The Rotne–Prager–Yamakawa (RPY) approximation is a commonly used approach to model the hydrodynamic interactions between small spherical particles suspended in a viscous fluid at a low Reynolds number. However, when the particles overlap, the RPY tensors lose their positive definiteness, which leads to numerical problems in the Brownian dynamics simulations as well as errors in calculations of the hydrodynamic properties of rigid macromolecules using bead modelling. These problems can be avoided by using regularizing corrections to the RPY tensors; so far, however, these corrections have only been derived for equal-sized particles. Here we show how to generalize the RPY approach to the case of overlapping spherical particles of different radii and present the complete set of mobility matrices for such a system. In contrast to previous ad hoc approaches, our method relies on the direct integration of force densities over the sphere surfaces and thus automatically provides the correct limiting behaviour of the mobilities for the touching spheres and for a complete overlap, with one sphere immersed in the other one. This approach can then be used to calculate hydrodynamic properties of complex macromolecules using bead models with overlapping, different-sized beads, which we illustrate with an example.

We report a self-similar behaviour of solutions (obtained numerically) of the Navier–Stokes equations behind a single-blade rotor. That is, the helical vortex core in the wake of a rotating blade is self-similar as a function of its age. Profiles of vorticity and azimuthal velocity in the vortex core are characterized, their similarity variables are identified and scaling laws of these variables are given. Solutions of incompressible three-dimensional Navier–Stokes equations for Reynolds numbers up to $Re= 2000$ are considered.

Gravity waves generated by an object moving at constant speed at the water surface form a specific pattern commonly known as the Kelvin wake. It was proved by Lord Kelvin that such a wake is delimited by a constant angle ${\simeq }19. 4{7}^{\circ } $. However a recent study by Rabaud and Moisy based on the observation of airborne images showed that the wake angle seems to decrease as the Froude number $Fr$ increases, scaling as $F{r}^{- 1} $ for large Froude numbers. To explain such observations they make the strong hypothesis that an object of size $b$ cannot generate wavelengths larger than $b$. Without the need of such an assumption and modelling the moving object by an axisymmetric pressure field, we analytically show that the angle corresponding to the maximum amplitude of the waves scales as $F{r}^{- 1} $ for large Froude numbers, whereas the angle delimiting the wake region outside which the surface is essentially flat remains constant and equal to the Kelvin angle for all $Fr$.

We show how a fully nonlinear variational method can be used to design a more nonlinearly stable laminar shear flow by quantifying the effect of manipulating the boundary conditions of the flow. Using the example of plane Couette flow, we demonstrate that by forcing the boundaries to undergo spanwise oscillations in a certain way, it is possible to increase the critical disturbance energy for the onset of turbulence by 41 %. If this is sufficient to ensure laminar flow (i.e. ambient noise does not exceed this increased threshold), nearly four times less energy is consumed than in the turbulent flow which exists in the absence of imposed spanwise oscillations.

In this work we quantitatively assess, via instabilities, a Navier–Stokes-order (small-Knudsen-number) continuum model based on the kinetic theory analogy and applied to inelastic spheres in a homogeneous cooling system. Dissipative collisions are known to give rise to instabilities, namely velocity vortices and particle clusters, for sufficiently large domains. We compare predictions for the critical length scales required for particle clustering obtained from transient simulations using the continuum model with molecular dynamics (MD) simulations. The agreement between continuum simulations and MD simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers ($K{n}_{u} $) at the time of cluster detection reveals $K{n}_{u} \gg 1$ due to the presence of large velocity gradients associated with vortices. Although kinetic-theory-based continuum models are based on a small-$Kn$ (i.e. small-gradient) assumption, our findings suggest that, similar to molecular gases, Navier–Stokes-order (small-$Kn$) theories are surprisingly accurate outside their expected range of validity.

Direct numerical simulation (DNS) of elastically induced turbulent flows has posed great challenges to researchers engaged in developing first-principle models and simulations that can predict faithfully the complex spatio-temporal dynamics of polymeric flows. To this end, DNS of elastically induced turbulent flow states in the Taylor–Couette (TC) flow are reported here with the aim of paving the way for a mechanistic understanding of this new class of flows. Specifically, the DNS not only faithfully reproduce the key feature of elastically induced turbulent flows, namely, substantial excitation of fluid motion at the smallest temporal and spatial scales, but also for the first time demonstrate the existence of three distinct flow regions in the gap for the inertio-elastic turbulence state: (i) a fluid-inertia (or outflow-) dominated inner-wall region; (ii) a fluid-elasticity (or inflow-) dominated outer-wall region; and (iii) an inflow/outflow core region. Based on this observation, a simple mechanism for the inertio-elastic turbulence in the TC flow has been postulated.

