Wave loads estimation and structural strength evaluation are the fundamental work at the ship design stage. The hydroelastic responses and slamming strength issues are also concerned especially for large-scale high-speed ships sailing in harsh waves. To accurately predict the wave-induced motions and loads acting on the ship sailing in regular waves, a fully coupled 3D time-domain nonlinear hydroelasticity theory is developed in this paper. The vibration modal characteristics of the flexible hull structure derived by the 3D finite element method (FEM) and simplified 1D nonuniform Timoshenko beam theory are firstly described. The hydrostatic restoring force and hydrodynamic wave force are calculated on the real-time wetted surface of hull to address geometric nonlinearity due to the steep wave and large amplitude motions. The bow slamming and green water loads acting on the ship in severe regular waves are estimated by the momentum impact method and dam-breaking method, respectively. Moreover, a small-scaled segmented ship model is designed, constructed, and tested in a laboratory wave basin to validate the hydroelasticity algorithm. The results predicted by theoretical and experimental approaches are systemically compared and analyzed. Finally, future work for predictions of ship hydroelasticity and slamming loads in irregular waves is prospected.
The evaluation of the ship seakeeping performance and structural strength is a key element in the design, optimization, and operation guidance of seagoing ships. Fundamental to this is the predictions of the environment loads acting on ships when sailing in seaways [
Ship advancing in harsh wave conditions: (a) seagoing warship; (b) deck view of severe bow slamming.
The hydroelasticity is a phenomenon concerned with the mutual interactions among inertial, hydrodynamic, and elastic forces. The concept of hydroelasticity was first proposed in the field of aerodynamics by Heller and Abramson [
Transient impulsive loads, induced by bottom slamming, flare slamming, and green water on deck, should be considered during hydroelastic analysis especially for ships with pronounced flare bow sailing in harsh waves or at high speed. Severe slamming loads can result in not only local structure damages but also transient global whipping responses. Up to now, considerable efforts have been made on predicting the slamming loads acting on ships [
Although remarkable achievements have been made on developing the ship hydrodynamics and hydroelasticity theory, physical experiments constitute an indispensable tool in the field of naval architecture and ocean engineering. Experiments do not only reflect what really happens on the ship, but are also used to validate the numerical algorithm [
This paper aims at presenting a fully coupled 3D time-domain nonlinear hydroelasticity theory to estimate the motions and wave loads acting on ships sailing in severe regular waves. Small-scaled segmented model tests are also conducted to validate the numerical algorithm. The nonlinear behavior of ship motion and wave load responses under different conditions is comprehensively analyzed based on the numerical and experimental results. Moreover, our recent work regarding theoretical and experimental investigation on ship hydroelasticity and slamming loads in both long-crested and short-crested irregular waves is prospected.
The dry mode analysis of the flexible hull structure is the fundamental work prior to ship hydroelastic loads computation. Real hull girder is a highly nonlinear vibration system due to the complexity of the hull structure and external environment loads. The 3D finite element method (FEM) has good accuracy in the simulation of the vibration response of complex hull structure; it is, however, complex, expensive, and time-consuming [
The hull structure is assumed to be homogeneous, isotropic, and linear elastic. According to the knowledge of 3D structural finite element (FE) theory, the dynamic governing equation of flexible hull structure is expressed as follows:
The vibration characteristic equation of hull structure is expressed as follows:
The
The overall modal shape of elastic structural system with first
Based on the principle of mode superposition, the nodal displacement matrix is expressed as
Substituting Equation (
The 1D nonuniform hull girder is dispersed into a number of beam elements, and each element can be regarded as a uniform Timoshenko beam. The vibration mode characteristics of the overall 1D hull girder can be solved by transfer matrix method (TMM). The acquiring of vertical vibration mode of hull girder by TMM theory is described below, which is also applicable for hull horizontal vibration mode analysis. The vertical vibration equation of a simplified uniform beam element, with the influence of shearing deformation and moment of inertia considered, is expressed as follows:
The vertical displacement of the element can be further decomposed into space-dependent term and time-dependent term:
By substituting Equation (
The solution of Equation (
According to beam theory, the sectional rotation angle
As observed from (
