Stall is a complex phenomenon in aircraft that must be suppressed during flight. As a novel passive control method, bionic leading-edge protuberances (LEPs) have attracted widespread interest, particularly for delaying stall. Bionic protuberances at the leading edge of airfoils have been designed to control stall and increase the stability of unmanned aerial vehicles during operation, and it is the flow control mechanism associated with this application that is investigated in this study. First, numerical simulations are conducted to obtain the aerodynamic characteristics of original and bionic airfoils based on the S1223 large-lift airfoil. Next, the impact of the LEP amplitude is investigated. Finally, a vortex definition parameter, the Liutex vector, is utilized to analyze the influence of LEPs on vortices. The results show that bionic LEPs inspired by those on humpback whale flippers can improve the aerodynamic performance of airfoils under the extreme conditions that exist after stall, resulting in an ∼22% increase in the lift–drag ratio. LEPs are found to segment the flow field near the wing surface. The flow becomes bounded between adjacent protuberance structures, significantly inhibiting the development of flow separation and providing a drag reduction effect. This study thus provides a new approach for improving aircraft performance.

## NOMENCLATURE

*C*_{d}drag coefficient

*C*_{dB}drag coefficient of modified airfoil

*C*_{l}lift coefficient

*C*_{lB}lift coefficient of modified airfoil

*e*_{a}approximate relative error

*e*_{ext}extrapolated similarity error

- GCI
_{fine} convergence index of fine grid

*h*grid size (m)

*k*_{d}relative drag coefficient

*k*_{l}relative lift coefficient

*N*number of cells

*p*apparent order

*S*strain rate (s

^{−1})

### Greek

## I. INTRODUCTION

Unmanned aerial vehicles (UAVs) are mostly operated under remote control and have a wide range of applications.^{1} During an actual flight, the attitude of an UAV changes to enable it to perform various high-mobility actions, and the approximate range of Reynolds numbers for an UAV during operation is between 3 × 10^{4} and 5 × 10^{5}. The lift force abruptly drops and an airfoil appears to stall when the angle of attack (AOA) increased to a critical value. In various complex working environments, the operating conditions of wind turbines,^{2,3} pump turbines,^{4} aircraft,^{5} and other pieces of machinery may reach or even exceed stall operating conditions. A stall not only reduces operating performance and efficiency, but also induces periodic shedding of separation vortices, which can cause component vibrations, affect operational stability, and even cause damage and accidents. Therefore, it is necessary to control airfoil stalls, thereby improving the safety margins and flow separation behavior of airfoils.

There are two main types of stall control: active and passive. Active control is used to restrain flow separation and improve aerodynamic performance by introducing external energy and coupling this with the flow. Examples include synthetic jets,^{6,7} blowing and suction flow control,^{8} and plasma flow control.^{9,10} However, active control methods have disadvantages, including a relatively complex structure, high energy consumption, and difficulties in installation. Most such control methods are limited to the research stage, and only a few have been applied in practice. Passive control does not involve any external energy injection, but instead controls the flow by changing the airfoil geometry, such as through the use of zigzag strips, grooves, and slotted airfoils. In this case, it is necessary to optimize the parameters, positional distribution, and other structural factors to obtain satisfactory performance without loss due to drag loss.^{11,12} According to Laffane *et al.*,^{13} adding vortex generators to an airfoil could reduce drag at high AOA, and enhance the maximum lift coefficient. For the overall performance of the airfoil, the lift–drag ratio could also be raised. Chamorro *et al.*^{14} discovered that installing riblets on wind turbine blades can minimize surface friction. Inspired by the pop-up of feathers on bird wings, Fang *et al**.*^{15} added flexible flaps to an airfoil, thereby effectively improving its aerodynamic performance and reducing the fluctuations in its lift coefficient.

The leading-edge protuberances (LEPs) of humpback whale flippers are the reasons for the high mobility and rotation ability of these animals, as shown in Fig. 1. The vortices generated by the LEPs generates lift forces.^{16} Numerical simulations of the application of wave-shaped protuberances to wings according to the bionic principle^{17} have shown that these structures do indeed improve aerodynamic characteristics.

In wind tunnel experiments, it was found that adding LEPs to an idealized airfoil would postpone the stall AOA by around 40% and reduce the drag coefficient.^{18} Post *et al.*^{19} compared the aerodynamic performances of various wings using oil surface flow visualization. For an airfoil with LEPs, the lift coefficient changed steadily without abrupt stall. Zhang *et al.*^{20} investigated flows in different sections of LEPs using particle image velocimetry. They found that the flow separation, transition, reattachment, and turbulent separation of flow in the trough section occurred in advance at small AOA. At high AOA, the attachment flow was maintained in the peak section, which improved the stall performance of the airfoil. A mathematical model for airflow simulation was constructed by Van Nierop *et al.*^{21} to investigate the impact on aerodynamic performance of LEP parameters, particularly the amplitude. Their study demonstrated that the amplitude had a significant effect on the maximum lift coefficient. The appropriate parameters of the LEPs must be selected according to the conditions of the airfoil, the application scenario, etc.^{22} Studies have found that appropriate selection of the main parameters of the LEPs, such as their width and length, can lead to significant changes in airfoil performance.^{23} When the amplitude of bionic LEPs is excessively large, it may degrade performance. According to Yan *et al.*,^{24} for airfoils with larger protuberance amplitude, stall occurred earlier than for the original airfoil, and while the lift–drag ratio was reduced dramatically, the lift coefficient decreased overall. Zhang *et al.*^{25} studied delta-wing UAVs with protuberances of various amplitudes and found that the optimal performance at small AOA was achieved with those of the largest amplitude However, the lift force was reduced compared with that of the baseline wing. Stall was reached earliest in the case of the largest-amplitude protuberances. The results of the above studies indicate that further investigation of bionic airfoils is required.

Custodio^{26} visualized the flow field around an airfoil with a wingspan four times the wavelength of its protuberance structure through staining experiments and found that the flow field exhibited a doubly periodic phenomenon. Fan *et al**.*^{27} showed through dye visualization experiments that there was a transition to doubly periodic flow on the suction side of an airfoil with LEPs. They also found that the interaction of two vortex pairs enhanced momentum exchange near the wall, thereby improving the stall phenomenon. Weber *et al.*^{28} reported that in the trough sections of LEPs, the pressure gradient was increased. As the AOA increased, multiple vortex structures appeared behind the protuberances. Kim *et al**.*^{29} used particle image velocimetry to analyze the underlying flow mechanisms responsible for the inhibition of flow separation by LEPs. They found that the hemispherical separation bubble at the trough formed a vortex pair and inhibited flow separation along the spanwise direction downstream. Cai^{30} examined the effects of LEPs by combining numerical simulations with experiments. The LEP-induced downstream flow vortices enhanced the flow in the boundary layer. The attached flow at the protuberance peak was maintained further downstream and played a role in the flow field segmentation. The studies by Pérez-Torró and Kim^{31} and Wei *et al.*^{32} came to a similar conclusion.

Thus, streamwise vortices generated between adjacent peaks are considered to be a possible mechanism for the effectiveness of LEPs. Vortex pairs generated by the LEPs mix downstream, bringing energy to the wall. By exerting an influence on the flow pattern within the boundary layer, this reduces drag and provides better post-stall characteristics. The influence on the wing of the vortices generated by LEPs is to some extent consistent with the effects of a passive wing fence, which can prevent the flow along the spanwise direction and thereby improve aerodynamic performance.^{33} At the stall AOA, LEPs on airfoils could improve performance with fewer drag penalties.

To ensure stable flight, particularly in the event of sudden gusts, it is necessary to study how a UAV wing achieves maximum aerodynamic performance. The flow mechanism of LEPs is complex, and additional experimental and simulation data are required to provide a firm basis for further investigation. Therefore, the aim of the present study is to explore the role of LEPs in wing stall and analyze the impact of changes in LEP parameters on stall control. Aerodynamic performance and vortex patterns are examined by numerical simulations of unmodified and bionic airfoils. Because the effect of the protuberance structure on stall control is related to the vortex structure around an airfoil, the behavior and distribution of the streamwise vortices are analyzed using an advanced vortex identification method, so as to provide a theoretical basis for the application of bionic LEPs to UAVs.

## II. MODEL AND NUMERICAL SIMULATION METHOD

### A. Airfoil models and grid generation

*x*

_{p},

*y*

_{p}) are the coordinates of the basic section, $(xp\u2032,yp\u2032)$ are those of the modified peak section,

*x*

_{q}is the abscissa at the maximum airfoil thickness, and

*A*is the amplitude. Here,

*p*= 1, 2, …,

*q*.

The appropriate wavelength and amplitude combination was picked according to the results of a previous study.^{34} For the bionic airfoil of types A1, the wavelength was set to 0.2*C*, and the amplitude to *A* = 0.02*C*. For the bionic airfoils of types B1, B2, and B3, the wavelength was set to 0.2*C*. To explore the influence of LEP amplitude, the amplitudes were set to *A* = 0.01*C*, 0.02*C*, and 0.04*C*, respectively. Figure 3 shows the section lines of different airfoils.

Figure 4 shows contour diagrams of the original and modified airfoils. The dimensions of each airfoil type are listed in Table I.

Model . | Type . | Chord C (mm)
. | Span l (mm)
. | Amplitude A (mm)
. | Wavelength λ (mm)
. | Sweep angle (°) . | Dihedral angle (°) . |
---|---|---|---|---|---|---|---|

Original | A | 115 | 268.5 | ⋯ | ⋯ | ⋯ | ⋯ |

A1 | 115 | 268.5 | 2.25 | 22.5 | ⋯ | ⋯ | |

Modified | B | 115 | 268.5 | ⋯ | ⋯ | 12.6 | 10 |

B1 | 115 | 268.5 | 1.125 | 22.5 | 12.6 | 10 | |

B2 | 115 | 268.5 | 2.25 | 22.5 | 12.6 | 10 | |

B3 | 115 | 268.5 | 4.5 | 22.5 | 12.6 | 10 |

Model . | Type . | Chord C (mm)
. | Span l (mm)
. | Amplitude A (mm)
. | Wavelength λ (mm)
. | Sweep angle (°) . | Dihedral angle (°) . |
---|---|---|---|---|---|---|---|

Original | A | 115 | 268.5 | ⋯ | ⋯ | ⋯ | ⋯ |

A1 | 115 | 268.5 | 2.25 | 22.5 | ⋯ | ⋯ | |

Modified | B | 115 | 268.5 | ⋯ | ⋯ | 12.6 | 10 |

B1 | 115 | 268.5 | 1.125 | 22.5 | 12.6 | 10 | |

B2 | 115 | 268.5 | 2.25 | 22.5 | 12.6 | 10 | |

B3 | 115 | 268.5 | 4.5 | 22.5 | 12.6 | 10 |

The calculation domain was sufficiently large to ensure adequate flow development. Considering calculational resources and time, the calculation domain inlet was taken to be 15*C* from the airfoil, the outlet to be 25*C* from the airfoil, and the spanwise width to be twice the wingspan length (Fig. 5).

