An improved discrete velocity method for nonequilibrium calculations of diatomic molecular rotations

Abstract

The present invention discloses an improved discrete velocity method for the calculation of diatomic molecular rotational non-equilibrium, which relates to the field of numerical simulation of rarefied gas dynamics. The method comprises: initializing the flow field according to the actual working conditions, using the Boltzmann equation based on the Rykov model as the control equation, solving the collisionless Boltzmann equation to obtain the unit interface distribution function, calculating the macro equation flux corresponding to the local solution of the collisionless Boltzmann equation and the macro equation flux corresponding to the Navier-Stokes equation considering the molecular rotation effect, weighting the two types of fluxes according to the rarefaction of the gas to obtain the macro flux, solving the macro adjoint equation to obtain the estimated macro quantity and the estimated equilibrium distribution function, solving the microscopic control equation to update the distribution function, integrating the distribution function to update the macro quantity and judge the convergence. The present invention can achieve accurate prediction of the full-domain non-equilibrium flow field of diatomic molecular gas on the basis of maintaining the simplicity of the traditional semi-implicit discrete velocity method.

Description

Technical Field

[0001] The present invention relates to the field of numerical simulation of rarefied gas dynamics, and more particularly to an improved discrete velocity method for non-equilibrium calculation of diatomic molecule rotation. Background Art

[0002] The phenomenon of non-equilibrium flow across multiple flow domains is widely present in aerospace engineering, such as hypersonic near-space vehicles, re-entry capsules, and aircraft engines. Due to the thin atmosphere in near-space, vehicles can fly at relatively high Mach numbers, making it a strategic priority for various countries today. However, when a vehicle flies at high speed, the gas around the windward area, such as the leading edge, is highly compressed, resulting in a small molecular mean free path and a high collision frequency, exhibiting the characteristics of a continuous flow domain. In areas such as the trailing edge, the gas is in a highly rarefied state due to the "vacuum" effect. The study of near-space vehicles presents many difficulties due to the highly non-equilibrium and unstable flow fields surrounding them and the fact that they span multiple flow domains. Currently, due to the difficulty and high cost of experiments, numerical simulation is the main method used to study such problems.

[0003] Traditional macroscopic methods, such as the Euler equations or the Navier-Stokes equations, adopt the continuity assumption and assume that the flow field is always in equilibrium. They ignore the non-equilibrium effects caused by gas discontinuity and are completely unsuitable for numerical simulation of rarefied flow problems.

[0004] The Boltzmann equation, derived from the kinetic theory of gases, describes the temporal evolution of the distribution function of gas molecules. Compared to the Navier-Stokes equations, it breaks free from the limitation of the continuous medium assumption. Therefore, the Boltzmann equation is theoretically capable of describing flow problems from continuous flow to rarefied flow.

[0005] Solving the Boltzmann equation requires integration over the particle's velocity space. During simulation, the particle velocity space is usually truncated to a certain region and discretized into a finite number of velocity integration points. The resulting governing equation is usually called the discrete velocity Boltzmann equation (DVBE), and the numerical method for solving the DVBE is usually called the discrete velocity method (DVM).

[0006] To achieve better stability, traditional discrete velocity methods typically employ a semi-implicit discretization scheme for the collision term of the Boltzmann equation. This involves using macroscopic quantities at the current moment to estimate the equilibrium distribution function at the next moment, which is then used to calculate the collision term. This approach also ignores the effects of particle collisions at cell interfaces, requiring a physical mesh size smaller than the mean free path of particle collisions to ensure accurate results. Consequently, this approach suffers from low computational efficiency and poor accuracy in continuous and near-continuous flow regimes where particle collisions are frequent.

[0007] The improved discrete velocity method achieves an accurate estimation of the equilibrium state by introducing a macroscopic adjoint equation, thereby enabling a fully implicit discretization of the collision term of the Boltzmann equation. It also accounts for particle collision effects when calculating the macroscopic flux at the cell interface, resolving the aforementioned issue of mesh size being limited by the molecular mean free path. This method offers high accuracy and computational efficiency in full-scale simulations. However, this method currently only supports the numerical simulation of monatomic gas flows and has yet to address the numerical simulation of diatomic gas flows. In practical engineering applications, air, the primary research subject, consists primarily of diatomic molecules, oxygen and nitrogen. Furthermore, the rotational excitation temperatures of oxygen and nitrogen are 2.08 K and 2.07 K, respectively, indicating that the rotational degrees of freedom of molecules should be considered in most numerical simulations. In particular, in thermal nonequilibrium flows involving shock waves, the translational and rotational temperatures of certain regions within the flow field are not equal and should not be represented by the same temperature. Therefore, there is an urgent need to develop an improved discrete velocity method that can account for the rotational degrees of freedom of molecules to achieve more realistic numerical simulations of full-scale gas flows. Summary of the Invention

[0008] To solve the above problems, the present invention provides an improved discrete velocity method for non-equilibrium calculation of diatomic molecule rotation.

