We report an investigation of inverse energy cascade in steady-state two-dimensional turbulence by direct numerical simulation (DNS) of the two-dimensional Navier–Stokes equation, with small-scale forcing and large-scale damping. We employed several types of damping and dissipation mechanisms in simulations up to 20482 resolution. For all these simulations we obtained a wavenumber range for which the mean spectral energy flux is a negative constant and the energy spectrum scales as k−5/3, consistent with the predictions of Kraichnan (Phys. Fluids, vol. 439, 1967, p. 1417). To gain further insight, we investigated the energy cascade in physical space, employing a local energy flux defined by smooth filtering. We found that the inverse energy cascade is scale local, but that the strongly local contribution vanishes identically, as argued by Kraichnan (J. Fluid Mech., vol. 47, 1971, p. 525). The mean flux across a length scale ℓ was shown to be due mainly to interactions with modes two to eight times smaller. A major part of our investigation was devoted to identifying the physical mechanism of the two-dimensional inverse energy cascade. One popular idea is that inverse energy cascade proceeds via merger of like-sign vortices. We made a quantitative study employing a precise topological criterion of merger events. Our statistical analysis showed that vortex mergers play a negligible direct role in producing mean inverse energy flux in our simulations. Instead, we obtained with the help of other works considerable evidence in favour of a ‘vortex thinning’ mechanism, according to which the large-scale strains do negative work against turbulent stress as they stretch out the isolines of small-scale vorticity. In particular, we studied a multi-scale gradient (MSG) expansion developed by Eyink (J. Fluid Mech., vol. 549, 2006a, p. 159) for the turbulent stress, whose contributions to inverse cascade can all be explained by ‘thinning’. The MSG expansion up to second order in space gradients was found to predict well the magnitude, spatial structure and scale distribution of the local energy flux. The majority of mean flux was found to be due to the relative rotation of strain matrices at different length scales, a first-order effect of ‘thinning’. The remainder arose from two second-order effects, differential strain rotation and vorticity gradient stretching. Our findings give strong support to vortex thinning as the fundamental mechanism of two-dimensional inverse energy cascade.

We present the results of a combined experimental and theoretical investigation of the normal impact of hydrophobic spheres on a water surface. Particular attention is given to characterizing the shape of the resulting air cavity in the low Bond number limit, where cavity collapse is driven principally by surface tension rather than gravity. A parameter study reveals the dependence of the cavity structure on the governing dimensionless groups. A theoretical description based on the solution to the Rayleigh–Besant problem is developed to describe the evolution of the cavity shape and yields an analytical solution for the pinch-off time in the zero Bond number limit. The sphere's depth at cavity pinch-off is also computed in the low Weber number, quasi-static limit. Theoretical predictions compare favourably with our experimental observations in the low Bond number regime, and also yield new insight into the high Bond number regime considered by previous investigators. Discrepancies are rationalized in terms of the assumed form of the velocity field and neglect of the longitudinal component of curvature, which together preclude an accurate description of the cavity for depths less than the capillary length. Finally, we present a theoretical model for the evolution of the splash curtain formed at high Weber number and couple it with the underlying cavity dynamics.

The optimal energy growth of perturbations sustained by a zero pressure gradient turbulent boundary is computed using the eddy viscosity associated with the turbulent mean flow. It is found that even if all the considered turbulent mean profiles are linearly stable, they support transient energy growths. The most amplified perturbations are streamwise uniform and correspond to streamwise streaks originated by streamwise vortices. For sufficiently large Reynolds numbers two distinct peaks of the optimal growth exist, respectively scaling in inner and outer units. The optimal structures associated with the peak scaling in inner units correspond well with the most probable streaks and vortices observed in the buffer layer, and their moderate energy growth is independent of the Reynolds number. The energy growth associated with the peak scaling in outer units is larger than that of the inner peak and scales linearly with an effective turbulent Reynolds number formed with the maximum eddy viscosity and a modified Rotta–Clauser length based on the momentum thickness. The corresponding optimal perturbations consist of very large–scale structures with a spanwise wavelength of the order of 8δ. The associated optimal streaks scale in outer variables in the outer region and in wall units in the inner region of the boundary layer, in which they are proportional to the mean flow velocity. These outer streaks protrude far into the near wall region, having still 50% of their maximum amplitude at y+ = 20. The amplification of very large–scale structures appears to be a robust feature of the turbulent boundary layer: optimal perturbations with spanwise wavelengths ranging from 4δ to 15δ can all reach 80% of the overall optimal peak growth.

