1. Introduction
Flapping propeller phenomena have continued to be the focus of many researchers for many years due to their positive contribution to aerodynamic performance [
1]. This effect can also be found in nature, such as in birds and insects, as well as in manmade apparatus, such as airplanes. Flapping phenomena could be used in the application of airplane construction elements, such as wings or propellers, to increase effectivity, power, and flight time. The propeller, from the perspective of aerodynamics, can be viewed as a spinning wing; therefore, the cause of flapping [
2] is identical for the thrust generation mechanics of both wing and propeller elements.
Due to the high complexity of a flapping foil, the object of investigation can be simplified into two dimensions, as researchers investigate the lift, drag, and thrust for 2D models and compare these results with flapping profiles.
Researchers Knoller and Betz [
2,
3] were the first to investigate flapping foil and to explain flapping-wing thrust generation mechanics.
One of the first comprehensive investigations of a flapping wing was reported by [
3], who demonstrated equations for the theoretical efficiency of flapping motion and the effect of angular oscillations.
Thrust-producing harmonically oscillating foils are studied through force and power measurements by [
4]. In this research, the data were obtained using digital particle image velocimetry. The authors found that conditions of high efficiency are associated with the formation of a moderately strong vortex on alternating sides of the foil, which interacts with the leading-edge vorticity and creates a reverse Karman street.
A group of researchers, Benkherouf et al., Chen and Liu, and Xiao and Zhu [
5,
6,
7], provided a 2D foil-flapping numerical investigation using Navier–Stokes equations-based fluid flow modeling.
Gopalkrishnan et al. and Streitlien and Triantafyllou [
8,
9] gave three basic types of interactions of a harmonically oscillating wing with vortices in the wake: (1) an optimal interaction of newly generated vortices, with the general vortices shed by the wing, resulting in +C in the generation of more powerful vortices in the reverse Karman vortex street; (2) a destructive interaction of new vortices with those shed by the wing, resulting in the generation of weaker vortices in the reverse Karman street; and (3) the interaction of vortex pairs with an opposite sign shed from the wing, leading to the generation of a wide wake composed of vortex pairs that are shed at an angle to the freestream.
Since then, various types of device that use a flapping-foil motion have been proposed. One possible use is demonstrated by ship propellers, as a flapping propeller foil generates thrust for the ship’s motion and propulsion [
10,
11,
12,
13].
In [
14], a review of a wide flapping foil was given. The authors observed that a 2D flapping-foil investigation cannot fully explain the flapping-structure mechanics due to the complicated 3D flow patterns. In this paper, they focused on high-speed RPM airplane propellers with a relatively low
k (reduced frequency):
, where
f indicates the frequency (Hz),
c is profile length (m), and
represents flow speed (m/s).
In the case of an insect’s flight, the wing-beating frequencies obtained from physical tests showed that they are significantly lower than the resonant frequencies, as shown in [
15]. The dragonfly flutter frequency is about 16% of the first resonant frequency (dragonfly flapping frequency of 27 Hz is about 16% of the fundamental natural frequency). Studies of elastic wings with a large flapping amplitude in [
16,
17] showed that the highest efficiency was achieved when the wing-flapping frequency was lower than the first wing resonance.
These articles describe the mechanisms that control the dynamics of fluttering wings, but it is clear that the basic details of the problem of fluid–structure interaction are still poorly understood. More specifically, the basic phase dynamics, which determine the instantaneous shape of the wing and lead first to an increase in the traction force fluctuation, remains unexplained [
18].
It has been reported that wing flexibility allows for flight energy consumption to be reduced [
19]. For real flight conditions, simulation results show that these energy advantages are at a maximum when the ratio of flutter to first resonant frequency is ω∕ω1 ≈ 0.35 [
19,
20].
A comprehensive analysis of flapping-wing mechanics is given in [
21], where the structural dynamics and flight dynamics of small birds as well as those of micro air vehicles are demonstrated. The primary focus of this source is explaining the mechanics of flapping-wing aerodynamics at low Reynolds numbers.
