Shock and Vibration

Volume 2019, Article ID 4610524, 13 pages

https://doi.org/10.1155/2019/4610524

## Grid-Free Modelling Based on the Finite Particle Method for Incompressible Viscous Flow Problems

Naval Architecture and Ocean Engineering College, Dalian Maritime University, Dalian 116026, China

Correspondence should be addressed to Xin Chang; moc.kooltuo@ueh_nixgnahc

Received 18 January 2019; Revised 24 March 2019; Accepted 15 April 2019; Published 8 May 2019

Academic Editor: Mohammad Rafiee

Copyright © 2019 Yu Lu 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

In this paper, we present a grid-free modelling based on the finite particle method for the numerical simulation of incompressible viscous flows. Numerical methods of interest are meshless Lagrangian finite point scheme by the application of the projection method for the incompressibility of the Navier–Stokes flow equations. The moving least squares method is introduced for approximating spatial derivatives in a meshless context. The pressure Poisson equation with Neumann boundary condition is solved by the finite particle method in which the fluid domain is discretized by a finite number of particles. Also, a continuous particle management has to be done to prevent particles from moving into configurations problematic for a numerical approximation. With the proposed finite particle technique, problems associated with the viscous free surface flow which contains the study on the liquid sloshing in tanks with low volumetric fluid type, solitary waves movement, and interaction with a vertical wall in numerical flume as well as the vortex patterns of the ship rolling damping are circumvented. These numerical models are investigated to validate the presented grid-free methodology. The results have revealed the efficiency and stability of the finite particle method which could be well handled with the incompressible viscous flow problems.

#### 1. Introduction

In the last several decades, various meshless techniques [1–6] have been developed as alternatives to traditional grid-based methods such as finite volume method [7, 8] and finite element method [9, 10] as well as boundary element method [11, 12], which have attracted plenty of research fellow for the potential interest and requirement. Compared to the traditional mesh generation method which is of consumption and difficulty and even hardly solving the thorny problem with the large deformation viscous flow or within complex geometry boundary, the grid-free modelling reflects superior ability to complete simulation and adapt it locally by constructing space arbitrary point distribution without using any mesh to connect them. This approach could well put an end to distortion occurrence in the numerical grid solution. And, the other significant consideration factor of computation effort, especially for the situation of sophisticated time-dependent geometries, the meshless method shall balance better accuracy and less time. Therefore, the satisfactory performance of the meshless or particle method has been exploited and demonstrated to be conveniently implemented and computationally efficient.

Amongst the numerous grid-free and particle methods, the longest established meshless method is smoothed particle hydrodynamics that was originally developed in the late 1970s [13, 14], which has been extensively studied, explored, and conducted in many fields. With its wide attention in the area of computational fluid dynamics, large numbers of applications have been implemented which range over free surface flows [15], viscous flows [16, 17], multiphase flows [18, 19], geophysical flows [20, 21], the coastal engineering [22], turbulence modelling [23], viscoelastic flows [24, 25], and free-surface viscoelastic flows [26]. Though there are several advantages of SPH over grid-based methods, such as its Lagrangian and adaptive nature for the issue on large deformation and complex inconstant flows or easy programming, this meshless method is still confronted with numerical shortcomings which involve inconsistency and instability as well as incompatible boundary conditions treatment. The approach for enforcing the incompressibility condition in governing flow equations, namely, the incompressible SPH (ISPH), has been explored [27, 28]. In addition, an enhanced ISPH-SPH coupled method has also been proposed and studied for the simulation of incompressible fluid-elastic structure interactions [29]. Also, a great many studies have been devoted to improve the performance of this particle method, and the related research progress can refer to the literature cites [30, 31].

As another Lagrangian meshless method, moving-particle semi-implicit method (MPS) has been adopted in solving the hydrodynamics phenomenon, where the computational domain is similarly discrete and reconstructed by particles without the use of mesh in traditional methods. Hereby, there exist a number of studies on the flow problems resolved by MPS [32, 33], especially for the violent free-surface flows accompanied by large deformations and fragmentations, and it appears to be an appropriate candidate due to the particle characteristics [34, 35]. However, the unphysical pressure fluctuations in numerical computation procedure have also bothered the development of this meshless method.

Keeping this in mind and with the purpose to gain some insight into the real capabilities of meshless approach, a technique known as the finite particle method [36] has been extensively brought into effect for the numerical solution of a wide range of problems in computational mechanics. And, similar meshless discretization technique such as the finite-volume particle method (FVPM) has been employed for convergence analysis for compressible as well as incompressible flows [37, 38]. In addition, the validations of the finite particle method have been conducted through classical benchmark tests with exact analytical solutions, in terms of accuracy as well as conservation properties [39, 40]. The truly meshless method presented in references [41–43] has been developed and implemented for the flow problems with complicated and rapidly changing geometry [44], free-surface flows [45], and multiphase flows [46]. In the finite particle method, particles which schlep the density and pressure as well as necessary physical characteristics move with fluid velocities. Similarly, boundaries are also approximated by a finite number of boundary particles, where the boundary conditions are also applied to these particles. Meanwhile, these points need not be regularly distributed and can even be quite arbitrary. Hence, the numerical treatment technique makes the finite particle method valid and efficient in the flow problems resolution.

