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ISSN : 1229-3431(Print)
ISSN : 2287-3341(Online)

Journal of the Korean Society of Marine Environment and Safety Vol.30 No.6 pp.658-667
DOI : https://doi.org/10.7837/kosomes.2024.30.6.658

A Numerical Study of the Effect of Skegs on the Course-keeping Ability of a Catamaran

Michael^*, Jun-Taek Lim^**, Gyoung-Woo Lee^***, Kwang-Cheol Seo^***†

^*Graduate School of Mokpo National Maritime University, Mokpo 58628, Korea
^**Graduate School of Mokpo National Maritime University, Mokpo 58628, Korea
^***Department of Naval Architecture & Ocean Engineering, Mokpo National Maritime University, Mokpo 58628, Korea

* First Author : michael.navarc@gmail.com

^† Corresponding Author : kcseo@mmu.ac.kr, 061-240-7303

Received July 9, 2024 Review October 18, 2024 Accepted October 28, 2024

Abstract

Maneuverability is a critical hydrodynamic performance metrics of a ship. This study numerically evaluated the course-keeping ability of a 24 m shallow draft catamaran, equipped with skegs installedon it, was numerically to assess whether the skegsthese appendages improved its maneuverability. Computational fluid dynamics simulations were conducted to simulate virtual captive model tests. Pure-sway and pure-yaw motions were assessed as part of harmonic tests conducted in the planar motion mechanism were at the design speed. An empirical formula was proposed for to evaluate the course-keeping ability (as measured quantified by a criterion, termed the C index) was suggested and assess the maneuverability. The results demonstrated that the initial condition of the bare hull had exhibited a relatively poor C index, and the installation of Skeg 1 did not address resolve this issue, judging by the negative value of C. On the other handConversely, Skegs 2, 3, and 4 yielded positive C index values, could address this problem, as judged by thesthey yielded, which indicatinged that the hull could yaw or change its heading more easilyeffectively, as opposed to drifting in sway motions.

Key Words : Skeg , Computational fluid dynamics (CFD) , Hydrodynamic coefficient , Course-keeping ability , Planar motion mechanism (PMM)

수치해석을 이용한 쌍동선의 직진성능에 대한 Skeg의 영향에 관한 연구

마이클^*, 임준택^**, 이경우^***, 서광철^***†

^*국립목포해양대학교 대학원
^**국립목포해양대학교 대학원
^***국립목포해양대학교 조선해양공학과 교수

초록

선박의 조종성능은 조선해양공학 분야에서 매우 중요한 유체역학적 성능 중 하나입니다. 본 연구에서는 저수심에서 운용되는 24m급 쌍동선을 대상으로 직진성능 분석에 대한 연구를 진행하였습니다. Skeg의 설치를 통해 조종성능을 개선할 수 있는지 여부를 확인하기 위해, 전산유체역학(CFD) 수치해석 시뮬레이션을 활용하여 가상의 포획 모형테스트를 진행하였습니다. 시뮬레이션은 PMM(Planar Motion Mechanism) 조화 시험 중 Pure-sway 및 Pure-Sway motion, 2가지 시험을 진행하였으며, 선박의 선회성능은 직진성능 지수(C)의 경험식을 통해 확인하였습니다, 결론적으로, Skeg가 없는 기존 선체는 직진성능이 상대적으로 저조하며, Skeg 1을 적용해도 부정적인 C 지수 값을 보여 항로유지 능력이 개선되지 않았다. 하지만 Skeg 2, 3 또는 4를 적용하였을 때, 긍정적인 C 지수 값을 보여 항로유지 능력이 개선 되는 것을 보이며, 이는 Sway motion의 드리프트 경향에 비해 선체가 Yaw 또는 방향을 더 바꾸기 쉽다는 것을 나타낸다.

키워드 : 스케그 , 수치해석 , 유체역학계수 , 직진성능 , PMM(Planar Motion Mechanism)

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1. Introduction

Electric vessels have become popular in recent years, designed and targeted for mostly inland vessels. These vessels are designed to operate in relatively shallow draft, like rivers, or lakes. Due to the limitations of water depth, the vessels are mostly designed with a relatively more flat-type bottom, which allows better clearance of the bottom section and the riverbed. On the negative side, flat-type bottom hull form can be considered as poor in maneuverability and yaw stability due to low restoring moment when performing turning ability.

