A method and system for continuously shaping the leading edge curvature of a two-dimensional blade profile of a low-pressure turbine

Abstract

The application discloses a low-pressure turbine two-dimensional blade profile front edge curvature continuous modeling method and system, relates to the turbine blade profile parameterization modeling technical field, and comprises the following steps: acquiring a blade profile line and representing the same as a parameter curve, and setting a connecting point candidate interval at the front end of the blade profile; determining a front edge point, a front edge dimension parameter, a front edge structure angle and two side half wedge angles, and generating a front edge feature section sharing the same front edge point according to the same; selecting a candidate blade profile connecting point from the candidate interval, establishing a Bezier transition section between the end point of the front edge feature section and the candidate blade profile connecting point, and making the two ends of the transition section meet position continuity, tangential continuity and curvature continuity respectively; performing a geometric feasibility and curvature quality check on the transition section, determining the blade profile connecting point and the corresponding transition section according to the check result; and sequentially splicing the reserved blade profile, the transition section and the front edge feature section, and outputting complete blade profile coordinates. The application can keep the front edge engineering features and realize the curvature continuous connection of the front edge and the blade profile.

Description

Technical Field

[0001] This application relates to the field of parametric modeling technology for aero-engine turbine blade profiles, specifically to a method and system for continuous modeling of the leading edge curvature of a two-dimensional blade profile for a low-pressure turbine. Background Technology

[0002] The leading edge of a two-dimensional blade profile for low-pressure turbines has a small geometric scale and a high curvature level. Significant differences in tangential direction and curvature are typically observed between the pressure and suction sides of the blade in the leading-edge region. The leading-edge radius or semi-elliptical axis, the leading-edge configuration angle, and the semi-wedge angles on both sides determine the local thickness, geometric bluntness, windward direction, and lateral extension range of the leading edge. Changes in curvature at the junction of the leading edge and the blade body affect the geometric compliance of the complete blade profile, the stability of mesh generation, and the geometric reliability of subsequent aerodynamic analyses.

[0003] In existing two-dimensional blade profile designs for low-pressure turbines, circular and elliptical arcs offer advantages such as fewer parameters and clear engineering meaning, allowing direct control of the leading-edge radius, elliptical semi-axis, leading-edge construction angle, and the semi-wedge angles on both sides. However, given the pressure and suction sides of the blade, once the endpoint positions and tangential directions of a standard circular or elliptical arc are determined by the leading-edge parameters, their endpoint curvature is also determined, making it difficult to achieve continuous curvature connection with the predetermined blades on both sides simultaneously. If the connection points from the common leading-edge point to the predetermined blades on the pressure and suction sides are fitted using a single high-degree-of-freedom curve, the blade connection conditions may negatively influence the leading-edge body, causing unexpected changes in the leading-edge dimensions, construction angle, semi-wedge angle, and local curvature distribution. When a large number of control points, control point coordinates, weights, or nodal parameters are simultaneously solved as free variables, parameter coupling, approximate equivalent solutions, local bulges, folds, and curvature oscillations can easily occur, making it difficult to maintain the clear engineering meaning of the leading-edge parameters.

[0004] Furthermore, the leading edge shapes that can be represented by standard circular and elliptical arcs are determined by a small number of scale and angle parameters, making it difficult to further express non-elliptical local shapes or specific curvature distributions proposed by users according to design needs. Directly moving the control points of free curves in an unconstrained manner can easily alter the common leading edge point, leading edge scale, construction angle, semi-wedge angle, and end conditions of the characteristic segment, potentially generating local curvature spikes or oscillations. Therefore, a new parametric modeling method is needed. This method should first generate leading edge characteristic segments using parameters with clear engineering significance, ensuring that the leading edge body is not excessively affected by the predetermined blade connection conditions. Then, an independent low-dimensional transition segment should be set between the leading edge characteristic segment and the predetermined blade, satisfying the end position, tangential, and curvature conditions using a small number of parameters with clear geometric functions, and checking the curvature changes throughout the transition segment. When the current combination of leading edge parameters cannot form a satisfactory transition with the predetermined blade, the cause of incompatibility should be identified and graded adjustments should be performed. Summary of the Invention

[0005] This application addresses the technical problems in the prior art where it is difficult to simultaneously achieve continuous connection of position, tangential direction and curvature between the leading edge feature segment and the predetermined blade body, and where overall fitting of a single high-degree-of-freedom curve can easily lead to the destruction of leading edge engineering parameters, parameter coupling and curvature oscillation. It provides a method and system for continuous leading edge curvature modeling of a two-dimensional blade profile for low-pressure turbines.

[0006] To achieve the above objectives, this application adopts the following technical solution: In a first aspect, this application provides a method for continuous modeling of the leading edge curvature of a two-dimensional blade profile for a low-pressure turbine, comprising the following steps: S1, Obtain the blade profile of the two-dimensional blade of the low-pressure turbine, the blade profile includes the pressure side blade profile and the suction side blade profile, represent the blade profile as a parametric curve with at least second-order differentiability, and set the connection point candidate interval defined by the normalized blade arc length at the front end of the pressure side blade and the suction side blade respectively. S2, determine the leading edge point, leading edge dimensional parameters, leading edge construction angle, pressure-side half-wedge angle, and suction-side half-wedge angle; determine the leading edge construction direction based on the leading edge construction angle, wherein the leading edge construction direction is the direction from the leading edge point to the center of the circle or the center of the ellipse; based on the leading edge point, the leading edge dimensional parameters, the leading edge construction direction, the pressure-side half-wedge angle, and the suction-side half-wedge angle, generate a pressure-side leading edge feature segment and a suction-side leading edge feature segment that share the same leading edge point; S3, starting from the candidate connection point interval near the leading edge, select multiple positions as candidate blade connection points according to a preset search step size; for each candidate blade connection point, extract the position, tangential direction, and curvature of the candidate blade connection point; establish a transition segment with the endpoint of the leading edge feature segment and the candidate blade connection point as the two ends, so that the transition segment satisfies positional continuity, tangential continuity, and curvature continuity with the leading edge feature segment and the blade profile at both ends respectively; S4, Perform geometric feasibility and curvature quality checks on the transition section, and determine the blade connection points and corresponding transition sections that meet the requirements from the candidate blade connection points based on the results of the geometric feasibility and curvature quality checks; S5. Based on the determined blade connection point and corresponding transition section, the retained pressure-side blade, pressure-side transition section, pressure-side leading edge feature section, suction-side leading edge feature section, suction-side transition section and retained suction-side blade are sequentially spliced ​​together to output the complete two-dimensional blade coordinates of the low-pressure turbine.

[0007] Furthermore, in S2, the pressure-side leading edge feature segment and the suction-side leading edge feature segment are generated using a circular leading edge pattern. The specific process is as follows: Using the leading edge point as the starting point of the arc, the leading edge dimensional parameter as the radius, and the leading edge construction direction as the direction of the center of the circle, the pressure side half-wedge angle and the suction side half-wedge angle are used to determine the end point of the pressure side leading edge feature segment and the end point of the suction side leading edge feature segment, respectively. The pressure-side arc segment and the suction-side arc segment are formed from the leading edge point to the two arc endpoints, respectively, and serve as the pressure-side leading edge feature segment and the suction-side leading edge feature segment.

