A curvature-weighted spectral precursor to dissipation in decaying three-dimensional turbulence: robustness across initial conditions and viscosity effects

Satori Tsuzuki

TL;DR

This study generalizes a curvature-weighted spectral precursor based on the curl of vorticity, $| abla imesoldsymbol{oldsymbol{oldsymbol{ abla}}}|^2(k)=k^4E(k)$, to a diverse set of freely decaying 3D turbulent flows beyond the Taylor–Green vortex. Using pseudo-spectral DNS across multiple initial conditions, the authors show a robust temporal ordering where the curvature-weighted peak scale time $t_k$ precedes the dissipation peak time $t_oldsymbol{ ightarrow}$, which in turn precedes the peak scale of the nonlinear flux $t_oldsymbol{ ightarrow}$, i.e., $t_k<t_oldsymbol{ ightarrow}<t_oldsymbol{ ightarrow}$. They address potential cutoff artifacts with explicit spike inspection and higher-resolution references, demonstrating that the precursor persists under moderate-to-low viscosity but can weaken under strong viscous damping. The results support the curvature-weighted diagnostic as a practical, general early-warning marker for imminent dissipation and a tool for guiding online monitoring or adaptive resolution in simulations, while outlining limitations and avenues for extending the approach to forced, anisotropic, or magnetized turbulence. The work therefore provides a principled, robust predictor of small-scale development during turbulent transients with potential for real-time application in computational fluid dynamics.

Abstract

We investigate the robustness of a curvature-weighted spectral precursor to dissipation in freely decaying three-dimensional incompressible turbulence. Building on our recent work in Physical Review Fluids on the Taylor--Green vortex, we analyze direct numerical simulations using the curl-of-vorticity spectrum $|\nabla\times \boldsymbolω|^2(k)$, equivalent to a $k^4$-weighted energy spectrum for solenoidal flow. Extending the study across multiple initial conditions -- multi-mode ABC flows, a randomized low-wavenumber ABC field, the Taylor--Green vortex, and the Kida--Pelz flow -- we find a consistent temporal ordering: the characteristic time associated with the advance and saturation of the peak wavenumber of $|\nabla\times \boldsymbolω|^2(k)$ precedes the dissipation-peak time, which in turn precedes the characteristic time associated with the peak scale of the nonlinear energy-flux spectrum. We further probe viscosity effects in Taylor--Green turbulence: the precursor persists at lower viscosity when adequate resolution is employed, but weakens and can break at higher viscosity, consistent with stronger viscous damping of curvature-dominated small-scale content. Throughout, we use explicit inspection of curvature-weighted spectra to distinguish physical peak evolution from cutoff-proximate artifacts. These results establish robustness across initial conditions and clarify the practical role of viscosity and resolution for deploying curvature-weighted spectral precursors in decaying turbulence.

A curvature-weighted spectral precursor to dissipation in decaying three-dimensional turbulence: robustness across initial conditions and viscosity effects

TL;DR

This study generalizes a curvature-weighted spectral precursor based on the curl of vorticity,

, to a diverse set of freely decaying 3D turbulent flows beyond the Taylor–Green vortex. Using pseudo-spectral DNS across multiple initial conditions, the authors show a robust temporal ordering where the curvature-weighted peak scale time

precedes the dissipation peak time

, which in turn precedes the peak scale of the nonlinear flux

, i.e.,

. They address potential cutoff artifacts with explicit spike inspection and higher-resolution references, demonstrating that the precursor persists under moderate-to-low viscosity but can weaken under strong viscous damping. The results support the curvature-weighted diagnostic as a practical, general early-warning marker for imminent dissipation and a tool for guiding online monitoring or adaptive resolution in simulations, while outlining limitations and avenues for extending the approach to forced, anisotropic, or magnetized turbulence. The work therefore provides a principled, robust predictor of small-scale development during turbulent transients with potential for real-time application in computational fluid dynamics.

Abstract

We investigate the robustness of a curvature-weighted spectral precursor to dissipation in freely decaying three-dimensional incompressible turbulence. Building on our recent work in Physical Review Fluids on the Taylor--Green vortex, we analyze direct numerical simulations using the curl-of-vorticity spectrum

, equivalent to a

-weighted energy spectrum for solenoidal flow. Extending the study across multiple initial conditions -- multi-mode ABC flows, a randomized low-wavenumber ABC field, the Taylor--Green vortex, and the Kida--Pelz flow -- we find a consistent temporal ordering: the characteristic time associated with the advance and saturation of the peak wavenumber of

precedes the dissipation-peak time, which in turn precedes the characteristic time associated with the peak scale of the nonlinear energy-flux spectrum. We further probe viscosity effects in Taylor--Green turbulence: the precursor persists at lower viscosity when adequate resolution is employed, but weakens and can break at higher viscosity, consistent with stronger viscous damping of curvature-dominated small-scale content. Throughout, we use explicit inspection of curvature-weighted spectra to distinguish physical peak evolution from cutoff-proximate artifacts. These results establish robustness across initial conditions and clarify the practical role of viscosity and resolution for deploying curvature-weighted spectral precursors in decaying turbulence.

