The triple decomposition of the velocity gradient tensor as a standardized real Schur form

Abstract

The triple decomposition of a velocity gradient tensor provides an analysis tool in fluid mechanics by which the flow can be split into a sum of irrotational straining flow, shear flow, and rigid body rotational flow. In 2007, Kolář formulated an optimization problem to compute the triple decomposition [V. Kolář, “Vortex identification: New requirements and limitations,” Int. J. Heat Fluid Flow 28, 638–652 (2007)], and more recently, the triple decomposition has been connected to the Schur form of the associated matrix. We show that the standardized real Schur form, which can be computed by state of the art linear algebra routines, is a solution to the optimization problem posed by Kolář. We also demonstrate why using the standardized variant of the real Schur form makes computation of the triple decomposition more efficient. Furthermore, we illustrate why different structures of the real Schur form correspond to different alignments of the coordinate system with the fluid flow and may, therefore, lead to differences in the resulting triple decomposition. Based on these results, we propose a new, simplified algorithm for computing the triple decomposition, which guarantees consistent results.