Computing exact coherent states in channels starting from the laminar profile: A resolvent-based approach

We present an iterative method to compute traveling wave exact coherent states (ECS) in Couette and Poiseuille flows starting from an initial laminar profile. The approach utilizes the resolvent operator for a two-dimensional, three-component streamwise-averaged mean and exploits the underlying physics of the self-sustaining process. A singular value decomposition of the resolvent operator is used to obtain the mode shape for a single streamwise-varying Fourier mode. The self-interaction of the single mode is computed and used to generate an updated mean velocity input to the resolvent operator. The process is repeated until a nearly neutrally stable mean flow that self-sustains is obtained, as defined by suitable convergence criteria; the results are further verified with direct numerical simulation. The approach requires the specification of only two unknown parameters: the wave speed and amplitude of the mode. It is demonstrated that within as few as three iterations, the initial one-dimensional laminar field can be transformed into three-dimensional ECS.