Analysis and Application of Perfectly Matched Layer Absorbing Boundary Conditions for Computational Aeroacoustics

The Perfectly Matched Layer (PML) was originally proposed by Berenger as an absorbing boundary condition for Maxwell\u27s equations in 1994 and is still used extensively in the field of electromagnetics. The idea was extended to Computational Aeroacoustics in 1996, when Hu applied the method to Euler\u27s equations. Since that time much of the work done on PML in the field of acoustics has been specific to the case where mean flow is perpendicular to a boundary, with an emphasis on Cartesian coordinates. The goal of this work is to further extend the PML methodology in a two-fold manner: First, to handle the more general case of an oblique mean flow, where mean velocities strike the boundary at an arbitrary angle, and second, to adapt the equations for use in a cylindrical coordinate system. These extensions to the PML methodology are effectively carried out in this dissertation. Perfectly Matched Layer absorbing boundary conditions are presented for the linearized and nonlinear Euler equations in two dimensions. Such boundary conditions are presented in both Cartesian and cylindrical coordinates for the case of an oblique mean flow. In Cartesian coordinates, the PML equations for the side layers and corner layers of a rectangular domain will be derived independently. The approach used in the formation of side layer equations guarantees that the side layers will be perfectly matched at the interface between the interior and PML regions. Because of the perfect matching of the side layers, the equations are guaranteed to be stable. However, a somewhat different approach is used in the formation of the corner layer equations. Therefore, the stability of linear waves in the corner layer is analyzed. The results of the analysis indicate that the proposed corner equations are indeed stable. For the PML equations in cylindrical coordinates, there is no need for separate derivations of side and corner layers, and in this case, the stability of the equations is achieved through an appropriate space-time transformation. As is shown, such a transformation is needed for correcting the inconsistencies in phase and group velocities which can negatively affect the stability of the equations. After this correction has been made, the cylindrical PML can be implemented without risk of instability. In both Cartesian and cylindrical coordinates, the PML for the linearized Euler equations are presented in primitive variables, while conservation form is used for the nonlinear Euler equations. Numerical examples are also included to support the validity of the proposed equations. Specifically, the equations are tested for a combination of acoustic, vorticity and entropy waves. In each example, high-accuracy solutions are obtained, indicating that the PML conditions are effective in minimizing boundary reflections

REVIEW OF NUMERICAL SCHEMES AND BOUNDARY CONDITIONS APPLIED TO WAVE PROPAGATION PROBLEMS

As a review framework, the present study describes the application and performance of different numerical schemes for Computational Aeroacoustics (CAA) of simple wave propagation problems. The current approach aims to simulate pulse propagation on the near field by the use of different spatial and temporal numerical schemes for the full and Linearized Euler Equations (LEE) in a dimensional and dimensionless formulation. Comparisons of processing time, residual error and quality of results are present and discussed shedding light to the relevant parameters which play important role in aeroacoustics. The investigation is focused on different Gaussian pulse propagation cases in unbounded and bounded domains which is solved by using optimized spatial and temporal schemes for reducing dissipative and dispersive errors. The numerical results are compared with the exact analytical solutions when available, showing good agreement

An efficient discontinuous Galerkin method for aeroacoustic propagation

An efficient discontinuous Galerkin formulation is applied to the solution of the linearized Euler equations and the acoustic perturbation equations for the simulation of aeroacoustic propagation in two-dimensional and axisymmetric problems, with triangular and quadrilateral elements. To improve computational efficiency, a new strategy of variable interpolation order is proposed in addition to a quadrature-free approach and parallel implementation. Moreover, an accurate wall boundary condition is formulated on the basis of the solution of the Riemann problem for a reflective wall. Time discretization is based on a low dissipation formulation of a fourth-order, low storage Runge-Kutta scheme. Along the far-field boundaries a perfectly matched layer boundary condition is used. For the far-field computations, the integral formulation of Ffowcs Williams and Hawkings is coupled with the near-field solver. The efficiency and accuracy of the proposed variable order formulation is assessed for realistic geometries, namely sound propagation around a high-lift airfoil and the Munt problem

Comparison of outflow boundary conditions for subsonic aeroacoustic simulations

Aeroacoustics simulations require much more precise boundary conditions than classical aerodynamics. Two classes of non-reflecting boundary conditions for aeroacoustics are compared in the present work: characteristic analysis based methods and Tam and Dong approach. In characteristic methods, waves are identified and manipulated at the boundaries while Tam and Dong use modified linearized Euler equations in a buffer zone near outlets to mimic a non-reflecting boundary. The principles of both approaches are recalled and recent characteristic methods incorporating the treatment of transverse terms are discussed. Three characteristic techniques (the original NSCBC formulation of Poinsot and Lele and two versions of the modified method of Yoo and Im) are compared to the Tam and Dong method for four typical aeroacoustics problems: vortex convection on a uniform flow, vortex convection on a shear flow, acoustic propagation from a monopole and from a dipole. Results demonstrate that the Tam and Dong method generally provides the best results and is a serious alternative solution to characteristic methods even though its implementation might require more care than usual NSCBC approaches

