Two-dimensional Euler flows, in the plane or on simple surfaces, possess a material invariant, namely the scalar vorticity normal to the surface. Consequently, flows with piecewise-uniform vorticity remain that way, and moreover evolve in a way which is entirely determined by the instantaneous shapes of the contours (interfaces) separating different regions of vorticity – this is known as ‘contour dynamics’. Unsteady vorticity contours or interfaces often grow in complexity (lengthen and fold), either as a result of vortex interactions (like mergers) or ‘filamentation’. In the latter, wave disturbances riding on a background, equilibrium contour shape appear to inevitably steepen and break, forming filaments, repeatedly– and perhaps endlessly. Here, we revisit the onset of filamentation. Building upon previous work and using a weakly nonlinear expansion to third order in wave amplitude, we derive a universal, parameter-free amplitude equation which applies (with a minor change) both to a straight interface and a circular patch in the plane, as well as circular vortex patches on the surface of a sphere. We show that this equation possesses a local, self-similar form describing the finite-time blow up of the wave slope (in a re-scaled long time proportional to the inverse square of the initial wave amplitude). We present numerical evidence for this self-similar blow-up solution, and for the conjecture that almost all initial conditions lead to finite-time blow up. In the full contour dynamics equations, this corresponds to the onset of filamentation.

The wave kinetic equation has become an important tool in different fields of physics. In particular, for surface gravity waves, it is the backbone of wave forecasting models. Its derivation is based on the Hamiltonian dynamics of surface gravity waves. Only at the end of the derivation are the non-conservative effects, such as forcing and dissipation, included as additional terms to the collision integral. In this paper, we present a first attempt to derive the wave kinetic equation when the dissipation/forcing is included in the deterministic dynamics. If, in the dynamical equations, the dissipation/forcing is one order of magnitude smaller than the nonlinear effect, then the classical wave action balance equation is obtained and the kinetic time scale corresponds to the dissipation/forcing time scale. However, if we assume that the nonlinearity and the dissipation/forcing act on the same dynamical time scale, we find that the dissipation/forcing dominates the dynamics and the resulting collision integral appears in a modified form, at a higher order.

Planar linear flows are a one-parameter family, with the parameter $\hat {\alpha }\in [-1,1]$ being a measure of the relative magnitudes of extension and vorticity; $\hat {\alpha } = -1$, $0$ and $1$ correspond to solid-body rotation, simple shear flow and planar extension, respectively. For a neutrally buoyant spherical drop in a hyperbolic planar linear flow with $\hat {\alpha }\in (0,1]$, the near-field streamlines are closed for $\lambda \gt \lambda _c = 2 \hat {\alpha } / (1 - \hat {\alpha })$, $\lambda$ being the drop-to-medium viscosity ratio; all streamlines are closed for an ambient elliptic linear flow with $\hat {\alpha }\in [-1,0)$. We use both analytical and numerical tools to show that drop deformation, as characterized by a non-zero capillary number ($Ca$), destroys the aforementioned closed-streamline topology. While inertia has previously been shown to transform closed Stokesian streamlines into open spiralling ones that run from upstream to downstream infinity, the streamline topology around a deformed drop, for small but finite $Ca$, is more complicated. Only a subset of the original closed streamlines transforms to open spiralling ones, while the remaining ones densely wind around a configuration of nested invariant tori. Our results contradict previous efforts pointing to the persistence of the closed streamline topology exterior to a deformed drop, and have important implications for transport and mixing.

