Impedance boundary conditions for acoustic time harmonic wave propagation in viscous gases

We present Helmholtz or Helmholtz like equations for the approximation of the time-harmonic wave propagation in gases with small viscosity, which are completed with local boundary conditions on rigid walls. We derived approximative models based on the method of multiple scales for the pressure and the velocity separately, both up to order 2. Approximations to the pressure are described by the Helmholtz equations with impedance boundary conditions, which relate its normal derivative to the pressure itself. The boundary conditions from first order on are of Wentzell type and include a second tangential derivative of the pressure proportional to the square root of the viscosity, and take thereby absorption inside the viscosity boundary layer of the underlying velocity into account. The velocity approximations are described by Helmholtz like equations for the velocity, where the Laplace operator is replaced by \nabla \Div, and the local boundary conditions relate the normal velocity component to its divergence. The velocity approximations are for the so-called far field and do not exhibit a boundary layer. Including a boundary corrector, the so called near field, the velocity approximation is accurate even up to the domain boundary. The boundary conditions are stable and asymptotically exact, which is justified by a complete mathematical analysis. The results of some numerical experiments are presented to illustrate the theoretical foundation

High order asymptotic expansion for the acoustics in viscous gases close to rigid walls

We derive a complete asymptotic expansion for the singularly perturbed problem of the acoustic wave propagation inside gases with small viscosity, this for the non-resonant case in smooth bounded domains in two dimensions. Close to rigid walls the tangential velocity shows a boundary layer of size $O(\sqrt{\eta})$ where $\eta$ is the dynamic viscosity. The asymptotic expansion based on the technique of multiscale expansion is in powers of $\sqrt{\eta}$ and takes into account curvature effects. The terms of the velocity and pressure expansion are defined independently by partial differential equations, where the normal component of velocities or the normal derivative of the pressure, respectively, are prescribed on the boundary. The asymptotic expansion is rigorously justified with optimal error estimates

Impedance boundary conditions for acoustic time‐harmonic wave propagation in viscous gases in two dimensions

We present impedance boundary conditions for the viscoacoustic equations for approximative models that are in terms of the acoustic pressure or in terms of the macroscropic acoustic velocity. The approximative models are derived by the method of multiple scales up to order 2 in the boundary layer thickness. The boundary conditions are stable and asymptotically exact, which is justified by a complete mathematical analysis. The models can be discretized by finite element methods without resolving boundary layers. In difference to an approximation by asymptotic expansion for which for each order 1 PDE system has to be solved, the proposed approximative are solutions to one PDE system only. The impedance boundary conditions for the pressure of first and second orders are of Wentzell type and include a second tangential derivative of the pressure proportional to the square root of the viscosity and take thereby absorption inside the viscosity boundary layer of the underlying velocity into account. The conditions of second order incorporate with curvature the geometrical properties of the wall. The velocity approximations are described by Helmholtz‐like equations for the velocity, where the Laplace operator is replaced by ∇div, and the local boundary conditions relate the normal velocity component to its divergence. The velocity approximations are for the so‐called far field and do not exhibit a boundary layer. Including a boundary corrector, the so‐called near field, the velocity approximation is accurate even up to the domain boundary. The results of numerical experiments illustrate the theoretical foundations

Impedance boundary conditions for acoustic time harmonic wave propagation in viscous gases in two dimensions

We present impedance boundary conditions for the viscoacoustic equations for approximative models that are in terms of the acoustic pressure or in terms of the macroscropic acoustic velocity. The approximative models are derived by the method of multiple scales up to order 2 in the boundary layer thickness. The boundary conditions are stable and asymptotically exact, which is justified by a complete mathematical analysis. The models can be discretized by finite element methods without resolving boundary layers. In difference to an approximation by asymptotic expansion for which for each order 1 PDE system has to be solved, the proposed approximative are solutions to one PDE system only. The impedance boundary conditions for the pressure of first and second orders are of Wentzell type and include a second tangential derivative of the pressure proportional to the square root of the viscosity and take thereby absorption inside the viscosity boundary layer of the underlying velocity into account. The conditions of second order incorporate with curvature the geometrical properties of the wall. The velocity approximations are described by Helmholtz-like equations for the velocity, where the Laplace operator is replaced by \nabla \textrmDiv , and the local boundary conditions relate the normal velocity component to its divergence. The velocity approximations are for the so-called far field and do not exhibit a boundary layer. Including a boundary corrector, the so-called near field, the velocity approximation is accurate even up to the domain boundary. The results of numerical experiments illustrate the theoretical foundations

High-order asymptotic expansion for the acoustics in viscous gases close to rigid walls

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Asymptotic analysis for acoustics in viscous gases close to rigid walls

We derive a complete asymptotic expansion for the singularly perturbed problem of acoustic wave propagation inside gases with small viscosity. This derivation is for the non-resonant case in smooth bounded domains in two dimensions. Close to rigid walls the tangential velocity exhibits a boundary layer of size where η is the dynamic viscosity. The asymptotic expansion, which is based on the technique of multiscale expansion is expressed in powers of and takes into account curvature effects. The terms of the velocity and pressure expansion are defined independently by partial differential equations, where the normal component of velocities or the normal derivative of the pressure, respectively, are prescribed on the boundary. The asymptotic expansion is rigorously justified with optimal error estimates

BALANCED AVERAGING OF BILINEAR SYSTEMS WITH APPLICATIONS TO STOCHASTIC CONTROL

Abstract. We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics can be split into a fast and a slow subspace and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity. 1

On impedance conditions for circular multiperforated acoustic liners

Background The acoustic damping in gas turbines and aero-engines relies to a great extent on acoustic liners that consists of a cavity and a perforated face sheet. The prediction of the impedance of the liners by direct numerical simulation is nowadays not feasible due to the hundreds to thousands repetitions of tiny holes. We introduce a procedure to numerically obtain the Rayleigh conductivity for acoustic liners for viscous gases at rest, and with it define the acoustic impedance of the perforated sheet. Results The proposed method decouples the effects that are dominant on different scales: (a) viscous and incompressible flow at the scale of one hole, (b) inviscid and incompressible flow at the scale of the hole pattern, and (c) inviscid and compressible flow at the scale of the wave-length. With the method of matched asymptotic expansions we couple the different scales and eventually obtain effective impedance conditions on the macroscopic scale. For this the effective Rayleigh conductivity results by numerical solution of an instationary Stokes problem in frequency domain around one hole with prescribed pressure at infinite distance to the aperture. It depends on hole shape, frequency, mean density and viscosity divided by the area of the periodicity cell. This enables us to estimate dissipation losses and transmission properties, that we compare with acoustic measurements in a duct acoustic test rig with a circular cross-section by the German Aerospace Center in Berlin. Conclusions A precise and reasonable definition of an effective Rayleigh conductivity at the scale of one hole is proposed and impedance conditions for the macroscopic pressure or velocity are derived in a systematic procedure. The comparison with experiments show that the derived impedance conditions give a good prediction of the dissipation losses