Kelvin–Helmholtz instability (KHI) occurs at the interface of two fluids, in which the heavier fluid flows at the bottom. In the present numerical work, the KHI was analyzed by solving the modification of two-dimensional (2-D) incompressible Navier–Stokes equations. To capture the interface, the Cahn–Hilliard equation was implemented into the Navier–Stokes equations. The phenomena around the KHI of two and three-component fluids were investigated numerically by using radial basis function (RBF) combined with the domain decomposition method (DDM) in a primitive variable formulation. Here DDM is able to solve the large scale problem. On the other hand the calculation accuracy decreases with the increase of the number of the subdomains. For the above reason, in the present works, the domain was partitioned into 15 x 15 subdomains in order to reduce the decrease in accuracy due to the domain division. Next, fractional step method was used to solve the modification of the Navier–Stokes equations.

The numerical results indicate that the procedure used in the present work can easily handle the KHI problem under the variations of the interface thickness, density ratios, and initial velocity differences. Moreover, the interface evolutions obtained from the present method agree well with those of the finite difference method. The effects of the interface thickness, density ratio and magnitude of velocity difference on the KHI were also investigated. The decrease of the interface thickness produces a non-smooth concentration profile, and the increase of the interface thickness produces too much surface diffusion. It was found also that the increase of the density ratio reduces the growth of KHI. The interface rolls up are strongly affected by the initial horizontal velocity difference. Finally, the present study also shows that the RBF method is a reliable method to solve the KHI on the complex domains.

开尔文-亥姆霍兹不稳定性（KHI）发生在两种流体的界面处，其中较重的流体在底部流动。在当前的数值工作中，通过求解二维（2-D）不可压缩的Navier-Stokes方程的修正来分析KHI。为了捕获界面，将Cahn–Hilliard方程实现为Navier–Stokes方程。通过在原始变量公式中使用径向基函数（RBF）与域分解方法（DDM）结合，对二元和三元流体KHI周围的现象进行了数值研究。DDM可以解决大规模问题。另一方面，计算精度随着子域数量的增加而降低。由于上述原因，在本作品中，该域被划分为15 x 15个子域，以减少由于域划分而导致的准确性下降。接下来，分数步法被用来解决Navier–Stokes方程的修改。

数值结果表明，在界面厚度，密度比和初始速度差的变化下，本工作中使用的程序可以轻松解决KHI问题。而且，从本方法获得的界面演化与有限差分法的界面演化很好地吻合。还研究了界面厚度，密度比和速度差大小对KHI的影响。界面厚度的减小产生了非光滑的浓度分布，而界面厚度的增大产生了太多的表面扩散。还发现密度比的增加降低了KHI的生长。界面卷起受到初始水平速度差的强烈影响。最后，