Energy-containing eddies (energy-eddies) are the elementary structures of wall turbulence that carry most of the kinetic energy and momentum. Despite the consensus that energy-eddies can self-sustain at each relevant length scale, their precise origin and spatial evolution are currently not well understood. In this study, we examine the spatial evolution of energy-eddies by quenching them at the inflow of a turbulent channel flow. Our study shows that the eddies involved in the energy cascade cannot be sustained without the energy-eddies. The streamwise velocity spectra of the evolving flow start to recover at a spanwise wavelength of $\lambda _z^+ \simeq 100$, equal to the near-wall spacing of streaks in the buffer layer located at $y^+ \simeq 15$, whereas there are no active vortical motions in the streamwise vorticity spectra until the energy at the streak location is re-established. Hence, the present study demonstrates that in a spatially evolving flow, the formation of near-wall streaks is the primary process necessary in the recovery of energy-eddies.

Using the theory of wave turbulence for rapidly rotating incompressible fluids derived by Galtier (Phys. Rev. E, vol. 68, 2003, 015301), we find the locality conditions that the solutions of the kinetic equation must satisfy. We show that the exact anisotropic Kolmogorov–Zakharov spectrum satisfies these conditions, which justifies the existence of this constant (positive) energy flux solution. Although a direct cascade is predicted in the transverse ($\perp$) and parallel ($\parallel$) directions to the rotation axis, we show numerically that in the latter case some triadic interactions can have a negative contribution to the energy flux, while in the former case all interactions contribute to a positive flux. Neglecting the parallel energy flux, we estimate the Kolmogorov constant at $C_K \simeq 0.749$. These results provide theoretical support for recent numerical and experimental studies.

When a cylinder is mounted on an elastic support within a current, vortex-induced vibrations (VIV) may occur down to a Reynolds number (Re) close to $20$, based on the body diameter ($D$) and inflow velocity ($U$), i.e. below the critical value of $47$ reported for the onset of flow unsteadiness when the body is fixed. The impact of a forced rotation of the elastically mounted cylinder on the system behaviour is explored numerically for $Re \leqslant 30$, over wide ranges of values of the rotation rate (ratio between body surface velocity and $U$, $\alpha \in [0,5]$) and reduced velocity (inverse of the oscillator natural frequency non-dimensionalized by $D$ and $U$, $U^\star \in [2,30]$). The influence of the rotation is not monotonic, but the most prominent effect uncovered in this work is a substantial enhancement of the subcritical-Re, flow-induced vibrations beyond $\alpha =2$. This enhancement is twofold. First, the rotation results in a considerable expansion of the vibration/flow unsteadiness region in the $({Re},U^\star )$ domain, down to $Re=4$. Second, the elliptical orbits described by the rotating body are subjected to a major amplification, with a transition from VIV to responses whose magnitude tends to increase unboundedly with $U^\star$, even though still synchronized with flow unsteadiness. The emergence of such galloping-like oscillations close to the onset of vibrations disrupts the scenario of gradual vibration growth with Re, as amplitudes larger than $10$ body diameters may be observed at $Re=10$.

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.

In this paper, we present a generic approach of a dynamical data-driven model-order reduction technique for three-dimensional fluid–structure interaction problems. A low-order continuous linear differential system is identified from snapshot solutions of a high-fidelity solver. The reduced-order model uses different ingredients, such as proper orthogonal decomposition, dynamic mode decomposition and Tikhonov-based robust identification techniques. An interpolation method is used to predict the capsule dynamics for any values of the governing non-dimensional parameters that are not in the training database. Then a dynamical system is built from the predicted solution. Numerical evidence shows the ability of the reduced model to predict the time evolution of the capsule deformation from its initial state, whatever the parameter values. Accuracy and stability properties of the resulting low-order dynamical system are analysed numerically. The numerical experiments show very good agreement, measured in terms of modified Hausdorff distance between capsule solutions of the full-order and low-order models, in the case of both confined and unconfined flows. This work is a first milestone to move towards real-time simulation of fluid–structure problems, which can be extended to nonlinear low-order systems to account for strong material and flow nonlinearities. It is a valuable innovation tool for rapid design and for the development of innovative devices.

