Linear and nonlinear Granger causality analysis of turbulent flows*

This work presents a novel nonlinear extension of multivariate Granger causality analysis to enable the study of second-order (and if needed, higher-order) nonlinear interactions between energetically-dominant coherent structures in turbulent flows. We focus in particular on turbulent flow through a square duct, which features a secondary mean flow containing pairs of counter-rotating streamwise vortices located near the duct corners (Prandtl's secondary flow of the second kind). This secondary mean flow has velocities significantly smaller in magnitude than the streamwise mean flow component. We apply the proposed causality analysis framework upon  temporal coefficients of proper orthogonal decomposition modes, which are obtained from direct numerical simulation data. The analysis suggests that secondary flow fluctuations are the principal driver for both the formation of the near-wall and near-corner streamwise streaks, and for the motion of these structures towards and away from the corner. We further distinguish between linear and nonlinear causal mechanisms present in the system.

This work was supported by Air Force Office of Scientific Research grant FA9550-22-1-0109, National Science Foundation grant CBET-2238770 and the European Research Council under the  Caust grant ERCAdG-101018287.