Prediction of ship power and speed performance based on RANS method

Abstract

This paper adopts virtual tank experimental technology to study ship rapidity performance. Based on RANS method, a new method is put forward to predict the speed and power performance of full scale ship. The open water performance of propeller model, the resistance performance of ship model and the self-propulsion performance of ship model are numerically calculated. Through the analysis of rapidity numerical results of fishing ship and the change law of wake flow field, some conclusions can be drawn. The numerical prediction accuracy of propeller open water efficiency has a direct influence on total thrust efficiency. The interference between hull, propeller and rudder leads to the formation of a lower pressure zone on stern hull surface and rudder surface. The axial velocity and radial velocity are affected by the interference between the flow of hull stern and propeller, the absolute value of axial velocity on blade leaf is smaller than that on blade back, and the radial velocity induces vortices on the blade tip of the propeller. The increase of propeller rotation speed nearly has no effect on the free surface wave pattern. The prediction results deviations of full scale ship speed and propeller rotation speed between RANS method and EFD method are 0.04 kn and 0.4 r/min, and therefore they are negligible. The proposed method can provide reference for engineering application.

Keywords

RANS method EFD method power performance self-propulsion factor speed prediction

1. Introduction

In the field of ship hydrodynamics, rapidity is a vital performance index. The prediction of power and speed performance of full scale ship is always an important study subject in the field of naval architecture and academia. The traditional prediction method of the power and speed performance of full scale ship is based on model test, including ship model resistance test, propeller model open water test and ship model self-propulsion test [15]. The primary advantages of traditional method are high reliability and good practicability. But its disadvantages are also obvious, such as high cost, long experimental period, difficult to analyze and understand the microscopic flow mechanism and difficult to give quick response to variation of design parameters. At present, there are mainly two kinds of resistance conversion methods between model ship and full scale ship, 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 through the equivalent of two residual resistance coefficients. The three-dimensional method, according to Hugh’s view, divides the total resistance into friction resistance, viscous resistance and wave-making resistance. The total resistance conversion of the model ship and full scale ship is realized through the equivalent of two wave-making resistance coefficients.

As technology progresses and the computer performance updates, the Computational Fluid Dynamics (CFD) gains rapid development. On this background, the virtual tank experimental technology, the core of which is CFD technology, has got a wide application in the field of design and prediction of ship’s comprehensive hydrodynamic performance. This method has the advantages of low cost, short period, perfect visual interface for flow field information, quick response for the deviation of any design input. Nowadays, the virtual tank experimental technology has made considerable progress. The CFD technology for ship model resistance performance and propeller model open water performance is gradually becoming mature, and its prediction accuracy has met the requirement of engineering application [5,7,14]. A significant achievement has been made in the field of directly solving ship model self-propulsion performance based on Reynolds Averages Navier Stokes (RANS) method [11,12], but since the interference between ship, propeller and rudder, the flow field is very complicated and the numerical accuracy needs to be increased. At present, for the simulation of ship model self-propulsion performance, the most common approach is to combine the RANS method with potential flow method. The propeller is substituted by a volume force based on RANS method, and the obtained flow field is taken as input data. Then the lifting line (surface) method is applied to numerically calculate the thrust and torque of the propeller. The self-propulsion point and self-propulsion factor can be obtained by repeated iteration [1–3,9,10,16,17].

Based on RANS method, this paper conducts the numerical simulation of propeller model open water performance, ship model resistance performance and ship model self-propulsion to predict the power and speed performance of the full scale ship. Constant ship forward speed and variable propeller rotation speed are followed in the numerical simulation of ship model self-propulsion performance. The numerical results can be used to study the mechanism of ship resistance and propulsion characteristics, to numerically analyze the change of the free surface wave pattern for the resistance simulation and the variation of flow field information for self-propulsion simulation. It shows that the prediction results of power and speed performance obtained from RANS method achieves high accuracy and matches well with Experimental Fluid Dynamics (EFD) method. And the feasibility and practicability of using the virtual tank experimental technology to solve ship resistance and propulsion characteristics are proved.

2. Calculation method

2.1. RANS equation

Taking the viscosity and incompressibility into consideration, the RANS method is applied to numerically calculate ship rapidity. Its continuity equation(mass conservation equation) is expressed as below [4,18]: $\begin{matrix} (1) & \frac{\partial ρ}{\partial t} + \frac{\partial (ρ u)}{\partial x} + \frac{\partial (ρ v)}{\partial y} + \frac{\partial (ρ w)}{\partial z} = 0 \end{matrix}$ Where, ρ is the fluid density, ( $u, v, w$ ) is the three components of the velocity of flow along three axes ( $x, y, z$ ) of the Cartesian coordinate system.

There will be free surface flow in the simulation of both ship model resistance performance and ship model self-propulsion performance. The free surface flow is treated as two-phase flow (water and air), the interface between water and air is the free surface. The water and air comply with the conservation of mass. The VOF method is applied and its continuity equation of volume fraction is expressed as below: $\begin{matrix} (2) & \sum_{q = 1}^{p} [\frac{\partial}{\partial t} (α_{q} ρ_{q}) + \nabla \cdot (α_{q} ρ_{q} \vec{V})] = 0 p = 2 \end{matrix}$ Where, $\vec{V}$ is the velocity vector, ρ is density, α is volume fraction, subscript p and q respectively stands for water and air, every control volume in the computational domain is filled with water and air, the sum of their volume fraction equals 1, namely: $α_{p} + α_{q} = 1$ . The two-phase flow shall comply with the Reynolds averaged N–S equation (momentum conservation equation) as follows [4,18]: $\begin{matrix} (3) & \frac{\partial u_{i}}{\partial t} + u_{j} \frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial \overline{u_{i}^{'} u_{j}^{'}}}{\partial x_{j}} = - \frac{1}{ρ} \frac{\partial p}{\partial x_{i}} + ν \frac{\partial^{2} u_{i}}{\partial x_{j} \partial x_{j}} + f_{i} \end{matrix}$ Where, ( $ρ, p, f_{i}, t, ν, u_{i}, u_{i}^{'}, \partial \overline{u_{i}^{'} u_{j}^{'}} / \partial x_{j}$ ) respectively stands for fluid density, fluid static pressure, mass force, time, fluid kinematic viscosity coefficient, time averaged velocity, fluctuation velocity and Reynolds stress term.

