The receptivity to localized surface roughness of a swept-wing boundary layer is studied by direct numerical simulation (DNS) and computations using the parabolized stability equations (PSEs). The DNS is laid out to reproduce wind tunnel experiments performed by Saric and coworkers, where micron-sized cylinders were used to trigger steady crossflow modes. The amplitudes of the roughness-induced fundamental crossflow wave and its superharmonics obtained from nonlinear PSE solutions agree excellently with the DNS results. A receptivity model using the direct and adjoint PSEs is shown to provide reliable predictions of the receptivity to roughness cylinders of different heights and chordwise locations. Being robust and computationally efficient, the model is well suited as a predictive tool of receptivity in flows of practical interest. The crossflow mode amplitudes obtained based on both DNS and PSE methods are 40 % of those measured in the experiments. Additional comparisons between experimental and PSE data for various disturbance wavelengths reveal that the measured disturbance amplitudes are consistently larger than those predicted by the PSE-based receptivity model by a nearly constant factor. Supplementary DNS and PSE computations suggest that possible natural leading-edge roughness and free-stream turbulence in the experiments are unlikely to account for this discrepancy. It is more likely that experimental uncertainties in the streamwise location of the roughness array and cylinder height are responsible for the additional receptivity observed in the experiments.

The behaviour of a gravity current propagating into a two-layer stratified ambient fluid is described in detail. A comprehensive description is given of the different flow regimes, with particular emphasis on the front condition linking the thickness of the gravity current to its speed of propagation and the transfer of energy to upstream disturbances in the form of internal bores and nonlinear solitary waves. Hydraulic theory analogous to that of two-layer flow over topography (Baines, J. Fluid Mech., vol. 146, 1984, pp. 127–167) is extended to the gravity current problem to classify frontal behaviour into the following regimes: Type I, subcritical currents; Type II, currents that generate upstream undular bores; Type III, currents that generate an upstream monotonic bore connected by a rarefaction; Type IV, supercritical fronts with a large-amplitude trapped solitary-wave-like disturbance; and Type V, supercritical gravity currents. Over 200 two-dimensional Boussinesq–Euler simulations spanning a range of gravity current properties demonstrate good agreement, for both the behavioural regime and the front condition , with hydraulic theory that extends original work by Rottman & Simpson (Q. J. R. Meteorol. Soc., vol. 115, 1989, pp. 941–963) to arbitrary ambient layer thickness, and uses an improved closure for the upstream bore that correctly predicts the behaviour in the limit of large bore amplitude. In addition, the energy balance is analysed, and it is shown that the energy transfer from the gravity current to upstream disturbances is significant, and consistent with the hydraulic theory. The results demonstrate a clear connection to the problem of upstream resonance in two-layer flow over topography, and have significant implications for interpreting field observations of nonlinear internal waves generated by atmospheric density currents and coastal river plumes.

The structure of zonal jets arising in forced-dissipative, two-dimensional turbulent flow on the -plane is investigated using high-resolution, long-time numerical integrations, with particular emphasis on the late-time distribution of potential vorticity. The structure of the jets is found to depend in a simple way on a single non-dimensional parameter, which may be conveniently expressed as the ratio , where and are two natural length scales arising in the problem; here may be taken as the r.m.s. velocity, is the background gradient of potential vorticity in the north–south direction, and is the rate of energy input by the forcing. It is shown that jet strength increases with , with the limiting case of the potential vorticity staircase, comprising a monotonic, piecewise-constant profile in the north–south direction, being approached for . At lower values, eddies created by the forcing become sufficiently intense to continually disrupt the steepening of potential vorticity gradients in the jet cores, preventing strong jets from developing. Although detailed features such as the regularity of jet spacing and intensity are found to depend on the spectral distribution of the forcing, the approach of the staircase limit with increasing is robust across a variety of different forcing types considered.

Oceanic large- and meso-scale flows are nearly balanced in forces between Earth’s rotation and density stratification effects (i.e. geostrophic, hydrostatic balance associated with small Rossby and Froude numbers). In this regime advective cross-scale interactions mostly drive energy toward larger scales (i.e. inverse cascade). However, viscous energy dissipation occurs at small scales. So how does the energy reservoir at larger scales leak toward small-scale dissipation to arrive at climate equilibrium? Here we solve the linear instability problem of a balanced flow in a rotating and continuously stratified fluid far away from any boundaries (i.e. an interior jet). The basic flow is unstable not only to geostrophic baroclinic and barotropic instabilities, but also to ageostrophic instabilities, leading to the growth of small-scale motions that we hypothesize are less constrained by geostrophic cascade behaviours in a nonlinear regime and thus could escape from the inverse energy cascade. This instability is investigated in the parameter regime of moderate Rossby and Froude numbers, below the well-known regimes of gravitational, centrifugal, and Kelvin–Helmholtz instability. The ageostrophic instability modes arise with increasing Rossby number through a near-degeneracy of two unstable modes with coincident phase speeds. The near-degeneracy occurs in the neighbourhood of an identified criterion for the non-integrability of the ‘isentropic balance equations’ (namely with the absolute vertical vorticity and the horizontal strain rate associated with the basic flow), beyond which development of an unbalanced component of the flow is expected. These modes extract energy from the basic state with large vertical Reynolds stress work (unlike geostrophic instabilities) and act locally to modify the basic flow by reducing the isopycnal Ertel potential vorticity gradient near both its zero surface and its critical surface (phase speed equal to basic flow speed).

