Low Speed Course-Keeping and Zigzag Manoeuvres of the KSUPRAMAX Bulk Carrier in Waves

1 Low Speed Course-Keeping and Zigzag Manoeuvres of the KSUPRAMAX Bulk Carrier in Waves Dong-Jin Kim * , Kunhang Yun, Chang-Seop Kwon, Yeon-Gyu Kim, Seung-Hyun Hwang Korea Research Institute of Ships and Ocean Engineering, Daejeon, Republic of KOREA Abstract. Manoeuvring performances of ships in waves are significantly different from those in calm water. Reliable prediction techniques on the ship’s manoeuvrabilities in waves are required at the early design stage for its safe operations in actual seas. In particular, attentions should be paid to the ship’s course-keeping and changing abilities which become more important in rough waves. In this study, course-keeping and zigzag manoeuvres of a ‘KSUPRAMAX’ model ship which has a supramax class bulk carrier hull form are experimentally investigated. The full-scale ship length is 192 m, a 1/64-scaled model ship is constructed for free-running model tests in Ocean Engineering Basin of Korea Research Institute of Ships and Ocean Engineering (KRISO). At first, turning circle tests are repeatedly conducted in long-crested irregular waves with the model propulsion point corresponded to the full-scale design speed of 14.5 knots. Equivalent regular waves that show similar turning circle trajectories and approach speeds compared to irregular waves are searched. Next, course-keeping tests are performed in irregular and equivalent regular waves with wave incident angles of 0 to 180 degrees at speeds lower than 5 knots. Low speed course-keeping paths, check helms, and drift angles in equivalent regular waves are close to those in irregular waves. 3-DoF manoeuvring simulation models are constructed based on captive model tests and empirical methods. Since the wave drift forces are equal to sums of hydrodynmaic forces acting on the hull, the propeller, and the rudder during coursekeeping manoeuvres, wave drift forces are identified from course-keeping simulations. Zigzag tests are also conducted in equivalent regular waves with approaching conditions which are the same as those in coursekeeping tests, zigzag behaviors are simulated by the present models including wave drift forces identified. Course-keeping and changing abilities are discussed with considerations of wave drift forces which are varied with wave incident angles. Keywords: KSPURAMAX bulk carrier, Free-running model test, Course-keeping ability, Zigzag manoeuvre, Wave drift force 1 Introduction Ship’s manoeuvrabilities are conventionally evaluated in calm water, which are different from those in a seaway due to environmental loads. At an early design stage, it is necessary to predict manoeuvring performance of a ship in waves as well as in calm water for its safe operations in seas. Moreover, installed engine powers of new-built ships are recently decreased to meet EEDI requirements which aim to reduce greenhouse gas emissions. Attentions should be paid to check whether ships have sufficient manoeuvrabilities for operating in adverse weather conditions. In adverse conditions, ship advance speeds are decreased, manoeuvrabilities of ships are remarkably different from those in calm water. Since ships should overcome violent environmental loads and keep or change their courses by their own steering systems, course-keeping and changing abilities of ships are of particular importance in such situations. As in conventional IMO standard manoeuvring tests, course-keeping and changing abilities of ships can be evaluated though zigzag manoeuvring performances. In addition, in adverse weather condition under strong environmental loads, asymmetric behaviors of ships such as hull drifts and check helms for keeping straight courses need to be investigated in detail. There are some previous studies on course–keeping or zigzag abilities of ships in waves through experimental or numerical