We investigate the influence of free-stream vortical disturbances on the evolution and instability of an incompressible laminar boundary layer, focusing on components of sufficiently long wavelength, which are known to penetrate into the boundary layer to generate streamwise elongated streaks. The free-stream disturbance is assumed to be sufficiently strong (but still of small amplitude) that the induced streaks acquire an O(1) streamwise velocity in the region where the boundary-layer thickness becomes comparable with the spanwise wavelength of the perturbation. The formation and evolution of the streaks are governed by the nonlinear unsteady boundary-region equations supplemented by appropriate upstream and far-field boundary conditions. This initial-boundary-value problem is solved for the special case where the free-stream disturbance is modelled by a pair of oblique vortical modes with the same frequency but opposite spanwise wavenumbers. Nonlinearity is found to inhibit the response. The nonlinear interaction alters the mean-flow profile appreciably, the shape of which is in quantitative agreement with experimental measurements. Wall-normal inflection points are detected in the instantaneous streamwise velocity profiles. The secondary stability analysis indicates that in the presence of free-stream disturbance with an intensity of 2.8%, the resulting streaky boundary layer becomes inviscidly unstable. The characteristic frequency, phase and group velocities, and growth rate of unstable sinuous modes are found to be in broad agreement with recent experiments. The present theoretical framework allows in principle a quantitative relation to be established between the characteristics of free-stream turbulence and the secondary instability, and this relation may be exploited to develop an efficient and physics-based approach for predicting bypass transition.

We experimentally investigate the dynamics of surface waves excited by oscillations from a cylindrical sidewall. Particle-imaging-velocimetry measurements with fluorescent particles were used to determine the flow patterns near the sidewall of the cylindrical fluid container and to identify the locations of the evolving air–water interfaces. The high-frequency wall oscillations created four jets that originate at the cylindrical sidewall. Four vortex streets shed from the jets propagate from the sidewall to the centre of the container and subsequently excite a low-frequency gravity wave. The interaction between this gravitational surface wave and the high-frequency capillary waves was found to be responsible for creating droplet splash at the water surface. This phenomenon was first described as ‘Long-Xi’ or ‘dragon wash’ in ancient China. The physical processes for generating the droplet ejection, including the circular capillary waves, azimuthal waves, streaming jets and low-frequency gravity waves, are described in this paper.

The attenuation of three-dimensional wavepackets of streaks and Tollmien–Schlichting (TS) waves in a transitional boundary layer using feedback control is investigated numerically. Arrays of localized sensors and actuators (about 10–20) with compact spatial support are distributed near the rigid wall equidistantly along the spanwise direction and connected to a low-dimensional (r = 60) linear quadratic Gaussian controller. The control objective is to minimize the disturbance energy in a domain spanned by a number of proper orthogonal decomposition modes. The feedback controller is based on a reduced-order model of the linearized Navier–Stokes equations including the inputs and outputs, computed using a snapshot-based balanced truncation method. To account for the different temporal and spatial behaviour of the two main instabilities of boundary-layer flows, we design two controllers. We demonstrate that the two controllers reduce the energy growth of both TS wavepackets and streak packets substantially and efficiently, using relatively few sensors and actuators. The robustness of the controller is investigated by varying the number of actuators and sensors, the Reynolds number and the pressure gradient. This work constitutes the first experimentally feasible simulation-based control design using localized sensing and acting devices in conjunction with linear control theory in a three-dimensional setting.

Results of direct numerical simulations of the development of a breaking internal gravity wave are presented. The wave was forced by the imposition of an appropriate bottom boundary shape (a two-dimensional cosine hill) within a density-stratified domain having a uniform upstream velocity and density gradient. The focus is on turbulence generation and maintenance within the turbulent patch generated by the wave breaking. Pathlines, density contours, temporal and spatial spectra, and second moments of the velocity and density fluctuations and turbulent kinetic energy balance terms obtained from the data averaged over the span in the mixed zone are all used in the analysis of the flow. Typical Reynolds numbers, based on the vertical scale of the breaking region and the upstream velocity, were around 6000 and the fully resolved computations yielded sufficient resolution to capture the fine-scale transition processes as well as the subsequent fully developed turbulence. It is shown that globally, within the turbulent patch, there is an approximate balance in the production, dissipation and transport processes for turbulent kinetic energy, so that the patch remains quasi-steady over a significant time. Although it is far from being axially homogeneous, with turbulence generation occurring largely near the upstream bottom part of the patch where the mean velocity shear is particularly large, it has features not dissimilar to those of a classical turbulent wake.

