The preference for particles to accumulate at specific regions in the near-wall part is a widely observed phenomenon in wall-bounded turbulence. Unlike small particles more frequently found in low-speed streaks, finite-size particles can accumulate in either low-speed or high-speed streaks. However, mechanisms and influencing factors leading to the different preferential concentration locations still need to be clarified. The present study conducts particle-resolved direct numerical simulations of particle-laden turbulent channel flows to provide a better understanding of this seemingly puzzling behaviour of preferential accumulation. These simulations cover different particle-to-fluid density ratios, particle volume fractions, particle sizes and degrees of sedimentation intensity. We find that the large particle size is the crucial factor that results in particles accumulating in high-speed streaks. Large particles not only are difficult to be conveyed by the quasi-streamwise vortices to low-speed streaks but also can escape from the near-wall region before moving spanwisely out from high-speed streaks. The sedimentation effect allows particles to gather closer to the channel wall and stay longer in the near-wall regions, reinforcing the sweeping mechanism of quasi-streamwise vortices that transport particles from high- to low-speed streaks. As a result, sedimenting particles tend to accumulate in the low-speed streaks.

We experimentally explored the effect of single-sidewall cooling on Rayleigh–Bénard (RB) convection. Canonical RB was also studied to aid insight. The scenarios shared tank dimensions and bottom and top wall temperatures; the single sidewall cooling had the top wall temperature. Turbulence was explored at two canonical Rayleigh numbers, $Ra=1.6\times 10^{10}$ and $Ra=2\times 10^9$ under Prandtl number $Pr=5.4$. Particle image velocimetry described vertical planes parallel and perpendicular to the sidewall cooling. The two $Ra$ scenarios reveal pronounced changes in the flow structure and large-scale circulation (LSC) due to the sidewall cooling. The density gradient induced by the sidewall cooling led to asymmetric descending and ascending flows and irregular LSC. Flow statistics departed from the canonical case, exhibiting lower buoyancy effects, represented by an effective Rayleigh number with effective height dependent on the distance from the lateral cooling. Velocity spectra show two scalings, $\varPhi \propto f^{-5/3}$ Kolmogorov (KO41) and $\varPhi \propto f^{-11/5}$ Bolgiano (BO59) in the larger $Ra$; the latter was not present in the smaller set-up. The BO59 scaling with sidewall cooling appears at higher frequencies than its canonical counterpart, suggesting weaker buoyancy effects. The LSC core motions allowed us to identify a characteristic time scale of the order of vortex turnover time associated with distinct vortex modes. The velocity spectra of the vortex core oscillation along its principal axis showed a scaling of $\varPhi _c \propto f^{-5/3}$ for the single sidewall cooling, which was dominant closer there. It did not occur in the canonical case, evidencing the modulation of LSC oscillation on the flow.

We investigate the concentration fluctuations of passive scalar plumes emitted from small, localised (point-like) steady sources in a neutrally stratified turbulent boundary layer over a rough wall. The study utilises high-resolution large-eddy simulations for sources of varying sizes and heights. The numerical results, which show good agreement with wind-tunnel studies, are used to estimate statistical indicators of the concentration field, including spectra and moments up to the fourth order. These allow us to elucidate the mechanisms responsible for the production, transport and dissipation of concentration fluctuations, with a focus on the very near field, where the skewness is found to have negative values – an aspect not previously highlighted. The gamma probability density function is confirmed to be a robust model for the one-point concentration at sufficiently large distances from the source. However, for ground-level releases in a well-defined area around the plume centreline, the Gaussian distribution is found to be a better statistical model. As recently demonstrated by laboratory results, for elevated releases, the peak and shape of the pre-multiplied scalar spectra are confirmed to be independent of the crosswind location for a given downwind distance. Using a stochastic model and theoretical arguments, we demonstrate that this is due to the concentration spectra being directly shaped by the transverse and vertical velocity components governing the meandering of the plume. Finally, we investigate the intermittency factor, i.e. the probability of non-zero concentration, and analyse its variability depending on the thresholds adopted for its definition.

