High-resolution simulations of gravity currents in the lock-exchange configuration are conducted to study the flow within the head. The simulations exhibit the geometric features of the head as reported in the laboratory experiments and numerical simulations, and provide more detailed information on the flow within the head of a gravity current. The flow in the lower part of the head, where the lobes and clefts are forming at the leading edge, is qualitatively different from but interconnected to the flow in the upper part of the head, where steepening bulges are protruding from the upright surface above the clefts. Interestingly, regions of positive and negative streamwise vorticity are observed not only in the lower part of the head but also in the upper part of the head at staggered spanwise locations. We have shown that both the streamwise vorticity at the leading edge of the lobes in the lower part of the head and the streamwise vorticity at the steepening bulges in the upper part of the head are contributed from the twisting of spanwise vorticity into the streamwise direction, due to the geometric features of the lobes and the steepening bulges, and contributed from the baroclinic production of vorticity. Our results from visualization using tracers indicate that the ambient fluid ingested in and rising from the clefts is being swept towards the leading edge of a gravity current before being carried upwards from the leading edge to the upright surface above the left and right neighbouring lobes. Furthermore, the heavy fluid inside a lobe may descend towards the bottom boundary, move forward towards the leading edge and outwards towards the neighbouring clefts, and ultimately be carried upwards to the upright surface above the left and right neighbouring lobes. With the knowledge that the erosive power of a gravity current is concentrated in the head region, it is plausible that the bed material, once resuspended by a gravity current, may be lifted up away from the bottom boundary and be dispersed in both the streamwise and spanwise directions. The present study complements existing findings in the literature and provides new insights into the three-dimensional flow field within the head of a gravity current.

We analyze a dataset from a numerically simulated, temporally evolving turbulent wake (Zhou, Phys. Rev. Fluids, vol. 7, 2022, 104802) that exhibits spontaneous anisotropic layering under strong stratification, alongside significant spatiotemporal variability within the flow. This analysis focuses on the irreversible flux coefficient, $\varGamma$, defined as the ratio between turbulent potential and kinetic energy dissipation rates. We find that the volume-averaged $\varGamma$ initially rises, reaches a plateau between 0.45 and 0.49 when the layering dynamics become dominant, and then decreases as viscosity plays a larger role. These peak $\varGamma$ values are consistent with those from prior simulations under strongly stratified conditions. Such efficient mixing occurs when the Ozmidov to Thorpe length scale ratio is between 0.37 and 0.52, consistent with numerical and field data reported by Mashayek et al. (J. Fluid Mech., vol. 826, 2017, pp. 522–552). To account for the coexistence of dynamically distinct regions within the flow, we perform conditional sampling of $\varGamma$ against a locally defined gradient Richardson number, ${\textit {Ri}}$. This reveals a flux-gradient relation between $\varGamma$ and ${\textit {Ri}}$ that remains largely consistent over time. This relation features a large, approximately constant value of $\varGamma$ for ${\textit {Ri}}$ values greater than one, echoing the ‘constant-power’ scenario postulated by Balmforth et al. (J. Fluid Mech., vol. 355, 1998, pp. 329–358).

Understanding the effect of intricate surface wettability conditions on microswimmers is crucial for precisely navigating them across narrow microcirculatory networks. Here, we adopt the spherical squirmer model and Navier slip condition to delineate the microswimmer locomotion under a Poiseuille flow in a slit microchannel. Through a combined analytical–numerical approach utilizing bispherical coordinates and the superposition technique, we resolve the slip-modulated simultaneous hydrodynamic interaction with substrate boundaries. Phase portraits reveal that slip significantly alters propulsion mechanisms, destabilizing centreline stable oscillations of pullers beyond a threshold slip length. Superhydrophobic surfaces suppress near-wall rheotaxis states but preserve centreline focusing, facilitating slip-assisted directed transport without surface accumulation. Under strong background flows, subcritical Hopf bifurcation emerges for pullers at a critical slip length, transitioning dynamics from coexisting stable and unstable states to purely unstable behaviour. Contrastingly, for pushers, slip causes a transition from unstable to either stable or fixed-amplitude oscillations. Increased slip length reduces hydrodynamic repulsion on pullers from the walls by enhancing rotational velocity near the walls, whereas it counteracts the torque that causes unstable oscillations of pushers. Three-dimensional analysis of the trajectories reveals the significant role of the out-of-plane orientation of the microswimmer in its transitions between different swimming states. The presented regime maps offer parametric combinations for specific motion behaviours, guiding the development of smart microfluidic drug delivery systems and preventing biofilm deposition in biomedical devices.

