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

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. References L. Allen and T. J. Bridges. Numerical exterior algebra and the compound matrix method. Numer. 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Firedrake: An Automated Finite Element System

Version of Firedrake in the Firedrake paper. This release is specifically created to document the version of Firedrake used in a particular set of experiments. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to http://www.firedrakeproject.org/publications.html</span

firedrakeproject/firedrake: an automated finite element system

This release is specifically created to document the version of firedrake used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.htmlThis release is specifically created to document the version of firedrake used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html20181204.

firedrake: an automated finite element system

Version of Firedrake in the Firedrake paper. This release is specifically created to document the version of Firedrake used in a particular set of experiments. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to http://www.firedrakeproject.org/publications.htm