The concept of statistical stability is central to Malkus's 1956 attempt to predict the mean profile in shear flow turbulence. Here we discuss how his original attempt to assess this – an Orr–Sommerfeld (OS) analysis on the mean profile – can be improved by considering a cumulant expansion of the Navier–Stokes equations. Focusing on the simplest non-trivial closure (commonly referred to as CE2) that corresponds to the quasilinearized Navier–Stokes equations, we develop an extended OS analysis that also incorporates information about the fluctuation field. A more practical version of this – minimally extended OS analysis – is identified and tested on a number of statistically steady and, therefore, statistically stable turbulent channel flows. Beyond the concept of statistical stability, this extended stability analysis should also improve the popular approach of mean flow linear analysis in time-dependent shear flows by including more information about the underlying flow in its predictions as well as for other flows with additional physics such as convection.

Based on a generalized local Kolmogorov–Hill equation expressing the evolution of kinetic energy integrated over spheres of size $\ell$ in the inertial range of fluid turbulence, we examine a possible definition of entropy and entropy generation for turbulence. Its measurement from direct numerical simulations in isotropic turbulence leads to confirmation of the validity of the fluctuation relation (FR) from non-equilibrium thermodynamics in the inertial range of turbulent flows. Specifically, the ratio of probability densities of forward and inverse cascade at scale $\ell$ is shown to follow exponential behaviour with the entropy generation rate if the latter is defined by including an appropriately defined notion of ‘temperature of turbulence’ proportional to the kinetic energy at scale $\ell$.

In this paper, we study the spreading of droplets of density-matched granular suspensions on the surface of a solid. Bidispersity of the particle size distribution enriches the conclusions drawn from monodisperse experiments by highlighting key elements of the wetting dynamics. In all cases, the relation between the dynamic contact angle and the velocity of the contact line is similar to that for a simple fluid, despite the complexity introduced by the presence of particles. We extract from this relation an apparent wetting viscosity of the suspensions that differs from that measured in the bulk. Dimensional analysis supported by experimental measurements yields an estimate of the size of the region inside the droplet where the value of the dynamic contact angle depends on a balance of viscous dissipation and capillary stresses. How particle size compares with this viscous cut-off length seems crucial in determining the value of the apparent wetting viscosity. With bimodal blends, the particle size ratio can be used to show the effects of the local structure and volume fraction at the contact line, both impacting the value of the corresponding wetting viscosity.

We study the effect of insoluble surfactants on the impact of surfactant-free droplets onto surfactant-laden thin liquid films via a fully three-dimensional direct numerical simulation approach that employs a hybrid interface-tracking/level-set method, and by taking into account surfactant-induced Marangoni stresses due to gradients in interfacial surfactant concentration. Our numerical predictions for the temporal evolution of the surfactant-free crown are validated against the experimental work by Che & Matar (Langmuir, vol. 33, 2017, pp. 12140–12148). We focus on the ‘crown-splash regime’, and we observe that the crown dynamics evolves through various stages: from the growth of linear modes (through a Rayleigh–Plateau instability) to the development of nonlinearities leading to primary and secondary breakup events (through droplet shedding modulated by an end-pinching mechanism). We show that the addition of surfactants does not affect the wave selection via the Rayleigh–Plateau instability. However, the presence of surfactants plays a key role in the late stages of the dynamics as soon as the ligaments are driven out from the rim. Surfactant-induced Marangoni stresses delay the end-pinching mechanisms to result in longer ligaments prior to their capillary singularity. Our results indicate that Marangoni stresses bridge the gap between adjacent protrusions promoting the adjacent protrusions' collision and the merging of ligaments. Finally, we demonstrate that the addition of surfactants leads to surface rigidification and consequently to the retardation of the flow dynamics.

