The multipolar spherical vortex solutions to the Euler and Navier–Stokes equations in background cylindrical flow with swirl admit an additional background divergent radial flow with arbitrary time-dependent amplitude. In this case the radial wavenumber $k$, fundamental frequency $\omega$ and overall amplitude $U$ of the multipolar mode superposition become time-dependent and related functions. Assumption of an additional constraint, as a constitutive equation defining the time evolution of the spatially homogeneous divergence of the background flow, is required for the time evolution of the total flow to be completely evaluated from the initial conditions. It is found that flow compression implies an increase of the absolute values of the fundamental frequency $\omega$ and overall velocity amplitude $U$ of the oscillations.

We examine the mechanisms responsible for the onset of the three-dimensional mode B instability in the wake behind a circular cylinder. We show that it is possible to explicitly account for the stabilising effect of spanwise viscous diffusion and then demonstrate that the remaining mechanisms involved in this short-wavelength instability are preserved in the limit of zero wavelength. Using the resulting simplified equations, we show that perturbations in different fluid particles interact only through the in-plane viscous diffusion which turns out to have a destabilising effect. We also show that in the presence of viscous diffusion, the closed trajectories which had been conjectured to play a crucial role in the onset of the mode B instability are not actually a prerequisite for the growth of mode B type perturbations. We combine these observations to identify the three essential ingredients for the development of the mode B instability: (i) the amplification of perturbations in the braid regions due to the stretching mechanism; and the spreading of perturbations through (ii) viscous diffusion, and (iii) cross-flow advection which transports fluid between the two braid regions on either side of the cylinder. Finally, we develop a simple criterion that allows the prediction of the regions where three-dimensional short-wavelength perturbations are amplified by the stretching mechanism. The approach used in our study is general and has the potential to give insights into the onset of three-dimensionality via short-wavelength instabilities in other flows.

The asymptotic analysis of steady azimuthally invariant electromagnetically driven flows occurring in a shallow annular layer of electrolyte undertaken in Part 1 of this study (McCloughan & Suslov, J. Fluid Mech., vol. 980, 2024, A59) predicted the existence of a two-tori flow state that has not been detected previously. In Part 2 of the study we confirm its existence by numerical time integration of the governing equations. We observe a hysteresis, where the type of solution obtained for the same set of governing parameters depends on the choice of the initial conditions and the way the governing parameters change, which is fully consistent with the analytic results of Part 1. Subsequently, we perform a linear stability analysis of the newly obtained steady state and deduce that the experimentally observed anti-cyclonic free-surface vortices appear on its background as a result of a centrifugal (Rayleigh-type) instability of the interface separating two counter-rotating toroidal structures that form the newly found flow solution. The quantitative characteristics of such instability structures are determined. It is shown that such structures can only exist in sufficiently thin layers with the depth not exceeding a certain critical value.

In this paper we discuss the dynamics of vorticity at partial-slip boundaries. We consider the total vector circulation, which includes both the total vorticity of the fluid and the slip velocity at the boundary (the interface vortex sheet). The generation of vector circulation is an inviscid process, which does not depend on either viscosity or the slip length at the boundary. Vector circulation is generated by the inviscid relative acceleration between the fluid and the solid, due to either tangential pressure gradients or tangential acceleration of the partial-slip wall. While the slip length does not affect the creation of vector circulation, it governs how vector circulation is distributed between the total vorticity of the fluid and the interface vortex sheet. Specifically, the partial-slip boundary condition prescribes the ratio between boundary vorticity and the strength of the interface vortex sheet, and the viscous boundary flux transfers vector circulation between the interface vortex sheet and the fluid interior to maintain this condition. The interaction between a vortex ring and a partial-slip wall is examined to highlight various aspects of this formulation. For the head-on collision, the quantity of vector circulation diffused into the fluid as secondary vorticity increases as the slip length is decreased, resulting in a stronger secondary vortex and increased rebound of the vortex ring. For the oblique interaction, the extent to which the vortex ring connects to the boundary increases as the slip length is increased.

