A Stability Analysis Method for Two-Dimensional Rarefied Gas Flow Based on Shakhov Model

Abstract

The invention discloses a stability analysis method for two-dimensional rarefied gas flow based on the Shakhov model, which relates to the field of aerodynamic research and comprises: selecting the Shakhov model as an initial model; obtaining a model about a reduced distribution function; processing to obtain a small disturbance equation; obtaining the relationship between a disturbance reduced distribution function and a disturbance macroscopic quantity; obtaining a stability equation; obtaining a diffuse reflection boundary condition corresponding to the disturbance; obtaining a target equation and a target boundary condition; solving eigenvalues ​​and eigenvectors based on the target equation, the target boundary condition and given parameters, and obtaining an analysis result of the stability of the rarefied flow based on the solution results of the eigenvalues ​​and the eigenvectors; the method adopts the Shakhov model to consider the heat flow effect, and the Prandtl number changes with the temperature, thereby correcting and solving the problem that the prior art can only analyze the gas flow with the Prandtl number of 1.

Description

Technical Field

[0001] The invention relates to the field of aerodynamic research, and in particular to a two-dimensional rarefied gas flow stability analysis method based on a Shakhov model. Background Art

[0002] For near-space high-speed aircraft, boundary layer transition causes the friction and heat flow on its surface to increase by 3 to 5 times, seriously affecting flight performance and safety. Unlike conventional aircraft, near-space high-speed aircraft have the characteristics of flying in a large airspace and a wide speed range. In a large airspace flight environment, the local rarefied gas effect becomes an important factor affecting the transition. In addition, high Mach number flight conditions have a great influence on the transport characteristics of the gas, such as gas viscosity and thermal conductivity. However, the existing rarefied flow stability analysis method is based on the BGK model, which does not consider the heat flow effect and can only study the flow with a Prandtl number of 1. Summary of the invention

[0003] In order to overcome the shortcomings of the prior art, the present invention provides a stability analysis method for two-dimensional rarefied gas flow based on the Shakhov model. The Shakhov model takes into account the thermal flow effect, and the Prandtl number changes with temperature. The problem that the prior art (BGK model) can only analyze the gas flow with a Prandtl number of 1 is corrected and solved.

[0004] To achieve the above-mentioned object of the invention, the present invention provides a method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model, the method comprising:

[0005] Step 1: Select the Shakhov model as the initial model;

[0006] Step 2: introducing a reduced distribution function to eliminate the dependence of the distribution function in the initial model on the molecular velocity in the vertical direction, and obtaining a model about the reduced distribution function;

[0007] Step 3: Under the condition that the rarefied gas satisfies the parallelism hypothesis and the small perturbation hypothesis, the instantaneous quantity in the model about the reduced distribution function is rewritten into the form of the sum of the basic flow and the perturbation quantity; under the condition that the basic flow satisfies the model about the reduced distribution function, the small perturbation equation is obtained;

[0008] Step 4: Process the perturbation reduced distribution function in the small perturbation equation by moment calculation to obtain the relationship between the perturbation reduced distribution function and the perturbation macroscopic quantity;

[0009] Step 5: Write the disturbance in the small disturbance equation as the product of the shape function and the wave function to obtain the stability equation;

[0010] Step 6: Using diffuse reflection boundary conditions, derive the diffuse reflection boundary conditions corresponding to the disturbance under the assumption of small disturbance;

[0011] Step 7: Based on the relationship between the disturbance reduced distribution function and the disturbance macroscopic quantity, the stability equation and the diffuse reflection boundary conditions of the disturbance are discretized to obtain the target equation and target boundary conditions;

[0012] Step 8: Solve the eigenvalues ​​and eigenvectors based on the target equation, the target boundary conditions and given parameters, and obtain the analysis results of the stability of the rarefied flow based on the solution results of the eigenvalues ​​and eigenvectors.

