A finite element method for calculating lift coefficient of airfoil based on potential flow theory

Abstract

The present invention discloses a finite element method for calculating the lift coefficient of an airfoil based on the potential flow theory, and belongs to the field of calculation and analysis of the aerodynamic characteristics of an aircraft airfoil. The present invention comprises: given geometric information of an airfoil; determining an airflow region according to the geometric information of the airfoil; creating a gap in the airflow region, wherein there is no airflow in the gap; using the airflow region excluding the gap region as a solution domain; using the superposition of uniform incoming flow and pure annular flow as an unperturbed flow field to determine the boundary conditions of the solution domain; given a series of circulation values, combining the solution domain and boundary conditions, using the finite element method to solve the potential flow control equation for each circulation value to obtain the corresponding velocity potential; using the Kutta condition to determine the true circulation value, and then calculating the lift coefficient of the airfoil accordingly. The present invention makes it possible to obtain the lift coefficient of an airfoil by solving the potential flow control equation in a finite element framework, and the present invention has high solution efficiency and simple operation.

Description

Technical Field

[0001] The invention belongs to the field of calculation and analysis of aerodynamic characteristics of aircraft airfoils, and specifically relates to a finite element method for calculating the lift coefficient of an airfoil based on potential flow theory. Background Art

[0002] The fundamental reason why an aircraft can overcome its own gravity and fly in the air is that the lift is large enough and the drag is small enough. The wing is the main component of the aircraft to obtain lift. For convenience, when studying the aerodynamic characteristics of the wing, the three-dimensional wing is generally simplified into a two-dimensional airfoil. The airfoil is the cross-section of the wing, which is usually streamlined and preferably produces greater lift in the airflow while bearing less drag. The airfoil is not only one of the determining factors of the wing shape, but also has a profound impact on the aerodynamic performance of the aircraft. Therefore, airfoil research occupies an important position in aviation technology.

[0003] As an efficient lift shape, the basic characteristic of the airfoil is that it can generate a large lift, that is, it has a large lift coefficient c. l . The lift coefficient of an airfoil can be obtained by experiment or predicted by numerical simulation. The most direct simulation method is computational fluid dynamics (CFD), which solves the Navier-Stokes equations numerically. CFD has high prediction accuracy, but because it solves a (highly) nonlinear set of equations, it is computationally intensive and usually requires high computer performance and long computing time. A simplified method is to make reasonable restrictions on the flow properties, such as the incompressible, irrotational, and inviscid flow assumed by potential flow theory. The main method for simulating the lift coefficient of an airfoil based on potential flow theory is the vortex panel method (VPM), which uses a series of vortices to generate streamlines that match the shape of the airfoil. VPM is a classic method in aerodynamics with a simple principle and easy to understand. However, it is very cumbersome in numerical implementation and has a certain degree of arbitrariness. For example, VPM has one more control equation than unknown quantity, so one control equation must be discarded when solving it, but there is no consensus on which one to discard.

[0004] VPM does not solve the Laplace equation, the basic governing equation of potential flow theory. If you want to obtain the lift coefficient by solving the Laplace equation, you must introduce circulation, otherwise you will get a lift-free flow. However, even if the influence of circulation is considered, because the flow field contains an impermeable object such as the airfoil, the solution domain is not a simply connected domain and a definite solution cannot be obtained. Therefore, it is necessary to take into account the influence of circulation in an appropriate manner and add additional definite solution conditions to obtain the lift coefficient of the airfoil by solving the governing equation of potential flow. Summary of the invention

[0005] In view of the deficiencies in the prior art, an object of the present invention is to provide a finite element method for calculating the lift coefficient of an airfoil based on potential flow theory.

