Gravity currents are an important buoyancy-driven flow in environmental, geophysical and industrial settings. Turbulence and mixing is commonplace in these flows, but is typically overlooked in theoretical models and predictions. Sher & Woods (J. Fluid Mech., vol. 784, 2015, pp. 130–162) have quantified the velocity and density structure in turbulent gravity currents by combining high-quality experimental data with new theory. Their insights are set to stimulate significant advances in the area.

Modal decompositions of unperturbed and acoustically driven injector flows from shear coaxial jets are implemented using dynamic-mode decomposition, which is a natural approach in the search for collective oscillatory behaviour in nonlinear systems. Previous studies using proper orthogonal decomposition had revealed the most energetic pairs of coherent structures in injector flows. One of the difficulties in extracting lower-energy coherent structures follows from the need to differentiate robust flow constituents from noise and other irregular facets of a flow. The identification of robust features is critical for applications such as flow control as well, since only they can be used for the tasks. A dynamic-mode decomposition based algorithm for this differentiation is introduced and used to identify different classes of robust dynamic modes. They include (1) background modes located outside the injector flow that decay rapidly, (2) injector modes – including those presented in earlier studies – located in the vicinity of the flow, (3) modes that persist under acoustic driving, (4) modes responding linearly to the driving and, most interestingly, (5) a mode whose density exhibits antiphase oscillatory behaviour in the observation plane and that appears only when $J$, the outer-to-inner-jet momentum flux ratio, is sufficiently large; we infer that this is a projection of a mode rotating about the symmetry axis and born via a spontaneous symmetry breaking. Each of these classes of modes is analysed as $J$ is increased, and their consequences for the flow patterns are discussed.

We experimentally study the cavities forming in the wake of rigid spheres when submerging into a stratified, two-layer system of immiscible, quiescent liquids comprising a thin layer of oil above a deep pool of water. The results obtained for our two-layer system are compared with data from the literature for the corresponding type of cavities formed when spheres enter a homogeneous liquid that is not covered by an oil layer. The discussion and the data analysis reveal that the oil coating acquired by the spheres while propagating through the thin oil layer, before entering the pool of water underneath, substantially affects qualitative and quantitative aspects of the dynamics associated with the cavity formation. In particular, we observe the formation of a ripple-like pattern on the cavity walls which is not known to exist when spheres enter a homogeneous liquid. The data analysis suggests that the ripple patterns form as a consequence of a two-dimensional instability arising due to the shear between the oil layer coating the spheres and the ambient water.

Self-preservation (SP) solutions on the axis of a turbulent round jet are derived for the transport equation of the second-order structure function of the turbulent kinetic energy ($k$), which may be interpreted as a scale-by-scale (s.b.s.) energy budget. The analysis shows that the mean turbulent energy dissipation rate, $\overline{{\it\epsilon}}$, evolves like $x^{-4}$ ($x$ is the streamwise direction). It is important to stress that this derivation does not use the constancy of the non-dimensional dissipation rate parameter $C_{{\it\epsilon}}=\overline{{\it\epsilon}}u^{\prime 3}/L_{u}$ ($L_{u}$ and $u^{\prime }$ are the integral length scale and root mean square of the longitudinal velocity fluctuation respectively). We show, in fact, that the constancy of $C_{{\it\epsilon}}$ is simply a consequence of complete SP (i.e. SP at all scales of motion). The significance of the analysis relates to the fact that the SP requirements for the mean velocity and mean turbulent kinetic energy (i.e. $U\sim x^{-1}$ and $k\sim x^{-2}$ respectively) are derived without invoking the transport equations for $U$ and $k$. Experimental hot-wire data along the axis of a turbulent round jet show that, after a transient downstream distance which increases with Reynolds number, the turbulence statistics comply with complete SP. For example, the measured $\overline{{\it\epsilon}}$ agrees well with the SP prediction, i.e. $\overline{{\it\epsilon}}\sim x^{-4}$, while the Taylor microscale Reynolds number $Re_{{\it\lambda}}$ remains constant. The analytical expression for the prefactor $A_{{\it\epsilon}}$ for $\overline{{\it\epsilon}}\sim (x-x_{o})^{-4}$ (where $x_{o}$ is a virtual origin), first developed by Thiesset et al. (J. Fluid Mech., vol. 748, 2014, R2) and rederived here solely from the SP analysis of the s.b.s. energy budget, is validated and provides a relatively simple and accurate method for estimating $\overline{{\it\epsilon}}$ along the axis of a turbulent round jet.

