Normality-based analysis of multiscale velocity gradients and energy transfer in direct and large-eddy simulations of isotropic turbulence

TL;DR

The paper advances a normality-based decomposition of the velocity gradient tensor to distinguish normal straining, pure shear, and rigid rotation at multiple scales in isotropic turbulence. By applying a Gaussian-filter framework, it derives a multiscale expansion of the interscale energy transfer, decomposed into scale-local and scale-nonlocal contributions, and identifies shear layers as the main drivers of forward energy transfer and bottleneck backscatter in subinertial ranges. DNS data reveal that backscatter originates predominantly from strain–vorticity covariance associated with shear layers, while forward transfer is governed by normal straining and shear-vorticity interactions; LES results show that a mixed closure reproduces DNS-filtered behavior, whereas an eddy-viscosity closure mimics unfiltered DNS at a lower Reynolds number, explaining artificial bottlenecks. Altogether, the framework clarifies the structural mechanisms of the energy cascade and provides guidance for designing and evaluating LES closures that faithfully capture multiscale flow features, particularly the role of small-scale shear layers and Burgers-type structures.

Abstract

Symmetry-based analyses of multiscale velocity gradients highlight that strain self-amplification (SS) and vortex stretching (VS) drive forward energy transfer in turbulent flows. By contrast, a strain-vorticity covariance mechanism produces backscatter that contributes to the bottleneck effect in the subinertial range of the energy cascade. We extend these analyses by using a normality-based decomposition of filtered velocity gradients in forced isotropic turbulence to distinguish contributions from normal straining, pure shearing and rigid rotation at a given scale. Our analysis of direct numerical simulation (DNS) data illuminates the importance of shear layers in the inertial range and (especially) the subinertial range of the cascade. Shear layers contribute significantly to SS and VS and play a dominant role in the backscatter mechanism responsible for the bottleneck effect. Our concurrent analysis of large-eddy simulation (LES) data characterizes how different closure models affect the flow structure and energy transfer throughout the resolved scales. We thoroughly demonstrate that the multiscale flow features produced by a mixed model closely resemble those in a filtered DNS, whereas the features produced by an eddy viscosity model resemble those in an unfiltered DNS at a lower Reynolds number. This analysis helps explain how small-scale shear layers, whose imprint is mitigated upon filtering, amplify the artificial bottleneck effect produced by the eddy viscosity model in the inertial range of the cascade. Altogether, the present results provide a refined interpretation of the flow structures and mechanisms underlying the energy cascade and insight for designing and evaluating LES closure models.

Figures (10)

• Figure 1: (a) Symmetry-based scale-local and scale-non-local contributions to interscale energy transfer in forced isotropic turbulence. The symbols represent direct numerical simulation (DNS) datasets at Taylor-scale Reynolds numbers of $Re_\lambda \approx 315$ (DNS315) and $Re_\lambda \approx 400$ (DNS400) and the curves represent the $Re_\lambda \approx 400$ results of Joh2020Joh2021. The shaded region captures the bottom of the inertial range for DNS400. (b) Energy spectra for the unfiltered and filtered velocity fields in DNS400 as well as LES cases that employ eddy viscosity (Vis400) and mixed (Mix400) models at $Re_\lambda \approx 400$. The filtered DNS and LES cases employ a filter width of $2\ell/\eta = 48$. The dotted line represents the inertial range scaling, $E(k) = 1.6 \left\langle \mathit{\varPhi} \right\rangle^{2/3} k^{-5/3}$, and the inset depicts a linear--log plot of the compensated energy spectra. Technical details of the simulations are described in § \ref{['sec:sims']}.
• Figure 2: Partitioning of filtered velocity gradients for the DNS cases, where shearing is represented using $S_{\ell,\gamma}^2 = \mathit{\varOmega}_{\ell,\gamma}^2 = \tfrac{1}{2}A_{\ell,\gamma}^2$. The horizontal dashed lines represent the unfiltered partitioning in the high-$Re_\lambda$ limit, the vertical dotted line represents the typical thickness of small-scale shear layers, $\delta_\gamma = 9\eta$, and the shaded region approximates the inertial range for DNS400 as $50 \leq 2\ell/\eta \leq 150$.
• Figure 3: (a) Partitioning of filtered velocity gradients for the LES cases. The solid curves represent the filtered DNS400 partitioning and the horizontal dashed lines represent the unfiltered partitioning in the high-$Re_\lambda$ limit. The lower limit of the filter width axis represents the LES filter width, $2\ell_{LES}/\eta = 48$, and the shaded region approximates the inertial range. (b) Partitioning for Vis400 replotted as a function of $2\ell_*/\eta_*$, where $\ell_* = \sqrt{\ell^2 - \ell_{LES}^2}$ is the complementary filter width and $\eta_* \approx 15 \eta$ is an effective Kolmogorov scale. The solid curves represent the partitioning produced by a DNS at $Re_\lambda \approx 61$, which has a Kolmogorov scale of approximately $\eta_*$. The vertical dotted line represents the typical thickness of small-scale shear layers, $\delta_\gamma = 9\eta$.
• Figure 4: Vortical flow structures associated with rigid rotation and shear vorticity for an unfiltered ($\ell = 0$) DNS at $Re_\lambda \approx 61$ (a,b), DNS400 filtered at $2\ell/\eta = 48$ (c,d) and Vis400 (e,f) and Mix400 (g,h) at the LES filter width, $2\ell_{LES}/\eta = 48$. The greyscale visualizations depict the strengths of rigid rotation, $\omega_{\ell,\varphi}^2$, and shear vorticity, $\omega_{\ell,\gamma}^2$, normalized by the spatially averaged vorticity strength, $\langle \omega_\ell^2 \rangle$, and the isosurfaces represent $\omega_{\ell,\varphi}^2 / \langle \omega_\ell^2 \rangle = 2$ (red) and $\omega_{\ell,\gamma}^2 / \langle \omega_\ell^2 \rangle = 2$ (blue).
• Figure 5: Normality-based contributions to interscale energy transfer for the DNS cases. The contributions represent multiscale SS (a), VS (b), strain--vorticity covariance (c) and aggregates across these three mechanisms (d). The vertical dotted lines represent the typical thickness of small-scale shear layers, $\delta_\gamma = 9\eta$, and the shaded regions capture the bottom of the inertial range for DNS400.