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In this work we examine the turbulence maintained in a Restricted Nonlinear(RNL) model of plane Couette flow. This model is a computationally efficientapproximation of the second order statistical state dynamics (SSD) obtained bypartitioning the flow into a streamwise averaged mean flow and perturbationsabout that mean, a closure referred to herein as the RNL$_\infty$ model. TheRNL model investigated here employs a single member of the infinite ensemblethat comprises the covariance of the RNL$_\infty$ dynamics. The RNL system haspreviously been shown to support self-sustaining turbulence with a mean flowand structural features that are consistent with DNS. This paper demonstratesthat the RNL system's self-sustaining turbulent state is supported by a smallnumber of streamwise varying modes, which form the natural support for theself-sustaining process maintaining RNL turbulence. Remarkably, truncation ofthe RNL system's support to a single streamwise varying mode can suffice tosustain the turbulent state. The close correspondence between RNL simulationsand DNS that has been previously observed along with the results presented heresuggest that the fundamental mechanisms underlying wall-turbulence can beanalyzed using these highly simplified RNL systems.

A minimal model of self-sustaining turbulence