Numerical Analysis of Scale Effects on Resistance and Propulsion Performance of the K-SUPRAMAX in Low-Speed Condition

1 Numerical Analysis of Scale Effects on Resistance and Propulsion Performance of the K-SUPRAMAX in Low-Speed Condition Jae-Hyeon An1, Kwang-Jun Paik1, * , Myeong-Min Kim1, Seung-Hyun Hwang2 1 Department of Naval Architecture and Ocean Engineering, Inha University, Incheon, South Korea 2 Korea Research Institute of Ships & Ocean engineering(KRISO), Daejeon, South Korea Abstract. In this study, numerical simulations at full-scale were conducted for the 66K DWT K-SUPRAMAX ship to perform a fundamental investigation on resistance and self-propulsion performance under low-speed conditions in calm water, taking scale effects into account. Based on the full-scale CFD analysis at a low speed of 4 knots, it was found that the residuary resistance coefficient decreases compared to the model-scale due to scale effects, thereby influencing the evaluation of full-scale resistance performance. The Propeller open water (POW) performance was found to be consistent with the ITTC-1978 extrapolation results. In the full-scale analysis, it was confirmed that there is no significant difference in POW performance between the design and low rotational speeds due to the fully turbulent flow. Accordingly, the self-propulsion performance at low speed was evaluated based on the design rotational speed. In addition, the wake fraction estimated using the ITTC1978 method tends to overpredict the full-scale 𝑊𝑛 as speed decreases. However, applying the 𝑊𝑇𝑆 extracted from full-scale self-propulsion CFD analysis improves the agreement with the actual full-scale 𝑊𝑛 value. At low speeds, compared to the design speed, the effective wake fraction increases and the thrust deduction factor decreases, resulting in improved hull efficiency. Nevertheless, a reduction in propeller rotational speed leads to a decrease in open water efficiency Keywords: Low speed, Full-scale, Scale effect, K-SUPRAMAX, Calm water, Resistance and Propulsion 1 Introduction Due to the recent acceleration of global warming, efforts to regulate CO2 emissions have been continuously pursued. The IMO (International Maritime Organization) introduced the EEDI (Energy Efficiency Design Index) to enhance the energy efficiency of ships and thereby reduce CO₂ emissions [1]. In line with this, environmental regulations have been strengthened through the promotion of high-efficiency shipbuilding and the implementation of operational guidelines aimed at improving navigational efficiency. High-efficiency shipbuilding includes the development of eco-friendly hull forms, high-performance propulsion systems, and energy recovery devices. Operational strategies, such as operating ships at a reduced speed of about 4 knots, have proven to be effective methods for lowering CO₂ emissions and fuel consumption [2], [3], [4]. Moreover, it offers the advantage of requiring no retrofitting of existing ships. To meet the EEDI requirements, reducing ship speed has been adopted as an effective means of decreasing CO₂ emissions. Accordingly, accurate estimation of the required power has become increasingly important, and CFD-based self-propulsion simulations have emerged as a useful tool in this context. However, under low-speed conditions, unlike at the design speed, the decrease in 𝜂𝑂 and the increase in 𝑊𝑇𝑆 may result in deviations in selfpropulsion performance. Therefore, it is necessary to investigate changes in self-propulsion characteristics under low-speed conditions. In general, full-scale self-propulsion performance is estimated by applying the ITTC-1978 method to model-scale results. However, discrepancies may arise because the 𝑡 and 𝜂𝑅 do not account for scale effects. Thus, a fundamental study is required to evaluate the applicability of the ITTC method under low-speed conditions and to compare self-propulsion performance at design and low speeds, with the aim of improving the accuracy of full-scale self-propulsion predictions. As a preceding study on full-scale self-propulsion estimation, Calm water resistance analyses were conducted for the KVLCC2 hull form at four different scales, and it was confirmed that both CVP and CR tend to decrease as the scale approaches full-scale [5]. Regarding changes in 𝑊𝑛, For the KVLCC2, the difference in 𝑊𝑛 between model and full scales was analyzed, and wake contraction effects caused by increased Reynolds number