We model the advective Marangoni spreading of insoluble surfactant at the free surface of a viscous fluid that is confined within a two-dimensional rectangular cavity. Interfacial deflections are assumed small, with contact lines pinned to the walls of the cavity, and inertia is neglected. Linearising the surfactant transport equation about the equilibrium state allows a modal decomposition of the dynamics, with eigenvalues corresponding to decay rates of perturbations. Computation of the family of mutually orthogonal two-dimensional eigenfunctions reveals singular flow structures near each contact line, resulting in spatially oscillatory patterns of shear stress and a pressure field that diverges logarithmically. These singularities at a stationary contact line are associated with dynamic compression of the surfactant monolayer. We show how they can be regularised by weak surface diffusion. Their existence highlights the need for careful treatment in computations of unsteady advection-dominated surfactant transport in confined domains.

We study the three-dimensional hydrodynamic interaction of a pair of identical, initially spherical capsules freely suspended in a simple shear flow under Stokes flow conditions. The capsules are filled with a Newtonian liquid (same density and viscosity as the suspending fluid). Their membranes satisfy the neo-Hookean constitutive law. We consider the rarely studied case where the capsule centres are initially located in (or near) the plane defined by the flow direction and the vorticity vector, i.e. in two different shear planes. The motion and deformation of the capsules are modelled by means of a boundary integral technique to compute the flows, coupled to a finite element method to calculate the force exerted by the membranes on the fluids. We follow the motion and deformation of the capsules as they are convected towards each other after a sudden start of the flow. Our main finding is that, depending on their initial position and deformability, the two capsules may oscillate slowly about the flow gradient axis, get nearer to each other at each oscillation to finally interact strongly and separate. This minuet motion had not been identified previously. We identify the regions of space where either simple crossing or minuet occurs. This phenomenon has a marked influence on the irreversible trajectory drift of two capsules after crossing: the minuet process leads to a significant trajectory displacement along the flow gradient when none was expected, based on the previous studies where the two capsules had a significant relative velocity.

Understanding how turbulence enhances irreversible scalar mixing in density-stratified fluids is a central problem in geophysical fluid dynamics. While isotropic overturning regions are commonly the focus of mixing analyses, we here investigate whether significant mixing may arise in anisotropic statically stable regions of the flow. Focusing on a single forced direct numerical simulation of stratified turbulence, we analyse spatial correlations between the vertical density gradient $\partial \rho /\partial z$ and the dissipation rates of kinetic energy $\epsilon$ and scalar variance $\chi$, the latter quantifying scalar mixing. The domain is characterized by relatively well-mixed density layers separated by sharp stable interfaces that are correlated with high vertical shear. While static instability is most prevalent within the mixed layers, much of the scalar mixing is localized to the intervening interfaces, a phenomenon not apparent if considering local static instability or $\epsilon$ alone. While the majority of the domain is characterized by the canonical flux coefficient $\varGamma \equiv \chi /\epsilon =0.2$, often assumed in ocean mixing parametrizations, extreme values of $\chi$ within the statically stable interfaces, associated with elevated $\varGamma$, strongly skew the bulk statistics. Our findings suggest that current parametrizations of turbulent mixing may be biased by undersampling, such that the most common, but not necessarily the most significant, mixing events are overweighted. Having focused here on a single simulation of stratified turbulence, it is hoped that our results motivate a broader investigation into the role played by stable density interfaces in mixing, across a wider range of parameters and forcing schemes representative of ocean turbulence.

Compressible mixing layer instabilities are of importance to a wide range of environmental and industrial applications. Past studies have focused on either ideal-gas or real-fluid thermodynamic regimes of single-species mixing layers. However, mixing layers of binary mixtures at supercritical conditions, commonly encountered in fuel injection systems, introduce additional complexities due to the added compositional degree-of-freedom. Moreover, the effect of strong variations in thermodynamic response functions across the Widom line on the binary mixing layer stability remains poorly understood. Thus, the objective of this study is to examine the coupling between the hydrodynamic instability and the real-fluid thermodynamics across the Widom line and its effects on the overall binary mixing layer dynamics. To this end, we develop a linear stability analysis of the full binary-species compressible transport equations coupled with the PC-SAFT equation of state. Analysis shows the existence of a novel instability mechanism that arises from juxtapositioning of the Widom-line transition and the hydrodynamic inflection point. This novel thermodynamically induced instability mechanism has the net effect of destabilizing the binary mixing layer at lowering supercritical conditions towards the critical pressure point. This is in contrast to previous stability analyses of supercritical single-species mixing layers, where increasing pressure destabilizes the flow due to its effect on reducing the density stratification. The discovered thermodynamically induced instability mechanism of binary mixing flows highlights the need for an extension of classical instability criteria to incorporate the effect of strong variations in the thermodynamic response functions across the Widom line on mixing layer instability.

