We investigate enstrophy variations by collapse of point vortices in an inviscid flow and, in particular, focus on the enstrophy dissipation that is a significant property characterising two-dimensional (2-D) turbulent flows. To reveal the vortex dynamics causing the enstrophy dissipation, we consider the dynamics of point vortices, whose vorticity is concentrated on points and dynamics on the inviscid flow, governed by the point-vortex system. The point-vortex system has self-similar collapsing solutions, which are expected to be a key to understand the enstrophy dissipation, but the collapsing process cannot be described by solutions to the 2-D Euler equations. We thus consider the 2-D filtered-Euler equations, which are a regularised model of the 2-D Euler equations, and their point-vortex solutions. The preceding studies (Gotoda and Sakajo, J. Nonlinear Sci. 2016, vol. 26, pp. 1525–1570, Gotoda and Sakajo, SIAM J. Appl. Math. 2018, vol. 78, 2105–2128) have proven that there exist three point-vortex solutions to the filtered model such that they converge to self-similar collapsing orbits in the three point-vortex system and dissipate the enstrophy at the event of collapse in the zero limit of the filter parameter. In this study, we numerically show that the enstrophy dissipation by the collapse of point vortices could occur for the four and five vortex problems in a filtered model. Moreover, we show the detailed convergence process of the point vortices for gradually decreasing filter parameters, which provides a new insight for the three vortex problem. In addition, numerical computations suggest that the enstrophy dissipation is caused by collapse of separated point vortices with the negative interactive energy.

Viscoplastic fluids exhibit yield stress, beyond which they flow viscously, while at lower stress levels they behave as solids. Despite their fundamental biological and medical importance, the hydrodynamics of swimming in viscoplastic environments is still evolving. In this study, we investigate the swimming of an ellipsoidal squirmer and the associated tracer diffusion in a Bingham viscoplastic fluid. The results illustrate that neutral squirmers in viscoplastic fluids experience a reduction in swimming speed and an increase in power dissipation as the Bingham number increases, with swimming efficiency peaking at moderate Bingham numbers. As the aspect ratio of a squirmer increases, ellipsoidal squirmers exhibit significantly higher swimming speeds in viscoplastic fluids. The polar and swirling modes can either enhance or reduce swimming speed, depending on the specific scenarios. These outcomes are closely related to the confinement effects induced by the yield surface surrounding the swimmer, highlighting how both swimmer shape and swimming mode can significantly alter the yield surface and, in turn, modify the swimming hydrodynamics. In addition, this study investigates the influence of viscoplasticity on swimmer-induced diffusion in a dilute suspension. The plasticity enforces the velocity far from the swimmer to be zero, thus breaking the assumptions used in Newtonian fluids. The diffusivity reaches its maximum at intermediate aspect ratios and Bingham numbers, and increases with the magnitude of the squirmer’s dipolarity. These findings are important to understand microscale swimming in viscoplastic environments and the suspension properties.

Bubble–particle collisions in turbulence are key to the froth flotation process that is widely employed industrially to separate hydrophobic from hydrophilic materials. In our previous study (Chan et al., 2023 J. Fluid Mech. 959, A6), we elucidated the collision mechanisms and critically reviewed the collision models in the no-gravity limit. In reality, gravity may play a role since, ultimately, separation is achieved through buoyancy-induced rising of the bubbles. This effect has been included in several collision models, which have remained without a proper validation thus far due to a scarcity of available data. We therefore conduct direct numerical simulations of bubbles and particles in homogeneous isotropic turbulence with various Stokes, Froude and Reynolds numbers, and particle density ratios using the point-particle approximation. Generally, turbulence enhances the collision rate compared with the pure relative settling case by increasing the collision velocity. Surprisingly, however, for certain parameters the collision rate is lower with turbulence compared with without, independent of the history force. This is due to turbulence-induced bubble–particle spatial segregation, which is most prevalent at weak relative gravity and decreases as gravitational effects become more dominant, and reduced bubble slip velocity in turbulence. The existing bubble–particle collision models only qualitatively capture the trends in our numerical data. To improve on this, we extend the model by Dodin & Elperin (2002 Phys. Fluids 14, 2921–2924) to the bubble–particle case and found excellent quantitative agreement for small Stokes numbers when the history force is negligible and segregation is accounted for.

