An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows

In this paper, an incompressible smoothed particle hydrodynamics (SPH) method is presented to solve unsteady free-surface flows. Both Newtonian and viscoelastic fluids are considered. In the case of viscoelastic fluids, both the Maxwell and Oldroyd-B models are investigated. The proposed SPH method uses a Poisson pressure equation to satisfy the incompressibility constraints. The solution algorithm is an explicit predictor-corrector scheme and employs an adaptive smoothing length based on density variations. To alleviate the numerical difficulties encountered when fluid is highly stretched, an artificial stress term is incorporated into the momentum equation which reduces the risk of unrealistic fractures in the material. Two challenging test cases, the impacting drop and the jet buckling problems, are solved to demonstrate the capability of the proposed scheme in handling viscoelastic flows with complex free surfaces. The jet buckling test case was solved for a wide range of Weissenberg numbers. It was shown that in all cases the method is stable and fairly accurate and agrees well with the available data.     

Numerical study of PPE source term errors in the incompressible SPH models

In a recent work by Gui et al. 13 , an incompressible SPH model was presented that employs a mixed pressure Poisson equation (PPE) source term combining both the density‐invariant and velocity divergence‐free formulations. The present work intends to apply the model to a wider range of fluid impact situations in order to quantify the numerical errors associated with different formulations of the PPE source term in incompressible SPH (ISPH) models. The good agreement achieved between the model predictions and the documented data is taken as a further demonstration that the mixed source term formulation can accurately predict the fluid impact pressures and forces, both in the magnitude and in the spatial and temporal patterns. Furthermore, an in‐depth numerical analysis using either the pure density‐invariant or velocity divergence‐free formulation has revealed that the pure density‐invariant formulation can lead to relatively large divergence errors while the velocity divergence‐free formulation may cause relatively large density errors. As compared with these two approaches, the mixed source term formulation performs much better having the minimum total errors in all test cases. Although some recent studies found that the weakly compressible SPH models perform somewhat better than the incompressible SPH models in certain fluid impact problems, we have shown that this could be largely caused by the particular formulation of PPE source term in the previous ISPH models and a better formulation of the source term can significantly improve the accuracy of ISPH models. Copyright © 2014 John Wiley & Sons, Ltd.     

Improvement of stability in moving particle semi‐implicit method

Moving particle semi‐implicit (MPS) method is one of the particle methods, which can be used to analyze incompressible free surface flow without surface tracking by a mesh or a scalar quantity. However, MPS causes unphysical numerical oscillation of pressure with high frequencies. We proposed a new formulation for the source term of Poisson equation of pressure. The proposed source term consists of three parts, one main part and two error‐compensating parts. With proper selection of the coefficients for the error‐compensating parts, we can suppress the unphysical pressure oscillation. Smoother pressure distributions are obtained in hydrostatic pressure and dam break problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Incompressible SPH simulation of water entry of a free‐falling object

In this paper an incompressible smoothed particle hydrodynamics (Incom‐SPH) model is used to simulate the interactions between the free surface flow and a moving object. Incom‐SPH method is a two‐step semi‐implicit hydrodynamic formulation of the SPH algorithm and is capable of accurately treating the free surface deformations and impact forces during the solid–fluid interactions. For a free‐falling object, its motion is tracked by an additional Lagrangian algorithm based on Newton's law to couple with the Incom‐SPH program. The developed model is employed to investigate the water entry of a free‐falling wedge. The accuracy of the computations is validated by the good agreement in predicting the relevant hydrokinematic and hydrodynamic parameters. Finally, a numerical test is performed to study the influence of spatial resolution on the water entry features. The Incom‐SPH modeling coupled with the solid–fluid interaction algorithm could provide a promising computational tool to predict the slamming problems in coastal and offshore engineering. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimized incompressible smoothed particle hydrodynamics methods and validations