The dynamics of symmetric instability and two-dimensional inertia–gravity waves in a baroclinic geostrophic flow undergoing frontogenesis is analysed. A frontogenetic strain associated with a balanced deformation field drives an ageostrophic circulation and temporal variations in the basic state that significantly affect the properties of perturbations to the background flow. For stable stratification, perturbations to the basic state result in symmetric instability or inertia–gravity waves, depending on the sign of the Ertel potential vorticity and the magnitude of the Richardson number of the geostrophic flow. The kinetic energy (KE) of both types of motion is suppressed by frontogenetic strain due to the vertical shear in the ageostrophic circulation. This is because the perturbation streamlines tilt with the ageostrophic shear causing the disturbances to lose KE via shear production. The effect can completely dampen symmetric instability for sufficiently strong strain even though the source of KE for the instability (the vertical shear in the geostrophic flow) increases with time. Inertia–gravity waves in a baroclinic flow undergoing frontogenesis simultaneously lose KE and extract KE from the deformation field as they decay. This is because the horizontal velocity of the waves becomes rectilinear, resulting in a Reynolds stress that draws energy from the balanced flow. The process is most effective for waves of low frequency and for a geostrophic flow with low Richardson number. However, even in a background flow that is initially strongly stratified, frontogenesis leads to an exponentially fast reduction in the Richardson number, facilitating a rapid energy extraction by the waves. The KE transferred from the deformation field is ultimately lost to the unbalanced ageostrophic circulation through shear production, hence the inertia–gravity waves play a catalytic role in loss of balance. Given the large amount of KE in low-frequency inertia–gravity waves and the ubiquitous combination of strain and baroclinic geostrophic currents in the ocean, it is estimated that this mechanism could play a significant role in the removal of KE from both the internal wave and mesoscale eddy fields.

The origin of the secondary vortex street, observed in the far wake in the flow past a circular cylinder, is investigated. The Reynolds number, based on the diameter of the cylinder, is 150. The von Kármán vortex street, which originates in the near wake, decays exponentially downstream of the cylinder. Beyond the region of decay, a broad band of frequencies are selectively amplified, leading to the formation of a secondary vortex street consisting of packets of large-scale vortex structures. The streamwise location of the onset of the instability, frequency of the generation of packets and their convection speed are estimated via direct numerical simulation (DNS). Global linear stability analysis of the time-averaged flow reveals the presence of unstable convective modes that travel at almost the same speed and have a structure similar to the packet-like disturbances as observed in the DNS. Sensitivity analysis of the global convective modes to structural perturbations is carried out to locate the region of the wake that is most significant in generating the modes responsible for the appearance of the secondary vortex street. This information is utilized to control the flow. By placing a ‘slip’ splitter plate along the wake centre line, in the overlap region of the direct and the adjoint modes, the oscillations in the far wake are significantly reduced, though the oscillations related to the primary vortex shedding in the near wake are not. It is also found that suppression of the primary vortex shedding leads to annihilation of the secondary vortex street as well. Linear stability analysis of the steady-state flow does not yield any modes that can explain the appearance of the secondary vortex street. The steady and time-averaged wake profiles, for the flow, are compared to bring out the differences in the two. The effect of free-stream oscillations on the evolution of the secondary vortex street is investigated. By reducing the amplitude of inlet excitation, a gradual transition from ordered shedding in the far wake to the appearance of a broad-band spectrum of frequencies, as in the unforced wake, is observed. All the computations have been carried out using a stabilized finite element method.

The paper describes an asymptotic flow of a viscous fluid in an infinite annular cylindrical cavity as the distance from the flow source tends to infinity. If the driving flow near the source is axisymmetric then the asymptotic pattern is cellular; otherwise it is typically not. Boundary conditions are derived to match the asymptotic axisymmetric flow with that near the source. For a narrow cavity, the asymptotic solutions for the axisymmetric and three-dimensional flows are obtained analytically. For any gap, the flow is described by a numerical solution of an eigenvalue problem. The least decaying mode corresponds to azimuthal wavenumber .