◆ Helmho| tz's Theorems For inviscid and incompressible flows 1. The strength of a vortex filament is constant along its length 2. A vortex filament cannot end in a fluid; it must extend to the boundary of the fluid or form a closed path o Concept of lift distribution along the span of a finite wing 1. L(1): lift per unit span at location y1
Helmholtz’s Theorems For inviscid and incompressible flows 1. The strength of a vortex filament is constant along its length. 2. A vortex filament cannot end in a fluid; it must extend to the boundary of the fluid or form a closed path. Concept of lift distribution along the span of a finite wing 1. : lift per unit span at location ( ) 1 L y 1 y
2. With the different location in the span direction, the chord and attack angle may be different, that means C, a will be different 3, Concept of geometric twist washout and washin a has a distribution along the span direction 4. Concept of aerodynamic twist a,so has a distribution along the span direction 5. As there is different distribution of the chord of the airfoil, geometric angle of attack, and zero lift angle of attack, the lift per unit span at location y2, L(2) will be different from L(u)
2. With the different location in the span direction, the chord and attack angle may be different, that means will be different. c, 3. Concept of geometric twist. washout and washin. has a distribution along the span direction 4. Concept of aerodynamic twist. has a distribution along the span direction L=0 5. As there is different distribution of the chord of the airfoil, geometric angle of attack, and zero lift angle of attack, the lift per unit span at location , will be different from 2 y ( ) 2 L y ( ) 1 L y
6. As the lift per unit span is proportional to the circulation, so, the circulation is also a function of y 7. The lift distribution goes to zero at the wing tips Front view L=L(v)=pooVoor() of wing b 8. The calculation of the lift distribution L'(y) [or the circulation distribution r(y)] is one of the central problems of the finite-wing theory
6. As the lift per unit span is proportional to the circulation, so, the circulation is also a function of y 7. The lift distribution goes to zero at the wing tips. 8. The calculation of the lift distribution [or the circulation distribution ] is one of the central problems of the finite-wing theory. L( y) ( y)
5.3 Prandtl's Classical Lifting-line Theory iMportance of the Prandtl's lifting-line theory o bound vortex and free vortex o Replacement of the finite wing with a bound vortex y b Free- trailing vortex Replace finite wing with bound vortex Free-trailing vortex Finite wing Horseshoe vortex
5.3 Prandtl’s Classical Lifting-line Theory Importance of the Prandtl’s lifting-line theory bound vortex and free vortex Replacement of the finite wing with a bound vortex