Modeling flow statistics using the linearized Navier-Stokes equations

We develop a model for second order statistics of turbulent channel flow using an associated linear stochastically forced input-output system. The correlation operator of the velocity fields is computed by solving the appropriate Lyapunov equations of a Galerkin approximation of the original system. We use a variety of excitation force correlations and show the dependence of the velocity fields statistics on them. By using certain excitation correlations, we are able to closely match flow statistics computed from Direct Numerical Simulation (DNS) of channel flow. The implications of these result for the proper weight selection in optimal control problems for channel flow are discussed.