[0006] The object of the present invention is achieved through the following technical solution: A finite element method for calculating the lift coefficient of an airfoil based on potential flow theory comprises the following steps:

[0007] Given the geometric information of the airfoil;

[0008] Determine an airflow region according to geometric information of the airfoil, wherein a boundary of the airflow region is far enough from the airfoil so that the flow at the boundary is not affected by the airfoil;

[0009] Creating a narrow gap in the airflow region extending from the trailing edge of the airfoil to the boundary of the airflow region to simulate a region where there is a jump in velocity potential, wherein there is no airflow in the gap;

[0010] The airflow area excluding the gap area is taken as the solution domain;

[0011] The superposition of the uniform incoming flow and the pure annular flow is used as the unperturbed flow field to determine the boundary conditions of the solution domain;

[0012] Given a series of circulation values, combined with the solution domain and boundary conditions, the potential flow control equation is solved by the finite element method for each circulation value to obtain the velocity potential;

[0013] The airfoil trailing edge velocity is obtained according to the velocity potential, the circulation corresponding to the airfoil trailing edge velocity satisfying the Kutta condition is taken as the true circulation, and the lift coefficient of the airfoil is calculated according to the true circulation.

[0014] Furthermore, the step of creating a narrow gap extending from the trailing edge of the airfoil to the boundary of the airflow region in the airflow region to simulate a region where the velocity potential jumps is specifically:

[0015] The circulation Γ is the path integral of the airflow velocity V along a closed curve, as shown in the following formula:

[0016] Γ=∮Vds (1)

[0017] The air velocity V is the gradient of the velocity potential Φ, as shown in the following formula:

[0018] V=▽Φ (2)

[0019] According to the gradient theorem, we get:

[0020] Γ=∮Vds =∮▽Φds=Φ(x2)-Φ(x1) (3)

[0021] Where x1 and x2 represent the starting point and end point of the path integral respectively; since the integral is performed along a closed curve, the starting point and end point are the same point, so x1 = x2 = x * ; If Γ≠0, then the velocity potential Φ at point x * There is a jump at , and the jump value is the cycle Γ;

[0022] The region where the velocity potential jumps is the region where the unknown quantity Φ to be solved in the finite element is discontinuous; to simulate the discontinuous region, a narrow gap extending from the trailing edge of the airfoil to the boundary of the airflow region is created in the fluid domain, and there is no airflow in the gap.

[0023] Furthermore, the airflow area is set as a circular area.

[0024] Furthermore, the superposition of the uniform incoming flow and the pure annular flow is used as the unperturbed flow field to determine the boundary conditions of the solution domain as follows:

[0025] The superposition of uniform incoming flow and pure annular flow is taken as the undisturbed initial flow field;

[0026] The velocity potential Φ1 of the pure circulation is expressed in the polar coordinate system (r, θ) as:

[0027] Φ1=[Γ / (2π)]θ (4)

[0028] Transforming equation (4) into the rectangular coordinate system (x, y), the velocity potential of the pure circulation is:

[0029] Φ1=[Γ / (2π)]arccos(x / r),y≥0 (5-1)

[0030] Φ1=[Γ / (2π)](-arccos(x / r)+2π), y<0 (5-2)

[0031] To consider the airfoil angle of attack α, let the velocity of the uniform incoming flow be V ∞ , the velocity potential Φ2 of the uniform incoming flow in the rectangular coordinate system (x, y) is:

[0032] Φ2=V ∞ cosα•x+V ∞ sinα•y (6)

[0033] The uniform incoming flow and pure circulation are superimposed as the undisturbed flow field, and the velocity potential Φ of the undisturbed flow field is:

[0034] Φ=V ∞ cosα•x+V ∞ sinα·y+[Γ / (2π)]arccos(x / r),y≥0 (7-1)

[0035] Φ=V ∞ cosα•x+V ∞ sinα·y+[Γ / (2π)](-arccos(x / r)+2π),y<0 (7-2)

[0036] According to equations (7-1) and (7-2), the stacking velocity potential Φ is continuous in the circular region except for the ray x≥0, y=0, and there is a jump on the ray; let the stacking velocity potential on both sides of the ray be Φ + , Φ - , we get |Φ + -Φ - |=Γ;

[0037] The center of the circular area is taken as the origin of the polar coordinate system and the rectangular coordinate system, and the radius of the circular area is denoted as R; the airflow at the boundary of the circular area is not disturbed, and the velocity potential on the boundary is denoted as , then the upper and lower boundaries of the circular area are:

[0038] , the upper boundary of the circular area (8-1)

[0039] , the lower boundary of the circular area (8-2)

[0040] The boundary conditions of the gap are as follows:

[0041] , the upper boundary of the gap (9-1)

[0042] , the lower boundary of the gap (9-2)

[0043] The airfoil boundary satisfies the no-penetration condition; assuming the outer normal vector of the airfoil boundary is n, the boundary condition is:

[0044] , airfoil boundary (10).

