Investigated in this paper is the stability of the gravity-driven flow of a liquid layer laden with soluble surfactant down a heated incline. A mathematical model incorporating variations in surface tension with surfactant concentration and temperature has been formulated. A linear stability analysis is carried out both asymptotically for small wavenumbers and numerically for arbitrary wavenumbers. An expression for the critical Reynolds number has been derived which accounts for thermocapillary and solutocapillary effects, and reduces to known documented results for special cases. Also, a nonlinear reduced model has been derived using weighted residuals, and solved numerically to simulate the instability of the equilibrium flow and the development of permanent surface waves that arise. The nonlinear simulations were found to be in good agreement with the linear stability analysis.

Turbulent superstructures, i.e. large-scale flow structures in turbulent flows, play a crucial role in many geo- and astrophysical settings. In turbulent Rayleigh–Bénard convection, for example, horizontally extended coherent large-scale convection rolls emerge. Currently, a detailed understanding of the interplay of small-scale turbulent fluctuations and large-scale coherent structures is missing. Here, we investigate the resolved kinetic energy and temperature variance budgets by applying a filtering approach to direct numerical simulations of Rayleigh–Bénard convection at high aspect ratio. In particular, we focus on the energy transfer rate between large-scale flow structures and small-scale fluctuations. We show that the small scales primarily act as a dissipation for the superstructures. However, we find that the height-dependent energy transfer rate has a complex structure with distinct bulk and boundary layer features. Additionally, we observe that the heat transfer between scales mainly occurs close to the thermal boundary layer. Our results clarify the interplay of superstructures and turbulent fluctuations and may help to guide the development of an effective description of large-scale flow features in terms of reduced-order models.

Mach reflection in steady supersonic flow with two incident shock waves that reflect at the same point on the reflecting surface has been studied recently. Under some conditions we have pre-Mach reflection, where the first incident shock wave produces Mach reflection, the reflected shock wave of which intersects the second incident shock wave, leading to a type I shock interference structure. In this study, we find that a critical condition exists to have a double solution of this shock interference, i.e. we may either have type I interference or type II interference. However, numerical simulation shows that, for inverted Mach reflection, the double solution domain is below the theoretical one and for usual Mach reflection, the double solution domain is above the theoretical one. This discrepancy is found to be due to secondary Mach waves on the initial segment of the slipline of the Mach reflection, thus demonstrating for the first time a case where the transition criterion is modified by secondary Mach waves developed over the primary flow structure.

Comprehensive linear stability study of flow in an annular layer of electrolyte driven by the action of the Lorentz force is conducted following the analysis of steady axisymmetric solutions of Suslov et al. (J. Fluid Mech., vol. 828, 2017, pp. 573–600). It is shown that an experimentally observed instability in the form of anticyclonic moving vortices reported in Pérez-Barrera et al. (Magnetohydrodynamics, vol. 51 (2), 2015, pp. 203–213) develops on a background of the basic flow consisting of two tori with the opposite azimuthal vorticity components. It is found that, while the background flow is driven electromagnetically, the appearance of vortices is purely due to hydrodynamic effects: shear of the flow and centrifugal inertial forcing. The current study has also revealed that the unstable two-torus basic flow has a stable single-torus counterpart, both emanating from a saddle-node bifurcation of steady states when the Lorentz force is sufficiently strong. The transition from a one-torus to two-torus flow at weaker forcing is abrupt and leads to the appearance of vortices as soon as it occurs. The ranges of layer depths and Reynolds numbers for which vortices develop on a steady background are determined. Subsequently, weakly nonlinear amplitude expansion is used to find an approximate unsteady solution beyond the saddle-node bifurcation.

