The nonlinear dynamics of a helical vortex disturbed by a long-wave-instability mode is studied by direct numerical simulation. Vortex reconnection or self-reconnection of the helical vortex is shown to play a crucial role depending on the pitch of the helical vortex. For the larger pitch, a vortex ring is created after the vortex reconnection; it detaches from the remaining helical vortex, whose pitch is doubled. A vortex ring is also created for the smaller pitch; however, it forms a linked system with the remaining vortex. The topological constraint due to this linkage forces strong interaction between the different parts of the helical vortex, leading to turbulent transition.

We investigate two-dimensional (2-D) axisymmetric flow in toroidal geometry, with a focus on a transition between 2-D three-component flow and 2-D two-component flow. This latter flow state allows a self-organization of the system to a quiescent dynamics, characterized by long-living coherent structures. When these large-scale structures orient in the azimuthal direction, the radial transport is reduced. Such a transition, if it can be triggered in toroidally confined fusion plasmas, is beneficial for the generation of zonal flows and should consequently result in a flow field beneficial for confinement.

We study the energy stability of pressure-driven laminar magnetohydrodynamic flow in a rectangular duct with a transverse homogeneous magnetic field and electrically insulating walls. For sufficiently strong fields, the laminar velocity distribution has a uniform core and convex Hartmann and Shercliff boundary layers on the walls perpendicular and parallel to the magnetic field. The problem is discretized by a double expansion in Chebyshev polynomials in the cross-stream coordinates. The linear eigenvalue problem for the critical Reynolds number depends on the streamwise wavenumber, Hartmann number and the aspect ratio. We consider the limits of small and large aspect ratios in order to compare with stability models based on one-dimensional base flows. For large aspect ratios, we find good numerical agreement with results based on the quasi-two-dimensional approximation. The lift-up mechanism dominates in the limit of a zero streamwise wavenumber and provides a linear dependence between the critical Reynolds and Hartmann numbers in the duct. As the aspect ratio is reduced away from unity, the duct results converge to Orr's original energy stability result for spanwise uniform perturbations imposed on the plane Poiseuille base flow. We also examine different possible symmetries of eigenmodes as well as the purely hydrodynamic case in the duct geometry.

This paper reviews 70 years of cricket ball swing literature and reassesses the results in light of new measurements. A comparison of ball tracking data with published experimental results shows that current understanding does not explain the behaviour observed in professional cricket matches. Descriptions of cricket ball boundary layer aerodynamics are updated with new results which show that the seam acts more like a series of vortex generators than a trip and that there is a laminar separation bubble on the seam side of the ball. Previous results for reverse swing are consolidated and compared with new data, showing that different magnitudes of swing occur depending on the condition of both sides of the ball. Variation in pressure and temperature should be included when the Reynolds number, a non-dimensional ball speed, is calculated, while humidity can be neglected. Studies on the effect of wind speed and direction are summarised and results considering the effect of free-stream turbulence are compared with new measurements. Throughout the paper, recommendations for future work are suggested. These include quantitative study of the effect of surface defects and roughness, an assessment of whether atmospheric turbulence can affect swing and investigation into the effect of backspin on swing.

Large eddy simulation of flow past a cricket ball with its seam at $30^\circ$ to the free stream is carried out for $5 \times 10^4 \le Re \le 4.5 \times 10^5$. Three regimes of flow are identified on the basis of the time-averaged swing force coefficient ($\bar {C}_Z$) – no swing (NS), conventional swing (CS, $\bar {C}_Z>0$) and reverse swing (RS, $\bar {C}_Z<0$). The effect of seam on the boundary layer is investigated. Contrary to the popular belief, the boundary layer does not transition to a turbulent state in the initial stages of CS. The seam energizes the laminar boundary layer and delays its separation. The delay is significantly larger in a region near the poles, whose extent increases with an increase in $Re$ causing $\bar {C}_Z$ to increase. Here $\bar {C}_Z$ assumes a near constant value in the later stage of CS. The boundary layer transitions to a turbulent state via formation of a laminar separation bubble (LSB) in the equatorial region and directly, without a LSB, in the polar region. The extent of the LSB shrinks while the region of direct transition near the poles increases with an increase in $Re$. A LSB forms on the non-seam side of the ball in the RS regime. A secondary vortex is observed in the wake bubble. While it exists on the non-seam side for the entire range of $Re$ considered, the mixing in the flow introduced by the seam causes it to disappear beyond a certain $Re$ on the seam side. The pressure difference between the seam and non-seam sides sets up wing-tip-like vortices. Their polarity reverses with the switch from the CS to RS regime.

