Study on the Estimation Method of Wind Resistance Considering Self-Induced Wind by Ship Advance Speed

Abstract

A numerical analysis of the wind load for the purpose of evaluating the wind resistance acting on a ship and the validity of the wind profile applied to determine the wind load coefficient were conducted. Through the evaluation of estimation results by a wind tunnel test, CFD analysis, and present semi-empirical formulae, it was recognized that the difference in estimation of ship resistance due to wind could not be ignored. In order to identify the main causes of the difference, extensive analyses were performed for a container, tanker, and LNG carrier. In particular, the estimation results for a container ship with two islands showed unreliable results. The main reason for the difference is that each method reflects the wind speed in the vertical direction differently, and the wind profile applied when considering the self-induced wind effect is not a uniform wind profile. In the calculation of wind resistance by self-induced wind, wind resistance estimation results differed by about 1.5% to 3.4% depending on the application of uniform or non-uniform wind profile. The total wind resistance acting on the vessel shall be divided into wind resistance from a stationary vessel without speed and wind resistance caused by the forward speed of the vessel in no wind conditions. Therefore, it is reasonable to apply a uniform wind profile to estimate wind resistance caused by the ship’s forward speed, while a wind profile that reflects the effect of changes in the ship’s vertical speed should be applied to estimate the wind resistance caused by the ship’s forward speed.

1. Introduction

IMO’s Marin Environmental Protection Committee (MEPC) is continuing its efforts to reduce greenhouse gas emissions from ships. As part of this policy, all ships newly built since 1 January 2013 are required to obtain an International Energy Efficiency Certificate and Energy Efficiency Design Index (EEDI). The MEPC adopted amendments to MARPOL Annex VI to significantly strengthen the Energy Efficiency Design Index (EEDI) “phase 3” requirements, and that amendment came into force on 1 April 2022. The demand for eco-friendly ship development and related regulations has gradually strengthened in recent years. For this reason, the need for scientific performance analysis is becoming more important than ever before, along with the demand for ship performance improvement itself. Accurate evaluation of performance and continuous development of eco-friendly vessels is considered the best way to ultimately improve the shipbuilder’s competitiveness in winning orders and is believed to increase customer reliability on ship performance. According to these market demands for eco-friendly ships, continuous efforts to improve ship performance have been conducted by developments of hull forms, propellers, energy-saving devices, etc.

In general, the ship’s performance verification stage can be divided into three major categories as follows. The first stage is to use the numerical simulation approach during the optimization process. The second stage is to perform the model test at the towing tank facility when the hull form, propeller, and rudder designs are completed. The last stage is to verify the performance, as well known, through the official sea trial test. A ship encounters diverse and complex sea conditions such as wind, waves, tides, currents, water temperature, and water depth during the sea trial. Therefore, various external forces acting on ships should be eliminated using scientific analysis methods to evaluate the speed–power performance accurately.

This paper deals with only wind-induced results among the resistance added by external environments, which is one of the critical external forces. There are two methods for estimating the additional resistance due to wind. The first method is to apply the wind tunnel test results or existing empirical formulas. Most empirical formulas [1,2] provide the data inferred from wind tunnel test results for various ship types. These wind tunnel test results used in the proposed empirical formulas did not correctly reflect the recent shape of the superstructure, and at the same time, it has a weakness in that the change of the wind speed along the vertical direction at the actual sea condition was also not considered. For this reason, it is not easy to obtain an accurate wind resistance coefficient for modern design ships. A direct way to solve the problem of empirical formulas is to use the wind tunnel test results. When it is difficult, a numerical calculation method can be considered as an alternative approach. Before using numerical calculation results, it is necessary to check the validity of the results. As a method to verify the validity, it is desirable to evaluate the difference between the numerical calculation and wind tunnel test results for a similar vessel and then perform numerical calculations for the target ship. Since wind resistance cannot be evaluated by the above model test or numerical analysis in the initial stage in which the shape of the superstructure of the ship is not determined, it is a reasonable alternative to apply the existing empirical formulas widely used so far such as the Blendermann chart [1], Isherwood and Fujiwara’s regressions [3,4], JTTC guideline [5] and ISO 15016:2015 methods [6].

On the other hand, depending on these methods, there is a significant difference in the added resistance due to wind evaluated during the sea trial, which affects the ship’s speed performance. The reason is that there is a difference in the wind coefficients evaluated or inferred from each method. In this paper, the main characteristics of evaluation methods are identified. In addition, the effect of wind speed distributions along the vertical direction above the waterline is also investigated. The problems of the existing methods are discussed, and alternative solutions are suggested.

