Coherence resonance (CR) is a phenomenon in which the response of a stable nonlinear system to external noise exhibits a peak in coherence at an intermediate noise amplitude. We report the first experimental evidence of CR in a hydrodynamic system, a low-density jet capable of undergoing both supercritical and subcritical Hopf bifurcations. By applying noise to the jet in its unconditionally stable regime, we find that, for both types of bifurcation, the coherence factor peaks at an intermediate noise amplitude and increases as the stability boundary is approached. We also find that the autocorrelation function decays differently between the two types of bifurcation, indicating that CR can reveal information about the nonlinearity of a system even before it bifurcates to a limit cycle. We then model the CR dynamics with a stochastically forced van der Pol oscillator calibrated in two different ways: (i) via the conventional method of measuring the amplitude evolution in transient experiments and (ii) via the system-identification method of Lee et al. (J. Fluid Mech., vol. 862, 2019, pp. 200–215) based on the Fokker–Planck equation. We find better experimental agreement with the latter method, demonstrating the deficiency of the former method in identifying the correct form of system nonlinearity. The fact that CR occurs in the unconditionally stable regime, prior to both the Hopf and saddle-node points, implies that it can be used to forecast the onset of global instability. Although demonstrated here on a low-density jet, CR is expected to arise in almost all nonlinear dynamical systems near a Hopf bifurcation, opening up new possibilities for the development of global-instability precursors in a variety of hydrodynamic systems.

A new time scale for turbulence modulation by particles is introduced. This time scale is inversely proportional to the number density and the radius of particles, and can be regarded as a counterpart of the phase relaxation time, an important time scale in cloud physics, which characterizes the interaction between turbulence and cloud droplets by condensation–evaporation. Scaling analysis and direct numerical simulations of dilute inertial particles in homogeneous isotropic turbulence suggest that turbulence modulation by particles with a fixed mass-loading parameter can be expressed as a function of the Damköhler number, which is defined as the ratio of the turbulence large-eddy turnover time to the new time scale.

Plane Poiseuille flow, the pressure-driven flow between parallel plates, shows a route to turbulence connected with a linear instability to Tollmien–Schlichting (TS) waves, and another route, the bypass transition, that can be triggered with finite-amplitude perturbation. We use direct numerical simulations to explore the arrangement of the different routes to turbulence among the set of initial conditions. For plates that are a distance $2H$ apart, and in a domain of width $2\unicode[STIX]{x03C0}H$ and length $2\unicode[STIX]{x03C0}H$, the subcritical instability to TS waves sets in at $Re_{c}=5815$ and extends down to $Re_{TS}\approx 4884$. The bypass route becomes available above $Re_{E}=459$ with the appearance of three-dimensional, finite-amplitude travelling waves. Below $Re_{c}$, TS transition appears for a tiny region of initial conditions that grows with increasing Reynolds number. Above $Re_{c}$, the previously stable region becomes unstable via TS waves, but a sharp transition to the bypass route can still be identified. Both routes lead to the same turbulent state in the final stage of the transition, but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence compete.

The nonlinear character of the primary bifurcation is investigated for the flow around a flexibly mounted circular cylinder. We have considered the cases in which the cylinder can oscillate in the transverse direction only and in both transverse and in-line directions. Low and high values of mass ratio ($m^{\ast }=5$ and 50) were studied, and reduced velocity ($V_{r}$) values are chosen inside ($V_{r}=9$) and outside ($V_{r}=5$ and $V_{r}=13$) the lock-in range for low Reynolds numbers. For each combination of $m^{\ast }$ and $V_{r}$, a global linear stability analysis was applied to find the critical Reynolds number $Re_{c}$ of the fluid–structure system. For $V_{r}$ in the lock-in range, the values of $Re_{c}$ were noticeably less than the critical Reynolds number of the flow around a fixed circular cylinder ($Re_{c_{0}}\cong 47$). On the other hand, for $V_{r}$ outside the lock-in range, the values of $Re_{c}$ were close to $Re_{c_{0}}$. Next, nonlinear analyses were performed in the vicinity of $Re_{c}$ for each case. Subcritical character (with hysteresis) was observed for $V_{r}$ in the lock-in range, while for $V_{r}$ outside the lock-in region the bifurcations were found to be supercritical (without hysteresis). This shows that when the coupling between the structure and flow is strong, due to the proximity of the natural frequencies of the isolated systems, it significantly changes both the linear and nonlinear responses observed.