[0009] The technical solutions adopted in the present invention include:

[0010] (1) Initialize the flow field according to the actual working conditions to obtain the initial equilibrium distribution function and use it as the initial distribution function;

[0011] (2) Calculate the distribution function f i n , macroscopic conservation quantity The spatial derivative of .

[0012] (3) From the local solution of the collisionless Boltzmann equation, according to the distribution function f i n and its spatial derivative to calculate the unit interface distribution function

[0013] (4) According to the unit interface distribution function Calculating the macroscopic flux corresponding to the local solution of the collisionless Boltzmann equation

[0014] (5) Calculate the macroscopic flux corresponding to the Navier-Stokes equation considering the molecular rotation effect

[0015] (6) Solve the macroscopic adjoint equation, and according to the macroscopic equation flux corresponding to the local solution of the collisionless Boltzmann equation The macroscopic equation flux corresponding to the Navier-Stokes equation considering the molecular rotation effect Calculate the estimated macroscopic conservation quantity And calculate the estimated equilibrium distribution function

[0016] (7) Solve the Boltzmann-Rykov equation and calculate the equilibrium distribution function according to the estimated equilibrium distribution function and unit interface distribution function Update the distribution function f i n+1 ;

[0017] (8) Update the macroscopic quantities and judge the convergence. If it does not converge, repeat the above steps from step (2) until the convergence conditions are met to obtain the macroscopic quantities and distribution functions of the flow field.

[0018] The improved discrete velocity method for calculating the nonequilibrium effects of diatomic molecular rotation uses the Boltzmann-Rykov equation to simultaneously consider both the translational and rotational motions of the molecule. The Boltzmann-Rykov equation is expressed as follows:

[0019]

[0020]

[0021] Where f represents the distribution function, f * represents the equilibrium distribution function, f t and f r They represent the equilibrium distribution function of translation and rotation respectively, ξ represents the particle velocity, Z represents the number of rotational collisions (i.e., the ratio of the relaxation rates of elastic collisions to inelastic collisions of particles), t represents time, and τ represents the collision time.

[0022] The macroscopic conservation quantities include gas density ρ, momentum per unit volume of gas ρu, total energy per unit volume of gas ρE and rotational energy per unit volume of gas ρE rThe macroscopic conservation quantities and stress P, translational heat flow q t and the rotational heat flux q r The calculation formula is as follows:

[0023]

[0024] P = ∫ccfdΕ;

[0025]

[0026]

[0027] Where W represents the macroscopic conservation quantity, P represents stress, u represents macroscopic velocity, c = ξ - u represents the thermal motion velocity of molecules, ζ represents the rotation velocity of molecules, and dE = dξdζ represents the integral differential element in velocity space. The heat flow q is the sum of the translational heat flow and the rotational heat flow, that is, q = q t +q r . is the microscopic moment vector, that is

[0028] The unit interface distribution function ignores the collision effect and is calculated using the upwind format. The expression is as follows:

[0029]

[0030] Where, represents the unit interface distribution function, and represents the distribution function on the left and right sides of the unit interface, n ij represents the unit external normal vector of element i at the interface with element j, x ij is the center coordinate of the interface between unit i and unit j. The distribution functions on the left and right sides of the unit interface are obtained by interpolating the unit center distribution function using limiters.

[0031] The macroscopic adjoint equation is obtained by momenting the Boltzmann-Rykov equation in the particle velocity space, and the expression is as follows:

[0032]

[0033]

[0034] Where F represents the macroscopic flux, and S represents the source term generated after the introduction of rotational freedom.

[0035] The flux of the macroscopic adjoint equation is obtained by solving the Boltzmann-Rykov equation locally at the cell interface, and the expression is as follows:

[0036]

[0037]

[0038] Where, F c and F v denote the inviscid flux and viscous flux respectively, F DVM represents the macroscopic equation flux contributed by the local solution of the collisionless Boltzmann equation, F NS It represents the macro equation flux contributed by the Navier-Stokes equation considering the molecular rotation effect. p is the local physical time step, determined by the CFL condition.