It is proposed that critical balance – a scale-by-scale balance between the linear propagation and nonlinear interaction time scales – can be used as a universal scaling conjecture for determining the spectra of strong turbulence in anisotropic wave systems. Magnetohydrodynamic (MHD), rotating and stratified turbulence are considered under this assumption and, in particular, a novel and experimentally testable energy cascade scenario and a set of scalings of the spectra are proposed for low-Rossby-number rotating turbulence. It is argued that in neutral fluids the critically balanced anisotropic cascade provides a natural path from strong anisotropy at large scales to isotropic Kolmogorov turbulence at very small scales. It is also argued that the k−2⊥ spectra seen in recent numerical simulations of low-Rossby-number rotating turbulence may be analogous to the k−3/2⊥ spectra of the numerical MHD turbulence in the sense that they could be explained by assuming that fluctuations are polarised (aligned) approximately as inertial waves (Alfvén waves for MHD).

The roles of the Coriolis force and the convection associated with the fluid motion in the formation of columnar eddies in freely decaying homogeneous rotating turbulence at a moderate Rossby number are studied by direct numerical simulation of the Navier–Stokes equations in a periodic box. The simulated field is compared with a series of artificial fields generated by switching off the nonlinear and viscous terms in the Navier–Stokes equation at given instants. The comparison shows that, without the nonlinear convection effect, the Coriolis force cannot sustain the substantial growth in the direction parallel to the rotational axis of the length scale defined on the basis of the two-point correlation of the square of the vorticity, i.e. cannot sustain the formation of the columnar eddies. The length scale characterizes well the intuitive impression from visualization of flow obeying the dynamics with or without the nonlinear effect. It is shown that the lack of substantial growth is insensitive to the scale of the eddies, the box size and viscosity, at least in the case studied here.

The zero-pressure-gradient turbulent boundary layer over a flat plate was perturbed by a short strip of two-dimensional roughness elements, and the downstream response of the flow field was interrogated by hot-wire anemometry and particle image velocimetry. Two internal layers, marking the two transitions between rough and smooth boundary conditions, are shown to represent the edges of a ‘stress bore’ in the flow field. New scalings, based on the mean velocity gradient and the third moment of the streamwise fluctuating velocity component, are used to identify this ‘stress bore’ as the region of influence of the roughness impulse. Spectral composite maps reveal the redistribution of spectral energy by the impulsive perturbation – in particular, the region of the near-wall peak was reached by use of a single hot wire in order to identify the significant changes to the near-wall cycle. In addition, analysis of the distribution of vortex cores shows a distinct structural change in the flow associated with the perturbation. A short spatially impulsive patch of roughness is shown to provide a vehicle for modifying a large portion of the downstream flow field in a controlled and persistent way.

The analysis of surface oscillations of liquid drops allows measurements of the surface tension and viscosity of the liquid. For small oscillations of spherical drops with a free surface, classical formulae by Rayleigh and Lamb relate these quantities to the frequency and damping of the oscillations. In many cases, however, the drop's surface is covered by a surface film, typically an oxide layer or a surfactant, exhibiting a rheological behaviour different from the bulk fluid. It is the purpose of this paper to investigate how such surface properties influence the oscillation spectrum of a spherical drop. For small bulk shear viscosity, the cases of small, finite and large surface viscosities are discussed, and the onset of aperiodic motion as a function of the surface parameters is also derived.

