Many granular surface flows occur as shear flows of finite thickness, over erodible beds composed of the same granular material. Such beds may be fragile, and offer no more resistance to erosion than to sustained shear. Or they may be brittle, and offer instead an excess resistance to erosion. To take this contrast into account, new basal boundary conditions are proposed. Their implications for parallel flows down infinite slopes are then examined for three different cases: stationary flows; starting; and stopping transients. For all three cases, flow behaviour is altered significantly when beds present an excess resistance to erosion. For stationary flows, non-unique velocity profiles are obtained, implying hysteresis or history-dependence. For starting transients, a power law growth of the flow thickness is predicted, instead of the jump to finite or infinite depth that would otherwise occur. For stopping transients, flows start to decelerate with a finite basal shear rate, even over erodible substrates. Analytical solutions to the corresponding free and moving boundary problems are obtained, and checked against numerical results. Model predictions are then compared with experimental measurements. Overall, good agreement is obtained. In particular, the model describes well the very different erosional responses observed for fragile and brittle beds.

In a three-layer system with equal upper and lower layer thicknesses that are sufficiently thin and with the same density difference across each interface, breathers have been shown to exist using fully nonlinear governing equations. These breathers are well modelled by theoretical solutions of the mKdV equation, provided the interfaces between the layers do not cross a critical depth. The soliton-like characteristics of fully nonlinear breathers, in particular how two breathers interact, have yet to be explored. Using numerical simulations, this study addresses this shortcoming by studying fully nonlinear overtaking collisions of two breathers in a three-layer symmetric stratification. We apply the fully nonlinear and strongly dispersive FDI-3s internal wave equations, based on a variational principle, in a three-layer system. When the amplitude is small, the analytic breathers fit the wave shapes of the overtaking collision breathers. We find that the larger the upper and lower layer thicknesses are, provided they are below the critical thickness, the more the breathers behave like solitons. We show that an overtaking collision of two breathers is close to elastic.

We present an investigation of the resonance conditions governing triad interactions of cylindrical internal waves, i.e. Kelvin modes, described by Bessel functions. Our analytical study, supported by experimental measurements, is performed both in confined and unconfined axisymmetric domains. We are interested in two conceptual questions: can we find resonance conditions for a triad of Kelvin modes? What is the impact of the boundary conditions on such resonances? In both the confined and unconfined cases, we show that sub-harmonics can be spontaneously generated from a primary wave field if they satisfy at least a resonance condition on their frequencies of the form $\omega _0 = \pm \omega _1 \pm \omega _2$. We demonstrate that the resulting triad is also spatially resonant, but that the resonance in the radial direction may not be exact in confined geometries due to the prevalence of boundary conditions – a key difference compared with Cartesian plane waves.

When two small fluid drops are sufficiently close, the van der Waals force overcomes surface tension and deforms the surfaces into contact, initiating coalescence. The dynamics of surface deformation across an inviscid gap falls into two distinct regimes (Stokes and inertial–viscous) characterized by the forces that balance the van der Waals attraction at leading order (viscosity, and both inertia and viscosity). The previously studied Stokes regime holds for very viscous drops but fails for less viscous drops as inertia becomes significant before contact is reached. We show that the subsequent inertial–viscous dynamics is self-similar as contact is approached, with the gap width decreasing as $t{'^{3/8}}$ and the radial scale of the deformed region decreasing as $t{'^{1/2}}$ as $t{'}\to 0$, for time until contact $t'$. The self-similar behaviour is universal and is the generic asymptotic behaviour observed in time-dependent simulations. The unique self-similar gap profile of the inertial–viscous regime suggests new initial conditions for the coalescence of the drops after contact.

Plenoptic particle image velocimetry and surface pressure measurements were used to analyse the early development of leading-edge vortices (LEVs) created by a flat-plate wing of aspect ratio 2 rolling in a uniform flow parallel to the roll axis. Four cases were constructed by considering two advance coefficients, $J=0.54$ and 1.36, and two wing radii of gyration, $R_g/c=2.5$ and 3.25. In each case, the wing pitch angle was articulated such as to achieve an angle of attack of $33^{\circ }$ at the radius of gyration of the wing. The sources and sinks of vorticity were quantified for a chordwise rectangular control region, using a vorticity transport framework in a non-inertial coordinate system attached to the wing. Within this framework, terms associated with Coriolis acceleration provide a correction to tilting and spanwise convective fluxes measured in the rotating frame and, for the present case, have insignificant values. For the baseline case ($J=0.54, R_g/c=3.25$), three distinct spanwise regions were observed within the LEV, with distinct patterns of vortex evolution and vorticity transport mechanisms in each region. Reducing the radius of gyration to $R_g/c=2.5$ resulted in a more stable vortex with the inboard region extending over a broader spanwise range. Increasing advance ratio eliminated the conical vortex, resulting in transport processes resembling the mid-span region of the baseline case. Although the circulation of the LEV system was generally stronger at the larger advance coefficient, the shear-layer contribution was diminished.