### B. Turbulence model and numerical scheme setting

*k*-

*ω*model was adopted because of its good fitting effect for boundary layer flow separation. The transmission equation and turbulent kinetic energy equation were taken as follows:

*S*is the absolute value of the vorticity,

*μ*is the turbulent dynamic viscosity, and

*β**,

*σ*

_{ω}, and

*β*are constants. The expressions for the blending functions

*F*

_{1}and

*F*

_{2}are as follows:

The Reynolds number was 1.80 × 10^{5}, and the average chord length for calculating the aerodynamic coefficient was selected as the characteristic length. Air was selected as the flow medium in the numerical simulation. The inlet flow velocity for this study was 30 m/s. Both temperature variations and compressibility were ignored. The outlet was set as free flow. The walls on both sides were set as symmetric surfaces, and the wing surfaces were set as nonslip walls. For a bionic airfoil with LEP structures on the leading edge, it is very important to accurately solve for the boundary layer flow state on the LEP surface. Therefore, the pressure and velocity were determined using the SIMPLEC algorithm.

A structured grid method was used to divide the flow field. The grid height of the wing surface was adjusted to 1 × 10^{−5} m to ensure that the *y*^{+} value met the requirements of the calculation model. The grid near the wing surface was encrypted to ensure that the details of the boundary layer flow changes could be captured. Bionic LEPs control airflow by changing its direction and magnitude, and therefore it was necessary to simulate the flow at an LEP with a dense grid structure. The specific grid structure and local grid amplification are shown in Fig. 6.

### C. Grid-independence test

*h*is given by

*v*

_{i}is the space occupied by grid

*i*in the domain and

*N*is the number of independent grids in the domain.

*r*was ∼1.3. For structured grids, the grid refinement requirements need to be satisfied globally such that

*h*

_{1}<

*h*

_{2}<

*h*

_{3}. The apparent order

*p*was then calculated as follows:

*ɛ*

_{21}=

*ϕ*

_{2}−

*ϕ*

_{1}and

*ɛ*

_{32}=

*ϕ*

_{3}−

*ϕ*

_{2}. The calculated apparent order was the same as that chosen for the study, and thus the grid could be regarded as satisfactory. The extrapolated $\varphi ext21$ and $\varphi ext32$ were calculated as follows:

The numbers of grid nodes in the three groups using the Richardson extrapolation method for verification were 14.45 × 10^{6}, 6.42 × 10^{6}, and 2.95 × 10^{6}. The lift and drag coefficients were determined as representative airfoil performance parameters for grid evaluation. Table II lists the variations in airfoil performance with increasing number of grid nodes. From a comparison of the required calculational accuracy and the available calculational resources, the grid with 6.42 × 10^{6} nodes was selected for the study.

