Aerodynamic optimization design of low reynolds number aerofoils based on induced laminar separation

At low Reynolds numbers, the design of aerofoils with a high lift-to-drag ratio (RLD) is crucial for enhancing the aerodynamic performance of Micro Air Vehicles (MAVs). However, the transition from laminar to turbulent flow plays a dominant role in the aerodynamic performance of aerofoils at Low-Reynolds-Numbers (LRN). The laminar separation bubbles formed in laminar flow alter the effective aerofoil shape, reducing aerodynamic efficiency. Additionally, predicting the shape and location of the laminar separation bubbles becomes challenging due to its sensitivity to the flight environment. To design aerofoils suitable for low Reynolds number MAV applications, we optimize a high RLD aerofoil by inducing transition to improve the effective aerofoil shape in the flow field. The study reveals that excessive curvature can suppress the formation of laminar separation bubbles at the leading edge, redirecting separation to the midsection. This results in reduced aerofoil drag and enables a high lift distribution at the leading edge.


Introduction
Micro Unmanned Aerial Vehicles (MAVs) have garnered widespread attention across various industries due to their small size, lightweight, relatively low manufacturing costs, good concealment, and ease of portability [1,2] .As electronic devices continue to integrate and miniaturize, UAVs are transitioning towards micro-scale development.However, with the reduction in size, flight conditions are moving towards low Reynolds numbers (Re < 1 e5).
One of the most significant characteristics in low Reynolds number environments is the laminar separation bubbles, as shown in Figure 1.Since the discovery of laminar separation bubbles, extensive research has been conducted [3][4][5][6] : in low Reynolds numbers, the flow over the wing surface initially experiences laminar flow.In the laminar flow process, the fluid will gradually separate under the influence of adverse pressure gradients, subsequently transitioning into turbulent flow.It then reattaches to the surface in a turbulent manner, and the transition location forms a laminar separation bubble (LSB), as illustrated in the diagram below.
The presence of laminar separation bubbles alters the effective shape of the aerofoil, impacting aerodynamic performance by typically increasing drag and consequently reducing aerodynamic efficiency in a detrimental manner.The existence of laminar separation bubbles has a significant impact on the performance of equipment.Understanding the physical properties of laminar separation bubbles and possible methods to control them is a prerequisite for the effective design of these aerodynamic devices.After the separation of the laminar boundary layer flow, an extremely unstable separated shear plane is formed, leading to a transition to turbulence within the separated shear plane.Enhanced momentum transfer in turbulent flow typically facilitates reaffixment, and then a turbulent boundary layer is formed downstream.From the point of view of time homogenization, there is a "dead zone" region beneath the separated shear plane after separation, altering the effective external shape.The generation of laminar separation bubbles generally has adverse effects on aerofoil performance [7] .From a design perspective, Reynolds-Averaged Navier-Stokes (RANS) numerical simulations represent a reasonable trade-off between computational accuracy and cost [8] .Common turbulence models include the one-equation SA turbulence model, two-equation k-ε model, k-ω model [9] , and a low Reynolds number modification model for the k-ω SST model, as well as the three-equation k-kl-ω model.For optimization, stable, reliable, and fast-solving techniques are crucial for optimizing efficiency and results.
Common aerofoil description methods include the Bezier curve, B-spline curve, Non-Uniform Rational B-spline (NURBS) surfaces, Hicks-Henne bump functions [10] , Trigonometric functions, Specific discipline functions, basis vectors, and FFD.Parametric description methods have undergone significant development, and optimizing low Reynolds numbers requires consideration of their specificity.

