Transient Growth in Shear Flows: Linearity vs Nonlinearity

TL;DR

This paper investigates the nonlinear transient growth in shear flows, revealing a new optimal disturbance that surpasses linear predictions in triggering turbulence, thus bridging the gap between linear and nonlinear transition theories.

Contribution

It introduces a fully nonlinear transient growth analysis that identifies a more effective disturbance for turbulence initiation, challenging traditional linear approaches.

Findings

Discovered a nonlinear optimal disturbance more efficient than linear ones.

Showed the nonlinear disturbance can significantly differ from linear predictions.

Bridged the gap between linear transient growth and nonlinear boundary calculations.

Abstract

Two approaches to the problem of transition to turbulence of shear flows are popular in the literature. The first is the linear one of transient growth which focuses on the likely form of the most 'dangerous' (lowest energy) turbulence-triggering disturbances. The second is the nonlinear calculation of the laminar-turbulent boundary which instead focuses on their typical amplitudes. We look to bridge the gap between these two perspectives by considering the fully nonlinear transient growth problem to estimate both the form and amplitude of the most dangerous disturbance. We thereby discover a new nonlinear optimal disturbance which outgrows the well-known linear optimal for the same initial energy and is crucially much more efficient in triggering turbulence. The conclusion is then that the most dangerous disturbance can differ markedly from what traditional linear transient growth analysis predicts.