Nomenclature

2.0 Multi-rotor’s geometrical and performance parameters

Their cross-sections of propellers are aerofoils that are placed uniformly in the spanwise direction, as in Fig. 2(a). Thus, the propeller blades have geometric characteristics similar to aerofoils, including the trailing edge, leading edge, chord, thickness and more, as depicted in Fig. 2(b) [Reference Loureiro, Oliveira, Hallak, de Souza Bastos, Rocha, Delmonte and de Castro Lemonge21, Reference Chevula, Chillamcharal and Maddula22]. The number of blades of a multi-rotor can vary based on its design consideration. They typically utilise two blades for stability, control and overall weight consistency of the structure [Reference Varki, Yeter and Doğru23]. The geometric pitch or theoretical pitch, actual or effective pitch, and twist angle known as pitch angle are illustrated in Fig. 2 and described in Refs (Reference Loureiro, Oliveira, Hallak, de Souza Bastos, Rocha, Delmonte and de Castro Lemonge21 and Reference Chevula, Chillamcharal and Maddula22).

Figure 2. Propeller geometry and performance parameter; propeller cross-section at different local radius and twist along the span (a); propeller’s geometry parameter (b); geometric pitch, effective pitch and advance ratio (c).

One of the parameters that can be used for the performance of the propeller is the propeller efficiency, which can be calculated from the relation (Equation(1)) and is defined as the ratio of the output power (which is the driving power) to the input power (which is the mechanical power) [Reference Deters, Kleinke and Selig13].

(1) \begin{align}{{\rm{\eta }}_{\rm{P}}} = \frac{{{\rm{T}}{{\rm{V}}_0}}}{{{{\rm{P}}_{\rm{m}}}}}.\end{align}

where T is the thrust, V₀ is the free-stream velocity and P_m is the required mechanical power, which can be calculated from the following relation [Reference Deters, Kleinke and Selig13]:

(2) \begin{align}{{{P}}_{{m}}}{{\;}} = {{\;}}2{{\pi nQ}}.\end{align}

where Q is the torque and n is the rotational speed (in rpm). In the propeller efficiency equation, C_P is the power coefficient and C_T is the thrust coefficient, which can be calculated from the following relations [Reference Deters, Kleinke and Selig13]:

(3) \begin{align}{{{C}}_{{T}}}{{\;}} = {{\;}}\frac{{{T}}}{{{{\rho }}{{{n}}^2}{{{D}}^4}}}, \\[-28pt] \nonumber \end{align}

(4) \begin{align}{{{C}}_{{P}}}{{\;}} = {{\;}}\frac{{{{{P}}_{{m}}}}}{{{{{\rho }}^3}{{{D}}^5}}} \\[4pt]\nonumber \end{align}

Also, the torque coefficient, C_Q, can be defined as the following relation [Reference Mahmuddin24].

(5) \begin{align}{{{C}}_{{Q}}} = {{{Q}} \over {{{\rho }}{{{n}}^2}{{{D}}^5}}}.\end{align}

In the above relations, ρ is the flow density and D is the propeller diameter. The advance ratio, J, is calculated from the relation (Equation (6)). Due to the low value of the speed, the advanced ratio in this research is very close to zero [Reference Mahmuddin24].

(6) \begin{align}{{J\;}} = {{\;}}\frac{{{{{V}}_0}}}{{{{nD}}}}.\end{align}

Using Equations (1) to (4) and Equation (5), another relation for the propeller efficiency could be obtained:

(7) \begin{align}{{{\eta }}_{{P}}} = {{\;J}}\frac{{{{{C}}_{{T}}}{{\;}}}}{{{{{C}}_{{P}}}{{\;}}}}\end{align}

3.0 Geometry of insect-inspired propeller

Regarding the experimental test that has been done on DJI phantom-3 propeller in Ref. (Reference Deters, Kleinke and Selig13) and the existence of the experimental data, it is a suitable choice to consider as the base propeller for the present study, in order to validate the numerical simulation. This propeller has diameter of 0.24 m [25] as shown in Fig. 3.

Figure 3. Image of DJI Phantom-3 propeller (a), and its schematic (b) [26].