We present high-resolution numerical investigations of heat transport by two-dimensional (2D) turbulent Rayleigh–Bénard (RB) convection over the Rayleigh number range $1{0}^{8} \leqslant Ra\leqslant 1{0}^{10} $ and the Prandtl number range $0. 7\leqslant Pr\leqslant 10$. We find that there exists strong counter-gradient local heat flux with magnitude much larger than the global Nusselt number $Nu$ of the system. Two mechanisms for generating counter-gradient heat transport are identified: one is due to the bulk dynamics and the other is due to the competition between the corner-flow rolls and the large-scale circulation (LSC). While the magnitude of the former is found to increase with increasing Prandtl number, that of the latter maximizes at medium $Pr$. We further reveal that the corner–LSC competition leads to the anomalous $Nu$–$Pr$ relation in 2D RB convection, i.e. $Nu(Pr)$ minimizes, rather than maximizes as in the three-dimensional cylindrical case, at $Pr\approx 2\sim 3$ for moderate $Ra$.

A numerically exact full Navier–Stokes counterpart of the asymptotic nonlinear solution in Hagen–Poiseuille flow proposed by Smith & Bodonyi (Proc. R. Soc. A, vol. 384, 1982, pp. 463–489) is discovered. The solution takes the form of a spiral travelling wave, with a novel feature being a strong induced component of swirl. Our solution shows excellent quantitative agreement with the asymptotic theory at Reynolds numbers of the order of $1{0}^{8} $.

The rotating-disk boundary layer is generally considered as an example of a flow that displays a robust transition from laminar to turbulent régimes. By taking into account disks of finite radius, Healey (J. Fluid Mech., vol. 663, 2010, pp. 148–159) has predicted a stabilizing effect of the boundary condition, but Imayama et al. (J. Fluid Mech., vol. 716, 2013, pp. 638–657) were unable to confirm this prediction experimentally. Following these contradictory results, the present experimental investigation revisits the rotating-disk boundary layer, without any artificially imposed excitation, and studies in further detail the dynamics prevailing in the region closely surrounding the edge of the disk, as well as the flow beyond the disk. Azimuthal mean velocities and fluctuation amplitudes are recorded with small steps in radial and axial directions for a wide range of disk sizes. An objective criterion is used to define the onset of fluctuations consistently over a large data set. Two distinct mechanisms for the onset of fluctuations are identified. In particular, it is found that the flow over the edge of the disk acts as a strong source of fluctuations. Explanations and suggestions for a possible reconciliation of previous studies are given.

We examine the dynamics of a thin film flowing under gravity down the exterior of a vertically aligned inner cylinder, with a co-aligned, concentric cylinder acting as an outer electrode; the space between the outer cylinder and the film is occupied by an inviscid gas. The stability of the interface is studied when it is subjected to an electric field, applied by imposing a potential difference between the two cylinders. Leaky-dielectric theory is used in conjunction with asymptotic reduction, in the large-conductivity limit, to derive a single, two-dimensional evolution equation for the interfacial location, which accounts for gravity, capillarity, and electrostatic effects. A linear stability analysis is carried out which shows that non-axisymmetric modes become more dominant with increasing electric field strength. Our fully two-dimensional numerical solutions of the evolution equation demonstrate qualitative agreement between the trends observed in the nonlinear regime and those predicted by linear theory. These numerical solutions also show that, depending on the electric field strength and the relative proximity of the outer electrode, the interface either remains spatially uniform, or exhibits either axisymmetric or, importantly, non-axisymmetric travelling waves. The effect of wave formation on the interfacial area is investigated in connection with the use of electric fields to control thin film flows to enhance heat and mass transfer rates.

The dynamics of an inverted flag are investigated experimentally in order to find the conditions under which self-excited flapping can occur. In contrast to a typical flag with a fixed leading edge and a free trailing edge, the inverted flag of our study has a free leading edge and a fixed trailing edge. The behaviour of the inverted flag can be classified into three regimes based on its non-dimensional bending stiffness scaled by flow velocity and flag length. Two quasi-steady regimes, straight mode and fully deflected mode, are observed, and a limit-cycle flapping mode with large amplitude appears between the two quasi-steady regimes. Bistable states are found in both straight to flapping mode transition and flapping to deflected mode transition. The effect of mass ratio, relative magnitude of flag inertia and fluid inertia, on the non-dimensional bending stiffness range for flapping is negligible, unlike the instability of the typical flag. Because of the unsteady fluid force, a flapping sheet can produce elastic strain energy several times larger than a sheet of the deformed mode, improving the conversion of fluid kinetic energy to elastic strain energy. According to the analysis of the leading-edge vortex formation process, the time scale of optimal vortex formation correlates with efficient conversion to elastic strain energy during bending.