The turbulence behaviour of current-dominated pulsating flows has been investigated. Direct numerical simulations have been carried out for Stokes lengths over a range of $l_s^+=5\unicode{x2013}26$, and amplitudes spanning 90 % of the current-dominated regime, about a mean flow of $\overline {Re}=6275$. The results show that the turbulence response in intermediate and low-frequency pulsations is governed by a multistage turbulent–turbulent transition process, which bears a strong similarity to the multistage response of non-periodic acceleration. During the early acceleration period, the flow enters a pretransition stage, in which a new laminar perturbation boundary layer forms at the wall, and the streamwise velocity streaks are stretched. If the low-speed streaks destabilise prior to the deceleration period, then the flow enters a transition stage in which the perturbation boundary layer undergoes a bypass-like transition process. A unique feature of pulsating flows is the ongoing mechanism of turbulence decay, which initiates during the deceleration period and constitutes the main transient turbulence mechanism for much of the cycle. For high-frequency pulsations, the perturbation boundary layer fails to reach the pretransition stage prior to the deceleration period. Instead, the flow alternates between two inertial stages which are characterised by two layers of amplified viscous force; one growing at the wall, and one detached and moving towards the core.

The numerical investigation focuses on the flow patterns around a rectangular cylinder with three aspect ratios ($L/D=5$, $10$, $15$) at a Reynolds number of $1000$. The study delves into the dynamics of vortices, their associated frequencies, the evolution of the boundary layer and the decay of the wake. Kelvin–Helmholtz (KH) vortices originate from the leading edge (LE) shear layer and transform into hairpin vortices. Specifically, at $L/D=5$, three KH vortices merge into a single LE vortex. However, at $L/D=10$ and $15$, two KH vortices combine to form a LE vortex, with the rapid formation of hairpin vortex packets. A fractional harmonic arises due to feedback from the split LE shear layer moving upstream, triggering interaction with the reverse flow. Trailing edge (TE) vortices shed, creating a Kármán-like street in the wake. The intensity of wake oscillation at $L/D=5$ surpasses that in the other two cases. Boundary layer transition occurs after the saturation of disturbance energy for $L/D=10$ and $15$, but not for $L/D=5$. The low-frequency disturbances are selected to generate streaks inside the boundary layer. The TE vortex shedding induces the formation of a favourable pressure gradient, accelerating the flow and fostering boundary layer relaminarization. The self-similarity of the velocity defect is observed in all three wakes, accompanied by the decay of disturbance energy. Importantly, the decrease in the shedding frequency of LE (TE) vortices significantly contributes to the overall decay of disturbance energy. This comprehensive exploration provides insights into complex flow phenomena and their underlying dynamics.

Superhydrophobic surfaces dramatically reduce skin friction of overlying liquid flows. These surfaces are complex and numerical simulations usually rely on models to reduce this complexity. One of the simplest consists of finding an equivalent boundary condition through a homogenisation procedure, which in the case of channel flow over oriented riblets, leads to the presence of a small spanwise component in the homogenised base flow velocity. This work aims at investigating the influence of such a three-dimensionality of the base flow on stability and transition in a channel with walls covered by oriented riblets. Linear stability for this base flow is investigated: a new instability region, linked to cross-flow effects, is observed. Tollmien–Schlichting waves are also retrieved but the most unstable are three-dimensional. Transient growth is also affected as oblique streaks with non-zero streamwise wavenumber become the most amplified perturbations. When transition is induced by Tollmien–Schlichting waves, after an initial exponential growth regime, streaky structures with large spanwise wavenumber rapidly arise. Modal mechanisms appear to play a leading role in the development of these structures and a secondary stability analysis is performed to retrieve successfully some of their characteristics. The second scenario, initiated with cross-flow vortices, displays a strong influence of nonlinearities. The flow develops into large quasi-spanwise-invariant structures before breaking down to turbulence. Secondary stability on the saturated cross-flow vortices sheds light on this stage of transition. In both cases, cross-flow effects dominate the flow dynamics, suggesting the need to consider the anisotropicity of the wall condition when modelling superhydrophobic surfaces.

