A multiphase compressible model for the simulation of multiphase flows

A compressible model able to manage incompressible two-phase flows as well as compressible motions is proposed. After a presentation of the multiphase compressible concept, the new model and related numerical methods are detailed on fixed structured grids. The presented model is a 1-fluid model with a reformulated mass conservation equation which takes into account the effects of compressibility. The coupling between pressure and flow velocity is ensured by introducing mass conservation terms in the momentum and energy equations. The numerical model is then validated with four test cases involving the compression of an air bubble by water, the liquid injection in a closed cavity filled with air, a bubble subjected to an ultrasound field and finally the oscillations of a deformed air bubble in melted steel. The numerical results are compared with analytical results and convergence orders in space are provided.     

The adaptive Lagrangian particle method for macroscopic and micro–macro computations of time-dependent viscoelastic flows

We propose a new numerical technique, referred to as the Adaptive Lagrangian Particle Method (ALPM), for computing time-dependent viscoelastic flows using either a differential constitutive equation (macroscopic approach) or a kinetic theory model (micro–macro approach). In ALPM, the Eulerian finite element solution of the conservation equations is decoupled from the Lagrangian computation of the extra-stress at a number of discrete particles convected by the flow. In the macroscopic approach, the extra-stress carried by the particles is obtained by integrating the constitutive equation along the particle trajectories. In the micro–macro approach, the extra-stress is computed by solving along the particle paths the stochastic differential equation associated with the kinetic theory model. At each time step, ALPM automatically enforces that all elements of the mesh have a number of Lagrangian particles ranging within a user-specified interval. Results are given for the start-up flow between highly eccentric rotating cylinders, using the FENE and FENE-P dumbbell models for dilute polymer solutions.     

Development and sensitivity analysis of a Lagrangian particle model for long range dispersion

A new long range dispersion model of Lagrangian particle type (MILORD) has been recently developed at the “Istituto di Cosmogeofisica”. Its capabilities have been tested by comparing its predictions with the Cs-137 air concentrations recorded over Europe by many laboratories during the Chernobyl accident. MILORD sensitivity to variations in its parametrizations has been studied and, in order to ascertain the accuracy of the simulations, some widely used statistical indexes and graphs have been computed. The study has made possible the accomplishment of a set of model input parameters able to produce the best agreement between observations and predictions. It is shown that this model was able to reconstruct with a good accuracy the main characteristics of the radioactive cloud spread over Europe.     

An adaptative augmented Lagrangian method for three-dimensional multimaterial flows

The direct numerical simulation of incompressible multimaterial flows, based on predictor/corrector and volume of fluid (VOF) approaches is presented. An original adaptative augmented Lagrangian method is proposed to solve the predictor solution, satisfying at the same time the conservation equations as well as the incompressibility constraint. This algorithm is based on an Uzawa optimisation technique. The corrector solution is obtained with a projection method on a divergence free subspace. Several examples of two- and three-dimensional flows are proposed to illustrate the ability of the method to deal with unsteady, multimaterial problems.     

Extended Lagrangian particle-in-cell (ELPIC) code for inhomogeneous compressible flows

ELPIC, a macroparticle code for modeling complex nonstationary inhomogeneous compressible flows is described and demonstrated. It operates with Langrangian-type finite-sized rectangular particles with adjustable sizes. The dimensions of the particles smoothly adjust to peculiarities of the flow in order to minimize overlapping and gaps between the particles. The particles can divide in rarefaction waves to smoothly cover the simulated flow area. The particles carry mass, which is constant between divisions, and a number of chemical and thermodynamical properties of the substance they represent, including the index of the equation of state, chemical composition, mechanical properties, etc. The ELPIC approach combines the essential advantages of both Eulerian and Langrangian approaches, and overcomes the difficulties encountered by Nishiguchi and Yabe in their well-known code SOAP, based on conceptually similar principles.     

Evaluation of three lattice Boltzmann models for multiphase flows in porous media

A free energy (FE) model, the Shan–Chen (S–C) model, and the Rothman and Keller (R–K) model are studied numerically to evaluate their performance in modeling two-dimensional (2D) immiscible two-phase flow in porous media on the pore scale. The FE model is proved to satisfy the Galilean invariance through a numerical test and the mass conservation of each component in the simulations is exact. Two-phase layered flow in a channel with different viscosity ratios was simulated. Comparing with analytical solutions, we see that the FE model and the R–K model can give very accurate results for flows with large viscosity ratios. In terms of accuracy and stability, the FE model and the R–K model are much better than the S–C model. Co-current and countercurrent two-phase flows in complex homogeneous media were simulated and the relative permeabilities were obtained. Again, it is found that the FE model is as good as the R–K model in terms of accuracy and efficiency. The FE model is shown to be a good tool for the study of two-phase flows with high viscosity ratios in porous media.     

