Layout and Rotation Effect on Aerodynamic Performance of Multi-Rotor Ducted Propellers

Highlights

What are the main findings?

• Rotational consistency (co-rotation vs. counter-rotation) is the dominant factor affecting aerodynamic performance, with co-rotation significantly enhancing thrust and efficiency.
• The duct mitigates inter-propeller aerodynamic interference, altering the amplitude and phase of the sinusoidal performance shifts caused by the rotational phase gap.

What is the implication of the main finding?

• Optimizing rotational consistency, particularly using co-rotation, is a critical strategy that plays a crucial role in maximizing the performance of multi-rotor ducted propeller systems.
• This research offers a validated analytical method (ART-ANOVA) and reveals core flow mechanisms, providing a framework to guide the design and optimization of advanced low-altitude aircraft propulsion systems.

Abstract

Multi-rotor ducted propellers, which integrate the high-efficiency characteristics of ducted propellers with the layout flexibility and safety advantages of distributed propulsion, are extensively utilized in the propulsion systems of low-altitude transport systems and large-scale unmanned aerial vehicles. This study numerically investigates the effects of spanwise distance, streamwise distance, rotational consistency, and rotational phase gap on the unsteady aerodynamic characteristics of multi-rotor ducted propellers under hovering conditions. A parameterized numerical computation model and an Aligned Rank Transform Analysis of Variance (ART-ANOVA) method suitable for small datasets exhibiting regular patterns were developed. Initially, numerical simulations investigated the aerodynamic performance of multi-rotor ducted propeller models with varying layout parameters. The aerodynamic coefficients of the propellers monotonically decrease as the layout spacing increases; however, the change trends differ. Aerodynamic interference reduces the airflow velocity and influences the distribution of high-pressure zones, consequently impacting thrust and efficiency. Subsequently, this paper examined the coupled effects of two rotational characteristics. The relationship between propeller aerodynamic performance and rotational phase gap exhibits distinct trigonometric function characteristics. The presence of the duct mitigates the mutual interference between blades, thereby altering the amplitude and phase of these characteristics. Finally, an ART-ANOVA method was employed to quantify the main and interaction effects, revealing that rotational consistency has a dominant influence on all aspects of aerodynamic performance. Insights into aerodynamic performance are crucial for advancing low-altitude transport systems that utilize ducted propeller propulsion systems.

1. Introduction

Compared to open propellers, ducted propellers can generate higher hovering lift by leveraging the Bernoulli effect without significantly increasing power. The duct encloses the propeller, suppressing tip vortices, increasing its operational speed, and reducing noise [1,2,3]. In recent years, ducted propellers have been utilized as propulsion systems on multiple electric Vertical Take-Off and Landing (eVTOL) or low-altitude transportation systems [4,5,6,7]. Ducted propellers as propulsion systems typically adopt a multi-rotor configuration, which differs from the extensively studied single-rotor ducted propellers used as standalone Unmanned Aerial Vehicles [3].

The aerodynamic performance of distributed propulsion systems in aircraft closely connects to the design of the fuselage and wings. Extensive research has addressed the mutual interactions arising from coupling effects between distributed propellers and multi-rotor ducted propellers, their impact on overall aerodynamic characteristics, and the corresponding optimization methodologies [8,9,10,11,12,13,14,15,16,17,18,19].

Zhou et al. combined an actuator disk model with an overset grid method to solve for the aerodynamic performance of a tiltable distributed ducted system located at the wing trailing edge. They determined the variation patterns of duct aerodynamic performance under different installation angles and tip speed ratios, where inflow velocity plays a decisive role [20]. Ma et al. proposed a momentum source calculation and optimization method for multiple duct units based on a distributed propulsion configuration with ducted fans positioned between upper and lower wings. This method allowed for quick aerodynamic calculations and designs for multi-duct systems. However, the mutual interference effects between ducts require additional optimization and comprehensive research [21]. Guo et al. performed parametric modeling of distributed ducted fans and conducted steady-state approximate solutions using the Multiple Reference Frame (MRF) model. They established an optimization framework for geometry, evaluated aerodynamic performance under different duct cross-sections, and discovered how layout parameters of duct spacing affect aft flow separation via inlet pressure using flow field analysis [22]. Xue et al. calculated circulation distribution and established an aerodynamic interference computational model for leading-edge distributed propellers and wings based on a propeller wake model. They evaluated and optimized the wing lift-to-drag ratio under unsteady wake effects [13]. Zhang et al. investigated the steady aerodynamic–propulsion coupling and vector thrust characteristics for a tandem-configuration distributed ducted propulsion aircraft. Deflection of the propulsion unit leads to increased thrust and normal force, which is more pronounced in ducted fan propulsion systems [23]. However, studies on overall aerodynamic characteristics often neglect the unsteady characteristics between ducted propellers.

