This study introduces a novel approach to investigate the Reynolds analogy in complex flow scenarios. It is shown that the total mechanical energy $\mathit {B}$, viz. the sum of kinetic energy and pressure work, and the field $\Gamma =\theta ^2/2$ (where $\theta$ is the transported passive scalar) are governed by two equations that are similar in form, when time-averaged for statistically stationary flows. For fully developed channel flows the integral energy balance links the mean bulk velocity and scalar with the volume averages of the respective dissipation rates, allowing the assessment of the Reynolds analogy in terms of the dissipation fields. This approach is tested on direct numerical simulation data of rough-wall turbulent channel flow at two different roughness Reynolds numbers, namely $k^+=15$ and $k^+=90$. For a unit Prandtl number, the same qualitative behaviour is observed for the mean wall-normal distributions of the budget-equation terms of $B$ and $\Gamma$, the latter being larger than the corresponding terms in the mechanical-energy budget. The Reynolds decomposition of the flow into temporal mean and stochastic parts reveals that roughness primarily affects the mean-flow dissipation. For the $k^+=90$ case, the analysis shows that attached-flow and high-shear regions dominate the integral mean scalar and momentum transfer and exhibit the greatest differences between the mean mechanical and scalar dissipation rates. In contrast, well-mixed regions, sheltered by large roughness elements, contribute similarly and minimally to the integral scalar and momentum transfer.

Inspired by laboratory experiments showing internal waves generated by a plume impinging upon a stratified fluid layer (Ansong & Sutherland. 2010 J. Fluid Mech. 648, 405–434), we perform large eddy simulations in three dimensions to examine the structure and source of internal waves emanating from the top of a plume that rises vertically into stratification whose strength ranges over two orders of magnitude between different simulations. Provided the plume is sufficiently energetic to penetrate into the stratified layer, internal waves are generated with frequencies in a relatively narrow band moderately smaller than the buoyancy frequency. Through adaptations of ray theory including viscosity and use of dynamic mode decomposition, we show that the waves originate from within the turbulent flow rather than at the turbulent/non-turbulent interface between the fountain top and the surrounding stratified fluid.

We report the first shock-tube experiments on Richtmyer–Meshkov instability at a single-mode light–heavy interface accelerated by a strong shock wave with Mach number higher than 3.0. Under the proximity effect of the transmitted shock and its induced secondary compression effect, the interface profile is markedly different from that in weakly compressible flows. For the first time, the validity of the compressible linear theory and the failure of the impulsive model in predicting the linear amplitude evolution in highly compressible flows are verified through experiments. Existing nonlinear and modal models fail to accurately describe the perturbation evolution, as they do not account for the shock proximity and secondary compression effects on interface evolution. The shock proximity effect manifests mainly in the early stages when the transmitted shock remains close to the interface, while the effect of secondary compression manifests primarily at the period when interactions of transverse shocks occur at the bubble tips. Based on these findings, we propose an empirical model capable of predicting the bubble evolution in highly compressible flows.

The dynamics of evolving fluid films in the viscous Stokes limit is relevant to various applications, such as the modelling of lipid bilayers in cells. While the governing equations were formulated by Scriven (1960), solving for the flow of a deformable viscous surface with arbitrary shape and topology has remained a challenge. In this study, we present a straightforward discrete model based on variational principles to address this long-standing problem. We replace the classical equations, which are expressed with tensor calculus in local coordinates, with a simple coordinate-free, differential-geometric formulation. The formulation provides a fundamental understanding of the underlying mechanics and translates directly to discretization. We construct a discrete analogue of the system using Onsager's variational principle, which, in a smooth context, governs the flow of a viscous medium. In the discrete setting, instead of term-wise discretizing the coordinate-based Stokes equations, we construct a discrete Rayleighian for the system and derive the discrete Stokes equations via the variational principle. This approach results in a stable, structure-preserving variational integrator that solves the system on general manifolds.