Density segregation of multi-component granular mixtures in a dense, gravity-driven flow over a rough and bumpy periodic chute surface is studied using theory and simulations. An existing theoretical model for predicting the steady-state concentration field of each species in a binary mixture using the forces acting on the particles is generalised for multi-component mixtures in this work. In addition, the rheological model for binary mixtures is also extended to multi-component mixtures. In contrast to the percolation velocity-based empirical segregation models that do not account for the rheology and need prior knowledge of the velocity field, the present approach accounts for the inter-coupling of rheology with segregation. The momentum balance equations are solved, along with the mixture rheological model as well as the multi-component density segregation model, to obtain concentration fields using an iterative numerical method. The theoretical predictions are compared with discrete element method (DEM) simulations for ternary, quaternary and quinary granular mixtures differing in density. The steady-state profiles for the concentration of different species as well as other flow properties predicted from the theory are in excellent agreement with the DEM simulation results for a variety of compositions over a range of inclination angles for different density ratios. Through this work, we take the first essential step towards proposing a generalised particle force-based segregation theory for multi-component mixtures differing in size and/or density.

The statistics of breaking wave fields are characterised within a novel multi-layer framework, which generalises the single-layer Saint-Venant system into a multi-layer and non-hydrostatic formulation of the Navier–Stokes equations. We simulate an ensemble of phase-resolved surface wave fields in physical space, where strong nonlinearities, including directional wave breaking and the subsequent highly rotational flow motion, are modelled, without surface overturning. We extract the kinematics of wave breaking by identifying breaking fronts and their speed, for freely evolving wave fields initialised with typical wind wave spectra. The $\varLambda (c)$ distribution, defined as the length of breaking fronts (per unit area) moving with speed $c$ to $c+{\rm d}c$ following Phillips (J. Fluid Mech., vol. 156, 1985, pp. 505–531), is reported for a broad range of conditions. We recover the $\varLambda (c) \propto c^{-6}$ scaling without wind forcing for sufficiently steep wave fields. A scaling of $\varLambda (c)$ based solely on the root-mean-square slope and peak wave phase speed is shown to describe the modelled breaking distributions well. The modelled breaking distributions are in good agreement with field measurements and the proposed scaling can be applied successfully to the observational data sets. The present work paves the way for simulations of the turbulent upper ocean directly coupled to a realistic breaking wave dynamics, including Langmuir turbulence, and other sub-mesoscale processes.

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of oblique, spatially intermittent turbulent structures. In plane Couette flow, these emerge from uniform turbulence via a spatio-temporal intermittent process in which localised quasi-laminar gaps randomly nucleate and disappear. For slightly lower Reynolds numbers, spatially periodic and approximately stationary turbulent–laminar patterns predominate. The statistics of quasi-laminar regions, including the distributions of space and time scales and their Reynolds-number dependence, are analysed. A smooth, but marked transition is observed between uniform turbulence and flow with intermittent quasi-laminar gaps, whereas the transition from gaps to regular patterns is more gradual. Wavelength selection in these patterns is analysed via numerical simulations in oblique domains of various sizes. Via lifetime measurements in minimal domains, and a wavelet-based analysis of wavelength predominance in a large domain, we quantify the existence and nonlinear stability of a pattern as a function of wavelength and Reynolds number. We report that the preferred wavelength maximises the energy and dissipation of the large-scale flow along laminar–turbulent interfaces. This optimal behaviour is due primarily to the advective nature of the large-scale flow, with turbulent fluctuations playing only a secondary role.

We derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier–Stokes turbulence and show that uncertainty is increased by strain rate compression and decreased by strain rate stretching. We use three different direct numerical simulations (DNS) of non-decaying periodic turbulence and identify a similarity regime where (a) the production and dissipation rates of uncertainty grow together in time, (b) the parts of the uncertainty production rate accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (c) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy, (d) the uncertainty energy spectrum's evolution is self-similar if normalised by the uncertainty's average uncertainty energy and characteristic length and (e) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (a), (b) and (c) imply that the average uncertainty energy grows exponentially in this similarity time range. The Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on random large-scale sweeping of dissipative eddies. In the two DNS cases of statistically stationary turbulence, this exponential growth is followed by an exponential of exponential growth, which is, in turn, followed by a linear growth in the one DNS case where the Navier–Stokes forcing also produces uncertainty.

