Influence of Roll Motion on Added Resistance in Waves in Full and Ballast Load Conditions

1 Influence of Roll Motion on Added Resistance in Waves in Full and Ballast Load Conditions Saori Yokota, Masaru Tsujimoto and Mariko Kuroda 1 1 National Maritime Research Institute, National Institute of Maritime, Port, and Aviation Technology, Tokyo, Japan Abstract. As movements to improve the safety of marine transportation and reduce Greenhouse Gas (GHG) emissions in the shipping industry have increased, interest in ship performance in actual seas has also increased. To correctly evaluate ship performance in actual seas, accurate estimation of the external force acting on a ship is important. So far, many formulas for estimating the added resistance in heading waves have been proposed, and the accuracy has been improved steadily. There are also several methods for estimating the added resistance in oblique waves, but there is no estimation formula that takes into account the roll motion. The authors proposed a practical correction method to improve the accuracy of the added resistance in oblique waves by considering the roll motion. In this method, the frequency response of the added resistance in waves can be expressed, and it was confirmed that the proposed equation considering the roll motion was more consistent with the experimental value. The authors conducted a tank test to investigate the effect of different load conditions on the added resistance in waves considering the roll motion. In this paper, tank tests are conducted to investigate the effect of the roll motion on the added resistance in waves of a full ship in a different load condition. Calculated results and tank tests results in regular waves are compared. In the full load condition, the peak frequency of the roll amplitude is in long waves and the effect on the added resistance in waves is small. In the ballast load condition, the peak frequency of roll amplitude is in shorter waves than that of the full load condition, and the effect on the added resistance in waves is shown. When the experimental results of the roll amplitude are used in the estimation of the added resistance in waves, it is confirmed that the estimation accuracy is improved. Keywords: added resistance in waves, roll motion, model test, tank test, oblique waves, KVLCC2. 1. Introduction As movements to improve the safety of marine transportation and reduce Greenhouse Gas (GHG) emissions in the shipping industry have increased, interest in ship performance in actual seas has also increased. In order to correctly evaluate ship performance in actual seas, accurate estimation of the external force acting on a ship is important. Resistance due to waves, namely, added resistance in waves, is one of the main components of external force in actual seas, and thus must be estimated accurately. In estimations of the added resistance in regular waves, the wave reflection component of the added resistance is generally appended to the radiation and diffraction components. The radiation and diffraction components of the added resistance are expressed by the formula derived by Maruo[1]. A semi-empirical formula for the wave reflection component has been proposed by Fujii and Takahashi[2], and a practical correction method was presented by Tsujimoto et al.[3] The added resistance in waves estimated by this method has been verified many times, and the accuracy of estimations has also been studied by comparison with onboard monitoring data. The estimation accuracy of this method has been demonstrated, especially for head waves, and the method has now become a general estimation method. It is necessary to estimate the added resistance in waves with high accuracy not only for head waves, but also for oblique waves. Study of the added resistance in oblique waves was began with study of computational evaluation by Hosoda’s[4] , and the results of calculations by the estimation formula and test results were compared by Maruo and Iwase[5] and Fujii and Takahashi[2]. Several recent reports have examined the relationship between the added resistance in waves and the roll motion of a ship. Yoshida et al.[6] conducted experimental research on the added resistance in oblique waves of a large blunt ship and noted the relationship between the roll motion and the added resistance. They confirmed that the added resistance becomes large at the frequency where the roll amplitude has its peak value. In addition, a test by Valanto et al.[7] confirmed the effect of the roll motion on the added resistance in beam and quartering waves. 16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th – 23rd 2025 2 These results suggest the possibility that the contribution of the roll motion to the added resistance in waves is not negligible, depending on the type of ship and its load condition. Therefore, the authors proposed a practical correction method to improve the accuracy of the added resistance in oblique waves by considering the roll motion[8]. It has been confirmed that this proposed method improves the accuracy of estimation of the added resistance in oblique waves of fine ships. This paper describes the effect of the roll motion to the added resistance in waves and the comparison of the test results and estimation results in the full and ballast load conditions. 