CN103942366A - Continuous-curvature airfoil profile represented on basis of four rational Bezier curves, and generation method for continuous-curvature airfoil profile - Google Patents

Abstract

The invention discloses a continuous-curvature airfoil profile represented on the basis of four rational Bezier curves, and a generation method for the continuous-curvature airfoil profile. The front edge portion of an upper molded line and the tail edge portion of the upper molded line are represented by two rational Bezier curves, the front edge portion of a lower molded line and the tail edge portion of the lower molded line are represented by the other two rational Bezier curves, and the four rational Bezier curves are sequentially connected by adjusting the positions and the weights of control vertexes near the splicing points so that the continuous-curvature airfoil profile can be generated. Parameters in a function have clear geometrical significance, the expected airfoil profile or an expected airfoil profile family can be generated by adjusting parameter values, the curvature radius of the front edge of the airfoil profile can be controlled, the position and the curvature of the highest point of the upper molded line are matched with those of the tail portion of the airfoil profile so that the closed airfoil profile with the thick tail edge can be generated, and reverse design is achieved; the four rational Bezier curves are made to approach an existing airfoil profile by adjusting the parameter values, and therefore approximate expression of the existing airfoil profile can be obtained, and the continuous-curvature airfoil profile can be applied to forward direction aerodynamic optimization design.

Description

Continuous aerofoil profile and the generation method thereof of curvature representing based on four sections of reasonable Bézier curves

Technical field

The present invention relates to the blade of turbomachine or wing and preparation method thereof, the especially aerofoil profile of blade or wing and generation method thereof

Background technology

At present, the aerofoil profile molded line of blade or wing is generally determined by coordinate database, provides a series of coordinate datas, and the image lattice then representing by smooth curve connection data in order generates aerofoil profile molded line with the method.This kind of molded line existence deficiency in many ways that method generates: the molded line that (1) generates is difficult to ensure the overall continuity of curvature; (2) number of parameters in molded line expression formula is generally more, also there is no clear and definite geometric meaning simultaneously, and in the time that parameter value changes, image can be difficult to the variation of precognition, and air foil shape is even beyond expression; (3) be difficult to for all available similar smooth curve connections of polytype aerofoil profile data; (4) the aerofoil profile line of this kind of method generation only can represent an aerofoil profile, can not represent a family of aerofoil sections.

Summary of the invention

In order to overcome the deficiencies in the prior art in blade or air-foil making, the invention provides continuous aerofoil profile and the generation method thereof of curvature that reasonable Bézier curve that four sections of number of times of a kind of use are not less than 3 times represents, in the time that the parameter value in function changes, a new aerofoil profile has just generated, in function, the geometric meaning of parameter is very clear and definite, by adjusting parameter value, can generate aerofoil profile by the direction of expection, realize reverse design.

The technical solution adopted for the present invention to solve the technical problems is: be not less than the reasonable Bézier curve C of 3 times with four sections of number of times ₀, C ₁, C ₂and C ₃by splicing and combining into a Curve of wing that curvature is continuous, each section of reasonable Bézier curve is by function

$C_j\left(t\right)=\frac{\underset{i=0}{\overset{n_j}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{P}_i^j{B}_{{i,n}_j}\left(t\right)}{\underset{i=0}{\overset{n_j}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{B}_{i,n_j}\left(t\right)},$ 0≤t≤1,n _j≥3,j＝0,1,2,3

Represent, or represent by its algebraic transformation formula, or represent by its coordinate transform formula, or represent with its parametric equation, or represent with its polar coordinates type.

In formula , be n) _jinferior reasonable Bézier curve C _jcontrol vertex, it is curve C _jpower corresponding to control vertex, C _j(t) be curve C _jthe point that upper parametric t is corresponding.Connecting method is: four sections of curves connect successively, C ₀one end P ₀ ⁰be set to the trailing edge point of molded line in aerofoil profile, the other end with C ₁one end P ₀ ¹splicing, C ₁the other end with C ₂one end P ₀ ²splicing, C ₂the other end with C ₃one end P ₀ ³splicing, C ₃the other end be set to the trailing edge point of molded line under aerofoil profile, and by C ₁with C ₂splice point be set to the leading edge point of aerofoil profile.

Be that curvature is continuous in order to make the represented aerofoil profile of build-up curve, each section of reasonable Bézier curve need meet: P ₀ ⁰with horizontal ordinate is identical; with P ₁ ¹three point on a straight line; with P ₁ ²three point on a straight line and horizontal ordinate are identical; with P ₁ ³three point on a straight line; Curve C _jwith C _j+1curvature identical (j=0,1,2) in stitching portion.

