# Resolvent-based estimation of space-time flow statistics

## TL;DR

This paper introduces a resolvent-based method to estimate space-time flow statistics from limited data, capturing the physics of turbulent flows for improved modeling and control.

## Contribution

It develops a novel approach that infers nonlinear forcing statistics to predict unknown flow statistics without assuming low-rank structures, suitable for complex turbulent flows.

## Key findings

- Successfully predicts flow statistics in Ginzburg-Landau model
- Accurately estimates turbulent channel flow data
- Demonstrates applicability to high-rank turbulent flows

## Abstract

We develop a method to estimate space-time flow statistics from a limited set of known data. While previous work has focused on modeling 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 modeling 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 rank-1 assumption, 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 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.