The conditions for the appearance of instabilities in systems of helical vortices constitute an intriguing problem that still remains partly unsolved. The experimental study of Felli, Camussi & Di Felice (J. Fluid Mech., this issue, vol. 682, 2011, pp. 5–53) has shed new light on some of the basic mechanisms governing the instability mechanisms.

In the present study the mechanisms of evolution of propeller tip and hub vortices in the transitional region and the far field are investigated experimentally. The experiments involved detailed time-resolved visualizations and velocimetry measurements and were aimed at examining the effect of the spiral-to-spiral distance on the mechanisms of wake evolution and instability transition. In this regard, three propellers having the same blade geometry but different number of blades were considered. The study outlined dependence of the wake instability on the spiral-to-spiral distance and, in particular, a streamwise displacement of the transition region at the increasing inter-spiral distance. Furthermore, a multi-step grouping mechanism among tip vortices was highlighted and discussed. It is shown that such a phenomenon is driven by the mutual inductance between adjacent spirals whose characteristics change by changing the number of blades.

We consider the steady-state propagation of a high-Reynolds-number gravity current of height h and density ρc on the bottom of a horizontal channel of height H filled with ambient fluid of density ρa(<ρc), usually known as Benjamin's current problem. The objective is to derive an analytical result for the speed of propagation, U, in the form of the dimensionless Froude number, Fr(a) = U/(g′h)1/2). Here g′ = (ρc/ρa − 1)g is the reduced gravity-driving effect (g being the gravity acceleration) and a = h/H is the depth (thickness) ratio of the layer of the current to that of the ambient fluid into which the current propagates. The analysis is performed in a frame of reference attached to the current; in this frame the current is a motionless slug. The original analysis of Benjamin assumes that the speed of the ambient in the domain above the parallel-horizontal main part of the current (behind the head) is independent of the vertical coordinate z, but here we assume that a small u′(z) fluctuation about the depth-averaged speed u exists. Then, we impose the balances of volume flux, flow-force (momentum flux) and global energy conservation, for a control volume attached to the current. We show that this gives a unique analytical result for Fr as a function of a = h/H. We recall that the original counterpart solution FrB(a) of Benjamin does not satisfy the above-mentioned energy conservation condition, i.e. the system displays energy dissipation (except for the half-depth current case a = 1/2). The present dissipationless-flow Fr(a) result is valid for any a ≤ 1/2, i.e. currents of at most half-depth of the channel height. On the other hand, in agreement with Benjamin's solution, gravity currents of more than half-depth of the channel height require an energy source and are impossible in normal conditions. The new Fr(a) is slightly smaller than Benjamin's FrB(a) result for 0 < a < 1/2, and the difference vanishes at a = 1/2 and a → 0 (a current of finite height in a very deep ambient).

Excitation of Görtler vortices in a boundary layer over a concave wall by free-stream vortical disturbances is studied theoretically and numerically. Attention is focused on disturbances with long streamwise wavelengths, to which the boundary layer is most receptive. The appropriate initial-boundary-value problem describing both the receptivity process and the development of the induced perturbation is formulated for the generic case where the Görtler number GΛ (based on the spanwise wavelength Λ of the disturbance) is of order one. The impact of free-stream disturbances on the boundary layer is accounted for by the far-field boundary condition and the initial condition near the leading edge, both of which turn out to be the same as those given by Leib, Wundrow & Goldstein (J. Fluid Mech., vol. 380, 1999, p. 169) for the flat-plate boundary layer. Numerical solutions show that for a sufficiently small GΛ, the induced perturbation exhibits essentially the same characteristics as streaks occurring in the flat-plate case: it undergoes considerable amplification and then decays. However, when GΛ exceeds a critical value, the induced perturbation exhibits (quasi-) exponential growth. The perturbation acquires the modal shape of Görtler vortices rather quickly, and its growth rate approaches that predicted by local instability theories farther downstream, indicating that Görtler vortices are excited. The amplitude of the Görtler vortices excited is found to decrease as the frequency increases, with steady vortices being dominant. Comprehensive quantitative comparisons with experiments show that the eigenvalue approach predicts the modal shape adequately, but only the initial-value approach can accurately predict the entire evolution of the amplitude. An asymptotic analysis is performed for GΛ ≫ 1 to map out distinct regimes through which a perturbation with a fixed spanwise wavelength evolves. The centrifugal force starts to influence the generation of the pressure when x* ~ ΛRΛG−2/3Λ, where RΛ denotes the Reynolds number based on Λ. The induced pressure leads to full coupling of the momentum equations when x* ~ ΛRΛGΛ−2/5. This is the crucial regime linking the pre-modal and modal phases of the perturbation because the governing equations admit growing asymptotic eigensolutions, which develop into fully fledged Görtler vortices of inviscid nature when x* ~ ΛRΛ. From this position onwards, local eigenvalue formulations are mathematically justified. Görtler vortices continue to amplify and enter the so-called most unstable regime when x* ~ ΛRΛGΛ, and ultimately approach the right-branch regime when x* ~ ΛRΛG2Λ.

