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.

Interfaces and inhomogeneous turbulence

“…In an intermediate convective-diffusive regime, there is a balance between the accumulative sheltering effect of the shear and the influence of viscosity. Beyond the initial stage of vortical penetration into the shear, the evolution of interfacial disturbances is studied using the nonlinear parabolized stability equations (PSE) developed by Cheung & Zaki [20]. Unlike linear stability analysis, this PSE formulation accounts for non-parallel effects in developing boundary layers, and also finite deformation of the interface, energy exchange between instability waves and the distorion of the base flow owing to the non-linear disturbance field.…”

The effect of a low-viscosity near-wall film on bypass transition in boundary layers

“…Two wall-film-to-free-stream viscosity ratios were investigated, µ BT = 0.5 and 0.2, in addition to a reference single-fluid case. While this choice of viscosity ratios is motivated by previous linear studies, it should be noted that a lower viscosity film is not always guaranteed to delay transition: For example, the film can al-ter the receptivity of the boundary layer to free-stream disturbances (Zaki & Saha 2009); the viscosity mismatch at the two-fluid interface can lead to new instability mechanisms (Yih 1967); and predictions based on linear theory become inaccurate when finite interface displacements are taken into consideration (Cheung & Zaki 2010). Therefore, only direct numerical simulations can provide a complete account of the influence of a wall film on the full transition process.…”

A nonlinear PSE method for two-fluid shear flows with complex interfacial topology

“…While the earlier study of Cheung and Zaki [7] also used the PSE method to capture interfacial deformations, the current work includes capabilities which were not possible in the previous study. Since the earlier approach relies on a coordinate transformation to track the position of the interface, it is limited to flows where the interface remains single-valued.…”

“…The extension of the NPSE approach to handle two-fluid shear layers was first addressed by Cheung and Zaki [7]. Their work demonstrated the importance of accounting for nonlinear interactions and the mean-flow distortion when examining the development of interfacial disturbance waves.…”

“…However, because of the assumptions made in linear theory, the results of these studies are generally applicable only during the early stages of the development of instability waves, when the amplitude of the interfacial deformation remains minimal. Nonlinear effects, such as interactions between instability waves, and the distortion of the mean flow due to the disturbances, are typically ignored, despite their importance to the waves' evolution [7].…”