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Calculation of low Mach number acoustics : a comparison of MPV, EIF and linearized Euler equations

Sabine Roller, Thomas Schwartzkopff, Roland Fortenbach, Michael Dumbser, Claus-Dieter Munz (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The calculation of sound generation and propagation in low Mach number flows requires serious reflections on the characteristics of the underlying equations. Although the compressible Euler/Navier-Stokes equations cover all effects, an approximation via standard compressible solvers does not have the ability to represent acoustic waves correctly. Therefore, different methods have been developed to deal with the problem. In this paper, three of them are considered and compared to each other. They...

Calculation of low Mach number acoustics: a comparison of MPV, EIF and linearized Euler equations

Sabine Roller, Thomas Schwartzkopff, Roland Fortenbach, Michael Dumbser, Claus-Dieter Munz (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The calculation of sound generation and propagation in low Mach number flows requires serious reflections on the characteristics of the underlying equations. Although the compressible Euler/Navier-Stokes equations cover all effects, an approximation via standard compressible solvers does not have the ability to represent acoustic waves correctly. Therefore, different methods have been developed to deal with the problem. In this paper, three of them are considered and compared to each other....

Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition

Eliane Bécache, Jeronimo Rodríguez, Chrysoula Tsogka (2009)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly...

Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition

Eliane Bécache, Jeronimo Rodríguez, Chrysoula Tsogka (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always...

Generalized combined field integral equations for the iterative solution of the three-dimensional Helmholtz equation

Xavier Antoine, Marion Darbas (2007)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper addresses the derivation of new second-kind Fredholm combined field integral equations for the Krylov iterative solution of tridimensional acoustic scattering problems by a smooth closed surface. These integral equations need the introduction of suitable tangential square-root operators to regularize the formulations. Existence and uniqueness occur for these formulations. They can be interpreted as generalizations of the well-known Brakhage-Werner [A. Brakhage and P. Werner, Arch....

Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

Roberto Triggiani (2008)

Applicationes Mathematicae

This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators. Part I: Let λ i be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in ℝⁿ, with Neumann homogeneous boundary conditions on Γ = tial Ω. Let φ i j j = 1 ℓ i be the corresponding linearly independent (normalized) eigenfunctions...

Low Mach number limit for viscous compressible flows

Raphaël Danchin (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropic fluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes equations...

Low Mach number limit for viscous compressible flows

Raphaël Danchin (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropic fluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes...

Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition

Eric Luneville, Jean-Francois Mercier (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We study the time-harmonic acoustic scattering in a duct in presence of a flow and of a discontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuous one leads to still open modeling questions, as in particular the singularity of the solution at the abrupt transition and the choice of the right unknown to formulate the scattering problem. To address these questions we propose a mathematical approach based on variational formulations set in weighted Sobolev spaces. Considering...

Modeling of the resonance of an acoustic wave in a torus

Jérôme Adou, Adama Coulibaly, Narcisse Dakouri (2013)

Annales mathématiques Blaise Pascal

A pneumatic tyre in rotating motion with a constant angular velocity Ω is assimilated to a torus whose generating circle has a radius R . The contact of the tyre with the ground is schematized as an ellipse with semi-major axis a . When ( Ω R / C 0 ) ≪ 1 and ( a / R ) ≪ 1 (where C 0 is the velocity of the sound), we show that at the rapid time scale R / C 0 , the air motion within a torus periodically excited on its surface generates an acoustic wave h . A study of this acoustic wave is conducted and shows that the mode associated to...

Multiple spatial scales in engineering and atmospheric low Mach number flows

Rupert Klein (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this...

Multiple spatial scales in engineering and atmospheric low Mach number flows

Rupert Klein (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of...

Nouvelles formulations intégrales pour les problèmes de diffraction d’ondes

David P. Levadoux, Bastiaan L. Michielsen (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We present an integral equation method for solving boundary value problems of the Helmholtz equation in unbounded domains. The method relies on the factorisation of one of the Calderón projectors by an operator approximating the exterior admittance (Dirichlet to Neumann) operator of the scattering obstacle. We show how the pseudo-differential calculus allows us to construct such approximations and that this yields integral equations without internal resonances and being well-conditioned at all frequencies....

Nouvelles formulations intégrales pour les problèmes de diffraction d'ondes

David P. Levadoux, Bastiaan L. Michielsen (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present an integral equation method for solving boundary value problems of the Helmholtz equation in unbounded domains. The method relies on the factorisation of one of the Calderón projectors by an operator approximating the exterior admittance (Dirichlet to Neumann) operator of the scattering obstacle. We show how the pseudo-differential calculus allows us to construct such approximations and that this yields integral equations without internal resonances and being well-conditioned at all...

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