Wave rider rapid analysis design method based on bending shock wave theory

Abstract

The invention discloses a wave rider rapid analysis design method based on a bending shock wave theory, and relates to near space hypersonic flight vehicle design. Two of a wave rider outlet capture molded line, a wave rider horizontal projection molded line and a shock wave outlet molded line are randomly specified according to design requirements, a geometrical relationship between design curves is established, and an analytical expression of a wave rider three-dimensional front edge molded line and a multiplied three-dimensional shock wave is solved. Taylor series expression of streamlines in an osculating plane is rapidly obtained by using a flow field analysis method based on a bending shock wave theory, and waverider parameterization expression considering incoming flow conditions and shock wave shapes is realized through coordinate transformation. According to the plot ratio requirement, the upper surface of the wave rider is constructed to be a plane or a convex surface, and rapid analysis design of the wave rider based on the bending shock wave theory is achieved. The design efficiency is effectively improved while the design precision of the wave rider is guaranteed, a higher lift-drag ratio and better wide-area performance can be obtained by selecting a proper design curve, and rapid optimization design of the wave rider in engineering is facilitated.

Description

Technical Field

[0001] The present invention relates to the technical field of near-space hypersonic aircraft design, and in particular to a rapid analytical design method of a waverider based on bending shock wave theory. Background Art

[0002] Near-space vehicles have great military and civilian value due to their advantages such as large airspace, ultra-high speed, long distance and high precision. With the increase of flight Mach number, the shock wave separation of hypersonic vehicles with conventional aerodynamic layout becomes more and more obvious, and a "lift-to-drag ratio barrier" appears. In order to achieve the aerodynamic requirements of high lift-to-drag ratio under hypersonic flight conditions, a waverider configuration with a leading edge appendage shock wave has become an ideal candidate for the aerodynamic layout design of hypersonic vehicles. The concept of waverider was first proposed by Nonweiler in 1959. He designed a wedge-guided waverider based on a two-dimensional wedge flow field. Subsequently, Jones et al. designed a cone-guided waverider based on axisymmetric conical flow. In addition, more general three-dimensional flow fields are also used for waverider design. However, the shock wave shape in the above method is not controllable and it is difficult to meet the increasingly complex flight-engine integrated design requirements. Therefore, a type of tangent theory for spanwise design of waveriders has gradually been developed. In 1990, Sobieczky et al. first proposed the tangential cone design theory, using a series of conical flow fields in the tangential plane to approximate three-dimensional supersonic flow. Subsequently, Sobieczky et al. proposed the tangential axisymmetric theory, in which the basic flow field in the tangential plane is obtained by scaling the same axisymmetric basic flow field. On this basis, Rodi further developed the tangential flow field theory, which allows different axisymmetric basic flow fields to be selected in each tangential plane according to design needs. Since then, domestic and foreign scholars have conducted extensive research on waverider design based on the tangential theory.

[0003] The design of waverider includes three basic curves: the horizontal projection profile of the waverider, the capture profile of the waverider outlet and the shock wave outlet profile. The existing waverider design methods usually select two of them as input and numerically solve the third curve through discrete processing. At present, some scholars have studied the geometric relationship and parameterization of the design curves, but the proposed methods are only applicable to the tangent cone theory, and the analytical expression of the design curve cannot be obtained. The design space and accuracy of the waverider are obviously limited. After determining the design curve, the waverider profile is obtained by streamline tracing in the basic flow field, which leads to the second problem of waverider design, that is, the calculation method of flow parameters in the basic flow field. The characteristic line method, as a traditional supersonic flow field calculation method, needs to be applied to the entire basic flow field before streamline tracing. It is not only complex to program, but also has poor stability, which will bring a large computational load. On the other hand, as a point-by-point iterative grid numerical method, the characteristic line method cannot obtain an analytical solution of the flow field parameters. As a numerical interpolation technology, the streamline tracing technology also makes it impossible to parametrically express the waverider profile, which is not conducive to the optimization design of the waverider. Therefore, a rapid analytical design method for waverider based on bending shock wave theory is proposed, which can effectively improve the design efficiency while ensuring the design accuracy. Summary of the invention

[0004] The purpose of the present invention is to provide a rapid analytical design method for waveriders based on the bending shock wave theory in view of the above-mentioned status of the prior art, and to specify certain key curves of the waverider, such as the exit capture profile, the horizontal projection profile and the shock wave exit profile, as required, and solve the three-dimensional leading edge profile of the waverider and the three-dimensional bending shock wave ridden by it; by selecting appropriate design curves, a higher lift-to-drag ratio and better wide-range performance are obtained, thereby broadening the design space of the waverider and effectively improving the design efficiency while ensuring the design accuracy of the waverider.

[0005] In order to achieve the above-mentioned object of the invention, the present invention adopts the following technical scheme.

