Koopman decomposition is a nonlinear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman-invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on a multi-task feature learning to extract the most informative Koopman-invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity-promoting extensions of EDMD/KDMD. Furthermore, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship-airwake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analysed, with an emphasis on characterizing transient dynamics. The results are compared with existing theoretical expositions and numerical approximations.

Cavitating flow in the wake of a wedge-shaped bluff body is examined to understand the role of the presence of high void-fraction regions in the near-wake region on the process of vortex formation and shedding. Previous studies have noted that developed cavitation forming in the wake of bluff bodies typically leads to an increase in the vortex shedding rate, peaking at a particular cavitation number. Further reduction in cavitation number leads to a return to lower shedding rates as the cavity grows into a super-cavity. The underlying flow processes that lead to this phenomenon are explored using traditional flow visualisation combined with time-resolved void-fraction flow fields based on X-ray densitometry. These measurements allow us to relate the compressibility of the near-wake bubbly flow to the underlying flow processes. Specifically, we use proper orthogonal decomposition (POD) of the void-fraction fields to show that the increased rate of vortex shedding is associated with a pulsating mode of the void-fraction flow field, compared with a sinusoidal variation corresponding to the lower void-fraction shedding processes similar to that of the non-cavitating wake. The pulsating mode becomes more pronounced when the wake void fraction increases with decreasing cavitation number, with the maximum shedding occurring near the point that the wake flow becomes locally supersonic. The important influence of flow compressibility on the wake dynamics is confirmed through the examination of the effect of non-condensable gas injection.

This paper examines two-dimensional liquid curtains ejected at an angle to the horizontal and affected by gravity and surface tension. The flow is, to leading order, shearless and viscosity, negligible. The Froude number is large, so that the radius of the curtain's curvature exceeds its thickness. The Weber number is close to unity, so that the forces of inertia and surface tension are almost perfectly balanced. An asymptotic equation is derived under these assumptions, and its steady solutions are explored. It is shown that, for a given pair of ejection velocity/angle, infinitely many solutions exist, each representing a steady curtain with a stationary capillary wave superposed on it. These solutions describe a rich variety of behaviours: in addition to arching downwards, curtains can zigzag downwards, self-intersect and even rise until the initial supply of the liquid's kinetic energy is used up. The last type of solutions corresponds to a separatrix between upward- and downward-bending curtains – in both cases, self-intersecting (such solutions are meaningful only until the first intersection, after which the liquid just splashes down). Finally, suggestions are made as to how the existence of upward-bending curtains can be tested experimentally.

Boussinesq and non-Boussinesq gravity currents produced from a finite volume of heavy fluid propagating into an environment of light ambient fluid on unbounded uniform slopes in the range $0\,^\circ \le \theta \le 12\,^\circ$ are reported. The relative density difference $\epsilon = (\rho _1-\rho _0)/\rho _0$ is varied in the range $0.05 \le \epsilon \le 0.15$ in this study, where $\rho _1$ and $\rho _0$ are the densities of the heavy and light ambient fluids, respectively. Our focus is on the influence of the relative density difference on the deceleration phase of the propagation. In the early deceleration phase, the front location history follows the power relationship ${(x_f+x_0)}^2 = {(K_I B)}^{1/2} (t+t_{I})$, where $(x_f+x_0)$ is the front location measured from the virtual origin, $K_I$ an experimental constant, $B$ the total buoyancy, $t$ the time and $t_I$ the $t$ intercept. The dimensionless constant $K_I$ is influenced by the slope angle and the relative density difference. In the late deceleration phase for the gravity currents on the steeper slopes in this study ($12\,^\circ$, $9\,^\circ$ and $6\,^\circ$), an ‘active’ head separates from the body of the current and the front location history follows the power relationship ${(x_f+x_0)}^{8/3} = K_{VS} {B}^{2/3} V^{2/9}_0 {\nu }^{-1/3} ({t+t_{VS}})$, where $K_{VS}$ is an experimental constant, $V_0$ the total volume of heavy fluid, $\nu$ the kinematic viscosity of fluid and $t_{VS}$ the $t$ intercept. The dimensionless constant $K_{VS}$ is shown to be influenced by the slope angle but not significantly influenced by the relative density difference. In the late deceleration phase for the gravity currents on the milder slopes in this study ($3\,^\circ$ and $0\,^\circ$), the gravity currents maintain an integrated shape without violent mixing with the ambient fluid and the front location history follows the power relationship ${(x_f+x_0)}^{4} = K_{VM} {B}^{2/3} V^{2/3}_0 {\nu }^{-1/3} ({t+t_{VM}})$, where $K_{VM}$ is an experimental constant and $t_{VM}$ the $t$ intercept. The dimensionless constant $K_{VM}$ is shown to be influenced by both the slope angle and the relative density difference. While the influence of the relative density difference on $K_{VM}$ is carried along for the gravity currents on the milder slopes in the late deceleration phase, the relative density difference interestingly has no significant influence on $K_{VS}$ for the gravity currents on the steeper slopes in the late deceleration phase. Our results suggest that the non-Boussinesq gravity currents on the milder slopes may remain non-Boussinesq ones in the late deceleration phase while the non-Boussinesq gravity currents on the steeper slopes may have become Boussinesq ones in the late deceleration phase.