The work examines upper-ocean response to time-varying winds within the Ekman paradigm. Here, in contrast to the earlier works we assume the eddy viscosity to be both time and depth dependent. For self-similar depth and time dependence of eddy viscosity and arbitrary time dependence of wind we find an exact general solution to the Navier–Stokes equations which describes the dynamics of the Ekman boundary layer in terms of the Green’s function. Two basic scenarios (a periodic wind and an increase of wind ending up with a plateau) are examined in detail. We show that accounting for the time dependence of eddy viscosity is straightforward and that it substantially changes the ocean response, compared to the predictions of the models with constant-in-time viscosity. We also examine the Stokes–Ekman equations taking into account the Stokes drift created by surface waves with an arbitrary spectrum and derive the general solution for the case of a linearly varying with depth eddy viscosity. Stability of transient Ekman currents to small-scale perturbations has never been examined. We find that the Ekman currents evolving from rest quickly become unstable, which breaks down the assumed horizontal uniformity. These instabilities proved to be sensitive to the model of eddy viscosity, they have small (${\sim}10^{2}~\text{m}$) spatial scales and can be very fast compared to the inertial period, which suggests spikes of dramatically enhanced mixing localized in the vicinity of the water surface. This picture is incompatible with the Ekman paradigm and thus prompts radical revision of the Ekman-type models.

Jet flows induced by pitching cantilever plates provide a power-efficient solution for fluid acceleration and cooling enhancement. In such applications, the time-averaged (mean) properties of the induced jet flows are of great importance. We report a combined experimental and numerical study on the three-dimensional (3-D) characteristics of the mean jet downstream of a harmonically pitching cantilever plate in a quiescent fluid. These characteristics are then correlated with the transient 3-D vortex structures emanated from the trailing and side edges. Our particle image velocimetry and 3-D numerical simulations reveal that the mean induced jet has two distinct regions – a shrinking region immediately downstream of the trailing edge followed by an abrupt expansion region – separated by a necking point. We investigate the transient 3-D wake vortex evolution downstream of the plate to help elucidate the physics underlying the geometry of the mean jet. Our observations suggest that the breakdown of the shed vortex structure and reorientation of the consequent substructures are the primary factors governing the shape of the jet. These factors in turn are controlled primarily by the plate width and the amplitude of the oscillations. The results presented in this study improve our understanding of the complicated 3-D geometry of the induced mean jet in oscillating plates and facilitate optimal design of devices that operate based on this principle, such as piezoelectric fans.

In searching for the mechanisms of boundary layer bypass transition, Schubauer & Klebanoff (NACA-TR-1289, 1956) conducted an extensive search for negative skin-friction events during the laminar-to-turbulent transition in a zero-pressure-gradient, smooth flat-plate boundary layer (ZPGSFPBL) under a wide range of free-stream turbulence intensity levels. They concluded that no evidence of incipient flow separation could be found in any part of the transition region. Although it has been known that extremely rare backflow events can occur in a fully turbulent ZPGSFPBL and in fully developed turbulent pipe flow, the Schubauer–Klebanoff conclusion regarding the total absence of negative skin friction in ZPGSFPBL transition has remained unchallenged over the past six decades. Here we report our discovery of negative skin-friction events during the bypass transition in an extensively researched representative ZPGSFPBL case, and in pipe flows with no-slip and slip boundary conditions under circumferential mode inlet disturbance, respectively. The peak probability of such events is located in the late stage of transition between the streamwise stations where the minimum and maximum mean skin friction are attained. The magnitude of the peak probability is substantially larger than the corresponding probability in the downstream fully turbulent region: in the case of no-slip pipe flow, the difference is two orders of magnitude and is primarily due to an enhanced number density of the backflow events coupled with a notable increase in the wall footprint size of such events. For both the boundary layer and pipe flows considered here, and in both the transitional and fully turbulent regions, the incipient negative skin friction is found, through instantaneous flow visualization and conditional averaging, to be induced by the head element of a reverse hairpin vortex.

We experimentally study the transition from droplet to wave regimes in microfluidic liquid–liquid multiphase flows having large differences in viscosity. A unified approach based on periodic pattern analysis is employed to study relationships between dispersed and separated flow regimes, including dripping, jetting, capillary waves, inertial waves and core–annular flows over a wide range of flow rates and viscosity contrasts. We examine the morphology and dynamics of each flow regime based on wavelength, frequency and velocity of repeating unit cells to elucidate their connections and to develop predictive capabilities based on dimensionless control parameters. We demonstrate in particular that pattern selection is contingent upon the propagation velocity of droplets and waves at the transition. We also investigate microfluidic wave breaking phenomena with the formation of ligaments and droplets from wave crests in both capillary and inertial wave regimes. This work expands conventional multiphase flow regimes observed in microchannels and shows new routes to disperse highly viscous materials using interfacial waves dynamics in confined microsystems.