This work is a parametric study of the fluxes of heat and salt across unsheared and sheared double-diffusive interfaces using one-dimensional-turbulence (ODT) simulations. It is motivated by the need to understand how these fluxes scale with parameters related to the fluid molecular properties and background shear. Comparisons are made throughout with previous models and available measurements. In unsheared interfaces, ODT simulations show that the dimensionless heat flux Nu scales with the stability parameter Rρ, Rayleigh number Ra and Prandtl number Pr as Nu ~ (Ra/Rρ)0.37±0.03 when Pr varies from 3 to 100 and as Nu ~ (Ra/Rρ)0.31Pr0.22±0.04 when Pr varies from 0.01 to 1. Here Ra/Rρ can be seen as the ratio of destabilizing and stabilizing effects. The simulation results also indicate that the ratio of salt and heat fluxes Rf is independent of Pr, scales with the Lewis number Le as Rf ~ Le0.41±0.04 when Rρ is large enough and deviates from this expression for low values of Rρ, when the interface becomes heavily eroded. In sheared interfaces, the simulations show three flow regimes. When the Richardson number Ri ≪ 1, shear-induced mixing dominates, the heat flux scales with the horizontal velocity difference across the interface and Rf = Rρ. Near Ri ~ 1 the heat and salt fluxes are seen to increase abruptly as the shear increases. The flow structure and scaling of the fluxes are similar to those of unsheared interfaces when Ri ≫ 1.

Vortex structures made by impulsively translating low aspect-ratio plates are studied experimentally using defocusing digital particle image velocimetry. The investigation of translating plates with a 90° angle of attack is important since it is a fundamental model for a better understanding of drag-based propulsion systems. Rectangular flat-rigid, flexible and curved-rigid thin plates with the same aspect ratio are studied in order to develop qualitative and quantitative understanding of their vortex structures and hydrodynamic forces. We find that the vortex formation processes of all three cases are drastically different from each other. The interaction of leading-edge vortices and tip flow near the tip region is an important mechanism to distinguish vortex patterns among these three cases. The drag trends of three cases are correlated closely with vortex structure and circulation. The initial peak of hydrodynamic force in the flexible plate case is not as large as the initial peak of the flat and curved rigid plate cases during the acceleration phase. However, after the initial peak, the flexible plate generates a large force comparable to that of the flat-rigid plate case in spite of its deformed shape, which results from the slow development of the vortex structure.

High-elasticity simulations of flows through a two-dimensional (2D) 4 : 1 abrupt contraction and a 4 : 1 three-dimensional square–square abrupt contraction were performed with a finite-volume method implementing the log-conformation formulation, proposed by Fattal & Kupferman (J. Non-Newtonian Fluid Mech., vol. 123, 2004, p. 281) to alleviate the high-Weissenberg-number problem. For the 2D simulations of Boger fluids, modelled by the Oldroyd-B constitutive equation, local flow unsteadiness appears at a relatively low Deborah number (De) of 2.5. Predictions at higher De were possible only with the log-conformation technique and showed that the periodic unsteadiness grows with De leading to an asymmetric flow with alternate back-shedding of vorticity from pulsating upstream recirculating eddies. This is accompanied by a frequency doubling mechanism deteriorating to a chaotic regime at high De. The log-conformation technique provides solutions of accuracy similar to the thoroughly tested standard finite-volume method under steady flow conditions and the onset of a time-dependent solution occurred approximately at the same Deborah number for both formulations. Nevertheless, for Deborah numbers higher than the critical Deborah number, and for which the standard iterative technique diverges, the log-conformation technique continues to provide stable solutions up to quite (impressively) high Deborah numbers, demonstrating its advantages relative to the standard methodology. For the 3D contraction, calculations were restricted to steady flows of Oldroyd-B and Phan-Thien–Tanner (PTT) fluids and very high De were attained (De ≈ 20 for PTT with ϵ = 0.02 and De ≈ 10000 for PTT with ϵ = 0.25), with prediction of strong vortex enhancement. For the Boger fluid calculations, there was inversion of the secondary flow at high De, as observed experimentally by Sousa et al. (J. Non-Newtonian Fluid Mech., vol. 160, 2009, p. 122).

