Riming, the process whereby ice crystals get coated by impacting supercooled liquid droplets, is one of the dominant processes leading to precipitation in mixed-phase clouds. How a settling crystal collides with very small water droplets has been mostly studied in laminar conditions. The present numerical study aims at providing further insight on how turbulent flow motion affects the riming of ice crystals. We model the crystals as narrow oblate ellipsoids, smaller than the Kolmogorov elementary scale. By neglecting the effect of fluid inertia on the motion of the crystals and droplets, and using direct numerical simulations of the Navier–Stokes equations in a moderately turbulent regime, over a range of kinetic energy dissipation $1~\text{cm}^{2}~\text{s}^{-3}\lesssim \unicode[STIX]{x1D700}\lesssim 256~\text{cm}^{2}~\text{s}^{-3}$ , we determine the collision rate between disk-shaped ice crystals and very small liquid water droplets. Whereas differential settling plays the dominant role in determining the collision rate at small turbulence intensity, the role of turbulence becomes more important at the large values of $\unicode[STIX]{x1D700}$ simulated, an effect that can be partly attributed to the increased role of inertia. We always find that collisions occur with a large probability on the rim of the ellipsoids, a phenomenon that can be explained to a large extent by kinematic considerations. The difference in the settling velocity of crystals and droplets induces a strong asymmetry in the probability of collision between the faces of the ellipsoids. Our results shed light on the physical mechanisms involved in the riming of ice crystals in clouds.

The onset of convection in a narrow cylindrical annulus heated from below and rotating about its vertical axis of symmetry is considered in the case when the rigid cylindrical walls are thermally insulating. An analytical expression is derived for the Rayleigh number $R$ for onset of convection as a function of rotation rate and azimuthal wavenumber. The critical value $R_c$ for high rotation rates is much lower than the corresponding value in an extended layer. At finite amplitudes the convection flow generates a differential rotation which is antisymmetric with respect to the middle of the layer and is prograde near the outer wall for low rotation rates, but changes sign for higher values of the rotation parameter.

Passive scalar transport in turbulent channel flow subject to spanwise system rotation is studied by direct numerical simulations. The Reynolds number $Re=U_{b}h/\unicode[STIX]{x1D708}$ is fixed at 20 000 and the rotation number $Ro=2\unicode[STIX]{x1D6FA}h/U_{b}$ is varied from 0 to 1.2, where $U_{b}$ is the bulk mean velocity, $h$ the half channel gap width and $\unicode[STIX]{x1D6FA}$ the rotation rate. The scalar is constant but different at the two walls, leading to steady scalar transport across the channel. The rotation causes an unstable channel side with relatively strong turbulence and turbulent scalar transport, and a stable channel side with relatively weak turbulence or laminar-like flow, weak turbulent scalar transport but large scalar fluctuations and steep mean scalar gradients. The distinct turbulent–laminar patterns observed at certain $Ro$ on the stable channel side induce similar patterns in the scalar field. The main conclusions of the study are that rotation reduces the similarity between the scalar and velocity field and that the Reynolds analogy for scalar-momentum transport does not hold for rotating turbulent channel flow. This is shown by a reduced correlation between velocity and scalar fluctuations, and a strongly reduced turbulent Prandtl number of less than 0.2 on the unstable channel side away from the wall at higher $Ro$. On the unstable channel side, scalar scales become larger than turbulence scales according to spectra and the turbulent scalar flux vector becomes more aligned with the mean scalar gradient owing to rotation. Budgets in the governing equations of the scalar energy and scalar fluxes are presented and discussed as well as other statistics relevant for turbulence modelling.

