Significant advances have been made in applying ideas from nonlinear dynamical systems theory to flows which exhibit sequences of bifurcations in the transition to turbulence. Moreover, the recent discoveries of finite-amplitude states in linearly stable flows holds great promise for a breakthrough in our understanding transition in shear flows. Tsukahara, Tillmark & Alfredsson (J. Fluid Mech., 2010, this issue, vol. 648, pp. 5–33) study a novel variant of a classical shear flow by adding global rotation. The competition between the induced body force and shear-induced instabilities leads to the discovery of a rich and beautiful tapestry of transition sequences.

Flow states in plane Couette flow in a spanwise rotating frame of reference have been mapped experimentally in the parameter space spanned by the Reynolds number and rotation rate. Depending on the direction of rotation, the flow is either stabilized or destabilized. The experiments were made through flow visualization in a Couette flow apparatus mounted on a rotating table, where reflected flakes are mixed with the water to visualize the flow. Both short- and long-time exposures have been used: the short-time exposure gives an instantaneous picture of the turbulent flow field, whereas the long-time exposure averages the small, rapidly varying scales and gives a clearer representation of the large scales. A correlation technique involving the light intensity of the photographs made it possible to obtain, in an objective manner, both the spanwise and streamwise wavelengths of the flow structures. During these experiments 17 different flow regimes have been identified, both laminar and turbulent with and without roll cells, as well as states that can be described as transitional, i.e. states that contain both laminar and turbulent regions at the same time. Many of these flow states seem to be similar to those observed in Taylor–Couette flow.

A numerical study is carried out of the injection of a very viscous liquid of small electrical conductivity at a constant flow rate through an orifice in a metallic plate under the action of an electric field. The conditions under which the injected liquid can form an elongated meniscus with a thin jet emanating from its tip are investigated by computing the flow, the electric field and the transport of electric charge in the meniscus and a leading region of the jet. A stationary solution is found only for values of the flow rate above a certain minimum. At moderate values of the applied field, this minimum flow rate decreases when the applied field or the conductivity of the liquid increase. The electric shear stress acting on the surface of the liquid is not able to drive the liquid into the jet at flow rates smaller than the minimum while, for any flow rate higher than the minimum, the transfer of electric current to the surface may occur in a slender region of the jet where charge relaxation effects are small and the field induced by the electric charge of the jet is important. At high values of the applied field, the flow rate must be higher than another minimum, which increases with the applied field, in order for the viscous stress to balance the strong electric stress acting on the meniscus. The two conditions taken together determine lower and upper bounds for the applied field at a given flow rate, but the value of the applied field at which a stationary jet is first established when this parameter is gradually increased is higher than the lower bound, leading to hysteresis. When the liquid is electrosprayed in a surrounding dielectric fluid, the viscous shear stress that this fluid exerts on the surface of the jet eventually balances the electric shear stress and stops the continuous stretching of the jet. A fraction of the conduction current is left in the jet when the effect of the outer liquid comes into play in the region where this current is transferred to the surface, and no stationary solution is found above a maximum flow rate that decreases when the viscosity of the outer liquid increases or the applied field decreases. Order of magnitude estimates of the electric current and the conditions in the current transfer region are worked out.

Vibrations acting on a fluid with density gradient induced by temperature variations can cause relative flows. High-frequency vibration leads to the appearance of time-averaged (mean) flows (or streaming flows), which can essentially affect heat and mass transfer processes. This phenomenon is most pronounced in the absence of other external forces (in particular, static gravity). In this work, an extensive experimental and computational study of thermal vibrational convection in a reduced-gravity environment of a parabolic flight is performed. The transient evolution of the temperature field in a cubic cell subjected to translational vibration is investigated by optical digital interferometry. The mean flow structures previously reported in numerical studies are confirmed. The transition from four-vortex flow to a pattern with a large diagonal vortex and two small vortices is observed in the transient state. The experiments reveal a significant enhancement of heat transfer by vibrational mean flows with increasing the vibrational strength. Three-dimensional direct numerical simulation with real microgravity profile and two-dimensional numerical modelling based on averaging approach provide a very good agreement with the experimental results. The influence of residual gravity on heat transfer and bifurcation scenario is first investigated numerically and correlated with the experimental data. It is demonstrated that gravity effects on non-uniformly heated fluids can be reproduced in weightlessness by applying vibrations to the system.

