We present an exact solution for periodic travelling waves in two-dimensional, infinitely deep and constant vorticity flows, in the absence of the effects of gravity or surface tension. The shape of the free surface is the same as for Crapper’s celebrated capillary waves in an irrotational flow, but the flow beneath the wave, which is also explicit, is completely different. This confirms a conjecture made by Dyachenko & Hur (J. Fluid Mech., vol. 878, 2019b, pp. 502–521; Stud. Appl. Maths, vol. 142 (2), 2019c, pp. 162–189) and Hur & Vanden-Broeck (Eur. J. Mech. (B/Fluids), 2020, to appear), based on numerical and asymptotic evidence.

We study the impact between a plate and a drop of non-colloidal solid particles suspended in a Newtonian liquid, paying specific attention to the case when the particle volume fraction, $\unicode[STIX]{x1D719}$, is close to – or even exceeds – the critical volume fraction, $\unicode[STIX]{x1D719}_{c}$, at which the steady effective viscosity of the suspension diverges. We use a specific concentration protocol together with an accurate determination of $\unicode[STIX]{x1D719}$ for each drop, and we measure the deformation $\unicode[STIX]{x1D6FD}$ for different liquid viscosities, impact velocities and particle sizes. At low volume fractions, $\unicode[STIX]{x1D6FD}$ is found to follow closely an effective Newtonian behaviour, which we determine by documenting the low-deformation limit for a highly viscous Newtonian drop and characterizing the effective shear viscosity of our suspensions. By contrast, whereas the effective Newtonian approach predicts that $\unicode[STIX]{x1D6FD}$ vanishes at $\unicode[STIX]{x1D719}_{c}$, a finite deformation is observed for $\unicode[STIX]{x1D719}>\unicode[STIX]{x1D719}_{c}$. This finite deformation remains controlled by the suspending liquid viscosity and increases with increasing particle size, which suggests that the dilatancy of the particle phase is a key factor in the dissipation process close to and above $\unicode[STIX]{x1D719}_{c}$.

We derive weakly dispersive Boussinesq equations using a $\unicode[STIX]{x1D70E}$ coordinate for the vertical direction, employing a series expansion in powers of $\unicode[STIX]{x1D70E}$. We restrict attention initially to the case of constant still-water depth $h$ in order to simplify subsequent analysis, and consider equations based on expansions about the bottom elevation $\unicode[STIX]{x1D70E}=0$, and then about a reference elevation $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}}$ in order to improve linear dispersion properties. We use a perturbation analysis, suggested recently by Madsen & Fuhrman (J. Fluid Mech., vol. 889, 2020, A38), to show that the resulting models are not subject to the trough instability studied there. A similar analysis is performed to develop a model for interfacial waves in a two-layer fluid, with comparable results. We argue, by extension, that a necessary condition for eliminating trough instabilities is that the model’s nonlinear dispersive terms should not contain still-water depth $h$ and surface displacement $\unicode[STIX]{x1D702}$ separately.

The complex hierarchical texture covering the lotus leaf is at the source of two of its extraordinary properties. While its water-repellent properties are now emblematic, the lotus is much less known for its extreme slipperiness. And for good reason: until the recent work of Martouzet et al. (J. Fluid Mech., vol. 892, 2020, R2), the effect of slippage on drop impact dynamics had never been demonstrated. This remarkable study unveils a complex interplay between wetting and friction, with counter-intuitive consequences. Hierarchical structures, which minimize the contact between the substrate and the droplets, are less efficient at repelling viscous liquids than simpler systems, because of the slip! A clever and original approach, based on a scaling analysis of the spreading time, is used to disentangle the different physical phenomena occurring during drop impact. This is an important step towards a better understanding of the crucial problem of drop impact dynamics on both wetting and non-wetting substrates.

