Rotating convection is considered on the tilted $f$-plane where gravity and rotation are not aligned. For sufficiently large rotation rates, $\Omega$, the Taylor–Proudman effect results in the gyroscopic alignment of anisotropic columnar structures with the rotation axis giving rise to rapidly varying radial length scales that vanishes as $\Omega ^{-1/3}$ for $\Omega \rightarrow \infty$. Compounding this phenomenon is the existence of viscous (Ekman) layers adjacent to the impenetrable bounding surfaces that scale as $\Omega ^{-1/2}$. In this investigation, these constraints are relaxed upon utilising a non-orthogonal coordinate representation of the fluid equations where the upright coordinate aligns with rotation axis. This exposes the problem to asymptotic perturbation methods that permit: (i) relaxation of the constraints of gyroscopic alignment; (ii) the filtering of Ekman layers through the uncovering of parameterised velocity pumping boundary conditions; and (iii) the development of reduced quasi-geostrophic systems valid in the limit $\Omega \rightarrow \infty$. Linear stability investigations reveal excellent quantitative agreement between results from parameterised or unapproximated mechanical boundary conditions. For no-slip boundaries, it is demonstrated that the associated Ekman pumping alters convective onset through an enhanced destabilisation of large spatial scales. The range of unstable modes at a fixed thermal forcing is thus significantly extended with a direct dependence on $\Omega$. This holds true even for geophysical and astrophysical regimes characterised by extreme values of the non-dimensional Ekman number $E$. The nonlinear regime is explored via the global heat and momentum transport of single-mode solutions to the quasi-geostrophic systems which indicate $O(1)$ changes which do not scale with the size of $E$.

Recognising that surfactants can impede the drag reduction resulting from superhydrophobic surfaces (SHS), we investigate the impact of spatio–temporal fluctuations in surfactant concentration on the drag-reduction properties of SHS. We model the unsteady transport of soluble surfactant in a channel flow bounded by two SHS. The flow is laminar, pressure driven and the SHS are periodic in the streamwise and spanwise directions. We assume that the channel length is much longer than the streamwise period, the streamwise period is much longer than the channel height and spanwise period, and bulk diffusion is sufficiently strong for cross-channel concentration gradients to be small. By combining long-wave and homogenisation theories, we derive an unsteady advection–diffusion equation for surfactant-flux transport over the length of the channel, which is coupled to a quasi-steady advection–diffusion equation for surfactant transport over individual plastrons. As diffusion over the length of the channel is typically small, the surfactant flux is governed by a nonlinear wave equation. In the fundamental case of the transport of a bolus of surfactant, we predict its propagation speed and describe its nonlinear evolution via interaction with the SHS. The propagation speed can fall below the average streamwise velocity as the surfactant adsorbs and rigidifies the plastrons. Smaller concentrations of surfactant are advected faster than larger ones, so that wave-steepening effects can lead to shock formation in the surfactant-flux distribution. Our asymptotic results reveal how unsteady surfactant transport can affect the spatio–temporal evolution of the slip velocity, drag reduction and effective slip length in SHS channels.

The coherent vortical structures in turbulent flow through a strong 16 : 1 three-dimensional contraction are studied using time-resolved volumetric measurements. Visualization using the vorticity magnitude criterion shows the emergence of long, stretched cylindrical vortices aligned with the mean flow. This alignment is quantified by probability density functions (p.d.f.s) of the direction cosines. We propose two measures to quantify the alignment, the peak height in the probability and a coefficient from the moment of the p.d.f., both of which reaffirm the strong streamwise alignment. The root mean square streamwise vorticity grows within the contraction to become 4.5 times larger than the transverse component, at the downstream location where the contraction ratio is $C=11$. The characteristic vortices become as long as the measurement volume, or more than 4 times the integral scale at the entrance to the contraction. We also characterize the vorticity enhancement along individual vortices, measuring 65 % strengthening over the distance where $C$ goes from 4 to 11. The prevalence of these coherent structures is estimated from 700 000 measured volumes, showing that near the outlet, it is more likely to have one or two of these structures present than none.

