Development of Virtual Disk Method for Propeller Interacting with Free Surface

Abstract

As the environmental regulations of the International Maritime Organization (IMO) become more stringent, the accurate prediction of ship propulsion performance has become essential. Under ballast conditions where the draft is shallow, the propeller approaches the free surface, causing complex phenomena such as ventilation and surface piercing, which reduce propulsion efficiency. The conventional virtual disk (VD) method cannot adequately capture these free-surface effects, leading to deviations from model propeller results. To resolve this, a correction formula that accounts for the advance ratio (J) and submergence ratio ($h/D$) has been proposed in previous studies. In this study, the correction formula was simplified and implemented in a CFD environment using a field function, enabling dynamic adjustment of body force based on time-varying submergence depth. A comparative analysis was conducted between the conventional VD, modified VD, and model propeller using POW and self-propulsion simulations for an MR tanker and SP598M propeller. The improved method was validated in calm and regular wave conditions. The results showed that the modified VD method closely matched the performance trends of the model propeller, especially in free surface-interference conditions (e.g., $h/D$ < 0.5). Furthermore, additional validations in wave-induced self-propulsion confirmed that the modified VD method accurately reproduced the reductions in wake fraction and thrust deduction coefficient, unlike the overestimations observed with the conventional VD. These results demonstrate that the modified VD method can reliably predict propulsion performance under real sea states and serve as a practical tool in the early design stage.

1. Introduction

The International Maritime Organization (IMO) has been continuously reinforcing its strategies to reduce greenhouse gas (GHG) emissions generated by the maritime industry. In July 2023, the IMO adopted the “2023 IMO Strategy on Reduction of GHG Emissions from Ships,” aiming to achieve net-zero GHG emissions from international shipping by 2050. The intermediate targets are set to reduce emissions by at least 20% by 2030 and by at least 70% by 2040. Additionally, the IMO recommends replacing at least 5% of total energy usage with zero- or low-emission technologies, fuels, or energy sources by 2030 [1]. As concrete measures, the Energy Efficiency Existing Ship Index (EEXI) and the Carbon Intensity Indicator (CII) have been introduced for existing vessels. Notably, the CII provides a quantitative rating (A to E) based on the annual CO₂ emissions per transport work, enabling an objective assessment of operational efficiency. As such, accurate prediction and optimization of ship propulsion systems have become essential not only for environmental compliance but also for ensuring commercial competitiveness.

To ensure the effectiveness of such international regulations, it is crucial to develop technologies capable of accurately predicting ship performance under actual operating conditions. Reliable performance prediction requires the integration of multiple factors, including waves, currents, vessel motions, and propeller–hull interactions. In particular, the accurate estimation of propeller performance, which governs propulsion efficiency, cannot be achieved by conventional methods that predict only open water thrust. Therefore, establishing advanced numerical analysis techniques based on Computational Fluid Dynamics (CFD) has become increasingly important for detailed propulsion performance evaluation.

With the rapid development of CFD techniques in recent years, studies have expanded beyond calm water conditions to include realistic sea states involving wave environments. CFD simulations based on Reynolds-Averaged Navier–Stokes (RANS) equations are particularly advantageous for predicting nonlinear fluid–structure interactions under wave conditions. Tezdogan et al. [2] demonstrated that CFD predictions under regular waves show strong correlation with experimental data, validating the reliability of CFD as a tool for simulating realistic ocean conditions [3].

Accurate prediction of ship propulsion performance depends primarily on the precise calculation of thrust and torque generated by the propeller. Traditionally, experimental evaluations using model propellers have been widely employed, and in CFD environments, detailed modeling of actual propeller geometries has also been commonly utilized. However, such approaches require extremely fine mesh resolutions and small time-steps, leading to high computational costs and excessively long simulation times. As an alternative, the Virtual Disk (VD) method has been proposed. In this approach, the actual propeller geometry is replaced by a uniformly distributed momentum source over a circular disk. The fundamental assumption is that the mean thrust and torque produced by the propeller can be represented without explicitly resolving blade-scale flow structures. Within this framework, the governing equations are solved with additional momentum source terms in the axial and tangential directions, derived either from propeller open-water characteristics or from analytical distributions such as Goldstein’s optimum circulation. By avoiding the need for detailed blade geometry in the computational grid, the VD method substantially reduces computational cost. Due to this efficiency, the VD method has been widely employed in early-stage ship design, parametric studies of hull–propeller interactions, and performance assessments under calm and wave conditions. Nevertheless, because free-surface effects such as ventilation and partial submergence are not inherently captured, modifications are required to extend its applicability to shallow-draft and ballast conditions. Recently, the VD method has also been applied to more complex scenarios, including combined wave conditions, asymmetric hull forms, and the presence or absence of energy-saving devices (ESDs). Tokgoz et al. [4] applied the VD method under wave conditions to perform self-propulsion analysis and demonstrated good agreement with experimental wake and thrust results [5,6].