2.2. Turbulence model

Table 1
Turbulence model constant

$α_{1}$ $β_{1}$ $σ_{k 1}$ $σ_{ω 1}$ $α_{2}$ $β_{2}$ $σ_{k 2}$ $σ_{ω 2}$ $β^{*}$

$5 / 9$ $3 / 40$ 0.85 0.5 0.44 0.0828 1 0.856 0.09

$α_{1}$	$β_{1}$	$σ_{k 1}$	$σ_{ω 1}$	$α_{2}$	$β_{2}$	$σ_{k 2}$	$σ_{ω 2}$	$β^{*}$
$5 / 9$	$3 / 40$	0.85	0.5	0.44	0.0828	1	0.856	0.09

Standard K–ε model, developed from turbulence flow, is a turbulence calculation model aimed at high Reynolds number, but it has some limitation at low Reynolds number flow calculation, such as near-wall inadequate turbulence developing flow. Standard K–ω turbulence model takes low Reynolds number, compressibility and shear flow into consideration, it is widely used to deal with boundary layer problems at various pressure gradients. SST K–ω turbulence model is a hybrid model which is widely used in engineering, and it combines the advantages of far field calculation of Standard K–ε turbulence model and the advantages of near-wall turbulence flow calculation of Standard K–ω model. SST K–ω turbulence model takes the transportation characteristics of the turbulence shear stress into consideration in the calculation of turbulence viscous coefficient, and can accurately predict the flow separation area and flow separation points resulted from the adverse pressure gradient. Therefore, SST K–ω turbulence model has great advantages in studying and predicting the complex turbulence with flow separation, and it has high applicability and reliability. The transportation equations of its turbulent kinetic energy K and turbulent dissipation ω are as follows [13,19]: $\begin{array}{rcl} (4) & \frac{\partial k}{\partial t} + \overline{u_{j}} \frac{\partial k}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(υ + σ_{k} υ_{t}) \frac{\partial k}{\partial x_{j}}] + P_{k} - β^{*} k ω \\ \frac{\partial ω}{\partial t} + \overline{u_{j}} \frac{\partial ω}{\partial x_{j}} = \frac{\partial}{\partial x_{j}} [(υ + σ_{ω} υ_{t}) \frac{\partial ω}{\partial x_{j}}] + α S^{2} - β ω^{2} \\ (5) & + 2 (1 - F_{1}) σ_{ω 2} \frac{1}{ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}} \end{array}$ Where, υ is fluid dynamic viscosity; $P_{k} = min [μ_{t} \frac{\partial \overline{u_{i}}}{\partial x_{j}} (\frac{\partial \overline{u_{i}}}{\partial x_{j}} + \frac{\partial \overline{u_{j}}}{\partial x_{i}}), 10 \cdot β^{*} ρ k ω]$ ; $υ_{t} = \frac{α_{1} k}{max (α_{1} ω, Ω F_{2})}$ ; $σ_{k} = \frac{1}{F_{1} / σ_{k 1} + (1 - F_{1}) / σ_{k 2}}$ ; $S = \sqrt{2 S_{i j} S_{i j}}$ ; $S_{i j} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})$ ; $F_{1}$ and $F_{2}$ are composite functions, their definitions are as follows: $F_{1} = tanh (Φ_{1}^{4})$ ; $F_{2} = tanh (Φ_{2}^{2})$ ; $Φ_{1} = min [max (\frac{\sqrt{k}}{β^{*} ω y}, \frac{500 υ}{y^{2} ω}), \frac{4 ρ σ_{ω 2} k}{C D_{k ω} y^{2}}]$ ; $Φ_{2} = max [\frac{2 \sqrt{k}}{β^{*} ω y}, \frac{500 υ}{y^{2} ω}]$ ; $C D_{k ω} = max [2 ρ σ_{ω 2} \frac{1}{ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}}, 10^{- 10}]$ . Each constant (coefficient) for the transportation equations of k and ω can be obtained from the equation $φ = φ_{1} F_{1} + φ_{2} (1 - F_{1})$ . Where $φ_{1}$ is the constant (coefficient) of the Standard K–ε turbulence model equation, $φ_{2}$ is the constant (coefficient) of the Standard K–ω turbulence model equation. The detailed values are listed in Table 1.

Table 2

Parameters of ducted propeller

Nomenclature	Unit	Full scale	Model
$D_{P}$	m	1.9600	0.2000
$C_{0.7 R}$	m	0.7428	0.0758
$t_{0.7 R}$	m	0.0245	0.0025
$A_{E} / A_{0}$	–	0.6100
${(P / D_{p})}_{0.7 R}$	–	1.2000
Z	–	4
$d_{hub} / D_{P}$ (–)	–	0.2550
R	–	Right-handed
L	m	0.9800	0.1000
M	–	19A

2.3. Numerical analysis method

In order to solve the governing equations, the finite volume method is used for the space discretization [18]. 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 PISO (Pressure Implicit with Splitting of Operators) algorithm is applied to the velocity–pressure coupling (Issa, 1986) [8]. 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 (Hirt and Nichols, 1981) [6] and volume fraction discretization, respectively.

3. Numerical simulation of propeller open water performance

3.1. Parameters of propeller

The ducted propeller is selected to carry out the numerical simulation of propeller open water performance. The size of the ducted propeller used in numerical calculation is same with the EFD method, with a scale ratio of 9.8. The parameters of full scale propeller and model propeller are listed in Table 2. The geometry parameters of the propeller in Table 2 are defined as follows: ( $D_{P}$ , $C_{0.7 R}$ , $t_{0.7 R}$ , $A_{E} / A_{0}$ , ${(P / D_{p})}_{0.7 R}$ , Z, $d_{hub} / D_{P}$ , R, L, M) is respectively the propeller diameter, the chord length, the maximum thickness, the disk ratio, the pitch ratio, the propeller blade number, the hub diameter ratio, the rotation direction, the ducted length and the ducted type.

3.2. Calculation model and meshing

The numerical calculation model for ducted propeller is created by Pro/ENGINEER software with the parameters mentioned above. Pro/ENGINEER (Pro/E) is a famous CAD/CAE/CAM software, developed by Parametric Technology Corporation (PTC) in American. It integrates 3D modeling design, process, analysis and plot of product etc. The numerical calculation model and the experimental model are shown in Fig. 1.

Fig. 1.

Model of ducted propeller (a) Calculation model (b) Experimental model.

The accuracy and time of the numerical calculation is determined by the grids quality and the size of the computational domain, so the key point is to select a suitable size of the computational domain and generate high quality grids. This paper adopted trimmed mesh to mesh the computational domain. Trimmed mesh breaks through the limitation of unstructured mesh in low filling rate and low accuracy in solving viscosity problem. Multiple reference frame is applied to numerically calculate the propeller open water performance. The whole computational domain is divided into stationary domain and rotation domain, and the interface between them is defined as Interior. The whole computational domain is a cylinder, with a diameter of 10D (D is the diameter of propeller model) and a length of 16D. The distance between the propeller disk and the velocity entrance is 4D. The rotation domain is a smaller cylinder inside the duct to ensure that only the propeller itself is rotating while the duct is stationary. The length of rotation domain is 0.8D, the diameter is 1.02D. According to the needs of grid refinement, the computational domain is divided into several different blocks, and the degree of grid refinement in each block is different. The new block is generated by cutting and refining original block. Any body grid can be cut into any number of body grids. The grids between blocks are gradually changed. Since the complexity of 3-D surface of the propeller blade, hub and duct, a block is used to realize the grids refinement near the propeller. The grids size of blade, hub and duct surface is 0.015D; the grids size of the outer boundary of stationary domain is 0.12D. The boundary layer is generated around the propeller wall surface. The height of the first layer grids is 0.001 m, and the value of dimensionless y+ is between 30 and 500. The meshing of the computational domain and the boundary condition settings are shown in Fig. 2.

