Wing boundary simulation method based on wing profile curved object plane normal, terminal equipment and storage medium

Abstract

The invention discloses a wing boundary simulation method based on an airfoil curved object plane normal, a terminal device and a storage medium, after wing curved boundary information is obtained, the integration degree of a curved unit is judged, the arc length is determined according to different integration degrees, then a Gaussian integration point is determined according to the arc length, and finally a numerical integration is obtained. The accuracy of numerical integration calculation is greatly improved, and then the accuracy of simulation result calculation is indirectly improved.

Description

Technical Field

[0001] The present invention relates to the simulation technology of aircraft wings, in particular to a wing boundary simulation method, a terminal device and a storage medium based on the normal of the airfoil curved surface. Background Art

[0002] The airfoil, commonly known as the wing section or blade section, is one of the core factors affecting the comprehensive performance of an aircraft. The airfoil is the basic element for the aerodynamic surface shape design of aircraft wings, tails, helicopter rotors, propellers, and wind turbine blades. It directly affects the aerodynamic performance of the aircraft, including key aerodynamic characteristics such as lift, drag, and pitching moment. The aerodynamic characteristics of the airfoil can also directly affect the maneuverability and stability of the aircraft. A good airfoil design can provide sufficient control moments to ensure the stability and maneuverability of the aircraft under various flight conditions, thereby improving flight safety.

[0003] In addition, the airfoil is also closely related to the energy conservation, emission reduction, and consumption reduction of the aircraft. Optimizing the airfoil design is one of the effective measures to reduce drag and energy consumption and reduce fuel consumption. The airfoil design determines the lift-to-drag ratio of the aircraft [Li Yunpeng, Han Yongzhi. Research progress on the design of lift augmentation devices based on laminar wings [J]. Advances in Aeronautical Science and Engineering, 2021, 12(4): 1-11.], that is, the ratio of lift to drag. The larger the lift-to-drag ratio, the higher the flight efficiency, and it can provide a longer range under limited fuel conditions while reducing environmental pollution (Allison, E., Kroo, I., Sturdza, P., Suzuki, Y., & Martins-Rivas, H. (2010). Aircraft conceptual design with natural laminar flow. In Proceedings of the 27th Congress of the International Council of the Aeronautical Sciences (pp. 428–436). Nice, France: ICAS.). An excellent airfoil design can achieve a high lift-to-drag ratio, effectively reduce fuel consumption, reduce carbon emissions, increase the range and endurance time, which is crucial for improving the flight efficiency and economy of the aircraft.

[0004] In order to obtain excellent airfoils with strong stability and good drag reduction effect economically and efficiently, computational fluid dynamics methods are usually adopted in engineering applications to achieve the design, simulation, and optimization of airfoils. Computational fluid dynamics is a numerical simulation method that discretizes the physical space into a computational space composed of grids and then uses time and space numerical formats to simulate the complex flow problems during the flight of the wing. The process of discretizing the physical space is called mesh generation. The commonly used meshes in engineering applications include structured meshes, unstructured meshes, and Cartesian meshes. Structured meshes have a high generation efficiency in simple geometric shapes, but it will become increasingly difficult to generate meshes in complex situations. The generation of Cartesian meshes is simple and can provide high computational accuracy in regular computational domains, but in complex geometric shapes, the situation of mesh and boundary mismatch is likely to occur, and special treatment methods such as the immersed boundary method are required, which increases the complexity of implementation. Unstructured meshes have strong automated generation capabilities and can flexibly adapt to any complex geometric shape. In recent decades, they have been widely used in practical engineering problems. The numerical formats on unstructured meshes include Finite Volume (FV), Spectral Volume (SV), Discontinuous Galerkin (DG), PNPM, and Correction Procedure via Reconstruction (CPR), etc. Among them, the derivation and implementation process of the finite volume format are relatively simple and it is easy to be extended to high-order accuracy. In recent years, the proposed Compact Finite Volume (CFV) format and the mature shock capturing technology have continuously enhanced the influence of the finite volume format.

[0005] In order to enable the unstructured mesh finite volume numerical simulation method to truly reflect the flight state of the wing, two aspects need to be particularly noted. One is that for the unstructured meshes generated in engineering applications, the mesh edges are all line segments, while the airfoil generally has a curved boundary. Using line segments to replace the curved boundary makes the numerical shape of the wing slightly different from the actual shape, which may lead to inaccurate numerical simulation results. Therefore, high-order curved boundary representation of the airfoil is required. The other is that in order to improve the computational accuracy and solution efficiency, high-precision numerical formats are often used for solving in engineering applications. However, high-precision formats are often used for the calculation inside the computational domain, and it is often difficult to obtain high-order accuracy for the numerical format on the curved boundary of the airfoil. And the values at the airfoil boundary directly affect the calculation results of the entire computational domain, that is, if the numerical simulation of the airfoil curved boundary cannot reach high-order accuracy, the overall accuracy will surely be affected.

