A particle method for two‐phase flows with large density difference

A new numerical approach for solving incompressible two‐phase flows is presented in the framework of the recently developed Consistent Particle Method (CPM). In the context of the Lagrangian particle formulation, the CPM computes spatial derivatives based on the generalized finite difference scheme and produces good results for single‐phase flow problems. Nevertheless, for two‐phase flows, the method cannot be directly applied near the fluid interface because of the abrupt discontinuity of fluid density resulting in large change in pressure gradient. This problem is resolved by dealing with the pressure gradient normalized by density, leading to a two‐phase CPM of which the original singlephase CPM is a special case. In addition, a new adaptive particle selection scheme is proposed to overcome the problem of ill‐conditioned coefficient matrix of pressure Poisson equation when particles are sparse and non‐uniformly spaced. Numerical examples of Rayleigh–Taylor instability, gravity current flow, water‐air sloshing and dam break are presented to demonstrate the accuracy of the proposed method in wave profile and pressure solution. Copyright © 2015 John Wiley & Sons, Ltd.     

A 3D fully Lagrangian Smoothed Particle Hydrodynamics model with both volume and surface discrete elements

A 3D fully Lagrangian smoothed particle hydrodynamics (SPH) model has been developed adopting a particle approximation, which considers both volume and surface elements at boundaries. The model is based on the main principles of the 2D model of Ferrand et al. (2012) and on the spatial reconstruction schemes used in SPH–arbitrary Lagrangian–Eulerian modeling to treat boundaries (Marongiu et al., 2007). This model is conceived to represent free surface flows and their interactions with solid structures. It is validated on a 2D water jet impact over a plate, a sloshing tank test case, and two experimental 3D dam break fronts, which interact with fixed obstacles. The results are compared with the available measurements and analytical solutions. We finally provide supplementary inter‐comparisons using another SPH numerical model, based on the semi‐analytical approach.Copyright © 2013 John Wiley & Sons, Ltd.     

A wall boundary treatment using analytical volume integrations in a particle method

Numerical treatment of complicated wall geometry has been one of the most important challenges in particle methods for computational fluid dynamics. In this study, a novel wall boundary treatment using analytical volume integrations has been developed for two-dimensional (2D) incompressible flow simulations with the moving particle semi-implicit method. In our approach, wall geometry is represented by a set of line segments in 2D space. Thus, arbitrary-shaped boundaries can easily be handled without auxiliary boundary particles. The wall's contributions to the spatial derivatives as well as the particle number density are formulated based on volume integrations over the solid domain. These volume integrations are analytically solved. Therefore, it does not entail an expensive calculation cost nor compromise accuracy. Numerical simulations have been carried out for several test cases including the plane Poiseuille flow, a hydrostatic pressure problem with complicated shape, a high viscous flow driven by a rotating screw, a free-surface flow driven by a rotating cylinder and a dam break in a tank with a wedge. The results obtained using the proposed method agreed well with analytical solutions, experimental observations or calculation results obtained using finite volume method (FVM), which confirms that the proposed wall boundary treatment is accurate and robust.     

An accurate and stable multiphase moving particle semi‐implicit method based on a corrective matrix for all particle interaction models

The Lagrangian moving particle semi‐implicit (MPS) method has potential to simulate free‐surface and multiphase flows. However, the chaotic distribution of particles can decrease accuracy and reliability in the conventional MPS method. In this study, a new Laplacian model is proposed by removing the errors associated with first‐order partial derivatives based on a corrected matrix. Therefore, a corrective matrix is applied to all the MPS discretization models to enhance computational accuracy. Then, the developed corrected models are coupled into our previous multiphase MPS methods. Separate stabilizing strategies are developed for internal and free‐surface particles. Specifically, particle shifting is applied to internal particles. Meanwhile, a conservative pressure gradient model and a modified optimized particle shifting scheme are applied to free‐surface particles to produce the required adjustments in surface normal and tangent directions, respectively. The simulations of a multifluid pressure oscillation flow and a bubble rising flow demonstrate the accuracy improvements of the corrective matrix. The elliptical drop deformation demonstrates the stability/accuracy improvement of the present stabilizing strategies at free surface. Finally, a turbulent multiphase flow with complicated interface fragmentation and coalescence is simulated to demonstrate the capability of the developed method.     