Fig. 2.

Meshing and size of Computational domain and boundary condition (a) Meshing and size of Computational domain (b) Boundary condition.

3.3. Grid convergence analysis

The open water model of ducted propeller is chosen to analyze the convergence of the mesh. The computational grid system (Grid-1) is generated through the grid partition method introduced in Section 3.2. 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 ducted propeller 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 2404800 ( $167 \times 120 \times 120$ ), 852550 ( $118 \times 85 \times 85$ ) and 298800 ( $83 \times 60 \times 60$ ), respectively.

When advance coefficient $J_{m}$ is 0.9 and the rotation speed is set to be $N_{m} = 18$ r/s, the same with EFD method, the calculation results of thrust coefficient $K_{T m}$ and torque coefficient $10 K_{Q m}$ generated by ducted propeller for three kinds of grids are shown in Table 3.

Table 3
Results of thrust coefficient and torque coefficient of ducted propeller

$J_{m} = 0.9$ Grid-1 Grid-2 Grid-3

$K_{T m}$ 0.0709 0.0722 0.0741

$10 K_{Q m}$ 0.3157 0.3221 0.3308

$J_{m} = 0.9$	Grid-1	Grid-2	Grid-3
$K_{T m}$	0.0709	0.0722	0.0741
$10 K_{Q m}$	0.3157	0.3221	0.3308

According to the numerical calculation results in Table 3, grid convergence calculation is conducted to thrust coefficient $K_{T m}$ and torque coefficient $10 K_{Q m}$ , respectively, the result is shown in Table 4. The subscript T and Q refer to the grid convergence calculation results of thrust coefficient $K_{T m}$ and torque coefficient $10 K_{Q m}$ , respectively. The formulas of grid convergence rate $R_{G}$ , accuracy order $P_{G}$ and correction factor $C_{G}$ are as follows. $\begin{array}{l} (6) & ε_{i j} = S_{G i} - S_{G j} \\ (7) & R_{G} = ε_{21} / ε_{32} \\ (8) & P_{G} = ln (ε_{32} / ε_{21}) / ln r_{G} \\ (9) & C_{G} = (r_{G}^{P_{G}} - 1) / (r_{G}^{P_{Gest}} - 1) \end{array}$ Where: $S_{G i}$ refers to thrust coefficient $K_{T m}$ (or torque coefficient $10 K_{Q m}$ ) of the Grid-i systems, $P_{Gest}$ refers to estimate for the limiting order of accuracy. Grid convergence rate $R_{G}$ are obtained from calculation result, then the grid convergence conditions are determined. All three kinds of grids meet the monotonic convergence condition as $0 < R_{G T}$ , $R_{G Q} < 1$ . The Richardson extrapolation method is used to estimate both errors and uncertainties. Correction factor $C_{G}$ meets the formula $| 1 - C_{G} | ⩾ 0.25$ , according to the research on the verification of uncertainty validity of Stern et al. [20,21], the improved numerical error $δ_{G}$ , the corrected numerical result $S_{C}$ , the numerical uncertainty $U_{G}$ and the corrected numerical uncertainty $U_{G C}$ are calculated, the corresponding formulas are as follows: $\begin{array}{l} (10) & δ_{G} = C_{G} ε_{21} / (r_{G}^{P_{G}} - 1) \\ (11) & S_{C} = S_{G 1} - δ_{G} \\ (12) & U_{G} = [2 | 1 - C_{G} | + 1] \times ε_{21} / (r_{G}^{P_{G}} - 1) for | 1 - C_{G} | ⩾ 0.125 \\ (13) & U_{G C} = [| 1 - C_{G} |] \times ε_{21} / (r_{G}^{P_{G}} - 1) for | 1 - C_{G} | ⩾ 0.25 \end{array}$

According to the grid convergence calculation results based on ducted propeller thrust coefficient $K_{T m}$ and torque coefficient $10 K_{Q m}$ respectively in Table 4, it shows that the improved numerical error $δ_{G T}$ and the corrected numerical uncertainty $U_{G C T}$ are all smaller than numerical uncertainty $U_{G T}$ , the improved numerical error $δ_{G Q}$ and the corrected numerical uncertainty $U_{G C Q}$ are also all smaller than the numerical uncertainty $U_{G Q}$ . So it can be concluded that the grid system (Grid-1) utilized by the paper meets the requirement of numerical error and uncertainty.

Table 4

Calculation results of grid convergence

$R_{G T}$	$P_{G T}$	$C_{G T}$	$δ_{G T}$	$U_{G T} \times 10$	$U_{GCT} \times 10$	$S_{C T}$
0.6842	1.0949	0.4615	0.0013	0.0585	0.0152	0.0696
$R_{G Q}$	$P_{G Q}$	$C_{G Q}$	$δ_{G Q}$	$U_{G Q} \times 10$	$U_{GCQ} \times 10$	$S_{C Q}$
0.7356	0.8858	0.3594	0.0064	0.4063	0.1141	0.3093

3.4. Analysis of numerical calculation results

For the simulation of the propeller open water performance, the rotation speed is set to be $N_{m} = 18$ r/s, the same with EFD method. The variation of advance coefficient $J_{m}$ is adjusted by changing the flow velocity $V_{m}$ and $J_{m}$ ranges from 0.1 to 0.9. The numerical results can be converted to non-dimensional form through the formulas below: $\begin{array}{l} Thrust coefficient: K_{T m} = \frac{T_{m}}{ρ_{m} N_{m}^{2} D_{p m}^{4}} \\ Torque coefficient: K_{Q m} = \frac{Q_{m}}{ρ_{m} N_{m}^{2} D_{p m}^{5}} \\ Propeller open water efficiency: η_{m} = \frac{J_{m} K_{T m}}{2 π K_{Q m}} \end{array}$

Fig. 3.

Open water performance curve of model propeller and full scale propeller (a) Open water performance curve of Model propeller (b) Open water performance curve of Full scale propeller.

The propeller open water performance curve is drawn as Fig. 3(a). It’s obvious that thrust coefficient $K_{T m}$ , torque coefficient $K_{Q m}$ , propeller open water efficiency $η_{m}$ matches well with the EFD results. For different advance coefficients, thrust coefficient $K_{T m}$ and propeller open water efficiency $η_{m}$ is a little smaller than the experimental results, while torque coefficient $K_{Q m}$ , a little bigger than the experimental results. With the increase of the advance coefficient $J_{m}$ , the accuracy of the numerical prediction slightly decreases. When $J_{m} = 0.9$ , the deviation of propeller open water efficiency reaches the biggest, 2.07%.

The prediction of full scale propeller open water performance can be carried out by ITTC-1978 method through scale effect correction, and the detailed formulas are listed in Appendix. The open water performance curve of full scale propeller is shown in Fig. 3(b).

