Analysis and Evaluation of Aerodynamic Noise Characteristics of Toroidal Propeller

Abstract

Toroidal propellers, a new type of drone propellers capable of significantly reducing noise, offer new possibilities for future low-altitude flying platforms. In this study, a numerical model was established to analyze the aerodynamic noise of the toroidal propeller under normal atmospheric conditions. The aerodynamic calculations for the toroidal and benchmark propellers were performed to obtain noise source information using a transient large eddy simulation. The hybrid computational aeroacoustic method was employed to calculate the noise spectrum at different speeds and locations. Moreover, an experimental system for propellers was designed and built within an anechoic chamber to investigate the aerodynamic performance and noise characteristics at various speeds. The noise reduction effect of the toroidal propeller compared to the benchmark propeller was analyzed, taking into account the sound pressure level, lift coefficient, and figure of merit. At the same thrust level, the lift coefficient of the toroidal propeller increased by 187% relative to the benchmark propeller. The radial and axial sound pressure levels decreased by 5.2 dB(A) and 19.6 dB(A), respectively. The toroidal propeller significantly reduced noise while improving aerodynamic performance. This study provides a theoretical basis, experimental methods, and data support for the aerodynamic and noise calculations of toroidal propellers. It has significant engineering implications for the development of low-noise propellers for drones.

1. Introduction

Due to their unique advantages, multi-propeller UAVs play an important role in military patrols, field exploration, disaster search and rescue, etc. [1]. Propellers, as the main noise source of UAVs, exhibit complex aerodynamic noise characteristics under varying working conditions [2]. High-speed rotating propellers generate a large amount of noise, including thickness noise, load noise, and blade-vortex interference noise [3]. The results of psychoacoustic experiments conducted by the NASA Langley Research Center in 2017 showed that humans were more sensitive to the noise produced by small multi-propeller UAVs than to other traffic noise [4]. Prolonged exposure to propeller noise can have adverse effects on health [5].

These issues limit the broader application of drones in urban environments and military fields. Traditional approaches to the development of propeller structures for noise suppression have reached a bottleneck. Special leading-edge and trailing-edge structures have been shown to be effective in reducing aerodynamic noise. Kim et al. [6] studied the noise reduction mechanism of leading-edge serrated propellers through numerical simulation and found that the serrated structure altered the pressure pulsation on the leading-edge surface, leading to sound source truncation and noise reduction. Chaitanya et al. [7] investigated the sensitivity of serrated leading-edge parameters to aerodynamic noise. Yang et al. [8] examined the influence of the structural parameters and configurations of trailing edge serrations on both the aerodynamic performance and noise characteristics of drones under forward flight conditions, and identified the optimal parameters through a comprehensive performance comparison. The British company Westland [9] launched a "BERP" propeller that integrated forward sweep, backward sweep, and taper. This propeller demonstrated a significant suppression effect on high-speed pulse noise while improving hovering efficiency. However, these complex surface shapes of the propeller resulted in a loss of aerodynamic lift. The structural optimization efforts mentioned above were primarily focused on noise reduction, with less emphasis on enhancing aerodynamic performance.

To improve both the aerodynamic performance and noise characteristics of the propeller, Sebastian T. et al. [10] introduced a toroidal propeller configuration. This design reduced noise levels while maintaining the thrust and weight characteristics of traditional UAV propellers. As an innovative configuration, its unique shape facilitates aerodynamic noise reduction while preserving the propeller’s aerodynamic efficiency. Ye Liyu et al. [11] incorporated geometric parameters such as the wheelbase and inclination angle, and developed a mathematical method for describing the geometric shape of the toroidal propeller configuration, enabling the rapid 3D modeling of toroidal propellers. Hannah Jansen [12] employed numerical methods to evaluate the aerodynamic performance of three-blade, four-blade, and toroidal propellers, examining how variations in the minor axis length and the number of blades in toroidal propellers affect the aerodynamic performance. Most research on the toroidal propeller configuration has focused on structural optimization and parameterization. However, a comprehensive evaluation of the toroidal propeller configuration, including noise, is lacking. It is essential to obtain the noise characteristics of toroidal propellers and assess their overall performance before optimizing the structure of the configuration.