The hydrodynamics of a self-propelling swimmer undergoing intermittent S-start swimming are investigated extensively with varying duty cycle $DC$, swimming period $T$, and tailbeat amplitude $A$. We find that the steady time-averaged swimming speed $\bar {U}_x$ increases directly with $A$, but varies inversely with $DC$ and $T$, where there is a maximal improvement of $541.29\,\%$ over continuous cruising swimming. Our results reveal two scaling laws, in the form of input versus output relations, that relate the swimmer's kinematics to its hydrodynamic performance: swimming speed and efficiency. A smaller $DC$ causes increased fluctuations in the swimmer's velocity generation. A larger $A$, on the other hand, allows the swimmer to reach steady swimming more quickly. Although we set out to determine scaling laws for intermittent S-start swimming, these scaling laws extend naturally to burst-and-coast and continuous modes of swimming. Additionally, we have identified, categorized and linked the wake structures produced by intermittent S-start swimmers with their velocity generation.

Surfactant-like impurities are omnipresent in multiphase emulsions and may substantially affect the motion of small droplets by altering their interfacial properties. Usually these surfactants are soluble in the bulk and undergo adsorption–desorption onto the interface which modifies their surface concentration and hence their overall influence on droplet motion. Yet, the impact of the bulk solubility and transport of surfactants on droplet dynamics, especially in the presence of bounding walls, remains poorly understood. As such, in this article, we assess the impact of bulk soluble surfactants on the settling of a spherical drop towards a plane wall. We consider coupled bulk and interfacial transport of surfactants, mediated by adsorption–desorption processes and construct a semi-analytical framework for arbitrary values of ‘bulk interaction parameter’, which dictates the strength of adsorption–desorption kinetics compared with bulk diffusion. Our results indicate that while mass exchange between the bulk and the interface can remobilize the drop, a finite bulk diffusion rate restricts this process and therefore slows down the drop. This also results in bulk concentration depletion near the south pole and accumulation near the north pole, the extent of which becomes strongly asymmetric with an enhanced intensity of depletion, as the drop approaches the wall. Presence of the wall and bulk solubility are found to aid each other towards remobilizing the drop by aptly modifying the interfacial concentration. Our results may provide fundamental insights into the kinetics of surfactant-laden drops, with potential applications in food and pharmaceutical industries, separation processes, etc.

The propagation of the gravity current generated from a moving source of buoyancy is of interest in deep-sea mining and related technologies. The study by Ouillon et al. (J. Fluid Mech., vol. 924, 2021, A43) elucidated some salient patterns of the flow concerning a source close to the bottom on the basis of direct numerical simulation on a supercomputer. Here, we present a simple box model that provides further insights and useful analytical approximations for this gravity-current flow system. We show that this flow is very different from that produced by a moving source at the top, studied by Hogg et al. (J. Fluid Mech., vol. 539, 2005, pp. 349–385). The model confirms that the main governing parameter is the ratio $a$ of speed of source to that of buoyancy propagation. The model points out dependency also on the front-jump Froude number (which implies dependency on the height of the ambient fluid). For a sufficiently large $a >a_{crit}$, a supercritical regime appears in which the gravity current forms a wedge behind the moving source; in the subcritical regime, the upstream propagation attains a maximum $x_m$ at time $t_m$. The model predicts the value $a_{crit}$, the distance and time $x_m$ and $t_m$ in the subcritical case, and the shape of the wedge in the supercritical case, without any adjustable constant. Comparisons with the numerical data show fair agreement.