An extended turbulent state can coexist with the stable laminar state in pipe flows. We focus here on short pipes with additional discrete symmetries imposed. In this case, the boundary between the coexisting basins of attraction, often called the edge of chaos, is the stable manifold of an edge state, which is a lower-branch travelling wave solution. We show that a low-dimensional submanifold of the edge of chaos can be constructed from velocity data using the recently developed theory of spectral submanifolds (SSMs). These manifolds are the unique smoothest nonlinear continuations of non-resonant spectral subspaces of the linearized system at stationary states. Using very low-dimensional SSM-based reduced-order models, we predict transitions to turbulence or laminarization for velocity fields near the edge of chaos.

The first long-lived turbulent structures observable in planar shear flows take the form of localized stripes, inclined with respect to the mean flow direction. The dynamics of these stripes is central to transition, and recent studies proposed an analogy to directed percolation where the stripes’ proliferation is ultimately responsible for the turbulence becoming sustained. In the present study we focus on the internal stripe dynamics as well as on the eventual stripe expansion, and we compare the underlying mechanisms in pressure- and shear-driven planar flows, respectively, plane-Poiseuille and plane-Couette flow. Despite the similarities of the overall laminar–turbulence patterns, the stripe proliferation processes in the two cases are fundamentally different. Starting from the growth and sustenance of individual stripes, we find that in plane-Couette flow new streaks are created stochastically throughout the stripe whereas in plane-Poiseuille flow streak creation is deterministic and occurs locally at the downstream tip. Because of the up/downstream symmetry, Couette stripes, in contrast to Poiseuille stripes, have two weak and two strong laminar turbulent interfaces. These differences in symmetry as well as in internal growth give rise to two fundamentally different stripe splitting mechanisms. In plane-Poiseuille flow splitting is connected to the elongational growth of the original stripe, and it results from a break-off/shedding of the stripe's tail. In plane-Couette flow splitting follows from a broadening of the original stripe and a division along the stripe into two slimmer stripes.

Boundary layers of Novec649, a low-global-warming potential fluid of interest for low-grade heat recovery, are investigated numerically by means of linear stability theory, direct numerical simulation (DNS) and large-eddy simulations (LES). This organic vapour is of interest in organic Rankine cycle (ORC) turbines and realistic thermodynamic conditions are selected. Under these conditions, the vapour behaves as a dense gas and, due to its high molecular complexity, real-gas effects occur. In addition, the fluid exhibits large and highly variable heat capacities and density- as well as temperature-dependent transport properties. More specifically we report the first direct and LES of transitional and turbulent boundary layers of Novec649 at high-subsonic conditions $M=0.9$. A controlled transition is performed by using oblique modes determined by linear stability theory extended to dense gases. An oblique-type transition is obtained as in low-speed air flows, where sinuous streaks develop by the lift-up mechanism and break down into turbulence. In the turbulent state, the profiles of dynamic flow properties (velocities, turbulent intensities, turbulent kinetic energy budgets) are little affected by the gas properties and remain very close to incompressible DNS, despite the high-subsonic flow speed. The fluctuations levels for thermodynamic properties have been quantified with respect to air flows. Notwithstanding a drastic reduction, genuine compressibility effects are present. For example, the fluctuating Mach number and the acoustic mode are characteristic of high-speed flows. The influence of forcing frequency and amplitude on the established turbulent state has been investigated using LES. An analysis of integral quantities shows a slow relaxation towards a canonical equilibrium turbulent state for all cases due to the high Reynolds numbers typical of dense gas flows. Overall the present DNS constitutes a valuable reference not only for forthcoming experiments but also for future studies of free-stream transition and loss mechanisms in ORC turbines.

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of spatially intermittent turbulent structures. In plane shear flows, these appear as a regular pattern of alternating turbulent and quasi-laminar flow. Both the physical and the spectral energy balance of a turbulent–laminar pattern in plane Couette flow are computed and compared to those of uniform turbulence. In the patterned state, the mean flow is strongly modulated and is fuelled by two mechanisms: primarily, the nonlinear self-interaction of the mean flow (via mean advection), and secondly, the extraction of energy from turbulent fluctuations (via negative spectral production, associated with an energy transfer from small to large scales). Negative production at large scales is also found in the uniformly turbulent state. Important features of the energy budgets are surveyed as a function of $Re$ through the transition between uniform turbulence and turbulent–laminar patterns.

