Spectrum of the Curl of Vorticity as a Precursor to Dissipation in 3D Taylor--Green Turbulence

Abstract

Predicting when a three-dimensional turbulent flow reaches its dissipation peak is essential for both theory and adaptive algorithms in simulations and experiments. Using direct numerical simulations (DNSs) of the Taylor--Green vortex (TGV) at resolutions of $256^3$--$1024^3$, we introduce and test a small-scale weighted diagnostic: the spectrum of $|\nabla \times \boldsymbol{\omega}|^2$ (with $\boldsymbol{\omega}=\nabla \times \mathbf{u}$), which, for incompressible flow, is equivalent to a $k^4$-weighted energy spectrum. We show that the peak wavenumber of this spectrum, $k_{{\rm peak}}[\,|\nabla \times \boldsymbol{\omega}|^2\,]$, advances rapidly to intermediate-small scales and then levels off before the dissipation rate $\varepsilon(t)=\sum_k 2\nu k^2 E(k)$ reaches its maximum. Across all resolutions, we observe robust temporal ordering $t_k<t_\varepsilon<t_\Pi$, where $t_k$ marks the onset of the rapid rise of $k_{{\rm peak}}[\,|\nabla \times \boldsymbol{\omega}|^2\,]$, $t_\varepsilon$ is the time of the maximal $\varepsilon(t)$, and $t_\Pi$ is when the cumulative flux $|\Pi(K)|$ attains its largest peak scale. This early-warning signal correlates with the morphological transition to filament-dominated structures visible in $Q$-criterion isosurfaces and is consistent with integral-scale trends ($L_{{\rm int}},\lambda,\eta$). The diagnostic is simple to compute from standard DNS data and highlights the incipient formation of high-curvature structures, where viscosity acts most strongly.