A dilute magnetic emulsion under the combined action of a uniform external magnetic field and a small amplitude oscillatory shear is studied using numerical simulations. We consider a three-dimensional domain with a single ferrofluid droplet suspended in a non-magnetizable Newtonian fluid. We present results of droplet shape and orientation, viscoelastic functions and bulk emulsion magnetization as functions of the shear oscillation frequency, magnetic field intensity and orientation. We also investigate how the magnetic field induces mechanical anisotropy by producing internal torques in oscillatory conditions. We found that, when the magnetic field is parallel to the shear plane, the droplet shape is mostly independent of the shear oscillation frequency. Regarding the viscometric functions, we show how the external magnetic field modifies the storage and loss moduli, especially for a field aligned to the main velocity gradient. The bulk emulsion magnetization is studied in the same fashion as the viscoelastic functions of the oscillatory shear. We show that the in-phase component of the magnetization with respect to the shear rate reaches a saturation magnetization, at the high frequencies limit, dependent on the magnetic field intensity and orientation. On the other hand, we found a non-zero out-of-phase response, which indicates a finite emulsion magnetization relaxation time. Our results indicate that the magnetization relaxation is closely related to the mechanical relaxation for dilute magnetic emulsions under oscillatory shear.

We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space $(Re,\theta )$ with $Re=\sqrt {Re_c^2+Re_M^2}$ and $\theta =\arctan (Re_M/Re_c)$, where $Re_c,Re_M$ are independent Reynolds numbers based on the plate speed $U_c$ and the volume flow rate per unit span, respectively. The database comprises direct numerical simulations (DNS) at $Re=4000,6000$, wall-resolved large-eddy simulations at $Re = 10\,000, 20\,000$, and some wall-modelled large-eddy simulations (WMLES) up to $Re=10^{10}$. Attention is focused on the transition (from Couette-type to Poiseuille-type flow), defined as where the mean skin-friction Reynolds number on the bottom wall $Re_{\tau,b}$ changes sign at $\theta =\theta _c(Re)$. The mean flow in the $(Re,\theta )$ plane is modelled with combinations of patched classical log-wake profiles. Several model versions with different structures are constructed in both the Couette-type and Poiseuille-type flow regions. Model calculations of $Re_{\tau,b}(Re,\theta )$, $Re_{\tau,t}(Re,\theta )$ (the skin-friction Reynolds number on the top wall) and $\theta _c$ show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as $\theta$ is increased at fixed $Re$, $Re_{\tau,t}$ passes through a peak at approximately $\theta = 45^{\circ }$, while $Re_{\tau,b}$ increases monotonically. Near the bottom wall, the flow laminarizes as $\theta$ passes through $\theta _c$ and then re-transitions to turbulence. As $Re$ increases, $\theta _c$ increases monotonically. The transition from Couette-type to Poiseuille-type flow is accompanied by the rapid attenuation of streamwise rolls observed in pure Couette flow. A subclass of flows with $Re_{\tau,b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the $Re\to \infty$ limit, giving $\theta _c \to 45^\circ$ as $Re\to \infty$.

