Glycerol is a hygroscopic fluid that spontaneously absorbs water vapour from the atmosphere. For applications involving glycerol, care must be taken to avoid exposure to humidity, since its viscosity decreases quickly as water is absorbed. We report experimental measurements of the viscosity of glycerol in a parallel-plate rheometer where the outer interface is exposed to atmosphere. The measurements decrease with time as water is absorbed from the atmosphere and transported throughout the glycerol via diffusion and advection. Measured viscosities drop faster at higher relative humidities, confirming the role of hygroscopicity in the transient viscosities. The rate of viscosity decrease shows a non-monotonic relationship with the rheometer gap height. This behaviour is explained by considering the transition from diffusion-dominated transport in the narrow-gap regime to the large-gap regime where transport is dominated by inertia-driven secondary flows. Numerical simulations of the water absorption and transport confirm this non-monotonic behaviour. The experimental viscosity measurements show unexpectedly fast decreases at very small gap heights, violating the parallel-plate, axisymmetric model. We propose that this drop-off may be due to misalignment in the rheometer that becomes non-negligible for small gaps. Theoretical considerations show that secondary flows in a misaligned rheometer dominate the typical secondary inertial flows in parallel-plate rheometers at small gaps. Finally, simulations in a misaligned parallel-plate system demonstrate the same sharp drop-off in viscosity measurements at small gap heights. This modelling can be used to estimate the gap height where misalignment effects dominate the transient glycerol viscosity measurements.

Understanding the transport of particles immersed in a carrier fluid (bedload transport) is still an exciting challenge. Among the different types of gas–solid flows, when the dynamics of solid particles is essentially dominated by collisions between them, kinetic theory can be considered as a reliable tool to derive continuum approaches from a fundamental point of view. In a recent paper, Chassagne et al. (J. Fluid Mech., vol. 964, 2023, A27) proposed a two-fluid model based on modifications to a classical kinetic theory model (Garzó & Dufty, Phys. Rev. E, vol. 59, 1999, pp. 5895–5911). First, in contrast to the classical model, the model proposed by Chassagne et al. takes into account the interparticle friction not only in the radial distribution function but also through an effective restitution coefficient in the rate of dissipation term of granular temperature. As a second modification, at the top of the bed where the volume fraction is quite small, the model accounts for the saltation regime in the continuum framework. The theoretical results derived from the model agree with discrete simulations for moderate and high densities and they are also consistent with experiments. Thus, the model proposed by Chassagne et al. (J. Fluid Mech., vol. 964, 2023, A27) helps provide a better understanding of the combined impact of friction and inelasticity on the macroscopic properties of granular flows.

We derive explicit formulae for the mean profiles of passive scalars (either temperature or concentration of a diffusing substance), and their respective wall fluxes (either heat or mass fluxes), in forced turbulent convection, as a function of the Reynolds and Prandtl numbers. Direct numerical simulation data for turbulent flow within a smooth straight pipe of circular cross-section, at friction Reynolds number ${{Re}}_{\tau }=1140$, in the range of Prandtl numbers from ${{Pr}}=0.00625$ to ${{Pr}}=16$, are used to infer the proper analytical form of the eddy diffusivity. This is leveraged to derive accurate predictive formulae for the mean passive scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling laws result for the thickness of the conductive (diffusive) layer, and for the Nusselt number, which significantly extend the predictive envelope of classical formulae.

Six consecutive solitary waves with identical wave height and separation time are generated to study the flow structures during the uprush–downwash interactions in the swash zone. Using particle image velocimetry, the cross-shore velocity fields are captured. Two different wave conditions are examined with different wave-height-to-water-depth ratios, i.e. $H_o/h=0.11$ and 0.22. The uprush–downwash interaction reaches quasi-steady state from the third solitary wave for both cases. For the former case, a weak non-stationary hydraulic jump appears during the downwash flow for all the six consecutive waves. The weak hydraulic jump evolves into a momentarily ‘stationary’ broken bore when the next wave arrives. For the latter case, the larger wave height generates stronger wave breaking. No non-stationary hydraulic jump is observed as the duration of downwash flow is relatively short. The flow reverses to the onshore direction before the downwash Froude number reaches the hydraulic jump condition. The temporal and spatial evolution of turbulence structure at the quasi-steady state is quantified using the spatial spectral analysis, the integral length scale and turbulence eddy viscosity. The results suggest that the large-scale energy generated during the uprush–downwash interaction modified the slope of the turbulence energy spatial spectrum in the inertial subrange from $-$5/3 to $-$1 in the larger length scale region, indicating the energy cascade depends not only on the dissipation rate, but also on the turbulent kinetic energy from the large-scale turbulence structure because of the large-scale energy injection in the inertial subrange.

