High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

In this paper, we present a class of high‐order accurate cell‐centered arbitrary Lagrangian–Eulerian (ALE) one‐step ADER weighted essentially non‐oscillatory (WENO) finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two‐dimensional unstructured triangular meshes. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space–time Galerkin predictor allows the schemes to be high order accurate also in time by using an element‐local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained either (i) as an arithmetic average among the states surrounding the node, as suggested by Cheng and Shu, or (ii) as a solution of multiple one‐dimensional half‐Riemann problems around a vertex, as suggested by Maire, or (iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional Harten–Lax–van Leer Riemann solver recently proposed by Balsara et al. Once the vertex velocity and thus the new node location have been determined by the node solver, the local mesh motion is then constructed by straight edges connecting the vertex positions at the old time level tⁿ with the new ones at the next time level t^{n + 1}. If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unnecessary, as the ALE fluxes already properly take into account the rezoned geometry. In this sense, our scheme falls into the category of direct ALE methods. Furthermore, the geometric conservation law is satisfied by the scheme by construction. We apply the high‐order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic magnetohydrodynamic equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of the full approximation storage method to the numerical simulation of two-dimensional steady incompressible viscous multiphase flows

In recent years multigrid algorithms have been applied to increasingly difficult systems of partial differential equations and major improvements in both speed of convergence and robustness have been achieved. Problems involving several interacting fluids are of great interest in many industrial applications, especially in the process and petro-chemical sectors. However, the multifluid version of the Navier–Stokes equations is extremely complex and represents a challenge to advanced numerical algorithms. In this paper, we describe an extension of the full approximation storage (FAS) multigrid algorithm to the multifluid equations. A number of special issues had to be addressed. The first was the development of a customised, non-linear, coupled relaxation scheme for the smoothing step. Automatic differentiation was used to facilitate the coding of a robust, globally convergent quasi-Newton method. It was also necessary to use special inter-grid transfer operators to maintain the realisability of the solution. Algorithmic details are given and solutions for a series of test problems are compared with those from a widely validated, commercial code. The new approach has proved to be robust; it achieves convergence without resorting to specialised initialisation methods. Moreover, even though the rate of convergence is complex, the method has achieved very good reduction factors: typically five orders of magnitude in 50 cycles. © 1998 John Wiley & Sons, Ltd.     

A simple incompressible flux splitting for sharp free surface capturing

This paper first applies a flux vector‐type splitting method based on the numerical speed of sound for computing incompressible single and multifluid flows. Here, a preconditioning matrix based on Chorin's artificial compressibility concept is used to modify the incompressible multifluid Navier–Stokes equations to be hyperbolic and density or volume fraction‐independent. The current approach can reduce eigenvalues disparity induced from density or volume fraction ratios and enhance numerical stability. Also, a simple convection‐pressure flux‐splitting method with high‐order essentially nonoscillatory‐type primitive variable extrapolations coupled with monotone upstream‐centered schemes for conservation laws‐type volume fraction recompressed reconstruction is used to maintain the preservation of sharp interface evolutions in multifluid flow simulations. Benchmark tests including a solid rotation test of a notched two‐dimensional cylinder, the evolution of spiral and rotational shapes of deformable circles, a dam breaking problem, and the Rayleigh–Taylor instability were chosen to validate the current incompressible multifluid methodology. An incompressible driven cavity was also chosen to check the robustness of the proposed method on the computation of single fluid incompressible flow problems. Copyright © 2011 John Wiley & Sons, Ltd.     

关于二相流、多相流、多流体模型和非牛顿流等概念的探讨

本文分析了单相流、二相流和多相流等概念上的差异,也分析了单流体模型、双流体模型和多流体模型等概念上的差异,指出前面三种概念是按流动介质的客观物理构成划分的,而后者是按主观采用的研究方法划分的．目前这些概念在使用中存在一些混乱,如二相流与多相流,多相流与多流体模型等．本文还研究了扩散模型、非牛顿流模型和颗粒流模型等,指出前两种模型在分类上属于单流体模型,分析了非牛顿流模型、扩散模型和双（多）流体模型的特点和应用范围,最后,以泥石流为例讨论了以上概念的应用．      

Effects of Atomic Mixing in Inertial Confinement Fusion by MultifluidInterpenetration Mix Model

