We use a generalized version of the equation of motion for a thin film of liquid on a solid, horizontal substrate as a model system to study the formation of singularities in space dimensions greater than one. Varying both the exponent controlling long-ranged forces, as well as the exponent of the nonlinear mobility, we predict the structure of the singularity as the film thickness goes to zero. The spatial structure of rupture may be either ‘pointlike’ (approaching axisymmetry) or ‘quasi-one-dimensional’, in which case a one-dimensional singularity is unfolded into two or higher space dimensions. The scaling of the profile with time may be either strictly self-similar (the ‘regular’ case) or discretely self-similar and perhaps chaotic (the ‘irregular’ case). We calculate the phase boundaries between these regimes, and confirm our results by detailed comparisons with time-dependent simulations of the nonlinear thin film equation in two space dimensions.

This paper examines two-dimensional liquid curtains ejected at an angle to the horizontal and affected by gravity and surface tension. The flow is, to leading order, shearless and viscosity, negligible. The Froude number is large, so that the radius of the curtain's curvature exceeds its thickness. The Weber number is close to unity, so that the forces of inertia and surface tension are almost perfectly balanced. An asymptotic equation is derived under these assumptions, and its steady solutions are explored. It is shown that, for a given pair of ejection velocity/angle, infinitely many solutions exist, each representing a steady curtain with a stationary capillary wave superposed on it. These solutions describe a rich variety of behaviours: in addition to arching downwards, curtains can zigzag downwards, self-intersect and even rise until the initial supply of the liquid's kinetic energy is used up. The last type of solutions corresponds to a separatrix between upward- and downward-bending curtains – in both cases, self-intersecting (such solutions are meaningful only until the first intersection, after which the liquid just splashes down). Finally, suggestions are made as to how the existence of upward-bending curtains can be tested experimentally.

We present a series of experiments to explore the dynamics of particle-laden fountains rising through a stratified environment with zero buoyancy flux at the source. We find that the ratio $U$ between the particle sedimentation speed $V_s$ and the characteristic fountain velocity $(M_0N^{2})^{1/4}$, where $M_0$ is the initial momentum flux and $N$ the frequency of the ambient stratification, has a profound effect on the structure of the fountain and the dispersal of the particles. In a mono-disperse particle fountain, when the settling speed of the particles is small in comparison to the characteristic fountain speed ($U\ll 1$) the flow initially behaves in an analogous fashion to a single-phase fountain, forming an intrusion at a height of approximately 0.5 of the maximum fountain height. As the fluid–particle mixture spreads out, the particles gradually sediment to the tank floor. The intruding fluid subsequently rises and forms a new intrusion at its neutral buoyancy height. Some of the particles are carried up from the original intrusion with the rising fluid. This leads to the formation of a sedimenting column of particles with a characteristic radius. We observe a transition in the behaviour of the particle fountains in the vicinity of $U\sim 0.1$, with the particles now separating from the fluid near the top of the fountain. The separation of the particles leads to a reduction in the steady-state height of the particle-laden fountain, while the fluid in the fountain continues upwards until reaching its neutral buoyancy height and forming an intrusion above the fountain top. We compare the experimental data with two models of turbulent fountains to help rationalise the trends observed as a function of the dimensionless fall speed $U$. We briefly consider the dynamics of poly-disperse particle fountains and relate their dynamics to the regimes observed in their mono-disperse counterparts. We discuss the implications of this work for the dispersal of different sized particles from submarine volcanic eruptions.