Unsteady aerodynamic interference between propellers is significant in distributed propulsion, affecting overall lift performance. The position and rotational state have a remarkable effect on the intensity of this interference. Power modules exhibit distinct coupling characteristics in distributed propulsion as well. Layout parameters and the operational status of the modules also impact overall propulsive performance. Nguyen et al. proposed an aeroelastic wing employing distributed propulsion and investigated the regulatory effect of differential thrust on overall aircraft control [24]. Perry et al. calculated the aerodynamic characteristics of distributed ducts fixed at the wing trailing edge using the actuator disk method and conducted experimental validation based on the Particle Image Velocimetry method. They quantitatively described the complex interactions between ducts and wing segments, as well as between the ducts themselves, under inflow conditions, also focusing on the impact of differential thrust [25]. De Vries et al. adapted wind tunnel experiments to study the aerodynamic characteristics and flow field distribution of three closely arranged side-by-side propellers, analyzing the effects of factors such as rotational direction, angle of attack, relative position, thrust difference, and phase angle on system performance [26]. Stokkermans et al. and Alvarez, utilizing wind tunnel experiments and the Vortex Particle Method, respectively, conducted detailed research on the influence of propeller layout on aerodynamic interference. The correlation between downwash and geometric position is decisive [27,28]. Li et al. proposed an equivalent analysis criterion for ducted propellers, enabling size scaling for different configurations of ducted propellers that produce the same hovering thrust at the same power consumption [29]. In low-altitude flight scenarios, the hovering state is crucial during flight operations [30]. During eVTOL flight, the aerodynamic characteristics in the transition from hovering to forward flight state are complex, and the drag impact introduced by the ducts becomes more prominent [31,32]. Studying how ducted propellers’ layout and rotational characteristics affect aerodynamic performance, especially the unsteady characteristics during significant hovering operations, is crucial.

This paper thoroughly evaluates the factors influencing unsteady aerodynamic characteristics of ducted propellers as propulsion systems for low-altitude aircraft and investigates the hovering state’s performance patterns and flow mechanisms. Based on an equivalent ducted propeller analysis criterion, this study proposes a series of multi-rotor ducted propellers with different layouts but identical power consumption and thrust output. Using a validated Unsteady Reynolds-Averaged Navier–Stokes (URANS) numerical method, the influence patterns of two-dimensional layout parameters and two types of rotational characteristics on aerodynamic performance are analyzed, along with an explanation of the underlying flow mechanisms. The research structure is organized as follows: Section 2 details the research model of multi-rotor ducted propellers, clarifying the group settings for layout parameters and rotational characteristics. Section 3 analyzes the influence patterns of spanwise and streamwise layout parameters, focusing on the coupling effects of rotational consistency and rotational phase gap. It also depicts changes in flow field characteristics. It employs the Aligned Rank Transform Analysis of Variance (ART-ANOVA) method to determine rotational characteristics’ main and interaction effects on various aerodynamic performances. Section 4 summarizes the research findings.

2. Research Model and Numerical Methods

2.1. Model Setup

The literature [29] designed a baseline configuration for a ducted propeller and proposed equivalent multi-rotor ducted propellers, as demonstrated in Figure 1a. The geometric structures of the two ducts are merged on the intersecting side using a Boolean operation (blue region), and redundant geometric structures within the internal flow field of the duct are removed (red region). In the baseline configuration, the duct inner diameter D_D11 is 0.749 m, the gap between the duct and propeller L_DP11 is 0.003 m, and the duct length L_D11 is 0.35 m. Detailed geometric parameters can be found in the literature [29].

This paper adopts the baseline model and geometric merging method illustrated in Figure 1a to propose multi-rotor ducted propellers with different layouts and rotational characteristics as research objects for design and analysis. The multi-rotor ducted propeller system comprises two sets of ducted propellers. The fundamental principle of variation is that the left duct and propeller remain unchanged. In contrast, the position of the right duct and propeller and the rotational direction and initial phase angle are adjusted. First, the layout parameters between the two ducts are discretized into spanwise and streamwise dimensions. The ratio γ of the propeller axle spacing to the duct inner wall diameter represents the spanwise parameter, with geometric structures at different γ values shown in Figure 1b. The streamwise parameter is represented by the ratio ξ of the streamwise distance difference between the two sets of ducts to the duct length, with geometric structures at different ξ values presented in Figure 1c. When studying the influence of the spanwise parameter, ξ is set to zero. Likewise, γ adopts 1.10 when analyzing the influence of the streamwise parameter. This approach allows for a better understanding of the independent effects of the two parameters and ensures the geometric independence of the internal flow field. Rotational characteristics include rotational consistency (co-rotation vs. counter-rotation) and rotational phase gap. Co-rotation means both propellers rotate clockwise (CW), while counter-rotation means the right propeller rotates counterclockwise (CCW). The rotational phase gap represents the difference in initial phase angles between the two sets of propellers, as demonstrated in Figure 1d. When studying rotational characteristics, the value ranges for each parameter are listed in

Table 1. Parameter ranges for the research model.