We examine the effect of prescribed wall-driven oscillations of a flexible tube of arbitrary cross-section, through which a flow is driven by prescribing either a steady flux at the downstream end or a steady pressure difference between the ends. A large-Womersley-number large-Strouhal-number regime is considered, in which the oscillations of the wall are small in amplitude, but sufficiently rapid to ensure viscous effects are confined to a thin boundary layer. We derive asymptotic expressions for the flow fields and evaluate the energy budget. A general result for the conditions under which there is zero net energy transfer from the flow to the wall is provided. This is presented as a critical inverse Strouhal number (a dimensionless measure of the background flow rate) which is expressed only in terms of the tube geometry, the fluid properties and the profile of the prescribed wall oscillations. Our results identify an essential component of a fundamental mechanism for self-excited oscillations in three-dimensional collapsible tube flows, and enable us to assess how geometric and flow properties affect the stability of the system.

In Part 1 of this work, we derived general asymptotic results for the three-dimensional flow field and energy fluxes for flow within a tube whose walls perform prescribed small-amplitude periodic oscillations of high frequency and large axial wavelength. In the current paper, we illustrate how these results can be applied to the case of flow through a finite-length axially non-uniform tube of elliptical cross-section – a model of flow in a Starling resistor. The results of numerical simulations for three model problems (an axially uniform tube under pressure–flux and pressure–pressure boundary conditions and an axially non-uniform tube with prescribed flux) with prescribed wall motion are compared with the theoretical predictions made in Part 1, each showing excellent agreement. When upstream and downstream pressures are prescribed, we show how the mean flux adjusts slowly under the action of Reynolds stresses using a multiple-scale analysis. We test the asymptotic expressions obtained for the mean energy transfer E from the flow to the wall over a period of the motion. In particular, the critical point at which E = 0 is predicted accurately: this point corresponds to energetically neutral oscillations, the condition which is relevant to the onset of global instability in the Starling resistor.

Direct numerical simulation (DNS) is used to explore similarities and differences between fully developed supersonic turbulent plane channel and axisymmetric non-swirling pipe flow bounded by isothermal walls. The comparison is based on equal friction Mach number, friction Reynolds number, Prandtl number, ratio of specific heats and viscosity exponent. The channel half-width and pipe radius are chosen to define the Reynolds numbers. To what extent and why mean flow quantities, second-order turbulence statistics and terms in the Reynolds stress equations coincide or diverge in both flows are investigated. The role of the fluctuating pressure in causing characteristic differences among correlations involving pressure fluctuations is identified via a Green-function-based analysis of the pressure field.

Geometrical and dynamical properties of turbulent flows have been investigated by streamline segment analysis. Starting from each grid point, a streamline segment is defined as the part of its streamline bounded by the two adjacent extremal points of the velocity magnitude. Physically the streamline segments can be extended into a more meaningful concept, namely the streamtube segments, which are non-overlapping and space filling. This decomposition of the flow allows for new insights into vector-related statistics in turbulence. According to the variation of velocity, the streamline segments can be sorted into positive and negative segments. The overall properties of turbulent flows can be newly understood and explained from the statistics of these segments with simple structures; for instance, the negative skewness of the velocity derivative becomes naturally a kinematic outcome. Furthermore, from direct numerical simulations conditional statistics of pressure and kinetic energy dissipation along the streamline segments are evaluated and discussed.

