In this paper, we use asymptotic theory and numerical methods to study resonant triad interactions among discrete internal wave modes at a fixed frequency ($\omega$) in a two-dimensional, uniformly stratified shear flow. Motivated by linear internal wave generation mechanisms in the ocean, we assume the primary wave field as a linear superposition of various horizontally propagating vertical modes at a fixed frequency $\omega$. The weakly nonlinear solution associated with the primary wave field is shown to comprise superharmonic (frequency $2\omega$) and zero frequency wave fields, with the focus of this study being on the former. When two interacting primary modes $m$ and $n$ are in triadic resonance with a superharmonic mode $q$, it results in the divergence of the corresponding superharmonic secondary wave amplitude. For a given modal interaction $(m, n)$, the superharmonic wave amplitude is plotted on the plane of primary wave frequency $\omega$ and Richardson number $Ri$, and the locus of divergence locations shows how the resonance locations are influenced by background shear. In the limit of weak background shear ($Ri\to \infty$), using an asymptotic theory, we show that the horizontal wavenumber condition $k_m + k_n = k_q$ is sufficient for triadic resonance, in contrast to the requirement of an additional vertical mode number condition ($q = |m-n|$) in the case of no shear. As a result, the number of resonances for an arbitrarily weak shear is significantly larger than that for no shear. The new resonances that occur in the presence of shear include the possibilities of resonance due to self-interaction and resonances that occur at the seemingly trivial limit of $\omega \approx 0$, both of which are not possible in the no shear limit. Our weak shear limit conclusions are relevant for other inhomogeneities such as non-uniformity in stratification as well, thus providing an understanding of several recent studies that have highlighted superharmonic generation in non-uniform stratifications. Extending our study to finite shear (finite $Ri$) in an ocean-like exponential shear flow profile, we show that for cograde–cograde interactions, a significant number of divergence curves that start at $Ri\to \infty$ will not extend below a cutoff $Ri$$\sim O(1)$. In contrast, for retrograde–retrograde interactions, the divergence curves extend all the way from $Ri\to \infty$ to $Ri = 0.5$. For mixed interactions, new divergence curves appear at $\omega = 0$ for $Ri\sim O(10)$ and extend to other primary wave frequencies for smaller $Ri$. Consequently, the total ($\text {cograde} + \text {retrograde} + \text {mixed}$) number of resonant triads is of the same order for small $Ri\approx 0.5$ as in the limit of weak shear ($Ri\to \infty$), although it attains a maximum at $Ri\sim O(10)$.

A data-driven approach for the identification of local turbulent-flow states and of their dynamics is proposed. After subdividing a flow domain in smaller regions, the $K$-medoids clustering algorithm is used to learn from the data the different flow states and to identify the dynamics of the transition process. The clustering procedure is carried out on a two-dimensional (2-D) reduced-order space constructed by the multidimensional scaling (MDS) technique. The MDS technique is able to provide meaningful and compact information while reducing the dimensionality of the problem, and therefore the computational cost, without significantly altering the data structure in the state space. The dynamics of the state transitions is then described in terms of a transition probability matrix and a transition trajectory graph. The proposed method is applied to a direct numerical simulation dataset of an incompressible boundary layer flow developing on a flat plate. Streamwise–spanwise velocity fields at a specific wall-normal position are referred to as observations. Reducing the dimensionality of the problem allows us to construct a 2-D map, representative of the local turbulence intensity and of the spanwise skewness of the turbulence intensity in the observations. The clustering process classifies the regions containing streaks, turbulent spots, turbulence amplification and developed turbulence while the transition matrix and the transition trajectories correctly identify the states of the process of bypass transition.

