The incipient cavitation of a pair of unequal strength counter-rotating vortices undergoing the long-wavelength Crow instability is examined with high-speed video, acoustic measurements and volumetric particle tracking velocimetry. This work expands upon the previous studies of Chang & Ceccio ( J. Acoust. Soc. Am. , vol. 130, 2011, pp. 3209–3219) and Chang et al. ( Phys. Fluids , vol. 24, 2012, 014107). Volumetric velocimetry results presented by Knister et al. ( J. Fluid Mech. , 2026) were used to predict the core pressures of the stretched secondary vortices. These data are combined with free-stream nuclei measurements to predict the rates of cavitation inception, which compared well with the directly measured inception rates. The acoustic emissions of incipient cavitation events are also related to the vortex properties and the nuclei content of the water. The reduced pressure in the stretched vortices is shown to be related primarily to the reduction in the core radius of the secondary vortex and not due to axial jetting or straining. The measured vortex dynamics indicates that the process leading to the pressure drop in the secondary vortex core is a transient process but not more rapid than the development of the Crow instability. In conclusion, these results show that a relatively simple model of cavitation inception in a stretched secondary vortex captures the essential physics connecting the nuclei population and the underlying vortical flow field, enabling prediction of the resulting observed inception rates. These results also indicate that the reduced pressures in vortical flow leading to inception are primarily due to the reduction in the core of the vortices and not due to substantive axial jetting. The pressure drop in the vortex cores is accordingly a transient process, but despite the appearances of cavitation inception it does not proceed faster than the development of the Crow instability in the secondary vortices.

We investigate the stability of the flow past two side-by-side square cylinders (at Reynolds number 200 and gap ratio 1) using tools from dynamical systems theory. The flow is highly irregular due to the complex interaction between the flapping jet emanating from the gap and the vortices shed in the wake. We first perform spectral proper orthogonal decomposition (SPOD) to understand the flow characteristics. We then conduct Lyapunov stability analysis by linearising the Navier–Stokes equations around the irregular base flow and find that it has two positive Lyapunov exponents. The covariant Lyapunov vectors (CLVs) are also computed. Contours of the time-averaged CLVs reveal that the footprint of the leading CLV is in the near-wake, whereas the other CLVs peak further downstream, indicating distinct regions of instability. SPOD of the two unstable CLVs is then employed to extract the dominant coherent structures and oscillation frequencies in the tangent space. For the leading CLV, the two dominant frequencies match closely with the prevalent frequencies in the drag coefficient spectrum and correspond to instabilities due to vortex shedding and jet-flapping. The second unstable CLV captures the subharmonic instability of the shedding frequency. Global linear stability analysis (GLSA) of the time-averaged flow identifies a neutral eigenmode that resembles the leading SPOD mode of the first CLV, with a very similar structure and frequency. However, while GLSA predicts neutrality, Lyapunov analysis reveals that this direction is unstable, exposing the inherent limitations of the GLSA when applied to chaotic flows.

Contrary to accepted turbulence folklore, which holds that no mathematical relation exists between the Navier–Stokes equations (NSEs) and the multifractal model (MFM) of Parisi and Frisch, we develop a theory that reconciles the MFM with Leray’s weak solutions of Navier–Stokes analysis. From a combination of Euler invariant scaling and the NSEs set in a three-dimensional box of side $L$ , we also derive the Paladin–Vulpiani scale $\eta_{h,pav}$ which is related to the Reynolds number Re by $L\eta _{h,\textit{pa}v}^{-1} = \textit{Re}^{1/(1+h)}$ , and which acts as a mediator between the two theories. This is achieved by considering $L^{2m}$ -norms of the velocity gradient to find a correspondence between $m$ and the local scaling exponent $h$ in the multifractal model. The parameter $m$ acts as if it were the sliding focus control on a telescope which allows us to zoom in and out on different structures. The range $1 \leqslant m \leqslant \infty$ is equivalent to $-{{ {2}/{3}}} \leqslant h_{\textit{min}} \leqslant {{{1}/{3}}}$ , which lies precisely in the region where Bandak et al. ( Phys. Rev. E , 2022, vol. 105, p. 065113; Phys. Rev. Lett. , 2024, vol. 132, p. 104002) have suggested that thermal noise makes the NSEs inadequate and generates spontaneous stochasticity. The implications of this are discussed.