[0045] Furthermore, the use of the finite element method to solve the potential flow control equation for each circulation value includes:

[0046] The governing equation of potential flow theory is a Laplace equation with velocity potential Φ as the basic unknown, as shown below:

[0047] ▽ 2 Φ = 0 (11).

[0048] Further, obtaining the airfoil trailing edge velocity according to the velocity potential comprises:

[0049] The airflow corresponding to the velocity potential obtained by finite element solution must satisfy the Kutta condition. For an airfoil with a finite angle at the trailing edge, the airflow velocity at the trailing edge is 0.

[0050] According to the Kutta condition, the real circulation value Γ is determined * , including the following steps:

[0051] According to Γ0=V ∞ παc estimates the circulation Γ0 corresponding to the current airfoil, where c represents the chord length of the airfoil, V ∞ Indicates the velocity of uniform incoming flow;

[0052] According to the estimated circulation value Γ0, a series of circulation values ​​{Γ i}; where {Γ i The minimum value in} is not greater than 0.5Γ0, the maximum value is not less than 2Γ0, and the values ​​are uniformly taken in the interval, and the difference between adjacent values ​​is not greater than 0.2Γ0, so as to improve the search range and search accuracy;

[0053] For {Γ i Each cycle value Γ in i , using finite element method to solve the corresponding velocity potential Φ i , and then use the following formula to calculate the velocity vector V at the trailing edge of the airfoil i TE :

[0054] V i TE = ▽Φ i (x TE ,y TE ) (15)

[0055] Among them, (x TE ,y TE ) represents the coordinates of the trailing edge point;

[0056] Compare all vectors V i TE length, choose the smallest V i TE , the minimum V i TE The corresponding circulation is the real circulation Γ * .

[0057] Further, the lift coefficient of the airfoil is calculated according to the true annulus as follows:

[0058] Get the real circulation Γ * After that, the lift coefficient c of the airfoil is obtained by the following formula l :

[0059] c l =2Γ* / (V ∞ c) (16)

[0060] Where c is the chord length of the airfoil, V ∞ Indicates the velocity of the uniform incoming flow.

[0061] The present invention also provides an electronic device, comprising a memory and a processor, wherein the memory is coupled to the processor; wherein the memory is used to store program data, and the processor is used to execute the program data to implement the finite element method for calculating the lift coefficient of an airfoil based on potential flow theory.

[0062] The present invention also provides a computer-readable storage medium on which a computer program is stored. When the program is executed by a processor, the finite element method for calculating the lift coefficient of an airfoil based on potential flow theory is implemented.

[0063] The present invention also provides a finite element device for calculating the lift coefficient of an airfoil based on potential flow theory, comprising one or more processors for implementing the finite element method for calculating the lift coefficient of an airfoil based on potential flow theory.

[0064] The beneficial effects of the present invention are as follows: the present invention proposes a finite element method for calculating the lift coefficient of an airfoil based on the potential flow theory, the core of which is to propose an original method for taking into account the influence of circulation in the potential flow. The present invention combines the Kutta conditions to make it possible to obtain the lift coefficient of an airfoil by solving the potential flow control equation in a finite element framework. The technical solution adopted by the present invention, i.e., using the finite element method to solve the potential flow control equation, has two major advantages: first, the potential flow control equation is linear, so the solution efficiency is high; second, the finite element method is a mature method for solving partial differential equations, which can be implemented with the help of commercial software, so the operation is simple. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 It is a flow chart for the implementation of the present invention;

[0066] Figure 2 It is a schematic diagram of the present invention showing the airfoil attack angle α by adjusting the airflow direction;

[0067] Figure 3 It is a schematic diagram of the jump of velocity potential in the flow field of the present invention;

[0068] Figure 4 A schematic diagram of boundary conditions required when the finite element method of the present invention is used to solve the potential flow control equation to obtain the lift coefficient of the airfoil;

[0069] Figure 5 This is the flow field velocity diagram corresponding to the real circulation of the present invention. DETAILED DESCRIPTION

[0070] Exemplary embodiments will be described in detail herein, examples of which are shown in the accompanying drawings. When the following description refers to the drawings, the same numbers in different drawings represent the same or similar elements unless otherwise indicated. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present invention. Instead, they are merely examples of devices and methods consistent with some aspects of the present invention as detailed in the appended claims.

