Short laminar separation bubbles can develop on a flat plate due to an externally imposed pressure gradient. Here, these bubbles are computed by means of direct numerical simulations. Laminar–turbulent transition occurs in the bubble, triggered by small disturbance input with fixed frequency, but varying amplitude, to keep the bubbles short. The forcing amplitudes span a range of two orders of magnitude. All resulting bubbles differ with respect to their mean flow, linear-stability characteristics and distance between transition and mean reattachment locations. Mechanisms responsible for these differences are analysed in detail. Switching off the disturbance input or reducing it below a certain, very small threshold causes the short bubble to grow continuously. Eventually, it no longer exhibits typical characteristics of a short laminar separation bubble. Instead, it is argued that bursting has occurred and the bubble displays characteristics of a long-bubble state, even though this state was not a statistically steady state. This hypothesis is backed by a comparison of numerical results with measurements. For long bubbles, the transition to turbulence is not able to reattach the flow immediately. This effect can lead to the bursting of a short bubble, which remains short only when sufficiently large disturbances are convected into the bubble. Large-scale spanwise-oriented vortices at transition are observed for short but not for long bubbles. The failure of the transition process to reattach the flow in the long-bubble case is ascribed to this difference in transitional vortical structures.

Through a comparison between experiments and numerical simulations, we have examined the dynamics of a cloud of spheres at a small but finite Reynolds number. The cloud is seen to flatten and to transition into a torus, which further widens and eventually breaks up into droplets. While this behaviour bears some similarity to that observed at zero inertia, the underlying physical mechanisms differ. Moreover, the evolution of the cloud deformation is accelerated as inertia is increased. Two inertial regimes in which macro-scale inertia and micro-scale inertia become successively dominant are clearly identified.

The mean velocity profile scaling and the vorticity structure of a stably stratified, initially turbulent wake of a towed sphere are studied numerically using a high-accuracy spectral multi-domain penalty method model. A detailed initialization procedure allows a smooth, minimum-transient transition into the non-equilibrium (NEQ) regime of wake evolution. A broad range of Reynolds numbers, Re = UD/ν ∈ [5 × 103, 105] and internal Froude numbers, Fr = 2U/(ND) ∈ [4, 64] (U, D are characteristic velocity and length scales, and N is the buoyancy frequency) is examined. The maximum value of Re and the range of Fr values considered allow extrapolation of the results to geophysical and naval applications.

At higher Re, the NEQ regime, where three-dimensional turbulence adjusts towards a quasi-two-dimensional, buoyancy-dominated flow, lasts significantly longer than at lower Re. At Re = 5 × 103, vertical fluid motions are rapidly suppressed, but at Re = 105, secondary Kelvin–Helmholtz instabilities and ensuing turbulence are clearly observed up to Nt ≈ 100. The secondary motions intensify with increasing stratification strength and have significant vertical kinetic energy.

These results agree with existing scaling of buoyancy-driven shear on Re/Fr2 and suggest that, in the field, the NEQ regime may last up to Nt ≈ 1000. At a given high Re value, during the NEQ regime, the scale separation between Ozmidov and Kolmogorov scale is independent of Fr. This first systematic numerical investigation of stratified turbulence (as defined by Lilly, J. Atmos. Sci. vol. 40, 1983, p. 749), in a controlled localized flow with turbulent initial conditions suggests that a reconsideration of the commonly perceived life cycle of a stratified turbulent event may be in order for the correct turbulence parametrizations of such flows in both geophysical and operational contexts.