The effects of atomic-level mixing are systemically investigated in a multifluid interpenetration mix model, and results are compared with the single-fluid model's simulations and experimental data. It is shown that increasing the model free parameter α, shock Mach number, and the initial density discontinuity makes the mix length and fraction of mixing particle increase, resulting in the lower shock temperatures compared with the results of single-fluid model without mixing. Recent high-compressibility direct-drive spherical implosions on OMEGA are simulated by the interpenetration mix model. The calculations with atomic mixing between fuel and shell match quite well with the observations. Without considering any mixing, the calculated neutron yields and ion temperatures are overpredicted; while inclusion of the interpenetration mix model with the adjustable parameter α could fit the simulated neutron yields and ion temperatures well with experimental data.     

多介质流体力学两步欧拉方法的模型封闭性方法

描述多介质流体力学两步欧拉方法拉氏步中方程的不封闭问题.分析已存在的模型封闭性方法的优缺点,给出根据声速特征的自适应的模型封闭性方法.数值例子表明方法合理可靠,并说明压力弛豫过程在数值计算中的作用.     

高密度比多介质可压缩流动的PPM方法

介绍一种基于流体体积分数模型的界面捕捉方法,数值模拟高密度比及含强激波的多介质可压缩流动.将PPM方法应用到多介质体积分数形式的Euler方程组,用双波近似求解一般刚性气体状态方程Riemann问题,可方便处理界面强剪切的滑移线问题,并给出了水下激波和气泡相互作用以及多相Richtmyer Meshkov不稳定性的计算结果.     

Effects of Atomic Mixing in Inertial Confinement Fusion by Multifluid Interpenetration Mix Model

The effects of atomic-level rnixing are systemically investigated in a multifluid interpenetration mix model ,and results are compared with the single-fluid model＇s simulations and experimental data. It is shown that increasing the model free parameter α, shock Mach number, and the initial density discontinuity makes the mix length and fraction of mixing particle increase, resulting in the lower shock temperatures compared with the results of single-fluid model without mixing. Recent high-compressibility direct-drive spherical implosions on OMEGA are simulated by the interpenetration mix modal. The calculations with atomic mixing between fuel and shell match quite well with the observations. Without considering any mixing, the calculated neutron yields and ion temperatures are overpredicted; while inclusion of the interpenetration mix model with the adjustable parameter α could fit the simulated neutron yields and ion temperatures well with experimental data.     

高压作用下不同形状气腔压缩过程的二维数值计算

 采用三阶精度的Parabolic Piecewise Method（PPM）方法和Volume of Fluid（VOF）方法相结合,运用Lagrange-Remapping算法,对高压作用下不同形状气腔压缩过程进行了二维高精度数值计算,给出了复杂形状的多流体流场之间的相互作用。此方法可以用来处理界面两边高压力比、高密度比流动以及强剪切滑移运动等问题。     

Numerical simulation of compressible multifluid flows using an adaptive positivity-preserving RKDG-GFM approach

The Runge-Kutta discontinuous Galerkin method together with a refined real-ghost fluid method is incorporated into an adaptive mesh refinement environment for solving compressible multifluid flows, where the level set method is used to capture the moving material interface. To ensure that the Riemann problem is exactly along the normal direction of the material interface, a simple and efficient modification is introduced into the original real-ghost fluid method for constructing the interfacial Riemann problem, and the initial conditions of the Riemann problem are obtained directly from the solution polynomials of the discontinuous Galerkin finite element space. In addition, a positivity-preserving limiter is introduced into the Runge-Kutta discontinuous Galerkin method to suppress the failure of preserving positivity of density or pressure for the problems involving strong shock wave or shock interaction with material interface. For interfacial cells in adaptive mesh refinement, the data transfer between different grid levels is achieved by using a L² projection approach along with the least squares fitting. Various numerical cases, including multifluid shock tubes, underwater explosions, and shock-induced collapse of a underwater air bubble, are computed to assess the capability of the present adaptive positivity-preserving RKDG-GFM approach, and the simulated results show that the present approach is quite robust and can provide relatively reasonable results across a wide variety of flow regimes, even for problems involving strong shock wave or shock wave impacting high acoustic impedance mismatch material interface.