可压缩瑞利泰勒不稳定流动的数值模拟研究

Numerical simulation of compressible Rayleigh-Taylor instability

罗腾飞

11849158

硕士

力学

王建春

2020-05-28

2020-07-08

哈尔滨工业大学

深圳

瑞利泰勒（RT） 不稳定性是自然界中一种常见的界面不稳定性现象，在很多自然现象和工程应用中广泛存在。百年来，瑞利泰勒不稳定性一直是流体力学中一个尚未完全解决的基本问题。随着计算能力的迅速发展，数值模拟已经成为研究瑞利泰勒不稳定性问题的主要方法之一。为研究可压缩性对瑞利泰勒不稳定性的影响，我们采用高阶中心紧致差分格式，数值模拟了不同等温马赫数和阿德伍德数的等温初始平衡状态的二维和三维单模瑞利泰勒不稳定性。本文的主要研究结果如下：我们研究了不同阿德伍德数（$A_t$）下可压缩性对二维单模瑞利泰勒不稳定性的影响。在等温平衡状态下，可压缩性会使得初始时刻的密度场存在分层，初始密度分层对瑞利泰勒不稳定性起稳定作用，而流动的膨胀压缩效应起到失稳作用。可压缩性对RT不稳定性的影响取决于这两种效应的竞争。在阿德伍德数较小的情况下，界面两侧的密度差较小，上下层的密度分布基本对称。此时，初始密度分层起主导作用，膨胀压缩效应影响不大。随着阿德伍德数的增加，初始密度分层的稳定效应减弱，而膨胀压缩效应引起的失稳效应逐渐变得显著。在中等阿德伍德数下，不同马赫数的气泡和尖钉的流动结构有很大的不同。可压缩性对气泡速度的影响在阿德伍德数等于0.5（$A_t=0.5$）时便已发生了逆转，压缩性促进了气泡势流阶段的速度。高阿德伍德数下，压缩性对气泡速度的影响很大。$A_t=0.9$时，势流增长阶段的气泡厚度近似为时间的二次函数，平均气泡加速度几乎与马赫数的平方成正比。我们研究了三维单模不可压缩瑞利泰勒不稳定性。在再加速阶段及此阶段之前，二维和三维模拟中的气泡以及尖钉发展基本一致，只是涡结构上存在差异。在再加速阶段之后，三维模拟与二维模拟出现了差异。在中等以及高雷诺数之下，低阿德伍德数的三维瑞利泰勒不稳定性会进入一个均速加速的循环阶段，这是由气泡或尖钉顶端不断形成新的气泡或尖钉结构导致的。直到尖钉或气泡结构变得很弱，这个阶段结束。在我们模拟的参数范围内，雷诺数越大，均速加速的循环次数越多。在中等阿德伍德数下，雷诺数增大依旧促进尖钉的发展，但对气泡有一定的抑制作用，这种影响主要表现在再加速及之后的阶段。我们对比了三维和二维瑞利泰勒不稳定性中，可压缩性对RT不稳定性发展的影响。在混合厚度增长方面，可压缩性对RT不稳定性的影响基本是一致的，但是在涡结构等细节方面，可压缩性对三维和二维瑞利泰勒不稳定性的影响会有差异。

Rayleigh-Taylor (RT) instability is a common phenomenon of interface instability in nature, and widely exists in many natural phenomena and engineering applications. In the past hundred years, Rayleigh-Taylor instability has been actively studied as a fundamental problem of fluid dynamics. With rapid development of computational methods, numerical simulation becomes one of the important methods to study this problem. In order to study the effect of compressibility on Rayleigh-Taylor instability, we numerically simulated the nonlinear evolution of two-dimensional and three-dimensional single-mode RT instability for isothermal background stratification with different isothermal Mach numbers and Atwood numbers $(A_t)$ using a high-order central compact finite difference scheme. The main research results of this thesis are as follows.The effect of compressibility on two-dimensional single-mode Rayleigh-Taylor instability at different Atwood number is studied. It is found that the initial density stratification caused by compressibility plays a stabilizing role, while the expansion-compression effect of flow plays a destabilizing role. The overall impact of compressibility on the evolutions of RT instability depends on competition between two effects.For the case of small Atwood number, the density difference between the two sides of the interface is small, and the density distribution of the upper and lower layers is nearly symmetrical. The initial density stratification plays a dominant role, and the expansion-compression effect has little influence. With the increase of Atwood number, the stabilization effect of initial density stratification decreases, and the instability caused by expansion-compression effect becomes more significant. The flow structures of bubbles and spikes are quite different at medium Atwood number. The effect of compressibility on the bubble velocity has been reversed at $A_t = 0.5 $. The bubble velocity in the potential flow stage increases with the increase of compressibility. The effect of compressibility on the bubble velocity is strong at large $A_t$. The bubble height is approximately a quadratic function of time at potential flow growth (PFG) stage. The average bubble acceleration is nearly proportional to the square of Mach number at $A_t=0.9$. We study three-dimensional single-mode incompressible Rayleigh-Taylor instability. Bubble velocity and spike velocity are basically the same in two-dimensional and three-dimensional simulations before and during the reacceleration stage. However, the vortex structures are different. After the reacceleration stage, there are some differences between 3D simulation and 2D simulation. At medium and high Reynolds numbers with small Atwood number, the three-dimensional RT instability enters a uniform-acceleration state after reacceleration state until the spike or bubble structure becomes very weak. This stage is caused by the continuous formation of new bubble or spike structure at the top of bubble or spike. At medium Atwood number, the increase of Reynolds number promotes the development of spike. But it has a certain inhibition effect on bubble during and after the reacceleration state.We compare the effects of compressibility on the development of three-dimensional and two-dimensional Rayleigh-Taylor instability. It is found that the influence of compressibility on RT instability is basically the same for bubble and spike velocities. However, with respect to vortex structures or other details, the effect of compressibility is different.

瑞利泰勒不稳定性, 可压缩性, 单模, 阿德伍德数

Rayleigh-Taylor instability compressible single-mode Atwood number

中文

联合培养

南方科技大学

罗腾飞. 可压缩瑞利泰勒不稳定流动的数值模拟研究[D]. 深圳. 哈尔滨工业大学,2020.