The solution of the Poisson's equation used by the incompressible smoothed particle hydrodynamics (ISPH) methods for estimating the pressure field is expensive in CPU time. The CPU time, consumed by the inversion of the operator ∇(1/ρ∇) and the estimation of the right hand side of the Poisson's equation, increases with the number N of particles used in a purely Lagrangian framework. In this work, this default of ISPH methods is overcome by solving the Poisson's equation on a Cartesian grid. This SPH-mesh coupling is equivalent to the particle in cell method. In a first step, in order to analyze its efficiency, the optimized version of two ISPH methods (divergence free and density invariant) is compared with the standard weakly compressible SPH method through two benchmarks of incompressible bidimensional flows characterized by the Reynolds number Re, Lamb-Oseen vortex (10 ≤Re≤ 100) and lid-driven cavity flow (100 ≤Re≤ 1000). In a second step, the numerical results obtained by the three SPH methods are compared to laboratory experimental data of a dam break flow in order to show the performance of the SPH-mesh coupling in a practical and complex flow problem. As in the configuration of the experimental setup, the numerical results are obtained for a Reynolds number Re = 3.8 10⁶.     

An improved KGF‐SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows

The kernel gradient free (KGF) smoothed particle hydrodynamics (SPH) method is a modified finite particle method (FPM) which has higher order accuracy than the conventional SPH method. In KGF‐SPH, no kernel gradient is required in the whole computation, and this leads to good flexibility in the selection of smoothing functions and it is also associated with a symmetric corrective matrix. When modeling viscous incompressible flows with SPH, FPM or KGF‐SPH, it is usual to approximate the Laplacian term with nested approximation on velocity, and this may introduce numerical errors from the nested approximation, and also cause difficulties in dealing with boundary conditions. In this paper, an improved KGF‐SPH method is presented for modeling viscous, incompressible fluid flows with a novel discrete scheme of Laplacian operator. The improved KGF‐SPH method avoids nested approximation of first order derivatives, and keeps the good feature of ‘kernel gradient free’. The two‐dimensional incompressible fluid flow of shear cavity, both in Euler frame and Lagrangian frame, are simulated by SPH, FPM, the original KGF‐SPH and improved KGF‐SPH. The numerical results show that the improved KGF‐SPH with the novel discrete scheme of Laplacian operator are more accurate than SPH, and more stable than FPM and the original KGF‐SPH. Copyright © 2015 John Wiley & Sons, Ltd.     

不可压缩机翼绕流的有限谱法计算

结合有限谱QUICK格式求解不可压缩粘性流问题。这一格式用于模拟不同攻角下的NACA1200机翼绕流问题。利用体积力,提出了将流场速度从0加速到来流速度的方法。区别于传统的压力梯度为零的边界条件,推导出一个更精确的压力边界条件。为使速度散度保持为零,在泊松方程中给速度散度一个特殊的处理。这一成果说明了有限谱法不但具有很高的精度,而且能灵活地和其他格式一起构造出新的格式,从而成功地应用到复杂流场不可压缩流动的数值计算中。     

Incompressible smoothed particle hydrodynamics simulation of multifluid flows

This paper presents an incompressible SPH (ISPH) technique to simulate multifluid flows. The SPH method is a mesh‐free particle modeling approach that can treat free surfaces and multi‐interfaces in a simple and efficient manner. The ISPH method employs an incompressible hydrodynamic formulation to solve the fluid pressure that ensures a stable pressure field. Two multifluid ISPH models are proposed following different interface treatments: the coupled ISPH model does not distinguish the different fluid phases and applies the standard ISPH technique across the interface, whereas the decoupled ISPH model first treats each fluid phase separately and then couples the different phases by applying pressure and shear stress continuities across the interface. The two proposed models were used to investigate a gravity underflow with a low density ratio in a Generalized Reservoir Hydrodynamics (GRH) flume and a horizontal lock exchange flow with a high density ratio. Comparisons with data and relevant numerical error analysis indicated that the decoupled model performed well in cases of both low and high density ratios because of the accurate treatment of interface boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