techniques. Hirayama and Kim (1994) performed zigzag model tests of a tanker in long and short crest irregular waves generated in a towing tank, simulations by manoeuvring models including wave drift forces are compared with model test results [1]. Yasukawa and Adnan (2006) experimentally investigated motions and wave drift forces of a container model in regular head and beam waves, effects of drift angles on wave drift forces * Correspondence to: [email protected] 16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 202 5 Ann Arbor, MI, USA, October 19th – 23rd 2025 2 were discussed, Yasukawa (2008) validated zigzag and stopping manoeuvring simulations through comparison with free-running model tests [2][3]. In a research by Sprenger et al. (2017), free-running turn and zigzag model tests of a KVLCC2 tanker and a DTC container were performed in four European institutes and were analyzed within the framework of SHOPERA project [4]. Zhang et al. (2017) carried out two-time scale simulations on turn and zigzag manoeuvres of a S175 container in regular waves and compared them with model test results [5]. Wang et al. (2018) investigated standard zigzag manoeuvring characteristics of an ONRT model in regular waves by using CFD solver [6]. Sanada et al. (2018) carried out course-keeping and manoeuvring model tests of an ONRT in various regular waves, in particular, time-averaged and harmonic amplitudes of ship motions and rudder angles during course-keeping tests are analyzed in various wave incident angles [7]. Kim et al. (2022) performed freerunning CFD calculations of a KCS model and analyzed low-speed course-keeping and turning abilities of the model in different wave incident angles [8]. Suzuki et al. (2023, 2024) carried out free-running course-keeping model tests of a KVLCC1 tanker in regular short waves and compared them with numerical simulations considering steady wave drift forces and moments obtained by captive model tests and CFD [9][10]. In this study, course-keeping and zigzag abilities of a ‘KSUPRAMAX’ bulk carrier in waves are investigated by free-running model tests. At first, 35° turning circle tests are carried out in long-crested irregular and various regular waves. The significant height and the peak period of long-crested irregular waves are selected with consideration of adverse weather conditions defined in an IMO minimum propulsion power guideline [11]. Equivalent regular waves are determined so that model ship approach speeds and trajectories in irregular waves are similar to those in regular waves. It is also confirmed that low speed course-keeping abilities depending on wave incident angles in irregular waves are similar to those in equivalent regular waves. By using steady state variables of the hull, the propeller, and the rudder in course-keeping tests, wave drift forces and moments depending on incident angles are identified through modular type manoeuvring models. Low speed 10°/10° starboard and port zigzag tests are conducted in equivalent regular waves with various incident angles. Zigzag abilities in waves are significantly different from those in calm water, and are varied depending on wave incident angles, i.e. wave drift forces and moments. Zigzag behaviors depending on wave incident angles are analyzed with considerations of wave drift forces identified. 