This empirical study aims to characterize the dynamical behaviour of sprays using time-series analysis of the size–velocity data acquired using a phase Doppler particle analyser. The prime motivation of this analysis is to capture the spatio-temporal correlations using time-series modelling paradigms that provide valuable new insights into spray dynamics. As a first step, we study long-held assumptions, especially on stationarity and time unsteadiness. We show that air-blast sprays have increased drop size as well as velocity ordering near the edge of the spray. Analysis of the inter-particle time of the droplets shows non-Poisson behaviour where droplets that are closely spaced in time are also closely spaced in the size and velocity coordinates. Temporal auto-correlation and partial auto-correlation calculations reveal the presence of inherent correlated features in the spray. This correlation is stronger and short lived in an air-blast spray and weaker but more persistent in a pressure swirl spray. These correlations render the probability density function (p.d.f.) estimate obtained from standard methods inaccurate; therefore, we propose a technically correct way of estimating the p.d.f. using a suitable downsampling and averaging method. Statistical analysis of residuals (from appropriate autoregressive integrated moving average time-series models) uncovers an interesting feature of spray data pertaining to heteroskedasticity (stochastically changing variance) of the diameter series. In order to account for heteroskedasticity, appropriate generalized autoregressive conditional heteroskedasticity models are developed. Finally, we present a utilitarian view of these results as an empirically consistent boundary condition implementation tool for computational fluid dynamics (CFD).

We study the spreading of Newtonian viscous (aqueous glycerin solution) and viscoelastic (aqueous polymer solution) drops on solid substrates with different wettabilities. For drops of the same zero-shear viscosity, we find in the early stages of spreading that viscoelastic drops (i) spread faster and (ii) their contact radius shows a different power law vs time than Newtonian drops. We argue that the effect of viscoelasticity is only observable for experimental time scales of the order of or larger than the internal relaxation time of the viscoelastic polymer solution. We attribute this behaviour to the shear thinning of the viscoelastic polymer solution. When approaching the contact line, the shear rate increases and the steady-state viscosity of the viscoelastic drop is lower than that of the Newtonian drop. We support our experimental findings with a simple (first-order) perturbation model that qualitatively agrees with our findings.

Capillary folding is the process of folding planar objects into three-dimensional (3-D) structures using capillary force. It has many important industrial applications, e.g. the fabrication of microelectromechanical systems. In this work, we propose a 3-D model for the capillary folding of thin elastic sheets with pinned contact lines. The energy of the system consists of interfacial energies between the different phases and the elastic energy of the sheet. The latter is described by the nonlinear Koiter's model, which can accommodate large deformations of the sheet. From the energy, we derive the governing equations for the static system using a variational approach. We then develop a numerical method to find equilibrium solutions via a relaxation dynamics. At each time step, we evolve the sheet by using the subdivision element method, and update the droplet surface by minimizing a squared area functional using linear finite elements. Qualitatively, numerical solutions for the equilibrium configurations of the sheet–droplet system agree well with those observed in experiments. Furthermore, we identify the critical bendabilities and present bifurcation diagrams for the folding of a triangular sheet. The results exhibit rich and fully 3-D behaviours that were not captured by previous two-dimensional models. Our results provide new insights into the nonlinear process of capillary folding, and may contribute to the advancement of microfabrication techniques.

High-resolution, long-time three-dimensional simulations are presented for slow, pressure-driven flow of a periodic emulsion of deformable drops through a dense, simple cubic array of solid spheres (one drop and one particle per periodic cell). The drops, covered with insoluble, non-diffusive surfactant, are large compared with pores, and they squeeze with high resistance, very closely coating the solids to overcome surface tension and lubrication effects. The solid volume fraction is 50 %, the emulsion concentration $c_{em}$ in the pore space is 36 % or 50 %, the drop-to-medium viscosity ratio $\lambda$ is 0.25 to 4. The contamination measure $\beta \leq 0.1$ keeps the linear surfactant model (assumed in most of the work) physically relevant. The boundary-integral solution requires extreme resolutions (tens of thousands of boundary elements per surface) achieved by multipole acceleration with special desingularizations, combined with flow-biased surfactant transport algorithms for numerical stability. The time-periodic regime is typically attained after a few squeezing cycles; the motion period is used in the extrapolation scheme to evaluate critical capillary numbers $Ca_{crit}$ demarcating squeezing from trapping. Due to Marangoni stresses, even light ($\beta =0.05$) to moderate ($\beta =0.1$) contaminations significantly reduce the average drop-phase migration velocity (up to 2.8 times, compared with clean drops), especially at small $\lambda =0.25$. In contrast, $Ca_{crit}$ is weakly sensitive to contamination and levels off completely at $\beta =0.05$. At $\lambda =0.25$ and $c_{em}=0.36$, the average drop-phase velocities are much different for lightly and moderately contaminated emulsions, except for near-critical squeezing when they become the same. Nonlinear surfactant models (Langmuir, Frumkin) are used to validate the linear model.

