A complete electrohydrodynamic axisymmetric model of coaxial electrosprays under constant electrical permittivities and conductivities, and strict immiscibility of the liquids, is presented. The numerical solution of the model is experimentally validated, and a detailed portrait of the complex physics begotten by the large spectrum of variables involved is described in a choice of representative cases. The appearance of surface charge in both the outer and the inner free interfaces, and some of their interesting features, are revealed. Among those, the possibility of the appearance of a limited segment of opposite charge (negative, if the polarity of the applied potential is positive, or vice versa) in the inner free surface at the transition region illustrates the unexpected features exhibited by this complex system. This has fundamental physicochemical implications in the production of core–shell formulations or the use of the system in analytical chemistry, among a wide variety of applications.

Direct numerical simulations (DNS) of turbulent channel flow over hexagonally packed hemispheres were performed for friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}=200{-}600$. For cases at $Re_{\unicode[STIX]{x1D70F}}=400$, the inner-scaled roughness height $k^{+}=20$ was maintained while the spacing between roughness elements was varied from $d/k=2$ to 4. Two additional rough-wall cases were performed at $Re_{\unicode[STIX]{x1D70F}}=200$ and $600$, where $k^{+}=20$ and $d/k=4$ were fixed to investigate the $Re$ trends. For each case, wall shear stress was extracted from DNS by integrating the stress tensor over the rough surfaces. Spherical harmonics were employed to investigate the detailed spectral behaviour of the wall shear stress. Flow visualization near roughness elements was used to assist physical interpretations of the dominant flow features observed for various roughness characteristics. Analysis of amplitude modulation was applied to investigate the interactions between the ‘cell-averaged’ wall shear stress and outer, large-scale structures. A universal signal was obtained by removing the effects of outer, large-scale motions, based on the model proposed by Mathis et al. (J. Fluid Mech., vol. 715, 2013, pp. 163–180). Pre-multiplied spectra of the universal wall shear stress showed distinct behaviours at smaller scales for the ‘k-type’ roughness ($d/k=3{-}4$) compared to ‘d-type’ roughness ($d/k=2$), whereas the spectra at larger scales appeared similar for both types of roughness. A scaling relation between the variance of universal wall shear stress and averaging cell dimensions was found for both ‘k-type’ and ‘d-type’ roughness, which can be useful in designing candidate wall models used in large-eddy simulation.

We study the compressible flow dynamics of two-dimensional, steady detonation wave propagation in a high explosive (HE) confined by aluminium (Al) or stainless steel (SS), outside of which is an air layer. We examine how the thickness of the confinement affects the subsonic detonation driving zone structure (DDZ) and the detonation speed $(D_{0})$, demonstrating a strong dependence on whether the oblique shock-driven flow in the confiner is supersonic, as for SS, or subsonic, as for Al. A characteristic path analysis is used to examine the information flow from the material boundaries through the supersonic flow regions in both the HE and confiner that can impact the sonic surfaces bounding the subsonic flow regions. It is shown that the nature of gas-dynamic wave reflection off the SS–air or Al–air boundary can significantly influence the DDZ and $D_{0}$.

The aerodynamic characteristics of a hovering rhinoceros beetle are numerically and theoretically investigated. Its wing kinematics is measured using high speed cameras and used for numerical simulations of flow around a flapping rhinoceros beetle in hovering flight. The numerical results show that the aerodynamic forces generated (especially for lift) and power required by the hind wing during a quasi-periodic state are quite different from those during the first stroke. This indicates that the wing–wake interaction significantly affects the aerodynamic performance of the hind wing during the quasi-periodic state. Also, twisting of the hind wing along the wing span direction does not contribute much to total force generation as compared to that of the flat wing, and the role of elytron and body on the aerodynamic performance is quite small, at least for the present hovering flight. Based on a previous model (Wang et al., J. Fluid Mech., vol. 800, 2016, pp. 688–719), we suggest an improved predictive aerodynamic model without any ad hoc model constants for a rigid and flat hind wing by considering the effect of the wing–wake interaction in hovering flight. In this model, we treat the wake as a steady or unsteady non-uniform downwash motion and obtain its magnitude by combining a quasi-steady blade element theory with an inviscid momentum theory. The lift and drag forces and aerodynamic power consumption predicted by this model are in excellent agreement with those obtained from numerical simulations.

