Stereo Particle Image Velocimetry Measurements of the Wake Fields Behind a Panamax Bulker Ship Model Under the Ballast Condition

Journal of Marine Science and Engineering

Article Stereo Particle Image Velocimetry Measurements of the Wake Fields behind a Panamax Bulker Ship Model under the Ballast Condition

Tiecheng Wu 1,2 , Wanzhen Luo 1,2,*, Dapeng Jiang 1,2,*, Rui Deng 1,2 and Yulong Li 1,2

1 School of Marine Engineering and Technology, Sun Yat-sen University, Zhuhai 518000, China; [email protected] (T.W.); [email protected] (R.D.); [email protected] (Y.L.) 2 Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai 519000, China * Correspondence: [email protected] (W.L.); [email protected] (D.J.); Tel.: +86-187-4514-7797 (W.L.) Received: 23 April 2020; Accepted: 26 May 2020; Published: 30 May 2020

Abstract: Particle image velocimetry is applied in this study to measure the wake flow field of a Panamax Bulker ship model under the ballast condition. This investigation revealed that the Froude number is 0.167. The time-averaged velocity, turbulent fluctuations, turbulent kinetic energy (TKE), Reynolds stresses, and vorticity information were measured to perform a comparison with the design condition. The time-averaged velocity contours indicated that the ballast and design conditions have distinct hook-like axial velocity contours; however, they appeared at different positions. The big difference under the ballast condition is that the top of the propeller disk area is near the free surface and a region with strong root mean square velocity fluctuation is formed near the free surface. The TKE, the Reynolds stresses, and the hub cap vortex (Hcv) are all affected by the turbulent velocity fluctuation region under the ballast condition. A strong bilge vortex (Bv) is produced when the water flows through the U-shaped stern for the design and ballast conditions.

Keywords: stereo particle image velocimetry; ballast condition; hub cap vortex; bilge vortex; wake ﬁelds

1. Introduction Bulk carriers, oil tankers, and container ships are three main types of ships and they occupy a very important position in the international shipping industry. As of 1 January 2018, there are more than 94,171 merchant ships that totals almost 1.92 billion dead weight tons (DWT). Of these, 42.5% are bulk carriers, 29.2% are oil tankers, and 13.1% are container ships [1]. By having detailed measurements combined with an accurate knowledge and an insight into the flow characteristics of ships, this forms the basis for the theory and numerical simulations verification of the hull optimization and the propeller design. The particle image velocimetry (PIV) method has an extensive use on measuring the wake of ships and propellers. Due to the coupled interaction of the turbulent boundary layer, 3D flow separation, and the orbital motion of water particles, the flow characteristics around the ships are very complex [2,3]. In addition, the flow characteristics are affected by the hull form, stern type, and loading conditions. For bulk carriers, the ballast condition accounts for almost 50% of the total operation time [4]. Our previous paper [2] described in detail the measurement of the wake fields based on the PIV under the design-loading condition, and most of the literature focuses on the flow characteristics around the hulls under the design conditions [5–13]. The flow characteristics around the ship under the ballast condition is different from that under the design condition with the change

J. Mar. Sci. Eng. 2020, 8, 397; doi:10.3390/jmse8060397 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 397 2 of 17 in the draft, inclination angle, and the float status. Hence, the literature review of this paper focuses on the hydrodynamic performance, especially the flow field measurement, for the ballast conditions. Lee et al. [3] applied a 2D-PIV system to study the wake field of a Korea Research Institute of Ships & Ocean engineering (KRISO) container ship (KCS) with Lpp = 1.422 m under the design and ballast conditions in a circulating water channel. The PIV measurement results show that the intensity of the longitudinal counter-rotating vortices and the location of the vortex core are significantly different under the two conditions. Wu et al. [14,15] used the LaVision underwater stereoscopic particle image velocimetry (SPIV) system to measure the phase averaged wake fields in the wave of the second variant of KRISO very large crude carrier (KVLCC2) hull under the design and ballast conditions. At the same time, the corresponding numerical study was carried out using computational fluid dynamics (CFD) Ship-Iowa V4.5 code. The experimental and numerical results show that the ballast condition has a different axial velocity distribution, a hook-shaped velocity contour, and vortex distribution characteristics under the design condition. Park et al. [4] applied experimental and numerical methods to study the added resistance in the wave under the full-load, ballast, and two intermediate conditions. Luo et al. [16] conducted a numerical study on the viscous flow field around a KCS ship with Lpp = 7.2786 m under the ballast conditions. The influence of the different loading conditions on the surrounding flow field of the ship is described by the wave elevation contours, the nominal flow field distribution, and the volume fraction of the water on the hull surface at the bow and stern. Guo et al. [17] used the Dantec underwater SPIV system to measure the nominal wake field of a KCS with a Lpp = 4.3671 m for off-design loading conditions in a towing tank, which corresponds to numerical studies that have been simultaneously carried out. The experimental and numerical results show that the draft and inclination angle of the ships are different under the different loading conditions, which leads to a different inflow velocity and flow direction in the stern region, and different wake field distribution characteristics. Compared with the extensive research literature on the hydrodynamic performance of ships under the design condition, there are few research articles for the ballast condition, and there are fewer articles that provides detailed measurements and the analysis of a wake flow for a ship under the ballast condition. Although some studies have been carried out for this problem, there is still some important flow field information such as the velocity distribution, turbulence characteristic parameters, vortex distribution characteristics, and the spatial distribution of the wake vortices that need to be further investigated. In the present study, the wake fields under the ballast condition were studied by a towed underwater SPIV system. In the process of the experimental study, the time-averaged velocity, turbulent fluctuations, TKE, Reynolds stresses, and vorticity information are the main concerns. Initially, we identified the time-averaged velocities, and then analyzed the TKE and Reynolds stresses. The vorticity information was analyzed based on the parameters of vorticity, lambda 2-criterion, planar 2nd invariant Q, the swirling strength, and the evolution of the bilge vortex (Bv) and the propeller hub cap vortex (Hcv). In addition, these characteristics are compared to our previous study of the wake field under the design condition, which reveals the loading condition influence on the flow field distribution. The research content of this paper is as follows. Section2 describes the experimental set up. The discussion of measurement results under the ballast condition are illustrated in detail in Section3. In particular, Sections 3.1–3.3 report the time-averaged velocity fields. The turbulence characteristics and the vorticity information of the flow fields are described in Sections 3.4 and 3.4, respectively. Section4 delivers the conclusions for this study.

2. Experimental Set Up

2.1. Testing Model and Conditions The test model is a 76000 DWT Panamax Bulker, which has a scale of 1:45 that is manufactured with ﬁberglass. The stern of the ship model is painted matte black in case of the laser sheet reﬂection. J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 3 of 16

system is shown in Figure 1, which is located at the intersection of the hull center line, base line, and the Forward Perpendicular (FP). The directions of the xyz-axis, lines plans, and the geometric model are shown in Figure 1. The main dimensions of the model under the ballast condition are in Table 1.

Table 1. Main parameters of the test model.

Principal Hull Data Model Full Scale Scale λ 45

J. Mar. Sci. Eng. 2020, 8, 397 Length Over All (m) Loa 5.00 225 3 of 17 Length Between Perpendiculars (m) Lpp 4.82 217 Breadth (m) B 0.72 32.25 Meanwhile, as described in the previousDepth (m) paper [2], turbulenceD stimulators0.45 were20.10 installed according to the international towingForward tank Perpendicular conference (ITTC)Draft (FP/m) rules [18T].FP The0.112 origin of the5.04 global xyz coordinate system is shown in FigureAfter 1Perpendicular, which is located Draft at the(AP/m) intersection TAP of the0.160 hull center7.20 line, base line, and the Forward Perpendicular (FP). TheSpeed directions (m/s) of the xyz-axis,V0 lines 1.150 plans, and7.716 thegeometric model are shown in Figure1. The mainPropeller dimensions Radius of the(m) model underR the0.075 ballast condition3.375 are in Table1.

Figure 1. Lines plan and the geometric model.model.

