We analyse the stability of the interface between two immiscible fluids both flowing in the horizontal direction in a thin cell with vertically varying gap width. The dispersion relation for the growth rate of each mode is derived. The stability is approximately determined by the sign of a simple expression, which incorporates the density difference between the fluids and the effect of surface tension in the along- and cross-cell directions. The latter arises from the varying channel width: if the non-wetting fluid is in the thinner part of the channel, the interface is unstable as it will preferentially migrate into the thicker part. The density difference may suppress or complement this effect. The system is always stable to sufficiently large wavenumbers owing to the along-flow component of surface tension.

Particle–wall interactions have broad biological and technological applications. In particular, some artificial microswimmers capitalize on their translation–rotation coupling near a wall to generate directed propulsion. Emerging biomedical applications of these microswimmers in complex biological fluids prompt questions on the impact of non-Newtonian rheology on their propulsion. In this work, we report some intriguing effects of shear-thinning rheology, a ubiquitous non-Newtonian behaviour of biological fluids, on the translation–rotation coupling of a particle near a wall. One particularly interesting feature revealed here is that the wall-induced translation by rotation can occur in a direction opposite to what might be intuitively expected for an object rolling on a solid substrate. We elucidate the underlying physical mechanism and discuss its implications on the design of micromachines and bacterial motion near walls in complex fluids.

Viscoplastic fluids can hold bubbles/particles stationary by balancing the buoyancy stress with the yield stress – the key parameter here is the yield number $Y$, the ratio of the yield stress to the buoyancy stress. In the present study, we investigate a suspension of bubbles in a yield-stress fluid. More precisely, we compute how much is the gas fraction $\phi$ that could be held trapped in a yield-stress fluid without motion. Here the goal is to shed light on how the bubbles feel their neighbours through the stress field and to compute the critical yield number for a bubble cloud beyond which the flow is suppressed. We perform two-dimensional computations in a full periodic box with randomized positions of the monosized circular bubbles. A large number of configurations are investigated to obtain statistically converged results. We intuitively expect that for higher volume fractions, the critical yield number is larger. Not only here do we establish that this is the case, but also we show that short-range interactions of bubbles increase the critical yield number even more dramatically for bubble clouds. The results show that the critical yield number is a linear function of volume fraction in the dilute regime. An algebraic expression model is given to approximate the critical yield number (semi-empirically) based on the numerical experiment in the studied range of $0\le \phi \le 0.31$, together with lower and upper estimates.

We derive an accurate estimate for the diffusive evaporation rates of multiple droplets of different sizes and arbitrary contact angles placed on a horizontal substrate. The derivation, which is based on a combination of Green's second identity and the method of reflections, simply makes use of the solution for the evaporation of a single droplet. The theoretical results can serve as a guide for future computational and experimental studies on the collective evaporation of arrays of droplets, as well as similar multi-body, diffusion-dominated transport problems.

The problem of unidirectional shoaling of a water-wave field with a narrow energy spectrum is treated by using a new Alber equation. The stability of the linear stationary solution to small non-stationary disturbances is analysed; and numerical solutions for its subsequent long-distance evolution are presented. The results quantify the physics which causes the gradual decay in the probability of freak-wave occurrence, when moving from deep to shallow coastal waters.

We investigate the large-scale signature of the random switches between two mirrored turbulent wake states of flat-backed bodies. A direct numerical simulation (DNS) of the flow around an Ahmed body at a Reynolds number ($Re$) of 10 000 is considered. Using proper orthogonal decomposition (POD), we identify the most energetic modes of the velocity field and build a low-dimensional model based on the first six fluctuating velocity modes capturing the characteristics of the flow dynamics during and between switches. In the absence of noise, the model produces random switches with characteristic time scales in agreement with the simulation and experiments. This chaotic model suggests that random switches are triggered by the increase of the vortex shedding activity. However, the addition of noise results in a better agreement in the temporal spectra of the coefficients between the model and the simulation.

