In order to study the interactions between fluid and elastic structure (such as marine lifeboat falling down and ship), this paper presents a new CFD method on hydroelastic waterentry problem of freefalling elastic wedge, which can more conveniently handle moving solid boundaries. In the CFD solver, a surface capturing method and the Cartesian cut cell mesh are employed to deal with the moving free surface and solid boundaries, respectively. On the other hand, in structural analysis, the finite element method and lathbeam structural model are introduced to calculate the elastic response. Furthermore, based on the current CFD and structural solver, a particular data transfer method and coupling strategy are presented for the fluidstructure interaction. Finally, by comparing numerical results with experimental data, the present method is validated to be available and feasible for hydroelastic waterentry problem and further successfully adopted to analyze the motion characteristics of freefalling elastic wedge.
Interactions between fluid and elastic structure, especially water entry problem of elastic body, are of concern and practical importance in many ocean and marine engineering applications, such as seaplane landing on sea, ship slamming, sloshing, greenwater impacting on deck, and marine lifeboat falling down. For the water entry problem, it is complicated and difficult to numerically study moving solid bodies and free surface and the challenge is further higher if global motion and elastic deformation of solid body need to be calculated from the fluidstructure interaction.
The water entry problem has been widely studied by many scientists and engineers using various methods since the initial study of von Karman [
Therein boundary element method is based on the theory assumption of potential flow, so the rotational motion near free surface cannot be well simulated. Thus in order to better predict the force and motion of body and achieve fluid information near free surface simultaneously, a CFD method should be applied in water entry problem. Furthermore, compared with the traditional CFD method ([
This paper presents surface capturing method and Cartesian cut cell mesh to treat moving free surface and solid boundaries. In the CFD solver, incompressible Euler equations are presented as governing equations for a variable density fluid and the location of free surface can be captured as a contact discontinuity in the density field. Furthermore, finite volume method is applied in numerical spatial discretization. Therein Roe’s approximate Riemann solver is adopted to evaluate numerical flow flux and dualtime stepping technique with artificial compressibility method is used for time advancing. On the other hand, in the structural solver finite element method and lathbeam structural model are used to analyze structural response of elastic wedge. Particularly based on the geometry of Cartesian cut cell mesh and lathbeam element here a particular data transfer method on solid boundary is deduced and presented. Furthermore, the flow field, structural response, and global motion are calculated in a coupled manner, which not only computes hydrodynamic force on solid boundary, but also allows the feedback of elastic deformation and global motion into the CFD solver. Finally, some test cases of water entry for various kinds of freefalling elastic wedge are numerically simulated and the calculated results show the feasibility and availability of the present method. Furthermore, by this method the global motion and local response of elastic wedge are studied and analyzed during waterentry phase.
For the 2D, incompressible, unsteady, inviscid fluid system with a variable density field, the
In the present study, various kinds of boundary conditions can be classified.
Outlet or open boundary: a zero gradient condition is applied in the velocity and density and the pressure at this boundary is fixed to be static pressure, which allows fluid to enter or leave the computational domain freely according to the local flow velocity and direction.
Solid body boundary: the nopenetration condition can be applied in velocity and the density is assumed to have a zero normal gradient. For pressure boundary condition, here the momentum equations (
In the present study, Cartesian cut cell mesh is employed for spatial discretization, which can be generated by cutting solid bodies out of a background Cartesian mesh. Thus fluid cell, solid cell and cut cell are accordingly created, which include various storage data (such as identification of cell type, geometry, and fluid variables). For more information, refer to Coirier and Powell [
Based on the Cartesian cut cell mesh, here a cellcentral finite volume method is applied in numerical discretization. In this numerical scheme, Roe’s approximate Riemann solver is adopted to calculate the numerical flux on each edge of fluid cell where fluid variables are reconstructed by using a piecewise linear upwind scheme. In the linear representation, a leastsquare method [
For the bottom plate with cylindrical bending of elastic wedge structure, a lathbeam model can be used to analyze structural deformation. Compared with EulerBernoulli beam, due to bilateral constraints, the crosssection of lathbeam model cannot freely deform and keep the original shape. On this basis, only if by introducing
Initially, the global (fixed) and local (moving with body) coordinate systems are created. Then, lathbeam structure is divided into many beam elements with equal length, which can support bending and tensioncompression effects simultaneously. Here in the local coordinate system, the nodal displacement vector
Next, in local coordinate system the corresponding mass and stiffness matrix of lathbeam elements are generated and then load vectors of element nodes are calculated from the fluid pressure on solid boundary. Subsequently, the mass matrix, stiffness matrix, and load vector are transformed into global coordinate system. For the waterentry model of elastic wedge, the elastic deformations basically take place on sloping sides; therefore here on the corner of sloping and straight sides the rigid connection is selected as boundary constraint of lathbeam structure. Then the global dynamical equation without structural damping can be written as
Finally, by means of numerical Newmark method, the nodal vector of displacement, velocity, and acceleration are computed and fed back to CFD solver and meanwhile local structural response on elastic boundary can be obtained.
