Shock and Vibration

Volume 2018, Article ID 7890892, 16 pages

https://doi.org/10.1155/2018/7890892

## Bow Flare Water Entry Impact Prediction and Simulation Based on Moving Particle Semi-Implicit Turbulence Method

College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Jingzheng Yao; moc.361@ueh_gnehzgnijoay

Received 4 December 2017; Revised 19 January 2018; Accepted 3 April 2018; Published 21 May 2018

Academic Editor: Mickaël Lallart

Copyright © 2018 Fenglei Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Prandtl’s mixing length method and the -epsilon method are introduced into the Moving Particle Semi-Implicit (MPS) method for the purpose of modeling turbulence effects associated with water entries of two-dimensional (2D) bow flare section. The presented numerical method is validated by comparing its numerical prediction with experimental data and other numerical results obtained from the Boundary Element Method (BEM). The time histories of the pressure and the vertical slamming force acting on the dropping ship section subjected to various conditions with different dropping velocity and inclined angles are analyzed. The results show that both the pressure and the vertical slamming force are in good agreement with the experimental data.

#### 1. Introduction

For container ships the bow flare slamming can easily happen in rough seas; the slamming impact may lead to a series of instantaneous vibration and structure damage.

The slamming calculation theory is firstly proposed by Von Karman [1] and Wagner [2], which is mainly used to solve the 2D water entry problem. Wagner’s theory is focused on flat plate water entry problem; based on the ideal incompressible fluid potential flow theory the linearized Bernoulli equation is used to calculate the pressure. Based on 2D vee-wedge theory, Bisplinghoff and Doherty [3] derived the unsteady state force using the flow pattern about an expanding prism. Zhao and Faltinsen [4] improved Wagner’s theory to solve symmetric wedge water entry problem not considering the fluid separation; the body surface and nonlinear free surface conditions are simulated by numerical analysis; by this way the wedges of dead rise angle from 4° to 81° water entry problems are computed. In order to solve arbitrary shape water entry problem, Zhao et al. [5] proposed Boundary Element Method (BEM) to realize the practical numerical analysis; the slamming pressure is calculated by solving the nonlinear Bernoulli equation. This method can satisfy the moving boundary condition accurately when calculating the velocity potential.

When the dead rise angle is large, the original Wagner method overestimates the slamming pressure; in order to solve this problem, Korobkin [6] modified Logvinovich model (MLM), and the body boundary surface condition is solved by the Taylor expansion term of the velocity potential of the original Wagner model. Further, Korobkin [6] and Tassin et al. [7] expanded MLM to 3D calculation method to solve slamming problem.

On aspect of experiments, Marintek Sintef Group used a 30° dead rise angle wedge to accomplish the water entry impact experiment; the slamming impact is recorded in time domain; the results show that the pressure reached the maximum value before the flow separation, and after the flow separation the pressure rapidly attenuated at the separation point. Chuang [8] studied the air cushion effect from 1° to 15° wedge; when the maximum pressure appeared at the oblique elevation angle equal to 3°, the air between the wedge and the surface of the water more easily escapes than the smaller inclined angle. Aarsnes (1996) designed a bow flare section water entry experiment; four pressure sensors and two force sensors were installed on the model; the section dropped at different heights and inclined to simulate the real slamming phenomenon; the pressures and the vertical slamming forces were recorded in time domain. In this paper the numerical results will be compared with this experiment data.

In this paper the main numerical method is Moving Particle Semi-Implicit method, which is a meshless method assuming the fluid as particles. Meshless method is a new numerical solution; the representative method is Smoothed Particle Hydrodynamic Method, which is firstly proposed by Lucy [9]. Nowadays the typical meshless method includes many different realization solutions. Diffuse Element Method (DEM) is proposed by Nayroles et al. [10], which is a method for function approximation, by which the partial differential equations and fluid dynamic problems can be solved. Belytschko et al. (1994) used an element-free Galerkin method to solve elasticity and heat conduction problems, in which moving least-squares interpolants are used to construct the trial and test functions for the variation principle, and the moving least-squares interpolants and the choices of the weight functions are also discussed in detail. Batina [11] applied clouds of points in CFD algorithms, which are not required to be connected. This method is named Hp-clouds gridless CFD method, which offers great potential for solving viscous flows about complex fluid calculations. Oñate et al. [12] proposed a procedure named “Finite Point Method” to solve convection-diffusion and fluid flow problem. Based on reproducing kernel and wavelet analysis a multiple scale method was developed by Liu and Chen [13], which can be used to simulate the wave numbers corresponding to spatial variables or the frequencies corresponding to the temporal variables. The Radial Basis Function (RBF) collocation method is a numerical approach for the solution of partial differential equations (PDE), in which no mesh is required. To transform the Finite Element Method to obtain a method which satisfies the meshless definitions, Idelsohn et al. [14] proposed Meshless Finite Element Method (MFEM). Hao et al. [15] introduced a shape function and derivatives to reproduce any order of monomial based on particle method, the approximation is constructed at individual discrete points where an approximation is desired, and this method is Moving Particle Finite Element Method (MPFEM). Zhu et al. [16] combined boundary integral equation method and moving least-squares approximation to form a direct meshless boundary integral equation method.

