We provide a rigorous analysis of the self-similar solution of the temporal turbulent boundary layer, recently proposed by Biau (2023 Comput. Fluids 254, 105795), in which a body force is used to maintain a statistically steady turbulent boundary layer with periodic boundary conditions in the streamwise direction. We derive explicit expressions for the forcing amplitudes which can maintain such flows, and identify those which can hold either the displacement thickness or the momentum thickness equal to unity. This opens the door to the first main result of the paper, which is to prove upper bounds on skin friction for the temporal turbulent boundary layer. We use the Constantin–Doering–Hopf bounding method to show, rigorously, that the skin-friction coefficient for periodic turbulent boundary layer flows is bounded above by a uniform constant which decreases asymptotically with Reynolds number. This asymptotic behaviour is within a logarithmic correction of well-known empirical scaling laws for skin friction. This gives the first evidence, applicable at asymptotically high Reynolds numbers, to suggest that Biau’s self-similar solution of the temporal turbulent boundary layer exhibits statistical similarities with canonical, spatially evolving, boundary layers. Furthermore, we show how the identified forcing formula implies an alternative, and simpler, numerical implementation of periodic boundary layer flows. We give a detailed numerical study of this scheme presenting direct numerical simulations up to a momentum Reynolds number of $\textit{Re}_\theta = 2000$ and implicit large-eddy simulations up to $\textit{Re}_\theta = 8300$, and show that these results compare well with data from canonical spatially evolving boundary layers at equivalent Reynolds numbers.

We examine the dispersion of prolate spheroidal microswimmers in pressure-driven channel flow, with the emphasis on a novel anomalous scaling regime. When time scales corresponding to swimmer orientation relaxation, and diffusion in the gradient and flow directions, are all well separated, a multiple scales analysis leads to a closed form expression for the shear-enhanced diffusivity, $D_{\it{eff}}$, governing the long-time spread of the swimmer population along the flow (longitudinal) direction. This allows one to organize the different $D_{\it{eff}}$-scaling regimes as a function of the rotary Péclet number (${\it{{\it{Pe}}}}_r)$, where the latter parameter measures the relative importance of shear-induced rotation and relaxation of the swimmer orientation due to rotary diffusion. For large ${\it{{\it{Pe}}}}_r$, $D_{\it{eff}}$ scales as $O({\it{{\it{Pe}}}}_r^4D_t)$ for $1 \leqslant \kappa \lesssim 2$, and as $O({\it{{\it{Pe}}}}_r^{ {10}/{3}}D_t)$ for $\kappa = \infty$, with $D_t$ being the intrinsic translational diffusivity of the swimmer arising from a combination of swimming and rotary diffusion, and $\kappa$ being the swimmer aspect ratio; $\kappa = 1$ for spherical swimmers. For $2 \lesssim \kappa \lt \infty$, the swimmers collapse onto the centreline with increasing ${\it{{\it{Pe}}}}_r$, leading to an anomalously reduced longitudinal diffusivity of $O({\it{{\it{Pe}}}}_r^{5-C(\kappa )}D_t)$. Here, $C(\kappa )\!\gt \!1$ characterizes the algebraic decay of swimmer concentration outside an $O({\it{{\it{Pe}}}}_r^{-1})$ central core, with the anomalous exponent $(5-C)$ governed by large velocity variations occasionally sampled by swimmers outside this core. Here, $C(\kappa )\gt 5$ for $\kappa \gtrsim 10$, leading to $D_{\it{eff}}$ eventually decreasing with increasing ${\it{{\it{Pe}}}}_r$, in turn implying a flow-independent maximum, at a finite ${\it{{\it{Pe}}}}_r$, for the rate of slender swimmer dispersion.