In the theory of interaction noise from contra-rotating propellers with many blades, the usual far-field radiation formulae can be re-cast as a double integral, over a source surface, which can be evaluated asymptotically solely in terms of the contributions from critical points. The paper shows that these critical points have a particularly interesting physical meaning. They relate to locations on an event line, running between hub and tip, that represent the locus of the wake–blade interactions at a fixed point in time. The event line rotates at the speed of the spinning interaction tone but does not coincide with the radial variation in either the wake location or the rear blade leading edge. At the precise critical locations on the event line, it is shown that the Mach number of the event line is unity in the direction of the observer (the sonic condition) and the tangent to the event line – at a fixed time – is normal to a line drawn between it and the observer (the normal-edge condition). The zero-mode case is also considered, for which we show that, even though the event line rotates at infinite speed, there can still exist locations that satisfy the sonic and normal-edge conditions. The paper also discusses the physical meaning of the lower-order boundary solutions from the hub and tip.

The hydrodynamic quantification of superhydrophobic slipperiness has traditionally employed two canonical problems – namely, shear flow about a single surface and pressure-driven channel flow. We here advocate the use of a new class of canonical problems, defined by the motion of a superhydrophobic particle through an otherwise quiescent liquid. In these problems the superhydrophobic effect is naturally measured by the enhancement of the Stokes mobility relative to the corresponding mobility of a homogeneous particle. We focus upon what may be the simplest problem in that class – the rotation of an infinite circular cylinder whose boundary is periodically decorated by a finite number of infinite grooves – with the goal of calculating the rotational mobility (velocity-to-torque ratio). The associated two-dimensional flow problem is defined by two geometric parameters – namely, the number $N$ of grooves and the solid fraction $\unicode[STIX]{x1D719}$. Using matched asymptotic expansions we analyse the large-$N$ limit, seeking the mobility enhancement from the respective homogeneous-cylinder mobility value. We thus find the two-term approximation,

$$\begin{eqnarray}\displaystyle 1+{\displaystyle \frac{2}{N}}\ln \csc {\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D719}}{2}}, & & \displaystyle \nonumber\end{eqnarray}$$

$N=1,2,4,8,\ldots$

$N$

A statistical description of flow regions with negative streamwise velocity is provided based on simulations of turbulent plane channels in the Reynolds number range $547\leqslant Re_{\unicode[STIX]{x1D70F}}\leqslant 2003$. It is found that regions of backflow are attached and their density per surface area – in wall units – is an increasing function of $Re_{\unicode[STIX]{x1D70F}}$. Their size distribution along the three coordinates reveals that, even though in the mean they appear to be circular in the wall-parallel plane, they tend to become more elongated in the spanwise direction after reaching a certain height. Time-tracking of backflow regions in a $Re_{\unicode[STIX]{x1D70F}}=934$ simulation showed they convect downstream at the mean velocity corresponding to $y^{+}\approx 12$, they seldom interact with other backflow events, their statistical signature extends in the streamwise direction for at least $300$ wall units, and they result from a complex interaction between regions of high and low spanwise vorticity far beyond the viscous sublayer. This could explain why some statistical aspects of these near-wall events do not scale in viscous units; they are dependent on the $Re_{\unicode[STIX]{x1D70F}}$-dependent dynamics further away from the wall.

We study the dynamical system of a two-dimensional, forced, stratified mixing layer at finite Reynolds number $Re$, and Prandtl number $Pr=1$. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications, in a domain of fixed, finite width and height. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well known, if the minimum gradient Richardson number of the flow, $Ri_{m}$, is less than a certain critical value $Ri_{c}$, the flow is linearly unstable to Kelvin–Helmholtz instability in both cases. Using Newton–Krylov iteration, we find steady, two-dimensional, finite-amplitude elliptical vortex structures – i.e. ‘Kelvin–Helmholtz billows’ – existing above $Ri_{c}$. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying $Re$. In particular, when $Re$ is sufficiently high we find that finite-amplitude Kelvin–Helmholtz billows exist when $Ri_{m}>1/4$ for the background flow, which is linearly stable by the Miles–Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slow-growing, linear instability of the background profiles at finite $Re$, which complicates the dynamics.

