Cavitation and Induced Excitation Force of Ice-Class Propeller Blocked by Ice

Journal of Marine Science and Engineering

Article Cavitation and Induced Excitation Force of Ice-Class Propeller Blocked by Ice

Pei Xu 1, Chao Wang 1 , Liyu Ye 1,*, Chunyu Guo 1, Weipeng Xiong 1 and Shen Wu 2

1 College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China; [email protected] (P.X.); [email protected] (C.W.); [email protected] (C.G.); [email protected] (W.X.) 2 National Key Laboratory on Ship Vibration and Noise, China Ship Scientiﬁc Research Center, Wuxi 214082, China; [email protected] * Correspondence: [email protected]; Tel.: +86-18-845-596-576

Abstract: The presence of broken ice in the flow field around a propeller causes severe blade erosion, shafting, and hull vibration. This study investigates the performance of the propeller of a ship sailing in the polar regions under the propeller–ice non-contact condition. To this end, we construct a test platform for the propeller-induced excitation force due to ice blockage in a large circulating water channel. The hydrodynamic load of the propeller, and the cavitation and propeller- induced fluctuating pressure, were measured and observed by varying the cavitation number and ice–propeller axial distance under atmospheric pressure and decompression conditions. The results show that the fluctuation range of the blade load increases with a decrease in cavitation number and

ice–propeller axial distance. The decrease in the cavitation number leads to broadband characteristics in the frequency-domain curves of the propeller thrust coefﬁcient and blade-bearing force. Under the

Citation: Xu, P.; Wang, C.; Ye, L.; combined effects of ice blockage and proximity, propeller suction, the circumfluence zone around Guo, C.; Xiong, W.; Wu, S. Cavitation the ice, and the Pirouette effect, propeller–hull vortex cavitation is generated between the ice and and Induced Excitation Force of propeller. The decrease in cavitation number leads to a sharp increase in the amplitude of the Ice-Class Propeller Blocked by Ice. J. high-order frequency of the propeller-induced fluctuating pressure. Mar. Sci. Eng. 2021, 9, 674. https:// doi.org/10.3390/jmse9060674 Keywords: ice-class propeller; blockage condition; hydrodynamic load; cavitation; fluctuating pressure Academic Editors: Francesco Salvatore, Fabio Di Felice and Francisco Alves Pereira 1. Introduction Received: 11 May 2021 When polar-class ships navigate in frigid zones, broken fragments of ice frequently Accepted: 15 June 2021 Published: 19 June 2021 sink along the hull and gradually flow against its surface. The broken ice slowly approaches the propeller and interacts with it. This induces extreme loads on the latter, leading to a

Publisher’s Note: MDPI stays neutral large cavitation area on the back of the blade and reducing its performance. Blockage of the with regard to jurisdictional claims in propeller’s intake by ice aggravates the non-uniformity of the wake field. Consequently, published maps and institutional affil- the propeller begins to generate more severe periodic excitation forces. The blades bear iations. the excitation force and transmit it to the hull, resulting in fatigue damage to the main components of the ship, difficulties in onboard operations, equipment failure, poor crew comfort and reduced safety, etc. [1,2]. Therefore, the study of cavitation and the induced excitation force of ice-class propellers under ice blockage is of significance to ship and marine engineering. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. Research on the cavitation performance of ice-class propellers mainly involves ex- This article is an open access article perimentation. For example, Lindroos and Bjorkestam [3] first simulated the cavitation distributed under the terms and phenomenon of a propeller with ice–propeller interactions in a cavitation tunnel. They conditions of the Creative Commons placed a flat plate in front of a ducted propeller to simulate the ice blockage effect. The Attribution (CC BY) license (https:// results showed that the blocked flow increased the vibrations, cavitation, thrust, and torque creativecommons.org/licenses/by/ of the propeller. They also demonstrated the importance of cavitation for studying the 4.0/). ice–propeller interactions. Walker et al. [4] discussed the influence of cavitation on the

J. Mar. Sci. Eng. 2021, 9, 674. https://doi.org/10.3390/jmse9060674 https://www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2021, 9, 674 2 of 25

hydrodynamic load of an R-class propeller under blockage conditions and concluded that blocked flow can increase the propeller thrust and reduce the total thrust and efficiency of the system. Cavitation can reduce the average thrust and torque. With a decrease in cavitation number, the load of the propeller will became unstable. The variation law of propeller hydrodynamic load with cavitation is analyzed emphatically. Doucet et al. [5] conducted an experimental study on cavitation erosion of open and ducted propellers under a blockage condition in a cavitation tunnel. The test results showed that given the same test conditions, the ducted propeller experienced more erosion than the open propeller. For both propellers, the amount of face erosion increased with increasing advance coefficient. The influence of cavitation on the erosion of two ice-class propellers was analyzed. Atlar et al. [6] and Sampson et al. [7,8] carried out blockage experiments on an ice-podded propulsor in a cavitation tunnel using unfrozen model ice. Thus, they demonstrated the variation curves of propeller hydrodynamics under different advance coefficients, cavitation numbers, axial distances, and depth of model ice recess. The results showed that blockage effect of ice blocks leads to the generation of sheet cavitation, tip vortex cavitation, and cloud cavitation. Cavitation changed the thrust and torque of the propeller, produced severe vibration and noise, and exposed the propeller and related equipment to potential fatigue-related hazards. They verified that the cavitation effect is an important factor in studying the mechanism of ice–propeller interaction. Wu et al. [9] conducted an experiment on the influence of parameters related to ice blockage, i.e., the axial and vertical distances between the propeller and ice, on the hydrodynamic performance of the propeller. The propeller hydrodynamics did not change in any discernible manner with the ice–propeller distance even under heavy loads during cavitation. The cavitation effect of the blades reduced the degree of influence of the ice blockage effect. The above study focuses on the influence of cavitation on the hydrodynamic load of the whole propeller under the blockage condition; however, it does not analyze the change in the single blade load in the blocked or unblocked regions or the influence of the change in the single blade surface cavitation on the hydrodynamic load. To summarize, the mechanisms of cavitation and load change under sea-ice blockage are not clearly understood. Moreover, there are a few significant results related to the induced excitation force of ice-class propellers. Numerical simulation methods such as CFD were the primary mode of analysis. For example, Wu et al. [10] simulated the unsteady cavitation and hydrodynamic performance of conventional propellers using the Reynolds-averaged Navier–Stokes (RANS) equations. They considered the influence of different ice axial positions and pressure environments in their simulation. The study mainly analyzed the fluctuating amplitude of the propeller thrust coefficient. The results showed that the order of the propeller cavitation excitation force significantly increased owing to ice blockage. The frequency of excitation moved to a higher order. Wang et al. [2,11] established a numerical model of the propeller-induced excitation force under ice–propeller interactions using the overlapping grid method. They analyzed the changes in the bearing force of the propeller and fluctuating pressure around the propeller with different advance coefficients. The results showed that the fluctuating amplitude of the bearing force gradually increased and that of the fluctuating pressure decreased with an increase in the advance coefficient under ice-blockage conditions. However, they did not analyze the variation in the propeller fluctuating pressure under different positions of ice at the bottom of the ship and under heavy load cavitation conditions, and the influence of ice–propeller interaction on ship vibration has not been revealed. In view of this, we used a scaled model instead of a purely numerical model. To conduct the experiment, a test platform for induced excitation force of propeller under ice-blockage condition was built in a large circulating water tank. We study the influence of ice-blockage parameters, ice–propeller axial distance and cavitation number, on the propeller hydrodynamic performance, cavitation, and excitation force under different J.J. Mar.Mar. Sci.Sci. Eng.Eng. 20212021,, 99,, xx FORFOR PEERPEER REVIEWREVIEW 33 ofof 55

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operatingoperating conditions.conditions. TheThe modelmodel testtest providedprovided datadata toto predictpredict thethe propellerpropeller performanceperformance operatinginin aa polarpolar conditions. environment.environment. The model test provided data to predict the propeller performance in a polar environment. 2.2. TestTest ModelModel andand TestTest EquipmentEquipment 2. Test Model and Test Equipment 2.1.2.1. PropellerPropeller ModelModel 2.1. Propeller Model TheThe modelmodel forfor thisthis studystudy waswas aa 1:16.461:16.46 scaledscaled andand locallylocally correctedcorrected versionversion ofof thethe iceice‐‐ classclassThe propellerpropeller model ofof for thethe this CanadianCanadian study was CoastCoast a 1:16.46 GuardGuard scaled RR‐‐classclass and icebreakericebreaker locally [6,12,13]. corrected[6,12,13]. TheThe version diameterdiameter of the ofof ice-class propeller of the Canadian Coast Guard R-class icebreaker [6,12,13]. The diameter thethe fullfull‐‐scalescale propellerpropeller isis 4.1154.115 m.m. Therefore,Therefore, thethe modelmodel hadhad aa diameterdiameter DD ofof 0.250.25 m,m, fourfour of the full-scale propeller is 4.115 m. Therefore, the model had a diameter D of 0.25 m, four blades,blades, aa propellerpropeller areaarea ratioratio ofof 0.699,0.699, aa pitchpitch ratioratio ofof 0.775,0.775, andand aa hubhub diameterdiameter ratioratio ofof blades, a propeller area ratio of 0.699, a pitch ratio of 0.775, and a hub diameter ratio of 0.368. 0.368.0.368. TheThe modelmodel waswas mademade ofof anan AlAl alloy.alloy. BecauseBecause thethe objectiveobjective ofof thethe studystudy requiredrequired The model was made of an Al alloy. Because the objective of the study required testing testingtesting thethe loadload onon aa singlesingle blade,blade, eacheach bladeblade ofof thethe propellerpropeller modelmodel waswas designeddesigned andand the load on a single blade, each blade of the propeller model was designed and fabricated fabricatedfabricated separatelyseparately (Figure(Figure 1).1). FigureFigure 22 illustratesillustrates thethe propellerpropeller modelmodel afterafter bladeblade instalinstal‐‐ separately (Figure1). Figure2 illustrates the propeller model after blade installation. lation.lation.

FigureFigureFigure 1. 1.1.Single SingleSingle blades. blades.blades.

FigureFigureFigure 2. 2.2.Propeller PropellerPropeller model modelmodel photograph photographphotograph and andand three-dimensional threethree‐‐dimensionaldimensional schematic schematicschematic diagram. diagram.diagram.

2.2.2.2.2.2. Test TestTest Equipment EquipmentEquipment TheTheThe propeller propellerpropeller cavitation cavitationcavitation and andand induced inducedinduced excitation excitationexcitation force forceforce under underunder ice-blockage iceice‐‐blockageblockage conditions conditionsconditions werewerewere measured measuredmeasured in inin a aa large largelarge circulating circulatingcirculating water waterwater channel channelchannel of ofof the thethe China ChinaChina Ship ShipShip Science ScienceScience Research ResearchResearch Center.Center.Center. FigureFigure3 33 illustrates illustratesillustrates thethe shape shape ofof of thethe the testtest test sectionsection section ofof of thethe the channel.channel. channel. TheThe The circulatingcirculating circulating tanktank tankatat thethe at CenterCenter the Center isis thethe is largestlargest the largest cavitationcavitation cavitation experimentalexperimental experimental facilityfacility facility inin ChinaChina in China andand offersoffers and offerslowlow turbuturbu low‐‐ turbulencelencelence andand backgroundbackground and background noise.noise. noise. TheThe facilityfacility The facility cancan bebe can usedused be toto used measuremeasure to measure hydrodynamichydrodynamic hydrodynamic perforperfor‐‐ performance,mance,mance, pressurepressure pressure fluctuations,fluctuations, fluctuations, andand noise,noise, and and noise,and makemake and cavitationcavitation make cavitation observationsobservations observations forfor allall typestypes for all ofof typesunderwaterunderwater of underwater vehiclesvehicles vehicles andand wholewhole and shipship whole modelsmodels ship with modelswith thrusters.thrusters. with thrusters. ItIt alsoalso hashas It also equipmentequipment has equipment toto measmeas‐‐ toureure measure frequencyfrequency frequency responsesresponses responses inin thethe interactionsinteractions in the interactions ofof offshoreoffshore of engineeringengineering offshore engineering structuresstructures structuresandand fluids.fluids. andMoreover,Moreover, fluids. itit Moreover, cancan achieveachieve it canthethe experimentsexperiments achieve the under experimentsunder thethe numbernumber under ofof the realreal number shipship cavitation,cavitation, of real ship butbut cavitation,thethe conventionalconventional but the cavitation conventionalcavitation tunneltunnel cavitation andand circulatingcirculating tunnel and waterwater circulating channelchannel water areare difficultdifficult channel to areto meet.meet. difficult TheThe to meet. The length, width, and height of the working section of the large circulating

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length,water channelwidth, and were height 10.5 mof ×the2.2 working m × 2.0 section m, the of pressure the large was circulating 10–400 kPa,water the channel water werevelocity 10.5 was m × 1.0–15.02.2 m × 2.0 m/s, m, the pressure non-uniformity was 10–400 of velocity kPa, the was water less velocity than 1.0%, was 1.0–15.0 and the m/s,minimum the non cavitation‐uniformity number of velocity was 0.07 was (the less water than velocity1.0%, and was the 15 minimum m/s, and cavitation the top pressure num‐ ber10 kPa)was 0.07 [14]. (the water velocity was 15 m/s, and the top pressure 10 kPa) [14].

