This paper presents a numerical investigation of the turbulence transition phenomenon in the wake of wall-mounted prisms. Large-eddy simulations are performed at $Re = 1\times 10^3 {-}5\times 10^3$ for prisms with a range of aspect ratio (height to width) from $0.25$ to $1.5$, and depth ratios (length to width) between $1$ and $4$. The results show that the wake irregularity is enhanced with increasing depth ratio, evidenced by higher turbulent kinetic energy (${\approx}90\,\%$) near the leading edge, and the onset of irregular, unsteady vortex shedding. This is attributed to interactions between Kelvin–Helmholtz instability (KHI) of the shear layer and large-scale vortex shedding, and it is induced by an unsteady shear layer, resembling flapping-like motion. These interactions elevate the flow momentum due to increased turbulence intensity and mixing, contributing to the wake transition phenomenon. To this end, this study defines the role of depth ratio in the transition phenomenon by showing that increasing depth ratio (e.g. from $1$ to $4$) leads to earlier onset of KHIs in the shear layer. These instabilities intensify with depth ratio, resulting in stronger interactions between shear layer and large-scale vortex shedding. Specifically, KHI-induced vortices interact more frequently with large-scale wake structures for higher depth ratio prisms, exciting larger flow fluctuations and irregular wake patterns. This interaction alters the frequency and coherence of vortex shedding, revealing a complex coupling mechanism that drives the transition to turbulence.

The transport of particles in elastoviscoplastic (EVP) fluids is of significant interest across various industrial and scientific domains. However, the physical mechanisms underlying the various particle distribution patterns observed in experimental studies remain inadequately understood in the current literature. To bridge this gap, we perform interface-resolved direct numerical simulations to study the collective dynamics of spherical particles suspended in a pressure-driven EVP duct flow. In particular, we investigate the effects of solid volume fraction, yield stress, inertia, elasticity, shear-thinning viscosity, and secondary flows on particle migration and formation of plug regions in the suspending fluid. Various cross-streamline migration patterns are observed depending on the rheological parameters of the carrier fluid. In EVP fluids with constant plastic viscosity, particles aggregate into a large cluster at the duct centre. Conversely, EVP fluids with shear-thinning plastic viscosity induce particle migration towards the duct walls, leading to formation of particle trains at the corners. Notably, we observe significant secondary flows ($O(10^{-2})$ compared to the mean velocity) in shear-thinning EVP suspensions, arising from the interplay of elasticity, shear-thinning viscosity and particle presence, which further enhances corner-ward particle migration. We elucidate the physical mechanism by which yield stress augments the first normal stress difference, thereby significantly amplifying elastic effects. Furthermore, through a comprehensive analysis of various EVP suspensions, we identify critical thresholds for elasticity and yield stress necessary to achieve particle focusing at the duct corners.

A collection of secondary instability calculations in streaky boundary layers is presented. The data are retrieved from well-resolved numerical simulations of boundary layers forced by free-stream turbulence (FST), considering different geometries and FST conditions. The stability calculations are performed before streak breakdown, taking place at various $Rey_x$ the Reynolds number based on the streamwise coordinate. Despite the rich streak population of various sizes, it is found that breaking streaks have similar aspect ratios, independently of the streamwise position where they appear. This suggests that wider streaks will break down further downstream than thinner ones, making the appearance of secondary instabilities somewhat independent of the streak’s wavelength. Moreover, the large difference in the integral length scale among the simulations suggests that this aspect ratio is also independent of the FST scales. An explanation for this behaviour is provided by showing that these breaking streaks are in the range of perturbations that can experience maximum transient growth according to optimal disturbance theory. This could explain why, at a given streamwise position, there is a narrow spanwise wavelength range where streak breakdown is more likely to occur.

