Numerical Calculation of 1P Aerodynamic Loads on Aviation Propellers

To accurately predict the 1P aerodynamic loads of aviation propellers, this paper established a mathematical model of aviation propeller 1P aerodynamic loads based on the coupling method of blade element theory and momentum theory. Correction methods such as the Prandtl tip correction method and the propeller root correction method were implemented to further improve calculation accuracy. A 1P aerodynamic load calculation procedure was developed based on the mathematical model by using the Matlab software. 1P aerodynamic loads of a three-blade propeller were predicted for three different angles including 3 °, 9 °, and 12°. The numerical calculation results show that the calculated aerodynamic characteristic parameters of individual propeller blades obtained based on the propeller 1P aerodynamic load mathematical model deviate less than 6% from the CFD simulation results, and regular periodic pulsations are observed. The numerical calculations in this paper show that the propeller 1P aerodynamic load calculation procedure developed based on this model can accurately predict the propeller 1P aerodynamic load, which can provide some reference for the study of aviation propeller aerodynamic characteristics.


Introduction
Propeller airplanes play an important role in the current air transportation field.However, 1P aerodynamic loads exist when a propeller aircraft is maneuvering such as turning, pitching and pulling up.The 1P aerodynamic load will adversely affect the maneuvering stability of the aircraft, for example, it will cause the aircraft to produce the pitch phenomenon, head up or head down, and it will also cause the aircraft to produce the deflection, so the 1P aerodynamic load has a non-negligible effect on the aerodynamic performance of the propeller.In addition, 1P aerodynamic loads reduce the operation life of propeller hubs and bearings; therefore, accurate prediction of 1P aerodynamic loads is also necessary for the structure design of propellers [1][2].
In order to study propeller 1P aerodynamic loads, a great deal of research has been done on propellers subjected to oblique inflow.Gonzalez-Martino [3] et al. established a calculation procedure based on the HOST theory and modified the procedure to calculate parameters such as the overall performance of the AI-PX7 rotor.Park D [4] et al. used the method of combining the theory of blade element-momentum and induced velocity for the design, wind tunnel tests and CFD calculations of the EAV-3 propeller, and the reliability of the methodology was verified by comparing the calculations with the experiments.Garcia A J [5], et al. calculated the aerodynamic characteristics of ERICA tilt-rotor aircraft under different operating conditions using the CFD method and verified the accuracy of the CFD method by comparing it with experimental data.Higgins R J [6] et al. conducted a numerical study of the propeller during yaw.The relationship between yaw inflow angle and propeller thrust azimuth change was accurately predicted by using CFD methods.Yang X [7] et al. established a fast computational procedure based on the BET model to predict the propeller performance of a V/STOL aircraft during hovering, cruising, and transitional states.Qu Yuchi [8] et al. derived the relationship between 1P load and measured strain by the propeller strain test method and verified it experimentally.
In summary, numerous researchers and scholars have utilized diverse research techniques to investigate the aerodynamic load of propeller 1P.However, most experimental methods require significant investment in terms of testing funds and time, while the research efficiency of CFD simulation remains low.After considering the time and computational costs, this paper established a mathematical model for propeller 1P aerodynamic load based on the BEM theory.To refine this model, correction methods such as the Prandtl tip correction method and the propeller root flow correction method were implemented.Using the newly developed model, a rapid calculation procedure for propeller 1P aerodynamic load using Matlab software was developed.To validate our approach, we compare our numerical calculations with high-precision CFD results.

Mathematical Models Establishment
Mathematical modeling of propeller 1P aerodynamic load with oblique inflow based on the blademomentum theory.
Figure 1 illustrates the coordinate system with the decomposition of the incoming flow velocity. represents the angle between the incoming velocity V and the X-axis of the rotational axis of the paddle disk.Consequently, the incoming velocity V can be decomposed into the axial component V X and the tangential component V YZ of the paddle disk, as specified in equations ( 1) and ( 2), respectively.The tangential component V YZ can be further decomposed into V YZ⊥ and V YZ∥ , according to equations ( 3) and (4), respectively.In these equations,  denotes the azimuthal angle. sin V ia represents the axially induced velocity of each segment of the propeller blade, dL indicates the lift perpendicular to the resultant flow V R , and dD indicates the drag along V R .The expressions for propeller thrust and torque, derived from the blade element theory, are given in equations ( 6) and ( 7), respectively: ( ) ( ) Based on the momentum theory combined with Bernoulli's equation and the momentum conservation theorem, the equations for thrust and torque are given as ( 8) and ( 9) respectively: ( ) V disc represents the overall velocity through the propeller, which varies with azimuth and is described by the expression (10): The axially induced velocity V ia is a function of the azimuth angle  when there is an inflow angle, and the relationship between the induced velocity V ia and the azimuth angle is described by introducing the inflow model developed by Peter & Pitt [9][10].The expression of this inflow model is given as (11): Where, V ia,0 is the induced velocity at the center of the propeller, which is constant within a ring; and  is the wake deflection angle at the center of the ring, as shown in expression (12): The above equation is solved iteratively by joining the equations to find the required one-week sweep of the blade element to produce the thrust dT and the torque dQ.Repeat the above calculation steps at each location of the blade span and integrate each dT and dQ to obtain the aerodynamic performance parameters of the propeller in the case of an inflow angle.