The turbulent wake behind a flat-back Ahmed body is investigated using stacked stereoscopic particle image velocimetry. The wake is disturbed by a steady jet from the centre of the base and the effects are quantified for key blowing rates. The unactuated wake exhibits bistable dynamics in the horizontal plane that are completely subdued for the optimal blowing case, yielding a base drag reduction of 9 %. The three-dimensional mean wake is reconstructed and used to evaluate the wake mass fluxes whose equilibrium determines the recirculation length. The results for the unactuated wake show that up to 80 % of replenishment fluid flux entering the recirculation bubble from the free-stream flow is provided through the low-pressure side of the base, where the symmetry-breaking shear layer roll-up occurs near the base. For the optimal blowing configuration, where the wake becomes symmetric, the flux of wake replenishment is severely reduced. This flow configuration results in elongated shear layers on all sides, which terminate the bubble with a roll-up of reduced intensity at a further downstream location. The dominant cause of bubble growth and the accompanying drag reduction is attributed to the momentum of the base blowing, and the new regime is referred to as the ‘favourable momentum regime’. Similar trends are observed when the model is at $5^{\circ }$ yaw where a reduction of drag and yaw-induced asymmetry are obtained. Proper orthogonal decomposition of the wake reveals the coherent structures related to the bistable flow and the symmetric wake under optimal blowing coefficient.

To study two-dimensional dispersive waves propagating through turbulent flows, a new and less restrictive fast waves approximation is proposed using a multiscale setting. In this ansatz, large and small scales of the turbulence are treated differently. Correlation lengths of the random small-scale turbulence components can be considered negligible in the wave packet propagating frame. Nevertheless, the large-scale flow can be relatively strong, to significantly impact wavenumbers along the propagating rays. New theoretical results, numerical tools and proxies are derived to describe ray and wave action distributions. All model parameters can be calibrated robustly from the large-scale flow component only. We illustrate our purpose with ocean surface gravity waves propagating in different types of surface currents. The multiscale solution is demonstrated to efficiently document wave trapping effects by intense jets.

Trapped surface waves have been observed in a swimming pool trapped by, and rotating around, the cores of vortices. To investigate this effect, we have numerically studied the free-surface response of a Lamb–Oseen vortex to small perturbations. The fluid has finite depth but is laterally unbounded. The numerical method used is spectrally accurate, and uses a novel non-reflecting buffer region to simulate a laterally unbounded fluid. While a variety of linear waves can arise in this flow, we focus here on surface gravity waves. We investigate the linear modes of the vortex as a function of the perturbation azimuthal mode number and the vortex rotation rate. We find that at low rotation rates, linear modes decay by radiating energy to the far field, while at higher rotation rates modes become nearly neutrally stable and trapped in the vicinity of the vortex. While trapped modes have previously been seen in shallow water surface waves due to small perturbations of a bathtub vortex, the situation considered here is qualitatively different owing to the lack of an inward flow and the dispersive nature of non-shallow-water waves. We also find that for slow vortex rotation rates, trapped waves propagate in the opposite direction to the vortex rotation, whereas, above a threshold rotation rate, waves corotate with the flow.

The first two parts of this study showed by experiment and by large-eddy simulation that, in mixing layers formed of processions of spanwise-oriented vortex structures, there is a fundamental change in the dynamics of the large-structure growth at what has been called the ‘mixing’ transition. This third part examines the relationship of the post-transition ‘organised’ flow to the mixing layer of classical turbulence theory. Further large-eddy simulations are presented which, like some of the experiments reported in Part 1, have captured the mixing layer in both its classical and organised turbulent states, thereby allowing them to be characterised separately. The simulation results are then used to calibrate model-free integral analyses of the dynamics of the mixing layer's spatial growth. It is shown that the organised and classical flows are alternative self-preserving turbulent flow states involving fundamentally different exchanges of mass, momentum and energy, one or other of which emerges naturally in particular realisations depending on the initial conditions.