2. Wind Resistance Estimation

The added wind resistance can be calculated using the following equation:

$${\mathrm{R}}_{\mathrm{A}\mathrm{A}}={{\displaystyle \frac{1}{2}}\mathsf{\rho }}_{\mathrm{A}}·{\mathrm{C}}_{\mathrm{A}\mathrm{A}}({\mathsf{\psi }}_{\mathrm{W}\mathrm{R}})·{\mathrm{A}}_{\mathrm{X}\mathrm{V}}·{{\mathrm{V}}_{\mathrm{W}\mathrm{R}}}^2$$

(1)

Here, ${\mathsf{\rho }}_{\mathrm{A}}$ is the air density, ${\mathrm{C}}_{\mathrm{A}\mathrm{A}}\left({\mathsf{\psi }}_{\mathrm{W}\mathrm{R}}\right)$ is the wind resistance coefficient depending on wind direction, ${\mathsf{\psi }}_{\mathrm{W}\mathrm{R}}$ is the relative wind direction, A_XV is the area of maximum transverse section exposed to the wind, and V_WR is the relative wind speed.

2.1. Empirical Methods

This section briefly describes the estimation methods of the wind resistance coefficient based on empirical methods and discusses the main features of each technique. The practical formula-based methods can be applied to estimating the wind resistance in cases where wind tunnel test results or CFD calculation results are not available. One way is to use the methods described ISO 15016:2015(E) [6]. In the JTTC guideline [5], the wind resistance coefficient in a headwind ${\mathrm{C}}_{\mathrm{A}\mathrm{A}0}$ and the wind direction coefficient k(ψ) are proposed based on wind tunnel test results for various ship types, such as tanker, cargo, container, and passenger ships. The wind resistance coefficient for each wind direction ${\mathrm{C}}_{\mathrm{A}\mathrm{A}}\left(\mathsf{\psi }\right)$ can be obtained by multiplying ${\mathrm{C}}_{\mathrm{A}\mathrm{A}0}$ and $\mathrm{k}\left(\mathsf{\psi }\right)$. One drawback of this method is that it is insufficient to obtain highly accurate values since two regression curves are only used to estimate ${\mathrm{C}}_{\mathrm{A}\mathrm{A}0}$ regardless of the type of ships. Blendermann [1] presented a diagram of the wind resistance coefficient for 22 different vessels based on wind tunnel test results under uniform flow conditions. Isherwood (1973) [3] proposed a set of formulas based on the results of wind tunnel tests performed under the uniform wind profile. The results of the wind tunnel test were represented by an empirical formula capable of estimating wind resistance for each wind direction using multiple regression techniques. For the purpose of classifying the shape of the upper structure of the ship, eight shape parameters were introduced and applied to the wind load coefficient estimation formulas.

In 2006, Fujiwara improved the existing wind resistance estimation method proposed [4] in 1998 by dividing the ship’s upper shape structure into accommodation, funnel, and other structures above deck. This method, as recommended by ISO 15016:2015(E), has been adopted and used as one of the reasonable methods. On the other hand, one of the relatively latest wind resistance calculation methods is considered to be the ISO 15016:2015(E) [6] based on the wind tunnel test results of STA-JIP [7]. The first advantage of ISO 15016:2015(E) [6] is that it is based on wind tunnel test results for various ship kinds, such as tanker ships, LNG carriers, car carriers, passenger ships, and general cargo ships. The second advantage is that a non-uniform wind profile, which is similar to the actual sea condition, was applied to the wind tunnel test instead of the uniform wind profile. For this reason, it is expected that the accuracy of wind resistance estimation results could be improved by the test results obtained under the non-uniform wind profile condition, as well as the direct application of test results for similar ships.

2.2. Wind Tunnel Test

The most accurate method for calculating wind resistance is to directly obtain the wind resistance coefficient through the wind tunnel test. Figure 1 shows the installation of the model for performing the model test. Wind resistance for each direction can be measured by rotating the model. The material of this model is generally made of wood or high-density polyurethane. The structure above the deck of the ship is manufactured in a simplified shape according to the scale ratio. In addition, the model surface has sufficient roughness to consider the effects of peeling points and friction resistance.

On the other hand, it is necessary to measure the wind resistance of the model ship by reproducing the wind velocity distribution similar to the condition experienced by the ship’s superstructure. The advantage of the wind tunnel test is that accurate wind resistance coefficients can be obtained by testing with the same shape as an actual ship. In terms of cost and time, it is true that it is burdensome to conduct wind tunnel tests for all projects. As an alternative way to solve this problem, there is a way to consider the difference between the CFD results of similar ships and the wind tunnel test results in the CFD results of the target ship. Through this process, it is expected that reliable CFD results for the subject ship can be obtained even if the wind tunnel test is not performed directly.

The wind tunnel test introduced in this paper was performed by Force Technology in Denmark. The wind tunnel is a boundary-layer wind tunnel (WT2) with 1.8~2.3 M height, 2.6 M width, and 20.8 M length. The max. flow velocity is 24.0 m/s. For the tests, a representative ocean wind velocity and turbulence profile were simulated in the wind tunnel, corresponding to the NPD wind profile described by NORSOK [8].