Stencil adaptive diffuse interface method for simulation of two-dimensional incompressible multiphase flows

Diffuse interface method is becoming a more and more popular approach for simulation of multiphase flows. As compared to other solvers, it is easy to implement and can keep conservation of mass and momentum. In the diffuse interface method, the interface is not considered as a sharp discontinuity. Instead, it treats the interface as a diffuse layer with a small thickness. This treatment is similar to the shock-capturing method. To have a fine resolution around the interface, one has to use very fine mesh in the computational domain. As a consequence, a large computational effort will be needed. To improve the computational efficiency, this paper incorporates the efficient 5-points stencil adaptive algorithm [1] into the diffuse interface method with local refinement around the interface and then applies the developed method to simulate two-dimensional incompressible multiphase flows. Three cases are chosen to test the performance of the method, including Young-Laplace law for a 2D drop, drop deformation in the shear flow and viscous finger formation. The method is well validated through the comparison with theoretical analysis or earlier results available in the literature. It is shown that the method can obtain accurate results at much lower cost, even for problems with moving contact lines. The improvement of computational efficiency by the stencil adaptive algorithm is demonstrated obviously.     

Eulerian and Lagrangian particle transport with drag

The linear Boltzmann equation is formulated in relative velocities against a moving background, and coupled to the Eulerian and Lagrangian continuity equations through the material divergence. Material acceleration terms are expanded locally, effective drag coefficients and moving sources are defined, and incorporated in the transport equation in consistent fashion. Standard multigroup representations for the differenced material motion and drag terms are obtained. Specific applications are presented and contrasted. The representation in relative velocities serves as a useful alternative to a formulation using effective cross sections defined against a moving background.     

Cluster integration method in Lagrangian particle dynamics

An efficient and robust approach is proposed in order to conduct numerical simulations of collisional particle dynamics in the Lagrangian framework. Clusters of particles are made of particles that interact or may interact during the next global time-step. Potential collision partners are found by performing a test move, that follows the patterns of a hard-sphere model. The clusters are integrated separately and the collisional forces between particles are given by a soft-sphere collision model. However, the present approach also allows longer range inter-particle forces. The integration of the clusters can be done by any one-step ordinary differential equation solver, but for dilute particle systems, the variable step-size Runge-Kutta solvers as the Dormand and Prince scheme [J. Comput. Appl. Math. 6 (1980) 19] are superior. The cluster integration method is applied on sedimentation of 5000 particles in a two-dimensional box. A significant speed-up is achieved. Compared to a traditional discrete element method with the forward Euler scheme, a speed-up factor of three orders of magnitude in the dilute regime and two orders of magnitude in the dense regime were observed. As long as the particles are dilute, the Dormand and Prince scheme is ten times faster than the classical fourth-order Runge-Kutta solver with fixed step size.     

Using augmented Lagrangian particle swarm optimization for constrained problems in engineering">Using augmented Lagrangian particle swarm optimization for constrained problems in engineering

The comparatively new stochastic method of particle swarm optimization (PSO) has been applied to engineering problems especially of nonlinear, non-differentiable, or non-convex type. Its robustness and its simple applicability without the need for cumbersome derivative calculations make PSO an attractive optimization method. However, engineering optimization tasks often consist of problem immanent equality and inequality constraints which are usually included by inadequate penalty functions when using stochastic algorithms. The simple structure of basic particle swarm optimization characterized by only a few lines of computer code allows an efficient implementation of a more sophisticated treatment of such constraints. In this paper, we present an approach which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrange multiplier method, for constraint handling where ill conditioning is a far less harmful problem and the correct solution can be obtained even for finite penalty factors. We describe the basic PSO algorithm and the resulting method for constrained problems as well as the results from benchmark tests. An example of a stiffness optimization of an industrial hexapod robot with parallel kinematics concludes this paper and shows the applicability of the proposed augmented Lagrange particle swarm optimization to engineering problems.     

