An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows

A three-dimensional, incompressible, multiphase particle-in-cell method is presented for dense particle flows. The numerical technique solves the governing equations of the fluid phase using a continuum model and those of the particle phase using a Lagrangian model. Difficulties associated with calculating interparticle interactions for dense particle flows with volume fractions above 5% have been eliminated by mapping particle properties to an Eulerian grid and then mapping back computed stress tensors to particle positions. A subgrid particle, normal stress model for discrete particles which is robust and eliminates the need for an implicit calculation of the particle normal stress on the grid is presented. Interpolation operators and their properties are defined which provide compact support, are conservative, and provide fast solution for a large particle population. The solution scheme allows for distributions of types, sizes, and density of particles, with no numerical diffusion from the Lagrangian particle calculations. Particles are implicitly coupled to the fluid phase, and the fluid momentum and pressure equations are implicitly solved, which gives a robust solution.     

A numerical source of small-scale number-density fluctuations in Eulerian–Lagrangian simulations of multiphase flows

Eulerian–Lagrangian simulations of multiphase flow are known to suffer from two errors that can introduce small-scale fluctuations in the number-density of the dispersed phase. These errors can be reduced by increasing the number of particles in the simulation. Here, we present results to demonstrate that a third error exists that can also generate small-scale number-density fluctuations. In contrast to the two known errors, this error cannot be lowered by increasing the number of particles. Analysis shows that this error is caused by spatial variation at the subgrid scale in the interpolation error of the fluid velocity to the particle location. If the particle velocity divergence is zero, the particle concentration error remains at the subgrid scale. However, if particles preferentially accumulate either due to their inertia or due to divergence of the underlying fluid-velocity field, this error manifests as number-density fluctuations on the grid scale. The only mechanism of reducing these errors is through higher-order accurate interpolation. By studying two model problems, estimates for the errors are derived. These estimates are shown to be quite accurate for simulations of shock and expansion waves interacting with particles.     

The shallow water equations in Lagrangian coordinates

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge–Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations.     

Modeling material responses by arbitrary Lagrangian Eulerian formulation and adaptive mesh refinement method

In this paper we report an efficient numerical method combining a staggered arbitrary Lagrangian Eulerian (ALE) formulation with the adaptive mesh refinement (AMR) method for materials modeling including elastic–plastic flows, material failure, and fragmentation predictions. Unlike traditional AMR applied on fixed domains, our investigation focuses on the application to moving and deforming meshes resulting from Lagrangian motion. We give details of this numerical method with a capability to simulate elastic–plastic flows and predict material failure and fragmentation, and our main focus of this paper is to create an efficient method which combines ALE and AMR methods to simulate the dynamics of material responses with deformation and failure mechanisms. The interlevel operators and boundary conditions for these problems in AMR meshes have been investigated, and error indicators to locate material deformation and failure regions are studied. The method has been applied on several test problems, and the solutions of the problems obtained with the ALE–AMR method are reported. Parallel performance and software design for the ALE–AMR method are also discussed.     

A fixed-mesh method for incompressible flow–structure systems with finite solid deformations

A fixed-mesh algorithm is proposed for simulating flow–structure interactions such as those occurring in biological systems, in which both the fluid and solid are incompressible and the solid deformations are large. Several of the well-known difficulties in simulating such flow–structure interactions are avoided by formulating a single set of equations of motion on a fixed Eulerian mesh. The solid’s deformation is tracked to compute elastic stresses by an overlapping Lagrangian mesh. In this way, the flow–structure interaction is formulated as a distributed body force and singular surface force acting on an otherwise purely fluid system. These forces, which depend on the solid elastic stress distribution, are computed on the Lagrangian mesh by a standard finite-element method and then transferred to the fixed Eulerian mesh, where the joint momentum and continuity equations are solved by a finite-difference method. The constitutive model for the solid can be quite general. For the force transfer, standard immersed-boundary and immersed-interface methods can be used and are demonstrated. We have also developed and demonstrated a new projection method that unifies the transfer of the surface and body forces in a way that exactly conserves momentum; the interface is still effectively sharp for this approach. The spatial convergence of the method is observed to be between first- and second-order, as in most immersed-boundary methods for membrane flows. The algorithm is demonstrated by the simulations of an advected elastic disk, a flexible leaflet in an oscillating flow, and a model of a swimming jellyfish.     

