A parameterization method for natural laminar flow airfoils of green supersonic civil transport

Abstract

The application provides a parameterization method suitable for a green supersonic civil aircraft natural laminar wing profile, comprising: given a wing profile surface coordinate point to be fitted; a modified function is constructed by using a constructor satisfying a constraint condition; the Bernstein polynomial is modified by using the modified function to obtain a type function; the type function is multiplied by a class function to obtain a general expression of a parameterization curve; and undetermined coefficients in the general expression of the parameterization curve are determined by using a least square fitting method, so that a wing profile surface curve is fitted by using the wing profile surface coordinate points. The parameterization method realizes high-precision fitting of a supersonic natural laminar wing profile and other wing profiles with sharp changes in leading edges without increasing the number of control parameters and without significantly reducing the fitting accuracy of the middle and rear sections of the wing profile, and improves the efficiency and precision of drag reduction design of a supersonic civil aircraft natural laminar wing.

Description

Technical Field

[0001] This invention belongs to the field of aircraft design technology, specifically relating to a parameterization method for natural laminar flow airfoils suitable for green supersonic civil aircraft. Background Technology

[0002] Efficient, convenient, and comfortable travel is a timeless pursuit for humankind. Supersonic aircraft, with their speed advantage of nearly twice that of existing aircraft, have become one of the main directions for future civil aircraft development. However, first-generation supersonic aircraft, such as the Concorde, had a significantly lower lift-to-drag ratio compared to subsonic aircraft, only around 7, resulting in high fuel consumption and expensive fares. Therefore, drag reduction design for supersonic aircraft to improve their cruise efficiency has become a crucial part of the design of the next generation of green supersonic aircraft. Natural laminar flow wing design is an important approach to drag reduction in supersonic aircraft, requiring the flow to be accelerated to a high speed within a very short region at the airfoil's leading edge to suppress flow instabilities on its surface. This characteristic places extremely high demands on the fitting accuracy of parametric methods at the airfoil's leading edge.

[0003] After years of development, the most widely used airfoil parameterization methods today include: the CST parameterization method, which uses class functions and shape functions to describe the geometry; the Hicks-Henne parameterization method, which parameterizes the changes in airfoil camber and thickness; and the PARSEC parameterization method and spline parameterization method, which use multiple parameters with explicit geometric meanings to describe the airfoil shape. Each of these parameterization methods has its own advantages and disadvantages, and their application scenarios vary considerably. Among these methods, the CST parameterization method has the advantage of ensuring high fitting accuracy while minimizing the appearance of wavy shapes, and has become one of the most popular airfoil parameterization methods currently.

[0004] Research on the CST parameterization method has been ongoing for a long time, and numerous researchers and institutions have made improvements to it. Michiel et al. developed the CSRT method, which uses the superposition of B-spline basis functions and Bernstein polynomials, significantly improving the local control and fitting capabilities of the CST parameterization method. However, this method increases the number of control variables in the B-spline basis function part on top of the original number of control variables. Stephen et al. improved the fitting accuracy of the airfoil's local area by optimizing the exponent of the shape function. However, the fitting accuracy of existing commonly used parameterization methods for the airfoil leading edge remains low, failing to meet the accuracy requirements for airfoil leading edge fitting in research fields such as supersonic natural laminar flow airfoil design and supercritical airfoil design. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a parameterization method suitable for natural laminar flow airfoils of green supersonic civil aircraft, which can effectively solve the above-mentioned problems.

[0006] The technical solution adopted in this invention is as follows:

[0007] This invention provides a parameterization method for natural laminar flow airfoils suitable for green supersonic civil aircraft, comprising the following steps:

[0008] Step 1: Given the coordinate points of the airfoil surface to be fitted. , Represents the x-axis of the airfoil surface; Represents the longitudinal coordinate of the airfoil surface;

[0009] Step 2: Use a constructor that satisfies the constraints. The correction function is constructed. ;

[0010] Step 3, using the correction function By modifying the Bernstein polynomial, we obtain the type function. ;

[0011] Step 4, convert the type function With class functions Multiplying them together yields the general expression for the parameterized curve;

[0012] Step 5, using the airfoil surface coordinate points from Step 1 The least squares fitting method is used to determine the undetermined coefficients in the general expression of the parameterized curve, thereby obtaining a curve with a clear mathematical expression, which is the coordinate point of the airfoil surface. The fitted airfoil surface curve.

