## 6 Results
### Control of VIV with Symmetrical Lorentz Force  
In order to describe the differences between the VIV behavior of the cylinder in shear flow and that in uniform flow, the periodical variation of vortex-induced vibration for different shear rate \(K\) at the steady VIV with \(\mathrm{Re}=150\) in a shedding cycle are exhibited in Figure 4 by the shaded vorticity contours, where the red refers to the negative vortex, and blue the positive. The cross-hairs mark the equilibrium position of cylinder for shear rate \(K=0\). For convenience, the times at \(\mathrm{t}=0\mathrm{T}/4\), \(1\mathrm{T}/4\), \(2\mathrm{T}/4\) and \(3\mathrm{T}/4\) in all the \(K\) cases are denoted by A, B, C and D, respectively, where T denotes one period of the cylinder oscillation. The symmetrical flow field is broken due to the background vorticity, which is generated by shear flow that also causes the increase of upper vortex strength and the decrease of lower vortex strength. With the effect of shear flow, the separation point on the upper side of cylinder moves downstream due to the increase of fluid momentum on the upper side of cylinder while the separation point on the lower side of cylinder. The vortex street inclines toward the lower side and the inclination of vortex streets increase with the increasing shear rate \(K\), as does the distance of two vortex rows. The equilibrium position of vibration cylinder shifts to the lower side due to the shear flow. The variation of \(C_{dF}\sim C_{IF}\) phase diagram with shear rate \(K\) is shown in Figure 5. The curve moves down due to the shear flow, which means the lift points to the lower side of cylinder. Moreover, the absolute value of lift increases with the increase of shear rate, as do the amplitudes of drag and lift, which lead to the separation of points A and C.  
The periodic variations of vorticity of VIV with \(K=0.2\) under the Lorentz force control with different values of interaction parameter \(N\) are exhibited in Figure 6 by the shaded vorticity contours, where the red represents the negative vortex, and blue the positive. The initial position of the cylinder is marked with cross-hairs. For convenience, the times at t = 0T/4, 1T/4, 2T/4 and 3T/4 in all the \(N\) cases are denoted by A, B, C and D, respectively, where T denotes one period of the cylinder oscillation. The interaction parameters are \(N=0\), \(0.8\) and \(1.3\) for the three columns in Figure 6. From the figure, the symmetry of flow field is broken as the background vorticity, which is generated by shear flow,  
Figure 4: The periodical variation of flow field with different shear rate \(K\) where A, B, C and D represent the times at t = 0T/4, 1T/4, 2T/4 and 3T/4, respectively: (**a**) \(K=0\); (**b**) \(K=0.1\); and (**c**) \(K=0.2\). Figure 5: The variation of \(C_{dF}\sim C_{IF}\) phase diagram with shear rate \(K\). increases the strength of upper vortex and decreases the strength of lower vortex. The vortex street slants toward the lower side due to the background vorticity, which is shown in the instance \(N=0\). It is obvious that the vortex shedding and the cylinder vibration are suppressed under the influence of symmetrical Lorentz force and the control effects increase with the increase of interaction parameter \(N\). Despite the control, it is still observed that the front stagnation point moves to the upper side of cylinder and the wake shifts to the lower side, which generally occurs with the effect of shear incoming flow. The distributions of shear stress \(\mathcal{C}_{\tau}^{\theta}\) with different values of interaction parameter \(N\) for the different stages of one cycle are depicted in Figure 7, where the shear rate \(K=0.2\). The symmetrical curve is broken due to the different strength of shear layers on the upper and lower sides, which is generated by shear flow. The boundary layer fluid is hastened by Lorentz force, leading to the increase of shear stress. In addition, the strength of shear stress increases with the increase of interaction parameter \(N\).