Since the introduction of the logarithmic law of the wall more than 80 years ago, the equation for the mean velocity profile in turbulent boundary layers has been widely applied to model near-surface processes and parameterize surface drag. Yet the hypothetical turbulent eddies proposed in the original logarithmic law derivation and mixing length theory of Prandtl have never been conclusively linked to physical features in the flow. Here, we present evidence that suggests these eddies correspond to regions of coherent streamwise momentum known as uniform momentum zones (UMZs). The arrangement of UMZs results in a step-like shape for the instantaneous velocity profile, and the smooth mean profile results from the average UMZ properties, which are shown to scale with the friction velocity and wall-normal distance in the logarithmic region. These findings are confirmed across a wide range of Reynolds number and surface roughness conditions from the laboratory scale to the atmospheric surface layer.

Here we provide a theoretical framework revealing that the radius $R_{d}$ of the top droplet ejected from a bursting bubble of radius $R_{b}$ and $Bo\leqslant 0.05$ can be expressed as $R_{d}/R_{b}=K_{b}(1-(Oh/Oh_{c}^{\prime })^{1/2})$ for $Oh\lesssim Oh_{c}^{\prime }$ or as $R_{d}\approx 18\,\unicode[STIX]{x1D707}_{l}^{2}/(\unicode[STIX]{x1D70C}_{l}\unicode[STIX]{x1D70E})$ for $Oh\gtrsim Oh_{c}^{\prime }$, with the numerically fitted constants $K_{b}\approx 0.2$, $Oh_{c}^{\prime }\approx 0.03$, $Oh=\unicode[STIX]{x1D707}_{l}/\sqrt{\unicode[STIX]{x1D70C}_{l}\,R_{b}\,\unicode[STIX]{x1D70E}}\ll 1$ the Ohnesorge number, $Bo=\unicode[STIX]{x1D70C}_{l}\,g\,R_{b}^{2}/\unicode[STIX]{x1D70E}$ the Bond number, and $\unicode[STIX]{x1D70C}_{l}$, $\unicode[STIX]{x1D707}_{l}$ and $\unicode[STIX]{x1D70E}$ indicating the liquid density, dynamic viscosity and interfacial tension coefficient, respectively. These predictions, which do not only have solid theoretical roots but are also much more accurate than the usual 10 % rule used in the context of marine spray generation via whitecaps for $R_{b}\lesssim 1$ mm, agree very well with both experimental data and numerical simulations for the values of $Oh$ and $Bo$ investigated. Moreover, making use of a criterion which reveals the mechanism that controls the growth rate of capillary instabilities, we also explain here why no droplets are ejected from the tip of the fast Worthington jet for $Oh\gtrsim 0.04$. In addition, our results predict the generation of submicron-sized aerosol particles with diameters below 100 nm and velocities ${\sim}\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D707}_{l}$ for bubble radii $10~\unicode[STIX]{x03BC}\text{m}\lesssim R_{b}\lesssim 20~\unicode[STIX]{x03BC}\text{m}$, within the range found in natural conditions and in good agreement with experiments – a fact suggesting that our study could be applied in the modelling of sea spray aerosol production.

We investigate the stability of katabatic slope flows over an infinitely wide and uniformly cooled planar surface subject to a downslope uniform ambient wind aloft. We adopt an extension of Prandtl’s original model for slope flows (Lykosov & Gutman, Izv. Acad. Sci. USSR Atmos. Ocean. Phys., vol. 8 (8), 1972, pp. 799–809) to derive the base flow, which constitutes an interesting basic state in stability analysis because it cannot be reduced to a single universal form independent of external parameters. We apply a linear modal analysis to this basic state to demonstrate that for a fixed Prandtl number and slope angle, two independent dimensionless parameters are sufficient to describe the flow stability. One of these parameters is the stratification perturbation number that we have introduced in Xiao & Senocak (J. Fluid Mech., vol. 865, 2019, R2). The second parameter, which we will henceforth designate the wind forcing number, is hitherto uncharted and can be interpreted as the ratio of the kinetic energy of the ambient wind aloft to the damping due to viscosity and the stabilising effect of the background stratification. For a fixed Prandtl number, stationary transverse and travelling longitudinal modes of instabilities can emerge, depending on the value of the slope angle and the aforementioned dimensionless numbers. The influence of ambient wind forcing on the base flow’s stability is complicated, as the ambient wind can be both stabilising as well as destabilising for a certain range of the parameters. Our results constitute a strong counterevidence against the current practice of relying solely on the gradient Richardson number to describe the dynamic stability of stratified atmospheric slope flows.