[0039] The macroscopic flux contributed by the local solution of the collisionless Boltzmann equation is calculated by the unit interface distribution function without considering molecular collisions. It is obtained by calculating the moment in the particle velocity space.

[0040] The macro equation flux contributed by the Navier-Stokes equation considering the molecular rotation effect is directly calculated from the macro quantity.

[0041] The microscopic control equation, i.e. the fully implicit discretized Boltzmann-Rykov equation, is expressed as follows:

[0042]

[0043] Where, subscript i represents unit i, subscript ij represents the interface between unit i and unit j, superscript n and n+1 represent the time step, j∈N(i) represents the unit around unit i, V i represents the volume of unit i, n ij represents the unit external normal vector of unit i on the interface with unit j, S ij represents the interface area between unit i and unit j, represents the equilibrium distribution function of unit i at time n+1 estimated by the macroscopic adjoint equation, Represents the distribution function on the interface between unit i and unit j at time n+1.

[0044] Preferably, the equation can be solved using flux splitting and the LU-SGS (Lower-Upper Symmetric Gauss-Seidel) implicit method.

[0045] The macroscopic adjoint equation is discretized using the finite volume method, and the discretized expression is as follows:

[0046]

[0047] Where, Represents the increment of macroscopic conserved quantity.

[0048] The inviscid flux can be directly calculated using a traditional CFD flux scheme, preferably the Lattice Boltzmann Flux Scheme (LBFS). The viscous flux is obtained by central difference, and the source term is calculated based on the physical quantity at the cell center.

[0049] Optionally, the LU-SGS implicit algorithm can be used to speed up the convergence when solving the macroscopic control equations.

[0050] Compared with the prior art, the present invention has at least the following effects: (1) Compared with the improved discrete velocity method for monatomic gas molecules, the present invention takes into account the rotational non-equilibrium effect of diatomic molecules during simulation, which is closer to the real atmospheric conditions and can better simulate thermal non-equilibrium flow problems. (2) It overcomes the defects of the traditional semi-implicit discrete velocity method, while maintaining the simplicity of calculation, and improves the solution accuracy and efficiency in continuous flow and near-continuous flow domains. (3) Compared with the traditional semi-implicit discrete velocity method, by solving the macroscopic adjoint equation to predict the equilibrium state, the collision term is fully implicitly discretized, which accelerates the convergence speed. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 1 is a schematic diagram of the steps of an improved discrete velocity method for calculating the non-equilibrium effect of diatomic molecule rotation provided by an embodiment of the present invention;

[0052] Figure 2 3 is a graph showing the equilibrium temperature and rotational temperature distribution of a shock tube at t=0.2s under different Knudsen numbers provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0053] The following describes the embodiments of the present invention in detail with reference to the accompanying drawings. The embodiments are provided for illustrative purposes only and are not to be construed as limiting the present invention. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without creative work fall within the scope of protection of the present invention.

[0054] The present invention provides an improved discrete velocity method for non-equilibrium calculation of diatomic molecular rotation, such as Figure 1 As shown, the method includes the following steps:

[0055] (1) Initialize the flow field according to the actual working conditions to obtain the initial equilibrium distribution function and use it as the initial distribution function;

[0056] An improved discrete velocity method suitable for calculating the non-equilibrium effect of diatomic molecular rotation provided by an embodiment of the present invention gives an initial flow field according to actual working conditions, calculates the equilibrium distribution function based on the macroscopic quantities of the initial flow field, and uses the equilibrium distribution function as the initial value of the distribution function.

[0057] Optionally, for complex flow problems, the results obtained by solving macroscopic equations can be used for initialization.

[0058] Among them, the microscopic control equation, namely the Boltzmann-Rykov equation, is expressed as follows:

[0059]

[0060]

[0061] Where f represents the distribution function, f * represents the equilibrium distribution function, f t and f r They represent the equilibrium distribution functions of translation and rotation respectively, ξ represents the particle velocity, Z represents the number of rotational collisions (i.e., the ratio of the relaxation rates of elastic collisions to inelastic collisions of particles), t represents time, and τ represents the collision time.