This work investigates the spatio-temporal evolution of coherent structures in the wake of a generic high-speed train, based on a three-dimensional database from large eddy simulation. Spectral proper orthogonal decomposition (SPOD) is used to extract energy spectra and energy ranked empirical modes for both symmetric and antisymmetric components of the fluctuating flow field. The spectrum of the symmetric component shows overall higher energy and more pronounced low-rank behaviour compared with the antisymmetric one. The most dominant symmetric mode features periodic vortex shedding in the near wake, and wave-like structures with constant streamwise wavenumber in the far wake. The mode bispectrum further reveals the dominant role of self-interaction of the symmetric component, leading to first harmonic and subharmonic triads of the fundamental frequency, with remarkable deformation of the mean field. Then, the stability of the three-dimensional wake flow is analysed based on two-dimensional local linear stability analysis combined with a non-parallelism approximation approach. Temporal stability analysis is first performed for both the near-wake and the far-wake regions, showing a more unstable condition in the near-wake region. The absolute frequency of the near-wake eigenmode is determined based on spatio-temporal analysis, then tracked along the streamwise direction to find out the global mode growth rate and frequency, which indicate a marginally stable global mode oscillating at a frequency very close to the most dominant SPOD mode. The global mode wavemaker is then located, and the structural sensitivity is calculated based on the direct and adjoint modes derived from a local spatial analysis, with the maximum value localized within the recirculation region close to the train tail. Finally, the global mode shape is computed by tracking the most spatially unstable eigenmode in the far wake, and the alignment with the SPOD mode is computed as a function of streamwise location. By combining data-driven and theoretical approaches, the mechanisms of coherent structures in complex wake flows are well identified and isolated.

Reynolds-averaged models for solving the Navier–Stokes equations are implicitly based on Kolmogorov's theory for describing energy transfers between the different turbulent scales, which means that all the energy produced at large scales is transferred at a constant rate to the smallest turbulent scales where it is dissipated. As a result, these models use a single scale to describe the turbulence spectrum, which in cases of non-equilibrium turbulence does not provide an adequate description of the transfers actually observed. This is particularly the case for wall-bounded flows at high Reynolds numbers, such as turbulent channel flows. Taking up an approach developed by Schiestel (2007 Modeling and Simulation of Turbulent Flows, ISTE Ltd and John Wiley & Sons), which aims to define a Reynolds-averaged Navier–Stokes model transporting several scales of turbulence, a two-scale Reynolds stress model (RSM) was developed in order to take into account the interactions between the inner and outer regions of wall-bounded flows. The results obtained with the model are compared with the direct numerical simulations (DNS) of Lee & Moser (J. Fluid Mech., vol. 860, 2019, pp. 886–938) in a turbulent channel for several friction Reynolds numbers up to $Re_{\tau }=5200$, for which partial integrations in spectral space were carried out, highlighting distinct behaviours between small and large scales of turbulence. The model developed provides an accurate description of the contributions at small and large scales and thus reproduces the high-Reynolds-number effects observed in DNS data. In addition, comparisons with the DNS data served to validate a large part of the closure relations used for the various terms in the two-scale RSM.

The unsteady flow physics of wind-turbine wakes under dynamic forcing conditions are critical to the modelling and control of wind farms for optimal power density. Unsteady forcing in the streamwise direction may be generated by unsteady inflow conditions in the atmospheric boundary layer, dynamic induction control of the turbine or streamwise surge motions of a floating offshore wind turbine due to floating-platform oscillations. This study seeks to identify the dominant flow mechanisms in unsteady wakes forced by a periodic upstream inflow condition. A theoretical framework for the problem is derived, which describes travelling-wave undulations in the wake radius and streamwise velocity. These dynamics encourage the aggregation of tip vortices into large structures that are advected along in the wake. Flow measurements in the wake of a periodically surging turbine were obtained in an optically accessible towing-tank facility, with an average diameter-based Reynolds number of 300 000 and with surge-velocity amplitudes of up to 40 % of the mean inflow velocity. Qualitative agreement between trends in the measurements and model predictions is observed, supporting the validity of the theoretical analyses. The experiments also demonstrate large enhancements in the recovery of the wake relative to the steady-flow case, with wake-length reductions of up to 46.5 % and improvements in the available power at 10 diameters downstream of up to 15.7 %. These results provide fundamental insights into the dynamics of unsteady wakes and serve as additional evidence that unsteady fluid mechanics can be leveraged to increase the power density of wind farms.