Even without breaking or wind influence, ocean surface waves are observed to produce turbulence in the water, possibly influencing ocean surface dynamics and air–sea interactions. Based on the water-side free-surface simulations, recent studies suggest that such turbulence is produced through the interaction between the waves and the near-surface Eulerian current associated with the viscous attenuation of waves. To clarify the dynamical role of the air–water interface in the turbulence production, the attenuating interfacial gravity waves were simulated directly using a newly developed two-phase wave-resolving numerical model. The air–water coupling enhanced the wave energy dissipation through the formation of a strong shear at the air-side viscous boundary layer. This led to an enhancement of the wave-to-current momentum transfer and the formation of the down-wave Eulerian mean sheared current, which is favourable for the CL2 instability responsible for the production of Langmuir circulations. As a result, the water-side turbulence grew stronger compared with the corresponding free surface (water-only) wave-resolving simulation. The evolution of the wave-averaged field was well reproduced with the Craik–Leibovich equation with the upper boundary condition provided with the virtual wave stress based on linear theory. The wave energy dissipation by air–water coupling plays a significant role in the quantitative understanding of the wave-induced turbulence at the laboratory and field scales.

The velocity gradient tensor can be decomposed into normal straining, pure shearing and rigid rotation tensors, each with distinct symmetry and normality properties. We partition the strength of turbulent velocity gradients based on the relative contributions of these constituents in several canonical flows. These flows include forced isotropic turbulence, turbulent channels and turbulent boundary layers. For forced isotropic turbulence, the partitioning is in excellent agreement with previous results. For wall-bounded turbulence, the partitioning collapses onto the isotropic partitioning far from the wall, where the mean shearing is relatively weak. By contrast, the near-wall partitioning is dominated by shearing. Between these two regimes, the partitioning collapses well at sufficiently high friction Reynolds numbers and its variations in the buffer layer and the log-law region can be reasonably modelled as a function of the mean shearing strength. Altogether, our results highlight the expressivity and broad applicability of the velocity gradient partitioning as advantages for turbulence modelling.

The peculiar migration and rotational dynamics of non-spherical particles in non-Newtonian flows stem from the interplay between fluid rheology and fluid inertial effects. In this paper, the cross-flow migration of a neutrally buoyant oblate spheroid (aspect ratio $AR = 0.5$) immersed in the elasto-inertial duct flow is investigated by particle-resolved simulations with the immersed boundary method. Different from spherical particles, due to the orientation-dependent lift force, the oblate spheroid migrates in an oscillating manner in the duct. The travelling period for particles reaching the duct centreline undergoes a non-monotonic change with elastic number, revealing the existence of a critical elastic number governing the migrating efficiency of oblate particles within the present flow system. For the particle rotation and orientation, the present results indicate that the particle rotation rate and orbit drifting rate are both hindered by the fluid elasticity. With increasing the fluid elasticity, three different orientation modes – log-rolling mode, kayaking-like mode and longside-flow alignment mode – are observed successively during the elasto-inertial migration of the oblate spheroid. Potentially, the present results could be used to design the rheology-based controlling strategy for guiding particles to achieve optimal focusing and orientation in microfluidic applications without the need for external forcing fields.

