*2.1. Governing Equations*

The governing equations are the non-dimensional incompressible Navier–Stokes equations and the continuity equation.

$$\frac{\partial \overrightarrow{\boldsymbol{\mu}} \prime}{\partial t} + (\overrightarrow{\boldsymbol{\mu}}\_0 \cdot \nabla) \overrightarrow{\boldsymbol{\mu}} \prime + (\overrightarrow{\boldsymbol{\mu}} \prime \cdot \nabla) \overrightarrow{\boldsymbol{\mu}}\_0 + (\overrightarrow{\boldsymbol{\mu}} \prime \cdot \nabla) \overrightarrow{\boldsymbol{\mu}} \prime = -\nabla p \prime + \frac{1}{Re} \nabla^2 \overrightarrow{\boldsymbol{\mu}} \prime \tag{1}$$

$$
\nabla \cdot \overrightarrow{\boldsymbol{\mu} \prime} = 0,\tag{2}
$$

where <sup>∇</sup> is the gradient operator, <sup>∇</sup><sup>2</sup> is the Laplacian, *Re* is the Reynolds number, <sup>→</sup> *u*<sup>0</sup> = (*u*0, *v*0) and *p*<sup>0</sup> are the velocity and the pressure of Blasius solutions, and <sup>→</sup> *u* = (*u*, *v*, *w*) and *p* are the disturbance velocity and pressure of vortex structures. In this paper, *Re* = *U*<sup>∞</sup> · δ/υ = 2000, where *U*<sup>∞</sup> is the free stream velocity of Blasius basic flow, δ is the upstream thickness of the boundary layer in Blasius basic flow, and υ is the kinematics viscosity. The velocity of Blasius basic flow <sup>→</sup> *u*<sup>0</sup> = (*u*0, *v*0) can be obtained through the Falkner–Skan equation, *f* + *f f* = 0.

#### *2.2. Numerical Methods*

The direct numerical simulations of Equations (1) and (2) were implemented as follows: a third-order mixed explicit-implicit scheme was employed for the time discretization, and the space discretization combined the higher-accuracy compact finite differences of non-uniform meshes with the Fourier spectral expansion. The nonlinear terms were approximated by a fifth-order upwind compact difference scheme for non-uniform meshes. The treat of pressure terms was approximated by a third-order center finite difference scheme with five points. The viscous terms were approximated by a fifth-order compact difference scheme for non-uniform meshes. Detailed numerical methods and verifications of simulation accuracy were given in Reference [11].

The computational time step was 0.01. Owing to the limitation of computational capacity, the range of directions *x*, *y*, *z* was limited to 84, 4, and 12, respectively. The number of Fourier modes was 16, which implies that the number of collocation points in the *z*-direction was 32, and the numbers of mesh points in the *x-* and *y*-directions were 240 and 150, respectively. Uniform and non-uniform meshes were applied in the *x-* and *y*-directions, respectively. The node coordinate *y*(*k*) in the *y*-direction can be expressed by Equation (3) below, which is used to refine the mesh in the near-wall region.

 $y(\mathbf{k}) = 4[1 - \tanh\frac{300 - 2\mathbf{k}}{149}]/\tanh 2 - c\_y(150 - \mathbf{k})/150$ 
$$\mathbf{c}\_y = 4(1 - \tanh\frac{300}{149})/\tanh 2$$

## *2.3. Boundary Layer*

Boundary layer conditions were as follows:

Inflow boundary conditions, *x* = 0, <sup>→</sup> *u* = 0, and ∂p/∂x = 0; Outflow boundary conditions, *x* = 84, non-reflecting boundary condition, ∂p/∂x = 0; Boundary conditions at the top wall, *y* = 4, ∂ → *u* /∂*y* = 0, p = 0; Boundary condition at the bottom wall, *y* = 0, ∂p/∂y = 0, if 0 ≤ *t* ≤ 15, *v* is shown above, *t* > 15, → *u* = 0.

To verify the independence of results on mesh and time-step sizes, the numbers of mesh points in the *x*- and *y*-directions were raised to 480 and 200, respectively, and the computational time step was reduced to 0.005. The maximum absolute values of the velocity of the vortex structures *u*, *v*, and *w* with two kinds of meshes and time steps were compared, as shown in Figure 2. Lines represent the simulation results of the grid and time step used in this paper, and circles represent the simulation results of the raised grid and reduced time step, *t* < 25; there is almost no difference in the comparison of simulation results. Because the simulation efficiency was too low when the number of grids was increased and the time step was reduced, a total of more than one million grids were used in this paper, and the time step was set as 0.01.

**Figure 2.** Mesh and time-step independence verification.

## **3. Numerical Results and Analyses**
