the center position of the propeller. It can be observed that the pressure coefficient of the RDT is generally smaller than

Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 17

the one of the ducted propeller at different sections. In addition, the overall pressure difference on the propeller’s pressure and suction sides for the RDT is also smaller compared to the ducted propeller, which explains the lower thrust production. It is noted that for r/R = 0.2 and 0.4, namely the areas near the propeller blade root, the discrepancy for the pressure difference between the pressure and the suction sides is mainly situated near the Appl. Sci. 2021, 11, 4919 propeller’s leading edge, where the highest pressure can be found. However, for r/R10 = of0.6 17 and 0.8, namely near the propeller’s blade tip, the discrepancy for the pressure difference extends almost along the entire chord length.

TableTable 3. 3. ContributionContribution of of pressure pressure an andd viscosity viscosity to to the the thrust thrust and and torque torque on on the the propeller. propeller.

Advance DPDP RDT RDT Advance Coefficient Contributor Contributor Coefﬁcient ThrustThrust (N) (N) Torque Torque (N·m) (N·m) Thrust Thrust (N) (N) Torque Torque (N·m) (N·m) PressurePressure 97.41 97.41 3.87 3.87 90.5 90.5 3.56 3.56 JJ = = 0.1 0.1 ViscousViscous −−2.052.05 0.43 0.43 −−1.991.99 0.4 0.4 Pressure 46.57 2.01 41.45 1.83 J = 0.7 Pressure 46.57 2.01 41.45 1.83 J = 0.7 Viscous −2.16 0.44 −2.11 0.41 Viscous −2.16 0.44 −2.11 0.41

(a) (b)

FigureFigure 7. 7. ComparisonComparison of the pressurepressure distributiondistribution on on the the propeller’s propeller’s (a ()a suction) suction side side for for the the DP, DP, (b) suction(b) suction side side for thefor RDT,the RDT,(c) pressure (c) pressure side forside the for DP, the and DP, ( dand) pressure (d) pressure side for side the for RDT the atRDT J = 0.1.at J = 0.1.

To quantitatively understand the pressure difference on the propeller surface of the DP and the RDT, the pressure distribution on the blade at various radial cross-sections is presented in Figure8, in which the pressure coefﬁcient is given as a function of the normalized chord-wise direction z. Coordinate 0 represents the center position of the propeller. It can be observed that the pressure coefﬁcient of the RDT is generally smaller than the one of the ducted propeller at different sections. In addition, the overall pressure difference on the propeller’s pressure and suction sides for the RDT is also smaller compared to the ducted Appl. Sci. 2021, 11, 4919 11 of 17

propeller, which explains the lower thrust production. It is noted that for r/R = 0.2 and 0.4, namely the areas near the propeller blade root, the discrepancy for the pressure difference between the pressure and the suction sides is mainly situated near the propeller’s leading Appl. Sci. 2021, 11, x FOR PEER REVIEWedge, where the highest pressure can be found. However, for r/R = 0.6 and 0.8, namely11 of 17

near the propeller’s blade tip, the discrepancy for the pressure difference extends almost along the entire chord length.

(a) (b)

Figure 8. 8. ComparisonComparison of of the the pressu pressurere distribution distribution onon the the blade’s blade’s radial radial cross-sections cross-sections at: ( at:a) r/R (a) = r/R 0.2; =(b 0.2;) r/R (b =) 0.4; r/R (c =) r/R 0.4; =(c 0.6;) r/R and = 0.6;(d) andr/R = ( d0.8) r/R at J == 0.80.1. at J = 0.1.

