Short-wave analysis of 3D and 2D instabilities in a driven cavity

The short-wave asymptotic approximation of inviscid instabilities proposed by Bayly (\textit{Phys. Fluids} \textbf{31}, 1988) and Lifschitz \& Hameiri (\textit{Phys. Fluids A} \textbf{3}, 1991) is here applied to the dominant (three-dimensional) instability of two-dimensional flow in either an open or a closed driven cavity, and compared to the structural sensitivity obtained by direct-adjoint computation. The comparison shows that the structural sensitivity of the eigenmode is indeed localized around the critical streamline identified by short-wave asymptotics, and that the latter provides a reasonably good expression of even the first unstable eigenvalue at critical Reynolds number. Curiously enough, the same approximation appears also to apply with success to the two-dimensional instability of the same flow, despite the absence of a large spanwise wavenumber to be used as an expansion parameter. The theoretical justification of this extension, and the importance of phase quantization along the trajectory, will be discussed.