Load Distribution Prediction and Design Models for Ship Special Mechanical Devices and Equipment under Non-linear Sea Conditions

Ships are generally composed of hull structure, power plant, power plant systems, piping systems, and oil and water tanks and equipment compartments, etc., with a solid structural composition, good sea performance and loading and transportation capacity; these functions require some high parameters of the load-bearing equipment, high-specification electromechanical equipment, such as “special equipment” to support the protection [1-3]. Ship special equipment used in harsh environments, complex operating conditions, and the risk of hidden danger, once the problem directly affects the ship’s life safety, is major high-risk equipment [4-6]. With the continuous expansion of the cruise range of the ship, the possibility of encountering adverse sea conditions has greatly increased. When the ship with a large external drift speed in bad sea conditions, easy to be affected by the violent impact of the bang load and the hull movement will produce significant nonlinear phenomenon, high sea state of the hull of a large movement in the nonlinear phenomenon is more obvious [7-10]. A large number of experiments have proved that the time-record results of hull motion and wave loading in severe sea states have obvious nonlinearities [11-12]. The large motion of the ship in a nonlinear sea state will generate a large inertia load, and the waves surging on the deck will also have an impact on the ship equipment, threatening the safety of the ship equipment [13-15]. Therefore, how accurately predict the motion loads of ships under severe sea state provides a certain reference value for ship navigation and the safety guarantee of special mechanical devices and equipment.

In order to improve the accuracy of forecasting large ship motions, nonlinear factors in the motions need to be taken into account. Literature [16] shows that the structural rules of a ship specify various loads corresponding to the ship in the most severe sea state to provide safety, but most of its assessments are based on linear wave resistance coding and linear statistical prediction preparation formulas, which are unable to adapt to the complex sea state, so fiber grating sensors are introduced to measure in detail the numerical calculations of different ships in both regular and irregular waves, which are used to assess the structural strength of the ship. Literature [17] discusses the advantages and disadvantages of numerical prediction methods in the structural assessment of ships subjected to steady state or transient excitation loads and also evaluates the uncertainty in predicting wave-induced loads, as well as probabilistic methods for evaluating the long-term response and fatigue analyses to lay the foundation for the development of wave load prediction for ships. Literature [18] proposes a large-scale model measurement technique with better applicability and versatility to predict the short-term motion and load response of a ship under the corresponding sea state by measuring its wave resistance and wave load characteristics in different hydrodynamics, in addition to the long-term extreme prediction of the motion load based on the numerical results of the short-term prediction. Literature [19] designs a relatively simplified nonlinear method to predict the pendulum motion, pitching motion and vertical bending moment of a container ship on the basis of the three-dimensional time-domain panel method, including an algorithm for accurately solving the dipping surface under the incident wave profile and an estimation method for the correction of the object’s motion conditions, and the experiments show that the proposed method can efficiently estimate the wave-induced nonlinear response.

Some scholars have predicted the equipment loads under nonlinear sea state to assess the safety risk of marine special equipment use. Literature [20] used a three-dimensional, fully nonlinear time domain model based on the boundary element method to analyze the nonlinear dynamic response of the suspended loads of lifting vessels, to study the dynamic characteristics of the underwater loads of offshore lifting vessels, and to use this as a basis for designing techniques to control or damp the unexpected motion of underwater loads. Literature [21] studied the dynamics and robust control problems of the wave compensation system of marine robotic arms under complex sea conditions, introduced nonlinear differential and integral sliding mode control strategies to improve the stability of the dynamic control system of marine robotic arms, and also provided useful references for the wave compensation systems of other marine equipment. Literature [22] establishes an electromechanical-hydraulic coupling dynamics model of the shipboard gun that includes the influence of the follower system and adopts a bilaterally symmetric tooth arc arrangement scheme to reduce the influence of the ship’s base motion excitation on the accuracy of the shipboard gun, which provides theoretical support for the relevant design of the shipboard gun.

In this paper, the second-order Stokes wave and Ochi-Hubble six-parameter bimodal wave spectrum are combined to achieve the environmental modelling of nonlinear mixed sea state, and then numerical simulation is carried out based on the fluid-structure coupling method combining the Computational Fluid Dynamics (CFD) and the Finite Element Method (FEM) to complete the prediction of load distribution of the ship’s special mechanical devices and equipments under the nonlinear sea state. Taking the pump-jet thruster as an example, the optimal design of load distribution is carried out according to the prediction results, and it is verified whether the designed load distribution meets the requirements.

