A coordinate analytical generating method of S-shaped unit chord airfoil
By generating expressions for the mid-curve and thickness variation laws of S-shaped unit chord airfoils, the problem of generating S-shaped airfoil coordinates was solved, enabling precise control of characteristic variables and performance optimization.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- JIANGSU UNIV
- Filing Date
- 2025-06-16
- Publication Date
- 2026-07-14
AI Technical Summary
Existing technologies cannot effectively generate coordinate analysis of S-shaped airfoils and lack the ability to control elements such as leading edge radius and centerline leading edge angle, thus failing to meet the performance consistency requirements of bidirectional axial flow fluid machinery.
A coordinate analysis method for generating S-shaped unit chord airfoils is provided. By using a general expression describing the variation law of the mid-curve and thickness, the exact expressions of the mid-curve and thickness are generated, and the coordinates of discrete points are calculated. Combined with the tilt angle, the coordinates of discrete points on the upper and lower sides of the airfoil are generated.
It achieves precise control and flexible adjustment of S-shaped airfoils, solves the problem of missing coordinate generation for S-shaped airfoils, and promotes the optimization of hydraulic or aerodynamic performance.
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Figure CN120724608B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of axial flow impeller design technology, and in particular to a coordinate analytical generation method for an S-shaped unit chord airfoil. Background Technology
[0002] Axial flow pumps, axial flow fans, and axial flow compressors dominate applications requiring high flow rates and low head, and are widely used in key sectors such as energy, shipbuilding, and aviation. The axial flow impeller is the core component of axial flow fluid machinery, and its hydraulic or aerodynamic performance directly determines the overall efficiency of the machinery. An axial flow impeller is composed of airfoils stacked according to a specific pattern across its cross-sections. The geometry of the airfoils directly governs the energy conversion patterns and performance within the impeller. The geometry of the airfoils is determined by airfoil coordinates, and the accurate and efficient generation of these coordinates is crucial for the intelligent optimization of the airfoil's hydraulic or aerodynamic performance.
[0003] In the field of airfoil research, characteristic variables such as camber, thickness, and position are typically normalized to the airfoil chord length, resulting in a unit chord length airfoil. Currently, the main methods for generating airfoil coordinates include shape function perturbation, parametric methods, and analytical methods. Among these, analytical methods provide a mathematical description of airfoil coordinates through defined functional expressions, accurately determining the airfoil coordinates corresponding to different characteristic variables, and thus rapidly generating airfoils with different geometric shapes. Comparatively, analytical methods offer significant advantages such as zero error, high computational efficiency, and clear physical meaning of characteristic variables, making them the most ideal method for airfoil coordinate generation.
[0004] In certain specialized applications, axial-flow fluid machinery requires bidirectional operation with high consistency in performance during both directions. Examples include bidirectional ballast pumps in the attitude active control system of marine engineering equipment and bidirectional turbines in tidal power generation systems. To meet these requirements, such axial-flow impellers must employ an S-shaped impeller, meaning their airfoil is S-shaped. The geometry of the S-shaped airfoil is centrally symmetrical about the midpoint of the chord, and its leading-edge radius and trailing-edge radius are both greater than zero and equal, exhibiting an S-shaped double-rounded-head characteristic.
[0005] Existing methods for generating airfoil coordinates, such as the NACA airfoil analytical formula and the Zhukowski transform airfoil analytical formula, can generate coordinates for conventional airfoils with rounded noses and pointed tails. However, they cannot be directly applied to relatively complex S-shaped airfoils, and their flexibility is low, lacking the ability to control elements such as leading edge radius and centerline leading edge angle. Summary of the Invention
[0006] This invention proposes a coordinate analytical generation method for S-shaped unit chord length airfoils, which solves the technical problem of the lack of coordinate analytical generation methods for S-shaped airfoils, expands the automatic modeling method for S-shaped airfoils, and can promote the development of hydraulic or aerodynamic performance optimization technology for S-shaped airfoils.
[0007] To achieve the above objectives, the present invention provides a method for analytically generating the coordinates of an S-shaped unit chord airfoil, comprising:
[0008] Determine the general expressions for describing the variation law of the mid-curvature of the airfoil and the general expressions for describing the variation law of the airfoil thickness, and obtain the mid-curvature characteristic variables and the thickness characteristic variables;
[0009] The values of the mid-arc feature variable and the thickness feature variable are respectively assigned to generate the exact expressions for the mid-arc and the thickness.
[0010] Discretized abscissa points are generated within a preset chord length range, and the ordinate of the mid-arc line and the ordinate of the thickness corresponding to each abscissa point are calculated based on the exact expression.
