The interaction between the flow in a channel with an obstruction on the bottom and an elastic sheet representing the ice covering the liquid is considered for the case of steady flow. The mathematical model based on the velocity potential theory and the theory of thin elastic shells fully accounts for the nonlinear boundary conditions at the elastic sheet/liquid interface and on the bottom of the channel. The integral hodograph method is employed to derive the complex velocity potential of the flow, which contains the velocity magnitude at the interface in explicit form. This allows one to formulate the coupled ice/liquid interaction problem and reduce it to a system of nonlinear equations in the unknown magnitude of the velocity at the interface. Case studies are carried out for a semi-circular obstruction on the bottom of the channel. Three flow regimes are studied: a subcritical regime, for which the interface deflection decays upstream and downstream; an ice supercritical and channel subcritical regime, for which two waves of different lengths may exist; and a channel supercritical regime, for which the elastic wave is found to extend downstream to infinity. All these regimes are in full agreement with the dispersion equation. The obtained results demonstrate a strongly nonlinear interaction between the elastic and the gravity wave near the first critical Froude number where their lengths approach each other. The interface shape, the bending moment and the pressure along the interface are presented for wide ranges of the Froude number and the obstruction height.

Evaporation of multi-component liquid mixtures in confined geometries, such as capillaries, is crucial in applications such as microfluidics, two-phase cooling devices and inkjet printing. Predicting the behaviour of such systems becomes challenging because evaporation triggers complex spatio-temporal changes in the composition of the mixture. These changes in composition, in turn, affect evaporation. In the present work, we study the evaporation of aqueous glycerol solutions contained as a liquid column in a capillary tube. Experiments and direct numerical simulations show three evaporation regimes characterised by different temporal evolutions of the normalised mass transfer rate (or Sherwood number $Sh$), namely $Sh (\tilde{t} ) = 1$, $Sh \sim 1/\sqrt {\tilde{t} }$ and $Sh \sim \exp (-\tilde{t} )$, where $\tilde {t}$ is a normalised time. We present a simplistic analytical model that shows that the evaporation dynamics can be expressed by the classical relation $Sh = \exp ( \tilde{t} )\,\mathrm {erfc} ( \sqrt {\tilde{t} })$. For small and medium $\tilde{t}$, this expression results in the first and second of the three observed scaling regimes, respectively. This analytical model is formulated in the limit of pure diffusion and when the penetration depth $\delta (t)$ of the diffusion front is much smaller than the length $L(t)$ of the liquid column. When $\delta \approx L$, finite-length effects lead to $Sh \sim \exp (-\tilde{t} )$, i.e. the third regime. Finally, we extend our analytical model to incorporate the effect of advection and determine the conditions under which this effect is important. Our results provide fundamental insights into the physics of selective evaporation from a multi-component liquid column.

Plane Couette flow at Reynolds number $Re=1200$ (based on the channel half-height and half the velocity difference between the top and bottom plates) is investigated with a spatial domain designed to retain only two spanwise integral length scales. In this system, the computation of invariant solutions that are physically representative of the turbulent state has been understood to be challenging. To address this challenge, our approach is to employ an accurate reduced-order model with 600 degrees of freedom (Cavalieri & Nogueira, Phys. Rev. Fluids, vol. 7, 2022, L102601). Using the two-scale energy budget and the temporal cross-correlation of key observables, it is first demonstrated that the model contains most of the multi-scale physical processes identified recently (Doohan et al., J. Fluid Mech., vol. 913, 2021, A8); i.e. the large- and small-scale self-sustaining processes, the energy cascade for turbulent dissipation, and an energy-cascade mediated small-scale production mechanism. Invariant solutions of the reduced-order model are subsequently computed, including 96 equilibria and 43 periodic orbits. It is found that none of the computed equilibrium solutions are able to reproduce an accurate energy balance associated with the multi-scale dynamics of the turbulent state. Incorporation of unsteadiness into invariant solutions is seen to be essential for a sensible description of the multi-scale turbulent dynamics and the related energetics, at least in this type of flow, as periodic orbits with a sufficiently long period are mainly able to describe the complex spatio-temporal dynamics associated with the known multi-scale phenomena.

