Oscillatory flows have the potential to overcome long-standing limitations encountered when using steady flows for inertial focusing due to low particle Reynolds numbers. The periodic displacement generated by oscillatory flow increases the total path length travelled by a suspended particle with no net displacement within a channel or the need for increased channel length. The effects of unsteady inertial forces on inertial focusing, however, have not been thoroughly examined. Here, we present a combined theoretical and experimental study on the effect of Womersley number on inertial focusing in planar pulsatile flows. The migration velocity for a small and weakly inertial particle was evaluated for different oscillation frequencies using the method of matched asymptotics. Using experiments in a custom-built microchannel, we show that oscillatory flows are remarkably efficient for inertial focusing, even at low particle Reynolds numbers. For appropriately selected oscillation amplitude and frequency, inertial focusing is achieved in only a fraction of the channel length (1 % to 10 %) compared to what would be required in a steady flow. We show that the Womersley number influences the equilibrium focusing position and the overall focusing efficiency. In fact, above a critical Womersley number, inertial focusing does not occur despite increasing particle Reynolds number. Lastly, the application of oscillatory inertial focusing for the direct measurement of particle migration velocity is demonstrated.

This paper addresses peristaltic flow induced in a non-axisymmetric annular tube by a periodic small-amplitude wave of arbitrary shape propagating axially along its inner surface, assumed to be a circular cylinder. The study is motivated by recent in vivo experimental observations pertaining to the flow of cerebrospinal fluid along the perivascular spaces of cerebral arteries. The analysis employs the lubrication approximation, describing low-Reynolds-number peristaltic flow in the long-wavelength approximation. Closed-form analytic expressions are derived for the average pumping rate in infinitely long tubes and also in tubes of finite length. Consideration is also given to the transverse motion arising in non-axisymmetric tubes. For small-amplitude waves, the solution is reduced to the integration of a parameter-free Stokes-flow problem, which is solved for relevant cross-sectional shapes, with closed-form analytical results derived for thin canals.

A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a ‘pure’ point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter $A$ in the hybrid solutions approaches the limits $A\to 0,\infty$. While $A\to 0$ reproduces the input equilibrium, $A\to \infty$ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by $A$ as a ‘Liouville link’. In some cases, the emergent point vortex equilibrium as $A\to \infty$ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a ‘Liouville chain’. We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717) and by Krishnamurthy et al. (J. Fluid Mech., vol. 874, 2019, R1) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (Proc. R. Soc. A, vol. 476, 2020, 20200310) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the two-dimensional Euler equation.

We prove a new explicit inequality for the non-dimensional flow force constant, significantly improving the Benjamin and Lighthill conjecture about irrotational steady water waves. As a corollary, we prove a bound for the wave amplitude in terms of the Bernoulli constant. We show that the amplitude decays as $r^{-2}$ when $r \to +\infty$, where $r$ is the non-dimensional Bernoulli constant. We explain that the latter limit corresponds to deep water waves and the bound for the amplitude is sharp. In terms of physical parameters the result states that the amplitude $a$ of an arbitrary Stokes wave is bounded by $C m^2 g/Q^2$, where $m$ is the relative mass flux, $g$ is the gravitational constant, $Q$ is the total head and $C$ is an absolute constant given explicitly. In particular, this implies that $a < C c^2 g^{-1}$, where $c$ is the wave speed. The latter inequality is valid for all Stokes waves, irrespective of wavelength or amplitude, including extreme waves.

The characteristics of acoustic tones near the nozzle of jets are investigated for Mach numbers between ${M}=0.50$ and 2 using large-eddy simulations. Peaks are observed in all cases. They are tonal for ${M}\geq 0.75$ and broaden for lower Mach numbers. They are associated with the azimuthal modes $n_{\theta }=0$ to $n_{\theta }^{max}$, with $n_{\theta }^{max}=8$ for ${M}=0.75$ and 1 for ${M}=2$, for example. Their frequencies do not appreciably vary with the nozzle-exit boundary-layer thickness and turbulence and fall in the frequency bands predicted for the upstream-propagating guided jet waves using a vortex-sheet model. For all azimuthal modes, the peak intensities are highest for the first radial guided jet mode. They increase approximately as ${M}^8$ for ${M}\leq 1$ and as ${M}^3$ for ${M}\geq 1$, following the scaling laws of jet noise, suggesting that they mainly result from a band-pass filtering of the upstream-travelling sound waves by the guided jet modes. In support of this, the Mach number variations of the peak width and sharpness are explained by the eigenfunctions of the guided waves. Moreover, it appears that, for high subsonic Mach numbers, among the waves possibly resonating in the potential core, only those close to the cutoff frequencies of the guided jet modes can contribute to the near-nozzle peaks. Finally, the peaks are detectable in the far field for large radiation angles. For ${M}= 0.90$, for instance, they emerge for angles higher than $135^\circ$.