Maneuverability is one of the most important hydrodynamic performances of a ship. Yaw checking and course-keeping abilities are some of the maneuvering criteria that need to be checked due to the requirements from the International Maritime Organization (IMO) through the Maritime Safety Committee (MSC) rules of 137(76), which summarized the standards for ship maneuverability (MSC, 2002a;MSC, 2002b).

Skegs are well-known to improve the course-keeping ability because the presence of skegs tackles main issues in the hydrodynamic interactions of the hull and water by increasing the lateral resistance area, changing the steering force, and drift angle, which affects the tendency of the hull to yaw, or sway.

In general, since the design stage, free-running model tests (FRMT) are commonly used to check the maneuverability of the ship. However, the complexity of hull and rudder interactions may lead to many design configurations that require producing a lot of ship models for the experiments. Therefore, a numerical simulation can be an alternative to model tests.

Skegs are mostly applied on a towed barge which commonly not equipped with any control or maneuvering system. The effect of skeg is studied on the still water and head waves by numerical simulations of Computational Fluid Dynamics (CFD) to evaluate the directional stability (Lee and Lee, 2020). The results show that an improvement of course-keeping ability criteria has been made after the skeg is installed. A total of 12 designs of skeg with various span and chord configurations are evaluated (Piaggio et al., 2020a) numerically to see the interaction and distribution of the forces acting on hull and skeg (Piaggio et al., 2020b). A typical skeg is also installed into different type of hull-forms to evaluate the directional stability by computing the sway force and yaw moment during drift and steady yaw (Im et al., 2015). It has proved that the skeg contributed to the improvement of course-keeping ability by changing the shape of a vortex generated in aft section, which influenced by the skeg position and number.

Virtual simulation of oblique towing test and planar motion mechanism are performed using CFD to derive the hydrodynamic coefficients of maneuvering. The coefficients are then utilized to model the turning circle ability and zig-zag maneuver which experimentally verified (Hajivand and Mousavizadegan, 2015). In completed design of hulls and rudder, static drift, static rudder, rudder-drift, pure-sway, pure yaw, yaw-drift, and yaw-rudder as the part of static and harmonic test of virtual captive tests are performed using CFD (Liu et al., 2018). The simulations are conducted to get the hydrodynamic derivatives which later are used to model the turning circle and zig-zag test. Abkowitz model along with virtual captive model tests are used to mathematically model a maneuvering performance. The existing results of hydrodynamic coefficients are then compared to see the reliability of the virtual captive model tests. Besides, the course-keeping ability criteria is also calculated using the empirical formula with the hydrodynamic coefficients as the input parameters (Khan et al., 2022). As a replacement for conventional towing tank facilities, a CFD solve was utilized to perform the static drift, prue sway, pure yaw, and combined drift-yaw tests. The results shows that the hydrodynamic forces and moments are agree well with the experimental results (Zhu et al., 2022).

The absence of rudder and propeller designs is a challenge to assess the maneuverability of the hull-form and skeg’s impact. Therefore, a course-keeping ability criterion notated as C, is used to compare the course-keeping ability of the bare hull and skegs-installed conditions. The course-keeping ability criteria (Rawson and Tupper, 2001) is a term that is used to see the tendency of the hull-form to sway or yaw. Notated as positive and negative values, the hull is considered to have good criteria if the C value is positive, which indicates that the ship is easier to yaw than sway, making it better to change and keep its heading angle. If the C value is negative, the tendency of the vessel to sway is bigger, indicating that the ship is more likely to sway or drift away and hard to keep its heading back (Traintafyllou, 2003).

