We investigate the properties of almost limiting short-crested gravity waves with harmonic resonance for various incident angles. When the incident angle is less than 47.5°, the enclosed crest angle in non-resonant limiting waves is 90°, which corresponds to that in standing waves. In contrast, when the incident angle exceeds 47.5°, the enclosed crest angle in non-resonant limiting waves is 120°, which corresponds to that in two-dimensional progressive waves. The enclosed crest angle is 90° in resonant limiting waves for all incident angles. The crest becomes flatter than the trough in resonant limiting waves if the fundamental mode has a different sign from its harmonic resonant mode. Bifurcation of short-crested waves is also investigated.

Experimental studies have demonstrated that spermatozoa synchronize their flagella when swimming in close proximity. In a Newtonian fluid, it was shown theoretically that such synchronization arises passively due to hydrodynamic forces between the two swimmers if their waveforms exhibit a front–back geometrical asymmetry. Motivated by the fact that most biological fluids possess a polymeric microstructure, here we address synchronization in a viscoelastic fluid analytically. Using a two-dimensional infinite sheet model, we show that the presence of polymeric stresses removes the geometrical asymmetry constraint and therefore even symmetric swimmers synchronize. Such synchronization occurs on asymptotically faster time scales than in a Newtonian fluid, and the swimmers are seen to be driven into a stable in-phase conformation minimizing the energy dissipated in the surrounding fluid.

Turbulent solutions of the two-dimensional Navier–Stokes equations are a paradigm for the chaotic space–time patterns and equilibrium distributions of turbulent geophysical and astrophysical ‘thin’ flows on large horizontal scales. Here we investigate how homogeneous, stationary two-dimensional turbulence varies with the Reynolds number (Re) in stationary solutions with large-scale, random forcing and viscous diffusion, also including hypoviscous diffusion to limit the inverse energy cascade. This survey is made over the computationally feasible range in Re ≫ 1, approximately between 1.5 × 103 and 5.6 × 106. For increasing Re, we witness the emergence of vorticity fine structure within the filaments and vortex cores. The energy spectrum shape approaches the forward-enstrophy inertial-range form k−3 at large Re, and the velocity structure function is independent of Re. All other statistical measures investigated in this study exhibit power-law scaling with Re, including energy, enstrophy, dissipation rates and the vorticity structure function. The scaling exponents depend on the forcing properties through their influences on large-scale coherent structures, whose particular distributions are non-universal. A striking result is the Re independence of the intermittency measures of the flow, in contrast with the known behaviour for three-dimensional homogeneous turbulence of asymptotically increasing intermittency. This is a consequence of the control of the tails of the distribution functions by large-scale coherent vortices. Our analysis allows extrapolation towards the asymptotic limit of Re → ∞, fundamental to geophysical and astrophysical regimes and their large-scale simulation models where turbulent transport and dissipation must be parameterized.

The statistics of velocity differences between pairs of heavy inertial point particles suspended in an incompressible turbulent flow is studied and found to be extremely intermittent. The problem is particularly relevant to the estimation of the efficiency of collisions among heavy particles in turbulence. We found that when particles are separated by distances within the dissipative subrange, the competition between regions with quiet regular velocity distributions and regions where very close particles have very different velocities (caustics) leads to a quasi bi-fractal behaviour of the particle velocity structure functions. Contrastingly, we show that for particles separated by inertial-range distances, the velocity-difference statistics can be characterized in terms of a local roughness exponent, which is a function of the scale-dependent particle Stokes number only. Results are obtained from high-resolution direct numerical simulations up to 20483 collocation points and with millions of particles for each Stokes number.