A model-scale experimental investigation of an installed jet–pylon–wing configuration was conducted at the University of Southampton, with the scope to study the effect a pylon has on noise generation and to clarify its impact on the fluctuating wall-pressure load. The set-up consisted of two single-stream nozzles, a baseline axisymmetric annular nozzle and a partially blocked annular pylon nozzle. The nozzles were tested first isolated and then installed next to a NACA4415 aerofoil ‘wing’ at a single nozzle–wing position. The jet Mach number was varied between $0.5 \leq M_{{j}} \leq 0.8$ and measurements were performed both under static and in-flight ambient flow conditions up to ${M_{{f}} = 0.2}$. The jet flow-field qualification was carried out using a single-velocity-component hot-wire anemometer probe. The pressure field on the wing surface was investigated using two miniature wall-pressure transducers that were flush-mounted in the streamwise and spanwise directions within the pressure side of the wing. A linear ‘flyover’ microphone array was used to record the noise radiated to the far field. The unsteady pressure data were analysed in both time and frequency domains using multi-variate statistics, highlighting a far-field noise reduction provided by the presence of the pylon only in the installed case. Furthermore, the wake field generated behind the pylon is seen to significantly modify the wall-pressure fluctuations, particularly at streamwise locations close to the pylon trailing edge.

When a volatile drop impacts on a superheated solid, air drainage and vapour generation conspire to create an intermediate gas layer that delays or even prevents contact between the liquid and the solid. In this article, we use high-speed synchronized reflection interference and total internal reflection imaging to measure the short-time dynamics of the intermediate gas film and to probe the transition between levitation and contact. We observe that the substrate temperature strongly affects the vertical position of the liquid–gas interface and that the dynamic Leidenfrost transition is influenced by both air and vapour drainage (i.e. gas drainage), and evaporation, the latter giving rise to hitherto unreported vertical oscillations of the gas film that can trigger liquid–solid contact. We first derive scaling relations for the height of the gas film trapped under the drop's centreline, called the dimple height, and the minimum film thickness at short times. The former is set by a competition between gas drainage and liquid inertia, similarly as for isothermal impacts, while the latter strongly depends on the vapour production. The gas pressure, at the location where the minimum thickness is reached, is determined by liquid inertia and vapour production and ultimately balanced by the increasing interfacial curvature, determining the minimum thickness. We show that, in the low impact velocity limit, the transient stability of the draining gas film remarkably makes dynamic levitation less demanding than static levitation. We characterize the vertical gas film oscillations by measuring their frequency and monitoring their occurrence in the parameter space spanned by surface temperature and impact velocity. Finally, we model the occurrence of these oscillations and account for their frequency through a hydrodynamic mechanism.

We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number $Sc$ (ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including $Sc$), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at $Sc = 1$), and (ii) an oscillatory instability (absent at $Sc = 1$). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from $Sc$) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with $Sc = 1$, monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as $Sc$ moves away from unity. Neutral stability boundaries on the plane of $Sc$ and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.

The presence of ambient air in liquid-slamming events plays a crucial role in influencing the shape of the liquid surface prior to the impact, and the distribution of loads created upon impact. We study the effect of trapped air on impact loads in a simplified geometry, by slamming a horizontal flat disc onto a stationary water bath at a well-controlled velocity. We show how air trapping influences pressure peaks at different radial locations on the disc, how the pressure impulses are affected and how local pressure impulses differ from those obtained from area-integrated (force) impulses at impact. More specifically, we find that the air layer causes a gradual buildup of the load before the peak value is reached, and show that this buildup follows inertial scaling. Further, the same localised pressure impulse at the disc centre is found to be lower than the corresponding (area-integrated) force impulse on the entire disc. While the (area-integrated) force impulses are close to the classical result of Batchelor (An Introduction to Fluid Dynamics, Cambridge University Press, 1967, § 6.10) and Glasheen & McMahon (Phys. Fluids, vol. 8, issue 8, 1996, pp. 2078–2083), the localised pressure impulses at the disc centre, where the trapped air layer is at its thickest, lie closer to the theoretical estimation by Peters et al. (J. Fluid Mech., vol. 724, 2013, pp. 553–580) for an air-cushioned impact.

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.

