Journal of Marine Science and Engineering

Article Experimental and Numerical Research on the Influence of Stern Flap Mounting Angle on Double-Stepped Planing Hull Hydrodynamic Performance

Jin Zou 1, Shijie Lu 1,* , Yi Jiang 2, Hanbing Sun 1,* and Zhuangzhuang Li 1

1 College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China; [email protected] (J.Z.); [email protected] (Z.L.) 2 China Ship Scientific Research Center, Wuxi 214000, China; [email protected] * Correspondence: [email protected] (S.L.); [email protected] (H.S.); Tel.: +86-188-451-42429 (S.L.); +86-137-966-29437 (H.S.)


Received: 30 June 2019; Accepted: 19 September 2019; Published: 1 October 2019


Abstract: In the current hydrodynamic research relating to planing hulls, the stern flap and steps are generally considered to be two independent resistance reduction measures. Limited research has focused on the coupled effects of flaps and steps. Therefore, experimental and numerical simulation methods are carried out in this paper to explore the influence of the flap mounting angle coupled with the steps. A series of model towing tests were implemented for a double-stepped planing hull with 2◦, 3◦ and 4.5◦ flap angles. The test results show that, as the mounting angle increased, the low speed resistance performance was improved and the porpoising critical speed was delayed, with a slight resistance cost. Based on the tests, a numerical simulation method was established with volume Froude numbers ranging from 0.88 to 5.20. The simulated hull flow field showed good agreement with the testing data. The simulation results suggest a cavity induces the negative pressure after the steps; the cavity core region is the air phase, and this expands with the air–water mixture flow. The cavity also causes wetted surface reduction and pressure distribution changes. Finally, comparisons of cavities after-steps and load coefficients of different planing surfaces among models were considered. Numerical results analysis gave distinct interpretations for the experimental phenomenon of porpoising critical speed increasing with a slight resistance increment.

Keywords: experimental and numerical research; stern flap; stepped planing hull; cavity; porpoising; wetted surface; hydrodynamic performance

1. Introduction The stern flap, also known as the stern baffle, is always shaped as a flat plate with a high aspect ratio, fixed or angle adjustable, and attached to the rear of a high speed transom stern hull. During the navigation process, the hydrodynamic lift generated by the stern flap can affect the motion attitude of the hull, to a certain extent [1]. As a result, the pressure distribution at the hull bottom and the wave-making property of the wake field are changed, while the resistance performance is also improved [2]. The stern flap can also be introduced to the planing hull, regulating the motion attitude to a specific attack angle for optimal hydrodynamic performance [3,4]. The research on planing crafts with a stern flap shows that it can improve the resistance performance, and also enhance the motion stability performance at high speed, for instance by inhibiting the inception of porpoising [5,6]. As an independent resistance reducing method from the stern flap, steps are introduced to improve the resistance and efficiency of the planing hull [7–10]. At present, research into stepped planing

J. Mar. Sci. Eng. 2019, 7, 346; doi:10.3390/jmse7100346 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2019, 7, 346 2 of 25 hulls has mainly been carried out through model tests conducted by different researchers focusing on resistance performance, cavity formation processes, and influencing factors of the double-stepped planing hull. Garland has carried out tests on NSWC (Naval Surface Warfare Center, USA) series models with steps to investigate the effects of the stepped planing hull [11]. A comparison of the resistance and motion attitude of the modified and non-stepped hull was carried out, along with investigations into the cavity shape after the step. Lotf et al. also investigated a one-stepped planing hull, and validated their simulations with experimental works [12]. A series of model tests on a double-stepped planing hull were carried out to research the resistance performance influencing factors of the hull. Besides the numerical method, other research has also been conducted focusing on the stepped planing hull [13–15]. As a derivative high speed planing hull type, the stepped planing hull has obvious resistance advantages in the planing state, as compared with the same series of non-stepped hull. However, in low speed segments, the stepped hull has an obvious trim and resistance peak [16,17]. As discussed above, the stern flaps can improve the resistance and navigation stability performance of the planing hull. Therefore, in order to improve the resistance and seakeeping performance, as well as increase the porpoising inception speed as far as possible, stern flaps can also be introduced to the stepped planing hull attitude adjustment. The research mentioned above covers multiple aspects of the stern flap and stepped planing hull. However, in the current hydrodynamics research on planing hulls, stern flaps and steps are generally treated as two independent resistance reduction measures. Relatively few researchers have focused on the effects of stern flaps and steps coupling [17–19]. The coupling effects of the flaps and steps are inevitably related to the motion and resistance characteristics of the hull. Therefore, in the current paper, the focus is on the double-stepped planing hull with different mounting angles. By combining experimental and numerical methods, the effects of the stern flaps on reducing low speed resistance and inhibiting porpoising of the double-stepped planing hull are investigated. Based on the experimental and numerical simulation results, the flap action mechanism and influence on hydrodynamic performance, coupled with the effects of the steps, are discussed and summarized. In Section2, a brief description of the hull and the towing test arrangement is introduced, and the experimental results and analysis of the experiment are also shown in detail. In Section3, the numerical method is developed; the boundary conditions and mesh generation process are carried out. In Section4, the numerical simulation results are validated. Further, based on the numerical analysis and results, the action mechanism and influence of the flaps coupled with the steps on hydrodynamic performance are discussed. In Section5, the conclusions are briefly summarized.

2. Towing Test

2.1. Geometrical Description of the Hull Model Teng [19] discussed the height of steps and the longitudinal step position of series DSPBM (double-stepped planing boat model) models. Based on that work, we selected DSPBM as the parent ship, and combined with the performance and research experience, an optimized model was introduced. The geometry of the hull and step configuration are shown in Figure1. It can be seen that two V-type steps are introduced to the planing hull bottom along the longitudinal direction. Hence, the planing bottom is divided into three segments, which are identified as the first, second and third planing faces, abbreviated as P1, P2 and P3 from the bow to rear. In addition, to study the effects of stern flap mounting angle on the stepped planing hull, two stern flaps with a length of 4.5% L were mounted to both sides of the bottom. The lifting surface was connected to the planing face, and the chine line was adapted to the flap at the bilge, so as to remain consistent with the hull. During the test, the stern flap mounting angle is defined as the spatial acute angle between the surface of P3 and the stern flap. The flap was mounted in a fixed form with angles of 2◦, 3◦ and 4.5◦. The principal dimensions of the hull are defined as follows: The overall length L was 2370 mm, and the beam between the chine line B J. Mar. Sci. Eng. 2019, 7, 346 3 of 25

L, referring to the set value of the parent hull, which has both appropriate resistance and stability performance [19]. The exact longitudinal center of gravity X g was 700 mm. The rest of the main characteristics are given in Table 1.

Table 1. Main dimensions of the hull and flap.

Main Feature Symbol Value Length overall (mm) L 2370 Beam between chine line (mm) B 646 Longitudinal step-A (from rear) (mm) LA 1000 Longitudinal step-B (from rear) (mm) LB 600 Longitudinal center of gravity (from rear) (mm) X g 700 J. Mar. Sci. Eng. 2019, 7, 346 3 of 25 Height of steps (mm) H 8 Deadrise angle/ (°) β1 18 was 646 mm.Angle As for between the longitudinal steps & middle center line of plane/ gravity, (°) it has a significantβ2 impact80 on both resistance and stabilityMounting performance. angle Therefore,s of stern flap/ the hull(°) gravity center was setβ at 0.3 L, referring2, 3, 4.5 to the set value of the parentAverage hull, which width has projection both appropriate of stern flap resistance (mm) and stabilityBf performance138 [19]. The exact longitudinalLength center ofof gravitystern flapX g(mm)was 700 mm. The rest of the mainL characteristicsf 100 are given in Table1.

Figure 1. Model of the double-stepped planing hull. Figure 1. Model of the double-stepped planing hull. Table 1. Main dimensions of the hull and flap. 2.2. Experimental Setup Main Feature Symbol Value The towingLength tank test overall of the (mm) double-stepped planing hull modelL was carried2370 out in the towing tank of the ChinaBeam Special between Vehicle chine Research line (mm) Institute (also called theB NO. 605 646Subsidiary Research Institute, AviationLongitudinal Industry of step-A China (from Group). rear) Th (mm)e main dimensionsL were:A length1000 510 m × width 6.5 m × depth 6.8 m.Longitudinal As the corollary step-B carriage (from rear) towing (mm) system can reachL Ba speed range600 of 0.1 m/s to 22 m/s, with a stableLongitudinal speed error center under of 0.1%, gravity the (frommaximum rear) (mm)length towingX g tank in China700 adequately met the requirementsHeight of theof test. steps Figure (mm) 2 shows a setup schematic overviewH of the experimental8 test. The Deadrise angle/(◦) β1 18 Angle between steps & middle line plane/(◦) β2 80 Mounting angles of stern flap/(◦) β 2, 3, 4.5 Average width projection of stern flap (mm) Bf 138 Length of stern flap (mm) Lf 100

2.2. Experimental Setup The towing tank test of the double-stepped planing hull model was carried out in the towing tank of the China Special Vehicle Research Institute (also called the NO. 605 Subsidiary Research Institute, Aviation Industry of China Group). The main dimensions were: length 510 m width 6.5 m depth 6.8 m. As the corollary carriage towing system can reach a speed × × range of 0.1 m/s to 22 m/s, with a stable speed error under 0.1%, the maximum length towing tank in China adequately met the requirements of the test. Figure2 shows a setup schematic overview of the experimental test. The double-stepped planing hull model was fixed to the carriage with the pitch and heave set to two degrees of freedom. In detail, the measuring device and sensors were arranged as follows: J. Mar. Sci. Eng. 2019, 7, 346 4 of 25

J. Mar.double-stepped Sci. Eng. 2019, 7 ,planing 346 hull model was fixed to the carriage with the pitch and heave set to two4 of 25 degrees of freedom. In detail, the measuring device and sensors were arranged as follows:

Figure 2. Schematic view of the towing test set up. Figure 2. Schematic view of the towing test set up. Two guide rods were introduced and mounted in the front and back of the model, preventing Two guide rods were introduced and mounted in the front and back of the model, preventing yaw and roll motions. The towing point at both broadsides aligned with the gravity center, avoiding yaw and roll motions. The towing point at both broadsides aligned with the gravity center, avoiding a longitudinal movement being produced during the towing process. During the towing tests, a a longitudinal movement being produced during the towing process. During the towing tests, a resistance dynamometer was mounted on the carriage to measure the resistance, a cable-extension resistance dynamometer was mounted on the carriage to measure the resistance, a cable-extension displacement sensor was mounted at the gravity center, measuring the value of sinkage, and the trim displacementangle was measured sensor was by an mounted angle sensor at the attached gravity to center, the foredeck. measuring All devices the value used of for sinkage, data acquisition and the trim anglehave was regularly measured been by calibrated. an angle sensorAfter these attached device to and the foredeck.sensor arrangements All devices were used set for up, data to acquisitionmonitor havethe regularly wave-making been characteristics, calibrated. After two thesecameras device were and mounted sensor before arrangements and after the were model. set up, to monitor the wave-makingThe towing characteristics, test was strictly two conducted cameras werefollowing mounted the ITTC before Recommended and after the model.Procedures and GuidelinesThe towing [20,21]. test During was strictly each run, conducted the experimental following data the ITTCsuch as Recommended resistance, sinkage Procedures and trim and Guidelinesacquisition [20 began,21]. Duringafter a steady each run, speed the had experimental been reached. data As such for asthe resistance, data post-processing, sinkage and the trim acquisitionexperimental began data after is the a steadymean value speed derived had been from reached. an integration As for of the instantaneous data post-processing, measured the experimentalvalues over data the issame the meanmeasuring value interval, derived with from th ane integrationzero measurements of the instantaneous being subtracted measured from valuesthe overaverage the same values. measuring In the model interval, tests, with three the zerodiffer measurementsent stern flap mounting being subtracted angle conditions from the averagewere values.considered In the for model β = 2°, tests, 3° and three 4.5°. di ffCalmerent water stern towing flap mounting tests were angle carried conditions out at speeds were ranging considered from for 0.44 to 6.13 in Froude numbers. For each angle condition, when the porpoising phenomenon β = 2◦, 3◦ and 4.5◦. Calm water towing tests were carried out at speeds ranging from 0.44 to 6.13 in Froudeoccurred, numbers. the towing For each test process angle condition, was terminated. when the porpoising phenomenon occurred, the towing test process was terminated. 2.3. Experimental Results and Analysis 2.3. ExperimentalIn this paper, Results a series and Analysisof tests were carried out under different conditions of heave, pitch, and total resistance, at different speeds. In addition, according to the camera records, the hull wave- In this paper, a series of tests were carried out under different conditions of heave, pitch, and total making and motion attitudes were also observed. This information was advantageous to the auxiliary resistance, at different speeds. In addition, according to the camera records, the hull wave-making and analysis of the results. Test data dimensionless processing was carried out as follows: motion attitudes were also observed. This information was advantageous to the auxiliary analysis of Non-dimensional resistance is defined by the resistance to weight ratio (R/Δ). Non-dimensional thespeed results. is defined Test data by dimensionlessvolume based on processing the Froude was number, carried following out as follows: the equation: Non-dimensional resistance is defined by the resistance to weight ratio (R/∆). Non-dimensional V , Fr∇ = speed is defined by volume based on the Froude number,1/3 following the equation: (1) g ()∇ where V is the test speed, and ∇ is the displacementV of the model. Non-dimensional sinkage is Fr = q (1) ∇ defined by the equation: g( )1/3 δ ∇ δ = exp , (2) where V is the test speed, and is the displacement ofT the model. Non-dimensional sinkage is defined ∇ bywhere the equation: δexp is the experimental trim angle and sinkage value, and T is the static average draft. In δexp addition, the positive value of trim angle θ andδ δ= represent, the increase of the up-pitch and heave. (2) T where δexp is the experimental trim angle and sinkage value, and T is the static average draft. In addition, the positive value of trim angle θ and δ represent the increase of the up-pitch and heave. J. Mar. Sci. Eng. 2019, 7, 346 5 of 25 J. Mar. Sci. Eng. 2019, 7, 346 5 of 25

DuringDuring the the towing towing test, test, models models with with flap flap mounting mounting anglesangles of 22°,◦, 3° 3◦ andand 4.5° 4.5 ◦werewere carried carried out, out, respectively.respectively. In order In order to facilitate to facilitate the the discussion discussion in thisin this paper, paper, the the series series models models are are named named M1, M1, M2 M2 and M3 withand M3 the with increasing the increasing angle of angle the sternof the flap.stern Inflap. Figure In Figure3, the 3, non-dimensional the non-dimensional resistance, resistance, trim, trim, and heaveand are heave plotted are asplotted functions as functions of the volumeof the volume Froude Fr number,oude number, individually. individually It can. It becan seen be seen that, that, after increasingafter increasing the stern flapthe angle,stern flap the trimangle, and the heave trim ofand the modelheave of were the both model reduced. were Simultaneously,both reduced. beforeSimultaneously, porpoising, the before maximum porpoising, Froude the number maximum that Froude M1 could number reach that was M1 5.20, could while reach the was maximum 5.20, velocitywhile of the M2 maximum increased tovelocityFr = 5.69; of M2 and increased M3 to 6.13. to Fr Thus,▽ = 5.69; for the and double-stepped M3 to 6.13. Thus, planing for the boat double- studied 5 in thisstepped paper, planing increasing boat thestudied mounting in this anglepaper, stern increasing flap angle the mounting can clearly angle inhibit stern the flap porpoising angle can clearly motion. inhibit the porpoising motion.

0.3

0.25

0.2

Δ 0.15 No flaps R/ M1 0.1 M2 M3 0.05

0 01234567Fr ▽ (a)

14 No flaps 12 M1 10 M2 8 M3

° /

θ 6

4

2

0 01234567 Fr ▽ (b)

0.8

0.6

No flaps 0.4 M1 δ M2 0.2 M3

0 01234567

-0.2 Fr▽

(c)

FigureFigure 3. Comparisons 3. Comparisons of experimentalof experimental results results with with increasing increasing speed.speed. ( (aa)) Non-dimensional Non-dimensional resistance; resistance; (b) trim(b) trim angle; angle; (c) non-dimensional(c) non-dimensional sinkage. sinkage. J. Mar. Sci. Eng. 2019, 7, 346 6 of 25

Table2 shows the comparison of the dimensionless experimental resistance among di fferent models. The resistance increasing rate is also compared in Table2, reflecting the e ffects of flap mounting angle on resistance. A positive value indicates increasing resistance and a negative value indicates reducing resistance. A symbol of “–” indicates when porpoising occurred. At that time, there existed no stable resistance value. It can be observed that when the Froude number was under 3.94, the stern flap played a positive role in improving the resistance performance. Both M2 and M3 achieved a certain resistance reduction effect in this segment of speed. The resistance reduction of M3 was more obvious. At the speed of Fr = 0.88, M3 reached a resistance reducing benefit of 8.9%, while at the 5 resistance hump, the value was 4.9%. With the increase of speed, under the condition of Fr > 3.94, the 5 stern flap had a limited effect on the hull resistance performance. The resistance increased to a slight extent. Compared with the experimental data of M1, the maximum resistance increase was just 1.7%.

Table 2. Comparisons of experimental resistance between different models.

M1 M2 M3 Fr ∇ R /∆ R /∆ (R R )/R R /∆ (R R )/R 1 2 2− 1 1 3 3− 1 1 0.44 0.0073 0.0069 5.5% 0.0076 4.1% − 0.88 0.0715 0.0709 0.8% 0.0652 8.9% − − 1.31 0.1971 0.1940 1.6% 0.1875 4.9% − − 2.19 0.2370 0.2340 1.3% 0.2263 4.5% − − 2.63 0.2106 0.2070 1.7% 0.2024 3.9% − − 3.07 0.1891 0.1879 0.6% 0.1824 3.5% − − 3.50 0.1799 0.1781 1.0% 0.1736 3.5% − − 3.94 0.1780 0.1777 0.2% 0.1741 2.2% − − 4.38 0.1803 0.1827 1.3% 0.1816 0.7% 4.82 0.1924 0.1941 0.9% 0.1921 0.2% − 5.20 0.2025 0.2057 1.6% 0.2060 1.7% 5.69 – 0.2187 – 0.2286 – 6.13 – – – 0.2534 –

According to the experimental data and observations, the stern flap improved the fluid field around the double-stepped planing hull and the longitudinal movement generated by the hydrodynamic lifting force, which not only optimized the double-stepped planing hull motion attitude but also extended the porpoising critical speed. With the flaps’ and steps’ coupled effects, the resistance performance was impacted by the distribution and state of the cavity after the steps, as well as the wetted surface. Specifically, in the medium and low speed segment, the resistance performance was improved, while in the high speed segment, the porpoising inception speed was delayed, with a slight resistance cost.

3. Numerical Simulation Setup According to the experimental data analysis and observations above, the tested planing hull coupled flaps and double steps show favorable resistance performance at medium and low speed. Furthermore, they also play a positive role in inhibiting porpoising, with a slight resistance cost. However, the mechanism and process of the coupling effect deserve in-depth analysis and research. For this reason, the numerical simulations were carried out based on the model tank towing test. The Froude number ranged from 0.88 to 5.20.

3.1. Mathematical and Numerical Models As mentioned, the numerical simulations were carried out at the whole speed segment, where the Froude number ranged from 0.88 to 5.20. The speed distribution and stern flap mounting angles were consistent with the trial conditions. In order to establish the solving system of the incompressible viscous flow field surrounding the hull, FVM (finite volume method) software CFX was adapted for use in this paper. As for the simulation of incompressible viscous flow, the RANS J. Mar. Sci. Eng. 2019, 7, 346 7 of 25

(Reynold-Averaged Navier–Stokes) equation method was adopted for solving the Navier–Stokes (N–S) equations. The continuity equations and Reynolds average from the N–S equations are written as:

∂ρ ∂(ρui) + = 0, (3) ∂t xi

∂(ρu ) ∂(ρuiuj) ∂p ∂ ∂u i + = + ( i ) + µ ρui0u0j ρgi, (4) ∂t ∂xi −∂xi ∂xj ∂xj − where ui, uj represent the components of the velocity vector, p is the pressure, ρ is the density of the

fluid, µ is the coefficient of dynamic viscosity, and ρui0u0j is the Reynolds stress. The SST (shear stress transfer) k-ω turbulence model was introduced to close the system, and the wall function was used for the near wall treatment. The coupling of pressure and velocity was realized with the SIMPLE algorithm. Concretely, the transport equations for k and ω are written as:

∂ ∂ ∂ ∂k (ρk) + (ρkui) = (Γk ) + Gk Yk, (5) ∂t ∂xi ∂xi ∂xi −

∂ ∂ ∂ ∂ω (ρω) + (ρωui) = (Γω ) + Gω Yω + Dω, (6) ∂t ∂xi ∂xi ∂xi − where Gk and Gω represent the turbulence kinetic energy generation items, and Yk and Yω represent the turbulence dissipation rate of k and ω, respectively. As for the free surface, the VOF (volume of fluid) method was introduced to simulate the evolution procedure. At the initial moment, the free-surface height was defined as Hw, while the mesh height was Z. Further, the volume fraction of air–water was defined as:

V = step(Z Hw), (7) air −

Vwater = 1 V , (8) − air If Z-Hw > 0, then Vair = 1. This indicates a total air phase mesh; if Z-Hw < 0, then Vwater = 1. This situation indicates a total water phase mesh. When Z-Hw = 0, Vair = Vwater = 0.5, it represents the distribution here is a free surface. Thus, the air–water phase in the whole flow domain is decided. In the process of finding the discrete flow field by using the finite volume method, due to the strong directivity of the convection, a high resolution scheme was adopted to discretize the convection term. In the towing test, the heave and trim angle were measured to describe the motion. Thus, during the numerical simulation process, in order to remain consistent with the towing test setups, 2-DOF (2 degrees of freedom, heave and pitch) motions were released. The following force and moment equilibrium motion equations were introduced to solve the resistance, and motion of the hull:

d2→X →F= m , (9) dt2    →  → d  dθ  M = I , (10) dt dt  where I is the gravity center inertia mass matrix of the hull. →F and M→ are the forces and moments calculated by the equations below: Z →F = ([τ] p[I]) →ndS →G, (11) s − • − J. Mar. Sci. Eng. 2019, 7, 346 8 of 25

Z     M→ = →r →r →τ p[I] →ndS, (12) J. Mar. Sci. Eng. 2019, 7, 346 G 8 of 25 s − × − ·  wherewhere[τ],[pτ[]I，],p and[I ] ,→G and, respectively, G , respectively, represent represent the shear the shea stress,r stress, pressure, pressure, and and gravity. gravity.S represents S represents the   hull surface, and the vectors →r and →r G represent the displacements of the mesh nodes and gravity the hull surface, and the vectors r and rG represent the displacements of the mesh nodes and center. The main dynamic solver simulating procedure is shown in Figure4, and the numerical process gravity center. The main dynamic solver simulating procedure is shown in Figure 4, and the is mainly carried out as follows: numerical process is mainly carried out as follows:

FigureFigure 4. 4.Solving Solving procedure procedure ofof thethe algorithm.algorithm.