2.3. CFD Method

The application of CFD technology in estimating wind resistance coefficients has been actively increasing in recent years. In particular, a number of papers on the application of CFD have been published. The detailed descriptions of the technical information of the CFD technology used in this study are described. The coordinate system uses right-handed coordinates. The forward direction of the ship is the x-axis, the vertical direction is the z-axis, and the y-axis is positive to the starboard side. The origin is at the intersection of the centerline, the waterline plane, and the aft of the forward perpendicular (L_pp/2). The wind direction is set counterclockwise based on the wind coming from the ship’s bow. Details for the coordinate system are shown in Figure 2.

The average wind speed profile, which represents the wind intensity along the vertical direction, is very complex. However, it can usually be expressed in a simple form. Common examples include logarithmic profile, power-law profile, and NPD (Norwegian Petroleum Directorate) profile.

$$U\left(H\right)=U\left(H_R\right){\left({\displaystyle \frac{H}{H_R}}\right)}^{\alpha }$$

(2)

In this study, the wind speed distribution is simplified in the form of the power-law presented in Equation (2) and implemented in a numerical model using field function, as shown in Figure 3. Here, $U\left(H_R\right)$ is the wind speed at the reference height ($H_R)$ of 10 m, H is the height from the waterline, and $\alpha$ is the power-law index. The reference height for the wind resistance coefficients, $H_R$ is selected as the corresponding height for the wind resistance coefficient from wind tunnel tests (normally 10 m).

A commercial finite volume (FV)-based CFD code, Star-CCM+, is used. The trimmer mesh with a prism layer is adopted to generate the computational domain. It provides a robust and efficient way to generate a high-quality mesh for simple shapes and complex geometries. Figure 4 illustrates mesh distributions for a modern container ship as an example. The mesh resolution is much more refined around the model body to achieve an accurate solution. In the simulation, the trimmer mesh was used to generate the computational domain and regions near the target vessel, which was composed of 20 million cells. It provides a robust and efficient method of producing a high-quality mesh for both simple and complex structures. The domain size is determined in consideration of the spatial area size proposed by Hall et al. (2003) [9], Blocken et al. (2003) [10], and Franke et al. (2004) [11] to avoid flow disturbances and blockage effects as described in Figure 5. Hall et al. (2003) [9] discuss the evaluation of computational fluid dynamics modeling of gas dispersion near buildings and complex terrain, focusing on uncertainties and applicability. Blocken et al. (2003) [10] discuss the importance of considering the pedestrian wind environment when designing buildings, covering topics such as wind pressure on buildings, statistical concepts applied to wind loading of structures, and wind tunnel study of wind velocities in passages between and through buildings. Franke et al. (2004) [11] provide recommendations for using Computational Wind Engineering in wind engineering tasks, focusing on statistically steady simulations of pedestrian wind in built urban areas, emphasizing the importance of choosing an appropriate turbulence model, selecting the computational domain, and determining the spatial resolution of the numerical solution to ensure reliable results.

As shown in Figure 5, the upstream boundary is set to a velocity inlet, while the downstream boundary is set to a pressure outlet. The bottom is treated as a slip-wall. The y⁺ range is applied less than 60. Simulations are performed only on the model scale. The air density and viscosity used in the computation are 1.18415 kg/m³ and 1.56 × 10⁻⁵ m²/s, respectively. For the turbulence model, the Shear Stress Transport k-ω (SST k-ω) model is used. The convection and diffusion terms are discretized by the second-order upwind and second-order central difference scheme. For the pressure-velocity coupling, the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm is adopted. To prevent the flow disturbance and blockage effect, the domain size was determined, as shown in Figure 5. The upstream of the lateral sides was set to be the velocity inlet, whereas the downstream boundary was set to be the pressure outlet. The bottom was treated as a slip-wall condition.

3. Comparison of Wind Resistance Coefficients

In this chapter, wind resistance coefficients for the crude oil tanker (COT), liquefied natural gas carrier (LNGC), and container vessel (CONT) are estimated using the methods described in Chapter 2, and the results are compared. The structure features above the waterline, and main particulars for subject vessels are presented in Figure 6 and

Table 1. Principal particulars of subject vessels.

COT		Design	Ballast
Length between perpendiculars	LBP (m)	264.0	264.0
Breadth	B (m)	48.00	48.00
Mean draft	TM (m)	16.00	7.80
Longitudinal windage area	AT (m2)	801.00	1195.00
Transverse windage area	AS (m2)	2295.00	4550.00
LNGC		Design	Ballast
Length between perpendiculars	LBP (m)	280.00	280.00
Breadth	B (m)	47.80	47.80
Mean draft	TM (m)	11.65	9.50
Longitudinal windage area	AT (m2)	1390.00	1490.00
Transverse windage area	AS (m2)	6350.00	6950.00
CONT		Ballast	Ballast
Length between perpendiculars	LBP (m)	317.00	286.00
Breadth	B (m)	45.80	48.20
Mean draft	TM (m)	8.05	7.10
Longitudinal windage area	AT (m2)	2070.00	2060.00
Transverse windage area	AS (m2)	6480.00	6430.00
Remark	-	Single-island type	Two-island type

, respectively. As shown in Figure 6a, the accommodation and funnel are located near the stern region without additional equipment in the COT case. In addition, there are noticeable differences in the area exposed above the waterline depending on the design and ballast draught conditions.