Recently, subcritical transition to turbulence in the quasi-two-dimensional (quasi-2-D) shear flow with strong linear friction (Camobreco et al., J. Fluid Mech., vol. 963, 2023, R2) has been demonstrated by the 2-D mechanism at $Re = 71\,211$, and the nonlinear Tollmien–Schlichting (TS) waves related to the edge state were approached independently of initial optimal disturbances. For 2-D plane Poiseuille flow, transition to the fully developed turbulence requires that the Reynolds number is several times larger than the critical Reynolds number $Re_c$ (Markeviciute & Kerswell, J. Fluid Mech., vol. 917, 2021, A57). In this paper, we observed the subcritical transitional flow in 2-D plane Poiseuille flow driven by the nonlinear TS waves by both linear and nonlinear optimal disturbances ($Re < Re_c$) with different quantitative edge states. The nonlinear optimal disturbances could trigger the sustained subcritical transitional flow for $Re \geqslant 2400$. The initial energy for nonlinear optimal disturbance is more efficient than the linear optimal disturbance in reaching the subcritical transitional flow for $2400 \leqslant Re \leqslant 5000$. Moreover, the initial energy of linear optimal disturbance is larger than the energy of its edge state. The nonlinear TS waves along the edge state are formed by the nonlinear optimal disturbances to trigger transitional flow, which agrees well with the main conclusions of Camobreco et al. (J. Fluid Mech., vol. 963, 2023, R2), while the required $Re$ of 2-D plane Poiseuille flow is much smaller.

This study reports the first time-resolved particle image velocimetry characterization of a planar two-phase mixing layer flow, whose velocity field is measured simultaneously in gas and liquid streams. Two parallel air and water flows meet downstream of a splitter plate, giving rise to an initially spanwise invariant configuration. The aim is to elucidate further the mechanisms leading to the flow breakup in gas-assisted atomization. The complete experimental characterization of the velocity field represents a database that could be used in data-driven reduced-order models to investigate the global behaviour of the flow system. After the analysis of a selected reference case, a parametric study of the flow behaviour is performed by varying the liquid ($Re_l$) and gas ($Re_g$) Reynolds numbers, and as a consequence also the gas-to-liquid dynamic pressure ratio ($M$), shedding light on both time-averaged (mean) and unsteady velocity fields. In the reference case, it is shown that the mean flow exhibits a wake region just downstream of the splitter plate, followed by the development of a mixing layer. By increasing both $Re_l$ and $Re_g$, the streamwise extent of the wake decreases and eventually vanishes, the flow resulting in a pure mixing layer regime. The spectral analysis of the normal-to-flow velocity fluctuations outlines different flow regimes by variation of the governing parameters, giving more insights into the global characteristics of the flow field. As a major result, it is found that at high $Re_g$ and $M$ values, the velocity fluctuations are characterized by low-frequency temporal oscillations synchronized in several locations within the flow field, which suggest the presence of a global mode of instability. The proper orthogonal decomposition of velocity fluctuations, performed in both gas and liquid phases, reveals finally that the synchronized oscillations are associated with a low-frequency dominant flapping mode of the gas–liquid interface. Higher-order modes correspond to interfacial wave structures travelling with the so-called Dimotakis velocity. For lower gas Reynolds numbers, the leading modes describe higher frequency fingers shedding at the interface.

This study introduces an analytical solution for the laminar entrance flow in circular pipes, aiming to confirm the occurrence of velocity overshoot. Velocity overshoot is characterised by the maximum axial velocity appearing near the pipe wall instead of the central axis. Similar to the previous studies, the analytical solution is derived from the parabolised Navier–Stokes equation; however, the specific approach used in linearising the momentum equation has not been attempted before. The accuracy of this analytical solution has been verified through a comprehensive comparison with various published experimental data. The existence of velocity overshoot at a short distance from the inlet, which is evident in numerous numerical calculations based on the full Navier–Stokes equations and corroborated by recent magnetic resonance (MR) velocimetry experiments, is identified analytically for the first time. The parabolised Navier–Stokes equation has inherent self-similarity with respect to the Reynolds number, implying that $Re$ is incorporated into the dimensionless variables rather than serving as an independent flow parameter. According to both MR velocimetry measurements and the present analytical solution, the self-similarity is not valid immediately following the pipe inlet, and this becomes more evident as $Re$ decreases; hence, the analytical solution derived from the parabolised Navier–Stokes equation cannot accurately predict the evolution of the velocity profile within this region near the pipe inlet.

