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.

We study the preference of two-phase displacements systematically by theoretical derivations and numerical simulations via a non-uniform pore doublet model. All the most important impact factors, including viscosity ratio, capillary number, wetting conditions and boundary conditions, have been considered, and finally a complete phase diagram for preferential flow has been obtained. The simple treatment for the dual-permeability media has been validated, and further, with a few necessary corrections the phase diagram is applicable for disordered permeable media in general. These results help us to understand the occurrence and manipulation of preferential flow in heterogeneous permeable media.

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.

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.

We examine Lagrangian transport by a nonlinearly evolving vertical mode-1 internal tide in non-uniform stratification. In a companion paper (Sutherland & Dhaliwal, J. Fluid Mech., 2022, in press) it was shown that a parent internal tide can excite successive superharmonics that superimpose to form a solitary wave train. Despite this transformation, here we show that the collective forcing by the parent wave and superharmonics is effectively steady in time. Thus we derive relatively simple formulae for the Stokes drift and induced Eulerian flow associated with the waves under the assumption that the parent waves and superharmonics are long compared with the fluid depth. In all cases, the Stokes drift exhibits a mixed mode-1 and mode-2 vertical structure with the flow being in the waveward direction at the surface. If the background rotation is non-negligible, the vertical structure of the induced Eulerian flow is equal and opposite to that of the Stokes drift. This flow periodically increases and decreases at the inertial frequency with maximum magnitude twice that of the Stokes drift. When superimposed with the Stokes drift, the Lagrangian flow at the surface periodically changes from positive to negative over one inertial period. If the background rotation is zero, the induced Eulerian flow evolves non-negligibly in time and space for horizontally modulated waves: the depth below the surface of the positive Lagrangian flow becomes shallower ahead of the peak of the amplitude envelope and becomes deeper in the lee of the peak. These predictions are well-captured by fully nonlinear numerical simulations.

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.

The drag force on a spherical intruder in dense granular shear flows is studied using discrete element method simulations. Three regimes of the intruder dynamics are observed depending on the magnitude of the drag force (or the corresponding intruder velocity) and the flow inertial number: a fluctuation-dominated regime for small drag forces; a viscous regime for intermediate drag forces; and an inertial (cavity formation) regime for large drag forces. The transition from the viscous regime (linear force-velocity relation) to the inertial regime (quadratic force-velocity relation) depends further on the inertial number. Despite these distinct intruder dynamics, we find a quantitative similarity between the intruder drag in granular shear flows and the Stokesian drag on a sphere in a viscous fluid for intruder Reynolds numbers spanning five orders of magnitude. Beyond this first-order description, a modified Stokes drag model is developed that accounts for the secondary dependence of the drag coefficient on the inertial number and the intruder size and density ratios. When the drag model is coupled with a segregation force model for intruders in dense granular flows, it is possible to predict the velocity of gravity-driven segregation of an intruder particle in shear flow simulations.

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.

Hot-wire measurements in a Mach 3.5 quiet tunnel were made in the wake of a roughness patch on a flat plate. These measurements were used to determine mode shapes and frequencies of the dominant instabilities leading to boundary-layer transition. The egg-crate roughness pattern is an analytic function described by a sinusoidal equation, similar to an array of discrete elements that are positioned in a spanwise and streamwise grid, but containing both protuberances and dimples. This is an intermediate configuration towards understanding the underlying physics of a pseudorandom distributed roughness, and ultimately, the underlying physics of roughness-induced boundary-layer transition. The roughness pattern had a wavelength of 6.25 mm, with a nominal amplitude of 272 ${\rm \mu}{\rm m}$ (0.49 times the boundary layer thickness at the first row of protuberances). The roughness was positioned near the leading edge of the flat plate and contained 3.5 wavelengths in the streamwise direction and 7.5 wavelengths in the spanwise direction. The dominant instability was centred near 74 kHz at a free stream unit Reynolds number of $12.9\times 10^{6}\,{\rm m}^{-1}$ and resembled an antisymmetric mode downstream of each of the protuberances in the roughness patch. Computations using linear stability analysis based on the plane-marching parabolized stability equations (PSE) showed limited agreement with measurements when comparing the growth of the wake instability. Better agreement with the measurements was observed when considering the modification of first mode waves by the egg-crate roughness patch and the solution of the three-dimensional harmonic linearized Navier–Stokes equations was used as the in-flow to the PSE. The agreement confirms the significance of disturbance growth both upstream of and above a finite length roughness patch and the effect on the growth of instabilities in the wake.