[0013] Preferably, in step 1, the given coordinate points of the airfoil surface to be fitted The coordinates include the coordinates of the upper and lower surfaces of the airfoil, arranged in the order from the trailing edge of the lower surface to the leading edge and then to the trailing edge of the upper surface.

[0014] Preferably, the constructor The constraints that need to be satisfied are: Within the interval, constructor An expression that satisfies equation (1):

[0015] (1)

[0016] This constraint ensures that the constructor... exist The inner part is a monotonically increasing convex function.

[0017] Preferably, constructor This includes: the form of multiplying an exponential function and a linear function, expressed as equation (2); the form of a quadratic polynomial function, expressed as equation (3); and the form of composing a sine function and a polynomial function, expressed as equation (4).

[0018] (2)

[0019] (3)

[0020] (4)

[0021] in: All of these are constants, given by the designer.

[0022] Preferably, a constructor that satisfies the constraints is used. The constructed correction function The expression is as shown in equation (5):

[0023] (5)

[0024] Preferably, a correction function is used. By modifying the Bernstein polynomial, the resulting function is of type [missing information]. The expression is as shown in equation (6):

[0025] (6)

[0026] in:

[0027] The order of the polynomial is given by the designer.

[0028] Undetermined coefficients in a representative function;

[0029] The value is ,represent A value taken from the middle.

[0030] Preferably, the general expression for the parametric curve obtained in step 4 includes the general expression for the parametric curve of the airfoil's upper surface. General expression for parametric curves of lower surface of airfoil As shown in equation (7):

[0031] (7)

[0032] in:

[0033] Represents the upper surface profile function of the airfoil;

[0034] Represents the lower surface profile function of the airfoil;

[0035] The ordinate of the point at the trailing edge of the upper surface of the airfoil;

[0036] The ordinate of the point at the trailing edge of the lower surface of the airfoil;

[0037] The representative class function is expressed as in equation (8):

[0038] (8)

[0039] in:

[0040] and This is the exponential term of a function.

[0041] The present invention provides a parameterization method for natural laminar flow airfoils of green supersonic civil aircraft, which has the following advantages: The present invention provides a parameterization method LE-CST for natural laminar flow airfoils of green supersonic civil aircraft. By constructing a correction function to correct the Bernstein polynomial in the CST parameterization method, the present invention addresses the problem of insufficient fitting accuracy of the CST parameterization method for the leading edge of some airfoils, thereby improving the fitting accuracy of airfoil shape in the design of natural laminar flow airfoils for supersonic civil aircraft and transonic airfoil design. Attached Figure Description

[0042] Figure 1 A flowchart illustrating the parameterization method for airfoils proposed in this invention is shown.

[0043] Figure 2 The diagram shows the Bernstein polynomial function curve used for the 8th order polynomial function.

[0044] Figure 3 This invention demonstrates the correction function used. Schematic diagram of the curved shape;

[0045] Figure 4 The correction function used in this invention is shown. Schematic diagram of the curved shape;

[0046] Figure 5 Show when When multiplying a linear function by an exponential function, the parameters are... Assign the value 2 and construct the correction function. The function curve obtained by correcting the Bernstein polynomial using this correction function;

[0047] Figure 6 The display adopts Figure 5 The function curve shown is the type function obtained. A comparison of the fitting accuracy of the constructed LE-CST (function1) parameterization method and the CST parameterization method for Clark Y airfoil;

[0048] Figure 7 The display adopts Figure 5 The function curve shown is the type function obtained. A comparison of the fitting accuracy of the constructed LE-CST (function1) parameterization method and the CST parameterization method for the RAE5214 airfoil;

[0049] Figure 8 Show when When the function is a composite of a sine function and a quadratic polynomial function, the parameters are... Assign a value of 0.8 and construct a correction function. The function curve obtained by correcting the Bernstein polynomial using this correction function;

[0050] Figure 9 The display adopts Figure 8 The function curve shown is the type function obtained. A comparison of the fitting accuracy of the constructed LE-CST(function2) parameterization method and the CST parameterization method for Clark Y airfoil;

[0051] Figure 10 The display adopts Figure 8 The function curve shown is the type function obtained. A comparison of the fitting accuracy of the constructed LE-CST (function2) parameterization method and the CST parameterization method for the RAE5214 airfoil;

[0052] Figure 11 The demonstration uses a correction function. A comparison of the fitting accuracy of the LE-CST parameterization method (obtained after modifying the Bernstein polynomial) and the CST parameterization method for a certain supersonic natural laminar flow airfoil.