Therefore, the relationship between the left side boundary condition (at stern side) and right side boundary condition (at bow side) of hull girder is written as follows:
Since the hull girder can be regarded as a beam with two free ends, the sectional VBM and VSF acting on the two ends are zero. Therefore, we obtain
Due to the fact that
Once the natural frequency
When the coupled effect between horizontal bending and twisting is not considered, the horizontal vibration mode of hull girder can be obtained in a same manner as vertical bending vibration. While for torsional deformation of hull girder, the relationship between the left side boundary condition (at stern side) and right side boundary condition (at bow side) of hull girder is written as follows:
According to the boundary condition of two free ends (
A 3D time-domain nonlinear hydroelasticity algorithm for ship motion and load predictions is developed in this section. In the hydroelasticity algorithm, the potential flow theory-based 3D boundary element method (BEM) and structural response-based 3D FEM (or simplified 1D Timoshenko beam theory) are combined to estimate the external loads acting on the elastic hull surface. The calculation of hydrostatic restoring force, wave excitation force, wave diffraction force, and radiation force are performed on the instantaneous wetted surface of elastic hull. The impact loads caused by bow slamming and green water on deck are included in the time-domain hydroelastic governing equation. The fourth order Runge–Kutta method is used to solve the time-domain nonlinear differential equation and the hull hydroelastic responses are obtained by the modal superposition principle.
In the potential flow theory, the fluid is assumed to be ideal, that is, incompressible, inviscid, continuous, and irrotational. The boundary conditions of fluid domain for the fluid-flexible structure interaction (FFSI) issue are illustrated in Figure
Boundary conditions of the hull and fluid domain.
The ship is advancing in waves at a constant speed of
Definition of coordinate systems: (a) top view; (b) side view.
Coordinate System 1: the space-fixed system
Coordinate System 2: the plane movement system
Coordinate System 3: the body-fixed system
For consistency, the relationship between values in these three coordinate systems is expressed as follows:
Since the fluid is ideal, the fluid velocity vector can be expressed by the gradient of the velocity potential:
The steady velocity potential corresponding to the steady wave-making flow component is ignored in this study. The disturbance velocity potential around the ship can be further decomposed into incident wave potential, diffraction wave potential, and radiation wave potential in the plane movement system
The incident wave potential
For the convenience of formulism, the diffraction potential
In this study, the source distribution method is used to solve the flow field velocity potential. The velocity potential can be obtained by integrating the distributed source over the body surface:
According to the hull body surface boundary condition, the distributed source strength should satisfy the following boundary integral equation:
The Green functions for both finite water deep and infinite water deep conditions are employed for calculation. The Hess-Smith panel element method is used to solve the distributed source strength in Equation (
The nonlinear effects of hydrodynamic forces for the ship sailing in severe waves at high speed are obvious. Therefore, the hydrostatic and hydrodynamic forces, including the hydrostatic restoring force, incident wave force, diffraction wave force, and radiation force, are calculated in real-time to take into consideration the nonlinear effects attributed to body geometry.
The real-time hydrostatic restoring force acting on the elastic hull surface is calculated as follows:
The incident wave force acting on the instantaneous wetted elastic hull surface is expressed as
The diffraction wave force acting on the instantaneous wetted elastic hull surface is expressed as
The radiation force acting on the instantaneous wetted elastic hull surface is expressed as
The hydrodynamic coefficients in real-time are computed based on the instantaneous static wetted body surface condition by using frequency-domain Green function method. The real-time hydrodynamic coefficients are expressed as follows:
As aforementioned, the potential flow theory ignores the fluid viscosity. In fact, the roll damping of the ship is associated with strong nonlinear behavior and should be considered when the ship sailing in beam or oblique waves. The roll damping coefficient
In this study, Muller’s method is used to determine the roll damping coefficient [
Generally, Wagner theory and momentum impact theory are the classical methods in use to acquire the slamming pressure and slamming force, respectively. In this study, the momentum impact theory is adopted to calculate the sectional impact force. In the theory, the slamming force is determined by the momentum changing rate of the fluid around the bow. Thus, the sectional slamming force is obtained as follows:
The selected hull sections for 2D slamming estimation are shown in Figure
Selected sections for slamming load estimation.