ToTo examine the the thrust thrust difference difference on on the the du ductct for for the the DP DP and and th thee RDT, RDT, the pressure distributionsdistributions at at three three representative representative sections sections along along the the longitudinal direction direction of of the duct areare studied, studied, in in which which the the rim rim is is not not included included for for the the RDT. RDT. The locations of these sections areare situated situated at at the the propeller’s propeller’s leading leading edge, edge, middle middle plane, plane, and and trailing trailing edge, as indicated inin Figure Figure 99a.a. The ﬁguresfigures exhibit the pressure coefﬁcientcoefficient asas aa functionfunction ofof positionposition alongalong thethe duct’sduct’s normalized chord-wisechord-wise direction,direction, where where position position 0 corresponds0 corresponds to theto the location location of theof thepropeller. propeller. It can It becan seen be seen that thethat pressure the pressure distribution distribution on the on duct’s the du exteriorct’s exterior side is side nearly is nearlythe same the for same both for the both DP the and DP the and RDT the and RDT that and the that pressure the pressure discrepancy discrepancy is mainly is situatedmainly situatedon the interior on the sideinterior of the side duct. of the Near duct. the Ne leadingar the edge,leading the edge, pressure the pressure difference difference between betweenthe DP and the the DP RDT and isthe not RDT so obvious; is not so however, obvious; at however, the middle at planethe middle and the plane trailing and edge, the trailing edge, the pressure difference for the ducted propeller is significantly larger than that of the RDT, resulting in a larger thrust. It can also be noted that at the interior side of the duct there are two sudden changes in pressure, which are caused by the rim gap. There is always a steep change in pressure near the blade location (Coordinate 0) for the DP because of the huge pressure gradient between the propeller’s pressure and suction sides. However, for the RDT, the pressure gradient in the rim gap along the axial direction remains almost constant, which is expected for Couette flow.

Appl. Sci. 2021, 11, 4919 12 of 17

the pressure difference for the ducted propeller is signiﬁcantly larger than that of the RDT, resulting in a larger thrust. It can also be noted that at the interior side of the duct there are two sudden changes in pressure, which are caused by the rim gap. There is always a steep change in pressure near the blade location (Coordinate 0) for the DP because of the huge pressure gradient between the propeller’s pressure and suction sides. However, Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 17 for the RDT, the pressure gradient in the rim gap along the axial direction remains almost constant, which is expected for Couette ﬂow.

(a) (b)

Figure 9. The locations of the different sections on the duct (a) and comparisons of pressure distributions in the section at the propeller’s ( b) leading edge (Sec-L), ( c) middle plane (Sec-M), and ( d) trailing edge (Sec-T) between the RDT and the ducted propeller (DP) at J = 0.1. ducted propeller (DP) at J = 0.1. 3.2. Tip Leakage Flow in the Ducted Propeller 3.2. Tip Leakage Flow in the Ducted Propeller The tip clearance between the propeller tip and the duct inner wall is necessary for The tip clearance between the propeller tip and the duct inner wall is necessary for the the regular operation of the DP. However, it can pose some flow features, such as compli- regular operation of the DP. However, it can pose some flow features, such as complicated cated vortical structures, which are usually undesirable. These vortical structures are of- vortical structures, which are usually undesirable. These vortical structures are often ten generated by the tip leakage flow and its interaction with surrounding boundaries and generated by the tip leakage flow and its interaction with surrounding boundaries and can can cause rotational instabilities and blockage in the flow passage, which affects the over- cause rotational instabilities and blockage in the flow passage, which affects the overall all performance [17]. performance [17]. Figure 10 displays the velocity vector field at the blade tip region with the pressure contours as background for the DP and the RDT at J = 0.1 and J = 0.5 at Sec-M, as defined in Figure 9a. It is shown in Figure 10a,b that when approaching the tip clearance region, the boundary layer on the duct’s inner surface starts to separate due to an adverse pressure gradient, and the separation occurs earlier at J = 0.1 than at J = 0.5, as a heavier loading condition means a larger pressure gradient. On the other hand, the flow on the propeller tip separates at the pressure side of the propeller and interacts with the separated boundary layer on the duct’s interior side. It can be clearly observed that vortical structures are formed in this region and, with increasing fluid momentum, these vortices are forced to

Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 17

Appl. Sci. 2021, 11, 4919 move to the propeller suction side and become smaller. In contrast, as shown in Figure13 of 17 10c,d, the flow field near the blade tip for the RDT is less complex. According to their features, You [17] roughly classified vortical structures into three categories, namely the tip leakage vortex, the tip separation vortex, and the induced vortex. TheFigure tip leakage10 displays vortex the is velocity strongly vector influe fieldnced at by the the blade pressure tip region difference with thebetween pressure the propeller’scontours as pressure background and forsuction the DPsides, and and the the RDT induced at J = 0.1 vortex and Jis = generated 0.5 at Sec-M, alongside as defined the predominantin Figure9a. Ittip is leakage shown invortex. Figure The 10 tipa,b separation that when vortex approaching is formed the by tip the clearance separated region, flow the boundary layer on the duct’s inner surface starts to separate due to an adverse pressure from the propeller tip. As a result, these vortical structures interact with each other and gradient, and the separation occurs earlier at J = 0.1 than at J = 0.5, as a heavier loading cause viscous loss in development, which influences the performance of the propeller. condition means a larger pressure gradient. On the other hand, the flow on the propeller The Q-criterion for vortex identification [18] is introduced to help visualize the vor- tip separates at the pressure side of the propeller and interacts with the separated boundary tical structures in the tip gap flow. Figure 11 displays the three characteristic vortical struc- layer on the duct’s interior side. It can be clearly observed that vortical structures are formed tures at J = 0.1 and J = 0.5. As expected, the vortex scale at J = 0.5 is smaller than the one at in this region and, with increasing fluid momentum, these vortices are forced to move to J = 0.1, and it can also be observed that in the downstream direction from the trailing edge the propeller suction side and become smaller. In contrast, as shown in Figure 10c,d, the the tip separation vortex starts to interact with the tip leakage vortex. flow field near the blade tip for the RDT is less complex.

(a) (b)

Figure 10. Velocity ﬁeldsfields nearnear thethe bladeblade tiptip for:for: ((aa)) DPDP atat JJ == 0.1; (b) DP at J = 0.5; ( c) RDT at J = 0.1; 0.1; and and ( (d)) RDT RDT at at J J = = 0.5. 0.5.

According to their features, You [17] roughly classified vortical structures into three categories, namely the tip leakage vortex, the tip separation vortex, and the induced vortex. The tip leakage vortex is strongly influenced by the pressure difference between the propeller’s pressure and suction sides, and the induced vortex is generated alongside the predominant tip leakage vortex. The tip separation vortex is formed by the separated flow from the propeller tip. As a result, these vortical structures interact with each other and cause viscous loss in development, which influences the performance of the propeller. The Q-criterion for vortex identification [18] is introduced to help visualize the vortical structures in the tip gap flow. Figure 11 displays the three characteristic vortical structures at J = 0.1 and J = 0.5. As expected, the vortex scale at J = 0.5 is smaller than the one at J = 0.1, and it can also be observed that in the downstream direction from the trailing edge the tip separation vortex starts to interact with the tip leakage vortex.

3.3. Gap Flow in the RDT There is no tip leakage flow in the RDT as no tip clearance exists between the propeller tip and duct. However, the gap formed by the rim surface and duct interior surface in the (aRDT) can cause interesting flow phenomena as well. The( gapb) flow in the RDT resembles the flow of a viscous fluid between two concentric cylinders, with one rotating and the other one static. The major influence of the gap on the flow field is the creation of a significant friction torque, which is mainly related to the width of the gap, the dimensions of the inner cylinder, and the Reynolds number of the gap flow. In the work of Bilgen Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 17

move to the propeller suction side and become smaller. In contrast, as shown in Figure 10c,d, the flow field near the blade tip for the RDT is less complex. According to their features, You [17] roughly classified vortical structures into three categories, namely the tip leakage vortex, the tip separation vortex, and the induced vortex. The tip leakage vortex is strongly influenced by the pressure difference between the propeller’s pressure and suction sides, and the induced vortex is generated alongside the predominant tip leakage vortex. The tip separation vortex is formed by the separated flow from the propeller tip. As a result, these vortical structures interact with each other and cause viscous loss in development, which influences the performance of the propeller. The Q-criterion for vortex identification [18] is introduced to help visualize the vortical structures in the tip gap flow. Figure 11 displays the three characteristic vortical structures at J = 0.1 and J = 0.5. As expected, the vortex scale at J = 0.5 is smaller than the one at J = 0.1, and it can also be observed that in the downstream direction from the trailing edge the tip separation vortex starts to interact with the tip leakage vortex.

