Comparing local energy cascade rates in isotropic turbulence using structure function and filtering formulations

TL;DR

This study compares two definitions of local energy cascade rates in isotropic turbulence using DNS data, revealing inverse cascade phenomena linked to vortex structures and clarifying the interpretation of these measures.

Contribution

It provides a direct comparison of structure function and filtering-based cascade rates, clarifies their physical interpretation, and explores their relationship with flow structures in turbulence.

Findings

Evidence of inverse cascade when large-scale rotation dominates

Inverse cascade is stronger in vortex compression regions

Spatial inverse cascade events are associated with large-scale vortices

Abstract

Two common definitions of the spatially local rate of kinetic energy cascade at some scale $\ell$ in turbulent flows are (i) the cubic velocity difference term appearing in the generalized Kolmogorov-Hill equation (GKHE) (structure function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number of 1250. While observations of negative subfilter-scale energy flux (backscatter) have in the past led to debates regarding interpretation and relevance of such observations, we argue that the interpretation of the local structure function-based cascade rate definition is unambiguous since it arises from a divergence term in scale space. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered viscous dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. We find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ``vortex compression'' $R>0$, $Q>0$ quadrant where $R$ and $Q$ are velocity gradient invariants. Qualitatively similar, but quantitatively much weaker trends are observed for the conditionally averaged subfilter scale energy flux. Flow visualizations show consistent trends, namely that spatially the inverse cascade events appear to be located within large-scale vortices, specifically in subregions when $R$ is large.