[0008] Furthermore, in S2, the pressure-side leading edge feature segment and the suction-side leading edge feature segment are generated using an elliptical leading edge pattern. The specific process is as follows: The ellipse shape is determined by the first and second semi-axes in the leading edge dimensional parameters, the major axis direction of the ellipse is determined by the leading edge construction direction, and the end point of the pressure-side leading edge feature segment and the end point of the suction-side leading edge feature segment are determined by the pressure-side semi-wedge angle and the suction-side semi-wedge angle, respectively. An elliptical arc segment on the pressure side and an elliptical arc segment on the suction side are formed from the leading edge point to the endpoints of the two elliptical arcs, respectively, serving as the leading edge feature segment on the pressure side and the leading edge feature segment on the suction side.

[0009] Furthermore, in S2, the pressure-side leading edge feature segment and the suction-side leading edge feature segment are generated using a general irregular leading edge pattern. The specific process is as follows: Using the elliptical leading edge determined by the leading edge scale parameters, the leading edge construction angle, the pressure side half-wedge angle, and the suction side half-wedge angle as a reference ellipse, an initial leading edge feature segment is constructed using a fifth-order Bézier curve on both the pressure side and the suction side, such that the initial leading edge feature segment has the same position, first derivative, and second derivative as the reference ellipse at both ends. While maintaining the position of the common leading edge point, the tangential direction of each side at the common leading edge point determined by the reference ellipse, the position and tangential direction of the endpoints of the leading edge feature segments on each side, and ensuring that the pressure side curvature and the suction side curvature are equal at the common leading edge point after being converted to the same circumferential direction of the complete airfoil, the internal control points of the fifth-order Bézier curve are adjusted under constraints to obtain the pressure side leading edge feature segment and the suction side leading edge feature segment.

[0010] Furthermore, the constrained adjustment is implemented based on the control point adjustment amount or curvature optimization target given by the user; The constrained adjustment, while maintaining the reference ellipse's scale parameters, the leading edge construction angle, and the reference engineering features defined by the half-wedge angles on both sides, forms a user-defined non-elliptical local shape and adjusts the internal curvature distribution and endpoint curvature of the leading edge feature segments on both sides. The constrained adjustment does not change the common leading edge point, the endpoints of the leading edge feature segments on both sides, or their tangential directions; After adjustment, the curvature of the corresponding leading edge feature segment endpoint is recalculated, and the endpoint curvature is used as the starting curvature condition of the corresponding transition segment.

[0011] Furthermore, in S3, each transition segment is a quintic Bézier curve with six control points, where the six control points are P0 to P5, respectively. P0 is the endpoint of the leading edge feature segment, and P5 is the blade connection point. After the positions, tangential directions, and curvatures of P0 and P5 are determined, the curve is scaled using the first derivative of the starting point. l 0. Endpoint first derivative scale l 1. Tangential component of the second derivative at the starting point m 0 and the tangential component of the second derivative at the endpoint m 1 Four scalar parameters determine the positions of P1, P2, P3, and P4:

[0012] Where t0 and t1 are the unit tangent vectors at P0 and P5, respectively, and n0 and n1 are the unit normal vectors at P0 and P5 determined by the parameterized direction, respectively. k 0 and k 1 represents the signed curvature at points P0 and P5, respectively. l 0>0 and l 1>0.

[0013] Furthermore, in S4, the curvature quality check includes: The curvature of the blade profile, the curvature of the leading edge feature segment, and the curvature of the transition segment are converted into signed curvatures. The signed curvatures of the pressure side and the suction side are converted to the same circumferential direction of the complete blade profile. The normalized curvature reverse change, the maximum relative curvature overshoot, and the root mean square deviation of the curvature matching at the blade connection end are calculated respectively. Preferred limits and maximum allowable limits are set respectively. When the normalized curvature inverse change, the maximum relative curvature overshoot, and the root mean square deviation of the blade connection end curvature matching all do not exceed the corresponding maximum allowable limit, the curvature quality of the transition section is deemed acceptable.

[0014] Furthermore, in S4, the geometric feasibility check includes: determining whether the transition section has self-intersection, whether it has foldback, whether it exceeds the preset local geometric boundary, and whether the pressure side and suction side maintain a positive local airfoil thickness after combination; when all the above checks are passed, the transition section is determined to meet the geometric feasibility requirements.

[0015] Further, in S4, when no transition segment satisfying the geometric feasibility requirements and with acceptable curvature quality is obtained within the candidate connection point interval, the incompatibility cause is identified based on the unmet geometric feasibility condition or the curvature quality evaluation value exceeding the maximum allowable limit, and at least one of the following adjustments is performed: Expand the candidate interval for the corresponding side connection point, so that the candidate interval extends along the blade in a direction away from the leading edge; Shorten the corresponding side leading edge feature segment; adjust the corresponding side half-wedge angle; adjust the leading edge structural angle; After each adjustment, the affected leading edge feature segments are regenerated and the corresponding transition segments are solved. If the required transition section cannot be obtained after the aforementioned adjustments, the current leading edge parameter combination is returned as incompatible with the blade profile, and the complete blade coordinates are not output.

[0016] Secondly, this application also provides a continuous leading edge curvature modeling system for a two-dimensional blade profile of a low-pressure turbine, comprising: The blade acquisition module is used to acquire the pressure side blade profile and the suction side blade profile of the two-dimensional blade of the low-pressure turbine, represent the blade profile as a parametric curve that is at least second-order differentiable, and set the connection point candidate interval defined by the normalized blade arc length at the front end of the pressure side blade and the suction side blade, respectively. The leading edge generation module is used to determine the leading edge point, leading edge scale parameters, leading edge construction angle, pressure-side half-wedge angle, and suction-side half-wedge angle; determine the leading edge construction direction based on the leading edge construction angle, wherein the leading edge construction direction is the direction from the leading edge point to the center of the circle or the center of the ellipse; and generate pressure-side leading edge feature segments and suction-side leading edge feature segments that share the same leading edge point based on the leading edge point, the leading edge scale parameters, the leading edge construction direction, the pressure-side half-wedge angle, and the suction-side half-wedge angle. The transition segment establishment module is used to select a position from the candidate connection point interval as a candidate blade connection point, extract the position, tangential direction and curvature of the candidate blade connection point; and establish a transition segment with the endpoint of the leading edge feature segment and the candidate blade connection point as the two ends, so that the transition segment satisfies positional continuity, tangential continuity and curvature continuity with the leading edge feature segment and the blade profile at both ends, respectively. The quality inspection module is used to perform geometric feasibility and curvature quality checks on the transition section, and to determine the blade connection points and corresponding transition sections that meet the requirements from the candidate blade connection points based on the results of the geometric feasibility and curvature quality checks. The blade profile output module is used to sequentially splice the retained pressure-side blade, pressure-side transition section, pressure-side leading edge feature section, suction-side leading edge feature section, suction-side transition section and retained suction-side blade based on the determined blade connection point and corresponding transition section, and output the complete two-dimensional blade profile coordinates of the low-pressure turbine.