Recent clinical trials have shown the efficacy of a passive acoustic device used during shock wave lithotripsy (SWL) treatment. The device uses the far-field acoustic emissions resulting from the interaction of the therapeutic shock waves with the tissue and kidney stone to diagnose the effectiveness of each shock in contributing to stone fragmentation. This paper details simulations that supported the development of that device by extending computational fluid dynamics (CFD) simulations of the flow and near-field pressures associated with shock-induced bubble collapse to allow estimation of those far-field acoustic emissions. This is a required stage in the development of the device, because current computational resources are not sufficient to simulate the far-field emissions to ranges of O(10 cm) using CFD. Similarly, they are insufficient to cover the duration of the entire cavitation event, and here simulate only the first part of the interaction of the bubble with the lithotripter shock wave in order to demonstrate the methods by which the far-field acoustic emissions resulting from the interaction can be estimated. A free-Lagrange method (FLM) is used to simulate the collapse of initially stable air bubbles in water as a result of their interaction with a planar lithotripter shock. To estimate the far-field acoustic emissions from the interaction, this paper developed two numerical codes using the Kirchhoff and Ffowcs William–Hawkings (FW-H) formulations. When coupled to the FLM code, they can be used to estimate the far-field acoustic emissions of cavitation events. The limitation of the technique is that it assumes that no significant nonlinear acoustic propagation occurs outside the control surface. Methods are outlined for ameliorating this problem if, as here, computational resources cannot compute the flow field to sufficient distance, although for the clinical situation discussed, this limitation is tempered by the effect of tissue absorption, which here is incorporated through the standard derating procedure. This approach allowed identification of the sources of, and explanation of trends seen in, the characteristics of the far-field emissions observed in clinic, to an extent that was sufficient for the development of this clinical device.

We investigate the in-line and cross-flow vortex-induced vibrations of a long cylindrical tensioned beam, with length to diameter ratio L/D = 200, placed within a linearly sheared oncoming flow, using three-dimensional direct numerical simulation. The study is conducted at three Reynolds numbers, from 110 to 1100 based on maximum velocity, so as to include the transition to turbulence in the wake. The selected tension and bending stiffness lead to high-wavenumber vibrations, similar to those encountered in long ocean structures. The resulting vortex-induced vibrations consist of a mixture of standing and travelling wave patterns in both the in-line and cross-flow directions; the travelling wave component is preferentially oriented from high to low velocity regions. The in-line and cross-flow vibrations have a frequency ratio approximately equal to 2. Lock-in, the phenomenon of self-excited vibrations accompanied by synchronization between the vortex shedding and cross-flow vibration frequencies, occurs in the high-velocity region, extending across 30% or more of the beam length. The occurrence of lock-in disrupts the spanwise regularity of the cellular patterns observed in the wake of stationary cylinders in shear flow. The wake exhibits an oblique vortex shedding pattern, inclined in the direction of the travelling wave component of the cylinder vibrations. Vortex splittings occur between spanwise cells of constant vortex shedding frequency. The flow excites the cylinder under the lock-in condition with a preferential in-line versus cross-flow motion phase difference corresponding to counter-clockwise, figure-eight orbits; but it damps cylinder vibrations in the non-lock-in region. Both mono-frequency and multi-frequency responses may be excited. In the case of multi-frequency response and within the lock-in region, the wake can lock in to different frequencies at various spanwise locations; however, lock-in is a locally mono-frequency event, and hence the flow supplies energy to the structure mainly at the local lock-in frequency.

Results are presented from an experimental study on the dynamics of pairs of vortices, in which the axial velocity within each core differs from that of the surrounding fluid. Co- and counter-rotating vortex pairs at moderate Reynolds numbers were generated in a water channel from the tips of two rectangular wings. Measurement of the three-dimensional velocity field was accomplished using stereoscopic particle image velocimetry, revealing significant axial velocity deficits in the cores. For counter-rotating pairs, the long-wavelength Crow instability, involving symmetric wavy displacements of the vortices, could be clearly observed using dye visualisation. Measurements of both the axial wavelength and the growth rate of the unstable perturbation field were found to be in good agreement with theoretical predictions based on the full experimentally measured velocity profile of the vortices, including the axial flow. The dye visualisations further revealed the existence of a short-wavelength core instability. Proper orthogonal decomposition of the time series of images from high-speed video recordings allowed a precise characterisation of the instability mode, which involves an interaction of waves with azimuthal wavenumbers m = 2 and m = 0. This combination of waves fulfils the resonance condition for the elliptic instability mechanism acting in strained vortical flows. A numerical three-dimensional stability analysis of the experimental vortex pair revealed the same unstable mode, and a comparison of the wavelength and growth rate with the values obtained experimentally from dye visualisations shows good agreement. Pairs of co-rotating vortices evolve in the form of a double helix in the water channel. For flow configurations that do not lead to merging of the two vortices over the length of the test section, the same type of short-wave perturbations were observed. As for the counter-rotating case, quantitative measurements of the wavelength and growth rate, and comparison with previous theoretical predictions, again identify the instability as due to the elliptic mechanism. Importantly, the spatial character of the short-wave instability for vortex pairs with axial flow is different from that previously found in pairs without axial flow, which exhibit an azimuthal variation with wavenumber m = 1.