[0006] The object discretized by the finite volume scheme is the conservation law equation in integral form. Therefore, one of the keys to achieving high-order accuracy lies in the high-order numerical integration that matches the format accuracy, such as the flux area integral and the source term volume integral. On straight-sided elements, the implementation of high-order numerical integration is relatively simple, and accurate results can be obtained according to the integration points and weight coefficients in the literature (C. Ollivier-Gooch, A. Nejat, K. Michalak, Obtaining and Verifying High-Order Unstructured Finite Volume Solutions to the Euler Equations, AIAA Journal 47 (2009) 2105–2120. https: / / doi.org / 10.2514 / 1.40585.). However, in the presence of curved-sided elements, especially in the problem of the curved boundary of an airfoil, the numerical integration process faces many challenges. There are two common current treatment methods. The first is the isoparametric element method commonly used in the finite element field. By constructing the mapping relationship between the physical space and the isoparametric space, the equivalent calculation on the isoparametric element is realized. However, this method requires defining the distribution of parameter points, solving and storing the Lagrangian interpolation coefficients and the Jacobian, so the process is slightly cumbersome, especially in high-order isoparametric elements, which will further increase the computational complexity. Krivodonova and Berger proposed a new curved element treatment method for the DG format in 2006. It is based on the calculation of straight-sided elements and uses the normal on the curved surface at the integration points, significantly reducing the complexity of curved element treatment, but the overall accuracy cannot be guaranteed. Li and Nishikawa followed this method in their work and achieved high-order accuracy in the DG / FV and EB3 (Edge-Based Third-Order) formats, but the accuracy of the calculation results is not as good as that of the isoparametric method, and the accuracy can be further improved. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to provide a wing boundary simulation method, a terminal device, and a storage medium based on the normal of the curved airfoil surface, so that the unstructured grid finite volume numerical simulation method can truly reflect the flight state of the wing in view of the deficiencies of the prior art.

[0008] To solve the above technical problem, the technical solution adopted by the present invention is: a wing boundary simulation method based on the normal of the curved airfoil surface, including the following steps: S1. Obtain the information of the curved boundary of the wing; S2. Determine the integration degree of the curved element based on the wing curved boundary information. If the integration degree is 3, use the Simpson's rule to calculate the arc length L; if the integration degree is 4, use the Simpson's 3 / 8 rule to calculate the arc length L. S3. Use the Newton iteration method to calculate the Gaussian integration points on the wing curved boundary. S4. Calculate the normal direction of the Gaussian integration points on the wing curved boundary. S5. Substitute the normal direction of the Gaussian integration points into the formula of the straight boundary discretization method to calculate the numerical integration: ; where, represents the average value of the wing velocity or temperature within the curved element, represents the wing element volume, represents the number of element faces, represents the number of element area integration points, is the area integration coefficient, represents the convective and viscous fluxes at the area integration point k, represents the number of element volume integration points, represents the volume integration coefficient, represents the heat source at the volume integration point q.

[0009] During the simulation of the aircraft wing boundary, inaccurate numerical integration will lead to inaccurate calculation of the wing flow field, and thus incorrect simulation results. After obtaining the wing curved boundary information, the present invention determines the integration degree of the curved element, determines the arc length according to different integration degrees, and then determines the Gaussian integration points according to the arc length, and finally obtains the numerical integration, greatly improving the accuracy of the numerical integration calculation and reducing the calculation complexity.

[0010] The expression of the arc length L of the wing boundary curve calculated using the Simpson's rule on the interval is: ; where, , n is the number of subintervals of , , , are the first-order derivatives of the airfoil curve function f(x) at , , respectively, , , represent the abscissas at point a, the i-th integration point, and point b respectively.

[0011] The arc length L of the wing boundary curve calculated using the Simpson's rule on the interval The expression for the arc length L on it is: ; Among them, , n is the number of sub - intervals of

[0012] In step S3, the specific implementation process of using the Newton iteration method to solve the Gaussian integration points on the airfoil curved boundary includes: 1) Set two parameters: ; , and are the positions of the two integration points respectively, , are the target arc lengths corresponding to the positions of the two integration points, and are the arc lengths from the starting point to the positions of the two integration points respectively; 2) When and are greater than the error tolerance, update the positions of the two integration points using the following formula: and ; 3) Repeat the above step 2) until and are less than or equal to the error tolerance, and output the updated positions of the integration points.