We analytically derive an amplitude equation for the weakly nonlinear evolution of the linearly most amplified response of a non-normal dynamical system. The development generalizes the method proposed in Ducimetière et al. (J. Fluid Mech., vol. 947, 2022, A43), in that the base flow now arbitrarily depends on time, and the operator exponential formalism for the evolution of the perturbation is not used. Applied to the two-dimensional Lamb–Oseen vortex, the amplitude equation successfully predicts the nonlinearities to weaken or reinforce the transient gain in the weakly nonlinear regime. In particular, the minimum amplitude of the linear optimal initial perturbation required for the amplitude equation to lose a solution, interpreted as the flow experiencing a bypass (subcritical) transition, is found to decay as a power law with the Reynolds number. Although with a different exponent, this is recovered in direct numerical simulations, showing a transition towards a tripolar state. The simplicity of the amplitude equation and the link made with the sensitivity formula permits a physical interpretation of nonlinear effects, in light of existing work on Landau damping and on shear instabilities. The amplitude equation also quantifies the respective contributions of the second harmonic and the spatial mean flow distortion in the nonlinear modification of the gain.

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 derive interface models for three-dimensional Rayleigh–Taylor instability (RTI), making use of a novel asymptotic expansion in the non-locality of the fluid flow. These interface models are derived for the purpose of studying universal features associated with RTI such as the Froude number in single-mode RTI, the predicted quadratic growth of the interface amplitude under multi-mode random perturbations, the optimal (viscous) mixing rates induced by the RTI and the self-similarity of horizontally averaged density profiles and the remarkable stabilization of the mixing layer growth rate which arises for the three-fluid two-interface heavy–light–heavy configuration, in which the addition of a third fluid bulk slows the growth of the mixing layer to a linear rate. Our interface models can capture the formation of small-scale structures induced by severe interface roll-up, reproduce experimental data in a number of different regimes and study the effects of multiple interface interactions even as the interface separation distance becomes exceedingly small. Compared with traditional numerical schemes used to study such phenomena, our models provide a computational speed-up of at least two orders of magnitude.

We introduce maps of Cauchy–Green strain tensor eigenvalues to barycentric coordinates to quantify and visualize the full geometry of three-dimensional deformation in stationary and non-stationary fluid flows. As a natural extension of Lagrangian coherent structure diagnostics, which provide separate scalar fields and a one-dimensional quantification of fluid deformation, our barycentric mapping visualizes the role of all three Cauchy–Green eigenvalues (or rates of stretching) in a single plot through a novel stretching coordinate system. The coordinate system is based on the distance from three distinct limiting states of deformation that correspond with the dimension of the underlying invariant manifolds. One-dimensional axisymmetric deformation (sphere to rod deformation) corresponds to one-dimensional unstable manifolds, two-dimensional axisymmetric deformation (sphere to disk deformation) corresponds to two-dimensional unstable manifolds and the rare three-dimensional isometric case (sphere to sphere translation and rotation) corresponds to shear-free elliptic Lagrangian coherent structures (LCSs). We provide methods to visualize the degree to which fluid deformation approximates these limiting states, and tools to quantify differences between flows based on the compositional geometry of invariant manifolds in the flow. We also develop a simple analogue for bilinearly representing and plotting both rates of stretching and rotation as a single vector. As with other LCS techniques, these diagnostics define frame-indifferent material features in the flow. We provide multiple computed examples of LCS and momentum transport barriers, and show advantages over other coherent structure diagnostics.

This paper sets out to investigate the vortex flow of spinnaker yacht sails, which are low-aspect-ratio highly cambered wings used to sail downwind. We tested three model-scale sails with the same sections but different twists over a range of angles of attack in a water tunnel at a Reynolds number of 21 000. We measured the forces with a balance and the velocity field with particle image velocimetry. The sails experience massively separated three-dimensional flow and leading-edge vortices convect at half of the free-stream velocity in a turbulent shear layer. Despite the massive flow separation, the twist of the sail does not change the lift curve slope, in agreement with strip theory. As the angle of attack and the twist vary, flow reattachment might occur in the time-average sense, but this does not necessarily result in a higher lift to drag ratio as the vorticity field is marginally affected. Finally, we investigated the effect of secondary vorticity, vortex stretching and diffusion on the vorticity fluxes. Overall, these results provide new insights into the vortex flow and associated force generation mechanism of wings with massively separated flow.