Inertial sedimentation of a cloud of cylinders released within a confined fluid-filled cell is experimentally investigated. Various cylinder numbers, $N_c$, aspect ratios, $\xi$, solid-to-fluid density ratios, $\rho _c / \rho _{\!f}$, and settling velocities corresponding to moderate Reynolds numbers are examined. The parameters correspond to two distinct path regimes for isolated cylinders: oscillatory trajectories for higher-density cylinders and rectilinear sedimentation for lower-density cylinders. In both cases, we observe the formation of subgroups (termed objects of class $N$) composed of $N$ cylinders in contact, as well as their recombination due to splitting or merging. Depending on the parameters, specific distributions of class-$N$ objects are found. In addition, beyond the formation of individual objects, large-scale vertical columnar structures emerge, made of densely packed objects and alternating regions of ascending and descending fluid. These structures, driven by complex interactions between local clustering and global flow organisation, which persist throughout the sedimentation process, are highly sensitive to $\xi$. Despite its inner complex dynamics, the group is observed to sediment as a collective entity, with a constant velocity exceeding that of an isolated cylinder. This velocity may be predicted from multi-scale information. Fluctuating velocities of the objects are further analysed. Different mechanisms for horizontal and vertical components are identified. Horizontal fluctuations are related to intrinsic particle mobility, while vertical fluctuations are attributed to strong wakes and vertical streams. Both fluctuations are mainly influenced by the cylinders’ aspect ratio, which also affects the structural and spatial distribution of the objects.

Identifying self-similarity is key to understanding and modelling a plethora of phenomena in fluid mechanics. Unfortunately, this is not always possible to perform formally in highly complex flows. We propose a methodology to extract the similarity variables of a self-similar physical process directly from data, without prior knowledge of the governing equations or boundary conditions, based on an optimisation problem and symbolic regression. We analyse the accuracy and robustness of our method in five problems which have been influential in fluid mechanics research: a laminar boundary layer, Burger’s equation, a turbulent wake, a collapsing cavity and decaying turbulence. Our analysis considers datasets acquired via both numerical and wind tunnel experiments. The algorithm recovers the known self-similarity expressions in the first four problems and generates new insights into single length scale theories of homogeneous turbulence.

In this work, the correlations between streamwise velocity and temperature fluctuations are investigated in compressible turbulent channel flows from the perspective of coherent structures. The intense fluctuation structures and quadrant-event structures of both velocity and temperature have been identified, extracted separately and compared. Analyses show that although their structure sizes are similar in the whole channel, high correlation only exists in the near-wall region with a high overlapping rate of the instantaneous structures. The hierarchy of the temperature structures are passively formed following the dynamic process of the velocity such as ejections, which contributes to the remaining correlation in the outer layer. However, this passive scalar property cannot provide the production mechanism in the outer layer according to the budget analysis after scale decomposition, and the interscale energy transfer progress is also different from the velocity fluctuation field. Therefore, the temperature structures deviate from the velocity structures in the outer layer and cannot be carried by the following dynamic process of the velocity such as sweeps, passively, which can be found from the conditional averaged structures. All of these findings provide a new perspective for understanding the velocity–temperature relationship in compressible channel flows.

The interaction of near-inertial waves (NIWs) with submesoscale vorticity filaments is explored using theory and simulations. We study three idealised set-ups representative of submesoscale flows allowing for $O(1)$ or greater Rossby numbers. First, we consider the radiation of NIWs away from a cyclonic filament and develop scalings for the decay of wave energy in the filament. Second, we introduce broad anticyclonic regions that separate the cyclonic filaments mimicking submesoscale eddy fields and analyse the normal modes of this system. Third, we extend this set-up to consider the vertical propagation and the radiation of NIW energy. We identify a key length scale $L_m$, dependent on the strength of the filament, stratification and vertical scale of the waves, that when compared with the horizontal scales of the background flow determines the NIW behaviour. A generic expression for the vertical group velocity is derived that highlights the importance of horizontal gradients for vertical wave propagation. An overarching theme of the results is that NIW radiation, both horizontally and vertically, is most efficient when $L_m$ is comparable to the length scales of the background flow.

Elastic turbulence can lead to increased flow resistance, mixing and heat transfer. Its control – either suppression or promotion – has significant potential, and there is a concerted ongoing effort by the community to improve our understanding. Here we explore the dynamics of uncertainty in elastic turbulence, inspired by an approach recently applied to inertial turbulence in Ge et al. (J. Fluid Mech., vol. 977, 2023, A17). We derive equations for the evolution of uncertainty measures, yielding insight on uncertainty growth mechanisms. Through numerical experiments, we identify four regimes of uncertainty evolution, characterised by (i) rapid transfer to large scales, with large-scale growth rates of $\tau ^{6}$ (where $\tau$ represents time), (ii) a dissipative reduction of uncertainty, (iii) exponential growth at all scales and (iv) saturation. These regimes are governed by the interplay between advective and polymeric contributions (which tend to increase uncertainty), viscous, relaxation and dissipation effects (which reduce uncertainty) and inertial contributions. In elastic turbulence, reducing Reynolds number increases uncertainty at short times, but does not significantly influence the growth of uncertainty at later times. At late times, the growth of uncertainty increases with Weissenberg number, with decreasing polymeric diffusivity and with the logarithm of the maximum length scale, as large flow features adjust the balance of advective and relaxation effects. These findings provide insight into the dynamics of elastic turbulence, offering a new approach for the analysis of viscoelastic flow instabilities.