The calculation of propeller aerodynamic noise is primarily based on the FW-H equation derived from the acoustic analogy method. Wang Yang et al. [13] experimentally measured the aerodynamic forces on the rotor surface, obtained the precise aerodynamic loads, and then used the FW-H method to calculate the rotor noise. By combining the FW-H method with CFD technology, Xie Futian et al. [14] analyzed the directivity of rotor aerodynamic noise using the FW-H equation. HUANG et al. [15] derived the Navier–Stokes equations to analyze the aerodynamic characteristics of propellers with varying shape parameters and used the FW-H equation to study their noise performance. In terms of aerodynamic interactions between propellers and airfoils, Zhe Yang et al. [16] employed large eddy simulations with adaptive mesh refinement to predict the turbulent near-field flow, while the FW-H method was used to assess the noise emission to the far field. Daniel Lindblad et al. [17] coupled the FW-H method with a regularized boundary element method (BEM) to compute noise scattering by solid bodies.

Building on the previous research on aerodynamic calculations, this paper investigates the aerodynamic and noise performance of the toroidal propeller independently designed by our research group. The suitability of the proposed simulation method and the accuracy of the simulation results were verified through a comparison with the experimental data. The noise and aerodynamic performance of the toroidal propeller were comprehensively compared with those of the benchmark propeller, demonstrating the toroidal propeller’s low noise and high aerodynamic efficiency. The results provide a method for predicting the noise performance of toroidal propellers and offer design guidelines for the optimization of low-noise propellers.

2. Numerical and Experimental Methods

2.1. Numerical Methods

This paper investigates the toroidal propeller, with structural parameters obtained from the previous DOE parametric design conducted by our team. The propeller has a diameter of 208 mm. The two-dimensional streamlined airfoil section of the toroidal propeller was constructed by superimposing the thickness of the mid-arc line. The two mid-arc lines were defined using third-order Bézier curves, and a thickness distribution function was introduced to describe the normalized maximum thickness of the unit airfoil per unit length. Once the mid-arc line description and thickness distribution were defined, the unit airfoil section was generated. The parent section, which served as the source for the airfoil sections of the entire toroidal propeller, was a unit streamlined airfoil located at the origin, with a constant chord length of one unit. By varying the chord length, torsion angle, and normalized maximum thickness, the parent section was stretched and twisted to generate new sections. To ensure the smoothness of the toroidal propeller’s three-dimensional model and optimize its aerodynamic performance, five newly generated sections were used to control the shape of the propeller. In contrast to other propellers composed of sections stacked along radial lines, the toroidal propeller was formed by sweeping these five airfoil sections along a circular trajectory.

Sweeping involves taking a two-dimensional object as a cross-section and moving it along a trajectory that follows certain rules to form a solid. The sweeping process of the toroidal propeller was constrained by using upper and lower trajectories, both of which were fitted with non-uniform rational B-spline (NURBS) curves. A set of points on the curve surface was used to inversely solve for the coordinates of the control points, and the entire curve surface was then fitted and expressed using the obtained control point coordinates. To ensure the smoothness of the model’s surface, the sweeping trajectory and rule were defined such that the normal direction of the trajectory aligned with the plane of the cross-section. The airfoil section and the upper and lower trajectory curves are shown in Figure 1, while the final three-dimensional model of the toroidal propeller obtained after sweeping is shown in Figure 2.

The aerodynamic and noise performance of the toroidal propeller were compared with that of a two-blade propeller (8330) having the same diameter parameters. To calculate the aerodynamic noise, a flow field simulation was first conducted. The lift coefficient and figure of merit were determined from steady-state simulations, which provided lift and torque data under various speed conditions, used to evaluate the aerodynamic performance. Based on these steady-state simulations, specific speed conditions were solved transiently to obtain the pressure pulsation data on the propeller surface, which served as input for the aerodynamic noise simulation. Through sound propagation calculations, the aerodynamic noise at the monitoring point was determined.

