## 3 Mathematical Model and Numerical Calculation
### Computational Domain and Grid Generation  
The calculation area was a two-dimensional rectangular area with a size of \((30D+d)\times 16D\), as shown in Figure 2. The entire domain included five boundaries: Inflow, outflow, top, bottom, and cylinders. The inflow velocity was considered to be uniform and constant. The distance from the cylinder to the outlet boundary was sufficiently long, and, thus, the zero gradient boundary condition had negligible influence on the near-wake region of the cylinder flow. According to the hypothesis, the outflow was fully developed, and the derivative of velocity along the flow direction was zero. Therefore, the exit boundary was the pressure outlet, the flow velocity normal gradient was zero, and the pressure was zero. The top and bottom were fixed wall boundaries. When the cylinder was fixed, and it was considered that the cylinder wall was a non-slip wall boundary condition. To match the dynamic mesh technique of topological change, a moving wall boundary condition was applied for the cylinder surface when the cylinder was elastically supported in this work.  
In order to describe the interaction force when the fluid flows around the cylinder with roughness strips for the cylinder with diameter \(D\) and length \(L\), the following formula was used to calculate the lift and drag coefficients:  
\[C_{d}\ =\ \frac{F_{fluid,x}}{\frac{1}{2}\rho U^{2}DL} \tag{5}\]  
Figure 2: Computational domain \((30D+d)\times 16D\).  
\[C_{l}\ =\ \frac{F_{fluid,y}}{\frac{1}{2}\rho U^{2}DL} \tag{6}\]  
The wake shedding frequency of a stationary flow around a circular cylinder is described by the dimensionless Strouhal number (\(St\)). The definition of \(St\) is as follows:  
\[St\ =\ \frac{f_{vs}D}{U} \tag{7}\]  
where \(f_{vs}\) is the frequency of vortex shedding.  
Two-dimensional structured grids were used to calculate the static flow around two tandem cylinders. In order to capture the main characteristics of the flow field, it was necessary to encrypt the local mesh in the near-wall region with a large gradient of parameter variation. The verification of grid independence achieved good results, and, finally, the computation was carried out at medium density. The FIV of two staggered cylinders was calculated. In order to reduce the calculation error caused by grid distortion, the dynamic mesh technique of topological change was employed, and the calculation area near each cylinder with roughness strips was divided into \(2D\times 2D\) square mesh sub-blocks. The cylinder was located in the centre of the sub-blocks, and the mesh density was increased. When FIV occurred, the mesh in the sub block moved along the cylinder wall without any mesh deformation. The GGI (Generalized Grid Interface) interface was used to connect the mesh sub-blocks with the upstream and downstream computational regions, and the topological mesh deformation technique was used to simulate the FIV of two cylinders. In addition, in order to converge within a reasonable time and obtain numerical solutions that meet the accuracy requirements, three different kinds of meshes (coarse, medium and fine) were selected to verify the grid independence. The calculation results in Table 2 show that the difference between the lift coefficient (\(C_{l}\)) and the drag coefficient (\(C_{d}\)) of the cylinder was very small with the change of the grid density, and the grid independence verification achieved good results. Therefore, the flow-induced vibration of two cylinders with roughness strips was simulated by using medium meshes. The computational area of the medium density grid is shown in Figure 3.