[0053] Figure 12 The display adopts A comparison of the leading-edge pressure distribution of an airfoil obtained by fitting a supersonic natural laminar flow airfoil using the constructed LE-CST parameterization method with a reference airfoil.

[0054] Figure 13The image shows a comparison of the leading-edge pressure distribution of an airfoil obtained by fitting a supersonic natural laminar flow airfoil using the conventional CST parameterization method, and that of a reference airfoil. Detailed Implementation

[0055] The exemplary embodiments of the present invention are described in more detail below with reference to the accompanying drawings. Exemplary embodiments of the present disclosure are shown in the drawings to provide a more thorough understanding of the present disclosure and to fully convey the scope of the disclosure to those skilled in the art. It should be understood that the present invention may be implemented in various forms and should not be limited to the embodiments described herein. It should be noted that, without conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0056] This invention provides a parameterization method (LE-) suitable for natural laminar flow airfoils of green supersonic civil aircraft.

[0057] CST (Leading-edge Enhanced CST) corrects the Bernstein polynomial in the CST parameterization method by constructing a correction function. This primarily addresses the problem of insufficient fitting accuracy of the CST parameterization method for the leading edge of certain airfoils, improving the fitting accuracy of airfoil shapes in designs such as supersonic natural laminar flow airfoils and transonic airfoils. This invention effectively solves the problem that current parameterization methods cannot accurately fit the leading edge of supersonic natural laminar flow airfoils with drastic geometric changes when there are few control variables. This parameterization method can also be extended to the design of propeller airfoils, supercritical airfoils, and other airfoil designs where high leading-edge fitting accuracy is also required.

[0058] refer to Figure 1 This invention provides a parameterization method suitable for natural laminar flow airfoils of green supersonic civil aircraft, called the LE-CST parameterization method. The process of this method is as follows:

[0059] Step 1: Given the coordinate points of the airfoil surface to be fitted. , Represents the x-axis of the airfoil surface; Represents the longitudinal coordinate of the airfoil surface;

[0060] Specifically, given the coordinate points of the airfoil surface to be fitted The coordinates include the coordinates of the upper and lower surfaces of the airfoil, arranged in the order from the trailing edge of the lower surface to the leading edge and then to the trailing edge of the upper surface.

[0061] According to embodiments of this disclosure, the required airfoil geometry can be given, with data in .dat format. The data file contains two columns, representing the x and y coordinates of all coordinate points on the upper and lower surfaces of the airfoil, respectively. The data are arranged in order from the trailing edge of the lower surface of the airfoil to the leading edge of the airfoil and then to the trailing edge of the upper surface of the airfoil.

[0062] The parameterization method proposed in this invention performs fitting on the upper and lower surfaces of the airfoil separately. Therefore, the data of the airfoil geometry to be fitted in step 1 should include the horizontal and vertical coordinates of all coordinate points on the upper and lower surfaces of the airfoil.

[0063] Specifically, airfoil data is generally stored in coordinate point format. Except for some airfoils that possess analytical expressions, most airfoil data consists of coordinate points. This invention addresses airfoils such as supersonic laminar flow airfoils, which exhibit drastic leading-edge geometry changes during the design process and lack analytical expressions. It constructs a general airfoil analytical expression using modified shape functions and class functions. Subsequently, for the airfoil to be fitted, the undetermined coefficients in the general expression are determined by least-squares fitting on the upper and lower surfaces of the airfoil. This allows for the use of explicit mathematical expressions to describe the airfoil instead of discrete coordinate points during the design process.

[0064] Step 2: Use a constructor that satisfies the constraints. The correction function is constructed. ;

[0065] Specifically, the constructor The constraints that need to be satisfied are: Within the interval, constructor An expression that satisfies equation (1):

[0066] (1)

[0067] This constraint ensures that the constructor... exist The inner part is a monotonically increasing convex function.

