A Stable, Perfectly Matched Layer for Linearized Euler Equations in Unsplit Physical Variables

Hu, Fang Q.

doi:10.1006/jcph.2001.6887

Cited by 263 publications

(183 citation statements)

References 29 publications

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“…The point source is implemented numerically as a Gaussian monopole of width w = ∆x + ∆y. A PML nonreflecting boundary [40] is applied at the bottom of the domain, and periodic boundary conditions connect the upstream and downstream ends of the domain at x = L ± . The impedance boundary condition is implemented by modifying the incoming characteristic at the boundary y = 0, as described in (11), with the velocity derivative ∂v ′ /∂t given by the modified boundary condition (9) and the mass-spring-damper modelled using (10); this requires storage of four extra quantities (v s , ξ, ν and η) at each location along the boundary.…”

Section: Test Case and Numerical Implementationmentioning

confidence: 99%

“…A small oscillatory point mass source of strength Q ′ = δ(x)δ(y +y s ) sin(ωt) generates small perturbations to this steady flow, and the wall at y = 0 responds as a mass-springdamper boundary governed by (2). The governing equations (19), together with a Perfectly Matched Layer (PML) absorbing boundary [40] along the bottom boundary at y = −H, and the Myers (5) or modified (9) boundary conditions enforced in terms of characteristics (22) along the top boundary y = 0, lead to ∂q ∂t = −F x ∂q ∂x − F y ∂q ∂y − σ y q − σ y F x ∂q ∂x + H, …”

Section: Appendix B Details Of the Two-dimensional Test Casementioning

confidence: 99%

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Time-domain implementation of an impedance boundary condition with boundary layer correction

Gabard

2016

Journal of Computational Physics

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A time-domain boundary condition is derived that accounts for the acoustic impedance of a thin boundary layer over an impedance boundary, based on the asymptotic frequency-domain boundary condition of Brambley [ , AIAA J. 49(6), pp. 1272[ -1282. A finite-difference reference implementation of this condition is presented and carefully validated against both an analytic solution and a discrete dispersion analysis for a simple test case. The discrete dispersion analysis enables the distinction between real physical instabilities and artificial numerical instabilities. The cause of the latter is suggested to be a combination of the real physical instabilities present and the aliasing and artificial zero group velocity of finite-difference schemes. It is suggested that these are general properties of any numerical discretization of an unstable system. Existing numerical filters are found to be inadequate to remove these artificial instabilities as they have a too wide pass band. The properties of numerical filters required to address this issue are discussed and a number of selective filters are presented that may prove useful in general. These filters are capable of removing only the artificial numerical instabilities, allowing the reference implementation to correctly reproduce the stability properties of the analytic solution.

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Section: Test Case and Numerical Implementationmentioning

confidence: 99%

Section: Appendix B Details Of the Two-dimensional Test Casementioning

confidence: 99%

Time-domain implementation of an impedance boundary condition with boundary layer correction

Gabard

2016

Journal of Computational Physics

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“…The stable PML formulation proposed by Hu [19] for a uniform flow is used here. The modified field equations in the PML regions are…”

Section: Perfectly Matched Layersmentioning

confidence: 99%

A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow

Gabard

2014

Journal of Computational Physics

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The effect of flow over an acoustic liner is generally described by the Myers impedance condition. The use of this impedance condition in time-domain numerical simulations has been plagued by stability issues, and various ad hoc techniques based on artificial damping or filtering have been used to stabilise the solution. The theoretical issue leading to the ill-posedness of this impedance condition in the time domain is now well understood. For computational models, some trends have been identified, but no detailed investigation of the cause of the instabilities in numerical simulations has been undertaken to date. This paper presents a dispersion analysis of the complete numerical model, based on finite-difference approximations, for a twodimensional model of a uniform flow above an impedance surface. It provides insight into the properties of the instability in the numerical model and clarifies the parameters that influence its presence. The dispersion analysis is also used to give useful information about the accuracy of the acoustic solution.Comparison between the dispersion analysis and numerical simulations shows that the instability associated with the Myers condition can be identified in the numerical model, but its properties differ significantly from that of the continuous model. The unbounded growth of the instability in the continuous model is not present in the numerical model due to the wavenumber aliasing inherent to numerical approximations. Instead, the numerical instability includes a wavenumber component behaving as an absolute instability. The trend previously reported that the instability is more likely to appear with fine grids is explained. While the instability in the numerical model is heavily dependent on the spatial resolution, it is well resolved in time and is not sensitive to the time step. In addition filtering techniques to stabilise the solutions are considered and it is found that, while they can reduce the instability in some cases, they do not represent a systematic or robust solution in general.

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“…There are three main categories for NRBCs, including characteristic boundary conditions (CBCs) [17,18], absorbing boundary condition based on the perfectly matched layer (PML) concept [19,20] and based on the sponge layer concept [21,22]. These types of boundary conditions are successfully implemented to have stable numerical computing and realistic results based on the Navier-Stokes equations [17,19,21] and LBM [18,20,22].…”

Section: Introductionmentioning

confidence: 99%

Implementation of a curved wall- and an absorbing open-boundary condition for the D3Q19 lattice Boltzmann method for simulation of incompressible fluid flows

Ezzatneshan

2018

Scientia Iranica

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A Stable, Perfectly Matched Layer for Linearized Euler Equations in Unsplit Physical Variables

Cited by 263 publications

References 29 publications

Time-domain implementation of an impedance boundary condition with boundary layer correction

Time-domain implementation of an impedance boundary condition with boundary layer correction

A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow

Implementation of a curved wall- and an absorbing open-boundary condition for the D3Q19 lattice Boltzmann method for simulation of incompressible fluid flows

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