were * Correspondence to: [email protected] 16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th – 23rd 2025 2 confirmed [6]. Furthermore, studies observed that the inflow velocity increases at full-scale, and when comparing across various speeds, the inflow velocity decreases at lower speeds, leading to an increase in the 𝑊𝑛 [7], [8]. In fullscale POW performance analysis, A comparative analysis of model-scale and full-scale results was conducted for the KP505 propeller. Their study showed that, compared to the model-scale, the influence of viscosity decreases at full-scale, resulting in an increase in the 𝐾𝑇 and a decrease in the 𝐾𝑄. They also confirmed that the full-scale POW performance remains consistent regardless of wall y+ values [9]. For the full-scale selfpropulsion performance analysis, simulations were conducted on the KCS hull form, and the self-propulsion factors between model and full scales were compared [10]. In a similar prior study, full-scale self-propulsion simulations were performed for a large crude oil tanker, and a comparative analysis was carried out between the self-propulsion factors derived from the ITTC-1978 method and those obtained through full-scale CFD analysis [11]. Research on performance evaluation under low-speed conditions remains relatively limited. In addition, the scale effect caused by the Reynolds number difference between model and full-scale leads to significant changes in flow characteristics, such as rapid pressure recovery near the stern and increased axial velocity into the propeller. As a result, it becomes challenging to accurately predict full-scale performance based on model test results using the Froude similarity law. In this study, full-scale numerical simulations were conducted for the low-speed, K-SUPRAMAX ship to perform a fundamental investigation on resistance and self-propulsion performance under low-speed conditions in calm water, considering scale effects. The flow characteristics at both design and low speeds were analyzed, and the scale effects were evaluated by comparing the model-scale numerical results, full-scale performance estimated using the ITTC extrapolation method and directly computed resistance and self-propulsion performance at fullscale. 2 Numerical method 2.1 Target ship and propeller The K-SUPRAMAX, a 66K bulk carrier designed by the KRISO, was selected as the target ship, with the KP1306 propeller chosen as the corresponding propulsion system. The geometry of the selected ship and propeller is presented in Figure 1, while their principal specifications are provided in Table 1 and Table 2 for the ship and propeller, respectively. (a) (b) Figure 1. Target ship and propeller model geometry: (a) 66k DWT K-SUPRMAX; (b) KP1306 propeller. Table 1. Principal particulars of the target ship Particulars Unit Full-scale Model-scale Scale ratio, 𝜆 - 1 24 Length, 𝐿𝑃𝑃 m 192 8 Length, 𝐿𝑊𝐿 m 104.5 4.35 Breath, 𝐵 m 36 1.50 Draft, 𝑇 m 11.2 0.47 Displacement, ∇ m3 65,028 4.704 Design speed, 𝑉 knots 14.5 3 Table 2. Principal particulars of the target propeller Particulars Unit Full-scale Model-scale Scale ratio, 𝜆 - 1 24 Diameter, 𝐷𝑃 m 6 0.25 rps, 𝑛 rps 2 17 Pitch ratio, 𝑃/𝐷𝑃 - 0.727 Thickness of ratio, 𝑡/𝐷𝑃 - 0.012 Chord ratio, 𝑐/𝐷𝑃 - 0.266 2.2 Governing equation The RANS (Reynolds-Averaged Navier-Stokes) equations and the continuity equation were employed as governing equations to simulate three-dimensional incompressible viscous flow. This formulation, incorporating fluid viscosity, is expressed in Equations (1) and (2). The flow around the hull was assumed to be incompressible and inviscid, except at the hull surface. 𝜕𝑢𝑖  𝜕𝑥𝑖 =0 (1) 𝜕𝑢𝑖  𝜕𝑡+𝑢𝑗 𝜕𝑢𝑖  𝜕𝑥𝑗 =−1 𝜌𝜌 𝑥𝑗 +1 𝜌𝜕2 𝑥𝑗 (𝜇𝜕𝑢𝑖  𝜕𝑥𝑗 −𝜌𝑢𝑖, 𝑢𝑗, ) (2) where 𝜌, 𝜈, 𝑃, 𝑥𝑖, (𝑥, 𝑦, 𝑧) and 𝑢𝑖(𝑢, 𝑣, 𝑤) correspond to the density of the fluid, kinematic viscosity, pressure, spatial coordinates, and velocity components along each coordinate axis, respectively. The Realizable k-ε turbulence model, which is widely used in resistance analysis and ship self-propulsion simulations, was employed to analyze the flow characteristics around the ship. This model effectively controls grid resolution near wall boundaries and provides relatively accurate numerical results, making it a preferred choice in the field of ship hydrodynamics. The SIMPLE algorithm was used to couple velocity and pressure, and the Volume of Fluid (VOF) method was applied to account for free-surface interactions induced by the movements of the propeller and rudder. 