Abstract
In this paper, a nonlinear flapping equation for large inflow angles and flap angles is established by analyzing the aerodynamics of helicopter blade elements. In order to obtain a generalized flap equation, the Snel stall model was first applied to determine the lift coefficient of the helicopter rotor. A simulation experiment for specific airfoils was then conducted to verify the effectiveness of the Snel stall model as it applies to helicopters. Results show that the model requires no extraneous parameters compared to the traditional stall model and is highly accurate and practically applicable. Based on the model, the relationship between the flapping angle and the angle of attack was analyzed, as well as the advance ratio under the dynamic stall state.
1 Introduction
The helicopter, an aircraft lifted and propelled by movable blades, is utilized often in both military and civilian fields, favored for unique characteristics including hovering ability and flexible manipulation [1] . The movement of a helicopter blade can be categorized, roughly, in one of two ways: in-plane motion (lag motion) and out-of-plane motion (flap movement) [2] . The primary focus of this paper is the flap motion of the rigid blade; specifically, the focus is on an equation that effectively reflects flap movement.
Many previous researchers have addressed this problem with notable achievements including the work of Leishman [3] and Johnson [4] . These studies, however, use linear approximation methods to obtain flap equations, based on assumptions of small inflow angle and flapping angle. This approach conveniently solves engineering problems inherent to small angle situations, but is not applicable to large angles. In an effort to remedy this, the present study first analyzes the aerodynamics of the blade element, then establishes a more general, nonlinear flap equation for both small angles and large angles.
Solving flap equations is no simple task. Accurate determination of the lift coefficient and drag coefficient under different conditions is crucial and rather difficult to achieve. For a small advance ratio and angle of attack, the lift coefficient can be obtained using a linear relation to advance ratio, and the drag coefficient can be obtained using a second-order polynomial fitting model in terms of angle of attack [5] . Under large advance ratio and angle of attack, the rotor readily falls into a stall condition, and the simple approximation relationship does not accurately calculate the lift coefficient or drag coefficient. Basically, stall conditions make coefficients nearly impossible to obtain.
Researchers Barakos and Spentzos used the CFD method to calculate the lift coefficient and drag coefficient under stall conditions in a previous study [6, 7]. Their method, which does show high accuracy, requires very lengthy calculation time and is thus not conducive to real-time computation or simulation. In order to build a model that satisfies both accuracy and real-time requirements, two semi-empirical models, the Leishman-Beddoes model, proposed by Leishman, Beddoes, Crouse et al. [8-10], and the ONERA model, proposed by Tran, Petot, McAlister~ et al. [11-13], both utilize experimental, static data to accurately determine lift coefficient and drag coefficient. These require in-practice parameters, however, which affects the practicability of the model.
To address all the issues described above, the Snel stall model, typically applied to wind turbine rotors, is applied for the first time in this study to determine the lift coefficient of helicopter blades. The effectiveness of the proposed method is then verified by simulation experiments. The quasi-static stall model (proposed by Leishman), is utilized to solve the drag coefficient problem.
In this study, a flap equation is established first, and the Snel dynamic stall model is applied to a helicopter for calculating lift coefficient under dynamic stall conditions. Results from an experiment to test the effectiveness of the model are described. Finally, the flap equation is solved alongside the model, using boundary value conditions, and the relationship between flapping angle, advance ratio, and angle of attack is also analyzed.
2 Materials and methods
2.1 Flap equation
2.1.1 Velocity and force analysis of blade element
Blade element theory (BET) is adopted first to establish the flap equation for the rotor. The blade's cross-section at a radial position y, with azimuth angle ϕ and pitch angle θ, and the radial length of the section dr, are selected for analysis as shown in Fig. 1.
The velocities of the blade element are shown in Fig. 1(a), where UR is the radial velocity, and UT and UP are tangential and perpendicular to the disk plane, respectively. The velocities are then expressed as follows:
where Ω is the rotation rate of the blades, R is the blade radius, μ is the advance ratio, and VC is the climb velocity. λ is the inflow ratio, and it is assumed that the inflow is a uniform (or linear) inflow. The pitch angle is θ = θ0 + θ1ccos ϕ + θ1s sin ϕ. According to the relationship of velocities shown in Fig. 1(a), the inflow angle is
The majority of the flap model, based on the assumption of small flap angle and inflow angle, is a linearization model that uses the approximate relationships sin β ≈ β and cos β ≈ 1, with arctan φ ≈ φ.
