Acoustical response of shear layers

In the present work the linear acoustical response of shear layers is investigated for two different geometrical configurations. Both theoretical modelling and experiments are carried out. The first studied configuration is a sudden area expansion in a duct with mean flow. Here, a shear layer, separating a region with mean flow from a region where the fluid is quiescent, is formed downstream of the area discontinuity. Theoretical modelling for this configuration is done by means of a modal analysis method. The geometry is split into a duct upstream and a duct downstream of the area expansion. The acoustical field in both ducts is found as an expansion of eigenmodes by solving a generalized eigenvalue problem, derived from the linearized Euler equations for conservation of mass and momentum. Here, a discretization in the transverse direction of the duct is employed. By mode matching, a procedure in which continuity of massand momentum flux at the interface between the two ducts is applied, the aeroacoustical behaviour is found in the form of a scattering matrix. This scattering matrix relates the modes propagating away from the area discontinuity to the modes propagating towards the area discontinuity. The influence of the mean flow profile, and in particular the application of an acoustical Kutta condition at the edge, on the scattering at the area expansion is investigated. A relatively small influence is observed, and a smooth transition from the case where a Kutta condition is not imposed to the case where it is imposed is seen. Scattering results for plane waves have also been compared to results of an alternative model proposed by Boij and Nilsson, as well as to experimental data of Ronneberger. The alternative model considers an area expansion in a rectangular duct and an infinitely thin shear layer. The experimental data of Ronneberger are obtained for an expansion in a cylindrical tube. In this context, the scaling rule, proposed by Boij and Nilsson, for the comparison of results obtained in a rectangular and a cylindrical geometry, is examined. For this purpose both modal analysis calculations are carried out for rectangular and cylindrical geometry. The scaling rule appears to be reasonably valid in a wide Strouhal number range. However, it is found that a deviation can occur around a certain critical Strouhal number. Here, a specific behaviour of the scattering is found, which depends on the ratio of the upstream and downstream duct heights or duct radii. Furthermore, comparison of results of the modal analysis method with those of the alternative model and the experimental data provided by Ronneberger shows fairly good correspondence. Also, an improved prediction of the experimental results by the modal analysis method is obtained in some cases when accounting for the non-uniform mean flow profile. The second configuration studied is that of a shear layer formed in a rectangular orifice in a wall due to the presence of mean grazing flow. Experimentally, the acoustical response of such a shear layer in an orifice is investigated by means of a multi-microphone impedance tube set-up. Care was taken to remain in the regime of linear perturbations. The acoustical behaviour is expressed as a change of orifice impedance due to the grazing flow. Here, the real and imaginary part of the difference of the non-dimensional impedance with flow and the non-dimensional impedance without flow are scaled to the Mach number and the Helmholtz number respectively. The obtained quantities are denoted as the non-dimensional scaled resistance due to the mean flow, respectively the non-dimensional scaled reactance due to the mean flow. This procedure was originally proposed by Golliard on basis of the theory of Howe. The influence of the boundary layer characteristics is investigated. For this purpose boundary layer characterization is performed by means of hot-wire measurements. The Strouhal number dependency of the non-dimensional scaled resistance and reactance due to the flow shows an oscillatory shape. When the Strouhal number is based on the phase velocity of the hydrodynamic instability in the shear layer, rather than the mean flow velocity, these oscillations coincide for different boundary layer flows. The phase velocity is deduced from the shear layer profiles, measured with a hot-wire, using the spatial instability analysis for parallel flows by Michalke. For laminar boundary layer flows the amplitudes of the oscillations increase with decreasing boundary layer thickness. Furthermore, the oscillating behaviour appears to vanish around a Strouhal number, at which the shear layer becomes stable. Since the instability of a shear layer depends on the Strouhal number based on its momentum thickness, the ratio of shear layer momentum thickness and orifice width determines the number of observed oscillations in impedance. The influence of the edge geometry of the orifice on the impedance with grazing flow is examined. It was found that the amplitudes of oscillations in the impedance increase when using sharp edges. Especially the downstream edge of the orifice is important. Similar to the configuration of an area expansion in a duct, theoretical modelling for the aeroacoustical response of a shear layer in an orifice is done by means of the modal analysis method. In this case the considered geometry is that of two parallel rectangular ducts, of which one carries mean flow. The orifice is represented by an interconnection between the two ducts. This geometry is split into five ducts, in each of which the acoustic field is solved as an expansion of eigenmodes. Mode matching at the relevant interfaces gives the acoustical behaviour in terms of a scattering matrix, which relates the modes propagating away from the orifice to the modes propagating towards the orifice. From the scattering matrix an orifice impedance is calculated. Modal analysis results are compared with those from an analytical model, proposed by Howe, which considers the low Helmholtz number, low Mach number acoustical response of an infinitely thin shear layer in an orifice in a thin wall separating unbounded uniform grazing flows. Qualitatively, good correspondence is seen between the two models. However, an unexpected influence of the geometrical ratios in the duct configuration on the non-dimensional scaled resistance and reactance is present in case of the modal analysis method. Also, the results both obtained by Howe’s theory and by the modal analysis method seem to display non-physical behaviour, especially in the high Strouhal number limit. For non-uniform grazing flow over an orifice convergence of the modal analysis method is a problematic issue. However, a tentative comparison with experimental results shows that the model at least qualitatively predicts the behaviour of the impedance with grazing flow fairly well. In particular, the oscillations in Strouhal number dependency of the non-dimensional scaled resistance and reactance due to the flow, and the related influence of the boundary layer thickness of the flow, are accurately predicted.