2.1.1. Numerical Calculation Method of Aerodynamic Noise

A steady-state numerical simulation of the toroidal propeller flow field was conducted using the Reynolds-averaged Navier–Stokes equations, with the integral form given as follows:

$${\displaystyle \frac{\partial }{\partial t}}{\displaystyle \underset{\mathrm{\Omega }}{\iiint }UdV}+{\displaystyle \underset{d\mathrm{\Omega }}{\iint }F(U)}·ndS={\displaystyle \underset{d\mathrm{\Omega }}{\iint }G(U)}·ndS$$

(1)

The turbulence model adopted was the SST k-ω model, which is suitable for numerical calculations of turbulence with varying intensities. The definitions of each term in the governing equations were referenced from the literature [18]. The transient simulation methods for the flow field include the unsteady RANS method, discrete eddy simulation (DES), and large eddy simulation (LES). The LES method can more accurately simulate the turbulent pulsations near the wall, providing a more precise prediction of blade surface pressure fluctuations [19]. LES separates large-scale and small-scale vortices in turbulence using a filter function, decomposing transient variables into solvable scale variables and sub-grid scale variables:

$$\varphi =\tilde{\varphi }+{\varphi }^{\prime }$$

(2)

The filter variable is defined as follows:

$$\tilde{\varphi }(x)={\displaystyle \int \tilde{\varphi }(x^{\prime })}G(x,x^{\prime })d{x}^{\prime }$$

(3)

where $G(x,x^{\prime })$ is the filter function. The continuity equation and momentum equation are spatially filtered to obtain the large eddy simulation control equation:

$${\displaystyle \frac{\partial {\tilde{u}}_j}{\partial x_i}}=0$$

(4)

$${\displaystyle \frac{\partial {\tilde{u}}_i}{\partial t}}+{\displaystyle \frac{\partial }{x_j}}({\tilde{u}}_i{\tilde{u}}_j)=-{\displaystyle \frac{\partial \tilde{p}}{x_j}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[v\left({\displaystyle \frac{\partial {\tilde{u}}_i}{\partial x_j}}+{\displaystyle \frac{\partial {\tilde{u}}_j}{\partial x_i}}-{\displaystyle \frac{2}{3}}{\delta }_{ij}{\displaystyle \frac{\partial {\tilde{u}}_k}{\partial x_k}}\right)\right]-{\displaystyle \frac{\partial {\tau }_{ij}}{\partial x_j}}$$

(5)

When performing transient simulations, the sliding grid method was employed. The simulation time step and number of steps were determined based on the rotational speed. The time required for the blade to rotate 1° was taken as the step length, and the total simulation time corresponded to the time for the toroidal propeller to complete seven full rotations.

Currently, numerical methods for solving aerodynamic noise are divided into the direct noise calculation (DNC) method [20] and the hybrid method [21]. The direct method requires an extremely fine grid to capture near-wall flow, which leads to excessive computational cost and time, making it unsuitable for complex turbulent aerodynamic noise calculations. By using the FW-H equation for sound propagation, the calculation speed of aerodynamic noise can be significantly improved. In the hybrid method, CFD and FW-H are combined: sound source terms at the boundary are calculated and extracted via transient simulations. The acoustic expansion is obtained through the Lighthill acoustic analogy method and substituted into the FW-H equation to calculate the far-field noise propagation.

The FW-H equation reorganizes the N-S equations into a wave equation using a generalized function, which can be expressed as follows [22]:

$$\left({\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2}{\partial t^2}}-{\displaystyle \frac{{\partial }^2}{\partial x_i^2}}\right)p^{\prime }(x_i,t)={\displaystyle \frac{\partial }{\partial t}}\left\{\left[{\rho }_0{v}_n+\rho (u_n-v_n)\right]\delta (f)\right\}-{\displaystyle \frac{\partial }{\partial x_i}}\left\{\left[-P_{ij}^{\prime }·n_{ij}+\rho (u_n-v_n)\right]\delta (f)\right\}+{\displaystyle \frac{{\partial }^2}{\partial x_i\partial x_j}}[T_{ij}H(f)]$$

(6)

In the formula, $p^{\prime }(x_i,t)$ represents the acoustic pressure at the observation point at time t; $\rho$, $u_i$, $P_{ij}$ represent density, velocity, and stress tensors, respectively; ${\delta }_{ij}$ is the Kronecker symbol; $n_j$ is the unit disturbance velocity on the control surface; $v_n$ is the motion velocity of the control surface; subscript 0 represents the undisturbed quantity, and the prime sign represents the disturbance quantity. $H(f)$ is the Heaviside step function, and $\delta (f)$ is the Dirac function. The three terms on the right-hand side of Formula (6) represent monopole noise, dipole noise, and quadrupole noise, respectively. For the aerodynamic noise generated by the propellers on UAVs, the influence of quadrupole noise is typically neglected. In this paper, only monopole and dipole noise are considered in the calculation of the aerodynamic noise of the toroidal propeller.

