We consider the evolution of the bulk bubble-size distribution $N(a,t)$ of large bubbles (Weber number ${\textit {We}}\gg 1$) under free-surface entrainment described generally by an entrainment size distribution $I(a)$ with power-law slope $\gamma$ and large-radius cutoff $a_{max}$. Our main focus is the interaction between turbulence-driven fragmentation and free-surface entrainment, and, for simplicity, we ignore other mechanisms such as degassing, coalescence and dissolution. Of special interest are the equilibrium bulk distribution $N_{eq}(a)$, with local power-law slope $\tilde {\beta }_{eq}(a)$, and the time scale $\tau _c$ to reach this equilibrium after initiation of entrainment. For bubble radii $a\ll a_{max}$, we find two regimes for the dependence of $N_{eq}(a)$ on the entrainment distribution. There is a weak injection regime for $\gamma \ge -4$, where $\tilde {\beta }_{eq}(a)=-10/3$ independent of the entrainment distribution; and a strong injection regime for $\gamma <-4$, where the power-law slope depends on $\gamma$ and is given by $\tilde {\beta }_{eq}(a)=\gamma +2/3$. The weak regime provides a general explanation for the commonly observed $-10/3$ power law originally proposed by Garrett et al. (J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163–2171), and suggests that different weak entrainment mechanisms can all lead to this result. For $a\sim a_{max}$, we find that $N_{eq}(a)$ exhibits a steepening deviation from a power law due to fragmentation and entrainment, similar to what has been observed, but here absent other mechanisms such as degassing. The evolution of $N(a,t)$ to $N_{eq}(a)$ is characterised by the critical time $\tau _c \propto C_f \varepsilon ^{-1/3} {a_{max}}^{2/3}$, where $\varepsilon$ is the turbulence dissipation rate and $C_f$ is a new constant that quantifies the dependence on the daughter size distribution in a fragmentation event. For typical breaking waves, $\tau _c$ can be quite small, limiting the time $t\lesssim \tau _c$ when direct measurement of $N(a,t)$ might provide information about the underlying entrainment size distribution.

In a sheared suspension, the chaotic motion of the particles disperses the suspending liquid and drives an exponential and broadly distributed growth of the elongation of its material lines. This paper addresses experimentally and theoretically the consequences of this complex advection on the mixing of an initially segregated blob of diffusive dye, at large Péclet number and down to the finest scales of mixing. As the suspension is sheared, the combined action of the mean and fluctuating components of the flow stretches and folds the blob into a multi-scale lamellar structure. At short time, overlaps between the lamellae are scarce and the probability distribution concentration can be understood from the local stretching statistics only. At intermediate times, dispersion limits the volume within which lamellae deploy; overlaps become abundant and their statistics determine the concentration distribution. At longer times, the dye becomes homogeneous at the particle scale and the concentration distribution is set by dispersion only. Predictions for both the concentration distribution and the transitions between these successive stages of mixing are provided and compared to the experimental results.

Oscillatory flow features are common in the unstart of hypersonic mixed-compression intakes and can be classified as low-amplitude or high-amplitude oscillatory unstarted flows. The low-amplitude oscillatory unstarted flow is driven by the shear layer from shock interactions ahead of the cowl, while the high-amplitude oscillatory unstarted flow is driven by the separation caused by shock–boundary-layer interaction on the ramp. While previous studies have observed these flow features and reported their associated frequency, there is no simple criterion available for predicting which mode will occur, and there is a lack of consensus on the appropriate frequency scaling parameter. We study a mixed-compression hypersonic intake in a hypersonic wind tunnel by varying the internal contraction ratio and the throttling ratio to observe various kinds of unstart regimes. Two significant conclusions emerge from considering the results for high-throttling-ratio conditions $(TR > 0.55)$ from the current as well as previous studies. Firstly, the actual shock-on-lip condition at the cowl corresponding to the unthrottled condition, as observed from schlieren images, demarcates the boundary between the two modes of oscillatory unstart flows upon throttling. Secondly, a suitable length scale $(l^*)$, defined as the extent of the subsonic region in the unstarted flow (as observed from the experimental schlieren images), gives the appropriate frequency scaling parameter ($f^* = a_0/4l^*$ where $a_0$ is the stagnation acoustic speed).