The flow from a supersonic nozzle when operated at off-design conditions exhibits a wide variety of complex flow structures. The prominent ones are the shock reflections inside the jet. The two main types of shock interactions are the regular reflection (RR) and Mach reflection (MR). The prominent characteristic change in the shock pattern when an RR transforms into an MR is the appearance of the Mach stem with a subsonic region downstream of it. The estimation of the Mach stem height gives the size of the subsonic domain and is a direct method to predict the transition from RR to MR or vice versa. The present study is carried out to estimate the size of the Mach stem in an inviscid jet in the overexpanded regime. The analytical methods employed here are the extension of the techniques developed to estimate the Mach stem size in the wedge flows by Li & Ben-Dor (J. Fluid Mech., vol. 341, 1997a, pp. 101–125), Mouton & Hornung (AIAA J., vol. 45, issue 8, 2007, pp. 1977–1987) and Bai & Wu (J. Fluid Mech., vol. 818, 2017, pp. 116–140). The results from the analytical formulation have been compared with the high resolution computational and experimental results. The analytical method reveals that the open jet has a unique, stable MR configuration and forms an upper limit for achievable Mach stem height for the wedge flows. Apart from the estimation of Mach stem height, the growth rates of Mach stem where the RR–MR transition takes place are also calculated and compared with the ones corresponding to wedge flows.

The unsteady effects of transitioning surface tension, $\gamma (t)$, on the dynamics of capillary imbibition in channels of arbitrary shape are analytically investigated with a focus on rectangular and circular channels. With proper scaling, two unsteady models for $\gamma (t)$ are defined and used to highlight this transient behaviour. The convoluted dynamics at the flow front are correctly captured in the governing equations, which are rigorously analysed using unsteady eigenfunction expansion. Then, the final solution and data are obtained by employing the Runge–Kutta fourth-order scheme elegantly applied simultaneously to two derived nonlinear ordinary differential equations. Ultimately, the results are more accurate compared with previous studies. Dynamics and kinematic similarity between rectangular and circular channels are investigated and discussed and the conditions for equivalence in both channels are highlighted. Using a small parameter ($\epsilon$) that stretches the time scale, we successfully use a robust asymptotic analysis to develop and capture the long-time dynamics. Ultimately, we recover the Lucas–Washburn regime analysed in Washburn (Phys. Rev., vol. 17, 1921, pp. 273–283), Lucas (Kolloidn. Z., vol. 23, 1918, pp. 15–22) for steady surface tension where the variations of depth and rate with time result in $h\thicksim t^{1/2}$ and $v \thicksim t^{-1/2}$. In the end, the three forces, namely the inertia, $F_{v}$, the viscous, $F_{\mu }$, and the surface tension, $F_{\gamma }$, are briefly analysed and used to highlight three main distinct regimes. We show that at early times, $F_{v}/F_{\gamma } \thicksim 1$, whereas at a long time, $F_{\mu }/F_{\gamma } \thicksim -1$.

We simulated laminar flow through a hexagonal sphere pack driven by a sinusoidal volume force using direct numerical simulation. We vary two independent parameters, the Hagen and Womersley numbers, representing the amplitude and frequency of the forcing. First, we determine for which regions in the parameter space nonlinear effects have to be considered. We judge the presence of nonlinear effects from the departure of the superficial velocity and kinetic energy from a linear behaviour as well as from the presence of higher harmonics in the discrete Fourier transform of the velocity field. We discuss the asymptotic behaviour of the onset of nonlinearity in the limits of low and high Womersley number, and we delineate approximately the parameter region that can be described using the linear theory. Second, we document the changes of instantaneous velocity fields with Hagen and Womersley numbers. We show that the onset of nonlinearity is accompanied by a loss of fore–aft symmetry of the flow, and subsequently, we employ the deviation from this symmetry to quantify the strength of nonlinear effects in the instantaneous velocity fields. Based on this analysis, we demonstrate that for higher Womersley numbers, the strongest nonlinear effects occur during the deceleration of the superficial velocity; consequently, the development of the nonlinearity is not in phase with the superficial velocity. Finally, we describe the leading-order nonlinear effects in the frequency domain and the interaction among the nonlinear Fourier modes that leads to a temporal variation in the strength of nonlinear effects.

We employ reconnection-capable, vortex filament methods and finite-volume, Navier–Stokes flow solvers to investigate the topological and helicity dynamics of vortex links for medium and high Reynolds numbers. Our vortex-dynamical model is based on discretization of vortex tubes into bundles of numerical analogues of vortex lines. Due to their nearly singular nature, the numerical vortex lines have topological writhe but not twist. By means of our reconnecting vortex tube model, it is shown that the helicity of a vortex link is conserved during the unknotting process. The dynamics of linking and writhe topological measures indicate that most of the initial linking becomes writhe during the post-reconnection evolution. The helicity spectra of the vortex link present alternating-sign helicity fluctuations at all (potential flow) scales up to the vortex core. At pre-reconnection times, these fluctuations are damped by Biot–Savart vortex stretching and helicity becomes single signed. The post-reconnection spectra indicate an inverse helicity cascade corresponding to the creation of a homogenized vortex blob, a process reminiscent of coherent structure formation in turbulence. An accompanying Navier–Stokes calculation of vortex link dynamics at Reynolds numbers $Re=1500$ confirms the post-reconnection transformation of linking into different topological measures, the pre-reconnection damping of helicity spectra fluctuations and the spectral shift to low wavenumbers at post-reconnection times. Due to viscous dissipation action, however, this shift is accompanied by progressive reduction of helicity peak values.

