A three-dimensional RANS method to numerically solve ship’s wave-making resistance compared with the Rankine source method

Abstract

According to the forming reason and classification of the hull resistance, this paper puts forward a three dimensional RANS method to solve the wave-making resistance. The obtained numerical results are compared with the results obtained from numerical program based on Rankine source boundary element method and the results obtained from Experimental fluid dynamics (EFD) method. The conclusion is drawn that the three dimensional RANS method matches well with EFD method, and its accuracy is higher than Rankine source method. From the results of the used methods it appears that the value of shape factor $(1 + k)$ for three dimensional RANS method is slightly increased with the increase of Froude number, the free surface wave system of three dimensional RANS conforms to the characteristics of the Kelvin wave system, the height of wave crest for three dimensional RANS method is bigger than Rankine source method, while the wave trough is lower, and the variation tendency of wave profile based on three dimensional RANS method is almost the same with Rankine Source method. The proposed method is proved to be appropriate for engineering application.

Keywords

Three dimensional RANS method Rankine source method EFD method wave resistance shape factor wave pattern

1. Introduction

Resistance performance is of great importance among a vessel’s various performances (e.g. stability, sea keeping, maneuverability etc.). The prediction of hull resistance is always being the main research subject in ship engineering and academia for a long time. Especially, the description and simulation of free surface wave has been paid much attention.

At present, model test, based on its high accuracy and good practicability, is the most commonly used method to obtain the hull resistance, but it also has obvious weakness of high economy cost and time cost. Besides, there is scale effect between the model ship and full scale ship. There are mainly two kinds of resistance conversion method, two-dimensional method and three-dimensional method. The two-dimensional method, according to Froude’s view, divides the total resistance into friction resistance and residual resistance. The total resistance conversion of the model ship and full scale ship is realized according to the equivalent of two residual resistance coefficients. The three-dimensional method, according to Hughes’s view, divides the total resistance into friction resistance, viscous resistance and wave-making resistance, and it defines the ratio of the viscous resistance to friction resistance into a constant, the shape factor k. The total resistance conversion of the model ship and full scale ship is realized according to the equivalent of two wave-making resistance coefficients. In terms of three-dimensional method, it is important to determine the shape factor k and solve the wave-making resistance accurately. But influenced by unstable water flow in towing tank and the interference of wave, the accuracy of form factor k may be affected to some extent. Therefore, model test has some limitation for the measure of wave-making resistance. Using theoretical method to solve the hull resistance has been a long-standing way. But limited by viscosity of water, non-linearity of free surface and complexity of 3-D hull surface, it is rather difficult to accurately solve hull resistance by using theoretical method, and the wave-making theoretical research has went through a progress from linearity to non-linearity. In 1964, great attention was focused on Hess Smith, who applied panel method to solve 3-D infinite fluid field and brought extensive use of panel method in ship dynamics [7]. Dawson proposed a method to calculate linear wave-making resistance based on Rankine sources in 1977. Dawson’s method had great flexibility and was easy to be generalized. His method made it possible to progress linear wave-making theory to non-linear wave-making theory by perturbation theory [6,12,16,18]. The application of potential flow theory to solve wave-making resistance has limitation because it can’t take viscosity of water, wave breaking, vortex separation and slapping into consideration. With the development of technology and computer performance, the rise of Computational Fluid Dynamics (CFD) overcame the limitation of potential theory and made it more realistic to solve free-surface viscous flow past a complex 3-D surface. The numerical simulation is easy to be carried out and is very economical. The numerical results calculated by an experienced CFD engineer can match well with the experimental results based on Experimental Fluid Dynamics (EFD). Characterized by high accuracy and extensive applicability, CFD has become one of the important research methods in the field of ship resistance performance study [2,5,10,14,15,19].

This paper applies both the viscous flow and potential flow to solve wave-making resistance. Starting from viscous N-S equation, this paper puts forward a three dimensional RANS method to solve the wave-making resistance. The meshing methods for both the double-body and free-surface model are studied in detail. The numerical results have satisfactory accuracy and match well with the results calculated by the numerical development program based on potential theory Rankine Sources, and the results obtained by EFD method. The numerical wave pattern conforms to Kelvin wave and coincides with the wave pattern of Rankine sources. The precision meets the requirements of engineering application and the proposed method for the study of ship wave-making resistance is feasible.

2. The Rankine source method

Rankine source method is a numerical computation method based on the well-known potential flow theory [17]. The numerical computation program for solving hull wave-making resistance is compiled under the following assumptions: the ship is assumed to sail on the free surface in a uniform speed U as shown in Fig. 1, the X-axis and Y-axis are located in undisturbed free surface, the origin of the co-ordinate system is located in an undisturbed free surface at amidship, the direction of X-axis is the same as flow direction and the Z-axis is vertically upwards.

Fig. 1.

The schematic diagram of the co-ordinate system.