FigureFigure 3. 3. TestTest section section of of the the large large circulating circulating water water channel. channel.

FigureFigure 4 illustratesillustrates thethe ice-blockage ice‐blockage device device and and the the force-measuring force‐measuring device. device. A A deflector deflec‐ torwas was installed installed in the in the upstream upstream direction. direction. A C-frame A C‐frame and and a fixing a fixing plate plate were were tightly tightly connected con‐ nectedto clamp to clamp and fix and the fix model the model ice. Before ice. Before the model the model test, we test, tested we thetested reliability the reliability of the iceof theclamber ice clamber to ensure to ensure the model the model ice did ice not did detach not detach and fall. and Thefall. heightThe height of the of icethe chamber’s ice cham‐ ber’sdriving driving mechanism mechanism was was adjusted adjusted by a by vertical a vertical linear linear module, module, with with a range a range of of motion motion of ofh =h 0–70= 0–70 mm. mm.The The axial axial distance distance was was adjusted adjusted by a by horizontal a horizontal linear linear module, module, with awith range a rangeof motion of motion of L = of 0–400 L = 0–400 mm. mm. The The entire entire driving driving device device was was connected connected to to the the circulating circulat‐ ingwater water channel’s channel’s mounting mounting plate plate through through the the mobile mobile device device mounting mounting bracket. bracket. TheThe dynamometer dynamometer was was powered powered by by a a motor. motor. It It was was used used to to rotate rotate the the propeller propeller shaft shaft viavia the the bevel bevel gear. gear. A A force force-balance‐balance device device was was installed installed on on the the right right side side of of the the sliding sliding bearingbearing to to measure measure the the thrust thrust and and torque torque of of the the whole whole propeller. propeller. The The thrust thrust and and torque torque werewere in in the the ranges ranges of of 0–1500 0–1500 N Nand and 0–50 0–50 N∙m, N· m,respectively. respectively. The Thehub huband andblade blade were were ma‐ chinedmachined separately. separately. The Thehub hubwas washollow hollow and andequipped equipped with witha five a‐component five-component force force bal‐ ance.balance. The The key key blade blade was was fixedly fixedly connected connected to tothe the force force balance. balance. The The roots roots of of the the other other bladesblades were were equipped withwith counterweightcounterweight structures, structures, which which were were fixedly fixedly connected connected to theto thehub hub screw screw to realize to realize the the overall overall dynamic dynamic balance balance of the of hub.the hub. The The force force and thrustand thrust values val of‐ uesthe of five-component the five‐component force balanceforce balance were inwere the in ranges the ranges of 0–800 of 0–800 N, 0–800 N, 0–800 N, and N, 0–30 and 0–30 N·m, N0–30∙m, 0–30 N·m, N and∙m, 0–10and 0–10 N·m, N respectively.∙m, respectively. The unfrozen model ice used in this experiment is nylon model. The length, width, and thickness of the model ice were 250 mm × 200 mm × 90 mm, respectively. In the process of ice position adjustment, the centerlines of the propeller shaft and propeller disk were used as the vertical and axial reference positions, respectively. The distances between the bottom face of the model ice and the center of the propeller shaft, and the ice surface near the side of the propeller and the propeller disk were defined as the vertical and axial relative positions of the ice–propeller, respectively. The origin of the coordinate system is at the center of the propeller disk. The positive direction of the X-axis is the direction from the pod to the propeller, the positive direction of the Z-axis is upward, and the direction of the Y-axis is determined according to the right-hand rule, as shown in Figure4. For the propeller fluctuating-pressure test, a fluctuating pressure transducer was attached to the flat plate directly above the propeller model. The vertical distance between the transducer and the center line of the propeller shaft was 250 mm. The point of inter- section between the center line of the propeller shaft and the propeller radiation reference

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line was taken as the center of the prism. The axial and transverse dimensions of the prism J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 5 of 7 were 0.15D [15]. Five pressure transducers were arranged at the four corners and center of the prism (Figure5).

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FigureFigure 4. 4.Schematic Schematic diagram diagram of of the the test test platform platform for for propeller-induced propeller‐induced excitation excitation force. force.

The unfrozen model ice used in this experiment is nylon model. The length, width, and thickness of the model ice were 250 mm × 200 mm × 90 mm, respectively. In the pro‐ cess of ice position adjustment, the centerlines of the propeller shaft and propeller disk were used as the vertical and axial reference positions, respectively. The distances be‐ tween the bottom face of the model ice and the center of the propeller shaft, and the ice surface near the side of the propeller and the propeller disk were defined as the vertical and axial relative positions of the ice–propeller, respectively. The origin of the coordinate system is at the center of the propeller disk. The positive direction of the X‐axis is the direction from the pod to the propeller, the positive direction of the Z‐axis is upward, and the direction of the Y‐axis is determined according to the right‐hand rule, as shown in Figure 4. For the propeller fluctuating‐pressure test, a fluctuating pressure transducer was at‐ tached to the flat plate directly above the propeller model. The vertical distance between the transducer and the center line of the propeller shaft was 250 mm. The point of inter‐ section between the center line of the propeller shaft and the propeller radiation reference

line was taken as the center of the prism. The axial and transverse dimensions of the prism FigureFigurewere 5.0.15 5.Installation InstallationD [15]. Five of of ﬂuctuating fluctuatingpressure pressuretransducers pressure transducer. transducer. were arranged at the four corners and center of the prism (Figure 5). 3.3. Experimental Experimental Method Method 3.1.3.1. Similarity Similarity Criteria Criteria TheThe simulation simulation of of ice–propeller–ﬂow ice–propeller–flow interaction interaction must must meet meet the the similarity similarity criteria, criteria, i.e., i.e., geometricgeometric similarity, similarity, motion motion similarity, similarity, viscous viscous force force similarity, similarity, and and cavitation cavitation similarity similarity of theof the propeller. propeller. (1)(1) Geometric Geometric similarity similarity TheThe scale scale of of the the propeller propeller model, model, i.e., i.e., 1:16.46, 1:16.46, ensured ensured the geometric the geometric similarity similarity between be‐ thetween full-scale the full propeller‐scale propeller and the and model. the model. (2) Motion similarity The dimensionless advance coefficient J of the model is the same as that of the actual propeller, which ensures it has the same motion as the full‐scale propeller. The dimen‐ sionless coefficients of the thrust and torque of the propeller and the single blade were defined as follows: V J  (1) nD T K  (2) T nD24

Q K  (3) Q nD25

T K  X (4) TX_ BLADE nD24

T K  T (5) TT_ BLADE nD24

Q K  X (6) QX_ BLADE nD25

Q K  Y (7) QY_ BLADE nD25

Q K  Z (8) QZ_ BLADE nD25

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(2) Motion similarity The dimensionless advance coefficient J of the model is the same as that of the actual propeller, which ensures it has the same motion as the full-scale propeller. The dimensionless coefficients of the thrust and torque of the propeller and the single blade were defined as follows: V J = (1) nD T K = (2) T ρn2D4 Q K = (3) Q ρn2D5 T K = X (4) TX_BLADE ρn2D4 T K = T (5) TT_BLADE ρn2D4 Q K = X (6) QX_BLADE ρn2D5 Q K = Y (7) QY_BLADE ρn2D5 Q K = Z (8) QZ_BLADE ρn2D5 where V is the inflow velocity; n is the propeller rotational speed; D is the propeller diameter; T and Q are the propeller thrust and torque, respectively; KT and KQ are the propeller thrust and torque coefficients, respectively; ρ is the density of water; TX is the axial thrust of blade in the X direction; TT is the tangential force of blade; KTX_BLADE and KTT_BLADE are the axial thrust and tangential force coefficients of the blade, respectively; QX, QY, and QZ are the torques of blade in the X, Y, and Z directions, respectively; and KQX_BLADE, KQY_BLADE, and KQZ_BLADE are the torque coefficients of the blade in the three directions, respectively. (3) Viscous force similarity Because the condition of the Reynolds number cannot be satisfied in the model test, the Reynolds number Rn(0.75R) of the blade section at a chord length of 0.75R was required to exceed the critical Reynolds number, i.e., q 2 2 L0.75R V + (0.75πnD) Rn = > 3.0 × 105 (9) (0.75R) ν

where L0.75R is the chord length of the blade section at 0.75R, and ν is the kinematic viscosity coefﬁcient of water. (4) Cavitation similarity The cavitation number of the rotational speed of the ﬂow passing through the propeller disk and at a radius of 0.8R directly above the propeller axis is equal to that of the full-scale ship [14,16], i.e.,

p + ρ g(h − 0.4D ) − p = a s s s v Full-scale : σns(0.8R) 2 (10) 0.5ρs(0.8πnsDs)

p − pv = 0(0.8R) Model : σnm(0.8R) 2 (11) 0.5ρm(0.8πnmDm)

σns(0.8R) = σnm(0.8R) (12) J. Mar. Sci. Eng. 2021, 9, 674 7 of 25

where Pa is the atmospheric pressure, Pv is the vaporization pressure of water, hs is the center depth of the full-scale propeller shaft, and P0(0.8R) is the static pressure of the 0.8R radial blade section at the 12 o’clock position. ρs and ρm are the densities of seawater and fresh water, respectively; Ds and Dm are the diameters of the full-scale propeller and model propeller, respectively; and ns and nm are the rotational speeds of the full-scale propeller and model propeller, respectively.

3.2. Test Method and Operating Conditions Before the test, the pod package and the guide rail of the ice blockage device were ﬁxed to the model mounting bracket. The bracket was hoisted to the test section of the circulating water channel. The position of the connecting rod on the mounting bracket was adjusted to make the centerline of the pod parallel to the axis of this section. The guide rail and the propeller shaft center line were regarded as the same line from top to bottom in the circulating water channel. The centerline of the ﬁxed pod-package propeller shaft was approximately 750 mm above the center of the circulating water channel. After adjusting the relative position of the ice and the propeller, two groups of full-scale ship operating conditions of the experimental ice-class propeller model were selected [17,18] (Table1) to test the performance of the propeller under ice-blockage conditions.

Table 1. Operating conditions of full-scale ship.

Cavitation Number of Advance Full-Scale Propeller Full-Scale Ship Rotational Speed at 0.8R Coefﬁcient J Rotational Speed ns(r/s) Speed Vs(m/s) σ = 0.33 0.325 162 3.61 σ = 0.44 0.281 141 2.72

The pressure of the circulating water channel was adjusted according to the specified cavitation number of the rotational speed. The fluid velocity in the test section was adjusted according to the advance coefficient and rotational speed. Changes in the axial dynamic force, cavitation, and excitation force of the propeller with varying ice–propeller axial distances under atmospheric pressure, cavitation numbers of rotational speed σ = 0.44 and 0.33 were measured respectively. Table2 presents the operating conditions.

Table 2. Operating conditions of the test.

