Path and wake instabilities of buoyancy-driven oblate spheroidal bubbles with a prescribed shape rising freely in a viscous fluid otherwise at rest are studied using global stability analysis, following the technique recently developed for a coupled fluid $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}+$ body system by Tchoufag, Fabre & Magnaudet (J. Fluid Mech. vol. 740, 2014, pp. 278–311). The essential role of the wake on the path instability is evidenced by comparing the shape of the global stability diagram with that obtained in the case of a fixed bubble. However, dramatic differences are also found, since the critical curve of the coupled system mostly involves low- and high-frequency oscillating modes, whereas that of a fixed bubble only involves stationary modes. Comparison of the present predictions with results obtained through direct numerical simulation is achieved in several regimes, confirming the predictions of the linear approach but also highlighting some of its limitations when the system successively encounters several unstable modes.

A new set of numerical simulations of turbulent plane Couette flow in a large box of dimension ( $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}20\pi h,\, 2h,\, 6\pi h$ ) at Reynolds number $(\mathit{Re}_{\tau }) =125$ , 180, 250 and 550 is described and compared with simulations at lower Reynolds numbers, Poiseuille flows and experiments. The simulations present a logarithmic near-wall layer and are used to verify and revise previously known results. It is confirmed that the fluctuation intensities in the streamwise and spanwise directions do not scale well in wall units. The scaling failure occurs both near to and away from the wall. On the contrary, the wall-normal intensity scales in inner units in the near-wall region and in outer units in the core region. The spectral ridge found by Hoyas & Jiménez (Phys. Fluids, vol. 18, 2003, 011702) for the turbulent Poiseuille flow can also be seen in the present flow. Away from the wall, very large-scale motions are found spanning through all the length of the channel. The statistics of these simulations can be downloaded from the webpage of the Chair of Fluid Dynamics.

Two-dimensional steady gravity waves with vorticity are considered on water of finite depth. The dispersion equation is analysed for general vorticity distributions, but under assumptions valid only for unidirectional shear flows. It is shown that for these flows (i) the general dispersion equation is equivalent to the Sturm–Liouville problem considered by Constantin & Strauss (Commun. Pure Appl. Math., vol. 57, 2004, pp. 481–527; Arch. Rat. Mech. Anal., vol. 202, 2011, pp. 133–175), (ii) the condition guaranteeing bifurcation of Stokes waves with constant wavelength is fulfilled. Moreover, a necessary and sufficient condition that the Sturm–Liouville problem mentioned in (i) has an eigenvalue is obtained.

This paper demonstrates that the shallow water semigeostrophic equations arise from a degenerate second-order Hamilton principle of very special structure. The associated Euler–Lagrange operator factors into a fast and a slow first-order operator; restricting to the slow part yields the geostrophic momentum approximation as balanced dynamics. While semigeostrophic theory has been considered variationally before, this structure appears to be new. It leads to a straightforward derivation of the geostrophic momentum approximation and its associated potential vorticity law. Our observations further affirm, from a different point of view, the known difficulty in generalizing the semigeostrophic equations to the case of a spatially varying Coriolis parameter.

We present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier–Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions cause rapid disruption of the interior roll-streak structure and formation of a turbulent spot, whose growth or decay depends on the Reynolds number and the choice of perturbation.

We present results of direct numerical simulation of incompressible fluid flow over a thick bed of mobile spherically shaped particles. The algorithm is based upon the immersed-boundary technique for fluid–solid coupling and uses a soft-sphere model for the solid–solid contact. Two parameter points in the laminar flow regime are chosen, leading to the emergence of sediment patterns classified as ‘small dunes’, while one case under turbulent flow conditions leads to ‘vortex dunes’ with significant flow separation on the lee side. The wavelength, amplitude and propagation speed of the patterns extracted from the spanwise-averaged fluid–bed interface are found to be consistent with available experimental data. The particle transport rates are well represented by available empirical models for flow over a plane sediment bed in both the laminar and the turbulent regimes.

