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 $|\nabla \times \boldsymbolω|^2$ (with $\boldsymbolω=\nabla \times \mathbf{u}$), which, for incompressible flow, is equivalent to a $k^4$-weighted energy spectrum. We show that the peak wavenumber of this spectrum, $k_{\rm peak}[\,|\nabla \times \boldsymbolω|^2\,]$, advances rapidly to intermediate-small scales and then levels off before the dissipation rate $\varepsilon(t)=\sum_k 2νk^2 E(k)$ reaches its maximum. Across all resolutions, we observe robust temporal ordering $t_k<t_\varepsilon<t_Π$, where $t_k$ marks the onset of the rapid rise of $k_{\rm peak}[\,|\nabla \times \boldsymbolω|^2\,]$, $t_\varepsilon$ is the time of the maximal $\varepsilon(t)$, and $t_Π$ is when the cumulative flux $|Π(K)|$ 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 ($L_{\rm int},λ,η$). The diagnostic is simple to compute from standard DNS data and highlights the incipient formation of high-curvature structures, where viscosity acts most strongly.

Figures (6)

• Figure 1: $Q$-criterion isosurfaces (colored by $|\mathbf{u}|$) at selected times for the $512^3$ run ($t=0,\,3,\,5.2,\,6.5,\,9,\,11.2,\,15,\,20$). The morphological transition to filament-dominated structures occurs prior to the dissipation peak, consistent with the spectral precursor identified in Fig. \ref{['fig:fig_timeval_kpeak']}.
• Figure 2: Main result. Time evolution of (red) $k_{\mathrm{peak}}[\,|\nabla\times\boldsymbol{\omega}|^2(k)\,]$, (blue) $k_{\mathrm{peak}}(|\Pi(K)|)$, and (green) dissipation $\varepsilon(t)=\sum_k 2\nu k^2 E(k)$ for $N^3=256^3,512^3,1024^3$. Dashed verticals indicate $t_k$ (red), $t_\varepsilon$ (green), and $t_\Pi$ (blue). The consistent ordering $t_k < t_\varepsilon < t_\Pi$ is evident and almost grid independent.
• Figure 3: Integral scales: $L_{\mathrm{int}}(t)=\frac{\pi}{2\sum_k E(k)}\sum_k E(k)/k$, $\lambda(t)=\sqrt{\sum_k E(k)/\sum_k k^2 E(k)}$, $\eta(t)=(\nu^3/\varepsilon)^{1/4}$, and $\varepsilon(t)$. In Fig. \ref{['fig:fig_timeval_kpeak']}, the peak wavenumber $k_{\mathrm{peak}}[\,|\nabla\times\boldsymbol{\omega}|^2\,]$ increases rapidly and levels off by $t\simeq 6.5$, preceding the minimum of $\eta$ (i.e., the dissipation peak).
• Figure 4: Spectral evolution at $1024^3$. (a) $E(k)$, (b) $D(k)=2\nu k^2E(k)$, and (c) $|\nabla\times\boldsymbol{\omega}|^2(k)$ plotted for all saved times $t\in[0,20]$ with color encoding $t$. Only shells with mask = 1 (spherical two-thirds analysis cut) are shown. The high-$k$ front of $|\nabla\times\boldsymbol{\omega}|^2$ advances first and then levels off earlier than that of $D(k)$, thereby anticipating the dissipation peak (cf. Fig. \ref{['fig:fig_timeval_kpeak']}).
• Figure 5: Resolution metrics for $512^3$: time series of $R_\lambda(t)$ and $k_{\max}\eta(t)$ (Nyquist definition). The spectra are calibrated to $K_0=1/8$ and $\varepsilon(t)$ is computed from $D(k)=2\nu k^2E(k)$.