Test Axial Distance Vertical Cavitation Number Advance Rotational Speed Conditions L/D Distance h/D Rotational Speed at 0.8R Coefﬁcient J Speed nm(r/s) Vm(m/s) Case 1 0 0 Open-water condition 0.281 19.36 1.36 Case 2 7/16 Atmospheric pressure 0.281 19.36 1.36 1/8, 2/8, 3/8, Case 3 7/16 σ = 0.44 0.281 19.36 1.36 4/8, 5/8, 6/8, Case 4 5/16 σ = 0.44 0.281 19.36 1.36 7/8, 1 Case 5 7/16 σ = 0.33 0.325 22.15 1.8

4. Results and Analysis 4.1. Hydrodynamic Load Analysis of Propeller Taking cases 1 to 3 in Table2 as examples, the influence of the ice–propeller axial distance on the hydrodynamic performance of the propeller was analyzed. Figures6–9 illustrate the mean value and time-domain comparison curves of the propeller and single blade hydrodynamic performance with different ice–propeller axial distances under atmospheric pressure and cavitation number of rotational speed, σ = 0.44. These values are compared with the open-water experimental values of the propeller with the same advance coefficient. As indicated in Figure6, at atmospheric pressure and under blockage conditions, the mean values of the thrust and torque coefficients increase with the decrease in L/D. J. Mar. Sci. Eng. 2021, 9, 674 8 of 25

The smaller the L/D, the more distinct the increase. There are two main reasons for the increase in propeller thrust coefficient and torque coefficient. One is the blockage effect; this is because of the blockage effect of model ice on the inflow decreases the inflow velocity in front of the propeller, increase the angle of attack of the blade section, and increases the propeller thrust. When the blade rotates, the separation of the fluid at the back of the blade will also intensify, resulting in an increase in the torque of the propeller. Nonetheless, the wall effect (or boundary effect) shows that the flow velocity between the ice and the propeller increases, which makes the flow field between the propeller and the ice occur in a high-speed wake area, thereby increasing the thrust and torque of the propeller [4,19,20]. When σ = 0.44, the mean values of the propeller thrust and torque coefficients also increase with the decrease in L/D, which is higher than the atmospheric pressure. The main reason is that the blade back begins to cavitate at its local position as the cavitation number decreases; however, in the first stage of cavitation development, the lift coefficient continues to increase [21]. Compared with the open-water condition, the J. Mar. Sci. Eng. 2021, 9, x FOR PEERthrust REVIEW and torque coefficients of the propeller increased under the two blockage conditions,9 of 11 but the propeller efficiency decreased (Figure6c). In Figure6c, at atmospheric pressure, the propeller efficiency decreases with a decrease in L/D. The main reason is that the smaller thepropeller,L/D, the the more higher obvious the angle the of ice-blockage attack of the affects, blade section; and the this smaller reduces the the inflow lift drag velocity ratio in frontis, which of the leads propeller, to a reduction the higher in propeller the angle efficiency. of attack ofWhen the bladeσ = 0.44, section; the propeller this reduces efficiency the lift dragremains ratio nearly is, which constant leads toeven a reduction as L/D changes; in propeller however, efficiency. it is still When lessσ than= 0.44, the the propeller propeller efficiencyefficiency remains at atmospheric nearly pressure. constant even as L/D changes; however, it is still less than the propeller efficiency at atmospheric pressure.

(a) (b)

(c)

Figure 6. Variation of KT,10KQ and η with L/D: (a) KT; (b) 10KQ; (c) η. Figure 6. Variation of KT,10KQ and η with L/D:(a) KT;(b) 10KQ;(c) η.

As indicated in Figure 7, in the three states, the axial thrust is approximately 1.8 times the tangential force. The torque in the Y direction is approximately 3.7 times the torque in the X direction and 6.4 times the torque in the Z direction. The force and torque changes in different directions are of the same order of magnitude. At atmospheric pressure and σ = 0.44, and under blockage conditions, the variation of the five‐component force of a single blade with L/D is the same as that of the thrust and torque coefficients of the propeller. The hydrodynamic performance of the blade at σ = 0.44 is greater than that at atmospheric pressure. Compared with the open‐water condition, at atmospheric pressure and σ = 0.44, and under blockage conditions, the axial thrust and tangential force coefficients and the torque coefficients in the X and Y directions of the single blade increased significantly. Under the same conditions, the torque coefficient in the Z direction of the single blade is different. At atmospheric pressure, when L/D > 5/8, the change in L/D had little impact on the torque coefficient in the Z direction of the single blade, which remained almost

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unchanged. When σ = 0.44, the torque in the Z direction gradually increased with a de‐ crease in L/D.

(a) (b)

(e)

Figure 7. Variation of five‐component force of single blade with L/D: (a) KTX_BLADE; (b) KTT_BLADE; (c) 10KQX_BLADE; (d) Figure 7. Variation of ﬁve-component force of single blade with L/D:(a) KTX_BLADE;(b) KTT_BLADE;(c) 10KQX_BLADE; 10KQY_BLADE; (e) 10KQZ_BLADE. (d) 10KQY_BLADE;(e) 10KQZ_BLADE.

BasedAs indicated on Figures in Figure 6 and7, 7, in the the time three‐domain states, the variation axial thrust curves is of approximately the axial thrust 1.8 coeffi times‐ cientthe tangential and torque force. coefficient The torque in the in Y the directionY direction of the is approximatelysingle blade during 3.7 times propeller the torque stabili in‐ zationthe X direction for L/D = and1/8, 3/8, 6.4 times and 7/8 the under torque atmospheric in the Z direction. conditions The were force extracted. and torque To changesanalyze thein different load variation directions law in are detail, of the the same time‐ orderdomain of variation magnitude. curve At of atmospheric the six periods pressure of the and σ = 0.44, and under blockage conditions, the variation of the ﬁve-component force

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blade is illustrated in Figure 8. Meanwhile, for L/D = 1/8, the time‐domain curves of the single blade in the three states are compared in Figure 9. J. Mar. Sci. Eng. 2021, 9, 674 10 of 25 Figure 8 illustrates the time‐domain curves of the thrust coefficient and torque coef‐ ficient in the Y direction of a single blade at different distances between the ice and the propeller under atmospheric pressure. As shown in Figure 8, when L/D is fixed, the peak ofamplitude a single of blade the thrust with L coefficient/D is the sameand torque as that coefficient of the thrust in the and Y direction torque coefficients of the single of theblade propeller. in each cycle The is hydrodynamic almost equal, performanceand the variation of the range blade is small, at σ = whereas 0.44 is greater the fluctua than‐ thattion atrange atmospheric of the wave pressure. trough Compared is notably different. with the open-water The main reason condition, for this at atmosphericdifference is pressurethat the ice–propeller and σ = 0.44, interaction and under process blockage is complex, conditions, and the the axial hydrodynamic thrust and load tangential of the forceblade coefficientsin the unblocked and the region torque is coefficients sensitive to in this the interaction.X and Y directions The reason of the is singlealso closely blade increasedrelated to significantly.the periodic Undervortex theshedding same conditions, of ice [19,22]. the As torque L/D coefficientdecreased, in the the peakZ direction ampli‐ oftude the of single the thrust blade coefficient is different. and At torque atmospheric coefficient pressure, in the Y when directionL/D of> 5/8,the single the change blade inincreased,L/D had the little fluctuation impact on range the torque near coefficientthe peak increased, in the Z direction and the of amplitude the single blade,phase whichchanged remained to a certain almost extent. unchanged. The primary When causeσ = 0.44, of the the amplitude torque in the phaseZ direction change gradually is the in‐ increasedcreasing blockage with a decrease range of in theL/ iceD. as L/D decreases as well as the change in the flow field structureBased around on Figures the propeller.6 and7, the The time-domain latter results variation in a change curves in the of position the axial of thrust the maxi co- ‐ efficientmum pressure and torque difference coefficient between in the bladeY direction back and of thethe singleblade surface blade during [10,13]. propeller The am‐ stabilizationplitude of the for waveL/D trough= 1/8, of 3/8, the and thrust 7/8 coefficient under atmospheric and torque conditions coefficient were in the extracted. Y direction To analyzeof the blade the loadalso exhibited variation lawa larger in detail, fluctuation the time-domain range as L/ variationD decreased. curve It can of the be six concluded periods ofthat the the blade stronger is illustrated the blockage in Figure effect8. Meanwhile,of ice, the wider for L /theD =fluctuation 1/8, the time-domain range of the curves blade ofload, the and single the blade more in unstable the three the states load areis. compared in Figure9.

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(a) (b) of the thrust and torque coefficient in the Y direction are different in each rotation cycle of Figure 8. Time-domainTime‐domain curves of KTX__BLADE andand 10 10KQYK_BLADE_ withwith different different L/D L(atmospheric/D (atmospheric pressure pressure under under blockage blockage con‐ a singleTX blade,BLADE and the absoluteQY BLADE value and amplitude in the wave troughs are not substan‐ conditions):ditions): (a) K (aTX) _KBLADE; (b) 10;(KbQY) 10_BLADEK . . TX_BLADE tially greaterQY_BLADE than those in the atmospheric pressure. The trend in Figure 9 led us to conclude that the peak shape of the thrust coefficient and torque coefficient in the Y direction of the single blade in the open‐water working condition was regular and the time‐domain curve was smoother. The curve gradually be‐ comes unsmooth in the trough. When the ice and propeller interacted, whether its atmos‐ pheric pressure or decompression, the absolute values and amplitudes of the axial thrust coefficient and torque coefficient in the Y direction of the single blade changed signifi‐ cantly. In the same vein, the amplitude phase also changed to a certain extent. At atmos‐ pheric pressure, two peaks and one trough appeared in the main peaks of the thrust coef‐ ficient and torque coefficient in the Y direction, and there are many peaks and troughs in the main trough, and the main troughs of each period of the blade are different. When σ = 0.44, the periodic variation regularity of the thrust coefficient and torque coefficient in the Y direction with time is similar to that of atmospheric pressure, but the curve becomes more unsmooth. The variation range of the peak and trough on the main peak of the thrust coefficient(a) and torque coefficient in the Y direction is larger,(b) and the amplitude is greater than that of the atmospheric pressure. However, the change rule of the main wave troughs Figure 9. Time‐domain curves of KTX_BLADE and 10KQY_BLADE (Open‐water conditions, atmospheric pressure and σ = 0.44 Figure 9. Time-domain curves of KTX_BLADE and 10KQY_BLADE (Open-water conditions, atmospheric pressure and σ = 0.44 under blockage conditions): (a) KTX_BLADE; (b) 10KQY_BLADE. under blockage conditions): (a) KTX_BLADE;(b) 10KQY_BLADE.

4.2. CavitationFigure8 illustrates Analysis of the Propeller time-domain curves of the thrust coefﬁcient and torque coef- ﬁcientBy in taking the Y casedirection 4 in Table of a 2 single as an bladeexample, at different i.e., the rotational distances speed between cavitation the ice number and the σ = 0.44, vertical distance h/D = 5/16, propeller rotational speed n = 19.36 r/s, and inflow velocity V = 1.36 m/s and recording the cavitation shape on the propeller through strobo‐ scopic video, we analyzed the influence of the change in the ice–propeller axial distance on the blade surface cavitation. The axial distance was L/D = 1, 5/8, 3/8, and 1/8. Figure 10 presents images of the experiment.

(a) (b)

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propeller under atmospheric pressure. As shown in Figure8, when L/D is fixed, the peak amplitude of the thrust coefficient and torque coefficient in the Y direction of the single blade in each cycle is almost equal, and the variation range is small, whereas the fluctuation range of the wave trough is notably different. The main reason for this difference is that the ice–propeller interaction process is complex, and the hydrodynamic load of the blade in the unblocked region is sensitive to this interaction. The reason is also closely related to the periodic vortex shedding of ice [19,22]. As L/D decreased, the peak amplitude of the thrust coefficient and torque coefficient in the Y direction of the single blade increased, the fluctuation range near the peak increased, and the amplitude phase changed to a certain extent. The primary cause of the amplitude phase change is the increasing blockage range of the ice as L/D decreases as well as the change in the flow field structure around the propeller. The latter results in a change in the position of the maximum pressure difference between the blade back and the blade surface [10,13]. The amplitude of the wave trough of the thrust coefficient and torque coefficient in the Y direction of the blade also exhibited a larger fluctuation range as L/D decreased. It can be concluded that the stronger the blockage effect of ice, the wider the fluctuation range of the blade load, and the more unstable the load is. The trend in Figure9 led us to conclude that the peak shape of the thrust coefficient and torque coefficient in the Y direction of the single blade in the open-water working condition was regular and the time-domain curve was smoother. The curve gradually becomes unsmooth in the trough. When the ice and propeller interacted, whether its atmospheric pressure or decompression, the absolute values and amplitudes of the axial thrust coefficient and torque coefficient in the Y direction of the single blade changed significantly. In the same vein, the amplitude phase also changed to a certain extent. At atmospheric pressure, two peaks and one trough appeared in the main peaks of the thrust coefficient and torque coefficient in the Y direction, and there are many peaks and troughs in the main trough, and the main troughs of each period of the blade are different. When σ = 0.44, the periodic variation regularity of the thrust coefficient and torque coefficient in the Y direction with time is similar to that of atmospheric pressure, but the curve becomes more unsmooth. The variation range of the peak and trough on the main peak of the thrust coefficient and torque coefficient in the Y direction is larger, and the amplitude is greater than that of the atmospheric pressure. However, the change rule of the main wave troughs of the thrust and torque coefficient in the Y direction are different in each rotation cycle of a single blade, and the absolute value and amplitude in the wave troughs are not substantially greater than those in the atmospheric pressure.

