Influence of Propeller Parameters on the Aerodynamic Performance of Shrouded Coaxial Dual Rotors in Hover

Abstract

A numerical method is used to evaluate the influence of propeller parameters on the aerodynamic performance of shrouded coaxial dual rotors in hover. Compared with the open-rotor configuration, the shrouded rotors reduce the tip vortex, resulting in a higher thrust and figure of merit (FoM). The varying inflow distribution over the inlet lip of the shroud introduces a different working condition for the propellers, thereby providing an opportunity for propeller parameter optimization. The pitch length, chord length, and tip clearance are chosen as the design parameters to improve the aerodynamic performance of the rotors. When the pitch lengths of the upper and lower propellers increase, the shrouded rotors outperform the original ones in both the coefficient of thrust (C_t) and in terms of the (FoM). The results show that increasing the chord length improves the C_t but that the (FoM) is almost unaffected. Moreover, smaller tip clearance can significantly improve the aerodynamic performance of the shrouded rotors. Results presented in this work provide insights into the parametric design and optimization of shrouded rotors.

1. Introduction

In recent years, sustained growth has taken place in the micro air vehicle (MAV) market owing to the compactness and portability that popularized them from military service to commercial use. The widely used MAVs, the Reynolds number of which is between 10⁴ and 10⁵ [1], are typically of an open multi-rotor configuration. Compared with conventional aircraft, such a low Reynolds number implies a higher viscosity force and a lower inertial force [2], leading to problems such as flow separation and transition that decrease the aerodynamic efficiency of propellers. Furthermore, the flow separation near the blade tip, pronounced viscous effect, and propeller sensitivity to the flow conditions are also raising challenges in the study of low-Reynolds-number fluid dynamics. Such a situation will hinder the improvement in the figure of merit ($FoM$), thus adversely affecting the aerodynamic performance of the MAV [3].

From the perspective of safety, the exposed blades of a MAV impose potential risks, limiting its use in urban and indoor environments. Once the MAV control system fails, the swiftly rotating propellers can cause grievous harm to objects and people nearby. Consequently, MAVs are subject to strict regulations within urban areas, thereby limiting their wider deployment.

One effective way to overcome the problems is adding a shroud, which is an outer casing around the propeller. Since 1930, the aircraft industry has been studying and applying shrouds to improve aircraft loading capacity [4]. For a small MAV, enclosing the propellers with a shroud can effectively improve the aerodynamic performance due to the reduced tip vortex [5]. Implementing a shroud also yields other benefits, for instance, protecting the propellers from physical damage, facilitating a good security setup, and diminishing the noise emitted by the propellers. Over the last few years, researchers have attempted to optimize the geometry of shrouded rotors, including the geometry of the shroud, the blade number, and the position of the propellers. Huo et al. [6,7] investigated the influence of the length and angle of the shroud diffusion part on the aerodynamic performance of the shrouded rotors through both experimental and simulation methods. Pereira et al. [8,9] systematically compared the influence of different geometric parameters and configurations of the shroud and propeller on the aerodynamic performance of the shrouded rotors. Gao et al. [10] optimized the diameter of a ducted fan through the technique for order of preference by similarity to the ideal solution (TOPSIS). Lakshminarayan et al. [11] explored the impact of different propeller configurations on the aerodynamic performance of the shrouded MAV and compared the efficiency of propellers of different diffuser lengths and angles. Lucas [12] also optimized the geometry configuration of propellers and improved the design accuracy based on the blade element moment theory. The effect of the diffuser length was further studied by Misiorowski [13]. Many novel experimental methods have also been developed to enhance the aerodynamic performance. Akturk et al. [14] used a planar particle image velocimeter system to investigate the near-duct aerodynamic performance in hover and edgewise flight conditions. Anzalotta et al. [15] conducted a comprehensive analysis of the flow field and the interaction between the upper and lower propellers utilizing an advanced computational solver named RotUNS.

