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Towards the modeling of the ditching of a ground-effect wing ship within the framework of the SPH method

T

H. Chenga, F.R. Minga, , P.N. Sunb, P.P. Wanga, A.M. Zhanga, ⁎

a b

⁎

College of Shipbuilding Engineering, Harbin Engineering University, Harbin, China Ecole Centrale Nantes, LHEEA res. dept. (ECN and CNRS), Nantes, France

ARTICLE INFO

ABSTRACT

Keywords: Ditching WIG ship Free-surface flows Six DOF Smoothed particle hydrodynamics

Ditching often takes place for a ground-effect wing (WIG) ship. During the ditching, the extreme load developed by water impacts may cause damages to structures, posing a great threat to the safety of crew and passengers. In the paper, a weakly compressible smoothed particle hydrodynamics (SPH) model combined with enhanced numerical techniques has been adopted to tackle the ditching problems. In order to handle the motion of a rigid body in the three-dimensional ditching problems, the six degrees-of-freedom (6-DOF) equations of motion are incorporated into the SPH scheme. The accuracy of the SPH model is validated through two benchmarks, respectively, the two-dimensional wedge water entry and the three-dimensional stone-skipping. The former is aimed to validate the prediction of pressures during the free surface impact while the latter is a good case for testing the coupling motions of the rigid body. Furthermore, the ditching of the real scale WIG ship under different conditions is simulated with the established SPH model, through which some useful conclusions are drawn.

1. Introduction The ground-effect wing (WIG) ship is a new-developing ship type, aiming to take the advantages of surface ships and airplanes. The WIG ship usually has a higher velocity than common surface ships, a larger deadweight than airplanes and a stronger adaptability to the environment. Hence, it owns a great prospect in both civilian and military applications. In recent years, more and more researches have been focused on the WIG ship, and have made some desirable research results [1–4]. However, few people are acquainted with the landing process of the WIG ship which is mainly accomplished on the sea especially in the case of mechanical failure. In that case, the landing process becomes a ditching. The landing process can be divided into two stages, i.e., the impact stage and the slide stage. The first stage is transient and violent characterized by high-pressure, posing a great threat to the safety of structures and passengers. In the latter stage, the WIG ship may be bounced up again like a skipping stone. This type of bounce-up motion is complex and hard to maneuver, thus it should be avoided during the ditching. Von Karman is a pioneer for the study of the ditching problem [5]. In the study, the airplane ditching problem was simplified to a wedge water entry, and the impact load was predicted through a theoretical study. After that, the theoretical analysis of the similarity flow induced by wedge entry was conducted by Wagner [6] who was the first ⁎

researcher accounting for nonlinearities in the impact. On these bases, many other scholars have made numerous improvements and innovations [7–12]. These theoretical methods are suitable for dealing with the water entry of structures with regular geometry, while for the modeling of the water impact involving a complex geometry like the WIG ship, there are still some limitations. In the 1950s, a series of experiments for the ditching was carried out [13,14]. The hydrodynamic characteristics of a scaled airplane model were investigated under different conditions, and some valuable conclusions were obtained aiming at defining an optimum ditching scheme. To investigate the effect of air, Okada et al. [15] carried out the experiments in terms of the water entry of a plate with different deadrise angles. In that work, by introducing the characteristic angles, i.e., βT = 1° and βW = 3°, the free surface impact problem can be divided into three types: (a) the trapped-air type (0° < β < βT), (b) the transitional type (β < βT < βW), and (c) the Wagner type (βW < β). It was found the Wagner-type pressure pattern dominates as the impact angle increases. Recently, the water entry of a flat plate with different horizontal velocities and deadrise angles component was investigated experimentally by Iafrati et al. [16,17]. In these works, the valuable results in terms of hydrodynamic pressures, structure strains were reported, and they have become reliable references for numerical validations, see e.g. [18–21].

Corresponding authors. E-mail addresses: [email protected] (F.R. Ming), [email protected] (A.M. Zhang).

https://doi.org/10.1016/j.apor.2018.09.014 Received 25 April 2018; Received in revised form 18 September 2018; Accepted 19 September 2018 Available online 03 December 2018 0141-1187/ © 2018 Elsevier Ltd. All rights reserved.