The reshocked Richtmyer–Meshkov instability (RMI) is examined in three different configurations via shock-tube experiments: RMI at a single-mode interface with a planar reshock (configuration I); RMI at a flat interface with a sinusoidal reshock (configuration II); RMI at a single-mode interface with a sinusoidal reshock (configuration III). The sinusoidal reshock is created by an incident shock reflecting off a sine-shaped wall surface. For all three configurations, the initial conditions of the experiment are specially set such that the interface evolution is at the linear stage when the reshock arrives. It is found that the amplitude of the reshocked interface increases linearly with time for all three configurations. For configuration I, the post-reshock perturbation growth depends heavily on the pre-reshock amplitude and growth rate, which can be predicted by a modified Mikaelian model (Phys. Rev. A, vol. 31, 1985, pp. 410–419). For configuration II, velocity perturbation associated with the non-uniform rippled reshock plays an important role in the instability growth. For configuration III, the post-reshock instability growth is much quicker (lower) than in configuration I when the sinusoidal reshock is in phase (out of phase) with the interface. A major reason is that for the in-phase (anti-phase) case, the velocity perturbation gives rise to an instability growth with an identical (opposite) direction to the pressure perturbation. A linear theory is developed that takes velocity perturbation, pressure perturbation and pre-reshock growth rate into account, which gives a reasonable prediction of the growth of the reshocked RMI in configurations II and III.

We consider high Reynolds number supersonic flow over a compression ramp in the triple-deck formulation. Previous studies of compression-ramp stability have shown rapid growth of high-frequency disturbances in initial-value computations; however, no physical or numerical origin has yet been identified robustly. By considering linear perturbations to steady compression-ramp solutions, we show that instabilities observed in previous studies do not have a growth rate that is described by the integral eigenrelation of Tutty & Cowley (J. Fluid Mech., vol. 168, 1986, pp. 431–456) for a (long-wave) Rayleigh instability. We solve both the temporal and spatial instability problems in the limit of asymptotically large wavenumber $K$ (or equivalently frequency) and show that the growth rate of the instability remains $o(K)$, being dominated by higher-order terms in the expansion at moderate ramp angles.

Engineering flow systems operating under low pressures and/or at the micro/nano scale generally include a physically adsorbed gas layer next to the surface. In this paper, we develop a scattering kernel that accounts for the effect of adsorption, arising from van der Waals interactions, on the dynamics of molecules impinging on solid smooth surfaces. In the limit of low bulk density, surface adsorption becomes negligible and the scattering kernel recovers consistently the Cercignani–Lampis model, which best describes molecular collisions with a clean, smooth surface. In the limit of high bulk density, a dense adsorbed molecular layer forms next to the surface and its presence is picked up by the Maxwell model with complete diffuse reflection, which better captures the multiple collisions suffered by molecules. A weight coefficient based on the Langmuir adsorption isotherm is incorporated into the modelling to handle the transition between these two limiting conditions of low and high densities. The proposed model is validated against high-fidelity molecular dynamics simulations that are performed for a variety of gas–surface combinations and adsorbed molecular layers with different densities. It is shown that the proposed model very well captures the scattering patterns of beams of gas molecules at different velocities impinging on surfaces, as well as momentum and energy accommodation coefficients in the entire range of explored conditions.

We present measurements of turbulent drag reduction (DR) in boundary layers at high friction Reynolds numbers in the range of $4500 \le Re_\tau \le 15\ 000$. The efficacy of the approach, using streamwise travelling waves of spanwise wall oscillations, is studied for two actuation regimes: (i) inner-scaled actuation (ISA), as investigated in Part 1 of this study, which targets the relatively high-frequency structures of the near-wall cycle, and (ii) outer-scaled actuation (OSA), which was recently presented by Marusic et al. (Nat. Commun., vol. 12, 2021) for high-$Re_\tau$ flows, targeting the lower-frequency, outer-scale motions. Multiple experimental techniques were used, including a floating-element balance to directly measure the skin-friction drag force, hot-wire anemometry to acquire long-time fluctuating velocity and wall-shear stress, and stereoscopic particle image velocimetry to measure the turbulence statistics of all three velocity components across the boundary layer. Under the ISA pathway, DR of up to 25 % was achieved, but mostly with net power saving (NPS) losses due to the high-input power cost associated with the high-frequency actuation. The low-frequency OSA pathway, however, with its lower input power requirements, was found to consistently result in positive NPS of 5–10 % for moderate DRs of 5–15 %. The results suggest that OSA is an attractive pathway for energy-efficient DR in high-Reynolds-number applications.