[0006] The rapid analytical design method of waverider based on bending shock wave theory includes the following steps:

[0007] 1) Specify the design curve: According to the design requirements, specify any two of the waverider exit capture profile, waverider horizontal projection profile and shock wave exit profile, and specify the curved shock wave profile in the axisymmetric basic flow field;

[0008] 2) Establishing geometric relationships and solving analytical expressions: Establishing geometric relationships between the waverider exit capture profile, the waverider horizontal projection profile, and the shock wave exit profile, solving the analytical expressions of the waverider 3D leading edge profile and the 3D curved shock wave it rides on;

[0009] 3) Obtain streamline expression: Given the incoming flow conditions, according to the shock wave angle and shock wave curvature of the three-dimensional curved shock wave at the three-dimensional leading edge profile of the waverider, the Taylor series expression of the streamlines in each tangent surface is quickly obtained using the flow field analysis method based on the curved shock wave theory, and the parametric expression of the lower surface of the waverider considering the incoming flow conditions and the shock wave shape is realized through coordinate transformation;

[0010] 4) Constructing the waverider shape: According to the volume ratio requirements, the upper surface of the waverider is constructed to be a plane or a convex surface, which together with the lower surface of the waverider generated by step 3) and the trailing edge surface of the waverider constitute the waverider shape, thereby realizing a rapid analytical design of the waverider based on the bending shock wave theory.

[0011] In step 2), the geometric relationship between the waverider exit capture profile, the waverider horizontal projection profile and the shock wave exit profile is established, and the analytical expression of the waverider three-dimensional leading edge profile and the three-dimensional curved shock wave is solved. The specific process is:

[0012] 2.1) Define the waverider exit capture profile (FCT), waverider horizontal projection profile and shock wave exit profile as follows:

[0013] (1)

[0014] (2)

[0015] (3)

[0016] Where w is the half-length of the waverider;

[0017] The curved shock wave profile in the axisymmetric basic flow field is defined as follows:

[0018] (4)

[0019] 2.2) Arrange the geometric information related to the shock wave exit profile:

[0020] The center of curvature A at any shock point D on the SWPC can be determined by the derivative of s(y):

[0021] (5)

[0022] The corresponding radius of curvature r AD That is the distance between point D and point A:

[0023] (6)

[0024] The shock wave A0D in the kissing surface AD ​​is obtained by scaling the bending shock wave in the axisymmetric basic flow field, and its equation satisfies:

[0025] (7)

[0026] Where k is the scaling factor, which is related to the size of the cut surface (i.e. the radius of curvature r AD )Related:

[0027] (8)

[0028] The equation of shock wave A0D in Cartesian coordinate system can be expressed as:

[0029] (9)

[0030] Among them, α is the inclination angle of the kissing surface, which is obtained by the following formula:

[0031] (10)

[0032] 2.3) The analytical expression of the three-dimensional leading edge profile (LEPC) of the waverider is derived when the shock wave exit profile (SWPC) and the waverider exit capture profile (FCT) are given:

[0033] The function of the projection AD of the shock wave A0D in the exit section is expressed as:

[0034] (11)

[0035] By solving equation (11) and equation (1) together, we can obtain the coordinates of the projection point B of the waverider front point B0 in the exit section:

[0036] (12)

[0037] Point B and point B0 satisfy Substituting equation (12) into equation (9) yields the Cartesian coordinates of point B0:

[0038] (13)

[0039] here, represents the inverse function of the bending shock wave equation c(x). Project point B0 onto the horizontal plane (let ) to get point B on PPC 01 .

[0040] Substituting equations (3) and (5) into equation (13), we can obtain the analytical expression of the three-dimensional leading edge profile (LEPC) of the waverider when the shock wave exit profile (SWPC) and the waverider exit capture profile (FCT) are specified:

[0041] (14)

[0042] Among them, f(y), s(y) and is the design input, is the design output. Repeat the above process along SWPC (which means that the variable y D By changing the value within the range [-w, w]), we can obtain the LEPC of the waverider and its projection PPC in the horizontal plane.

[0043] 2.4) The analytical expression of the three-dimensional leading edge profile (LEPC) of the waverider is derived when the shock wave exit profile (SWPC) and the horizontal projection profile of the waverider are given:

[0044] By solving equation (2) and equation (9) together, we can obtain the Cartesian coordinates of point B0:

[0045] (15)

[0046] Project point B0 back to the exit section (let ) can obtain point B on the waverider exit capture line FCT.

[0047] Substituting equations (3) and (5) into equation (15), we can obtain the analytical expression of LEPC when SWPC and PPC are specified:

[0048] (16)

[0049] Where p(y), s(y) and c(x) are design inputs, is the design output. Repeat the above process along SWPC (which means that the variable y D By changing the range of [-w, w]), the LEPC of the waverider and its projection FCT in the exit cross section can be obtained.

[0050] 2.5) The analytical expression of the shock wave exit profile (SWPC) is derived for the given waverider exit capture profile (FCT) and waverider horizontal projection profile (PPC):

[0051] According to equations (1) and (2), the three-dimensional leading edge profile of the waverider (LEPC) is obtained in the form of a parametric equation:

[0052] (17)

[0053] in, For any leading edge point B0 on ​​the LEPC, the kissing plane where it is located intersects with the SWPC at the shock point D. Obviously, the position of point D is determined by the angle α of the kissing plane and the distance between point B and point D.

[0054] The tangent line of LEPC at point B0 intersects the tangent line of FCT at point B and the tangent line of SWPC at point D at point F. Here, the tangent line segments BF and DF are both located within the exit cross section. The sweep angle λ of LEPC at point B0 is determined by the following formula:

[0055] (18)

[0056] here, is the unit tangent vector of LEPC at point B0, which is calculated by equation (17) as follows:

[0057] (19)

[0058] is the unit velocity vector of the free flow, which can be simply taken as (1,0,0).

