A regularised shadowing method for sensitivity analysis of turbulence flows*

There is increasing demand for optimisation using scale-resolving methods, such as direct and large-eddy simulations. However, standard methods to evaluate the gradient of a time-averaged objective function to a control parameter, for example the adjoint method, fail for chaotic turbulent flows due to the so-called “butterfly effect”. A promising method for calculating the gradient (or sensitivity) is the Least-Squares Shadowing (LSS) method. However, for systems that are not uniformly hyperbolic, the resulting system matrix has a very large condition number, which affects the accuracy of the results. To address this challenge, we introduce a new regularised non-intrusive LSS method. By applying Tikhonov regularisation, we can make the system well-conditioned. The optimal regularisation parameter is obtained by finding the maximum curvature of the L-curve. This innovation improves the accuracy of the method and significantly reduces the standard deviation of the computed gradient. It therefore extends the applicability of LSS to a broader range of turbulent flow systems.

We apply the proposed method to the Lorenz 96 system and the Kolmogorov flow. The results demonstrate its effectiveness and stability. By calculating covariant Lyapunov vectors (CLVs), we analyse the hyperbolicity of the system. The correlations between the hyperbolicity and the regularisation indicate that regularisation is activated when the system deviates from hyperbolicity, i.e. when the CLVs tend to align.