2. Estimation Method and Test Result 2.1. Calculation method considering roll motion In the conventional calculation method, the added resistance in regular waves RAW comprises the added resistance mainly due to ship motion RAWM and its correction RAWR, which is so called added resistance due to wave reflection. RAWM is calculated by Maruo’s theorem[1]. RAWR is introduced as a correction term for RAWM from the viewpoint of accuracy[3] and is calculated according to Eq. (1). RAWR comprises the bluntness coefficient, Bf, the coefficient of draft and frequency, α d, and the effect of advance speed, 1+ α U. The coefficient of advance speed, CU, can be determined by tank tests or the empirical formula. In this paper, CU in head waves is determined by the tank tests in regular head waves and CU in oblique waves is determined by the empirical formula. 2 1(1 ) 2 AWR a max f d U R g BB ρζ α α = + (1) () UU r CF αα = (2) where ρ is the fluid density, g is the gravitational acceleration, ζ a is the amplitude of regular waves, Bmax is the ship breadth, α is the angle between the ship heading and regular waves (0 deg. is defined as head waves) and Fr is the Froude number. Considering that the added resistance in waves is affected by the roll motion, the added resistance due to roll motion, RAWRoll, is expressed in separately as Eq. (3). AW AWM AWR AW Roll RR R R=++ (3) For estimation of RAWRoll , the concept of Gerritsma & Beukelman’s method[9] is expanded, with further details provided in our 2021 publication[8]. RAWRoll is expredssed in Eq. (4). 22 44 4/ e xx a AW Roll pp k Bk RdL ωφ = (4) where k is the circular wave number, ω e is the encounter circular frequency of waves, φ a is the amplitude of the roll angle, Lpp is the length between perpendiculars, B44 is the roll damping coefficient and kxx is the radius of lateral inertia. 2.2. Test results In order to examine the relationship between RAWRoll and the roll motion, the experimental results and the estimated results are compared. In this paper, the object ship is selected as KVLCC2 and the model is shown in Fig. 1. The principal dimensions of the ship are shown in Table 1. This ship is not equipped with bilge keels. The experiments were carried out at the Actual Sea Model Basin[10]. The length of the model is about 4.4 m, and the test speed is Fr=0.142. 3 Fig. 1 Model of KVLCC2. Table 1 Principal dimensions of KVLCC2. Item Unit Value KVLCC2 Condition - Full Ballast Length between perpendiculars (Lpp) m 320.0 Ship breadth (Bmax) m 58.0 Midship draft (dm) m 20.8 10.0 Aft draft (da) m 20.8 11.4 Fore draft (df) m 20.8 8.6 Transverse metacentric height (GM) m 5.71 17.8 Non-dimensional lateral radius of gyration (kxx/Bmax) - 0.40 0.35 Non-dimensional longitudinal radius of gyration (kyy/Lpp) - 0.25 0.26 Bluntness coefficient in head waves (Bf) - 0.41 0.24 Coefficient of advance speed in head waves (CU) - 14.9 12.2 The experimental and estimated results of the frequency response of the roll amplitude and the added resistance in oblique waves are shown in Fig. 2 and Fig.3, respectively in the full and ballast load conditions. In these figures, λ is the wave length, k is the circular wave number and KAW is the non-dimensional coefficient of the added resistance in waves shown in Eq. (5). In order to ensure the reproducibility, experiments are taken twice at each wave length. 22 4 AW AW a max pp R KgB L ρζ = (5) The roll amplitude in the experiments is selected for the wave height equivalent to 3 m in full scale, which is the same as in the calculation. In (a-2), (b-2) and (c-3) in Fig.2 and Fig. 3, the symbols represent the experimental results and the solid line represents the calculation results. The roll motion is calculated using the extinction parameters φ m and ∆ φ m obtained by the free roll tests. The quadratic coefficients a and b in Eq. (8) are obtained using these parameters[11]. Table 2 shows the quadratic coefficients of a and b at the speed of each test of the added resistance in waves. 1mii φ φφ + ∆=− (6) 1 2 ii m φφ φ + + = (7) 2 m mm ab φ φφ ∆= + (8) where i denotes the number of a peak or trough in the free roll tests and the unit of φ i is degree. The relationship between the roll damping coefficient B44 and the quadratic coefficients is shown in Eq. (9). 44 2 180 m g GM B ab φ ρφ πω π ∇ = +   (9) where ωφ is the roll natural frequency and ∇ is the displacement volume of the ship. 4 Table 2 Quadratic coefficients in roll motion of KVLCC2 Ship (load condition) T φ (Fr=0) [s] a (Fr=0.142) [-] b (Fr=0.142) [1/deg.] KVLCC2 (Full) 19.5 0.020 0.005 KVLCC2 (Ballast) 11.0 0.098 0.005 The estimation of the roll motion in relation to B44 has been investigated using full-scale ship data[12]. Comparison on the standard deviation of the roll motion between estimated results and onboard monitoring data show in good agreement. In this case, the roll damping coefficients are determined by the tank test results and the parameters related to the center of gravity are determined by the designed load condition. Focusing on the result of full load condition shown in Fig. 2, the peak of the frequency response function of the roll amplitude is in long waves from the estimated results. Therefore, the roll effect of the added resistance in waves appears in long waves. Although the estimation result shows in good agreement with the test