The literature review shows that a more detailed explanation of the propeller stiffness effect on its aerodynamic performance, which includes the limitations of the engine, model and materials, is needed.
The air combat aircraft model (class f2d) is regulated by the rules of the FAI (World Air Sports Federation). This is a combat category aircraft model in which multiple pilots simultaneously control the model in a circle. The aircraft are light in weight and very short from nose to tail in order to maneuver quickly in the air. Each has a 2.5 m crepe paper streamer attached to the rear of the aircraft by a 3 m string. Each pilot may attack the other’s aircraft at the streamer only, in an attempt to cut the streamer with their model’s propeller or wing leading edge. The action is so fast that the new observer cannot frequently see the actual cuts of the streamers. The advantage of this for the pilot is that their model flies faster than others.
During the last decade, phenomena have been observed by a number of pilots—who use propellers with the same shape but with production processes achieving stiffness deviations—in which the additional noise in typical propeller operation conditions with an increase in F2d models accelerated to 0.5–0.8 s, or 3%.
The model performance triangle given in
Figure 1 shows that all three elements, namely, the engine, propeller, and model speed, affect each other because, by changing one component, the other also subsequently changes. For example, using a powerful engine with a high RPM (revolutions per minute) effect model, fly speed and the propeller’s effectivity establish a new balance between engine power, RPM, propeller effectivity, and model speed. The observed increase in model speed with propellers of relatively lower stiffness indicates an additional energy source affecting the propeller, or new behavior acting during operation conditions.
Hypothesis: an increase in the model speed can be achieved by flapping-propeller behavior.
Investigation novelty: the connection between non-steady fluid dynamics behavior and structural stiffness remains complex and ambiguous.
This paper deals with the propellers of class F2 aircraft and their dynamic characteristics. The purpose of the paper is to investigate the influence of the propeller’s construction stiffness on its modal eigenvalues and the effect of the modal eigenvalues on the air model horizontal speed. While the propeller forms were the same for all cases, they differed in matrix stiffness, as the amount of plasticizer in epoxy resin changed. Later, propellers with varying degrees of stiffness were tested with the same air model and engine. In the first part of this article, the physical experiments that were performed are described (
Figure 2): (1) an F2 aircraft test with three types of propellers; (2) experiments on the dynamic parameters of the propellers considered. The second part deals with theoretical research.
3. Results
Due to the high complexity of oscillating propeller analysis, at first a propeller static thrust analysis with v = 0 m/s and with an engine speed of 28,600 RPM was performed. At these boundary conditions, the physical model showed 1.3 kg or 12.75 N of thrust. The CFD model showed ≈ 2.1 percent less static thrust for all tested propellers, and
Figure 12 illustrates the streamline distribution on the blade’s surfaces. This verification of the numerical CFD model shows high precision and confirms the possibility of using it for the oscillating propeller analysis.
Investigation of propeller force and drag contribution along the propeller blade given in
Figure 13:
As illustrated in
Figure 13, the thrust increased by an increase in
R and at the 0.75
R of the propeller blade, creating the highest thrust. Later, the thrust started to decrease, while drag force
Fx increased in the whole
R range. This implies that higher propeller effectiveness is distributed close to the blade tips. Additionally, the highest blade-flapping amplitudes were found here, and the interaction of these two factors is of considerable importance in highly effective propeller design.
Effect of a flapping-propeller blade. A comparison of a transient rigid propeller with a flapping propeller and a transition turbulence Gamma Theta model, which allowed us to simulate laminar, laminar-to-turbulent, and turbulence states in a fluid flow, was made. The Gamma-Re model does not intend to model the physics of the problem but attempts to fit a wide range of experiments and transition methods for its formulation.