First,First, the N–Sthe N–S equation equation system system and theand continuity the continui equationty equation were were solved solved based based on the on initial the initial meshes inputted.meshes After inputted. that, theAfter shear that, stress the shear field stress and pressurefield and field pressure around field the around hull surface the hull were surface integrated were to calculateintegrated the to forces calculate and the moments forces and acting moments on the acti hull,ng on as the described hull, as indescri Equationsbed in Equations (7) and (8). (7) Third,and by solving(8). Third, Equations by solving (5) andEquations (6), the (5) parameters and (6), the of parameters velocity, motion of velocity, acceleration, motion acceleration, and displacement and weredisplacement calculated and were obtained. calculated Subsequently, and obtained. the Subs hullequently, positing the and hull mesh positing node and could mesh then node be updated. could then be updated. With updated hull position and mesh nodes, a loop computation was carried out to With updated hull position and mesh nodes, a loop computation was carried out to solve the fluid solve the fluid field. The termination criteria of this algorithm were variations of forces and moments field. The termination criteria of this algorithm were variations of forces and moments stabilizing to stabilizing to approach 0, which suggests that the hull has achieved an equilibrium state and the approach 0, which suggests that the hull has achieved an equilibrium state and the numerical process numerical process can be terminated. can be terminated. 3.2. Boundary Conditions 3.2. Boundary Conditions To simulate the flow field around the double-stepped planing hull, a CFD (Computational Fluid Dynamics)To simulate simulation the flow fieldbased around on the therigid double-stepped body motion technique planing hull,was applied, a CFD (Computational and the calculating Fluid Dynamics)domain simulationwas primarily based established. on the rigidConcerning body motionthe symmetry technique of the was flow, applied, a symmetrical and the model calculating and domaindomain was were primarily introduced, established. as shown Concerning in Figure the5. The symmetry dimensions of the have flow, a significant a symmetrical impact model on the and domaincomputation were introduced, accuracy and as time shown consumption. in Figure5 Hence,. The dimensionsin this study, have the domain a significant was established impact on with the computationa total length accuracy of about and 6 L time, which consumption. extends 1 L before Hence, the in front this study,of the model, the domain and 4 wasL after established the rear. The with a totalwidth length of the of domain about 6wasL, whichabout 1.5 extends L. Water 1 Ldepthbefore under the the front free of surface the model, was 1 andL, and 4 La lengthafter the of 0.5 rear. TheL width extends of theabove. domain The initial was about attitude 1.5 ofL. Waterthe hull depth was upright under the floating free surface, which was is consistent 1 L, and awith length the of 0.5 Linitialextends model above. attitude The in initial the tank attitude trial. of the hull was upright floating, which is consistent with the initial model attitude in the tank trial. J. Mar. Sci. Eng. 2019, 7, 346 9 of 25 J. Mar. Sci. Eng. 2019, 7, 346 9 of 25

FigureFigure 5. 5.Computational Computational domain. domain.

TheThe boundary boundary conditions conditions were were specified specified asas follows:follows: The The top, top, bottom, bottom, and and side side planes planes of ofthe the domaindomain were were defined defined as free-slipas free-slip walls. walls. The The plane plane in front in front of the of bow the wasbow consideredwas considered the fluid the velocityfluid inlet,velocity where inlet, the velocitywhere the was velocity set as thewas trialset as speeds. the trial On speeds. the opposing On the opposing side of the side domain, of the domain, the plane the was consideredplane was the considered outlet of pressure.the outlet Asof forpressure. the hull, As afor rigid the andhull, no-slip a rigid walland boundaryno-slip wall condition boundary was considered.condition Duewas toconsidered. the symmetry Due to principle, the symmetry the longitudinal principle, the center longitudinal plane was center specified plane was as aspecified symmetry plane.as a Furthermore,symmetry plane. the Furthermore, volume fraction the volume function fraction of water function and of air water at both and the air inletat both and the outlet inlet and plane outlet plane was applied to determine the location of the free surface. was applied to determine the location of the free surface.

3.3.3.3. Mesh Mesh Generation Generation and and Dependency Dependency Analysis Analysis As shown in Figure 6, the whole calculating domain was discretized by a series of structured As shown in Figure6, the whole calculating domain was discretized by a series of structured meshes, and the mesh cells approached the free surface and the hull, as the meshes were refined to meshes, and the mesh cells approached the free surface and the hull, as the meshes were refined to accurately capture the flow. Moreover, boundary layer meshes were also carried out around the hull accurately capture the flow. Moreover, boundary layer meshes were also carried out around the hull body. As a result, a high resolution inflation layer mesh with dimensionless y+ was near the hull body.bottom As a result,wall around a high resolution70, while inflationthe y+ was layer around mesh with 50 at dimensionless the bow. The y+ wholewas near hull the near hull bottomwall walldimensionless around 70, while y+ was the around y+ was 50~3 around00. In 50addition, at the bow. a mesh The independence whole hull near study wall was dimensionless conducted to y+ wasdecide around an 50~300.appropriate In addition, mesh size. a mesh independence study was conducted to decide an appropriate meshJ. Mar. size.As Sci. forEng. the2019 ,mesh 7, 346 size, a finer mesh always indicates more precise results, but conversely,10 of 25it sometimes increases the computational consumption and time cost, due to the large number of mesh elements. Therefore, in order to determine the mesh size with acceptable numerical accuracy and mesh number, a mesh convergence study was carried out, based on a Froude number of 5.20. The mesh at the steps and on hull bottom were gradually refined since the flow around the planing surface has direct influence on the hydrodynamic characteristic of the step. The initial minimum mesh size adopted in the current investigation was 6‰ L. Four mesh plans were made according the refinement ratio of 2 [22]. The total mesh number of each plan and the calculated resistance error (also defined by R/Δ, comparing with the experimental data) were plotted and are shown in Figure 7. As shown in the figure, when the mesh was as fine as Grid3, further refinement would lead to a large increment of element number and hence require more computational time. However, the difference between grid plan 3 and plan 4 was not significant. Therefore, Grid3 was selected as the optimum mesh plan.

FigureFigure 6. 6.Surface Surface meshmesh distribution on on the the hull. hull.

5 AsThe for total the meshmesh number size, a finerwas about mesh 9.42 always × 10 indicates. During the more computing precise course, results, the but physical conversely, time it −3 sometimesstep was increasesset at 5 × 10 the s, computational and the total simulation consumption time and step time number cost, was due 2000. to the large number of mesh elements. Therefore, in order to determine the mesh size with acceptable numerical accuracy and mesh number, a mesh convergence study was carried out, based on a Froude number of 5.20. The mesh 0.3 18 at the steps and on hull bottom were gradually refined since the flow around the planing surface 0.275 16 has direct influence on0.25 the hydrodynamic characteristic of the step. The initial minimum mesh size 14 0.2 0.209 12 0.169 0.165 10 0.15

Error 8 12.737 0.1 6 9.421 4

0.05 6.173 6.891 (1x10^5) Number Mesh 2 0 0 Grid1 Grid2 Grid3 Grid4 Mesh number Error

Figure 7. Results of the mesh dependency study.

4. Results and Discussion

4.1. Validation of the Numerical Method The evaluation of the numerical simulation was implemented by comparing the wave-making profile between the model test and numerical solution at different velocities. This was also the case for the numerical and experimental resistance values. The numerical method showed an accurate simulation of these characteristics with better anastomosis. Five wave-making profile snapshots of different conditions of model M1 are listed in Figure 8, ranging from a Froude number of 0.88 to 5.20. At the Froude number of 0.88, the hull trim was small, and the bow generated a water-pushing effect. A wave was generated along the hull, with the wave trough distributed at the midship, while the peak was at the hull stern. With the speed increasing to the Froude number of 2.19, the trim was intensified. An enormous wake flow crest formed at the longitudinal mid-section near the hull stern. At the Froude number of 3.07, the hull stern was lifted by hydrodynamic and aerodynamic lift. The hull reached the planing condition gradually, with trim angle decreasing distinctly. The wake flow crest also departed from the stern and the crest value also showed a slight decrease. The hull had already entered the planing condition at the Froude number of 3.94. The trim angle decrement was slight, and the wave-making kept extending toward the downstream. With the Froude number J. Mar. Sci. Eng. 2019, 7, 346 10 of 25

J. Mar. Sci. Eng. 2019, 7, 346 10 of 25 adopted in the current investigation was 6% L. Four mesh plans were made according the refinement ratio of √2 [22]. The total mesh number of each plan and the calculated resistance error (also defined by R/∆, comparing with the experimental data) were plotted and are shown in Figure7. As shown in the figure, when the mesh wasFigure as fine 6. asSurface Grid3, mesh further distribution refinement on the wouldhull. lead to a large increment of element number and hence require more computational time. However, the difference between grid The total mesh number was about 9.42 × 105. During the computing course, the physical time plan 3 and plan 4 was not significant. Therefore, Grid3 was selected as the optimum mesh plan. step was set at 5 × 10−3 s, and the total simulation time step number was 2000.

0.3 18 0.275 16 0.25 14 0.2 0.209 12 0.169 0.165 10 0.15

Error 8 12.737 0.1 6 9.421 4

0.05 6.173 6.891 (1x10^5) Number Mesh 2 0 0 Grid1 Grid2 Grid3 Grid4 Mesh number Error

FigureFigure 7. 7.Results Results ofof the mesh dependency dependency study. study.

