Symmetry analysis in linear hydrodynamic stability theory: Classical and new modes in linear shear

Abstract

We present a symmetry classification of the linearised Navier-Stokes equations for a two-dimensional unbounded linear shear flow of an incompressible fluid. The full set of symmetries is employed to systematically derive invariant ansatz functions. The symmetry analysis grasps three approaches. Two of them are existing ones, representing the classical normal modes and the Kelvin modes, while the third is a novel approach and leads to a new closed-form solution of traveling modes, showing qualitatively different behaviour in energetics, shape, and kinematics when compared to the classical approaches. The last modes are energy conserving in the inviscid case. They are localized in the cross-stream direction and periodic in the streamwise direction. As for the kinematics, they travel at constant velocity in the cross-stream direction, while in the streamwise direction they are accelerated by the base flow. In the viscous case, the modes break down due to damping of high wavenumber contributions.