In this article, we study numerically the dispersion of colloids in a two-dimensional cellular flow in the presence of an imposed mean salt gradient. Owing to the additional scalar, the colloids do not follow exactly the Eulerian flow field, but have a (small) extra-velocity proportional to the salt gradient, $\boldsymbol {v}_{dp}=\alpha \boldsymbol {\nabla } S$, where $\alpha$ is the phoretic constant and $S$ the salt concentration. We study the demixing of an homogenous distribution of colloids and how their long-term mean velocity $\boldsymbol {V_m}$ and effective diffusivity $D_{eff}$ are influenced by the phoretic drift. We observe two regimes of colloids dynamics depending on a blockage criterion $R=\alpha G L/\sqrt {4 D_cD_s}$, where $G$ is the mean salt gradient amplitude, $L$ the length scale of the flow and $D_c$ and $D_s$ the molecular diffusivities of colloids and salt. When $R<1$, the mean velocity is strongly enhanced with $V_m \propto \alpha G \sqrt {Pe_s}$, ${Pe}_s$ being the salt Péclet number. When $R > 1$, the compressibility effect due to the phoretic drift is so strong that a depletion of colloids occurs along the separatrices inhibiting cell-to-cell transport.

In this paper we study the generation of Tollmien–Schlichting waves initiated by vibrations of a wall where the wall is coated with a thin liquid film in a transonic flow regime. Motion of fluids are described by the two-dimensional Navier–Stokes equations assuming the Reynolds number is large. To find asymptotic solutions of the transonic boundary layer, we conduct an inspection analysis on the affine transformations of the triple-deck model for a subsonic flow and the unsteady full potential equations, with the intention of obtaining the order quantity of the free-stream Mach number in the transonic flow. We construct a modified triple-deck model for the transonic flow by considering the scalings of the perturbations that lead to the viscous–inviscid interaction problem for the flow in a subsonic regime. In particular, we are interested in the region where the subsonic scalings become invalid as the flow approaches transonic regime. We assume the wall oscillates in the vertical direction to the oncoming flow and these vibrations are periodic in time. We outline the process where the flow in the boundary layer converts the wall vibration perturbations into the instability modes which are measured by the receptivity coefficient. The viscous–inviscid interaction problem describes the stability of the boundary layer on the lower branch of the neutral curve. We show that the governing equations for the air viscous sublayer and the film flow are quasi-steady. The equation describing the inviscid layer of the airflow is unsteady and its referred to as the unsteady Kàrmàn–Guderley equation. The influence of the film surface tension is expressed through normal shear stress condition at the interface. We present an analytic formula for the amplitude of the Tollmien–Schlichting waves that are formed in the boundary layer. We analysed our model with different values of surface tension, initial film thickness and Kàrmàn–Guderley parameter. Depending on the value of these parameters, the initial amplitude of the instability waves may grow or decay.

The wind energy industry relies on computationally efficient engineering-type models to design wind farms. Typically these models do not account for the effect of atmospheric stratification in either the boundary layer or the free atmosphere. This study proposes a new analytical model for fully developed wind-turbine arrays in conventionally neutral atmospheric boundary layers frequently encountered in nature. The model captures the effect of the free-atmosphere stratification, Coriolis force, wind farm layout and turbine operating condition on the wind farm performance. The model is developed based on the physical insight derived from large-eddy simulations. We demonstrate that the geostrophic drag law (GDL) for flow over flat terrain can be extended to flow over fully developed wind farm arrays. The presence of a vast wind farm significantly increases the wind farm friction velocity compared with flow over flat terrain, which is modelled by updated coefficients in the GDL. The developed model reliably captures the vertical wind speed profile inside the wind farm. Furthermore, the power production trends observed in simulations are reliably reproduced. The wind farm performance, normalized by the geostrophic wind speed, decreases as the free-atmosphere thermal stability increases or the Coriolis force decreases. In addition, we find that the optimal wind farm performance is obtained at a lower thrust coefficient than the Betz limit, which indicates that optimal operating conditions for turbines in a wind farm are different than for a single turbine.