Parameter	Range	Step
γ(Case-1D2P)	1.02~1.2	0.02
γ(Case-2D2P)	1.2~2	0.1
ξ	0~1	1/7
Consistency	CW-CCW and CW-CW	/
Phase Gap	0°~120°	15°

.

2.2. Numerical Methods

The ducted propeller’s typical flow field constitutes a low-Reynolds-number, strongly coupled transient flow of rotating machinery. Therefore, this study employs the URANS equations, solved using the finite volume method (FVM) in the commercial software Fluent 2024 R2. The shear-stress transport (SST) k-ω turbulence model characterizes viscosity by representing turbulence through Reynolds stress terms while applying temporal/statistical averaging to NS equations. Compared to large eddy simulation (LES) or direct numerical simulation (DNS), this model significantly enhances computational efficiency. In conventional single-propeller rotation, geometric central symmetry ensures consistent inflow conditions across blade phases. Thus, hover analysis typically utilizes the MRF model for steady-state modeling of FVM control volume elements. However, for multi-rotor ducted propellers, distinct phase-dependent inflow variations occur during rotation 10. Consequently, MRF results lack global credibility, necessitating the use of the sliding mesh model (SMM) for high-fidelity transient solutions. During computation, the converged MRF solutions provided high-accuracy initial conditions for SMM calculations, addressing convergence challenges. At maximum operational speed, the tip Mach number reaches 0.7, placing it within the transonic regime. To address gas compressibility, a density-based solver and total energy model were activated to meet the ideal gas compressibility requirements as specified by Sutherland’s law.

2.3. Mesh Setup

The setup of the computational domain is illustrated in Figure 2. The computational domain includes one external stationary fluid domain and two internal rotating fluid domains, matched through four pairs of non-conformal interfaces at the top and bottom to achieve data exchange in the sliding mesh. The cylindrical fluid domain on the outer circumferential side of the duct has a radius of 5D_D11. The distance from the inlet to the duct is 6D_D11, and the distance from the outlet to the duct is 10D_D11. Pressure inlet and pressure outlet boundary conditions were adopted for the computation, balancing the needs of both steady-state and unsteady calculations. The computational domain was discretized using a polyhedral unstructured mesh, with a mesh curvature angle set to 10° and proximity for narrow geometries set to 3. The mesh is refined near the propeller tips and in the duct wake region. The y+ value in the internal rotating domain is not greater than one, while the wall y+ value in the external static domain is not greater than three.

Table 2. Mesh quantity and size configuration.

Configuration	Number of Cells	Min. Cell Size (Propeller)	Min. Cell Size (Duct)	Wake Refinement Size
Case-1D2P	25.3 million	0.00015 m	0.0014 m	0.012 m~0.048 m
Case-2D2P	19.3 million	0.00015 m	0.0014 m	0.012 m~0.048 m

lists detailed mesh quantities and size settings. This solver and mesh setup’s independence and computational accuracy have been thoroughly validated [29].

2.4. Aerodynamic Characteristics

In this study, the dimensionless thrust coefficient, power coefficient, and figure of merit are used to define the aerodynamic performance of different components. For cases with varying configurations and geometric dimensions, data were averaged to a single ducted propeller for calculation. Under this premise, the total thrust and total power of each component were averaged based on propeller count, with dimensionless coefficients computed using reference values from a single ducted propeller. Consequently, aerodynamic performance parameters of different configurations exhibit comparable ranges.

Thrust Coefficient:

$$C_T={\displaystyle \frac{T}{\mathsf{\rho }{n}^2{D}^4}},$$

(1)

Power Coefficient:

$$C_P={\displaystyle \frac{Q}{\rho n^3{D}^5}},$$

(2)

FoM:

$$\mathrm{F}\mathrm{o}\mathrm{M}={\displaystyle \frac{C_T^{\frac{3}{2}}}{\sqrt{2}{C}_P}},$$

(3)

Pressure Coefficient:

$$C_p={\displaystyle \frac{8p}{\rho n^2{D}^2}},$$

(4)

Vorticity Magnitude:

$$\mathsf{\omega }=\sqrt{{\left({\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{y}}}-{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{z}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}-{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{x}}}-{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{y}}}\right)}^2,}$$

(5)

where T represents thrust, ρ denotes air density, n is the propeller rotational speed, D is the propeller diameter, Q signifies propeller torque, P is propeller power, p indicates pressure, and u, v, w are the velocity components of V in the x, y, z directions, respectively.

3. Results and Discussion

3.1. Influence of Spanwise Layout Parameter on Hovering Aerodynamic Performance

In the study of layout parameters, the rotational characteristics of the model were set to co-rotation with a rotational phase gap of 0°. The demarcation point between Case-1D2P and Case-2D2P is γ = 1.2. Furthermore, when γ is greater than or equal to 1.1 for the baseline configuration, the expansion ratio no longer changes; thus, the diameter and rotational speed of the equivalent multi-rotor ducted propeller remain constant. The research in this paper selects the rotational speed expressed as the blade tip Ma of the linear velocity, which is uniformly set to 0.4573. At this speed, lift loss due to severe aerodynamic interference exists in Case-1D2P [29]. The geometric dimensions and rotational speed designs for different γ values are listed in

Table 3. Geometric and operating condition parameters for different γ values.