Large-eddy simulations are analysed to determine the influence of suspended canopies, such as those formed in macroalgal farms, on ocean mixed layer (OML) deepening and internal wave generation. In the absence of a canopy, we show that Langmuir turbulence, when compared with wind-driven shear turbulence, results in a deeper OML and more pronounced internal waves beneath the OML. Subsequently, we examine simulations with suspended canopies of varying densities located in the OML, in the presence of a background geostrophic current. Intensified turbulence occurs in the shear layer at the canopy’s bottom edge, arising from the interaction between the geostrophic current and canopy drag. Structures resembling Kelvin–Helmholtz (KH) instability emerge as the canopy shear layer interacts with the underlying stratification, radiating internal waves beneath the OML. Both intensified turbulence and lower-frequency motions associated with KH-type structures are critical factors in enhancing mixing. Consequently, the OML depth increases by up to a factor of two compared with cases without a canopy. Denser canopies and stronger geostrophic currents lead to more pronounced KH-type structures and internal waves, stronger turbulence and greater OML deepening. Additionally, vertical nutrient transport is enhanced as the OML deepens due to the presence of the canopy. Considering that the canopy density investigated in this study closely represents offshore macroalgal farms, these findings suggest a mechanism for passive nutrient entrainment conducive to sustainable farming. Overall, this study reveals the intricate interactions between the suspended canopy, turbulent mixing and stratification, underscoring their importance in reshaping OML characteristics.

Experiments are presented to explore the non-axisymmetric instabilities of spreading films of aqueous suspensions of Carbopol and Xanthan gum floating on a bath of perfluoropolyether oil. The experimental observations are compared against theoretical predictions exploiting a shallow-film model in which the viscoplastic rheology is captured by the Herschel–Bulkley constitutive law. With this model, we construct axisymmetric base states that evolve from the moment that the film floats onto the bath, out towards long times at which spreading becomes self-similar, and then test their linear stability towards non-axisymmetric perturbations. In the geometry of a thinning expanding film, we find that shear thinning does not drive a loss of axisymmetry at early times (when the degree of expansion is small), but when the film has expanded in radius by a factor of two or so, shear-thinning hoop stresses drive non-axisymmetric instabilities. Unstable modes possess relatively low angular wavenumber, and the loss of symmetry is not particularly dramatic. When the oil in the bath is replaced by salty water, the experiments are completely different, with dramatic non-axisymmetric patterns emerging from interfacial effects.

We perform direct numerical simulations of turbulent channel flows. Secondary motions are produced by applying a streamwise-homogeneous, spanwise-heterogeneous roughness pattern of spanwise period $\Lambda _s$ to the walls of the channel; their time evolution is observed. Notice that, owing to the geometry, the secondary motions are streamwise-invariant at any instant of time, so that no spatial development is seen. Once the secondary motions reach a statistically steady state, the roughness pattern is suddenly removed, so that the secondary motions decay. The time needed for the secondary motions to vanish is then measured; in doing so, we distinguish between the streamwise-momentum pathways and the cross-sectional circulatory motions that compose the secondary motions. Larger values of $\Lambda _s$ are generally associated with a longer time scale for the decay of the momentum pathways, although this might not hold true for $\Lambda _s/h\gt 4$ (where $h$ is the channel half-height). The value of such a time scale for the circulatory motions, instead, saturates for $\Lambda _s/h \geqslant 2$; this may be related to the observed spatial confinement of said circulatory motions. For specific values of $\Lambda _s$ ($2 \leqslant \Lambda _s/h \leqslant 4$), the volume-averaged energy associated with the momentum pathways undergoes an unexpected transient growth with respect to its value at the beginning of the decay. This might indicate that structures of such a specific size are able to self-sustain as postulated by Townsend (The Structure of Turbulent Shear Flow, 2nd edition, 1976, ch. 7.19); the evidence we gather in this respect is however inconclusive. Finally, the present data suggest that most of the energy of the momentum pathways is produced by the circulatory motions transporting the mean (spanwise-averaged) velocity.