In the case of LNGC, as shown in Figure 6b, a cargo dome and an engine machinery room are additionally located in front of the accommodation. Therefore, it is thought that the size and shape of the Cargo dome and engine machine room will have a meaningful effect on wind resistance. On the other hand, the influence of the draft difference is expected to be insignificant. As shown in Figure 6c, the main features of container ships can be divided into two types: single-island and two-island. The types surveyed in the present study are (1) a single-island type applied to a mid-sized container ship class smaller than 8000 TEU and (2) a two-island type applied to a large-sized container ship class larger than 16,000 TEU. In the single-island CONT, the accommodation and funnel are usually located in the stern region, as shown in the lower part of Figure 6c. On the other hand, the main characteristic of the two-island CONT is that the accommodation and funnel are separated back and forth from the midship, as shown in the upper part of Figure 6c. In addition, a lashing bridge system is installed to fasten the container boxes, which causes additional resistance.

Figure 7 indicates longitudinal wind resistance coefficients C_X estimated using seven different methods for the COT. The x-axis represents the absolute wind direction, and the y-axis is the wind resistance coefficient defined below:

$$C_X={\displaystyle \frac{F_X}{\frac{1}{2}{\rho }_A{V}_{H_R}^2{A}_T}}$$

(3)

Here, ${\rho }_A$ is the air density, $V_{H_R}^2$ is the wind speed at the reference height of 10 m, A_T is the frontal area exposed to the wind, and F_X is the wind resistance acting on structures above the waterline.

The results based on the JTTC chart [5] show that C_X = ${(\mathrm{C}}_{\mathrm{A}\mathrm{A}}\left(\mathsf{\psi }\right)$) values tend to be larger than those of other methods. The main reason for the difference is that, regardless of ship type, only two curves are applied to obtain the wind resistance coefficient in a headwind (${\mathrm{C}}_{\mathrm{A}\mathrm{A}0}$). In contrast, the wind resistance coefficients with Blendermann chart [1] are relatively small compared to other methods. In addition, wind resistance coefficients estimated through Isherwood’s regression formula [3] have significant differences from the wind tunnel test results in the 30- to 60-degree ranges of the wind direction. The wind resistance coefficient estimated through Fujiwara’s regression formula [4] and published in ISO 15016:2015(E) [6] shows more accurate results than the previous method, but it is still different from the present wind tunnel test results considering the dynamic pressure effect. The main reason for this difference is what kind of wind profile is applied in the wind tunnel test. In order to obtain more realistic wind resistance results from the regression formula, the results of the wind tunnel test conducted under a uniform wind profile should be included in the regression formula. Meanwhile, the CFD simulation results confirmed that the Cx distribution results were well matched to the wind tunnel test results performed under the non-uniform wind profile. For this reason, the CFD analysis approach could be considered as an alternative for wind tunnel testing.

Figure 8 shows C_X distributions for the LNGC obtained through five different methods, which are similar except for the results by Isherwood’s regression formula [3]. It is believed that the effects of structures, such as the cargo hold, engine machinery room, and deck store, are not properly reflected. On the other hand, like the COT case, CFD analysis results show the best consensus with wind tunnel test results. Figure 9 illustrates C_X distributions for the CONT, including the single-island and the two-island types, respectively. Here, ISO 15016:2015(E) [6] data are a wind resistance coefficient for the single-island CONT. First, wind resistance coefficients for the single-island CONT are noticeably scattered into two large groups. CFD analysis and ISO 15016:2015(E) [6] results are consistent with the wind tunnel test results.

However, C_X distributions estimated using regression formulas proposed by Fujiwara and Isherwood and Blendermann’s chart [1] differ from the wind tunnel test results. The main reason is that the effective dynamic pressure due to wind speed change along the vertical direction is not considered correctly. In addition, it seems that the height of the superstructure of the ship used in the estimation formulas is quite a bit higher than that of present conventional ships.

The difference in wind resistance coefficients between the wind tunnel test and other estimation methods for the two-island CONT is more pronounced. In particular, C_X distributions between 0 and 60 degrees, which are critical wind directions during the sea trial, show a significant difference. The reason for this is that existing methods were proposed based on wind tunnel test data for the single-island CONT. As shown in Figure 10, however, the accommodation and funnel are separated in the two-island CONT. As a result, additional pressure is applied to the funnel. Unfortunately, existing regression methods do not take these factors into account. Therefore, it is desirable to use wind tunnel test or CFD analysis data to evaluate the quantitative wind resistance for the two-island CONT. Apart from this, ISO 15016:2015(E) [6] should derive reasonable wind coefficients for two island CONT vessels and supplement the results by reflecting them in the present proposed method.

