Vortex events in Euler and Navier–Stokes simulations with smooth initial conditions

We present high-resolution numerical simulations of the Euler and Navier–Stokes equations for a pair of colliding dipoles. We study the possible approach to a finite-time singularity for the Euler equations, and contrast it with the formation of developed turbulence for the Navier–Stokes equations. We present numerical evidence that seems to suggest the existence of a blow-up of the inviscid velocity field at a finite time (${t}_{s} $) with scaling $\vert u\vert _{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1/ 2} $, $\vert \omega \vert _{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1} $. This blow-up is associated with the formation of a ${k}^{\ensuremath{-} 3} $ spectral range, at least for the finite range of wavenumbers that are resolved by our computation. In the evolution toward ${t}_{s} $, the total enstrophy is observed to increase at a slower rate, $\Omega \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 3/ 4} $, than would naively be expected given the behaviour of the maximum vorticity, ${\omega }_{\infty } \ensuremath{\sim} \mathop{ ({t}_{s} \ensuremath{-} t)}\nolimits ^{\ensuremath{-} 1} $. This indicates that the blow-up would be concentrated in narrow regions of the flow field. We show that these regions have sheet-like structure. Viscous simulations, performed at various $\mathit{Re}$, support the conclusion that any non-zero viscosity prevents blow-up in finite time and results in the formation of a dissipative exponential range in a time interval around the estimated inviscid ${t}_{s} $. In this case the total enstrophy saturates, and the energy spectrum becomes less steep, approaching ${k}^{\ensuremath{-} 5/ 3} $. The simulations show that the peak value of the enstrophy scales as ${\mathit{Re}}^{3/ 2} $, which is in accord with Kolmogorov phenomenology. During the short time interval leading to the formation of an inertial range, the total energy dissipation rate shows a clear tendency to become independent of $\mathit{Re}$, supporting the validity of Kolmogorov’s law of finite energy dissipation. At later times the kinetic energy shows a ${t}^{\ensuremath{-} 1. 2} $ decay for all $\mathit{Re}$, in agreement with experimental results for grid turbulence. Visualization of the vortical structures associated with the stages of vorticity amplification and saturation show that, prior to ${t}_{s} $, large-scale and the small-scale vortical structures are well separated. This suggests that, during this stage, the energy transfer mechanism is non-local both in wavenumber and in physical space. On the other hand, as the spectrum becomes shallower and a ${k}^{\ensuremath{-} 5/ 3} $ range appears, the energy-containing eddies and the small-scale vortices tend to be concentrated in the same regions, and structures with a wide range of sizes are observed, suggesting that the formation of an inertial range is accompanied by transfer of energy that is local in both physical and spectral space.