In harsh wave conditions, the incident wave may overtop the bow and run up onto the deck after the occurrence of bow slamming event. In this study, the green water load on deck is estimated by 1D dam-breaking model. This method treats the green water flow as the fluid subject to a sudden collapse of a dam, the phenomenon of which is illustrated in Figure
Dam-breaking modeling for green water load estimation: (a) dam-breaking flow model; (b) green water on the deck.
The green water pressure on deck
The time-domain motion governing equation of the elastic hull in regular waves, including both rigid motion and elastic distortion, can be expressed as
The structural damping has significant influence on high-frequency hydroelastic vibrations of the ship [
The diagonal element in the structural damping matrix is obtained by:
The fourth order Runge–Kutta method is employed to solve the time-domain nonlinear differential equation of Equation (
In order to validate the hydroelasticity theory algorithm, a segmented model was designed and manufactured according to a bow-flare ship to allow the conduct of the corresponding physical experiment. The experiment was conducted in a laboratory wave basin for regular head wave tests. The experimental details including model design, tank facility, and testing procedure are reported as follows.
The design of scaled model hydrodynamic test is guided by Froude’s similitude law. A 1:50 scaled model is designed according to geometric similarity, kinematic similarity, dynamic similarity, and vertical bending vibration modal similarity principles. The longitudinal distribution of weight and stiffness of model is also designed to be similar with the prototype. Main dimensions of the prototype and model ship are listed in Table
Main dimensions of the ships.
Item | Prototype | Model |
---|---|---|
Scale | 1:1 | 1:50 |
Overall length (m) | 313 | 6.26 |
Waterline length (m) | 292 | 5.84 |
Moulded breadth (m) | 39.5 | 0.79 |
Depth (m) | 25.5 | 0.51 |
Draft (m) | 10 | 0.20 |
Displacement (t) | 71,875 | 0.575 |
The model has 20 stations and is cut into seven parts of segments by divisions of 2nd, 4th, 6th, 8th, 10th, and 12th. A flexible steel backbone is used to connect the discontinuous segments. The backbone is elaborately designed with varying cross section so as to match the natural frequency and longitudinal stiffness distribution of the prototype in vertical bending vibration mode. The fluid forces subjected by the segmented hulls are fully transferred to the continuous backbone beam. Gaps of 15 mm wide are provided between adjacent segments to prevent their contact due to elastic deformation of backbone in waves. The gaps are sealed by latex rubber for waterproofing. The sectional wave and slamming loads at cut divisions are measured by strain gauges mounted on the backbone surface. An array of pressure sensors is mounted on the hull surface at the bow-flare area to measure the bow slamming pressures. Three accelerometers are mounted on bow, middle, and aft area of deck to measure the vertical acceleration. A set of propulsion system is installed at stern area from divisions 13th to 20th. The overall arrangement of model setup is illustrated in Figure
Overview of the model arrangement.
The model hull shell is made by Fiberglass-Reinforced Plastics (FRP). The backbone beams are made by hollow rectangular and cylindrical steel tubes, and they are fixed on the segment’s bases rigidly by aluminum fixing plate. The backbone beam is calibrated by applying known static loads prior to the assembly of segments. The ballast iron is installed at the desired positions to adjust the model’s weight distribution and moment of inertia. View of the segmented model before tank measurement is shown in Figure
View of the segmented model: (a) disassembled segments; (b) backbone beam calibration; (c) assembled model.