[0071] The terms used in the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The singular forms "a", "the" and "the" used in the present invention and the appended claims are also intended to include plural forms unless the context clearly indicates other meanings. It should also be understood that the term "and / or" used herein refers to and includes any or all possible combinations of one or more associated listed items.

[0072] It should be understood that although the terms first, second, third, etc. may be used in the present invention to describe various information, these information should not be limited to these terms. These terms are only used to distinguish the same type of information from each other. For example, without departing from the scope of the present invention, the first information may also be referred to as the second information, and similarly, the second information may also be referred to as the first information. Depending on the context, the word "if" as used herein may be interpreted as "at the time of" or "when" or "in response to determining".

[0073] The present invention is described in detail below in conjunction with the accompanying drawings. In the absence of conflict, the features of the following embodiments and implementations can be combined with each other.

[0074] like Figure 1 As shown, an embodiment of the present invention provides a finite element method for calculating the lift coefficient of an airfoil based on potential flow theory, comprising the following steps:

[0075] Step 1: Given the geometric information of the airfoil

[0076] The airfoil is an impenetrable object, so there is no airflow in the space occupied by the airfoil. Given the geometric information of the airfoil, such as the coordinates of a series of points sufficient to determine the shape of the airfoil, it is convenient to exclude the airfoil area from the airflow area later. The geometric information of the classic airfoil can be obtained through professional websites such as http: / / airfoiltools.com / airfoil / index. The airfoil angle of attack α will be reflected in the subsequent steps by adjusting the airflow direction, so the airfoil does not need to be rotated, such as Figure 2 shown.

[0077] Step 2: Determine the airflow area

[0078] An airflow region is defined around the airfoil region. The airflow region should be large enough so that the airflow at the boundary is not affected by the airfoil. Considering the computational cost of the subsequent solution, the airflow region should not be too large. For the convenience of applying boundary conditions later, the present invention defines the airflow region as a circle. For example, a circle with the midpoint of the airfoil chord as the center and a radius of three times the airfoil chord length is a suitable airflow region.

[0079] Step 3: Create gaps in the airflow area

[0080] In order to simulate lift flow under the potential flow assumption, the influence of circulation must be considered. Circulation is the path integral of the fluid velocity along a closed curve, usually expressed as Γ, that is:

[0081] Γ=∮Vds (1)

[0082] In potential flow theory, the air flow velocity V is the gradient of the velocity potential Φ, that is:

[0083] V=▽Φ (2)

[0084] According to the gradient theorem, we have:

[0085] Γ=∮Vds =∮▽Φds=Φ(x2)-Φ(x1) (3)

[0086] In the above formula, x1 and x2 represent the starting point and end point of the path integral. Since the integral is performed along a closed curve, the starting point and the end point are actually the same point. This means that if Γ≠0, then the velocity potential Φ has a jump at this point, and the jump value is exactly the circulation Γ, as Figure 3 In actual viscous airflow, the starting vortex is formed and shed from the trailing edge of the airfoil, so the trailing edge of the airfoil and its downstream can be regarded as a region where the velocity potential jumps.

[0087] The area where the velocity potential jumps is the area where the unknown quantity Φ to be solved in the finite element is discontinuous. In order to simulate this discontinuity, the present invention creates a narrow gap downstream of the trailing edge of the airfoil, and there is no airflow in the gap. In order to reduce the simulation error, the gap should be as narrow as possible, and the gap width can be taken as 1 or 2 finite element grid sizes.

[0088] Step 4: Determine the solution domain

[0089] The circular area delineated in step 2 (excluding the airfoil area) excluding the narrow gap in step 3 is the solution domain of the subsequent finite element problem.