It should be noted that the majority of the studies above are conducted on micro-shrouded rotors, which are significantly smaller in size than conventional aircraft. At such low Reynolds numbers, the flow velocity of the shroud inlet is smaller, leading to a less pronounced laminar-to-turbulent transition at the shroud lip. Secondly, many investigations concentrated on analyzing the flow field of the shroud, including the significant reduction in vortex near the wall of the shroud [14,15,16]. Despite being subjected to distinct flow conditions compared to open propellers, the design and optimization of the propellers inside the shroud have received relatively little attention. Therefore, this paper mainly focuses on the design and optimization of shrouded dual rotors, utilizing numerical simulations to enhance the overall aerodynamic performance.

2. Method

2.1. Shrouded Coaxial Dual-Rotor Model

Figure 1a shows the model of the MAV with two mirror-symmetry and contra-rotating propellers. The diameters of the propellers are both 7 inches with a pitch length of 4 inches. The airfoil of the blade is NACA HS1712. The radial distribution of chord lengths is governed by a modified form of the Bessel equation, which is commonly used in the fluid dynamics community to describe oscillatory phenomena with cylindrical or spherical symmetry [17,18]. The equation is presented as follows:

$$c\left(r\right)=C_0{\left(1-r\right)}^3+\alpha r{\left(1-r\right)}^2+\beta r^2\left(1-r\right)+\gamma r^3$$

(1)

where $c\left(r\right)$ is the chord length at the radial position of $r$ and $\alpha , \beta ,$ and $\gamma$ are chord-length distribution parameters. Specifically, the propeller parameter $\gamma$, controlling the chord length at the tip of the propeller [19], is investigated in the last section. Additional geometric parameters are shown in

Table 1. Geometry parameters of the propeller.

Geometry Parameter	Range
Radius of propellers, R	3.5 inches
Pitch length	4 inches
Number of blades	3
Tip clearance	1.2% R

.

A simplified model is unitized in the simulation, as shown in Figure 1b. The inner diameter ($D_{\mathrm{in}}$) is taken as 7.08 inches to maintain tip clearance, while the outer diameter ($D_{\mathrm{out}}$) is taken as 9 inches.

2.2. Blade Element Momentum (BEM) Theory

As a general theory for the design of propellers, the BEM theory attempts to divide the blade into small segments, which are called the blade elements [12]. When the interaction between adjacent blades can be ignored, each element can be simplified to a two-dimensional airfoil. As a result, the aerodynamic forces on each element are calculated using airfoil data, such as thrust and drag coefficients, at different angles of attack and local flow conditions. The inflow angle of the upper propeller is taken as ${\varphi }_1$ at the radial distance of r, as shown in Figure 2. ${\varphi }_1$ is defined by the axial flow velocity $U_{z1}$ and the rotating velocity $U_{\mathrm{m}1}=\omega ·r$.

Hence, the inflow angle of the upper propeller (${\varphi }_1$) is:

$${\varphi }_1={\tan }^{-1}\left(\frac{U_{\mathrm{z}1}}{U_{\mathrm{m}1}}\right).$$

(2)

The relative velocity of the upper propeller is:

$$U_{\mathrm{up}}=\sqrt{U_{\mathrm{z}1}^2+U_{\mathrm{m}1}^2}.$$

(3)

To determine the optimal angle of attack for a blade element, it is necessary to calculate the inflow angle of the element based on the axial velocity. We acknowledge that the airflow velocity of the lower propeller is accelerated by the upper propeller by a factor of ${\phi }_{12}$, which represents the incoming flow speed ratio:

$${\phi }_{12}=\frac{U_{\mathrm{z}2}}{U_{\mathrm{z}1}}.$$

(4)

Since the distance between the propellers is small and the shroud cross section is constant in this region, the speed loss and pressure increase are neglected. In other words, the wake-flow velocity of the upper propeller is regarded as the same as the inflow velocity of the lower propeller. Hence, the inflow angle of the lower propeller can be calculated as follows:

$${\varphi }_2={\tan }^{-1}\left(\frac{U_{\mathrm{z}2}}{U_{\mathrm{m}2}+U_c}\right)={\tan }^{-1}\left(\frac{{\phi }_{12}·U_{\mathrm{z}1}}{\omega ·r+U_c}\right),$$

(5)

where “$U_c$” represents the component of the induced velocity in the rotating direction. It should be pointed out that the inflow of the lower propeller is as turbulent as the wake from the upper propeller. Such airflow may not meet all the requirements of the BEM theory. To address the complexities of the wake, a simplification of the inflow condition will be proposed in Section 4.3.