In this paper we study finite particle Reynolds number effects up to $Re_p=50$ on the dynamics of small spherical bubbles and solid particles in an isotropic turbulent flow. We consider direct numerical simulations of light pointwise particles with various expressions of the drag force to account for finite $Re_p$ and the type of particle. Namely, we consider the Stokes drag law, the Schiller and Neumann relation and the Mei law. We show that an effective Stokes number, based on the mean value of the drag coefficient to account for the inertial effects involved in the drag law, gives a quasi-self-similar evolution of the variances of the bubble acceleration and of the forces exerted on the particle. This allows us to provide a satisfactory prediction of these quantities using Tchen's theory at finite particle Reynolds number. Based on these relations, we can specify the conditions under which the total inertial force (sum of the added mass and the Tchen contributions) is negligible compared with the drag force. Thus, for particles of very small dimensions, the fluid inertia force is negligible, provided the density ratio is of order 1 or larger. However, when the particle inertia becomes consequential, the threshold value of the density ratio increases significantly. Although this corresponds to the limit of the validity of the model, this draws attention to the fact that, for large Stokes numbers, the added mass and fluid inertia forces could play a more important role than is usually attributed to them.

Viscous drops, subject to a linear flow in an immiscible viscous fluid, deform. When the resulting drop shape is simple the problem can be addressed by an asymptotic approach or by approximating the deformation using known simple shapes. When the resulting deformation is more complex, the problem is usually addressed numerically. In this paper, we address the problem of drops that are deforming in an axisymmetric compressional (bi-axial extensional) flow. Yielded shapes are flat drops, flat drops with dimples and toroidal drops. The latter two are highly unstable. We propose to approximate the solution of this problem, approximating the shapes by using generalized Cassini ovals, defined herein. The analysis reproduced the branches with shapes of stationary stable flat drops and stationary unstable toroidal drops, available from numerical calculation. Furthermore, it predicts the point of loss of stability of the flat drop to exhibit the transition branch that leads into the formation of the toroidal shapes, and shows that this branch shows stationary, yet unstable, flat drops with ever growing dimples up to collapse.

The theoretical existence of the granular monoclinal wave, based on the Saint-Venant equations for flowing granular matter, was reported recently by Razis et al. (J. Fluid Mech., vol. 843, 2018, pp. 810–846). The present paper focuses on the mathematical interpretation of its behaviour, treating the equation of motion that describes any granular waveform as a dynamical system, taking also into consideration the Froude number offset $\varGamma$ introduced by Forterre & Pouliquen (J. Fluid Mech., vol. 486, 2003, pp. 21–50). The critical value of the Froude number below which stable uniform flows are observed is determined directly from the stability analysis of the aforementioned dynamical system. It is shown that the granular monoclinal wave, represented as a heteroclinic orbit in phase space, can be categorized into two classes: (i) the mild class, for which the exact form of the waveform can be approximated by the non-viscous (first-order) adaptation of the granular Saint-Venant equations, and (ii) the steep class, for a description of which a second-order (viscous) term in the Saint-Venant equations is absolutely needed to capture the dynamics of the wave. The mathematical criterion that distinguishes the two classes is the changing sign of the trace of the Jacobian matrix evaluated at the fixed point corresponding to the waveform's lower plateau.

The role of the different helical components of the magnetic and velocity fields in the inverse spectral transfer of magnetic helicity is investigated through Fourier shell-to-shell transfer analysis. Magnetic helicity transfer analysis is performed on chosen data from direct numerical simulations of homogeneous isothermal compressible magnetohydrodynamic turbulence, subject to both a large-scale mechanical forcing and a small-scale helical electromotive driving. The root mean square Mach number of the hydrodynamic turbulent steady state taken as initial condition varies from 0.1 to about 11. Three physical phenomena can be distinguished in the general picture of the spectral transfer of magnetic helicity towards larger spatial scales: local inverse transfer (LIT), non-local inverse transfer (NLIT) and local direct transfer (LDT). A shell decomposition allows these three phenomena to be associated with clearly distinct velocity scales: the LDT is driven by large-scale velocity shear and associated with a direct magnetic energy cascade; the NLIT is mediated by small-scale velocity fluctuations which couple small- and large-scale magnetic structures; and the LIT by the intermediate spatial scales of the velocity field. The helical decomposition shows that like-signed helical interactions and interactions with the compressive velocity field are predominant. The latter has a high impact on the LDT and on the NLIT, but plays no role for the LIT. The locality and relative strength of the different helical contributions are mainly determined by the triad helical geometric factor, derived here in the compressible case.

