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

Reporting period: 2016-01-31 to 2018-01-30

Trailing-edge flows arise in many technological applications, such as aircraft wings in aeronautics and mixing devices in chemical engineering. Two important and fundamental processes take place near the trailing edge, namely the acoustic radiation and receptivity, which refer to generation of sound by fluctuations within the flows and excitations of instability waves by ambient disturbances, respectively. This project will investigate (a) generation of instability waves in the wake as free-stream acoustic and vortical disturbances impinge on the trailing edge; (b) radiation of sound when Tollmien-Schlichting (T-S) waves in the upstream boundary layer propagate through, and interact with, the trailing-edge flow. Both processes will be analysed mathematically by developing a Local Scattering Theory, which we recently proposed as an appropriate framework for describing the coupling of distinct characteristic motions in a region of strong inhomogenuity. Furthermore, with radiation and receptivity being described properly, we will move on to investigate the so-called acoustic feedback loops, in which instability waves and acoustic waves are generated from each other, leading to self-sustained oscillations. First-principle theories will be developed to predict the tones (frequencies) of the oscillations.

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.

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.

1. Feb/2016 - Sep/2016: Acoustic radiation due to scattering of Tollmien-Schlichting (T-S) waves by surface imperfections

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

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

In this project, we have completed the description and extension of the local scattering framework, which can be used to describe the scattering of the instability of the wall layer (lower deck), including the receptivity and transmission of the instability modes and the acoustic radiation due to the scattering process. The key quantities of interest, such as the receptivity and transmission coefficients, and the directivity and intensity of the sound wave, are predicated for the first time on the basis of first principles. In addition to being an elegant mathematical formulation, this approach is computationally efficient, and thus can be used for systematically parametric study. Trailing edge receptivity and radiation underpin some remarkable flow phenomena of practical importance such as mixing of reactants, jet and airframe noise as well as transition to turbulence. The improved understanding of, and the ability to predict, receptivity and radiation are important for developing effective flow control strategies. The computational efficiency of our approach makes it an appealing tool.