Control of Vortex Dynamics using Invariants

TL;DR

This paper introduces a novel control approach for vortex dynamics using invariants and Koopman eigenfunctions, enabling manipulation of chaotic vortex behavior and mixing in a 4-vortex system.

Contribution

It develops a Koopman-based invariant control method for vortex dynamics, applying it to chaotic vortex systems to modify their states and enhance mixing.

Findings

Controlling invariants can transition vortex states from chaos to quasiperiodicity.

The invariant control method effectively manipulates vortex behavior.

Demonstrated improved mixing in a 4-vortex system.

Abstract

Vortex-dominated flows are ubiquitous in engineering, and the ability to efficiently manipulate the dynamics of these vortices has broad applications, from wake shaping to mixing enhancement. However, the strongly nonlinear behavior of the vortex dynamics makes this a challenging task. In this work, we investigate the control of vortex dynamics by using a change of coordinates from the Biot-Savart equations into well-known invariants, such as the Hamiltonian, linear, and angular impulses, which are Koopman eigenfunctions. We then combine the resulting model with model predictive control to generate control laws that force the vortex system using "virtual cylinders". The invariant model is beneficial as it provides a linear, global description of the vortex dynamics through a recently developed Koopman control scheme for conserved quantities and invariants. The use of this model has not been well studied in the literature in the context of control. In this paper, we seek to understand the effect of changing each invariant individually or multiple invariants simultaneously. We use the 4-vortex system as our primary test bed, as it is the simplest configuration that exhibits chaotic behavior. We show that by controlling to specific invariant quantities, we can modify the transition from chaotic to quasiperiodic states. Finally, we computationally demonstrate the effectiveness of invariant control on a toy example of tracer mixing in the 4-vortex system.