A parameterized design method of s-shaped reversible airfoil

Abstract

The application provides a parameterized design method of S-shaped reversible airfoil, including S1, four-elliptic-arc is adopted to be smoothly connected in sequence from leading edge to trailing edge to form S-shaped middle camber line, discrete coordinates of the middle camber line are obtained through a segmented function definition containing 9 core parameters and uniform sampling; S2, an improved Joukowsky symmetric thickness distribution function is adopted to define half-thickness distribution through 4 core parameters and auxiliary parameters, and half-thickness values are calculated by substituting the sampling points; S3, S-shaped reversible airfoil profile is generated through the way of symmetrically superimposing thickness distribution on the middle camber line. The S-shaped reversible airfoil is defined by 13 parameters, the parameter space is simple and intuitive, and the finally generated airfoil profile has smooth continuity, and can be directly used for subsequent numerical analysis without additional fairing treatment.

Description

Technical Field

[0001] This application relates to the field of fluid machinery technology, and in particular to a parametric design method for an S-shaped reversible airfoil. Background Technology

[0002] The S-shaped reversible airfoil is the core design unit of reversible fluid machinery impellers. Its core geometric feature is the bidirectional S-shaped curvature of its mid-curve, which maintains stable hydrodynamic performance under both forward and reverse flow conditions, making it a key design element for bidirectional fluid dynamic equipment. Currently, the mainstream construction methods for S-shaped reversible airfoils in the industry are twofold: one is to splice existing conventional airfoils in reverse, and the other is to superimpose the thickness distribution along the S-shaped mid-curve. The S-shaped mid-curve is often obtained using Bezier curve fitting and polynomial function construction, with the shape controlled by multiple control points or polynomial coefficients. The thickness distribution often adopts the thickness distribution of conventional symmetrical airfoils, superimposed along the mid-curve to ultimately generate the complete S-shaped reversible airfoil profile.

[0003] However, existing technologies have three major drawbacks in addressing the specific structural requirements of S-shaped reversible airfoils: First, the design parameters of existing Bezier curves and polynomial function methods are not directly related to the key geometric features of S-shaped reversible airfoils. Designers cannot directly control the curvature, inflection point position, and curvature change of the arc in the S-shape through design parameters. They can only rely on repeated attempts with a large number of control points, which consumes a lot of time and effort in the construction process and has poor engineering practicality.

[0004] Secondly, the S-shaped mid-curve constructed by some methods cannot guarantee a smooth and continuous curve at the curve splicing point, and is prone to problems such as abrupt curvature changes and angle breaks in the profile, which will have an adverse effect on the hydrodynamic performance of the airfoil.

[0005] Third, existing thickness distribution functions cannot independently control the geometry of the airfoil's leading and trailing edges. Due to the bidirectional operating characteristics of the S-shaped reversible airfoil, the leading and trailing edges alternately face the incoming flow. Existing construction methods cannot be specially designed according to the different requirements of forward and reverse operating conditions, thus preventing the airfoil from achieving optimal bidirectional performance.

[0006] Therefore, there is a need to provide an improved technical solution that addresses the shortcomings of the existing technology. Summary of the Invention

[0007] The purpose of this application is to provide a parametric design method for S-shaped reversible airfoils, the core objective of which is: 1. Establish a method for constructing the S-shaped mid-arc based on four-segment elliptical arcs to achieve a strong correlation between design parameters and airfoil geometric features, allowing designers to directly and accurately control the core features of the S-shaped reversible airfoil, such as camber, inflection point, and curvature, without the need for repeated trial and error. 2. Ensure that the S-shaped arc is smooth and continuous, and that the first derivative of the four elliptical arcs exists and is continuous at the splicing point, thereby avoiding the problems of abrupt curvature changes and geometric distortions from the root. 3. Establish a thickness distribution construction method that adapts to bidirectional working requirements, realize independent control of the leading and trailing edge shapes of the airfoil, and at the same time take into account the hydrodynamic requirements of the forward and reverse flow of the S-shaped reversible airfoil.

[0008] The S-shaped reversible airfoil construction method of the present invention follows a standardized process of dimensionless design → S-shaped mid-arc construction → thickness distribution construction → airfoil superposition generation. All steps can be directly reproduced, and finally a complete S-shaped reversible airfoil ...

