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Stochastic Structural Stability Theory for roll/streak formation in forced and free wall bounded shear flow turbulence

Stochastic Structural Stability Theory (SSST) employs an ensemble mean stochastic turbulence model (STM) for perturbation dynamics coupled to the streamwise mean flow dynamics to obtain an autonomous, deterministic, nonlinear dynamical system for evolving an approximate statistical mean turbulent state. In this work SSST is applied to the problem of understanding growth, maintenance, and breakdown of the roll/streak structures that occur in association with forced and free turbulence in neutrally stratified wall bounded shear flow. Roll structures in the cross-stream/spanwise plane and associated streamwise streaks are shown using this analysis to grow through an emergent coherent eddy/mean-flow interaction. In this interaction incoherent Reynolds stresses arising from mechanically forced turbulence 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 exponential growth of the combined roll/streak/turbulence complex. The dominant turbulent eddy structures involved in producing the roll/streak/turbulence complex instability are non-modal perturbations with the form of oblique waves. Over a range of parameters this instability equilibrates nonlinearly to form finite amplitude roll/streak coherent structures. However, for sufficiently high levels of forced turbulence the roll/streak complex becomes structurally unstable and this instability leads to a time dependent state with realistic turbulent statistics that rapidly evolves toward a minimal representation of wall bounded shear turbulence in which the dynamics are limited to interaction of the streamwise mean flow with a single streamwise wave structure. In this naturally emergent minimal turbulence model the streamwise mean flow is supported by Reynolds stress torque organized from the perturbation field while the perturbation field is supported by the universal parametric non-normal growth process inherent to time dependent dynamical systems. This analysis shows how the roll/streak structure recruits supporting oblique waves from the background turbulence and how for sufficiently high levels of forced turbulence the interaction between these waves and the roll/streak can result in transition to and maintenance of a turbulent state.