Medium reynolds number single element airfoil for general aviation and uavs

Abstract

The application relates to a medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles. The airfoil is evolved and optimized on the basis of an original airfoil by using a computational fluid dynamics method and a CST parameterization method; the leading edge radius of the airfoil is 1.15% C, the maximum thickness of the airfoil is 15.00% C, the maximum thickness position is 31.13% C, the maximum camber of the airfoil is 4% C, and the maximum camber position is 43.70% C; wherein C is a chord length. Compared with the original airfoil, the airfoil has a lower drag coefficient under the same lift coefficient, and the corresponding lift-drag ratio is also improved; the performance is more excellent within the set lift coefficient range.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of air vehicle airfoil design, in particular to a medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles. BACKGROUND

[0002] At a medium Reynolds number (the Reynolds number is between 1.0x10 6 and 1.0x10 7 ), the transition position of the airfoil surface is relatively forward, and the drag characteristics are poor. According to statistics, when the Reynolds number of the incompressible flat plate flow is 1.0x10 6 , the turbulent friction drag coefficient is 3.5 times that of the laminar friction drag coefficient, and when the Reynolds number is 1.0x10 7 , the turbulent friction drag coefficient is 7 times that of the laminar friction drag coefficient, so the benefit of using a laminar airfoil will be greater and greater as the Reynolds number increases. For unmanned aerial vehicles in a medium Reynolds number working condition of 1.0x10 6 -1.0x10 7 , a laminar airfoil is generally used, or a laminar airfoil is not used based on special design considerations.

[0003] NASA first proposed the concept of a laminar airfoil and developed the NACA six-digit airfoil, which uses an improved theoretical method for design, and the thickness distribution is determined by the required drag coefficient, critical Mach number and maximum lift coefficient. To this day, its improved NACA 6A series airfoils are still widely used. In the 1970s, NASA began to develop natural laminar flow series airfoils. Natural laminar flow technology is a passive flow control method, the main idea of which is to control the characteristics of the flow boundary layer through a well-designed geometric shape, maintain a pressure distribution conducive to transition delay. The natural laminar flow airfoil can effectively reduce the friction drag of the airfoil at the same Reynolds number and improve the aerodynamic efficiency, and is one of the current research hotspots in the field of aviation.

[0004] By optimizing the curvature distribution of the upper surface of the airfoil and reducing the adverse pressure gradient of the leading edge of the airfoil, the transition position of the airfoil can be moved backward. However, this will cause the laminar airfoil to be more prone to flow separation, and thus the maximum use lift coefficient of this type of airfoil is low, so the natural laminar flow airfoil generally has good low resistance performance at a not too high lift coefficient. The cruise lift coefficient of the high-performance subsonic laminar flow airfoil is generally about 1.0, such as the airfoil LRN-1015 of the Global Hawk, which uses a lift coefficient of about 1.1. SUMMARY

[0005] Therefore, it is necessary to provide a medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles in view of the above technical problems.

[0006] A medium Reynolds number single-element airfoil suitable for general aircraft and unmanned aerial vehicles, the leading edge radius of the airfoil is 1.15%C, the maximum thickness of the airfoil is 15.00%C, the maximum thickness position is 31.13%C, the maximum camber of the airfoil is 4%C, and the maximum camber position is 43.70%C; wherein C is the chord length.

[0007] The geometric coordinate expressions of the airfoil upper surface and the airfoil lower surface are as follows:

[0008]

[0009]

[0010] wherein y u and y l are the longitudinal coordinates of the upper surface and the lower surface of the medium Reynolds number single-element airfoil respectively, the similarity function C(x) is C(x)=x N1 (1-x) N2 , N1 and N2 are two empirical parameters, A ui and A li are the undetermined coefficients of the upper surface and the lower surface of the airfoil respectively, S i (x) is the Bernstein polynomial, x is the chord length position, y TEu and y TEu are the original airfoil upper surface and lower surface longitudinal coordinates respectively.