Parametric modeling approach
Optimizing low Reynolds number aerofoils must consider the adverse effects of laminar separation bubbles.Some studies suggest that in aerofoil shape analysis, the transition location from laminar to turbulent flow is extremely sensitive even to very subtle shape deformations.Changes in the transition location, coupled with alterations in the effective external shape caused by laminar separation bubbles, complicate the optimization of aerofoil designs.This sensitivity necessitates careful consideration in selecting the parametric description and optimization algorithm for aerofoils.The use of control points for shape constraints during the optimization process results in smoother shape variations, mitigating the drastic changes in shape that could lead to significant differences in solution outcomes and, consequently, significant interference in the analysis of optimization results by optimization algorithms.The choice for parameterizing aerofoil descriptions adopts the Bezier-PARSEC description method proposed by Derksen et al. [11] , denoted as BP3434.The PARSEC variables are used as parameters of the BP parameterization, defining four independent Bezier curves.These four curves are used to describe the curvature of the aerofoil in terms of camber and the distribution of thickness along the chord, both in the leading and trailing edges.
Parameters include the radius of the leading edge (R le ), the sweep angle of the trailing edge (α te ), the wedge angle of the trailing edge (β te ), the vertical displacement of the trailing edge (Z te ), the direction of the leading edge (γ le ), the position of maximum camber (X c , Y c ), curvature peak curvature (K c ), the position of maximum thickness (X t , Y t ), maximum thickness (K t ), half-thickness of the trailing edge (d zte ), as well as several Bezier related parameter variables, as shown in Figure 2. The aerofoil, for example, mainly involves the BP3333 and BP3434 description methods.BP3333 divides the thickness function and curvature function into two parts, using the maximum camber position as the boundary for the curvature function and the maximum thickness position as the boundary for the thickness function.It employs four 3rd-order Bezier curves to describe these sections, expressed as:  The most distinctive difference between BP3434 and BP3333 descriptions lies in the segmentation of the thickness and curvature functions into front and rear sections.The front half utilizes a 3rd-order Bezier curve, while the rear half uses a 4th-order Bezier curve.This description method introduces an additional control point at the trailing edge to allow for more deformations in the rear section, covering a broader range of shapes.For a 4th-order Bezier curve, the expression is: The schematic diagram of the aerofoil description is shown in Figure 2(b).
Research indicates that this method can effectively fit various publicly available aerofoil shapes, demonstrating good aerofoil coverage to meet the requirements of the parameter description in this study.To better fit the aerodynamic characteristics of low Reynolds number aerofoils, the main design parameters include maximum thickness and position, maximum thickness tangent control, maximum camber and position, trailing-edge angle and trailing-edge reflex control, leading-edge curvature control, and trailing-edge height.To account for the actual processing of the wing trailing edge, a 2 mm trailingedge cut thickness is set at the root chord, as this thickness, based on processing experience, ensures a relatively low level of difficulty in processing common composite materials and maximizes processing accuracy.

Simulation for the incident flow
The high sensitivity of laminar separation bubbles requires high precision in computational solutions.To reduce the introduction of solution errors, direct solving is chosen instead of using surrogate models for higher-fidelity optimization calculations.Considering the trade-off between solution accuracy and computational cost, the numerical simulations of RANS are selected.This basic equation is satisfied: Where U represents the vector of velocity, and f represents the vector of body force.The k equation and ω equation of the SST model are as follows: Grid independence verification primarily focuses on global grid size and the maximum line size on the aerofoil surface.Results for different grid scales are shown in Figure 3: In the figure, experimental data points near the cruising condition are selected as a basis for comparing resistance results under different grids.Grid A has 64,000 cells, Grid B has 80,000 cells, and both A and B grid solutions slightly exceed experimental values.Grids C and D have 112,000 and 158,000 cells, respectively, with results closely matching the experimental data.The grid size and quantity of Grid C are deemed sufficient for the solution, and with further grid growth, the solution results show little difference.Therefore, the grid size of Grid C is selected as the template for the optimization solution.An unstructured grid is employed, with a maximum surface grid size of 80 mm, a maximum line grid size of 0.5 mm, front and rear edge grid sizes of 0.1 mm, and a line grid growth rate of 1.05, with a chord length of 150 mm.
Regarding optimization algorithms, to ensure global optimization to address the aerodynamic variations caused by laminar separation bubbles at low Reynolds numbers, the mature NAGA-II genetic algorithm is adopted as the optimization intelligent algorithm.This algorithm introduces an elite retention strategy to enhance the efficiency of using process data during optimization.It performs Pareto sorting on the characteristic targets of optimization results for a comprehensive evaluation.The population size is set at 30, with 30 evolution iterations, retaining five elites in each iteration.The probability of crossover is set at about 30%, and the probability of mutation is set at about 20%.

