A stepped airfoil and method of designing the same

Abstract

This application relates to the field of airfoil design technology, specifically to a stepped airfoil and its design method. A recessed step with a smooth curved surface is provided in the middle of the lower surface of the airfoil. Along the leading and trailing edges of the stepped airfoil, the center point P of the recessed step is located between 0.25C and 0.8C, the depth D of the recessed step ranges from 0.005C to 0.02C, and the aspect ratio L / W of the recessed step is (2:1) to (4:1), where C is the chord length of the stepped airfoil, L is the length of the recessed step, and W is the width of the recessed step, with W ranging from 0.02C to 0.05C. This stepped airfoil can achieve a lift-to-drag ratio improvement of over 15% at the design angle of attack and exhibits good performance gains within the design angle of attack range. The design method, combining the Hua Luogeng optimization method and the coordinate rotation method, can achieve multi-parameter optimization without calculating gradients, effectively solving multi-parameter airfoil optimization problems with fast convergence speed and high computational efficiency.

Description

Technical Field

[0001] This application relates to the field of airfoil design technology, specifically to a stepped airfoil and its design method. Background Technology

[0002] Optimizing the aerodynamic performance of airfoils is a core technical issue in the fields of aerospace and wind energy utilization.

[0003] Cavity steps are a passive flow control technique that uses specific geometrically shaped depressions on the airfoil surface to effectively improve boundary layer characteristics, delay flow separation, and thus enhance the airfoil's lift-to-drag ratio. However, traditional cavity step airfoil designs rely heavily on empirical formulas and extensive wind tunnel testing, resulting in long development cycles and high costs. Summary of the Invention

[0004] This application provides a stepped airfoil and its design method, which can efficiently and quickly optimize the airfoil, thereby improving the lift-to-drag ratio of the obtained airfoil and providing good performance gains within the design angle of attack range.

[0005] In a first aspect, this application provides a stepped airfoil having an upper surface and a lower surface, wherein a recessed step with a smooth curved surface is provided in the middle of the lower surface; Along the direction from the leading edge to the trailing edge of the stepped airfoil, the center point P of the recessed step is located between 0.25C and 0.8C, the depth D of the recessed step ranges from 0.005C to 0.02C, and the aspect ratio L / W of the recessed step is (2:1) to (4:1), where C is the chord length of the stepped airfoil, L is the length of the recessed step, W is the width of the recessed step, and W ranges from 0.02C to 0.05C.

[0006] In a preferred embodiment, a recessed step is provided on the lower surface of the NACA4412 airfoil as a reference.

[0007] In a preferred embodiment, the center point P of the recessed step is located between 0.5°C and 0.7°C.

[0008] In a preferred embodiment, the contour curve of the recessed step is generated using a cosine function, specifically y(ξ)=D×sin2(πξ)×[1 0.3×sin(2πξ)], where ξ = (x - x_start) / L, and x_start is the starting position of the concave step near the leading edge.

[0009] In a preferred embodiment, the local angle of the cosine function profile is: .

[0010] In a preferred embodiment, the width W of the recessed step is 0.03C.

[0011] Secondly, this application provides a design method, comprising the following steps: S1. Determine the baseline values ​​of the first and second parameters of the concave step, and use the Hua Luogeng optimization method to find the optimal value of the third parameter in this round. S2. Based on the optimal value of the third parameter and the baseline value of the first parameter, the second parameter is optimized using the Hua Luogeng optimization method to obtain the optimal value of the second parameter in this round; S3. Based on the optimal values ​​of the third parameter and the second parameter, the Hua Luogeng optimization method is used to optimize the first parameter to obtain the optimal value of the first parameter in this round. S4. Check whether the convergence of the optimal values ​​of the first parameter, the second parameter, and the third parameter obtained in this round meets the convergence requirement; if it does, end the optimization algorithm; if it does not, use the optimal values ​​of the first parameter, the second parameter, and the third parameter obtained in this round as the reference values ​​of the first parameter, the second parameter, and the third parameter in the next round of optimization, and repeat steps S1 to S4. Wherein, the first parameter, the second parameter, and the third parameter are any one of the depth of the recessed step, the aspect ratio of the recessed step, and the position of the center point of the recessed step, respectively.

[0012] In a preferred embodiment, the convergence requirement means that the convergence accuracy of the first parameter, the second parameter, and the third parameter is simultaneously less than a set threshold.

