{'fraction_non_alphanumeric': 0.01002004008016032, 'fraction_numerical': 0.0, 'mean_word_length': 5.2677824267782425, '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}}
Application of the method of integral relations to laminar boundary layers in three dimensions

The method of integral relations, which has been used successfully for the solution of a series of boundary layer problems in two dimensions, is extended to apply to three dimensional compressible boundary layer flows with and without separation. The essential steps in reducing the equations of motion to quasi-incompressible form are discussed and an outline of the method of integral relations as applied in three dimensions is presented. Analytical representations of the streamwise shear stress function θ and the cross flow velocity component V are given in terms of the streamwise velocity component U. Different functions are employed for attached and separated flows reflecting the different physical restrictions for each case. The system of hyperbolic partial differential equations obtained from applications of the method are solved by the method of lines. The method is first applied to two model incompressible problems with known solutions, namely parabolic flow over a flat plate and flow over a yawed cylinder. It is then applied to flow past a plate-cylinder combination previously solved by a finite difference method. Finally, it is applied in its compressible formulation to calculate supersonic laminar boundary layer flow over a swept back wedge. This flow field was recently investigated experimentally and the observed data are compared with results calculated by the method.