[0013] In step S4, the normal direction calculation formula for the Gaussian integration points on the wing curved boundary is: ; Among them, , are the first - order derivatives of the wing boundary curve at the two integration points respectively.

[0014] As an inventive concept, the present invention also provides a terminal device, including a memory, a processor, and a computer program stored on the memory; the processor executes the computer program to implement the steps of the above - mentioned method.

[0015] As an inventive concept, the present invention also provides a computer - readable storage medium, on which a computer program / instructions are stored; when the computer program / instructions are executed by a processor, the steps of the above - mentioned method are implemented.

[0016] As an inventive concept, the present invention also provides a computer program product, including computer program / instructions; when the computer program / instructions are executed by a processor, the steps of the above - mentioned method are implemented.

[0017] Compared with the prior art, the beneficial effects of the present invention are: 1. The method of the present invention does not require defining the distribution of parameter points, and the calculation process is simple; 2. After obtaining the wing curved boundary information, the present invention judges the integration degree of the curved element, determines the arc length according to different integration degrees, then determines the Gauss integration points according to the arc length, and finally obtains the numerical integration, which greatly improves the accuracy of the numerical integration calculation, and indirectly improves the accuracy of the simulation result calculation. Description of the Drawings

[0018] Figure 1 is the NACA0012 airfoil of quadrilateral mesh; Figure 2 is the schematic diagram of spatial discretization of high-order finite volume format; Figure 3 is the schematic diagram of the straight-edge element calculation method based on the normal direction of the curved surface; (a) curved boundary and curved normal, (b) straight boundary and curved normal; Figure 4 is the flow chart of the method of the embodiment of the present invention; Figure 5 is the distribution of three sets of meshes near the airfoil surface; (a) the sparsest, (b) medium, (c) the densest; Figure 6 is the negative pressure coefficient (-C p ) on the airfoil surface obtained by different methods at different angles of attack; (a) 3-degree angle of attack, (b) 4-degree angle of attack, (c) 5-degree angle of attack; Figure 7 is the entropy error distribution of the wall element; (a) the sparsest, (b) the densest; Figure 8 is the flow-around mesh of the viscous NACA0012 airfoil; (a) the sparsest (14200 mesh quantity), (b) medium (15620 mesh quantity), (c) the densest (17040 mesh quantity); Figure 9 Reconstructed airfoil curve and mesh near the wall at local positions of three sets of meshes; (a) sparsest position 1, (b) medium position 1, (c) densest position 1, (d) sparsest position 2, (e) medium position 2, (f) densest position 2; Figure 10 is the pressure coefficient C p obtained by different methods; (a) the sparsest, (b) medium, (c) the densest; Figure 11 is the integrated drag coefficient C D obtained by different methods; (a) the sparsest, (b) medium, (c) the densest; Among them, Figure 6 、 Figure 7 、 Figure 9 、 Figure 10 the abscissa of the horizontal axis is the abscissa of the wing boundary point, Figure 9The vertical axis represents the ordinate of the wing boundary points. Detailed implementation mode

[0019] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention. Embodiment

[0020] The expression of the airfoil curved boundary requires using a curve function to represent the geometric model boundary of the airfoil, which involves the mapping from the geometric model to the computational model.

[0021] The mapping from the geometric model to the computational model requires three steps: generating a grid, constructing a coordinate system, and calculating coordinates. Taking the NACA0012 airfoil shown in Figure 1 as an example, the three steps will be described below.

[0022] 1. Generating a grid As shown in Figure 1 , first, points are densely distributed on the geometric model boundary of the NACA0012 airfoil for the discretization of the geometric boundary. These points are called grid points, and the adjacent grid points are connected to form grid edges. After the grid edges on the geometric boundary are formed, a grid generation software can be used to generate a grid for the external computational region of the NACA0012 airfoil. As shown in Figure 1 , a quadrilateral grid is generated, and a triangular grid can also be generated. Although the grid edges of the unstructured grid are all composed of line segments, the curved boundary of the airfoil can be accurately expressed through the airfoil high-order curved boundary representation method (Gao H, Wang Z, Liu Y. A study of curved boundary representations for 2D high-order Euler solvers [J]. Journal of Scientific Computing. 2010, 44: 323–336.), which is the basis for constructing the airfoil element-free high-order accuracy format and performing high-order numerical integration.

[0023] 2. Constructing a coordinate system It is necessary to construct the XOY plane. Set the leading edge point of the NACA0012 airfoil as point O. Along the chord line of the NACA0012 airfoil, that is, the straight line connecting the leading edge and the trailing edge, set the x-axis, and the direction from the leading edge to the trailing edge is the positive direction of the x-axis. Pass through point O and set the direction perpendicular to the x-axis as the y-axis, and the upward direction is the positive direction of the y-axis.