The present work uses a reduced-order model to study the motion of a buoyant vortex ring with non-negligible core size. Buoyancy is considered in both non-Boussinesq and Boussinesq situations using an axisymmetric contour dynamics formulation. The density of the vortex ring differs from that of the ambient fluid, and both densities are constant and conserved. The motion of the ring is calculated by following the boundary of the vortex core, which is also the interface between the two densities. The velocity of the contour comes from a combination of a specific continuous vorticity distribution within its core and a vortex sheet on the core boundary. An evolution equation for the vortex sheet is derived from the Euler equation, which simplifies considerably in the Boussinesq limit. Numerical solutions for the coupled integro-differential equations are obtained. The dynamics of the vortex sheet and the formation of two possible singularities, including singularities in the curvature and the shock-like profile of the vortex sheet strength, are discussed. Three dimensionless groups, the Atwood, Froude and Weber numbers, are introduced to measure the importance of physical effects acting on the motion of a buoyant vortex ring.

The mild-slope equations, devised to approximate surface wave propagation over water of slowly varying depth, have hitherto been based on either the velocity potential formulation or the streamfunction formulation. By using a more general version of the governing equations, a single framework is developed that relates the existing mild-slope equations and provides new examples and derivations.

Experiments were performed to further investigate a quadratic interaction between stationary and travelling cross-flow modes that was revealed in the previous experiment by Corke et al. (J. Fluid Mech., vol. 856, 2018, pp. 822–849) in the boundary layer on a sharp right-circular cone at an angle of attack at Mach 6.0. As with the previous experiment, passive discrete roughness was applied near the cone tip, just upstream of Branch I of the linear stability neutral curve for stationary cross-flow modes. The passive roughness consisted of indentations (dimples) that were evenly spaced azimuthally to excite a specific azimuthal wavenumber. A plasma actuator was located just downstream of the discrete roughness array. This was designed to produce an azimuthally uniform unsteady disturbance with a frequency that was at the centre of the band of most amplified travelling cross-flow modes. Measurements consisted of off-wall azimuthal profiles of mean and fluctuating total pressure at different axial locations. Spectra of total-pressure fluctuations verified the receptivity of the boundary layer to the unsteady excitation. This affected the azimuthal and streamwise development of the stationary cross-flow modes, with a general effect to move the transition location upstream by approximately 16 %. The quadratic interaction between the stationary and travelling cross-flow modes was further enhanced by the excitation of the travelling cross-flow mode. This was particularly evident by an enlarged band of azimuthal wavenumbers over which a significant triple phase locking existed. The significantly enlarged range of wavenumber sum and difference interactions offered a mechanism for rapid spectral broadening that could account for the hastened transition by the addition of unsteady disturbances.

The nonlinear reflection of a finite-width plane internal gravity wave incident onto a uniform slope is addressed, relying on the inviscid theory of Thorpe (J. Fluid Mech., vol. 178, 1987, pp. 279–302) for pure plane waves. The aim of this theory is to determine the conditions under which the incident and the reflected waves form a resonant triad with the second-harmonic wave resulting from their interaction. Thorpe’s theory leads to an indeterminacy of the second-harmonic wave amplitude at resonance. In waiving this indeterminacy, we show that the latter amplitude has a finite behaviour at resonance, increasing linearly from the slope. We investigate the influence of background rotation and find similar results with a weaker growth rate. We then adapt the theory to the case of an incident plane wave of finite width. In this case, nonlinear interactions are confined to the area where the incident and reflected finite-width waves superpose, implying that the amplitude of the second-harmonic wave is bounded at resonance. We find good agreement with the results of numerical simulations in a vertical plane as long as the dissipated power of the incident and reflected waves remain smaller than the power transferred to the second-harmonic wave. This is the case for small slope angles. As the slope angle increases, the focusing of the reflected wave enhances viscous effects and dissipation eventually dominates over nonlinear transfer. We finally discuss the relevance of laboratory experiments to assess the validity of the theoretical results.