The measured sections, testTable draft, 1. Main and parametersthe test conditions of the test for model. the wake field measurements are listed in Table 2. The experimentalPrincipal Hullsection Data of the flow fields is Model the propeller Full Scale plane, which is located at X/Lpp = 0.0199. Scale λ 45 Length Over All (m) Loa 5.00 225 Table 2. PIV test conditions under the ballast condition. Length Between Perpendiculars (m) Lpp 4.82 217 Breadth (m) B 0.72 32.25 Testing Conditions Parameters Depth (m) D 0.45 20.10 Forward PerpendicularTest Section Draft (FP /m) TPropellerFP 0.112 plane 5.04 After PerpendicularFore Draft Draft (FP/m) (AP/m) TAP 0.1120.160 7.20 SpeedAfter (mDraft/s) (AP/m) V0 0.1601.150 7.716 PropellerFluid Radius Medium (m) R Water0.075 3.375 Test Velocity (m/s) 1.150 The measured sections, test draft,Waves and the test conditionsSteady surface for the wake field measurements are listed in Table2. The experimental section of the flow fields is the propeller plane, which is located at X2.2./Lpp Facility= 0.0199. and SPIV System Tests were conducted in the towing tank, which is 180 m long, 7 m wide, and 3.5 m deep with a Table 2. PIV test conditions under the ballast condition. Dantec towed underwater 2D–3C stereoscopic PIV system. The submersible PIV system consists of the submersible camera housing,Testing the Conditions mirror modules, and Parameters the submersible light sheet unit; the submersible camera housing and Test submersible Section light sheet Propeller unit plane are assembled in the water proof section. The mirror modules areFore assembled Draft (FP in/m) the sections 0.112full of water and this ensures that the optical image can be acquired inAfter a single Draft (APwater/m) medium. The 0.160 facility is described in our previous Fluid Medium Water Test Velocity (m/s) 1.150 Waves Steady surface

2.2. Facility and SPIV System Tests were conducted in the towing tank, which is 180 m long, 7 m wide, and 3.5 m deep with a Dantec towed underwater 2D–3C stereoscopic PIV system. The submersible PIV system consists of the submersible camera housing, the mirror modules, and the submersible light sheet unit; the submersible J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 4 of 16

experimental studies [2]. The experimental setup of the SPIV measurement is shown in Figure 2. The specifications of the SPIV system are as follows. The resolution of the CCD is 2048 × 2048 pixels. The field of view (FOV) is 320 mm width and 280 mm height. The adopted laser supply is an Neodymium- doped: Yttrium Aluminum Garnet (Nd: YAG) laser with an energy of 200 MJ, a frequency of 15 Hz, a laser beam duration of 4 ns, a wavelength of 532 nm, and the thickness of the pulse laser sheet is 2 mm. The velocity vectors were calculated using the adaptive correlation method (Dantec Dynamics, 2015) with an initial 128 × 128 pixel interrogation window and a final 32 × 32 pixel interrogation window. An overlap of 50% among neighboring interrogation windows was used during the correlation analysis process. A total of 250 instantaneous velocity fields were measured with a single towing carriage run to acquire the time-averaged velocity fields. The final spacing of the vector grid 2 J.of Mar. one Sci. velocity Eng. 2020 vector, 8, 397 in the time-averaged velocity map was 2.45 × 2.45 mm . The tracer particles4 of 17 used are polyamide seeding particles. The diameter of the seeding particles is 50 μm. The calibration process and vector field calculation process of the SPIV system are described in detail in previous camerastudies housing[2]. and submersible light sheet unit are assembled in the water proof section. The mirror modulesIn the are paper, assembled the uncertainty in the sections and convergence full of water analysis and this was ensures basically that the the same optical with image that canin the be acquiredreference in[10]. a single The root water-mea medium.n-square The of facilitythe velocity is described as well in as our Reynolds previous stresses experimental are second studies-order [2]. Thesmall experimental quantities compared setup of theto the SPIV average measurement velocity iswhich shown means in Figure that the2. The uncertainty specifications and convergence of the SPIV system are as follows. The resolution of the CCD is 2048 2048 pixels. The field of view (FOV) is of the root-mean-square of the velocity and Reynolds stresses× require more instantaneous velocity 320samples. mm widthBecause and of 280the mmlimit height. of the towing The adopted tank length, laser supplywe can isonly an Neodymium-doped:take 250 images per run. Yttrium The Aluminumsimilar PIV test Garnet of the (Nd: Seoul YAG) National laser withUniversity an energy towing of 200tank MJ, (110 a m frequency long, 8 m of wide, 15 Hz, and a 3.5 laser m deep) beam durationand the IIHR of 4 ns, towing a wavelength tank of the of 532 University nm, and of the Iowa thickness (100 m of thelong, pulse 3 m laser wide, sheet and is 3 2m mm. deep) The also velocity took vectorsalmost were 250 images, calculated the using measurement the adaptive accuracy correlation of method these images (Dantec can Dynamics, meet the 2015) requirements with an initial of 128 128 pixel interrogation window and a final 32 32 pixel interrogation window. An overlap engineering× and scientific research. In this paper, the× detailed uncertainty analysis method and ofconvergence 50% among analy neighboringsis process interrogation of the present windows SPIV was will used not during be the described correlation repeatedly. analysis process. This is Ademonstrated total of 250 instantaneous in a previous velocitypaper [2]. fields The were method measured of convergence with a single analysis towing has carriage referred run to to Falchi acquire et theal. [10] time-averaged and Bendat velocity and Piersol fields. [20] The [21]. final The spacing distribution of the vector of the grid measurement of one velocity errors vector is used in the to time-averaged velocity map was 2.45 2.45 mm2. The tracer particles used are polyamide seeding determine a 95% confidence interval × that contains the true displacement. The uncertainty of the particles.velocity is The2.5%. diameter The uncertainty of the seeding of the second particles-order is 50 statisticalµm. The moments calibration of processthe statistical and vector samples field is calculation17%. process of the SPIV system are described in detail in previous studies [2].

Figure 2. Experimental setup of SPIV measurement: ( a)) photograph of SPIV setup in the towing tank; ((b)) schematicschematic diagramdiagram ofof SPIVSPIV setup;setup; ((cc)) close-upclose-up ofof the position of the propeller plane and PIVPIV system; ((d)) close-upclose-up of the position of the propellerpropeller plane,plane, freefree surface,surface, andand laserlaser sheet.sheet.

3. ResultsIn the and paper, Discussion the uncertainty and convergence analysis was basically the same with that in the reference [10]. The root-mean-square of the velocity as well as Reynolds stresses are second-order small In order to facilitate the comparative study of the flow field of the Panama bulk carrier under quantities compared to the average velocity which means that the uncertainty and convergence of the the ballast and design load conditions. Hereby description, the entire discussion includes results root-mean-square of the velocity and Reynolds stresses require more instantaneous velocity samples. from previous work of the authors under design condition [2]. Because of the limit of the towing tank length, we can only take 250 images per run. The similar PIV test of the Seoul National University towing tank (110 m long, 8 m wide, and 3.5 m deep) and the IIHR 3.1. Time-Averaged Velocity Fields towing tank of the University of Iowa (100 m long, 3 m wide, and 3 m deep) also took almost 250 images, the measurementThe time-averaged accuracy characteristics of these images of the can velocity meet the information requirements are of analy engineeringzed under and the scientiﬁc ballast 1 250 research.condition Inin this paper,section. the The detailed time-averaged uncertainty velocity analysis component method ui and can convergencebe defined as: analysis process, uui  i, n of the present SPIV will not be described repeatedly. This is demonstrated in a previous250 papern1 [2]. Thewhere method the index of convergence i =1, 2, 3 indicates analysis the has velocity referred components to Falchi et u al., v [,10 and] and w in Bendat the x, and y and Piersol z direction [19,20]., The distribution of the measurement errors is used to determine a 95% conﬁdence interval that contains the true displacement. The uncertainty of the velocity is 2.5%. The uncertainty of the second-order statistical moments of the statistical samples is 17%.

