Abstract
In recent years, the parabolized stability equations (PSE) approach has gained popularity for transition studies in boundary-layer flows. In this paper, the physical origin of ill-posedness of PSE in the primitive-variable form is studied by spectral analysis. In subsonic flows, a continuous spectrum representing the upstream-propagating acoustic waves is shown to be responsible for the ill-posedness of PSE. In supersonic flows, acoustic waves no longer contribute to ill-posedness despite the existence of a subsonic region within the boundary layer. In this case, ill-posedness is caused by some discrete modes with upstream influence. Semi-discretized models of PSE are similarly studied. It is shown that successful implementation of PSE to physical problems can only be achieved through manipulation of the spectra of the PSE operator. A generalized condition for the stable marching solution is devised.
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