[0013] Among them, in the embodiment of the present invention, the initial model is:

[0014] ;

[0015] in, f Represents the distribution function of particles in time, physical space and velocity space, f =( t , r , ξ ), r represents the particle direction, r =( x , y ), x Indicates the flow direction, y represents the normal direction, ξ represents the particle speed, ξ =( ξ x , ξ y , ξ z ), ξ x , ξ y and ξ z are the velocity components of the particle velocity in the streamwise direction, normal direction and spanwise direction, respectively. τ represents the relaxation time, t Indicates time, f s represents the particle equilibrium distribution function.

[0016] Among them, in the embodiment of the present invention, the reduced distribution function is:

[0017] ;

[0018] ;

[0019] in, g( t , x , y , ξ x , ξ y )and h ( t , x , y , ξ x , ξ y ) is the reduced distribution function, g ( t , x , y , ξ x , ξ y )= g , h ( t , x , y , ξ x , ξ y )= h ;

[0020] On the Reduced Distribution Function g and h The equation corresponding to the model is:

[0021] ;

[0022] ;

[0023] in, h s and g s is the particle Shakhov equilibrium state reduced distribution function, calculated as;

[0024] ;

[0025] ;

[0026] Where Pr is the Prandtl number, c is the characteristic velocity of the particle, c=(c x ,c y ), c x is the velocity component of the particle’s characteristic velocity in the stream direction, c y is the velocity component of the particle’s characteristic velocity in the normal direction, q represents the heat flow vector, q=(q x ,q y ), qx is the component of the heat flux vector in the streamwise direction, q y is the component of the heat flux vector in the normal direction, p Indicates pressure, R represents the particle equilibrium constant, T Indicates temperature, D represents the spatial dimension, g e represents the particle reduced equilibrium distribution function, K represents the internal freedom degree of the molecule;

[0027] ;

[0028] in, ρ is the density, u =( u , v ) represents the macroscopic speed, u is the flow velocity, v is the normal velocity.

[0029] In the embodiment of the present invention, under the condition that the rarefied gas satisfies the parallelism assumption and the small disturbance assumption, the instantaneous quantity in the model of the reduced distribution function is rewritten in the form of the sum of the basic flow and the disturbance quantity, which is expressed as:

[0030] ;

[0031] in, t Indicates time, x Indicates the flow direction, y represents the normal direction, ξ x and ξ y are the velocity components of the particle velocity in the streamwise direction and the normal direction, μ is the viscosity, μ c ( y ) is the basic flow of viscosity, is the viscosity disturbance, p c ( y ) is the basic flow of pressure, is the density perturbation, u c ( y ) is the basic quantity of stream velocity, is the disturbance of the streamwise velocity, v is the normal velocity, is the perturbation of the normal velocity, g and h is the reduced distribution function, forg The basic amount, for g The disturbance amount, for h The basic amount, for h The disturbance amount, h s and g s is the particle Shakhov equilibrium reduced distribution function, for g s The basic amount, for g s The disturbance amount, for h s The basic amount, for h s The amount of disturbance;

[0032] The small perturbation equation is:

[0033] ;

[0034] ;

[0035] in, r represents the particle direction, μ c is the basic quantity of viscosity, is the pressure disturbance, p c is the basic quantity of pressure, is the viscosity disturbance.

[0036] In the embodiment of the present invention, the disturbance amount in the small disturbance equation is written in the form of the product of the shape function and the wave function, which is expressed as:

[0037] ;

[0038] in, e is the natural base, i is an imaginary number, a is the wave number, is the frequency, is the density perturbation, is the characteristic function of density, is the disturbance of the streamwise velocity, is the characteristic function of the streamwise velocity, is the perturbation of the normal velocity, is the characteristic function of the normal velocity, is the characteristic function of viscosity, is the characteristic function of pressure, for g The characteristic function of for h The characteristic function of for g s The characteristic function of for h s The superscripts ~ and ^ denote the disturbance and characteristic function, respectively;

[0039] The stability equation is obtained as:

[0040] ;

[0041] .

