Splashes from impacts of drops on liquid pools are ubiquitous and generate secondary droplets important for a range of applications in healthcare, agriculture and industry. The physics of splash continues to comprise central unresolved questions. Combining experiments and theory, here we study the sequence of topological changes from drop impact on a deep, inviscid liquid pool, with a focus on the regime of crown splash with developing air cavity below the interface and crown sheet above it. We develop coupled evolution equations for the cavity–crown system, leveraging asymptotic theory for the cavity and conservation laws for the crown. Using the key coupling of sheet and cavity, we derive similarity solutions for the sheet velocity and thickness profiles, and asymptotic prediction of the crown height evolution. Unlike the cavity whose expansion is opposed by gravitational effects, the axial crown rise is mostly opposed by surface tension effects. Moreover, both the maximum crown height and the time of its occurrence scale as ${\textit {We}}^{5/7}$. We find our analytical results to be in good agreement with our experimental measurements. The cavity–crown coupling achieved enables us to obtain explicit estimates of the crown splash spatio-temporal unsteady dynamics, paving the way to deciphering ultimate splash fragmentation.

We study the instability of a dusty simple shear flow where the dust particles are distributed non-uniformly. A simple shear flow is modally stable to infinitesimal perturbations. Also, a band of particles remains unaffected in the absence of any background flow. However, we demonstrate that the combined scenario – comprising a simple shear flow with a localized band of particles – can exhibit destabilization due to their two-way interaction. The instability originates solely from the momentum feedback from the particle phase to the fluid phase. Eulerian–Lagrangian simulations are employed to illustrate the existence of this instability. Furthermore, the results are compared with a linear stability analysis of the system using an Eulerian–Eulerian model. Our findings indicate that the instability has an inviscid origin and is characterized by a critical wavelength below which it is not persistent. We have observed that increasing particle inertia dampens the unstable modes, whereas the strength of the instability increases with the strength of the coupling between the fluid and particle phases.

Simulating complex gas flows from turbulent to rarefied regimes is a long-standing challenge, since turbulence and rarefied flow represent contrasting extremes of computational aerodynamics. We propose a multiscale method to bridge this gap. Our method builds upon the general synthetic iterative scheme for the mesoscopic Boltzmann equation, and integrates the $k$–$\omega$ model in the macroscopic synthetic equation to address turbulent effects. Asymptotic analysis and numerical simulations show that the macroscopic–mesoscopic coupling adaptively selects the turbulence model and the laminar Boltzmann equation. The multiscale method is then applied to opposing jet problems in hypersonic flight surrounding by rarefied gas flows, showing that the turbulence could cause significant effects on the surface heat flux, which cannot be captured by the turbulent model nor the laminar Boltzmann solution alone. This study provides a viable framework for advancing understanding of the interaction between turbulent and rarefied gas flows.

We investigate the effect of external oscillatory forcing on evolving two-dimensional (2-D) gravity currents, resulting from the well-known lock-exchange set-up, by superimposing a horizontally uniform oscillating pressure gradient. This pressure gradient generates a 2-D horizontally uniform laminar oscillating flow over the flat no-slip bottom that interacts with the evolving gravity current. We explore the effect of the velocity amplitude of the applied oscillating flow and its period of oscillations on the behaviour of the evolving gravity currents. A key element introduced by the external forcing is the Stokes boundary layer near the no-slip bottom wall generating differential advection near the bottom wall when the propagation direction of the gravity current and the oscillating externally imposed flow are in the same direction. This results in a phenomenon that we refer to as lifting of the gravity current, which clearly distinguishes the oscillatory forced gravity current from the freely evolving case. This phenomenon induces fine-scale density structures when the externally imposed flow is opposite to the propagation direction of the gravity current a semi-period later. We have explored the effect of lifting on the current propagation and the density structure of the gravity current front. Three separate regimes are distinguished for the evolution of the density structure in the front of the gravity current depending on the period of forcing, including a regime reminiscent of tidally forced estuarine flows. The present study shows the existence of significant effects of an oscillatory forcing on the dynamics, advection and density distribution of gravity currents.

Linear non-modal analyses are performed to study the mechanism of how deformable free surfaces influence very-large-scale motions (VLSMs) in turbulent open channel flows. The mean velocity and eddy viscosity profiles obtained from direct numerical simulations are used in the generalised Orr–Sommerfeld and Squire equations to represent background turbulence effects. Solutions of surface-wave eigenmodes and shear eigenmodes are obtained. The results indicate that at high Froude numbers, free surfaces enhance the maximum transient growth rate of VLSMs through surface-wave eigenmodes. We then analyse the energy budget equation to reveal the underlying mechanism. For streamwise-uniform motions, the energy growth rate is enhanced by an energy production term associated with the correlation between the streamwise velocity, which is generated by the lifting-up effect of streamwise vortices composed of shear eigenmodes, and the vertical velocity, which is induced by a spanwise standing wave composed of surface-wave eigenmodes. For streamwise-varying motions, the energy growth rate is enhanced by a standing wave moving with a pair of vortices that travel at a speed approximately equal to the projection of the mean surface velocity along the wavenumber vector direction. Finally, an analytical expression of the energy production term is derived to provide the initial conditions for the maximum transient growth and explain the weak free-surface effect observed at large spanwise wavenumbers and low Froude numbers. The results demonstrate a linear non-modal mechanism in interactions between free surfaces and VLSMs in open channel flows.