The regular wave experiments were conducted in the towing tank of Harbin Engineering University (HEU). The tank has a dimension of 108 m long, 7 m wide, and 3.5 m deep. Both long-crested regular and irregular waves can be generated by a single flap type hydraulically driven wave-maker. There is a wave absorbing beach at the opposite side of the wave-maker. A resistive wave probe is installed in situ near the wave-maker to measure the surface elevation of the actual generated waves. Another wave probe is installed onboard the carriage to measure the encountered waves suffered by the model. View of the tank facilities is shown in Figure
View of the tank facility: (a) model overview; (b) wave tank; (c) wave-maker.
The model’s motions are measured by an in-housed developed 5-DOF (i.e., heave, pitch, roll, sway, and surge) motion measurement device, which is installed on the measurement bridge of towing carriage. The model is attached to the 5-DOF motion measurement device by two heave sticks. The motion measurement device guides the model sailing heading and acts as speed reference during model running. The self-propelled model’s forward speed is achieved by its four propellers. The scheme of model setup on the 5-DOF measurement device is shown in Figure
Model experimental setup.
The regular wave testing scheme is determined by ship speed, wave states (wave height, wavelength, or frequency), and wave heading angle. The regular head wave tests are conducted for both zero-speed and forward speed conditions in this study. The testing schemes involved are summarized in Table
Summarize of experimental scheme.
Speed ( |
Wave height ( |
Wavelength/ship length ( | ||
---|---|---|---|---|
Prototype (knots) | Model (m/s) | Prototype (m) | Model (mm) | |
0, 5, 12, 15, 18, 21, 24, 30 | 0, 0.364, 0.873, 1.091, 1.309, 1.528, 1.746, 2.182 | 4 | 80 | 0.4, 0.6, 0.8, 0.9, 1.0, 1.1, 1.2, 1.5, 2.0, 2.5 |
8 | 160 | 0.8, 0.9, 1.0, 1.1, 1.2 | ||
12 | 240 | 0.8, 0.9, 1.0, 1.1, 1.2 | ||
16 | 320 | 0.8, 0.9, 1.0, 1.1, 1.2 | ||
19 | 380 | 0.8, 0.9, 1.0, 1.1, 1.2 |
The tests were conducted by a team of experienced researchers and staff in the towing tank laboratory of HEU. The testing procedure for each set of running measurement is described as follows: The carriage and model are situated at the start side (opposite the wave-maker) of the towing tank at first. The generated waves will propagate towards the model once the wave-maker started working. The carriage will start running when the waves are about to reach the model. The start time of carriage is comprehensively determined by the model testing speed and wave frequency so as to reduce the influence of wave reflection and also achieve maximum effective testing measurement distance. The data collector will start to record the model’s responses prior to model running. During the acceleration phase of carriage, the model is towed by ropes tied on its centerline at bow and stern. When the carriage reaches the desired speed, the ropes will be released. Then the model forward speed is achieved and maintained by its propellers. When the carriage is about to approach the end of the running, the ropes are recovered and tightened, after which the carriage starts deceleration. Finally, the carriage and model will return to the start point and wait for calm water and prepare for the next running measurement.
The necessary details during the conduct of numerical simulation and experimental measurement are reported in this section. The vertical vibration mode characteristics are reported at first. Then overview of the obtained typical numerical and experimental results in regular waves is presented.
For simplification, the flexible hull girder is described by a 1D nonuniform Timoshenko beam. The combination of 1D beam theory with 3D BEM turned out to be applicable and reliable for ship hydroelastic analysis as a compromise between the calculation accuracy and efficiency [
The normalized principal mode shapes in the 1st∼3rd orders. (a) Vertical displacement. (b) Rotation angle. (c) VBM. (d) VSF.
Impact hammer test was performed on the model to identify the actual natural frequency of model vibration in wetted condition. The test is performed by hitting the bow of model in calm water and recording the subsequent stress decaying curve. The corresponding frequency-domain information can be obtained by applying the fast Fourier transformation (FFT) on the recorded stress decaying curve. The measured vertical bending stress decay time series at the six different measurement stations (i.e., 2nd, 4th, 6th, 8th, 10th, and 12th) and the corresponding frequency-domain results are presented in Figure
Hammer test results for the vertical vibration mode at the model scale: (a) measured time series; (b) corresponding frequency-domain results.
