Two-dimensional Euler flows, in the plane or on simple surfaces, possess a material invariant, namely the scalar vorticity normal to the surface. Consequently, flows with piecewise-uniform vorticity remain that way, and moreover evolve in a way which is entirely determined by the instantaneous shapes of the contours (interfaces) separating different regions of vorticity – this is known as ‘contour dynamics’. Unsteady vorticity contours or interfaces often grow in complexity (lengthen and fold), either as a result of vortex interactions (like mergers) or ‘filamentation’. In the latter, wave disturbances riding on a background, equilibrium contour shape appear to inevitably steepen and break, forming filaments, repeatedly– and perhaps endlessly. Here, we revisit the onset of filamentation. Building upon previous work and using a weakly nonlinear expansion to third order in wave amplitude, we derive a universal, parameter-free amplitude equation which applies (with a minor change) both to a straight interface and a circular patch in the plane, as well as circular vortex patches on the surface of a sphere. We show that this equation possesses a local, self-similar form describing the finite-time blow up of the wave slope (in a re-scaled long time proportional to the inverse square of the initial wave amplitude). We present numerical evidence for this self-similar blow-up solution, and for the conjecture that almost all initial conditions lead to finite-time blow up. In the full contour dynamics equations, this corresponds to the onset of filamentation.

We study the response of a flexible prism with a square cross-section placed in cross-flow through a series of experiments conducted at increasing flow velocities. We show that as the reduced velocity (a dimensionless flow velocity that also depends on the natural frequency of the structure) is increased, the prism undergoes vortex-induced vibration (VIV) in its first mode, which then transitions to VIV in the second mode and then third mode. In these ranges, the shedding frequency is synchronised with the oscillation frequency, and the oscillations are mainly in the transverse (cross-flow – CF) direction. As we keep increasing the reduced velocity, we observe a linear increase in the amplitude of the torsional oscillations of the prism, resembling a torsional galloping. This increase in the torsional oscillations then causes an increase in the amplitudes of the CF and inline (IL) oscillations while the third structural mode is still excited in the CF direction. A transition to oscillations in the fourth structural mode is observed at higher reduced velocities, which reduces the CF and IL amplitudes, while the torsional oscillations reach a plateau. After this plateau is reached in the torsional oscillations, galloping is observed in the CF oscillations of the response, which results in large-amplitude oscillations in both the CF and IL directions. The CF galloping response at these higher reduced velocities is accompanied by a torsional VIV response and the shedding frequency is synchronised with the frequency of the torsional oscillations.

Liquid metal flows are important for many industrial processes, including liquid metal batteries (LMBs), whose efficiency and lifetime can be affected by fluid mixing. We experimentally investigate flows driven by electrical currents in an LMB model. In our cylindrical apparatus, we observe a poloidal flow that descends near the centreline for strong currents, and a poloidal flow that rises near the centreline for weak currents. The first case is consistent with electrovortex flow, which is an interaction between current and its own magnetic field, whereas the second case is consistent with an interaction between current and the external field, which drives Ekman pumping. Notably, we also observe an intermediate case where the two behaviours appear to compete. Comparing results with Frick et al. (2022 J. Fluid Mech. 949, A20), we test prior estimates of the scaling of flow speed with current to predict the observed reversal. Based on these data, we propose two different ways to apply the Davidson et al. (1999 J. Fluid Mech. 245, 669–699) poloidal suppression theory that explain both experimental results simultaneously: either taking the wire radius into account to scale the Lorentz force, or taking viscous dissipation into account to scale the swirl velocity, following Herreman et al. (2021 J. Fluid Mech. 915, A17).

