*3.3. Mean Flow Profile and Neutral Curve*

The black dashed lines in Figure 6a,b represent the velocity *u*<sup>0</sup> of Blasius basic flow at *x* = 64 for the P–N model and at *x* = 56 for the N–P model. The red solid lines in Figure 6a,b represent the mean value of the stream-wise disturbance velocity within a local region adding to Blasius basic flow; the local region was 52 < *x* < 76, −6 < *z* < 6 for the P–N model and 44 < *x* < 68, −6 < *z* < 6 for the N–P model. As can be seen, the mean velocity profile deformations in the area of the vortex structures could easily be discerned. On one hand, because of the presence of stream-wise disturbance velocity, the shear stress close to the wall increased. On the other hand, under the conditions of different initial disturbances at the walls in the laminar boundary flow, the mean velocity profiles all became plump after a period of evolution. Although vortex structures were only in their initial stages, the velocity profiles had a tendency to evolve into the turbulent mean velocity profiles. Furthermore, the mean velocity profiles were plumper at *y* < 0.3, which indicates that the stream-wise high-speed disturbance velocity mainly distributed near the wall region.

The green solid lines in Figure 6a,b represent the mean value of the stream-wise disturbance velocity within a local scope adding to Blasius basic flow; the local region was 52 < *x* < 76, −3 < *z* < 3 for the P–N model and 44 < *x* < 68, −3 < *z* < 3 for the N–P model. The black solid lines in Figure 6a,b represent the mean value of the stream-wise disturbance velocity within a local region adding to Blasius basic flow; the local region was 52 < *x* < 76, −1.5 < *z* < 1.5 for the P–N model and 44 < *x* < 68, −1.5 < *z* < 1.5 for the N–P model. It can be seen from Figure 6a,b that the mean velocity profiles had inflection points in the local scope of −1.5 < *z* < 1.5; the flow stability in this scope would, thus, be altered. According to the theory of linear stability, the solution of a three-dimensional T–S wave is obtained by solving the eigenvalue problem of Orr–Sommerfeld equations.

$$
\overrightarrow{\boldsymbol{\mu}}\prime = a\_0 \boxed{\boldsymbol{\mu}}(\boldsymbol{y}) \left[ e^{i(\boldsymbol{ax} + \boldsymbol{\beta}\boldsymbol{z} - \boldsymbol{\omega}\prime)} + \text{c.c.} \right. \tag{7}
$$

where *c* · *c* is the conjugate complex, α = α*<sup>r</sup>* + *i*α*<sup>i</sup>* is the stream-wise wave number, α*<sup>r</sup>* is the real part, α*<sup>i</sup>* is the imaginary part, β is the span-wise wave number, ω is the frequency, *a*<sup>0</sup> is the initial amplitude, → *u*(*y*) = ' *u*(*y*), *v*(*y*), *w*(*y*) ( is the eigenvalue velocity, and α*<sup>i</sup>* = 0 represents the points on the neutral curve. For each span-wise wave number β, the corresponding unstable T–S wave with maximum local growth rate −α*<sup>i</sup>* exists.

**Figure 6.** Mean stream-wise velocity and flow stability: (**a**) *t* = 114; (**b**) *t* = 95; (**c**) neutral curve; (**d**) growth rate.

T–S waves inside the neutral curve are unstable. The black solid line in Figure 6c is the neutral curve at *x* = 64, and the black dotted line is the neutral curve at *x* = 56 in the Blasius boundary layer. The frequency of the neutral curve at *x* = 64 was less than that at *x* = 56. The red solid line in Figure 6c is the neutral curve based on the mean value of the stream-wise disturbance velocity added to the Blasius basic flow within the local region 52 < *x* < 76, −1.5 < *z* < 1.5 of the P–N model. The range of the neutral curve (red solid line) was obviously larger than that of the black solid line. The red dashed line in Figure 6c is the neutral curve based on the mean value of the stream-wise disturbance velocity added to the Blasius basic flow within the local region 44 < *x* < 68, −1.5 < *z* < 1.5 of the N–P model. The range of the neutral curve (red dashed line) was obviously larger than that of the black dashed line. Due to the existence of the vortex structures, the neutral curve range of the N–P mode was relatively the largest.

The black solid and dashed lines in Figure 6d correspond to the maximum local growth rates −α*<sup>i</sup>* at *x* = 64 and *x* = 56 in the Blasius boundary layer. The red solid line in Figure 6d is the maximum local growth rate −α*<sup>i</sup>* based on the mean value of the stream-wise disturbance velocity added to the Blasius basic flow within the local region 52 < *x* < 76, −1.5 < *z* < 1.5 of the P–N model. The amplitude of the maximum local growth rate −α*<sup>i</sup>* of the red solid line was obviously larger than that of the black solid line. The red dashed line in Figure 6d is the maximum local growth rate −α*<sup>i</sup>* based on the mean value of the stream-wise disturbance velocity added to the Blasius basic flow within the local region 44 < *x* < 68, −1.5 < *z* < 1.5 of the N–P model. Also, the amplitude of the maximum local growth rate −α*<sup>i</sup>* of the red dashed line was larger than that of the black dashed line. The amplitude of the maximum local

growth rate −α*<sup>i</sup>* of the N–P model was relatively the largest. Growth rates of the T–S waves and profile characteristics of the mean flow had the ability to promote each other. The self-sustaining structures in the logarithmic region of the boundary agree with the results obtained by Yang [11].