We present a theoretical model of the temperature distribution in the boundary layer region close to the seabed. Using a perturbation expansion, multiple scales and similarity variables, we show how free-surface waves enhance heat transfer between seawater and a seabed with a solid, horizontal, smooth surface. Maximum heat exchange occurs at a fixed frequency depending on ocean depth, and does not increase monotonically with the length and phase speed of propagating free-surface waves. Close agreement is found between predictions by the analytical model and a finite-difference scheme. It is found that free-surface waves can substantially affect the spatial evolution of temperature in the seabed boundary layer. This suggests a need to extend existing models that neglect the effects of a wave field, especially in view of practical applications in engineering and oceanography.

The linear stability of the incompressible flow in an infinitely extended cavity with rectangular cross-section is investigated numerically. The basic flow is driven by a lid which moves tangentially, but at yaw with respect to the edges of the cavity. As a result, the basic flow is a superposition of the classical recirculating two-dimensional lid-driven cavity flow orthogonal to a wall-bounded Couette flow. Critical Reynolds numbers computed by linear stability analysis are found to be significantly smaller than data previously reported in the literature. This finding is confirmed by independent nonlinear three-dimensional simulations. The critical Reynolds number as a function of the yaw angle is discussed for representative aspect ratios. Different instability modes are found. Independent of the yaw angle, the dominant instability mechanism is based on the local lift-up process, i.e. by the amplification of streamwise perturbations by advection of basic flow momentum perpendicular to the sheared basic flow. For small yaw angles, the instability is centrifugal, similar as for the classical lid-driven cavity. As the spanwise component of the lid velocity becomes dominant, the vortex structures of the critical mode become elongated in the direction of the bounded Couette flow with the lift-up process becoming even more important. In this case the instability is made possible by the residual recirculating part of the basic flow providing a feedback mechanism between the streamwise vortices and the streamwise velocity perturbations (streaks) they promote. In the limit when the basic flow approaches bounded Couette flow the critical Reynolds number increases very strongly.

This study contributes to an improved understanding of the stability of the boundary layer generated at the bottom of a propagating surface wave of small but finite amplitude such that both a second harmonic component and a steady streaming component, which are superimposed on the main oscillatory flow, assume significant values. A linear stability analysis of the laminar flow is made to determine the conditions leading to transition and turbulence appearance. The Reynolds number of the phenomenon is assumed to be large and a ‘momentary’ criterion of stability is used. The results show that, at a given location, the laminar regime becomes unstable when the flow close to the bottom reverses its direction from the onshore to the offshore direction and the Reynolds number exceeds a first critical value $R_{\delta ,c1}$. However, close to the critical condition, the flow is expected to relaminarize during the other phases of the cycle. Only when the Reynolds number is increased does turbulence tend to appear also after the passage of the wave trough when the flow close to the bottom reverses from the offshore to the onshore direction. When the Reynolds number is further increased and becomes larger than a second ‘threshold’ value, the growth rate of the perturbations becomes positive over the entire wave period. The obtained results suggest the existence of four different flow regimes: the laminar regime, the disturbed laminar regime, the intermittently turbulent regime and the fully developed turbulent regime.

Two models based on convolutional neural networks are trained to predict the two-dimensional instantaneous velocity-fluctuation fields at different wall-normal locations in a turbulent open-channel flow, using the wall-shear-stress components and the wall pressure as inputs. The first model is a fully convolutional neural network (FCN) which directly predicts the fluctuations, while the second one reconstructs the flow fields using a linear combination of orthonormal basis functions, obtained through proper orthogonal decomposition (POD), and is hence named FCN-POD. Both models are trained using data from direct numerical simulations at friction Reynolds numbers $Re_{\tau } = 180$ and 550. Being able to predict the nonlinear interactions in the flow, both models show better predictions than the extended proper orthogonal decomposition (EPOD), which establishes a linear relation between the input and output fields. The performance of the models is compared based on predictions of the instantaneous fluctuation fields, turbulence statistics and power-spectral densities. FCN exhibits the best predictions closer to the wall, whereas FCN-POD provides better predictions at larger wall-normal distances. We also assessed the feasibility of transfer learning for the FCN model, using the model parameters learned from the $Re_{\tau }=180$ dataset to initialize those of the model that is trained on the $Re_{\tau }=550$ dataset. After training the initialized model at the new $Re_{\tau }$, our results indicate the possibility of matching the reference-model performance up to $y^{+}=50$, with $50\,\%$ and $25\,\%$ of the original training data. We expect that these non-intrusive sensing models will play an important role in applications related to closed-loop control of wall-bounded turbulence.

