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Chapter 5 Incompressible Flow over Finite Wings

Chapter 5 Incompressible Flow over Finite Wings. Yanjie Li Harbin Institute of Technology Shenzhen Graduate School. Outline . Downwash and Induced Drag Vortex filament, Biot-Savart Law and Helmholtz’s Theorems Prandtl’s Classical Lifting-Line Theory

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Chapter 5 Incompressible Flow over Finite Wings

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  1. Chapter 5Incompressible Flow over Finite Wings Yanjie Li Harbin Institute of Technology Shenzhen Graduate School

  2. Outline • Downwash and Induced Drag • Vortex filament, Biot-Savart Law and Helmholtz’s Theorems • Prandtl’s Classical Lifting-Line Theory • A Numerical Nonlinear Lifting-Line Method

  3. A Question • Are the aerodynamic coefficients of finite wings the same as those for the airfoil shape from which the wing is made?? • No!! • Why?

  4. Downwash and Induced Drag(1) • The flow over an airfoil is two-dimensional The flow over the finite wing is three-dimensional

  5. Downwash and Induced Drag(2) • The leaked flow around the tips creates a trailing vortex

  6. Downwash and Induced Drag(3) • Downwash These wing-tip vortices of the wing induce a small velocity component in the downward direction, which is called Downwash, denoted by

  7. Downwash and Induced Drag(4) • Downwash has two important effects on the local airfoil section: • Effective angle of attack • Induced drag • The tilting backward of the lift vector • A net pressure imbalance, a type of “pressure drag”

  8. Note: A difference in nomenclature Two-dimensional: Three-dimensional: Road map for Chapter 5

  9. Vortex Filament • Curved vortex filament Curved vortex filament Straight vortex filament

  10. Biot-Savart Law Biot-Savart Law: The Biot-Savart law is a general result of potential theory. The law is similar to that of electromagnetic fields induced by electrical currents.

  11. Apply Boit-Savart Law to a straight vortex filament Biot-Savart Law The direction is downward +

  12. For the semi-infinite vortex filament in Figure 5.10

  13. Helmholtz’s Vortex Theorem: Lift distribution New concepts: Washout and Washin

  14. Prandtl’s Classical Lifting-Line Theory Replace the finite wing with a bound vortex , a vortex filament of strength that is somehow bound to a fixed location in a flow.

  15. Horseshoe vortex

  16. The velocity at any point along the bound vortex induced by the trailing semi-infinite vortices • The downwash approach an infinite value at the tips!! How to solve this problem??

  17. Lifting–Line Theory • Replace one single horseshoe vortex by a large number of horseshoe vortices with a different length of the bound vortex, but with all the bound vortices coincident with a single line---- lifting line

  18. Consider an infinitesimally small segment of the lifting line located at the coordinate .The circulation at is . The change in circulation over the segment is An infinite number of horseshoe vortex with a vanishingly small strength The velocity at induced by the entire semi-infinite trailing vortex located at is

  19. Our central problem is to calculate for a given finite wing Thin airfoil theoretical value Lift slope Fundamental equation of Prandtl’s theory

  20. Special Case: Elliptical Lift Distribution Circulation distribution given by Eq. 4.26 Constant

  21. A more useful expression for Aspect ratio

  22. + The induced drag coefficient

  23. Constant Constant The chord must vary elliptically along the span • Drag due to lift • The Induced drag coefficient is inversely proportional to aspect ratio

  24. Elliptical circulation distribution, Elliptical lift distribution, Elliptical platform Constant downwash It provides a reasonable approximation for the induced drag coefficient for an arbitrary finite wing

  25. General Lift Distribution Consider the transformation the elliptical circulation distribution A Fourier sine series approximation for general circulation distribution Differentiating the circulation distribution Applying the Prandtl’s lifting-line theory

  26. Ea. (4.26) +

  27. The induced drag coefficient The induced angle of attack + Defining The elliptical lift distribution yields the minimum induced drag

  28. The minimum induced drag but expensive to manufacture Not optimum lift distribution But easy to manufacture Trade-off design a taper ratio

  29. A Numerical Nonlinear Lifting-Line Method The classical Prandtl lifting line theory assumes that a linear variation of versus . In fact, It is nonlinear.

  30. Consider the most general case of a finite wing of given platform and geometric twist, with different airfoil at different spanwise stations. Assume that we have all the experiment data for the lift curve of the airfoil sections. A numerical method: Simpson’s Rule Three-point estimation of the derivative Singularity

  31. Compare the classical solution of Prandtls’ with the numerical method. The latter has an excellent agreement Compare the numerical solution with exiting experimental data obtained by Univ. of Maryland. The numerical lift-line solution at high angle of attack agrees with experiment within 20 percent or much closer. Therefore the numerical solution gives reasonable preliminary engineering results for the high-angle-of-attack region. However, the flow is three dimensional. The basic assumptions of lifting-line theory, classical and numerical cannot properly account for such three dimensional flow.

  32. To be continued

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