The vortex-induced vibration of a flexible filament attached behind a stationary cylinder is studied in the two-dimensional, laminar flow regime. We explore the response of the filament for a wide range of flexibility and inertia. Lock-in with a large number of normal modes of the filament, each in a different regime of reduced speed, is observed. Reduced speed is the free-stream speed of the incoming flow non-dimensionalized with the first natural frequency of the structure and the diameter of the cylinder. Several branches, based on response of the filament, are identified and the contributions of various structural modes along these branches are quantified. Contribution from a particular structural mode increases significantly during lock-in, accompanied by a large amplitude of vibration. The transition between different branches is found to be hysteretic and intermittent. The flow exhibits a variety of vortex-shedding patterns, including the $\mathsf {2P+2S}$ mode. The modes of shedding show a systematic variation with amplitude and frequency. The map of vortex-shedding patterns in the amplitude–frequency plane resembles the corresponding map for forced vibration of a rigid cylinder. The transformation of wake from one mode of shedding to another is explained phenomenologically. Variation of rate of energy transfer between the fluid and filament with space and time is analysed to determine optimal placement of transducers for harvesting energy.

This study performs global stability/receptivity analyses of hypersonic flows over a swept blunt body with infinite span. For the first time, we obtain the characteristics of the leading attachment-line mode to the variation of sweep angles from $20^{\circ }$ to $70^{\circ }$. The global eigenfunctions exhibit the characteristics of the attachment-line instability at the leading edge. At the same time, cross-flow (at small sweep angles) or second Mack mode (at larger sweep angles) dominates further downstream. We establish an adjoint-based bi-orthogonal eigenfunction system to address the receptivity problem of such flows to any external forces and boundary perturbations. The receptivity analyses indicate that the global modes are the most responsive to external forces and surface perturbations applied in the vicinity of the attachment line, regardless of the sweep angles. It is also proven that the present global extension of the bi-orthogonal eigenfunction system can be successfully applied to complex hypersonic flows.

Liquid-infused surfaces (LISs) are a promising technique for reducing friction, fouling and icing in both laminar and turbulent flows. Previous work has demonstrated that these surfaces are susceptible to shear-driven drainage. Here, we report a different failure mode using direct numerical simulations of a turbulent channel flow with liquid-infused longitudinal grooves. When the liquid–liquid surface tension is small and/or grooves are wide, we observe travelling-wave perturbations on the interface with amplitudes larger than the viscous sublayer of the turbulent flow. These capillary waves induce a roughness effect that increases drag. The generation mechanism of these waves is explained using the theory developed by Miles for gravity waves. Energy is transferred from the turbulent flow to the LIS provided that there is a negative curvature of the mean flow at the critical layer. Given the groove width, the Weber number and an estimate of the friction Reynolds number, we provide relations to determine whether a LIS behaves as a smooth or rough surface in a turbulent flow.

Instability and transition in the boundary layer on a slender cone ($60^{\circ }$ apex angle) rotating in still fluid are investigated using hot-wire anemometry as well as through linear stability analysis. In contrast to broad cones (including the disk), where a cross-flow instability dominates the transition and different studies report similar transition Reynolds numbers, the reported transition Reynolds numbers on slender cones are scattered. The present experiments provide quantitative experimental datasets and the stability and transition are evaluated based on both the Reynolds number and a Görtler number. The results consistently show that the instability development depends on the Görtler number rather than the Reynolds number and that transition starts at a well-defined Görtler number, whereas the transition Reynolds number depends on the rotational rate. The measured disturbance that first grows in the laminar region has a frequency approximately the same as or twice the rotational rate of the cone, which according to the stability analysis corresponds to the critical frequency of a slightly inclined vortex structure with respect to the cone axis or an axisymmetric vortex structure. These structures are similar to those observed in the flow visualisations of Kobayashi & Izumi (J. Fluid Mech., vol. 127, 1983, pp. 353–364) and considered as being due to a centrifugal instability.

We present experimentally obtained time-averaged vertical temperature profiles $\theta (z)$ in horizontal convection (HC) in water (Prandtl number $Pr \simeq 6$), which were measured near the heating and cooling plates that are embedded in the bottom of HC samples. Three HC rectangular samples of different sizes but the same aspect ratio $\varGamma \equiv L:W:H = 10:1:1$ ($L$, $W$ and $H$ are the length, width and height of the sample, respectively) were used in the experiments, which allowed us to study HC in a Rayleigh-number range $2 \times 10^{10} \lesssim {Ra} \lesssim 9 \times 10^{12}$. The measurements revealed that above the cooling plate, the mean temperature profiles have a universal scaling form $\theta (z/\lambda _c)$ with $\lambda _c$ being a $Ra$-dependent thickness of the cold thermal boundary layer (BL). The $\theta (z/\lambda _c)$-profiles agree well with solutions to a laminar BL equation in HC, which is derived under assumption that the large-scale horizontal velocity achieves its maximum near the plate and vanishes in the bulk. Above the heating plate, the mean temperature field has a double-layer structure: in the lower layer, the $\theta$ profiles scale with the hot thermal BL thickness $\lambda _h$, while in the upper layer, they again scale with $\lambda _c$. Both scaling forms are in good agreement with the solutions to the BL equation with a proper parameter choice.

