Influence of centrifugal forces on the development of wave packets in boundary layers with a uniformly valid model

Abstract

The behavior of three-dimensional wave packets in the boundary layer on curved surfaces is analyzed in this study based on a modification of the triple-deck theory referred to as the “criss-cross” interaction model. The equations of the criss-cross interaction describe a particular type of boundary layer instability mode that possesses underlying properties of both the Tollmien–Schlichting waves and Taylor–Görtler vortices. Previous analysis of the criss-cross interaction regime suggests a possibility for upstream propagation of perturbations in the boundary layer and possible absolute instability of the flow. However, these results cannot be considered as conclusive because the initial-value problem for the criss-cross interaction equations is ill-posed. In a recent work [Turkyilmazoglu M, Ruban AI. A uniformly valid well-posed asymptotic approach to the inviscid–viscous interaction theory. Stud Appl Math 2002;108:161–85] a regularized non-asymptotic model to describe criss-cross interaction has been proposed. Whereas in the original version of the theory, perturbations have an unbounded growth rate as the longitudinal wave number ∣ k∣ → ∞, in the new model of [Turkyilmazoglu M, Ruban AI. A uniformly valid well-posed asymptotic approach to the inviscid–viscous interaction theory. Stud Appl Math 2002;108:161–85], as physically expected the amplification rate remains bounded for both spatially growing and temporally growing waves. A Fourier transform method is used in the present study to solve the linearized equations for the flow over concave roughness and humps and it is found that disturbances develop and are convected downstream as wave packets. The behavior of the wave packets is consistent with convective instability, and the upstream influence is no longer present at large times.