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With the development of computer technologies, numerical models have gradually become an important tool for the study of complex hydrodynamic problems. During a ditching process, significant free surface deformations and splashing jets are involved. Many challenges are encountered in traditional mesh-based Eulerian models where the free surface detection can be problematic unless very fine meshes are adopted. Hence, in the paper, the smoothed particle hydrodynamics (SPH), that is a mesh-free Lagrangian particle method, is adopted. The SPH method was originally developed for astrophysics problems [22,23], and then applied to the study of free surface flows [24–27]. With the recent advancements in theory and numerical techniques, the SPH method has been recognized as a predictive tool in a variety of industrial applications [28–37]. The water entry of a structure, e.g. the ditching of an airplane, is one of its most important applications [38]. Previously, Oger et al. [39] studied the water entry of a wedge under different conditions by using a weakly compressible SPH method. In that work, the pressure prediction of the numerical results shows good agreements with experimental measurements. Some other scholars have also applied the SPH method to the simulation of ditching problems. Previously, a combined SPH-FEM method was selected by Groenenboom et al. [18] to deal with the ditching problem, where the effect of air was discussed. The study shows the air is responsible for only a slight discrepancy in time between experiment and simulation, yet its effects on structural responses within the investigated range of test conditions are minor. In the present paper, the SPH method combined with enhanced numerical techniques has been applied toward the modeling of the ditching of a real scale WIG ship. As the simulations are carried out for a large deadrise angle, the effect of the air is ignored. The WIG ship studied in the paper is counted as a rigid body whose motions are in six degrees of freedom (6-DOF). The coupling algorithm between the fluid and the solid body is much easier in the mesh-free SPH method since both the fluid and the solid body are described in a Lagrangian way and the problems related to mesh distortions or grid refinements are avoided. Before the study of the WIG ship ditching, two numerical examples are firstly simulated to validate the feasibility and accuracy of the SPH model. The first one is the water entry of a two-dimensional wedge, with which impact loads and local pressure evolutions are validated against experimental data. The second case is a simplified problem resembling the skipping of a three-dimensional stone. In this case, we mainly focus on the kinematic characteristics of the stone. Finally, the simulation of the WIG ship ditching is carried out. The effect of the ditching velocities on the impact loads and motion trails of the WIG ship is discussed. The paper is organized as follows: in Section 2, the SPH model applied in the paper is briefly presented. The numerical techniques for boundary treatment, fluid-solid coupling and the stabilization of pressure fields, etc. are briefly introduced. In Section 3, the wedge water entry and the stone skipping are simulated and validated, demonstrating the sufficient accuracy and stability of the established SPH model. Finally, in Section 4, the dynamic characteristics of the WIG ship ditching on the calm water under different velocities are simulated and analyzed, and useful conclusions are obtained.

where ρ, p, v refer to the density, the pressure and the velocity respectively. g represents the gravitational acceleration and ν is the constant kinematic viscosity of the fluid. As the fluid is treated as barotropic and assumed to be weakly compressible, the Tait equation of state can be combined to solve Eq. (1) and Eq. (2). Here a linearized form of the Tait equation of state has been adopted and it is written as [40]:

p = c02 (

where ρ0 is the reference density of the fluid when its pressure is zero and c0 refers to the sound speed which is assumed to be a constant inside the fluid. As pointed out in [41], within the weakly compressibility region, the SPH solution can be regarded as the combination of two parts, an incompressible solution and a solution for the acoustic problem and the latter is believed to generate a negligible effect to the former part when the sound speed is large enough. In this work, c is determined by:

c0 > 10 max(Umax ,

2.2.1. Basic equations Smoothed particle hydrodynamics (SPH) is a Lagrangian meshless method in which the continuum fluid is discretized into a set of individual particles possessing physical variables, e.g., pressure, density and velocity, etc. The governing equation introduced in the last section can be solved based on the particle approximations [43]. Specifically, an arbitrary function f(r0) and its spatial derivative ∇f(r0) at a general position denoted by the vector can be approximated using the kernel interpolation as [24]:

f (r0 )

1

p+g+

r , h) dr

f (r ) W (r0

r , h) dr =

(5)

f (r ) W (r0

r , h) dr

(6)

where h refers to the smoothing length of the kernel function W which has to satisfy several properties as described in [43]. r denotes the position all over the compact domain Ω of the kernel function. In this paper, the renormalized Gaussian kernel proposed in [44] is applied:

W (r0

r , h) =

d (e

(q / h ) 2

0

C ) q 3h q > 3h

(7)

−9

where q = |r0 − r|, C = e . αd is a coefficient satisfying the normalization property and it is written as: d

(1) 2v

f (r ) W (r0

f (r0)

In the present work, the fluid evolution is solved based on the Navier–Stokes equations as follows:

dv = dt

(4)

0)

2.2. SPH formulations

2.1. Governing equations

·v

pmax /

where Umax and pmax refer to the maximum expected velocity and pressure [20], respectively. On this condition, the maximum density variation inside the fluid is believed to be less than 1% and the weakly compressible hypothesis is satisfied [42]. For the problems characterized by a strong impact, the sound speed should be considered to reflect the real compressibility of the fluid. But in that way, the time step restricted by the sound speed will become very small. Recently, in Marrone et al. [20], for evaluating a proper sound speed, a technique based on a theoretical approximation of the maximum pressure, reaching a compromise between the accuracy and efficiency, has been proposed. However, in the present work, we mainly focus on the kinematic character of the body, thus the sound speed is set to be larger than 10 times of the maximum speed of the fluid. For the impact problem with a small deadrise angle, the technique of [20] has to be adopted.