In this thesis, we consider numerical simulation of incompressible viscous flow problems by the finite particle method. Numerical methods of interest are meshless Lagrangian particle methods. This approach is in some sense natural for fluid flows, since the governing equations themselves can be derived by considering small fluid particles which move with the flow. The incompressible Navier–Stokes equations are resolved by the projection method. This requires the solution of Poisson problems in each time step in which the spatial derivatives generated can be approximated by the moving least squares (MLS) method. As for the Neumann boundary condition, it is settled by placing stabilized finite points. In detail, the particle data management of the remeshing technique is adopted to circumvent the problems associated with irregular particle distribution. The proposed finite particle method is tested for three benchmark problems associated with incompressible viscous flows containing liquid sloshing of tanks with low volumetric fluid type, solitary waves movement, and interaction with a vertical wall in numerical flume and vortex patterns for the cross sections of ship rolling damping, in which the capability of finite particle method is demonstrated as a meshless method to solve the problems on incompressible viscous flow accurately and robustly.

The paper is organized as follows. In the next section, the theoretical principle of the finite particle method is expounded firstly. The application of the moving least squares (MLS) method for derivatives approximation is introduced in Section 2. We describe the explanation and discretization of the incompressible Navier–Stokes equations in Section 3. And, the particle data management as well as the resolution of time step has been explored in Section 4. Then, the numerical tests are presented in Section 5. The paper ends up with concluding remarks eventually.

#### 2. Derivatives Approximation by the Moving Least Squares (MLS) Method

##### 2.1. Moving Least Squares (MLS) Method

The moving least squares (MLS) method was developed as an approach to reconstruct the function for approximation in the combination form of polynomial basis, weighted with respect to the values of the scattered interpolation particles. Since the weights move through over the domain but with staid polynomial basis, this method is also designated by the name of moving weighted least squares (MWLS) method.

Firstly, let a function be given on a domain . Consider a point cloud . The function values are given at the data points by . The task is to approximate the function value at an arbitrary point , i.e., in the first place, we are confronted with a meshless interpolation problem.

*u* is assigned to be the function to be sought at control point *x* and its approximation function ; then, there iswhere the operator acts on a function and yields a new polynomial function which is constructed only from the values of at the neighbouring data points . denotes the coefficient vector, is the basis function vector, and is the number of basis functions. For the condition of 2D, the canonical basis is

While the canonical basis in 3D can be as follows:

The next step of the MLS method is to assign weights to the points via a distance weight function . The distance weight function is chosen smooth and decaying with increasing distance. The weight function used in this calculation takes the distance weight function and the specific expression asor splineswhere is a positive distance between any two particles in Euclidian norm, since it is rotationally invariant. The radius of the support domain *h*_{0} determines , the number of neighbouring particles around to be used for MLS approximation.

In the moving least squares (MLS) method, for the aim of minimum error square sum, a specific point is considered and the functionalshould be minimized, and the polynomial is defined by the coefficient vector , which minimizes the moving least squares distance at the data points . Since the weights are taken with respect to the point , for every point , one obtains a different polynomial and thus a different coefficient vector .

The coefficient vector is obtained by the normal equationwhere

Hence, for each point , a small linear system could be solved with matrix .

##### 2.2. Approximation of the Derivatives Scheme by MLS

In the process of the finite point method, the derivative term of velocity and pressure arises in the numerical scheme. The MLS method with distance weight function is implemented to approximate derivatives for the pressure Poisson equation. While taking the weight function, such as equations (4) and (5), the MLS approximation can be differentiated at data points. The resulting expression will be a linear combination of function values at the data points .

Approximating function equation (1) by MLS, we can find that the coefficient vector is the solution to the systemwhere the matrix isand the right hand side is

Note that both and depend on . Consequently, as well as the basis vector depends on . In the following, we omit the -dependence in the expressions. For equation (9), the coefficient vector is calculated as

First of all, we are going to take the first-order derivative of both sides of equation (9); that is,where the prime denotes a differential operator (such as ), and thus,Then, the second-order derivative of both sides of equation (12) is taken; it yieldswhere the double prime denotes the operator applied twice, and then,

According to equation (14), the first derivative approximation of the MLS function can be obtained as

Similarly, using equations (14) and (16), we obtain the MLS function’s second derivative approximation as

Therefore, the spatial derivatives could be approximated by the above approach described in this subsection. The final sparse linear algebraic equations for the unknown pressure values are solved by employing an iterative scheme known as the stabilized biconjugate gradient (BiCgStab) method [47].

