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.
Highlights
In the last several decades, various meshless techniques [1,2,3,4,5,6] 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
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]. ough 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. e approach for enforcing the incompressibility condition in governing flow equations, namely, the incompressible SPH (ISPH), has been explored
Summary
In the last several decades, various meshless techniques [1,2,3,4,5,6] 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. Is 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. E 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. E 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. Approximating function equation (1) by MLS, we can find that the coefficient vector a is the solution to the system. According to equation (14), the first derivative approximation of the MLS function can be obtained as ( u)′ b′T · a + bT · a′ (17). Erefore, the spatial derivatives could be approximated by the above approach described in this subsection. e 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]
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