The interaction of a shock wave and a water droplet embedded with a vapour cavity is experimentally investigated in a shock tube for the first time. The vapour cavity inside the droplet is generated by decreasing the surrounding pressure to the saturation pressure, and an equilibrium between the liquid phase and the gas phase is obtained inside the droplet. Direct high-speed photography is adopted to capture the evolution of both the droplet and the vapour cavity. The formation of a transverse jet inside the droplet during the cavity-collapse stage is clearly observed. Soon afterwards, at the downstream pole of the droplet, a water jet penetrating into the surrounding air is observed during the cavity-expansion stage. The evolution of the droplet is strongly influenced by the evolution of the vapour cavity. The phase change process plays an important role in vapour cavity evolution. The effects of the relative size and eccentricity of the cavity on the movement and deformation of the droplet are presented quantitatively.

Turbulent flow over a surface with streamwise-elongated rough and smooth stripes is studied by means of direct numerical simulation (DNS) in a periodic plane open channel with fully resolved roughness. The goal is to understand how the mean height of roughness affects the characteristics of the secondary flow formed above a spanwise heterogeneous rough surface. To this end, while the statistical properties of roughness texture as well as the width and spacing of the rough stripes are kept constant, the elevation of the smooth stripes is systematically varied in different simulation cases. Utilizing this variation, three configurations – representing protruding, recessed and an intermediate type of roughness – are analysed. In all cases, secondary flows are present and the skin friction coefficients calculated for all the heterogeneous rough surfaces are meaningfully larger than what would result from the area-weighted average of those of homogeneous smooth and rough surfaces. This drag increase appears to be linked to the strength of the secondary flow. The rotational direction of the secondary motion is shown to depend on the relative surface elevation. The present results suggest that this rearrangement of the secondary flow is linked to the spatial distribution of the spanwise-wall-normal Reynolds stress component, which carries opposing signs for protruding and recessed roughness.

We prove a new rigorous upper bound on the vertical heat transport for Bénard–Marangoni  convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma\gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu\leqslant \text{const.}\times Ma^{2/7}(\ln Ma)^{-1/7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluid’s surface and new estimates for the coupling between temperature and vertical velocity.

We consider the size spectrum of entrained bubbles under strong free-surface turbulence (SFST). We investigate the entrainment bubble-size spectrum per unit (mean) interface area, ${\mathcal{N}}_{a}^{E}(r)$, with dimension length$^{-3}$, and develop a physical/mechanistic model for ${\mathcal{N}}_{a}^{E}(r)$ through energy arguments. The model obtains two distinct regimes of ${\mathcal{N}}_{a}^{E}(r)$, separated by bubble-size scale $r_{0}$. For bubble radius $r>r_{0}$, the effects of gravity $g$ dominate those of the surface tension force $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}$, and ${\mathcal{N}}_{a}^{E}(r)\propto g^{-1}\unicode[STIX]{x1D716}^{2/3}r^{-10/3}$, where $\unicode[STIX]{x1D716}$ is the turbulence dissipation rate. For $r<r_{0}$, surface tension is more important and ${\mathcal{N}}_{a}^{E}(r)\propto (\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C})^{-1}\unicode[STIX]{x1D716}^{2/3}r^{-4/3}$. From the model, we show that $r_{0}\approx r_{c}=1/2\sqrt{\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}g}$, the capillary length scale, and not the generally assumed Hinze scale $r_{H}$. For an air–water interface and Earth gravity, $r_{c}\approx$ 1.5 mm. The model provides an $\unicode[STIX]{x1D716}$–$r$ entrainment regime map that identifies a critical dissipation rate $\unicode[STIX]{x1D716}_{cr}$ (constant for given $g$ and $\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}$) above which there is appreciable air entrainment, thus separating SFST and weak FST. We confirm the theoretical model and its predictions using two-phase, high-fidelity direct numerical simulations of a canonical FST flow using the conservative volume-of-fluid method: the respective power laws of ${\mathcal{N}}_{a}^{E}(r)\propto r^{-10/3}$ and $r^{-4/3}$ for $r>r_{0}$ and $r<r_{0}$; the value $r_{0}=r_{c}$; the scaling ${\mathcal{N}}_{a}^{E}(r)\propto \unicode[STIX]{x1D716}^{2/3}$; and the predictions of the $\unicode[STIX]{x1D716}$–$r$ entrainment regime map.