[0024] 3. Calculate coordinates The construction of the numerical format in numerical form is related to the grid point coordinates. For each grid point in the calculation model, the grid coordinates of the grid point in the XOY plane are calculated according to the grid scale .

[0025] After completing the above three steps, a high-order accurate numerical format for the airfoil curved element can be constructed

[0026] The following introduces the discretization method of the straight-edge element of the airfoil curved element

[0027] (1) High-order accurate unstructured finite volume format The high-order accurate unstructured finite volume format needs to solve the Navier-Stokes (NS) equation in integral form. The integral conservation law equation is

[0028] where u is the conserved variable, s is the source term, F c and F v are the convective and viscous fluxes respectively. The above equation can be transformed by Gauss's theorem into

[0029] where is the average value of the conserved variable on the control volume , and are the convective and viscous fluxes along the outer normal direction of the cell face respectively. According to the discretization scheme of the high-order finite volume format, on the control volume cell i, the semi-discrete form of the equation can be expressed as

[0030] Combined with Figure 1 it can be seen that represents the number of cell faces of the control volume cell i, and represent the number of area integration and volume integration points respectively, and and are the area integration and volume integration coefficients. According to different format accuracy requirements, the number, position and integration weight coefficients of the area integration and volume integration points will change

[0031] In addition, in the formula represents the numerical flux at the integration point k, which consists of two parts: convection and viscosity :

[0032] The convective numerical flux can be calculated using the Roe scheme:

[0033] where is the Jacobian based on the Roe-averaging method, and are the left and right values of the variables at the integration points respectively. In contrast, the calculation of the viscous numerical flux depends on the variable gradients at the interface and can be calculated using the α-damping scheme:

[0034] In the formula, α is a constant, which is taken as in the calculation of the embodiments of the present invention, and are the length of the line connecting the reference points of element i and its adjacent element j on the face and the unit vector of the line respectively:

[0035] (2) Discretization method for straight-sided elements When integrating the airfoil curved elements, the isoparametric elements or directly the straight-sided elements are usually used for calculation. The mathematical derivation process of the isoparametric element method is rigorous, and it can accurately obtain the flux surface integral and the source term volume integral on the curved element, providing a basis for the realization of high-order accuracy, but the overall process is slightly cumbersome. Based on this situation, Krivodonova and Berger proposed a simplified scheme for the DG format, avoiding the construction of isoparametric elements and the calculation and storage of Jacobians.

[0036] Such as Figure 3As shown in the figure, for the curved-edge elements of the airfoil, they can be directly calculated according to the straight-edge elements, that is, on the curved boundary, the curved boundary is regarded as a straight edge, and the corresponding boundary length is directly solved according to formula (7), and then the high-order finite volume format on the curved boundary of the airfoil is obtained according to formula (3). This method is easy to implement and does not require excessive special treatment of the curved elements. Moreover, from the results obtained in the literature (L. Krivodonova, M. Berger, High-order accurate implementation of solid wall boundary conditions in curved geometries, Journal of Computational Physics 211 (2006) 492–512. https: / / doi.org / 10.1016 / j.jcp.2005.05.029.), when this method is adopted in the DG format, high-order accurate results are obtained in the inviscid problem.

[0037] When using the straight-edge element discretization method to calculate the numerical flux at the curved Gauss integration points, the velocity component still needs to be calculated using the outer normal of the curved boundary at the fitting position. However, based on the high-order curved boundary expression of the airfoil, the straight-edge element discretization method cannot clearly select the outer normal of the airfoil surface of the curved boundary. Krivodonova did not reconstruct the high-order expression of the curved surface, but completely based on the grid information of the straight edge, constructed an approximate circular arc with two element vertices located on the curved boundary, and calculated the average value of the circular arc radius according to the adjacent vertices on the left or right, so as to obtain the outer normal at the projection point of the Gauss integration point on the straight boundary on the approximate circular arc curved boundary.

[0038] In Li's work, the curved element processing idea of Krivodonova and Berger was adopted for the DG / FV format. However, different from this, Li used the cubic spline interpolation method to construct the curved boundary expression, thus overcoming the problem of directly approximating the real curved boundary with a circular arc, and also obtained high-order accurate results in the curved boundary problem. However, after analysis, there are still three key problems in this idea. First, Li did not mention in his work how to determine the calculation position of the outer normal vector of the curved boundary based on the reconstructed standard cubic spline interpolation curve. Second, using the standard cubic spline interpolation curve is not an ideal choice. Even after improving it with the piecewise idea, there are still obvious geometric errors in the reconstructed curve compared with the improved polynomial curve. Finally, this curved boundary treatment scheme has only been verified in the high-order accurate DG and DG / FV formats at present. Whether it is still effective within the framework of the high-order FV format needs to be further verified.