[0011] The medium Reynolds number single-element airfoil suitable for general aircraft and unmanned aerial vehicles is evolved and optimized based on the original airfoil using the computational fluid dynamics method and the CST parameterization method; the leading edge radius of the airfoil is 1.15%C, the maximum thickness of the airfoil is 15.00%C, the maximum thickness position is 31.13%C, the maximum camber of the airfoil is 4%C, and the maximum camber position is 43.70%C; wherein C is the chord length. Compared with the original airfoil, the airfoil has lower drag coefficient at the same lift coefficient, and the corresponding lift-drag ratio is also improved; the performance is more excellent within the set lift coefficient range. BRIEF DESCRIPTION OF DRAWINGS

[0012] Figure 1 is the appearance comparison of the medium Reynolds number single-element airfoil and the original airfoil in one embodiment;

[0013] Figure 2 is the comparison of the aerodynamic performance of the airfoils before and after optimization in another embodiment, wherein (a) is the lift-drag characteristic curve, and (b) is the lift coefficient and lift-drag ratio curve;

[0014] Figure 3 is the comparison of the transition position of the airfoils before and after optimization in another embodiment;

[0015] Figure 4 For another embodiment, the optimized pre and post pressure distribution comparison, where (a) is the optimized pre and post pressure distribution comparison when the lift coefficient is 0.6, and (b) is the optimized pre and post pressure distribution comparison when the lift coefficient is 1.0. DETAILED DESCRIPTION

[0016] In order to make the purposes, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and not intended to limit the present application.

[0017] The medium Reynolds number single-element airfoil suitable for general aircraft and unmanned aerial vehicles proposed in the present application is referred to as GYW-NLF-015 airfoil.

[0018] The medium Reynolds number mentioned in the present application refers to: the Reynolds number is between 1.0x10 6 and 1.0x10 7 .

[0019] In one embodiment, a medium Reynolds number single-element airfoil suitable for general aircraft and unmanned aerial vehicles is provided, the leading edge radius of the airfoil is 1.15%C, the maximum thickness of the airfoil is 15.00%C, the maximum thickness position is 31.13%C, the maximum camber of the airfoil is 4%C, and the maximum camber position is 43.70%C; wherein C is the chord length.

[0020] The geometric coordinate expressions of the upper surface of the airfoil and the lower surface of the airfoil are as follows:

[0021]

[0022]

[0023] wherein, y u and y l are the longitudinal coordinates of the upper surface and the lower surface of the medium Reynolds number single-element airfoil respectively, the class function C(x) is: C(x) = x N1 ·(1-x) N2 , N1 and N2 are two empirical parameters, and as preferred, N1 and N2 are respectively 0.5 and 1, A ui and A li are the undetermined coefficients of the upper surface and the lower surface of the airfoil respectively, x is the chord length position, y TEu and y TEu are the original longitudinal coordinates of the upper surface and the lower surface of the airfoil respectively, S i (x) is the Bernstein polynomial, wherein, N is the order of the Bernstein polynomial.

[0024] The profile comparison between the medium Reynolds number single-element airfoil suitable for general aircraft and unmanned aerial vehicles and the original airfoil is shown as follows Figure 1 .

[0025] Specifically, the medium Reynolds number single-element airfoil is based on the disclosed NLF-1015 airfoil (original airfoil), and is obtained by evolutionary optimization using the computational fluid dynamics method and the CST parameterization method for flight conditions with a Reynolds number of 1.0×10 6 ~1.0×10 7 . The design Reynolds number of the airfoil is 2.0×10 6 , the airfoil has good aerodynamic efficiency in the range of 0.4~1.5 lift, the drag coefficient is the smallest in the range of 0.8~1.2 lift coefficient, the drag coefficient of the airfoil is lower than that of the original airfoil at the same lift coefficient, the drag is reduced by 5%, and the lift-drag ratio is also improved.

