We present a thin-film viscoplastic fluid model of a mountain range, where a uniform fluid layer is deformed by a vertical backstop moving at constant speed. This represents a simplification of the geometry in subduction zones, where an overlying tectonic plate scrapes sediments off the underthrusting plate, thereby forming an accretionary wedge. By using a viscoplastic rheology, we aim to generalise Newtonian models that capture the effective viscous behaviour of rock at large length scales. The model system is characterised by the dimensionless Bingham number, which is the ratio of the yield stress to characteristic shear stress. At low and high Bingham numbers and at early and late times the system is found to be asymptotically self-similar, which we confirm by solving the governing equations numerically. In addition, we test the high-Bingham-number results experimentally using ultrasound gel as the working fluid. The experiments reproduce many features of the theoretically predicted behaviour. The size of the observed wedge grows in the manner predicted, and the fluid surface profile is found to collapse to a universal shape. The viscoplastic fluid wedge exhibits features of both viscous continuum and Coulomb models of accretionary wedges, suggesting that a viscoplastic rheology may provide quantitative insights into the dynamics of real accretionary wedges found in convergent tectonic settings.

We develop a model for the interaction of a fluid flowing above an otherwise static particle bed, with generally the particles being entrained or detrained into the fluid from the upper surface of the particle bed, and thereby forming a fully two phase fluidized cloud above the particle bed. The flow in this large-scale fluidized region is treated as a two-phase flow, whilst the key processes of entrainment and detrainment from the particle bed are treated by examining the local dynamical force balances on the particles in a thin transition layer at the interface between the fully fluidized region and the static particle bed. This detailed consideration leads to the formation of an additional macroscopic boundary condition at this interface, which closes the two-phase flow problem in the bulk fluidized region above. We then introduce an elementary model of the well-known helicopter brownout problem, and use the theory developed in the first part of the paper to fully analyse this model, both analytically and numerically.

Supersonic shear layers experience instabilities that generate significant adverse effects; in complex configurations, these instabilities have global impacts as they foster compounding complications with other independent flow features. We consider the flow near the exit of a dual-stream rectangular nozzle. The supersonic core and sonic bypass streams mix downstream of a splitter plate trailing edge (SPTE) just above an adjacent deck. We perform two-dimensional direct numerical simulations with laminar inflow conditions to parametrically explore the influence of active flow control, considering various actuation angles and locations. The goal is to alleviate the prominent tone associated with vortices shed at the SPTE; these vortices initiate an unsteady shock system that affects the entire flow field through a shock-induced separation and the downstream evolution of plume shear layers. Resolvent analysis is performed on the baseline flow. The identified optimal response guides the placement of steady-blowing actuators. Since the resolvent analysis fundamentally investigates the input–output dynamics of a system, it is also utilized to uncover actuation-induced changes in the forcing–response dynamics. Spectral analysis shows that the baseline flow fluctuating energy is concentrated in the shedding instability. Actuating at optimal angles based on location disperses this energy into various flow features; this affects the shedding itself, and the structure and unsteadiness of the shock system, and thus the response of the deck and nozzle wall boundary layers and the plume. The resolvent analysis indicates, and Navier–Stokes solutions confirm, that favourable control is obtained by either indirectly or directly mitigating the baseline instability.

A hydrodynamic theory of premixed flame propagation within closed vessels is developed assuming the flame is much thinner than all other fluid dynamic lengths. In this limit, the flame is confined to a surface separating the unburned mixture from burned combustion products, and propagates at a speed determined from the analysis of its internal structure. Unlike freely propagating flames that propagate under nearly isobaric conditions, combustion in a closed vessel results in continuous increases in pressure, burning rate and flame temperature, and a progressive decrease in flame thickness. The flame speed is shown to depend on the voluminal stretch rate, which measures the deformation of a volume element of the flame zone, and on the rate of pressure rise. Both effects are modulated by pressure-dependent Markstein numbers that depend on heat release and mixture properties while capturing the effects of temperature-dependent transport and stoichiometry. The model applies to flames of arbitrary shape propagating in general flows, laminar or turbulent, within vessels of general configurations. The main limitation of hydrodynamic flame theories is the assumption that variations inside the flame zone due to chemistry or turbulence, which could potentially alter its internal structure, are physically unresolved. Nonetheless, the theory, deduced from physical first principles, identifies the various mechanisms involved in the combustion process as demonstrated in detailed discussions of planar flames propagating in rectangular channels and spherically expanding flames in spherical vessels. It also enables the construction of instructive models to numerically simulate the evolution of multi-dimensional and corrugated flames under confinement.