The instability of the hub vortex observed in wind turbine wakes has recently been studied by Iungo et al. (J. Fluid Mech., vol. 737, 2013, pp. 499–526) via local stability analysis of the mean velocity field measured through wind tunnel experiments. This analysis was carried out by neglecting the effect of turbulent fluctuations on the development of the coherent perturbations. In the present paper, we perform a stability analysis taking into account the Reynolds stresses modelled by eddy-viscosity models, which are calibrated on the wind tunnel data. This new formulation for the stability analysis leads to the identification of one clear dominant mode associated with the hub vortex instability, which is the one with the largest overall downstream amplification. Moreover, this analysis also predicts accurately the frequency of the hub vortex instability observed experimentally. The proposed formulation is of general interest for the stability analysis of swirling turbulent flows.

Taylor–Couette (TC) flow is used to probe the hydrodynamical (HD) stability of astrophysical accretion disks. Experimental data on the subcritical stability of TC flow are in conflict about the existence of turbulence (cf. Ji et al. (Nature, vol. 444, 2006, pp. 343–346) and Paoletti et al. (Astron. Astroph., vol. 547, 2012, A64)), with discrepancies attributed to end-plate effects. In this paper we numerically simulate TC flow with axially periodic boundary conditions to explore the existence of subcritical transitions to turbulence when no end plates are present. We start the simulations with a fully turbulent state in the unstable regime and enter the linearly stable regime by suddenly starting a (stabilizing) outer cylinder rotation. The shear Reynolds number of the turbulent initial state is up to $Re_s \lesssim 10^5$ and the radius ratio is $\eta =0.714$ . The stabilization causes the system to behave as a damped oscillator and, correspondingly, the turbulence decays. The evolution of the torque and turbulent kinetic energy is analysed and the periodicity and damping of the oscillations are quantified and explained as a function of shear Reynolds number. Though the initially turbulent flow state decays, surprisingly, the system is found to absorb energy during this decay.

On the basis of a two-point similarity analysis, the well-known power-law variations for the mean kinetic energy dissipation rate $\overline{\epsilon }$ and the longitudinal velocity variance $\overline{u^2}$ on the axis of a round jet are derived. In particular, the prefactor for $\overline{\epsilon } \propto (x-x_0)^{-4}$ , where $x_0$ is a virtual origin, follows immediately from the variation of the mean velocity, the constancy of the local turbulent intensity and the ratio between the axial and transverse velocity variance. Second, the limit at small separations of the two-point budget equation yields an exact relation illustrating the equilibrium between the skewness of the longitudinal velocity derivative $S$ and the destruction coefficient $G$ of enstrophy. By comparing the latter relation with that for homogeneous isotropic decaying turbulence, it is shown that the approach towards the asymptotic state at infinite Reynolds number of $S+2G/R_{\lambda }$ in the jet differs from that in purely decaying turbulence, although $S+2G/R_{\lambda } \propto R_{\lambda }^{-1}$ in each case. This suggests that, at finite Reynolds numbers, the transport equation for $\overline{\epsilon }$ imposes a fundamental constraint on the balance between $S$ and $G$ that depends on the type of large-scale forcing and may thus differ from flow to flow. This questions the conjecture that $S$ and $G$ follow a universal evolution with $R_{\lambda }$ ; instead, $S$ and $G$ must be tested separately in each flow. The implication for the constant $C_{\epsilon 2}$ in the $k-\overline{\epsilon }$ model is also discussed.

The characteristics of a turbulent boundary layer overlying a complex roughness topography were explored with stereo particle-image velocimetry measurements in the wall-normal–spanwise $(y\mbox{--}z)$ plane. The roughness under consideration was replicated from a turbine blade damaged by deposition of foreign materials containing a broad range of topographical scales arranged in a highly irregular manner. The single-point turbulence statistics displayed strong spanwise heterogeneity, in particular spanwise-alternating low- and high-momentum flow pathways in the mean flow marked by enhanced Reynolds stresses and turbulent kinetic energy. The spanwise regions between high- and low-momentum flow pathways were occupied by swirling motions, suggesting the generation and sustainment of turbulent secondary flows due to the spanwise heterogeneity of the complex roughness under consideration. Similar observations were recently reported for more ordered spanwise roughness transitions by Nugroho, Hutchins & Monty (Intl J. Heat Fluid Flow vol. 41, 2013, pp. 90–102) and Willingham et al. (Phys. Fluids vol. 26, 2014, 025111).

