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.     

An SPH model for free surface flows with moving rigid objects

This paper presents a computational model for free surface flows interacting with moving rigid bodies. The model is based on the SPH method, which is a popular meshfree, Lagrangian particle method and can naturally treat large flow deformation and moving features without any interface/surface capture or tracking algorithm. Fluid particles are used to model the free surface flows which are governed by Navier–Stokes equations, and solid particles are used to model the dynamic movement (translation and rotation) of moving rigid objects. The interaction of the neighboring fluid and solid particles renders the fluid–solid interaction and the non‐slip solid boundary conditions. The SPH method is improved with corrections on the SPH kernel and kernel gradients, enhancement of solid boundary condition, and implementation of Reynolds‐averaged Navier–Stokes turbulence model. Three numerical examples including the water exit of a cylinder, the sinking of a submerged cylinder and the complicated motion of an elliptical cylinder near free surface are provided. The obtained numerical results show good agreement with results from other sources and clearly demonstrate the effectiveness of the presented meshfree particle model in modeling free surface flows with moving objects. Copyright © 2013 John Wiley & Sons, Ltd.     

固体介质中SPH方法的拉伸不稳定性问题研究进展

光滑粒子流体动力学法(smoothed particle hydrodynamics, SPH)是一种基于核估计的无网格Lagrange数值方法.它用粒子方程离散流体动力学的连续方程, 既可以处理有限元难于处理的大变形和严重扭曲问题, 又可以处理有限差分法不易处理的自由边界和材料界面的问题, 在固体力学中的冲击、爆炸和裂纹模拟中具有广阔的发展前景.但是, 该算法的拉伸不稳定性(tensile instability)问题是它在固体力学领域中应用的最大障碍.对SPH稳定性分析表明, 算法不稳定性的条件仅与应力状态和核函数的2阶导数有关.目前, 应力点法(stress points)、Lagrange核函数法、人工应力法(artificialstress)、修正光滑粒子法(corrective smoothed particle method, CSPM)和守恒光滑法(conservativesmoothing)以及其他一些方法成功地改善了SPH的拉伸不稳定性, 但是每一种方法都不能彻底解决SPH的拉伸不稳定性问题.本文介绍了SPH法的方程和Von Neumann稳定性分析的思想, 以及国内外在这几个方面的研究成果及其最新进展, 同时指出目前研究中存在的问题和研究的方向.      

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.     

含液颗粒材料液固耦合分析的离散颗粒模型及特征线SPH法

对含液颗粒材料流固耦合分析建议了一个基于离散颗粒模型与特征线SPH法的显式拉格朗日-欧拉无网格方案。在已有的用以模拟固体颗粒集合体的离散颗粒模型[1]基础上,将颗粒间间隙内的流体模型化为连续介质,对其提出并推导了基于特征线的SPH法。数值例题显示了所建议方案在模拟颗粒材料与间隙流相互作用的能力和性能以及间隙流体对颗粒结构承载能力及变形的影响。     

圆柱形汇聚激波诱导 Richtmyer-Meshkov不稳定的 SPH 模拟

汇聚激波诱导不同物质界面的Richtmyer-Meshkov(RM)不稳定现象在惯性约束核聚变领域有重要的学术意义和工程背景.基于网格离散的宏观流体力学方法由于数值扩散问题往往需要高阶精度算法才能准确追踪界面演化,且对大变形和破碎合并等复杂界面追踪也极为困难.光滑粒子流体动力学(smoothed particlehydrodynamics,SPH)方法采用纯拉格朗日算法,可以有效克服上述难点.但经典SPH算法需采用人工黏性处理强间断,在激波间断处往往会出现严重的非物理振荡,对于涉及强冲击不稳定性问题,很难达到理想的模拟效果.本文采用基于HLL黎曼求解器的SPH算法,实现了对强激波和大密度比物质界面的有效分辨和追踪.一维数值校核证明了代码的可靠性、健壮性,并进一步模拟了二维圆柱形汇聚冲击波冲击四边形轻/重气界面诱导的RM不稳定性问题,与已有实验结果进行了对比,发现模拟结果与实验结果吻合.通过分析界面演化过程中的密度及压力变化,发现本文所采用的方法可准确地追踪激波与界面作用的复杂界面和波系演化规律.研究结果为进一步理解和解释汇聚冲击条件下的RM不稳定性机理奠定了基础.      