[0026] The NLF-1015 airfoil is a special medium Reynolds number natural laminar flow airfoil for high-altitude long-endurance unmanned aerial vehicles designed by Somers et al., and the design Reynolds number is 7.0×10 5 ~2.0×10 6 , the airfoil has good aerodynamic efficiency in the range of 0.4~1.5 lift, and of course the aerodynamic gain of this type of airfoil is greater under higher Reynolds number conditions compared with traditional low Reynolds number airfoils.

[0027] In the above-mentioned medium Reynolds number single-element airfoil suitable for general aircraft and unmanned aerial vehicles and the original airfoil, the airfoil is obtained by evolutionary optimization using the computational fluid dynamics method and the CST parameterization method based on the original airfoil; the leading edge radius of the airfoil is 1.15%C, the maximum thickness of the airfoil is 15.00%C, the maximum thickness position is 31.13%C, the maximum camber of the airfoil is 4%C, and the maximum camber position is 43.70%C; wherein C is the chord length. The drag coefficient of the airfoil is lower than that of the original airfoil at the same lift coefficient, and the lift-drag ratio is also improved; the performance is more excellent in the set lift coefficient range.

[0028] In one embodiment, when the Reynolds number is 2.0×10 6 , the airfoil is optimized according to a preset optimization target using an eight-order CST parameterization method to obtain a medium Reynolds number single-element airfoil; the preset optimization target is the minimum drag coefficient in the range of 0.8~1.2 lift coefficient, and the airfoil thickness and the nose-down moment are limited to be unchanged.

[0029] Specifically, the CST method divides the order according to different orders of Bernstein polynomials, and the eight-order CST parameterization method is used in this embodiment.

[0030] When the Reynolds number is 2.0×10 6At this time, the preset optimization target is set as the minimum drag coefficient when the lift coefficient is in the range of 0.8-1.2, and the wing profile thickness is limited to be unchanged in order to maintain the structural strength, and the nose-down moment is set to be unchanged and large in order to facilitate the aircraft trimming; a set of suitable variables x1 is found to satisfy the preset optimization target, wherein the preset optimization target expression is:

[0031] Objective min Cd Cl＝0.8 (x1)

[0032] min Cd Cl＝1.2 (x1)

[0033] s.t.Cm Cl＝0.8 (x1)≥-0.22

[0034] Cm Cl＝1.2 (x1)≥-0.22

[0035] MaxThickness>0.15C

[0036] Wherein, MaxThickness is the maximum thickness of the wing profile, x1 is the amplitude of the up-and-down fluctuation of the wing profile, C is the chord length; Cd Cl＝0.8 , Cd Cl＝1.2 are the drag coefficients when the lift coefficient is 0.8 and 1.2 respectively; Cm Cl＝0.8 , Cm Cl＝1.2 are the moment coefficients when the lift coefficient is 0.8 and 1.2 respectively.

[0037] In one embodiment, the upper surface of the wing profile and the lower surface of the wing profile are each described by nine parameters; U1, U2, U3, U4, U5, U6, U7, U8, U9 are parameters for describing the upper surface of the wing profile; D1, D2, D3, D4, D5, D6, D7, D8, D9 are parameters for describing the lower surface of the wing profile; the range of the parameters is:

[0038] Parameter Parameter range Parameter Parameter range U1 0.18～0.34 D1 -0.14～-0.08 U2 0.24～0.44 D2 -0.047～-0.027 U3 0.08～0.14 D3 -0.14～-0.08 U4 0.54～0.98 D4 0.014～0.026 U5 -0.44～-0.24 D5 -0.09～-0.05 U6 0.7～1.3 D6 0.01～0.014 U7 -0.25～-0.13 D7 0.05～0.09 U8 0.38～0.68 D8 0.1～0.2 U9 0.27～0.47 D9 0.29～0.49