[0068] In this invention, the constructor It can be a function of various types, and needs to satisfy the conditions shown in equation (1) to construct the function. The function curve diagram is shown below. Figure 4 As shown, the function type can be a quadratic polynomial function, an exponential function, a trigonometric function, etc. This invention provides the following three different construction formats, which can effectively improve the fitting accuracy of the airfoil leading edge, including: the form of multiplying an exponential function and a linear function, the expression of which is Equation (2); the form of a quadratic polynomial function, the expression of which is Equation (3); and the form of composing a sine function and a polynomial function, the expression of which is Equation (4).

[0069] (2)

[0070] (3)

[0071] (4)

[0072] in: All of these are constants, given by the designer.

[0073] It should be noted that the above three constructors The selection of parameters must ensure that The new constructor is obtained by strictly satisfying the conditions of equation (1) and only adjusting the parameters in the constructor. With correction function It remains within the scope of the claims of this patent.

[0074] According to embodiments of this disclosure, the selected constructor... It must be ensured that it is in The interval must satisfy the expression as shown in equation (1) to ensure that within the interval... The inner part is a monotonically increasing upward convex function. This criterion stems from the modification strategy of the Bernstein polynomial curve. The expression of the Bernstein polynomial is shown in equation (9). The curves of each function of the Bernstein polynomial are plotted in sequence, as follows: Figure 2 .

[0075] (9)

[0076] Depend on Figure 2 It can be seen that the peak positions of these polynomials are uniformly distributed within the interval [0,1]. Each undetermined coefficient in the CST parameterization method, which uses Bernstein polynomials as the type function, is... The CST parameterization method achieves maximum fitting accuracy at the peak value of its corresponding Bernstein polynomial. This characteristic leads to insufficient fitting accuracy of the airfoil leading edge by the CST parameterization method. Therefore, this invention employs a correction function to alter the distribution of these peak values, bringing them closer to the airfoil leading edge.

[0077] The present invention performs a correction function in steps 2 and 3. The construction of the polynomial and the modification of the Bernstein polynomial are then performed. Differentiating any term in the Bernstein polynomial yields the result shown in equation (10):

[0078] (10)

[0079] It is not difficult to see exist The value at this location is 0, that is exist The peak value is reached at this point. Therefore, as long as a correction function is found that satisfies the condition that the first derivative is greater than 1 near the leading edge and close to 1 near the middle and later segments, the problem can be solved. The function curve diagram is as follows: Figure 3 This allows for improved fitting accuracy of the airfoil's leading edge without excessive loss of fitting accuracy in the middle and rear sections. To achieve this, this invention proposes a function construction method: employing a function constructor that satisfies the constraints. The constructed correction function The expression is as shown in equation (5):

[0080] (5)

[0081] This correction function From quadratic polynomial functions and constructors The combination yields the result.

[0082] For ease of description, the constructors of equations (2), (3), and (4) can be described as follows: , respectively represented as: constructor , , The corresponding correction function They are represented as follows: , , Therefore, it can be represented in the following way:

[0083] Multiplying an exponential function by a linear function:

[0084]

[0085] Quadratic polynomial function:

[0086]

[0087] Composition of sine and polynomial functions:

[0088]

[0089] Step 3, using the correction function By modifying the Bernstein polynomial, we obtain the type function. ;

[0090] Specifically, using the correction function The Bernstein polynomial is modified by summing Bernstein polynomials of different orders to obtain the type function. The expression is as shown in equation (6):

[0091] (6)

[0092] in:

[0093] The order of the polynomial is given by the designer based on actual needs;

[0094] The undetermined coefficients in the representative function need to be determined by using the least squares fitting method on the airfoil coordinate points.

[0095] The value is ,represent A value taken from the middle.

[0096] Therefore, type function Bernstein polynomials of different orders and undetermined coefficients The summation of multiplication. Since the parameterization method of this invention performs fitting on the upper and lower surfaces of the airfoil separately, the upper surface shape function is expressed as... The lower surface type function is expressed as .