2

Load distribution of ship specialised machinery in non-linear sea state

2.1

Ship specialised machinery

Ship special machinery refers to the special mechanical devices that can meet the needs of a ship’s special performance, including the rocking device, pitch paddle device, ship lift, and special conveyor devices. With the ship’s comfort, safety, economy, and other requirements, the focus has gradually shifted to equipping civilian ships.

2.2

Non-linear sea state

Under severe sea conditions, the environmental loads such as wave force, wind force and current force to which the ship is subjected when sailing at high speed have significant uncertainty, coupling and time-varying nature, which makes the ship motion process show obvious nonlinear characteristics.

Therefore, non-linear sea state usually refers to irregular variations and interactions of natural factors such as waves, currents, winds, etc., which may lead to a significant increase in the six-degree-of-freedom motions of the ship (longitudinal swing, transverse swing, plumb swing, transverse rocking, longitudinal rocking, and bow rocking), which may result in complex dynamic loads on specialised mechanical installations and equipments on board the ship.

2.3

Characteristics of load distribution in non-linear sea state

The load distribution in a nonlinear sea state is mainly characterized by the following characteristics:

Dynamic variability

In a non-linear sea state, the wave force and wind force to which the ship is subjected are often irregular and changing, which leads to a non-linear motion response of the ship. Therefore, the special mechanical devices and equipment on the ship must be able to adapt to dynamically changing load conditions to ensure normal operation under various circumstances.

Uneven structural stress

The complex motion environment in a non-linear sea state will lead to uneven stress distribution in the ship structure. This requires shipping special mechanical devices and equipments to take into account the maximum stresses on different parts during design and how to disperse these stresses through reasonable structural design and material selection to avoid structural damage or fatigue damage.

Vibration and shock absorption requirements

The violent movement of the ship under severe sea conditions will generate large vibrations and impacts, which is a severe test for the shipboard equipment. Therefore, these energies need to be absorbed by installing shock absorbers and vibration isolation systems to protect sensitive equipment from damage.

Energy management optimisation

Under a non-linear sea state, the energy consumption pattern of the ship will change, especially in the propulsion and stabilization systems. Therefore, an intelligent energy management system is needed to optimize the distribution and usage efficiency of energy to ensure the safety of the ship’s energy supply during a long voyage.

Control system adaptability

In order to address the challenges posed by non-linear sea conditions, the ship’s control system must be highly adaptable and flexible. This means that the control system needs to be able to detect environmental changes in real-time and quickly adjust the ship’s speed, heading, and attitude, while ensuring stability and safety.

3

Predictive Modelling of Load Distribution of Ship’s Special Installations in Non-Linear Sea State

3.1

Modelling of non-linear sea state

In order to achieve the prediction of the load distribution of special mechanical devices and equipment of ships under nonlinear sea conditions, this paper introduces the second-order Stokes wave model for nonlinear wave simulation analysis [23]. At the same time, this paper uses the Ochi-Hubble six-parameter wave spectrum, which covers a variety of wave spectral shapes, including surges related to the growth and decay of storms, to simulate the sea state. In order to put the ship’s special mechanical devices and equipment in a nonlinear mixed sea state, the bimodal spectrum of the Ochi-Hubble six-parameter wave spectrum family is chosen in this paper [24].

3.1.1

Second-order Stokes waves

The free surface wave height η(x,t) of the second-order S nonlinearity can be expressed as the sum of the first-order term η⁽¹⁾(x,t), the second-order term η⁽²⁾(x,t) ......: Tokes waves (1) $η (x, t) = η^{(1)} (x, t) + η^{(2)} (x, t) + \dots$ \[\eta (x,t)={{\eta }^{(1)}}(x,t)+{{\eta }^{(2)}}(x,t)+\ldots \]

For an irregular sea state characterised by a specific wave spectrum S_n(ω) (ω denotes the angular frequency), the expression for the first-order term of wave height η⁽¹⁾(x,t) can be written as: (2) $η^{(1)} (x, t) = Re \sum_{n = 0}^{N} c_{n} expi (ω_{n} t - k_{n} x + ε_{n})$ \[{{\eta }^{(1)}}(x,t)=\operatorname{Re}\sum\limits_{n=0}^{N}{{{c}_{n}}}\operatorname{expi}\left( {{\omega }_{n}}t-{{k}_{n}}x+{{\varepsilon }_{n}} \right)\]

Where N tends to infinity, Re denotes the real part of the complex number, and i denotes the imaginary unit. For each cosine wave, c_n denotes its complex-valued amplitude, ω_n denotes the angular frequency, k_n denotes the wave number, ε_n is the phase angle, which is uniformly distributed in [0,2π] , and ω_n and k_n satisfy the linear dispersion relation: (3) $ω_{n}^{2} = g k_{n} \tanh (k_{n} d)$ \[{{\omega }_{n}}^{2}=g{{k}_{n}}\tanh \left( {{k}_{n}}d \right)\]

Where g and d denote gravitational acceleration and water depth, respectively.