[0017] Compared with the prior art, this application has the following beneficial effects: This application provides a method for continuous leading-edge curvature modeling of a two-dimensional airfoil for low-pressure turbines. The method separates the leading-edge feature segment and the transition segment into independent control components: the leading-edge feature segment is directly generated from the leading-edge point, dimensional parameters, construction angle, and half-wedge angle, maintaining the determinism of the leading-edge engineering features; the transition segment only handles the geometric coordination between the endpoints of the leading-edge feature segment and the candidate blade connection point. By establishing positional continuity, tangential continuity, and curvature continuity connections at both ends with the leading-edge feature segment and the blade profile respectively, the leading-edge engineering features are not affected by the blade connection conditions. Based on this, by performing geometric feasibility and curvature quality checks on the transition segment, blade connection points that meet the requirements are selected from the candidate interval. This avoids parameter coupling, local bulging, and folding caused by high-degree-of-freedom fitting, enabling the output complete airfoil to achieve a continuous curvature transition with the predetermined blade while maintaining the leading-edge engineering features.

[0018] Furthermore, by performing geometric feasibility checks on the transition section, including self-intersection, folding, local geometric boundaries, and airfoil thickness, infeasible solutions are eliminated from the geometric constraint level. When no suitable transition section is found within the candidate interval for the connection point, hierarchical adjustments are made based on the incompatibility reasons, such as expanding the candidate interval, shortening the leading edge feature segment, or adjusting the half-wedge angle and construction angle. After each adjustment, the leading edge feature segment is regenerated and the transition section is solved until a connection scheme that meets the geometric feasibility and curvature quality requirements is obtained. This hierarchical adjustment mechanism enables automatic compatibility matching between the leading edge parameters and the given airfoil, avoiding repeated trial and error caused by direct failure when parameter combinations are incompatible, and improving the stability, controllability, and design efficiency of the complete airfoil output results. Attached Figure Description

[0019] To more clearly illustrate the technical solutions of the embodiments of this application, 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 this application and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0020] Figure 1 A flowchart of a method for continuous modeling of the leading edge curvature of a two-dimensional blade profile for low-pressure turbines provided in this application; Figure 2 A schematic diagram illustrating the definitions of the leading edge point, leading edge dimensional parameters, leading edge construction angle, pressure-side half-wedge angle, and suction-side half-wedge angle provided in this application; Figure 3 A schematic diagram of the segmented structure of the pressure-side blade, pressure-side Bessel transition section, pressure-side leading edge characteristic section, common leading edge point, suction-side leading edge characteristic section, suction-side Bessel transition section, and suction-side blade provided for this application; Figure 4 A schematic diagram of the control points and constrained adjustment directions of the fifth-order Bezier feature segments on both sides of the general irregular leading edge provided in this application; Figure 5 A schematic diagram of the control points and end constraints of the fifth-order Bezier transition section provided for this application; Figure 6 A schematic diagram illustrating the connection point search, curvature quality evaluation, and parameter adjustment process provided for this application; Figure 7 Two-dimensional leaf shape and curvature distribution diagram of the elliptical leading edge embodiment provided in this application; Figure 8 A comparison diagram of the local geometry and curvature distribution before and after optimization of the suction-side leading edge in the general irregular leading edge embodiment provided in this application. Detailed Implementation

[0021] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0022] See Figure 1 This application provides a method for continuous modeling of the leading edge curvature of a two-dimensional blade profile for low-pressure turbines, comprising the following steps: S1, Obtain the blade profile of the two-dimensional blade of the low-pressure turbine, the blade profile includes the pressure side blade profile and the suction side blade profile, represent the blade profile as a parametric curve with at least second-order differentiability, and set the connection point candidate interval defined by the normalized blade arc length at the front end of the pressure side blade and the suction side blade respectively. In this step, the pressure-side and suction-side blade profiles of the low-pressure turbine's two-dimensional airfoil are obtained. The input pressure-side and suction-side blade profiles can be represented using Bézier curves, non-uniform rational B-spline (NURBS) curves, spline curves, piecewise polynomial curves, or ordered discrete coordinates.

[0023] Specifically, for the blade profile represented by ordered discrete coordinates, duplicate points and obvious outliers are first removed. Then, a smooth reconstruction is performed based on the arc length parameter, ensuring that the reconstruction error is within a preset geometric tolerance range. The smooth reconstruction can be performed using cubic spline interpolation or NURBS curve least squares fitting. The curvature continuity of the reconstructed curve is numerically checked to ensure that it meets at least the requirement of second-order differentiability.

[0024] Furthermore, for any blade profile B(τ), its normalized blade arc length is defined as:

[0025] In the formula, t 0 corresponds to the parameter of the blade profile near the leading edge on the input side. t 1 represents the parameter of the other end of the blade profile input on this side. s represents the cumulative arc length from the front end of the corresponding blade profile to the current point, and S represents the total arc length of the corresponding input blade profile.

[0026] Candidate connection point intervals defined by the normalized blade arc length are set at the leading edges of both the pressure-side and suction-side blades. The candidate connection point intervals and search step sizes are set independently for the pressure and suction sides to accommodate the differences in the leading-edge geometry of the two blades. The leading edge of the candidate interval should avoid unstable regions in the input blade curvature calculation, while the trailing edge is used to limit the replacement range of the main blade by the new leading-edge local region. Except for the portion replaced by the new leading-edge local region, the rest of the blade remains unchanged.

[0027] S2, determine the leading edge point, leading edge dimensional parameters, leading edge construction angle, pressure-side half-wedge angle, and suction-side half-wedge angle; determine the leading edge construction direction based on the leading edge construction angle, wherein the leading edge construction direction is the direction from the leading edge point to the center of the circle or the center of the ellipse; based on the leading edge point, the leading edge dimensional parameters, the leading edge construction direction, the pressure-side half-wedge angle, and the suction-side half-wedge angle, generate a pressure-side leading edge feature segment and a suction-side leading edge feature segment that share the same leading edge point; like Figure 3 As shown, the leading edge local region of the present invention sequentially includes, along the complete airfoil boundary, a retained pressure-side airfoil, a pressure-side transition section, a pressure-side leading edge feature section, a common leading edge point, a suction-side leading edge feature section, a suction-side transition section, and a retained suction-side airfoil. The leading edge feature section is used to maintain the engineering features defined by the leading edge point, leading edge dimensional parameters, leading edge construction angle, and the two lateral half-wedge angles. The transition section is used to coordinate the position, tangential, and curvature conditions between the endpoint of the leading edge feature section and the predetermined airfoil connection point.