[0009] A parametric design method for an S-shaped reversible airfoil includes the following steps: S0. Establish the basic coordinate system; To ensure the universality of the construction method, the airfoil chord length is dimensionless and set to 1. The coordinates of the airfoil leading edge are set to (0,0) and the coordinates of the trailing edge are set to (1,0). The chord is a horizontal straight line from the leading edge to the trailing edge. All geometric parameters are dimensionless values ​​relative to the chord length, and can be scaled proportionally according to the chord length required by the actual engineering. S1, Construct the S-shaped arc; The S-shaped arc of this application is formed by smoothly splicing four elliptical arcs from the front edge to the rear edge. The four elliptical arcs are the first elliptical arc, the second elliptical arc, the third elliptical arc, and the fourth elliptical arc, respectively. The three splicing points are the first splicing point, the second splicing point, and the third splicing point, respectively. Furthermore, the first derivative of each of the four elliptical arcs exists and is continuous at all splicing points, ensuring that the arc is smooth, continuous, and without any sharp angles. Core parameter definitions (9): This represents the maximum camber value at the front of the airfoil. This represents the maximum camber value at the rear of the airfoil. This refers to the relative position of the maximum curvature at the front. The relative position of the intersection point of the middle arc and the chord (the inflection point of the middle arc); This refers to the relative position of the maximum camber at the rear of the airfoil; , , , These are the shape factors corresponding to the four elliptical arcs. , , The value range is 0~1, and This ensures that the maximum camber and inflection point are within the effective chordal range of the airfoil. , The values ​​range from -0.05 to 0.05, with one positive and one negative, which avoids fluid separation, energy loss, and performance degradation caused by excessive curvature, ensuring high efficiency and stability of the reversible impeller in both forward and reverse operation. Meanwhile, and A positive and a negative curve can create an S-shaped bidirectional curvature in the mid-curve, enabling the airfoil to have matching curvature characteristics and flow field adaptability in both forward and reverse flow, thus meeting the core design requirements for bidirectional operation of reversible fluid machinery. , , , The value range is 0 to 1, and it must satisfy the following conditions: = Setting the shape factor to 0-1 ensures a smooth change in the curvature of the four elliptical arcs, avoiding pathological structures such as abrupt curvature changes, angled curves, and geometric distortions. = This ensures that the slopes of the tangents to the second and third elliptical arcs are consistent at the inflection points, guaranteeing a smooth and continuous mid-arc. The piecewise function expression for the S-shaped mid-arc, defined based on the above nine parameters, is as follows: (1); In the interval Within, uniform sampling is performed at equal step sizes, and the x-coordinate of each sampling point is... Substituting into the piecewise function above, the ordinate of the corresponding mid-arc is calculated. Finally, the discrete coordinate set of the middle arc is obtained. Complete the construction of the arc in the S-shape.

[0010] S2, structural thickness distribution; To address the bidirectional operation requirement of S-shaped reversible airfoils, this invention employs an improved Joukowsky symmetric thickness distribution function to achieve independent control of the airfoil's leading and trailing edge shapes. Only four parameters are needed to fully define the airfoil's thickness distribution, adapting to the geometric requirements of bidirectional flow in S-shaped reversible airfoils. Core parameter definitions (4): The maximum relative thickness of the airfoil is 0.03 to 0.25. Within this range, the strength of the airfoil structure and the flow loss can be balanced, avoiding the increase of drag due to excessive thickness and the lack of stiffness due to insufficient thickness, thus meeting the requirements for bidirectional stable operation of the reversible impeller. The relative position of the maximum thickness; This is the thickness distribution leading edge shape factor, with a value ranging from 0 to 2. The thickness distribution trailing edge shape factor ranges from 0 to 2. This range ensures a relatively uniform thickness distribution at the leading and trailing edges of the airfoil, preventing excessively large or small radii of curvature and reducing flow losses. Simultaneously, it can be achieved through... and The thickness distribution at the leading and trailing edges is adjusted independently without interference. Based on the above four parameters, the half-thickness distribution function of the airfoil is expressed as follows: (2); In the formula, This is a cutoff function used to divide the thickness distribution into two parts, with the maximum thickness as the dividing point. Output 1 when ≥1, otherwise output 0, to achieve independent control of the thickness distribution in the front and back segments; , , , These are auxiliary parameters, calculated directly from the core design parameters, and require no additional setting. ; ; ; ; The x-coordinates of all sampling points generated in step S1 Substituting into formula (2), the half-thickness value corresponding to each sampling point is calculated. The thickness distribution design was completed.