[0039] In one embodiment, the initial values of the nine parameters for describing the upper surface of the wing profile and the nine parameters for describing the lower surface of the wing profile are:

[0040]

[0041] In one embodiment, the coordinate values are taken from the trailing edge of the wing in a counterclockwise direction, and there are 100 points. When the airfoil chord length is 1, the coordinates of the upper surface of the airfoil are as follows: X, Y represent the discrete point coordinate values of the upper and lower surfaces of the airfoil in a two-dimensional coordinate system, wherein the X, Y values of the upper surface of the airfoil are as follows: (0.99663, 0.0039496), (0.9863428, 0.007334), (0.9671225, 0.0135585), (0.9449418, 0.0203742), (0.921445, 0.0270589), (0.8972054, 0.0334176), (0.8723359, 0.0395015), (0.8469136, 0.0454105), (0.8210849, 0.05122), (0.7950418, 0.056957), (0.7689691, 0.0626006), (0.7429974, 0.0680994), (0.7171886, 0.0733909), (0.6915445, 0.0784183), (0.6660276, 0.0831381), (0.6405796, 0.0875238), (0.6151381, 0.0915649), (0.5896482, 0.0952648), (0.5640746, 0.0986357), (0.5384081, 0.1016935), (0.5126631, 0.1044505), (0.4868866, 0.106911), (0.4611411, 0.1090657), (0.4354967, 0.1108891), (0.4100221, 0.1123414), (0.3847659, 0.113372), (0.3597582, 0.1139252), (0.3350053, 0.113947), (0.3104953, 0.1133879), (0.2862074, 0.1122054), (0.2621247, 0.1103703), (0.2382492, 0.1078608), (0.2146219, 0.1046624), (0.1913257, 0.1007651), (0.1685043, 0.0961586), (0.1463375, 0.0908271), (0.1250189, 0.0847455), (0.1047328, 0.0778861), (0.0856605, 0.0702454), (0.0680464, 0.0619104), (0.0522987, 0.0531749), (0.0389579,0.0446056)、(0.0283517,0.0368291)、(0.0202871,0.0301528)、(0.0142371,0.024512)、(0.009679,0.019684)、(0.0062333,0.0154455)、(0.0036603,0.0116206)、(0.0018236,0.0080877)、(0.000649,0.0047793)、(0.0000815,0.0016801)；.

