1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS
Introduction to Lifting Line Theory
April 11, 2011
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk

2. NECESSARY TOOL
Return to vortex filament, which in general maybe curved
General treatment accomplished with Biot-Savart Law
Electromechanical Analogy:
Think of vortex filament as a wire carrying an electrical current I
The magnetic field strength, dB, induced at point P by segment dl is:

3. EXAMPLE APPLICATIONS Case 1 Case 2
Case 1: Biot-Savart Law applied between ± ∞
Case 2: Biot-Savart Law applied between fixed point A and ∞
Case 1
Case 2

4. BIOT-SAVART LAW

5. EXAMPLE APPLICATIONS
Case 1:

6. HELMHOLTZ’S VORTEX THEOREMS
The strength of a vortex filament is constant along its length
A vortex filament cannot end in a fluid; it must extend to boundaries of fluid (which can be ± ∞) or form a closed path
Note: Statement that “vortex lines do not end in the fluid” is kinematic, due to definition of vorticity, w, (or x in Anderson) and totally general
We will use Helmholtz’s vortex theorems for calculation of lift distribution which will provide expressions for induced drag
L’=L’(y)=r∞V∞G(y)

7. CONSEQUENCE: ENGINE INLET VORTEX

8. CHAPTER 4: AIRFOIL Each is a vortex line G4 G7
One each vortex line G1=constant
Strength can vary from line to line
Along airfoil, g=g(s)
Integrations done:
Leading edge to
Trailing edge G4 G7 G1
Side view
Entire airfoil has G
z/c
x/c

9. CHAPTER 5: WINGS

10. PRANDTL’S LIFTING LINE THEORY
Replace finite wing (span = b) with bound vortex filament extending from y = -b/2 to y = b/2 and origin located at center of bound vortex (center of wing)
Helmholtz’s vorticity theorem: A vortex filament cannot end in a fluid
Filament continues as two free vorticies trailing from wing tips to infinity
This is called a ‘Horseshoe Vortex’

11. PRANDTL’S LIFTING LINE THEORY
Trailing vorticies induce velocity along bound vortex with both contributions in downward direction (w is in negative z-direction)
Contribution from left trailing vortex
(trailing from –b/2)
Contribution from right trailing vortex
(trailing from b/2)
This has problems: It does not simulate downwash distribution of a real finite wing
Problem is that as y → ±b/2, w → ∞
Physical basis for solution: Finite wing is not represented by uniform single bound vortex filament, but rather has a distribution of G(y)

12. PRANDTL’S LIFTING LINE THEORY
Instead of G=constant
We need a way to let G=G(y)
Represent wing by a large number of horseshoe vorticies, each with different length of bound vortex, but with all bound vorticies coincident along a single line
This line is called the Lifting Line
Circulation, G, varies along line of bound vorticies
Also have a series of trailing vorticies distributed over span
Strength of each trailing vortex = change in circulation along lifting line

13. PRANDTL’S LIFTING LINE THEORY
dG1
Example shown here will use 3 horseshoe vorticies

14. PRANDTL’S LIFTING LINE THEORY
dG1
dG2

15. PRANDTL’S LIFTING LINE THEORY
dG1
dG2
dG3

16. PRANDTL’S LIFTING LINE THEORY
dG1
dG2
dG3
Represent wing by a large number of horseshoe vorticies, each with different length of bound vortex, but with all bound vorticies coincident along a single line
This line is called the Lifting Line
Circulation, G, varies along line of bound vorticies
Also have a series of trailing vorticies distributed over span
Strength of each trailing vortex = change in circulation along lifting line
Example shown here uses 3 horseshoe vorticies
→ Consider infinite number of horseshoe vorticies superimposed on lifting line

17. PRANDTL’S LIFTING LINE THEORY
Infinite number of horseshoe vorticies superimposed along lifting line
Now have a continuous distribution such that G = G(y), at origin G = G0
Trailing vorticies are now a continuous vortex sheet (parallel to V∞)
Total strength integrated across sheet of wing is zero

18. PRANDTL’S LIFTING LINE THEORY
Circulation at y is G(y)
Change in circulation over dy is dG = (dG/dy)dy
Strength of trailing vortex at y = dG along lifting line
Consider arbitrary location y0 along lifting line
Segment dx will induce velocity at y0 given by Biot-Savart law
Velocity dw at y0 induced by semi-infinite trailing vortex at y is:

19. PRANDTL’S LIFTING LINE THEORY
Total velocity w induced at y0 by entire trailing vortex sheet can be found by integrating from –b/2 to b/2:
Equation gives value of
downwash at y0 due to
all trailing vorticies

20. SUMMARY SO FAR
We’ve done a lot of theory so far, what have we accomplished?
We have replaced a finite wing with a mathematical model
We did same thing with a 2-D airfoil
Mathematical model is called a Lifting Line
Circulation G(y) varies continuously along lifting line
Obtained an expression for downwash, w, below the lifting line
We want is an expression so we can calculate G(y) for finite wing (WHY?)
Calculate Lift, L (Kutta-Joukowski theorem)
Calculate CL
Calculate aeff
Calculate Induced Drag, CD,i (drag due to lift)

21. FINITE WING DOWNWASH
Recall: Wing tip vortices induce a downward component of air velocity near wing by dragging surrounding air with them ai
Equation for induced angle of attack
along finite wing in terms of G(y)

22. EFFECTIVE ANGLE OF ATTACK, aeff, EXPRESSION
aeff seen locally by airfoil
Recall lift coefficient
expression (Ref, EQ: 4.60)
a0 = lift slope = 2p
Definition of lift coefficient and Kutta-Joukowski
Related both expressions
Solve for aeff

23. COMBINE RESULTS FOR GOVERNING EQUATION
Effective angle of attack
(from previous slide)
Induced angle of attack
(from two slides back)
Geometric angle of attack = Effective angle of attack + Induced angle of attack

24. PRANDTL’S LIFTING LINE EQUATION
Fundamental Equation of Prandtl’s Lifting Line Theory
In Words: Geometric angle of attack is equal to sum of effective angle of attack plus induced angle of attack
Mathematically: a = aeff + ai
Only unknown is G(y)
V∞, c, a, aL=0 are known for a finite wing of given design at a given a
Solution gives G(y0), where –b/2 ≤ y0 ≤ b/2 along span

25. WHAT DO WE GET OUT OF THIS EQUATION?
Lift distribution
Total Lift and Lift Coefficient
Induced Drag

26. ELLIPTICAL LIFT DISTRIBUTION
For a wing with same airfoil shape across span and no twist, an elliptical lift distribution is characteristic of an elliptical wing planform

27. SPECIAL SOLUTION: ELLIPTICAL LIFT DISTRIBUTION
Points to Note:
At origin (y=0) G=G0
Circulation varies elliptically with distance y along span
At wing tips G(-b/2)=G(b/2)=0
Circulation and Lift → 0 at wing tips

28. SPECIAL SOLUTION: ELLIPTICAL LIFT DISTRIBUTION
Elliptic distribution
Equation for downwash
Coordinate transformation → q
See reference for integral
Downwash is constant over span for an elliptical lift distribution
Induced angle of attack is constant along span
Note: w and ai → 0 as b → ∞

29. SPECIAL SOLUTION: ELLIPTICAL LIFT DISTRIBUTION
We can develop a more
useful expression for ai
Combine L definition for elliptic profile with previous result for ai
Define AR because it
occurs frequently
Useful expression for ai
Calculate CD,i
CD,i is directly proportional to square of CL
Also called ‘Drag due to Lift’

30. SUMMARY: TOTAL DRAG ON SUBSONIC WING
Profile Drag
Profile Drag coefficient relatively constant with M∞ at subsonic speeds
Also called drag due to lift
May be calculated from
Inviscid theory:
Lifting line theory
Look up
(Infinite Wing)

31. SUMMARY Induced drag is price you pay for generation of lift
CD,i proportional to CL2
Airplane on take-off or landing, induced drag major component
Significant at cruise (15-25% of total drag)
CD,i inversely proportional to AR
Desire high AR to reduce induced drag
Compromise between structures and aerodynamics
AR important tool as designer (more control than span efficiency, e)
For an elliptic lift distribution, chord must vary elliptically along span
Wing planform is elliptical
Elliptical lift distribution gives good approximation for arbitrary finite wing through use of span efficiency factor, e

32. WHAT IS NEXT?
Lots of theory in these slides → Reinforce ideas with relevant examples
We have considered special case of elliptic lift distribution
Next step: develop expression for general lift distribution for arbitrary wing shape
How to calculate span efficiency factor, e
Further implications of AR and wing taper
Swept wings and delta wings
New A380:
Wing is tapered and swept