Unstable Periodically Forced Navier-Stokes Solutions---Towards Nonlinear First-Principle Reduced-Order Modeling of Actuator Performance

TL;DR

This paper introduces a novel method for computing unstable periodically forced solutions of the linearized Navier-Stokes equations, enabling improved modal analysis of unsteady flows for reduced-order modeling.

Contribution

It presents a new technique to compute higher-order unstable modes in linearized Navier-Stokes flows using complex eigenvalue shifts and iterative growth rate estimation.

Findings

Successfully applied to wake flows around various bluff bodies.

Facilitates flow control via periodic wake actuation.

Supports development of Navier-Stokes based Galerkin models.

Abstract

We advance the computation of physical modal expansions for unsteady incompressible flows. Point of departure is a linearization of the Navier-Stokes equations around its fixed point in a frequency domain formulation. While the most amplified stability eigenmode is readily identified by a power method, the technical challenge is the computation of more damped higher-order eigenmodes. This challenge is addressed by a novel method to compute unstable periodically forced solutions of the linearized Navier-Stokes solution. This method utilizes two key enablers. First, the linear dynamics is transformed by a complex shift of the eigenvalues amplifying the flow response at the given frequency of interest. Second, the growth rate is obtained from an iteration procedure. The method is demonstrated for several wake flows around a circular cylinder, a fluidic pinball, i.e. the wake behind a cluster of cylinders, a wall-mounted cylinder, a sphere and a delta wing. The example of flow control with periodic wake actuation and forced physical modes paves the way for applications of physical modal expansions. These results encourage Galerkin models of three-dimensional flows utilizing Navier-Stokes based modes.