The total velocity potential ϕ is the sum of the double-body velocity potential $φ_{d}$ and wave-making perturbed velocity potential $φ_{w}$ , namely, $\begin{matrix} (1) & ϕ = φ_{d} + φ_{w} . \end{matrix}$

The total velocity potential ϕ needs to satisfy the Laplace equation and the following boundary conditions: no wave radiation condition ahead, solid boundary condition and the dynamic and kinematic conditions on the free surface. The detailed expressions are described as follows: $\begin{array}{l} (2) & \nabla^{2} ϕ = \nabla^{2} (φ_{d} + φ_{w}) = 0, \\ (3) & \nabla ϕ \cdot \vec{n} = \nabla (φ_{d} + φ_{w}) \cdot \vec{n} = 0, \\ (4) & g ζ + \frac{1}{2} \nabla ϕ \cdot \nabla ϕ = \frac{1}{2} U^{2}, z = ζ (x, y), \\ (5) & ϕ_{x} ζ_{x} + ϕ_{y} ζ_{y} - ϕ_{z} = 0, z = ζ (x, y), \end{array}$ $\vec{n}$ is the normal vector of the hull surface, and ζ in the dynamic and kinematic boundary equations is the wave height. By eliminating ζ from $\vec{n}$ , we can obtain, $\begin{matrix} (6) & \frac{1}{2} ϕ_{x} {(\nabla ϕ \cdot \nabla ϕ)}_{x} + \frac{1}{2} ϕ_{y} {(\nabla ϕ \cdot \nabla ϕ)}_{y} + g ϕ_{z} = 0, z = ζ (x, y) . \end{matrix}$

The flow fluid of the double-body is symmetric about the still water surface, the boundary conditions of which can be expressed as follows: $\begin{matrix} (7) & {ϕ_{d}}_{z} = 0, z = 0 . \end{matrix}$

Compared with double-body velocity potential $φ_{d}$ , the wave-making perturbed velocity potential $φ_{w}$ is small quantity. Substituting formula (1) and (7) into formula (6), ignoring the higher order term of $φ_{w}$ , the linear boundary condition of free surface can be obtained. $\begin{matrix} (8) & φ_{d l}^{2} φ_{w l l} + 2 φ_{d l} φ_{d l l} φ_{w l} + g φ_{w z} = - φ_{d l}^{2} φ_{d l l}, z = 0, \end{matrix}$ where l is the differential of velocity potential along streamline on the symmetric plane $z = 0$ . Rankine sources can be distributed on hull surface $S_{B}$ and undisturbed free surface $S_{F}$ based on Rankine source boundary element method, then, $\begin{array}{l} (9) & φ_{d} = U x - \iint_{S_{B}} σ_{B} \frac{1}{r} d s, \\ (10) & φ_{w} = - \iint_{S_{F}} σ_{F} \frac{1}{r_{1}} d s - \iint_{S_{B}} Δ σ_{B} \frac{1}{r} d s, \\ (11) & r = \sqrt{{(x - x_{1})}^{2} + {(y - y_{1})}^{2} + {(z - z_{1})}^{2}}, \\ (12) & r_{1} = \sqrt{{(x - x_{1})}^{2} + {(y - y_{1})}^{2} + z^{2}}, \end{array}$ where r, $r_{1}$ is the distances between source points $(x_{1}, y_{1}, z_{1})$ , $(x_{1}, y_{1}, 0)$ and field point $(x, y, z)$ respectively. The flow past the double-body is obtained from the numerical solution of a boundary value problem subject to the Neumann boundary condition on the double-body hull. If the hull surface $S_{B}$ is divided into $N_{B}$ surface elements, the discretized system of linear equations for double-body velocity potential $φ_{d}$ is shown as follows, $\begin{matrix} (13) & \begin{matrix} - 2 π σ_{B} (i) + \sum_{j = 1, j \neq i}^{N_{B}} \iint_{Δ S_{j}} σ_{B} (j) \frac{\partial}{\partial n} (\frac{1}{r}) d s \\ = U \cdot \vec{n} (i = 1, 2, \dots, N_{B}) . \end{matrix} \end{matrix}$

The source intensity $σ_{B}$ on each control point of the hull surface can be obtained by solving the system of linear equations above. And then the velocity potential $φ_{d}$ and the streamline distribution past the double-body flow can be calculated.