4. Numerical calculation of ship model resistance

4.1. Parameters of ship

A fishing vessel is taken as the calculation model with the scale ratio of 9.8. The parameters of the numerical model are the same with the EFD method and are listed in Table 5. The wet surface area in Table 5 includes the wet surface area of an appendage rudder; the parameters of this rudder are not introduced in detail in this paper. The geometry parameters of the ship in Table 5 are defined as follows: ( $L_{p p}$ , $L_{W L}$ , B, T, S, $C_{B}$ , $C_{W}$ , $C_{p}$ ) is respectively the length between fore and aft perpendiculars, the length of waterline, the breadth of ship, the design draft, the wet surface area, the block coefficient and the water plane coefficient and the diamond coefficient. In addition, the waterline plan is shown in Fig. 4.

Table 5
Ship particulars of the fishing vessel

Nomenclature Unit Full scale Model

$L_{p p}$ m 27.40 2.7959

$L_{W L}$ m 30.24 3.0857

B m 6.00 0.6122

T m 2.10 0.2143

S $m^{2}$ 243.40 2.5344

$C_{B}$ – 0.6517

$C_{W}$ – 0.9339

$C_{P}$ – 0.6850

Nomenclature	Unit	Full scale	Model
$L_{p p}$	m	27.40	2.7959
$L_{W L}$	m	30.24	3.0857
B	m	6.00	0.6122
T	m	2.10	0.2143
S	$m^{2}$	243.40	2.5344
$C_{B}$	–	0.6517
$C_{W}$	–	0.9339
$C_{P}$	–	0.6850

Fig. 4.

Waterline plan of the ship.

4.2. Meshing and selection of the computational domain

Based the parameters mentioned above, Auto CAD software and Pro/ENGINEER software are used to create the calculation model (rudder is included). The calculation model is shown as Fig. 5.

Fig. 5.

Ship model.

Considering the symmetry of the flow, only half of the flow field and hull form is created. 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 $Z_{A}$ , wavelength $λ_{w}$ and velocity. The height of the topsides is greater than the maximum height of the ship wave which is determined by $Z_{A} = v^{2} / 2 g$ [22], and the wavelength can be obtained based on $λ_{w} = 2 π c^{2} / g$ . Since the existence of free surface, the wave surface upheaves at bow. The length of upstream part is taken bigger than $λ_{w}$ . 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 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 Fig. 6. The detail of water plane grids and midship section grids can be easily seen in Fig. 6. L is the length of waterline.

Fig. 6.

Meshing and size of Computational domain (a) Meshing of water plane (b) Meshing of midship section.

Since the numerical calculation of ship resistance performance takes the wave resistance into consideration, the computational model shall consider the effect of free surface. To accurately track the free surface wave pattern, the mesh need to be refined both for the region just above and below the water plane and the region where Kelvin wave system will occur. The height of the first layer mesh at the level of water plane is taken as 0.002L. The grids after refinement for Kelvin wave system region is shown as Fig. 6(a). In addition, for some regions where the velocity gradient is big, the grids also need to be refined to gain the detailed information of the flow field. The whole computational domain is meshed by trimmed mesh. According to the needs of grid refinement, the computational domain is divided into several different blocks, and the degree of grid refinement in each block is different. The new block is generated by cutting and refining original block. Any body grid can be cut into any number of body grids. The grids between blocks are gradually changed. The fore part and aft part as well as the region around the hull wall surface all use blocks to realize the grids refinement. The grids size of fore part and aft part is 0.003L and the maximum size of the hull wall surface grids is 0.01L. There are in total 8 layers of mesh in the boundary layer. The height of the innermost grids is taken as 1 mm, fulfilling the requirement of dimensionless y+. To maintain high consistency between numerical simulation and EFD method, the boundary conditions for the computational domain are set as Fig. 7. Besides, the numerical damping is defined on boundaries to eliminate the reflection of ship wave.

Fig. 7.

Boundary conditions setting of the Computational domain.

4.3. Analysis of numerical calculation results

The ship resistance at different Froude numbers is numerically calculated. The curves of frictional resistance coefficient $C_{F m} = \frac{2 R_{f m}}{ρ V_{m}^{2} S_{m}}$ , total resistance coefficient $C_{T m} = \frac{2 R_{T m}}{ρ V_{m}^{2} S_{m}}$ and residual resistance coefficient $C_{R m} = C_{T m} - C_{F m}$ are shown in Fig. 8. Where, $R_{f m}$ is the frictional resistance of ship model, $R_{T m}$ is the total resistance of ship model. It’s obvious that the numerical results of $C_{F m}$ , $C_{T m}$ and $C_{R m}$ match well with the EFD results and the deviation is very small. With the increase of Reynolds number, the frictional resistance coefficient $C_{F m}$ decreases and the value of $C_{F m}$ calculated by RANS method is a little smaller than 1957 ITTC Tabletfrictionalresistancecoefficient. With the increase of the Froude number, the deviation of total resistance coefficient $C_{T m}$ and residual resistance coefficient $C_{R m}$ between numerical results and EFD results slightly enlarges. The maximum deviation of $C_{T m}$ and $C_{R m}$ is respectively 1.54% and 1.72%. It can be concluded that using RANS method to solve ship resistance performance is feasible.

Fig. 8.

Resistance coefficient curve of ship model (a) Curve of $C_{F m}$ and Re (b) Curve of $C_{T m}$ and $F n$ (c) Curve of $C_{R m}$ and $F n$ .

The experimental results of ship model resistance test and numerical results of RANS method are converted by two-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 through the equivalent of two residual resistance coefficients. For fishing vessels, the roughness allowance $C_{F} = 0.6 \times 10^{- 3}$ , the detailed conversion formulas are listed in Appendix. The converted full scale frictional resistance coefficient $C_{F s}$ , total resistance coefficient $C_{T s}$ and effective power $P_{E}$ are drawn in Fig. 9.

Fig. 9.

Curve of full scale ship resistance coefficient and effective power (a) Curve of $C_{F s}$ and Re (b) Curve of $C_{T s}$ and $F n$ (c) Curve of $P_{E}$ and $V_{s}$ .

4.4. Analysis of wave pattern with free surface

To study the characteristics and change law of ship wave, the free surface wave pattern both for RANS method and EFD method are compared at $F n = 0.239$ and $F n = 0.329$ , and Fig. 10 shows the free surface wave in detail. For the EFD method, the ship wave swells at bow and the free surface wave pattern conforms to Kelvin wave system. With the increase of Froude number, the disturbance of wave surface enhances and the wave-making region spreads outward. For the RANS method, the highest wave crest occurs at bow and stern area, and the ship wave at fore and aft shoulder area presents obvious characteristics of Kelvin wave system. With the increase of Froude number, the disturbance region at bow and stern and two sides of hull becomes bigger and the free surface wave pattern spreads outwards. The wave crest becomes higher and the wave trough becomes lower. At the same time, ship wave obviously moves backwards. The variation tendency of free surface wave pattern for both RANS method and EFD method is almost the same and totally conforms to the characteristics of Kelvin wave system, has good engineering applicability.

Fig. 10.

Free surface wave pattern at various speed (a) Wave pattern based on RANS method at $F n = 0.239$ (b) Wave pattern based on EFD method at $F n = 0.239$ (c) Wave pattern based on RANS method at $F n = 0.329$ (d) Wave pattern based on EFD method at $F n = 0.329$ .