Turbulence beneath a free surface leaves characteristic long-lived signatures on the surface, such as upwelling ‘boils’, near-circular ‘dimples’ and elongated ‘scars’, easily identifiable by eye, e.g. in riverine flows. In this paper, we analyse data from direct numerical simulations to explore the connection between these surface signatures and the underlying vortical structures. We investigate dimples, known to be imprints of surface-attached vortices, and scars, which have yet to be extensively studied, by analysing the conditional probabilities that a point beneath a signature is within a vortex core as well as the inclination angles of sub-signature vorticity. The analysis shows that the probability of vortex presence beneath a dimple decreases from the surface down through the viscous and blockage layers. This vertical variation in probability is approximately a Gaussian function of depth and depends on the dimple’s size and the bulk turbulence properties. Conversely, the probability of finding a vortex beneath a scar increases sharply from the surface to a peak at the edge of the viscous layer, regardless of scar size. The probability distributions of the angle between the vorticity vector and the vertical axis also show a clear pattern about vortex orientation: a strong preference for vertical alignment below dimples and an equally strong preference for horizontal alignment below scars. Our findings corroborate previous studies that tie dimples to surface-attached vertical vortices. Moreover, they suggest that scars can be defined as imprints of horizontal vortices that are located approximately a quarter of the Taylor microscale beneath the free surface.

Particle image velocimetry is used to study the control of swirl momentum, delivered through an orifice formed by a physically rotating tube of finite length, relevant to the evolution of vortex rings produced at a Reynolds number ${Re}\approx 1000$ based on the average discharge velocity, for swirl numbers ${S} \in [0, 1]$. Experiments without discharge, reinforced with complimentary numerical predictions, reveal the presence of an intriguing secondary flow pattern in the rotating tube, preventing attainment of a solid-body-like swirl distribution. Nevertheless, it is found that fully established rings produced in this way, following discharge once conditions in the tube have reached a steady state, exhibit similar characteristics to rings formed by an otherwise solid-body rotating initial condition as explored computationally by Ortega-Chavez et al. (2023, J. Fluid Mech. 967, A16). Namely, opposite-signed vorticity forms due to vortex tilting, which subsequently interacts with the ring, promoting vorticity cancellation and vortex ring breakdown. A key feature of the experimental work is that partially established vortex rings, produced before a steady-state rotating tube condition is reached, show unique characteristics. Their creation, a short time after the onset of tube rotation: (i) facilitates more efficient delivery of swirl momentum to the vortex core area; (ii) maintains a low level of swirl in the ring bubble’s central region which would otherwise promote the formation of opposite-signed vorticity and vortex breakdown.

The formation process of a vortex pair generated by a two-dimensional starting jet has been investigated numerically over a range of Reynolds numbers from 500 to 2000. The effects of stroke ratio and nozzle configuration are examined. Only a single vortex pair can be observed in the vorticity field generated by small stroke ratios less than 10 while the leading vortex pair formed by larger stroke ratios eventually disconnects from the trailing jet. The formation numbers (13.6 and 9.3) for a straight nozzle and an orifice nozzle have been identified by the circulation criterion and they are further analysed by four other criteria. Using the contraction coefficient, formation numbers can be transformed into a universal value at about 16.5 for both nozzles. The effect of Reynolds number on the formation number is found to be within 12 % for parallel flow cases but it will increase up to 27 % for non-parallel flow cases due to shear-layer instability. A modified contraction-based slug model is proposed, and it can accurately predict the total invariants (e.g. circulation, hydrodynamic impulse and kinetic energy) shedding from the nozzle edge. Analytical estimation of the formation number is further conducted by matching the predicted total invariants to the Pierrehumbert model of steady vortex pairs. By assuming that pinch-off starts when the vortex pair achieves the steady state, two analytical models are proposed in terms of vortex impulse and translational velocity. The latter appears to be more appropriate to predict the formation number for two-dimensional flows.

