Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 1. Energy spectra

TL;DR

This paper introduces a spectral decomposition method for streamwise turbulence energy in boundary layers, revealing conditions for the appearance of the $k_x^{-1}$ scaling and analyzing turbulence structure contributions across Reynolds numbers.

Contribution

It develops universal spectral filters based on coherence analysis to decompose turbulence energy into wall-attached and detached parts, clarifying the conditions for $k_x^{-1}$ scaling in boundary layers.

Findings

A $k_x^{-1}$ scaling region appears only at high Reynolds numbers ($Re_ au extgreater 80,000).

A broad outer spectral peak exists even at low Reynolds numbers.

Spectral contributions from non-attached turbulence structures can obscure the $k_x^{-1}$ scaling.

Abstract

In wall-bounded turbulence, a multitude of coexisting turbulence structures form the streamwise velocity energy spectrum from the viscosity- to the inertia-dominated range of scales. Definite scaling-trends for streamwise spectra have remained empirically elusive, although a prominent school of thought stems from the works of Perry and Abell (J. Fluid Mech., vol. 79, 1977, pp. 785-799) and Perry et al. (J. Fluid Mech., vol. 165, 1986, pp. 163-199), which were greatly inspired by the attached-eddy hypothesis of Townsend (The Structure of Turbulent Shear Flow, Cambridge University Press, 1976). In this paper, we re-examine the turbulence kinetic energy of the streamwise velocity component in the context of the spectral decompositions of Perry and coworkers. Two universal spectral filters are derived via spectral coherence analysis of two-point velocity signals, spanning a Reynolds number range $Re_\tau \sim \mathcal{O}(10^3)$ to $\mathcal{O}(10^6)$ and form the basis for our decomposition of the logarithmic-region turbulence into stochastically wall-detached and wall-attached portions of energy. The latter is composed of scales larger than a streamwise/wall-normal ratio of $\lambda_x/z \approx 14$. If the decomposition is accepted, a $k_x^{-1}$ scaling region can only appear for $Re_\tau \geq 80\,000$, at a wall-normal position of $z^+ = 100$. Following Perry and co-workers, it is hypothesized that spectral contributions from turbulence structures other than attached eddies obscure a $k_x^{-1}$ scaling. When accepting the idea of different spectral contributions it is furthermore shown that a broad outer-spectral peak is present even at low $Re_\tau$.