We utilize resolvent and weakly nonlinear analyses in combination with direct numerical simulations (DNS) to identify mechanisms for oblique transition in a Mach $5$ hypersonic flow over an adiabatic slender double wedge. Even though the laminar separated flow is globally stable, resolvent analysis demonstrates significant amplification of unsteady external disturbances to the linearized flow equations. These disturbances are introduced upstream of the separation zone and they lead to the appearance of oblique waves further downstream. We demonstrate that the large amplification of oblique waves arises from the growth of fluctuation shear stress due to streamline curvature of the laminar base flow in the separated shear layer. This is in contrast to the attached boundary layers, where no such mechanism exists. We also use a weakly nonlinear analysis to show that the resolvent operator associated with linearization around the laminar base flow governs the evolution of steady reattachment streaks that arise from quadratic interactions of unsteady oblique waves. These quadratic interactions generate vortical excitations in the reattaching shear layer which lead to the formation of streaks in the recirculation zone and their subsequent amplification, breakdown and transition to turbulence downstream. Our analysis of the energy budget shows that deceleration of the base flow near reattachment is primarily responsible for amplification of steady streaks. Finally, we employ DNS to examine latter stages of transition to turbulence and demonstrate the predictive power of a weakly nonlinear input–output framework in uncovering triggering mechanisms for oblique transition in separated high-speed boundary layer flows.

Fast agglomeration by emulsion binders to capture fine, hydrophobic particles has been developed in the past few years as an alternative to froth flotation by small air bubbles. This new method consists of mixing a particle suspension and saltwater-filled droplets covered with semi-permeable oil layers. These droplets expand due to an osmotic flux of water caused by the presence of salt inside the droplets. To better understand the physics underlying this novel particle capture method, we investigate binary interactions between droplets and particles. The current work examines the dynamics of a rigid spherical particle and a semi-permeable spherical drop that expands due to osmosis in an external, pure-extensional flow field. The droplet is governed by an expansion-diffusion problem, which is coupled to the set of dynamical equations governing the relative particle trajectory. By performing multiple trajectory simulations, we calculate transient collision efficiencies, which can be used to determine the collision kernel for population dynamics. We also use these simulations to better understand the evolution of the microstructure by determining the transient behaviour of the pair distribution function. Our results indicate that the presence of drop expansion increases the collision efficiency of the system, especially for very small particles, which are the most difficult to capture by froth flotation. Moreover, although the presence of slow salt diffusion inside the drops can mitigate this improvement, the contribution of expansion to the collision efficiency may still be considerable, even in the absence of hydrophobic or other attractive forces.

Low-order thermoacoustic models are qualitatively correct, but typically, they are quantitatively inaccurate. We propose a time-domain bias-aware method to make qualitatively low-order models quantitatively (more) accurate. First, we develop a Bayesian ensemble data assimilation method for a low-order model to self-adapt and self-correct any time that reference data become available. Second, we apply the methodology to infer the thermoacoustic states and heat-release parameters on the fly without storing data (real time). We perform twin experiments using synthetic acoustic pressure measurements to analyse the performance of data assimilation in all nonlinear thermoacoustic regimes, from limit cycles to chaos, and interpret the results physically. Third, we propose practical rules for thermoacoustic data assimilation. An increase, reject, inflate strategy is proposed to deal with the rich nonlinear behaviour; and physical time scales for assimilation are proposed in non-chaotic regimes (with the Nyquist–Shannon criterion) and in chaotic regimes (with the Lyapunov time). Fourth, we perform data assimilation using data from a higher-fidelity model. We introduce an echo state network to estimate in real time the forecast bias, which is the model error of the low-fidelity model. We show that: (i) the correct acoustic pressure, parameters, and model bias can be inferred accurately; (ii) the learning is robust as it can tackle large uncertainties in the observations (up to 50 % of the mean values); (iii) the uncertainty of the prediction and parameters is naturally part of the output; and (iv) both the time-accurate solution and statistics can be inferred successfully. Data assimilation opens up new possibility for real-time prediction of thermoacoustics by combining physical knowledge and experimental data synergistically.