Although poroelastic clusters in nature, such as bristled wings and plumed seeds, exhibit remarkable flight performances by virtue of their porous structure, the effects of another key feature, elasticity, on aerodynamic loading remain elusive. For a poroelastic cluster, we investigate the aerodynamic effects of elastic deformation that occurs through the collective rearrangement of many elastic components and the fluid-dynamic interactions between them. As a simple two-dimensional model, an array of multiple cylinders which are individually and elastically mounted is employed with diverse values of porosity and elasticity. Under a uniform free stream, the poroelastic cluster enlarges its frontal area and augments the total drag force in the quasi-steady state; this is in contrast to the general reconfiguration of fixed elastic structures, which tends to reduce the frontal area and drag. The rearrangement of the poroelastic cluster is dominated by the virtual fluid barrier that develops in a gap between the elastic components, interrupting the flow penetrating between them. The effects of this hydrodynamic blockage on changes in the frontal area and drag force are analysed in terms of porosity and elasticity, revealing the fluid-dynamic mechanism underlying the appearance of peak drag at an intermediate porosity. Moreover, to represent the coupled effects of porosity and elasticity on the rearrangement, a scaled elastic energy is derived through a consideration of the energy balance.

A depth-averaged model for turbulent open-channel flows with breaking roll waves on a sloping smooth bottom is derived under an assumption of independence between the wall turbulence and the roller turbulence. The model includes four variables – the water depth, the average velocity, and two variables called enstrophy, the shear enstrophy and the roller enstrophy – which take into account the deviations of the velocity with respect to its depth-averaged value due to shear effect and roller turbulence, respectively. The four equations of the model are the mass, momentum, energy and shear enstrophy balance equations, with the mathematical structure of the Euler equations of compressible fluids, with an additional transport equation and with source terms. The system is hyperbolic. The roller enstrophy is created by shocks. A non-zero value of the roller enstrophy indicates a breaking wave and a turbulent roller. The model is solved by a fast and well-known numerical scheme, with an explicit finite-volume method in one step. The model is used to simulate periodic and natural roll waves with a good agreement with existing experimental results. There is no parameter to calibrate in the model, which gives it a predictive character.

Three-dimensional flows over low-aspect-ratio rectangular flat plates (${A{\kern-4pt}R} = 1.00$–$1.50$) are investigated using tomographic and planar particle image velocimetry techniques. The chord-based Reynolds number is $5400$, and the angle of attack is fixed at $6^\circ$. This study reveals for the first time the interplay between spanwise fluid transport and downwash, both originating from the tip effects. Spanwise fluid transport promotes the formation and subsequent coherent development of leading-edge vortices, whereas downwash stabilizes the flow. Specifically, two mechanisms related to spanwise fluid transport are revealed. First, the spanwise fluid transport enhances the intensity of the reversed flow, promoting the shear layer roll-up and vortex shedding. Second, the near-wall spanwise flow interacts with the shed C-shape vortices, thereby strengthening the vortex heads. In particular, through these interactions, spanwise fluid transport can sustain the coherence of the C-shape vortices until the vortex heads split in a regular fashion. Consequently, the C-shape vortices are transformed into novel Þ-shape vortices for the plates of ${A{\kern-4pt}R} \leq 1.25$, which supplements the previously discovered transformation from C-shape to M-shape vortices for larger ${A{\kern-4pt}R}$ plates. Downstream of this novel vortex-splitting transformation, two fundamental processes contribute to the formation of hairpin vortices. The above comprehensive understanding of complete vortex evolution routine provides valuable insights into the tip effects on the formation of three-dimensional flows over low-${A{\kern-4pt}R}$ plates.

This paper examines the rigid body motion of a spheroid sedimenting in a Newtonian fluid with a spatially varying viscosity field. The fluid is at zero Reynolds number, and the viscosity varies linearly in space in an arbitrary direction with respect to the external force. First, we obtain the correction to the spheroid's rigid body motion in the limit of small viscosity gradients, using a perturbation expansion combined with the reciprocal theorem. Next, we determine the general form of the particle's mobility tensor relating its rigid body motion to an external force and torque. The viscosity gradient does not alter the force/translation and torque/rotation relationships, but introduces new force/rotation and torque/translation couplings that are determined for a wide range of particle aspect ratios. Finally, we discuss results for the spheroid's rotation and centre-of-mass trajectory during sedimentation. A steady orientation arises at long time whose value depends on the viscosity gradient direction and particle shape. These results are significantly different than when no viscosity gradient is present, where the particle stays at its initial orientation for all times. We summarize the observations for prolate and oblate spheroids for different viscosity gradient directions and provide plots for the orientation and centre-of-mass trajectory versus time. We also provide guidelines to extend the analysis when the viscosity gradient exhibits a more complicated spatial behaviour.