We carry out a weakly nonlinear analysis of the centrifugal instability for a columnar vortex in a rotating fluid, and compare the results to those of the semi-linear model derived empirically by Yim et al. (J. Fluid Mech., vol. 897, 2020, A34). The asymptotic analysis assumes that the Reynolds number is close to the instability threshold so that the perturbation is only marginally unstable. This leads to two coupled equations that govern the evolutions of the amplitude of the perturbation and of the mean flow under the effect of the Reynolds stresses due to the perturbation. These equations differ from the Stuart–Landau amplitude equation or coupled amplitude equations involving a mean field that have been derived previously. In particular, the amplitude does not saturate to a constant as in the supercritical Stuart–Landau equation, but decays afterwards reflecting the instability disappearance when the mean flow tends toward a neutrally stable profile in the direct numerical simulations (DNS). These equations resemble those of the semi-linear model except that the perturbation in the weakly nonlinear model keeps at leading order the structure of the eigenmode of the unperturbed base flow. The predictions of the weakly nonlinear equations are compared to those of the semi-linear model and to DNS for the Rossby number $Ro=-4$ and various Reynolds numbers and wavenumbers. They are in good agreement with the DNS when the growth rate is sufficiently small. However, the agreement deteriorates and becomes only qualitative for parameters away from the marginal values, whereas the semi-linear model continues to be in better agreement with the DNS.

Flow transitions are an important fluid-dynamic phenomena for many reasons, including the direct effect on the aerodynamic forces acting on the body. In the present study, two-dimensional (2-D) and three-dimensional (3-D) wake transitions of a NACA0012 airfoil are studied for angles of attack in the range $0^\circ \leq \alpha \leq 20^\circ$ and Reynolds numbers $500 \leq {\textit {Re}} \leq 5000$. The study uses water-channel experiments and 2-D and 3-D numerical simulations based on the nodal spectral-element method, level-set function-based immersed-interface method and Floquet stability analysis. The different wake states are categorised based on the time-instantaneous wake structure, non-dimensional frequency and aerodynamic force coefficients. The wake states and transition boundaries are summarised in a wake regime map. The critical angle of attack and Reynolds number for the supercritical Hopf bifurcation (i.e. steady to periodic wake transition) varies as $\alpha _1 {\sim} {\textit {Re}}^{-0.65}$, while the critical angle of attack for the onset of three dimensionality varies as $\alpha _{3D} {\sim} {\textit {Re}}^{-0.5}$. Over the entire Reynolds number range, the transition to 3-D flow occurs through a mode C (subharmonic) transition. Beyond this initial transition, further instabilities of the 2-D periodic base flow arise and are investigated. For instance, at $ {\textit {Re}}=2000$ and $\alpha _{3D,2}=11.0^\circ$, mode C coexists together with modes related to modes A and QP seen in a stationary circular cylinder wake. In contrast, at $ {\textit {Re}}=5000$ and $\alpha _{3D,2}=8.0^\circ$, the dominant mode C coexists with mode QP. Three-dimensional simulations well beyond critical angles indicate that 2-D vortex-street transitions are approximately maintained in the fully saturated 3-D wakes in a spanwise-averaged sense.

We study the dynamics of an initially axisymmetric and vertical Lamb–Oseen vortex in a stratified-rotating fluid under the complete Coriolis force on the $f$ plane, i.e. in the presence of a background rotation both along the vertical and horizontal directions. By a combination of direct numerical simulations and asymptotic analyses for small horizontal background rotation, we show that a critical layer appears at the radius where the angular velocity of the vortex is equal to the buoyancy frequency when the Froude number is larger than unity. This critical layer generates a vertical velocity that is invariant along the vertical and which first increases linearly with time and then saturates at an amplitude scaling like $Re^{1/3}/\widetilde{Ro}$, where $Re$ is the Reynolds number and $\widetilde {Ro}$ the non-traditional Rossby number based on the horizontal component of the background rotation. In turn, a quasi-axisymmetric anomaly of vertical vorticity is produced at the critical radius through the non-traditional Coriolis force. Below a critical $\widetilde {Ro}$ depending on $Re$, the Rayleigh's inflectional criterion is satisfied and a shear instability is subsequently triggered rendering the vertical vorticity fully non-axisymmetric. The decay of the angular velocity is then enhanced until it is everywhere lower than the buoyancy frequency. A theoretical criterion derived from the Rayleigh condition predicts well the instability. It shows that this phenomenon can occur even for a large non-traditional Rossby number $\widetilde {Ro}$ for large $Re$. Hence, the non-traditional Coriolis force might have much more impact on geophysical vortices than anticipated by considering the order of magnitude of $\widetilde {Ro}$.

