On new stability modes of plane canonical shear flows using symmetry classification

Abstract

In the work of Nold and Oberlack [Phys. Fluids 25, 104101 (2013)], it was shown that three different instability modes of the linear stability analysis perturbing a linear shear flow can be derived in the common framework of Lie symmetry methods. These modes include the normal-mode, the Kelvin mode, and a new mode not reported before. As this was limited to linear shear, we now present a full symmetry classification for the linearised Navier-Stokes equations which are employed to study the stability of an arbitrary plane shear flow. If viscous effects for the perturbations are neglected, then we obtain additional symmetries and new Ansatz functions for a linear, an algebraic, an exponential, and a logarithmic base shear flow. If viscous effects are included in the formulation, then the linear and a quotient-type base flow allow for additional symmetries. The symmetry invariant solutions derived from the new and classical generic symmetries for all different flow types naturally lead to algebraic growth and decay for all cases except for two linear base flow cases. In turn this leads to the formulation of a novel eigenvalue problem in the analysis of the transition to turbulence for the respective flows, all of which are very distinct from the classical Orr-Sommerfeld eigenvalue problems.