The differences in wind resistance coefficients according to each estimation method, including the wind tunnel test, for Tanker, LNGC, and CONT were investigated. The distribution of wind resistance coefficients by the wind tunnel test and CFD analysis is confirmed to match well, but the deviation in terms of quantitative prediction shows some differences. In particular, there is a limitation in predicting the wind resistance coefficient for a CONT with a two-island superstructure.

In conclusion, if it is necessary to estimate the quantitative value of the wind resistance coefficient, the wind tunnel test is considered a reasonable approach to obtain the correct wind resistance coefficient. In case the wind tunnel test results are not available, the CFD approach could be the next best option. On the other hand, since the aforementioned effective dynamic effects cannot be included in other empirical methods, such as Blenderman charts [1] and Isherwood and Fujiwara’s regression equations [3,4], it is considered a reasonable approach to apply correction coefficients to consider the dynamic effects. The proposed correction coefficient increases the original wind load distribution according to the ship’s upper structure height.

4. Added Wind Resistance Prediction by Each Methodology

Added resistance due to wind considering wind speed profile can be obtained in Equation (4). The first term on the right-hand side of the equation means the wind resistance caused by the relative wind speed. It is determined by the absolute wind speed and advancing ship speed. The second term indicates the air resistance naturally caused by a ship’s forward speed in a calm sea. Thus, the added resistance due to wind means the first term, excluding the second term. In this case, the value of $C_X^{\prime }\left(0\right)$ on the second term also includes the wind speed profile effect. Here, ${\rho }_A$ is the air density, $C_X^{\prime }\left(\mathsf{\psi }\right)$ is the wind resistance coefficient considering the wind speed profile, ${\mathsf{\psi }}_R$ is the relative wind direction, ${\mathrm{A}}_T$ is the frontal projection area, ${\mathrm{V}}_R$ is the relative wind speed, and ${\mathrm{V}}_S$ is the vessel speed.

Meanwhile, when estimating the added wind resistance using C_X values obtained from Blendermann chart [1] and Isherwood, and Fujiwara’s regression formulas [3,4], it can be calculated as Equation (5). Here, ${\mathrm{C}}_{\mathrm{X}}\left(\mathsf{\psi }\right)$ and ${\mathrm{C}}_{\mathrm{X}}\left(0\right)$ are values obtained under the uniform flow condition; $\mathsf{\alpha }$ of Equation (5) is a correction coefficient that reflects the effect of the effective dynamic pressure that can be calculated as Equation (6). Here, n is the power function index, ${\mathrm{H}}_{\mathrm{R}}$ is the reference height of 10 m, and H is the maximum height of the ship structure.

$${\mathrm{R}}_{\mathrm{A}\mathrm{A}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{A}}{\mathrm{C}}_{\mathrm{X}}^{\prime }\left({\mathsf{\psi }}_{\mathrm{R}}\right){\mathrm{A}}_{\mathrm{T}}{\mathrm{V}}_{\mathrm{R}}^2}{2}}-{\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{A}}{\mathrm{C}}_{\mathrm{X}}^{\prime }\left(0\right){\mathrm{A}}_{\mathrm{T}}{\mathrm{V}}_{\mathrm{S}}^2}{2}}$$

(4)

$${\mathrm{R}}_{\mathrm{A}\mathrm{A}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{A}}{\mathrm{C}}_{\mathrm{X}}\left(\mathsf{\psi }\right){\mathrm{A}}_{\mathrm{T}}{\mathrm{V}}_{\mathrm{R}}^2}{2}}\mathsf{\alpha }-{\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{A}}{\mathrm{C}}_{\mathrm{X}}\left(0\right){\mathrm{A}}_{\mathrm{T}}{\mathrm{V}}_{\mathrm{S}}^2}{2}}\mathsf{\alpha }$$

(5)

$$\mathsf{\alpha }={\displaystyle \frac{\mathrm{n}}{\mathrm{n}+2}}{\left({\displaystyle \frac{\mathrm{H}}{{\mathrm{H}}_{\mathrm{R}}}}\right)}^{{\displaystyle \frac{2}{\mathrm{n}}}}$$

(6)

Figure 11 shows the ratio of added wind resistance (%R_AA) to total resistance at three different Beaufort (BF) scales in the ballast draught condition for the COT. In Figure 11, the x-axis represents the absolute incident angle of the wind, and the y-axis represents the %R_AA defined as Equation (7). Here, R_AA is the added wind resistance and ${\mathrm{R}}_{\mathrm{T}\mathrm{S}}$ is the total resistance of the ship in a calm sea. In Figure 11, the maritime conditions applied to the comparative calculation assumed Beaufort 2, 3, and 4 which are the most frequently encountered sea conditions during the official sea trial.

$${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}={\displaystyle \frac{{\mathrm{R}}_{\mathrm{A}\mathrm{A}}}{{\mathrm{R}}_{\mathrm{T}\mathrm{S}}}}$$

(7)

As shown in Figure 11, the results of %R_AA estimated by the JTTC chart [5] are predicted to be relatively large values, while the results using ISO 15016:2015(E) [6] data and the Blendermann chart [1] tend to have small values compared to the results of the wind tunnel test data. The forward speed and total resistance of the ship applied to the comparison calculation are 14.5 Knots and 741 KN. The added wind resistance estimation result (%Raa) by Isherwood’s regression equation is relatively well matched with the wind tunnel test result when the absolute wind angle is within 0 to 30 degrees but shows an increased value than the wind tunnel test result in absolute wind angle of 30 to 120 degrees. Meanwhile, the results of the added resistance estimation by Fujiwara’s regression analysis [4] or CFD analysis showed that the defined %Raa by Equation (7) was similar to those results obtained by the wind tunnel test. The range of the error ratio (%E) for each method is summarized in

Table 2. Comparisons of error ratio (%E), COT.