Work as n _jtechnical scheme when=3 (j=0,1,2,3) is: each section of 3 reasonable Bézier curves are by function

$C_j\left(t\right)=\frac{\underset{i=0}{\overset{3}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{P}_i^j{B}_{i,3}\left(t\right)}{\underset{i=0}{\overset{3}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{B}_{i,3}\left(t\right)},$ 0≤t≤1,j＝0,1,2,3

Represent, or represent by its algebraic transformation formula, or represent by its coordinate transform formula, or represent with its parametric equation, or represent with its polar coordinates type.

Be that curvature is continuous in order to make the represented aerofoil profile of build-up curve, each section of reasonable Bézier curve need meet (see figure 1): P ₀ ⁰with P ₃ ³horizontal ordinate is identical; P ₂ ⁰, P ₃ ⁰(P ₀ ¹) and P ₁ ¹three point on a straight line; P ₂ ¹, P ₃ ¹(P ₀ ²) and P ₁ ²three point on a straight line and horizontal ordinate are identical; P ₂ ², P ₃ ²(P ₀ ³) and P ₁ ³three point on a straight line; Curve C _jwith C _j+1curvature identical (j=0,1,2) in stitching portion.Due to for 3 reasonable Bézier curve C _j, can be by power corresponding control vertex under the prerequisite that does not change curve shape be adjusted into 1, 1( with separate and by unique definite), therefore the curvilinear function in the technical program can be reduced to

$C_j\left(t\right)=\frac{{P_0}^j{B}_{\mathrm{0,3}}\left(t\right)+v_1^j{P_1}^j{B}_{\mathrm{1,3}}\left(t\right)+v_2^j{P_2}^j{B}_{\mathrm{2,3}}\left(t\right)+{P_3}^j{B}_{\mathrm{3,3}}\left(t\right)}{B_{\mathrm{0,3}}\left(t\right)+v_1^j{B}_{\mathrm{1,3}}\left(t\right)+v_2^j{B}_{\mathrm{2,3}}\left(t\right)+B_{\mathrm{3,3}}\left(t\right)},$ 0≤t≤1,j＝0,1,2,3

Thereby, according to the curvature at two end points places of every section of curve, can determine by following formula is unique with

$v_1^j=\frac{4}{3}{\left(\frac{{\left(c_0^j\right)}^2{c}_1^j}{k_j^2{k}_{j+1}}\right)}^{\frac{1}{3}},v_2^j=\frac{4}{3}{\left(\frac{c_0^j{\left(c_1^j\right)}^2}{k_j{k}_{j+1}^2}\right)}^{\frac{1}{3}}$

$c_0^j=\frac{1}{2}\frac{\left|\begin{array}{ccc}{x}_0^j& x_1^j& x_2^j\\ y_0^j& y_1^j& y_2^j\\ 1& 1& 1\end{array}\right|}{\sqrt{{(x_1^j-x_0^j)}^2+{(y_1^j-y_0^j)}^2}},$ $c_1^j=\frac{1}{2}\frac{\left|\begin{array}{ccc}{x}_1^j& x_2^j& x_3^j\\ y_1^j& y_2^j& y_3^j\\ 1& 1& 1\end{array}\right|}{\sqrt{{(x_3^j-x_2^j)}^2{(y_3^j-y_2^j)}^2}}$

Here k _j+1represent curve C _jwith C _j+1the curvature (j=0,1,2) of stitching portion, k ₀represent the curvature of molded line trailing edge point in aerofoil profile, k ₄represent the curvature of molded line trailing edge point under aerofoil profile.

In the aerofoil profile generating function providing in the present invention, parameter is selected as the control vertex coordinate of reasonable B é zier function and the curvature of splice point, is 3 times at four curves, and P ₂ ⁰, P ₃ ⁰(P ₀ ¹) and P ₁ ¹3 place straight lines and P ₂ ², P ₃ ²(P ₀ ³) and P ₁ ³when 3 place straight lines are all parallel to x axle, remove the parameter that is mutually related, 23 final separate parameters are: parameter 1(P ₀ ⁰with P ₃ ³horizontal ordinate), parameter 2 (P ₀ ⁰ordinate), parameter 3 (P ₁ ⁰horizontal ordinate), parameter 4 (P ₁ ⁰ordinate), parameter 5 (P ₂ ⁰horizontal ordinate), parameter 6 (P ₂ ⁰, P ₃ ⁰, P ₀ ¹, P ₁ ¹ordinate), parameter 7 (P ₃ ⁰, P ₀ ¹horizontal ordinate), parameter 8 (P ₁ ¹horizontal ordinate), parameter 9 (P ₂ ¹ordinate), parameter 10(P ₂ ¹, P ₃ ¹, P ₀ ², P ₁ ²horizontal ordinate), parameter 11 (P ₁ ²ordinate), parameter 12(P ₂ ²horizontal ordinate), parameter 13(P ₂ ², P ₃ ², P ₀ ³, P ₁ ³ordinate), parameter 14(P ₀ ³, P ₃ ²horizontal ordinate), parameter 15(P ₁ ³horizontal ordinate), parameter 16(P ₂ ³horizontal ordinate), parameter 17(P ₂ ³ordinate), parameter 18(P ₃ ³ordinate), parameter 19(P ₀ ⁰place curvature), parameter 20(P ₃ ⁰or P ₀ ¹place curvature), parameter 21(P ₃ ¹or P ₀ ²place curvature), parameter 22(P ₃ ²or P ₀ ³place curvature), parameter 23(P ₃ ³place's curvature).