[0042] The values of the lower surface of the airfoil X, Y are as follows: (0.0000924, -0.0012267), (0.0010486, -0.0041061), (0.0029425, -0.0068247), (0.0056443, -0.0093489), (0.0091273, -0.0117269), (0.0134854, -0.0140231), (0.0189404, -0.0163018), (0.0258795, -0.0186272), (0.0349195, -0.0210639), (0.0469421, -0.023663), (0.0628201, -0.026408), (0.0826311, -0.0291475), (0.1052596, -0.0316464), (0.1293437, -0.0337365), (0.1540872, -0.0353583), (0.1791811, -0.0365146), (0.2045363, -0.0372356), (0.2301324, -0.0375645), (0.2559556, -0.0375506), (0.2819781, -0.0372425), (0.3081534, -0.0366827), (0.3344224, -0.0359024), (0.3607223, -0.0349185), (0.3869984, -0.0337334), (0.4132122, -0.0323383), (0.4393471, -0.0307173), (0.4654107, -0.0288524), (0.4914309, -0.02673), (0.5174557, -0.0243439), (0.5435459, -0.0217006), (0.5697709, -0.0188218), (0.5961918, -0.0157487), (0.6228213, -0.0125472), (0.649489, -0.0093235), (0.6759307, -0.0062042), (0.7020838, -0.0032869), (0.7279813, -0.0006451), (0.7536865, 0.0016675), (0.779269, 0.0036159), (0.8047876, 0.0051815), (0.8302679, 0.0063574), (0.8556726, 0.0071391), (0.8808643, 0.0075107), (0.9049802, 0.0077427), (0.9291279, 0.0078502), (0.9532846, 0.0078832), (0.9774413, 0.0078722), (1.001558, 0.0078502), (1.0256957, 0.0077822), (1.0499324, 0.0076822), (1.0739091, 0.0075822), (1.0976588, 0.0074822), (1.1219295, 0.0073822), (1.1456582, 0.0072822), (1.169937, 0.0071822), (1.1936717, 0.0070822), (1.2175504, 0.0069822), (1.2419701, 0.0068822), (1.2659398, 0.0067822), (1.2899701, 0.0066822), (1.3139904, 0.0065822), (1.3379701, 0.0064822), (1.3619904, 0.0063822), (1.3859701, 0.0062822), (1.4099904, 0.0061822), (1.4339701, 0.0060822), (1.4579904, 0.0059822), (1.4819701, 0.0058822), (1.5059904, 0.0057822), (1.5299701, 0.0056822), (1.5539904, 0.0055822), (1.5779701, 0.0054822), (1.6019904, 0.0053822), (1.6259701, 0.0052822), (1.6499904, 0.0051822), (1.6739701, 0.0050822), (1.6979904, 0.0049822), (1.7219701, 0.0048822), (1.7459904, 0.0047822), (1.7699701, 0.0046822), (1.7939904, 0.0045822), (1.8179701, 0.0044822), (1.8419904, 0.0043822), (1.8659701, 0.0042822), (1.8899904, 0.0041822), (1.9139701, 0.0040822), (1.9379904, 0.0039822), (1.9619701, 0.0038822), (1.9859904, 0.0037822), (1.990.0074265), (0.9293744, 0.0067986), (0.9517429, 0.0055099), (0.9720858, 0.0034705), (0.9898278, 0.000717), (1, -0.0014099).

[0043] In the verification embodiment, the NLF-1015 airfoil is taken as the basis, and the Reynolds number is 2.0 x 10 6 For a period of time, the computational fluid dynamics method and the CST parameter optimization method are used for optimization to obtain a medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles. The comparison of the aerodynamic performance of the medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles and the original airfoil is shown in Table 1. It can be seen that the drag coefficient of the airfoil is reduced at the same lift coefficient on the basis of the original airfoil. Considering that the original airfoil is a designed low-drag airfoil, a 5% reduction in resistance indicates that the optimization has achieved good results, and it can be considered that the optimized airfoil is more excellent than the original airfoil in the set lift coefficient range.

[0044] Table 1 Comparison of airfoil aerodynamic performance before and after optimization

[0045] Before optimization After optimization Gain Before optimization After optimization <![CDATA[Cd Cl＝0.8 ]]> 0.00543 0.00514 5.3％ Cm Cl＝0.8 ]]> -0.2030 -0.2089 Cd Cl＝1.2 ]]> 0.00602 0.00574 4.9％ Cm Cl＝1.2 ]] -0.2081 -0.2161

[0046] Figure 2 For further comparison and analysis of the aerodynamic performance of the airfoils before and after optimization, it can be seen that the drag coefficient of the optimized airfoil is reduced in the lift coefficient range of 0.5-1.3, and the corresponding lift-drag ratio is also improved. Further analysis of the transition position of the airfoil on the upper surface under the same lift coefficient condition shows that the transition position of the optimized airfoil on the upper surface is later than that of the original airfoil before the lift coefficient is 1.2, which is the reason for the reduction of the airfoil resistance and the increase of the aerodynamic efficiency under the same lift coefficient. Figure 3

[0047] The pressure distribution on the surface of the airfoil with a lift coefficient of 0.6 and 1.0 before and after optimization is analyzed as shown in Figure 4 , wherein (a) is the comparison of pressure distribution before and after optimization when the lift coefficient is 0.6, and (b) is the comparison of pressure distribution before and after optimization when the lift coefficient is 1.0. It can be seen from Figure 4 that the negative pressure peak position of the optimized airfoil is obviously moved backward, which increases the range of the upper surface pressure gradient, which is also the reason for the delay of the airfoil transition position.

