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\title{problems of Chapter 4-1\\(Oblique shock and expansion waves)}
\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 $20^{th}$, 2023}}
 \end{center}}


4.1 Consider an oblique shock wave with wave angle equal to $35^o$. Upstream of the wave $p_1=2000 lb/ft^2$,  $T_1=520^oR$, and $V_1=3355 ft/s$.  Calculate $p_2,T_2,V_2$,  and the flow deflection angle.



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4.2 Consider a wedge with a half angle of $10^o$ flying at Mach 2. Calculate the ratio of total pressure across the shock wave emanating from the leading edge of the wedge.



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4.3 Calculate the maximum surface pressure (in units of Newton per square meter) that can be achieved on the forward face of a wedge flying at Mach 3 at standard sea level conditions ($p_1=1.01\times10^5$ N/m$^2$) with an attached shock wave.


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4.4 In the flow past a compression corner, the upstream Mach number and pressure are 3.5 and 1 atm, respectively. Downstream of the corner, the pressure is 5.48 atm. Calculate the deflection angle of the corner.



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4.5 Consider a $20^o$ half angle wedge in a supersonic flow at Mach 3 at standard sea level ($p_1=2116lb/ft^2=1atm$ and $T_1=519^oR=288K$). Calculate the wave angle, and the surface pressure, temperature, and Mach number.



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4.10 Consider the flow past a $30^o$ expansion corner, as sketched in Fig. 4.32.The upstream conditions are $M_1=2, p_1=3$ atm, and $T_1=400$ K, calculate the following downstream conditions: $M_2, p_2, T_2, T_{02}$, and $p_{02}$. 


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4.11 For a given Prandtl-Meyer expansion, the upstream Mach number is 3 and the pressure ratio across the wave is $p_2/p_1=0.4$. Calculate the angle of the forward and the rearward Mach lines of the expansion fan relative to the free stream direction.



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4.12 Consider a supersonic flow with upstream Mach number of 4 and pressure of 1 atm. This flow is first expanded around an expansion corner with $\theta=15^o$, and then compressed through a compression corner with equal angle $\theta=15^o$, so that it is returned to its original upstream direction. Calculate the Mach number and pressure downstream of the compression corner.




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4.14 Consider a supersonic flow past a compression corner with $\theta= 20^o$. The upstream properties are $M_1 = 3$ and $p_1 = 2116$lb/ft$^2$. A Pitot tube is inserted in the flow downstream of the corner. Calculate the value of pressure measured by the Pitot tube.








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4.20 The flow of a chemically reacting gas is sometimes approximated by the use of relations obtained assuming a calorically perfect gas, such as in this chapter, but using an “effective gamma,” a ratio of specific heats less than 1.4. 

Consider the Mach 3 flow of chemically reacting air, where the flow is approximated by a ratio of specific heats equal to 1.2. If this gas flows over a compression corner with a deflection angle of 20 degrees, calculate the wave angle of the oblique shock. Compare this result with that for ordinary air with a ratio of specific heats equal to 1.4. What conclusion can you make about the general effect of a chemically reacting gas on wave angle?



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4.21For the two cases treated in Problem 4.20, calculate and compare the pressure ratio (shock strength) across the oblique shock wave. What can you conclude about the effect of a chemically reacting gas on shock strength?\\


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