We investigate spanwise-coherent structures in the turbulent flow around airfoils, motivated by their connection with trailing-edge noise. We analyse well-resolved large-eddy simulations (LES) of the flow around NACA 0012 and NACA 4412 airfoils, both at a Reynolds number of 400 000 based on the chord length. Spectral proper orthogonal decomposition performed on the data reveals that the most energetic coherent structures are hydrodynamic waves, extending over the turbulent boundary layers around the airfoils with significant amplitudes near the trailing edge. Resolvent analysis was used to model such structures, using the mean field as a base flow. We then focus on evaluating the dependence of such structures on the domain size, to ensure that they are not an artefact of periodic boundary conditions in small computational boxes. To this end, we performed incompressible LES of a zero-pressure-gradient turbulent boundary layer, for three different spanwise sizes, with the momentum-thickness Reynolds number matching those near the airfoils trailing edge. The same coherent hydrodynamic waves were observed for the three domains. Such waves are accurately modelled as the most amplified flow response from resolvent analysis. The signature of such wide structures is seen in non-premultiplied spanwise wavenumber spectra, which collapse for the three computational domains. These results suggest that the spanwise-elongated structures are not domain-size dependent for the studied simulations, indicating thus the presence of very wide structures in wall-bounded turbulent flows.

We perform a scaling analysis of the terms composing the Ffowcs-Williams and Hawkings (FWH) equation, which rules the propagation of noise generated by a rigid body in motion. Our analysis extends the seminal work of Lighthill (Proc. R. Soc. Lond. A, vol. 211, 1952, pp. 564–587) and the dimensional analysis of classical sources (monopole, dipole and quadrupole) considering all the FWH integral terms. Scaling properties are analysed in light of perfect/imperfect similarity when laboratory-scale data are used for full-scale predictions. As a test case we consider a hydrodynamic example, namely a laboratory-scale ship propeller. The data, obtained numerically in a previous study, were post-processed according to the scaling analysis presented herein. We properly scale the speed of sound to obtain perfect similarity and quantify the error with respect to the imperfect scaling. Imperfect similarity introduces errors in the acoustic response related both to the linear terms and to the nonlinear terms, the latter of great importance when the wake is characterized by robust and organized vorticity. Successively, we analyse the effect of a free surface, often present in hydrodynamic applications. We apply the method of images to the FWH equation. The free surface may generate a frequency-dependent constructive/destructive interference. The analysis of an archetypal acoustic field (monopole) provides robust explanation of these interference effects. Finally, we find that imperfect similarity and the absence of a free surface may introduce errors when model-scale data are used to obtain the full-scale acoustic pressure. The error is small for microphones placed in the near field and becomes relevant in the far field because of the nonlinear terms.

The theoretical foundations of the boundary integral method are considered for inviscid monochromatic internal waves, and an analytical approach is presented for the solution of the boundary integral equation for oscillating bodies of simple shape: an elliptic cylinder in two dimensions, and a spheroid in three dimensions. The method combines the coordinate stretching introduced by Bryan and Hurley in the frequency range of evanescent waves, with analytic continuation to the range of propagating waves by Lighthill's radiation condition. Not only are the waves obtained for arbitrary oscillations of the body, with application to radial pulsations and rigid vibrations, but also the distribution of singularities equivalent to the body, allowing later inclusion of viscosity in the theory. Both a direct representation of the body as a Kirchhoff–Helmholtz integral involving single and double layers together, and an indirect representation involving a single layer alone, are considered. The indirect representation is seen to require a certain degree of symmetry of the body with respect to the horizontal and the vertical. As the surface of the body is approached the single- and double-layer potentials exhibit the same discontinuities as in classical potential theory.