In the generalized quasilinear approximation (GQLA) (Marston et al., Phys. Rev. Lett., vol. 116, 2016, 214501), a threshold wavenumber ($k_0$) in the direction of translational symmetry segregates the total into large- ($l$) and small-scale ($h$) fields. While the governing equation for the large-scale field is fully nonlinear, that for the small scales is linearized with respect to the large-scale field. In addition, some nonlinear triad interactions are omitted in the GQLA. Herein, the GQLA is applied to two-dimensional planar Rayleigh–Bénard convection (RBC). A scale separation between the large-scale convection rolls and small-scale turbulent fluctuations is typical in RBC. The present work explores the efficacy of GQLA in capturing the scale-by-scale energy transfer processes in RBC. The initial condition for the GQLA simulations was either the statistically stationary state obtained in direct numerical simulation (DNS) or random fluctuations superimposed on the linear conductive temperature profile and $\boldsymbol {u}=0$. The GQLA simulations can capture the convection rolls for $k_0$ larger than or equal to the dominant wavenumber for thermal driving of the flow ($k_0 \ge k_{\hat {Q}}$). Additionally, the GQLA emulates the fully nonlinear dynamics for $k_0$ larger than or equal to the first harmonic of the convection-roll wavenumber ($k_0 \ge 2k_{roll}$). In the intermediate regime with $2k_{roll} > k_0 \ge k_{\hat {Q}}$, the dynamics captured in GQLA simulations is different from the DNS. In DNS, two primary energy transfer processes dominate: (i) the energy transfer to/from the convection rolls and (ii) the scale-by-scale inverse kinetic energy and forward thermal energy cascades mediated by the convection rolls. The fully nonlinear dynamics is emulated by GQLA when these energy transfer processes are faithfully reproduced. Utilizing the framework of altering the triad interactions in GQLA, an additional intrusive calculation, including target triad interactions, is performed here to study their influence. This intrusive calculation shows that the convection rolls are not captured in GQLA for $k_0 < k_{\hat {Q}}$ because of the exclusion of the $h \to l \to l$ and $l \to h \to h$ triad interactions in GQLA. The inclusion of these triad interactions in the intrusive calculation yields the convection rolls, and the reproduced dynamics is similar to that of the intermediate GQLA regime with $2k_{roll} > k_0 \ge k_{\hat {Q}}$.

Shock refraction in a gas–liquid interface is ubiquitous in nature and engineering. This study investigates the shock refraction phenomena in air–water interfaces for various inclination angles. The interface inclination angles are achieved using a tiltable vertical shock tube. The time-resolved schlieren images are compared with numerical simulations performed using the BlastFoam solver in the OpenFOAM software. The stiffened gas equation of state is used to model water in the simulations. The shock polar analysis using modified shock relations for a stiffened gas is used to elucidate the refraction patterns. A regular refraction pattern with a reflected shock wave and a bound precursor refraction with a regular reflection are observed experimentally for the first time in an air–water system. Further, a new free precursor refraction pattern with a Mach reflection is observed. The transition criteria and the corresponding boundaries for each refraction pattern are demarcated in the ($M_S, \theta _w^c$)-plane. The refraction sequence and the range for various incident shock strength regimes are also identified.