5. Numerical simulation of ship model self-propulsion

5.1. Calculation model and meshing

Since the symmetry is not available for the numerical simulation of ship model self-propulsion, the entire hull surface is created and the width of the computational domain is twice of the original one. Except the width, other dimensions of the computational domain still keep the original values. Refer to the meshing method for the simulation of propeller open water performance and ship model resistance performance, trimmed mesh is applied to the new computational domain. The blocks are used to realize the refinement of grids adjacent to hull wall surface, propeller, rudder and free surface, and the block for refinement of grids adjacent to propeller is set as rotation zone. Figure 11 shows the calculation model of ship self-propulsion with grids on the surface of hull, propeller and rudder. Figure 12 shows the experimental model of ship self-propulsion.

The definition of boundary conditions is referred to the numerical simulation of ship model resistance performance and propeller open water performance. The initial condition for calculation is also uniform inflow.

Fig. 11.

Grids model of ship self-propulsion (a) the grids of whole ship model (b) the grids of ship bow model (c) the grids of ship stern model.

Fig. 12.

Experimental model of ship self-propulsion.

5.2. Ship model self-propulsion based on RANS method

The RANS method is used to numerically simulate ship model self-propulsion performance. Constant ship forward speed and variable propeller rotation speed are adopted in the simulation to obtain the self-propulsion factor. The detailed calculation process is as follows (Note: the unit for propeller rotation speed is r/s):

(1) Based on the characteristics of the calculation model, estimate the self-propulsion point rotation speed $N_{1}$ for each forward speed $V_{m}$ of the ship model.

(2) Two rotation speeds $N_{0}$ and $N_{2}$ are respectively selected in appropriate range before and after the estimated self-propulsion point rotation speed ( $N_{0} < N_{1} < N_{2}$ , to make sure that the self-propulsion point is included in the range from $N_{0}$ to $N_{2}$ ). In this paper, the interval between different rotation speeds is taken as 0.25 r/s, namely, $N_{0} + 0.25 = N_{1}$ , $N_{1} + 0.25 = N_{2}$ .

(3) Based on the numerical calculation results of each rotation speed ( $N_{0}$ , $N_{1}$ , $N_{2}$ ), the propeller thrust ( $T_{0}$ , $T_{1}$ , $T_{2}$ ), propeller torque ( $Q_{0}$ , $Q_{1}$ , $Q_{2}$ ) and ship model resistance under self-propulsion condition ( $R_{0}$ , $R_{1}$ , $R_{2}$ ) at each rotation speed can be obtained. The compelling force ( $Z_{0}$ , $Z_{1}$ , $Z_{2}$ ) at each rotation speed can be calculated according to the formula $Z = R - T$ . And then the curves of compelling force ( $Z_{0}$ , $Z_{1}$ , $Z_{2}$ ), propeller thrust ( $T_{0}$ , $T_{1}$ , $T_{2}$ ), propeller torque ( $Q_{0}$ , $Q_{1}$ , $Q_{2}$ ) versus rotation speed ( $N_{0}$ , $N_{1}$ , $N_{2}$ ) can be plotted.

(4) Calculate the self-propulsion point compelling force $F_{D}$ at forward speed $V_{m}$ , $F_{D}$ is the correction value to compensate the difference of the frictional resistance coefficient between ship model and full scale ship, $F_{D} = \frac{1}{2} ρ_{m} V_{m}^{2} S_{m} (C_{F m} - C_{F s} - Δ C_{F})$ , where $C_{F m}$ is the frictional resistance coefficient obtained from the numerical simulation of ship model resistance performance, $C_{F s}$ is the converted full scale frictional resistance coefficient. For fishing vessels, the roughness allowance is taken as $C_{F} = 0.6 \times 10^{- 3}$ . Based on the calculated $F_{D}$ , the interpolation is made in the curves drawn in step (3) according to $Z = F_{D}$ to obtain the self-propulsion point rotation speed $N_{0 m}$ , propeller thrust $T_{0 m}$ , and propeller torque $Q_{0 m}$ .

(5) Based on the ship forward speed, obtained self-propulsion point rotation speed N, propeller thrust and propeller torque, calculates the dimensionless parameters: $J_{0 m} = V_{m} / (N_{0 m} D_{p m})$ , $K_{T 0 m} = T_{0 m} / (ρ_{m} N_{0 m}^{2} D_{p m}^{4})$ and $K_{Q 0 m} = Q_{0 m} / (ρ_{m} N_{0 m}^{2} D_{p m}^{5})$ , where $D_{p m}$ is the diameter of propeller model.

(6) According to constant thrust method $K_{T 0 m} = K_{T m}$ , assuming that the thrust coefficient of self-propulsion propeller $K_{T 0 m}$ and the thrust coefficient of open water propeller $K_{T m}$ is equal, and then $J_{m}$ , $K_{Q m}$ and $η_{m}$ can be obtained. The effective wake fraction and relative rotative efficiency is respectively $w_{0 m} = 1 - J_{m} / J_{0 m}$ and $η_{R 0 m} = K_{Q m} / K_{Q 0 m}$ .

(7) The ship model resistance obtained from the numerical simulation of ship model resistance performance is $R_{m}$ (corresponding ship speed is $V_{m}$ ). The thrust deduction fraction is $t_{0 m} = (T_{0 m} - R_{m} + F_{D}) / T_{0 m}$ and the hull efficiency and total thrust efficiency is respectively $η_{H 0 m} = (1 - t_{0 m}) / (1 - w_{0 m})$ and $η_{D 0 m} = η_{R 0 m} η_{H 0 m} η_{m}$ .

5.3. Analysis of the self-propulsion factor

For the ship model self-propulsion performance, the numerical results of RANS method and experimental results of EFD method at different Froude numbers are drawn in Fig. 13 for compelling force Z, propeller thrust T, propeller torque Q and rotation speed N.

Fig. 13.

Curve of ship model self-propulsion performance (a) Curve of self-propulsion performance based on RANS method at $F n = 0.239$ (b) Curve of self-propulsion performance based on EFD method at $F n = 0.239$ (c) Curve of self-propulsion performance based on RANS method at $F n = 0.269$ (d) Curve of self-propulsion performance based on EFD method at $F n = 0.269$ (e) Curve of self-propulsion performance based on RANS method at $F n = 0.299$ (f) Curve of self-propulsion performance based on EFD method at $F n = 0.299$ (g) Curve of self-propulsion performance based on RANS method at $F n = 0.329$ (h) Curve of self-propulsion performance based on EFD method at $F n = 0.329$ .

Fig. 13.

(Continued.)

