The formation of swirling vortex rings and their early time evolution, resulting from the controlled discharge of an incompressible, Newtonian fluid into a stationary equivalent fluid bulk, is explored for weak to moderate swirl number $S \in [0, 1]$. Two practically realisable inlet conditions are investigated with swirl simultaneously superposed onto a linear momentum discharge; the corresponding circulation based Reynolds number is 7500. The results obtained reveal that for $S > 1/2$, the addition of swirl promotes the breakdown of the leading primary vortex ring structure, giving rise to the striking feature of significant negative azimuthal vorticity generation in the region surrounding the primary vortex ring core, whose strength scales with ${S}^2$. Through a nonlinear interaction with the vortex breakdown, the radius of the primary toroidal vortex core is rapidly increased; consequently, the self-induced propagation velocity of the leading ring decreases with $S$ and vortex stretching along the circular primary vortex core increases counteracting viscous diffusion effects. The latter governs the evolution of the peak vorticity intensity and the swirl velocity magnitude in the primary ring core, the circulation growth rate of the primary ring, as well as the vorticity intensity of the trailing jet and hence its stability. This combination of effects leads to an increased dimensionless kinetic energy for the primary ring with increasing $S$ and results in an almost linearly decreasing circulation based formation number, $F$.

Vortex shedding in the wake of a cylinder in uniform flow can be suppressed via the application of a porous coating; however, the suppression mechanism is not fully understood. The internal flow field of a porous coated cylinder (PCC) can provide a deeper understanding of how the flow within the porous medium affects the wake development. A structured PCC (SPCC) was three-dimensionally printed using a transparent material and tested in water tunnel facilities using flow visualisation and tomographic particle image velocimetry at outer-diameter Reynolds numbers of $Re = 7 \times 10^{3}$ and $7.3 \times 10^{4}$, respectively. The internal and near-wall flow fields are analysed at the windward and mid-circumference regions. Flow stagnation is observed in the porous layer on the windward side and its boundary is shown to fluctuate with time in the outermost porous layer. This stagnation region generates a quasi-aerodynamic body that influences boundary layer development on the SPCC inner diameter, that separates into a shear layer within the porous medium. For the first time via experiment, spectral content within the separated shear layer reveals vortex shedding processes emanating through single pores at the outer diameter, providing strong evidence that SPCC vortex shedding originates from the inner diameter. Velocity fluctuations linked to this vortex shedding propagate through the porous layers into the external flow field at a velocity less than that of the free stream. The Strouhal number linked to this velocity accurately predicts the SPCC vortex shedding frequency.

The critical effect of the windward interior angles of elastically mounted trapezoidal bodies on a galloping instability is numerically investigated in this paper using two methodologies of high-fidelity computational fluid dynamics simulations and data-driven stability analysis using the eigensystem realization algorithm. A micro exploration of the dynamical response is processed to understand the mechanism underpinning the structural amplification at the initial stage of the galloping instability and the competition between wake and structural modes. It is observed that very small changes in the windward interior angle of an isosceles-trapezoidal body can provoke or suppress galloping – indeed, a small decrease or increase (low to $1^\circ$) of the windward interior angle from a right angle ($90^\circ$) can result in a significant enhancement and complete suppression, respectively, of the galloping oscillations. This supports our hypothesis that the contraction and/or expansion (viz., fore-aft tapering and/or widening) of the cross-section in the streamline direction has potential influences on galloping triggering from the geometrical perspective. The data-driven stability analysis is also applied to verify and analyse this phenomenon from the perspective of modal analysis. The experimental measurements are also conducted in the wind tunnel to support this hypothesis.

We examine the equilibrium configurations of a nematic liquid crystal with an immersed body in two dimensions. A complex variables formulation provides a means for finding analytical solutions in the case of strong anchoring. Local tractions, forces and torques on the body are discussed in a general setting. For weak (finite) anchoring strengths, an effective boundary technique is proposed which is used to determine asymptotic solutions. The energy-minimizing locations of topological defects on the body surface are also discussed. A number of examples are provided, including circular and triangular bodies, and a Janus particle with hybrid anchoring conditions. Analogies to classical results in fluid dynamics are identified, including d'Alembert's paradox, Stokes’ paradox and the Kutta condition for circulation selection.

