Solving the three-dimensional boundary layer equations carries theoretical significance and practical applications, which also poses substantial challenges due to its inherent complexity. In this paper, the laminar boundary layer equations for the symmetry plane of three-dimensional bodies are derived in an orthogonal curvilinear coordinate system associated with the principal curvatures. The derivation of the boundary layer equations is based not only on the common symmetric properties of the flow, as given by Hirschel et al. (Three-Dimensional Attached Viscous Flow, 2014, Academic Press, pp. 183–187), but also incorporates the geometric symmetry properties of the body. The derived equations are more representative and simplified. Notably, these equations can degenerate to a form consistent with or equivalent to the commonly used boundary layer equations for special bodies such as flat plates, cones and spheres. Furthermore, for hypersonic flows, the crossflow velocity gradient at the boundary layer edge on the symmetry plane is derived based on Newtonian theory. Subsequently, this parameter can provide the necessary boundary condition needed for solving the boundary layer equations using existing methods. Finally, as examples, the equations developed in this paper are solved using the difference-differential method for several typical three-dimensional blunt shapes that appeared on hypersonic vehicles. They prove to be useful in the analysis and interpretation of boundary layer flow characteristics in the symmetry plane of blunt bodies.

Free transverse oscillations of an elastically mounted circular cylinder with low mass-damping are studied with a focus on the effects of the cylinder end condition on structural oscillations and vortex shedding. While the top end of the cylinder pierces the free surface of a water channel, the lower end is changed to have three end conditions: an attached endplate, an endplate unattached from the cylinder at varying gaps and no endplate. All three response branches are examined, with the reduced velocity sweeping $2 \le {U^\ast } \le 15$ (corresponding to a Reynolds number range of $2000 \le Re \le 14\;700$). Although the cylinder oscillations are unaffected by the end condition in the initial and upper branches, they show significant dependency on the end condition in the lower branch. When the endplate is attached or unattached with a small gap, the upper-to-lower branch transition occurs with a sudden decrease in oscillation amplitude, after which the lower branch maintains a near-constant amplitude. For larger gaps or no endplate, with increasing reduced velocity, the oscillation amplitude decreases gradually from its peak without any discernible sign of transition between the upper and lower branches. The three-dimensional effects of the gap are the basis for these differences in oscillation. Atop the strong tip vortex, a low-magnitude streamwise velocity region develops downstream of the cylinder, which delays upper-to-lower branch transition over the cylinder span that sees this low-velocity region. With increasing gap, this low-velocity region and the delay in transition spread over larger spanwise extents, forcing the overall body to oscillate at larger amplitudes.

We numerically study the influence of a soluble surfactant on the microjetting mode of the liquid–liquid flow focusing configuration. The surfactant adsorbs on the interface next to the feeding capillary and accumulates in front of the emitted jet, significantly lowering the surface tension there. The resulting Marangoni stress substantially alters the balance of the tangential stresses at the interface but does not modify the interface velocity. The global stability analysis at the minimum flow rate stability limit shows that the Marangoni stress collaborates with soluto-capillarity to stabilize the microjetting mode. Our analysis unveils the noticeable effect of the Marangoni stress associated with the surface tension perturbation. Surfactant diffusion and desorption hardly affect the stability limit. Transient numerical simulations show how subcritical and supercritical base flows respond to a spatially localized initial perturbation. Our parametric study indicates that the minimum flow rate ratio depends on the adsorption constant and the surfactant concentration through the product of these two variables. The surfactant stabilizing effect increases with the outer stream flow rate. We show that surfactants not only stabilize the microemulsion resulting from the jet breakup in hydrodynamic focusing, but also allow for the reduction of droplet size. Our findings advance the fundamental understanding of the complex role of surfactants in tip streaming via hydrodynamic focusing. In particular, our results contradict the common assumption that adding surfactant favours tip streaming simply because it reduces the meniscus tip surface tension.

Nonlinear steady solutions of the barotropic quasi-geostrophic equation in basins, gulfs and channels on a $\beta$-plane are presented. The domains are rectangular with arbitrary aspect ratios. The two-dimensional solutions assume a linear relationship between the potential vorticity $q$ and the stream function $\psi$. The sign of the slope in the linear $q\unicode{x2013}\psi$ relationship defines two broad sets of solutions. For a positive slope, the solutions in a closed basin correspond to the inertial gyres derived by Fofonoff in 1954. The negative slope solutions consist of normal modes that can be resonant. For gulfs and channels, the conditions at the open boundaries are almost arbitrary flows entering or leaving the domain. Such conditions allow a great variety of solutions in the interior, characterised mainly by arrays of vortices with alternate signs. Several examples are presented and discussed.