Dry lakes covered with a salt crust organised into beautifully patterned networks of narrow ridges are common in arid regions. Here, we consider the initial instability and the ultimate fate of buoyancy-driven convection that could lead to such patterns. Specifically, we look at convection in a deep porous medium with a constant throughflow boundary condition on a horizontal surface, which resembles the situation found below an evaporating salt lake. The system is scaled to have only one free parameter, the Rayleigh number, which characterises the relative driving force for convection. We then solve the resulting linear stability problem for the onset of convection. Further exploring the nonlinear regime of this model with pseudo-spectral numerical methods, we demonstrate how the growth of small downwelling plumes is itself unstable to coarsening, as the system develops into a dynamic steady state. In this mature state we show how the typical speeds and length scales of the convective plumes scale with forcing conditions, and the Rayleigh number. Interestingly, a robust length scale emerges for the pattern wavelength, which is largely independent of the driving parameters. Finally, we introduce a spatially inhomogeneous boundary condition – a modulated evaporation rate – to mimic any feedback between a growing salt crust and the evaporation over the dry salt lake. We show how this boundary condition can introduce phase locking of the downwelling plumes below sites of low evaporation, such as at the ridges of salt polygons.

We present a series of experiments to explore the dynamics of particle-laden fountains rising through a stratified environment with zero buoyancy flux at the source. We find that the ratio $U$ between the particle sedimentation speed $V_s$ and the characteristic fountain velocity $(M_0N^{2})^{1/4}$, where $M_0$ is the initial momentum flux and $N$ the frequency of the ambient stratification, has a profound effect on the structure of the fountain and the dispersal of the particles. In a mono-disperse particle fountain, when the settling speed of the particles is small in comparison to the characteristic fountain speed ($U\ll 1$) the flow initially behaves in an analogous fashion to a single-phase fountain, forming an intrusion at a height of approximately 0.5 of the maximum fountain height. As the fluid–particle mixture spreads out, the particles gradually sediment to the tank floor. The intruding fluid subsequently rises and forms a new intrusion at its neutral buoyancy height. Some of the particles are carried up from the original intrusion with the rising fluid. This leads to the formation of a sedimenting column of particles with a characteristic radius. We observe a transition in the behaviour of the particle fountains in the vicinity of $U\sim 0.1$, with the particles now separating from the fluid near the top of the fountain. The separation of the particles leads to a reduction in the steady-state height of the particle-laden fountain, while the fluid in the fountain continues upwards until reaching its neutral buoyancy height and forming an intrusion above the fountain top. We compare the experimental data with two models of turbulent fountains to help rationalise the trends observed as a function of the dimensionless fall speed $U$. We briefly consider the dynamics of poly-disperse particle fountains and relate their dynamics to the regimes observed in their mono-disperse counterparts. We discuss the implications of this work for the dispersal of different sized particles from submarine volcanic eruptions.

