## 4 Reynolds Number Effect on VIV Characteristics and the Underlying Mechanism
### Time-Frequency Characteristics of Dynamic Aerodynamic Forces
#### 4.2.6 Contribution Coefficients of the Local Aerodynamic Forces to the Overall VIF  
The contribution of the local aerodynamic force at each measurement point on the main beam surface to the VIF is determined by the \(C_{\text{P,rms}}\) at each point and the \(C_{\text{cor}}\) between the local aerodynamic force and the VIF. The contribution coefficient \(C_{\text{aero,i}}\) of the local aerodynamic force at each measurement point to the VIF is defined as  
\[C_{\text{aero,i}}=C_{\text{P,rms,i}}C_{\text{cor,i}} \tag{9}\]  
where \(C_{\text{P,rms,i}}\) represents the RMS value of the wind pressure coefficients at the \(i\)-th measurement point and is obtained from the wind pressure time history analysis, \(C_{\text{cor,i}}\) is the correlation coefficient between the local aerodynamic force at the \(i\)-th point and the overall VIF, and \(C_{\text{aero,i}}\) is the contribution coefficient of the local aerodynamic force at the \(i\)-th point to the overall VIF. When \(C_{\text{aero,i}}\) is positive, the local aerodynamic force at the \(i\)-th measurement point promotes the VIF. When \(C_{\text{aero,i}}\) is negative, the local aerodynamic force suppresses the VIF. Figure 14 shows the distribution characteristics of the contribution coefficients on the surface of the main beam at different Reynolds numbers. As shown in Figure 14a, the contribution coefficients on the upper surface differ across Reynolds numbers. At \(Re\) = 3.2 \(\times\) 10\({}^{4}\) and 4.7 \(\times\) 10\({}^{4}\), the distribution characteristics of \(C_{\text{aero}}\) are similar in the upstream and downstream regions, showing a trend of first increasing and then decreasing, with relatively large values. For example, at \(Re\) = 3.2 \(\times\) 10\({}^{4}\), \(C_{\text{aero}}\) reaches maxima at \(D_{\text{d}}\) = 0.15 and \(D_{\text{d}}\) = 0.88, with values of 0.11 and 0.18, respectively. At \(Re\) = 7.1 \(\times\) 10\({}^{4}\) and 8.3 \(\times\) 10\({}^{4}\), the distribution characteristics of \(C_{\text{aero}}\) are similar, with no significant changes and relatively small values. For example, at \(Re\) = 7.1 \(\times\) 10\({}^{4}\), the maximum \(C_{\text{aero}}\) occurs at \(D_{\text{d}}\) = 0.88, with a value of 0.38. As the Reynolds number increases from 3.2 \(\times\) 10\({}^{4}\) to 8.3 \(\times\) 10\({}^{4}\), \(C_{\text{aero}}\) generally decreases, and the distribution characteristics undergo significant changes, indicating a noticeable Reynolds number effect on the contribution coefficients of local aerodynamic forces to the VIF on the upper surface. As shown in Figure 14b, at \(Re\) = 3.2 \(\times\) 10\({}^{4}\), the \(C_{\text{aero}}\) values of the local aerodynamic forces to the VIF on the lower surface of the main beam are predominantly positive. At \(Re\) = 4.7 \(\times\) 10\({}^{4}\), \(C_{\text{aero}}\) in the regions \(D_{\text{d}}\) = 0.03-0.13 and \(D_{\text{d}}\) = 0.92-0.99 is negative, which indicates that the local aerodynamic forces suppress the occurrence of VIV, whereas in other regions, \(C_{\text{aero}}\) is positive. At \(Re\) = 7.1 \(\times\) 10\({}^{4}\) and 8.3 \(\times\) 10\({}^{4}\), \(C_{\text{aero}}\) is nearly zero across the entire lower surface, indicating that there is no significant effect on VIV. There are clear differences in the distribution characteristics of \(C_{\text{aero}}\) at different Reynolds numbers. At \(Re\) = 3.2 \(\times\) 10\({}^{4}\), 4.7 \(\times\) 10\({}^{4}\), and 7.1 \(\times\) 10\({}^{4}\), \(C_{\text{aero}}\) first decreases  
Figure 14: Contribution coefficient distributions of the local aerodynamic force to the VIF at different Reynolds numbers. (**a**) Upper surface. (**b**) Lower surface. and then increases in the \(D_{\rm d}\) = 0.01-0.20 region and decreases in the \(D_{\rm d}\) = 0.79-0.99 region, which is distinct from the distribution at \(Re\) = 8.3 \(\times\) 10\({}^{4}\). As the Reynolds number increases, the changes in the contribution coefficients become more complex.