[0062] The equilibrium distribution function of the translation and the equilibrium distribution function of the rotation are expressed as follows:

[0063]

[0064]

[0065] Where ρ represents density, K represents the number of rotational degrees of freedom, u represents macroscopic velocity, c = ξ-u represents the thermal motion velocity of molecules, ζ represents the rotation velocity of molecules, and q t and q r They represent translational heat flow and rotational heat flow respectively. The heat flow q is the sum of translational heat flow and rotational heat flow, that is, q = q t +q r ,λ t ,λ r and λ eq and the translation temperature T t , rotation temperature T r and equilibrium temperature T eq Related, that is, λ t,r,eq =1 / 2RT t,r,eq , σ, ω0 and ω1 are relevant parameters in the model and are related to the gas properties.

[0066] (2) Calculate the distribution function f i n , macroscopic conservation quantity The spatial derivative of .

[0067] The distribution function f i n , macroscopic conservation quantity The spatial derivative of is calculated based on the distribution function of the adjacent unit centers, macroscopic conservation quantities, and coordinates.

[0068] (3) From the local solution of the collisionless Boltzmann equation, according to the distribution function f i n and its spatial derivative to calculate the unit interface distribution function

[0069] The unit interface distribution function ignores molecular collisions and is calculated using the upwind format. The expression is as follows:

[0070]

[0071] Where, represents the unit interface distribution function, and represents the distribution function on the left and right sides of the unit interface, n ij represents the unit external normal vector of element i at the interface with element j, x ij is the center coordinate of the interface between unit i and unit j. The distribution functions on the left and right sides of the unit interface can be obtained by interpolating the unit center distribution function using limiters. The limiters can be selected according to the requirements.

[0072] (4) According to the unit interface distribution function Calculating the macroscopic flux corresponding to the local solution of the collisionless Boltzmann equation

[0073] The macroscopic equation flux corresponding to the local solution of the collisionless Boltzmann equation is By considering the unit interface distribution function without considering molecular collision The moment is obtained in the velocity space, and the calculation formula is as follows:

[0074]

[0075] Where dΕ=dξdζ represents the integral differential element in the velocity space.

[0076] (5) Calculate the macroscopic flux corresponding to the Navier-Stokes equation considering the molecular rotation effect

[0077] The macroscopic equation flux corresponding to the Navier-Stokes equation considering the molecular rotation effect is According to the macroscopic adjoint equation considering the rotation effect, the macroscopic adjoint equation and the viscous flux F v , inviscid flux F c , the expression of the source term S is as follows:

[0078]

[0079] F NS =F c +F v ;

[0080] F c =[ρu,ρuu+pI,(ρE+p)u,ρE r u] T ;

[0081]

[0082]

[0083]

[0084] Where W represents the macroscopic conservation quantity, ρu represents the momentum of gas per unit volume, ρE represents the total energy of gas per unit volume, and ρE r Represents the rotational energy per unit volume of gas, E r,eq represents the rotational energy at equilibrium, p represents the pressure, R represents the gas molecular constant, and I represents the unit tensor. represents the viscous stress tensor, represent heat flux, translational heat flux, and rotational heat flux, respectively, and are expressed as follows:

[0085]

[0086]

[0087]

[0088]

[0089] Where μ represents the viscosity coefficient, which is only related to the translation temperature and is expressed as follows:

[0090]

[0091] Where, T ref and μ ref denote the reference temperature and reference viscosity, respectively, and ω denotes the coefficient related to the intermolecular collision model.

[0092] (6) Solve the macroscopic adjoint equation, and according to the macroscopic equation flux corresponding to the local solution of the collisionless Boltzmann equation The macroscopic equation flux corresponding to the Navier-Stokes equation considering the molecular rotation effect Calculate the estimated macroscopic conservation quantity And calculate the estimated equilibrium distribution function

[0093] The macroscopic adjoint equation is spatially discretized using the finite volume method and is expressed as follows:

[0094]

[0095]

[0096]

[0097] Where, represents the increment of macroscopic conservation quantity, j∈N(i) represents the unit around unit i, represents the macroscopic flux at the unit interface, V i represents the volume of unit i, S ij Represents the area of ​​the interface between unit i and unit j. Δt p is the local physical time step, determined by the CFL condition.

[0098] The inviscid flux of the macroscopic adjoint equation is directly calculated using a traditional CFD equation flux scheme, preferably the Lattice Boltzmann Flux Scheme (LBFS). The viscous flux is obtained by central difference, and the source term is calculated from the physical quantity at the cell center.

[0099] Optionally, the LU-SGS implicit method can be used to solve the macroscopic adjoint equation to speed up convergence.

[0100] The estimated equilibrium state distribution function at the new moment Using estimated macroscopic conservation quantities It is calculated by the equilibrium distribution function expression in step (1).