Shallow-water wave data usually do not follow a linear Gaussian ocean model due to bottom effects. The linear Gaussian ocean model can be corrected by adding a second-order term, and the expression for the second-order term η⁽²⁾(x,t) can be written as: (4) $\begin{matrix} η^{(2)} (x, t) = Re \sum_{m = 0}^{N} \sum_{n = 0}^{N} c_{m} c_{n} (r_{m n} \exp i (ω_{m} t - k_{m} x + ε_{m} + ω_{n} t - k_{n} x + ε_{n}) \\ + q_{m n} \exp i (ω_{m} t - k_{m} x + ε_{m} - ω_{n} t + k_{n} x - ε_{n})) \end{matrix}$ \[\begin{align} & {{\eta }^{(2)}}(x,t)=\operatorname{Re}\sum\limits_{m=0}^{N}{\sum\limits_{n=0}^{N}{{{c}_{m}}}}{{c}_{n}}\left( {{r}_{mn}}\exp i\left( {{\omega }_{m}}t-{{k}_{m}}x+{{\varepsilon }_{m}}+{{\omega }_{n}}t-{{k}_{n}}x+{{\varepsilon }_{n}} \right) \right. \\ & \left. +{{q}_{mn}}\exp i\left( {{\omega }_{m}}t-{{k}_{m}}x+{{\varepsilon }_{m}}-{{\omega }_{n}}t+{{k}_{n}}x-{{\varepsilon }_{n}} \right) \right) \end{align}\]

Where r_mn and q_mn are quadratic transfer functions. are given by equations (5) and (6), respectively: (5) $\begin{array}{l} r_{m n} = - (\frac{1}{g}) (\frac{(\frac{1}{4 ω_{m} ω_{n}}) 2 (ω_{m} + ω_{n}) (ω_{n}^{2} ω_{m}^{2} - k_{n} k_{m} g^{2}) + ω_{n} (ω_{m}^{4} - g^{2} k_{m}^{2}) + ω_{m} (ω_{n}^{4} - g^{2} k_{n}^{2})}{{(ω_{m} + ω_{n})}^{2} \cosh ((k_{m} + k_{n}) d) - g (k_{m} + k_{n}) \sinh ((k_{m} + k_{n}) d)}) \\ \times (ω_{m} + ω_{n}) \cosh ((k_{m} + k_{n}) d) - (\frac{1}{4 g ω_{m} ω_{n}}) (k_{m} k_{n} g^{2} - ω_{n}^{2} ω_{m}^{2}) + (\frac{1}{4 g}) (ω_{m}^{2} + ω_{n}^{2}) \end{array}$ \[\begin{align} & {{r}_{mn}}=-\left( \frac{1}{g} \right)\left( \frac{\left( \frac{1}{4{{\omega }_{m}}{{\omega }_{n}}} \right)2\left( {{\omega }_{m}}+{{\omega }_{n}} \right)\left( \omega _{n}^{2}\omega _{m}^{2}-{{k}_{n}}{{k}_{m}}{{g}^{2}} \right)+{{\omega }_{n}}\left( \omega _{m}^{4}-{{g}^{2}}k_{m}^{2} \right)+{{\omega }_{m}}\left( \omega _{n}^{4}-{{g}^{2}}k_{n}^{2} \right)}{{{\left( {{\omega }_{m}}+{{\omega }_{n}} \right)}^{2}}\cosh \left( \left( {{k}_{m}}+{{k}_{n}} \right)d \right)-g\left( {{k}_{m}}+{{k}_{n}} \right)\sinh \left( \left( {{k}_{m}}+{{k}_{n}} \right)d \right)} \right) \\ & \times \left( {{\omega }_{m}}+{{\omega }_{n}} \right)\cosh \left( \left( {{k}_{m}}+{{k}_{n}} \right)d \right)-\left( \frac{1}{4g{{\omega }_{m}}{{\omega }_{n}}} \right)\left( {{k}_{m}}{{k}_{n}}{{g}^{2}}-\omega _{n}^{2}\omega _{m}^{2} \right)+\left( \frac{1}{4g} \right)\left( \omega _{m}^{2}+\omega _{n}^{2} \right) \\ \end{align}\] (6) $\begin{array}{l} q_{m} = - (\frac{1}{g}) (\frac{(\frac{1}{4 ω_{m} ω_{n}}) 2 (ω_{m} - ω_{n}) (ω_{n}^{2} ω_{m}^{2} + k_{n} k_{m} g^{2}) - ω_{n} (ω_{m}^{4} - g^{2} k_{m}^{2}) + ω_{m} (ω_{n}^{4} - g^{2} k_{n}^{2})}{{(ω_{n} - ω_{m})}^{2} \cosh (| k_{m} - k_{n} | d) - g | k_{n} - k_{m} | \sinh (| k_{n} - k_{m} | d)}) \\ \times (ω_{m} - ω_{n}) \cosh (k_{n} - k_{m} | d) - (\frac{1}{4 g ω_{m} ω_{n}}) (k_{m} k_{n} g^{2} + ω_{n}^{2} ω_{m}^{2}) + (\frac{1}{4 g}) (ω_{m}^{2} + ω_{n}^{2}) \end{array}$ $\begin{array}{*{35}{l}} {{q}_{m}}=-\left( \frac{1}{g} \right)\left( \frac{\left( \frac{1}{4{{\omega }_{m}}{{\omega }_{n}}} \right)2\left( {{\omega }_{m}}-{{\omega }_{n}} \right)\left( \omega _{n}^{2}\omega _{m}^{2}+{{k}_{n}}{{k}_{m}}{{g}^{2}} \right)-{{\omega }_{n}}\left( \omega _{m}^{4}-{{g}^{2}}k_{m}^{2} \right)+{{\omega }_{m}}\left( \omega _{n}^{4}-{{g}^{2}}k_{n}^{2} \right)}{{{\left( {{\omega }_{n}}-{{\omega }_{m}} \right)}^{2}}\cosh \left( \left| {{k}_{m}}-{{k}_{n}} \right|d \right)-g\left| {{k}_{n}}-{{k}_{m}} \right|\sinh \left( \left| {{k}_{n}}-{{k}_{m}} \right|d \right)} \right) \\ \times \left( {{\omega }_{m}}-{{\omega }_{n}} \right)\cosh \left( {{k}_{n}}-{{k}_{m}}|d \right)-\left( \frac{1}{4g{{\omega }_{m}}{{\omega }_{n}}} \right)\left( {{k}_{m}}{{k}_{n}}{{g}^{2}}+\omega _{n}^{2}\omega _{m}^{2} \right)+\left( \frac{1}{4g} \right)\left( \omega _{m}^{2}+\omega _{n}^{2} \right) \\ \end{array}$