2.3. Experimental Test Rig

The experimental tests were conducted utilizing the test rig shown in Figure 3. Both the shrouded and open rotors were fixed on the test table to closely replicate hovering conditions. This setup ensured precise measurements of thrust and power results. The thrust data are acquired through measurement with a force sensor integrated into the apparatus, while power is determined by calculating the product of voltage and current. Subsequently, the $C_t$ is represented as follows:

$$C_t=\frac{T}{\rho \pi R^2{U}_{tip}^2},$$

(6)

and $C_p$ is represented as:

$$C_p=\frac{P}{\rho \pi R^2{U}_{tip}^3}.$$

(7)

3. Numerical Simulation Setup

3.1. Domain Division

As shown below in Figure 4, the whole simulation domain is divided into three cylindrical sub-domains, including one stationary domain and two rotating domains. These three domains are defined as the “stationary domain”, “rotating domain 1”, and “rotating domain 2”, respectively. The upper and lower propellers are enclosed by the “rotating domain 1” and “rotating domain 2”, respectively, while the shroud is enclosed by the “stationary domain”. The geometrical features and specific dimensions of each domain are presented in detail in

Table 2. Geometry parameters of three domains.

Parameter	Rotating Domain 1	Rotating Domain 2	Stationary Domain
Diameter	1.011D	1.011D	6D
Height	0.056D	0.056D	10D
Position	3.2D from the inlet	3D from the inlet	/

.

Figure 4 also presents the arrangement of 48 observation points which are evenly distributed among three horizontal planes along the axis. These points are strategically located to capture the details of inflow velocity which are decomposed into axial and rotational components. The three horizontal planes are positioned at $z_1=c$, $z_2=2c$, and $z_3=3c$ from the base plane, with “$c$” defined as $0.06D$.

3.2. Mesh

The numerical simulations adopt an unstructured mesh throughout all domains, as shown in Figure 5. Both the rotating domains consist of 22 × 96 × 583 mesh points in the axial, radial, and rotation directions, respectively. The background cylindrical domain exhibits 247 points along the axial direction. The boundary layers are divided into five slices, with a total thickness of 1 mm. This thickness is selected to ensure that the nondimensional wall distance, y+, is larger than 30, which is considered a suitable value for resolving the turbulent boundary layer. A grid convergence study is performed for all cases to ensure that the results are not significantly affected by the resolution of the computational mesh.

3.3. Solver

Some fundamental characteristics of the shrouded rotors, such as inflow velocity and turbulence, are first identified. The inlet speed is assigned an empirical value of 0.1 m/s and the relative pressure at the outlet boundary is set as 0 pa, which represents the hovering state of the shrouded rotors. The outer cylinder surface of the domain region is set as an opening boundary condition, enabling counterflow through the interface. The propeller and shroud surfaces are set as non-slip walls where the standard wall function was employed. To capture the turbulence characteristics near propellers, the standard k-epsilon turbulent model is selected. The model employs a preconditioned dual-time scheme to solve the compressible RANS equations, as described by Li et al. [20] and Jiang [21]. It is particularly effective in predicting boundary layer flows, where the turbulence intensity increases with distance from the wall, making it well-matched for analyzing the flow near the shroud wall.

The investigation is made with the rotation speed ($\omega$) ranging from 4000 to 10,000 RPM and with a step of 1000 RPM. The corresponding tip Mach number varies from 0.136 to 0.271 and the Reynolds number varies from 18,470 to 36,941. The Reynolds number of the propeller is defined as

$$Re=\frac{c_{0.75R}·0.75\omega R}{\nu }$$

(8)

where $c_{0.75R}$ refers to the chord length in the radial position of 0.75R [22]. The thrust and power data of the shrouded and open rotors are obtained in the simulations and the coefficients of thrust and power are calculated. With a transient calculation scheme, the simulations are completed on a 52-core Intel Xeon 4.0 GHz processor and the total CFD simulation time took 21 × 24 h.

