A stochastic framework for uncertainty quantification and data-enhanced modeling of turbulent flows

Abstract

The dynamics of wall-bounded turbulent flows are typically analyzed using the linearized Navier-Stokes equations around base profiles, which either describe a steady-state solution or a long-time averaged mean of a simulation- or experiment-based flow field. Deterministic or stochastic forcing can then be used to compensate for the neglected nonlinear terms and evaluate the input-output features of the linearized dynamics. While uncertainty in the base profile and inputs challenging the effectiveness of linearized models for analysis and flow control, it also presents an opportunity for improving the accuracy of the linearized dynamics in capturing spatio-temporal flow features. We demonstrate how modeling such sources of uncertainty can enable physical discovery and statistical modeling of quantities of interest in two applications involving the analysis and control of complex fluid flows. First, we show how uncertainty quantification with highly structured, random base flow perturbations can uncover transition trends in channel flows. To this end, we build upon an input-output framework for the analysis of linear systems subject to additive and multiplicative sources of uncertainty to provide easily verifiable conditions for the stability and frequency response of the flow dynamics. We then show how stochastic dynamical modeling of additive sources of uncertainty can provide a data-driven refinement of control-oriented models of wind turbine wakes whereby predictions of turbulence intensity variations behind wind the turbines and thrust force and power generation can be improved in accordance with large-eddy simulations of the flow over a cascade of turbines.