In this paper, a concept of a floating elastic wave energy converter consisting of a disk-shaped elastic plate is proposed. The floating plate is moored to the seabed through a series of power take-off (PTO) units. A theoretical model based on the linear potential flow theory and eigenfunction matching method is developed to study the hydroelastic characteristics and evaluate wave power absorption of the device. The PTO system is simulated as a discrete PTO, and moreover, it is also modelled as a continuum PTO to represent the case when the PTO system is composed of a large number of PTO units. The continuum PTO approximation is tested against the discrete PTO simulation for accuracy. Two methods are proposed to predict the wave power absorption of the device. After running convergence analysis and model validation, the present model is employed to do a multiparameter impact analysis. The device adopting a continuum PTO system is found to capture wave power efficiently in an extensive range of wave frequencies. For the continuum PTO system, it is theoretically possible to adopt optimised PTO damper and stiffness/mass to guarantee the absorption of 100 % of the energy flux available in one circular component of the plane incident wave.

Plane Poiseuille flow of a Bingham fluid in a channel armed with a superhydrophobic (SH) lower wall is analysed via a semi-analytical model, accompanied by complementary direct numerical simulations (DNS). The SH surface represents a groovy structure with air trapped inside its cavities. Therefore, the fluid adjacent to the wall undergoes stick–slip conditions. The model is developed based on introducing infinitesimal wall-induced perturbations into the motion equations, followed by Fourier series expansions, and solving the resulting equations as a boundary value problem. The Navier slip law accounts for the slip at the liquid/air interface (assuming the Cassie state). The presented analysis is fairly comprehensive, covering the creeping and inertial regimes for thick channels (via the semi-analytical and DNS solutions). The main dimensionless numbers are the Reynolds ($Re$), Bingham ($Bi$) and slip ($b$) numbers, as well as the groove periodicity length ($\ell$) and the slip area fraction ($\varphi$). By increasing $Bi$, the perturbation and slip velocity fields grow. As $Re$ increases, the perturbation and slip velocity fields become asymmetric. For certain flow parameters, an unyielded plug zone may appear on the SH wall liquid/air interface, while its formation is accelerated by inertial effects. The results classify the regimes of creeping and inertial flows via predicting the onset of the unyielded plug zone formation at the SH wall.

The topology of a laminar separation bubble (LSB) on a sphere in the critical regime is investigated via experiments at five turbulent intensities: $T_u=0.06\,\%$, $0.42\,\%$, $0.71\,\%$, $1.00\,\%$ and $1.36\,\%$. The drag crisis occurs at a lower $Re$ and becomes gradual with increasing $T_u$. The flow is devoid of the LSB at the onset of the critical regime. It forms on a small part of the sphere and not at all azimuthal locations, early in the critical regime. The LSB forms at more azimuthal locations with increasing $Re$. This azimuthal expansion of the LSB is accompanied by intermittency for a small range of $Re$. Towards the end of the critical regime, an axisymmetric LSB forms on the sphere at all time instants. A model is proposed to estimate the azimuthal extent and distribution of the LSB from the mean force coefficients of a flow state. The model predicts that the LSB forms as multiple segments for a large part of the critical regime. During the spatial growth of the LSB with $Re$ in the critical regime, some of its fragments relocate to alternate locations. Moderate increase in $T_u$ ($0.42\,\% \leq T_u \leq 0.71\,\%$) leads to rich dynamics with several intermittent flow states. However, fewer intermittent states are observed beyond a certain $T_u$ (${\geq }1.00\,\%$).

