**4. Conclusions**

A direct numerical simulation of the nonlinear evolution of vortex structures in a laminar boundary layer induced by different local micro-vibrations on the wall was carried out. The characteristics of vortex structures were studied in detail, and the conclusions are summarized below.

With the evolution of time, the amplitude of vortex structures undergoes a process of linear increase and nonlinear rapid increase, and the amplitude of the stream-wise velocity disturbance is the relative maximum. The growth rate of vortex structures in the P–N model is slightly less than that in the N–P model, and the growth of the P–N model vortex structure is most affected by the vertical velocity and boundary layer shear. The growth of vortex structures in the N–P model is greatly influenced by the derivative term of the stream-wise disturbance velocity and span-wise disturbance velocity. In the core region of vortex structures, the inflection point of the velocity profile exists, which leads to the expansion of the range of neutral curves, as well as the increased growth rate of T–S waves. The change in mean flow profile further induces or promotes the growth or formation of vortex structures. These results will help to understand and further study how coherent structures in turbulence come into being and evolve.

**Author Contributions:** W.C. designed the scheme and wrote the program, organized paper; Z.J. helped to analyze the data; Q.Z. contributed some figures.

**Funding:** This work was supported by the National Key R&D Program of China (2018YFC0810506) and the Key R&D Program of Zhenjiang (SH2017049, BK20161472).

**Conflicts of Interest:** The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