[0028] In this step, the leading edge point can be specified directly by the user, or it can be determined based on existing airfoil reference points and lead-out lengths. For example... Figure 2 As shown, the leading edge construction direction is defined as the direction from the leading edge point to the center of the circle or ellipse. When the global x-axis is taken as the positive direction of the leaf axis and the counterclockwise direction is taken as the positive direction of the angle, the leading edge construction angle β is the directed angle rotated from the global x-axis to the leading edge construction direction. The leading edge construction angle β is used to determine the unit vector e of the leading edge construction direction. β And its counterclockwise normal direction is taken as the unit vector e. n :

[0029] Based on the determined leading edge point, leading edge scale parameters, leading edge structural direction, pressure-side half-wedge angle, and suction-side half-wedge angle, pressure-side leading edge feature segments and suction-side leading edge feature segments sharing the same leading edge point are generated.

[0030] This invention supports circular leading edge mode, elliptical leading edge mode and general irregular leading edge mode.

[0031] As one embodiment, a circular leading-edge pattern is used to generate the pressure-side leading-edge feature segment and the suction-side leading-edge feature segment. Specifically, the process is as follows: using the leading-edge point as the starting point of the arc, the leading-edge dimensional parameter as the radius, and the leading-edge construction direction as the direction of the center of the circle; the pressure-side half-wedge angle and the suction-side half-wedge angle are used to determine the endpoint positions of the pressure-side and suction-side leading-edge feature segments, respectively; and pressure-side and suction-side circular arc segments are formed from the leading-edge point to the endpoints of the two arcs, serving as the pressure-side and suction-side leading-edge feature segments, respectively.

[0032] In the circular leading edge mode, let the radius of the circle be... r The center of the circle is P c =P LE + r e β Then the parametric curve of the circular leading edge is expressed as:

[0033] The pressure side and suction side use parameter directions with opposite signs or pre-defined parameters. The half-wedge angle on the pressure side and the half-wedge angle on the suction side are the angles between the leading edge construction direction and the tangential direction at the endpoint of the corresponding leading edge feature segment, respectively, and the parameter range of the two arcs is determined accordingly. The two arcs can have different extension lengths, but share the same leading edge point, center, and radius.

[0034] In another embodiment, an elliptical leading edge pattern is used to generate the pressure-side leading edge feature segment and the suction-side leading edge feature segment. Specifically, the process is as follows: using the leading edge point as the starting point of the elliptical arc, the elliptical shape is determined by the first and second semi-axes in the leading edge scale parameters; the direction of the major axis of the ellipse is determined by the leading edge construction direction; the endpoint positions of the pressure-side and suction-side leading edge feature segments are determined by the pressure-side and suction-side semi-wedge angles, respectively; and pressure-side and suction-side elliptical arc segments are formed from the leading edge point to the endpoints of the two elliptical arcs, respectively, serving as the pressure-side and suction-side leading edge feature segments.

[0035] Let the semi-axis along the leading edge structural direction and its normal direction be a and b, respectively, and let the center of the ellipse be P. c =P LE +a e β The leading edge of the ellipse is represented as:

[0036] If one half-wedge angle is oh Then the absolute parameter position of the endpoint of the corresponding elliptic arc i ω Determined based on the angle between the ellipse tangent and the leading edge structural direction:

[0037] The parameter span of an elliptical arc is usually not equal to the half-wedge angle. The corresponding parameter positions are obtained by using their respective half-wedge angles on the pressure side and suction side, and the leading edge feature segments on both sides are extracted from the same ellipse.

[0038] As another embodiment, the pressure-side leading edge feature segment and the suction-side leading edge feature segment are generated using a general irregular leading edge pattern. Specifically, the process is as follows: using an elliptical leading edge determined by the leading edge scale parameters, the leading edge construction angle, the pressure-side half-wedge angle, and the suction-side half-wedge angle as a reference ellipse, initial leading edge feature segments are constructed using a fifth-order Bézier curve on each of the pressure and suction sides. The initial leading edge feature segments are made to have the same position, first derivative, and second derivative as the reference ellipse at both ends. While maintaining the position of the common leading edge point, the tangential direction of each side at the common leading edge point determined by the reference ellipse, and the position and tangential direction of the endpoints of the leading edge feature segments on each side, and ensuring that the pressure-side curvature and suction-side curvature are equal at the common leading edge point after being converted to the same circumferential direction of the complete airfoil, the internal control points of the fifth-order Bézier curve are adjusted under constraints to obtain the pressure-side leading edge feature segment and the suction-side leading edge feature segment.

[0039] like Figure 4 As shown, the control points and constrained adjustment directions of the fifth-order Bezier feature segments on both sides of the general irregular leading edge are as follows: the pressure side and the suction side respectively adopt the control points Q j,0 To Q j,5 A definite quintic Bézier curve, where j=p represents the pressure side and j=s represents the suction side.

[0040] The parametric equation of the quintic Bézier curve is:

[0041] Suppose that a certain side of the reference ellipse feature segment is parameterized according to the direction from the common leading edge point to the endpoint of the feature segment, and its two ends are E(0) and E(1) respectively, with corresponding first derivatives E′(0) and E′(1) and second derivatives E″(0) and E″(1) respectively. Then the control points of the initial quintic Bézier curve satisfy: Q0=E(0), Q1=Q0+E′(0) / 5, Q2=2Q1 Q0+E′′(0) / 20,Q5=E(1),Q4=Q5 E′(1) / 5,Q3=2Q4 Q5+E′′(1) / 20.

[0042] The resulting initial quintic Bézier curve has the same position, first derivative, and second derivative as the reference ellipse at both ends.

[0043] Furthermore, the constrained adjustment is implemented based on the user-given control point adjustment amount or curvature optimization target. While maintaining the reference ellipse's scale parameters, the leading edge construction angle, and the baseline engineering features defined by the semi-wedge angles on both sides, the constrained adjustment forms a user-defined non-elliptical local shape and adjusts the internal curvature distribution and endpoint curvature of the leading edge feature segments on both sides. The constrained adjustment does not change the common leading edge point, the endpoints of the leading edge feature segments on both sides, or their tangential directions.

[0044] S3, starting from the candidate connection point interval near the leading edge, select multiple positions as candidate blade connection points according to a preset search step size; for each candidate blade connection point, extract the position, tangential direction, and curvature of the candidate blade connection point; establish a transition segment with the endpoint of the leading edge feature segment and the candidate blade connection point as the two ends, so that the transition segment satisfies positional continuity, tangential continuity, and curvature continuity with the leading edge feature segment and the blade profile at both ends respectively; After the leading edge feature segment is generated, a transition segment is established between the endpoint of each leading edge feature segment and the corresponding candidate blade connection point. The transition segment is parameterized according to the direction from the endpoint of the leading edge feature segment to the candidate blade connection point.

[0045] In some embodiments of the present invention, each transition segment is a six-control-point quintic Bézier curve, and the parametric equation of the transition segment is:

[0046] A six-control-point quintic Bézier curve can be represented as a quintic clamped NURBS curve with all control point weights equal to 1, no internal nodes, and a repetition degree of 6 for both end nodes. A preferred embodiment of the present invention directly employs the quintic Bézier form to avoid simultaneously solving for the control point weights and node positions.