High-resolution direct numerical simulations (DNS) were carried out to investigate the nonlinear breakdown process of a three-dimensional wave packet initiated by a short-duration pulse in a flared cone boundary layer at Mach 6 and zero angle of attack. For these simulations the cone geometry of the flared cone experiments conducted in the Boeing/AFOSR Mach 6 Quiet Tunnel (BAM6QT) at Purdue University was considered. The computational domain covered a large extent of the cone in the azimuthal direction to allow for a wide range of azimuthal wavenumbers ($k_{c}$) and to include shallow instability waves with small azimuthal wavenumbers. The simulation results indicated that the wave packet development was dominated by axisymmetric and shallow (small $k_{c}$) second-mode waves for a large downstream extent. Towards the downstream end of the computational domain a rapid broadening of the disturbance amplitude spectra was observed, which is an indication that the wave packet reached the strongly nonlinear stages. The disturbance spectra of the nonlinear regime, and the downstream amplitude development of the dominant disturbance wave components, provided conclusive evidence that the so-called fundamental breakdown was the dominant nonlinear mechanism. Furthermore, contours of the time-averaged Stanton number exhibited ‘hot’ streaks within the wave packet on the surface of the cone. Hot streaks have also been observed in the Purdue flared cone experiments using temperature sensitive paint (TSP) and in numerical investigations using DNS. The azimuthal streak spacing obtained from the wave packet simulation agrees well with that observed in the Purdue quiet tunnel experiments.

Resolvent analysis is performed to identify the origin of two-dimensional transonic buffet over an airfoil. The base flow for the resolvent analysis is the time-averaged flow over a NACA 0012 airfoil at a chord-based Reynolds number of 2000 and a free-stream Mach number of 0.85. We reveal that the mechanism of buffet is buried underneath the global low-Reynolds-number flow physics. At this low Reynolds number, the dominant flow feature is the von Kármán shedding. However, we show that with the appropriate forcing input, buffet can appear even at a Reynolds number that is much lower than what is traditionally associated with transonic buffet. The source of buffet is identified to be at the shock foot from the windowed resolvent analysis, which is validated by companion simulations using sustained forcing inputs based on resolvent modes. We also comment on the role of perturbations in the vicinity of the trailing edge. The present study not only provides insights on the origin of buffet but also serves a building block for low-Reynolds-number compressible aerodynamics in light of the growing interests in Martian flights.

The Richtmyer–Meshkov instability of a helium layer surrounded by air is studied in a semi-annular convergent shock tube by high-speed schlieren photography. The gas layer is generated by an improved soap film technique such that its boundary shapes and thickness are precisely controlled. It is observed that the inner interface of the shocked light gas layer remains nearly undisturbed during the experimental time, even after the reshock, which is distinct from its counterpart in the heavy gas layer. This can be ascribed to the faster decay of the perturbation amplitude of the transmitted shock in the helium layer and Rayleigh–Taylor stabilization on the inner surface (light/heavy) during flow deceleration. The outer interface first experiences ‘accelerated’ phase inversion owing to geometric convergence, and later suffers a continuous deformation. Compared with a sole heavy/light interface, the wave influence (interface coupling) inhibits (promotes) growth of instability of the outer interface.

Experimental and numerical studies over the past two decades indicate that as the Reynolds number becomes large the turbulent boundary layer is increasingly composed of zones of uniform streamwise momentum, segregated by narrow regions of high shear. Recent experimental evidence suggests that passive scalar fields (for example, temperature) in turbulent boundary layers at high Reynolds number show similar characteristics; namely, large uniform temperature zones (UTZs) separated by narrow regions of high gradient, which we term thermal fissures (TFs). Herein, a model informed by analysis of the mean scalar transport equation, and that leverages the dynamical model recently developed by the authors (Cuevas Bautista et al., J. Fluid Mech., vol. 858, 2019, pp. 609–633), is formulated to predict passive scalar transport using the UTZ/TF concept. First, a finite number of TFs are distributed across the boundary layer. In analogy with the aforementioned dynamical model, the wall-normal positions of the TFs and their characteristic temperatures are then perturbed to generate independent ensembles, from which statistical moments are computed. The model successfully reproduces the statistical profiles of the temperature field as well as the streamwise turbulent heat flux. Lastly, the Prandtl number dependency of the empirically chosen parameters is investigated. It is concluded that the higher-order statistics, especially the kurtosis, produced by the model are sensitive to the Prandtl number, while the mean temperature and turbulent heat flux do not show noticeable Prandtl number dependency.