where k_n and ω_n satisfy the linear dispersion relation.

Adding the first-order term and the second-order term, the wave height η(x,t) of the 2nd-order linear irregular wave is obtained , and let x = 0, we get: (7) $\begin{matrix} η (0, t) = η^{(1)} (t) + η^{(2)} (t) = Re \sum_{n = 0}^{N} c_{n} \exp i (ω_{n} t + ε_{n}) \\ + Re \sum_{m = 0}^{N} \sum_{n = 0}^{N} c_{m} c_{n} r_{m n} \exp i (ω_{m} t + ε_{m} + ω_{n} t + ε_{n}) \\ + q_{m n} \exp (i (ω_{m} t + ε_{m} - ω_{n} t - ε_{n})) \end{matrix}$ \[\begin{align} & \eta \left( 0,t \right)={{\eta }^{\left( 1 \right)}}\left( t \right)+{{\eta }^{\left( 2 \right)}}\left( t \right)=\operatorname{Re}\sum\limits_{n=0}^{N}{{{c}_{n}}}\exp i\left( {{\omega }_{n}}t+{{\varepsilon }_{n}} \right) \\ & +\operatorname{Re}\sum\limits_{m=0}^{N}{\sum\limits_{n=0}^{N}{{{c}_{m}}}}{{c}_{n}}{{r}_{mn}}\exp i\left( {{\omega }_{m}}t+{{\varepsilon }_{m}}+{{\omega }_{n}}t+{{\varepsilon }_{n}} \right) \\ & +{{q}_{mn}}\exp \left( i\left( {{\omega }_{m}}t+{{\varepsilon }_{m}}-{{\omega }_{n}}t-{{\varepsilon }_{n}} \right) \right) \end{align}\]

