## 4 Reynolds Number Effect on VIV Characteristics and the Underlying Mechanism
### Time-Frequency Characteristics of Dynamic Aerodynamic Forces
#### 4.2.5 Correlation Coefficient Between the Local Aerodynamic Forces and the Total VIF  
The correlation coefficient \(C_{\text{cor,i}}\) between the local aerodynamic force at each measurement point on the main beam surface and the total VIF reflects both their frequency and phase characteristics [38] and is defined as  
\[C_{\text{cor,i}}=\frac{Cov(F_{\text{aero}}(t),p_{\text{i}}(t))}{D(F_{\text{aero }}(t))D(p_{\text{i}}(t))} \tag{7}\]  
where \(C_{\text{cor,i}}\) represents the correlation coefficient between the pressure at the \(i\)-th measurement point and the VIF. \(F_{\text{aero}}(t)\) represents the VIF time history, and \(p_{\text{i}}(t)\) is the pressure time history at the \(i\)-th measurement point. The VIF is obtained through pressure integra  
Figure 12: Distribution characteristics of the wind pressure coefficient at the dominant frequency at different Reynolds numbers. (**a**) Upper surface. (**b**) Lower surface. tion. \(D(F_{\rm aero}(t))\) and \(D(p_{\rm t}(t))\) denote the standard deviation of \(F_{\rm aero}(t)\) and \(p_{\rm t}(t)\), respectively. \(Cov(F_{\rm aero}(t),p_{\rm t}(t))\) denotes the covariance, which can be defined as  
\[Cov(F_{\rm aero}(t),p_{\rm i}(t))=E[(F_{\rm aero}(t)-E(F_{\rm aero}(t)))(p_{\rm i }(t)-E(p_{\rm i}(t)))] \tag{8}\]  
where \(E\) is the mathematical expectation. Figure 13 shows the distribution characteristics of the correlation coefficients between the local aerodynamic forces at each measurement point and the total VIF at different Reynolds numbers. On the basis of Figure 13a, in the range \(D_{\rm d}\) = 0.01-0.64 on the upper surface of the main beam, \(C_{\rm cor}\) exhibits large fluctuations and higher values when \(Re\) = 3.2 \(\times\) 10\({}^{4}\) and 4.7 \(\times\) 10\({}^{4}\). However, when \(Re\) = 7.1 \(\times\) 10\({}^{4}\) and 8.3 \(\times\) 10\({}^{4}\), the \(C_{\rm cor}\) values along the surface change more steadily and are smaller. In the range \(D_{\rm d}\) = 0.64-0.99, the \(C_{\rm cor}\) values for all four Reynolds numbers show an increasing trend followed by a decrease, with the maximum point being at the same location as the maximum point of the upper surface. There are significant differences in the distribution characteristics of the correlation coefficients on the upper surface of the main beam at different Reynolds numbers. Overall, \(C_{\rm cor}\) decreases with increasing Reynolds numbers. For \(Re\) = 3.2 \(\times\) 10\({}^{4}\) and 7.1 \(\times\) 10\({}^{4}\), the correlation is positive over the entire upper surface, indicating that local aerodynamic forces on the upper surface contribute to VIV. Negative correlations are observed for the \(D_{\rm d}\) = 0.59-0.62 region when \(Re\) = 4.7 \(\times\) 10\({}^{4}\) and the \(D_{\rm d}\) = 0.13-0.16 and \(D_{\rm d}\) = 0.95-0.99 regions when \(Re\) = 8.3 \(\times\) 10\({}^{4}\), meaning that local aerodynamic forces in these regions suppress VIV. According to Figure 13b, the distribution characteristics of \(C_{\rm cor}\) on the lower surface of the main beam are similar for \(Re\) = 3.2 \(\times\) 10\({}^{4}\), 4.7 \(\times\) 10\({}^{4}\), and 7.1 \(\times\) 10\({}^{4}\). In the \(D_{\rm d}\) = 0.01-0.20 region, \(C_{\rm cor}\) first decreases but then increases. Afterward, \(C_{\rm cor}\) remains relatively stable along the lower surface, decreasing in the \(D_{\rm d}\) = 0.79-0.99 region, with \(C_{\rm cor}\) being negative in parts of the lower slanted belly plate. This finding indicates that in the first three Reynolds number tests, local aerodynamic forces in parts of the lower slanted belly plate suppress VIV. The \(C_{\rm cor}\) values differ significantly across the first three Reynolds number tests. As the Reynolds number increases from 3.2 \(\times\) 10\({}^{4}\) to 7.1 \(\times\) 10\({}^{4}\), the \(C_{\rm cor}\) values first decrease and then increase in the \(D_{\rm d}\) = 0.01-0.20 region.