Effect of Skeg on the Wave Drift Force and Directional Stability of a Barge

Journal of Marine Science and Engineering

Article Eﬀect of Skeg on the Wave Drift Force and Directional Stability of a Barge Using Computational Fluid Dynamics

Chunki Lee 1 and Sangmin Lee 2,* 1 Division of Navigation Science, Korea Maritime and Ocean University, Busan 49112, Korea; [email protected] 2 Division of Marine Industry Transportation Science and Technology, Kunsan National University, Jeonbuk 54150, Korea * Correspondence: [email protected]; Tel.: +82-63-469-1814

Received: 28 September 2020; Accepted: 24 October 2020; Published: 27 October 2020

Abstract: To accurately estimate the navigation safety of a barge in still water and waves, computational fluid dynamics (CFD) was used to calculate the hydrodynamic force on the barge and simulate the flow field to comprehensively study the effect of skeg for wave drift force and directional stability of the barge. With respect to the wave drift force, the steady surge force and steady sway force are found to decrease due to the skeg effect in both short and long wavelength regions. The overall steady yaw moment tends to be reduced by the skeg. As the drift angle increased, the sway force and yaw moment also increased. The changing trend in the sway force varied with the angular velocity, depending on the installation of the skeg, and it was noted that the yaw moment value significantly increased because of the skeg. Owing to the effect of the skeg installed on the barge, the yaw damping lever became larger, while the sway damping lever became smaller regardless of waves. It was confirmed that the directional stability was improved both in still water and head waves.

Keywords: skeg eﬀect; wave drift force; directional stability; barge; CFD

1. Introduction In the stern towing method where a tug pulls a barge from the stern side, the directional stability of the barge is one of the most important factors in the operation of the tug-barge system. While navigating tug and barge, the yaw motion of the barge should be reduced to ensure a safe towing operation because it limits the steering performance of the tug. Therefore, it is necessary to accurately estimate and evaluate the directional stability of the barge to ensure its safe navigation. Studies have been conducted to prove that installing a skeg is an effective method to improve the stability of the barge in still water. Inoue et al. [1] conducted various experiments on different types of skegs; they reported that the directional stability of the barge is greatly affected by the type of skeg and the installation conditions. Experiments showed that directional stability could be improved by installing the knuckle-type skeg to a bare-hull barge, having a yaw amplitude that was 10 times the vessel width [2]. The study on the directional stability of the barge in still water was conducted by experimental [3] and numerical methods [4], and the characteristics of the maneuvering performance of the barge were analyzed. Previous studies have focused on the optimal arrangement and shapes of skeg for a specific barge in still water. To estimate the maneuvering performance of the vessel, it is necessary to determine the derivatives of the hydrodynamic force that includes the angular velocity and drift angle as variables. Most of the existing studies have been able to calculate this using the empirical equations of the vessel or obtaining the coefficients through experiments. Numerical computation based on the potential theory