[0059] From this, the size of BF can be calculated: (20)

[0060] The size of BD is calculated in the tangent plane. Given the distribution of the scaling factor k, combined with equation (7), we can obtain:

[0061] (twenty one)

[0062] The angle γ between the tangent segments BF and DF can be calculated by the following formula:

[0063] (twenty two)

[0064] The following angle relationships exist within triangle BDF:

[0065] (twenty three)

[0066] The angle β between the tangent line BF and the horizontal direction can be derived from formula (1):

[0067] (twenty four)

[0068] At this point, the position information of the shock point D in the tangent plane where the leading edge point B0 is located can be calculated as follows:

[0069] (25)

[0070] Substituting equations (1), (2), (20) to (24) into equation (25), we get the analytical expression of SWPC as follows:

[0071] (26)

[0072] Among them, f(y), p(y) and c(y) are design inputs, (y D , z D ) is the design output. The above process is repeated along the LEPC (which means that the variable (varies within [-w, w]), the shock wave exit profile SWPC can be obtained.

[0073] In step 3), given the incoming flow conditions, according to the shock wave angle and shock wave curvature of the three-dimensional curved shock wave at the three-dimensional leading edge profile of the waverider, the Taylor series expression of the streamlines in each tangent surface is quickly obtained by using the flow field analysis method based on the curved shock wave theory, and the parameterized expression of the lower surface of the waverider considering the incoming flow conditions and the shock wave shape is realized by coordinate transformation. The specific process is as follows:

[0074] 3.1) Given the incoming flow conditions, according to the shock angle and shock curvature of the three-dimensional curved shock wave at the three-dimensional leading edge of the waverider, the oblique shock wave relationship, the first-order curved shock wave equations and the second-order curved shock wave equations are solved to obtain the aerodynamic parameters, the first-order derivatives and the second-order derivatives of the aerodynamic parameters immediately after the leading edge of the waverider is adjacent to the curved shock wave:

[0075] The first-order bending shock wave equations are as follows:

[0076] (27)

[0077] Where, coefficients A, B, E, C and G can be calculated from the incoming flow conditions and shock wave angle, and subscripts 1 and 2 represent the upstream and downstream flow parameters of the shock wave, respectively. P is the pressure gradient (the derivative of the pressure along the streamline), D is the streamline curvature (the derivative of the airflow angle along the streamline), is the orthogonal vorticity, S a is the curvature of the shock wave surface, S b is the vertical curvature of the shock flow, which is defined as follows:

[0078] (28)

[0079] Where p is pressure, δ is stream angle, ρ is density, V is stream velocity, ω is vorticity, is the shock wave angle, s is the streamline direction, is the shock wave direction, and Y is the ordinate in the tangent plane coordinate system.

[0080] The second-order bending shock wave equations are as follows:

[0081] (29)

[0082] The coefficients in the formula are also expressions of the incoming flow conditions and shock wave angles. is the second-order pressure gradient (the second-order derivative of pressure along the streamline), is the streamline curvature gradient (the second-order derivative of the airflow angle along the streamline), is the orthogonal vorticity gradient, is the curvature derivative of the shock wave surface, is the vertical curvature derivative of the shock flow, which is defined as follows:

[0083] (30)

[0084] 3.2) When the derivatives of the aerodynamic parameters along the streamlines at the leading edge shock wave point are known, the streamlines behind the curved shock wave in the tangent plane can be quickly solved using the following Taylor series:

[0085] (31)

[0086] Where X is the horizontal coordinate in the tangent plane coordinate system, the subscript 0 represents the parameters at the leading edge of the waverider (i.e., the starting point of the streamline), and each coefficient is constructed by the aerodynamic parameters and their derivatives of various orders:

[0087] (32)

[0088] 3.3) According to the angle of the tangent plane, the streamlines in each two-dimensional tangent plane are restored to the real three-dimensional Cartesian coordinate system, and the parameterized expression of the lower surface of the waverider body considering the incoming flow conditions and the shock wave shape is obtained:

[0089] (33)

[0090] Among them, the variable From 0 to The above process is repeated along the LEPC (which means that the variable y D The streamlines in each tangent plane are combined to form a complete lower surface of the waverider.

[0091] Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

[0092] The present invention establishes the geometric relationship between the waverider design curves, solves the analytical expressions of the three-dimensional leading edge profile of the waverider and the three-dimensional curved shock wave it rides under different inputs, improves the flexibility of the waverider design, and allows the designer to freely select the design curve input as needed. By freely selecting the design curve input, the actual needs of different projects are met, and a waverider with specific geometric features and better aerodynamic performance is obtained. The present invention uses a flow field analytical method based on the curved shock wave theory to quickly obtain the Taylor series expression of the streamlines within the tangent surface, and realizes the parametric expression of the waverider surface considering the incoming flow conditions and the shock wave shape, so as to obtain a waverider with specific geometric features and better aerodynamic performance. Compared with the discrete processing of the design curve and the numerical solution of the characteristic line method of the basic flow field by the traditional waverider design method, the analytical expression derived by the present invention and the curved shock wave theory used effectively improve the efficiency while ensuring the accuracy, which is conducive to the rapid optimization design of the waverider in engineering. The present invention is an efficient, flexible and accurate design method, which is suitable for various actual engineering needs. BRIEF DESCRIPTION OF THE DRAWINGS

[0093] Figure 1 Schematic diagram of the rapid analytical design method of waveriders based on bending shock wave theory.

