## 2 Methodology Details
### Computational Model and Boundary Conditions  
As a preliminary work concerning the proposed strategy and relevant enhanced-mixing mechanisms, the two-dimensional compressible N-S equations are solved. The two-dimensional simulation strategy is acceptable in the present study since the computational model is based on the inner flow of a scramjet engine, and a great deal of research with two-dimensional models has already been performed to reveal the relevant flow mechanisms in inner engine flows [31; 32]. Moreover, the compressibility effects on mean velocity field and turbulent statistics are researched, employing the two-dimensional method since it is the first step towards understanding the realistic three-dimensional flow, which is much more complex. The simulation in this paper is based on the model of [PERSON] [33], where the length of the computational domain is 200 mm and the width is 40 mm, corresponding to the height of the inlet on both the upper and the lower sides is 20 mm. The low-speed inlet on the lower side is defined as inlet 1 and the high-speed inlet on the upper side is defined as inlet 2, as shown Figure 1. To enhance the mixing effect of the incoming flow and to simulate the random disturbance of the actual combustor, white noise disturbance is added to both the upstream and the downstream incoming flows. Unlike the previous approach of generating a shock wave through a wedge shock generator, a wall jet (\(16<x<17\)) at the upper boundary is used in the present work to generate a controllable shock wave with a jet nozzle, and the throat width is 1 mm. The incoming flow parameters for this calculation are listed in Table 1.  
The convective Mach number is defined as  
\[M_{c}=\frac{u_{1}-u_{2}}{a_{1}+a_{2}} \tag{9}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & _Ma_ & _V/(m\(\cdot\)s\({}^{-1}\))_ & _P/kPa_ & _T/K_ & _Mc_ \\ \hline inlet1 & 1.7 & 658.4 & 101 & 346.6 & \\ \hline inlet2 & 5.6 & 986.3 & 101 & 76.7 & 0.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Inflow parameters. Figure 1: Schematic of calculation model and flow field structure. (The red solid line represents the position of the jet shock wave in the flow field, the area between the blue dotted lines is the mixing layer, and the blue arrow represents the incoming flow and the wall jet respectively). and the subscripts 1 and 2 represent the lower and the upper layers of the flow field, respectively. The local sound velocity can be obtained from the relation:  
\[a=\sqrt{\gamma RT} \tag{10}\]  
As for the selection of the boundary condition, the pressure inlet condition is set in the inlet, outlet, and jet nozzle; the slip wall is applied to upper and lower wall. The jet nozzle is set just for the choked state, which is marked by the superscript '', and the nozzle flow will reach the maximum at this state. The choked flow calculation formula is as follows. \[\frac{p^{*}}{{p_{0}}^{*}}=0.528 \tag{11}\]  
\(p^{*}\) and \({p_{0}}^{*}\) are the static and the total pressure of the choked flow, respectively. The total pressure of the jet nozzle is set to fluctuate in the form of a sine curve around 0.02 times the total pressure of the upper incoming flow. \[p_{0}^{*}=p_{02}\ [A\sin(2\pi ft)+\zeta] \tag{12}\]  
\(p_{02}\) is the total pressure of the upper incoming flow. Here, we set \(A=0.01\) and \(\zeta=0.02\), and \(f\) takes different values in different cases to control the frequency of the jet. This paper sets a case of a mixing layer and two cases of jet shock interference, where \(f\) is taken as \(2f_{0}\) and \(0.2f_{0}\), respectively, and \(f_{0}\) is the characteristic frequency of an unstable mode of the free mixing layer, the value of which is \(3\times 10^{4}\) Hz.