Accuracies and conservation errors of various ghost fluid methods for multi-medium Riemann problem

Since the (original) ghost fluid method (OGFM) was proposed by Fedkiw et al. in 1999 [5], a series of other GFM-based methods such as the gas–water version GFM (GWGFM), the modified GFM (MGFM) and the real GFM (RGFM) have been developed subsequently. Systematic analysis, however, has yet to be carried out for the various GFMs on their accuracies and conservation errors. In this paper, we develop a technique to rigorously analyze the accuracies and conservation errors of these different GFMs when applied to the multi-medium Riemann problem with a general equation of state (EOS). By analyzing and comparing the interfacial state provided by each GFM to the exact one of the original multi-medium Riemann problem, we show that the accuracy of interfacial treatment can achieve “third-order accuracy” in the sense of comparing to the exact solution of the original mutli-medium Riemann problem for the MGFM and the RGFM, while it is of at most “first-order accuracy” for the OGFM and the GWGFM when the interface approach is actually near in balance. Similar conclusions are also obtained in association with the local conservation errors. A special test method is exploited to validate these theoretical conclusions from the numerical viewpoint.     

多介质流动数值计算中的界面处理方法

 求解Riemann问题得到界面接触间断的流动状态,并以此构造带状区域的虚拟流体状态,对于多维问题设计了一种方便有效的算法。同时求解耦合的守恒形式欧拉方程组和非守恒界面捕捉方程,并用Level-Set函数捕捉界面,数值计算采用高分辨率MWENO格式。最后对可压缩多介质流动问题进行了数值模拟。     

FIVER: A finite volume method based on exact two-phase Riemann problems and sparse grids for multi-material flows with large density jumps

A robust finite volume method for the solution of high-speed compressible flows in multi-material domains involving arbitrary equations of state and large density jumps is presented. The global domain of interest can include a moving or deformable subdomain that furthermore may undergo topological changes due to, for example, crack propagation. The key components of the proposed method include: (a) the definition of a discrete surrogate material interface, (b) the computation of a reliable approximation of the fluid state vector on each side of a discrete material interface via the construction and solution of a local, exact, two-phase Riemann problem, (c) the algebraic solution of this auxiliary problem when the equation of state allows it, and (d) the solution of this two-phase Riemann problem using sparse grid tabulations otherwise. The proposed computational method is illustrated with the three-dimensional simulation of the dynamics of an underwater explosion bubble.     

可压缩多介质流体数值模拟中的Level-Set间断跟踪方法

针对可压缩多介质流体的数值模拟,发展了一种Level-Set间断追踪技术,用LS(Level-Set)函数追踪激波和捕捉界面,用Riemann问题解构造带状区域内的虚拟流体状态,对物理量的外推方法、间断附近虚拟流体的构造、间断推进速度的计算等问题进行了研究.最后对可压缩多介质流体一维和二维守恒律方程组进行数值模拟,数值计算采用通量重构的高精度WENO格式,计算结果令人满意.     

模拟多介质界面问题的虚拟流体方法综述

虚拟流体方法为模拟具有清晰物质界面的多介质流动问题提供了一种简便途径.尤其基于多介质Riemann问题解的修正虚拟流体方法及其变体，能够真实考虑到界面附近非线性波的相互作用和物质性质的影响，可以有效解决各种界面强间断等挑战性难题，具有巨大的工程应用潜力.文章重点回顾了虚拟流体方法的发展历史，总结和对比了各种代表性版本在模拟可压缩多介质流时的界面条件定义方式和多维推广方式，并介绍了该方法的设计原则和精度分析方面的研究进展.文章还回顾了该方法在其他更广泛和更具挑战性典型科学问题中的最新应用进展，并对方法的优势和特点进行了总结.      

A RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in Lagrangian coordinate

In this paper, Runge-Kutta Discontinuous Galerkin (RKDG) finite element method is presented to solve the one-dimensional inviscid compressible gas dynamic equations in Lagrangian coordinate. The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method. A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method. For multi-medium fluid simulation, the two cells adjacent to the interface are treated differently from other cells. At first, a linear Riemann solver is applied to calculate the numerical flux at the interface. Numerical examples show that there is some oscillation in the vicinity of the interface. Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical flux at the interface, which suppress the oscillation successfully. Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

运动激波和气泡串相互作用的初步数值模拟

通过对激波和流体界面相互作用诱导的大变形界面演化的数值模拟,验证Level set方法精确模拟多个流体界面的有效性.采用2阶迎风TVD求解欧拉方程得到流场解,采用5阶WENO求解Level set方程追踪多流体界面,采用GFM方法处理流体内界面.利用文[1]的计算结果校核本文程序.在此基础上,对运动激波和气泡串相互作用过程进行了初步数值模拟,得到了不同时刻运动激波和圆管内的两个气泡作用后的演化图象,包括压力和密度等值线分布.计算结果表明:针对推广后的多界面Level set方程,该方法仍可高质量地捕捉多个流体界面.     

