On a class of unsteady, non-parallel, three-dimensional disturbances to boundary-layer flows

Steady, spatial, algebraically growing eigenfunctions are now known to occu r in several important classes of boundary-layer flow, including two-dimens ional hypersonic boundary layers and more recently in Blasius boundary laye rs subject to three-dimensional linearized disturbances, and in more genera l three-dimensional boundary layers. These spatial eigensolutions are parti cularly important and intriguing, given that they exist within the broad li mits of the classical steady boundary-layer approximation, and as such are independent of Reynolds number. In this paper we make the natural extension to these previous (stability) a nalyses by incorporating the effects of unsteadiness into the model for tre ating disturbances to a quite general class of similarity-type boundary-lay er flows. The flow disturbances are inherently non-parallel, but this effec t is properly incorporated into the analysis. A further motivation for this paper is that Duck et al. (1999, 2000) have s hown that by permitting a spanwise component of flow within a boundary laye r of the appropriate form (in particular, growing linearly with the spanwis e coordinate), it is found that new families of solutions exist-even the Bl asius boundary layer has a three-dimensional 'cousin'. Therefore a further aim of this paper is to assess the stability of the different solution bran ches, using the ideas introduced in this paper, to give some clues as to wh ich of the solutions may be encountered experimentally. Several numerical methods are presented for tackling various aspects of the problem. It is shown that when algebraically growing, steady eigensolution s exist, their effect remains important in the unsteady context. We show ho w even infinitesimal, unsteady flow perturbations can provoke extremely lar ge-amplitude flow responses, including in some cases truly unstable flow di sturbances which grow algebraically downstream without bound in the linear context. There are some interesting parallels suggested therefore regarding mechanisms perhaps linked to bypass transition in an important class of bo undary-layer flows.