Reviewing recent studies that has been published, this study aims to utilize the CFD to see the skegs effect to the course-keeping ability of a catamaran. A virtual planar motion mechanism as the part of virtual captive tests will be performed on with and without skegs condition. Mathematical model of hydrodynamic forces and moments approach is used which resulting in hydrodynamic coefficients. The hydrodynamic coefficients regressed from the model will be used for calculation of course-keeping ability criteria, which will be compared and evaluated to see the maneuverability performance of the catamaran with and without skegs.

2. Methodology

In general, Computational Fluid Dynamics (CFD) software will be used in this study, where the setup is designed to simulate the real conditions of captive model tests (ITTC, 2021).

2.1 Target Vessel

A 24 m-long shallow draft catamaran became the target vessel of this study. With a relatively flat-bottom design, the vessel’s maneuverability and course-keeping ability become a concern since the design stage. The hull-form is designed with a relatively rounded streamline at the stern section, followed by a modified axe bow in the bow region. With a design speed of 10 knots, the vessel has a speed Froude Number of 0.336. The bare hull’s principal dimensions and linesplan are detailed and summarized in Table 1, and Fig. 1 as follows.

2.2 Skegs Configurations

The study is conducted to see the performance of different types of skegs towards the maneuvering performance of course-keeping ability, to get the smallest size as possible considering the additional resistance due to skegs installment. Four different skegs are considered and installed in the centerline of each demi-hull, making in total of 5 study cases in this study. The skeg installment and the hydrostatics parameters are mentioned in Table 2 and Fig. 2.

3. Mathematical Models of Maneuvering

Mathematical models are widely used in the ship’s design process to simulate the motions of maneuverability of a vessel in terms of hydrodynamic coefficients. These hydrodynamic coefficients of maneuver are widely used in Guidance, Navigation, and Control (GNC) algorithm design, especially in the case of autonomous ships.

Typically, mathematical models of ship maneuvering motion can be classified into two main approaches (MSC, 2002b). The first approach is called the response model, which expresses the direct relationship between the input and output control of the ship. For this approach, the input may vary depending on the case, such as rudder deflection, or wave disturbance, to see the output, which is the ship’s heading. The second approach is called the hydrodynamic force model, which is based on the forces that are applied in some limited motions. By knowing the hydrodynamic forces applied in some motion, the hydrodynamic coefficients relative to the motion can be derived and considered.

Due to the limitations of the design stage, namely the absence of rudders and propellers, the first approach is not a proper choice since direct measurement of input and output cannot be performed. Therefore, the hydrodynamic force model is used to derive the considered coefficients related to the course-keeping ability.

3.1 Fluid Force and Moments

Several models are categorized concerning of the complexity of the mathematical models, such as whole-ship models (Abkowitz, 1964), and the Maneuvering Modeling Group (MMG) model (Yasukawa and Yoshimura, 2014) are widely used. Due to the absence of rudders and other control devices, the Abkowitz model mentioned in equations (1) and (2) is used in this study.

\begin{array}{l} Y = Y_{e} + Y_{\dot{υ}} \dot{υ} + Y_{\dot{r}} \dot{r} + Y_{u} u + Y_{υ} υ + Y_{r} r + Y_{δ} δ \\ + Y_{u u} u^{2} + Y_{δ u} δ u + Y_{υ u} υ u + Y_{r u} r u \\ + Y_{υ υ υ} υ^{3} + Y_{r r r} r^{3} + Y_{δ δ δ} δ^{3} + Y_{υ r δ} υ r δ \\ + Y_{υ u u} υ u^{2} + Y_{r u u} r u^{2} + Y_{δ u u} δ u^{2} \\ + Y_{r r δ} r^{2} δ + Y_{r r υ} r^{2} υ + Y_{υ υ r} υ^{2} r + Y_{υ υ δ} υ^{2} δ + Y_{δ δ r} δ^{2} r + Y_{δ δ υ} δ^{2} υ \end{array}