3. Results and Discussion In order to facilitate the comparative study of the ﬂow ﬁeld of the Panama bulk carrier under the ballast and design load conditions. Hereby description, the entire discussion includes results from previous work of the authors under design condition [2]. J. Mar. Sci. Eng. 2020, 8, 397 5 of 17

3.1. Time-Averaged Velocity Fields The time-averaged characteristics of the velocity information are analyzed under the ballast 250 1 P condition in this section. The time-averaged velocity component ui can be defined as: ui = 250 ui,n, n=1 where the index i = 1, 2, 3 indicates the velocity components u, v, and w in the x, y and z direction, respectively. The time-averaged and towing speed of the ship model U are dealt with a dimensionless method. The y-axes and z-axes are normalized as Y/R and Z/R. The dimensionless axial flow velocity u/U, the cross-flow vector, cross-flow velocity S/U, and the streamline in the propeller region under the ballast and design conditions shown in Figures3 and4a–d illustrates the results of the following parameters: the dimensionless transverse velocity v/U, vertical velocity w/U, whole-flow velocity L/U; and average kinetic energy AKE/U2. The whole-flow velocity L ((u2 + v2 + w2)1/2) and the cross-flow velocity S ((v2 + w2)1/2) conduct dimensionless processing with the towing speed of the ship model U. The average kinetic energy (AKE = 1/2(u2 + v2 + w2)) conducts dimensionless processing with the square of the towing speed U2. Under the ballast condition, the ship is a little trim by the stern and the draft is smaller than the design condition. Influenced by the change of the loading condition, the characteristics of the hook-like velocity contour can be found in Figure3a, the distinct hook-like contours (labelled A) appear at the position of u/U = 0.4 (u/U = 0.3 under the design condition) [2], and the vertex of the hook-like contour is at the position of Z/R = 0.6 and Z/R = 0.25 under the design and ballast condition, respectively. The definition of hook-like characteristic, which was first proposed by Kim [6], is caused by flows with less average kinetic energy (Figure4d, labeled D) being transferred through an intense longitudinal bilge vortex towards the center of the hull, which reduces its velocity and creates local hook-like structures. The bilge vortices formed by the flow passing through the stern of the U-shaped ship move down, which is almost in parallel with the hub cap vortices. The position of the hub cap vortex changes a little; however, the rotating range is larger than the design condition. The cross-flow vector and streamline clearly capture and illustrate the characteristics of the bilge vortex and the hub cap vortex. Figure3c shows the contours of S/U. The bilge vortex foci (labelled F1) are at Y/R = 0.5, Z/R = 0.05, and the hub cap vortex foci (labelled F2) are at Y/R = 0.215, Z/R = 0.425 ± ± − under the design condition. The bilge vortex foci (labelled F1) are at Y/R = 0.75, Z/R = 0.25, and the ± − hub cap vortex foci (labelled F2) are Y/R = 0.215, Z/R = 0.55 under the ballast condition. ± − As the streamline shown in Figure3d, on the starboard side of the ship, the bilge vortex flows counter-clockwise and the hub cap vortex flows clockwise. On the port side of the ship, the bilge vortex flows clockwise and the hub cap vortex flows counter-clockwise. The shape of the bilge vortex is “round” under the design condition. Under the ballast condition, the bilge vortex is in the shape of “ears”, and the range of the hub cap vortex is larger than that under the design condition. In addition, the vortex foci of the twin vortex structure and the two saddle points of the propeller disk (labelled S1 and S2) are captured. The saddle point S1 is the intersection point of the upwash flow shear layer and the downwash flow shear layer. This is the point where the cross-section velocity S/U = 0 (S1 at Y/R = 0.1,Z/R = 0.05, and Y/R = 0.05, Z/R = 0.25 for the design and ballast condition, respectively). ± − ± − The saddle point S2 is the point where the left-wash flow and the right-wash flow separated from the up/down-wash flow shear layer and this is the point where the cross-section velocity S/U = 0 (S2 at Y/R = 0.4,Z/R = 0.65, and Y/R = 0.35, Z/R = 0.95 for the design and ballast condition, respectively). ± − ± − It can be seen in Figure4a,b that the positive and negative velocities represent the rotation direction of the bilge vortex and the hub cap vortex. This is because the ship is under the ballast condition and the top of the propeller disk area is near the free surface. The propeller disk area produces the near free surface layer disturbed by the free surface waveform. The dimensionless transverse velocity v/U and the dimensionless vertical velocity w/U are smaller than the dimensionless axial velocity u/U. These are greatly affected by the free surface disturbance. Therefore, the distribution of the dimensionless transverse velocity v/U and the dimensionless vertical velocity w/U in the near free surface layer is chaotic. Figure4c,d shows that the hook-like region of the wake field in the propeller J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 5 of 16

respectively. The time-averaged and towing speed of the ship model U are dealt with a dimensionless method. The y-axes and z-axes are normalized as Y/R and Z/R. The dimensionless axial flow velocity u/U, the cross-flow vector, cross-flow velocity S/U, and the streamline in the propeller region under the ballast and design conditions shown in Figure 3 and Figure 4a-d illustrates the results of the following parameters: the dimensionless transverse velocity v/U, vertical velocity w/U, whole-flow velocity L/U; and average kinetic energy AKE/U2. The whole-flow velocity L ((u2+v2+w2)1/2) and the cross-flow velocity S ((v2+w2)1/2) conduct dimensionless processing with the towing speed of the ship model U. The average kinetic energy (AKE=1/2(u2+v2+w2)) conducts dimensionless processing with the square of the towing speed U2. Under the ballast condition, the ship is a little trim by the stern and the draft is smaller than the design condition. Influenced by the change of the loading condition, the characteristics of the hook-like velocity contour can be found in Figure 3a, the distinct hook-like contours (labelled A) appear at the position of u/U = 0.4 (u/U = 0.3 under the design condition) [2], and the vertex of the hook-like contour is at the position of Z/R = 0.6 and Z/R = 0.25 under the design and ballast condition, respectively. The definition of hook-like characteristic, which was first proposed by Kim [6], is caused by flows with less average kinetic energy (Figure 4d, labeled D) being transferred through an intense longitudinal bilge vortex towards the center of the hull, which reduces its velocity and creates local hook-like structures. The bilge vortices formed by the flow passing through the stern of the U-shaped ship move down, which is almost in parallel with the hub cap vortices. The position of the hub cap vortex changes a little; however, the rotating range is larger than the design condition. The cross-flow vector and streamline clearly capture and illustrate the J.characteristics Mar. Sci. Eng. 2020 of, 8the, 397 bilge vortex and the hub cap vortex. Figure 3c shows the contours of S/U6. ofThe 17 bilge vortex foci (labelled F1) are at Y/R = ±0.5, Z/R = 0.05, and the hub cap vortex foci (labelled F2) are at Y/R = ±0.215, Z/R = −0.425 under the design condition. The bilge vortex foci (labelled F1) are at diskY/R = area ±0.75, has Z/R lower =-0.25, average and the kinetic hub energy cap vortex (AKE) foci (labelled (labelled D F2) in Figure are Y/R4d) = characteristics±0.215, Z/R = − due0.55 tounder the inﬂuencethe ballast of condition. the bilge vortex and the hub cap vortex.

J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 6 of 16

Figure 3. The dimensionless axial flow velocity u/U, the cross-flow vector, cross-flow velocity S/U, and the streamline in the propeller region: left. The ballast conditions; right. The design condition.