We consider the axisymmetric, radial extrusion of Newtonian and shear-thinning, power-law fluids from a cylindrical source, which displace an ambient inviscid fluid of equal density. In unconfined geometries, the upper and lower fluid interfaces are stress free, and the flow is dominated by extensional stresses everywhere. In a layer of extruded shear-thinning fluid, a radially growing viscosity field, associated with a radially decaying velocity field, causes the current to bulge near the cylindrical source, with the thickness of the layer growing without bound over time. In contrast, with a Newtonian fluid, the thickness of the fluid layer never exceeds the height of the cylindrical source. We compute numerical solutions to this system, and find similarity solutions describing its late-time behaviour for values of the rheological power-law exponent $1\leqslant n\leqslant 3/2$ . We also consider extrusion between parallel plates, in which the shear-thinning fluid displaces the inviscid fluid and fills the cell completely up to a grounding line, beyond which it separates from the boundaries to extend freely. In this case, we find similarity solutions for values of the power-law exponent $n \geqslant 1$ .

The orientation dynamics of a spheroidal magnetic particle in a viscous fluid subject to a rotating magnetic field is analysed for realistic two-parameter models for the magnetic moment. It is shown that the equations can be mapped onto those for a spherical magnetic particle in a steady magnetic field subject to shear flow. Time evolution equations for the azimuthal and meridional angles of the orientation vector are derived from the condition that the sum of the hydrodynamic and magnetic torques is zero in the viscous limit. One parameter is $\omega ^{\dagger}$ , the ratio of the magnetic field frequency and the particle viscous relaxation rate. For the non-hysteretic Langevin model, the second parameter is the ratio of the saturation moment $m_s$ and the susceptibility $\chi$ times the magnetic field $H,\!(m_s/\chi\! H)$ . There is parallel corotation of the particle with the field for $\omega ^{\dagger} \lt \omega _b^{\dagger}$ , and parallel slip relative to the magnetic field for high $\omega ^{\dagger} \gt \omega _b^{\dagger}$ , where $\omega _b^{\dagger}$ is the breakdown frequency. For the hysteretic Stoner–Wohlfarth model, the second parameter is $h$ , the ratio of the Zeeman energy and the anisotropy energy due to the misalignment between the moment and the particle axis. There are three states, parallel corotation for low $\omega ^{\dagger}$ , precessed corotation for high $\omega ^{\dagger}$ and low $h$ , where the orientation precesses relative to the axis of rotation of the magnetic field, and parallel slip at high $\omega ^{\dagger}$ and high $h$ .

This study experimentally investigates bubble size evolution and void fraction redistribution in an unexplored, coalescence-dominated regime of a decaying turbulent bubbly flow. The flow is generated downstream of a regenerative pump in a duct, with bulk Reynolds number ( Re ) $\sim \mathcal{O}(10^5)$ , Taylor-scale Reynolds number ( Re $_\lambda$ ) $\sim \mathcal{O}(10^3)$ and void fraction ( $\phi$ ) $\sim \mathcal{O}(1\,\%)$ , where the inlet turbulence is extremely intense (turbulence intensity $\gt 30\,\%$ ) but decays rapidly along the duct. Shadowgraph imaging and particle shadow velocimetry are used for measurements. The experimentally obtained turbulent dissipation in the duct flow decays as $\varepsilon \sim \mathcal{L}^{-2}$ , where $\mathcal{L}$ is the axial position, in close agreement with the homogeneous isotropic turbulence prediction of $\varepsilon \sim \mathcal{L}^{-2.2}$ . High-speed imaging and statistical analysis reveal that bubble coalescence dominates over breakup across most of the domain, leading to monotonic growth in the Sauter mean diameter ( $d_{32}$ ) and progressive broadening of the bubble size distribution. The normalised extreme-to-mean diameter ratio ( $\mathcal{D}$ ) increases axially and asymptotically from ${\sim} 1.9$ (breakup regime) and saturates at ${\sim} 2.2$ (coalescence regime), indicating the emergence of a quasi-self-similar bubble size distribution. The probability density function of the bubble diameter exhibits a dual power-law tail with exponents $-10/3$ and $-3/2$ near the duct inlet. However, after a few hydraulic diameters, a single $-3/2$ power-law scaling emerges, indicating a regime of pure coalescence in which all bubbles are smaller than the Hinze scale. The cumulative distribution plotted against $d/d_{32}$ shows that the slope decreases and the distribution width increases with both axial position and void fraction $(\phi )$ . Although classical Hinze scaling gives $d_{\textit{H}} \propto \mathcal{L}^{0.9}$ , our theory for $d_{32}$ and $d_{99.8}$ (99.8th percentile bubble diameter) in a pure-coalescence regime predicts the slower law $\propto \mathcal{L}^{0.5}$ , which our experimental results confirm – indicating negligible breakup and sub-Hinze growth. Concurrently, in contrast to current models, transient $\phi$ profiles evolve from nearly uniform to sharply core-peaked Gaussian distributions in the developing regime, with increasing centreline values and decreasing near-wall values, due to lift-force reversal. These results provide the first spatially resolved characterisation of coalescence-dominated bubbly flows at high Re , advancing the design of industrial systems as in nuclear cooling and multiphase forming processes (e.g. paper manufacturing, chemical reactors).