The self-propulsion point compelling force $F_{D}$ is calculated at different Froude numbers. According to $Z = F_{D}$ , Fig. 13 can be used to interpolate to obtain the self-propulsion point rotation speed $N_{0 m}$ , propeller thrust $T_{0 m}$ and propeller torque $Q_{0 m}$ , and then the self-propulsion factor can be calculated. The ship model self-propulsion factors for RANS method and EFD method are listed in Table 6. Through the interpolation results of Fig. 13, it can be concluded that the self-propulsion point rotation speed N of RANS method is bigger than EFD method at any Froude number. It can be seen from Table 6 that the self-propulsion factors obtained from RANS method match well with the EFD results in general. At any Froude number, the advance coefficient $J_{m}$ , open water efficiency $η_{m}$ , relative rotative efficiency $η_{R 0 m}$ , hull efficiency $η_{H 0 m}$ and total thrust efficiency $η_{D 0 m}$ obtained from RANS method are always smaller than the results of EFD method. When $F n = 0.329$ , the deviation of total thrust efficiency $η_{D 0 m}$ is the biggest, 2.71%. The reason for this deviation is mainly comes from the deviation of the numerical result of propeller open water efficiency $η_{m}$ .

Table 6

Self-propulsion factor of ship model

Nomenclature		$J_{m}$	$η_{m}$	$w_{0 m}$	$t_{0 m}$	$η_{R 0 m}$	$η_{H 0 m}$	$η_{D 0 m}$
$F n = 0.239$	RANS	0.648	0.579	0.312	0.455	0.792	1.068	0.490
$F n = 0.239$	EFD	0.654	0.590	0.309	0.451	0.794	1.071	0.502
$F n = 0.269$	RANS	0.620	0.574	0.329	0.475	0.782	1.074	0.482
$F n = 0.269$	EFD	0.624	0.585	0.327	0.472	0.784	1.078	0.494
$F n = 0.299$	RANS	0.590	0.566	0.330	0.451	0.819	1.074	0.498
$F n = 0.299$	EFD	0.594	0.576	0.328	0.447	0.823	1.077	0.511
$F n = 0.329$	RANS	0.569	0.558	0.330	0.435	0.842	1.068	0.502
$F n = 0.329$	EFD	0.572	0.568	0.328	0.430	0.849	1.071	0.516

5.4. Analysis of flow filed information

This paper applies the visualization technology to study the change law of fluid field of ship model self-propulsion and the mechanism of ship rapidity. Take the numerical results of ship model self-propulsion at different rotation speeds $N_{0 m}$ and constant $F n = 0.299$ for example. When the rotation speed is $N_{0 m} = 9.25$ r/s, the free surface wave pattern and streamline at stern is shown in Fig. 14. It can be seen from Fig. 14(a) that the free surface wave swells at bow and obvious fluctuation appears in the water. Figure 14(b) shows the detailed streamline passing through the stern, the direction of the streamline is changed because of the existence of ducted propeller.

Fig. 14.

Free surface wave pattern and streamline at stern $N_{0 m} = 9.25$ r/s (a) Free surface wave pattern (b) Streamline at stern.

As $F n = 0.299$ , the free surface wave pattern of ship model self-propulsion at propeller rotation speed $N_{0 m} = 9.25$ r/s and $N_{0 m} = 9.5$ r/s also conforms to the characteristics of Kelvin wave system. The free surface wave patterns at different rotation speeds are almost the same. With the increase of rotation speed, the difference between free surface wave patterns can be ignored. In addition, the pressure distribution on hull surface of ship model self-propulsion at propeller rotation speed $N_{0 m} = 9.25$ r/s and $N_{0 m} = 9.5$ r/s are shown in Fig. 15. The conclusion can be drawn from Fig. 15 that the numerically calculated hull surface pressure at $N_{0 m} = 9.5$ r/s is bigger than that at $N_{0 m} = 9.25$ r/s. The reason for this is that, the increase of ducted propeller’s rotation speed $N_{0 m}$ changed the flow field around the hull. The pressure distribution on hull surface is changed and the magnitude of the pressure becomes bigger. The interference between hull, propeller and rudder leads to the formation of a lower pressure zone on stern hull surface and rudder surface.

Fig. 15.

Pressure distribution on hull surface at various propeller rotation speed (a) Pressure distribution on hull surface at $N_{0 m} = 9.25$ r/s (b) Pressure distribution on stern hull surface at $N_{0 m} = 9.25$ r/s (c) Pressure distribution on hull surface at $N_{0 m} = 9.5$ r/s (d) Pressure distribution on stern hull surface at $N_{0 m} = 9.5$ r/s.

For the ship model self-propulsion at $F n = 0.299$ and $N_{0 m} = 9.25$ r/s, the distribution diagram of axial velocity and radial velocity on the propeller disk is shown in Fig. 16. In Fig. 16(a), since the complex surface of the stern hull and the interference between stern hull surface and propeller, in the upper region most close to the stern hull surface, the axial velocity at the center of propeller disk appears as lower pressure area, and the absolute value of axial velocity on blade leaf is smaller than that on blade back (the axial velocities on blade back and leaf are all negative value). The blade back is located in low pressure zone and the suction surface is formed on it. The velocity along the radius of propeller is radial velocity. In Fig. 16(b), since the interference between stern hull surface and propeller, a clear vortex system is formed at the upper left of the center of the propeller disk. The vortices are induced on the blade tip of the propeller, the reason is that a negative velocity region and a positive velocity region are respectively formed around blade back and blade leaf on the blade tip of the propeller, and the area of negative velocity region is larger than that of positive velocity region.

Fig. 16.

Distribution diagram of axial velocity and radial velocity on the propeller disk (a) Distribution diagram of axial velocity (b) Distribution diagram of radial velocity.

6. Prediction of power and speed performance of full scale ship

6.1. Numerical prediction method

Since the numerical simulation of propeller model open water performance, ship model resistance performance and ship model self-propulsion performance have been finished, and the self-propulsion factor is obtained. The power and speed performance of full scale ship can be predicted by the following numerical prediction method:

Assuming that the thrust reduction fraction of full scale ship is equal with ship model: $t_{s} = t_{0 m}$

Assuming that the relative rotative efficiency of full scale ship is equal with ship model: $η_{R s} = η_{R 0 m}$

Assuming that the wake fraction of full scale ship is equal with ship model: $w_{s} = w_{0 m}$

Full scale ship hull efficiency: $η_{H s} = \frac{1 - t_{s}}{1 - w_{s}}$

Full scale propeller load coefficient: $\frac{K_{T}}{J^{2}} = \frac{S_{s}}{2 D_{P s}^{2}} \times \frac{C_{T s}}{(1 - t_{s}) {(1 - w_{s})}^{2}}$

Based on the constant load coefficient method, the open water characteristics curve of full scale propeller can be used to obtain the full scale advance coefficient $J_{s}$ , open water torque coefficient $K_{Q S}$ and open water efficiency $η_{s}$ , and then full scale total thrust efficiency can be obtained:

$η_{D s} = η_{H s} η_{R s} η_{s}$

Full scale propeller rotation speed r/s: $N_{s} = \frac{V_{s} (1 - w_{s})}{J_{s} D_{P s}}$

The delivered power of full scale propeller: $P_{D s} = P_{E s} / η_{D s}$ . $P_{E s}$ is the effective power obtained by the resistance prediction of full scale ship. Based on the obtained $P_{D s}$ and $N_{s}$ , the curves of $P_{D s}$ and $N_{s}$ versus $V_{s}$ can be plotted. And then the full scale ship speed $V_{s}$ and full scale propeller rotation speed $N_{s}$ can be obtained by interpolation through the propeller thrust power $P_{D T}$ provided by main engine.