One-step hybrid remapping algorithm for multi-material arbitrary Lagrangian–Eulerian methods

In this paper, a new flux-based one-step hybrid remapping method for multi-material arbitrary Lagrangian–Eulerian (ALE) approach is introduced. In the vicinity of material interfaces, the swept region is intersected with pure material polygons in the Lagrangian mesh to construct the material fluxes. Far from interfaces, the fluxes are constructed in a standard swept-region manner without intersections. This method is conservative, second-order accurate and linearity-preserving (in case of straight material interfaces), and faster than method based on intersections, as shown on selected numerical examples.     

Comparisons of Lagrangian and Eulerian PDF methods in simulations of non-premixed turbulent jet flames with moderate-to-strong turbulence-chemistry interactions

Transported probability density function (PDF) methods have been applied widely and effectively for modelling turbulent reacting flows. In most applications of PDF methods to date, Lagrangian particle Monte Carlo algorithms have been used to solve a modelled PDF transport equation. However, Lagrangian particle PDF methods are computationally intensive and are not readily integrated into conventional Eulerian computational fluid dynamics (CFD) codes. Eulerian field PDF methods have been proposed as an alternative. Here a systematic comparison is performed among three methods for solving the same underlying modelled composition PDF transport equation: a consistent hybrid Lagrangian particle/Eulerian mesh (LPEM) method, a stochastic Eulerian field (SEF) method and a deterministic Eulerian field method with a direct-quadrature-method-of-moments closure (a multi-environment PDF-MEPDF method). The comparisons have been made in simulations of a series of three non-premixed, piloted methane–air turbulent jet flames that exhibit progressively increasing levels of local extinction and turbulence-chemistry interactions: Sandia/TUD flames D, E and F. The three PDF methods have been implemented using the same underlying CFD solver, and results obtained using the three methods have been compared using (to the extent possible) equivalent physical models and numerical parameters. Reasonably converged mean and rms scalar profiles are obtained using 40 particles per cell for the LPEM method or 40 Eulerian fields for the SEF method. Results from these stochastic methods are compared with results obtained using two- and three-environment MEPDF methods. The relative advantages and disadvantages of each method in terms of accuracy and computational requirements are explored and identified. In general, the results obtained from the two stochastic methods (LPEM and SEF) are very similar, and are in closer agreement with experimental measurements than those obtained using the MEPDF method, while MEPDF is the most computationally efficient of the three methods. These and other findings are discussed in detail.     

On a new defect of shock-capturing methods

This paper proposes an explanation and a cure (or avoidance) to the new defect found of Eulerian shock-capturing methods in “A note on the conservative schemes for the Euler equations” by Tang and Liu [H. Tang, Tiegang Liu, A note on the conservative schemes for the Euler equations, J. Comput. Phys. 218 (2006) 451–459]. The latter gives a numerical investigation using several popular high resolution conservative schemes applied to Riemann problems of inviscid, compressible, perfect gas flows in Eulerian and Lagrangian coordinates with an initial high density ratio as well as a high pressure ratio. The results show that these methods work very inefficiently when applied to such problems and may give inaccurate numerical results, especially in shock location (or speed), even with a very fine grid.We have found that in problems of this type a strong rarefaction wave (SRW) is present adjacent to a contact line. Godunov averaging over the wave then produces large errors which, when the wave is strong, also persist for a long time. The cumulative error is thus very large which violates the strength of the contact line adjacent to it which, in turn, affects the speed and hence the location of the shock on the other side of the contact. We confirm this numerically using a method based on the unified coordinates with the shock-adaptive Godunov scheme plus contact strength preserving. The method, when applied to the Examples 2.1 and 2.2 of Tang and Liu [H. Tang, Tiegang Liu, A note on the conservative schemes for the Euler equations, J. Comput. Phys. 218 (2006) 451–459], produces high quality results even for comparatively coarse grids.     