[0042] Among them, in the embodiment of the present invention, the perturbation reduced distribution function in the small perturbation equation is processed by moment calculation to obtain the relationship between the perturbation reduced distribution function and the perturbation macroscopic quantity, specifically:

[0043] ;

[0044] in: ;

[0045] in, is the density perturbation, ξ represents the particle speed, is the disturbance of the streamwise velocity, is the perturbation of the normal velocity, is the macroscopic energy disturbance, E c is the basic quantity of macroscopic energy, is the disturbance to the heat flow, is the disturbance of the normal heat flow, v c is the basic quantity of normal velocity, Q qxu is the first intermediate quantity, Q qxv is the second intermediate quantity, Q qyv is the third intermediate quantity, Q qyu It is the fourth intermediate quantity.

[0046] Among them, in the embodiment of the present invention, the diffuse reflection boundary condition corresponding to the disturbance is:

[0047] ;

[0048] ;

[0049] in, and represents the reduced perturbation distribution function of particles on the boundary, r w represents the position on the particle boundary, and represents the equilibrium perturbation distribution function on the particle boundary, ρ w represents the boundary density, u w represents the macroscopic velocity of the disturbance on the boundary, T w represents the temperature on the boundary, n w represents the normal vector of the boundary, ξ represents the particle velocity;

[0050] in:

[0051] ;

[0052] ;

[0053] in, is the equilibrium distribution function g e The disturbance amount, is the equilibrium distribution function g e The basic amount, is the density perturbation, ρ c is the basic quantity of density, ξ x and y are the velocity components of the particle velocity in the streamwise direction and the normal direction, R is the gas constant, u c is the basic quantity of stream velocity, is the disturbance of the streamwise velocity, v c is the basic quantity of normal velocity, is the perturbation of the normal velocity, is the equilibrium distribution function h e The disturbance amount, is the equilibrium distribution function h e The basic amount, T c The basic quantity that represents temperature, Represents the temperature disturbance.

[0054] In this embodiment of the present invention, the discrete form of the stability equation is:

[0055] c1-c2-c3-c4-c5-c6-c7;

[0056] Among them, c1 to c7 are all intermediate quantities;

[0057] in:

[0058] ;

[0059] ;

[0060] ;

[0061] ;

[0062] ;

[0063] ;

[0064] ;

[0065] ;

[0066] Where: N represents the number of physical space points, j Indicates the current point in physical space. ξ y,k Indicates the point in velocity space k The normal discrete particle velocity, Represents a point in physical space j Speed ​​Space Point k The reduced perturbation distribution function of ξ x,k represents the streamwise particle velocity at the velocity space point, Represents physical space m Speed ​​Space Point k The reduced perturbation distribution function of E jm represents the coefficients of the Chebyshev derivative matrix at different physical space points, W p represents the weight coefficient, Q represents a velocity space point, and Represents physical space j Speed ​​Space Point p The reduced perturbation distribution function of Represents physical space j The basic density quantity, Represents a point in physical space jFlow to macro velocity, ξ x,p Represents a velocity space point p The particle velocity of the stream, Represents a velocity space point p The normal particle velocity is K is the intramolecular degree of freedom, , G ρ , A g1 , G u , G v , G E , A g1qx , A g1qy and G p They are the fifth to twelfth intermediate quantities respectively.

[0067] in:

[0068] ;

[0069] d 1 to d 6 are all intermediate quantities;

[0070] in:

[0071] ;

[0072] ;

[0073] ;

[0074] ;

[0075] ;

[0076] ;

[0077] Among them, the Chebyshev derivative matrix ( D N ) E jm for:

[0078] ;

[0079] in, , , and are the coefficients in the Chebyshev derivative matrix.

[0080] A h1 =P c / μ c , A h1 ,H ρ , H u , H v , H E , A h1qx , A h1qy They are the thirteenth to nineteenth intermediate quantities respectively;

[0081] ;

[0082] ;

[0083] ;

[0084] ;

[0085] ;

[0086] .

[0087] in: Represents a point in physical space m Speed ​​Space Point k The reduced perturbation distribution function of ξ x,k represents the streamwise particle velocity at the velocity space point, Represents physical space j Speed ​​Space Point k The reduced perturbation distribution function of .