Wave interaction with graded metamaterials exhibits the phenomenon of rainbow reflection, in which broadband wave signals slow down and separate into their frequency components before being reflected. This phenomenon has been qualitatively understood by describing the wave field in the metamaterial using the local Bloch wave approximation (LBWA), which locally represents the wave field as a superposition of propagating wave solutions in the cognate infinite periodic media (so-called Bloch waves). We evaluate the performance of the LBWA quantitatively in the context of two-dimensional linear water-wave scattering by graded arrays of surface-piercing vertical barriers. To do this, we implement the LBWA numerically so that the Bloch waves in one region of the graded array are coupled to Bloch waves in adjacent regions. This coupling is computed by solving the scattering of Bloch waves across the interface between two semi-infinite arrays of vertical barriers, where the barriers in each semi-infinite array can have different submergence depths. Our results suggest that the LBWA accurately predicts the free surface amplitude across a wide range of frequencies, except those just above the cutoff frequencies associated with each of the vertical barriers in the array. This highlights the importance of decaying Bloch modes above the cutoff in rainbow reflection.

The traditional approximation neglects the cosine components of the Coriolis acceleration, and this approximation has been widely used in the study of geophysical phenomena. However, the justification of the traditional approximation is questionable under a few circumstances. In particular, dynamics with substantial vertical velocities or geophysical phenomena in the tropics have non-negligible cosine Coriolis terms. Such cases warrant investigations with the non-traditional setting, i.e. the full Coriolis acceleration. In this manuscript, we study the effect of the non-traditional setting on an isothermal, hydrostatic and compressible atmosphere assuming a meridionally homogeneous flow. Employing linear stability analysis, we show that, given appropriate boundary conditions, i.e. a bottom boundary condition that allows for a vertical energy flux and non-reflecting boundary at the top, the atmosphere at rest becomes prone to a novel unstable mode. The validity of assuming a meridionally homogeneous flow is investigated via scale analysis. Numerical experiments were conducted, and Rayleigh damping was used as a numerical approximation for the non-reflecting top boundary. Our three main results are as follows: (i) experiments involving the full Coriolis terms exhibit an exponentially growing instability, yet experiments subjected to the traditional approximation remain stable; (ii) the experimental instability growth rate is close to the theoretical value; (iii) a perturbed version of the unstable mode arises even under sub-optimal bottom boundary conditions. Finally, we conclude our study by discussing the limitations, implications and remaining open questions. Specifically, the influence on numerical deep-atmosphere models and possible physical interpretations of the unstable mode are discussed.

We show that Kolmogorov scale-by-scale equilibrium in the intermediate layer of a fully developed turbulent channel flow is only achieved asymptotically around the Taylor length and, therefore, not in an inertial range. Furthermore, we analyse scale-by-scale turbulence production and interscale turbulence energy transfer in terms of alignments/anti-alignments of fluctuating velocities, straining/compressive relative motions, forward/inverse interscale transfer/cascade and homogeneous/non-homogeneous interscale transfer rate contributions. We also propose leading order scalings for second- and third-order two-point statistics, including the extremum interscale turbulence energy transfer rate and a second-order anisotropic structure function, which acts as a scale-by-scale Reynolds shear stress and determines the scale-by-scale (two-point) turbulence production rate.

We study the onset of the three-dimensional mode A instability in the near wake behind a circular cylinder. We show that long-wavelength perturbations organise in a time-shifting pattern such that the in-plane velocity in each streamwise slice corresponds to the base flow solution at shifted times. This observation introduces an additional unifying characteristic for certain mode A type instabilities. We then analyse the mechanisms which control the growth or decay of these perturbations and highlight the crucial role played by the tilting mechanism which operates via non-local interactions in a manner similar to Biot–Savart induction. We characterise its domain of influence using a Green's function-based approach which allows us to rationalise the non-trivial dependence of the growth rate on the spanwise wavenumber. We connect this behaviour to the subtle balance between the local growth of the perturbations as they are swept along by the flow and the feedback on the perturbations that are generated during the next period of the time-periodic base flow. Finally, we discuss generalisations of our findings to other types of flows.