2.1.2. Aerodynamic Simulation Model

A flow field calculation model was developed for the toroidal propeller and its surrounding flow domain, which consisted of both a rotating domain and a stationary domain. As shown in Figure 3, the rotating domain fully enclosed the toroidal propeller. The diameter of the stationary domain was eight times the size of the rotating domain, and its height was ten times the size of the rotating domain.

The flow field calculation model was meshed. If the mesh size is too large, the error in the subsequent noise performance simulation will be significant; conversely, a smaller mesh size leads to higher computational costs and time. The mesh size for the rotating domain surface was 4 mm, with a total of 2.9 million cells, while the mesh size for the stationary domain surface was 50 mm, with 1.3 million cells. The mesh quality was improved by refining cells with a quality lower than 0.1. The resulting mesh for the toroidal propeller blade surface is shown in Figure 4. Based on the surface y+ values obtained from the steady-state simulation, as depicted in Figure 5, the maximum value did not exceed 10, indicating that the mesh division met the calculation requirements for the model.

The steady-state simulation employed the k-omega SST model. The upper and lower boundaries of the static domain were defined as pressure inlet and outlet, respectively, while both the static domain and the surface of the toroidal propeller were set as wall boundaries. The rotating coordinate system method was applied in the steady-state simulation, with spatial discretization based on the least squares cell gradient and second-order upwind turbulent kinetic energy. The center of rotation and speed conditions were specified, and the steady-state simulation was conducted to obtain the lift and figure of merit of the toroidal propeller for subsequent aerodynamic performance evaluation. The steady-state simulation results served as initial values for unsteady numerical calculations of the flow field. A large eddy simulation (LES) model was employed to better capture small pressure fluctuations on the blade surface [23], which were then used as input for the subsequent aerodynamic noise simulation.

2.1.3. Aerodynamic Noise Prediction Model

To establish a noise prediction model for toroidal propellers, it is necessary to first generate the acoustic mesh and perform load transfer preprocessing. Compared to the boundary element method, the finite element method generates acoustic grids for all areas, including the propeller, resulting in a much larger mesh size than that produced by the boundary element method. Therefore, the boundary element method was selected for acoustic meshing. For acoustic calculations, it is generally required that at least six mesh elements fit within the shortest wavelength of the sound wave [24]. The maximum mesh size can be calculated as one-sixth of the shortest wavelength:

$$L\le {\displaystyle \frac{c}{6f_{\max }}}$$

(7)

where c is the speed of sound in the flow field. $f_{\max }$ is the maximum noise frequency. L is the maximum grid size. Figure 6 is the obtained acoustic boundary element grid.

The surface pressure pulsation data obtained from the transient simulation were located at the center points of the grid. After being imported into the acoustic boundary element solver, the noise source information was transferred to the nodes of the acoustic grid. Due to the large number of nodes, the computational resources and time required for the numerical solution were prohibitive. To address this, adjacent nodes were integrated, resulting in a finite number of compact fan sound sources, as shown in Figure 7. The fan source model and the acoustic boundary element mesh were then used for subsequent sound propagation calculations. It was assumed that there were no reflective surfaces in the external environment, and the noise reflection from the rotating shaft was neglected.

The acoustic boundary element mesh and the processed fan sound source model were imported into the sound propagation solver. Far-field monitoring points were set up to extract sound pressure level signals at various directions and positions. A 2000 mm × 2000 mm vertical and horizontal dot matrix was created on the cross-section of the toroidal propeller and the longitudinal section of the propeller hub, with the center of the toroidal propeller as the origin. Directional field points were arranged at 10° intervals on circles with radii of 0.5 m and 1 m, both centered at the origin, as shown in Figure 8.

The sliding grid method was used to control the grid movement in the rotating domain. The speed, boundary conditions, and other settings were consistent with those used in the aerodynamic performance simulation. The sampling frequency for the noise simulation was determined based on the Nyquist sampling theorem, and the simulation time step was calculated as follows:

$$f_s={\displaystyle \frac{1}{\mathrm{\Delta }t}}\ge 2f_{\max }$$

(8)

The sound pressure level at the layout points was calculated using the above model. Since the geometric dimensions of the benchmark and toroidal propellers were consistent, the same procedure was applied to the benchmark propeller, and the sound pressure levels at the corresponding layout points were calculated. The results for both propellers were processed, and the comprehensive evaluation of the toroidal propeller was conducted by comparing its aerodynamic and noise performance with that of the benchmark propeller.

