Advective dispersion of solutes in long thin axisymmetric channels is important to the analysis and design of a wide range of devices, including chemical separation systems and microfluidic chips. Despite extensive analysis of Taylor dispersion in various scenarios, most studies focus on long-term dispersion behaviour and cannot capture the transient evolution of the solute zone across the spatial variations in the channel. In the current study, we analyse the Taylor–Aris dispersion for arbitrarily shaped axisymmetric channels. We derive an expression for solute dynamics in terms of two coupled ordinary differential equations, which allow prediction of the time evolution of the mean location and axial (standard deviation) width of the solute zone as a function of the channel geometry. We compare and benchmark our predictions with Brownian dynamics simulations for a variety of cases, including linearly expanding/converging channels and periodic channels. We also present an analytical description of the physical regimes of transient positive versus negative axial growth of solute width. Finally, to further demonstrate the utility of the analysis, we demonstrate a method to engineer channel geometries to achieve desired solute width distributions over space and time. We apply the latter analysis to generate a geometry that results in a constant axial width and a second geometry that results in a sinusoidal axial variance in space.

Invariant solutions of the Navier–Stokes equations play an important role in the spatiotemporally chaotic dynamics of turbulent shear flows. Despite the significance of these solutions, their identification remains a computational challenge, rendering many solutions inaccessible and thus hindering progress towards a dynamical description of turbulence in terms of invariant solutions. We compute equilibria of three-dimensional wall-bounded shear flows using an adjoint-based matrix-free variational approach. To address the challenge of computing pressure in the presence of solid walls, we develop a formulation that circumvents the explicit construction of pressure and instead employs the influence matrix method. Together with a data-driven convergence acceleration technique based on dynamic mode decomposition, this yields a practically feasible alternative to state-of-the-art Newton methods for converging equilibrium solutions. We compute multiple equilibria of plane Couette flow starting from inaccurate guesses extracted from a turbulent time series. The variational method outperforms Newton(-hookstep) iterations in converging successfully from poor initial guesses, suggesting a larger convergence radius.

Compound bubbles with a liquid coating in another continuous immiscible bulk phase are ubiquitous in a wide range of natural and industrial processes. Their formation, rise and ultimate bursting at the air–liquid interface play crucial roles in the transport and fate of natural organic matter and contaminants. However, the dynamics of compound bubbles has not received considerable attention until recently. Here, inspired by our previous work (Yang et al., Nat. Phys., vol. 19, 2023, pp. 884–890), we investigate the entrainment of daughter oil droplets in bulk water produced by a bursting oil-coated bubble. We document that the size of the entrained daughter oil droplet is affected by the oil coating fraction and the bulk liquid properties. We rationalize this observation by balancing the viscous force exerted by the extensional flow produced by bubble bursting with the capillary force resisting the deformation of the oil coating, and considering the subsequent end-pinching process which finally entrains the daughter oil droplets. We propose a scaling analysis for the daughter oil droplet size that well captures the experimental results for a wide range of oil coating fractions and Ohnesorge numbers of the bulk liquid. In addition, we discuss the non-monotonic variation of daughter droplet size with the Ohnesorge number, and show the eventual absence of daughter droplets because of the strong viscous effect in the high-Ohnesorge-number regime. Our findings may advance the fundamental understanding of compound bubble bursting and provide guidance and modelling constraints for bubble-mediated contaminant transport in liquids.

Solute–surface interactions have garnered considerable interest in recent years as a novel control mechanism for driving unique fluid dynamics and particle transport with potential applications in fields such as biomedicine, the development of microfluidic devices and enhanced oil recovery. In this study, we will discuss dispersion induced by the diffusioosmotic motion near a charged wall in the presence of a solute concentration gradient. Here, we introduce a plug of salt with a Gaussian distribution at the centre of a channel with no background flow. As the solute diffuses, the concentration gradient drives a diffusioosmotic slip flow at the walls, which results in a recirculating flow in the channel; this, in turn, drives an advective flux of the solute concentration. This effect leads to cross-stream diffusion of the solute, altering the effective diffusivity of the solute as it diffuses along the channel. We derive theoretical predictions for the solute dynamics using a multiple-time-scale analysis to quantify the dispersion driven by the solute–surface interactions. Furthermore, we derive a cross-sectionally averaged concentration equation with an effective diffusivity analogous to that from Taylor dispersion. In addition, we use numerical simulations to validate our theoretical predictions.