The interaction between turbulence in a minimal supersonic channel and radiative heat transfer is studied using large-eddy simulation. The working fluid is pure water vapour with temperature-dependent specific heats and molecular transport coefficients. Its line spectra properties are represented with a statistical narrow-band correlated-k model. A grey gas model is also tested. The parallel no-slip channel walls are treated as black surfaces concerning thermal radiation and are kept at a constant temperature of 1000 K. Simulations have been performed for different optical thicknesses (based on the Planck mean absorption coefficient) and different Mach numbers. Results for the mean flow variables, Reynolds stresses and certain terms of their transport equations indicate that thermal radiation effects counteract compressibility (Mach number) effects. An analysis of the total energy balance reveals the importance of radiative heat transfer, compared to the turbulent and mean molecular heat transport.

The global-scale interior magnetic field Bi needed to account for the Sun's observed differential rotation can be effective only if confined below the convection zone in all latitudes including, most critically, the polar caps. Axisymmetric solutions are obtained to the nonlinear magnetohydrodynamic equations showing that such polar confinement can be brought about by a very weak downwelling flow U ~ 10−5cms−1 over each pole. Such downwelling is consistent with the helioseismic evidence. All three components of the magnetic field B decay exponentially with altitude across a thin, laminar ‘magnetic confinement layer’ located at the bottom of the tachocline and permeated by spiralling field lines. With realistic parameter values, the thickness of the confinement layer ~10−3 of the Sun's radius, the thickness scale being the magnetic advection–diffusion scale δ = η/U where the magnetic (ohmic) diffusivity η ≈ 4.1 × 102cm2s−1. Alongside baroclinic effects and stable thermal stratification, the solutions take into account the stable compositional stratification of the helium settling layer, if present as in today's Sun, and the small diffusivity of helium through hydrogen, χ ≈ 0.9 × 101cm2s−1. The small value of χ relative to η produces a double boundary-layer structure in which a ‘helium sublayer‘ of smaller vertical scale (χ/η)1/2δ is sandwiched between the top of the helium settling layer and the rest of the confinement layer. Solutions are obtained using both semi-analytical and purely numerical, finite-difference techniques. The confinement-layer flows are magnetostrophic to excellent approximation. More precisely, the principal force balances are between Lorentz, Coriolis, pressure-gradient and buoyancy forces, with relative accelerations negligible to excellent approximation. Viscous forces are also negligible, even in the helium sublayer where shears are greatest. This is despite the kinematic viscosity being somewhat greater than χ. We discuss how the confinement layers s at each pole might fit into a global dynamical picture of the solar tachocline. That picture, in turn, suggests a new insight into the early Sun and into the longstanding enigma of solar lithium depletion.

The liquid lining in small human airways can become unstable and form liquid plugs that close off the airways. Direct numerical simulations are carried out on an airway model to study this airway instability and the flow-induced stresses on the airway walls. The equations governing the fluid motion and the interfacial boundary conditions are solved using the finite-volume method coupled with the sharp interface method for the free surface. The dynamics of the closure process is simulated for a viscous Newtonian film with constant surface tension and a passive core gas phase. In addition, a special case is examined that considers the core dynamics so that comparisons can be made with the experiments of Bian et al. (J. Fluid Mech., vol. 647, 2010, p. 391). The computed flow fields and stress distributions are consistent with the experimental findings. Within the short time span of the closure process, there are large fluctuations in the wall shear stress. Furthermore, dramatic velocity changes in the film during closure indicate a steep normal stress gradient on the airway wall. The computational results show that the wall shear stress, normal stress and their gradients during closure can be high enough to injure airway epithelial cells.