The direct numerical simulation database from Shin et al. (J. Fluid Mech., vol. 823, 2017, pp. 1–25) is used to study three-dimensional vortical and very-large-scale coherent structures in a turbulent round jet at a Reynolds number of $7300$. In particular, horseshoe vortices and their role in the formation of very-large-scale coherent structures in the jet near and intermediate fields are assessed. The swirling strength criterion together with conditional averaging are used to visualize volumetric vortical structures. It is shown that, similar to wall-bounded turbulent flows, the turbulent jet is populated with symmetric and asymmetric horseshoe-like vortices, which induce high-momentum and low-momentum regions in the flow. However, unlike what is found for wall-bounded flows, inverse horseshoe-like vortices are common in the turbulent jet. They prevail in the shear region around the potential core in the jet near field and contribute to the mixing of the potential core in the jet. In the jet near field, groups of axially aligned horseshoe structures induce long streaky structures, which are periodic in the azimuthal and streamwise directions. In the jet intermediate field, very-large-scale motions (VLSMs) consisting of high-momentum regions, flanked on either side by low-momentum regions, are found to be associated with groups of horseshoe vortices. Instantaneous three-dimensional flow fields suggest that horseshoe vortices tend to concatenate and form organized spiral as well as axially aligned coherent VLSMs. A detection scheme is introduced to identify and average over these VLSMs. This conditional averaging reveals that spiral VLSMs and axially aligned VLSMs constitute $72\,\%$ and $28\,\%$ of the total VLSMs, respectively.

In the vicinity of walls, turbulence is anisotropic. Since the classical hypotheses of Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941, pp. 299–303), Obukhov (Izv. Akad. Nauk SSSR Geogr. Geofiz, vol. 13, 1949, pp. 58–69) and Corrsin (J. Appl. Phys., vol. 22, 1951, pp. 469–473) require small-scale turbulence to be isotropic, they have only limited relevance to wall-bounded turbulent flows. Here, we put forward a hypothesis whereby small-scale near-wall statistics, when suitably normalized, are independent of the type of flow as well as of the Reynolds and Péclet numbers. The relatively large amount of available wall turbulence direct numerical simulations data, related mainly to second-order statistics, in a channel flow and a boundary layer provides good support for the independence with respect to the Reynolds number. To fully validate the hypothesis, more data are required for higher-order statistics as well as for other wall flows and for different surface conditions.

Highly resolved laboratory measurements of the airflow over wind-generated waves are examined using a novel wave growth diagnostic that quantifies the presence of Miles’ critical layer mechanism of wind-wave growth. The wave growth diagnostic is formulated based on a linear stability analysis, and results in growth rates that agree well with those found by a pressure reconstruction method as well as other, less direct, methods. This finding, combined with a close agreement between the airflow measurements and the predictions of linear stability (critical layer) theory, demonstrate that the Miles’ critical layer mechanism can cause significant wave growth in young (wave age $c/u_* = 6.3$, where $c$ is the wave phase speed, and $u_*$ the friction velocity) wind-forced waves.

Horizontally extended turbulent convection, termed mesoscale convection in natural systems, remains a challenge to investigate in both experiments and simulations. This is particularly so for very low molecular Prandtl numbers, such as occur in stellar convection and in the Earth's outer core. The present study reports three-dimensional direct numerical simulations of turbulent Rayleigh–Bénard convection in square boxes of side length $L$ and height $H$ with the aspect ratio $\varGamma =L/H$ of 25, for Prandtl numbers that span almost 4 orders of magnitude, $10^{-3}\le Pr \le 7$, and Rayleigh numbers $10^5 \le Ra \le 10^7$, obtained by massively parallel computations on grids of up to $5.36\times 10^{11}$ points. The low end of this $Pr$-range cannot be accessed in controlled laboratory measurements. We report the essential properties of the flow and their trends with the Rayleigh and Prandtl numbers, in particular, the global transport of momentum and heat – the latter decomposed into convective and diffusive contributions – across the convection layer, mean vertical profiles of the temperature and temperature fluctuations and the kinetic energy and thermal dissipation rates. We also explore the degree to which the turbulence in the bulk of the convection layer resembles classical homogeneous and isotropic turbulence in terms of spectra, increment moments and dissipative anomaly, and find close similarities. Finally, we show that a characteristic scale of the order of the mesoscale seems to saturate to a wavelength of $\lambda \gtrsim 3H$ for $Pr\lesssim 0.005$. We briefly discuss possible implications of these results for the development of subgrid-scale parameterization of turbulent convection.

