## 6 Results
### Control of VIV with Symmetrical Lorentz Force
Moreover, the lift-drag curve (a Figure 8 shape at \(N=0\)) is switched to a teardrop shape by increasing the interaction parameter \(N\). For \(N=0\), the drag at Time C is larger than that at Time A due to the fact that the strength of upper vortex is larger than that of lower vortex with the effect of shear flow (shown in Figure 6) and then the pressure \(C_{pF}^{\theta}\) on the cylinder leeward at Time C is smaller than that at Time A (shown in Figure 9), which leads to Point A and Point C separating in the lift-drag phase diagram. The flow field on the cylinder leeward is gradually dominated by the upper vortex with the increase of Lorentz force. At Time C, the shear layer of upper side is accelerated under the effect of pressure side which leads to the increase of the strength of upper vortex, then the decrease of pressure \(C_{pF}^{\theta}\) on the cylinder leeward and subsequently the increase of drag. The distance between Point A and Point C of lift-drag curve is increased, which leads to the lift-drag phase diagram switching to a teardrop shape gradually. Furthermore, the curve shifts from the left to the right, which implicates the increase of drag. The additional lift still exists under the effect of shear flow. The evolution of the cylinder displacement from still to steady vibration, then to a new steady state with symmetric Lorentz force (\(N=0.8\)) for \(K=0.2\) is shown in Figure 6.2 The release of transverse  
Figure 6.3 Lift–drag phase diagrams with different values of interaction parameter \(N\) for shear rate \(K=0.2\). Figure 6.4 Distributions of \(C_{pV}^{\theta}\) on cylinder surface with \(K=0.2\) for different values of interaction parameter \(N\). confinement occurs at \(t=446\), which leads to the displacement of cylinder increasing gradually under the effect of lift. Moreover, the equilibrium position which separates from the point \(l/a=0\) and shifts to the lower side. When \(t=640\), the steady vibration is reached. When the symmetric Lorentz force is initiated at time \(t_{2}=650\), a well-developed VIV is performed, and the cylinder displacement will decrease over time. However, the equilibrium position shifts because shear flow still exists. The variation of vorticity of VIV before and after the application of symmetric Lorentz force (\(N=0.8\)) for \(K=0.2\) is shown in Figure 13 where time \(B_{i}\) in this figure corresponds to that in Figure 12 while the cylinder arrives at the lowermost position. The cylinder starts to vibrate with the effect of lift as the confinement of cylinder is released. With the energy transferring from the fluid to the cylinder, there is an increase in the cylinder oscillations. The corresponding vortex patterns are described as \(B_{1}\sim B_{4}\). The cylinder vibrates steadily as the total cylinder energy develops into an equilibrium state. The corresponding vortex pattern is described as \(B_{5}\). When the symmetric Lorentz force is attached on the cylinder at time \(t_{2}=650\), the cylinder vibration and the fluid separation are suppressed at some extent (\(B_{6}\sim B_{8}\)) while the wake vortex is inclined to the lower side. Figure 6.5 Evolution of the cylinder displacement with the symmetric Lorentz force (\(N=0.8\)) for \(K=0.2\). Figure 6.6 _Cont._  
Time evolutions of lift-drag phase diagram for VIV in a developing and then suppressed process with \(N=0.8\) and \(K=0.2\) are shown in Figure 6.7 The closed curve \(A_{1}B_{1}C_{1}D_{1}A_{1}\), which represents the stationary cylinder, is turned right at \(180^{\circ}\) as the cylinder starts to vibrate. Then, Point A and Point C are separated until VIV is well-established. However, Point A and Point C do not coincide any longer, where the phase diagram \(C_{dF}\sim C_{IF}\) is depicted by the closed-curve \(A_{5}B_{5}C_{5}D_{5}A_{5}\). The drag \(C_{dF}\) that is generated by the field Lorentz force increases with the application of symmetric Lorentz, though the total drag \(C_{d}\) decreases with the wall Lorentz force effect [41]. The phase diagram therefore shifts to the right significantly.