SPLASH program for three dimensional fluid dynamics with free surface boundaries

This paper describes a three dimensional computer program SPLASH that solves Navier-Stokes equations based on the Arbitrary Lagrangian Eulerian (ALE) finite element method. SPLASH has been developed for application to the fluid dynamics problems including the moving boundary of a liquid metal cooled Fast Breeder Reactor (FBR). To apply SPLASH code to the free surface behavior analysis, a capillary model using a cubic Spline function has been developed. Several sample problems, e.g., free surface oscillation, vortex shedding development, and capillary tube phenomena, are solved to verify the computer program. In the analyses, the numerical results are in good agreement with the theoretical value or experimental observance. Also SPLASH code has been applied to an analysis of a free surface sloshing experiment coupled with forced circulation flow in a rectangular tank. This is a simplified situation of the flow field in a reactor vessel of the FBR. The computational simulation well predicts the general behavior of the fluid flow inside and the free surface behavior. Analytical capability of the SPLASH code has been verified in this study and the application to more practical problems such as FBR design and safety analysis is under way.     

Performance of high‐resolution TVD schemes for 1D dam‐break simulations

Abstract

The performance of high‐resolution total variation diminishing (TVD) schemes for simulating dam‐break problems are presented and evaluated. Three robust and reliable first‐order upwind schemes, namely FVS, Roe and HLLE schemes, are extended to six second‐order TVD schemes using two different approaches, the Sweby flux limiter approach and the direct MUSCL‐Hancock slope limiter. For idealized dam‐break flows, comparisons of the simulated results with the exact solutions show that the flux vector splitting (FVS) scheme coupled with the direct MUSCL‐Hancock (DMH) slope limiter approach has the best numerical performance among the presented schemes. Application of the FVS‐DMH scheme to a dam‐break experiment with sloping dry bed shows that the simulated water depths agree well with the measured.     

Non‐linear sloshing in partially liquid filled containers with baffles

A finite element method is used for computing the non‐linear sloshing response of liquid in a two‐dimensional rigid rectangular tank with rigid baffles. The potential formulation is considered for the liquid domain and a mixed Eulerian–Langrangian scheme is adopted. The solution is obtained by the Galerkin method. The fourth‐order Runge–Kutta method is employed to advance the solution in the time domain. A regridding technique is applied to the free surface of the liquid, which effectively eliminates the numerical instabilities without the use of artificial smoothing. Through the comparison with the available results for the rectangular tank without baffle, the validity of the present formulation is checked and then extended to the solution of tanks with rigid baffles. The effects of baffle parameters such as position, dimension and numbers on the non‐linear sloshing response are examined. The present numerical solution procedure is also applied to the non‐linear sloshing problems in a circular cylindrical container with annular baffle. Copyright © 2006 John Wiley & Sons, Ltd.     

A semi-implicit stabilized particle Galerkin method for incompressible free surface flow simulations

A new particle Galerkin method is introduced to solve the Naiver-Stokes equations in a Lagrangian fashion. The present method aims to suppress key numerical instabilities observed in the strong form Lagrangian particle methods such as smoothed particle hydrodynamics (SPH), incompressible SPH, and moving particle semi-implicit for incompressible free surface flow simulations. It is well-known that strong form Lagrangian particle methods usually rely on ad hoc particle stabilization techniques based on particle shifting, artificial viscosity, or density-invariant condition due to some formulation inconsistency issues. In the present method, we introduce a momentum-consistent velocity smoothing algorithm which is used to combine with the second-order rotational incremental pressure-correction scheme to stabilize the pressure field as well as to enforce the consistency of Neumann boundary condition. To further impose slip-free or nonslip boundary conditions for the fluid flow, a penalty method which is free of ghost or dummy particles is developed. Finally, a particle insertion-deletion adaptive scheme is proposed when the violent fluid flow is considered. Four numerical examples are studied to validate the accuracy and stability of the present method.     