We study the free surface of a turbulent flow, in particular the relation between the statistical properties of the wrinkled surface and those of the velocity field beneath it. Channel flow turbulence is generated using an active grid. Through a judicial choice of the stirring protocol the anisotropy of the subsurface turbulence can be controlled. The largest Taylor Reynolds number obtained is Reλ = 258. We characterize the homogeneity and isotropy of the flow and discuss Taylor's frozen turbulence hypothesis, which applies to the subsurface turbulence but not to the surface. The surface gradient field is measured using a novel laser-scanning device. Simultaneously, the velocity field in planes just below the surface is measured using particle image velocimetry (PIV). Several intuitively appealing relations between the surface gradient field and functionals of the subsurface velocity field are tested. For an irregular flow shed off a vertical cylinder, we find that surface indentations are strongly correlated with both vortical and strain events in the velocity field. For fully developed turbulence this correlation is dramatically reduced. This is because the large eddies of the subsurface turbulent flow excite random capillary–gravity waves that travel in all directions across the surface. Therefore, the turbulent surface has dynamics of its own. Nonetheless, it does inherit both the integral scale, which determines the predominant wavelength of the capillary–gravity surface waves, and the (an)isotropy from the subsurface turbulence. The kinematical aspects of the surface–turbulence connection are illustrated by a simple model in which the surface is described in terms of waves originating from Gaussian wave sources that are randomly sprinkled on the moving surface.

This paper presents state of the art three-dimensional numerical simulations of the Rayleigh–Bénard convection in a supercritical fluid. We consider a fluid slightly above its critical point in a cube-shaped cell heated from below with insulated sidewalls; the thermodynamic equilibrium of the fluid is described by the van der Waals equation of state. The acoustic filtering of the Navier–Stokes equations is revisited to account for the strong stratification of the fluid induced by its high compressibility under the effect of its own weight. The hydrodynamic stability of the fluid is briefly reviewed and we then focus on the convective regime and the transition to turbulence. Direct numerical simulations are carried out using a finite volume method for Rayleigh numbers varying from 106 up to 108. A spatiotemporal description of the flow is presented from the convection onset until the attainment of a statistically steady state of heat transfer. This description concerns mainly the identification of the vortical structures in the flow, the distribution of the Nusselt numbers on the horizontal isothermal walls, the structure of the temperature field and the global thermal balance of the cavity. We focus on the influence of the strong stratification of the fluid on the penetrability of the convective structures in the core of the cavity and on its global thermal balance. Finally, a comparison with the case of a perfect gas, at the same Rayleigh number, is presented.

In this paper, we examine the generation of pressure and lift forces in a random soft fibrous media layer that is confined between two planar surfaces, an infinite horizontal lower boundary and a horizontal inclined upper boundary, in the lubrication limit where the characteristic thickness of the fibre layer H ≪ L the length of the inclined surface. The model for the fibre layer is a Brinkman equation and the Darcy permeability Kp is described by the widely used Carman–Kozeny equation for random porous media. Two cases are considered: (a) an inclined upper boundary which slides freely on top of a stationary fibre layer which is firmly attached to the lower boundary and (b) an inclined stationary upper boundary with an attached fibre layer in which the horizontal lower boundary slides freely in its own plane beneath it. Superficially, the problems appear equivalent to the classical problem for a slider bearing where the solutions for the pressure distribution and lift force are independent of which boundary is moving. In this problem there is an optimum compression ratio k = h1/h2 = 2.2, where h1 and h2 are the heights at the leading and trailing edges, for maximum lift force. However, this symmetry is lost if the intervening space is filled with a soft porous fibrous material since the Brinkman equation is not invariant under a transformation of coordinates in which the inherently unsteady problem in case (a) is transformed to a steady reference frame in which the inclined upper boundary is stationary and the horizontal boundary with the adhered fibre layer moves below it. Although in the steady reference frame case (a) now appears to resemble case (b), the solutions are strikingly different and depend critically on the value of the dimensionless fibre interaction layer thickness . For α ≪ 1 the solutions for both cases approach the classical solution for a slider bearing. For α ≫ 1 we show, using asymptotic analysis that the solutions diverge dramatically. In case (a) the pressure and lift force increase as α2 and asymptotically approach a limiting behaviour for large values of α, first predicted in Feng and Weinbaum (J. Fluid Mech., vol. 422, 2000, p. 288), while in case (b) the pressure and lift force decay as α−2 since the inclined upper boundary is screened by the fibre layer and the amount of fluid dragged through the fluid gap decreases as α increases and vanishes for α ≫ 1. The solution in case (a), where the inclined upper boundary moves, is of particular interest since it reveals the potential to generate enormous lift forces using commercially available inexpensive soft porous materials provided the lateral leakage at the edge of the planform can be eliminated through the use of a channel with impermeable sidewalls as first proposed in the work by Wu, Andreopolous and Weinbaum (Phys. Rev. Lett., vol. 93, 2004, p. 194501). The behaviour is illustrated for both a toboggan sliding in such a channel and a larger planform that might be useful in commercial transportation.