We propose a simple method to identify unstable parameter regions in general inviscid unidirectional shear flow stability problems. The theory is applicable to a wide range of basic flows, including those that are non-monotonic. We illustrate the method using a model of Jupiter's alternating jet streams based on the quasi-geostrophic equation. The main result is that the flow is unstable if there is an interval in the flow domain for which the reciprocal Rossby Mach number (a quantity defined in terms of the zonal flow and potential vorticity distribution), surpasses a certain threshold or ‘hurdle’. The hurdle height approaches unity when we can take the hurdle width to greatly exceed the atmosphere's intrinsic deformation length, as holds on gas giants. In this case, the Kelvin–Arnol’d sufficient condition of stability accurately detects instability. These results improve the theoretical framework for explaining the stable maintenance of Jupiter and Saturn's jets over decadal time scales.

This study explores the implementation of an online control strategy based on dynamic mode decomposition in the context of flow control. The investigation is conducted mainly with a fixed Reynolds number of $Re = 100$, focusing on the flow past a circular cylinder constrained between two walls to mitigate vortex shedding. The control approach involves the activation of two synthetic jets on the cylinder through blowing and suction. Velocity fluctuations in the wake, specifically in the $x$-direction, are harnessed to ascertain the mass flow rate of the jets using the linear quadratic regulator and online dynamic mode decomposition. The study systematically assesses the control performance across various configurations, including different values of the input penalty factor ${\boldsymbol {R}}$, varying numbers of probes and distinct probe arrangement methods. The synthetic jets prove effective in stabilising the separation bubble, and their interaction with the unsteady wake leads to a notable reduction in drag force, its fluctuations and the amplitude of the lift force. Specifically, the mean and standard deviation of the drag coefficient witness reductions of $7.44\,\%$ and $96.67\,\%$, respectively, and the standard deviation of the lift coefficient experiences an impressive reduction of $85.18\,\%$. The robustness of the proposed control method has also been tested on two more complicated cases, involving unsteady incoming flows with multiple frequency components. Comparatively, the methodology employed in this paper yields results akin to those obtained through deep reinforcement learning in terms of control effectiveness. However, a noteworthy advantage lies in the substantial reduction of computational resource consumption, highlighting the efficiency of the proposed approach.

We present a Mach 15 air flow over a blunt two-dimensional wedge simulated using the direct molecular simulation method. As electronically excited states are not modelled, the resulting air mixture around the wedge contains the electronic ground states only, namely ${\rm N}_2(\text {X}^1 \varSigma _g^{+})$, ${\rm O}_2(\text {X}^3 \varSigma _g^{-})$, ${\rm NO}(\text {X}^2\varPi _r)$, ${\rm N}(^4{\rm S})$ and ${\rm O}(^3{\rm P})$. All the potential energy surfaces (PESs) that are used to model the various interactions between air particles are ab initio, with two notable exceptions, namely ${\rm N}_2+{\rm NO}$ and ${\rm O}_2+{\rm NO}$. At the selected free-stream conditions, strong vibrational non-equilibrium is observed in the shock layer. The flow is characterized by significant chemical activity, with near-complete oxygen dissociation, considerable formation of NO and minimal molecular nitrogen dissociation. Complex mass diffusion kinetics, driven by composition, temperature and pressure gradients, are identified in the shock layer. All these physical phenomena are directly coupled to, and responsible for, the mechanics of the gas flow and are all solely traceable to the PESs’ inputs, without the need for any thermochemical models, mixing rules or constitutive laws for transport properties. Because the flow is entirely at near-continuum conditions, it is a gas-phase thermophysics benchmark that is useful to enhance the fidelity of continuum models used in computational fluid dynamics of hypersonic flows.