2.2. Experimental Methods

A propeller noise performance test bench was built in an anechoic chamber to measure the sound pressure levels of the toroidal and benchmark propellers at different distances and angles. This bench integrated acoustic sensors, speed sensors, electric regulators, and other equipment. As shown in Figure 9, the propeller under test was fixed to the platform in a perpendicular direction using connectors and support rods. The speed was adjusted by the motor speed regulator to meet the test conditions. Microphone sensors were arranged at specific distances and angles to measure aerodynamic noise, and a sound pressure level spectrum was obtained to describe the noise characteristics. The physical models of the toroidal and benchmark propellers are shown in Figure 10.

The sound pressure level directivity indicates that the noise radiates unevenly in all directions. Analyzing the directivity differences of propellers with different configurations provides guidance for the subsequent design of noise-reduction propellers. The acoustic array monitoring points were evenly arranged on a circular plane with a fixed radius centered on the propeller, spaced every 20°, and the sound pressure levels at these points were measured by microphone sensors fixed on a circular bracket. Since both propeller configurations were symmetrical, the distribution of the sound pressure levels was also symmetrical. The overall sound pressure level directivity was obtained by measuring on one side of the propeller. A total of 11 microphone sensors were arranged in the range of 0 to 180°, as shown in Figure 11a. The microphone sensors were fixed parallel to the ground on a custom-made steel ring, and the height of the steel ring was adjusted so that the sensors faced the center of the propeller. Wind shields were placed in front of the sensors to reduce the impact of airflow pressure pulsation during propeller rotation on the measurements. Figure 11b shows the sound pressure level directivity test bench.

3. Results and Discussion

3.1. Analysis of Aerodynamic Characteristics of Toroidal Propeller

The lift and figure of merit of both the toroidal propeller and the benchmark propeller at different speeds were calculated based on the results of the aerodynamic performance simulation. The figure of merit is defined as the ratio of induced power to actual power during the propeller’s hovering state. It represents the hovering efficiency of the propeller and is related to the lift and drag coefficients. A higher figure of merit indicates greater hovering efficiency, and the expression is as follows:

$$FM={\displaystyle \frac{C_T^{3/2}}{\sqrt{2}{C}_P}}$$

(9)

where $C_T$ is the lift coefficient and $C_P$ is the power coefficient. The expression is as follows:

$$C_T={\displaystyle \frac{T}{\rho A{\mathrm{\Omega }}^2{R}^2}}$$

(10)

$$C_P={\displaystyle \frac{P}{\rho A{\mathrm{\Omega }}^3{R}^3}}$$

(11)

In the formula, $T$ is the propeller lift; $\mathrm{A}=\pi R^2$ is the disc area, $R$ is the propeller radius; $\mathrm{\Omega }$ is the disc speed; power is $\mathrm{P}=Q\mathrm{\Omega }$; and the torque coefficient is numerically equal to the power coefficient. The air density at room temperature is 1.225 kg/m³, and all units are in the International System of Units (SI). The lift coefficient represents the propeller’s ability to generate lift. As the propeller radius decreases and the speed reduces, the lift increases, resulting in a higher lift coefficient.

Table 1. Comparison of the lift between toroidal propeller and benchmark propeller at different speeds.

Speed/r/min	Lift/N
	Toroidal Propeller	Benchmark Propeller	Percentage Increase
3000	1.38	0.46	200%
4000	2.47	0.83	197%
5000	3.90	1.33	193%

and Figure 12 present a comparison of the lift between the toroidal propeller and the benchmark propeller, as obtained through a simulation at 3000, 4000, and 5000 r/min.

Table 2. Comparison of the figure of merit between toroidal propeller and benchmark propeller at different speeds.

Speed/r/min	Figure of Merit
	Toroidal Propeller	Benchmark Propeller	Percentage Increase
3000	0.502	0.468	7.3%
4000	0.518	0.481	7.7%
5000	0.521	0.502	3.8%

and Figure 13 present a comparison of the figure of merit between the toroidal propeller and the benchmark propeller, as obtained through a simulation at 3000, 4000, and 5000 r/min.