[0011] S3, S-type reversible airfoil with closed profile; By symmetrically superimposing thickness distribution along the mid-arc line, the upper and lower surface profiles of the airfoil are generated, ultimately forming a closed S-shaped reversible airfoil profile. Based on the mid-arc line coordinates and half-thickness values, the discrete coordinates of the upper and lower surfaces of the airfoil are calculated respectively, using the following formulas: (3); (4); Finally, all the upper surface coordinates Sort the horizontal coordinates from 0 to 1 to form the upper surface profile from the leading edge to the trailing edge; then sort all the lower surface coordinates... Arrange the horizontal coordinates from 1 to 0 to form the lower surface profile from the trailing edge to the leading edge; then connect the upper surface profile with the lower surface profile to form a complete closed S-shaped reversible airfoil profile, thus completing the entire construction process of the S-shaped reversible airfoil.

[0012] Supplementary explanation of formula (1): Formula (1) is an original formula based on the standard elliptical arc equation with its center point located at the origin. It is adjusted through translation, scaling of the major and minor axes, and smooth connection of the four elliptical arcs at the splicing points. Nine parameters precisely control the continuity of the four elliptical arcs. , , , The shape factor enables a wide range of S-shaped curve designs. The first formula will be used as an example for illustration. It is the arc length and semi-axis length of the ellipse. Let be the length of the minor semi-axis of the ellipse arc, and let be the coefficient of . and By controlling the lengths of the major and minor semi-axis of the ellipse separately, the shape of the ellipse can be changed, and finally, the shape factor can be used. The shape of the first elliptical arc can be adjusted. The ordinate of the center point of the elliptical arc is used to translate the elliptical arc along the y-axis, so that the maximum camber of the airfoil's leading edge reaches the design camber value. ; The x-coordinate of the center point of the elliptical arc is used to translate the elliptical arc along the x-axis, ensuring that the termination boundary of the first elliptical arc reaches the first splicing point, while simultaneously setting the derivative at this point to 0 to guarantee smooth continuity with the second elliptical arc at the first splicing point. Furthermore, the above parameters and operations ensure that the starting point of the elliptical arc is located at the origin, guaranteeing the rationality of the arc in the S-shape.

[0013] Formula (1) divides the chord length interval It is divided into four segments, and the x-coordinates of the splicing points are respectively , , ,by Let's take an example to verify: 1) Continuity verification: when When the square root term in the first and second parts of formula (1) is 1, then: The first formula, ; The second formula, ; The common point of the first and second formulas coincides, proving that continuity holds.

[0014] 2) Verification that the first derivative is 0: Differentiate the piecewise function formula for the arc in the first S-shaped segment: ; when When the above derivative term is zero, that is... ; Similarly, differentiating the piecewise function formula for the arc in the second S-shaped segment, when... When the derivative of the formula is zero, Therefore, at the splicing point of the two elliptical arcs The first derivatives at each point are all 0, satisfying the condition that the first derivatives are continuous and zero, thus ensuring that the middle arc is smooth and continuous at that point.

[0015] Supplementary explanation of formula (2): Formula (2) is based on the classical Joukowsky symmetric thickness distribution, i.e. The fixed index of 0.5 was changed to an independently adjustable shape factor (a fixed index of 0.5 cannot independently adjust the leading and trailing edge bluntness, and is only applicable to unidirectional conventional airfoils), and a truncation function was introduced to achieve functional separation of the leading and trailing edge segments with the maximum thickness as the dividing point. and The thickness distribution of the leading and trailing edges is adjusted independently and matched. , , , Auxiliary parameters ensure that the thickness distribution curve of the airfoil is continuous, smooth, and has an accurate peak value at the maximum thickness position.