In above-mentioned 23 parameters, there are 15 parameter correspondences the geometrical property of aerofoil profile: parameter 1-2(P ₀ ⁰coordinate) corresponding trailing edge upper side position, parameter 1-4(P ₀ ⁰and P ₁ ⁰coordinate) determine afterbody contract angle, the coordinate of camber line peak in parameter 6-7 correspondence, parameter 10(P ₂ ¹, P ₃ ¹, P ₀ ², P ₁ ²horizontal ordinate) corresponding leading edge point horizontal ordinate, parameter 1,16-18(P ₂ ³and P ₃ ³coordinate) determine afterbody contract angle, parameter 19-23 correspondence the curvature of relevant position.

The generation method of the continuous aerofoil profile of the curvature that represents with four sections of reasonable Bézier curves is: first determine above-mentioned respective function, or its algebraic transformation formula, or its coordinate transform formula, or its parametric equation, or the expression of its polar coordinates type, and then respectively by this respective function, or its algebraic transformation formula, or its coordinate transform formula, or its parametric equation, or the expression of its polar coordinates type generates aerofoil profile.The method of determining expression is to parameter assignment, then can obtain respectively a Curve of wing that curvature is continuous by four sections of reasonable Bézier curve combination producings, given different parameter class value can generate dissimilar aerofoil profile, also can allow four reasonable Bézier curves approach existing aerofoil profile by adjusting parameter value, provide the approximate expression of existing aerofoil profile.

Beneficial effect of the present invention: just can control by setup parameter variation range and generate aerofoil profile leading edge radius-of-curvature, the position of upper molded line peak and the adjustment of curvature, afterbody contract angle and thickness, produce dissimilar aerofoil profile, the aerofoil profile curvature of generation is continuous, there will not be any rough phenomenon.

Brief description of the drawings

Below in conjunction with accompanying drawing and example, the present invention is further described.

Fig. 1 is the symbol diagram of four sections of three reasonable Bézier curves and each control vertex thereof.

Fig. 2 is one of aerofoil profile image embodiment generating with four three reasonable Bézier curves.

Fig. 3 is with two of the aerofoil profile image embodiment of four three reasonable Bézier curves generations.

Fig. 4 is with three of the aerofoil profile image embodiment of four three reasonable Bézier curves generations.

Fig. 5 is with four of the aerofoil profile image embodiment of four three reasonable Bézier curves generations.

Fig. 6 is with five of the aerofoil profile image embodiment of four three reasonable Bézier curves generations.

Fig. 7 is with six of the aerofoil profile image embodiment of four three reasonable Bézier curves generations.

Fig. 8 is with seven of the aerofoil profile image embodiment of four three reasonable Bézier curves generations.

Fig. 9 is with eight of the aerofoil profile image embodiment of four three reasonable Bézier curves generations.

Embodiment

For four three reasonable B é zier functions of each group, provide some expressions with definite parameter value as embodiment, and draw corresponding aerofoil profile image.

One of embodiment: give respectively lower train value by the control vertex of four three reasonable B é zier functions and power

P ₀ ⁰=(1.00，0.00)，P ₁ ⁰=(0.69，0.04)，P ₂ ⁰=(0.42，0.06)，P ₃ ⁰=(0.29，0.06)，

$\begin{array}{c}{\mathrm{\ω}}_0^0=1.00,{\mathrm{\ω}}_1^0=1.14,{\mathrm{\ω}}_2^0=1.46,{\mathrm{\ω}}_3^0=1.00\\ {P_0}^1=\left(\mathrm{0.29,0.06}\right),{P_1}^1=\left(\mathrm{0.26,0.06}\right),{P_2}^1=\left(\mathrm{0.00,0.06}\right),{P_3}^1=\left(\mathrm{0.00,0.00}\right),\\ {\mathrm{\ω}}_0^1=1.00,{\mathrm{\ω}}_1^1=1.53,{\mathrm{\ω}}_2^1=1.10,{\mathrm{\ω}}_3^1=1.00\\ {P_0}^2=\left(\mathrm{0.00,0.00}\right),{P_1}^2=(0.00,-0.06),{P_2}^2=(0.26,-0.06),{P_3}^2=(0.29,-0.06),\\ {\mathrm{\ω}}_0^2=1.00,{\mathrm{\ω}}_1^2=1.10,{\mathrm{\ω}}_2^2=1.53,{\mathrm{\ω}}_3^2=1.00\\ {P_0}^3=(0.29,-0.06),{P_1}^3=(0.42,-0.06),{P_2}^3=(0.69,-0.04),{P_3}^3=\left(\mathrm{1.00,0.00}\right),\\ {\mathrm{\ω}}_0^3=1.00,{\mathrm{\ω}}_1^3=1.46,{\mathrm{\ω}}_2^3=1.14,{\mathrm{\ω}}_3^3=1.00\end{array}$

In substitution mapping software, and merge image, generate NACA0012 aerofoil profile shown in Fig. 2.