Turbulent drag reduction (DR) through streamwise travelling waves of the spanwise wall oscillation is investigated over a wide range of Reynolds numbers. Here, in Part 1, wall-resolved large-eddy simulations in a channel flow are conducted to examine how the frequency and wavenumber of the travelling wave influence the DR at friction Reynolds numbers $Re_\tau = 951$ and $4000$. The actuation parameter space is restricted to the inner-scaled actuation (ISA) pathway, where DR is achieved through direct attenuation of the near-wall scales. The level of turbulence attenuation, hence DR, is found to change with the near-wall Stokes layer protrusion height $\ell _{0.01}$. A range of frequencies is identified where the Stokes layer attenuates turbulence, lifting up the cycle of turbulence generation and thickening the viscous sublayer; in this range, the DR increases as $\ell _{0.01}$ increases up to $30$ viscous units. Outside this range, the strong Stokes shear strain enhances near-wall turbulence generation leading to a drop in DR with increasing $\ell _{0.01}$. We further find that, within our parameter and Reynolds number space, the ISA pathway has a power cost that always exceeds any DR savings. This motivates the study of the outer-scaled actuation pathway in Part 2, where DR is achieved through actuating the outer-scaled motions.

The effect of surface vibrations on the pressure-gradient-driven flows in channels has been studied. The analysis considered monochromatic waves and laminar flows. The effectiveness of the vibrations was gauged by determining the pressure gradient correction required to maintain the same flow rate as without vibrations. Waves propagating upstream always increase pressure losses. Flow response to waves propagating downstream is more complex and changes as a function of the flow Reynolds number. Such waves reduce losses if the Reynolds number $Re <\ \sim\!\!100$, but these waves must be sufficiently fast to reduce pressure losses for larger Re values. In general, the supercritical waves, i.e. waves faster than the reference flow, reduce pressure losses with the magnitude of reduction increasing monotonically with the wave phase speed and wavenumber. The need for an external pressure gradient is eliminated if sufficiently short and fast waves are used. Generally, the subcritical waves, i.e. waves with velocities similar to the reference flow, increase pressure losses. This increase changes somewhat irregularly as a function of the wave phase speed and wavenumber forming local maxima and minima. These waves can reduce pressure losses only if the Reynolds number becomes large enough. It is shown that subcritical waves with very small amplitudes but matching the natural flow frequencies produce significant pressure losses.

This study aims at establishing a model for close-contact melting (CCM) of shear-thinning fluids. We presented a theoretical framework for predicting the variation of liquid melt film thickness and motion of unmelted solid for both Carreau and power-law fluids. We identified the appropriate energy equation considering the convective effect and derived an analytical temperature profile across the liquid film. Using the lubrication approximation, force equilibrium relationships and the corresponding numerical approaches were built. By using laser interferometry and photographic recording methods, we found excellent agreement between numerical solutions and experimental results for Carreau liquids, revealing that the convective effect weakens heat transfer and melting rate. We identified the critical liquid film thickness that determines three situations of CCM in the theoretical model for Carreau fluids. Numerical prediction demonstrated that the CCM of Carreau fluids can be almost equivalent to that of power-law fluids if the initial film thickness is greater than the critical value. Finally, approximate analytical models were developed for both Carreau and power-law models. For the applicability of the approximate analytical solutions, we derived two- and three-dimensional dimensionless phase diagrams of validity range and identified a key dimensionless group $(\varLambda Re)^{4/3}{Re}\left [3\ln (Ste+1)\right ]^{1/3}{Pe}^{-1/3}$, where $\varLambda$ is dimensionless characteristic time, Re is Reynolds number, Ste is Stefan number and Pe is Peclect number. The reliability of the approximate solutions was verified by comparing with the numerical results. These approximate solutions enable convenient and low-cost computational prediction of the dynamic CCM process of shear-thinning fluids.

The Perron–Frobenius operator (PFO) is adapted from dynamical-system theory to the study of turbulent channel flow. It is shown that, as long as the analysis is restricted to the system attractor, the PFO can be used to differentiate causality and coherence from simple correlation without performing interventional experiments, and that the key difficulty remains the collection of enough data to populate the operator matrix. This is alleviated by limiting the analysis to two-dimensional projections of the phase space, and developing a series of indicators to choose the best parameter pairs from a large number of possibilities. The techniques thus developed are applied to the study of bursting in the inertial layer of the channel, with emphasis on the process by which bursts are reinitiated after they have decayed. Conditional averaging over phase-space trajectories suggested by the PFO shows, somewhat counter-intuitively, that a key ingredient for the burst recovery is the development of a low-shear region near the wall, overlaid by a lifted shear layer. This is confirmed by a computational experiment in which the control of the mean velocity profile by the turbulence fluctuations is artificially relaxed. The behaviour of the mean velocity profile is thus modified, but the association of low wall shear with the initiation of the bursts is maintained.