Parameter . | C_{l}
. | C_{d}
. |
---|---|---|

N_{1}, N_{2}, N_{3} (million) | 14.45, 6.42, 2.95 | 14.45, 6.42, 2.95 |

ϕ_{1}, ϕ_{2}, ϕ_{3} | 1.6156, 1.6181, 1.6193 | 0.2101, 0.2110, 0.2127 |

r_{21}, r_{32} | 1.31, 1.30 | 1.31, 1.30 |

ɛ_{21}, ɛ_{32} | 0.0423, 0.0217 | 0.0163, 0.0285 |

p | 0.3732 | 0.3674 |

$\varphi ext21$ | 1.6119 | 0.2087 |

$ea21$ | 0.154% | 0.455% |

$eext21$ | 0.229% | 0.582% |

$GCIfine21$ | 0.286% | 0.848% |

Parameter . | C_{l}
. | C_{d}
. |
---|---|---|

N_{1}, N_{2}, N_{3} (million) | 14.45, 6.42, 2.95 | 14.45, 6.42, 2.95 |

ϕ_{1}, ϕ_{2}, ϕ_{3} | 1.6156, 1.6181, 1.6193 | 0.2101, 0.2110, 0.2127 |

r_{21}, r_{32} | 1.31, 1.30 | 1.31, 1.30 |

ɛ_{21}, ɛ_{32} | 0.0423, 0.0217 | 0.0163, 0.0285 |

p | 0.3732 | 0.3674 |

$\varphi ext21$ | 1.6119 | 0.2087 |

$ea21$ | 0.154% | 0.455% |

$eext21$ | 0.229% | 0.582% |

$GCIfine21$ | 0.286% | 0.848% |

Numerical simulations of the S1223 airfoil at attack angles of 0°, 8°, 12°, 16°, 18°, and 20° were performed using a grid of 6.42 × 10^{6} nodes. Figure 7 shows a comparison of the results with experimental data at the same Reynolds number.^{35} The numerical simulation was simplified by calculating the AOA separately, which gave results different from the experimental conditions. During the experiment, the AOA was varied continuously to obtain the lift coefficient. The flow history affects the experimental conditions. Although the numerical simulation result for the stall point differed from the experimental data, the curve fitting was satisfactory, and the relative error was within a reasonable range.

## III. CALCULATION RESULTS AND ANALYSIS

### A. Performance comparison between bionic airfoil and original airfoil

The aerodynamic performances of various airfoils at different AOA obtained from the numerical simulations are shown in Fig. 8. At small AOA, the lift coefficients are almost the same for all airfoils and they differ only slightly with increasing AOA until stall occurs. The drag coefficients of the modified airfoils of types B and B2 are significantly lower than those of the original airfoils of types A and A1. The performance trends of different airfoils vary similarly with the AOA, and as the AOA increases, a stall occurs for each airfoil.

However, the AOA and performance variations after stall differ between the airfoils. The peak lift coefficient of the bionic airfoil of type A1 is lower than that of the original airfoil of type A, with a 2° drop in the forward stall AOA, but the stall state of the type A1 airfoil is not evident, and the lift coefficient variation is relatively minimal after the stall. When stall occurs, the lift coefficient will decrease by 50%. Bionic LEPs can effectively enhance the performance of an airfoil after stall, preventing the problems caused by the sudden drop in lift. For the type B modified airfoil, there is a sharp increase in drag coefficient when it enters the stall stage, while the type B2 bionic airfoil with LEPs exhibits excellent performance. The LEPs not only improve the aerodynamic performance, but also inhibit stall development. When the AOA exceeds 16°, the lift coefficient of the modified airfoil increases by 20%, indicating that the LEPs have significantly improved the airfoil function.

Working conditions at AOA of 12°, 18°, and 22° are selected for comparison, and the flow characteristics around the bionic airfoil and their influence on the stall are analyzed. Figure 9 shows pressure nephograms and limiting streamlines of the airfoil suction surface. Transverse red dashed lines represent the boundary between the flow attachment and flow separation zones, and longitudinal dashed red lines represents the changes in the streamline caused by the dihedral angle. It can be seen that at 12° AOA, the addition of LEPs has a slight effect compared with the original airfoil (type A), although the lift and drag coefficients change only slightly. As the airfoil enters stall, there is strong flow separation at the root. However, the effect of the bionic LEPs is not obvious. Compared with the original airfoil (type A), the flow separation zone of the type A1 bionic airfoil is significantly larger. The flow attachment zone at the wingtip of the original airfoil (type A) remains large at an AOA of 18°, owing to the wingtip effect, which corresponds to the lift coefficient curve presented in Fig. 8(a). At high AOA, the boundary between the flow separation and attachment zones of the bionic airfoil exhibits a wavy variation. The amplitude of the flow boundary curve on the suction surface is greater than that of the LEP structure. The effects of the LEPs will be amplified in the flow direction.

The limiting streamline provides a representation of the flow state of the airfoil. The flow separation moves continuously to the leading edge, and the area of the separation zone increases. LEPs interfere with the streamline and delay the advance of the separation boundary to the leading edge. Significant periodic and symmetrical changes occur near the boundary between the flow attachment and separation zones of the type B2bionic airfoil. When flow separation occurs, the peak section of an LEP divides the flow field. The fluid separates from the peak section to both sides and gathers at the trough section, forming a pair of reversely rotating backflow vortices within a single-wavelength range. Flow separation begins from the position of the LEP trough section corresponding to the trailing edge, whereas the flow adheres to the airfoil at the corresponding section, thereby delaying flow separation.

