{'fraction_non_alphanumeric': 0.02263374485596708, 'fraction_numerical': 0.00205761316872428, 'mean_word_length': 5.486666666666666, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Singularity formation in the strongly nonlinear wide-vortex/Tollmien–Schlichting-wave interaction equations

A combined numerical/analytical study of the wide-vortex/wave interaction equations, describing boundary-layer instability, is presented. Depending on the obliqueness β of the wave input, different solution properties are obtained. For β = 1, oscillations in the wave amplitude lead to the evolution of a strongly three-dimensional mean flow, while for β = 2 the interaction is characterized by the development of a singularity in the wave pressure amplitude. This latter behaviour is modelled using an approximate form for the mean flow skin friction and the resulting amplitude equation is analysed using a combination of numerical and asymptotic techniques. A simple method is described for determining the singularity location for a given spanwise wavenumber, and the asymptotic behaviour of the pressure amplitude as the singularity is approached is deduced.