The root systems of mangroves, a tree species found in intertidal tropical and subtropical coastal zones, provide a natural barrier that dissipates wave energy effectively and reduces sediment erosion. Here, we use a combination of experiments and numerical simulations to examine the wake and drag characteristics of porous arrays of cylinders, which serve as simplified models of mangrove root networks. Optimal arrangements of the arrays are obtained by coupling Navier–Stokes simulations with a multi-objective optimization algorithm, which seeks configurations that minimize wake enstrophy and maximize drag on the porous structure. These optimal configurations are investigated using particle image velocimetry, and the internal and external flows around the porous arrays are analysed using a combination of proper orthogonal decomposition and Lagrangian particle tracking. Large variations in drag and enstrophy are observed by varying the relative positions of the cylinders, which indicates that the geometrical arrangement of porous arrays plays a prominent role in determining wake and drag characteristics. A sensitivity analysis suggests that enstrophy is more sensitive than drag to specific cylinder placement, and depends on distinctive flow patterns that develop in the interior due to interactions among neighbouring cylinders. Arrays with higher drag involve a combination of larger projected frontal area and minimal flux through the interior, leading to increased wake enstrophy, which is unfavourable for particle deposition and erosion. Based on the analysis of characteristics associated with the optimal arrays, several manually designed arrays are tested, and they display the expected behaviour with regard to drag and enstrophy.

It is known that inertial lift forces can lead to particle focusing in channel flows; yet oscillatory straining effects have also been suggested as a mechanism for particle focusing in wavy channels. To explore the synergy between these two mechanisms, we analytically and experimentally investigate the focusing behaviour of rigid neutrally buoyant particles in a wavy channel. We decompose the particle-free channel flow into a primary Poiseuille flow and secondary eddies induced by the waviness. We calculate the perturbation of the particle on the particle-free flow and the resulting lateral lift force exerted on the particle using the method of matched asymptotic expansions. This yields a zeroth-order lift force arising from the Poiseuille flow and a first-order lift force due to the waviness of the channel. We further incorporate the inertial lift force into the Maxey–Riley equation and simulate particle trajectories in wavy channels. The interactions between the zeroth-order lift force and the particle-free flow largely determine the focusing locations. Experiments in wavy channels with varying amplitudes at channel Reynolds numbers ranging from 5 to 250 are consistent with the predictions of the focusing locations, which are mainly governed by the channel Reynolds number as well as the competition between the inertial lift and the oscillatory straining effects.

Advective dispersion of solutes in long thin axisymmetric channels is important to the analysis and design of a wide range of devices, including chemical separation systems and microfluidic chips. Despite extensive analysis of Taylor dispersion in various scenarios, most studies focus on long-term dispersion behaviour and cannot capture the transient evolution of the solute zone across the spatial variations in the channel. In the current study, we analyse the Taylor–Aris dispersion for arbitrarily shaped axisymmetric channels. We derive an expression for solute dynamics in terms of two coupled ordinary differential equations, which allow prediction of the time evolution of the mean location and axial (standard deviation) width of the solute zone as a function of the channel geometry. We compare and benchmark our predictions with Brownian dynamics simulations for a variety of cases, including linearly expanding/converging channels and periodic channels. We also present an analytical description of the physical regimes of transient positive versus negative axial growth of solute width. Finally, to further demonstrate the utility of the analysis, we demonstrate a method to engineer channel geometries to achieve desired solute width distributions over space and time. We apply the latter analysis to generate a geometry that results in a constant axial width and a second geometry that results in a sinusoidal axial variance in space.