The propagation of small-amplitude modes in an inviscid but sheared subsonic mean flow inside a duct is considered. For isentropic flow in a circular duct with zero swirl and constant mean flow density the pressure modes are described in terms of the eigenvalue problem for the Pridmore-Brown equation with Myers' locally reacting impedance boundary conditions.The key purpose of the paper is to extend the results of the numerical study of the spectrum for the case of lined ducts with uniform mean flow in Rienstra (Wave Motion, vol. 37, 2003b, p. 119), in order to examine the effects of the shear and wall lining. In the present paper this far more difficult situation is dealt with analytically. The high-frequency short-wavelength asymptotic solution of the problem based on the WKB method is derived for the acoustic part of the spectrum. Owing to the stiffness of the governing equations, an accurate numerical study of the spectral properties of the problem for mean flows with strong shear proves to be a non-trivial task which deserves separate consideration.The second objective of the paper is to gain theoretical insight into the properties of the hydrodynamic part of the spectrum. An analysis of hydrodynamic modes both in the short-wavelength limit and for the case of the narrow duct is presented. For simplicity, only the hard-wall flow configuration is considered.

The propagation of small-amplitude modes in an inviscid but sheared mean flow inside a duct is studied numerically. For isentropic flow in a circular duct with zero swirl and constant mean flow density the pressure modes are described in terms of the eigenvalue problem for the Pridmore-Brown equation. Since for sufficiently high Helmholtz and wavenumbers, which are of great interest for the applications, the field equation is inherently stiff, special care is taken to insure the stability of the numerical algorithm designed to tackle this problem. The accuracy of the method is checked against the well-known analytical solution for the uniform flow. The numerical method is shown to be consistent with the analytical predictions at least for the Helmholtz numbers up to 100 and the circumferential wavenumber as large as 50, typical Mach numbers being up to 0.65.In order to gain further insight into the possible structure of the modal solutions and to get an independent verification of the robustness of the numerical scheme, comparison to the asymptotic solution of the problem based on the WKB method is performed. The asymptotic solution is also used as a benchmark for computations with high Helmholtz numbers, where numerical solutions of other authors are not available.The bulk of the analysis concentrates on the influence of the wall lining. The proposed numerical procedure is adapted in order to include Ingard-Myers boundary conditions. In parallel with this, the WKB solution is used to check the numerical predictions of the typical behaviour of the axial wavenumber in the complex plane, when the wall impedance varies in the complex plane.Numerical analysis of the problem with zero mean flow at the wall and acoustic lining shows that the use of Ingard-Myers condition in combination with an appropriate slipstream approximation instead of the actual no-slip mean flow profile gives valid results in the limit of vanishing boundary-layer thickness, although the boundary layer must be very thin in some cases.

Since early manned space flight orographically forced cloud patterns have been described in terms of the single isolated shock structure of shallow-water flow or, equivalently, compressible fluid flow. Some of these observations show, behind an initial “bow wave,” a series of almost parallel wave crests. This paper considers the simplest extension of shallow-water theory that retains not only nonlinear steepening of waves but includes departures from hydrostatic balance, and thus wave dispersion, showing that the single shocks of shallow-water theory are transformed into multiple parallel finite-amplitude wave crests. The context of the discussion is the forced Kadomtsev–Petviashvili equation from classical ship wave dynamics, which plays the same role in two-dimensional near-critical fluid flow as the more familiar Korteweg–de Vries equation in one-dimensional flow. The drag and flow regimes in near-critical flow over isolated orography are described in terms of the three governing parameters of the flow: the deviation of the flow speed from critical, the strength of nonhydrostatic effects, and the strength of orographic forcing.

This paper reports experimental observations of finite amplitude interfacial waves forced by a surface-mounted obstacle towed through a two-layer fluid both when the fluid is otherwise at rest and when the fluid is otherwise rotating as a solid body. The experimental apparatus is sufficiently wide so that sidewall effects are negligible even in near-critical flow when the towing speed is close to the interfacial long-wave speed and the transverse extent of the forced wavefield is large. The observations are modelled by a simple forced Benjamin–Davis–Acrivos equation and comparison between integrations of both linear and nonlinear problems shows the fundamental nonlinearity of the near-critical flow patterns. In both the experiments and integrations rotation strongly confines the wavefield to extend laterally over distances only of order of the Rossby radius and also introduces finite-amplitude sharply pointed lee waves in supercritical flow.

This paper presents analytical and numerical results for separated stratified inviscid flow over and around an isolated mountain in the limit of small Froude number. The vertical density profile consists of a lower strongly stratified layer whose depth is just less than that of the mountain. It is separated from a semi-infinite upper stably stratified layer by a thin, highly stable, inversion layer. The paper aims to provide, for this particular profile, a thorough analysis of the three-dimensional separated flow over a mountain top with strong stratification. The Froude numbers F and F I of the lower layer and the interface are small with F I F 1, but the upper-layer Froude number is arbitrary. The flow at each height in the lower layer is governed by the twodimensional Euler equations and moves horizontally around the mountain. It is given by a modification of a previous model using Kirchhoff free-streamline theory for the separated flow region downstream of the mountain. The pressure variations associated with the lower-layer flow are of the same order as the dynamic head and induce significant displacements of the inversion layer. When the inversion is near the top of the mountain these deflections are of the same order as the height of the projecting part of the mountain top and combine with the flow over the mountain top to excite vertically propagating internal waves in the upper layer. The resultant pressure field, vertical stream surface displacements, and surface streamlines in the upper layer are described consistently in the hydrostatic limit. Many of the features of the upper flow, including the perturbations of the critical dividing streamlines, are similar to those in flows with uniform stable stratification at low Froude number. Comparisons are made with experiments and approximate models for these summit flows based on the assumption that the dividing streamlines have small vertical displacement.