Answers to Study Questions

1. Why must expansions in supersonic flow take place over a finite region in space?
   ans:    A discontinuous expansion would result in large gradients and thus an increase in entropy. An isentropic change can only occur gradually.
    
2. What is an expansion fan?
   ans:    An infinite collection of Mach waves that produce the expansion.
    
3. What is a Mach wave?
   ans:    A infinitely weak discontinuity where the flow properties change infinitesimally. The normal component of the upstream Mach number is 1.0.
    
4. How can the Mach number downstream of an expansion be determined if the upstream Mach number and the flow turning angle are known?
   ans:    The Prandtl-Meyer relation tells us that n(M₂) = n(M₁) + q. Thus with M₁ and q known, n₂ = n(M₂) can be calculated and finally M₂ deduced from a table or graph of the Prandtl-Meyer function.
    
5. How are the pressure and temperature computed on the downstream side of the expansion fan?
   ans:    Through use of the isentropic relations and knowledge of the Mach number on either side of the expansion fan.
    
6. Does the total pressure change across the expansion fan? Explain your answer.
   ans:    No. The expansion is isentropic and thus the total pressure remains constant across the fan.
    
7. Explain how lift is generated in supersonic flow.
   ans:    Through the compression and expansion processes as the flow is turned to go around the airfoil.
    
8. Does the Kutta condition apply in a supersonic flow? Explain your answer.
   ans:    No. The flow angles on the upper and lower airfoil surface near the trailing edge need not be equal. Equalization will occur immediately downstream of the trailing edge vi shock and expansion waves.
    
9. Why do two-dimensional airfoils in supersonic flow produce drag?
   ans:    Due to the losses in total pressure as the flow crosses the shock waves.
    
10. Define wave drag.
    ans:    Drag due to the production of shock waves in supersonic flow.
     
11. Indicate whether each of the following quantities goes up, down, or remains the same as the flow crosses an expansion fan. Mach number, velocity, density, pressure, temperature, entropy, stagnation temperature, stagnation pressure.
    ans:    Mach number goes up, velocity goes up, density goes down, pressure goes down, temperature goes down, entropy stays the same, stagnation temperature stays the same, stagnation pressure stays the same.