We study experimentally the dynamics of a water droplet on a tilted and vertically oscillating rigid fibre. As we vary the frequency and amplitude of the oscillations the droplet transitions between different modes: harmonic pumping, subharmonic pumping, a combination of rocking and pumping modes, and a combination of pumping and swinging modes. We characterize these responses and report how they affect the sliding speed of the droplet along the fibre. The swinging mode is explained by a minimal model making an analogy between the droplet and a forced elastic pendulum.

We compute the flow induced by the steady translation of a rigid sphere along the axis of a large cylindrical container filled with a low-viscosity fluid set in rigid-body rotation, the sphere being constrained to spin at the same rate as the undisturbed fluid. The parameter range covered by the simulations is similar to that explored experimentally by Maxworthy (J. Fluid Mech., vol. 40, 1970, pp. 453–479). We describe the salient features of the flow, especially the internal characteristics of the Taylor columns that form ahead of and behind the body and the inertial wave pattern, and determine the drag and torque acting on the sphere. Torque variations are found to obey two markedly different laws under rapid- and slow-rotation conditions. The corresponding scaling laws are predicted by examining the dominant balances governing the axial vorticity distribution in the body vicinity. Results for the drag agree well with the semi-empirical law proposed for inertialess regimes by Tanzosh & Stone (J. Fluid Mech., vol. 275, 1994, pp. 225–256). This law is found to apply even in regimes where inertial effects are large, provided that rotation effects are also large enough. Influence of axial confinement is shown to increase dramatically the drag in rapidly rotating configurations, and the container length has to be approximately a thousand times larger than the sphere for this influence to become negligibly small. The reported simulations establish that this confinement effect is at the origin of the long-standing discrepancy existing between Maxworthy's results and theoretical predictions.

We consider the role of gravity in solute transport when a thin droplet evaporates. Under the physically relevant assumptions that the contact line is pinned and the solutal Péclet number, ${Pe}$, is large, we identify two asymptotic regimes that depend on the size of the Bond number, ${Bo}$. When ${Bo} = O(1)$ as ${Pe}\rightarrow \infty$, the asymptotic structure of solute transport follows directly from the surface-tension-dominated regime, whereby advection drives solute towards the contact line, only to be countered by local diffusive effects, leading to the formation of the famous ‘coffee ring.’ In the distinguished limit in which ${Bo} = O({Pe}^{4/3})$ as ${Pe}\rightarrow \infty$, this interplay between advection and diffusion takes place alongside that between surface tension and gravity. In each regime, we perform a systematic asymptotic analysis of the solute transport and compare our predictions to numerical simulations. We identify the effect of gravity on the nascent coffee ring, providing quantitative predictions of the size, location and shape of the solute mass profile. In particular, for a fixed Péclet number, as the effect of gravity increases, the coffee ring is diminished in height and situated further from the contact line. Furthermore, for certain values of ${Bo}$, ${Pe}$ and the evaporation time, a secondary peak may exist inside the classical coffee ring. The onset of this secondary peak is linked to the change in type of the critical point in the solute mass profile at the droplet centre. Both the onset and the peak characteristics are shown to be independent of ${Pe}$.

We study numerically the characteristics of fluid–fluid displacement in simple mixed-wet porous micromodels using a dynamic pore network model. The porous micromodel consists of distinct water-wet and oil-wet regions, whose fractions are varied systematically to yield a variety of displacement patterns over a wide range of capillary numbers. We find that the impact of mixed-wettability is most prominent at low capillary numbers, and it depends on the complex interplay between wettability fraction and the intrinsic contact angle of the water-wet regions. For example, the fractal dimension of the displacement pattern is a monotonically increasing function of wettability fraction in flow cells with strongly water-wet clusters, but it becomes non-monotonic with respect to wettability fraction in flow cells with weakly water-wet clusters. Additionally, mixed-wettability also manifests itself in the injection pressure signature, which exhibits fluctuations especially at low wettability fraction. Specifically, preferential filling of water-wet regions leads to reduced effective permeability and higher injection pressure, even at vanishingly small capillary numbers. Finally, we demonstrate that scaling analyses based on a weighted average description of the overall wetting state of the mixed-wet system can effectively capture the variations in observed displacement pattern morphology.