2 Free-running model test 2.1 Model ship and test basin Free-running model tests are carried out with a 1/64-scaled model ship of ‘KSUPRAMAX’ in this study. The full-scale ship is a 66,000 DWT supramax bulk carrier, which were used in recent CFD or model test studies on mainly its added resistances in waves [12][13][14]. Main particulars of full-scale and model ships are summarized in Table 1. Figure 1 shows the body plan and the scaled model ship constructed. The ship geometry and test data can be downloaded from KRISO website [15], otherwise, can be provided via the corresponding author’s email. Table 1. Main particulars of full-scale and model ships (design draft) Particular Full-scale Model Scale ratio (λ) 1 64 Length between perpendiculars (L) [m] 192.0 3.000 Breadth (B) [m] 36.0 0.563 Fore / aft draft ( TF / TA) [m] 11.2 / 11.2 0.175 / 0.175 Displacement (∇) [m3] 65028.2 0.248 LCB (fwd, +) [m] 5.773 0.090 Yaw radius of gyration (kzz) [L] 0.250 0.238 Metacentric height (GM) [m] - 0.080 Propeller diameter (DP) [m] 6.0 0.094 Rudder lateral area (AR) [m2] 45.91 0.0112 3 Figure 1. Body plan (left) and 1/64-scaled model ship (right) of KSUPRAMAX bulk carrier Free-running model tests are conducted in Ocean Engineering Basin of Korea Research Institute of Ships and Ocean Engineering (KRISO) as described in Figure 2. The basin is 56 m in length, 30 m in beam, and 4.5 m in depth. Wave makers are installed along short and long ends of the basin. For present model tests, regular and longcrested irregular waves are generated by short end wave makers. Ship motions, propulsion and steering signals are measured by onboard PC mounted on the model, and all synchronized signals are sent to the ground control PC wirelessly. Three prisms are fixed on the model deck. They are traced by three total station devices on the ground so that the 3-dimensional ship positions can be measured automatically, which have been updated compared to the previous 2-dimensional measurement system [16][17]. Figure 3 shows the coordinate system used in the present study. O-XY is a space-fixed coordinate, X-coordinate is set toward the model ship heading when each test starts. o-xy is a body-fixed coordinate, whose origin is located on the midship of the ship. Figure 2. Free-running model test in KRISO Ocean Engineering Basin Figure 3. Coordinate system 4 2.2 Test program Free-running turning circle and zigzag tests are carried out in calm water at the model propulsion points of 19.0 and 8.0 RPS which correspond to the full-scale speeds of 14.5 (design speed) and 5.0 knots, respectively. 35° portside turning circle tests are conducted in long-crested irregular waves and several kinds of regular waves at 19.0 RPS. Based on above turning circle test results, one specific regular wave condition is selected. At the propulsion point of 8.0 RPS, course-keeping and 10/10 zigzag tests are performed in irregular and selected regular waves with varying wave incident angles. Total program is summarized in Table 2. Table 2. Free-running model test program (SB: starboard, PS: port side) 3 Search of ‘equivalent regular waves’ corresponding to the adverse condition 3.1 Turning circle tests at 19.0 RPS (corr. full-scale 14.5 knots in calm water) To maintain the manoeuvrability of ships in adverse weather conditions, IMO interim guideline for determining minimum propulsion power of ships was proposed [11]. In the guideline, adverse weather condition is defined as Table 3. For the present full-scale KSUPRAMAX bulk carrier, the significant wave height and peak period can be determined as Table 4. Wind loads are not considered in this study. Table 3. Adverse weather condition (IMO, 2021) Test Wave RPS [/s] Wave incident angle [°] (e.g. 180°: head wave) 35° SB, PS Turn Calm water 19.0 (corr. 14.5 knots) - 35° SB, PS Turn 10°/10° SB, PS Zigzag 8.0 (corr. 5.0 knots) 35° PS Turn Irregular (Hs= 4.5m, Tp=12.0s) 19.0 180 Regular (H= 2.589~4.500 m, T= 8.520~12.000 s) Course-keeping Irregular (Hs= 4.5m, Tp=12.0s) 8.0 180, 150, 120, 90, 60, 30, 0 Regular (H= 2.878 m, T= 9.780 s) 10°/10° SB, PS Zigzag Irregular (Hs= 4.5m, Tp=12.0s) 8.0 180, 150, 90, 30, 0 Regular (H= 2.878 m, T= 9.780 s) Ship length [m] Significant wave height [m] Peak wave period [s] Mean wind speed [m/s] L > 250 6.0 7.0 ~ 15.0 22.6 200 ≤ L ≤ 250 Linearly interpolated depending on ship length L < 200 4.5 7.0 ~ 15.0 19.0 5 Table 4. Long-crested irregular wave condition in the present study No. Approach speed [knots] 𝐻𝐷 720−360 [L] 𝜇𝐷 720−360 [°] 1 11.5 2.19 30 2 11.1 2.11 37 3 11.2 1.78 19 4 11.1 1.94 27 5 11.4 1.84 30 Mean 11.3 1.97 29 Std. dev. 0.2 0.18 6 Figure 4. 