We consider the flow of a thermoviscous fluid within a vertical tube which is heated from below, modelling a scenario where a fluid melts, flows and eventually drips due to a temperature-dependent viscosity. To do so, we develop a two-dimensional axisymmetric model comprising three regions, a solid granular upper region (modelled as a region of hydrostatic pressure), a middle highly viscous ‘crust’ region which flows and a lower cavity region within which the material can drip. New material is continuously added to the top, yet the highly viscous middle region can slow mass transfer from the top region to the cavity if it becomes too thick or does not drip fast enough. In the limit of a tall, thin geometry, akin to what is often seen in industrial applications, the resulting model comprises a moving boundary problem governed by an energy equation with a Stefan condition, both subject to a non-local radially averaged convective term. We carry out numerical simulations and an asymptotic analysis of the model in this tall, thin limit, for a variety of physically relevant parameter regimes. Our results reveal a variety of qualitatively different behaviours, and enable us to explore how various parameter regimes influence the salient features of the flow, including the ‘crust’ thickness and the flux of material through the lower moving boundary.

We derive a theory for material surfaces that maximally inhibit the diffusive transport of a dynamically active vector field, such as the linear momentum, the angular momentum or the vorticity, in general fluid flows. These special material surfaces (Lagrangian active barriers) provide physics-based, observer-independent boundaries of dynamically active coherent structures. We find that Lagrangian active barriers evolve from invariant surfaces of an associated steady and incompressible barrier equation, whose right-hand side is the time-averaged pullback of the viscous stress terms in the evolution equation for the dynamically active vector field. Instantaneous limits of these barriers mark objective Eulerian active barriers to the short-term diffusive transport of the dynamically active vector field. We obtain that in unsteady Beltrami flows, Lagrangian and Eulerian active barriers coincide exactly with purely advective transport barriers bounding observed coherent structures. In more general flows, active barriers can be identified by applying Lagrangian coherent structure (LCS) diagnostics, such as the finite-time Lyapunov exponent and the polar rotation angle, to the appropriate active barrier equation. In comparison to their passive counterparts, these active LCS diagnostics require no significant fluid particle separation and hence provide substantially higher-resolved LCS and Eulerian coherent structure boundaries from temporally shorter velocity data sets. We illustrate these results and their physical interpretation on two-dimensional, homogeneous, isotropic turbulence and on a three-dimensional turbulent channel flow.

We study the deformation and dewetting of liquid films under impinging gas jets using experimental, analytical and numerical techniques. We first derive a reduced-order model (a thin-film equation) based on the long-wave assumption and on appropriate decoupling of the gas problem from that for the liquid. The model not only provides insight into relevant flow regimes, but is also used in conjunction with experimental data to guide more computationally prohibitive direct numerical simulations of the full governing equations. A unique feature of our modelling solution is the use of an efficient iterative procedure in order to update the interfacial deformation based on stresses originating from computational data. We show that both gas normal and tangential stresses are equally important for achieving accurate predictions. The interplay between these techniques allows us to study previously unreported flow features. These include finite-size effects of the host geometry, with consequences for flow and vortex formation inside the liquid, as well as the specific individual contributions from the non-trivial gas flow components on interfacial deformation. Dewetting phenomena are found to depend on either a dominant gas flow or contact line motion, with the observed behaviour (including healing effects) being explained using a bifurcation diagram of steady-state solutions in the absence of the gas flow.

Direct numerical simulation of polymer-induced flow relaminarization of turbulent spanwise-rotating plane Couette flow (RPCF) is reported for the first time. Specifically, the reverse transition pathway from a Newtonian turbulent RPCF to a fully relaminarized drag enhanced viscoelastic flow has been elucidated. Evidently, this transition occurs gradually by weakening and eventual elimination of small-scale vortices as the Weissenberg number ($Wi$) is enhanced, paving the way for a two-dimensional laminar flow consisting of large-scale and highly organized roll cells. The influence of polymer additives on convective momentum exchange by large-scale roll cells and small-scale turbulent vortices, namely, the drag reduction (DR) realized by elimination of turbulent vortices and the significant drag enhancement (DE) that results from polymer roll cell interactions has been identified as the mechanism of DE. The observed vortical changes point to a universal mechanism for the coupling of polymer chains and turbulent vortices in wall-bounded viscoelastic DE and DR flows.