For the waterentry model of freefalling elastic wedge, the global motion and local deformation of elastic body should be calculated from the fluidstructure interaction, so here the data transfer method on elastic boundary and fluidstructure coupling strategy are discussed, respectively.
In order to study and analyze the fluidstructure interaction, it is necessary to transfer the fluid pressure to the element node and then feed the global motion and structural deformation back to CFD solver.
In this paper, Cartesian cut cell mesh in CFD solver and lathbeam element in structural solver are different from each other. So here a particular data transfer method on solid boundary is deduced and demonstrated, which includes transfer of distributional fluid pressure to structural nodes and feedback of structural response on CFD boundary condition.
Based on respective geometries of Cartesian cut cell mesh and lathbeam element, there should be many kinds of position relations between the solid boundary
Different position relations of cut cell and structural elements; (a) full inclusion of
Firstly, local coordinate system
Next, by using the coordinates
Then by combining (
Finally, according to the equivalence principle about total force and moment, the distributional fluid pressure on
Here among many kinds of position relations of Cartesian cut cell mesh and lathbeam element as mentioned in Section
In Figure
Firstly based on the shape function of structural element
Here it should be noted that
Meanwhile, gradient of normal velocity
Finally, the tangential velocity, average normal velocity and its gradient can be obtained from (
In the present study, the fluidstructure interaction of waterentry model for freefalling elastic wedge includes the coupling calculation of fluid with global body motion and local structural deformation.
On the one hand, in order to ensure enough numerical stability, a subiteration coupling approach for the global velocity of wedge proposed by Kleefsman et al. [
On the other hand, compared with military missile at high speed, ship and ocean engineering structures usually enter water with mediumlow velocity and therefore the deformation of elastic boundary is relatively small and its reaction to the fluid field can be neglected. Thus in order to increase the computational efficiency, here a twoway “weak” coupling method as shown in Figure
Flow chart of iteration procedure for the fluidstructure coupled solution strategy.
In this section, the current numerical method for hydroelastic analysis on waterentry problems of freefalling wedge should be tested by comparing calculated results with experimental data of Sun et al. [
Based on the parameters of water tank in Sun’s experiment, 2D CFD computational domain was a square of 0.8 m × 1.45 m and the water depth was 1.1 m. The material properties of elastic wedges were 1.059 × 10^{3} kg/m^{3} density, 2.67 × 10^{9} kg/(ms^{2}) elastic module, and 0.357 passion ratio. Furthermore, geometric parameters of three different wedges were in Table
Geometric parameters of three different elastic wedges.
Number  Parameter  Unit  Value 

1  Length  m  0.6 
Width  m  0.2  
Thickness  mm  1.5  
Mass  kg  7.7870  
Deadrise angle  45°  


2  Length  m  0.6 
Width  m  0.2  
Thickness  mm  1.5  
Mass  kg  7.4780  
Deadrise angle  20°  


3  Length  m  0.6 
Width  m  0.2  
Thickness  mm  3.0  
Mass  kg  7.7890  
Deadrise angle  20° 
Sketch of elastic model and description of geometric parameters.
In the numerical calculation, the zero physical time (
Global mesh (a) and local grid (b) near solid boundary.
In Figure
Time history of global acceleration and local strain of observation point on the sloping edge for three different elastic wedges, (a) 1.5 mm thickness, 7.787 kg mass, and 45° deadrise angle; (b) 1.5 mm thickness, 7.478 kg mass, and 20° deadrise angle; (c) 3.0 mm thickness, 7.789 kg mass, and 20° deadrise angle.
By using the present method to study waterentry model 1 of Table
Free surface profile and velocity vectors for a freefalling elastic wedge with 1.5 mm thickness, 7.787 kg mass, and 45° deadrise angle; (a)
At
In order to study the global dynamic behaviors and local structural responses of elastic wedge, here, based on the geometric parameters of model 1 in Table
Firstly, by means of the current numerical method with or without elastic deformation, the global hydrodynamic acceleration and velocity versus elastic and rigid wedge model were calculated and compared as shown in Figure
Time history of global hydrodynamic acceleration and velocity versus elastic and rigid wedge model with 7.787 kg mass and 45° deadrise angle.