As the streamline shown in Figure 3d, on the starboard side of the ship, the bilge vortex flows counter-clockwise and the hub cap vortex flows clockwise. On the port side of the ship, the bilge vortex flows clockwise and the hub cap vortex flows counter-clockwise. The shape of the bilge vortex is "round" under the design condition. Under the ballast condition, the bilge vortex is in the shape of "ears", and the range of the hub cap vortex is larger than that under the design condition. In addition, the vortex foci of the twin vortex structure and the two saddle points of the propeller disk (labelled S1 and S2) are captured. The saddle point S1 is the intersection point of the upwash flow shear layer and the downwash flow shear layer. This is the point where the cross-section velocity S/U = 0 (S1 at Y/R = ±0.1, Z/R =−0.05, and Y/R = ±0.05, Z/R = −0.25 for the design and ballast condition, respectively). The saddle point S2 is the point where the left-wash flow and the right-wash flow separated from the up/down-wash flow shear layer and this is the point where the cross-section velocity S/U = 0 (S2 at Y/R = ±0.4, Z/R = −0.65, and Y/R = ±0.35, Z/R = −0.95 for the design and ballast condition, respectively). It can be seen in Figure 4a,b that the positive and negative velocities represent the rotation direction of the bilge vortex and the hub cap vortex. This is because the ship is under the ballast condition and the top of the propeller disk area is near the free surface. The propeller disk area produces the near free surface layer disturbed by the free surface waveform. The dimensionless transverse velocity v/U and the dimensionless vertical velocity w/U are smaller than the dimensionless axial velocity u/U. These are greatly affected by the free surface disturbance. Therefore, the distribution of the dimensionless transverse velocity v/U and the dimensionless vertical velocity w/U Figurein the near 3. The free time-averaged surface layer characteristicsis chaotic. Fig ofure the 4c,d velocity shows information that the hook in- thelike propeller region of plane: the wake field( ain) dimensionless the propeller axial disk flowarea velocity has loweru/U ;( averageb) the cross-flow kinetic energy vector; (AKE) (c) dimensionless (labelled D cross-flow in Figure 4h) characteristicsvelocity S/U due;(d) to streamline. the influence For (a –ofd ),theLeft bilge. The vortex ballast and condition; the hubRight cap. Thevortex. design condition.

FigureFigure 4.4. TheThe time-averageddimensionless characteristicstransverse velocity of the v/U velocity, vertical information velocity w/U in the, whole propeller-flow velocity plane under L/U, ballastand average condition: kinetic (a )energy dimensionless AKE/U2 under transverse the ballast velocity conditions.v/U;(b) dimensionless vertical velocity w/U; (c) dimensionless whole-ﬂow velocity L/U;(d) dimensionless average kinetic energy AKE/U2. 3.2. Time-Averaged Velocity at Z/R = 0

Figure 5 shows the time-averaged velocity u/U, v/U, w/U, S/U, L/U, and AKE/U2 profiles at Z/R = 0.00 in the propeller region under the ballast condition. The measurement results are as follows. The dimensionless axial velocity u/U and the whole-flow velocity L/U show an oscillatory profile from Y/R = 0 to Y/R = 1. The hook-like distributions of time-averaged velocity u/U and L/U first increase from Y/R = 0–0.5, then decrease from Y/R = 0.5–0.8, and finally increase from Y/R = 0.8–1.0. The velocity v/U increases first and then decreases at Y/R = 0.55. The velocity w/U shows a gradually decreasing trend from Y/R = 0 to Y/R = 1. Because of the larger range of the hook-like velocity profile under the ballast condition than that under the design condition, the distribution of S/U presents concave in the range Y/R = 0.5–1.0. It has a minimum velocity at Y/R = 0.85 with a magnitude of S/U

J. Mar. Sci. Eng. 2020, 8, 397 7 of 17

3.2. Time-Averaged Velocity at Z/R = 0 Figure5 shows the time-averaged velocity u/U, v/U, w/U, S/U, L/U, and AKE/U2 profiles at Z/R = 0.00 in the propeller region under the ballast condition. The measurement results are as follows. The dimensionless axial velocity u/U and the whole-flow velocity L/U show an oscillatory profile from Y/R = 0 to Y/R = 1. The hook-like distributions of time-averaged velocity u/U and L/U first increase from Y/R = 0–0.5, then decrease from Y/R = 0.5–0.8, and finally increase from Y/R = 0.8–1.0. The velocity v/U increases first and then decreases at Y/R = 0.55. The velocity w/U shows a gradually decreasing J. Mar. Sci. Eng. 2020trend, 7, fromx FOR Y/R PEER= 0 to REVIEW Y/R = 1. Because of the larger range of the hook-like velocity profile under the 7 of 16 ballast condition than that under the design condition, the distribution of S/U presents concave in the = 0.065. The lowrange kinetic Y/R = 0.5–1.0. energy It has AKE/U a minimum2 appears velocity at in Y/R the= 0.85 range with a Y/R magnitude = 0– of1.0S/U because= 0.065. The of low the disturbance kinetic energy AKE/U2 appears in the range Y/R = 0–1.0 because of the disturbance caused by the caused by the hookhook-like-like velocity velocity contours contours and the bilge and vortex. the bilge vortex.

2 Figure 5. TimeFigure-averaged 5. Time-averaged velocity velocity and andthe theAKE/UAKE/U2 profilesprofiles at Z at/R =Z/R0.00 = under 0.00 the under ballast condition.the ballast condition. 3.3. Time-Averaged Velocity at Each Radius 3.3. Time-AveragedIn Velocity this section, at the Each velocity Radius and the AKE/U2 profiles with several radii (r/R = 0.4, 0.6, 0.8, 1.0) under the ballast condition are illustrated. Figure6 shows the velocities u/U, v/U, w/U, S/U, L/U, and AKE/U2. 2 In this section,The circumferential the velocity angle andθ runs the clockwise AKE/U fromprofiles 0◦ (top) towith 180◦. several When r/R radii= 0.4 and (r/R 0.6, = as 0.4, shown 0.6, in 0.8, 1.0) under the ballast conditionFigure6, are the positive illustrated. and negative Figure values 6 shows of the time-averaged the velocities velocity u/Uv,/U v/Uand, ww/U/U represent, S/U, L/U the , and AKE/U2. direction of these parameters. The time-averaged velocity u/U, S/U, L/U, and AKE/U2 are basically The circumferentialpositive angle and decrease θ runs with clockwise respect to θ ,from which 0° is the(top) same to with 180°. that When under the r/R design = 0.4 condition. and 0.6, as shown in Figure 6, the positiveFor r/R = 0.6 and, the negative cross-section values velocity S /ofU has the some time slight-averaged fluctuations. velocity The dimensionless v/U and transverse w/U represent the velocity v/U and w/U are basically negative. The "absolute" values of v/U and w/U with respect to the 2 direction of thesecircumferential parameters. angle θ firstThe increase time- andaveraged then decrease. velocity At r/R = u/U,0.8, the S/U, distributions L/U, and of u/U, AKE/U S/U, L/U, are basically positive and decreaseand AKE/U 2withwith respectrespect to θ toshow θ, a which decreasing is trendthe followedsame with by an increasingthat under and decreasing the design trend. condition. For r/R = 0.6, the crossThe "absolute"-section values velocity of v/U and S/Uw/ Uhaswith some respect slight to the circumferential fluctuations. angle Theθ first dimensionless increases and transverse then it decreases with a relatively irregular change. However, the trend of S/U and AKE/U2 are not velocity v/U andsignificant w/U are and arebasically relatively stable.negative. When rThe/R = 1.0,"absolute"u/U, S/U, L/U values, and AKE of/U2 v/Uare increasing and w/U except with for respect to the circumferentialθ =angle0–30◦ and θ first 160–180 increase◦ with a slight and fluctuation then decrease. (u/U, L/U). The A dimensionlesst r/R = 0.8, transverse the distributions velocity v/U of u/U, S/U, exhibits strong fluctuations. The dimensionless vertical velocity w/U has a relatively stable distribution. L/U, and AKE/U2 with respect to θ show a decreasing trend followed by an increasing and decreasing trend. The "absolute" values of v/U and w/U with respect to the circumferential angle θ first increases and then it decreases with a relatively irregular change. However, the trend of S/U and AKE/U2 are not significant and are relatively stable. When r/R = 1.0, u/U, S/U, L/U, and AKE/U2 are increasing except for θ = 0–30°and 160–180°with a slight fluctuation (u/U, L/U). The dimensionless transverse velocity v/U exhibits strong fluctuations. The dimensionless vertical velocity w/U has a relatively stable distribution.

