Numerical Simulation of Ship-added Resistance Based on a Numerical Wave Tank

The wave resistance increase of a ship during its actual voyage can affect its speed and safety. To design ships with excellent sailing performance, it is necessary to accurately predict the wave resistance increase of a ship in waves, as well as the motion response and wave resistance prediction of ships in waves, which involves complex hydrodynamic problems. This paper first establishes a physical model of the KCS vessel, then utilizes the numerical wave tank platform independently developed by our university. Based on viscous flow theory, it calculates the resistance of the KCS under calm water conditions. Using three-dimensional potential flow methods and spectral analysis, it computes the wave-induced resistance values for the KCS under six-degree sea states, analyzes the wave-induced resistance performance of the KCS, and ultimately, based on the calculation results of calm water resistance and wave-induced resistance combined with propeller efficiency parameters, calculates the effective power of the engine.


Introduction
When ships navigate in waves, they experience an increase in resistance compared to when they are in calm water, a phenomenon known as added resistance.This phenomenon significantly reduces a ship's forward speed, leading to excess fuel consumption and the risk of stalling.The added resistance has a negative impact on ship safety, in addition to affecting economic efficiency, the extra fuel consumption and emissions also have serious implications for the marine environment, contradicting the principles of green shipping.
Havelock [1] first introduced the concept of added resistance in 1937, but the calculations differed significantly from experimental results due to the neglect of diffraction effects.Maruo [2] (1963) found that although the flow around the ship's hull is complex, the flow field away from the ship's body is relatively simple.Based on Havelock's ideas and using momentum and energy conservation principles, Maruo derived a far-field calculation formula for added resistance.Yishan D. and Debo H. [3] (1993) provided a concise derivation of the added resistance formula, starting from the near-field formula and directly deriving the far-field formula.Liu S. [4] (2011) and others also applied Maruo's method to calculate added resistance, which showed good agreement with experimental values, confirming the reliability of the calculation method.Xie Yunping X. and Wei Y. [5] (2013) conducted research on the added resistance of high-speed displacement ships using a three-dimensional numerical wave tank based on the Volume of Fluid (VOF) method and Computational Fluid Dynamics (CFD) technology.Si C. [6] (2017) combined the weakly nonlinear potential flow theory with Maruo's added resistance theory and analyzed the added resistance of ships under headwave conditions.
After years of research both domestically and internationally, many scholars have explored the mechanism of added resistance using different methods, including model tests (Experimental Fluid Dynamics or EFD), theoretical calculations based on potential flow theory, and CFD numerical simulations based on viscous flow theory.The use of CFD method to calculate wave resistance has become an important tool in ship design due to the long and expensive test period required for physical pool testing, which extends the ship design period.Viscous flow methods require high computational performance and consume a lot of computational time [7], while added resistance virtual test tanks based on potential flow theory can effectively address these issues [8].
This paper, based on our university's independently developed numerical wave tank platform, conducted the following research: 1. Introduced the fundamental theory of using CFD numerical simulations.
2. Selecting the KCS ship type, in the numerical water pool software, the ship pitching, heeling and wave resistance RAO under different wave frequencies are calculated.The convergence of the calculation is analyzed with different mesh densities, free surface mesh density, and time steps.
3. Under convergence conditions, conducted research on added resistance and motion RAO, obtained the resistance values, and assessed the minimum installed power for the ship.

Three-dimensional Potential Flow Theory with Ship Speed
For ship hydrodynamics problems involving three-dimensional potential flow in the time domain, the coordinate system is as shown in Figure 1.This coordinate system is known as the body-fixed coordinate system of the ship, with the origin located at the projection of the ship's stern onto the horizontal plane.The X-axis points in the positive direction of the ship's heading, and the Z-axis is oriented vertically upwards.The normals on the ship's surface point towards the interior of the ship.Linear wave assumptions are made, neglecting nonlinearity in the free surface and wave behavior.In this translational coordinate system, we solve for both steady-state and unsteady velocity potentials to obtain the matching boundary integral equation.The total velocity potential  is decomposed into the basic velocity potential b  , perturbation velocity potential d  , and incident potential I  , as shown in equation ( 1): The basic velocity potential is expressed as b Ux      , which is a time-independent term.We separate the terms related to time and denote the remaining time-dependent term as the unsteady potential, represented as ( , , , ) x y z t  .
Assuming that the fluid in the flow field is an ideal fluid, the ship's motion in the frequency domain follows the Euler equations and the continuity equation.We define that there is no rotation in the fluid, which means that the fluid velocity potential is given by  . Additionally, the entire fluid domain satisfies the Lagrange integral and the Laplace equation: , , , 0 In the equations (2) x, y, z represent the motion coordinates, port coordinates, and vertical coordinates, respectively.t is time, ( , , , ) x y z t  is the unsteady velocity potential, V is velocity, P is pressure,  is fluid density, g is gravitational acceleration, and C(t) is the Lagrange integral.
By using the Taylor expansion boundary element method to solve the boundary integral equation, you can obtain information about the pressure distribution and velocity potential within the flow field.This information can then be used to calculate the forces acting on the fluid.

Dimensionless Scaling Methods for Simulation Experiments
(1) swinging motion: , where 3  is Pendulum Amplitude, a  is Incident Wave Amplitude.
(3) Added resistance: aw , where Wave Amplitude, B is Ship Width, L is Ship Length.