Cavitating flow over a circular cylinder is investigated over a range of cavitation numbers ($\unicode[STIX]{x1D70E}=5$ to $0.5$) for both laminar (at Reynolds number $(Re)=200$) and turbulent (at $Re=3900$) regimes. We observe non-cavitating, cyclic and transitional cavitation regimes with reduction in free-stream $\unicode[STIX]{x1D70E}$. The cavitation inside the Kármán vortices in the cyclic regime, is significantly altered by the onset of ‘condensation front’ propagation in the transitional regime. At the transition, an order of magnitude jump in shedding Strouhal number ($St$) is observed as the dominant frequency shifts from periodic vortex shedding in the cyclic regime, to irregular–regular vortex shedding in the transitional regime. In addition, a peak in pressure fluctuations, and a maximum in $St$ versus $\unicode[STIX]{x1D70E}$ based on cavity length are observed at the transition. Shedding characteristics in each regime are discussed using dynamic mode decomposition. A numerical method based on the homogeneous mixture model, fully compressible formulation and finite rate mass transfer developed by Gnanaskandan & Mahesh (Intl J. Multiphase Flow, vol. 70, 2015, pp. 22–34) is extended to include the effects of non-condensable gas (NCG). It is demonstrated that the condensation fronts observed in the transitional regime are supersonic (referred to as ‘condensation shocks’). In the presence of NCG, multiple condensation shocks in a given cycle are required for complete cavity condensation and detachment, as compared to a single condensation shock when only vapour is present. This is explained by the reduction in pressure ratio across the shock in the presence of NCG, effectively reducing its strength. In addition, at $\unicode[STIX]{x1D70E}=0.85$ (near transition from the cyclic to the transitional regime), the presence of NCG suppresses the low frequency irregular–regular vortex shedding. Vorticity transport at $Re=3900$, in the transitional regime, indicates that the region of attached cavity is nearly two-dimensional, with very low vorticity, affecting Kármán shedding in the near wake. Majority of vortex stretching/tilting and vorticity production is observed following the cavity trailing edge. In addition, the boundary-layer separation point is found to be strongly dependent on the amounts of vapour and gas in the free stream for both laminar and turbulent regimes.

Hysteresis phenomena in forced gravity–capillary waves on deep water where the minimum phase speed $c_{min}=23~\text{cm}~\text{s}^{-1}$ are experimentally investigated. Four kinds of forcings are considered: two-dimensional/three-dimensional air-blowing/air-suction forcings. For a still-water initial condition, as the forcing speed increases from zero towards a certain target speed ($U$), there exists a certain critical speed ($U_{crit}$) at which the transition from linear to nonlinear states occurs. When $U<U_{crit}$, steady linear localized waves are observed (state I). When $U_{crit}<U<c_{min}$, steady nonlinear localized waves, including steep gravity–capillary solitary waves, are observed (state II). When $U\approx c_{min}$, periodic shedding phenomena of nonlinear localized depressions are observed (state III). When $U>c_{min}$, steady linear non-local waves are observed (state IV). Next, with these state-II, III and IV waves as new initial conditions, as the forcing speed is decreased towards a certain target speed ($U_{final}$), a certain critical speed ($U_{crit,2}$) is identified at which the transition from nonlinear to linear states occurs. When $U_{crit,2}<U_{final}<U_{crit}$, relatively steeper steady nonlinear localized waves, including steeper gravity–capillary solitary waves, are observed. When $U_{final}<U_{crit,2}$, linear state-I waves are observed. These are hysteresis phenomena, which show jump transitions from linear to nonlinear states and from nonlinear to linear states at two different critical speeds. For air-blowing cases, experimental results are compared with simulation results based on a theoretical model equation. They agree with each other very well except that the experimentally identified critical speed ($U_{crit,2}$) is different from the theoretically predicted one.