result, the estimated result in (c-2) of Fig. 2 is excessive result compared to the test result. The frequency response functions of roll phase shown in (a-3), (b-3) and (c-3) of Fig. 2 are equivalent to the experimental result and the estimation in all wave directions. (a-1) Added resistance in bow waves (a-2) Roll amplitude in bow waves (a-3) Roll phase in bow waves (b-1) Added resistance in beam waves (b-2) Roll amplitude in beam waves (b-3) Roll phase in beam waves (c-1) Added resistance in quartering waves (c-2) Roll amplitude in quartering waves (c-3) Roll phase in quartering waves Fig. 2 Frequency response function for added resistance (left), roll amplitude (middle) and roll phase (right) in full load condition. Next, (a-2) and (b-2) in Fig.3 show that the peak of the roll amplitude in ballast load condition appears in shorter waves than that of full load condition. The estimation results of the peak frequency is in good agreement with the test results. However, the estimation results are overestimated in terms of the roll amplitude compared to the test results. Thus the added resistance in waves is overestimated. (c-2) in Fig. 3 shows that estimation result of the roll amplitude is in good agreement with the test result. (c-1) in Fig. 3 shows that the roll effect on the added resistance in waves shows small in quartering waves in both the estimation and test results. The frequency response functions of roll phase shown in (a-3), (b-3) and (c-3) of Fig. 3 are equivalent to the experimental result and the estimation in all wave directions. 5 From these results shown in Fig. 2 and Fig. 3, it is found that the estimation of the roll amplitude is overestimated. Therefore, the case where the experimental results are used for the roll amplitude is examined in estimation of added resistance in waves. Fig. 4 shows the result of estimating the added resistance in waves by interpolating the experimental results of the roll amplitude in ballast load condition. The lines in (a-2’) and (b-2’) of Fig. 4 are interpolated experimental results. The lines in (a-1’) and (b-1’) of Fig. 4 are the added resistance in waves calculated from the interpolation results. As a result of calculations using the interpolated value, the added resistance in waves shows practical results without excess value. However, compared to the test results of the added resistance in waves, calculated results are still some overestimated. (a-1) Added resistance in bow waves (a-2) Roll amplitude in bow waves (a-3) Roll phase in bow waves (b-1) Added resistance in beam waves (b-2) Roll amplitude in beam waves (b-3) Roll phase in beam waves (c-1) Added resistance in quartering waves (c-2) Roll amplitude in quartering waves (c-3) Roll phase in quartering waves Fig. 3 Frequency response function for added resistance (left), roll amplitude (middle) and roll phase (right) in ballast load condition. 6 (a-1’) Added resistance in bow waves (a-2’) Roll amplitude in bow waves (b-1’) Added resistance in beam waves (b-2’) Roll amplitude in beam waves Fig. 4 Frequency response function and test interpolation for added resistance (left) and roll amplitude (right) in ballast load condition. From the previous research[8], the roll effect on added resistance in waves is significant in fine ships. Therefore the results of this study are characterized as the damping component of the roll motion. In the case of a large full ship like the target ship, the roll amplitude is overestimated since the roll amplitude is smaller than the fine ship. Therefore, it is considered that the influence of the parameters in Eq. (4) requires to examine. 3. Conclusions In this study, the estimation of the added resistance in waves considering the roll motion is discussed by using a ship model of KVLCC2. The results are as follows. In the full load condition, the peak frequency of the roll amplitude of the estimation shows good agreement with the experimental result, but the roll amplitude overestimates in the estimation. The peak frequency of the roll amplitude is in long waves and the effect on the added resistance in waves is small. In the ballast load condition, the peak frequency of the roll amplitude appears in shorter waves than that of the full load condition and the effect on the added resistance in waves is larger than in the full load condition. The accuracy of the estimation of the roll amplitude affects that of the added resistance in waves. When the experimental results are used in the estimation of the added resistance in waves, it is found that the accuracy is improved. Acknowledgments This research was supported by JSPS KAKENHI Grant Number JP24K07913. The authors would like to express our heartfelt gratitude to Mr. Daisuke Wako, Mr. Ryohei Fukasawa, Ms. Akiko Sakurada, Mr. Tatsuya Hamada and Ms. Azumi Kaneko of National Maritime Research Institute for conducting the experiments together in this research. 7 References [1] H. Maruo: Resistance in Waves, Research on Seakeeping Qualities of Ships in Japan, The Society of Naval Architects of Japan, Vol. 8, 1963, pp. 67-102. [2] H. Fujii and T. Takahashi: “Experimental Study on the Resistance Increase of a Ship in Regular Oblique Waves,” Proc. of 14th ITTC, Vol. 4, pp. 351-360. [3] M. Tsujimoto, K. Shibata, M. Kuroda and K. 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