[0088] The discrete form corresponding to the diffuse reflection boundary condition is:

[0089] ;

[0090] Among them, a1, a2, a3 and a4 are all intermediate quantities;

[0091] in:

[0092] ;

[0093] ;

[0094] ;

[0095] ;

[0096] in, ;

[0097] Among them, b1, b2, b3 and b4 are all intermediate quantities;

[0098] in:

[0099] ;

[0100] ;

[0101] ;

[0102] ;

[0103] in, W p is the integral weight coefficient.

[0104] In this embodiment of the present invention, by changing the Knudsen number, Prandtl number and wave number, the eigenvalues ​​and eigenvectors of the system under specific parameters are calculated, the eigenvalues ​​correspond to the growth rate and frequency of the disturbance, and the eigenvectors correspond to the spatial distribution of the disturbance.

[0105] Among them, in the embodiment of the present invention, the step 5 is specifically: using the canonical modulus analysis method, the disturbance quantity in the small perturbation equation is written in the form of the product of the shape function and the wave function to obtain the stability equation.

[0106] Among them, in an embodiment of the present invention, the step 7 is specifically: using a preset method based on the relationship between the disturbance reduced distribution function and the disturbance macroscopic quantity, the stability equation and the diffuse reflection boundary conditions of the disturbance are discretized to obtain the target equation and the target boundary conditions; the preset method includes the Chebyshev spectral configuration point method and the Gauss-Hermite velocity space discretization method.

[0107] In this embodiment of the present invention, the method further includes:

[0108] Step 9: Based on the analysis results of the stability of the rarefied flow and the shape parameters of the aircraft model, obtain the transition prediction results of the aircraft model.

[0109] Among them, this method corrects and solves the problem that the existing technology (BGK model) can only analyze the gas flow with a Prandtl number of 1, and obtains more reliable analysis results of the stability of the rarefied flow. Then, based on the reliable analysis results of the stability of the rarefied flow, a reliable transition prediction result of the actuator model is obtained.

[0110] One or more technical solutions provided by the present invention have at least the following technical effects or advantages:

[0111] This method adopts the Shakhov model to consider the thermal flow effect. The Prandtl number changes with temperature. The correction solves the problem that the existing technology can only analyze the gas flow with a Prandtl number of 1. This method enables the stability analysis method suitable for rarefied flow to have the ability to analyze the flow with variable Prandtl number, providing a more reliable technology for studying the flow stability of the boundary layer of high-altitude aircraft. BRIEF DESCRIPTION OF THE DRAWINGS

[0112] The drawings described herein are used to provide a further understanding of the embodiments of the present invention, constitute a part of the present invention, and do not constitute a limitation on the embodiments of the present invention;

[0113] Figure 1 This is a flow chart of a method for analyzing rarefied gas flow stability based on the Shakhov model in one embodiment of the present application;

[0114] Figure 2 It is a schematic diagram comparing the results obtained by the stability analysis method based on the Shakhov model and the stability analysis method based on the Navier-Stokes equation considering the slip boundary condition in one embodiment of the present application. DETAILED DESCRIPTION

[0115] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present invention and the features in the embodiments can be combined with each other without conflict.

[0116] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those within the scope of this description. Therefore, the protection scope of the present invention is not limited to the specific embodiments disclosed below.

[0117] Embodiment 1;

[0118] Please refer to Figure 1 The present invention provides a method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model, the method comprising:

[0119] Step 1: Select the Shakhov model as the initial model;

[0120] Step 2: introducing a reduced distribution function to eliminate the dependence of the distribution function in the initial model on the molecular velocity in the vertical direction, and obtaining a model about the reduced distribution function;

[0121] Step 3: Under the condition that the rarefied gas satisfies the parallelism hypothesis and the small perturbation hypothesis, the instantaneous quantity in the model about the reduced distribution function is rewritten into the form of the sum of the basic flow and the perturbation quantity; under the condition that the basic flow satisfies the model about the reduced distribution function, the small perturbation equation is obtained;