To elucidate the attenuation mechanism of wall-bounded turbulence due to heavy small particles, we conduct direct numerical simulations (DNS) of turbulent channel flow laden with finite-size solid particles. When particles cannot follow the swirling motions of wall-attached vortices, vortex rings are created around the particles. These particle-induced vortices lead to additional energy dissipation, reducing the turbulent energy production from the mean flow. This mechanism results in the attenuation of turbulent kinetic energy, which is more significant when the Stokes number of particles is larger or particle size is smaller under the condition that the volume fraction of particles is fixed. Moreover, we propose a method to quantitatively predict the degree of turbulence attenuation without using DNS data by estimating the additional energy dissipation rate in terms of particle properties.

The physical fidelity of turbulence models can benefit from a partial resolution of fluctuations, but doing so often comes with an increase in computational cost. To explore this trade-off in the context of wall-bounded flows, this paper introduces a framework for turbulence-resolving integral simulations (TRIS) with the goal of efficiently resolving the largest motions using a two-dimensional, three-component representation of the flow defined by instantaneous wall-normal integrals of velocity and pressure. Self-sustaining turbulence with qualitatively realistic large-scale structures is demonstrated for TRIS on an open-channel (half-channel) flow configuration using moment-of-momentum integral equations derived from Navier–Stokes with relatively simple closure approximations. Evidence from direct numerical simulations (DNS) suggests that TRIS can theoretically resolve $35\,\%{-}40\,\%$ of the turbulent skin friction enhancement for friction Reynolds numbers between $180$ and $5200$, without a noticeable decrease or increase as a function of Reynolds number. The current implementation of TRIS can match this resolution while simulating one flow through time in ${\sim}1$ minute on a single processor, even for very large Reynolds numbers. The framework facilitates a detailed apples-to-apples comparison of predicted statistics against data from DNS. Comparisons at friction Reynolds numbers of $395$ and $590$ show that TRIS generates a relatively accurate representation of the flow, while highlighting discrepancies that demonstrate a need for improving the closure models. The present results for open-channel flow represent a proof of concept for TRIS as a new approach for wall-bounded turbulence modelling, motivating extension to more general flow configurations such as boundary layers on immersed objects.

An analysis is presented of the suspensions of small, electrified particles in a gas. Two limits of interest for the electrodynamic particulate suspension technique are considered, corresponding to large and small values of the ratio $t_{coll}/t_s$ of the mean time between particle collisions to the viscous adaptation time required for the particles to reach their terminal velocities. The effect of the particle inertia can be neglected when this ratio is large, and only the distribution of particle charges at each point of the suspension needs to be computed. The way this distribution approaches an equilibrium form, determined elsewhere in the continuum regime when the mean free path of the particles is small compared with the suspension size, is described, as well as the connection between continuum regime and quasi-neutrality of the suspension. In the opposite case when $t_{coll}/t_s$ is small, the inertia of the particles plays an important role, and the joint distribution of particle charges and velocities is required. A Boltzmann equation is proposed for this distribution function, taking advantage of the fact that the charges of the particles have little effect on the redistribution of momentum and energy in the collisions. The equilibrium distribution function in the continuum regime is computed approximately, and hydrodynamic equations for the particle phase analogous to the Euler equations for a monoatomic gas are derived. The simplification of these equations when the particle inertia is negligible at the scale of the suspension is worked out.