We present the results of an experimental study of buoyancy-driven exchange flows in a vertical pipe, where the lower fluid is Newtonian of low viscosity and the upper fluid has a yield stress. The fluids are initially separated by a gate valve, opened at time $\hat {t}=0$. The fluids are miscible, but away from the diffusive limit. For a sufficiently large ratio $Y$, of the yield stress to the buoyancy stress, no sustained fluid motions arise: the flow is stable. For smaller $Y$ numbers an exchange flow results. Commonly, the less dense fluid penetrates upwards in a central finger, displacing the upper fluid downwards around the walls of the pipe. Three regimes are classified: helical finger, disconnected finger and slug flow. The transition between regimes is governed by increasing relevance of inertial to viscous stresses, in balancing buoyancy. The disconnected finger and slug flow regimes are associated with yielded fluid at the interface and early growth of instabilities. Helical fingers are viscous dominated and evolve slowly until late in the experiments. The scenarios studied represent an idealised set-up for the industrial process of plug cementing. The regimes identified are helpful for industrial process design.

This work presents models for the behaviour of both upstream- and downstream-travelling waves in screeching elliptical jets. Proper orthogonal decomposition is performed on experimental velocity data in both the major and minor axis planes, for an aspect ratio $AR=2$ converging elliptical jet operating at nozzle pressure ratios of $2.6$ and $3.4$. From this decomposition, the radial and axial structure of the guided-jet mode (GJM) and the Kelvin–Helmholtz instability are educed. Linear-stability analysis (LSA) is performed using both the experimentally obtained mean flow, and one obtained using Reynolds-averaged Navier–Stokes (RANS) at matched conditions. It is shown that the wavenumber predicted by LSA for both waves are within the range of experimentally observed wavenumbers. Furthermore, the model accurately predicts the structure of these waves at multiple axial locations, using either the experimental or RANS mean flow. Most critically, it is demonstrated that the GJM is only predicted to be neutrally stable at the screech frequency for a relatively limited streamwise domain, the size and location of which is dependent on the nozzle pressure ratio. A comparison with the amplitude envelope for the GJM extracted from the experimental measurements indicates that the maximum fluctuations associated with the GJM are collocated with this region of the flow that is predicted to support the GJM. While there have been extensive discussions about the frequency dependence of the GJM, this is the first demonstration that its existence is highly dependent on streamwise position within the flow.

We consider the direct numerical simulation of the flow over a forward-facing step protruding in a turbulent boundary layer. Proper orthogonal decomposition (POD) is applied to the velocity field in different regions using Fourier modes in the spanwise direction. The upstream flow is characterized by a structure with a spanwise modulation of the order of the step height, the origin of which is consistent with a centrifugal instability. The structure is associated with ejections over the step of low-speed fluid from the upstream recirculation, and organized into streaks through the action of strong spanwise motions along the step face. The spanwise-averaged instantaneous momentum deficit created by the ejections is directly related to the maximal shear rate at the edge of the step, and is well correlated with the dynamics of the downstream recirculation. The most energetic patterns consist of three-dimensional motions with a large spanwise wavelength located in the shear layer developing at the edge of the step, as well as two-dimensional fluctuations downstream of the reattachment. A linear model based on the interaction of the mean flow with the dominant POD modes is able to recover the main frequencies of the fluctuations at these wavenumbers. Including the time variations of the ejections into the model yields temporal spectra that resemble qualitatively those computed from the simulation. The results suggest that the global dynamics of the flow are at least partly driven by linear mechanisms and depend on the characteristic structure identified in the upstream region close to the step.