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= 0.065. The low kinetic energy AKE/U2 appears in the range Y/R = 0–1.0 because of the disturbance caused by the hook-like velocity contours and the bilge vortex.

Figure 5. Time-averaged velocity and the AKE/U2 profiles at Z/R = 0.00 under the ballast condition.

3.3. Time-Averaged Velocity at Each Radius

In this section, the velocity and the AKE/U2 profiles with several radii (r/R = 0.4, 0.6, 0.8, 1.0) under the ballast condition are illustrated. Figure 6 shows the velocities u/U, v/U, w/U, S/U, L/U, and AKE/U2. The circumferential angle θ runs clockwise from 0° (top) to 180°. When r/R = 0.4 and 0.6, as shown in Figure 6, the positive and negative values of the time-averaged velocity v/U and w/U represent the direction of these parameters. The time-averaged velocity u/U, S/U, L/U, and AKE/U2 are basically positive and decrease with respect to θ, which is the same with that under the design condition. For r/R = 0.6, the cross-section velocity S/U has some slight fluctuations. The dimensionless transverse velocity v/U and w/U are basically negative. The "absolute" values of v/U and w/U with respect to the circumferential angle θ first increase and then decrease. At r/R = 0.8, the distributions of u/U, S/U, L/U, and AKE/U2 with respect to θ show a decreasing trend followed by an increasing and decreasing trend. The "absolute" values of v/U and w/U with respect to the circumferential angle θ first increases and then it decreases with a relatively irregular change. However, the trend of S/U and AKE/U2 are not significant and are relatively stable. When r/R = 1.0, u/U, S/U, L/U, and AKE/U2 are increasing except for θ = 0–30°and 160–180°with a slight fluctuation (u/U, L/U). The dimensionless transverse velocityJ. Mar. v/U Sci. Eng.exhibits2020, 8 , strong 397 fluctuations. The dimensionless vertical velocity w/U has a relatively8 of 17 stable distribution.

Figure 6. Time-averaged velocity and the average kinetic energy (AKE) proﬁles at each radius in the propeller region under the ballast condition.

3.4. Velocity Fluctuations The turbulence characteristic of the flow fields is expressed as the root mean square velocity fields, s 250 = = 1 P ( )2 which can be computed as: ui RMS ui0 250 ui,n ui . Figure7 shows the root mean square − n=1 − velocity fields under the ballast and design conditions. The root mean square velocity is dealt with a dimensionless method. Figure7a–c displays the results of the dimensionless axial root mean square velocity uRMS/U, the dimensionless transverse root mean square velocity vRMS/U, and the dimensionless vertical root mean square velocity wRMS/U, respectively. The white dotted line is the position where r/R = 1.0 and the light blue wavy line is near the free surface zone under the ballast condition. As depicted in Figure7, the distribution of the root mean square velocity in the three directions is similar under the ballast and design conditions, which is either hook-shaped or U-shaped. Compared with that under the design condition, the areas with a strong fluctuation are mainly distributed at the end of the shaft, the range of r/R = 0.9–1.5, and the end of the hub cap. In addition, the top edge of the propeller is tangent to the free surface under the ballast condition, and a strong root mean square velocity fluctuation region is formed near the free surface. Figure8 presents the TKE in the propeller region under the ballast and design conditions, Figure9 presents the Reynolds stresses in the propeller region under the ballast condition. The Reynolds stresses 250 = 1 P ( )( ) = ( + + ) tensor and TKE can be defined as: ui0u0j 250 ui,n ui uj,n uj and TKE 1/2 u0u0 v0v0 w0w0 , n=1 − − respectively, where i, j = 1, 2, 3 indicate the velocity fluctuations u0, v0, and w0 in the x, y and z direction, respectively, ui,n or uj, n represents the n-th instantaneous velocity data. The TKE and the Reynolds stresses are dealt with the dimensionless method. Figure9a–c displays the dimensionless Reynolds 2 2 2 normal stress u0u0/U , v0v0/U , and w0w0/U , respectively. Figure9d–f depicts the dimensionless shear 2 2 2 stress u0v0/U , u0w0/U , and v0w0/U , respectively, and the white dotted line shows the position of r/R = 1.0. J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 8 of 16

Figure 6. Time-averaged velocity and the average kinetic energy (AKE) profiles at each radius in the propeller region under the ballast condition.

3.4. Velocity Fluctuations The turbulence characteristic of the flow fields is expressed as the root mean square velocity 1 250  2 fields, which can be computed as: ui RMS u i () u i, n  u i . Figure 7 shows the root mean 250 n1 square velocity fields under the ballast and design conditions. The root mean square velocity is dealt with a dimensionless method. Figure 7a–c displays the results of the dimensionless axial root mean square velocity uRMS/U, the dimensionless transverse root mean square velocity vRMS/U, and the dimensionless vertical root mean square velocity wRMS/U, respectively. The white dotted line is the position where r/R = 1.0 and the light blue wavy line is near the free surface zone under the ballast condition. As depicted in Figure 7, the distribution of the root mean square velocity in the three directions is similar under the ballast and design conditions, which is either hook-shaped or U- shaped. Compared with that under the design condition, the areas with a strong fluctuation are mainly distributed at the end of the shaft, the range of r/R = 0.9–1.5, and the end of the hub cap. In addition, the top edge of the propeller is tangent to the free surface under the ballast condition, and J. Mar. Sci. Eng. 2020, 8, 397 9 of 17 a strong root mean square velocity fluctuation region is formed near the free surface.

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TKE1 2( u u   v  v   w  w  ) , respectively, where i, j = 1, 2, 3 indicate the velocity fluctuations u , v , and w in the x, y and z direction, respectively, ui,n or uj, n represents the n-th instantaneous velocity data. The TKE and the Reynolds stresses are dealt with the dimensionless method. Figure 9a-c Figure 7. RootRoot mean mean square square velocity velocity fields ﬁelds in the in propeller the propeller region: region: Left. The (a) dimensionlessballast conditions; axial Right root. displays the dimensionless Reynolds normal stress u u/ U 2 , v v/ U 2 , and w w/ U 2 , respectively. meanThe design square condition. velocity uRMS/U;(b) dimensionless transverse root mean square velocity vRMS/U; u v/ U 2 u w/ U 2 v w/ U 2 Figure(c) 9 dimensionlessd-f depicts the vertical dimensionless root mean shear square stress velocity wRMS/,U . For (a–c),, andLeft. The ballast, respectively, condition; and the whiteFigRighture dotted. The8 presents design line shows condition.the TKE the in position the propeller of r/R region= 1.0. under the ballast and design conditions, Figure 9 presents the Reynolds stresses in the propeller region under the ballast condition. The Reynolds 1 250  stresses tensor and TKE can be defined as: ui u j( u i,, n  u i )( u j n  u j ) and 250 n1

Figure 8. TheThe turbulent kinetic energy (TKE) in the propeller region: LLefteft. The The ballast ballast conditions; conditions; RRightight.. The The design design condition. condition.

Figure 9. The Reynolds stresses in the propeller region under the ballast condition.

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J. Mar.Figure Sci. Eng. 8.2020 The, 8 turbulent, 397 kinetic energy (TKE) in the propeller region: Left. The ballast conditions;10 of 17 Right. The design condition.

Figure 9.FigureThe Reynolds 9. The Reynolds stresses in stresses the propeller in the regionpropeller under region the ballastunder condition.the ballast (condition.a) Reynolds normal 2 2 2 stress u0u0/U ;(b) Reynolds normal stress v0v0/U ;(c) Reynolds normal stress w0w0/U ;(d) Reynolds 2 2 2 shear stress u0v0/U ;(e) Reynolds shear stress u0w0/U ;(f) Reynolds shear stress v0w0/U .