Linear and nonlinear mechanisms governing the growth of second-mode waves are analysed using a newly derived disturbance energy conservation equation that highlights the physical processes responsible for fluctuation energy production, flux-based transport and destruction. Axisymmetric direct numerical simulations (DNS) data from a Mach 6 hypersonic boundary layer, simulated over a $3^\circ$ half-angle sharp cone at zero angle of attack, is used as a reference. A Legendre polynomial-based forcing methodology is used to trigger transition in the DNS over a range of various amplitude levels and different forcing frequency content. Closure of the disturbance energy budgets is demonstrated numerically using the DNS data. The terms responsible for the amplification of the disturbance are identified, and nonlinear attenuation effects are discussed. We show that the interaction between the entropy/velocity fluctuations and the base temperature gradient governs the second-mode growth in the linear regime. Energy production occurs in the critical layer due to non-isentropic processes and accumulates in acoustic form below the relative sonic line through downward transport. At higher forcing amplitudes, nonlinear spectral broadening is observed, with simultaneous thermoviscous diffusion attenuating the disturbance energy growth. This effect is responsible for the non-monotonic streamwise variation of the wall-pressure spectrum. Phase speed and growth rate analyses, informed by linear stability theory (LST), reveal wave steepening effects preceding this nonlinear attenuation effect. The disturbance energy is observed to match the LST predictions at lower forcing amplitudes, deviating, as expected, at higher amplitudes.

Inertia–gravity waves are scattered by background flows as a result of Doppler shift by a non-uniform velocity. In the Wentzel–Kramers–Brillouin regime, the scattering process reduces to a diffusion in spectral space. Other inhomogeneities that the waves encounter, such as density variations, also cause scattering and spectral diffusion. We generalise the spectral diffusion equation to account for these inhomogeneities. We apply the result to a rotating shallow-water system, for which height inhomogeneities arise from velocity inhomogeneities through geostrophy, and to the Boussinesq system for which buoyancy inhomogeneities arise similarly. We compare the contributions that height and buoyancy variations make to the spectral diffusion with the contribution of the Doppler shift. In both systems, we find regimes where all contributions are significant. We support our findings with exact solutions of the diffusion equation and with ray tracing simulations in the shallow-water case.

Particle image velocimetry is used to study the control of swirl momentum, delivered through an orifice formed by a physically rotating tube of finite length, relevant to the evolution of vortex rings produced at a Reynolds number ${Re}\approx 1000$ based on the average discharge velocity, for swirl numbers ${S} \in [0, 1]$. Experiments without discharge, reinforced with complimentary numerical predictions, reveal the presence of an intriguing secondary flow pattern in the rotating tube, preventing attainment of a solid-body-like swirl distribution. Nevertheless, it is found that fully established rings produced in this way, following discharge once conditions in the tube have reached a steady state, exhibit similar characteristics to rings formed by an otherwise solid-body rotating initial condition as explored computationally by Ortega-Chavez et al. (2023, J. Fluid Mech. 967, A16). Namely, opposite-signed vorticity forms due to vortex tilting, which subsequently interacts with the ring, promoting vorticity cancellation and vortex ring breakdown. A key feature of the experimental work is that partially established vortex rings, produced before a steady-state rotating tube condition is reached, show unique characteristics. Their creation, a short time after the onset of tube rotation: (i) facilitates more efficient delivery of swirl momentum to the vortex core area; (ii) maintains a low level of swirl in the ring bubble’s central region which would otherwise promote the formation of opposite-signed vorticity and vortex breakdown.

Three-dimensional eddy-resolving simulations are used to study the structure of turbulent boundary layers developing in an open channel where an array of sparse roughness elements in the form of partially burrowed mussels is placed on the smooth, flat channel bed starting at a certain streamwise location. The identical mussels are oriented with their major axis parallel to the mean flow. Their positions are randomised while making sure that the mussels are close to uniformly distributed inside the array. The turbulent flow approaching the leading edge of the array is fully developed. The protruding parts of the mussels play the role of sparse roughness elements and generate a rough-bed, internal boundary that is characterised by zero net flow exchange but non-zero local flow exchange due to active filtering. A double-averaging technique is used to obtain an equivalent, width- and time-averaged, boundary layer over a ‘flat’ rough surface containing no mussels. The paper discusses the effects of varying the mussel array density, protruding height of the mussels and filtering discharge on the spatial growth of the two-dimensional boundary layer. With proper scaling, the profiles of the (double-averaged) streamwise velocity are close to self-similar inside the inertial layer (e.g. for h $\lt$ z $\lt$ $\delta$, where h is the height of the protruding part of the mussels and $\delta$ is the boundary layer thickness) starting some distance from the leading edge of the array. The scaled turbulent kinetic energy and concentration profiles associated with the scalar advected through the excurrent syphons are also found to be self-similar above the vertical location where the maximum value is reached. An analytical model containing three subzones is proposed for the streamwise velocity in between the bed (z $=$ 0) and z $=$ $\delta$. The velocity profile inside the inertial region contains a law-of-the-wall component supplemented by a law-of-the-wake component. The scaling coefficient of the law-of-the-wake component is found to be larger than typical values used to describe velocity variation in turbulent boundary layers developing in a surrounding flow with close-to-uniform free stream velocity. The equivalent roughness height for this particular type of boundary layers developing over sparse roughness elements increases monotonically with h and the mussel array density, $\rho$N. The paper also discusses the effect of the mussel bed density on the average refiltration fraction and the phytoplankton removal efficiency of the mussel bed.