Thermoacoustic devices represent a significant future opportunity in the fields of energy generation and refrigeration. A key component of this type of device is the regenerator, where the conversion between acoustic energy and thermal energy takes place. This conversion occurs due to an externally imposed temperature gradient on the walls of the regenerator channels. Hence, this paper concerns the physics of sound waves in the proximity of such walls. It establishes a new analytical framework which clarifies the disturbance energy conservation in thermoacoustic devices. In this framework, a thermoacoustic production term is proposed to quantify the generation or consumption of disturbance energy originating from the temperature gradient. An extended disturbance energy flux term is identified to account for wave growth or decay through the regenerator. The disturbance energy balance relation states that the disturbance energy flux equals the thermoacoustic production less the viscous and thermal dissipation resulting from gradients of fluctuating velocity and temperature. The analytical framework is implemented in an axisymmetric cylindrical domain; the two-dimensional nature of this work helps to uncover that the wave always decays in the region close to the wall. A dimensional analysis is conducted to identify the controlling parameters, namely the Womersley, Helmholtz and Prandtl numbers. A parametric study of the Womersley and Helmholtz numbers is conducted to showcase the new analytical methodology; the results make it possible to optimize the geometry, wave properties and working conditions of a thermoacoustic device according to the preference of its efficiency, loss and output.

Large-eddy simulations of a skewed bump are performed with the bump at angles of $10^{\circ }$ and $50^{\circ }$ with respect to the free stream. The wake has a large separation bubble and unsteady vortex structures that are highly sensitive to the bump angle. The $10^{\circ }$ bump has secondary separation with forward-facing flow in the main separation bubble. The Reynolds stress suggests large-scale motions in the wake, but are insufficient to understand the shedding dynamics. The quasi-periodic shedding cycle is reconstructed using spectral proper orthogonal decomposition. For the $10^{\circ }$ case, opposite-sign streamwise vortices are alternately advected downstream in the wake, but only a single vortex is distinguishable in the $50^{\circ }$ case. The wake streamwise velocity pulses twice each cycle in the $10^{\circ }$ bump, but once per cycle in the $50^{\circ }$ bump. The separation bubble and wake oscillate in the spanwise direction. Conditional averaging shows that the vortices are associated with low streamwise velocity and high turbulence. Surface pressures on the bump have negatively skewed distributions in regions on the upstream face and in the separation bubble. In the separation bubble, fluctuations associated with the shedding cycle contribute to the negative skewness.

Collisions between a stream of drops and a continuous jet of a different liquid are experimentally investigated. In contrast to previous studies, our work focuses on the effects of liquid miscibility and wettability on the collision outcomes. Thus, miscible and immiscible liquids providing total and partial wetting are used. We show that, as long as the jet surface tension is smaller than the drop surface tension, the drops can be encapsulated by the jet, providing the so-called drops-in-jet structure. The transitions between the different regimes remain similar in nature with a capillary fragmentation responsible for the jet break-up and an inertial fragmentation causing the drops (and then possibly the jet) to break up. The dimensionless numbers proposed in the literature to model the inertial fragmentation thresholds do not bring the results obtained with different liquids at the same critical value. We explain the reason via a detailed analysis of the collisions, accounting for the drop and jet extensions and their kinetics. The drop fragmentation is found to occur during the recoil phase, leading us to propose a new dimensionless parameter that successfully reproduced all our experimental data obtained with immiscible liquids. Finally, we demonstrate that the most dramatic change of the collision outcomes is produced by using drops that totally wet the jet. In this case, the encapsulation of the drops cannot be achieved, constituting a true limit to some applications based on the solidification of the drops-in-jet structure.

