Liquid metal buoyant flow around two differentially heated horizontal cylinders in the presence of a uniform vertical magnetic field is investigated experimentally. While magneto-convection in pipes or ducts has been studied theoretically and experimentally in recent years, data for heat transfer at immersed obstacles are rare and, to our knowledge, detailed experimental investigations on this fundamental magnetohydrodynamic problem do not exist. In the present work, two horizontal cylinders inserted into an adiabatic rectangular cavity filled with gallium–indium–tin are kept at constant temperatures to establish a driving temperature gradient in the surrounding liquid metal. The buoyancy-driven flow, quantified by the Grashof number $Gr$, is varied in the range ${10^{6} \leq Gr \leq ~5\times 10^{7}}$. With increasing magnetic field, expressed via the Hartmann number $Ha$, different flow regimes are identified from measurements for $0 \leq Ha \leq ~3000$. The effect of the electromagnetic force primarily consists in suppressing turbulence and damping the convective flow. The heat transfer is quantified in terms of the non-dimensional Nusselt number $Nu$, and its dependence on $Gr/{Ha}^{2}$, which is identified as the important group governing the flow, is discussed.

In the present study, we perform direct numerical simulations of compressible turbulent boundary layers at free stream Mach numbers $2\unicode{x2013}6$ laden with dilute phase of spherical particles to investigate the Mach number effects on particle transport and dynamics. Most of the phenomena observed and well-recognized for inertia particles in incompressible wall-bounded turbulent flows – such as near-wall preferential accumulation and clustering beneath low-speed streaks, flatter mean velocity profiles, and trend variation of the particle velocity fluctuations – are identified in the compressible turbulent boundary layer as well. However, we find that the compressibility effects are significant for large inertia particles. As the Mach number increases, the near-wall accumulation and the small-scale clustering are alleviated, which is probably caused by the variations of the fluid density and viscosity that are crucial to particle dynamics. This can be affected by the fact that the forces acting on the particles with viscous Stokes number greater than 500 are modulated by the comparatively high particle Mach numbers in the near-wall region. This is also the reason for the abatement of the streamwise particle velocity fluctuation intensities with the Mach numbers.

We study the stability of unsteady particle-laden flows in long, tilted water columns in batch settling mode, where the quasi-steady assumption of base flow no longer holds for the fast settling of particles. For this purpose, we introduce a settling time scale in the momentum and transport equations to solve the unsteady base flow, and utilise non-modal analysis to examine the stability of the disturbance flow field. The base flow increases in magnitude as the settling speed decreases and attains its maximum value when the settling speed becomes infinitesimal. The time evolution of the disturbance flow energy experiences an algebraic growth caused by the lift-up mechanism of the wall-normal disturbance, followed by an exponential growth owing to the shear instability of the base flow. The streamwise and spanwise wavenumbers corresponding to the peak energy gain are identified for both stages. In particular, the flow instability is enhanced as the Prandtl number increases, which is attributed to the sharpening of the particle-laden interface. On the other hand, the flow instability is suppressed by the increase in settling speed, because less disturbance energy can be extracted from the base flow. There exists an optimal tilted angle for efficient sedimentation, where the particle-laden flow is relatively stable and is accompanied by a smaller energy gain of the disturbance.

The understanding of the entrainment mechanism of synthetic jets can help optimise the synthetic jet actuators in engineering applications. It is generally believed that vortex rings or strong velocity fluctuations in the near field of the synthetic jet are responsible for its enhanced entrainment. However, in recent years, it has been found that the enhanced entrainment of the synthetic jet may be caused by the instability or the vortex ring breakdown in the transition region. To shed new light on this issue, synthetic jets with different Reynolds numbers and dimensionless stroke lengths are investigated with time-resolved two-dimensional particle image velocimetry. Based on the analyses of velocity triple-decomposition, Fourier mode decomposition and phase-averaged $\lambda _{ci}D/U_0$ field, the streamwise positions of the vortex ring breakdown are determined for the synthetic jets, and the entrainment coefficient can be divided into three components, i.e. the coherent turbulent kinetic energy production, the random turbulent kinetic energy production and the shape of the velocity profile. It is found that the entrainment coefficient is dominated by the component related to the random turbulent kinetic energy production, and reaches its peak value at the position of vortex ring breakdown. The results obtained in different cases show a strong correlation between vortex ring breakdown and entrainment enhancement. From the perspective of instantaneous snapshot, the mechanism of vortex ring breakdown enhanced entrainment is revealed, that is, vortex ring breakdown enhanced the small-scale vortex near the turbulent/non-turbulent interface, resulting in an increase of enstrophy production, and thus enhanced local entrainment.

