Causal analysis of a turbulent shear flow model

We perform causal analysis on the low-dimensional Galerkin model for shear flow developed by Moehlis et al. ( New J. Phys ., vol. 6, 2004, 56). Our method integrates both equation-based analysis and the proposed Galerkin-based Granger causality (GGC) to investigate the effect of the nonlinear terms on the dynamics. Two types of quadratic interactions are identified: a fully triadic interaction and a modulated two-mode coupling. The propagation of these interactions through the nonlinear dynamics leads to a directed cause-and-effect network. Furthermore, the relative importance of each mode amplitude on the dynamics of the target mode is quantified. This analysis provides a deeper understanding of the nonlinear dynamics and distills control opportunities. To demonstrate the applicability of the proposed GGC to realistic flows where Galerkin projection is impractical, a turbulent lid-driven cavity flow is further studied. We foresee applications of the proposed causal analysis framework as valuable tools for Galerkin modelling – guiding investigations of modal causality, prediction uncertainty, model-order reduction and control design.