Problem $12.$ Shock tubes are often used for ignition experiments, for example, on biofuels. The schematics of a shock tube is shown in Fig. $1.$ Upon rupture of the diaphragm separating the high$-$pressure driver section and the low$-$pressure driven section, an incident shock wave is generated, which propagates in the air initially at rest $($region $1)$ towards the end$-$wall of the shock tube. A time moment when the incident shock approaches the end$-$wall is shown on the top schematics of Fig. $2.$ Region $2$ is the region behind the incident shock. The incident shock then reflects from the end$-$wall and the reflected shock propagates back, as shown on the bottom schematics of Fig. $2.$ Region $3$ is the region between the reflected shock and the end wall.

The setup includes a pressure transducer and a thermocouple near the wall to monitor parameters in Region $3.$ After the shock reflects from the wall, a pressure $\backslash (\backslash$left$($p$\_\{3\}\backslash$right$)\backslash )$ of $50\mathrm{.}88$ atm and a temperature $\backslash (\backslash$left$($T$\_\{3\}\backslash$right$)\backslash )$ of $1397$ K were measured. With the aid of high$-$speed imaging techniques, the velocity of the reflected shock $\backslash (\backslash$left$($V$\_\{$s r$\}\backslash$right$)\backslash )$ was measured to be $\backslash (426\mathrm{.}4\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )($lab frame$).$

Figure $1$: Schematics of a shock tube $($Problem

Ignition Section $12).$

Figure $2$: Schematics of the flow near the end$-$wall of a shock tube before shock reflection $($top$)$ and after shock reflection $($bottom$).($Problem $12)$

Assume that the air behaves as an ideal gas with the following properties: $\backslash (\backslash$gamma$=1\mathrm{.}4\backslash )$ and $\backslash ($ R$=\backslash )\backslash (287\backslash$mathrm$\{$~J$\}/(\backslash$mathrm$\{$kg$\}\backslash$cdot $\backslash$mathrm$\{$K$\})\backslash ).$

$($a$)$ Calculate the Mach number $\backslash ($ M$\_\{$s r$\}\backslash )$ of the reflected shock.

$($b$)$ Estimate the Mach number $\backslash ($ M$\_\{$s$\}\backslash )$ of the incident shock. $($Hint: Try the trial$-$and$-$error approach between $\backslash ($ M$\_\{$s$\}=2\mathrm{.}7\backslash )$ and $\backslash ($ M$\_\{$s$\}=2\mathrm{.}85\backslash )).$

$($c$)$ Find the parameters near the end$-$wall before the incident shock arrives $(\backslash ($ p$\_\{1\}\backslash )$ and $\backslash ($ T$\_\{1\}\backslash )).$