Cilia perform various functions, including sensing, locomotion, generation of fluid flows and mass transport, serving to underpin a vast range of biological and ecological processes. However, analysis of the mass transport typically fails to resolve the near-field dynamics around individual cilia, and therefore overlooks the intricate role of power/recovery strokes of ciliary motion. Selvan et al. (2023, Phys. Rev. Fluids 8, 123103) observed that the flow field due to a point torque (i.e. a rotlet) accurately resolves both the near- and far-field characteristics of a single cilium’s flow in a semi-infinite domain. In this paper, we calculate the mass transport between a no-slip boundary and an adjacent fluid, as a model system for nutrient exchange with ciliated tissues. We develop a Langevin model in the presence of a point torque (i.e. a single cilium) to examine the nutrient flux from a localised surface source. This microscopic transport model is validated using a macroscopic continuum model, which directly solves the advection–diffusion equation. Our findings reveal that the flow induced by a point torque can enhance the particles’ transport, depending on their diffusivity and the magnitude of the point torque. Additionally, the average mass transport affected by a single cilium can be enhanced or diminished by the presence of an externally imposed linear shear flow, with a strong dependence on the alignment of the cilium. Taken together, this framework serves as a useful minimal model for examining the average nutrient exchange between ciliated tissues and fluid environments.

The Liebau effect generates a net flow without the need for valves. For the Liebau effect pumping phenomenon to occur, the pump must have specific characteristics. It needs tubes with different elastic properties and an actuator to provide energy to the fluid. The actuator periodically compresses the more flexible element. Furthermore, asymmetry is a crucial factor that differentiates between two pumping mechanisms: impedance pumping and asymmetric pumping. In this work, a model based on the fluid dynamics of an asymmetric valveless pump under resonant conditions is proposed to determine which parameters influence the pumped flow rate. Experimental work is used to validate the model, after which each of the parameters involved in the pump performance is dimensionlessly analysed. This highlights the most significant parameters influencing the pump performance such as the actuator period, length tube ratio and tube diameters. The results point out ways to increase a valveless asymmetric pump’s net-propelled flow rate, which has exciting applications in fields such as biomedicine. The model also allows for predicting the resonance period, a fundamental operating parameter for asymmetric pumping.

The dynamics of small-scale structures in free-surface turbulence is crucial to large-scale phenomena in natural and industrial environments. Here, we conduct experiments on the quasi-flat free surface of a zero-mean-flow turbulent water tank over the Reynolds number range $Re_{\lambda } = 207$–312. By seeding microscopic floating particles at high concentrations, the fine scales of the flow and the velocity-gradient tensor are resolved. A kinematic relation is derived expressing the contribution of surface divergence and vorticity to the dissipation rate. The probability density functions of divergence, vorticity and strain rate collapse once normalised by the Kolmogorov scales. Their magnitude displays strong intermittency and follows chi-square distributions with power-law tails at small values. The topology of high-intensity events and two-point statistics indicate that the surface divergence is characterised by dissipative spatial and temporal scales, while the high-vorticity and high-strain-rate regions are larger, long-lived, concurrent and elongated. The second-order velocity structure functions obey the classic Kolmogorov scaling in the inertial range when the dissipation rate on the surface is considered, with a different numerical constant than in three-dimensional turbulence. The cross-correlation among divergence, vorticity and strain rate indicates that the surface-attached vortices are strengthened during downwellings and diffuse when those dissipate. Sources (sinks) in the surface velocity fields are associated with strong (weak) surface-parallel stretching and compression along perpendicular directions. The floating particles cluster over spatial and temporal scales larger than those of the sinks. These results demonstrate that, compared with three-dimensional turbulence, in free-surface turbulence the energetic scales leave a stronger imprint on the small-scale quantities.