4.2. Cavitation Analysis of Propeller By taking case 4 in Table2 as an example, i.e., the rotational speed cavitation number σ = 0.44, vertical distance h/D = 5/16, propeller rotational speed n = 19.36 r/s, and inflow velocity V = 1.36 m/s and recording the cavitation shape on the propeller through strobo- scopic video, we analyzed the influence of the change in the ice–propeller axial distance on the blade surface cavitation. The axial distance was L/D = 1, 5/8, 3/8, and 1/8. Figure 10 presents images of the experiment. When L/D = 1, as shown in Figure 10a, sheet and tip vortex cavitation occur at 0.9R of the leading edge at the back of the blade. Because of the large distance between the ice and the propeller, blockage due to ice has little influence on the wake field of the propeller disk. Therefore, when the blade is in the blocked region, the area of the blade sheet and tip vortex cavitation is small. With a decrease in L/D, when L/D = 3/8, the blade back cavitates more severely in the blocked region. The area of sheet cavitation increases, whereas the change in the tip vortex cavitation is small. Similarly, the blade back cavitates more severely when L/D = 1/8, and a large area with sheet cavitation emerges in the form of cloud cavitation at the trailing edge close to the blade tip position. In other words, the presence of ice affects the shape of cavitation in the blocked region, i.e., it results in an increase in the blade cavitation area. Meanwhile, propeller–hull vortex cavitation occurs between the ice J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 12 of 27

of the thrust and torque coefficient in the Y direction are different in each rotation cycle of a single blade, and the absolute value and amplitude in the wave troughs are not substan‐ tially greater than those in the atmospheric pressure.

(a) (b) J. Mar. Sci. Eng. 2021 9 Figure 9. Time, , 674‐domain curves of KTX_BLADE and 10KQY_BLADE (Open‐water conditions, atmospheric pressure and σ = 0.44 12 of 25 under blockage conditions): (a) KTX_BLADE; (b) 10KQY_BLADE.

4.2. Cavitation Analysis of Propeller and the propeller. This and the blade sheet cavitation together form a larger area of sheet By taking case 4 in Table 2 as an example, i.e., the rotational speed cavitation number cavitation, and the whole body emerges along the blade’s trailing edge as an agglomerate σ = 0.44, vertical distance h/D = 5/16, propeller rotational speed n = 19.36 r/s, and inflow shape. To further analyze the generation, movement, and collapse of the propeller–hull velocity V = 1.36 m/s and recording the cavitation shape on the propeller through strobo‐ vortex cavitation between the ice and the propeller, we take L/D = 1/8. In this case, the four scopic video, we analyzed the influence of the change in the ice–propeller axial distance processeson the blade of blade surface in cavitation. the unblocked The axial region, distance rotation was into L/D the= 1, blocked5/8, 3/8, and region, 1/8. Figure the blocked 10 region,presents exit images from of the the blocked experiment. region are analyzed (Figure 11).

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(a) (b)

FigureFigure 10. 10.Variation Variation inin blade cavitation cavitation with with different different L/DL (/σ D= 0.44):(σ = ( 0.44):a) L/D (=a 1;) L(b/)D L/=D 1;= 5/8; (b) (cL)/ LD/D= = 5/8;3/8; (d ()c )L/LD/ =D 1/8.= 3/8; (d) L/D = 1/8. When L/D = 1, as shown in Figure 10a, sheet and tip vortex cavitation occur at 0.9R of theFigure leading 11 illustratesedge at the the back cavitation of the blade. morphology Because of the large blade distance at four differentbetween positions.the ice Theand blockage the propeller, effect blockage of ice was due relatively to ice has small little wheninfluence the on blade the waswake in field the of unblocked the propeller region. Meanwhile,disk. Therefore, the blade when mainly the blade produced is in the stable blocked tip region, vortex the cavitation area of the but blade no propeller–hull sheet and vortextip vortex cavitation. cavitation When is small. the blade With rotates a decrease into thein L blocked/D, when region, L/D = 3/8, as depicted the blade in backFigure cav 11‐ b, propeller–hullitates more severely vortex in cavitation the blocked appears, region. The extending area of sheet from cavitation the leading increases, edge to whereas the front surfacethe change of the in ice.the tip As vortex the blade cavitation rotates, is small. the propeller–hull Similarly, the vortexblade back cavitation cavitates gradually more migratesseverely from when the L/D leading = 1/8, edgeand a of large the bladearea with to the sheet trailing cavitation edge (Figure emerges 11 c).in the When form the of end ofcloud the propeller–hull cavitation at the vortex trailing cavitation edge close reaches to the the blade blade’s tip position. trailing edge, In other it combines words, the with thepresence blade sheetof ice cavitationaffects the andshape emerges of cavitation along in the the trailing blocked edge region, as an i.e., agglomerate it results in shape an (Figureincrease 11 ind). the blade cavitation area. Meanwhile, propeller–hull vortex cavitation occurs between the ice and the propeller. This and the blade sheet cavitation together form a larger area of sheet cavitation, and the whole body emerges along the blade’s trailing edge as an agglomerate shape. To further analyze the generation, movement, and collapse of the propeller–hull vortex cavitation between the ice and the propeller, we take L/D = 1/8. In this case, the four processes of blade in the unblocked region, rotation into the blocked region, the blocked region, exit from the blocked region are analyzed (Figure 11).

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(a) (b)

FigureFigure 11. 11.Change Change process process of of the the propeller–hull propeller–hull vortex vortex cavitationcavitation during blade blade rotation rotation (σ (σ= =0.44): 0.44): (a) ( aUnblocked) Unblocked region; region; (b)( Rotationb) Rotation into into the the blocked blocked region; region; (c ()c Blocked) Blocked region; region; ( (dd)) ExitExit fromfrom the blocked region. region.

TheFigure primary 11 illustrates cause of the the cavitation propeller–hull morphology vortex of cavitation the blade betweenat four different the ice andposi‐ the propellertions. The is blockage the decrease effect of in iceL/ wasD, which relatively led small to the when stronger the blade the was blockage in the unblocked effect on the propeller,region. Meanwhile, a smaller axial the blade inflow mainly velocity, produced a smaller stable actual tip vortex advance cavitation coefficient, but no and propel a larger‐ hydrodynamicler–hull vortex loadcavitation. on the When blade, the and blade the rotates hydrodynamic into the blocked interaction region, between as depicted the blade in andFigure the ice11b, is propeller–hull continuously vortex increased cavitation [4,7,9,13 appears,]. Considering extending the from effect the of leading propeller edge suction, to thethe flow front of surface water of into the the ice. propeller As the blade disk rotates, from both the propeller–hull sides of the ice vortex was cavitation bound to increasegrad‐ underually migrates the blockage from condition.the leading Simultaneously, edge of the blade an to accelerationthe trailing edge zone (Figure was formed 11c). When on the bottomthe end surface of the propeller–hull of the ice, and vortex an upward cavitation roll reaches was formed the blade’s at the trailing end [10 edge,]. Because it com‐ ice isbines a bluff with body, the blade the recirculation sheet cavitation zone and on emerges its upper along surface the trailing is related edge as to an its agglomer length and‐ velocity.ate shape Meanwhile, (Figure 11d). a recirculation zone may also appear [19,22]. The fluid with these typesThe of motion primary is cause strongly of the coupled propeller–hull between vortex the ice cavitation and the propeller, between the which ice and may the form apropeller flow hysteresis is the decrease point, i.e., in L a/D fluid, which layer led with to the a stronger flow velocity the blockage of zero effect [23,24 on]. the This pro layer‐ waspeller, induced a smaller at the axial flow inflow stagnation velocity, point a smaller by the actual rotation advance of the coefficient, propeller itself, and a forming larger a vortex.hydrodynamic Then, under load theon the continuous blade, and action the hydrodynamic of the Pirouette interaction effect [25 between], the pressure the blade of the flowand fieldthe ice near is continuously the tip of the increased blade decreases, [4,7,9,13]. and Considering this vortex the extends effect of topropeller the propeller suction, and finallythe flow reaches of water the into blade. the propeller When the disk pressure from both drops sides below of the the ice saturated was bound vapor to increase pressure, propeller–hull vortex cavitation is formed. In the Pirouette effect, the rotation of the vortex core is constantly accelerated, which leads to an increased vortex strength, in the process of

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under the blockage condition. Simultaneously, an acceleration zone was formed on the bottom surface of the ice, and an upward roll was formed at the end [10]. Because ice is a bluff body, the recirculation zone on its upper surface is related to its length and velocity. Meanwhile, a recirculation zone may also appear [19,22]. The fluid with these types of motion is strongly coupled between the ice and the propeller, which may form a flow hysteresis point, i.e., a fluid layer with a flow velocity of zero [23,24]. This layer was in‐ duced at the flow stagnation point by the rotation of the propeller itself, forming a vortex. Then, under the continuous action of the Pirouette effect [25], the pressure of the flow field near the tip of the blade decreases, and this vortex extends to the propeller and finally reaches the blade. When the pressure drops below the saturated vapor pressure, propel‐ J. Mar. Sci. Eng. 2021, 9, 674 ler–hull vortex cavitation is formed. In the Pirouette effect, the rotation of the vortex14 core of 25 is constantly accelerated, which leads to an increased vortex strength, in the process of stretching and thinning of a large vortex, similar to the tornado phenomenon. The exist‐ encestretching of the and propeller thinning‐hull of vortex a large cavitation vortex, similar makes to theblade tornado hydrodynamic phenomenon. load The unstable, existence as shownof the propeller-hull in Figure 9. vortex cavitation makes blade hydrodynamic load unstable, as shown in Figure9. 4.3. Analysis of Propeller‐induced Excitation Force 4.3. AnalysisTaking cases of Propeller-Induced 2, 3, and 5 in Table Excitation 2 as the Force research object, the excitation force induced by theTaking propeller cases with 2, 3, different and 5 in axial Table gaps2 as between the research the ice object, and the propeller excitation at force atmospheric induced pressureby the propeller and σ = with0.44 and different σ = 0.33 axial was gaps tested. between the ice and the propeller at atmospheric pressure and σ = 0.44 and σ = 0.33 was tested. 4.3.1. Atmospheric Condition 4.3.1. Atmospheric Condition At atmospheric pressure, the time‐domain curves of the propeller‐ and single blade‐ bearingAt forces atmospheric under pressure,different blockage the time-domain conditions curves were of transformed the propeller- into and the single frequency blade-‐ domainbearing curves forces underby the fast different Fourier blockage transform conditions (FFT) (Figure were transformed12). According into to thethe frequency-main fluc‐ tuatingdomain frequency curves by in the Figure fast Fourier12, the fluctuating transform (FFT) amplitude (Figure of 12each). According order of the to propeller the main bearingfluctuating force frequency with varying in Figure distance 12, the between fluctuating the ice amplitude and the ofpropeller each order was of extracted the propeller and bearing force with varying distance between the ice and the propeller was extracted and plotted as a column figure (Figure 13). Table 3 lists the fluctuating amplitudes of each plotted as a column figure (Figure 13). Table3 lists the fluctuating amplitudes of each order order of the single‐blade bearing force. of the single-blade bearing force.