[0013] In a preferred embodiment, the step further includes: S5. Determine the starting position, ending position, and depth of the recessed step based on the obtained first parameter, second parameter, and third parameter, and connect the starting position of the recessed step with a smooth curved surface.

[0014] In a preferred embodiment, the contour curve of the recessed step adopts a cosine function, specifically y(ξ)=D×sin2(πξ)×[1 0.3×sin(2πξ)], where ξ = (x - x_start) / L, and x_start is the starting position of the concave step near the leading edge.

[0015] This application has the following beneficial effects: The recessed stepped structure on the lower surface of the airfoil can form a stable recirculation zone inside the step. This 'trapping' effect can reduce the effective curvature of the lower surface and increase the pressure difference between the upper and lower surfaces. At the same time, the smooth contour transition avoids flow separation caused by sharp edges, thus achieving the dual effect of increased lift and reduced drag.

[0016] The stepped airfoil obtained in this application can achieve a lift-to-drag ratio improvement of more than 15% at the design angle of attack, and can exhibit good performance gains within the design angle of attack range.

[0017] The Hua Luogeng optimization method can reduce the search range to 61.8% of the original range in each iteration within the search interval of a unimodal function, offering advantages such as fast convergence speed and low computational cost. Combined with the coordinate rotation method, multi-parameter optimization can be achieved without calculating gradients, effectively solving multi-parameter airfoil optimization problems with fast convergence speed and high computational efficiency. Attached Figure Description

[0018] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0019] Figure 1 A schematic diagram of a stepped airfoil provided in an embodiment of this application; Figure 2 for Figure 1 An enlarged schematic diagram of the recessed steps in the middle; Figure 3 for Figure 1 A schematic diagram of a local angle of the recessed steps; Numbering on the map: 1-Leading edge; 2-Leading edge; 3-Mid-arc line; 4-Chord line; 5-Concave step. Detailed Implementation

[0020] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. The components of the embodiments of the present invention described and labeled in the accompanying drawings can generally be arranged and designed in various different configurations.

[0021] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0022] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0023] In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, or the orientation or positional relationship commonly used when the product of this invention is in use. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this invention. In addition, the terms "first," "second," "third," etc., are only used to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0024] Furthermore, terms such as "horizontal," "vertical," and "sag" do not imply that components must be absolutely horizontal or suspended, but rather that they can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal relative to "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted.

[0025] In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0026] The following detailed description of some embodiments of the present invention is provided in conjunction with the accompanying drawings. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0027] The Hua Luogeng optimization method, also known as the 0.618 method or the golden section method, is a highly efficient one-dimensional search algorithm promoted by Academician Hua Luogeng in the 1960s. This method utilizes the golden ratio φ = ( The special property of -1) / 2≈0.618 means that within the search interval of a unimodal function, each iteration can reduce the search range to 61.8% of the original range, which has the advantages of fast convergence speed and small computational cost.

[0028] The coordinate rotation method decomposes a multidimensional optimization problem into a series of one-dimensional optimization subproblems, searching sequentially along each coordinate direction. Combined with the Hua Luogeng optimization method, it can achieve multi-parameter optimization without calculating gradients, and is suitable for situations where the objective function has a complex or non-differentiable analytical expression.

[0029] The following is combined Figures 1-3 The embodiments of this application will be further described below.

[0030] In this embodiment, the NACA4412 airfoil is selected as the reference airfoil. This airfoil belongs to the NACA four-digit series and has good lift characteristics and a moderate drag level. Its specific characteristic parameters are as follows: Maximum curvature: 4% of chord length, located at 40% of chord length; Maximum thickness: 12% of chord length; Design lift coefficient: approximately 0.4 (zero angle of attack); Typical applications: general aviation aircraft wings, small wind turbine blades.

[0031] like Figure 1 As shown, the NACA4412 airfoil has an upper surface and a lower surface, with a leading edge 1 at the front end and a trailing edge 2 at the rear end. Between the leading edge 1 and the trailing edge 2, there is a mid-curve 3 and a chord 4.

[0032] A recessed step 5 with a smooth curved surface is provided in the middle of the lower surface of the NACA4412 airfoil. The center point P of the recessed step 5 is located between 0.25C and 0.8C along the direction from the leading edge 1 to the trailing edge 2 of the stepped airfoil. The depth D of the recessed step 5 ranges from 0.005C to 0.02C. The aspect ratio L / W of the recessed step 5 is (2:1) to (4:1). Wherein, C is the length of the chord 4 of the stepped airfoil, i.e., the chord length; L is the length of the recessed step 5, i.e., the length of the recessed step 5 in the chord direction, that is, the dimension parallel to the chord length direction of the airfoil; W is the width of the recessed step 5, i.e., the dimension along the length direction of the airfoil, and the value of W ranges from 0.02C to 0.05C.