Boussinesq and non-Boussinesq gravity currents produced from a finite volume of heavy fluid propagating into an environment of light ambient fluid on unbounded uniform slopes in the range $0\,^\circ \le \theta \le 12\,^\circ$ are reported. The relative density difference $\epsilon = (\rho _1-\rho _0)/\rho _0$ is varied in the range $0.05 \le \epsilon \le 0.15$ in this study, where $\rho _1$ and $\rho _0$ are the densities of the heavy and light ambient fluids, respectively. Our focus is on the influence of the relative density difference on the deceleration phase of the propagation. In the early deceleration phase, the front location history follows the power relationship ${(x_f+x_0)}^2 = {(K_I B)}^{1/2} (t+t_{I})$, where $(x_f+x_0)$ is the front location measured from the virtual origin, $K_I$ an experimental constant, $B$ the total buoyancy, $t$ the time and $t_I$ the $t$ intercept. The dimensionless constant $K_I$ is influenced by the slope angle and the relative density difference. In the late deceleration phase for the gravity currents on the steeper slopes in this study ($12\,^\circ$, $9\,^\circ$ and $6\,^\circ$), an ‘active’ head separates from the body of the current and the front location history follows the power relationship ${(x_f+x_0)}^{8/3} = K_{VS} {B}^{2/3} V^{2/9}_0 {\nu }^{-1/3} ({t+t_{VS}})$, where $K_{VS}$ is an experimental constant, $V_0$ the total volume of heavy fluid, $\nu$ the kinematic viscosity of fluid and $t_{VS}$ the $t$ intercept. The dimensionless constant $K_{VS}$ is shown to be influenced by the slope angle but not significantly influenced by the relative density difference. In the late deceleration phase for the gravity currents on the milder slopes in this study ($3\,^\circ$ and $0\,^\circ$), the gravity currents maintain an integrated shape without violent mixing with the ambient fluid and the front location history follows the power relationship ${(x_f+x_0)}^{4} = K_{VM} {B}^{2/3} V^{2/3}_0 {\nu }^{-1/3} ({t+t_{VM}})$, where $K_{VM}$ is an experimental constant and $t_{VM}$ the $t$ intercept. The dimensionless constant $K_{VM}$ is shown to be influenced by both the slope angle and the relative density difference. While the influence of the relative density difference on $K_{VM}$ is carried along for the gravity currents on the milder slopes in the late deceleration phase, the relative density difference interestingly has no significant influence on $K_{VS}$ for the gravity currents on the steeper slopes in the late deceleration phase. Our results suggest that the non-Boussinesq gravity currents on the milder slopes may remain non-Boussinesq ones in the late deceleration phase while the non-Boussinesq gravity currents on the steeper slopes may have become Boussinesq ones in the late deceleration phase.

A simple principle is formulated, which makes it possible to explain and, in some cases, to predict the shape of secondary flows of Prandtl's second kind that arise in turbulent flows in straight pipes of non-circular cross-section. The motion along the no-slip wall is directed in the direction of reduction of pressure. Therefore, the direction of the secondary motion along the perimeter of the pipe cross-section can be determined by the location of the mean wall pressure local extrema. Velocity fluctuations along a curved wall cause the mean wall pressure change, the greater the change, the greater the curvature of the wall. Thus, the location of local wall pressure extrema is consistent with the wall curvature. In a number of cases, an approximate picture of secondary flows can be predicted a priori using only symmetry considerations and a wall curvature analysis. The conditions for the formation of a secondary flow change on a free surface where, in addition to pressure, Reynolds stresses act on the fluid particles, and the result of their interaction is unknown in advance. A turbulent flow in a rectangular channel is considered, in which one of the walls (upper) is shear-free. Application of the formulated principle dictates the presence of secondary flows in the upper corners, with a downward motion along the solid sidewalls. This contradicts the popular point of view about the shape of secondary flows in open channels, according to which the motion along the sidewalls is directed upward towards the corner and then towards the centre of the channel along the free surface. The paper presents the direct numerical simulation (DNS) results showing that along with the large-scale secondary flow, which generally agrees with the traditional picture, there is a smaller-scale single-vortex secondary flow with the opposite direction of rotation in the vicinity of each upper corner. The presence of a no-slip wall is an important element of the described mechanism of secondary flow formation. In this regard, the question logically arises about the possibility of the occurrence of secondary flows in the corners between two free boundaries. A series of DNS of turbulent flows in a square duct with two solid and two free adjacent walls have been carried out. It was found that the Reynolds stress forces in the vicinity of the corner between free boundaries are potential, are neutralized by the pressure gradient, and therefore do not create a secondary flow.