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(b) (a)

Figure 12. FrequencyFrequency-domain‐domain curves of propeller and blade bearing force under blockage condition (J = 0.281, 0.281, atmospheric atmospheric pressure): (a) KT curve in frequency‐domain; (b) 10KQ curve in frequency‐domain; (c) KTX_BLADE, KTT_BLADE curves in fre‐ pressure): (a) KT curve in frequency-domain; (b) 10KQ curve in frequency-domain; (c) KTX_BLADE, KTT_BLADE curves in quency‐domain; (d)10KQX_BLADE, 10KQY_BLADE, 10KQZ_BLADE curves in frequency‐domain. frequency-domain; (d)10KQX_BLADE, 10KQY_BLADE, 10KQZ_BLADE curves in frequency-domain.

FigureFigure 1212 illustrateillustrate the the frequency frequency-domain‐domain curves of the the bearing bearing forces forces of of the the propeller propeller andand single single blade under the blockage conditioncondition ( (LL//D = 1/8). Figure Figure 12a,b 12a,b led led us us to to conclude conclude that,that, at at atmospheric atmospheric pressure, pressure, the the main main fluctuating ﬂuctuating frequencies frequencies of the of the propeller propeller thrust thrust co‐ efficient are the first‐order shaft frequency (rotational speed n = 19.36 r/s, 19.36 Hz) and the first‐ (four‐blade propeller, 76.98 Hz), second‐, and third‐order blade frequencies. The blade passage frequency (BPF) had the largest fluctuating amplitude, followed by the sec‐ ond‐order blade frequency (2BPF). This amplitude was gradually attenuated. The varia‐ tion trend of each order of the frequency‐domain curve of the propeller torque coefficient was different from that of the thrust coefficient. The fluctuating amplitudes of the first‐ order shaft frequency (APF), second‐order shaft frequency (2APF), BPF, and fifth‐order shaft frequency (5APF) were the most distinguishable. The fluctuating amplitude of higher‐order quantities after 5APF can be ignored. The main reason for the apparent fluc‐ tuating amplitude of 5APF is that the propeller torque is mainly due to the resistance of the propeller airfoil during its rotation. In the process of ice‐blocking the propeller, the wake behind the ice may form a flow related to the resistance direction of the airfoil, thereby increasing the lateral force and bending moment of the propeller. At the same time, the instability of the propeller’s forward inflow velocity results in very severe vibra‐ tions at higher frequencies; that is, the fluctuating amplitude of 5APF increases. The in‐ crease in fluctuating amplitude at 5APF is closely related to the relative position of the ice–propeller, inflow velocity, rotational speed of the propeller, and shape of the ice. Fig‐ ure 12c,d illustrate the frequency‐domain curves of the single‐blade‐bearing force. As ob‐ served in the figures, the five‐component force coefficients of the single blade have differ‐ ent amplitudes at the integral multiples of the APF. The amplitudes at APF, 2APF, and 3APF are the most distinct, and they gradually attenuate subsequently; however, the fluc‐ tuating amplitude increases again near the third‐order blade frequency (3BPF). By com‐ paring the fluctuating amplitude of the single blade five‐component force bearing forces, we observed that the pulsation amplitudes of the axial thrust coefficient and torque coef‐ ficient in the Y direction of the single blade are larger than those of other directional forces and torques. Figure 13 highlights the variation of the main‐order fluctuating amplitude of the pro‐ peller bearing force with different L/D (1/8, 3/8, 5/8, and 7/8). An investigation of Figure 14 revealed that the BPF amplitude of the propeller thrust coefficient was significantly larger than that of the other orders. The closer the ice was to the propeller disk, the greater the difference in amplitude between each order. However, the 5APF amplitude of propel‐ ler torque coefficient is the largest. With the decrease in the L/D, the APF, BPF, and2BPF

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coefficient are the first-order shaft frequency (rotational speed n = 19.36 r/s, 19.36 Hz) and the first- (four-blade propeller, 76.98 Hz), second-, and third-order blade frequencies. The blade passage frequency (BPF) had the largest fluctuating amplitude, followed by the second-order blade frequency (2BPF). This amplitude was gradually attenuated. The variation trend of each order of the frequency-domain curve of the propeller torque coefficient was different from that of the thrust coefficient. The fluctuating amplitudes of the first-order shaft frequency (APF), second-order shaft frequency (2APF), BPF, and fifth- order shaft frequency (5APF) were the most distinguishable. The fluctuating amplitude of higher-order quantities after 5APF can be ignored. The main reason for the apparent fluctuating amplitude of 5APF is that the propeller torque is mainly due to the resistance of the propeller airfoil during its rotation. In the process of ice-blocking the propeller, the wake behind the ice may form a flow related to the resistance direction of the airfoil, thereby increasing the lateral force and bending moment of the propeller. At the same time, the instability of the propeller’s forward inflow velocity results in very severe vibrations at higher frequencies; that is, the fluctuating amplitude of 5APF increases. The increase in fluctuating amplitude at 5APF is closely related to the relative position of the ice–propeller, inflow velocity, rotational speed of the propeller, and shape of the ice. Figure 12c,d illustrate the frequency-domain curves of the single-blade-bearing force. As observed in the figures, the five-component force coefficients of the single blade have different amplitudes at the integral multiples of the APF. The amplitudes at APF, 2APF, and 3APF are the most distinct, and they gradually attenuate subsequently; however, the fluctuating amplitude increases again near the third-order blade frequency (3BPF). By comparing the fluctuating amplitude of the single blade five-component force bearing forces, we observed that the pulsation amplitudes of the axial thrust coefficient and torque coefficient in the Y direction of the single blade are larger than those of other directional forces and torques. Figure 13 highlights the variation of the main-order fluctuating amplitude of the propeller bearing force with different L/D (1/8, 3/8, 5/8, and 7/8). An investigation of Figure 14 revealed that the BPF amplitude of the propeller thrust coefficient was signifi- J. Mar. Sci. Eng. 2021, 9, x FOR PEERcantly REVIEW larger than that of the other orders. The closer the ice was to the propeller17 disk, of 19 the greater the difference in amplitude between each order. However, the 5APF amplitude of propeller torque coefficient is the largest. With the decrease in the L/D, the APF, BPF, andof the 2BPF propeller of the thrust propeller coefficient thrust showed coefficient a gradually showed increasing a gradually trend, increasing and the variation trend, and thetrend variation of the BPF trend was of the most BPF wassignificant. the most When significant. the L/D was When small the (e.g.,L/D Lwas/D =1/8), small the (e.g., L/amplitudeD =1/8), of the the amplitude 3BPF increased of the sharply. 3BPF increased As the L/ sharply.D decreased, As the theL BPF/D decreased,amplitude of the the BPF amplitudepropeller torque of the propellercoefficient torque slowly coefficient increased, slowlywhereas increased, the APF and whereas 5APF the amplitudes APF and did 5APF amplitudesnot show any did distinguished not show any changes. distinguished The 2APF changes. amplitude The 2APF first amplitudedecreased and first then decreased in‐ andcreased. then increased.Therefore, Therefore,the smaller thethe smallerL/D, the the strongerL/D, thethe stronger blockage the by blockageice, and the by more ice, and thedistinct more the distinct increase the in increase the BPF in amplitude the BPF amplitude of the propeller of the bearing propeller force. bearing force.

(a) (b)

FigureFigure 13. 13.Variation Variation of of ﬂuctuating fluctuating amplitude of of propeller propeller bearing bearing force force with with L/DL /(JD = (0.281,J = 0.281, atmospheric atmospheric pressure): pressure): (a) Fluctuating amplitude of KT; (b) Fluctuating amplitude of 10KQ. (a) Fluctuating amplitude of KT;(b) Fluctuating amplitude of 10KQ. Table 3. Variation of fluctuating amplitude of single blade‐bearing force with varying L/D.m (J = 0.281, atmospheric pres‐ sure).

APF 2APF L/D 1/8 3/8 5/8 7/8 1/8 3/8 5/8 7/8 KTX_BLADE 0.0053 0.0034 0.0024 0.0016 0.0050 0.0030 0.0019 0.0012 KTT_BLADE 0.0022 0.0018 0.0016 0.0014 0.0011 0.0007 0.0005 0.0004 10KQX_BLADE 0.0044 0.0030 0.0026 0.0024 0.0040 0.0024 0.0015 0.0011 10KQY_BLADE 0.0217 0.0128 0.0085 0.0054 0.0223 0.0127 0.0082 0.0051 10KQZ_BLADE 0.0030 0.0012 0.0012 0.0013 0.0037 0.0015 0.0009 0.0006 3APF 3BPF L/D 1/8 3/8 5/8 7/8 1/8 3/8 5/8 7/8 KTX_BLADE 0.0049 0.0028 0.0018 0.0012 0.0032 0.0003 0.0001 0.0002 KTT_BLADE 0.0015 0.0009 0.0007 0.0005 0.0010 0.0001 0 0 10KQX_BLADE 0.0042 0.0025 0.0018 0.0013 0.0026 0.0003 0.0001 0.0002 10KQY_BLADE 0.0210 0.0114 0.0075 0.0048 0.0085 0.0009 0.0003 0.0005 10KQZ_BLADE 0.0035 0.0014 0.0009 0.0006 0.0021 0.0002 0.0001 0.0001

As in Table 3, when L/D is fixed, the APF, 2APF, and 3APF amplitudes of the single blade‐bearing force do not change significantly, but the 3BPF amplitude decreases signifi‐ cantly. In addition to the torque coefficient in the Z direction, as L/D decreased, the APF, 2APF, and 3APF amplitudes of the other force and torque coefficients of the single blade showed a gradually increasing trend, whereas the 3BPF amplitude showed a variable trend. However, the 3BPF fluctuating amplitude was the largest when L/D was small (e.g., L/D = 1/8). With the increase in L/D, the 3BPF amplitude rapidly decreased and gradually tended to zero. Therefore, we concluded that the main fluctuating frequencies of the blade‐bearing force were APF, 2APF, and 3APF. As P1 is located above the ice when L/D = 1/8, the BPF, 2BPF, 3BPF, and 4BPF ampli‐ tudes of the five measurement points induced by the propeller for L/D = 3/8, 5/8, and 7/8

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Table 3. Variation of fluctuating amplitude of single blade-bearing force with varying L/D.m (J = 0.281, atmospheric pressure).

APF 2APF L/D 1/8 3/8 5/8 7/8 1/8 3/8 5/8 7/8

KTX_BLADE 0.0053 0.0034 0.0024 0.0016 0.0050 0.0030 0.0019 0.0012 KTT_BLADE 0.0022 0.0018 0.0016 0.0014 0.0011 0.0007 0.0005 0.0004 10KQX_BLADE 0.0044 0.0030 0.0026 0.0024 0.0040 0.0024 0.0015 0.0011 10KQY_BLADE 0.0217 0.0128 0.0085 0.0054 0.0223 0.0127 0.0082 0.0051 10KQZ_BLADE 0.0030 0.0012 0.0012 0.0013 0.0037 0.0015 0.0009 0.0006 3APF 3BPF J. Mar. Sci.L Eng./D 2021, 9, x FOR PEER1/8 REVIEW 3/8 5/8 7/8 1/8 3/8 5/8 7/818 of 20

KTX_BLADE 0.0049 0.0028 0.0018 0.0012 0.0032 0.0003 0.0001 0.0002 KTT_BLADE 0.0015 0.0009 0.0007 0.0005 0.0010 0.0001 0 0 10KQX_BLADE 0.0042were analyzed 0.0025 to study 0.0018 the influence 0.0013 of different 0.0026 L/D values 0.0003 on the 0.0001 fluctuating 0.0002pressure 10K 0.0210 0.0114 0.0075 0.0048 0.0085 0.0009 0.0003 0.0005 QY_BLADE at different measurement points. Figure 14 presents the test results. 10KQZ_BLADE 0.0035 0.0014 0.0009 0.0006 0.0021 0.0002 0.0001 0.0001

(a) (b)

Figure 14.14. VariationVariation of of ﬂuctuating fluctuating pressure pressure at measuringat measuring point point with withL/D L(/JD= ( 0.281,J = 0.281, atmospheric atmospheric pressure): pressure): (a) BPF; (a) (BPF;b) 2BPF; (b) (2BPF;c) 3BPF; (c) (3BPF;d) 4BPF. (d) 4BPF.

