The evolution of two- and three-dimensional small-amplitude disturbances in the laminar part of a laminar separation bubble is investigated in detail. We apply a combination of local linear stability theory, results from different experimental measurement campaigns and direct numerical simulations to identify two different discrete eigenmodes in the laminar part of the bubble. A stable eigenmode, the outer mode, governs unsteady oscillations in the upstream part of the bubble. However, this perturbation is quickly overtaken by an unstable eigenmode, the inner mode, which eventually leads to transition of the detached shear layer. Such a behaviour is observed due to an acceleration region with a favourable pressure gradient preceding the adverse-pressure-gradient region. The flow is stable in the acceleration region, in which the outer mode is only moderately damped, while the inner mode is strongly damped. At the onset of instability for the unstable eigenmode upstream of separation, both viscous Tollmien–Schlichting and inviscid Kelvin–Helmholtz instability mechanisms contribute to amplification, while deeper inside the bubble only the inviscid mechanism is active. If the explicit forcing is moved to a region downstream of the favourable pressure gradient, only the unstable eigenmode appears. The same behaviour is found for two-dimensional and weakly oblique waves.

The instabilities and transitions of flow in a vertical cylindrical cavity with heated bottom, cooled top and insulated sidewall are investigated by linear stability analysis. The stability boundaries for the axisymmetric flow are derived for Prandtl numbers from 0.02 to 1, for aspect ratio () equal to 1, 0.9, 0.8, 0.7, respectively. We found that there still exists stable non-trivial axisymmetric flow beyond the second bifurcation in certain ranges of Prandtl number for , and 0.8, excluding the case. The finding for is that very frequent changes of critical mode (azimuthal Fourier mode) of the second bifurcation occur when the Prandtl number is changed, where five kinds of steady modes and three kinds of oscillatory modes are presented. These multiple modes indicate different flow structures triggered at the transitions. The instability mechanism of the flow is explained by kinetic energy transfer analysis, which shows that the radial or axial shear of base flow combined with buoyancy mechanism leads to the instability results.

The paper presents the results of direct numerical simulations of the fall of a single freely moving sphere in a vertical circular tube. Most results are obtained for the solid–fluid density ratio . The parametric investigation is carried out depending on the Galileo number defined in Jenny, Dušek & Bouchet J. Fluid Mech., vol. 508, 2004, pp. 201–239. A qualitatively new scenario is found, as compared to that of an unconfined sphere. The primary bifurcation making the sphere deviate from a vertical fall along the tube axis at a constant velocity is of Hopf type. It sets in at a Galileo number (between 155 and 160) similar to that for an unconfined sphere. We find evidence for two stages of the primary regime: a planar trajectory at and a helical one (at and 170). At these Galileo numbers, the regime is perfectly periodic, with a slow period corresponding to a Strouhal number only slightly above 0.01. The dynamics is identified as a periodic wake–wall interaction. The helical regime is found to give way directly to chaos between and . This transition is associated with the onset of vortex shedding in the wake of the falling sphere and with a complex interaction between the unsteady wake and the wall marked by intermittent wake extinction. The effect of density ratio is partly investigated at by considering three density ratios: 2, 3 and 5. A significant change of behaviour is found between the ratios 3 and 5.

Tornadoes and other intense atmospheric vortices are known to occasionally transition to a flow structure with multiple vortices within their larger circulations. This phenomenon has long been ascribed to fluid dynamical instability of the inner-core circulation, and many previous studies have diagnosed low-wavenumber unstable modes in tornado-like vortices that resemble the observed structures. However, relatively few of these studies have incorporated the strong vertical motions of the inner-core circulation into the stability analysis, and no stability analyses have been performed using a complete, frictionally driven secondary circulation with strong radial inflow near the surface. Stability analyses are presented using the complete circulations generated from idealized simulations of tornado-like vortices. Fast-growing unstable modes are found that are consistent with the asymmetric structures present in these simulations. Attempts to correlate the structures and locations of these modes with instability conditions for vortices with axial jets derived by Howard & Gupta and by Leibovich & Stewartson produce only mixed results. Analyses of perturbation energy growth show that interactions between eddy fluxes and the radial shear of the azimuthal wind contribute very little to the growth of the dominant modes. Rather, the radial shear of the vertical wind and the vertical shear of the vertical wind (corresponding to deformation in the axial direction) are the primary energy sources for perturbation growth. Relatively weak axisymmetric instabilities are also identified that have some similarity to symmetric oscillations that have been observed in tornadoes.