A novel approach is presented for identifying disturbance sources in wall-bounded shear flows. The underlying approach models the flow state, as measured by sensors embedded in the flow, as a mixture of disturbance sources. The degenerate unmixing estimation technique is adopted as a blind source separation technique to recover the separate sources and their unknown mixing process. The efficiency of this approach stems from its ability to isolate any, a priori unknown, number of sources, using two sensors only. Furthermore, by adding a single additional sensor, the method is expanded to also determine the propagation velocity vector of each of the isolated sources, based on sensor readings from three sensors appropriately located in the flow field. Theoretical guidelines for locating the sensors are provided. The power of the method is demonstrated via computer simulations and wind-tunnel experiments. The numerical study considers disturbances comprising discrete Tollmien–Schlichting waves and wave packets. Linear stability theory is used to model source mixtures acquired by sensors placed in a Blasius boundary layer. The experimental study investigates the flow over a flat plate, with hot wires as sensors, and a loudspeaker and plasma actuators as source generators. Based on numerical and experimental demonstrations, it is believed that the new approach should prove useful in various applications, including active control of boundary layer transition from laminar to turbulent flow.

This paper presents laboratory measurements of surface transport due to non-breaking and breaking deep-water focusing surface wave packets and examines the dependence of the transport on the wave packet bandwidth, $\varDelta$. This extends the work of Deike et al. (J. Fluid Mech., vol. 829, 2017, pp. 364–391) and Lenain et al. (J. Fluid Mech., vol. 876, 2019, p. R1), where similar numerical and laboratory experiments were conducted, but the bandwidth was held constant. In this paper, it is shown that the transport is strongly affected by the bandwidth. A model for the horizontal length scale of the breaking region is proposed that incorporates the bandwidth, central frequency, the linear prediction of the slope at focusing and the breaking threshold slope of the wave packet. This is then evaluated with data from archived and new laboratory experiments, and agreement is found. Furthermore, the horizontal length scale of the breaking region implies modifications to the model of the energy dissipation rate from Drazen et al. (J. Fluid Mech., vol. 611, 2008, pp. 307–332). This modification accounts for differing trends in the dissipation rate caused by the bandwidth in the available laboratory data.

Time-dependent flows are notoriously challenging for classical linear stability analysis. Most progress in understanding the linear stability of these flows has been made for time-periodic flows via Floquet theory focusing on time-asymptotic stability. However, little attention has been given to the transient intracyclic linear stability of periodic flows since no general tools exist for its analysis. In this work, we explore the potential of using the recent framework of the optimally time-dependent (OTD) modes (Babaee & Sapsis, Proc. R. Soc. Lond. A, vol. 472, 2016, 20150779) to extract information about both the transient and the time-asymptotic linear stability of pulsating Poiseuille flow. The analysis of the instantaneous OTD modes in the limit cycle leads to the identification of the dominant instability mechanism of pulsating Poiseuille flow by comparing them with the spectrum and the eigenmodes of the Orr–Sommerfeld operator. In accordance with evidence from recent direct numerical simulations, it is found that structures akin to Tollmien–Schlichting waves are the dominant feature over a large range of pulsation amplitudes and frequencies but that for low pulsation frequencies these modes disappear during the damping phase of the pulsation cycle as the pulsation amplitude is increased beyond a threshold value. The maximum achievable non-normal growth rate during the limit cycle was found to be nearly identical to that in plane Poiseuille flow. The existence of subharmonic perturbation cycles compared with the base flow pulsation is documented for the first time in pulsating Poiseuille flow.

The goal of this study is to elucidate the effect the particle moment of inertia (MOI) has on the dynamics of spherical particles rising in a quiescent and turbulent fluid. To this end, we performed experiments with varying density ratios $\varGamma$, the ratio of the particle density and fluid density, ranging from $0.37$ up to $0.97$. At each $\varGamma$ the MOI was varied by shifting mass between the shell and the centre of the particle to vary $I^*$ (the particle MOI normalised by the MOI of a particle with the same weight and a uniform mass distribution). Helical paths are observed for low, and ‘three-dimensional (3-D) chaotic’ trajectories at higher values of $\varGamma$. The present data suggest no influence of $I^*$ on the critical value for this transition $0.42<\varGamma _{{crit}}<0.52$. For the ‘3-D chaotic’ rise mode, we identify trends of decreasing particle drag coefficient ($C_d$) and amplitude of oscillation with increasing $I^*$. Due to limited data it remains unclear if a similar dependence exists in the helical regime as well. Path oscillations remain finite for all cases studied and no ‘rectilinear’ mode is encountered, which may be the consequence of allowing for a longer transient distance in the present compared with earlier work. Rotational dynamics did not vary significantly between quiescent and turbulent surroundings, indicating that for the present configuration these are predominantly wake driven.