[0090] Step 5: Set boundary conditions

[0091] Generally, only the lift coefficient of the airfoil in the uniform incoming flow is considered, so in the finite element simulation, the initial flow field should be the uniform incoming flow. In the present invention, since the influence of circulation needs to be introduced, the superposition of the uniform incoming flow and the pure annular flow is taken as the undisturbed initial flow field.

[0092] The velocity potential of a pure circulation, i.e. a vortex, is usually expressed in polar coordinates (r, θ) as:

[0093] Φ1=[Γ / (2π)]θ (4)

[0094] Converted to the rectangular coordinate system (x, y), the velocity potential of pure circulation is:

[0095] Φ1=[Γ / (2π)]arccos(x / r),y≥0 (5-1)

[0096] Φ1=[Γ / (2π)](-arccos(x / r)+2π), y<0 (5-2)

[0097] Considering the airfoil angle of attack α, let the velocity of the uniform incoming flow be V ∞ , then the velocity potential of the uniform incoming flow can be written in the rectangular coordinate system (x, y) as:

[0098] Φ2=V ∞ cosα•x+V ∞ sinα•y (6)

[0099] The uniform incoming flow and pure annular flow are superimposed as the undisturbed flow field, and the velocity potential Φ of the undisturbed flow field is:

[0100] Φ=V ∞ cosα•x+V ∞ sinα·y+[Γ / (2π)]arccos(x / r),y≥0 (7-1)

[0101] Φ=V ∞ cosα•x+V ∞ sinα·y+[Γ / (2π)](-arccos(x / r)+2π),y<0 (7-2)

[0102] From the simple calculation of equations (7-1) and (7-2), we can know that the superposition velocity potential Φ is continuous everywhere in the circular region except for the ray x≥0, y=0, and there is a jump on this ray. Let the superposition velocity potential on both sides of the ray be Φ + , Φ - , obviously |Φ + -Φ - |=Γ. This shows that by superimposing the uniform incoming flow and the pure annular flow, an undisturbed initial flow field that satisfies the lift flow condition (see step 3 for details) can be created. Next, the boundary conditions of the finite element problem are given, such as Figure 4 shown.

[0103] The center of the circular area in step 2 is taken as the origin of the polar coordinate system and the rectangular coordinate system, and the radius of the circular area is denoted as R. Since the boundary of the circular area is far enough from the airfoil, the airflow at the boundary of the circular area is not disturbed, and the velocity potential on the boundary is denoted as , then the velocity potential of the upper and lower boundaries of the circular area can be written as:

[0104] , the upper boundary of the circular area (8-1)

[0105] , the lower boundary of the circular area (8-2)

[0106] In addition to the boundary conditions of the circular area, the boundary conditions on the gap and the boundary conditions on the airfoil in step 3 must also be given. For the gap, the boundary conditions are as follows:

[0107] , the upper boundary of the gap (9-1)

[0108] , the lower boundary of the gap (9-2)

[0109] The airfoil boundary should satisfy the no-penetration condition. Let the outer normal vector of the airfoil boundary be n, then the boundary condition can be written as:

[0110] , airfoil boundary (10)

[0111] Step 6: Given a series of circulation quantities, use the finite element method to solve the potential flow governing equations

[0112] The governing equation of potential flow theory is a Laplace equation with velocity potential Φ as the basic unknown:

[0113] ▽ 2 Φ = 0 (11)

[0114] Combine the boundary conditions (8-1)(8-2)(9-1)(9-2)(10) and the uniform flow velocity V ∞ By assigning specific values ​​to the airfoil angle of attack α and circulation Γ, the potential flow control equation (11) can be solved by the finite element method to obtain the corresponding velocity potential Φ.

[0115] In this step, given different annularity Γ, different flow fields passing through the same airfoil can be obtained in the same uniform incoming flow. However, the real airflow passing through the airfoil must satisfy the Kutta condition, that is, the airflow on the upper and lower surfaces of the airfoil should smoothly leave the trailing edge of the airfoil. This shows that for an airfoil with a finite angle at the trailing edge (the angle is not 0), the airflow velocity at the trailing edge should be 0, that is, the trailing edge of the airfoil is a stagnation point.

