On the application of en-methods to three-dimensional boundary-layer flows

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Extension of the e n -method from two-dimensional to three-dimensional boundary-layer flows has not been straightforward. Confusion has centred on whether to use temporal or spatial stability theories, conversion between the two approaches, and the choice of integration path. The aim of this study is to clarify the confusion about the direction and magnitude of maximum growth in convectively unstable three-dimensional non-parallel boundary layers. To this end, the time-asymptotic response of the boundary layer to an impulsive point excitation is considered. Since all frequencies and all wavenumbers are excited by an impulsive point source, the most amplified component of the response is equivalent to the result of maximizing the growth over arbitrary choices of harmonic point excitation; the standard e n -approach. The impulse response is calculated using a spatial steepest-descent method, which is distinct from the earlier Cebeci–Stewartson method. It is necessary to allow both time and spanwise distance to become complex during integration, but with the constraint that both are real at the end point. This method has been applied to the two-dimensional Blasius boundary layer, for which validation of the method is more straightforward, and also to a three-dimensional Falkner–Skan–Cooke (with non-zero pressure gradient and sweep) boundary layer. Dimensional frequencies and spanwise wavenumbers of propagating components are kept constant (although not necessarily real), as is physically relevant to steady flows with spatial inhomogeneity in the chordwise direction only. With this method a spatial approach is taken without having to make a priori choices about the value of disturbance frequency or wavenumber. Further, purely by choosing a downstream observation point, it is possible to find the maximum-amplitude component directly without having to calculate the entire impulse response (or wave packet). If the flow is susceptible to more than one convective instability mode, provided the modes are separated in the frequency–wavenumber space, separate n-factors can be calculated for each mode. Wave-packet propagation in the Ekman layer (a strictly parallel three-dimensional boundary layer) is also discussed to draw comparisons between the conditions for maximum growth in parallel and non-parallel boundary layers.