We compare the properties of the turbulence induced by the breakdown of Kelvin–Helmholtz instability (KHI) at high Reynolds number in two classes of stratified shear flows where the background density profile is given by either a linear function or a hyperbolic tangent function, at different values of the minimum initial gradient Richardson number ${{Ri}}_0$. Considering global and local measures of mixing defined in terms of either the irreversible mixing rate $\mathscr {M}$ associated with the time evolution of the background potential energy, or an appropriately defined density variance dissipation rate $\chi$, we find that the proliferation of secondary instabilities strongly affects the efficiency of mixing early in the flow evolution, and also that these secondary instabilities are highly sensitive to flow perturbations that are added at the point of maximal (two-dimensional) billow amplitude. Nevertheless, mixing efficiency does not appear to depend strongly on the far field density structure, a feature supported by the evolution of local horizontally averaged values of the buoyancy Reynolds number ${Re}_b$ and gradient Richardson number ${Ri}_g$. We investigate the applicability of various proposed scaling laws for flux coefficients $\varGamma$ in terms of characteristic length scales, in particular discussing the relevance of the overturning ‘Thorpe scale’ in stratified turbulent flows. Finally, we compare a variety of empirical model parameterizations used to compute diapycnal diffusivity in an oceanographic context, arguing that for transient flows such as KHI-induced turbulence, simple models that relate the ‘age’ of a turbulent event to its mixing efficiency can produce reasonably robust mixing estimates.

Drop coalescence occurs through the rapid growth of a liquid bridge that connects the two drops. At early times after contact, the bridge dynamics is typically self-similar, with details depending on the geometry and viscosity of the liquid. In this paper we analyse the coalescence of two-dimensional viscous drops that float on a quiescent deep pool; such drops are called liquid lenses. The analysis is based on the thin-sheet equations, which were recently shown to accurately capture experiments of liquid lens coalescence. It is found that the bridge dynamics follows a self-similar solution at leading order, but, depending on the large-scale boundary conditions on the drop, significant corrections may arise to this solution. This dynamics is studied in detail using numerical simulations and through matched asymptotics. We show that the liquid lens coalescence can involve a global translation of the drops, a feature that is confirmed experimentally.

It has been commonly observed on open waters that ducklings/goslings follow their mothers in a highly organized formation. The questions arise: (1) why are they swimming in formation? (2) what is the best swimming formation? (3) how much energy can be preserved by each individual in formation swimming? To address these questions, we established a simplified mathematical and numerical model and calculated the wave drag on a group of waterfowl in a swimming formation. We observed two new and interesting findings: wave-riding and wave-passing. By riding the waves generated by a mother duck, a trailing duckling can obtain a significant wave-drag reduction. When a duckling swims at the ‘sweet point’ behind its mother, a destructive wave interference phenomenon occurs and the wave drag of the duckling turns positive, pushing the duckling forward. More interestingly, this wave-riding benefit could be sustained by the rest of the ducklings in a single-file line formation. Starting from the third one in a queue, the wave drag of individuals gradually tended towards zero, and a delicate dynamic equilibrium was achieved. Each individual under that equilibrium acted as a wave passer, passing the waves’ energy to its trailing one without any energy losses. Wave-riding and wave-passing are probably the principal reasons for the evolution of swimming formation by waterfowl. This study is the first to reveal the reasons why the formation movement of waterfowl can preserve individuals’ energy expenditure. Our calculations provide new insights into the mechanisms of formation swimming.