Fluid interface detection technique based on neighborhood particles centroid deviation (NPCD) for particle methods

Particle‐based CFD methods are powerful approaches to investigate free surface, multiphase flows, and fluid structure interaction problems because of their ability of tracking moving fluid interface even with huge deformations or fragmentation and merging. However, many fluid interface particle detection techniques are simple to implement but with low accuracy or provide relatively good detection results at complicated implementation cost or higher computational time. In case of incompressible flow simulation methods solving the Poisson equation of pressure, such as the moving particle semi‐implicit method, boundary particles detection techniques' accuracy affects precision and stability of pressure computation and interaction between fluid phases. In the present work, a new fluid interface particle detection technique is proposed to improve the accuracy of the boundary particles detection and keep the implementation easy. Denominated as the neighborhood particles centroid deviation technique, it is a two‐criteria technique based on the particle number density and the neighborhood particles weighted geometric center deviation. Compared with other techniques, the proposed neighborhood particles centroid deviation technique shows the best results by eliminating false interface particles inside the fluid domain and keeping the interface particles layer thin and regular. As a result, relatively stable pressure time histories and more consistent pressure and velocity fields are achieved. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.     

流固耦合不可压物质点法及其在晃动问题中的应用

作为一种混合拉格朗日欧拉法,物质点法在流固耦合问题中具有重要的应用前景。对于自由液面的流动问题,基于物质点法框架已建立了弱可压物质点法和完全不可压物质点法,但在处理流固耦合问题时遇到了困难。弱可压物质点法由于采用可压缩状态方程,导致求解时间步长过小,压力振荡严重,产生了非物理的飞溅现象;完全不可压物质点法基于投影算法和不可压条件,消除了弱可压物质点法的压力振荡,提高了时间步长,但难以处理移动边界问题。基于变分形式的投影算法提出了一种新型流固耦合不可压物质点法,得到了体积加权的压力泊松方程PPE(Pressure Poisson Equation),解决了完全不可压物质点法无法处理不规则边界和移动边界的问题。采用流固耦合不可压物质点法研究了运动刚体容器中的液体晃动问题,并与已有实验和数值结果进行对比,验证了算法的正确性和精度。     

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.     

Simulation of compressibility of entrapped air in an incompressible free surface flow using a pressure-based method for unified equations

A pressure-based method is developed to solve the unified conservation laws for incompressible and compressible fluids. A polytropic law is used to model the compressibility of a gas and decouple the energy equation. The pressure field is calculated by solving a single-pressure Poisson equation for the entire flow domain. The effects of the compressibility of the gas are reflected in the source term of the Poisson equation. The continuities of pressure and normal velocity across a material interface are achieved without any additional treatment along the interface. To validate the developed method, the oscillation of a water column in a closed tube due to the compression and expansion of air in the tube is simulated. The computed time history of the pressure at the end wall of the tube is in good agreement with other computational results. The free drop of a water column in a closed tank is simulated. The time history of the pressure at the center of the bottom of the tank shows good agreement with other reported results. The developed code is applied to simulate the air cushion effect of entrapped air in a dam break flow. The computed result is in good agreement with other experimental and computational results until the air is entrapped. As the entrapped air pocket undergoes rapid pulsation, the pressure field of water around the air pocket oscillates synchronously.     

Space potential particles to enhance the stability of projection-based particle methods

Particle methods have been seldom verified by a Karman vortex simulation, which is commonly performed as a typical benchmark in computational fluid dynamics. This is mainly due to a difficulty in suppression of occurrence of unphysical voids manifested usually in a strong vortex on account of definition of free surface by the Lagrangian tracking framework with inconsistency in volume conservation. This paper presents a simple and effective scheme as a free-surface boundary condition of projection-based particle methods, namely the MPS (moving particle semi-implicit) and Incompressible SPH (ISPH) methods to handle the free surface with consistency in volume conservation. The new scheme is introduced into the Poisson pressure equation (PPE) with consideration of a potential in void space as space potential particle (SPP), to reproduce physical motions of particles around free surface through a particle–void interaction. The enhancing effect of the newly proposed SPP scheme is shown by simulating a few numerical tests, including a whirling water flow, a two-phase surfacing flow, and a set of Karman vortex simulations.     

