## Periodic Reporting for period 1 - STAR (An Extended Local Scattering Theory for Acoustic-radiation and Receptivity of Trailing-edge Flows)

Période du rapport: 2016-01-31 au 2018-01-30

Although the proposed research is primarily of fundamental nature, the theoretical progress would be of considerable interest to aeronautic industry. Moreover, the study on the feedback loops reveals the mechanism of the airfoil tonal noise, which provides potential opportunities for noise reduction strategy.

The overall objectives of this project are twofold. Firstly, it provides a theoretical framework to predict the receptivity and acoustic radiation of trailing-edge flows, which is of high practical relevance to a variety of engineering applications. Secondly, from the methodology point of view, this project combines sophisticated asymptotic methods with highly accurate numerical computations, enabling us to tackle a greater range of complex problems, for which numerical or analytical method alone would be inadequate.

This is an extension of the recently-developed local scattering framework (Ref 1). The surface imperfection can be a roughness, a suction/injection slot, or a trailing edge. The triple-deck formalism is employed, and in order to accommodate the acoustic radiation in the potential stream, we include the second-order expansion in the upper deck. A transmission coefficient, defined as the ratio of the T-S wave amplitudes downstream of the scatter to that upstream, is introduced, which leads to a complete description of the unsteady perturbation field and reduces the whole system to a generalized eigenvalue problem. The acoustic pressure of a suction-induced radiation is shown in Fig. 1. Results are published in Ref 2.

2. Sep/2016 - June/2017: Receptivity of the trailing-edge flow to free-stream acoustic wave.

We describe the generation of the varicose wake mode behind a trailing edge by freestream acoustic waves by extending the local scattering theory. As shown by the sketch in Fig. 2, the mean flow in the vicinity of the trailing edge is again described by triple-deck formalism. The oncoming perturbation is the Stokes wave induced by the freestream sound wave, while the downstream perturbation is one of the near-wake modes. In the near wake, three instability modes, which are of purely viscous, viscous-inviscid-interactive (V-I-I) and inviscid long-wavelength Rayleigh (LWR) nature, may emerge depending on the frequency of the sound; their evolutions are shown in Fig. 3. The asymptotic behaviours of all modes are analyzed, and the receptivity coefficients are obtained by solving the local scattering framework. A preliminary version of this work has been published in a conference proceeding (Ref 3), and the whole version has been submitted to J Fluid Mech.

3. July/2017-Dec/2017: Generation of the wake modes behind the trailing edge by oncoming T-S wave

We consider the transmission of an oncoming T-S wave through the trailing-edge region of a thin flat plate; the T-S wave excites the varicose instability modes in the wake. The frequencies of the T-S waves are assumed to correspond to the lower unstable branch, which are higher than those considered in the receptivity problem, and the near-wake instability modes are found to be purely inviscid. Although the previously formulated LWR mode appears in the downstream limit, and a near-wake regime is formed upstream of its onset, where the lower deck has not yet split into two layers; the asymptotic structure is illustrated in Fig. 4. By employing the local scattering framework, we obtain the perturbation field in the vicinity of the trailing edge. This work will be submitted to J Fluid Mech.

4. Jan/2018-Jan/2018: Mechanism of tonal noise: acoustic feedback loop

We establish a model problem to exhibit the acoustic feedback-loop phenomenon, as shown in Fig. 5. The model is a semi-infinite flat plate with two separated humps, and the feedback loop is described as follows: (i) when an oncoming T-S wave is scattered by the downstream hump, an acoustic wave is emitted to the far field; (ii) the upstream propagating branch then interacts with the upstream hump, leading to generation of the T-S wave in the boundary layer; (iii) the T-S wave propagates downstream to enhance the existing T-S wave in general. Regime (i) is described by a similar way as that in Ref2, but we take the high-frequency limit, and regime (ii) is studied by employing the local scattering framework. The feedback loop operates only if a synchronization condition, i.e. the phase and amplitude of the original T-S mode match with those of regenerated one, is satisfied. The system is unstable at a synchronization frequency when the regenerated mode has an amplitude larger than that of the original one. This work is in preparation for publication.

References

1 Wu X., Dong M. J. Fluid Mech. 2016. 794: 68-108.

2 Dong M., Wu X. AIAA-2016-3194. 2016

3 Dong M., Wu X. AIAA-2017-4022. 2017.