[0116] According to the above Kutta conditions, the true circulation value Γ can be determined * The specific steps are as follows:

[0117] ①According to the lift coefficient formula (12) in the thin wing theory, the Kutta-Zhukovsky theorem (13) and the two-dimensional airfoil lift coefficient calculation formula (14), the circulation corresponding to the current airfoil is roughly estimated to be Γ0=V ∞ παc, where c represents the chord length of the airfoil.

[0118] c l =2πα (12)

[0119] L = ρ ∞ V ∞ Γ (13)

[0120] c l =2L / (ρ ∞ V ∞ 2 c) (14)

[0121] ② According to the estimated circulation value Γ0, a series of circulation values ​​{Γ i To ensure that the search range is large enough and the search accuracy is high enough, it is recommended to i The minimum value in} is no more than 0.5Γ0, and the maximum value is no less than 2Γ0, and the values ​​are taken evenly within this range, and the difference between adjacent values ​​is no more than 0.2Γ0.

[0122] ③For {Γ i Each cycle value Γ in i , using finite element method to solve the corresponding velocity potential Φ i , and then use the following formula to calculate the velocity vector V at the trailing edge of the airfoil i TE :

[0123] V i TE = ▽Φ i (x TE ,y TE ) (15)

[0124] Where (x TE ,y TE ) represents the coordinates of the trailing edge point.

[0125] ④Compare all vectors V i TE length, find the V with the shortest length i TE , and its corresponding circulation is the real circulation Γ * .

[0126] In the present invention, due to the introduction of the gap at the trailing edge of the airfoil, there is no unique trailing edge point. Therefore, in actual execution, the velocity vector V of the two points where the gap intersects with the airfoil should be calculated in the above step ④. i1 TE and V i2 TE , and finally find the V with the smallest total length i1 TE and V i2 TE The corresponding circulation value.

[0127] Step 7: Calculate the lift coefficient

[0128] Get the real circulation Γ * After that, the lift coefficient of the airfoil can be obtained by the following formula:

[0129] c l =2Γ * / (V ∞ c) (16)

[0130] In addition, according to the real circulation Γ * The corresponding velocity potential Φ * The air flow velocity field can be obtained, such as Figure 5 shown.

[0131] V * =▽Φ * (17)

[0132] According to the air flow velocity field V * , other useful information about the airfoil can be further obtained, such as the pressure distribution on the airfoil surface.

[0133] The present invention introduces circulation by cleverly setting boundary conditions, and uses the Kutta conditions as the solution conditions, and successfully predicts the lift coefficient of the airfoil using potential flow theory in a finite element framework. Compared with CFD, this method does not solve the highly nonlinear Navier-Stokes equations, but solves the linear Laplace equations, so the calculation efficiency is greatly improved. Compared with VPM, this method uses a standard finite element program for solution, and the steps are clear and concise, so it is easier to master. In summary, compared with the existing technology, the present invention has stronger practicality.

[0134] An embodiment of the present invention also provides an electronic device, including a memory and a processor, wherein the memory is coupled to the processor; wherein the memory is used to store program data, and the processor is used to execute the program data to implement the finite element method for calculating the lift coefficient of an airfoil based on potential flow theory.

[0135] An embodiment of the present invention further provides a computer-readable storage medium having a computer program stored thereon, and when the program is executed by a processor, the finite element method for calculating the lift coefficient of an airfoil based on potential flow theory is implemented.

[0136] The computer-readable storage medium may be an internal storage unit of any device with data processing capability described in any of the aforementioned embodiments, such as a hard disk or a memory. The computer-readable storage medium may also be any device with data processing capability, such as a plug-in hard disk, a smart media card (SMC), an SD card, a flash card, etc. equipped on the device. Furthermore, the computer-readable storage medium may also include both an internal storage unit of any device with data processing capability and an external storage device. The computer-readable storage medium is used to store the computer program and other programs and data required by any device with data processing capability, and may also be used to temporarily store data that has been output or is to be output.

[0137] The present invention also provides a finite element device for calculating the lift coefficient of an airfoil based on potential flow theory, comprising one or more processors for implementing the finite element method for calculating the lift coefficient of an airfoil based on potential flow theory.

[0138] The implementation process of the functions and effects of each unit in the above-mentioned device is specifically described in the implementation process of the corresponding steps in the above-mentioned method, and will not be repeated here.