[0016] The technical solution of this application has the following beneficial effects: This invention is achieved through , , The S-shaped reversible airfoil can be fully defined with only 13 geometrically meaningful parameters (9 controlling the mid-curve and 4 controlling the thickness), resulting in a concise and intuitive parameter space. All design parameters directly correspond to the core geometric features of the S-shaped reversible airfoil, such as camber, inflection point, thickness, and leading and trailing edge shapes. Designers can precisely control the shape of the S-shaped reversible airfoil by adjusting these parameters, eliminating the need for repeated trial and error with numerous control points. The resulting airfoil profile exhibits smooth continuity and can be directly used for subsequent numerical analysis without additional smoothing. By independently controlling the leading and trailing edge shape factors, this method can simultaneously address the hydrodynamic requirements of both forward and reverse flow in the S-shaped reversible airfoil, solving the problem that existing methods cannot independently match the bidirectional performance of airfoils. The constructed airfoil exhibits greater applicability in reversible fluid machinery. Attached Figure Description

[0017] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. Wherein: Figure 1 This is the overall flowchart of the present invention.

[0018] Figure 2 This is a schematic diagram illustrating the principle of the arc in the S-shaped structure.

[0019] Figure 3 This is a schematic diagram of the thickness distribution of the improved Joukowsky airfoil according to the present invention.

[0020] Figure 4 The diagram shows the S-shaped reversible airfoil of Example 1.

[0021] Figure 5 The diagram shows the S-shaped reversible airfoil of Example 2.

[0022] Figure 6 This is a line drawing of the S-shaped reversible airfoil in Example 3. Detailed Implementation

[0023] The present application will now be described in detail with reference to the accompanying drawings and embodiments. Various examples are provided by way of explanation and not by way of limitation. In fact, those skilled in the art will recognize that modifications and variations can be made to the present application without departing from the scope or spirit thereof. For example, a feature shown or described as part of one embodiment may be used in another embodiment to produce yet another embodiment. Therefore, it is desirable that the present application encompass such modifications and variations that fall within the scope of the appended claims and their equivalents.

[0024] In the following description, the terms "first / second / third" are used merely to distinguish similar objects and do not represent a specific order of objects. It is understood that "first / second / third" may be interchanged in a specific order or sequence where permitted, so that the embodiments of this application described herein can be implemented in an order other than that illustrated or described herein.

[0025] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. The terminology used herein is for the purpose of describing embodiments of this disclosure only and is not intended to limit this disclosure.

[0026] A parametric design method for an S-shaped reversible airfoil includes the following steps: S0. Establish the basic coordinate system; To ensure the universality of the construction method, the airfoil chord length is dimensionless and set to 1. The coordinates of the airfoil leading edge are set to (0,0) and the coordinates of the trailing edge are set to (1,0). The chord is a horizontal straight line from the leading edge to the trailing edge. All geometric parameters are dimensionless values ​​relative to the chord length, and can be scaled proportionally according to the chord length required by the actual engineering. S1, Construct the S-shaped arc; The S-shaped arc of this application is formed by smoothly splicing four elliptical arcs from the front edge to the rear edge. The four elliptical arcs are the first elliptical arc, the second elliptical arc, the third elliptical arc, and the fourth elliptical arc, respectively. The three splicing points are the first splicing point, the second splicing point, and the third splicing point, respectively. Furthermore, the first derivative of each of the four elliptical arcs exists and is continuous at all splicing points, ensuring that the arc is smooth, continuous, and without any sharp angles. Core parameter definitions (9): This represents the maximum camber value at the front of the airfoil. This represents the maximum camber value at the rear of the airfoil. This refers to the relative position of the maximum curvature at the front. The relative position of the intersection point of the middle arc and the chord (the inflection point of the middle arc); This refers to the relative position of the maximum camber at the rear of the airfoil; , , , These are the shape factors corresponding to the four elliptical arcs. , , The value range is 0~1, and This ensures that the maximum camber and inflection point are within the effective chordal range of the airfoil. , The values ​​range from -0.05 to 0.05, with one positive and one negative, which avoids fluid separation, energy loss, and performance degradation caused by excessive curvature, ensuring high efficiency and stability of the reversible impeller in both forward and reverse operation. Meanwhile, and A positive and a negative curve can create an S-shaped bidirectional curvature in the mid-curve, enabling the airfoil to have matching curvature characteristics and flow field adaptability in both forward and reverse flow, thus meeting the core design requirements for bidirectional operation of reversible fluid machinery. , , , The value range is 0 to 1, and it must satisfy the following conditions: = Setting the shape factor to 0-1 ensures a smooth change in the curvature of the four elliptical arcs, avoiding pathological structures such as abrupt curvature changes, angled curves, and geometric distortions. = This ensures that the slopes of the tangents to the second and third elliptical arcs are consistent at the inflection points, guaranteeing a smooth and continuous mid-arc. The piecewise function expression for the S-shaped mid-arc, defined based on the above nine parameters, is as follows: (1); In the interval Within, uniform sampling is performed at equal step sizes, and the x-coordinate of each sampling point is... Substituting into the piecewise function above, the ordinate of the corresponding mid-arc is calculated. Finally, the discrete coordinate set of the middle arc is obtained. Complete the construction of the arc in the S-shape.