Beaufort Scale		BF2	BF3	BF4
Error ratio, %E (%)	Wind Tunnel	-	-	-
	JTTC Chart [5]	−0.5~+0.9	−0.7~+1.9	−0.8~+3.9
	ISO 15016:2015(E) [6]	−0.8~+0.1	−1.5~+0.1	−2.8~+0.2
	Blendermann [1]	−0.7~+0.3	−1.4~+0.5	−2.5~+0.6
	Isherwood [3]	−0.1~+0.2	−0.1~+1.2	−0.1~+3.3
	Fujiwara [4]	−0.4~+0.3	−0.7~+0.5	−1.3~+0.9
	CFD	−0.1~+0.1	−0.2~+0.3	−0.2~+0.5

. Here, %E is defined as Equation (8).

$$\mathrm{\%}\mathrm{E}={\displaystyle \frac{{(\mathrm{R}}_{{\mathrm{A}\mathrm{A}}_{\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h} \mathrm{m}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{d}}}-{\mathrm{R}}_{{\mathrm{A}\mathrm{A}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d} \mathrm{t}\mathrm{u}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l} \mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t} \mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}})}{{\mathrm{R}}_{{\mathrm{A}\mathrm{A}}_{\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{d} \mathrm{t}\mathrm{u}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l} \mathrm{t}\mathrm{e}\mathrm{s}\mathrm{t} \mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}}}}}\times 100$$

(8)

As presented in Table 2, the %E range obtained using CFD analysis results showed the best agreement with the wind tunnel test results. Conversely, the most significant difference compared with the wind tunnel test results occurred in the case of applying the wind resistance coefficients extracted from the JTTC chart [5]. Even if the wind resistance coefficients proposed by ISO 15016:2015(E) [6] are recent results, it seems unreasonable to conclude that the result is more consistent with the wind tunnel test results, particularly from a quantitative point of view, compared to other existing methods.

In the same way, the magnitudes of added wind resistance are compared in the ballast draught condition for the LNGC. The speed and total resistance of LNGC selected for the comparative review are 19.5 Knots and 1324 KN, respectively.

As shown in Figure 12, it can be seen that the ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ by Isherwood’s regression formula is significantly smaller than the ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ by other estimation methods. The main cause of these results seems to be the difference due to the fact that the ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ estimated through Isherwood’s regression formula [3] does not take into account the effects of the cargo hold, engine mechanical room, and deck store. On the other hand, it was confirmed that ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ by Fujiwara’s regression formula [4] and ISO 15016:2015(E) [6] method were somewhat in good agreement with the wind tunnel test results. The quantitative aspect shows that the ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ by CFD analysis method matches well the wind tunnel test result, as in the results of the COT discussed above. The range of %E according to each method is summarized in

Table 3. Comparisons of error ratio (%E), LNGC.

Beaufort Scale		BF2	BF3	BF4
Error ratio, %E (%)	Wind Tunnel	-	-	-
	ISO 15016:2015(E) [6]	−0.3~+0.2	−0.5~+0.9	−0.9~+2.0
	Isherwood [3]	−0.8~+0.6	−1.4~+0.8	−2.5~+1.1
	Fujiwara [4]	+0.0~+0.5	+0.0~+0.8	−0.1~+1.3
	CFD	−0.2~+0.4	−0.4~+0.4	−0.6~+0.3

. As shown in the summarized results, it may be seen that the results of Fujiwara’s regression formula [4] are closer to the wind tunnel test results than the results of ISO 15016:2015(E) [6].

For the single-island container ship, ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ distributions and %E ranges are investigated, and results are described in Figure 13 and

Table 4. Comparisons of error ratio (%E), single-island CONT.

Beaufort Scale		BF2	BF3	BF4
Error ratio, %E (%)	Wind Tunnel	-	-	-
	ISO 15016:2015(E) [6]	−0.7~+0.8	−1.3~+1.0	−2.3~+1.2
	Blendermann [1]	−1.3~+1.0	−2.4~+1.6	−4.3~+2.1
	Isherwood [3]	−0.9~+0.5	−1.7~+0.8	−3.0~+1.1
	Fujiwara [4]	−1.5~+1.2	−2.7~+1.8	−4.7~+2.3
	CFD	−0.4~+0.3	−0.7~+0.5	−1.3~+1.1

. In addition to the COT and LNGC results described above, ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ analysis by CFD was found to be closest to the wind tunnel test results even on single-island and two-island container ships and is considered sufficiently reliable results from a quantitative perspective. Except for the CFD method, it would be difficult to say that the results of other methods are similar to those of the wind tunnel test. In addition, as the Beaufort scale increased, it was confirmed that the difference between other estimation methods and wind tunnel tests increased. Among the methods of estimating results by regression or chart, the ISO 15016:2015(E) [6] method is considered an alternative estimation method similar to the wind tunnel test result. Figure 14 and

Table 5. Comparisons of error ratio (%E), two-island CONT.