If the free surface is divided into surface elements, the discretized system of linear equations for perturbation velocity potential $φ_{w}$ fulfilling the solid boundary condition and linear free-surface condition is as shown below, $\begin{array}{l} (14) & \sum_{j = 1}^{N_{F}} σ_{F} (j) C_{F B} (i, j) + \sum_{j = 1}^{N_{B}} Δ σ_{B} (j) C_{B B} (i, j) = 0 (i = 1, 2, \dots, N_{B}), \\ \sum_{j = 1}^{N_{F}} σ_{F} (j) C_{F F} (i, j) + \sum_{j = 1}^{N_{B}} Δ σ_{B} (j) C_{B F} (i, j) - 2 π g σ_{F} (i) \\ (15) & = B (i) (i = 1, 2, \dots, N_{F}), \\ (16) & C_{F B} (i, j) = - \iint_{S_{F}} \frac{\partial}{\partial n} (\frac{1}{r_{1}}) d s \cdot \vec{n} (i), \\ (17) & C_{B B} (i, j) = - \iint_{S_{B}} \frac{\partial}{\partial n} (\frac{1}{r}) d s \cdot \vec{n} (i), \\ (18) & C_{B F} (i, j) = φ_{d l}^{2} (i) C L_{B} (i, j) + 2 φ_{d l} (i) φ_{d l l} (i) L_{B} (i, j), \\ (19) & C_{F F} (i, j) = φ_{d l}^{2} (i) C L_{F} (i, j) + 2 φ_{d l} (i) φ_{d l l} (i) L_{F} (i, j), \\ (20) & B (i) = - φ_{d l}^{2} (i) φ_{d l l} (i), \\ (21) & C L_{B} (i, j) = \sum_{n = 1}^{N - 1} e_{n} L_{B} (i - n, j), \\ (22) & C L_{F} (i, j) = \sum_{n = 1}^{N - 1} e_{n} L_{F} (i - n, j), \end{array}$ where $e_{n}$ is the upstream finite difference operator at point N. Since the system of linear equations above is fully populated and asymmetric, and that the diagonal is not dominant, the Gaussian elimination method is used to obtain the discretized source intensity on the hull surface and still water surface. Then the perturbation velocity potential can be calculated. The pressure distribution over the hull surface can be calculated from Bernoulli equation fulfilling free-surface condition. The wave-making resistance can be obtained by integrating the pressure over the total hull surface. $\begin{array}{l} P - P_{\infty} = \frac{1}{2} ρ (U^{2} - 2 g z - φ_{d x}^{2} - φ_{d y}^{2} - φ_{d z}^{2} \\ (23) & - 2 φ_{d x} φ_{w x} - 2 φ_{d y} φ_{w y} - 2 φ_{d z} φ_{w z}), \\ (24) & R_{w} = \sum_{i = 1}^{N_{B}} {P (i) - P_{\infty}} {\vec{n}}_{x i} Δ S_{i}, \end{array}$ where ${\vec{n}}_{x i}$ is the x-component of outward unit normal vector of the surface element, $Δ S_{i}$ is the area of the surface element. The corresponding wave height can be described as, $\begin{matrix} (25) & ζ (x, y) = \frac{1}{2 g} (U^{2} - φ_{d x}^{2} - φ_{d y}^{2} - 2 φ_{d x} φ_{w x} - 2 φ_{d y} φ_{w y}) . \end{matrix}$

3. Three dimensional RANS method

3.1. RANS equation

The three dimensional RANS method is based on the non-linearity (such as viscosity, incompressibility and wave-breaking etc.) of free surface to solve ship wave-making resistance. It’s continuity equation (mass conservation equation) and Reynolds averaged N-S equation (momentum conservation equation) are expressed as below [4,21] $\begin{array}{l} (26) & \frac{\partial ρ}{\partial t} + \frac{\partial (ρ u)}{\partial x} + \frac{\partial (ρ v)}{\partial y} + \frac{\partial (ρ w)}{\partial z} = 0, \\ (27) & \frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial \overline{u_{i}^{'} u_{j}^{'}}}{\partial x_{j}} = - \frac{1}{ρ} \frac{\partial p}{\partial x_{i}} + ν \frac{\partial^{2} u_{i}}{\partial x_{j} \partial x_{j}} + f_{i}, \end{array}$ where $(u, v, w)$ is the three components of the velocity of flow along three axes of the Cartesian coordinate system, $(ρ, p, f_{i}, t, ν, u_{i}, u_{i}^{'}, \partial \overline{u_{i}^{'} u_{j}^{'}} / \partial x_{j})$ is respectively fluid density, fluid static pressure, mass force, time, kinematic coefficient of viscosity, time averaged velocity, fluctuation velocity and Reynolds stress term.

3.2. Turbulence model

Reynolds stress is of vital importance to solve the hull resistance accurately [8]. The turbulence model is applied to make the Reynolds averaged equation closed. The turbulence model includes zero equation turbulence model, one equation turbulence model, two equation turbulence model and Reynolds stress turbulence model, and the Reynolds stress turbulence model establishes Reynolds stress term directly to make equation closed. The most widely used RNG turbulence model of the two equation turbulence model is chosen to make the Reynolds averaged equation closed. The RNG $K - ε$ turbulence model is obtained by performing viscous correction for Standard $K - ε$ turbulence model. The RNG $K - ε$ turbulence model, which adds terms, improves the ε equation and takes the effect of vortices on turbulence flow into consideration, can simulate complex flow better such as, jet impingement, secondary flow and rotational flow. The transportation equation of RNG $K - ε$ turbulence model is as follows [24,25] $\begin{array}{l} (28) & \frac{\partial}{\partial t} (ρ k) + \frac{\partial}{\partial x_{i}} (ρ k u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{j}}] + G_{k} - ρ ε, \\ \frac{\partial}{\partial t} (ρ ε) + \frac{\partial}{\partial x_{i}} (ρ ε u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{j}}] \\ (29) & + C_{1 ε} \frac{ε}{k} G_{k} - C_{2 ε} \frac{ε^{2}}{k} - R_{ε}, \end{array}$ where $μ_{t} = ρ C_{μ} \frac{k^{2}}{ε}$ , $G_{k} = - ρ \overline{u_{i}^{'} u_{j}^{'}} \frac{\partial u_{j}}{\partial x_{i}}$ , $R_{ε} = \frac{C_{μ} ρ η^{3} (1 - η / η_{0})}{1 + β η^{3}} \frac{ε^{2}}{k}$ , $η \equiv S k / ε$ , $S = \sqrt{2 S_{i j} S_{i j}}$ , $S_{i j} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})$ , its turbulence model constants are listed in Table 1.