[0027] S2, structural thickness distribution; To address the bidirectional operation requirement of S-shaped reversible airfoils, this invention employs an improved Joukowsky symmetric thickness distribution function to achieve independent control of the airfoil's leading and trailing edge shapes. Only four parameters are needed to fully define the airfoil's thickness distribution, adapting to the geometric requirements of bidirectional flow in S-shaped reversible airfoils. Core parameter definitions (4): The maximum relative thickness of the airfoil is 0.03 to 0.25. Within this range, the strength of the airfoil structure and the flow loss can be balanced, avoiding the increase of drag due to excessive thickness and the lack of stiffness due to insufficient thickness, thus meeting the requirements for bidirectional stable operation of the reversible impeller. The relative position of the maximum thickness; This is the thickness distribution leading edge shape factor, with a value ranging from 0 to 2. The thickness distribution trailing edge shape factor ranges from 0 to 2. This range ensures a relatively uniform thickness distribution at the leading and trailing edges of the airfoil, preventing excessively large or small radii of curvature and reducing flow losses. Simultaneously, it can be achieved through... and The thickness distribution at the leading and trailing edges is adjusted independently without interference. Based on the above four parameters, the half-thickness distribution function of the airfoil is expressed as follows: (2); In the formula, This is a cutoff function used to divide the thickness distribution into two parts, with the maximum thickness as the dividing point. Output 1 when ≥1, otherwise output 0, to achieve independent control of the thickness distribution in the front and back segments; , , , These are auxiliary parameters, calculated directly from the core design parameters, and require no additional setting. ; ; ; ; The x-coordinates of all sampling points generated in step S1 Substituting into formula (2), the half-thickness value corresponding to each sampling point is calculated. The thickness distribution design was completed.

[0028] S3, S-type reversible airfoil with closed profile; By symmetrically superimposing thickness distribution along the mid-arc line, the upper and lower surface profiles of the airfoil are generated, ultimately forming a closed S-shaped reversible airfoil profile. Based on the mid-arc line coordinates and half-thickness values, the discrete coordinates of the upper and lower surfaces of the airfoil are calculated respectively, using the following formulas: (3); (4); Finally, all the upper surface coordinates Sort the horizontal coordinates from 0 to 1 to form the upper surface profile from the leading edge to the trailing edge; then sort all the lower surface coordinates... Arrange the horizontal coordinates from 1 to 0 to form the lower surface profile from the trailing edge to the leading edge; then connect the upper surface profile with the lower surface profile to form a complete closed S-shaped reversible airfoil profile, thus completing the entire construction process of the S-shaped reversible airfoil.

[0029] Example 1 like Figure 1 As shown, a parametric design method for an S-shaped reversible airfoil includes the following steps: S0. Dimensionlessize the airfoil chord length to 1, which can be scaled to any chord length according to actual needs; set the airfoil leading edge coordinates to (0,0) and trailing edge coordinates to (1,0), and the chord line to be a horizontal straight line from the leading edge to the trailing edge; all geometric parameters are dimensionless values ​​relative to the chord length. The 13 core parameters in formulas (1) and (2) are: , , , , , , , , , , , .

[0030] S1, in the chordal interval Within the range, uniform sampling is performed at a step size of 0.005, generating a total of 201 sampling points. The horizontal coordinate of each sampling point is substituted into the piecewise function of the arc (formula (1)) to calculate the corresponding vertical coordinate, forming a discrete coordinate set of the S-shaped arc.