This paper presents a numerical investigation of the turbulence transition phenomenon in the wake of wall-mounted prisms. Large-eddy simulations are performed at $Re = 1\times 10^3 {-}5\times 10^3$ for prisms with a range of aspect ratio (height to width) from $0.25$ to $1.5$, and depth ratios (length to width) between $1$ and $4$. The results show that the wake irregularity is enhanced with increasing depth ratio, evidenced by higher turbulent kinetic energy (${\approx}90\,\%$) near the leading edge, and the onset of irregular, unsteady vortex shedding. This is attributed to interactions between Kelvin–Helmholtz instability (KHI) of the shear layer and large-scale vortex shedding, and it is induced by an unsteady shear layer, resembling flapping-like motion. These interactions elevate the flow momentum due to increased turbulence intensity and mixing, contributing to the wake transition phenomenon. To this end, this study defines the role of depth ratio in the transition phenomenon by showing that increasing depth ratio (e.g. from $1$ to $4$) leads to earlier onset of KHIs in the shear layer. These instabilities intensify with depth ratio, resulting in stronger interactions between shear layer and large-scale vortex shedding. Specifically, KHI-induced vortices interact more frequently with large-scale wake structures for higher depth ratio prisms, exciting larger flow fluctuations and irregular wake patterns. This interaction alters the frequency and coherence of vortex shedding, revealing a complex coupling mechanism that drives the transition to turbulence.

Coherent structures over two distinct, organized wall perturbations – a transverse sinusoidal bump with and without small-scale longitudinal grooves – are studied using direct numerical simulations. Large-scale spanwise rollers (SRs) form via shear layer rollup past the bump peak, enveloping a large separation bubble (SB) for both a smooth wall (SW) and a grooved wall (GW). In a GW, small-scale alternatingly spinning jets emanating from the crests’ corners merge with the shear layer, altering the SRs compared with SRs in a SW. The underlying coherence of the highly turbulent SRs is educed via phase-locked ensemble averaging. Coherent vorticity contours of SRs are ellipses tilted downward, hence causing co-gradient Reynolds stress. The limited streamwise length of SB precludes SR tumbling, unlike in a free shear layer. The coherent field reveals minibubbles attached to the bump’s downstream wall with circulation opposite to that of the SB – they are larger, stronger and more numerous in GW than in SW – reducing skin friction. Compared with SW, the swirling jets in GW increase coherent production while decreasing incoherent production. Additionally, the jets push the SRs to travel faster and farther before reattachment. The SB experiences two different modes of oscillation due to high-frequency advection of the shear layer SR and low-frequency breathing of the SB, where the former dominates in GW and the latter in SW. Negative production is caused by counter-rotating vortex dipoles inducing flow ejections (for both SW and GW) and single vortices penetrating the grooves – both occurring in the region of flow acceleration.

The turbulent boundary layer is a region where both preferential dissipation of energy and the production of significant vorticity arises as a consequence of the strong velocity gradients. Previous work has shown that, following a Reynolds decomposition, the purely fluctuating component of the enstrophy production is the dominant term. Near the wall this varies in a complex manner with height. In this study, we additionally decompose the strain rate and vorticity terms into normal and non-normal components using a Schur decomposition and are able to explain all these features in terms of contributions at different heights from constituents involving different combinations of normal and non-normal quantities. What is surprising about our results is that, while the mean shear and the action of larger-scale structures should mean that non-normal effects are of over-riding importance at the wall, the most important individual term involves the fluctuating normal strain rate in the transverse direction. In part, this is because of a strong correlation between this term and the non-normal vorticity with a transverse axis, but it is also the case that individual components of the purely non-normal enstrophy production are negative in the mean. Hence, a local strain rate that is orthogonal to the direction of the dominant mean and fluctuating shear plays a crucial role in amplifying vorticity that is yet to have developed a local component. These conclusions support the emphasis in the control literature on the transverse velocity components at the wall.

