## 5 Conclusions and Summary  
Numerical studies on vortex-induced transverse vibration of an isolated cylinder and two tandemly arranged cylinders near a plane wall with \(Re=150\), \(m^{*}=8/\pi\), \(\zeta=0\), \(e/D=0.1,\ 1,\ 2,\ L/D=1.5\), and \(U_{r}=2-9\) have been conducted. The self-developed DFD method is adopted to avoid the application of a remeshing technique when the gap between the cylinder and the wall is very small. In addition, a bounce-back model is adopted when the cylinder collides with the wall. The research has important significance for the design of near-wall cylindrical structures in engineering practices. The main conclusions of this study are summarized as follows:  
(1) For the one-isolated-cylinder case, collision with the wall occurs at a larger reduced velocity and a higher time-averaged lift coefficient \(C_{L,\ mean}\) is acquired for a small \(e/D\). For each of the three \(e/D\), the predominant frequency of cylinder vibration \(f_{d}\) coincides with that of the hydrodynamic force \(f_{cl}\) at the most \(U_{r}\) and deviation occurs at large \(U_{r}\). The minimal and maximal gaps, \(e_{min}\) and \(e_{max}\), increase with increasing \(e/D\) and a larger \(e_{max}\) always accompanies a smaller \(e_{min}\) in the lock-in region. The positive vortices shed from the gap-side surface are suppressed and their intensity gradually becomes stronger as \(e/D\) increases;  
(2) For the two-inline-cylinders case, the response amplitude of the front cylinder is larger than that of the rear one when \(e/D\) is small. As \(e/D\) increases, significant vibrations are excited, especially for the rear cylinder, leading to the amplitude of the rear cylinder being larger than that of the front one when \(f_{d}/f_{n}\) approach unity. The amplitude of the rear cylinder increases with \(e/D\). For a small \(e/D\), the vortices shed from the gap-side surface are also suppressed by the plane wall and only a single and elongated freestream-side vortex exists at small \(U_{r}\). However, 2P mode can be seen at the lock-in region when \(e/D\) increases, even though the gap-side vortices damped out away from the cylinder;  
(3) Compared to the one-isolated-cylinder case, the vibrations of the cylinders are suppressed and the collision with the wall only occurs for the rear cylinder when \(e/D\) is small. In addition, the vibration amplitudes of the front and rear cylinders are both larger and the lock-in region is obviously broadened.  
In this paper, the VIV of rigid cylinders near a plane wall is investigated and the above conclusions are obtained without considering the cylinders' deformation, which will be considered in our future work. Additionally, the Reynolds number in all the computational cases is 150, which is in the 2D laminar flow region, and the future extension is towards three-dimensional high-Reynolds-number flow simulations.  
Conceptualization, H.Q.; methodology, T.P. and Y.Z.; software, T.P. and C.Z.; validation, Y.Z. and H.J.; formal analysis, T.P.; investigation, T.P. and H.J.; resources, T.P. and W.Z.; data curation, T.P. and Z.S.; writing--original draft preparation, H.Q. and T.P.; writing--review and editing, H.Q. and Z.S.; visualization, Z.S. and H.J.; supervision, H.Q.; project administration, H.Q.; funding acquisition, H.Q. and T.P. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (No. 12102154), the Foundation of State Key Laboratory of Aerodynamics of China Aerodynamics Research and Development Center (No.RAL202302-1), and the Nature Science Foundation of Yangzhou (No. YZ2024164).  
Not applicable.  
Data Availability Statement: Data is contained within the article.  
The authors declare no conflicts of interest.