We study the stratified gas in a rapidly rotating centrifuge as a model for the Earth's atmosphere. Based on methods of perturbation theory, it is shown that in certain regimes, internal waves in the gas centrifuge have the same dispersion relation to leading order as their atmospheric siblings. Assuming an air filled centrifuge with a radius of around 50 cm, the optimal rotational frequency for realistic atmosphere-like waves is around 10 000 revolutions per minute. Using gases of lower heat capacities at constant pressure, such as xenon, the rotational frequencies can be even halved to obtain the same results. Similar to the atmosphere, it is feasible in the gas centrifuge to generate a clear scale separation of wave frequencies and therefore phase speeds between acoustic waves and internal waves. In addition to the centrifugal force, the Coriolis force acts in the same plane. However, its influence on axially homogeneous internal waves appears only as a higher-order correction. We conclude that the gas centrifuge provides an unprecedented opportunity to investigate atmospheric internal waves experimentally with a compressible working fluid.

The linear convective instability of a mixture of dry air, water vapour and liquid water, with a stable unsaturated layer residing on an unstable saturated layer, is studied. It may serve as a prototype model for understanding the instability that causes mixing at the top of stratocumulus cloud or fog. Such a cloud-clear air interface is modelled as an infinitely thin saturation interface where radiative and evaporative cooling take place. The interface position is determined by the Clausius–Clapeyron equation, and can undulate with the evolution of moisture and temperature. In the small-amplitude regime two physical mechanisms are revealed. First, the interface undulation leads to the undulation of the cooling source, which destabilizes the system by superposing a vertical dipole heating anomaly on the convective cell. Second, the evolution of the moisture field induces non-uniform evaporation at the interface, which stabilizes the system by introducing a stronger evaporative cooling in the ascending region and vice versa in the descending region. These two mechanisms are competing, and their relative contribution to the instability is quantified by theoretically estimating their relative contribution to buoyancy flux tendency. When there is only evaporative cooling, the two mechanisms break even, and the marginal stability curve remains the same as the classic two-layer Rayleigh–Bénard convection with a fixed cooling source.

This research is motivated by the desire to control the solids distribution during the drying of a film containing particles of two different sizes. A variety of particle arrangements in dried films has been seen experimentally, including a thin layer of small particles at the top surface. However, it is not fully understood why this would occur. This work formulates and solves a colloidal hydrodynamics model for (i) diffusion alone and (ii) diffusion plus excluded volume diffusiophoresis, to determine their relative importance in affecting the particle arrangement. The methodology followed is to derive partial differential equations (PDEs) describing the motion of two components in a drying film. The diffusive fluxes are predicted by generalising the Stokes–Einstein diffusion coefficient, with the dispersion compressibility used to produce equations valid up until close packing. A further set of novel equations incorporating diffusiophoresis is derived. The diffusiophoretic mechanism investigated in this work is the small particles being excluded from a volume around the large particles. The resulting PDEs are scaled and solved numerically using a finite volume method. The model includes the chemical potentials of the particles, allowing for incorporation of any interaction term. The relative magnitudes of the fluxes of the differently sized particles are compared using scaling arguments and via numerical results. The diffusion results, without any inter-particle interactions, predict stratification of large particles to the top surface. Addition of excluded volume diffusiophoresis introduces a downwards flux on the large particles, that can result in small-on-top stratification, thus providing a potential explanation of the experimental observations.