A geometrical predictor–corrector advection scheme and its application to the volume fraction function

We present a multidimensional Eulerian advection method for interfacial and incompressible flows in two-dimensional Cartesian geometry. In the scheme we advect the grid nodes backwards along the streamlines to compute the pre-images of the grid lines. These pre-images are approximated by continuous, piecewise-linear lines. The enforcement of the discrete version of the incompressibility constraint is a very important issue to determine correctly the flux polygons and to reduce considerably the integration, discretization and interpolation numerical errors. The proposed method compares favorably with other previous multidimensional advection methods as long as the initial interface line is well reconstructed. Conversely, we show that when the interface is very fragmented the total numerical error is completely dominated by the reconstruction error and in these conditions it is very difficult to assess which advection scheme is the most reliable one.     

Eulerian–Lagrangian multiscale methods for solving scalar equations – Application to incompressible two-phase flows

The present article proposes a new hybrid Eulerian–Lagrangian numerical method, based on a volume particle meshing of the Eulerian grid, for solving transport equations. The approach, called Volume Of Fluid Sub-Mesh method (VOF-SM), has the advantage of being able to deal with interface tracking as well as advection–diffusion transport equations of scalar quantities. The Eulerian evolutions of a scalar field could be obtained on any orthogonal curvilinear grid thanks to the Lagrangian advection and a redistribution of particles on the Eulerian grid. The Eulerian concentrations result from the projection of the volume and scalar informations handled by the particles. The particle velocities are interpolated from the Eulerian velocity field. The VOF-SM method is validated on several scalar interface tracking and transport problems and is compared to existing schemes within the literature. It is finally coupled to a Navier–Stokes solver and applied to the simulation of two free-surface flows, i.e. the two-dimensional buckling of a viscous jet during the filling of a square mold and the three-dimensional dam-break flow in a tank.     

A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows

A momentum exchange-based immersed boundary-lattice Boltzmann method is presented in this Letter for simulating incompressible viscous flows. This method combines the good features of the lattice Boltzmann method (LBM) and the immersed boundary method (IBM) by using two unrelated computational meshes, an Eulerian mesh for the flow domain and a Lagrangian mesh for the solid boundaries in the flow. In this method, the non-slip boundary condition is enforced by introducing a forcing term into the lattice Boltzmann equation (LBE). Unlike the conventional IBM using the penalty method with a user-defined parameter or the direct forcing scheme based on the Navier–Stokes (NS) equations, the forcing term is simply calculated by the momentum exchange of the boundary particle density distribution functions, which are interpolated by the Lagrangian polynomials from the underlying Eulerian mesh. Numerical examples show that the present method can provide very accurate numerical results.     

自适应多分辨率方法在反应多相流数值模拟中的应用

将自适应多分辨率方法应用到刚性气体状态方程CJ模型的数值模拟,采用多相流问题的守恒锐利界面格式,通过level-set方法和虚拟流体方法来追踪和处理界面,能够很好地处理时间尺度较大的界面交互问题.利用金字塔型数据结构和多分辨率自适应方法,提高算法的存储效率和计算效率.给出一维和二维的数值算例,证明该算法在反应多相流数值模拟中的稳定性和高效性.     

Flux-corrected transport. III. Minimal-error FCT algorithms

This paper presents an error analysis of numerical algorithms for solving the convective continuity equation using flux-corrected transport (FCT) techniques. The nature of numerical errors in Eulerian finite-difference solutions to the continuity equation is analyzed. The properties and intrinsic errors of an “optimal” algorithm are discussed and a flux-corrected form of such an algorithm is demonstrated for a restricted class of problems. This optimal FCT algorithm is applied to a model test problem and the error is monitored for comparison with more generally applicable algorithms. Several improved FCT algorithms are developed and judged against both standard flux-uncorrected transport algorithms and the optimal algorithm. These improved FCT algorithms are found to be four to eight times more accurate than standard non-FCT algorithms, nearly twice as accurate as the original SHASTA FCT algorithm, and approach the accuracy of the optimal algorithm.     