3.1.2

Ochi-Hubble six-parameter wave spectrum

The Ochi-Hubble six-parameter wave spectrum can be decomposed into two parts, each containing three parameters in its expression. The total spectrum is obtained by adding the two three-parameter spectra. The expression is given below:Ochi-Hubble: (8) $S (ω) = \frac{1}{4} \sum_{j = 1}^{2} \frac{{(((4 λ_{j} + 1) / 4) ω_{m i}^{4})}^{2}}{Γ (λ_{j})} \frac{H_{5 j}^{2}}{ω^{4 λ_{j} + 1}} \exp (- (\frac{4 λ_{j} + 1}{4}) {(\frac{ω_{m j}}{ω})}^{4})$ \[S\left( \omega \right)=\frac{1}{4}\sum\limits_{j=1}^{2}{\frac{{{\left( \left( {\left( 4{{\lambda }_{j}}+1 \right)}/{4}\; \right){{\omega }_{mi}}^{4} \right)}^{2}}}{\Gamma \left( {{\lambda }_{j}} \right)}}\frac{{{H}_{5j}}^{2}}{{{\omega }^{4{{\lambda }_{j}}+1}}}\exp \left( -\left( \frac{4{{\lambda }_{j}}+1}{4} \right){{\left( \frac{{{\omega }_{mj}}}{\omega } \right)}^{4}} \right)\]

Where H_s1, ω_m1 and λ₁ denote the meaningful wave height, modal frequency and spectral shape parameter of the low-frequency part, and H_s2, ω_m2 and ω_m2 denote the meaningful wave height, modal frequency and spectral shape parameter of the high-frequency part, respectively.

The six parameters in Eq. 1 can be converted into meaningful wave heights H_s(m), and thus a six-parameter wave spectrum family consisting of 11 wave spectra is obtained as shown in Table 1. Among these 11 wave spectra, wave spectrum parameter No. 1 is considered to be the “most probable wave spectrum” representing a particular ocean, and the remaining wave spectra parameters from No. 2 to No. 11 are considered to be “95 per cent plausible” representing a particular ocean.

Table 1.

Values of six parameters

	No.	H_s1	H_s2	ω_m1	ω_m2	λ₁	λ₂
Most likely	1	0.84H_s	0.54H_s	0.70e^-0.046Hs	1.15e^-0.039Hs	3.00	1.54e^-0.062Hs
95% credibility	2	0.95H_s	0.31H_s	0.70e^-0.046Hs	1.50e^-0.046Hs	1.35	2.48e^-0.102Hs
	3	0.65H_s	0.76H_s	0.61e^-0.039Hs	0.94e^-0.036Hs	4.95	2.48e^-0.102Hs
	4	0.84H_s	0.54H_s	0.93e^-0.056Hs	1.50e^-0.046Hs	3.00	2.77e^-0.112Hs
	5	0.84H_s	0.54H_s	0.41e^-0.016Hs	0.88e^-0.026Hs	2.55	1.82e^-0.089Hs
	6	0.90H_s	0.44H_s	0.81e^-0.052Hs	1.60e^-0.033Hs	1.80	2.95e^-0.105Hs
	7	0.77H_s	0.64H_s	0.54e^-0.039Hs	0.61	4.50	1.95^e-0082Hs
	8	0.73H_s	0.68H_s	0.70e^-0.046Hs	0.99e^-0.039Hs	6.40	1.78e^-0.069Hs
	9	0.92H_s	0.39H_s	0.70e^-0.046Hs	1.37e^-0.039Hs	0.70	1.78e^-0.069Hs
	10	0.84H_s	0.54H_s	0.74e^-0.052Hs	1.30e^-0.039Hs	2.65	2.65e^-0.085Hs
	11	0.84H_s	0.54H_s	0.62e^-0.039Hs	1.03e^-0.030Hs	2.60	0.53e^-0.069Hs

The meaningful wave height of the waves in this study, H_s = 7.5m, is substituted into the above 11 wave spectra, and it is observed that the wave spectrum of No. 3 has an obvious double peak, as shown in Fig. 1. That is, this wave spectrum has both low-frequency swell and high-frequency wind waves, which meets the requirements of nonlinear mixed sea state.

3.2

CFD-FEM based load distribution prediction model

In this paper, a coupled fluid-structure calculation method combining Computational Fluid Dynamics (CFD) and Finite Element Method (FEM) is used, combined with the established nonlinear sea state model, to simulate the loading condition of special mechanical devices and equipment of the ship, so as to achieve the prediction of the load distribution, and to provide a reference for the further optimisation of the design [25].

