Available online at ScienceDirect. Procedia Engineering 126 (2015 )

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1 Available online at ScienceDirect Procedia Engineering 6 (05 ) th International Conference on Fluid Mechanics, ICFM7 Simulate wave body interaction based on the incompressible SPH method Xi-Peng Lv a, Xing Zheng a, *, Ni-Bo Zhang a, Kang-Ning Niu a a College of Shipbuilding Engineering, Harbin Engineering University, Harbin 5000, China Abstract The smoothed particle hydrodynamics method (SPH) has been proved to have an excellent ability in dealing with issues related to the wave. Compared with the SPH method, the incompressible smoothed particle hydrodynamics (ISPH) method has solved the pressure by solving the Poisson equation, rather than by solving the equation of state. The ISPH method presents a good approach to capture the free surface particle. With the respect of the SPH method, the ISPH method shows more stability and accurate of the numerical results than the traditional SPH method. This paper presents an ISPH method for two dimensional wave-body coupled problems, especially for the move structure. In most conventional ISPH techniques, the boundary is fixed; in this work, the move boundary is applied in the ISPH method. Then the model is applied to two problems: () interaction between a fixed rectangular structure and a liner wave; () two dimensional floating rectangular structure coupled with the nonlinear wave. The final results show agreement with the experiment data and phenomena. Crown 05 Copyright The Authors. 05 Published by by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM). Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM) Keywords: ISPH;moving boundary treatment; wave-body coupled problem. Introduction Marine structures are commonly used in ocean exploitation; the interactions between waves and floating bodies have been studied extensively in naval, ocean and coastal engineering. Violent pressures from wave impact can cause severe damage or collapse of coastal structures such as breakwaters, offshore oil platforms, and ships, especially when hurricanes and tsunamis occur. Usually, it is difficult to measure the impact pressure in natural violent waves, and to do fully reproduce real phenomena with laboratory measurements because of the scale effect of the physical models. Therefore, numerical studies of wave impact on coastal structures are widely carried out by * Corresponding author. Tel.: ; fax: address: zhengxing@hrbeu.edu.cn Crown Copyright 05 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of The Chinese Society of Theoretical and Applied Mechanics (CSTAM) doi:0.06/j.proeng

2 Xi-Peng Lv et al. / Procedia Engineering 6 ( 05 ) researchers and engineers either to understand the physical processes involved or to provide practical design parameters Most traditional numerical simulation techniques for moving bodies are grid-based, which encounter some difficulties when treating a large free surface deformation and a moving boundary. Longuet-Higgins and Cokelet [] introduced the Mixed Eulerian-Lagrangian (MEL) method based on BEM to deal with the instantaneous free surface and the wet body boundary. Jung [] simulated the coupled interaction between a wave and a free-rolling rectangular structure using FVM. In these models, moving structures in a Lagrangian formulation need to be appropriately described in a grid, and the special treatment of the moving boundary is inevitable..as a purely Lagrangian method, the Smoothed Particle Hydrodynamics (SPH) has been used in many areas of solid and fluid dynamics, Monaghan [3] utilized WCSPH to investigate the entry of a box traveling down a slope into water. Omidvar [4] enhanced SPH efficiency with the variable particle mass technique to model a 3D floating body motion. Bing Ren [5]using WCSPH method to analyse the wave-induced motions of a freely floating body. To improve the pressure stability and model performance, the incompressible SPH (ISPH) method was proposed. In the ISPH method, pressure is not a thermodynamic variable obtained from the equation of state. It is obtained by solving a pressure Poisson equation (PPE) derived from a semi-implicit algorithm. Therefore, both computational accurancy and stability have been improved for many documented ISPH applications. This paper presents an incompressible smoothed particle hydrodynamics (ISPH) method for wave-structure problem. Considering the floating body motion, the moving boundary is introduced to the ISPH model. Due to the support domain of the kernel function will be cut off when the particle is near the boundary. There are several different means to handle the boundary, like Shao [6] adopted the ghost particle to overcome the truncation by solid boundary, Lee et al.[7] introduced the uniform distribution ghost particle to handling the solid boundary. Due to the corner of the structure will have an over mirroing region, and then information of moving boundary is different from fixed,we sign up a new ISPH model which combine the boundary handling ways supposed by Lo and Shao [8] and combine the motion of the body with the boundary condition to calculate the wave-body coupled problem.. ISPH methodology.. ISPH In the method of incompressible SPH, the Lagrangian form of the Navier-Stokes equation is written as follows u 0 () Du p g u () Dt The computation of the ISPH method is composed of two basic steps. (a) Prediction step Computing forces by considering gravitational term and viscose term in Eq. (), it can get the temporary particle positions r * and velocities u* u u u (3) * t * u g u (4) * ( ) t r r u t (5) * t * where u t, r t is the velocity and position on t step, t is the time step, u * is the particle velocity change during the prediction step. (b) Correction step The velocity change during the correction step is u ** Pt t (6) where is density during the prediction step, pt is the pressure on t step. Velocity and position on t step is u t, r t, which can be defined as