Turbulent flames in practical devices are subject to a superposition of broadband turbulence and narrowband harmonic flow oscillations. In such cases, flames have a superposition of space–time correlated wrinkles, superposed with broadband turbulent disturbances that interact nonlinearly. This paper extends our prior experimental work to characterise and quantify these flame dynamics. We extract ensemble-averaged flame edge and velocity by ensemble-averaging the instantaneous data at the same phase with respect to the forcing cycle. This paper shows that the ensemble-averaged spatio-temporal dynamics of the flame changes significantly with turbulence intensity. From a spatial viewpoint, the ensemble-averaged flame at weak turbulence intensities exhibits clear cusps and a large ratio between curvature in concave and convex regions. In contrast, at high turbulence intensities, the concave and convex parts of the ensemble-averaged flame are nearly symmetric. From a temporal viewpoint, increasing turbulence intensity monotonically suppresses higher harmonics of the forcing frequency that are manifestations of flame nonlinearities. Taken together, these both point to the interesting observation that the ensemble-averaged flame exhibits increasingly linear dynamics with increasing turbulence intensities, in contrast to its very strong nonlinear behaviours at weak turbulence intensities and juxtaposed with the increasingly nonlinear nature of its instantaneous dynamics with increasing turbulence intensity. In addition, prior studies have shown clear coherent modulation of turbulent flame speed correlated with coherent curvature modulation and that this relationship could be quantified via a ‘turbulent Markstein number’, $M_{T}$. We develop correlations for $M_{T}$ showing how it scales with turbulent and narrowband disturbance quantities, such as turbulent flame brush thickness and convective length scale.

We investigate the effects of thermal boundary conditions and Mach number on turbulence close to walls. In particular, we study the near-wall asymptotic behaviour for adiabatic and pseudo-adiabatic walls, and compare to the asymptotic behaviour recently found near isothermal cold walls (Baranwal et al. 2022. J. Fluid Mech. 933, A28). This is done by analysing a new large database of highly-resolved direct numerical simulations of turbulent channels with different wall thermal conditions and centreline Mach numbers. We observe that the asymptotic power-law behaviour of Reynolds stresses as well as heat fluxes does change with both centreline Mach number and thermal condition at the wall. Power-law exponents transition from their analytical expansion for solenoidal fields to those for non-solenoidal field as the Mach number is increased, though this transition is found to be dependent on the thermal boundary conditions. The correlation coefficients between velocity and temperature are also found to be affected by these factors. Consistent with recent proposals on universal behaviour of compressible turbulence, we find that dilatation at the wall is the key scaling parameter for these power-law exponents, providing a universal functional law that can provide a basis for general models of near-wall behaviour.

Debris flows are a growing natural hazard as a result of climate change and population density. To effectively assess this hazard, simulating field-scale debris flows at a reasonable computational cost is crucial. We enhance existing debris flow models by rigorously deriving a series of depth-averaged shallow models with varying complexities describing the behaviour of grain–fluid flows, considering granular mass dilatancy and pore fluid pressure feedback. The most complete model includes a mixture layer with an upper fluid layer, and solves for solid and fluid velocity in the mixture and for the upper fluid velocity. Simpler models are obtained by assuming velocity equality in the mixture or single-layer descriptions with a virtual thickness. Simulations in a uniform configuration mimicking submarine landslides and debris flows reveal that these models are extremely sensitive to the rheology, the permeability (grain diameter) and initial volume fraction, parameters that are hard to measure in the field. Notably, velocity equality assumptions in the mixture hold true only for low permeability (corresponding to grain diameter $d=10^{-3}$ m). The one-layer models’ results can strongly differ from those of the complete model, for example, the mass can stop much earlier. One-layer models, however, provide a rough estimate of two-layer models when permeability is low, initial volume fraction is far from critical and the upper fluid layer is very thin. Our work with uniform settings highlights the need of developing two-layer models accounting for dilatancy and for an upper layer made either of fluid or grains.

An analytical theory is developed that describes acoustic microstreaming produced by the interaction of an oscillating gas bubble with a viscoelastic particle. The bubble is assumed to undergo axisymmetric oscillation modes, which can include radial oscillation, translation and shape modes. The oscillations of the particle are excited by the oscillations of the bubble. No restrictions are imposed on the ratio of the bubble and the particle radii to the viscous penetration depth and the separation distance, as well as on the ratio of the viscous penetration depth to the separation distance. Capabilities of the developed theory are illustrated by computational examples. The shear stress produced by the acoustic microstreaming on the particle’s surface is calculated. It is shown that this stress is much higher than the stress predicted by Nyborg’s formula (1958 J. Acoust. Soc. Am. 30, 329–339), which is commonly used to evaluate the time-averaged shear stress produced by a bubble on a rigid wall.