An interface treating technique for compressible multi-medium flow with Runge–Kutta discontinuous Galerkin method

The high-order accurate Runge–Kutta discontinuous Galerkin (RKDG) method is applied to the simulation of compressible multi-medium flow, generalizing the interface treating method given in Chertock et al. (2008) [9]. In mixed cells, where the interface is located, Riemann problems are solved to define the states on both sides of the interface. The input states to the Riemann problem are obtained by extrapolation to the cell boundary from solution polynomials in the neighbors of the mixed cell. The level set equation is solved by using a high-order accurate RKDG method for Hamilton–Jacobi equations, resulting in a unified DG solver for the coupled problem. The method is conservative if we include the states in the mixed cells, which are however not used in the updating of the numerical solution in other cells. The states in the mixed cells are plotted to better evaluate the conservation errors, manifested by overshoots/undershoots when compared with states in neighboring cells. These overshoots/undershoots in mixed cells are problem dependent and change with time. Numerical examples show that the results of our scheme compare well with other methods for one and two-dimensional problems. In particular, the algorithm can capture well complex flow features of the one-dimensional shock entropy wave interaction problem and two-dimensional shock–bubble interaction problem.     

一种固定网格上拉格朗日点追踪的快速算法

提出一种基于非结构网格颗粒追踪的改进算法.在已有算法的基础上,发展适用于包含任意多边形单元界面的混合网格算法,对不共面的空间多边形界面进行虚拟三角化,实现基于任意混合网格的拉格朗日追踪.数值试验证明方法可行可靠.通过对固定容器内喷入的不同数量颗粒进行追踪的数值测试及与文献对比,显示算法高效,在颗粒数为10000时,效率提高约47%.     

Identification of Regions with Inhomogeneous Particle Behavior in Dilute Gas‐solid Flows

The behavior of the particulate phase in a highly turbulent gas flow has been investigated in a vertical channel. Variations of the flow configuration (1. Flow past a cylinder, 2. flow past a wall‐mounted obstacle and 3. flow around a horizontally injected jet) have been subject to both experiments and numerical simulations. The velocity vector field of the solid phase has been measured by digital particle image velocimetry (DPIV). The measurements have been focused on particle‐obstacle collisions and crossflow in the vicinity of the jet nozzle using the lately developed twinpeak detection method. By application of this method regions of highly inhomogeneous particle behavior could be detected mainly upstream of the flow perturbation. Numerical results have been obtained by an Eulerian‐Lagrangian method on boundary‐fitted grids. Particle‐particle interactions as well as interphase exchange of momentum have been taken into account. The simulation results showed to be well in accordance with the experimental results.     

基于Riemann解的二维流体力学Lagrange有限点无网格方法

在高维流体力学计算中,对于多介质大变形等一类问题,采用有网格方法常遇到较大的困难.针对二维问题,研究了一种无网格方法——Lagrange有限点方法:在求解区域上设置适当的离散点集,视其中每一点为流体力学Lagrange点;对于点集的任一点,确定邻点集合,并基于该点同邻点集合的联系,应用Godunov方法将流体力学Lagrange方程进行离散;考虑到算法的稳健性,方法中可设置较多邻点并采用最小二乘法.将该方法应用于典型的数值算例,取得了良好效果.     

Euler calculations with embedded Cartesian grids and small-perturbation boundary conditions

This study examines the use of stationary Cartesian mesh for steady and unsteady flow computations. The surface boundary conditions are imposed by reflected points. A cloud of nodes in the vicinity of the surface is used to get a weighted average of the flow properties via a gridless least-squares technique. If the displacement of the moving surface from the original position is typically small, a small-perturbation boundary condition method can be used. To ensure computational efficiency, multigrid solution is made via a framework of embedded grids for local grid refinement. Computations of airfoil wing and wing-body test cases show the practical usefulness of the embedded Cartesian grids with the small-perturbation boundary conditions approach.     

Particle filtering with path sampling and an application to a bimodal ocean current model

This paper introduces a recursive particle filtering algorithm designed to filter high dimensional systems with complicated non-linear and non-Gaussian effects. The method incorporates a parallel marginalization (PMMC) step in conjunction with the hybrid Monte Carlo (HMC) scheme to improve samples generated by standard particle filters. Parallel marginalization is an efficient Markov chain Monte Carlo (MCMC) strategy that uses lower dimensional approximate marginal distributions of the target distribution to accelerate equilibration. As a validation the algorithm is tested on a 2516 dimensional, bimodal, stochastic model motivated by the Kuroshio current that runs along the Japanese coast. The results of this test indicate that the method is an attractive alternative for problems that require the generality of a particle filter but have been inaccessible due to the limitations of standard particle filtering strategies.     