The influence of ambient fluid stratification on buoyant miscible jets and plumes is studied theoretically and experimentally. Given a fixed set of jet/plume parameters, and an ambient fluid stratification sandwiched between top and bottom homogeneous densities, a theoretical criterion is identified to show how step-like density profiles constitute the most effective mixers within a broad class of stable density transitions. This is assessed both analytically and experimentally, respectively by establishing rigorous a priori estimates on generalized Morton–Taylor–Turner (MTT) models (Morton et al., Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23; Fischer et al., Mixing in Inland and Coastal Waters. Academic, 1979), and by studying a critical phenomenon determined by the distance between the jet/plume release height with respect to the depth of the ambient density transition. For fluid released sufficiently close to the background density transition, the buoyant jet fluid escapes and rises indefinitely. For fluid released at locations lower than a critical depth, the buoyant fluid stops rising and is trapped indefinitely. A mathematical formulation providing rigorous estimates on MTT models is developed along with nonlinear jump conditions and an exact critical-depth formula that is in good quantitative agreement with the experiments. Our mathematical analysis provides rigorous justification for the critical trapping/escaping criteria, first presented in Caulfield & Woods (J. Fluid Mech., vol. 360, 1998, pp. 229–248), within a class of algebraic density decay rates. Further, the step-like background stratification is shown to be the most efficient mixing profile amongst a broad family of stably stratified profiles sharing the same density transition within a fixed distance. Finally, the analysis uncovers surprising differences between the Gaussian and top-hat profile closures concerning initial mixing of the jet and ambient fluid.

We identify a benchmark problem simple enough that it can be solved both by an eddy-viscosity model and by direct numerical simulation: this is the linear response of a turbulent flow’s mean-velocity profile to an external volume force. An example of such a force was found in a study of the perturbation induced by bottom topography by Luchini & Charru (J. Fluid Mech., vol. 656, 2010, pp. 337–341). On the other hand, a modification of the method by Quadrio & Luchini (Proceedings of the IX European Turbulence Conference, Southampton, UK, 2002, pp. 715–718) and Luchini et al. (Phys. Fluids, vol. 18, 2006, 121702) to compute the linear impulse response of a wall-bounded turbulent flow allows the response to a volume force to be computed directly. The comparison exhibits significant differences and suggests that there might be fundamental obstacles to designing an eddy-viscosity model that provides the correct result.

Direct numerical simulation data of turbulent channel flow ($Re_{{\it\tau}}=930$) are used to investigate the statistics of long motions of streamwise velocity fluctuations ($u$), and the interaction of these structures with the near-wall disturbances, which is facilitated by their associated large-scale circulations. In the log layer, the negative-$u$ structures are organized into longer streamwise extent (${>}3{\it\delta}$) in comparison to the positive-$u$ counterparts. Near the wall, the footprint of negative-$u$ structures is relatively narrow in comparison to the footprint of positive-$u$ structures. This difference is due to the opposite spanwise motions in the vicinity of the footprints, which are either congregative or dispersive depending on the circulation of the outer roll cells. Conditional sampling of the footprints shows that the spanwise velocity fluctuations ($w$) are significantly enhanced by the dispersive motions of high-speed structures. On the other hand, the near-wall congregative motions of negative-$u$ structures generate relatively weak $w$ but intense negative-$u$ regions due, in part, to the spanwise collective migration of near-wall streaks. The concentrated near-wall regions of negative-$u$ upwell during the merging of the outer long scales – an effect that is demonstrated using statistical analysis of the merging process. This leads to a reduction of the convection speed of downstream negative-$u$ structures and thus promotes the merging with upstream ones. These top-down and bottom-up interactions enhance the spatial coherence of long negative-$u$ structures in the log region.