Induced-charge electro-osmotic (ICEO) flow caused by an alternating electric field applied around an infinitely long, ideally polarizable, uncharged circular cylinder in a binary electrolyte with unequal cation and anion diffusion coefficients is analysed. The thin-Debye-layer and weak-field approximations are invoked to compute the time-averaged, or rectified, quadrupolar ICEO flow around the cylinder. The inequality of ionic diffusion coefficients leads to transient ion concentration gradients, or concentration polarization, in the electroneutral bulk electrolyte outside the Debye layer. Consequently, the electric potential in the bulk is non-harmonic. Further, the concentration polarization alters the electro-osmotic slip at the surface of the cylinder and generates body forces in the bulk, both of which affect the rectified ICEO flow. Predictions for the strength of the rectified flow for varying ratio of ionic diffusion coefficients are in reasonable agreement with available experimental data. Our work highlights that an inequality in ionic diffusion coefficients – which all electrolytes possess to some extent – is an important factor in modelling ICEO flows.

Rayleigh–Bénard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimensional (2-D) RB convection and the other one three-dimensional (3-D) RB convection with a rotating axis parallel to the plate, which for strong rotation mimics 2-D RB convection. For the 2-D simulations, we explore the parameter range of Rayleigh numbers $Ra$ from $10^{7}$ to $10^{9}$ and Prandtl numbers $Pr$ from $1$ to $100$. The effect of the width-to-height aspect ratio $\varGamma$ is investigated for $1\leqslant \varGamma \leqslant 128$. We show that zonal flow, which was observed, for example, by Goluskin et al. (J. Fluid. Mech., vol. 759, 2014, pp. 360–385) for $\varGamma =2$, is only stable when $\varGamma$ is smaller than a critical value, which depends on $Ra$ and $Pr$. The regime in which only zonal flow can exist is called the first regime in this study. With increasing $\varGamma$, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger $\varGamma$, in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. How many convection rolls form (or in other words, what the mean aspect ratio of an individual roll is), depends on the initial conditions and on $Ra$ and $Pr$. For instance, for $Ra=10^{8}$ and $Pr=10$, the aspect ratio $\varGamma _r$ of an individual, statistically stable convection roll can vary in a large range between $16/11$ and $64$. A convection roll with a large aspect ratio of $\varGamma _r = 64$, or more generally already with $\varGamma _r \gg 10$, can be seen as ‘localized’ zonal flow, and indeed carries over various properties of the global zonal flow. For the 3-D simulations, we fix $Ra=10^{7}$ and $Pr=0.71$, and compare the flow for $\varGamma =8$ and $\varGamma = 16$. We first show that with increasing rotation rate both the flow structures and global quantities like the Nusselt number $Nu$ and the Reynolds number $Re$ increasingly behave like in the 2-D case. We then demonstrate that with increasing aspect ratio $\varGamma$, zonal flow, which was observed for small $\varGamma =2{\rm \pi}$ by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), completely disappears for $\varGamma =16$. For such large $\varGamma$, only convection roll states are statistically stable. In-between, here for medium aspect ratio $\varGamma = 8$, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2-D case.

In many applications, conventional aerofoils are subject to a number of simultaneous motions that complicate the prediction of flow separation. The purpose of this work is to evaluate the impact of a large-amplitude free-stream oscillation on the timing of vortex formation for a simultaneously surging and pitching wing. Experimental flow field measurements were obtained on a NACA 0012 aerofoil over a wide range of surge amplitudes ($1.50 \leq \lambda \leq 2.25$) and reduced frequencies ($0.1 \leq k \leq 0.3$). Particular attention was paid to how various mechanisms of flow separation, specifically the velocity induced by the trailing wake and unsteady effects in the boundary layer, were impacted by a change in the properties of the surge motion. In the regime where $k \leq 0.3$, a change in the surge kinematics primarily manifested as a change in the relative strength of the trailing wake. Boundary layer unsteadiness was found to have a negligible influence on the timing of vortex formation in the same flow regime. Thus, the timing of leading-edge vortex formation was well predicted by a combination of an unsteady inviscid flow solver and a quasi-steady treatment of the boundary layer, a promising result for low-order predictions of vortex behaviour in unsteady aerofoil flows.