[0094] Figure 2 Schematic diagram of the geometric relationship of the waverider design curve, where (a) is the horizontal projection plane and (b) is the exit section.

[0095] Figure 3 Schematic diagram of flow in the flow field, where (a) is the axisymmetric basic flow field and (b) is the flow field in the tangent plane.

[0096] Figure 4 Design a curved perspective view for the three-dimensional waverider.

[0097] Figure 5 Schematic diagram for solving the shock wave exit profile.

[0098] Figure 6 The schematic diagram of the waverider structure obtained by the design method of the present invention is shown in the following example, wherein (a) is a front view, (b) is a bottom view, (c) is a rear view, and (d) is a half-section view.

[0099] The figure numbers are as follows: 1 represents the three-dimensional leading edge profile of the waverider, 2 represents the waverider outlet capture profile (i.e., FCT line), 3 represents the waverider horizontal projection profile (i.e., PPC line), 4 represents the shock wave outlet profile (i.e., SWPC line), 5 represents the center of curvature of the kissing surface, 6 represents the kissing surface, 7 represents the curved shock wave in the kissing surface, 8 represents the curved shock wave in the horizontal projection plane, 9 represents the streamline in the kissing surface, 10 represents the streamline in the horizontal projection plane, 11 represents the three-dimensional curved shock wave surface, 12 represents the trailing edge line of the lower surface of the waverider, 13 represents the curved shock wave profile in the axisymmetric basic flow field, 14 represents the waverider symmetry plane, 15 represents the upper surface of the waverider, 16 represents the trailing edge surface of the waverider, and 17 represents the lower surface of the waverider. DETAILED DESCRIPTION

[0100] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the following embodiments will further illustrate the present invention in conjunction with the accompanying drawings. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. On the contrary, the present invention encompasses any substitutions, modifications, equivalent methods and schemes made on the essence and scope of the present invention as defined by the claims. Further, in order to enable the public to have a better understanding of the present invention, some specific details are described in detail in the following detailed description of the present invention. For those skilled in the art, the present invention can be fully understood without the description of these details.

[0101] like Figures 1 to 6 As shown in FIG. 1 , a rapid analytical design method for a waverider based on bending shock wave theory includes the following steps:

[0102] (1) According to the design requirements, any two of the waverider exit capture profile 2 (i.e., FCT line), the waverider horizontal projection profile 3 (i.e., PPC line), and the shock wave exit profile 4 (i.e., SWPC line) are arbitrarily specified, and the curved shock wave profile 13 in the axisymmetric basic flow field is specified;

[0103] (2) Establish the geometric relationship between the waverider exit capture profile 2, the waverider horizontal projection profile 3 and the shock wave exit profile 4, and solve the analytical expression of the waverider three-dimensional leading edge profile 1 (i.e., LEPC line) and the three-dimensional curved shock wave 11 it rides on;

[0104] (3) Given the incoming flow conditions, according to the shock wave angle and shock wave curvature of the three-dimensional curved shock wave 11 at the three-dimensional leading edge profile 1 of the waverider, the Taylor series expression of the streamlines in each tangent surface is quickly obtained by using the flow field analysis method based on the curved shock wave theory, and the parameterized expression of the lower surface 17 of the waverider considering the incoming flow conditions and the shock wave shape is realized by coordinate transformation;

[0105] (4) According to the volume ratio requirement, the upper surface 15 of the waverider is constructed to be a plane or a convex surface, which together with the lower surface 17 of the waverider generated in step (3) and the trailing edge surface 16 of the waverider constitute the outer shape of the waverider, thereby realizing a rapid analytical design of the waverider based on the bending shock wave theory.

[0106] In step (2), the geometric relationship between the waverider exit capture profile 2, the waverider horizontal projection profile 3 and the shock wave exit profile 4 is established, and the analytical expression of the waverider three-dimensional leading edge profile 1 and the three-dimensional curved shock wave 11 is solved. The specific process is:

[0107] (2.1) Figure 2 As shown, the waverider exit capture profile 2, the waverider horizontal projection profile 3 and the shock wave exit profile 4 are defined as follows:

[0108] (1)

[0109] Where w is the half-length of the waverider, that is, half of the maximum width of the waverider in the y direction; z is the coordinate in the vertical direction, and f(y) is a function of y, which represents the coordinate of z at a given y value.

[0110] Waverider horizontal projection line 3:

[0111] (2)

[0112] Where p(y) is a function of y, which represents the coordinate of x at a given y value; w is the half-length of the waverider;

[0113] Shock wave exit profile 4:

[0114] (3)

[0115] Where z represents the vertical coordinate, s(y) is a function of y, representing the coordinate of z at a given y value. w is the half-length of the waverider;

[0116] like Figure 3 As shown in Figure (a) in FIG. 1 , the curved shock wave profile 13 in the axisymmetric basic flow field is defined as follows:

[0117] (4)

[0118] (2.2) Arrange the geometric information related to the shock wave exit profile 4:

[0119] like Figure 2 In Figure (b), the center of curvature A at any shock point D on the shock wave exit profile 4 can be determined by the derivative of s(y):

[0120] (5)

[0121] The corresponding radius of curvature r AD That is the distance between point D and point A:

[0122] (6)

[0123] The center of curvature 5 of the tangent surface is the center of curvature of the curved shock wave 7 in the tangent surface 6. When the center of curvature A of any shock point D on the shock exit profile 4 is determined, the center of curvature of the tangent surface 6 is also determined. The center of curvature 5 of the tangent surface changes with the change of the shock point D on the shock exit profile 4, and is located at the center of a circle passing through the shock point D and being tangent to the shock exit profile 4 at that point.