4. TheResults total and mesh Discussion number was about 9.42 105. During the computing course, the physical time 3 × step was set at 5 10− s, and the total simulation time step number was 2000. 4.1. Validation of× the Numerical Method 4. ResultsThe and evaluation Discussion of the numerical simulation was implemented by comparing the wave-making profile between the model test and numerical solution at different velocities. This was also the case 4.1. Validation of the Numerical Method for the numerical and experimental resistance values. The numerical method showed an accurate simulationThe evaluation of these of characteristics the numerical with simulation better anastomosis. was implemented Five wave-making by comparing profile the snapshots wave-making of profiledifferent between conditions the model of model test M1 and are numerical listed in Figure solution 8, ranging at different from velocities.a Froude number This was of 0.88 also to the 5.20. case for the numericalAt the Froude and number experimental of 0.88, the resistance hull trim values. was small, The and numerical the bow methodgenerated showed a water-pushing an accurate simulationeffect. A wave of these was characteristics generated along with the betterhull, with anastomosis. the wave trough Five wave-makingdistributed at the profile midship, snapshots while of difftheerent peak conditions was at the of hull model stern. M1 With are listedthe speed in Figure increa8sing, ranging to the fromFroude a Froudenumber number of 2.19, the of 0.88trim towas 5.20. intensified. An enormous wake flow crest formed at the longitudinal mid-section near the hull stern. At the Froude number of 0.88, the hull trim was small, and the bow generated a water-pushing At the Froude number of 3.07, the hull stern was lifted by hydrodynamic and aerodynamic lift. The effect. A wave was generated along the hull, with the wave trough distributed at the midship, while hull reached the planing condition gradually, with trim angle decreasing distinctly. The wake flow the peak was at the hull stern. With the speed increasing to the Froude number of 2.19, the trim crest also departed from the stern and the crest value also showed a slight decrease. The hull had wasalready intensified. entered An the enormous planing condition wake flow at the crest Froude formed number at the of longitudinal 3.94. The trim mid-section angle decrement near the was hull stern.slight, At theand Froude the wave-making number of 3.07,kept theextending hull stern toward was liftedthe downstream. by hydrodynamic With the and Froude aerodynamic number lift. The hull reached the planing condition gradually, with trim angle decreasing distinctly. The wake flow crest also departed from the stern and the crest value also showed a slight decrease. The hull had already entered the planing condition at the Froude number of 3.94. The trim angle decrement was slight, and the wave-making kept extending toward the downstream. With the Froude number increased to 5.20, the hull kept planing condition. The trim angle decrement was not obvious, while the hull side wave-making extended to the back slightly. J. Mar. Sci. Eng. 2019, 7, 346 11 of 25 increased to 5.20, the hull kept planing condition. The trim angle decrement was not obvious, while the hull side wave-making extended to the back slightly. By comparing the experimental phenomenon with the simulation results, it can be seen that the numerical calculation simulated the flow field accurately. All of the details of the flow field were well J.demonstrated, Mar. Sci. Eng. 2019 and, 7, 346 the flow field change of the model test and simulation results were11 also of 25 synchronous.

(a)

(b)

(c)

(d)

(e)

Figure 8. Comparison between experimental and computational flowflow fieldfield at different different speeds, M1. ( a) Fr ▽ = 0.88;0.88; ( (bb)) FrFr▽ == 2.19;2.19; (c ()c )FrFr▽ = =3.07;3.07; (d ()d Fr) Fr▽ = 3.94;= 3.94; (e) ( eFr) ▽Fr = 5.20.= 5.20. 5 5 5 5 5 ByThe comparing computed theand experimentalnumerical erro phenomenonrs are summarized withthe in Table simulation 3. results, it can be seen that the numerical calculation simulated the flow field accurately. All of the details of the flow field were well demonstrated,Table 3. Comparison and thebetween flow experimental field change and of computational the model test resistance and simulation coefficients. results were also synchronous. The computed and numerical errors are summarized in Table3. J. Mar. Sci. Eng. 2019, 7, 346 12 of 25

J. Mar. Sci. Eng. 2019, 7, 346 12 of 25 Table 3. Comparison between experimental and computational resistance coefficients. Table 3. Comparison between experimental and computational resistance coefficients. M1 M2 M3 Fr M1 M2 M3 Fr∇ ∇ Cmpt.Cmpt. Exp. Exp. Err. Cmpt.Cmpt. Exp.Exp. Err.Err. Cmpt.Cmpt. Exp.Exp. Err.Err. 0.880.88 0.068 0.068 0.072 0.072 5.62% 0.0690.069 0.0710.071 2.69%2.69% 0.0650.065 0.065 0.065 0.36%0.36% 1.311.31 0.191 0.191 0.197 0.197 3.24% 0.1890.189 0.1940.194 2.77%2.77% 0.1840.184 0.188 0.188 1.85%1.85% 2.192.19 0.240 0.240 0.237 0.237 –1.22%1.22% 0.2250.225 0.2340.234 3.96%3.96% 0.2230.223 0.226 0.226 1.38%1.38% − 2.632.63 0.206 0.206 0.211 0.211 2.20% 0.2050.205 0.2070.207 1.00%1.00% 0.1970.197 0.202 0.202 2.55%2.55% 3.073.07 0.170 0.170 0.189 0.189 9.87% 0.1770.177 0.1880.188 5.76%5.76% 0.1670.167 0.182 0.182 8.20%8.20% 3.503.50 0.172 0.172 0.180 0.180 4.23% 0.1670.167 0.1780.178 6.53%6.53% 0.1640.164 0.174 0.174 5.65%5.65% 3.94 0.162 0.178 8.80% 0.166 0.178 6.68% 0.158 0.174 9.22% 3.94 0.162 0.178 8.80% 0.166 0.178 6.68% 0.158 0.174 9.22% 4.38 0.164 0.180 9.02% 0.165 0.183 9.70% 0.161 0.172 11.44% 4.384.82 0.164 0.169 0.180 0.192 12.40%9.02% 0.1720.165 0.1940.183 11.63%9.70% 0.1710.161 0.192 0.172 10.82%11.44% 4.825.20 0.169 0.168 0.192 0.203 12.40%16.97% 0.1730.172 0.2060.194 15.89%11.63% 0.1770.171 0.206 0.192 14.16%10.82% 5.20 0.168 0.203 16.97% 0.173 0.206 15.89% 0.177 0.206 14.16%

It canIt can be be seen seen from from Table Table3 and 3 and Figure Figure9 that, 9 that under, under most most conditions, conditions, thethe currentcurrent numerical simulationssimulations underestimate underestimate the the test test data. data. The The numerical numerical errorerror of the the resistance resistance between between the the simulation simulation andand test test data data of of M1, M1, M2 M2 and and M3 M3 varies varies fromfrom 0.36% to to 15.89%. 15.89%. However, However, the the resistance resistance curve curve trend trend is is synchronoussynchronous forfor thethe wholewhole speedspeed segment.segment. In In the the low low speed speed segment, segment, the the computational computational and and experimentalexperimental result result diff erencedifference is limited. is limited. With With the increase the increase of speed, of speed, the proportion the proportion of splash of resistancesplash amongresistance the total among resistance the total increases resistance gradually. increases The meshgradually. precision The ismesh not enoughprecision to is simulate not enough the splash to phenomenonsimulate the accurately. splash phenomenon Therefore, accurately. the error increases Therefore, gradually. the error However,increases gradually. by vertically However, comparing by thevertically data of each comparing model, athe consistent data of each variation model, trend a co ofnsistent the resistance variation can trend be observed.of the resistance Moreover, can bothbe theobserved. resistance Moreover, humps at both the speed the resi Froudestance numbershumps at ofthe 2.19 speed and Froude 5.20 of numbers the three of models 2.19 and are 5.20 accurately of the predictedthree models by the numerical are accurately simulation. predicted Generally, by the the numerical numerical simulation. simulation Generally, results reflect the thenumerical resistance characteristicssimulation results of the reflect double-stepped the resistance planing characte hullristics well, of and the double-stepped the resistance di planingfference hull and well, variation and trendthe between resistance di differencefferent models and variation can also trend be captured between precisely. different models can also be captured precisely.

0.3 0.25

0.25 0.2

0.2 0.15 Δ 0.15Δ R/ R/ M1 Exp. 0.1 M2 Exp. 0.1 M1 Cmpt. M2 Cmpt. 0.05 0.05

0 0 0123456 0123456 Fr▽ Fr ▽ (a) (b)

0.25

0.2

0.15 Δ R/ 0.1 M3 Exp. M3 Cmpt. 0.05

0 0123456 Fr▽ (c)

FigureFigure 9. Comparison 9. Comparison between between experimental experimental and andcomputational computational resistance coefficients. coefficients. (a ()a Resistance) Resistance curvescurves of M1; of M1; (b) ( resistanceb) resistance curves curves of of M2; M2; (c ()c resistance) resistance curves curves ofof M3.M3. J. Mar. Sci. Eng. 2019, 7, 346 13 of 25 J. Mar. Sci. Eng. 2019, 7, 346 13 of 25

It canIt can be be seen seen in in Figure Figure 10 10 that, that, comparedcompared with the the corresponding corresponding simulati simulationon results, results, the the testing testing datadata of theof trimthe trim and sinkageand sinkage are underestimated are underestimated in most in cases.most cases. With theWith increase the increase of speed, of thespeed, calculated the sinkagecalculated values sinkage increase values gradually increase and gradually they exceed and thethey test exceed data atthe the test speed data intervalat the speed of Froude interval number of 2~3.Froude Both thenumber trim and2~3. sinkageBoth the discrepancies trim and sinkage between discrepancies the experimental between and the calculated experimental data increaseand graduallycalculated with data the increase speed. gradually However, with a good the speed agreement. However, still cana good befound agreement between still can the be two found groups of curves.between Thethe two numerical groups of simulation curves. The accurately numerical predicts simulation the accurately trend of the predicts tworesults. the trend Therefore, of the two the validityresults. of Therefore, the numerical the validity method of is approved.the numerical The method results is have appr aoved. certain The credibility results have and cana certain be used forcredibility further flow and field can be analysis. used for further flow field analysis.

1 14

0.8 12

10 0.6 M1 Sinkage Exp. M1 sinkage Cmpt. 8 0.4 M1 Trim Exp. ° δ / M1 Trim Cmpt.

θ 6 0.2 4

0 2 0123456 -0.2 0 Fr▽

(a)

1 14

0.8 12

10 0.6 M2 Sinkage Exp.

° / M2 Sinkage Cmpt. 8 θ

0.4 M2 Trim Exp. δ M2 Trim Cmpt. 6 0.2 4

0 2 0123456 -0.2 0 Fr▽

(b)

Figure 10. Cont. J. Mar. Sci. Eng. 2019, 7, 346 14 of 25 J. Mar. Sci. Eng. 2019, 7, 346 14 of 25

0.8 12

10 0.6 M3 Sinkage Exp. M3 Sinkage Cmpt. 8 0.4 M3 Trim Exp.

° M3 Trim Cmpt. 6 δ / θ 0.2 4

0 2 01234567

-0.2 Fr 0 ▽ (c)

FigureFigure 10. 10.Comparison Comparison between experimental experimental and and comput computationalational trim trim and and sinkage sinkage coefficients. coefficients. (a) (a) TrimTrim and and sinkage sinkage curves curves ofof M1; ( b)) trim trim and and sinkage sinkage curves curves of ofM2; M2; (c) (trimc) trim and and sinkage sinkage curves curves of M3. of M3.