J. Mar. Sci. Eng. 2020, 8, 844; doi:10.3390/jmse8110844 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 844 2 of 19 has limitations when it comes to precisely calculating the effects of interference and viscous effects between the attachment and hull, which affect maneuvering performance. Furthermore, in case the ship’s speed is not high enough to affect the wave-breaking phenomenon, or because of the efficiency of numerical calculation, the ship’s maneuverability is evaluated without taking into account the influence of free surface [5,6]. However, an actual barge is mostly operated in waves on the sea. The sway force and yaw moment in waves are different from those in still water, and the variation is significant enough to affect the maneuvering performance of the barge. Therefore, it is necessary to estimate accurately the variation in the sway force and yaw moment because of the waves. Studies on the maneuvering performance of the vessel in waves have been conducted experimentally and theoretically. Hirano et al. [7], Ueno et al. [8], and Yasukawa [9,10] conducted an experimental analysis on a free-running model test for the turning circle and zigzag test. Recently, a free-running test was performed to analyze the effects of wave lengths, heights, and directions on the ship turning trajectories [11]. Tello Ruiz et al. [12] conducted a study to investigate the effect of waves on the ship’s maneuverability in a finite water depth mainly experimentally and to some extent numerically. Skejic and Faltinsen [13] analyzed ship maneuvering in waves using a unified seakeeping and maneuvering two-time scale model based on the strip theory. Seo and Kim [14] solved the ship maneuvering problem in waves using the time-domain ship motion program based on the Rankine panel method. Zhang et al. [15] calculated the maneuvering motion using maneuvering modeling group (MMG) model and solved the seakeeping problem based on the time domain Rankine panel method. The mean second-order wave loads were included in the MMG model as the wave drift forces. Chillcce and el Moctar [16] presented a numerical method for maneuvering in waves based on nonlinear equations of motion derived with respect to an inertial coordinate system. The still water maneuvering forces were calculated based on hydrodynamic coefficients obtained by RANS planar motion mechanism (PMM) tests, while wave induced forces were calculated using the Rankine boundary element method. The ships used for the maneuverability evaluation in the waves mentioned above were self-propulsive ones, and most of the numerical calculations were based on potential theory. Few studies have been conducted on towed barge, of which directional stability is an essential requirement in waves. Nonlinear factors must be considered to estimate the sway force and yaw moment acting on a barge when a skeg is attached to the stern side of the barge, or when there is the influence of external forces such as waves. The computational fluid dynamics (CFD) technique can precisely estimate the hydrodynamic force on a vessel in waves containing nonlinear and viscous effects, and it can be used very effectively to evaluate maneuvering performance by analyzing the behavior of the vessel and obtaining the related maneuvering derivatives. In this study, numerical simulations were performed using CFD to calculate the hydrodynamic forces acting on the barge in still water and head waves for its safe navigation in sea areas. Firstly, the steady wave drift force, which was a nonlinear second order force, was analyzed in consideration of the skeg effect. Next, the static drift motion test and pure yaw motion test were performed to obtain the maneuvering hydrodynamic derivatives. Based on these results, the directional stability of the barge in still water and waves considering the skeg effect were evaluated.

2. Numerical Calculation

2.1. Computational Method STAR-CCM +, a commercial CFD program based on the ﬁnite volume method, was used in this study to perform the numerical simulation. The continuity and momentum equations for incompressible ﬂow analysis are as follows: ∂(ρui) = 0 (1) ∂xi J. Mar. Sci. Eng. 2020, 8, 844 3 of 19

∂(ρui) ∂ ∂p ∂τij + ρuiuj + ρu0iu0 j = + (2) ∂t ∂xj −∂xi ∂xj where τij represents the mean viscous stress tensor components: ! ∂ui ∂uj τij = µ + (3) ∂xj ∂xi where ρ is the density, ui denotes the averaged Cartesian components of the velocity vector, ρu0iu0 j denotes the Reynolds stresses, and µ is the dynamic viscosity. Temporal discretization of the unsteady term used a first-order temporal scheme, and a second-order scheme was applied to discretize convection terms. The SIMPLE method was used in the velocity-pressure coupling. In the turbulence model, the realizable k-ε model was used and all y+ wall treatment was applied. The k-ε model is used extensively in industrial applications and has advantages in terms of the cost of CPU computation time [17]. For the stability and efficiency of the numerical computation, Sung and Park [18] utilized the realizable k-ε model to conduct a dynamic test on a vessel. In implicit unsteady simulations, the time step is determined by the flow properties rather than the Courant number [17]. At least 100 time steps per encounter period were used for the simulation in waves as recommended by ITTC [19]. A minimum cell size of 0.0125% of wavelength and 0.0625% of wave height were used in the horizontal and vertical directions, respectively. A free surface was solved by applying the volume of fluid (VOF) method. The dynamic fluid body interaction (DFBI) function was used to simulate the hull motion [20], with the barge free in heave and pitch. The fifth-order Stokes wave was selected for the numerical simulation. A regular wave with a wave amplitude ζa = 0.5 m in the head direction was analyzed, and the directional stability characteristics were analyzed at wavelengths of 0.5L, 1.0L, and 1.5L (where L denotes the length between perpendiculars). These characteristics were then compared to the characteristics in still water. To confirm the effect of the skeg on the directional stability, numerical computations were performed when the skeg was installed and when the skeg was not installed (bare hull).

2.2. Computational Condition A 91.5 m barge, which was the size of an actual vessel, was used as a subject vessel when conducting the numerical simulations to analyze the characteristics of hydrodynamic forces acting on the barge in still water and waves. The barge used in the numerical calculation is a round-type model frequently accessed in practice. Table1 shows the main speciﬁcations of the barge, and the basic shapes of the barge are shown in Figure1. The speed of the barge U was set to 7 knots (Froude number Fn = 0.12), and the numerical computations were conducted in still water and waves at Reynolds 8 number Rn = 3.218 10 . × Table 1. Main speciﬁcations of the barge.