The multipolar spherical vortex solutions to the Euler and Navier–Stokes equations in background cylindrical flow with swirl admit an additional background divergent radial flow with arbitrary time-dependent amplitude. In this case the radial wavenumber $k$, fundamental frequency $\omega$ and overall amplitude $U$ of the multipolar mode superposition become time-dependent and related functions. Assumption of an additional constraint, as a constitutive equation defining the time evolution of the spatially homogeneous divergence of the background flow, is required for the time evolution of the total flow to be completely evaluated from the initial conditions. It is found that flow compression implies an increase of the absolute values of the fundamental frequency $\omega$ and overall velocity amplitude $U$ of the oscillations.

We develop a simple model which describes the repeated injection and extraction of hydrogen in a permeable water-saturated rock which has the form of an anticline. We demonstrate that the flow is controlled by the dimensionless ratio of the square of the buoyancy speed to the product of the two-dimensional volume injection rate and the injection–extraction frequency, and we explore the cases in which this ratio is large and small. Over the first few cycles, the volume of hydrogen in the system gradually builds up since during the extraction phase, some of the water eventually reaches the extraction well, and in our model the system ceases to extract fluid for the remainder of this extraction phase. After many cycles, there is sufficient hydrogen in the system that a quasi-equilibrium state develops in which the mass of fluid injected matches the mass extracted over the course of a cycle. We show that in this equilibrium, the ratio between the mass of gas remaining in the aquifer at the end of the extraction phase, known as the cushion gas, to the mass of gas injected, known as the working gas, decreases if either the flow rate or frequency of the cycles decrease or the buoyancy speed increases, leading to more efficient storage.

This study investigates the heating issue associated with a V-shaped blunt leading edge (VBLE) in a hypersonic flow. The heat flux generation on the VBLE is highly correlated with the shock interaction configurations in the crotch region, determined by the relative position of the triple point T and the curved shock (CS). The primary Mach reflection (MR), accompanied by a series of secondary shock–shock interactions and shock wave–boundary layer interactions, can produce extremely high heating peaks on the crotch. To configure the shock wave structures and reduce the heat flux, a shock-controllable design approach is developed based on the simplified continuity method. The strategy involves the inverse design of the crotch sweep path, according to the location of the triple point and the contour of the CS. The comparisons between the pre-designed shock configurations and the numerical results demonstrate the reliability of the design approach across various free stream Mach numbers ranging from 6 to 10. A VBLE model designed with the shock configuration of regular reflection from the same family (sRR) at a free stream Mach number of 8 is examined. Under the design conditions, the outermost heat flux peak is reduced by 80 % compared with the baseline case. The heating reduction capabilities of the model under varying free stream Mach numbers and sideslip angles are also evaluated, confirming its robustness under undesigned operating scenarios.

Fragmentation of a fluid body into droplets underlies many contamination and disease transmission processes where pathogens are transported in a liquid phase. An important class of such processes involves formation of a fluid ligament and its destabilization into droplets. Inertial detachment (Gilet & Bourouiba, J. R. Soc. Interface, vol. 12, 2015, 20141092) is one of these modes: upon impact on a sufficiently compliant substrate, the substrate's motion can transfer its impulse to a contaminated sessile drop residing on it. The fragmentation of the sessile drop is efficient at producing contaminated ejected droplets with little dilution. Inertial detachment, particularly from substrates of intermediate wetting, is also interesting as a fundamental fragmentation process on its own merit, involving the asymmetric stretching of the sessile drop under impulsive axial forcing with one-sided pinning due to the substrate's intermediate wetting. Our experiments show that the radius, $R_{tip}$, of the tip drop ejected become insensitive to the Bond number value for $Bo>1$. Here, $Bo$ quantifies the inertial effects via the relative axial impulsive acceleration compared with capillarity. The time, $t_{tip}$, of tip-drop breakup is also insensitive to $Bo$. Combining experiments, theory and validated numerics, we decipher the selection of $R_{tip}$ and its sensitivity to the surface-wetting and substrate foot dynamics. Using asymptotic theory in the large $Bo$ limit for which the thin-film/slender-jet approximations hold, we derive a reduced physical model that predicts $R_{tip}$ consistent with our experiments. Finally, we discuss how pathogen physical properties (e.g. wetting and buoyancy) within the sessile drop determine their distribution in the tip and secondary fragmentation droplets.

