It is generally accepted that the evolution of the deep-water surface gravity wave spectrum is governed by quartet resonant and quasi-resonant interactions. However, it has also been reported in both experimental and computational studies that non-resonant triad interactions can play a role, e.g. generation of bound waves. In this study, we investigate the effects of triad and quartet interactions on the spectral evolution, by numerically tracking the contributions from quadratic and cubic terms in the dynamical equation. In a finite time interval, we find that the contribution from triad interactions follows the trend of that from quartet resonances (with comparable magnitude) for most wavenumbers, except that it peaks at low wavenumbers with very low initial energy. This result reveals two effects of triad interactions. (1) The non-resonant triad interactions can be connected to form quartet resonant interactions (hence exhibiting the comparable trend), which is a reflection of the normal form transformation applied in wave turbulence theory of surface gravity waves. (2) The triad interactions can fill energy into the low-energy portion of the spectrum (low wavenumber part in this case) on a very fast time scale, with energy distributed in both bound and free modes at the same wavenumber. We further analyse the latter mechanism using a simple model with two initially active modes in the wavenumber domain. Analytical formulae describing the distribution of energy in free and bound modes are provided, along with numerical validations.

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.

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.

We consider the dynamics of a liquid film with a pinned contact line (for example, a drop), as described by the one-dimensional, surface-tension-driven thin-film equation $h_t + (h^n h_{xxx})_x = 0$, where $h(x,t)$ is the thickness of the film. The case $n=3$ corresponds to a film on a solid substrate. We derive an evolution equation for the contact angle $\theta (t)$, which couples to the shape of the film. Starting from a regular initial condition $h_0(x)$, we investigate the dynamics of the drop both analytically and numerically, focusing on the contact angle. For short times $t\ll 1$, and if $n\ne 3$, the contact angle changes according to a power law $\displaystyle t^{\frac {n-2}{4-n}}$. In the critical case $n=3$, the dynamics become non-local, and $\dot {\theta }$ is now of order $\displaystyle {\rm{e}}^{-3/(2t^{1/3})}$. This implies that, for $n=3$, the standard contact line problem with prescribed contact angle is ill posed. In the long time limit, the solution relaxes exponentially towards equilibrium.

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.

A complete analytical solution procedure is proposed here to work out the mixed boundary value problems associated with the oblique wave scattering problem involving either a complete elastic porous plate or a permeable membrane in both the cases of finite and infinite depth water in a two-layer fluid. Problems for two different velocity potentials with a phase difference are described in the upper half-planes. They are redefined as the solution potentials for the problems in the quarter-plane. A couple of novel integro-differential relations are constructed to connect the solution potentials of the redefined problems with auxiliary wave potentials. The subsequent potentials are solutions to relatively simpler boundary value problems for the modified Helmholtz equation, with structural boundary conditions of the Neumann type. The generalised orthogonal relation is then used to address the auxiliary wave potential problems analytically. The solution wave potentials are then derived in terms of these auxiliary wave potentials with the aid of the integro-differential relations. Further, explicit analytical expressions are derived for the significant hydrodynamic quantities such as energy reflection and transmission coefficients corresponding to the surface mode (SM) and interface mode (IM), respectively. Moreover, the deflection of the flexible porous structures is derived analytically. The scattering quantities in both SM and IM are presented graphically against the wavenumber and angle of incidence for various values of non-dimensional parameters involved in the structures.