Improvements in Robustness and Versatility of Blade Element Momentum Theory for UAM/AAM Applications

Abstract

This study proposes an improved formulation of the blade element momentum theory (BEMT) to enhance its robustness and versatility for urban/advanced air mobility (UAM/AAM) applications. A new velocity factor was introduced to eliminate numerical singularity issue under low inflow velocity conditions. The BEMT framework was further extended and modified to account for non-axial inflow and descent flight conditions. The proposed approach was validated for an isolated propeller case by comparing the results with wind tunnel test data and the computational fluid dynamics (CFD) based on both the overset mesh and sliding mesh methods. The improved BEMT provided reliable accuracy even in low inflow velocity conditions where basic BEMT fails to converge, and yielded reasonable performance predictions with respect to the sliding mesh results. The practicality of the method was confirmed through further application studies such as analyzing on the tilt propeller of single-seated UAM along its mission profile and constructing a propeller performance database for the lift and propulsion propellers of a lift and cruise type 5-seated UAM. The improved BEMT exhibited satisfactory engineering-level accuracy for various flight conditions, with prediction errors within 14% of the CFD results. The results and observations indicate that the proposed BEMT framework is suitable for use in the early design stages, performance analysis, and construction of a performance database, for distributed propulsion aircraft, such as eVTOL and UAM/AAM.

1. Introduction

Recently, the development and research of the aircraft with multiple propellers (or rotors), such as distributed propulsion aircraft, electric-powered vertical and take-off and landing (eVTOL) aircraft, and multi-copters, is growing rapidly [1,2,3]. As the number of propellers increases, accurate calculation of propeller performance becomes an important requirement for improving the design and performance analysis results. With aid of recent advances in both computing power and cost, propeller performance is increasingly being calculated and investigated using computational fluid dynamics (CFD) [4,5,6]. However, in the early stages of design or for specific applications, it is still preferable to obtain propeller performance quickly rather than carry out a time-consuming CFD analysis involving simulation of the entire flow field. To meet the above requirements, the blade element momentum theory (BEMT) has been widely used and proven to be useful in efficiently predicting propeller performance [7,8,9,10].

In the case of distributed propulsion eVTOL aircraft for advanced air mobility (AAM) or urban air mobility (UAM) applications, the prediction of not only the propeller performance, but also the aerodynamic interaction and interference induced by the propellers, could also have a significant influence on the design and performance analysis results under different flow conditions that may be encountered along the mission profile [11,12]. Although detailed analysis and accurate prediction of aerodynamic interference effects can be performed using CFD at the preliminary or detailed design stage, low-fidelity methods are suitable for parametric studies or design space exploration at the initial sizing or conceptual design stage. In addition, many sub-processes or disciplines within a multi-disciplinary design optimization framework, such as performance analysis, dynamic modeling and flight control, and structural design, prefer to quickly obtain aerodynamic and flight load data under the influence of interference without investing significant resources and time. From this perspective, BEMT is an attractive option employing for the early stage of design of a modern aircraft with multiple propellers (i.e., AAM/UAM aircraft and drones), and reliable prediction capability of BEMT is required to achieve high quality of final design.

The BEMT can also be useful for the design of a propeller/rotor blade geometry, which requires the exploration of a large design space of design variables [13,14,15,16]. This usually involves analyzing the propeller performance for a number of geometries corresponding to different combinations of design variables at various flow conditions. The optimal design solution is then sought and determined from the surrogate model for the response surface of the analysis results, or Bayesian optimization or genetic algorithm with addition of analysis results based on infill sampling criteria. The large number of calculations required to obtain propeller performance generally makes it difficult to design based on high fidelity analysis methods such as CFD. The fast turnaround design framework can provide the flexibility and feasibility to respond to changes in design conditions and objective functions that are frequently encountered in the early design phase.

Conceptually, the BEMT combines the momentum theory with the blade element theory (BET) to calculate the propeller performance by considering the induced velocity. In the BEMT, the propeller/rotor blade is first divided into a finite number of elements along the radial direction, each segment being called a blade element. The BEMT calculates the aerodynamic forces and moments acting on each blade element, simultaneously satisfying the momentum theory and the BET. These are then numerically integrated for all the blade elements, resulting in overall performance values. The BEMT has been confirmed to give reasonable results for conditions where the propeller is operating at or near design conditions. However, it is also known that analyses for off-design conditions or for high-loading conditions can lead to a deterioration in accuracy [17,18]. When applied to the analysis of UAM/AAM aircraft, which experience a variety of flight states such as hover, climb, transition, cruise, and descent, the BEMT may suffer from a reduction in the accuracy of the results depending on the analysis condition, as the analyses are performed for a wide range of flow conditions. If the inaccurate results are used in the design process, the number of design iterations may increase, leading to an increase in overall development cost and time. From an engineering perspective, it is practically important to alleviate or eliminate the potential sources of inaccuracy in the BEMT, or to improve the generality of the BEMT.

There are several factors that contribute to the reduction in the accuracy and limit the conditions under which it can be used. To overcome the potential shortcomings, ongoing efforts are being made to improve the BEMT to increase its accuracy and adaptability. The basic formulation of the BEMT only considers the case of axial inflow, where the direction of inflow is aligned with the rotational axis of the propeller (zero inflow angle); therefore, the accuracy will inevitably deteriorate for cases with a non-zero inflow angle. In addition, the basic formulation becomes invalid when the inflow velocity is equal to or less than zero and suffers from a numerical instability for conditions where the inflow velocity is close to zero [19,20].

To enable the analysis of the nonzero inflow angle condition, the improved BEMT has been proposed, where the BEMT is calculated individually for each azimuth angle of the propeller [19,21,22,23]. It represents that the propeller disk is divided in both the radial and azimuth directions, each element being called a cell, taking into account the inflow velocity according to the azimuth angle and radial section. Therefore, the 3-axis (i.e., 6-component) forces and moments are calculated based on the forces in the axial and rotational directions obtained for each cell, whereas the basic BEMT formulation only calculates the thrust and torque of the propeller. To obtain the 6-component, a coordinate transformation must be performed from the local coordinate system to the global coordinate system, considering the azimuth and the incidence angles for each cell.

The BET predicts the changes in the velocity at the disk plane due to the propeller operation. It is categorized into two types: correlation and mathematical models. These models are critical to the accuracy of the BET because they directly determine the effective angle of attack on the airfoil corresponding to each blade element. Correlation models are a way to directly calculate the induced velocity using inflow models. They are based on curve-fit equations derived from experimental data under specific flow conditions, or correlation equations obtained from the momentum theory. Mathematical models, on the other hand, obtain the induced velocity that satisfies the momentum theory through an iterative calculation process. The BET with this model is generally referred to as a BEMT. The BET combined with correlation models may only be suitable for the specific conditions for which the inflow model was originally intended [24,25,26]. In contrast, due to the greater versatility of mathematical models based on momentum theory, the BEMT applied to these models is widely used for general applications.