光滑粒子动力学方法的发展与应用

光滑粒子动力学(smoothed particle hydrodynamics,SPH)是一种拉格朗日型无网格粒子方法,已经成功地应用到了工程和科学的众多领域.SPH使用粒子离散及代表所模拟的介质,并且基于粒子体系估算和近似介质运动的控制方程.本文分析和综述了SPH模拟方法的发展历程、数值方法与应用进展.介绍了SPH方法的基本思想;从连续性、边界处理、稳定性和计算效率4个方面阐述了SPH方法的研究现状;介绍了SPH方法近年来在可压缩流动、不可压缩流动以及弹塑性材料高速变形与失效方面的一些典型应用;并对SPH方法的发展与应用进行了预测与展望.      

Numerical simulation of landslide impulsive waves by incompressible smoothed particle hydrodynamics

An incompressible‐smoothed particle hydrodynamics (I‐SPH) formulation is presented to simulate impulsive waves generated by landslides. The governing equations, Navier–Stokes equations, are solved in a Lagrangian form using a two‐step fractional method. Landslides in this paper are simulated by a submerged mass sliding along an inclined plane. During sliding, both rigid and deformable landslides mass are considered. The present numerical method is examined for a rigid wedge sliding into water along an inclined plane. In addition solitary wave generated by a heavy box falling inside water, known as Scott Russell wave generator, which is an example for simulating falling rock avalanche into artificial and natural reservoirs, is simulated and compared with experimental results. The numerical model is also validated for gravel mass sliding along an inclined plane. The sliding mass approximately behaves like a non‐Newtonian fluid. A rheological model, implemented as a combination of the Bingham and the general Cross models, is utilized for simulation of the landslide behaviour. In order to match the experimental data with the computed wave profiles generated by deformable landslides, parameters of the rheological model are adjusted and the numerical model results effectively match the experimental results. The results prove the efficiency and applicability of the I‐SPH method for simulation of these kinds of complex free surface problems. Copyright © 2007 John Wiley & Sons, Ltd.     

A correction for balancing discontinuous bed slopes in two‐dimensional smoothed particle hydrodynamics shallow water modeling

In this paper, a smoothed particle hydrodynamics (SPH) numerical model for the shallow water equations (SWEs) with bed slope source term balancing is presented. The solution of the SWEs using SPH is attractive being a conservative, mesh‐free, automatically adaptive method without special treatment for wet‐dry interfaces. Recently, the capability of the SPH–SWEs numerical scheme with shock capturing and general boundary conditions has been used for predicting practical flooding problems. The balance between the bed slope source term and fluxes in shallow water models is desirable for reliable simulations of flooding over bathymetries where discontinuities are present and has received some attention in the framework of Finite Volume Eulerian models. The imbalance because of the source term resulting from the calculation of the the water depth is eradicated by means of a corrected mass, which is able to remove the error introduced by a bottom discontinuity. Two different discretizations of the momentum equation are presented herein: the first one is based on the variational formulation of the SWEs in order to obtain a fully conservative formulation, whereas the second one is obtained using a non‐conservative form of the free‐surface elevation gradient. In both formulations, a variable smoothing length is considered. Results are presented demonstrating the corrections preserve still water in the vicinity of either 1D or 2D bed discontinuities and provide close agreement with 1D analytical solutions for rapidly varying flows over step changes in the bed. The method is finally applied to 2D dam break flow over a square obstacle where the balanced formulation improves the agreement with experimental measurements of the free surface. Copyright © 2012 John Wiley & Sons, Ltd.     