[0122] Step 4: Process the perturbation reduced distribution function in the small perturbation equation by moment calculation to obtain the relationship between the perturbation reduced distribution function and the perturbation macroscopic quantity;

[0123] Step 5: Write the disturbance in the small disturbance equation as the product of the shape function and the wave function to obtain the stability equation;

[0124] Step 6: Using diffuse reflection boundary conditions, derive the diffuse reflection boundary conditions corresponding to the disturbance under the assumption of small disturbance;

[0125] Step 7: Based on the relationship between the disturbance reduced distribution function and the disturbance macroscopic quantity, the stability equation and the diffuse reflection boundary conditions of the disturbance are discretized to obtain the target equation and target boundary conditions;

[0126] Step 8: Solve the eigenvalues ​​and eigenvectors based on the target equation, the target boundary conditions and the given parameters, and obtain the analysis results of the stability of the rarefied flow based on the eigenvalue and eigenvector solution results. That is, by changing the Knudsen number, Prandtl number and wave number, calculate the eigenvalues ​​and eigenvectors of the two-dimensional rarefied gas flow under specific parameters, the eigenvalues ​​correspond to the growth rate and frequency of the disturbance, and the eigenvectors correspond to the spatial distribution of the disturbance.

[0127] The following is a detailed introduction to this method:

[0128] S1: Select the Shakhov model as the initial model.

[0129] In an optional embodiment of the present specification, the Shakhov initial model is:

[0130] ; (1)

[0131] S2: In order to reduce computer storage and improve computing efficiency, a reduced distribution function is introduced:

[0132] ; (2)

[0133] ; (3)

[0134] In order to eliminate the dependence of the initial model on the molecular velocity in the vertical direction, the model equation corresponding to the reduced distribution function is obtained as follows:

[0135] , ; (4)

[0136] in:

[0137] ; (5)

[0138] ; (6)

[0139] ; (7)

[0140] S3: Under the condition that the rarefied gas satisfies the parallelism assumption, the instantaneous macroscopic quantity and the reduced distribution function in the reduced distribution function equation are written in the form of the sum of the basic flow and the disturbance quantity, and the disturbance model is simplified, wherein the parallelism assumption means that the flow physical quantity only depends on y in the Cartesian coordinate system and is independent of x and z.

[0141] Specifically, the instantaneous macroscopic quantity and the reduced distribution function are expressed as follows:

[0142] ; (8)

[0143] According to the equation of the reduced distribution function satisfied by the basic flow and ignoring the high-order terms, the small perturbation equation can be expressed as:

[0144] ; (9)

[0145] ; (10)

[0146] S4: According to the method of finding the moment of the distribution function, the relationship between the perturbation reduced distribution function and the perturbation macroscopic quantity is determined:

[0147] ; (11)

[0148] in:

[0149] ; (12)

[0150] S5: Performing canonical modal analysis on the disturbance in the disturbance model, and obtaining the product of the shape function and the wave function in the form of:

[0151] ; (13)

[0152] In an optional embodiment of the present specification, the function of the stability model is expressed as follows:

[0153] ; (14)

[0154] ; (15)

[0155] S6: Using diffuse reflection boundary conditions, derive the corresponding perturbation diffuse reflection boundary conditions under the assumption of small perturbation:

[0156] ; (16)

[0157] ; (17)

[0158] in:

[0159] ; (18)

[0160] ; (19)

[0161] S7: discretize the stability model and the preset boundary conditions using a specified method to obtain a target equation and a target boundary condition. The specified method in this specification includes a Chebyshev spectral collocation point method and a Gauss-Hermite velocity space discretization method.