Convective plumes emanating from fixed buoyant sources such as volcanoes, hot springs and oil spills are common in the atmosphere and the ocean. Most of what we know about their dynamics comes from scaling laws, laboratory experiments and numerical simulations. A plume grows laterally during its ascent mainly due to the process of turbulent entrainment of fluid from the environment into the plume. In an unstratified system, nothing hampers the vertical motion of the plume. By contrast, in a stratified system, as the plume rises, it reaches and overshoots the neutral buoyancy height – due to the non-zero momentum at that height. This rising fluid is then dense relative to the environment and slows down, ceases to rise and falls back to the height of the intrusion. For buoyant plumes occurring in the ocean or atmosphere, the rotation of the Earth adds an additional constraint via the conservation of angular momentum. In fact, the effect of rotation is still not well understood, and we addressed this issue in the study reported here. We looked for the steady states of an axisymmetric model in both the rotating and non-rotating cases. At the non-rotating limit, we isolated two regimes of convection depending on the buoyancy flux/momentum flux ratio at the base of the plume, in agreement with scaling laws. However, the inclusion of rotation in the model strongly affects these classical convection patterns: the lateral extension of the plume is confined at the intrusion level by the establishment of a geostrophic balance, and non-trivial swirl speed develops in and around the plume.

We report experimental observations of two canonical surface wave patterns – ship waves and ring waves – skewed by sub-surface shear, thus confirming effects predicted by recent theory. Observed ring waves on a still surface with sub-surface shear current are strikingly asymmetric, an effect of strongly anisotropic wave dispersion. Ship waves for motion across a sub-surface current on a still surface exhibit striking asymmetry about the ship’s line of motion, and large differences in transverse wavelength for upstream versus downstream motion are demonstrated, all of which is in good agreement with theoretical predictions. Neither of these phenomena can occur on a depth-uniform current. A quantitative comparison of measured versus predicted average phase shift for a ring wave is grossly mispredicted by no-shear theory, but in good agreement with predictions for the measured shear current. A clear difference in wave frequency within the ring wave packet is observed in the upstream versus downstream direction for all shear flows, while wave dispersive behaviour is identical to that for quiescent water for propagation normal to the shear current, as expected. Peak values of the measured two-dimensional Fourier spectrum for ship waves are shown to agree well with the predicted criterion of stationary ship waves, with the exception of some cases where results are imperfect due to the limited wavenumber resolution, transient effects and/or experimental noise. Experiments were performed on controlled shear currents created in two different ways, with a curved mesh and beneath a blocked stagnant-surface flow. Velocity profiles were measured with particle image velocimetry, and surface waves with a synthetic schlieren method. Our observations lend strong empirical support to recent predictions that wave forces on vessels and structures can be greatly affected by shear in estuarine and tidal waters.

We calculate the optimal upper and lower bounds, subject to the assumption of streamwise invariance, on the long-time-averaged mechanical energy dissipation rate $\varepsilon$ within the flow of an incompressible viscous fluid of constant kinematic viscosity $\nu$ and depth $h$ driven by a constant surface stress $\tau=\rho u^2_\star$, where $u_\star$ is the friction velocity. We show that $\varepsilon \leq \varepsilon_{\max}=\tau^2/(\rho^2\nu)$, i.e. the dissipation is bounded above by the dissipation associated with the laminar solution $\bu=\tau(z+h)/(\rho\nu) \xvec$, where $\xvec$ is the unit vector in the streamwise $x$-direction.

By using the variational ‘background method’ (due to Constantin, Doering and Hopf) and numerical continuation, we also generate a rigorous lower bound on the dissipation for arbitrary Grashof numbers $G$, where $G=\tau h^2/(\rho \nu^2)$. Under the assumption of streamwise invariance as $G \rightarrow \infty$, for flows where horizontal mean momentum balance and total power balance are imposed as constraints, we show numerically that the best possible lower bound for the dissipation is $\varepsilon \geq \varepsilon_{\min}=7.531 u_\star^3/h$, a bound that is independent of the flow viscosity. This scaling (though not the best possible numerical coefficient) can also be obtained directly by applying the same imposed constraints and restricting attention to a particular, analytically tractable, class of mean flows.

The first 27 terms in a series expansion for the profile of a solitary wave are computed. From this, series expansions for the wave amplitude, mass and potential energy are obtained. A previous study indicated that the partial sums of these series converged for small- to medium-amplitude waves and that the diagonal Padé approximants converged for waves of all amplitudes. The data derived here show that this is not the case and that apparent convergence of Padé approximants can be misleading.