(1)

\begin{array}{l} N = N_{e} + N_{\dot{υ}} \dot{υ} + N_{\dot{r}} \dot{r} + N_{u} u + N_{υ} υ + N_{r} r + N_{δ} δ \\ + N_{u u} u^{2} + N_{δ u} δ u + N_{υ u} υ u + N_{r u} r u \\ + N_{υ υ υ} υ^{3} + N_{r r r} r^{3} + N_{δ δ δ} δ^{3} + N_{υ r δ} υ r δ \\ + N_{υ u u} υ u^{2} + N_{r u u} r u^{2} + N_{δ u u} δ u^{2} \\ + N_{r r δ} r^{2} δ + N_{r r υ} r^{2} υ + N_{υ υ r} υ^{2} r + N_{υ υ δ} υ^{2} δ + N_{δ δ r} δ^{2} r + N_{δ δ υ} δ^{2} υ \end{array}

(2)

Where Y symbolizes the sway force, and N symbolizes the yaw moment, followed by subscripts u, v, and r symbolize surge, sway, and yaw velocity, and δ for the rudder deflection. The dot superscript (·) symbolizes the acceleration. The coupled motion is also considered in this force model, as example subscript uv is the coupled motion due to the surge (u) and sway velocity (v) in the ship’s motion (Triantafyllou, 2003).

3.2 Course-keeping Ability Criteria (C)

Course-keeping ability criteria (C) can be calculated based on the hydrodynamic coefficients derived from the hydrodynamic force model in the ship’s motion by following equation (Triantafyllou, 2003).

C = {Y^{'}}_{υ} ({N^{'}}_{r} - m^{'} {x^{'}}_{G} U^{'}) + {N^{'}}_{υ} (m^{'} U^{'} - {Y^{'}}_{r})

(3)

Where the superscript (‘) symbolizes the non-dimensional factor of each hydrodynamic coefficient. Y’v, Y’r, N’v, N’r are the hydrodynamic coefficients that need to be obtained from the hydrodynamic force model, which shows the relation of sway force (F), and yaw moment (N) to the sway velocity (v) and yaw velocity or rate (r). Followed by m’, x’G, and U’ which show the non-dimensional of the ship’s mass, Longitudinal Center of Gravity (LCG), and forward surge speed. This criterion is used to see the tendency of the hull to sway or yaw, notated by the positive or negative value of the C coefficient.

3.3 Virtual Captive Model Tests

The following coefficients can be derived from the virtual captive model tests, or Planar Motion Mechanism (PMM). In most cases, a carriage is set into a towing tank to enable different kinds of motions, which later the forces and moments acted on the hull are experimentally measured and the hydrodynamic coefficients can be derived from the mathematical models, based on the motions limitations.

Harmonic tests can be performed, which require more complex equipment in the towing tank, usually equipped with the PMM equipment, to perform the limited motions of pure sway, pure yaw, pure roll, combined sway and yaw, yaw with drift, yaw with rudder deflection, or other different combinations.

However, the tests mentioned above required high computational costs due to the facility and equipment required for the tests, along with the ship’s model production and time limitations. Therefore, an alternative of Computational Fluid Dynamics (CFD) commercial software is used to simulate the PMM tests. The numerical simulations are set to imitate the real conditions of PMM tests in the real situation of a towing tank facility.

4. Motions Modeling

The CFD simulations are used to model the motions related to the coefficients that need to be obtained in the equation (3). It can be clearly seen there is no coupled motion required to be performed, a harmonic test of pure-sway and pure-yaw motions are the only tests that are required for this study. Pure-sway motion needs to be performed to obtain the sway-related coefficients, namely Y’v and N’v, and pure-yaw motion needs to be performed to obtain the yaw-related coefficients, Y’r and N’r.

4.1 Pure Sway Motions

Hydrodynamic forces’ relation towards the coefficients mentioned in equations (1) and (2) can be easily simplified due to the limited motions of pure sway. Illustrated in Fig. 3, with the absence of rudder and deflection, gives a zero value of δ related coefficient, followed by no coupled motions in pure-sway, giving a zero value of coupled coefficients such as vu, ru, and 2^nd order of v²r, r²v, etc. Lastly, due to the absence of the ship’s heading in this pure-sway motion, all related coefficient of yaw (r) is omitted.

