Resolvent-based Estimation and Control of Aerodynamic Flows

Abstract

Aerodynamic flows are prevalent across numerous engineering fields and play a crucial role in addressing critical societal challenges. In particular, chaotic fluctuations in wake flows can significantly degrade the performance of aerodynamic systems and generate noise. For example, wake flows are closely linked to upstream separated flow, which can reduce the efficiency of aircraft and cars. Additionally, wake perturbations contribute to noise generation in wind turbines and can negatively impact the performance of downstream turbines. Therefore, reducing wake perturbations is essential for enhancing efficiency and delivering broader societal benefits. However, predicting and controlling such flows is challenging due to their nonlinear behavior, significant computational demands, and our incomplete understanding of the underlying physics. To address these challenges, we employ resolvent-based approaches to estimate and mitigate flow perturbations. Our estimators and controllers are derived using both operator-based and data-driven approaches. The operator-based approach, which relies on the linearized Navier-Stokes operator, offers low computational costs without requiring a priori model reduction and incorporates time-colored statistics of the nonlinear terms from the Navier-Stokes equations, interpreted as forcing on the linear dynamics, to partially capture the impact of nonlinearity. The data-driven approach, which uses training data from nonlinear simulations, avoids the need to construct linearized Navier-Stokes operators. This allows for the construction of kernels even for globally unstable flows and naturally incorporates the colored statistics of the nonlinear terms. For both approaches, the Wiener-Hopf formalism is employed to ensure optimal causality in the estimator and controller, enhancing real-time estimation and control performance. As a first case, we apply these resolvent-based tools to the flow over a backward-facing step at Re = 500 using an implementation within the incompressible flow solver, NEK5000. By imposing random external forcing upstream, we generate unsteadiness at the downstream target and then achieve effective estimation and control of downstream fluctuations in both the linear and nonlinear regimes. Next, we develop a new implementation of the resolvent-based estimation and control tools within the compressible flow solver CharLES for high-performance computing environments. With this implementation, we apply the resolvent-based tools to both laminar and turbulent flows over a NACA 0012 airfoil at Reynolds numbers of 5,000 and 23,000, respectively. For the laminar case, we introduce random upstream perturbations to disrupt the periodic vortex shedding, leading to chaotic fluctuations, which are then successfully predicted and suppressed by the resolvent-based estimator and controller. For the turbulent case, the resolvent-based estimator accurately captures the wake fluctuations present in the spanwise-averaged, spanwise-Fourier-transformed, and the mid-span-plane flows using a small number of shear-stress sensors on the surface of the airfoil.