The interactions between shear-free turbulence in two regions (denoted as + and − on either side of a nearly flat horizontal interface are shown here to be controlled by several mechanisms, which depend on the magnitudes of the ratios of the densities, ρ+/ρ−, and kinematic viscosities of the fluids, μ+/μ−, and the root mean square (r.m.s.) velocities of the turbulence, u0+/u0−, above and below the interface. This study focuses on gas–liquid interfaces so that ρ+/ρ− ≪ 1 and also on where turbulence is generated either above or below the interface so that u0+/u0− is either very large or very small. It is assumed that vertical buoyancy forces across the interface are much larger than internal forces so that the interface is nearly flat, and coupling between turbulence on either side of the interface is determined by viscous stresses. A formal linearized rapid-distortion analysis with viscous effects is developed by extending the previous study by Hunt & Graham (J. Fluid Mech., vol. 84, 1978, pp. 209–235) of shear-free turbulence near rigid plane boundaries. The physical processes accounted for in our model include both the blocking effect of the interface on normal components of the turbulence and the viscous coupling of the horizontal field across thin interfacial viscous boundary layers. The horizontal divergence in the perturbation velocity field in the viscous layer drives weak inviscid irrotational velocity fluctuations outside the viscous boundary layers in a mechanism analogous to Ekman pumping. The analysis shows the following. (i) The blocking effects are similar to those near rigid boundaries on each side of the interface, but through the action of the thin viscous layers above and below the interface, the horizontal and vertical velocity components differ from those near a rigid surface and are correlated or anti-correlated respectively. (ii) Because of the growth of the viscous layers on either side of the interface, the ratio uI/u0, where uI is the r.m.s. of the interfacial velocity fluctuations and u0 the r.m.s. of the homogeneous turbulence far from the interface, does not vary with time. If the turbulence is driven in the lower layer with ρ+/ρ− ≪ 1 and u0+/u0− ≪ 1, then uI/u0− ~ 1 when Re (=u0−L−/ν−) ≫ 1 and R = (ρ−/ρ+)(v−/v+)1/2 ≫ 1. If the turbulence is driven in the upper layer with ρ+/ρ− ≪ 1 and u0+/u0− ≫ 1, then uI/u0+ ~ 1/(1 + R). (iii) Nonlinear effects become significant over periods greater than Lagrangian time scales. When turbulence is generated in the lower layer, and the Reynolds number is high enough, motions in the upper viscous layer are turbulent. The horizontal vorticity tends to decrease, and the vertical vorticity of the eddies dominates their asymptotic structure. When turbulence is generated in the upper layer, and the Reynolds number is less than about 106–107, the fluctuations in the viscous layer do not become turbulent. Nonlinear processes at the interface increase the ratio uI/u0+ for sheared or shear-free turbulence in the gas above its linear value of uI/u0+ ~ 1/(1 + R) to (ρ+/ρ−)1/2 ~ 1/30 for air–water interfaces. This estimate agrees with the direct numerical simulation results from Lombardi, De Angelis & Bannerjee (Phys. Fluids, vol. 8, no. 6, 1996, pp. 1643–1665). Because the linear viscous–inertial coupling mechanism is still significant, the eddy motions on either side of the interface have a similar horizontal structure, although their vertical structure differs.

We discuss the analogy of the behaviour of films and drops of liquid on a rotating horizontal cylinder on the one hand and substrates with regular one-dimensional wettability patterns on the other hand. On the basis of the similarity between the respective governing long-wave equations, we show that a drop of partially wetting liquid on a rotating cylinder undergoes a depinning transition when the rotation speed is increased. The transition occurs via a sniper bifurcation, as in a recently described scenario for drops depinning on heterogeneous substrates.

This paper investigates the vortex-induced vibrations (VIV) of a spring-mounted circular cylinder. We compute analytically the leading-order equations describing the nonlinear interaction of the fluid and structure modes by carrying out an asymptotic analysis of the Navier–Stokes equations close to the threshold of instability of the fluid-only system. We show that vortex-shedding can occur at subcritical Reynolds numbers as a result of the coupled system being linearly unstable to the structure mode. We also show that resonance occurs when the frequency of the nonlinear limit cycle matches the natural frequency of the cylinder, the displacement being then in phase with the flow-induced lift fluctuations. Using an extension of this model meant to encompass the effect of the low-order added-mass and damping forces induced by the displaced fluid, we show that the amount of energy that can be extracted from the flow can be optimized by an appropriate choice of the structural parameters. Finally, we suggest a possible connection between the present ‘exact’ model and the empirical wake oscillator model used to study VIV at high Reynolds numbers. We show that for the low Reynolds numbers considered here, the effect of the structure on the fluid can be represented by a first coupling term proportional to the cylinder acceleration in the fluid equation, and by a second term of lower magnitude, which can stem either from an integral term or from a term proportional to the third derivative of the cylinder position.