The aeroacoustic source mechanism of a deep rectangular cavity, which has an aspect ratio of $D/L=2.632$ and is subjected to a turbulent boundary layer of $\theta /L=0.0345$ at a Mach number of 0.2, is investigated by using a high-order accurate large-eddy simulation. The primary aim of this study is to provide an improved understanding of the fluid–acoustic coupling mechanism that triggers a self-sustained acoustic resonance in a deep cavity. Various analysis methods, which include Doak's momentum potential theory that allows for the separation of hydrodynamic and acoustic components, are used to provide highly detailed investigations and findings. The vortex dynamics near the cavity opening region is investigated as the potential primary source of noise generation. In addition, the noise generation mechanism is quantitatively explained by the onset of the separation region near the downstream corner that ensues from the synchronised shear layer–wall interaction. The current work extensively focuses on the fluid–acoustic coupling mechanism, and it is found that the acoustic resonance favourably modulates the hydrodynamic fluctuation near the upstream corner of the cavity. Furthermore, the current study also suggests that nonlinear interactions between fundamental acoustic resonance and higher harmonics are plausible. Based on the discussions provided in this paper, a semi-empirical model to predict the critical free stream velocity at which a strong fluid–acoustic coupling occurs as a function of cavity geometry and inflow boundary-layer property is proposed.

The hydrodynamic response of an axisymmetric jet placed at various positions in a standing wave oriented normally to the jet is investigated. At the velocity and pressure nodes the axisymmetric ($m=0$) and first azimuthal ($m={\pm }1$) modes are excited, respectively, through manipulation of the jet exit boundary conditions. At positions between the nodes, both the $m=0$ and $m={\pm }1$ modes are simultaneously excited resulting in asymmetric forcing due to the phase difference between the transverse and longitudinal acoustic fluctuations. This leads to the asymmetric formation of vortices in the near field and bifurcation into two or more momentum streams further downstream. The dominant momentum stream is deflected in the direction of the velocity node. It is shown that the asymmetric response can be well approximated by a superposition of the boundary conditions at the pressure and velocity nodes where the contributions from each mode are proportional to the acoustic pressure and velocity. A method is proposed to characterize the bifurcation behaviour statistically via moments of the probability density functions constructed from profiles of streamwise momentum. The jet symmetry and momentum spreading are shown to be proportional to the magnitude of the transverse acoustic velocity. Finally, the streamwise velocity is reconstructed as a superposition of Gaussian profiles providing a robust method to characterize the number of individual momentum streams which also shows that each of the streams behave self-similarly.

Thermally driven vertical convection (VC) – the flow in a box heated on one side and cooled on the other side, is investigated using direct numerical simulations with Rayleigh numbers over the wide range of $10^7\le Ra\le 10^{14}$ and a fixed Prandtl number $Pr=10$ in a two-dimensional convection cell with unit aspect ratio. It is found that the dependence of the mean vertical centre temperature gradient $S$ on $Ra$ shows three different regimes: in regime I ($Ra \lesssim 5\times 10^{10}$), $S$ is almost independent of $Ra$; in the newly identified regime II ($5\times 10^{10} \lesssim Ra \lesssim 10^{13}$), $S$ first increases with increasing $Ra$ (regime $\textrm {{II}}_a$), reaches its maximum and then decreases again (regime $\textrm {{II}}_b$); and in regime III ($Ra\gtrsim 10^{13}$), $S$ again becomes only weakly dependent on $Ra$, being slightly smaller than in regime I. The transition from regime I to regime II is related to the onset of unsteady flows arising from the ejection of plumes from the sidewall boundary layers. The maximum of $S$ occurs when these plumes are ejected over approximately half of the area (downstream) of the sidewalls. The onset of regime III is characterized by the appearance of layered structures near the top and bottom horizontal walls. The flow in regime III is characterized by a well-mixed bulk region owing to continuous ejection of plumes over large fractions of the sidewalls, and, as a result of the efficient mixing, the mean temperature gradient in the centre $S$ is smaller than that of regime I. In the three different regimes, significantly different flow organizations are identified: in regime I and regime $\textrm {{II}}_a$, the location of the maximal horizontal velocity is close to the top and bottom walls; however, in regime $\textrm {{II}}_b$ and regime III, banded zonal flow structures develop and the maximal horizontal velocity now is in the bulk region. The different flow organizations in the three regimes are also reflected in the scaling exponents in the effective power law scalings $Nu\sim Ra^\beta$ and $Re\sim Ra^\gamma$. Here, $Nu$ is the Nusselt number and $Re$ is the Reynolds number based on maximal vertical velocity (averaged over vertical direction). In regime I, the fitted scaling exponents ($\beta \approx 0.26$ and $\gamma \approx 0.51$) are in excellent agreement with the theoretical predictions of $\beta =1/4$ and $\gamma =1/2$ for laminar VC (Shishkina, Phys. Rev. E., vol. 93, 2016, 051102). However, in regimes II and III, $\beta$ increases to a value close to 1/3 and $\gamma$ decreases to a value close to 4/9. The stronger $Ra$ dependence of $Nu$ is related to the ejection of plumes and the larger local heat flux at the walls. The mean kinetic dissipation rate also shows different scaling relations with $Ra$ in the different regimes.