The physics associated with various cavitation regimes about a hydrofoil is investigated in a variable-pressure water tunnel using high-speed photography and synchronised force measurements. Experiments were conducted on a relatively stiff stainless steel hydrofoil at a chord-based Reynolds number, $Re=0.8\times 10^{6}$ for cavitation numbers, $\unicode[STIX]{x1D70E}$, ranging from 0.2 to 1.2, with the hydrofoil experiencing sheet, cloud and supercavitation regimes. The NACA0009 model of tapered planform was vertically mounted in a cantilevered configuration to a six-component force balance at an incidence, $\unicode[STIX]{x1D6FC}$, of $6^{\circ }$ to the oncoming flow. Tip deformations and cavitation behaviour were recorded with synchronised force measurements utilising two high-speed cameras mounted underneath and to the side of the test section. Break-up and shedding of an attached cavity was found to be due to either interfacial instabilities, re-entrant jet formation, shockwave propagation or a complex coupled mechanism, depending on $\unicode[STIX]{x1D70E}$. Three primary shedding modes are identified. The Type IIa and IIb re-entrant jet-driven oscillations exhibit a non-linear dependence on $\unicode[STIX]{x1D70E}$, decreasing in frequency with decreasing $\unicode[STIX]{x1D70E}$ due to growth in the cavity length, and occur at higher $\unicode[STIX]{x1D70E}$ values (Type IIa: 0.4–1.0; Type IIb: 0.7–0.9). Shockwave-driven Type I shedding occurs for lower $\unicode[STIX]{x1D70E}$ values (0.3–0.6) with the oscillation frequency being practically independent of $\unicode[STIX]{x1D70E}$. The Type IIa oscillations locked in to the first sub-harmonic of the hydrofoil’s first bending mode in water which has been modulated due to the reduced added mass of the vapour cavity. Supplementary movies are available with the online version of the paper.

Three amplification mechanisms present in turbulent jets, namely lift-up, Kelvin–Helmholtz and Orr, are characterized via global resolvent analysis and spectral proper orthogonal decomposition (SPOD) over a range of Mach numbers. The lift-up mechanism was recently identified in turbulent jets via local analysis by Nogueira et al. (J. Fluid Mech., vol. 873, 2019, pp. 211–237) at low Strouhal number ($St$) and non-zero azimuthal wavenumbers ($m$). In these limits, a global SPOD analysis of data from high-fidelity simulations reveals streamwise vortices and streaks similar to those found in turbulent wall-bounded flows. These structures are in qualitative agreement with the global resolvent analysis, which shows that they are a response to upstream forcing of streamwise vorticity near the nozzle exit. Analysis of mode shapes, component-wise amplitudes and sensitivity analysis distinguishes the three mechanisms and the regions of frequency–wavenumber space where each dominates, finding lift-up to be dominant as $St/m\rightarrow 0$. Finally, SPOD and resolvent analyses of localized regions show that the lift-up mechanism is present throughout the jet, with a dominant azimuthal wavenumber inversely proportional to streamwise distance from the nozzle, with streaks of azimuthal wavenumber exceeding five near the nozzle, and wavenumbers one and two most energetic far downstream of the potential core.

A gravity-driven water film falling down an ice sheet is considered within the framework of a long-wave approximation. The integral-boundary-layer method, modified with the account of the phase transition, is adopted to describe the evolution of both the free surface of a water film and the interface between the ice and water. A set of governing equations consisting of five coupled nonlinear partial differential equations is established. The linear instability analysis of the uniform base flow is performed, and the result is in good agreement with the Orr–Sommerfeld analysis of the linearized Navier–Stokes equations. The phase transition at the interface between the ice and water plays a role in stabilizing the system linearly with long-wavelength perturbations. The nonlinear solutions of the steady travelling waves are constructed numerically. The phase transition tends to suppress the dispersion of the interfacial wave. Comparisons to direct numerical simulation of the Navier–Stokes equations, which are performed with an extended marker and cell method, show a remarkable agreement. The integral-boundary-layer method captures the water film thickness and the topography of the ice sheet satisfactorily. The phase transition is observed to enhance the backflow phenomenon in the capillary region of the solitary-like interfacial wave.