[0047] For the plane parametric curve C( u )=[ x ( u ), y ( u The signed curvature of ] is:

[0048] The curvature signs of the leading edge feature segments, transition segments, and blade profiles on the pressure and suction sides are determined according to the same circumferential direction of the complete blade boundary. For curves generated using different parameter directions, their signed curvature is first converted to the unified circumferential direction using a side-specific sign factor, and then the curvature is compared.

[0049] Let A be the endpoint of a certain leading edge feature segment, B be the candidate blade connection point, t0 and t1 be the unit tangent vectors at both ends, n0 and n1 be the unit normal vectors determined according to the uniform bypass direction of the complete blade shape, and n0 and n1 be the signed curvatures at both ends. k 0 and k 1. The Bessel transition section should meet the following endpoint conditions:

[0050]

[0051]

[0052] Let the six control points be P0 to P5, then control point P1 is determined according to the following formula:

[0053] Control point P2 is determined according to the following formula:

[0054] Control point P4 is determined according to the following formula:

[0055] Control point P3 is determined according to the following formula:

[0056] In the formula, P0=A, P5=B; l 0>0 and l 1>0 represents the first derivative scales of the starting and ending points, respectively. m 0 and m 1 represents the tangential components of the second derivatives of the starting and ending points, respectively. The above control point structure directly satisfies the following conditions: the two ends coincide, the tangential directions are consistent, and the curvatures are equal. The solution process only requires determining four scalar parameters within preset upper and lower bounds.

[0057] like Figure 5 As shown in the diagram, the transition section control points and end constraints illustrate the geometric relationship between the aforementioned control points and end conditions.

[0058] In some embodiments of the present invention, candidate positions are selected sequentially from the end of the candidate connection point interval near the leading edge point according to a first search step size for coarse search; with the candidate position obtained by coarse search as the center, an interval containing the candidate position and its adjacent coarse search candidate positions is selected as a candidate sub-interval; within the candidate sub-interval, a second search step size smaller than the first search step size is used for refined search, and the blade connection point is determined according to the refined search results.

[0059] S4, Perform geometric feasibility and curvature quality checks on the transition section, and determine the blade connection points and corresponding transition sections that meet the requirements from the candidate blade connection points based on the results of the geometric feasibility and curvature quality checks; like Figure 6 As shown, the process for connection point search, curvature quality evaluation, and parameter adjustment is as follows: For each candidate transition segment, a geometric feasibility check is first performed. The geometric feasibility check includes: determining whether the transition segment has self-intersection, whether it has folds back, whether it exceeds the preset local geometric boundary, and whether it maintains a positive local airfoil thickness after the pressure side and suction side are combined; when all the above checks are passed, the transition segment is determined to meet the geometric feasibility requirements.

[0060] For candidate transition sections that meet the geometric feasibility requirements, further curvature quality checks are performed. The curvature of the blade profile, the curvature of the leading edge feature section, and the curvature of the transition section are converted into signed curvatures. After converting the signed curvatures of the pressure side and suction side to the same circumferential direction of the complete blade profile, the normalized curvature reverse change, the maximum relative curvature overshoot, and the root mean square deviation of the curvature matching at the blade connection end are calculated, and preferred limits and maximum allowable limits are set for each.

[0061] Specifically, let the signed curvatures at both ends of the transition segment be respectively... k 0 and k 1. Using the absolute value of the curvature difference between the two ends and the preset lower limit of the curvature scale. k min The larger value in the range is used as the curvature normalization scale:

[0062] Let the target direction of curvature change at both ends be d, and set the allowable reverse slope tolerance based on the average rate of change of curvature over the entire segment. e s :

[0063] In the formula, L is the total arc length of the transition section. or s This is the reverse slope tolerance coefficient. e minThis is the lower limit of the preset curvature change rate scale.

[0064] Normalized curvature inverse change R rev Used to evaluate the cumulative magnitude of reverse changes in the curvature of the transition segment that exceed the allowable fluctuation range.

[0065]

[0066] Maximum relative curvature overshoot R over Used to evaluate the maximum extent by which the curvature of the transition section exceeds the target curvature at the blade connection point along the target change direction:

[0067] Root mean square deviation of blade connection end curvature matching R tail This is used to evaluate the degree of matching between the actual curvature and the reference curvature of the transition section near the blade junction. Based on the curvature of the leading-edge characteristic segment endpoint and the blade junction point, as well as the curvature variations in their neighborhoods, a constrained cubic interpolation method is used to construct the reference curvature. k ref ( x ),in x = / L In the pre-defined rear section near the blade connection point. x ≥ x t Inside:

[0068] In the formula, N t This is the preset number of curvature sampling points within the rear interval.

[0069] When the normalized curvature reverses its change R rev The maximum relative curvature overshoot R over Root mean square deviation of the curvature matching with the blade connection end R tail When all three evaluation parameters do not exceed the corresponding maximum allowable limit, the curvature quality of the transition segment is deemed acceptable. When all three evaluation parameters do not exceed the corresponding preferred limit, the transition segment is deemed to have met the preferred curvature quality standard.

[0070] The above-mentioned full-section curvature quality evaluation allows for local non-monotonic variations with limited amplitude, and does not require all discrete curvature points to be mathematically absolutely strict monotonic. Its purpose is to limit obvious curvature reversal changes with engineering impact, target curvature overshoot, and rapid curvature correction near the blade connection end, thereby quantifying the full-section curvature quality that is difficult to identify by the profile appearance and improving the geometric reliability of subsequent aerodynamic analysis.

[0071] After applying a preset window to locally smooth the discrete curvature of the transition segment, the number of internal curvature extrema exceeding a preset threshold can be counted to identify multiple consecutive reverse changes or local curvature oscillations. The number of curvature extrema serves as an auxiliary diagnostic measure and does not independently determine the feasibility of the transition segment.

[0072] For candidate transition segments with acceptable curvature quality, a comprehensive curvature quality value is constructed using the normalized sum of squares of three curvature quality evaluation metrics relative to their respective optimum limits:

[0073] In the formula, the superscript pref indicates the preferred limit value for the corresponding evaluation quantity. The selection of candidate results first compares geometric feasibility and curvature quality level, then compares the comprehensive curvature quality value. Among candidate results with the same curvature quality level and similar comprehensive curvature quality values, the smoothness of curvature change, transition segment length, normal deviation of the transition segment relative to the line connecting the two endpoints or the preset reference curve, and the forward movement of the control polygon along the parameter direction are further compared. When the difference in the comprehensive curvature quality values ​​of multiple candidate results does not exceed the preset tolerance, and all auxiliary shape evaluation quantities meet their corresponding limits, the candidate connection point with the smaller normalized blade arc length is preferentially selected.

[0074] Based on the results of geometric feasibility and curvature quality checks, blade connection points that meet the requirements and their corresponding transition sections are determined from the candidate blade connection points.

