This paper reviews results demonstrating the benefits of studying wall-bounded shear flows using dynamics for the evolution of the statistical state of the turbulent system. The statistical state dynamics (SSD) approach used in this work employs a second order closure which isolates the interaction between the streamwise mean and the equivalent of the perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean together with nonlinear interactions between the mean and the perturbation covariance. This dynamical restriction, in which explicit perturbation-perturbation nonlinearity is removed from the perturbation equation, results in a simplified dynamics referred to as the restricted nonlinear (RNL) dynamics. RNL systems in which an ensemble of a finite number of realizations of the perturbation equation share the same mean flow provide tractable approximations to the equivalently infinite ensemble RNL system. The infinite ensemble system, referred to as the S3T, introduces new analysis tools for studying turbulence. The RNL with a single ensemble member can be alternatively viewed as a realization of RNL dynamics. RNL systems provide computationally efficient means to approximate the SSD, producing self-sustaining turbulence exhibiting qualitative features similar to those observed in direct numerical simulations (DNS) despite its greatly simplified dynamics. Finally we show that RNL turbulence can be supported by as few as a single streamwise varying component interacting with the streamwise constant mean flow and that judicious selection of this truncated support, or 'band-limiting', can be used to improve quantitative accuracy of RNL turbulence. The results suggest that the SSD approach provides new analytical and computational tools allowing new insights into wall-turbulence.
Another way of including nonlinear effects was proposed via a statistical state dynamics approach, in which they are restricted to the interaction of the streamwise average flow and perturbations (see the review by REF ).
A statistical state dynamics approach to wall-turbulence
Phil. Trans. R. Soc. A, 375, 20160081 (2017)
Mathematics Physics Medicine