Parasitic torque on struts
This document describes the parasitic torque loss due to drag on the struts of a vertical-axis wind turbine (VAWT). In vertical-axis turbines, radial struts connect the blades to the central shaft, and as they rotate through the fluid, they experience drag forces that oppose the rotation and reduce the net power output. This parasitic torque must be accounted for in performance calculations.
1 Physical Description
We consider a vertical-axis tidal turbine with
$$N$$
identical radial struts, each of length $$l$$
, constant diameter $$d$$
, and drag coefficient $$C_d$$
. The turbine rotates at a constant angular velocity $$\Omega$$
in a uniform flow of speed $$v$$
.
As the struts rotate, each point along the strut experiences a relative velocity that combines the tangential velocity due to rotation and the incoming flow velocity. The drag force on the struts acts to resist the rotation, creating a parasitic torque that reduces the net torque available from the turbine.
2 Derivation of Parasitic Torque
2.1 Relative Velocity
At radius
$$r$$
along a strut, the tangential speed due to rotation is:
$$U_t = \Omega r$$
The total relative velocity magnitude seen by the strut is approximated by:
$$U_{\text{rel}} \approx \sqrt{(\Omega r)^2 + 0.5v^2}$$
where
$$v$$
is a representative flow speed at the strut reference location.
This approximation neglects the azimuthal variation of the relative flow and the coupling between tangential and free-stream velocities. The free-stream contribution is therefore reduced by a factor 0.5, representing the azimuth-averaged projection of the drag force onto the tangential direction, which is the only component contributing to torque. The resulting expression defines an effective relative velocity suitable for estimating the mean parasitic drag and torque on the strut.
2.2 Drag Force on a Differential Element
The differential drag force on a small segment
$$\mathrm{d}r$$
of the strut is given by:
$$\mathrm{d}F_D = \frac{1}{2} \rho C_d d \, U_{\text{rel}}^2 \, \mathrm{d}r$$
where
$$\rho$$
is the fluid density and $$d$$
is the strut diameter (projected width for drag). Only the tangential component of the drag force contributes to torque, and in this simplified azimuth-averaged model, we assume the tangential projection factor is unity.
2.3 Differential Torque
The differential torque about the rotation axis is obtained by multiplying the drag force by the moment arm
$$r$$
:
$$\mathrm{d}Q = r \, \mathrm{d}F_D = \frac{1}{2} \rho C_d d \, r \, U_{\text{rel}}^2 \, \mathrm{d}r$$
2.4 Radial Integration
To obtain the total torque on a single strut, we integrate along the strut from
$$r=0$$
to $$r=l$$
:
$$Q_{\text{strut}} = \frac{1}{2} \rho C_d d \int_0^l r \left( (\Omega r)^2 + 0.5v^2 \right) \mathrm{d}r$$
Expanding the integrand:
$$Q_{\text{strut}} = \frac{1}{2} \rho C_d d \int_0^l \left( \Omega^2 r^3 + 0.5v^2 r \right) \mathrm{d}r$$
Evaluating the integral:
$$Q_{\text{strut}} = \frac{1}{2} \rho C_d d \left[ \frac{\Omega^2 l^4}{4} + \frac{v^2 l^2}{4} \right]$$
Simplifying:
$$Q_{\text{strut}} = \frac{\rho C_d d}{8} \left( \Omega^2 l^4 + v^2 l^2 \right)$$
2.5 Total Parasitic Torque for All Struts
For
$$N$$
identical struts, the total parasitic torque is:
$$T_{\text{parasitic}} = N \, Q_{\text{strut}} = N \frac{\rho C_d d}{8} \left( \Omega^2 l^4 + v^2 l^2 \right)$$
This can also be written as:
$$T_{\text{parasitic}} = \frac{N \rho C_d d l^2}{8} \left( \Omega^2 l^2 + v^2 \right)$$
3 Implementation in Ashes
In Ashes, the parasitic torque on struts is computed using the formula derived above. The calculation requires the following parameters:
- $$N$$: total number of struts (struts per blade$$\times$$number of blades)
- $$\rho$$: fluid density (water or air)
- $$d$$: strut diameter
- $$C_d$$: drag coefficient for a circular cross-section (typically around 1.0 for a cylinder)
- $$l$$: strut length (equal to rotor radius)
- $$\Omega$$: rotational speed [rad/s]
- $$v$$: representative flow speed at reference position
The parasitic torque is computed at each time step and subtracted from the aerodynamic or hydrodynamic torque generated by the blades to obtain the net torque available at the shaft. This ensures that the power losses due to strut drag are properly accounted for in the turbine performance calculations.
4 Remarks
The derivation presented here makes several simplifying assumptions:
- The struts are assumed to have constant diameter along their length
- The drag coefficient is assumed constant and independent of Reynolds number
- The relative velocity is azimuth-averaged, neglecting variations as the strut rotates
- The flow is assumed uniform across the rotor swept area
Despite these simplifications, the formula provides a reasonable estimate of the parasitic torque losses for preliminary design and performance assessment of vertical-axis turbines.