Analysis of optimization results
The post-optimized aerofoil exhibits relatively small thickness, significant camber, and a segmented distribution in its external shape.Using the maximum camber position as the dividing point, the front and rear sections appear relatively flat, while the transition position features a pronounced curvature.The external contour and the distribution of pressure points on the aerofoil surfaces at various angles of attack are illustrated in Figure 4.The figure presents the distribution of pressure points on the aerofoil surface at angles of attack of 1°, 3°, 5.5° (maximum R LD point), and 8° in the incident flow.On the negative pressure side, a distinct stepwise distribution becomes apparent with increasing angles of attack.This phenomenon occurs after the maximum camber position and moves towards the leading edge as the angle of attack increases.The pressure field contour and laminar separation bubble locations are depicted in Figure 5.

Figure 5 (d).
At an incident flow angle of 8°, the laminar separation bubble continues to advance, and trailing edge separation becomes more pronounced.Due to the significant camber of the aerofoil, separation vortices occur on the pressure surface at small angles of attack, leading to pressure losses and a relatively small lift coefficient.With the increase of the angle of attack, the leading-edge stagnation point moves downward, resulting in gradual flow attachment on the pressure surface, and a flat pressure distribution occurs before the maximum camber position.Laminar separation bubbles begin to appear on the suction surface, initially close to the midsection, approximately at 45.3% of the chord length.The initial laminar separation bubbles are relatively small, with a length of only 4.67% of the chord length, causing minimal changes to the effective pressure distribution.

Mechanism of induced LSB
Subsequently, the laminar separation bubbles begin to shift forward.When the angle of attack increases to 5.5°, the laminar separation bubbles are located at approximately 42% of the chord length, and its shape changes, with a length twice that of the 3° angle of attack, reaching the maximum R LD .At this point, turbulent separation starts at the trailing edge.When the angle of attack increases to 8°, the laminar separation bubbles approach the maximum camber position, and turbulent separation at the trailing edge becomes more pronounced.
After passing through the maximum camber position, the incoming flow struggles to maintain a laminar state, and the transition location, as well as the shape of the LSBs, are influenced by the maximum camber position.Conventional aerofoil shapes show a smoother transition in the external shape, causing the laminar separation bubbles' position to be closer to the leading edge for thin aerofoils.In contrast, the aerofoil optimized by using the BP3434 description method exhibits a distinctive external shape, particularly with a large smooth transition curvature at the transition position.
The significant transition curvature plays an inducing role in separation.Conventional thin aerofoils often experience separation starting from the leading edge, with the laminar separation bubbles moving aft with increasing angle of attack.The optimized aerofoil improves the pressure distribution by inducing the formation of laminar separation bubbles.The original publicly available aerofoil has relatively flat transitions on both sides of surfaces.As thickness increases, the position of laminar separation bubbles also gradually moves aft.The segmented characteristics represented by the optimized aerofoil indicate that the laminar separation bubbles' position is influenced by the external shape, deviating from conventional conclusions.Overall, both the lift coefficient and R LD are significantly improved.
Enhancing the R LD effectively reduces power consumption during level flight.Increasing the lift coefficient can potentially lower cruise speed, improving the efficiency of the propulsion system.Additionally, a reduced cruise angle of attack can lower the risk of stalling.