[0097] Step 4, convert the type function With class functions Multiplying them together yields the general expression for the parameterized curve;

[0098] This step employs a parametric equation construction method that multiplies the type function and class function while considering the airfoil trailing edge thickness, resulting in a general expression for the parametric curves, including a general expression for the parametric curves of the airfoil's upper surface. General expression for parametric curves of lower surface of airfoil As shown in equation (7):

[0099] (7)

[0100] in:

[0101] Represents the upper surface profile function of the airfoil;

[0102] Represents the lower surface profile function of the airfoil;

[0103] The ordinate of the point at the trailing edge of the upper surface of the airfoil;

[0104] The ordinate of the point at the trailing edge of the lower surface of the airfoil;

[0105] The representative class function is expressed as in equation (8):

[0106] (8)

[0107] in:

[0108] and The exponent term is a function-like term, typically taken as 0.5 and 1 respectively.

[0109] Step 5, using the airfoil surface coordinate points from Step 1 The least squares fitting method is used to determine the undetermined coefficients in the general expression of the parameterized curve, thereby obtaining a curve with a clear mathematical expression, which is the coordinate point of the airfoil surface. The fitted airfoil surface curve is the parametric fitting result of the airfoil geometry.

[0110] For ease of comparison, this manual uses the 8th-order CST parameterization method to verify the effect of the parameterization methods under two different correction functions, and shows the fitting effect on Clark Y airfoil, RAE5214 airfoil and a certain supersonic natural laminar flow airfoil respectively.

[0111] Case 1: When When the product of an exponential function and a linear function is given by equation (2), it satisfies the condition in equation (1). Constraints, including the parameters Assigning the value 2 and constructing the correction function. This correction function is expressed as the correction function. Using this correction function The polynomial curve obtained after modifying the Bernstein polynomial is as follows: Figure 5 As shown. The correction capability of this type of correction function is positively correlated with the base of the exponential function; the larger the base, the higher the fitting accuracy for the leading edge, but the worse the fitting accuracy for the middle and rear segments. The fitting effect of the parameterized method LE-CST(function1) on the airfoil after correction is shown in the figure. Figure 6 and Figure 7 As shown, where Figure 6 To assess the fitting effect of the Clark Y airfoil, Figure 7 The image shows the fitting effect on the RAE5214 airfoil. It can be seen that the fitting accuracy for the leading edges of both airfoils is significantly improved compared to the original version. Although the fitting accuracy for a portion of the lower surface of the RAE5214 airfoil decreases slightly, it still meets the requirements for wind tunnel test model fabrication.

[0112] Case 2: When When a sine function is combined with a quadratic polynomial function, it becomes the expression of equation (4), which satisfies the condition in equation (1). The constraints will Medium parameters Assign a value of 0.8 and construct a correction function. This correction function is expressed as the correction function. Using this correction function The polynomial curve obtained after modifying the Bernstein polynomial is as follows: Figure 8 As shown. The modified parametric method LE-CST(function2) fits the two airfoils as follows. Figure 9 and Figure 10 As shown, where Figure 9 To assess the fitting effect of the Clark Y airfoil, Figure 10 The figure shows the fitting effect on the RAE5214 airfoil. As can be seen from the figure, the fitting accuracy for the leading edges of both airfoils is significantly improved compared to the previous version. Although the fitting accuracy for a portion of the lower surface of the RAE5214 airfoil decreases slightly, it still meets the requirements for wind tunnel test model fabrication.

[0113] Case 3: Select the same correction function as in Case 2 The parameterization method obtained using this correction function was used to fit the supersonic natural laminar flow experimental airfoil designed by the research group, with a parameterization order of 12. The fitting accuracy of this method is similar to that of the CST parameterization method. Figure 11 As shown, flow field calculations were performed on the airfoils fitted by the two methods, and their pressure distributions were compared. The airfoil fitted using the parameterization method proposed in this invention still retains the "zero pressure gradient" pressure distribution pattern required by the original supersonic natural laminar flow airfoil, as shown in the figure. Figure 12 As shown. However, the pressure distribution of the airfoil obtained by fitting using the CST method exhibits more severe oscillations, such as... Figure 13 As shown, it no longer possesses the "zero pressure gradient" characteristic required for a natural laminar flow airfoil. The flow on the airfoil surface rapidly transitions into turbulence at the leading edge, thus failing to achieve a large-scale natural laminar flow on the surface, resulting in a significant increase in its frictional drag.