In this study, we experimentally investigate the stress field around a gradually contaminated bubble as it moves straight ahead in a dilute surfactant solution with an intermediate Reynolds number ($20 \lt {{\textit{Re}}} \lt 220$) and high Péclet number. Additionally, we investigate the stress field around a falling sphere unaffected by surface contamination. A newly developed polarisation measurement technique, highly sensitive to the stress field in the vicinity of the bubble or the sphere, was employed in these experiments. We first validated this method by measuring the flow around a solid sphere sedimenting in a quiescent liquid at a terminal velocity. The measured stress field was compared with established numerical results for ${{\textit{Re}}} = 120$. A quantitative agreement with the numerical results validated this technique for our purpose. The results demonstrated the ability to determine the boundary layer. Subsequently we measured a bubble rising in a quiescent surfactant solution. The drag force on the bubble, calculated from its rise velocity, was set to transiently vary from that of a clean bubble to a solid sphere within the measurement area. With the intermediate drag force between clean bubble and solid sphere, the stress field in the vicinity of the bubble front was observed to be similar to that of a clean bubble, and the structure near the rear was similar to that of a solid sphere. Between the front and rear of the bubble, the phase retardation exhibited a discontinuity around the cap angle at which the boundary conditions transitioned from no slip to slip, indicating an abrupt change in the flow structure. A reconstruction of the axisymmetric stress field from the phase retardation and azimuth obtained from polarisation measurements experimentally revealed that stress spikes occur around the cap angle. The cap angle (stress jump position) shifted as the drag on the bubble increased owing to surfactant accumulation on its surface. Remarkably, the measured cap angle as a function of the normalised drag coefficient quantitatively agreed with the numerical results at intermediate ${{\textit{Re}}} = 100$ of Cuenot et al. (1997 J. Fluid Mech. 339, 25–53), exhibiting only a slight deviation from the curve predicted by the stagnant cap model at low ${\textit{Re}}$ (creeping flow) proposed by Sadhal & Johnson (1983 J. Fluid Mech. 126, 237–250).

This study examines the reflection of a rightward-moving shock (RMS) over expansion waves, dividing the reflection structure into three components. The first component analyses the pre- and post-interaction parts of the expansion waves, categorising primary flow patterns into four types with defined transition criteria, visualised through Mach contours. The second component investigates the curved perturbed shock. Through numerical simulations, the influence of increasing shock strength on the flow structures is displayed. A triple point forms for an RMS of the first family, and the Mach stem height increases with the increase of shock strength. When the RMS is strong enough, a vortex forms in the near-wall region, which acts like a wedge to distort the near-foot part of the RMS. The third component, the near-foot region, is analysed using a one-dimensional Riemann problem approach. The calculated wave speeds are used to mark waves in Mach contours for eight cases. The position of the waves indicates that the left-going shock for an RMS of the first family or the right-going shock for an RMS of the second family corresponds to the foot of the RMS. This can explain the finding that the right-hand side of an RMS of the first family or the left-hand side of an RMS of the second family is disturbed. The regions to have different wave patterns solved from the one-dimensional Riemann problem are displayed in the original Mach number–shock speed Mach number plane.