The wall shear stress (WSS) vector field provides a signature for near-wall convective transport, and can be scaled to obtain a first-order approximation of the near-wall fluid velocity. The near-wall flow field governs mass transfer problems in convection-dominated open flows with high Schmidt number, in which case a flux at the wall will lead to a thin concentration boundary layer. Such near-wall transport is of particular interest in cardiovascular flows whereby haemodynamics can initiate and progress biological events at the vessel wall. In this study we consider mass transfer processes in pulsatile blood flow of abdominal aortic aneurysms resulting from complex WSS patterns. Specifically, the Lagrangian surface transport of a species released at the vessel wall was advected in forward and backward time based on the near-wall velocity field. Exposure time and residence time measures were defined to quantify accumulation of trajectories, as well as the time required to escape the near-wall domain. The effect of diffusion and normal velocity was investigated. The trajectories induced by the WSS vector field were observed to form attracting and repelling coherent structures that delineated species distribution inside the boundary layer consistent with exposure and residence time measures. The results indicate that Lagrangian WSS structures can provide a template for near-wall transport.

A theoretical study is presented of the flow of viscoplastic fluid through a Hele-Shaw cell that contains various kinds of obstructions. Circular and elliptical blockages of the cell are considered together with stepwise contractions or expansions in slot width, all within the simplifying approximation of a narrow gap. Specific attention is paid to the flow patterns that develop around the obstacles, particularly any stagnant plugged regions, and the asymptotic limits of relatively small or large yield stress. Periodic arrays of circular contractions or expansions are studied to explore the interference between obstructions. Finally, viscoplastic flow through a cell with randomly roughened walls is examined, and it is shown that constructive interference of local contractions and expansions leads to a pronounced channelization of the flow. An optimization algorithm based on minimization of the pressure drop is derived to construct the path of the channels in the limit of relatively large yield stress or, equivalently, relatively slow flow.

Fluid outflow is considered, from a binary system of two point sources. The sources inject fluid of a lower density than the surrounding medium, and there is a sharp interface separating the two fluids. The overall geometry is taken to be axisymmetric around the line joining the two sources. Numerical solutions are presented for the shape of the interface in unsteady flow, and compared with a linearized solution based on small deformation of the interface from its initial spherical configuration. In addition, a novel spectral method is developed for the solution of the Boussinesq viscous flow problem, accounting exactly for the presence of the two sources and modelling the interface as a narrow region in which fluid mixing is possible. Bipolar outflow jets are seen to be possible.

This experimental study explores and quantifies mixing characteristics associated with a gaseous round jet injected perpendicularly into cross-flow for a range of flow and injection conditions. The study utilizes acetone planar laser-induced fluorescence imaging to determine mixing metrics in both centreplane and cross-sectional planes of the jet, for a range of jet-to-cross-flow momentum flux ratios ($2\leqslant J\leqslant 41$), density ratios ($0.35\leqslant S\leqslant 1.0$) and injector configurations (flush nozzle, flush pipe and elevated nozzle), all at a fixed jet Reynolds number of 1900. For the majority of conditions explored, there is a direct correspondence between the nature of the jet’s upstream shear layer instabilities and structure, as documented in detail in Getsinger et al. (J. Fluid Mech., vol. 760, 2014, pp. 342–367), and the jet’s mixing characteristics, consistent with diffusion-dominated processes, but with a few notable exceptions. When quantified as a function of distance along the jet trajectory, mixing metrics for jets in cross-flow with an absolutely unstable upstream shear layer and relatively symmetric counter-rotating vortex pair cross-sectional structure tend to show better local molecular mixing than for jets with convectively unstable upstream shear layers and generally asymmetric cross-sectional structures. Yet the spatial evolution of mixing with downstream distance can be greater for a few specific convectively unstable conditions, apparently associated with the initiation and nature of shear layer rollup as a trigger for improved mixing. A notable exception to these trends concerns conditions where the equidensity jet in cross-flow has an upstream shear layer that is already absolutely unstable, and the jet density is then reduced in comparison with that of the cross-flow. Here, density ratios below unity tend to mix less well than for equidensity conditions, demonstrated to result from differences in the nature of higher-density cross-flow entrainment into lower-density shear layer vortices.