The structure of free-surface flows in a straight compound channel was investigated in a laboratory flume, consisting of a central smooth-bed main channel (MC) and two adjacent rough-surface floodplains (FPs). The experiments covered both uniform and non-uniform flow conditions, with the latter generated by imposing an imbalance in the discharge distribution between MC and FPs at the flume entrance. The non-uniform cases involved transverse currents directed from MC to FPs and vice versa. The focus of the study was on assessing the effects of transverse currents on: (i) transverse shear layer and horizontal Kelvin–Helmholtz-type coherent structures (KHCSs) forming at the interfaces between MC and FPs; (ii) helical secondary currents (SCs) developing across the channel due to topography-induced flow heterogeneity; and (iii) turbulent large- and very-large-scale motions (VLSMs). Transverse currents can entirely displace the shear layer over FP or in MC, but they do not alter the KHCSs to the same degree, resulting in a mismatch between shear layer extent and KHCS length scales. KHCSs emerge once dimensionless velocity shear exceeds a critical value above which KHCS length scales increase with the shear. Three well-established SC cells, which are induced by turbulence anisotropy, are observed in uniform flow and non-uniform flow with transverse currents towards FP. They are replaced by a single cell in the presence of a transverse mean flow towards MC. The spectral signatures of VLSMs are visible at the upstream section of the flume but they quickly disappear along the flow, being suppressed by simultaneous development of KHCSs and SCs.

Microfluidic systems are usually fabricated with soft materials that deform due to the fluid stresses. Recent experimental and theoretical studies on the steady flow in shallow deformable microchannels have shown that the flow rate is a nonlinear function of the pressure drop due to the deformation of the upper soft wall. Here, we extend the steady theory of Christov et al. (J. Fluid Mech., vol. 841, 2018, pp. 267–286) by considering the start-up flow from rest, both in pressure-controlled and in flow-rate-controlled configurations. The characteristic scales and relevant parameters governing the transient flow are first identified, followed by the development of an unsteady lubrication theory assuming that the inertia of the fluid is negligible, and that the upper wall can be modelled as an elastic plate under pure bending satisfying the Kirchhoff–Love equation. The model is governed by two non-geometrical dimensionless numbers: a compliance parameter $\unicode[STIX]{x1D6FD}$, which compares the characteristic displacement of the upper wall with the undeformed channel height, and a parameter $\unicode[STIX]{x1D6FE}$ that compares the inertia of the solid with its flexural rigidity. In the limit of negligible solid inertia, $\unicode[STIX]{x1D6FE}\rightarrow 0$, a quasi-steady model is developed, whereby the fluid pressure satisfies a nonlinear diffusion equation, with $\unicode[STIX]{x1D6FD}$ as the only parameter, which admits a self-similar solution under pressure-controlled conditions. This simplified lubrication description is validated with coupled three-dimensional numerical simulations of the Navier equations for the elastic solid and the Navier–Stokes equations for the fluid. The agreement is very good when the hypotheses behind the model are satisfied. Unexpectedly, we find fair agreement even in cases where the solid and liquid inertia cannot be neglected.

The dynamics and control of a vortex pair in ground effect are investigated in a planar, incompressible and laminar setting. The evolution of the vortices obtained numerically shows vortex rebound as a consequence of the separation of the boundary layer induced at the wall by the vortices. An optimal control approach is developed and employed for vortex Reynolds numbers of 200 and 1000 in order to identify the optimal Dirichlet boundary condition at the wall to counteract this rebound and allow for an increased lateral displacement of the vortex, similarly to the inviscid evolution of the flow, which features hyperbolic trajectories. The work is primarily a conceptual approach to deal with aircraft separation distances in airport airspace by moving the vortices laterally, away from the runway but may also apply to the control of coherent structures in wall bounded turbulence. The most efficient control is able to double the lateral position and yields mostly vertical in and outflow at the wall. An optimal horizon time is found, equal to 5 characteristic time units of the vortex system, beyond which control is not able to further displace the vortices. The control is shown to delay the separation of the boundary layer at the origin of vortex rebound by applying suction ahead of the vortex, and to generate a vorticity flux at the wall, leading to a pusher vortex of sign opposite to that of the primary vortex, that attenuates the effect of the no-slip boundary condition at the wall by pushing the vortex outward.