4.2.4.2. Cavity Cavity and and WettedWetted SurfaceSurface Analysis

4.2.1.4.2.1. Cavity Cavity Behind Behind thethe Step DuringDuring thethe navigation,navigation, with the the increase increase of of spee speed,d, the the cavity cavity after after both both steps steps gradually gradually takes takes shape.shape. To To study studythe the formationformation process process and and action action me mechanismchanism of ofthe the cavity, cavity, the the fluid fluid field field distribution distribution andand morphology morphology evolution evolution under step-B step-B of of M1 M1 along along the the longitudinal longitudinal center center plane plane at different at different speeds speeds isis extracted. extracted. As As shownshown inin Figure 11a,11a, the the color color scale scale values values air air volume volume fractions fractions 0 and 0 and 1 represent 1 represent the the completecomplete water water phase phase and and gas gas phase, phase, respectively. respectively. It It can can be be observed observedthat that atatthe theFroude Froude numbernumber ofof 1.31, only1.31, the only water the water phase phase exists exists after theafter step. the step A small. A small amount amount of water–air of water–air mixture mixture appears appears at Frat Fr=▽1.75, = 1.75, and with the increase of the speed, the component of the gas phase increases gradually. At5 the and with the increase of the speed, the component of the gas phase increases gradually. At the Froude Froude number of 2.63, a cavity with a core region entirely in the gas phase is formed. The cavity number of 2.63, a cavity with a core region entirely in the gas phase is formed. The cavity extends extends along the longitudinal direction, presenting a sharp wedge-shape with a gas-phase along the longitudinal direction, presenting a sharp wedge-shape with a gas-phase composition that composition that decreases gradually. With Fr▽ continuing to increase, the cavity keeps expanding decreases gradually. With Fr continuing to increase, the cavity keeps expanding to the rear at the to the rear at the speed of Froude5 number 5.20, shown in Figure 11b. The longitudinal length of the speedcavity of gas-phase Froude number core region 5.20, reache showns its in maximum Figure 11 atb. about The longitudinal 2.5 times H. length of the cavity gas-phase core region reaches its maximum at about 2.5 times H. We now discuss the cavity formation and evolution process according to the above analysis. To discuss the formation mechanism, a typical flow field around step-B of M1 was further analyzed at a Froude number of 4.38. Figure 12 shows the pressure distribution of the bottom at the longitudinal center section. The inflow flows through the step, and a low-pressure area is primarily formed behind it, while a distinct high-pressure area is formed at the region whereFr the▽ = 1.31 flow scuds the step and contacts the planing surface again. Due to the effect of two adjacent pressure difference areas, a backflow is developed, as shown in Figure 13. Simultaneously, the flow at both sides of the hull is pushed to the Fr▽ = 1.75 hull bottom along the step. With increasing speed to a certain extent, the terminal of the step is exposed to the air with the hull lifting. As a result, a suction effect is generated by the pressure difference, Fr▽ = 2.19 and the air is pumped into the hull bottom in a cyclone formation. The cavity, as stated, is in essence generated by the low-pressure induced cyclone. Fr▽ = 2.63

Fr▽ = 3.07 J. Mar. Sci. Eng. 2019, 7, 346 14 of 25

0.8 12

10 0.6 M3 Sinkage Exp. M3 Sinkage Cmpt. 8 0.4 M3 Trim Exp.

° M3 Trim Cmpt. 6 δ / θ 0.2 4

0 2 01234567

-0.2 Fr 0 ▽ (c)

Figure 10. Comparison between experimental and computational trim and sinkage coefficients. (a) Trim and sinkage curves of M1; (b) trim and sinkage curves of M2; (c) trim and sinkage curves of M3.

4.2. Cavity and Wetted Surface Analysis J. Mar. Sci. Eng. 2019, 7, 346 15 of 25 J. Mar. Sci. Eng. 2019, 7, 346 15 of 25 4.2.1. Cavity Behind the Step During the navigation, with the increase of speed, the cavity after both steps gradually takes Fr▽ = 3.50 shape. To study the formation process and action mechanism of the cavity, the fluid field distributionFr▽ = 3.50 and morphology evolution under step-B of M1 along the longitudinal center plane at different speeds is extracted. As shown in Figure 11a, the color scale values air volume fractions 0 and 1 representFr▽ =the 3.94 Fr▽ = 3.94 complete water phase and gas phase, respectively. It can be observed that at the Froude number of

1.31, only the water phase exists after the step. A small amount of water–air mixture appears at Fr▽ Fr▽ = 4.38 = 1.75, and with the increase of the speed, the component of the gas phase increases gradually. Fr At▽ =the 4.38 Froude number of 2.63, a cavity with a core region entirely in the gas phase is formed. The cavity J. Mar. Sci. Eng. 2019, 7, 346 Fr▽ = 5.20 15 of 25 extends along the longitudinal direction, presenting a sharp wedge-shape with a gas-phaseFr▽ = 5.20 J. Mar. Sci. Eng. 2019, 7, 346 (a) 15 of 25 composition that decreases gradually. With Fr▽ continuing to increase, the( acavity) keeps expanding to the rear at the speed of Froude number 5.20, shown in Figure 11b. The longitudinal length of the cavity gas-phase core region reaches its maximum at about 2.5 times H. Fr▽ = 3.50

Fr▽ = 3.94

(b) (b) Fr▽ = 4.38

Figure 11. Cavity morphology and distributionFr▽ = 1.31 of M1. (a) Cavity evolution with different velocities Figure 11. Cavity morphology and distribution of M1. (a) Cavity evolution with different velocities of step-B, M1; (b) cavity distribution on the hull bottom, Fr▽ = 5.20, M1. Fr▽ = 5.20 of step-B, M1; (b) cavity distribution on the hull bottom, Fr▽ = 5.20, M1. (a) Fr▽ = 1.75 We now discuss the cavity formation and evolution process according to the above analysis. To We now discuss the cavity formation and evolution process according to the above analysis. To discuss the formation mechanism, a typical flow field around step-B of M1 was further analyzed at a discuss the formation mechanism, a typical flow field around step-B of M1 was further analyzed at a Froude number of 4.38. Figure 12 showsFr the▽ = 2.19pressure distribution of the bottom at the longitudinal Froude number of 4.38. Figure 12 shows the pressure distribution of the bottom at the longitudinal center section. The inflow flows through the step, and a low-pressure area is primarily formed behind center section. The inflow flows through the step, and a low-pressure area is primarily formed behind it, while a distinct high-pressure area is formed at the region where the flow scuds the step and it, while a distinct high-pressure area isFr ▽formed = 2.63 at the region where the flow scuds the step and contacts the planing surface again. Due to the effect of two adjacent pressure difference areas, a contacts the planing surface again. Due to the effect of two adjacent pressure difference areas, a backflow is developed, as shown in Figure 13. Simultaneously, the flow at both sides of the hull is backflow is developed, as shown in FigureFr▽ = 13. 3.07 Simu ltaneously, the flow(b) at both sides of the hull is pushed to the hull bottom along the step. With increasing speed to a certain extent, the terminal of pushed to the hull bottom along the step. With increasing speed to a certain extent, the terminal of the step is exposed to the air with the hull lifting. As a result, a suction effect is generated by the theFigure step 11.is exposedCavity morphology to the airFigure with and 11. distributionthe Cavity hull morphology lifting. of M1. As (a anda) Cavityresult, distribution evolutiona suction of M1. with effect (a di) ffCavityiserent generated velocitiesevolution by with ofthe different velocities pressure difference, and theof air step-B, is pumped M1; (b) intocavity the distribution hull bottom on thein a hull cyclone bottom, formation. Fr▽ = 5.20, The M1. cavity, as pressurestep-B, M1;difference, (b) cavity and distribution the air is pumped on the hull into bottom, the hullFr bottom= 5.20, in M1. a cyclone formation. The cavity, as stated, is in essence generated by the low-pressure induced5 cyclone. stated, is in essence generated by the low-pressure induced cyclone. We now discuss the cavity formation and evolution process according to the above analysis. To discuss the formation mechanism, a typical flow field around step-B of M1 was further analyzed at a Froude number of 4.38. Figure 12 shows the pressure distribution of the bottom at the longitudinal center section. The inflow flows through the step, and a low-pressure area is primarily formed behind it, while a distinct high-pressure area is formed at the region where the flow scuds the step and contacts the planing surface again. Due to the effect of two adjacent pressure difference areas, a backflow is developed, as shown in Figure 13. Simultaneously, the flow at both sides of the hull is pushed to the hull bottom along the step. With increasing speed to a certain extent, the terminal of

the step is exposed to the air with the hull lifting. As a result, a suction effect is generated by the Figurepressure 12. Local difference, pressure distribution and the air behind is pumped the step into of M1,the hullFr▽ = bottom4.38. in a cyclone formation. The cavity, as FigureFigure 12. 12.Local Local pressure pressure distribution distribution behindbehind the step step of of M1, M1, FrFr▽ = =4.38.4.38. stated, is in essence generated by the low-pressure induced5 cyclone.

Figure 13. Velocity vector distribution of the mid-plane section M1, Fr▽ = 4.38. Figure 13. Velocity vector distribution of the mid-plane section M1, Fr▽ = 4.38. Figure 13. Velocity vector distribution of the mid-plane section M1, Fr = 4.38. Figure 12. Local pressure distribution behind5 the step of M1, Fr▽ = 4.38. 4.2.2. Influence of Steps on the Wetted Surface To determine the influence of a double step on the wetted surface of the planing bottom, the distribution of the wetted area of the hull M1 under different speeds was extracted. The forms of water volume fraction were used to represent the air–water mixture situation. In order to provide a direct observation of the wetted surface evolution, a bottom view of the hull is displayed in Figure 14. The color scale values 0 and 1 represent the complete gas phase and water phase state, respectively. A value between 0–1 represents corresponding weight gas–water mixtures of the volume fraction. It can be observed that the wetted surface changes are mainly caused by two factors: one is the change Figure 13. Velocity vector distribution of the mid-plane section M1, Fr▽ = 4.38. of the air–water mixture component caused by the cavity, and the other factor is the shape change of the wetted surface caused by the navigation state. J. Mar. Sci. Eng. 2019, 7, 346 16 of 25

4.2.2. Influence of Steps on the Wetted Surface To determine the influence of a double step on the wetted surface of the planing bottom, the distribution of the wetted area of the hull M1 under different speeds was extracted. The forms of water volume fraction were used to represent the air–water mixture situation. In order to provide a direct observation of the wetted surface evolution, a bottom view of the hull is displayed in Figure 14. The color scale values 0 and 1 represent the complete gas phase and water phase state, respectively. A value between 0–1 represents corresponding weight gas–water mixtures of the volume fraction. It can be observed that the wetted surface changes are mainly caused by two factors: one is the change of the air–water mixture component caused by the cavity, and the other factor is the shape change of the wetted surface caused by the navigation state. For the first factor, with the increase of speed, the component of the gas phase increases continuously. Especially at the planing surface of P3, the gas phase increases and forms a relatively complete air cavity. The cavity near the step terminal bulges to the rear, while the middle is sunken to the bow. Correspondingly, the wetted surface is concentrated at the stern and draws close to the middle plane. The wetted surface distribution on planing surface P2 is similar, but clear wetted surface is observed. For the second factor, with the hull lifting and trim variation caused by increasing speed, the stagnation line (the intersecting line of the bow and water surface) collapses gradually. This factor mainly affects the wetted surface variation on planing surface P2. For further study of the performance of the wetted surface reduction, the wetted surface calculation equation was introduced. The bottom wetted area was computed by integrating the water volume fraction over bottom S. The equation is written as: A = Vds wet water , (13) s where Awet represents the wetted area and Vwater represents the water volume fraction at each differential element. The ratio of the calculated wetted area and bottom area Awet/A was introduced to represent the wetted surface reducing effect. The result is plotted as a function of the Froude number and is shown in Figure 15. The wetted surface reduces rapidly with the increase of speed. After the Froude numberJ. Mar. Sci. exceeds Eng. 2019 4.38,, 7, 346the ratio remains at a fairly stable value of 0.23. The ratio 16 of 25 suggests an obvious effect of the step, which also indicates the action mechanism of the double-step in this study.