The mathematical models can be further divided into direct and indirect approaches. The direct method applies only the induced velocity (v_i) based on propeller performance and flow conditions [27,28,29], whereas the indirect method calculates the velocity factors (a_a and a_t) to determine the induced velocity [17,18,19,20]. In general, the indirect method is employed because induced velocity in the direct method is sensitive to the initial value and changes in the inflow angle. To improve the versatility of the direct method, models that consider the nonaxial inflow and wake skew angle have been proposed [28,29]. The velocity factors, which represent the ratios of the induced velocity to the inflow velocity, can be obtained by combining the momentum theory with the BET. The denominator of the axial velocity factor (a_a) is the vertical inflow velocity at the disk plane. Therefore, flight conditions where the vertical inflow velocity is zero or close to zero such as around hovering or static thrust case, allow division by zero and can lead to a mathematical and numerical singularity [19,20]. Studies on improvement to ensure the robustness of the axial velocity factor by adding the correction factors has been conducted [30,31,32].

From a UAM/AAM perspective, it is essential to analyze a wide range of flow conditions that reflect their mission profiles. To respond to this need, previous studies have introduced improvements of BEMT for propeller analysis in such aircraft [33,34,35]. However, these studies have primarily focused on validating results under limited conditions, with confirmation on the general knowledge of the limitations of BEMT. Comprehensive evaluations and in-depth analyses on the applicability of BEMT from a broader UAM/AAM perspective are rather rare and limited.

Although recent studies have attempted to improve either the accuracy or the versatility of BEMT for broader application in propeller aerodynamics analysis, most BEMT models remain accurate and robust only under a limited range of conditions, which limits their versatility. These limitations make it difficult to establish an efficient analysis method that can be applied across diverse flow conditions and geometries. This study proposes a formulation that alleviates these issues and improves both the robustness and versatility of BEMT. A new expression for the axial velocity factor is proposed to eliminate the singularities in the axial velocity component, which leads to enhance the robustness of BEMT under conditions of low inflow velocity. Additionally, the BEMT formulation is expanded to apply BEMT to a broader range of flow conditions. First, a formulation that can handle general, non-zero inflow angles is introduced, along with a vector-based approach for calculating BEMT. Second, an approach is devised to estimate the induced velocity for descent flight conditions at a specific range, in which the inflow velocity calculation is replaced with a determination based on curve-fit and correlation equations. The improved BEMT is validated using an isolated propeller of a quad-tilt propeller (QTP) aircraft as a benchmark case. The results of the improved BEMT are compared with those of CFD simulations, including Reynolds averaged Navier-Stokes (RANS) simulations with the sliding mesh method (SMM) using STAR-CCM+ (2020.3 version) [36] and RANS simulations combined with the actuator disk (ADM) and actuator surface (ASM) methods using OpenFOAM (v4.1) [37]. Through this comparison, the predictive performance of the improved BEMT is evaluated. Finally, a comprehensive evaluation is conducted under various aerodynamic conditions to assess the practicality and versatility of the improved BEMT for use in UAM/AAM applications.

2. Mathematical Models

2.1. Basic Formulation of Blade Element Momentum Theory

As mentioned above, the BEMT combines BET with momentum theory. BET divides the blade into discrete blade elements and calculates the propeller performance by summing the results for all the elements. The sectional forces (dF_a, dF_t) exerted on each element is calculated based on the flow conditions experienced by each element, assuming a two-dimensional airfoil. These forces are calculated by transforming the aerodynamic forces (dL, dD) acting on each element, which are obtained from an aerodynamic data table of two-dimensional airfoil based on the effective angle of attack (α_eff). Here, the subscript “a” denotes the axial direction along the propeller axis, and “t” denotes the tangential direction in the propeller plane. A schematic illustration of these forces and the relevant flow variables, including the inflow angle (ϕ_in) and induced velocities (v_a, v_t) representing the wake shedding, is provided in Figure 1.

Equations (1) and (2) are the expressions for calculating the thrust (T) and torque (Q) of the propeller by summing the force and moment of the blade elements, where B is the number of blades, W is the effective velocity, ρ is the density, and c is the chord length of the blade element. In this study, the aerodynamic data (c_l, c_d) for a 2D airfoil is constructed in advance and used to obtain the aerodynamic performances of the element corresponding to the effective angle of attack, Reynolds number (Re), and Mach number via table lookup.

$$T={\displaystyle \sum dT}={\displaystyle \sum d{F}_a}={\displaystyle \sum B\frac{1}{2}\rho W^2\left[c_l\cos {\varphi }_{\mathrm{in}}-c_d\sin {\varphi }_{\mathrm{in}}\right]cdr}$$

(1)

$$Q={\displaystyle \sum dQ}={\displaystyle \sum d{F}_{t}r}={\displaystyle \sum B\frac{1}{2}\rho W^2\left[c_l\sin {\varphi }_{\mathrm{in}}+c_d\cos {\varphi }_{\mathrm{in}}\right]crdr}$$

(2)

The momentum theory is used to account for the change in the flow velocity as the fluid passes through disk plane, i.e., the generation of induced velocity, due to the momentum and kinetic energy added to the flow by the propeller.

The flow passing through the disk plane forms a streamtube along the flow direction, as illustrated in Figure 2. According to momentum theory, the increase in the momentum of the flow in the streamtube is equal to the thrust of the propeller. By applying Bernoulli’s principle, an expression for the induced velocity (v_i) at the propeller disk is derived. In general, the BEMT is applied to the performance analysis of a forward propeller under the assumption of an axisymmetric streamtube, and its application is limited to the flow condition with only low asymmetry for validity.

This leads to a formulation for a differential streamtube in the form of an annulus corresponding to the blade element. Thus, the velocity and loads depend on the radial direction. By applying the conservation of axial and angular momentum, the differential thrust (dT) and torque (dQ) on each blade element can be expressed as follows, as discussed by Manwell [38].

$$\begin{array}{l}dT=\rho \left(2\pi rdr\right)W_{a}2\left(W_a-U_a\right)F\\ =4\rho \pi r{U}_a^2\left(1+a_a\right)a_{a}Fdr\end{array}$$

(3)

$$\begin{array}{l}dQ=\rho \left(2\pi rdr\right)W_{a}2\left(r\omega -W_t\right)rF\\ =4\rho \pi r^3\omega U_a\left(1+a_a\right)a_{t}Fdr\end{array}$$

(4)

where W_a and W_t are the effective axial and tangential velocities, respectively. a_a and a_t are the axial and tangential velocity factors, respectively, which are defined as follows:

$$a_a={\displaystyle \frac{W_a-U_a}{U_a}}={\displaystyle \frac{v_a}{U_a}}$$

(5)

$$a_t={\displaystyle \frac{r\omega -W_t}{r\omega }}={\displaystyle \frac{v_t}{r\omega }}$$

(6)

F is the tip/hub loss functions, which is expressed as

$$F_{\mathrm{tip}}={\displaystyle \frac{2}{\pi }}{\cos }^{-1}\left(\exp \left(-{\displaystyle \frac{B\left(R-r\right)}{2r\sin {\varphi }_{\mathrm{in}}}}\right)\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} r > 0.8R$$