The motions are set into some amplitude along with the frequency, and finally resulting in simplified equations of sway force-related coefficients as follows:

Y = Y_{\dot{υ}} \dot{υ} + Y_{υ} + Y_{υ υ υ} υ^{3}

(4)

N = N_{\dot{υ}} \dot{υ} + N_{υ} + N_{υ υ υ} υ^{3}

(5)

Equations (4) and (5) summarized the sway force acted on the hull during the pure-sway motions, where the coefficient of Yv can be obtained, along with the Nv from the yaw moment.

4.2 Pure Yaw Motions

Just like the pure-sway motions, in this motion, the motions are limited to 1 degree of freedom which is yaw. The remaining coefficients of Yr and Nr can be obtained from this motion test. The yaw motions are set into some frequency and amplitude where the bank effect and interference of the tank walls can be minimized, as per the guidance of ITTC (ITTC, 2021;ITTC, 2014). The same phenomenon happens in limited yaw motions, where there are no angle deflections and coupled motions.

Y = Y_{\dot{r}} \dot{r} + Y_{r} + Y_{r r r} r^{3}

(6)

N = N_{\dot{r}} \dot{r} + N_{r} + N_{r r r} r^{3}

(7)

Unlike the pure sway motions, now, the sway velocity of v, can be omitted, and the heading angle of yaw velocity or rate r can be calculated.

As illustrated in Fig. 4, the simplified equation is obtained, and the yaw-related coefficients can be calculated.

4.3 Least Square Regression

The force and moments-related coefficients mentioned in equations (4), (5), (6), and (7) can be rewritten into the matrix form to calculate the course-keeping ability criteria mentioned in equations (3). Where a simple function and expansion of matrix notation (Rawlings et al., 1998) can be defined as follows:

\begin{array}{l} [\begin{matrix} Y_{\dot{υ}} \\ Y_{υ} \\ Y_{υ υ υ} \end{matrix}] = {({[\dot{υ} υ υ^{3}]}^{T} [\dot{υ} υ υ^{3}])}^{- 1} [\dot{υ} υ υ^{3}] [Y] \\ [\begin{matrix} N_{\dot{υ}} \\ N_{υ} \\ N_{υ υ υ} \end{matrix}] = {({[\dot{υ} υ υ^{3}]}^{T} [\dot{υ} υ υ^{3}])}^{- 1} [\dot{υ} υ υ^{3}] [N] \end{array}

(8)

\begin{array}{l} [\begin{matrix} Y_{\dot{r}} \\ Y_{r} \\ Y_{r r r} \end{matrix}] = {({[\dot{υ} υ υ^{3}]}^{T} [\dot{υ} υ υ^{3}])}^{- 1} [\dot{υ} υ υ^{3}] [Y] \\ [\begin{matrix} N_{\dot{r}} \\ N_{r} \\ N_{r r r} \end{matrix}] = {({[\dot{r} r r^{3}]}^{T} [\dot{r} r r^{3}])}^{- 1} [\dot{r} r r^{3}] [N] \end{array}

(9)

With CFD simulations, all parameters on the right side of equations (8) and (9) can be obtained. Sway acceleration of $\dot{υ}$ , sway velocity of v, can be obtained in the pure-sway motions simulation, along with the sway force of Y, and yaw moment N. While the remaining parameters related to yaw can be obtained from the pure-yaw motions simulation.

Lastly, the results of hydrodynamic coefficients on the left side of equations (8) and (9) can be regressed and calculated. A matrix inverse computation code is created using the means of MATLAB R2013b software.

All parameters and hydrodynamic forces, moments, and coefficients are to be non-dimensionalized to calculate the course-keeping ability criteria. Therefore, the following equations are used as the non-dimensional factor of each parameter considered (Traintafyllou, 2003).

\begin{array}{l} m^{'} = \frac{m}{(0.5 ρ L^{3})} {x^{'}}_{G} = \frac{x_{G}}{L} u^{'} = \frac{U}{U} \\ Y^{'} = \frac{Y}{(0.5 ρ U^{2} L^{2})} N^{'} = \frac{N}{(0.5 ρ U^{2} L^{3})} \\ υ^{'} = \frac{\dot{υ}}{(U^{2} / L)} υ^{'} = \frac{υ}{U} {\dot{r}}^{'} = \frac{\dot{r}}{(U^{2} / L^{2})} r = \frac{r}{(u / L)} \end{array}

(10)

5. Numerical Simulations

With the development of computational analysis, a numerical simulation could be performed in order to simulate the PMM test. The idea is to reduce the computational cost and time for the experiment test. In this study, commercial Computational Fluid Dynamics (CFD) software namely Simcenter Star-CCM+ code (Simcenter, 2020) is used.

