Analyse : Multiple streamtube model, [pg2], [code] | single streamtube | Glauert empirical formula | about Naca airfoils data | finite aspect ratio on airfoil, [code] | dimensionless analysis on Darrieus rotor efficiency

Multiple streamtubes momentum model for Darrieus rotor

In the multiple streamtube model (Strickland 1976 version), the wind flow over the rotor is divided into multiple streamtubes as shown in the figure above. The tube cross section area remains the same along the path from upstream to downstream even though it is expected to expand as the wind velocity is slowed down. Also, the path of the wind remains straight even though it is expected to be deflected slightly when being ‘hit’ by the moving blade. A more realistic flow to illustrate both point is shown in the figure below. Anyway, these illogical flow field assumptions made it possible to achieve a simple analysis but fortunately still with acceptable accuracy.

Streamline from fixed wake vortex model
(Note the two deflection directions when the wind got hit by the blade twice)

The wind start blowing from the undisturbed far upstream with uniform velocity U. Wind velocity in each streamtube start decreasing from U at different rate until it reaches the blade circular rotation path (upwind semicircle) where the velocity is U’. Then these velocities remain constant when the wind moves thru the blade circular path even though it is expected to decrease in here too. After the leaving the blade circular path (downwind semicircle), the wind speeds start decreasing again at different rate from each other. At far wake downstream the velocities finally settle down. For each streamtube, the total upstream velocity decrement is equal to the total downstream decrement as proved by the Bernoulli equation in the Betz limit (look at links section)

Each of the streamtube experience continuous momentum loss and two pulses of aerodynamic forces per revolution. The rate of momentum change (loss) in the wind as it moves over the circular blade path can be calculated if the velocity U’ is known. Also, the aerodynamic forces acting on the blade (at certain position and rpm) can estimated from lift and drag data if the velocity U’ is known. To solve this, the streamtube model equate the momentum rate to the force to find the velocity U’. In more precisely, the component of aerodynamic forces in the wind direction or the thrust force is equalised to the momentum loss. In a streamtube, the forces only act while the blade is rotating in it (once upwind and once downwind) and for the rest (most) of the time, the forces are zeros. So, the streamtube model averaged up all these forces (zeros and two nonzero) to find an averaged value, which can be assumed to act in that streamtube all the time.

The simple velocity field is solved when the U’ velocities are found. From the U’ velocities, the aerodynamics forces acting on the blade in every angular position can be found. Finally, these values can be averaged to find the important average torque and power per revolution.

Some contradictions arisen from this model. First the constant area streamtube with increasing wind speed means the mass flowrate increased, i.e. continuity equation not satisfied. Anyway, the mass flowrate is taken with wind speed U’ at the rotor swept area. Second, the aerodynamics forces parallel to the wind direction or crosswind force exist but no momentum change occurs in that direction. However, the upwind blade experience crosswind force in the direction opposite to the downwind blade, so the effects of these two forces can be assumed as sort of cancelled out.

The following is the equations for multiple streamtube model for simple STRAIGHT BLADED Darrieus rotor only. All the values are normalised to reduce repetitive work and to greatly improve understanding (of the result). So instead of solving for the dimensional wind speed U’ directly, the U’ is normalised to a less direct but useful dimensionless variable, the induction factor as defined below.

Last updated at November 6, 2002
Comments are welcomed