3 66 Xi-Peng Lv et al. / Procedia Engineering 6 ( 05 ) u u u (7) t * ** ut ut rt r t t (8) Combined Eq.() and Eq.(6), it can get the pressure Poisson equation, which can be shown as u* P t t (9) We can get the pressure value by solving the pressure Poisson equation... Moving solid boundary handling Different from the traditional SPH method, in this model, we follow the treatment used by Koshizuka et al. [9] arranging a real particles on the wall. And the wall particle can avoid the fluid particle move through the boundary, and it can be balanced the pressure of the fluid particle. So that in this ISPH model the body can be seen as consisted of some real particles. The pressure Poisson equation (is solved on these wall particles using a homogeneous Neumann boundary condition. This computationally robust procedure was initially established by Koshizuka in the MPS method and was revised in the ISPH approach. So that it is very easy to find the force of the body. After obtaining the external force and moment, the structure motion can be easily calculated by using Newton s second law as equations: du F g (0) dt M dw dt body M () I We can get the boundary total velocity includes translation and rotation velocity easily. Simultaneously, the mirror particles should be placed outside the fluid region to complete the support domain of the kernel function, so that we can calculate the density more accuracy, which is the same as the SPH method, and helped. As the boundary will have a corner such as the figure shows. We follow the ways supposed by Lo and Shao. The relationship between velocity of mirror particle and real particle are as equation shows: w n v (* v v ) n () j w i v v t (3) t j where vw is the boundary displacement velocity of the moving solid, and n and t correspond to the normal and tangential directions of the solid boundary. vi and v j are the velocity of the fluid particle and mirror particle. Equations () and (3) describe non-penetration and free-slip boundary conditions, respectively. i 3.Numerical tests 3.The interaction between fixed rectangular box and wave In this part, the parameters of wave and the objects are come from the Bing Ren. The numerical water pool is 0m long and.8 m high with water depth of. m. In this model The breadth of the box is 0.8m, and the height is 0.4 m. The rectangular box is fixed and semi-immersed, which centroid is located at point (4.6 m,. m). The model of this part are shown as Fig.

Xi-Peng Lv et al. / Procedia Engineering 6 ( 05 ) 660 664 663 Fig. the sketch of the fixed rectangular in the pool The motion of the wave maker obtain Eq.(4), and starting at the point x=0.

And an artificial damping layer is arranged at the downstream boundary of the pool to prevent the reflection waves When the H=0.06m and T=.s,The relative wave height depth is 0.

So we can compare the results calculate by ISPH method with the analytical solution using the frequency domain method obtained by Mei and Black [0].