This paper first uses a low-speed stereoscopic particle image velocimetry (SPIV) system to measure the convergent statistical quantities of the flow field and then simultaneously measure the time-resolved flow field and the wall mass transfer rate by a high-speed SPIV system and an electrochemical system, respectively. We measure the flow field and wall mass transfer rate under upstream pipe Reynolds numbers between 25 000 and 55 000 at three specific locations behind the orifice plate. Moreover, we apply proper orthogonal decomposition (POD), stochastic estimation and spectral analysis to study the properties of the flow field and the wall mass transfer rate. More importantly, we investigate the large-scale coherent structures’ effects on the wall mass transfer rate. The collapse of the wall mass transfer rates’ spectra by the corresponding time scales at the three specific positions of orifice flow suggest that the physics of low-frequency wall mass transfer rates are probably the same, although the flow fields away from the wall are quite different. Furthermore, the spectra of the velocity reconstructed by the most energetic eigenmodes agree well with the wall mass transfer rate in the low-frequency region, suggesting that the first several energetic eigenmodes capture the flow dynamics relevant to the low-frequency variation of the wall mass transfer. Stochastic estimation results of the velocity field associated with large wall mass transfer rate at all three specific locations further reveal that the most energetic coherent structures are correlated with the wall mass transfer rate.

Binary droplet collisions exhibit a wide range of outcomes, including coalescence and stretching separation, with a transition between these two outcomes arising for high Weber numbers and impact parameters. Our experimental study elucidates the effect of viscosity on this transition, which we show exhibits inertial (viscosity-independent) behaviour over an order-of-magnitude-wide range of Ohnesorge numbers. That is, the transition is not always shifted towards higher impact parameters by increasing droplet viscosity, as it might be thought from the existing literature. Moreover, we provide compelling experimental evidence that stretching separation only arises if the length of the coalesced droplet exceeds a critical multiple of the original droplet diameters (3.35). Using this as a criterion, we provide a simple but robust model (without any arbitrarily chosen free parameters) to predict the coalescence/stretching-separation transition.

A high-order transition route from inertial to elasticity-dominated turbulence (EDT) in Taylor–Couette flows of polymeric solutions has been discovered via direct numerical simulations. This novel two-step transition route is realized by enhancing the extensional viscosity and hoop stresses of the polymeric solution via increasing the maximum chain extension at a fixed polymer concentration. Specifically, in the first step inertial turbulence is stabilized to a laminar flow much like the modulated wavy vortex flow. The second step destabilizes this laminar flow state to EDT, i.e. a spatially smooth and temporally random flow with a $-3.5$ scaling law of the energy spectrum reminiscent of elastic turbulence. The flow states involved are distinctly different to those observed in the reverse transition route from inertial turbulence via a relaminarization of the flow to elasto-inertial turbulence in parallel shear flows, underscoring the importance of polymer-induced hoop stresses in realizing EDT that are absent in parallel shear flows.