The practical implementation of machine learning in flow control is limited due to its significant training expenses. In the present study the convolutional neural network (CNN) trained with the data of the restricted nonlinear (RNL) model is used to predict the normal velocity on a detection plane at $y^+=10$ in a turbulent channel flow, and the predicted velocity is used as wall blowing and suction for drag reduction. An active control test is carried out by using the well-trained CNN in direct numerical simulation (DNS). Substantial drag reduction rates up to 19 % and 16 % are obtained based on the spanwise and streamwise wall shear stresses, respectively. Furthermore, we explore the online control of wall turbulence by combining the RNL model with reinforcement learning (RL). The RL is constructed to determine the optimal wall blowing and suction based on its observation of the wall shear stresses without using the label data on the detection plane for training. The controlling and training processes are conducted synchronously in a RNL flow field. The control strategy discovered by RL has similar drag reduction rates with those obtained previously by the established method. Also, the training cost decreases by over thirty times at $Re_{\tau }=950$ compared with the DNS-RL model. The present results provide a perspective that combining the RNL model with machine learning control for drag reduction in wall turbulence can be effective and computationally economical. Also, this approach can be easily extended to flows at higher Reynolds numbers.

The global linear stability analysis for the magnetohydrodynamic liquid metal flow past an insulated sphere subjected to a constant streamwise magnetic field is investigated in the range of the Reynolds number $Re\leq 400$ and the interaction number $N\leq 40$ coupled with direct numerical simulations, where $N$ stands for strength of the electromagnetic force. The stability of the steady axisymmetric base flow to independent time-azimuthal modes is discussed. Five critical curves associated with various wake transitions are obtained in the $\{Re, N\}$ phase diagram. These critical curves reveal the stabilising effect of a weak magnetic field, the destabilising effect of a strong magnetic field and re-stabilising effect of a much stronger magnetic field. To explore the impact of the magnetic field on flow instability, a sensitivity analysis utilizing an adjoint method is performed for the first regular bifurcation. Sensitivity functions of growth rate to base-flow modifications and Lorentz force are defined to identify the region that has the most significant influence on flow instability, such as the recirculation region responsible for the stabilising effect at a weak magnetic field and the shear layer region responsible for the destabilising effect at a strong magnetic field. Furthermore, a competition between the stabilising and shear destabilising effects of the magnetic field is discussed. This analysis provides valuable insights into the non-monotonic effect of the magnetic field on flow instability.

The scaling relations mapping the turbulence statistics in compressible turbulent boundary layers (TBLs) onto their incompressible counterparts are of fundamental significance for turbulence modelling, such as the Morkovin scaling for velocity fields, while for pressure fluctuation fields, a corresponding scaling relation is currently absent. In this work, the underlying scaling relations of pressure fluctuations about Mach number ($M$) contained in their generation mechanisms are explored by analysing a series of direct numerical simulation data of compressible TBLs over a wide Mach number range $(0.5\leq M \leq 8.0)$. Based on the governing equation of pressure fluctuations, they are decomposed into components according to the properties of source terms. It is notable that the intensity of the compressible component, predominantly originating from the acoustic mode, obeys a monotonic distribution about the Mach number and wall distance; further, the intensity of the rest of the pressure components, which are mainly generated by the vorticity mode, demonstrates a uniform distribution consistent with its incompressible counterpart. Moreover, the coupling between the two components is negligibly weak. Based on the scaling relations, semiempirical models for the fluctuation intensity of both pressure and its components are constructed. Hence, a mapping relation is obtained that the profiles of pressure fluctuation intensities in compressible TBLs can be mapped onto their incompressible counterparts by removing the contribution from the acoustic mode, which can be provided by the model. The intrinsic scaling relation can provide some basic insight for pressure fluctuation modelling.