Stabilisation of a shadowing-based method for sensitivity analysis of turbulent flows using minimal energy optimal control

Shape optimisation and control of turbulent flows obtained from scale-resolving simulations require accurate sensitivities. However, traditional tangent/adjoint methods fail in turbulent systems due to the so-called butterfly effect. Shadowing-based methods overcome this, and the recently developed augmented shadowing method [1] improves efficiency by filtering high-frequency modes—reducing the number of positive Lyapunov exponents (LEs) through artificial forcing in the tangent equation.

Inspired by this, we propose a control-based algorithm to compute sensitivities of time-averaged quantities in chaotic systems. A control term is added to the tangent equation, and an optimisation problem with quadratic cost function is solved analytically. Our approach can be interpreted as a generalised shadowing method, in which the control term serves to selectively stabilise only the unstable modes. In the minimum energy limit, the method mirrors the positive LEs to negative while preserving the negative LE spectrum.

We apply the method to the Kuramoto–Sivashinsky equation, the Lorenz 96 model, and the Kolmogorov flow, and demonstrate accuracy, robustness, and efficiency.



[1] Fang, L., & Papadakis, G. (2025). An augmented shadowing algorithm for calculating the sensitivity of time-average quantities of chaotic systems. Journal of Computational Physics, 114030.