2.3 Computation domain and grid system For the full-scale resistance and self-propulsion simulations, the full-domain mesh system shown in Figure 2(a) was employed. To more accurately capture the motion of the ship, an overset mesh was also applied. All boundaries were defined as velocity inlet. The number of computational cells was approximately 6.93M for the resistance analysis. In the case of the self-propulsion simulation, fine mesh refinement was applied in the stern region to match the mesh resolution in the propeller rotation zone, resulting in a total of approximately 9.12M cells. The boundary conditions used for the full-scale POW analysis are illustrated in Figure 2(b). The inlet was set as a velocity inlet, the outlet as a pressure outlet, and the side boundaries were defined as symmetry. For modeling the rotating region, the sliding mesh method was applied with a rotation angle increment of 2 degrees to ensure convergence. The total number of cells used in this setup was approximately 3.09M. (a) (b) Figure 2. Computational domain & Boundary conditions: (a) Resistance and self-propulsion; (b) POW. 4 3 Validation of numerical simulation (Model-scale) Prior to conducting the full-scale simulations, numerical validation was performed using the model ship. As part of the verification, a GCI test was carried out at the design speed, and the results are summarized in Table 3. Both resistance and motions (sinkage and trim) demonstrated grid convergence within 1%. In addition, resistance analyses were conducted at six different speeds, and as shown in Table 4, the numerical results exhibited good agreement with the KRISO experimental data, with errors in both resistance and motion remaining within 4%. A simulation was conducted under low-speed conditions using the model ship. When comparing the flow fields at the design speed and low speed, it was clearly observed that the inflow velocity toward the stern significantly decreased under low-speed conditions (Figure 3(a)), resulting in a thicker boundary layer compared to that at the design speed (Figure 3(b)). Furthermore, analysis of the resistance coefficients at various speeds revealed that frictional resistance became the dominant component at lower speeds, leading to a decreasing trend in the difference between CTM and CVM (Figure 3(c)). Table 3. GCI of resistance coefficients and motions in calm water at model scale (Fr = 0.172) Case Number of grids CTM (x102) GCI21(%) Sinkage (σ/LPPx103) GCI21(%) Trim (deg) GCI21(%) Coarse 1.93M 3.969 0.78 0.1460 0.88 0.1859 0.30 Medium 3.84M 3.992 0.1470 0.1864 Fine 7.68M 4.007 0.1476 0.1867 Table 4. Comparison of CTM and ship motion at various speeds Particulars Velocity(knots) 𝐶𝑇𝑀∙103 𝜎/𝐿𝑝𝑝∙102 τ [deg] EFD (KRISO) 10 3.975 0.0703 0.0830 12 3.928 0.1023 0.1240 14 3.969 0.1401 0.1750 14.5 4.021 0.1513 0.1900 15 4.105 0.1635 0.2060 16 4.333 0.1875 0.2370 CFD (Present) 10 3.900 (1.89%) 0.0675 (1.80%) 0.0815 (3.99%) 12 3.884 (1.10%) 0.0990 (1.32%) 0.1224 (3.23%) 14 3.914 (1.38%) 0.1379 (1.80%) 0.1719 (1.61%) 14.5 3.992 (0.73%) 0.1470 (1.88%) 0.1864 (2.82%) 15 4.062 (1.04%) 0.1615 (1.29%) 0.2034 (1.25%) 16 4.295 (0.88%) 0.1869 (1.33%) 0.2338 (0.31%) 5 (a) (b) (c) Figure 3. Comparing the flow fields at the design speed and low speed: (a) Velocity vector; (b) Boundary thickness; (c) Variation of resistance components according to ship speed. In the model ship self-propulsion analysis, the direct rotation method was applied. As shown in Table 5, the numerical results exhibited good agreement with the experimental self-propulsion factors at the design speed, with deviations within 5%. The self-propulsion factors were analyzed under both the design and low-speed conditions. As illustrated in Figures 4(a) and 4(b), the stern CP distribution shows that, under low-speed conditions, an increase in stern pressure results in a tendency for the WTM to increase. Furthermore, the influence of propeller suction decreased at low speed, leading to a reduced low-pressure distribution around the propeller and a subsequent decrease in the 𝑡. Additionally, as the speed decreased, the advance coefficient was reduced, causing a drop in 𝜂𝑂. This, in turn, led to a decrease in 𝜂𝐷 compared to that at design speed. These findings demonstrate the effectiveness of the modelscale numerical simulation in evaluating self-propulsion performance under low-speed steady conditions. Table 5. Comparison of self-propulsion factors for model-scale KRISO (EFD) Present (CFD) Present (CFD) Vs (knots) 14.5 4 𝑊𝑇𝑀 0.437 0.418 (-4.3%) 0.468 𝑊𝑇𝑆 0.352 0.339 𝑡 0.209 0.201 (-3.8%) 0.184 𝜂H 1.217 1.234 𝜂R 1.012 1.006 (-0.6%) 1.022 𝜂O 0.561 0.536 𝜂D 0.686 0.676 6 (a) (b) Figure 4. Stern CP distribution of the model-scale at different ship speeds: (a) 14.5 knots; (b) 4.0 knots. 