*3.1. Disturbance Amplitude of Vortex Structures*

Figure 2 shows the evolution of the maximum disturbance amplitude (*A*) of the vortex structures as they originated from the P–N model and N–P model and propagated downstream in the boundary layer. The maximum disturbance amplitude (*A*) is defined as

$$A = \sqrt{|\mu\nu|\_{\text{max}}^2 + |\upsilon\nu|\_{\text{max}}^2 + |\upsilon\nu|\_{\text{max}}^2}. \tag{4}$$

At *t* = 15, when the wall local forced vibrations stop, the maximum amplitude of vortex structures derived by the P–N and N–P models was almost 0.1, which is obviously much greater than the initial forced vibration amplitude of 0.02. For *t* < 65, the maximum amplitude of vortex structures derived by the N–P model showed almost no difference compared to that of the P–N model; in this range, the maximum amplitude of the two models gradually rose from 0 to about 0.35. The maximum amplitude of the N–P model was almost unchanged from *t* = 65 to *t* = 80, and showed rapid growth for *t* > 80. On the other hand, however, the maximum amplitude of *v* using the N–P model obviously showed a rapid growth for *t* > 65.

It can also be seen from Figure 3 that the largest contribution to the maximum amplitude was that of |*u*|max, with *u* being the stream-wise velocity, while the second and third contributions were those of |*w*|max and |*v*|max, respectively. In addition, as |*u*|max > 0.6, both |*w*|max and |*v*|max increased gradually.

**Figure 3.** Maximum disturbance amplitude: (**a**) positive–negative (P–N) model; (**b**) negative–positive (N–P) model.

In order to study the reasons for the rapid growth of velocity disturbance in the stream-wise direction, the governing equation for the velocity and pressure disturbances of vortex structures in the *x*-direction was written in the following form:

$$\begin{array}{l} \frac{\partial \boldsymbol{u} \boldsymbol{\nu}}{\partial t} = \left( -\frac{\partial \boldsymbol{\nu}}{\partial \mathbf{x}} \right) + \left( -\boldsymbol{\nu} \boldsymbol{\nu} \frac{\partial \boldsymbol{u} \boldsymbol{\nu}}{\partial \mathbf{x}} - \boldsymbol{\nu} \boldsymbol{\Gamma} \frac{\partial \boldsymbol{u} \boldsymbol{\nu}}{\partial \mathbf{x}} - \boldsymbol{\nu} \boldsymbol{\nu} \frac{\partial \boldsymbol{L}\_0}{\partial \mathbf{x}} \right) + \left( -\boldsymbol{\nu} \boldsymbol{\nu} \frac{\partial \boldsymbol{u} \boldsymbol{\nu}}{\partial \boldsymbol{y}} - \boldsymbol{V}\_0 \frac{\partial \boldsymbol{u} \boldsymbol{\nu}}{\partial \boldsymbol{y}} - \boldsymbol{\nu} \boldsymbol{\nu} \frac{\partial \boldsymbol{L}\_0}{\partial \boldsymbol{y}} \right) \\ + \left( -\boldsymbol{\nu} \boldsymbol{\nu} \frac{\partial \boldsymbol{u} \boldsymbol{\nu}}{\partial \mathbf{z}} \right) + \frac{1}{\text{Re}} \left( \frac{\partial^2 \boldsymbol{u} \boldsymbol{\nu}}{\partial \mathbf{x}^2} + \frac{\partial^2 \boldsymbol{u} \boldsymbol{\nu}}{\partial \boldsymbol{y}^2} + \frac{\partial^2 \boldsymbol{u} \boldsymbol{\nu}}{\partial \boldsymbol{z}^2} \right) \end{array} \tag{5}$$

where *u*, *v*, *w* are the components of the velocity of the vortex structures, *p* is the pressure disturbance, and *u*<sup>0</sup> and *v*<sup>0</sup> are the stream-wise and vertical velocity components of the Blasius solution. The various terms appearing in Equation (5) can be grouped in the following form, where *b*, *c*, and *d* are all nonlinear terms, *c* = *c*1 + *c*2 + *c*3, and *e* is the viscous term:

$$\begin{cases} a = \iota\nu \times 0.1\\ b = \left(-\iota\nu \frac{\partial\mu}{\partial x} - \iota\eta\_0 \frac{\partial\mu}{\partial x} - \iota\nu \frac{\partial u\_0}{\partial x}\right) \\ c = \left(-\iota\nu \frac{\partial\mu}{\partial y} - \upsilon\_0 \frac{\partial\mu}{\partial y} - \upsilon\nu \frac{\partial u\_0}{\partial y}\right) \\ c1 = -\upsilon\nu \frac{\partial\mu}{\partial y}, c2 = -\upsilon\_0 \frac{\partial\mu}{\partial y}, c3 = -\upsilon\nu \frac{\partial u\_0}{\partial y} \\ d = \left(-\iota\nu \frac{\partial\mu}{\partial z}\right) \\ c = \frac{1}{\text{Re}}\left(\frac{\partial^2\mu}{\partial x^2} + \frac{\partial^2\mu}{\partial y^2} + \frac{\partial^2\mu}{\partial z^2}\right) \end{cases} \tag{6}$$

The evolution of grouped terms in time is shown in Figure 4. In particular, Figure 4a shows the evolution trends of *a*, *b*, *c*, *d*, and *e* at the position of maximum stream-wise velocity disturbance with *u* > 0 for the P–N model; the characteristics of *a* were similar to those shown in Figure 3a. Compared with the viscous term *e*, the nonlinear terms including *b*, *c*, and *d* contributed to the acceleration of the growth of *u*, while term *b* contributed a little. For *t* < 55, it can be seen that *c* was the most significant term for the disturbance amplitude growth.

**Figure 4.** *Cont*.

**Figure 4.** Evolution in time of the terms in the governing equations: (**a**) *u* > 0, P–N model; (**b**) *u* > 0, P–N model; (**c**) *u* < 0, P–N model; (**d**) *u* < 0, P–N model; (**e**) *u* > 0, N–P model; (**f**) *u* < 0, N–P model.

Figure 4b shows the evolution trends of *c*, *c*1, *c*2, and *c*3, whereby *c* was almost equal to *c*3, as *c*3 is the product of −*v* and the mean shear rate of boundary layer ∂*u*0/∂*y*. Since ∂*u*0/∂*y* > 0, vertical velocity disturbance at the position of maximum stream-wise velocity disturbance was of course negative. However, for *t* > 55, term *d* becomes important for the disturbance amplitude growth; the span-wise disturbance velocity *w* and the derivative term of the stream-wise disturbance velocity ∂*u*/∂*z* increase significantly.

Figure 4c shows the evolution trends of *a*, *b*, *c*, *d*, and *e* at the position of maximum stream-wise disturbance velocity with *u* < 0 for the P–N model. |*u*|max was less than that in Figure 3a, which means that the strength of the stream-wise high-speed disturbance velocity was greater than that of the stream-wise low-speed disturbance velocity. Nonlinear terms including *b*, *c*, and *d* can promote further reduction of the stream-wise disturbance velocity, while viscous term *e* can prevent it. It can be seen that *c* was the most significant term for the disturbance amplitude growth, while term *d* seemed unimportant for |*u*|max with *u* < 0. Figure 4d shows the evolution trends of *c*, *c*1, *c*2, and *c*3, whereby *c*3 was almost equal to *c*, and, due to *c*3 < 0, vertical disturbance velocity at the position of maximum stream-wise disturbance velocity was of course positive.

Figure 4e shows the evolution trends of *a*, *b*, *c*, *d*, and *e* at the position of maximum stream-wise disturbance velocity with *u* > 0 for the N–P model. For *t* < 50, it can be seen that *c* was the most significant term for the disturbance amplitude growth. Although |*e*| in Figure 4e was greater than that in Figure 4a, term *d* in Figure 4e rose very fast for *t* > 50, which was the main reason for the growth of the vortex structures in the N–P model. It can be inferred that terms *w* and ∂*u*/∂*z* of the vortex structures of the N–P model were relatively greater than those of the P–N model.

Compared to Figure 4c, the stream-wise low-speed disturbance velocity strength of the N–P model shown in Figure 4f was greater than that of the P–N model. Term *c* was the key factor for the growth of low-speed disturbance velocity. In contrast, terms *c* and *d* were the only important factors for the growth of high-speed disturbance velocity.
