## 2 Analytic Model Development
### The Conservation Equations  
For inviscid flow, the integral form of the conservation equations is given as following:  
\[\frac{\partial}{\partial t}\iiint_{v}\rho dv+\iint_{s}\rho\overset{\rightarrow} {u}\overset{\rightarrow}{d}\overset{\rightarrow}{S}=0 \tag{1}\]  
\[\frac{\partial}{\partial t}\iint_{v}\rho\overset{\rightarrow}{u}dv+\iint_{s }\left(\rho\overset{\rightarrow}{u}\overset{\rightarrow}{d}\overset{ \rightarrow}{S}\right)\overset{\rightarrow}{u}=-\iint_{s}Pd\overset{ \rightarrow}{S}+\iiint_{v}\rho\overset{\rightarrow}{f}dv \tag{2}\]  
\[\frac{\partial}{\partial t}\iint_{v}\rho\left(e+\frac{u^{2}}{2}\right)dv+ \iint_{s}\rho\left(e+\frac{u^{2}}{2}\right)\overset{\rightarrow}{u} \overset{\rightarrow}{d}\overset{\rightarrow}{S}=-\iint_{s}P\overset{ \rightarrow}{u}\overset{\rightarrow}{d}\overset{\rightarrow}{S}+\overset{ \rightarrow}{Q} \tag{3}\]  
For the chamber control volume, the gas velocity is negligible. These conservation equations are represented as the following:  
\[\frac{\mathrm{d}\rho_{\mathrm{c}}}{\mathrm{d}t}=-\frac{\rho_{\mathrm{i}}u_{ \mathrm{i}}A_{\mathrm{i}}}{V_{\mathrm{c}}} \tag{4}\]  
\[\frac{dT_{\mathrm{c}}}{dt}=\frac{1}{\rho_{\mathrm{c}}}\Bigg{(}\frac{\dot{Q}-C _{\mathrm{v}}T_{\mathrm{i}}\rho_{\mathrm{i}}uA_{\mathrm{t}}-0.5\rho_{\mathrm{i }}u^{3}A_{\mathrm{t}}}{C_{\mathrm{v}}V_{\mathrm{c}}}-T_{\mathrm{c}}\frac{d\rho _{\mathrm{c}}}{dt}\Bigg{)} \tag{5}\]  
Figure 1: The structure of plasma synthetic jet actuator (PSJA).  
However, limiting by the throat geometry, the maximum gas velocity is the local sound speed. That means that the flow condition has two modes: To be checked and to be unchocked. The criterion for judgment is as follows:  
\[\left\{\begin{array}{l}P_{\rm i}+0.5\rho_{\rm i}u^{2}\geq 1.89P_{\infty}\\ \gamma\geq 1\end{array}\right. \tag{11}\]  
If flow parameters meet this criterion, the flow is considered to be choked. The pressure at the throat outlet face is calculated as follows:  
\[P_{\rm o}=\frac{1}{1.89}\left(P_{\rm i}+0.5\rho_{\rm i}u^{2}\right) \tag{12}\]  
In recovery stage, the gas flows from the outside environment to the chamber. At this time, the parameters over the throat inlet control face are determined by the gas state parameters in the throat; the parameters over the throat outlet control face are determined by the gas state parameters in the external environment. The corresponding functions are presented as follows:  
\[\left\{\begin{array}{l}\rho_{\rm i}=\rho_{\rm t}\\ P_{\rm i}=P_{\rm c}\\ T_{\rm i}=\frac{R_{\rm i}}{R\rho_{\rm i}}\\ \rho_{\rm o}=\rho_{\rm o}\varepsilon(\lambda)\\ P_{\rm i}=P_{\rm o}\pi(\lambda)-0.25\rho_{\rm o}u^{2}\\ T_{\rm i}=\frac{P_{\rm o}}{R\rho_{\rm o}}\end{array}\right. \tag{13}\]  
In the above functions, \(\lambda\) indicates the velocity coefficient, which is calculated by the following functions:  
\[\left\{\begin{array}{l}a_{\rm cr}\ =\sqrt{\frac{2\gamma}{\gamma+1}RT_{t}^{*}} \\ T_{t}^{*}=T_{\rm t}+\frac{u^{2}}{2C_{\rm p}}\\ \lambda=\frac{|u|}{a_{\rm cr}}\end{array}\right. \tag{14}\]  
\(\varepsilon(\lambda)\) and \(\pi(\lambda)\) are calculated as follows:  
\[\begin{array}{l}\varepsilon(\lambda)\ =\left(1-\frac{\gamma-1}{\gamma+1} \lambda^{2}\right)^{\frac{1}{\gamma-1}}\\ \pi(\lambda)=\left(1-\frac{\gamma-1}{\gamma+1}\lambda^{2}\right)^{\frac{ \gamma}{\gamma-1}}\end{array} \tag{15}\]