The basic wave field resulting from Holmboe's instability is studied both numerically and experimentally. Comparisons between the direct numerical simulations (DNS) and laboratory experiments result in Holmboe waves that are similar in their appearance and phase speed. However, different boundary conditions result in mean flows that display gradual variations either temporally (in the simulations) or spatially (in the experiments). These differences are found to affect the evolution of the dominant wavenumber and amplitude of the wave field. The simulations exhibit a nonlinear ‘wave coarsening’ effect, whereby the energy is shifted to lower wavenumbers in discrete merging events. This process is typically found to result from either ejections of mixed fluid away from the density interface or vortex pairing. In the experiments, energy is transferred to lower wavenumbers by the ‘stretching’ of the wave field by a gradually varying mean velocity. This stretching results in a reduction of wave amplitude compared with the DNS.

A study investigating the flow around a cylinder rolling or sliding on a wall has been undertaken in two and three dimensions. The cylinder motion is specified from a set of five discrete rotation rates, ranging from prograde through to retrograde rolling. A Reynolds number range of 20–500 is considered. The effects of the nearby wall and the imposed body motion on the wake structure and dominant wake transitions have been determined. Prograde rolling is shown to destabilize the wake flow, while retrograde rotation delays the onset of unsteady flow to Reynolds numbers well above those observed for a cylinder in an unbounded flow.

Two-dimensional simulations show the presence of two recirculation zones in the steady wake, the lengths of which increase approximately linearly with the Reynolds number. Values of the lift and drag coefficient are also reported for the steady flow regime. Results from a linear stability analysis show that the wake initially undergoes a regular bifurcation from a steady two-dimensional flow to a steady three-dimensional wake for all rotation rates. The critical Reynolds number Rec of transition and the spanwise wavelength of the dominant mode are shown to be highly dependent on, but smoothly varying with, the rotation rate of the cylinder. Varying the rotation from prograde to retrograde rolling acts to increase the value of Rec and decrease the preferred wavelength. The structure of the fully evolved wake mode is then established through three-dimensional simulations. In fact it is found that at Reynolds numbers only marginally (~5%) above critical, the three-dimensional simulations indicate that the saturated state becomes time dependent, although at least initially, this does not result in a significant change to the mode structure. It is only at higher Reynolds numbers that the wake undergoes a transition to vortex shedding.

An analysis of the three-dimensional transition indicates that it is unlikely to be due to a centrifugal instability despite the superficial similarity to the flow over a backward-facing step, for which the transition mechanism has been speculated to be centrifugal. However, the attached elongated recirculation region and distribution of the spanwise perturbation vorticity field, and the similarity of these features with those of the flow through a partially blocked channel, suggest the possibility that the mechanism is elliptic in nature. Some analysis which supports this conjecture is undertaken.

Stability characteristics of aerofoil flows are investigated by linear stability analysis of time-averaged velocity profiles and by direct numerical simulations with time-dependent forcing terms. First the wake behind an aerofoil is investigated, illustrating the feasibility of detecting absolute instability using these methods. The time-averaged flow around an NACA-0012 aerofoil at incidence is then investigated in terms of its response to very low-amplitude hydrodynamic and acoustic perturbations. Flow fields obtained from both two- and three-dimensional simulations are investigated, for which the aerofoil flow exhibits a laminar separation bubble. Convective stability characteristics are documented, and the separation bubble is found to exhibit no absolute instability in the classical sense; i.e. no growing disturbances with zero group velocity are observed. The flow is however found to be globally unstable via an acoustic-feedback loop involving the aerofoil trailing edge as a source of acoustic excitation and the aerofoil leading-edge region as a site of receptivity. Evidence suggests that the feedback loop may play an important role in frequency selection of the vortex shedding that occurs in two dimensions. Further simulations are presented to investigate the receptivity process by which acoustic waves generate hydrodynamic instabilities within the aerofoil boundary layer. The dependency of the receptivity process to both frequency and source location is quantified. It is found that the amplitude of trailing-edge noise in the fully developed simulation is sufficient to promote transition via leading-edge receptivity.