We present a new Eulerian framework for the computation of turbulent compressible multiphase channel flows, specifically to assess turbulence modulation by dispersed particulate matter in dilute concentrations but with significant mass loadings. By combining a modified low-dissipation numerical scheme for the carrier gas phase and a quadrature-based moment method for the solid particle phase, turbulent statistics of the fluid phase and fluctuations of the particle phase may be obtained as both are resolved as coupled fields. Using direct numerical simulations, we demonstrate how this method effectively resolves the turbulent statistics, kinetic energy, skin friction drag, particle mass flow rate and interphase drag for moderate-Reynolds-number channel flows for the first time. Validation of our approach to the turbulent particle-free flow and the turbulent particle-laden flow proves the applicability of the carrier flow low-dissipation scheme to simulate relatively low-Mach-number compressible flows and of the quadrature-based moment method to simulate the particle phase as an Eulerian field. This study also rationalises the computed interphase drag modulation and total Reynolds shear stress results using a simplified analytical approach, revealing how the particle migration towards the wall can affect the drag between the two phases at different Stokes numbers and particle loadings. Furthermore, we show the effect of near-wall particle accumulation on the particle mass flow rate. Using our Eulerian approach, we also explore the complex interplay between the particles and turbulent fluctuations by capturing the preferential clustering of particles in turbulence streaks. This interplay leads to turbulence modulations similar to recent observations reported in prior computational works using Lagrangian simulations. Our study extends the applicability of the Eulerian approach to accurately study particle–fluid interactions in compressible turbulent flows by explicitly calculating the energy equations for both the particle phase and the carrier fluid motion. Since the formulation is compressible and includes energy equations for both the particle and carrier flow fields, future studies for compressible flows involving heat and mass transfer may be simulated using this methodology.

Researchers have long debated which spatial arrangements and swimming synchronisations are beneficial for the hydrodynamic performance of fish in schools. In our previous work (Seo and Mittal, Bioinsp. Biomim., Vol. 17, 066020, 2022), we demonstrated using direct numerical simulations that hydrodynamic interactions with the wake of a leading body -caudal fin carangiform swimmer could significantly enhance the swimming performance of a trailing swimmer by augmenting the leading-edge vortex (LEV) on its caudal fin. In this study, we develop a model based on the phenomenology of LEV enhancement, which utilises wake velocity data from direct numerical simulations of a leading fish to predict the trailing swimmer’s hydrodynamic performance without additional simulations. For instance, the model predicts locations where direct simulations confirm up to 20 % enhancement of thrust. This approach enables a comprehensive analysis of the effects of relative positioning, phase difference, flapping amplitude, Reynolds number and the number of swimmers in the school on thrust enhancement. The results offer several insights regarding the effect of these parameters that have implications for fish schools as well as for bio-inspired underwater vehicle applications.

This work explores the use of a shallow surface hump for passive control and stabilisation of stationary crossflow (CF) instabilities. Wind tunnel experiments are conducted on a spanwise-invariant swept-wing model. The influence of the hump on the boundary layer stability and laminar–turbulent transition is assessed through infrared thermography and particle image velocimetry measurements. The results reveal a strong dependence of the stabilisation effect on the amplitude of the incoming CF disturbances, which is conditioned via discrete roughness elements at the wing leading edge. At a high forcing amplitude, weakly nonlinear stationary CF vortices interact with the hump and result in an abrupt anticipation of transition, essentially tripping the flow. In contrast, at a lower forcing amplitude, CF vortices interact with the hump during linear growth. Notable stabilisation of the primary CF disturbance and considerable transition delay with respect to the reference case (i.e. without hump) is then observed. The spatial region just downstream of the hump apex is shown to be key to the stabilisation mechanism. In this region, the primary CF disturbances rapidly change spanwise orientation and shape, possibly driven by the pressure gradient change-over caused by the hump and the development of CF reversal. The amplitude and shape deformation of the primary CF instabilities are found to contribute to a long-lasting suboptimal growth downstream of the hump, eventually leading to transition delay.

