Boundary-layer receptivity due to distributed surface imperfections of a deterministic or random nature

Abstract

Acoustic receptivity of a Blasius boundary layer in the presence of distributed, two-dimensional surface irregularities is investigated analytically. It is shown that, out of the entire spatial spectrum of the surface irregularities, only a narrow band of Fourier components can lead to an efficient conversion of the acoustic input at any given frequency to an unstable eigenmode of the boundary-layer flow. The location and the width of this most receptive band of wave numbers is fixed by the requirement of a relative detuning of O(R sup−3/8inf1.b. ) or less with respect to the instability wave number at the lower-branch station for the frequency under consideration. Surface imperfections in the form of discrete-mode waviness in this range of wave numbers then lead to initial instability amplitudes which are larger by a factor of O(R sup3/8inf1.b. ) than the amplitudes resulting from a single, isolated roughness element of streamwise extent comparable with the instability wavelength at the lower-branch location. In contrast, random irregularities which are spatially homogeneous in nature, and also possess a continuous spectrum in the streamwise direction, lead to instability amplitudes that are intermediate to those caused by the periodic and isolated irregularities, respectively, being, in fact, of the same order as the geometrical mean of the amplitudes in the latter two cases. A physical explanation for these asymptotic scalings is given, in addition to providing an analytical expression for the expected value of the instability amplitude for an ensemble of statistically irregular surfaces with random phase distributions. The duality between the localized and distributed receptivity analyses is also discussed.