To achieve the objectives of this study, several numerical simulations are set to get the data of hydrodynamic forces, moments, and velocity-related parameters. In total, 10 simulations are to be performed, which are summarized in Table 3.

5.1 Methodology and Setup

Reynolds-Averaged Navier Stokes is chosen as the governing equation in this study (Rosenfeld and Kwak, 1991). Followed by the Volume of Fluid (VOF) as the free surface model, with zero velocity of flat waves. The time model is implicitly unsteady, with k-ε as the turbulence model. Multiphase of water and air is set to simulate the real conditions of tests, followed by Dynamic Fluid Body Interaction (DFBI) to simulate the ship’s motion. The Planar Motion Mechanism (PMM) type of motion is set into motion modeling, where the surge velocity, pure sway, pure-yaw frequency, and amplitude mentioned in Table 3 are set. In total, 5 degrees of freedom are activated, since the roll is disabled.

5.2 Computational Domain and Boundaries

The computational domain is designed as following the ITTC guidance of CFD application, and captive model test, considering the frequency of motion and the speed of the vessel. The details of boundary conditions and dimensions are notated in terms of the ship’s overall length (L), as summarized in Table 4.

The simulations are planned to be performed at least 2.5 cycles of motion, where the first 0.5 cycles are considered as unsteady turns. Therefore, with calculation, 13L is chosen as the length of the domain, which is capable of providing the space required for at least 3 cycles (60s).

5.3 Mesh Configurations

The static tests of the captive model test are used to reduce computational time. The vessel in bare hull condition is towed in a straight line without any heading deflections. The surge force (X) is evaluated to see which grid size is suitable for the simulations. In this study, a y⁺ value of y⁺>40 is used since the simulations are convergence within this range, as shown in Fig. 5.

Grid Convergence Index (GCI) is used to evaluate the grid independence test (Richardson, 1927). The results are summarized in Table 5 and presented in Fig. 6, in which the medium size of the mesh is chosen due to a reasonable difference of 0.98% to the zero-grid spacing, and a GCI index of 0.24%. Resulting in 6.9-7.1 million cells for bare hull and skegs installed configurations. The computational time takes about 5-6 days to finish for each case. Volume refinement is applied in the free surface area to capture the wave produced along the maneuvering motions, as illustrated in Fig. 7.

6. Results and Discussions

As mentioned in Table 3, 10 simulations are performed for each hull configuration. The results are illustrated in Fig. 8 for pure-sway motions and Fig. 9 for pure-yaw motions, and summarized in Table 6. It can be seen that the wave height is slightly higher for the skeg installed configuration, in the pure sway motions.

Finally, with a simple code using MATLAB R2013b commercial software, the hydrodynamic coefficients can be derived. Considering the previously calculated non-dimensional parameters such as m’, x’G, and u’, the course-keeping ability criteria of C can be calculated by the formula mentioned in equations (3).

The results are shown in Table 7, where the bare hull condition in fact has an unstable course-keeping performance by having a C index of –1.474. This can be known as the C index is negative, which can be concluded that the hull tends to sway rather than yaw. This phenomenon could be described as the hull having difficulty changing its heading by yawing and is easier to drift away in the sway axis or to its breadth perspective. This is categorized as poor maneuverability since the disturbance of waves in the sea condition especially acting from the beam could drift away the hulls, and it takes more effort to keep the heading back to the preferable condition, due to the poor maneuverability performance in terms of course-keeping.

