Linear and nonlinear instability waves in spatially developing two-phase mixing layers

Abstract

Two-phase laminar mixing layers are susceptible to shear-flow and interfacial instabilities, which originate from infinitesimal disturbances. Linear stability theory has successfully described the early stages of instability. In particular, parallel-flow linear analyses have demonstrated the presence of mode competition, where the dominant unstable mode can vary between internal and interfacial modes, depending on the flow parameters. However, the dynamics of two-phase mixing layers can be sensitive to additional factors, such as the spreading of the mean flow. In addition, beyond the early linear stage, the amplitude of the instability waves becomes finite and nonlinear effects become appreciable. As a result, an accurate description of the evolution of the mixing layer must account for nonlinear interactions including the generation of higher harmonics of the instability waves and the modification of the mean flow. These effects are investigated herein using the framework of the nonlinear parabolized stability equations. The formulation includes nonparallel effects, nonlinear modal interactions, a coupled mean flow correction, and finite amplitude deformation of the interface. Mode competition between liquid and interfacial modes is investigated. We demonstrate that nonparallelism and streamwise evolution of the flow can significantly alter the predictions of locally parallel, linear stability analyses. This is followed by a discussion on nonlinear interactions of two- and three-dimensional instability waves. It is shown that nonlinear effects can serve dual purposes. On one hand, they can be a limiting mechanism for the growth of instability waves. On the other hand, they can destabilize high frequency, linearly stable modes, and thus lead to the generation of smaller scale features in the flow.