Two of embodiment: give respectively lower train value by the control vertex of four three reasonable B é zier functions and power

P ₀ ⁰=(1.00，2e-04)，P ₁ ⁰=(0.76，0.02)，P ₂ ⁰=(0.52，0.05)，P ₃ ⁰=(0.36，0.05)，

$\begin{array}{c}{\mathrm{\ω}}_0^0=1.00,{\mathrm{\ω}}_1^0=2.95,{\mathrm{\ω}}_2^0=2.14,{\mathrm{\ω}}_3^0=1.00\\ {P_0}^1=(0.,\mathrm{36,0.05}),{P_1}^1=\left(\mathrm{0.11,0.05}\right),{P_2}^1=\left(\mathrm{0.00,0.02}\right),{P_3}^1=\left(\mathrm{0.00,0.00}\right),\\ {\mathrm{\ω}}_0^1=1.00,{\mathrm{\ω}}_1^1=1.17,{\mathrm{\ω}}_2^1=1.66,{\mathrm{\ω}}_3^1=1.00\\ {P_0}^2=\left(\mathrm{0.00,0.00}\right),{P_1}^2=(0.00,-0.02),{P_2}^2=(0.11,-0.05),{P_3}^2=(0.36,-0.05),\\ {\mathrm{\ω}}_0^2=1.00,{\mathrm{\ω}}_1^2=1.66,{\mathrm{\ω}}_2^2=1.17,{\mathrm{\ω}}_3^2=1.00\\ {P_0}^3=(0.36,-0.05),{P_1}^3=(0.52,-0.05),{P_2}^3=(0.76,-0.02),{P_3}^3=(1.00,-2e-04),\\ {\mathrm{\ω}}_0^3=1.00,{\mathrm{\ω}}_1^3=2.14,{\mathrm{\ω}}_2^3=2.94,{\mathrm{\ω}}_3^3=1.00\end{array}$

In substitution mapping software, and merge image, generate NACA63A010 aerofoil profile shown in Fig. 3.

Three of embodiment: give respectively lower train value by the control vertex of four three reasonable B é zier functions and power

P ₀ ⁰=(1.00,1.8e-03)，P ₁ ⁰=(0.79,0.09)，P ₂ ⁰=(0.50,0.09)，P ₃ ⁰=(0.49,0.09)，

$\begin{array}{c}{\mathrm{\ω}}_0^0=1.00,{\mathrm{\ω}}_1^0=3.07,{\mathrm{\ω}}_3^0=5.86,{\mathrm{\ω}}_3^0=1.00\\ {P_0}^1=\left(\mathrm{0.49,0.09}\right),{P_1}^1=\left(\mathrm{0.26,0.09}\right),{P_2}^1=\left(\mathrm{0.00,0.05}\right),{P_3}^1=\left(\mathrm{0.00,0.00}\right),\\ {\mathrm{\ω}}_0^1=1.00,{\mathrm{\ω}}_1^1=1.13,{\mathrm{\ω}}_2^1=1.18,{\mathrm{\ω}}_3^1=1.00\end{array}$

P ₀ ²=(0.00，0.00)，P ₁ ²=(0.00,-0.05)，P ₂ ²=(0.27,-0.09)，P ₃ ²=(0.49,-0.09)，

$\begin{array}{c}{\mathrm{\ω}}_0^2=1.00,{\mathrm{\ω}}_1^2=1.18,{\mathrm{\ω}}_2^2=1.13,{\mathrm{\ω}}_3^2=1.00\\ {P_0}^3=(0.49,-0.09),{P_1}^3=(0.50,-0.09),{P_2}^3=(0.79,-0.09),{P_3}^3=(1.00,-1.8e-03),\\ {\mathrm{\ω}}_0^3=1.00,{\mathrm{\ω}}_1^3=5.88,{\mathrm{\ω}}_2^3=3.07,{\mathrm{\ω}}_3^3=1.00\end{array}$

In substitution mapping software, and merge image, generate NACA16018 aerofoil profile shown in Fig. 4.