The three-dimensional flow field past an I-shaped dual-step cylinder has been obtained by numerical integration of the Navier–Stokes equations at Reynolds number ($Re_D$) 150. The I-shaped cylinder consisted of two large-diameter (D) cylinders with a small-diameter (d) cylinder in between. With a view to exploring the vortex dynamics and structural loads, simulations were performed for eight different lengths $l$ of the small cylinder, varied from $l/D=10$ to 0.2 for a fixed diameter ratio $D/d=2$. When the length of the small cylinder is sufficiently large, the wake behind the I-shaped cylinder is similar to the wake behind the single-step cylinder (Tian et al., J. Fluid Mech., vol. 891, 2020, A24). As the small cylinder length decreases, the enhanced interactions between the two steps make the present wake deviate from the wake of the single-step cylinder, leading to four different wake modes distinguished by different combinations of vortex cells. The physical formation mechanisms were analysed in terms of the vortex dynamics. Besides the wake flow, the streamwise vortices around the I-shaped step cylinder were also investigated. A pair of edge vortices and a junction vortex were identified for $l/D \geq ~1$. When the gap between the two steps becomes too small, $l/D \leq ~0.2$, the junction vortex disappears, and only a pair of edge vortices exists. Varying the distance between the two steps strongly affects the structural loads (drag and lift) along the I-shaped cylinder. The dependence of the loads on $l/D$ was readily explained by the different wake modes.

The current study characterizes the attenuation of instabilities in steady and unsteady shear layers by investigating shear-thinning flows downstream of a confined axisymmetric sudden expansion. Flow fields were captured using particle image velocimetry. Tested fluids exhibited approximate power-law indices of 1, 0.81, 0.61 and 0.47 and measurements were performed at mean throat-based Reynolds numbers of ${Re_m} = 4800$ and 14 400. Unsteady flows were tested at a Strouhal number and amplitude-to-mean velocity ratio of $St = 0.15$ and $\lambda = 0.95$, respectively. For unsteady shear layers, shear-layer roll-up regardless of shear-thinning strength was evidenced by collapse of average circulation over time. For steady shear layers, consistent shear-layer behaviour regardless of shear-thinning strength was evidenced by similar shear-layer trajectories and by growth rates in vorticity thickness. However, vorticity fields of the unsteady and steady shear layers, standard deviations of shear-layer trajectory, thickness of steady shear layers and Reynolds shear-stress spectra of the steady shear layers reveal an attenuation of shear-layer instabilities not captured by Reynolds number. Specifically, shear-layer instabilities exhibit increased diffusion with increasing shear-thinning strength and, in the case of steady shear layers, shear-thinning strength is shown to promote shear-layer stabilization. Also, evidenced by vorticity fields and through Reynolds shear-stress spectra, instabilities frequently coalesce into large rollers, a result that would suggest the presence of an inverse eddy cascade. The behaviour of shear-thinning fluids is shown to stabilize shear layers through attenuating shear-layer instabilities, complementing observations from previous studies showing how shear-thinning fluids promote turbulence in the dominant flow direction.

The concept of vortex lock-in for a single circular cylinder in an oscillating flow, induced through acoustic forcing, is revisited. Multiple cylinder diameters are investigated over a Reynolds number range between 500 and 7200. The lock-in behaviour is investigated quantitatively through hot-wire anemometry and planar particle image velocimetry measurements. The results corroborate previous findings describing the frequency range over which vortex lock-in occurs. It is found that the cylinder location in a standing wave (pressure node or velocity node) had a significant influence on the lock-in behaviour. A novel scaling which captures the onset of vortex lock-in is proposed which demonstrates that the Strouhal number is important in predicting the amplitude of the velocity fluctuations required to induce lock-in. Velocity fields also reveal the existence of bimodal vortex shedding during lock-in. This is confirmed using snapshot proper orthogonal decomposition which demonstrates that symmetric and alternate shedding modes are simultaneously present during lock-in and that symmetric shedding is inherent to the near wake region only. Reduced-order reconstruction of the instantaneous velocity fields confirmed that features associated with the forcing frequency control the shear layer roll-up up to $x/d=2.1$ while the influence of the asymmetric mode is simply to skew the trajectory of the vortex pair. Since vortex roll-up and the cylinder wake ends at $x/d=2.1$, the emergence of spectral content at $0.5f_e$ is attributed to a wavelength doubling measured between the vortical structures in the flow field.