[0011] The tilt angle at each discrete point is obtained based on the exact expression of the middle arc.
[0012] By combining the abscissa point, the ordinate of the mid-arc line, the ordinate of the thickness, and the tilt angle, discrete point coordinates on the upper and lower sides of the airfoil are generated through analytical calculation.
[0013] Preferably, the condition satisfied by the variation law of the middle arc is:
[0014] The mid-arc line is centrally symmetrical about the midpoint of the chord, and the first half of the mid-arc line has a downward arch shape within the preset horizontal coordinate range.
[0015] Preferably, the characteristic variable of the middle arc includes: the ratio of the maximum vertical height of the middle arc to the chord length. The ratio of the horizontal distance from the point of maximum curvature of the first half of the arc to the leading edge of the airfoil to the chord length. The angle α between the tangent of the mid-curve at the leading edge of the airfoil and the positive x-axis;
[0016] in, The range of values is The range of values is
[0017] Preferably, the angle α between the tangent of the mid-arc line at the leading edge of the airfoil and the positive x-axis is described in the... After being determined, the range of values for α is as follows: The unit is radians.
[0018] Preferably, the thickness variation pattern is as follows:
[0019] The thickness distribution of the airfoil is axially symmetric about the midpoint of the chord, and the maximum thickness value is obtained at the midpoint of the chord.
[0020] Preferably, the thickness characteristic variables include:
[0021] The ratio of the maximum inscribed circle diameter to the chord length of the airfoil The ratio of the radius of the inscribed circle to the chord length at the leading edge of the airfoil And the adjustment angle γ according to the variation law of airfoil thickness.
[0022] Preferably, the ratio of the maximum inscribed circle diameter to the chord length of the airfoil is... The range of values is
[0023] Preferably, the ratio of the radius of the inscribed circle at the leading edge of the airfoil to the chord length is... And the value of the adjustment angle γ for the variation law of airfoil thickness is to be determined. The value is determined after being taken, wherein the... The range of values is The range of values for γ is The unit is radians.
[0024] Preferably, the conditions for generating the discretized abscissa points are:
[0025] The distribution density of the abscissa points is higher at the leading and trailing edges of the airfoil than in the middle region, and the total number of abscissa points is not less than 100.
[0026] Preferably, the discrete point coordinates of the upper and lower sides of the airfoil are generated through analytical calculation, including:
[0027]
[0028]
[0029] In the formula, X u Y represents the x-coordinate of discrete points on the upper side of the airfoil; u X represents the ordinate of discrete points on the upper side of the airfoil; l Y represents the x-coordinate of discrete points on the underside of the airfoil; l X represents the ordinate of the discrete points on the underside of the airfoil, X represents the number of discretized x-coordinate points, and Y represents the ordinate of the discrete points on the underside of the airfoil. t Y is the ordinate of the thickness, Φ is the inclination angle at each discrete point of the middle arc, and Y... c The point is the ordinate of the middle arc.
[0030] Compared with the prior art, the present invention has the following advantages and technical effects:
[0031] This invention provides general expressions for describing the mid-curve and thickness variation laws of S-shaped unit chord airfoils, enabling precise control of characteristic variables such as maximum relative camber, maximum relative thickness, and leading edge relative radius, as well as effective adjustment of the mid-curve and thickness variation laws. It also possesses the function of generating coordinates of S-shaped unit chord airfoils with different geometric shapes, solving the technical problem of the lack of coordinate generation methods for S-shaped airfoils, expanding the automatic modeling method for S-shaped airfoils, and promoting the development of S-shaped airfoil hydraulic or aerodynamic performance optimization technology. Attached Figure Description
[0032] 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. In the drawings:
[0033] Figure 1 This is a flowchart illustrating a method for generating coordinates of an S-shaped unit chord airfoil according to an embodiment of the present invention.
[0034] Figure 2 This is a data map of 100 abscissa points generated within the [0,1] chord length range of this invention embodiment;
[0035] Figure 3 This is a scatter plot of the coordinate points of the arc line in the airfoil according to an embodiment of the present invention;
[0036] Figure 4 This is a scatter plot of the airfoil thickness coordinates according to an embodiment of the present invention;
[0037] Figure 5 This is an airfoil shape diagram corresponding to the airfoil coordinates generated by the analysis in an embodiment of the present invention. Detailed Implementation
[0038] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.
[0039] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.
[0040] This embodiment proposes a method for analytically generating the coordinates of an S-shaped unit chord airfoil, such as... Figure 1 ,include:
[0041] Determine the general expressions for describing the variation law of the mid-curvature of the airfoil and the general expressions for describing the variation law of the airfoil thickness, and obtain the mid-curvature characteristic variables and the thickness characteristic variables;
[0042] Values are assigned to the mid-curve feature variable and the thickness feature variable respectively, generating exact expressions for the mid-curve and the thickness.