Fr▽ = 1.31

Fr▽ = 2.63

J. Mar. Sci. Eng. 2019, 7, 346 17 of 25

Fr▽ = 3.94 Fr▽ = 5.20

Figure 14. Contour of the water–air phase distribution of different velocities, M1. Figure 14. Contour of the water–air phase distribution of different velocities, M1. For the first factor, with the increase of speed, the component of the gas phase increases continuously. Especially at the planing surface of P3, the gas phase0.95 increases and forms a relatively complete air 0.90 cavity. The cavity near the step terminal bulges to the rear, while the middle is sunken to the bow.

Correspondingly, the wetted surface is concentrated0.75 at the stern0.75 and draws close to the middle plane. The wetted surface distribution on planing surface P2 is similar, but clear wetted surface is observed. 0.63 /A For the second factor, with the hull lifting and trim variation0.55 caused by increasing speed, the stagnation wet 0.53 line (the intersecting line of the bow and water surface)A collapses gradually. This factor mainly affects the wetted surface variation on planing surface P2. 0.43 0.35 0.37 For further study of the performance of the wetted surface reduction, the wetted0.30 surface calculation 0.24 0.23 equation was introduced. The bottom wetted area was computed by integrating the water0.23 volume 0.15 fraction over bottom S. The equation is written as: 123456 Fr▽

A = V ds, (13) Figure 15. Ratiowet ofx the calculatedwater wetted area and bottom area curve with the speed, M1. s 4.3. Hydrodynamic Characteristics of the Stern Flap where Awet represents the wetted area and Vwater represents the water volume fraction at each differential element. The ratio of the calculatedDuring navigation, wetted area the and stern bottom flap is, area in essence,Awet/A wasa planing introduced plane. Hydrodynamic to represent lift is generated the wetted surface reducingunder etheffect. flap. The Meanwhile, result is plotted due to asthe a functionmounting of angle, the Froude when the number fluid flows and is through the flap a shown in Figure 15. Theretardative wetted surface effect occurs reduces in rapidlythe flow with field. the As increasea result, the of speed. pressure After and themovement Froude distribution adjust number exceeds 4.38, theand ratio vary. remains The flow at afield fairly of stablestern flap value M3 of at 0.23. the Froude The ratio number suggests of 5.20 an obviousis taken as an example for effect of the step, whichfurther also indicates analysis. the As action shown mechanism in Figure of16, the the double-step incoming flow in this velocity study. vector line direction shows deflection at the connection of the hull bottom and stern flap. The pressure distribution of the flow field is automatically changed by this effect, shaping a distinct high-pressure region. In addition, a bottom view of the flap is shown in Figure 17. A high-pressure region forms in the shape of a belt. This region has a relatively uniform distribution in the transverse direction. The region covers most of the anterior part of the stern flap in the longitudinal direction.

Figure 16. Pressure and velocity vector distributions at the center plane of the stern flap. J.J. Mar. Mar. Sci. Sci. Eng. Eng. 2019 2019, ,7 7, ,346 346 1717 ofof 2525

Fr▽ = 5.20 Fr▽ = 5.20

J. Mar. Sci. Eng. 2019, 7, 346 17 of 25 FigureFigure 14. 14. Contour Contour of of the the water–air water–air phase phase distr distributionibution of of different different velocities, velocities, M1. M1.

0.95 0.95 0.900.90

0.75 0.75 0.750.75

0.630.63 /A

/A 0.55 wet 0.55 0.53 wet

A 0.53 A 0.430.43 0.35 0.37 0.35 0.37 0.30 0.30 0.24 0.23 0.24 0.23 0.230.23 0.15 0.15 123456 123456Fr▽ Fr▽

Figure 15. Ratio of the calculated wetted area and bottom area curve with the speed, M1. FigureFigure 15. 15. Ratio Ratio of of the the calculated calculated wetted wetted area area and and bottom bottom area area curve curve with with the the speed, speed, M1. M1. 4.3. Hydrodynamic Characteristics of the Stern Flap 4.3.4.3. Hydrodynamic Hydrodynamic Characteristics Characteristics of of the the Stern Stern Flap Flap During navigation, the stern flap is, in essence, a planing plane. Hydrodynamic lift is generated DuringDuring navigation, navigation, the the stern stern flap flap is, is, in in essence, essence, a a planing planing plane. plane. Hydrodynamic Hydrodynamic lift lift is is generated generated under the flap. Meanwhile, due to the mounting angle, when the fluid flows through the flap a underunder thethe flap.flap. Meanwhile,Meanwhile, duedue toto thethe mountingmounting angle,angle, whenwhen thethe fluidfluid flowsflows throughthrough thethe flapflap aa retardative effect occurs in the flow field. As a result, the pressure and movement distribution adjust retardativeretardative effect effect occurs occurs in in the the flow flow field. field. As As a a re result,sult, the the pressure pressure and and movement movement distribution distribution adjust adjust and vary. The flow field of stern flap M3 at the Froude number of 5.20 is taken as an example for further andand vary. vary. The The flow flow field field of of stern stern flap flap M3 M3 at at the the Froude Froude number number of of 5.20 5.20 is is taken taken as as an an example example for for analysis. As shown in Figure 16, the incoming flow velocity vector line direction shows deflection furtherfurther analysis.analysis. AsAs shownshown inin FigureFigure 16,16, thethe incoincomingming flowflow velocityvelocity vectorvector lineline directiondirection showsshows atdeflection thedeflection connection at at the the connection ofconnection the hull of bottomof the the hull hull and bottom bottom stern and and flap. stern stern The flap. flap. pressure The The pressure pressure distribution di distributionstribution of the of of flow the the flow flow field is automaticallyfieldfield is is automatically automatically changed changed bychanged this eby ffbyect, this this shaping effect, effect, shap shap a distinctinging a a distinct distinct high-pressure high-pressure high-pressure region. region. region. In addition, In In addition, addition, a bottom a a viewbottombottom of the view view flap of isof the shownthe flap flap inis is Figureshown shown 17in in .Figure Figure A high-pressure 17. 17. A A high-pressure high-pressure region forms region region in theformsforms shape in in the the of shape ashape belt. of Thisof a a belt. belt. region hasThisThis a relatively region region has has uniform a a relatively relatively distribution uniform uniform in distribution distribution the transverse in in the the direction. transverse transverse The direction. direction. region coversThe The region region most covers covers of the most anteriormost partofof ofthe the the anterior anterior stern part flappart of inof the the stern longitudinalstern fl flapap in in the the direction. longitudinal longitudinal direction. direction.

Figure 16. Pressure and velocity vector distributions at the center plane of the stern flap. J. Mar. Sci.Figure Eng.Figure 2019 16. 16., 7Pressure, Pressure346 andand velocityvelocity vector distributions distributions at at the the center center plane plane of ofthe the stern stern flap. flap. 18 of 25

FigureFigure 17. 17.Pressure Pressure distribution distribution ofof the flap flap surface, bottom bottom view. view.

As mentionedAs mentioned previously, previously, the sternthe stern flaps haveflaps anhave obvious an obvious influence influence on the pressureon the pressure and movement and distribution.movement Clarifying distribution. the Clarifying characteristics the characteristics of stern flaps of isstern important flaps is important in the analysis in the of analysis the flap of actionthe mechanism,flap action in mechanism, order to evaluate in order theto evaluate hydrodynamic the hydrodynamic characteristics characteristics of the stern of the flap. stern A flap. series A of series of non-dimensional evaluation coefficients were applied during our data process. The non- dimensional pressure coefficient Cp and torque coefficient CMΔ are specified in Equations (14) and (15):

PP- ref C = , p 1 (14) ρV 2 2 ref

= M C Δ , (15) M BΔ

where P is the local absolute pressure, Pref is the reference pressure, or atmospheric pressure, Vref represents the velocity of the inlet flow, ρ represents the water density and M is the movement generated by the stern flap, referring to the transom axis of the gravity center. Figure 18 shows the influence of the stern flap mounting angle on pressure distribution along the centerline of the stern plate at a Froude number of 5.20. The non-dimensional pressure coefficient Cp is plotted with a lateral axis of the ratio between longitudinal coordinates of the pressure monitor point and hull length, L. The first monitor point is located at the trailing edge of step-B, while the last monitor point is located at the trailing edge of the flap. The positive coordinates point to the direction of the bow. It can be seen that there are two pressure peaks along the longitudinal direction. The frontal peak is near step-B. As discussed in Section 4.2, this is mainly caused by the flow scudding over the step and making contact with the planing surface again. The latter peak represents the local high-pressure caused by the retardation effect of the flap. Comparing the peak values among M1, M2 and M3, the peak shows a positive correlation relationship with the increasing mounting angle. The peak value and longitudinal range of the high-pressure region gradually increase with an increasing mounting angle, suggesting that the lift is also magnified. J. Mar. Sci. Eng. 2019, 7, 346 18 of 25 non-dimensional evaluation coefficients were applied during our data process. The non-dimensional pressure coefficient Cp and torque coefficient CM∆ are specified in Equations (14) and (15):

P Pre f C = − , (14) p 1 2 2 ρVre f

M C = , (15) M∆ B∆ where P is the local absolute pressure, Pref is the reference pressure, or atmospheric pressure, Vref represents the velocity of the inlet flow, ρ represents the water density and M is the movement generated by the stern flap, referring to the transom axis of the gravity center. Figure 18 shows the influence of the stern flap mounting angle on pressure distribution along the centerline of the stern plate at a Froude number of 5.20. The non-dimensional pressure coefficient Cp is plotted with a lateral axis of the ratio between longitudinal coordinates of the pressure monitor point and hull length, L. The first monitor point is located at the trailing edge of step-B, while the last monitor point is located at the trailing edge of the flap. The positive coordinates point to the direction of the bow. It can be seen that there are two pressure peaks along the longitudinal direction. The frontal peak is near step-B. As discussed in Section 4.2, this is mainly caused by the flow scudding over the step and making contact with the planing surface again. The latter peak represents the local high-pressure caused by the retardation effect of the flap. Comparing the peak values among M1, M2 and M3, the peak shows a positive correlation relationship with the increasing mounting angle. The peak value and longitudinal range of the high-pressure region gradually increase with an increasing mounting angle,J. Mar. suggesting Sci. Eng. 2019 that, 7, 346 the lift is also magnified. 19 of 25

0.12

M1

0.08 M2

M3

0.04 p C

0

-0.04

-0.08 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 X/L

Figure 18. Pressure distribution on the center line of the flaps with different mounting angles (positive Figure 18. Pressure distribution on the center line of the flaps with different mounting angles (positive directions point to the bow). directions point to the bow). During navigation, the motion attitude of the hull is mainly adjusted by the hydrodynamic During navigation, the motion attitude of the hull is mainly adjusted by the hydrodynamic liftinglifting force force and and trimming trimming moment. moment. The The torque torque coecoeffifficients of of flaps flaps with with different different mounting mounting angles angles areare plotted plotted in Figurein Figure 19 19,, as as a functiona function of of the the Froude Froude number.number. The minimum minimum trim trim moment moment is islocated located at the speed point of Fr =▽ 1.31. After that, with the increase of speed, the trim moment increases at the speed point of 5Fr = 1.31. After that, with the increase of speed, the trim moment increases gradually.gradually. Moreover, Moreover, the the mounting mounting angleangle increaseincrease makes the the trim trim moment moment generated generated by by the the flap flap increase.increase. Synchronously, Synchronously, not onlynot only the trimthe momenttrim mome butnt also but its also increasing its increasing amplitude amplitude increase increase obviously withobviously the speed, with rising the speed, under rising the high under speed the segment.high speed Correspondingly, segment. Correspondingly, the motion the attitude motion adjustment attitude abilityadjustment is improved ability by is enlargingimproved theby enlarging mounting the angle. mounting angle.