We carry out direct numerical simulations of two-dimensional turbulence with forcing at different wavenumbers and resolutions up to 327682 grid points. In the absence of large-scale drag, a state is reached where enstrophy is quasi-stationary while energy is growing. In the enstrophy cascade range the energy spectrum has the form E(k) = εω2/3k−3, without any logarithmic correction, where εω is the enstrophy dissipation and is of the order of unity. However, is varying between different simulations and is thus not a perfect constant. This variation can be understood as a consequence of large-scale dissipation intermittency, following the argument by Landau (Landau & Lifshitz, Fluid Mechanics, 1959, Pergamon). In the presence of a large-scale drag, we obtain a slightly steeper spectrum. When forcing is applied at a scale which is somewhat smaller than the computational domain, no vortices are formed, and the statistics remain close to Gaussian in the enstrophy cascade range. When forcing is applied at a smaller scale, long-lived coherent vortices form at larger scales than the forcing scale, and intermittency measures become very large at all scales, including the scales of the enstrophy cascade. We conclude that the enstrophy cascade with a k−3-spectrum is a robust feature of the two-dimensional Navier–Stokes equations. However, there is a complete lack of universality of higher-order statistics of vorticity increments in the enstrophy cascade range.

Experimental studies of laminar plumes descending under gravity into stably stratified environments have shown the existence of a critical injection velocity beyond which the plume exhibits a bifurcation to a coiling instability in three dimensions or a sinuous instability in a Hele-Shaw flow. In addition, flow visualization has shown that, prior to the onset of the instability, a stable base flow is established in which the plume penetrates to a depth significantly smaller than the neutral buoyancy depth. Moreover, the fresh water that is viscously entrained by the plume recirculates within a ‘conduit’ whose boundary with the background stratification appears sharp. Beyond the bifurcation, the buckling plume takes the form of a travelling wave of varying amplitude, confined within the conduit, which disappears at the penetration depth. To determine the mechanisms underlying these complex phenomena, which take place at a strikingly low Reynolds number but a high Schmidt number, we study here a two-dimensional arrangement, as it is perhaps the simplest system which possesses all the key experimental features. Through a combination of numerical and analytical approaches, a scaling law is found for the plume's penetration depth within the base flow (i.e. the flow where the instability is either absent or artificially suppressed), and the horizontal cross-stream velocity and concentration profile outside the plume are determined from an asymptotic analysis of a simplified model. Direct numerical simulations show that, with increasing flow rate, a sinuous global mode is destabilized giving rise to the self-sustained oscillations as in the experiment. The sinuous instability is shown to be a consequence of the baroclinic generation of vorticity, due to the strong horizontal gradients at the edge of the conduit, a mechanism that is relevant even at very low Reynolds numbers. Despite the strength of this instability, the penetration depth is not significantly affected by it, instead being determined by the properties of the plume in the vicinity of the source. This scenario is confirmed by a local stability analysis. A finite region of local absolute instability is found near the source for sinuous modes prior to the onset of the global instability. Sufficiently far from the source the flow is locally stable. Near the onset of the global instability, varicose modes are also found to be locally, but only convectively, unstable.

The hysteretic behaviour of a freely vibrating cylinder, near the low-Reynolds-number end of synchronization/lock-in, in the laminar regime is investigated. Computations are carried out using a stabilized finite-element method. The flow remains two-dimensional in this Reynolds number regime. This is verified via comparison of two- and three-dimensional computations. The cylinder is free to undergo crossflow as well as in-line vibrations. The combined effect of mass ratio (1 ≤ m* ≤ 100) and blockage (0.25% ≤ B ≤ 12.5%) is studied in detail. The existence of a critical mass ratio (m*cr = 10.11), below which hysteresis disappears for an unbounded flow situation, is identified. For higher mass ratio the hysteretic behaviour is observed for all blockage. However, the hysteresis loop width is found to vary with B; its variation with m* and B is studied. The concept of critical blockage Bcr is introduced. For B ≤ Bcr the response of the cylinder is virtually the same as that in an unbounded flow domain. The variation of Bcr with m* is investigated. Furthermore, Bcr is found to vary non-monotonically with m* for m* ≤ m*cr and is almost constant for m* ≥ 20. The effect of damping, as well as restricting the cylinder to undergo transverse vibrations only, on the hysteresis behaviour is studied. The transverse-only motion leads to a larger hysteresis loop width compared with the transverse and the in-line motion of the cylinder. An attempt is made to explain this by comparing the results from forced vibrations.