A 3D hydroelastic analysis code was in-house developed using FORTRAN language based on the nonlinear time-domain hydroelasticity theory established in Section
Hydrodynamic grid model.
Typical numerical simulation results for the full-scale ship sailing at a forward speed of 5 knots in the harsh head wave condition (wave height
Simulation results for rigid body motion and elastic deformation of the ship in regular head waves: (a) heave motion; (b) pitch motion; (c) the 1st order vibration; (d) the 2nd order vibration; (e) comparison of vertical displacement; (f) zoom of vertical displacement in short term.
The calculation results for VSF and VBM at amidships (the station 10) of the ship in regular head waves are shown in Figures
Simulation results for sectional loads at amidships of the ship in regular head waves: (a) VSF at amidships; (b) VBM at amidships; (c) zoom of VSF in short term; (d) zoom of VBM in short term.
The experimental results for the same condition scheme as that presented in Section
The measured incident waves: (a) in situ wave measurement; (b) onboard encountered wave measurement.
The heave and pitch motions at COG of the model in the above wave condition measured by the 5-DOF measurement device are shown in Figure
The measured ship vertical motion: (a) heave motion; (b) pitch motion.
The measured time series of VBM loads at different stations (i.e., 2nd, 4th, 6th, 8th, 10th, and 12th) are summarized in Figure
The measured ship sectional VBM at different stations: (a) station 2; (b) station 4; (c) station 6; (d) station 8; (e) station 10; (f) station 12.
The extreme values (includes peak and valley values) of total VBM at different stations are extracted based on the time series during a steady run range of 20∼45 s. The amplitude values (includes hogging and sagging values) of linear wave-frequency VBM at different stations are obtained by Fourier’s filter and read from the corresponding filtered time series. Moreover, the ratio coefficient of the extreme value of total load to the amplitude value of wave-frequency load is calculated. The distribution of extreme values of total load and amplitude values of wave-frequency load for VBM along ship length is shown in Figure
Distribution of sectional VBM loads along ship length: (a) the peak and amplitude values of loads; (b) the ratio of the extreme value to amplitude value.
The bow slamming and green water phenomena recorded by video cameras are shown in Figure
View of bow slamming and green water on deck events: (a) bow emergence from water; (b) bow impact event; (c) deck view of bow slamming; (d) green water on the deck.
The results for ship motions and loads in different conditions obtained by theoretical and experimental methods are compared in this section. The numerical algorithm is also well validated by comparing with the model testing results.
The RAOs for ship sailing in head wave conditions at a forward speed of 5 knots are compared between calculation and measurement. The wave frequency in numerical simulation ranges from 0.02 rad/s to 2 rad/s with a step of 0.02 rad/s. The testing wave frequency is selected at points
Comparison of motion RAOs for the head wave at 5 knots: (a) heave; (b) pitch.
The comparative full-scale pitch and heave time series between calculation and measurement in different wave height conditions (
Comparison of time series for vertical motion responses for
Comparison of time series for vertical motion responses for
Figure
Comparison of VBM responses for the head wave at 5 knots: (a) RAO of VBM amidships; (b) VBM distribution along the ship for
The comparison of time series of sectional VBM amidships (at station 10) between calculation and measurement for different wave height conditions (
Comparison of time series for sectional VBM amidships: (a) VBM for
The comparison of different VBM load components (low-frequency wave loads and high-frequency slamming loads) at amidships for schemes (
Comparison of time series for VBM loads components for
Comparison of time series for VBM loads components for
The comparative bow-flare slamming pressures at positions of sensor nos. 1 and 2 (Figure
Comparison of time series for slamming pressure at
Although this paper presents detailed theoretical and experimental approaches for the prediction of ship hydroelasticity in regular waves, the prediction of ship hydroelasticity in irregular waves will be more insightful and practical. Accurately prediction of motion and loads performance of the ship in irregular waves is of great importance for full-scale ship design and evaluation. Our research project is still in progress. We are devoted to investigating ship hydroelasticity and slamming loads in both regular and irregular waves theoretically and experimentally. In our recent work, the theoretical and experimental investigations of ship hydroelasticity in both long-crested and short-crested irregular waves are undertaken, which is briefly prospected as follows.