3.2.1

Control equations

Fluid control equations

When the compressibility of the fluid is not considered and the surface tension of the fluid is not taken into account, the continuum equation with the N-S equation is used to describe the viscous fluid motion in ocean engineering problems: (9) $\nabla \cdot u = 0$ \[\nabla \cdot u=0\] (10) $\frac{\partial ρ u}{\partial t} + \nabla \cdot (ρ u u) = \nabla \cdot μ \nabla u - \nabla p + ρ g a$ \[\frac{\partial \rho u}{\partial t}+\nabla \cdot \left( \rho uu \right)=\nabla \cdot \mu \nabla u-\nabla p+\rho ga\]

Where: u is the velocity vector. ρ is the fluid density. p is the pressure. DD is the gravity vector. μ is the viscosity coefficient, which is the sum of the dynamic viscosity coefficient μ_fluid and the eddy viscosity coefficient μ_t.

2. Structural control equations

It is assumed that the structure is a linear elastic material, which does rigid body motion and deformation relative to the original equilibrium position under the action of external loads such as waves, and its structural equations of motion are obtained by the finite element method: (11) $m \ddot{x} + c \dot{x} + k x = F (t)$ \[m\ddot{x}+c\dot{x}+kx=F\left( t \right)\]

Where: m is the structural mass matrix. c is the structural damping matrix. k is the structural stiffness matrix. x is the node displacement matrix. x is the equivalent node force matrix synthesised by various external forces.

For linear elastic materials, the stress-strain relationship is linear and given by Hooke’s law: (12) $σ = D_{ε}$ \[\sigma ={{D}_{\varepsilon }}\]

where: is the material tangent coefficient, is the stress, and is the strain.

3.2.2

Numerical methods

Turbulence modelling

When solving engineering problems, the continuous equation and the N–S equation are usually time-averaged, and then the N–S equation becomes the Reynolds time-averaged N–S(RANS) equation. When solving the RANS equation, due to the introduction of eddy viscosity coefficient μ_t , will lead to the equation is not closed, need to turbulence model for calculation. In marine engineering calculations, the SSTk–ω model adds a cross-diffusion term in the equation and considers the effect of shear stress in the turbulence toughness coefficient coefficients, which results in good computational stability, high computational efficiency and accuracy, so the turbulence model used in the calculations is the SSTk–ω model. The transport equation of the SSTk–ω model is: (13) $\frac{\partial}{\partial t} ((ρ k) + \frac{\partial}{\partial x_{i}} (ρ k u_{i}) = \frac{\partial}{\partial x_{j}} (Γ_{k} \frac{\partial k}{\partial x_{j}}) + {\tilde{G}}_{k} + Y_{k} + S_{k}$ \[\frac{\partial }{\partial t}(\left( \rho k \right)+\frac{\partial }{\partial {{x}_{i}}}\left( \rho k{{u}_{i}} \right)=\frac{\partial }{\partial {{x}_{j}}}\left( {{\Gamma }_{k}}\frac{\partial k}{\partial {{x}_{j}}} \right)+{{\tilde{G}}_{k}}+{{Y}_{k}}+{{S}_{k}}\] (14) $\frac{\partial}{\partial t} (ρ ω) + \frac{\partial}{\partial x_{i}} (ρ ω u_{i}) = \frac{\partial}{\partial x_{j}} (Γ_{ω} \frac{\partial ω}{\partial x_{j}}) + G_{ω} + Y_{ω} + S_{ω}$ \[\frac{\partial }{\partial t}(\rho \omega )+\frac{\partial }{\partial {{x}_{i}}}\left( \rho \omega {{u}_{i}} \right)=\frac{\partial }{\partial {{x}_{j}}}\left( {{\Gamma }_{\omega }}\frac{\partial \omega }{\partial {{x}_{j}}} \right)+{{G}_{\omega }}+{{Y}_{\omega }}+{{S}_{\omega }}\]

Where k and ω denote the turbulent kinetic energy and turbulent dissipation rate, respectively. u_i is the velocity component. ${\tilde{G}}_{k}$ \[{{\tilde{G}}_{k}}\] is the term generated by the turbulent kinetic energy k due to the mean velocity. G_ω is the unit dissipation term. Γ_k and Γ_ω are the effective diffusion terms of k and ω, respectively. Y_k and Y_ω are turbulent kinetic energy dissipation terms for k and ω, respectively. S_k and S_ω are custom source terms.

Numerical wave generation method

Compared to linear waves, non-linear waves have steeper peaks and flatter valleys, presenting an asymmetric curve that is more consistent with actual waves in reality. In this paper, the second-order Stokes wave model and Ochi-Hubble six-parameter wave spectrum are selected for wave simulation.