Free-stream turbulence (FST) induced boundary layer transition is an intricate physical process that starts already at the leading edge (LE) with the LE receptivity process dictating how the broad spectrum of FST scales is received by the boundary layer. The importance of the FST integral length scale, apart from the turbulence intensity, has recently been recognized in transition prediction but a systematic variational study of the LE pressure gradient has still not been undertaken. Here, the LE pressure gradient is systematically varied in order to quantify its effect on the transition location. To this purpose, we present a measurement technique for accurate determination of flat-plate boundary layer transition location. The technique is based on electret condenser microphones which are distributed in the streamwise direction with high spatial resolution. All time signals are acquired simultaneously and post-processed giving the full intermittency distribution of the flow over the plate in a few minutes. The technique is validated against a similar procedure using hot-wire anemometry measurements. Our data clearly shows that the LE pressure gradient plays a decisive role in the receptivity process for small integral length scales, at moderate turbulence intensities, leading to variations in the transitional Reynolds number close to 40 %. To our knowledge, this high sensitivity of LE pressure gradient to transition has so far not been reported and our experiments were therefore partly repeated using another LE to ensure set-up independence and result repeatability.

We experimentally investigate the surface drag characteristics of a staggered-distributed cube array and its interaction with the turbulent structure of the overlying flow. Instantaneous maps of the pressure field, inferred from in-plane velocity data are used to estimate the forces acting on a target roughness element. Coupled statistics of the force in combination with conditional flow analysis and extended proper orthogonal decomposition (POD) of the pressure field, based on the velocity POD modes, elucidate the relevant mechanisms responsible for surface drag generation. The results show that turbulent motions, at different scales, leave an imprint on the pressure field. Specifically, positive and negative fluctuations are generally associated with flow regions experiencing a local deceleration and acceleration, respectively. Although large-scale motions were found to be the single greatest contributor to the fluctuating pressure field, their direct influence on the surface drag fluctuations appears to be mitigated by the relative size of the considerably smaller roughness obstacles. We hypothesise that a pressure wave induced by the passage of alternating high- and low-momentum regions evenly affects the flow field over a broad region, coupling the forces on the windward and leeward sides of the cube, which, in turn, partially cancel each other out. Uncorrelated, intermediate and small-scale pressure events are thus more important to the overall drag fluctuations. While the direct influence of the large-scale structures on the surface drag may be smaller than expected, the results suggest that they are still significant for the role they play in modulating the small-scale pressure events in the canopy region.

In this work a mathematical model based on interface energy is proposed within the framework of surface continuum mechanics to study the dynamics of encapsulated bubbles. The interface model naturally induces a residual stress field into the bulk of the bubble, with possible expansion or shrinkage from a stress-free configuration to a natural equilibrium configuration. The significant influence of interface area strain and the coupled effect of stretch and curvature is observed in the numerical simulations based on constrained optimization. Due to the bending rigidity related to additional terms, the dynamic interface tension can become negative, but not due to the interface area strain. The coupled effect of interface strain and curvature term observed is new and plays a dominant role in the dominant compression behaviour of encapsulated bubbles observed in the experiments. The present model is validated by fitting the experimental data of $1.7\,\mathrm {\mu }$m, $1.4\,\mathrm {\mu }$m and $1\,\mathrm {\mu }$m radii bubbles by calculating the optimized parameters. This work also highlights the role of interface parameters and natural configuration gas pressure in estimating the size-independent viscoelastic material properties of encapsulated bubbles with interesting future developments.

The steady dense granular flow in a vertical channel bounded by flat frictional walls in one horizontal direction and with periodic boundary conditions in the other horizontal and vertical directions is studied using the discrete element method. The shape of the scaled velocity profile is characterized quantitatively by a universal exponential function, and the ratio of the maximum and slip velocities is independent of the average volume fraction $\bar {\phi }$ and the channel width $W$. For sufficiently wide channels, the maximum and slip velocities increase proportional to $\sqrt {W}$, and the thickness of the shearing zones increases proportional to $W$. There are four zones in the flow, each with distinct dynamical properties. There is no shear in the plug zone at the centre, but there is particle agitation, and the volume fraction $\phi$ is lower than the random close packing volume fraction $\phi _{rcp}$. In the adjoining dense shearing zone, $\phi$ is greater than the volume fraction for arrested dynamics $\phi _{ad} = 0.587$, and the granular temperature and shear rate depend on the particle stiffness. Adjacent to the dense shearing zone is the loose shearing zone where $\phi < \phi _{ad}$. Here, the properties do not depend on the particle stiffness, and the constitutive relations are well described by hard-particle models. The rheology in the loose shearing zone is similar to the dense flow down an inclined plane. There is high shear and a sharp decrease in $\phi$ in the wall shearing zone of thickness about two particle diameters, where the particle angular velocity is different from the material rotation rate due to the presence of the wall.