[0033] The position of the center point of the recessed step is set as the first parameter, and the position of the center point of the recessed step 5 relative to the front edge 1 is located.

[0034] The depth D of the recessed step is set as the second parameter to determine the maximum depth of the recessed step 5.

[0035] The length and width L / W of the recessed step are set as the third parameter, denoted as R.

[0036] The step airfoil is analyzed in the chord direction as follows: In the leading edge region (between 0 and 0.25°C), the boundary layer is extremely thin and sensitive to pressure peaks; therefore, indentations are prohibited. In the trailing edge restricted area (between 0.8°C and 0.25°C), the boundary layer is thick or has separated; therefore, indentations are prohibited.

[0037] The center point P of the concave step 5 is set between 0.25°C and 0.8°C, with the theoretically optimal position between 0.5°C and 0.6°C, where the boundary layer develops moderately and the effect is best. The boundary layer in the region between 0.6°C and 0.8°C is thicker, and the effect is weakened but still usable.

[0038] This embodiment uses the lift-to-drag ratio (L / D) as the optimization objective function: L / D = CL / CD, where CL is the lift coefficient and CD is the drag coefficient. The lift-to-drag ratio is a comprehensive indicator of an airfoil's aerodynamic efficiency; a higher lift-to-drag ratio indicates that the airfoil consumes less energy to generate the same lift. The optimization objective is to maximize the lift-to-drag ratio under the design angle of attack (4°).

[0039] The step airfoil design method provided in this embodiment combines the Hua Luogeng optimization method with the coordinate transformation method, and includes the following steps: S1. Determine the baseline values ​​of the first and second parameters of the concave step, and use the Hua Luogeng optimization method to find the optimal value of the third parameter in this round. S2. Based on the optimal value of the third parameter and the baseline value of the first parameter, the second parameter is optimized using the Hua Luogeng optimization method to obtain the optimal value of the second parameter in this round; S3. Based on the optimal values ​​of the third parameter and the second parameter, the Hua Luogeng optimization method is used to optimize the first parameter to obtain the optimal value of the first parameter in this round. S4. Check whether the convergence of the optimal values ​​of the first parameter, the second parameter, and the third parameter obtained in this round meets the convergence requirements; if it does, end the optimization algorithm; if it does not, use the optimal values ​​of the first parameter, the second parameter, and the third parameter obtained in this round as the reference values ​​of the first parameter, the second parameter, and the third parameter in the next round of optimization, and repeat steps S1 to S4. The first parameter, the second parameter, and the third parameter are any one of the following: the depth of the recessed step, the aspect ratio of the recessed step, and the position of the center point of the recessed step, respectively.

[0040] Specifically, you can follow these steps: Assume the overall objective function is And assume that the position reference value is P0, the depth reference value is D0, and the aspect ratio reference value is R0.

[0041] S1. With fixed depth reference value D0 and aspect ratio reference value R0, the position parameter P is optimized using the Hua Luogeng optimization method, transforming the original ternary function into a univariate unimodal function with respect to p: ; Will Substituting into the Hua Luogeng optimization method process, the convergence accuracy is set to... Golden ratio ; First iteration, calculate the test points: , , Calculate the function value: calculate , calculate , Comparison and range scaling: Scenario A: If This indicates that the maximum value is biased to the left. According to the 0.618 rule, the new search interval is updated to... At the same time, the original It will be directly used as the right test point for the next iteration, without needing to recalculate its function value.

[0042] Scenario B: If This indicates that the maximum value is biased to the right. The new search range is updated to... At the same time, the original It will be used directly as the left test point for the next iteration.

[0043] Termination conditions: Repeat the scaling process described above until, in a certain iteration, the interval length satisfies... . At this point, the key point of the last interval is taken as the optimal parameter for this round of optimization: .

[0044] This step led to the discovery of... P*.

