Dynamical slowdown, bottlenecks, and multiscaling in Voigt-regularised turbulence

Abstract

We investigate bottleneck formation in turbulence using the Voigt-regularised SABRA model and DNS of the corresponding Voigt-Navier-Stokes (NSV) equations. The Voigt regularisation introduces a scale-dependent slowdown of nonlinear interactions without enhancing dissipation, providing a natural setting to study the interplay between nonlinear transfer and thermalised behaviour. We find three distinct spectral regimes: an inertial range at $k<k_I$, an intermediate equilibrium-like range associated with partial thermalisation for $k_I<k<k_{II}$, and a high-$k$ thermal regime for $k>k_{II}$, where the Voigt contribution dominates the conserved invariant. The crossover to the high-$k$ regime occurs at $k_{II}\sim 1/α$, while $k_I$ marks the onset of thermalised behaviour. Equal and multi-time statistics reveal a progressive suppression of intermittency and a tendency towards Gaussianity at small scales, together with a transition from dynamic multiscaling in the turbulent regime to simple scaling in the equilibrium ranges. The shell model resolves these three regimes over a broad range of scales, while DNS of the corresponding NSV equations reproduces the same qualitative trends, including bottleneck formation, delayed cascade completion, reduced intermittency, and a tendency towards Gaussianity at small scales. We find that bottleneck formation might be associated with scale-dependent dynamical slowdown and incipient thermalisation, rather than being purely dissipative in origin. We provide strong evidence that, in the regime where the regularization parameter $α$ is much smaller than the dissipation length scale, the Voigt model reproduces the same inertial-range turbulent regime and turbulence statistics as the Navier-Stokes (NS) equations. This provides evidence that the Voigt model constitutes an excellent practical approximation to the NS equations for small $α$.