This paper examines the Reynolds number ($Re$) dependence of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) which develops over a two-dimensional rough wall with a view to ascertaining whether this type of boundary layer can become independent of $Re$. Measurements are made using hot-wire anemometry over a rough wall that consists of a periodic arrangement of cylindrical rods with a streamwise spacing of eight times the rod diameter. The present results, together with those obtained over a sand-grain roughness at high Reynolds number, indicate that a $Re$-independent state can be achieved at a moderate $Re$. However, it is also found that the mean velocity distributions over different roughness geometries do not collapse when normalised by appropriate velocity and length scales. This lack of collapse is attributed to the difference in the drag coefficient between these geometries. We also show that the collapse of the $U_{\unicode[STIX]{x1D70F}}$-normalised mean velocity defect profiles may not necessarily reflect $Re$-independence. A better indicator of the asymptotic state of $Re$ is the mean velocity defect profile normalised by the free-stream velocity and plotted as a function of $y/\unicode[STIX]{x1D6FF}$, where $y$ is the vertical distance from the wall and $\unicode[STIX]{x1D6FF}$ is the boundary layer thickness. This is well supported by the measurements.

Non-modal mechanisms underlying transient growth of propagating acoustic waves and non-propagating vorticity perturbations in an unbounded compressible shear flow are investigated, making use of closed form solutions. Propagating acoustic waves amplify mainly due to two mechanisms: growth due to advection of streamwise velocity that is typically termed as the lift-up mechanism leading for large Mach numbers to an almost linear increase in streamwise velocity with time and growth due to the downgradient irrotational component of the Reynolds stress leading to linear growth of acoustic wave energy for large times. Synergy between these mechanisms along with the downgradient solenoidal component of the Reynolds stress produces large and robust energy amplification.

On the other hand, non-propagating vorticity perturbations amplify due to kinematic deformation of vorticity by the mean flow. For weakly compressible flows, an initial vorticity perturbation abruptly excites acoustic waves with exponentially small amplitude, and the energy gained by vorticity perturbations is transferred back to the mean flow. For moderate Mach numbers, a strong coupling between vorticity perturbations and acoustic waves is found with the energy gained by vorticity perturbations being transferred to acoustic waves that are abruptly excited by the vortex.

Calculation of the optimal perturbations for a viscous flow shows that for low Mach numbers, acoustic wave excitation by vorticity perturbations and the subsequent growth of acoustic waves leads to robust energy growth of the order of Reynolds number, while for large Mach numbers, synergy between the lift-up mechanism and the downgradient solenoidal component of the Reynolds stress dominates the growth and leads to a comparable large amplification of streamwise velocity.

Flow past a NACA 65 blade at chord-based Reynolds number 138 500 is studied using stability analysis, generalized (spatially weighted) transient growth analysis and direct numerical simulations (DNS). The mechanisms of transition on various sections of the blade observed in previous work by Zaki et al. (J. Fluid Mech., vol. 665, 2010, pp. 57–98) are examined, with a focus on the pressure side around the leading edge. In this region, the linearly most energetic perturbation has spanwise wavenumber $40\unicode[STIX]{x03C0}$ (five boundary-layer thicknesses) and is tilted against the mean shear to take advantage of the Orr mechanism. In a DNS, the nonlinear development of this optimal perturbation induces $\unicode[STIX]{x1D6EC}$ structures, which are further stretched to hairpin vortices before breaking down to turbulence. At higher spanwise wavenumber, e.g. $120\unicode[STIX]{x03C0}$, a free-stream optimal perturbation is obtained upstream of the leading edge, in the form of streamwise vortices. During its nonlinear evolution, this optimal perturbation tilts the mean shear and generates spanwise periodic high- and low-speed streaks. Then through a nonlinear lift-up mechanism, the low-speed streaks are lifted above the high-speed ones. This layout of streaks generates a mean shear with two inflectional points and activates secondary instabilities, namely inner and outer instabilities previously reported in the literature.

In a stable background density gradient, initially turbulent flows eventually evolve into a state dominated by low-Froude-number dynamics and frequently also contain persistent pattern information. Much empirical evidence has been gathered on these latter stages, but less on how they first got that way, and how information on the turbulence generator may potentially be encoded into the flow in a robust and long-lasting fashion. Here an experiment is described that examines the initial stages of evolution in the vertical plane of a turbulent grid-generated wake in a stratified ambient. Refractive-index-matched fluids allow optically based measurement of early ($Nt<2$) stages of the flow, even when there are strong variations in the local density gradient field. Suitably averaged flow measures show the interplay between internal wave motions and Kelvin–Helmholtz-generated vortical modes. The vertical shear is dominant at the wake edge, and the decay of horizontal vorticity is observed to be independent of $\mathit{Fr}$. Stratified turbulence, originating from Kelvin–Helmholtz instabilities, develops up to non-dimensional time $Nt\approx 10$, and the scale separation between Ozmidov and Kolmogorov scales is independent of $\mathit{Fr}$ at higher $Nt$. The detailed measurements in the near wake, with independent variation of both Reynolds and Froude numbers, while limited to one particular case, are sufficient to show that the initial turbulence in a stratified fluid is neither three-dimensional nor universal. The search for appropriately generalizable initial conditions may be more involved than hoped for.