We employ kinetic theory, extended to incorporate the influence of velocity correlations, friction and particle stiffness, and a model for rate-independent, elastic components of the stresses at volume fractions larger than a critical value, in an attempt to reproduce the results of discrete-element numerical simulations of steady, fully developed, dissipative, collisional shearing flows over and within inclined, erodible, fragile beds. The flows take place between vertical, frictional sidewalls at different separations with sufficient total particle flux so that differently inclined, erodible beds result. Numerical solutions of the spanwise-averaged differential equations of the theory and the associated boundary conditions are seen to be capable of reproducing profiles of stresses, solid volume fraction, average velocity and the strength of the particle velocity fluctuations, both in the rapid collisional flow above the bed and in the slower creeping flow within the bed. The indication is that extended kinetic theory has the unique ability to faithfully describe steady, inhomogeneous, granular shearing flows, ranging from dilute to extremely dense, using balances of momentum and energy and employing boundary conditions that are associated with the balances, with a small number of physically determined, microscopic parameters.

The flow within an oscillatory boundary layer, which approximates the flow generated by propagating sea waves of small amplitude close to the bottom, is simulated numerically by integrating the Navier–Stokes and continuity equations. The bottom is made up of spherical particles, free to move, which mimic sediment grains. The approach allows one to fully resolve the flow around the particles and to evaluate the forces and torques that the fluid exerts on their surface. Then, the dynamics of sediments is explicitly computed by means of the Newton–Euler equations. For the smallest value of the flow Reynolds number presently simulated, the flow regime turns out to fall in the intermittently turbulent regime such that turbulence appears when the free-stream velocity is close to its largest value but the flow recovers a laminar-like behaviour during the remaining phases of the cycle. For the largest value of the Reynolds number, turbulence is significant during almost the whole flow cycle. The evaluation of the sediment transport rate allows one to estimate the reliability of the empirical predictors commonly used to estimate the amount of sediments transported by sea waves. For large values of the Shields parameter, the sediment flow rate during the accelerating phases does not differ from that observed during the decelerating phases. However, for relatively small values of the Shields parameter, the amount of moving particles depends not only on the bottom shear stress but also on flow acceleration. Moreover, the numerical results provide information on the role that turbulent eddies have on sediment dynamics.

The problem of optimal initial disturbances in thermal wind shear is revisited and extended to include non-hydrostatic effects. This systematic study compares transient and modal growth rates of submesoscale instabilities over a large range of zonal and meridional wavenumbers, aspect ratios and different Richardson number regimes. Selection criteria were derived to remove spurious and unresolved instability modes that arise from the eigenvalue problem and we generalize the study of the hydrostatic Eady problem by Heifetz & Farrell (J. Atmos. Sci., vol. 60, 2003; J. Atmos. Sci., vol. 64 (12), 2007, pp. 4366–4382; Q. J. R. Meteorol. Soc., vol. 134 (635), 2008, pp. 1627–1633) to thin fronts, characterized by large aspect ratios. Such fronts are commonly found at the early stages of frontogenesis, for example, in the ocean mesoscale eddies and near the eye wall of hurricanes. In particular, we show that transient energy growth rates are up to two orders of magnitude larger than modal counterparts for a wide range of Richardson number and that the effects of transient energy gain become even greater when non-hydrostatic effects become important and/or for large Richardson numbers. This study also compares the dominant energy pathways contributing to the energy growth at short and long times. For symmetric modes, we recover the inertia–gravity instability described in Xu et al. (J. Atmos. Sci., vol. 64 (6), 2007, pp. 1764–1781). These mechanisms are shown to be the most powerful mediator of vertical transport when compared with the fastest growing baroclinic and symmetric modes. These results highlight the importance of transient processes in the ocean and the atmosphere.