Conclusion
Through optimization design, we have obtained an aerofoil with a high lift coefficient and R LD .Compared to conventional aerofoils, the optimized aerofoil exhibits segmented features, with the maximum camber position as the boundary, where both the front and rear surfaces are relatively flat.The front portion features a laminar attached flow, resulting in a smoother flow, while the transition occurs in the rear section.Studies indicate that inducing laminar separation bubbles can be achieved by altering the suction surface curvature, improving the R LD at the cruising point.

2 Figure 1 .
Figure 1.Laminar flow separation and transition (LSB).After the separation of the laminar boundary layer flow, an extremely unstable separated shear plane is formed, leading to a transition to turbulence within the separated shear plane.Enhanced momentum transfer in turbulent flow typically facilitates reaffixment, and then a turbulent boundary layer is formed downstream.From the point of view of time homogenization, there is a "dead zone" region beneath the separated shear plane after separation, altering the effective external shape.The generation of laminar separation bubbles generally has adverse effects on aerofoil performance[7] .From a design perspective, Reynolds-Averaged Navier-Stokes (RANS) numerical simulations represent a reasonable trade-off between computational accuracy and cost[8] .Common turbulence models include the one-equation SA turbulence model, two-equation k-ε model, k-ω model[9] , and a low Reynolds number modification model for the k-ω SST model, as well as the three-equation k-kl-ω model.For optimization, stable, reliable, and fast-solving techniques are crucial for optimizing efficiency and results.Common aerofoil description methods include the Bezier curve, B-spline curve, Non-Uniform Rational B-spline (NURBS) surfaces, Hicks-Henne bump functions[10] , Trigonometric functions, Specific discipline functions, basis vectors, and FFD.Parametric description methods have undergone significant development, and optimizing low Reynolds numbers requires consideration of their specificity.

Figure 2 (
Figure 2 (a).Description method for BP3333.Figure 2 (b).Description method for BP3434.In the figure, subscripts represent the positions of control points, and the schematic diagram of the aerofoil description is shown in the diagram.The most distinctive difference between BP3434 and BP3333 descriptions lies in the segmentation of the thickness and curvature functions into front and rear sections.The front half utilizes a 3rd-order Bezier curve, while the rear half uses a 4th-order Bezier curve.This description method introduces an additional control point at the trailing edge to allow for more deformations in the rear section, covering a broader range of shapes.For a 4th-order Bezier curve, the expression is:

Figure 3 .
Figure 3. Grid independence validation, comparison of solutions with different grid scales.In the figure, experimental data points near the cruising condition are selected as a basis for comparing resistance results under different grids.Grid A has 64,000 cells, Grid B has 80,000 cells, and both A and B grid solutions slightly exceed experimental values.Grids C and D have 112,000 and 158,000 cells, respectively, with results closely matching the experimental data.The grid size and quantity of Grid C are deemed sufficient for the solution, and with further grid growth, the solution results show little difference.Therefore, the grid size of Grid C is selected as the template for the optimization solution.An unstructured grid is employed, with a maximum surface grid size of 80 mm, a maximum line grid size of 0.5 mm, front and rear edge grid sizes of 0.1 mm, and a line grid growth rate of 1.05, with a chord length of 150 mm.Regarding optimization algorithms, to ensure global optimization to address the aerodynamic variations caused by laminar separation bubbles at low Reynolds numbers, the mature NAGA-II genetic algorithm is adopted as the optimization intelligent algorithm.This algorithm introduces an elite

Figure 4 (
Figure 4 (a).Aerofoil shape illustration, including surface distribution of pressure at various angles of attack.

Figure 5 (
Figure 5 (a).At an incident flow angle of 1°, the laminar-separation bubble appears on the pressure surface.

Figure 5 ( 6 Figure 5
Figure 5 (b).At an incident flow angle of 3°, the LSB appears on the suction surface near the midchord.