Description Full Scale Length between perpendiculars, L (m) 91.5 Breadth, B (m) 27.5 Draft, T (m) 4.0 Volume, ∆ (m3) 10,846.0 Block coeﬃcient, CB 0.923 Length of skeg, Ls (m) 12.810 Breadth of skeg, Bs (m) 1.375 Height of skeg, Hs (m) 3.766 J. Mar. Sci. Eng. 2020, 8, 844 4 of 19

Figure 1. Shapes of barge model.

Figure2 shows the computational domain and coordinate system that was used in this study. It is a Cartesian coordinate system in which the forward direction is the +x-axis, the port direction of the barge is the +y-axis, and the opposite direction of gravity is the +z-axis. X, Y, and N are the surge force, sway force, and yaw moment, respectively. u, v, and r denote the surge, sway, and yaw velocities, respectively. The numerical simulation was conducted by selecting a computational domain 1.5L, 2.5L, 2.0L, 3.0T, and 10.0T away from the fore perpendicular of the barge, aft perpendicular of the barge, center line to the transverse direction, water surface to the vertical direction, and water surface to the seabed, respectively. The inlet, top, bottom, and side boundaries were selected as velocity inlets. The hull surface was selected as a no-slip boundary condition. Numerical computation was performed using the VOF wave damping functionality of the software with a damping length of 1.0L to avoid wave reﬂection into the domain.

Figure 2. Coordinate system (left) and computational domain (right).

The space and hull surface grids were created using the trimmed mesh and prism layer, respectively. To implement the flow more precisely in the head, stern, free surface, and divergent regions, the grid size was created to be denser than the base size, using volumetric control. The number of grids used in the numerical simulation was about 1.5 million, and the mean y+ values for λ/L = 1.0 at β = 10◦ was 181 and 208 at r0 = 0.3. y+ is the dimensionless wall distance, λ is the wavelength, β is the drift angle, and r0 denotes the nondimensionalized angular velocity by (rL/U). Drift angle β is the J. Mar. Sci. Eng. 2020, 8, 844 5 of 19 difference between heading and actual course. The detailed grid convergence uncertainty is described in Section 3.1. In this study, the drift motion test, which is a static test, and the pure yaw motion test, which is a dynamic test, were simulated based on the planar motion mechanism (PMM) to obtain the derivative value of the hydrodynamic force for evaluating directional stability. Velocity-dependent derivatives of the maneuvering hydrodynamic force,Yv = ∂Y∂v and Nv = ∂N∂v, were calculated from the drift motion test, and rotary derivatives Yr = ∂Y∂r and Nr = ∂N∂r were calculated from the pure yaw motion test. The drift angle β was set to 5◦, 10◦, and 15◦. The direction of the water flow into the barge was set to the port side of the barge. The nondimensionalized angular velocity r0 was set to 0.1, 0.3, and 0.5. In the pure yaw motion test, the sway force (Y) and yaw moment (N) at the position where the yaw angle is at the minimum, i.e., the lateral displacement is at its maximum, are computed. The PMM test procedure and analysis method were performed by referring to the study by Kim et al. [21]. Table2 shows the test conditions for PMM simulation.

Table 2. Test conditions for planar motion mechanism (PMM) simulation.

Static Drift Simulation Fn Drift Angle, β Wavelength, λ Wave Amplitude, ζa Hydrodynamic Derivatives (-)(deg)(L)(m)(-)

0.12 5, 10, 15 0.5, 1.0, 1.5 0.5 Yv, Nv Pure Yaw Simulation Fn Angular Lateral Wave- Wave Hydrodynamic Velocity, r0 Displacement length, λ Amplitude, ζa Derivatives (-)(-)(m)(L)(m)(-) 0.12 0.1, 0.3, 0.5 100.0 0.5, 1.0, 1.5 0.5 Yr, Nr

3. Results and Discussion

3.1. Mesh Sensitivity Analysis This study used the grid convergence index (GCI) method based on the Richardson extrapolation to estimate the grid convergence uncertainty of the numerical simulations. Park et al. [22], Tezdogan et al. [17], and Demirel et al. [23] have applied the GCI method to evaluate numerical uncertainty. The veriﬁcation process using the GCI method was followed according to the approach described by Celik et al. [24]. The numerical convergence ratio R is calculated as follows: ε R = 21 (4) ε32