Table 1
RNG $K - ε$ turbulence model constant

$σ_{ε}$ $σ_{k}$ $C_{1 ε}$ $C_{2 ε}$ $C_{μ}$ $η_{0}$ β

1.3 1.0 1.42 1.68 0.0845 4.38 0.012

$σ_{ε}$	$σ_{k}$	$C_{1 ε}$	$C_{2 ε}$	$C_{μ}$	$η_{0}$	β
1.3	1.0	1.42	1.68	0.0845	4.38	0.012

3.3. Numerical analysis method

There are kinds of discretization methods for solving the governing equations. The commonly used discretization methods include finite volume method, finite-element method and spectrum method, etc. The methods of flow field calculation include SIMPLE algorithm, SIMPLEC algorithm, SIMPLER algorithm and PISO algorithm. In order to solve the governing equations, the finite volume method is used for space discretization [21]. The flow domain is subdivided into a finite number of cells and these equations are changed into algebraic form via the discretization process. The convective terms are discretized using the Second order upwind scheme. The diffusion terms utilize the central difference scheme. The viscosity flow field is solved using PISO (Pressure Implicit with Splitting of Operators) algorithm [11]. The PISO algorithm has obvious advantages in solving transient problems. In the case of the free-surface flow computations, the 2nd order backward implicit formulation and the Geo-Reconstruct scheme are applied to the VOF [9] and volume fraction discretization, respectively. In addition, the precision of hull resistance is affected by boundary layer, especially the grid heights of the first boundary layer. The calculation formula of the first layer of the boundary layer is as follows, $y_{p} = (y^{+} μ) / (ρ C_{μ}^{1 / 4} k_{p}^{1 / 2})$ . Where, $y_{p}$ is the height of the first layer, ρ is the fluid density, μ is the kinematic viscosity coefficient, $k_{p}$ is the turbulent kinematic energy at point P, $y^{+}$ is the dimensionless height of the first layer ( $y^{+} \approx 30 \sim 300$ ), $C_{μ}$ is a constant.

3.4. Three dimensional calculation method

Based on RANS equations and three-dimensional resistance classification method, a three dimensional RANS method is proposed to solve the double-body flow and free-surface flow, obtaining the hull’s form factor and wave-making resistance. The three dimensional RANS method is divided into five steps as follows.

The viscous double-body flow is numerically solved to obtain the double-body total resistance $C_{t 1}$ .

Solve the non-viscous double-body flow to get double-body non-viscous total resistance $C_{t 2}$ .

According to the double-body total resistance coefficient $C_{t 1}$ and double-body non-viscous total resistance coefficient $C_{t 2}$ , the corrected double-body viscous total resistance coefficient $C_{t 3}$ can be obtained, $C_{t 3} = C_{t 1} - C_{t 2}$ . The shape factor can be calculated by $(1 + k) = C_{t 3} / C_{f 1}$ , where $C_{f 1}$ is the equivalent friction resistance coefficient obtained based on 1957 ITTC plate friction resistance coefficient. Since there is no free surface, three dimensional RANS method breaks through the limitation of the inflow velocity, unstable flow and wave disturbance, replacing the traditional methods such as 15th ITTC method and Prohaska method, a tank test method, in solving the form factor $(1 + k)$ [3].

Considering the viscosity and nonlinearity of free surface, the free-surface model flow can be numerically solved to obtain the total resistance $R_{t}$ and friction resistance $R_{f}$ .

According to three-dimensional resistance division method, the total resistance $R_{t}$ is divided into friction resistance $R_{f}$ , viscous resistance $R_{v}$ and wave-making resistance $R_{w}$ . Based on the calculation results of double-body and free-surface model, the wave-making resistance $R_{w}$ can be obtained, $R_{w} = R_{t} - (1 + k) R_{f}$ .

4. Calculation model, meshing and convergence analysis

The famous Wigley hull and S60 hull are selected as calculation models to conduct the verification of the numerical modeling. Wigley hull is a mathematic hull form, its surface can be expressed by a mathematical equation $y = B / 2 * (1 - 4 x^{2} / L^{2}) * (1 - z^{2} / T^{2})$ . The model parameters for Wigley hull are set as $L = 2.0 m$ , $B / L = 0.1$ , $T / L = 0.0625$ . For S60 hull form, the block coefficient is selected as $C_{b} = 0.6$ , others are chosen as $L = 2.0 m$ , $B = 0.267 m$ , $T = 0.107 m$ [1,20].