As demonstrated in Figure8, the distribution of the TKE is also U-shaped; however, the interval is wider and the U-shaped distribution is concentrated in the kinetic energy change layer, which is the transition from low kinetic energy to high kinetic energy. Compared with that under the design condition, the changing layer with strong gradient of the turbulent kinetic energy under the ballast condition is outward-transferring to 0.9R. Due to the influence of the loading condition, the shaft is no longer in the hydrodynamic downstream state under the ballast condition. Therefore, the TKE at the end of the shaft in the non-downstream state has a large amplitude. As demonstrated in Figure9, 2 2 the axial Reynolds normal stress u0u0/U , the transverse Reynolds normal stress v0v0/U , and the 2 vertical Reynolds normal stress w0w0/U have the characteristic of anisotropy. The influence of the shallow draft at the stern under the ballast condition makes the propeller disk area have a turbulent area of about 0.25R near the free surface. Due to the influence of the rooster tail near the free surface, the TKE and the Reynolds stress in the turbulent region have a large amplitude. The maximum value of p p the axial Reynolds normal stress u u 0.07U, the transverse Reynolds normal stress v v 0.05U, 0 0 ≈p 0 0 ≈ and the vertical Reynolds normal stress w0w0 0.016U in the non-interference region. The contours ≈ 2 of the TKE are similar with the axial Reynolds normal stress u0u0/U . The maximum TKE occurs in the high shear zone and the velocity gradient of the axial velocity component is large. This is because the axial Reynolds normal stress is much larger than the maximum of the transverse and vertical Reynolds normal stress. J. Mar. Sci. Eng. 2020, 8, 397 11 of 17

2 2 2 The distribution of the Reynolds shear stress u0v0/U , u0w0/U , and v0w0/U is similar to the Reynolds normal stress and the TKE, which is U-shaped, except that the U-shape distribution of 2 v0w0/U is not obvious. The propeller disk is also affected by the U-shaped stern and it has a U-shaped shear layer. However, due to the influence of the draft under the ballast condition, the underwater shape of the U-shaped stern is different from that under the design condition; thus, resulting in a different U-shaped shear layer. The shear layer under the ballast condition covers the bilge vortex and hub cap vortex in a wider range. The Reynolds shear stress u v /U2 is negative, where ∂u/∂Y is p 0 0 decreasing and it has a maximum value of u v 0.05U. The Reynolds shear stress u w /U2 has 0 0 ≈ − p 0 0 similar behavior; however, it is correlated with ∂u/∂Z and it attains a maximum of u0w0 0.01U. 2 ≈ The vertical Reynolds shear stress v0w0/U represents the turbulent shear characteristics of the y-z section of the propeller disk. Compared with that under the design condition, the U-shaped shear layer in the outer region of the two vortices is slightly weaker than that under the design condition. In addition, the turbulent region near the free surface also has a great influence on the Reynolds shear stress in the range of Z/R = 0.75–1.0R. Figure 10 depicts the root mean square velocity profiles at Z/R = 0.00 under the ballast condition, which is a quantitative expression of the distribution of the turbulent fluctuating velocity field. The results show that the dimensionless turbulent fluctuating velocity first decreases, then it increases, and it finally decreases from Y/R = 0 to Y/R = 1. Figure 11 displays the turbulence statistics at Z/R = 0.00 2 2 2 under the ballast condition. The dimensionless Reynolds normal stress u0u0/U , v0v0/U , w0w0/U , and TKE/U2 show an overall decreasing distribution from Y/R = 0 to Y/R = 1. The dimensionless 2 2 Reynolds shear stress u0v0/U is increasing. The dimensionless Reynolds shear stress u0w0/U and 2 v0w0/U show a relatively flat trend of change. As illustrated in Figure 11 on the right for the close 2 2 up of the turbulence statistics profiles, the dimensionless Reynolds normal stress u0u0/U , v0v0/U , 2 2 J. Mar. Sci. Eng.TKE 2020/U, ,7 and, x FOR the ReynoldsPEER REVIEW shear stress u0v0/U have been acutely fluctuating. 11 of 16

Figure 10. The root mean square velocity proﬁles at Z/R = 0.00 of the propeller region under the Figure 10. The root mean square velocity profiles at Z/R = 0.00 of the propeller region under the ballast ballast condition. condition.

Figure 11. Turbulence statistics profiles at Z/R = 0.00 under the ballast condition: Left. The global distribution of the turbulence statistics profiles; Right. Close up of the turbulence statistics profiles).

Figure 12 presents the turbulence statistics profiles at different radius in the propeller region under the ballast condition and the interval is 10°. Among them, the 0° circumferential angle is directly above the propeller disk surface and the circumferential angle gradually increases clockwise to 180°. The red area is near the free surface wave core turbulent area, the yellow area is the boundary area affected by the core area, and the green frame is the local high-turbulence area in the propeller disk. In the circumferential distribution area with a radius r/R = 0.4, the TKE and Reynolds stress at the end of the shaft show a relatively stable distribution trend with increasing circumferential angle, and the peak distribution appears in the corresponding area at the end of the shaft. In the region with r/R = 0.6–0.8, which is located in the low-kinetic-energy region, the TKE and Reynolds stress are relatively stable as the circumferential angle increases. In the circumferential distribution area with r/R = 1.0–1.4, the curves of the points taken from the circumferential distribution are in the average kinetic energy change layer region. With the increase in the circumferential angle, the dimensionless TKE and the Reynolds normal stress present a relatively violent fluctuation distribution as a whole. In addition, the near free surface wave core turbulent region in red and the boundary region affected by the yellow core have a large amplitude and a violent fluctuation. The shape of the stern has typical longitudinal curvature and cross-sectional curvature (see Figure 1). The anisotropic turbulent flow of the wake has a strong correlation with the longitudinal curvature and cross-sectional curvature of the stern as well as the flow gathering and dispersion around the stern. The local geometry in the stern

J. Mar. Sci. Eng. 2020, 8, 397 12 of 17 J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 12 of 17

J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 12 of 16 Figure 11. Turbulence statistics profilesprofiles atat ZZ/R/R == 0.000.00 underunder thethe ballastballast condition:condition: LeftLeft.. The global region of the ship is different at different span wise and circumferential angle positions, and thus the distribution of the turbulence statistics profiles;profiles; RightRight.. CloseClose up of the turbulence statisticsstatistics profiles).profiles). average velocity and turbulence characteristic are different at various span wise and circumferential angle positions.FigureFigure 1212 Figurepresents presents 6 theand the turbulence Figuresturbulence 10 statistics– 12statistics in the profiles articleprofiles at illustra di atff erentdifferentte radiusthe averageradius in the in propellervelocity the propeller and region turbulence region under characteristictheunder ballast the condition ballast parameters condition and at the different interval and the is span interval 10◦. and Among is circumferential 10°. them, Among the 0 ◦ them,circumferential angle the positions. 0° circumferential angle The is directlytrend angle is above closely is relatedthedirectly propeller to theabove flow disk the nature, surfacepropeller and and disk it the should surface circumferential be and pointed the circumferential angle out graduallythat it also angle increases includes gradually clockwise measurement increases to 180 ◦clockwiseuncertainty. The red factors.areato 180°. isBut near The the thered main freearea factor surface is near is wavethe freeflow core surface factor, turbulent waveand area,uncertainty core the turbulent yellow has area, arealittle isthe effect the yellow boundary on the area overall is area the a distributionboundaryffected by characteristicsthearea core affected area, and by and thedistribution the core green area, frame trends. and isthe the green local frame high-turbulence is the local area high in-turbulence the propeller area disk. in the propeller disk. In the circumferential distribution area with a radius r/R = 0.4, the TKE and Reynolds stress at the end of the shaft show a relatively stable distribution trend with increasing circumferential angle, and the peak distribution appears in the corresponding area at the end of the shaft. In the region with r/R = 0.6–0.8, which is located in the low-kinetic-energy region, the TKE and Reynolds stress are relatively stable as the circumferential angle increases. In the circumferential distribution area with r/R = 1.0–1.4, the curves of the points taken from the circumferential distribution are in the average kinetic energy change layer region. With the increase in the circumferential angle, the dimensionless TKE and the Reynolds normal stress present a relatively violent fluctuation distribution as a whole. In addition, the near free surface wave core turbulent region in red and the boundary region affected by the yellow core have a large amplitude and a violent fluctuation. The shape of the stern has typical longitudinal curvature and cross-sectional curvature (see Figure 1). The anisotropic turbulent flow of the wake has a strong correlation with the longitudinal curvature and cross-sectional curvature of the stern as well as the flow gathering and dispersion around the stern. The local geometry in the stern region of the ship is different at different span wise and circumferential angle positions, and thus the average velocity and turbulence characteristic are different at various span wise and circumferential angle positions. Figure 6 and Figures 10–12 in the article illustrate the average velocity and turbulence characteristic parameters at different span and circumferential angle positions. The trend is closely related to the flow nature, and it should be pointed out that it also includes measurement uncertainty factors. But the main factor is the flow factor, and uncertainty has little effect on the overall distribution characteristics and distribution trends.