Super-resolution of turbulence is a term used to describe the prediction of high-resolution snapshots of a flow from coarse-grained observations. This is typically accomplished with a deep neural network and training usually requires a dataset of high-resolution images. An approach is presented here in which robust super-resolution can be performed without access to high-resolution reference data, as might be expected in an experiment. The training procedure is similar to data assimilation, wherein the model learns to predict an initial condition that leads to accurate coarse-grained predictions at later times, while only being shown coarse-grained observations. Implementation of the approach requires the use of a fully differentiable flow solver in the training loop to allow for time-marching of predictions. A range of models are trained on data generated from forced, two-dimensional turbulence. The networks have reconstruction errors which are similar to those obtained with ‘standard’ super-resolution approaches using high-resolution data. Furthermore, the methods are comparable to the performance of standard data assimilation for state estimation on individual trajectories, outperforming these variational approaches at initial time and remaining robust when unrolled in time where performance of the standard data-assimilation algorithm improves.

We develop a simple model which describes the repeated injection and extraction of hydrogen in a permeable water-saturated rock which has the form of an anticline. We demonstrate that the flow is controlled by the dimensionless ratio of the square of the buoyancy speed to the product of the two-dimensional volume injection rate and the injection–extraction frequency, and we explore the cases in which this ratio is large and small. Over the first few cycles, the volume of hydrogen in the system gradually builds up since during the extraction phase, some of the water eventually reaches the extraction well, and in our model the system ceases to extract fluid for the remainder of this extraction phase. After many cycles, there is sufficient hydrogen in the system that a quasi-equilibrium state develops in which the mass of fluid injected matches the mass extracted over the course of a cycle. We show that in this equilibrium, the ratio between the mass of gas remaining in the aquifer at the end of the extraction phase, known as the cushion gas, to the mass of gas injected, known as the working gas, decreases if either the flow rate or frequency of the cycles decrease or the buoyancy speed increases, leading to more efficient storage.

We study experimentally the onset of Faraday waves near the end walls of a rectangular vessel containing two stably stratified fluid layers, subject to horizontal oscillations. These subharmonic waves (SWs) are excited, because the horizontal inertial forcing drives a harmonic propagating wave which displaces the interface in the vertical direction at the end walls. We find that the onset of SWs is regulated by a balance between capillary and viscous forces, where the rate of damping is set by the Stokes layer thickness at the wall rather than the wavelength of the SWs. We model the onset of SWs with a weakly damped Mathieu equation and find that the dimensional critical acceleration scales as $\nu _m^{1/2} \omega ^{3/2}$, where $\nu _m$ is the mean viscosity and $\omega$ is the frequency of forcing, in excellent agreement with the experiment over a wide range of parameters.

We present a theory that quantifies the interplay between intrapore and interpore flow variabilities and their impact on hydrodynamic dispersion. The theory reveals that porous media with varying levels of structural disorder exhibit notable differences in interpore flow variability, characterised by the flux-weighted probability density function (PDF), $\hat {\psi }_\tau (\tau ) \sim \tau ^{-\theta -2}$, for advection times $\tau$ through conduits. These differences result in varying relative strengths of interpore and intrapore flow variabilities, leading to distinct scaling behaviours of the hydrodynamic dispersion coefficient $D_L$, normalised by the molecular diffusion coefficient $D_m$, with respect to the Péclet number $Pe$. Specifically, when $\hat {\psi }_\tau (\tau )$ exhibits a broad distribution of $\tau$ with $\theta$ in the range of $(0, 1)$, the dispersion undergoes a transition from power-law scaling, $D_L/D_m \sim Pe^{2-\theta }$, to linear scaling, $D_L/D_m \sim Pe$, and eventually to logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$, as $Pe$ increases. Conversely, when $\tau$ is narrowly distributed or when $\theta$ exceeds 1, dispersion consistently follows a logarithmic scaling, $D_L/D_m \sim Pe\ln (Pe)$. The power-law and linear scaling occur when interpore variability predominates over intrapore variability, while logarithmic scaling arises under the opposite condition. These theoretical predictions are supported by experimental data and network simulations across a broad spectrum of porous media.