4. Results and Discussion

4.1. Model Validation

Figure 6a,b presents a comparison of the surface pressure and tip vortex between a shrouded and open propeller. The results indicate that the shrouded propeller generates a smaller tip vortex compared to the open propeller. However, it is worth noting that despite the existence of the shroud, a few vortices persist on the tip of the propeller. This phenomenon can be attributed to the tip clearance, which introduces a narrow gap between the blades. A minor airflow is pushed through the gap due to the pressure differential between the two blade surfaces. Additionally, the existence of the small vortices also suggests a high-quality flow field, which is capable of capturing the tiny tip vortex. The observed flow field is clean, with a distinct resolution of the tip vortex near the blades.

Figure 7a,b shows the thrust and power data of the shrouded and open rotors obtained from the experimental measurement and numerical simulations. The thrust difference between the simulation and experimental data is less than 5.6%, which verifies the accuracy of the simulation results. Interestingly, the experimental power results consistently exhibit higher values than the simulation results. This observation is likely attributable to motor loss; numerical simulations compute power as speed times torque while experimental measurements derive it from voltage multiplied by current, encompassing losses in the rotational motors. Moreover, it is worth noting that the total thrust is related to the square of rotation speed and the power is related to the cubic rotation speed. Therefore, according to the definition of $FoM$:

$$FoM=\frac{C_t^{3/2}/2}{C_p}=\frac{{\left(T/\rho \pi R^2{U}_{tip}^2\right)}^{3/2}}{\left(P/\rho \pi R^2{U}_{tip}^3\right)}=Const,$$

(9)

it is apparent that the $FoM$ does not change with the Reynolds number (rotating speed), which is consistent with the results in Figure 8. The $FoM$ of the open rotors fluctuates around 0.7 by less than 4.68%. The $FoM$ of the shrouded rotors varies around 0.83 and the fluctuation magnitude does not exceed 2.46% except at a low rotation speed (4000–5000 RPM).

4.2. Comparison of Aerodynamic Performance between the Shrouded and Open Rotors

In our examination of thrust comparison, it becomes evident that the total thrust generated by shrouded rotors surpasses that of open rotors across all rotational speeds. Nevertheless, a closer examination of thrust components, as depicted in Figure 9, reveals that the propellers enclosed in the shroud yield a reduced thrust in comparison to the open rotors. In other words, the shroud reduces the thrust produced by the propellers but the thrust loss is compensated by the thrust generated itself. This combination of thrust results in the shrouded rotors producing a higher overall thrust than that of the open rotors.

To further explore the distribution of shroud thrust, the velocity contour of the vertical section of the shroud is presented in Figure 10. The integral of pressure along the lip and the non-lip part is calculated separately and shown in

Table 3. Comparison of the velocity and pressure in different zones of the shrouded and open rotors.

Zone	Relative Pressure (Integral)/N	Averaged Inflow Velocity 2 of Shrouded Rotors/m/s	Averaged Inflow Velocity of Open Rotors/m/s
Lip 1	2.32	15.792	8.378
Non-lip	−0.22	19.206	23.323
Total	2.10	/	/

¹ The lip and non-lip parts of the shroud are defined in Figure 4. ² The averaged inflow velocity is calculated by dividing the quantity of flow divided by the cross-section square, which is measured at the entrance of each part.

. Clearly, the thrust of the shroud is primarily concentrated near the lip section, while the non-lip section provides a downward drag, which suggests the potential significance of the lip part. We can also see from Figure 10 that the inflow velocity of shroud rotors increases significantly compared with the case of open rotors. The increase in inflow velocity can be explained by the contraction of the cross-section of the inside shroud wall, which acts like a Venturi tube. Airflow tends to accelerate when passing through a narrow tube. Open rotors, on the other hand, do not incorporate a shroud that functions as an accelerator tube, resulting in a lower inflow velocity. Furthermore, the increased inflow velocity in the shroud can further increase the inflow angle and angle of attack, which may put the propellers inside the shroud in an unfavorable working condition. This could also explain the lower thrust produced by the propellers inside the shroud compared to open propellers.

Lakshminarayan et al. [11] also found that the shroud can accelerate the airflow above the upper propeller in the numerical simulation of a shrouded rotor within a similar Reynolds number range. Interestingly, they concluded that the effect of the shroud on the flow condition of the rotors was limited. Consequently, they suggested that the design of the open propeller could be directly applied to the shrouded propeller. However, the results in the current simulation indicate a significant increase in the inflow velocity of the shrouded rotors. We believe that the influence of the inflow cannot be ignored, and thus, there remains a large room for the optimization of the shrouded propeller design.