多相图像分割Vese-Chan模型连续最大流方法

目的 多相图像分割是图像处理与分析的重要问题,变分图像分割的Vese-Chan模型是多相图像分割的基本模型,由于该模型使用较少的标签函数构造区域划分的特征函数,具有求解规模小的优点。图割（graph cut,GC）算法可将上述能量泛函的极值问题转化为最小割/最大流问题求解,大大提高了计算效率。连续最大流（continuous max-flow,CMF）方法是经典GC算法的连续化表达,不仅具备GC算法的高效性,且克服了经典GC算法由于离散导致的精度下降问题。本文提出基于凸松弛的多相图像分割Vese-Chan模型的连续最大流方法。方法 根据划分区域编号的二进制表示构造两类特征函数,将多相图像分割转化为多个交替优化的两相图像分割问题。引入对偶变量将Vese-Chan模型转化为与最小割问题相对应的连续最大流问题,并引入Lagrange乘子设计交替方向乘子方法（alternating direction method of multipliers,ADMM）,将能量泛函的优化问题转化为一系列简单的子优化问题。结果 对灰度图像和彩色图像进行数值实验,从分割效果看,本文方法对于医学图像、遥感图像等复杂图像的分割效果更加精确,对分割对象和背景更好地分离;从分割效率看,本文方法减少了迭代次数和运算时间。在使用2个标签函数的分割实验中,本文方法运算时间加速比分别为6.35%、10.75%、12.39%和7.83%;在使用3个标签函数的分割实验中,运算时间加速比分别为12.32%、15.45%和14.04%;在使用4个标签函数的分割实验中,运算时间加速比分别为16.69%和20.07%。结论 本文提出的多相图像分割Vese-Chan模型的连续最大流方法优化了分割效果,减少了迭代次数,从而提高了计算效率。     

多相Chan-Vese模型的直接对偶方法

多相图像分割的变分模型采用水平集函数定义不同区域的特征函数,其极值问题需要迭代求解一系列动态演化方程,计算效率低。较快的方法是对离散的二值标记函数凸松弛后设计对偶方法或Split Bregman方法,并结合阈值化技术得到分割结果。提出一种无需凸松弛和阈值化的快速分割方法—直接对偶方法(DDM)。DDM利用二值标记函数的二值特性,并根据KKT条件得到原变量的二值解析解和对偶变量的简单迭代格式。该方法首先应用到两相Chan-Vese模型,然后拓展到多相Chan-Vese模型。实验结果表明,DDM比梯度降方法、对偶方法和Split Bregman方法分割效果好、计算效率高。     

Some aspects of the modeling at different scales of multiphase flows

We present the various levels of possible modeling for multiphase flows: coupling of fluid equations in different domains with a free boundary; coupling (in the same domain) of a fluid equation and a kinetic (Vlasov or Vlasov–Boltzmann) equation; coupling (in the same domain) of two (or more) fluid equations. We briefly present the mathematical results relative to the passage from one of these approaches to another approach, and we give some ideas of how to use those different models on a specific practical example.     

A particle system for interactive visualization of 3D flows

We present a particle system for interactive visualization of steady 3D flow fields on uniform grids. For the amount of particles we target, particle integration needs to be accelerated and the transfer of these sets for rendering must be avoided. To fulfill these requirements, we exploit features of recent graphics accelerators to advect particles in the graphics processing unit (GPU), saving particle positions in graphics memory, and then sending these positions through the GPU again to obtain images in the frame buffer. This approach allows for interactive streaming and rendering of millions of particles and it enables virtual exploration of high resolution fields in a way similar to real-world experiments. The ability to display the dynamics of large particle sets using visualization options like shaded points or oriented texture splats provides an effective means for visual flow analysis that is far beyond existing solutions. For each particle, flow quantities like vorticity magnitude and A2 are computed and displayed. Built upon a previously published GPU implementation of a sorting network, visibility sorting of transparent particles is implemented. To provide additional visual cues, the GPU constructs and displays visualization geometry like particle lines and stream ribbons.     

A control-volume finite element method for dilute gas-solid particle flows

The formulation of a co-located equal-order Control-Volume-based Finite Element Method (CVFEM) for the solution of two-fluid models of 2-D, planar or axisymmetric, incompressible, dilute gas-solid particle flows is presented. The proposed CVFEM is formulated by borrowing and extending ideas put forward in earlier CVFEMs for single-phase flows. In axisymmetric problems, the calculation domain is discretized into torus-shaped elements and control volumes: in a longitudinal cross-sectional plane, or in planar problems, these elements are three-node triangles, and the control volumes are polygons obtained by joining the centroids of the three-node triangles to the midpoints of the sides. In each element, mass-weighted skew upwind functions are used to interpolate the convected scalar dependent variables and the volume concentrations. An iterative variable adjustment algorithm is used to solve the discretized equations. The capabilities of the proposed CVFEM are illustrated by its application to two test problems and one demonstration problem, using a simple two-fluid model for dilute gas-solid particle flows. The results are quite encouraging.     