The charged liquid micro-jet issued from a Taylor cone may develop a special type of non-axisymmetric instability, usually referred to in the literature as a whipping mode. This instability usually manifests itself as a series of fast and violent lashes of the charged jet, which makes its characterization in the laboratory difficult. Recently, we have found that this instability may also develop when the host medium surrounding the Taylor cone and the jet is a dielectric liquid instead of air. When the oscillations of the jet occur inside a dielectric liquid, their frequency and amplitude are much lower than those of the oscillations taking place in air. Taking advantage of this fact, we have performed a detailed experimental characterization of the whipping instability of a charged micro-jet within a dielectric liquid by recording the jet motion with a high-speed camera. Appropriate image processing yields the frequency and wavelength, among the other important characteristics, of the jet whipping as a function of the governing parameters of the experimental set-up (flow rate and applied electric field) and liquid properties. Alternatively, the results can be also written as a function of three dimensionless numbers: the capillary and electrical Bond numbers and the ratio between an electrical relaxation and residence time.

A complementary experimental and numerical investigation was performed to study the three-dimensional flow structures and interactions of a finite-span synthetic jet in a crossflow at a chord-based Reynolds number of 100,000 and a 0° angle of attack. Six blowing ratios in the range of 0.2–1.2 were considered. Experiments were conducted on a finite wing with a cross-sectional profile of NACA 4421, where particle-image velocimetry data were collected at the centre jet. To complement the experiments, three-dimensional numerical simulations were performed, where the numerical set-up matched not only the physical parameters (e.g. free stream) but also the physical dimensions (e.g. orientation and location of the jet. For the low blowing ratio cases, spatial non-uniformities developed, due to the finite span of the slit, which led to the formation of small and organized secondary structures or a streak-like pattern in the mean flow. On the other hand, for the high blowing ratio range, turbulent vortical structures were dominant, leading to larger spanwise structures, with a larger spanwise wavelength. Moreover, the phase-locked flow fields exhibited a train of counter-rotating coherent vortices that lifted off the surface as they advected downstream. In the mid-blowing ratio range, combined features of the low range (near the slit) and high range (in downstream locations) were found, where a pair of counter-rotating vortices issued in the same jet cycle collided with each other. In all cases, the spanwise extent of the secondary coherent structures reduced with downstream distance with a larger decrease at higher blowing ratios. Similar observations were made in earlier studies on finite-span synthetic jets in quiescent conditions.

The structure of wall pressure fluctuations beneath a turbulent boundary layer interacting with a normal shock wave at Mach number M∞ = 1.3 is studied exploiting a direct numerical simulation database. Upstream of the interaction, in the zero-pressure-gradient region, pressure statistics compare well with canonical low-speed boundary layers in terms of fluctuation intensities, space–time correlations, convection velocities and frequency spectra. Across the interaction zone, the root-mean-square wall pressure fluctuations attain very large values (in excess of 162 dB), with a maximum increase of about 7 dB from the upstream level. The two-point wall pressure correlations become more elongated in the spanwise direction, indicating an increase of the pressure-integral length scales, and the convection velocities (determined from space–time correlations) are reduced. The interaction qualitatively modifies the shape of the frequency spectra, causing enhancement of the low-frequency Fourier modes and inhibition of the higher ones. In the recovery region past the interaction, the pressure spectra collapse very accurately when scaled with either the free-stream dynamic pressure or the maximum Reynolds shear stress, and exhibit distinct power-law regions with exponent −7/3 at intermediate frequencies and −5 at high frequencies. An analysis of the pressure sources in the Lighthill's equation for the instantaneous pressure has been performed to understand their contributions to the wall pressure signature. Upstream of the interaction the sources are mainly located in the proximity of the wall, whereas past the shock, important contributions to low-frequency pressure fluctuations are associated with long-lived eddies developing far from the wall.

The Rayleigh–Taylor instability of an inclined buoyant cylinder of one very viscous fluid rising through another is examined through linear stability analysis, numerical simulation and experiment. The stability analysis represents linear eigenmodes of a given axial wavenumber as a Fourier series in the azimuthal direction, allowing the use of separable solutions to the Stokes equations in cylindrical polar coordinates. The most unstable wavenumber k∗ is long-wave if both the inclination angle α and the viscosity ratio λ (internal/external) are small; for this case, k∗ ∝ max{α, (λ ln λ−1)1/2} and thus a small angle in experiments can have a significant effect for λ ≪ 1. As α increases, the maximum growth rate decreases and the upward propagation rate of disturbances increases; all disturbances propagate without growth if the cylinder is sufficiently close to vertical, estimated as α ≳ 70°. Results from the linear stability analysis agree with numerical calculations for λ = 1 and experimental observations. A point-force numerical method is used to calculate the development of instability into a chain of individual plumes via a complex three-dimensional flow. Towed-source experiments show that nonlinear interactions between neighbouring plumes are important for α ≳ 20° and that disturbances can propagate out of the system without significant growth for α ≳ 40°.