[0045] S2. With fixed position parameter P* and aspect ratio parameter R0, the Hua Luogeng optimization method is used to optimize the depth parameter D. Referring to the steps in S1, the optimal value of the depth parameter D* for this round is obtained. S3. With fixed position parameters P* and depth parameters D*, the aspect ratio parameter R is optimized using the Hua Luogeng optimization method. Referring to the steps in S1, the optimal value R* of the aspect ratio parameter in this round is obtained. S4. After completing one cycle, check the convergence. If the obtained P*, D*, and R* all meet the convergence requirements, stop the optimization. Otherwise, use the obtained P*, D*, and R* as new baseline values ​​and repeat steps S1 to S4 until the result that meets the convergence requirements is obtained.

[0046] The convergence criterion is that the change in parameters is less than a set threshold of 0.001 or the change in the objective function is less than 1 / 100 of the threshold.

[0047] After determining the center point P, depth D, and aspect ratio R of the recessed step, the concave profile of recessed step 5 is smoothed using a cosine function to avoid sharp edges. Specifically, y(ξ) = D × sin2(πξ) × [1 [0.3×sin(2πξ)], where ξ = (x - x_start) / L, and x_start is the starting position of the concave step near the leading edge. This function has zero slope at both ends (ξ=0 and ξ=1), ensuring a smooth connection with the original airfoil surface.

[0048] If the edges of a concave step have physically sharp angles, they can easily induce mesh singularities in flow field numerical simulations. Furthermore, at the aerodynamic level, they can trigger premature forced transition of the boundary layer and even lead to unsteady airflow separation. Therefore, evaluating the angular characteristics of the cosine smooth transition curve is crucial. The local angle of the cosine function profile is: , like Figure 3 As shown, the angular feature indicators include: Equivalent leading edge angle: characterizes the front half of the concave step. The geometric angle corresponding to the average descent slope is calculated using the following formula: ; Equivalent trailing edge angle: characterizes the rear half of the concave step. The geometric angle corresponding to the average rising slope. Due to the symmetry of the standard cosine curve, its value is equal to the equivalent angle of the leading edge: ; Maximum local angle: occurs at the steepest point of the profile, i.e., the inflection point of the cosine curve. The instantaneous angle at that location is the critical point where the flow field is most prone to separation: ; Average equivalent angle: The arithmetic mean of the leading edge equivalent angle and the trailing edge equivalent angle. Due to the symmetry of this geometric model, this value is numerically equal to... or .

[0049] In this embodiment, the base width W = 0.03C (3% chord length) is assumed, then the step length L = W × R. The positional relationship of the step on the airfoil is as follows: Step start position: x_start = P - L / 2, The end position of the steps: x_end = P + L / 2, The center position of the step: x_center = P.

[0050] The optimal design parameters obtained through iterative optimization using the Hua Luogeng optimization method are shown in the table below:

[0051] Optimal design angular feature analysis results: The equivalent leading edge angle is approximately 4.8°, the equivalent trailing edge angle is approximately 4.8°, the average equivalent angle is approximately 4.8°, and the maximum local angle is 14.75°.

[0052] Angular feature analysis shows that the maximum local angle of the cosine profile occurs at 1 / 4 of the length at both ends of the step. This angle value is moderate and will not cause abrupt flow separation. The symmetrical angle distribution at the leading and trailing edges is conducive to the formation of a stable recirculation zone.

[0053] Compared to the baseline NACA 4412 airfoil, the optimized recessed stepped airfoil shows the following performance improvements:

[0054] To verify the aerodynamic robustness of the stepped airfoil under off-design conditions, the lift-to-drag ratio characteristics of the baseline and optimized airfoils within the 0°–12° angle of attack range were extracted and compared. The data show that the optimized airfoil did not exhibit performance degradation across the entire angle of attack range, and its aerodynamic benefits exhibited a typical segmented characteristic: The optimal operating angle of attack range is 4°~8°. The lift-to-drag ratio remains at a high level. Within this range, the adverse pressure gradient at the trailing edge of the reference airfoil gradually becomes apparent, and the step structure achieves the best matching state in terms of flow control. It can delay the migration of the boundary layer separation line to the greatest extent, significantly increase aerodynamic lift and suppress pressure drag, and exhibit the greatest comprehensive aerodynamic benefits.

[0055] In the low angle of attack range of 0° to 4°, the performance improvement gradually increases. Within this range, the boundary layer itself is in a relatively stable adhesion state. As the angle of attack increases, the stepped structure begins to function as a micro-vortex generator. The induced vorticity effectively replenishes the kinetic energy of the bottom layer of the boundary layer, resulting in an upward trend in the relative improvement of the lift-to-drag ratio.