Improvement of the pressure algorithm of the stand-alone PDF method to treat unsteady three-dimensional turbulent reacting flows

This paper deals with the particle-mesh probability density function (PDF) method. It shows how an existing but less precise pressure algorithm for the stand-alone method can be improved. The present algorithm is able to handle the general case of an unsteady three-dimensional turbulent reacting flow. The transport equation of the joint PDF of velocity and composition is solved with a particle method. Open boundary conditions are realized and for statistical reasons a simple but effective particle splitting procedure is applied.

Based on a simple configuration, the properties of the presented improved pressure algorithm are analysed. It is shown which numerical condition must be taken care of so that the algorithm is able to correct the particle positions such that the normalization condition is fulfilled as accurately as specified.

To verify the algorithm the combustion of a methane–air mixture enclosed in an open simulation volume is calculated. It is shown that the simple particle splitting algorithm works very effectively in the studied case. The behaviour of the improved pressure algorithm is examined by different calculations. To analyse the convergence of the algorithm, the particle number per cell and the grid spacing are varied. To demonstrate the accuracy, a statistically stationary inflow/outflow configuration is used and the numerical solution is compared to an analytical one. For a less symmetric test case, the previous unsteady combustion problem is simulated, including an additional mean velocity in one direction.

The presented improved pressure algorithm provides the opportunity to calculate unsteady three-dimensional turbulent reacting flows with a stand-alone method, and offers an alternative to the complex hybrid finite-volume/particle PDF method.     

高超声速层流绕三维压缩拐角的数值模拟

通过数值模拟研究了高超声速来流绕过压缩拐角的层流分离三维流动特性.数值方法采用三维N-S方程,结合2阶精度Roe格式以及分区结构网格有限体积法进行离散.数值模拟的空间激波结构与实验纹影结果符合较好;激波/边界层干扰区内3条纵向线上的计算压力分布与实验结果进行了对比分析,计算获得在三维楔侧面存在低压力区,与实验结果反映的规律一致,计算结果表明低压力区是由楔体侧缘尖端发起的二次涡的抽吸作用造成的.此外,在楔体后端尾流区的低压沿边界层内的亚声速区往上游传递了一定距离.      

特定边界跟踪中角点检测研究

针对边界跟踪算法中在角点附近丢失边界信息的问题,对待检测边界中角点的判别和检测方法进行了研究。分析了边界搜索中所扩展的邻域的半径和角点位置的关系,提出了角点存在判据,给出了角点存在区间。提出了寻区间法角点检测算法,用5个已知边界点之间的关系,判断出角点存在的邻域,通过选取合适的邻域半径,使角点存在区间向角点收敛。对测试图像和人侧面轮廓图像的边界利用所提出的算法进行跟踪,测试图像的边界点从19个增长为37个,人侧面轮廓图像边界点从13个增长为21个,新算法准确地检测到轮廓角点和后续种子点,使边界跟踪能够反映所有的边界信息。     

An implicit immersed boundary method for three-dimensional fluid–membrane interactions

We present an implicit immersed boundary method for the incompressible Navier–Stokes equations capable of handling three-dimensional membrane–fluid flow interactions. The goal of our approach is to greatly improve the time step by using the Jacobian-free Newton–Krylov method (JFNK) to advance the location of the elastic membrane implicitly. The most attractive feature of this Jacobian-free approach is Newton-like nonlinear convergence without the cost of forming and storing the true Jacobian. The Generalized Minimal Residual method (GMRES), which is a widely used Krylov-subspace iterative method, is used to update the search direction required for each Newton iteration. Each GMRES iteration only requires the action of the Jacobian in the form of matrix–vector products and therefore avoids the need of forming and storing the Jacobian matrix explicitly. Once the location of the boundary is obtained, the elastic forces acting at the discrete nodes of the membrane are computed using a finite element model. We then use the immersed boundary method to calculate the hydrodynamic effects and fluid–structure interaction effects such as membrane deformation. The present scheme has been validated by several examples including an oscillatory membrane initially placed in a still fluid, capsule membranes in shear flows and large deformation of red blood cells subjected to stretching force.     