4.1. Meshing based on Rankine source method

The hull surface and free surface are numerically discretized into numbers of surface elements. The Rankine source method is applied to distributing sources on surface elements to simulate the actual wave motion. Mesh numbers of surface elements has some effect to the calculation results. Theoretically, the free surface is infinity, but for numerical discretization, it’s acceptable and reasonable to take only the perturbated free surface near the hull into consideration. The wave making has no effect to the further away upstream of the free surface. The effect of wave-making to the downstream and two sides of the free surface are gradually decreasing; therefore the size of the free surface area must be selected properly to avoid unexpected effect to the results of the flow field past the hull. The size of the free surface area should be selected according to the principal particulars of the ship. Based on the experience of meshing for Rankine source method, this paper selects the size of the computational domain as follows, the half breadth $0.75 L$ , the length of upstream part $0.5 L$ , the length of downstream part $1.5 L$ . The grids at bow and stern are specially refined. The surface of Wigley hull is meshed as twenty multiplied by eight surface elements, while the S60 hull is meshed as twenty multiplied by six surface elements. The mesh of free surface is divided into fifty multiplied by sixteen surface elements, and it’s the same for Wigley hull and S60 hull. The mesh for hull and free surface is shown as Fig. 2 and Fig. 3.

Fig. 2.

Mesh of Wigley hull and free surface.

Fig. 3.

Mesh of S60 hull and free surface.

4.2. Computational domain selection for RANS method

Considering the symmetry of the flow, the numerical model is created just for half of the flow field for both Wigley hull and S60 hull. Since the ship length and incoming flow velocity for both hull are the same, the computational domain is also taken as the same. The visualized software Auto CAD and mesh generation software ICEM are used to create the calculation model of double-body flow and free-surface model flow. Except that the calculation model of double-body does not take the part above the still water surface into consideration, other information is the same with the calculation model of free-surface.

Based on the wave theory, the size of the cubic computational domain for the calculation model of free-surface flow is selected according to the relationship of wave height, wavelength and velocity. The height of computation domain, above water plane, is greater than the maximum height of the ship wave which is determined by $Z_{A} = v^{2} / 2 g$ , and the wavelength can be obtained based on $λ_{w} = 2 π c^{2} / g$ [24]. Since the existence of free surface, the wave surface upheaves at bow. The length of upstream part is taken bigger than $λ_{w}$ . By applying the User Defined Function, a numerical program is used to control the variation of pressure with the depth of the water to improve the efficiency and stability of the process of numerical calculation. Taking the effect of fore and aft wave system, the length of downstream part is chosen big enough to make the flow filed to fully develop. In this paper, the length of upstream part is taken as 4 meters, and the length of downstream part is 12 meters (bigger than 8 times of the maximum ship wave length). In order to minimize the reflection of the ship wave by the domain boundary in the width direction, suitable width must be selected according to the length of computing domain and Kelvin angle. The water depth is chosen bigger than $λ_{w}$ , therefore the influence of water depth can be ignored. The size of the computational domain is shown in Table 2.

Table 2
Size of the computational domain

The length of design waterline (m) The length of the upstream part (m) The length of the downstream part (m) Depth of water (m) Half breadth (m) The height above waterline (m)

2 4 12 4 5 2

The length of design waterline (m)	The length of the upstream part (m)	The length of the downstream part (m)	Depth of water (m)	Half breadth (m)	The height above waterline (m)
2	4	12	4	5	2

4.3. Mesh generation and boundary conditions for RANS method

Since there is no free surface, the mesh for the double-body flow need not consider the part above the still water surface. To take the wave-making resistance into consideration, the mesh need to be refined near the water plane to track the actual wave profile. In addition, considering the complexity of the 3-D hull surface, the mesh adjacent to the hull surface need also be refined within boundary layer. The height of the innermost mesh is taken as 2 mm. There are in total 8 layers of mesh in the boundary layer and the enlargement ratio of the layer is 1.1. The mesh of the computational domain is divided into two parts. Structured mesh is applied to the inner part and unstructured mesh to the outer part. The interface for structured mesh and unstructured mesh is treated using ICEM software to realize the mesh reconstruction. By this way, the efficiency of data exchange between two kinds of mesh can be significantly increased and the calculation time can be reduced. The calculation model and mesh generation of Wigley hull and S60 hull is shown as Fig. 4, Fig. 5, Fig. 6 and Fig. 7, and the boundary conditions with double-body and free-surface model flow are also shown in Fig. 6 and Fig. 7.

Fig. 4.

Calculation model of Wigley hull.

Fig. 5.

Calculation model of S60 hull.

Fig. 6.

Mesh of the double-body flow.

Fig. 7.

Mesh of free surface flow.