We present a direct numerical simulation (DNS) study of pseudo-turbulence in buoyancy-driven bubbly flows for a range of Reynolds ($150\leqslant Re\leqslant 546$) and Atwood ($0.04\leqslant At\leqslant 0.9$) numbers. We study the probability distribution function of the horizontal and vertical liquid velocity fluctuations and find them to be in quantitative agreement with the experiments. The energy spectrum shows a $k^{-3}$ scaling at high $Re$ and becomes steeper on reducing $Re$. To investigate spectral transfers in the flow, we derive the scale-by-scale energy budget equation. Our analysis shows that, for scales smaller than the bubble diameter, the net transfer because of the surface tension and the kinetic energy flux balances viscous dissipation to give $k^{-3}$ scaling of the energy spectrum for both low and high $At$.

Observational and computational studies of inertia-dominated wall turbulence with unstable thermal stratification have demonstrated that the inclination angle of large-scale motions (LSMs) increases with increasing buoyancy (as characterized by the Monin–Obukhov stability variable $\unicode[STIX]{x1D701}_{z}$). The physical implications of this structural steepening have received relatively less attention. Some authors have proposed that LSMs thicken – yet remain attached to the wall – with increasing buoyancy (Salesky & Anderson, J. Fluid Mech., vol. 856, 2018, pp. 135–168), while others have presented evidence that the upstream edge of an LSM remains anchored to the wall while its downstream edge lifts away from the wall (Hommema & Adrian, Boundary-Layer Meteorol., vol. 106, 2003, pp. 147–170). Using a suite of large-eddy simulations (LES) of unstably stratified turbulent channel flow, we demonstrate that buoyancy acts to lift LSMs away from the wall, leaving a wedge of fluid beneath with differing momentum. We develop a prognostic model for LSM inclination angle that accounts for this observed structure, where the LSM inclination angle $\unicode[STIX]{x1D6FE}$ is the sum of the inclination angle observed in a neutrally stratified wall-bounded turbulent flow, $\unicode[STIX]{x1D6FE}_{0}\approx 12^{\circ }{-}15^{\circ }$, and the stability-dependent inclination angle of the wedge $\unicode[STIX]{x1D6FE}_{w}(\unicode[STIX]{x1D701}_{z})$. Reported values of $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D701}_{z})$ from the literature, LES results and atmospheric surface layer observations are found to be in good agreement with the new model for $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D701}_{z})$.

Shock–shock interaction structures and a newly discovered dynamic instability in granular streams resulting from such interactions are presented. Shock waves are generated by placing two similar triangular wedges in a gravity-driven granular stream. When the shock waves interact, grains collapse near the centre region of the wedges and a slow-moving concentrated diamond-shaped streak of grains is formed that grows as the inclination of the channel is increased. The diamond streak, under certain geometric conditions, is found to become unstable and start oscillating in the direction transverse to the mainstream. When the wedges are placed too close to each other, the granular flux of the incoming stream is unable to pass through the small gap, resulting in the formation of a single bow shock enveloping both the wedges. Experiments are performed for a wide range of flow speeds, wedge angles and wedge separations to investigate the interaction zone. We discuss a possible mechanism for the formation of the central streak and the associated dynamic instability observed for specific physical parameters.

Nonlinear three-dimensional dynamo equilibrium solutions of viscous-resistive magneto-hydrodynamic equations are continued to formally infinite magnetic and hydrodynamic Reynolds numbers. The external driving mechanism of the dynamo is a uniform shear, which constitutes the base laminar flow and cannot support any kinematic dynamo. Nevertheless, an efficient subcritical nonlinear instability mechanism is found to be able to generate large-scale coherent structures known as streaks, for both velocity and magnetic fields. A finite amount of magnetic field generation is identified at the self-consistent asymptotic limit of the nonlinear solutions, thereby confirming the existence of an effective nonlinear dynamo action at astronomically large Reynolds numbers.