Appl. Sci. 2021, 11, 4919 14 of 17

and Boulos [19], empirical formulations for the torque coefficient for a flow without an (a)axial pressure gradient were developed based on experimental(b) results. As the flow regime changes from laminar to turbulent flow at different working conditions, the functional relation for the torque coefficient has to be adjusted. Therefore, only the formulas applied to the present work are considered and are listed as follows:

ρωR (R − R ) Re = i o i (11) r µ 0.3 Ro − Ri −0.2 4 CMr = 0.065 Rer (Rer > 10 ) (12) (c) Ri (d) 2 4 Figure 10. Velocity fields near the blade tip for: (a) DP at J = 0.1; (Mb)r DP= 0.5at Jπρω = 0.5; R(ci) LCRDTMr at J = 0.1; and (d) RDT at J = 0.5. (13)

(a) (b)

Figure 11. Vortical structures along the propeller tip region at (a) J = 0.1 and (b) J = 0.5.

where Rer is the Reynolds number for the radial gap, ρ is the fluid density, ω is the angular velocity, Ri is the radial gap inner radius, Ro is the radial gap outer radius, Ro − Ri is the gap width, µ is the dynamic viscosity, CMr is the torque coefficient, Mr is the torque of the radial gap, and L is the length of the rim. A comparison of the torque coefficient between the simulations in this paper and the empirical method in Equations (9)–(11) is provided in Table4. It can be seen that the torque calculated by the empirical formulas is smaller than the one of the numerical simulations. This may be attributed to the existence of an axial pressure gradient, as it was also observed in the study of Cao [20] that the rim torque with a pressure gradient was larger than without. It is assumed that the interaction between the radial gap flow and axial gap flow is enhanced by the pressure gradient, which makes the flow regime more complex, resulting in the production of a larger torque.

Table 4. Comparison of the rim torque coefﬁcient between empirical correlations and simulations.

2 Methods Ktr × 10 Numerical 0.57 Empirical 0.37

In the study of Yamada [21], the effects of the axial pressure gradient on the torque coefﬁcient were examined under different working conditions. It was found that when the characteristic Reynolds number for the axial pressure gradient is above 5 × 103 the torque coefﬁcient increases steadily with the increase in the pressure gradient. In the present study, the Reynolds number is much higher than the critical value, and the corresponding results Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 17

Appl. Sci. 2021, 11, 4919 15 of 17

are observed, as shown in Figure 12. When the propeller loading is heavy, as mentioned Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 17 above, the pressure difference is more signiﬁcant and the rim torque, due to gap ﬂow, is larger.

Figure 12. Torque of the rim under different loading conditions.

The velocity profile in the gap along the axial and tangential direction is shown in Figure 13. A small velocity gradient with respect to the wall is observed near the rim outer surface, which is caused by the centrifugal force as the rim rotates. In the axial direction, the flow is pressure driven and due to the location of the rim ends the flow runs in the opposite direction to the main flow. However, the velocity profile in the tangential direction is quite different from the one in the axial direction; the flow is viscosity based, as the Figure 12. Torque ofof thethe rimrim underunder differentdifferent loadingloading conditions.conditions. influence of the pressure difference along this direction is insignificant compared to the shearThe stress. velocity profileprofile in the gapgap alongalong thethe axialaxial andand tangentialtangential directiondirection isis shownshown inin Figure 1313.. A small velocity gradient with respectrespect toto the wall is observed nearnear thethe rim outer Table 4. Comparison of the rim torque coefficient between empirical correlations and simulations. surface, whichwhich isis caused caused by by the the centrifugal centrifugal force forc ase as the the rim rim rotates. rotates. In theIn the axial axial direction, direction, the flow is pressure driven and due to the location of the rim ends the flow runs in the opposite the flow is pressureMethods driven and due to the location of the rimKtr ends× 102 the flow runs in the direction to the main flow. However, the velocity profile in the tangential direction is quite opposite directionNumerical to the main flow. However, the velocity profile0.57 in the tangential direc- different from the one in the axial direction; the flow is viscosity based, as the influence of tion is quite differentEmpirical from the one in the axial direction; the flow0.37 is viscosity based, as the the pressure difference along this direction is insignificant compared to the shear stress. influence of the pressure difference along this direction is insignificant compared to the shear stress.

Table 4. Comparison of the rim torque coefficient between empirical correlations and simulations.

