Compressible Flow Design Problem Set 2 — Two-Dimensional Supersonic Inlets  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Problem #1
Consider that you are faced with designing a two-dimensional supersonic diffuser. The main challenge in designing a supersonic diffuser is to minimize the stagnation pressure losses while achieving a high compression ratio. This can be done by compressing the flow through a series of equal strength oblique shocks preceding a normal shock. To minimize the geometric complexity of the inlet, it is here desired to use only three shocks to decelerate the flow to subsonic conditions:
The inlet is designed for a flight Mach number of $2.4$ and such that the pressure increases twofold across each oblique shock. Find the $x,~y$ coordinates of points A and B in the inlet depicted above for a diffuser height $h$ fixed to 1.0 m.
Problem #2
A two-dimensional supersonic diffuser is to be designed as shown below for a Mach number of $2.5$. The ratio $h_2/h_1$ is to be chosen so that the diffuser will barely swallow the initial shock, and the ratio $l/h_1$ is to be selected so as to obtain the wave pattern shown.
diffuser.png  ./download/file.php?id=1745&sid=434085a20bdd058bd5c01e920325679f  ./download/file.php?id=1745&t=1&sid=434085a20bdd058bd5c01e920325679f
(a)  Determine $h_2/h_1$ and $l/h_1$
(b)  Neglecting friction compare the overall stagnation-pressure ratio of this diffuser with the stagnation pressure ratio of a diffuser in which a normal shock occurs at Mach number $2.5$.
Problem #3
After obtaining a Ph.D. from Pusan National University, you are hired by the Korean Agency for Defense Development (ADD) in Daejeon. Your first design project consists of developing a suitable mixed-compression inlet for a supersonic combustion ramjet (scramjet) engine. In a mixed-compression inlet, some of the compression process takes place externally and some internally. Since the scramjet engine is intended to power a non-reusable hypersonic cruise missile, the geometrical complexity is desired to be kept to a minimum to keep the manufacturing costs low. For this reason, the compression process consists of only two oblique shocks, as depicted in the figure below:
scramjetinlet-scaled.png  ./download/file.php?id=1881&sid=434085a20bdd058bd5c01e920325679f  ./download/file.php?id=1881&t=1&sid=434085a20bdd058bd5c01e920325679f
For an inlet exit temperature fixed to $1066$ K, design the inlet such that it yields optimal performance (i.e. minimal stagnation pressure loss) for a flight Mach number of 7.5 at an altitude of 15 km. Specifically, perform the following tasks:
(a)  Find the compression ratio and percent stagnation pressure loss through the inlet
(b)  Find the angles $\delta$ and $\theta$
(c)  Find the Mach number at the inlet exit
(d)  Find the $x,~y$ coordinates of points A and B as a function of the inlet height $h$
Assume frictionless and constant-$\gamma$ flow.
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