An analytical approximation is derived for the flow field in the vicinity of a transition zone between electrolytes of different mobility in isotachophoretic transport through a channel. Due to the difference in electroosmotic mobility and electric field on both sides of the transition zone, the flow field consists of a superposition of electroosmotic and pressure-driven flow. The corresponding convective ion transport inherently reduces the resolution of isotachophoretic separation processes. The derived analytical result is adequate for both wide and narrow transition zones and valid in the limit of thin electric double layers, relevant for most situations where isotachophoresis is employed. In this way, it complements and generalizes the results obtained for wide transition zones in the lubrication approximation. The analysis is extended to multiple sample zones with ions of different electrophoretic mobility, a scenario characteristic for applications in the field of analytical chemistry. The results are validated by comparison to finite-element calculations accounting for the transport of different ionic species governed by the coupled Nernst–Planck and Stokes equations, both for situations with only a single transition zone as well as for several transition zones. Excellent agreement is obtained between the analytical and the numerical results for realistic parameter values encountered in ITP experiments. This suggests using the analytical expression for the flow field in the framework of numerical studies of species transport in ITP experiments, since the time-consuming computation of the velocity field is essentially eliminated. The latter is successfully demonstrated using an iterative procedure, numerically solving the Nernst–Planck equation for a given flow field, and using the resulting concentration fields as an input for the derived analytical expression.

Recently, Deloncle, Billant & Chomaz (J. Fluid Mech., vol. 599, 2008, p. 229) and Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239) have performed numerical simulations of the nonlinear evolution of the zigzag instability of a pair of counter-rotating vertical vortices in a stratified fluid. Both studies report the development of a small-scale secondary instability when the vortices are strongly bent if the Reynolds number Re is sufficiently high. However, the two papers are at variance about the nature of this secondary instability: it is a shear instability according to Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) and a gravitational instability according to Waite & Smolarkiewicz (J. Fluid Mech., vol. 606, 2008, p. 239). They also profoundly disagree about the condition for the onset of the secondary instability: ReF2h > O(1) according to the former or ReFh > 80 according to the latter, where Fh is the horizontal Froude number. In order to understand the origin of these discrepancies, we have carried out direct numerical simulations of the zigzag instability of a Lamb–Chaplygin vortex pair for a wide range of Reynolds and Froude numbers. The threshold for the onset of a secondary instability is found to be ReF2h ≃ 4 for Re ≳ 3000 and ReFh = 80 for Re ≲ 1000 in agreement with both previous studies. We show that the scaling analysis of Deloncle et al. (J. Fluid Mech., vol. 599, 2008, p. 229) can be refined to obtain a universal threshold: (Re − Re0)F2h ≃ 4, with Re0 ≃ 400, which works for all Re. Two different regimes for the secondary instabilities are observed: when (Re − Re0)F2h ≃ 4, only the shear instability develops while when (Re − Re0)F2h ≫ 4, both shear and gravitational instabilities appear almost simultaneously in distinct regions of the vortices. However, the shear instability seems to play a dominant role in the breakdown into small scales in the range of parameters investigated.

The steady, coaxial flow in which two immiscible, incompressible fluids of differing densities move past each other slowly in a vertical cylindrical tube has a continuum of possibilities due to the arbitrariness of the interface between the fluids. By invoking the presence of surface tension to at least restrict the shape of any interface to that of a circular arc or full circle, we consider the following question: which flow will maximise the exchange when there is only one dividing interface Γ? Surprisingly, the answer differs fundamentally from the better-known co-directional two-phase flow situation where an axisymmetric (concentric) core-annular solution always optimises the flux. Instead, the maximal flux state is invariably asymmetric either being a ‘side-by-side’ configuration where Γ starts and finishes at the tube wall or an eccentric core-annular flow where Γ is an off-centre full circle in which the more viscous fluid is surrounded by the less viscous fluid. The side-by-side solution is the most efficient exchanger for a small viscosity ratio β ≲ 4.60 with an eccentric core-annular solution optimal otherwise. At large β, this eccentric solution provides 51% more flux than the axisymmetric core-annular flow which is always a local minimiser of the flux.

