PML absorbing boundary conditions for the linearized and nonlinear Euler equations in the case of oblique mean flow

For the case of uniform mean flow in an arbitrary direction, perfectly matched layer (PML) absorbing boundary conditions are presented for both the linearized and nonlinear Euler equations. Although linear perfectly matched side layers with an oblique mean flow have been studied in previous works, we propose in the present paper a construction of corner layer equations that are dynamically stable. Stability issues are investigated by examining the dispersion relations of linear waves supported by the corner layer equations. For increased efficiency, a pseudo mean flow is included in the derivation of the PML equations for the nonlinear case. Numerical examples are given to support the validity of the proposed equations. Specifically, the linear PML formulation is tested for the case of acoustic, vorticity, and entropy waves traveling with an oblique mean flow. The nonlinear formulation is tested with an isentropic vortex moving diagonally with a constant velocity. Copyright © 2008 John Wiley & Sons, Ltd.     

Perfectly matched layer boundary condition for two-dimensional Euler equations in generalized coordinate system

In the past, perfectly matched layer (PML) equations have been constructed in Cartesian and spherical coordinates. In this article, the focus is on the development of a PML absorbing technique for treating numerical boundaries, especially those with unbounded domains, in a generalized coordinate system for a flow in an arbitrary direction. The PML equations for two-dimensional Euler equations are developed in split form through a space–time transformation involving a complex variable transformation with the application of a pseudo-mean-flow in the PML domain. A numerical solver is developed using conventional numerical schemes without employing any form of filtering or artificial dissipation to solve the governing PML equations for two-dimensional Euler equations in a generalized coordinate system. Physical domains of arbitrary shapes are considered and numerical simulations are carried out to validate and demonstrate the effectiveness of the PML as an absorbing boundary condition in generalized coordinates.     

Radiation and Wall Boundary Conditions for Computational Aeroacoustics: A Review

A review of unsteady computational boundary conditions for computational aeroacoustics (CAA) problems is presented. This review is meant to serve as a general overview of previous work on solid wall, radiation and outflow boundary conditions that have been proposed and used in CAA calculations. Both the physical nature of the boundary condition problem as well as the numerical considerations affecting their implementation are discussed.     

Numerical boundary conditions for unsteady transonic flow calculations

In calculations of transonic flows it is necessary to limit the domain of computation to a size that is manageable by computers. At the boundary of the computational domain, boundary conditions are required to ensure a unique solution. Since wave solutions exist in the unsteady transonic flow field, incorrect boundary conditions may result in spurious reflections from the computational boundary. This may introduce errors into the solution. To prevent the spurious reflections, absorbing boundary conditions are often used on the computational boundary. In this paper we describe a method to derive absorbing boudary conditions for transonic calculations. We demonstrate both theoretically and numerically that the use of the absorbing boundary conditions will reduce the spurious reflections in the calculation.     

Complex frequency shifted convolution PML for FDTD modelling of elastic waves

The perfectly matched layer (PML) is nowadays considered as the best optimum absorbing boundary condition available. However, the PML with the classical stretching tensor has certain limitations. Strangely, these limitations have rarely been addressed in elastic wave modelling. For example, substantial reflections occur when strong evanescent waves are propagating parallel to the interface. To circumvent problems like this, the complex frequency shifted stretching tensor has been introduced in electromagnetic modelling. In this paper we show that the convolution PML with this stretching tensor as used in electromagnetic modelling can be adapted for elastic wave modelling. Numerical results of a model where the presence of evanescent waves is predominant show that the PML based on the complex frequency shifted stretching tensor can improve the performance of the absorbing boundary layer considerably.     

Absorbing boundary conditions for elastic waves in transversely isotropic media

In the numerical simulation of elastic wave propagation in the solid, it is essential to introduce absorbing boundary conditions to limit the large or unbounded domain of computation. In this paper, the absorbing boundaries for transversely isotropic media are composed of simple first-order partial differential operators, and each of the operators can perfectly absorb a plane wave outgoing at a certain angle. To test the absorbing ability, the reflection coefficient formulas for the quasi-P and quasi-S wave on the absorbing boundary are derived based on the potential functions theory of the elastic wave. Numerical examples show that the absorbing effect is good. The boundary conditions given here have a practical meaning.Supported by National Natural Science Foundation of China.     

Multi-directional and multi-time step absorbing layer for unbounded domain

Variant techniques are proposed for reproducing the elastic wave propagation in an unbounded medium such as the infinite elements, the absorbing boundary conditions or the perfect matched layers. Here, a simplified approach is adopted by considering absorbing layers characterized by the viscous Rayleigh matrix as studied by Semblat et al. [16] and Rajagopal et al. [14]. Here, further improvements to this procedure are provided. First, we start by establishing the strong form for the elastic wave propagation in a medium characterized by the Rayleigh matrix. This strong form will be used for deriving optimal conditions for damping out in the most efficient way the incident waves while minimizing the spurious reflected waves at the interface between the domain of interest and the Rayleigh damping layer. A procedure for designing the absorbing layer is proposed by targeting a performance criterion expressed in terms of logarithmic decrement of the wave amplitude in the layer thickness. Second, the GC subdomain coupling method, proposed by Combescure and Gravouil [9], is introduced for enabling the choice of any Newmark time integration schemes associated with different time steps depending on subdomains. When wave propagation is predicted by an explicit time integrator, the subdomain strategy is of great interest because it enables a different time integrator for the absorbing layer to be adopted. An external coupling software, based on the GC method, is used to carry out multi=time step explicit/implicit co-computations, making interact in time an explicit FE code (Europlexus) for the domain of interest, with an implicit FE code (Cast3m) handling the absorbing boundary layers. The efficiency of the approach is shown in 1D and 2D elastic wave propagation problems.     