To date, ship motion and load responses in random waves are usually predicted on the assumption that the incident waves are long-crested irregular waves [
According to the linear superposition theory, the 3D irregular wave field is composed of a number of regular waves with different frequencies and different spreading directions:
A nonlinear hydroelasticity theory is established to predict ship motions and loads in irregular waves with instantaneous wetted surface condition, nonlinear wave forces, slamming, and green water loads considered. The flow field velocity potential around the elastic hull is solved by the Rankine panel method to consider ship forward speed effects. The fourth order Runge–Kutta algorithm is used to solve the nonlinear governing equation in time domain. The nonlinear hydroelastic governing differential equation of the ship sailing in irregular waves is expressed as follows:
The real-time hydrostatic restoring force acting on the elastic hull surface in 3D irregular waves is calculated by integrating the pressure over the instantaneous wetted hull surface:
Considering the 3D effects of the flexible hull structure and irregular wave field, the incident wave force is obtained by integrating the contribution of the component wave at different frequencies and different spreading directions. Thus, the time-domain convolution integral method is used to obtain the incident wave force induced by short-crested irregular waves:
The diffraction wave force of the ship in short-crested irregular waves can be obtained in the same manner:
The radiation wave force of the ship in short-crested irregular waves is obtained by the indirect frequency-domain method. To take the wave memory effects into account, the radiation wave force is expressed as
The slamming loads
The tank model experiments were conducted in three different tanks using the small-scale segmented model, described in Section
Towing tank for long-crested regular and irregular wave measurement: (a) towing tank; (b) deep ocean basin; (c) high speed long tank.
To allow the experimental investigation of ship hydroelasticity and slamming loads in short-crested irregular waves, a large-scale segmented model was constructed at a scale of 1/25 according to the same prototype [
Sea trial by the large-scale model in short-crested irregular waves: (a) model overview; (b) backbone system; (c) wave buoy; (d) bow slamming event.
This paper focuses on theoretical and experimental investigations of hydroelastic responses and slamming loads behavior of large ships sailing in regular waves. This study also lays a foundation and provides fundamentals for the ship structure design and strength evaluation. The following conclusions can be made from this study: The developed fully coupled 3D time-domain nonlinear hydroelasticity theory, which combines 3D BEM, 3D FEM(or 1D nonuniform Timoshenko beam model), 2D momentum impact model, and 1D dam-breaking model, is an effective tool in the estimation of the nonlinear wave and slamming loads of the bow-flare ship sailing in regular waves. Although the 3D FEM has good accuracy in the simulation of the vibration response of the complex hull structure, it is complex, expensive, and time-consuming. The 1D nonuniform Timoshenko beam turned out to be applicable and reliable for ship modal analysis as a compromise between the calculation accuracy and efficiency. The established segmented model and tank experimental system are reliable and capable for the measurement of global motions, wave loads, slamming loads, and impact pressure acting on the ship sailing in waves. The numerical results are successfully validated by the tank testing results. The time series of motions are smooth and stationary even for the ship sailing in high wave states. However, the sectional loads comprise considerable high-frequency components besides the wave-frequency component especially for high speed or high wave state conditions. Therefore, the slamming loads should be concerned during the ship structure design and operational guidance.
The expressions of
The transfer matrix
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant no. 51709118), the China Postdoctoral Science Foundation (Grant nos. 2017M622696 and 2017M612669), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant no. 2017KQNCX004), the Funds for Economic Development of Guangdong, China (Grant no. GDME-2018B003), and the Science and Technology Program of Guangzhou, China (Grant no. 201804010482).