The effects of the $A + B \rightarrow C$ chemical reaction on miscible viscous fingering in a radial source flow are analysed using linear stability theory and numerical simulations. This flow and transport problem is described by a system of nonlinear partial differential equations consisting of Darcy's law for an incompressible fluid coupled with nonlinear advection–diffusion–reaction equations. For an infinitely large Péclet number ($Pe$), the linear stability equations are solved using spectral analysis. Further, the numerical shooting method is used to solve the linearized equations for various values of $Pe$ including the limit $Pe \rightarrow \infty$. In the linear analysis, we aim to capture various critical parameters for the instability using the concept of asymptotic instability, i.e. in the limit $\tau \rightarrow \infty$, where $\tau$ represents the dimensionless time. We restrict our analysis to the asymptotic limit $Da^{\ast }$$(= Da \tau ) \rightarrow \infty$ and compare the results with the non-reactive case ($Da = 0$) for which $Da^{\ast } = 0$, where $Da$ is the Damköhler number. In the latter case, the dynamics is controlled by the dimensionless parameter $R_{Phys} = -(R_{A} - \beta R_{B})$. In the former case, for a fixed value of $R_{Phys}$, the dynamics is determined by the dimensionless parameter $R_{Chem} = -(R_{C} - R_{B} - R_{A})$. Here, $\beta$ is the ratio of reactants’ initial concentration and $R_{A}$, $R_{B}$ and $R_{C}$ are the log-viscosity ratios. We perform numerical simulations of the coupled nonlinear partial differential equations for large values of $Da$. The critical values $R_{Phys, c}$ and $R_{Chem, c}$ for instability decrease with $Pe$ and they exhibit power laws in $Pe$. In the asymptotic limit of infinitely large $Pe$ they exhibit a power-law dependence on $Pe$ ($R_{Chem, c} \sim Pe^{-1/2}$ as $Pe \rightarrow \infty$) in both the linear and nonlinear regimes.

Cavitating flow in the wake of a wedge-shaped bluff body is examined to understand the role of the presence of high void-fraction regions in the near-wake region on the process of vortex formation and shedding. Previous studies have noted that developed cavitation forming in the wake of bluff bodies typically leads to an increase in the vortex shedding rate, peaking at a particular cavitation number. Further reduction in cavitation number leads to a return to lower shedding rates as the cavity grows into a super-cavity. The underlying flow processes that lead to this phenomenon are explored using traditional flow visualisation combined with time-resolved void-fraction flow fields based on X-ray densitometry. These measurements allow us to relate the compressibility of the near-wake bubbly flow to the underlying flow processes. Specifically, we use proper orthogonal decomposition (POD) of the void-fraction fields to show that the increased rate of vortex shedding is associated with a pulsating mode of the void-fraction flow field, compared with a sinusoidal variation corresponding to the lower void-fraction shedding processes similar to that of the non-cavitating wake. The pulsating mode becomes more pronounced when the wake void fraction increases with decreasing cavitation number, with the maximum shedding occurring near the point that the wake flow becomes locally supersonic. The important influence of flow compressibility on the wake dynamics is confirmed through the examination of the effect of non-condensable gas injection.

The instability and transition to turbulence and its evolution in pulsatile flows, which involve reverse flows and unsteady flow separations, is the primary focus of this experimental work. A piston driven by a programmable DC servo motor was used to set-up a water flow system and provide the pulsation characteristics. Time-resolved particle image velocimetry data were acquired in a refractive index matching set-up by using a continuous wave laser and a high-frame-rate digital camera. The position of the piston was continuously recorded by a laser proximity sensor. Five different experiments were carried out with Reynolds numbers in the range of 535–4825 and Womersley numbers from 11.91 to 23.82. The non-stationarity of the data was addressed by incorporating trend removal methods involving low- and high-pass filtering of the data, and using empirical mode decomposition together with the relevant Hilbert–Huang transform to determine the intrinsic mode functions. This latter method is more appropriate for nonlinear and non-stationary cases, for which traditional analysis involving classical Fourier decomposition is not directly applicable. It was found that transition to turbulence is a spontaneous event covering the whole near-wall region. The instantaneous vorticity profiles show the development of a large-scale ring-like attached wall vortical layer (WVL) with smaller vortices of higher frequencies than the pulsation frequency superimposed, which point to a shear layer Kelvin–Helmholtz (K–H) type of instability. Inflectional instability leads to flow separation and the formation of a major roll-up structure with the K–H vortices superimposed. This structure breaks down in the azimuthal direction into smaller turbulence patches with vortical content, which appears to be the prevailing structural content of the flow at each investigated Reynolds number (Re). At higher Re numbers, the strength and extent of the vortices are larger and substantial disturbances appear in the free stream region of the flow, which are typical of pipe flows at transitional Re numbers. Turbulence appears to be produced at the locations of maximum or minimum vorticity within the attached WVL, in the ridges between the K–H vortices around the separated WVL and the upstream side of the secondary vortex where the flow impinges on the wall. This wall turbulence breaks away into the middle section of the pipe, at approximately $Re \ge 2200$, by strong eruptions of the K–H vortices.