Empirical evidence is provided that within the inertial sublayer (i.e. logarithmic region) of adiabatic turbulent flows over smooth walls, the skewness of the vertical-velocity component $S_w$ displays universal behaviour, being a positive constant and constrained within the range $S_w \approx 0.1\unicode{x2013}0.16$, regardless of flow configuration and Reynolds number. A theoretical model is then proposed to explain this behaviour, including the observed range of variations of $S_w$. The proposed model clarifies why $S_w$ cannot be predicted from down-gradient closure approximations routinely employed in large-scale meteorological and climate models. The proposed model also offers an alternative and implementable approach for such large-scale models.

Capsules, which are potentially active fluid droplets enclosed in a thin elastic membrane, experience large deformations when placed in suspension. The induced fluid–structure interaction stresses can potentially lead to rupture of the capsule membrane. While numerous experimental studies have focused on the rheological behaviour of capsules until rupture, there remains a gap in understanding the evolution of their mechanical properties and the underlying mechanisms of damage and breakup under flow. We here investigate the damage and rupture of bioartificial microcapsules made of ovalbumin reticulated with terephthaloyl chloride and placed in simple shear flow. We characterize damage by identifying how the surface shear modulus of the capsule membrane changes over time. Rupture is then characterized by comparing the number and size distribution of capsules before and after exposure to shear, while varying the shear rates and time during which capsules are sheared. Our findings reveal how the percentage of ruptured capsules increases with their size, exposure time to shear, and the ratio of viscous to elastic forces at rupture.

The velocity gradient tensor can be decomposed into normal straining, pure shearing and rigid rotation tensors, each with distinct symmetry and normality properties. We partition the strength of turbulent velocity gradients based on the relative contributions of these constituents in several canonical flows. These flows include forced isotropic turbulence, turbulent channels and turbulent boundary layers. For forced isotropic turbulence, the partitioning is in excellent agreement with previous results. For wall-bounded turbulence, the partitioning collapses onto the isotropic partitioning far from the wall, where the mean shearing is relatively weak. By contrast, the near-wall partitioning is dominated by shearing. Between these two regimes, the partitioning collapses well at sufficiently high friction Reynolds numbers and its variations in the buffer layer and the log-law region can be reasonably modelled as a function of the mean shearing strength. Altogether, our results highlight the expressivity and broad applicability of the velocity gradient partitioning as advantages for turbulence modelling.

Turbulent flow induced by elastorotational instability in viscoelastic Taylor–Couette flow (TCF) with Keplerian rotation is analogous to a turbulent accretion disk destabilized by magnetorotational instability. We examine this novel viscoelastic Keplerian turbulence via direct numerical simulations (DNS) for the shear Reynolds number ($Re$) ranging from $10^2$ to $10^4$. The observed characteristic flow structure consists of penetrating streamwise vortices with axial length scales much smaller than the gap width, distinct from the classic centrifugally induced Taylor vortices, which have axial lengths of the gap width. These intriguing vortices persist for the wide $Re$ range considered and give rise to intriguing scaling behaviour in key flow quantities. Specifically, the characteristic axial length of the penetrating vortices is shown to scale as $Re^{-0.22}$; the angular momentum transport scales as $Re^{0.42}$; the kinetic and elastic boundary-layer thicknesses based on angular velocity and hoop stress near the inner cylinder wall scale as $Re^{-0.48}$ and $Re^{-0.49}$, respectively. This implies that the viscoelastic Keplerian turbulence belongs to the classical turbulent regime of TCF with the Prandtl–Blasius-type boundary layer. Furthermore, we present an analytical relation between the viscous and elastic dissipation rates of kinetic energy and the angular momentum transport and in turn demonstrate its validity using our DNS data. This study has paved the way for future research to explore astrophysics-related Keplerian turbulence and angular momentum transport via the scaling relations of the analogous TCF of dilute polymeric solutions.