[0124] like Figure 3 As shown in the middle figure (b), the curved shock wave 7 in the kissing surface 6 is obtained by scaling the curved shock wave in the axisymmetric basic flow field, and its equation satisfies:

[0125] (7)

[0126] Where k is a scaling factor, which is related to the size of the cut surface 6 (i.e., the radius of curvature r AD )Related:

[0127] (8)

[0128] The equation of the bending shock wave 7 in the tangent plane in the Cartesian coordinate system can be expressed as:

[0129] (9)

[0130] Wherein, α is the inclination angle of the kissing surface 6, which is obtained by the following formula:

[0131] (10)

[0132] (2.3) The analytical expression of the waverider three-dimensional leading edge profile 1 is derived when the shock wave exit profile 4 and the waverider exit capture profile 2 are given:

[0133] like Figure 2 As shown in Figure (b), the function of the projection AD of the curved shock wave 7 (A0D) in the tangent plane in the exit section is expressed as:

[0134] (11)

[0135] By solving equation (11) and equation (1) together, we can obtain the coordinates of the projection point B of the waverider front point B0 in the exit section:

[0136] (12)

[0137] Point B and point B0 satisfy Substituting equation (12) into equation (9) yields the Cartesian coordinates of point B0:

[0138] (13)

[0139] here, represents the inverse function of the bending shock wave equation c(x). Project point B0 onto the horizontal plane (let ) to obtain point B on the horizontal projection line 3 of the waverider 01 .

[0140] Substituting equations (3) and (5) into equation (13), we can obtain the analytical expression of the three-dimensional leading edge profile 1 of the waverider when the shock wave exit profile 4 and the waverider exit capture profile 2 are specified:

[0141] (14)

[0142] Among them, f(y), s(y) and is the design input, is the design output. Repeat the above process along the shock wave exit profile 4 (which means that the variable y D By changing within [-w, w]), the three-dimensional leading edge profile 1 of the waverider and its projection 3 in the horizontal plane can be obtained.

[0143] (2.4) The analytical expression of the waverider three-dimensional leading edge profile 1 is derived when the shock wave exit profile 4 and the waverider horizontal projection profile 3 are given:

[0144] By solving equation (2) and equation (9) together, we can obtain the Cartesian coordinates of point B0:

[0145] (15)

[0146] Project point B0 back to the exit section (let ) can obtain point B on the waverider exit capture profile 2.

[0147] Substituting equations (3) and (5) into equation (15), we can obtain the analytical expression of the three-dimensional leading edge profile 1 of the waverider when the shock wave exit profile 4 and the horizontal projection profile 3 of the waverider are specified:

[0148] (16)

[0149] Where p(y), s(y) and c(x) are design inputs, is the design output. Repeat the above process along the shock wave exit profile 4 (which means that the variable y DBy changing within [-w, w]), the three-dimensional leading edge profile 1 of the waverider and its projection 2 in the exit section can be obtained.

[0150] The curved shock wave 8 in the horizontal projection plane is the projection of the curved shock wave corresponding to the horizontal projection profile 3 (ie, the PPC line) of the waverider on the horizontal plane, reflecting the flow field characteristics of the waverider in the horizontal direction.

[0151] (2.5) The analytical expression of the shock wave exit profile 4 is derived when the waverider exit capture profile 2 and the waverider horizontal projection profile 3 are given:

[0152] According to equations (1) and (2), the three-dimensional leading edge profile of the waverider in the form of a parametric equation is obtained:

[0153] (17)

[0154] in, .like Figure 4 As shown, for any leading edge point B0 on ​​the three-dimensional leading edge profile 1 of the waverider, the tangent plane 6 where it is located intersects with the shock wave exit profile 4 at the shock point D. Obviously, the position of the shock point D is determined by the angle α of the tangent plane 6 and the distance between point B and point D.

[0155] The tangent line of the waverider three-dimensional leading edge profile 1 at point B0 intersects the tangent line of the waverider exit capture profile 2 at point B and the tangent line of the shock wave exit profile 4 at point D at point F. Here, the tangent segments BF and DF are both located within the exit cross section. The sweep angle λ of the waverider three-dimensional leading edge profile 1 at point B0 is determined by the following formula:

[0156] (18)

[0157] here, is the unit tangent vector of the three-dimensional leading edge profile 1 of the waverider at point B0, which is calculated by equation (17) as follows:

[0158] (19)

[0159] is the unit velocity vector of the free flow, which can be simply taken as (1,0,0).