Selecting the wings of insects was a significant challenge in this research. One of the key requirements is the availability of a perfectly captured wing planform. In other words, a high-quality image from a top view that accurately depicts the final shape of the wing’s planform is necessary. However, obtaining such an image is difficult due to various factors. Many insects have extremely small dimensions, rapid movement and unique rest positions of wings, making it rare to find high-quality images that showcase the planform characteristics and features. This issue has been a fundamental challenge in choosing insect wings. For this reason, insects with these characteristics have been excluded due to the lack of suitable images. Consequently, this scarcity of suitable images adds to the challenge of wing selection. In the selection process of insect wings, one crucial factor to consider is the simplicity of the borderlines that define its shape. This holds particularly true when utilising design software for three-dimensional modeling, and depends on the level of expertise involved.

Figure 4 is a summarised classification of winged insects. Winged insects are divided into two parts: Paleoptera and Neoptera. Paleoptera, like dragonflies, do not have the ability to fold their wings, whereas Neoptera show this ability and are divided into two parts: exopterygota and endopterygota. Among the insects classified in Fig. 4, considering some concerns including the availability of images to use for the initial steps of modeling, some specific features of the species, and clarity of available images, several species from the chart in Fig. 4 have been chosen for further studies. Figure 5 is designed and illustrated in order to show the family order of insect and that special insect whose wings are selected. In the internal cells of this graph the insects are related to the family shown and in the external cells are that special insect whose wings are selected are illustrated. In the centre of Fig. 5, six selected insect species are placed. These selected species belong to the Hemiptera, Orthoptera, Neuroptera, Hymenoptera, Mantodea and Odonata. Odonata has more than 6000 species and is divided into two subcategories of dragonflies and mayflies [Reference Samways and Deacon27]. Although they are different in appearance, dimensions and the way the wings are positioned in the resting phase, all Odonata species have a pair of forewings and a pair of hindwings. The other selected wing is from the Hemiptera family order, which has more than 80,000 known species. Aphids and field crickets are some examples of this order. Usually there are wings in Hemiptera, however, some of them are wingless or lose their wings. Neuroptera is an order of insects with about 6000 known species. Silky wings and spoon wings are examples of Neuroptera. Their dimensions vary from small to large, with a wing length of more than 100 mm [Reference Flindt28, Reference Gibb29]. Neuroptera usually have four large, membranous and unequal wings, and when resting, the wings are placed behind the insects like a tent [Reference Gibb29]. Mantodea has more than 2400 species. It is the next selected insect family order for inspiration. Mantises are the largest family of this order. The size of wings in mantises can be big or small, or there are no wings at all. If there are wings, mantises have two pairs of wings. Orthoptera is an order of insects with about 20,000 to 250,000 known species. This order of insects includes crickets and grasshoppers. These insects do not have the ability to fold their wings while resting, thus the wings are tangent to the body. Hymenoptera is one of the largest orders of insect. More than 120,000 species of these insects are known. Hymenoptera usually have two pairs of wings, and the forewings are larger than the hindwings. Bees and ants are insects of this order.

Figure 4. Insect’s family chart [Reference Zakeri Nazar, Askari and Masdari40].

Figure 5. Selected insect’s species for capturing their wings [Reference Zakeri Nazar, Askari and Masdari40].

The selection of each insect and the reason for this selection was one of the most important and challenging parts of this research. One of the most important activities carried out in this regard was the review of previous related studies as we talked about in the last review [Reference Moslem, Masdari, Fedir and Moslem20], insect wings have special aerodynamic qualities that help them fly higher and quieter. Accordingly, and based on previous studies and investigations, there was a need to investigate aerodynamic performance in the continuation of research and studies conducted in this field. Therefore, we included families of insects that had been previously investigated in the review list. In addition, as an innovation, insects from new families were also selected, the results of which indicate their good performance. Some insects such as Orthoptera and Mantodea have not been studied before, and the reason for choosing these insects is to expand more references for inspiration in this field.

• • Odonata:

  The insect selected from the dragonfly species is “Matrona Basilaris”. One of the main reasons for choosing this wing is the multiplicity of studies that have been done on dragonflies from various aspects such as wing structure and behaviour, section, flight and aerodynamics [26–Reference Norberg35]. Also, the planform of these wings, especially the shape of the leading edge, is simple and is a suitable choice for three-dimensional modeling.