Shear-induced self-diffusion is a fundamental mode of transport in granular flows. Yet its critical behaviour and dependence on the particle solid fraction are still unclear. Here, we rationalize these dependencies by performing two-dimensional pressure-imposed numerical simulations of dense non-Brownian frictional suspensions. Our results, combined with existing numerical data on inertial granular flows, show that the shear-induced diffusion coefficients of both systems can be captured by a single function of the distance to jamming. They further show that the grain diffusive behaviour is underpinned by a specific random walk process, having a constant elementary step length driven at a frequency that increases with the solid fraction. The proposed scaling laws pave the way for a better understanding of mixing processes in granular media.

Regular reflection (RR) to Mach reflection (MR) transitions (${\rm RR}\leftrightarrow {\rm MR}$) on long wedges in steady supersonic flows have been well studied and documented. However, in a short wedge where the wedge length is small, the transition prediction becomes really challenging owing to the interaction of the expansion fan emanating from the trailing edge of the wedge with the incident shock and the triple/reflection point. The extent of this interaction depends on the distance between the wedge trailing edge and the symmetry line (Ht). This distance is a geometric combination of the distance of the wedge leading edge from the symmetry line $(H)$, the wedge angle ($\theta$) and the wedge length $(w)$. In the present study, we used the method of characteristics to model the complete wave interactions which accurately predicted shock curvatures and the reflection configurations for all ranges of the incoming flow Mach number. In the case of short wedges, the transition criterion strongly depends on the wedge length, which can be so adjusted even to eliminate the ${\rm RR}\rightarrow {\rm MR}$ transitions till the wedge angle reaches the no-reflection domain. Transition lines for both the detachment criterion and von Neumann criterion are also drawn to investigate the dual solution domain, and the reflection configurations were verified experimentally for the first time on short wedges. By using proper input configuration parameter ($w/H$), various types of shifts in the dual solution domain for short wedges are studied and categorised into three types, namely Type I, Type II and Type III.

We discuss the applicability of quasilinear-type approximations for a turbulent system with a large range of spatial and temporal scales. We consider a paradigm fluid system of rotating convection with vertical and horizontal temperature gradients. In particular, the interaction of rotation with the horizontal temperature gradient drives a ‘thermal wind’ shear flow whose strength is controlled by the horizontal temperature gradient. Varying this parameter therefore systematically alters the ordering of the shearing time scale, the convective time scale and the correlation time scale. We demonstrate that quasilinear-type approximations work well when the shearing time scale or the correlation time scale is sufficiently short. In all cases, the generalised quasilinear approximation systematically outperforms the quasilinear approximation. We discuss the consequences for statistical theories of turbulence interacting with mean gradients.

We study the autophoretic motion of a spherical active particle interacting chemically and hydrodynamically with its fluctuating environment in the limit of rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamic fields can be expressed in terms of integrals. The resulting boundary-domain integral equations provide a direct way of obtaining the traction on the particle, requiring the solution of linear integral equations. An exact solution for the chemical and hydrodynamic problems is obtained for a particle in an unbounded domain. For motion near boundaries, we provide corrections to the unbounded solutions in terms of chemical and hydrodynamic Green's functions, preserving the dissipative nature of autophoresis in a viscous fluid for all physical configurations. Using this, we give the fully stochastic update equations for the Brownian trajectory of an autophoretic particle in a complex environment. First, we analyse the Brownian dynamics of particles capable of complex motion in the bulk. We then introduce a chemically permeable planar surface of two immiscible liquids in the vicinity of the particle and provide explicit solutions to the chemo-hydrodynamics of this system. Finally, we study the case of an isotropically phoretic particle hovering above an interface as a function of interfacial solute permeability and viscosity contrast.

