A direct coupling analysis method of hydroelastic responses of VLFS in complicated ocean geographical environment *

Jun Ding, You-sheng Wu, Ye Zhou, Zhi-wei Li, Chao Tian, Xue-feng Wang, Zheng-wei Zhang,Xiao-long Liu

1. China Ship Scientific Research Center, Wuxi 214082, China 2. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240,China

Abstract: In this paper a direct coupling analysis method (DCAM) of hydroelastic responses of a very large floating structures (VLFS)in complicated geographical environment is presented. In this method the three-dimensional hydroelasticity theory of floating bodies is combined with the shallow water wave theory, to allow for proper description of the influence of uneven seabed and sheltering effect of islands on the hydroelastic responses of a VLFS deployed near island and reefs in shallow sea. This method and the numerical procedures were verified and validated by the predictions and the model test results of a 3-module VLFS and an 8-module VLFS in two simulated shallow sea regions with different seabed topography.

Key words: Direct coupling analysis method, very large floating structures (VLFS), hydroelasticity, complicated environment* Project supported by the National Key Research and Development Program of China (Grant No. 2017YFB0202701),the Ministry of Science and Technology with the Research Project (Grant No. 2013CB36102) and the Ministry of Industry and Information Technology with the research project (Grant No. [2016]22).

Introduction

In 1895, the imagination of large floating island was firstly described in a science fiction novel, written by Jules Verne. In 1924 Armstrong patented a Sea Station to be used as “Aero Plane Supply and Navigating Station” that was of the length 335 m,width 100 m and displacement 50 000 t. The concepts were investigated more seriously for military applications by the US in the 1940s and a demonstration project was built and tested successfully in 1943. Since the early 1970s the technology for very large floating structures has developed continually,while the social needs have resulted in many different applications of the technology .

During the past three decades the research on very large floating structures (VLFS) gained great progress in Japan, USA etc.[1-5]. Japan launched a Mega-Float project in 1995. This project investigated the major technical problems and the possibility of building a large floating airport. Later a 5-module VLFS of the length 1 000 m, width 60 m and depth 3 m was constructed and deployed in Tokyo Bay.Meanwhile, the research project of Mobile Offshore Base was carried out in USA, aimed to provide a movable, pre-positioned floating naval base for cargos transferring from airplanes and container ships to the shore. Several concept designs of typical applications of MOB were investigated. The project was accomplished at the beginning of 21st century. The research achievements have provided the guidelines for future design of floating ocean base.

Comparing with an ordinary ship, much larger dimensions of a VLFS make its natural frequencies drop down to the wave frequency region. In the assessment of wave responses and structural safety of a VLFS, the hydroelastic effects naturally are of great concern. The three-dimensional hydroelasticity theory[6]established in early 1980s and its further develop- ments[7-14]have then been widely employed to predict the motions, wave loads and structural responses of VLFSs. This includes the VLFS project supported by National Science Foundation of America and carried out in University of Hawaii at Manoa from 1989 to 1991, the VLFS project supported by National Science Foundation of China and carried out in Shanghai Jiao Tong University from 2001 to 2005,and the project “Response and Structure Safety of VLFS in Complicated Environment” supported by the State Key Fundamental Research Program of China and carried out in CSSRC from 2013 to 2017, etc..

However, most publications in this field were for VLFSs in open sea. If a VLFS is deployed near islands and reefs in complicated geographical environment, the wave conditions, wave loads and the hydroelastic responses of a VLFS will be quite different from in open sea. Many researchers have recently shifted their focus to floating-body motion over an actual coastal seabed and some numerical models or methods[15-18]have been developed to provide a reference for the analysis of dynamic responses, design and construction of the floating bodies near seashores and islands.

A VLFS deployed in shallow water near islands and reefs, will encounter waves with quite different characteristics from those described by a usual wave spectrum in open and deep sea. To allow for proper description of the environment influence on the hydroelastic responses of such a VLFS, a direct coupling analysis method of hydroelastic responses of a VLFS in complicated geographical environment was proposed. The brief description of this method, the examples of numerical predictions and the comparisons with the model test results of two multi-module VLFSs are presented in this paper.