• • Hymenoptera:

  The next selected wing is from the Hymenoptera species, which many studies on the flight and wing of this species has been done [Reference Dudley and Ellington38, Reference Golding, Ennos and Edmunds39]. The insect selected from this species with the scientific name “Apis mellifera” is a honey bee. It is one of the fastest flying insects that has a simple planform line and shape.

• • Neuroptera:

  The wing of the “Crealeon prolongaveruntus” insect from the Neuroptera species is the third selected wing. In general, the insects of this species are similar in appearance to a suborder of dragonflies in Odonata family, and the wing of the selected insect is almost similar to the wings of dragonflies, with the difference that its width is less. Also, its simple shape and planform line is one of the main reasons for choosing this wing.

• • Hemiptera:

  The wing of the “pyrops clavatus” insect from Hemiptera has been chosen as the fourth wing because it has a large wing surface and a simple planform line.

• • Orthoptera and Mantodea:

  There is no research on the wings of Orthoptera species and Mantodea and is the reason for choosing the insects “Phymateus karschi” and “Raptrix perspicua” from these species. In addition to simple shape and similar planform line to other selected wings, there is an interest to investigate their wing potential for inspiration in modeling for use in small multi-rotor interiors.

To create a precise three-dimensional model of a propeller blade, it is crucial to determine three essential factors: the chord distribution, twist distribution and aerofoil of the propeller section. To achieve the geometrical properties for each insect, its right forewing was selected. Considering the top view of each insect wing, the closed curve that includes the perimeter of the wing and defines the general shape of the wing has drawn accurately, similar to Fig. 6. This closed curve is the wing planform. However, it should be noted that insect wings like that shown in Fig. 6 may have some broken or torn parts on the leading or trailing edge, which for making modeling easier, these parts are not considered during the drawing of the planform curve, like what is shown in Fig. 6 from a closer view (see red circle).

Figure 6. The process of wing planform capturing, in a case, facing with scratches on the edge of the wing of Hemiptera.

To ensure a fair comparison, all insect-inspired propellers needed to be scaled and dimensionless proportionately to the base propeller radius, as depicted in Fig. 7(a). The planform curves drawn in the previous step are placed in a grid. Therefore, for all insect wings, their tips are exactly located at the position of r/R equal to 1 of grid, and their roots are located at the position of r/R equal to 0 of grid. For the simulations that will be explained in the next section, by scaling, all wings have a radius of 0.12 m, similar to the base propeller. Once the scaling of the insect-inspired propellers was accomplished, the chord at each r/R (radius ratio) could be obtained from the chord distributions shown in Fig. 7. Further description regarding the modeling of the present insect-inspired propellers is presented in Refs (Reference Zakeri Nazar, Askari and Masdari40 and Reference Zakeri Nazar, Askari and Masdari41).

The three-dimensional geometry modeling of the propellers was executed using SolidWorks software, incorporating specific twist and chord values for each section’s aerofoil. After merging the sections, the final solid geometries of the propellers were obtained, as depicted in Fig. 8. It should be acknowledged that the mentioned three-dimensional modeling of the insect wings may contain some modeling errors and simplifications. However, these would only yield a negligible margin of error. The twist distribution, taking into account previous studies [Reference Deters, Kleinke and Selig13, Reference Moslem, Masdari, Fedir and Moslem20], starts at 17.72 degrees from the root to approximately 30% of the propeller radius. From that point to the tip, the twist decreases linearly to 4.7 degrees. Since there is a lack of information regarding the geometry and aerofoil specifications of the DJI Phantom-3, EPPLER E63 aerofoil, shown in Fig. 7(h), is considered for our insect-inspired propellers. This aerofoil is widely used in research on small-scale propellers and is known to be a suitable choice for low Reynolds number flows [Reference Ning and Hu16, Reference Moslem, Masdari, Fedir and Moslem20]. This aerofoil is utilised for all sections and propellers in this study. For three-dimensional modeling of other propellers, the diameter is hold fixed, equal to the base propeller. Also, considering that the geometry of the hub does not have a significant effect on the results of this research, the hub shape is considered constant for all propellers.