The nonlinear triad interaction of two opposing gravity waves with almost identical frequencies and one much longer acoustic-gravity wave is studied for non-resonance, as well as for exact resonance conditions. For non-resonance conditions the previously known results for a ‘bound’ acoustic-gravity wave are recovered. For resonance, or near-resonance conditions, where all three waves are ‘free waves’, the interaction is recurrent and the amplitude of the free acoustic-gravity wave turns out to be much larger than that known for the bound wave. The results for the recurrent evolution are given analytically, in terms of Jacobian elliptic functions and elliptic integrals.

In a recent study on a coupled laser system, Thévenin et al. (Phys. Rev. Lett., vol. 107, 2011, 104101) reported the first experimental evidence of phase trapping, a partially synchronous state characterized by frequency locking without phase locking. To determine whether this state can arise in a hydrodynamic system, we reanalyse the data from our recent experiment on a periodically forced self-excited low-density jet (J. Fluid Mech., vol. 726, 2013, pp. 624–655). We find that this jet exhibits the full range of phase dynamics predicted by model oscillators with weak nonlinearity. These dynamics include (i) phase trapping between phase drifting and phase locking when the jet is forced far from its natural frequency and (ii) phase slipping during phase drifting when it is forced close to its natural frequency. This raises the possibility that similar phase dynamics can be found in other similarly self-excited flows. It also strengthens the validity of using low-dimensional nonlinear dynamical systems based on a universal amplitude equation to model such flows, many of which are of industrial importance.

Time-resolved stereoscopic PIV was used to investigate the curvature-induced structures downstream of a $90{{}^\circ} $ bend at Reynolds numbers between $20\times 1{0}^{3} $ and $115\times 1{0}^{3} $. Data were taken at three downstream locations to investigate the evolution of the structures. Snapshot proper orthogonal decomposition (POD) analysis shows that the most energetic structure is not the well-known Dean motion but a bimodal single cell structure with alternating direction of rotation, called the ‘swirl switching’ mode. The strengths of the Dean motion and the swirl-switching structures are similar, indicating that the difference in energy is related to their duration of occurrence, where the Dean motion is associated with a comparatively rapid transition between the two states in the swirl switching mode.

Two new families of exact coherent states are found in plane Poiseuille flow. They are obtained from the stationary and the travelling-wave mirror-symmetric solutions in plane Couette flow by a homotopy continuation. They are characterized by the mirror symmetry inherited from those continued solutions in plane Couette flow. The first family arises from a saddle-node bifurcation and the second family bifurcates by breaking the top–bottom symmetry of the first family. We find that both families exist below the minimum saddle-node-point Reynolds number known to date (Waleffe, Phys. Fluids, vol. 15, 2003, pp. 1517–1534).

Through theory and numerical simulations in an axisymmetric geometry, we examine evolution of a symmetric intrusion released from a cylindrical lock in stratified fluid as it depends upon the ambient interface thickness, $h$, and the lock aspect ratio ${R}_{c} / H$, in which ${R}_{c} $ is the lock radius and $H$ is the ambient depth. Whereas self-similarity and shallow-water theory predicts that intrusions, once established, should decelerate shortly after release from the lock, we find that the intrusions rapidly accelerate and then enter a constant-speed regime that extend between $2{R}_{c} $ and $5{R}_{c} $ from the gate, depending upon the relative interface thickness ${\delta }_{h} \equiv h/ H$. This result is consistent with previously performed laboratory experiments. Scaling arguments predict that the distance, ${R}_{a} $, over which the lock fluid first accelerates increases linearly with ${R}_{c} $ if ${R}_{c} / H\ll 1$ and ${R}_{a} / H$ approaches a constant for high aspect ratios. Likewise in the constant-speed regime, the speed relative to the rectilinear speed, $U/ {U}_{\infty } $, increases linearly with ${R}_{c} / H$ if the aspect ratio is small and is of order unity if ${R}_{c} / H\gg 1$. Beyond the constant-speed regime, the intrusion front decelerates rapidly, with power-law exponent as large as $0. 7$ if the relative ambient interface thickness, ${\delta }_{h} \lesssim 0. 2$. For intrusions in uniformly stratified fluid (${\delta }_{h} = 1$), the power-law exponent is close to $0. 2$. Except in special cases, the exponents differ significantly from the $1/ 2$ power predicted from self-similarity and the $1/ 3$ power predicted for intrusions from partial-depth lock releases.