SPH固壁边界处理方法的改进

提出一种适用于光滑质点水动力学（SPH）方法的改进的边界处理方法。在这种方法中,边界粒子的压力可通过其周围的流体粒子的压力插值得到,从而改进了耦合边界法在边界上压力不准的问题。运用这种改进的边界处理方法模拟了二维方形水箱中的非线性晃荡问题以及二维楔形体自由入水问题。模拟结果与实验结果吻合较好,证明了此改进的边界处理方法是有效的。     

A novel method to calculate the pressure interaction between dust and fluid using SPH

Unstable behavior of smoothed particle hydrodynamics (SPH) dust particles, such as clumping or fingering under certain conditions, has been reported by several researchers who have conducted studies on dusty fluid SPH. The simulation results in this study show that this instability is numerical, and the instability is mainly attributable to the ill‐interpolated pressure gradient in the interaction term between 2 phases. In this paper, we introduce a new method to calculate the pressure force interaction term between dust and fluid particles. The key idea is to first interpolate the pressure gradient at SPH fluid particles and then use the values to calculate the pressure gradient at SPH dust particles, in a consecutive manner. To compare the new method with the existing method, we first conducted an interpolation of pressure gradient at hydrostatic equilibrium under gravity to estimate any error. The results show that the new method is more accurate. We then conducted additional numerical tests, namely, dust‐liquid counterflow, sedimentation in a confined tank, and sedimentation in the presence of turbulence. The unphysical unstable behavior of SPH dust particles such as clumping or fingering was significantly reduced in the new method. The results also show that the instability becomes more significant when using the existing method especially for the case when simulating a flow with relatively high concentration of dust or for the case in which inertia dominates the dynamics of dust particles. Especially, in those cases, the existing method should be avoided, and the newly proposed method is highly recommended.     

An improved SPH model for multiphase flows with large density ratios

This paper presents a new smoothed particle hydrodynamics (SPH) model for simulating multiphase fluid flows with large density ratios. The new SPH model consists of an improved discretization scheme, an enhanced multiphase interface treatment algorithm, and a coupled dynamic boundary treatment technique. The presented SPH discretization scheme is developed from Taylor series analysis with kernel normalization and kernel gradient correction and is then used to discretize the Navier‐Stokes equation to obtain improved SPH equations of motion for multiphase fluid flows. The multiphase interface treatment algorithm involves treating neighboring particles from different phases as virtual particles with specially updated density to maintain pressure consistency and a repulsive interface force between neighboring interface particles into the pressure gradient to keep sharp interface. The coupled dynamic boundary treatment technique includes a soft repulsive force between approaching fluid and solid particles while the information of virtual particles are approximated using the improved SPH discretization scheme. The presented SPH model is applied to 3 typical multiphase flow problems including dam breaking, Rayleigh‐Taylor instability, and air bubble rising in water. It is demonstrated that inherent multiphase flow physics can be well captured while the dynamic evolution of the complex multiphase interfaces is sharp with consistent pressure across the interfaces.     

A kernel gradient free (KGF) SPH method

The finite particle method (FPM) is a modified SPH method with high order accuracy while retaining the advantages of SPH in modeling problems with free surfaces, moving interfaces, and large deformations. In both SPH and FPM, kernel gradient is necessary in kernel and particle approximation of a field function and its derivatives. In this paper, a new FPM is presented, which only involves kernel function itself in kernel and particle approximation. The kernel gradient is not necessary in the whole computation, and this approach is thus referred to as a kernel gradient free (KGF) SPH method. This is helpful when a kernel function is not differentiable or the resultant kernel gradient is not sufficiently smooth, and thus it is more general in selecting a kernel function. Moreover, different from the original FPM with an asymmetric corrective matrix, in the new FPM, the resultant corrective matrix is symmetric, and this is advantageous in particle approximations. A series of numerical examples have been conducted to show the efficiencies of KGF‐SPH including one‐dimensional mathematical tests of polynomial functions with equal or variable smoothing length and two‐dimensional incompressible fluid flow of shear cavity. It is found that KGF‐SPH is comparable with FPM in accuracy and is flexible as SPH. Copyright © 2015 John Wiley & Sons, Ltd.