Stochastic Structural Stability Theory applied to roll/streak formation in boundary layer shear flow

TL;DR

This paper applies Stochastic Structural Stability Theory (SSST) to understand how roll and streak structures form in boundary layer shear flows due to free-stream turbulence, revealing a linear instability mechanism that leads to their formation and stabilization.

Contribution

The work demonstrates that SSST can model the formation of roll/streak structures as a linear instability driven by turbulence-mean flow interactions, providing new insights into bypass transition.

Findings

Roll/streak structures arise as a linear instability in the turbulence-mean flow interaction.

Optimal oblique wave perturbations support the instability of the roll/streak/turbulence complex.

The instability bifurcates at a critical turbulence excitation parameter, leading to finite-amplitude stable streaks.

Abstract

Stochastic Structural Stability Theory (SSST) provides an autonomous, deterministic, nonlinear dynamical system for evolving the statistical mean state of a turbulent system. In this work SSST is applied to the problem of understanding the formation of the roll/streak structures that arise from free-stream turbulence (FST) and are associated with bypass transition in boundary layers. Roll structures in the cross-stream/spanwise plane and associated streamwise streaks are shown to arise as a linear instability of interaction between the FST and the mean flow. In this interaction incoherent Reynolds stresses arising from FST are organized by perturbation streamwise streaks to coherently force perturbation rolls giving rise to an amplification of the streamwise streak perturbation and through this feedback to an instability of the combined roll/streak/turbulence complex. The dominant turbulent perturbation structures involved in supporting the roll/streak/turbulence complex instability are non-normal optimal perturbations with the form of oblique waves. The cooperative linear instability giving rise to the roll/streak structure arises at a bifurcation in the parameter of STM excitation parameter. This structural instability eventually equilibrates nonlinearly at finite amplitude and although the resulting statistical equilibrium streamwise streaks are inflectional the associated flows are stable. Formation and equilibration of the roll/streak structure by this mechanism can be traced to the non-normality which underlies interaction between perturbations and mean flows in modally stable systems.