Paper Structure (30 sections, 54 equations, 18 figures, 2 tables)

This paper contains 30 sections, 54 equations, 18 figures, 2 tables.

Table of Contents

Introduction
Governing equations and numerical method
Governing equations
Pseudo-spectral direct numerical simulations
Isotropic spectra, transfer, and flux
Initial conditions and simulation set
Initial conditions
Simulation parameters and data output
Spectral diagnostics and characteristic times
Isotropic spectra and calibration of the energy scale
Spectral quantities: $E(k)$, $D(k)$, $\Pi(K)$, and $|\nabla\times\boldsymbol{\omega}|^2(k)$
Integral and small-scale measures: $L_{\mathrm{int}}$, $\lambda$, $\eta$, $R_\lambda$, and $k_{\max}\eta$
$k$-metrics for spectral "peak scales": peak, centroid, and quantiles
Characteristic times: $t_\varepsilon$, $t_k$, and $t_\Pi$
Inspection for high-$k$ locking and cutoff effects
...and 15 more sections

Figures (18)

Figure 1: $Q$-criterion isosurfaces (colored by $|\mathbf{u}|$) at selected times for the $512^3$ Multi-ABC run. Panels (a)--(h) show $t=0,\,8.55,\,17.1,\,25.65,\,34.2,\,42.75,\,51.3,\,59.85$. Panels (i)--(m) show additional snapshots at characteristic times extracted from spectral diagnostics: $t_k=22.6$, $t_c=27.0$, $t_{95}=26.85$, $t_\varepsilon=26.6$, and $t_\Pi=28.95$ (see Sec. \ref{['secIV:diagnostics']} for definitions). The isosurface level and rendering settings are the same as in Ref. TsuzukiTGVPrecursorPRF.
Figure 2: $Q$-criterion isosurfaces (colored by $|\mathbf{u}|$) at selected times for the $512^3$ Multi-ABC (Asymmetric) run. Panels (a)--(h) show $t=0,\,8.55,\,17.1,\,25.65,\,34.2,\,42.75,\,51.3,\,59.85$. Panels (i)--(m) annotate snapshots at characteristic times: $t_k=27.0$, $t_\varepsilon=30.0$, and $t_\Pi=53.75$ (Sec. \ref{['secIV:diagnostics']}). Here $t_c$ and $t_{95}$ coincide ($t_c=t_{95}=25.9$), and the corresponding snapshot is shown in panel (j). The isosurface level and rendering settings are the same as in Ref. TsuzukiTGVPrecursorPRF.
Figure 3: $Q$-criterion isosurfaces (colored by $|\mathbf{u}|$) at selected times for the $512^3$ ABC+random-phase low-$k$ run. Panels (a)--(h) show $t=0,\,17.1,\,34.2,\,51.3,\,68.4,\,85.5,\,102.6,\,119.7$. Panels (i)--(m) show snapshots at characteristic times: $t_k=48.95$, $t_c=50.15$, $t_{95}=49.6$, $t_\varepsilon=51.3$, and $t_\Pi=68.45$ (see Sec. \ref{['secIV:diagnostics']} for definitions). The isosurface level and rendering settings are the same as in Ref. TsuzukiTGVPrecursorPRF.
Figure 4: $Q$-criterion isosurfaces (colored by $|\mathbf{u}|$) at selected times for the $512^3$ Taylor--Green vortex (TGV) run. Panels (a)--(h) show $t=0,\,2.85,\,5.7,\,8.55,\,11.4,\,14.25,\,17.1,\,19.95$. Panels (i)--(m) show snapshots at characteristic times: $t_k=6.55$, $t_c=9.15$, $t_{95}=9.05$, $t_\varepsilon=9.0$, and $t_\Pi=11.2$ (see Sec. \ref{['secIV:diagnostics']} for definitions). The isosurface level and rendering settings are the same as in Ref. TsuzukiTGVPrecursorPRF.
Figure 5: $Q$-criterion isosurfaces (colored by $|\mathbf{u}|$) at selected times for the $512^3$ Kida--Pelz run. Panels (a)--(h) show $t=0,\,2.85,\,5.7,\,8.55,\,11.4,\,14.25,\,17.1,\,19.95$. Panels (i)--(l) show additional snapshots around the dissipation episode, including the dissipation- and flux-related times $t_\varepsilon=3.4$ and $t_\Pi=7.15$ (Sec. \ref{['secIV:diagnostics']}). The $512^3$ Kida--Pelz run is shown here to highlight the rapid emergence of fine-scale structures and the resolution sensitivity of peak picking in curvature-weighted spectra; the characteristic times reported for Kida--Pelz in Table \ref{['tab:times_summary']} are extracted from the $1024^3$ reference simulation (Sec. \ref{['subsec:spike-inspection']}). The isosurface level and rendering settings are the same as in Ref. TsuzukiTGVPrecursorPRF. The snapshot labels (e.g., $t_k$, $t_c$, $t_{95}$) correspond to the $512^3$ visualization; $t_k$ differs from the $1024^3$ reference value due to peak-locking issues, whereas $t_{\varepsilon}$, $t_{c}$, $t_{95}$ were confirmed unchanged at $1024^{3}$ (Table \ref{['tab:aux_reaching_times']}).
...and 13 more figures