The application of cavitation-induced shock waves generated at low driving frequencies, known as power ultrasound, is essential for a wide range of fields, such as sonochemistry, lithotripsy, nanomaterials, emulsions and casting, to name but a few. In this paper, we present measurements of the shock wave pressures emitted by cavitating bubbles in water, under ultrasonic excitation produced by an immersed probe oscillating at 24 kHz. A broad-spectrum fibre-optic hydrophone calibrated in the range of 1–30 MHz was used for this purpose. Spectral analysis of the data reveals a consistent resonance peak at a very narrow range of frequencies (3.27–3.43 MHz). Results were confirmed using real-time analysis of high-speed recordings. By eliminating other possible sources, we propose that this new peak might be associated with shock wave emissions from collapsing bubbles. Spatial maps obtained by collating individual shock wave pressures highlight the effect of pressure shielding with increasing input power, attributed to a cloud of bubbles surrounding the probe. This work contributes towards the elucidation of the key properties of cavitation-driven shock waves and the underlying mechanisms, essential in controlling the effectiveness of the external processing conditions on various physical, chemical and biological systems.

Direct numerical simulations (DNS) were carried out to investigate laminar-turbulent transition for a blunt (right) cone ($7^\circ$ half-angle) at Mach 5.9 and zero angle of attack. First, (linear) stability calculations were carried out by employing a high-order Navier–Stokes solver and using very small disturbance amplitudes in order to capture the linear disturbance development. Contrary to standard linear stability theory (LST) results, these investigations revealed a strong ‘linear’ instability in the entropy-layer region for a very short downstream distance for oblique disturbance waves with spatial growth rates far exceeding those of second-mode disturbances. This linear instability behaviour was not captured with conventional LST and/or the parabolized stability equations (PSE). Secondly, a nonlinear breakdown simulation was performed using high-fidelity DNS. The DNS results showed that linearly unstable oblique disturbance waves, when excited with large enough amplitudes, lead to a rapid breakdown and complete laminar-turbulent transition in the entropy layer just upstream of the second-mode instability region.

We calculate analytically the self-similar Stokes flow driven by an externally applied electric field in a multiple coaxial Taylor cone consisting of an arbitrary number of immiscible leaky-dielectric or dielectric fluids. The proposed conical solutions open new avenues for innovative technological applications, some of which are briefly discussed.

Coating of filtration membranes with a polymer brush holds great promise for efficiently preventing the deposition of fouling particles. The polymer chains are compressed by incoming particles, carried with the permeation flow towards the membrane, and consequently exert a repulsive force that acts to keep the particles away from the membrane surface. Here, we theoretically investigate the effect of a polymer brush coating on the permeation-induced hydrodynamic force, $F_{h}$, pulling a particle towards the membrane, and its balance with the steric repulsion exerted by the compressing brush, resisting the particle's approach. Lubrication theory yields an ordinary differential equation for the pressure, from which $F_{h}$ is calculated numerically. Further, an asymptotic analysis is performed for the limiting cases of a dilute or dense brush, providing analytic expressions that demonstrate how brush properties affect $F_{h}$. Finally, the equilibrium position of a particle is evaluated by considering a balance between the opposing forces. The results provide an upper boundary for the brush properties, beyond which the brush is barely compressed under conditions typical of membrane filtration processes. Further increasing the brush density or thickness only decreases the total system permeance, resulting in increased energy consumption. The results shed light on the mechanisms by which a polymer brush affects the forces acting on a foulant particle, providing quantitative measures for assessing the potential efficacy of brush coatings.

