Resolvent-based estimation of space-time flow statistics

We develop a method to estimate space-time flow statistics from a limited set of known data. While previous work has focused on modelling spatial or temporal statistics independently, space-time statistics carry fundamental information about the physics and coherent motions of the flow and provide a starting point for low-order modelling and flow control efforts. The method is derived using a statistical interpretation of resolvent analysis. The central idea of our approach is to use known data to infer the statistics of the nonlinear terms that constitute a forcing on the linearized Navier-Stokes equations, which in turn imply values for the remaining unknown flow statistics through application of the resolvent operator. Rather than making an a priori assumption that the flow is dominated by the leading singular mode of the resolvent operator, as in some previous approaches, our method allows the known input data to select the most relevant portions of the resolvent operator for describing the data, making it well suited for high-rank turbulent flows. We demonstrate the predictive capabilities of the method, which we call resolvent-based estimation, using two examples: the Ginzburg-Landau equation, which serves as a convenient model for a convectively unstable flow, and a turbulent channel flow at low Reynolds number.