[0162] The discretized form of the target stability equation is:

[0163] c1-c2-c3-c4-c5-c6-c7; (20)

[0164] Among them, c1 to c7 are all intermediate quantities;

[0165] in:

[0166] ;

[0167] ;

[0168] ;

[0169] ;

[0170] ;

[0171] ;

[0172] ;

[0173] in:

[0174] ;(twenty one)

[0175] d 1 to d 6 are all intermediate quantities;

[0176] in:

[0177] ;

[0178] ;

[0179] ;

[0180] ;

[0181] ;

[0182] ;

[0183] The discrete form corresponding to the diffuse reflection boundary condition is:

[0184] ;(twenty two)

[0185] Among them, a1, a2, a3 and a4 are all intermediate quantities;

[0186] in:

[0187] ;

[0188] ;

[0189] ;

[0190] ;

[0191] in, ;(twenty three)

[0192] Among them, b1, b2, b3 and b4 are all intermediate quantities;

[0193] in:

[0194] ;

[0195] ;

[0196] ;

[0197] ;

[0198] in, W p is the integral weight coefficient.

[0199] S8: The discrete stability equations and discrete boundary conditions can be written in the following form:

[0200] ; (twenty four)

[0201] Where: M j,k represents a function containing the Knudsen number, N j,k Velocity as a function of wave number, L j,k represents the function of the Mach number, E jm Spectral method coefficients, and Represents the reduced distribution function of different velocity space points at the same physical space point, W l Represents the weight coefficient.

[0202] Equation (24) is a homogeneous system of equations. Therefore, in general, there are only meaningless zero solutions. When the Mach number, Knudsen number, wave number or circular frequency in the equation satisfies a certain relationship, the equation may have a non-zero solution. In classical stability theory, this relationship is called the characteristic relationship, and the parameters that satisfy the characteristic relationship are called eigenvalues. The solution of the equation is called the characteristic function, and the problem of solving the eigenvalue and characteristic function is called the eigenvalue problem. Therefore, the stability problem can be transformed into the solution of the eigenvalue problem, as follows:

[0203] ; (25)

[0204] in: represents the coefficient matrix, represents the feature vector, represents the identity matrix, Represents the eigenvalue.

[0205] The eigenvalues ​​of the generalized eigenvalue problem above can be determined by the following determinant:

[0206] ; (26)

[0207] S8: The implicit restart Arnoldi algorithm is used to solve the generalized eigenvalue problem above, and the eigenvalue and special value vector can be obtained. Each eigenvalue corresponds to a perturbation mode. When the imaginary part of the eigenvalue is greater than zero, the mode is a growth mode. When the imaginary part of the eigenvalue is less than zero, the mode is a decay mode. When the imaginary part of the eigenvalue is equal to zero, the mode is neutrally stable. The spatial distribution characteristics of the disturbance can be analyzed through the characteristic function.

[0208] In this embodiment of the present invention, the method further includes:

[0209] Step 9: Based on the analysis results of the stability of the rarefied flow and the shape parameters of the aircraft model, obtain the transition prediction results of the aircraft model.

[0210] Among them, this method corrects and solves the problem that the existing technology (BGK model) can only analyze the gas flow with a Prandtl number of 1, and obtains more reliable analysis results of the stability of the rarefied flow. Then, based on the reliable analysis results of the stability of the rarefied flow, a reliable transition prediction result of the actuator model is obtained.

[0211] Among them, the method of predicting the transition of the aircraft model based on the analysis results of the stability of the rarefied flow and the aircraft model shape parameters in the present invention can adopt the existing method, and the embodiments of the present invention are not specifically limited, such as CN117408189A and CN114036869A.

[0212] Furthermore, the applicant also used a compressible flat plate Couette flow example to verify the method in this specification during the research process. The calculation conditions used in the verification are shown in Table 1, which is a calculation condition table. The applicant used a linear stability method (NS+Slip BC) based on the Navier-Stokes (NS) equation considering the slip boundary condition and the method of the present invention to calculate the phase velocity spectrum of the compressible flat plate Couette flow in the near-continuous state, in order to verify the correctness of the method of the present invention.

[0213] Table 1 Calculation conditions

[0214] Mach number Ma Knudsen number Kn Flow direction wave number α 2.0 0.001 2.0

[0215] Figure 2 The calculation results of the phase velocity spectrum of the near-continuum flow using NS+Slip BC and the method of the present invention are shown in Figure 2. Figure 2 It can be seen from the figure that the results obtained by the two calculation methods are in good agreement, thus verifying the correctness of the method in the present invention.

