Blade Element Analysis

Whatever progress the airplane might make, the helicopter will come to be taken up by advanced students of aeronautics.

Thomas Edison (circa, 1920)

3,1 Introduction

The blade element theory (BET) forms the basis of most modem analyses of helicopter rotor aerodynamics because it provides estimates of the radial and azimuthal distributions of blade aerodynamic loading over the rotor disk. The BET assumes that each blade section acts as a quasi-2-D airfoil to produce aerodynamic forces (and moments). Tip loss and other empirical factors may be applied to account for three-dimensional effects. Rotor performance can be obtained by integrating the sectional airloads at each blade ele­ment over the length of the blade and averaging the result over a rotor revolution. Therefore, unlike the simple momentum theory, the BET can be used as a basis to help design the rotor blades in terms of the blade twist, the planform distribution and perhaps also the airfoil shape to provide a specified overall rotor performance.

The idea of the BET was apparently first suggested by Drzewiecki (1892, 1920) for the analysis of airplane propellers, although Glauert (1935) indicates that in the nineteenth century F. W. Lanchester also made contributions to solving the problem – see Lanchester (1915). At the beginning of the twentieth century, there was considerable scientific debate about the proper theoretical aerodynamic analysis of propellers and helicopter rotors, mainly between Stefan Drzewiecki and Louis Breguet – see Liberatore (1998). The principles of blade element theory assume no mutual influence of adjacent blade elements sections; these sections are idealized as 2-D airfoils. However, the effects of a nonuniform “induced inflow” across the blade (its source from the rotor wake) is accounted for through a modification to the angle of attack (AoA) at each blade element. Unless we make some simple analytic assumption for the distribution of induced velocity over the disk, such as a uniform or linear distribution, the blade element calculation is quite a formidable undertaking because it must more precisely represent the highly nonuniform velocity field induced by the vortical wake trailed from each blade, as well as to account for the influence of all the blades and possibly airframe components. However, if the induced velocity can be calculated, or even approximated, then the net thrust and power and other forces and moments acting on the rotor can be readily obtained.

In an extension to the basic approach, the BET and momentum theories were linked together by Reissner (1910,1937, 1940), de Bothezat (1919), and Glauert (1935) to define the induced velocity or induced AoA distribution. A similar approach for hovering helicopter rotors was developed by Gustafson & Gessow (1946) and Gessow (1948), and the approach is reviewed in Gessow & Myers (1952). The BET was extended to explicitly include the influence of the vortical wake (and the other blades) through an induced AoA component

as calculated by means of the Biot-Savart law. These basic ideas were first pursued at the beginning of the twentieth century by Joukowski – see Tokaty (1971), Glauert (1922), Bienen & von Karman (1924), and Lock et al. (1925). Betz (1919, and appendix therein by Prandtl) and Goldstein (1929) developed a prescribed vortex wake theory for lifting propellers. This work was later extended by Theodorsen (1948). The early work with the technique as it applies to propellers is reviewed by Glauert (1935). Knight & Hefner (1937) were among the first to apply blade element and prescribed vortex wake principles to the calculation of the helicopter rotor problem. Coleman et al. (1945), Castles & De Leeuw (1954), and Castles & Durham (1956) later extended this work to helicopters operating in forward flight.

Figure 3.1 shows a sketch of the flow environment and aerodynamic forces at repre­sentative blade element on the rotor. The aerodynamic forces are assumed to arise solely from the velocity and AoA normal to the leading edge of the blade section. The effect of the radial component of velocity, Ur, on the lift is usually ignored in accordance with the independence principle; see Jones & Cohen (1957). However, the Ur component will affect the drag on the blade in forward flight and should be included in this case. The measured 2-D aerodynamic characteristics of the airfoil as a function of AoA can be assumed for

the purposes of calculating the. resultant lift and pitching moment on each blade element. Such results are available in the published literature for a large number of airfoils and over a wide range of operating conditions (see Chapter 7). The inflow angle of attack, ф, arises, primarily because of the velocity induced by the rotor and its wake. Therefore, the induced velocity serves to modify the direction of the relative flow velocity vector and so alters the AoA at each blade element from its 2-D value. This inflow velocity also inclines the local lift vectors, which by definition act perpendicular to the resultant velocity vector at the blade element and, therefore, provides a source of induced drag (drag resulting from lift) and is the source of induced power required at the rotor shaft.