This paper focuses on the linear evolution of Mack instability modes in a hypersonic boundary layer over a flat plate that is partially coated by a compliant section. The compliant section is a thin, flexible membrane covering on a porous wall consisting of micro holes. The instability pressure could induce a vibration of the membrane, leading to a feedback to the boundary-layer fluids through the transverse velocity fluctuation. Such a process is formulated by an admittance boundary condition for the boundary-layer perturbation, which is dependent on the thickness and tension of the membrane, the properties of the porous wall, and the frequency of the Mack-mode perturbation. Using this admittance condition, the impact of the compliant coating on the Mack growth rate is studied systematically by solving the compressible Orr–Sommerfeld equations. It is found that the compliant coating could suppress the Mack instability with a frequency band in the neighbourhood of the most unstable frequency, and the stabilising frequency band widens as the membrane thickness and tension decrease, indicating a more favourable effect of a softer membrane. For a Mack mode with a specified dimensional frequency – since its dimensionless frequency, normalised by the local boundary-layer thickness and oncoming velocity, increases as it propagates downstream – the second-mode frequency band usually appears in downstream locations, and so does the stabilising effect of the membrane. Thus it is favourable to apply a compliant panel at a downstream region. In this situation, the solid–compliant junction could produce an additional scattering effect on the evolution of the Mack mode due to the sudden change of its boundary condition. The scattering effect is quantified by a transmission coefficient defined by the equivalent amplitude of the compliant-wall perturbation to the solid-wall perturbation, which can be obtained by the harmonic linearised Navier–Stokes (HLNS) approach. If the admittance is weak, then the transmission coefficient can also be predicted by an analytical solution based on the residue theorem. It is found that most of the second modes are suppressed by the scattering effect as long as the argument of the admittance is in the interval $[150^\circ, 210^\circ ]$, agreeing with most of the physical situation. The analytical predictions agree well with the HLNS calculations when the modulus of the admittance is less than $O(0.1)$.

This is an ocean motivated study which investigates the impacts of sinusoidal bottom topography on baroclinic instability of zonal vertically sheared flows in the two-layer quasigeostrophic model. The corresponding linear stability problem is solved by assuming Fourier-mode solutions in both the zonal and meridional directions. In the presence of variable topographic features, the Fourier modes become coupled due to phase shifts in the wavevectors. The spectral discretisation method used in this work retains the primary relationship between different Fourier modes; thus, the linear stability eigenproblem can be solved for any periodic topography. Moreover, this method does not need any additional assumptions, such as considering small-amplitude or large-scale bottom irregularities, as in some previous studies. In this work, the eigenproblem is solved for a range of topographic amplitudes and wavenumbers, and their effects on the growth rates and shapes of the most unstable eigenmodes are discussed. In general, both the zonal and meridional variations in topography tend to suppress the baroclinic instability. However, it is found that only meridionally varying topography affects the magnitudes of the fastest growth rates. In this instance, unstable modes appear to form two clusters well separated in the zonal wavenumber axis and growth rate maxima occur at two distinct zonal wavenumbers. Dependencies of the characteristics of these clusters on the values of topography amplitude and ridge width are reviewed. Finally, doubly periodic numerical simulations are used to verify the results from the linear stability analysis.

This is the first of a two-part paper. We formulate a data-driven method for constructing finite-volume discretizations of an arbitrary dynamical system's underlying Liouville/Fokker–Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centres) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We illustrate the method on the Lorenz equations (a three-dimensional ordinary differential equation) saving a fluid dynamical application for Part 2 (Souza, J. Fluid Mech., vol. 997, 2024, A2).

We have developed a parameter-free, two-phase, volume-averaged approach to predictively describe the spin-up flow of dilute, cluster-free ferrofluids excited by low-frequency rotating magnetic fields. Predictive validation of the model was performed through a thorough comparison with local velocity profile measurements, and it demonstrated its ability to capture the spin-up flow dynamics without the need for parameter tuning by carefully delineating the validity domain of the ferrofluid dilutedness conditions. To gain insight into the underlying flow mechanisms, we performed a systematic parametric analysis examining the effects of the induced magnetic field, the dipolar interactions between magnetic nanoparticles and the demagnetizing field. How these mechanisms shape the flow of dilute ferrofluids excited by low-frequency rotating fields in a standard spin-up flow geometry has been addressed using probabilistic nanoparticle orientational dynamics, combining Faxén's laws and the Smoluchowski equation to describe the transport of particle magnetic moments. Our findings revealed that the induced magnetic field is the primary driving force of ferrofluid spin-up flow. The dipole interactions and demagnetizing field, on the other hand, contribute only as secondary phenomena to the overall flow behaviour. Furthermore, we have discussed the potential extension of the two-phase approach, in particular with respect to the formation of chain-like aggregates under the influence of strong magnetic fields. Overall, our study provides valuable insights into the complex dynamics of ferrofluid flow and contributes to a comprehensive understanding of the key mechanisms governing the spin-up flow of dilute ferrofluids excited by low-frequency rotating magnetic fields.