This paper presents a numerical study on the flow around two tandem circular cylinders beneath a free surface at a Reynolds number of $180$. The free-surface effects on the wake dynamics and hydrodynamic forces are investigated through a parametric study, covering a parameter space of gap ratios from $0.20$ to $2.00$, spacing ratios from $1.50$ to $4.00$ and Froude numbers from $0.2$ to $0.8$. A jet-like flow accompanied by a shear layer of positive vorticity separating from the free surface is formed in the wake at small gap ratios, which significantly alters the wake pattern through its dynamic behaviours. At shallow submergence depths, the three-dimensional wake transitions from mode B to mode A as the distance between the cylinders increases. As submergence depth increases, the wavy deformation of the primary vortex cores disappears in the wake, and the flow transitions to a two-dimensional state. Higher Froude numbers can extend the effect of the free surface to deeper submergence depths. The critical spacing ratio tends to be larger at higher Froude numbers. Furthermore, the free-surface deformation is examined. The free-surface profile typically comprises a hydraulic jump immediately ahead of the upstream cylinder, trapped waves in the vicinity of the two tandem cylinders and well-defined travelling waves on the downstream side. The frequencies of the waves cluster around the vortex shedding frequency, indicating a close association between the generation of waves and the vortex shedding process.

We explore the application of the reference map technique, originally developed for Eulerian simulation of solid mechanics, in Lagrangian kinematics of turbulent flows. Unlike traditional methods based on explicit particle tracking, the reference map facilitates the calculation of flow maps and gradients without the need for particles. This is achieved through an Eulerian update of the reference map, which records the take-off positions of fluid particles. This approach is found to be mathematically equivalent to the work of Leung (J. Comput. Phys., vol. 230, issue 9, 2011, pp. 3500–3524), who computed the flow map of simple two-dimensional flows using an Eulerian approach. We discuss important modifications necessary for its first application to complex three-dimensional turbulent flows, including the conservative, low-dissipation update of the flow map and the treatment of periodic boundary conditions. We first demonstrate the accuracy of finite-time Lyapunov exponent (FTLE) calculations based on the reference map against the standard particle-based approach in a two-dimensional Taylor–Green vortex. Then we apply it to turbulent channel flow at $Re_\tau =180$, where Lagrangian coherent structures identified as ridges of the backward-time FTLE are found to bound vortical regions of flow, consistent with Eulerian coherent structures from the $Q$-criterion. The reference map also proves suitable for material surface tracking despite not explicitly tracking particles. This capability can provide valuable insights into the Lagrangian landscape of turbulent momentum transport, complementing Eulerian velocity field analysis. The evolution of initially wall-normal material surfaces in the viscous sublayer, buffer layer and log layer sheds light on the Reynolds stress-generating events from a Lagrangian perspective.

