Sensitivity computation of statistically stationary quantities in turbulent flows

Abstract: It is well-known that linearized perturbation methods for sensitivity analysis, such as tangent or adjoint equation-based, finite difference and automatic differentiation are not suitable for turbulent flows. The reason is that turbulent flows exhibit chaotic dynamics, leading to the norm of an infinitesimal perturbation to the state growing exponentially in time. As a result, these conventional methods cannot be used to compute the derivatives of long-time averaged quantities to control or design inputs. The ensemble-based approaches and shadowing-based approaches to circumvent the problems of the conventional methods in chaotic systems, also suffer from computational impracticality and lack of consistency guarantees, respectively. We introduce the space-split sensitivity, or the S3 algorithm, which is a Monte-Carlo approach to the chaotic sensitivity computation problem. In this work, we derive the S3 algorithm under simplifying assumptions on the dynamics and present a numerical validation on a low-dimensional example of chaos.