(7)

$$F_{\mathrm{hub}}={\displaystyle \frac{2}{\pi }}{\cos }^{-1}\left(\exp \left(-{\displaystyle \frac{B\left(r-r_{\mathrm{hub}}\right)}{2r\sin {\varphi }_{\mathrm{in}}}}\right)\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{for} r \le 0.2R$$

(8)

The induced velocity distributed along the blade radial direction due to the vortex shed from the blade tip causes a loss of sectional lift; however, the basic formulation of the BEMT ignores this effect. The loss function (F) proposed by Prandtl [39] (see Equations (7) and (8)) is applied to take into account this loss in the expressions of the differential thrust and torque derived from momentum theory. By substituting Equations (3) and (4) on the left-hand side of Equations (1) and (2), respectively, the expression for the velocity factors can be obtained as follows:

$$a_a={\left[{\displaystyle \frac{4F}{\sigma c_n}}{\sin }^2{\varphi }_{\mathrm{in}}-1\right]}^{-1}$$

(9)

$$a_t={\left[{\displaystyle \frac{4F}{\sigma c_t}}\sin {\varphi }_{\mathrm{in}}\cos {\varphi }_{\mathrm{in}}+1\right]}^{-1}$$

(10)

where σ is the solidity of the propeller defined as

$$\sigma ={\displaystyle \frac{Bc}{2\pi r}}$$

(11)

The inflow angle ϕ_in is determined by

$$\tan {\varphi }_{\mathrm{in}}={\displaystyle \frac{U_a\left(1+a_a\right)}{r\omega \left(1-a_t\right)}}$$

(12)

An iterative procedure for solving the axial and tangential velocity factors that satisfies the above formulations can be established as shown schematically in Figure 3. The tolerance (ε) as the convergence criteria of the iterative procedures for the inflow angle is set to be 0.0001. This provided the propeller performance, substituting into Equations (1) and (2).

2.2. Improvement of Formulation for Velocity Factors

For the standard formulation of the BEMT, there is a possibility that mathematical singularity and numerical instability issues may arise in the calculation process for the conditions in the vicinity of hover flight or static thrust (U_a ≈ 0). The present study addresses these issues by modifying the formulation to a more general representation. To eliminate the fundamental cause of singularity, the axial velocity factor given in Equation (5) is redefined by using the tangential velocity (U_t), as shown in Equation (14). In addition, the tangential velocity factor (a_t) is also defined by using the tangential velocity (U_t) given in Equation (13).

$$a_a={\displaystyle \frac{W_a-U_a}{U_t}}={\displaystyle \frac{v_a}{U_t}}$$

(13)

$$a_t={\displaystyle \frac{U_t-W_t}{U_t}}={\displaystyle \frac{v_t}{U_t}}$$

(14)

By substituting Equations (13) and (14) into Equations (3) and (4), respectively, the expressions for differential thrust and torque can be modified, as shown in Equations (15) and (16).

$$\begin{array}{l}dT=\rho \left(2\pi rdr\right)W_{a}2\left(W_a-U_a\right)F\\ =\rho \left(2\pi rdr\right)U_t\left(1-a_t\right)\tan {\varphi }_{\mathrm{in}}2a_a{U}_{t}F\\ =4\rho \pi rF{U}_t^2\tan {\varphi }_{\mathrm{in}}\left(1-a_t\right)a_{a}dr\end{array}$$

(15)

$$\begin{array}{l}dQ=\rho \left(2\pi rdr\right)W_{a}2\left(U_t-W_t\right)rF\\ =\rho \left(2\pi rdr\right)U_t\left(1-a_t\right)\tan {\varphi }_{\mathrm{in}}2a_t{U}_{t}rF\\ =4\rho \pi r^{2}F{U}_t^2\tan {\varphi }_{\mathrm{in}}\left(1-a_t\right)a_{t}dr\end{array}$$

(16)

By substituting Equations (15) and (16) into Equations (1) and (2), respectively, the velocity factors can be rewritten as

$$a_a=\left(1-a_t\right){\left[{\displaystyle \frac{8\pi r}{Bc}}{\displaystyle \frac{F}{c_n}}\sin {\varphi }_{\mathrm{in}}\cos {\varphi }_{\mathrm{in}}\right]}^{-1}$$

(17)

$$a_t={\left[{\displaystyle \frac{8\pi r}{Bc}}{\displaystyle \frac{F}{c_t}}\sin {\varphi }_{\mathrm{in}}\cos {\varphi }_{\mathrm{in}}+1\right]}^{-1}$$

(18)

By modifying the velocity factors, the inflow angle ϕ_in is redetermined as

$$\tan {\varphi }_{\mathrm{in}}={\displaystyle \frac{\left(U_a/U_t+a_a\right)}{\left(1-a_t\right)}}$$

(19)

The loss factors given in Equations (7) and (8) also lead to singularity problem when ϕ_in ≤ 0. In the case of axisymmetric flow with forward and hover flight conditions, the authors have observed that the application of the loss factors results in a slight improvement in the accuracy of the results. In the cases of asymmetric flow conditions or the blade with the large twist, however, the loss factor is divergent due to the ϕ_in ≤ 0 condition in the blade elements. In this study, the loss factor is set to be negligible (F = 1) for the condition that is ϕ_in ≤ 0 for the robustness of the BEMT.

2.3. Formulation for Non-Axial Inflow Condition

When the propeller is operating under a tilt-angle or cant-angle condition relative to the inflow direction, the angle between the rotational axis of the propeller and the inflow direction leads to asymmetry in the streamtube, as illustrated in Figure 2. Therefore, for three-dimensional flow, an asymmetric streamtube is assumed and the momentum theory is applied along an individual streamline. The propeller is regarded as an actuator disk which is divided into cells in both the rotational and radial directions. The differential thrust and torque for each cell are calculated over the entire disk plane, and they are numerically integrated to obtain the total thrust and torque. To apply the BET formula for each cell, Equations (1) and (2) are rewritten as follows:

$$T={\displaystyle \sum ^{2\pi }{\displaystyle \sum ^{R}dT}}={\displaystyle \sum ^{2\pi }{\displaystyle \sum ^{R}B\frac{1}{2}\rho W^2{c}_{n}cdr}}\cdot d\theta /2\pi$$

(20)

$$Q={\displaystyle \sum ^{2\pi }{\displaystyle \sum ^{R}dQ}}={\displaystyle \sum ^{2\pi }{\displaystyle \sum ^{R}B\frac{1}{2}\rho W^2{c}_{t}crdr}}\cdot d\theta /2\pi$$

(21)

To enable the application for the three-dimensional flow case, the inflow velocity at each cell is converted into a local coordinate system, and the local forces and moments calculated in the local coordinates of each cell are converted back into the global coordinate system. The unit vectors in the axial (n), radial (l), and rotational (m) directions of each cell are defined, as shown in Figure 2. The unit vector (${\overrightarrow{n}}_{\mathrm{cell}}$) in the axial direction of the cell is identical to the axial vector (${\overrightarrow{n}}_{\mathrm{cen}}$) in the propeller center (see Equation (22)). The radial unit vector (${\overrightarrow{l}}_{\mathrm{cell}}$) is the unit vector of the distance vector (${\overrightarrow{d}}_{\mathrm{cell}}$) from the propeller center point (P_cen) to the cell point (P_cell) as given in Equations (23) and (24). The rotational unit vector (${\overrightarrow{m}}_{\mathrm{cell}}$) is calculated as given in Equation (25). The vectors for the propeller rotating in the opposite direction can be obtained by reversing the order in the cross products of two vectors in Equation (25).