Recently, a non-local eddy diffusivity model for the turbulent scalar flux was proposed to improve the local model and was validated using direct numerical simulation (DNS) of homogeneous isotropic turbulence with an inhomogeneous mean scalar (Hamba, J. Fluid Mech., vol. 950, 2022, A38). The non-local eddy diffusivity was assumed to be proportional to the two-point velocity correlation that was expressed in terms of the energy spectrum. Because the Fourier transform of velocity in the homogeneous directions was used to define the energy spectrum, it is not yet understood whether the proposed model can be applied to inhomogeneous turbulence. Thus, this study aimed to improve the non-local model using the scale-space energy density instead of the energy spectrum. First, the scale-space energy density based on filtered velocities was examined using the DNS data of homogeneous isotropic turbulence to obtain its simple form corresponding to the Kolmogorov energy spectrum. Subsequently, the two-point velocity correlation was expressed in terms of the scale-space energy density. Using these expressions, a new non-local eddy diffusivity model was proposed and validated using the DNS data. The one-dimensional non-local eddy diffusivity obtained from the new model agrees with the DNS value. The temporal behaviour of the three-dimensional non-local eddy diffusivity was improved compared with the previous model. Because the scale-space energy density was already examined in turbulent channel flow, it is expected that the new non-local model can also be applied to inhomogeneous turbulence and is useful for gaining insight into turbulent scalar transport.

We study experimentally the aeroelastic instability boundaries and three-dimensional vortex dynamics of pitching swept wings, with the sweep angle ranging from 0$^\circ$ to 25$^\circ$. The structural dynamics of the wings are simulated using a cyber-physical control system. With a constant flow speed, a prescribed high inertia and a small structural damping, we show that the system undergoes a subcritical Hopf bifurcation to large-amplitude limit-cycle oscillations (LCOs) for all the sweep angles. The onset of LCOs depends largely on the static characteristics of the wing. The saddle-node point is found to change non-monotonically with the sweep angle, which we attribute to the non-monotonic power transfer between the ambient fluid and the elastic mount. An optimal sweep angle is observed to enhance the power extraction performance and thus promote LCOs and destabilize the aeroelastic system. The frequency response of the system reveals a structural-hydrodynamic oscillation mode for wings with relatively high sweep angles. Force, moment and three-dimensional flow structures measured using multi-layer stereoscopic particle image velocimetry are analysed to explain the differences in power extraction for different swept wings. Finally, we employ a physics-based force and moment partitioning method to correlate quantitatively the three-dimensional vortex dynamics with the resultant unsteady aerodynamic moment.

Linear stability theory (LST) is often used to model the large-scale flow structures in the turbulent mixing region and near pressure field of high-speed jets. For perfectly expanded single round jets, these models predict the dominance of azimuthal wavenumbers $m=0$ and $m = 1$ helical modes for the lower frequency range, in agreement with empirical data. When LST is applied to twin-jet systems, four solution families appear following the odd/even behaviour of the pressure field about the symmetry planes. The interaction between the unsteady pressure fields of the two jets also results in their coupling. The individual modes of the different solution families no longer correspond to helical motions, but to flapping oscillations of the jet plumes. In the limit of large jet separations, when the jet coupling vanishes, the eigenvalues corresponding to the $m=1$ mode in each family are identical, and a linear combination of them recovers the helical motion. Conversely, as the jet separation decreases, the eigenvalues for the $m=1$ modes of each family diverge, thus favouring a particular flapping oscillation over the others and preventing the appearance of helical motions. The dominant mode of oscillation for a given jet Mach number $M_j$ and temperature ratio $T_R$ depends on the Strouhal number $St$ and jet separation $s$. Increasing both $M_j$ and $T_R$ independently is found to augment the jet coupling and modify the $(St,s)$ map of the preferred oscillation mode. Present results predict the preference of two modes when the jet interaction is relevant, namely varicose and especially sinuous flapping oscillations on the nozzles’ plane.