γ	D/m	n/rpm	Tip Ma
1.02	0.5318	5588.79	0.4573
1.04	0.5313	5594.04	0.4573
1.06	0.5304	5603.04	0.4573
1.08	0.5298	5609.55	0.4573
≥1.10	0.5296	5611.54	0.4573

.

The variation in thrust coefficients for the components along the spanwise distance is illustrated in Figure 3a. The magenta region {1} represents γ < 1.10, where geometric dimensions and rotational speed change correspond to the Case-1D2P configuration interval. The pale-yellow region {2} represents 1.10 < γ < 1.20, where geometric dimensions are constant, corresponding to the Case-1D2P configuration interval. The light blue region {3} represents γ > 1.20, corresponding to the Case-2D2P configuration interval. As the spanwise distance increases, the propeller thrust decreases gradually, though the rate of decrease slows down. Conversely, the duct thrust initially fluctuates within a small range and then, after transitioning to the Case-2D2P configuration, rapidly increases to a new plateau of fluctuation. In regions {1} and {3}, the total thrust coefficient exceeds 0.1795, which is considered a relatively optimal range for the dimensional parameters of the multi-rotor ducted propeller.

Figure 3b indicates the efficiency and duct thrust of the ducted propeller at various spanwise distances. It is evident from the figure that the duct thrust coefficient curve and the overall efficiency curve almost coincide in region {2} and region {3} and also exhibit the same trend in region {1}. This trend suggests that in the aerodynamic performance of the ducted propeller when the independent variable is the spanwise distance difference, the change in duct thrust may dominate the change in overall efficiency. Focusing solely on overall efficiency, it is clear that the hovering efficiency in region {3} is generally higher than in region {1} under similar thrust conditions. This improvement is primarily due to an increase in duct-augmented lift and a reduction in propeller power.

The variations in the propeller $C_T$, $C_T$, and FoM with spanwise distance, as shown in Figure 3c, exhibit highly consistent trends. The rate of decrease for all three coefficients is significant in regions {1} and {2}, diminishing in region {3}. This behavior is directly linked to the physical transition in the aerodynamic interference mechanism. As γ becomes large, the aerodynamic characteristics of the Case-2D2P approach those of the baseline Case-1D1P. Conversely, as γ approaches 1, substantial aerodynamic interference causes the coefficients to stabilize at another distinct value.

This macroscopic “S-shaped” transition from a strong to a weak interference regime can be effectively described by a phenomenological model. Moreover, the Boltzmann function form aptly captures this transitional behavior. Therefore, it was selected to establish empirical formulas relating the aerodynamic coefficients to the spanwise distance. The selection of this function is based on its characteristic shape as a practical mathematical tool, rather than an assertion of underlying statistical thermodynamics mechanisms. The primary purpose is to serve as computationally efficient, lightweight surrogate models for future design and optimization frameworks. The $R^2$ for these models exceeds 0.98, indicating their ability to reproduce the macroscopic trends revealed by CFD accurately. The precision of the formula parameters is determined by the fitting confidence to ensure the reproducibility of this study.

Propeller $C_T$:

$$\mathrm{y}=0.09821+{\displaystyle \frac{0.0035}{\left(1+{\mathrm{e}}^{\frac{\left(\mathrm{x}-0.636\right)}{0.057}}\right)}}.$$

(6)

Propeller $C_P$:

$$\mathrm{y}=0.06151+{\displaystyle \frac{0.00097}{\left(1+{\mathrm{e}}^{\frac{\left(\mathrm{x}-0.652\right)}{0.039}}\right)}}.$$

(7)

Propeller FoM:

$$\mathrm{y}=0.3539+{\displaystyle \frac{0.012}{\left(1+{\mathrm{e}}^{\frac{\left(\mathrm{x}-0.63\right)}{0.055}}\right)}}.$$

(8)

The change in duct thrust originates from differences in pressure distribution characteristics. The pressure coefficient distribution characteristics at the duct leading edge are shown in Figure 4a. In Case-1D2P, a significant high-pressure region is at the center of the duct’s leading edge on the intersecting side. The distribution range of this high-pressure region changes with the geometric variations in the intersecting side under different spanwise layouts. At non-intersecting circumferential positions, the distribution of the low-pressure region also migrates, mainly manifested as a reduction in the circumferential extent of the blue extremely low-pressure region.