The VD method performs well when the propeller is deeply submerged; however, when the propeller operates near the free surface, physical limitations can lead to significant discrepancies between the predicted and actual propulsive performance. This is particularly evident under ballast conditions, where the ship’s draft is reduced and the submergence depth of the propeller becomes shallower. Consequently, complex phenomena such as ventilation, surface piercing, and distorted particle acceleration occur. These free-surface interactions are difficult to capture with conventional VD models, often resulting in overestimation or underestimation of thrust and torque compared to experimental measurements. Ha [7] reported that, under ballast conditions, the conventional VD method failed to reproduce actual propeller performance accurately, as observed in self-propulsion tests [8].

To address this issue, recent studies have proposed correction models that use advance ratio (J) and submergence ratio ($h/D$) as key parameters. Eom et al. [9] developed a correction model compatible with CFD implementation, which was applied as a field function in STAR-CCM+. This correction dynamically adjusts the body force distribution based on real-time variations in propeller submergence, thereby enabling time-dependent correction of thrust and torque. Generally, under calm water conditions, there is a minimal difference in performance between the model propeller and the conventional VD method. However, when the submergence ratio ($h/D$) falls below 0.5, the discrepancy becomes significant. The conventional VD method does not account for unsteady flow effects caused by free-surface interactions, such as cavitation and flow separation, which can lead to inaccurate performance predictions [7,10]. In contrast, the corrected VD method, incorporating the model, has been shown to produce results that closely match experimental thrust and torque distributions [11].

In parallel, several numerical improvements have been proposed to enhance the precision and applicability of the VD method. Youn et al. [12] conducted numerical studies on small fishing vessels of various sizes and quantitatively demonstrated the impact of submergence depth variations on self-propulsion performance under wave conditions. Kang et al. [13] emphasized the need for incorporating free-surface interference effects into early-stage design and suggested the potential of body-force-based methods, such as the VD approach, as practical design tools. Zhou et al. [10] further analyzed cavitation and ventilation phenomena near the free surface using CFD coupled with the Volume of Fluid (VOF) method and systematically examined the influence of submergence depth and inflow velocity, highlighting the limitations of conventional VD models [14].

This study aims to overcome the limitations of the conventional virtual disk method in accounting for free-surface interactions and to develop and validate an improved body-force method that can more accurately predict actual propeller performance. To this end, performance differences between the model propeller and the conventional VD method are quantitatively analyzed, and a simplified correction model based on the key variables, advance ratio (J) and submergence ratio ($h/D$), is implemented as a field function within the CFD framework. The correction model is formulated to allow for real-time dynamic adjustment based on time-dependent submergence variation.

First, open water (POW) simulations including vertical motion are conducted under calm-water conditions to quantitatively compare the performance of the model propeller and the improved Virtual Disk (VD) method. Subsequently, the same approach is applied to self-propulsion analysis under ballast conditions in waves to evaluate its applicability and reliability in realistic operating scenarios. Through this procedure, the proposed method is assessed in terms of both accuracy and applicability. The VD approach inherently reduces computational demand compared with fully discretized propeller CFD, since the propeller geometry is replaced by a body-force model and fine blade-resolving meshes are not required. Owing to this characteristic, the VD method has been widely utilized in numerical analyses as an efficient alternative. In this study, a correction scheme is introduced into the VD framework to enhance accuracy under free-surface interference conditions, and the focus is placed on extending and validating its applicability.

2. Materials and Numerical Methods

2.1. Ship and Propeller Specifications

The target ship and propeller selected in this study were the MR tanker and SP598M. The specifications of the MR tanker and SP598M used in numerical analysis and model tests are shown in

Table 1. Specifications of the model ship (MR tanker, Daejeon, Republic of Korea).