In this paper, we study the transition from regular to Mach reflection (RR → MR) in the dual solution domain due to the influence of an upstream disturbance, by considering the transition as an evolutionary rather than an abrupt process. From numerical simulation, we observe for the early stage of transition a multiple interaction structure, composed of a triple-shock structure, a type VI shock interaction and a shock/slipline interaction. In the end, we observe a pure unsteady MR structure. Under self-similar assumption of the triple point for the first stage and including additional Mach waves over the slipline for the last stage, we develop an idealized unsteady model to obtain the evolution of the Mach stem height and the time taken for the Mach stem to stabilize. The triple point is found to move at a nearly constant speed in the multiple interaction stage which occupies about one quarter of the transition time. In the pure unsteady MR stage, which occupies the rest of transition, the speed of the triple point drops nonlinearly until the Mach stem stabilizes.

We study convective motion in vertically vibrated three-dimensional granular beds by comparing the predictions of a model based on a hydrodynamic description to Navier–Stokes order with experimental results obtained using positron emission particle tracking (PEPT). The three-dimensional conservation equations relating mass, momentum and energy are solved using the finite element (FE) method for a viscous vibrofluidized bed by using only observable system parameters such as particle number, size, mass and coefficients of restitution. The mean velocity profiles from the viscous model show reasonable agreement with the experimental results at relatively low altitudes for the range of experimental values studied, though the velocity fields at higher altitudes were systematically underestimated by the model. We confirm that the convection rolls are influenced by the sidewall coefficient of restitution and demonstrate the scaling relationships that operate, where increasing amplitude of vibration leads to a reduction in the angular velocity of the rolls.

In this article, we present a modelling approach for rapid dynamic wetting based on the phase field theory. We show that in order to model this accurately, it is important to allow for a non-equilibrium wetting boundary condition. Using a condition of this type, we obtain a direct match with experimental results reported in the literature for rapid spreading of liquid droplets on dry surfaces. By extracting the dissipation of energy and the rate of change of kinetic energy in the flow simulation, we identify a new wetting regime during the rapid phase of spreading. This is characterized by the main dissipation to be due to a re-organization of molecules at the contact line, in a diffusive or active process. This regime serves as an addition to the other wetting regimes that have previously been reported in the literature.

We report on an experimental study of cavitation bubble dynamics within sub-millimetre-sized narrow gaps. The gap height is varied, while the position of the cavitation event is fixed with respect to the lower gap wall. Four different sizes of laser-induced cavitation bubbles are studied using high-speed photography of up to 430,000 frames per second. We find a strong influence of the gap height, H, on the bubble dynamics, in particular on the collapse scenario. Also, similar bubble dynamics was found for the same non-dimensional gap height η = H/Rx, where Rx is the maximum radius in the horizontal direction. Three scenarios are observed: neutral collapse at the gap centre, collapse onto the lower wall and collapse onto the upper wall. For intermediate gap height the bubble obtains a conical shape 1.4 < η < 7.0. For large distances, η > 7.0, the bubble no longer feels the presence of the upper wall and collapses hemispherically. The collapse time increases with respect to the expansion time for decreasing values of η. Due to the small scales involved, the final stage of the bubble collapse could not be resolved temporally and numerical simulations were performed to elucidate the details of the flow. The simulations demonstrate high-speed jetting towards the upper and lower walls and complex bubble splitting for neutral collapses.

The oceanic forced wave beneath a moving atmospheric disturbance is amplified by Proudman resonance. When modified by the Earth's rotation this classical resonance only occurs if the disturbance time scale is smaller than the inertial period. With or without Coriolis effects, free transients generated by storm forced waves obliquely crossing step changes in water depth at particular angles are shown to resonate by exciting a range of long barotropic free waves. Rotationally influenced slow atmospherically forced waves crossing a vertical coast at a critical angle lead to a form of subcritical resonance, which occurs only when the component of the disturbances' phase velocities along the coast matches that of a free Kelvin wave (KW). In a rotating ocean, transients generated by disturbances crossing a step at a particular angle are shown to excite a free double Kelvin wave (DKW). This new type of resonance only occurs for sufficiently large steps and disturbances with time scale greater than the inertial period. A storm crossing a step shelf can result in the excitation of an infinite set of edge waves, a single KW, a unique DKW and a first-mode continental shelf wave, depending on the topography and the disturbance time scale, translation speed and incident angle. The study of resonances and wave mode excitations generated by storms crossing a coast or a continental shelf may contribute to understanding how a particular combination of the storm characteristics can result in destructive coastal events with time scales encompassing the typical meteotsunami period band (tens of minutes) and storm surges with periods of several hours or days.