Four of embodiment: give respectively lower train value by the control vertex of four three reasonable B é zier functions and power

P ₀ ⁰=(1.00，0.00)，P ₁ ⁰=(0.69,0.06)，P ₂ ⁰=(0.45,0.06)，P ₃ ⁰=(0.43,0.06)，

$\begin{array}{c}{\mathrm{\ω}}_0^0=1.00,{\mathrm{\ω}}_1^0=2.02,{\mathrm{\ω}}_2^0=2.54,{\mathrm{\ω}}_3^0=1.00\\ {P_0}^1=\left(\mathrm{0.43,0.06}\right),{P_1}^1=\left(\mathrm{0.18,0.06}\right),{P_2}^1=\left(\mathrm{0.00,0.02}\right),{P_3}^1=\left(\mathrm{0.00,0.00}\right),\\ {\mathrm{\ω}}_0^1=1.00,{\mathrm{\ω}}_1^1=1.13,{\mathrm{\ω}}_2^1=1.36,{\mathrm{\ω}}_3^1=1.00\\ {P_0}^2=\left(\mathrm{0.00,0.00}\right),{P_1}^2=(0.00,-0.04),{P_2}^2=(0.20,-0.06),{P_3}^2=(0.35,-0.06),\\ {\mathrm{\ω}}_0^2=1.00,{\mathrm{\ω}}_1^2=0.59,{\mathrm{\ω}}_2^2=0.54,{\mathrm{\ω}}_3^2=1.00\end{array}$

P ₀ ³=(0.35,-0.06)，P ₁ ³=(0.58,-0.06)，P ₂ ³=(0.79,0.01)，P ₃ ³=(1.00,0.00)，

${\mathrm{\ω}}_0^3=1.00,{\mathrm{\ω}}_1^3=0.90,{\mathrm{\ω}}_2^3=0.75,{\mathrm{\ω}}_3^3=1.00$

In substitution mapping software, and merge image, generate RAE822 aerofoil profile shown in Fig. 5.

Five of embodiment: give respectively lower train value by the control vertex of four three reasonable B é zier functions and power

P ₀ ⁰=(1.00，2.5e-03)，P ₁ ⁰=(0.64,0.04)，P ₂ ⁰=(0.56,0.05)，P ₃ ⁰=(0.39,0.05)，

$\begin{array}{c}{\mathrm{\ω}}_0^0=1.00,{\mathrm{\ω}}_1^0=4.84,{\mathrm{\ω}}_2^0=2.09,{\mathrm{\ω}}_3^0=1.00\\ {P_0}^1=\left(\mathrm{0.39,0.05}\right),{P_1}^1=\left(\mathrm{0.20,0.05}\right),{P_2}^1=\left(\mathrm{0.00,0.03}\right),{P_3}^1=\left(\mathrm{0.00,0.00}\right),\\ {\mathrm{\ω}}_0^1=1.00,{\mathrm{\ω}}_1^1=1.53,{\mathrm{\ω}}_2^1=1.81,{\mathrm{\ω}}_3^1=1.00\end{array}$

P ₀ ²=(0.00，0.00)，P ₁ ²=(0.00,-0.03)，P ₂ ²=(0.20,-0.05)，P ₃ ²=(0.39,-0.05)，

${\mathrm{\ω}}_0^2=1.00,{\mathrm{\ω}}_1^2=1.81,{\mathrm{\ω}}_2^2=1.53,{\mathrm{\ω}}_3^2=1.00$

P ₀ ³=(0.39,-0.05)，P ₁ ³=(0.56,-0.05)，P ₂ ³=(0.64,-0.04)，P ₃ ³=(1.00,-2.5e-03)，

${\mathrm{\ω}}_0^3=1.00,{\mathrm{\ω}}_1^3=2.09,{\mathrm{\ω}}_2^3=4.84,{\mathrm{\ω}}_3^3=1.00$

In substitution mapping software, and merge image, generate SC (2)-0010 supercritical airfoil shown in Fig. 6.

Six of embodiment: give respectively lower train value by the control vertex of four three reasonable B é zier functions and power

P ₀ ⁰=(1.00，3.3e-03)，P ₁ ⁰=(0.92,0.02)，P ₂ ⁰=(0.63,0.07)，P ₃ ⁰=(0.39,0.07)，

${\mathrm{\ω}}_0^0=1.00,{\mathrm{\ω}}_1^0=2.68,{\mathrm{\ω}}_2^0=2.54,{\mathrm{\ω}}_3^0=1.00$

P ₀ ¹=(0.39,0.07)，P ₁ ¹=(0.26,0.07)，P ₂ ¹=(0.00,0.05)，P ₃ ¹=(0.00，0.00)，

${\mathrm{\ω}}_0^1=1.00,{\mathrm{\ω}}_1^1=2.78,{\mathrm{\ω}}_2^1=2.48,{\mathrm{\ω}}_3^1=1.00$

P ₀ ²=(0.00，0.00)，P ₁ ²=(0.00,-0.05)，P ₂ ²=(0.25,-0.07)，P ₃ ²=(0.35,-0.07)，

${\mathrm{\ω}}_0^2=1.00,{\mathrm{\ω}}_1^2=2.55,{\mathrm{\ω}}_2^2=2.91,{\mathrm{\ω}}_3^2=1.00$

P ₀ ³＝(0.35，-0.07)，P ₁ ³＝(0.58，-0.07)，P ₂ ³＝(0.87，0.02)，P ₃ ³＝(1.00，-2.7e-03)，

${\mathrm{\ω}}_0^3=1.00,{\mathrm{\ω}}_1^3=2.76,{\mathrm{\ω}}_2^3=2.45,{\mathrm{\ω}}_3^3=1.00$

In substitution mapping software, and merge image, generate SC (2)-0414 supercritical airfoil shown in Fig. 7.