[0114] This invention provides a parameterization method, LE-CST, suitable for natural laminar flow airfoils of green supersonic civil aircraft. This method has the following advantages: First, by constructing a correction function through a constructor and modifying the Bernstein polynomial, the fitting capability for the airfoil's leading edge can be significantly improved while maintaining the original parameterization order and without sacrificing the fitting accuracy for the middle and rear sections of the airfoil. Second, by multiplying the type function and class function and using least squares fitting to determine the undetermined coefficients, this method employs a method of determining the undetermined coefficients through least squares fitting. The resulting geometric curves possess the advantage of ensuring high fitting accuracy while minimizing the appearance of wavy shapes. The proposed methods for constructing various correction functions can also meet research needs with different fitting accuracy requirements.

[0115] The parameterization method proposed in this invention achieves high-precision fitting of airfoil shapes with dramatic leading-edge changes, such as supersonic natural laminar flow airfoils, without increasing the number of control parameters or significantly reducing the fitting accuracy of the mid-to-rear section of the airfoil. This improves the efficiency and accuracy of drag reduction design for supersonic civil aircraft natural laminar flow wings, contributing to the development of next-generation green supersonic civil aircraft. Furthermore, the parameterization method proposed in this invention can also be extended to the design of airfoils with dramatic leading-edge geometric changes, such as propeller airfoils and transonic airfoils.

[0116] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A parameterization method suitable for green supersonic civil transport natural laminar flow airfoils, characterized in that, comprising the steps of: Step 1, given the coordinate points of the airfoil surface to be fitted , represent the airfoil surface abscissa; represent the airfoil surface ordinate; Step 2, constructor satisfying constraints Constructing the modified function ; The constructor The constraints to be satisfied are that the constructor satisfies an expression as in equation (1) within the interval satisfies an expression as in equation (1) within the interval （1） By this constraint, it is guaranteed that the constructor In is a monotonically increasing upper convex function; Constructors that satisfy the constraint condition are used The constructed correction function The expression is as formula (5): （5） Step 3, using a correction function The Bernstein polynomials are modified to obtain the type function , the expression is as formula (6): （6） wherein: The order of the polynomial is given by the designer. Undetermined coefficients in a representative function; the value of represents a certain value taken in the interval Step 4, multiply the type function with the class function to get the parametric curve general expression; The expression is as formula (8): （8） wherein: and is a class function exponential term; Step 5, using the airfoil surface coordinate points in step 1 The undetermined coefficients in the general expression of the parameterized curve are determined by using the least square fitting method, so as to obtain the curve with explicit mathematical expression, i.e. the airfoil surface curve fitted by the airfoil surface coordinate points in step 1.

2. The parameterization method suitable for natural laminar flow airfoils of green supersonic civil transport according to claim 1, characterized in that, In Step 1, given the airfoil surface coordinate points to be fitted , including the airfoil upper surface coordinate points and the airfoil lower surface coordinate points, arranged in the order from the airfoil lower surface trailing edge to the airfoil leading edge to the airfoil upper surface trailing edge.

3. The parameterization method suitable for natural laminar flow airfoils of green supersonic civil transport according to claim 1, characterized in that, Constructor includes: a form in which an exponential function is multiplied by a linear function, which is expressed as Equation (2); a form of a quadratic polynomial function, which is expressed as Equation (3); a form in which a sine function is compounded with a polynomial function, which is expressed as Equation (4); （2） （3） （4） where: are constants given by the designer.

4. The parameterization method suitable for natural laminar flow airfoils of green supersonic civil transport according to claim 1, characterized in that, The general expression of the parametric curve obtained in Step 4 includes the general expression of the parametric curve on the upper surface of the airfoil and the general expression of the parametric curve on the lower surface of the airfoil as equation (7): （7） wherein: wherein: Represents the upper surface profile function of the airfoil; Represents the lower surface profile function of the airfoil; The ordinate of the point at the trailing edge of the upper surface of the airfoil; The ordinate represents the point at the trailing edge of the lower surface of the airfoil.

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