Vortex structures can more clearly demonstrate the changing process of the flow mechanism at the duct leading edge, as demonstrated in Figure 4b. After the leading edges of the two ducts are merged geometrically using a Boolean operation, a distinct set of intersection features is created. This merging results in abrupt changes in curvature, which causes instability in the airflow perpendicular to the surface. As a result, the transition pattern of the boundary layer on the duct’s leading edge is affected, leading to flow separation. Consequently, two spanwise vortices are formed across the ducts on the intersecting side. As γ increases, the correlation between the vortex ring, characterized by a specific iso-vorticity surface, and the geometric intersection features weakens. The pair of spanwise vortices continuously approach each other circumferentially, and the thickness of the vortices increases. Once the two ducts are separate in Case-2D2P, the spanwise vortices break down, and aerodynamic interference becomes limited to a small local area. This process describes the entire sequence of vortex generation to dissipation as the spanwise distance increases.

In ducted propellers, the variation in vortex structures significantly impacts pressure distribution, primarily based on Bernoulli’s principle. The spanwise vortices facilitate large-scale, strong momentum transport between the two ducts, which hinders the streamwise velocity of the airflow. As the streamwise velocity decreases, pressure gradually increases, resulting in changes to the distribution range of the high-pressure region. As a result, the boundary of the high-pressure region changes in response to the distribution of the spanwise vortices. This change is observed as an expansion in the spanwise direction and a contraction in the circumferential direction, ultimately leading to the complete disappearance of the high-pressure region once the two ducts are fully separated. These alterations in the high-pressure area illustrate how the spanwise vortices evolve within the flow field.

The change in pressure can reveal the cause of propeller thrust variation, as illustrated in Figure 5a. The high-pressure area beneath the propeller gradually decreases in size. Below the propeller, the “vortex-pressure difference” mechanism still exists, but the trend is the opposite. The vortex structure below the intersecting side of the propeller and duct is demonstrated in Figure 5b. After shedding, the propeller tip vortices develop along the inner wall of the duct to the trailing edge of the intersecting side and then couple with the wake.

As γ increases, the streamwise distance on the intersecting side becomes longer, and the vorticity of the tip vortex at the time of coupling decreases. This leads to pronounced oscillation characteristics in the wake on the intersecting side when γ is small. As γ gradually increases, the wake oscillates to stable. In this situation, lateral transport is gradually weakening.

3.2. Influence of Streamwise Layout Parameter on Hovering Aerodynamic Performance

The variation in component thrust coefficients with streamwise distance is shown in Figure 6a. As the streamwise distance increases, the propeller thrust monotonically decreases. However, this non-linear trend differs from the behavior demonstrated in the spanwise variation. Conversely, the duct thrust rapidly increases to a higher level and then fluctuates around this level, forming a “plateau” characteristic. Influenced by the propeller’s changing trend, the total thrust reaches its maximum when ξ increases by 1/7, after which the change is predominantly a decrease. The changes in the overall efficiency and duct thrust of the ducted propeller at different streamwise distances are demonstrated in Figure 6b. The two aerodynamic coefficients exhibit little overlap, suggesting that the impact of duct thrust on overall efficiency is minimal. Figure 6c demonstrates how propeller aerodynamic performance varies with streamwise distance.

At different ξ values, the pressure distribution at the duct leading edge also shows significant differences, as depicted in Figure 7a. When ξ increases to 1/7, the distribution range of the high-pressure region on the intersecting side of the duct leading edge decreases circumferentially but not significantly. However, as ξ further increases, the high-pressure region almost disappears from the intersecting side of the duct leading edge. This aligns with the “plateau” characteristic exhibited by the thrust coefficient. The leading-edge vortex structure shows more pronounced differences, as seen in Figure 7b. At ξ = 1/7, the spanwise vortex almost vanishes. However, as ξ increases, an evident reverse flow phenomenon appears at the leading edge of the lower duct. This leads to the generation and gradual strengthening of streamwise vortices, which have different impacts on the intersecting and non-intersecting sides of the lower duct. Since the obstruction effect of streamwise vortices on the streamwise airflow velocity through lateral transport is not as high as that of spanwise vortices, the pressure coefficient distribution on the intersecting side does not increase significantly. However, to some extent, the reverse flow increases the air mass flow through the lower duct, enhancing the total air momentum. This results in the expansion of the low-pressure region on the non-intersecting side, and the balance between the intersecting and non-intersecting sides maintains the duct thrust within a relatively high range.

Figure 8a shows the pressure change below the propeller. It is noteworthy that, in addition to the continued shrinking trend of the high-pressure region below the propeller, the continuous sub-high pressure region (yellow) is also truncated by a sub-low pressure region when ξ is high. Figure 8b illustrates that the dissipation distance of the wake on the intersecting side has increased. Tip vortex coupling and lateral momentum transport, which should occur at specific streamwise locations, are shifted downstream with increasing ξ, causing pressure distribution changes.

