On the receptivity of aerofoil tonal noise: an adjoint analysis

Abstract

(Received ?; revised ?; accepted ?. -To be entered by editorial office)For moderate-to-high Reynolds numbers, aerofoils are known to produce substantial levels of acoustic radiation, known as tonal noise, which arises from a complex interplay between laminar boundary-layer instabilities, trailing-edge acoustic scattering and upstream receptivity of the boundary layers on both aerofoil surfaces. The resulting acoustic spectrum is commonly characterised by distinct equally-spaced peaks encompassing the frequency range of convectively-amplified instability waves in the pressuresurface boundary-layer. In this work, we assess the receptivity and sensitivity of the flow by means of global stability theory and adjoint methods which are discussed in light of the spatial structure of the adjoint global modes as well as the wave-maker region. It is found that for the frequency range corresponding to acoustic tones the direct global modes capture the growth of instability waves on the suction surface and the near wake together with acoustic radiation into the far field. Conversely, it is shown that the corresponding adjoint global modes, which capture the most receptive region in the flow to external perturbations, have compact spatial support in the pressure surface boundarylayer, upstream of the separated flow region. Furthermore, we find that the relative spatial amplitude of the adjoint modes is higher for those modes whose real frequencies correspond to the acoustic peaks. Finally, the analysis of the wave-maker region points at the pressure surface near 30% of chord as the preferred zone for the placement of actuators for flow control of tonal noise.

Transition, intermittency and phase interference effects in airfoil secondary tones and acoustic feedback loop

“…The forcing mode, upper-left plot with white-blue contours, highlights the most receptive location, where minimal disturbances result in larger transient growth by the system. This mode is related to the adjoint operator of the LNS equations and it is introduced in the context of airfoil secondary tones by Fosas de Pando et al (2017). In the present results, the leading-edge region on the suction side shows the higher values of forcing modes.…”

“…Prior to this work, the stability analyses were limited a parallel flow assumption, and only a discussion of the dominant tonal frequency was presented. By means of adjoint and resolvent analyses, Fosas de Pando, Schmid & Lele (2014a) and Fosas de Pando, Schmid & Sipp (2017) also identified sensitive regions of the flow which are prone to close the feedback loop mechanism. Hence, linear stability theory has proven to be an important methodology to investigate the generation of airfoil secondary tones and the feedback loop mechanism.…”

Modal Analysis of a Laminar-Flow Airfoil under Buffet Conditions at Re = 500,000

“…We next analyse the global instability of the 2D mean flow with spanwise wavenumber β = 0. Considering the flow around a NACA 0012 airfoil at Re = 200,000 and M = 0.4, Fosas De Pando et al [41] report a region in the spectrum that is dominated by distinct equally-spaced frequencies (tonal noise) around a maximum peak at St ≈ 7 that corresponds to a stable global mode. An impulse response analysis by [42] showed the vivid interaction between suction and pressure side at that frequency and suggested a feedback mechanism due to pressure waves that are scattered at the trailing edge and form upstream moving acoustic waves.…”

Mesh Adaptation Using Adjoint Methods and Reduced-Order Models for Large Eddy Simulation

“…Another solution is to apply the checkpointing technique [12] in which optimalselected primal states are stored and then used to re-solve the primal problem locally. Such techniques have been employed to solve unsteady adjoint problem in sensitivity analyses of dynamical systems [13], cylinder flows [14], and tonal noise [15]. However, this requires significant computational effort due to the re-computation.…”