Direct numerical simulation (DNS) and dynamic mode decomposition (DMD) are used to study the shear layer characteristics of a jet in a crossflow. Experimental observations by Megerian et al. (J. Fluid Mech., vol. 593, 2007, pp. 93–129) at velocity ratios ($R=\overline{v}_{j}/u_{\infty }$) of 2 and 4 and Reynolds number ($Re=\overline{v}_{j}D/{\it\nu}$) of 2000 on the transition from absolute to convective instability of the upstream shear layer are reproduced. Point velocity spectra at different points along the shear layer show excellent agreement with experiments. The same frequency ($St=0.65$) is dominant along the length of the shear layer for $R=2$, whereas the dominant frequencies change along the shear layer for $R=4$. DMD of the full three-dimensional flow field is able to reproduce the dominant frequencies observed from DNS and shows that the shear layer modes are dominant for both the conditions simulated. The spatial modes obtained from DMD are used to study the nature of the shear layer instability. It is found that a counter-current mixing layer is obtained in the upstream shear layer. The corresponding mixing velocity ratio is obtained, and seen to delineate the two regimes of absolute or convective instability. The effect of the nozzle is evaluated by performing simulations without the nozzle while requiring the jet to have the same inlet velocity profile as that obtained at the nozzle exit in the simulations including the nozzle. The shear layer spectra show good agreement with the simulations including the nozzle. The effect of shear layer thickness is studied at a velocity ratio of 2 based on peak and mean jet velocity. The dominant frequencies and spatial shear layer modes from DNS/DMD are significantly altered by the jet exit velocity profile.

Motivated by the fabrication of microstructured optical fibres, a model is presented for the extension under gravity of a slender fluid cylinder with internal structure. It is shown that the general problem decouples into a two-dimensional surface-tension-driven Stokes flow that governs the transverse shape and an axial problem that depends upon the transverse flow. The problem and its solution differ from those obtained for fibre drawing, because the problem is unsteady and the fibre tension depends on axial position. Solutions both with and without surface tension are developed and compared, which show that the relative importance of surface tension depends upon both the parameter values and the geometry under consideration. The model is compared with experimental data and is shown to be in good agreement. These results also show that surface-tension effects are essential to accurately describing the cross-sectional shape.