$${\overrightarrow{n}}_{\mathrm{cell}}={\overrightarrow{n}}_{\mathrm{cen}}=n_x\widehat{i}+n_y\widehat{j}+n_k\widehat{k}$$

(22)

$${\overrightarrow{l}}_{\mathrm{cell}}={\overrightarrow{d}}_{\mathrm{cell}}/\left|{\overrightarrow{d}}_{\mathrm{cell}}\right|$$

(23)

$${\overrightarrow{d}}_{\mathrm{cell}}=\left(x_{\mathrm{cell}}-x_{\mathrm{cen}}\right)\widehat{i}+\left(y_{\mathrm{cell}}-y_{\mathrm{cen}}\right)\widehat{j}+\left(z_{\mathrm{cell}}-z_{\mathrm{cen}}\right)\widehat{k}$$

(24)

$${\overrightarrow{m}}_{\mathrm{cell}}={\overrightarrow{n}}_{\mathrm{cell}}\times {\overrightarrow{l}}_{\mathrm{cell}}$$

(25)

The components of inflow velocity, U_a and U_t, corresponding to each cell are computed by applying the inner product of the velocity vectors.

$$\left(U_a,U_t\right)=\left({\overrightarrow{V}}_{\infty }\cdot {\overrightarrow{n}}_{\mathrm{cell}},r\omega +{\overrightarrow{V}}_{\infty }\cdot {\overrightarrow{m}}_{\mathrm{cell}}\right)$$

(26)

Once the differential forces (dF_n, dF_m) of each cell are obtained by performing the iterative procedure depicted in Figure 3, the contribution to forces and moment components in the global coordinate system are calculated using Equations (27) and (28). Here, dF_n denotes the force in the axial direction, which is negative dF_a, and dF_m denotes the force in the rotational direction, which is dF_t. In Equation (27), the index i takes the values x, y, z, while in Equation (28), the indices (i, j, k) follow the cyclic permutations of (x, y, z); that is, (x, y, z), (y, z, x), (z, x, y).

$$F_i={\displaystyle \sum _{2\pi }{\displaystyle \sum _{R}d{F}_x}}={\displaystyle \sum _{2\pi }{\displaystyle \sum _R\left(d{F}_n{n}_i+d{F}_m{m}_i\right)}}$$

(27)

$$M_i={\displaystyle \sum _{2\pi }{\displaystyle \sum _{R}d{M}_i}}={\displaystyle \sum _{2\pi }{\displaystyle \sum _R\left(d{F}_k{l}_j-d{F}_j{l}_k\right)}}$$

(28)

2.4. Treatment for Descent Flights

The concept of the streamtube shown in Figure 2 may not be applicable or valid for conditions of negative axial inflow (U_a < 0). The propeller would encounter the situations or phenomena such as the turbulent wake state, vortex ring state, or windmill brake state. The reliability of the BEMT formulation described above is not guaranteed because the momentum theory is invalid. The analysis of the descent flight condition is replaced with BET combined with the inflow model, for which simple models are used for determining the induced velocity under the complicated flow state. Although this approach is BET with correlation-based inflow models, for consistency, the method and results will also be referred to as “BEMT” or “improved BEMT” hereafter. An equation for the induced velocity for the range of descent flight is considered, and the effective angle of attack is calculated using the equation. For the range of −2 ≤ U_a/v_h ≤ 0, corresponding to the turbulent wake and vortex ring states, the empirical correlation proposed by Young [40] is applied. This equation was obtained by curve fit equations to existing experimental data as given in Equation (29). Here, κ is induced power factor, and v_h is the axial induced velocity under the hovering condition (U_a = 0). In general, κ is a correction factor to match the measured performance. Johnson [41] obtained and proposed a value of κ as 1.17 by applying a linear inflow distribution and demonstrated that this choice also provides reasonable results even for nonuniform inflow conditions. In the windmill brake state (i.e., U_a/v_h ≤ −2), the correlation equation derived from the momentum theory, as given in Equation (30) [19], is applied because the formation of the streamtube can be assumed.

$${\displaystyle \frac{v_a}{v_h}}=\kappa +k_1\left({\displaystyle \frac{U_a}{v_h}}\right)+k_2{\left({\displaystyle \frac{U_a}{v_h}}\right)}^2+k_3{\left({\displaystyle \frac{U_a}{v_h}}\right)}^3+k_4{\left({\displaystyle \frac{U_a}{v_h}}\right)}^4\hspace{1em}\hspace{1em},\mathrm{for} -2 U_a/v_h \le 0$$

(29)

where κ = 1.17, k₁ = −1.125, k₂ = −1.372, k₃ = −1.718, k₄ = −0.655

$${\displaystyle \frac{v_a}{v_h}}=-\left({\displaystyle \frac{U_a}{2v_h}}\right)-\sqrt{{\left({\displaystyle \frac{U_a}{2v_h}}\right)}^2-1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em},\mathrm{for} U_a/v_h \le -2$$

(30)

3. Validation

The isolated propeller of QTP unmanned aerial vehicle developed by Korea Aerospace Research Institute (KARI) was selected as a validation case. The results from BEMT of the present study were compared with those from full CFD and actuator method, as well as wind tunnel test data. During the development of the QTP, wind tunnel tests (WTT) were carried out by KARI [42,43] to obtain the aerodynamic performance (i.e., three-component forces and moments) of the isolated propeller. KARI also performed the CFD simulations using an overset mesh (OSM) techniques based on an unstructured overlapping grid system [44]. As shown in Figure 4, the analysis geometry consists of three blades and a nacelle. For the purposes of analysis, the support strut and connecting rods present in the wind tunnel testing were removed, and the geometry of the air intake was simplified. Additionally, the nacelle is also excluded in the BEMT analysis. The basic specifications are summarized in

Table 1. Specifications of QTP propeller.

Diameter [m]	Solidity, σ	No. of Blades	No. of Airfoil [45]
1.1	0.10861	3	4 (SF30, SF25, SF18, SF12)

, and the chord and twist distributions of the blade can be referred to [43]. The blade of the QTP propeller consists of four airfoils, which are designed to enhance aerodynamic performance at low Reynolds numbers [45], as shown in Figure 4. The aerodynamic data (c_l, c_d) for each airfoil were generated using RANS simulation as a function of Re and applied to the present analysis. Due to the small diameter of the QTP propeller and its low M_tip, the aerodynamic data were generated based on Re, which disregards the compressibility effect. The reference point for the moment was set to be a center of the propeller disk for both the wind tunnel test and analyses.