The present paper analyses the vortical structures in the wake of a naval propeller operating underneath a free surface using detached-eddy simulation. We investigate the flow topology for several loading conditions and compare it with analogous observations behind a propeller operating in open water. We show that the wake topology is similar to that observed in open water only for low-loading conditions. For mild blade loading, the free surface's presence seems to stabilize the flow. On the contrary, for high blade loading, the mutual interaction between the vortex system and the free surface leads to vortex breakdown that overshadows the multiple pairing mechanisms observed in open-water conditions.

Linear modal instabilities of flow over untapered wings with aspect ratios $AR=4$ and 8, based on the NACA 0015 profile, have been investigated numerically over a range of angles of attack, $\alpha$, and angles of sweep, $\varLambda$, at chord Reynolds numbers $100\le Re\le 400$. Laminar base flows have been generated using direct numerical simulation and selective frequency damping, as appropriate. Several families of unstable three-dimensional linear global (TriGlobal) eigenmodes have been identified and their dependence on geometric parameters has been examined in detail at $Re=400$. The leading global mode A is associated with the peak recirculation in the three-dimensional laminar separation bubble formed on the wing and becomes unstable when recirculation reaches $\textit {O}(10\,\%)$. On unswept wings, this mode peaks in the midspan region of the wake and moves towards the wing tip with increasing $\varLambda$, following the displacement of peak recirculation; its linear amplification leads to wake unsteadiness. Additional amplified modes exist at nearly the same and higher frequencies compared to mode A. The critical $Re$ has been identified and it is shown that amplification increases with increasing sweep, up to $\varLambda \approx 10^\circ$. At higher $\varLambda$, all global modes become less amplified and are ultimately stable at $\varLambda =30^\circ$. An increase in amplification of the leading mode with sweep was not observed over the $AR=4$ wing, where tip vortex effects were shown to dominate.

The aerodynamics of a stationary wing in a turbulent wake are investigated. Force and velocity measurements are used to describe the unsteady flow. Various wakes are studied with different dominant frequencies and length scales. In contrast to the pre-stall angles of attack, the time-averaged lift increases substantially at post-stall angles of attack as the wing interacts with the von Kármán vortex street and experiences temporal variations of the effective angle of attack. At an optimal offset distance from the wake centreline, the time-averaged lift becomes maximum despite of small amplitude oscillations in the effective angle of attack. The stall angle of attack can reach 20° and the maximum lift coefficient can reach 64 % higher than that in the freestream. Whereas large velocity fluctuations at the wake centreline cause excursions into the fully attached and separated flows during the cycle, small-amplitude oscillations at the optimal location result in periodic shedding of leading edge vortices. These vortices may produce large separation bubbles with reattachment near the trailing-edge. Vorticity roll-up, strength and size of the vortices increase with increasing wavelength and period of the von Kármán vortex street, which also coincides with an increase in the spanwise length scale of the incident wake, and all contribute to the remarkable increase in lift.

Multipolar spherical solutions to the three-dimensional steady vorticity equation are provided. These solutions are based on the separation of radial and angular contributions in terms of the spherical Bessel functions and vector spherical harmonics, respectively. In this set of multipolar vortex solutions, the Hicks–Moffatt swirling vortex is categorized as a vortex of degree ${\ell }=1$ and therefore as a vortex dipole. This swirling vortex is the three-dimensional dipole in spherical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry. The three-dimensional dipole solution admits two linearly superposable solutions. The first one is a Trkalian flow and the second one is a cylindrical solid-body rotation with swirl. The higher ${\ell }>1$ multipolar vortices found are either vanishing-helicity vortices or Trkalian flow vortices. The multipolar Trkalian flows admit two circular polarizations given by the sign of the wavenumber $k$. It is also found that piecewise vortex solutions, consisting of interior rotational and exterior potential flow domains, satisfying velocity continuity conditions at the vortex boundary, are possible in the general multipolar Trkalian spherical vortex. A particular polarized dipole solution in three-dimensional cylindrical geometry, consisting as well in the superposition of a Trkalian flow and a rigid motion, is also analysed. This swirling vortex may be interpreted as the three-dimensional dipole in cylindrical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry.

