# Thrust Augmentation of Micro-Resistojets by Steady Micro-Jet Blowing into Planar Micro-Nozzle
## 3 Numerical Setup and Methodology
### Validation, Boundary Conditions, and Test Matrix  
The present numerical methodology has been previously validated in [4]. Without going into details of the previous work, a numerical methodology was proposed to predict the performance of VLMs, which combined a 1D model with 3D CFD computations of the micro-nozzle flow. Using the experimental data provided in [32], the methodology allowed us to estimate the flow conditions at the micro-nozzle entrance for a given mass flow rate and wall temperature, which defined the boundary conditions in the 3D CFD computations performed using the same numerical approach and setup previously described in Section 3.1. In particular, at inlet conditions of 505.58 K temperature, 5 \(\times\) 10\({}^{-6}\) kg/s mass flow rate, and 500 backward discharge pressure, 3D CFD simulations predicted a thrust force of 4.72 mN in comparison with 5.20 mN measured in experiments, with an error of about 9%. The estimation of severe viscous losses gave suggestion for the present work, with the attempt to mitigate viscous losses by means of steady micro-jet blowing into the micro-nozzle. Consequently, the same reference operating condition has been used, i.e., a water vapor flow at inlet temperature of 505.58 K, a mass flow rate of 5 \(\times\) 10\({}^{-6}\) kg/s, and a backward discharge pressure of 500 Pa, which defined the baseline configuration. Furthermore, the water vapor flow, resulting from the operation in VLM mode, has been compared to the nitrogen gaseous flow entering the micro-nozzle at 293 K and a mass flow rate of 5 \(\times\) 10\({}^{-6}\) kg/s. in relation to operation in CG mode. The operating condition of the secondary injection is defined by the injection angle \(\alpha_{jet}\), with respect to the micro-nozzle axis, the momentum co-efficient \(c_{\mu}\), and the number of micro-jets, i.e., single jet mode or dual jet mode. In the present study, the injection of the secondary flow was always set perpendicular to the jet cross section plane, i.e., \(\alpha_{jet}\) = 75\({}^{\circ}\). Instead, the overall analysis was conducted by varying the jet velocity magnitude \(U_{jet}\) at 150 m/s, 300 m/s, and 600 m/s, corresponding to different levels of \(c_{\mu}\). At first, the water vapor flow was investigated on a narrow range of operating conditions and micro-jet blowing configurations with sonic injection at jet 1 and sub-sonic injection at jet 2, as reported in Table 5. Hence, the analysis was extended to the nitrogen gas flow at same mass flow rate of the baseline configuration without secondary injection and similar Reynolds number at the throat section. With the aim to provide a more accurate  
\begin{table} \begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Test Case** \\ **Name** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Refinement** \\ **Level** \\ \end{tabular} } & \multirow{2}{*}{**Cell Number**} & \multirow{2}{*}{\begin{tabular}{c} **Grid** \\ **Spacing 1** \\ **[\(\upmu\)m]** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Spacing** \\ **Factor** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **GCI** \\ **Parameter,** \\ **\(\theta_{yy}\) 2 [\(\upmu\)m]** \\ \end{tabular} } & \multirow{2}{*}{**GCI [\%]**} & \multirow{2}{*}{
\begin{tabular}{c} **Computational** \\ **Cost 3 [h]** \\ \end{tabular} } \\ \hline
3 DGIS\_1 & Fine & 607,720 & 10 & 1 & 22.47 & \(\text{GCI}_{12}\) = 0.67 & 105.34 \\
3 DGIS\_2 & Intermediate & 516,360 & 15 & 1.5 & 23.33 & \(\text{GCI}_{23}\) = 5.26 & 87.51 \\
3 DGIS\_3 & Coarse & 493,520 & 20 & 2 & 16.31 & - & 80.58 \\ \hline \hline \end{tabular}   
* *insight of the impact of micro-jet blowing, the analysis has been extended to a wider range of actuation conditions, as summarized in Table 6. In particular, the Mach number of the jet 1 varied from 0.43 (sub-sonic injection) to 0.86 (transonic injection) up to 1.71 (super-sonic injection). Instead, the Mach number of the jet 2 ranged from 0.43 to 0.87.