[0043] Discretized abscissa points are generated within a preset chord length range, and the ordinate of the mid-arc line and the ordinate of the thickness corresponding to each abscissa point are calculated based on the exact expression.
[0044] The tilt angle at each discrete point is obtained based on the exact expression of the middle arc.
[0045] By combining the horizontal coordinate point, the vertical coordinate of the mid-arc line, the vertical coordinate of the thickness, and the tilt angle, discrete point coordinates on the upper and lower sides of the airfoil are generated through analytical calculation.
[0046] This embodiment provides a general expression for describing the mid-curve and thickness variation law of an S-shaped unit chord airfoil, realizing precise control of characteristic variables such as maximum relative camber, maximum relative thickness, and leading edge relative radius, as well as effective adjustment of the mid-curve variation law and thickness variation law. It also has the function of generating coordinate analysis of S-shaped unit chord airfoils with different geometric shapes.
[0047] Specifically, the general expression describing the variation law of the arc in an airfoil is as shown in equation ①:
[0048] ①
[0049] In the formula, x is the horizontal axis variable; y c The vertical coordinate variable of the middle arc; It is the ratio of the maximum vertical height of the middle arc to the chord length; α is the ratio of the horizontal distance from the point of maximum curvature of the first half of the arc (corresponding to the x-coordinate range of [0, 0.5]) to the leading edge of the airfoil to the chord length; α is the angle between the tangent of the arc at the leading edge of the airfoil and the positive x-axis; sign is the sign function, when x < 0, sign(x) = -1, when x = 0, sign(x) = 0, and when x > 0, sign(x) = 1.
[0050] Furthermore, the condition that the variation law of the middle arc line satisfies is:
[0051] The mid-arc line is centrally symmetrical about the midpoint of the chord, and the first half of the mid-arc line has a downward arch shape within the preset horizontal coordinate range.
[0052] Furthermore, the characteristic variables of the mid-arc line include: the ratio of the maximum vertical height of the mid-arc line to the chord length.
[0053] The ratio of the horizontal distance from the point of maximum curvature of the first half of the arc to the leading edge of the airfoil to the chord length. The angle α between the tangent of the mid-curve at the leading edge of the airfoil and the positive x-axis;
[0054] in, The range of values is The range of values is
[0055] Specifically, the angle between the tangent of the mid-arc line at the leading edge of the airfoil and the positive x-axis is described in...
[0056] After being determined, the range of values for α is as follows: The unit is radians.
[0057] Furthermore, the general expression describing the variation law of airfoil thickness is as shown in Equation ②:
[0058] ②
[0059] In the formula, x is the horizontal axis variable; y t The thickness is the ordinate variable; This is the ratio of the maximum inscribed circle diameter of the airfoil to the chord length; γ is the ratio of the radius of the inscribed circle at the leading edge of the airfoil to the chord length; γ is the adjustment angle for the variation of airfoil thickness.
[0060] Furthermore, the thickness variation pattern is as follows:
[0061] The thickness distribution of the airfoil is axially symmetric about the midpoint of the chord, and the maximum thickness value is obtained at the midpoint of the chord.
[0062] Specifically, the thickness characteristic variables include:
[0063] The ratio of the maximum inscribed circle diameter to the chord length of the airfoil The ratio of the radius of the inscribed circle to the chord length at the leading edge of the airfoil And the adjustment angle γ according to the variation law of airfoil thickness.
[0064] Furthermore, the ratio of the maximum inscribed circle diameter to the chord length of the airfoil The range of values is
[0065]
[0066] Specifically, the ratio of the radius of the inscribed circle to the chord length at the leading edge of the airfoil. And the value of the adjustment angle γ for the variation law of airfoil thickness is to be determined. The value is determined after being taken, wherein the... The range of values is The range of values for γ is The unit is radians.
[0067] Furthermore, values are assigned to the mid-curve feature variable and the thickness feature variable respectively to generate the exact expressions for the mid-curve and the thickness.
[0068] Specifically, based on the value ranges of the mid-curve characteristic variable and the thickness characteristic variable, values are assigned to the mid-curve characteristic variable and the thickness characteristic variable, respectively, and these values are substituted into equations ① and ② to generate the exact expressions y for the mid-curve and thickness. c =f1(x), y t = f2(x).
[0069] Furthermore, the conditions for generating discretized abscissa points are:
[0070] The distribution density of the abscissa points is higher at the leading and trailing edges of the airfoil than in the middle region, and the total number of abscissa points is not less than 100.