Here, it is obtained by ε = φ φ and ε = φ φ , each of which represents the difference in 21 2 − 1 32 3 − 2 the solution value between medium-fine and coarse-medium, respectively. φ1, φ2, or φ3 each represent the solution calculated for the fine, medium, or coarse mesh. The convergence ratio calculated as described is classified as follows: (i) monotonic convergence (0 < R < 1), (ii) oscillatory convergence ( 1 < R < 0), (iii) oscillatory − divergence (R < 1), and (iv) monotonic divergence (1 < R). − The apparent order p of the method can be obtained as follows:

ln(ε32/ε21) p = (5) ln(r32) where r32 is the grid reﬁnement factor r32 = hcoarse/hmedium, i.e., √2 used in this study. J. Mar. Sci. Eng. 2020, 8, 844 6 of 19

The extrapolated values are calculated as shown in Equation (6):

p r φ φ 32 32 2 − 3 φext = p (6) r 1 32 − In addition, the approximate relative error and extrapolated relative error are obtained by the following equations, respectively:

32 φ2 φ3 ea = − (7) φ2 32 φext φ2 e32 = − (8) ext 32 φext Last, the medium-grid convergence index is calculated as follows:

32 32 1.25ea GCI = p (9) med r 1 32 − The number of meshes for the grid convergence study is shown in Table3 when the skeg was not installed. The sway force (Y) and yaw moment (N) were calculated using the number of coarse, medium, and ﬁne grids, and the results are shown in Table4. As shown in the calculation results, the numerical uncertainty for the medium mesh using the GCI method were estimated to be 3.12% and 9.37%, respectively. According to the results of the grid convergence uncertainty study, it was concluded that eﬃcient numerical simulation is possible using the medium mesh. Therefore, the medium mesh was selected as the base size in this study.

Table 3. Mesh numbers for each mesh conﬁguration.

Mesh Conﬁguration Total No. of Cells Coarse 894,673 Medium 1,549,265 Fine 3,542,882

Table 4. Grid convergence study for Y and N at β = 5◦ , λ/L = 1.0.

Y (N) N (N m) · φ 78,122.5 3,206,399.7 1 − − φ 79,280.9 3,244,955.9 2 − − φ 82,076.5 3,290,778.9 3 − − R 0.414 0.841 p 2.542 0.498 32 φext 77,302.9 3,001,826.2 32 − − ea 3.53% 1.41% 32 eext 2.56% 8.09% GCI32 med 3.12% 9.37%

3.2. Analysis of Steady Wave Drift Force

Table5 shows the linear surge force X0, sway force Y0, and yaw moment N0 of the barge with and without skeg in still water at the drift angle of 15◦. X0, Y0, and N0 became dimensionless by the barge speed (U), barge length (L), and draft (T) as shown below.

X X0 = (10) 1 2 2 ρU LT J. Mar. Sci. Eng. 2020, 8, 844 7 of 19

Y Y0 = (11) 1 2 2 ρU LT N N0 = (12) 1 2 2 2 ρU L T

As indicated in Table5, X0 and Y0 values increase due to the installation of an addition such as skeg, and N0 value decreases because of the skeg eﬀect.

Table 5. Comparison of X0 , Y0, and N0 at β = 15◦ in still water.

Without Skeg With Skeg

X0 0.07768 0.08098 Y 0.13355 0.13915 0 − − N 0.03873 0.02914 0 − −

However, the steady wave drift force, which is the second order force in waves, shows a different tendency. Figure3 shows calculation results for cases with and without skeg under conditions with the drift angle of 15◦, wave height 1 m, and λ/L = 0.5, 0.7, 1.0, 1.2, and 1.5 in head seas to compare the steady wave drift force. The steady wave drift force is obtained by subtracting the results in still water from the results in waves. x The steady surge force FD also means added resistance caused by waves. This wave drift force shows the largest value around λ/L = 1.2, and this distribution shows a similar trend as published in other research results [25,26]. In a short wavelength of λ/L = 0.5 and a long wavelength of λ/L = 1.5, the steady surge force is found to decrease due to the skeg effect. The steady sway force shows the largest value at a short wavelength of λ/L = 0.5 regardless of the installation of skeg. This seems to be due to the reflected wave effect in short waves as discussed in other studies [25,27]. It can be seen that skeg effect plays the role of reducing the steady sway x force in both short and long wavelength areas in the same manner as FD. It can be presumed that skeg is effective in suppressing drift motions on longitudinal and lateral directions. The changes in hydrodynamic force by skeg are very small at λ/L = 1.0 and 1.2. The overall steady yaw moment tends to be reduced by the skeg. The highest reduction amount = z = occurs at λ/L 0.7, and the skeg rather increases ND at a short wavelength of λ/L 0.5. It is shown that the skeg is effective in reducing not only yaw moment N0, which is the first order force, but also the z steady yaw moment ND value, which is a nonlinear fluid force. It is assumed that this effect reduces sway damping lever to improve directional stability of the barge.