An experimental campaign was conducted to examine the impact of an abrupt change in surface geometry on hypersonic boundary-layer instability waves. The primary test configuration consisted of a $5^{\circ }$ half-angle, nominally sharp cone with a $15^{\circ }$ half-angle flare attachment. Tests were conducted at Mach 6 with the unit Reynolds number varying from $3.0\times 10^6$ to $4.9\times 10^6\ \textrm {m}^{-1}$. The $10^{\circ }$ compression was sufficient to create a small separation region at the cone–flare junction at these conditions. Ultra-high-speed schlieren (822 kHz) revealed the propagation of second-mode disturbances with frequencies between 200 and 300 kHz within the upstream boundary layer; when these reached the separation region, radiation of disturbance energy along the separation shock was observed. Tests conducted at low unit Reynolds numbers demonstrated inhibited instability growth (compared to the straight-cone case) through the separation region and the development of low-frequency (${\sim }75\ \textrm {kHz}$) instabilities within the separated shear layer. At higher Reynolds numbers, however, the corner interaction was found to cause rapid breakdown near reattachment, leading to earlier transition than for a straight cone. Analysis of the schlieren images using spectral proper orthogonal decomposition provided a global picture of the structure and development of the second-mode and shear-generated instabilities.

The phenomena of self-sustained shock-wave oscillations over conical bodies with a blunt axisymmetric base subject to uniform high-speed flow are investigated in a hypersonic wind tunnel at Mach number $M = 6$. The flow and shock-wave dynamics is dictated by two non-dimensional geometric parameters presented by the three length scales of the body, two of which are associated with the conical forebody and one with the base. Time-resolved schlieren imagery from these experiments reveals the presence of two disparate states of shock-wave oscillations in the flow, and allows for the mapping of unsteadiness boundaries in the two-parameter space. Physical mechanisms are proposed to explain the oscillations and the transitions of the shock-wave system from steady to oscillatory states. In comparison with the canonical single-parameter problem of shock-wave oscillations over spiked-blunt bodies reported in literature, the two-parameter nature of the present problem introduces distinct elements to the flow dynamics.

The dynamics of intense vorticity is investigated by means of synchronisation experiments in direct numerical simulations of isotropic turbulence. By imposing similar dynamics above the dissipative range, the same structures of intense vorticity appear in two independent turbulent flows, showing that intense vorticity synchronises to large-scale dynamics. Remarkably, this synchronisation takes place despite the presence of chaos, and affects mostly the intense vorticity, but not so much the weak vorticity background, which remains comparatively asynchronous. These results pinpoint the role of large-scale dynamics in the formation of intense vorticity structures, the so-called ‘worms’, and rule out the possibility that they emerge primarily due to interactions within the dissipative range, and then grow or coalesce into elongated structures. The stretching of the vorticity vector by the large-scale rate-of-strain tensor is identified as the mechanism responsible for the synchronisation of intense vorticity, supporting the extended view of vortex stretching as a fundamental inter-scale mechanism in turbulence.

Extreme cavitation scenarios, such as water column separations in hydraulic systems during transient processes caused by large cavitation bubbles, can lead to catastrophic destruction. In the present paper, we study the onset criteria and dynamics of large cavitation bubbles in a tube. A new cavitation number $Ca_2 = {l^*}^{-1} Ca_0$ is proposed to describe the maximum length $L_{max}$ of the cavitation bubble, where $l^*$ is a non-dimensional length of the water column indicating its slenderness, and $Ca_0$ is the classic cavitation number. Combined with the onset criteria for acceleration-induced cavitation ($Ca_1<1$, Pan et al., Proc. Natl Acad. Sci. USA, vol. 114, 2017, pp. 8470–8474), we show that the occurrence of large cylindrical cavitation bubbles requires both $Ca_2<1$ and $Ca_1<1$ simultaneously. We also establish a Rayleigh-type model for the dynamics of large cavitation bubbles in a tube. The bubbles collapse at a finite end speed, and the time from the maximum bubble size to collapse is $T_c=\sqrt {2}\sqrt {lL_{max}}\sqrt {{\rho }/{p_\infty }}$, where $l$ is the length of the water column, $L_{max}$ is the maximum bubble length, $\rho$ is the liquid density and $p_{\infty }$ is the reference pressure in the far field. The analytical results are validated against systematic experiments using a modified ‘tube-arrest’ apparatus, which can decouple acceleration and velocity. The results in the current work can guide design and operation of hydraulic systems encountering transient processes.