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of oblique, spatially intermittent turbulent structures. In plane Couette flow, these emerge from uniform turbulence via a spatio-temporal intermittent process in which localised quasi-laminar gaps randomly nucleate and disappear. For slightly lower Reynolds numbers, spatially periodic and approximately stationary turbulent–laminar patterns predominate. The statistics of quasi-laminar regions, including the distributions of space and time scales and their Reynolds-number dependence, are analysed. A smooth, but marked transition is observed between uniform turbulence and flow with intermittent quasi-laminar gaps, whereas the transition from gaps to regular patterns is more gradual. Wavelength selection in these patterns is analysed via numerical simulations in oblique domains of various sizes. Via lifetime measurements in minimal domains, and a wavelet-based analysis of wavelength predominance in a large domain, we quantify the existence and nonlinear stability of a pattern as a function of wavelength and Reynolds number. We report that the preferred wavelength maximises the energy and dissipation of the large-scale flow along laminar–turbulent interfaces. This optimal behaviour is due primarily to the advective nature of the large-scale flow, with turbulent fluctuations playing only a secondary role.

This investigation characterises the time response and the transient turbulence dynamics undergone by rapidly decelerating turbulent pipe flows. A series of direct numerical simulations of decelerating flows between two steady Reynolds numbers were conducted for this purpose. The statistical analyses reveal that rapidly decelerating turbulent flows undergo four coherent, unambiguous transitional stages: inertial (stage I), a dramatic change of sign in the viscous force associated with the decay of the viscous shear stress at the wall together with a mild turbulence decay in the viscous sublayer; friction recovery (stage II), a recovery in viscous force and progressive decay in the turbulent inertia at the near-wall region; turbulence decay (stage III), a balanced decay in both turbulent inertia and viscous force at the near-wall and overlap regions; core relaxation (stage IV), slow turbulence decay at the core region. The FIK identity derived by Fukagata, Iwamoto and Kasagi (Phys. Fluids, vol. 14, 2002, L73–L76) was used to understand further how the flow dynamics influence the time response of the skin friction coefficient ($C_f$). The results show that although $C_f$ plateaus during the fourth stage, the turbulent contribution keeps decaying, undershoots and finally recovers to attain its final steady value. The time evolution of the azimuthal vorticity ($\omega _\theta$) flux reveals that as the flow is decelerated, a layer of negative $\omega _\theta$ is produced at the wall during the flow excursion. As time progresses, this negative vorticity propagates in the wall-normal direction, attenuating the pre-existing vorticity and producing a decay in the turbulence levels.

Recent findings on the Reynolds-number-dependent behaviour of near-wall turbulence in terms of the ‘foot-printing’ of outer large-scale structures call for a new modelling development. A two-scale framework was proposed to couple a local fine-mesh solution with a global coarse-mesh solution by He (Intl J. Numer. Meth. Fluids, vol. 86, 2018, pp. 655–677). The methodology was implemented and demonstrated by Chen & He (J. Fluid Mech, vol. 933, 2022, p. A47) for a canonical turbulent channel flow, where the mesh-count scaling with Reynolds number is potentially reduced from $O(R{e^2})$ for a conventional wall-resolved large-eddy simulation (WRLES) to $O(R{e^1})$. The present work extends the two-scale method to turbulent boundary layers. A two-dimensional roughness element is used to trip a turbulent boundary layer. It is observed that large-scale disturbances originating at the trip have a much shorter lifetime and weaker foot-printing signatures on near-wall flow compared to those long streaky coherent structures in well-developed wall-bounded turbulent flows. Modal analyses show that the impact of trip-induced large scales can be adequately captured by a locally embedded fine-mesh block. For the tripped turbulent boundary layer, a Chebyshev block-spectral mapping is adopted to propagate source terms from the local fine-mesh blocks to the global coarse-mesh domain, driving to a target solution for the upscaled equations. The computed mean statistics and energy spectra are in good agreement with corresponding experimental data, WRLES and direct numerical simulation (DNS) results. The overall mesh count–$Re$ scaling is estimated to reduce from $O(R{e^{1.8}})$ for the full wall-resolved LES to $O(R{e^{0.9}})$ for the present two-scale solution.

The flow past a cylinder in proximity to a plane wall is investigated numerically for small gap ratios. Three vortex dynamic processes associated with different hairpin vortex generation mechanisms are identified for the first time, and the wake-induced turbulent transition is analysed. The vortex shedding is suppressed at $G/D = 0.1$, while the spanwise vortex is generated via a Kelvin–Helmholtz instability and evolves into hairpin vortices. For $G/D= 0.3$, the upper and lower rollers alternatively shedding from the cylinder, interact with the secondary vortex. The split secondary vortex merges with the upper roller and results in a new vortex downstream, which develops into hairpin vortices. When $G/D = 0.9$, the secondary vortex interacts with the lower roller and then evolves into hairpin vortices. A tertiary vortex induced by the secondary vortex is observed, rotating in the opposite direction to the secondary vortex the wake-induced transitions share the same route. The velocity fluctuations deviate from the optimal growth theory in the pre-transitional region. In the transitional region low-frequency disturbances penetrate the sheltering edge to generate streaks where the disturbance energy declines. In the turbulent region the logarithmic layer is formed, indicating that the turbulent equilibrium is established.

