Competitions between Rayleigh-Taylor instability and Kelvin-Helmholtz instability with continuous density and velocity profiles

In this research, competitions between Rayleigh-Taylor instability (RTI) and Kelvin-Helmholtz instability (KHI) in two-dimensional incompressible fluids within a linear growth regime are investigated analytically. Normalized linear growth rate formulas for both the RTI, suitable for arbitrary density ratio with continuous density profile, and the KHI, suitable for arbitrary density ratio with continuous density and velocity profiles, are obtained. The linear growth rates of pure RTI (γ_RT), pure KHI (γ_{KH), and combined RTI and KHI (γtotal) are investigated, respectively. In the pure RTI, it is found that the effect of the finite thickness of the density transition layer (Lρ}) reduces the linear growth of the RTI (stabilizes the RTI). In the pure KHI, it is found that conversely, the effect of the finite thickness of the density transition layer increases the linear growth of the KHI (destabilizes the KHI). It is found that the effect of the finite thickness of the density transition layer decreases the "effective" or "local" Atwood number (A ) for both the RTI and the KHI. However, based on the properties of γ_{RT∝√A and γKH∝√1-A² , the effect of the finite thickness of the density transition layer therefore has a completely opposite role on the RTI and the KHI noted above. In addition, it is found that the effect of the finite thickness of the velocity shear layer (Lu}) stabilizes the KHI, and for the most cases, the combined effects of the finite thickness of the density transition layer and the velocity shear layer (L_ρ=L_u) also stabilize the KHI. Regarding the combined RTI and KHI, it is found that there is a competition between the RTI and the KHI because of the completely opposite effect of the finite thickness of the density transition layer on these two kinds of instability. It is found that the competitions between the RTI and the KHI depend, respectively, on the Froude number, the density ratio of the light fluid to the heavy one, and the finite thicknesses of the density transition layer and the velocity shear layer. Furthermore, for the fixed Froude number, the linear growth rate ratio of the RTI to the KHI decreases with both the density ratio and the finite thickness of the density transition layer, but increases with the finite thickness of the velocity shear layer and the combined finite thicknesses of the density transition layer and the velocity shear layer (L_ρ=L_u). In summary, our analytical results show that the effect of the finite thickness of the density transition layer stabilizes the RTI and the overall combined effects of the finite thickness of the density transition layer and the velocity shear layer (L_ρ=L_u) also stabilize the KHI. Thus, it should be included in applications where the transition layer effect plays an important role, such as the formation of large-scale structures (jets) in high energy density physics and astrophysics and turbulent mixing.