Recent findings on the Reynolds-number-dependent behaviour of near-wall turbulence in terms of the ‘foot-printing’ of outer large-scale structures call for a new modelling development. A two-scale framework was proposed to couple a local fine-mesh solution with a global coarse-mesh solution by He (Intl J. Numer. Meth. Fluids, vol. 86, 2018, pp. 655–677). The methodology was implemented and demonstrated by Chen & He (J. Fluid Mech, vol. 933, 2022, p. A47) for a canonical turbulent channel flow, where the mesh-count scaling with Reynolds number is potentially reduced from $O(R{e^2})$ for a conventional wall-resolved large-eddy simulation (WRLES) to $O(R{e^1})$. The present work extends the two-scale method to turbulent boundary layers. A two-dimensional roughness element is used to trip a turbulent boundary layer. It is observed that large-scale disturbances originating at the trip have a much shorter lifetime and weaker foot-printing signatures on near-wall flow compared to those long streaky coherent structures in well-developed wall-bounded turbulent flows. Modal analyses show that the impact of trip-induced large scales can be adequately captured by a locally embedded fine-mesh block. For the tripped turbulent boundary layer, a Chebyshev block-spectral mapping is adopted to propagate source terms from the local fine-mesh blocks to the global coarse-mesh domain, driving to a target solution for the upscaled equations. The computed mean statistics and energy spectra are in good agreement with corresponding experimental data, WRLES and direct numerical simulation (DNS) results. The overall mesh count–$Re$ scaling is estimated to reduce from $O(R{e^{1.8}})$ for the full wall-resolved LES to $O(R{e^{0.9}})$ for the present two-scale solution.

The main objective of this work is to develop a unified framework that can be used as a lens to quantitatively assess and augment a wide range of coarse-grained models of turbulence, namely large eddy simulations (LES), hybrid Reynolds-averaged/LES methods and wall-modelled (WM)LES. Taking a turbulent channel flow as an example, optimality is assessed in the wall-resolved limit, the hybrid RANS–LES limit and the WMLES limit, via projections at different resolutions suitable for these approaches. These optimal a priori estimates are shown to have similar characteristics to existing a posteriori solutions reported in the literature. Consistent accuracy metrics are developed for scale-resolving methods using the optimal solution as a reference, and evaluations are performed. We further characterise the slip velocity in WMLES in terms of the near-wall under-resolution and develop a universal scaling relationship. Insights from the a priori tests are used to augment existing slip-based wall models. Various a posteriori tests reveal superior performance over the dynamic slip wall model. Guidance for the development of improved slip-wall models is provided, including a target for the dynamic procedure.

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.

This paper addresses the issue of actuator selection for active flow control by proposing a novel method built on top of a reinforcement learning agent. Starting from a pre-trained agent using numerous actuators, the algorithm estimates the impact of a potential actuator removal on the value function, indicating the agent's performance. It is applied to two test cases, the one-dimensional Kuramoto–Sivashinsky equation and a laminar bidimensional flow around an airfoil at $Re=1000$ for different angles of attack ranging from $12^{\circ }$ to $20^{\circ }$, to demonstrate its capabilities and limits. The proposed actuator-sparsification method relies on a sequential elimination of the least relevant action components, starting from a fully developed layout. The relevancy of each component is evaluated using metrics based on the value function. Results show that, while still being limited by this intrinsic elimination paradigm (i.e. the sequential elimination), actuator patterns and obtained policies demonstrate relevant performances and allow us to draw an accurate approximation of the Pareto front of performances versus actuator budget.

The spatial structure and time evolution of tornado-like vortices in a three-dimensional cavity are studied by topological analysis and numerical simulation. The topology theory of the unsteady vortex in the rectangular coordinate system (Zhang, Zhang & Shu, J. Fluid Mech., vol. 639, 2009, pp. 343–372) is generalized to the curvilinear coordinate system. Two functions $\lambda (q_1,t)$ and $q(q_1,t)$ are obtained to determine the topology structure of the sectional streamline pattern in the cross-section perpendicular to the vortex axis and the meridional plane, respectively. The spiral direction of the sectional streamlines in the cross-section perpendicular to the vortex axis depends on the sign of $\lambda (q_1,t)$. The types of critical points in the meridional plane depend on the sign of $q(q_1,t)$. The relation between the critical points of the streamline pattern in the meridional plane and that in the cross-section perpendicular to the vortex axis is set up. The flow in a three-dimensional rectangular cavity is numerically simulated by solving the three-dimensional Navier–Stokes equations using high-order numerical methods. The spatial structures and the time evolutions of the tornado-like vortices in the cavity are analysed with our topology theory. Both the bubble type and spiral type of vortex breakdown are observed. They have a close relationship with the vortex structure in the cross-section perpendicular to the vortex axis. The bubble-type breakdown has a conical core and the core is non-axisymmetric in the sense of topology. A criterion for the bubble type and the spiral type based on the spatial structure characteristic of the two breakdown types is provided.

