Stretching in continuum mechanics is naturally described using the Cauchy–Green strain tensors. These tensors quantify the Lagrangian stretching experienced by a material element, and provide a powerful way to study processes in turbulent fluid flows that involve stretching such as vortex stretching and alignment of anisotropic particles. Analysing data from a simulation of isotropic turbulence, we observe preferential alignment between rods and vorticity. We show that this alignment arises because both of these quantities independently tend to align with the strongest Lagrangian stretching direction, as defined by the maximum eigenvector of the left Cauchy–Green strain tensor. In particular, rods approach almost perfect alignment with the strongest stretching direction. The alignment of vorticity with stretching is weaker, but still much stronger than previously observed alignment of vorticity with the eigenvectors of the Eulerian strain rate tensor. The alignment of strong vorticity is almost the same as that of rods that have experienced the same stretching.

We provide some estimates for the wave height of a two-dimensional travelling gravity water wave from pressure measurements at the flat bed. The approach is applicable without limitations on the wave amplitude. It improves the classical estimates available if one relies on the hydrostatic approximation or on the linear theory of waves of small amplitude.

We study experimentally the growth of an isolated gas bubble in a slightly supersaturated water–CO2 solution at 6 atm pressure. In contrast to what was found in previous experiments at higher supersaturation, the time evolution of the bubble radius differs noticeably from existing theoretical solutions. We trace the differences back to several combined effects of the concentration boundary layer around the bubble, which we disentangle in this work. In the early phase, the interaction with the surface on which the bubble grows slows down the process. In contrast, in the final phase, before bubble detachment, the growth rate is enhanced by the onset of density-driven convection. We also show that the bubble growth is affected by prior growth and detachment events, though they are up to 15 min apart.

Unexpectedly, under certain conditions, sessile drops from different but completely miscible liquids do not always coalesce instantaneously upon contact: the drop bodies remain separated in a temporary state of non-coalescence, connected through a thin liquid bridge. Here we investigate the transition between the states of instantaneous coalescence and temporary non-coalescence. Experiments reveal that it is barely influenced by viscosities and absolute surface tensions. The main system control parameters for the transition are the arithmetic means of the three-phase angles, $\overline{\Theta }_{a}$ , and the surface tension differences $\Delta \gamma $ between the two liquids. These relevant parameters can be combined into a single system parameter, a specific Marangoni number $\widetilde{M}=3\Delta \gamma /(2\overline{\gamma }\overline{\Theta }_{a}^2)$ . This $\widetilde{M}$ universally characterizes the coalescence transition behaviour as a function of both the physicochemical liquid properties and the shape of the liquid body in the contact region. The transition occurs at a certain threshold value $\widetilde{M}_t$ and is sharp within the experimental resolution. The experimentally observed threshold value of $\widetilde{M}_t\approx 2$ agrees quantitatively with values obtained by simulations assuming realistic material parameters. The simulations indicate that the absolute value of $\widetilde{M}_t$ very weakly depends on the molecular diffusivity.

Direct numerical simulations of turbulent channel flow at the matched friction Reynolds number of 590, comparing the effect of no-slip versus shear-stress boundary conditions, reveal that the outer flow of wall turbulence, in accord with Townsend’s outer-layer similarity hypothesis, remains largely independent of the viscous sublayer. First- and second-order statistics, including spectra, agree closely from the buffer region out to the centre of the channel. Higher-order statistics also appear to obey the hypothesised similarity, although the influence of boundary conditions is more pronounced than in the lower-order statistics. The statistical agreement in the outer layer, in spite of the structural differences in the viscous sublayer, support Townsend’s idea that the primary effect of the wall is not the no-slip condition, but the impermeability condition imposed by a solid wall.

Inspired by recent calculations by Thoraval et al. (Phys. Rev. Lett., vol. 108, 2012, p. 264506) relating to droplet impact, this paper presents an analysis of the perturbations to the free surface caused by small surface tension and viscosity in steady Helmholtz flows. In particular, we identify the regimes in which appreciable vorticity can be shed from the boundary layer to the bulk flow.

Triad interactions, involving a set of wave-vectors $\{\pm \boldsymbol {k}, \pm \boldsymbol {p}, \pm \boldsymbol {q}\}$, with $ \boldsymbol {k} + \boldsymbol {p}+ \boldsymbol {q}=0$, are considered, and the results of triad truncation are compared with the results of exact Euler evolution starting from the same initial conditions. The essential two-dimensionality of the triad interaction is used to separate the problem into two parts: a nonlinear two-dimensional flow problem in the triad plane, and a linear problem of ‘passive scalar’ type for the evolution of the component of velocity perpendicular to this plane. Several examples of triad evolution are presented in detail, and the marked contrast with Euler evolution is demonstrated. It is known that energy and helicity are conserved under triad truncation; it is shown that the ‘in-plane’ energy and enstrophy are also conserved. However, it is also shown that, in general, the evolution of the vorticity under triad truncation cannot be represented as transport by any divergence-free velocity field, with the consequence that the detailed topology of the vorticity field is not conserved under this truncation.

Lord Kelvin’s result that waves behind a ship lie within a half-angle $\phi _{\mathit{K}}\approx 19^{\circ }28'$ is perhaps the most famous and striking result in the field of surface waves. We solve the linear ship wave problem in the presence of a shear current of constant vorticity $S$ , and show that the Kelvin angles (one each side of wake) as well as other aspects of the wake depend closely on the ‘shear Froude number’ $\mathit{Fr}_{\mathit{s}}=VS/g$ (based on length $g/S^2$ and the ship’s speed $V$ ), and on the angle between current and the ship’s line of motion. In all directions except exactly along the shear flow there exists a critical value of $\mathit{Fr}_{\mathit{s}}$ beyond which no transverse waves are produced, and where the full wake angle reaches $180^\circ $ . Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed $90^\circ $ . On the other hand, the angle of maximum wave amplitude scales as $\mathit{Fr}^{-1}$ ( $\mathit{Fr}$ based on size of ship) when $\mathit{Fr}\gg 1$ , a scaling virtually unaffected by the shear flow.

In this work, the scaling statistics of the dissipation along Lagrangian trajectories are investigated by using fluid tracer particles obtained from a high-resolution direct numerical simulation with $\mathit{Re}_{\lambda }=400$. Both the energy dissipation rate $\epsilon $ and the local time-averaged $\epsilon _{\tau }$ agree rather well with the lognormal distribution hypothesis. Several statistics are then examined. It is found that the autocorrelation function $\rho (\tau )$ of $\ln (\epsilon (t))$ and variance $\sigma ^2(\tau )$ of $\ln (\epsilon _{\tau }(t))$ obey a log-law with scaling exponent $\beta '=\beta =0.30$ compatible with the intermittency parameter $\mu =0.30$. The $q{\rm th}$-order moment of $\epsilon _{\tau }$ has a clear power law on the inertial range $10<\tau /\tau _{\eta }<100$. The measured scaling exponent $K_L(q)$ agrees remarkably with $q-\zeta _L(2q)$ where $\zeta _L(2q)$ is the scaling exponent estimated using the Hilbert methodology. All of these results suggest that the dissipation along Lagrangian trajectories could be modelled by a multiplicative cascade.

We consider the interaction of two identical disks freely falling side by side in a fluid at rest for Reynolds numbers ranging from 100 to 300, corresponding to rectilinear and oscillatory paths. For the three aspect ratios of the disks investigated, we observed that the bodies always repel one another when the horizontal distance between their centres of gravity is less than 4.5 diameters. They never come closer for distances spanning between 4.5 and 6 diameters. Beyond the latter distance, the disks appear indifferent to each other. For both rectilinear and periodic paths, the repulsion effect is weak, leading to an overall horizontal drift lower than 3 % of the vertical displacement. We propose a model for the repulsion coefficient Cr, which decreases with the separation distance between the bodies and is inversely proportional to the aspect ratio of the bodies, Cr thus being stronger for the thicker ones. Furthermore, in the case of the oscillatory paths, we show that the effect of the interaction reduces to the repulsion effect, since the characteristics of the oscillatory motion of each disk appear unaffected by the presence of the companion disk and no synchronization is observed between the paths, nor between the wakes, of the two disks.

We study the Marangoni propulsion of a spheroidal particle located at a liquid–gas interface. The particle asymmetrically releases an insoluble surface-active agent and so creates and maintains a surface tension gradient leading to the self-propulsion. Assuming that the surface tension has a linear dependence on the concentration of the released agent, we derive closed-form expressions for the translational speed of the particle in the limit of small capillary, Péclet and Reynolds numbers. Our derivations are based on the Lorentz reciprocal theorem, which eliminates the need to develop the detailed flow field.

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.