Premixed turbulent flames, encountered in power generation and propulsion engines, are an archetype of a randomly advected, self-propagating surface. While such a flame is known to exhibit large-scale intermittent flapping, the possible intermittency of its small-scale fluctuations has been largely disregarded. Here, we experimentally reveal the inner intermittency of a premixed turbulent V-flame, while clearly distinguishing this small-scale feature from large-scale outer intermittency. From temporal measurements of the fluctuations of the flame, we find a frequency spectrum that has a power-law subrange with an exponent close to $-2$, which is shown to follow from Kolmogorov phenomenology. Crucially, however, the moments of the temporal increment of the flame position are found to scale anomalously, with exponents that saturate at higher orders. This signature of small-scale inner intermittency is shown to originate from high-curvature, cusp-like structures on the flame surface, which have significance for modelling the heat release rate and other key properties of premixed turbulent flames.

Experiments are reported that explore the onset of motion of bubbles in a model yield stress fluid, Carbopol gel. Starting from a trapped spherical bubble in a gel, the yielding limit for the bubble motion is obtained by gradually expanding the bubble via a stepwise decrease in pressure. Our results show that at the yielding limit bubbles are longer and thinner when they are in a higher concentrated gel. This is suggestive of a link between the shape and size of the bubbles at the onset of motion and the rheology of the material, in particular elastic behaviour below the yielding point. Particular attention has been paid to investigating the dynamic response of gel during the bubble growth. Subjecting the bubble to a periodic change in the pressure confirms the irreversibility of the gel deformation and its hysteresis, which are hallmarks of nonlinear viscoelastic behaviour of the gel before yielding. In this context, the periodic expansion and contraction of the bubbles leave residual deformation (stresses) in the gel which facilitates the liberation of bubbles.

Responses of a geometry-induced separation bubble (GISB) and a pressure-induced separation bubble (PISB) at enhanced levels of free-stream turbulence (FST) have experimentally been investigated for a comparative study using the particle image velocimetry (PIV) technique. The outlines of separation bubbles based on the dividing streamlines are self-similar for different levels of FST and Reynolds numbers. The spectral analyses of the time-resolved PIV data show that the vortex shedding frequency of a separated shear layer remains unchanged for the GISB cases even with an enhanced level of FST. In contrast, it is different for the PISB cases. We propose a criterion that determines whether the frequency will remain the same even for the cases with FST. Linear stability analyses reveal that the inviscid-inflectional instability dominates the transition process, and the linear stages of transition are not completely bypassed even at an enhanced level of FST. The most amplified frequencies, while scaling with the displacement thickness and the boundary layer edge velocity, collapse in a single curve for all the cases. Furthermore, measurements in the spanwise plane show that the streamwise velocity streak/Klebanof mode at an enhanced level of FST is not a general flow feature for all types of separation bubbles. However, at an enhanced level of FST for the PISB case, the boundary layer streaks are found to distort the two-dimensional vortex structure associated with the Kelvin–Helmholtz instability, eventually leading to a three-dimensional $\varLambda$-like structure in the spanwise plane.

We carry out a weakly nonlinear analysis of the centrifugal instability for a columnar vortex in a rotating fluid, and compare the results to those of the semi-linear model derived empirically by Yim et al. (J. Fluid Mech., vol. 897, 2020, A34). The asymptotic analysis assumes that the Reynolds number is close to the instability threshold so that the perturbation is only marginally unstable. This leads to two coupled equations that govern the evolutions of the amplitude of the perturbation and of the mean flow under the effect of the Reynolds stresses due to the perturbation. These equations differ from the Stuart–Landau amplitude equation or coupled amplitude equations involving a mean field that have been derived previously. In particular, the amplitude does not saturate to a constant as in the supercritical Stuart–Landau equation, but decays afterwards reflecting the instability disappearance when the mean flow tends toward a neutrally stable profile in the direct numerical simulations (DNS). These equations resemble those of the semi-linear model except that the perturbation in the weakly nonlinear model keeps at leading order the structure of the eigenmode of the unperturbed base flow. The predictions of the weakly nonlinear equations are compared to those of the semi-linear model and to DNS for the Rossby number $Ro=-4$ and various Reynolds numbers and wavenumbers. They are in good agreement with the DNS when the growth rate is sufficiently small. However, the agreement deteriorates and becomes only qualitative for parameters away from the marginal values, whereas the semi-linear model continues to be in better agreement with the DNS.

We examine the effect of Dean number on the inertial focusing of spherical particles suspended in flow through curved microfluidic ducts. Previous modelling of particle migration in curved ducts assumed the flow rate was small enough that a leading-order approximation of the background flow with respect to the Dean number produces a reasonable model. Herein, we extend our model to situations involving a moderate Dean number (in the microfluidics context) while the particle Reynolds number remains small. Variations in the Dean number cause a change in the axial velocity profile of the background flow which influences the inertial lift force on a particle. Simultaneously, changes in the cross-sectional velocity components of the background flow directly affect the secondary flow induced drag. In keeping the particle Reynolds number small, we continue to approximate the inertial lift force using a regular perturbation while capturing the subtle effects from the modified background flow. This approach pushes the limits at which a regular perturbation is applicable to provide some insights into how variations in the Dean number influence particle focusing. Our results illustrate that, as the extrema in the background flow move towards the outside of edge of the cross-section with increasing Dean number, we observe a similar shift in the stable equilibria of some, but not all, particle sizes. This might be exploited to enhance the lateral separation of particles by size in a number of practical scenarios.