[0075] In some embodiments of the present invention, when no transition segment satisfying the geometric feasibility requirements and with acceptable curvature quality is obtained within the candidate interval of the connection point, the incompatibility cause is identified based on the unmet geometric feasibility conditions or the curvature quality evaluation value exceeding the maximum allowable limit, and at least one of the following adjustments is performed: expanding the candidate interval of the corresponding side connection point so that the candidate interval extends along the direction away from the leading edge of the blade; shortening the leading edge feature segment of the corresponding side; adjusting the half-wedge angle of the corresponding side; adjusting the leading edge construction angle. After each adjustment, the affected leading edge feature segment is regenerated and the corresponding transition segment is solved. If a transition segment satisfying the requirements still cannot be obtained after the adjustments, the current leading edge parameter combination is returned as incompatible with the blade profile, and the complete blade profile coordinates are not output.

[0076] As an optional extension, when keeping the leading edge engineering parameters unchanged and the six-control-point quintic Bézier transition section still cannot meet the requirements, the transition section can be extended to a quintic NURBS curve with more control points and at least one internal node to increase the degree of freedom for local shape adjustment. This extension is only an optional implementation and is not included in the aforementioned preferred hierarchical adjustment steps; the preferred transition section of the present invention is still a low-dimensional six-control-point quintic Bézier curve.

[0077] S5. Based on the determined blade connection point and corresponding transition section, the retained pressure-side blade, pressure-side transition section, pressure-side leading edge feature section, suction-side leading edge feature section, suction-side transition section and retained suction-side blade are sequentially spliced ​​together to output the complete two-dimensional blade coordinates of the low-pressure turbine.

[0078] After obtaining satisfactory transition sections on both the pressure and suction sides, the portions of the original blade profile replaced by the new leading edge local area are deleted, and the curves are spliced ​​together according to a unified bypass direction. The output coordinates are resampled globally using arc length, and local densification is performed near the common leading edge point, the endpoints of the leading edge feature segments on both sides, and the connection points between the two blades.

[0079] In some embodiments of the present invention, a continuous leading edge curvature modeling system for a two-dimensional blade profile of a low-pressure turbine is also provided, comprising: The blade acquisition module is used to acquire the pressure side blade profile and the suction side blade profile of the two-dimensional blade of the low-pressure turbine, represent the blade profile as a parametric curve that is at least second-order differentiable, and set the connection point candidate interval defined by the normalized blade arc length at the front end of the pressure side blade and the suction side blade, respectively. The leading edge generation module is used to determine the leading edge point, leading edge scale parameters, leading edge construction angle, pressure-side half-wedge angle, and suction-side half-wedge angle; determine the leading edge construction direction based on the leading edge construction angle, wherein the leading edge construction direction is the direction from the leading edge point to the center of the circle or the center of the ellipse; and generate pressure-side leading edge feature segments and suction-side leading edge feature segments that share the same leading edge point based on the leading edge point, the leading edge scale parameters, the leading edge construction direction, the pressure-side half-wedge angle, and the suction-side half-wedge angle. The transition segment establishment module is used to select a position from the candidate connection point interval as a candidate blade connection point, extract the position, tangential direction and curvature of the candidate blade connection point; and establish a transition segment with the endpoint of the leading edge feature segment and the candidate blade connection point as the two ends, so that the transition segment satisfies positional continuity, tangential continuity and curvature continuity with the leading edge feature segment and the blade profile at both ends, respectively. The quality inspection module is used to perform geometric feasibility and curvature quality checks on the transition section, and to determine the blade connection points and corresponding transition sections that meet the requirements from the candidate blade connection points based on the results of the geometric feasibility and curvature quality checks. The blade profile output module is used to sequentially splice the retained pressure-side blade, pressure-side transition section, pressure-side leading edge feature section, suction-side leading edge feature section, suction-side transition section and retained suction-side blade based on the determined blade connection point and corresponding transition section, and output the complete two-dimensional blade profile coordinates of the low-pressure turbine.

[0080] The functions of each module in the above system embodiment correspond to the steps in the method embodiment. For technical details and beneficial effects, please refer to the foregoing method embodiment, which will not be repeated here.

[0081] Two specific embodiments are given below to illustrate the application effect of the method of the present invention on actual two-dimensional blade profiles of low-pressure turbines. In the embodiments, coordinates, chord lengths, half-shaft lengths, control point adjustments, and geometric deviations are all measured in millimeters, and curvature is expressed as the reciprocal of millimeters (mm). - ¹) Calculate.

[0082] Example 1: Elliptical leading edge A set of two-dimensional blade profiles for a low-pressure turbine was selected as input. The pressure side and suction side were each represented by 88 ordered coordinate points, and the blade profile normalization scale was 100 mm. The leading edge feature segment adopted an elliptical leading edge pattern, with semi-axial lengths of 4.468 mm and 2.0 mm along the leading edge construction direction and its normal direction, respectively. The leading edge construction angle was 44°, the pressure side semi-wedge angle was 10.5°, and the suction side semi-wedge angle was 11.5°.

[0083] Candidate intervals for connection points were set at the leading edges of the blades on both the pressure and suction sides, and the connection points and transition sections on both sides were solved independently. The normalized blade arc lengths selected for the connection points on the pressure and suction sides were 0.120 and 0.034, respectively. The resulting transition sections on both sides satisfied positional continuity, tangential continuity, and curvature continuity, with zero geometric constraint violations and no self-intersections or folds.

[0084] In this embodiment, the preferred limits for the normalized curvature inverse change, the maximum relative curvature overshoot, and the root mean square deviation of the blade connection end curvature matching are set to 0.08, 0.05, and 0.08, respectively, and the maximum allowable limits are set to 0.15, 0.10, and 0.15, respectively.

[0085] The normalized curvature inverse change, maximum relative curvature overshoot, and root mean square deviation of the blade connection curvature matching in the pressure-side transition section were 0.0566, 0.0237, and 0.0216, respectively; the corresponding three evaluation values ​​for the suction side were 0.0247, 0.0344, and 0.0158, respectively. All three evaluation values ​​for both the pressure and suction sides were below their respective optimal limits.

[0086] like Figure 7As shown, both the pressure-side and suction-side transition sections meet the preferred curvature quality standards and can be spliced ​​with the corresponding leading-edge feature sections and the retained blade body to form a complete two-dimensional blade shape. This embodiment demonstrates that, while maintaining the elliptical leading-edge dimensions, construction angle, and half-wedge angles on both sides, the present invention can achieve continuous curvature connection between the leading edge and a predetermined blade body through independent connection point search on both sides and four-parameter low-dimensional transition section solution, and can quantitatively evaluate the curvature quality of the entire transition section.

[0087] Example 2: General irregular leading edge Based on the above-described elliptical leading edge embodiment, the pressure-side elliptical leading edge feature segment and its transition segment remain unchanged. The first semi-axis, second semi-axis, leading edge construction angle, and semi-wedge angle of the suction-side reference ellipse remain 4.468 mm, 2.0 mm, 44°, and 11.5°, respectively, and the suction-side reference elliptical arc is converted into a fifth-order Bezier leading edge feature segment. The common leading edge point, the endpoints of the suction-side leading edge feature segment, and their tangential directions remain consistent before and after the conversion, and the curvature of the common leading edge point remains unchanged.