Figure 3. Cont. J. Mar. Sci. Eng. 2020, 8, 844 8 of 19

Figure 3. (a) Steady surge force; (b) steady sway force; (c) steady yaw moment according to wavelength

at β = 15◦.

3.3. Static Drift Simulation In this section, numerical simulation using CFD was carried out to achieve the maneuvering derivatives of the barge in drift motion, and the results are as follows: Figures4 and5 show the time history for the sway force and yaw moment acting on the barge in still water and waves (λ/L = 1.0). It can be seen that the sway force value is slightly increased by the skeg, and the yaw moment value is decreased. Ten times of the encounter wave period was used to average the sway force and yaw moment.

Figure 4. Time history of sway force (left) and yaw moment (right) for still water and λ/L = 1.0 without skeg.

Figure 5. Time history of sway force (left) and yaw moment (right) for still water and λ/L = 1.0 with skeg.

The nondimensionalized sway force (Y0) and yaw moment (N0) at each drift angle are shown in Figures6 and7. As shown in the ﬁgure, as the drift angle increases, the negative value of the sway J. Mar. Sci. Eng. 2020, 8, 844 9 of 19 force and yaw moment increases as well. The greatest sway force occurs in the short wavelength, and the smallest occurs in still water, regardless of the installation of skeg. Although the sway force does not change signiﬁcantly with the installation of the skeg, the yaw moment shows a relatively large change because of the installation of the skeg. In other words, it was found that the negative value of the yaw moment became smaller with the installation of the skeg.

Figure 6. Sway force in drift motion for barge without skeg (left) and with skeg (right).

Figure 7. Yaw moment in drift motion for barge without skeg (left) and with skeg (right).

A transverse velocity component v is calculated as follows:

v = Usin β (13) where U is the ship’s velocity and β is the drift angle. Based on the sway velocity v, hydrodynamic derivatives Yv0 and Nv0 are produced. It can be seen that Yv0 is not much affected by the skeg, and Nv0 is smaller. It is assumed that the sway damping lever (lv0 = Nv0 /Yv0 ) value is decreased, which can contribute to improving the stability of the barge. Figure8 shows wave patterns around a barge without a skeg in still water and in waves with a wavelength of λ/L = 1.0. It can be seen that unbalanced flow fields are generated on the port and starboard sides with a drift angle of 10◦. J. Mar. Sci. Eng. 2020, 8, 844 10 of 19

Figure 8. Wave patterns at β = 10◦:(a) still water; and (b) λ/L = 1.0.

Figures9 and 10 show the axial velocity, pressure distribution, and streamline of the barge in the cases where the skeg was not installed and where they were installed, respectively. The 3D axial velocity is shown in the range of 0.32L to 0.58L from the midship. The pressure distribution on the − − hull surface by subtracting the hydrostatic pressure component is shown in Figure 10. The pressure side is the port side of the barge that directly receives the ﬂuid ﬂow, and the suction side is the starboard side, which is on the opposite side of the port.

Figure 9. Cont. J. Mar. Sci. Eng. 2020, 8, 844 11 of 19

Figure 9. Axial velocity for barge at β = 10◦, λ/L = 1.0 : (a) without skeg; and (b) with skeg.

Figure 10. Streamline and pressure distribution for barge at β = 10◦, λ/L = 1.0 :(a) without skeg; and (b) with skeg.

