Wavetrains in diverging mixing layers

Abstract

It is generally accepted that a linear stability theory, together with a slowly diverging base flow, can describe many of the characteristics of coherent structures in free shears flow. In this dissertation we model these two-dimensional instability waves as they travel in a slightly inhomogeneous steady and viscous unstable base flow. These unstable and inviscid wave packets are analysed using linear stability theory. The analysis is performed by separating the physical flow in two parts. In the first part, the instability waves are evolving in a parallel mixing layer and their solution serves of initial conditions for the second part of the flow. The parallel flow analysis leads to the receptivity of the flow to both pulse-type and periodic excitations. This part of the study is done by solving the initial-value problem completely and studying its long-time behaviour, which is a wave packet. We then repeat the same analysis with some modifications and arrive at the receptivity of the flow for sinusoidal excitations. We find that a shear layer is very receptive to high-frequency disturbances that are generated near the center line of the layer. The second part of the solution is concerned with the evolution of the wave packets on longer space-time scales which are associated with non-parallel effects arising from the spreading of the mixing layer. The solution in this part of the physical flow is handled by extending Whitham's kinematic wave theory, and the ray equations for instability waves are derived for physical and propagation spaces using a WKB J expansion. Our high-frequency ansatz also leads us to the derivation of a very simple complex amplitude equation. While the rays obtained represent characteristics in the complex plane along which the complex frequency of our disturbances is conserved (steady base flow), the amplitude equation expresses the conservation of the volume integrals of a complex wave action density subject to a certain flux and a source term. The amplitude equation was rendered easily tractable due to a transformation of our dependent variables and their practical projections on the cross and propagation spaces. Different methods, (steepest descent, ray-tracing, and fully numerical solution) are used to solve the ray equations, and comparisons are made among them. The results presented are obtained for the piecewise linear profile of Rayleigh and the general tanh profile. The very good agreement among all the methods of solution reveals the validity of the method of characteristics in the complex plane, (ie complex rays). Finally we perform some calculations for spatially varying shear layers and and study their implications in the development of spatial instability modes. We discover that when starting with a convectively unstable base velocity profile it is possilble to interrupt the development of spatial instability modes by allowing the base velocity profile to vary slowly and become absolutely unstable. However the reverse is not true. That is to say that in a base flow that is initially absolutely unstable, one does not observe spatial modes, even after the base flow is permitted to assume slowly a convectively unstable profile.