• Sections 5.3.1-5.7 of Anderson.

Problems from Anderson
5.4
5.5

Supplemental Problems

1. Consider an untwisted rectangular wing of aspect ratio 8, whose cross-section is a NACA-0012 airfoil. Use the lifting line theory to compute the performance of this wing when it is flying at an angle of attack of 3 degrees. Specifically, do the following:

   1. Solve Eq. 5.51 for the first three non-zero expansion coefficients, A₁, A₃, A₅ (why are A₂ and A₄ zero?). To do this you will need to choose three stations along the wing in which to enforce Eq. 5.51. Use q₀ = p/6, p/3, p/2 as the three stations. You can solve the resulting 3 by 3 linear system either by hand or by using a computer.
   2. With the A_n known, compute the lift and induced drag coefficients as well as the span efficiency, i.e. C_L, C_Di, and e.

   To answer the following two problems, download the lifting line program from the course website. It should run on any Intel-based pc. You can run the program on the computers in the CAD lab or on any other computer that is convenient for you. When you run the program, specify that the lifting line equation be enforced at 10 points along the span.

2. Design an improved rectangular wing of aspect ratio 8 by introducing twist to unload the wingtips (washout). The program allows you to specify the twist in terms of the difference in angle of attack between the tip and root (a negative value denotes washout). A linear variation in the twist angle is assumed between the root and tip. Find the washout that maximizes the span efficiency. You probably want to make a plot of span efficiency versus washout to do this convincingly.
3. Design an improved wing of aspect ratio 8 using taper this time instead of washout. The code allows you to input the taper ratio directly. Find the taper ratio that maximizes the span efficiency (again use a plot to convey the results). Which wing works better, the twisted rectangular wing or the tapered wing?

Study Questions

1. How does the downwash velocity vary along the span if the wing carries an elliptical lift distribution?
    
2. Why does the induced drag coefficient for an elliptical load distribution depend on the lift coefficient?
    
3. How is the wing aspect ratio defined?
    
4. How does the induced drag coefficient for an elliptical load distribution depend on the wing aspect ratio?
    
5. Explain why induced drag dominates viscous drag when the aircraft is flying near the stall speed.
    
6. What fraction of total drag is the induced drag for a typical aircraft in cruise flight?
    
7. Why do high performance sailplanes use wings with aspect ratios 2-3 times higher than conventional aircraft?
    
8. What planform shape gives rise to an elliptical load distribution (assuming no twist)?
    
9. How is Eq. 5.51 solved for the Fourier expansion coefficients?
    
10. What is the meaning of the span efficiency parameter, e?
     
11. What load distribution leads to minimum induced drag? Explain your answer.
     
12. Why are elliptical wing planforms not too common?
     
13. What are the advantages of using a tapered wing? Are there any disadvantages ?
     
14. Which parameter typically has a larger impact on the induced drag: span efficiency or aspect ratio? Explain your answer.
     
15. Why is the lift curve slope for a wing always less than the lift curve slope for the airfoil that it uses for its cross-section?
     
16. Why does the far wake of a real aircraft consist of a simple vortex pair instead of a continuous vortex sheet? Does this present an inconsistency in Prandtl's lifting line theory?
     
17. How does a delta wing generate lift at subsonic speeds?
     
18. How can a delta wing produce lift at angles of attack in excess of 30 degrees? Roughly at what angle of attack does a delta wing ßtall"?
     
19. Why isn't the delta wing capable of produce high lift coefficients?
     
20. Why is the lift to drag ratio for a delta wing is subsonic flight fairly low?