2. Theoretical background

d = dt

(3)

0)

= [hd

d/2 (1

10e 9)]

1

(8)

where d is the spatial dimension of the problem. If we discretize the continuum field into particles, Eq. (5) and Eq. (6) can be written in a particle approximation form where the integration is transformed as the summation, namely:

(2) 371

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mj

f (ri)

f (rj ) W (ri

rj , h )

(9)

j

j =1 N

mj

f (ri ) j=1

f (r j )

W (ri

i

to reduce the spurious density/pressure fluctuation. This scheme has been widely validated in different applications [55,56]. Recently, a δ+SPH scheme which combined the δ-SPH model with the shifting algorithm was established by Sun et al. [57] aiming for a good distribution in both pressure and particle spacing within the fluid domain. Another alternative way to reduce the pressure noise is to obtain the pressure field by solving the Poisson equation, which results in an incompressible SPH model (ISPH) [58–61]. In the paper, we seek for 3D simulations with a large particle quantity in the framework of the WCPSH model. Comparing the computational cost in the different techniques for smoothing the pressure field, we find that the Shepard density filter is the cheapest one since it can be applied every 20-time steps according to:

rj , h )

(10)

j

where the subscript i denotes the i-th particle and the subscript j represents the j-th particle within the kernel function of the particle i. m is the mass and m/ρ represents the volume of a particle. ∇iW(ri − rj, h) is the gradient of the kernel function respect to the particle i. h is determined as 1.2Δx in 2D problems while 1.0Δx in 3D cases where Δx is the initial particle spacing. According to Eq. (9) and Eq. (10), the governing equations can be rewritten as:

d i = dt dvi = dt

mj vij·

i

W (ri

rj, h)

(11)

j

mj ( j

pi

2 i

+

pj 2 j

)

i

W (ri

i

rj , h) + g

mj ( j

pi

2 i

+

pj 2 j

+

ij) i

W (ri

rj , h ) + g

c0 h ij

=

j

0

rj, h)

mj

(15)

j

In spite of good results have been achieved for fluid–body interactions [30,62,63], it is still elusive for the SPH method to model a complex moving boundary, especially for three-dimensional problems. The mirror particle boundary [50] is a feasible approach to model the moving boundary by generating the mirror images of the fluid particles [39,64]. However, it is difficult to be implemented for complicated geometries in three dimension. In the paper, the fixed ghost boundary [48] has been adopted. This technique can also be called the dummy particle boundary, in which a simple way for interpolating the physical variables from the fluid particles to the ghost particles was proposed by Adami et al. [65]. It has been proved to be a pragmatic and robust boundary condition even for three-dimensional problems [31]. The key idea of this method is to arrange enough dummy (ghost) particles inside the boundary area to complete the missing support of the kernel function. The dummy particle is deemed to be the fluid particle participating in the calculation of the continuity equation (Eq. (11)) and the momentum equation (Eq. (13)), which means it will exert a force upon the fluid particles nearby and prevent the fluid particles from penetrating the boundary. For the dummy boundary, the pressure information of the dummy particles (denoted by the subscript w) is obtained through the Shepard interpolation from the neighboring fluid particles (denoted by the subscript f) as [65]:

(13)

vij· rij < 0 vij· rij > 0

r j , h ) mj

2.3. Algorithm for the fluid–body interaction

where the artificial viscosity term Πij is given by: ij

N W (ri j=1 N W (ri j=1

As the concerned problems in this work are characterized by a short duration, the problem related to the non-physical diffusions of the free surface particle is not very relevant. In this way, a good balance between the algorithmic simplicity, numerical accuracy and computational costs can be obtained.

(12)

where vij = vi vj . In order to reduce the errors arising from the particle inconsistency problem [43], Eq. (11) and Eq. (12) are written in an anti-symmetrized form and a symmetrized form respectively. As pointed out in [45], for the impact problem, the fluid viscosity leads to a negligible effect to the final result. Thus, the laminar viscosity term in the momentum equation can be replaced by an artificial viscosity to alleviate the particle disorder and therefore the unphysical oscillation is reduced. In the present work, an artificial viscosity term proposed by Monaghan [46] is adopted. Then, Eq. (12) can be rewritten as:

dvi = dt

=

(14)

where α is a constant coefficient which has been set to be 0.05 in this paper. νij equals to vij·rij /(rij2 + 2 ) where η2 = 0.01h2 and rij = ri − rj. 2.2.2. Suppressing the pressure noise The pressure noise in the classic weakly compressible SPH is the main problem reducing its accuracy in the evaluation of the pressure load. There are several numerical techniques that have been published to tackle this problem. The main purpose of the smoothing activities is to remove the unphysical noise in the pressure, making the flow filed more stable. The basic idea of these activities is a kind of numerical diffusion, which has a small influence on the fluid–body interactions [47–49]. A common scheme to reduce the spurious pressure oscillation is to apply a density filter [50]. For most applications see e.g. [47,50], the Shepard kernel interpolation has been applied. However, it may encounter problems when applied to a longtime simulation (e.g., the water wave propagation) leading to the non-physical diffusion of the free surface particles [51]. Another feasible way to smooth the density/ pressure field is called the Rusanov flux [52]. This method has been improved and adopted in some published articles [53,54], where good results especially for the pressure evaluation were obtained. However, similar to the density filtering technique, the Rusanov flux introduces the non-physical energy flux across the free-surface because of the kernel truncation across this boundary, which confines its application in some longtime simulations [30,40]. The well-known δ-SPH scheme was proposed in [30,40] where the problem related to the free surface is avoided. In this model, a δ-term is added into the continuity equation

pw =

f

pf Wwf +

f

f f

(g

Wwf

a w )·rwf Wwf (16)

where the gravity acceleration g and the local acceleration on the dummy particle a w (the acceleration due to the body motion under external forces) have been both considered, see more in Marrone et al. [48] and Adami et al. [65]. Then, based on the equation of state (Eq. (3)), the density and the mass of the dummy particles can be calculated as follows:

= pw /c02 + m w = w V0 w

0

(17)

where V0 represents the volume of the the dummy particles. In twodimensional cases, V0 equals Δx2 and for three-dimensional cases, it equals Δx3.

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Fig. 1. The earth-fixed system and the body-fixed system.

In the present SPH scheme, the acceleration of the body is obtained based on the calculation of the real-time flow field pressure, see the right side of Eq. (18). In Eq. (18), the expression on the red side indicates the fluid inertia force exerted on the body. The position of the body is then updated based on the external force including the fluid inertia force and the gravity force. Based on the new position of the body, we can calculate the fluid particle acceleration once again. This procedure is justified by the use of a very small time-step required by the weakly compressible assumption. Detailed discussion can be found in [30] where a complete algorithm able to compute fully coupled viscous fluid–solid interactions is described. To describe the motion of the rigid bodies, the earth-fixed system (o-xyz) and the body-fixed system (ob-xbybzb) are both defined (Fig. 1). In the earth-fixed system, the total force Ffluid–body and the moment Tfluid–body exerted on the rigid body are firstly calculated as follows [62]:

F fluid–body =

f fluid

w body

mf m w (

f fluid pf

w body

mf m w (

T fluid–body =

× [(

2 f

+

pw

+

2 w

fw )

f

pf

+

2 f

pw 2 w

r f + rw

+

fw )

f

W (rf

Fig. 2. The coupling algorithm between the SPH scheme and the rigid body motion.

rw, h)

ro )

2

W (r f

rw, h)] (18)

Then, for the total force and the moment, the transformation from the earth-fixed system to the body-fixed system can be obtained as [66]:

Fbfluid–body

fluid–body ( ) F fluid–body T

=

Tbfluid–body

y

( )= A1 =

cos

A2 = cos B1 = sin B2 =

sin

x x

x

cos

(19)

y

sin

sin

z

x

cos

y

B1

B2

sin

x

cos

y

z z

cos

+ sin

+ sin

+ cos z

x x

x

+ cos

sin

sin sin x

y y

y

sin

cos

sin

z z

y

sin

z

by)

= Fbx fluid–body

M (Uby

Ubz

bx

+ Ubx

bz )

= Fby fluid–body

M (Ubz

Ubx

by

+ Uby

bx )

= Fbz fluid–body

Ix

bx

(Iy

Iz )

by

bz

fluid–body = Tbx

Iy

by

(Iz

Ix )

bz

bx

fluid–body = Tby

Iz

bz

(Ix

Iy )

bx

by

= Tbzfluid–body

z

(21)

1 sin

x

tan

cos x = 0 0 sin x /cos

y

cos

tan

y

y

sin x cos x /cos

y

x

bx by bz

(22)

Note that the term cosφy in Eq. (22) cannot be 0, otherwise the calculation will be terminated because of the singularity. But in the present paper, the parameter φy refers to the pitch angle of the body and it is below 90° for all the cases during the simulation, thus the singularities are avoided. For the larger pitch angle case, as pointed out in [66], the quaternion-based approach should be applied to obtain numerical stability. In the SPH scheme, the rigid body is also discretized into the individual particles. Moreover, the motion of these particles can be

z

cos

+ Ubz

y

sin

z

bz

x

y

A2

cos

x

cos

A1 sin

sin

z

Uby

where M is the total mass of the rigid body, Ix, Iy and Iz the moments of inertia with respect to the gravity center of the rigid body in the three main axes of the body-fixed system, Ubx, Uby and Ubz the translational velocities on the body centroid and Ωbx, Ωby and Ωbz the rotational velocities. Note that as the rigid body performs 6-DOF (Degree of Freedom) motions with respect to the body-fixed system, the moments of inertia are constant all the time. The dots above the variables U and Ω refer to the time derivative operation. According to the rotational velocities, the Euler angles φx, φy and φz are updated as follows [66]:

where φ = (φx, φy, φz) refer to the Euler angles between the earth-fixed system and the body-fixed system. The transformation matrix, Γ(φ), namely the Rodrigues formula [67] is given by:

cos

M (Ubx

(20)

To update the position of the rigid body, the linear and the angular acceleration of the rigid body should be calculated as:

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rwox rwoy = rwoz

decomposed into two parts, i.e., translation and rotation. According to the principle of rigid body kinematics, the equation of the motion can be written as [68]:

abw =

b

× (rbw

DUb D b + × (rbw Dt Dt

rbo) +

b

× (vbw

Ub)

w

w

1(

( )

vbw ) a bw

(24)

