Comparison of temporal and spatial direct numerical simulation of compressible boundary-layer transition

Two models are used in the direct numerical simulation of transitional boundary-layer flows, the spatial model and the temporal model. The spatial approach is the closest realization of a transition experiment but is significantly more expensive than the temporal approach, which employs a local parallel flow assumption in its formulation. Because of the parallel flow assumption, a temporal approach cannot take into account the nonparallel effects of boundary-layer flows properly. The consequence of this defect has never been addressed adequately. In this study, the results from various temporal approaches are compared extensively to those obtained by our two newly developed spatial direct numerical simulation codes for nonlinear subharmonic transition and oblique wave breakdown at Mach numbers 1.6 and 4.5. A recently developed new temporal approach that can include some nonparallel effects in the computation is also introduced. The use of our new formulation is essential in obtaining quantitative agreement with the spatial approach at high Mach numbers, whereas the standard temporal approach does not give satisfactory results.