耗散粒子动力学处理复杂固体壁面的一种有效方法

耗散粒子动力学(dissipative particle dynamics,DPD)作为一种介观尺度拉格朗日型粒子方法,已经成功地应用于微纳米流动和生化科技的研究中.复杂固体壁面的处理和壁面边界条件的实施一直是DPD方法发展及应用的一个障碍.提出了处理复杂固体壁面的一种新的方法.复杂固体区域通过冻结随机分布并且达到平衡状态的DPD粒子代表;所冻结的DPD粒子位于临近流动区域的一个截距内;在靠近固体壁面的流动区域中设置流动反弹层,当流动DPD粒子进入此流动层后反弹回流动区域.应用这种固体壁面处理方法对简单流动区域的Poiseuille流动和复杂多孔介质内的流动进行了分析.研究表明,这种新的固体壁面处理方法能够有效模拟复杂固体区域,准确实施壁面边界条件.     

Large-eddy simulation of evaporating spray in a coaxial combustor

Large-eddy simulation of an evaporating isopropyl alcohol spray in a coaxial combustor is performed. The Favre-averaged, variable density, low-Mach number Navier-Stokes equations are solved on unstructured grids with dynamic subgrid scale model to compute the turbulent gas-phase. The original incompressible flow algorithm for LES on unstructured grids by [Mahesh et al., J. Comp. Phys. 197 (2004) 215–240] is extended to include density variations and droplet evaporation. An efficient particle-tracking scheme on unstructured meshes is developed to compute the dispersed phase. Experimentally measured droplet size distribution and size-velocity correlation near the nozzle exit are used as the inlet conditions for the spray. The predictive capability of the LES approach on unstructured grids together with Lagrangian droplet dynamics models to capture the droplet dispersion characteristics, size distributions, and the spray evolution is examined in detail. The mean and turbulent quantities for the gas and particle phases are compared to experimental data to show good agreement. It is shown that for low evaporation rates considered in the present study, a well resolved large-eddy simulation together with simple subgrid models for droplet evaporation and motion provides good agreement of the mean and turbulent quantities for the gas and droplet phases compared to the experimental data. This work represents an important first step to assess the predictive capability of the unstructured grid LES approach applied to spray vaporization. The novelty of the results presented is that they establish a baseline fidelity in the ability to simulate complex flows on unstructured grids at conditions representative of gas-turbine combustors.     

Two-dimensional finite element mesh generation algorithm for electromagnetic field calculation

Two-dimensional finite element mesh generation algorithm for electromagnetic field calculation is proposed in this paper to improve the efficiency and accuracy of electromagnetic calculation. An image boundary extraction algorithm is developed to map the image on the geometric domain. Identification algorithm for the location of nodes in polygon area is proposed to determine the state of the node. To promote the average quality of the mesh and the efficiency of mesh generation, a novel force-based mesh smoothing algorithm is proposed. One test case and a typical electromagnetic calculation are used to testify the effectiveness and efficiency of the proposed algorithm. The results demonstrate that the proposed algorithm can produce a high-quality mesh with less iteration.     

High-order compact finite-difference methods on general overset grids

This work investigates the coupling of a very high-order finite-difference algorithm for the solution of conservation laws on general curvilinear meshes with overset-grid techniques originally developed to address complex geometric configurations. The solver portion of the algorithm is based on Padé-type compact finite-differences of up to sixth-order, with up to 10th-order filters employed to remove spurious waves generated by grid non-uniformities, boundary conditions and flow non-linearities. The overset-grid approach is utilized as both a domain-decomposition paradigm for implementation of the algorithm on massively parallel machines and as a means for handling geometric complexity in the computational domain. Two key features have been implemented in the current work; the ability of the high-order algorithm to accommodate holes cut in grids by the overset-grid approach, and the use of high-order interpolation at non-coincident grid overlaps. Several high-order/high-accuracy interpolation methods were considered, and a high-order, explicit, non-optimized Lagrangian method was found to be the most accurate and robust for this application. Several two-dimensional benchmark problems were examined to validate the interpolation methods and the overall algorithm. These included grid-to-grid interpolation of analytic test functions, the inviscid convection of a vortex, laminar flow over single- and double-cylinder configurations, and the scattering of acoustic waves from one- and three-cylinder configurations. The employment of the overset-grid techniques, coupled with high-order interpolation at overset boundaries, was found to be an effective way of employing the high-order algorithm for more complex geometries than was previously possible.     