Beaufort Scale		BF2	BF3	BF4
Error ratio, %E (%)	Wind Tunnel	-	-	-
	ISO 15016:2015(E) [6]	−2.2~+1.7	−4.2~+2.6	−7.7~+3.4
	Blendermann [1]	−3.5~+2.5	−6.5~+3.9	−12.0~+5.1
	Isherwood [3]	−3.3~+2.2	−6.3~+3.3	−11.4~+4.4
	Fujiwara [4]	−3.6~+2.8	−6.8~+4.2	−12.1~+5.6
	CFD	−0.7~+0.5	−1.2~+0.8	−2.1~+1.0

show the ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ distribution and %E ranges for two-island container ships with accommodation located near midship. The ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ and %E mean a difference between a result by each estimation method and a result by a wind tunnel test. An important feature shown in Figure 14 is that the CFD analysis results are relatively consistent with the wind tunnel test results, but the results of other estimation methods are significantly different from the wind tunnel test or CFD results. In particular, in the case of two-island container ships, there is a problem with accuracy in using other estimation methods except for wind tunnel test and CFD analysis. As shown in Table 5, in the case of the Blendermann chart [1], the maximum error (%E) identified in this study is 12%. Therefore, it is recommended to accurately evaluate additional wind resistance using a wind tunnel test or CFD method.

From this point of view, especially for two-island container ships, the present ISO 15016:2015(E) [6] method should be revised as soon as possible to improve wind resistance.

5. Considerations on an Air Resistance Component Caused by an Advancing Ship

The added resistance due to wind in the actual sea condition can be calculated by excluding the self-induced wind resistance caused by the ship’s advancement at a certain speed from the total wind resistance acting on the ship. The formulas for added resistance due to wind are described by Equations (4) and (5). The issue to be examined in these two formulas is that the concept of effective dynamic pressure (wind profile) is applied equally to evaluating the total wind speed resistance acting on the ship and the self-induced wind resistance caused by the advance speed of the ship itself. The meanings of C(x′) and C(x) applied to Equations (4) and (5) are explained as follows. Wind Resistance Coefficient (Cx’) of Equation (4) is the result of including the wind speed profile effect, and wind resistance Coefficient (Cx) applied to Equation (5) is the result of applying a uniform wind profile excluding the wind speed profile effect. However, the wind resistance in Equation (5) is the wind resistance included wind speed profile effect by adaption of the correction factor (α) specified in Equation (6). As a result, both the wind resistance results of Equations (4) and (5) include an effective dynamic pressure (wind profile).

Therefore, predicting the wind resistance due to the forward speed of the ship in accordance with the second term of Equations (4) and (5) means that the effective dynamic pressure effect is included in the self-induced wind resistance calculation result. However, the self-induced wind resistance caused by the ship’s forward speed itself in the no-wind condition is not related to the effective dynamic pressure. For this reason, the self-induced wind resistance results calculated by the second term of Equations (4) and (5) may differ from the actual phenomenon. Through this study, it is considered a physically valid approach to apply a uniform wind profile for the reasonable evaluation of self-induced wind resistance that occurs when a ship moves forward at an arbitrary speed. For this reason, Equations (4) and (5) can be replaced by Equations (9) and (10), which are more relevant to actual physical phenomena.

$${\mathrm{R}}_{\mathrm{A}\mathrm{A}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{A}}{\mathrm{C}}_{\mathrm{X}}^{\prime }\left(\mathsf{\psi }\right){\mathrm{A}}_{\mathrm{T}}{\mathrm{V}}_{\mathrm{R}}^2}{2}}-{\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{A}}{\mathrm{C}}_{\mathrm{X}}\left(0\right){\mathrm{A}}_{\mathrm{T}}{\mathrm{V}}_{\mathrm{S}}^2}{2}}$$

(9)

$${\mathrm{R}}_{\mathrm{A}\mathrm{A}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{A}}{\mathrm{C}}_{\mathrm{X}}\left(\mathsf{\psi }\right){\mathrm{A}}_{\mathrm{T}}{\mathrm{V}}_{\mathrm{R}}^2}{2}}\mathsf{\alpha }-{\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{A}}{\mathrm{C}}_{\mathrm{X}}\left(0\right){\mathrm{A}}_{\mathrm{T}}{\mathrm{V}}_{\mathrm{S}}^2}{2}}$$

(10)

CFD simulations are performed under uniform wind speed conditions for each ship to investigate their effects. Wind resistance coefficients are shown in Figure 15, Figure 16 and Figure 17, with the results considering the wind speed profile. The effects caused by these expressions can be evaluated using the C_X value at the heading angle of 0 degrees. As a reference, however, C_X distributions of all wind directions are plotted together for reference.