For comparison and validation purposes, the present analysis referred to the CFD results (STAR-CCM+ and OpenFOAM) from the previous study [46,47,48]. The STAR-CCM+ simulations employed the sliding mesh method (SMM) to carry out a full CFD analysis, resolving the unsteady flow around the rotating blades in detail. In contrast, the OpenFOAM simulations adopted a more computationally efficient approach by combining actuator methods with the PIMPLE (PISO + SIMPLE) solver. The time step size for the unsteady calculation was set as the time interval for the propeller to rotate by 1° of the azimuth angles (ψ). Considering the computational cost and resource constraints, a body-fitted grid with y⁺ of approximately 30–50 was generated for the surface to accompany the wall function. The total number of volume cells was 9.64 × 10⁶ for the STAR-CCM⁺ simulation and 5.12 × 10⁶ for the OpenFOAM simulation. All simulations were conducted in parallel using 20–40 cores of the Intel Xeon E5-2640v4 2.4 GHz CPUs. Detailed numerical schemes and solver settings for both STAR-CCM+ and OpenFOAM analyses can be found in References [46,47].

For validation cases of axisymmetric inflow velocity, the calculation results for the hover (static), climb (forward), and descent (backward) flights are compared with those of the wind tunnel test and CFD. The inflow velocity was considered from −30 m/s to 30 m/s with a fixed rotational speed of 1800 rpm, corresponding to −4 < U_a/v_h < 4. The collective pitch angle is fixed at 15°. The thrust and torque results with respect to forward speed are plotted in Figure 5 for both hover and forward flights. Hereafter, the results obtained by the improved BEMT in the present study are denoted as “BEMT” in all cases, including descent and non-symmetric inflow conditions, for which the basic BEMT is not applicable. The terms “improved BEMT” and “basic BEMT” are used in cases such as the hover and forward flight conditions to distinguish between the two comparable methods. The results denoted by a reference number in the legend are from previous studies or wind tunnel tests, whereas the others are the results obtained in the present study for validation and comparison purposes. In the figures, F and M denote the force and moment components, respectively. The x-axis is defined in the direction opposite to the thrust, i.e., F_x represents the negative of the thrust (F_x = −T). The torque and M_x are defined to be in the same direction along the x-axis by this coordinate definition.

The comparison of results identifies improvements in robustness, which is attributed to the modified formulation for the velocity factors. The basic formulation of BEMT was unable to provide a converged solution for low-speed inflow conditions. During the calculation process, the axial velocity factor (a_a) of the basic BEMT becomes a value greater than 1 (v_a > U_a, U_a < 9 m/s) in these conditions, resulting in an unphysical and large value of the right-hand side of Equation (12), causing the inflow angle (ϕ_in) to diverge. Therefore, the application of basic formulation is rather limited, especially for the low advance ratio or static thrust condition. For the cases where the axial velocity factor (a_a) is less than 1, the results of the basic BEMT and improved BEMT of the present study show good agreement with the wind tunnel test data and CFD results (both OSM and SMM). The ADM approach underpredicts performance compared to the SMM and improved BEMT results at low-speed inflow conditions, which indicates an inherent limitation of the method under high disk-loading conditions.

Table 2. Comparison of calculation resource and time.

Solver	CPU	Cores	Times
OSM (In-house code)	Intel Xeon Gold 6130 2.1 GHz	224	69 h
SMM (STAR-CCM+)	Intel Xeon E5-2640v4 2.4 GHz	40	140.8 h
ASM (OpenFOAM)	Intel Xeon E5-2640v4 2.4 GHz	20	18 h
ADM (OpenFOAM)	Intel Xeon E5-2640v4 2.4 GHz	20	7.4 h
BEMT	Intel Core i7-11700kf 3.6 GHz	1	1 s

summarizes the calculation time and computational resources used in each analysis method for a collective pitch angle of 15° and inflow velocity of 12 m/s. The unsteady simulations were conducted with a time interval for 1° of propeller rotation, and the duration of the simulation time corresponds to 10 revolutions. The comparison indicates that the improved BEMT provides results of reasonable accuracy for engineering use and is considerably more time efficient compared to RANS-based aerodynamic solvers. This is primarily due to the fact that the BEMT requires calculations of only a few algebraic equations derived from the analytic formulation with quasi-steady state assumptions, without considering the solution of the governing equation of fluid motion for the entire flow field around the propeller. The ADM and ASM also seem more efficient than the full CFD (OSM and SMM), because they do not directly resolve the surface of blades.

Figure 6 shows the plots of thrust and torque in the descent flight (−4 < U_a/v_h < 0). The rotational speed was fixed at 1800 rpm and the collective pitch angles of 10°, 15°, and 20° were considered. The results of the improved BEMT for the descent flight are in reasonably good agreement with the CFD (SMM) results. A comparison with the CFD results reveals that the BEMT slightly overpredicts the thrust and torque in the region corresponding to the turbulent wake and vortex ring states (−2 ≤ U_a/v_h < 0), and underpredicts the thrust in the windmill brake state (U_a/v_h ≤ −2). The BEMT provides the results with a maximum error of approximately 10% at an inflow velocity of −30 m/s. Due to the significant computational costs and time required for unsteady CFD simulations of a descent flight condition to capture and simulate the complex flow field around until equilibrium state is reached, the BEMT can be regarded as efficient and reasonable.

To confirm the validity of the BEMT for the non-symmetric flow cases, the results of three-component forces and moments are compared with the wind tunnel test data and CFD results, as shown in Figure 7. Two cases with a fixed rotational speed of 1800 rpm were considered for analysis. The forward speed and collective pitch angle are 32 m/s and 25° for the first case, and 24 m/s and 20° for the second case. The definition of three axes, rotational direction of the propeller, and yaw angle of the inflow are shown in Figure 4. It should be noted that, in the BEMT calculation for non-axial inflow, only the velocity components in the rotational and axial directions are considered, and the influence of the radial velocity component is therefore ignored. Therefore, the F_z and M_z appear to be “0”, as shown in Figure 7. In general, these values are relatively small compared to other force and moment components, as indicated by the wind tunnel data and CFD results.

The thrust and torque (F_x and M_x, respectively) results obtained from the BEMT shows reasonably good agreement with the wind tunnel data and CFD results, as shown in Figure 7. However, the BEMT reveals the tendency of underprediction of the y-axis force (F_y) at high yaw angles. As previously mentioned, it is noted that the BEMT does not consider or calculate the forces and moments exerted on the nacelle. In contrast to the BEMT, the wind tunnel test data and CFD results include forces and moments acting on the nacelle. Lee et al. [47] performed a CFD analysis of two different configurations for the forward speed condition of 24 m/s, with and without the nacelle, to confirm how the nacelle affects the analysis results, as shown in Figure 7c,d. A minor change in F_x (thrust) and M_x (torque) was observed due to the nacelle, while a substantial difference in F_y was identified due to side force acting on the nacelle. As illustrated in Figure 8, both the ADM and ASM results from OpenFOAM show a noticeable pressure coefficient difference between the left (negative y-direction) and right (positive y-direction) sides of the nacelle, resulting in a significant force difference in the y-axis direction. Therefore, the BEMT results for F_y closely correspond to the ASM results for the case of an isolated propeller without the nacelle, as shown in Figure 7c,d. This indicates that the majority of the discrepancy in F_y can be attributed to the inclusion of the force acting on the nacelle in the wind tunnel data. This confirms that the improvements made for the BEMT formulation proposed in this study are reasonable and appropriate for calculating the performance of a propeller at various flow conditions. It provides accurate and useful results for engineering use with relatively very small analysis time.