#### 3. Discretization of the Incompressible Flow Governing Equations

##### 3.1. Incompressible Flow Governing Equations

In the Lagrangian form, the incompressible Navier–Stokes equations can be written aswhere is the Cartesian components of the velocity field, is the Cartesian components of the position vector, is the time, is the fluid density, is the pressure, is the kinematic viscosity and , is the dynamic viscosity, and is the source term (normally the gravity ). denotes the total or material time derivative of a function .

##### 3.2. Numerical Discretization by the Projection Method

The projection method which can be referred in reference [48] is employed in the finite particle method for the solution of equations (19) and (20) in time, which is completed in two fractional steps. The first step is to explicitly calculate the particle update position and the intermediate velocity from the momentum conservation equation (20):

The second step is to correct the intermediate speed and get the speed at the next time step by solving the equation

Equation (22) is observed, where the pressure gradient term is added and the new arising velocity can be handled by the incompressibility of the fluid:

Meanwhile, the pressure is calculated implicitly with the following Poisson equation for pressure deduced from equations (22) and (23); that is

The boundary condition for is obtained by projecting equation (22) on the outward unit normal vector to the boundary . Thus, the pressure Neumann boundary condition is written as

In the projection method, the particle positions change only in the first step. The intermediate velocity, the pressure, and the final divergence free velocity are all computed on the new particle positions. The spatial derivatives appearing in the above equations are approximated by the moving least squares (MLS) method proposed above which will directly be used to solve the pressure Poisson equation (24). Eventually, we use this mathematical model to numerically simulate the incompressible viscous flow problems.

#### 4. Treatment of Particle Data and Time Step Size

##### 4.1. Particle Data Management

In the finite particle method, points move with the fluid velocity. Since the velocity field is the solution itself, one cannot predict in advance where the particles will move. Hence, even if the particles are well distributed initially, they will in general cluster in some places and become scarce in others. The former can in best case result in an unnecessarily high local resolution and thus computational effort and in the worst case, yield numerical instabilities. The latter can spoil the accuracy and cause problems in the point neighbourhood search.

In order to remedy these problems, a constant particle data management has to be done. Particles too close to each other have to be removed, or better merged, while new particles have to be inserted into large holes and gaps. This requires on the one hand efficient detection procedures for nearby particles and holes and on the other hand, correct interpolation of the field data, when particles are merged and inserted.

When finding some points’ shortest distances closer than a given distance *d*_{min} by looping over all points, removing close points by the method of interpolating using all neighbouring points is worth effort. As for the condition with too many largest holes generating during flow, searching for all the Voronoi cell of point x_{i} in 2D or in 3D and constructing the full Voronoi diagram [49] are the core missions. Once those largest holes are identified by Voronoi diagrams, new points will be inserted and participant calculation is done.

##### 4.2. Determination of Time Step

For numerical stability, several time step constraints must be satisfied, including a Courant–Friedrichs–Lewy (CFL) condition,where is the maximum value of velocity for a given problem. And, additional constraints due to the hydrodynamical force acting on the particle ,

#### 5. Numerical Examples

In this section, three numerical examples concerning free deformation of a viscous fluid patch with free surface are presented to test the performance of the finite particle method proposed in this paper to simulate incompressible viscous fluid flow. The numerical results are compared with the available experimental data for validating the presented grid-free methodology.

##### 5.1. Liquid Sloshing of Tanks with Low Volumetric Fluid Type

The liquid sloshing of tanks is a typical strongly nonlinear flow problem, accompanied by traveling wave, breaking wave, splashing and high-speed lashing, and some other complicated flow phenomena. Especially for the high-pressure impacting force on the tank structure, this condition may result in local structural deformation or damage; therefore, it is well worth intensively studying. This numerical example is used for the simulation of liquid sloshing of tanks with low volumetric fluid by the proposed finite particle method, analysis of sharp motion characteristics and large deformation of the fluid flow, periodic slamming pressure on tank walls, and comparison with test results.

For consistency with the experiment condition in reference [50], liquid sloshing of tanks with low volumetric fluid type of the same liquid capacity (40%) but with two different sway excitation periods is selected and numerically simulated. Model I has the period of excitation 1.5 s and Model II has excitation period 1.3 s, while the resonant period of tank is found to be 1.3 s. At the same time, the pressure monitoring point O is set in the calculation to compare with the measured pressure acting on the tank bulkhead in the test, quantitatively analysing the accuracy of the numerical calculation results. The calculation model of the tank which is shown in Figure 1 takes a two-dimensional rectangular chamber with a width of 0.6 m and a height of 0.3 m.