When a horizontal cylinder is placed on a vertically moving belt coated with a thin layer of viscous fluid, experiments reveal that, at a specific belt velocity, the cylinder can be levitated at a fixed height while rotating around its own axis at an a priori unknown rate. We develop and solve a model for this experiment, using a combination of asymptotic analysis and direct numerical simulation. We obtain a relationship between the belt speed and cylinder rotation rate, which we successfully compare with experimental results.

Response modes computed via linear resolvent analysis of a turbulent mean-flow field have been shown to qualitatively capture characteristics of the observed turbulent coherent structures in both wall-bounded and free shear flows. To make such resolvent models predictive, the nonlinear forcing term must be closed. Strategies to do so include imposing self-consistent sets of triadic interactions, proposing various source models or through turbulence modelling. For the latter, several investigators have proposed using the mean-field eddy viscosity acting linearly on the fluctuation field. In this study, a data-driven approach is taken to quantitatively improve linear resolvent models by deducing an optimal eddy-viscosity field that maximizes the projection of the dominant resolvent mode to the energy-optimal coherent structure educed using spectral proper orthogonal decomposition (SPOD) of data from high-fidelity simulations. We use large-eddy simulation databases for round isothermal jets at subsonic, transonic and supersonic conditions and show that the optimal eddy viscosity substantially improves the agreement between resolvent and SPOD modes, reaching over 90 % agreement at those frequencies where the jet exhibits a low-rank response. We then consider a fixed model for the eddy viscosity and show that with the calibration of a single constant, the results are generally close to the optimal one. In particular, the use of a standard Reynolds-averaged Navier–Stokes eddy-viscosity resolvent model, with a single coefficient, provides substantial agreement between SPOD and resolvent modes for three turbulent jets and across the most energetic wavenumbers and frequencies.

Trees are known to reduce the wind momentum efficiently. Yet, firm quantitative estimates of their contribution to the land surface drag have remained elusive, partly because trees have complex shapes that consist of elastic multi-scale elements. This structural complexity makes trees inherently difficult to scale for wind tunnel studies. Here, we test a new method for quantifying the drag force on a solitary mature tree in its natural environment. The method is based on the application of mass and momentum conservation over a control volume that encloses the tree. For this control volume, the drag force is estimated through the momentum deficit in the wake. For the characterisation of the heterogeneous and high-gradient wind field in the wake, spatially distributed measurements of the wind vector were acquired using three synchronously scanning wind lidar instruments in a vertical plane encompassing the wake. The resulting drag force estimate is compared to a reference measurement from a tree-mounted sensor at the base of the stem. We find that the drag force in both methods shows a dependence on the wind speed raised to an exponent of 1.8 and that the drag force, based on the momentum deficit method, is consistently underestimated by 1 %–10 %. Potential reasons for this bias are discussed in light of the accuracy of both methods. The relatively close agreement between the two methods indicates that scanning Doppler lidar measurements can be used to determine the drag force on complex objects in their natural environment, such as trees.