Studies of Kelvin–Helmholtz (KH) instability have typically modelled the initial mean flow as an isolated stratified shear layer. However, geophysical flows frequently exhibit multiple layers. As a step towards understanding these flows, we examine the case of two adjacent stratified shear layers using both linear stability analysis and direct numerical simulations. With sufficiently large layer separation, the characteristics of instability and mixing converge towards the familiar KH turbulence, and similarly when the separation is near zero and the layers add to make a single layer, albeit with a reduced Richardson number. Here, our focus is on intermediate separations, which produce new and complex phenomena. As the separation distance $D$ increases from zero to a critical value $D_c$, approximately half the thickness of the shear layer, the growth rate and wavenumber both decrease monotonically. The minimum Richardson number is relatively low, potentially inducing pairing, and shear-aligned convective instability (SCI) is the primary mechanism for transition. Consequently, mixing is relatively strong and efficient. When $D\sim D_c$, billow length is increased but growth is slowed. Despite the modest growth rate, mixing is strong and efficient, engendered primarily by secondary shear instability manifested on the braids, and by SCI occurring on the eyelids. Shear-aligned vortices are driven in part by buoyancy production; however, shear production and vortex stretching are equally important mechanisms. When $D>D_c$, neighbouring billow interactions suppress the growth of both KH instability and SCI. Strength and efficiency of mixing decrease abruptly as $D_c$ is exceeded. As turbulence decays, layers of marginal instability may arise.

We consider the motion at large Reynolds number of an incompressible fluid around a thin circular disk of finite radius rotating in its plane. The disk is placed in a large tank filled with an initially stagnant fluid. Then it is brought into rotation about its centre with constant angular velocity. Due to viscosity, a layer of fluid adjacent to the disk gets involved in the circumferential motion. This activates centrifugal forces; the fluid particles start to deviate in the radial direction. When they cross the edge of the disk, a thin jet is formed. Firstly, we solve the classical boundary-layer equations for the flow in the boundary layer in the direct neighbourhood of the disk surface and in the jet. The solution is found to develop a discontinuity at the ‘head’ of the jet where the radial and circumferential velocity components experience a jump. This type of discontinuity, called a pseudo-shock, was previously observed by Ruban & Vonatsos (J. Fluid Mech., vol. 614, 2008, pp. 407–424). Then, we investigate the internal structure of the pseudo-shock. We find that the fluid motion is described by the Euler equations in the leading-order approximation. Their solution shows that, as the jet penetrates the stagnant fluid, it ejects the fluid from the boundary layer into the surrounding area. Analysis of the inviscid region outside the boundary layer reveals that the ejected fluid returns back to the boundary layer through the ‘entrainment process’. Finally, we conclude this paper with the study of the wake in the vicinity of the disk rim.

We investigate the instabilities and transition mechanisms of Boussinesq stratified boundary layers on sloping boundaries when subjected to oscillatory body forcing parallel to the slope. We examine idealized forms of boundary layers on hydraulically smooth abyssal slopes in tranquil mid- to low-latitude regions, where low-wavenumber internal tides gently heave isopycnals up and down adiabatic slopes in the absence of mean flows, high-wavenumber internal tides, shelf breaks, resonant tide–bathymetry interactions (critical slopes) and other phenomena associated with turbulence ‘hot spots’. In non-rotating low-Reynolds-number flow, increased stratification on the downslope phase has a relaminarizing effect, while on the upslope phase we find transition-to-turbulence pathways arise from shear production triggered by gravitational instabilities. When rotation is significant (low slope Burger numbers) we find that boundary layer turbulence is sustained throughout the oscillation period, resembling stratified Stokes–Ekman layer turbulence. Simulation results suggest that oscillating boundary layers on smooth slopes at low Reynolds number ($\textit {Re}\leqslant 840$), unity Prandtl number and slope Burger numbers greater than unity do not cause significant irreversible turbulent buoyancy flux (mixing), and that flat-bottom dissipation rate models derived from the tide amplitude are accurate within an order of magnitude.

This study theoretically establishes a flow rule for a granular flow down a rough inclined plane, capable of determining granular rheology in the presence of strong non-local effects resulting from grain cooperativity. To describe the non-local rheology, a Landau–Ginzburg model is formulated in terms of the fluidisation parameter represented by the granular inertial number. The exact solutions of the inertial-number field are solved and provide physical insights into the evolution of the internal rheology and the flow arrest process controlled by the flow height. Through asymptotic analysis in the regime dominated by strong non-locality, the exact solutions are further reduced to yield an analytical flow rule for the mean flow velocity. A comparison between the prediction of the flow rule and experimental data from the literature for sand grains determines the underlying rheology law and the relevant rheological parameters. Thus, the proposed flow rule serves as an effective tool for inferring granular rheology from strongly non-local inclined flow data, surpassing the limitations of the classical flow rule deduced from the local rheology framework.