Mirela Darau

Boundary-layer thickness effects of the hydrodynamic instability along an impedance wall

The Ingard―Myers condition, modelling the effect of an impedance wall under a mean flow by assuming a vanishingly thin boundary layer, is known to lead to an ill-posed problem in time domain. By analysing the stability of a linear-then-constant mean flow over a mass-spring-damper liner in a two-dimensional incompressible limit, we show that the flow is absolutely unstable for h smaller than a critical h c and convectively unstable or stable otherwise. This critical h c is by nature independent of wavelength or frequency and is a property of liner and mean flow only. An analytical approximation of h c is given, which is complemented by a contour plot covering all parameter values. For an aeronautically relevant example, h c is shown to be extremely small, which explains why this instability has never been observed in industrial practice. A systematically regularised boundary condition, to replace the Ingard―Myers condition, is proposed that retains the effects of a finite h, such that the stability of the approximate problem correctly follows the stability of the real problem.

The critical layer in sheared flow

Critical layers arise as a singularity of the linearized Euler equations when the phase speed of the disturbance is equal to the mean flow velocity. They are usually ignored when estimating the sound field, with their contribution assumed to be negligible. It is the aim of this paper to study fully both numerically and analytically a simple yet typical sheared ducted flow in order to distinguish between situations when the critical layer may or may not be ignored. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing for exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. It is found that the critical layer contribution decays algebraically in the constant flow part, with an additional contribution of constant amplitude when the source is in the boundary layer, an additional contribution of constant amplitude is excited, representing the hydrodynamic trailing vorticity of the source. This immediately triggers, for thin boundary layers, the inherent convective instability in the flow. Extra care is required for high frequencies as the critical layer can be neglected only together with the pole beneath it. For low frequencies this pole is trapped in the critical layer branch cut.

A new approximation of the dispersion relations occurring in the sound-attenuation problem of turbofan aircraft engines

The dispersion relations, appearing in the analysis of the stability of a gas flow in a straight acoustically-lined duct with respect to perturbations produced by a time harmonic source, beside the wave number and complex frequency contain the solution of a boundary value problem of the Pridmore-Brown equation depending on the wave number and frequency. For this reason, in practice the dispersion relations are rarely simple enough for carried out the zeros. The determination of zeros of these dispersion relations is crucial for the prediction of the perturbation attenuation or amplification. In this paper an approximation of the dispersion relations is given. Our approach preserves the general character of the mean flow, the general Pridmore-Brown equation and its only in the shear flow that we replace the exact solution of the boundary value problem with its Taylor polynomial approximate. In this way new approximate dispersion relations are obtained which zeros can be found by computer.