## 3 Theoretical Formulation and Numerical Modeling
### Computational Mesh  
As shown in Figure 2, in order to reduce the calculation errors near the wall, the whole computational zone is divided into two parts: the dynamic mesh zone and the accompanying moving zone. The energy transfer between the two zones is achieved by the interpolation method. The accompanying moving zone is discretized with the structured grid, which is of a size small enough to meet the requirement of \(y^{+}\approx 1\). The boundary layer moves together with the instantaneously started cylinder. In order to adapt the complex boundary conditions, the dynamic mesh zone is discretized with unstructured grid. The local remeshing method is used to get over the problems of poor grid quality. It is done by deleting the nodes that are near the boundary, or inserting new nodes to form new grids. Different initial grid numbers and different time steps are selected for the independent verification calculation. The lift and drag coefficients of the smooth cylinder with a started speed of 1.1 m/s, calculated with different numbers of initial grids and different time steps, are shown in Tables 1 and 2. The relative errors in the brackets are calculated by using the results from the previous set as reference values. As can be seen from Table 1, when the number of initial grids is 60711, the relative errors of the drag coefficient and the lift coefficient are 0.52% and 1.16%, respectively. It is seen from Table 2 that the errors become relatively stable as the iteration time step reduces to 0.0001 s. In consideration of CPU time consumption, the number of grids is chosen to be 60,711, and iterative calculations are performed in time steps of 0.0001 s. The lift and drag coefficients are defined as follows:  
\[\Delta C_{D}=\frac{2f_{D}}{\rho V^{2}D} \tag{12}\]  
\[C_{L}=\frac{2f_{L}}{\rho V^{2}D} \tag{13}\]  
of the wake. \(B\) is the intersection of the recirculation zone and the \(x\) axis, and the reflow velocity at this point is equal to zero. \(S\) is the separation point of the boundary layer. \(L\) represents the length of the wake, and \(a\) and \(b\) stand for the location of the center of the vortex. Figure 4 demonstrates the comparison of the wake flow between the present solutions and the results of Reference [43] for \(\text{Re}=5000\). After a period of time, we can see that the wake flow is separated from the cylinder, and a pair of symmetric vortices is formed, which we called the main eddies. As time goes on, a pair of secondary eddies are formed near the separation point. The special phenomenon was also observed by [PERSON] and [PERSON] [43]. Figure 5 shows the velocity distribution at different moments on the axis for \(\text{Re}=550\). Figure 6 depicts the length of the wake and the coordinate of the center of the eddy. Moreover, we can see that the present results agree well with those in Reference [43]. This indicated that the dynamic mesh method which is used to investigate the unsteady viscous incompressible flow in this paper is accurate and reliable.  
Figure 4: The comparison of the wake flow between the present solutions and the results of Reference [43] for \(\text{Re}=5000\).  
Figure 3: Schematic diagram describing the parameters of the wake.