Table 3

Convergence analysis of grids

$S_{G 1} \times 10^{3}$	$S_{G 2} \times 10^{3}$	$S_{G 3} \times 10^{3}$	$R_{G}$	$P_{G}$	$C_{G}$	$δ_{G} \times 10^{3}$	$U_{G} \times 10^{3}$	$U_{G C} \times 10^{3}$	$S_{C} \times 10^{3}$
5.510	5.692	5.963	0.672	1.149	0.489	0.182	0.753	0.190	5.328

4.4. Convergence analysis for RANS method

The free surface model of Wigley hull is chosen to analyze the convergence of the mesh. The computational grid system (Grid-1) is generated according to the grid partition method introduced in Section 4.3. At the same time, the other two grid systems (Grid-2, Grid-3) are applied. Grid-1, Grid-2 and Grid-3 are the fine, medium and coarse grid systems, respectively, with constant refinement ratio ( $r_{G} = \sqrt{2}$ ). The first layer mesh height of the hull surface in all three grid systems meets the requirements of the wall function. The number of cells of Grid-1, Grid-2 and Grid-3 is 2203432 ( $292 \times 77 \times 98$ ), 781770 ( $206 \times 55 \times 69$ ) and 279006 ( $146 \times 39 \times 49$ ), respectively. Since $F n = 0.3$ , the computational results of convergence analysis are shown in Table 3, where $S_{G i}$ is the computational values of total resistance coefficient of the Grid-i systems. Convergence ratio ( $R_{G}$ ), order of accuracy ( $P_{G}$ ) and correction factor ( $C_{G}$ ) are defined as follows: $\begin{array}{l} (30) & ε_{i j} = S_{G i} - S_{G j}, \\ (31) & R_{G} = ε_{21} / ε_{32}, \\ (32) & P_{G} = ln (ε_{32} / ε_{21}) / ln r_{G}, \\ (33) & C_{G} = (r_{G}^{P_{G}} - 1) / (r_{G}^{P_{G est}} - 1), \end{array}$ where $P_{G est}$ is the estimate for the limiting order of accuracy. Since $0 < R_{G} < 1$ , the results meet the monotonic convergence condition. The Richardson extrapolation is used to estimate both errors and uncertainties. For the correction factor ( $C_{G}$ ) meets the requirements ( $| 1 - C_{G} | ⩾ 0.125$ and $| 1 - C_{G} | ⩾ 0.25$ ), according to the research on valid estimation of uncertainties by Stern et al. [22,23], the improved numerical error ( $δ_{G}$ ), the corrected numerical result ( $S_{C}$ ), the numerical uncertainty ( $U_{G}$ ) and the numerical uncertainty of corrected value ( $U_{G C}$ ) are calculated. The $δ_{G}$ , $S_{C}$ , $U_{G}$ and $U_{G C}$ are defined as follows: $\begin{array}{l} (34) & δ_{G} = C_{G} ε_{21} / (r_{G}^{P_{G}} - 1), \\ (35) & S_{C} = S_{G 1} - δ_{G}, \\ (36) & U_{G} = [2 | 1 - C_{G} | + 1] \times ε_{21} / (r_{G}^{P_{G}} - 1) for | 1 - C_{G} | ⩾ 0.125, \\ (37) & U_{G C} = [| 1 - C_{G} |] \times ε_{21} / (r_{G}^{P_{G}} - 1) for | 1 - C_{G} | ⩾ 0.25 . \end{array}$

The convergence ratio ( $R_{G}$ ) meets the monotonic convergence condition and both the improved numerical error ( $δ_{G}$ ) and the numerical uncertainty of corrected value ( $U_{G C}$ ) are smaller than the numerical uncertainty ( $U_{G}$ ), which indicate that the grid system (Grid-1) utilized by the paper meets the requirements of numerical error and uncertainty.

5. Numerical calculation results

5.1. Numerical calculation results for Wigley hull

According to the calculation results of three dimensional RANS method and Rankine source method, the relationship of form factor $(1 + k)$ and Froude number is shown in Fig. 8(a). Since the three dimensional RANS method does not take free surface into consideration, it has no limitation to the inflow velocity. The form factor $(1 + k)$ based on the three dimensional RANS method is related to Froude number. From the results of the three-dimensional RANS method it appears that the value of $(1 + k)$ slightly increases with the increase of the Froude number. The relationship of resistance coefficient and Reynolds number is shown in Fig. 8(b). The resistance coefficient decreases with the increase of the Reynolds number. The numerical results achieve reasonable accuracy and match well with the plate friction resistance coefficient results calculated based on 1957 ITTC formula. Figure 8(c) shows that the wave-making resistance coefficients of both the three dimensional RANS method and Rankine source method match well with the experimental results in general. It’s obvious that three dimensional RANS method has better accuracy than Rankine source method, the reason for this can be attributed to taking the viscosity of water and non-linearity of free surface into consideration in three dimensional RANS method. Also, the conclusion can be drawn from Fig. 8(d) that the total resistance coefficient $C_{t}$ based on the three dimensional RANS method matches well with the EFD results. With the increase of the Froude number, the deviation of total resistance coefficient $C_{t}$ between numerical results and EFD results slightly enlarges. When $F n = 0.34$ , the deviation of total resistance coefficient $C_{t}$ is the biggest, 2.99%. The experimental results in Fig. 8 is obtained from model test in the towing tank of Osaka University [13].

Fig. 8.

Curves of $(1 + k)$ , $C_{f}$ , $C_{w}$ , $C_{t}$ and $F n$ , $R e$ .