An experimental investigation focusing on the effect of dissipation on the evolution of the Benjamin–Feir instability is reported. A series of wave trains with added sidebands, and varying initial steepness, perturbed amplitudes and frequencies, are physically generated in a long wave flume. The experimental results directly confirm the stabilization theory of Segur et al. (J. Fluid Mech., vol. 539, 2005, pp. 229–271), i.e. dissipation can stabilize the Benjamin–Feir instability. Furthermore, the experiments reveal that the effect of dissipation on modulational instability depends strongly on the perturbation frequency. It is found that the effect of dissipation on the growth rates of the sidebands for the waves with higher perturbation frequencies is more evident than on those of waves with lower perturbation frequencies. In addition, numerical simulations based on Dysthe’s equation with a linear damping term included, which is estimated from the experimental data, can predict the experimental results well if the momentum integral of the wave trains is conserved during evolution.

Vesicles exposed to the human circulatory system experience a wide range of flows and Reynolds numbers. Previous investigations of vesicles in fluid flow have focused on the Stokes flow regime. In this work the influence of inertia on the dynamics of a vesicle in a shearing flow is investigated using a novel level-set computational method in two dimensions. A detailed analysis of the behaviour of a single vesicle at finite Reynolds number is presented. At low Reynolds numbers the results recover vesicle behaviour previously observed for Stokes flow. At moderate Reynolds numbers the classical tumbling behaviour of highly viscous vesicles is no longer observed. Instead, the vesicle is observed to tank-tread, with an equilibrium angle dependent on the Reynolds number and the reduced area of the vesicle. It is shown that a vesicle with an inner/outer fluid viscosity ratio as high as 200 will not tumble if the Reynolds number is as low as 10. A new damped tank-treading behaviour, where the vesicle will briefly oscillate about the equilibrium inclination angle, is also observed. This behaviour is explained by an investigation on the torque acting on a vesicle in shear flow. Scaling laws for vesicles in inertial flows have also been determined. It is observed that quantities such as vesicle tumbling period follow square-root scaling with respect to the Reynolds number. Finally, the maximum tension as a function of the Reynolds number is also determined. It is observed that, as the Reynolds number increases, the maximum tension on the vesicle membrane also increases. This could play a role in the creation of stable pores in vesicle membranes or for the premature destruction of vesicles exposed to the human circulatory system.

A review of existing theories for flows of inertial particles suspended in an unbounded sheared viscous fluid is presented first. A comparison between theoretical predictions and numerical simulation results is made for Stokes numbers from to in dilute and dense flows. Both particle agitation and anisotropy coefficients are examined, showing that neither of them is able to give satisfactory results in dense flows. A more precise calculation of collisional contributions to the balance law of the particle stress tensor is presented. Results of the corresponding theory are in very good agreement with numerical simulations both in dilute and dense flows.

Microscopic holographic PIV performed in an optically index-matched facility resolves the three-dimensional flow in the inner part of a turbulent channel flow over a rough wall at Reynolds number . The roughness consists of uniformly distributed pyramids with normalized height of . Distributions of mean flow and Reynolds stresses agree with two-dimensional PIV data except very close to the wall () owing to the higher resolution of holography. Instantaneous realizations reveal that the roughness sublayer is flooded by low-lying spanwise and groove-parallel vortical structures, as well as quasi-streamwise vortices, some quite powerful, that rise at sharp angles. Conditional sampling and linear stochastic estimation (LSE) reveal that the prevalent flow phenomenon in the roughness sublayer consists of interacting U-shaped vortices, conjectured in Hong et al. (J. Fluid Mech., 2012, doi:10.1017/jfm.2012.403). Their low-lying base with primarily spanwise vorticity is located above the pyramid ridgeline, and their inclined quasi-streamwise legs extend between ridgelines. These structures form as spanwise vorticity rolls up in a low-speed region above the pyramid’s forward face, and is stretched axially by the higher-speed flow between ridgelines. Ejection induced by interactions among legs of vortices generated by neighbouring pyramids appears to be the mechanism that lifts the quasi-streamwise vortex legs and aligns them preferentially at angles of to the streamwise direction.

