Resolvent-based estimation and control of a laminar airfoil wake

TL;DR

This paper introduces a resolvent-based estimation and control framework for predicting and reducing vortex shedding in a laminar airfoil wake, offering computational efficiency and improved accuracy over traditional methods.

Contribution

The authors develop a novel resolvent-based approach that efficiently estimates and controls flow fluctuations, accommodating nonlinear forcing with colored-in-time statistics, and demonstrate its effectiveness in flow control.

Findings

Achieves approximately 3% and 30% prediction error for clean and noisy flows.

Reduces turbulent kinetic energy in the wake by 98% with four actuators.

Operates at lower computational cost compared to Kalman filter and LQG controller.

Abstract

We develop an optimal resolvent-based estimator and controller to predict and attenuate unsteady vortex shedding fluctuations in the laminar wake of a NACA 0012 airfoil at an angle of attack of 6.5 degrees, chord-based Reynolds number of 5000, and Mach number of 0.3. The resolvent-based estimation and control framework offers several advantages over standard methods. Under equivalent assumptions, the resolvent-based estimator and controller reproduce the Kalman filter and LQG controller, respectively, but at substantially lower computational cost using either an operator-based or data-driven implementation. Unlike these methods, the resolvent-based approach can naturally accommodate forcing terms (nonlinear terms from Navier-Stokes) with colored-in-time statistics, significantly improving estimation accuracy and control efficacy. Causality is optimally enforced using a Wiener-Hopf formalism. We integrate these tools into a high-performance-computing-ready compressible flow solver and demonstrate their effectiveness for estimating and controlling velocity fluctuations in the wake of the airfoil immersed in clean and noisy freestreams, the latter of which prevents the flow from falling into a periodic limit cycle. Using four shear-stress sensors on the surface of the airfoil, the resolvent-based estimator predicts a series of downstream targets with approximately 3% and 30% error for the clean and noisy freestream conditions, respectively. For the latter case, using four actuators on the airfoil surface, the resolvent-based controller reduces the turbulent kinetic energy in the wake by 98%.