[0160] From this, the size of BF can be calculated:

[0161] (20)

[0162] like Figure 3 As shown in Figure (b), the size of BD is calculated within the kissing plane 6. Given the distribution of the scaling factor k, combined with formula (7), we can get:

[0163] (twenty one)

[0164] like Figure 5 As shown, the angle γ between the tangent segments BF and DF can be calculated by the following formula:

[0165] (twenty two)

[0166] The following angle relationships exist within triangle BDF:

[0167] (twenty three)

[0168] The angle β between the tangent line BF and the horizontal direction can be derived from formula (1):

[0169] (twenty four)

[0170] At this point, the position information of the shock point D in the kissing surface 6 where the leading edge point B0 is located can be calculated as follows:

[0171] (25)

[0172] Substituting equations (1), (2), (21) to (24) into equation (25), the analytical expression of the shock wave exit profile 4 is obtained as follows:

[0173] (26)

[0174] Among them, f(y), p(y) and c(y) are design inputs, (y D , z D ) is the design output. The above process is repeated along the waverider three-dimensional leading edge profile 1 (which means that the variable (varies within [-w, w]), the shock wave exit profile 4 can be obtained.

[0175] In step (3), given the incoming flow conditions, according to the shock wave angle and shock wave curvature of the three-dimensional curved shock wave 11 at the three-dimensional leading edge profile 1 of the waverider, the Taylor series expression of the streamlines in each tangent surface is quickly obtained by using the flow field analysis method based on the curved shock wave theory, and the parameterized expression of the lower surface 17 of the waverider considering the incoming flow conditions and the shock wave shape is realized by coordinate transformation. The specific process is as follows:

[0176] (3.1) Given the incoming flow conditions, according to the shock angle and shock curvature of the three-dimensional curved shock wave 11 at the three-dimensional leading edge profile 1 of the waverider, the oblique shock wave relationship, the first-order curved shock wave equations and the second-order curved shock wave equations are solved to obtain the aerodynamic parameters, the first-order derivatives and the second-order derivatives of the aerodynamic parameters of the three-dimensional leading edge profile 1 of the waverider immediately adjacent to the curved shock wave 11:

[0177] The first-order bending shock wave equations are as follows:

[0178] (27)

[0179] In the formula, coefficients A, B, E, C and G can be calculated from the incoming flow conditions and shock wave angle. Subscripts 1 and 2 represent the upstream and downstream flow parameters of the shock wave, respectively. P is the pressure gradient (the derivative of the pressure along the streamline), D is the streamline curvature (the derivative of the airflow angle along the streamline), is the orthogonal vorticity, S a is the curvature of the shock wave surface, S b is the vertical curvature of the shock flow, which is defined as follows:

[0180] (28)

[0181] Where p is pressure, δ is stream angle, ρ is density, V is stream velocity, ω is vorticity, is the shock wave angle, s is the streamline direction, is the shock wave direction, and Y is the ordinate in the tangent plane coordinate system.

[0182] The second-order bending shock wave equations are as follows:

[0183] (29)

[0184] The coefficients in the formula are also expressions of the incoming flow conditions and shock wave angles. is the second-order pressure gradient (the second-order derivative of pressure along the streamline), is the streamline curvature gradient (the second-order derivative of the airflow angle along the streamline), is the orthogonal vorticity gradient, is the curvature derivative of the shock wave surface, is the vertical curvature derivative of the shock flow, which is defined as follows:

[0185] (30)

[0186] (3.2) Figure 3 As shown in Figure (b), when the derivatives of the aerodynamic parameters along the streamlines at the leading edge shock point B0 are known, the streamlines 9 in the kissing surface after the curved shock wave 7 in the kissing surface 6 can be quickly solved by the following Taylor series:

[0187] (31)

[0188] Where X is the abscissa in the tangent plane coordinate system, subscript 0 represents the parameter at the leading edge point B0 of the waverider (i.e., the starting point of streamline 9 in the tangent plane), and each coefficient is constructed by the aerodynamic parameters and their derivatives of various orders:

[0189] (32)

[0190] The streamlines 10 in the horizontal projection plane are streamlines formed by the airflow in the flow field behind the curved shock wave 8 in the horizontal projection plane. The flow path of the airflow passing through the waverider is displayed on the horizontal plane. Similar to the streamlines 9 in the tangent plane, the streamlines 10 in the horizontal projection plane are also obtained by the flow field analysis method, but they are calculated on the horizontal plane.

[0191] (3.3) According to the angle of the tangent plane, a series of streamlines in each two-dimensional tangent plane are restored to the real three-dimensional Cartesian coordinate system, and the parameterized expression of the lower surface 17 of the waverider body considering the incoming flow condition and the shock wave shape is obtained:

[0192] (33)

[0193] Among them, the variable From 0 to Repeat the above process along the waverider three-dimensional front edge profile 1 (which means that the variable y D The streamlines in each tangent plane are combined to form a complete lower surface of the waverider 17.

[0194] See also Figure 6 The waverider generated by the waverider rapid analytical design method based on the bending shock wave theory is composed of the waverider lower surface 17, the waverider upper surface 15 and the waverider trailing edge surface 16. The waverider lower surface trailing edge line 12 defines the rear end boundary of the waverider lower surface. The waverider three-dimensional leading edge profile 1 is the front end boundary, which is derived from the waverider outlet capture profile 2 (FCT line) and the waverider horizontal projection profile 3 (PPC line), and has an important influence on the aerodynamic performance. The waverider outlet capture profile 2 determines the airflow captured by the waverider, and the waverider horizontal projection profile 3 determines the horizontal size and shape of the waverider, and together with the waverider three-dimensional leading edge profile 1, the waverider upper surface 15 and the waverider lower surface 17, determines the external shape. The waverider lower surface trailing edge line 12 connects the waverider lower surface 17 and the waverider trailing edge surface 16. The waverider symmetry plane 14 is a virtual plane, and all components maintain symmetry on it. The waverider upper surface 15 and the waverider lower surface 17 constitute the waverider main body structure.

