Edward J. Brambley

Well-Posed Boundary Condition for Acoustic Liners in Straight Ducts with Flow

DOI: 10.2514/1.J050723 The Myers boundary condition for acoustics within flow over an acoustic lining has been shown to be ill-posed, leading to numerical stability issues in the time domain and mathematical problems with stability analyses. This paper gives a modification (for flat or cylindrical straight ducts) to make the Myers boundary condition well posed, and indeed more accurate, by accounting for a thin inviscid boundary layer over the lining and correctly deriving the boundary condition to first order in the boundary-layer thickness. The modification involves two integraltermsovertheboundarylayer.The firstmaybewrittenintermsofthemass,momentum,andkinetic-energy thicknessesoftheboundarylayer,whichareshownto physicallycorrespondtoamodifiedboundarymass,modified grazingvelocity,andatensionalongtheboundary. Thesecondintegral termisrelatedtothecriticallayerwithinthe boundary layer. A time domain version of the new boundary condition is proposed, although not implemented. The modified boundary condition is validated against high-fidelity numerical solutions of the Pridmore-Brown equation for sheared inviscid flow in a cylinder. Absolute instability boundaries are given for certain examples, though convective instabilities appear to always be present at certain frequencies for any boundary-layer thickness.

Stability and acoustic scattering in a cylindrical thin shell containing compressible mean flow

We consider the stability of small perturbations to a uniform inviscid compressible flow within a cylindrical linear-elastic thin shell. The thin shell is modelled using Flugges equations, and is forced from the inside by the fluid, and from the outside by damping and spring forces. In addition to acoustic waves within the fluid, the system supports surface waves, which are strongly coupled to the thin shell. Stability is analysed using the Briggs-Bers criterion, and the system is found to be either stable or absolutely unstable, with absolute instability occurring for sufficiently small shell thicknesses. This is significantly different from the stability of a thin shell containing incompressible fluid, even for parameters for which the fluid would otherwise be expected to behave incompressibly (for example, water within a steel thin shell). Asymptotic expressions are derived for the shell thickness separating stable and unstable behaviour. We then consider the scattering of waves by a sudden change in the duct boundary from rigid to thin shell, using the Wiener-Hopf technique. For the scattering of an inbound acoustic wave in the rigid-wall section, the surface waves are found to play an important role close to the sudden boundary change. The solution is given analytically as a sum of duct modes. The results in this paper add to the understanding of the stability of surface waves in models of acoustic linings in aeroengine ducts. The oft-used mass-spring-damper model is regularized by the shell bending terms, and even when these terms are very small, the stability and scattering results are quite different from what has been claimed for the mass-spring-damper model. The scattering results derived here are exact, unique and causal, without the need to apply a Kutta-like condition or to include an instability wave. A movie is available with the online version of the paper.

Sound transmission in strongly curved slowly varying cylindrical ducts with flow

In this paper we consider the propagation of acoustic waves on top of an inviscid steady flow along a curved hollow or annular duct with hard or lined walls. The curvature of the duct centreline (which is not restricted to being small) and the wall radii vary slowly along the duct, allowing application of an asymptotic multiple-scales analysis. The modal wavenumbers and mode shapes are determined locally as modes of a torus with the same local curvature, while the amplitude of the modes evolves as the mode propagates along the duct. The duct modes are found explicitly at each axial location using a pseudospectral numerical method. Unlike the case of a straight duct carrying uniform flow, there is a fundamental asymmetry between upstream and downstream propagating modes, with some mode shapes tending to be concentrated on either the inside or outside of the bend depending on the direction of propagation, curvature and steady-flow Mach number. The interaction between the presence of wall lining and curvature is also significant; for instance, in a representative case it is found that the curvature causes the first few acoustic modes to be more heavily damped by the duct boundary than would be expected for a straight duct. Using ray theory, we suggest explanations of these features. For the lowest azimuthal-order modes, three distinct regimes are found in which the modes are localized in different parts of the duct cross-section. This phenomenon is explained by a balance between whispering-gallery effects along the duct and refraction by the steady flow. At the opposite extreme, strongly spinning modes are investigated, and are seen to be due to a different whispering-gallery effect across the duct cross-section.