We explore the drawing of an axisymmetric viscoelastic tube subject to inertial and surface tension effects. We adopt the Giesekus constitutive model and derive asymptotic long-wave equations for weakly viscoelastic effects. Intuitively, one might imagine that the elastic stresses should act to prevent hole closure during the drawing process. Surprisingly, our results show that the hole closure at the take-up point is enhanced by elastic effects for most parameter values. However, the opposite is true if the tube has a sufficiently large hole size at the inlet nozzle of the device or if the axial stretching is sufficiently weak. We explain the physical mechanism underlying this phenomenon by examining how the second normal stress difference induced by elastic effects modifies the hole evolution process. We also determine how viscoelasticity affects the stability of the drawing process and show that elastic effects are always destabilising for negligible inertia. On the other hand, our results show that if the inertia is non-zero, elastic effects can be either stabilising or destabilising depending on the parameters.

We present results of three-dimensional direct numerical simulations of turbulent Rayleigh–Bénard convection of dilute polymeric solutions for Rayleigh number ($Ra$) ranging from $10^6$ to $ 10^{10}$, and Prandtl number $Pr=4.3$. The viscoelastic flow is simulated by solving the incompressible Navier–Stokes equations under the Boussinesq approximation coupled with the finitely extensible nonlinear elastic Peterlin constitutive model. The Weissenberg number ($Wi$) is either $Wi=5$ or $Wi=10$, with the maximum chain extensibility parameter $L=50$, corresponding to moderate fluid elasticity. Our results demonstrate that both heat transport and momentum transport are reduced by the presence of polymer additives in the studied parameter range. Remarkably, the specific parameters used in the current numerical study give similar heat transfer reduction values as observed in experiments. We demonstrate that polymers have different effects in different regions of the flow. The presence of polymers stabilises the boundary layer, which is found to be the primary cause of the overall heat transfer reduction. In the bulk region, the presence of polymers slows down the flow by increasing the effective viscosity, enhances the coherency of thermal plumes, and suppresses the small-scale turbulent fluctuations. For small $Ra$, the heat transfer reduction in the bulk region is associated with plume velocity reduction, while for larger $Ra$, it is caused by the competing effects of suppressed turbulent fluctuations and enhanced plume coherency.

This study investigates the strong influence of a splitter plate on two- and three-dimensional wake transitions of a circular cylinder. Direct numerical simulations and Floquet analyses are conducted over a parameter space including Reynolds numbers (Re) of 10–480 and non-dimensional plate lengths (L/D) of 0–6. With the increase in L/D, the critical Re for the onset of vortex shedding (Recr2D) increases monotonically. The delayed onset of vortex shedding with elongation of the body is physically explained. The critical Re for the onset of three-dimensionality (Recr3D) and the three-dimensional wake instability modes and structures are also significantly altered by the splitter plate. Compared with an isolated cylinder, the Recr3D for L/D = 1 is significantly reduced via a long wavelength mode, whereas the Recr3D for L/D = 2–6 is significantly increased via other modes. For each L/D, with increasing Re over the wake transition process, the spanwise wavelength of the wake structure gradually decreases, and the wake structure becomes increasingly chaotic. The strong influence of the splitter plate on the formation of the primary vortices and three-dimensional wake structures alter the hydrodynamic characteristics strongly. In particular, optimal lift reduction is achieved at L/D ∼ 1. In addition, the existence/absence of a hysteresis effect at the onset of three-dimensionality is identified by three methods. Among which, the method involving the Landau equation may be contaminated by initial transients induced by stable Floquet modes and may thus lead to a false conclusion on the existence/absence of hysteresis.