[0139] For the device embodiment, since it basically corresponds to the method embodiment, the relevant parts can refer to the partial description of the method embodiment. The device embodiment described above is only schematic, wherein the units described as separate components may or may not be physically separated, and the components displayed as units may or may not be physical units, that is, they may be located in one place, or they may be distributed on multiple network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the scheme of the present invention. Ordinary technicians in this field can understand and implement it without paying creative work.

[0140] Those skilled in the art will readily appreciate other embodiments of the present application after considering the description and practicing the contents disclosed herein. The present application is intended to cover any variations, uses or adaptations of the present application, which follow the general principles of the present application and include common knowledge or customary technical means in the art that are not disclosed in the present application. The description and examples are intended to be exemplary only.

[0141] It should be understood that the present application is not limited to the exact construction that has been described above and shown in the drawings, and that various modifications and changes may be made without departing from the scope thereof.

Claims

1. A finite element method for calculating the lift coefficient of an airfoil based on potential flow theory, characterized in that: The following steps are involved: Given the geometric information of the airfoil; Determine an airflow region according to geometric information of the airfoil, wherein a boundary of the airflow region is far enough from the airfoil so that the flow at the boundary is not affected by the airfoil; Creating a narrow gap in the airflow region extending from the trailing edge of the airfoil to the boundary of the airflow region to simulate a region where there is a jump in velocity potential, wherein there is no airflow in the gap; The airflow area excluding the gap area is taken as the solution domain; The superposition of the uniform incoming flow and the pure annular flow is used as the unperturbed flow field to determine the boundary conditions of the solution domain; Given a series of circulation values, combined with the solution domain and boundary conditions, the potential flow control equation is solved by the finite element method for each circulation value to obtain the velocity potential; The airfoil trailing edge velocity is obtained according to the velocity potential, the circulation corresponding to the airfoil trailing edge velocity satisfying the Kutta condition is taken as the true circulation, and the lift coefficient of the airfoil is calculated according to the true circulation; The method of creating a narrow gap extending from the trailing edge of the airfoil to the boundary of the airflow region in the airflow region to simulate an area where the velocity potential jumps is specifically: The circulation Γ is the path integral of the airflow velocity V along a closed curve, as shown in the following formula: Γ=∮Vds (1) Among them, ∮ is the integral symbol, ds represents the path differential element; The airflow velocity V is the gradient of the velocity potential Φ. The symbol ▽ is used to represent the gradient. The airflow velocity V is shown in the following formula: V=▽Φ (2) According to the gradient theorem, we get: Γ=∮Vds=∮▽Φds=Φ(x2)-Φ(x1) (3) Among them, x1 and x2 represent the starting point and end point of the path integral respectively; since the integral is performed along a closed curve, the starting point and the end point are the same point, and we can obtain x1 = x2 = x * ; If Γ≠0, then the velocity potential Φ at point x * There is a jump at , and the jump value is the cycle Γ; The region where the velocity potential jumps is the region where the unknown quantity Φ to be solved in the finite element is discontinuous; to simulate the discontinuous region, a narrow gap extending from the trailing edge of the airfoil to the boundary of the airflow region is created in the fluid domain, and there is no airflow in the gap.

2. The finite element method for calculating the lift coefficient of an airfoil based on potential flow theory according to claim 1, characterized in that: The airflow area is set as a circular area.

