Local scattering theory and the role of an abrupt change in boundary-layer instability and acoustic radiation

摘要

When a boundary-layer instability mode propagates through a region of stream wise rapid distortion, the ensuing scattering causes two consequences of physical interest. First, the amplitude of the instability mode may be suppressed or energized, through which the instability and transition is influenced. Second, if the flow is compressible, a sound wave of substantial intensity can be radiated to the far field. This is a rather generic mechanism, and one of important circumstances that it operates is when an oncoming Tollmien-Schliching (T-S) wave is scattered by the mean-flow distortion caused by a local suction slot. This paper focuses on this issue by proposing an extension of the Local Scattering Theory. In this framework, a transmission coefficient, defined as the ratio of the T-S wave amplitude downstream of the scatter to that upstream, is introduced to characterize the effect of a local scatter on boundary-layer instability. The mathematical formulation is based on the triple-deck formalism, and the extension is realized by retaining the second-order terms in the upper-deck equations in order to accommodate the acoustic field. The extended theory describes not only the impact of a steady local suction on the hydrodynamic instability but also acoustic radiation. Numerical computations indicate that (1) a suction/injection slot would suppress/enhance the oncoming T-S waves, (2) the acoustic waves radiated by the scattering have similar directivities and (3) the intensity of the sound depends on the mass flux of the suction/inject, the frequency of the oncoming T-S wave and the Reynolds number.