The lift and drag forces are produced by the flow at the blade sections, L and D, which are normal and parallel to U.
2.1.2 Establishment of flap equation
This paper uses a rigid blade to build the blade flap equation, focusing primarily on the moment of inertia and the aerodynamic torque related to the center of the disk plane produced by the flapping movement. The forces acting on blade elements are primarily inertial force FI, centrifugal force FC, and aerodynamic force FA, the direction of which is opposite the flapping motion and radially outward, normal to the blade, as shown in Fig. 2.
For a blade with an offset flap hinge, the dimensionless flap-hinge offset is e and the radial position is R. The inertial force is
The differential form of the flap equation at radial position r0 is:
where
2.2 Snel dynamic stall model
Under increased angle of attack and Mach number, flow separation and stall occur on the airfoil surface, impacting the distribution of the lift. In other words, the relationship between lift coefficient, angle of attack, and Mach number becomes irregular, rendering it impossible to obtain the coefficients of a simple polynomial.
Calculating the load of a helicopter blade under dynamic stall conditions is highly challenging. Vortex theory and Navier-Stokes numerical calculation, which both show reasonable accuracy, are commonly applied to solve this problem. However, the complexity of these calculation processes affects their real-time performance [14] . A simplified stall model that is quick (without sacrificing accuracy) is necessary to remedy this.
The lift coefficient under steady state is easy to obtain through experimentation, so a semi-empirical model is often adopted based on data obtained from experiments. Many semi-empirical dynamic stall models have been applied to aircraft research -- the two most commonly utilized are the Beddoes-Leishman model and the ONERA model. The Beddoes-Leishman model mainly focuses on simplifying calculations, deriving a series of ordinary differential equations to calculate the lift coefficient. The ONERA semi-empirical model uses both a first and second-order differential equation to describe the unstable behavior of a rotor. The inviscid contribution is described by the first-order differential equation, and the second-order differential equation calculates stall delay plus lift increments. These two models often contain some of the necessary parameters, but determination of these parameters is typically based on dynamic or static measurement data for specific airfoils, which makes the models all together rather inconvenient. The Snel stall model is adopted in this paper to remedy this [15].
The Snel stall model, first proposed by H. Snel [16, 17], is typically applied to the elastic stall effect of wind turbines. The model, which is based on lift coefficient under steady state, consists of two parts: one describing the forcing frequency response, and one describing the higher frequency of a self-excited nature. The lift coefficient is expressed as follows:
where ∆Cl,1 and ∆Cl,2 are obtained by first and second-order differential equations, respectively [18] . The differential equations are:
where time is a constant parameter
According to the Snel model, C10 can be denoted:
where ∆ = 2π sin(α - αZ) - Cl(steady), αZ is the zero lift angle of attack, Cl(steady) is the lift coefficient in the steady state, and F1(ϕ) is the frequency of the forcing term of the first-order differential equation, obtained as follows:
According to the description by V. K. Truong and H. Snel, C21 can be written as follows:
C20 is expressed:
The forcing term of the first-order differential equation concerns the difference between lift coefficient and potential flow and steady state:
According to the model, it is possible to determine the lift coefficient even in a stall condition. The magnitude of drag is small compared to lift, according to J. G. Leishman [19] . As for the definition of the drag coefficient, an approximate expression is effective:
where αS is the threshold value for the stall, and the values of a0, a1, and a2 can be obtained dependent on the specific airfoil.
3 Results and discussion
Each blade is, effectively, a series of infinitely thin cross sections at different positions in the radial direction. Here, the blade is divided into 100 pieces, n = 100, and the blade element ∆r at radial position r0 is selected for analysis.