关于气动声学数值计算的方法与进展

气动声学数值计算是近年才出现的研究领域。本文介绍了气动声学数值计算的方法和有关的问题、边界条件的处理以及计算非线性声波的数值方法和进展。讨论了计算气动声学(CAA)的特性及其与计算流体力学(CFD)的差异，指出气动声学数值方法的关键是建立能保持色散关系的差分方程和正确处理无反射边界条件。对于非线性声波传播的问题，为了得到正确的解，应注意提高差分格式对短波的分辨能力，同时发展能抑制“伪”振荡(短波)而对长波基本不起作用的数值方法。     

横观各向同性介质中弹性波的吸收边界条件

在数值求解固体中的弹性波动问题时，常需引入吸收边界条件来限制大范围或无边界的求解区域，使数值计算得以顺利进行。本文通过合成简单的一阶偏微分算子，给出了横观各向同性介质中弹性波的吸收边界条件，其中每个单一的算子均可完全吸收沿某一角度出射的平面波。文中还基于弹性波的势函数理论，导出了准Ｐ波和准Ｓ波在吸收边界处的反射系数公式，用以检验其吸收能力。本文所给出的吸收条件，形式简单，且算例表明吸收效果良好，因     

Effective non-reflective boundary for Lamb waves: Theory,finite element implementation,and applications

This article presents a new approach to designing non-reflective boundary (NRB) for inhibiting Lamb wave reflections at structural boundaries. Our NRB approach can be effectively and conveniently implemented in commercial finite element (FE) codes. The paper starts with a review of the state of the art: (a) the absorbing layers by increasing damping (ALID) approach; and (b) the Lysmer–Kuhlemeyer absorbing boundary conditions (LK ABC) approach is briefly presented and its inadequacy for Lamb wave applications is explained. Hence, we propose a modified Lysmer–Kuhlemeyer approach to be used in the NRB design for Lamb wave problems; we call our approach MLK NRB. The implementation of this MLK NRB was realized using the spring–damper elements which are available in most commercial FE codes. Optimized implementation parameters are developed in order to achieve the best performance for Lamb wave absorption. Our MLK NRB approach is compared with the state of the art ALID and LK ABC methods. Our MLK NRB shows better performance than ALID and LK ABC for all Lamb modes in the thin-plate structures considered in our examples. Our MLK NRB approach is also advantageous at low frequencies and at cut-off frequencies, where extremely long wavelengths exist. A comprehensive study with various design parameters and plate thicknesses which illustrates the advantages and limitations of our MLK NRB approach is presented. MLK NRB applications for both transient analysis in time domain and harmonic analysis in frequency domain are illustrated. The article finishes with conclusions and suggestions for future work.     

On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid

For incompressible Navier–Stokes equations in primitive variables, a method of setting absorbing outflow boundary conditions on an artificial boundary is considered. The advection equations used on the outflow boundary are convenient for finite difference (FD) methods, where a weak formulation of a problem is inapplicable. An unsteady viscous incompressible Navier–Stokes flow in a channel with a moving damper is modeled. An accurate comparison and analysis of numerical and mechanical situations are carried out for a variety of boundary conditions and Reynolds numbers. The proposed outflow conditions provide that the problem with Dirichlet boundary conditions should be solved on each time step.     

二维稳态辐射声场的光滑有限元-完美匹配层解法

针对外场声学有限元计算精度偏低的问题, 将光滑有限元技术引入到二维稳态辐射声场预测中, 提出了光滑有限元-完美匹配层解法. 该解法采用完美匹配层截断声场计算域, 并将其离散为等参四边形单元, 采用指数吸收函数实现完美匹配层内参数坐标和笛卡尔坐标的映射关系, 采用光滑声压梯度技术计算辐射声场刚度矩阵, 将形函数梯度的域内积分转换为形函数域边界积分. 某汽车二维声腔辐射声场的数值分析结果表明, 与标准有限元-完美匹配层相比, 光滑有限元-完美匹配层解法在完美匹配层内的声波吸收效果更好, 在计算域内的数值计算精度更高, 具有良好的工程应用前景.      

Recent advances of computational aeroacoustics

Computational aeroacoustics (CAA) is an interdiscipline of aeroacoustics and computational fluid dynamics (CFD) for the investigation of sound generation and propagation from various aeroacoustics problems. In this review, the foundation and research scope of CAA are introduced firstly. A review of the early advances and applications of CAA is then briefly surveyed, focusing on two key issues, namely, high order finite difference scheme and non-reflecting boundary condition. Furthermore, the advances of CAA during the past five years are highlighted. Finally, the future prospective of CAA is briefly discussed.     