The internal flow within an evaporating sessile droplet has intriguing fluid mechanics important to various microfluidics applications. In the present study, a phenomenon is observed through numerical methods wherein the buoyancy-driven flow structure inside a droplet on a non-wetting substrate transitions from an axisymmetric toroidal vortex flow to a non-axisymmetric single vortex flow with increase in the substrate temperature. As the axisymmetric nature of the droplet flow field and evaporation characteristics are broken, the internal velocity accelerates significantly. The transition, which is attributed to a flow instability inside the droplet, is more prone to occur as the droplet volume or the contact angle increases. The onset of the flow transition is analysed as the amplification of a small perturbation, thereby establishing a correlation between the flow instability and the Rayleigh number (Ra). Specifically, when Ra exceeds some critical value, the onset of the flow transition is observed, which explains the effects of substrate temperature and droplet volume on the internal flow. Next, the influence of the droplet contact angle on the critical Ra was investigated, and the underlying reasons were analysed. Finally, we discuss the heat transfer efficiency within the droplet and analyse why the internal flow tends to transition to a non-axisymmetric flow pattern from an energy minimization perspective.

An experimental study has been conducted on the near wake of a 6 : 1 spheroid, in both uniform and stratified backgrounds. The pitch angle, $\theta$, was varied from $0^\circ \text { to }20^\circ$. When $\theta = 0^\circ$, stratification decreases the characteristic wake element spacing so a characteristic Strouhal number ($St$) increases from 0.32 to 0.4. However, a similar measure scaled on wake momentum thickness shows the wake spacing to converge on those measured for other bluff and streamlined bodies. There is an apparent effect of Reynolds number, which changes the location of separation lines and hence the initial wake thickness. When $\theta > 0^\circ$, the wake is a combination of the usual drag wake together with a collection of streamwise vortices that have separated from the body, and this wake geometry can evolve in ways that are measurably different from the zero incidence case. These differences may be limited to the near wake, as the later evolution appears to converge with previous bluff- and streamlined bodies, with normalised wake height, $L_V = 0.5$ and centreline velocity, $\bar {u}_0 = 0.3$ at $Nt = 10$, as the early wake enters the non-equilibrium regime with similar values to previously studied stratified wakes. In the presence of density stratification, the inclined wake itself generates large-scale internal wave undulations with time scale $2{\rm \pi} /N$, even when the background stratification is not strong and a body-based Froude number is $O(10)$. The geometry and strengths of the primary streamwise vortices are not symmetric, mirroring previous results from experiments and computations in the literature.

Turbulent flow around curved tandem cylinders has been studied for the first time, by means of direct numerical simulation. The convex configuration was used, with a nominal gap ratio of $L/D = 3$ and a Reynolds number of 3900. Along the span, the flow regimes vary from alternating overshoot/reattachment to co-shedding. Three distinct Strouhal numbers coexist in the flow that are tied directly to different tandem cylinder flow regimes. This result differs substantially from convex curved tandem cylinders at a transitional Reynolds number, where only a single dominant frequency is found. All regimes exhibit some degree of instability, so that the flow can be considered multistable. A mode switch from alternating overshoot/reattachment to symmetric reattachment is found. Complex interactions are observed between the primary instability, the shear layer instability and the flow mode alterations. As opposed to previous investigations with single and tandem straight cylinders in the subcritical flow regime, our results indicate that there may be direct feedback from the primary instability to the shear layer instability. The downdraft region in the gap exhibits slow meandering, and may travel upstream and amplify the shear layer instability, causing early transition in the gap shear layer. This downdraft is governed by the slow modulations of the vortex formation region in the lower gap, meaning that the vortex dynamics of this region may indirectly influence the shear layer instability higher up in the gap.