6.2. Analysis of numerical prediction results

The full scale propeller load coefficient $K_{T} / J^{2}$ , advance coefficient $J_{s}$ , open water efficiency $η_{s}$ , full scale total thrust efficiency $η_{D s}$ , full scale propeller rotation speed $N_{s}$ , and full scale propeller delivered power $P_{D s}$ obtained by both RANS method and EFD method are listed in Table 7. The numerical results of RANS method match well with the experimental results of EFD method and the deviation is small. At any Froude number, compared with the experimental results of EFD method, propeller load coefficient $K_{T} / J^{2}$ , rotation speed $N_{s}$ and delivered power $P_{D s}$ calculated by RANS method are bigger, while advance coefficient $J_{s}$ , open water efficiency $η_{s}$ , relative rotative efficiency $η_{R s}$ , hull efficiency $η_{H s}$ and total thrust efficiency $η_{D s}$ are smaller. What’s more, the numerical precision of propeller open water efficiency $η_{s}$ is not so good, and its deviation directly affects the prediction accuracy of total thrust efficiency $η_{D s}$ .

Table 7
Self-propulsion factor of full scale ship

Nomenclature $K_{T} / J^{2}$ $J_{s}$ $η_{s}$ $η_{D s}$ $N_{s}$ $P_{D s}$

$F n = 0.239$ RANS 0.643 0.648 0.581 0.492 133.7 92.7

EFD 0.633 0.654 0.592 0.503 133.2 90.7

$F n = 0.269$ RANS 0.750 0.620 0.576 0.484 153.3 143.4

EFD 0.744 0.625 0.587 0.496 152.8 140.5

$F n = 0.299$ RANS 0.883 0.590 0.568 0.500 178.7 233.7

EFD 0.874 0.594 0.578 0.512 178.3 229.8

$F n = 0.329$ RANS 0.989 0.569 0.560 0.504 203.8 355.5

EFD 0.985 0.572 0.570 0.518 203.4 349.8

Nomenclature	$K_{T} / J^{2}$	$J_{s}$	$η_{s}$	$η_{D s}$	$N_{s}$	$P_{D s}$
$F n = 0.239$	RANS	0.643	0.648	0.581	0.492	133.7	92.7
EFD	0.633	0.654	0.592	0.503	133.2	90.7
$F n = 0.269$	RANS	0.750	0.620	0.576	0.484	153.3	143.4
EFD	0.744	0.625	0.587	0.496	152.8	140.5
$F n = 0.299$	RANS	0.883	0.590	0.568	0.500	178.7	233.7
EFD	0.874	0.594	0.578	0.512	178.3	229.8
$F n = 0.329$	RANS	0.989	0.569	0.560	0.504	203.8	355.5
EFD	0.985	0.572	0.570	0.518	203.4	349.8

The numerical results in Table 7 can be used to predict the speed performance of full scale ship. Under the condition of no wind and no wave, propeller thrust power provided by main engine $P_{D T} = 232$ kW, the curves of $P_{D s}$ and $N_{s}$ versus $V_{s}$ are plotted in Fig. 17. The prediction results of ship speed performance by RANS method are as follows: ship speed 9.98 kn, propeller rotation speed 178.3 r/min. The results match well with EFD method, the difference of ship speed is just 0.04 kn and the difference of propeller rotation speed is just 0.4 r/min. The feasibility and accuracy of RANS method in solving ship speed and power performance is well proved.

Fig. 17.

Power and speed performance curve of full scale ship (a) Power and speed performance based on RANS method (b) Power and speed performance based on EFD method.

7. Conclusions

This paper adopts RANS method to study ship speed and power performance, and propeller open water performance, ship model resistance performance and ship model self-propulsion performance are numerically calculated. The self-propulsion factor can be obtained from the numerical results and then the speed and power performance of full scale ship can be predicted. The change of the free surface wave pattern for the resistance simulation and the variation of flow field information for the self-propulsion simulation are analyzed. The following conclusions are drawn:

(1) The numerical prediction results of propeller open water performance shows that the prediction accuracy fulfills the requirement of engineering application. But the numerical precision decreases with the increase of advance coefficient. When $J_{m} = 0.9$ , the deviation of open water efficiency reaches 2.07%.

(2) The prediction results of ship model resistance by RANS method match well with the EFD method. The flow field information can also be easily visualized to reveal the relationship between the hull lines and flow field. Compared with traditional EFD method, RANS method costs less time and resources.

(3) As for the study of ship wave-making characteristics, RANS method also achieves high consistency with EFD method. The numerical results of RANS method show that: the biggest wave crest occurs at bow and stern, apparent Kelvin wave system appears at the fore and aft shoulders. With the increase of Froude number, the wave crest becomes higher and the wave trough lower, the disturbance area at bow, stern and ship sides enlarges, free surface wave-making region spreads outwards, and the ship wave obviously moves backwards.

(4) The RANS method is also be used to conduct the numerical prediction of ship model self-propulsion performance to obtain the self-propulsion factor. The numerical results of RANS method show good agreement with EFD results. The numerical precision of propeller open water efficiency $η_{m}$ directly affects the prediction accuracy of total thrust efficiency $η_{D 0 m}$ .

(5) The flow field of ship model self-propulsion is analyzed to study the mechanism of ship rapidity. The change of ducted propeller’s rotation speed of Nchanged the flow field around the hull. The stern streamline and the pressure distribution on hull surface are changed. The interference between hull, propeller and rudder leads to the formation of a lower pressure zone on stern hull surface and rudder surface. The numerical results also indicate that the increase of propeller rotation speed nearly has no effect to the free surface wave pattern. In the upper region most close to the stern hull surface, the axial velocity at the center of propeller disk appears as lower pressure area, and the absolute value of axial velocity on blade leaf is smaller than that on blade back. The blade back is located in low pressure zone and the suction surface is formed on it. The radial velocity induces vortices on the blade tip of the propeller, the reason is that a negative velocity region and a positive velocity region are respectively formed around blade back and blade leaf on the blade tip of the propeller, and the area of negative velocity region is larger than that of positive velocity region. Besides, since the interference between stern hull surface and propeller, a clear vortex system is formed at the upper left of the center of the propeller disk.

(6) Under the condition of no wind and no wave, the speed performance of full scale ship is numerically calculated by RANS method. When the propeller thrust power $P_{D T} = 232$ kW, the numerical prediction results of RANS method are as follows: ship speed is 9.98 kn, propeller rotation speed is 178.3 r/min. The results match well with EFD method, the difference of ship speed is just 0.04 kn and the difference of propeller rotation speed is just 0.4 r/min. The feasibility and accuracy of RANS method is well proved and it can be used to solve the speed and power performance of practical engineering application.

Footnotes

Extrapolation Methods

Acknowledgements

The first author would like to acknowledge the China Ship Scientific Research Center for experiments support, and also 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.

References

K.Y.

Chao, Numerical propulsion simulation for the KCS container ship, in: Proceedings of CFD Workshop, Tokyo, 2005, Tokyo, Japan.

J.E.