Active matter exploits motion to induce changes in shape and conformation via external input. In this paper, we establish theoretically that viscous liquid droplets containing magnetic nanoparticles with frozen-in magnetic moments, sitting on a solid substrate and surrounded by an ambient gas phase, can deform and migrate under the influence of a magnetic torque. The effect arises because the collective rotation of the magnetic nanoparticles at the liquid–gas interface tilts the droplet away from a symmetric configuration, breaks the reflection symmetry with respect to the centre axis, and leads to a left–right asymmetry of the contact angles. A sufficiently strong magnetic torque leads the contact angles to overcome hysteresis effects leading the droplet to migrate. We develop a general framework to explain how symmetry-breaking affects droplet migration. Thus previous results of droplet spreading and migration can be recovered as special cases. Such droplets can be employed as agents in active surfaces and can move against gravity, chemical and thermal gradients, providing a mechanism that could be utilized by both industry and medicine.

In this paper, we analyse the instability of a thin-film suspension of micro-swimmers subjected to an attractant gradient. The imposed attractant gradient introduces a preferential swimming direction for the swimmers, resulting in an anisotropic orientation field. By modelling the swimmers as force dipoles, the study by Kasyap & Koch (Phys. Rev. Lett., vol. 108, issue 3, 2012, 038101, J. Fluid Mech., vol. 741, 2014, pp. 619–657) found an instability arising from the coupling between the active stress associated with an anisotropic orientation field and perturbations to the swimmer density field. In this work we begin by presenting a stability analysis that calculates the modification of this instability in the presence of an interface. We then detail the presence of a new mode of instability that arises solely from the deformation of the interface mediated by the active stress in the suspension. The resulting hydrodynamic instabilities in the system are observed to be sensitive to the direction of the attractant gradient relative to the interface. Furthermore, we show that the coupling between the two modes involving a jump in interfacial viscous stresses and normal stress differences within the suspension allows for an instability to manifest in a suspension of chemotactic pullers, previously thought to be unconditionally stable. We then use a long-wave theory to write down a set of nonlinear equations governing the film evolution. The numerical solution of the system using the long-wave theory aids in explaining the mechanism associated with the instability in addition to validating the predictions from the linear stability theory.

In this article, we present an entrainment-based model for predicting the flow and power output of finite-length wind farms. The model is an extension of the three-layer approach of Luzzatto-Fegiz & Caulfield (Phys. Rev. Fluids, vol. 3, 2018, 093802) for wind farms of infinite length, and assumes dependence of key flow quantities, such as the wind farm bulk velocity, on the streamwise distance from the farm entrance. To assist our analysis and validate the proposed model, we undertake a series of large-eddy simulations with different turbine spacing arrangements and layouts. Comparisons are also made with the top-down model with entrance effects of Meneveau (J. Turbul., vol. 13, 2012, N7) and data from the literature. The finite-length entrainment model is shown to be capable of capturing the power drop between contiguous rows of turbines as well as describing the advection and turbulent transport of kinetic energy in both the entrance and fully developed regions. The fully developed regime is approximated only deep in the wind farm, after approximately 15 rows of turbines. Our data suggest that for the cases considered in this study, the empirical coefficients that can be used to describe turbulent entrainment and transfers above the wind farm exhibit little dependence on the farm layout and may be considered constant for modelling purposes. However, the flow field within the wind farm layer can be strongly modulated by the turbine density (spacing) as well as the array layout, and to that extent it can be argued that they are both primary factors determining the wind farm power output.