This work investigates the spatio-temporal evolution of coherent structures in the wake of a generic high-speed train, based on a three-dimensional database from large eddy simulation. Spectral proper orthogonal decomposition (SPOD) is used to extract energy spectra and energy ranked empirical modes for both symmetric and antisymmetric components of the fluctuating flow field. The spectrum of the symmetric component shows overall higher energy and more pronounced low-rank behaviour compared with the antisymmetric one. The most dominant symmetric mode features periodic vortex shedding in the near wake, and wave-like structures with constant streamwise wavenumber in the far wake. The mode bispectrum further reveals the dominant role of self-interaction of the symmetric component, leading to first harmonic and subharmonic triads of the fundamental frequency, with remarkable deformation of the mean field. Then, the stability of the three-dimensional wake flow is analysed based on two-dimensional local linear stability analysis combined with a non-parallelism approximation approach. Temporal stability analysis is first performed for both the near-wake and the far-wake regions, showing a more unstable condition in the near-wake region. The absolute frequency of the near-wake eigenmode is determined based on spatio-temporal analysis, then tracked along the streamwise direction to find out the global mode growth rate and frequency, which indicate a marginally stable global mode oscillating at a frequency very close to the most dominant SPOD mode. The global mode wavemaker is then located, and the structural sensitivity is calculated based on the direct and adjoint modes derived from a local spatial analysis, with the maximum value localized within the recirculation region close to the train tail. Finally, the global mode shape is computed by tracking the most spatially unstable eigenmode in the far wake, and the alignment with the SPOD mode is computed as a function of streamwise location. By combining data-driven and theoretical approaches, the mechanisms of coherent structures in complex wake flows are well identified and isolated.

Investigations are conducted on the effect of wall proximity on the flow around a cylinder under an axial magnetic field, using the electrical potential probe technology to measure the velocity of liquid metal flow. The study focused on the impact of the inlet velocity of the fluid, the magnetic field and wall proximity on the characteristics of velocity fields, particularly on the vortex-shedding mode. Based on different magnitudes of the magnetic field and the distance from the cylinder to the duct wall, three types of vortex-shedding modes are identified, (I) shear layer oscillation state, (II) quasi-two-dimensional vortex-shedding states and (III) transition of the magnetohydrodynamic to hydrodynamic Kármán street. The transitions between these modes are analysed in detail. The experimental results show that the weak wall-proximity effect leads to the formation of the Kármán vortex street, while a reverse Kármán vortex street and secondary vortices emerge under a strong wall-proximity effect. It is noticed that the Kelvin–Helmholtz instability drives vortex shedding under regime I, leading to an increase in the Strouhal number (St) with stronger magnetic fields. Additionally, under a strong axial magnetic field, the wall-proximity effect (‘Shercliff layer effect’) promotes the instability of shear layers on both sides of the cylinder. These unique coupling effects are validated by variations in modal coefficients and energy proportions under different vortex-shedding regimes using the proper orthogonal decomposition method.

Spanwise vortex instability and the growth of secondary hairpin-like vortical structures in the wake of an oscillating foil are investigated numerically at Reynolds number 8000 in a range of chord-based Strouhal number ($0.32 \le St_c \le 0.56$). The phase-offset ($\phi$) between the heaving and pitching motion is $\phi = 90^\circ$. The wake at the lowest $St_c$ (0.32) is characterized by a single system of streamwise hairpin-like structures that evolve from the core vorticity outflux of the secondary leading edge vortex (LEV) over the foil boundary. The primary LEV features spanwise dislocations, but it does not reveal substantial changes advecting downstream. Increasing $St_c$ beyond 0.32 reveals that the transition in spanwise instability characterizes the deformation of primary LEV cores, which subsequently transforms to hairpin-like secondary structures. At higher $St_c$, stronger trailing edge vortices (TEVs) grow in close proximity to the primary LEVs, which contributes to an enhanced localized vortex compression and tilting near dislocations. This phenomenon amplifies the undulation amplitude of primary LEVs, eventually leading to vortex tearing. The larger circulation of TEVs with increasing $St_c$ provides an additional explanation for an accelerated vortex compression that coincides with a faster transition of spanwise LEV instability to secondary hairpin-like structures in the wake.