The cubic interactions in a discrete system of four weakly nonlinear waves propagating in a conservative dispersive medium are studied. By reducing the problem to a single ordinary differential equation governing the motion of a classical particle in a quartic potential, the complete explicit branches of solutions are presented, either steady, periodic, breather or pump, thus recovering or generalizing some already published results in hydrodynamics, nonlinear optics and plasma physics, and presenting some new ones. Various stability criteria are also formulated for steady equilibria. Theory is applied to deep-water gravity waves for which models of isolated quartets are described, including bidirectional standing waves and quadri-directional travelling waves, steady or not, resonant or not.

Equations describing mushy systems, in which solid and liquid are described as a single continuum, have been extensively studied. Most studies of mushy layers have assumed them to be ‘ideal’, such that the liquid and solid were in perfect thermodynamic equilibrium. It has become possible to simulate flows of passive porous media at the pore scale, where liquid and solid are treated as separate continua. In this contribution, we study the simplest possible mushy layers at the pore scale, modelling a single straight cylindrical pore surrounded by a cylindrical annulus representing the solid matrix. Heat and solute can be exchanged at the solid–liquid boundary. We consider harmonic temperature and concentration perturbations and examine their transport rates due to advection and diffusion and the melting and solidification driven by this transport. We compare the results of this numerical model with those of a one-dimensional ideal mushy layer and with analytical solutions valid for ideal mushy layers for small temperature variations. We demonstrate that for small values of an appropriately defined Péclet number, the results of the pore-scale model agree well with ideal mushy layer theory for both transport rates and melting. As this Péclet number increases, the temperature and concentration profiles with radius within the pore differ significantly from constant, and the behaviour of the pore-scale model differs significantly from that of an ideal mushy layer. Some effects of mechanical dispersion arise naturally in our pore-scale model and are shown to be important at high Péclet number.

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

The aeroacoustic feedback loops in high-speed circular jets that impinge on a large flat plate are investigated via acoustic measurements and schlieren visualizations. In the present experiments, the nozzle pressure ratio ranges from 1.39 to 2.20, the corresponding ideally expanded jet Mach number $M_j$ is from 0.70 to 1.12 and the nozzle-to-plate distance ($H$) is from 4.0$D$ to 6.0$D$, where $D$ is the nozzle exit diameter. The results of acoustic measurements show that the strongest tones are generated in a limited frequency band. The empirical dispersion relations obtained from the fluctuating greyscales along the jet centreline of time-resolved schlieren images have good agreement with the dispersion relations from the vortex-sheet model. The coherent flow structures at tonal frequencies are extracted by spectral proper orthogonal decomposition and are analysed in detail. For the $M_j<0.82$ jets, the upstream-propagating guided jet mode is progressively confined to the potential core of jets with increasing tonal frequency, which provides the first direct experimental support for theoretical results. The evolution in the structures of acoustic resonance loops is studied along a single frequency stage of axisymmetric impinging tones. When the acoustic resonance between the upstream- and downstream-propagating guided jet modes is formed at tonal frequencies, the impinging tones are intenser. Slightly underexpanded impinging jets can simultaneously produce impingement tones and screech tones. Shock-cell structures have modulatory effects on the downstream-propagating Kelvin–Helmholtz wavepacket and the upstream- and downstream-propagating guided jet modes. Due to the interaction between the flow structures at the frequencies of impinging and screech tones, tones of axisymmetric modes can be produced outside the frequency ranges in which the axisymmetric upstream-propagating guided jet modes are supported by jets.

Flow physics vary in different regimes across the full Mach number range, with our knowledge being particularly poor about the hypersonic regime. An Eulerian realization of the particles on demand method, a kinetic model formulated in the comoving reference frame, is proposed to simulate hypersonic compressible flows. The present model allows for flux evaluation in different reference frames, in this case rescaled and shifted by local macroscopic quantities, i.e. fluid speed and temperature. The resulting system of coupled hyperbolic equations is discretized in physical space with a finite volume scheme ensuring exact conservation properties. Regularization via Grad expansion is introduced to implement distribution function and flux transformation between different reference frames. It is shown that the proposed method possesses Galilean invariance at a Mach number up to $100$. Different benchmarks including both inviscid and viscous flows are reproduced with the Mach number up to $198$ and pressure ratio up to $10^5$. Finally, the new model is demonstrated to be capable of simulating hypersonic reactive flows, including one-dimensional and two-dimensional detonations. The developed methodology opens up possibilities for the simulation of the full range of compressible flows, without or with chemical reactions, from the subsonic to hypersonic regimes, leading to enhanced understanding of flow behaviours across the full Mach number range.