水中空气隔层衰减冲击波性能研究

 根据鱼雷对舰船毁伤的特点,建立了多介质水下爆炸数值模型,提出了交界面处理及广义光滑长度计算方法,编制了多介质水下爆炸计算程序,并对水下爆炸空气隔层衰减冲击波的性能进行了定量分析。结果表明：无论是接触爆炸还是非接触爆炸,空气隔层均可有效衰减冲击波,并且接触爆炸中空气隔层衰减冲击波的效果更好,最大可使冲击压力峰值降低约55%;空气隔层厚度与爆炸厚度之比为1时便可达到较好的衰减冲击波效果,继续增加空气隔层厚度对衰减冲击波效果影响不大。研究结果可为舰船结构防护及防雷舱的结构设计提供参考。     

An RKDG finite element method for the one-dimensional inviscid compressible gas dynamics equations in a Lagrangian coordinate

In this paper,Runge-Kutta Discontinuous Galerkin(RKDG) finite element method is presented to solve the onedimensional inviscid compressible gas dynamic equations in a Lagrangian coordinate.The equations are discretized by the DG method in space and the temporal discretization is accomplished by the total variation diminishing Runge-Kutta method.A limiter based on the characteristic field decomposition is applied to maintain stability and non-oscillatory property of the RKDG method.For multi-medium fluid simulation,the two cells adjacent to the interface are treated differently from other cells.At first,a linear Riemann solver is applied to calculate the numerical ?ux at the interface.Numerical examples show that there is some oscillation in the vicinity of the interface.Then a nonlinear Riemann solver based on the characteristic formulation of the equation and the discontinuity relations is adopted to calculate the numerical ?ux at the interface,which suppresses the oscillation successfully.Several single-medium and multi-medium fluid examples are given to demonstrate the reliability and efficiency of the algorithm.     

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.     

水下多点爆炸条件下的冲击波载荷特性

针对水下多爆源起爆的实战背景,开展了两点同时起爆条件下冲击波载荷特性的数值模拟研究。基于自研的多相可压缩流体计算程序,采用高精度的数值格式对流体控制方程进行离散求解。将数值模型计算的自由场水下爆炸的结果与理论结果比较,初步验证了数值模型计算的准确性与可靠性。利用该模型计算了典型工况下水下两点起爆工况,计算结果表明:两爆源对称面上压力相比单爆源线性叠加后的峰值压力增加12%~16%;两爆源垂直截面之间的压力存在双峰现象;而对于两垂直截面之外的测点压力也存在双峰现象,第1个峰值压力与单爆源线性叠加的峰值相等,第2个峰值压力要远低于单爆源线性叠加的峰值,峰值压力下降幅度可高达30%左右。研究结果能够为水下武器防护设计与威胁评估提供参考。     

An interface capturing method for the simulation of multi-phase compressible flows

A novel finite-volume interface (contact) capturing method is presented for simulation of multi-component compressible flows with high density ratios and strong shocks. In addition, the materials on the two sides of interfaces can have significantly different equations of state. Material boundaries are identified through an interface function, which is solved in concert with the governing equations on the same mesh. For long simulations, the method relies on an interface compression technique that constrains the thickness of the diffused interface to a few grid cells throughout the simulation. This is done in the spirit of shock-capturing schemes, for which numerical dissipation effectively preserves a sharp but mesh-representable shock profile. For contact capturing, the formulation is modified so that interface representations remain sharp like captured shocks, countering their tendency to diffuse via the same numerical diffusion needed for shock-capturing. Special techniques for accurate and robust computation of interface normals and derivatives of the interface function are developed. The interface compression method is coupled to a shock-capturing compressible flow solver in a way that avoids the spurious oscillations that typically develop at material boundaries. Convergence to weak solutions of the governing equations is proved for the new contact capturing approach. Comparisons with exact Riemann problems for model one-dimensional multi-material flows show that the interface compression technique is accurate. The method employs Cartesian product stencils and, therefore, there is no inherent obstacles in multiple dimensions. Examples of two- and three-dimensional flows are also presented, including a demonstration with significantly disparate equations of state: a shock induced collapse of three-dimensional van der Waal’s bubbles (air) in a stiffened equation of state liquid (water) adjacent to a Mie-Grüneisen equation of state wall (copper).     

A ghost fluid,level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection

In this paper, we present the development of a sharp numerical scheme for multiphase electrohydrodynamic (EHD) flows for a high electric Reynolds number regime. The electric potential Poisson equation contains EHD interface boundary conditions, which are implemented using the ghost fluid method (GFM). The GFM is also used to solve the pressure Poisson equation. The methods detailed here are integrated with state-of-the-art interface transport techniques and coupled to a robust, high order fully conservative finite difference Navier–Stokes solver. Test cases with exact or approximate analytic solutions are used to assess the robustness and accuracy of the EHD numerical scheme. The method is then applied to simulate a charged liquid kerosene jet.     