Simulation of surface tension in 2D and 3D with smoothed particle hydrodynamics method

The methods for simulating surface tension with smoothed particle hydrodynamics (SPH) method in two dimensions and three dimensions are developed. In 2D surface tension model, the SPH particle on the boundary in 2D is detected dynamically according to the algorithm developed by Dilts [G.A. Dilts, Moving least-squares particle hydrodynamics II: conservation and boundaries, International Journal for Numerical Methods in Engineering 48 (2000) 1503–1524]. The boundary curve in 2D is reconstructed locally with Lagrangian interpolation polynomial. In 3D surface tension model, the SPH particle on the boundary in 3D is detected dynamically according to the algorithm developed by Haque and Dilts [A. Haque, G.A. Dilts, Three-dimensional boundary detection for particle methods, Journal of Computational Physics 226 (2007) 1710–1730]. The boundary surface in 3D is reconstructed locally with moving least squares (MLS) method. By transforming the coordinate system, it is guaranteed that the interface function is one-valued in the local coordinate system. The normal vector and curvature of the boundary surface are calculated according to the reconstructed boundary surface and then surface tension force can be calculated. Surface tension force acts only on the boundary particle. Density correction is applied to the boundary particle in order to remove the boundary inconsistency. The surface tension models in 2D and 3D have been applied to benchmark tests for surface tension. The ability of the current method applying to the simulation of surface tension in 2D and 3D is proved.     

The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary

The Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is a particularly simple linear transport equation. Its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly. By a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall, and that approach a linearly increasing position space density far from the wall. Various aspects of these approximate solutions are discussed. In particular we find that the extrapolated asymptotic density reaches zero at a distance x_M beyond the wall. We find x_M=1.46 in units of the velocity persistence length of the Brownian particle. This study was motivated by certain problems in the theory of diffusion-controlled reactions, and the results might be used to test approximate theories employed in that field.     

Efficient quadrature-free high-order spectral volume method on unstructured grids: Theory and 2D implementation

An efficient implementation of the high-order spectral volume (SV) method is presented for multi-dimensional conservation laws on unstructured grids. In the SV method, each simplex cell is called a spectral volume (SV), and the SV is further subdivided into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions. In the traditional implementation, Gauss quadrature formulas are used to approximate the flux integrals on all faces. In the new approach, a nodal set is selected and used to reconstruct a high-order polynomial approximation for the flux vector, and then the flux integrals on the internal faces are computed analytically, without the need for Gauss quadrature formulas. This gives a significant advantage over the traditional SV method in efficiency and ease of implementation. For SV interfaces, a quadrature-free approach is compared with the Gauss quadrature approach to further evaluate the accuracy and efficiency. A simplified treatment of curved boundaries is also presented that avoids the need to store a separate reconstruction for each boundary cell. Fundamental properties of the new SV implementation are studied and high-order accuracy is demonstrated for linear and non-linear advection equations, and the Euler equations. Several well known inviscid flow test cases are utilized to show the effectiveness of the simplified curved boundary representation.     

A reduced basis method for electromagnetic scattering by multiple particles in three dimensions

We consider the development of efficient and fast computational methods for parametrized electromagnetic scattering problems involving many scattering three dimensional bodies. The parametrization may describe the location, orientation, size, shape and number of scattering bodies as well as properties of the source field such as frequency, polarization and incident direction. The emphasis is on problems that need to be solved rapidly to accurately simulate the interaction of scattered fields under parametric variation, e.g., for design, detection, or uncertainty quantification. For such problems, the use of a brute force approach is often ruled out due to the computational cost associated with solving the problem for each parameter value.In this work, we propose an iterative reduced basis method based on a boundary element discretization of few reference scatterers to resolve the computationally challenging large scale problem. The approach includes (i) a computationally intensive offline procedure to create a selection of a set of snapshot parameters and the construction of an associated reduced basis for each reference scatterer and (ii) an inexpensive online algorithm to generate the surface current and scattered field of the parametrized configuration, for any choice of parameters within the parameter domains used in the offline procedure. Comparison of our numerical results with directly measured results for some benchmark configurations demonstrate the power of our method to rapidly simulate the interacting electromagnetic fields under parametric variation of the overall multiple particle configuration.