We investigate experimentally the effect of double diffusion on the dynamics of initially neutrally buoyant warm and salty turbulent jets discharged horizontally into stationary cooler freshwater ambient. Jets over a range of source Reynolds numbers and source temperature/salinity combinations are examined. In all cases, we observed sinking jet trajectories and the formation of salt fingers along the lower surface of the jet. Increasing the source concentration of both scalar properties led to more pronounced jet sinking trajectories, and to a reduction in the distance between the source and the onset point of salt fingers, demonstrating the significance of the double-diffusive processes. We propose that is it the differential double-diffusive fluxes across the jet–ambient turbulent/non-turbulent interfaces that causes the build-up of negative buoyancy and hence the sinking motion. In addition, we make predictions on the onset point of the salt fingers based on the balance between diffusive processes and the jet entrainment, and compare them with the experimental observations.

A new analytical model is derived based on physical concepts and conservation laws, in order to evaluate the post-shock gas velocity, the gas density and the spray dispersion topology during the interaction of a shock wave and a water spray in a one-dimensional configuration. The model is validated against numerical simulations over a wide range of incident Mach numbers $M_s$ and particle volume fractions $\tau _{v,0}$. Two regimes of shock reflection have been identified depending on $M_s$, where the reflected pressure expansion propagates either opposite to the incident shock-wave direction for weak incident Mach numbers or along with it for strong Mach numbers. The numerical simulations reveal the presence of a particle number-density peak for $M_s > 2$ and with particle diameters of the order of ${O}(10)\ \mathrm {\mu } \textrm {m}$. The formation of the number-density peak is discussed and a necessary condition for its existence is proposed for the first time.

The evolution of the potent second-mode instability in hypersonic boundary layers (HBLs) is examined holistically, by tracking its linear and nonlinear evolution, followed by its role in initiating transition and eventual breakdown of the HBL into a fully turbulent state. Linear stability theory is utilized to first identify the features of the second-mode wave after $FS$-synchronization. These are then employed in separate linearly and nonlinearly forced two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations (DNS). The nonlinear 2-D DNS shows saturation of the fundamental frequency, and the resulting superharmonics induce tightly braided ‘rope-like’ patterns near the generalized inflection point (GIP). The instability exhibits a second region of growth constituted by the fundamental frequency downstream of the primary envelope, which is absent in the linear scenario. Subsequent fully 3-D DNS identify this region as crucial in amplifying oblique instabilities riding on the 2-D second-mode ‘rollers’. This results in lambda vortices below the GIP, which are detached from the rollers in the inner boundary layer. Streamwise vortex-stretching results in a localized peak in length scales inside the HBL, eventually forming hairpin vortices. Spectral analyses track the transformation of harmonic peaks into a turbulent spectrum. The appearance of oblique modes at the fundamental frequency suggests that fundamental resonance is the most dominant mechanism of transition. The bispectrum reveals coupled nonlinear interactions between the fundamental and its superharmonics leading to spectral broadening, as well as traces of subharmonic resonance. The global forms of the fundamental and subharmonic modes show that the former disintegrate at the location of spanwise breakdown, beyond which the latter amplify. Statistical analyses of the near-wall flow field indicate an increase in large-scale ‘splatting’ motions immediately following transition, resulting in extreme skin-friction events, which equilibrate as turbulence sets in. Fundamental resonance results in complete breakdown of streamwise streaks in the lower log-layer, ultimately resulting in a fully turbulent HBL.