Embedded vortices in turbulent wall-bounded flow over a flat plate, generated by a passive rectangular vane-type vortex generator with variable angle β to the incoming flow in a low-Reynolds-number flow (Re = 2600 based on the inlet grid mesh size L = 0.039 m and free stream velocity U∞ = 1.0 ms−1), have been studied with respect to helical symmetry. The studies were carried out in a low-speed closed-circuit wind tunnel utilizing stereoscopic particle image velocimetry (SPIV). The vortices have been shown to possess helical symmetry, allowing the flow to be described in a simple fashion. Iso-contour maps of axial vorticity revealed a dominant primary vortex and a weaker secondary one for 20° ≤ β ≤ 40°. For angles outside this range, the helical symmetry was impaired due to the emergence of additional flow effects. A model describing the flow has been utilized, showing strong concurrence with the measurements, even though the model is decoupled from external flow processes that could perturb the helical symmetry. The pitch, the vortex core size, the circulation and the advection velocity of the vortex all vary linearly with the device angle β. This is important for flow control, since one thereby can determine the axial velocity induced by the helical vortex as well as the swirl redistributing the axial velocity component for a given device angle β. This also simplifies theoretical studies, e.g. to understand and predict the stability of the vortex and to model the flow numerically.

The growth of perturbations on a capillary jet issuing from a circular nozzle in the Rayleigh regime is experimentally investigated. Electrohydrodynamic sinusoidal stimulation is employed to this end, along with two independent methods to obtain growth rates of the linear regime with the best accuracy so far. The first method exploits the correlation between the stimulation voltage and the breakup time measured with the help of stroboscopic images of the jet. The second method is an analysis of the spatial evolution of perturbations through a local jet-shadow-width photometry, with careful avoidance of the initial transient and the final nonlinear stages. Experiments conducted with ink allow the application of both methods, as the liquid is opaque. They give consistent results, with very small statistical errors, with respect to the expected theoretical dispersion relation, once the dynamic surface tension is adjusted. The adjusted value is in accordance with an estimate made from drop-dynamics experiments also reported here. By dealing with a simpler liquid (aqueous solution of NaNO3), we are able to compare results from the first method against the theoretical predictions without adjustment of any parameter. The agreement is again excellent. Possible sources of systematic errors in this kind of measurements are identified and procedures for avoiding them are designed.

Fully three-dimensional computations of flow through a long pipe demand a huge number of degrees of freedom, making it very expensive to explore parameter space and difficult to isolate the structure of the underlying dynamics. We therefore introduce a ‘2+ε-dimensional’ model of pipe flow, which is a minimal three-dimensionalization of the axisymmetric case: only sinusoidal variation in azimuth plus azimuthal shifts are retained; yet the same dynamics familiar from experiments are found. In particular the model retains the subcritical dynamics of fully resolved pipe flow, capturing realistic localized ‘puff-like’ structures which can decay abruptly after long times, as well as global ‘slug’ turbulence. Relaminarization statistics of puffs reproduce the memoryless feature of pipe flow and indicate the existence of a Reynolds number about which lifetimes diverge rapidly, provided that the pipe is sufficiently long. Exponential divergence of the lifetime is prevalent in shorter periodic domains. In a short pipe, exact travelling-wave solutions are found near flow trajectories on the boundary between laminar and turbulent flow. In a long pipe, the attracting state on the laminar–turbulent boundary is a localized structure which resembles a smoothened puff. This ‘edge’ state remains localized even for Reynolds numbers at which the turbulent state is global.