The electroosmotic flow (EOF) fields in the vicinity of solids with high dielectric permittivity are studied for the case of charge-asymmetric electrolyte solutions. Corresponding solutions of the coupled Poisson–Nernst–Planck and Navier–Stokes equations are obtained analytically and numerically. When a direct-current (DC) electric field is applied to a high-permittivity uncharged sphere, a net EOF develops that translates into a non-zero electrophoretic mobility of the sphere, although it does not carry any charge. Similarly, a DC field acting on a channel in a high-permittivity material results in a net flow through the channel, although the solid is uncharged. Such phenomena are expected to occur frequently whenever high-permittivity solids are immersed in charge-asymmetric electrolyte solutions and do not rely on special scenarios such as ion crowding. Also, the net flow velocities are very significant for realistic values of the electric field strength. The derived scaling relationships even predict giant net flow velocities through nanochannels of the order of metres per second for practically relevant scenarios.

When a partially miscible binary mixture is quenched below its critical temperature, it transitions from its single-phase to a two-phase region, undergoing phase separation. The processes of formation and coalescence of droplets are driven by diffusive and convective phenomena, taking place isotropically in the system. The application of an external force field, which exerts a different contribution on the two species, breaks the symmetry of phase separation, leading to the segregation of two equilibrated phases separated by a single interface. This study investigates the dynamics of phase segregation under an external force. The effects of various force magnitudes, captured by the Bond number, in both high- and low-viscosity mixtures, distinguished by different fluidity numbers, are quantified via numerical simulations by using the phase field model. The intricate dynamics of formation, floating and coalescence of droplets towards complete segregation are described along with the quantification of the segregation time, revealing different patterns for high and low Bond numbers. Results show that in none of the cases, formation and floating can be regarded as strictly serial processes. A universal scaling between segregation time, Bond number, fluidity number and domain size is not possible, with a power-law dependence emerging only under the diffusion-dominated regime.

Predicting the transient flow fields that develop when a shock wave passes through an area expansion is a fundamental problem in compressible fluid mechanics and significant in many engineering applications. Experiments, large eddy simulations and geometrical shock dynamics are used to study the mechanism by which a normal shock wave that expands across an area expansion evolves into a uniform normal shock far downstream of it. This study analyses shock waves with moderate Mach numbers of 1.1–1.8 that expand at area ratios of up to 5. As the shock wave propagates into the expanded region, it experiences rapid deceleration, forming a non-uniform shock front. Impinging on the walls of the larger cross-section region, the shock wave reflects and generates a complex and highly transient shock pattern near the expansion region. We have found that as the shock front propagates further downstream, a laterally moving shock wave that intersects the shock front at a triple point reverberates laterally between the walls. This process effectively evens out the flow behind the incident shock front, thus reducing the variation of properties behind it. The extended duration of this process leads to significant pressure fluctuation behind the shock front. The results show that the evolution of the shock front can be scaled using the expanded region height and the velocity of the shock wave far downstream of the expansion. The results enabled the formulation of a simple empirical relation, allowing us to predict the shock velocity far downstream of gradual and abrupt area expansions.

The injection of ${\rm CO}_2$ into depleted reservoirs carries the potential for significant Joule–Thomson cooling, when dense, supercritical ${\rm CO}_2$ is injected into a strongly under-pressured reservoir. The resulting low temperatures around the wellbore risk causing thermal fracturing of the well/near-well region or causing freezing of pore waters or formation of gas hydrates which would reduce injectivity and jeopardise well and reservoir integrity. These risks are particularly acute during injection start-up when ${\rm CO}_2$ is in the gas stability field. In this paper we present a model of non-isothermal single-phase flow in the near-wellbore region. We show that during radial injection, with fixed mass injection rate, transient Joule–Thomson cooling can be described by similarity solutions at early times. The positions of the ${\rm CO}_2$ and thermal fronts are described by self-similar scaling relations. We show that, in contrast to steady-state flow, transient flow causes slight heating of ${\rm CO}_2$ and reservoir gas either side of the thermal front, as pressure diffuses into the reservoir. The scaling analysis here identifies the parametric dependence of Joule–Thomson cooling. We present a sensitivity analysis which demonstrates that the primary controls on the degree of cooling are reservoir permeability, reservoir thickness, injection rate and Joule–Thomson coefficient. The analysis presented provides a computationally efficient approach for assessing the degree of Joule–Thomson cooling expected during injection start-up, providing a complement to complex, fully resolved numerical simulations.

