Some results are presented about the study of airloads of the helicopter rotor blades, the aerodynamic characteristics of airfoil sections, the physical features, and the techniques for modeling the unsteady effects found on airfoil operating under nominally attached flow conditions away from stall. The unsteady problem was approached on the basis of Theodorsen's theory, where the aerodynamic response (lift and pitching moment) is considered as a sum of noncirculatory and circulatory parts. The noncirculatory or apparent mass accounts for the pressure forces required to accelerate the fluid in the vicinity of the airfoil. The apparent mass contributions to the forces and pitching moments, which are proportional to the instantaneous motion, are included as part of the quasi-steady result.

The most important component of the helicopter is the main rotor for which there is a great deal of activity in developing new and improved mathematical models that predict the flow physics. A high tip speed gives the rotor a high level of stored rotational kinetic energy and reduces the rotor torque required for a given power, but there are two important factors that work against the use of a high tip speed: compressibility effects and noise.

The additional effects of compressibility on the overall rotor profile power requirements, when the tip of the advancing blade approaches and exceeds the drag divergence Mach number, were estimated using blade element theory combined with the airfoil section characteristics [

The classical unsteady aerodynamic theories describing the observed behavior have formed the basis for many types of rotor analysis. The tools for the analysis of 2D, incompressible, and unsteady aerodynamic problems were extended to compressible flows, being a basis for developing linearized unsteady aerodynamic models applicable to compressible flows [

The helicopter rotor airfoil must assure a high maximum lift coefficient, a high drag divergence Mach number, a good lift-to-drag ratio over a wide range of Mach number, and a low pitching moment. At high angles of attack, the adverse pressure gradients produced on the upper surface of the airfoil result in a progressive increase in the thickness of the boundary layer and cause some deviation from the linear lift versus angle of attack behavior. On many airfoils, the onset of flow separation and stall occurs gradually with increasing angle of attack, but on some airfoils (those with sharp leading edges) the flow separation may occur suddenly.

The region of the rotor disk affected by compressibility effects is shown in Figure

Helicopter rotor blade in forward flight.

The angular or rotational speed of the rotor is denoted by

The increment in profile power

The aerodynamic behavior of airfoils in the high AoA regime is important for predicting the adverse effects produced in the reverse flow regime on the rotor. In the reverse flow region, the direction of the relative flow vector changes from the trailing edge toward the leading edge of the airfoil. While the fundamental process of the blade wake and tip vortex formation is similar to that found with a fixed wing, one difference with helicopter tip vortices is that they are curved, and so they experience a self-induced effect. Another complication with helicopter rotors is that the wakes and tip vortices from other blades can lie close to each other and to the plane of blade rotation, and so they have large induced effects on the blade lift distribution.

If the wake is assumed to be undistorted in the tip path plane and no wake contraction occurs in the radial direction (Figure

Tip vortex trajectory.

One important parameter used in the description of unsteady aerodynamics and unsteady airfoil behavior is the reduced frequency,

Helicopter main rotor.

The approach to modeling of unsteady aerodynamic effects through an extension of steady, 2D thin airfoil theory gives a good level of analysis of the problem and provides considerable insight into the physics responsible for the underlying unsteady behavior. The Laplace’s equation for incompressible flow is eliptic; therefore, the unsteady aerodynamic theories cannot be obtained in a corresponding analytical form.

The rate of change of the impulse vector, in general, is not in the direction of the acceleration of the body. The external force

Solid-fluid surface.

If the surface is represented by a scalar function of position and time,

The fluid force acting on a rigid body of arbitrary shape translating with a velocity

The unit vector

The component of the vector

On the other hand,

Rotor blade element.

In the Cartesian coordinate system, the vectors

The oscillatory motion of the airfoil can be decomposed into contributions associated with angle of attack which is equivalent to a pure plunging motion (Figure

Plunge velocity.

Pitch rate.

A plunge velocity

For a pitch rate imposed about an axis at “

The problem of finding the airloads on an oscillating airfoil was solved by Theodorsen, who gave a solution to the unsteady airloads on a 2D harmonically oscillated airfoil in inviscid, incompressible flow, with the assumption of small disturbances [

For a general motion, where an airfoil of chord

Bessel functions.

The Hankel functions in above expression are

Theodorsen’s function.

It could be appreciated that

This effect can be seen if a pure oscillatory variation in angle of attack is considered, that is,

These loops are circumvented in a counterclockwise direction such that the lift is lower than the steady value, when

For a harmonic variation in

The first term inside the brackets is the circulatory term, and the second term is the apparent mass contribution, which is proportional to the reduced frequency and leads the forcing by a phase angle of

The normalized lift amplitude is

Normalized lift amplitude.

Phase angle.

At lower values of reduced frequency, the circulatory terms dominate the solution. At higher values of reduced frequency, the apparent mass forces dominate.

For a harmonic plunging motion such as that contributed by blade flapping, the forcing is

For harmonic pitch oscillations, additional terms involving pitch rate

Von Karman and Sear analyzed the problem of a thin airfoil moving through a sinusoidal vertical gust field, where the gust can be considered as an upwash velocity that is uniformly convected by the free stream. The forcing function in this case is

In terms of Bessel functions, Sears’s function is given by

Sears function.

The kinematics of the pitching and plunging airfoil of a typical blade element is the resultant of a combination of forcing from collective and cyclic blade pitch, twist angle, elastic torsion, blade flapping velocity, and elastic bending. At low angles of attack with fully attached flow, the various sources of unsteady effects manifest primarily as moderate amplitude and phase variations relative to the quasi-steady airloads. At higher angles of attack when time-dependent flow separation from the airfoil may be involved, the dynamic stall may occur. The amplitude and phase effects produced by the stalled airloads can lead to various aeroelastic problems on the helicopter rotor that may limit its performance. The need to control the aerodynamic forces on the rotor requires that the pitch of each blade be changed individually as the blades rotate about the shaft.

The first flap frequency of a helicopter rigid blade is about

When a wing’s angle of attack is increased rapidly, it can momentarily generate a higher maximum lift coefficient than it could if the angle of attack was increased slowly. This overshoot can be related to the change in angle of attack during the time required for the air to travel one chord length. The dynamic overshoot is attributed to two effects (for the airfoils that stall first at the leading edge): the delay in the separation of the boundary layer and the momentary existence of a vortex shed at the leading edge after the boundary layer does separate. The delay in separation corresponds to the finite time required for the aft edge of the separation bubble to move forward to its bursting position. On the other hands, the airfoil can generate high lift as a result of a vortex that is shed at the leading edge at the instant of stall. The vortex travels back over the top of the airfoil carrying with it a low pressure wave that accounts for the very large lift coefficient. Airfoils that stall first at the trailing edge also exhibit a dynamic overshoot but considerably less than those airfoils that have leading edge stall.