4 Results 4.1 Evaluation of full-scale resistance performance Based on the validated numerical simulation of the model-scale, the mesh system was extended to the fullscale model. To verify the mesh convergence of the full-scale simulation, a Grid Convergence Index (GCI) test was conducted under the design speed condition. The GCI method was employed to quantitatively assess the degree of numerical convergence [12]. The convergence test was performed based on mesh resolution, with the mesh conditions for resistance analysis categorized as fine, medium, and coarse. As shown in Table 6, the GCI test results indicated that both resistance and motion exhibited convergence errors of less than 1%, confirming that the results were within acceptable limits. Therefore, the medium mesh was selected as the full-scale resistance mesh configuration for this study. Table 6. GCI of resistance coefficients and motion in calm water at full scale (Fr = 0.172) Case Number of grids CTS (x103) GCI21(%) Sinkage (σ/LPPx103) GCI21(%) Trim (deg) GCI21(%) Coarse 3.50M 2.521 0.5 0.1439 0.3 0.1970 0.2 Medium 6.93M 2.536 0.1455 0.1972 Fine 13.7M 2.544 0.1460 0.1980 A numerical resistance analysis was performed for the full-scale ship in calm water. As shown in Figure 5, the flow field exhibited a Kelvin wave pattern consistent with that observed in previous studies [13] (Figure 5(a)). When comparing the wave profiles based on medium mesh resolutions for both the model and full-scale ships, satisfactory agreement was observed (Figure 5(b)). As shown in Figure 6, the CP distributions along the stern and sides of the hull also demonstrated similar trends. 7 (a) (b) Figure 5. Comparison of wave between the model and full-scale ship: (a) Wave elevation; (b) Wave profile. (a) (b) Figure 6. Comparison of Cp between the model and full-scale ship: (a) Model-scale [13]; (b) Full-scale (Present). To evaluate the scale effect at full-scale, the hull boundary layer distributions of the model and full-scale ships were compared. As shown in Figure 7, the boundary layer thickness was found to be reduced in full-scale case, which is attributed to the increased inflow velocity. As confirmed in Figure 7(c), the increased inflow velocity results in higher inflow values for the full-scale ship compared to the model-scale when examining the radial inflow velocity distribution on the wake plane. This led to wake contraction in the nominal wake field at full-scale, and the 𝑊𝑛 increased from 0.533 on the model-scale to 0.353 in the full-scale, as shown in Figure 7(d). In the stern CP distribution, the increased inflow velocity and reduced boundary layer thickness in the full-scale case resulted in a decrease in pressure loss at the stern (Figure 8). Consequently, the pressure difference between the bow and stern was reduced compared to the model ship, suggesting a potential reduction in viscous pressure resistance at full-scale. 8 (a) (b) (c) (d) Figure 7. Evaluation of scale effects between model and full-scale ships: (a) Boundary thickness of model; (b) Boundary thickness of full; (c) Vx/V According to wake radius; (d) Wake field. (a) (b) Figure 8. Comparison of stern CP distributions between the model and full-scale ship: (a) Model-scale; (b) Full-scale. 9 A full-scale resistance analysis was performed under calm water and low-speed conditions across six different ship speeds to examine the variation in resistance components. As shown in Figure 9, the analysis of the full-scale 𝑊𝑛 revealed a slight increase in the low-velocity region beneath the hub as the ship speed decreased, while no significant difference was observed in the vortex region. As presented in Table 7, the 𝑊𝑛 under low-speed conditions increased by approximately 9.3% compared to the design speed. (a) (b) (c) (d) Figure 9. Comparison of full-scale wake fields at different ship speeds: (a) 14.5knots; (b) 10.0knots; (c) 7.0knots; (d) 4.0knots. Table 7. Comparison of 𝑊𝑛(𝐹𝑢𝑙𝑙−𝑠𝑐𝑎𝑙𝑒) at different ship speeds Vs (knots) 𝑊𝑛(𝐹𝑢𝑙𝑙−𝑠𝑐𝑎𝑙𝑒) 14.5 0.3532 10.0 0.3621 7.0 0.3711 4.0 0.3862 In addition, a comparison was made between the full-scale wake fraction obtained through CFD and that extrapolated from model-scale results As shown in Equation (3), which is used for extrapolating the full-scale nominal wake fraction, 𝑊𝑛(𝑀𝑜𝑑𝑒𝑙−𝑠𝑐𝑎𝑙𝑒) refers to the nominal wake fraction extracted from the model-scale resistance analysis, and 𝑊𝑇𝑀 represents the effective wake fraction obtained from the model-scale self-propulsion analysis. 