The force balance of rotating Rayleigh–Bénard convection regimes is investigated using direct numerical simulation on a laterally periodic domain, vertically bounded by no-slip walls. We provide a comprehensive view of the interplay between governing forces both in the bulk and near the walls. We observe, as in other prior studies, regimes of cells, convective Taylor columns, plumes, large-scale vortices (LSVs) and rotation-affected convection. Regimes of rapidly rotating convection are dominated by geostrophy, the balance between Coriolis and pressure-gradient forces. The higher-order interplay between inertial, viscous and buoyancy forces defines a subdominant balance that distinguishes the geostrophic states. It consists of viscous and buoyancy forces for cells and columns, inertial, viscous and buoyancy forces for plumes, and inertial forces for LSVs. In rotation-affected convection, inertial and pressure-gradient forces constitute the dominant balance; Coriolis, viscous and buoyancy forces form the subdominant balance. Near the walls, in geostrophic regimes, force magnitudes are larger than in the bulk; buoyancy contributes little to the subdominant balance of cells, columns and plumes. Increased force magnitudes denote increased ageostrophy near the walls. Nonetheless, the flow is geostrophic as the bulk. Inertia becomes increasingly more important compared with the bulk, and enters the subdominant balance of columns. As the bulk, the near-wall flow loses rotational constraint in rotation-affected convection. Consequently, kinetic boundary layers deviate from the expected behaviour from linear Ekman boundary layer theory. Our findings elucidate the dynamical balances of rotating thermal convection under realistic top/bottom boundary conditions, relevant to laboratory settings and large-scale natural flows.

Apparent inconsistency between (i) experimental and direct numerical simulation (DNS) data that show the significant influence of differential diffusion on the turbulent burning rate and (ii) recent complex-chemistry DNS data that indicate mitigation of the influence of differential diffusion on conditioned profiles of various local flame characteristics at high Karlovitz numbers, is explored by analysing new DNS data obtained from lean hydrogen–air turbulent flames. Both aforementioned effects are observed by analysing the same DNS data provided that the conditioned profiles are sampled from the entire computational domain. On the contrary, the conditioned profiles sampled at the leading edge of the mean flame brush do not indicate the mitigation, but are significantly affected by differential diffusion phenomena, e.g. because reaction zones are highly curved at the leading edge. This observation is consistent with a significant increase in the computed turbulent burning velocity with decreasing Lewis number, with all the results considered jointly being consonant with the leading point concept of premixed turbulent combustion. The concept is further supported by comparing DNS data obtained by allowing for preferential diffusion solely for a single species, either atomic or molecular hydrogen.

In droplet impacts, transitions between coalescence and bouncing are determined by complex interplays of multiple mechanisms dominating at various length scales. Here we investigate the mechanisms and governing parameters comprehensively by experiments and scaling analyses, providing a unified framework for understanding and predicting the outcomes when using different fluids. Specifically, while bouncing had not been observed in head-on collisions of water drops under atmospheric conditions, it was found in our experiments to appear on increasing the droplet diameter sufficiently. Contrarily, while bouncing was always observed in head-on impacts of alkane drops, we found it to disappear on decreasing the diameter sufficiently. The variations are related to gas draining dynamics in the inter-droplet film and suggest an easier means for controlling bouncing as compared to alternating the ambient pressure usually sought. The scaling analysis further shows that for a given Weber number, enlarging droplet diameter or fluid viscosities, or lowering surface tension contributes to a larger characteristic minimum thickness of the gas film, thus enhancing bouncing. The key dimensionless group $(O{h_{g,l}},\;O{h_l},\;{A^\ast })$ is identified, referred to as the two-phase Ohnesorge number, the Ohnesorge number of liquid and the Hamaker constant, respectively. Our thickness-based model indicates that as ${h^{\prime}_{m,c}} > 21.1{h_{cr}}$, where ${h^{\prime}_{m,c}}$ is the maximum value of the characteristic minimum film thickness $({h_{m,c}})$ and ${h_{cr}}$ is the critical thickness, bouncing occurs in both head-on and off-centre collisions. That is, when $1.2O{h_{g,l}}/(1 - 2O{h_l}) > \sqrt[3]{{{A^\ast }}}$, a fully developed bouncing regime occurs, thereby yielding a lower coalescence efficiency. The transitional Weber number is found universally to be 4.