圆柱形水下爆炸实验容器壁部强度设计研究

 基于薄壁壳理论和水下爆炸理论,对圆柱形水下爆炸实验容器在爆炸冲击波作用下弹性范围内的壁部应变进行了理论分析和实验研究。导出了圆柱形水下爆炸实验容器在爆炸冲击波作用下壁部弹性应变与容器直径、壁厚及内部爆炸药量之间的关系,并对计算结果进行了实验验证。实验表明公式求解结果与实验结果具有较好的一致性。     

爆炸作用驱动液体抛撒初始阶段数值仿真

 针对灭火战斗部引战配合设计,研究爆炸作用驱动液体（水基灭火剂）抛撒初始阶段的特性。依据典型灭火战斗部的结构建立数值仿真模型,采用LS-DYNA软件进行仿真计算,分别获得了加载系数（驱动装药与液体的质量比）对壳体破裂时间和壳体破裂初期液体/空气界面运动速度的影响规律,以及液体/空气界面后续的运动特性。进行了同条件下爆炸驱动水基灭火剂抛撒实验,得到了不同加载系数条件下壳体破裂初期液体/空气界面的运动速度。实验结果与数值仿真结果吻合良好,表明所采用的数值仿真方法具有正确性和有效性,对相关问题的研究具有重要参考价值。     

Mesh refinement algorithms in an unstructured solver for multiphase flow simulation using discrete particles

This study developed spray-adaptive mesh refinement algorithms with directional sensitivity in an unstructured solver to improve spray simulation for internal combustion engine application. Inadequate spatial resolution is often found to cause inaccuracies in spray simulation using the Lagrangian–Eulerian approach due to the over-estimated diffusion and inappropriate liquid–gas phase coupling. Dynamic mesh refinement algorithms adaptive to fuel sprays and vapor gradients were developed in order to increase the grid resolution in the spray region to improve simulation accuracy. The local refinement introduced the coarse-fine face interface that requires advanced numerical schemes for flux calculation and grid rezoning with moving boundaries. To resolve the issue in flux calculation, this work implemented the refinement/coarsening algorithms into a collocated solver to avoid tedious interpolations in solving the momentum equations. A pressure correction method was applied to address unphysical pressure oscillations due to the collocation of pressure and velocity. An edge-based algorithm was used to evaluate the edge-centered quantities in order to account for the contributions from all the cells around an edge at the coarse-fine interface. A quasi-second-order upwind scheme with strong monotonicity was also modified to accommodate the coarse-fine interface for convective fluxes. To resolve the issue related to grid rezoning, rezoning was applied to the initial baseline mesh only and the new locations of the refined grids were obtained by interpolating the updated baseline mesh. The time step constraints were also re-evaluated to account for the change resulting from mesh refinement. The present refinement algorithm was used in simulating fuel sprays in an engine combustion chamber. It was found that the present approach could produce the same level of results as those using the uniformly fine mesh with substantially reduced computer time. Results also showed that this approach could alleviate the artifacts related to the Lagrangian discrete modeling of spray drops due to insufficient spatial resolution.     

Numerical simulation of cavitating flow around a slender body with slip boundary condition

In this paper, we perform a numerical simulation of the cavitating flow around an underwater hemispherical-head slender body running at a high speed. For the first time, the slip boundary condition is introduced into this problem, and we find that the slip boundary condition has a big influence on the cavitation in the flow-separation zone. By simulating the cavitating flow under different cavitation numbers, we demonstrate that the slip boundary condition can effectively reduce the intensity of cavitation, as represented by the length of cavitation bubbles. The present paper provides a new method for utilization of new surface materials to control the cavitation on the underwater moving objects.     

三角形网格上多介质流动问题界面处理方法

多介质流动问题的求解一般是在结构网格上实现,而三角形网格对于复杂计算区域具有更好的适应性,本文结合rGFM方法,给出三角形网格上多介质流动问题界面处理方法.利用level-set方法跟踪界面,在界面处构造Riemann问题,得到界面处流体准确的流动状态.通过定义界面边界条件,将多介质流动问题转化为单介质流动问题,利用高精度RKDG方法求解.采用多个算例验证该方法的稳健性和有效性,结果表明该方法能准确捕捉界面和激波的位置,保持界面清晰.     

Ghost fluid method for strong shock impacting on material interface

It is found that the original ghost fluid method (GFM) as put forth by Fedkiw et al. [J. Comp. Phys. 152 (1999) 457] does not work consistently and efficiently using isentropic fix when applied to a strong shock impacting on a material interface. In this work, the causes for such inapplicability of the original GFM are analysed and a modified GFM is proposed and developed for greater robustness and consistency. Numerical tests also show that the modified GFM has the property of reduced conservation error and is less problem-related.     

柱形装药水中爆炸气泡的形态演变

 采用高速摄影方法在实验水箱中获得了长径比为3.35~6.75的柱形PETN炸药水中爆炸气泡脉动的图像,进而结合真实的爆轰过程和Rayleigh气泡运动方程,研究了气泡的形态演变规律。研究结果表明,柱形装药在水中爆炸时,形成的初始气泡的形状为非球对称形,这种非对称特征随长径比的增加而增大。气泡表面的运动也表现出明显的非球对称特征,气泡表面在装药径向的膨胀运动呈指数衰减,在装药轴向两端的膨胀运动更接近于分段线性衰减。气泡表面的不对称运动与柱形装药水下爆炸的能量输出结构不均有关,与冲击波的有效能量分布规律是相似的。