[0071] Specifically, using the discretization strategy shown in Equation ③, the x-coordinate points within the chord length range of [0,1] are generated:
[0072] ③
[0073] Where, x i Let be the i-th x-coordinate point in X; n is the total number of x-coordinate points in X.
[0074] Furthermore, determine the ordinate points of the mid-curve and thickness:
[0075] Substitute the generated X into the exact expressions for the mid-curve and thickness y, respectively. c =f1(x), y t In f2(x), find the ordinate point Y of the middle arc. c The ordinate point Y of the thickness t .
[0076] Furthermore, find the angle of inclination at each discrete point on the mid-arc:
[0077] The exact expression y of the generated mid-arc line c =f1(x) Using the method shown in Equation ④, the inclination angle at each discrete point on the middle arc is obtained:
[0078] ④
[0079] Where Φ is the inclination angle at each discrete point of the mid-arc; φ i Let be the i-th tilt angle in Φ.
[0080] Furthermore, the generated X and Y c and Y t Substituting Φ into equation ⑤, we can find the analytical solution for the airfoil coordinates:
[0081] ⑤
[0082] In the formula, X u Y represents the x-coordinate of discrete points on the upper side of the airfoil; u X represents the ordinate of discrete points on the upper side of the airfoil; l Y represents the x-coordinate of discrete points on the underside of the airfoil; l X represents the ordinate of the discrete points on the underside of the airfoil, X represents the number of discretized x-coordinate points, and Y represents the ordinate of the discrete points on the underside of the airfoil. t Y is the ordinate of the thickness, Φ is the inclination angle at each discrete point of the middle arc, and Y... c The point is the ordinate of the middle arc.
[0083] Thus, the coordinate analysis generation of the S-shaped unit chord airfoil was achieved.
[0084] This embodiment provides a general expression describing the mid-curve and thickness variation laws of an S-shaped unit chord airfoil. It achieves precise control of characteristic variables such as maximum relative camber, maximum relative thickness, and leading edge relative radius, as well as effective adjustment of the mid-curve and thickness variation laws. It has the function of generating coordinate analysis for S-shaped unit chord airfoils with different geometric shapes, solves the technical problem of the lack of coordinate analysis generation methods for S-shaped airfoils, expands the automatic modeling method for S-shaped airfoils, and can promote the development of S-shaped airfoil hydraulic or aerodynamic performance optimization technology.
[0085] To more clearly illustrate the technical solution of the present invention, specific embodiments are provided below for description:
[0086] (1) Determine the general expression describing the variation law of the arc in the airfoil, as shown in Equation ①;
[0087] (2) Determine the general expression describing the variation law of airfoil thickness, as shown in Equation ②;
[0088] (3) Value the feature variables to generate precise expressions for the arc and thickness. Based on the value range of the arc feature variable given in step (1) and the value range of the thickness feature variable given in step (2), value the arc feature variable and the thickness feature variable. The value results are as follows: Substituting these values into equations ① and ② respectively, we can then generate the precise expressions for the mid-curve and thickness, y. c =f1(x), y t =f2(x);
[0089] (4) Generate the x-coordinate points within the chord length range of [0,1]. Take n = 100 and use equation ③ to generate the x-coordinate points X within the chord length range of [0,1]. The result is as follows: Figure 2As shown, the distribution of the generated abscissa points clearly exhibits the characteristic of being dense at both ends and sparse in the middle of an airfoil;
[0090] (5) Find the ordinate points of the mid-arc line and thickness, and substitute the X generated in step (4) into the exact expressions y of the mid-arc line and thickness generated in step (3). c =f1(x), y t =f2(x), find the ordinate Y of the middle arc. c The ordinate point Y of the thickness t Discrete points (X, Y) of the mid-curve of the airfoil c Changes such as Figure 3 As shown, the thickness discrete points (X, Y) t Changes such as Figure 4 As shown, the change in the mid-curve is clearly symmetrical about the midpoint of the wing chord, and the first half of the mid-curve has a downward arching shape, which corresponds to... The value of α is exactly the same as the above values. The thickness variation of the airfoil is axially symmetric about the midpoint of the chord and reaches its maximum value at the midpoint of the chord. The value of γ is exactly the same as the values mentioned above;
[0091] (6) Find the inclination angle at each discrete point on the middle arc, using the exact expression y of the middle arc generated in step (3). c =f1(x) The inclination angle at each discrete point on the middle arc is obtained by using the method shown in Equation ④.
