Қанат элементтері теориясы: Пропеллерлер мен роторлардың күштерін есептеу әдісі

Blade element theory (BET) is a mathematical process originally designed by William Froude (1878), David W. Taylor (1893) and Stefan Drzewiecki (1885) to determine the behavior of propellers. It involves breaking a blade down into several small parts then determining the forces on each of these small blade elements. These forces are then integrated along the entire blade and over one rotor revolution in order to obtain the forces and moments produced by the entire propeller or rotor. One of the key difficulties lies in modelling the induced velocity on the rotor disk. Because of this the blade element theory is often combined with momentum theory to provide additional relationships necessary to describe the induced velocity on the rotor disk, producing blade element momentum theory. At the most basic level of approximation a uniform induced velocity on the disk is assumed:

Alternatively the variation of the induced velocity along the radius can be modeled by breaking the blade down into small annuli and applying the conservation of mass, momentum and energy to every annulus. This approach is sometimes called the Froude–Finsterwalder equation. If the blade element method is applied to helicopter rotors in forward flight it is necessary to consider the flapping motion of the blades as well as the longitudinal and lateral distribution of the induced velocity on the rotor disk. The most simple forward flight inflow models are first harmonic models.

While the momentum theory is useful for determining ideal efficiency, it gives a very incomplete account of the action of screw propellers, neglecting among other things the torque. In order to investigate propeller action in greater detail, the blades are considered as made up of a number of small elements, and the air forces on each element are calculated. Thus, while the momentum theory deals with the flow of the air, the blade element theory deals primarily with the forces on the propeller blades. The idea of analyzing the forces on elementary strips of propeller blades was first published by William Froude in 1878. Again, in 1907, Lanchester published a somewhat more advanced form of the blade element theory without knowledge of previous work on the subject. The simple blade element theory is usually referred to, however, as the Drzewiecki theory, for it was Drzewiecki who put it into practical form and brought it into general use. Also, he was the first to sum up the forces on the blade elements to obtain the thrust and torque for a whole propeller and the first to introduce the idea of using airfoil data to find the forces on the blade elements. In the Drzewiecki blade element theory the propeller is considered a warped or twisted airfoil, each segment of which follows a helical path and is treated as a segment of an ordinary wing. It is usually assumed in the simple theory that airfoil coefficients obtained from wind tunnel tests of model wings (ordinarily tested with an aspect ratio of 6) apply directly to propeller blade elements of the same cross sectional shape. The air flow around each element is considered two dimensional and therefore unaffected by the adjacent parts of the blade. The independence of the blade elements at any given radius with respect to the neighbouring elements has been established theoretically and has also been shown to be substantially true for the working sections of the blade by special experiments made for the purpose. It is also assumed that the air passes through the propeller with no radial flow (i. e., there is no contraction of the slipstream in passing through the propeller disc) and that there is no blade interference.

According to momentum theory, a velocity is imparted to the air passing through the propeller, and half of this velocity is given the air by the time it reaches the propeller plane. This increase of velocity of the air as it passes into the propeller disc is called the inflow velocity. It is always found where there is pressure discontinuity in a fluid. In the case of a wing moving horizontally, the air is given a downward velocity, as shown in Fig. 4., and theoretically half of this velocity is imparted in front of and above the wing, and the other half below and behind. This induced downflow is present in the model wing tests from which the airfoil coefficients used in the blade element theory are obtained; the inflow indicated by the momentum theory is therefore automatically taken into account in the simple blade element theory. However, the induced downflow is widely different for different aspect ratios, being zero for infinite aspect ratio. Most model airfoil tests are made with rectangular wings having an arbitrarily chosen aspect ratio of 6, and there is no reason to suppose that the downflow in such a test corresponds to the inflow for each element of a propeller blade. In fact, the general conclusion drawn from an exhaustive series of tests, in which the pressure distribution was measured over 12 sections of a model propeller running in a wind tunnel, is that the lift coefficient of the propeller blade element differs considerably from that measured at the same angle of attack on an airfoil of aspect ratio 6. This is one of the greatest weaknesses of the simple blade element theory. Another weakness is that the interference between the propeller blades is not considered. The elements of the blades at any particular radius form a cascade similar to a multiplane with negative stagger, as shown in Fig. 4. Near the tips where the gap is large the interference is very small, but in toward the blade roots it is quite large. In actual propellers there is a tip loss which the blade element theory does not take into consideration. The thrust and torque forces as computed by means of the theory are therefore greater for the elements near the tip than those found by experiment. In order to eliminate scale effect, the wind tunnel tests on model wings should be run at the same value of Reynolds number (scale) as the corresponding elements in the propeller blades. Airfoil characteristics measured at such a low scale as, for example, an air velocity of 30 m. p. h. with a 3 in. chord airfoil, show peculiarities not found when the tests are run at a scale comparable with that of propeller elements. The standard propeller section characteristics given in Figs. 11, 12, 13, and 14 were obtained from high Reynolds number tests in the Variable Density Tunnel of the N. A. C. A., and, fortunately, for all excepting the thickest of these sections there is very little difference in characteristics at high and low Reynolds numbers. These values may be used with reasonable accuracy as to scale for propellers operating at tip speeds well below the speed of sound in air, and therefore relatively free from any effects of compressibility. The poor accuracy of the simple blade element theory is very well shown in a report by Durand and Lesley, in which they have computed the performance of a large number of model propellers (80) and compared the computed values with the actual performances obtained from tests on the model propellers themselves. In the words of the authors: The divergencies between the two sets of results, while showing certain elements of consistency, are on the whole too large and too capriciously distributed to justify the use of the theory in this simplest form for other than approximate estimates or for comparative purposes. The airfoils were tested in two different wind tunnels and in one of the tunnels at two different air velocities, and the propeller characteristics computed from the three sets of airfoil data differ by as much as 28%, illustrating quite forcibly the necessity for having the airfoil tests made at the correct scale. In spite of all its inaccuracies the simple blade element theory has been a useful tool in the hands of experienced propeller designers. With it a skilful designer having a knowledge of suitable empirical factors can design propellers which usually fit the main conditions imposed upon them fairly well in that they absorb the engine power at very nearly the proper revolution speed. They are not, however, necessarily the most efficient propellers for their purpose, for the simple theory is not sufficiently accurate to show slight differences in efficiency due to changes in pitch distribution, plan forms, etc.

Many modifications to the simple blade element theory have been suggested in order to make it more complete and to improve its accuracy. Most of these modified theories attempt to take into account the blade interference, and, in some of them, attempts are also made to eliminate the inaccuracy due to the use of airfoil data from tests on wings having a finite aspect ratio, such as 6. The first modification to be made was in the nature of a combination of the simple Drzewiecki theory with the Froude momentum theory.