Figure 7. Schematic of: base propeller (a), Mantodea (b), Neuroptera (c), Orthoptera (d), Odonata (e), Hymenoptera (f), Hemiptera (g), Eppler E63 aerofoil (h).

4.0 Numerical simulation methodology and validation

The governing equations of fluid dynamics include the continuity equation, momentum equation and energy equation, which are given below and are fully explained in Refs (Reference Anderson47–Reference Andersson, Andersson, Håkansson, Mortensen, Sudiyo and van Wachem49).

1. 1. Continuity equation or law of conservation of mass:

   Equation (8) is general and used in compressible and incompressible fluids [Reference Anderson47].

   (8) \begin{align}\frac{{\partial \rho }}{{\partial t}} + \nabla .(\rho \overrightarrow V ) = o.\end{align}

2. 2. The momentum equation

   The momentum equation can be written in three directions – x, y, z – and is available in Ref. (Reference Anderson47).

   (9) \begin{align}\rho \left[ {\frac{{\partial \overrightarrow V }}{{\partial t}} + (\overrightarrow {V.} \nabla )\overrightarrow V } \right] = - \nabla p + \nabla .\overline{\overline \tau } + \rho \overrightarrow f .\end{align}

   The second term on the right side of Equation (2) is the viscous stress tensor, and the third term on this side is the body forces. The viscous stress tensor is given in Equation (10) [Reference GarofanoSoldado, SanchezCuevas, Heredia and Ollero48].

   (10) \begin{align}{\tau _{ij}} = \mu \left( {\frac{{\partial {V_i}}}{{\partial {x_j}}} + \frac{{\partial {V_j}}}{{\partial {x_i}}}} \right) + \lambda (\nabla .\overrightarrow V ){\delta _{ij}}.\end{align}

   And ${\delta _{ij}}$ which is called Kronecker dela.

3. 3. The energy equation

   The third equation is the energy equation, which was not calculated in this study and its equation is fully presented in the Ref. (Reference Anderson47).

Figure 8. Three-dimensional models of bio-inspired propellers, Mantodea (a) [Reference Ábrahám42], Neuroptera (b) [43], Orthoptera (c) [Reference Constant and Pham44], Odonata (d) [Reference Descouens45], Hymenoptera (e), Hemiptera (f) [Reference Seehausen46].

Governing equation

The general form of Navier-Stokes equations is as Equation (11), and the new terms appearing in this equation are turbulence effects. To solve this equation, the Reynolds stresses in the third term on the right should be modeled [Reference Andersson, Andersson, Håkansson, Mortensen, Sudiyo and van Wachem49].

(11) \begin{align}\begin{array}{l}\dfrac{\partial }{{\partial t}}(\rho {u_i}) + \dfrac{\partial }{{\partial {x_j}}}(\rho {u_i}{u_j}) = \\[10pt] - \dfrac{{\partial p}}{{\partial {x_i}}} + \dfrac{\partial }{{\partial {x_j}}}\left[ {\mu \left( {\dfrac{{\partial {u_i}}}{{\partial {x_j}}} + \dfrac{{\partial {u_j}}}{{\partial {x_i}}} - \dfrac{2}{3}{\delta _{ij}}\dfrac{{\partial {u_i}}}{{\partial {x_i}}}} \right)} \right]\\[10pt] + \dfrac{\partial }{{\partial {x_j}}}( - \rho \overline {{{u'}_i}{{u'}_j}} )\end{array}\end{align}

The general form of the instantaneous Navier-Stokes equations is as Equation (3), and the new terms appearing in this equation are turbulence effects. To solve this equation, the Reynolds stresses in the third term on the right should be modeled [Reference Andersson, Andersson, Håkansson, Mortensen, Sudiyo and van Wachem49].