As shown in Figures9 and 10, it can be seen that the axial velocity of the flow is rapid owing to the shedding of the aft-body side vortex at the end of the raked stern on the pressure side when the skeg is not installed. However, when the skeg is installed, it can be seen that the axial velocity is accelerating both outside the suction side skeg and inside the pressure side skeg. Analyzing the raked stern of the suction side (Figure 10), the streamline that has passed rapidly over the main bottom end is gradually slowing down towards the aft peak. This change in velocity causes the pressure distribution on the raked stern to form like a staircase. This distribution shows that the pressure increases from the pressure side skeg to the suction side skeg. It is estimated that the pressure distribution is changed by the flow velocity, and the modified pressure acts on the suction side skeg to reduce the negative yaw moment value. Such a change in the flow field seems to improve the directional stability by directing the stern to the starboard side. J. Mar. Sci. Eng. 2020, 8, 844 12 of 19

3.4. Pure Yaw Simulation Figure 11 shows the PMM test results for pure yaw simulation. In order to make it easier to understand the various wave conditions, the wave patterns are shown in Figure 11 under each condition of still water, and λ/L = 0.5, 1.0, 1.5. The locations of the barge during the simulation 1 1 3 were T, T + 4 T, T + 2 T, and T + 4 T. Here, T is the cycle of the PMM test. Figure 11 shows the result of a case where r0 = 0.3. As indicated in the ﬁgure, it is estimated that captive sinusoidal motion is well realized.

Figure 10. Cont. J. Mar. Sci. Eng. 2020, 8, 844 13 of 19

1 Figure 11. Wave patterns around the barge at r0 = 0.3:(a) T, still water; (b) T + 4 T, λ/L = 0.5 and; 1 3 (c) T + 2 T, λ/L = 1.0; (d) T + 4 T, λ/L = 1.5.

The nondimensionalized sway force (Y0) and yaw moment (N0) at each angular velocity of the barge are shown in Figures 12 and 13.

Figure 12. Sway force in pure yaw motion for barge without skeg (left) and with skeg (right).

Figure 13. Yaw moment in pure yaw motion for barge without skeg (left) and with skeg (right).

As shown in the figure, as the angular velocity increases, the negative value of the sway force gradually increases when the skeg is not installed, while the positive value of the sway force increases when the skeg is installed. When the skeg is not installed, the sway force acting on the bow part is larger than the force acting on the stern part during turning with the angular velocity r, so the negative value is indicated. However, when the skeg is installed, the sway force acting on the stern part is larger than the force acting on the bow part due to the skeg effect, and it seems that the sign becomes the positive value with the opposite sign. In this study, the simulation was conducted on a barge with round bow shape. If the bow shape is a box type, different results are expected. In the future, a comparative study will be conducted on barges with different bow shapes. Furthermore, the sway force showed a J. Mar. Sci. Eng. 2020, 8, 844 14 of 19 slight difference, depending on the wavelength. That is, when the skeg was not installed, it showed the smallest value in the still water and the short wavelength region and the largest value in the long wavelength region. However, when the skeg was installed, it showed the largest sway force in the still water and the smallest sway force in the long wavelength region. It was found that the negative value in the yaw moment was significantly increased when the skeg was installed, compared to the case when the skeg was not installed. In addition, in the case of the pure yaw simulation, a rapid change in fluid flow appeared from the ship’s bow and stern direction during changing yaw angle. The changing flow pattern results in unstable prediction [21]. It is noted that a study on CFD analysis with high mesh resolution in low time step needs to be carried out. Figures 14 and 15 show the axial velocity, pressure distribution, and streamline for the barge when the skeg was not installed and when they were installed, respectively. The pressure side is the port side of the barge that directly receives the fluid flow, and the suction side is the starboard side. As shown in Figure 14, when the skeg is not installed, the axial velocity shows a slightly faster distribution at the end of the suction side. However, when the skeg is installed, it is observed that the axial velocity is very fast inside the suction side skeg. The aft-body bilge flows on the pressure side are observed to be shedding without moving into the raked stern. On the other hand, the aft-body bilge flows on the suction side can be seen moving over the raked stern and moving out to the aft peak. Because of such a change in the flow velocity, the pressure distribution is modified and finally the negative yaw moment value increases.

Figure 14. Axial velocity for barge at r0 = 0.3, λ/L = 1.0 : (a) without skeg; and (b) with skeg. J. Mar. Sci. Eng. 2020, 8, 844 15 of 19

Figure 15. Streamline and pressure distribution for barge at r0 = 0.3, λ/L = 1.0 :(a) without skeg; and (b) with skeg.

3.5. Analysis of Directional Stability The equation for evaluating the directional stability index that is derived from the maneuvering motion equation is shown in (14) [28,29].