[0088] Under the aforementioned fixed conditions, the constrained control degrees of freedom of the suction-side Bessel leading edge feature segment are optimized. The four control adjustment quantities correspond sequentially to the tangential movement of the control points adjacent to the common leading edge point, the tangential adjustment of the next control point, the local normal movement of the middle control point, and the tangential movement of the control points adjacent to the endpoint of the leading edge feature segment. Their values ​​are approximately 0.0010 mm, ... 0.0087 mm The maximum adjustment amounts are 0.0009 mm and 0.0017 mm, with the largest adjustment being less than 0.009 mm. The maximum normal deviation of the optimized Bezier leading edge relative to the reference ellipse is 0.0024 mm, indicating that the adjustment mainly changes the local curvature distribution of the leading edge, while the overall geometric change of the leading edge is relatively small.

[0089] like Figure 8 As shown, the comparison of the geometry and curvature distribution of the suction-side leading edge before and after optimization is as follows: the curvature of the common leading edge point remains unchanged. 1.117 mm - ¹, the curvature of the endpoint of the characteristic segment on the suction side is changed from 0.1249 adjusted to 0.1262 mm - ¹, the change is approximately 1.0%. The curvature of the optimized leading edge feature segment changes smoothly, without producing any new obvious curvature peaks or valleys, nor does it exhibit self-intersection, foldback, or abnormal curvature reversal.

[0090] Using the optimized curvature of the suction-side leading edge feature segment endpoint as the starting curvature condition for the corresponding transition segment, the blade connection point is re-searched and the transition segment is solved. The normalized blade arc length at the suction-side connection point is adjusted from 0.0340 to 0.0344. After optimization, the curvature reversal exceeding the allowable fluctuation range in the transition segment is slightly reduced, and its evaluation value decreases from 0.0247 to 0.0245; the maximum relative amplitude exceeding the target curvature of the blade connection point decreases from 0.0344 to 0.0335. The curvature matching deviation near the blade connection end increases slightly from 0.0158 to 0.0162, but is still significantly lower than the preferred limit of 0.08. Therefore, all three optimized evaluation values ​​are still within the preferred range.

[0091] After optimization, the overall curvature quality value of the suction-side transition section decreased from 0.606 to 0.583, a reduction of approximately 3.8%, with the maximum relative curvature overshoot decreasing by approximately 2.5%. The position, tangential, and curvature conditions at the endpoints of the suction-side leading edge characteristic section and the blade connection point were all satisfied, with a maximum position error of 0 mm. The tangential and curvature errors were both within the numerical calculation accuracy range, and no self-intersection or folding occurred in the leading edge characteristic section or the transition section.

[0092] This embodiment demonstrates that the general irregular leading edge pattern can be constrained and customized according to the user-preset local shape while maintaining the reference ellipse scale, construction angle, half-wedge angle, and end geometry. It can also be numerically optimized with curvature connection quality as the objective, and re-achieve curvature continuity with the given blade body. The above-described embodiments are merely preferred embodiments of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various improvements and substitutions without departing from the principles of the present invention, and these improvements and substitutions should also be considered within the scope of protection of the present invention. Therefore, the scope of protection of this invention should be determined by the scope of the claims.

Claims

1. A method for continuous leading edge curvature modeling of a two-dimensional blade profile for low-pressure turbines, characterized in that, Includes the following steps: S1, Obtain the blade profile of the two-dimensional blade of the low-pressure turbine, the blade profile includes the pressure side blade profile and the suction side blade profile, represent the blade profile as a parametric curve with at least second-order differentiability, and set the connection point candidate interval defined by the normalized blade arc length at the front end of the pressure side blade and the suction side blade respectively. S2, determine the leading edge point, leading edge dimensional parameters, leading edge construction angle, pressure-side half-wedge angle, and suction-side half-wedge angle; determine the leading edge construction direction based on the leading edge construction angle, wherein the leading edge construction direction is the direction from the leading edge point to the center of the circle or the center of the ellipse; based on the leading edge point, the leading edge dimensional parameters, the leading edge construction direction, the pressure-side half-wedge angle, and the suction-side half-wedge angle, generate a pressure-side leading edge feature segment and a suction-side leading edge feature segment that share the same leading edge point; S3, starting from the candidate connection point interval near the leading edge, select multiple positions as candidate blade connection points according to a preset search step size; for each candidate blade connection point, extract the position, tangential direction, and curvature of the candidate blade connection point; establish a transition segment with the endpoint of the leading edge feature segment and the candidate blade connection point as the two ends, so that the transition segment satisfies positional continuity, tangential continuity, and curvature continuity with the leading edge feature segment and the blade profile at both ends respectively; S4, Perform geometric feasibility and curvature quality checks on the transition section, and determine the blade connection points and corresponding transition sections that meet the requirements from the candidate blade connection points based on the results of the geometric feasibility and curvature quality checks; S5. Based on the determined blade connection point and corresponding transition section, the retained pressure-side blade, pressure-side transition section, pressure-side leading edge feature section, suction-side leading edge feature section, suction-side transition section and retained suction-side blade are sequentially spliced ​​together to output the complete two-dimensional blade coordinates of the low-pressure turbine.

2. The method for continuous leading edge curvature modeling of a two-dimensional blade profile for low-pressure turbines according to claim 1, characterized in that, In S2, the pressure-side leading edge feature segment and the suction-side leading edge feature segment are generated using a circular leading edge pattern. The specific process is as follows: Using the leading edge point as the starting point of the arc, the leading edge dimensional parameter as the radius, and the leading edge construction direction as the direction of the center of the circle, the pressure side half-wedge angle and the suction side half-wedge angle are used to determine the end point of the pressure side leading edge feature segment and the end point of the suction side leading edge feature segment, respectively. The pressure-side arc segment and the suction-side arc segment are formed from the leading edge point to the two arc endpoints, respectively, and serve as the pressure-side leading edge feature segment and the suction-side leading edge feature segment.

3. The method for continuous leading edge curvature modeling of a two-dimensional blade profile for low-pressure turbines according to claim 1, characterized in that, In S2, the pressure-side leading edge feature segment and the suction-side leading edge feature segment are generated using an elliptical leading edge pattern. The specific process is as follows: The ellipse shape is determined by the first and second semi-axes in the leading edge dimensional parameters, the major axis direction of the ellipse is determined by the leading edge construction direction, and the end point of the pressure-side leading edge feature segment and the end point of the suction-side leading edge feature segment are determined by the pressure-side semi-wedge angle and the suction-side semi-wedge angle, respectively. An elliptical arc segment on the pressure side and an elliptical arc segment on the suction side are formed from the leading edge point to the endpoints of the two elliptical arcs, respectively, serving as the leading edge feature segment on the pressure side and the leading edge feature segment on the suction side.