4. Application Examples

This section will describe application examples of the improved BEMT that was established and validated above. As the first case, the CFD simulations of an isolated propeller of an optionally piloted personal air vehicle (OPPAV) at several operating conditions along its mission profile, which were performed by KARI [49,50], were considered, and the results were compared to evaluate the practicality of the improved BEMT. In the second case, the improved BEMT was employed to generate a database of propeller performance, which is constructed for dynamic modeling in the flight simulator of UAM aircrafts with distributed propulsion [51,52]. The advantages and limitations of the improved BEMT were examined by comparing it with CFD results. In the final case, the improved BEMT was applied to a commercially available T-motor propeller. The propeller’s geometry was reconstructed through reverse engineering based on 3D scanning to demonstrate the applicability of the BEMT to real-world propeller configurations.

4.1. Performance of Tilt Propeller of OPPAV Along Flight Schedule

As shown in Figure 9, KARI has developed and performed flight tests an OPPAV, a single-seat electric-powered distributed propulsion aircraft. During the development process, KARI performed a numerical analysis of the tilt propeller in various flight conditions, including the hover, transition, and forward flights, considering its mission profile (see Figure 10) [49,50]. For the same conditions, an analysis of propeller performance was performed using the improved BEMT to compare with the CFD results of KARI.

The specifications of the tilt propeller are listed in

Table 3. Specifications of tilt propeller for OPPAV.

Radius [m]	Solidity, σ	No. of Blades	No. of Airfoil
0.75	0.1428	3	1 (Clark Y)

. The blade is made of the Clark Y airfoil. Further details, including the chord length and twist angle distributions, can be found in literature [53]. The aerodynamic data (c_l, c_d) of the airfoil used for the present study were generated by using RANS analysis as a function of the Reynolds and Mach numbers.

The tilt angle is defined as 0° and 90° at the forward flight and hovering condition, respectively. The analyses for conditions corresponding to the transition flight conditions were performed. The results were compared with those of CFD provided by Kang [49]. The results of the BEMT in the forward flight condition were compared with those of the CAMRAD II provided by Paek et al. [50]. CAMRAD II is a widely used commercial program for rotorcraft analysis and design. However, it should be noted that it uses a BET approach with inflow model and wake models for aerodynamic analysis.

Figure 10 shows the mission profile of the OPPAV. The tilt propeller of the OPPAV is used in all segments, therefore an understanding of the performance at each segment is required. Kang conducted an analysis using STAR-CCM+, based on an incompressible solver and a sliding mesh method, and investigated the aerodynamic characteristics of the tilt propeller in the transition segment [49]. The required aerodynamic forces generated from the propeller were determined by considering the lift and drag generated by the wings. Therefore, the tilt angle, collective pitch angle, and rotational speed of the tilt propeller were calculated at several forward speeds along the standard schedule of the transition flight segment, as shown in Figure 11.

Figure 12 shows the thrust and torque of the tilt propeller according to tilt angles from 90° to 0° at 15° intervals. The result of the improved BEMT exhibits reasonable agreement with the STAR-CCM+ result [49] from an engineering perspective. While the prediction accuracy was found to be slightly lower for tilt angles of 60° and 45°, it has a similar level of accuracy to ADM. However, such flight conditions are typically encountered during the short duration of actual flight operations, and the resulting discrepancies are expected to have minor impact on overall design and evaluations in early design stages. A good accuracy in predicting performance is observed for flight conditions of small tilt angles at high flight speeds as well as high tilt angles at the hover and low flight speeds. At a tilt angle of 90°, which corresponds to a hovering condition, the ADM tends to underpredict the aerodynamic performance compared to the SMM, whereas the improved BEMT slightly overpredicts. This result is consistent with the previous observations in the validation case, suggesting that the characteristics remain consistent regardless of the configuration.

The radial velocity component on the blade becomes greater when both forward flight speed and tilt angle are higher. The both improved BEMT and ADM are based on BET for analysis and cannot take into account the effect of radial velocity component, leading to low accuracy under this condition. As stated in the validation case for QTP, the forces and moments resulting from radial velocity are neglected. This results in F_z and M_z being “0”, as shown in Figure 13. A comparison of the results indicates that the nonzero F_y and M_y values obtained from the improved BEMT are found to agree reasonably well with the trend of the ADM results. However, the improved BEMT does exhibit a certain degree of underprediction for F_y in comparison to the results of STAR-CCM+ (CFD). This observation is attributed to the difference in taking into account the force exerted on the nacelle, as discussed for the QTP results previously.

For forward flight conditions, the results are compared with those obtained using CAMRAD II [50], a commercial comprehensive analysis software for helicopters and rotorcrafts. The forward speeds (V_∞) of 160 and 200 km/h, with a fixed rotation speed of 1650 rpm, were considered for analysis. For the 160 km/h case, the following collective pitch angles were selected: 23°, 24°, 26°, 28°, 30°, 34°, and 36°. For the 200 km/h case, i.e., the maximum speed, the following angles were selected: 29°, 30°, 32°, 34°, 36°, 38°, and 40°. As illustrated in Figure 14, the power-thrust curve from the improved BEMT is compared with the CAMRAD II results. The results clearly show the consistent trends in thrust to power. A high degree of prediction accuracy is achieved in the case of 200 km/h while the thrust and power tended to be slightly over-predicted in the case of relatively low speed (i.e., 160 km/h).

The comparisons with reference data confirm that the improved BEMT is capable of calculating the performance of the propeller efficiently. In the initial stages of an aircraft design, it is highly desirable to obtain aerodynamic performance quickly; therefore, rapid analysis tools such as CAMRAD II and BEMT are employed. These tools can respond to repetitive tasks that may arise during design iterations or optimization processes. The applications of the improved BEMT to the design and optimization of propellers, the performance evaluation, and the construction of an aerodynamic database for distributed propulsion aircraft are expected to provide significant advantages in terms of cost, time, and design reliability.

4.2. Generation of Database of Propeller Performance for Lift-Cruise Type UAM

The demonstration project, known as the K-UAM Grand Challenge, has been conducted by the Ministry of Land, Infrastructure, and Transport (KAIA) through a public-private partnership to support the commercialization of Korean UAM [54,55]. As part of this project, a flight simulator is being developed to train unskilled UAM pilots and to verify various operational scenarios that may occur in urban environments [51,52]. For this simulator development of the lift-cruise (LC) type, as shown in Figure 15, taking into account the flight characteristics of the UAM, an aerodynamic database (aeroDB) for the flight simulator was constructed by separating the hover and forward flight modes.