The effects of the evolution of vortices on the aeroacoustics generated by a hovering wing are numerically investigated by using a hybrid method of an immersed boundary–finite difference method for the three-dimensional incompressible flows and a simplified model based on the Ffowcs Williams-Hawkings acoustic analogy. A low-aspect-ratio ($AR=1.5$) rectangular wing at low Reynolds ($Re=1000$) and Mach ($M=0.04$) numbers is investigated. Based on the simplified model, the far-field acoustics is shown to be dominated by the time derivative of the pressure on the wing surface. Results show that vortical structure evolution in the flow fields, which is described by the divergence of the convection term of the incompressible Navier–Stokes equations in a body-fixed reference frame, determines the time derivative of the surface pressure and effectively the far-field acoustics. It dominates over the centrifugal acceleration and Coriolis acceleration terms in determining the time derivative of the surface pressure. The position of the vortex is also found to affect the time derivative of the surface pressure. A scaling analysis reveals that the vortex acoustic source is scaled with the cube of the flapping frequency.

Most of the existing theories on electrophoresis are based on the consideration of a weak applied electric field and ions as point charges, which create a mean electric potential and neglect ion–solvent interactions. These theories cannot demonstrate the dependence of electrophoretic mobility on the applied electric field (nonlinear electrophoresis), reversal in mobility with increasing ion concentration and/or surface charge density or counterion saturation in the electric double layer. In this study we consider a modified electrokinetic model to analyse nonlinear electrophoresis by taking into account the finite ion size effects and ion–ion electrostatic correlations. In this approach, the mean-field-based model is extended to capture the many-body phenomena by considering the non-local electrostatic contribution in the ion free energy functional and the ion–ion hydrodynamic steric interactions are incorporated through the volume exclusion effect in the electrochemical potential. The viscosity of the medium is considered to vary with the local ionic volume fraction. Stronger correlations for multivalent counterions create ion layering, charge density oscillation and mobility reversal. Such phenomena are captured by the present continuum model. The ion crowding attenuates the growth of the electrophoretic mobility with the electric field. At a higher range of the imposed electric field, the ion concentration in the electric double layer enhances, which modifies both the overscreening and ion crowding processes.

The impact of a chemical reaction, $A+B \rightarrow C$, on the stability of a miscible radial displacement in a porous medium is established. Our study involves a comprehensive analysis employing both linear stability analysis and nonlinear simulations. Through linear stability analysis, the onset of instability for monotonic as well as non-monotonic viscosity profiles corresponding to the same end-point viscosity are discussed and compared. We establish a $(R_b,R_c)$ phase plane for a wide range of Damköhler number ($Da$) and Péclet number ($Pe$) into stable and unstable regions. Here, $R_b=\ln (\mu _B/ \mu _A)$ and $R_c=\ln (\mu _C/ \mu _A)$ and $\mu _{i}$ is the viscosity of fluid $i$ $\in \{A$, $B$, $C$}. The stable zone in the $(R_b, R_c)$ phase plane contracts with increased $Da$ and $Pe$ but never vanishes. It exists even for $Da \rightarrow \infty$. Interestingly, we obtain a $Da$ independent stable region in the neighbourhood of $R_c=R_b$ where no transition occurs in stability despite changes in reaction rate. The study allows us to acquire knowledge about the transition of the stability for varying $Da, Pe$ and different reactions classified using $R_b, R_c$.

Colloidal gels have strong industrial relevance as they can behave as liquids or solids. The latter allows them to support a buoyant weight against gravity. However, the system is intrinsically out of equilibrium, which means that the colloids must eventually settle out of the suspension. The process of settling has been captured theoretically, but the presence of a delay time during which the gel appears relatively unaffected by gravity has not. Here, we modify existing frameworks to capture this delay, by treating the gel as a continuum with viscous response that is based on the local bond density. We can solve our model numerically to obtain the evolution of the colloid density profile and recover qualitatively the accumulation of a dense layer on top of the settling gel, as is observed experimentally in depletion gels. This numerical study is complemented by a theoretical analysis that allows us to identify an emergent time and length scale that set the dynamics of the gel. Our model provides a solid foundation for future studies that incorporate hydrodynamic erosion and tackle industrially relevant geometries.