4 Xi-Peng Lv et al. / Procedia Engineering 6 ( 05 ) Fig. the sketch of the fixed rectangular in the pool The motion of the wave maker obtain Eq.(4), and starting at the point x=0.m,s is the distance of the wave maker. H is the height of the wave.and T is the periodic of the wave. And an artificial damping layer is arranged at the downstream boundary of the pool to prevent the reflection waves When the H=0.06m and T=.s,The relative wave height depth is 0.54 and the relative wave height H/L=0.07, we can consider the incoming wave as a linear wave. So we can compare the results calculate by ISPH method with the analytical solution using the frequency domain method obtained by Mei and Black [0]. We choose the horizontal force of the rectangular box and the wave elevation at the position x=4.6m while the box is not placed in the pool to compare with the analytical solution. The pool in ISPH method is consisted of 95 fluid particles, and 500 boundary particle. (4) Fig. the result of the wave simulation Fig.3 the wave elevation of x=4.6m Fig.4 the result of fixed rectangular interaction with wave Fig.5 the horizontal force of the fixed rectangular In this case the wave can be seen as a liner wave, the computed wave elevation is almost identical with the analytical solution, and the horizontal force have a good coincidence with the analytical solution from the Fig., Fig.3, Fig.4 and Fig.5. It means the wave conducted by ISPH method is reliable. And the pressure on the boundary is accurate. 3. Free floating box in waves In this section, refers to the experiment about the floating body coupled with waves which has been done by Bing Ren in Dalian University of Technology, the depth of the numerical pool changed to 0.4m to research the free floating box coupled with the regular wave. The breadth of the box is 0.3m and the length of the box is 0.m.The initial position of the centroid of the box is (.0m, 0.4m) and the box can have sway, heave and roll. In this case, the H is 0.04m, T is still.s. The figure.6 indicates that the position and the wave free surface computed by ISPH method are in good agreement with the experiment results.

664 Xi-Peng Lv et al. / Procedia Engineering 6 ( 05 ) 660 664 Fig.

5 664 Xi-Peng Lv et al. / Procedia Engineering 6 ( 05 ) Fig.6 The position of the floating box and the free surface calculated by ISPH (down figure),compare with experimental photos (up), (a)t=t 0 +T/4, (b)t=t 0 +T/, (c)t=t 0 +3T/4, (d)t=t 0 +T. 4.Conclusions This paper applied the ISPH method in to the wave-body coupled problem. In this ISPH model, we can simulate the floating body motion and the free surface of wave very well. Acknowledgements This work is sponsored by The National Natural Science Funds of China( , 57904)Foundational Research Funds for the central Universities (HEUCDZ0HEUCF03) and Defense Pre Research Funds program (940A4007CB058), to which the authors are most grateful. References []J.H. Jung, H.S. Yoon, H.H. Chun, I. Lee, H. Park Numerical simulation of wave interacting with a free rolling body Int J Naval Archit Ocean Eng, (03) []M.S. Longuet-Higgins, E.D. Cokelet The deformation of steep surface waves on water: I. A numerical method of computation Proc R Soc A Math Phys Eng Sci, 350 (976) 6 [3] Monaghan JJ, Kos A, Issa N. Fluid motion generated by impact.j Water w Port Coast Ocean Eng (003);9:50 9. [4]Omidvar P, Stansby PK, Rogers BD.SPH for 3D floating bodies using variable mass particle distribution. Int J Numer Methods Fluids; 7 (03)47 5 [5] Bing Ren,Ming He. Nonlinear simulations of wave-induced motions of a freely floating body using WCSPH method. Applied Ocean Research, 50(05) - [6]Sonfdong Shao. Incompressible SPH method for simulating Newtonian and non-newtonian flows with a free surface. Advances in Water Resources.(003) [7] Lee et al., E.S. Lee, C. Moulinec, R. Xu, D. Violeau, D. Laurence, P. Stansby Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. Journal of Computational Physics, 7 (008), [8] Lo EYM, Shao SD. Simulation of near-shore solitary wave mechanics by an incompressible SPH method. Appl Ocean Res 00;4: [9] Koshizuka S, Nobe A, Oka Y. Numerical analysis of breaking waves using the moving particle semi-implicit method. Int J Numer Meth Fluids, 6 (998) [0] C.C. Mei, J.L. Black Scattering of surface waves by rectangular obstacles in waters of finite depth J Fluid Mech, 38 (969) 499 5

Experimental Validation of the Computation Method for Strongly Nonlinear Wave-Body Interactions

Experimental Validation of the Computation Method for Strongly Nonlinear Wave-Body Interactions by Changhong HU and Masashi KASHIWAGI Research Institute for Applied Mechanics, Kyushu University Kasuga