In non-uniform stratification, horizontally propagating internal waves with the vertical structure of a single mode self-interact to excite superharmonics. Baker & Sutherland (J. Fluid Mech., vol. 891, 2020, R1) showed that a vertical mode-1 parent wave of sufficiently small amplitude dominantly excites a vertical mode-1 superharmonic with double the horizontal wavenumber. Through theory, assuming a parent wave of sufficiently small amplitude, they showed that the superharmonics grew and decayed periodically due to the parent forcing frequency being off-resonant with the natural frequency of the superharmonic. Here, we extend this theory to allow for larger parent wave amplitudes and/or stronger resonant forcing, as would occur at lower latitudes, where the influence of background rotation is small. The resulting coupled system of nonlinear ordinary differential equations is shown to well predict the evolution of the internal tide as determined in fully nonlinear numerical simulations. With strong nonlinear forcing, successive superharmonics grow to non-negligible amplitudes in what we call the ‘superharmonic cascade’. The phase relationship between the superharmonics is such that when superimposed, the internal tide transforms into a solitary wave train, consistent with the predictions of well-established shallow-water models, particularly that of the Ostrovsky equation, which is an extension of the Korteweg–de Vries equation accounting for background rotation. This work thus gives new insight into internal solitary wave generation. The model equations have less restrictive assumptions than models based upon shallow-water theory, and because they are quickly solved, these provide a potentially powerful new tool to examine the nonlinear evolution of the internal tide.

The flow past a bullet-shaped blunt body moving in a pipe is investigated through global linear stability analysis (LSA) and direct numerical simulation. A cartography of the bifurcation curves is provided thanks to LSA, covering the range of parameters corresponding to Reynolds number $Re = [50\unicode{x2013}110]$, confinement ratio $a/A = [ 0.01\unicode{x2013}0.92]$ and length-to-diameter ratio $L/d = [2\unicode{x2013}10]$. Results show that the first bifurcation is always a steady bifurcation associated to a non-oscillating eigenmode with azimuthal wavenumber $m=\pm 1$ leading to a steady state with planar symmetry. For weakly confined cases ($a/A<0.6$), the second bifurcation is associated to an oscillating mode with azimuthal wavenumber $m=\pm 1$, as in the unconfined case. On the other hand, for the strongly confined case ($a/A>0.8$), a destabilization of non-oscillating modes with $|m| =2,3$ and a restabilization of the $m=\pm 1$ eigenmodes are observed. The aspect ratio $L/d$ is shown to have a minor influence for weakly confined cases and almost no influence for strongly confined cases. Direct numerical simulation is subsequently used to characterize the nonlinear dynamics. The results confirm the steady bifurcation predicted by LSA with excellent agreement for the threshold Reynolds. For weakly confined cases, the second bifurcation is a Hopf bifurcation leading to a periodic, planar-symmetric state in qualitative accordance with LSA predictions. For more confined cases, more complex dynamics is obtained, including a steady state with $|m|=3$ geometry and aperiodic states.

Pulsatile flow in a straight pipe is a model system for unsteady internal flows in industrial engineering and physiology. In some parameter regimes, the laminar flow is susceptible to helical perturbations, whose transient energy growth scales exponentially with the Reynolds number (Re). In this paper, we link the transient growth of these perturbations to the instantaneous linear instability of the laminar flow. We exploit this link to study the effect of the waveform on turbulence transition by performing linear stability and transient growth analyses of flows driven with different waveforms. We find a higher-energy growth in flows driven with longer low-velocity phases as well as with steeper deceleration and acceleration phases. Finally, we perform direct numerical simulations and show that cases with larger transient growth transition faster to turbulence and exhibit larger turbulence intensities. However, these same cases are also more prone to relaminarisation once turbulence has been established. This highlights that, in pulsatile flows, the linear mechanisms responsible for turbulence transition are distinctly different from the nonlinear mechanisms sustaining turbulence.