0.05

0.04 M1 M2 0.03 M3 Δ M C 0.02

0.01

0.00 0123456 Fr ▽ Figure 19. Moment coefficient of stern flaps with different mounting angles.

4.4. Influence of the Mounting Angle on the Stepped Planing Hull As discussed in Section 4.2, salutatory changes were introduced to the stepped hull bottom. When the inflow flows through the steps, cavities are formed on account of the pressure difference. The cavities cover the hull bottom, which decreases the wetted surface area, and reduces the friction resistance. Therefore, the size of the cavity is an important factor that affects the resistance performance of stepped planing hulls. In this section, the models of M1, M2 and M3 with different mounting angles are examined and compared to evaluate the evolution process and mechanism of cavitation under different angle configurations. As shown in Figure 20, a comparison of the cavity morphology after step-B has been carried out. The cavity morphologies are grouped by different velocity cases in which the Froude number is 2.63, 3.50, 4.38 and 5.20. It can still be observed that the core region of the cavity is dominated by the gas phase, while the periphery of the core region is a mixture of water and gas. As the speed increases, J. Mar. Sci. Eng. 2019, 7, 346 19 of 25

0.12

M1

0.08 M2

M3

0.04 p C

0

-0.04

-0.08 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 X/L

Figure 18. Pressure distribution on the center line of the flaps with different mounting angles (positive directions point to the bow).

During navigation, the motion attitude of the hull is mainly adjusted by the hydrodynamic lifting force and trimming moment. The torque coefficients of flaps with different mounting angles are plotted in Figure 19, as a function of the Froude number. The minimum trim moment is located at the speed point of Fr▽ = 1.31. After that, with the increase of speed, the trim moment increases gradually. Moreover, the mounting angle increase makes the trim moment generated by the flap increase. Synchronously, not only the trim moment but also its increasing amplitude increase J. Mar.obviously Sci. Eng. 2019 with, 7 the, 346 speed, rising under the high speed segment. Correspondingly, the motion attitude19 of 25 adjustment ability is improved by enlarging the mounting angle.

0.05

0.04 M1 M2 0.03 M3 Δ M C 0.02

0.01

0.00 0123456 Fr ▽ FigureFigure 19. 19.Moment Moment coe coefficientfficient of of stern stern flapsflaps with different different mounting mounting angles. angles.

4.4.4.4. Influence Influence of theof the Mounting Mounting Angle Angle on on the the Stepped Stepped Planing Planing HullHull AsAs discussed discussed in in Section Section 4.2 4.2,, salutatory salutatory changeschanges werewere introduced to to the the stepped stepped hull hull bottom. bottom. WhenWhen the the inflow inflow flows flows through through the the steps, steps, cavitiescavities are formed on on account account of of the the pressure pressure difference. difference. TheThe cavities cavities cover cover the the hull hull bottom, bottom, which which decreases decreases thethe wettedwetted surface area, area, and and reduces reduces the the friction friction resistance.resistance. Therefore, Therefore, the sizethe ofsize the of cavity the iscavity an important is an important factor that factor affects that the affects resistance the performanceresistance of steppedperformance planing of stepped hulls. In planing this section, hulls. the In this models section, of M1, the M2models and M3of M1, with M2 di andfferent M3 mounting with different angles aremounting examined angles and compared are examined to evaluate and compared the evolution to evaluate process the evolution and mechanism process and of cavitation mechanism under of differentcavitation angle under configurations. different angle configurations. As shownAs shown in Figurein Figure 20 20,, a comparisona comparison of of the the cavitycavity morphology after after step-B step-B has has been been carried carried out. out. TheThe cavity cavity morphologies morphologies are are grouped grouped by by di differentfferent velocity velocity cases in in which which the the Froude Froude number number is 2.63, is 2.63, 3.50, 4.38 and 5.20. It can still be observed that the core region of the cavity is dominated by the gas 3.50, 4.38J. Mar. and Sci. 5.20.Eng. 2019 It,can 7, 346 still be observed that the core region of the cavity is dominated by the20 of gas 25 phase,phase, while while the the periphery periphery of of the the core core region region is is aa mixturemixture ofof water and and gas. gas. As As the the speed speed increases, increases, the cavitythe cavity expands expands toward toward the rear.the rear. As forAs diforff differenerent modelst models at at the the same same speed, speed, when when in inthe the crossingcrossing modemode (Fr = (Fr2.63),▽ = 2.63), the cavitiesthe cavities are inare the in the initial initial shaping shaping stage stage where where the the cavity cavity shape shape and and sizesize areare not 5 significantlysignificantly different different from from each each other. other. With With the the increase increase of theof the speed, speed, after after the the hull hull enters enters the the planingplaning state (Frstate (Fr3.50),▽ ≥ 3.50), the cavitation the cavitation expanding expanding rate rate varies varies in eachin each model. model. The The expanding expanding rate rate ofof M3M3 isis the 5 ≥ most rapid,most rapid, and M2 and is M2 second, is second, while while M1 M1 is the is the minimum. minimum. Therefore, Therefore, after after the the hull hull enters enters thethe planingplaning state, the cavity expands with the increasing flap mounting angle. Similarly, the expanding of the state, the cavity expands with the increasing flap mounting angle. Similarly, the expanding of the cavity reduces the wetted surface and friction resistance, which indicates that the resistance increase cavity reduces the wetted surface and friction resistance, which indicates that the resistance increase caused by the flap mounting angle increasing is small in scale. caused by the flap mounting angle increasing is small in scale.

(a) Fr▽ = 2.63 (b) Fr▽ = 3.50

(c) Fr▽ = 4.38 (d) Fr▽ = 5.20

FigureFigure 20. Comparisons 20. Comparisons of air of cavitiesair cavities between between diff differenterent models models at at various various speeds. speeds.

To study the influence of stern flap mounting angle on the hydrodynamic performance of the hull curves, the load coefficient is plotted as a function of the Froude number. The curves are divided into three groups based on the difference of the planing surface. The load coefficient is defined as: Δ = n CΔ , (16) n Δ

where Δn is the load on each planing surface. The subscript n is valued as 1, 2, and 3, representing different planing surfaces. CΔ1,2,3 correspond to P1, P2 and P3, as defined in Figure 1. As is shown in Figure 21, the load coefficient curve of different models has basically the same trend in every specific planing surface. The load of the planing surfaces is mainly affected by the flap mounting angle in the following two modes. The first mode directly affects the load on the planing surface by changing the motion attitude. This mode mainly acts on P1 and is reflected in the curves of CΔ1. Due to the increase of the stern flap mounting angle, the trim and heave during navigation are both reduced. This will inevitably lead to the increase of the wetted surface at P1, and correspondingly, the load coefficient CΔ1 also increases with the increasing mounting angle. J. Mar. Sci. Eng. 2019, 7, 346 20 of 25

To study the influence of stern flap mounting angle on the hydrodynamic performance of the hull curves, the load coefficient is plotted as a function of the Froude number. The curves are divided into three groups based on the difference of the planing surface. The load coefficient is defined as:

∆ C = n , (16) ∆n ∆ where ∆n is the load on each planing surface. The subscript n is valued as 1, 2, and 3, representing different planing surfaces. C∆1,2,3 correspond to P1, P2 and P3, as defined in Figure1. As is shown in Figure 21, the load coefficient curve of different models has basically the same trend in every specific planing surface. The load of the planing surfaces is mainly affected by the flap mounting angle in the J. Mar. Sci. Eng. 2019, 7, 346 21 of 25 following two modes.

0.30 0.50

0.25 0.40

0.20 0.30 2 1 Δ Δ

0.15 C C 0.20 M1 0.10 M1 M2 0.10 M2 M3 0.05 M3

0.00 0.00 0123456 0123456 Fr Fr▽ ▽ (a) (b)

0.70

0.60

0.50 3 Δ C 0.40

M1 0.30 M2 M3 0.20 0123456 Fr▽ (c)

FigureFigure 21. 21.Load Load coe coefficientfficient of of di differentfferent planing planing surfaces. surfaces. (a) Load coefficient coefficient of of P1; P1; (b ()b load) load coefficient coefficient of P2;of P2; (c) load(c) load coe coefficientfficient of of P3. P3.

TheThe first other mode mode directly affects aff theects planing the load surface on the load planing indirectly surface by changing by changing the shape the motion of the cavity. attitude. This mode acts on P2 and P3 and is reflected in the curves of CΔ2 and CΔ3. Combined with the cavity This mode mainly acts on P1 and is reflected in the curves of C∆1. Due to the increase of the stern flap mountingmorphology angle, comparison the trim and in Figure heave during20, due navigationto the increase are of both the reduced.mounting Thisangle, will the inevitably cavity expands lead to to the hull rear, resulting in a decrease of the wetted surface and the load coefficient. Thus, with the the increase of the wetted surface at P1, and correspondingly, the load coefficient C∆1 also increases increase of the mounting angle, an opposite trend is obtained in CΔ2 and CΔ3 from the load coefficient with the increasing mounting angle. variation trend of CΔ1. The other mode affects the planing surface load indirectly by changing the shape of the cavity. Furthermore, among the curves from CΔ1 to CΔ2, the uniformity of the load distribution before This mode acts on P2 and P3 and is reflected in the curves of C and C . Combined with the cavity and after the gravity center is improved with the increasing flap∆2 angle. As∆3 discussed above, after the morphology comparison in Figure 20, due to the increase of the mounting angle, the cavity expands hull enters the planing state, the cavity expands with the increasing flap mounting angle. The main to thecause hull should rear, resulting be that inthe a decreaseflaps reduce of the the wetted trim angle surface by and improving the load the coe ffiplaningcient. Thus,surface with load the increasedistribution, of the mounting which in turn angle, leads an to opposite the cavity trend length is obtained increasing. in AsC∆ 2a andresult,C∆ once3 from the the hull load is disturbed coefficient variationfrom its trend equilibrium of C∆1. position, the planing surfaces can provide a fairly effective restoring movement. Under the action of the restoring movement, the hull can rapidly return back to the equilibrium position, which suggests that increasing the stern flap mounting angle can achieve the delaying of porpoising.