The collective hydrodynamics in fish schools and bird flocks, which includes self-organization of multiple dynamic bodies, is complex and lacks sufficient exploration. In this paper, we study the performance of multiple self-propelled foils in tandem formation, whose flapping motions are asynchronous with a phase difference. It is shown that a compact formation, in which all of the foils perform like a complete anguilliform swimmer, can be spontaneously formed by multiple foils via hydrodynamic interactions. Both velocity enhancement and energy saving can be achieved by multiple foils in anguilliform-like swimming. Furthermore, such anguilliform-like swimming behaviour can be observed over a wide range of parameters, including the number of foils, the phase difference, the initial distance, the heaving amplitude and the pitching amplitude. The results obtained here may provide some light on understanding the self-organization behaviour of biological collectives.

In this work, we carried out direct numerical simulations in large channel domains and studied the kinematics and dynamics of fully localised turbulent bands at Reynolds number $Re=750$. Our results show that the downstream end of the band features fast streak generation and travels into the adjacent laminar flow, whereas streaks at the upstream end decay continually and more slowly. This asymmetry is responsible for the transverse growth of the band. We particularly investigated the mechanism of streak generation at the downstream end, which drives the growth of the band. We identified a spanwise inflectional instability associated with the local mean flow near the downstream end, and our results strongly suggest that this instability is responsible for the streak generation and ultimately for the growth of the band. Based on our study, we propose a possible self-sustaining mechanism of fully localised turbulent bands at low Reynolds numbers in channel flow.

In three-dimensional turbulence there is on average a cascade of kinetic energy from the largest to the smallest scales of the flow. While the dominant idea is that the cascade occurs through the process of vortex stretching, evidence for this is debated. Here we show theoretically and numerically that vortex stretching is in fact not the main contributor to the average cascade. The main contributor is the self-amplification of the strain-rate field, and we provide several arguments for why its role must not be conflated with that of vortex stretching. Numerical results, however, indicate that vortex stretching plays a more important role during fluctuations of the cascade about its average behaviour. We also resolve a paradox regarding the differing role of vortex stretching on the energy cascade and energy dissipation rate dynamics.

We report laboratory experiments of long-crested irregular water surface waves propagating over a shoal. For a sufficiently shallow shoal we find that the surface elevation can have a local maximum of skewness and kurtosis above the shallower part of the shoal close to the edge on the incoming side, and a local minimum of skewness over the downward slope on the lee side of the shoal. We find that the horizontal fluid velocity can have a local maximum and minimum of skewness at the same locations as those for the surface elevation. However, the kurtosis of the horizontal fluid velocity can have a local maximum over the downward slope on the lee side of the shoal, different from the location of the maximum of kurtosis of the surface elevation.

Thermoacoustic instabilities are one of the most challenging problems faced by gas turbine and rocket motor manufacturers. The key instability mechanism is described by the Rayleigh criterion. The Rayleigh criterion does not directly show how to alter a system to make it more stable. This is the objective of sensitivity analysis. Because thermoacoustic systems have many design parameters, adjoint sensitivity analysis has been proposed to obtain all the sensitivities with one extra calculation. Although adjoint sensitivity analysis can be carried out in both the time and the frequency domain, the frequency domain is more natural for a linear analysis. Perhaps surprisingly, the Rayleigh criterion has not yet been rigorously derived and comprehensively interpreted in the frequency domain. The contribution of this theoretical paper is threefold. First, the Rayleigh criterion is interpreted in the frequency domain with integral formulae for the complex eigenvalue. Second, the first variation of the Rayleigh criterion is calculated both in the time and frequency domain, both with and without Lagrange multipliers (adjoint variables). The Lagrange multipliers are physically related to the system’s observables. Third, an adjoint Rayleigh criterion is proposed. The paper also points out that the conclusions of Juniper (Phys. Rev. Fluids, vol. 3, 2018, 110509) apply to the first variation of the Rayleigh criterion, not to the Rayleigh criterion itself. The mathematical relations of this paper can be used to compute sensitivities directly from measurable quantities to enable optimal design.