The overall aeroDB was constructed based on the component build-up method. This method combines the three-component forces and moments of each subcomponent (airframe, control surfaces, and propellers) to obtain the total aerodynamic performance. This approach was chosen to address the considerable computational cost that would be required for carrying out the analyses of the full configuration, including all the propellers, at every data point across a wide range of flow and operating conditions. During the initial stage of developing the flight simulator, the propeller’s aerodynamic performance database (propDB) was first generated using the improved BEMT and then refined using CFD.

The propDB was constructed for each propeller by calculating its performance under various flight conditions, such as angle of attack (AOA), side-slip angle (AOS), freestream velocity, and rotational speed. Further details can be found in references [51,52]. Because the LC type UAM has both lift and propulsion propellers, propDBs were generated separately for the eight lift propellers and two propulsion propellers. The lift propellers are used to produce required lift during hover and transition flight, while the propulsion propellers are used to generate required thrust for forward flight, as shown in Figure 15.

The lift propellers consist of two blades and are installed with cant angles of ±5° about the x-axis to maintain stability for attitude control, as shown in Figure 16. The propulsion propellers, each consisting of five blades, are mounted in front of the V-tail and function as tractor propellers to provide forward thrust, as illustrated in Figure 16. Both types of propellers are controlled by rotational speed and have a radius of 1.33 m. All propellers are designed to meet their respective operational requirements using the Clark Y airfoil. The same aerodynamic data of the airfoil used in the analysis of the OPPAV tilt propeller were used.

The propDB for the two types of propellers was constructed according to the flight modes: hovering and forward flight. The propeller performance was first calculated with respect to the propeller axis. Then, the three-component forces and moments were converted to those corresponding to the global coordinate system illustrated in Figure 16. The range of analysis condition includes only low-speed conditions (0, 2, and 10 m/s) for the hover mode, while both low-speed and high-speed conditions (40, 70, and 85 m/s) were considered for the forward flight mode, as summarized in

Table 4. Analysis condition of propeller database.

		Hover Flight	Forward Flight
Velocity [m/s]		0/2/10	0/2/10/40/70/85
AOA [°]		−90/−60/−30/−20~20 (∆ = 4)/ 30/60/90	−30/−20~20 (∆ = 4)/30
AOS [°]		−180/−150/−120/−90/−60/−30 /−20~20 (∆ = 4)/ 30/60/90/120/150/180
RPM [rev/min]	Lift	0/1000/1200/1400	0/600/800/1000/1200
	Propulsion	0/800/1000/1200	0/800/1000/1200

. The AOA and AOS were determined based on the attitude angle and flight path angle. Since both types of propellers are controlled by rotational speed, RPM was included as an independent variable in the propDB. The lift and propulsion propellers have nominal rotational speeds of 1200 rpm and 1000 rpm, respectively. The range of RPM for analysis was determined to include these values and cover the flight conditions corresponding to mission profile.

The trends in the results of the improved BEMT were evaluated under flight conditions in which the two types of propellers are primarily operated. Figure 17 depicts the performance of the lift propeller in hover mode, where it rotates clockwise at a cant angle of 5°. The propDB under a freestream velocity of 10 m/s was generated with respect to the AOA, AOS, and RPM. Figure 17a illustrates the inflow directions in the hover mode, including climb and descent flight conditions, from the perspective of the lift propeller. The arrows show the freestream velocity vectors and indicate the flow directions across all the flight conditions considered in this analysis. The direction of the incoming flow is determined by AOA and AOS, which leads to significant differences in the performance results. In addition, due to the presence of the cant angle, asymmetric results can be observed as the AOS varied. The thrust, torque, and power increase with RPM. The effects of AOA and AOS on performance were found to be consistent across the different RPM conditions.

Figure 18 shows the performance results of the propulsion propeller in forward flight mode. These results correspond to a freestream velocity of 70 m/s (i.e., cruise speed). Figure 18a shows that the analysis of the propulsion propeller in forward flight mode is conducted within an AOA and AOS range of ±30°. This is in contrast to the hover flight condition depicted in Figure 17a, which indicates that distinct aeroDB should be constructed for each flight mode, given the unique inflow characteristics. The variation in performance with increasing RPM is consistent with the results for the lift propeller. However, the performance with respect to in AOA and AOS shows a different tendency. Increasing the inflow angles (AOA and AOS) leads to an increase in thrust, torque, and power. This is attributed to the increase in the effective angle of attack (α_eff) on the advancing side of the propeller, which results from a decrease in the axial velocity component and an increase in the tangential flow velocity component, thereby enhancing thrust and torque. Conversely, thrust and torque contributions of the retreating side decrease due to reduction in the tangential component of the inflow velocity and α_eff. Nevertheless, the advancing side’s contributions outweighs those of the retreating side, resulting in an overall increase in performance as the incidence angle increases. Since there is no cant angle, the performance is symmetrical with respect to positive and negative changes in AOA and AOS.

The results of the BEMT are compared with those of the SMM and ADM. The lift propeller results are compared for hover, climb, and descent flight conditions. The analysis conditions were defined as follows: a rotational speed of 1200 rpm (nominal), freestream velocities of 0, 2, and 10 m/s, and angles of attack of 0° (hover), −90° (climb), and 90° (descent). Figure 19 shows the comparison of the results. At hovering conditions (V_∞ = 0 m/s), the differences between the BEMT and SMM in thrust and torque were 10.7% and 4.7%, respectively, while the corresponding differences with ADM were 18.5% and 14.7%. The BEMT tends to overpredict at hovering conditions, as observed in the QTP and OPPAV tilt propeller analyses. This consistent result would be attributed to the limitations of the BEMT in taking into account for three-dimensional flow effects, which become more significant as the blade radius increases. A previous study [56] has also observed and reported a similar overprediction result with large blades, which is considered to have contributed to the present observations. As with previous observations, the ADM results again show an under-prediction in hovering conditions, reflecting the inherent limitations of the method.

In descent flight conditions, the BEMT predicted a slight decrease in thrust and an increase in torque compared to those in hover, whereas the SMM predicted an increase in thrust and a decrease in torque. These opposing trends result in a smaller difference in thrust between the BEMT and SMM and a slightly larger difference in torque. For the climb flight condition, the results are found to be similar to those observed in hover. The BEMT is found to provide reasonable aerodynamic data within the operational range of the lift propeller, as the overall differences between the BEMT and SMM (full CFD) were within 11%.

The results of propulsion propeller are evaluated for transition and cruise flight conditions. The freestream velocities were set at 40 m/s and 70 m/s for transition and cruise flight conditions, respectively. For the transition flight case, the rotational speed was set to 800 rpm, and the analysis was performed for AOA of −8°, −4°, 0°, 4°, and 8°. For the cruise flight case, the rotational speed was set to the nominal value of 1000 rpm, and the analysis was performed for an AOA of −4°, 0°, 4°, 8°, and 12°. Figure 20 presents the results of three-component forces and moments, as well as the corresponding results from SMM and ADM. As previously observed and discussed, F_y and M_y were calculated to be zero. Since these components were also small in the SMM and ADM results, they were excluded from the comparison. In the case of transition flight conditions, the BEMT showed a maximum difference of 5.8% in F_x compared to the SMM, indicating good agreement. However, in the case of cruise conditions, the difference in F_x increased. At an AOA of 0°, the difference was 11.2%, and the deviation increased slightly with increasing AOA. The results for the propulsion propeller results are found to be reasonable overall, as were the results for lift propeller.