The vortex pair generated by the LEPs of the type B2 bionic airfoil prevent the vortex formation that occurs at the wing root of the type B modified airfoil at a large AOA. When the AOA reaches 22°, the suction surface flow becomes complex, and the limiting streamline shape is more disordered in the case of the type A1 bionic airfoil. However, the LEPs effectively inhibit the development of flow separation, improve airfoil performance after stall, and reduce the area of the flow separation zone. Comparison between the streamline morphologies at AOA of 18° and 22° shows that the wingtip effect and flow vortices caused by the protuberances interact with each other, effectively restraining large-scale flow separation. Flow attachment vortices appears alternately in the saddle part, modifying the airfoil flow structure and it more complex.

From an analysis of the flow field for the bionic airfoil with LEPs, it can be seen that the LEPs play a significant role in dividing the flow field. This causes the flow separation zones to gradually transition from front and back distributions to the spanwise direction distribution of the original airfoil, which is reflected when the flow field structure changes. Because the LEP is generated by a sine curve, it exhibits periodicity and symmetry. A cross-section of the type B2 bionic airfoil at a single wavelength is selected here for an analysis of streamlines. The distances from the selected sections to the wing root are *x* = 0.0911, 0.1025, and 0.1139 m.

Figure 10 shows the cross-sectional streamlines of a single wavelength of the type B2 bionic airfoil. The cross-sections at *x* = 0.0911 and 0.1139 m from the wing root are the trough sections. It is worth noting that the area and intensity of the flow separation zone in the peak section are smaller than those in the trough section. For small AOA, the flow development is stable, adhering to the suction surface. At an AOA of 18°, a trailing-edge vortex starts to form in the trough section, and its interaction with the backflow vortex generated in the separation zone leads to a decrease in airfoil performance. However, the attached flow is still maintained in the peak section. The airfoil attains a complete stall state at large AOA, and flow separation occurs at the front edge owing to the increased AOA. Moreover, the flow cannot be completely bounded between adjacent protuberances. However, the adjacent protuberance section still plays a role in suppressing flow separation.

The flow separation vortex structure is significantly smaller than that corresponding to the peak section, and the flow separation point is delayed. This also corresponds to the wavy shape of the boundary line for flow separation in Fig. 9. At the airfoil trailing edge, there is an increase in the size of the vortex structure with increasing AOA. The fluid on the trough section surface is squeezed by the fluid on both sides of the peak section and as a result of the limiting streamline and pressure distribution, and the flow obtains a specific amount of energy to resist the effect of an inverse pressure gradient. Thus, flow separation maintains a certain degree of attachment, and separation at the trailing edge is decreased significantly.

### B. Effect of LEP amplitude on aerodynamic performance

Figure 8 has shown that for the type B2 bionic airfoil, the drag coefficient at large AOA is significantly lower than that of the type B modified airfoil. To further examine how LEPs affect the performance of the airfoil, three models with different protuberance amplitudes based on the type B modified airfoil were numerically simulated. A comparison between aerodynamic coefficients for the different amplitudes is presented in Fig. 11. At small AOA, bionic airfoils with different LEP amplitudes exhibited similar trends in lift coefficient development. The type B2 bionic airfoil has a maximum lift coefficient of ∼1.68 as the AOA reaches 16°, which is 15.56% higher compared with type B. The lift coefficient of the type B3 bionic airfoil decreases the most, dropping rapidly to ∼1.2 after stall. For the type B1 and B2 bionic airfoils, the lift coefficients also decrease. However, these decreases are less significant than that of the type B3 bionic airfoil. The growth trends of the drag coefficients of the bionic airfoils with three different amplitudes are similar to that of the type B modified airfoil. After the AOA exceeds 12°, the drag coefficient decreases for all AOA, and the drag reduction function of the bionic LEPs is obvious compared with the smooth leading edge airfoil. In addition, the drag coefficients of all the airfoils increase rapidly after stall.