[0195] The above embodiments are only preferred embodiments of the present invention and cannot be considered to limit the scope of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of the present invention.

Claims

1. A rapid analytical design method for waveriders based on bending shock wave theory, characterized by The following steps are involved: 1) Specify the design curve: According to the design requirements, specify any two of the waverider exit capture profile, waverider horizontal projection profile and shock wave exit profile, and specify the curved shock wave profile in the axisymmetric basic flow field; 2) Establishing geometric relationships and solving analytical expressions: Establishing geometric relationships between the waverider exit capture profile, the waverider horizontal projection profile, and the shock wave exit profile, solving the analytical expressions of the waverider 3D leading edge profile and the 3D curved shock wave it rides on; 3) Obtain streamline expression: Given the incoming flow conditions, according to the shock wave angle and shock wave curvature of the three-dimensional curved shock wave at the three-dimensional leading edge profile of the waverider, the Taylor series expression of the streamlines in each tangent surface is quickly obtained using the flow field analysis method based on the curved shock wave theory, and the parametric expression of the lower surface of the waverider considering the incoming flow conditions and the shock wave shape is realized through coordinate transformation; 4) Constructing the waverider shape: According to the volume ratio requirements, the upper surface of the waverider is constructed to be a plane or a convex surface, which together with the lower surface of the waverider generated by step 3) and the trailing edge surface of the waverider constitute the waverider shape, thereby realizing a rapid analytical design of the waverider based on the bending shock wave theory.

2. The rapid analytical design method of waverider based on bending shock wave theory as claimed in claim 1, characterized in that In step 2), the geometric relationship is established and the analytical expression is solved. The specific process includes: (1) Define the waverider exit capture profile, waverider horizontal projection profile and shock wave exit profile respectively; (2) Arrange the geometric information related to the shock wave exit profile, including the center of curvature, radius of curvature, and shock wave equation within the tangent plane; (3) Derive the analytical expression of the three-dimensional leading edge profile of the waverider when the shock wave exit profile and the waverider exit capture profile are given; (4) Derive the analytical expression of the three-dimensional leading edge profile of the waverider when the shock wave exit profile and the horizontal projection profile of the waverider are given; (5) Derive the analytical expression for the shock wave exit profile when the waverider exit capture profile and the waverider horizontal projection profile are given.

3. The rapid analytical design method of waverider based on bending shock wave theory as claimed in claim 1, characterized in that In step 2), the geometric relationship between the waverider exit capture profile, the waverider horizontal projection profile and the shock wave exit profile is established, and the analytical expression of the waverider three-dimensional leading edge profile and the three-dimensional curved shock wave is solved. The specific process is: 2.1) Define the waverider exit capture profile, waverider horizontal projection profile and shock wave exit profile as follows: (1) (2) (3) Where w is the half-length of the waverider; The curved shock wave profile in the axisymmetric basic flow field is defined as follows: (4) 2.2) Arrange the geometric information related to the shock wave exit profile: The center of curvature A at any shock point D on the shock wave exit profile is determined by the derivative of s(y): (5) The corresponding radius of curvature r AD That is the distance between point D and point A: (6) The shock wave A0D in the kissing surface AD ​​is obtained by scaling the bending shock wave in the axisymmetric basic flow field, and its equation satisfies: (7) Where k is the scaling factor, which is related to the size of the cut surface, i.e. the radius of curvature r AD Related: (8) The equation of shock wave A0D in Cartesian coordinate system is expressed as: (9) Among them, α is the inclination angle of the kissing surface, which is obtained by the following formula: (10) 2.3) Derive the analytical expression of the three-dimensional leading edge profile of the waverider when the shock wave exit profile and the waverider exit capture profile are given: The function of the projection AD of the shock wave A0D in the exit section is expressed as: (11) Solving equation (11) and equation (1) together, we can obtain the coordinates of the projection point B of the waverider front point B0 in the exit section: (12) Point B and point B0 satisfy Substituting equation (12) into equation (9) yields the Cartesian coordinates of point B0: (13) here, represents the inverse function of the bending shock wave equation c(x); project point B0 onto the horizontal plane (let ) is the point B on the horizontal projection line of the waverider 01 ; Substituting equations (3) and (5) into equation (13), we can obtain the analytical expression of the three-dimensional leading edge profile of the waverider when the shock wave exit profile and the waverider exit capture profile are specified: (14) Among them, f(y), s(y) and is the design input, is the design output; repeat the above process along the shock wave exit profile, the variable y D Change within [-w, w], that is, obtain the three-dimensional leading edge line of the waverider and its projection in the horizontal plane, the horizontal projection line of the waverider; 2.4) Derive the analytical expression of the three-dimensional leading edge profile of the waverider when the shock wave exit profile and the horizontal projection profile of the waverider are given: Solving equation (2) and equation (9) together gives the Cartesian coordinates of point B0: (15) Project point B0 back to the exit section, let , that is, point B on the capture line of the waverider exit is obtained; Substituting equations (3) and (5) into equation (15), we can obtain the analytical expression of the three-dimensional leading edge profile of the waverider when the shock wave exit profile and the horizontal projection profile of the waverider are specified: (16) Where p(y), s(y) and c(x) are design inputs, is the design output; repeat the above process along the shock wave exit profile, the variable y D By changing within [-w, w], the three-dimensional leading edge profile of the waverider and its projection in the exit section are obtained; 2.5) Derive the analytical expression of the shock wave exit profile when the waverider exit capture profile and the waverider horizontal projection profile are given: According to equations (1) and (2), the three-dimensional leading edge profile of the waverider is obtained in the form of a parametric equation: (17) in, ; For any leading edge point B0 on ​​the three-dimensional leading edge line of the waverider, the kissing surface where it is located intersects with the shock wave exit line at the shock wave point D; obviously, the position of point D is determined by the angle α of the kissing surface and the distance between point B and point D; The tangent line of the waverider 3D leading edge profile at point B0 intersects with the tangent line of the waverider exit capture profile at point B and the tangent line of the shock wave exit profile at point D at point F; here, the tangent segments BF and DF are both located in the exit cross section; the sweep angle λ of the waverider 3D leading edge profile at point B0 is determined by the following formula: (18) here, is the unit tangent vector of the three-dimensional leading edge of the waverider at point B0, which is calculated by equation (17) as follows: (19) is the unit velocity vector of the free flow, which is simply taken as (1,0,0); From this, calculate the size of BF: (20) The size of BD is calculated in the tangent plane. Given the distribution of scaling factor k, combined with equation (7), we get: (21) The angle γ between the tangent segments BF and DF is calculated by the following formula: (22) The following angle relationships exist within triangle BDF: (23) The angle β between the tangent line BF and the horizontal direction is derived from equation (1): (24) At this point, the position information of the shock point D in the kissing plane where the leading edge point B0 is located is calculated as follows: (25) Substituting equations (1), (2), (20) to (24) into equation (25), the analytical expression of the shock wave exit profile is obtained as follows: (26) Among them, f(y), p(y) and c(y) are design inputs, (y D , z D ) is the design output; repeat the above process along the three-dimensional leading edge profile of the waverider, and the variable By varying within [-w, w], we obtain the shock wave exit profile.