The stability and postcritical behaviour of a horizontal flag undergoing gravity-induced deformation and periodic contact with a nearby horizontal rigid wall are experimentally investigated. The results elucidate the combined effects of gravity and contact on flutter, and reveal design principles for application to triboelectric energy harvesting. By varying the free-stream velocity, flag thickness and distance between the flagpole and the wall, the dynamics of the flag are classified into quasistatic equilibrium, flutter, partial contact and saturated contact modes. Considering the significance of gravitational effects, a new dimensionless flow velocity is proposed to identify the distribution of the dynamic modes, and its definition varies according to whether the wall is placed above or below the flag. The critical conditions for transitions between the dynamic modes are determined from the balance of fluid dynamic and gravitational effects. The distance from the flagpole to the wall is found to be more critical for transitions in the lower-wall configuration than in the upper-wall configuration. The peak contact force as well as the oscillation amplitude and frequency at postequilibrium exhibits remarkably different trends depending on the location of the wall. The peak contact force imposed on the wall by the fluttering flag weakens as the distance to the wall increases in the case of an upper wall, whereas it becomes stronger in the case of a lower wall.

A new resolvent-based method is developed to predict the space–time properties of the flow field. To overcome the deterioration of the prediction accuracy with increasing distance between the measurements and predictions in the resolvent-based estimation (RBE), the newly proposed method utilizes the RBE to estimate the relative energy distribution near the wall rather than the absolute energy directly estimated from the measurements. Using this extra information from RBE, the new method modifies the energy distribution of the spatially uniform and uncorrelated forcing that drives the flow system by minimizing the norm of the cross-spectral density tensor of the error matrix in the near-wall region in comparison with the RBE-estimated one, and therefore it is named as the resolvent-informed white-noise-based estimation (RWE) method. For validation, three time-resolved direct numerical simulation (DNS) datasets with the friction Reynolds numbers $Re_\tau = 180$, 550 and 950 are generated, with various locations of measurements ranging from the near-wall region ($y^+ = 40$) to the upper bound of the logarithmic region ($y/h \approx 0.2$, where h is the half-channel height) for the predictions. Besides the RWE, three existing methods, i.e. the RBE, the $\lambda$-model and the white-noise-based estimation (WBE), are also included for the validation. The performance of the RBE and scale-dependent model ($\lambda$-model) in predicting the energy spectra shows a strong dependence on the measurement locations. The newly proposed RWE shows a low sensitivity on $Re_{\tau }$ and the measurement locations, which may range from the near-wall region to the upper bound of the logarithmic region, and has a high accuracy in predicting the energy spectra. The RWE also performs well in predicting the space–time properties in terms of the correlation magnitude and the convection velocity. We further utilize the new method to reconstruct the instantaneous large-scale structures with measurements from the logarithmic region. Both the RWE and RBE perform well in estimating the instantaneous large-scale structure, and the RWE has smaller errors in the estimations near the wall. The structural inclination angles around $15^\circ$ are predicted by the RWE and WBE, which generally recover the DNS results.

The turbulent external flow around a three-dimensional stepped cylinder is studied by means of direct numerical simulations with the adaptive mesh refinement technique. We give a broad perspective of the flow regimes from laminar to turbulent wake at $Re_D=5000$, which is the highest ever considered for this flow case. In particular, we focus on the intermediate Reynolds number $Re_D=1000$ that reveals a turbulent wake coupled with a stable cylinder shear layer (subcritical regime). This flow shows a junction dynamics similar to the laminar $Re_D=150$, where no hairpin vortex appears around the edges, and just two horseshoe vortices are visible. A new stable vortex in the form of a ring, which coils around the rear area, is also identified. In the turbulent wake, the presence of three wake cells is pointed out: the large and small cylinder cells together with the modulation region. However, the modulation dynamics varies between the subcritical and turbulent regimes. A time-averaged, three-dimensional set of statistics is computed, and spatially coherent structures are extracted via proper orthogonal decomposition (POD). The POD identifies the (long-debated) connection between the N-cell and the downwash behind the junction. Furthermore, as the Reynolds number increases, the downwash phenomenon becomes less prominent. Eventually, a reduced-order reconstruction with the most energetically relevant modes is defined to explain the wake vortex interactions. This also serves as a valuable starting point for simulating the stepped cylinder wake behaviour within complex frameworks, e.g. fluid–structure interaction.