The interactions between a vortex ring and a gas bubble released at the axis are studied numerically, which shed light on understanding more complicated bubble–turbulence coupling. We fix the Reynolds number at $Re_{\tau }=7500$ and consider various Weber numbers in the range of $We=130\unicode{x2013}870$. We find that the translating speed of the vortex ring is substantially lower than the case of the vortex ring without a bubble. It is explained with two different mechanisms, depending on the Weber number. In the low-$We$ range, the reduction of translating speed of the ring is due to the capture of bubbles into the ring core, leading to significant changes in the vorticity distribution within the core. In the high-$We$ range, the repeatedly generated secondary vortex rings perturb the primary one, which bring about an earlier flow transition, thereby reducing the translating speed of the vortex ring. On the other hand, the evolution of a gas bubble is also affected by the presence of the vortex ring. In the low-$We$ range, we observe binary breakup of the bubble after it is captured by the primary vortex ring. In contrast, in the high-$We$ range, it is interesting to find that the bubble experiences sequentially stretching, spreading and breakup stages. In the high-$We$ range, the numbers of smaller bubbles predicted by the classical Rayleigh–Plateau instability of a stretched cylindrical bubble agree well with our numerical simulations. Consistent with the previous experiments, this number keeps unchanged at 16 as $We$ further increases. An additional comparison is made between two higher Reynolds numbers, indicating that the finer eddies in a vortical field with a higher Reynolds number tend to tear the bubble into more fragments.

A slip asymmetry can break the fore–aft symmetry of the local hydrodynamic force distribution on the surface of an otherwise no-slip or uniform-slip particle. Here, we use the Lorentz reciprocal theorem to demonstrate that such asymmetry, even in a fractional amount, can qualitatively alter the swimming characteristics of a self-propelled spherical squirmer, markedly different from those of no-slip or uniform-slip squirmers. Unlike the usual tangential squirming by the thrust-providing B1 mode and the type-determining B2 mode, we discover two unique features for a stick-slip squirmer. First, the squirmer can acquire a swimming velocity U without the B1 mode but simply by a symmetric extensile/contractile squirming from the B2 mode, which is able to reverse the swimming direction of the squirmer. Second, a stresslet $\boldsymbol{\mathsf{S}}$ can also be induced by a unidirectional squirming from the B1 mode, capable of inverting the squirmer's stresslet from extensile type to contractile type or vice versa to change the squirmer from puller to pusher or in a reverse manner. We further show that the two squirming modes can reinforce or compete with each other to enhance or diminish U and $\boldsymbol{\mathsf{S}}$ due to interplays between the asymmetric squirming forces on the stick and the slip faces. A phase diagram is also established to categorize a variety of newly emerging swimming states, such as an enhanced/degraded puller/pusher and a backward puller/pusher, depending on the relative strength of the squirming modes β = B2/B1, the direction of the stick-slip polarity and the degree of the slip disparity. As a result of such cooperative and competitive natures, a stick-slip squirmer can swim more or less efficiently than no-slip and uniform-slip ones. These distinctive features arising from stick-slip disparity can not only be made geometrically tuneable for steering the motion of a squirmer, but also provide new means for making efficient artificial microswimmers using amphiphilic Janus particles.