Elastoinertial turbulence (EIT) is a chaotic state that emerges in the flows of dilute polymer solutions. Direct numerical simulation (DNS) of EIT is highly computationally expensive due to the need to resolve the multiscale nature of the system. While DNS of two-dimensional (2-D) EIT typically requires $O(10^6)$ degrees of freedom, we demonstrate here that a data-driven modelling framework allows for the construction of an accurate model with 50 degrees of freedom. We achieve a low-dimensional representation of the full state by first applying a viscoelastic variant of proper orthogonal decomposition to DNS results, and then using an autoencoder. The dynamics of this low-dimensional representation is learned using the neural ordinary differential equation (NODE) method, which approximates the vector field for the reduced dynamics as a neural network. The resulting low-dimensional data-driven model effectively captures short-time dynamics over the span of one correlation time, as well as long-time dynamics, particularly the self-similar, nested travelling wave structure of 2-D EIT in the parameter range considered.

We review some of the processes leading to dispersion and mixing in porous media, exploring the differences between the travel time distribution of fluid particles within a pore throat and between pore throats of different size within the porous layer. A recent paper of Liu et al. has combined a model of these travel time distributions with a continuous time random walk to quantify the dispersion as a function of the Peclet number. We describe some further problems relating to dispersive mixing of tracer which may be amenable to this approach, including dispersion caused by macroscopic lenses of different permeability, dispersion of tracer which partitions between the fluid and matrix and the effects of buoyancy on mixing.

We derive leading-order governing equations and boundary conditions for a sheet of viscous fluid retracting freely under surface tension. We show that small thickness perturbations about a flat base state can lead to regions of compression, where one or both of the principal tensions in the sheet becomes negative, and thus drive transient buckling of the sheet centre-surface. The general theory is applied to the simple model problem of a retracting viscous disc with small axisymmetric thickness variations. Transient growth in the centre-surface is found to be possible generically, with the dominant mode selected depending on the imposed initial thickness and centre-surface perturbations. An asymptotic reduction of the boundary conditions at the edge of the disc, valid in the limit of large normalised thickness perturbations, reduces the centre-surface evolution equation to an ordinary differentional equation (ODE) eigenvalue problem. Analysis of this eigenvalue problem leads to insights such as how the degree of transient buckling depends on the imposed thickness perturbation and which thickness perturbation gives rise to the largest transient buckling.

We investigate viscous dissipation in linear flows driven by small-amplitude longitudinal librations in rotating fluid spheres focusing on the rapid rotation regime applicable to planets. Viscous coupling can resonate with inertial modes in the bulk of the fluid when the frequency of the forcing is within the range $(0,2\Omega _0)$, where $\Omega _0$ is the mean angular velocity of the sphere. We solve the linearised equations of motion with a semi-spectral numerical method and with an asymptotic expansion exploiting the small Ekman number, $E$, which quantifies the strength of viscous forces relative to the Coriolis force. Our results confirm that the dominant contribution to the dissipation occurs in the Ekman boundary layer with leading-order scaling $E^{1/2}$. When the forcing frequency coincides with that of an inertial mode, dissipation is reduced by as much as 9 % compared with boundary layer theory alone. The percentage-wise reduction is independent of $E$ and the frequency width of the reduction envelope scales as $E^{1/2}$. At non-resonant frequencies conic shear layers develop in the bulk interior and, together with the Ekman layer bulge at critical latitude, slightly enhance dissipation. We confirm critical latitude bulge and shear layer contributions to the overall dissipation scale as $E^{4/5}$ and $E^{6/5}$ respectively, becoming negligible compared with dissipation in the main boundary layer as $E\rightarrow 0$. The frequencies at which the dissipation enhancement from critical latitude effects is maximised are displaced from the inviscid limit periodic orbit frequencies by a factor that scales with $E^{0.23}$.