The spatial evolution of small-amplitude unsteady disturbances of an axisymmetric swirling jet is examined theoretically. The slow axial divergence of the jet mean flow is accounted for by using the method of multiple scales and a consistent solution for both the mean flow and unsteady disturbance is derived. Previous work by Lu & Lele (1999) has considered the leading-order analysis, in which the modal eigenvalues are determined from locally parallel theory, but the key feature of our analysis is the solution of the next-order secularity condition for the axial variation of the wave-envelope amplitude.

The swirling jet profile sustains two types of instability waves: the Kelvin–Helmholtz instability associated with axial shear, and a centrifugal instability which arises due to a decrease in circulation with radial distance. The evolution of the disturbance axial wavenumber and envelope amplitude with downstream distance is calculated. Numerical results show that the growth of the centrifugal mode is significantly curtailed as a result of a rapidly decaying envelope amplitude. The shear instability is significantly more amplified by the addition of swirl.

The general solution for the disturbance envelope amplitude breaks down at so-called turning points. This is found to occur for a series of neutral propagating modes. A rescaling in the vicinity of the turning point shows that the amplitude in this region is governed by a parabolic cylinder equation. The modal amplitude is seen to decay very significantly through this turning point, even though the mode is neutral to leading order.

The structure of shock-waves in gas mixtures is studied. The separation of component velocities and temperatures is described. Velocity overshoot is never found to exist. Other effects, namely, temperature overshoot and undershoot and velocity undershoot are shown to exist in a manner which is self-consistent with the derivation of the governing equations.

We use some well-known properties of the Taylor-Goldstein equation to generate a set of stably stratified shear flows with known singular neutral-mode solutions. The novel feature of the analysis is that it includes such solutions for flows in which, proceeding upwards from a rigid boundary, the Brunt-Väisälä frequency and the flow shear do not change sign and are monotonically decreasing functions of height. Such profiles are much closer to the conditions met in work on the atmospheric boundary layer than the more frequently used inflected flow profiles.

The theoretical work reported herein studies the free-surface profile, the flow structure, and the pressure distribution of a finite-amplitude solitary wave on shallow water with uniform vorticity. The kinematic problem for the stream function is formulated employing the vertical coordinate and the free surface as the independent variables of the Poisson equation with variable coefficients that are functions of the Hamiltonian of the rotational solitary wave. The exact solution of the boundary-value kinematic problem for the stream function is derived in the form of a power series complemented by a recurrence relation. The dynamic problems for the Hamiltonian and the free surface are solved globally in the Boussinesq–Rayleigh approximation. To find angles enclosed by the branches of the solution at critical points and points of bifurcation the surface streamline is also treated locally by an exact topological solution. The complete analysis of the four-dimensional Hamiltonian maps presented in §4 specifies critical values of the Froude number and the vorticity for five flow regimes: the emergence of the solitary wave, the flow separation near the bottom, the flow separation near the crest, the critical regime for an instability, and the formation of a limiting configuration. The streamlines of the recirculating flow are obtained as a single-eddy bifurcation that preserves continuity of all derivatives on the boundary streamline. The eddy separated near the crest forms the limiting configuration by blocking the upstream current. The results are compared with weakly nonlinear theory, with numerical simulations and with field observations with satisfactory agreement.

Separating and reattaching flows in a two-dimensional compression corner were investigated experimentally at a Mach number of 7·0 and Reynolds numbers (based on the distance from the leading edge to the corner) of 4·75 × 106, 9·51 × 106 and 1·55 × 107. Heat-transfer measurements and Pitot traverses in the upstream boundary layer showed that the boundary layer had become fully turbulent at the start of the interactions. Increases in the Reynolds number gave increases in the length of separated shear layers and decreases in the corner angle required for incipient, separation. The reattachment pressure coefficients gave good agreement with the criterion of Batham (1969).

We study the motion of a fluid within a rigid spherical container subject to small-amplitude periodic rotations. The sphere is divided into two equal portions by an impermeable stretched elastic membrane whose boundary is attached to the container wall. The model aims to represent in a simplified fashion the dynamics of a vitreous membrane subject to microsaccadic movements of the human eye, assuming the vitreous to be liquefied. The vitreous is modelled as a Newtonian, incompressible fluid in irrotational motion and the problem is linearized, taking advantage of the hypothesis of small-amplitude eye rotations. Results show that, due to the presence of the fluid, the natural frequencies of oscillation of the membrane decrease significantly with respect to the case of a free membrane. Moreover, oscillations of a stretched membrane are found to be resonantly excited by rotations of the sphere with frequencies which are typical of microsaccadic eye movements. This study suggests the possibility that oscillations of vitreous membranes may induce the development of large tensile stresses capable of producing a retinal detachment. Such a conclusion will have to be further substantiated by more refined analyses accounting for further effects, such as nonlinearity and the possible viscoelastic behaviour of the vitreous located on one or both sides of the membrane.