1. Wave environment

Numerical wave simulation becomes an important tool to predict the design waves for a floating structure in the sea near islands and reefs because there has been lack of recorded wave statistics.Various numerical models can be used to simulate the wave characteristics near islands and reefs. These models can be classified into three categories: (1)Mega-scale models used to deal with wave growth and propagation in a zone of several hundreds of kilometers, based on a third-generation spectral action balance equation with the source terms describing shallow water wave distortions. Examples are those adopted in software packages like Wave Action Model (WAM) , WW3 (Wave Watch III) and SWAN .(2) Middle-scale models used to simulate the wave refraction, diffraction and dispersion characteristics in a zone of several kilometers including shallow water near shore or islands, based on one of the semithree-dimensional theories, including the Boussinesq equation , the mild-slope equation and Green-Naghdi equation . These are adopted in software FUNWAVE,MIKE21, etc.. (3) Local-scale models used to describe the interactions of waves with the floating structure and the shallow seabed in a zone around the VLFS. Usually one of the viscous flow or potential flow solvers can be used with the input wave condition obtained from the middle-scale models.

To predict the wave parameters at the deployment location of a VLFS near islands and reefs, these three sets of numerical wave models have to be used to simulate the wave propagation from the far field deep water in the mega-scale zone (Wave Watch III based on wave action density balance theory) to the near field shallow water (FUNWAVE based on Boussinesq equation) around the structure in the local-scale zone. As an example, Ref. [10] used this procedure to simulate the waves around an Island C shown in Fig. 1. The calculated results were validated by comparison with the data obtained from the on-site long-term measurements by three buoys as shown in Fig. 2. The measurements have been carried out by CSSRC starting from June 2014 and forth.

Fig. 1 (Color online) Sketch of the mega-, middle- and local-scale zones

Fig. 2 (Color online) Layout sketch of the buoys (P1-P3) near islands

The waves generated by typhoon “Kalmaegi”and passing by the islands in 2014 were simulated from 14 to 21 September 2014 for 191 h. The space resolution of the wind input data was defined by 0.1°interval along the longitude and latitude direction,while the time resolution was one hour. The calculated significant wave heights at P1 were compared with the on-site measurement results, which are shown in Fig.3(a). Fairly good agreement has been achieved, with the maximum discrepancy less than 30%. Figure 3(b)presents the distribution of significant wave height at 4:00, in 16 September 2014.

Fig. 3 (Color online) Simulation results of waves generated by the typhoon “Kalmaegi”

The main characteristics of wave height decay at the position P1, due to the existence of islands and reefs can be obtained by numerical simulation and represented by the “wave height ratio” Hs(P1)/ Hs(P2),where Hs(P1), Hs(P2) denote the characteristic wave heights at P1, P2 respectively. It appears that due to the sheltering effect of the island, the wave height ratios in the east, south-east, south and south-west wind directions are smaller than 0.56.

2. A direct coupling analysis method of hydroelastic responses of a VLFS in complicated geographical environment

In the proposed direct coupling method, the time domain three-dimensional hydroelasticity theory of floating bodies is combined with the shallow water wave theory, to properly represent the influence of complicated geographical environment on the hydroelastic responses of a VLFS deployed near island and reefs. An imaginary vertical closed boundary from the free surface to the seabed surrounding the floating body is introduced, which divides the computation domain to the inner and outer regions (Fig. 4). The notations of all the boundaries of these two regions are shown in Table 1. In the inner region, a Local-scale model, i.e., a potential flow solver with the Rankine sources distributed over the whole boundaries of the inner region is adopted. In the outer region, a Middle-scale model, for example the Boussinesq equation is employed to solve the wave evolution problems. On the imaginary boundary the continuity relations, including the wave elevation and the vertical distribution of fluid velocity are used to match the fluid motions in the outer and the inner regions.

Fig. 4 Computational domain of direct coupling method

Table 1 Notations of the boundaries

For a VLFS moored in the sea, its dynamic behavior and the fluid motions in both outer and inner regions are observed in a fixed coordinate system Oxyz with x- axis pointing from stern to bow of the VLFS, x- y plane on the undisturbed water surface,pointing upwards and passing through the equilibrium position of the gravity center of the VLFS.