According to the lift table of the toroidal propeller and the benchmark propeller at different speeds, under the same speed condition, the lift provided by the toroidal propeller was increased by up to 200% compared to the benchmark propeller, and the figure of merit was increased by up to 7.7%. At the same speed, the lift and figure of merit of the toroidal propeller were higher than those of the benchmark propeller. Compared to the benchmark propeller, the toroidal propeller had a better aerodynamic performance.

3.2. Analysis of the Noise Spectrum Characteristics of Toroidal Propeller

All noise performance experiments were conducted in a closed anechoic chamber during late hours. The ambient noise level was significantly lower than the aerodynamic noise produced by the toroidal propeller, and its influence on the results was deemed negligible. The noise spectrum was obtained through both numerical calculations and experimental methods. If the noise spectrum at a specific point in the sound field is known, the total sound pressure level at that point can be expressed as follows:

$$OSPL=20\lg {\displaystyle \frac{\sqrt{\frac{1}{f_{\max }-f_{\min }}{\displaystyle {\int }_{f_{\min }}^{f_{\max }}{p}^2(f)df}}}{p_r}}$$

(12)

$p(f)$ is the sound pressure value at frequency f in the spectrum, and the trapezoidal integration method and 1/3 octave bands were used to integrate the above formula.

Based on the spectrum derived from both the simulation and experimental results, as shown in Figure 14, the noise generated by the toroidal propeller consisted of broadband noise and discrete noise components. The pressure pulsations on the surface of the propeller contributed to a low-level broadband noise, which spanned a wide range of frequencies. In contrast, the discrete noise was primarily due to rotational effects, with distinct peaks occurring at multiples of the blade passing frequency. The blade passing frequency is related to the rotational speed and the number of blades, as follows:

$$f={\displaystyle \frac{nZ}{60}}i (i = 1,2,3\dots )$$

(13)

where n is the speed; Z is the number of blades; and i is the blade passing frequency order. The shaft frequency is solely dependent on the rotational speed. For the toroidal propeller, which has two blades, the shaft frequency is half that of the blade passing frequency.

At a rotational speed of 5000 r/min, the shaft frequency of the toroidal propeller was 83.3 Hz, and the blade frequency was 166.6 Hz. Based on both simulation predictions and experimental results, the noise sound pressure level reached its maximum at the first-order blade frequency. Within the 1000 Hz range, including the fifth-order blade frequency, the peak values of discrete noise exhibited a decreasing trend with increasing order. Additionally, between the peak sound pressure levels of each blade order, lower sound pressure peaks were observed, corresponding to the shaft frequencies of the toroidal propeller.

Figure 14 presents the predicted and experimental results at 5000 r/min for the 0° azimuth of the toroidal propeller disk cross-section.

Table 3. Simulation and experimental errors of noise sound pressure levels at different distances.

Distance/m	Frequency/Hz	Sound Pressure Level/dB(A)
		Simulation	Test	Error
1	166.6	50.08	49.14	−1.91%
	333.3	42.74	42.56	−0.42%
	0–1000	53.41	57.37	6.90%
0.5	166.6	60.88	61.67	1.28%
	333.3	52.18	50.22	−3.90%
	0–1000	63.52	64.59	1.65%

compares the sound pressure levels at the first- and second-order blade frequencies within the 1000 Hz range. At a distance of 1 m, the error between the predicted and experimental total sound pressure levels for the toroidal propeller was 6.90%. At a distance of 0.5 m, this error reduced to 1.7%. Generally, the closer the observation point, the more accurate the predicted sound pressure levels. The errors for the first- and second-order blade frequency sound pressure levels at 1 m did not exceed 2%. In contrast, the error for the second-order blade frequency sound pressure level was higher at closer distances. This discrepancy can be attributed to the limitations of the hybrid numerical simulation method, which struggles to capture high-frequency and smaller turbulent pulsations. However, since high-frequency and small-scale vortices have minimal impact on the overall noise performance of the toroidal propeller, the resulting errors are deemed small and acceptable. Furthermore, the method’s lower computational cost and reduced calculation time make it a viable approach. Overall, the noise prediction model demonstrated a high accuracy and provided a reliable estimate of the aerodynamic noise performance of the toroidal propeller.

