An objective measure of unsteadiness

Unsteadiness lies at the heart of turbulent fluid dynamics, eddy formation and instabilities in flows, thus making it central to both understanding and controlling fluid systems. In this work, we present an objective measure for the unsteadiness of a time-dependent velocity field, the deformation unsteadiness, derived from a spatio-temporal variational principle, allowing for a frame-independent assessment of the unsteadiness of a given flow field. Additionally, as an application of our main result, we define an objective analogue of the classical $Q$ -criterion based on extremisers of unsteadiness minimisation. We apply our results to several examples of analytical flows as well as simulated flow data sets in two and three dimensions. In particular, we apply our newly derived vortex criterion to several explicit, time-dependent solutions of the Navier–Stokes equation and compare the results with existing vortex criteria. We give a physical interpretation of the deformation unsteadiness and discuss future research directions.