The accurate prediction of turbulent mixing induced by Rayleigh–Taylor (R–T), Richtmyer–Meshkov (R–M) and Kelvin–Helmholtz (K–H) instabilities is very important in understanding natural phenomena and improving engineering applications. In applications, the prediction of mixing with the Reynolds-averaged Navier–Stokes (RANS) equation remains the most widely used method. The RANS method involves two aspects, i.e. physical modelling and model coefficients. Generally, the latter is determined empirically; thus, there is a lack of universality. In this paper, inspired by the well-known Reynolds decomposition, we propose a methodology to determine the model coefficients with the following three steps: (i) preset a set of analytical RANS solutions by fully using the knowledge of mixing evolutions; (ii) simplify the differential RANS equations to algebraic equations by imposing the preset solutions to RANS equations; (iii) solve the algebraic equations approximately to give the values of the entire model coefficients. The specific application of this methodology in the widely used K–L mixing model shows that, using the same set of model coefficients determined from the current methodology, the K–L model successfully predicts the mixing evolutions in terms of different physical quantities (e.g. temporal scalings and spatial profiles), density ratios and problems (e.g. R–T, R–M, K–H and reshocked R–M mixings). It is possible to extend this methodology to other turbulence models characterised with self-similar evolutions, such as K-$\epsilon$ mixing models.

Water particle orbits are key elements in the Lagrange wave formulation. The stochastic Miche implementation of the Lagrange model is a linear Gaussian two-dimensional or three-dimensional space–time model which exhibits typical nonlinear wave characteristics when transformed to Eulerian coordinates. This paper investigates the statistical relation between the degree of front–back asymmetry of individual waves and the orbit orientation for the particle located at the wave maximum at the point of observation. It is shown that, in the Lagrangian model with statistical front–back symmetry, for individual waves there is a clear connexion between the degree of individual wave asymmetry and the orientation of the randomly deformed elliptic orbit: a steep front correlates with upward tilt, a steep back is correlated with a downward tilt. This holds both for waves observed in time and in space, and the dependence is stronger for large amplitude waves than for smaller ones. The dependence is strongly dependent on the depth and on the significant steepness and spectral width. Inclusion of the average Stokes drift has a moderate effect on the dependence. For models with forced front–back asymmetry there is both a systematic dependence and a statistical correlation between asymmetry and tilt; for large amplitude waves the systematic relation dominates. The conclusions are based on Fourier simulations of Gauss–Lagrange waves of first and second order with a wind–sea Pierson–Moskowitz spectrum and a narrow swell JONSWAP spectrum.

We study Taylor–Couette flow of a glycerol–water mixture containing a wide range of concentration (0–2000 ppm) of xanthan gum, which induces both shear-thinning and viscoelasticity, in order to assess the effect of the changes in rheology on various flow instabilities. For each suspension, the Reynolds number (the ratio of inertial to viscous forces) is slowly increased to a peak value of around 1000, and the flow is monitored continuously using flow visualisation. Shear-thinning is found to suppress many elasticity-controlled instabilities that have been observed in previous studies of viscoelastic Taylor–Couette flow, such as diwhirls and disordered oscillations. The addition of polymers is found to reduce the critical Reynolds number for the formation of Taylor vortices, but delay the onset of wavy flow. However, in the viscoelastic regime (${\geq }1000\ \textrm {ppm}$ concentration), the flow becomes highly unsteady soon after the formation of Taylor vortices, with substantial changes in the waviness with Reynolds number, which are shown to be highly repeatable. Vortices are found to suddenly merge as the Reynolds number increases, with the number of mergers increasing with polymer concentration. These abrupt changes in wavelength are highly hysteretic and can occur in both steady and wavy regimes. Finally, the vortices in moderate and dense polymer solutions are shown to undergo a gradual drift in both their size and position, which appears to be closely linked to the splitting and merger of vortices.

