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\title{Problems of Ch3 Part 1}
\author{Yi-Chao XIE}
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\textcolor{red}{Relation between the units:
\begin{center}
 1 ft=0.3048m;1lb=0.454 kg; 1lb/ft$^2$=47.89N/m$^2$=47.89 Pa; $1^o$R=5/9K\\
 \textcolor{red}{\bf {Due October $15^{th}$, 2024}}
 \end{center}}
 
3.1 At a given point in the high-speed flow over an airplane wing, the local Mach number, pressure and temperature are 0.7, 0.9 atm and 250 K, respectively. Calculate the values of $p_0$,$T_0$,$p^*$,$T^*$, and $a^*$ at this point.


\vspace{9cm}
3.2 At a given point in a supersonic wind tunnel, the pressure and temperature are $5\times10^4$ N/m$^2$ and 200 K, respectively. The total pressure at this point is $1.5\times10^6$ N/m$^2$. Calculate the local Mach number and total temperature.



\vspace{9cm}
3.3 At a point in the flow over a high speed missile, the local velocity and temperature are 3000 ft/s and $500^oR$, respectively. Calculate the Mach number and the characteristic Mach number $M^*$ at this point.



\vspace{11cm}
3.4 Consider a normal shock wave in air. The upstream conditions are given by $M_1$=3, $p_1=1 $ atm, and $\rho_1=$1.23kg/m$^3$. Calculate the downstream values of $p_2, T_2, \rho_2, M_2, u_2, p_{02},$ and $T_{02}$.



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3.5 Consider a Pitot static tube mounted on the nose of an experimental airplane. A Pitot tube measures the total pressure at the tip of the probe (hence sometimes called the Pitot pressure), and a Pitot static tube combines this with a simultaneous measurement of the free stream static pressure. The Pitot and free-stream static measurements are given below for three different flight conditions. Calculate the free stream Mach number at which the airplane is flying for each of the three different conditions:\\
a. Pitot pressure=$1.22\times10^5 N/m^2$, static pressure=$1.01\times10^5$ N/m$^2$;\\
b. Pitot pressure=7222lb/ft$^2$, static pressure=2116 lb/ft$^2$;\\
c. Pitot pressure=13107lb/ft$^2$, static pressure=1020 lb/ft$^2$.\\


\vspace{8cm}
3.6 Consider compression of air by means of (a) a shock compression and (b) isentropic compression. Starting from the same initial condition of $p_1$ and $v_1$, plot to scale the $pv$ diagram for both compression processes on the same graph. From the comparison, what can you say about the effectiveness of shock versus isentropic compression?



\vspace{11cm}
3.7 During the entry of the Apollo vehicle into the Earth's atmosphere, the Mach number at a given point on the trajectory was Ma=38 and the atmosphere temperature was 270K. Calculate the temperature at the stagnation point of the vehicle, assuming a calorically perfect gas with $\gamma=1.4$. Do you think this is an accurate calculation? If not, why? If not, is your answer an overestimate or underestimate?


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