Direct numerical simulations are performed to investigate the interaction between a stably stratified jet and internal gravity waves from an adjacent shear layer with mild stratification. Results from two simulations are presented: one with the jet located far from the shear layer (far jet) and the other with the shear layer right on top of the jet (near jet). The near jet problem is motivated by velocity and stratification profiles observed in equatorial undercurrents. In the far case, internal waves excited by the Kelvin–Helmholtz (K-H) rollers do not penetrate the jet. They are reflected and trapped in the region between the shear layer and the jet and lead to little dissipation. In the near case, internal waves with wavelength larger than that of the K-H rollers are found in and below the jet. Pockets of hot fluid, associated with horseshoe vortices that originate from the shear layer, penetrate into the jet region, initiate turbulence and disrupt the internal wave field. Coherent patches of enhanced dissipation moving with the mean velocity are observed. The dissipation in the stably stratified near jet is large, up to three orders of magnitude stronger than that in the propagating wave field or the jet of the far case.

The dynamical relevance of vortex tubes and vortex sheets in a wall-bounded supersonic turbulent flow at Mach number M = 2 and Reynolds number Reθ ≈ 1350 is quantitatively analysed. The flow in the viscous sublayer and in the buffer region is characterized by intense, elongated vorticity tongues forming a shallow angle with respect to the wall, whose characteristic length is O(200) wall units and whose size in the cross-stream direction is O(50) wall units. The formation of vortex tubes takes place starting from y+ ≈ 10, and it is mainly associated with the roll-up and the interaction of vortex sheets. The analysis of the non-local dynamical effect of tubes and sheets suggests that the latter have a more important collective effect, being closely associated with low-speed streaks, and being responsible for a substantial contribution to the mean momentum balance and to the production of turbulence kinetic energy and enstrophy.

Experiments and numerical simulations have been performed to investigate the deformation and break-up of a cloud of rigid fibres falling under gravity through a viscous fluid in the absence of inertia and interfacial tension. The cloud of fibres is observed to evolve into a torus that subsequently becomes unstable and breaks up into secondary droplets which themselves deform into tori in a repeating cascade. This behaviour is similar to that of clouds of spherical particles, though the evolution of the cloud of fibres occurs more rapidly. The simulations, which use two different levels of approximation of the far-field hydrodynamic interactions, capture the evolution of the cloud and demonstrate that the coupling between the self-motion and hydrodynamically induced fluctuations are responsible for the faster break-up time of the cloud. The dynamics of the cloud are controlled by a single parameter which is related to the self-motion of the anisotropic particles. The experiments confirm these findings.

We have used viscous fluids in simple laboratory experiments to explore the dynamics of grounding lines between marine ice sheets and the freely floating ice shelves into which they develop. We model the ice sheets as shear-dominated gravity currents, and the ice shelves as extensional gravity currents having zero shear to leading order. We consider the flow of viscous fluid down an inclined plane into a dense inviscid ‘ocean’. A fixed flux of fluid is supplied at the top of the plane, which is at ‘sea level’. The fluid forms a gravity current flowing down and attached to the plane for some distance before detaching to form a freely floating extensional current. We have derived a mathematical model of the flow that incorporates a new dynamic boundary condition for the position of the grounding line, where the gravity current loses contact with the solid base. The grounding line initially advances and eventually reaches a steady position. Good agreement between our theoretical predictions and experimental measurements and observations gives confidence in the fundamental assumptions of our model.

The effects of the ambient fluid on granular flow dynamics are poorly understood and commonly ignored in analyses. In this article, we characterize and quantify these effects by combining theoretical and experimental analyses. Starting with the mixture theory, we derive a set of two-phase continuum equations for studying a compressible granular flow composed of homogenous solid particles and a Newtonian ambient fluid. The role of the ambient fluid is then investigated by studying the collapse and spreading of two-dimensional granular columns in air or water, for different solid particle sizes and column aspect (height to length) ratios, in which the front speed is used to describe the flow. The combined analysis of experimental measurements and numerical solutions shows that the dynamics of the solid phase cannot be explained if the hydrodynamic fluid pressure and the drag interactions are not included in the analysis. For instance, hydrodynamic fluid pressure can hold the reduced weight of the solids, thus inducing a transition from dense-compacted to dense-suspended granular flows, whereas drag forces counteract the solids movement, especially within the near-wall viscous layer. We conclude that in order to obtain a realistic representation of gravitational granular flow dynamics, the ambient fluid cannot be neglected.