In many tidal embayments, complex patterns of channels and shoals are observed. To gain a better understanding of these features, an idealized model, that describes the interaction of water motion, sediment transport and bed evolution in a semi-enclosed, rectangular basin, is developed and analysed. To explain the initial formation of channels and shoals, two-dimensional perturbations superposed on a laterally uniform equilibrium bottom are studied. These perturbations evolve due to convergences of various residual suspended sediment fluxes: a diffusive flux, a flux related to the bed topography, an advective flux resulting from internally generated overtides and an advective flux due to externally prescribed overtides. For most combinations of these fluxes, perturbations start to grow if the bottom friction is strong enough. Their growth is mainly a result of convergences of diffusive and topographically induced sediment fluxes. Advective contributions due to internally generated overtides enhance this growth. If only diffusive sediment fluxes are considered, the underlying equilibrium is always unstable. This can be traced back to the depth dependence of the deposition parameter. Contrary to the results of previous idealized models, the channels and shoals always initiate in the shallow, landward areas. This is explained by the enhanced generation (compared to that in previous models) of frictional torques in shallow regions. The resulting initial channel–shoal formation compares well with results found in complex numerical model studies. The instability mechanism and the location of the initial formation of bottom patterns do not change qualitatively when varying parameters. Changes are mainly related to differences in the underlying equilibrium profile due to parameter variations.

Double-diffusive instabilities are often invoked to explain enhanced transport in stably stratified fluids. The most-studied natural manifestation of this process, fingering convection, commonly occurs in the ocean's thermocline and typically increases diapycnal mixing by 2 orders of magnitude over molecular diffusion. Fingering convection is also often associated with structures on much larger scales, such as thermohaline intrusions, gravity waves and thermohaline staircases. In this paper, we present an exhaustive study of the phenomenon from small to large scales. We perform the first three-dimensional simulations of the process at realistic values of the heat and salt diffusivities and provide accurate estimates of the induced turbulent transport. Our results are consistent with oceanic field measurements of diapycnal mixing in fingering regions. We then develop a generalized mean-field theory to study the stability of fingering systems to large-scale perturbations using our calculated turbulent fluxes to parameterize small-scale transport. The theory recovers the intrusive instability, the collective instability and the γ-instability as limiting cases. We find that the fastest growing large-scale mode depends sensitively on the ratio of the background gradients of temperature and salinity (the density ratio). While only intrusive modes exist at high density ratios, the collective and γ instabilities dominate the system at the low density ratios where staircases are typically observed. We conclude by discussing our findings in the context of staircase-formation theory.

Regions of the ocean's thermocline unstable to salt fingering are often observed to host thermohaline staircases, stacks of deep well-mixed convective layers separated by thin stably stratified interfaces. Decades after their discovery, however, their origin remains controversial. In this paper we use three-dimensional direct numerical simulations to shed light on the problem. We study the evolution of an analogous double-diffusive system, starting from an initial statistically homogeneous fingering state, and find that it spontaneously transforms into a layered state. By analysing our results in the light of the mean-field theory developed in Part 1 (Traxler et al., J. Fluid Mech. doi:10.1017/jfm.2011.98, 2011), a clear picture of the sequence of events resulting in the staircase formation emerges. A collective instability of homogeneous fingering convection first excites a field of gravity waves, with a well-defined vertical wavelength. However, the waves saturate early through regular but localized breaking events and are not directly responsible for the formation of the staircase. Meanwhile, slower-growing, horizontally invariant but vertically quasi-periodic γ-modes are also excited and grow according to the γ-instability mechanism. Our results suggest that the nonlinear interaction between these various mean-field modes of instability leads to the selection of one particular γ-mode as the staircase progenitor. Upon reaching a critical amplitude, this progenitor overturns into a fully formed staircase. We conclude by extending the results of our simulations to real oceanic parameter values and find that the progenitor γ-mode is expected to grow on a time scale of a few hours and leads to the formation of a thermohaline staircase in about one day with an initial spacing in the order of 1–2 m.