UCLA

External flows over wings is a traditional flow of interest in aerodynamics. Over the last century, research efforts were dedicated to studying the wake patterns that form in the flows around finite wings. Among others, we can pinpoint the wing tip vortex, the separation region that develops under adverse pressure gradient, and the coherent vortical structures. Thanks to these past efforts, we were able to significantly extend our knowledge in aerodynamics, which paved the way for an impressive evolution of aircraft designs in the last century. Over the years, commercial flight became an ordinary asset in our society. More recently, small and micro air vehicles have also reached the market, being operated by individuals who hold no necessary knowledge of the complexity of the Navier-Stokes equations. The advanced knowledge currently held on flight physics has played a fundamental role in the development of aircraft designs, however, there is still room for improvement. In post-stall flow conditions, the aerodynamic performance of the wing decays considerably, making it challenging to sustain flight at high incidence angles. To enable flight in such flow conditions, it is important to develop physics-based flow control strategies capable of improving the overall aerodynamic performance of the wing and its flight stability. The main implications are reduced fuel (and energy) consumption during flight, increased aircraft range, improvements in safety and productivity of air travel, attenuation of the acoustic signature, as well as enabled capability of aircraft to fly in challenging external environments.

Towards this goal, many studies have been performed to analyze and control flows over airfoils in spanwise periodic configurations. These may also be called infinite-span wings. These studies were fundamental to revealing important aspects of flow physics. However, in reality, the flows around wings are three-dimensional (3-D). In addition, modern aircraft wings are usually tapered and swept. The 3-D vortex dynamics of flows over wings has a significant influence on aircraft design, by reducing the overall lift, generating induced drag, and increasing flow unsteadiness. Thus, it is important to develop strategies to control the vortex dynamics that encompass the knowledge of the 3-D characteristics of the flow over finite, swept, and tapered wings. Especially for post-stall flow conditions, the wake dynamics around tapered swept wings is largely unexplored. It is still a challenge to understand how the wing geometry relates to the vortex formation for different aspect and taper ratios, as well as angles of attack and sweep. To design control strategies to improve aerodynamic performance for finite, swept, and tapered wings, we must go beyond the sole characterization of flow structures. In fact, the identification of perturbation dynamics is called for to modify the flow field.

Three-dimensional flow control is challenging due to the high-dimensional and nonlinear nature of the flow dynamics of the wakes. Thus, it is necessary to find an appropriate actuation setup for the problem that can alter the base flow behavior. This effort can be guided by modal analysis methods. In our work, we have used resolvent analysis, a method based on the singular value decomposition (SVD) of the resolvent, which is a linear operator constructed using the Navier-Stokes equations linearized with respect to the base flow. For unsteady flows, the statistically converged time-averaged flow field is used as a base flow to construct the resolvent operator. The strength of the resolvent operator through this approach is the capability to find optimal forcing modes which amplify outputs in the flow field and give insights into the perturbed flow through the spatial response modes. The challenge within resolvent analysis is the SVD computation for large-scale resolvent operators that are generated for high-dimensional flow fields, such as three-dimensional and turbulent flows.

By taking advantage of low-rank approximation of the resolvent operator, recent developments using randomized numerical linear algebra have accelerated the computation of the dominant resolvent modes. With reduced computational costs, these efforts have enabled the use of resolvent analysis in turbulent flows over spanwise periodic airfoils and expanded its applicability to triglobal problems and higher Reynolds number flows. With the randomized algorithm, we can use resolvent analysis to uncover the dynamics of 3-D flows over finite, swept, and tapered wings, supporting flow control efforts to improve their overall aerodynamic performance.

The present study has shown how to develop a 3-D resolvent-based flow control over finite wings. We initiate by studying flows over finite wings and the effects of wing sweep and taper on the post-stall wake dynamics through direct numerical simulations (DNS). We consider laminar flows at chord-based Reynolds numbers of 400 and 600 with weak compressible effects at a freestream Mach number 0.1. The flows are studied around wings at angles of attack between 14 and 30 degrees with semi aspect ratios ranging from 1 to 4, sweep angles up to 50 degrees, and taper ratios from 0.27 to 1. Following a comprehensive characterization of the wake dynamics through DNS results, we extend our knowledge of the dynamics of flow perturbations by analyzing the triglobal resolvent modes. Through the identification of the optimal harmonic perturbations that can be amplified in the flow field, we develop 3-D active flow control that is shown to significantly modify the wake structures around the wings. This comprehensive investigation provides novel and unique insights that reveal the flow structures that can be amplified in the wake and modify their dynamics in post-stall flow conditions.