A model for real-time on-surface flows

Simulating fluid flows for visualization purposes is known to be one of the most challenging fields of the computer graphics domain. While rendering vast liquid areas has been widely addressed this last decade, few papers have tackled the problematic of on-surface flows, even though real-time applications such as drive simulators or video games could greatly benefit from such methods. We present a novel empirical method for the animation of liquid droplets lying on a flat surface, the core of our technique being a simulation operating on a 2D grid which is implementable on GPU. The wetted surface can freely be oriented in space and is not limited to translucent materials, the liquid flow being governed by external forces, the viscosity parameter and the presence of obstacles. Furthermore, we show how to simply incorporate in our simulation scheme two enriching visual effects, namely absorption and ink transport. Rendering can be achieved from an arbitrary view point using a GPU image based raycasting approach and takes into account the refraction and reflection of light. Even though our method doesn’t benefit from the literature of fluid mechanics, we show that convincing animations in terms of realism can be achieved in real-time.     

Self-ordered particle trains in inertial microchannel flows

Controlling the transport of particles in flowing suspensions at microscale is of interest in numerous contexts such as the development of miniaturized and point-of-care analytical devices (in bioengineering, for foodborne illnesses detection, etc.) and polymer engineering. In square microchannels, neutrally buoyant spherical particles are known to migrate across the flow streamlines and concentrate at specific equilibrium positions located at the channel centerline at low flow inertia and near the four walls along their symmetry planes at moderate Reynolds numbers. Under specific flow and geometrical conditions, the spherical particles are also found to line up in the flow direction and form evenly spaced trains. In order to statistically explore the dynamics of train formation and their dependence on the physical parameters of the suspension flow (particle-to-channel size ratio, Reynolds number and solid volume fraction), experiments have been conducted based on in situ visualizations of the flowing particles by optical microscopy. The trains form only once particles have reached their equilibrium positions (following lateral migration). The percentage of particles in trains and the interparticle distance in a train have been extracted and analyzed. The percentage of particles organized in trains increases with the particle Reynolds number up to a threshold value which depends on the concentration and then decreases for higher values. The average distance between the surfaces of consecutive particles in a train decreases as the particle Reynolds number increases and is independent of the particles size and concentration, if the concentration remains below a threshold value related to the degree of confinement of the suspension flow.     

A novel multiphase active contour model for inhomogeneous image segmentation

The problem of image segmentation has been investigated with a focus on inhomogeneous multiphase image segmentation. Intensity inhomogeneity is an undesired phenomenon that represents the main obstacle for magnetic resonance (MR) and natural images segmentation. The complex images usually contain an arbitrary number of objects. This paper presents a new multiphase active contour model method for simultaneous regions classification of MR images and natural images without bias field correction. In this model, a simple and effective initialization method is taken to speed up the curve evolution toward final results; a new multiphase level set method is proposed to segment the multiple regions. This model not only extracts multiple objects simultaneously, but also provides smooth and accurate boundaries of the objects. The results for experiments on several synthetic and real images demonstrate the effectiveness and accuracy of our model.     

Modeling the infiltration of multiphase fluid flows in a layered porous medium

Multiphase fluid flows in a layered porous medium are studied with allowance for capillary and gravitation forces. The developed computational algorithms have been tested on a number of test problems of two- and three-phase flows. The suggested approach can be used to deal with applied environmental problems of soil and ground water contamination by petroleum products and other agents.     

An ISPH scheme for numerical simulation of multiphase flows with complex interfaces and high density ratios

Multiphase problems with high density ratios and complex interfaces deal with numerical instabilities and require accurate considerations for capturing the multiphase interfaces. An Incompressible Smoothed Particle Hydrodynamics (ISPH) scheme is presented to simulate such problems. In order to keep the present scheme simple and stable, well-established formulations are used for discretizing the spatial derivatives and a repulsive force is applied at the multiphase interface between particles of different fluids to maintain the interface sharpness. Special considerations are included to overcome the difficulties to model severe physical discontinuities at the interface and surface tension effects are taken into account. Different particle shifting schemes are also tested for a range of problems. Several two phase flows are investigated and the presented scheme is validated against both analytical and numerical solutions. A detailed study is also carried out on the influence of the repulsive force in an ISPH scheme showing that this simple treatment efficiently enhances the interface capturing features. The comparisons indicate that the proposed scheme is robust and capable of simulating a wide range of multiphase problems with complex interfaces including low to high ratios for density and viscosity.