As indicated in Table3, in when FigureL/ D14,is the fixed, amplitudes the APF, of 2APF, the fluctuating and 3APF amplitudes pressures at of the the BPF single of theblade-bearing five measurement force do points not change were much significantly, larger than but the those 3BPF at the amplitude other frequencies; decreases signifi- as the frequencycantly. In addition increased, to the the amplitude torque coefficient of the fluctuating in the Z direction, pressure as presentedL/D decreased, a gradually the APF, de‐ creasing2APF, and trend. 3APF At amplitudes 4BPF, the amplitude of the other rapidly force and decreased torque coefficientsand tended of to the zero. single In Figure blade 14a, the amplitude of the fluctuating pressure undergoes subtle changes at the five meas‐ urement points. However, the P3 fluctuating pressure is the extreme value, and the exist‐ ence of the maximum and minimum values is closely related to L/D. The P3 fluctuating pressure was at its maximum when L/D was large (e.g., L/D = 5/8 and 7/8) and minimum when L/D was small (e.g., L/D = 3/8). By comparing the pressure of each measurement point, we obtained the law of variation of pressure near the hull surface. When L/D = 5/8 and 7/8, the pressure at P3 in the axial direction was the largest, followed by P2, whereas that at P1 was the smallest. The main reason for this trend is that ice could not effectively blocked the flow field around the propeller at large L/D values [4,9,21]. Owing to the in‐ fluence of the propeller wake, pressure at P3 at the propeller exhaust was higher than that at P1 at the intake. The pressure at P3 in the radial direction was the largest, followed by P4, whereas that at P5 was the smallest, which was due to the impact of water flow driven by the

J. Mar. Sci. Eng. 2021, 9, 674 17 of 25

showed a gradually increasing trend, whereas the 3BPF amplitude showed a variable trend. However, the 3BPF fluctuating amplitude was the largest when L/D was small (e.g., L/D = 1/8). With the increase in L/D, the 3BPF amplitude rapidly decreased and gradually tended to zero. Therefore, we concluded that the main fluctuating frequencies of the blade-bearing force were APF, 2APF, and 3APF. As P1 is located above the ice when L/D = 1/8, the BPF, 2BPF, 3BPF, and 4BPF amplitudes of the five measurement points induced by the propeller for L/D = 3/8, 5/8, and 7/8 were analyzed to study the influence of different L/D values on the fluctuating pressure at different measurement points. Figure 14 presents the test results. As indicated in Figure 14, the amplitudes of the fluctuating pressures at the BPF of the five measurement points were much larger than those at the other frequencies; as the frequency increased, the amplitude of the fluctuating pressure presented a gradually decreasing trend. At 4BPF, the amplitude rapidly decreased and tended to zero. In Figure 14a, the amplitude of the fluctuating pressure undergoes subtle changes at the five measurement points. However, the P3 fluctuating pressure is the extreme value, and the existence of the maximum and minimum values is closely related to L/D. The P3 fluctuating pressure was at its maximum when L/D was large (e.g., L/D = 5/8 and 7/8) and minimum when L/D was small (e.g., L/D = 3/8). By comparing the pressure of each measurement point, we obtained the law of variation of pressure near the hull surface. When L/D = 5/8 and 7/8, the pressure at P3 in the axial direction was the largest, followed by P2, whereas that at P1 was the smallest. The main reason for this trend is that ice could not effectively blocked the flow field around the propeller at large L/D values [4,9,21]. Owing to the influence of the propeller wake, pressure at P3 at the propeller exhaust was higher than that at P1 at the intake. The pressure at P3 in the radial direction was the largest, followed by P4, whereas that at P5 was the smallest, which was due to the impact of water flow driven by the clockwise of the propeller. The fluctuating pressure at different measurement points varied in a similar manner as that in the full scale hull-propeller–rudder system [26], which indirectly validates the test results. When L/D was smaller than the maximum (e.g., L/D = 3/8), the blockage effect of ice had a significant influence on the flow field structure at both the intake and the exhaust of the propeller, which changes the pressure at each measurement point. In addition, by comparing the influence of the L/D ratio on the fluctuating pressure at each measurement point, the amplitude of the fluctuating pressure at each measurement point presents a gradually increasing trend with a decrease in L/D. The smaller the L/D, the greater the amplitude of the increase. In Figure 14b, the law of variation of the 2BPF amplitude at each measurement point is as follows: the pressure at P3 is the largest at different L/D values. At no measurement point did the amplitude of the fluctuating pressure demonstrate a gradual increasing trend with a decrease in L/D. However, the amplitude at the measurement points was the largest when L/D was the smallest. In Figure 14c,d, because the amplitudes of the fluctuating pressure at 3BPF and 4BPF are much less than the 2BPF, the influence on the pressure near the hull surface is relatively small. Therefore, we did not analyze the amplitudes at those frequencies.

4.3.2. Rotational Speed Cavitation Number σ = 0.44 At the rotational speed cavitation number of σ = 0.44, the frequency-domain curves and the fluctuating amplitude of each order of the propeller- and single blade-bearing force under different ice blockage conditions were analyzed. Figures 15 and 16 present the results. In Figure 15, compared with the trend under atmospheric pressure, at σ = 0.44, the main fluctuating frequency in the frequency-domain curve of the propeller- and blade- bearing forces does not change. However, broadband characteristics appear near the 3BPF of the propeller thrust coefficient and blade-bearing force. The fluctuating amplitude of the frequency in this broadband was larger. Particularly, the amplitude near the 3BPF in the frequency-domain curve of the propeller thrust coefficient was larger than that of J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 19 of 21

clockwise of the propeller. The fluctuating pressure at different measurement points var‐ ied in a similar manner as that in the full scale hull‐propeller–rudder system [26], which indirectly validates the test results. When L/D was smaller than the maximum (e.g., L/D = 3/8), the blockage effect of ice had a significant influence on the flow field structure at both J. Mar. Sci. Eng. 2021, 9, 674 18 of 25 the intake and the exhaust of the propeller, which changes the pressure at each measure‐ ment point. In addition, by comparing the influence of the L/D ratio on the fluctuating pressure at each measurement point, the amplitude of the fluctuating pressure at each measurementthe BPF. The main point reason presents for a this gradually trend is increasing that the blockage trend with and a proximity decrease effectsin L/D of. The ice smalleron the propeller the L/D, the are greater clear when the amplitudeL/D is small. of the The increase. propeller–hull In Figure vortex 14b, the cavitation law of varia may‐ tionappear of the between 2BPF amplitude the ice and at the each propeller. measurement This and point the is bladeas follows: sheet the cavitation pressure combine at P3 is theto form largest a larger at different area of L/ sheetD values. cavitation At no (Figuremeasurement 11). The point broadband did the characteristicsamplitude of the of fluctuatingthe frequency-domain pressure demonstrate curve induced a gradual by sheet increasing cavitation trend bursting with a decrease are distinct. in L However,/D. How‐ ever,the broadband the amplitude characteristics at the measurement of the propeller points was torque the in largest the frequency-domain when L/D was the curve smallest. are Innot Figure obvious. 14c,d, The because main reasonsthe amplitudes for this areof the that fluctuating the pitch pressure of propeller at 3BPF blade and is relatively 4BPF are muchsmall, less the thrustthan the pressure 2BPF, isthe relatively influence large, on the the pressure rotational near speed the is hull relatively surface fast, is relatively the shear small.force is Therefore, relatively we small, did not and analyze the shear the force amplitudes induced at bythose cavitation frequencies. bursting is not very strong. In addition, compared with the atmospheric pressure (Figure 12), the decrease in 4.3.2.the cavitation Rotational number Speed also Cavitation leads to Number a clear decrease σ = 0.44 in the fluctuating amplitude of the BPF of theAt propeller the rotational thrust speed coefficient. cavitation The decreasenumber of in σ the= 0.44, amplitude the frequency of the main‐domain fluctuating curves andfrequency the fluctuating of the propeller amplitude torque of coefficient each order and of thethe singlepropeller blade-bearing‐ and single force blade is not‐bearing clear. forceTherefore, under we different do not ice perform blockage a separate conditions analysis were of analyzed. the amplitude Figures of 15 the and main 16 present fluctuating the results.frequency of the single blade-bearing force below.

(a) (b)

Figure 15. Frequency-domainFrequency‐domain curves of propeller and blade bearing force under blockage condition (J = 0.281, σ σ = 0.44): (a) KT, 10KQ curves in the frequency‐domain; (b) KTX_BLADE,10KQY_BLADE curves in the frequency‐domain. (a) KT, 10KQ curves in the frequency-domain; (b) KTX_BLADE,10KQY_BLADE curves in the frequency-domain.

InAs Figure shown 15, in compared Figure 16, with when theσ =trend 0.44, under the APF, atmospheric BPF, 2BPF, pressure, and 3BPF at σ amplitudes= 0.44, the mainof the fluctuating propeller thrust frequency coefficient in the showfrequency a gradually‐domain increasing curve of the trend propeller with a‐ decreaseand blade in‐ bearingL/D, which forces is does different not change. from that However, under atmospheric broadband characteristics pressure. The BPFappear amplitude near the of 3BPF the ofthrust the propeller coefficient thrust is not atcoefficient its maximum and blade at different‐bearingL/ Dforce.values. The The fluctuating maximum amplitude fluctuating of theamplitude frequency of eachin this order broadband is closely was related larger. to Particularly,L/D. Compared the amplitude with the near trend the under 3BPF the in theatmospheric frequency pressure‐domain conditioncurve of the (Figure propeller 13), thethrust decrease coefficient in the was cavitation larger than number that underof the BPF.this condition The main leads reason to for a decrease this trend in is the that BPF the and blockage 2BPF amplitudes and proximity of the effects thrust of coefficient ice on the propellerand an increase are clear in when the APF L/D amplitude, is small. The whereas propeller–hull the 3BPF vortex amplitude cavitation in the may broadband appear betweenincreases the significantly ice and the when propeller.L/D isThis large. and As theL blade/D decreased, sheet cavitation the BPF combine amplitude to form of the a largerpropeller area torque of sheet coefficient cavitation increased (Figure gradually, 11). The broadband which was characteristics consistent with of the the trend frequency under‐ domainatmospheric curve pressure. induced However, by sheet cavitation compared bursting with the are trend distinct. under However, the atmospheric the broadband pressure, characteristicsthe BPF amplitude of the of propeller the torque torque coefficient in the underfrequency this‐ conditiondomain curve decreased are not significantly. obvious. The mainTo reasons analyze for the this variation are that of the each pitch order of amplitudepropeller blade of the is propeller-induced relatively small, fluctuating the thrust pressure at σ = 0.44 and the influence of decompression on the fluctuating pressure at any given measurement point, the fluctuating pressures at that measurement point under atmospheric pressure and decompression are compared in Figure 17. The atmospheric pressure data are provided in Figure 14. J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 20 of 22

pressure is relatively large, the rotational speed is relatively fast, the shear force is rela‐ tively small, and the shear force induced by cavitation bursting is not very strong. In ad‐ dition, compared with the atmospheric pressure (Figure 12), the decrease in the cavitation number also leads to a clear decrease in the fluctuating amplitude of the BPF of the pro‐ peller thrust coefficient. The decrease in the amplitude of the main fluctuating frequency of the propeller torque coefficient and the single blade‐bearing force is not clear. There‐ J. Mar. Sci. Eng. 2021, 9, 674 fore, we do not perform a separate analysis of the amplitude of the main fluctuating19 offre 25‐ quency of the single blade‐bearing force below.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW(a) (b) 21 of 23

Figure 16. VariationVariation of of fluctuating ﬂuctuating amplitude amplitude of of propeller propeller bearing bearing force force with with L/LD/ (DJ =( J0.281,= 0.281, σ = σ0.44):= 0.44): (a) Fluctuating (a) Fluctuating am‐ plitude of KT; (b) Fluctuating amplitude of 10KQ. amplitude of KT;(b) Fluctuating amplitude of 10KQ.