Seven of embodiment: give respectively lower train value by the control vertex of four three reasonable B é zier functions and power

P ₀ ⁰=(1.00,-9.8e-03)，P ₁ ⁰=(0.81,0.03)，P ₂ ⁰=(0.46,0.03)，P ₃ ⁰=(0.37,0.03)，

${\mathrm{\ω}}_0^0=1.00,{\mathrm{\ω}}_1^0=0.90,{\mathrm{\ω}}_2^0=0.94,{\mathrm{\ω}}_3^0=1.00$

P ₀ ¹=(0.37,0.03)，P ₁ ¹=(0.15,0.03)，P ₂ ¹=(0.00,0.03)，P ₃ ¹=(0.00，0.00)，

${\mathrm{\ω}}_0^1=1.00,{\mathrm{\ω}}_1^1=0.39,{\mathrm{\ω}}_2^1=0.52,{\mathrm{\ω}}_3^1=1.00$

P ₀ ²=(0.00，0.00)，P ₁ ²=(0.00,-0.02)，P ₂ ²=(0.10,-0.03)，P ₃ ²=(0.35,-0.03)，

${\mathrm{\ω}}_0^2=1.00,{\mathrm{\ω}}_1^2=0.69,{\mathrm{\ω}}_2^2=0.54,{\mathrm{\ω}}_3^2=1.00$

P ₀ ³=(0.35,-0.03)，P ₁ ³=(0.89,-0.03)，P ₂ ³=(0.72,0.04)，P ₃ ³=(1.00,-0.01)，

${\mathrm{\ω}}_0^3=1.00,{\mathrm{\ω}}_1^3=0.44,{\mathrm{\ω}}_2^3=0.28,{\mathrm{\ω}}_3^3=1.00$

In substitution mapping software, and merge image, generate SC (2)-0606 supercritical airfoil shown in Fig. 8.

Eight of embodiment: give respectively lower train value by the control vertex of four three reasonable B é zier functions and power

P ₀ ⁰=(1.00,-9.5e-03)，P ₁ ⁰=(0.76,0.07)，P ₂ ⁰=(0.42,0.07)，P ₃ ⁰=(0.38,0.07)，

${\mathrm{\ω}}_0^0=1.00,{\mathrm{\ω}}_1^0=1.73,{\mathrm{\ω}}_2^0=1.98,{\mathrm{\ω}}_3^0=1.00$

P ₀ ¹=(0.38,0.07)，P ₁ ¹=(0.23,0.07)，P ₂ ¹=(0.00,0.05)，P ₃ ¹=(0.00，0.00)，

${\mathrm{\ω}}_0^1=1.00,{\mathrm{\ω}}_1^1=2.16,{\mathrm{\ω}}_2^1=2.28,{\mathrm{\ω}}_3^1=1.00$

P ₀ ²=(0.00，0.00)，P ₁ ²=(0.00,-0.05)，P ₂ ²=(0.25,-0.07)，P ₃ ²=(0.36,-0.07)

${\mathrm{\ω}}_0^2=1.00,{\mathrm{\ω}}_1^2=2.71,{\mathrm{\ω}}_2^2=2.93,{\mathrm{\ω}}_3^2=1.00$

P ₀ ³=(0.36,-0.07)，P ₁ ³=(0.59,-0.07)，P ₂ ³=(0.86,0.02)，P ₃ ³=(1.00,-0.016)，

${\mathrm{\ω}}_0^3=1.00,{\mathrm{\ω}}_1^3=2.85,{\mathrm{\ω}}_2^3=2.57,{\mathrm{\ω}}_3^3=1.00$

In substitution mapping software, and merge image, generate SC (2)-0714 supercritical airfoil shown in Fig. 9.