Flow around curved tandem cylinders in the convex configuration has been studied by means of direct numerical simulations, for a Reynolds number of 500 and a nominal gap ratio of 3.0. Spanwise variation of flow regimes, as well as curvature-induced axial velocity, leads to an exceedingly complex vortex dynamics in the wake. Both parallel and oblique vortex shedding are observed. Oblique shedding is connected to repeated occurrences of dislocations. The dislocations are caused by two main mechanisms: frequency differences in the upper part of the curved geometry and shedding of gap vortices into the lower near wake. Both types of dislocations are closely associated with a mode switch in the gap. In parts of the gap, there is low-frequency quasi-periodic asymmetry of the gap vortices, where the flow is biased to one side of the gap for intervals of several wake vortex shedding periods. The switch from side to side is associated with a surge of the vertical velocity, and the frequency of the switch is similar to that of long-term variation of the recirculation length in the lower gap.

This work investigates the statistical response of short and long laminar separation bubbles to external flow parameters, such as Reynolds number, free-stream turbulence intensity and streamwise pressure gradient, known to govern bubble formation and characteristics. A parametric experimental campaign has been performed using particle image velocimetry on a flat plate to provide a comprehensive database for the characterization of separation-induced transition in both short and long separation bubbles. The proper orthogonal decomposition (POD) was applied to the data set of all dividing streamlines commonly used to identify a laminar separation bubble. This provides an optimal state-space basis for the data-driven classification of the state of a laminar separation bubble, with the leading modes capturing the change in length and height of the laminar separation bubble in response to changes in the flow parameters. When projected onto the POD subspace constituted by the first three leading modes, the normalized data from the present study and the results from prior investigations not used in the modal analysis collapse on the same trajectory in the low-dimensional space. The present POD basis can be therefore adopted for the description of the general response of the time-mean shape of a laminar separation bubble to changes in the main influencing parameters. A well-defined pattern was observed in the case of short laminar separation bubbles in the reduced-order space defined by the first three POD coefficients, whereas a higher dispersion in the long-bubble regime indicates an increased sensitivity of long bubbles to the external flow characteristics.

Diapycnal mixing plays a key role in the thermohaline circulation of the deep ocean. Field observations have suggested that this mixing is intensified over the rough topography along the boundaries of the ocean. In this study, we experimentally explore the transport of salinity and tracer in a horizontally stirred, stratified fluid with a steady vertical buoyancy flux. The mechanical mixing occurs either uniformly across the tank or near a sidewall. To compare uniform and boundary mixing, first we explore the steady state dynamics and find that, in both cases, the surfaces of constant density are horizontal. With uniform mixing, vertical transport of buoyancy occurs uniformly throughout the tank while, with boundary mixing, transport is confined to the turbulent region and the interior space remains quiescent and does not mix or have any net movement. We then explore the transient evolution of the stratification when the source of buoyancy is removed from the base of the system. In the boundary mixing case, the resulting divergence of the turbulent diffusive flux in the boundary region leads to a reduction in the buoyancy of the fluid in the boundary region, and a net upflow develops in the boundary region. In turn, this drives a downwelling in the interior. Vertical gradients in the rate of downwelling lead to stretching of isopycnals and, together, these processes enable the interior stratification to evolve. The experiments highlight that, with boundary mixing, the main transport of buoyancy occurs near the boundaries even though the interior is stratified; independent measurements of tracer mixing in the interior show that this fluid may be quiescent and stratified even though there is a large flux being transported in the boundary region. This has important implications for the interpretation of mixing data in the ocean. In a companion paper, (Li and Woods Part 2), we explore the interaction of a net upwelling with such boundary mixing.