In this paper, we develop a theory to establish that the thermo-osmotic (TOS) effects, induced by the application of an axial temperature gradient, lead to a massive enhancement in liquid transport in nanochannels grafted with charged polyelectrolyte (PE) brushes. We quantify the TOS transport by quantifying the induced electric field and the induced TOS flow field. The different components of the electric field, namely the ionic component, the thermal component and the osmotic component, as well as the contributions of different ions to these components, are quantified. Furthermore, we express the TOS velocity as a combination of chemiosmotic (COS), thermal and electro-osmotic (EOS) components. The COS and the thermal components augment each other and the overall strength and direction of the TOS flow are dictated by the direction and the relative strength of the EOS component. Most importantly, we compare the cases of brush-grafted nanochannels with those of the brush-free nanochannels of identical surface charge densities: the TOS transport is massively augmented in the brush-grafted nanochannels attributed to the combination of the localization of the electric double layer (EDL) (and hence any body force that depends on the EDL charge density) away from the nanochannel wall (i.e. the location of the maximum drag force) and the presence of a possible molecular slip (experienced by the liquid) along the brush surface.

We report a numerical study of Rayleigh–Bénard convection through random porous media using pore-scale modelling, focusing on the Lagrangian dynamics of fluid particles and heat transfer for varied porosities $\phi$. Due to the interaction between the porous medium and the coherent flow structures, the flow is found to be highly heterogeneous, consisting of convection channels with strong flow strength and stagnant regions with low velocities. The modifications of flow field due to porous structure have a significant influence on the dynamics of fluid particles. Evaluation of the particle displacement along the trajectory reveals the emergence of anomalous transport for long times as $\phi$ is decreased, which is associated with the long-time correlation of Lagrangian velocity of the fluid. As porosity is decreased, the cross-correlation between the vertical velocity and temperature fluctuation is enhanced, which reveals a mechanism to enhance the heat transfer in porous-media convection.

Granular systems are of a discrete nature. Nevertheless, it can be advantageous to describe their dynamics by means of continuum mechanical methods. The numerical solution of the corresponding hydrodynamic equations is, however, difficult. Therefore, previous numerical simulations are typically geared towards highly specific systems and frequently restricted to two dimensions or mild driving conditions. Here, we present the first robust general simulation scheme for granular hydrodynamics in three dimensions which is not bound to the above limitations. The performance of the simulation scheme is demonstrated by means of three applications which have been proven as notoriously difficult for numerical hydrodynamic description. Although, by construction, our numerical method covers grain-inertia flows, the presented examples demonstrate that it produces reliable results even in the jammed or high density limit.

The dynamics of a vortex ring advecting towards and interacting with a solid wall with a coaxial aperture is governed by the aperture-to-ring radius ratio, $R_a/R_i$, which is parametrically explored herein through a series of numerical simulations for ring Reynolds numbers ranging from $Re=1000$ to $3000$. For $R_a/R_i \lessapprox 0.9$, the interaction largely resembles that of a vortex ring impacting a solid wall (impact regime), whereas, for $R_a/R_i \gtrapprox 1.3$, the ring passes through the aperture, with the influence of the interaction diminishing as $R_a/R_i$ increases (slip-through regime). When the aperture radius is approximately equal to that of the ring, however, an interesting phenomenon is observed, wherein the hydrodynamic impulse of the vortex ring is enhanced up to an additional 11 % at the highest considered Reynolds number when comparing with a free vortex ring that experiences no collision (herein termed the ‘vortex nozzle’ effect). Detailed investigation of the ‘vortex nozzle’ illustrates that the impulse enhancement is a consequence of two complementary effects: (i) fluid originating along the impact side of the wall is entrained into the ring, increasing its radius and volume; and (ii) the circulation loss during the interaction with the aperture tip is minimized due to the vortex core enveloping the aperture tip. In addition to ring impulse enhancement, the ‘vortex nozzle’ regime also exhibits the greatest volumetric flow rate through the nozzle and the highest loading on the structure, which may have practical engineering applications in smart material-based energy harvester designs.

A simple model is presented for the evolution of a dipolar vortex propagating horizontally in a vertical-slice model of a weakly stratified inviscid atmosphere, following the model of Flierl & Haines (Phys. Fluids, vol. 6, 1994, pp. 3487–3497) for a modon on the ${\rm beta}$-plane. The dipole is assumed to evolve to remain within the family of Lamb–Chaplygin dipoles but with varying radius and speed. The dipole loses energy and impulse through internal wave radiation. It is argued, and verified against numerical solutions of the full equations, that an appropriately defined centre vorticity for the dipole is closely conserved throughout the flow evolution. Combining conservation of centre vorticity with the requirement that the dipole energy loss balances the work done on the fluid by internal wave radiation gives a model that captures much of the observed dipole decay. Similar results are noted for a cylindrical dipole propagating along the axis of a rotating fluid when the dipole axis is perpendicular to the axis of rotation and for a spherical vortex propagating horizontally in a weakly stratified fluid. The model extends to fluids of small viscosity and so provides an estimate for the relative importance of wave drag and dissipation in dipole decay.