Hydrodynamic stability of compressible boundary layers is strongly influenced by the Mach number ($M$), Prandtl number ($Pr$) and thermal wall boundary condition. These effects manifest on the flow stability via the flow–thermodynamics interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow–thermodynamics interactions are examined using linear analysis and direct numerical simulations in the following parameter regime: $0.5 \leq M \leq 8$; and $0.5 \leq Pr \leq 1.3$. For the adiabatic wall boundary condition, increasing Prandtl number has a destabilizing effect. In this work, we characterize the behaviour of production, pressure–strain correlation and pressure dilatation as functions of the Mach and Prandtl numbers. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite. The Prandtl number influence on instability is explicated in terms of the base flow profile with respect to the different perturbation mode shapes.

The linear dynamics of perturbations developing in an infinite channel with compliant walls is investigated for pulsatile flow conditions. Two-dimensional modal perturbations are considered for Womersley-type pulsating base flows and the wall motion is only allowed in the normal direction. It is found that the flow dynamics is mainly governed by four control parameters: the Reynolds number $Re$, the reduced velocity $V_R$, the Womersley number $Wo$ and the amplitude of the base-flow modulation $\tilde Q$. Linear stability analyses are carried out within the framework of Floquet theory, implementing an efficient approach for removing spurious eigenmodes. The characteristics of flow-based (Tollmien–Schlichting) and wall-based (both travelling-wave flutter and divergence) modes are investigated over a large control-parameter space. It is shown that travelling-wave flutter (TWF) modes are predominantly influenced by the reduced velocity and that the Reynolds number has only a marginal effect. The critical reduced velocity (corresponding to onset of linear instability) is demonstrated to depend both on the Womersley number and modulation amplitude for a given set of wall parameters. Similarly to the steady flow case, the Tollmien–Schlichting (TS) mode is also found to be only weakly affected by the flexibility of the wall in pulsatile flow conditions. Finally, the classification given by Benjamin (J. Fluid Mech., vol. 16, 1963, pp. 436–450) is found to be too restrictive in the case of pulsatile base flows. In particular, a new type of mode is identified that shares characteristics of two distinct Floquet eigenmodes: TS and TWF modes. Due to coupling of the different Floquet harmonics, a phenomenon specific to time-periodic base flows, this two-wave mode exhibits a beating over the intracyclic dynamics.

Symmetry relations are derived for the added mass and damping of structures where the shape is unchanged by rotation about the vertical axis through an angle $\theta = 2 {\rm \pi}/N$ with the integer $N\ge 3$. For this type of structure, the added mass and damping for horizontal translation are the same for all directions, as in the case of axisymmetric structures. The same symmetry applies to rotations about horizontal axes. The principal application is to offshore structures and other bodies floating on the free surface or submerged, but the same symmetry relations apply more generally to unsteady body motions in an ideal fluid.

When a freely suspended liquid film ruptures, it retracts spontaneously under the action of surface tension. If the film is surrounded by air, the retraction velocity is known to approach the constant Taylor–Culick velocity. However, when surrounded by an external viscous medium, the dissipation within that medium dictates the magnitude of the retraction velocity. In the present work, we study the retraction of a liquid (water) film in a viscous oil ambient (two-phase Taylor–Culick retractions), and that sandwiched between air and a viscous oil (three-phase Taylor–Culick retractions). In the latter case, the experimentally measured retraction velocity is observed to have a weaker dependence on the viscosity of the oil phase as compared with the configuration where the water film is surrounded completely by oil. Numerical simulations indicate that this weaker dependence arises from the localization of viscous dissipation near the three-phase contact line. The speed of retraction only depends on the viscosity of the surrounding medium and not on that of the film. From the experiments and the numerical simulations, we reveal unprecedented regimes for the scaling of the Weber number ${We}_{f}$ of the film (based on its retraction velocity) or the capillary number ${Ca}_{s}$ of the surroundings versus the Ohnesorge number ${Oh}_{s}$ of the surroundings in the regime of large viscosity of the surroundings (${Oh}_{s} \gg 1$), namely ${We}_{f} \sim {Oh}_{s}^{-2}$ and ${Ca}_{s} \sim {Oh}_{s}^{0}$ for the two-phase Taylor–Culick configuration, and ${We}_{f} \sim {Oh}_{s}^{-1}$ and ${Ca}_{s} \sim {Oh}_{s}^{1/2}$ for the three-phase Taylor–Culick configuration.