[0031] In this embodiment, the middle arc line is The maximum camber at the front of the airfoil reaches 0.015. The point intersects with the chord, at The maximum camber value at the rear of the airfoil reaches -0.015, forming a smooth, continuous S-shaped mid-arc without abrupt curvature changes. The airfoil profile is symmetrical about the center point (0.5,0), and its forward and reverse working capabilities are the same. Figure 2 This is a schematic diagram illustrating the principle of constructing the S-shaped mid-arc. The construction principle of the S-shaped mid-arc, the splicing relationship of the four elliptical arcs, and the core geometric features in this embodiment are all described in... Figure 2 It is presented in its entirety.

[0032] S2. Substitute the abscissas of the 201 sampling points into the improved Joukowsky thickness distribution function (Formula (2)) to calculate the half-thickness value of each sampling point. In this embodiment, the maximum relative thickness of the airfoil is 0.05, and the maximum thickness position is located at the midpoint of the chord, i.e., x=0.5. The shape factor values ​​of the leading edge and trailing edge are the same, and the thickness distribution is symmetrical to adapt to the same working conditions of bidirectional flow. For the thickness distribution law used in this embodiment, please refer to [reference needed]. Figure 3 This is a schematic diagram of the thickness distribution obtained using the improved Joukowsky thickness distribution function of this invention; for the effect of adjusting the leading and trailing edge shape factors and the airfoil profile characteristics, see [link to relevant documentation]. Figure 4 .

[0033] S3. Complete the calculation of coordinate points on the upper and lower surfaces of the airfoil and generate the airfoil profile. The specific steps are as follows: 1) Calculate the coordinates of the upper and lower surfaces corresponding to the 201 sampling points according to the formulas for the upper and lower surface coordinates (formulas (3) and (4)); 2) Sort the coordinates of the upper surface from 0 to 1 and the coordinates of the lower surface from 1 to 0, and splice them together to form a closed S-shaped reversible airfoil profile.

[0034] The final S-shaped reversible airfoil profile constructed in this embodiment is shown in [reference]. Figure 4 A schematic diagram of the S-shaped reversible airfoil in Example 1. The profile is smooth and continuous with no abrupt changes in curvature, making it suitable for impeller design in reversible fluid machinery such as bidirectional axial flow pumps and reversible fans.

[0035] Example 2 This embodiment is designed for heavy load conditions. The S-shaped reversible airfoil parametric design method described in this invention is used to construct the airfoil. The construction steps are the same as in Embodiment 1, except that the camber and thickness parameters are adjusted to verify the universality of this design method.

[0036] A parametric design method for an S-shaped reversible airfoil includes the following steps: Step S0 is the same as in Example 1; The 13 core parameters in formulas (1) and (2) are: , , , , , , , , , , , .

[0037] S1. Following the same sampling step size and calculation method as in Example 1, in the chordal interval... Within the range, uniform sampling is performed at a step size of 0.005, generating a total of 201 sampling points. The horizontal coordinate of each sampling point is substituted into the piecewise function of the arc (formula (1)) to calculate the corresponding vertical coordinate, forming a discrete coordinate set of the S-shaped arc.

[0038] This embodiment increases... The absolute value of the airfoil camber increases the working load of the airfoil, while the relative thickness t increases the structural strength of the airfoil to meet the design requirements of high loads. The maximum camber and inflection points remain symmetrically arranged, and the mid-curve is smooth and continuous at each splicing point. The construction principle of the S-shaped mid-curve, the splicing relationship of the four elliptical arcs, and the core geometric features of this embodiment are all described in... Figure 2 It is presented in its entirety.

[0039] S2. Substitute the abscissas of the 201 sampling points into the improved Joukowsky thickness distribution function (Formula (2)) to calculate the half-thickness value of each sampling point. In this embodiment, the maximum relative thickness of the airfoil is 10%, and the thickness distribution is symmetrical, which is suitable for the working conditions of bidirectional flow. For the thickness distribution law used in this embodiment, please refer to [reference needed]. Figure 3 This is a schematic diagram of the improved Joukowsky thickness distribution function of this invention; for the effect of adjusting the leading and trailing edge shape factors and the airfoil profile characteristics, see [link to relevant documentation]. Figure 5 .

