On the absolute instability of the attachment-line and swept-Hiemenz boundary layers

Recently it has been shown that flow over a rotating-disk is absolutely unstable. In this paper we investigate the absolute instability of the related swept-Hiemenz and attachment-line boundary-layer flows. The linearized stability equations are obtained and the eigenvalues of the dispersion relation are found by solving the full stability equations in Fourier-transform space using a spectral method. Unlike previous work on this problem, no quasi-parallel approximation has been made and all the terms appearing in the stability equations have been retained. We were unable to locate branch points satisfying the Briggs-Bers criterion for the attachment-line boundary layer suggesting that this flow is only convectively unstable. However, for the swept-Hiemenz boundary layer our results show that this flow becomes absolutely unstable (in the chordwise direction), starting from the leading-edge extending up to a chordwise position of approximately 310 for some particular spanwise Reynolds numbers. It is found that the retention of all the terms in the full system of equations leads to results which are more unstable, in terms of absolute instability, than the Orr-Sommerfeld system studied by others. The techniques used here apply equally to other non-parallel two- and three-dimensional boundary-layer flows.