The topology of a laminar separation bubble (LSB) on a sphere in the critical regime is investigated via experiments at five turbulent intensities: $T_u=0.06\,\%$, $0.42\,\%$, $0.71\,\%$, $1.00\,\%$ and $1.36\,\%$. The drag crisis occurs at a lower $Re$ and becomes gradual with increasing $T_u$. The flow is devoid of the LSB at the onset of the critical regime. It forms on a small part of the sphere and not at all azimuthal locations, early in the critical regime. The LSB forms at more azimuthal locations with increasing $Re$. This azimuthal expansion of the LSB is accompanied by intermittency for a small range of $Re$. Towards the end of the critical regime, an axisymmetric LSB forms on the sphere at all time instants. A model is proposed to estimate the azimuthal extent and distribution of the LSB from the mean force coefficients of a flow state. The model predicts that the LSB forms as multiple segments for a large part of the critical regime. During the spatial growth of the LSB with $Re$ in the critical regime, some of its fragments relocate to alternate locations. Moderate increase in $T_u$ ($0.42\,\% \leq T_u \leq 0.71\,\%$) leads to rich dynamics with several intermittent flow states. However, fewer intermittent states are observed beyond a certain $T_u$ (${\geq }1.00\,\%$).

This paper describes a study of spatially accelerating turbulent flow based on the direct numerical simulation of a flow with longitudinally accelerating moving walls to create a relative acceleration between the fluid and the wall without inducing streamline curvature. The results show a broad similarity to those of previous investigations of spatial acceleration, albeit with some differences. A new interpretation has been proposed considering this spatially accelerating flow to be characterised by the formation of a new boundary layer superimposed on the pre-existing turbulent flow. This is followed by the transition of the flow in response to the development of this new boundary layer. This can be seen as an extension of the transition theory for temporally accelerating turbulent flows (He & Seddighi, J. Fluid Mech., vol. 715, 2013, pp. 60–102). The existing turbulent structures act as disturbances for the new boundary layer similar to the role of free-stream turbulence in bypass transition. This boundary layer modulates the pre-existing near-wall structures, amplifying and elongating the streaks. Some streaks eventually become unstable in a sinuous mechanism reminiscent of streak breakdown in near-wall turbulence, resulting in the formation localised turbulent spots which spread until the entire wall is covered in new turbulence. This interpretation naturally splits the flow into a new boundary layer region and a core (or free-stream) flow with interactions between the two dominating a significant length of the flow development and potentially offers a new explanation for the slow evolution of the turbulent stresses observed previously.

Machine Learning (ML) is becoming increasingly popular in fluid dynamics. Powerful ML algorithms such as neural networks or ensemble methods are notoriously difficult to interpret. Here, we introduce the novel Shapley additive explanations (SHAP) algorithm (Lundberg & Lee, Advances in Neural Information Processing Systems, 2017, pp. 4765–4774), a game-theoretic approach that explains the output of a given ML model in the fluid dynamics context. We give a proof of concept concerning SHAP as an explainable artificial intelligence method providing useful and human-interpretable insight for fluid dynamics. To show that the feature importance ranking provided by SHAP can be interpreted physically, we first consider data from an established low-dimensional model based on the self-sustaining process (SSP) in wall-bounded shear flows, where each data feature has a clear physical and dynamical interpretation in terms of known representative features of the near-wall dynamics, i.e. streamwise vortices, streaks and linear streak instabilities. SHAP determines consistently that only the laminar profile, the streamwise vortex and a specific streak instability play a major role in the prediction. We demonstrate that the method can be applied to larger fluid dynamics datasets by a SHAP evaluation on plane Couette flow in a minimal flow unit focussing on the relevance of streaks and their instabilities for the prediction of relaminarisation events. Here, we find that the prediction is based on proxies for streak modulations corresponding to linear streak instabilities within the SSP. That is, the SHAP analysis suggests that the break-up of the self-sustaining cycle is connected with a suppression of streak instabilities.