Implementation of higher-order absorbing boundary conditions for the Einstein equations

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition B_L of order L=l yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions B_L with L<l, which include the widely used freezing-Psi_0 boundary condition that imposes the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in Class. Quantum Grav

Implementation of higher-order absorbing boundary conditions for the Einstein equations

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge-Wheeler-Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers l=2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition B_L of order L=l yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions B_L with L<l, which include the widely used freezing-Psi_0 boundary condition that imposes the vanishing of the Newman-Penrose scalar Psi_0.Comment: 25 pages, 9 figures. Minor clarifications. Final version to appear in Class. Quantum Grav

Numerical Simulation of the Generation of Axisymmetric Mode Jet Screech Tones

An imperfectly expanded supersonic jet, invariably, radiates both broadband noise and discrete frequency sound called screech tones. Screech tones are known to be generated by a feedback loop driven by the large scale instability waves of the jet flow. Inside the jet plume is a quasi-periodic shock cell structure. The interaction of the instability waves and the shock cell structure, as the former propagates through the latter, is responsible for the generation of the tones. Presently, there are formulas that can predict the tone frequency fairly accurately. However, there is no known way to predict the screech tone intensity. In this work, the screech phenomenon of an axisymmetric jet at low supersonic Mach number is reproduced by numerical simulation. The computed mean velocity profiles and the shock cell pressure distribution of the jet are found to be in good agreement with experimental measurements. The same is true with the simulated screech frequency. Calculated screech tone intensity and directivity at selected jet Mach number are reported in this paper. The present results demonstrate that numerical simulation using computational aeroacoustics methods offers not only a reliable way to determine the screech tone intensity and directivity but also an opportunity to study the physics and detailed mechanisms of the phenomenon by an entirely new approach

전산공력음향학에서 Perfectly Matched Layer의 안정적인 흡수조건에 관한 연구

학위논문 (석사)-- 서울대학교 대학원 : 기계항공공학부 우주항공공학전공, 2016. 2. 이수갑.In Computational Aeroacoustics, non-reflective boundary conditions such as radiation or absorbing boundary conditions are critical issues in that they can affect the whole solutions of computation. Among these types of boundary conditions, Perfectly Matched Layer boundary condition which has been widely used in Computational Electromagnetics and Computational Aeroacoustics is developed by augmenting the additional term by an absorption function in the original governing equations so as to stably absorb the outgoing waves. Even if Perfectly Matched Layer is perfectly non-reflective boundary condition analytically, spurious waves at the interface or instability could be shown since the analysis is performed in the discretized space. Hence, the study is focused on factors that affect these numerical instability and accuracy with particular numerical schemes. First, stability analysis preserving the dispersion relation is carried out in order to achieve the stability limit of time-step size. Then, through mathematical approach, stable absorption coefficient and PML width are suggested. In order to validate the prediction of analysis condition, numerical simulations are performed in generalized coordinate system as well as Cartesian coordinate system.Chapter 1. Introduction 1 1.1 BACKGROUND 1 1.2 MOTIVATION 2 1.3 SCOPE OF PRESENT STUDY 3 Chapter 2. Governing Equations 5 2.1 LINEARIZED EULER EQUATIONS 5 2.2 DERIVATION OF PML EQUATIONS 6 2.2.1 Complex Change of Variables 6 2.2.2 Space-time Transformation 7 2.2.3 Stable PML Equations 9 Chapter 3. Numerical methodology 14 3.1 OPTIMIZED NUMERICAL METHOD 14 3.1.1 Fourier Analysis of High-order Spatial Discretization 14 3.1.2 Optimized Time Discretization Scheme 17 3.2 NUMERICAL STABILITY ANALYSIS 19 Chapter 4. Non-Reflective PML Conditions 24 4.1 END CONDITION OF PML BOUNDARY 24 4.2 ANALYTICAL APPROACH ON ABSORPTION COEFFICIENT 28 4.2.1 Maximum Absorption Coefficient 28 4.2.2 Minimum Absorption Coefficient 34 Chapter 5. Numerical Tests 38 5.1 STABILITY ANALYSIS RESULTS 39 5.1.1 Sound Propagating in Low Mach number Uniform Flow 40 5.1.2 Sound Propagating in High Mach number Uniform Flow 40 5.2 ACCURACY ANALYSIS RESULTS 42 5.2.1 Sound Propagating in Cartesian Grid System 42 5.2.2 Sound Propagating in Curvilinear Grid System 44 Chapter 6. Concluding Remarks 51 References 52 Abstract in Korean 56Maste