The appearance of divergent streamlines and subsequent formation of free vortices in Newtonian fluid flows through microfluidic flow-focusing geometries is discussed in this work. The micro-geometries are shaped like a cross-slot but comprise three entrances and one exit. The divergent flow and subsequent symmetric vortical structures arising near the centreline of the main inlet channel are promoted even under creeping flow conditions, and are observed experimentally and predicted numerically above a critical value of the ratio of inlet velocities (VR). As VR is further increased these free vortices continue to grow until a maximum size is reached due to geometrical constraints. The numerical calculations are in good agreement with the experimental observations and we probe numerically the effects of the geometric parameters and of inertia on the flow patterns. In particular, we observe that the appearance of the central recirculations depends non-monotonically on the relative width of the entrance branches and we show that inertia enhances the appearance of the free vortices. On the contrary, the presence of the walls in three-dimensional geometries has a stabilizing effect for low Reynolds numbers, delaying the onset of these secondary flows to higher VR. The linearity of the governing equations for creeping flow of Newtonian fluids was invoked to determine the flow field for any VR as a linear combination of the results of three other independent solutions in the same geometry.

Fabrication of functionalized surfaces using polymer brushes is a relatively simple process and parallels the presence of glycocalyx filaments coating the luminal surface of our vasculature. In this paper, we perform atomistic-like simulations based on dissipative particle dynamics (DPD) to study both polymer brushes and glycocalyx filaments subject to shear flow, and we apply mean-field theory to extract useful scaling arguments on their response. For polymer brushes, a weak shear flow has no effect on the brush density profile or its height, while the slip length is independent of the shear rate and is of the order of the brush mesh size as a result of screening by hydrodynamic interactions. However, for strong shear flow, the polymer brush is penetrated deeper and is deformed, with a corresponding decrease of the brush height and an increase of the slip length. The transition from the weak to the strong shear regime can be described by a simple ‘blob’ argument, leading to the scaling , where is the critical transition shear rate and is the grafting density. Furthermore, in the strong shear regime, we observe a cyclic dynamic motion of individual polymers, causing a reversal in the direction of surface flow. To study the glycocalyx layer, we first assume a homogeneous flow that ignores the discrete effects of blood cells, and we simulate microchannel flows at different flow rates. Surprisingly, we find that, at low Reynolds number, the slip length decreases with the mean flow velocity, unlike the behaviour of polymer brushes, for which the slip length remains constant under similar conditions. (The slip length and brush height are measured with respect to polymer mesh size and polymer contour length, respectively.) We also performed additional DPD simulations of blood flow in a tube with walls having a glycocalyx layer and with the deformable red blood cells modelled accurately at the spectrin level. In this case, a plasma cell-free layer is formed, with thickness more than three times the glycocalyx layer. We then find our scaling arguments based on the homogeneous flow assumption to be valid for this physiologically correct case as well. Taken together, our findings point to the opposing roles of conformational entropy and bending rigidity – dominant effects for the brush and glycocalyx, respectively – which, in turn, lead to different flow characteristics, despite the apparent similarity of the two systems.

We investigate experimentally and theoretically the streamwise transport and dispersion properties of steady quasi-two-dimensional plane turbulent jets discharged vertically from a slot of width into a fluid confined between two relatively close rigid boundaries with gap . We model the evolution in time and space of the concentration of passive tracers released in these jets using a one-dimensional time-dependent effective advection–diffusion equation. We make a mixing length hypothesis to model the streamwise turbulent eddy diffusivity such that it scales like , where is the streamwise coordinate, is the jet width, is the maximum time-averaged vertical velocity. Under these assumptions, the effective advection–diffusion equation for , the horizontal integral of the ensemble-averaged concentration, is of the form , where is time, (the advection parameter) and (the dispersion parameter) are empirical dimensionless parameters which quantify the importance of advection and dispersion, respectively, and is the source momentum flux. We find analytical solutions to this equation for in the cases of a constant-flux release and an instantaneous finite-volume release. We also give an integral formulation for the more general case of a time-dependent release, which we solve analytically when tracers are released at a constant flux over a finite period of time. From our experimental results, whose concentration distributions agree with the model, we find that and , for both finite-volume releases and constant-flux releases using either dye or virtual passive tracers. The experiments also show that streamwise dispersion increases in time as . As a result, in the case of finite-volume releases more than 50 % of the total volume of tracers is transported ahead of the purely advective front (i.e. the front location of the tracer distribution if all dispersion mechanisms are ignored and considering a ‘top-hat’ mean velocity profile in the jet); and in the case of constant-flux releases, at each instant in time, approximately 10 % of the total volume of tracers is transported ahead of the advective front.

This paper extends Wagner theory for the ideal, incompressible normal impact of rigid bodies that are nearly parallel to the surface of a liquid half-space. The impactors considered are three-dimensional and have an oblique impact velocity. A formulation in terms of the displacement potential is used to reveal the relationship between the oblique and corresponding normal impact solutions. In the case of axisymmetric impactors, several geometries are considered in which singularities develop in the boundary of the effective wetted region. We present the corresponding pressure profiles and models for the splash sheets.