Shock propagation through a bubbly liquid contained in a deformable tube is considered. Quasi-one-dimensional mixture-averaged flow equations that include fluid–structure interaction are formulated. The steady shock relations are derived and the nonlinear effect due to the gas-phase compressibility is examined. Experiments are conducted in which a free-falling steel projectile impacts the top of an air/water mixture in a polycarbonate tube, and stress waves in the tube material and pressure on the tube wall are measured. The experimental data indicate that the linear theory is incapable of properly predicting the propagation speeds of finite-amplitude waves in a mixture-filled tube; the shock theory is found to more accurately estimate the measured wave speeds.

An experimental study is presented on the spatial structure of the internal wave field emitted by a horizontally oscillating sphere in a uniformly stratified fluid. The limits of linear theory and the nonlinear features of the waves are considered as functions of oscillation amplitude. Fourier decomposition is applied to separate first harmonic waves at the fundamental frequency and higher harmonic waves at multiples of this frequency. For low oscillation amplitude, of 10% of the sphere radius, only the first harmonic is significant and the agreement between linear theory and experiment is excellent. As the oscillation amplitude increases up to 30% of the radius, the first harmonic becomes slightly smaller than its linear theoretical prediction and the second and third harmonics become detectable. Two distinct cases emerge depending on the ratio Ω between the oscillation frequency and the buoyancy frequency. When Ω > 0.5, the second harmonic is evanescent and localized near the sphere in the plane through its centre perpendicular to the direction of oscillation, while the third harmonic is negligible. When Ω < 0.5, the second harmonic is propagative and appears to have an amplitude that exceeds the amplitude of the first harmonic, while the third harmonic is evanescent and localized near the sphere on either side of the plane through its centre perpendicular to the direction of oscillation. Moreover, the propagative first and second harmonics have radically different horizontal radiation patterns and are of dipole and quadrupole types, respectively.

The aim of this work was to investigate the flow evolution with time of fluid between two parallel disks and the corresponding pressure variations at the centre of the lower disk that occur subsequent to an impact-loading situation arising from dropping a mass onto the upper disk from a chosen height. During the event a fixed amount of energy is dissipated in the fluid between the disks through the action of friction. Therefore, this fundamental system may be regarded as a constant energy one, as distinct from one in which the upper disk is moving at a constant velocity, or is acted upon by a constant force. A test cell was set up to conduct the investigation, for which the separation between the disks was monitored, together with the pressure at the centre of the lower disk, over the duration of the experiment (about 8–10 ms). Glycerine was used as the test fluid. The equation of motion, based on a self-similarity approach, was reduced to a simpler (quasi-steady linear or QSL) form. Measured values of disk separation, velocity and acceleration were substituted as inputs into the full QSL model and two limiting cases, namely an inviscid and a viscous model. The QSL model provided excellent comparisons between the pressure measurements and data generated by a commercial computational fluid dynamics software package, throughout the duration of a typical experiment. The inviscid and viscous models achieved good correlations with measurements for the initial impact (during which disk accelerations exceeding 2 km s−2 occurred) and towards the end of the event, that were characterized by a small and much larger pressure rise, respectively. The former feature appears not to have been previously reported and is likely to typify that which would be observed in impact systems involving squeeze films.

We analyse small-amplitude oscillations of a weakly viscous electrically conducting liquid drop in a strong uniform DC magnetic field. An asymptotic solution is obtained showing that the magnetic field does not affect the shape eigenmodes, which remain the spherical harmonics as in the non-magnetic case. A strong magnetic field, however, constrains the liquid flow associated with the oscillations and, thus, reduces the oscillation frequencies by increasing effective inertia of the liquid. In such a field, liquid oscillates in a two-dimensional (2D) way as solid columns aligned with the field. Two types of oscillations are possible: longitudinal and transversal to the field. Such oscillations are weakly damped by a strong magnetic field – the stronger the field, the weaker the damping, except for the axisymmetric transversal and inherently 2D modes. The former are overdamped because of being incompatible with the incompressibility constraint, whereas the latter are not affected at all because of being naturally invariant along the field. Since the magnetic damping for all other modes decreases inversely with the square of the field strength, viscous damping may become important in a sufficiently strong magnetic field. The viscous damping is found analytically by a simple energy dissipation approach which is shown for the longitudinal modes to be equivalent to a much more complicated eigenvalue perturbation technique. This study provides a theoretical basis for the development of new measurement methods of surface tension, viscosity and the electrical conductivity of liquid metals using the oscillating drop technique in a strong superimposed DC magnetic field.