Two-dimensional (2-D) quadrant analysis is generally used for investigating flow and sediment dynamics around a rigid structure in open channel flows. Given that particle distribution around rigid obstacles is not spatially uniform and changes in time, while vortices evolve to become three-dimensional (3-D) structures, 2-D quadrant analysis might be unsuitable to completely determine the sediment transport. Hence, 3-D quadrant and 3-D octant analyses should be considered, using the 3-D instantaneous velocity data and relative 3-D bursting process to define sediment transport surrounding the submerged square and circular cylinders. The turbulent kinetic energy (TKE), transition probabilities, occurrence probabilities, stress fraction and angles of inclination of 3-D bursting events are considered to quantify the coherent structures surrounding the cylinders and their interaction with bed particles. Experiments were conducted at the Hydraulics and Water Resources Engineering Laboratory, School of Infrastructure, Indian Institute of Technology, Bhubaneswar, Odisha, India, and velocity data were recorded at different cross-sections around the submerged cylinders using an acoustic Doppler velocimeter. Results show that the TKE is greater for internal ejection, external ejection, and internal sweep, external sweep in the upstream of the circular and square cylindrical structures. On the other part, the TKE is significant for internal ejection, external ejection, and internal sweep, external sweep in the downstream of the aligned square cylindrical structures, which justifies the highest scour depth that occurred upstream of the circular and square cylindrical structures and downstream of the aligned square cylindrical structure. The transition probability of the bursting events was determined using the Markov process from the measured velocity data to investigate the consecutive occurrence of bursting events. Further, the importance of sweeps and ejections on sediment erosion surrounding the cylinders within the scour hole at various stages of its development was investigated via 3-D quadrant analysis of the bursting occurrences. The outcomes show that external sweep and internal ejection events are active mechanisms for bed particle transport surrounding the cylinder. The maximum transition probability values are found around aligned cylindrical structures in comparison with the circular and square cylindrical structures in the transverse direction. This depicts the formation of a trailing vortex on both sides of the aligned square cylindrical object. The results reveal that the effect of inclination angles with respect to the water flow is greater for internal ejection and external sweep from upstream to downstream within the scour hole surrounding the cylindrical structures at various phases of development as horseshoe vortices and downflow develop upstream of the cylindrical structures while trailing vortices and wake vortices form at the top and downstream of the cylindrical structures. Internal and external ejection have a higher stress fraction than an internal and external sweep for square cylinders with alignment angles of 0°, 20° and circular cylinders over underdeveloped and developed scoured beds, respectively. With the higher percentage of fractional contributions for internal sweeps, the external sweep is predicted close to the cylindrical objects in comparison with the internal ejection and external ejection events because of the formation and warping of the horseshoe vortex close to the cylindrical objects, suggesting a significant probability of 3-D bursting occurrences with sediment movement near the cylindrical structures.

The Marangoni flow induced by an insoluble surfactant on a fluid–fluid interface is a fundamental problem investigated extensively due to its implications in colloid science, biology, the environment and industrial applications. Here, we study the limit of a deep liquid subphase with negligible inertia (low Reynolds number, $Re\ll {1}$), where the two-dimensional problem has been shown to be described by the complex Burgers equation. We analyse the problem through a self-similar formulation, providing further insights into its structure and revealing its universal features. Six different similarity solutions are found. One of the solutions includes surfactant diffusion, whereas the other five, which are identified through a phase-plane formalism, hold only in the limit of negligible diffusion (high surface Péclet number $Pe_s\gg {1}$). Surfactant ‘pulses’, with a locally higher concentration that spreads outward, lead to two similarity solutions of the first kind with a similarity exponent $\beta =1/2$. On the other hand, distributions that are locally depleted and flow inwards lead to similarity of the second kind, with two different exponents that we obtain exactly using stability arguments. We distinguish between ‘dimple’ solutions, where the surfactant has a quadratic minimum and $\beta =2$, from ‘hole’ solutions, where the concentration profile is flatter than quadratic and $\beta =3/2$. Each of these two cases exhibits two similarity solutions, one valid prior to a critical time $t_*$ when the derivative of the concentration is singular, and another one valid after $t_*$. We obtain all six solutions in closed form, and discuss predictions that can be extracted from these results.