State estimation of turbulent near-wall flows based on wall measurements is one of the key pacing items in model-based flow control, with low-Re channel flow providing the canonical testbed. Model-based control formulations in such settings are often separated into two subproblems: estimation of the near-wall flow state via skin friction and pressure measurements at the wall, and (based on this estimate) control of the near-wall flow field fluctuations via actuation of the fluid velocity at the wall. In our experience, the turbulent state estimation sub-problem has consistently proven to be the more difficult of the two. Though many estimation strategies have been tested on this problem (by our group and others), none have accurately captured the turbulent flow state at the outer boundary of the buffer layer (5 ≤ y+ ≤ 30), which is deemed to be an important milestone, as this is the approximate range of the characteristic near-wall turbulent structures, the accurate estimation of which is important for the control problem. Leveraging the ensemble Kalman filter (an effective variant of the Kalman filter which scales well to high-dimensional systems), the present paper achieves at least an order of magnitude improvement (in the near-wall region) over the best results available in the published literature on the estimation of low-Reynolds number turbulent channel flow based on wall information alone.

Growth of optimal transient perturbations to an Oseen vortex column into the nonlinear regime is studied via direct numerical simulation (DNS) for Reynolds number, Re (≡ circulation/viscosity), up to 10000. An optimal bending-wave transient mode is obtained from linear analysis and used as the initial condition. (DNS of a vortex column embedded in finer-scale turbulence reveals that optimal modes are preferentially excited during vortex–turbulence interaction.) Tilting of the optimal mode's radial vorticity perturbation into the azimuthal direction and its concomitant stretching by the column's strain field produces positive Reynolds stress, hence kinetic energy growth. Modes experiencing the largest growth are those with initial vorticity localized at a ‘critical radius’ outside the core, such that this perturbation vorticity resonantly induces core waves. Resonant forcing leads to growth of perturbation energy concentrated within the core. Moderate-amplitude (~5%) perturbations cause significant distortion of the core and generate secondary filament-like spiral structures (‘threads’) outside the core. As the mode evolves into the nonlinear regime, radially outward self-advection of thread dipoles accelerates growth arrest by removing the perturbation from the critical radius and disrupting resonant forcing. With increasing Re, the evolving vorticity patterns become more chaotic, more turbulent-like (finer scaled, contorted vorticity), and persist longer. This suggests that at typical Re (~106), nonlinear transient growth may indeed be able to break up, hence induce rapid decay of, column vortices – highly relevant for addressing the aircraft wake hazard crisis and the looming air traffic capacity crisis. In addition, we discover a regenerative transient growth scenario in which threads induce secondary perturbations closer to the vortex column. A parent–offspring regenerative mechanism is postulated and verified by DNS. There is a clear trend towards stronger regenerative growth with increasing Re. These results, showing an important role of transient growth in turbulent vortex decay, are highly relevant to the prediction and control of vortex-dominated flows.

Large-scale mean flows often emerge in turbulent fluids. In this work, we formulate a stability theory, the stochastic structural stability theory (SSST), for the emergence of jets under external random excitation. We analytically investigate the structural stability of a two-dimensional homogeneous fluid enclosed in a channel and subjected to homogeneous random forcing. We show that two generic competing mechanisms control the instability that gives rise to the emergence of an infinitesimal jet: advection of the eddy vorticity by the mean flow that is shown to be jet forming and advection of the vorticity gradient of the jet by the eddies that is shown to hinder the formation of the mean flow. We show that stochastic forcing with small streamwise coherence and an amplitude larger than a certain threshold leads to the emergence of jets in the channel through a bifurcation of the non-linear SSST system.

Receptivity, disturbance growth and breakdown to turbulence in Görtler flow are studied by spatial direct numerical simulation (DNS). The boundary layer is exposed to free-stream vortical modes and localized wall roughness. We propose a normalization of the roughness-induced receptivity coefficient by the square root of the Görtler number. This scaling removes the dependence of the receptivity coefficient on wall curvature. It is found that vortical modes are more efficient at generating Görtler vortices than localized roughness. The boundary layer is most receptive to zero- and low-frequency free-stream vortices, exciting steady and slowly travelling Görtler modes. The associated receptivity mechanism is linear and involves the generation of boundary-layer streaks, which soon evolve into unstable Görtler vortices. This connection between transient and exponential amplification is absent on flat plates and promotes transition to turbulence on curved walls. We demonstrate that the Görtler boundary layer is also receptive to high-frequency free-stream vorticity, which triggers steady Görtler rolls via a nonlinear receptivity mechanism. In addition to the receptivity study, we have carried out DNS of boundary-layer transition due to broadband free-stream turbulence with different intensities and frequency spectra. It is found that nonlinear receptivity dominates over the linear mechanism unless the free-stream fluctuations are concentrated in the low-frequency range. In the latter case, transition is accelerated due to the presence of travelling Görtler modes.