Numerical wave cancellation method

In order to eliminate the influence of the reflected waves from the object and the reflected waves at the pool exit on the entrance boundary of the velocity wave making, the wave cancellation area will be set up in both the pool entrance area and the exit area as shown in Fig. 2. The numerical pool inlet wave cancellation area is shown in Fig. 2, the forced wave cancellation method is used, and the wave is forced by adding the source term in the momentum equation.

The source term added to the momentum equation in its inlet dissipation region is: (15) $q_{φ} = - γ ρ (φ - φ^{*})$ \[{{q}_{\varphi }}=-\gamma \rho \left( \varphi -{{\varphi }^{*}} \right)\]

Where: γ is the damping coefficient, γ is the numerical solution of the momentum equation, and φ* is the numerical solution of the theoretical calculation.

Damping dissipation is used in the numerical pool outlet dissipation zone in Fig. 2 to enhance wave dissipation by adding damping in the vertical direction of wave motion. The additional damping source term in the momentum equation is: (16) $S_{z}^{d} = ρ (f_{1} + f_{2} u_{3}) \frac{e^{k}}{e} u_{3}$ \[S_{z}^{d}=\rho \left( {{f}_{1}}+{{f}_{2}}{{u}_{3}} \right)\frac{{{e}^{k}}}{e}{{u}_{3}}\] (17) $k = {(\frac{x - x_{s d}}{x_{e d} - x_{s d}})}^{n_{d}}$ \[k={{\left( \frac{x-{{x}_{sd}}}{{{x}_{ed}}-{{x}_{sd}}} \right)}^{{{n}_{d}}}}\]

Where: u₃ is the velocity component in the vertical direction, f₁ and f₂ Cre the linear and nonlinear damping terms, respectively, and n_d is the damping coefficient along the wave propagation direction, and x_sd and x_ed are the coordinates of the start position and the end position of the damping region, respectively.

3.2.3

Coupling methods

In the coupling calculation, a two-way coupling method between CFD and FEM is used, and the CFD-FEM two-way coupling process is shown in Fig. 3. t₀ in the figure is the initial moment, and Δt indicates the time increment within each coupling. At the beginning, the pressure on the surface of the ship’s special device is calculated using CFD and transferred to the special device model in the finite element module using the data mapping method. Under the action of pressure load, the finite element nodes of the ship’s special device produce changes in velocity and acceleration, which lead to deformation of the fluid-structure coupling interface. Afterward, the deformed node data are transferred to the CFD calculation programme for the update of the interface. The next step will have the pressure and velocity fields calculated by CFD, as well as the velocity and acceleration of the nodes calculated by FEM. The data mapping in the coupling calculation is crucial because the mesh discretisation between CFD and FEM is different, and the mesh nodes do not correspond to each other. The data mapping in the calculation is carried out using shape function interpolation.

4

Numerical simulation and design implementation of load distribution

4.1

Load distribution analysis of special mechanical devices on ships

In this paper, the pump jet propeller, a special mechanical device of a ship, is taken as a specific research object, and numerical simulation is carried out by using the CFD-FEM two-way fluid-solid coupling method to predict the load distribution of the ship’s special mechanical device under nonlinear sea state.

Under the wave condition of wave height H=6m, wavelength to length ratio λ/Lpp=0.87, and sailing with 20kn speed to meet the waves, the CFD-FEM two-way fluid-structure coupling method is used to calculate the load on the pump jet propeller when the hull of the ship and the waves are banging, and the pressure time course of the equidistant measurement points P1~P4 is shown in Fig. 4.

As can be seen from Fig. 4, the load at measurement point P4 is seriously affected by the thumping action of the ship’s hull, and its peak value exceeds 3.1 kPa. The pressure peaks at measurement points P1 to P4 are more obtuse and decrease with the height of the measurement point, which is due to the fact that the pump-jet propellers are tilted inward, and the angle of their inclination rise is negative.

The loads at measurement point P4 obtained by using the two-way fluid-solid coupling method of test and CFD-FEM are shown in Fig. 5. As can be seen from Fig. 5, the hull thumping cycles of the two ships are consistent, and the amplitude of the load change triggered by the thumping action is similar.

4.2

Load Distribution Design

4.2.1

Design elements

The load distribution in the pump-jet propeller is mainly expressed as the distribution of blade velocity moment, i.e., the pressure difference between the blade pressure surface and the suction surface.