3.3. Influence Patterns and Flow Mechanisms of Rotational Characteristics on Hovering Aerodynamic Performance

In distributed propeller propulsion, the effects of differences in rotational direction and rotational phase gap are significant, with the impact of directional differences being greater [26]. The multi-rotor ducted propeller in the Case-1D2P configuration is quite sensitive to rotational consistency (co-rotation vs. counter-rotation), whereas Case-2D2P is not sensitive to rotational characteristics [29]. This paper focuses on Case-1D2P with γ = 1.02 and ξ = 0, employing numerical simulation methods to study the aerodynamic performance under the coupled effects of rotational consistency and rotational phase gap. Case-2P consists only of propellers without ducts and is a control group to compare the influence of ducts on the propellers. All aerodynamic coefficients are presented in Figure 9. In Figure 9, blue represents counter-rotating configurations, and red represents co-rotating configurations. The dashed line represents the ensemble average of the coefficients across all simulated phase gaps, serving as a phase-independent performance baseline.

Figure 9a–c represent all aerodynamic coefficients related to the duct of Case-1D2P under different rotational characteristics. Their values fluctuate significantly and unpredictably as the rotational phase gap changes. Furthermore, under the influence of duct thrust, the values of these three sets of coefficients are generally higher in co-rotating conditions than in counter-rotating conditions. Figure 9d–f represent the propeller-related aerodynamic coefficients for Case-1D2P under different rotational characteristics. A clear sinusoidal periodic fluctuation is observable in the trends of these coefficients with varying rotational phase gaps. This phenomenon has a strong physical basis. As the propellers rotate, the blades of one propeller periodically traverse the proximity zone of the other. This periodic passage results in periodic unsteady aerodynamic interactions between the propellers, which naturally leads to a periodic response in the aerodynamic performance coefficients. Therefore, a sinusoidal function is the most suitable option to model this inherent periodicity. This characteristic is especially evident under co-rotating conditions, where the R² value for sinusoidal fitting ranges between 0.98 and 0.99. Peaks consistently appear at a rotational phase gap of 0°, and troughs consistently appear at a rotational phase gap of 60°. Regarding counter-rotating conditions, although the periodicity is weak, with variations ranging from 0.64 to 0.84, the fundamental trend remains consistent. Moreover, the peaks shift to the vicinity of a 90° rotational phase gap. At the same time, troughs appear at a rotational phase gap of 30°. In contrast, the Case-2P does not exhibit a clear sinusoidal pattern, as shown in Figure 9g–i, indicating that the duct structure plays a crucial role in stabilizing and regularizing these periodic interactions.

The empirical formulas relating the aerodynamic coefficients to the rotational phase gap for the co-rotating condition are listed below. These models are designed to capture the primary periodic dependency and can help evaluate the potential of phase control strategies, as outlined below.

Propeller Thrust Coefficient ($R^2\approx 0.985$):

$$\mathrm{y}=0.101140+1.60\times {10}^{-4}\ast \sin \left({\displaystyle \frac{\mathsf{\pi }\ast \left(\mathrm{x}+34\right)}{61}}\right).$$

(9)

Propeller Power Coefficient ($R^2\approx 0.984$):

$$\mathrm{y}=0.062430+5\times {10}^{-5}\ast \sin \left({\displaystyle \frac{\mathsf{\pi }\ast \left(\mathrm{x}+27\right)}{55}}\right).$$

(10)

Propeller FoM ($R^2\approx 0.980$):

$$\mathrm{y}=0.36432+6.0\times {10}^{-4}\ast \sin \left({\displaystyle \frac{\mathsf{\pi }\ast \left(\mathrm{x}+40\right)}{66}}\right).$$

(11)

Figure 9 shows that in Case-1D2P, the influence of rotational characteristics on aerodynamic performance is more pronounced on the propellers than on the duct. A similar pattern is observed in the flow field. Figure 10 illustrates the formations of the leading-edge vortex based on various rotational features. Rotational consistency and the rotational phase gap have minimal effect on the leading edge spanwise vortices. This indicates that the flow field differences at the duct leading edge are determined by the geometric structure resulting from layout parameters and are not affected by rotational characteristics.

The vortex structures below the propeller and at the duct intersecting side trailing edge under different rotational characteristics are shown in Figure 11. It is noticeable that under counter-rotating conditions, the shed wake on the intersecting side exhibits stable symmetric characteristics, and the rotational phase gap has almost no effect on the vortex structure. In contrast, under co-rotating conditions, the wake on the intersecting side possesses highly prominent oscillation characteristics, and the influence of the rotational phase gap on the vortex structure is quite significant. Previous sections have discussed how tip vortices form along the inner wall of a duct until they interact with the trailing edge of the intersecting side and the wake that has been shed. This interaction creates a complex three-dimensional structure due to the comprehensive nature of the tip vortex’s vorticity vector characteristics. To clarify the coupling mechanism, the coupled flow field in a hypothetical 2D cross-section is illustrated schematically, as illustrated in Figure 11.

Under counter-rotating conditions, the vorticity vector characteristics on both sides of the wake are mirror-distributed, and their magnitudes are equal. Therefore, the induced effects from the two sets of tip vortices are also symmetric. This symmetric induced effect continues to develop along the streamwise position, forming multiple coupling regions of wake and tip vortices. Under the combined effect, the unique stable symmetric flow field of counter-rotation is formed.

