Nonlinear acoustics in a viscothermal boundary layer over an acoustic lining

Sound within aircraft engines can be 120dB-160dB, pushing the validity of linearized governing equations. Moreover, some components of sound within a visco-thermal mean flow boundary layer over an acoustic lining may be amplified by a factor of ~100 (~40dB) in a typical aircraft engine compared with the sound outside the boundary layer, which may be expected to trigger nonlinear effects within the boundary layer. This is in addition to the well-known nonlinear effects within the holes of the perforated lining facing sheet. This paper presents a mathematical investigation into the effects of weak nonlinearity on the acoustics within a thin parallel mean flow boundary layer in flow over an acoustic lining in a cylindrical duct. (This is the first investigation of nonlinear acoustics in a boundary layer flow over a non-rigid surface, to our knowledge.) The analysis combines the effects of sheared mean flow, viscosity, and nonlinearity into an effective impedance boundary condition. In certain cases, a surprisingly large acoustic streaming effect is found that escapes the mean flow boundary layer and pervades well out into the interior of the duct

Nonlinear sound propagation in 2D curved ducts : a multimodal approach

A method for studying weakly nonlinear acoustic propagation in 2D ducts of general shape - including curvature and variable width - is presented. The method is based on a local modal decomposition of the pressure and velocity in the duct. A pair of nonlinear ODEs for the modal amplitudes of the pressure and velocity modes is derived. To overcome the inherent instability of these equations, a nonlinear admittance relation between the pressure and velocity modes is presented, introducing a novel `nonlinear admittance' term. Appropriate equations for the admittance are derived which are to be solved through the duct, with a radiation condition applied at the duct exit. The pressure and velocity are subsequently found by integrating an equation involving the admittance through the duct. The method is compared, both analytically and numerically, against published results and the importance of nonlinearity is demonstrated in ducts of complex geometry. Comparisons between ducts of differing geometry are also performed to illustrate the effect of geometry on nonlinear sound propagation. A new 'nonlinear reflectance' term is introduced, providing a more complete description of acoustic reflection that also takes into account the amplitude of the incident wave

The impedance boundary condition for acoustics in swirling ducted flow

The acoustics of a straight annular lined duct containing a swirling mean flow is considered. The classical Ingard–Myers impedance boundary condition is shown not to be correct for swirling flow. By considering behaviour within the thin boundary layers at the duct walls, the correct impedance boundary condition for an infinitely thin boundary layer with swirl is derived, which reduces to the Ingard–Myers condition when the swirl is set to zero. The correct boundary condition contains a spring-like term due to centrifugal acceleration at the walls, and consequently has a different sign at the inner (hub) and outer (tip) walls. Examples are given for mean flows relevant to the interstage region of aeroengines. Surface waves in swirling flows are also considered, and are shown to obey a more complicated dispersion relation than for non-swirling flows. The stability of the surface waves is also investigated, and as in the non-swirling case, one unstable surface wave per wall is found

Acoustics in a two-deck viscothermal boundary layer over an impedance surface

The acoustics of a mean flow boundary layer over an impedance surface or acoustic lining is considered. By considering a thick mean flow boundary layer (possibly due to turbulence), the boundary-layer structure is separated asymptotically into two decks, with a thin weakly viscous mean flow boundary layer and an even thinner strongly viscous acoustic sublayer, without requiring a high frequency. Using this, analytic solutions are found for the acoustic modes in a cylindrical lined duct. The mode shapes in each region compare well with numerical solutions of the linearized compressible Navier–Stokes equations, as does a uniform composite asymptotic solution. A closed-form effective impedance boundary condition is derived, which can be applied to acoustics in inviscid slipping flow to account for both shear and viscosity in the boundary layer. The importance of the boundary layer is demonstrated in the frequency domain, and the new boundary condition is found to correctly predict the attenuation of upstream-propagating cut-on modes, which are poorly predicted by existing inviscid boundary conditions. Stability is also investigated, and the new boundary condition is found to yield good results away from the critical layer. A time-domain formulation of a simplified version of the new impedance boundary condition is proposed