非等温黏弹性复杂流动的改进SPH方法模拟

非等温黏弹性流体广泛存在于自然界和工业生产中,准确预测黏弹性流体的非等温流动机理和复杂流变特性有着重要的应用价值.文章提出一种改进的光滑粒子流体动力学(smoothed particle hydrodynamics,SPH)方法对非等温黏弹性复杂流动进行了数值模拟,其中流体的黏弹特性通过eXtended Pom-Pom本构模型来表征.为了提高模拟结果的精度,采用了一种核函数梯度的修正算法;为了灵活地施加边界条件,发展了边界粒子和虚拟粒子相联合的边界处理方法;为了消除流动过程中的拉伸不稳定性,施加了粒子迁移技术.运用改进SPH方法数值模拟了液滴撞击固壁和F型腔注塑成型问题,通过与Basilisk软件得到的结果进行比较验证了改进SPH方法求解非等温黏弹性流体的有效性.通过利用不同粒子初始间距进行计算,评价了改进SPH方法的数值收敛性.研究了非等温流动相较于等温流动的不同流动特征,深入分析了不同热流变参数对流动过程的影响.数值结果表明,文章提出的改进SPH方法可稳定、准确地描述非等温黏弹性复杂流动的传热机理、复杂流变特性和自由面变化特性.     

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.     

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.      

Superquadric DEM-SPH coupling method for interaction between non-spherical granular materials and fluids

The research on the coupling method of non-spherical granular materials and fluids aims to predict the particle–fluid interaction in this study. A coupling method based on superquadric elements is developed to describe the interaction between non-spherical solid particles and fluids. The discrete element method (DEM) and the smoothed particle hydrodynamics (SPH) are adopted to simulate granular materials and fluids. The repulsive force model is adopted to calculate the coupling force and then a contact detection method is established for the interaction between the superquadric element and the fluid particle. The contact detection method captures the shape of superquadric element and calculates the distance from the fluid particle to the surface of superquadric element. Simulation cases focusing on the coupling force model, energy transfer, and large-scale calculations have been implemented to verify the validity of the proposed coupling method. The coupling force model accurately represents the water entry process of a spherical solid particle, and reasonably reflects the difference of solid particles with different shapes. In the water entry process of multiple solid particles, the total energy of the water entry process of multiple solid particles tends to be stable. The collapse process of the partially submerged granular column is simulated and analyzed under different parameters. Therefore, this coupling method is suitable to simulate fluid–particle systems containing solid particles with multiple shapes.     

An SPH multi‐fluid model based on quasi buoyancy for interface stabilization up to high density ratios and realistic wave speed ratios

We introduce a smoothed particle hydrodynamics (SPH) concept for the stabilization of the interface between 2 fluids. It is demonstrated that the change in the pressure gradient across the interface leads to a force imbalance. This force imbalance is attributed to the particle approximation implicit to SPH. To stabilize the interface, a pressure gradient correction is proposed. In this approach, the multi‐fluid pressure gradients are related to the (gravitational and fluid) accelerations. This leads to a quasi‐buoyancy correction for hydrostatic (stratified) flows, which is extended to nonhydrostatic flows. The result is a simple density correction that involves no parameters or coefficients. This correction is included as an extra term in the SPH momentum equation. The new concept for the stabilization of the interface is explored in 5 case studies and compared with other multi‐fluid models. The first case is the stagnant flow in a tank: The interface remains stable up to density ratios of 1:1000 (typical for water and air), in combination with artificial wave speed ratios up to 1:4. The second and third cases are the Rayleigh‐Taylor instability and the rising bubble, where a reasonable agreement between SPH and level‐set models is achieved. The fourth case is an air flow across a water surface up to density ratios of 1:100, artificial wave speed ratios of 1:4, and high air velocities. The fifth case is about the propagation of internal gravity waves up to density ratios of 1:100 and artificial wave speed ratios of 1:4. It is demonstrated that the quasi‐buoyancy model may be used to stabilize the interface between 2 fluids up to high density ratios, with real (low) viscosities and more realistic wave speed ratios than achieved by other weakly compressible SPH multi‐fluid models. Real wave speed ratios can be achieved as long as the fluid velocities are not very high. Although the wave speeds may be artificial in many cases, correct and realistic wave speed ratios are essential in the modelling of heat transfer between 2 fluids (eg, in engineering applications such as gas turbines).     