An experimental investigation of droplet formation induced by an external electric field in a T-shaped microfluidic device is presented. The effect of electric field is reported for scenarios where the hydrodynamics is known to be governed by the cumulative effect of hydrodynamic pressure and interfacial tension acting on the liquid–liquid interface. Experiments reveal that the electrohydrodynamic phenomena transforms the droplet formation mechanism by inducing pinning of the dispersed phase to the channel wall, leading to a significant decrease in the droplet filling time and hence a decrease in the size of droplets generated. The experimental observations are used to formulate a correlation between the droplet size, applied electric field, fluid properties and flow parameters. A mechanistic explanation of droplet formation process using a mathematical model is also presented. Simulations reveal that the droplets are formed primarily due to normal electric stress acting on the liquid–liquid interface. The electric stress results in a distinct feature of pinning and early onset of neck formation of the emerging dispersed phase, leading to a reduction in the size of the droplet formed for the same hydrodynamic conditions. The findings reported demonstrate that an applied electric field has the potential to produce relatively smaller-sized droplets than that possible through hydrodynamics alone.

We study the spreading of viscous fluid injected under an elastic sheet, which is driven by gravity and by elastic bending and tension forces and resisted by viscous forces. The injected fluid forms a large blister and spreads outwards analogously to a viscous gravity current or a capillary droplet. The relative strengths of the three driving forces are determined by how the horizontal length scales of the system compare with three key transition length scales. Bending is dominant on small length scales, tension is dominant on intermediate length scales and gravity is dominant on large length scales. We show how to use the method of matched asymptotic expansions to predict the spreading rate and thickness profile of the blister of fluid in the seven possible asymptotic regimes, for both two-dimensional and axisymmetric geometries. Consideration of different physical effects at the fluid front increases the number of regimes yet further.

The small-scale physics within the first centimetres above the wavy air–sea interface are the gateway for transfers of momentum and scalars between the atmosphere and the ocean. We present an experimental investigation of the surface wind stress over laboratory wind-generated waves. Measurements were performed at the University of Delaware's large wind-wave-current facility using a recently developed state-of-the-art wind-wave imaging system. The system was deployed at a fetch of 22.7 m, with wind speeds from 2.19 to $16.63\ \textrm {m}\ \textrm {s}^{-1}$. Airflow velocity fields were acquired using particle image velocimetry above the wind waves down to $100\ \mathrm {\mu }\textrm {m}$ above the surface, and wave profiles were detected using laser-induced fluorescence. The airflow intermittently separates downwind of wave crests, starting at wind speeds as low as $2.19\ \textrm {m}\ \textrm {s}^{-1}$. Such events are accompanied by a dramatic drop in tangential viscous stress past the wave's crest, and a gradual regeneration of the viscous sublayer upon the following (downwind) crest. This contrasts with non-airflow separating waves, where the surface viscous stress drop is less significant. Airflow separation becomes increasingly dominant with increasing wind speed and wave slope $a k$ (where $a$ and $k$ are peak wave amplitude and wavenumber, respectively). At the highest wind speed ($16.63\ \textrm {m}\ \textrm {s}^{-1}$), airflow separation occurs over nearly 100 % of the wave crests. The total air–water momentum flux is partitioned between viscous stress and form drag at the interface. Viscous stress (respectively form drag) dominates at low (respectively high) wave slopes. Tangential viscous forcing makes a minor contribution (${\sim }3\,\%$) to wave growth.

The phase behaviour of fluids at capillary conditions differs from that in bulk media. Therefore, understanding the thermodynamics of solvents in confined media is essential for modelling non-isothermal and non-isobaric engineering applications in porous media – including enhanced oil and heavy-oil recovery. The Thomson equation states that pore sizes have control over the boiling points of liquids in capillary channels. As pore spaces get smaller, boiling temperatures become lower than the normal boiling temperatures of the same liquids. The objective of this paper is to inspect this phenomenon by physically measuring the boiling points of different liquids at capillary conditions and comparing them with the values at bulk conditions and boiling temperatures obtained from the Thomson equation. Several types of microfluidic chips were used as capillary media to observe the phase-change behaviour of heptane, heptane–decane mixture and naphtha. Additionally, vaporization of water, heptane and decane was investigated in Berea sandstone, Indiana limestone, tight sandstone and shale. Pore size distribution analysis was performed to identify the pore diameter variations in each rock sample, and how the existence of extended nanopores in the rocks could impact the phase alteration.