The early stage of a gravity-driven flow resulting from the sudden removal of a floating body is investigated. Initially, the fluid is at rest, with a rigid, symmetric wedge floating on its surface. The study focuses on the initial evolution of the wedge-shaped depression formed on the water’s free surface. The fluid has finite depth, and the resulting flow is assumed to be governed by potential theory. The initial flow is described by a linear boundary-value problem, which is solved using conformal mapping and the theory of complex analytic functions. The behaviour of the flow velocity near the corner points of the fluid domain is analysed in detail. It is shown that the linear theory predicts a power-law singularity in the flow velocity at the vertex of the wedge-shaped depression, with the exponent depending on the wedge angle. As the cavity extends toward the bottom, the flow singularity at the vertex becomes stronger. The local flow near the vertex is shown to be self-similar at leading order in the short-time limit. At the other two corner points – where the initial free surface intersects the surface of the wedge – the linear theory predicts continuous velocities with singular velocity gradients. Theoretical predictions are compared with numerical results obtained using OpenFOAM. Good agreement is observed at short times, except in small vicinities of the corner points, where inner solutions are required. In practical applications, understanding the short-time behaviour of the depressions is important for predicting jet formation in regions of high surface curvature.

A model for galloping detonations is conceived as a sequence of very fast re-ignitions followed by long periods of evolution with quenched reactions. Numerical simulations of the one-dimensional Euler equations are conducted in this limit. While the mean speed and structure is found in reasonable agreement with Chapman–Jouguet theory, very strong pulsations of the lead shock appear, along with a train of rear-facing N-waves. These dynamics are analysed using characteristics. A closed-form solution for the lead shock dynamics is formulated, which is found in excellent agreement with numerics. The model relies on the presence of a single time scale of the process, the pulsation period, which controls the shock dynamics via the shock change equations and establishes a shock decay with a single time constant. These long periods of shock decay with known dynamics are punctuated by energy release events, with ‘kicks’ in the shocked speed controlled by the pressure increase and resulting lead shock amplification. Model predictions are found in excellent agreement with previous numerical results of pulsating detonations far from the stability limit.

This numerical investigation focuses on the mechanisms, flow topology and onset of Kelvin–Helmholtz instabilities (KHIs), that drive the leading-edge shear-layer destabilisation in the wake of wall-mounted long prisms. Large-eddy simulations are performed at ${\textit{Re}} = 2.5\times 10^3, 5\times 10^3$ and $1\times 10^4$ for prisms with a range of aspect ratio (AR, height-to-width) between $0.25$ and $1.5$, and depth ratios (DR, length-to-width) of $1{-}4$. Results show that shear-layer instabilities enhance flow irregularity and modulate spanwise vortex structures. The onset of KHI is strongly influenced by depth ratio, such that long prisms (${\textit{DR}}= 4$) experience earlier initiation compared with shorter ones (${\textit{DR}}= 1$). At higher Reynolds numbers, the onset of KHI shifts upstream towards the leading-edge, intensifying turbulence kinetic energy and increasing flow irregularity, especially for long prisms. The results further show that in this configuration, energy transfer from the secondary recirculation region contributes to the destabilisation of the leading-edge shear layer by reinforcing low-frequency modes. A feedback mechanism is identified wherein energetic flow structures propagate upstream through reverse boundary-layer flow, re-energising the leading-edge shear layer. Quantification using probability density functions reveals rare, intense upstream energy convection events, driven by this feedback mechanism. These facilitate the destabilisation process regardless of Reynolds number. This study provides a comprehensive understanding of the destabilisation mechanisms for leading-edge shear layers in the wake of wall-mounted long prisms.

Undulatory slender objects have been a central theme in the hydrodynamics of swimming at low Reynolds number, where the slender body is usually assumed to be inextensible, although some microorganisms and artificial microrobots largely deform with compression and extension. Here, we theoretically study the coupling between the bending and compression/extension shape modes, using a geometrical formulation of kinematic microswimmer hydrodynamics to deal with the non-commutative effects between translation and rotation. By means of a coarse-grained minimal model and systematic perturbation expansions for small bending and compression/extension, we analytically derive the swimming velocities and report three main findings. First, we revisit the role of anisotropy in the drag ratio of the resistive force theory, and generally demonstrate that no motion is possible for uniform compression with isotropic drag. We then find that the bending–compression/extension coupling generates lateral and rotational motion, which enhances the swimmer’s manoeuvrability, as well as changes in progressive velocity at a higher order of expansion, while the coupling effects depend on the phase difference between the two modes. Finally, we demonstrate the importance of often-overlooked Lie bracket contributions in computing net locomotion from a deformation gait. Our study sheds light on compression as a forgotten degree of freedom in swimmer locomotion, with important implications for microswimmer hydrodynamics, including understanding of biological locomotion mechanisms and design of microrobots.