𝑊𝑇𝑆 is the value estimated by applying the ITTC extrapolation method given in Equation (4). The 𝑡 and 𝑊𝑇𝑀 were obtained from the model self-propulsion analysis. The form factor (1+k), calculated as 1.218 using the Prohaska method during the model resistance analysis, was applied. In addition, the roughness allowance ΔCF was determined using Equation (5). 𝑘𝑠 was set to the ITTC-recommended value of 150E-06 [14]. Accordingly, a comparative analysis was conducted between the 𝑊𝑛(𝐹𝑢𝑙𝑙−𝑠𝑐𝑎𝑙𝑒) values obtained from the full-scale resistance CFD and those calculated using Equation (3). 16 ⚫ To perform the full-scale self-propulsion analysis, the numerical methods for both model-scale and fullscale simulations were verified and validated. The model-scale resistance simulation was validated through comparison with experimental results. The validated model-scale grid was then extended to the full-scale configuration, and the GCI test confirmed convergence within 1%, demonstrating the reliability of the full-scale grid system. In addition, due to the influence of scale effects, the full-scale CR exhibited a decreasing trend compared to the model-scale results. ⚫ The evaluation of full-scale POW performance revealed that, due to the increase in Reynolds number, KT tends to increase while KQ decreases compared to the model-scale. Furthermore, a comparison of POW performance between the full-scale design and low rotational speeds showed similar results regardless of the rotational speed. This indicates that the POW performance at the design rotational speed can be reliably used to extract self-propulsion factors under low-speed conditions. ⚫ The differences in self-propulsion performance between the ITTC extrapolation method and full-scale CFD were evaluated. Contrary to the full-scale CFD results, the 𝑊𝑇 from the ITTC-1978 method was found to increase as speed decreased. When the 𝑊𝑇𝑆 extracted from full-scale CFD was applied using wake field scaling, the resulting 𝑊𝑛(𝐹𝑢𝑙𝑙−𝑠𝑐𝑎𝑙𝑒) showed good agreement with that obtained from the ITTC1978 method. ⚫ Under low-speed conditions, a reduction in inflow velocity and stern pressure loss led to an increase in the 𝑊𝑇𝑆 and a decrease in the 𝑡. As the effective wake fraction increased, 𝜂𝐻 also improved. However, due to the relatively lower advance ratio at low speeds, 𝜂𝑂 was reduced. In this study, full-scale numerical simulations were conducted under calm water and low-speed conditions, and the flow characteristics induced by Reynolds number differences were analyzed. Based on the evaluation of fullscale resistance and POW performance, the self-propulsion characteristics were estimated, providing a foundation for future research on predicting full-scale self-propulsion performance under low-speed conditions. In future work, added resistance analysis in regular head waves under low-speed conditions will be conducted, and the required engine power will be estimated using the load variation method. Acknowledgments This research was supported by a grant from the “Development and demonstration of data platform for AIbased safe fishing vessel design (RS-2022-KS221571)” funded by Ministry of Oceans and Fisheries, Republic of Korea, and the “Development of evaluation technology for ship’s performance in extreme environment” of the Korea Research Institute of Ships and Ocean Engineering Endowment Project, which was funded by the Ministry of Oceans and Fisheries (PES5460). References [1] Ø . Buhaug, J.J. Corbett, Ø . Endresen, V. Eyring, J. Faber, S. Hanayama, D.S. Lee, D. Lee, H. Lindstad, A.Z. Markowska, A. Mjelde, D. Nelissen, J. Nilsen, C. Pålsson, J.J. Winebrake, W. Wu, and K. Yoshida. Second IMO GHG Study 2009. International Maritime Organization (IMO), London, UK, April 2009. [2] H.G. Lim, K.Y. Han, and J.W. Lee. A study on the application of EEXI to ships exempted from GHG reduction measures. Journal of the Korean Society of Marine Engineering, 26(4):349–357, 2023. [3] P. 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