[0216] Although the preferred embodiments of the present invention have been described, those skilled in the art may make other changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention.

[0217] Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model, characterized in that: The method comprises: Step 1: Select the Shakhov model as the initial model; Step 2: introducing a reduced distribution function to eliminate the dependence of the distribution function in the initial model on the molecular velocity in the vertical direction, and obtaining a model about the reduced distribution function; Step 3: Under the condition that the rarefied gas satisfies the parallelism hypothesis and the small perturbation hypothesis, the instantaneous quantity in the model about the reduced distribution function is rewritten into the form of the sum of the basic flow and the perturbation quantity; under the condition that the basic flow satisfies the model about the reduced distribution function, the small perturbation equation is obtained; Step 4: Process the perturbation reduced distribution function in the small perturbation equation by moment calculation to obtain the relationship between the perturbation reduced distribution function and the perturbation macroscopic quantity; Step 5: Write the disturbance in the small disturbance equation as the product of the shape function and the wave function to obtain the stability equation; Step 6: Using diffuse reflection boundary conditions, derive the diffuse reflection boundary conditions corresponding to the disturbance under the assumption of small disturbance; Step 7: Based on the relationship between the disturbance reduced distribution function and the disturbance macroscopic quantity, the stability equation and the diffuse reflection boundary conditions of the disturbance are discretized to obtain the target equation and target boundary conditions; Step 8: solving eigenvalues ​​and eigenvectors based on the target equation, the target boundary conditions and given parameters, and obtaining analysis results of the stability of the rarefied flow based on the solution results of the eigenvalues ​​and eigenvectors; The initial model is: ； in, f Represents the distribution function of particles in time, physical space and velocity space, f =( t , r , ξ ), r represents the particle direction, r =( x , y ), x Indicates the flow direction, y represents the normal direction, ξ represents the particle speed, ξ =( ξ x , ξ y , ξ z ), ξ x , ξ y and ξ z are the velocity components of the particle velocity in the streamwise direction, normal direction and spanwise direction, respectively. τ represents the relaxation time, t Indicates time, f s represents the particle equilibrium distribution function; The reduced distribution function is: ； ； in, g ( t , x , y , ξ x , ξ y )and h ( t , x , y , ξ x , ξ y ) is the reduced distribution function, g ( t , x , y , ξ x , ξ y )= g , h ( t , x , y , ξ x , ξ y )= h ; On the Reduced Distribution Function g and h The equation corresponding to the model is: ； ； in, h s and g s is the particle Shakhov equilibrium state reduced distribution function, calculated as; ； ； Where Pr is the Prandtl number, c is the characteristic velocity of the particle, c=(c x ,c y ), c x is the velocity component of the particle’s characteristic velocity in the stream direction, c y is the velocity component of the particle’s characteristic velocity in the normal direction, q represents the heat flow vector, q=(q x ,q y ), q x is the component of the heat flux vector in the streamwise direction, q y is the component of the heat flux vector in the normal direction, p Indicates pressure, R represents the particle equilibrium constant, T Indicates temperature, D represents the spatial dimension, g e represents the particle reduced equilibrium distribution function, K represents the internal freedom degree of the molecule; ； in, ρ is the density, u =( u , v ) represents the macroscopic speed, u is the flow velocity, v is the normal velocity; The diffuse reflection boundary condition corresponding to the disturbance is: ； ； in, and represents the reduced perturbation distribution function of particles on the boundary, r w represents the position on the particle boundary, and represents the equilibrium perturbation distribution function on the particle boundary, ρ w represents the boundary density, u w represents the macroscopic velocity of the disturbance on the boundary, T w represents the temperature on the boundary, n w represents the normal vector of the boundary, ξ represents the particle velocity; in: ； ； in, is the equilibrium distribution function g e The disturbance amount, is the equilibrium distribution function g e The basic amount, is the density perturbation, ρ c is the basic quantity of density, ξ x and y are the velocity components of the particle velocity in the streamwise direction and the normal direction, R is the gas constant, u c is the basic quantity of stream velocity, is the disturbance of the stream velocity, v c is the basic quantity of normal velocity, is the perturbation of the normal velocity, is the equilibrium distribution function h e The disturbance amount, is the equilibrium distribution function h e The basic amount, T c The basic quantity that represents temperature, Represents the temperature disturbance.