This numerical study focuses on the coherent structures and bypass transition mechanism of the Stokes boundary layer in the intermittently turbulent regime. In particular, the initial disturbance is produced by a temporary roughness element that is removed immediately after triggering a two-dimensional vortex tube under an inflection-point instability. The present study reveals a complete scenario of self-induced motion of a vortex tube after rollup from the boundary layer. The trajectory of the vortex tube is reasonably described based on the Helmholtz point-vortex equation. The three-dimensional transition of the vortex tube is attributed to the Crow instability, which leads to a sinusoidal disturbance that eventually evolves into a ring-like structure, especially for the weaker vortex. Further investigation demonstrates that three-dimensional or quasi-three-dimensional vortex perturbations in the free stream play a critical role in the boundary layer transition through a bypass mechanism, which is featured by the non-modal and explosive transient growth of the subsequent boundary layer instabilities. This transition scenario is found to be analogous to the oblique transition in the steady boundary layer, both of which are characterised by the formation of streaks, rollup of hairpin-like vortices and burst into turbulent spots. In addition, the streamwise propagation of turbulent spots is discussed in detail. To shed more light on the nature of the intermittently turbulent Stokes boundary layer, a conceptual model is proposed for the periodically self-sustaining mechanism of the turbulent spots based on the present numerical results and experimental evidence reported in the literature.

The response to librational forcing of a cube in rapid rotation about a diagonal axis is explored. In this orientation, the faces of the cube are all oblique to the rotation axis. The system supports inertial waves, which predominantly comprise beams emitted from the edges and vertices of the cube. Which ones emit and the resulting complicated pattern of three-dimensional reflections and subsequent focusing depend on the libration frequency. Direct numerical simulations of the Navier–Stokes flows with no-slip boundary conditions at low Ekman number ($\text{E}=10^{-7}$) and small libration amplitude ($\unicode[STIX]{x1D716}=10^{-7}$) exhibit complicated spatio-temporal structure that is remarkably well described by considerations of the inviscid reflections of wavebeams over the whole range of libration frequencies from zero to twice the mean rotation rate of the cube.

This paper considers the relationship between nonlinearly interacting helical flow disturbances and flame area response in a swirling premixed flame. The present study was performed to determine whether there are nonlinear mechanisms through which helical modes ($m_{u}\neq 0$) can lead to non-zero unsteady heat release rate oscillations. The results show that for single frequency content (at $\unicode[STIX]{x1D714}_{0}$), helical modes excite unsteady heat release rate response of $O(\unicode[STIX]{x1D716}^{3})$ and that two-frequency excitation (e.g. at $\unicode[STIX]{x1D714}_{0}$ and $2\unicode[STIX]{x1D714}_{0}$), leads to a response of $O(\unicode[STIX]{x1D716}^{2})$ at $\unicode[STIX]{x1D714}_{0}$. There are two mechanisms through which this can occur: First, helical flow disturbances can distort the time-averaged flame shape to have an azimuthal component that matches that of the incident disturbance, $\exp (im_{u}\unicode[STIX]{x1D703})$. Second, multiple helical modes can nonlinearly interact to cause axisymmetric unsteady flame wrinkling. The paper derives the various modal contributions in the incident velocity disturbance that satisfy these criteria. These results suggest that it is only the $m_{u}=0$ mode which controls the linear dynamics (e.g. instability inception conditions) of these flames (where $\unicode[STIX]{x1D716}\ll 1$), but that their nonlinear dynamics is also controlled by the $m_{u}\neq 0$ helical modes.

An exact set of equations describing deep-water irrotational surface gravity waves, originally proposed by Balk (Phys. Fluids, vol. 8 (2), 1996, pp. 416–420), and studied in the case of standing waves by Longuet-Higgins (J. Fluid Mech., vol. 423, 2000, pp. 275–291) and Longuet-Higgins (Proc. R. Soc. Lond. A, vol. 457 (2006), 2001, pp. 495–510), are analytically examined and put in a form more suitable for practical applications. The utility of this approach is its simplicity. The Lagrangian is a low-order polynomial in the Fourier coefficients, leading to equations of motion that are correspondingly of low degree. The structure of these equations is examined, and the existence of solutions is considered. To gain intuition about the system of equations, a truncated model is first examined. Linear stability analysis is performed, and the evolution of the fully nonlinear system is discussed. The theory is then applied to fully resolved permanent progressive deep-water waves and a simple method for finding the eigenvalues and eigenvectors of the normal modes of this system is presented. Potential applications of these results are then discussed.