[0039] The following introduces the straight-edge element discretization method based on the normal of the curved surface of the airfoil.

[0040] For the three key issues mentioned above, the example of the present invention, based on the airfoil boundary expressed by the high-order curved boundary expression method, improves the straight-edge element discretization method directly used on the airfoil curved element from three aspects, and proposes a straight-edge element discretization method based on the normal direction of the airfoil curved surface.

[0041] Regarding the curved boundary unit outer normal, Gauss integration points, and arc length required for solving the airfoil curved element integral in formulas (5), (3), and (7), the straight-edge element discretization method based on the normal direction of the airfoil curved surface proposed in the embodiment of the present invention has made three optimizations compared with the direct straight-edge element discretization of the airfoil curved boundary: First, an algorithm for calculating the arc length of the airfoil curved boundary is proposed. Instead of directly calculating the length of the straight edge, the arc length of the airfoil curved boundary is accurately obtained; second, a new algorithm for calculating Gauss integration points is proposed. The Gauss integration points are no longer selected on the straight edge, but on the airfoil curved boundary; third, a calculation method for the unit outer normal on the surface is proposed. Instead of calculating the surface normal on the straight edge, the surface normal of the Gauss integration points on the exact curved boundary is obtained.

[0042] (3) Airfoil Curved Boundary Arc Length Calculation Method The position of the Gauss integration points is determined by the arc length. This is relatively easy for straight-edge elements. Assuming in two-dimensional conditions, the two endpoints of the surface to be integrated are respectively , . If the number of integration points is 2, the positions of the two integration points and are: (8) If the number of integration points is 3, we have: (9) However, on the curved boundary, it is not easy to determine the position of the integration points according to the arc length of the unit curved edge. Taking the curve as an example, the arc length L on the interval can be obtained according to the following analytical formula: (10) However, when the expression of the curve is relatively complex, the exact integral of the arc length cannot be directly obtained and numerical methods need to be used. The currently more common calculation method is the trapezoidal method, that is, the curve is divided into n subintervals, and the width of each subinterval is: (11) At this time, according to the trapezoidal method, we have: (12) However, the integration degree of the trapezoidal method is only 1. When the grid points on the curved boundary are relatively sparse, low-order truncation errors will be introduced.

[0043] The embodiment of the present invention proposes a method for calculating the curved edge length of an airfoil curved element based on the Simpson and Simpson 3 / 8 rules. When using Simpson and Simpson 3 / 8, the integration degrees are higher than that of the trapezoidal method, which are 3 and 4 respectively, and the accuracy is higher. Therefore, it is more compatible with the finite volume format with third-order and fourth-order accuracies.

[0044] The Simpson rule approximates the integration region through a quadratic polynomial, but it is necessary to divide the interval into an even number of uniform sub-intervals: (13) At this time, the arc length L can be approximated as: (14) On this basis, the more accurate Simpson 3 / 8 rule requires dividing the interval into multiples of 3: (15) The arc length L is: (16) The following introduces the method for calculating the Gaussian integration points on the airfoil curved boundary.

[0045] After calculating the curved edge arc length of the airfoil curved element using the Simpson and Simpson 3 / 8 rules, based on this arc length, the Gaussian integration points on the surface can be found, that is, the calculation positions of the outer normal components on the surface are determined.

[0046] Based on the Simpson and Simpson 3 / 8 rules to calculate the curved edge arc length of the airfoil curved element, the Newton iteration method can be used to find the integration points. Taking two Gaussian integration points as an example, assuming the starting endpoint position of the curved edge is , the position of the integration point to be calculated is and . The target arc lengths corresponding to the positions of the two integration points are: (17) Set two parameters: (18) Where and are the arc lengths from the starting point to the current position respectively, and these two arc lengths can be calculated by using the Simpson and Simpson 3 / 8 rules. The error tolerance set in the embodiment of the present invention is , when and When the error tolerance is exceeded, perform Newton iteration: (19) Until the positions of two integration points that satisfy the error tolerance are found.

[0047] The following introduces the calculation method of the unit outer normal of the airfoil curved boundary.

[0048] Based on accurately obtaining the integration points on the curved boundary, the unit outer normal vector at the corresponding position can be directly obtained according to the reconstructed interpolation polynomial. According to Figure 2 the relative positions of the integration points shown, the tangential vectors at two integration points on the curved surface are respectively (20) The unit outer normals pointing outwards are respectively (21) The above three methods constitute the core of the high-order discrete method for straight-edge elements based on the direction of the curved surface of an object. From a process perspective, this method is relatively simple in implementation. Although it is necessary to use the Simpson and Simpson 3 / 8 rules with high integration degrees to determine the positions for calculating the outer normal vector on the curved surface of the object, it avoids the construction of high-order isoparametric elements and does not require separate storage of the Jacobian.