We provide scaling relations for the Nusselt number $Nu$ and the friction coefficient $C_{S}$ in sheared Rayleigh–Bénard convection, i.e. in Rayleigh–Bénard flow with Couette- or Poiseuille-type shear forcing, by extending the Grossmann & Lohse (J. Fluid Mech., vol. 407, 2000, pp. 27–56, Phys. Rev. Lett., vol. 86, 2001, pp. 3316–3319, Phys. Rev. E, vol. 66, 2002, 016305, Phys. Fluids, vol. 16, 2004, pp. 4462–4472) theory to sheared thermal convection. The control parameters for these systems are the Rayleigh number $Ra$, the Prandtl number $Pr$ and the Reynolds number $Re_S$ that characterises the strength of the imposed shear. By direct numerical simulations and theoretical considerations, we show that, in turbulent Rayleigh–Bénard convection, the friction coefficients associated with the applied shear and the shear generated by the large-scale convection rolls are both well described by Prandtl's (Ergeb. Aerodyn. Vers. Gött., vol. 4, 1932, pp. 18–29) logarithmic friction law, suggesting some kind of universality between purely shear-driven flows and thermal convection. These scaling relations hold well for $10^6 \leq Ra \leq 10^8$, $0.5 \leq Pr \leq 5.0$, and $0 \leq Re_S \leq 10^4$.

Ultrasonic standing wave technology offers an ideal platform for manipulating particles in microfluidics. We study how fluid viscoelasticity and acoustic boundary formation in micro-confinements affect ultrasound-induced perturbations. These perturbations influence acoustic energy density (AED) and consequently particle transport dynamics. Our approach combines theoretical, numerical and experimental methods. Using the Oldroyd-B model for viscoelastic fluids, we advance acoustic radiation force (ARF) formulations of Doinikov et al. (Phys. Rev. E, vol. 104, no. 6, 2021a; Phys. Rev. E, vol. 104, no. 6, 2021b) for particles much smaller than the acoustic wavelength. This improved approach allows us to decouple AED and acoustic contrast factor terms in the ARF expression. It also enables us to examine the effects of viscoelastic parameters: $\mu ^*$ (ratio of the viscosity of the viscoelastic fluid to that of base Newtonian fluid) and $De$ (product of fluid relaxation time and actuation frequency) on AED and particle migration. Remarkably, we show that increasing fluid elasticity or $De$ transitions viscoelastic fluids from the energy dissipation (relaxation) mode to the energy storage (frozen) mode, increasing AED. Conversely, increasing viscosity ($\mu ^*$) reduces AED. Thus, our findings suggest that elastic effects accelerate particle migration, while viscous effects decelerate it. Consequently, a viscoelastic fluid-filled micro-confinement acts as an energy dissipation device at low $De$ and an energy storage device at high $De$. Particle migration can be controlled by adjusting viscoelastic and acoustic parameters, at a fixed power input. Our theoretical and numerical findings are validated with our experimental data. Our study advances the fundamental understanding of particle migration in viscoelastic fluids under ultrasound, and can significantly impact future studies on particle/cell migration in bio-fluids.

We introduce a novel recursive procedure to a neural-network-based subgrid-scale (NN-based SGS) model for large eddy simulation (LES) of high-Reynolds-number turbulent flow. This process is designed to allow an SGS model to be applicable to a hierarchy of different grid sizes without requiring expensive filtered direct numerical simulation (DNS) data: (1) train an NN-based SGS model with filtered DNS data at a low Reynolds number; (2) apply the trained SGS model to LES at a higher Reynolds number; (3) update this SGS model with training data augmented with filtered LES (fLES) data, accommodating coarser filter size; (4) apply the updated NN to LES at a further higher Reynolds number; (5) go back to Step (3) until a target (very coarse) filter size divided by the Kolmogorov length scale is reached. We also construct an NN-based SGS model using a dual NN architecture whose outputs are the SGS normal stresses for one NN and the SGS shear stresses for the other NN. The input is composed of the velocity gradient tensor and grid size. Furthermore, for the application of an NN-based SGS model trained with one flow to another flow, we modify the NN by eliminating bias and introducing a leaky rectified linear unit function as an activation function. The present recursive SGS model is applied to forced homogeneous isotropic turbulence (FHIT) and successfully predicts FHIT at high Reynolds numbers. The present model trained from FHIT is also applied to decaying homogeneous isotropic turbulence and shows an excellent prediction performance.