A model is presented for the bouncing dynamics of a fluid-immersed sphere impacting normally a textured wall with micropillars. By taking into account the hydrodynamic and contact interactions between the smooth sphere and the textured wall, the complete motion of the sphere is recovered when approaching, colliding with and bouncing off the wall. We demonstrate that the critical Stokes number for the bouncing transition, $St_{c}$, is the sum of two contributions corresponding to dissipation prior to and during the collision, both contributions being critically influenced by the geometrical parameters of the model roughness. The experimental data obtained from interferometric measurements are found to be in agreement with the theoretical predictions. In the bouncing regime, the coefficient of restitution is also derived analytically and shows a linear evolution with the Stokes number, $St$, just above the bouncing transition, in agreement with the experimental data obtained very close to $St_{c}$.

We present findings on scalar flashes, turbulent spots and transition statistics in a direct numerical simulation of a 500 radius-long pipe flow with a radial-mode inlet disturbance. Transitional spots are found to contain two different types of eddies. The wall region consists primarily of ‘reverse hairpin vortices’. This unusual structure is related to the high-speed streaks arising from the prescribed inlet perturbation. The core region is populated by the normally observed ‘forward’ hairpin vortices. Number density of the reverse hairpins is quantified indirectly and conservatively by measuring the number density of negative skin-friction patches. During the late stages of transition, second-order statistics such as Reynolds stresses and the rate of dissipation of turbulent kinetic energy exhibit substantial overshoot. This is associated with mid-to-high frequency content in the energy spectra that exceeds the corresponding levels in fully developed turbulence. Flow visualizations reveal bursts of small-scale vortex motions, including the reverse hairpins, that probably account for the enhanced mid-to-high frequency spectral content. A passive scalar is injected at the centreline of the inlet plane, mimicking laboratory injection of dye through a needle, to investigate the mysterious phenomenon of turbulent scalar patches residing in fully developed turbulent pipe flow. At several hundred radii downstream of transition where the velocity field is genuine fully developed turbulence, the scalar patches retain persistent memory of events far upstream. Comparing the present flow with a similar pipe flow disturbed by a significantly different inlet condition suggests that the foregoing observations are insensitive to the form of the disturbances.

This work studies the effects of skin-friction drag reduction in a turbulent flow over a curved wall, with a view to understanding the relationship between the reduction of friction and changes to the total aerodynamic drag. Direct numerical simulations are carried out for an incompressible turbulent flow in a channel where one wall has a small bump; two bump geometries are considered, that produce mildly separated and attached flows. Friction drag reduction is achieved by applying streamwise-travelling waves of spanwise velocity (StTW). The local friction reduction produced by the StTW is found to vary along the curved wall, leading to a global friction reduction that, for the cases studied, is up to 10 % larger than that obtained in the plane wall case. Moreover, the modified skin friction induces non-negligible changes of pressure drag, which is favourably affected by StTW and globally reduces by up to 10 %. The net power saving, accounting for the power required to create the StTW, is positive and, for the cases studied, is one half larger than the net saving of the planar case. The study suggests that reducing friction at the surface of a body of complex shape induces further effects, a simplistic evaluation of which might lead to underestimating the total drag reduction.

The turbulent boundary layer developing under a turbulence-laden free stream is numerically investigated using the temporal boundary layer framework. This study focuses on the interaction between the fully turbulent boundary layer and decaying free-stream turbulence. Previous experiments and simulations of this physical problem have considered a spatially evolving boundary layer beset by free-stream turbulence. The state of the boundary layer at any given downstream position in fact reflects the accumulated history of the co-evolution of boundary layer and free-stream turbulence. The central aim of the present work is to isolate the effect of local free-stream disturbances existing at the same time as the ‘downstream’ boundary layer. The temporal framework used here helps expose when and how disturbances directly above the boundary layer actively impart change upon it. The bulk of our simulations were completed by seeding the free stream above boundary layers that were ‘pre-grown’ to a desired thickness with homogeneous isotropic turbulence from a precursor simulation. Moreover, this strategy allowed us to test various combinations of the turbulence intensity and large-eddy length scale of the free-stream turbulence with respect to the corresponding scales of the boundary layer. The relative large-eddy turnover time scale between the free-stream turbulence and the boundary layer emerges as an important parameter in predicting if the free-stream turbulence and boundary layer interaction will be ‘strong’ or ‘weak’ before the free-stream turbulence eventually fades to a negligible level. If the large-eddy turnover time scale of the free-stream turbulence is much smaller than that of the boundary layer, the interaction will be ‘weak’, as the free-stream disturbances will markedly decay before the boundary layer is able be altered significantly as a result of the free-stream disturbances. For a ‘strong’ interaction, the injected free-stream turbulence causes increased spreading of the boundary layer away from the wall, permitting large incursions of free-stream fluid deep within it.