4.3. Influence of Pitch Length of the Upper Propeller

The aerodynamic performance of the upper propeller, including the influence of pitch length and inflow velocity, is initially investigated. The airflow is projected on the two-dimensional airfoil plane and investigated using the BEM theory. Inflow velocity, which is correspondingly decomposed into axial and rotational components, plays a crucial role in providing insights into the flow conditions. Taking the middle cross-section of the hub ($z_0=0$) as the base plane, we investigate the radial distribution of the inflow velocity from three planes ($z_1=c$, $z_2=2c$, and $z_3=3c$) parallel to the base plane. It is worth mentioning that, when “c” is large, the planes would be far from the base plane, resulting in an airflow velocity too small to represent the velocity of the inflow. Conversely, for small values of $c$, the planes would be too close to the propeller, causing significant periodic interactions with the propeller. In other words, different positions along the axis, characterized by varying values of $c$, correspond to distinct airflow velocities. Consequently, considering the findings in Figure 10 and the relevant reference [10], $c$ is taken as 0.06D.

Figure 11 shows the distribution of axial velocity in the radial direction of three parallel planes. It is apparent that the airflow undergoes an acceleration for both the open and the shrouded propellers, with the acceleration being more pronounced in the case of shrouded propellers. In the position of “$z_1=c$”, the averaged inflow velocity of the shrouded propellers is 94.3% higher than that of the open propellers. In addition, the velocity at $r/R=1.2-1.6$ of the shrouded propellers exhibits a negative value, which indicates that the incoming flow outside the shroud is reversed. We can infer that the shroud draws the external air into the shroud through the lip, consequently causing the reversed velocity phenomenon.

The pitch angle of the upper blade is investigated and optimized based on the data and analysis above. At the rotation speed of 8000 RPM, which corresponds to the hovering condition, the velocity at the radial position of r is $v=2\pi nr/60$. Typically, it is customary to estimate the aerodynamic characteristics of a propeller at the 0.75R position [19]. Therefore, the inflow velocity is decomposed into the axial and rotational directions at this position: the axial velocity is $C_m=U_z{|}_{r/R=0.75}=15.76 \mathrm{m}/\mathrm{s}$ and the rotation velocity is $U_m=2\pi nr/60=57.19 \mathrm{m}/\mathrm{s}$. The Reynolds number is 29,605 in this working condition. In Figure 12, the variation in the thrust coefficient and lift–drag ratio of the HS1712 airfoil in this Reynolds number is plotted, which is drawn using the open-source code Airfoil Tools. The lift–drag ratio, denoted as $FoM$, is determined by dividing $C_t$ by $C_p$ of the airfoil.

According to Figure 12, the $FoM$ increases with the angle of attack from 0 to 5.8 degrees and decreases with it from 5.8 to 17 degrees. The lift–drag ratio reaches its maximum value of 62.4 at 5.8 degrees, beyond which the airfoil experiences a stall phenomenon at 17 degrees. As a result, the optimal angle of attack of the 2-D airfoil is taken as 5.8 degrees.

Figure 13 presents the radial distribution of pitch angles and angles of attack for open and shrouded rotors. The axial and rotational velocities are calculated at each profile plane from 0.1R to 1.0R, allowing the determination of the radial distribution of inflow angles. This enables the measurement of angles of attack:

$$\alpha =\theta -\varphi ,$$

where $\theta$ refers to the inflow angle and $\varphi$ refers to the pitch angle.

The disparity between the shrouded and open propellers becomes apparent, with the former operating at an unfavorable working condition. The angles of attack observed for shrouded propellers at various radial positions consistently remain below 2 degrees, exhibiting a deviation of 6.1 degrees from the optimal angle of attack (5.8 degrees). Conversely, the average angle of attack of the open propeller is slightly higher than the optimal angle of attack.

Consequently, the angle of attack of the shrouded propellers at the radial position of 0.75R is increased to 5.8 degrees, corresponding to a 7.5-inch propeller. The selection of 0.75R as the radial position is a customary practice for representing the aerodynamic characteristics of the propellers as the inflow velocity and angle at this position closely resemble the average velocity and angle. Following the adjustment, the radial distribution of pitch angles and angles of attack for the shrouded 7.5-inch propellers is presented in Figure 14.