The three-dimensional dynamic behaviour of a tip vortex generated by a NACA $66_2$-415 hydrofoil is investigated under wetted flow and cavitating conditions using time-resolved tomographic particle image velocimetry. Two main cavitation modes are studied, namely the breathing and double-helical modes. The time-averaged flow field consists of a system of two streamwise vortices for all three conditions. The shape of the tip vortex resembles that of the cavitation mode, instead of a circular cylinder. Multi-scale vortical structures are captured in the instantaneous flow field. The surface oscillation of the cavity contributes to the growth of Kelvin–Helmholtz instability over the tip vortex, leading to the onset of hairpin vortices. Stronger spanwise interaction between the tip vortex and flow separation of the hydrofoil is produced by cavitation, further intensifying the perturbation growth. The proper orthogonal decomposition analysis gives insight into the relationship between cavity oscillation and vortex instability. Two major types of unstable modes of the tip vortex are obtained, leading to serpentine centreline displacement and elliptical deformation motion. The selection of the dominant unstable mode is associated with cavity surface oscillation. For the wetted flow condition, the displacement mode dominates the growth of vortex perturbation, while for breathing mode cavitation, the most energetic unstable mode changes into an elliptical deformation pattern, the disturbance energy of which is negligible in the wetted flow condition. Consistency is found between the peak frequency of the deformation mode and the cavity resonance frequency, indicating the contribution of cavity oscillation to the disturbance growth and breakdown of the tip vortex.

This paper presents the effects of wing kinematics on both normal forward flight and escape flight of a dragonfly. A Navier–Stokes-based numerical model has been adopted, and results have been substantiated by experimental data. The wing kinematics of tethered specimens and the prescribed wing morphology of a free-flying dragonfly were used in the simulation. To shed light on the interplay between kinematics and aerodynamics, a parametric study of the kinematics has been conducted. It is found that in escape flight, the dragonfly generates additional lift while the thrust reduces and the overall efficiency drops. Compared with normal forward flight, the escape mode produces larger lift force peaks. When the kinematics change to facilitate escape flight, the aerodynamic forces are affected by not only the flapping kinematics but, in the case of the hindwing, the varied wing–wing vortex interactions. The direction of the resultant force on each wing changes according to the change of the mean of pitching angle and stroke plane angle. We found that in the studied configurations, the varied phasing of the wings has a marginal effect on the aerodynamics of the dragonfly. It reduces lift and increases thrust, and this force modulation is slightly more efficient when the local angle of attack also changes. On the other hand, the change of angle of attack played a major role in leading-edge vortex formations and directing the resultant forces of the wings. The results can be useful in developing flight control strategies for micro air vehicle design.

The linear stability of the two-dimensional Taylor–Green vortices, which is a spatially periodic array of vortices, in rotating stratified fluids is investigated by local and modal stability analysis. Five types of instability appear in general: the pure hyperbolic instability, the strato-hyperbolic instability, the rotational-hyperbolic instability, the centrifugal instability and the elliptic instability. The condition for each instability and the estimate of the growth rate, which are useful in interpreting numerical results, are obtained in the framework of local stability analysis. Realizability of an instability is introduced to predict whether an unstable mode corresponding to an unstable region found in the local stability analysis exists at finite Reynolds numbers. In the absence of stratification, the pure hyperbolic instability is dominant for weak rotation; it is stabilized for strong rotation. For strong anti-cyclonic rotation, the elliptic instability or the centrifugal instability becomes dominant depending on the parameter values; further stronger rotation stabilizes both instabilities. For strong cyclonic rotation, the rotational-hyperbolic instability or the elliptic instability becomes dominant, although the growth rate is smaller than the anti-cyclonic cases. Strong stratification changes the stability properties. The strato-hyperbolic instability occurs for weak rotation. The rotational-hyperbolic instability and the elliptic instability are weakened under cyclonic rotation, while the latter survives and extends the unstable range under anti-cyclonic rotation. The pure hyperbolic instability and the centrifugal instability are less affected by stratification. The mode structures of each instability are in good agreement with the corresponding solution to local stability equations, confirming the physical mechanism of the instability.