5. Conclusions In this paper, an experimental and numerical combined study for the double-stepped planing hull has been carried out, and the focus has been mainly on the hydrodynamic characteristics of the step and stern flap, as well as the coupling effect on the hull body. The cavity formation and evaluation process after the step have been captured with acceptable accuracy. Furthermore, the effects of coupled steps and stern flap mounting angles have been investigated, and their action mechanism and effects on resistance performance and porpoising inhibition have been discussed and J. Mar. Sci. Eng. 2019, 7, 346 21 of 25

Furthermore, among the curves from C∆1 to C∆2, the uniformity of the load distribution before and after the gravity center is improved with the increasing flap angle. As discussed above, after the hull enters the planing state, the cavity expands with the increasing flap mounting angle. The main cause should be that the flaps reduce the trim angle by improving the planing surface load distribution, which in turn leads to the cavity length increasing. As a result, once the hull is disturbed from its equilibrium position, the planing surfaces can provide a fairly effective restoring movement. Under the action of the restoring movement, the hull can rapidly return back to the equilibrium position, which suggests that increasing the stern flap mounting angle can achieve the delaying of porpoising.

5. Conclusions In this paper, an experimental and numerical combined study for the double-stepped planing hull has been carried out, and the focus has been mainly on the hydrodynamic characteristics of the step and stern flap, as well as the coupling effect on the hull body. The cavity formation and evaluation process after the step have been captured with acceptable accuracy. Furthermore, the effects of coupled steps and stern flap mounting angles have been investigated, and their action mechanism and effects on resistance performance and porpoising inhibition have been discussed and summarized. Based on the presented results and analysis, the main conclusions can be drawn as follows:

(1) The low-pressure region caused by the step saltation is the immediate induction of the cavity. With a speed increase, the cavity expands toward the rear, increasing the cavity coverage area. Conversely, the wetted surface area decreases. As for the distribution, the wetted surface is mainly concentrated in the post-median part of a planing surface. After the Froude number exceeds 4.38, the ratio of the wetted area and bottom area Awet/A remains at a fairly stable value of 0.23. The decrease of the wetted surface indicates the steps’ friction resistance reducing mechanism. (2) The stern flap is an effective resistance and motion regulating device. Enlarging the stern flap mounting angle within the variation range improves the resistance performance obviously. Moreover, it has a significant porpoising inhibition effect, with a slight cost of resistance during the high speed segment. The maximal porpoising critical speed extends with the increase in the mounting angle. (3) The connection position of the stern flap and hull is a high-pressure concentration area. With the increase of the mounting angle, the pressure value and the trim moment of the stern flap also increase. Correspondingly, the hull motion regulating ability is also improved. (4) Increasing the stern flap mounting angle reduces the trim, and intensifies the cavity expansion speed. The wetted surface area shrinks; the friction resistance is reduced. Therefore, the total resistance amplification is slight. In the meantime, the load among the planing surfaces distributes more uniformly with the increasing flap angle, which is also beneficial for inhibiting porpoising.

Author Contributions: Conceptualization, J.Z.; Formal analysis, Y.J. and S.L.; Investigation, S.L., Z.L. and Y.J.; Methodology, S.L., J.Z. and H.S.; Resources, Y.J. and Z.L.; Supervision, J.Z.; Validation, H.S.; Writing—Original draft, S.L.; Writing—Review and editing, S.L. and J.Z. Funding: The research was funded by Opening Foundation of State Key Laboratory of Ocean Engineering of China (1617), Key Laboratory Fund (614222303030917) and National Science of Foundations of China grant numbers 51509055. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A In the mesh independent study, the total resistance is the main studied and analyzed element. Furthermore, the domain dependence study is also conducted to confirm a sufficiently large and acceptable computational domain. J. Mar. Sci. Eng. 2019, 7, 346 22 of 25

The initial computational domain size is chosen referring to a similar stepped-planing hull simulation study [12]. The domain is established with a total length of about 6 L, which extends 1 L before the front of the model, and 4 L after the rear. The width of the domain is about 1.5 L. Water depth under the free surface is 1 L, and a length of 0.5 L extends above.

Appendix A.1. Width and Length Study of the Domain Due to the symmetry principle, the practical simulating width is about 3 L, while in the experiment, the width of the towing tank is 6.5 m (2.74 L). It indicates that the domain width is wide enough to simulate the experimental facility. The side wall is defined as free slip walls. With the initial domain setup, several speed conditions of M1 are simulated. The results of the wave making after the hull of model M1 at different speeds are shown in Figure A1. It can be seen that the wave making is slightly more toward the side wall. With the speed increase, the wave-making area is narrowed down from the side wall and is extended along the longitudinal direction. However, the wave making area is still J. Mar. Sci. Eng. 2019, 7, 346 23 of 25 distributed in the calculation domain. Therefore, the waves reflect off the side boundaries and have veryside limited boundaries influence. and have Additionally, very limited the influence. domain outlet,Additionally, the size the of domain the domain outlet, that the issize downstream of the fromdomain the hull, that is downstream enlarged to 4fromL, making the hull, the is resultsenlarged insignificant to 4 L, making for thisthe results size. Therefore, insignificant the for width this and lengthsize. Therefore, of the domain the width is accepted. and length of the domain is accepted.

Fr▽ = 2.63

Fr▽ = 4.38

Fr▽ = 5.20

Figure A1. Wave making after the hull of model M1 at different speeds. Figure A1. Wave making after the hull of model M1 at different speeds. Appendix A.2. Depth Study of the Domain Appendix A.2. Depth Study of the Domain The initial depth of the calculation domain is 1 L, while the towing tank depth is 6.8 m (2.87 L). To verifyThe domain initial depth depth of independence,the calculation domain a simulation is 1 L, while for the the hull towing model tank with depth the is initial6.8 m (2.87 depth, L). and To verify domain depth independence, a simulation for the hull model with the initial depth, and another calculation domain, where the domain depth matches the experimental facility, is carried out. another calculation domain, where the domain depth matches the experimental facility, is carried out. The remaining parts of the domain setup stay the same. Figure A2 below shows the comparison of domain depth.

J. Mar. Sci. Eng. 2019, 7, 346 23 of 25

side boundaries and have very limited influence. Additionally, the domain outlet, the size of the domain that is downstream from the hull, is enlarged to 4 L, making the results insignificant for this size. Therefore, the width and length of the domain is accepted.

Fr▽ = 2.63

Fr▽ = 4.38

Fr▽ = 5.20

Figure A1. Wave making after the hull of model M1 at different speeds.

Appendix A.2. Depth Study of the Domain J. Mar. Sci.The Eng. initial2019, 7 depth, 346 of the calculation domain is 1 L, while the towing tank depth is 6.8 m (2.8723 L of). 25 To verify domain depth independence, a simulation for the hull model with the initial depth, and another calculation domain, where the domain depth matches the experimental facility, is carried The remaining parts of the domain setup stay the same. Figure A2 below shows the comparison of out. The remaining parts of the domain setup stay the same. Figure A2 below shows the comparison domain depth. of domain depth.

J. Mar. Sci. Eng. 2019, 7, 346 24 of 25 J. Mar. Sci. Eng. 2019, 7, 346 24 of 25 Figure A2. Depth comparison between original and deeper domain. Figure A2. Depth comparison between original and deeper domain. Figure A2. Depth comparison between original and deeper domain. The calculated results between original and deeper domains are shown in Figures A3 and A4. The calculated results between original and deeper domains are shown in Figures A3 and A4. It It can beThe seen calculated that using results the deeperbetween domain original couldand deep slightlyer domains improve are shown the numerical in Figures accuracy A3 and A4. in drag It can be seen that using the deeper domain could slightly improve the numerical accuracy in drag predictioncan be seen and weakenthat using the the influence deeper ofdomain hull body could on slightly the domain improve bottom, the numerical which makes accuracy the simulation in drag prediction and weaken the influence of hull body on the domain bottom, which makes the simulation closerprediction to the experimentaland weaken the data. influenc However,e of hull bybody comparing on the domain the wave bottom, pattern which of makes the free the simulation surface and closer to the experimental data. However, by comparing the wave pattern of the free surface and numericalcloser to resistance the experimental data, the data. enlargement However, of theby co deepermparing domain the wave does pattern not show of obviousthe free surface improvement and numerical resistance data, the enlargement of the deeper domain does not show obvious withnumerical the mesh resistance increasing data, sharply. the Therefore,enlargement the of results the deeper of the initialdomain domain does depthnot show still satisfyobvious the improvement with the mesh increasing sharply. Therefore, the results of the initial domain depth still researchimprovement requirements. with the mesh increasing sharply. Therefore, the results of the initial domain depth still satisfy the research requirements. satisfy the research requirements.

Figure A3. Calculated resistance and numerical accuracy. FigureFigure A3. A3.Calculated Calculated resistance resistance andand numerical accuracy. accuracy.

Figure A4. Calculated wave-making of the original domain (upper half) and deeper domain (lower FigureFigure A4. A4.Calculated Calculated wave-making wave-making of of the the original original domaindomain (upper half) half) and and deeper deeper domain domain (lower (lower half) M1 at Fr▽ = 5.20. half)half) M1 M1 at Fr at Fr=▽5.20. = 5.20. 5 Appendix A.3. Conclusions Appendix A.3. Conclusions As discussed above, the domain dependence study shows good results for the original domain. As discussed above, the domain dependence study shows good results for the original domain. The width and length of the initial domain meet the accuracy demands for the simulation. Although The width and length of the initial domain meet the accuracy demands for the simulation. Although the calculating domain is physically shallower than the towing tank and the deepened domain can the calculating domain is physically shallower than the towing tank and the deepened domain can slightly improve the numerical accuracy, the original domain can also capture the flow field details slightly improve the numerical accuracy, the original domain can also capture the flow field details and trends well, and with less cost. Therefore, the initial domain is selected as the calculation domain. and trends well, and with less cost. Therefore, the initial domain is selected as the calculation domain. References References 1. Zheng, Y.; Dong, W.C.; Yao, C.B. Experimental study on resistance reduction of displacement type Deep- 1. Zheng, Y.; Dong, W.C.; Yao, C.B. Experimental study on resistance reduction of displacement type Deep- V hull model using stern flap. J. Shanghai Jiaotong Univ. 2011, 45, 475–479. V hull model using stern flap. J. Shanghai Jiaotong Univ. 2011, 45, 475–479. 2. Maki, A.; Arai, J.; Tsutsumoto, T.; Suzuki, K.; Miyauchi, Y. Fundamental research on resistance reduction 2. Maki, A.; Arai, J.; Tsutsumoto, T.; Suzuki, K.; Miyauchi, Y. Fundamental research on resistance reduction of surface combatants due to stern flaps. J. Mar. Sci. Technol. 2016, 21, 344–358. of surface combatants due to stern flaps. J. Mar. Sci. Technol. 2016, 21, 344–358. 3. Savitsky, D. Hydrodynamic design of planing hulls. Mar. Technol. 1964, 1, 71–95. 3. Savitsky, D. Hydrodynamic design of planing hulls. Mar. Technol. 1964, 1, 71–95. J. Mar. Sci. Eng. 2019, 7, 346 24 of 25

Appendix A.3. Conclusions As discussed above, the domain dependence study shows good results for the original domain. The width and length of the initial domain meet the accuracy demands for the simulation. Although the calculating domain is physically shallower than the towing tank and the deepened domain can slightly improve the numerical accuracy, the original domain can also capture the flow field details and trends well, and with less cost. Therefore, the initial domain is selected as the calculation domain.

References

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