Animals and bio-inspired robots can swim/fly faster near solid surfaces, with little to no loss in efficiency. How these benefits change with propulsor aspect ratio is unknown. Here we show that lowering the aspect ratio weakens unsteady ground effect, thrust enhancements become less noticeable, stable equilibrium altitudes shift lower and become weaker and wake asymmetries become less pronounced. Water-channel experiments and potential flow simulations reveal that these effects are consistent with known unsteady aerodynamic scalings. We also discovered a second equilibrium altitude even closer to the wall (${<}0.35$ chord lengths). This second equilibrium is unstable, particularly for high-aspect-ratio foils. Active control may therefore be required for high-aspect-ratio swimmers hoping to get the full benefit of near-ground swimming. The fact that aspect ratio alters near-ground propulsion suggests that it may be a key design parameter for animals and robots that swim/fly near a seafloor or surface of a lake.

The geometry of compound sessile drops at equilibrium on a flat substrate can exhibit a variety of complicated morphological configurations. In this paper, we first investigate the configuration boundaries of the compound sessile drops in a wide parameter space, where a specific configuration is not stable outside its boundaries. Then, we focus on the transitions among the axisymmetric configurations, i.e. encapsulation, lens and collars. The configuration transitions result from the variation of the wettability of the substrate and the volume ratio of the two component droplets. With the help of theoretical analysis and numerical simulations, we obtain previously unidentified criteria for the onset of configuration transition, identify the irreversible and reversible configuration transitions, reveal the dynamic behaviours of configuration transitions that are not accessible to theoretical analysis, and provide a further step towards the ultimate purpose of such work, which is the controllable reconfiguration of the compound sessile drops.

This study investigates the interactions between combustor and isolator and the role played by combustion on choking-induced unstart. Shock train unsteadiness and pressure fluctuations in non-reacting environments have been previously explained in terms of shock-boundary layer interaction and acoustic forcing but, when applied to scramjets, it is still unclear whether and how this picture is altered by combustion effects. The novel experimental set-up used in this study consists of a circular cross-section model scramjet, with optically accessible combustor and isolator, tested in a high-enthalpy hypersonic free stream at Mach 4.5. A comparison is made between cases in which flow choking is induced via thermal (combustion) and non-thermal (mass addition) mechanisms using time-resolved static wall pressure measurements, high-speed flow visualization and planar laser-induced fluorescence of the OH radical as diagnostic tools. Further details on the nature of the interactions observed were provided by experiments performed in a low-enthalpy $\textrm {CO}_2$ free stream at Mach 4 using planar laser scattering visualization. The results revealed remarkable qualitative similarities between unstart processes occurring at high and low enthalpy. At high enthalpy the similarities between thermal and non-thermal choking-induced unstart were both qualitative and quantitative, suggesting very limited effect of combustion on the dynamics of the isolator shock train. Isolator flow unsteadiness, on the other hand, drastically affected the propagation of the pseudo-normal shock in the combustor, but no significant feedback effect on the isolator behaviour was observed.

Energy transfer in turbulent fluids is non-Gaussian. We quantify non-Gaussian energy transfer between the atmosphere and bodies of water using a turbulent diffusion operator coupled with temporally self-affine velocity distributions and a recursive integration method that produce multifractal measures. The measures serve as input to a system of moment field equations (derived from Navier–Stokes) that generate and track high-frequency gravity waves that propagate through the water surface (as a result of the air–water interactions). The dimension of the support of the air–water turbulence produced by our methods falls within the range of theory and observation, and correspondingly, hindcast statistical measures of the water-wave surface such as significant water-wave height and wave period are well correlated to observational buoy data. Further, our recursive integration method can be used by spectral resolving phase-averaged models to interpolate temporal wind data to smaller scales to capture the non-Gaussian behaviour of the air–water interaction.