Effect Of Surface Tension On The Growth Of Rayleigh-Taylor Instability

Fluid instability is a common phenomen in fluids. It will help to understand fluids in depth to investigate the fluid instability. Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities are the most important interfacial instabilities among the classical instabilities. It exists in many fields, such as inertial confinement fusion, plasma, astrophysics and so forth. The evolution of RT instability can be divided into three stages. In the first stage the amplitude of disturbance is very small and perturbations grow exponentially in time and can be described by linearized equations. The second stage is called the nonlinear stage. In the nonlinear evolution stage the fluid interface forms a finger shape structure which consists of bubbles (the lighter fluid part penetrating into the heavier fluid) and spikes (the heavier fluid part penetrating into the lighter fluid). In the third stage the turbulent takes place.There are still debates about the description of bubble growth in the second stage. There are mainly two theoretical models based on the single mode. One is the velocity-potential model which established by Layzer. The model was early used to describe the evolution of a single-mode vacuum bubble. Later on, a new complex velocity potential was introduced by Zhang and Goncharov et al. and the Layzer’s model is generalized to arbitrary density ratio. In2003, Sohn poposed a simple velocity potential model, which is based on the complex velocity potential introduced by Zhang and Goncharov et al.. The predictions from the simple velocity potential model were found to be in good agreement with numerical results. Another nonlinear theoretical model is Zufiria’s model. The model was originally limited to the fluids with an infinite density ratio, and then Sohn generalized the model to the unstable system for arbitrary density ratio. Sohn compared the predictions based on Zufiria’s model with that based on the Layzer’s model and found that the predictions for the late time velocity of bubble based on the two models agree with each other qualitatively.Recently, using Layzer’s model, Sohn investigated the influences of surface tension and viscosity to the RT instability. But only the influence of surface tension or viscosity has not been investigated up to now. This will distinguish the influences of surface and viscosity to the growth of instability. In2011, Cao and Guo et al. studied the effect of viscosity on the growth of RT instability in Zufiria’s model. They found that the visvisity depresses the RT bubble growth rate.In the present thesis we have investigated the effect of surface tension on the incompressible RT instability in the frameworks of Zufiria’s model and Lyzers’s model based on the simple velocity potential. We derived the analytical expressions for bubble velocity and curvature of the bubble. It is shown that the surface tension reduces the asymptotic bubble velocity, consistent with the results based on Layzer’s model with the complex velocity potential. However, Layzer’s model with a simple velocity potential gives a terminal bubble velocity, which is smaller than that based on Layzer’s model with the complex velocity potential but is larger than that based on Zufiria’s model over the whole range of Bond and Atwood numbers. When the Atwood number A=1, the velocity of bubble from the simple velocity potential agrees with the value obtained from the complex velocity potential. Finally, the analytical results have been compared with the numerical simulation. It has been shown that the prediction for the late time bubble velocity from the Layzer’s simple potential flow is better than that from Zufiria’s model when Atwood number A=1.