Inflow boundary condition for DNS of turbulent boundary layers on supersonic blunt cones

For direct numerical simulation (DNS) of turbulent boundary layers, gen- eration of an appropriate inflow condition needs to be considered. This paper proposes a method, with which the inflow condition for spatial-mode DNS of turbulent boundary layers on supersonic blunt cones with different Mach numbers, Reynolds numbers and wall temperature conditions can be generated. This is based only on a given instant flow field obtained by a temporal-mode DNS of a turbulent boundary layer on a flat plate. Effectiveness of the method is shown in three typical examples by comparing the results with those obtained by other methods.     

Approximate nonreflecting inflow-outflow boundary conditions for the calculation of navier-stokes equations in internal flows

In this paper, the nonreflecting boundary conditions based upon fundamental ideas of the linear analysis are developed for gas dynamic equations, and the modified boundary conditions for Navier-Stokes equations are proposed as a substitute of the nonreflecting boundary conditions inside boundary layers near rigid walls. These derived boundary conditions are then applied to calculations both for the Euler equations and the Navier-Stokes equations to determine if they can produce acceptable results for the subsonic flows in channels. The numerical results obtained by an implicit second-order upwind difference scheme show the effectiveness and generality of the boundary conditions. Furthermore, the formulae and the analysis performed here may be extended to three dimensional problems. recommended by Prof. Cui Erjie     

Mathematically consistent boundary conditions and turbulence matching at block interfaces for computational aeroacoustics

The method of characteristics is used to implement the various boundary conditions (e.g. wall and interface) in a high-order computational aeroacoustic (CAA) code developed by the first author. Most characteristic methods do not satisfy Pfaff's condition (which needs to be satisfied for any mathematical relation to be valid). A mathematically consistent and valid method is used in this work to derive the characteristic boundary conditions. Also, a robust and efficient approach for the matching of turbulence quantities at multi-block interfaces is proposed. Various numerical simulation cases were run to validate the concepts. The computed results show that the proposed method is accurate, robust and is in excellent agreement with experimental data. The results also indicate that the matching of turbulence quantities is essential for accurate turbulent flow calculations.     

透射边界高频失稳机理及其消除方法——SH波动

用有限元法求解近场波动问题,须选取人工边界条件以实现对无限域稳定、高效的数值模拟. 该文探讨了SH波导 有限元数值模拟中透射边界引发的高频失稳问题. 从离散模型出发,分析了内节点与人工边界节点运动方程频散曲线之间的匹配关系,揭示了高频失稳的一种机理,即二者相互耦合所 得计算方案支持自发从人工边界向计算区域内行进的高频波动. 提出通过调整内节点运动方程以改变这一匹配关系,从而消除失稳的措施. 理论分析与数值结果表明该措施能有效地消除高频振荡失稳.     

Shocklets in compressible flows

The mechanism of shocklets is studied theoretically and numerically for the stationary fluid, uniform compressible flow, and boundary layer flow. The conditions that trigger shock waves for sound wave, weak discontinuity, and Tollmien-Schlichting （T-S） wave in compressible flows are investigated. The relations between the three types of waves and shocklets are further analyzed and discussed. Different stages of the shocklet formation process are simulated. The results show that the three waves in compressible flows will transfer to shocklets only when the initial disturbance amplitudes are greater than the certain threshold values. In compressible boundary layers, the shocklets evolved from T-S wave exist only in a finite region near the surface instead of the whole wavefront.     

通用有限元系统实现无限域波动仿真方法

在有限元模型中加入人工边界,可以进行无限域波动仿真。本文将介绍一种改进的Higdon人工边界,并通过适当的变换形式,将其在通用有限元软件上实现。最后通过数值实验验证了这种方法具有良好的稳定性和计算精度,并且实施简单方便,其他人工边界条件可参照本文方法,通过变换形式在通用有限元程序中实现。     

Effective Flux Boundary Conditions for Upscaling Porous Media Equations

We introduce a new algorithm for setting pressure boundary conditions in subgrid simulations of porous media flow. The algorithm approximates the flux in the boundary cell as the flux through a homogeneous inclusion in a homogeneous background, where the permeability of the inclusion is given by the cell permeability and the permeability of the background is given by the ambient effective permeability. With this approximation, the flux in the boundary cell scales with the cell permeability when that permeability is small, and saturates at a constant value when that permeability is large. The flux conditions provide Neumann boundary conditions for the subgrid pressure. We call these boundary conditions effective flux boundary conditions (EFBCs). We give solutions for the flux through ellipsoidal inclusions in two and three dimensions, assuming symmetric tensor permeabilities whose principal axes align with the axes of the ellipse. We then discuss the considerations involved in applying these equations to scale up problems in geological porous media. The key complications are heterogeneity, fluctuations at all length scales, and boundary conditions at finite scales.