[0049] The flow chart of the high-order discrete method for straight-edge elements based on the direction of the airfoil curved surface is as Figure 4 shown.

[0050] 1. First, read the information of the high-order precision airfoil curved boundary; 2. According to the requirements of the high-order format, judge the integration degree of the airfoil curved element. If the integration degree is 3, turn to 3; if the integration degree is 4, turn to 4; 3. Use the Simpson rule to calculate the arc length and turn to 5; 4. Use the Simpson 3 / 8 rule to calculate the arc length and turn to 5; 5. Use the Newton iteration method to solve the Gaussian integration points on the airfoil curved boundary; 6. Calculate the normal of the Gaussian integration points on the airfoil curved boundary; 7. Substitute the normal of the Gaussian integration points into the formula of the straight boundary discrete method to calculate the numerical integration; 8. End.

[0051] Adopt the high-order curved boundary airfoil expression method to test the calculation results of the discrete method for curved elements based on the fifth-degree polynomial curve.

[0052] The following introduces the example of the inviscid flow around the NACA0012 airfoil.

[0053] Basic situation of the example: This example is the flow around the NACA0012 airfoil, and the incoming flow Mach number is taken , four sets of quadrilateral meshes from sparse to dense are adopted, with the number of elements being 8000, 9400, 12720, and 14300 in sequence. The four sets of meshes are named Coa, Med, Fin, and Vfin respectively Figure 5 First shows the distribution of the first two sets and the last set of meshes near the airfoil surface

[0054] Accuracy test and correctness test Figure 6 shows the distribution of the airfoil surface pressure coefficient obtained by different methods in the third-order and fourth-order accuracy formats. The third-order format - cubic isoparametric element and the third-order format - quintic isoparametric element are respectively the discrete methods of cubic and quintic isoparametric element curved elements in the third order. The fourth-order format - cubic isoparametric element and the fourth-order format - quintic isoparametric element are respectively the discrete methods of cubic and quintic isoparametric element curved elements of the embodiments of the present invention in the fourth order. The third-order format - simplified curved edge element and the fourth-order format - simplified curved edge element are respectively the high-order discrete methods of straight edge elements based on the airfoil curved surface direction in the third order and fourth order of the embodiments of the present invention

[0055] It can be seen that Figure 6 shows the negative pressure coefficient. The reason is that when analyzing the lift, it is usually desired to see the negative pressure contribution on the upper surface of the airfoil. When using , the negative pressure contribution on the upper surface is positive. The positive pressure on the lower surface is negative, which can more clearly show the distribution of the lift contribution

[0056] From the obtained results, under three attack angle conditions, the results obtained by different curved element discrete methods in the third-order and fourth-order formats are very close. As the attack angle continues to increase, the negative pressure provided by the upper surface of the airfoil and the positive pressure provided by the lower surface both gradually increase, providing a higher lift for the airfoil

[0057] Figure 7 shows the entropy error distribution of the wall elements on the sparsest and densest meshes at the attack angle of

[0058] It can be seen that the entropy error distributions of the wall elements obtained under different curved element discrete schemes are almost the same. As the format accuracy improves and the mesh is refined, the entropy error will further decrease. Table 1 further shows under the attack angle, the error statistics of different methods on all wall elements

[0059] Table 1 Calculation accuracy statistics of the L2 error between different methods on all wall elements and the last two sets of dense meshes

[0060] ​The correctness of the improved polynomial curved boundary representation combined with the straight-edge element discretization method based on the normal direction of the curved boundary is demonstrated by the inviscid flow around the NACA0012 airfoil under three different angles of attack conditions.

[0061] The following presents the example of viscous flow around the NACA0012 airfoil.

[0062] Basic situation of the example: This part is about the problem of viscous flow around the NACA0012 airfoil, with the given incoming flow Mach number being , and the Reynolds number being . As Figure 8 shown, three sets of elements are used in this part, which are successively refined near the wing boundary, and the number of grids is 14200, 15620, and 17040 respectively. We name them the sparsest, medium, and densest respectively, and the height of the first-layer grid is 5×10 -4 .

[0063] In the case of high Reynolds number viscous problems, since the height of the first-layer grid is small, before presenting the specific calculation results of different methods, we first examine the relative position relationship between the wall curve reconstructed based on the fifth-degree polynomial and the grid near the wall at some regions of the airfoil on the three sets of grids.