We employ reconnection-capable, vortex filament methods and finite-volume, Navier–Stokes flow solvers to investigate the topological and helicity dynamics of vortex links for medium and high Reynolds numbers. Our vortex-dynamical model is based on discretization of vortex tubes into bundles of numerical analogues of vortex lines. Due to their nearly singular nature, the numerical vortex lines have topological writhe but not twist. By means of our reconnecting vortex tube model, it is shown that the helicity of a vortex link is conserved during the unknotting process. The dynamics of linking and writhe topological measures indicate that most of the initial linking becomes writhe during the post-reconnection evolution. The helicity spectra of the vortex link present alternating-sign helicity fluctuations at all (potential flow) scales up to the vortex core. At pre-reconnection times, these fluctuations are damped by Biot–Savart vortex stretching and helicity becomes single signed. The post-reconnection spectra indicate an inverse helicity cascade corresponding to the creation of a homogenized vortex blob, a process reminiscent of coherent structure formation in turbulence. An accompanying Navier–Stokes calculation of vortex link dynamics at Reynolds numbers $Re=1500$ confirms the post-reconnection transformation of linking into different topological measures, the pre-reconnection damping of helicity spectra fluctuations and the spectral shift to low wavenumbers at post-reconnection times. Due to viscous dissipation action, however, this shift is accompanied by progressive reduction of helicity peak values.

In this work, Joukowski's rotor wake model is considered for a two-blade rotor of radius $R_b$ rotating at the angular velocity $\varOmega _R$ in a normal incident velocity $V_{\infty }$. This model is based on a description of the wake by a limited number of vortices of core size $a$: a tip vortex of constant circulation $\varGamma$ for each blade and a root vortex of circulation $-2\varGamma$ on the rotation axis. Using a free-vortex method, we obtain solutions matching uniform interlaced helices in the far field that are steady in the frame rotating with the rotor for a large range of tip-speed ratios $\lambda = R_b \Omega _R/V_{\infty }$ and vortex strengths $\eta = \varGamma /(R_b^{2} \Omega _R)$. Solutions are provided for a two-bladed rotor for both helicopters and wind turbines. Particular attention is brought to the study of the solutions describing steep-descent helicopter flight regimes and large tip-ratio wind turbine regimes, for which the vortex structure is strongly deformed in the near wake and crosses the rotor plane. Both the geometry of the structure and its induced velocity field are analysed in detail. The thrust and the power coefficient of the solutions are also provided and compared to the momentum theory. The stability of the solutions is studied by monitoring the linear spatio-temporal development of a localized perturbation placed at different locations. Good agreements with the theoretical predictions for uniform helices and for point vortex arrays are demonstrated for the stability properties in the far wake. However, a more complex evolution is observed for the more deformed solutions when the perturbation is placed close to the rotor.

This study presents a class of steadily translating two-dimensional approximately circular vortices. The proposed solutions take the form of a superposition of two nearly concentric vorticity patches with zero net vorticity. Exact quasi-monopolar solutions of this type are found for propagation speeds that are much less than typical azimuthal velocities. While all V-states obtained are shown to be formally unstable, a large subset of configurations is characterized by very low growth rates. Fully nonlinear simulations reveal that such nearly stable eddies can propagate large distances from the point of origin while maintaining their structure and intensity. Therefore, the proposed solutions can serve as models of geophysical vortices that are known to drift relative to the ambient fluid, often exhibiting remarkable longevity and resilience to external perturbations.

The development of a wing-tip vortex of a rectangular, square-tipped wing having a NACA 0012 airfoil at a chord Reynolds number $Re_{c_{w}}=2\times 10^{5}$, under the effect of synthetic jet actuation, was experimentally studied. Five control configurations were considered: case C1 with momentum coefficient $C_{\unicode[STIX]{x1D707}}=0.001$ and actuation frequency $F^{+}=0.075$; case C2 with $C_{\unicode[STIX]{x1D707}}=0.001$ and $F^{+}=0.15$; case C3 with $C_{\unicode[STIX]{x1D707}}=0.001$ and $F^{+}=0.3$; case C4 with $C_{\unicode[STIX]{x1D707}}=0.001$ and $F^{+}=0.6$; and case C5 with $C_{\unicode[STIX]{x1D707}}=0.001$ and $F^{+}=1.2$. Under the most effective configuration, case C3, the vortex was stretched and appeared to be diffuse with a nearly 40 % decrease in the peak circumferential velocity and 50 % decrease in the core axial vorticity. The vortex core radius largely broadened suggesting that the lower-frequency control configuration allowed the synthetic jet to travel larger distances into the vortex bringing turbulent structures within its core resulting in increased mixing and subsequently a more diffuse vortex.