[0048] Note: Figures 1 to 4 The NLF-1015 airfoil is the original airfoil, and the GYW-NLF-015 airfoil is the medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles proposed in the application. ​

[0049] Any combination of the technical features in the above embodiments can be made. For the sake of brevity, the foregoing description has not described all possible combinations of the technical features in the above embodiments, however, as long as the combination of the technical features does not contradict, it should be considered within the scope of the present disclosure.

[0050] The above embodiments only express several implementation manners of the present application, and the description is relatively specific and detailed, but it should not be understood as a limitation on the scope of the patent. It should be pointed out that for ordinary skilled persons in the art, some modifications and improvements can be made without departing from the concept of the present application, and these are within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the appended claims.

Claims

1. A medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles, characterized by: The leading edge radius of the airfoil is 1.15%C, the maximum thickness of the airfoil is 15.00%C, the maximum thickness position is 31.13%C, the maximum camber of the airfoil is 4%C, the maximum camber position is 43.70%C; wherein C is the chord length; The geometric coordinate expressions of the upper and lower surfaces of the airfoil are: Among them, y u with y l are the ordinates of the upper and lower surfaces of the single-section airfoil with a medium Reynolds number, respectively. The class function C(x) is: C(x) = x N1 (1-x) N2 , N1 and N2 are two empirical parameters, A ui and A li are the unknown coefficients of the upper and lower surfaces of the airfoil, S i (x) is the Bernstein polynomial, x is the chord length and position, y is the TEu 、y TEu are the ordinates of the upper and lower surfaces of the original airfoil, respectively.

2. The medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles according to claim 1, characterized in that: At a Reynolds number of 2.0×10 6 When the airfoil is optimized according to the preset optimization target, the eighth-order CST parameterization method is used to obtain a single-section airfoil with a medium Reynolds number; the preset optimization target is to minimize the drag coefficient when the lift coefficient is in the range of 0.8 to 1.2, while limiting the airfoil thickness to remain unchanged and the nose-down moment to remain unchanged.

3. The medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles according to claim 2, characterized in that: The upper and lower surfaces of the airfoil are each described by nine parameters; U1, U2, U3, U4, U5, U6, U7, U8, and U9 are parameters used to describe the upper surface of the airfoil; D1, D2, D3, D4, D5, D6, D7, D8, and D9 are parameters used to describe the lower surface of the airfoil; among them, the range of U1 is: [0.18, 0.34], the range of U2 is: [0.24, 0.44], the range of U3 is: [0.08, 0.14], the range of U4 is: [0.54, 0.98], the range of U5 is: [-0.44, -0.24], the range of U6 is: [0.7, 1.3], and the range of U7 is: [-0. The range of D1 is [-0.14,-0.08], the range of D2 is [-0.047,-0.027], the range of D3 is [-0.14,-0.08], the range of D4 is [0.014,0.026], the range of D5 is [-0.09,-0.05], the range of D6 is [0.01,0.014], the range of D7 is [0.05,0.09], the range of D8 is [0.1,0.2], and the range of D9 is [0.29,0.49].

4. The medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles according to claim 3, characterized in that: The initial values ​​of the nine parameters used to describe the upper surface of the airfoil are 0.26, 0.34, 0.11, 0.76, -0.34, 1.0, -0.19, 0.53, and 0.37, respectively; the initial values ​​of the nine parameters used to describe the lower surface of the airfoil are -0.11, -0.037, -0.11, 0.020, -0.07, 0.012, 0.07, 0.15, and 0.39, respectively.