FigureFigure 12. 12. TurbulenceTurbulence statistics statistics proﬁles profiles at each at each radius radius in the propellerin the propeller region under region the ballast under condition. the ballast condition.

3.5. Vorticity Information Figure 13 shows the vorticity information under the ballast and design conditions. Figure 13a wv shows the contour maps of the vorticity (x) which can be computed as: Vorticity( x ) = , x yz and the positive value of the vorticity (x) indicates that the rotation direction of the vortex is counter clockwise and the negative value is clockwise. Figure 13b shows the contour maps of the swirling strength, and the swirl strength is defined as the imaginary part of the complex eigen-value of the velocity gradient tensor [22], and the square of the imaginary part can be computed as: 2 1 WVVWVW         ci -4 -  -   , Figure 13c shows the contour maps of the planar 2nd 2 y  z  y  z    y  z  invariant Q, the 2nd invariant Q of the velocity gradient matrix [23,24], which can be used to identify 2 WVVWVW    1     vortices, can be defined as Q -  -  +  . Figure 13d shows the contour y  z  y  z 2   y  z  maps of the lambda 2 criterion, detailed definitions and calculation formulas of the lambda 2 criterion can be found in the literature [25], The vorticity (x), swirling strength, 2nd invariant Q and lambda 2

J. Mar. Sci. Eng. 2020, 8, 397 13 of 17

In the circumferential distribution area with a radius r/R = 0.4, the TKE and Reynolds stress at the end of the shaft show a relatively stable distribution trend with increasing circumferential angle, and the peak distribution appears in the corresponding area at the end of the shaft. In the region with r/R = 0.6–0.8, which is located in the low-kinetic-energy region, the TKE and Reynolds stress are relatively stable as the circumferential angle increases. In the circumferential distribution area with r/R = 1.0–1.4, the curves of the points taken from the circumferential distribution are in the average kinetic energy change layer region. With the increase in the circumferential angle, the dimensionless TKE and the Reynolds normal stress present a relatively violent fluctuation distribution as a whole. In addition, the near free surface wave core turbulent region in red and the boundary region affected by the yellow core have a large amplitude and a violent fluctuation. The shape of the stern has typical longitudinal curvature and cross-sectional curvature (see Figure1). The anisotropic turbulent flow of the wake has a strong correlation with the longitudinal curvature and cross-sectional curvature of the stern as well as the flow gathering and dispersion around the stern. The local geometry in the stern region of the ship is different at different span wise and circumferential angle positions, and thus the average velocity and turbulence characteristic are different at various span wise and circumferential angle positions. Figures6 and 10–12 in the article illustrate the average velocity and turbulence characteristic parameters at different span and circumferential angle positions. The trend is closely related to the flow nature, and it should be pointed out that it also includes measurement uncertainty factors. But the main factor is the flow factor, and uncertainty has little effect on the overall distribution characteristics and distribution trends.

3.5. Vorticity Information Figure 13 shows the vorticity information under the ballast and design conditions. Figure 13a ∂w ∂v shows the contour maps of the vorticity (x) which can be computed as: Vorticity(x) = ωx = , ∂y − ∂z and the positive value of the vorticity (x) indicates that the rotation direction of the vortex is counter clockwise and the negative value is clockwise. Figure 13b shows the contour maps of the swirling strength, and the swirl strength is defined as the imaginary part of the complex eigen-value of the velocity gradient tensor [21,22], and the square of the imaginary part can be computed r 2 as:λ = 1 4 ∂W ∂V ∂V ∂W ∂V + ∂W , Figure 13c shows the contour maps of the planar 2nd ci 2 − ∂y ∂z − ∂y ∂z − ∂y ∂z invariant Q, the 2nd invariant Q of the velocity gradient matrix [23,24], which can be used to identify 2 vortices, can be defined as Q = ∂W ∂V ∂V ∂W 1 ∂V + ∂W . Figure 13d shows the contour maps − ∂y ∂z − ∂y ∂z − 2 ∂y ∂z of the lambda 2 criterion, detailed definitions and calculation formulas of the lambda 2 criterion can be found in the literature [24], The vorticity (x), swirling strength, 2nd invariant Q and lambda 2 conducts dimensionless processing by multiply R/U and R2/U2, respectively, where R is the propeller and U is the towing speed. The local minimum negative value of lambda-2 can be used to identify the vortex core, while the positive value indicates the flow field, and the shear flow may be displayed but not rotated. The local maximum positive value of the swirling strength and planar 2nd invariant Q can be used to identify the vortex core. A strong bilge vortex (Bv) will be produced when the water flows through the U-shaped stern. There is a propeller hub cap vortex (Hcv) under the bilge vortex, which is opposite to the rotation direction of the bilge vortex, and the hub cap vortex is close to the longitudinal section of the ship. Under the ballast condition, the ship has a trim by the stern, and the draft is smaller than that under the design condition. Influenced by the position of the ship, the bilge vortex formed by the U-shaped stern moves down, which is almost in parallel with the hub cap vortex. The position of the hub cap vortex changes a little bit; however, the rotating range is larger than that under the design condition. The lambda 2-criterion, the vorticity, and planar 2nd invariant Q all recognize the bilge vortices and the propeller hub cap vortex. The positive value of the vorticity (x) indicates that the local vortex rotates anticlockwise and vice versa. The bilge vortex generated by the bilge area in the middle of the hull rotates anticlockwise on the port side and on the starboard side. In addition, it is transmitted to the propeller disk, and it is located within the velocity contour range from u/U = 0.3 to J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 13 of 16 conducts dimensionless processing by multiply R/U and R2/U2, respectively, where R is the propeller and U is the towing speed. The local minimum negative value of lambda-2 can be used to identify the vortex core, while the positive value indicates the flow field, and the shear flow may be displayed but not rotated. The local maximum positive value of the swirling strength and planar 2nd invariant Q can be used to identify the vortex core. A strong bilge vortex (Bv) will be produced when the water flows through the U-shaped stern. There is a propeller hub cap vortex (Hcv) under the bilge vortex, which is opposite to the rotation direction of the bilge vortex, and the hub cap vortex is close to the longitudinal section of the ship. Under the ballast condition, the ship has a trim by the stern, and the draft is smaller than that under the design condition. Influenced by the position of the ship, the bilge vortex formed by the U-shaped stern moves down, which is almost in parallel with the hub cap vortex. The position of the hub cap vortex changes a little bit; however, the rotating range is larger than that under the design condition. The lambda 2-criterion, the vorticity, and planar 2nd invariant Q all recognize the bilge vortices and the propeller hub cap vortex. The positive value of the vorticity (x) indicates that the local vortex rotates anticlockwise and vice versa. The bilge vortex generated by the bilgeJ. area Mar. Sci.in Eng.the 2020middle, 8, 397 of the hull rotates anticlockwise on the port side and on the starboard14 of 17 side. In addition, it is transmitted to the propeller disk, and it is located within the velocity contour range from u/U = 0.3 to u/U = 0.5 (Figure 3). The hub cap vortex has the opposite rotation direction to the u/U = 0.5 (Figure3). The hub cap vortex has the opposite rotation direction to the bilge vortex and it is bilge vortex and it is generated by the hub at the installation position of the propeller on the shaft of generated by the hub at the installation position of the propeller on the shaft of the hull. In addition, the hull.it is transmitted In addition, to theit is propeller transmitted disk area to the along propeller with the wake.disk area The hubalong cap with vortex the is locatedwake. The along hub the cap vortexvelocity is located contour along range the of velocityu/U < 0.3 contour (Figure 3range). According of u/U to< 0.3 the (Fig contourure 3) of. theAccording vorticity andto the swirling contour of the vorticitystrength, and the bilge swirling vortex strength, on the propeller the bilge disk vortex appears on “round-like”the propeller under disk the appears design "round condition-like" and under is the design“ear-like” condition under the and ballast is "ear condition.-like" under the ballast condition.