A cylindrical liquid thread readily destabilizes into a series of drops due to capillary instability, which is also responsible for undesirable bead-on-fibre structures observed when coating a thin fibre. In this experimental study, we show how a falling liquid thread can be stabilized by internally distorting the cross-sectional shape using two vertically hung fibres. Below a critical flow rate $Q_c$, the dual-fibre system deforms the falling thread into a smooth column with a non-circular cross-section, thereby suppressing instability. Above $Q_{{c}}$, the cylindrical thread is left undeformed by the fibres and destabilizes into beads connected by a stable, non-cylindrical film. An empirical stability threshold is identified showing that flow delays the onset of capillary instability when compared with a marginally stable quasi-static coating. When the flow is unstable $Q>Q_c$, the bead velocity $v$ obeys a simple scaling law that is well supported by our experiments over a large parameter range. This suppression technique can be extended to other slender geometries, such as a ribbon, which shows similar qualitative results but exhibits a different stability threshold due to spontaneous dewetting about its short edge.

Aerothermodynamic characteristics of a sphere in a subsonic flow are calculated over a broad range of gas rarefaction by the direct simulation Monte Carlo method based on ab initio interatomic potentials and Cercignani–Lampis surface scattering kernel. Calculations of the drag and average energy transfer coefficients are performed for various noble gases in the range of Mach number from 0.1 to 1. The obtained results point out that the influence of the interatomic potential is weak in subsonic flows. A comparison of the present results with a linear theory shows that the numerical solutions at Mach number equal to 0.1 are close to those obtained from the linearized kinetic equation in the transitional and free-molecular regimes. In the near-continuum flow regime, the difference between the present solution and the linear theory is significant. To reveal the effects of the gas–surface accommodation, a few sets of the tangential momentum and normal energy accommodation coefficients are considered in simulations. It is shown that the effect of the accommodation coefficients on the sphere drag is not trivial, and, for non-diffuse scattering, the drag coefficient can be either larger or smaller than that for diffuse scattering. The effect of the sphere temperature is also investigated and the calculated values of the average energy transfer coefficient are used to find the Stanton number, recovery factor and adiabatic surface temperature. The numerical results for the sphere drag and energy transfer are compared with the semi-empirical fitting equations known from the literature.

The breaking and energy distribution of mode-1 depression internal solitary wave interactions with Gaussian ridges are examined through laboratory experiments. A series of processes, such as shoaling, breaking, transmission and reflection, are captured completely by measuring the velocity field in a large region. It is found that the maximum interface descent ($a_{max}$) during wave shoaling is an important parameter for diagnosing the type of wave–ridge interaction and energy distribution. The wave breaking on the ridge depends on the modified blockage parameter $\zeta _m$, the ratio of the sum of the upper layer depth and $a_{max}$ to the water depth at the top of the ridge. As $\zeta _m$ increases, the interaction type transitions from no breaking to plunging and mixed plunging–collapsing breaking. Within the scope of this experiment, the energy distribution can be characterized solely by $\zeta _m$. The transmission energy decreases monotonically with increasing $\zeta _m$, and there is a linear relationship between $\zeta _m^2$ and the reflection coefficient. The value of $a_{max}$ can be determined from the basic initial parameters of the experiment. Based on the incident wave parameters, the depth of the upper and lower layers, and the topographic parameters, two new simple methods for predicting $a_{max}$ on the ridge are proposed.

Direct numerical simulation is performed for flow separation over a bump in a turbulent channel. Comparisons are made between a smooth bump and one where the lee side is covered with replicas of shark denticles – dermal scales that consist of a slender base (the neck) and a wide top (the crown). As flow over the bump is under an adverse pressure gradient (APG), a reverse pore flow is formed in the porous cavity region underneath the crowns of the denticle array. Remarkable thrust is generated by the reverse pore flow as denticle necks accelerate the fluid passing between them in the upstream direction. Several geometrical features of shark denticles, including some that had not previously been considered hydrodynamically functional, are identified to form the two-layer denticle structure that enables and sustains the reverse pore flow and thrust generation. The reverse pore flow is activated by the APG before massive flow detachment. The results indicate a proactive, on-demand drag reduction mechanism that leverages and transforms the APG into a favourable outcome.