Misaligned wind turbine rotors redirect the wake, and manipulate the wake shape by introducing a counter-rotating vortex pair. This mechanism is of great interest for improving wind farm power output through static or dynamic misalignment. In this study, cross-plane stereo-particle image velocimetry measurements are used to characterize the wake evolution for tilt misalignment and verify differences with yaw misalignment. Blockage from the ground, shear in the velocity profile, turbulence levels, hub-vortices and tip-vortices are found to strongly affect wake evolution for a tilted wind turbine resulting in a non-symmetric behaviour for upwards deflecting or downwards deflecting tilt. The downwards deflection of a negatively tilted wind turbine is found to result in the most benefits for wake recovery and power availability downstream through increased wake-curling, faster wake-recovery, and downdraft of high-momentum flow.

In this experimental study, multiscale rough surfaces with regular (cuboid) elements are used to examine the effects of roughness-scale hierarchy on turbulent boundary layers. Three iterations have been used with a first iteration of large-scale cuboids onto which subsequent smaller cuboids are uniformly added, with their size decreasing with a power-law as the number increases. The drag is directly measured through a floating-element drag balance, while particle image velocimetry allowed the assessment of the flow field. The drag measurements revealed the smallest roughness iteration can contribute to nearly 7 $\%$ of the overall drag of a full surface, while the intermediate iterations are responsible for over $12\,\%$ (at the highest Reynolds number tested). It is shown that the aerodynamic roughness length scale between subsequent iterations varies linearly, and can be described with a geometrical parameter proportional to the frontal solidity. Mean and turbulent statistics are evaluated using the drag information, and highlighted substantial changes within the canopy region as well as in the outer flow, with modifications to the inertial sublayer (ISL) and the wake region. These changes are shown to be caused by the presence of large-scale secondary motions in the cross-plane, which itself is believed to be a consequence of the largest multiscale roughness phase (spacing between largest cuboids), shown to be of the same order of magnitude as the boundary-layer thickness. Implications on the classical similarity laws are additionally discussed.

Compound particles are a class of composite systems in which solid particles encapsulated in a fluid droplet are suspended in another fluid. They are encountered in various natural and biological processes, for e.g. nucleated cells, hydrogels, microcapsules etc. Generation and transportation of such multiphase structures in microfluidic devices is associated with several challenges because of the poor understanding of their structural stability in a background flow and the rheological characteristics of their dispersions. Hence, in this work, we analyse the flow in and around a concentric compound particle and investigate the deformation dynamics of the confining drop and its stability against breakup in imposed linear flows. In the inertia-less limit (Reynolds number, $Re \ll 1$) and assuming that the surface tension force dominates the viscous forces (low capillary number, $Ca$, limit), we obtain analytical expressions for the velocity and pressure fields up to ${O}(Ca)$ for a compound particle subjected to a linear flow using a domain perturbation technique. Simultaneously, we determine the deformed shape of the confining drop correct up to ${O}(Ca^2)$, facilitating the following. (i) Since ${O}(Ca^2)$ calculations account for the rotation of the anisotropically deformed interface, the reorientation dynamics of the deformed compound particles is determined. (ii) Calculations involving the ${O}(Ca^2)$ shape of the confining interface are found to be important for compound particles as ${O}(Ca)$ calculations make qualitatively different predictions in generalised extensional flows. (iii) An ${O}(Ca)$ constitutive equation for the volume-averaged stress for a dilute dispersion of compound particles was developed to study both shear and extensional rheology in a unified framework. Our analysis shows that the presence of an encapsulated particle always enhances all the measured rheological quantities such as the effective shear viscosity, extensional viscosity and normal stress differences. (iv) Moreover, linear viscoelastic behaviour of a dilute dispersion of compound particles is characterised in terms of complex modulus by subjecting the dilute dispersion to a small-amplitude oscillatory shear (SAOS) flow. (v) Various expressions pertaining to a suspension of particles, drops, and particles coated with a fluid film are also derived as limiting cases of compound particles.

