On the Lie symmetry analysis of three-dimensional perturbed shear flows

Abstract

The study presents symmetry classifications of the linearized Navier–Stokes equations, governing the three-dimensional incompressible plane shear flows. The linearization is done with respect to small perturbations. In the case of a two-dimensional shear flow with a linear profile, Nold and Oberlack (PoF, 2013) showed the existence of three different kinds of linear instability modes using the framework of Lie symmetry classification. Those perturbation modes are normal mode, kelvin mode, and a new type invariant mode. We have extended their analysis for a three-dimensional plane shear flow with linear as well as non-linear base profiles. The invariant ansatz functions are systematically derived employing the full set of symmetries. The analysis is done for both viscous and inviscid flows by considering the linear, exponential, and fractional shear flow profiles. In the derivation process, the set of infinitesimal generators for the generalized system is first obtained using the classical Lie symmetry analysis, and then, some additional symmetries are searched out for each sub-case. Further, the governing system of partial differential equations is converted into ordinary differential equations by using symmetries and invariant conditions. The most popular three-dimensional normal modes and the Orr–Sommerfeld equation are acquired by taking the general symmetry. Moreover, for each of the sub-cases, we have derived the possible exact solutions of the associated system, and the behaviors of the solutions are explored for different parameter ranges.