	Ship	Model
Condition	Ballast Condition
Scale ratio $(\lambda$)	36.667
Design Speed [knot]	14
LPP [m]	174	4.745
LWL [m]	173.21	4.724
B [m]	32.2	0.878
Draft TF [m]	6.517	0.178
Draft TA [m]	8.016	0.219
WSA [m2]	6511.6	4.843
DISV [m3]	30,338.2	0.615
L.C.B [m] (+: Forward)	4.773	0.130
CB	0.7452
CP	0.756
KG [m]	8.848	0.241

and

Table 2. Specifications of the model propeller (SP598M, Daejeon, Republic of Korea).

	Full Scale	Model Scale
Diameter [m]	6.6	0.245
Number of Blades	4
(P/D)mean	0.7604
AE/AO	0.5004
Hub ratio	0.1600
Section type	NACA66
Propeller Revolution	15 rps

, and their shapes are shown in Figure 1 and Figure 2.

2.2. Simulation Model and Method

In this study, the commercial numerical analysis software STAR CCM+ version 15.06 was used. The governing equations used were the continuity equation and the incompressible RANS equations to consider incompressible viscous flow. The realizable k–ε model was used as the turbulence model considering both the POW analysis and the calculation of the wave-induced forces. This model can reasonably reduce the number of grid points near the walls and perform relatively accurate calculations. In addition, it was adopted in this study owing to its robustness and computational efficiency in large-scale free-surface simulations. Although the realizable k–ε model may diffuse coherent vortices and underpredict tip-vortex intensity, it provides stable convergence and sufficient accuracy for propulsion performance evaluation under ballast and wave conditions. Since the primary objective of this work is to validate the free-surface-corrected virtual disk method within a practical CFD framework, the realizable k–ε model was considered an appropriate choice. The SIMPLE (Semi-Implicit Method for Pressure-Lined Equation) method was used to consider the inertia of velocity–pressure, and the DFBI (Dynamic Fluid Body Interaction) model was used for ship movement analysis. To observe the interaction with the free surface, the Volume of Fluid (VOF) method was used as the numerical model, and the overset mesh technique was utilized to consider the deformation of the grid system due to the vertical motion of the propeller and wave.

3. Propeller Open Water Test of Improved Virtual Disk Applying Modified Equation

3.1. Numerical Method

Figure 3 shows the grid conditions of POW analysis with heaving motion in calm water that we used to develop a virtual body method considering the interaction with a free surface. In order to verify the numerical model and grid, a comparison was carried out with the propeller-only performance of SP598M provided by Samsung Ship Model Basin (SSMB, Daejeon, Republic of Korea). The boundary conditions were set as symmetry conditions except for the inlet and outlet surfaces. A total of 2 million grids were used, and the grids of the free surface and the part where the propeller moves were made denser to observe the interaction with the free surface in detail. The results are shown in Figure 4 and

Table 3. Comparison of POW characteristics between SSMB and CFD (INHA Univ).

	SSMB (EFD)		Present (CFD, Virtual Disk)		Present (CFD, Model Propeller)
J	KT	10KQ	KT	10KQ	KT	10KQ
0.1	0.326	0.361	0.334	0.366	0.316	0.357
0.30	0.250	0.293	0.254	0.299	0.245	0.286
0.40	0.209	0.254	0.210	0.257	-	-
0.45	0.188	0.233	0.187	0.235	0.182	0.225
0.50	0.166	0.212	0.164	0.212	-	-
0.60	0.122	0.168	0.116	0.164	0.116	0.158
0.70	0.075	0.120	0.068	0.112	-	-

. The results showed good agreement within ±3% for all advance ratios except for 0.6 and 0.7, which were relatively small values and showed no significant difference.

3.2. Development of Modified Equation Considering Free Surface

To compare the individual performance of two models according to depth ratio, analysis was conducted for advance ratios (J) of 0.3, 0.45, and 0.6. The range of heave motion was set with submergence depth ratios ($h/D$) of 0 to 1, and the performance difference between the two models was observed. Regarding the change in propeller performance according to submergence depth, it was confirmed that both the virtual disk method and the model propeller show a significant decrease in performance at a specific submergence depth ratio ($h/D_c$), like previous studies. The original correction equation was simplified to be directly applicable in CFD by removing variables related to propeller angles that have minimal impact and by using thrust and torque values instead of $K_T$ and $K_Q$ values for real-time feedback control. The final improved correction equation is expressed as Equation (1).