Meta-analysis of curvature trends in asymmetric rolling

This paper investigates curvature in asymmetric rolling by combining existing results from eight experimental and nine numerical previous studies. These previous results are digitised and a linear regression model fitted which explains 65% of the variance in these data. It is found that conclusions from several previous studies are contradicted by other previous studies, and that there is no consensus on the fundamental mechanism of curvature generation in rolling. Results from an existing curvature-predicting analytic slab model are also compared with the previous results, and the agreement is shown to be adequate at best. Future work is clearly needed to enable accurate curvature prediction, and it is hoped that the evidence collected here will inform future investigations and models to ensure the relevant range of parameters are considered

Stabilisation of hydrodynamic instabilities by critical layers in acoustic lining boundary layers

Acoustics are considered in a straight cylindrical lined duct with an axial mean flow that is uniform apart from a boundary layer near the wall. Within the boundary layer, which may or may not be thin, the flow profile is quadratic and satisfies no-slip at the wall. Time-harmonic modal solutions to the linearized Euler equations are found by solving the Pridmore-Brown equation using Frobenius series. The Briggs–Bers criterion is used to ascertain the spatial stability of the modes, without considering absolute instabilities. The modes usually identified as hydrodynamic instabilities are found to interact with the critical layer branch cut, also known as the continuous spectrum. By varying the boundary-layer thickness, flow speed, frequency, and wall impedance, it is found that these spatial instabilities can be stabilized behind the critical layer branch cut. In particular, spatial instabilities are only found for a boundary layer thinner than a critical boundary-layer thickness hc . The behaviors observed for the uniform-quadratic sheared flow considered here are further compared to a uniform-linear sheared flow, and a uniform slipping flow under the Ingard–Myers boundary condition, where this process of stabilization is not observed. It is therefore argued that modeling a sufficiently smooth mean flow boundary layer is necessary to predict the correct stability of flow over a lined wall

The critical layer in quadratic flow boundary layers over acoustic linings

A straight cylindrical duct is considered containing an axial mean flow that is uniform everywhere except within a boundary layer near the wall, which need not be thin. Within this boundary layer the mean flow varies parabolically. The linearized Euler equations are Fourier transformed to give the Pridmore-Brown equation, for which the Green's function is constructed using Frobenius series. The critical layer gives a non-modal contribution from the continuous spectrum branch cut, and dominates the downstream pressure perturbation in certain cases, particularly for thicker boundary layers. The continuous spectrum branch cut is also found to stabilize what are otherwise convectively unstable modes by hiding them behind the branch cut. Overall, the contribution from the critical layer is found to give a neutrally stable non-modal wave when the source is located within the sheared flow region, and to decay algebraically along the duct as O(x−5/2) for a source located with the uniform flow region. The Frobenius expansion, in addition to being numerically accurate close to the critical layer where other numerical methods lose accuracy, is also able to locate modal poles hidden behind the branch cut, which other methods are unable to find; this includes the stabilized hydrodynamic instability. Matlab code is provided to compute the Green's function

The critical layer in quadratic flow boundary layers over acoustic linings

A straight cylindrical duct is considered containing an axial mean flow that is uniform everywhere except within a boundary layer near the wall, which need not be thin. Within this boundary layer the mean flow varies parabolically. The linearized Euler equations are Fourier transformed to give the Pridmore-Brown equation, for which the Greens function is constructed using Frobenius series. Inverting the spatial Fourier transform, the critical layer contribution is given as the non-modal contribution from integrating around the continuous spectrum branch cut. This contribution is found to be the dominant downstream contribution to the pressure perturbation in certain cases, particularly for thicker boundary layers. Moreover, the continuous spectrum branch cut is found to be involved in stabilizing what are otherwise convectively unstable modes by hiding them behind the branch cut, particularly for slower flows. Overall, the contribution from the critical layer is found to give a neutrally stable non-modal wave with a phase velocity equal to the mean flow velocity at the source when the source is located within the sheared-flow region, and to decay algebraically along the duct as O(x-5/2) for a source located with the uniform flow region. The Frobenius expansion, in addition to being numerically accurate close to the critical layer where other numerical methods loose accuracy, is also able to locate modal poles hidden behind the branch cut, which other methods are unable to find. Matlab code is provided to compute the Greens function