Interpolation of aerodynamic coefficients

To compute the Aerodynamic loads on a blade element, the  BEM algorithm  requires the lift  and drag coefficients . These are provided for different angles of attack and different Reynolds numbers in the  Airfoil polar file . 

1 Angle of attack interpolation

When the  BEM algorithm  requires aerodynamic coefficients for an angle of attack that has not been provided in the  Airfoil polar file , Ashes carries out a linear interpolation between the two closest available angles. For a linear interpolation, it is assumed that the aerodynamic coefficient, for example the lift coefficient 
$$C_L$$
can be expressed as
$$C_L(\alpha)=a\cdot\alpha+b$$

where 
$$\alpha$$
 is the angle of attack, and 
$$a$$
 and 
$$b$$
 are the slope and the intercept  (see https://en.wikipedia.org/wiki/Linear_interpolation ).
If the lift coefficients 
$$C_{L1}$$
and 
$$C_{L2}$$
are given at the angles of attack 
$$\alpha_1$$
and 
$$\alpha_2$$
, respectively, then the following equation applies:
$$C_L(\alpha) = C_{L1}+(\alpha-\alpha_1)\frac{C_{L2}-C_{L1}}{\alpha_2-\alpha_1}$$
Equation 1 Linear interpolation of the lift coefficient




If the polar is only provided for a number of angles of attack that does not extend from -180 to 180 degrees, the  Viterna extrapolation  is used to produce coefficients for the whole range of angles.


2 Reynolds number interpolation

It is also possible to provide polars for different Reyniolds numbers. In this case, Ashes will compute the Reynolds number at any given Blade aerodynamical station  and the  BEM algorithm  will use the aerodynamic coefficients based on that Reynolds number.
For example. to find the lift coefficient
$$C_L$$
for an angle of attack
$$\alpha$$
and a Reynolds number
$$Re$$
, the process is as follows:
  1. The BEM algorithms asks for the lift coefficient 
    $$C_L(\alpha, Re)$$
  2. Ashes finds the two polars with Reynolds numbers closest to 
    $$Re$$
  3. Within each of these two polars, Ashes finds the two angles of attack closest to 
    $$\alpha$$
     and performs a linear interpolation (as described in Equation 1 ). This produces two lift coefficients, 
    $$C_L(\alpha,Re_1)$$
     and 
    $$C_L(\alpha,Re_2)$$
    .
  4. A new interpolation is perfoemrd between these two lift coefficients. This interpolation can be either Linear  or Logarithmic  (as defined by the Polar interpolation scheme  in the Advanced  section of the  Aerodynamics  tab)

2.1 Linear interpolation

If a linear interpolation is selected, the aerodynamic coefficients are computed as described in Equation 1 . For example, the lift coefficient will be
$$C_L(\alpha,Re) = C_{L}(\alpha,Re_1)+(Re-Re_1)\frac{C_L(\alpha,Re_2)-C_L(\alpha,Re_1)}{Re_2-Re_1}$$

2.2 Logarithmic interpolation

With regards to the variation of aerodynamic coefficients with respect the Reynolds number,  Thoms (2016)  shows that aerodynamic coefficients exhibit a logarithmic dependency. In his work, it is shown that a good fit for the lift and drag coefficients is obtained when assuming the following functions:

$$C_L(Re)=a\cdot\log_{10}(Re)+b$$
and 
$$\log_{10}(C_D(Re))=a\cdot\log_{10}(Re)+b$$

This implies that the aerodynamic coefficients can be expressed as 
$$C_L(\alpha,Re) = C_{L}(\alpha,Re_1)+\log_{10}(Re/Re_1)\frac{C_L(\alpha,Re_2)-C_L(\alpha,Re_1)}{\log_{10}(Re_2/Re_1)}$$
and 

$$\log_{10}(C_D(\alpha,Re)) = \log_{10}(C_{D}(\alpha,Re_1))+\log_{10}(Re/Re_1)\frac{\log_{10}(C_D(\alpha,Re_2)/C_D(\alpha,Re_1))}{\log_{10}(Re_2/Re_1)}$$


Note: the moment coefficient is treated assuming the same behaviour as the lift coefficient