$$ModifiedT,Q\left(t\right)=VirtualDiskT,Q\left(t\right)\times \left[1-\left({\displaystyle \frac{∆T,Q_{max}\left(V_a\right(t\left)\right)}{h/D_c\left(V_a\right(t\left)\right)}}\right)\times \left({\displaystyle \frac{h}{D_c}}\left(V_a\left(t\right)\right)-{\displaystyle \frac{h}{D}}\left(t\right)\right)\times \alpha \right]$$

(1)

In Equation (1), $h/D\left(t\right)$, $V_a\left(t\right)$, $VirtualDiskT\left(t\right),$ and $Q\left(t\right)$ represent the time-varying values of propeller submersion depth, effective velocity, thrust, and torque obtained from numerical analysis. In addition, $h/D_c\left(V_a\right(t\left)\right)$, $∆T,{\mathrm{a}\mathrm{n}\mathrm{d}Q}_{max}\left(V_a\right(t\left)\right)$ are correction parameters derived from trend equations established in previous studies to reconcile differences between the two models. Specifically, $∆T$ indicates the instantaneous thrust difference between the conventional VD method and the model propeller, $Q_{max}$ denotes the maximum torque discrepancy within the tested advance ratio range, and $h/D_c$ corresponds to the submersion depth at which rapid performance degradation begins for a given effective velocity.

The switching function α was designed to activate the correction only when physically required. Specifically, when the instantaneous submergence depth $h/D\left(t\right)$ exceeds the critical submergence depth $h/D_c\left(V_a\left(t\right)\right)$ for the given advance ratio, the propeller remains sufficiently submerged and no correction is applied (α = 0). Conversely, when $h/D\left(t\right)$ falls bellow $h/D_c\left(V_a\left(t\right)\right)$, indicating that the propeller is operating closer to the free surface where rapid performance degradation can occur, the correction function is activated (α = 1). This formulation ensures that corrections are applied only in the relevant operating regime and remain inactive otherwise.

For the validation of the validity of Equation (1), the time-step values of thrust and torque measured using the virtual disk method in CFD were applied to the correction equations. The calculated values of thrust and torque using the modified formula were compared with the values of the model propeller, as shown in Figure 5. The values shown with bold lines are the result of applying a modified formula to the values measured at each time-step using the virtual disk method in CFD. The values shown in bold lines are the result of applying a modified formular to the values measured at each time-step using the virtual disk method in CFD. When comparing propeller performance according to submergence depth, it was observed that the result of applying the modified formula was in good agreement with the average value of the model propeller.

Unlike earlier correction models that relied on more complex formulations with multiple empirical or motion-related parameters, the present work simplifies the correction into thrust- and torque-based terms, enabling direct and efficient implementation within a CFD environment. This formulation allows real-time feedback and dynamic adjustment in response to instantaneous variations in submergence ratio ($h/D$). Moreover, the applicability of the proposed correction has been extended beyond calm-water open-water tests to self-propulsion simulations under both calm and regular wave conditions, thereby broadening the scope of validation.

3.3. Development of CFD Analysis Technique for Propulsion Performance Considering Free Surface Effect

The virtual disk method is a technique that generates the forces produced by a model propeller on a cylindrical disk without considering the shape of the model propeller. This is achieved by measuring the velocity and density using the inlet velocity plane and using those values to calculate the values of thrust and torque on the propeller surface. In this study, a field function was used to implement the body force propeller method, which is one of the virtual disk methods, through a user defined method to apply the improved modified formula variables consistently.

The distribution of forces follows Goldstein’s optimum method and can be expressed by Equations (2)–(5). In this equation, $f_{bx}$ represents the force in the x-axis direction, and $f_{b\theta }$ represents the force in the θ direction. $R_H$ denotes the hub radius of the propeller, and $R_P$ represents the radius of the propeller. Goldstein’s optimum distribution is shown in Figure 6. This force distribution was used to calculate the values of thrust and torque, which were then applied to the modified formula for evaluating the performance of the model propeller [15].

$$f_{bx}=A_x{r}^{*}\sqrt{1-r^{*}}$$

(2)

$$f_{b\theta }=A_{\theta }·{\displaystyle \frac{r^{*}\sqrt{1-r^{*}}}{r^{*}(1-{r^{\prime }}_h)+{r^{\prime }}_h}}$$

(3)

$$A_x={\displaystyle \frac{105}{8}}·{\displaystyle \frac{T}{\pi ∆(3R_H+4R_P)(R_P-R_H)}}$$