*C*

_{lB}and

*C*

_{dB}are the lift and drag coefficients, respectively, of the type B modified airfoil at the same AOA. The changes in bionic airfoil performance can thus compared with those of the modified airfoil.

As can be seen from Fig. 12, the relative lift coefficient indicates that the type B1 and B2 bionic airfoils exhibit a slight increase in lift compared with the type B modified airfoil before the AOA reaches 12°, whereas the type B3 bionic airfoil exhibits an initial increase, followed by a decrease at an AOA of 8°, with significant fluctuations. When the AOA reaches 16°, the relative lift coefficients of all three bionic airfoils increase by ∼15%. The performance of the type B3 bionic airfoil then quickly decreases to become poorer than that of the type B modified airfoil.

The relative drag coefficients in Fig. 12 indicate that the LEPs cause a slight increase in drag. When the AOA exceeds 12°, the relative drag coefficients decrease rapidly, with the type B1 bionic airfoil exhibiting the most significant change among the three airfoils. The type B3 bionic airfoil exhibits a drag reduction effect, but this is slightly less than those of the type B1 and B2 bionic airfoils. Overall, significant post-stall improvements are achieved with the three bionic airfoils. Change in the amplitude of the LEPs do not prevent the occurrence of stall, but the post-stall lift and drag values are greatly affected. Bionic LEPs have an obvious drag reduction effect and can prevent a rapid increase in the drag coefficient after airfoil stall.

As can be seen from Fig. 13, for an AOA of 12°, the flow fields of all the different airfoils vary periodically at the trailing edge, and the protuberances have a restraining effect. However, the flow separation zone of the type B3 bionic airfoil with large protuberance amplitude extends further toward the leading edge compared with the type B1 bionic airfoil with a small protuberance amplitude. When the AOA reaches 16°, a vortex pair structure with opposite directions of rotation is formed near the interface of the separation zones of the type B2 and B3 bionic airfoils and develops from there toward the airfoil trailing edge. The counter-rotating vortex pair causes the free fluid in the flow field to be pulled toward the suction surface. Thus, the reattached flow field delays the separation time. At an AOA of 22°, the influence of small-amplitude protuberances is minimal, because the interface of the flow separation zone has become smooth. The vortex on the type B1 bionic airfoil surface has disappeared, and the flow field near the wing root has formed a powerful vortex. At large AOA, although the separation zone of the type B2 bionic airfoil has advanced toward the leading edge, the LEPs still play a role in dividing the flow and reattaching it to the airfoil surface. On the type B3 bionic airfoil, the suction surface pressure increases rapidly increased, and the flow on most of the airfoil surface is disrupted and already in a separated state. This results in a significant decrease in performance and makes it difficult for the required lift to be provided. The surface streamline of the type B3 bionic airfoil is relatively chaotic. The high-pressure region on the upper surface has extended to the leading edge, forming a relatively complex flow-attachment vortex structure near the wing root. This is another reason for the decline in performance.

### C. Analysis of protuberances by vortex structure using Liutex

^{36,37}The definition of the Liutex vector comes from the mathematical derivation of a special form of the velocity gradient tensor that describes the strength and direction of the vortex. The Liutex vector represents the magnitude of the vortex structures and is not contaminated by tensile compression or shear terms.

