2. Consider the supersonic duct shown below.

1. Determine the point along the lower wall (x coordinate) where the flow first feels the effects of the corner at point A.
2. Determine the point along the lower wall (x coordinate) where the flow last feels the effects of the corner at point A. (Exclude secondary reflections off the upper wall for this part.)
3. Determine the Mach number and pressure (p/p_¥) for the point determined in (b).

3. Compute the lift and drag coefficients for the triangular airfoil shown below when M_¥ = 2.0, a = 3⁰, t/c = 0.2, and x_h/c = 0.55. Use the exact values of pressure behind shock waves and expansion fans (i.e. do not use the small disturbance approximation).

1. What is the required nozzle throat area?
2. What is the pressure at the throat?
3. What is the required nozzle exit area?
4. What are the temperature and velocity at the nozzle exit?
5. What changes in flow configuration (i.e. presence of shocks, expansion waves, etc.) will occur if the engine and nozzle are operated at a lower altitude where the pressure is 0.9 atm?

5. A wing has a zero Mach number lift curve slope of 8.34×10^-2 deg^-1 and an angle for zero lift of a_L0 = -1.5 deg. The wing planform area is 510 m².

1. What is the wing lift coefficient when it is flying at M = 0.55, a = 2.1 deg?
2. What angle of attack is required to produce a lift of 8.5×10⁵ N when the flight conditions are M = 0.6, p = 0.5 atm, T = 230 K?
3. The wing has a critical Mach number of M_cr = 0.71 when the angle of attack is a = 4 deg. What is the minimum pressure coefficient for the wing at a = 4 deg, M = 0?

6. The stagnation pressure is a key parameter in the design of a supersonic engine inlet. Consider the two engines sketched below. Engine A uses an ideal inlet that is shock-free at the design condition, M=2.7. Engine B uses a pair of oblique shocks to perform the compression. Answer the following using the tables at the back of the book if that will save you time. If you use the tables, do not bother to interpolate, just use the closest entry to the data that you have.
Note: you will need to use the energy equation in the form h_0,4 = h_0,3 + q for this problem. Also assume that inlet A will start with the throat designed to be sonic.

1. Compute p₀₃/p₀₁ for engine A.
2. Verify that the shock angles are correct as shown for engine B.
3. Compute p₀₃/p₀₁ for engine B.
4. Which inlet is better? Explain your answer.
5. The pressure at the inlet to the combustor (station 3) in engine A is 20 atm. Compute T, r, M, and u at this station.
6. The combustor in engine A adds heat in the amount q = 1.3c_pT₁ and the temperature at the nozzle exit (station 5) is observed to be T₅ = 1.37T₁. Compute the exit velocity (u₅)
7. Compute the thrust generated by engine A.

7. Consider the supersonic duct shown below.

8. Consider a supersonic engine inlet that uses an oblique shock to perform the initial stage of the compression as shown below. Although the width of the inlet (distance into the paper) is only 0.5 meters, assume that the flow can be approximated as being two-dimensional. Answer the following for the design flight condition of M_¥ = 2.0, r_¥ = 0.6 kg/m³, T = 240 K.

1. Let the conditions behind the oblique shock be denoted by subscript 2. What are the values of M₂, p₂, r₂, and p₀₂/p₀₁?
2. Does the oblique shock reflect off the wall at the point B as a normal or oblique shock? Explain your answer. Draw this shock on the sketch.
3. Let the conditions behind the second shock be denoted by subscript 3. What are the values of M₃, p₃, r₃, and p₀₃/p₀₂?
4. The engine requires a mass flow rate of 30 kg/s. What should the distance measured between points B and C be in order to supply the engine with the correct mass flux when is is flying at the conditions stated above?
5. What is the minimum permissible distance between points D and E for the conditions considered above?
6. Assume that the upper wall (containing points B and D) can move fore and aft. Which way should the wall be moved if the flight Mach number is increased to M = 2.5? Explain your answer.