Self-similar invariant solution in the near-wall region of a turbulent boundary layer at asymptotically high Reynolds numbers

TL;DR

This paper demonstrates the existence of a self-similar invariant solution in the near-wall region of turbulent boundary layers at very high Reynolds numbers, advancing the understanding of turbulence dynamics.

Contribution

It introduces a new invariant solution of the Navier-Stokes equations that captures the self-similar scaling of near-wall turbulence at high Reynolds numbers.

Findings

Solution persists up to Re=1,000,000

Evidence suggests the solution is asymptotically self-similar

Captures universal near-wall turbulent structures

Abstract

At sufficiently high Reynolds numbers, shear-flow turbulence close to a wall acquires universal properties. When length and velocity are rescaled by appropriate characteristic scales of the turbulent flow and thereby measured in \emph{inner units}, the statistical properties of the flow become independent of the Reynolds number. We demonstrate the existence of a wall-attached non-chaotic exact invariant solution of the fully nonlinear 3D Navier-Stokes equations for a parallel boundary layer that captures the characteristic self-similar scaling of near-wall turbulent structures. The branch of travelling wave solutions can be followed up to $Re=1,000,000$. Combined theoretical and numerical evidence suggests that the solution is asymptotically self-similar and exactly scales in inner units for Reynolds numbers tending to infinity. Demonstrating the existence of invariant solutions that capture the self-similar scaling properties of turbulence in the near-wall region is a step towards extending the dynamical systems approach to turbulence from the transitional regime to fully developed boundary layers.