As shown in Figure 16, when σ = 0.44, the APF, BPF, 2BPF, and 3BPF amplitudes of the propeller thrust coefficient show a gradually increasing trend with a decrease in L/D, which is different from that under atmospheric pressure. The BPF amplitude of the thrust coefficient is not at its maximum at different L/D values. The maximum fluctuating am‐ plitude of each order is closely related to L/D. Compared with the trend under the atmos‐ pheric pressure condition (Figure 13), the decrease in the cavitation number under this condition leads to a decrease in the BPF and 2BPF amplitudes of the thrust coefficient and an increase in the APF amplitude, whereas the 3BPF amplitude in the broadband increases significantly when L/D is large. As L/D decreased, the BPF amplitude of the propeller torque coefficient increased gradually, which was consistent with the trend under atmos‐ pheric pressure. However, compared with the trend under the atmospheric pressure, the BPF amplitude of the torque coefficient under this condition decreased significantly. To analyze the variation of each order amplitude of the propeller‐induced fluctuating pressure at σ = 0.44 and the influence of decompression on the fluctuating pressure at any (a) (b) given measurement point, the fluctuating pressures at that measurement point under at‐ mospheric pressure and decompression are compared in Figure 17. The atmospheric pres‐ sure data are provided in Figure 14.

(c) (d) Figure 17. VariationVariation of of fluctuating ﬂuctuating pressure pressure at at measuring measuring point point with with LL/D/ D(J (=J 0.281,= 0.281, atmospheric atmospheric pressure pressure and and σ = σ0.44):= 0.44): (a) BPF;(a) BPF; (b) (2BPF;b) 2BPF; (c) 3BPF; (c) 3BPF; (d) (4BPF.d) 4BPF.

InIn Figure 1717,, thethe amplitudes amplitudes of of BPF BPF of of the the fluctuating fluctuating pressure pressure at the at the five five measurement measure‐ mentpoints points are larger are thanlarger those than at those the other at the frequencies. other frequencies. The difference The difference between the between amplitude the amplitudeof the 2BPF of and the 3BPF 2BPF is and related 3BPF to isL /relatedD. Meanwhile, to L/D. Meanwhile, the fluctuating the fluctuating amplitude of amplitude the 4BPF of the 4BPF no longer tends to zero and is of the same order of magnitude as the BPF. When σ = 0.44, the amplitude of each order of the fluctuating pressure at P3 reached its maximum as L/D was varied. In the axial direction, P2 takes the second place and P1 is the smallest. In the radial direction, P5 takes the second place, and P4 is the smallest, which is different from the trend under atmospheric pressure. As L/D decreased, the am‐ plitude of the fluctuating pressure at other frequencies gradually increased, except for the 2BPF. Compared with the trend under atmospheric pressure, when L/D is small (e.g., L/D = 3/8 and 5/8), the cavitation number decreases. Thus, the amplitude of each order of the fluctuating pressure at the given measurement point is higher than that at atmospheric pressure. When L/D is large (e.g., L/D = 7/8), the amplitude of each order of the fluctuating pressure at the given measurement point is higher than that at atmospheric pressure be‐ cause of the decrease in the cavitation number, except for the BPF. In addition, in Figure 17, regardless of the variation of L/D, the decrease in cavitation number leads to a sharp increase in the amplitude of the fluctuating pressure at the 2BPF, 3BPF, and 4BPF. There‐ fore, reducing or preventing cavitation in the ice‐class propeller and improving the cavi‐ tation performance of the propeller will effectively reduce the fluctuating pressure in‐ duced by the propeller. This will in turn reduce the vibrations in the hull.

J. Mar. Sci. Eng. 2021, 9, 674 20 of 25

no longer tends to zero and is of the same order of magnitude as the BPF. When σ = 0.44, the amplitude of each order of the fluctuating pressure at P3 reached its maximum as L/D was varied. In the axial direction, P2 takes the second place and P1 is the smallest. In the radial direction, P5 takes the second place, and P4 is the smallest, which is different from the trend under atmospheric pressure. As L/D decreased, the amplitude of the fluctuating pressure at other frequencies gradually increased, except for the 2BPF. Compared with the trend under atmospheric pressure, when L/D is small (e.g., L/D = 3/8 and 5/8), the cavitation number decreases. Thus, the amplitude of each order of the fluctuating pressure at the given measurement point is higher than that at atmospheric pressure. When L/D is large (e.g., L/D = 7/8), the amplitude of each order of the fluctuating pressure at the given measurement point is higher than that at atmospheric pressure because of the decrease in the cavitation number, except for the BPF. In addition, in Figure 17, regardless of the variation of L/D, the decrease in cavitation number leads to a sharp increase in the amplitude of the fluctuating pressure at the 2BPF, 3BPF, and 4BPF. Therefore, reducing or J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 22 of 24 preventing cavitation in the ice-class propeller and improving the cavitation performance of the propeller will effectively reduce the fluctuating pressure induced by the propeller. This will in turn reduce the vibrations in the hull. 4.3.3. Rotational Speed Cavitation Number σ = 0.33 4.3.3. Rotational Speed Cavitation Number σ = 0.33 In this section, we analyze the frequency‐domain curves of the bearing forces of the propellerIn this and section, the single we analyze blade, the frequency-domainvariation in the fluctuating curves of amplitude the bearing of forces each order, of the propellerand the amplitude and the single of each blade, order the of variationthe induced in the fluctuating fluctuating pressure amplitude under of different each order, ice blockageand the amplitude conditions of at each the rotational order of the speed induced cavitation fluctuating number pressure of σ = 0.33 under (Figures different 18–20; ice Tableblockage 4). conditions at the rotational speed cavitation number of σ = 0.33 (Figures 18–20; TableAs4). shown in Figure 18, when σ = 0.33 and L/D = 1/8, the BPF amplitude (rotational speedAs n shown= 22.15 inr/s, Figure four‐blade 18, when propeller,σ = 0.33 frequency and L/D = =88.6 1/8, Hz) the of BPF the amplitudethrust coefficient (rotational and thespeed broadbandn = 22.15 characteristics r/s, four-blade near propeller, the 3BPF frequency (265.8 Hz) = 88.6are more Hz) of distinct. the thrust However, coefficient the amplitudesand the broadband of the APF characteristics and 2BPF do near not the change 3BPF significantly. (265.8 Hz) are The more variation distinct. trend However, of the mainthe amplitudes fluctuating of frequency the APF in and the 2BPF frequency do not‐domain change curve significantly. of the propeller The variation thrust coeffi trend‐ cientof the at main σ = 0.33 fluctuating is different frequency from that in theat σ frequency-domain = 0.44 (Figure 15a), curve and the of the amplitude propeller is thrustlarger coefficient at σ = 0.33 is different from that at σ = 0.44 (Figure 15a), and the amplitude than that near the 3BPF when σ = 0.44. The main fluctuating frequency of the propeller is larger than that near the 3BPF when σ = 0.44. The main fluctuating frequency of the torque coefficient and the single blade‐bearing force is still distinguishable, which is sim‐ propeller torque coefficient and the single blade-bearing force is still distinguishable, which ilar to the characteristics when σ = 0.44. However, the 3BPF of the bearing force in the is similar to the characteristics when σ = 0.44. However, the 3BPF of the bearing force in single blade has a more distinct broadband characteristic. the single blade has a more distinct broadband characteristic.

(a) (b)

Figure 18.18. Frequency-domainFrequency‐domain curves of propeller and blade bearingbearing forceforce underunder blockageblockage conditioncondition (JJ == 0.325, σσ = 0.33): (a) KT, 10KQ curves in the frequency‐domain; (b) KTX_BLADE, 10KQY_BLADE curves in the frequency‐domain. (a) KT, 10KQ curves in the frequency-domain; (b) KTX_BLADE, 10KQY_BLADE curves in the frequency-domain.

(a) (b) Figure 19. Variation of fluctuating amplitude of propeller bearing force with L/D (J = 0.325, σ = 0.33): (a) Fluctuating am‐ plitude of KT; (b) Fluctuating amplitude of 10KQ.

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 22 of 24

4.3.3. Rotational Speed Cavitation Number σ = 0.33 In this section, we analyze the frequency‐domain curves of the bearing forces of the propeller and the single blade, the variation in the fluctuating amplitude of each order, and the amplitude of each order of the induced fluctuating pressure under different ice blockage conditions at the rotational speed cavitation number of σ = 0.33 (Figures 18–20; Table 4). As shown in Figure 18, when σ = 0.33 and L/D = 1/8, the BPF amplitude (rotational speed n = 22.15 r/s, four‐blade propeller, frequency = 88.6 Hz) of the thrust coefficient and the broadband characteristics near the 3BPF (265.8 Hz) are more distinct. However, the amplitudes of the APF and 2BPF do not change significantly. The variation trend of the main fluctuating frequency in the frequency‐domain curve of the propeller thrust coeffi‐ cient at σ = 0.33 is different from that at σ = 0.44 (Figure 15a), and the amplitude is larger than that near the 3BPF when σ = 0.44. The main fluctuating frequency of the propeller torque coefficient and the single blade‐bearing force is still distinguishable, which is sim‐ ilar to the characteristics when σ = 0.44. However, the 3BPF of the bearing force in the J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEWsingle blade has a more distinct broadband characteristic. 23 of 25

In Figure 19, the BPF fluctuating amplitude of the propeller thrust coefficient is the largest when σ = 0.33. With a decrease in L/D, the BPF fluctuating amplitude presented a gradually increasing trend, whereas the variation trends of APF, 2BPF, and 3BPF were closely related to L/D. However, the 3BPF amplitudes of smaller axial distances (e.g., L/D = 1/8) were significantly larger than those of the larger ones. With a decrease in L/D, the BPF amplitude of the propeller torque coefficient demonstrated a gradually increasing trend, and the 5APF amplitude was mostly unchanged, but it is still the maximum. The variation of the APF and 2APF amplitude was closely related to L/D, and its trend was similar to that of σ = 0.44 (Figure 16). However, compared with σ = 0.44, the decrease in cavitation number led to an increase in the BPF fluctuating amplitude of the thrust coeffi‐ cient. This in turn led to an increase in the amplitude of the other frequencies, except the BPF in the torque coefficient. Table 4 lists the main fluctuating frequency amplitudes of the single‐blade bearing (a) (b) force with different L/D values when σ = 0.33. When L/D was fixed, the APF fluctuating J. Mar.Figure Sci. Eng. 18.2021 Frequency, 9, 674 ‐domainamplitude curves of of propeller the single and blade blade‐ bearing force force under was theblockage largest, condition the amplitudes (J = 0.325, σ of= the0.33): 2APF,21 of 25 (a) KT, 10KQ curves in the frequency3APF, and‐domain; 3BPF (decreased,b) KTX_BLADE, and10K QYthe_BLADE decrease curves inin cavitationthe frequency number‐domain. led to a more significant difference between the fluctuating amplitudes of each order, which is different from the trend under atmospheric pressure (the difference between σ = 0.44 and atmospheric pres‐ sure is not obvious, so take atmospheric pressure as an example). Meanwhile, compared with the atmospheric pressure, the decrease in cavitation number led to an increase in the APF and 3BPF fluctuating amplitudes of the single blade‐bearing force. The main fluctu‐ ating frequency amplitude of the single blade‐bearing force gradually increased with the decrease in L/D, and the law of variation became obvious.

Table 4. Variation of fluctuating amplitude of blade‐bearing force with L/D (J = 0.325, σ = 0.33).

APF 2APF L/D 1/8 3/8 5/8 7/8 1/8 3/8 5/8 7/8 KTX_BLADE 0.0060 0.0045 0.0029 0.0024 0.0046 0.0028 0.0017 0.0013 10KQY_BLADE 0.0216 0.0163 0.0105 0.0083 0.0179 0.0114 0.0070 0.0054

3APF 3BPF (a) (b) L/D 1/8 3/8 5/8 7/8 1/8 3/8 5/8 7/8 FigureKTX_BLADE 19. VariationVariation0.0026 of of fluctuating ﬂuctuating 0.0013 amplitude amplitude of0.0008 of propeller propeller bearing bearing0.0006 force force with with0.0033 L/LD/ (DJ =( J0.325,=0.0006 0.325, σ = σ0.33):= 0.33): (0.0004a) Fluctuating (a) Fluctuating0.0003 am‐ plitude of KT; (b) Fluctuating amplitude of 10KQ. 10amplitudeKQY_BLADE of KT;(b0.0104) Fluctuating amplitude0.0055 of 100.0035KQ. 0.0024 0.0088 0.0012 0.0011 0.0009

J. Mar. Sci. Eng. 2021, 9, x FOR PEER REVIEW 24 of 26

(a) (b)

Figure 20. VariationVariation of ﬂuctuatingfluctuating pressure atat measuringmeasuring point point with withL L//D ((σσ = 0.44 and σ σ == 0.33): 0.33): ( (aa)) BPF; BPF; ( (bb)) 2BPF; 2BPF; ( (cc)) 3BPF; 3BPF; (d) 4BPF.