At 5000 r/min, the noise spectrum of the toroidal propeller within the 2000 Hz range was predicted and measured at a point 90° from the center of the hub’s longitudinal section. The results are shown in Figure 15. Clear peaks in the sound pressure level were observed at each blade frequency. Within 1000 Hz, the predicted sound pressure levels closely matched the measured results. However, above 1500 Hz, the predicted levels began to attenuate and diverged from the experimental results. This discrepancy can be attributed to two main factors: first, the limitations of the prediction model in the high-frequency range, and, second, the influence of high-frequency electromagnetic noise generated by the motor, as well as noise from the interaction between the airflow and the test bench, which impacted the experimental measurements.

3.3. Analysis of the Directivity Characteristics of Toroidal Propeller Noise

The directivity of propeller noise is a key parameter in noise characteristic analysis. The sound pressure levels in all directions are typically represented using a polar coordinate system. In this study, the noise directivity of the toroidal propeller’s cross-section and the longitudinal section of the hub at distances of 0.5 m and 1 m from the center were analyzed. Figure 16 illustrates the relationship between the reference direction of the polar coordinate system for different sections and the position of the toroidal propeller.

The noise directivity distribution of the propeller disc cross-section at 1 m and 0.5 m from the center was predicted at 3000 r/min and 5000 r/min, respectively. At 3000 r/min, the noise directivity distribution for both propellers at 0.5 m and 1 m is shown in Figure 17. At 5000 r/min, the noise directivity distribution for both propellers at 0.5 m and 1 m is shown in Figure 18.

Under different speed conditions, the noise characteristics of both the toroidal propeller and the benchmark propeller exhibited uniform monopole characteristics across all directions of the propeller disk cross-section. The noise sound pressure level at the same distance from the rotor center was nearly identical, and the sound pressure level decreased progressively with increasing distance. Based on the noise directivity characteristics predicted by numerical calculations, the average sound pressure level and sound pressure level attenuation rate for both the toroidal propeller and the benchmark propeller at 3000 r/min and 5000 r/min are presented in

Table 4. Comparison of noise performance between toroidal propeller and benchmark propeller.

Sound Pressure Level	3000 r/min		5000 r/min
	Toroidal Propeller	Benchmark Propeller	Toroidal Propeller	Benchmark Propeller
0.5 m	46.14	41.96	63.24	50.53
1 m	33.98	35.59	53.27	43.22
Attenuation	12.16	6.37	9.96	7.31
Attenuation rate	26.3%	15.2%	15.7%	14.4%

.

When the speed of the toroidal propeller was the same as that of the benchmark propeller, the noise sound pressure level attenuation rate of the toroidal propeller was higher than that of the benchmark propeller within a certain range of distance. As a result, the noise energy of the toroidal propeller attenuated more rapidly compared to the benchmark propeller.

The noise directivity characteristics of the toroidal propeller and the benchmark propeller on the hub’s longitudinal section at 5000 r/min were predicted, as shown in Figure 19. The noise directivity results for the two propellers at the same distance on the longitudinal section of the hub differed significantly. The noise of the benchmark propeller exhibited clear dipole characteristics, whereas the dipole characteristics of the toroidal propeller blade were less pronounced. The largest difference in noise sound pressure levels between the toroidal and benchmark propellers occurred at the 0° and 180° positions. Conversely, at the 90° and 270° vertical positions, the noise from the toroidal propeller was lower than that of the benchmark propeller. The unique configuration of the toroidal propeller notably altered its noise characteristics, resulting in a significant noise reduction effect.

The noise directivity characteristics of the toroidal propeller and the benchmark propeller on the hub’s longitudinal section were tested at 2000 r/min, 3000 r/min, 4000 r/min, and 5000 r/min, as shown in Figure 20. The benchmark propeller exhibited a distinct "8"-shaped dipole characteristic in its noise directivity. Although the sound pressure level of the toroidal propeller was slightly higher at positions between 30° and −30° horizontally compared to the symmetrical positions, its vertical sound pressure levels were lower. Under all speed conditions, the vertical sound pressure level of the toroidal propeller was consistently lower than that of the benchmark propeller. For the toroidal propeller, the noise distribution on both sides of the plane was more uniform, demonstrating a better noise reduction effect.

3.4. Comprehensive Evaluation of Toroidal Propeller

According to the results in Section 3.1, at the same speed, the lift of the toroidal propeller was significantly higher than that of the benchmark propeller, indicating a superior aerodynamic performance. Under identical thrust conditions, the noise performance of the toroidal propeller was compared with that of the benchmark propeller. A comprehensive evaluation of the toroidal propeller was conducted, considering both its lift and noise characteristics.