Non-linear finite element analysis of large amplitude sloshing flow in two-dimensional tank

This paper is concerned with the accurate and stable finite element analysis of large amplitude liquid sloshing in two-dimensional tank under the forced excitation. The sloshing flow is formulated as an initial-boundary-value problem based upon the fully non-linear potential flow theory. The flow velocity field is interpolated from the velocity potential with second-order elements according to least square method, and the free surface conditions are tracked by making use of the direct time differentiation and the predictor–corrector method. Meanwhile, the liquid mesh is adapted such that the incompressibility condition is strictly satisfied. The accuracy and stability of the numerical method introduced are verified from the comparison with the existing reference solutions. As well, the numerical results are compared with those obtained by the linear theory with respect to the liquid fill height and the excitation amplitude. Copyright © 2004 John Wiley & Sons, Ltd.     

内置有水平挡板的矩形储液器非线性晃动分析

该文以内置有水平挡板的矩形储液器为研究对象,对储液器的非线性晃动问题展开研究。利用势流理论和虚功原理,推导了由水平挡板引起的储液器非线性阻尼比(非线性体现于波高有关)的计算公式,同时考虑水平挡板对储液器晃动频率的影响而对非线性阻尼比计算公式进行了修正。结合液体晃动的非线性分析理论,研究了水平挡板处于不同位置、挡板长度不同时储液器的液面波高与晃动力变化情况。利用Fluent软件进行了数值模拟,并与理论模型分析的结果进行对比。结果表明：当水平挡板靠近储液器底部或长度较小时,储液器内液体的非线性晃动现象较明显,利用非线性三阶模态方程推导得到的波高、晃动力与数值模拟结果较接近,而仅考虑一阶线性响应会明显低估储液器液面波高,但其求解的晃动力却与考虑非线性值的状况基本一致;随着挡板到自由液面距离的变小或挡板长度的增大,储液器液面波高、晃动力幅值减小,液体晃动呈线性变化,说明水平挡板靠近自由液面或增大其长度时能够提高储液器的阻尼比,进而也更能抑制液体的非线性晃动。同时水平挡板逐渐靠近自由液面或长度逐渐增大时,储液器晃动频率逐渐减小,减小的幅度分别可达到5.7%～28%。     

An augmented incompressible material point method for modeling liquid sloshing problems

The incompressible material point method was proposed for modeling the free surface flow problems based on the operator splitting technique which decouples the solution of the velocity and the pressure in our previous work. To further model the coupling problems between the incompressible fluid and the moving irregular solid bodies, an augmented incompressible material point method is proposed in this paper based on the energy minimization form of operator splitting technique. The interaction between the fluid and the solid is taken into account via the work done by the fluid pressure on the solid bodies. By minimizing the total work done by the fluid pressure, volume-weighted pressure Poisson equations are obtained. The proposed method is validated with liquid sloshing in a rectangular tank subjected to various base-excitations, and is then used to study the optimal height of baffles mounted on the bottom of the tank to mitigate the sloshing wave.     

SPH with the multiple boundary tangent method

In this article, we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics (SPH) method. Benchmark simulations using previously reported boundary treatments can suffer from particle penetration and may produce results that numerically blow up near solid boundaries. As well, current SPH boundary approaches do not properly treat curved boundaries in complicated flow domains. These drawbacks have been remedied in a new boundary treatment method presented in this article, called the multiple boundary tangent (MBT) approach. In this article we present two important benchmark problems to validate the developed algorithm and show that the multiple boundary tangent treatment produces results that agree with known numerical and experimental solutions. The two benchmark problems chosen are the lid‐driven cavity problem, and flow over a cylinder. The SPH solutions using the MBT approach and the results from literature are in very good agreement. These solutions involved solid boundaries, but the approach presented herein should be extendable to time‐evolving, free‐surface boundaries. Copyright © 2008 John Wiley & Sons, Ltd.     

A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh

A novel Lagrangian gradient smoothing method (L‐GSM) is developed to solve “solid‐flow” (flow media with material strength) problems governed by Lagrangian form of Navier‐Stokes equations. It is a particle‐like method, similar to the smoothed particle hydrodynamics (SPH) method but without the so‐called tensile instability that exists in the SPH since its birth. The L‐GSM uses gradient smoothing technique to approximate the gradient of the field variables, based on the standard GSM that was found working well with Euler grids for general fluids. The Delaunay triangulation algorithm is adopted to update the connectivity of the particles, so that supporting neighboring particles can be determined for accurate gradient approximations. Special techniques are also devised for treatments of 3 types of boundaries: no‐slip solid boundary, free‐surface boundary, and periodical boundary. An advanced GSM operation for better consistency condition is then developed. Tensile stability condition of L‐GSM is investigated through the von Neumann stability analysis as well as numerical tests. The proposed L‐GSM is validated by using benchmarking examples of incompressible flows, including the Couette flow, Poiseuille flow, and 2D shear‐driven cavity. It is then applied to solve a practical problem of solid flows: the natural failure process of soil and the resultant soil flows. The numerical results are compared with theoretical solutions, experimental data, and other numerical results by SPH and FDM to evaluate further L‐GSM performance. It shows that the L‐GSM scheme can give a very accurate result for all these examples. Both the theoretical analysis and the numerical testing results demonstrate that the proposed L‐GSM approach restores first‐order accuracy unconditionally and does not suffer from the tensile instability. It is also shown that the L‐GSM is much more computational efficient compared with SPH, especially when a large number of particles are employed in simulation.     

Momentum Conservative Schemes for Shallow Water Flows

We discuss the implementation of the finite volume method on a staggered grid to solve the full shallow water equations with a conservative approximation for the advection term. Stelling & Duinmeijer [15] noted that the advection approximation may be energy-head or momentum conservative, and if suitable which of these to implement depends upon the particular flow being considered. The momentum conservative scheme pursued here is shown to be suitable for 1D problems such as transcritical flow with a shock and dam break over a rectangular bed, and we also found that our simulation of dam break over a dry sloping bed is in good agreement with the exact solution. Further, the results obtained using the generalised momentum conservative approximation for 2D shallow water equations to simulate wave run up on a conical island are in good agreement with benchmark experimental data.     

液舱晃荡的CIP法数值模拟

数值模拟是求解液舱晃荡问题的重要方法。基于CIP方法,该文建立了有限差分法求解二维不可压缩流体的Navier-Stokes方程的数值模型,使用对时间积分的分步算法求解流场,利用THINC格式捕捉自由液面,将液舱晃荡问题视为包括气体、液体和固体的多相流问题。分别模拟了无隔板和带纵隔板的液舱晃荡。该模型可以模拟晃荡过程中自由面破碎、翻卷和对舱壁的冲击压力以及隔板附近涡的生成与脱落,模型在不同网格的计算结果一致,并将计算结果与试验结果进行了对比。结果表明：该模型可以较好地模拟液舱晃荡问题并且纵隔板对液舱晃荡有明显的抑制作用。     

An implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

In this work, a new comprehensive method has been developed which enables the solution of large, non‐linear motions of rigid bodies in a fluid with a free surface. The application of the modern Eulerian–Lagrangian approach has been translated into an implicit time‐integration formulation, a development which enables the use of larger time steps (where accuracy requirements allow it). Novel features of this project include: (1) an implicit formulation of the rigid‐body motion in a fluid with a free surface valid for both two or three dimensions and several moving bodies; (2) a complete formulation and solution of the initial conditions; (3) a fully consistent (exact) linearization for free surface flows valid for any boundary elements such that optimal convergence properties are obtained when using a Newton–Raphson solver. The proposed framework has been completed with details on implementation issues referring mainly to the computation of the complete initial conditions and the consistent linearization of the formulation for free surface flows. The second part of the paper demonstrates the mathematical and numerical formulation through numerical results simulating large free surface flows and non‐linear fluid structure interaction. The implicit formulation using a fully consistent linearization based on the boundary element method and the generalized trapezoidal rule has been applied to the solution of free surface flows for the evolution of a triangular wave, the generation of tsunamis and the propagation of a wave up to overturning. Fluid–structure interaction examples include the free and forced motion of a circular cylinder and the sway, heave and roll motion of a U‐shaped body in a tank with a flap wave generator. The presented examples demonstrate the applicability and performance of the implicit scheme with consistent linearization. Copyright © 2001 John Wiley & Sons. Ltd.     

Flow Rate Limitation of Steady Convective Dominated Open Capillary Channel Flows Through a Groove

An open capillary channel is a structure that establishes a liquid flow path when the capillary pressure caused by surface tension forces dominates in comparison to the hydrostatic pressure induced by gravitational or residual accelerations. To maintain a steady flow through the channel the capillary pressure of the free surface has to balance the pressure difference between the liquid and the surrounding constant pressure gas phase. Due to convective and viscous momentum transport the pressure along the flow path of the liquid decreases and causes the free surface to bend inwards. The maximum flow rate through the channel is reached when the free surface collapses and gas ingestion occurs near the outlet. This stability limit depends on the geometry of the channel and the properties of the liquid. In this paper we present an experimental setup which is used in the low-gravity environment of the Bremen Drop Tower. Experiments with convective dominated systems have been performed where the flow rate was increased up to the maximum value. In comparison to this we present a one-dimensional theoretical model to determine important characteristics of the flow, such as the free surface shape and the limiting flow rate. Furthermore we present an explanation for the mechanism of flow rate limitation for these flow conditions which is similar to the choking problem for compressible gas flows.     

Free‐slip boundary conditions for simulating free‐surface incompressible flows through the particle finite element method

The possibility of using free‐slip conditions within the context of the particle finite element method (PFEM) is investigated. For high Reynolds number engineering applications in which tangential effects at the fluid–solid boundaries are not of primary interest, the use of free‐slip conditions can alleviate the need for very fine boundary layer meshes. Two novel ways for the imposition of free‐slip conditions in the framework of the PFEM are presented. The proposed approach emphasizes robustness and simplicity, while retaining a sufficient level of generality. These two methods are then tested in the case of dam break and sloshing problems, and their respective advantages and drawbacks are discussed. It is also shown how the use of free‐slip conditions can indirectly improve mass conservation properties of the PFEM, even when coarse meshes are employed. Copyright © 2016 John Wiley & Sons, Ltd.     

Computation of Stokes Flow due to the Motion or Presence of a Particle in a Tube

The computation of Stokes flow due to the motion or presence of a rigid particle in a fluid-filled tube with arbitrary geometry is discussed with emphasis on the induced upstream to downstream pressure change. It is proposed that expressing the pressure change as an integral over the particle surface involving (a) the a priori unknown traction, and (b) the velocity of the pure-fluid pressure-driven flow, simplifies the numerical implementation and ameliorates the effect of domain truncation. Numerical computations are performed based on the integral formulation in conjunction with a boundary-element method for a particle translating and rotating inside a cylindrical tube with a circular cross-section. The numerical results are consistent with previous asymptotic solutions for small particles, and complement available numerical solutions for particular types of motion