Application of the Lagrangian CMC method in elliptic bluff body and swirl flames

The Lagrangian CMC method was implemented in the open source programme OpenFOAM and applied to turbulent nonpremixed bluff body and swirl flames. Lagrangian CMC is more efficient than Eulerian CMC with the number of Lagrangian flame groups much less than the number of computational cells for Eulerian CMC equations in general. It is based on the conditional flame structure depending on the residence time of the fuel of fixed Lagrangian identity from the nozzle. According to sensitivity study the injected fuel was divided into ten flame groups according to the injection sequence with the resulting conditional profiles between those by ISR and Eulerian CMC. Minor deviation from Eulerian CMC was attributed to the flame structure that is difficult to be characterised by the residence time only in elliptic recirculating flows of the bluff body and swirl flames. The Eulerian and Lagrangian CMC showed the same trend of deviation from measurements for conditional temperature, H₂O, OH, CO and H₂ mass fractions. The significant deviation of H₂ was due to uncertainty in the reaction chemistry, as observed in the previous works based on other reaction mechanisms for methane and methanol.     

Arbitrary Lagrangian Eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping

The dynamic modification of the computational grid due to element displacement, deformation and edge swapping is described here in terms of suitably-defined continuous (in time) alterations of the geometry of the elements of the dual mesh. This new interpretation allows one to describe all mesh modifications within the arbitrary Lagrangian Eulerian framework, thus removing the need to interpolate the solution across computational meshes with different connectivity. The resulting scheme is by construction conservative and it is applied here to the solution of the Euler equations for compressible flows in two spatial dimensions. Preliminary two dimensional numerical simulations are presented to demonstrate the soundness of the approach. Numerical experiments show that this method allows for large time steps without causing element invalidation or tangling and at the same time guarantees high quality of the mesh elements without resorting to global re-meshing techniques, resulting in a very efficient solver for the analysis of e.g. fluid-structure interaction problems, even for those cases that require large mesh deformations or changes in the domain topology.     

典型自由面流动问题的SPH-ALE数值模拟

应用基于黎曼解的SPH-ALE方法对两种典型自由面流动问题进行数值模拟,并提出一种一阶核函数修正压力计算方法,通过对临近边界的水粒子压力进行核积分,近似估算固壁边界压力.给出不同时刻的流场压力分布及自由液面演化过程,将计算结果与相关的试验值及数值解进行对比,分析结果表明：SPH-ALE方法较传统SPH方法在流场压力计算精度上有较大的改进,在处理强非线性自由面流动问题时能够达到较高的精度.     

Geometric gradient-flow dynamics with singular solutions

The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcy’s Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.     

Modeling Three-Dimensional Multiphase Flow Using a Level Contour Reconstruction Method for Front Tracking without Connectivity

Three-dimensional multiphase flow and flow with phase change are simulated using a simplified method of tracking and reconstructing the phase interface. The new level contour reconstruction technique presented here enables front tracking methods to naturally, automatically, and robustly model the merging and breakup of interfaces in three-dimensional flows. The method is designed so that the phase surface is treated as a collection of physically linked but not logically connected surface elements. Eliminating the need to bookkeep logical connections between neighboring surface elements greatly simplifies the Lagrangian tracking of interfaces, particularly for 3D flows exhibiting topology change. The motivation for this new method is the modeling of complex three-dimensional boiling flows where repeated merging and breakup are inherent features of the interface dynamics. Results of 3D film boiling simulations with multiple interacting bubbles are presented. The capabilities of the new interface reconstruction method are also tested in a variety of two-phase flows without phase change. Three-dimensional simulations of bubble merging and droplet collision, coalescence, and breakup demonstrate the new method's ability to easily handle topology change by film rupture or filamentary breakup. Validation tests are conducted for drop oscillation and bubble rise. The susceptibility of the numerical method to parasitic currents is also thoroughly assessed.