The problem of a suspension droplet falling under gravity was examined for polydisperse droplets composed of a mixture of particles with different densities and sizes. The study was conducted using both simulations based on oseenlet particle interactions and laboratory experiments. The hydrodynamic interactions of the particles within the suspension droplet allow a polydisperse collection of particles to fall as a coherent droplet, even for cases where the difference in particle terminal velocity would cause them to separate quickly from each other in the absence of hydrodynamic interactions. However, a gradual segregation phenomenon is observed in which particles with lower terminal velocity preferentially leave the suspension droplet by entering into the droplet tail, whereas particles with higher terminal velocity remain for longer periods of time within the droplet. When computations and experiments are performed for bidisperse mixtures, a point is eventually reached where all of the lighter/smaller particles are ejected into the droplet tail and the droplet continues to fall with only the heavier/larger particles.

The diffusive strip method (DSM) is a near-exact numerical method for mixing computations initially developed in two dimensions (Meunier & Villermaux, J. Fluid Mech., vol. 662, 2010, pp. 134–172). The method, which consists of following stretched material lines to compute the resulting scalar field a posteriori, is extended here to three-dimensional flows. We describe the procedure and its three-dimensional peculiarity, which relies on the Lagrangian advection of a triangulated surface from which the stretching rate is extracted to infer the scalar field. The method is first validated at moderate Péclet number against a classical pseudospectral method solving the advection–diffusion equation for a Batchelor vortex, and then applied to a simple Taylor–Couette experimental configuration with non-rotating boundary conditions at the top-end disk, bottom-end disk and outer cylinder. This motion, producing an elaborate although controlled steady three-dimensional flow, relies on Ekman pumping arising from the rotation of the inner cylinder. A recurrent two-cell structure is separated by the horizontal mid-plane and formed by stream tubes shaped as nested tori under laminar flow conditions. A scalar blob in the flow experiences a Lagrangian oscillating dynamics undergoing stretchings and compressions, driving the mixing process. The DSM enables the calculation of the blob elongation and scalar concentration distributions through a single variable computation along the advected blob surface, capturing the rich evolution observed in the experiments. Interestingly, the mixing process in this axisymmetric and steady three-dimensional flow leads to a linear growth of surfaces in time similar to the one obtained in a two-dimensional shear. The potentialities, limits and extension of the method to more general flows are finally discussed.

We propose a new unified model for the small, intermediate and large-scale evolution of freely decaying two-dimensional turbulence in the inviscid limit. The new model's centerpiece is a recent theory of vortex self-similarity (Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501), applicable to the intermediate range of scales spanned by an expanding population of vortices. This range is predicted to have a steep k−5 energy spectrum. At small scales, this gives way to Batchelor's (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k−3 energy spectrum, corresponding to the (forward) enstrophy (mean square vorticity) cascade or, physically, to thinning filamentary debris produced by vortex collisions. This small-scale range carries with it nearly all of the enstrophy but negligible energy. At large scales, the slow growth of the maximum vortex size (~t1/6 in radius) implies a correspondingly slow inverse energy cascade. We argue that this exceedingly slow growth allows the large scales to approach equipartition (Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169), ultimately leading to a k1 energy spectrum there. Put together, our proposed model has an energy spectrum ℰ(k, t) ∝ t1/3k1 at large scales, together with ℰ(k, t) ∝ t−2/3k−5 over the vortex population, and finally ℰ(k, t) ∝ t−1k−3 over an exponentially widening small-scale range dominated by incoherent filamentary debris.

Support for our model is provided in two parts. First, we address the evolution of large and ultra-large scales (much greater than any vortex) using a novel high-resolution vortex-in-cell simulation. This verifies equipartition, but more importantly allows us to better understand the approach to equipartition. Second, we address the intermediate and small scales by an ensemble of especially high-resolution direct numerical simulations.

We calculate optimal driving motions for a fin ray in a two-dimensional inviscid fluid, which is a model for caudal fin locomotion. The driving is sinusoidal in time, and consists of heaving, pitching and a less-studied motion called ‘shifting’. The optimal phases of shifting relative to heaving and pitching for maximum thrust power and efficiency are calculated. The optimal phases undergo jumps at resonant combinations of fin ray bending and shear moduli, and are nearly constant in regions between resonances. In two examples, pitching- and heaving-based motions converge with the addition of optimal shifting. Shifting provides an order-one increase in output power and efficiency.