## 6 Results
### Control of VIV with Symmetrical Lorentz Force
However, the shift of front stagnation point still exists and the shift amplitude of shear stress \(\mathcal{C}_{\tau}^{\theta}\) decreases with the increase of Lorentz force (interaction parameter \(N\)) on a cycle. With the Lorentz force applied, the pressure on the surface of vibrating cylinder is composed of three parts  
\[\mathcal{C}_{p}^{\theta}=P_{\theta}-P_{\infty}=\mathcal{C}_{pW}^{\theta}+ \mathcal{C}_{pF}^{\theta}+\mathcal{C}_{pV}^{\theta}\]  
where \(\mathcal{C}_{pW}^{\theta}\) is the pressure induced by the wall Lorentz force, \(\mathcal{C}_{pF}^{\theta}\) is the pressure induced by vortex shedding and \(\mathcal{C}_{pV}^{\theta}\) is the pressure affected by the inertial force. The effects of interaction parameter \(N\) on the distributions of \(\mathcal{C}_{pW}^{\theta}\) which has no relationship with flow field and cylinder vibration, are discussed firstly based on Figure 8 to examine the VIV control. The curve with positive values is symmetrical about the line \(\theta=180^{\circ}\). Therefore, the wall Lorentz force only causes the drag decrease, without the effect on the lift. Figure 6: Vorticity evolutions of VIV with different values of interaction parameter \(N\) for shear rate \(K=0.2\) where A, B, C and D represent the times at t = 0T/4, 1T/4, 2T/4 and 3T/4, respectively: (**a**) \(N=0\); (**b**) \(N=0.8\); and (**c**) \(N=1.3\). Figure 8: Distributions of pressure coefficient \(C_{pW}^{\theta}\) with Lorentz force controls. Figure 7: Distributions of shear stress \(C_{\tau}^{\theta}\) on the surface of cylinder with different values of interaction parameter \(N\) for shear rate \(K=0.2\): (**a**) Time A; (**b**) Time B; (**c**) Time C; and (**d**) Time D.  
The distributions of \(\mathcal{C}_{pF}^{\theta}\) induced by vortex shedding at different stages of one cycle with different values of interaction parameter \(N\) are depicted in Figure 9 where the shear rate \(K=0.2\), and A, B, C, and D correspond to those in Figure 6. As the stagnation point shifts to the upper side with the effect of shear flow, the pressure distribution shifts along the clockwise direction as well, which then leads to the increase and decrease of pressure on the upper and lower side, respectively. Obviously, these basic features of VIV in shear flow maintains for the small symmetrical Lorentz force. The pressure on the leeward decreases rapidly with the effect of Lorentz force, leading to the increase of drag and the decrease of the lift oscillating amplitude, in turn, as well as the decline of cylinder vibration. However, the curves are asymmetrical about \(\theta=180^{\circ}\) due to the shear flow even if Lorentz force is large enough. The inertial force \(\mathcal{C}_{pV}^{\theta}\), resulting from the acceleration of the cylinder, has influence on the pressure distributions as well. The distributions of \(\mathcal{C}_{pV}^{\theta}\) related with the inertial force for different values of interaction parameter \(N\) are depicted in Figure 10. When \(N=0\), the additional lift generated by the background vortex caused by the shear incoming flow leads to the acceleration \(\frac{d^{2}l(t)}{dt^{2}}\) at Time B being larger than that at Time D. Namely, the inertial force at Time B is larger than that at Time D. When \(N>0\), the inertial force decreases with the increase of the values of Lorentz force while the inertial force at Time B is still larger than that at Time D.  
Figure 6.0 Distribution of \(\mathcal{C}_{pF}^{\theta}\) on the surface of cylinder with different values of interaction parameter \(N\) for shear rate \(K=0.2\): (**a**) Time A; (**b**) Time B; (**c**) Time C; and (**d**) Time D.  
The lift-drag phase diagrams with different values of interaction parameter \(N\) for shear rate \(K=0.2\) are depicted in Figure 6.1 The lift-drag curve shrinks with the increase of the values of Lorentz force, which indicates the decrease of drag and lift vibration. Therefore, the amplitude of vibrating cylinder decreases.