A local scattering approach for the effect of abrupt changes on boundary-layer instability and transition: a numerical method

Abstract

We investigate the influence of abrupt changes on boundary-layer instability and transition. Such changes include local porous porous wall, suction/injection and surface roughness, and they may modify the boundary conditions and/or the mean flow thereby affecting instability and transition. However, if the change occurs over a relatively short scale comparable with, or even shorter than, the characteristic wavelength of the instability, the conventional local linear stability theory (LST) becomes invalid. Instead, the effect is exerted through scattering with the abrupt change acting as a local scatter. As an instability mode, e.g. Tollmien-Schlichting (T-S) wave, propagates through the region of abrupt change, it is scattered to acquire a different amplitude. A local scattering approach (LSA) should be formulated instead, in which a transmission coefficient, defined as the ratio of the T-S wave amplitude after the scatter to that before the scatter, is introduced to characterize the effect on instability and transition. A numerical method specifically for solving the local scattering problem is developed. In this method, the equations governing the perturbation are discretized by a five-point finite difference scheme, but most importantly physically correct boundary conditions are imposed at the outlet of the computation domain, unlike most conventional methods where artificial ones are used. This allows the computation to be performed in a fairly small domain. We have applied this method to three cases: local porous wall, semi-infinite porous wall and local suction. We have also performed full direct numerical simulations (DNS) to verify the prediction by LSA for the case of a local porous wall. The LSA is found to be effective and accurate, whereas LST does not work for the conditions of interest. In addition, the calculations show that a porous wall, finite or semi-infinite, enhances the T-S wave while suction significantly stabilizes the T-S wave.

A stability analysis of the compressible boundary layer flow over indented surfaces

“…This allows to estimate transmission coefficients of A T ≈ 3 and 10 for M1 and M2 modes, respectively. These values are consistent with those reported [9,10]. It should be noted that ω i < 0 in our work, whereas ω is real (i.e.…”

“…It should be noted that ω i < 0 in our work, whereas ω is real (i.e. ω i = 0) in the local scattering approach [9,10]. However, since ω i is 370 usually rather small, the transmission coefficient extracted here is likely to approximate that in [9,10], and provides a measure of the destabilizing effect of the indentation.…”

“…ω i = 0) in the local scattering approach [9,10]. However, since ω i is 370 usually rather small, the transmission coefficient extracted here is likely to approximate that in [9,10], and provides a measure of the destabilizing effect of the indentation. If that were effectively the case, the NLF wing would be then operating suboptimally (increasing drag and fuel consumption).…”

“…An alternative approach, based in the definition of a Local Scattering Problem, has been proposed recently in [9,10,11]. This method leads to an eigenvalue problem whose solution bridges the spatial behaviour much before and after the scatter location (indentation, bump, different materials junction, .…”