2. The method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model according to claim 1, characterized in that: Under the condition that the rarefied gas satisfies the parallelism assumption and the small disturbance assumption, the instantaneous quantity in the model of the reduced distribution function is rewritten as the sum of the basic flow and the disturbance quantity, which is expressed as: ； in, t Indicates time, x Indicates the flow direction, y represents the normal direction, ξ x and ξ y are the velocity components of the particle velocity in the streamwise direction and the normal direction, μ is the viscosity, μ c ( y ) is the basic flow of viscosity, is the viscosity disturbance, p c ( y ) is the basic flow of pressure, is the density perturbation, u c ( y ) is the basic quantity of stream velocity, is the disturbance of the streamwise velocity, v is the normal velocity, is the perturbation of the normal velocity, g and h is the reduced distribution function, for g The basic amount, for g The disturbance amount, for h The basic amount, for h The disturbance amount, h s and g s is the particle Shakhov equilibrium reduced distribution function, for g s The basic amount, for g s The disturbance amount, for h s The basic amount, for h s The amount of disturbance; The small perturbation equation is: ； ； in, r represents the particle direction, μ c is the basic quantity of viscosity, is the pressure disturbance, p c is the basic quantity of pressure, is the viscosity disturbance.

3. The method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model according to claim 2, characterized in that: The perturbation in the small perturbation equation is written as the product of the shape function and the wave function, which is expressed as: ； in, e is the natural base, i is an imaginary number, a is the wave number, is the frequency, is the density perturbation, is the characteristic function of density, is the disturbance of the streamwise velocity, is the characteristic function of the streamwise velocity, is the perturbation of the normal velocity, is the characteristic function of the normal velocity, is the characteristic function of viscosity, is the characteristic function of pressure, for g The characteristic function of for h The characteristic function of for g s The characteristic function of for h s The characteristic function of The stability equation is obtained as: ； 。 4. The method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model according to claim 2, characterized in that: The perturbation-reduced distribution function in the small perturbation equation is processed by moment calculation to obtain the relationship between the perturbation-reduced distribution function and the perturbation macroscopic quantity, which is: ； in: ; in, is the density perturbation, ξ represents the particle speed, is the disturbance of the stream velocity, is the perturbation of the normal velocity, is the macroscopic energy disturbance, E c is the basic quantity of macroscopic energy, is the disturbance to the heat flow, is the disturbance of the normal heat flow, v c is the basic quantity of normal velocity, Q qxu is the first intermediate quantity, Q qxv is the second intermediate quantity, Q qyv is the third intermediate quantity, Q qyu It is the fourth intermediate quantity.

5. The method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model according to claim 1, characterized in that: The step 5 is specifically as follows: using the canonical modulus analysis method, the disturbance quantity in the small disturbance equation is written in the form of the product of the shape function and the wave function to obtain the stability equation.

6. The method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model according to claim 1, characterized in that: The step 7 is specifically as follows: based on the relationship between the disturbance reduced distribution function and the disturbance macroscopic quantity, the stability equation and the diffuse reflection boundary condition of the disturbance are discretized by a preset method to obtain the target equation and the target boundary condition; the preset method includes the Chebyshev spectral collocation point method and the Gauss-Hermite velocity space discretization method.

7. The method for analyzing the stability of two-dimensional rarefied gas flow based on the Shakhov model according to claim 1, characterized in that: The method further comprises: Step 9: Based on the analysis results of the stability of the rarefied flow and the shape parameters of the aircraft model, obtain the transition prediction results of the aircraft model.

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