The drag of a rigid object is expected to increase with flow velocity. For wide ranges of velocities commonly encountered, the drag increases as the square of the relative velocity of the fluid. This strong dependence of the load on the velocity accounts for specific survival strategies adopted by passive living systems such as plants in wind or algae in marine environments: through elastic reconfiguration, the drag on plants is reduced when compared to a rigid configuration and the velocity exponent for the drag is typically found to be between 1 and 1.5. In this work, a membrane configuration is presented that exhibits a drag force that is almost independent of the free-stream velocity. This surprising result is shown to be remarkably robust as it is experimentally observed for a range of geometries. This study opens the way for the design of devices subjected to a drag that is independent of the flow velocity. This possibility constitutes a key point in various fields involving flexible structures that are towed or subjected to wind.

Direct numerical simulations are used to investigate the individual dynamics of large spherical particles suspended in a developed homogeneous turbulent flow. A definition of the direction of the particle motion relative to the surrounding flow is introduced and used to construct the mean fluid velocity profile around the particle. This leads to an estimate of the particle slipping velocity and its associated Reynolds number. The flow modifications due to the particle are then studied. The particle is responsible for a shadowing effect that occurs in the wake up to distances of the order of its diameter: the particle calms turbulent fluctuations and reduces the energy dissipation rate compared to its average value in the bulk. Dimensional arguments are presented to draw an analogy between particle effects on turbulence and wall flows. Evidence is obtained for the presence of a logarithmic sublayer at distances between the thickness of the viscous boundary layer and the particle diameter ${D}_{p} $. Finally, asymptotic arguments are used to relate the viscous sublayer quantities to the particle size and the properties of the outer turbulence. It is shown in particular that the skin-friction Reynolds number behaves as $R{e}_{\tau } \propto {({D}_{p} / \eta )}^{4/ 3} $.

In Taylor’s analysis of electrohydrodynamic drop deformation by a uniform electric field (Proc. R. Soc. Lond. A, vol. 291, 1966, pp. 159–166) inertia is neglected at the outset, resulting in fluid velocities that scale as the square of the applied-field magnitude. For large (i.e. millimetric) drops, with increasing field strength the Reynolds number predicted by this scaling may actually become large, suggesting the need for a complementary large-Reynolds-number investigation. Balancing viscous stresses and electrical shear forces in this limit reveals a different velocity scaling, with the $4/ 3$-power of the applied-field magnitude. For simplicity, we focus upon the flow about a spherical gas bubble. It is essentially confined to two boundary layers propagating from the poles to the equator, where they collide to form a radial jet. The transition occurs over a small deflection region about the equator where the flow is effectively inviscid. The deviation of the bubble shape from the original sphericity is quantified by the capillary number given by the ratio of a characteristic Maxwell stress to Laplace’s pressure. At leading order the bubble deforms owing to: (i) the surface distribution of the Maxwell stress, associated with the familiar electric-field profile; (ii) the hydrodynamic boundary-layer pressure, engendered here by centrifugal forces; and (iii) the intense pressure distribution acting over the narrow equatorial deflection zone, appearing on the bubble scale as a concentrated load. Remarkably, the unique flow topology and associated scalings allow the obtaining of a closed-form expression for the deformation through the mere application of integral mass and momentum balances. On the bubble scale, the concentrated pressure load is manifested in the appearance of a non-smooth equatorial dimple.

The self-excited global instability mechanisms existing in flat-plate laminar separation bubbles are studied here, in order to shed light on the causes of unsteadiness and three-dimensionality of unforced, nominally two-dimensional separated flows. The presence of two known linear global mechanisms, namely an oscillator behaviour driven by local regions of absolute inflectional instability and a centrifugal instability giving rise to a steady three-dimensionalization of the bubble, is studied in a series of model separation bubbles. These results indicate that absolute instability, and consequently a global oscillator behaviour, does not exist for two-dimensional bubbles with a peak reversed-flow velocity below $12\hspace{0.167em} \% $ of the free-stream velocity. However, the three-dimensional instability becomes active for recirculation levels as low as ${u}_{rev} \approx 7\hspace{0.167em} \% $. These findings suggest a route to the three-dimensionality and unsteadiness observed in experiments and simulations substantially different from that usually found in the literature of laminar separation bubbles, in which two-dimensional vortex shedding is followed by three-dimensionalization.