Mathematical Model Correction
2.2.1.Tip Correction.The Prandtl tip correction method [6] was implemented to accurately simulate the phenomenon observed in propeller tips, where lift gradually diminishes to zero.The Prandtl tip correction model is a function used to describe the distribution of the correction coefficient F prandtl at the propeller blade spanwise.
where, B represents the number of propeller blades,  denotes the inlet angle of the airfoil, R denotes the complete radius of the propeller, and r denotes the radius of the propeller at radial position.

Root Flow Correction.
In the case of propellers experiencing oblique inflow, too large an angle of attack may result in the fluid not properly adhering to the blade surface and separating at the root region of the propeller.In order to accurately consider this impact, the F cl function is established.

Implementation of Calculation Methods Based on Matlab.
The procedure for operating this propeller 1P aerodynamic load is described as follows, as illustrated in Figure 3.In the first step, start by defining the basic parameters of the propeller, such as the number of propeller blades, radius, hub radius, blade chord length, and twist distribution.In the second step, specify calculation conditions such as inflow angle, inflow velocity, and other operating conditions.In the third step, use Xfoil to generate the lift and drag coefficient database for the specific airfoil and import it into the Matlab procedure.Next, introduce the inflow model established by Peter & Pitt and iteratively solve for the induced velocity distribution on the propeller disc.In the fifth step, the interpolation solution yields parameters such as chord length and blade angle at any radius of the propeller blade.Next, the lift and drag coefficients from the database are called and interpolated to solve for the lift and drag coefficients at each position of the propeller airfoil.Then, apply the Prandtl tip correction model correct at the propeller blade tip, and also correct at the propeller root due to separation flow at the root region.Multiply the correction coefficients at different radii by the corresponding lift coefficients to obtain the corrected lift coefficients.In the final step, decompose the inflow velocity and iteratively solve for aerodynamic parameters such as propeller blade bending moment and tangential force based on the blade elementmomentum theory.

Validation and Analysis
To verify the accuracy of the procedure, the prediction results of the propeller 1P aerodynamic load at different incidence angles by procedure were compared with the CFD calculation data.The coefficient of thrust (CT), power coefficient (CP), blade bending moment (B), and tangential force (T) of individual propeller blade were calculated for incidence angles of 3, 9, and 12 degrees to assess the load distribution of each blade with respect to azimuthal angle.The formula for calculating the deviation between the CFD calculation results and the procedure prediction results is given as (15).
Where, e represents the deviation between the procedure value and the CFX data, W Pro. represents the prediction value of the procedure, including B Pro. for the procedure prediction bending moment, T Pro. for the procedure prediction tangential force, CT Pro. for the procedure prediction thrust coefficient, and CP Pro. for the procedure prediction power coefficient; W Cal. represents the CFX calculated value, including B Cal. for the CFX calculation bending moment, T Cal. for the CFX calculation tangential force, CT Cal. for the CFX calculation thrust coefficient, and CP Cal. for the CFX calculation power coefficient; M signifies the amplitude of pulsation in the CFX calculated value, which is defined as the difference between the peak and valley values obtained from the CFX calculation.
Figure 4 shows the comparison of bending moment, tangential force, thrust coefficient, and power coefficient values calculated by the procedure for one blade with CFX calculations at 3, 9, and 12 degrees of incidence angles.From this figure, it can be seen that in the case of oblique inflow, the aerodynamic parameters of individual propeller blade show regular pulsating periodic changes with azimuthal angle changes, the maximum value of 1P load occurs near 90 degrees, and the minimum value occurs near 270 degrees, which is in line with the actual stress situation of the propeller, and the pulsating amplitude of the aerodynamic parameters shows a tendency to increase with the increase of the incidence angle.The deviation between the procedure prediction and CFX calculation of the aerodynamic parameters of the propeller is shown in Table 1, and the deviation range is within 6%, which is satisfactory.

Conclusions
This paper introduces the development of a fast prediction method for propeller 1P aerodynamic loads, which is based on the theory of blade element-momentum.By utilizing this procedure developed using Matlab for various propeller operating conditions, the following conclusions can be drawn: (1) By comparing the results of the procedure and CFX calculations at inflow angles of 3°, 9°, and 12°, it is observed that the deviations in the aerodynamic performance prediction for individual propeller blades are within 6%, which is satisfactory.This verification confirms the suitability of the mathematical model for 1P aerodynamic load and the accuracy of the procedure.Furthermore, it provides a fast and reliable method for the prediction of 1P aerodynamic loads.
(2) Through the calculation of 1P aerodynamic loads by this method, it is concluded that the aerodynamic performance of propeller demonstrates significant periodic variations in pulsation with respect to the azimuthal angle.The maximum values occur approximately at 90 degrees, while the minimum values occur around 270 degrees.

Figure 1 .
Figure 1.Definition of propeller coordinate system and decomposition of incoming velocity. cos

Figure 2 .
Figure 2. Sketch of lift and drag on blade element.
Where, r denotes the radius at any radial position of the propeller.The corrected lift coefficients are obtained by multiplying both F prandtl and F cl by the corresponding wing lift coefficients, and all subsequent calculations are performed using the corrected lift coefficients.

Figure 4 .
Figure 4. Variation of aerodynamic parameters in different incidence angles.

Table 1 .
Deviation of propeller aerodynamic parameter at different incidence angles.