(4)

$$A_{\theta }={\displaystyle \frac{105}{8}}·{\displaystyle \frac{Q}{\pi ∆R_P(3R_H+4R_P)(R_P-R_H)}}$$

(5)

$${r^{\prime }}_h={\displaystyle \frac{R_H}{R_P}}\hspace{1em}\hspace{1em}{r}^{\prime \hspace{1em}}={\displaystyle \frac{r}{R_P}}\hspace{1em}\hspace{1em}{r}^{*}={\displaystyle \frac{r^{\prime }-{r^{\prime }}_h}{1-{r^{\prime }}_h}}$$

In the numerical analysis program, a field function was used to apply the Goldstein’s optimum distribution to the virtual cylinder created along the axis of the propeller. This allowed for specifying the forces at different r and θ positions from the center axis. Additionally, a composite feature was used in the user defined method to output specific values for the X, Y, and Z directions. The input values for each direction are shown in Equations (6) and (7). The source term distribution for each component is shown in Figure 7, which allows us to confirm the direction of the forces acting on each component in the virtual disk method.

$$\mathrm{X}\mathrm{Component}=\mathrm{tangential}\mathrm{direction}\mathrm{of}\mathrm{force}\mathrm{component}\left(f_{b\theta }\right)\times \mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{y}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\right[\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{a}\left]\right)$$

(6)

$$\mathrm{Y}\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\left(f_{b\theta }\right)\times \mathrm{s}\mathrm{i}\mathrm{n}\left(\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{y}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\right[\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{t}\mathrm{a}\left]\right)$$

(7)

$$\mathrm{Z}\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}=\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\left(f_{bx}\right)\times \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{y}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\left[\mathrm{r}\right]$$

(8)

To obtain the modified values of thrust and torque, Equation (1) was used to apply the modified formula to the $T\mathrm{a}\mathrm{n}\mathrm{d}Q\left(t\right)$ values in the existing virtual disk methods, which resulted in the modified $T\mathrm{a}\mathrm{n}\mathrm{d}Q\left(t\right)$ values. Then, Equations (4) and (5) were used to output the final modified $A_x\mathrm{a}\mathrm{n}\mathrm{d}{A}_{\theta }$ values on the disk. The Z component of the source term in the virtual disk method confirmed Figure 8 in the region where there is a difference in thrust. It was observed that the source term became smaller after the application of the modified formula compared to the existing virtual disk method.

To validate the applicability of the modified virtual disk method in a CFD program, we implemented the modified virtual disk method in STAR CCM+ and compared the obtained thrust and torque values with those of the model propeller. The results are shown in Figure 9. The modified virtual disk method showed good agreement with the model propeller values in most of the range except for the range of $h/D$ = 0–0.2 at an advance ratio of 0.3. By verifying the propeller open-water performance of the virtual disk method, we confirmed the possibility of using the modified virtual disk method to estimate the performance of a ship in waves.

4. Self-Propulsion Test Using the Modified Virtual Disk Method Under Wave Ballast Conditions

4.1. Resistance and Self-Propulsion Numerical Method Verification

To apply the final modified virtual disk method for the self-propulsion test, a step-by-step verification of the ship’s resistance and self-propulsion performance under calm water and waves is required.

To verify the ship’s resistance performance and the modified virtual disk method, a resistance analysis was conducted in ballast conditions. In addition, the resistance and effective power values were compared and verified with the results of the model ship tests with the same scale conducted at Busan National University (PNU, Busan, Republic of Korea) and the numerical analysis conducted at the Research Institute of Medium & Small Shipbuilding (RIMS, Busan, Republic of Korea). We confirmed that in all cases, the values obtained from the modified virtual disk method were very similar to the values from the PNU model test and the RIMS numerical simulations, with differences of less than 5%, as shown in

Table 4. Comparison of MR tanker resistance in different institutions (ballast conditions).

	MR Tanker (with ESD)		MR Tanker (W/O ESD)
Institution	PNU (EFD)	INHA (CFD)	RIMS (CFD)	INHA (CFD)
CTM $\times$ 103	4.527	4.640 (2.5%)	4.708	4.803 (2.0%)
CTS $\times$ 103	2.608	2.721 (4.3%)	2.772	2.884 (4.0%)
PE [Kw]	3254	3395 (4.3%)	3458	3598 (4.0%)

.