Stability of hexagonal patterns in Rayleigh–Bénard convection for shear-thinning fluids with temperature-dependent viscosity is studied in the framework of amplitude equations. The rheological behaviour of the fluid is described by the Carreau model and the relationship between the viscosity and the temperature is of exponential type. Ginzburg–Landau equations including non-variational quadratic spatial terms are derived explicitly from the basic hydrodynamic equations using a multiple scale expansion. The stability of hexagonal patterns towards spatially uniform disturbances (amplitude instabilities) and to long wavelength perturbations (phase instabilities) is analysed for different values of the shear-thinning degree $\alpha$ of the fluid and the ratio $r$ of the viscosities between the top and bottom walls. It is shown that the amplitude stability domain shrinks with increasing shear-thinning effects and increases with increasing the viscosity ratio $r$. Concerning the phase stability domain which confines the range of stable wavenumbers, it is shown that it is closed for low values of $r$ and becomes open and asymmetric for moderate values of $r$. With increasing shear-thinning effects, the phase stability domain becomes more decentred towards higher values of the wavenumber. Beyond the stability limits, two different modes go unstable: longitudinal and transverse modes. For the parameters considered here, the longitudinal mode is relevant only in a small region close to the onset. The nonlinear evolution of the transverse phase instability is investigated by numerical integration of amplitude equations. The hexagon–roll transition triggered by the transverse phase instability for sufficiently large reduced Rayleigh number $\epsilon$ is illustrated.

In this study large-eddy simulations of under-expanded supersonic impinging jets are performed to develop a better understanding of the characteristics of the acoustic and hydrodynamic waves. Time history, dispersion relation and autocorrelation of the velocity and pressure fluctuations are used to investigate the propagation velocity, time and length scales of the dominant flow structures in the shear layer and near field. The mechanism by which the initial high-frequency instabilities change to low-frequency coherent structures within a short distance is investigated utilising Mach energy norm and linear spatial instability analysis with streamwise varying mean flow profiles. It is shown that the hydrodynamic and acoustic wavepackets have different propagation velocities and length scales while having a similar dominant frequency. It is also observed that the hydrodynamic wavepackets form approximately one jet diameter downstream of the nozzle lip. No evidence has been found to support the ‘collective interactive’ mechanism proposed by Ho & Nosseir (J. Fluid Mech., vol. 105, 1981, pp. 119–142). The ‘vortex pairing’ proposed by Winant & Browand (J. Fluid Mech., vol. 63, 1974, pp. 237–255) is observed near the nozzle; however, it has an insignificant role in the sharp reduction of the most unstable frequency of disturbances. Nonetheless, both Mach energy norm and linear spatial instability analyses show that the most unstable frequency of disturbances decreases rapidly in a very short distance from the nozzle lip in the near-nozzle region through the spatial growth of instabilities where the linear instability analysis overpredicts the frequency of the most unstable instabilities downstream of the nozzle.

The effects of background turbulence on the entrainment process, as well as the nature of the interfacial region between two bodies of turbulent fluid, were examined through an investigation of the far wake of a circular cylinder that is subjected to free-stream turbulence. Simultaneous particle image velocimetry and planar laser induced fluorescence measurements were conducted 40 diameters downstream of the cylinder. Despite the availability of turbulent, rotational fluid in the background, the outer interface between the wake and the ambient fluid exhibits an enstrophy jump akin to the classical result of a turbulent/non-turbulent interface. This jump at the wake boundary persists even when the intensity of the background turbulence is greater than the turbulence intensity of the wake itself. Analysis on the structure of the wake boundary reveals that an increase in background turbulence intensity results in an increased interfacial surface area relative to the non-turbulent case. However, instead of the intuitive result of increased entrainment as a result of increased surface area, a reduction in mean entrainment mass flux is observed with increased background turbulence intensity. Through the analysis of the flux probability density functions, the reduction in mean entrainment can be attributed to a tip in balance of extreme entrainment and detrainment events to the detrainment side in the presence of background turbulence. Lastly, a scale by scale analysis of entrainment behaviour revealed that free-stream turbulence affects entrainment behaviour across all length scales and is not just limited to the energy containing scales.