Stability Analysis of Fluid Flows Using Lagrangian Perturbation Theory (LPT): Application to the Plane Couette Flow

Abstract

We present a new application of Lagrangian Perturbation Theory (LPT): the stability analysis of fluid flows. As a test case that demonstrates the framework we focus on the plane Couette flow. The incompressible Navier-Stokes equation is recast such that the particle position is the fundamental variable, expressed as a function of Lagrangian coordinates. The displacement due to the steady state flow is taken to be the zeroth order solution and the position is formally expanded in terms of a small parameter (generally, the strength of the initial perturbation). The resulting hierarchy of equations is solved analytically at first order. We find that we recover the standard result in the Eulerian frame: the plane Couette flow is asymptotically stable for all Reynolds numbers. However, it is also well established that experiments contradict this prediction. In the Eulerian picture, one of the proposed explanations is the phenomenon of “transient growth” which is related to the non-normal nature of the linear stability operator. The first order solution in the Lagrangian frame also shows this feature, albeit qualitatively. As a first step, and for the purposes of analytic manipulation, we consider only linear stability of 2D perturbations but the framework presented is general and can be extended to higher orders, other flows and/or 3D perturbations.