3.1 Materials of simulation experiment
OA212 airfoils (in uniform mass distribution,) are used for the following simulation experiment. The radius is R = 8.5 m, the dimensionless flap-hinge offset is e = 0.039, the mass distribution is m = 13.0465 kg/m,
3.2 Validating the Snel stall model
To validate the Snel stall model as it applies to flap motion, the linear inflow model proposed by Pitt and Peters is adopted [20], expressed as follows:
The red line in Fig. 3 denotes the lift coefficient measured by the experiment in the steady state, and the blue line represents the lift coefficient obtained by the Snel stall model. Fig. 3 makes clear that the lift coefficient obtained by the Snel stall model is almost ubiquitously equal to the data obtained by experimentation when angle of attack is small and linear. As angle of attack increases compared to the threshold value, the flow near the surface of the airfoil becomes unstable, causing the lift coefficient to first drop sharply, then stabilize.
The relationship between flap angle and azimuth angle, with different collective pitch θ0 under conditions of θ1c = 4°, θ1s = -0°, and μ = 0.4, was obtained by solving the flap equation using the boundary value combined with the Snel stall model and B-L model (Beddoes-Leishman) [21, 22], as shown in Fig. 4.
The relationship between flap angle and azimuth angle with differing collective pitch θ0 values under conditions of θ1c = 2°, θ1s = -6°, and μ = 0.6 (ignoring the effect of reverse flow) is shown in Fig. 5.
As shown in Figs. 4 and 5, the flap response curve obtained using the Snel stall model is almost consistent with the curve obtained using the classic B-L model, which verifies the credibility and effectiveness of the Snel stall model as applied to helicopter rotors. Further, the Snel model does not require known parameters, which enhances its applicability.
3.3 Analysis of flap response
Successful analysis of the rotor involves gathering different flap response curves by altering collective pitch θ0 under the conditions of θ1c = 4°, θ1s = -10°, and μ = 0.4, as shown in the figure below (ignoring the effect of reverse flow).
When θ0 is 0°, 6°, or 12°, as shown in Fig. 6, the flapping angle first increases then decreases. When the azimuth angle reaches about 300°, the flapping angle begins to increase. With further increase in collective pitch, the amplitude of the flapping angle is usually small, and the azimuth corresponding to the minimum and maximum flap angles tends to be larger, which implies the effect of lag under increased collective pitch.
When θ0 reaches 18° or 24°, the flapping angle first declines then increases, then begins to decline until the azimuth reaches 200° ~ 250°. The azimuth corresponding to minimum and maximum flap angles tends to increase under the influence of these two periods, which suggests the effect of lag on the flapping response is more apparent under increased collective pitch; conversely, the amplitude of the flapping angle tends to increase. In order to thoroughly analyze this phenomenon, the relationship between angle of attack and azimuth is detailed next.
Fig. 7 shows where angle of attack is larger under the conditions described above, suggesting that the amplitude of flapping angle tends to increase first, then decrease until the angle of attack reaches a specific value.
Fig. 8(a) shows that with constant collective pitch value and increasing advance ratio, the amplitude of the flap angle tends to shrink, and the azimuth corresponding to the minimum and maximum flap angle tends to grow (this also demonstrates the effects of lag). When the azimuth reaches π, the flap angle reaches a constant value. For ϕ ∈ [0π], the flap angle decreases with increase in advance ratio. However, for ϕ ∈ [π, 2π], the flap angle instead increases with increase in advance ratio.
As shown in Fig. 8(b), inflow angle is barely affected by advance ratio around ϕ = 180°, and decreases with increased advance ratio. The angle of attack is negatively related to the inflow angle, which suggests that the angle of attack increases with increased advance ratio. This information is valuable for pilots who can then change the angle of attack of a blade, and the lift, as necessary.
4 Conclusion
A flap equation suitable for both small and large angles was established in this study based on a thorough analysis of blade elements. The Snel dynamic stall model was applied to a helicopter for the first time here in order to calculate lift coefficient under dynamic stall conditions. To demonstrate the effectiveness of the Snel model, an OA212 airfoil was used for simulation experiments. By utilizing the model established under different conditions, the characteristics of the flap response of the helicopter rotor were also analyzed in depth. Results showed that the model has favorable applicability, accuracy, and effectiveness.
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