Froude’s momentum theory of propulsion

1. inviscid,
2. incompressible, and
3. irrotational;and both
4. the velocity and
5. the static pressure are uniform over each cross section of the disk and stream-tube.

To take into account conservation of angular momentum the slip-stream should be subject to a rotation. As such assumption 4. and 5. have to be relaxed [3].

$$ \mathrm{Q}=\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{L}=\frac{\mathrm{d} m}{\mathrm{d}t}\omega \cdot r^2 $$

Some assumptions on the flow should be done to solve the problem [3]

1. Just as the annular stream-tube control volumes used in the slip-stream rotation analysis were assumed to be noninteracting [assumption (1)], it is assumed that there is no interaction between the analyses of each blade element
2. The forces exerted on the blade elements by the flow stream are determined solely by the two-dimensional lift and drag characteristics of the blade element airfoil shape and orientation relative to the incoming flow.

The contribution of each annulus to the total force and tordque is

$$ \mathrm{d} T = \frac{1}{2} B \rho U^2 (C_L \cos \phi -C_D \sin \phi ) c \cdot \mathrm{d} r $$

$$ \mathrm{d} Q = \frac{1}{2} B \rho U^2 (C_L \sin \phi + C_D \cos \phi ) c \cdot r \cdot \mathrm{d} r $$

[1] Anders Ahlstrom, "Aeroelastic Simulation of Wind Turbine Dynamics", Doctoral Thesis, Royal Institute of Technology Sweden

[2] Lin Wanga,*, Xiongwei Liu b, Athanasios Koliosa, "State of the Art in the Aeroelasticity of Wind Turbine Blades: Aeroelastic Modelling", Renewable and Sustainable Energy Reviews, Volume 64, October 2016, pp 195-210 DOI:10.1016/j.rser.2016.06.007

[3] M. K. Rwigema "PROPELLER BLADE ELEMENT MOMENTUM THEORY WITH VORTEX WAKE DEFLECTION", 27TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

-- RobertoBernetti - 29 Nov 2017