The transition to turbulence in a plane Poiseuille flow of dilute polymer solutions is studied by direct numerical simulations of a finitely extensible nonlinear elastic fluid with the Peterlin closure. The range of Reynolds number ($Re$) $2000 \le Re \le 5000$ is studied but with the same level of elasticity in viscoelastic flows. The evolution of a finite-amplitude perturbation and its effects on the transition dynamics are investigated. A viscoelastic flow begins transition at an earlier time than its Newtonian counterparts, but the transition time appears to be insensitive to polymer concentration in the dilute and semi-dilute regimes studied. Increasing polymer concentration, however, decreases the maximum attainable energy growth during the transition process. The critical or minimum perturbation amplitude required to trigger transition is computed. Interestingly, both Newtonian and viscoelastic flows follow almost the same power-law scaling of $Re^\gamma$ with the critical exponent $\gamma \approx -1.25$, which is in close agreement with previous studies. However, a shift downward is observed for viscoelastic flow, suggesting that smaller perturbation amplitudes are required for the transition. A mechanism of the early transition is investigated by the evolution of wall-normal and spanwise velocity fluctuations and flow structure. The early growth of these fluctuations and the formation of quasi-streamwise vortices around low-speed streaks are promoted by polymers, hence causing an early transition. These vortical structures are found to support the critical exponent $\gamma \approx -1.25$. Once the transition process is completed, polymers play a role in dampening the wall-normal and spanwise velocity fluctuations and vortices to attain a drag-reduced state in viscoelastic turbulent flows.

Turbine–wake and farm–atmosphere interactions influence wind farm power production. For large offshore farms, the farm–atmosphere interaction is usually the more significant effect. This study proposes an analytical model of the ‘momentum availability factor’ to predict the impact of farm–atmosphere interactions. It models the effects of net advection, pressure gradient forcing and turbulent entrainment, using steady quasi-one-dimensional flow assumptions. Turbulent entrainment is modelled by assuming self-similar vertical shear stress profiles. We used the model with the ‘two-scale momentum theory’ to predict the power of large finite-sized farms. The model compared well with existing results of large-eddy simulations of finite wind farms in conventionally neutral boundary layers. The model captured most of the effects of atmospheric boundary layer (ABL) height on farm performance by considering the undisturbed vertical shear stress profile of the ABL as an input. In particular, the model predicted the power of staggered wind farms with a typical error of 5 % or less. The developed model provides a novel way of predicting instantly the power of large wind farms, including the farm blockage effects. A further simplification of the model to predict analytically the ‘wind extractability factor’ is also presented. This study provides a novel framework for modelling farm–atmosphere interactions. Future studies can use the framework to better model large wind farms.

Evaporating sessile droplets are critical to many industrial applications and are also ubiquitous in nature. Two predominant evaporation models have emerged in the literature, one-sided and diffusion-limited, with differing assumptions on the evaporation process. Both models are used widely, and their predictions can differ greatly from each other, but the physical mechanisms responsible for these differences are not yet well understood. Here, we develop a lubrication-theory-based model of a thin evaporating sessile droplet, and compare predictions from both evaporation models to elucidate the origins of the differences in their predictions. For the one-sided model, we derive expressions for the droplet lifetime, show that in certain parameter regimes the total evaporation rate is proportional to the droplet surface area, and demonstrate that the contact line is always warmer than the bulk of the droplet. Furthermore, we show that differences in the structures of the evaporation models near the contact line lead to qualitatively different behaviour of the apparent contact angles and interface temperature profiles. The fundamental understanding gained from this work is expected to be helpful in determining which evaporation model is most appropriate for describing experimental observations.

We numerically study the transverse flow-induced vibration (FIV) of elastically coupled tandem cylinders at Reynolds number $100$, using an in-house immersed boundary method-based solver in two-dimensional coordinates. While several previous studies considered tandem cylinders coupled through flow between them, a hitherto unexplored elastic coupling with fluid flow between them significantly influences FIV. We consider a wide range of gap ratio, reduced velocity, an equal mass ratio of both cylinders and zero damping. A systematic comparison between the classic elastically mounted tandem cylinders and elastically coupled cylinders is presented. The latter configuration exhibits two vibration modes, in-phase and out-of-phase, with corresponding natural frequencies approaching the Strouhal frequency of the system. We quantify variation of the following output variables with reduced velocity and gap ratios: cylinders’ displacement; fluid forces; amplitude spectral density of displacement and force signals; phase characteristics; energy harvesting potential; and discuss the wake characteristics using flow separation, pressure distribution, gap flow quantification, and dynamic mode decomposition characterization. The FIV response is classified into several regimes: initial desynchronization with and without gap vortices; final desynchronization; mixed mode; initial branch; lock-in; upper and lower branch; wake-induced vibration; galloping. We draw upon similarities of computed FIV characteristics with those of an isolated cylinder, in which the lower branch exhibits larger a amplitude than the upper branch. The elastically coupled cylinders show a galloping response similar to an isolated D-section cylinder. By invoking the elastic coupling, we demonstrate FIV suppression and augmentation for in-phase and out-of-phase systems. Our calculations show larger energy harvesting potential at reduced cost for elastically coupled cylinders.