We simulated the sedimentations of two unequal spheres with different densities in a square tube for Galileo numbers (Ga) from 5 to 25, resulting in a Reynolds number range of $0.8\leqslant Re_{T}\leqslant 17.3$ based on the terminal settling velocity. The sedimentation of spheres with different densities is dynamically more complex than that of identical spheres. At high Ga the spheres oscillate in the centreline plane of the tube, where they are initially released from rest. By contrast, the spheres move to the diagonal or reverse–diagonal plane of the tube at low Ga, reaching a steady or periodic state depending on the density difference between them. A phase diagram illustrates the transitions between different sedimentation behaviours depending on Ga and the density difference. A possible mechanism for these behaviours is also presented. Furthermore, we compare two-dimensional (2-D) and three-dimensional computations for our system to attain a better understanding of the hydrodynamic interactions between two unequal spheres at low but finite Reynolds number. Comparing relative trajectories, periods of oscillation and flow features shows that 2-D circular cylinders oscillate much more strongly and frequently than spheres under the same flow conditions. In particular, spheres do not have the discontinuity in period that arises in the 2-D case from the change in rotation sign of a heavy particle.

We present lattice Boltzmann pore-scale numerical simulations of solute transport and reaction in porous electrodes at a high Schmidt number, $Sc=10^{2}$. The three-dimensional geometry of real materials is reconstructed via X-ray computed tomography. We apply a volume-averaging upscaling procedure to characterise the microstructural terms contributing to the homogenised description of the macroscopic advection–reaction–dispersion equation. We firstly focus our analysis on its asymptotic solution, while varying the rate of reaction. The results confirm the presence of two working states of the electrodes: a reaction-limited regime, governed by advective transport, and a mass-transfer-limited regime, where dispersive mechanisms play a pivotal role. For all materials, these regimes depend on a single parameter, the product of the Damköhler number and a microstructural aspect ratio. The macroscopic dispersion is determined by the spatial correlation between solute concentration and flow velocity at the pore scale. This mechanism sustains reaction in the mass-transfer-limited regime due to the spatial rearrangement of the solute transport from low-velocity to high-velocity pores. We then compare the results of pre-asymptotic transport with a macroscopic model based on effective dispersion parameters. Interestingly, the model correctly represents the transport at short characteristic times. At longer times, high reaction rates mitigate the mechanisms of heterogeneous solute transport. In the mass-transfer-limited regime, the significant yet homogeneous dispersion can thus be modelled via an effective dispersion. Finally, we formulate guidelines for the design of porous electrodes based on the microstructural aspect ratio.

The Holmboe wave instability is one of the classic examples of a stratified shear instability, usually explained as the result of a resonance between a gravity wave and a vorticity wave. Historically, it has been studied by linear stability analyses at infinite Reynolds number, $Re$, and by direct numerical simulations at relatively low $Re$ in the regions known to be unstable from the inviscid linear stability results. In this paper, we perform linear stability analyses of the classical ‘Hazel model’ of a stratified shear layer (where the background velocity and density distributions are assumed to take the functional form of hyperbolic tangents with different characteristic vertical scales) over a range of different parameters at finite $Re$, finding new unstable regions of parameter space. In particular, we find instability when the Richardson number is everywhere greater than $1/4$, where the flow would be stable at infinite $Re$ by the Miles–Howard theorem. We find unstable modes with no critical layer, and show that, despite the necessity of viscosity for the new instability, the growth rate relative to diffusion of the background profile is maximised at large $Re$. We use these results to shed new light on the wave-resonance and over-reflection interpretations of stratified shear instability. We argue for a definition of Holmboe instability as being characterised by propagating vortices above or below the shear layer, as opposed to any reference to sharp density interfaces.