[0064] It can be seen that based on the improved fifth-degree polynomial curve, an accurate representation of the airfoil wall can be achieved, and in the case of a small height of the first-layer grid, the polynomial curve does not cross the first-layer grid.

[0065] In the post-processing of the numerical experiment, even though the high-order discretization method of straight-edge elements based on the direction of the curved airfoil surface can ensure the corresponding accuracy of the flow results, the calculation of the pressure and drag coefficients involves the actual integration point positions and the integration along the boundary surface. At this time, both the integration points and the numerical integration process are along the straight boundary, so there may be an obvious difference between the calculation results and the high-order discretization method of straight-edge elements based on the direction of the curved airfoil surface.

[0066] Therefore, the post-processing process of the high-order discretization method of straight-edge elements based on the direction of the curved airfoil surface is improved. Based on the process of finding the position of the normal vector outside the curved boundary according to the Simpson 3 / 8 rule, it is ensured that the calculation position of the pressure coefficient is located at C G2 , the integration of the drag coefficient is carried out along the curved boundary, and the arc length of the curved boundary calculated based on the Simpson 3 / 8 rule is . The calculation results obtained based on this method are named simplified curved-edge elements (improved).

[0067] Figure 9 shows the pressure coefficient C DVariation curves with the number of iterative steps, where the cubic isoparametric element and the quintic isoparametric element are the cubic and quintic optimized isoparametric element discretization methods of the embodiments of the present invention, the simplified curved-edge element is the high-order discretization method of the straight-edge element based on the airfoil curved surface direction in the embodiments of the present invention, the simplified curved-edge element (improved) is the result of improving the post-processing of the high-order discretization method of the straight-edge element based on the airfoil curved surface direction in the embodiments of the present invention, and the straight-edge element refers to the straight-edge element discretization method.

[0068] Figure 10 First, the pressure coefficient distributions of different methods on three sets of meshes are shown, where the cubic isoparametric element and the quintic isoparametric element are the cubic and quintic optimized isoparametric element discretization methods respectively, the simplified curved-edge element is the high-order discretization method of the straight-edge element based on the airfoil curved surface direction in the embodiments of the present invention, the simplified curved-edge element (improved) is the result of improving the post-processing of the high-order discretization method of the straight-edge element based on the airfoil curved surface direction in the embodiments of the present invention, and the straight-edge element refers to the straight-edge element discretization method.

[0069] Judging from the obtained results, when the calculation points of the pressure coefficient are selected on the curved boundary at this time, no obvious improvement is brought, and the obtained results are still close to the discretization method completely based on the straight-edge element, and there is an obvious deviation from the high-order discretization method of the straight-edge element based on the airfoil curved surface direction.

[0070] Different from the pressure coefficient, the integral of the drag coefficient obtained by the high-order discretization method of the straight-edge element based on the airfoil curved surface direction with the improved post-processing steps is significantly improved and is closer to the results of the optimized isoparametric element discretization method, especially on the denser meshes. We analyze that the possible reason for this phenomenon is that the pressure coefficient is only the result at each x position. Therefore, for the high Reynolds number airfoil flow, due to the higher aspect ratio of the wall mesh itself, the difference in shape and area between the straight-edge element and the curved-edge element is very small, and the result difference before and after improving the calculation position cannot be directly reflected. And the drag coefficient outputs the integral result along the entire airfoil surface, which will cause error accumulation.

[0071] Through the viscous NACA0012 airfoil flow problem, it is further illustrated that performing the integral process along the straight edge and the curved edge will cause a large deviation, and using the high-order discretization method of the straight-edge element based on the airfoil curved surface direction can obtain high precision and is more economical than the isoparametric element method in airfoil design. Embodiment

[0072] Embodiment 2 of the present invention provides a terminal device corresponding to Embodiment 1 above. The terminal device can be a processing device for a client, such as a mobile phone, a laptop computer, a tablet computer, a desktop computer, etc., to execute the method of the above embodiment.

[0073] The terminal device in this embodiment includes a memory, a processor, and a computer program stored on the memory; the processor executes the computer program on the memory to implement the steps of the method in Embodiment 1 above.

[0074] In some implementations, the memory can be a high-speed random access memory (RAM: Random Access Memory), and may also include non-volatile memory, such as at least one disk memory.

[0075] In other implementations, the processor can be various types of general-purpose processors such as a central processing unit (CPU) or a digital signal processor (DSP), which is not limited here. Embodiment

[0076] Embodiment 3 of the present invention provides a computer-readable storage medium corresponding to Embodiment 1 above, on which a computer program / instructions are stored. When the computer program / instructions are executed by a processor, the steps of the method in Embodiment 1 above are implemented.