According to the BEM theory, the angle of attack reaches the optimal angle when the pitch length increases to 7.5 inches. However, altering the pitch length also leads to changes in the inflow velocity above the propeller. The $FoM$ may not necessarily align with the theoretical predictions based on two-dimensional BEM analysis. Furthermore, the design and discussion of the combined BEM theory did not take the thrust of the shroud into consideration. Therefore, it is imperative to conduct further investigation through numerical simulations on the shrouded propellers. At a rotation speed of 8000 RPM, the pitch lengths of the propellers vary from 5 to 10 inches. As shown in Figure 15, the $C_t$ of the shrouded propellers of 10-inch pitch length is 2.53 times that of the 5-inch pitch length, while the $C_p$ is 3.73 times higher. The $FoM$ attains its maximum value at an 8-inch pitch length, according to which the optimal pitch length is determined. The 8-inch optimal pitch length is consistent with the prediction derived from the BEM theory.

The airflow velocity differs between the lower and upper propellers, suggesting the optimal pitch length determined for the upper propeller may not be applicable for the lower one. In the preceding simulation, a mirror-symmetry configuration of the lower and upper propellers is used, with identical pitch lengths set to balance the anti-torque. However, it can be found in Figure 11 that the flow conditions of two propellers are not entirely identical. The airflow speed of the lower propeller is different from that of the upper one. In the next section, we investigate the influence of the differential pitch-length design of the upper and lower propellers on the aerodynamic performance of the shrouded propellers.

4.4. Influence of Pitch Length of the Lower Propeller

This section studies the impact of the pitch length of the lower propeller on the $C_t$, $C_p$, and $FoM$. In the shrouded coaxial dual-rotor MAV, the lower propeller can generate the compensation torque and increase the thrust. It is also capable of straightening the airflow downstream of the upper propeller, making the wake more uniform and identical. Such airflow is advantageous for the flight control of the shrouded rotors [19].

Figure 16 shows that the airflow is accelerated through the contra-rotating upper and lower propellers, which are turbulent and cluttered. Such turbulent wake flow, which in turn leads to a significant dissipation of energy, is harmful to enhancing the aerodynamic performance of the shrouded propellers. Consequently, the inflow velocity of the lower propeller is decomposed for optimization purposes, employing the BEM theory alongside numerical simulations. By examining the inflow velocity of the lower propeller, it is possible to mitigate the dissipation of kinetic energy and improve the overall aerodynamic efficiency of the shrouded propellers.

The inflow of the lower propeller is simplified based on the simulation results, the process of which is similar to the upper propeller. It can be observed that the radial distribution of the inflow velocity at 0.3–0.85R is relatively uniform. Within this radial range, the BEM theory can be effectively employed to analyze the flow conditions.

With the middle cross-section of the lower propeller set as the base plane, we investigate the cross-section parallel to it at the axial distance of $z=1.0c$. Figure 17 shows the distribution of inflow velocity and inflow angles at various radial positions. It is observed that, except for the region near the blade hub (0.1–0.3R), the inflow angle decreases progressively in the radial direction. In the range of 0.3–0.85R, the inflow velocity increases as the radius increases, while it decreases in the range of 0.85–1.0R.

A better understanding of the inflow differences between the upper and lower propellers can be obtained by comparing the radial distribution of the inflow velocity in Figure 11a and Figure 17. The average flow velocity of the lower propeller is 22.1 m/s, which increases by 6.7% compared with that of the upper propeller. The increase in the inflow velocity, indicating the flow acceleration produced by the upper propeller, is minor but affects the flow condition of the lower propeller. The inflow angles at various radial positions ranging from 0.3 to 0.9R are calculated and the radial distribution of pitch angles ($\theta$) and angles of attack ($\alpha =\theta -\varphi$) of the upper and lower propellers are presented in Figure 18.