A combined numerical and analytical approach is used to study the low-frequency shock motions observed in shock/turbulent-boundary-layer interactions in the particular case of a shock-reflection configuration. Starting from an exact form of the momentum integral equation and guided by data from large-eddy simulations, a stochastic ordinary differential equation for the reflected-shock-foot low-frequency motions is derived. During the derivation a similarity hypothesis is verified for the streamwise evolution of boundary-layer thickness measures in the interaction zone. In its simplest form, the derived governing equation is mathematically equivalent to that postulated without proof by Plotkin (AIAA J., vol. 13, 1975, p. 1036). In the present contribution, all the terms in the equation are modelled, leading to a closed form of the system, which is then applied to a wide range of input parameters. The resulting map of the most energetic low-frequency motions is presented. It is found that while the mean boundary-layer properties are important in controlling the interaction size, they do not contribute significantly to the dynamics. Moreover, the frequency of the most energetic fluctuations is shown to be a robust feature, in agreement with earlier experimental observations. The model is proved capable of reproducing available low-frequency experimental and numerical wall-pressure spectra. The coupling between the shock and the boundary layer is found to be mathematically equivalent to a first-order low-pass filter. It is argued that the observed low-frequency unsteadiness in such interactions is not necessarily a property of the forcing, either from upstream or downstream of the shock, but an intrinsic property of the coupled system, whose response to white-noise forcing is in excellent agreement with actual spectra.

In the smallest capillaries, or in tubes with diameter D ≲ 8 μm, flowing red blood cells are well known to organize into single-file trains, with each cell deformed into an approximately static bullet-like shape. Detailed high-fidelity simulations are used to investigate flow in a model blood vessel with diameter slightly larger than this: D = 11.3 μm. In this case, the cells deviate from this single-file arrangement, deforming continuously and significantly. At the higher shear rates simulated (mean velocity divided by diameter U/D ≳ 50s−1), the effective tube viscosity is shear-rate insensitive with μeff/μplasma = 1.21. This matches well with the value μeff/μplasma = 1.19 predicted for the same 30% cell volume fraction by an established empirical fit of high-shear-rate in vitro experimental data. At lower shear rates, the effective viscosity increases, reaching μeff/μplasma ≈ 1.65 at the lowest shear rate simulated (U/D ≈ 3.7s−1). Because of the continuous deformations, the cell-interior viscosity is potentially important for vessels of this size. However, most results for simulations with cell interior viscosity five times that of the plasma (λ = 5), which is thought to be close to physiological conditions, closely match results for cases with λ = 1. The cell-free layer that forms along the vessel walls thickens from 0.3 μm for U/D = 3.7s−1 up to 1.2 μm for U/D ≳ 100s−1, in reasonable agreement with reported experimental results. The thickness of this cell-free layer is the key factor governing the overall flow resistance, and this in turn is shown to follow a trend expected for lubrication lift forces for shear rates between U/D ≈ 8s−1 and U/D ≈ 100s−1. Only in this same range are the cells near the vessel wall on average inclined relative to the wall, as might be expected for a lubrication mechanism. Metrics are developed to quantify the kinematics of this dense cellular flow in terms of the well-known tank-treading and tumbling behaviours often observed for isolated cells in shear flows. One notable effect of λ = 5 versus λ = 1 is that it suppresses treading rotation rates by a factor of about 2. The treading rate is found to scale with the velocity difference across the cell-rich core and is thus significantly slower than the overall shear rate in the flow, which is presumably why the flow is otherwise insensitive to λ. The cells in all cases also have a similarly slow mean tumbling motion, which is insensitive to cell-interior viscosity and decreases monotonically with increasing U/D.

We show that in a turbulent flow transporting suspended sediment, the unsaturated sediment flux q(x, t) can be described by a first-order relaxation equation. From a mode analysis of the advection–diffusion equation for the particle concentration, the relaxation length and time scales of the dominant mode are shown to be the deposition length HU/Vfall and deposition time H/Vfall, where H is the flow depth, U the mean flow velocity and Vfall the sediment settling velocity. This result is expected to be particularly relevant for the case of sediment transport in slowly varying flows, where the flux is never far from saturation. Predictions are shown to be in quantitative agreement with flume experiments, for both net erosion and net deposition situations.