Furthermore, in the case of skeg 1 installment gives an improvement, but still does not pass the requirement to have a positive C index value of –0.509. The improvement was made, but not sufficient enough to reach the goal of maneuverability standards. Better improvements can be proved by the positive value of C in the case of both skeg 2, skeg 3, and skeg 4, by having a C index value of 0.088, 1.087, and 1.622. As shown in Fig. 10, the skeg affects the sway force in pure-sway motion the most.

Summarized in Table 6 and illustrated in Fig. 10, in the initial condition of bare hull in pure sway motions, the amplitude of sway force in non-dimensional form is 1.308 x10^-2. However, with skeg 1, 2, 3, and 4 configurations, the amplitude is increased significantly to 2.009 x10^-2, 2.415 x10^-2, 2.848 x10^-2, 3.341 x10^-2, respectively. This value proves that it is now more difficult to sway away the hulls, or in other words, it requires more force to drift the hulls to sway axis.

Even though there are still increments in the yaw moment in pure-yaw motions, the skegs still acted positively to the course-keeping ability. The reason is that the increment is relatively small, where the bare hull required about 1.377 x10^-3 yaw moment to change its heading in pure-yaw motions, and the skegs installment only slightly increased the required moments to 1.417 x10^-3, 1.460 x10^-3, 1.534 x10^-3, and 1.690 x10^-3 for skeg 1, 2, 3, and 4, respectively. The increment is not a big issue compared to the effectiveness of skegs to reduce the tendency of the hull to sway.

Finally, in terms of course-keeping ability, an improvement is made by comparing the C criteria. The bare hull, which has a negative C value of –1.474, along with skeg 1 by –0.509, now has a positive value of 0.088, 1.087, and 1.622 after skegs 2, 3, and 4 are installed, respectively. This gives an improvement compared to the bare hull conditions, as summarized in Table 7.

7. Conclusions

This study is conducted to improve the course-keeping ability of a 24m shallow draft catamaran. As virtual captive tests require a high computational cost and complex facilities, numerical simulations of CFD are utilized. Two main motions are simulated, namely pure-sway and pure-yaw motions, to obtain each hydrodynamic parameter required to calculate the course-keeping ability criteria. The results are considered satisfactory, CFD can capture the difference of force and moments acting on the hull. Several conclusions are to be summarized as follows:

Computational Fluid Dynamics (CFD) can be utilized to perform the numerical simulations of captive model tests, by performing the pure-sway and pure-yaw motions as a part of a harmonic test of Planar Motion Mechanism (PMM). This could help the ship designers to consider the maneuverability since the design stage with relatively lower computational cost, compared to the towing tank or captive model test facility.
Skegs are effective in improving the course-keeping ability criteria of C. The bare hull condition has a negative C index of – 1.474, skeg 1 configuration improves the C index by a value of – 0.509, yet still not sufficient enough to pass the maneuverability requirement. Therefore, a bigger skeg in height is required, as skeg 2, 3, and 4 improves the C index to positive value, by 0.088, 1.087, and 1.622, respectively. This also proves that the bigger the skegs, the course-keeping ability performance is also improved. However, on the downside, the resistance can be increased, due to additional weight and wetted surface area.
The improvement can be achieved due to the higher lateral force needed to drift the hulls away. A significant increase was achieved in the sway force in the pure-sway motions, which has an impact on the course-keeping ability overall. The rise in sway force can be translated as the hull needing more force to be drifted away. In other words, the hulls can sustain more force from external disturbances such as winds, or waves.

For further consideration, the skeg positions can be varied as a part of the skeg design optimization. Also, the aspect of resistance needs to be considered since the design of skegs affected the resistance performance of the vessel.

Acknowledgement

This research was supported by “Regional Innovation Strategy (RIS)” through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (MOE) (2021RIS-002).

Figure

Fig. 1.

Target vessel’s linesplan.

Fig. 2.

Skegs Configurations.

Fig. 3.

Pure-sway motions.

Fig. 4.

Pure-yaw motions.

Fig. 5.

y+ studies.

Fig. 6.

Grid independence test.

Fig. 7.

Mesh configurations.

Fig. 8.

Pure Sway Test of Bare Hull (left) and Skeg 4 (right).