Rapid granular free-surface flows on inclined planes can develop secondary vortices aligned with the dominant flow direction. The reason for their formation remains a subject of research, but plausible mechanisms include instabilities driven by (i) dilatation/compressibility, (ii) normal stress differences and (iii) a self-induced Raleigh–Taylor instability caused by segregation of large–dense and small–light particles. In this paper, a set of novel experiments are performed with large and small particles (of the same bulk density), which form longitudinal stripes due to a combination of secondary recirculation and particle-size segregation. A conceptual model is formulated, in which large particles concentrate in the downwelling sections, small particles concentrate in the upwelling sections and a breaking-size-segregation wave separates the two pure phases from one another. In each secondary vortex, the breaking waves allow the large and small particles to continuously recirculate. Assuming that a series of counter-rotating vortices exist, it is shown that this internal cross-slope structure emerges naturally from solving the gravity-shear-driven segregation-advection equations. When viewed from above, this generates a series of alternating bands of large and small particles, that are sharply separated from one another and are aligned with the downslope direction. Each complete stripe (measured from centre to centre of each large band) is formed by two counter-rotating secondary vortices. Despite the apparent order of the steady-state stripes, it is shown that the individual large and small particle paths form complex interpenetrating co-rotating sub-vortices as they avalanche downslope.

Active filaments, such as microtubules with attached cargo-carrying motor proteins, are important dynamic structures for fluid transport in and around living cells. The mathematical models of active filaments appearing in the literature typically involve combinations of follower forces, compressive tangential forces, along the filament, and an opposite force on the fluid that generates an effective surface flow. In this paper, we present a comparative dynamical systems study of active filament models examining the differences in dynamic states that occur when actuation is through follower forces alone, or the effect of surface flows is also included. We consider cases where actuation is applied only at the filament tip, or distributed uniformly along the filament length. By varying actuation strength, we show that the first bifurcations that provide the transition between the upright, whirling and beating states appear in all models. At higher values of actuation, when beating becomes unstable, however, qualitative differences between the models emerge. Those with distributed actuation produce a single, time-dependent state, which for the surface flow model is reminiscent of a rotating helix that periodically changes handedness and rotation direction. Tip actuation, however, yields complex transitions that ultimately produce a chaotic state. We link the differences in dynamics between tip and distributed actuation to differences in their respective internal stress distributions – differences that appear as early as the first bifurcation, where they affect the shapes of the unstable modes.

This paper presents a theoretical and computational investigation into how a propagating three-dimensional vortex modifies ambient turbulence. Using rapid distortion theory and numerical simulations, the study explores both local and non-local changes in the external vorticity field resulting from fluid displacement and stretching. Cases involving structured and unstructured turbulence reveal that the vortex introduces permanent distortions along its path, and alters the far field turbulence through reflux effects. The findings extend classical models by quantifying the impact of vortex-induced strain and displacement on turbulence, offering new insights into turbulent–turbulent interfaces and the role of coherent structures in modulating external turbulent fields.

Sinking marine snow particles, composed primarily of organic matter, control the global export of photosynthetically fixed carbon from the ocean surface to depth. The fate of sedimenting particles is partly regulated by their encounters with suspended objects, which leads to mass accretion and potentially alters their buoyancy, and with bacteria that can colonise the particles and degrade them. Their collision rates are typically calculated using two types of models focusing either on direct (ballistic) interception with a finite interaction range, or advective-diffusive capture with zero interaction range. Yet, since many relevant marine encounter scenarios span across both regimes, quantifying such encounters remains challenging because the two models yield asymptotically different predictions at high Péclet numbers. We reconcile the two approaches by quantifying encounters in the general case using theoretical analysis and simulations. By solving the advection-diffusion equation in Stokes flow around a sphere to model mass transfer to a sinking particle by finite-sized objects, we determine a new formula for the Sherwood number as a function of the Péclet number and the ratio of particle sizes. Contrary to the common assumption, we find that diffusion still plays a significant role in generating encounters even at high Péclet numbers. We predict that at Péclet numbers as high as 10 6 the direct interception model underestimates the encounter rate by up to two orders of magnitude. This overlooked contribution of diffusion to encounters suggests that processes affecting the fate of marine snow may proceed at a rate much higher than previously thought.