In continuation of our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, R3; Chen & Sreenivasan, J. Fluid Mech., vol. 933, 2022a, A20 – together referred to as CS hereafter), we present a self-consistent Reynolds number asymptotics for wall-normal profiles of variances of streamwise and spanwise velocity fluctuations as well as root-mean-square pressure, across the entire flow region of channel and pipe flows and flat-plate boundary layers. It is first shown that, when normalized by peak values, the Reynolds number dependence and wall-normal variation of all three profiles can be decoupled, in excellent agreement with available data, sharing the common inner expansion of the type $\phi (y^+)=f_0(y^+)+f_1(y^+)/Re^{1/4}_\tau$, where $\phi$ is one of the quantities just mentioned, the functions $f_0$ and $f_1$ depend only on $y^+$, and $Re_\tau$ is the friction Reynolds number. Here, the superscript $+$ indicates normalization by wall variables. We show that this result is completely consistent with CS. Secondly, by matching the above inner expansion and the outer flow similarity form, a bounded variation $\phi (y^\ast )=\alpha _\phi -\beta _{\phi }y^{{\ast {1}/{4}}}$ is derived for the outer region where, for each $\phi$, the constants $\alpha _\phi$ and $\beta _{\phi }$ are independent of $Re_\tau$ and $y^\ast$ $\equiv y^+/Re_\tau$ – also in excellent agreement with simulations and experimental data. One of the predictions of the analysis is that, for asymptotically high Reynolds numbers, a finite plateau $\phi \approx \alpha _\phi$ appears in the outer region. This result sheds light on the intriguing issue of the outer shoulder of the variance of the streamwise velocity fluctuation, which should be bounded by the asymptotic plateau of approximately 10.

Motivated by wind blowing over water, we use asymptotic methods to study the evolution of short wavelength interfacial waves driven by the combined action of these flows. We solve the Rayleigh equation for the stability of the shear flow, and construct a uniformly valid approximation for the perturbed streamfunction, or eigenfunction. We then expand the real part of the eigenvalue, the phase speed, in a power series of the inverse wavenumber and show that the imaginary part is exponentially small. We give expressions for the growth rates of the Miles (J. Fluid Mech., vol. 3, 1957, pp. 185–204) and rippling (e.g. Young & Wolfe, J. Fluid Mech., vol. 739, 2014, pp. 276–307) instabilities that are valid for an arbitrary shear flow. The accuracy of the results is demonstrated by a comparison with the exact solution of the eigenvalue problem in the case when both the wind and the current have an exponential profile.

Motile bacteria play essential roles in biology that rely on their dynamic behaviours, including their ability to navigate, interact and self-organize. However, bacteria dynamics on fluid interfaces are not well understood. Swimmers adsorbed on fluid interfaces remain highly motile, and fluid interfaces are highly non-ideal domains that alter swimming behaviour. To understand these effects, we study flow fields generated by Pseudomonas aeruginosa PA01 in the pusher mode. Analysis of correlated displacements of tracers and bacteria reveals dipolar flow fields with unexpected asymmetries that differ significantly from their counterparts in bulk fluids. We decompose the flow field into fundamental hydrodynamic modes for swimmers in incompressible fluid interfaces. We find an expected force-doublet mode corresponding to propulsion and drag at the interface plane, and a second dipolar mode, associated with forces exerted by the flagellum on the cell body in the aqueous phase that are countered by Marangoni stresses in the interface. The balance of these modes depends on the bacteria's trapped interfacial configurations. Understanding these flows is broadly important in nature and in the design of biomimetic swimmers.