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.

Separate and joint droplets, clusters, and voids characteristics of sprays injected in a turbulent co-flow are investigated experimentally. Simultaneous Mie scattering and interferometric laser imaging for droplet sizing along with separate hotwire anemometry are performed. The turbulent co-flow characteristics are adjusted using zero, one or two perforated plates. The Taylor-length-scale-based Reynolds number varies from 10 to 38, and the Stokes number estimated based on the Kolmogorov time scale varies from 3 to 25. The results show that the mean length scale of the clusters normalized by the Kolmogorov length scale varies linearly with the Stokes number. However, the mean void length scale is of the order of the integral length scale. It is shown that the number density of the droplets inside the clusters is approximately 7 times larger than that in the voids. The ratios of the droplets number densities in the clusters and voids to the total number density are independent of the test conditions and equal 5.5 and 0.8, respectively. The joint probability density function of the droplets diameter and clusters area shows that the droplets with the most probable diameter are found in the majority of the clusters. It is argued that intensifying the turbulence broadens the range of turbulent eddy size in the co-flow which allows for accommodating droplets with a broad range of diameters in the clusters. The results are of significance for engineering applications that aim to modify the clustering characteristics of large-Stokes-number droplets sprayed into turbulent co-flows.

This paper introduces a new operator relevant to input–output analysis of flows in a statistically steady regime far from the steady base flow: the mean resolvent ${{\boldsymbol{\mathsf{R}}}}_0$. It is defined as the operator predicting, in the frequency domain, the mean linear response to forcing of the time-varying base flow. As such, it provides the statistically optimal linear time-invariant approximation of the input–output dynamics, which may be useful, for instance, in flow control applications. Theory is developed for the periodic case. The poles of the operator are shown to correspond to the Floquet exponents of the system, including purely imaginary poles at multiples of the fundamental frequency. In general, evaluating mean transfer functions from data requires averaging the response to many realizations of the same input. However, in the specific case of harmonic forcings, we show that the mean transfer functions may be identified without averaging: an observation referred to as ‘dynamic linearity’ in the literature (Dahan et al., J. Fluid Mech., vol. 704, 2012, pp. 360–387). For incompressible flows in the weakly unsteady limit, i.e. when amplification of perturbations by the unsteady part of the periodic Jacobian is small compared to amplification by the mean Jacobian, the mean resolvent ${{\boldsymbol{\mathsf{R}}}}_0$ is well-approximated by the well-known resolvent operator about the mean flow. Although the theory presented in this paper extends only to quasi-periodic flows, the definition of ${{\boldsymbol{\mathsf{R}}}}_0$ remains meaningful for flows with continuous or mixed spectra, including turbulent flows. Numerical evidence supports the close connection between the two resolvent operators in quasi-periodic, chaotic and stochastic two-dimensional incompressible flows.

Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength $\beta =a^*e^*E_\infty ^*/k_B^*T^*$, defined as the ratio of the product of the applied electric field magnitude $E_\infty ^*$ and particle radius $a^*$, to the thermal voltage $k_B^*T^*/e^*$, where $k_B^*$ is Boltzmann's constant, $T^*$ is the absolute temperature, and $e^*$ is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density $\sigma$ over a wide range of $\beta$. Here, $\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$, where $\sigma ^*$ is the dimensional surface charge density, and $\epsilon ^*$ is the permittivity of the electrolyte. For moderately charged particles ($\sigma ={O}(1)$), the electrophoretic velocity is linear in $\beta$ when $\beta \ll 1$, and its dependence on the ratio of the Debye length ($1/\kappa ^*$) to particle radius (denoted by $\delta =1/(\kappa ^*a^*)$) agrees with Henry's formula. As $\beta$ increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is $\delta$-dependent. For $\beta \gg 1$, the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all $\delta$. For highly charged particles ($\sigma \gg 1$) in the thin-Debye-layer limit ($\delta \ll 1$), our computations are in good agreement with recent experimental and asymptotic results.

