The function of a propulsive system is to produce a thrust. This can be
achieved only by imparting axial momentum to the fluid to force a backward
motion. The energy associated to the fluid is an inevitable loss.

The original theory, as first formulated by Rankine and Froude, excluded the viscous
effects, the rotation of the slipstream, and the uneven load distribution, with the
scope of evaluating the ideal efficiency of such a propulsive system (also called
*actuator disc*).

The rotor is degenerated into a disc perpendicular to the thrust, and is capable of
sustaining a pressure difference between its two sides, and of generating/imparting
linear momentum to the fluid that passes through it. The mechanism of thrust
generation requires the evaluation of the mass flow through a stream tube bounded by
the disc.

An important extension of the axial momentum theory is the introduction of an angular
momentum balance equation for the slipstream, to take into account the rotation
imparted to the fluid by the propulsive system.

The theory finds that the optimal
load on the disc is constant, and a small efficiency loss for lightly loaded
propellers (of the order of a few percent).

A basic result was that the efficiency
increases with the propeller diameter, and decreases with the increasing disc
loading.

The propeller should be as large and as lightly loaded as possible, because it is
more efficient to move a given mass of fluid through a large stream tube at low
speed, rather than through a small stream tube at high speed.

A more realistic interpretation of the propeller operation is the use of sectional
airfoil data to derive integral quantities for the entire propeller. This theory was
developed only when the airfoil theory was well understood and airfoil data were made
available by systematic wind tunnel experiments.

The blade element, or strip theory,
provides a means to determine forces and moments by assuming the blade as composed of
a number of aerodynamically independent cross-sections, whose characteristics are the
same as a blade at a proper angle of attack. Therefore the operation of a
cross-section is indirectly related to that of a two-dimensional airfoil. Its main
drawback is that experimental data are needed *a priori*.

Even when these data
are available the correlation between airfoil in axial flow and a rotating airfoil is
not straightforward, since it involves the prediction of the field of wake-induced
velocities. This problem is usually solved by defining an equivalent angle of attack
to be used in two-dimensional calculations.

### Combined Blade-Element and Momentum Theory

The combination of the blade element with the momentum theory makes it possible to
evaluate a field of induced velocity around the propeller, and therefore correct the
inflow conditions assumed in the primitive blade element theory. Also, the use of
correction terms for CD and CL from the finite aspect-ratio wing theory yields more
realistic terms for the forcing equations.

The induced velocities are not known until
the propeller loads are computed. With the loading available one can recompute the
field of induced velocities and thus iterate the process. Many problems have remained
to this day: the theory holds for not too loaded propellers; stall cannot be
predicted; hub and tip losses cannot be evaluated; unsteady and yawed operation are
beyond the theory.

#### 3-D Effects on Propellers

At a closer insight it has been found that the airfoil section locally has a
behavior significantly different than the rotating airfoil.

Some important aspects
are the following: 1) the lift coefficient increases at the blade hub; 2) the
lift-curve slope of the rotating airfoil is lower than the two-dimensional airfoil;
3) the rotating blade stalls at a higher angle of attack than the two-dimensional
airfoil. 4) the centrifugal effects are important for low aspect-ratio propellers,
negligible for slender propellers, helicopter rotors and wind turbines; 5) more
important are the Coriolis effects and the apparent pressure gradients due to
potential cross-flow; 6) the flow is essentially chordwise in regions of attached
boundary layers, and strongly radial beyond the separation line.

The description of the blade and its slipstream by vortex lines/surfaces removes some
of the limitations of the the BEM/momentum theory.

If the wake were to be known
exactly, then the field of induced velocities and other relevant quantities could be
computed more readily. This can be done, in fact, by appropriate manipulation of the
Biot-Savart equation for the inviscid vortex line. The use of
the *vortex theory* has opened the way to the solution of time-dependent
flows.

The method is quite flexible, since it allows the computation of single- and
multi-bladed rotors with arbitrary flight path, and free wake analysis in the
time-stepping mode.

One of the problems with the method is that the computational time increases linearly
(or exponentially) with the number of time steps (e.g. with the number of panels
released in the wake). A solution past the transient state (three or four rounds) may
easily require thousands of vortices.

CFD is the last word in rotor aerodynamics. Full potential, Euler and Navier-Stokes
methods have been exploding in the last decade, thanks also to the progress in
computer hardware. Virtually all problems have been attempted.

Full potential methods have been left behind due to the difficulty of modeling the
wake; Euler methods are limited by the lack of viscosity. Additional wake modeling
is needed is the grid has not enough resolution, but this problem is just contingent,
because limited by the current computer power.

The development of accurate CFD codes for a rotating blade is in itself a difficult
task, that requires lengthy computer programming and debugging sessions.

#### Euler Methods

Euler methods have been advanced as higher order schemes, requiring very fine grids,
have been developed (Strawn-Barth, 1993). In the most recent works it is proven that,
along with fine resolution and high order integration schemes, multi-block grids are
required to reduce the numerical diffusion of the vorticity. Typically, this occurs
at the blade tips. However, if the main interest is to compute the blade loads, a
multi-block grid is not needed, but the requirements on the resolution are still very
high without assumptions on the wake evolution.

#### Navier-Stokes

The complexity of most of the new CFD methods has led to a time lag in code
development. The designers of large scale codes must consider the manageability of a
new projects, which usually requires years of development, or dozens of
scientist-years.

State-of-the art numerical methods (see Landgrebe, 1994, for a
review) are likely to be fully exploited in the next generation of computer codes,
especially if higher-level symbolic languages will become available.