Based on these observations, it is clear that the angles of attack of the lower propeller are generally 1–2.5 degrees less than that of the upper propeller. However, the expected angles of attack of the lower propeller should be the same as the upper propeller according to the BEM theory. Given the Reynolds number of the lower propeller, which is 31,002, we assume the optimal angle of attack remains as 5.8 degrees in the radial position of 0.75R. Therefore, the angles of attack of the lower propeller should be increased, which corresponds to larger pitch angles and thus a larger pitch length. The pitch angles at corresponding radial positions are then increased by 1–2.5 degrees, ensuring the optimal angles of attack.

Following the increase in pitch lengths of the lower propeller, further numerical simulations are conducted to investigate the details of the flow field. The shrouded propellers of varying pitch lengths are tested under the speed of 8000 RPM. The output results, including $C_t$, $C_p$, and $FoM$, are shown in Figure 19. It is apparent that $C_t$ and $C_p$ are sensitive to the pitch length. With the increase in the pitch length of the lower propeller, the C_t and $C_p$ increase by 14.2% and 12.0%, respectively. It can be expected that, if the pitch lengths of both the upper and lower propellers increase within a range of 10 inches, the $C_t$ and $C_p$ exhibit an incremental change. However, the $FoM$ is less sensitive to the changes in the pitch length of the lower propeller, with a maximum fluctuation in $FoM$ of only 6.09%. The observed disparities between the BEM prediction and simulation results can be explained by the turbulent wake flow of the upper propeller. Such a flow condition may not satisfy the assumptions and requirements of the BEM theory, which assumes a steady and uniform inflow.

Based on the limited influence of pitch length difference on $FoM$, it can be suggested that there may not be a significant need to differentiate the design of the upper and lower propellers based solely on pitch length. Moreover, in cases where there is a substantial difference in pitch lengths between the upper and lower propellers, it could potentially result in an imbalance in torque. Therefore, maintaining the mirror symmetry of 8-inch pitch lengths can be beneficial for enhancing the aerodynamic performance of the propellers.

4.5. Influence of Chord Length

In the previous section concerning the pitch length, the upper and lower propellers in the shrouded rotors are investigated and optimized using the BEM theory. With numerical simulations, it is concluded that the optimal pitch lengths of upper and lower propellers are both taken as eight inches. However, it should be noted that the $C_t$ and $FoM$ of shrouded propellers are also influenced by other factors, including the chord length, tip clearance, and sweep angle of the propellers. These additional factors play a crucial role in determining the overall performance of the shrouded propellers. Therefore, numerical simulations that consider these factors in conjunction with pitch length are necessary to achieve an optimal design for shrouded propellers.

We first investigate the impact of different chord lengths on the aerodynamic performance of the shrouded propellers. To represent the chord length, we utilize the pre-set propeller parameter $\gamma$, which determines the tip chord length. Figure 20a,b illustrates the distribution of $C_t$, $C_p$, and $FoM$ as a function of chord length ($\gamma$). The chord length was divided into five groups for the numerical simulations, specifically $\gamma$ = 0.7, 1.2, 1.7, 2.2, and 2.7, which is taken in the numerical simulations.

Increasing the chord length of shrouded propellers leads to significant improvements in $C_t$ and $C_p$, while the influence on $FoM$ is relatively minor due to potential interference between the blades. It is apparently shown in Figure 20a that, with the increase in the chord length, both the C_t and Cp increase. In particular, the increase in $C_p$ demonstrates a linear relationship. The results are consistent with the propeller theory that the increasing chord length implies an increase in the propeller solidity, thereby increasing $C_t$ and $C_p$ simultaneously [23,24]. When $\gamma$ increases from 0.7 to 2.7, the $C_t$ experiences a 22.61% increase, while $C_p$ shows a 36.59% increase. Interestingly, the $FoM$ is not sensitive to changes in the chord length, remaining relatively stable at around 0.72. To be specific, the FoM decrease is less than 1%. This change may be explained by the increase in interference between the blades. The shed vortex from the trailing edge of the front blade may collide with the leading edge of the neighboring blade, leading to minor effects on the overall $FoM$.