Methods Ktr × 102 Numerical 0.57 Empirical 0.37

(a) (b)

FigureFigure 13. 13. VelocityVelocity distribution in the gap along the (a) axial direction and (b)) tangentialtangential direction.direction.

From Table Table 55,, the contribution of of pressure pressure and and viscosity viscosity to to the the torque torque production production on theon thepropeller propeller and and rim rimare arequite quite distinct. distinct. For the For propeller, the propeller, the pressure the pressure is dominant is dominant and contributesand contributes nearly nearly 90% to 90% the tototal the torque, total torque, while the while viscous the viscous stress is stress almost is fully almost respon- fully sibleresponsible for the torque for the generation torque generation on the rim, on thewhich rim, coincides which coincides with the observation with the observation in Figure (a) 12b.in Figure 12b. (b) Figure 13. Velocity distribution in the gap along the (a) axial direction and (b) tangential direction. Table 5. Contribution of pressure- and viscosity-basedviscosity-based torque onon thethe propellerpropeller andand rim.rim.

FromJ = 0 TableJ = 0 5, the contribution Propeller of (N·m) Propeller pressure (N and·m) viscosity to the Rim torque Rim (N·m) (N production· m) on the PressurepropellerPressure and rim are quite distinct.3.6 For 3.6 the propeller, the pressure≈0 is≈ 0dominant and contributesViscosityViscosity nearly 90% to the total0.4 torque, wh 0.4ile the viscous stress is almost 1.2 1.2 fully respon- sible for the torque generation on the rim, which coincides with the observation in Figure 12b. 4. Conclusions Conclusions

TableThe 5. Contribution present work of pressure- compares and the viscosity-b hydrodynamicased torque performance on the propeller of aand DP rim. and an RDT with the same ﬂow conﬁguration, i.e., a Ka-47 propeller inside a 19A duct by CFD. Three J = 0 Propeller (N·m) Rim (N·m) Pressure 3.6 ≈0 Viscosity 0.4 1.2

4. Conclusions

Appl. Sci. 2021, 11, 4919 16 of 17

sets of meshes are generated to perform a mesh sensitivity analysis and, based on the results, the fine mesh is used for the study as more detailed flow features can be resolved. Steady-state simulations are carried out for the DP and the RDT with the MRF approach handling the rotational movement of the propeller. The open water performance and the main flow features of the DP and the RDT are presented. From the results, the DP has an overall better performance than the RDT. The propeller’s thrust and duct thrust coefficients of the ducted propeller are larger than those of the RDT under all considered advance coefficients. However, in the RDT, the existence of the rim and induced gap flow has much more detrimental than beneficial effects on the hydrodynamic performance of the RDT, resulting in a significant decrease in efficiency compared to the DP. Therefore, it is important to carefully choose appropriate parameters for the gap for the consideration of performance improvement. The loss caused by the rim can be roughly evaluated with existing empirical formulas; however, due to over-simplification of the actual flow, the calculated results are underestimated. The tip leakage flow is an important phenomenon in the DP. Complicated flow features can be caused by mixing of the main flow and the pressure-induced flow. In the tip clearance region, the interaction between the boundary layer of the duct’s interior side and the propeller tip greatly promotes the formation of vortical structures. Three characteristic vortices are identified with the Q-criterion, namely the tip leakage vortex, tip separation vortex, and the induced vortex. On the other hand, the gap flow in the RDT is also representative. By investigating the velocity profile in the gap, the flow in the gap is dominated by pressure differences in the axial direction and is shear stress based in the tangential direction.

Author Contributions: Conceptualization, B.L. and M.V.; methodology, B.L.; software, M.V.; validation, B.L. and M.V.; formal analysis, B.L.; investigation, B.L.; resources, B.L. and M.V.; data curation, B.L.; writing—original draft preparation, B.L.; writing—review and editing, M.V.; visualization, B.L.; supervision, M.V.; project administration, M.V.; funding acquisition, M.V. All authors have read and agreed to the published version of the manuscript. Funding: The authors thank the China Scholarship Council (CSC) for their financial support for the first author (Grant No. 201806950010). Data Availability Statement: Data will be available upon request. Acknowledgments: The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation—Flanders (FWO) and the Flemish Government—Department EWI. Conflicts of Interest: The authors declare no conflict of interest.

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