A brief derivation is given of the acoustic wave equation describing the propagation of sound in a unidirectional shear flow. This equation has been solved exactly in only one instance, namely a linear velocity profile; in the present paper a second exact solution is given, for the exponential velocity profile, which represents a boundary layer with weak suction at a high Reynolds number. The acoustic wave equation has a critical layer where the Doppler shifted frequency vanishes, and this corresponds to a regular singularity; another regular singularity corresponds to the free stream and the sound field consists either of propagating waves or of surface waves, showing that the critical layer can act as an absorbing layer. Analytical continuation is used to cover the whole flow region, from the wall boundary layer to the free stream; the appropriate boundary, radiation and stability conditions are discussed, and the acoustic pressure is plotted as a function of distance from the wall for several combinations of frequency, wave number parallel to the wall and low Mach number free–stream velocity; the combination of solutions appropriate to rigid and impedance walls is also plotted. The solutions are expressible in terms of Bessel functions only when the critical layer is in the free stream; when the critical layer is in the boundary layer, or when there is no critical layer, the solutions require an extension of the Gaussian hypergeometric equation, in which one of the three singularities is irregular; its solutions are extensions of Gaussian hypergeometric and Mathieu functions, whose general properties are discussed elsewhere.

A hard landing incident in Pico Aerodrome (LPPI) involving an Airbus A320-200 aircraft is investigated using airborne observations and forecasts of the AROME (Applications of Research to Operations at Mesoscale) model. A second flight is also analyzed. The severity of the wind shear during both flights is quantified using the intensity factor “I” that is based on aerial data and recommended by ICAO (International Civil Aviation Organization). During Flight 1, 36% of the landing phase (below 2100 ft) occurred under “severe” wind shear conditions and 16% occurred under “strong” conditions. Upstream characteristics included southwest winds, stable stratification and a Froude number close to 1. According to the AROME model, these circumstances triggered the development of vertically propagating mountain waves, with maximum vertical velocities above 400 ft/min and exceeding 200 ft/min in the flight path. These conditions, together with the severe wind shear, may have caused the incident. During the second flight, a wake with lee vortices and reversed flow developed in the region of the flight path, which is consistent with a low upstream Froude number and/or with the flow regime diagram of previous studies. During the approach phase of this flight, “severe” wind shear conditions were absent, with “strong” ones occurring 4% of the time. It predominantly displayed “light” conditions during 68% of this phase. As a result of the comparison between “I” and the AROME turbulence indicators, preliminary thresholds are proposed for these indexes. Lastly, this study provides an objective verification of AROME wind forecasts, showing a good agreement with airborne observations for wind speeds above 10 kt, but a poor skill for weaker winds.

The propagation of sound is considered in a potential cylindrical vortex, with superimposed axial flow, by means of explicit analytical solutions. The sound waves are sinusoidal in time and in the axial and azimuthal directions; the convected wave equation leads to a radial dependence specified by an ordinary second-order differential equation, with two singularities, at the origin and at infinity. Both singularities are irregular, implying that the acoustic fields have an essential singularity. In the neighborhood of the vortex axis, the essential singularity of the acoustic field is specified by an exponential of the integrated Doppler shift; using the latter as a factor, the acoustic fields are specified by asymptotic expansions in ascending powers of the radius. In the neighborhood of the point at infinity, where the tangential mean flow velocity vanishes, the leading terms are outward or inward propagating cylindrical waves; these factors multiply asymptotic expansions in descending powers of the radius. The two pairs of solutions, around the vortex axis and the point at infinity, are valid in all space or overlapping regions, as far as the asymptotic expansions can be calculated. The case of an annular nozzle, with uniform axial flow, and potential swirl is used as an example; the eigenvalues are obtained for rigid wall boundary conditions and the corresponding eigenfunctions are plotted.

A wave propagating in a non-uniform flow can have a critical layer where it is absorbed, amplified, reflected, or converted to another mode, possibly exchanging energy with the mean flow. Two examples are the propagation of: (i) fan noise in the shear flow in the air inlet of a jet engine; (ii) turbine noise in the swirling flow in the jet exhaust. Both situations (i) and (ii) are included by considering wave propagation in an axisymmetric isentropic non-homentopic mean flow allowing for the simultaneous existence of shear, swirl, temperature and density gradients. This corresponds to coupled acoustic-vortical modes that have both continuous and discrete spectra. It is shown that a critical layer exists where the Doppler shifted frequency vanishes. A stability condition is obtained for the continuous spectrum for all frequencies, axial and azimuthal wavenumbers generalizing previous results from the homentropic to the isentropic non-isentropic case. The wave fields are calculated and plotted for a case of non-rigid body rotation, namely angular velocity of swirl proportional to the radius; this demonstrates the mode conversion between acoustic and vortical waves across the critical layer where the waves are absorbed because the pressure spectrum vanishes.