We examine the fluid flow forced by precession of a rotating cylindrical container using numerical simulations and experimental flow measurements with ultrasonic Doppler velocimetry. The analysis is based on the decomposition of the flow field into contributions with distinct azimuthal symmetry or analytically known inertial modes and the corresponding calculation of their amplitudes. We show that the predominant fraction of the kinetic energy of the precession-driven fluid flow is contained only within a few large-scale modes. The most striking observation shown by simulations and experiments is the transition from a flow dominated by large-scale structures to a more turbulent behaviour with the small-scale fluctuations becoming increasingly important. At a fixed rotation frequency (parametrized by the Reynolds number, $Re$) this transition occurs when a critical precession ratio is exceeded and consists of a two-stage collapse of the directly driven flow going along with a massive modification of the azimuthal circulation (the zonal flow) and the appearance of an axisymmetric double-roll mode limited to a narrow range of precession ratios. A similar behaviour is found in experiments which make it possible to follow the transition up to Reynolds numbers of $Re\approx 2\times 10^6$. We find that the critical precession ratio decreases with rotation, initially showing a particular scaling ${\propto }Re^{-({1}/{5})}$ but developing an asymptotic behaviour for $Re\gtrsim 10^5$ which might be explained by the onset of turbulence in boundary layers.

This study aims to quantify how turbulence in a channel flow mixes momentum in the mean sense. We applied the macroscopic forcing method (Mani & Park, Phys. Rev. Fluids, 2021, 054607) to direct numerical simulation (DNS) of a turbulent channel flow at $Re_\tau =180$ using two different forcing strategies that are designed to separately assess the anisotropy and non-locality of momentum mixing. In the first strategy, the leading term of the Kramers–Moyal expansion of the eddy viscosity is quantified, revealing all 81 tensorial coefficients that essentially characterise the local-limit eddy viscosity. The results indicate the following: (1) the eddy viscosity has significant anisotropy, (2) Reynolds stresses are generated by both the mean strain rate and mean rotation rate tensors associated with the momentum field and (3) the local-limit eddy viscosity generates asymmetric Reynolds stress tensors. In the second strategy, the eddy viscosity is quantified as an integration kernel revealing the non-local influence of the mean momentum gradient at each wall-normal coordinate on all nine components of the Reynolds stresses over the channel width. Our results indicate that while the shear component of the Reynolds stress is reasonably reproduced by the local mean gradients, other components of the Reynolds stress are highly non-local. These results provide an understanding of anisotropy and non-locality requirements for closure modelling of momentum transport in attached wall-bounded turbulent flows.

Consider the flow through a channel with grooved edges on one (or both) side(s). If heating is applied to the boundaries, thermal drift is the flow generated by the interaction of the groove and heating patterns. It is known that, if one side of a channel is smooth while the other is grooved, the application of heating forms a so-called ‘thermal drift engine’. Two thermal drift engines are activated if both surfaces are grooved, and these may reinforce or oppose each other. Carefully choosing these engines can lead to an intensification of the thermal drift. The interplay of two drift engines is explored using a horizontal slot with grooves that have a sinusoidal profile with a prescribed wavenumber $\alpha $. It is shown that the strength of the flow decreases proportional to $\alpha $ as $\alpha \to 0$ and proportional to ${\alpha ^{ - 1}}$ as $\alpha \to \infty $. We determine the value of $\alpha $ corresponding to the strongest flow and characterize how the conclusions should be modified if a uniform heating component is added to the heating pattern.

We investigate effect of porous insert located upstream of the separation edge of a backward-facing step (BFS) in early transitional regime as a function of Reynolds number. This is an example of hydrodynamic system that is a combination of separated shear flow with large amplification potential and porous materials known for efficient flow destabilisation. Spectral analysis reveals that dynamics of BFS is dominated by spectral modes that remain globally coherent along the streamwise direction. We detect two branches of characteristic frequencies in the flow and with Hilbert transform we characterise their spatial support. For low Reynolds numbers, the dynamics of the flow is dominated by lower frequency, whereas for sufficiently large Reynolds numbers cross-over to higher frequencies is observed. Increasing permeability of the porous insert results in decrease in Reynolds number value, at which frequency cross-over occurs. By comparing normalised frequencies on each branch with local stability analysis, we attribute Kelvin–Helmholtz and Tollmien–Schlichting instabilities to upper and lower frequency branches, respectively. Finally, our results show that porous inserts enhance Kelvin–Helmholtz instability and promote transition to oscillator-type dynamics. Specifically, the amplitude of vortical (BFS) structures associated with higher-frequency branch follows Landau model prediction for all investigated porous inserts.