Panel flutter is a dangerous aeroelastic instability of the skin panels of supersonic flight vehicles. Though the linear stability of panels in uniform flow has been studied in detail, the influence of the boundary layer is still an open question. Most studies of panel flutter in the presence of the boundary layer are devoted to the ($1/7$)th-power velocity law and yield a stabilising effect of the boundary layer. Recently, Vedeneev (J. Fluid Mech., vol. 736, 2013, pp. 216–249) considered arbitrary velocity and temperature profiles and showed that, for a generalised convex boundary layer profile, a decrease of the growth rates of ‘supersonic’ perturbations (responsible for single-mode panel flutter) is accompanied by destabilisation of ‘subsonic’ perturbations that are neutral in uniform flow. However, this result is not self-consistent, as the long-wave expansion for solutions of the Rayleigh equation was used, whereas subsonic perturbations, generally speaking, cannot be considered as long waves. More surprising results are obtained for the boundary layer profile with a generalised inflection point, where the effect of the layer is destabilising even for ‘supersonic’ perturbations, and such waves can also have short lengths. In order to overcome this inconsistency, in this paper, we solve the Rayleigh equation numerically and investigate the stability of short-wave perturbation of the elastic plate in the presence of the boundary layer. As before, two problem formulations are investigated. First, we study running waves in an infinite plate. Second, we analyse eigenmodes of the plate of large finite length and use Kulikovskii’s global instability criterion. Based on the results of calculations, we confirm that the effect of the boundary layer with a generalised inflection point can be essentially destabilising. On the other hand, for generalised convex boundary layers, calculations show that, unlike the prediction of the long-wave approximation, the finite plate is fully stabilised for sufficiently thick boundary layers.

Experimental evaluation of the radial momentum equation near the surface for an axisymmetrie turbulent wall jet is reported. The equation for flow contains six terms of magnitudes which cannot be neglected. The pressure gradient across the flow as well as along the flow is found to be of major importance. Thus, for a wall jet flow it is impossible to treat the radial and vertical momentum equations as independent of one another.

The so-called 2/15-law for two-point third-order velocity statistics in isotropic turbulence with helicity is computed for the first time from a direct numerical simulation of the Navier–Stokes equations in a $512^3$ periodic domain. This law is a statement of helicity conservation in the inertial range, analogous to the benchmark Kolmogorov 4/5-law for energy conservation in high-Reynolds-number turbulence. The appropriately normalized parity-breaking statistics, when measured in an arbitrary direction in the flow, disagree with the theoretical value of 2/15 predicted for isotropic turbulence. They are highly anisotropic and variable and remain so over long times. We employ a recently developed technique to average over many directions and so recover the statistically isotropic component of the flow. The angle-averaged statistics achieve the 2/15 factor to within about 7% instantaneously and about 5% on average over time. The inertial- and viscous-range behaviour of the helicity-dependent statistics and consequently the helicity flux, which appear in the 2/15-law, are shown to be more anisotropic and intermittent than the corresponding energy-dependent reflection-symmetric structure functions, and the energy flux, which appear in the 4/5-law. This suggests that the Kolmogorov assumption of local isotropy at high Reynolds numbers needs to be modified for the helicity-dependent statistics investigated here.

To aid the physical understanding of spiralling-columnar convection emerging in rapidly rotating spheres and spherical shells, two-dimensional thermal convection in a rapidly rotating annulus is investigated through the radial propagation properties of topographic Rossby waves. Two kinds of the boundaries containing the fluid in the axial direction are considered: a convex type modelling a spherical geometry and a concave type for comparison. The linear stability of a basic state with no motion and uniformly unstable stratification is examined and spirally elongated structures of critical convection are obtained for small Prandtl numbers. An analysis of the energy budget shows that a part of the kinetic energy generated in the region with slightly inclined boundaries is dynamically transferred and dissipates through viscosity in the region with strongly inclined boundaries. This indicates that the Rossby waves propagate from the region with slightly inclined boundaries to the region with strongly inclined boundaries. It is presented that the appearance of a spiral structure corresponds to an increase of the local radial wavenumber of the Rossby waves propagating in the radial direction. The flow patterns obtained using the dispersion relation of the Rossby waves coincide with those of the tailing part of the spiral structure obtained numerically. As the Prandtl number increases, the Rossby waves barely propagate because of strong viscous dissipation, and the flow pattern is localized in the region with slightly inclined boundaries. For convex boundaries with unstable stratification concentrating near the outer boundary and concave boundaries with unstable stratification confined near the inner boundary, the flow patterns tilt in the direction inverse to the case of uniform unstable stratification. The tilting direction of the flow pattern is not determined by the curvature of the boundaries considered but instead by the radial propagation direction of the Rossby waves excited by thermal convection.