Fig. 9.

Pure Yaw Test of Bare Hull (left) and Skeg 4 (right)

Fig. 10.

Hydrodynamic forces and moment of (a) Pure Sway Test (left) and (b) Pure Yaw Test (right).

Table

Table 1.

Target vessel’s principal dimensions

Principal Dimensions	Value	Units
Length overall	23.90	m
Breadth	8.50	m
Draft	0.70	m
Speed	10	knots
Froude number	0.336	-
Wetted surface area	114.777	m2
Displacement	42	ton

Table 2.

Skegs configurations

Case	Additional weight	Additional WSA	Height	Length
	kg	m2	cm	m
#1 Barehull	-	-	-	-
#2 Skeg 1	101.90	4.10	10	10
#3 Skeg 2	203.80	8.20	20	10
#4 Skeg 3	305.71	12.30	30	10
#5 Skeg 4	401.47	16.14	30	13

Table 3.

Numerical simulations scope of study

Condition	Test	Froude Number	Period	Amplitude
#1 Barehull	Pure Sway	0.34	20s	7.50m
Pure Yaw
#2 Skeg 1	Pure Sway
Pure Yaw
#3 Skeg 2	Pure Sway
Pure Yaw
#4 Skeg 3	Pure Sway
Pure Yaw
#5 Skeg 4	Pure Sway
Pure Yaw

Table 4.

Computational domain and boundaries

Part	Regions	Size	Motion
Domain regions
Inlet	Velocity inlet	13L	Stationary
Outlet	Pressure outlet	3L
Top	Velocity inlet	1.5L
Bottom	Velocity inlet	3L
Starboard	Velocity inlet	2L
Port	Velocity inlet	2L
Overset regions
Hull	Non-slip wall	1L	DFBI-PMM
Overset length	Overset mesh	1.2L
Overset breadth	Overset mesh	0.6L
Overset height	Overset mesh	0.2L

Table 5.

Grid Convergence Index (GCI)

Mesh	Cells (mil.)	X (N)	Ref. ratio	p	f0 (N)	GCI (%)
Fine	9.8	13,483	2	4.57	13,457	-
Medium	6.9	13,589	0.24
Coarse	4.9	14,092	1.22

Table 6.

Ship’s non dimensional parameters and amplitude of hydrodynamic forces and moments (non-dimensional)

Case	Ship’s Parameters	Pure-sway Motions	Pure-yaw Motions
	m’	x’ G	u’	Y’	N’	Y’	N’
	x10-3	x10-3		x10-2	x10-3	x10-3	x10-3
Barehull	6.003	-3.975	1	1.308	3.946	1.159	1.377
Skeg 1	6.017	4.142	1	2.009	4.376	0.950	1.417
Skeg 2	6.032	4.142	1	2.415	4.409	0.794	1.460
Skeg 3	6.047	4.142	1	2.848	4.920	1.061	1.534
Skeg 4	6.060	2.134	1	3.341	7.502	1.132	1.690

Table 7.

Hydrodynamic coefficients of sway and yaw, and course-keeping ability index (C) criteria check

Case	Sway Coefficients	Yaw Coefficients	Ability Index	Check
	Y′υ˙	N′υ˙	Y′υ	N′υ	Y′r˙	N′r˙	Y′r	N′r	C	(C>0)
	x10-3	x10-4	x10-2	x10-3	x10-4	x10-4	x10-4	x10-3	x10-5
Barehull	-6.990	4.952	-0.855	-4.102	-4.702	-2.084	-3.199	-1.548	-1.474	Unstable
Skeg 1	-6.344	4.804	-1.385	-3.855	-5.952	-2.301	0.549	-1.542	-0.509	Unstable
Skeg 2	-7.230	4.549	-1.623	-3.878	-6.769	-2.327	6.354	-1.594	0.088	Stable
Skeg 3	-8.285	6.036	-2.052	-4.016	-7.286	-2.375	12.674	-1.716	1.087	Stable
Skeg 4	-9.440	8.206	-1.953	-4.870	-9.767	-2.572	11.784	-1.919	1.622	Stable

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