The flow field around the propellers are simulated numerically using ANSYS FLUENT and Multiple Reference Frame (MRF) method. MRF is a suitable method when there is a rotating section, like propeller, and the flow around this part is desired to simulate [50]. Noting the recommendation of the MRF method in previous studies [Reference Garofano-Soldado, Sanchez-Cuevas, Heredia and Ollero51–Reference Lopez, Escobar and Pociña Pérez53] and also considering the subject of the current study, MRF method is selected. Figure 9 illustrates the geometry of the computational domain that contains two parts: the rotating domain, which encloses the propeller, and the larger stationary domain, which includes the rotating domain. Design Modeler is used for the generation of this geometry. The propeller is fully enclosed within the rotating domain, causing the propeller tips to be slightly distanced from the interface. To enhance accuracy, two cases of different static domain dimensions were considered and the results were compared ith experimental data conducted by Deters et al., as presented in Table 1. As a final step, by considering a compromise between the available computing resources, the number of simulation cases and the computational cost of each case by increasing H = 14D and Dsd = 4.41, it was observed that the thrust is obtained with less than 0.005 difference compared with the previous case (H = 10D and Dsd = 4). Consequently, the diameter of the rotating domain is 1.08 times the propeller’s diameter. On the other hand, the diameter and the height of the stationary domain cylinder is 4.14 times and 14 times the propeller’s diameter, respectively [Reference Chevula, Chillamcharal and Maddula22, Reference Garofano-Soldado, Sanchez-Cuevas, Heredia and Ollero51, Reference Kutty and Rajendran54].

Table 1. Static domain independency

Figure 9. Computational domains along with dimensions and boundary conditions.

For meshing of propeller and computational domain, ANSYS meshing was used. Considering the relatively small dimensions and the three-dimensional nature of the propeller, unstructured mesh and tetrahedral elements were utilised. Additionally, as viscosity effects were found to be predominant, boundary layer meshing and prism elements were employed. The calculations for Reynolds number and thickness of the turbulent boundary layer were conducted based on the chord and tangential speed at 75% of the propeller radius and standard temperature (see red lines at the insect’s wing planforms in Fig. 7) [Reference Deters, Ananda Krishnan and Selig3, Reference Schetz and Bowersox55]. Figure 10 presents the propeller’s mesh.

The boundary conditions between the rotation and stationary domains are defined and mentioned in Table 2. The turbulence intensity is considered 0.1 %, which is equivalent to the conditions available in the experimental condition whose data used in the present study [Reference Deters, Kleinke and Selig13]. Pressure outlet is chosen for its better convergence rate. The rotational speed is considered in the range of 4000 to 8000 rpm with a step of 1000 rpm.

Table 2. Boundary conditions

Figure 10. Mesh of base propeller (a), Mantodea (b), Neuroptera (c), Orthoptera (d), Odonata (e), Hymenoptera (f), Hemiptera (g).

The SST K-ω turbulence model is used for turbulence flow modeling in the present study. This model has been recommended and used by various researchers to analyse the aerodynamic performance of small propellers [Reference Yilmaz and Hu5, 50, Reference Schetz and Bowersox55, Reference Li, Yonezawa, Xu and Liu56]. Its governing equations represented below, and assumptions and constants are presented in these Refs (Reference Wilcox57, Reference Menter58).

(12) \begin{align}\frac{{\partial k}}{{\partial t}} + {U_j}\frac{{\partial k}}{{\partial {x_j}}} = {P_k} - {\beta ^*}k\omega + \left[ {(\nu + {\sigma _k}{\nu _T})\frac{{\partial k}}{{\partial {x_j}}}} \right],\end{align}

(13) \begin{align}\frac{{\partial \omega }}{{\partial t}} + {U_j}\frac{{\partial \omega }}{{\partial {x_j}}} = \alpha {S^2} - \beta {\omega ^2} + \frac{\partial }{{\partial {x_j}}}\left[ {(\nu + {\sigma _k}{\nu _T})\frac{{\partial \omega }}{{\partial {x_j}}}} \right]. \\[5pt] \nonumber \end{align}

Although there are higher computational costs with boundary layer meshing, this turbulence model is suitable for accurately modeling propeller turbulence. Typically, the average Y+ value is remained around one for all simulations. The solver for solution is pressure based with coupled algorithm and discretisation of all equations are based on second upwind scheme. Also for gradients, Least Squares Cell-Based is used. The solution settings and boundary conditions are summarised in Table 3. The solution process is repeated until the residuals reach the selected convergence criterion of 0.0001. Moreover, a convergence criterion of 0.0001 is also applied to the thrust and torque measurements. The number of iteration steps varies between 2000 and 3000.

Table 3. Solution setting, numerical method and boundary conditions

Figure 11. Validation of numerical method, thrust versus RPM and power coefficient versus RPM.