The elasto-inertial focusing and rotating characteristics of spheroids in a square channel flow of Oldroyd-B viscoelastic fluids are studied by the direct forcing/fictitious domain method. The rotational behaviours, changes in the equilibrium positions and travel distances are explored to analyse the mechanisms of spheroid migration in viscoelastic fluids. Within the present simulated parameters (1 ≤ Re ≤ 100, 0 ≤ Wi ≤ 2, 0.4 ≤ α ≤3), the results show that there are four kinds of equilibrium positions and six (five) kinds of rotational behaviours for the elasto-inertial migration of prolate (oblate) spheroids. We are the first to identify a new rotational mode for the migration of prolate spheroids. Only when the particles are initially located at a corner and wall bisector, some special initial orientations of the spheroids have an impact on the final equilibrium position and rotational mode. In other general initial positions, the initial orientation of the spheroid has a negligible effect. A higher Weissenberg number means the faster the particles migrate to the equilibrium position. The spheroid gradually changes from the corner (CO), channel centreline (CC), diagonal line (DL) and cross-section midline (CSM) equilibrium positions as the elastic number decreases, depending on the aspect ratio, initial orientation and rotational behaviour of the particles and the elastic number of the fluid. When the elastic number is less than the critical value, the types of rotational modes of the spheroids are reduced. By controlling the elastic number near the critical value, spheroids with different aspect ratios can be efficiently separated.

Research on the settling dynamics of snow particles, considering their complex morphologies and real atmospheric conditions, remains scarce despite extensive simulations and laboratory studies. Our study bridges this gap through a comprehensive field investigation into the three-dimensional (3-D) snow settling dynamics under weak atmospheric turbulence, enabled by a 3-D particle tracking velocimetry (PTV) system to record over a million trajectories, coupled with a snow particle analyser for simultaneous aerodynamic property characterization of four distinct snow types (aggregates, graupels, dendrites, needles). Our findings indicate that while the terminal velocity predicted by the aerodynamic model aligns well with the PTV-measured settling velocity for graupels, significant discrepancies arise for non-spherical particles, particularly dendrites, which exhibit higher drag coefficients than predicted. Qualitative observations of the 3-D settling trajectories highlight pronounced meandering in aggregates and dendrites, in contrast to the subtler meandering observed in needles and graupels, attributable to their smaller frontal areas. This meandering in aggregates and dendrites occurs at lower frequencies compared with that of graupels. Further quantification of trajectory acceleration and curvature suggests that the meandering frequencies in aggregates and dendrites are smaller than that of morphology-induced vortex shedding of disks, likely due to their rotational inertia, and those of graupels align with the small-scale atmospheric turbulence. Moreover, our analysis of vertical acceleration along trajectories elucidates that the orientation changes in dendrites and aggregates enhance their settling velocity. Such insights into settling dynamics refine models of snow settling velocity under weak atmospheric turbulence, with broader implications for more accurately predicting ground snow accumulation.

We investigate the dynamics of a columnar Taylor–Green vortex array under strong stratification, focusing on Froude numbers $0.125\leq Fr \leq 1.0$, with the aim of identifying and understanding the primary instabilities that lead to the vortices’ breakdown. Linear stability analysis reveals that the fastest-growing vertical wavenumber scales with $Fr^{-1}$, while the dimensionless growth rate remains approximately constant. The most unstable eigenmode, identified as the mixed hyperbolic mode by Hattori et al. (J. Fluid Mech., vol. 909, 2021, A4), bears significant similarities to the zigzag instability, first discovered by Billant & Chomaz (J. Fluid Mech., vol. 418, 2000, pp. 167–188). Direct numerical simulations further confirm that the zigzag instability is crucial in amplifying initial random perturbations to finite amplitude, with the flow structure and modal growth rate consistent with the linear stability analysis. In particular, the characteristic vertical length scale of turbulence matches that of the fastest-growing linear mode. These findings underscore the broader relevance of the zigzag instability mechanism beyond its initial discovery in vortex pairs, demonstrating its role in facilitating direct energy transfer from vertically uniform vortical motions to a characteristic vertical length scale proportional to $Fr$ in strongly stratified flows.