[0040] S3. Complete the calculation of coordinate points on the upper and lower surfaces of the airfoil and generate the airfoil profile. The specific steps are as follows: 1) Calculate the coordinates of the upper and lower surfaces corresponding to the 201 sampling points according to the upper and lower surface coordinate formulas (formulas (3) and (4)); 2) Sort the coordinates of the upper surface from 0 to 1 and the coordinates of the lower surface from 1 to 0, and splice them together to form a closed S-shaped reversible airfoil profile.

[0041] The final S-shaped reversible airfoil profile constructed in this embodiment is shown in [reference]. Figure 5 Schematic diagram of the S-shaped reversible airfoil in Example 2. The profile is smooth and continuous with no abrupt curvature changes, making it suitable for impeller design in reversible fluid machinery such as high-power bidirectional axial flow pumps and reversible fans.

[0042] Example 3 This embodiment addresses different operational capacity requirements under forward and reverse flow conditions. It employs the S-shaped reversible airfoil parametric design method described in this invention, with construction steps identical to those in Embodiment 1. By adjusting parameters such as camber, maximum thickness, and leading and trailing edge shape factors, the adaptability of this method to different operational capacity requirements under forward and reverse flow conditions is verified.

[0043] A parametric design method for an S-shaped reversible airfoil includes the following steps: Step S0 is the same as in Example 1; The 13 core parameters in formulas (1) and (2) are: 288、 , , , , 1. , 1. , , , 4.

[0044] S1. Following the same sampling step size and calculation method as in Example 1, in the chordal interval... Within the range, uniform sampling is performed at a step size of 0.005, generating a total of 201 sampling points. The horizontal coordinate of each sampling point is substituted into the piecewise function of the arc (formula (1)) to calculate the corresponding vertical coordinate, forming a discrete coordinate set of the S-shaped arc.

[0045] In this embodiment, the maximum camber at the front of the airfoil is greater than that at the rear, resulting in a higher lift coefficient in forward airflow compared to reverse airflow, thus enhancing its performance in forward airflow. Simultaneously, by adjusting the aforementioned core parameters, the airfoil profile is streamlined, reducing drag loss. The four elliptical arcs are smoothly joined, resulting in a distortion-free profile. For the construction principle and core geometric features of the mid-arc line in this embodiment, please refer to [link to relevant documentation]. Figure 2 .

[0046] S2. Substitute the abscissas of the 201 sampling points into the improved Joukowsky thickness distribution function (Formula (2)) to calculate the half-thickness value of each sampling point. In this embodiment, the maximum relative thickness, the location of the maximum thickness, and the leading and trailing edge shape factors of the thickness distribution are adjusted to reduce drag and vortex loss. Asymmetric leading and trailing edge shape factors are used to make the leading edge thinner and the trailing edge more rounded, thereby improving aerodynamic efficiency and flow stability in bidirectional conditions. The thickness distribution law used in this embodiment is described in [reference]. Figure 3 The effect of controlling the leading and trailing edge shape factors on airfoil profile characteristics is discussed in [reference needed]. Figure 6 .

[0047] S3. Complete the calculation of coordinate points on the upper and lower surfaces of the airfoil and generate the airfoil profile. The specific steps are as follows: 1) Calculate the coordinates of the upper and lower surfaces corresponding to the 201 sampling points according to the upper and lower surface coordinate formulas (formulas (3) and (4)); 2) Sort the coordinates of the upper surface from 0 to 1 and the coordinates of the lower surface from 1 to 0, and splice them together to form a closed S-shaped reversible airfoil profile.

[0048] The final S-shaped reversible airfoil profile constructed in this embodiment is shown in [reference]. Figure 6 Schematic diagram of the S-shaped reversible airfoil in Example 3. The profile is smooth and continuous, and the forward working capacity is greater than the reverse working capacity, making it suitable for impeller design of reversible fluid machinery with different working capacity requirements for forward and reverse flow.

