# Thrust Augmentation of Micro-Resistojets by Steady Micro-Jet Blowing into Planar Micro-Nozzle
## 2 Micro-Nozzle and Active Flow Control Configuration Using Secondary Injection
### Micro-Nozzle Performance Estimation  
The performance of the micro-nozzle have been analyzed in terms of thrust force \(T\) and specific impulse \(I_{sp}\) as follows:  
\[T\ =\ T_{jet}+T_{press}\ =\ \dot{m}_{tot}U_{exit}+(p_{exit}-p_{amb})A_{exit}\ =\ \dot{m}_{tot}U_{eff} \tag{3}\]  
\[I_{sp}\ =\ \frac{T}{\left(\dot{m}_{tot}\ g_{0}\right)} \tag{4}\]  
where \(U\) is the flow velocity, \(p\) is the static pressure, and \(g_{0}\approx 9.81\) m/s is the gravitational acceleration at sea level. In regards to the subscripts, \(jet\) and \(press\) refer to the jet thrust and the pressure thrust, and \(eff\) denotes the effective velocity at the micro-nozzle exit, while the subscripts \(exit\) and \(amb\) respectively refer to the exit and ambient discharge conditions. Furthermore, \(\dot{m}_{tot}\) is the total mass flow rate exiting from the micro-nozzle, defined as follows:  
\[\dot{m}_{tot}\ =\ \int_{A_{exit}}\rho\ (\mathbf{U}\times\mathbf{\hat{n}}_{A})\ dA\ =\ \dot{m}_{0}+\dot{m}_{jet,1}+\dot{m}_{jet,2} \tag{5}\]  
The thrust co-efficient \(c_{T}\) has been estimated to take account the feeding pressure as an indicator of the cost supported to ensure the operating condition, as follows:  
\[c_{T}\ =\ \frac{T}{(p_{inlet}A^{*})} \tag{6}\]  
where \(p_{inlet}\) is the total pressure at the micro-nozzle inlet, and \(A^{*}\) is the throat section. It is worth observing that \(c_{T}\) relates the thrust force to the cost of feeding the corresponding mass flow rate. Thus, if the mass flow rate of the core flow increases, the thrust will increase, too. However, secondary injection is provided at the same feeding pressure of the core flow by exploiting the pressure drop between the feeding pressure in the propellant storage upstream and the pressure of the expanding flow into the micro-nozzle divergent without any additional cost of feeding.  
The efficiency of the micro-jet blowing has been analyzed by comparing the micro-nozzle performance of each actuated condition, denoted by the subscript \(i\), to the ones resulting from the steady-state solution without actuation, denoted with the subscript \(0\). The last one also corresponds to the solution of the baseline test case. It is worth observing that the jet thrust contribution is the most relevant one, since it represents a quality term of the expansion process into the divergent section, which leads to the actual thrust force acting on the micro-thruster. On the other hand, the pressure thrust is a thrust loss term, since it increased as the quality of the expansion process reduced. Consequently, the performance analysis has been based on the estimation of the percent variation of the thrust co-efficient \(\Delta^{\%}\!c_{T}\), jet thrust \(\Delta^{\%}\!c_{T,jet}\), specific impulse related to the jet thrust \(\Delta^{\%}\!c_{ISP,jet}\), and mass flow rate \(\Delta^{\%}\!_{\dot{m}_{i}}\) for each actuated condition \(i\) as follows:  
\[\Delta^{\%}\!c_{T}\ =\ \frac{(c_{T,i}-c_{T,0})}{c_{T,0}}100 \tag{7}\]  
\[\Delta^{\%}\!c_{T,jet}\ =\ \frac{\left(T_{jet,i}-T_{\text{jet,0}}\right)}{T_{ \text{jet,0}}}100 \tag{8}\]  
\[\Delta^{\%}\!c_{ISP,jet}\ =\ \frac{\left(I_{sp,\ jet,i}-I_{sp,\ jet,0}\right)}{I_ {sp,jet,0}}\ 100 \tag{9}\]\[\Delta\%_{o_{\dot{m}}}\ =\ \frac{\left(\dot{m}_{tot,j}-\dot{m}_{tot,0}\right)}{ \dot{m}_{tot,0}}\ 100 \tag{10}\]  
The analysis has been extended by the computation of the displacement and the momentum thicknesses at the exit section, \(\delta^{*}\) and \(\theta\), respectively, as follows:  
\[\delta^{*}\ =\ \int_{0}^{\infty}\left(1-\frac{\rho U_{x}}{\left(\rho U_{x} \right)_{\infty}}\right)dy \tag{11}\]  
\[\theta\ =\ \int_{0}^{\infty}\frac{\rho U_{x}}{\left(\rho U_{x}\right)_{ \infty}}\left(1-\frac{U_{x}}{U_{x,\infty}}\right)dy \tag{12}\]  
where \(x\) and \(y\) denote the parallel and normal directions to the solid wall, and the subscript \(\infty\) refers to the undisturbed flow condition.