In Figure
On this basis, in order to further study the long global motion of freefalling wedge during the water entry phase, here the water entry model was numerically simulated for a long physical time and the results of wedge velocity were shown in the last picture of Figure
Figure
Pressure contour in fluid field for elastic wedge with 7.787 kg mass and 45° deadrise angle;
The time history and frequency analysis of local pressure and strain on the midpoint of wedge edge are shown in Figure
Time history and frequency analysis of local pressure and strain on the midpoint of wedge edge versus elastic and rigid model with 7.787 kg mass and 45° deadrise angle.
One is the pressure peak with high amplitude and narrow range in low frequency domain. Based on the effect of max pressure (frequency tends to zero) in Figure
The other is the pressure oscillation with approximately 72~85 Hz range and low amplitude in high frequency domain which includes two crests and one trough. From the modal analysis, the pressure oscillation possibly causes the midpoint to locally vibrate with 11th and 12th natural frequencies 73.13 and 86.822 Hz which are calculated by ANSYS software. Furthermore, because the two natural frequencies are very close, the vibration phenomenon of “beats” should appear on the midpoint of wedge edge. Here according to the properties of “beats” phenomenon, the theoretical frequencies versus amplitude profile
Thus the local strain of midpoint on wedge edge can be divided into main deformation versus 1st principal mode and local deformation of “beat” phenomenon versus 11th and 12th modes.
In order to evaluate the hydroelastic effect of freefalling wedge, here case 1 is numerically simulated without feedback loop of elastic deformation, which means that the local structural response is handled as just a “pure” postprocessing step. Then the results are shown and compared with twoway “weak” coupling method in Figures
Pressure distribution on elastic boundaries of a freefalling elastic wedge with 1.5 mm thickness, 7.787 kg mass, and 45° deadrise angle at initial water entry moments: (a)
Comparison of local pressure (a) and strain (b) on the midpoint of wedge edge with 45° deadrise angle versus two handling method on local elastic deformation (twoway “weak” coupling and “pure” postprocessing method).
At initial waterentry moments (from
In Figure
Time history of local structural deformation and global displacement on the vertex of wedge with 1.5 mm thickness, 7.787 kg mass and 45° deadrise angle.
In this paper, surface capturing method and Cartesian cut cell mesh are successfully developed to handle the interaction among fluid field, global motion, and local deformation of elastic body simultaneously for water entry model of freefalling wedge.
In finite element analysis, based on the different characteristic of fluid and structural mesh, a particular data transfer method on solid boundary is successfully deduced and presented. Furthermore, an effective fluidstructure coupled solution strategy is taken to solve the interaction among fluid field, global motion, and local deformation of elastic body.
Based on the present method, some test cases of water entry for freefalling elastic wedge are numerically simulated. By comparing with experimental data, the results show that the present method has a good ability to study hydroelastic waterentry problems of freefalling elastic wedge.
Through further analysis, the global hydrodynamic performance and physical phenomenon of water entry problem (such as free surface rising, pressure distribution, jetflow, and negative pressure) for elastic wedge are discussed in detail.
For the local structural response on elastic boundaries, especial “beats” phenomenon of elastic vibration versus certain model without structural damping are discussed. Furthermore, based on the results of vertex for elastic wedge, a conclusion has been made that, for some test cases of elastic wedge, simplified half model can be applied in the numerical simulation to enhance computational efficiency.
Furthermore, the method should be developed further on the following aspects.
Although the present method is available and accurate for the waterentry model of wedge, it has trouble in waterentry problems of solid body with complex shape because of the invalidation of lathbeam structural model and a more finite element model (shell or plate) should be applied. Furthermore, the current fluidstructure coupling strategy should be improved and gradually transformed into other advanced coupling approaches (such as twoway “tight” coupling method).
Because here, the structural finite element analysis is based on linear elastic theory, the present method is not suitable for water entry problems of elastic body with large deformation. Hence in future the geometric nonlinearity should be taken into account to extend the application fields.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to Dr Qian of the Manchester Metropolitan University in the UK for providing very useful theoretical help and many suggestions about the numerical method. This work was financially supported by the National Innovation Team Foundation under Grant no. 50921001 (China) and the National Natural Science Foundation of China (Grant no. 11202047).