Low-dimensional modelling of turbulence using the proper orthogonal decomposition: A tutorial

The proper orthogonal decomposition identifies basis functions or modes which optimally capture the average energy content from numerical or experimental data. By projecting the Navier-Stokes equations onto these modes and truncating, one can obtain low-dimensional ordinary differential equation models for fluid flows. In this paper we present a tutorial on the construction of such models. In addition to providing a general overview of the procedure, we describe two different ways to numerically calculate the modes, show how symmetry considerations can be exploited to simplify and understand them, comment on how parameter variations are captured naturally in such models, and describe a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently. We illustrate for the example of plane Couette flow in a minimal flow unit - a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence.