Resolvent-analysis-based design of airfoil separation control

Abstract

We combine three-dimensional (3D) large-eddy simulations (LES) and resolvent analysis to design active separation control techniques on a NACA 0012 airfoil. Spanwiseperiodic flows over the airfoil at a chord-based Reynolds number of 23, 000 and a freestream Mach number of 0.3 are considered at two post-stall angles of attack of 6 • and 9 • . Near the leading edge, localized unsteady thermal actuation is introduced in an open-loop manner with two tunable parameters of actuation frequency and spanwise wavelength. For the most successful control case that achieves full reattachment, we observe a reduction in drag by up to 49% and increase in lift by up to 54%. To provide physicsbased guidance for the effective choice of these control input parameters, we conduct global resolvent analysis 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. 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. 4 Case 6-1A ∆C D = −41%, ∆C L = +5.1% Case 6-2A ∆C D = −43%, ∆C L = +1.2% Case 6-4A ∆C D = −36%, ∆C L = +11% 6 Case 6-1B ∆C D = −40%, ∆C L = +7.8% Case 6-2B ∆C D = −44%, ∆C L = −2.2% Case 6-4B ∆C D = −41%, ∆C L = +0.3% 12 Case 6-1C ∆C D = −38%, ∆C L = +2.8% Case 6-2C ∆C D = −37%, ∆C L = +0.3% Case 6-4C ∆C D = −30%, ∆C L = −2.4% 15 Case 6-1D ∆C D = −2.2%, ∆C L = +0.4% Case 6-2D ∆C D = −2.8%, ∆C L = −0.4% Case 6-4D ∆C D = +2.8%, ∆C L = −4.1% 2 Case 9-0A ∆C D = −35%, ∆C L = +39% Case 9-1A ∆C D = −19%, ∆C L = +28% Case 9-2A ∆C D = −17%, ∆C L = +29% 5.5 Case 9-0B ∆C D = −35%, ∆C L = +16% Case 9-1B ∆C D = −38%, ∆C L = +47% Case 9-2B ∆C D = −37%, ∆C L = +53% 12 Case 9-0C ∆C D = −43%, ∆C L = +28% Case 9-1C ∆C D = −46%, ∆C L = +41% Case 9-2C ∆C D = −49%, ∆C L = +37% 16 Case 9-0D ∆C D = −1.7%, ∆C L = −3.5% Case 9-1D ∆C D = +1.9%, ∆C L = −3.0% Case 9-2D ∆C D = +0.7%, ∆C L = −7.4%

A tale of two airfoils: resolvent-based modelling of an oscillator versus an amplifier from an experimental mean

“…The mode shapes for intermediate frequencies have a mix of both linear mechanisms as seen for ω = 12 in figure 16. These results are consistent with Thomareis & Papadakis (2018) and Yeh & Taira (2019), who computed resolvent modes for a NACA 0012 airfoil at similar Reynolds numbers and angles of attack from time-averaged DNS and large-eddy simulation (LES) data, respectively. The latter study also identified two branches in the eigenspectrum of the LNS operator which were grouped into wake or shear layer modes.…”

“…where s is a complex number with nonzero real part. Yeh & Taira (2019) applied the discounted resolvent to flow around a NACA 0012 airfoil since they encountered unstable modes. Since we do not find unstable eigenmodes near the wake frequencies, we opt to not apply this procedure.…”

Resolvent Analysis of Compressible Laminar and Turbulent Cavity Flows

“…,q k ] contains a set of left singular vectorsq j called the response modes, and The resolvent gain is studied in the context of having an energy norm defined by E = ∫ S (|ρ| 2 + |ρu| 2 + |ρv| 2 + |ρw| 2 + |ê| 2 )dA = ||q|| 2 E in the present work. This norm can be related to the induced 2-norm of the resolvent operator H(q; ω, β) through a weight matrix W, such that ||q|| 2 E = ||Wq|| 2 2 [11,28]. The weight matrix W can be constructed based on the discretization scheme adopted in the numerical configuration.…”

“…Because of the weight matrix W used in the induced 2-norm evaluation, the resulting forcing and response modes shown later have been scaled by W −1 to represent the correct flow field. Details on the norms can be found in Yeh & Taira [11].…”

Resolvent-based study of compressibility effects on supersonic turbulent boundary layers

“…For example, under the correct scaling of the wall-parallel wavenumbers, the self-similar hierarchies of response modes give rise to self-similar families of vortical structures. However, there has been less work applying this operator based decomposition to fully-developed compressible flows, with the exception of the recent studies on compressible jet flows (Jeun et al 2016;Towne et al 2018;Schmidt et al 2018), subsonic aerofoils (Yeh & Taira 2019), turbulent Couette flows (Dawson & McKeon 2019), and hypersonic boundary layer transitions (Cook et al 2018;Dwivedi et al 2018).…”