FigureFigure 13. Time 13. Time-averaged-averaged vorticity vorticity information information in in the propellerpropeller region: regionLeft: Left. The. The ballast ballast condition; condition; RightRight. The. design The design condition condition.. (a) Contour maps of the vorticity; (b) Contour maps of the swirling strength; (c) Contour maps of the planar 2nd invariant; (d) Contour maps of the lambda 2 criterion.

In order to analyze the vortex evolution characteristics of Panama bulk carriers under ballast condition, CFD numerical simulation analysis was carried out using the formed calculation strategy [2,25]. Figure 14 is the vortex evolution of Panama bulk carrier under the ballast condition based on CFD. As shown in Figure 14b, the CFD numerical results agree well with the PIV measurement results, which verifies the accuracy of the numerical calculation. Figure 14a,c is the global evolution and close-up view of the vortex structure, under the ballast condition. In addition to the bilge vortex and hubcap vortex similar to the design one, there are also vortex structures near the free surface caused by fluctuations of free surface. In addition, the evolution trajectories of the bilge vortex and the hub cap vortex in ballast are different from the design ones. J. Mar. Sci. Eng. 2020, 7, x FOR PEER REVIEW 14 of 16

In order to analyze the vortex evolution characteristics of Panama bulk carriers under ballast condition, CFD numerical simulation analysis was carried out using the formed calculation strategy [2,26]. Figure 14 is the vortex evolution of Panama bulk carrier under the ballast condition based on CFD. As shown in Figure 14b, the CFD numerical results agree well with the PIV measurement results, which verifies the accuracy of the numerical calculation. Figure 14a,c is the global evolution and close-up view of the vortex structure, under the ballast condition. In addition to the bilge vortex and hubcap vortex similar to the design one, there are also vortex structures near the free surface caused by J.fluctuations Mar. Sci. Eng. of2020 free, 8, 397surface. In addition, the evolution trajectories of the bilge vortex and the hub15 of 17 cap vortex in ballast are different from the design ones.

Figure 14. The vortex evolution of Panama bulk carrier under the ballast condition based on CFD: (a) Figure 14. The vortex evolution of Panama bulk carrier under the ballast condition based on CFD: The global evolution of the vortex structure; (b). Comparison chart of numerical and PIV result; (c) (a) The global evolution of the vortex structure; (b). Comparison chart of numerical and PIV result; Close-up of the vortex structure. (c) Close-up of the vortex structure.

4. Conclusions4. Conclusions TheThe wake wake fields fields of of the the test test model model under the the ballast ballast condition condition is isexperimentally experimentally investigated investigated based basedon theon authors’the authors’ previous previous research, research, which which is under is under the design the design condition. condition. The wake The wake field datafield includes data includestime-averaged, time-averaged, turbulence turbulence characteristics, characteristics, and vorticity and vorticity information. information. When When comparing comparin theg resultsthe resultsunder under the designthe design condition, condition, it can it becan concluded be concluded as follows. as follows. (1) (1)The The time time-averaged-averaged velocity velocity contours contours indicate indicate that that the the ballast ballast and and design design conditions conditions have have distinctdistinct hook hook-like-like axial axial velocity velocity contours, contours, but but they they appear appear at at different different positions. The distinctdistinct hook-likehook- likeaxial axial velocity velocity contours contours appear appear at at the the position position of ofu /u/UU = =0.3 0.3 and and the the vertex vertex of of the the hook-like hook-like contour contour is at is atthe the position position of Zof/R Z/R= 0.6 = under0.6 under the designthe design condition. condition. Under Under the ballast the ballast condition, condition, the distinct the distinct hook-like hookaxial-like velocity axial velocity contours contours appear atap thepear position at the position of u/U = of0.4 u/U and = the 0.4 vertex and the of thevertex hook-like of the hook contour-like is at contourthe position is at the of position Z/R = 0.25. of Z/R The = 0.25. shape The of shape the bilge of the vortex bilge is vortex “round-like” is "round under-like" the under design the condition,design conditionbut it has, but an it “ear-like” has an "ear shape-like" under shape the ballastunder condition.the ballast Thecondition. big diff erenceThe big under differ theence ballast under condition the ballastis that condition the top ofis that the propellerthe top of disk the areapropeller is near disk the area free is surface. near the The free propeller surface. disk The areapropeller produces disk the areanear produces free surface the near layer free that surface is disturbed layer bythat the is freedisturbed surface by waveform. the free surface The vortex waveform. foci of the The twin vortex vortex focistructure of the twin and vortexthe two structure saddle points and the (S1 two and saddleS2) of the points propeller (S1 and disk S2) are of captured the propeller under disk the ballast are capturedand design under conditions. the ballast and Under design the designconditions. condition, Under thethe saddledesign pointcondition, S1 at theY/R saddle= 0.1 point, Z/R =S1 at0.05 , ± − Y/RS2 = at±0.1, Y/ RZ/R= =0.4, −0.05, and S2 Z at/R Y/R= =0.65. ±0.4, Under and Z/R the = ballast−0.65. Under condition, the ballast the saddle condition, point the S1 atsaddle Y/R = point0.025, ± − ± S1 Zat/ RY/R= =0.25, ±0.025 S2, atZ/R Y /=R =−0.25,0.35, S2 andat Y/R=/R= ±0.35,0.9. and /R= −0.9. − ± − (2) (2) In In comparison comparison to to the the design design condition, condition, the the areas areas with with strong strong fluctuation fluctuation are mainly are distributed mainly distributedat the end at ofthe the end shaft, of the the shaft, range the of range r/R = of0.9–1.5, r/R = 0.9 and-1.5, the and end the of end the of hub the cap. hub Incap. addition, In addition, the top theedge top edge of the of propeller the propeller is tangent is tangent to the to free the surface free surface under theunder ballast the ballast condition, condition, and a strong and a rootstrong mean square velocity fluctuation region is formed near the free surface. The TKE, Reynolds stresses, and the hub cap vortex are all affected by the turbulent velocity fluctuation region under the ballast condition. The interval is wider and the U-shaped distribution is concentrated in the kinetic energy change layer under the ballast condition. The shear layer under the ballast condition covers the bilge vortex and the hub cap vortex in a wider range. (3) A strong bilge vortex (Bv) is produced when the water flows through the U-shaped stern under the design and ballast conditions. Under the ballast condition, the bilge vortex formed by the U-shaped stern moves down, which is almost in parallel with the hub cap vortex. The position of the hub cap J. Mar. Sci. Eng. 2020, 8, 397 16 of 17 vortex changes a little bit; however, the rotating range is larger than that under the design condition. According to the contour of the vorticity and the swirling strength, the bilge vortex on the propeller disk appears “round” under the design condition and it resembles an “ear” under the ballast condition.

Author Contributions: The following statements should be used: “T.W., R.D. and W.L. conceived and designed the experiments; T.W. and W.L. performed the experiments; D.J. and Y.L. analyzed the data; D.J. and W.L. contributed reagents/materials/analysis tools; T.W. and W.L. wrote the paper”. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (Grant No. 51679053), the Guangdong Basic and Applied Basic Research Foundation (2019A1515110721), the China Postdoctoral Science Foundation (Grant No. 2019M663243) and the Fundamental Research Funds for the Central Universities (NO.20 lgpy52). And The APC was funded by the Fundamental Research Funds for the Central Universities (NO.20 lgpy52). Acknowledgments: We would like to thank Editage (www.editage.cn) for English language editing. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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