35° PS turns in irregular waves (left) and definition of drifting distance and angle (SIMMAN,2023) (right) At first, the model propulsion point is fixed to 19.0 RPS which corresponds to the full-scale 14.5 knots in calm water, 35° portside turning circle tests are carried out in long-crested irregular waves determined in Table 4. Five repetition test results are analyzed in Figure 4. Due to the wave drift forces and moments, turning circle trajectories are drifted in waves. Definition of trajectory drifting distance and angle are described on the right of Figure 4 [18]. Statistically, the present model ship is drifted approximately two times ship length during a 360 ° turn in irregula r waves. The approach speed is 14.5 knots in calm water, whereas the mean value of approach speeds is reduced by 11.3 knots in irregular waves. Wave energy flux per unit crest length (𝑃𝑊 [W/m]) are theoretically formulated in both regular and longcrested irregular waves, respectively, which are shown in Eqs. (1) and (2) [19]. 𝜌 is the density of water, 𝑔 is the gravitational acceleration, 𝜁𝑎 is the amplitude of the incident wave, 𝑐𝐺 is the group velocity. 𝑓 is the wave frequency, 𝑆(𝑓) is the spectral density function of incident irregular waves. 𝐻𝑆 and 𝑇𝐸 are the significant wave height and the energy period, respectively. 𝑇𝐸 is defined as Eq. (3) [20]. 𝑃𝑊(𝑟𝑒𝑔𝑢𝑙𝑎𝑟)=1 2𝜌𝑔𝜁𝑎2𝑐𝐺=1 32𝜋𝜌𝑔2𝐻2𝑇 (1) 𝑃𝑊(𝑖𝑟𝑟𝑒𝑔𝑢𝑙𝑎𝑟)=𝜌𝑔∫𝑐𝐺(𝑓)𝑆𝑖(𝑓)𝑑𝑓 ∞ 0=1 64𝜌𝑔2𝐻𝑆2𝑇𝐸 (2) Significant wave height [m] Peak wave period [s] 4.5 12.0 6 𝑇𝐸=𝑚−1/𝑚0 , 𝑚𝑛= ∫𝑓𝑛𝑆𝑖(𝑓)𝑑𝑓 ∞ 0 (3) When Eq. (1) equals to Eq. (2), ‘equivalent regular waves’ whose energy flux is identical to that of long-crested irregular waves are estimated as Table 5. Table 5. Equivalent regular wave condition In order to investigate the model ship operated in the equivalent regular waves in Table 5 shows manoeuvring performances which are similar to those in long-crested irregular waves, 35° turning circle tests are carried out in various regular waves as described in Table 6. ‘REG03’ is the estimated condition in Table 5. Regular wave heights are varied from 𝐻𝑆/2 to 𝐻𝑆 , periods are changed to 𝑇𝑍, 𝑇𝑀, and 𝑇𝑃, respectively. 𝑇𝑍, 𝑇𝑀, and 𝑇𝑃 are zero-crossing, mean, and peak periods on irregular waves energy spectrum. Results of 35° turning circle tests at 19.0 RPS in regular waves are analyzed in Figure 5. In the viewpoint of wave energy, ship approach speeds and trajectory drifting distances are focused on. Both are deeply related to wave drift forces and moments in longitudinal and lateral directions. REG03 shows turn trajectory drifting distance which is similar to that in irregular waves as anticipated, but the ship approach speed in REG03 is lower than that in irregular waves. The added resistance in REG03 is expected to be larger than that in irregular waves due to larger relative ship motions, although wave energies in both waves are theoretically identical to each other. Table 6. Regular wave conditions (REG01 to REG06) Figure 5. Approach speeds (left) and trajectory drifting distances (right) in REG01 to REG06 Interestingly, trajectory drifting distances in regular waves are proportional to wave slopes. To maintain the trajectory drifting distance, and to increase the ship approach speed, REG07 and REG08 are additionally generated as shown in Table 7. Wave slopes of REG07 and REG08 are the same as that of REG03. Regular wave height, 𝐻 Regular wave period, 𝑇 𝐻=𝐻𝑆/√2 𝑇=𝑇𝐸 Period Height 𝑇𝑍 (8.520s) 𝑇𝑀 (9.276s) 𝑇𝐸 (10.284s) 𝑇𝑃 (12.000s) 𝐻𝑆 / 2 (2.250m) - - REG05 - 𝐻𝑆 / √2 (3.182m) REG01 REG02 REG03 REG04 𝐻𝑆 (4.500m) - - REG06 - 7 Table 7. Additional regular wave conditions (REG07 and REG08) Figure 6. Approach speeds (left) and trajectory drifting distances (right) in REG07 and REG08 as well as REG03 Results of approach speeds and trajectory drifting distances are described in Figure 6. Both the approach speed and the trajectory drifting distance in ‘REG07’ are similar to those in irregular waves. Therefore, ‘REG07’ is determined as the equivalent regular waves. Turning trajectories in irregular and regular (REG07) waves are shown in Figure 7. Irregular waves (𝐻𝑆= 4.5m, 𝑇𝑃= 12.0s) Regular waves, REG07 (H= 2.878m, T= 9.780s) Figure 7. 