N N C = 0r 0v (14) Y r m − Y v 0 − 0 0 where m0 denotes the nondimensionalized mass. In this equation, the ﬁrst term indicates the yaw damping lever value (lr0), and the second term indicates the sway damping lever (lv0 ). If the yaw damping lever term increases and the sway damping lever term decreases, then it indicates that the towing stability is improved. The hydrodynamic derivatives in this study were obtained using the least square method. Figure 16 shows the derivative values of the maneuvering hydrodynamic forces obtained from the simulations. The blue and red colors indicate the calculation results for when the skeg was not installed and when they were installed, respectively. Yv0 was slightly aﬀected by the skeg, but Nv0 decreased both in still water and in waves because of the skeg. This change in yaw moment directly caused the sway damping lever to decrease (Figure 17), resulting in the increase of the directional stability index. Because of the skeg, the sway damping lever value is reduced by 23.6% in still water, 18.2% in the short wavelength region (λ/L = 0.5), 21.6% in the region where the wavelength is equal to the length of the barge (λ/L = 1.0), and 22.8% in the long wavelength region (λ/L = 1.5). J. Mar. Sci. Eng. 2020, 8, x FOR PEER REVIEW 16 of 19

J. Mar.J. Mar. Sci. Sci.Eng. Eng. 20202020, 8,, x8, FOR 844 PEER REVIEW 16 of 19 16 of 19

-0.45 -0.050 still water(w/o skeg) still water(w/o skeg) still water(with skeg) still water(with skeg) -0.50 -0.075 -0.45 wave(w/o skeg) -0.050 wave(w/o skeg) stillwave(with water(w/o skeg) skeg) stillwave(with water(w/o skeg) skeg) -0.100 -0.55 still water(with skeg) still water(with skeg) still water(with skeg) -0.50 -0.075

wave(w/o skeg) still water(w/o skeg) wave(w/o skeg) v

v wave(with skeg)

-0.60 wave(with skeg) -0.125 N'

Y' -0.100 -0.55 still water(with skeg) stillstill water(with water(w/o skeg)skeg) still water(w/o skeg) -0.150 -0.65 v v

-0.60 -0.125 N' Y' still water(with skeg) still water(w/o skeg) -0.70 -0.175 -0.65 -0.150

-0.75 -0.200 -0.70 0.5 1.0 1.5 -0.175 0.5 1.0 1.5 l/L l/L -0.75 -0.200 0.5 1.0 1.5 0.5 1.0 1.5 /L /L

0.4 0.2 still water(w/o skeg) still water(w/o skeg) still water(with skeg) still water(with skeg) 0.1 0.4 wave(w/o skeg) 0.2 wave(w/o skeg) 0.2 stillwave(with water(w/o skeg) skeg) stillwave(with water(w/o skeg) skeg) still water(with skeg) 0.0 still water(with skeg) still water(with skeg) 0.1 wave(w/o skeg) wave(w/o skeg) still water(w/o skeg)

0.2 r r 0.0 wave(with skeg) -0.1 wave(with skeg)

Y' N' 0.0 stillstill water(with water(w/o skeg)skeg) still water(w/o skeg) r r -0.2 0.0 -0.1 still water(with skeg) Y' -0.2 N' still water(w/o skeg) -0.3 -0.2 still water(with skeg) -0.2 -0.4 -0.4 0.5 1.0 1.5 -0.3 0.5 1.0 1.5 l/L l/L -0.4 -0.4 0.5 1.0 1.5 0.5 1.0 1.5 /L Figure 16. Hydrodynamic derivatives. /L

Figure 16. Hydrodynamic derivatives. Figure 16. Hydrodynamic derivatives.

0.9 0.35 still water(w/o skeg) still water(w/o skeg) still water(with skeg) still water(with skeg) 0.9 wave(w/o skeg) 0.350.30 wave(w/o skeg) 0.6 stillwave(with water(w/o skeg) skeg) stillwave(with water(w/o skeg) skeg) still water(with skeg) still water(with skeg) still water(with skeg) wave(w/o skeg) 0.300.25 wave(w/o skeg) 0.6

v still water(w/o skeg) 0.3 wave(with skeg) wave(with skeg)

l' l' still water(with skeg) 0.250.20 still water(w/o skeg) r

v still water(w/o skeg) 0.3 still water(with skeg) l' 0.0 l' 0.200.15 still water(w/o skeg) still water(with skeg) 0.0 -0.3 0.150.10 0.5 1.0 1.5 0.5 1.0 1.5 l/L l/L -0.3 0.10 0.5 1.0 1.5 0.5 1.0 1.5 Figure 17. Yaw damping lever (left) and sway damping lever (right). Figure 17/L . Yaw damping lever (left) and sway damping lever (/Lright).

