Reduced-order models of linearized channel flow using balanced truncation

Fluid systems often exhibit inherently low-dimensional behavior, even though the governing equations are complex and high-dimensional. At this time, full 3D discretizations of the Navier-Stokes equations are too computationally intensive to be used for control synthesis, so low-order models of the flow physics are desirable. This paper compares two different model-reduction procedures for the linearized flow in a plane channel: the method of Proper Orthogonal Decomposition and Galerkin projection, popular in the fluid mechanics community; and balanced truncation, a common method for model reduction of linear systems. Standard methods of computing balancing transformations are computationally intractable for systems of this size, so we use a numerical approximation of balanced truncation using empirical Gramians computed from simulations of the linearized and adjoint systems. For the channel flow considered here, the subspaces spanned by POD modes and balancing modes are very close, but reduced-order models from approximate balanced truncation perform much better than the standard POD models, for the same order of truncation.