We present a computational study of cluster-induced turbulence (CIT), where the production of fluid-phase kinetic energy results entirely from momentum coupling with finite-size inertial particles. A separation of length scales must be established when evaluating the particle dynamics in order to distinguish between the continuous mesoscopic velocity field and the uncorrelated particle motion. To accomplish this, an adaptive spatial filter is employed on the Lagrangian data with an averaging volume that varies with the local particle-phase volume fraction. This filtering approach ensures sufficient particle sample sizes in order to obtain meaningful statistics while remaining small enough to avoid capturing variations in the mesoscopic particle field. Two-point spatial correlations are computed to assess the validity of the filter in extracting meaningful statistics. The method is used to investigate, for the first time, the properties of a statistically stationary gravity-driven particle-laden flow, where particle–particle and fluid–particle interactions control the multiphase dynamics. Results from fully developed CIT show a strong correlation between the local volume fraction and the granular temperature, with maximum values located at the upstream boundary of clusters (i.e. where maximum compressibility of the particle velocity field exists), while negligible particle agitation is observed within clusters.

The equilibrium shape of liquid drops on elastic substrates is determined by minimizing elastic and capillary free energies, focusing on thick incompressible substrates. The problem is governed by three length scales: the size of the drop $R$ , the molecular size $a$ and the ratio of surface tension to elastic modulus $\gamma /E$ . We show that the contact angles undergo two transitions upon changing the substrate from rigid to soft. The microscopic wetting angles deviate from Young’s law when $\gamma /(Ea)\gg 1$ , while the apparent macroscopic angle only changes in the very soft limit $\gamma /(ER)\gg 1$ . The elastic deformations are worked out for the simplifying case where the solid surface energy is assumed to be constant. The total free energy turns out to be lower on softer substrates, consistent with recent experiments.

A recent hydrodynamic theory of liquid slippage on a solid substrate (Kirkinis & Davis, Phys. Rev. Lett., vol. 110, 2013, 234503) gives rise to a sequence of eddies (Moffatt vortices) that co-move with a moving contact line (CL) in a liquid wedge. The presence of these vortices is established through secular equations that depend on the dynamic contact angle $\alpha $ and capillary number Ca. The limiting case $\alpha \rightarrow 0$ is associated with the appearance of such vortices in a channel. The vortices are generated by the relative motion of the interfaces, which in turn is due to the motion of the CL. This effect has yet to be observed in experiment.

The fall velocity of a dense large ball in a suspension of neutrally buoyant non-Brownian particles subjected to horizontal oscillatory shear is studied. As the strain amplitude is increased, the velocity increases up to a maximum value before decreasing to the value that it would have in a resting suspension. The higher the frequency is, the stronger the effect is. The falling ball velocity can be largely increased in the presence of the oscillatory cross-shear flow. For instance, for a particle volume fraction of $\varPhi =0.47$ it reaches four times the value it has in the unsheared suspension. At small strain amplitudes, it turns out that the velocity of the falling ball is determined by a balance between the steady drag flow, which drives the apparent suspension viscosity toward a high value, and the oscillatory cross-shear, which lessens it. A simple model is proposed to explain the experimental observations at small strain amplitude. The velocity decrease observed at larger amplitude is not completely understood yet.

We prove that there are no three-dimensional bounded travelling gravity waves with constant non-zero vorticity on water of finite depth. The result also holds for gravity–capillary waves under a certain condition on the pressure at the surface, which is satisfied by sufficiently small waves. The proof relies on unique continuation arguments and Liouville-type results for elliptic equations.