Under co-rotating conditions, the vorticity vector characteristics on both sides of the wake and their magnitudes are identical. Therefore, a significant feature of the coupling is that the induced effect of the tip vortices on the wake is asymmetric. This asymmetric induction continues to develop along the streamwise position, forming multiple coupling regions of wake and tip vortices. Due to the shorter streamwise length of the intersecting side for the selected γ = 1.02, the tip vortex strength is high, forming the characteristic asymmetric oscillation of co-rotation.

The rotational phase gap alters the streamwise position of the tip vortices at the wake but only slightly affects the magnitude of their vorticity. However, it does not affect the vorticity vector characteristics, which possess complex induced effects, thereby exhibiting a less significant impact than the rotational consistency.

3.4. Main and Interaction Effects of Rotational Characteristics on Component Performance

The Aligned Rank Transform is a non-parametric statistical method for multi-factor experimental designs. Its core idea is to eliminate the influence of redundant effects through data alignment, thereby allowing hypothesis testing within the Analysis of Variance framework. ART-ANOVA outperforms parametric methods for non-normal distributions (e.g., heavy-tailed, skewed data) and effectively handles repeated measures designs, providing a flexible and robust solution for experimental data analysis [33,34,35,36,37,38].

While the CFD results in Section 3.3 illustrate performance trends, they do not readily disentangle the coupled effects of rotational characteristics. The ART-ANOVA method is therefore employed to move beyond qualitative observations and provide a rigorous statistical quantification. Its application here offers two key advantages not available from the CFD analysis alone:

• Quantification of effect dominance, which precisely determines the relative influence of rotational consistency and phase gap on various aerodynamic metrics;
• Exposure of interaction effects, which reveals the significant synergistic or antagonistic relationship between phase gap and rotational consistency that is difficult to isolate through direct flow field inspection.

This statistical rigor is crucial for developing a deeper understanding and providing robust design guidelines for multi-rotor systems. This paper employs the ART-ANOVA method for analysis, with rotational phase gap and rotational consistency as independent variables and various aerodynamic coefficients as dependent variables. The study reveals the influence mechanisms of the independent variables on the dependent variables, quantifying the independent and synergistic effects of main effects and interaction effects on the dependent variables, as shown in Figure 12.

Since the propeller model used in this paper is three-bladed, the period of the rotational phase gap is 120°. Based on data type conditions and the need for intuitive representation, the rotational phase gap range of 0° to 120° is divided into four intervals: tiny phase gap (0~30°), small phase gap (30~60°), medium phase gap (60~90°), and significant phase gap (90~120°). Two states of rotational consistency are also defined: co-rotating and counter-rotating. The ART-ANOVA method is used to analyze main and interaction effects, with specific steps including the following:

• Data alignment to eliminate interference from main effects and retain interaction information;
• Rank transformation converting original data to ranks to suit non-parametric testing;
• Variance decomposition solving for partial ${\eta }^2$ and pseudo $R^2$ to assess the main effect of the rotational phase gap, the main effect of rotational consistency, and the interaction effect of the rotational phase gap × rotational consistency;
• Significance tests to evaluate the strength of effects through F-values and p-values.

The ART-ANOVA method requires a relatively high data scale, so the data samples obtained from the simulations in Section 3.3 underwent fitting. Polynomial or Fourier fitting provided a way to augment the sample size while maintaining regularity and avoiding overfitting, focusing on achieving the highest variance. The relationship between propellers and the rotational phase gap exhibits significant sinusoidal characteristics. However, its fitting follows a consistent pattern and was not adopted in this analysis. This decision helps minimize the impact of data augmentation on the accuracy of the results.

Table 4. ART-ANOVA analysis of rotational characteristics for Case-1D2P.

Parameters	Effect Term	df	F-Value	Significance	Partial ${\mathit{\eta }}^2$	Pseudo ${\mathit{R}}^2$
Total Thrust Coefficient	Phase Gap	3	234.46	<0.001	0.26	0.873
	Consistency	1	2960.53	<0.001	0.60
	Phase Gap. × Consist.	3	94.89	<0.001	0.13
Propeller Thrust Coefficient	Phase Gap	3	1217.9	<0.001	0.65	0.672
	Consistency	1	5687.6	<0.001	0.74
	Phase Gap. × Consist.	3	1408.0	<0.001	0.68
Duct Thrust Coefficient	Phase Gap	3	512.15	<0.001	0.44	0.694
	Consistency	1	5814.59	<0.001	0.74
	Phase Gap. × Consist.	3	862.76	<0.001	0.57
Total Power Coefficient	Phase Gap	3	1082.9	<0.001	0.62	0.695
	Consistency	1	5991.6	<0.001	0.75
	Phase Gap. × Consist.	3	1001.0	<0.001	0.58
Propeller Efficiency	Phase Gap	3	936.3	<0.001	0.59	0.650
	Consistency	1	5988.2	<0.001	0.75
	Phase Gap. × Consist.	3	1690.6	<0.001	0.72
Total Efficiency	Phase Gap	3	327.13	<0.001	0.33	0.759
	Consistency	1	6008.58	<0.001	0.75
	Phase Gap. × Consist.	3	347.41	<0.001	0.34