We investigate the motion of weakly negatively buoyant spheres settling in surface gravity waves using laboratory experiments. The trajectories of the settling spheres are tracked over most of the water depth with simultaneous measurements of the background fluid flow. These experiments are conducted in the regime relevant for environmental and geophysical applications where both particle inertia and fluid inertia are important. Using these data, we show that the sphere motion is well described by the kinematic sum of the undisturbed fluid velocity and the particle terminal settling velocity as long as the fluid inertia is not too large. We show how this result can be understood in the context of an ad hoc Maxey–Riley–Gatignol-type equation where the drag on the particle is given by the Schiller–Naumann drag correlation. We also evaluate whether inertial particles experience enhanced settling in waves, finding that measurement uncertainties in the particle terminal settling velocity and the presence of Eulerian-mean flows do not allow the small percentage increase in the settling velocity to be measured. When the fluid inertia becomes large enough, we observe path instabilities caused by particle wake effects in both quiescent and wavy conditions. However, the particle velocity fluctuations associated with the path instabilities are unaffected by the background flow. The minimal influence of the wavy flow on the particle path instabilities is thought to be due to the large-scale separation between the waves and the particle.

The passive flight of a thin wing or plate is an archetypal problem in flow–structure interactions at intermediate Reynolds numbers. This seemingly simple aerodynamic system displays an impressive variety of steady and unsteady motions that are familiar from fluttering leaves, tumbling seeds and gliding paper planes. Here, we explore the space of flight behaviours using a nonlinear dynamical model rooted in a quasisteady description of the fluid forces. Efficient characterisation is achieved by identification of the key dimensionless parameters, assessment of the steady equilibrium states and linear analysis of their stability. The structure and organisation of the stable and unstable flight equilibria proves to be complex, and seemingly related factors such as mass and buoyancy-corrected weight play distinct roles in determining the eventual flight patterns. The nonlinear model successfully reproduces previously documented unsteady states such as fluttering and tumbling while also predicting new types of motions, and the linear analysis accurately accounts for the stability of steady states such as gliding and diving. While the conditions for dynamic stability seem to lack tidy formulae that apply universally, we identify relations that hold in certain regimes and which offer mechanistic interpretations. The generality of the model and the richness of its solution space suggest implications for small-scale aerodynamics and related applications in biological and robotic flight.

This paper explores the construction of quadratic Lyapunov functions for establishing the conditional stability of shear flows described by truncated ordinary differential equations, addressing the limitations of traditional methods like the Reynolds–Orr equation and linear stability analysis. The Reynolds–Orr equation, while effective for predicting unconditional stability thresholds in shear flows due to the non-contribution of nonlinear terms, often underestimates critical Reynolds numbers. Linear stability analysis, conversely, can yield impractically high limits due to subcritical transitions. Quadratic Lyapunov functions offer a promising alternative, capable of proving conditional stability, albeit with challenges in their construction. Typically, sum-of-squares programs are employed for this purpose, but these can result in sizable optimisation problems as system complexity increases. This study introduces a novel approach using linear transformations described by matrices to define quadratic Lyapunov functions, validated through nonlinear optimisation techniques. This method proves particularly advantageous for large systems by leveraging analytical gradients in the optimisation process. Two construction methods are proposed: one based on general optimisation of transformation matrix coefficients, and another focusing solely on the system’s linear aspects for more efficient Lyapunov function construction. These approaches are tested on low-order models of subcritical transition and a two-dimensional Poiseuille flow model with degrees of freedom nearing 1000, demonstrating their effectiveness and efficiency compared with sum-of-squares programs.