[0056] High angle of attack range: 8°~12°. The performance improvement is somewhat reduced but still better than the benchmark. After entering the high angle of attack range, the airfoil's suction surface faces a strong adverse pressure gradient globally, and the airflow undergoes large-area separation (approaching stall). At this point, the local flow field control capability of the step reaches its limit, resulting in a slight decrease in the relative percentage improvement; however, because the presence of the step changes the leading-edge flow structure and delays the occurrence of deep stall, its absolute performance is still better than the benchmark airfoil.

[0057] Furthermore, no performance degradation was observed across the entire angle of attack range.

[0058] This embodiment is based on Hua Luogeng's optimization method and combined with the coordinate rotation method to systematically optimize the geometric parameters of the concave step on the lower surface of the NACA 4412 airfoil. It can effectively solve the multi-parameter airfoil optimization problem, with fast convergence speed and high computational efficiency.

[0059] The optimal step position is located at 55% of the chord length, with a depth of 0.503% of the chord length and an aspect ratio of 4:1. The angular characteristics of the cosine function profile are moderate (maximum local angle of 14.75°), which is conducive to forming a stable flow control effect.

[0060] The optimized design can achieve an increase of approximately 18% in lift-to-drag ratio at the design angle of attack, and exhibits good performance gains across the 0° to 12° angle of attack range.

[0061] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A stepped airfoil, characterized in that, It has an upper surface and a lower surface, and a recessed step with a smooth curved surface is provided in the middle of the lower surface; Along the direction from the leading edge to the trailing edge of the stepped airfoil, the center point P of the recessed step is located between 0.25C and 0.8C, the depth D of the recessed step ranges from 0.005C to 0.02C, and the aspect ratio L / W of the recessed step is (2:1) to (4:1), where C is the chord length of the stepped airfoil, L is the length of the recessed step, W is the width of the recessed step, and W ranges from 0.02C to 0.05C.

2. The stepped airfoil according to claim 1, characterized in that, Based on the NACA4412 airfoil, a recessed step is provided on its lower surface.

3. The stepped airfoil according to claim 1, characterized in that, The center point P of the recessed step is located between 0.5°C and 0.7°C.

4. The stepped airfoil according to claim 1, characterized in that, The contour curve of the concave step is generated using a cosine function, specifically y(ξ)=D×sin2(πξ)×[1 0.3×sin(2πξ)], where ξ = (x - x_start) / L, and x_start is the starting position of the concave step near the leading edge.

5. The stepped airfoil according to claim 4, characterized in that, The local angle of the cosine function profile is: .

6. The stepped airfoil according to claim 1, characterized in that, The width W of the recessed step is 0.03C.

7. A design method for a stepped airfoil as described in claim 1, characterized in that, Includes the following steps: S1. Determine the baseline values ​​of the first and second parameters of the concave step, and use the Hua Luogeng optimization method to find the optimal value of the third parameter in this round. S2. Based on the optimal value of the third parameter and the baseline value of the first parameter, the second parameter is optimized using the Hua Luogeng optimization method to obtain the optimal value of the second parameter in this round; S3. Based on the optimal values ​​of the third parameter and the second parameter, the Hua Luogeng optimization method is used to optimize the first parameter to obtain the optimal value of the first parameter in this round. S4. Check whether the convergence of the optimal values ​​of the first parameter, the second parameter, and the third parameter obtained in this round meets the convergence requirement; if it does, end the optimization algorithm; if it does not, use the optimal values ​​of the first parameter, the second parameter, and the third parameter obtained in this round as the reference values ​​of the first parameter, the second parameter, and the third parameter in the next round of optimization, and repeat steps S1 to S4. Wherein, the first parameter, the second parameter, and the third parameter are any one of the depth of the recessed step, the aspect ratio of the recessed step, and the position of the center point of the recessed step, respectively.

8. The design method according to claim 7, characterized in that, The convergence requirement refers to the fact that the convergence accuracy of the first parameter, the second parameter, and the third parameter are all less than a set threshold.

9. The design method according to claim 7, characterized in that, It also includes the following steps: S5. Determine the starting position, ending position, and depth of the recessed step based on the obtained first parameter, second parameter, and third parameter, and connect the starting position of the recessed step with a smooth curved surface.

10. The design method according to claim 9, characterized in that, The contour curve of the recessed step adopts a cosine function, specifically y(ξ)=D×sin2(πξ)×[1 0.3×sin(2πξ)], where ξ = (x - x_start) / L, and x_start is the starting position of the concave step near the leading edge.

Images