Claims (6)

1. the continuous aerofoil profile of curvature representing with four sections of reasonable Bézier curves, is characterized in that: be not less than the reasonable Bézier curve C of 3 times by four sections of number of times ₀, C ₁, C ₂and C ₃splice and combine into a Curve of wing that curvature is continuous, each section of reasonable Bézier curve function

$C_j\left(t\right)=\frac{\underset{i=0}{\overset{n_j}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{P}_i^j{B}_{{i,n}_j}\left(t\right)}{\underset{i=0}{\overset{n_j}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{B}_{i,n_j}\left(t\right)},$ 0≤t≤1,n _j≥3,j＝0,1,2,3

Represent, or represent by its algebraic transformation formula, or represent by its coordinate transform formula, or represent with its parametric equation, or represent with its polar coordinates type; P in formula _i ^j(i=0,1 ... n, _j) be n _jinferior reasonable Bézier curve C _jcontrol vertex, it is curve C _jpower corresponding to control vertex, C _j(t) be curve C _jthe point that upper parametric t is corresponding;

Connecting method is: four sections of curves connect successively, C ₀one end P ₀ ⁰be set to the trailing edge point of molded line in aerofoil profile, the other end with C ₁one end P ₀ ¹splicing, C ₁the other end with C ₂one end P ₀ ²splicing, C ₂the other end with C ₃one end P ₀ ³splicing, C ₃the other end be set to the trailing edge point of molded line under aerofoil profile, and by C ₁with C ₂splice point be set to the leading edge point of aerofoil profile;

Be that curvature is continuous in order to make the represented aerofoil profile of build-up curve, each section of reasonable Bézier curve need meet: P ₀ ⁰with horizontal ordinate is identical; with P ₁ ¹three point on a straight line; with P ₁ ²three point on a straight line and horizontal ordinate are identical; with P ₁ ³three point on a straight line; Curve C _jwith C _j+1curvature identical (j=0,1,2) in stitching portion.

2. the continuous aerofoil profile of curvature that four sections of reasonable Bézier curves of use according to claim 1 represent, is characterized in that: by four sections of 3 reasonable Bézier curve C ₀, C ₁, C ₂and C ₃splice and combine into a Curve of wing that curvature is continuous, each section of reasonable Bézier curve is by function

$C_j\left(t\right)=\frac{\underset{i=0}{\overset{3}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{P}_i^j{B}_{i,3}\left(t\right)}{\underset{i=0}{\overset{3}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{B}_{i,3}\left(t\right)},$ 0≤t≤1,j＝0,1,2,3

Represent, or represent by its algebraic transformation formula, or represent by its coordinate transform formula, or represent with its parametric equation, or represent with its polar coordinates type;

Be that curvature is continuous in order to make the represented aerofoil profile of build-up curve, each section of reasonable Bézier curve need meet: P ₀ ⁰with P ₃ ³horizontal ordinate is identical; P ₂ ⁰, P ₃ ⁰(P ₀ ¹) and P ₁ ¹three point on a straight line; P ₂ ¹, P ₃ ¹(P ₀ ²) and P ₁ ²three point on a straight line and horizontal ordinate are identical; P ₂ ², P ₃ ²(P ₀ ³) and P ₁ ³three point on a straight line; Curve C _jwith C _j+1curvature identical (j=0,1,2) in stitching portion.

3. the continuous aerofoil profile of curvature that four sections of reasonable Bézier curves of use according to claim 1 and 2 represent, is characterized in that: by four sections of 3 reasonable Bézier curve C ₀, C ₁, C ₂and C ₃splice and combine into a Curve of wing that curvature is continuous, each section of reasonable Bézier curve is by function

$C_j\left(t\right)=\frac{{P_0}^j{B}_{\mathrm{0,3}}\left(t\right)+{\mathrm{\ω}}_1^j{P_1}^j{B}_{\mathrm{1,3}}\left(t\right)+{\mathrm{\ω}}_2^j{P_2}^j{B}_{\mathrm{2,3}}\left(t\right)+{P_3}^j{B}_{\mathrm{3,3}}\left(t\right)}{B_{\mathrm{0,3}}\left(t\right)+{\mathrm{\ω}}_1^j{B}_{\mathrm{1,3}}\left(t\right)+{\mathrm{\ω}}_2^j{B}_{\mathrm{2,3}}\left(t\right)+B_{\mathrm{3,3}}\left(t\right)},$ 0≤t≤1,j＝0,1,2,3

Represent, or represent by its algebraic transformation formula, or represent by its coordinate transform formula, or represent with its parametric equation, or represent with its polar coordinates type;

Be that curvature is continuous in order to make the represented aerofoil profile of build-up curve, each section of reasonable Bézier curve need meet: P ₀ ⁰with P ₃ ³horizontal ordinate is identical; P ₂ ⁰, P ₃ ⁰(P ₀ ¹) and P ₁ ¹three point on a straight line; P ₂ ¹, P ₃ ¹(P ₀ ²) and P ₁ ²three point on a straight line and horizontal ordinate are identical; P ₂ ², P ₃ ²(P ₀ ³) and P ₁ ³three point on a straight line; Curve C _jwith C _j+1curvature identical (j=0,1,2) in stitching portion.