4. The rapid analytical design method of waverider based on bending shock wave theory as claimed in claim 1, characterized in that In step 3), given the incoming flow conditions, according to the shock wave angle and shock wave curvature of the three-dimensional curved shock wave at the three-dimensional leading edge profile of the waverider, the Taylor series expression of the streamlines in each tangent surface is quickly obtained by using the flow field analysis method based on the curved shock wave theory, and the parameterized expression of the lower surface of the waverider considering the incoming flow conditions and the shock wave shape is realized by coordinate transformation. The specific process is as follows: 3.1) Given the incoming flow conditions, according to the shock angle and shock curvature of the three-dimensional curved shock wave at the three-dimensional leading edge of the waverider, the oblique shock wave relationship, the first-order curved shock wave equations and the second-order curved shock wave equations are solved to obtain the aerodynamic parameters, the first-order derivatives and the second-order derivatives of the aerodynamic parameters immediately after the leading edge of the waverider is adjacent to the curved shock wave: The first-order bending shock wave equations are as follows: (27) Where, coefficients A, B, E, C and G are calculated from the incoming flow conditions and shock wave angle, subscripts 1 and 2 represent the upstream and downstream flow parameters of the shock wave, respectively; P is the pressure gradient, D is the streamline curvature, is the orthogonal vorticity, S a is the curvature of the shock wave surface, S b is the vertical curvature of the shock flow, which is defined as follows: (28) in, p is pressure, δ is stream angle, ρ is density, V is stream velocity, ω is vorticity, is the shock wave angle, s is the streamline direction, is the shock wave direction, Y is the ordinate in the tangent plane coordinate system; The second-order bending shock wave equations are as follows: (29) The coefficients in the formula are also expressions of the incoming flow conditions and shock wave angles; is the second-order pressure gradient, is the streamline curvature gradient, is the orthogonal vorticity gradient, is the curvature derivative of the shock wave surface, is the vertical curvature derivative of the shock flow, which is defined as follows: (30) 3.2) When the derivatives of the aerodynamic parameters along the streamlines at the leading edge shock wave point are known, the streamlines behind the curved shock wave in the tangent plane can be quickly solved using the following Taylor series: (31) Where X is the horizontal coordinate in the tangent plane coordinate system, the subscript 0 represents the parameters at the leading edge of the waverider body, i.e., the starting point of the streamline, and each coefficient is constructed by the aerodynamic parameters and their derivatives of various orders: (32) 3.3) According to the angle of the tangent plane, the streamlines in each two-dimensional tangent plane are restored to the real three-dimensional Cartesian coordinate system, and the parameterized expression of the lower surface of the waverider body considering the incoming flow conditions and the shock wave shape is obtained: (33) Among them, the variable From 0 to ; Repeat the above process along the three-dimensional leading edge of the waverider, and the variable y D It varies within [-w, w] and combines the streamlines within each tangent plane to form a complete lower surface of the waverider.

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