*3.2. Stripe of Vortex Structures*

Figure 5a provides the velocity vector of the vortex structures for the P–N model at *t* = 15, when the local micro-vibration at the wall stops. Because the disturbance at the wall was symmetrical along the plane *z* = 0, only the results of 0 < *z* < 6 are shown. It can be seen that the initial form of the vortex structures was complex, whereby the vortex structures mainly concentrated near *z* = 0, *x* = 15, and showed complex three-dimensional (3D) vortices in space. The core region of disturbance velocity was at the position of *x* > 15 due to the influence of basic flow in the laminar boundary layer.

**Figure 5.** *Cont*.

**Figure 5.** Vectors and contours of vortex structures: (**a**) velocity vectors at *t* = 15, P–N model; (**b**) velocity vectors at *t* = 15, N–P model; (**c**) stream-wise high- and low-speed disturbance velocity, contour increment of 0.03, P–N model; (**d**) stream-wise high- and low-speed disturbance velocity, contour increment of 0.03, N–P model.

Figure 5b provides the velocity vector of the vortex structures at *t* = 15 of the N–P model. The contrast between Figure 4a,b shows the opposite nature of the vortices; however, in fact, there were differences in the amplitudes of the velocities. For example, at the position of *x* = 18.2, *y* = 0.309, *z* = 0, the velocities of the P–N model and N–P model are compared in Table 1.

**Table 1.** Velocity comparison. P–N—positive–negative; N–P—negative–positive.


Apart from the different spatial evolution in time, the stream-wise high-speed and low-speed disturbance velocities of different vortex structures exhibited different properties. The vortex structure amplitude of the P–N model increased to about 1.0 at *t* = 114, and that of the N–P model increased to about 1.0 at *t* = 95. Figure 5c provides the stream-wise disturbance velocity distributions of high-speed and low-speed fluids of the P–N model at *t* = 15, 55, and 114, at *y* = 0.1, 0.3, 0.5, 0.7, 0.9, and 1.1. Because the vortex disturbances were symmetrically distributed along plane *z* = 0, only the contour between planes of *z* = 0 and *z* = 6 is displayed; the contour increment is 0.03, and red lines represent the disturbance velocity of the high-speed fluid, *u* > 0. Green lines represent the disturbance velocity of the low-speed fluid, *u* < 0. At *t* = 15, high-speed and low-speed fluids mainly distributed in a local area near *x* =16–20, *y* = 0.3, *z* = 0, and the intensity of the low-speed fluid was greater than that of the high-speed fluid, with the former being below the latter. With the evolution of the vortex structure, at *t* = 55, the intensity and area of both the high-speed and low-speed fluids increased, but it seems that the area of the high-speed fluid was larger than that of the low-speed fluid. The low-speed fluid concentrated near the plane of *z* = 0, and the high-speed fluid existed in relatively large areas. At *t* = 144, the vortex structure was further inclined due to the shear action of the basic flow in the laminar boundary layer. At the same time, it can be seen that the high-speed fluid mainly concentrated near the wall, which may have led to an increase in the friction shear force at the wall region. The area occupied by the high-speed fluid was larger than that occupied by the low-speed fluid. The spatial range of the high-speed fluid increased more than that of the low-speed fluid in all directions.

Figure 5d provides the stream-wise disturbance velocity distributions of the high-speed and low-speed fluids of the N–P model at *t* = 15, 55, and 114, at *y* = 0.1, 0.3, 0.5, 0.7, 0.9, and 1.1. Compared with Figure 4c, at *t* = 15, although the high-speed and low-speed fluids mainly distributed in a small area near *x* = 16–20, *y* = 0.3, *z* = 0, the intensity of the high-speed flow was slightly greater than that of the low-speed flow, while the high-speed fluid was lower than the low-speed fluid. The low-speed fluid main distributed at the location near *y* = 0.5 and *x* = 20, and there was no high-speed fluid at the same position in Figure 5c. At *t* = 55, the high-speed fluid exceeded the low-speed fluid, and the elongation range of the high-speed fluid in the stream-wise direction was larger than that in Figure 4c. At *t* = 95, the amplitude of vortex structures rose rapidly to nearly 1.0, and the scales of the high-speed fluid and low-speed fluid were smaller than those in Figure 5c in the stream-wise direction; however, there were mainly high-speed stripes near the wall, and the characteristics of the high-speed fluid occupied a larger area than the low-speed fluid, similar to Figure 5c. Results of the vortex structure stripe agree with the results of Wall et al. [12].

The rough center positions of the high-speed and low-speed fluids in the plane of *y* = 0.3 of the P–N and N–P models are listed in Table 2. The forward speed in the stream-wise direction of the P–N model was approximately 0.45*U*∞, and that of the N–P model was approximately 0.525*U*∞.


**Table 2.** Center position evolution with time.
