Performance coefficients

This calculation generates dimensionless power \(C_p = C_p(\lambda, \phi)\), torque \(C_q = C_q(\lambda, \phi)\) and thrust \(C_t = C_t(\lambda, \phi)\) coefficients for the rotor as a function of tip speed ratio \(\lambda\) and blade pitch angle \(\phi\). For this calculation option the rotor speed \(\omega\) is held constant for all simulations while a uniform steady wind speed \(U\) [\(\bunit{m/s}\)] and/or pitch angle \(\phi\) [\(\bunit{rad}\)] is varied. The wind speed \(U\) [\(\bunit{m/s}\)] is back calculated using the nominal rotor radius \(R\) and tip speed ratio \(\lambda\) by \(U = R\omega \lambda\).

The rotor performance coefficients are then defined

\[ \begin{flalign} A &= \pi R^2 \label{eq:areaEqn} \\[1ex] C_t &= \frac{T}{\frac{1}{2} \rho A U^2} \label{eq:thrustCoeff} \\[1ex] C_q &= \frac{Q}{\frac{1}{2} \rho R A U^2} \label{eq:torqueCoeff} \\[1ex] C_p &= \frac{Q \omega}{\frac{1}{2} \rho A U^3} \label{eq:powerCoeff} \end{flalign} \]

where \(A\) is the flat rotor swept area, \(\rho\) is the air density, \(Q\) is the aerodynamic torque at the hub and \(T\) is the aerodynamic thrust at the hub.

The nominal rotor radius \(R\) is the sum of the radius of blade connection \(R_h\) defined by the hub geometry plus the flat (without bending or coning) blade length \(R_b\) measured along the body-fixed \(z_{b,b}\) axis and then projected into the flat rotor plane . Thus, the total distance is measured in the rotor plane (without coning) as shown in Figure 1. The rotor plane is spanned by the \(y_{b,h}\) and \(z_{b,h}\) hub body-fixed axes.

The user must specify the tip speed ratio range by defining the minimum and maximum bounds and also an interval the calculation will step through.

The user can also vary the pitch angle when computing rotor performance coefficients. The range of pitch angles is defined by specifying the minimum and maximum pitch angles as well as the pitch angle step to define the intermediate pitch angle values that should be used. If only one pitch angle should be analysed then the minimum and maximum should be set equal at the appropriate angle and the step set to 0.

An example set of inputs are provided below. The performance coefficients are calculated for a rotor where the pitch angle is fixed at -2.0 degrees and the rotor speed is set at 19.95 RPM.

"SteadyCalculationType": "PerformanceCoefficients"
"SteadyCalculation": {
  "TipSpeedRatioRange": {
    "Minimum": 2.0,
    "Maximum": 16.5,
    "Interval": 0.1
  },
  "PitchRange": {
    "Minimum": -0.0349065850398866,
    "Maximum": -0.0349065850398866,
    "Interval": 0.0
  },
  "RotorSpeed": 2.0943951023932,
}

In addition to the power, thrust and torque coefficients blade deflections are also reported to the user. These are output in the root axes coordinate system.

Assumptions

Only the rotor is simulated in the performance coefficient calculation. To ensure that the rotor loading is azimuth independent a range of features are disabled and cannot be enabled by the user.

The user may simulate blade flexibility. If blade flexibility is included then the power, thrust and torque and so on are computed including the affect of blade deflection. However the area \(A\) (see Equation \(\eqref{eq:areaEqn}\)) used to normalise the power, thrust and torque is based on the flat blade length as described above.

If the rotor is rigid, varying rotor speed only changes the Reynolds number, so aerofoil properties will also change if they have been defined to interpolate on Reynolds number. However, if flexibility is included then the rotational speed does matter, because loads on the blade depend on the rotational speed and this can deflect the blade and change the angle of attack. This leads to different aerodynamic loads and different performance coefficients. In summary, if flexibility is included it will be useful to run Performance Coefficients calculation with different rotational speeds and find the one that corresponds to the wind speed you are interested in.

Table of Contents

Performance coefficients

Assumptions