Next, for the hydrodynamic derivatives calculated from the pure yaw motion, Yr0 shows positive Figure 17. Yaw damping lever (left) and sway damping lever (right). ′ andNext, negative for values, the hydrodynamic respectively, because derivatives of the eff ect calculated of the skeg. fromNr0 shows the pure all negative yaw values,motion, but 푌 the푟 shows ′ positivevalue increasedand negative owing values, to the installation respectively, of the because skeg. It of can the be effect seen that of the yawskeg. damping 푁푟 shows lever all value negative values,isNext, greatly but for increasedthe the value hydrodynamic because increased of the owing skegderivatives e fftoect the in still installationcalculated water and offrom waves. the the skeg. pure It canyaw be motion, seen that the shows yaw positivedampingFinally, and lever negative Figure value 18 is values, showsgreatly therespective increased directionally, becau stabilitybecausese of index ofthe the skeg in stilleffect effect water of inthe and still skeg. waves, water where andshows waves. it is all found negative values,thatFinally, thebut directionalthe Figure value 18 stability increasedshows the of theowingdirectional barge to isthe generallystability installa index improvedtion ofin stillthe when waterskeg. the Itand skeg can waves, isbe installed. seen where that Thus, it the is found yaw the effect of skeg on the directional stability is very large, regardless of the influence of the wave. It was dampingthat the directional lever value stability is greatly of increasedthe barge becauseis generally of the improved skeg effect when in stillthe waterskeg is and installed. waves. Thus, the found that the directional stability somewhat decreased in waves compared to still water, regardless of effectFinally, of skeg Figure on the 18 directional shows the stability directional is very stability large, index regardless in still ofwater the influenceand waves, of where the wave. it is foundIt was whether the skeg was installed or not. In particular, as the directional stability is relatively worse in the thatfound the that directional the directional stability stability of the bargesomewhat is generally decreased improved in waves when compared the skeg to isstill installed. water, regardless Thus, the effectof whether of skeg the on skeg the directionalwas installed stability or not. is In very partic large,ular, regardless as the directional of the influence stability of is therelatively wave. Itworse was foundin the thatlong the wavelength directional region stability than somewhat the short decreasedwavelength in region, waves comparedthe operation to still of the water, barge regardless requires ofspecial whether attention the skeg in this was condition. installed or not. In particular, as the directional stability is relatively worse in the long wavelength region than the short wavelength region, the operation of the barge requires special attention in this condition.

J. Mar. Sci. Eng. 2020, 8, 844 17 of 19 long wavelength region than the short wavelength region, the operation of the barge requires special attention in this condition.

Figure 18. Directional stability index of barge without skeg and with skeg.

4. Conclusions This study estimated the navigational safety of a barge in still water and head waves using CFD. The hydrodynamic force acting on the barge was calculated for each drift angle and angular velocity, and the flow field was simulated to comprehensively study the effect of skeg on the barge’s directional stability and wave drift force. To summarize the contents related to the wave drift force, the steady sway force is largest in the short wavelength region, and the steady yaw moment does not show a clear tendency. The steady surge force and steady sway force are found to decrease due to the skeg effect in both short and long wavelength areas. The overall steady yaw moment tends to be reduced by the skeg. As the drift angle increased, the sway force and yaw moment also increased. The changing trend in the sway force according to the angular velocity was different, depending on the installation of the skeg, and it was noted that the yaw moment value significantly increased because of the skeg. Because of the effect of the skeg installed on the barge, the yaw damping lever became larger, while the sway damping lever became smaller. Therefore, it was confirmed that the directional stability was improved both in still water and waves. By understanding the effect of skeg on barge in still water and waves and considering their effect on the wave drift force and directional stability characteristics of barge, it is anticipated that the study will contribute to increasing the safety of barges by establishing comprehensive countermeasures and providing an effective response system to those who deal with actual marine navigation. We plan to verify the more common skeg effect by conducting the research on other shapes of barges.

Author Contributions: Conceptualization, C.L. and S.L.; software, S.L.; validation, C.L.; writing—original draft preparation, S.L.; writing—reviewing and editing, C.L.; supervision, S.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A3B03030193). Conﬂicts of Interest: The authors declare no conﬂict of interest. J. Mar. Sci. Eng. 2020, 8, 844 18 of 19

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