Figure 15 shows the C_X distribution corresponding to each wind profile condition for the COT. As a result, under the uniform wind speed condition, the air resistance coefficient at the heading angle under the uniform wind speed condition decreases by about 27%, where the C_X value changes from −0.930 to −0.679. As shown in Figure 18, the main reason is that the high-pressure area acting on the superstructure decreases compared to that under the wind speed profile condition. Figure 16 illustrates the ${\mathrm{C}}_{\mathrm{X}}$ distribution for the LNGC. The values at the wind direction of 0 degrees are −0.812 and −0.583 for each wind profile condition, respectively. So, the resistance coefficient decreases by about 28%, similar to that in the COT case. Figure 19 indicates the pressure distribution when the wind blows from the front, i.e., headwind. Again, as observed in the COT case, the overall pressure distribution tends to decrease. For reference, differences in the pressure distribution between port and starboard are observed by structure arrangements such as engine machinery room and deck store. The Cx distribution of the two-island container ship is shown in Figure 17; at the true wind angle of 0 degrees, the corresponding Cx values for each wind profile tend to decrease by about 59% from −1.387 to −0.875 according to CFD analysis. In the case of container ships, the ratio of wind resistance to the total resistance is a relatively higher overall resistance than other ships. It can be expected that the calculation results according to the second term on the right of Equations (9) and (10) may differ significantly depending on the applied wind profile. In Figure 20, the difference in pressure distribution for container ships with two islands is shown separately in the case of uniform and non-uniform profile conditions. The tendency is similar to that of other ships.

In this paper, the considering method for the air resistance component caused by the self-induced wind was discussed, and the wind resistance coefficient (Cx) with pressure distribution on the superstructure of the ship based on CFD result was evaluated. Through this study, it is found that C_X values under uniform wind conditions tend to decrease by about 30% compared to the values considering the wind profile. The effect of these Cx variations on added resistance by wind was investigated through Equation (4) and Equation (10). Here, the C_X value under the wind speed profile condition was applied to the second term on the right-hand side in Equation (4), and the C_X value under the uniform condition was used for the second term on the right-hand side in Equation (9). As shown in Figure 21, the ratio of added wind resistance is relatively large due to the difference in C_X values at the wind direction of 0 degrees when using the results under the uniform wind speed condition. These trends are similarly maintained regardless of absolute wind direction. The difference in ${\mathrm{\%}\mathrm{R}}_{\mathrm{A}\mathrm{A}}$ is relatively significant for the container ship compared to other vessels due to a larger portion of air resistance. Also, the effects of changes in ship speed were investigated by applying the same method. As shown in Figure 22, differences in the added wind resistance for the COT, LNGC, and CONT are approximately 1.4%, 1.5%, and 3.4%, regardless of changes in ship speed.

6. Conclusions

In this study, a new approach method based on CFD analysis results was proposed to make up for the shortcomings of semi-empirical formulae (e.g., ISO 15016:2015(E), Blenderman Chart [1], and Isherwood Chart [3], etc.) that have been widely applied to wind resistance estimation. It was confirmed that the results of the newly proposed method based on CFD were most similar to those of the wind tunnel test. In particular, in the case of Container vessels with two islands, the estimation result by CFD was −1.2~+0.8% compared to the wind tunnel test, while the estimation result by ISO15016:2015(E) [6] method showed −4.2~+2.6% error rate(%E) at Beaufort Scale 3. The main cause of the error is that the wind speed distribution according to the vertical direction of the upper structure of the ship is not uniform under the actual sea conditions, and also, the present methods based on semi-empirical formulae methods are not sufficient to reflect the vertical speed distribution. Another important finding in this paper is the validity of wind profiles applied to self-induced wind resistance estimation of existing methods. A reasonable method to calculate wind resistance by self-induced wind is to apply wind resistance coefficients derived under uniform wind profile conditions instead of adopting wind coefficients obtained from non-uniform wind profiles. The difference in wind load coefficients due to the application of uniform wind profile or non-uniform wind profile was estimated for COT, LNGC, and Container vessels with two islands. The estimated differences in wind load coefficient were 27% to 59% at headwind conditions. The effect of the wind coefficient by the proposed method on the total wind resistance was found to be about 1.4 to 3.4%. The basic philosophy of this study is that the total wind acting on a ship should be divided into the wind caused by the ship’s advanced speed and the wind blowing into a stationary ship under no wind conditions. To this end, the problems of the semi-empirical method for various existing ships were reviewed and discussed. In conclusion, it would be desirable to obtain the wind load coefficient using the CFD approach or wind tunnel test. In addition, in order to verify and supplement the validity of the current presentation method, it is considered that a wind tunnel test and CFD analysis for various additional ships are necessary.