4. The method for continuous leading edge curvature modeling of a two-dimensional blade profile for low-pressure turbines according to claim 1, characterized in that, In S2, the pressure-side leading edge feature segment and the suction-side leading edge feature segment are generated using a general irregular leading edge pattern. The specific process is as follows: Using the elliptical leading edge determined by the leading edge scale parameters, the leading edge construction angle, the pressure side half-wedge angle, and the suction side half-wedge angle as a reference ellipse, an initial leading edge feature segment is constructed using a fifth-order Bézier curve on both the pressure side and the suction side, such that the initial leading edge feature segment has the same position, first derivative, and second derivative as the reference ellipse at both ends. While maintaining the position of the common leading edge point, the tangential direction of each side at the common leading edge point determined by the reference ellipse, the position and tangential direction of the endpoints of the leading edge feature segments on each side, and ensuring that the pressure side curvature and the suction side curvature are equal at the common leading edge point after being converted to the same circumferential direction of the complete airfoil, the internal control points of the fifth-order Bézier curve are adjusted under constraints to obtain the pressure side leading edge feature segment and the suction side leading edge feature segment.

5. The method for continuous leading edge curvature modeling of a two-dimensional blade profile for low-pressure turbines according to claim 4, characterized in that, The constrained adjustment is implemented based on the control point adjustment amount or curvature optimization target given by the user; The constrained adjustment, while maintaining the reference ellipse's scale parameters, the leading edge construction angle, and the reference engineering features defined by the half-wedge angles on both sides, forms a user-defined non-elliptical local shape and adjusts the internal curvature distribution and endpoint curvature of the leading edge feature segments on both sides. The constrained adjustment does not change the common leading edge point, the endpoints of the leading edge feature segments on both sides, or their tangential directions; After adjustment, the curvature of the corresponding leading edge feature segment endpoint is recalculated, and the endpoint curvature is used as the starting curvature condition of the corresponding transition segment.

6. The method for continuous leading edge curvature modeling of a two-dimensional airfoil for a low-pressure turbine according to claim 1, characterized in that, In S3, each transition segment is a quintic Bézier curve with six control points, where the six control points are P0 to P5, P0 being the endpoint of the leading edge feature segment and P5 being the blade connection point. After the positions, tangential directions, and curvatures of P0 and P5 are determined, the curve is scaled using the first derivative of the starting point. λ 0. Endpoint first derivative scale λ 1. Tangential component of the second derivative at the starting point μ 0 and the tangential component of the second derivative at the endpoint μ 1 Four scalar parameters determine the positions of P1, P2, P3, and P4: Where t0 and t1 are the unit tangent vectors at P0 and P5, respectively, and n0 and n1 are the unit normal vectors at P0 and P5 determined by the parameterized direction, respectively. κ 0 and κ 1 represents the signed curvature at points P0 and P5, respectively. λ 0>0 and λ 1>0.

7. The method for continuous leading edge curvature modeling of a two-dimensional blade profile for low-pressure turbines according to claim 1, characterized in that, In S4, the curvature quality check includes: The curvature of the blade profile, the curvature of the leading edge feature segment, and the curvature of the transition segment are converted into signed curvatures. The signed curvatures of the pressure side and the suction side are converted to the same circumferential direction of the complete blade profile. The normalized curvature reverse change, the maximum relative curvature overshoot, and the root mean square deviation of the curvature matching at the blade connection end are calculated respectively. Preferred limits and maximum allowable limits are set respectively. When the normalized curvature inverse change, the maximum relative curvature overshoot, and the root mean square deviation of the blade connection end curvature matching all do not exceed the corresponding maximum allowable limit, the curvature quality of the transition section is deemed acceptable.

8. The method for continuous leading edge curvature modeling of a two-dimensional airfoil for a low-pressure turbine according to claim 1, characterized in that, In S4, the geometric feasibility check includes: determining whether the transition section has self-intersection, whether it has foldback, whether it exceeds the preset local geometric boundary, and whether the pressure side and suction side maintain a positive local airfoil thickness after combination; when all the above checks are passed, the transition section is determined to meet the geometric feasibility requirements.

9. The method for continuous leading edge curvature modeling of a two-dimensional airfoil for a low-pressure turbine according to claim 1, characterized in that, In S4, when no transition segment satisfying the geometric feasibility requirements and with acceptable curvature quality is obtained within the candidate connection point interval, the incompatibility cause is identified based on the unmet geometric feasibility conditions or the curvature quality evaluation value exceeding the maximum allowable limit, and at least one of the following adjustments is performed: Expand the candidate interval for the corresponding side connection point, so that the candidate interval extends along the blade in a direction away from the leading edge; Shorten the corresponding side leading edge feature segment; adjust the corresponding side half-wedge angle; adjust the leading edge structural angle; After each adjustment, the affected leading edge feature segments are regenerated and the corresponding transition segments are solved. If the required transition section cannot be obtained after the aforementioned adjustments, the current leading edge parameter combination is returned as incompatible with the blade profile, and the complete blade coordinates are not output.

10. A continuous leading edge curvature modeling system for a two-dimensional blade profile of a low-pressure turbine, characterized in that, include: The blade acquisition module is used to acquire the pressure side blade profile and the suction side blade profile of the two-dimensional blade of the low-pressure turbine, represent the blade profile as a parametric curve that is at least second-order differentiable, and set the connection point candidate interval defined by the normalized blade arc length at the front end of the pressure side blade and the suction side blade, respectively. The leading edge generation module is used to determine the leading edge point, leading edge scale parameters, leading edge construction angle, pressure-side half-wedge angle, and suction-side half-wedge angle; determine the leading edge construction direction based on the leading edge construction angle, wherein the leading edge construction direction is the direction from the leading edge point to the center of the circle or the center of the ellipse; and generate pressure-side leading edge feature segments and suction-side leading edge feature segments that share the same leading edge point based on the leading edge point, the leading edge scale parameters, the leading edge construction direction, the pressure-side half-wedge angle, and the suction-side half-wedge angle. The transition segment establishment module is used to select a position from the candidate connection point interval as a candidate blade connection point, extract the position, tangential direction and curvature of the candidate blade connection point; and establish a transition segment with the endpoint of the leading edge feature segment and the candidate blade connection point as the two ends, so that the transition segment satisfies positional continuity, tangential continuity and curvature continuity with the leading edge feature segment and the blade profile at both ends, respectively. The quality inspection module is used to perform geometric feasibility and curvature quality checks on the transition section, and to determine the blade connection points and corresponding transition sections that meet the requirements from the candidate blade connection points based on the results of the geometric feasibility and curvature quality checks. The blade profile output module is used to sequentially splice the retained pressure-side blade, pressure-side transition section, pressure-side leading edge feature section, suction-side leading edge feature section, suction-side transition section and retained suction-side blade based on the determined blade connection point and corresponding transition section, and output the complete two-dimensional blade profile coordinates of the low-pressure turbine.

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