2.1 The outer region analysis

The wave evolution in the outer region is solved by employing the non-linear Boussinesq equation[19].The governing equations are as follows:

Continuity of mass

Continuity of momentum

where h in the equations is the local water depth.Only two unknowns, namely the horizontal velocityatand the wave elevation = ( x , y , z , t)η η , make the wave evolution problem being described in a two-dimensional way.On SBTO, SFOthe seabed and free surface boundary conditions have to be satisfied respectively. The boundary condition on Scis defined in the section 2.3.

2.2 The inner region analysis and the time domain hydroelasticity theory

To account for the effect of complex seabed profile on the hydroelastic responses of the VLFS in the inner region, the time domain three-dimensional hydroelasticity theory and the Rankine source method are employed[20].

By introducing the principal coordinates, the displacement =(u , v, )w u of the VLFS is expressed as the aggregation of the principal modes( =1,2, , )of the structure in vacuum r …m

For a floating body with zero forward speed, the flow field may be represented by unsteady potential.

The generalized equations of motion in time domain may be represented in terms of the principal coordinate vectorin the matrix form

where [ a] , [ b] and [ c] are matrices of generalized modal inertial, modal damping and modal stiffness of the dry structure. [ ]A , [ C] are the matrices of generalized hydrodynamic inertia and restoring coefficients respectively. [ K (T )] is the radiation impulse response function (or retardation function ). [ ]A ,[ K (t )] are defined by the integrals of1kψ ,kχ .Whereasare the generalized incident wave and diffraction wave exciting force vectors respectively.{ ( )}

DF t are defined by the integrals of,over the mean wetted surface SBof the body.is the additional restoring coefficients induced by the mooring system[21]. In linear theory its element is

where Hiis a 3×3 matrix representing the increments of the three mooring force components induced by a unit displacement in each of the three directions of the i-th mooring line connection point on the floating structure.,1kψ andkχ satisfy the Laplace equation. By denoting either of0φ ,Dφ ,1kψ orkχ as φ?, and distributing the Rankine sources over the whole boundary of the inner region

The radiation and diffraction flows satisfy the fluid-structure interface boundary condition on the mean wetted surface of the body SB, seabed condition on SBTIand free surface condition on SFI:

where n is the unit normal vector at0r on the boundary.

2.3The matching conditions on the imaginary boundary

(1) The boundary condition on SCfor solving the incident wave potential0φ in the inner region

The outer region solutions of Eqs. (1), (2)provide (η v on the imaginary boundary C S .The vertical distribution of the horizontal and vertical velocity components v(x, w(on C S relate to the solutions of α c v are as follows[19]:

(2) The boundary conditions onCS for solving the diffraction and radiation wave potentialsDφ , ψ1kand=1,2, )… in the inner region

To allow for total transmission of the diffraction and radiation waves on SCfrom the inner region to the outer region, an additional imaginary vertical closed boundary SC′surrounding SCwith certain distance is introduced. The diffraction and radiation waves satisfy the boundary conditions of numerical wave absorbers on it. The effects of their diffraction on the uneven seabed and reefs in the outer region are to be included in the following iteration of Boussinesq solutions in the outer region.

(3) The continuity relations of the fluid motions on the imaginary boundary outwards from the inner region

The boundary condition on SCfor solving the Boussinesq equations in the outer region only need. After solving Eqs. (3)-(12) and obtaining the total velocity potential φ ( r, t) in the inner region,they may be easily calculated, and are denoted asfor the boundary conditions on SCfor the outer region analysis.

2.4 The procedures of the solution

The numerical procedures coupling the shallowwater wave propagation with the three-dimensional hydroelastic analysis of a VLFS are performed in time domain.

(1) At a time step tn, with the incident wave on SI, the known boundary conditions on S, SBTOand SFO, and the dataon SCprescribed by the inner region solutions at the time step tn-1, Boussinesq equations are solved to producein the outer region, and their values on SCdenoted as.

