1. 1 MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Finite Wings: General Lift Distribution Summary April 18, 2011 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. 2 SUMMARY: PRANDTL’S LIFTING LINE THEORY (1/2) Fundamental Equation of Prandtl’s Lifting Line Theory Geometric angle of attack, , is equal to sum of effective angle of attack,  eff, plus induced angle of attack,  i Equation gives value of Downwash, w, at y 0 Equation for induced angle of attack,  i, along finite wing

3. 3 SUMMARY: PRANDTL’S LIFTING LINE THEORY (2/2) Lift distribution per unit span given by Kutta-Joukowski theorem Total lift, L Lift coefficient, C L Induced drag, D i Induced drag coefficient, C D,i

4. 4 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:  =  eff +  i Only unknown is  (y) –V ∞, c, ,  L=0 are known for a finite wing of given design at a given a –Solution gives  (y 0 ), where –b/2 ≤ y0 ≤ b/2 along span

5. 5 WHAT DO WE GET OUT OF THIS EQUATION? 1.Lift distribution 2.Total Lift and Lift Coefficient 3.Induced Drag

6. 6 GENERAL LIFT DISTRIBUTION (§5.3.2) Circulation distribution Transformation –At  =0, y=-b/2 –At  = , y=b/2 Circulation distribution in terms of  suggests a Fourier sine series for general circulation distribution N terms –now as many as we want for accuracy An’s are unkowns, however must satisfy fundamental equation of Prandtl’s lifting- line theory

7. 7 GENERAL LIFT DISTRIBUTION (§5.3.2) General circulation distribution Lifting line equation Finding d  /dy Transform to  Last integral has precise form for simplification

8. 8 GENERAL LIFT DISTRIBUTION (§5.3.2) Evaluated at a given spanwise location,  0 is specified Givens: –b: wingspan –c(  0 ): chord at the given location for evaluation –The zero lift angle of attack,  L=0 (  0 ), for the airfoil at this specified location Note that the airfoil may vary from location to location, and hence the zero lift angle of attack may vary from location to location Can put twist into the wing –Geometric twist –Aerodynamic twist This is one algebraic equation with N unknowns written at  0 Must choose N different spanwise locations to write the equation to give N independent equations

9. 9 WING TWIST

10. 10 GENERAL LIFT DISTRIBUTION (§5.3.2) General expression for lift coefficient of a finite wing Substitution of expression for  (  ) and transformation to  Integral may be simplified C L depends only on leading coefficient of the Fourier series expansion (however must solve for all An’s to find leading coefficient A 1 )

11. 11 GENERAL LIFT DISTRIBUTION (§5.3.2) General expression for induced drag coefficient Substitution of  (  ) and transformation to  Expression contains induced angle of attack,  i (  ) Expression for induced angle of attack Can be mathematically simplified Since  0 is a dummy variable which ranges from 0 to  across the span of wing, it can simply be replaced with 

12. 12 GENERAL LIFT DISTRIBUTION (§5.3.2) Expression for induced drag coefficient Expression for induced angle of attack Substitution of  i (  ) in C D,i Mathematical simplification of integrals More simplifications leads to expression for induced drag coefficient

13. 13 GENERAL LIFT DISTRIBUTION (§5.3.2) Repeat of expression for induced drag coefficient Repeat of expression for lift coefficient Substituting expression for lift coefficient into expression for induced drag coefficient Define a span efficiency factor, e, and note that e ≤ 1 –e=1 for an elliptical lift distribution

14. 14 VARIOUS PLANFORMS FOR STRAIGH WINGS Elliptic Wing Rectangular Wing Tapered Wing crcr ctct

15. 15 INDUCED DRAG FACTOR,  (e=1/(1+  ))

16. 16 SPECIAL CASE: Elliptical Wings → Elliptical Lift Distribution

17. 17 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

18. 18 SUMMARY: ELLIPTICAL LIFT DISTRIBUTION (1/2) Points to Note: 1.At origin (y = 0)  =  0 2.Circulation and Lift Distribution vary elliptically with distance, y, along span, b 3.At wing tips  (-b/2) =  (b/2) = 0 –Circulation and Lift → 0 at wing tips y/b /0/0

19. 19 SPECIAL SOLUTION: ELLIPTICAL LIFT DISTRIBUTION Elliptic distribution Equation for downwash Coordinate transformation →  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  i → 0 as b → ∞

20. 20 SUMMARY: ELLIPTICAL LIFT DISTRIBUTION Downwash is constant over span for an elliptical lift distribution Induced angle of attack is constant along span for an elliptical lift distribution Total lift Alternate expression for induced angle of attack, expressed in terms of lift coefficient Induced drag coefficient For an elliptic lift distribution, the chord must vary elliptically along the span → the wing planform is elliptical in shape

21. 21 SPECIAL SOLUTION: ELLIPTICAL LIFT DISTRIBUTION C D,i is directly proportional to square of C L Also called ‘Drag due to Lift’ We can develop a more useful expression for  i Combine L definition for elliptic profile with previous result for  i Define AR because it occurs frequently Useful expression for  i Calculate C D,i