## 2 Experimental Setup
### Molecular Tagging Velocimetry Configuration  
The MTV configuration adopted for this work is illustrated in Figure 3 and briefly described here; more details are available in [PERSON] et al. [14]. A nitrous oxide (N\({}_{2}\)O) seed gas is premixed with the pulse gas (compressed air) and dispersed around the jet region before the pulse. A _write_ 10-ns 193-nm ArF excimer laser pulse decomposes molecules of N\({}_{2}\)O via photodissociation and creates NO tracers with a predetermined regular spatial pattern, such as a horizontal line across the jet. Then, a _read_ tunable dye laser with a wavelength of 226.186 nm and a pulse duration of 10 ns is triggered within a short time interval dt from the _write_ laser on a cross section of the flow to induce fluorescence of the NO tracer molecules formed through the _write_ step. Both lasers operate at 10 Hz, and the beam energy at the jet location is 5 mJ/pulse and 0.5 mJ/pulse for the _write_ and the _read_ laser, respectively. The location of displaced tracers was recorded with a QImaging QClick CCD camera coupled with a LaVision IRO image intensifier and a Nikon UV-Nikkor 105 mm f/4.5 lens. A long-pass filter (230-nm cutoff from Layertec) mounted on the lens transmits the fluorescent signal and rejects the scattered laser light. The camera was translated on a motorized linear stage for precise alignment at three locations downstream of the nozzle: 6.45 mm (close to the nozzle), 47.01 mm (in the middle of the free jet), and 90.42 mm (right above the Venturi). The FOV is 1392 \(\times\) 1000 pixel (85 \(\times\) 61 mm\({}^{2}\)) with a 61 um/pixel magnification. The camera does not allow for a very short interframe gap, and therefore, we were not able to acquire an image _read_ pair for each _write_ pulse. Instead, for each measurement station, the initial _read_ image (with \(dt\approx 0\)) was recorded first, and then a series of deformed profiles (at \(\Delta t\)) was acquired. Figure 4: (**Top figure**): measured showerhead pressure vs. time during the discharge transient. (**Middle figure**): time sequence of the phase-averaged MTV frames recorded at three different heights downstream of the nozzle at 10 Hz frequency during the 0.8 s gas pulse. One, two, and five phases are acquired at the (**top**) (nozzle), (**middle,bottom**) (Venturi) measurement stations, respectively. (**Lower figure**): shadowgraphy images averaged over 10 ms show the evolution of the Mach disk below the nozzle from \(t=0.05\) to 0.85 s.  
Based on the high-speed shadowgraphy data, the jet flow was found to be highly repeatable from pulse to pulse following the initial valve opening. This repeatability enabled the acquisition of phase-averaged data across the pulse duration. To enhance image contrast and facilitate the automatic detection of the bright MTV line by our algorithm, between four and seven instantaneous frames (each from independent runs/gas pulses) were averaged at each measurement location and phase. Due to the comparatively higher cost of both acquisition and processing for MTV measurements, the number of such runs was limited. The camera captures ten frames per second (one frame every 100 ms), and five phases with 11 ms delay are acquired by shifting the laser pulses in time using a digital delay/pulse generator. Figure 4 shows the time sequence of the phase-averaged MTV frames during a 0.8 s gas pulse with a 138 psig (10.5 bar) starting pressure in the reservoir. One, two, and five phases were acquired at the top (6.45 mm below the nozzle), middle (0.34 mm above the middle plane), and bottom (4.28 mm above the Venturi) measurement stations, respectively. The number of phases studied for each location was adapted between one and five to limit the number of experimental runs. The timing between the _write_ and _read_ pulses (\(\Delta t\)) was also adapted for each location; it was 5 us, 6 us, and 8 us for the top, middle, and bottom measurement station, respectively.