The intrinsic uncertainty of fluid properties, including the equation-of-state, viscosity and thermal conductivity, on boundary layer stability has scarcely been addressed. When a fluid is operating in the vicinity of the Widom line (defined as the maximum of isobaric specific heat) in supercritical state, its properties exhibit highly non-ideal behavior, which is an ongoing research field leading to refined and more accurate fluid property databases. Upon crossing the Widom line, new mechanisms of flow instability emerge, feasibly leading to changes in dominating modes that yield turbulence. The present work investigates the sensitivity of three-dimensional boundary layer modal instability to these intrinsic uncertainties in fluid properties. The uncertainty, regardless of its source and the fluid regimes, gives rise to distortions of all profiles that constitute the inputs of the stability operator. The effect of these distortions on flow stability is measured by sensitivity coefficients, which are formulated with the adjoint operator and validated against linear modal stability analysis. The results are presented for carbon dioxide at a representative supercritical pressure of approximately 80 bar. The sensitivity to different inputs of the stability operator across various thermodynamic regimes shows an immense range of sensitivity amplitude. A balancing relationship between the density gradient and its perturbation leads to a quadratic effect across the Widom line, provoking significant sensitivity to distortions of the second derivative of the pressure with respect to the density, $\partial ^2 p/\partial \rho ^2$. From an application-oriented point of view, one important question is whether the correct baseflow profiles can be meaningfully analysed by the simplified ideal-fluid model. The integrated modal disturbance growth – the N factor calculated with different partly idealised models – indicates that the answer depends strongly on the thermodynamic regime investigated.

We consider the flow of a volume $\mathcal {V} = q t^\alpha$ of viscous fluid injected into a gap $H$ between two horizontal plates ($q$ and $\alpha$ are positive constants, $t$ is time). When the viscosity of the displaced fluid is negligible, the injected fluid forms a slug in contact with both plates connected (at a moving grounding line) to a gravity current (GC) with a downward-inclined interface. Hutchinson et al. (J. Fluid Mech., 598, 2023, pp. A4–1–13) considered a constant source ($\alpha = 1$) of Newtonian fluid at the center of an axisymmetric gap; the flow, governed by the parameter $J$ (the height ratio of the unconfined GC to $H$), admits a similarity solution. Here, the self-similar flow theory is (a) extended to rectangular geometry and power-law fluids, and (b) simplified. Similarity appears when $\alpha = n/(n+1)$ (two-dimensional) and $\alpha = 2n/(n+1)$ (axisymmetric), with propagation $\sim t^\beta$, where $\beta /\alpha = 1$ and $1/2$, respectively, and $n-1$ is the power of the shear in the viscosity law ($n=1$ for Newtonian fluid). The flow is governed by a single parameter $J$, representing the above-mentioned ratio. For small $J$, the GC is mostly unconfined; for large $J$, almost all the injected fluid is in contact with both boundaries of the gap. For given geometry and $n$, we solve one ordinary differential equation (ODE) for the reduced thickness over the reduced length $0\lt y \leqslant 1$, with a singular-regular condition at $y=1$. The details of the confined GC, functions of $J$, follow by simple formulae.

The present study aims to provide an understanding of the influence of an afterbody on the flow-induced vibration (FIV) of cylinders. This is achieved through experimental and numerical investigations into the FIV response of a reverse-D-cross-section cylinder of aspect ratio $AR=5$. By carefully monitoring the point of flow separation to show it always occurs at the sharp top and bottom edges and never further upstream, it is demonstrated that vortex-induced vibration (VIV) can occur without an afterbody. However, for other aspect ratios, an afterbody does play a crucial role in determining the type of fluid forces responsible for sustaining VIV from low to moderate Reynolds numbers in the range $100$–$4700$. For a cylinder without an afterbody, it is found that the viscous force originating from the presence of strong compact vortices forming close to the leeward side of the cylinder is responsible for sustaining strong transverse vibration. In contrast, for a cylinder with an afterbody, the dominant force component depends on the size of the afterbody. In cylinders with a small afterbody, such as a reverse-D semi-circular cylinder, the viscous force dominates, while in cylinders with a larger afterbody such as a circular cylinder, the pressure force dominates.