How linear is wall-bounded turbulence?

Javier Jiménez

doi:10.1063/1.4819081

Abstract

The relevance of Orr's inviscid mechanism to the transient amplification of disturbances in shear flows is explored in the context of bursting in the logarithmic layer of wall-bounded turbulence. The linearized problem for the wall normal velocity is first solved in the limit of small viscosity for a uniform shear and for a channel with turbulent-like profile, and compared with the quasiperiodic bursting of fully turbulent simulations in boxes designed to be minimal for the logarithmic layer. Many properties, such as time and length scales, energy fluxes between components, and inclination angles, agree well between the two systems. However, once advection by the mean flow is subtracted, the directly computed linear component of the turbulent acceleration is found to be a small part of the total. The temporal correlations of the different quantities in turbulent bursts imply that the classical model, in which the wall-normal velocities are generated by the breakdown of the streamwise-velocity streaks, is a better explanation than the purely autonomous growth of linearized bursts. It is argued that the best way to reconcile both lines of evidence is that the disturbances produced by the streak breakdown are amplified by an Orr-like transient process drawing energy directly from the mean shear, rather than from the velocity gradients of the nonlinear streak. This, for example, obviates the problem of why the cross-stream velocities do not decay once the streak has broken down.

Highlights

There is widespread agreement that turbulence requires the nonlinearity of the Navier–Stokes equations
The linearized problem for the wall normal velocity is first solved in the limit of small viscosity for a uniform shear and for a channel with turbulent-like profile, and compared with the quasiperiodic bursting of fully turbulent simulations in boxes designed to be minimal for the logarithmic layer
The best-known examples are the inflection-point linearized instabilities of the mean velocity profile,[3] which are known to represent well many of the properties of the largescale structures in fully nonlinear free-shear flows, especially forced ones.[4]

Summary

INTRODUCTION

There is widespread agreement that turbulence requires the nonlinearity of the Navier–Stokes equations. Usually known as lift-up, the cross-shear velocities deform the mean profile to create perturbations of the streamwise velocity component, which do not disappear as the cross-stream perturbations decay (see Figure 2(b)) Because it results in a net creation of turbulent energy, the lift-up was soon implicated in the maintenance of wall-bounded turbulence,[17] and in the formation of the streamwise-velocity streaks.[18]. Since the perturbation energy associated with the Orr mechanism is transient, even in the inviscid limit, we will be led to consider intermittent flow events that we will denote as “bursts.” That term was originally introduced in turbulence to describe the fluid eruptions observed near the wall in the early visualizations of turbulent boundary layers,[18,20] which were hypothesized to be reflections of the occasional break-up of the near-wall streaks mentioned above.

The equations of motion