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

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 two-dimensional 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.

Surface-Waves, Stability, and Scattering for a Lined Duct with Flow

We consider a straight cylindrical duct with a steady subsonic axial flow and a reacting boundary (e.g. an acoustic lining). The wave modes are separated into ordinary acoustic duct modes, and surface modes confined to a small neighbourhood of the boundary. Many researchers have used a mass-spring-damper boundary model, for which one surface mode has previously been identified as a convective instability; however, we show the stability analysis used in such cases to be questionable. We investigate instead the stability of the surface modes using the Briggs-Bers criterion for a Flugge thin-shell boundary model. For modest frequencies and wavenumbers the thin-shell has an impedance which is effectively that of a mass-spring-damper, although for the large wavenumbers needed for the stability analysis the thin-shell and mass-spring-damper impedances diverge, owing to the thin shells bending stiffness. The thin shell model may therefore be viewed as a regularization of the mass-spring-damper model which accounts for nonlocally-reacting effects. We find all modes to be stable for realistic thin-shell parameters, while absolute instabilities are demonstrated for extremely thin boundary thicknesses. The limit of vanishing bending stiffness is found to be a singular limit, yielding absolute instabilities of arbitrarily large temporal growth rate. We propose that the problems with previous stability analyses are due to the neglect of something akin to bending stiffness in the boundary model. Our conclusion is that the surface mode previously identified as a convective instability may well be stable in reality. Finally, inspired by Rienstras recent analysis, we investigate the scattering of an acoustic mode as it encounters a sudden change from a hard-wall to a thin-shell boundary, using a Wiener-Hopf technique. The thin-shell is considered to be clamped to the hard-wall. The acoustic mode is found to scatter into transmitted and reflected acoustic modes, and surface modes strongly linked to the solid waves in the boundary, although no longitudinal or transverse waves within the boundary are excited. Examples are provided that demonstrate total transmission, total reflection, and a combination of the two. This thin-shell scattering problem is preferable to the mass-spring-damper scattering problem presented by Rienstra, since the thin-shell problem is fully determined and does not need to appeal to a Kutta-like condition or the inclusion of an instability in order to avoid a surface-streamline cusp at the boundary change.

A well-posed modified Myers boundary condition

The Myers boundary condition for acoustics within flow over an acoustic lining has been shown to be illposed, leading to numerical stability issues in the time domain and mathematical problems with stability analyses. This paper gives a modification to make the Myers boundary condition well-posed, by accounting for a thin inviscid boundary layer over the lining, and correctly deriving the boundary condition to first order in the boundarylayer thickness. The modification involves two integral terms over the boundary layer. The first may be written in terms of the mass, momentum, and kinetic energy thicknesses of the boundary layer, which are shown to physically correspond a modified boundary mass, modified grazing velocity, and a tension along the boundary. The second integral term is related to the critical layer within the boundary layer. The modified boundary condition is validated against high-fidelity numerical solutions of the Pridmore-Brown equation for sheared inviscid flow in a cylinder. Absolute instability boundaries are given for certain examples, though convective instabilities appear to always be present for any boundary layer thickness.

Viscous effects on the acoustics and stability of a shear layer over an impedance wall

The effect of viscosity and thermal conduction on the acoustics in a shear layer above an impedance wall is investigated numerically and asymptotically by solving the compressible linearised Navier-Stokes equations. It is found that viscothermal effects can be as important as shear, and therefore including shear while neglecting viscothermal effects by solving the linearised Euler equations is questionable. In particular, the damping rate of upstream propagating waves is found to be dramatically under-predicted by the LEE in certain instances. The effects of viscosity on stability are also found to be important. Short wavelength disturbances are stabilised by viscosity, greatly altering the characteristic wavelength and maximum growth rate of instability. For the parameters typical of aeroacoustic simulations considered here, the Reynolds number below which the flow stabilizes ranges from

Optimized finite-difference (DRP) schemes perform poorly for decaying or growing oscillations

10^5

Surface modes in sheared flow using the modified Myers boundary condition

to

Viscous boundary layer effects on the Myers impedance boundary condition

10^7