Oscillatory shear flows and a new 2D self-sustaining process

Abstract

A new self-sustaining process (SSP) is identified in an oscillating spatially-homogeneous shear flow using a nonlinear Kelvin mode model. This SSP consists of energetic spanwise-invariant 1D `sheets' and weaker spanwise-dependent 2D fluctuations. It is instantaneously 2D, with a time-dependent invariant direction in the flow-cross-shear plane, and has a varicose symmetry. The sheets oscillate in time and are linearly unstable generating fluctuations which then nonlinearly self-interact to reinforce the sheets. The linear instability of the sheet is shown to be due to a lift-up mechanism acting in tandem with `push-forward', which arises due to the `streamwise' shear of the sheets. As the Reynolds number $Re\rightarrow\infty$, both asymptotics and numerics show SSP lower branches with fluctuation velocity $|\boldsymbol{u}'|\sim Re^{-1/2}$. Implications are discussed for oscillatory wall-bounded flows.