高速碰撞数值计算中的SPH自适应粒子分布法

针对传统光滑粒子法在计算高速碰撞问题时会出现近邻粒子逸出核函数影响域而产生数值破坏这一缺陷,提出了一种根据粒子间距变化自动添加、合并粒子的SPH自适应粒子分布算法。采用该方法对Taylor碰撞和超高速碰撞问题进行了数值模拟,结果表明,该方法可以有效地消除计算中出现的数值破坏,提高计算精度。     

Displacement of yield-stress fluids in a fracture

We consider a displacement of several yield-stress fluids in a Hele-Shaw cell. The topic is relevant to the development of a model for the flow of multiple phases inside a narrow fracture with application to hydraulically fracturing a hydrocarbon-bearing underground formation. Existing models for fracturing flows include only pure power-law models without yield stress, and the present work is aimed at filling this gap. The fluids are assumed to be immiscible and incompressible. We consider fluid advection in a plane channel in the presence of density gradients. Gravity is taken into account, so that there can be slumping and gravitational convection. We use the lubrication approximation so that governing equations are reduced to a 2D width-averaged system formed by the quasi-linear elliptic equation for pressure and transport equations for volume concentrations of fluids. The numerical solution is obtained using a finite-difference method. The pressure equation is solved using an iterative algorithm and the Multigrid method, while the transport equations are solved using a second-order TVD flux-limiting scheme with the superbee limiter. This numerical model is validated against three different sets of experiments: (i) gravitational slumping of fluids in a closed Hele-Shaw cell, (ii) viscous fingering of fluids with a high viscosity contrast due to the Saffman–Taylor (S–T) instability in a Hele-Shaw cell at microgravity conditions, (iii) displacement of Bingham fluids in a Hele-Shaw cell with the development of fingers due to the S–T instability. Good agreement is observed between simulations and laboratory data. The model is then used to investigate the joint effect of fingering and slumping. Numerical simulations show that the slumping rate of yield-stress fluid is significantly less pronounced than that of a Newtonian fluid with the same density and viscosity. If a low-viscosity Newtonian fluid is injected after a yield-stress one, the S–T instability at the interface leads to the development of fingers. As a result, fingers penetrating into a fluid with a finite yield stress locally decrease the pressure gradient and unyielded zones develop as a consequence.     

人工黏性增强三维移动粒子半隐算法稳定性

通过综合比较移动粒子半隐方法(moving-particle semi-implicitmethod, MPS)各种稳定性计算方法, 考虑了多种可能导致计算不稳定的因素, 并首次提出加入人工黏性来抑制非物理压力振荡, 得到了较为稳定的三维MPS算法.采用各种稳定性方案对比模拟了三维立方流体旋转状态下的变形及三维静水压问题, 并进一步与商业软件的流体体积函数方法(volumeof fluid, VOF)模型计算结果对比, 验证了其正确性.结果表明:对三维情形, 单纯应用已有的稳定性算法仍难以满足MPS计算稳定性的要求, 而进一步辅以该文提出的人工黏性方法则可以在准确性及稳定性方面均取得较好的效果, 且计算结果显示, 通过该文提出的开关模式增加人工黏性并不影响压强泊松方程的求解精度, 因而不会对流场的求解产生负面作用.     