A theoretical decomposition of mean skin friction generation into physical phenomena across the whole profile of the incompressible zero-pressure-gradient smooth-flat-plate boundary layer is derived from a mean streamwise kinetic-energy budget in an absolute reference frame (in which the undisturbed fluid is not moving). The Reynolds-number dependences in the laminar and turbulent cases are investigated from direct numerical simulation datasets and Reynolds-averaged Navier–Stokes simulations, and the asymptotic trends are consistently predicted by theory. The generation of the difference between the mean friction in the turbulent and laminar cases is identified with the total production of turbulent kinetic energy (TKE) in the boundary layer, represented by the second term of the proposed decomposition of the mean skin friction coefficient. In contrast, the analysis introduced by Fukagata et al. (Phys. Fluids, vol. 14 (11), 2002, pp. 73–76), based on a streamwise momentum budget in the wall reference frame, relates the turbulence-induced excess friction to the Reynolds shear stress weighted by a linear function of the wall distance. The wall-normal distribution of the linearly-weighted Reynolds shear stress differs from the distribution of TKE production involved in the present discussion, which consequently draws different conclusions on the contribution of each layer to the mean skin friction coefficient. At low Reynolds numbers, the importance of the buffer-layer dynamics is confirmed. At high Reynolds numbers, the present decomposition quantitatively shows for the first time that the generation of the turbulence-induced excess friction is dominated by the logarithmic layer. This is caused by the well-known decay of the relative contributions of the buffer layer and wake region to TKE production with increasing Reynolds numbers. This result on mean skin friction, with a physical interpretation relying on an energy budget, is consistent with the well-established general importance of the logarithmic layer at high Reynolds numbers, contrary to the friction breakdown obtained from the approach of Fukagata et al. (Phys. Fluids, vol. 14 (11), 2002, pp. 73–76), essentially based on a momentum budget. The new decomposition suggests that it may be worth investigating new drag reduction strategies focusing on TKE production and on the nature of the logarithmic layer dynamics. The decomposition is finally extended to the pressure-gradient case and to channel and pipe flows.

We consider the problem of short-term prediction of rare, extreme water waves in irregular unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually intense waves is nonlinear wave focusing. Recent results have demonstrated that random localizations of energy, induced by the linear dispersive mixing of different harmonics, can grow significantly due to modulation instability. Here we show how the interplay between (i) modulation instability properties of localized wave groups and (ii) statistical properties of wave groups that follow a given spectrum defines a critical length scale associated with the formation of extreme events. The energy that is locally concentrated over this length scale acts as the ‘trigger’ of nonlinear focusing for wave groups and the formation of subsequent rare events. We use this property to develop inexpensive, short-term predictors of large water waves, circumventing the need for solving the governing equations. Specifically, we show that by merely tracking the energy of the wave field over the critical length scale allows for the robust, inexpensive prediction of the location of intense waves with a prediction window of 25 wave periods. We demonstrate our results in numerical experiments of unidirectional water wave fields described by the modified nonlinear Schrödinger equation. The presented approach introduces a new paradigm for understanding and predicting intermittent and localized events in dynamical systems characterized by uncertainty and potentially strong nonlinear mechanisms.

We numerically investigate margination of white blood cells and demonstrate the dependency on a number of conditions including haematocrit, the deformability of the cells and the Reynolds number. The approach, which is based on a mesoscopic hydrodynamic Helfrich-type model, reproduces previous results, e.g. a decreasing tendency for margination with increasing deformability and a non-monotonic dependency on haematocrit. The consideration of inertia effects, which may be of relevance in various parts of the cardiovascular system, indicates a decreasing tendency for margination with increasing Reynolds number. The effect is discussed by analysing inertial and non-inertial lift forces for single cells under different flow conditions and large-scale two-dimensional simulations of interacting red blood cells and white blood cells in an idealized blood vessel.

Experimental measurements of entrainment by turbulent fountains from circular sources in quiescent uniform environments are presented. Our results span almost four orders of magnitude in the source Froude number ($0.004\leqslant \mathit{Fr}_{0}\leqslant 25$) and thereby encompass the entrainment across all classes of fountain behaviour identified to date. We identify scalings for the entrained volume flux $Q_{E}$, in terms of $\mathit{Fr}_{0}$ and the source volume flux $Q_{0}$, within a number of distinct Froude-number bands corresponding to each class of fountain. Additionally we identify a distinct class of new behaviour, as yet unreported, for $\mathit{Fr}_{0}\lesssim 0.1$.

We numerically study turbulent Taylor–Couette flow (TCF) between two independently rotating cylinders and the transition to rotating plane Couette flow (RPCF) in the limit of infinite radii. By using the shear Reynolds number $Re_{S}$ and rotation number $R_{{\it\Omega}}$ as dimensionless parameters, the transition from TCF to RPCF can be studied continuously without singularities. Already for radius ratios ${\it\eta}\geqslant 0.9$ we find that the simulation results for various radius ratios and for RPCF collapse as a function of $R_{{\it\Omega}}$, indicating a turbulent behaviour common to both systems. We observe this agreement in the torque, mean momentum transport, mean profiles and turbulent fluctuations. Moreover, in TCF and RPCF for $R_{{\it\Omega}}>0$, the profiles in the central region are found to conform with inviscid neutral stability. Intermittent bursts, that have been observed in the outer boundary layer and have been linked to the formation of a torque maximum for counter-rotation, are shown to disappear as ${\it\eta}\rightarrow 1$. The corresponding torque maximum disappears as well. Instead, two new maxima of different origin appear for ${\it\eta}\geqslant 0.9$ and RPCF, a broad and a narrow one, in contrast to the results for smaller ${\it\eta}$. The broad maximum at $R_{{\it\Omega}}=0.2$ is connected with a strong vortical flow and can be reproduced by streamwise-invariant simulations. The narrow maximum at $R_{{\it\Omega}}=0.02$ only emerges with increasing $Re_{S}$ and is accompanied by an efficient and correlated momentum transport by the mean flow. Since the narrow maximum is of larger amplitude for $Re_{S}=2\times 10^{4}$, our simulations suggest that it will dominate at even higher $Re_{S}$.

A homogeneous mixture model is used to study cavitation over a circular cylinder at two different Reynolds numbers ($Re=200$ and 3900) and four different cavitation numbers (${\it\sigma}=2.0$, 1.0, 0.7 and 0.5). It is observed that the simulated cases fall into two different cavitation regimes: cyclic and transitional. Cavitation is seen to significantly influence the evolution of pressure, boundary layer and loads on the cylinder surface. The cavitated shear layer rolls up into vortices, which are then shed from the cylinder, similar to a single-phase flow. However, the Strouhal number corresponding to vortex shedding decreases as the flow cavitates, and vorticity dilatation is found to play an important role in this reduction. At lower cavitation numbers, the entire vapour cavity detaches from the cylinder, leaving the wake cavitation-free for a small period of time. This low-frequency cavity detachment is found to occur due to a propagating condensation front and is discussed in detail. The effect of initial void fraction is assessed. The speed of sound in the free stream is altered as a result and the associated changes in the wake characteristics are discussed in detail. Finally, a large-eddy simulation of cavitating flow at $Re=3900$ and ${\it\sigma}=1.0$ is studied and a higher mean cavity length is obtained when compared to the cavitating flow at $Re=200$ and ${\it\sigma}=1.0$. The wake characteristics are compared to the single-phase results at the same Reynolds number and it is observed that cavitation suppresses turbulence in the near wake and delays three-dimensional breakdown of the vortices.

A theoretical model is used to study wave energy attenuation and directional spreading of ocean wave spectra in the marginal ice zone (MIZ). The MIZ is constructed as an array of tens of thousands of compliant circular ice floes, with randomly selected positions and radii determined by an empirical floe size distribution. Linear potential flow and thin elastic plate theories model the coupled water–ice system. A new method is proposed to solve the time-harmonic multiple scattering problem under a multidirectional incident wave forcing with random phases. It provides a natural framework for tracking the evolution of the directional properties of a wave field through the MIZ. The attenuation and directional spreading are extracted from ensembles of the wave field with respect to realizations of the MIZ and incident forcing randomly generated from prescribed distributions. The averaging procedure is shown to converge rapidly so that only a small number of simulations need to be performed. Far-field approximations are investigated, allowing efficiency improvements with negligible loss of accuracy. A case study is conducted for a particular MIZ configuration. The observed exponential attenuation of wave energy through the MIZ is reproduced by the model, while the directional spread is found to grow linearly with distance. The directional spreading is shown to weaken when the wavelength becomes larger than the maximum floe size.