Understanding the generation of large-scale magnetic fields and flows in magnetohydro-dynamical (MHD) turbulence remains one of the most challenging problems in astrophysical fluid dynamics. Although much work has been done on the kinematic generation of large-scale magnetic fields by turbulence, relatively little attention has been paid to the much more difficult problem in which fields and flows interact on an equal footing. The aim is to find conditions for long-wavelength instabilities of stationary MHD states. Here, we first revisit the formal exposition of the long-wavelength linear instability theory, showing how long-wavelength perturbations are governed by four mean field tensors; we then show how these tensors may be calculated explicitly under the ‘short-sudden’ approximation for the turbulence. For MHD states with relatively little disorder, the linear theory works well: average quantities can be readily calculated, and stability to long-wavelength perturbations determined. However, for disordered basic states, linear perturbations can grow without bound and the purely linear theory, as formulated, cannot be applied. We then address the question of whether there is a linear response for sufficiently weak mean fields and flows in a dynamical (nonlinear) evolution, where perturbations are guaranteed to be bounded. As a preliminary study, we first address the nature of the response in a series of one-dimensional maps. For the full MHD problem, we show that in certain circumstances, there is a clear linear response; however, in others, mean quantities – and hence the nature of the response – can be difficult to calculate.

We present an experimental investigation aimed at understanding the effects of surface roughness on the time-mean drag coefficient ($\bar {C}_{D}$) of finite-span cylinders ($\text {span/diameter} = \text {aspect ratio}$, $0.51 \le AR \le 6.02$) freely rolling without slip on an inclined plane. While lubrication theory predicts an infinite drag force for ideally smooth cylinders in contact with a smooth surface, experiments yield finite drag coefficients. We propose that surface roughness introduces an effective gap ($G_{eff}$) resulting in a finite drag force while allowing physical contact between the cylinder and the plane. This study combines measurements of surface roughness for both the cylinder and the plane panel to determine a total relative roughness ($\xi$) that can effectively describe $G_{eff}$ at the point of contact. It is observed that the measured $\bar {C}_{D}$ increases as $\xi$ decreases, aligning with predictions of lubrication theory. Furthermore, the measured $\bar {C}_{D}$ approximately matches combined analytical and numerical predictions for a smooth cylinder and plane when the imposed gap is approximately equal to the mean peak roughness ($R_p$) for rough cylinders, and one standard deviation peak roughness ($R_{p, 1\sigma }$) for relatively smooth cylinders. As the time-mean Reynolds number ($\overline {Re}$) increases, the influence of surface roughness on $\bar {C}_{D}$ decreases, indicating that wake drag becomes dominant at higher $\overline {Re}$. The cylinder aspect ratio ($AR$) is found to have only a minor effect on $\bar {C}_{D}$. Flow visualisations are also conducted to identify critical flow transitions and these are compared with visualisations previously obtained numerically. Variations in $\xi$ have little effect on the cylinder wake. Instead, $AR$ was observed to have a more pronounced effect on the flow structures observed. The Strouhal number ($St$) associated with the cylinder wake shedding was observed to increase with $\overline {Re}$, while demonstrating little dependence on $AR$.

We derive a generalised asymptotic model for the flow of a thin fluid film over an arbitrarily parameterised non-axisymmetric curved substrate surface based on the lubrication approximation. In addition to surface tension, gravity and centrifugal force, our model incorporates the effects of the Coriolis force and disjoining pressure, together with a non-uniform initial condition, which have not been widely considered in existing literature. We use this model to investigate the impact of the Coriolis force and fingering instability on the spreading of a non-axisymmetric spin-coated film at a range of substrate angular velocities, first on a flat substrate, and then on parabolic cylinder- and saddle-shaped curved substrates. We show that, on flat substrates, the Coriolis force has a negligible impact at low angular velocities, and at high angular velocities results in a small deflection of fingers formed at the contact line against the direction of substrate rotation. On curved substrates, we demonstrate that, as the angular velocity is increased, spin-coated films transition from being dominated by gravitational drainage with no fingering to spreading and fingering in the direction with the greatest component of centrifugal force tangent to the substrate surface. For both curved substrates and all angular velocities considered, we show that the film thickness and total wetted substrate area remain similar over time to those on a flat substrate, with the key difference being the shape of the spreading droplet.