任意区域上的粒子均匀分布方法

给出了任意区域内粒子的Voronoi面积均匀划分的方法,对给定的区域边界点,先由边界点得到边界粒子,再利用分布函数结合Delaunay加点算法得到内部粒子,这种方法可以得到一般的区域上粒子的均匀分布。这里给出了一个流体力学中利用SPH方法模拟内爆问题的例子,算例表明,均匀分布可以得到更高的无网格方法数值模拟精度。     

基于ALE,CEL和SPH方法的球形破片高速冲击充液结构对比研究

为研究ALE,CEL和SPH方法在高速冲击流固耦合动力学数值分析中的差异性,开展球形破片高速冲击充液结构数值模拟研究。建立经文献资料验证的ALE,CEL和SPH三种动力学模型,研究了流体压力变化、形成的空腔尺寸、破片速度衰减变化和充液结构变形等模拟精度,并分析相应的计算成本。结果表明,ALE,CEL和SPH三种方法均能有效模拟破片高速冲击充液结构的流固耦合动力学过程;ALE方法预测的空腔尺寸精度较高;CEL方法预测的流体压力、破片速度衰减和充液结构变形精度较高;SPH方法预测的空腔尺寸、破片速度衰减精度较高;当网格尺寸一致时, SPH方法计算时长约为ALE和CEL方法的两倍,但SPH方法前后处理更加简便。     

Imposing accurate wall boundary conditions in corrective‐matrix‐based moving particle semi‐implicit method for free surface flow

Corrective matrix that is derived to restore consistency of discretization schemes can significantly enhance accuracy for the inside particles in the Moving Particle Semi‐implicit method. In this situation, the error due to free surface and wall boundaries becomes dominant. Based on the recent study on Neumann boundary condition (Matsunaga et al, CMAME, 2020), the corrective matrix schemes in MPS are generalized to straightforwardly and accurately impose Neumann boundary condition. However, the new schemes can still easily trigger instability at free surface because of the biased error caused by the incomplete/biased neighbor support. Therefore, the existing stable schemes based on virtual particles and conservative gradient models are applied to free surface and nearby particles to produce a stable transitional layer at free surface. The new corrective matrix schemes are only applied to the particles under the stable transitional layer for improving the wall boundary conditions. Three numerical examples of free surface flows demonstrate that the proposed method can help to reduce the pressure/velocity fluctuations and hence enhance accuracy further.     

A modified SPH method to model the coalescence of colliding non-Newtonian liquid droplets

This paper develops a modified smoothed particle hydrodynamics (SPH) method to model the coalescence of colliding non-Newtonian liquid droplets. In the present SPH, a van der Waals (vdW) equation of state is particularly used to represent the gas-to-liquid phase transition similar to that of a real fluid. To remove the unphysical behavior of the particle clustering, also known as tensile instability, an optimized particle shifting technique is implemented in the simulations. To validate the numerical method, the formation of a Newtonian vdW droplet is first tested, and it clearly demonstrates that the tensile instability can be effectively removed. The method is then extended to simulate the head-on binary collision of vdW liquid droplets. Both Newtonian and non-Newtonian fluid flows are considered. The effect of Reynolds number on the coalescence process of droplets is analyzed. It is observed that the time up to the completion of the first oscillation period does not always increase as the Reynolds number increases. Results for the off-center binary collision of non-Newtonian vdW liquid droplets are lastly presented. All the results enrich the simulations of the droplet dynamics and deepen understandings of flow physics. Also, the present SPH is able to model the coalescence of colliding non-Newtonian liquid droplets without tensile instability.     

三维溃坝流的光滑粒子动力学方法模拟

采用光滑粒子动力学SPH(Smoothed Particle Hydrodynamics)方法对三维溃坝流问题进行了数值模拟。为了逼真地模拟出坝内水体与壁面间相互作用而产生的水花飞溅、融合以及近壁面流动等现象,加入了混合长度形式的湍流模型。为了有效地防止粒子穿透固壁,提出了一种新型的适合三维数值模拟的固壁边界处理方法。应用SPH方法对三维溃坝流进行了数值模拟,并分别考虑了未添加障碍物和添加圆柱障碍物两种情形。计算结果表明,改进SPH方法能够精细地捕捉溃坝流在不同时刻的自由液面,并获得稳定而精确的数值结果。