A combination of cross-wire probes with an array of flush-mounted skin-friction sensors are used to study the three-dimensional conditional organisation of large-scale structures in a high-Reynolds-number turbulent boundary layer. Previous studies have documented the amplitude modulation of small-scale motions in response to conditionally averaged large-scale events, but the data are largely restricted to the streamwise component of velocity alone. Here, we report results based on all three components of velocity and find that the small-scale spanwise and wall-normal fluctuations ( $v$  and  $w$ ) and the instantaneous Reynolds shear stress ( $-{uw}$ ) are modulated in a very similar manner to that previously noted for the streamwise fluctuations ( $u$ ). The envelope of the small scale fluctuations for all velocity components is well described by the large-scale component of the $u$ fluctuation. These results also confirm the conditional existence of roll modes associated with the very large-scale or ‘superstructure’ motions.

The structure of a steady axisymmetric thermal plume rising through a very viscous fluid with strongly temperature-dependent viscosity of the form $\mu \propto \exp (-\gamma T)$ is investigated. An analytic asymptotic solution is derived for the fast-flowing core of the plume, which predicts that the excess centreline temperature decays exponentially as $\exp \{ - 12 \pi \kappa z/(\gamma A) \}$, where $\kappa $ is the thermal diffusivity, $z$ the height and $A$ the vertical heat flux. This rate of decay, which is found to be in good agreement with numerical simulations of the boundary-layer equations, is three times faster than that predicted by the oft-quoted model of Olson, Schubert and Anderson (J. Geophys. Res., vol. 98 (B4), 1993, pp. 6829–6844).

It is shown here that a subset of the implicit analytical shock solutions discovered by Becker and by Johnson can be inverted, yielding several exact closed-form solutions of the one-dimensional compressible Navier–Stokes equations for an ideal gas. For a constant dynamic viscosity and thermal conductivity, and at particular values of the shock Mach number, the velocity can be expressed in terms of a polynomial root. For a constant kinematic viscosity, independent of Mach number, the velocity can be expressed in terms of a hyperbolic tangent function. The remaining fluid variables are related to the velocity through simple algebraic expressions. The solutions derived here make excellent verification tests for numerical algorithms, since no source terms in the evolution equations are approximated, and the closed-form expressions are straightforward to implement. The solutions are also of some academic interest as they may provide insight into the nonlinear character of the Navier–Stokes equations and may stimulate further analytical developments.

The complete description of the lifetime of a droplet on a solid substrate evaporating in a ‘stick–slide’ mode is obtained. The unexpectedly subtle relationship between the lifetime of such a droplet and the lifetimes of initially identical droplets evaporating in the extreme modes (namely the constant contact radius and constant contact angle modes) is described and summarised in an appropriate master diagram. In particular, it is shown that the lifetime of a droplet is not, in general, constrained by the lifetimes of the extreme modes.

When a bubble rises to an interface between two immiscible liquids, it can pass through the interface, if this is energetically favourable, i.e. the bubble preferring the side of the interface with the lower air–liquid surface tension. Once the intermediate film between the bubble and the interface has drained sufficiently, the bubble makes contact with the interface, forming a triple-line and producing strong capillary waves which travel around the bubble and can pinch off a satellite on the opposite side, akin to the dynamics in the coalescence cascade. We identify the critical Ohnesorge numbers where such satellites are produced and characterize their sizes. The total transition time scales with the bubble size and differential surface tension, while the satellite pinch-off time scales with the capillary-inertial time of the pool liquid, which originally surrounds the bubble. We also use high-speed video imaging to study the motion of the neck of the contact. For low viscosity we show that it grows in time with a power-law exponent between 0.44 and 0.50, with a prefactor modified by the net sum of the three interfacial tensions. Increasing the viscosity of the receiving liquid drop drastically slows down the motion of the triple-line, when the Ohnesorge number exceeds ${\sim }$ 0.08. This differs qualitatively from the coalescence of two miscible drops of different viscosities, where the lower viscosity sets the coalescence speed. We thereby propose a strong resistance from the triple-line.