We present herein the derivation of a lubrication-mediated (LM) quasi-potential model for droplet rebounds off deep liquid baths, assuming the presence of a persistent dynamic air layer which acts as a lubricating pressure transfer. We then present numerical simulations of the LM model for axisymmetric rebounds of solid spheres and compare quantitatively to current results in the literature, including experimental data in the low-speed impact regime. In this regime the LM model has the advantage of being far more computationally tractable than direct numerical simulation (DNS) and is also able to provide detailed behaviour within the micro-metric thin lubrication region. The LM system has an interesting mathematical structure, with the lubrication layer providing a free-boundary elliptic problem mediating the drop and bath free-boundary evolutionary equations.

Microscopic irregularity (roughness) of bounding surfaces affects macroscopic dynamics of fluid flows. Its effect on bulk flow is usually quantified empirically by means of a roughness coefficient. A new approach, which treats rough surfaces (e.g. parallel plates) as random fields whose statistical properties can be inferred from measurements, is presented. The mapping of a random flow domain onto its deterministic counterpart, and the subsequent stochastic averaging of the transformed Stokes equations, yield expressions for the effective viscosity and roughness coefficient in terms of the statistical characteristics of the irregular geometry of the boundaries. The analytical nature of the solutions allows one to handle surface roughness characterized by short correlation lengths, a challenging feature for numerical stochastic simulations.

Radial unstable stratification is a potential source of turbulence in the cold regions of accretion disks. To investigate this thermal effect, here we focus on two-dimensional Rayleigh–Bénard convection in an annulus subject to radially dependent gravitational acceleration $g \propto 1/r$. Next to the Rayleigh number $Ra$ and Prandtl number $Pr$, the radius ratio $\eta$, defined as the ratio of inner and outer cylinder radii, is a crucial parameter governing the flow dynamics. Using direct numerical simulations for $Pr=1$ and $Ra$ in the range from $10^7$ to $10^{10}$, we explore how variations in $\eta$ influence the asymmetry in the flow field, particularly in the boundary layers. Our results show that in the studied parameter range, the flow is dominated by convective rolls and that the thermal boundary-layer (TBL) thickness ratio between the inner and outer boundaries varies as $\eta ^{1/2}$. This scaling is attributed to the equality of velocity scales in the inner ($u_i$) and outer ($u_o$) regions. We further derive that the temperature drops in the inner and outer TBLs scale as $1/(1+\eta ^{1/2})$ and $\eta ^{1/2}/(1+\eta ^{1/2})$, respectively. The scalings and the temperature drops are in perfect agreement with the numerical data.

The flow of a nematic liquid crystal in a Hele-Shaw cell with an electrically controlled viscous obstruction is investigated using both a theoretical model and physical experiments. The viscous obstruction is created by temporarily electrically altering the viscosity of the nematic in a region of the cell across which an electric field is applied. The theoretical model is validated experimentally for a circular cylindrical obstruction, demonstrating user-controlled flow manipulation of an anisotropic liquid within a heterogeneous single-phase microfluidic device.

The recent work of Siegelman & Young (Proc. Natl Acad. Sci. USA, vol. 120, issue 44, 2023, e2308018120) revealed two extreme states reached by the evolution of unforced and weakly damped two-dimensional turbulence above random rough topography, separated by a critical kinetic energy $E_\#$. The low- and high-energy solutions correspond to topographically locked and roaming vortices, surrounded by non-uniform and homogeneous background potential vorticity (PV), respectively. However, we found that these phenomena are restricted to the particular intermediate length scale where the energy was initially injected into the system. Through simulations initialized by injecting the energy at larger and smaller length scales, we found that the long-term state of topographic turbulence is also dependent on the initial length scale and thus the initial enstrophy. If the initial length scale is comparable to the domain size, the long-term flow field resembles the minimum-enstrophy state proposed by Bretherton & Haidvogel (J. Fluid Mech., vol. 78, issue 1, 1976, pp. 129–154), with very few topographically locked vortices; the long-term enstrophy is quite close to the minimum value, especially when the energy is no larger than $E_\#$. As the initial length scale becomes smaller, more vortices nucleate and become more mobile; the long-term enstrophy increasingly deviates from the minimum value. Simultaneously, the background PV tends to homogenization, even if the energy is below $E_\#$. These results complement the phenomenology of topographic turbulence documented by Siegelman & Young, by showing that the minimum-enstrophy and background PV homogenization states can be adequately approached by large- and small-scale initial fields, respectively, with relatively arbitrary energy.