3. The finite element method for calculating the lift coefficient of an airfoil based on potential flow theory according to claim 2, characterized in that: The superposition of the uniform incoming flow and the pure annular flow is used as the unperturbed flow field to determine the boundary conditions of the solution domain as follows: The superposition of uniform incoming flow and pure annular flow is taken as the undisturbed initial flow field; The velocity potential Φ1 of the pure circulation is expressed in the polar coordinate system (r, θ) as: Φ1=[Γ / (2π)]θ (4) Transforming equation (4) into the rectangular coordinate system (x, y), the velocity potential of the pure circulation is: Φ1=[Γ / (2π)]arccos(x / r),y≥0 (5-1) Φ1=[Γ / (2π)](-arccos(x / r)+2π),y<0 (5-2) To consider the airfoil angle of attack α, let the velocity of the uniform incoming flow be V ∞ , the velocity potential Φ2 of the uniform incoming flow in the rectangular coordinate system (x, y) is: Φ2=V ∞ cosα·x+V ∞ sinα·y (6) The uniform incoming flow and pure annular flow are superimposed as the undisturbed flow field, and the velocity potential Φ of the undisturbed flow field is: Φ=V ∞ cosα·x+V ∞ sinα·y+[Γ / (2π)]arccos(x / r),y≥0 (7-1) Φ=V ∞ cosα·x+V ∞ sinα·y+[Γ / (2π)](-arccos(x / r)+2π),y<0 (7-2) According to equations (7-1) and (7-2), the stacking velocity potential Φ is continuous in the circular region except for the ray x≥0, y=0, and there is a jump on the ray; let the stacking velocity potentials on both sides of the ray be Φ + , Φ - , we get |Φ + -Φ - | = Γ; The center of the circular area is taken as the origin of the polar coordinate system and the rectangular coordinate system, and the radius of the circular area is denoted as R; the airflow at the boundary of the circular area is not disturbed, and the velocity potential on the boundary is denoted as Then the upper and lower boundaries of the circular area are: The boundary conditions of the gap are as follows: The airfoil boundary satisfies the no-penetration condition; assuming the outer normal vector of the airfoil boundary is n, the boundary condition is:

4. The finite element method for calculating the lift coefficient of an airfoil based on potential flow theory according to claim 1, characterized in that: The method of solving the potential flow control equation for each circulation value by using the finite element method includes: The governing equation of potential flow theory is a Laplace equation with velocity potential Φ as the basic unknown, as shown below: Among them, the symbol Represents gradient finding.

5. The finite element method for calculating the lift coefficient of an airfoil based on potential flow theory according to claim 4, characterized in that: The obtaining of the airfoil trailing edge velocity according to the velocity potential comprises: The airflow corresponding to the velocity potential obtained by finite element solution must satisfy the Kutta condition. For an airfoil with a finite angle at the trailing edge, the airflow velocity at the trailing edge is 0. According to the Kutta condition, the real circulation value Γ is determined * , including the following steps: According to Γ0=V ∞ παc estimates the circulation Γ0 corresponding to the current airfoil, where c represents the chord length of the airfoil, V ∞ Indicates the velocity of uniform incoming flow; According to the estimated circulation value Γ0, a series of circulation values ​​{Γ i }; where {Γ i The minimum value in} is not greater than 0.5Γ0, the maximum value is not less than 2Γ0, and the values ​​are uniformly taken in the interval, and the difference between adjacent values ​​is not greater than 0.2Γ0, so as to improve the search range and search accuracy; For {Γ i Each cycle value Γ in i , using finite element method to solve the corresponding velocity potential Φ i , and then use the following formula to calculate the velocity vector V at the trailing edge of the airfoil i TE : Among them, (x TE ,y TE ) represents the coordinates of the trailing edge point; Compare all vectors V i TE length, choose the smallest V i TE , the minimum V i TE The corresponding circulation is the real circulation Γ * .

6. The finite element method for calculating the lift coefficient of an airfoil based on potential flow theory according to claim 1, characterized in that: The lift coefficient of the airfoil is calculated according to the true annulus as follows: Get the real circulation Γ * After that, the lift coefficient c of the airfoil is obtained by the following formula l : c l =2Γ * / (V ∞ c) (16) Where c is the chord length of the airfoil, V ∞ Indicates the velocity of the uniform incoming flow.

7. An electronic device comprising a memory and a processor, characterized in that: The memory is coupled to the processor; wherein the memory is used to store program data, and the processor is used to execute the program data to implement a finite element method for calculating the lift coefficient of an airfoil based on potential flow theory as described in any one of claims 1-6.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, a finite element method for calculating the lift coefficient of an airfoil based on potential flow theory as described in any one of claims 1 to 6 is implemented.

9. A finite element device for calculating the lift coefficient of an airfoil based on potential flow theory, characterized in that: It comprises one or more processors for implementing a finite element method for calculating the lift coefficient of an airfoil based on potential flow theory as described in any one of claims 1 to 6.

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