[0077] A computer-readable storage medium can be a tangible device that holds and stores instructions used by an instruction execution device. A computer-readable storage medium can be, for example, but not limited to, an electrical storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any combination of the above.

[0078] Those skilled in the art should understand that the embodiments of the present application can be provided as a method, a system, or a computer program product. Therefore, the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk memories, CD-ROMs, optical memories, etc.) containing computer-usable program codes. The solutions in the embodiments of the present application can be implemented in various computer languages. For example, object-oriented programming languages such as Java and interpreted scripting languages such as JavaScript.

[0079] The present application is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each flow and / or block in the flowchart and / or block diagram, as well as the combination of flows and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing devices generate for implementation in the process Figure 1means for the functions specified in one or more processes and / or blocks Figure 1 and / or in one or more blocks.

[0080] These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus, so that a series of operational steps are performed on the computer or other programmable apparatus to produce a computer-implemented process, and thus the instructions executed on the computer or other programmable apparatus provide means for implementing the functions specified in one or more processes Figure 1 and / or in one or more blocks. Figure 1 and / or steps for the functions specified in one or more blocks.

[0081] Although the preferred embodiments of the present application have been described, additional changes and modifications can be made by those skilled in the art once they learn of the basic inventive concept. Therefore, the appended claims are intended to be construed to cover the preferred embodiments as well as all changes and modifications that fall within the scope of the present application.

[0082] Obviously, those skilled in the art can make various changes and modifications to the present application without departing from the spirit and scope of the present application. Thus, if these modifications and variations of the present application fall within the scope of the claims of the present application and their equivalent technologies, the present application is also intended to include these modifications and variations.

Claims

1. A wing boundary simulation method based on the normal direction of the airfoil curved surface, characterized in that, It includes the following steps: S1. Obtain the wing curved boundary information; S2. Judge the integration degree of the curved element according to the wing curved boundary information. If the integration degree is 3, use the Simpson rule to calculate the arc length L; if the integration degree is 4, use the Simpson 3 / 8 rule to calculate the arc length L; S3. Use the obtained arc length L and the Newton iteration method to calculate the Gaussian integration points of the wing curved boundary; S4. Calculate the normal direction of the Gaussian integration points of the wing curved boundary; S5. Substitute the normal direction of the Gaussian integration points into the formula of the straight boundary discretization method to calculate the numerical integration: ； Among them, represents the average value of the wing speed or temperature within the curved element, represents the volume of the wing element, represents the number of element faces, represents the number of area integration points, is the area integration coefficient, represents the convective and viscous fluxes at the area integration point k, represents the number of volume integration points, represents the volume integration coefficient, represents the heat source at the volume integration point q.

2. The wing boundary simulation method based on the normal of the airfoil curved surface according to claim 1, wherein The expression for the arc length L of the wing boundary curve calculated using the Simpson's rule in the interval is as follows: ； Among them, , n is the number of subintervals of , , are respectively the first-order derivatives of the airfoil curve function f(x) at , , ; , , respectively represent the abscissas at point a, the i-th integration point, and point b.

3. The wing boundary simulation method based on the normal of the airfoil curved surface according to claim 1, wherein The expression for the arc length L of the wing boundary curve calculated using the Simpson's rule in the interval is as follows: ； Among them, , n is the number of sub-intervals of 4. The wing boundary simulation method based on the normal of the airfoil curved surface according to claim 1, wherein In step S3, the specific implementation process of using the Newton iteration method to solve the Gaussian integration points of the airfoil curved boundary includes: 1) Set two parameters: ; , and are the positions of two integration points respectively, , are the target arc lengths corresponding to the positions of the two integration points, and are the arc lengths from the starting point to the positions of the two integration points respectively; 2) When and are greater than the error tolerance, update the positions of the two integration points using the following formula: and ; 3) Repeat the above step 2) until and is less than or equal to the error tolerance, and output the updated integral point position.

5. The wing boundary simulation method based on the normal of the airfoil curved surface according to claim 1, characterized in that In step S4, the normal direction calculation formula of the Gaussian integration points of the wing curved boundary is: ； Among them, and are the first derivatives of the wing boundary curve at two integration points, respectively.

6. A terminal device, comprising a memory, a processor, and a computer program stored on the memory; characterized in that, The processor executes the computer program to implement the steps of the method according to any one of claims 1 to 5 above.

7. A computer-readable storage medium having a computer program / instructions stored thereon; characterized in that, When the computer program / instructions are executed by the processor, the steps of the method according to any one of claims 1 to 5 above are implemented.

8. A computer program product, comprising a computer program / instructions; characterized in that, When the computer program / instructions are executed by the processor, the steps of the method according to any one of claims 1 to 5 are implemented.

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