Blade Profile Drag

Momentum Disk theory assumes that induced velocity vi is produced uniformly over the disk without losses.  In reality rotor blades are required to be propelled around the disk to produce a sustained pressure difference and hence induce the required velocity. Additional power is required to propel the blades around the disk. This additional power term can be estimated using a relatively simple analysis.

the local velocity at the blade will be composed of the rotation and stream components,

for small tilt angles cos(αT) ≈ 1 and the area of blade component S = c .dr thus the elemental drag will be

The power required to drive this element of blade around the circuit will be

The power requirement to overcome this profile drag will be found by integrating these elemental components along the length of the blade. Complications arise due to blade motion, as blades experience different local velocities at different positions as they rotate around the disk. In this case an average power can be obtained by integrating the elemental power components over the angular extent of the disk path and then dividing by the length of this path.

As local blade velocity changes with both radius and angle, then it is likely that CD will also vary

due to Reynolds and Mach number effects. Analytical integration of  this CD variation is not practical due to the highly non-linear nature of the coefficient, so a simplified approach is used by assuming that an average CD applies at all locations. This average is normally evaluated by analysing the aerofoil section at 70% blade radius.

The profile power of the blade will then be found as,

One standard simplification that can be applied before integration, is to introduce advance ratio

 μ = U / ( Ω R ) and to use radial position fraction r/R.

After this substitution, the integration of the velocity components gives,

so that the profile power to drive one blade will be,

If there are N blades in the rotor then total profile power will be,

It is also standard practice to quote this in terms of rotor solidity , σ = ( c.N R ) / (π R2) = (c.N)/(π R)

Hence the final ideal theoretical result is,

In practice, there are additional losses in forward flight due to variation in CD and flow separations near the hub.  This can be accounted for by replacing the "3" with an empirically derived correction factor, k.

 From experimental investigation, k is in the range 4.5 to 4.7, depending on advance ratio.  A typical value of k=4.65 is commonly used.