Exact solutions are found for an $N$-fold rotationally symmetric, steadily rotating hollow vortex where a continuous real parameter governs its deformation from a circular shape and $N \ge 2$ is an integer. The vortex shape is found as part of the solution. Following the designation ‘V-states’ assigned to steadily rotating vortex patches (Deem & Zabusky, Phys. Rev. Lett., vol. 40, 1978, pp. 859–862) we call the analogous rotating hollow vortices ‘H-states’. Unlike V-states where all but the $N=2$ solution – the Kirchhoff ellipse – must be found numerically, it is shown that all H-state solutions can be written down in closed form. Surface tension is not present on the boundaries of the rotating H-states but the latter are shown to be intimately related to solutions for a non-rotating hollow vortex with surface tension on its boundary (Crowdy, Phys. Fluids, vol. 11, 1999a, pp. 2836–2845). It is also shown how the results here relate to recent work on constant-vorticity water waves (Hur & Wheeler, J. Fluid Mech., vol. 896, 2020, R1) where a connection to classical capillary waves (Crapper, J. Fluid Mech., vol. 2, 1957, pp. 532–540) is made.

The interaction of fluids with surface-mounted obstacles in canopy flows leads to strong turbulence that dominates dispersion and mixing in the neutrally stable atmospheric surface layer. This work focuses on intermittency in the Lagrangian velocity statistics in a canopy flow, which is observed in two distinct forms. The first, small-scale intermittency, is expressed by non-Gaussian and not self-similar statistics of the velocity increments. The analysis shows an agreement in comparison with previous results from homogeneous isotropic turbulence (HIT) using the multifractal model, extended self-similarity and velocity increments’ autocorrelations. These observations suggest that the picture of small-scale Lagrangian intermittency in canopy flows is similar to that in HIT and, therefore, they extend the idea of universal Lagrangian intermittency to certain inhomogeneous and anisotropic flows. Second, it is observed that the root mean square of energy increments along Lagrangian trajectories depends on the direction of the trajectories’ time-averaged turbulent velocity. Subsequent analysis suggests that the flow is attenuated by the canopy drag while leaving the structure function's scaling unchanged. This observation implies the existence of large-scale intermittency in Lagrangian statistics. Thus, this work presents a first empirical evidence of intermittent Lagrangian velocity statistics in a canopy flow that exists in two distinct senses and occurs due to different mechanisms.

A three-layer asymptotic structure for turbulent pipe flow is proposed revealing, in terms of intermediate variables, the existence of a Reynolds-number-invariant logarithmic region for the streamwise mean velocity and variance. The formulation proposes a local velocity scale (which is not the friction velocity) for the intermediate layer and results in two overlap layers. We find that the near-wall overlap layer is governed by a power law for the pipe for all Reynolds numbers, whereas the log law emerges in the second overlap layer (the inertial sublayer) for sufficiently high Reynolds numbers ($Re_{\tau }$). This provides a theoretical basis for explaining the presence of a power law for the mean velocity at low $Re_{\tau }$ and the coexistence of power and log laws at higher $Re_{\tau }$. The classical von Kármán ($\kappa$) and Townsend–Perry ($A_1$) constants are determined from the intermediate-scaled log-law constants; $\kappa$ shows a weak trend at sufficiently high $Re_{\tau }$ but falls within the commonly accepted uncertainty band, whereas $A_1$ exhibits a systematic Reynolds-number dependence until the largest available $Re_{\tau }$. The key insight emerging from the analysis is that the scale separation between two adjacent layers in the pipe is proportional to $\sqrt {Re_{\tau }}$ (rather than $Re_{\tau }$) and therefore the approach to an asymptotically invariant state can be expected to be slow.

The transient cavity dynamics during water entry of a heavy, non-rotating sphere impacting a rotating pool of liquid is studied experimentally, numerically and theoretically. We show that the pool rotation advances the transition of the cavity type – from deep seal to surface seal – marked by a reduction in the transitional Froude number. The role of the dimensionless rotational number $\mathcal {S} \equiv \omega R_0/U_0$ on the transient cavity dynamics is unveiled, where $R_0$ is the sphere radius, $\omega$ the angular speed of the liquid and $U_0$ the impact velocity. The rotating background liquid has two discernible effects on the cavity evolution. Firstly, an increase in the underwater pressure field due to centripetal effects; and secondly, a reduction in the pressure of airflow in the cavity neck near the water surface. The non-dimensional pinch-off time of the deep seal shows a robust $1/2$ power-law dependence on the Froude number, but with a reducing prefactor for increasing $\omega$. Our findings reveal that the effects of a rotating background liquid on the water entry can be traced back to the subtle differences in the initial stage splash and the near-surface cavity dynamics.