## 3 Results and Discussion
### Spatial Growth Rate  
Having obtained the gas-liquid interface, we can obtain the spatial growth rate with the method of [PERSON] [24]. Figure 9 shows the growth rate from molecular dynamics simulation compared with the theoretical results achieved by [PERSON] et al. [21]. In their theory, the single equation of the radial component of the pressure perturbation \(\hat{p}*\) is:  
\[\frac{d^{2}\hat{p}^{*}}{dy^{*2}}-[\frac{2\overline{u}_{y}^{*}\cos\phi}{\overline {u}^{*}\cos\phi-c^{*}}+\frac{\overline{\rho}_{y}^{*}}{\overline{\rho}^{*}}] \frac{d\hat{p}^{*}}{dy^{*}}-k^{*2}[1-(\frac{\overline{u}\cos\phi-c}{\overline {a}})]\hat{p}^{*}=0 \tag{20}\]  
where \(\overline{a}\) denotes the speed of sound and superscript \({}^{*}\) denotes a dimensional quantity. Through solving Equation (20), the spatial growth rate of the transcritical mixing layer can be obtained theoretically. The solution to Equation (20) involves the problem of finding the eigenvalues. Considering a free jet, the derivative of the elementary stream is zero as the boundary condition. Taking the frequency \(\omega*\) as the eigenvalue of the specific frequency given by the outside, the solution of the equation can be obtained by the shooting iteration procedure, where \(\omega*=k*c*\). In addition, the spatial growth rate \(\overline{a}\) is the image of \(\omega*\). The numerical iteration of the original differential equation is performed by ODE45 in MATLAB, and then the eigenvalue can be obtained by iteratively shooting by the Newton-Raphson method, so the value of the growth rate can be derived. For a more detailed solution process, please refer to Reference [21]. When the jet is subcritical, because the molecular effect on jet development is insignificant, the results of molecular simulation should be consistent with the theoretical results. As we can see in Figure 9, at the subcritical region, the molecular dynamics result and the theoretical result agree very well. However, the real supercritical flow is very complex, including the coupling between thermodynamic quantities and hydrodynamic quantities, and even turbulent flow processes. Because molecular dynamics spontaneously contains all physical processes,  
Figure 8: Mixing rate profile of jet of a subcritical jet under condition I (upper) and a supercritical jet under condition IV (lower). the coupling and interaction between thermodynamic variables occur spontaneously. As long as the input model is correct, the simulation results are very close to the real situation. However, linear stability theory cannot contain all the variables and their mutual couplings, and when the parameters change dramatically, especially at the critical point, the classical theory becomes very difficult to process, so it may lead to inaccurate results. When the condition approaches the supercritical state, the effect of the molecular scale on the jet development becomes more and more obvious. The difference between the molecular simulation result and the theoretical result will also be correspondingly larger. In Figure 9, the growth rate obtained by MD is reduced while the theoretical one continues to grow. This decrease is a result of the large density gradient between the jet and the ambient fluid, which suppresses the instability of the shear layer. The discrepancy near the critical point can be explained as follows. When the fluid approaches the critical point, there are areas of large density gradients on the surface of the jet, and their presence acts like a solid wall, actually making the jet more stable, not the other way around. Our simulation accurately captures this effect, so we obtain a small growth rate near the critical point, which is the opposite of the classical linear stability theory. However, the classic instability theory cannot capture this phenomenon. The growth rates obtained by MD deviate from the theoretical results most at the critical point.