(2) Employing the solutionαv and Eq. (13), the distribution of the horizontal and vertical velocity components on SCare obtained to define the inner region boundary condition on SC, denoted as

Fig. 5 Flow diagram for the procedures of the solution

(4) Equations (9)-(12) together with the wave absorbing condition forDφ ,1kψ andkχ on SCare used to solve the radiation and diffraction potentials in the inner region. The generalized hydrodynamic coefficients [ A] , the radiation impulse response function[ K ( t)] and the generalized diffraction wave exciting forces { ( )}

DF t at time tnmay then be determined,and Eq. (7) be solved for the principal coordinates. The total velocity potential φ ( r, t) of Eq.(4), and hence the wave elevationcη , the horizontal velocity along the lineon SCare calculated and denoted as(5) Comparewithon SC,If the mean square values of their magnitude differencesare not less than a prescribed criterionε,let,returntotheproce-dure (1), start a new cycle of iteration at the time step tn. If δη, δvare all smaller than ε, let,, turn to the next time step calculation with the incident wave at tn+1on SI, and the boundary conditionon SC.

Fig. 6 (Color online) Transverse pontoon and the bird eye's view of a 3-module VLFS with semi-submersible configurations

The flow diagram for the procedures of the solution is shown in Fig. 5.

3. Hydroelastic responses of VLFS near islands and reefs

3.1 Hydroelastic responses of a 3-module VLFS

A model of 3-module semi-submersible-typesemi-submersible-typeVLFS with 5 transverse lower hulls in each module as shown in Fig. 5 was tested in the State Key Laboratory of Coastal and Offshore Engineering of Dalian University of Technology (Fig. 8)[22]. The principal particulars of the VLFS are listed in Table 2.The complicated three-dimensional seabed topography and wave environment in the area around the model was simulated in the wave basin according to the measured seabed topography near an island, as shown in Fig. 7[23]. The model was tested in regular and irregular wave conditions with different wave frequencies and directions.

Fig. 7 (Color online) Connector models

Fig. 8 (Color online) Model test of the 3-module VLFS

For a multi-module VLFS, connectors between modules (C1-C4 in Fig. 6) have great influence on the global responses of the VLFS in waves, and usually suffer the largest internal loads. In the test not only the dynamic responses of the modules, also the connecting forces were measured. To simulate the connector with specified stiffness components in model test, the connector model composed of 2 L-shape mounting plates, 5 springs with spherical hinges at the both ends as shown in Fig. 7 were designed and manufactured.Two springs were arranged in each of x, z directions and one spring in y direction. The coefficients of model connector-stiffness were=2×,, corresponding to the prototype connector-stiffness,

Table 2 Principal particulars of the 3-module VLFS

Fig. 9 The environment of uneven water depth (m) and wave direction (88°) where the VLFS is supposed to be deployed

Fig. 10 Comparisons of predictions and test results of RAO of motions of module 1 and connecting forces of C1 in beam sea(88°)

Fig. 11 (Color online) The sketch (bird eye's view) of the 8-module VLFS model and the connectors

The motions and connecting forces of the 3-module VLFS in the sea of uneven seabed as shown in Fig. 9 were predicted by employing the Direct Coupling Analysis Method . The VLFS was modeled as rigid modules and flexible connectors (the RMFC model). The predicted transfer functions of the rigid body motions of module 1 and the connecting forces of Connector C1 are compared with the model test results in Fig. 10. To exhibit the influence of the seabed topography on the responses, the predicted results of the same VLFS in the sea with even seabed ofconstantwaterdepth30marealso provided for comparison. Obviously, the predictions with the uneven seabed effect well agree with the model test results. To assume an averaged constant water depth is not suitable for response predictions of a VLFS deployed in shallow sea of uneven seabed. The influence of the sea topography is very obvious in most wave frequencies.

Fig. 12 The seabed topography for model test of the 8-module VLFS

3.2 Hydroelastic responses of an 8-module VLFS

An 8-module semi-submersible-type VLFS (Fig.11) with its mooring system was tested from March to June 2017 in the State Key Laboratory of Ocean Engineering of Shanghai Jiao Tong University. The particulars of each module are similar to that shown in Table 2. The stiffness components of model connector in three directions are 4.60×106N/m, 6.55×106N/m and 9.78×105N/m respectively. The uneven seabed was simplified as a 2-slope two-dimensional topography as shown in Fig. 12. To simulate the effect of inhomogeneous wave directions, a model island was fixed between the slope-seabed and the VLFS model(Fig. 13).