The impact of a drop train, a series of identical liquid drops separated by a constant distance, on a liquid pool initially generates a long slender cavity. However, the cavity soon collapses and turns into a shallow funnel. Here we theoretically model the dynamic profile of the elongated cavity and the steady shape of the funnel, which are then shown to agree well with experiment. When the liquid inertia plays a dominant role, the cavity assumes a parabolic profile that depends only on the drop diameter and the centre-to-centre spacing of adjacent drops. We consider the capillary forces as well as the drop impact force to obtain the shape of the funnel that persists once the elongated cavity collapses. Our study allows for predicting the interfacial morphology by the impact of a drop train and designing impact conditions useful for semiconductor cleaning processes.

The limiting configuration of interfacial solitary waves between two homogeneous fluids consisting of a sharp $120^{\circ }$ angle with an enclosed bubble of stagnant heavier fluid on top is investigated numerically. We use a boundary integral equation method to compute the almost limiting profiles which are nearly self-intersecting and thus extend the work of Pullin & Grimshaw (Phys. Fluids, vol. 31, 1988, pp. 3550–3559) by obtaining the overhanging solutions for very small density ratios. To further study the local configuration of the limiting profile, we propose a reduced model that replaces the $120^{\circ }$ angle with two straight solid walls intersecting at the bottom of the bubble. Using a series truncation method, a one-parameter family of solutions depending on the angle between the two solid walls (denoted by $\gamma$) is found. When $\gamma = {2{\rm \pi} }/{3}$, it is shown that the simplified model agrees well with the near-limiting wave profile if the density ratio is small, and thus provides a good local approximation to the assumed limiting configuration. Interesting solutions for other values of $\gamma$ are also explored.

The flexural-gravity symmetric waves propagating in an ice channel with a lead of open water in the ice cover are investigated within the linear theory of hydroelasticity. The ice sheets are modelled as either elastic or as rigid plates. Deflections of the elastic ice sheets are described by the linear elastic plate equation. The flexural-gravity waves propagating along the channel and their dispersion relations are obtained by the normal mode method. It is shown that the dispersion relations for elastic ice approach the dispersion relations for rigid ice as the rigidity of ice increases or the wavelength decreases. The region in the plane ice thickness/wavenumber, where the ice cover can be treated as rigid with a prescribed accuracy, is determined. The effects of the ice thickness and width of the lead on the phase speeds are studied to determine critical speeds of a vehicle moving along the channel. It is shown that wave modes have several critical speeds, which may merge for some parameters of the channel. The presence of the lead significantly changes the characteristics of waves in an ice channel. Short waves, which propagate in a channel fully covered with ice, weakly depend on gravity and the presence of water in the channel. However, short waves propagating in the same channel but with a lead of open water are gravity-dominated waves localised in the lead. We conclude that flexural-gravity waves in an ice channel with a lead are less pronounced than in the channel completely covered by ice.

This work is dedicated to studying the similarity between wakes of wind turbines with different yaw angles and tip speed ratios under different turbulent inflows using large-eddy simulations with actuator surface models. Simulation results show that wake characteristics from cases with different yaw angles overlap with each other when normalized properly, which include the streamwise variations of the wake deflection, the centreline velocity deficit, the widths of the wakes, the standard deviations of instantaneous wake centre positions and the instantaneous wake widths. Different scalings are proposed for the streamwise velocity deficit and the transverse velocity. The similarities observed between cases with different yaw angles and the different scalings suggest that it is proper to decompose the wake of a yawed wind turbine into a streamwise wake and a lateral wake deflection, which is critical for developing analytical models. The mean of the instantaneous wake widths and the mean of the instantaneous centreline streamwise velocity are observed as being smaller than those of the time-averaged wake. These quantities are then related by using two analytical expressions proposed in this work. The observed similarities together with the proposed analytical expressions provide a better understanding of wakes of yawed wind turbines and can be employed to develop physics-based dynamic wake models.