To investigate turbulent properties and the developing mechanisms of the tidally induced bottom boundary layer in the linearly stratified ocean, numerical experiments have been executed with a non-hydrostatic three-dimensional model in the rotating frame, changing the temporal Rossby number Rot = |σ/f|, i.e. the ratio of the tidal frequency σ to the Coriolis parameter f. After the flow transitions to turbulence, the entire water column can be characterized by three layers: the mixed layer where density is homogenized and the flow is turbulent (z < zm); the stratified layer where the initial stratification remains and the flow is laminar (z > zt); and the interfacial layer between them where the flow is turbulent but the stratification remains (zm < z < zt). Turbulence is scaled by the frictional velocity uτ and the mixed-layer thickness zm (uτ and uτ/N where N is the buoyancy frequency) in the mixed (interfacial) layer, and has similarity. The mixed layer is thickened by the process where light water of the upper stratified layer is mixed with the lower unstratified layer water through the interfacial layer. As Rot approaches unity, i.e. near the critical latitude, the mixed layer develops more rapidly according to the following mechanism. As becomes Rot closer to unity, the current shear in the interfacial layer is intensified, since the difference of velocity becomes larger between the lower turbulent mixed and upper laminar stratified layers, and this leads to thickening of the interfacial layer. As a result, density deviation of the water entrained from above becomes larger, and this causes more rapid development of the mixed layer. In terms of the energy conversion from the eddy kinetic energy (EKE) to the potential energy (PE), the efficiency factor β which is the ratio of the conversion rate from EKE to PE to that from the tidal shear to EKE increased from 0.25% for Rot = 0.5 to 3.5% for Rot = 1.05 on average. When the time is normalized by the period required for the mixed layer to be thickened to the unstratified turbulent boundary layer δ = uτ/|f+σ|, the mixed layer development occurred in a similar manner in all cases. This similarity suggests the possibility of universal formulation for the turbulent tidal mixing under stratification.

In order to compare the bubble dynamics of various quantum liquids, we performed the visual observation of a sound-induced bubble in a normal liquid 4He, pure superfluid 4He, and superfluid 3He–4He liquid mixtures of saturated and unsaturated 3He concentrations. When an acoustic wave pulse was applied to these liquids under saturated vapour pressure, a macroscopic bubble was generated on the surface of a piezoelectric transducer. For all liquids, the size of the bubble increased, as a higher voltage was applied to the transducer at a fixed temperature. In the normal 4He we observed a primary bubble surrounded with many small bubbles which ascended upward together. In contrast to normal phase, only one bubble was generated in the superfluid 4He, and its shape proved to be highly irregular with an ill-defined surface. In the 3He saturated superfluid mixture, we also observed a solitary bubble but with a nearly perfect spherical shape. The bubble in this mixture expanded on the transducer surface, grew to a maximum size of the order of 1 mm and then started shrinking. As the bubble detached from the transducer with shrinking, we clearly detected an origination of the upward jet flow which penetrated the bubble. The jet velocity in the liquid mixture was approximately 102–103 times smaller than in water. At the final stage of the process we could sometimes observe a vortex ring generation. It is interesting that, though the bubble size and time scale of the phenomenon differ from those in water, the behaviour in the collapsing process had much in common with the simulation study of the vortex ring generation in water. In addition, for the mixture with the unsaturated 3He concentration of about 25% at 600 mK, the shape of the upward jet was observed distinctly, using more precise measurement with shadowgraph method.