Unstable shear layers in environmental and industrial flows roll up into a series of vortices, which often form complex nonlinear merging patterns such as pairs and triplets. These patterns crucially determine the subsequent turbulence, mixing and scalar transport. We show that the late-time, highly nonlinear merging patterns are predictable from the linearized initial state. The initial asymmetry between consecutive wavelengths of the vertical velocity field provides an effective measure of the strength and pattern of vortex merging. The predictions of this measure are substantiated using direct numerical simulations. We also show that this measure has significant implications in determining the route to turbulence and the ensuing turbulence characteristics.

An exact solution of a stable vortex tripole in two-dimensional (2-D) Euler flows is provided. The stable tripole is composed of an inner elliptical vortex and two small-amplitude lateral vortices. The non-vanishing vorticity field of this tripole, referred to as here as an embedded tripole because of the closeness of its vortices, is given in elliptical coordinates $(\unicode[STIX]{x1D707},\unicode[STIX]{x1D708})$ by the even radial and angular order-0 Mathieu functions $\text{Je}_{0}(\unicode[STIX]{x1D707})\text{ce}_{0}(\unicode[STIX]{x1D708})$ truncated at the external branch of the vorticity isoline passing through the two critical points closest to the vortex centre. This tripole mode has a rigid vorticity field which rotates with constant angular velocity equal to $\unicode[STIX]{x1D701}_{0}\text{Je}_{0}(\unicode[STIX]{x1D707}_{1})\text{ce}_{0}(0)/2$, where $\unicode[STIX]{x1D707}_{1}$ is the first zero of $\text{Je}_{0}^{\prime }(\unicode[STIX]{x1D707})$ and $\unicode[STIX]{x1D701}_{0}$ is a constant modal amplitude. It is argued that embedded 2-D tripoles may be conceptually regarded as the superposition of two asymmetric Chaplygin–Lamb dipoles, separated a distance equal to $2R$, as long as their individual trajectory curvature radius $R$ is much shorter than their dipole extent radius.

In elastic macromolecules, the value of the short-time diffusion coefficient depends on the choice of the point the displacement of which is tracked. On the other hand, the experimentally more relevant long-time diffusion coefficient is independent of the reference point, but its estimation usually requires computationally expensive Brownian dynamics simulations. Here we show how to obtain a precise estimate of the long-time diffusion coefficient of elastic macromolecules in a fast and robust manner, without invoking Brownian dynamics.

In this paper we contrast the early impact stage of a highly viscous drop onto a liquid versus a solid substrate. Water drops impacting at low velocities can rebound from a solid surface without contact. This dynamic is mediated through lubrication of a thin air layer between the liquid and solid. Drops can also rebound from a liquid surface, but only for low Weber numbers. Impacts at higher velocities in both cases lead to circular contacts which entrap an air disc under the centre of the drop. Increasing the drop viscosity produces extended air films for impacts on a smooth solid surface even for much larger velocities. These air films eventually break through random wetting contacts with the solid. Herein we use high-speed interferometry to study the extent and thickness profile of the air film for a large-viscosity drop impacting onto a viscous film of the same liquid. We demonstrate a unified scaling of the centreline height of the air film for impacts on both solid and liquid, when using the effective impact velocity. On the other hand, we show that the large-viscosity liquid film promotes air films of larger extent. Furthermore, the rupture behaviour becomes fundamentally different, with the air film between the two compliant surfaces being more stable, lacking the random wetting patches seen on the solid. We map the parameter range where these air films occur and explore the transition from gliding to ring contact at the edge of the drop dimple. After the air film ruptures, the initial contraction occurs very rapidly and for viscosities greater than 100 cSt the retraction velocity of the air film is ${\sim}0.3~\text{m}~\text{s}^{-1}$, independent of the liquid viscosity and impact velocity, in sharp contrast with theoretical predictions.