As shown in Figure 20, the amplitude of the fluctuating pressure at each measure‐ ment point shows a decreasing trend with an increase in frequency when σ = 0.33. More‐ over, the amplitude of the fluctuating pressure no longer tends to zero at 4BPF. As L/D was varied, the amplitude of each order of the fluctuating pressure at P3 reached its max‐ imum. The difference of amplitudes between P2 and P1 and between P4 and P5 were re‐ lated to the order of the fluctuating pressure. As L/D was decreased, the fluctuating pres‐ sure amplitude of each frequency at each measurement point presented a gradually in‐ creasing trend. The variation in L/D from 3/8 to 7/8 increased the amplitude of the fluctu‐ ating pressure at each frequency of the measurement point by 2–3 times. Therefore, the stronger the ice blockage effect, the greater the fluctuating pressure at the measurement point, and the greater the hull vibration. In addition, comparing the amplitude of the fluc‐ tuating pressure at the measurement point at σ = 0.44 and 0.33 revealed that a decrease in the cavitation number led to a significant increase in the amplitudes of the APF, 2BPF, and 3BPF of the fluctuating pressure at the given measurement point. The main reason for this increase is closely related to the inflow velocity, rotational speed, and cavitation number at σ = 0.33.

5. Discussion During the ice breaking voyage of polar ships, especially under tail icebreaking con‐ ditions, large pieces of sea ice will be close to the propeller and result in a blockage effect on the propeller. Owing to the complexity of the ice–propeller blockage condition, the related research is not satisfactory; furthermore, the influence factors and related mecha‐ nisms of hydrodynamic performance, as well as the cavitation and excitation force due to ice–propeller blockage are not well‐understood. The experimental data obtained in this study can therefore help us understand the influence of ice–propeller blockage condition on the performance of propellers. Ice blocked the inflow of the propeller, resulting in a decrease in the axial inflow of the propeller disk, an increase in the angle of attack in the local area of the blade, and an increase of the thrust of the propeller. When the blade ro‐ tates, the separation of the fluid at the back of the blade also intensifies, resulting in a small increase in the torque of the propeller; especially when the blockage effect is large, the cavitation of the propeller will occur and the bearing force will increase. According to this result, the angle of attack of the blade section—that is, pitch ratio of the propeller—must be reduced; increasing the propeller area ratio can reduce the occurrence of cavitation. It is worth noting that the propeller excitation force will induce total and local vibra‐ tions of the hull, especially the stern vibration of the hull. Usually, this kind of vibration

J. Mar. Sci. Eng. 2021, 9, 674 22 of 25

Table 4. Variation of ﬂuctuating amplitude of blade-bearing force with L/D (J = 0.325, σ = 0.33).

APF 2APF L/D 1/8 3/8 5/8 7/8 1/8 3/8 5/8 7/8

KTX_BLADE 0.0060 0.0045 0.0029 0.0024 0.0046 0.0028 0.0017 0.0013 10KQY_BLADE 0.0216 0.0163 0.0105 0.0083 0.0179 0.0114 0.0070 0.0054 3APF 3BPF L/D 1/8 3/8 5/8 7/8 1/8 3/8 5/8 7/8

KTX_BLADE 0.0026 0.0013 0.0008 0.0006 0.0033 0.0006 0.0004 0.0003 10KQY_BLADE 0.0104 0.0055 0.0035 0.0024 0.0088 0.0012 0.0011 0.0009

In Figure 19, the BPF fluctuating amplitude of the propeller thrust coefficient is the largest when σ = 0.33. With a decrease in L/D, the BPF fluctuating amplitude presented a gradually increasing trend, whereas the variation trends of APF, 2BPF, and 3BPF were closely related to L/D. However, the 3BPF amplitudes of smaller axial distances (e.g., L/D = 1/8) were significantly larger than those of the larger ones. With a decrease in L/D, the BPF amplitude of the propeller torque coefficient demonstrated a gradually increasing trend, and the 5APF amplitude was mostly unchanged, but it is still the maximum. The variation of the APF and 2APF amplitude was closely related to L/D, and its trend was similar to that of σ = 0.44 (Figure 16). However, compared with σ = 0.44, the decrease in cavitation number led to an increase in the BPF fluctuating amplitude of the thrust coefficient. This in turn led to an increase in the amplitude of the other frequencies, except the BPF in the torque coefficient. Table4 lists the main fluctuating frequency amplitudes of the single-blade bearing force with different L/D values when σ = 0.33. When L/D was fixed, the APF fluctuating amplitude of the single blade-bearing force was the largest, the amplitudes of the 2APF, 3APF, and 3BPF decreased, and the decrease in cavitation number led to a more significant difference between the fluctuating amplitudes of each order, which is different from the trend under atmospheric pressure (the difference between σ = 0.44 and atmospheric pressure is not obvious, so take atmospheric pressure as an example). Meanwhile, compared with the atmospheric pressure, the decrease in cavitation number led to an increase in the APF and 3BPF fluctuating amplitudes of the single blade-bearing force. The main fluctuating frequency amplitude of the single blade-bearing force gradually increased with the decrease in L/D, and the law of variation became obvious. As shown in Figure 20, the amplitude of the fluctuating pressure at each measurement point shows a decreasing trend with an increase in frequency when σ = 0.33. Moreover, the amplitude of the fluctuating pressure no longer tends to zero at 4BPF. As L/D was varied, the amplitude of each order of the fluctuating pressure at P3 reached its maximum. The difference of amplitudes between P2 and P1 and between P4 and P5 were related to the order of the fluctuating pressure. As L/D was decreased, the fluctuating pressure amplitude of each frequency at each measurement point presented a gradually increasing trend. The variation in L/D from 3/8 to 7/8 increased the amplitude of the fluctuating pressure at each frequency of the measurement point by 2–3 times. Therefore, the stronger the ice blockage effect, the greater the fluctuating pressure at the measurement point, and the greater the hull vibration. In addition, comparing the amplitude of the fluctuating pressure at the measurement point at σ = 0.44 and 0.33 revealed that a decrease in the cavitation number led to a significant increase in the amplitudes of the APF, 2BPF, and 3BPF of the fluctuating pressure at the given measurement point. The main reason for this increase is closely related to the inflow velocity, rotational speed, and cavitation number at σ = 0.33. J. Mar. Sci. Eng. 2021, 9, 674 23 of 25

5. Discussion During the ice breaking voyage of polar ships, especially under tail icebreaking conditions, large pieces of sea ice will be close to the propeller and result in a blockage effect on the propeller. Owing to the complexity of the ice–propeller blockage condition, the related research is not satisfactory; furthermore, the influence factors and related mechanisms of hydrodynamic performance, as well as the cavitation and excitation force due to ice–propeller blockage are not well-understood. The experimental data obtained in this study can therefore help us understand the influence of ice–propeller blockage condition on the performance of propellers. Ice blocked the inflow of the propeller, resulting in a decrease in the axial inflow of the propeller disk, an increase in the angle of attack in the local area of the blade, and an increase of the thrust of the propeller. When the blade rotates, the separation of the fluid at the back of the blade also intensifies, resulting in a small increase in the torque of the propeller; especially when the blockage effect is large, the cavitation of the propeller will occur and the bearing force will increase. According to this result, the angle of attack of the blade section—that is, pitch ratio of the propeller—must be reduced; increasing the propeller area ratio can reduce the occurrence of cavitation. It is worth noting that the propeller excitation force will induce total and local vibrations of the hull, especially the stern vibration of the hull. Usually, this kind of vibration can cause problems such as cabin vibration in excess of the standard, failure of machinery and equipment, and fatigue of structural members. The uneven wake field and generation of variable cavitation at the stern are the catalysts for the sharp increase in excitation force. In the process of ice blocked the propeller, the occurrence of cavitation on the blade leads to a significant increase in the fluctuating pressure and a change in the phase. This is especially so for heavy load propellers operating in the wake with significant non-uniformity; that is, when the blade enters the high wake area after the ice, cavitation will inevitably appear. As the blade leaves the high wake area, the cavitation disappears again. This time-varying unsteady cavitation leads to a sharp increase in the amplitude of the high-order frequency of the propeller-induced fluctuating pressure, which is the main reason for the larger excitation force of the ice-class propeller. Therefore, according to the test results, during the process of ice-class propeller design, the rake of the blade should be properly adjusted, and the distance between the tip of the blade and the stern plate should be increased to reduce the influence of propeller-induced excitation force. At the same time, when the cavitation of the propeller is considered, the influence of propeller-induced excitation force can also be reduced by increasing the skew of the propeller.

6. Conclusions We studied the propeller cavitation and induced excitation force under ice-blockage conditions in a large circulating water channel. The changes in the hydrodynamic load of the propeller and single blade, cavitation, and excitation force with the change in the ice–propeller axial distance and cavitation number were analyzed. The main conclusions are as follows: • At atmospheric pressure and σ = 0.44, and under blockage conditions, the smaller the ice–propeller axial distance, the larger the propeller thrust coefficient, torque coefficient, and blade five-component force coefficient. However, the propeller efficiency gradually decreased at atmospheric pressure and was almost unchanged at σ = 0.44. The propeller efficiency in the two blockage conditions was less than that in the open- water condition. Compared with the trend under atmospheric pressure, the decrease in the cavitation number led to an increase in the propeller thrust coefficient, torque coefficient, and blade five-component force coefficient, but a decrease in propeller efficiency. The fluctuation range of the blade load increased with the decrease in the axial space between the ice and the propeller and the cavitation number. • At the rotational speed cavitation number of σ = 0.44, and under blockage conditions, propeller–hull vortex cavitation is induced between the ice and the propeller when the axial distance and vertical distance of ice–propeller are 1/8D and 5/16 D, respec- J. Mar. Sci. Eng. 2021, 9, 674 24 of 25

tively. The propeller–hull vortex cavitation is the result of the combined effects of ice blockage, proximity, propeller suction, the circumfluence zone around ice, and the Pirouette effect. • Under atmospheric pressure and decompression in the blockage condition, the smaller the ice–propeller axial distance, the larger the amplitude of the BPF of the propeller bearing force and the APF of the single blade. Compared with the trend under atmospheric pressure, the decrease in the cavitation number led to the appearance of broadband characteristics near the 3BPF of the propeller thrust coefficient and blade- bearing force. The smaller the cavitation number, the more distinct the broadband characteristics. When the ice–propeller axial distance was small, the amplitude of the 3BPF of the propeller thrust coefficient and blade-bearing force sharply increased. • Under atmospheric pressure and decompression in the blockage condition, the BPF amplitude of the fluctuating pressure induced by the propeller was the largest. With a decrease in the ice–propeller axial distance, the larger the amplitude of the BPF, and the closer it is to the propeller disk, the faster the amplitude of the BPF increases. Compared with the trend under atmospheric pressure, the decrease in cavitation number leads to a sharp increase in the amplitude of the high-order frequency of the propeller-induced fluctuating pressure. Under atmospheric pressure, the fluctuating pressure at the measurement point behind the propeller is an extreme value, which is closely related to the ice–propeller axial distance. The decrease in the cavitation number led to maximization of the fluctuating pressure at the measurement point behind the propeller, which had the most obvious effect on the hull vibration. Therefore, the results obtained from model test are expected to provide some experimental guidance and technical support for improving the comprehensive design level of ice-class propeller efficiency and ice load-carrying capacity and improving the safety and rapidity of polar-class ships in frigid zones. Furthermore, this study focused on the analysis of changes in propeller cavitation and induced excitation force under ice-blockage conditions. During the test, the ice asymptotic motion was constrained, and the degree of freedom of ice was curtailed. In the future, an experimental study on the cavitation and induced excitation force of the propeller can be carried out when the ice in free motion is close to the propeller via suction.

Author Contributions: Writing—original draft preparation, P.X.; writing—revision and review, C.W.; data curation, L.Y.; formal analysis, C.G.; provided the experimental guidance, W.X. and S.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China, grant numbers 51809055 and 51909043. China Postdoctoral Science Foundation, grant numbers 2019M651266 and 2020M681082. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors really appreciate the anonymous reviewers for their valuable comments and suggestions that can improve the quality of this manuscript. Conﬂicts of Interest: The authors declare that there are no conﬂicts of interest regarding the publica- tion of this paper.

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