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This study focuses on validating a two-phase flow solver based on the modified Simple Coupled Level Set and Volume of Fluid method (Uchihashi et al. (2023)) through viscous Kelvin-Helmholtz instability simulations. Our numerical simulation results are compared with the ones given by Funada and Joseph (2001) to provide reliable predictions of interface behavior under the influence of viscosity. The primary goal is to accurately assess the solver's ability to replicate theoretical analysis of interface behaviors under various conditions. First, the wave between two fluids of identical density is calculated. In addition, the effect of surface tension is investigated. By comparing growth rates, numerical simulations obtain well-agreements with the analytical results on the effect of the fluid viscosity, the wave number, and the surface tension. Finally, fluid density is changed to an air-water system. When relative velocity U is smaller than the criteria of relative velocity U_c given by analytical solutions, the wave is not broken. However, waves are splashed into droplets in the condition of U>U_c. This result agrees with the analysis by Funada and Joseph (2001). These findings provide a robust framework for applying the solver to more complex two-phase flow problems, supporting advancements in numerical simulations of fluid interfaces.
The Kelvin-Helmholtz instability arises at the interface between two fluid layers with a tangential velocity discontinuity, significantly impacting the safety of systems in industrial applications. Despite extensive studies, many theoretical analyses neglect viscosity and frictional drag due to the complexity of solving the dispersion equation. However, including these effects is crucial, as viscosity alters the growth rate of the instability, while frictional drag influences energy dissipation and momentum transfer. In this study, these effects are incorporated into both finite and infinite flows. The assumption of a potential flow combined with a dissipation method is employed to derive the dispersion equation, providing a more tractable approach than direct calculation methods. The results indicate that, in the case of infinite flows, the frictional drag suppresses the growth of the instability for long waves (small wavenumber k) but enhances it for short waves (large k). In contrast, the surface tension dominates, while the frictional drag only slightly affects the growth rate in the case of finite flows bounded by solid walls.