Jet-like flows are ubiquitous in the atmosphere and oceans, and thus a thorough investigation of their behaviour in rotating systems is fundamental. Nevertheless, how they are affected by vegetation or, generally speaking, by obstructions is a crucial aspect which has been poorly investigated up to now. The aim of the present paper is to propose an analytical model developed for jet-like flows in the presence of both obstructions and the Coriolis force. In this investigation the jet-like flow is assumed homogeneous, turbulent and quasi-geostrophic, and with the same density as the surrounding fluid. Laws of momentum deficit, length scales, velocity scales and jet centreline are analytically deduced. These analytical solutions are compared with some experimental data obtained using the Coriolis rotating platform at LEGI-Grenoble (France), showing a good agreement.

The flow physics of turbulent boundary layers is investigated using spectral analysis based on the spanwise scale decomposition of the Reynolds stress transport equation, with data obtained from a direct numerical simulation of the turbulent boundary layer at $Re_\tau \simeq 2020$. Here, we extend the framework of Kawata & Alfredsson (Phys. Rev. Lett., vol. 120, 2018, p. 244501) for plane Couette flows to zero-pressure-gradient boundary layers. The equation contains three fundamental fluxes, which govern the Reynolds stress transport: (i) a scale flux of the interaction between small-scale and large-scale structures, and two spatial fluxes dominated by (ii) pressure and (iii) turbulent transport along the wall-normal direction. The scale flux reveals evidence of the inverse turbulent kinetic energy transfer, from small to large scales, occurring at the near-wall region, whereby for the scale flux of the Reynolds shear stress transport, the inverse transfer extends across the entire boundary layer. The wall-normal fluxes reveal the interactions occurring between scales at the buffer and logarithmic regions. In addition, there is interaction between the large-scale structures and the free stream flow occurring at the edge of the boundary layer, which was not observed in the Couette flow. Flow structures associated with inverse interscale transport of Reynolds shear stress are identified by applying conditional analysis to the spectrally decomposed velocity fields. While the inverse transport is interpreted as the net energy transfer from small-scale ejections ($Q2$) and sweeps ($Q4$) to the large-scale counterparts, conditional time estimates of the direct and inverse interscale transport reveal that both processes play a substantial role across a broad range of scales.

High-resolution data from the large-eddy simulation of the atmospheric boundary layer (ABL) over a vegetation canopy are used to investigate the interaction between the most energetic large-scale structures from the ABL and the smaller scales from the near-canopy region. First, evidence of amplitude modulation (AM) involving the three velocity components is demonstrated. A multi-scale analysis of the transport equation of both the turbulent kinetic energy (TKE) and Reynolds shear stress (RSS) is then performed using a multi-level filtering procedure. It is found that, on average, in the investigated region, large scales are a source of TKE for the small scales (e.g. forward scatter of TKE) through nonlinear interscale transfer ($T_r^L$) with a maximum at canopy top while they are a sink via turbulent transport ($T_t^L$). Close to the canopy, the small-scale RSS transport behaves the same while, above the roughness sublayer, $T_r^L$ and $T_t^L$ switch roles showing the existence of RSS backscatter. The standard deviation of the transfer terms shows that there are intense instantaneous forward and backscatter of both TKE and RSS via all the transfer terms. It is therefore demonstrated that there is a two-way coupling between the ABL and the near-canopy scales, the well-known top-down mechanism through TKE transfer being complemented by a bottom-up feedback through RSS transfer. This analysis is extended to several stability regimes, confirming the above conclusions and showing the increasing role of the large-scale wall-normal component in AM and TKE or RSS transfers when the flow becomes buoyancy driven.

Understanding the development and breakup of interfacial waves in a two-phase mixing layer between the gas and liquid streams is paramount to atomization. Due to the velocity difference between the two streams, the shear on the interface triggers a longitudinal instability, which develops to interfacial waves that propagate downstream. As the interfacial waves grow spatially, transverse modulations arise, turning the interfacial waves from quasi-two-dimensional to fully three-dimensional. The inlet gas turbulence intensity has a strong impact on the interfacial instability. Therefore, parametric direct numerical simulations are performed in the present study to systematically investigate the effect of the inlet gas turbulence on the formation, development and breakup of the interfacial waves. The open-source multiphase flow solver, PARIS, is used for the simulations and the mass–momentum consistent volume-of-fluid method is used to capture the sharp gas–liquid interfaces. Two computational domain widths are considered and the wide domain will allow a detailed study of the transverse development of the interfacial waves. The dominant frequency and spatial growth rate of the longitudinal instability are found to increase with the inlet gas turbulence intensity. The dominant transverse wavenumber, determined by the Rayleigh–Taylor instability, scales with the longitudinal frequency, so it also increases with the inlet gas turbulence intensity. The holes formed in the liquid sheet are important to the disintegration of the interfacial waves. The hole formation is influenced by the inlet gas turbulence. As a result, the sheet breakup dynamics and the statistics of the droplets formed also change accordingly.

