## 2 Configuration Descriptions
### Similitude and Engine-Scale Configuration  
Synthetic jets are completely characterised by the Reynolds number \(Re\), the dimensionless stroke length \(L_{0}/D\) and the velocity ratio \(M\). The lab-scale configuration is based on a similitude in order to be representative of a classical acoustic liner, with perforation diameters equal to \(D=1\) mm. To do so, the dimensions of the rig are obtained by multiplying that of the liner perforated plate by \(1/\chi=6.25\). The different relations linking the geometrical and flow parameters of both scales, through the parameter \(\chi\), are summarised in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Similitude Ratios** & & \(\chi=\) 1/6.25 \\ \hline
**Parameter \(X\)** & \(X_{lab}/X_{engine}\) & \(X_{lab}\) & \(X_{engine}\) \\ \hline \multicolumn{4}{c}{Geometrical parameters} \\ \hline \(D\) (mm) & \(1/\chi\) & 6.25 & \(1\) \\ \(L\) (mm) & \(1/\chi\) & 300 & \(48\) \\ \hline \multicolumn{4}{c}{Flow parameters} \\ \hline \(U_{0}\) (m.s\({}^{-1}\)) & \(\chi\) & 12.8 & 80 \\ \(\overline{W}\) (m.s\({}^{-1}\)) & \(\chi\) & 2.07 & 12.94 \\ \(f\) (Hz) & \(\chi^{2}\) & 12.8 & 500 \\ \hline \multicolumn{4}{c}{Heat transfer parameter} \\ \hline \(h\) (W.m\({}^{-2}\).K\({}^{-1}\)) & \(\chi\) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Similitude ratios between the lab and engine scales. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(U_{0}\) (m.s\({}^{-1}\)) & \(f\) (Hz) & \(K\) (mm) & \(Re\) & \(L_{0}/D\) & \(M\) \\ \hline
12.8 & 12.8 & 11 & 829.2 & 12.94 & 0.16 \\ \hline
5 & & & & & \\
4 & & & & & \\
6 & & & & & \\
4 & & & & & \\
3 & & & & & \\
2 & & & & & \\
5 & &  
+++ ==WARNING: Truncated because of repetitions==  
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The test section and perforated plate of the engine-scale configuration are similar to that of the lab scale, except for their dimensions, which are \(1/\chi=6.25\) smaller than that of the lab-scale configuration. The ducts and pistons below each of the perforations are replaced by a single cavity, the dimensions of which are chosen to get a quarter-wave resonator, corresponding to a resonance frequency \(f_{1/4}=500\) Hz. The choice of a quarter-wave resonator leads to a cavity length around 190 mm, which is too long for an acoustic liner. In real applications, a Helmholtz resonator would be privileged to target such a resonant frequency while maintaining the minimum footprint required in an aeronautical context. However, the quarter-wave resonator is much simpler to design and the choice of resonator does not impact the synthetic jet generation. Figure 3 shows the geometrical differences between both configurations. An acoustic signal is superimposed on the grazing flow at the inlet of the domain, characterised by its frequency \(f=500\) Hz and Sound Pressure Level \(SPL=148.5\) dB. Two main limitations of the similitude can be pointed out at this stage. First of all, it has been shown that the actuator geometry, especially that of the neck, can impact the formation of vortices [20]. Here, the lab-scale geometry does not exactly correspond to the classical cavity and neck configuration of a liner with a long hole geometry (Figure 3). The other important point is the jet triggering mechanism. In the lab scale, the jets are generated by the imposed hydrodynamics while in the engine scale they are due to the acoustics. This point raises another issue: the propagation of an acoustic wave along the liner takes some time, which implies that the wave does not impact all of the perforations and resonators at the same time. A direct consequence is that the different synthetic jets generated by the wave are not fully in phase. On the contrary, the experimental rig is based on the use of pistons that are all moved by the same engine, and are thus all perfectly in phase. Differences could therefore be expected if comparing detailed data obtained from the experimental rig and a classic acoustic liner. In that respect, [PERSON] et al.