When a fluid in a tube is occluded, one finds a static configuration in which the occluding free surface of the fluid is an equilibrium capillary surface spanning the tube. We extend known criteria for existence and non-existence of such a surface, leading to an explicit mathematically rigorous occlusion criterion for cylindrical tubes in a transverse body force field, depending on the force magnitude and contact angle. For any contact angle γ ≠ π/2, we provide further an explicit design of a tube section, which will not occlude in a downward gravity field, regardless of the field strength. In addition, we derive a precise analytic occlusion criterion for liquid partially filling a circular vessel spinning about its axis.

The spreading of a compound drop on a partially wetting solid substrate is numerically simulated using a diffuse-interface method. Compared with a simple drop, the spreading of a compound drop exhibits much more complex behaviour. Depending on the core–shell size ratio and the substrate wettability, various flow regimes are identified in which the interfacial morphology evolves in distinct ways. A phase diagram is constructed in the parameter space of the core–shell size ratio and the wetting angle. For relatively small inner drops, the outer interface does not rupture during the spreading and the inner drop either remains suspended and encapsulated or attaches onto the substrate. Otherwise, the compound drop spontaneously breaks up and releases the inner drop into the ambient fluid. Several breakup scenarios are observed depending on the location of the initial rupture. In some regimes, the wetting of the substrate by one fluid can entrap secondary drops of the other, which can either attach to the substrate or stay suspended. The viscosity ratio mainly affects the spreading rate and plays a minor role in the morphology evolution.

A single bubble is placed in a solid-body rotating flow of silicon oil. From the measurement of its equilibrium position, lift and drag forces are determined. Five different silicon oils have been used, providing five different viscosities and Morton numbers. Experiments have been performed over a wide range of bubble Reynolds numbers (0.7 ≤ Re ≤ 380), Rossby numbers (0.58 ≤ Ro ≤ 26) and bubble aspect ratios (1 ≤ χ ≤ 3). For spherical bubbles, the drag coefficient at the first order is the same as that of clean spherical bubbles in a uniform flow. It noticeably increases with the local shear S = Ro−1, following a Ro−5/2 power law. The lift coefficient tends to 0.5 for large Re numbers and rapidly decreases as Re tends to zero, in agreement with existing simulations. It becomes hardly measurable for Re approaching unity. When bubbles start to shrink with Re numbers decreasing slowly, drag and lift coefficients instantaneously follow their stationary curves versus Re. In the standard Eötvös–Reynolds diagram, the transitions from spherical to deformed shapes slightly differ from the uniform flow case, with asymmetric shapes appearing. The aspect ratio χ for deformed bubbles increases with the Weber number following a law which lies in between the two expressions derived from the potential flow theory by Moore (J. Fluid Mech., vol. 6, 1959, pp. 113–130) and Moore (J. Fluid Mech., vol. 23, 1965, pp. 749–766) at low- and moderate We, and the bubble orients with an angle between its minor axis and the direction of the flow that increases for low Ro. The drag coefficient increases with χ, to an extent which is well predicted by the Moore (1965) drag law at high Re and Ro. The lift coefficient is a function of both χ and Re. It increases linearly with (χ − 1) at high Re, in line with the inviscid theory, while in the intermediate range of Reynolds numbers, a decrease of lift with aspect ratio is observed. However, the deformation is not sufficient for a reversal of lift to occur.

We present an analytical model of the time-dependent, small-amplitude deformation of a free liquid surface caused by a spatially localized, axisymmetric, pulsed or continuous, acoustic or electromagnetic radiation pressure exerted on the surface. By exactly solving the unsteady Stokes equation, we predict the surface dynamics in all dynamic regimes, namely inertial, intermediate and strongly damped regimes. We demonstrate the validity of this model in all dynamic regimes by comparing its prediction to experiments consisting of optically measuring the time-dependent curvature of the tip of a hump created at a liquid surface by the radiation pressure of an acoustic pulse. Finally, we present a numerical scheme simulating the behaviour of a fluid–fluid interface subjected to a time-dependent radiation pressure and show its accuracy by comparing the numerical predictions with the analytical model in the intermediate and strongly damped regimes.