According to Townsend’s attached eddy hypothesis (AEH), a boundary layer flow is comprised of wall-attached eddies, but to extract the part of the flow whose statistical behaviours are well described by the AEH is not at all straightforward. The objective of this work is to extract the part of the flow that can be described by the AEH, and study the statistical behaviours of the other part, which cannot be described by the AEH. In this process, two types of eddies are identified in addition to the Kolmogorov-scale eddies, i.e. wall-attached eddies and wall-detached eddies. The statistical behaviours of the wall-attached eddies are shown to be very well described by the AEH, i.e. the eddies are wall-attached, self-similar and, importantly, Gaussian, whereas the wall-detached eddies cannot be modelled by the AEH. Specifically, a decomposition scheme is proposed following Townsend (The Structure of Turbulent Shear Flow, Cambridge University Press, 1976). We apply our decomposition scheme to three different flows, i.e. channel, boundary layer and atmospheric surface layer flows. The results are similar with only quantitative differences, suggesting possible universality in both the wall-attached eddies and the wall-detached eddies.

Body-inclusive large-eddy simulations of disk wakes are performed for a homogeneous fluid and for different levels of stratification. The Reynolds number is 5 × 104 and the Froude number ($Fr$) takes the values of $\infty$, 50, 10 and 2. In the axisymmetric wake of a disk with diameter $L_{b}$ in a homogeneous fluid, it is found that the mean streamwise velocity deficit ($U_{0}$) decays in two stages: $U_{0}\propto x^{-0.9}$ during $10<x/L_{b}<65$ and, subsequently, $U_{0}\propto x^{-2/3}$. Consequently, none of the simulated stratified wakes is able to exhibit the classical 2/3 decay exponent of $U_{0}$ in the interval before buoyancy effects set in. Stratification affects the wake within approximately one buoyancy time scale, after which, we find three regimes: weakly stratified turbulence (WST), intermediately stratified turbulence (IST) and strongly stratified turbulence (SST). WST begins when the turbulent Froude number ($Fr_{h}$) decreases to $O(1)$, spans $1\lesssim Nt_{b}\lesssim 5$ and, while the mean flow is strongly affected by buoyancy in WST, turbulence is not. During IST, which commences at $Nt_{b}\approx 5$ when $Fr_{h}=O(0.1)$, the mean flow has arrived into the non-equilibrium (NEQ) regime with $U_{0}\propto x^{-0.18}$, but the turbulence state is still in transition, as indicated by progressively increasing turbulence anisotropy. When $Fr_{h}\sim O(0.01)$ at $Nt_{b}\approx 20$, the wake transitions into SST, where the turbulent vertical Froude number ($Fr_{v}$) asymptotes to a $O(1)$ constant. There is strong anisotropy ($u_{z}^{\prime }\ll u_{h}^{\prime }$), and both $u_{h}^{\prime }$ and $U_{0}$ satisfy $x^{-0.18}$ decay, signifying the arrival of the NEQ regime for both turbulence and mean flow. Turbulence is patchy and temporal spectra are broadband in the SST wake. The wake height decreases as $L_{V}\sim O(U_{0}/N)$ in IST/SST. Energy budgets reveal that stratification prolongs wake life during WST/early-IST by both an energy transfer from mean potential energy to mean kinetic energy and reduction of turbulent production. In the late-IST/early-SST stages, production is enhanced and, additionally, there is injection from turbulent potential energy slowing down turbulent kinetic energy (TKE) decay. Only in the SST stage, when NEQ is realized for both the mean and turbulence, does the turbulent buoyancy flux become negative again, acting as a sink of TKE.

Although the evolution of premixed flames in turbulence has been frequently studied, it is not well understood how small flames interact with large-scale turbulent flow motion. Since this question is of practical importance for the occurrence of cycle-to-cycle variations in spark ignition engines, the objective of the present work is to fundamentally differentiate early flame kernel development from well-established turbulent flame configurations. For this purpose, a direct numerical simulation database consisting of three flames propagating in homogeneous isotropic turbulence (Falkenstein et al., Combust. Flame, 2019, Preprint, arXiv:1908.07556 (accepted)) is considered. The flames feature different ratios of the initially laminar flame diameter to the integral length scale. To quantify flame kernel development, the time evolution of flame topology and flame front geometry are analysed in detail. It is shown that some realizations of the early flame kernel are substantially influenced by high compressive strain caused by large-scale turbulent flow motion with characteristic length scales greater than the flame kernel size. As a result, the initial spherical kernel topology may become highly distorted, which is reflected in the stochastic occurrence of excessive curvature variance. Two mechanisms of curvature production resulting from early flame kernel/turbulence interactions are identified by analysis of the mean curvature balance equation. Further, it is shown that the curvature distribution of small flame kernels becomes strongly skewed towards positive curvatures, which is contrary to developed turbulent flames. Hence, the transition of ignition kernels to self-sustaining turbulent flames is very different in nature compared with the development of a statistically planar flame brush.