Active particles often swim in confined environments. The transport mechanisms, especially the global one as reflected by the Taylor dispersion model, are of great practical interest to various applications. For the active dispersion process in confined flows, previous analytical studies focused on the long-time asymptotic values of dispersion characteristics. Only several numerical studies preliminarily investigated the temporal evolution. Extending recent studies of Jiang & Chen (J. Fluid Mech., vol. 877, 2019, pp. 1–34; vol. 899, 2020, A18), this work makes a semi-analytical attempt to investigate the transient process. The temporal evolution of the local distribution in the confined-section–orientation space, drift, dispersivity and skewness, is explored based on moments of distributions. We introduce the biorthogonal expansion method for solutions because the classic integral transform method for passive transport problems is not applicable due to the self-propulsion effect. Two types of boundary condition, the reflective condition and the Robin condition for wall accumulation, are imposed respectively. A detailed study on spherical and ellipsoidal swimmers dispersing in a plane Poiseuille flow demonstrates the influences of the swimming, shear flow, initial condition, wall accumulation and particle shape on the transient dispersion process. The swimming-induced diffusion makes the local distribution reach its equilibrium state faster than that of passive particles. Although the wall accumulation significantly affects the evolution of the local distribution and the drift, the time scale to reach the Taylor regime is not obviously changed. The shear-induced alignment of ellipsoidal particles can enlarge the dispersivity but impacts slightly on the drift and the skewness.

In this work we show that horizontal gradients of temperature and salinity with compensating effects on density can drive thermohaline intrusion in the fluid layer below. Specifically, different types of double diffusive convection generate differential vertical fluxes from the top boundary, which then sustain horizontal temperature and salinity gradients within the bulk. Interleaving layers develop in the bulk and slope downward towards the cold fresh side, which are of the diffusive type. New layers emerge near the bottom boundary and shift the existing layers upward due to the density difference induced by the divergence of the vertical fluxes through the top surface. Detailed analyses reveal that the present intrusion is consistent with those in the narrow fronts, and both layer thickness and current velocity follow the corresponding scaling laws. Such intrusion process provides an extra path to transfer heat and salinity horizontally towards the cold and fresh side, but transfer the density anomaly towards the warm and salty side. These findings extend the circumstances where thermohaline intrusions may be observed.

Double-diffusive convection, in which a fluid is acted upon by two fields (such as temperature and salinity) that affect the density, has been widely studied in areas as diverse as the oceans and stellar atmospheres. Assuming classical Fickian diffusion for both heat and salt, the evolution of temperature and salinity are governed by parabolic advection–diffusion equations. In reality, there are small extra terms in these equations that render the equations hyperbolic (the Maxwell–Cattaneo effect). Although these corrections are nominally small, they represent a singular perturbation and hence can lead to significant effects when the underlying differences of salinity and temperature are large. In this paper, we investigate the linear stability of a double-diffusive fluid layer and show that amending Fick's law for the temperature, or the salinity, alone can lead to new modes of oscillation and to very large changes in the preferred wavelength of oscillatory convection at onset. In particular, the salt finger regime of classical double diffusion is here replaced by Maxwell–Cattaneo oscillations when the salt concentration is very high. The more complicated case when both laws are amended is left to a future paper, now in preparation.

Phase effect on the modal interaction of flow instabilities is investigated for laminar-to-turbulent transition in a flat-plate boundary-layer flow. Primary and secondary three-dimensional (3-D) oblique waves at various initial phase differences between these two instability modes. Three numerical methods are used for a systematic approach for the entire transition process, i.e. before the onset of transition well into fully turbulent flow. Floquet analysis predicts the subharmonic resonance where a subharmonic mode locally resonates for a given basic flow composed of the steady laminar flow and the fundamental mode. Because Floquet analysis is limited to the resonating subharmonic mode, nonlinear parabolised stability equation analysis (PSE) is conducted with various phase shifts of the subharmonic mode with respect to the given fundamental mode. The application of PSE offers insights on the modal interaction affected by the phase difference up to the weakly nonlinear stage of transition. Large-eddy simulation (LES) is conducted for a complete transition to turbulent boundary layer because PSE becomes prohibitively expensive in the late nonlinear stage of transition. The modulation of the subharmonic resonance with the initial phase difference leads to a significant delay in the transition location up to $\Delta Re_{x, tr} \simeq 4\times 10^5$ as predicted by the current LES. Effects of the initial phase difference on the spatial evolution of the modal shape of the subharmonic mode are further investigated. The mechanism of the phase evolution is discussed, based on current numerical results and relevant literature data.