## 6 Results
### Control of VIV with Asymmetrical Lorentz Force  
For a shear flow past a cylinder, the shift of front stagnation point and additional lift, pointing to lower side for \(K>0\), are induced under the effect of background vorticity, even if symmetrical Lorentz forces are applied. Therefore, the equilibrium position of cylinder controlled by symmetrical Lorentz forces departs from its initial position, as mentioned in Figure 12. However, the lift \(C_{IL}\) generated by asymmetrical Lorentz forces can be used to counteract the additional lift, so that the total lift \(C_{l}=0\) and then the cylinder shifts to the initial position.  
The evolution of the cylinder displacement before and after the application of asymmetric Lorentz force (upper surface \(N=3\), lower surface \(N=2\)) in a shear flow with \(K=0.2\) is shown in Figure 16. When the asymmetric Lorentz force is initiated at time \(t_{2}=650\) when the VIV is well-developed, the lift \(C_{IL}\) pointing to the upper side is generated, which can counteract the additional lift to suppress the vibration of cylinder and lead to the equilibrium position of cylinder shifting back to the upper side. Finally, the cylinder is steady on the initial position.  
The vorticity of VIV with the asymmetric Lorentz force is shown in Figure 17. The wake suppressed completely by the Lorentz force inclines to the lower side with the influence of shear flow.  
The distribution of shear stress \(\mathcal{C}_{\mathrm{ T}}^{\theta}\) on the surface of cylinder with the asymmetric Lorentz force is depicted in Figure 18. Due to the higher flow speed on the upper surface, the shear stress on the upper side prevails over that on the lower side obviously.  
Figure 16: Evolution of the cylinder displacement before and after application of Lorentz force (upper surface \(N=3\), lower surface \(N=2\)) in shear flow for \(K=0.2\).  
Figure 17: The vorticity of VIV with the Lorentz force (upper surface \(N=3\), lower surface \(N=2\)) for \(K=0.2\).  
The distribution of pressure coefficient \(\mathcal{C}^{\theta}_{pW}\) with the asymmetric Lorentz force is shown in Figure 19. The value of \(\mathcal{C}^{\theta}_{pW}\) is positive and the maximum is achieved at \(\theta=180^{\circ}\). Meanwhile, the \(\mathcal{C}^{\theta}_{pW}\) on the lower side is larger than that on the upper side. Thus, the wall Lorentz force results in the amplification of lift with the direction pointing to the upper side and the drag reduction (thrust generated).  
The distribution of \(\mathcal{C}^{\theta}_{pF}\) on the surface of cylinder with the asymmetric Lorentz force is depicted in Figure 20. With the effect of asymmetric Lorentz force, the curve moves down. The pressure on the upper surface prevails over that on the lower surface, while the decrease of pressure on the leeward side is sharper than that on the windward side. Therefore, the lift decreases and the drag increases with the effect of field Lorentz force, which is opposite with the effect of wall Lorentz force. However, the total effect is dominated by the wall Lorentz force.  
Figure 19: The distributions of pressure coefficient \(\mathcal{C}^{\theta}_{pW}\) with the asymmetric Lorentz force (upper surface \(N=3\), lower surface \(N=2\)) for shear rate \(K=0.2\).  
Figure 18: The distribution of shear stress \(\mathcal{C}^{\theta}_{\gamma}\) on the cylinder surface with the asymmetric Lorentz force (upper surface \(N=3\), lower surface \(N=2\)) for shear rate \(K=0.2\).