Numerically accurate computation of the conditional trajectories of the topological invariants in turbulent flows

The computation of the topological invariants of the velocity gradient tensor and of their conditional mean trajectories in incompressible turbulent flows is revisited. It is argued that probability conservation requires that the conditional mean trajectories should be closed when a statistically stationary wall-bounded or periodic domain is considered, and this is confirmed numerically for a turbulent channel. It is argued that previous reports of inward spiraling of the conditional trajectories are either due to incomplete statistics in inhomogeneous flows or to numerical errors.