5. The medium Reynolds number single-section airfoil suitable for general aircraft and unmanned aerial vehicles according to claim 1, characterized in that: Starting from the trailing edge of the wing, the coordinates are taken counterclockwise. When the chord length of the airfoil is 1, the coordinates corresponding to the upper surface of the airfoil are as follows: X and Y represent the coordinate values ​​of the discrete points on the upper and lower surfaces of the airfoil in the two-dimensional coordinate system, respectively. The values ​​of X and Y on the upper surface of the airfoil are as follows: (0.99663, 0.0039496), (0.9863428, 0.007334), (0.9671225, 0.0135585), (0.9449418, 0.0203742), (0.921445, 0.0270589), (0.8972054, 0.0334176), (0.8723359, 0.0395015), (0.8469136, 0.045 4105),(0.8210849,0.05122),(0.7950418,0.056957),(0.7689691,0.0626006),(0.7429974,0.0680994),(0.7171886,0.0733909),(0.6915445,0.0784183),(0.6660276,0.0831381),(0.6405796,0.0875238),(0.6151381,0.0915649),(0.5896482,0.0952648),(0.5640746,0.0986357),(0.53 84081,0.1016935),(0.5126631,0.1044505),(0.4868866,0.106911),(0.4611411,0.1090657),(0.4354967,0.1108891),(0.4100221,0.1123414),(0.3847659,0.113372),(0.3597582,0.1139252),(0.3350053,0.113947),(0.3104953,0.1133879),(0.2862074,0.1122054),(0.2621247,0.110 3703),(0.2382492,0.1078608),(0.2146219,0.1046624),(0.1913257,0.1007651),(0.1685043,0.0961586),(0.1463375,0.0908271),(0.1250189,0.0847455),(0.1047328,0.0778861),(0.0856605,0.0702454),(0.0680464,0.0619104),(0.0522987,0.0531749),(0.0389579,0.0446056),(0.0283517,0.0368291)、(0.0202871,0.0301528)、(0.0142371,0.024512)、(0.009679,0.019684)、(0.0062333,0.0154455)、(0.0036603,0.0116206)、(0.0018236,0.0080877)、(0.000649,0.0047793)、(0.0000815,0.0016801)；. The X and Y values ​​of the lower surface of the airfoil are as follows: (0.0000924,-0.0012267), (0.0010486,-0.0041061), (0.0029425,-0.0068247), (0.0056443,-0.0093489), (0.0091273,-0.0117269), (0.0134854,-0.0140231), (0.0189404,-0.0163018), (0.0258795,-0.0186272), (0.0349195,-0.0210639), (0.0469421,-0.023663). 28201,-0.026408),(0.0826311,-0.0291475),(0.1052596,-0.0316464),(0.1293437,-0.0337365),(0.1540872,-0.0353583),(0.1791811,-0.0365146),(0.2045363,-0.0372356),(0.2301324,-0.0375645),(0.2559556,-0.0375506),(0.2819781,-0.0372425),(0.3081534,-0.0366827),(0 .3344224,-0.0359024),(0.3607223,-0.0349185),(0.3869984,-0.0337334),(0.4132122,-0.0323383),(0.4393471,-0.0307173),(0.4654107,-0.0288524),(0.4914309,-0.02673),(0.5174557,-0.0243439),(0.5435459,-0.0217006),(0.5697709,-0.0188218),(0.5961918,-0.0157487) 、(0.6228213,-0.0125472),(0.649489,-0.0093235),(0.6759307,-0.0062042),(0.7020838,-0.0032869),(0.7279813,-0.0006451),(0.7536865,0.0016675),(0.779269,0.0036159),(0.8047876,0.0051815),(0.8302679,0.0063574),(0.8556726,0.0071391),(0.8808643,0.0075107),(0.9055707,0.0074265)、(0.9293744,0.0067986)、(0.9517429,0.0055099)、(0.9720858,0.0034705)、(0.9898278,0.000717)、(1,-0.0014099)。.

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