Through numerical calculations, the thrust level of the toroidal propeller at 3000 r/min was found to be nearly equivalent to that of the benchmark propeller at 5000 r/min, as shown in

Table 5. The thrust level.

	Speed Conditions (r/min)	Lift (N)
The toroidal propeller	3000	1.381
The benchmark propeller	5000	1.335

. Under the same thrust conditions, the noise spectrum of both the toroidal propeller and the benchmark propeller at distances of 0.5 m and 1 m from the center is presented in Figure 21.

Based on the noise spectrum below 2000 Hz at different distances, the peak noise sound pressure level at the blade frequency of the toroidal propeller was significantly lower than that of the benchmark propeller, while the broadband noise levels were similar. Above 2000 Hz, the broadband noise of the toroidal propeller was substantially lower than that of the benchmark propeller.

Table 6. The sound pressure levels of the two propellers.

	Sound Pressure Level/dB(A)
	0.5 m	1 m
Toroidal propeller	53.78	48.33
Benchmark propeller	60.23	55.09
Noise reduction percentage	12%	14%

presents the sound pressure levels of both propellers, calculated from the noise spectrum.

At the same thrust level, the noise sound pressure level of the toroidal propeller was reduced by 12% at a distance of 0.5 m and by 14% at 1 m, compared to the benchmark propeller. The toroidal propeller not only reduced discrete noise at the low-order blade frequencies but also attenuated high-frequency broadband noise, resulting in noise reduction across the entire frequency range.

At the same thrust level, the lift coefficient, figure of merit, axial sound pressure level, and radial sound pressure level of the two propellers were comprehensively evaluated, as shown in Figure 22. A propeller with a higher figure of merit and lift coefficient, along with a lower sound pressure level, exhibits better overall performance.

From the four-dimensional diagram, at the same thrust level, the figure of merit of the toroidal propeller was nearly identical to that of the benchmark propeller. However, the lift coefficient of the toroidal propeller was 187% higher than that of the benchmark propeller. In comparison with the benchmark propeller, the radial sound pressure level of the toroidal propeller decreased by 5.2 dB(A), and the axial sound pressure level decreased by 19.6 dB(A). Under the same thrust level, the toroidal propeller demonstrated a better aerodynamic performance and lower noise. The new configuration not only enhanced the aerodynamic performance but also significantly reduced the aerodynamic noise, resulting in an improved overall performance.

4. Conclusions

This paper presents a toroidal propeller designed for enhanced noise reduction. A numerical model was developed to calculate both the aerodynamic and noise performance, which was then validated through experimental testing. The results were comprehensively compared with those of a benchmark propeller. The aerodynamic performance of the toroidal propeller was obtained through steady-state numerical simulations, while the noise performance was predicted using unsteady simulations coupled with a noise prediction model. To validate the predictions, an anechoic chamber and noise performance test bench were constructed for experimental research. Based on the results, the following conclusions were drawn:

(1) This paper presents a toroidal propeller designed to achieve both high lift and low noise. At the same thrust level, the lift coefficient of the toroidal propeller was increased by 187% compared to the benchmark propeller, while the radial and axial sound pressure levels were reduced by 5.2 dB(A) and 19.6 dB(A), respectively. The toroidal propeller demonstrated superior aerodynamic and noise performance, achieving a significant noise reduction effect under the same lift conditions.

(2) According to the noise spectrum, the toroidal propeller exhibited discrete noise peaks at each blade frequency, while broadband noise was widely distributed across the frequency spectrum. Within a 1-meter range from the toroidal propeller, the error between the predicted and experimental noise sound pressure levels in the 0–1000 Hz range did not exceed 7%. Additionally, the error at the first two blade frequencies remained below 4%.

(3) According to the noise directivity performance diagram, the sound pressure level results for the toroidal and benchmark propellers in the cross-section of the propeller disc were similar. However, the noise from the toroidal propeller attenuated more rapidly with distance compared to the benchmark propeller. In the hub longitudinal section, significant differences in noise directivity were observed. The benchmark propeller displayed clear dipole characteristics, whereas the toroidal propeller exhibited a more uniform sound pressure level distribution in all directions.