The familiar process of bubbles generated via breaking waves in the ocean is foundational to many natural and industrial applications. In this process, large pockets of entrained gas are successively fragmented by the ambient turbulence into smaller and smaller bubbles. The key question is how long it takes for the bubbles to reach terminal sizes for a given system. Despite decades of effort, the reported breakup time from multiple experiments differs significantly. Here, to reconcile those results, rather than focusing on one scale, we measure multiple time scales associated with the process through a unique experiment that resolves bubbles’ local deformation and curvature. The results emphasize that the scale separation among various time scales is controlled by the Weber number, similar to how the Reynolds number determines the scale separation in single-phase turbulence, but shows a distinct transition at a critical Weber number.

Large eddy simulation of flow past a cricket ball with its seam at $30^\circ$ to the free stream is carried out for $5 \times 10^4 \le Re \le 4.5 \times 10^5$. Three regimes of flow are identified on the basis of the time-averaged swing force coefficient ($\bar {C}_Z$) – no swing (NS), conventional swing (CS, $\bar {C}_Z>0$) and reverse swing (RS, $\bar {C}_Z<0$). The effect of seam on the boundary layer is investigated. Contrary to the popular belief, the boundary layer does not transition to a turbulent state in the initial stages of CS. The seam energizes the laminar boundary layer and delays its separation. The delay is significantly larger in a region near the poles, whose extent increases with an increase in $Re$ causing $\bar {C}_Z$ to increase. Here $\bar {C}_Z$ assumes a near constant value in the later stage of CS. The boundary layer transitions to a turbulent state via formation of a laminar separation bubble (LSB) in the equatorial region and directly, without a LSB, in the polar region. The extent of the LSB shrinks while the region of direct transition near the poles increases with an increase in $Re$. A LSB forms on the non-seam side of the ball in the RS regime. A secondary vortex is observed in the wake bubble. While it exists on the non-seam side for the entire range of $Re$ considered, the mixing in the flow introduced by the seam causes it to disappear beyond a certain $Re$ on the seam side. The pressure difference between the seam and non-seam sides sets up wing-tip-like vortices. Their polarity reverses with the switch from the CS to RS regime.

The impact of a liquid droplet with another droplet or onto a solid surface are important basic processes that occur in many applications such as agricultural sprays and inkjet printing, and in nature such as pathogens transport by raindrops. We investigated the head-on collision of unequal-size droplets of the same liquid on wetting surfaces using the direct numerical simulations technique at different size ratios. The unsteady Navier–Stokes equations are solved and the liquid–gas interface is tracked using the geometric volume-of-fluid method. The numerical model is validated by comparing simulation results of two extreme cases of droplets bouncing with the experimental data from previous studies and the agreement is quite accurate. The validated model is employed to simulate droplets bouncing at several size ratios at different Weber numbers and Ohnesorge number. Two distinct regimes are identified, namely, the inertial regime, where the restitution coefficient is a constant value close to 0.3, the viscous regime, where the restitution coefficient declines. To understand the bouncing behaviour, the velocity field is analysed and an energy budget calculation is performed. The distribution of the sessile droplet energy is found to be important and the sessile droplet surface energy is calculated by its deformation characteristics such as crater depth. Finally, a scaling analysis is performed to rationalize the insensitivity of the coefficient of restitution in the inertial regime, and its decline in the viscous regime, at large size ratios.

This study presents a noise-robust closed-loop control strategy for wake flows employing model predictive control. The proposed control framework involves the autonomous offline selection of hyperparameters, eliminating the need for user interaction. To this purpose, Bayesian optimization maximizes the control performance, adapting to external disturbances, plant model inaccuracies and actuation constraints. The noise robustness of the control is achieved through sensor data smoothing based on local polynomial regression. The plant model can be identified through either theoretical formulation or using existing data-driven techniques. In this work we leverage the latter approach, which requires minimal user intervention. The self-tuned control strategy is applied to the control of the wake of the fluidic pinball, with the plant model based solely on aerodynamic force measurements. The closed-loop actuation results in two distinct control mechanisms: boat tailing for drag reduction and stagnation point control for lift stabilization. The control strategy proves to be highly effective even in realistic noise scenarios, despite relying on a plant model based on a reduced number of sensors.