Using data viewer package to plot the numerical calculation results of three dimensional RANS method for Wigley hull, the shape of free surface wave along the hull can be obtained. The free surface wave at $F n = 0.25$ and $F n = 0.3$ is respectively plotted in Fig. 9(a) and Fig. 9(b). It’s obvious that the wave profile for both the three dimensional RANS method and Rankine source method is nearly the same with the EFD result. The wave crest has tendency to move afterwards with the increase of $F n$ . Compare with the experimental results, the numerical wave profile of three-dimensional RANS method is better than Rankine source method for the most part of free surface at shipside. The wave crest of the three-dimensional RANS method is higher than Rankine source method, while the wave trough value is lower. For the first wave crest at bow, the wave height of three dimensional RANS method is very close to, but a little bigger than, the experimental result. While the height of first wave crest at bow for Rankine source method is much smaller than the experiment result. The reason for this is that Rankine source method does not take the viscosity of water and nonlinearity of free surface and the nonlinear term of free surface boundary condition into consideration. Compared with three dimensional RANS method, the wave height for Rankine source method is the wave height near the hull instead of the actual wave height on the hull surface, so the wave profile deviation exists, especially at bow. The experimental results in Fig. 9 are obtained from Ishikawajima-Harima Heavy Industries Co., Ltd (IHHI) [17].

Fig. 9.

Wave profile for Wigley hull at different $F n$ .

5.2. Numerical calculation results for S60 hull

Curves of shape factor $(1 + k)$ and resistance coefficient are drawn based on numerical results of S60 hull in Fig. 10. From the results of the three-dimensional RANS method it appears that the value of $(1 + k)$ slightly increases with the increase of the Froude number. The friction resistance coefficient coincides well with 1957 ITTC plate friction resistance coefficient, with fairly small deviation. The value of wave-making resistance coefficient for three dimensional RANS method is more accurate than that of Rankine source method with linear free surface boundary condition, and the same conclusion can be obtained for Wigley hull. The accuracy and feasibility of three dimensional RANS method to solve wave-making resistance is verified. The experimental results of ship S60 are from Ishikawajima-Harima Heavy Industries Co., Ltd (IHHI) [17].

Fig. 10.

Curves of $(1 + k)$ , $C_{f}$ , $C_{w}$ and $F n$ , $R e$ .

Using data viewer package, the numerical results of three dimensional RANS method at $F n = 0.22$ and $F n = 0.3$ are shown in Fig. 11. The phases of wave profile obtained from three different methods are the same in general. The numerical wave profile of three-dimensional RANS method is better than Rankine source method for the most part of the free surface at shipside. Compare with Rankine source method, the wave trough of three dimensional RANS method is lower, while the wave crest is higher. Due to the fact that the wave height for the Rankine source method is near the hull instead of the actual wave height on the hull surface, and the Rankine source method does not take the viscosity of water and nonlinearity of free surface and the nonlinear term of free surface boundary condition into consideration. It’s obvious that, for Rankine source method, the height of wave crest has a large error at bow. And the same conclusion can be obtained for Wigley hull. The experimental results are from Ship Research Institute (SRI) (see [20]).

Fig. 11.

Wave profile for S60 hull at different $F n$ .

5.3. Analysis of wave pattern with free surface

The free surface wave of three dimensional RANS method and Rankine source method are numerically analyzed to study the changing rule of wave-making characteristics. The numerical results for Wigley hull and S60 hull at $F n = 0.24$ and $F n = 0.3$ are selected to obtain the 3-D free surface wave pattern using data viewer package as shown in Fig. 12 and Fig. 13. Figure 12 and Fig. 13 show that the wave pattern conforms to the characteristics of Kelvin wave system, and there is obvious wave disturbance at bow and two sides of the hull. The variation tendency of free surface wave pattern based on three-dimensional RANS method is as follows. With the increase of Froude number, the disturbance region at bow and stern and two sides of hull becomes bigger and the wave-making region spreads outwards. At the same time, the Kelvin wave at fore and aft shoulders becomes obvious and the wave crest moves backwards. The variation tendency of 3-D free surface wave pattern for both three-dimensional RANS method and Rankine source method is almost the same and totally conforms to the characteristics of Kelvin wave system, so the three-dimensional RANS method has good engineering applicability.

Fig. 12.

Wave pattern of the Wigley hull at various speed. (a) wave pattern based on three dimensional RANS method at $F n = 0.24$ , (b) wave pattern based on Rankine source method at $F n = 0.24$ , (c) wave pattern based on three dimensional RANS method at $F n = 0.3$ , (d) wave pattern based on Rankine source method at $F n = 0.3$ .

Fig. 13.

Wave pattern of the S60 hull at various speed. (a) wave pattern based on three dimensional RANS method at $F n = 0.24$ , (b) wave pattern based on Rankine source method at $F n = 0.24$ , (c) wave pattern based on three dimensional RANS method at $F n = 0.3$ , (d) wave pattern based on Rankine source method at $F n = 0.3$ .