We present the results from numerical simulations of turbulent Rayleigh–Bénard convection for an aspect ratio (diameter/height) of 1.0, Prandtl numbers of 0.4 and 0.7, and Rayleigh numbers from to . Detailed measurements of the thermal and viscous boundary layer profiles are made and compared to experimental and theoretical (Prandtl–Blasius) results. We find that the thermal boundary layer profiles disagree by more than 10 % when scaled with the similarity variable (boundary layer thickness) and likewise disagree with the Prandtl–Blasius results. In contrast, the viscous boundary profiles collapse well and do agree (within 10 %) with the Prandtl–Blasius profile, but with worsening agreement as the Rayleigh number increases. We have also investigated the scaling of the boundary layer thicknesses with Rayleigh number, and again compare to experiments and theory. We find that the scaling laws are very robust with respect to method of analysis and they mostly agree with the Grossmann–Lohse predictions coupled with laminar boundary layer theory within our numerical uncertainty.

An experimental investigation involving space-filling fractal square grids is presented. The flow is documented using particle image velocimetry (PIV) in a water tunnel as opposed to previous experiments which mostly used hot-wire anemometry in wind tunnels. The experimental facility has non-negligible incoming free-stream turbulence (with 2.8 % and 4.4 % in the streamwise () and spanwise () directions, respectively) which presents a challenge in terms of comparison with previous wind tunnel results. An attempt to characterize the effects of the incoming free stream turbulence on the grid-generated turbulent flow is made and an improved wake-interaction length scale is proposed which enables the comparison of the present results with previous ones for both fractal square and regular grids. This length scale also proves to be a good estimator of the turbulence intensity peak location. Furthermore, a new turbulence intensity normalization capable of collapsing for various grids in various facilities is proposed. Comparison with previous experiments indicates good agreement in turbulence intensities, Taylor microscale, as well as various other quantities, if the improved wake-interaction length scale is used. Global and local isotropy of fractal-generated turbulence is assessed using the velocity gradients of the two-component (2C) two-dimensional (2D) PIV and compared with regular grid results. Finally, the PIV data appear to confirm the new dissipation behaviour previously observed in hot-wire measurements.

Quadratic nonlinear interactions between two colliding internal gravity wave beams in a uniformly stratified fluid, and the resulting radiation of secondary beams with frequencies equal to the sum and difference of those of the primary beams, are discussed. The analysis centres on oblique collisions, involving beams that propagate in different vertical planes. The propagation directions of generated secondary beams are deduced from kinematic considerations and the use of radiation conditions, thus extending to oblique collisions previously derived selection rules for plane collisions. Using small-amplitude expansions, radiated-beam profiles at steady state are also computed in terms of the characteristics of the colliding beams. It is pointed out that, for certain oblique collision configurations, radiated beams with frequency equal to the difference of the primary frequencies have unbounded steady-state amplitude. This resonance, which has no counterpart for plane collisions, is further analysed via the solution of an initial-value problem; ignoring dissipation, the transient resonant response grows in time like , a behaviour akin to that of forced waves at cut-off frequencies.

Both theoretical analysis and eddy-damped quasi-normal Markovian (EDQNM) simulations are carried out to investigate the different decay regimes of an initially non-self-similar isotropic turbulence. Breakdown of self-similarity is due to the consideration of a composite three-range energy spectrum, with two different slopes at scales larger than the integral length scale. It is shown that, depending on the initial conditions, the solution can bifurcate towards a true self-similar decay regime, or sustain a non-self-similar state over an arbitrarily long time. It is observed that these non-self-similar regimes cannot be detected, restricting the observation to time exponents of global quantities such as kinetic energy or dissipation. The actual reason is that the decay is controlled by large scales close to the energy spectrum peak. This theoretical prediction is assessed by a detailed analysis of triadic energy transfers, which show that the largest scales have a negligible impact on the total transfers. Therefore, it is concluded that details of the energy spectrum near the peak, which may be related to the turbulence production mechanisms, are important. Since these mechanisms are certainly not universal, this may at least partially explain the significant discrepancies that exist between experimental data and theoretical predictions. Another conclusion is that classical self-similarity theories, which connect the asymptotic behaviour of either the energy spectrum or the velocity correlation function and the turbulence decay exponent, are not particularly relevant when the large-scale spectrum shape exhibits more than one range.