We present experimental results of the generation of internal gravity waves by a turbulent buoyant plume impinging upon the interface between a uniform density layer of fluid and a linearly stratified layer. The wave field is observed and its properties are measured non-intrusively using axisymmetric Schlieren. In particular, we determine the fraction of the energy flux associated with the plume at the neutral buoyancy level that is extracted by the waves. On average, this was found to be approximately 4%. Within the limits of the experimental parameters, the maximum vertical displacement amplitude of waves were found to depend linearly upon the maximum penetration height of the plume beyond the neutral level. The frequency of the waves was found to lie in a narrow range relative to the buoyancy frequency. The results are used to interpret the generation of waves in the atmosphere by convective storms impinging upon the tropopause via the mechanical oscillator effect.

A numerical investigation of the disturbance amplification in a Mach 4.8 flat-plate boundary layer with a localized two-dimensional roughness element is presented. The height of the roughness is varied and reaches up to approximately 70% of the boundary-layer thickness. Simulations are based on a time-accurate integration of the compressible Navier–Stokes equations, with a small disturbance of fixed frequency being triggered via blowing and suction upstream of the roughness element. The roughness element considerably alters the instability of the boundary layer, leading to increased amplification or damping of a modal wave depending on the frequency range. The roughness is also the source of an additional perturbation. Even though this additional mode is stable, the interaction with the unstable mode in the form of constructive and destructive interference behind the roughness element leads to a beating and therefore transiently increased disturbance amplitude. Far downstream of the roughness, the amplification rate of a flat-plate boundary layer is recovered. Overall, the two-dimensional roughness element behaves as disturbance amplifier with a limited bandwidth capable of filtering a range of frequencies and strongly amplifying only a selected range.

There is a lack of rigour in the usual explanation for the scaling of the vertical velocity of shallow flows based on geometrical arguments and the continuity equation. In this paper we show, by studying shallow axisymmetric swirl flows, that the dynamics of the flow are crucial to determine the proper scaling. In addition, we present two characteristic scaling parameters for such flows: Reδ2 for the radial velocity and Reδ3 for the vertical velocity, where Re is the Reynolds number of the swirl flow and δ=H/L is the flow aspect ratio with H the fluid depth and L a typical horizontal length scale. This scaling contradicts the common assumption that the vertical velocity should scale with the primary motion proportional to the aspect ratio δ. Moreover, if this scaling applies, then the primary flow can be considered as quasi-two-dimensional. Numerical simulations of a decaying Lamb–Oseen vortex served to test the analytical results and to determine their range of validity. It was found that the primary flow can be considered as quasi-two-dimensional only if δRe1/2≲3 and δRe1/3≲1.

A submerged cylinder in a uniform stream flow is approximated by a horizontal doublet, following Lamb's classical method. A linear steady solution including surface tension effects is derived, showing that under certain conditions small-scale ripples are formed ahead of the cylinder, while a train of ‘gravity-like’ waves appear downstream. Surface tension effects and a dipole are included in the fully nonlinear unsteady non-periodic boundary-integral solver described by Tanaka et al. (J. Fluid Mech., vol. 185, 1987, pp. 235–248). Nonlinear effects are modelled by considering a flat free surface or the linear stationary solution as an initial condition for the fully nonlinear irrotational flow programme. Long-run computations show that these unsteady flows approach a steady solution for some parameters after waves have radiated away. In other cases the flow does not approach a steady solution. Interesting features at the free surface such as the appearance of ‘parasitic capillaries’ near the crest of gravity waves and the formation of capillary–gravity waves upstream of the cylinder are found.