Rn + 1/2 = Rn +

t F(Rn) 2

(28)

where R is the variable calculated in the numerical simulation. F is a computational function evaluating the time derivative of R . These values are then corrected in the next step:

Rn + 1/2 = Rn +

(25)

t F(Rn + 1/2) 2

(29)

At last, they are recalculated at the end of the time step:

where Γ−1 refers to the inversion of the matrix Γ. The velocity vw is used to calculate the velocity divergence in Eq. (11) in Eq. (14), and the acceleration a w is substituted into Eq. (16) to calculate the pressure of the moving body particles. To move the body, we perform an algorithm [69] which can keep the shape of the moving body strictly unchanged. Specifically, the position rw is updated using a translational of the mass center to ro and then a rotation with the variation of the Euler angle at the end of each time step:

rw = ro + rwo

(27)

In the paper, a prediction-corrector scheme [70] is adopted due to its efficiency and accuracy. In the SPH scheme, the predicting step is firstly carried out as follows:

where vbw and abw are the velocity and the acceleration of each particle with respect to the body-fixed system. At last, for vbw and vaw , the transformation from the body-fixed system to the earth-fixed system is written as:

(av ) =

rwoz

2.4. Time integration scheme

(23)

rbo )

rwox ) rwoy

where rwo refers to the local vector at the last time step t′ and δ = (t − t′) Substituting Eq. (27) into Eq. (26), the position of the rigid body particles can be updated. In summary, Fig. 2 presents the coupling process between the SPH scheme and the rigid body motion. In the earth-fixed system, the processes A1, A2 and A3 are accomplished using Eqs. (16), (18) and (26) respectively. Then, the total force and the moment are transformed from the earth-fixed system to the body-fixed system based on Eq. (19). In the body-fixed system, the 6-DOF rigid body motion equations are solved using Eq. (21), and the calculations of the velocity and acceleration of each body particles are based on Eqs. (23) and (24). At last, some movement information of the rigid body including the Euler angles variation is transformed from the body-fixed system to the earthfixed body. Specifically, the process B2 is accomplished using Eq. (25) and the process B3 corresponds to Eq. (22).

Fig. 3. Sketch of the 2-D wedge water entry.

vbw = Ub +

(

Rn + 1

= 2Rn + 1/2

(30)

Rn

For the stability purpose, the time step Δt must satisfy the CFL condition [71]:

t

CFL min

h ; cmax + |vmax |

h |a max |

(31)

where CFL = 0.25, vmax and amax refer to the maximum velocity and acceleration of both the fluid and rigid particles. Note the time steps could be small when the artificial sound speed is taken much higher than the fluid velocities. But this is critical for the present SPH model since the weakly compressible hypothesis should be satisfied. The Mach number can be maintained below 0.1 when the sound speed is ten times as the maximum fluid velocity so that the behavior of the artificial

(26)

where rwo = rw ro is a local vector pointing from the mass center o to the body particle w and it only changes when the body has a rotational motion. The local vector rwo at the time t is calculated also according to the Rodrigues formula [67] as follow:

Fig. 4. Snapshots of the evolution of the 2-D wedge water entry in the water tank. 374

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compressible fluid is sufficiently close the real fluid [43]. 3. Numerical validation 3.1. The two-dimensional wedge water entry The first numerical validation is the water entry of a wedge tested by Zhao et al. [72]. As shown in Fig. 3, the width of the wedge is D = 0.5 m and the deadrise angle is 30°. In the experiment, a free falling rig is used to drop the wedge, and the total weight of the drop rig is 241 kg. The vertical velocity of the wedge is measured through an optical sensor, and the contact force between solid and fluid is measured using two force transducers connected to the drop rig. In the numerical simulation, a two-dimensional SPH model is used to reduce the computation complexity. The width of the water tank is 3 m and the depth is 1.5 m. The density of the water is 1000 kg/m3. As shown in Fig. 3, a sponge layer proposed by Gong et al. [49] is implemented inside the fluid domain to eliminate the affection of the wave reflected from the boundary particles. For the fluid particle within the sponge layer, the time derivatives of the density and the velocity should be altered to:

Fig. 5. Time histories of the vertical contact forces during the impacting process for the different particle resolutions.

d i dt dvi dt

(

= 1

100

0.950

)

d i dt dvi dt

(32)

where λ is defined as λ = (s − d)/s, s refers to the thickness of the sponge layer and d the distance between the fluid particles i and the solid boundary wall. The wedge is initially placed above the free surface with a vertical velocity of 6.15 m/s. The initial particle spacing ranges from 0.002D to 0.008D in order to investigate the convergence of the algorithm. The pressure field images of the fluid domain at the different times are shown in Fig. 4. It can be clearly seen that the high-pressure zone is generated when the wedge penetrates the free surface at t = 15.2 ms, and the water jets have been formed symmetrically and separated from the inclined wall. By t = 42.1 ms, as the wedge moves down, the smooth water jets are fully developed and the pressure is spatially noise free. The sensitivity of the vertical contact force to the particle resolution is presented in Fig. 5. It can be seen that the numerical results converge well as the particle spacing decreases. The numerical results at the particle spacing 0.002D are compared to the experimental data given by Zhao et al. [72] as shown in Fig. 6. The discrepancy between the numerical results and the experimental data could be caused by several factors. The most important one is the three-dimension effect. In the numerical simulations, the three-dimensional wedge water entry in

Fig. 6. Time histories of the vertical contact forces during the impacting process compared with the experimental data in [72].