Test Ship Model
The simulation employs the KCS ship type as shown in Figure 3. and the hull grid diagram as shown in Figure 4.The ship parameters are as shown in Table 1.
Table 1.Ship parameters for the input of the simulation experiment.

Simulation Test Conditions
In the simulation experiments, wave frequencies are provided as a ratio of wavelength to ship length, as shown in Table 2. and all conditions are in headwave states. is between 0.8 and 1.2, both body surface grid cell count and free surface grid density have a significant impact on added resistance.When the / L  is greater than or equal to 1.4, the time step size has a certain influence on added resistance.As the body surface grid cell count, free surface grid density, and time step size increase, the added resistance values tend to stabilize, indicating convergence in the calculation results.

Static Water Experiment
Firstly, a simulated static water experiment is conducted on the KCS ship model, and the contour plot is shown in Figure 2. The calculation results are presented in Table 3. -0.608E+01 0.000E+00 0.546E+01

Study on Added Resistance and Motion RAO
Simulation experiments were conducted with 1190 body surface grid cells, a free surface grid density of 52, and a time step size of T/50 at a ship speed of 23 knots.The dimensionless pendular, pitching motion, and added resistance RAO virtual test results are shown in Figure 6.
(a) (b) (c) Figure 6.Shows the heave motion (a), pitch motion (b), and added resistance RAO (c) under navigational speed conditions.The added resistance reaches its maximum value when the / L  is 1.09.The ITTC two-parameter spectrum formula is as follows: is the peak frequency of the wave.1/3  is the significant wave height.ITTC wave power spectral density is shown in Figure 7.
The random sea waves can be decomposed into the superposition of multiple harmonics, each with a random amplitude, phase, and wavelength represented as: where i  is the rand phase.Figure 8. shows the time history of wave height.Based on the RAO results and in combination with the ITTC two-parameter spectrum, it can be analyzed that in sea state 6 with a peak period of 9.5 seconds, the significant added resistance is calculated to be 8.36×10 5 N.

Calculation of Minimum Installed Power
The formula for calculating the main engine power is given by equation ( 5): where B P is the main engine power, O  is the open water efficiency of the propeller, R  is the relative rotative efficiency, S  is the shafting transmission efficiency, and H  is the hull efficiency.The effective power E P is calculated using the formula (5).3733444 836000) 23 0.214 ( 54019966w=73496hp H   , the minimum installed power can be obtained by using the provided formulas and the calculated values for the various efficiency factors 122702hp B P  .

Calculated Results
According to the above experiment, we draw the following conclusions: (1) Under regular waves, as the frequency increases, the added resistance gradually increases.The maximum added resistance occurs at a wave frequency of 0.419 rad/s, which corresponds to a / L  of 1.09.Subsequently, as the wave frequency increases further, the added resistance gradually decreases.
(2) In the simulation experiment, based on RAO results and using the ITTC two-parameter spectrum, it was analyzed that in sea state 6 with a peak period of 9.5 seconds, the significant added resistance is calculated to be 8.36×10 5 N.
(3) In the simulation experiment, the total resistance during navigation was calculated to be 4,569,444 N, and the minimum installed power was calculated to be122702hp .

Conclusion
This article mainly introduces a simulation test method for wave increase resistance of ships based on the numerical pool platform developed by our academic institute, and conducts wave tests on the KCS standard ship type.This method considers the influence of wave-induced resistance and motion conditions, providing technical support for improving the energy efficiency of ships, reducing carbon emissions, and promoting green development in the shipbuilding industry.Specifically, the method in this article includes the following steps: 1. Calculate the resistance of KCS sailing under still water conditions based on viscous flow.Using three-dimensional potential flow method and spectrum analysis, calculate the wave increase resistance and motion conditions of KCS under a six-level sea state.
2. By combining the calculations of still water resistance and wave increase resistance with propeller efficiency parameters, calculate the effective power of the engine.This provides a numerical simulation basis for long-distance navigation of KCS and technical reference for engine selection.
Currently, this method is only applied to the KCS.In future research, the author plans to extend this method to more ship types for calculation, further improving the energy efficiency of vessels, reducing carbon emissions, and promoting green development in the shipbuilding industry.

Figure 1 .
Figure 1.Coordinate System Definition.Figure 2. Contour plot of static water test.

Figure 2 .
Figure 1.Coordinate System Definition.Figure 2. Contour plot of static water test.

Figure 5 .
(a)(b)(c) represent the added resistance RAO under different conditions of body surface grid cell count, free surface grid density, and time step size, respectively.When the / L

Figure 7 .
Figure 7. ITTC wave power spectral density.Figure 8. Time history of wave height.

Figure 8 .
Figure 7. ITTC wave power spectral density.Figure 8. Time history of wave height.

Table 2 .
Simulation test conditions.During the calculation process, the main factors affecting convergence are the number of body surface grid cells, free surface grid density, and time step size.For the KCS ship model, half-ship body surface grid cell counts were set to 1050, 1190, and 1320, while free surface and matching surface grid density parameters were set to 45, 52, and 57, respectively.Time step sizes of T/40, T/50, and T/60 were chosen for convergence analysis.The analysis results are shown in Figure5.Considering both computational accuracy and efficiency, a half-ship body surface grid cell count of 1190, a free surface grid density of 52, and a time step size of T/50 were selected for subsequent calculations.

Table 3 .
Result of static water experiment.