Before conducting resistance analysis in waves, a grid convergence test was performed to verify the effect of grid size on the results in numerical analysis [16]. The test was conducted under the conditions of a wavelength-to-ship-length ratio (λ/LPP) of 1.1 and a wave steepness (H/λ) of 0.01 at a speed of 14 knots. Coarse and fine grids were derived by increasing or decreasing the mesh size around the free surface and hull by $\sqrt{2}$ based in the base mesh. The numerical analysis results for wave resistance were compared with the experimental results from the PNU towing tank and were found to be very similar.

The conditions of the numerical analysis in waves included the application of a forcing zone to reduce the reflection of waves by the boundary and the use of boundary conditions based on the recommended settings in STAR CCM+. Pressure outlet conditions were applied to the outlet, wall conditions were applied to the side, and velocity inlet conditions were applied to the remaining areas. With these conditions, resistance analysis in waves was performed. The results of the grid convergence test using three different mesh sizes are shown in Figure 10, and it can be observed that the resistance coefficient increases as the number of cells increases.

The grid dependency analysis was performed using three mesh systems (G3, G2, and G1), and the results are summarized in

Table 5. Test cases for grid convergence (λ/LPP = 1.1, H/λ = 0.01, Vs = 14 knots).

Grid Name	Mesh	λ/Δx	H/Δz	Cell Number
G1	Fine	84	27	8.3 × 106
G2	Base	60	19	3.6 × 106
G3	Coarse	42	13	1.6 × 106

. Based on the variations in the added resistance coefficient with mesh refinement, the observed order of convergence and the Grid Convergence Index (GCI) were calculated. Figure 11 presents the changes in the added resistance coefficient with respect to mesh resolution, and the GCI analysis confirmed that the results obtained with the fine mesh (G1) satisfied grid independence. Therefore, the G1 mesh was adopted for all subsequent numerical simulations in this study.

The comparison of the added resistance coefficient and motion responses under wave conditions is presented in Figure 12. Numerical simulations were performed for five wavelength-to-ship-length ratios, λ/LPP = 0.6,0.9, 1.1, 1.3, and 1.7, and the results were compared with the first and second model tests conducted at Pusan National University (PNU). The numerical predictions showed overall good agreement with the experimental data, with the largest added resistance coefficient observed at λ/LPP = 1.1.

The results of the self-propulsion analysis conducted after the resistance verification are summarized in

Table 6. Comparison of self-propulsion factor in calm water.

Institution	SSMB (EFD)	PNU (EFD)	INHA (CFD, Virtual Disk)	INHA (CFD, Model Propeller)
Scale ratio, λ	26.906	36.667
wTM	0.388	0.410	0.413	0.400
wTS	0.336	0.306	0.326	0.316
t	0.231	0.193	0.199	0.192
ηH	1.158	1.162	1.188	1.180
ηR	1.020	1.014	1.001	1.001
ηO	0.653	0.676	0.667	0.670
ηD	0.771	0.797	0.792	0.792
RPM	83.46	84.01	83.71	84.08
PE, EHP [kW]	3275	3253	3395	3395
PD, DHP [kW]	4249	4082	4286	4288

. The present numerical analyses, performed using both the model propeller and the body force method, were compared with the self-propulsion test results obtained at the Samsung Heavy Industries (SSMB) towing tank and the PNU basin. In the model propeller analysis, the wake fraction (w_TM) was predicted to be lower and the thrust deduction factor (t) to be higher than the experimental values, leading to a decrease in hull efficiency (η_H). In contrast, the body force method produced relatively higher values of both w_TM and t, resulting in a slight difference in the propeller revolution rate. Nevertheless, the overall self-propulsion factors were consistent with the experimental measurements, and the deviations remained within an acceptable error range, confirming the validity of the numerical results.

4.2. Comparison of Self-Propulsion Elements in Wave

To identify the problem of the existing virtual disk method that does not consider the interaction with free surface, wave-induced self-propulsion analysis was conducted under ballast conditions using the existing virtual disk method. When using the existing virtual disk method, the variation in the wave-induced self-propulsion factors under ballast conditions was somewhat inaccurate. Typically, the wake fraction and thrust deduction coefficients decrease in comparison to those in calm water. This is shown on Figure 13. Finally, to validate the modified virtual disk method, we performed a self-propulsion analysis using the modified virtual disk method under ballast conditions. The analysis results showed that the thrust and torque output values displayed a better match with the model propeller when using the modified virtual disk method compared to the existing virtual disk method, as shown in Figure 14.