The natural frequencies for sloshing without coupling with lateral tank motions differ from the natural sloshing frequencies with coupling. A consequence is that the nonlinear multimodal sloshing theory for prescribed tank motion should be revised when studying the liquid sloshing dynamics in connection with marine structures in ocean waves. The needed revisions are done for a rectangular rigid tank with finite liquid depth. Steady-state resonant solutions of the constructed nonlinear modal equations are derived to analytically describe the coupled resonant sloshing and sway of a floating rigid body in regular incident deep water waves with two-dimensional flow conditions at the lowest coupled sloshing–sway natural frequency. The steady-state theoretical results are validated by comparing them with the model tests by Rognebakke & Faltinsen (J. Ship Res., vol. 47, issue 3, 2003, pp. 208–221). The occurrence of an instability frequency range is theoretically justified.

Particles of differing sizes are notoriously prone to segregation in shear driven flows under the action of gravity. This has important implications in many industrial processes, where particle-size segregation can lead to flow problems and reduced product quality, as well as longer product development and start-up times. Particle-size segregation also readily occurs in many hazardous geophysical mass flows (such as snow avalanches, debris flows and volcanic pyroclastic flows) and can lead to the formation of destructive bouldery flow fronts and significantly longer runouts. Although general theories exist to model particle-size segregation, the detailed functional dependence of the segregation flux on the shear rate, gravity, pressure, particle concentration, grain size and grain-size ratio is still not known. This paper describes refractive-index matched oscillatory shear-cell experiments that shed light on the segregation velocity in the two extreme cases of (i) a single large intruder rising up through a matrix of smaller grains, and (ii) a single small intruder percolating down through a matrix of large particles. Despite the sometimes markedly different time scales for segregation in these two situations, a unifying scaling law has been found that is able to collapse all the experimental data over a wide range of shear rates and grain-size ratios in the range $[1.17,4.17]$. The resulting functional form is easily generalizable to intermediate concentrations and can quantitatively capture laboratory experiments and numerical simulations with a $50\ {:}\ 50$ mix of large and small grains.

The evolution of the interfacial instabilities on a smooth and laminar liquid sheet formed by the oblique impingement of two liquid jets is investigated in the presence of external acoustic forcing. The forced liquid sheet exhibits different instability patterns, such as flapping flag, sinuous breakup and vibrating membrane structure, depending on the forcing frequency. The transition frequencies, where the instability pattern changes, are obtained using the linear theory of sheet instability and are shown to be increasing functions of the jet Weber number. These different instability patterns show distinct growth rate behaviours, which cannot be predicted using a unique theory such as the conventional linear theory of sheet breakup. The dynamics of liquid sheet breakup in the presence of forcing is further assessed with the help of established instability theories. Our measurements show that the sheet thickness distribution plays a major role in dictating the relative influence of different mechanisms on the instability characteristics. All the results are finally consolidated to arrive at the instability regime map on the Weber number versus frequency ($We$–$\tilde {\omega }$) space to serve as a guideline for adopting a suitable modelling approach.

Formation of a microlayer in nucleate boiling results in substantial changes to the heat-transfer dynamics of the process, and establishing a criterion to distinguish between the microlayer and contact-line regimes encountered is necessary to obtain a quantitative description of the nucleate-boiling phenomenon. In this work, we describe microlayer formation as a dewetting transition in the presence of phase change. We derive a specific criterion for modelling the transition based on a synthesis of existing theoretical, experimental and numerical data. As a result, a new formulation of the transition criterion is developed, and applied to (i) reference data from a dewetting experiment of a volatile liquid, and (ii) results derived from a high-resolution direct numerical simulation of nucleate boiling. In both cases, very good agreement is found for Jakob numbers $\lesssim$75, a range which covers many important boiling situations.

Spontaneous motion of liquid droplets can occur on hydrophobic, micro-structured, solid surfaces comprising a structural gradient. In this study, we examine such motion experimentally and explain our observations by invoking variable droplet–surface interactions (both actuation and resistance forces) arising from the structural gradient. The oscillatory motion of the droplet constitutes an integral aspect of the behaviour and this is incorporated into the overall modelling. The theoretical model features a truncated spheroid for the drop shape (flattened in the region of solid contact) coupled with the oscillatory and alternate leading and trailing motion of the contact line. Results from the model and experiments provide good qualitative and quantitative agreement. The component of the vertical oscillation is found to help overcome wetting hysteresis and actuate the motion, this being a key element for the completeness of the model.