We discuss the modal, linear stability analysis of generalized Couette–Poiseuille (GCP) flow between two parallel plates moving with relative speed in the presence of an applied pressure gradient vector inclined at an angle $0\leqslant \phi \leqslant 90^\circ$ to the plate relative velocity vector. All possible GCP flows can be described by a global Reynolds number $Re$, $\phi$ and an angle $0\leqslant \theta \leqslant 90^\circ$, where $\cos \theta$ is a measure of the relative weighting of Couette flow to the composite GCP flow. This provides a novel and uncommon group of generally three-dimensional base velocity fields with wall-normal twist, for which Squire's theorem does not generally apply, requiring study of oblique perturbations with wavenumbers $(\alpha,\beta )$. With $(\theta,\phi )$ fixed, the neutral surface $f(\theta,\phi ;Re,\alpha,\beta )=0$ in $(Re, \alpha,\beta )$ space is discussed. A mapping from GCP to plane Couette–Poiseuille flow stability is found that suggests a scaling relation $Re^*\alpha /k = H(\theta ^*)$ that collapses all critical parameters, where ${Re}^*= Re\,({\alpha _1}/{\alpha })\,({\sin \theta }/{\sin {\theta }^*})$ and $\tan \theta ^*=({\alpha _{1}}/\alpha )\tan \theta$, with $\alpha _1=\alpha \cos \phi +\beta \sin \phi$. This analysis does not, however, directly reveal global critical properties for GCP flow. The global $Re_{cr}(\theta,\phi )$ shows continuous variation, while $\alpha _{cr}(\theta,\phi )$ and $\beta _{cr}(\theta,\phi )$ show complex behaviour, including discontinuities owing to jumping of critical states across neighbouring local valleys (in $Re$) or lobes of the neutral surface. The discontinuity behaviour exists for all low $\phi$. For $\phi \gtrsim 21^\circ$, variations of $\alpha _{cr}(\theta )$ and $\beta _{cr}(\theta )$ are generally smooth and monotonic.

The first experimental results on pattern transitions in the co-rotation regime (i.e. the rotation ratio $\varOmega = \omega _o/\omega _i > 0$, where $\omega _i$ and $\omega _o$ are the angular speeds of the inner and outer cylinders, respectively) of the Taylor–Couette flow (TCF) are reported for a neutrally buoyant suspension of non-colloidal particles, up to a particle volume fraction of $\phi = 0.3$. While the stationary Taylor vortex flow (TVF) is the primary bifurcating state in dilute suspensions ($\phi \leq ~0.05$), the non-axisymmetric oscillatory states, such as the spiral vortex flow (SVF) and the ribbon (RIB), appear as primary bifurcations with increasing particle loading, with an overall de-stabilization of the primary bifurcating states (TVF/SVF/RIB) being found with increasing $\phi$ for all $\varOmega \geq ~0$. At small co-rotations ($\varOmega \sim 0$), the particles play the dual role of stabilization ($\phi < 0.1$) and destabilization ($\phi \geq ~0.1$) on the secondary/tertiary oscillatory states. The distinctive features of the ‘particle-induced’ spiral vortices are identified and contrasted with those of the ‘fluid-induced’ spirals that operate in the counter-rotation regime.

Linear unsteady aerofoil theory, while successfully used for the prediction of unsteady aerofoil lift for many decades, has yet to be proven adequate for predicting the propulsive performance of oscillating aerofoils. In this paper we test the hypothesis that the central shortcoming of linear small-amplitude models, such as the Garrick function, is the failure to account for the flow acceleration caused by aerofoil thrust. A new analytical model is developed by coupling the Garrick function to a cycle-averaged actuator disc model, in a manner analogous to the blade-element momentum theory for wind turbines and propellers. This amounts to assuming the Garrick function to be locally valid and, in combination with a global control volume analysis, enables the prediction of flow acceleration at the aerofoil. The new model is demonstrated to substantially improve the agreement with large-eddy simulations of an aerofoil in combined heave and pitch motion.