$2.$ Supersonic Two$-$Shock diffuser:

Consider the $2$D planar supersonic diffuser shown in Fig $2$ that is $1$ m long $($into the page$).$ This configuration uses a combination of oblique and normal shockwaves to decelerate the flow. At supersonic flight conditions, the inlet should be designed to reduce the flow to subsonic velocities with as small of a stagnation pressure drop as possible before passing through the rest of the engine. Consider an engine operating on an aircraft with a cruise Mach number of $\backslash ($ M$\_\{1\}=1\mathrm{.}8\backslash )$ at an altitude where the ambient conditions of the air are $15\mathrm{.}0$ kPa and $\backslash (215\backslash$mathrm$\{$~K$\}(\backslash$gamma$=1\mathrm{.}4\backslash )$ and $\backslash (\backslash$bar$\{$M$\}=29\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$kmol$\})\backslash ).$ The engine inlet needs to provide $\backslash (45\mathrm{.}0\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$s$\}\backslash )$ of air flow to the engine at the design condition. The pressure inside the engine $($i$.$e$.,$ the back pressure on the diffuser$)$ is adjustable such that after the oblique shock $($if one exists$),$ a normal shock is always located at the entrance to the subsonic diffuser section of the inlet. This is known as the "critical" operating condition, where, at the optimal condition, an oblique shock intersects with the top of the inlet cowl $($as illustrated in Fig. $2).$ For a shorter ramp length, the shock would enter the inlet. For a longer length the shock would interfere with the freestream flow, thereby creating potential problems for the aircraft.

ser.

$($a$)$ Assume that the flow deflection angle induced by the spike is $10$ degrees and that the inlet area into the diffuser is defined by the rectangle between the cowl lip and spike where the normal shock sits. Calculate the diffuser dimensions $\backslash ($ R$\_\{$s$\},$ R$\_\{$c$\}\backslash ),$ and $\backslash ($ L $\backslash )($i$.$e$.,$ the entrance area and length of the spike$)$ required at the design condition of $\backslash ($ M$\_\{1\}=1\mathrm{.}8\backslash ).$

$($b$)$ Determine the stagnation pressure loss across the system of shocks at these conditions.