Answers to Study Questions

1. How does the flow over a finite wing differ from the flow over a two-dimensional airfoil?
   ans:    Flow over a finite wing is three-dimensional with a non-zero component of velocity in the spanwise direction.
    
2. Describe the flow on a finite wing in the vicinity of the tips.
   ans:    There is a circulation of fluid from the high pressure region on the lower surface to the low pressure region on the upper surface.
    
3. Describe the flow downstream of the tips of a finite wing.
   ans:    A vortex is formed as the circulating fluid leaves the wing tip.
    
4. What is meant by downwash? How is it related to the trailing vortices?
   ans:    It is the downward component of velocity induced by the trailing vortices.
    
5. What is meant by the induced angle of attack?
   ans:    It is the rotation of the relative wind due to the downwash velocity.
    
6. Explain how the downwash modifies the effective angle of attack.
   ans:    The relative wind is shifted downward by the downwash velocity. This change results in an effective decrease in the angle of attack.
    
7. What is meant by induced drag?
   ans:    By virtue of the rotation of the relative wind by the induced angle of attack, the lifting force has a component in the direction of travel of the wing. This component is a drag force and is known as induced drag.
    
8. How can a drag force exist in an inviscid flow? Hint, use the 1st law of thermodynamics to answer this.
   ans:    The kinetic energy in the trailing vortices increases at a constant rate with time. Work must be done continually by the wing to supply this energy and the induced drag multiplied by the flight speed gives the required rate of work.
    
9. What is meant by profile drag?
   ans:    It is the combination of skin friction and pressure drag (due to flow separation).
    
10. Why is three-dimensional theory not needed to compute the profile drag of a finite wing?
    ans:    Because profile drag can be computed to a good approximation by idealizing the flow to be two-dimensional.
     
11. What is a vortex filament? How is it different from the line vortex studied earlier?
    ans:    It is a general curve in space along which the circulation (computed in a plane perpindicular to the line) is constant. It varies from a vortex line in that it may be curved and does not necessarily extend to infinity.
     
12. What information does the Biot-Savart law give regarding a vortex filament?
    ans:    It gives the velocity induced by a segment of the vortex filament at an arbitrary point in space.
     
13. How can the Biot-Savart law be used to compute the velocity induced by a semi-infinite straight vortex filament? What is the end result?
    ans:    By evaluating an integral along its length (from 0 to ¥). The end result is u = G/(4pr), where r is the perpindicular distance to the vortex.
     
14. State Hemholtz's first vortex law.
    ans:    The circulation around a vortex filament is constant along its length.
     
15. State Hemholtz's second vortex law.
    ans:    A vortex filament must formed a closed loop (or extend to infinity).
     
16. What is meant by washout?
    ans:    Geometric or aerodynamic twisting of the wing such that the wing tips fly at a lower angle of attack than the wing root.
     
17. What is meant by washin?
    ans:    Geometric or aerodynamic twisting of the wing such that the wing tips fly at a higher angle of attack than the wing root.
     
18. What is meant by aerodynamic twist?
    ans:    The use of variable camber along the span of the wing so that a_{L = 0} varies. It has roughly the same end effect as geometric twisting.
     
19. What factors can contribute to lift variation over the span of the wing?
    ans:    Variations in chord, geometric angle of attack, camber, and induced angle of attack.
     
20. What value of lift always exists at the wing tips?
    ans:    Zero.
     
21. What is a horseshoe vortex? Does it violate the Hemholtz laws? Explain your answer.
    ans:    A vortex filament in the shape of a vertically-stretched block letter U. It does not violate the Helmholtz laws provided that its legs extend to infinity or that they connect within a finite distance.
     
22. Why is a single horseshoe vortex not a good model for a finite wing?
    ans:    Because it results in a non-physical downwash distribution (infinite downwash at the wing tips).
     
23. What simple model did Prandtl adopt for a finite wing in his famous lifting line theory?
    ans:    An infinite collection of horseshoe vortices with spans ranging from zero to b.
     
24. What distinguishes Prandtl's lifting line model from a single horseshoe vortex?
    ans:    The use of multiple vortices so that the circulation distribution can vary in the spanwise direction.
     
25. How are the strength of the trailing vortices in Prandtl's lifting line model related to the lift distribution on the wing?
    ans:    The strength of the trailing vortices is equal to the change in circulation along the lifting line
     
26. Why must there be a continuous vortex sheet behind the airfoil according to Prandtl's lifting line theory?
    ans:    For every change in circulation along the wing there must be vortex shed into the wake in order to satisfy the Helmholtz vortex laws.
     
27. How is the Biot-Savart law used to compute the downwash velocity in Prandtl's lifting line theory?
    ans:    It is used to compute the induction of a semi-infinite vortex. The downwash is then the superposition of the entire collection of trailing vortices.
     
28. What type of equation does Prandtl's lifting line theory lead to? What is the unknown quantity to be solved for?
    ans:    An integro-differential equation. The unknown quantity is the circulation distribution.
     
29. How can the total lift be computed once the circulation distribution is determined?
    ans:    From the kutta-joukowsky law, L = rU_¥ ò_-b/2^b/2 G(y) dy.
     
30. How can the induced drag be computed once the circulation distribution is determined?
    ans:    From the relation D¢_i @ a_i L¢, which leads to D_i = rU_¥ ò_-b/2^b/2 a_iG(y) dy