Long time fate of two-dimensional incompressible high Reynolds number Navier-Stokes turbulence: A quantitative comparison between theory and simulation

TL;DR

This paper compares theoretical models and high-resolution simulations to understand the long-term behavior of two-dimensional high Reynolds number turbulence, confirming the relevance of patch vortex models and the sinh-Poisson relation.

Contribution

It provides the first systematic comparison between DNS results and statistical mechanics models of inviscid patch vortices for 2D turbulence.

Findings

Late time states match patch vortex model predictions.

Reduced circulation leads to sinh-Poisson vorticity-stream function relationship.

Confirms long-standing theoretical predictions with numerical evidence.

Abstract

Predicting the long time or late time states of two-dimensional incompressible, high Reynolds number, slowly decaying turbulence has been one of the long-standing problems. Using ``point vortices'' as ``inviscid'' building blocks, which do not respect incompressibility, statistical mechanical models conserving only total energy and zero total circulation result in the well-known sinh-Poisson relation between vorticity and stream function. On the other hand, statistical mechanics of ``inviscid patch'' vortices, which respects incompressibility by conserving regions of zero and nonzero vorticity, predicts a generalized relaxed state, which has never been systematically compared with direct numerical simulations (DNS). In this study, starting from highly packed regions of nonzero initial vorticity, we demonstrate using high resolution, high Reynolds number DNS that the late time states agree with predictions from patch vortex models. As total circulation is reduced or diluted, we show that late time states of our DNS systematically and unambiguously lead to the sinh-Poisson relationship between vorticity and stream function. We believe that our quantitative findings solve one of the long-standing problems in two-dimensional turbulence.