Considering the limited impact of chord length on the $FoM$ and its linear relationship with $C_t$ of the shrouded propellers, it can be inferred that during the design of propellers within a shroud, the chord length can be determined based on the desired thrust and power requirements. In other words, when designing the shrouded rotors, the desired $C_t$ and $C_p$ can be obtained from the expected thrust and power, which can then guide the evaluation of the chord lengths for current propellers. For instance, at the same rotation speed, a smaller chord length propeller exhibits a lower $C_p$, making it suitable for improving the cruising ability of shrouded propellers when the load requirement is not high. Conversely, when the same thrust is needed, a larger chord-length propeller operates at a lower rotation speed, reducing the noise at the same time. In practical shrouded MAV design, propellers with specific chord lengths can be selected based on the desired thrust and rotation speed.

4.6. Influence of Tip Clearance

Simulations show that smaller tip clearances result in higher aerodynamic efficiency for shrouded rotors. However, it is worth noting that in physical models of shrouded rotors, constraining the tip clearance within 1 mm can be challenging due to factors such as dual-rotor balancing and manufacturing errors. In the simulations discussed previously, a tip clearance of 0.5 mm was used, which is 0.1–0.3 mm smaller than that of the physical model. Therefore, we investigate the aerodynamic influence of the increase in tip clearance on the shrouded rotors. Since the tip clearance $\delta =\left(D_{in}-D_{prop}\right)/2$ and $D_{in}$ is a constant, the parameter $\delta$ is increased by decreasing of the diameter of the propellers.

Results of the simulation are presented in Figure 21, where the tip clearance is changed while keeping the pitch and chord lengths fixed. It is apparent that, as the tip clearance increases, both the $C_t$ and $FoM$ of the shroud propellers decrease significantly. The total $C_t$ decreases by 18.9% when the tip clearance expands to 3 mm, and decreases by 32.5% when the tip clearance expands to 5 mm. These findings highlight the substantial impact of tip clearance on the performance of shrouded propellers.

It should be noted that the decrease in total $C_t$ is primarily attributed to the decline in the $C_t$ of the shroud as the tip clearance increases. Specifically, when the tip clearance expands above 1 mm, the proportion of $C_t$ of the shroud drops from 44.1% to 30.5%. Furthermore, the $C_t$ of the shroud is lower than the combined $C_t$ of the upper and lower propellers. The total $C_t$ of the upper and lower propeller (excluding the $C_t$ of the shroud) is around 0.056 when the tip clearance changes, with less than 3% fluctuation observed across varying tip clearances. In addition, it can be seen from Figure 21b that the $C_p$ is generally constant when the tip clearance exceeds 1 mm. This observation also suggests that the aerodynamic performance of the propellers (excluding the shroud) is less affected by changes in tip clearance, as the $C_p$ is solely related to the propellers and not the shroud. Therefore, when the tip clearance is higher than 1 mm, the total $C_t$ of propellers is less sensitive to the tip clearance, while the $C_t$ of the shroud is greatly impacted.

5. Conclusions

Numerical simulations were made to investigate the propeller parameters of the aerodynamic performance of the shrouded coaxial dual rotors in hover. Experiments were first conducted to examine the accuracy of numerical simulations, where a good agreement was obtained between numerical and experimental results. Compared with the open rotors, the existence of the shroud increases the inflow velocity, decreases the tip vortex, and improves the $C_t$ and $FoM$ of the shrouded rotors.

The BEM theory was used to facilitate the optimization of the upper and lower shrouded propellers. The parameters under investigation include the pitch length, chord length, and tip clearance. The results show that an increased pitch length is preferred for the shrouded propellers due to the increase in inflow velocity that results in an increase in inflow angles. The $FoM$ reaches a maximum when the pitch lengths of the upper and lower propellers are 8 inches, which is generally consistent with the prediction using the BEM theory. While increasing the chord length results in an increase in $C_t$, this effect does not significantly impact the $FoM$. The tip clearance demonstrates a significant impact on the $C_t$ and FoM of the shrouded propellers. By increasing the tip distance from 0.5 to 3 mm, $C_t$ decreases by 43.2% and $FoM$ decreases by 33.6%.

In conclusion, we investigated the difference in $C_t$, $C_p$, and $FoM$ between shrouded and open rotors, based on which, the influence of pitch length, chord length, and tip clearance on the aerodynamic performance were optimized. It should be noted that additional parameters, such as the receding angle and hub diameter, also impact the aerodynamic performance. Therefore, future work will focus on a broader range of parameters to further improve the aerodynamic performance of the shrouded rotors.