[0101] (7) Solve the Boltzmann-Rykov equation and calculate the equilibrium distribution function according to the estimated equilibrium distribution function and unit interface distribution function Update distribution function

[0102] The microscopic control equation is expressed as follows:

[0103]

[0104] Introducing increments The above equation can be rewritten as:

[0105]

[0106]

[0107] Preferably, the equation can be solved using the LU-SGS implicit method to simplify the calculation. The expression is as follows:

[0108]

[0109]

[0110]

[0111] Among them, D i and D j The expression is as follows:

[0112]

[0113]

[0114] (8) Update the macroscopic quantities and judge the convergence. If it does not converge, repeat the above steps from step (2) until the convergence conditions are met to obtain the macroscopic quantities and distribution functions of the flow field.

[0115] The macroscopic conservation quantities, as well as the expressions for stress, translational heat flow, and rotational heat flow are as follows:

[0116]

[0117] P = ∫ccfdΕ;

[0118]

[0119]

[0120] Where P represents stress, represents the microscopic moment vector, that is Total energy per unit volume ρE and rotational energy per unit volume ρE r The expression is as follows:

[0121]

[0122]

[0123] Compare the updated macro-conservation quantity with the macro-conservation quantity of the previous step to determine whether it converges. If so, stop the iteration; otherwise, return to step (2) to continue the iteration.

[0124] To facilitate understanding of the improved discrete velocity method for calculating the rotational non-equilibrium effect of diatomic molecules proposed in the embodiment of the present invention, and to verify the accuracy and full-flow-domain characteristics of the present invention, this embodiment is analyzed and explained in detail through the following examples.

[0125] The one-dimensional nitrogen shock tube is a classic example for verifying the ability of numerical simulation methods to handle complex nonequilibrium flow problems. It places high demands on the stability, accuracy, and computational efficiency of the numerical simulation methods, and the existence of multiple reference solutions makes it an excellent example for verifying the effectiveness of this embodiment in simulating flow problems involving the rotational nonequilibrium effects of diatomic molecules.

[0126] This example first sets the initial working condition, and the physical field settings are as follows:

[0127] Different Knudsen numbers Kn = 0.001, 0.01, 0.1, 1, the rotational collision number Z = 2.4, and the global CFL number 0.8. The physical space is x∈[0,1], uniformly discretized into 200 grid cells, and the initial values ​​of the macroscopic physical quantities are as follows:

[0128]

[0129] The velocity space is [-10, 10], which is evenly discretized into 201 velocity points, and the Newton-Cotes integral is used to calculate their weights.

[0130] In this example, the hard sphere model is used to calculate the gas viscosity coefficient (ω = 1). The local molecular collision time and reference viscosity coefficient are expressed as follows:

[0131]

[0132]

[0133] Where, ρ ref represents the reference density, L represents the characteristic length, and Kn represents the Knudsen number. The reference temperature is T ref =1.

[0134] Figure 2 The distribution curves of equilibrium temperature and rotation temperature at t = 0.2s are shown for different Knudsen numbers. Figure 2 It can be seen that with the increase of the Knudsen number, the rarefaction effect is enhanced, and the gap between the equilibrium temperature and the rotational temperature gradually widens, indicating that the rotational non-equilibrium effect is also enhanced; the calculation results of the current algorithm are in good agreement with the results of the unified gas kinetic scheme (UGKS) and the multiscale discrete velocity method (MDVM) used as reference solutions, verifying that the present invention has the ability to simulate the flow containing the rotational non-equilibrium effect of diatomic molecules in the entire flow domain.

[0135] The above-described embodiments merely represent several preferred implementations of the present invention, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the patent. It should be noted that a person skilled in the art can make several improvements and substitutions without departing from the technical principles of the present invention, and such improvements and substitutions should also be considered within the scope of protection of this application. Therefore, the scope of protection of this patent application shall be based on the scope of protection of the claims.

