Reduced order modeling for fluid flows based on nonlinear balanced truncation

For a vast variety of fluid flows the dynamics are governed by the Navier-Stokes equations which are highly nonlinear. In particular, the corresponding Galerkin models involve a quadratic type nonlinearity. The latter incudes the Burgers' equation as well. In this paper, a computational algorithm for nonlinear balanced truncation of the Galerkin models is proposed. The method is based on Taylor series expansion in which the computation of successively higher order terms reduces to solving consecutively higher order systems of linear algebraic equations. This procedure is also used to compute the coordinate transformation that yields the balanced system. Formulas to compute the nonlinear controllability, observability functions and coordinate transformation are derived.