When two converging flows of unequal velocities and different flow directions come into contact, large-scale coherent structures are generated. For shallow conditions and small angles between the incoming streams, the mixing layer (ML) forming downstream of the confluence apex contains quasi-two-dimensional, Kelvin–Helmholtz (KH) vortices. Due to flow shallowness (e.g. stabilizing effect of bed friction), these vortices gradually lose their coherence at large distances from the ML origin. For large angles between the incoming streams, the spatial development of the shallow ML is more complex as strongly coherent, streamwise-oriented-vortical (SOV) cells form in the vicinity of the shallow ML and helical cells of secondary flow are generated due to curvature effects. The present paper uses three-dimensional eddy-resolving numerical simulations to study the effects of varying the angles between the two incoming channels and the downstream channel, the velocity ratio (VR) of the incoming flows and the flow depth on flow, turbulence structure and sediment entrainment mechanisms inside the ML and its surroundings. The simulations are performed for highly idealized conditions in which the ML develops in a wide channel (no interactions with the channel banks), over a flat bed and the flow depth is constant. Simulation results show that the SOV cells play an important role in the redistribution of the streamwise momentum. Some of the SOV cells are subject to bimodal oscillations in the lateral direction which induce strong interactions between the SOV cells and the ML vortices and sharply increase mixing. As for the case of a shallow ML developing between parallel streams, vortex pairing ceases some distance from the ML origin, the KH vortices start losing their coherence and the ML assumes an undulatory shape. The paper describes the effects of VR, angle between the incoming streams (α = 0° and 60°), planform geometry (symmetric vs. asymmetric confluences with α = 60°) and flow shallowness on the transverse shift of the ML centreline, ML width, dynamics of the KH vortices and on the formation, position and circulation of the SOV cells. In most of the α = 60° cases, the largest bed shear stresses are induced beneath the SOV cells rather than beneath the high-speed stream and the SOV cells play a major role in enhancing mixing between the two streams.

We study the reactive displacement of two miscible fluids in channel flows and establish a quantitative link between fluid stretching and chemical reactivity. At the mixing interface, the two fluids react according to the instantaneous irreversible bimolecular reaction $A + B \rightarrow C$. We simulate the advection–diffusion–reaction problem using a random walk based reactive particle method that is free of numerical dispersion. The relative contributions of stretching and diffusion to mixing-limited reaction is controlled by changing the Péclet number, and the channel roughness is also systematically varied. We observe optimal ranges of fluid stretching that maximize reactivity, which are captured by a Lagrangian stretching measure based on an effective time period that honours the stretching history. We show that the optimality originates from the competition between the enhanced mixing by fluid stretching and the mass depletion of the reactants. We analytically derive the spatial distribution of reaction products using a lamellar formulation and successfully predict the optimal ranges of fluid stretching, which are consistent across different levels of channel roughness. These findings provide a mechanistic understanding of how the interplay between fluid stretching, diffusion and channel roughness controls mixing-limited reactions in rough channel flows, and show how reaction hot spots can be predicted from the concept of optimal fluid stretching.

Direct numerical simulations are performed to investigate a stratified shear layer at high Reynolds number ($Re$) in a study where the Richardson number ($Ri$) is varied among cases. Unlike previous work on a two-layer configuration in which the shear layer resides between two layers with constant density, an unbounded fluid with uniform stratification is considered here. The evolution of the shear layer includes a primary Kelvin–Helmholtz shear instability followed by a wide range of secondary shear and convective instabilities, similar to the two-layer configuration. During transition to turbulence, the shear layers at low $Ri$ exhibit a period of thickness contraction (not observed at lower $Re$) when the momentum and buoyancy fluxes are counter-gradient. The behaviour in the turbulent regime is significantly different from the case with a two-layer density profile. The transition layers, which are zones with elevated shear and stratification that form at the shear-layer edges, are stronger and also able to support a significant internal wave flux. After the shear layer becomes turbulent, mixing in the transition layers is shown to be more efficient than that which develops in the centre of the shear layer. Overall, the cumulative mixing efficiency ($E^C$) is larger than the often assumed value of 1/6. Also, $E^C$ is found to be smaller than that in the two-layer configuration at moderate Ri. It is relatively less sensitive to background stratification, exhibiting little variation for $0.08 \leqslant Ri \leqslant 0.2$. The dependence of mixing efficiency on buoyancy Reynolds number during the turbulence phase is qualitatively similar to homogeneous sheared turbulence.