Hypersonic flow on a V-shaped blunt leading edge (VSBLE) at Mach 12 is numerically investigated. A series of self-induced shock–shock interactions and shock wave/boundary layer interactions (SWBLIs) cause extremely high heat flux peaks on the crotch of the VSBLE. Based on these shock structures, a simplified model that divides the flow field into inviscid and viscous areas is constructed. This model can quickly solve the shock interactions and predict the heating/pressure peaks with high accuracy. Furthermore, a shock control bump (SCB) is placed on the SWBLI region to split the strong incident shock into a weaker multi-wave system. The mechanism study of the SCB shows that the inviscid effect significantly reduces the heating/pressure peaks, and the viscous effect suppresses the SWBLI-induced separation. Finally, a VSBLE with SCBs is numerically investigated. The heat flux peak is reduced by 66 % compared to that without the SCBs. The robustness of the SCB under various working conditions is also evaluated. This paper provides an idea for the simplified solution of complex shock interactions and extends the application of the SCB as a thermal protection device in hypersonic flow for the first time.

Understanding the dynamics of staircases in salt fingering convection presents a long-standing theoretical challenge to fluid dynamicists. Although there has been significant progress, particularly through numerical simulations, there are a number of conflicting theoretical explanations as to the driving mechanism underlying staircase formation. The Phillips effect proposes that layering in stirred stratified flow is due to an antidiffusive process, and it has been suggested that this mechanism may also be responsible for salt fingering staircases. However, the details of this process, as well as mathematical models to predict the evolution and merger dynamics of staircases, have yet to be established. We generalise the theory of the Phillips effect to a three-component system (e.g. temperature, salinity, energy) and demonstrate a regularised nonlinear model of layering based on mixing length parametrisations. The model predicts both the inception of layering and its long-term evolution through mergers, while generalising, and remaining consistent with, previous results for double-diffusive layering based on flux ratios. Our model of salt fingering is formulated using spatial averaging processes, and closed by a mixing length parametrised in terms of the kinetic energy and the ratio of the temperature and salt gradients. The model predicts a layering instability for a bounded range of parameter values in the salt fingering regime. Nonlinear solutions show that an initially unstable linear buoyancy gradient develops into layers, which proceed to merge through a process of stronger interfaces growing at the expense of weaker ones. Our results indicate that these mergers are responsible for the characteristic increase of buoyancy flux through thermohaline staircases.

With the severity and frequency of significant weather events increasing, methods for alleviating unsteady wind loading for high-rise buildings are gaining interest. This study numerically investigates the three-dimensional flow structures around a canonical high-rise building immersed in an atmospheric boundary layer at different oncoming wind angles, using wall-resolved large eddy simulations. A synthetic jet located on the top surface is used as open-loop active actuation with the aim of suppressing the building's side-force fluctuations when exposed to oncoming wind variations. Three different frequencies of jet forcing are considered, all half an order of magnitude larger than the vortex shedding frequency. The behaviour of the synthetic jet and its effect on the building's unsteady side force, time-averaged flow fields and unsteady flow structures are investigated numerically. The synthetic jet actuation is found to reduce the side-force fluctuation of the building, enhance the downwash flow and successfully attenuate the antisymmetric vortex shedding. This was achieved to different extents across the range of oncoming wind angles considered and may motivate future attempts to explore experimental active control strategies for attenuation of unsteady wind loading.

Direct numerical simulations (DNSs) are performed to investigate the roughness effects on the statistical properties and the large-scale coherent structures in the turbulent channel flow over three-dimensional sinusoidal rough walls. The outer-layer similarities of mean streamwise velocity and Reynolds stresses are examined by systematically varying the roughness Reynolds number $k^{+}$ and the ratio of the roughness height to the half-channel height $k / \delta$. The energy transfer mechanism of turbulent motions in the presence of roughness elements with different sizes is explored through spectral analysis of the transport equation of the two-point velocity correlation and the scale-energy path display of the generalized Kolmogorov equation. The results show that, with increasing $k^+$, the downward shift of the mean streamwise velocity profile in the logarithmic region increases and the peak intensities of turbulent Reynolds stresses decrease. At an intermediate Reynolds number ($Re_{\tau }= 1080$), the length scale and intensity of the large-scale coherent structures increase for a small roughness ($k^{+}=10$), which leads to failure of the outer-layer similarity in rough-wall turbulence, and decrease for a large roughness ($k^{+}=60$), as compared with the smooth-wall case. The existence of the small roughness ($k^{+}=10$) enhances the mechanism of inverse energy cascade from the inner-layer small-scale structures to the outer-layer large-scale structures. Correspondingly, the self-sustaining processes of the outer-layer large-scale coherent structures, including turbulent production, interscale transport, pressure transport and spatial turbulent transport, are all enhanced, whereas the large roughness ($k^{+}=60$) weakens the energy transfer between the inner and outer regions.