[0092] (7) Find the analytical solution of the airfoil coordinates, and combine the X generated in step (4) and the Y obtained in step (5). c and Y t Substituting Φ obtained in step (6) into equation ⑤, we obtain the analytical solution for the airfoil coordinates, and the corresponding airfoil shape is as follows. Figure 5 As shown, it is obvious that its corresponding The values are exactly the same as those mentioned above.
[0093] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.
Claims
1. A method for analytically generating coordinates of an S-shaped unit chord airfoil, characterized in that, include: Determine the general expressions for describing the variation law of the mid-curvature of the airfoil and the general expressions for describing the variation law of the airfoil thickness, and obtain the mid-curvature characteristic variables and the thickness characteristic variables; The values of the mid-arc feature variable and the thickness feature variable are respectively assigned to generate the exact expressions for the mid-arc and the thickness. Discretized abscissa points are generated within a preset chord length range, and the ordinate of the mid-arc line and the ordinate of the thickness corresponding to each abscissa point are calculated based on the exact expression. The tilt angle at each discrete point is obtained based on the exact expression of the middle arc. By combining the abscissa point, the ordinate of the mid-arc line, the ordinate of the thickness, and the tilt angle, discrete point coordinates on the upper and lower sides of the airfoil are generated through analytical calculation. The characteristic variables of the mid-arc line include: the ratio of the maximum vertical height of the mid-arc line to the chord length. The ratio of the horizontal distance from the point of maximum curvature of the first half of the arc to the leading edge of the airfoil to the chord length. The tangent of the mid-curve at the leading edge of the airfoil and Angle in the positive direction of the axis ; in, The range of values is , The range of values is ; The thickness characteristic variables include: The ratio of the maximum inscribed circle diameter to the chord length of the airfoil The ratio of the radius of the inscribed circle at the leading edge of the airfoil to the chord length and the adjustment angle of the airfoil thickness variation law ; Based on the value ranges of the mid-curve characteristic variable and the thickness characteristic variable, values are assigned to the mid-curve and thickness characteristic variables. These values are then substituted into the general expressions describing the variation law of the airfoil's mid-curve and the general expressions describing the variation law of the airfoil's thickness, respectively, to generate the precise expressions for the mid-curve and thickness. , ; The exact expression of the generated mid-arc line beg The inclination angle at each discrete point on the middle arc is obtained using the following method: ； in, The angle of inclination at each discrete point on the middle arc; for The first in An angle of inclination; for The first in One horizontal coordinate point; The condition that the variation law of the middle arc line satisfies is: The mid-arc line is centrally symmetrical about the midpoint of the chord, and the first half of the mid-arc line has a downward arch shape within the preset horizontal coordinate range.
2. The coordinate analysis generation method for an S-shaped unit chord airfoil according to claim 1, characterized in that, The tangent of the mid-arc line at the leading edge of the airfoil and... Angle in the positive direction of the axis In the , Determine the value after obtaining it. The range of values is The unit is radians.
3. The coordinate analysis generation method for an S-shaped unit chord airfoil according to claim 1, characterized in that, The thickness variation pattern is as follows: The thickness distribution of the airfoil is axially symmetric about the midpoint of the chord, and the maximum thickness value is obtained at the midpoint of the chord.
4. The coordinate analysis generation method for an S-shaped unit chord airfoil according to claim 1, characterized in that, The ratio of the maximum inscribed circle diameter to the chord length of the airfoil The range of values is .
5. The coordinate analysis generation method for an S-shaped unit chord airfoil according to claim 4, characterized in that, The ratio of the radius of the inscribed circle to the chord length at the leading edge of the airfoil and the adjustment angle of the airfoil thickness variation law The value to be determined The value is determined after being taken, wherein the... The range of values is , The range of values is The unit is radians.
6. The coordinate analysis generation method for an S-shaped unit chord airfoil according to claim 1, characterized in that, The conditions for generating the discretized abscissa points are: The distribution density of the abscissa points is higher at the leading and trailing edges of the airfoil than in the middle region, and the total number of abscissa points is not less than 100.
7. The coordinate analysis generation method for an S-shaped unit chord airfoil according to claim 1, characterized in that, Discrete point coordinates on the upper and lower sides of the airfoil are generated through analytical calculation, including: ； ； ； ； In the formula, The x-coordinate of the discrete points on the upper side of the airfoil; The ordinate of the discrete points on the upper side of the airfoil; The x-coordinate of the discrete points on the underside of the airfoil; Let be the ordinate of the discrete points on the underside of the airfoil. To discretize the number of x-axis points, The vertical coordinate point is the thickness. Let be the inclination angle at each discrete point on the middle arc. The point is the ordinate of the middle arc.