Turbulent flows over rough beds with macroroughness elements of low relative submergence are characteristic of natural river systems. These flows exhibit highly three-dimensional structures, including large-scale coherent patterns, complex nonlinear interactions and significant drag induced by immobile boulders. In this study, large-eddy simulations are conducted of the flow through an array of boulders on a rough bed, based on experiments by Papanicolaou et al. (2012) Acta Geophys. 60 (6), 1502–1546. The analysis includes the instantaneous flow dynamics, the parameterisation of hydrodynamic roughness on the averaged velocity profile and the application of the double-averaged methodology. These upscaling approaches reveal the combined influence of wake turbulence and secondary currents (SCs), and provide insights into momentum and energy conservation mechanisms, which are critical for transport processes in fluvial environments. Results indicate that the boulder array reduces total fluid stress at the rough bed surface to $0.5 \rho u_*^2$, which can have important implications for sediment transport. Form-induced stresses, primarily originating in the boulder wakes, reach up to 37 % of total fluid stress, with peak values comparable to turbulent stresses at mid-boulder elevation. Form-induced kinetic energy (DKE) is shown to have the same magnitude as the turbulent kinetic energy (TKE), highlighting energy transfers from mean flow drag to DKE, then to TKE, before final dissipation. This study underscores the critical role of macroroughness in stress distribution, and the importance of the joint action of SCs and wake turbulence in driving form-induced stresses, which partially counterbalance drag dissipation.

Wall-resolved large-eddy simulation (LES) of a non-equilibrium turbulent boundary layer (TBL) is performed. The simulations are based on the experiments of Volino (2020a J. Fluid Mech. 897, A2), who reported profile measurements at several streamwise stations in a spatially developing zero pressure gradient TBL evolving through a region of favourable pressure gradient (FPG), a zero pressure gradient recovery and subsequently an adverse pressure gradient (APG) region. The pressure gradient quantified by the acceleration parameter $K$ was held constant in each of these three regions. Here, $K = -(\nu /\rho U_e^{3}) {\textrm d}P_e/{\textrm d}x$, where $\nu$ is the kinematic viscosity, $\rho$ is density, $U_e$ is the free stream velocity and ${\textrm d}P_e/{\textrm d}x$ is the streamwise pressure gradient at the edge (denoted by the subscript ‘$e$’) of the TBL. The simulation set-up is carefully designed to mimic the experimental conditions while keeping the computational cost tractable. The computational grid appropriately resolves the increasingly thinning and thickening of the TBL in the FPG and APG regions, respectively. The results are thoroughly compared with the available experimental data at several stations in the domain, showing good agreement. The results show that the computational set-up accurately reproduces the experimental conditions and the results demonstrate the accuracy of LES in predicting the complex flow field of the non-equilibrium TBL. The scaling laws and models proposed in the literature are evaluated and the response of the TBL to non-equilibrium conditions is discussed.

Based on data from pore-resolved direct numerical simulation of turbulent flow over mono-disperse random sphere packs, we evaluate the budgets of the double-averaged turbulent kinetic energy (TKE) and the wake kinetic energy (WKE). While TKE results from temporal velocity fluctuations, WKE describes the kinetic energy in spatial variations of the time-averaged flow field. We analyse eight cases which represent sampling points within a parameter space spanned by friction Reynolds numbers $Re_\tau \in [150, 500]$ and permeability Reynolds numbers $Re_K \in [0.4, 2.8]$. A systematic exploration of the parameter space is possible by varying the ratio between flow depth and sphere diameter $h/D \in \{ 3, 5, 10 \}$. With roughness Reynolds numbers of $k_s^+ \in [20,200]$, the simulated cases lie within the transitionally or fully rough regime. Revisiting the budget equations, we identify a WKE production mechanism via viscous interaction of the flow field with solid surfaces. The scaling behaviour of different processes over $Re_K$ and $Re_\tau$ suggests that this previously unexplored mechanism has a non-negligible contribution to the WKE production. With increasing $Re_K$, progressively more WKE is transferred into TKE by wake production. A near-interface peak in the TKE production, however, primarily results from shear production and scales with interface-related scales. Conversely, further above the sediment bed, the TKE budget terms of cases with comparable $Re_\tau$ show similarity under outer-scaling. Most transport processes relocate energy in the near-interface region, whereas pressure diffusion propagates TKE and WKE into deeper regions of the sphere pack.