When a fluid is injected into a porous medium saturated with an ambient fluid of a greater density, the injected fluid forms a plume that rises upwards due to buoyancy. In the near field of the injection point, the plume adjusts its speed to match the buoyancy velocity of the porous medium, either thinning or thickening to conserve mass. These adjustments are the dominant controls on the near-field plume shape, rather than mixing with the ambient fluid, which occurs over larger vertical distances. In this study, we focus on the plume behaviour in the near field, demonstrating that for moderate injection rates, the plume will reach a steady state, whereby it matches the buoyancy velocity over a few plume width scales from the injection point. However, for very small injection rates, an instability occurs in which the steady plume breaks apart due to the insurmountable density contrast with the surrounding fluid. The steady shape of the plume in the near field depends only on a single dimensionless parameter, which is the ratio between the inlet velocity and the buoyancy velocity. A linear stability analysis is performed, indicating that for small velocity ratios, an infinitesimal perturbation can be constructed that becomes unstable, whilst for moderate velocity ratios, the shape is shown to be stable. Finally, we comment on the application of such flows to the context of CO$_2$ sequestration in porous geological reservoirs.

We propose an interpretable deep learning (DL) model that extracts physical features from turbulence data. Based on a conditional generative adversarial network combined with a new decomposition algorithm for the Prandtl number effect, we developed a DL model that is capable of predicting the local surface heat flux very accurately using only the wall-shear stress information and Prandtl number as inputs in channel turbulence. The considered range of Prandtl number is $Pr = 0.001 \sim 7$, with a focus on the subrange of $Pr = 0.1 \sim 7$. Through an investigation of the gradient maps of the trained prediction model, we were able to identify the nonlinear physical relationship between the wall-shear stresses and heat flux, which is quite diverse depending on the Prandtl number. Furthermore, the decomposition algorithm, which is used to separate the Prandtl number dependent field from the common field of the surface heat flux, helps not only in learning for good prediction of an arbitrary Prandtl number but also in analysing the effect of the Prandtl number on the determination of the heat flux for the given turbulent flow fields. We demonstrate that a physical interpretation of a trained network is possible.

A relation among invariants of filtered velocity gradients with two different filter sizes is derived. Based on this relation and physical reasoning, it is shown analytically that strain self-amplification contributes more to energy transfer than vortex stretching in homogeneous turbulence, as observed in recent numerical investigations of homogeneous isotropic turbulence. We note that the invariant relation studied and hence the inequality between strain self-amplification and vortex stretching apply to all homogeneous flows, not restricted to isotropic turbulence.

The motion of a finite-size particle in the cuboidal lid-driven cavity flow is investigated experimentally for Reynolds numbers $100$ and $200$ for which the flow is steady. These steady three-dimensional flows exhibit chaotic and regular streamlines, where the latter are confined to Kolmogorov–Arnold–Moser (KAM) tori. The interaction between the moving wall and the particle creates global particle attractors. For neutrally buoyant particles, these attractors are periodic or quasi-periodic, strongly attracting and located in or near KAM tori of the flow. As the density mismatch between particle and fluid increases, buoyancy and inertia become important, and the attractors evolve from those for neutrally buoyant particles, changing their shape, position and attraction rates.

Numerically computed with high accuracy are periodic travelling waves at the free surface of a two-dimensional, infinitely deep, and constant vorticity flow of an incompressible inviscid fluid, under gravity, without the effects of surface tension. Of particular interest is the angle the fluid surface of an almost extreme wave makes with the horizontal. Numerically found are the following. (i) There is a boundary layer where the angle rises sharply from $0^\circ$ at the crest to a local maximum, which converges to $30.3787\ldots ^\circ$, independently of the vorticity, as the amplitude increases towards that of the extreme wave, which displays a corner at the crest with a $30^\circ$ angle. (ii) There is an outer region where the angle descends to $0^\circ$ at the trough for negative vorticity, while it rises to a maximum, greater than $30^\circ$, and then falls sharply to $0^\circ$ at the trough for large positive vorticity. (iii) There is a transition region where the angle oscillates about $30^\circ$, resembling the Gibbs phenomenon. Numerical evidence suggests that the amplitude and frequency of the oscillations become independent of the vorticity as the wave profile approaches the extreme form.