Particle collisions in turbulent flow are critical to particle agglomeration and droplet coalescence. The collision kernel can be evaluated by radial distribution function (RDF) and radial relative velocity (RV) between particles at small separations $r$. Previously, the smallest $r$ was limited to roughly the Kolmogorov length $\eta$ due to particle position uncertainty and image overlap. We report a new approach to measuring RDF and RV near contact ($r/a \approx 2.07$, where $a$ is particle radius). Three-dimensional particle tracking velocimetry using the four-pulse shake-the-box algorithm recorded short tracks with the interpolated midpoints registered as particle positions, avoiding image overlap and track mismatch. We measured RDF and RV of inertial particles in a one metre diameter isotropic air turbulence chamber with Taylor Reynolds number $Re_\lambda =324$, $a=12 - 16\ \mathrm {\mu }\textrm {m}$ $({\approx }0.12\eta )$ and Stokes number ${\approx }0.7$. At large $r$ the measured RV agrees with the literature, but when $r<20\eta$ the first moment of negative RV starts to increase, reaching 10 times higher values than direct numerical simulations of non-interacting particles. Likewise, RDF scales as $r^{-0.39}$ when $r>\eta$, reflecting the well-known scaling for polydisperse particles, but when $r\lessapprox \eta$, RDF scales as $r^{-6}$, yielding 1000 times higher near-contact RDF than simulations. Such RV enhancement and extreme clustering at small $r$ can be attributed to particle–particle interactions including hydrodynamic interactions, which are not well-understood. Uncertainty analysis substantiates the observed trends. This first-ever simultaneous RDF and RV measurement at small separations provides a clear glimpse into the clustering and relative velocities of particles in turbulence near-contact.

We investigate the dynamics of cohesive particles in homogeneous isotropic turbulence, based on one-way coupled simulations that include Stokes drag, lubrication, cohesive and direct contact forces. We observe a transient flocculation phase, followed by a statistically steady equilibrium phase. We analyse the temporal evolution of floc size and shape due to aggregation, breakage and deformation. Larger turbulent shear and weaker cohesive forces yield smaller elongated flocs. Flocculation proceeds most rapidly when the fluid and particle time scales are balanced and a suitably defined Stokes number is $O(1)$. During the transient stage, cohesive forces of intermediate strength produce flocs of the largest size, as they are strong enough to cause aggregation, but not so strong as to pull the floc into a compact shape. Small Stokes numbers and weak turbulence delay the onset of the equilibrium stage. During equilibrium, stronger cohesive forces yield flocs of larger size. The equilibrium floc size distribution exhibits a preferred size that depends on the cohesive number. We observe that flocs are generally elongated by turbulent stresses before breakage. Flocs of size close to the Kolmogorov length scale preferentially align themselves with the intermediate strain direction and the vorticity vector. Flocs of smaller size tend to align themselves with the extensional strain direction. More generally, flocs are aligned with the strongest Lagrangian stretching direction. The Kolmogorov scale is seen to limit floc growth. We propose a new flocculation model with a variable fractal dimension that predicts the temporal evolution of the floc size and shape.

We analyse a way to make flight vehicles harvest energy from homogeneous turbulence by fast tracking in the way that falling inertial particles do. Mean air speed increases relative to flight through quiescent fluid when turbulent eddies sweep particles and vehicles along in a productive way. Once swept, inertia tends to carry a vehicle into tailwinds more often than headwinds. We introduce a forcing that rescales the effective inertia of rotorcraft in computer simulations. Given a certain thrust and effective inertia, we find that flight energy consumption can be calculated from measurements of mean particle settling velocities and acceleration variances alone, without the need for other information. In calculations using a turbulence model, we optimize the balance between the work performed to generate the forcing and the advantages induced by fast tracking. The results show net energy reductions of up to approximately 10 % relative to flight through quiescent fluid and mean velocities up to 40 % higher. The forcing expands the range of conditions under which fast tracking operates by a factor of approximately ten. We discuss how the mechanism can operate for any vehicle, how it may be even more effective in real turbulence and for fixed-wing aircraft and how modifications might yield yet greater performance.