In this paper, considering the actual sailing conditions of the ship, it is decided to use the intermediate uniform load distribution form that takes into account the efficiency and cavitation of the pump jet thruster, i.e., the load is the largest at 0.2 chord length, and then it is uniformly distributed, and it gradually decreases in the form of 0.6 times the chord length. The blade load distribution of the designed pump jet thruster is shown in Fig. 6. Among them, Figs. (a)~(b) represents the load distribution at the front and rear rotor hubs and rims of the pump jet thruster, respectively, while Fig. (c) represents the guide vane load distribution, given the load distribution at the hubs and rims, and the load distribution at the other radius cross sections is obtained by line interpolation. The load distribution is represented from inside to outside in the figure, and different angles represent multiples of the chord length.

Meanwhile, after obtaining the no-thickness blade, the blade is thickened by using the NACA series of airfoil thickness distribution law to obtain the final pump-jet thruster load distribution design.

4.2.2

Analysis of design results

After obtaining the geometry of the pump-jet propeller, the pump-jet propeller is placed at the stern of the ship with a finned rudder for numerical self-propelled calculations and analysis of the results under a nonlinear sea state.

The surfaces of the forward and aft rotor blades experience pressure distributions at the design speed, as depicted in Fig. 7. As can be seen from Fig. 7, except for the leading edge where the pressure fluctuation is large due to the violent action of the blade and water (the maximum pressure difference between the pressure distribution on the surface of the front and rear rotor blades is 1199.5 kPa and 1616.8 kPa, respectively), the pressure distribution of the rest of the blade is smooth in the front and rear, and there is no prominent pressure fluctuation, which indicates that the load distribution of the blade is uniform, and there is no region with large pressure gradient, avoiding localised low-pressure areas and reducing cavitation. This indicates that the load distribution of the blade is uniform. There is no region with a large pressure gradient, which avoids the local low-pressure region and reduces the risk of cavitation. This indicates that the load design of the pump-jet thruster under a nonlinear sea state basically meets the design requirements.

5

Conclusion

In this paper, on the basis of nonlinear sea state modelling using second-order Stokes waves and bimodal Ochi-Hubble wave spectra, the CFD-FEM two-way fluid-structure coupling method is used to achieve the prediction and design of the load distribution of the ship’s special mechanical devices and equipments, and the design effect is examined.

This paper takes the pump-jet propeller as the specific research object, and under the wave conditions of wave height H=6m, wavelength to length ratio λ/Lpp=0.87, when the ship sailing against the waves at a speed of 20kn and the waves are thumped, CFD-FEM two-way fluid-solid coupling calculations are carried out for the equidistant measurement points P1~P4 on the pump-jet propeller. The results show that the load at measurement point P4 is seriously affected by the hull thumping, and its peak value exceeds 3.1 kPa. The pressure peaks at measurement points P1 to P4 are flatter and decrease with the height of the measurement point, which is due to the inward tilting of the pump-jet thruster and the negative angle of its inclination rise. The load condition of measurement point P4 obtained by the CFD-FEM two-way fluid-structure coupling method is similar to the experimental results in terms of the hull banging cycle and the amplitude of the load change triggered by the banging action. This indicates the reliability of the CFD-FEM two-way fluid-structure coupling method used in this paper.

Considering the actual sailing conditions of the ship, this paper decides to adopt the intermediate uniform load distribution form that takes into account the efficiency of the pump-jet thruster and cavitation, i.e., the load is maximum at 0.2 chord length and then uniformly distributed and gradually decreases at 0.6 times the chord length. In the load distribution of the designed pump jet thruster, except for the guide edge where the pressure fluctuation is large due to the violent action of the blade and water (the maximum pressure difference between the pressure distribution of the front and rear rotor blade surfaces is 1,199.5kPa and 1,616.8kPa, respectively), the rest of the blade position is smooth in the front and rear pressure distribution, and there is no prominent pressure fluctuation, which indicates that the load distribution of the blade is uniform, and there is no pressure gradient. This indicates that the load distribution of the blade is uniform without a large pressure gradient, which avoids the local low-pressure area and reduces the risk of cavitation. This shows that the load design of the pump-jet thruster under a nonlinear sea state basically meets the design requirements.

Language:: English

Publication timeframe:: 1 times per year
Journal Subjects:: Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other

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Load Distribution Prediction and Design Models for Ship Special Mechanical Devices and Equipment under Non-linear Sea Conditions

Peiyao Yang

Published Online: Mar 17, 2025

Received: Nov 09, 2024

Accepted: Feb 09, 2025

DOI: https://doi.org/10.2478/amns-2025-0170

KeywordsSecond-order Stokes waves, Bimodal Ochi-Hubble spectra, CFD-FEM method, Marine specialised machinery, Load distribution

© 2025 Peiyao Yang, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Second-order Stokes waves, Bimodal Ochi-Hubble spectra, CFD-FEM method, Marine specialised machinery, Load distribution