Turbulent shear flows driven by a combination of a pressure gradient and buoyancy forcing are investigated using direct numerical simulations. Specifically, we consider the set-up of a differentially heated vertical channel subject to a Poiseuille-like horizontal pressure gradient. We explore the response of the system to its three control parameters: the Grashof number $Gr$, the Prandtl number $Pr$, and the Reynolds number $Re$ of the pressure-driven flow. From these input parameters, the relative strength of buoyancy driving to the pressure gradient can be quantified by the Richardson number $Ri=Gr/Re^2$. We compare the response of the mixed vertical convection configuration to that of mixed Rayleigh–Bénard convection, and find a nearly identical behaviour, including an increase in wall friction at higher $Gr$, and a drop in the heat flux relative to natural convection for $Ri=O(1)$. This closely matched response is despite vastly different flow structures in the systems. No large-scale organisation is visible in visualisations of mixed vertical convection – an observation that is confirmed quantitatively by spectral analysis. This analysis, combined with a statistical description of the wall heat flux, highlights how moderate shear suppresses the growth of small-scale plumes and reduces the likelihood of extreme events in the local wall heat flux. Vice versa, starting from a pure shear flow, the addition of thermal driving enhances the drag due to the emission of thermal plumes.

The measurement of lift on symmetrically shaped obstacles immersed in low Reynolds number flow is the quintessential way to signal odd viscosity. For flow past cylinders, such a lift force does not arise if incompressibility and no-slip boundary conditions are fulfilled, whereas for spheres, a lift force has been found in Stokes flow, which is valid for cases where the Reynolds numbers are negligible and convection can be ignored. When considering the role of convection at low but non-zero Reynolds numbers, two hurdles arise, the Whitehead paradox and the breaking of axial symmetry, which are overcome by the method of matched asymptotic expansions and the Lorentz reciprocal theorem, respectively. We also consider the case where axial symmetry is preserved because the translation of the sphere is aligned with the axis of chirality of odd viscosity. We find that while lift vanishes, the interplay between odd viscosity and convection gives rise to a stream-induced torque.

Self-sustained thermoacoustic oscillations as observed in low-emission combustion- involved gas turbines and aero-engines involve complicated thermal fluid–acoustics interaction and rich nonlinear dynamics. Such pulsating oscillations are known as thermoacoustic instability. When it occurs, large-amplitude limit cycle oscillations (LCOs) of thermodynamic parameters are frequently observed. These LCOs could cause overheating, flame flashback, and even engine failures. Thus it is critical to understand and predict the generation mechanisms and nonlinear dynamics behaviours, and then develop corresponding control approaches to prevent or control the onset of such instabilities. In this work, we develop and extend the classical van der Pol oscillators by integrating a physics-informed neural networks (PINNs) algorithm with a modelled nonlinear Rijke-type thermoacoustic combustor. The theoretical Rijke tube system (with Galerkin expansion and modified King's law implemented) and a CFD simulation model are applied to provide ‘training/calibration data’ for the extended van der Pol (EVDP)-PINNs model. The optimized EVDP oscillators are confirmed to be capable of capturing the key nonlinear characteristics by comparing the transient growth behaviours of thermodynamic perturbations and LCO amplitude and frequency. Further investigations are conducted to obtain Hopf bifurcation and amplitude death (AD) characteristics. Comparison is then made to the coupled EVDP systems. Quite similar Hopf bifurcation features, but differences in regions of AD, are observed. In general, we demonstrate an applicable approach to intelligently ‘learn’ a nonlinear thermoacoustic system and to create reliable EVDP oscillator systems, which have great potential to contribute to the development and testing of control approaches, such as the coupling described in this work, which may replace costly experimental tests.