Furthermore, when comparing the self-propulsion coefficients, there was a tendency for the wake fraction and thrust deduction coefficient to decrease more in the conditions of waves compared to the calm conditions. This trend was observed when using the modified virtual disk method and showed similar characteristics to the self-propulsion coefficients calculated using the model propeller. Therefore, by showing similar propulsion efficiency to the model propeller and estimating the delivered power similarly, the effectiveness of the modified virtual disk method was confirmed in the ballast conditions considering the free surface. A comparison of self-propulsion coefficients for each propeller configuration method is shown in

Table 7. Self-propulsion element with modified virtual disk.

Condition	Calm		Wave (λ/LPP = 1.1, H/λ = 0.01)
Propeller	Model Propeller	Virtual Disk	Model Propeller	Virtual Disk	Virtual Disk (Modify)
Scale ratio, λ	36.667
Design Speed, Vs [kts]	14
wTM	0.400	0.413	0.328	0.424	0.326
wTS	0.316	0.326	0.272	0.331	0.276
t	0.192	0.199	0.178	0.202	0.188
ηH	1.180	1.188	1.129	1.194	1.122
ηR	1.001	1.001	0.998	1.002	1.002
ηO	0.670	0.667	0.619	0.570	0.614
ηD	0.792	0.792	0.697	0.682	0.690
RPM	84.08	83.71	101.21	100.02	101.43
PE, [kW]	3395	3395	5941	5941	5941
PD, [kW]	4288	4286	8518	8710	8609

.

5. Conclusions

When performing a ship’s self-propulsion analysis using the existing virtual disk method, the efficiency is reduced due to surface piercing and air ventilation occurring as the propeller gets closer to the free surface in the low draft Ballast condition. Therefore, when comparing the model propeller and the existing virtual disk method in the section considering free surface, the thrust generated by the existing virtual disk method is significantly higher. To address this problem, a modified formula was presented in a previous study [4] based on the effect of ship’s heave and pitch motion, propeller immersion depth, angle, and wave on both the model propeller and the virtual disk method. The modified formula was expressed as the advance ratio (J) and immersion depth ratio ($h/D$). Through propeller open-water tests, the thrust and torque values obtained by the virtual disk method were corrected using the modified formula, and good agreement was confirmed with experimental values and the model propeller’s values. The objective of this study was to apply the modified virtual disk method to predict ship self-propulsion under ballast conditions. To achieve this goal, we systematically verified and analyzed a ship’s motion and additional resistance in both calm and wave conditions. The error rate of all numerical analysis results obtained in calm and wave conditions was less than 5%.

The values of thrust and torque of the model propeller were compared with those of the modified virtual disk method. The comparison showed that the corrected values from the modified virtual disk method showed good agreement with those of the model propeller except for the range of $h/D$ = 0–0.2 at the advance ratio of 0.3. Subsequently, the propeller performance in the ballast conditions under waves was compared using a model propeller, the existing virtual disk method, and a modified virtual disk method.

Generally, the wake fraction and thrust deduction coefficient tended to decrease under waves compared to calm water conditions. However, the existing virtual disk method failed to accurately represent air ventilation in the region where the free surface interference occurs compared to the model propeller. This resulted in excessive thrust and torque due to the faster inflow velocity and showed an increase in wake fraction and thrust deduction coefficient under waves compared to calm water conditions. However, the modified virtual disk method showed a decreasing trend in the wake fraction and thrust deduction coefficient under waves compared to calm water conditions and showed a high level of agreement in thrust and torque trends with the model propeller. Therefore, this study demonstrated the feasibility of using the modified virtual disk (VD) method, which requires relatively low computational cost, to estimate propeller performance in waves for future self-propulsion tests.

While the present validation in waves was limited to regular head seas with a relatively low steepness (H/λ = 0.01), the results confirmed that the modified VD method can reliably capture free-surface effects under these simplified conditions. Furthermore, since all simulations were performed at model scale, potential scale effects should be considered when applying the results to full-scale ships. Free-surface ventilation and partial submergence are expected to scale nonlinearly and could amplify thrust and torque discrepancies at full scale. Future work will therefore extend the validation to higher steepness, irregular spectra, and oblique wave incidence, and address scale effects to further broaden the applicability of the proposed method to practical ship design and operation.