Ramp–cliff patterns visible in scalar turbulent time series have long been suspected to enhance the fine-scale intermittency of scalar fluctuations compared to longitudinal velocity fluctuations. Here, we use the wavelet transform modulus maxima method to perform a multifractal analysis of air temperature time series collected at a pine forest canopy top for different atmospheric stability regimes. We show that the multifractal spectra exhibit a phase transition as the signature of the presence of strong singularities corresponding to sharp temperature drops (respectively jumps) bordering the so-called ramp (respectively inverted ramp) cliff patterns commonly observed in unstable (respectively stable) atmospheric conditions and previously suspected to contaminate and possibly enhance the internal intermittency of (scalar) temperature fluctuations. Under unstable (respectively stable) atmospheric conditions, these ‘cliff’ singularities are indeed found to be hierarchically distributed on a ‘Cantor-like’ set surrounded by singularities of weaker strength typical of intermittent temperature fluctuations observed in homogeneous and isotropic turbulence. Under near-neutral conditions, no such a phase transition is observed in the temperature multifractal spectra, which is a strong indication that the statistical contribution of the ‘cliffs’ is not important enough to account for the stronger intermittency of temperature fluctuations when compared to corresponding longitudinal velocity fluctuations.

We use direct numerical simulation data to study interscale and interspace energy exchanges in the near field of a turbulent wake of a square prism in terms of a Kármán–Howarth–Monin–Hill (KHMH) equation written for a triple decomposition of the velocity field which takes into account the presence of quasi-periodic vortex shedding coherent structures. Concentrating attention on the plane of the mean flow and on the geometric centreline, we calculate orientation averages of every term in the KHMH equation. The near field considered here ranges between two and eight times the width $d$ of the square prism and is very inhomogeneous and out of equilibrium so that non-stationarity and inhomogeneity contributions to the KHMH balance are dominant. The mean flow produces kinetic energy which feeds the vortex shedding coherent structures. In turn, these coherent structures transfer their energy to the stochastic turbulent fluctuations over all length scales $r$ from the Taylor length $\unicode[STIX]{x1D706}$ to $d$ and dominate spatial turbulent transport of small-scale two-point stochastic turbulent fluctuations. The orientation-averaged nonlinear interscale transfer rate $\unicode[STIX]{x1D6F1}^{a}$ which was found to be approximately independent of $r$ by Alves Portela et al. (J. Fluid Mech., vol. 825, 2017, pp. 315–352) in the range $\unicode[STIX]{x1D706}\leqslant r\leqslant 0.3d$ at a distance $x_{1}=2d$ from the square prism requires an interscale transfer contribution of coherent structures for this approximate constancy. However, the near constancy of $\unicode[STIX]{x1D6F1}^{a}$ in the range $\unicode[STIX]{x1D706}\leqslant r\leqslant d$ at $x_{1}=8d$ which was also found by Alves Portela et al. (2017) is mostly attributable to stochastic fluctuations. Even so, the proximity of $-\unicode[STIX]{x1D6F1}^{a}$ to the turbulence dissipation rate $\unicode[STIX]{x1D700}$ in the range $\unicode[STIX]{x1D706}\leqslant r\leqslant d$ at $x_{1}=8d$ does require interscale transfer contributions of the coherent structures. Spatial inhomogeneity also makes a direct and distinct contribution to $\unicode[STIX]{x1D6F1}^{a}$, and the constancy of $-\unicode[STIX]{x1D6F1}^{a}/\unicode[STIX]{x1D700}$ close to $1$ would not have been possible without it either in this near-field flow. Finally, the pressure-velocity term is also an important contributor to the KHMH balance in this near field, particularly at scales $r$ larger than approximately $0.4d$, and appears to correlate with the purely stochastic nonlinear interscale transfer rate when the orientation average is lifted.