[0049] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A parametric design method for an S-shaped reversible airfoil, characterized in that: Includes the following steps: S1, Construct the S-shaped arc; The S-shaped mid-curve is constructed by smoothly splicing together four elliptical arcs from the front edge to the rear edge. The S-shaped mid-curve is defined using a piecewise function, expressed as follows: （1）； in, This represents the maximum camber value at the front of the airfoil. This represents the maximum camber value at the rear of the airfoil. This refers to the relative position of the maximum curvature at the front. The relative position of the intersection point of the middle arc and the chord (the inflection point of the middle arc); This refers to the relative position of the maximum camber at the rear of the airfoil; , , , These are the shape factors corresponding to the four elliptical arcs; In the interval Within, uniform sampling is performed at equal step sizes, and the x-coordinate of each sampling point is... Substituting into the piecewise function above, the ordinate of the corresponding mid-arc is calculated. Finally, the discrete coordinate set of the middle arc is obtained. Complete the construction of the arc in the S-shape; S2, structural thickness distribution; An improved Joukowsky symmetric thickness distribution function is used to achieve independent control of the leading and trailing edge shapes of the airfoil. The half-thickness distribution function expression of the airfoil is as follows: （2）； in, This represents the maximum relative thickness of the airfoil. The relative position of the maximum thickness; The thickness distribution leading edge shape factor; The thickness distribution trailing edge shape factor; For truncation function, when Output 1 if the value is greater than or equal to 1, otherwise output 0. , , , For auxiliary parameters; The x-coordinates of all sampling points generated in step S1 Substituting into formula (2), the half-thickness value corresponding to each sampling point is calculated. Complete the design of the thickness distribution; S3. A closed S-shaped reversible airfoil profile is generated by symmetrically superimposing thickness distribution along the mid-arc line.

2. The parametric design method for the S-shaped reversible airfoil according to claim 1, characterized in that: In step S1 , , The value range is 0~1, and This ensures that the maximum camber and inflection point are within the effective chordal range of the airfoil.

3. The parametric design method for the S-shaped reversible airfoil according to claim 1, characterized in that: In step S1, , The value ranges from -0.05 to 0.05, with one positive and one negative value, to avoid fluid separation, energy loss and performance degradation caused by excessive curvature, and to ensure high efficiency and stability of the reversible impeller when it works in both directions.

4. The parametric design method for the S-shaped reversible airfoil according to claim 1, characterized in that: In step S1, , , , The value range is 0~1, ensuring that the curvature of the four elliptical arcs changes smoothly; and it is required that... = This ensures that the middle arc is smooth and continuous.

5. The parametric design method for the S-shaped reversible airfoil according to claim 1, characterized in that: In step S2, The value range is 0.03~0.25, which meets the requirements for bidirectional stable operation of the reversible impeller; , The value ranges from 0 to 2 to ensure a relatively uniform distribution of thickness at the leading and trailing edges of the airfoil.

6. The parametric design method for the S-shaped reversible airfoil according to claim 1, characterized in that: In step S2, , , , Calculated directly from core design parameters: ； ； ； ； in, This represents the maximum relative thickness of the airfoil. The relative position of the maximum thickness; The thickness distribution leading edge shape factor; is the shape factor of the trailing edge of the thickness distribution.

7. The parametric design method for the S-shaped reversible airfoil according to claim 1, characterized in that: In step S3, based on the mid-arc coordinates and half-thickness value, the discrete coordinates of the upper and lower surfaces of the airfoil are calculated respectively, as shown in the following formula: （3）； （4）； in, , These represent the first and second surfaces of the airfoil, respectively. The ordinate of each sampling point; Indicates the first arc of the S-shaped curve The ordinates of each sampling point are calculated using formula (1); Represents the half-thickness distribution. The ordinates of each sampling point are calculated using formula (2).

8. The parametric design method for the S-shaped reversible airfoil according to claim 1, characterized in that: Before step S1, there is also step S0, establishing a basic coordinate system; The airfoil chord length is dimensionless and set to 1. The coordinates of the airfoil leading edge are set to (0,0) and the coordinates of the trailing edge are set to (1,0). The chord is a horizontal straight line from the leading edge to the trailing edge. All geometric parameters are dimensionless values ​​relative to the chord length.

9. The parametric design method for the S-shaped reversible airfoil according to claim 1, characterized in that: The parametric design method for the S-shaped reversible airfoil is applicable to the impeller blade design of reversible fluid machinery, which includes reversible water pump turbines, reversible fans, and reversible pumps.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the parametric design method for the S-shaped reversible airfoil as described in any one of claims 1 to 9.

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