Further examination was conducted to investigate the causes of the discrepancies between the results of the BEMT and the SMM. The smallest difference was observed under the transition flight condition for the propulsion propeller. At this condition, the rotational speed was set to 800 rpm, resulting in a blade tip Mach number (M_tip) of 0.33. Under cruise flight, the rotational speed was increased to 1000 rpm, corresponding to an M_tip of 0.41. For the lift propeller under hovering conditions, the analysis was conducted at 1200 rpm, resulting in an M_tip of 0.49 and a discrepancy of approximately 12% from the SMM results. These differences may not have been solely due to compressibility effects, because none of the operating conditions exceeded an M_tip of 0.6, above which compressibility is typically known to become significant. As the rotational speed increases, flow in radial direction develops along the blade surface due to centripetal and reactive centrifugal forces, which in turn enhances crossflow over the blade [57]. This crossflow, appearing as a velocity component perpendicular to the rotational and axial directions on the blade surface, was not considered in the BEMT formulation. Due to these limitations, the difference between the BEMT and SMM results is expected to increase at higher rotational speeds. This discrepancy lies within a range of potential differences that can arise between CFD analysis and flight test data. While this evaluation is subject to the specific context and assumptions of each application, such a margin of error is often considered acceptable in early-stage design processes where rapid estimation is prioritized. Although the BEMT tends to overpredict performance at higher rotational speeds or for the large radii, it can be still regarded as an appropriate and reasonable method for rapidly generating aerodynamic data for various engineering applications, such as design and performance evaluation, particularly due to its short computation time.

4.3. Performance Analysis of a Reverse-Engineered T-Motor Propeller

The T-Motor NS52x20 and NS57x22 carbon fibre propellers were analyzed to assess the practical applicability of the improved BEMT in analyzing commercially available propellers widely used in electric UAV systems. These propellers are a vertical take-off and landing propeller developed to target large drones, UAVs, and UAMs. The blade geometries were reverse-engineered using 3D scanning to obtain accurate shape data, which was then used for subsequent aerodynamic analyses. The geometries reconstructed through 3D scanning, which served as the basis for the analysis, are illustrated in Figure 21.

The diameters of the NS52x20 and NS57x22 propellers are 52 and 57 inches, respectively. The chord and twist distributions along the radial direction are shown to be different in these propellers. The subsequent extraction of the chord and twist distributions along the radial direction was conducted from the scanned geometry. Representative airfoil sections were selected at radial positions where significant variations were observed in the chord and twist gradients or in the airfoil geometry. The aerodynamic data of sectional airfoils were generated for six and eight airfoil sections of the NS52x20 and NS57x22 propellers, respectively. For each airfoil, the aerodynamic data (c_l, c_d) were generated from RANS simulations as functions of Reynolds and Mach numbers for each sectional airfoil. The BEMT simulations were performed, and the results were compared with the test report (TR) for propeller performance data [58] provided by the manufacturer to assess the accuracy of the present method.

The results of the performances and figure of merit (FM) under the hover conditions are plotted in Figure 22 for both NS52x20 (52-inch) and NS57x22 (57-inch) propellers. The BEMT analysis was conducted at the same RPM as those used in the TR data. For the NS52x20 propeller, the mean errors in thrust and torque (and thus power) are approximately 1.6% and 3.0%, respectively. In the case of the NS57x22 propeller, the mean errors observed were 1.3% in thrust and 2.5% in torque and power. The mean percentage error was calculated by comparing the BEMT-predicted values with the TR data at each RPM, and then taking the mean of all points. In both cases, the FM is underpredicted by approximately 2% on average, primarily due to the tendency of BEMT to overpredict torque in comparison to TR data. Nevertheless, the present accuracy is considered sufficient for the practical estimation of propeller performance in electric-powered UAV or eVTOL aircraft design.

5. Concluding Remarks

A BEMT formulation was proposed to enhance the robustness and versatility for UAM/AAM applications. A new velocity factor was introduced to improve numerical stability under low inflow conditions, and the BEMT framework was extended to enable propeller performance analysis under asymmetric inflow and descent flight conditions.

The improved BEMT was validated through the analysis of the QTP isolated propeller, and by comparing the results with those of wind tunnel tests and CFD analysis (OSM, SMM). The improved BEMT demonstrated good accuracy even under low inflow speed conditions where the basic BEMT formulation fails to converge. Although the BEMT does not account for the forces and moments induced by rotational velocity components under non-axisymmetric inflow, these were found to be relatively small in magnitude and have limited influence on overall performance. The improved BEMT was confirmed to reasonably predicts the dominant aerodynamic performance. In the case of a descent flight condition, the difference between the predicted performance and the SMM results was found to be less than 10%. In addition, the superior efficiency of the BEMT was confirmed by comparing the calculation time with that of CFD simulations.

The proposed method was applied to the analysis of the OPPAV tilt propeller. The improved BEMT provided results that were consistently comparable to ADM in terms of accuracy, although the difference increased slightly for flow conditions in which considerable radial velocity is expected. This makes it suitable for applying to early design stages, where rapid assessments of performance with reasonable accuracy are essential. In addition, the improved BEMT was used to construct a propeller performance database for the lift and propulsion propellers of the LC type UAM aircraft, intended for use in the UAM/AAM flight simulator. The improved BEMT was identified as a practical and efficient means of generating the necessary data for the various conditions that could be encountered during operational scenarios. In all evaluated cases, the prediction error relative to the SMM results remained within 14%, which is considered an acceptable level for many engineering applications. Therefore, the improved BEMT has a significant potential for applications in conceptual design, rapid performance analysis, and the construction of aerodynamic database for the distributed propulsion aircraft configurations, including eVTOL and UAM systems. In addition, the proposed BEMT was applied to commercially available T-Motor propellers, whose geometries were reconstructed through 3D scanning. The BEMT analysis, based on the scanned shapes, demonstrated high predictive accuracy when compared to the test report data provided by the manufacturer. These results confirm the practical utility of the proposed method when applied to the analysis and performance estimation of actual commercial propellers, thereby demonstrating its general applicability across various propeller types.

A well-known limitation of the BEMT is that it tends to overpredict performances in conditions and configuration where crossflow effects become significant. This limitation should be kept in mind when applying the method to propellers subjected to strong radial flow. Further studies for developing the techniques or methods to correct or account for the crossflow effects in the BEMT framework would be useful and meaningful.

In conclusion, the improved BEMT demonstrates a good compromise between accuracy and efficiency, as evidenced by its successful applications in analyzing the propellers of OPPAV and LC type UAM aircraft. Although its prediction capability diminishes under certain conditions due to inherent modeling limitations, it still remains an effective tool for conceptual design and early-stage aerodynamic assessments. Future work may aim to address these limitations by incorporating the effects of radial to further enhance its applicability.