The numerical results for Wigley hull and S60 hull at $F n = 0.24$ , $F n = 0.26$ , $F n = 0.28$ and $F n = 0.3$ are selected to obtain the contour of wave height using by data viewer package as shown in Fig. 14 and Fig. 15. The conclusion can be easily drawn from Fig. 14 and Fig. 15 that the wave pattern of three dimensional RANS method is similar with that of Rankine source method, and there is some difference for the value of wave height. The height of wave crest obtained from three-dimensional RANS method is greater than that of Rankine source method at fore and aft and two sides of hull, but the wave trough is lower, which matches well with Figs 9 and 11. Figure 14 shows that the area of wave trough contour of Wigley hull based on the three dimensional RANS method is smaller than the results of Rankine source method at any Fround number. And the conclusion can be drawn from Fig. 15 that the area of wave crest contour at midship of S60 hull based on the dimensional RANS method is smaller than the results of Rankine source method at any Fround number. But the variation tendency of 3-D free surface wave pattern for both three-dimensional RANS method and Rankine source method is almost the same. The wave crest of free surface at bow and stern is obvious. With the increase of Froude number, the values of wave crest and wave trough increases and the wave-making region at midship and stern enlarges. The Kelvin wave system can be clearly observed with the wave crest moving backwards. The numerical result of 3-D free surface wave pattern is verified.

Fig. 14.

Contour of wave height of the Wigley hull at various speed. (a) contour of wave height based on three dimensional RANS method at $F n = 0.24$ , (b) contour of wave height based on Rankine source method at $F n = 0.24$ , (c) contour of wave height based on three dimensional RANS method at $F n = 0.26$ , (d) contour of wave height based on Rankine source method at $F n = 0.26$ , (e) contour of wave height based on three dimensional RANS method at $F n = 0.28$ , (f) contour of wave height based on Rankine source method at $F n = 0.28$ , (g) contour of wave height based on three dimensional RANS method at $F n = 0.3$ , (h) contour of wave height based on Rankine source method at $F n = 0.3$ .

Fig. 14.

(Continued.)

Fig. 15.

Contour of wave height of the S60 hull at various speed. (a) contour of wave height based on three dimensional RANS method at $F n = 0.24$ , (b) contour of wave height based on Rankine source method at $F n = 0.24$ , (c) contour of wave height based on three dimensional RANS method at $F n = 0.26$ , (d) contour of wave height based on Rankine source method at $F n = 0.26$ , (e) contour of wave height based on three dimensional RANS method at $F n = 0.28$ , (f) contour of wave height based on Rankine source method at $F n = 0.28$ , (g) contour of wave height based on three dimensional RANS method at $F n = 0.3$ , (h) contour of wave height based on Rankine source method at $F n = 0.3$ .

Fig. 15.

(Continued.)

6. Conclusion

Based on the theoretical methods of both viscous flow and potential flow, this paper proposes an engineering calculation method to solve wave-making resistance. The new method is developed from RANS method according to the three dimensional classification mode of resistance. The double-body flow and free-surface flow for both Wigley hull and S60 hull are numerically calculated. The numerical results for wave-making resistance are compared with the results of Rankine source method and EFD method. The following conclusions are drawn.

The result of the wave-making resistance based on Rankine source boundary element method matches well with the result of three-dimensional RANS method and EFD method, which indicates that the developed numerical program has good accuracy.

Compared with the model test, there is no free surface for double-body flow. Therefore the double-body flow breaks through the limitation of the inflow velocity, unstable flow and wave disturbance. The shape factor $(1 + k)$ can be obtained for a much bigger range of inflow velocity. From the results of the three-dimensional RANS method it appears that the value of $(1 + k)$ slightly increases with the increase of the Froude number.

Considering the viscosity of water, non-linearity of free surface and complexity of 3-D hull surface, the biggest deviation of total resistance coefficient $C_{t}$ for Wigley hull is 2.99%, the friction resistance coefficient of Wigley hull and S60 hull based on three-dimensional RANS method coincides well with 1957 ITTC plate friction resistance coefficient, with fairly small deviation, and the numerical results of wave-making resistance of Wigley hull and S60 hull based on three-dimensional RANS method match well with the results of EFD method and Rankine source method with linear free surface boundary condition. The numerical accuracy of three-dimensional RANS method is higher than that of Rankine source method with linear free surface boundary condition.

The free surface wave pattern can reflect the characteristics of wave-making. The height of wave crest for three-dimensional RANS method is greater than Rankine method, but the wave trough values are lower. Besides, with the increase of Froude number, the variation tendency of wave-making pattern is obvious, the height of free surface wave at bow and stern increases, and the wave crest of Kelvin wave system at fore and aft shoulders obviously moves afterwards with the wave-making region at midship and stern spreading outwards. The variation tendency of free surface is almost the same with Rankine source method. Therefore the three dimensional RANS method can meet the requirements of engineering application.

Combined the accuracy of the double-body model in solving form factor $(1 + k)$ and the maturity of CFD in solving free surface flow past hull, three dimensional RANS method is feasible and reliable to study the wave-making resistance, and its accuracy meets the requirements of engineering applications.

Footnotes

Acknowledgements

The authors are grateful to the National R&D Special Fund for Public Welfare Industry (201003024), the Project funded by China Postdoctoral Science Foundation (2014M561234, 2015T80256), the Program funded by National Defense Basic Scientific Research (A0820132027) and the Doctoral Scientific Research Foundation of Liaoning Province (201501176) for their financial support.

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