^{38,39}In the present study, the changes in the flow around the airfoil before and after the addition of LEPs are examined by comparing vortex structures extracted by the Liutex method. The rigid rotational component of any part of the fluid in the flow field is represented by Liutex vector, which conforms to the right-hand rule. The magnitude of the Liutex vector is twice the angular velocity, expressed as follows:

^{39}

**r**is the real eigenvector of the velocity gradient tensor ∇

**V**,

**ω**is the vorticity,

*λ*

_{ci}is the imaginary part of ∇

**V**, and

*λ*

_{r}is the real eigenvalue of ∇

**V**.

The Liutex method directly defines the vortex from the theoretical derivation, and its size and direction have specific physical significance. The colors on the isosurfaces in Fig. 14 represent the changes in velocity in the flow direction. The vortex structure of the modified airfoil is not evident when the AOA is small, and a strong vortex structure appears only at the tip. A vortex appears in the wake of the bionic airfoil, and the direction of the vortex structure is the same as the flow direction, corresponding to the LEP trough section. Before the airfoil stalls, the vortex structure continues to develop in the flow direction with increasing AOA. Owing to the strong and complex flow, mixing with the incoming flow coming from the upstream direction, ellipsoidal vortex formation occurs at the trailing edge. Subsequently, the vortex structure continues to evolve, before gradually dissipating, and the extent of the flow separation zone gradually expands.

At an AOA of 18°, there is a significant difference in the flow separation state between the two types of airfoil. The bionic airfoils exhibit less severe separation than the modified airfoil (Fig. 15). The flow field is divided into numerous relatively independent zones, which inhibits the development of separation on the surface. The original airfoil enters a stall state, and there is a large area of flow separation on the airfoil as the AOA increases. The vortex structure in the flow direction behind the airfoil becomes complex, and the boundary between the flow attachment and flow separation zones gradually approaches the leading edge. At an AOA of 20°, the flow is deflected between adjacent protuberances on the bionic wing surface, as revealed by the vortex structure. Many broken vortex structures with irregular distributions are formed on the wing surface, indicating significant flow separation. When the AOA is 22°, the vortex structures near the bionic airfoil surface form again into a regular distribution. For the type B airfoil, the stall state deepens and the flow separation becomes more intense.

## IV. CONCLUSION

This work has used numerical simulation to examine the stall control effect of several bionic airfoils incorporating leading-edge modifications based on the structures on humpback whale flippers. The impacts of the modification technique and the amplitude of the protuberance structures on flow performance have been examined. Additionally, the vortex structure generated by the bionic LEPs has been analyzed using the Liutex vector, and the mechanisms by which LEPs exert flow control have been studied.

We have confirmed that the application of LEPs to airfoils can effectively increase the lift coefficient. On a modified airfoil, the LEPs can reduce the drag penalty after stall and significantly improve the aerodynamic properties of the airfoil. The lift coefficient can be increased by ∼11%. The protuberance structures minimize the risk of damage caused by stalling and broaden the operating range of the airfoil.

With the LEP modifications, it is possible to bring the center of gravity of the airfoil closer to the middle of the fuselage and improve stability during flight. The numerical simulations of bionic airfoils with various protuberance amplitudes have shown that an appropriate amplitude matching the flow condition can effectively improve lift and drag characteristics. Additionally, the LEPs significantly reduce the rapid increase in the drag coefficient that would otherwise occur following stall. They also increase the maximum relative lift coefficient by 16% and decrease the maximum relative drag coefficient by 17%.

Vortex structures have compared and analyzed using the Liutex vector. The results suggest that the bionic LEPs limit flow separation by cutting the flow into relatively independent regions. The presence of irregular broken vortices on the wing surface suggests that the LEPs cause significant disturbances to the boundary layer flow and promote momentum exchange. Moreover, the vortex structure and flow separation zone are bounded between adjacent protuberance sections, thereby effectively reducing flow separation.

## ACKNOWLEDGMENTS

This study was supported by the Natural Science Foundation of Heilongjiang Province (No. YQ2023E015).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

All authors have read and agreed to the published version of the manuscript.

**Xuntong Wei**: Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Deyou Li**: Conceptualization (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal). **Siqi Li**: Data curation (equal). **Hong Chang**: Formal analysis (equal); Investigation (equal). **Xiaolong Fu**: Methodology (equal). **Zhigang Zuo**: Methodology (equal). **Hongjie Wang**: Software (equal); Supervision (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.