The primitive Orr–Sommerfeld equation and its solution by finite elements

Abstract

The linear stability of parallel shear flows of incompressible viscous fluids is classically described by the Orr–Sommerfeld equation in the disturbance streamfunction. This fourth-order equation is obtained by eliminating the pressure from the linearized Navier–Stokes equation. Here we consider retaining the primitive velocity-pressure formulation, as is required for general multidimensional geometries for which the streamfunction is unavailable; this affords a uniform description of one-, two-, and three-dimensional flows and their perturbations. The Orr–Sommerfeld equation is here discretized using Python and scikit- fem, in classical and primitive forms with Hermite and Mini elements, respectively. The solutions for the standard test problem of plane Poiseuille flow show the primitive formulation to be simple, clear, very accurate, and better-conditioned than the classical.