One airfoil blade

1. 10 ms
2. 5 ms
3. 10 ms RPM 10
4. 10 ms - unsteady BEM
5. 5 ms - unsteady BEM
6. 10 ms RPM 10 - unsteady BEM
7. 10 ms pitch 20
8. 5 ms pitch 20
9. 10 ms RPM 10 pitch 20
10. 10 ms pitch 20 - unsteady BEM
    
11. 5 ms pitch 20 - unsteady BEM
12. 10 ms RPM 10 pitch 20 - unsteady BEM
13. 10 ms twist
14. 5 ms twist
15. 10 ms RPM twist
16. 10 ms twist - unsteady BEM
17. 5 ms twist - unsteady BEM
18. 10 ms RPM 10 twist - unsteady BEM

2 Model

• one variant has no twist (when the pitch is 0, the chordline of each blade station is in the rotor plane)
• one variant has twist, as shown in the table below

1. define initial axial and tangential induced velocities as 
   $$a = 0.33$$
    and 
   $$a_t = 0$$
   , respectively
2. the flow angle is defined as 
   $$\phi = \tan^{-1}\frac{(1-a)\cdot V}{(1+a_t)\cdot\omega\cdot(r+r_h)}$$
3. the angle of attack is obtained by substracting the twist 
   $$\theta$$
    and the pitch
   $$\gamma$$
   to the flow angle. The angle of attack is thus 
   $$\alpha = \phi-\theta-\gamma$$
4. the lift and drag coefficients 
   $$C_L$$
    and
   $$C_D$$
   are then looked up from the polars. Linear interpolation is used if the current angle of attack is not in the look-up table.
5. the blade solidity is calculated as 
   $$\sigma = \frac{cB}{2\pi(r+r_h)}$$
   , where 
   $$B = 1$$
   is the number of blades
6. new values for the induced velocities are computed as
   
   $$a = \frac{1}{\frac{4\sin^2\phi}{\sigma C_n}+1}$$
   and
   
   $$a_t = \frac{1}{\frac{4\sin\phi\cos\phi}{\sigma C_t}+1}$$
   
   where 
   $$C_n = C_L\cos\phi+C_D\sin\phi$$
   and 
   $$C_t = C_L\sin\phi - C_D\cos\phi$$
7. compare the induced velocities to the previous values. If the difference is larger than 0.001, repeat steps 2 to 6
8. the relative velocity is defined as 
   $$V_{rel}=\sqrt{((1-a)\cdot V)^2+((1+a_t)\cdot\omega\cdot(r+r_h))^2}$$
9. the lift and drag forces, for blade station 
   $$i$$
   are then
   
   $$F_{D,i} = \frac{1}{2}\rho cC_DV_{rel}^2$$
   and
   
   $$F_{L,i} = \frac{1}{2}\rho cC_LV_{rel}^2$$
   
   
10. the thust and torque forces are then
    
    $$F_{th,i} = F_{L,i}\cos\phi+F_{D,i}\sin\phi$$
    and
    
    $$F_{t,i} = -F_{D,i}\cos\phi + F_{L,i}\sin\phi$$

4 Results