35° PS turns in irregular waves (left) and in regular waves REG07 (right) Period Height 𝑇𝑀 (9.276s) (𝑇𝐸+𝑇𝑀)/2 (9.780s) 𝑇𝐸 (10.284s) (2.589m) REG08 - - (2.878m) - REG07 - 𝐻𝑆 / √2 (3.182m) - - REG03 8 3.2 Course-keeping manoeuvres at 8.0 RPS in REG07 (corr. full-scale 5.0 knots in calm water) Figure 8. Trajectories, check helms, drift angles, and ship speeds of course-keeping tests in irregular waves (left) and in regular waves REG07 (right) 9 In irregular waves and equivalent regular waves previously determined, low speed course-keeping tests are conducted with varying wave incident angles. Model propeller rotation rate is fixed to 8.0 RPS, which corresponds to the full-scale 5.0 knots in calm water. Wave incident angles are varied from 180° to 0° with the interval of 30°. Figure 8 shows course-keeping trajectories, time averaged check helms, drift angles, and ship speeds, which are also introduced in the previous study by Kim et al. (2024) [21]. Low speed course-keeping manoeuvres in equivalent regular waves are considerably similar to those in irregular waves, although time averaged values in irregular waves are quite varied depending on the encountered wave elevations. 4 Identification of wave drift forces and moments 4.1 3-DoF manoeuvring simulation model To identify wave drift lateral forces and moments depending on wave incident angles, 3-DoF modular type manoeuvring models are used. Horizontal plane manoeuvres of the ship are represented by Eq. (1). Subscript H, P, and R denote force and moment components acting on the hull, the propeller, and the rudder. 𝑚(𝑢󰇗−𝑣𝑟−𝑥𝐺𝑟2) =𝑋𝐻+𝑋𝑃+𝑋𝑅 𝑚(𝑣󰇗+𝑢𝑟+𝑥𝐺𝑟󰇗) =𝑌𝐻+𝑌𝑅 𝐼𝑧𝑧𝑟󰇗+𝑚𝑥𝐺(𝑣󰇗+𝑢𝑟)=𝑁𝐻+𝑁𝑅 (1) Hull, propeller, and rudder forces and moments can be formulated as Eqs. (2) to (4). 𝑋𝐻=𝑋𝑢󰇗𝑢󰇗+𝑋𝑢𝑢𝑢2+𝑋𝑣𝑣𝑣2+𝑋𝑣𝑟𝑣𝑟+𝑋𝑟𝑟𝑟2+𝑋𝑣𝑣𝑣𝑣𝑣4 𝑌𝐻=𝑌𝑣󰇗𝑣󰇗+𝑌𝑟󰇗𝑟󰇗+𝑌𝑣𝑣+𝑌𝑟𝑟+𝑌𝑣𝑣𝑣𝑣3+𝑌𝑟𝑟𝑟𝑟3+𝑌𝑣𝑣𝑟𝑣2𝑟+𝑌𝑣𝑟𝑟𝑣𝑟2 𝑁𝐻=𝑁𝑣󰇗𝑣󰇗+𝑁𝑟󰇗𝑟󰇗+𝑁𝑣𝑣+𝑁𝑟𝑟+𝑁𝑣𝑣𝑣𝑣3+𝑁𝑟𝑟𝑟𝑟3+𝑁𝑣𝑣𝑟𝑣2𝑟+𝑁𝑣𝑟𝑟𝑣𝑟2 (2) 𝑋𝑃=(1−𝑡)𝜌𝑛2𝐷𝑃 4∙𝐾𝑇 (3) 𝑋𝑅=−(1−𝑡𝑅)𝐹𝑁sin𝛿 𝑌𝑅= −(1+𝑎𝐻)𝐹𝑁cos𝛿 𝑁𝑅=−(𝑥𝑅+𝑎𝐻𝑥𝐻)𝐹𝑁cos𝛿 (4) In Eqs. (3) and (4), detailed characteristics of the propeller and the rudder are obtained by Eqs. (5) and (6). 𝐾𝑇=𝑓(𝐽) 𝐽= 𝑢𝑃 𝑛𝐷𝑃 𝑢𝑃=(1−𝑤𝑃)𝑢 , 𝑤𝑃=𝑤·exp (−𝐶𝑃𝛽𝑃 2) , 𝛽𝑃=𝛽−𝑥𝑃′𝑟′ ( 𝐶𝑃= 𝐶𝑃+ 𝑤ℎ𝑒𝑛 𝛽𝑃>0, 𝐶𝑃= 𝐶𝑃− 𝑤ℎ𝑒𝑛 𝛽𝑃<0 ) (5) 𝐹𝑁=1 2𝜌𝐴𝑅𝑈𝑅𝑓𝛼sin𝛼𝑅 UR=√𝑢𝑅+𝑣𝑅, αR=𝛿−tan−1(𝑣𝑅/𝑢𝑅) 𝑢𝑅=𝜖𝑢𝑃√𝜂[1+𝜅(√1+8𝐾𝑇 𝜋𝐽2−1)]2+(1−𝜂) 𝑣𝑅=𝛾𝑅(𝑣+𝑙𝑅𝑟) , 𝛽𝑅=𝛽−𝑙𝑅′𝑟′ ( 𝛾𝑅= 𝛾𝑅+ 𝑤ℎ𝑒𝑛 𝛽𝑅>0, 𝛾𝑅= 𝛾𝑅− 𝑤ℎ𝑒𝑛 𝛽𝑅<0 ) (7) Hydrodynamic coefficients in Eqs. (2) to (7) are obtained based on 1/26.087-scaled static HPMM tests [22] and empirical formulas. Present simulation models can predict calm water turn and zigzag behaviors of the model ship with high accuracy as shown in Figure 9. 16  Substituting identified wave drift forces and moments into manoeuvring models, zigzag manoeuvres are simulated in different wave incident angles. Zigzag simulations are in good agreements with freerunning model tests. Acknowledgments This research was supported by a grant from Korea Research Institute of Ships and Ocean Engineering Endowment Project of “Development of evaluation technology for ship’s performance in extreme environment” funded by Ministry of Oceans and Fisheries, Republic of Korea (PES5461). References [1] T. Hirayama and S.E. Kim. Manoeuvrability of a full ship in directional spectrum waves of short wave length. Journal of the Society of Naval Architects of Japan, 176:129-136, 1994. [2] H. Yasukawa and F.A. Adnan. Experimental Study on Wave-Induced Motions and Steady Drift Forces of an Obliquely Moving Ship. 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