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.

We study the three-dimensional clustering of velocity stagnation points, of nulls of the vorticity and of the Lagrangian acceleration, and of inertial particles in turbulent flows at fixed Reynolds numbers, but under different large-scale flow geometries. To this end, we combine direct numerical simulations of homogeneous and isotropic turbulence and of the Taylor–Green flow, with particle tracking velocimetry in a von Kármán experiment. While flows have different topologies (as nulls cluster differently), particles behave similarly in all cases, indicating that Taylor-scale neutrally buoyant particles cluster as inertial particles.

We numerically and theoretically investigate the Boussinesq Eady model, where a rapidly rotating density-stratified layer of fluid is subject to a meridional temperature gradient in thermal wind balance with a uniform vertically sheared zonal flow. Through a suite of numerical simulations, we show that the transport properties of the resulting turbulent flow are governed by quasigeostrophic (QG) dynamics in the rapidly rotating strongly stratified regime. The ‘vortex gas’ scaling predictions put forward in the context of the two-layer QG model carry over to this fully three-dimensional system: the functional dependence of the meridional flux on the control parameters is the same, the two adjustable parameters entering the theory taking slightly different values. In line with the QG prediction, the meridional heat flux is depth-independent. The vertical heat flux is such that turbulence transports buoyancy along isopycnals, except in narrow layers near the top and bottom boundaries, the thickness of which decreases as the diffusivities go to zero. The emergent (re)stratification is set by a simple balance between the vertical heat flux and diffusion along the vertical direction. Overall, this study demonstrates how the vortex-gas scaling theory can be adapted to quantitatively predict the magnitude and vertical structure of the meridional and vertical heat fluxes, and of the emergent stratification, without additional fitting parameters.

In this study, basing on the level-set and point-particle methods, we have developed a numerical methodology for simulating the dynamics of colloidal droplets under flow conditions in which the particle–particle, particle–interface and particle–fluid interactions are all taken into account efficiently. By using this methodology, we have determined the essential role of particle-laden interfaces in the deformation of colloidal droplets in simple shear flow with relatively low particle concentrations. Generally, adsorbed particles strongly enhance the deformability of the whole droplet, which is principally attributed to the particle-induced reduction of the effective surface tension. Systematic simulations are performed to reveal the detailed roles of interparticle interactions and particle surface coverage in the deformation of particle-covered droplets. Most importantly, we find the promotion effect of adsorbed particles on the droplet deformation cannot be completely included via the effective capillary number characterizing the particle-induced overall reduction of the effective surface tension, which is particularly obvious at high particle coverage. We propose two potential reasons for this surprising phenomenon, i.e. the convection-induced non-uniform distribution of adsorbed particles over the droplet surface and the particle-induced reduction of the droplet surface mobility, which have not been discussed yet in previous numerical and experimental studies of particle-covered droplets in shear flow.

We study analytically the dynamic response of membrane aerofoils subject to arbitrary, small-amplitude chord motions and transverse gusts in a two-dimensional inviscid incompressible flow. The theoretical model assumes linear deformations of an extensible membrane under constant tension, which are coupled aeroelastically to external aerodynamic loads using unsteady thin aerofoil theory. The structural and aerodynamic membrane responses are investigated for harmonic heave oscillations, an instantaneous change in angle of attack, sinusoidal transverse gusts and a sharp-edged gust. The unsteady lift responses for these scenarios produce aeroelastic extensions to the Theodorsen, Wagner, Sears and Küssner functions, respectively, for a membrane aerofoil. These extensions incorporate for the first time membrane fluid–structure interaction into the expressions for the unsteady lift response of a flexible aerofoil. The indicial responses to step changes in the angle of attack or gust profile are characterised by a slower lift response in short times relative to the classical rigid-plate response, while achieving a significantly higher asymptotic lift at long times due to aeroelastic camber. The unsteady lift for harmonic gusts or heaving motions follows closely the rigid plate lift responses at low reduced frequencies but with a reduced lift amplitude and greater phase lag. However, as the reduced frequency approaches the resonance of the fluid-loaded membrane, the lift response amplitude increases abruptly and is followed by a sharp decrease. This behaviour reveals a frequency region, controlled by the membrane tension coefficient, for which membrane aerofoils could possess substantial aerodynamic benefits over rigid aerofoils in unsteady flow conditions.

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.

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.

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.

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.