Using DNS Data to Validate the Steady-State 2D/3C Model of Turbulence in Plane Couette Flow

Given the consensus that turbulent flow is characterized by coherent structures and observations of streamwise-elongated structures in numerical simulations and experiments (in the near wall region), we model the mean behavior of fully turbulent plane Couette flow using a streamwise constant projection of the Navier Stokes (NS) equations. This projection results in a two dimensional/three component ($2D/3C$) model comprised of two equations; one in terms of the spanwise/wall normal stream function $\psi(y,z,t)$ with noise forcing, and the other in terms of the stream-wise velocity, $u(y,z,t)$, and $\psi(y,z,t)$. This model is nonlinear but analytically more tractable than the full NS equations and was previously shown to have a single globally stable solution. In the present work we use the steady state $2D/3C$ model to explain features of the turbulent velocity field obtained from DNS data by Kawamura \textit{et al.} with $Re_w=3000$ ($Re_{\tau}=52$).