MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

1. MAE 3241: AERODYNAMICS ANDFLIGHT 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: 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 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 • 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 Instead of G=constant We need a way to let G=G(y)

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 • 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 dG2 dG3

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 • 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: • 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

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