A viscous, lubrication-like response can be triggered in a thin film of fluid squeezed between a rigid flat surface and the tip of an incoming projectile. We develop a scaling for this viscous approach stage of fluid-mediated normal impact, applicable to soft impactors. Under the assumption of mediating fluid being incompressible, the impacting solid displays two limit regimes: one dominated by elasticity, and the other by inertia. The transition between the two is predicted by a dimensionless parameter, which can be interpreted as the ratio between two time scales that are the time that it takes for the surface waves to warn the leading edge of the impactor of the forthcoming impact, and the characteristic duration of the final viscous phase of the approach. Additionally, we elucidate why nearly incompressible solids feature (a) substantial ‘gliding’ prior to contact at the transition between regimes, (b) the largest size of entrapped bubble between the deformed tip of the impactor and the flat surface, and (c) a sudden drop in entrapped bubble radius past the transition between regimes. Finally, we argue that the above time scale ratio (a dimensionless number) can govern the different dynamics reported experimentally for a fluid droplet as a function of its viscosity and surface tension.

Numerous flying and swimming creatures use the ground effect to boost their propulsive performance, with the ‘ground’ referring to either a solid boundary or a free surface. While our knowledge of how a solid boundary affects biolocomotion is relatively comprehensive, the ground effect of a free surface is not fully understood. To address this limitation, we conduct a numerical investigation on the propulsion performance of a flapping plate under a free surface, subject to a range of control parameters. When the Froude number ($Fr$) is very low (i.e. little surface deformation), the effects of a free surface are similar to those of a solid boundary, with enhanced thrust and input power but little change in efficiency. However, as $Fr$ increases (i.e. more surface deformation), our results reveal an optimal $Fr$ of approximately 0.6, where the free surface induces a more streamlined flow around the flapping plate, effectively reducing the added mass. This results in a significant decrease in input power and greatly enhanced efficiency.

Internal solitary waves are a widely observed phenomenon in natural waters. Mathematically, they are fundamentally a nonlinear phenomenon that differs from the paradigm of turbulence, in that energy does not move across scales. Internal solitary waves may be computed from the Dubreil–Jacotin Long equation, which is a scalar partial differential equation that is equivalent to the stratified Euler equations. When a background shear current is present the algebraic complexity of the problem increases substantially. We present an alternative point of view for characterizing the situation with a shear current using Lagrangian (particle-like) models analysed with graph theoretic methods. We find that this yields a novel, data-centric framework for analysis that could prove useful well beyond the study of internal solitary waves.

Meandering designates the main manifestation of unsteady vortex dynamics observed in experiments. This study has the twofold objective to (i) develop a theoretical model describing vortex meandering and (ii) conduct a quantitative and objective evaluation of the model against experimental data. Based on an analogy with Brownian motion, we derive the theoretical model in the framework of linear response theory. Taking the form of a Langevin equation, our model explains meandering as the competition between external excitation by free-stream perturbations, counteracted by stabilising intrinsic vortex dynamics. As such, it contains the previous approaches to explaining the phenomenon as limiting cases, and clearly highlights their shortcomings. The statistical identification of characteristic regularities in experimental data as well as the assessment of their consistency with theoretical models are important problems in physics. For samples obtained from finite-length records of correlated data, these statistical characteristics are not unique and may show spurious behaviour merely induced by the finiteness of the sample. Statistical inference provides a systematic and quantitative methodology to objectively assess the reproducibility of statistical characteristics and to evaluate their consistency with theoretical models. Their systematic application to the analysis of vortex meandering has not been done before and provides statistical evidence for our proposed Brownian-motion-like model. That is, experimental vortex meandering constitutes the manifestation of a stationary Gauss–Markov random process, which implies that the dynamics admits an ergodic probability measure.

We analyse the effect of drop-deformation-induced change in streamline topology on the scalar transport rate (the Nusselt number $Nu$) in an ambient planar linear flow. The drop-phase resistance is assumed dominant, and the drop deformation is characterised by the capillary number ($Ca$). For a spherical drop ($Ca = 0$) in an ambient planar extension, closed streamlines lead to $Nu$ increasing with the Péclet number ($Pe$), from $Nu_0$, corresponding to purely diffusive transport, to $4.1Nu_0$, corresponding to a large-$Pe$ diffusion-limited plateau. For non-zero $Ca$, we show that the flow field consists of spiralling streamlines densely wound around nested tori foliating the deformed drop interior. Now $Nu$ increases beyond the aforementioned primary plateau, saturating in a secondary one that approaches $22.3Nu_0$ for $Ca \rightarrow 0$, $Pe\,Ca \rightarrow \infty$. The enhancement appears independent of the drop-to-medium viscosity ratio. We further show that this singular dependence, of the transport rate on drop deformation, is generic across planar linear flows; chaotically wandering streamlines in some of these cases may even lead to a tertiary enhancement regime.

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.