The method of associated matrices is used to obtain Galerkin type representations. Fundamental solutions are then obtained for the cases of a point body couple and a point microstretch force. A formula for calculating the total couple acting on a rigid body rotating axi-symmetrically in a microstretch fluid is deduced. A generalized reciprocal theorem is deduced. An application for a rigid sphere rotating in a microstretch fluid is discussed. The results of the application are represented graphically.

Optical distortion through a plane shear layer between two uniform streams of unequal temperatures is investigated using imaging of scalar mixing by planar laser-induced fluorescence (LIF). Simultaneous fluorescence intensity measurements of disodium fluorescein and rhodamine B (the fluorescence of the latter is temperature dependent) are used to extract the temperature distribution within the plane of a laser sheet. This planar imaging technique allows a direct measurement of the index of refraction field, which is used to determine optical distortions in the imaging plane. While a given ray within the sheet is refracted at a cumulative angle that depends on the local index of refraction gradient along its path, LIF measurements in the cross-stream plane of the flow show substantial optical distortions near the upstream and downstream edges of the primary vortices where the angle between the local temperature (or index) gradient and the wavevector of the incident light is large. These distortions are manifested by the appearance of low- and high-intensity streaks that are advected with the flow and whose characteristic spatial scales significantly diminish with the onset of small-scale (mixing) transition. The formation of secondary (streamwise) vortical structures leads to substantial spanwise variations in index of refraction and consequently to optical distortions that vary with the phase of the base flow and can exceed the distortions in the cross-stream plane. An important aim of the present research is the investigation of the effects of direct actuation of small-scale motions on the mixing and consequently on optical distortion within the shear layer. The actuation suppresses the formation of the large coherent vortical structures and leads to localized enhancement of dissipation and reduction in turbulent kinetic energy. The actuation enhances mixing across the entire width of the forced shear layer and leads to significant reduction of optical distortion, suggesting that active small-scale mixing can play a significant role in the mitigation of aero-optical effects through turbulent shear flows.

An experimental investigation of electrokinetic instabilities (EKIs) of non-dilute colloidal suspensions in microchannels is presented. The addition of charged colloidal particles to a solution can alter the solution's electrical conductivity and permittivity as well as the average particle electrophoretic mobility. In this work, a colloidal (500 nm polystyrene) volume fraction gradient is achieved at the intersection of a Y-shaped polydimethylsiloxane (PDMS) microchannel. The flow becomes unstable when the electroviscous stretching and folding of the conductivity and permittivity interfaces exceed the dissipative effects of viscous forces and particle diffusion. The suspension conductivity as a function of the particle volume fraction is presented. The critical conditions required for flow instability are measured along with a scaling analysis which shows that the flow becomes unstable due to a coupling of applied electric fields and the electrical conductivity and permittivity gradients in the flow. The flow becomes unstable at a critical electric Rayleigh number of Rae = 1.8 × 105 for a wide range of applied electric fields spanning three orders of magnitude and colloid volume fractions varying two orders of magnitude. EKIs of non-dilute colloidal suspensions may be important for applications such as the electrophoretic deposition of micropatterned colloidal assemblies, electrorheological devices and on-chip electrokinetic (EK) manipulation of colloids.

Field observations indicate that the height of submarine levees decays with distance from the channel either exponentially or according to a power law. This investigation clarifies the flow conditions that lead to these respective shapes, via a shallow water model for the overflow currents that govern the levee formation. The model is based on a steady state balance of sediment supply by the turbidity current, and sediment deposition onto the levee, with the settling velocity and the entrainment rate appearing as parameters. It demonstrates that entrainment of ambient fluid is the determining factor for the levee shape. For negligible entrainment rates, levee shapes tend to exhibit exponential profiles, while constant rates of entrainment or detrainment result in power law shapes. Interestingly, whether a levee has an exponential or a power law shape is determined by kinematic considerations only, viz. the balance laws for sediment mass and fluid volume. We find that the respective coefficients governing the exponential or power law decay depend on the settling speeds of the sediment grains, which in turn is a function of the grain size. Two-dimensional, unsteady Navier–Stokes simulations confirm the emergence of a quasi-steady state. The depositional behaviour of this quasi-steady state is consistent with the predictions of the shallow water model, thus validating the assumptions underlying the model, and demonstrating its predictive abilities.