Cause-and-effect of linear mechanisms sustaining wall turbulence

Despite the nonlinear nature of turbulence, there is evidence that part of the energy-transfer mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities, neutral modes, transient growth from non-normal operators and parametric instabilities from temporal mean flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in simplified physical models. Whether the flow follows any or a combination of them remains elusive. Here, we evaluate the linear mechanisms responsible for the energy transfer from the streamwise-averaged mean flow to the fluctuating velocities . To that end, we use cause-and-effect analysis based on interventions: manipulation of the causing variable leads to changes in the effect. This is achieved by direct numerical simulation of turbulent channel flows at low Reynolds number, in which the energy transfer from to is constrained to preclude a targeted linear mechanism. We show that transient growth is sufficient for sustaining realistic wall turbulence. Self-sustaining turbulence persists when exponential instabilities, neutral modes and parametric instabilities of the mean flow are suppressed. We further show that a key component of transient growth is the Orr/push-over mechanism induced by spanwise variations of the base flow. Finally, we demonstrate that an ensemble of simulations with various frozen-in-time arranged so that only transient growth is active, can faithfully represent the energy transfer from to as in realistic turbulence. Our approach provides direct cause-and-effect evaluation of the linear energy-injection mechanisms from to in the fully nonlinear system and simplifies the conceptual model of self-sustaining wall turbulence.This work was supported by the Coturb project of the European Research Council (ERC-2014.AdG-669505) during the 2019 Coturb Turbulence Summer Workshop at the Universidad Politécnica de Madrid. M.-A.N. acknowledges the support of the Hellenic Foundation for Research and Innovation, and the General Secretariat for Research and Technology (grant number 1718/14518). A.L.-D. acknowledges the support of the NASA Transformative Aeronautics Concepts Program (grant number NNX15AU93A) and the Office of Naval Research (grant number N000141712310) M.K. was supported by the Air Force Office of Scientific Research under grant FA9550-16-1-0319 and the Office of Naval Research under grant N00014-17-1-2341