Flows over a disc are studied in a wind tunnel over incidence angles between $0^\circ \text { and }36^\circ$, a Reynolds number of $2.7 \times 10^4$ and rotational speed ratios of $0\unicode{x2013}10$. Smoke-wire visualization, particle image velocimetry and hot-film anemometry are employed. Two vortex shedding modes originating from the upstream surface of the disc are observed. The first is dominant at incidence angles up to ${\sim }21^\circ$. Beyond $21^\circ$, the second mode dominates. It appears as a soliton on the vortices and has a shedding frequency nearly twice that of the first at the highest incidence angle. The Strouhal number monotonically increases with incidence angle from approximately 0.2 to 0.4. Spectral analysis of the hot-film measurements confirms the findings from flow visualization experiments. Flows over the spinning disc generally mimic the stationary disc flows; however, centrifugal forces lead to cross-stream instability features that may be attributed to spiral wave instabilities intrinsic to the boundary layers in rotating flows. Velocity measurements are used to construct streamline patterns to compare with the smoke streaklines. The unsteadiness of the flows results in large variances. Mean strain rates are extracted from velocity data, where the fixed disc case at normal incidence compares well with theoretical predictions. The unsteady boundary layer thickness over the fixed disc, however, is approximately twice that predicted by theory for steady flow, stemming from the dominance of large unsteady vortices. Limited comparisons are made of the Strouhal numbers from experiments and numerical calculations in the literature.

The formation mechanism for the stopping vortex ring (SVR) and its effects on the development of starting jets have been systematically investigated. The radial inward flow near the nozzle exit, arising from the pressure difference caused by the deceleration of starting jets, is considered to be the main contributing factor to the formation of the SVR. The formation process can generally be divided into (i) the rapid accumulation stage ($t_d^*\leq 1$) and (ii) the development stage ($t_d^*>1$), where $t_d^*$ is the formation time defined by the duration of the deceleration stage. For starting jets with different $(L/D)_d$, the final circulation value and circulation growth rate of the SVR can be scaled by $[(L/D)_d]^{-0.5}$ and $[(L/D)_d]^{-1.5}$, respectively. Here $(L/D)_d$ represents the stroke ratio during the deceleration stage. Analysing the temporal evolution of fluid parcels in the vicinity of the nozzle exit reveals that SVR entrains fluid from both inside and outside of the nozzle. Additionally, the influence of the SVR on the leading vortex ring and the trailing jet has been examined, with particular attention to its effects on the propulsive performance of the starting jet. The SVR affects the profiles of axial velocity and gauge pressure at the nozzle exit, thereby enhancing the generation of total thrust during the deceleration stage. Analysis has shown that depending on the deceleration rate, SVR can enhance the average velocity thrust by at least $10\,\%$ and compensate for up to a $60\,\%$ reduction in pressure thrust due to deceleration.

Three-dimensional vortex dynamics around two pitching foils arranged in side-by-side (parallel) configurations is numerically examined at a range of separation (gap) distances ($0.5c \leqslant y^* \leqslant 1.5c$). In-phase ($\phi =0$) and out-of-phase ($\phi ={\rm \pi}$) motions are considered for Strouhal numbers of $0.3$ and $0.5$ at a Reynolds number of $8000$. In this work, we show that the foil proximity effect, defined as the influence of one foil on the flow characteristics around the other, induces a spanwise instability in the braids of trailing-edge vortices (TEVs) during their roll-up. This is a newly identified instability that manifests itself in the form of secondary vortical structures with opposite circulation compared with the TEVs formed on the foils, which leads to the formation of double necking on the braids of the TEVs. We provide quantitative evidence linking the formation of these secondary structures to the braid instability. The first neck merges with the TEV, while the second neck detaches from the braid region and moves downstream independently. As the foil proximity effect intensifies (spacing between the foils decreases), secondary vortical structures, as well as the necks, become more prominent, leading to the emergence of three-dimensional wake features. Lastly, the influence of kinematics of the foils on three-dimensionality of the wake is investigated. At higher Strouhal numbers, broader regions of high strain are developed near the trailing edge, associated with the detachment of stronger structures from the braids of TEVs. The characterized instability demonstrates consistent properties for in-phase and out-of-phase motions, albeit with specific differences in dynamics of leading-edge vortices.