Choi,

J.H.

Kim,

H.G.

Lee,

B.J.

Choi and

D.H.

Lee, Computational predictions of ship-speed performance, Journal of Maritime Science and Technology 14(3) (2009), 322–333. doi:10.1007/s00773-009-0047-4.

J.E.

Choi,

K.-S.

Min,

J.H.

Kim,

S.B.

Lee and

H.W.

Seo, Resistance and propulsion characteristics of various commercial ships based on CFD results, Ocean Engineering 37 (2010), 549–566. doi:10.1016/j.oceaneng.2010.02.007.

A.J.

Chorin, A numerical method for solving incompressible viscous flow problems, Journal of Computational Physics 2 (1967), 12–26. doi:10.1016/0021-9991(67)90037-X.

J.J.

Gorski and

R.M.

Coleman, Computations of the KVLCC2M tanker under yawed conditions, in: Proceedings of CFD Workshop, Tokyo, Japan, 2005.

C.W.

Hirt and

B.D.

Nichols, Volume of fluid (VOF) method for the dynamics of free boundary, Journal of Computational Physics 39 (1981), 201–225. doi:10.1016/0021-9991(81)90145-5.

R.H.M.

Huijsmans,

Van ’t Veer and

Kashiwagi, Ship motion predictions: A comparison between a CFD based method, a panel method and measurements, in: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering-OMAE, V3, ASME 2010 29th International Conference on Ocean, Offshore and Arctic Engineering, OMAE2010, 2010, pp. 685–691.

R.I.

Issa, Solution of the implicitly discretised fluid flow equations by operator-splitting, Journal of Computational Physics 62 (1986), 40–65. doi:10.1016/0021-9991(86)90099-9.

Kawamura,

Miyata and

Mashimo, Numerical simulation of the flow about self-propelling tanker model, Journal of Maritime Science and Technology 2 (1997), 245–256. doi:10.1007/BF02491531.

10.

Kim,

I.R.

Park,

K.S.

Kim and

S.H.

Van, RANS computation of turbulent free surface flow around a self-propelled KLNG carrier, J Soc Naval Archit Korea 42(6) (2005), 583–592 (in Korean). doi:10.3744/SNAK.2005.42.6.583.

11.

Larsson and

Zou, Evaluation of resistance, sinkage and trim, self-propulsion and wave pattern predictions, in: A Workshop on Numerical Hydrodynamics. Proceedings, Vol. I, Gothenburg, Sweden, 2010.

12.

L.O.

Lubke, Numerical simulation of the flow around a propelled KCS container ship, in: Proceedings of CFD Workshop, Tokyo, Japan, 2005.

13.

F.R.

Menrer,

Kuntz and

Langtry, Ten years of industrial experience with the SST turbulence model, Turbulence heat and Mass transfer 4 (2003), 35–42.

14.

I.R.

Park,

Kim and

S.H.

Van, Analysis of resistance performance of modern commercial ship in hull form using a level-set method, Journal of the Society of Naval Architects of Korea 41(2) (2004), 79–89, (in Korean). doi:10.3744/SNAK.2004.41.2.079.

15.

Stenson,

Boss,

van den Berg,

Garcia-Gromez,

Gulbrandsen,

Hadjimikhalev,

Jensen,

Miranda,

O.-X.

Shen and

Tsutsumi, Report of the performance committee, in: Proceedings of the 21st International Towing Tank Conference, 1996.

16.

Stern,

H.T.

Kim and

H.C.

Chen, A viscous-flow approach to the computation of propeller-hull interaction, Journal of Ship Research 32(4) (1988), 246–262.

17.

Tahara,

R.V.

Wilson,

P.M.

Carrica and

Stern, RANS simulation of a container ship using a single-phase level-set method with overset grids and the prognosis for extension to a self-propulsion simulator, J Mar Sci Technol 11 (2006), 209–228. doi:10.1007/s00773-006-0231-8.

18.

H.K.

Versteeg and

Malalasekera, An Introduction to Computation Fluid Dynamics: The Finite Volume Method, Wigley, New York, 1995.

19.

D.C.

Wilcos, Multi scale model for turbulence flows, in: AIAA 24th Aeropace Sciences Meeting, American Institute of Aeronautics and Astronautics, 1986.

20.

R.V.

Wilson,

Shao,

Stern, Discussion: Criticisms of the “correction factor” verification method, Journal of Fluids Engineering 126 (2004), 704–706. doi:10.1115/1.1780171.

21.

Xing and

Stern, Factors of safety for Richardson extrapolation for industrial applications, IIHR Technical Report, 2008.

22.

Yakhot and

S.A.

Orzag, Renormalization group analysis of turbulence: Basic theory, Scient Comput 1 (1986), 3–11. doi:10.1007/BF01061452.

Prediction of ship power and speed performance based on RANS method

Abstract

Keywords

1. Introduction

2. Calculation method

2.1. RANS equation

2.2. Turbulence model

Table 1 Turbulence model constant α 1 β 1 σ k 1 σ ω 1 α 2 β 2 σ k 2 σ ω 2 β ∗ 5 / 9 3 / 40 0.85 0.5 0.44 0.0828 1 0.856 0.09

3. Numerical simulation of propeller open water performance

3.1. Parameters of propeller

3.2. Calculation model and meshing

Table 3 Results of thrust coefficient and torque coefficient of ducted propeller J m = 0.9 Grid-1 Grid-2 Grid-3 K T m 0.0709 0.0722 0.0741 10 K Q m 0.3157 0.3221 0.3308

4.1. Parameters of ship

Table 5 Ship particulars of the fishing vessel Nomenclature Unit Full scale Model L p p m 27.40 2.7959 L W L m 30.24 3.0857 B m 6.00 0.6122 T m 2.10 0.2143 S m 2 243.40 2.5344 C B – 0.6517 C W – 0.9339 C P – 0.6850

5.1. Calculation model and meshing

5.3. Analysis of the self-propulsion factor

6.1. Numerical prediction method

6.2. Analysis of numerical prediction results

Footnotes

Extrapolation Methods

Acknowledgements

References

Table 1
Turbulence model constant

$α_{1}$ $β_{1}$ $σ_{k 1}$ $σ_{ω 1}$ $α_{2}$ $β_{2}$ $σ_{k 2}$ $σ_{ω 2}$ $β^{*}$

$5 / 9$ $3 / 40$ 0.85 0.5 0.44 0.0828 1 0.856 0.09

Table 3
Results of thrust coefficient and torque coefficient of ducted propeller

$J_{m} = 0.9$ Grid-1 Grid-2 Grid-3

$K_{T m}$ 0.0709 0.0722 0.0741

$10 K_{Q m}$ 0.3157 0.3221 0.3308

Table 5
Ship particulars of the fishing vessel

Nomenclature Unit Full scale Model

$L_{p p}$ m 27.40 2.7959

$L_{W L}$ m 30.24 3.0857

B m 6.00 0.6122

T m 2.10 0.2143

S $m^{2}$ 243.40 2.5344

$C_{B}$ – 0.6517

$C_{W}$ – 0.9339

$C_{P}$ – 0.6850