lists the ART-ANOVA analysis results for the main and interaction effects of the rotational phase gap and rotational consistency on six aerodynamic parameters. Rotational consistency significantly influences all aerodynamic characteristics, with its main effect’s partial ${\eta }^2$ values ranging from 0.60 to 0.75. Furthermore, co-rotation can significantly improve the total thrust and efficiency coefficients (p < 0.001). However, the main effect of the rotational phase gap on thrust and efficiency is not sufficiently significant. For the duct and the overall aerodynamic coefficients affected by the duct, the partial ${\eta }^2$ of the rotational phase gap main effect ranges from 0.13 to 0.44, indicating a weaker main effect. In contrast, for propeller-related aerodynamic coefficients, the partial ${\eta }^2$ of the rotational phase gap main effect ranges from 0.59 to 0.65, indicating a more substantial main effect. A similar pattern is observed for the interaction effect of the rotational phase gap and rotational consistency (phase gap × consistency) on propeller aerodynamic coefficients, where the partial ${\eta }^2$ ranges from 0.58 to 0.72, indicating a strong interaction effect. Conversely, the interaction effect of phase gap × consistency on duct and overall aerodynamic coefficients has a partial ${\eta }^2$ ranging from 0.13 to 0.57, indicating a weaker interaction effect. This pattern of main and interaction effects originates from the differences in vortex structures formed by turbulent transport at the intersecting side trailing edge. By synchronizing the rotational phases of the propellers and ducts, co-rotation optimizes flow field synergy, thereby significantly enhancing lift (mean total lift increased by 0.05%) and efficiency (efficiency increased by 0.17%). Unsteady vortex structures are particularly prominent in counter-rotating configurations, whose aerodynamic performance is generally lower than co-rotating configurations.

4. Conclusions

This study investigated the influencing factors of unsteady aerodynamic interference in multi-rotor ducted propellers when used as hovering propulsion systems. A numerical computation model with parametric characterization and an ART-ANOVA method suitable for patterned data from small datasets were developed to evaluate the significance of the effects of rotational characteristics on aerodynamic performance. Based on the research results, the following conclusions are drawn:

• As the layout spacing increases, the aerodynamic interference between the two ducted propellers weakens. The various aerodynamic coefficients of the propellers monotonically decrease. The decreasing trend of propeller aerodynamic performance with increasing spanwise spacing conforms to a Boltzmann distribution, while the decrease with increasing streamwise distance exhibits significant non-linear characteristics. The thrust coefficient of the duct generally shows a plateau characteristic, rapidly increasing within specific intervals and fluctuating within a small range in other intervals;
• The core mechanism by which aerodynamic interference affects the performance of multi-rotor ducted propellers is that the lateral momentum transport generated by aerodynamic interference reduces the streamwise velocity. This alters the pressure through Bernoulli’s principle, affecting the distribution range of high-pressure regions. With changes in spanwise and streamwise layout parameters, aerodynamic interference produces varying trends of enhancement or reduction at the duct leading edge, the intersecting side trailing edge, and below the propellers, leading to corresponding changes in aerodynamic performance;
• The duct weakens the mutual interference between propeller blades on the intersecting side and alters the amplitude and phase of the propeller aerodynamic performance variation with the rotational phase gap. The aerodynamic performance of the propellers exhibits significant sinusoidal periodic fluctuation characteristics, which are particularly evident in co-rotating configurations. Rotational characteristics do not influence the leading-edge flow field but substantially impact the trailing-edge flow field. During co-rotation, the tip vortices on the intersecting side trailing edge couple asymmetrically, forming periodic oscillation characteristics, whereas during counter-rotation, the vortices couple symmetrically, forming a stable flow field;
• ART-ANOVA analysis shows that rotational consistency (co-rotation vs. counter-rotation) has a decisive main effect on all aerodynamic performances; co-rotating conditions can significantly improve thrust and efficiency. In contrast, the main effect of the rotational phase gap and the interaction effect of phase gap × consistency only significantly impact propeller aerodynamic performance.

This study employs a well-validated unsteady numerical method to conduct simulation research, analyzing the effects of layout parameters and rotational characteristics on hover aerodynamic performance, while revealing core flow mechanisms governing aerodynamic behavior. A recognized limitation lies in the phenomenological nature of the established empirical formulas, which are not derived from first principles. The primary objective is to develop computationally efficient, lightweight surrogate models for complex aerodynamic interference in multi-ducted fan systems, thereby facilitating design iteration and system optimization in future research. Subsequent work will delineate applicable boundaries of these empirical models while exploring more universal physical models grounded in deeper flow mechanisms.