An asymptotic matching modal model is established based on the singular perturbation method for predicting mode evolution in single- and dual-mode interfaces accelerated by a shock wave. The startup process is incorporated into the model to provide a complete description of the mode evolution after the shock impact. Through considering the feedback from high-order harmonic to the third-order harmonic, the model accuracy is improved and the model divergence is prevented. In addition, the model can evaluate the mutual-coupling effect on the amplitude variations of high-order harmonics besides the ‘beat modes’. To validate the model, experiments on both light–heavy and heavy–light interfaces subject to a shock wave are conducted, and both single- and dual-mode interfaces formed by the soap-film technique are involved. The interface profiles extracted from mode decomposition and predicted by the model show high consistency with the experimental counterparts. Good agreement of the mode amplitude growths between the experiments and theoretical predictions shows the superiority of the model, especially for the heavy–light interface.

In this paper, we simulate the process of two-dimensional axisymmetric fluid–structure coupling of a droplet impacting on a flexible disk. The effects of dimensionless disk stiffness (K = 0.1–1000), Weber number (We = 1–500) and contact angle (θ = 130° and 60°) on the dynamics of the droplet impacting on the flexible disk are analysed. The results indicate that there are five typical impact modes for a hydrophobic surface (θ = 130°) and four typical impact modes for a hydrophilic surface (θ = 60°) within the range of considered parameters. The analysis of spreading factor reveals that a part of the energy is transferred to the substrate, which is manifested as a weakening of the droplet spreading, when the substrate deforms downwards due to the droplet impact; the squeezing of the droplet causes a tendency to flow from the centre of the droplet to the edge, which is manifested as an enhancement of the droplet spreading, when the substrate recovers from the downward deformation. The effect of the substrate flexibility on the maximum spreading factor depends on the competition of the two mechanisms above. Based on this, a modified scaling law of βmax has been proposed by introducing the effective Weber number (Wem). The analysis of impact force demonstrates that the peak of the impact force is related to the deflection of the flexible substrate which is different from that of a rigid wall; and three typical processes of impact force variation have been summarised. In addition, unlike the rigid substrate scenario, there is an energy interaction between the droplet and the flexible substrate after impact occurs, which is classified as three typical energy transformation processes.

We study experimentally the onset of Faraday waves near the end walls of a rectangular vessel containing two stably stratified fluid layers, subject to horizontal oscillations. These subharmonic waves (SWs) are excited, because the horizontal inertial forcing drives a harmonic propagating wave which displaces the interface in the vertical direction at the end walls. We find that the onset of SWs is regulated by a balance between capillary and viscous forces, where the rate of damping is set by the Stokes layer thickness at the wall rather than the wavelength of the SWs. We model the onset of SWs with a weakly damped Mathieu equation and find that the dimensional critical acceleration scales as $\nu _m^{1/2} \omega ^{3/2}$, where $\nu _m$ is the mean viscosity and $\omega$ is the frequency of forcing, in excellent agreement with the experiment over a wide range of parameters.

We explore the interaction of natural convection and mechanical ventilation in a room where fresh air is supplied at low level and stale air is extracted at high level. Turbulent buoyant plumes rising from heat sources interact with this upward airflow and establish a steady-state stratification with a warm upper layer above a layer of the cold supply air. Adapting the volume balance model used in natural ventilation (Linden et al., J. Fluid Mech., vol. 212, 1990, pp. 309–335) leads to the prediction that the upper layer will vent from the room when the ventilation volume flux exceeds the volume flux in the plumes at the ceiling. However, our new laboratory experiments establish that there is still a stable two-layer stratification beyond this point of critical ventilation. Motivated by our observations, we propose that the kinetic energy flux supplied by the plume leads to turbulent mixing in the upper layer. We propose a new model of this mixing which is consistent with our experiments in both the over- and under-ventilated regimes. This has important implications for air recirculation in buildings with large ventilation flows, particularly hospital operating theatres and clean rooms.

Liquid flowing down a fibre readily destabilises into a train of beads, commonly called a bead-on-fibre pattern. Bead formation results from capillary-driven instability and gives rise to patterns with constant velocity and time-invariant bead frequency $f$ whenever the instability is absolute. In this study, we develop a scaling law for $f$ that relates the Strouhal number $St$ and capillary number $Ca$ for Ostwaldian power-law liquids with Newtonian liquids recovered as a limiting case. We validate our proposed scaling law by comparing it with prior experimental data and new experimental data using xanthan gum solutions to produce a low capillary number $Ca \leq 10^{-2}$ regime. The experimental data encompasses both Ostwaldian and Newtonian flow, as well as symmetric and asymmetric patterns, and we find the data collapses along the predicted trend across seven orders of magnitude in $Ca$. Our proposed scaling law is a powerful tool for studying and applying bead-on-fibre flows where $f$ is critical, such as heat and mass transfer systems.