Moving Particle Semi-Implicit Method is firstly presented by Koshizuka [17] to calculate incompressible fluid flow problem, which is a meshless numerical method based on Lagrange particles. Further, Koshizuka and Oka [17, 18] applied this method to simulate the dam breaking flow, and the results were compared with the experiments; in their research the kernel functions and the particle distance were discussed through comparison. Yoon [19] modified MPS method to combine the Lagrange particles with Euler particles; the typical advantage is that the arrangement is not necessary to be uniform; this is a meshless advection using flow-direction local-grid method (MAFL). Nowadays, MPS method is widely applied in different fields. Gotoh and Sakai [20] applied MPS method to simulate wave propagation problem; the breaking waves after flowing across the vertical wall were simulated. Chikazawa et al. [21] applied particle interaction models for differential operators to simulate the interaction between fluid and elastic plastic structures based on MPS method. Khayyer and Gotoh [22] proposed a corrected MPS method to track water surface in breaking waves; the formulations were revisited from the view of momentum conservation; modifications and corrections were made to ensure the momentum conservation in a particle-based calculation of viscous incompressible free surface flows. Zhang and Wan [23] investigated the roll motion of a two-dimensional floating body in low-amplitude regular waves through applying an improved MPS method; the efficiency and the accuracy in solving the interaction between the floating body and regular waves are improved. K. S. Kim and M.-H. Kim [24] extended MPS method to a multiphase system with multiple interfaces to solve the multifluid multi-interface problems; the results are compared with corresponding experiment results with three liquids. Xiang and Chen [25] proposed an interparticle force stabilization and consistency restoring MPS method to overcome the compressive instability that occurred under compressive stress states. A new and simple free surface detection criterion is proposed by Wang et al. [26] to enhance the free surface recognition in the MPS method. Sun et al. [27] used a weak coupling between MPS and BEM method to improve the computational efficiency for wave structure interaction simulation; better accuracy is obtained compared with published results. A multiphase MPS method is presented by Nabian and Farhadi [28] to develop a straightforward, robust, stable, and accurate mesh-free numerical technique for modeling the dynamic behavior of free surface, incompressible, multiphase granular flows. Wada et al. [29] proposed a distributed point source method (DPSM) and the least square Moving Particle Semi-Implicit (LS-MPS) method to handle the phenomenon including fluid dynamics within the droplet, and the rotation of the droplet is successfully reproduced numerically and its acceleration is discussed and compared with those in the literature. Nath et al. [30] solved the time-dependent Navier-Stokes equations in stream function-vorticity form for lid-driven cavity flows using a meshless method based on fundamental and particular solution (MFS-MPS). Xu and Jin [31] in their study presented a local rheology model to calculate the effective viscosity and shear stress in granular flows in the numerical method; the final wave front in the simulations showed good agreement with the relationship obtained from experiments.

MPS method can be applied in water entry problems; Shibata et al. [32] calculated the acceleration of a free-fall lifeboat at water entry in three dimensions, in which the large deformation of the water surface, the splash of water at water entry, and the rigid-fluid interactions are predicted numerically. Alam et al. [33] used MPS method to solve the unsteady Navier-Stokes equation for incompressible fluid flows with and without the surface tension effect; equilateral prism-shaped object models were simulated to fall onto the free surface of the water; the hydrodynamic behaviors of water splash with and without the surface tension effect were presented to show the comparison and differences. Zhang et al. [34] presented an in-house solver based on MPS method to predict the process of free-falling wedge impacting on water; the numerical pressures, free surface elevations, and velocities of wedge show agreement with experimental data.

Turbulence models can be applied in meshless calculation method, based on eddy viscosity assumption and Langevin process. Violeau et al. [35] modified SPH method to prove the effective application in reproducing a large variety of flows. Shao and Gotoh [36] combined Large Eddy Simulation (LES) model with SPH method to simulate the coupled motion between progressive wave and floating curtain-wall type breakwater. Large Eddy Simulation (LES) filtering procedure was modified as a sort of LES Lagrangian filtering by Mascio et al. [37]; the closure formulas were derived for the additional terms to show the features to be used in SPH method. Dalrymple and Rogers [38] applied a two-dimensional LES model to wave propagation and interaction with coastal defense. Shao [39, 40] applied two-equation - model to deal with turbulence and vortices during wave breaking and overtopping and it is coupled with the incompressible SPH numerical scheme. In order to reproduce cnoidal wave breaking on a slope under two different breaking conditions, spilling and plunging, Shao [39, 40] coupled - model with the incompressible SPH method to realize the calculation; the results are in good agreement with the experiment data. Based on SPH method a subparticle-scale model is applied to account for the effect of turbulence, which is proposed by Kazemi et al. [41] to simulate depth-limited turbulent open channel flows over hydraulically rough beds.