Fig. 7. Sketch of the wedge and the positions of the pressure transducers, refering to [73] (d = 50 mm).

Fig. 8. Time history of pressures at different pressure transducers compared with the experimental data in [73]. 375

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Fig. 9. Sketch of the 3-D skipping stones.

the experiment is simplified by solving a two-dimensional problem. Thus, the influence of the cross section on both sides of the wedge is ignored, which may bring errors especially when the wedge is fully submerged after 0.015 s as shown in Fig. 6. For the applications in the ditching problem, a validation of the code in terms of the pressure distribution and its time histories is also necessary. The pressures in this part are compared with the

experimental data in [73]. The sketch of the wedge and the positions of pressure transducers are shown in Fig. 7. Different from the previous one, the deadrise angle of the wedge in [73] is smaller which is 25° and the width is 1.2 m. The mass of the wedge is 94 kg. In the numerical simulation, the width and the depth of the water domain are 2 m and 1.5 m respectively. The slamming velocity of the body is 5.05 m/s. The initial particle spacing of SPH particles is set to be 0.00125 m. Fig. 8

Fig. 10. Images from the SPH simulation of the skipping stone with v = 3.5 m/s, α = 20°, β = 20°, ω = 65 rot/s. (The two-dimensional views of the numerical results are shown in the left column and the profiles of the experimental results [79] are denoted by brown dots; the three-dimensional views of the velocity fields images are shown in the right column.) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 376

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Fig. 11. Images from the SPH simulation of the skipping stone with v = 3.5 m/s, α = 35°, β = 20°, ω = 0 rot/s. (The two-dimensional views of the numerical results are shown in the left column and the profiles of the experimental results [79] are denoted by brown dots; the three-dimensional views of the velocity field images are shown in the right column.) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. The motion trail of the head of the plate compared with the experimental data extracted from [79].

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Fig. 13. The hydrodynamic lift force on the plate compared with the theoretical results obtained from [79].

3.2. The three-dimensional stone skipping In this section, we perform a three-dimensional simulation of the disk-water impact. Relevant experiments were firstly carried out by Clanet et al. [74]. In the experiment, the circular disk obliquely impacts on a fluid surface with the different incident angles β, attack angles α, translational velocities v and angular velocities ω as shown in Fig. 9. The radius of the circular stone is 2.5 cm, and the thickness is 2.75 mm. The density of the circular stone is about 2700 kg/m3. The experiments show that the projectile may ricochet off the water surface in some given conditions. Furthermore, a “magic angle” about 20° between the plate and the water surface is found which can minimize the lowest velocity for a bounce velocity. Anyway, it is observed in the experiments that the motion of the projectile is changeable and complex. The comparison of the numerical results with the experiment in terms of the motion trail for the stone skipping problems can help us validate the feasibility of the fluid-body interaction algorithm established in the paper. In fact, this problem has been simulated through the SPH

Fig. 14. The geometric model of the WIG ship.

shows the time history of the pressure at the different pressure transducers. It can be seen from the figure that with the increase of the height along the inclined wall, the peak value of the pressure decreases gradually. In general, the present results are in good agreement with the experimental data in [73], which validates the accuracy of the numerical model adopted in the paper.

Fig. 15. The discretization of the WIG ship model.

Fig. 16. Sketch of the ditching of the WIG ship.

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Fig. 17. Velocity field images in the three-dimensional view during the ditching of the WIG ship with u = 40 m/s, v = 5 m/s, α = 7°.

method by some scholars previously [75,76], where some valuable results have been obtained. In this paper, we simulate the motion of the circular disk bouncing from the free surface in three dimensions and with a higher resolution. Note that for the sake of observation and comparison, the SPH results will be presented by interpolating the particle data to Cartesian meshes [77]. In the numerical simulation, the body enters the fluid with the different attack angles and angular velocities. The SPH result with the parameters of v = 3.5 m/s, α = 20°, β = 20°, ω = 65 rot/s is shown in Fig. 10. It can be observed that the attack angle basically remains unchanged during the whole impacting process, and the stone ricochets off the water surface under this condition. Observing the three-dimensional views of the velocity field images in Fig. 10, the plate pushes fluid both sideways and upwards with a large velocity. Finally, a fluid cavity is generated behind the plate. In another condition with v = 3.5 m/s, α = 35°, β = 20°, ω = 0 rot/s, the effect of spin is investigated as shown in Fig. 11. It can be seen that without any rotation, the plate tumbles after impact and dives into the fluid domain. Thus, the stone spin velocity ω has a substantial influence on the stone skipping. According to [78], the spin movement can stabilize the stone during the impacting process through the Gyroscopic Effect. However, the large solid body in real engineering problems is not likely to have such a big spin velocity. Thus, the effect of the spin velocity will not be mentioned later. In conclusion, for the above two conditions, as shown in Fig. 12, the computational results match with the experimental data well, so the effectiveness of the numerical model used in the paper is verified. According to [79], the hydrodynamic lift force on the plate during the impact process can be approximated as:

FL =

1 2

wU

2S

wetted

sin( + ) b

impact process. In the following sections, the effect of the factors on the ditching of the WIG ship will be selectively discussed. As shown in Fig. 13, the SPH results of the hydrodynamics lift force on the plate are compared with the theoretical value [79]. The numerical results coincide with the theoretical solution satisfactorily, which proves the reliability of this method. 4. Results and discussion 4.1. Ditching of the WIG ship In this section, the ditching of the WIG ship is simulated using the present SPH model. We mainly focus on the impact force and the motion trail of the WIG ship during the ditching. As shown in Fig. 14, the WIG ship configuration presented in the paper is based on the DXF100 ground effect vehicle, which is mainly used for business operations. The model consists of the hull, the wing, the pontoon and the empennage, whose mass are 11,231 kg, 2254 kg, 3524 kg and 2436 kg respectively. The lengths of the hull in z and x directions are 13.3 m and 2.28 m respectively. The lengths of the wing in z and x directions are 4.30 m and 4.02 m respectively. The lengths of the empennage in z and x directions are 6.54 m and 1.37 m respectively. The length of the pontoon in x direction is 6 m. Since the dummy particle boundary [65] is applied in the paper, the boundary of the WIG ship should be discretized into at least three layers of dummy particles. We first build a complete geometric model in the reak scale and then mesh it using the hexahedron element with the edge equals to the particle spacing of the fluid. Lastly, the coordinate information of the grid node is extracted for the SPH simulation. The entire process is shown in Fig. 15, where the WIG ship model is discretized into 54,261 particles. In the numerical simulation, as shown in Fig. 16, the WIG ship is initially positioned above the free surface, then launched into the water with the horizontal and vertical velocities denoted by u and v and in the

(33)

where Swetted is the plate area immersed in the fluid domain. It can be seen that the lift forces depend on more than one factor during the 379

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Fig. 18. Pressure field images in the top view during the ditching of the WIG ship with u = 40 m/s, v = 5 m/s, α = 7°.

angle of attack denoted by α. The lengths of the water domain in x, y and z directions are 100 m, 5 m and 30 m respectively. The particle spacing is set to be 0.1 m, and the whole fluid domain is discretized into 14 million particles. The gravitational acceleration is 9.81 m/s2. The reference density of the water is 1000 kg/m3. An OpenMP parallel computing with 50 cores has been carried out for this problem. The time duration of the simulated ditching process is 2 s. Fig. 17 shows plots of the three-dimensional velocity field images during the ditching of the WIG ship with parameters as u = 40 m/s, v = 5 m/s, α = 7°. Note that the SPH results are demonstrated by interpolating the fluid quantities from particles to Cartesian grids and after that, and the free surface deformation can be extracted and easier for observation. At t = 0.1 s, the hull of the WIG ship impacts on the water firstly, and generates a cavity on the free surface. By t = 0.7 s, the ship continues sliding on the free surface, pushing the water around the body both sideways and upwards. The stern wave and its crushing can be observed within the fluid domain behind the ship. Then as shown in Fig. 17(c), after sliding a short distance, the WIG ship is bounced up again. At t = 1.8 s, the ship entries water for the second time, and the waves generated during the first slide phase are getting calm. Up to now, since the ship moves in a large velocity, only the hull and the pontoons are in contact with the water, generating three cavities

respectively. The middle cavity is the most violent and complex one with subsequent wave breaking due to the high Froude number, and it interacts with the side cavities. The pressure fields in the top view during the first slide phase of the ditching are depicted in Fig. 18. At t = 0.1 s, the WIG ship impacts on the free surface, and a high-pressure area is generated on the tail of the hull. As time goes on, more parts of the hull including the pontoon start to contact with water by t = 0.27 s. Note that during this period, the high-pressure area moves from the tail of the WIG ship to its head. At t = 0.5 s, the high-pressure area is mainly concentrated in the middle part of the hull. Then, at t = 0.7 s seen from Fig. 18(d), only a small part of the hull and pontoon is still in contact with water. During the first slide phase, the position of the high-pressure area in the water is changing all the time, leading to the complex motion trail of the WIG ship as shown in Fig. 19. The WIG ship moves like a stone-skipping accompanied with the angular variation. As shown in Fig. 20, the WIG ship moves in a clockwise direction firstly and then in an anti-clockwise direction, which is related to the load distribution as mentioned before. During the free glide phase after the rebound of the WIG ship, the angle decreases rapidly. As shown in Fig. 21, during the first slide phase, the horizontal velocity of the WIG ship keeps decreasing until ship hull leaves water. At the free glide phase, the horizontal velocity of the WIG ship remains constant, while the vertical velocity decreases linearly

Fig. 19. The motion trail of the WIG ship during ditching. I, II and III refer to the first slide phase, the glide phase and the second slide phase respectively.