It is well known that buoyancy suppresses, and can even laminarise, turbulence in upward heated pipe flow. Heat transfer seriously deteriorates in this case. A new direct numerical simulation model is established to simulate flow-dependent heat transfer in an upward heated pipe. The model shows good agreement with experimental results. Three flow states are simulated for different values of the buoyancy parameter $C$: shear turbulence, laminarisation and convective turbulence. The latter two regimes correspond to the heat transfer deterioration regime and the heat transfer recovery regime, respectively (Jackson & Li 2002; Bae et al., Phys. Fluids, vol. 17, issue 10, 2005; Zhang et al., Appl. Energy, vol. 269, 2020, 114962). We confirm that convective turbulence is driven by a linear instability (Su & Chung, J. Fluid Mech., vol. 422, 2000, pp. 141–166) and that the deteriorated heat transfer within convective turbulence is related to a lack of rolls near the wall, which leads to weak mixing between the flow near the wall and the centre of the pipe. Having surveyed the fundamental properties of the system, we perform a nonlinear non-modal stability analysis, which seeks the minimal perturbation that triggers a transition from the laminar state. Given the differences between shear and convective turbulence, we aim to determine how the nonlinear optimal (NLOP) changes as the buoyancy parameter $C$ increases. We find that at first, the NLOP becomes thinner and closer to the wall. Most importantly, the critical initial energy $E_0$ required to trigger turbulence keeps increasing, implying that attempts to trigger it artificially may not be an efficient means to improve heat transfer at larger $C$. At $C=6$, a new type of NLOP is discovered, capable of triggering convective turbulence from lower energy, but over a longer time. It is active only in the centre of the pipe. We next compare the transition processes, from linear instability and by the nonlinear non-modal excitation. At $C=4$, linear instability leads to a state that approaches a travelling wave solution or periodic solutions, while the minimal seed triggers shear turbulence before decaying to convective turbulence. Deeper into the parameter space for convective turbulence, at $C=6$, the new nonlinear optimal triggers convective turbulence directly. Detailed analysis of the periodic solution at $C=4$ reveals three stages: growth of the unstable eigenfunction, the formation of streaks, and the decay of the streaks. The stages of the cycle correspond to changes in the linear instability of the turbulent mean velocity profile. Unlike the self-sustaining process for classical shear flows, where the streak is disrupted via instability, here, decay of the streak is more closely linked to suppression of the linear instability of the mean flow, and hence suppression of the rolls. Flow visualisations at $C$ up to $10$ also show similar processes, suggesting that the convective turbulence in the heat transfer recovery regime is sustained by these three typical processes.

Time-resolved two-dimensional two-component particle image velocimetry measurements with high spatial resolution are carried out in a water tank agitated by four blades rotating at constant speed. Different blade geometries and rotation speeds are tested for the purpose of modifying turbulent flow conditions. In all cases where no baffles are used to break the rotation, the Zeman length is an order of magnitude smaller than the Taylor length. Compared with the cases with baffles which break the rotation, in the unbaffled cases turbulence production and/or mean advection are significant and the turbulence nonlinearity is dramatically reduced for the horizontal (i.e. normal to the axis of rotation) two-point turbulence fluctuating velocities. This nonlinearity reduction is manifest not only in the interscale turbulent energy transfer but also in the interspace turbulent energy transfer, which nearly vanishes. However, the nonlinearity is not reduced for the vertical two-point turbulence fluctuating velocities: the corresponding interscale turbulent transfer rate is in fact intensified, and its dependence on the two-point separation distance, as well as that of the corresponding interspace turbulent transfer rate, which does not vanish, is significantly modified. Even though non-homogeneities are very different for different blades and rotation speeds in the unbaffled cases, the horizontal fluctuating velocity's second-order structure function collapses with scalings which resemble predictions for homogeneous turbulence subject to strong rotation. The vertical fluctuating velocity's second-order structure function does not collapse for different blade geometries by neither these nor the Kolmogorov predictions.

The present work focuses on a specific bouncing behaviour as a spherical particle settles through a density interface in the absence of a neutral buoyant position. This behaviour was initially discovered by Abaid et al. (Phys. Fluids, vol. 16, issue 5, 2004, pp. 1567–1580) in salinity-induced stratification. Both experimental and numerical investigations are conducted to understand this phenomenon. In our experiments, we employ particle image velocimetry (PIV) to measure the velocity distribution around the particle and to capture the transient wake structure. Our findings reveal that the bouncing process begins after the wake detaches from the particle. The PIV results indicate that an upward jet forms at the central axis behind the particle following wake detachment. By performing a force decomposition procedure, we quantify the contributions from the buoyancy of the wake ($F_{sb}$) and the flow structure ($F_{sj}$) to the enhanced drag. It is observed that $F_{sb}$ contributes primarily to the enhanced drag at the early stage, whereas $F_{sj}$ plays a critical role in reversing the particle's motion. Furthermore, our results indicate that the jet is a necessary condition for the occurrence of the bouncing motion. We also explore the minimum velocities (where negative values denote the occurrence of bouncing) of the particle, while varying the lower Reynolds number $Re_l$, the Froude number $Fr$, and the upper Reynolds number $Re_u$, within the ranges $1 \leqslant Re_l\leqslant 125$, $115 \leqslant Re_u\leqslant 356$ and $2 \leqslant Fr\leqslant 7$. Our findings suggest that the bouncing behaviour is influenced primarily by $Re_l$. Specifically, we observe that the bouncing motion occurs below a critical lower Reynolds number $Re^\ast _{l}=30$ in our experiments. In the numerical simulations, the highest value for this critical number is $Re^\ast _{l}=46.2$, which is limited to the parametric ranges studied in this work.