Use of Resolvent Analysis for Design of Active Separation Control with Thermal-Based Actuators

This study aims to examine the use of a fundamental thermal input for separation control and provide a design guideline via resolvent analysis. The use of the thermal actuator is motivated by the interest in the use of thermal-energy-based actuators from the active flow control community. We conduct a series of numerical investigations to uncover the underlying control mechanism of thermal-energy-based actuators and examine their control effectiveness when a fundamental thermal energy source is introduced as the only external perturbation. We consider a thermal actuator model that introduces localized boundary actuation in the form of unsteady heat flux. This external thermal actuation is added to the right-hand-side of the energy equation in the compressible Navier--Stokes equations. We study how the localized thermal forcing affects the far-field and near-field of the surrounding flow. To provide design guidelines to active separation control, we perform resolvent analysis on the mean baseline flows and use it as a linear model to examine the energy amplification and flow response to different actuation frequencies and wavenumbers. Since the key to suppressing flow separation lies in the excitation of the instabilities in the shear layer that forms over the airfoil, we consider a free shear layer and perturb it with a fundamental thermal input in the first part of this study. In this model problem, local periodic heating is introduced at the trailing edge of a finite-thickness splitter plate. Two-dimensional direct numerical simulations are performed at the plate-thickness-based Reynolds number of 1000. We find that thermal actuation introduces low level of oscillatory surface vorticity flux and baroclinic torque at the actuation frequency in the vicinity of the trailing edge. The produced vortical perturbations can independently excite the fundamental instability that accounts for shear layer roll-up as well as the subharmonic instability that encourages the vortex pairing process farther downstream. We demonstrate that the nonlinear dynamics of a spatially developing shear layer can be modified by the local oscillatory heat flux as a control input. Next, we leverage the findings from the thermally perturbed free shear layer and extend the employment of the thermal actuation technique to the control of flow separation over an airfoil. The separated flows over a NACA 0012 airfoil at two post-stall angles of attack 6° and 9° and chord-based Reynolds number ReLc = 23,000 are considered. The thermal actuator is placed near the leading edge to introduce unsteady thermal forcing. Three-dimensional large-eddy simulations (LES) are employed in this investigation. Of particular interest is the influence of the frequency and spanwise wavelength of the thermal actuator introduced at the natural separation point. We observe that the thermal forcing is capable of reattaching the flow by encouraging the roll-up of the vortex sheet emanating from the leading edge. Discrete coherent spanwise vortices are formed and leads to the generation of low-pressure region over the top surface enabling lift enhancement. Under certain 2D actuation cases, the turbulent flow is completely laminarized and turns into a 2D flow. To further enhance the aerodynamic performance of the wing, we consider the trigger of vortex breakdown by introducing spanwise perturbation in the thermal forcing. It is found that spanwise variation in the forcing can enhance mixing past the mid chord and entrains free-stream momentum, benefitting from both the low-pressure core of the spanwise vortices and the recovery of the flow momentum in the aft portion of the wing. For the successful controlled case that achieves reattachment, we observe a significant reduction in drag by up to 49% and an increase in lift by up to 54%. The fluctuations in aerodynamic forces are also reduced by up to 84% with the unsteady thermal actuation. We also find that the excitation of shear-layer roll-up over the airfoil and the turbulent-laminar transition after the roll-up both play important roles to achieve effective separation control. Complementary to the parametric study using LES, we perform resolvent analysis on the baseline flows to obtain further physics-based guidance for the effective choice of these control input parameters. The global resolvent analysis is conducted on the baseline turbulent mean flows to identify the actuation frequency and wavenumber that provide high energy amplification. The present analysis also considers the use of a temporal filter to limit the time horizon for assessing the energy amplification to extend resolvent analysis to unstable base flows. We incorporate the amplification and response mode from resolvent analysis to provide a metric that quantifies momentum mixing associated with the modal structure. By comparing this metric from resolvent analysis and the LES results of controlled flows, we demonstrate that resolvent analysis can predict the effective range of actuation frequency as well as the global response to the actuation input. We demonstrated the effectiveness of the thermal actuation to suppress flow separation over the airfoil. Supported by the agreements between the results from resolvent analysis and LES, we believe that this study provides insights for the use of resolvent analysis in guiding future active flow control.