Fig. 20. Time history of the pitching angle of the WIG ship during ditching. 380

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under the gravity. The motion law of the WIG ship during the second sliding phase is similar to the first one. As for the acceleration, as shown in Fig. 22, it reaches the peak around t = 0.3 s for both the two directions, while the value in the vertical direction is slightly larger than it in the horizontal direction. And the acceleration peak in the first sliding phase is bigger than it in the second sliding phase. 4.2. Analysis of the fluid dynamics during the ditching During the ditching of the WIG ship, the water behind the body is characterized by a large Froude number. Despite the demonstration of the free surface in Fig. 17, the flow feature under the water surface has not been completely revealed. Thus in this section, a detailed study of the flow patterns behind the WIG ship has been performed at t = 0.7 s. The streamlines in the water are extracted and demonstrated in Fig. 23. Based on analyzing these streamlines in water, the flow patterns behind the WIG ship are characterized. The three-dimensional view of the SPH solution at t = 0.7 s is shown in Fig. 23 where the streamlines behind the WIG ship have been revealed. It can be clearly observed from Fig. 23 that the middle cavity generated by the hull has a relatively large velocity. In the domain A, the water in the center has a large kinetic energy, and it flows to the sides with less resistance. But in the domain B, as the velocity of the water in the center decreases, the water on both sides starts flowing back under the function of gravity. As mentioned above, there are two main flow patterns in the water behind the WIG ship, and they have a big influence on the shape of the wake flow.

Fig. 21. Time history of the velocity of the WIG ship during ditching.

4.3. Considering the effects of various parameters on the ditching In order to investigate the effect of the key factors on the ditching, two sets of conditions will be carried out in this section as follows: (1) Keeping α = 7°, v = 5 m/s unchanged, the horizontal velocities are set to be 30 m/s, 40 m/s, 50 m/s respectively. (2) Keeping α = 7°, u = 40 m/s unchanged, the vertical velocities are set to be 2.5 m/s, 5.0 m/s, 7.5 m/s respectively.

Fig. 22. Time history of the acceleration of the WIG ship during ditching.

The accelerations of the WIG ship in the vertical and horizontal directions during the ditching are presented as shown in Figs. 24–27,

Fig. 23. Three-dimensional view of streamlines in the water around the free surface.

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Fig. 24. The horizontal accelerations versus time for the different vertical velocities.

Fig. 26. The horizontal accelerations versus time for the different horizontal velocities.

and they are calculated according to:

where FH refers to the horizontal component of the contact force and FV the vertical component. Since the attack angle θ is usually small for the ditching, sin θ is much smaller than cos θ. Thus, the vertical component FV plays a more important role in the increase of the contact force F.

mj ajx

{ }

ax ay =

j solid

M mj ajy j solid

M

5. Conclusions

(34)

In the paper, a weakly compressible SPH model is adopted to tackle the hydrodynamic problem related to the ditching of the WIG ship. The fundamental theories of the SPH method are briefly introduced, and some effective numerical techniques have been applied for suppressing the pressure noise and improving the numerical stability. Furthermore, the 6-DOF equations of motion are incorporated into the SPH model to model the fluid–structure interaction in three dimensions. The simulations of the two-dimensional wedge water entry and the three-dimensional stone skipping are carried out to validate the feasibility and the accuracy of the present SPH model. After that, the ditching of a real-scale WIG ship under different velocities is simulated. The ditching characteristics of the ship are described and the fluid dynamics in the ditching area is analyzed. Finally, some useful conclusions are drawn. For the case with a large horizontal velocity (u > 40 m/s), the ditching involves different motion stages, like the

M indicates the total mass of the body. For all the cases, we only show the results in the first two seconds which can be divided roughly into two phases as shown in Fig. 24. In general, the acceleration peak in the first period is much larger than that in the second period. As shown in Figs. 24 and 25, with the increase of the vertical velocity of the WIG ship, the first peak value increases significantly while the second peak value change is not very noticeable. As shown in Figs. 26 and 27, the horizontal velocity has less influence on the impacting acceleration. Thus it can be seen that the maximum impacting acceleration of the WIG ship during the ditching mainly depends on the vertical velocity of the WIG ship rather than the horizontal velocity. As shown in Fig. 16, the contact force F exerted on the WIG ship can be decomposed into the horizontal and the vertical components:

F = FH sin

+ FV cos

(35)

Fig. 25. The vertical accelerations versus time for the different vertical velocities.

Fig. 27. The vertical accelerations versus time for the different horizontal velocities. 382

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slide stage and the glide stage, which are analogous to the stone skipping. When ditching with a certain angle of attack θ, the pitching angle of the WIG ship varies due to the variation of the hydrodynamic loading. The maximum acceleration of the WIG ship occurs during the first sliding stage and the acceleration increases with the increase of the vertical velocity significantly.

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