For all cases, a shear stress transportation fluid model with two options was used: the full turbulence and the Gamma Theta model (
Figure 14). The effect of air separation and laminar flow transition on turbulent flow showed an increase in propeller thrust and required power by ~3.6% due to flow separation occurring near propeller blade surfaces. The combination of the propeller-flapping effect with air flow separation is the key factor for modeling this type of propeller. For precise investigation, it is also necessary to take in to account the behavior of nonconstant propeller deformation from the center to the blade tip and the most effective propeller thrust (lift) and drag (required moment to rotate it at given RPM) generation area. Three propeller operation cases are given in
Figure 14 and
Figure 15.
Unsteady flow is a complicated analytical model. As a simplified model for harmonically pitching and plunging thin airfoils, Theodorsen delivered an expression for lift by assuming a planar wake and a trailing-edge Kutta condition in incompressible flow.
A general theoretical
Cl(t) lift coefficient equation is given [
21] as follows:
Here, C(k) is the complex-valued Theodorsen function with a magnitude of 1; it accounts for the attenuation of lift amplitude and the time lag in lift response from the real and imaginary parts. The first term is the steady-state lift; the second term is due to acceleration effects; and the last term evaluates circulatory effects, where c is half the cord length, α is the angle of attack, h is the plunge motion, and xp is the bending distance from the cord center.
The simplified model of unsteady flow for the flat plate is given in
Figure 16.
As illustrated in
Figure 17, the analytical model shows that the lift coefficient
Cl(t) had higher fluctuations compared with the CFX results. This could be explained by the specific flapping profile, which was taken from the physical test and later used for the CFX analysis.
Based on the results shown in
Figure 18, our main focus is on the propeller airflow parameters at point h
3 (
Figure 4).
As shown in
Figure 18, it can be observed that, at positions b and d, there was a TEV vortex formulation on the propeller blade (the blue zone at the top of the blade’s trailing edge), which affected lift force generation by the propeller blade. The authors of [
22] stipulate that the thrust and lift coefficient could continuously increase with a heaving amplitude up to a relatively large range (h/c < 1.5), where h represents amplitude and c is blade width. Amplitude is limited by other factors, such as the required propeller stiffness to start the engine by hand, and fatigue of the material. This study shows that a partial decrease in propeller stiffness positively affects propeller aerodynamic characteristics in horizontal flight. Moreover, this study can be extended to a full flight path of an F2D combat model, which consists of combinations of tight loops and straight flight segments.
The flapping propeller velocity field in limit points is shown in
Figure 18A–E.
4. Conclusions
Physical tests showed a significant effect of propeller stiffness on the speed of the F2D combat air model for the investigated propellers.
Experimental studies yielded the modal parameters for the three examined propellers. Analysis of the results shows that the frequency of the first resonant form of the investigated propellers varied from 524 to 542 Hz, respectively, and that the frequency of the second resonant form varied from 556 to 600 Hz. The damping coefficients of the first resonant form of the examined propellers varied from 0.31 to 0.51. Analyzing the amplitudes of the significant points of the examined propellers in the first resonant form, it was determined that the maximum amplitudes of the displacement of the propeller points can be found for the three propellers with the lowest damping coefficient.
Analysis of the significant point displacement results of the examined propellers during harmonic excitation showed that, when the excitation frequency was selected according to the aircraft engine speed, the maximum propeller edge displacement amplitudes ranged from 19.1 to 37.5 µm when the propeller attachment point amplitude was 2.2 µm. Evaluation of the results obtained when the harmonic excitation frequency corresponded to the first resonant frequency of the respective propeller revealed that the maximum amplitudes of the propeller edge displacement reached from 81.8 to 107.6 µm.
The change in one triangle participant parameter affected two others. The results of Physical test 1 in
Table 1 show that the best speed generated was that of prop. 2 because its spectrum density local maximum (blue color) was very close to the engine RPM (the engine RPM is 34,300 or 571 Hz when local peak is at 575 Hz).
The decrease in the initial propeller “C1” (prop. 1) stiffness by 3.28% increased the propeller’s effectiveness, and flight speed increased by 0.8%. The numerical investigation showed a significant propeller thrust increase when the propeller blade was flapping during its rotation of up to 3.6%, due to the non-steady flow separation and reattachment contribution regime near the trailing leading edge.