Velocity structures associated with large streamwise density gradients in an incompressible turbulent boundary layer (with air as the working fluid, $Pr=0.71$) are analysed experimentally using planar image velocimetry and aero-optic measurements. The resulting flow topologies for the velocity fluctuations associated with large negative and positive density gradients are in excellent agreement with results for coolings and heatings in time, respectively (Antonia & Fulachier, J. Fluid Mech., vol. 198, 1989, pp. 429–451). The current results are complimentary to those from Saxton-Fox et al. (AIAA J., vol. 57 (7), 2019, pp. 2828–2839), on the signature of the vertical velocity structures associated with large density gradients. In the present work, these structures are shown to exhibit a sign change, consistent with the scalar gradient, and are localised in the wall-normal direction with an average height of approximately $0.1\unicode[STIX]{x1D6FF}$, almost constant for increasing distance from the wall. The corresponding small-scale streamwise fluctuations also exhibit a consistent sign change, which is found to originate, on average, from upstream leaning structures. The emerging picture for the velocity field is then that of a multiscale phenomenon, where small-scale structures, responsible for large optical aberrations, are superimposed on the back of large-scale bulge-like structures that are known to populate the outer layers. The proposed conceptual model is consistent with early ideas of ‘typical’ eddies (Falco, Phys. Fluids, vol. 20 (10), 1977, pp. S124–S132), the hairpin vortex model and associated shear layers (Adrian et al., J. Fluid Mech., vol. 422, 2000, pp.1–54), as well as with notions of multiscale velocity organisation in shear layers (Klewicki & Hirschi, Phys. Fluids, vol. 16 (11), 2004, pp. 4163–4176; Saxton-Fox et al. 2019), and it provides new insight into the geometry of the small-scale velocity structures.

The present article aims at optimising the spread of a bifurcating jet: a jet that combines axisymmetric and helical forcing to achieve increased mixing in a preferential plane. Parekh et al. (Tech. Rep. TF-35, Stanford University, 1988) explained such a bifurcation as the result of nonlinear interaction between ring vortices (triggered by $m=0$ axisymmetric forcing), shifted off-axis in alternate directions (owing to $m=1$ helical forcing). Following this idea, we linearly optimise the periodic helical forcing to be applied at the inlet, in order to maximally displace the ring vortices of an axisymmetrically forced jet. Two norms are introduced for evaluating the effect of helical forcing onto the helical response: the standard ${\mathcal{L}}_{2}$-norm and a semi-norm reflecting the off-axis vortex displacement. The linear results show one dominant forcing mode over the entire Strouhal band studied ($0.35\leqslant St\leqslant 0.8$), with a large gain separation from suboptimals. The dominant forcing is mainly radial, independent of the chosen response norm, and provides a gain at least five times larger than what was achieved by previous ad hoc forcing strategies. Superposition of base flow and linear results show the alternate shifting and twisting provoked by the the small-amplitude helical forcing, which is an essential ingredient for triggering jet bifurcation. When tested in three-dimensional direct numerical simulations, low-amplitude helical forcing achieves efficient bifurcation at all Strouhal values studied. At high Strouhal numbers, an additional central branch emerges in the mean flow, leading to trifurcation. Across all frequencies, compared with ad hoc forcing strategies, the optimal forcing triggers a much stronger and robust spreading, by moving the bifurcation point upstream. As a result, bifurcating jets are observed over a much larger Strouhal band ($0.35\leqslant St\leqslant 0.8$) compared with the band where ad hoc forcing achieves bifurcation in our setting ($0.4\leqslant St\leqslant 0.5$).