Upon interaction with underwater shock waves, bubbles can collapse and produce high-speed liquid jets in the direction of the wave propagation. This work experimentally investigates the impact of laser-induced underwater impulsive shock waves, i.e. shock waves with a short, finite width, of variable peak pressure on bubbles of radii in the range 10–500 $\mathrm {\mu }$m. The high-speed visualisations provide new benchmarking of remarkable quality for the validation of numerical simulations and the derivation of scaling laws. The experimental results support scaling laws describing the collapse time and the jet speed of bubbles driven by impulsive shock waves as a function of the impulse provided by the wave. In particular, the collapse time and the jet speed are found to be, respectively, inversely and directly proportional to the time integral of the pressure waveform for bubbles with a collapse time longer than the duration of shock interaction and for shock amplitudes sufficient to trigger a nonlinear bubble collapse. These results provide a criterion for the shock parameters that delimits the jetting and non-jetting behaviour for bubbles having a shock width-to-bubble size ratio smaller than one. Jetting is, however, never observed below a peak pressure value of 14 MPa. This limit, where the pressure becomes insufficient to yield a nonlinear bubble collapse, is likely the result of the time scale of the shock wave passage over the bubble becoming very short with respect to the bubble collapse time scale, resulting in the bubble effectively feeling the shock wave as a spatially uniform change in pressure, and in an (almost) spherical bubble collapse.

The normal and shear viscosities of non-Brownian suspensions are measured by optical suspension imaging for particle volume fractions $\phi$ between $0.3\phi _m$ and $0.98\phi _m$, where $\phi _m$ is the jamming fraction. Two distinct refractive-index-matched suspensions, made with the same polymethyl methacrylate spherical particles dispersed in a mixture of water and Triton X-100, are studied. One is density-matched while the other one is negatively buoyant. They are both sheared in a Couette rheometer where the velocity and particle volume fraction fields are measured. The shear viscosity and the second particle normal stress $\varSigma _{22}^p$ are determined through the study of these profiles in the neutrally buoyant suspension, while the third particle normal stress $\varSigma _{33}^p$ is deduced from the analysis of the vertical $\phi$ profiles measured in the negatively buoyant suspension. Our results indicate that the shear viscosity decreases with shear stress $\varSigma _{12}$, and that this shear-thinning behaviour can be captured by the variation of $\phi _m$ with $\varSigma _{12}$. We show that $\varSigma _{33}^p$ is proportional to $\varSigma _{12}$, and that $\varSigma _{33}^p/\eta _0\dot {\gamma }$ is a function of only $\phi /\phi _m(\varSigma _{12})$. The values of $\varSigma _{22}^p$ deduced from the radial $\phi$ profiles are consistent with the results of Zarraga et al. (J. Rheol., vol. 44, 2000, pp. 185–220). We conclude by discussing our results in the framework of the $\mu (J)$ rheology for viscous numbers $J$ ranging from $2\times 10^{-4}$ to $3\times 10^1$. We obtain very good agreement with the results obtained by Boyer et al. for $J\lesssim 10^{-1}$ (Phys. Rev. Lett., vol. 107, 2011, 188301) and by Zarraga et al. for $J\gtrsim 10^{-1}$.

An experimental study of the dynamics and droplet production in three mechanically generated plunging breaking waves is presented in this two-part paper. In the present paper (Part 1), the dynamics of the three breakers are studied through measurements of the evolution of their free surface profiles during 10 repeated breaking events for each wave. The waves are created from dispersively focused wave packets generated with three wave maker motions that differ primarily by small changes in their overall amplitude. Breaker profiles are measured with a cinematic laser-induced fluorescence technique covering a streamwise region of approximately one breaker wavelength and over a time of 3.4 breaker periods. The aligned profile data is used to create spatio-temporal maps of the ensemble average surface height and the standard deviation of both the local normal distance and the local arc length relative to the instantaneous mean profile. It is found that the mean and standard deviation maps contain strongly correlated localized features and indicate that the transition from laminar to turbulent flow is a highly repeatable process. Regions of high standard deviation include the splash created by the plunging jet impact and subsequent splash impacts at the front of the breaking region as well as the site where the air pocket entrained under the plunging jet at the moment of jet tip impact comes to the surface and pops. In Part 2 (Erinin et al., J. Fluid Mech., vol. 967, 2023, A36), these features are used to interpret various features of the distributions of droplet number, diameter and velocity.