Instability of isolated boundary-layer streaks to spatially-developing travelling waves

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The spatial stability properties of an isolated low-speed streak embedded in a Blasius boundary layer are determined; the streak is generated by steady localised injection, whilst the disturbance is generated by a linear harmonic localised injection. Isolated streaks driven by short-scale spanwise forcing have comparable growth rates of both sinuous and varicose instabilities. These features have been discussed previously via DNS methods, but the novel aspect here is a treatment via a rationally-parabolised version of the Navier–Stokes equations in the high Reynolds number limit. The parabolic formulation allows for a more efficient and Reynolds number independent computation of fully three-dimensional non-parallel streaks their stability. We compute the non-parallel development of a perturbation by downstream marching from a time-harmonic disturbance generator (in tandem with the streamwise streak development), before comparing these results with bi-global eigenvalue calculations. The stability properties are well captured by a weakly non-parallel eigenvalue formulation of the boundary-region equations, provided that one is not in the vicinity of the disturbance generator. Further downstream, or at higher excitation frequencies we directly recover the long-wave limit of a two-dimensional Rayleigh stability problem. Even further downstream (or at even higher excitation frequencies, which is mathematically equivalent) we must return to a (two-dimensional) Rayleigh formulation as the streamwise wavelength of the disturbance becomes comparable to the boundary-layer thickness. For streaks that are comparable with those obtained experimentally our spatial growth rates and eigenmode shapes compare favourably with the experimentally-determined values. For the range of streaks considered, we demonstrate the sinuous mode retains the higher growth rate in the viscous stability problem. The experimentally-observed change over to a dominant varicose mode nearer the disturbance site is shown to be true only for frequencies that provoke an inviscid response.