Fig. 13 (Color online) The models of an island and the slopeseabed on the bottom of the wave basin for simulation of the inhomogeneous waves

Fig. 14 (Color online) The numerical simulation of the wave propagation and evolution from an irregular long crested incoming wave, where W1-W13 are the positions of wave elevation measurements

Fig. 15 Comparisons of the predicted and measured results of wave elevations at the 13 observation positions W1-W13.

Fig. 16 Comparisons of predictions and test results of motion transfer functions (RAO) of module 1

A long crested irregular incoming wave=) was scattered by the island and the uneven seabed to become short crested waves in 5.54 m,=10.7 s inhomogeneous directions as shown in Fig. 14. In the figure, W1-W13 denote the 13 positions for wave height measurements. Figure 15 exhibits the comparison of the predicted and measured results of wave elevation at the 13 observation positions W1-W13, and showed good agreement. Figures 16, 17 are the comparisons between the predictions and the test results of the transfer functions of motions of module1,and the connecting force components of C1, C7. It really shows that the present Direct Coupling Analysis Method may reasonably predict the hydroelastic responses of a very large floating structure deployed in shallow sea with complicated geographical environment near islands and reefs.

Fig. 17 Comparisons of predictions and test results of the connecting force transfer functions (RAO)

Fig. 18 (Color online) The dry modes that dominate the connecting forces of C7

The inhomogeneous incident wave and the effect of the uneven seabed makes the distribution of hydrodynamic loads acting on the 8 modules, the responses of the modules and the connecting forces between two adjacent modules rather complicated.Further investigation on Fig. 17 shows that the connecting forcexF is mainly determined by the responses of the first horizontal bending mode (the 14thmode) of the 8-module VLFS, while the connecting forcezF is dominated by the first two torque modes (the 17thand 19thmodes) and the first vertical bending mode (the 20thmode). These modes and corresponding principal coordinates are shown in Figs. 18, 19. Figure 20 provides the variations of transfer functions at three wave frequencies of the connecting forcesxF ,zF along the starboard of the 8-module VLFS from C1, C3,… to C13. The three frequencies are 0.897 rad/s (T =7 s), 0.483 rad/sand 0.209 rad/srepresenting the typical short, medium and long regular waves. The maximum values of the connecting forces for each connector among whole frequency region are also given in Fig. 20 for comparison. The forcesxF for connectors in the middle of the VLFS are always greater than the connectors located at the two end parts, but for forceszF , there is not very obvious regulation.

4. Conclusions and recommendations

With large size but relatively smaller global rigidities, the hydroelastic responses of a VLFS in waves are one of the major concerns, both in deep open sea and in shallow water near islands and reef.

Fig. 19 The corresponding principal coordinates

The direct coupling analysis method of hydroelastic responses of floating bodies in complicated geographical environment and the corresponding numerical approaches that may be used to predict the dynamic behaviors and structural safety of a VLFS in shallow sea near islands and reefs are briefly described in this paper. Model tests of a 3-module and a 8-module semi-submersible-type VLFSs in wave basins were performed. In the test of the 3-module VLFS, the bottom of the basin was modeled to simulate the measured seabed topography near an island. The wave basin for the 8-module tests was specially constructed to have a 2-slope bottom and an island model in front of the VLFS model to simulate the inhomogeneous incoming wave environment along the length of the huge body. The numerical predictions using the Direct Coupling Analysis Method are compared with the model test results and showed good agreement.

Fig. 20 Variation of transfer functions (RAO) of the connecting forces along the length of the VLFS from C1 to C13

The Direct Coupling Analysis Method has provided an applicable tool for the design and optimization of the structure of the modules, the connectors and the mooring system of a multi-module VLFS deployed in the complicated geographical environment near islands and reefs. Even though, the full-scale measurements are needed in the future for further validation. Moreover, the non-linear responses induced by the severe typhoon environment,impact/slamming loads, the influence of non-linear connector characteristics, the effect of non-linear mooring forces on the motion predictions and safety assessment of a multi-module VLFS are the topics that require more investigation.

Acknowledgment

The authors also gratefully acknowledge the contributions to this paper from Xue-kang Gu,Xiao-ming Cheng, Ye Lu, Ming-gang Tang, Yong-lin Ye and Jia-jun Hu of CSSRC.