A class of similarity solutions for two-dimensional unsteady flow in the neighbourhood of a front or rear stagnation point on a plane boundary is considered, and a wide range of possible behaviour is revealed, depending on whether the flow in the far field is accelerating or decelerating. The solutions, when they exist, are exact solutions of the Navier–Stokes equations, having a boundary-layer character analogous to that of the classical steady front stagnation point flow. The velocity profiles are obtained by numerical integration of a nonlinear ordinary differential equation. For the front-flow situation, the solution is unique for the accelerating case, but bifurcates for modest deceleration, while for sufficient rapid deceleration there exists a one-parameter family of solutions. For the rear-flow situation, a unique solution exists (remarkably!) for sufficiently strong acceleration, and a one-parameter family again exists for sufficient strong deceleration. Analytic results, which are consistent with the numerical results, are obtained in the limits of strong acceleration or deceleration, and for the asymptotic behaviour far from the boundary.

Three-dimensional direct numerical simulations are performed that give us an in-depth account of the evolution and structure of the double-diffusive interface. We examine the diffusive convection regime, which, in the oceanographically relevant case, consists of relatively cold fresh water above warm salty water. A ‘double-boundary-layer’ structure is found in all of the simulations, in which the temperature () interface has a greater thickness than the salinity () interface. Therefore, thin gravitationally unstable boundary layers are maintained at the edges of the diffusive interface. The -interface thickness ratio is found to scale with the diffusivity ratio in a consistent manner once the shear across the boundary layers is accounted for. The turbulence present in the mixed layers is not able to penetrate the stable stratification of the interface core, and the -fluxes through the core are given by their molecular diffusion values. Interface growth in time is found to be determined by molecular diffusion of the -interface, in agreement with a previous theory. The stability of the boundary layers is also considered, where we find boundary layer Rayleigh numbers that are an order of magnitude lower than previously assumed.

The spatially developing compressible plane mixing layer with a convective Mach number of 0.7 is investigated by direct numerical simulation. A pair of equal and opposite oblique instability waves is introduced to perturb the mixing layer at the inlet. The full evolution process of instability, including formation of -vortices and hairpin vortices, breakdown of large structures and establishment of self-similar turbulence, is presented clearly in the simulation. In the transition process, the flow fields are populated sequentially by -vortices, hairpin vortices and ‘flower’ structures. This is the first direct evidence showing the dominance of these structures in the spatially developing mixing layer. Hairpin vortices are found to play an important role in the breakdown of the flow. The legs of hairpin vortices first evolve into sheaths with intense vorticity then break up into small slender vortices. The later flower structures are produced by the instability of the heads of the hairpin vortices. They prevail for a long distance in the mixing layer until the flow starts to settle down into its self-similar state. The preponderance of slender inclined streamwise vortices is observed in the transversal middle zone of the transition region after the breakup of the hairpin legs. This predominance of streamwise vortices also persists in the self-similar turbulent region, though the vortices there are found to be relatively very weak. The evolution of both the mean streamwise velocity profile and the Reynolds stresses is found to have close connection to the behaviour of the large vortex structures. High growth rates of the momentum and vorticity thicknesses are observed in the transition region of the flow. The growth rates in the self-similar turbulence region decay to a value that agrees well with previous experimental and numerical studies. Shocklets occur in the simulation, and their formation mechanisms are elaborated and categorized. This is the first three-dimensional simulation that captures shocklets at this low convective Mach number.

Experiments on vertical turbulent flow with millimetric bubbles, under three gravity conditions, upward, downward and microgravity flows (, and ), have been performed to understand the influence of gravity upon the flow structure and the phase distribution. The mean and fluctuating phase velocities, shear stress, turbulence production, gas fraction and bubble size have been measured or determined. The results for flow obtained during parabolic flights are taken as reference for buoyant and flows. Three buoyancy numbers are introduced to understand and quantify the effects of gravity with respect to friction. We show that the kinematic structure of the liquid is similar to single-phase flow for flow whereas it deviates in and buoyant flows. The present results confirm the existence of a two-layer structure for buoyant flows with a nearly homogeneous core and a wall layer similar to the single-phase inertial layer whose thickness seems to result from a friction–gravity balance. The distributions of phase velocity, shear stress and turbulence are discussed in the light of various existing physical models. This leads to a dimensionless correlation that quantifies the wall shear stress increase due to buoyancy. The turbulent dispersion, the lift and the nonlinear effects of added mass are taken into account in a simplified model for the phase distribution. Its analytical solution gives a qualitative description of the gas fraction distribution in the wall layer.