4. the generation method of the continuous aerofoil profile of the curvature that represents with four sections of reasonable Bézier curves, is characterized in that: comprise the following steps

1) first determine each section of reasonable Bézier curve

$C_j\left(t\right)=\frac{\underset{i=0}{\overset{n_j}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{P}_i^j{B}_{{i,n}_j}\left(t\right)}{\underset{i=0}{\overset{n_j}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{B}_{i,n_j}\left(t\right)},$ 0≤t≤1,n _j≥3,j＝0,1,2,3

Frequency n _j, control vertex coordinate and the curvature at each splice point and upper and lower molded line trailing edge point place, or its algebraic transformation formula, or its coordinate transform formula, or its parametric equation, or the value of relevant parameter in its polar coordinates type; Be that curvature is continuous in order to make the represented aerofoil profile of build-up curve, in the time establishing each section of reasonable Bézier curve apex coordinate, need meet: P ₀ ⁰with horizontal ordinate is identical, with P ₁ ¹three point on a straight line; with P ₁ ²three point on a straight line and horizontal ordinate are identical, with P ₁ ³three point on a straight line, and determine P in each section of curve controlled summit by given curvature ₀ ^jor P ₁ ^jor P ₂ ^jand or or corresponding power (control vertex that these two power can not be corresponding identical);

2) determine again each section of power corresponding to other control vertexs of curve;

3) then by above-mentioned expression formula, or its algebraic transformation formula, or its coordinate transform formula, or its parametric equation, or its polar coordinates type, aerofoil profile generated.

5. the generation method of the continuous aerofoil profile of the curvature that represents with four sections of 3 reasonable Bézier curves, is characterized in that:

First determine each section of reasonable Bézier curve

$C_j\left(t\right)=\frac{\underset{i=0}{\overset{3}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{P}_i^j{B}_{i,3}\left(t\right)}{\underset{i=0}{\overset{3}{\mathrm{\Σ}}}{\mathrm{\ω}}_i^j{B}_{i,3}\left(t\right)},$ 0≤t≤1,j＝0,1,2,3

Frequency n _j, control vertex coordinate and the curvature at splice point and upper and lower molded line trailing edge point place, or its algebraic transformation formula, or its coordinate transform formula, or its parametric equation, or the value of relevant parameter in its polar coordinates type; Be that curvature is continuous in order to make the represented aerofoil profile of build-up curve, each section of reasonable Bézier curve need meet: P ₀ ⁰with P ₃ ³horizontal ordinate is identical; P ₂ ⁰, P ₃ ⁰(P ₀ ¹) and P ₁ ¹three point on a straight line, P ₂ ¹, P ₃ ¹(P ₀ ²) and P ₁ ²three point on a straight line and horizontal ordinate are identical, P ₂ ², P ₃ ²(P ₀ ³) and P ₁ ³three point on a straight line; And determined the power of any two control vertexs in each section of curve controlled summit by given curvature.Then determine each section of power corresponding to other control vertexs of curve, finally by above-mentioned expression formula, or its algebraic transformation formula, or its coordinate transform formula, or its parametric equation, or its polar coordinates type, generate aerofoil profile.

6. the generation method of the continuous aerofoil profile of the curvature that represents with four sections of 3 reasonable Bézier curves, is characterized in that:

First determine each section of reasonable Bézier curve

$C_j\left(t\right)=\frac{{P_0}^j{B}_{\mathrm{0,3}}\left(t\right)+{\mathrm{\ω}}_1^j{P_1}^j{B}_{\mathrm{1,3}}\left(t\right)+{\mathrm{\ω}}_2^j{P_2}^j{B}_{\mathrm{2,3}}\left(t\right)+{P_3}^j{B}_{\mathrm{3,3}}\left(t\right)}{B_{\mathrm{0,3}}\left(t\right)+{\mathrm{\ω}}_1^j{B}_{\mathrm{1,3}}\left(t\right)+{\mathrm{\ω}}_2^j{B}_{\mathrm{2,3}}\left(t\right)+B_{\mathrm{3,3}}\left(t\right)},$ 0≤t≤1,j＝0,1,2,3

Frequency n _j, control vertex coordinate and the curvature at splice point and upper and lower molded line trailing edge point place, or its algebraic transformation formula, or its coordinate transform formula, or its parametric equation, or the value of relevant parameter in its polar coordinates type; Be that curvature is continuous in order to make the represented aerofoil profile of build-up curve, each section of reasonable Bézier curve need meet: P ₀ ⁰with P ₃ ³horizontal ordinate is identical; P ₂ ⁰, P ₃ ⁰(P ₀ ¹) and P ₁ ¹three point on a straight line; P ₂ ¹, P ₃ ¹(P ₀ ²) and P ₁ ²three point on a straight line and horizontal ordinate are identical; P ₂ ², P ₃ ²(P ₀ ³) and P ₁ ³three point on a straight line; And determine in each section of curve and weigh by given curvature with value;

Then press above-mentioned expression formula, or its algebraic transformation formula, or its coordinate transform formula, or its parametric equation, or its polar coordinates type, aerofoil profile generated.