Claims

1. An improved discrete velocity method for nonequilibrium calculations of diatomic molecular rotations, characterized in that: The Boltzmann-Rykov equation is used as the control equation and the corresponding equilibrium distribution function is used to take into account the rotation of molecules and better simulate real gases. The macro equation flux is divided into two parts: the macro equation flux corresponding to the local solution of the collisionless Boltzmann equation and the macro equation flux corresponding to the Navier-Stokes equation considering the molecular rotation effect. This is to consider the collision effect of the unit interface and improve the accuracy and efficiency in continuous and near-continuous flow domains. The local solution of the collisionless Boltzmann equation is used to obtain the unit interface distribution function. The Navier-Stokes equation considering the molecular rotation effect is used to obtain its corresponding macro equation flux. The macro quantities are updated by calculating the integral moment of the distribution function. The specific steps are as follows: (1) Initialize the flow field according to the actual working conditions to obtain the initial equilibrium distribution function and use it as the initial distribution function; (2) Calculate the distribution function f i n , macroscopic conservation quantity The spatial derivative of (3) From the local solution of the collisionless Boltzmann equation, according to the distribution function f i n and its spatial derivative to calculate the unit interface distribution function (4) According to the unit interface distribution function Calculating the macroscopic flux corresponding to the local solution of the collisionless Boltzmann equation (5) Calculate the macroscopic flux corresponding to the Navier-Stokes equation considering the molecular rotation effect (6) Solve the macroscopic adjoint equation, and according to the macroscopic equation flux corresponding to the local solution of the collisionless Boltzmann equation The macroscopic equation flux corresponding to the Navier-Stokes equation considering the molecular rotation effect Calculate the estimated macroscopic conservation quantity And calculate the estimated equilibrium distribution function (7) Solve the Boltzmann-Rykov equation and calculate the equilibrium distribution function according to the estimated equilibrium distribution function and unit interface distribution function Update the distribution function f i n+1 ; (8) Update the macroscopic quantities and judge the convergence. If it does not converge, repeat the above steps from step (2) until the convergence conditions are met to obtain the macroscopic quantities and distribution functions of the flow field; The Boltzmann-Rykov equation is expressed as: Where f represents the distribution function, f * represents the equilibrium distribution function, f t and f r They represent the equilibrium distribution function of translation and rotation respectively, ξ represents the particle velocity, Z represents the number of rotational collisions, that is, the ratio of the relaxation rates of elastic collisions to inelastic collisions of particles, t represents time, and τ represents the particle collision time.

2. The improved discrete velocity method for rotational non-equilibrium calculation of diatomic molecules according to claim 1, characterized in that: The macroscopic conserved quantities include gas density, momentum per unit volume of gas, total energy per unit volume of gas, and rotational energy. The gas density, momentum per unit volume of gas, total energy per unit volume of gas, and rotational energy are calculated using the distribution function as follows: Where W represents the macroscopic conservation quantity, ρ represents the gas density, u represents the macroscopic velocity, ρE and ρE r are the total energy and rotational energy of the gas per unit volume, respectively, dΕ=dξdζ represents the integral differential element in the velocity space; is the microscopic moment vector, that is where ζ represents the rotational speed of the molecule.

3. The improved discrete velocity method for rotational non-equilibrium calculation of diatomic molecules according to claim 1, characterized in that: The evolution equation of its updated distribution function is: Where, subscript i represents unit i, subscript ij represents the interface between units i and j, superscript n and n+1 represent the time step, V i represents the volume of unit i, j∈N(i) represents the unit around unit i, n ij represents the unit external normal vector of unit i on the interface with unit j, S ij represents the area of ​​the interface between unit i and unit j, represents the equilibrium distribution function of unit i at time n+1 estimated by the macroscopic adjoint equation, represents the distribution function on the interface between unit i and unit j at time n+1; Introducing increments The distribution function evolution equation is rewritten as:

4. The improved discrete velocity method for rotational non-equilibrium calculation of diatomic molecules according to claim 1, characterized in that: The expression of the interface flux F is as follows: Where Δt p is the local physical time step, determined by the CFL condition, F DVM represents the macroscopic flux contributed by the local solution of the collisionless Boltzmann equation, F NS represents the macro equation flux contributed by the Navier-Stokes equation considering the molecular rotation effect, F c and F v denote the inviscid flux and viscous flux respectively, Represents the unit interface distribution function when particle collisions are not considered.

5. The improved discrete velocity method for rotational non-equilibrium calculation of diatomic molecules according to claim 1, characterized in that: The Navier-Stokes equation considering the molecular rotation effect is as follows: F NS =F c +F v ; F c =[ρu,ρuu+pI,(ρE+p)u,ρE r u] T ; Where W represents macroscopic variables, S represents source term, p represents pressure, I represents unit tensor, and E represents r,eq represents the rotational energy at equilibrium, R represents the gas molecule constant, represents the viscous stress tensor, They represent heat flux, translational heat flux, and rotational heat flux, respectively. K represents the rotational degree of freedom, and T eq represents the equilibrium temperature.

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