The meshless method is now widely applied in computational fluid dynamics (CFD) to promise the complex free surface flow. However, the increasing flow complexity requires appropriate approaches for taking account of turbulent effects. The above recently developed turbulence models adapted to the SPH method are presented; each proposal is tested and validated on the basis of laboratory data. Theoretical or numerical solutions are available in the general field of turbulence free surface incompressible flows; they gave satisfactory results, but some progress should still be made in different methods in the future.

Compared with the other methods, in MPS method the fluid is separated into moving particles and diffusion terms, which are calculated during the motions. Therefore, the numerical diffusion problem in Eulerian methods does not take place. In this method, incompressibility is presented by keeping the density of particles constant during the computational time. The equations of continuity and momentum are converted into equations of interaction between particles in which all interactions are limited to a specific distance and weighing of interaction between two particles with distance of is determined based on kernel function. The movement of each particle due to the interaction of neighboring particles is calculated using kernel weighing function approximation. Consequently, Laplacian gradient and divergence operators can be changed to consider the effect of moving particles.

In this study, MPS method is expanded to investigate the bow flare section of container ship water entry prediction. The effect of turbulence is considered using two turbulence models of Prandtl’s mixing length method and - method. The modified method is based on the equations updated by Khayyer et al. [42] to provide more stability and accuracy; the results are compared with those obtained by BEM method and experiment data to verify the feasibility of this proposed method.

#### 2. Realization of MPS Method

##### 2.1. Governing Equations

As presented by Kenyon and Seeger [43], for a Newtonian incompressible viscous fluid, the continuity and momentum equations can be presented as follows:where is velocity vector, is time, is fluid density, is pressure, is kinematic viscosity, and is body force vector.

For MPS method, as proposed by Koshizuka [17], the simulation process consists of two steps: the first step is explicit location modification of particles, and after the first step the number density is changed. The second step is implicit modification; the particle number density is modified to the initial value, so as to maintain the fluid incompressible. According to the derivative principle (2) can be written aswhere and are the velocity vectors of and steps after modification, is the intermediate velocity between two time steps, and are defined as

After the velocity modification the particle number density is changed; in order to maintain the density invariable, (1) can be written aswhere and are the particle number densities of and steps after modification, is the intermediate particle number density between two time steps, and and are defined as

Substituting (5) and (3) into (1) and (2), in the first explicit modification, the fluid is considered as compressible.

Based on (7), the pressure can be obtained through the following equation:where is the particle number density after the implicit modification; is the initial particle number density.

The pressure obtained from (8) satisfies the incompressible condition, according to pressure value to modify the velocities and locations implicitly, the particle number density resumes the initial value, and the incompressible condition is satisfied.

According to velocity increment the location and velocity can be modified aswhere is the location of particles, when the particle number density is obtained, Poisson’s equation (8) can be solved, and the velocity is calculated implicitly; at last the velocity and location are modified again as follows:

Through implicit pressure Poisson equation, the pressure for each particle can be presented aswhere is the initial particle number density and is the dimension number, and means the intermediate time step:

##### 2.2. Reynolds-Averaged Numerical Turbulence Models

In order to apply turbulent model in the MPS method, the - model can be used in the Eulerian numerical calculation. According to Prandtl’s mixing length theory, the eddy viscosity is written as

For MPS method the above equation can be presented as

The - equation is deployed in the model, for each particle the kinetic energy can be defined as , and the energy dissipation rate is defined as ; the eddy viscosity is written aswhere is empirical constant.

The turbulence energy can be expressed based on Lagrangian transport equation

In scalar advection-diffusion equation (16), is kinetic energy of particle , which is an energy source term, and is energy dissipation of particle , which is a sink term. The Schmidt number is equal to the ratio between the viscosity and energy diffusivity.

In continuous formulation the pressure and velocity can be written as Reynolds-averaged form; the Navier-Stokes equation is written as

Based on Boussinesq assumption the Reynolds stress tensor takes the following form:where is eddy viscosity, , is the turbulent fluctuation velocity, and where letters and here denote spatial coordinates, and the production of kinetic energy , so can be written as

In this equation is the scalar mean rate of strain, in water entry problem the jet flow is difficult to simulate, in this situation is overestimated in case of large , the turbulence anisotropy is bounded by , and in MPS method for particle the production of kinetic energy is written as

In MPS method the rate of strain is presented as

The energy dissipation rate is written as MPS form:

The constant coefficients are valued as follows, given by Shakibaeinia and Jin [44]:

.

After the implicit calculation velocities are obtained, the location of the particles should be modified as follows:

The MPS realization considering the turbulence effect is presented in Figure 1.