Practical Ship Hydrodynamics

44 Practical Ship Hydrodynamics
2.3.2 Momentum theory
Momentum theory models the propeller as a simple actuator disk accelerating
the flow in the axial direction by somehow creating a pressure jump in the
propeller plane. The propeller is then seen as a continuous circular disk with
infinite blades and AE/A0 D 1. The model is too crude to be of any value
in propeller design, but allows some valuable insight into the global mechanisms
of a propeller. The momentum theory regards inflow and outflow of the
propeller plane as the flow through a tube of varying cross-section, but always
of circular shape. Only the longitudinal velocity component is considered, i.e.
the velocity is a scalar quantity.
The inflow to the propeller is given by Ð uA Ð AA where AA is the crosssectional
area of the considered propeller plane. The propeller induces a
velocity jump to the outflow velocity uj and the cross-sectional area of the
‘flow tube’ is Aj. ThethrustTis the change in the momentum:
T D Ð uA Ð AA Ð ⊲uj uA⊳
Continuity requires Aj Ð uj D AA Ð uA, i.e. the flow contracts after the propeller
due to the higher velocity (Fig. 2.5).
The velocity in the propeller plane is the average between the velocities
far upstream and far downstream of the propeller in this model. Bernoulli’s
law couples the pressure to the velocity yielding qualitatively the distribution
showninFig.2.5.
The actuator disk yields an ideal efficiency for the propeller of:
p
u j
u A
i D 2uA
uj C uA
Ship
x
x
Figure 2.5 Momentum theory considers propeller flow as one-dimensional flow with sudden
pressure jump accelerating velocity from uA to uj