Method and apparatus for optimizing two-phase flow gas-kinetic algorithm, and device and storage medium

Abstract

Disclosed in the present invention are a method and apparatus for optimizing a two-phase flow gas-kinetic scheme algorithm, and a device and a storage medium. The method comprises: constructing a gas-kinetic scheme algorithm for phase-field solving; on the basis of Chapman-Enskog analysis, constructing a phase-field interface flux of the phase-field gas-kinetic scheme algorithm; constructing a gas-kinetic scheme algorithm for flow-field solving; on the basis of the Chapman-Enskog analysis, constructing a flow-field interface flux of the flow-field gas-kinetic scheme algorithm; and on the basis of a Cahn-Hilliard equation and a Navier-Stokes equation, implementing a two-phase flow numerical algorithm for coupling a phase field and a flow field. In the present invention, starting from a Boltzmann-BGK equation, the gas-kinetic scheme algorithm for coupling the phase field and the flow field is constructed within a finite volume framework, and by means of Chapman-Enskog expansion, numerical fluxes can be accurately recovered to macroscopic equations, thereby simplifying the computation process, and significantly improving the computation efficiency and accuracy. The method can effectively simulate interface dynamics and flow-field evolution in a complex two-phase flow system.

Description

Two-phase flow gas dynamics algorithm optimization method, apparatus, equipment and storage medium Technical Field

[0001] This invention belongs to the field of fluid mechanics, specifically relating to a two-phase flow gas dynamics algorithm optimization method, apparatus, equipment, and storage medium. Background Technology

[0002] Two-phase flow problems are widespread in nature and engineering applications, particularly in energy development and utilization, and carbon capture, utilization, and storage. Their complex physical phenomena and fundamental laws have a significant impact on engineering design and optimization. Currently, common numerical simulation methods for two-phase flows are mostly based on traditional macroscopic Navier-Stokes equations or discrete velocity-based mesoscopic methods such as the LBM. However, these methods have limitations in capturing two-phase interfaces and simulating complex flow fields, especially in accurately describing phase interfaces and simulating complex multiphase flows. The Gas Kinetic Scheme (GKS), a finite volume numerical method based on the mesoscopic Boltzmann equations, possesses advantages such as high accuracy, good stability, and strong mesh flexibility, and has been widely used in numerical simulations of compressible flows. However, existing GKS methods still have shortcomings when dealing with two-phase flows with significant density field variations, especially in applications involving interfacial dynamics, and cannot effectively capture and describe complex phase interfaces. This limitation restricts the widespread application of GKS methods in the field of multiphase flows. Summary of the Invention

[0003] In view of the shortcomings of the prior art, the purpose of this invention is to provide a two-phase flow gas kinetics algorithm optimization method, apparatus, device and storage medium to solve the problems existing in the prior art.

[0004] According to one aspect of this application, a two-phase flow gas kinetics algorithm optimization method is disclosed, the method comprising:

[0005] A gas kinetic scheme algorithm for phase field solution is constructed, wherein the gas kinetic scheme algorithm for phase field solution includes a continuous Boltzmann-BGK equation with source terms;

[0006] The phase field interface flux is constructed based on the gas kinetic scheme algorithm for solving the phase field using Chapman-Enskog analysis, wherein the phase field interface flux can be recovered to the Cahn-Hilliard phase field equation.

[0007] A gas kinetic scheme algorithm for solving flow fields is constructed, which includes continuous Boltzmann-BGK equations with source terms;

[0008] The flow field interface flux is constructed based on the gas kinetic scheme algorithm for solving the flow field using Chapman-Enskog analysis, wherein the flow field interface flux can be recovered to the Navier-Stokes equations.

[0009] Based on the Cahn-Hilliard equation and the Navier-Stokes equation, a numerical algorithm for two-phase flow is realized by coupling the phase field and the flow field.

[0010] In some embodiments, the gas kinetic scheme algorithm for solving the phase field includes a continuous Boltzmann-BGK equation with a source term, embodied in the following formula:

[0011] In the formula, c is the velocity of the particle distribution function g, and τ g Let G be the relaxation time, and G be the phase field source term.

[0012] In some embodiments, the phase-field interface flux of the gas kinetic scheme algorithm for solving the phase field is constructed based on Chapman-Enskog analysis, wherein the phase-field interface flux is recovered to the Cahn-Hilliard phase-field equations by:

[0013] Obtain the phase field equilibrium distribution function and phase field source term for each moment that satisfy the first condition;

[0014] The equilibrium distribution function and the moments of the source terms satisfy the following conditions;

[0015] In the formula, η is the relaxation time adjustment coefficient, and the mobility can be expressed as: parameter Set it to 0, B α C0 and C0 are coefficients to be determined. First, we use... The operator is used to obtain the expression for the phase field equilibrium distribution function;

[0016] The phase field equilibrium distribution function, based on the Chapman-Enskog representation, is as follows:

[0017] Taking the zeroth moment of the phase field equilibrium distribution function, we obtain the zeroth-order flux equation;

[0018] The zeroth-order flux equation is:

[0019] In the formula, F = <c α ,g>,F is the phase field interface flux of the gas kinetic scheme algorithm for phase field solution;

[0020] Substituting the phase field equilibrium distribution function and the moments of the phase field source terms into the zeroth-order flux equation yields the first-order phase field flux.

[0021] The first-order phase field flux is:

[0022] The first-order phase field flux is expanded using the Chapman-Enskog method to restore the phase field interface flux to the Cahn-Hilliard phase field equation.

[0023] In some embodiments, the gas kinetic scheme algorithm for solving the flow field includes a continuous Boltzmann-BGK equation with a source term, embodied in the following formula:

[0024] In the formula, τ f Let S be the relaxation time, S be the source term of the flow field, and c be the velocity of the particle distribution function f.

[0025] f eq =ρΓ(u,U) is the equilibrium distribution function;

[0026] ρ is the density of the gas, ρ = <f>;

[0027] The expression for Γ(u,U) is:

[0028] In some embodiments, the flow field source term is represented as:

[0029] The flow field interfacial flux is constructed based on the gas kinetic scheme algorithm for solving the flow field using Chapman-Enskog analysis, wherein the flow field interfacial flux is restored to the Navier-Stokes equations by:

[0030] Sure The moments of the equilibrium distribution function of the flow field and the moments of the source terms of the flow field at that time:

[0031] The moments of the equilibrium distribution function of the flow field are:

[0032] The moments of the flow field source terms are:

[0033] The zeroth and first moments of the gas kinetic scheme algorithm for solving the flow field are obtained based on Chapman-Enskog analysis.

[0034] The zeroth moment of the gas kinetic scheme algorithm for solving the flow field is:

[0035] The first moment of the gas kinetic scheme algorithm for solving the flow field is:

[0036] Define a new distribution function:

[0037] in,

[0038] Based on the new distribution function, the expression for the Boltzmann-BGK, the moments of the equilibrium distribution function of the flow field, and the moments of the source term of the flow field are determined.

[0039] The expression for Boltzmann-BGK is:

[0040] The moments of the equilibrium distribution function of the flow field are:

[0041] The moments of the flow field source terms are:

[0042] Substituting the equilibrium distribution function of the flow field and the moments of the source term of the flow field into the zero moment, we obtain the first-order flow flux.

[0043] Substituting the equilibrium distribution function of the flow field and the moments of the source term of the flow field into the first moment, the second-order flow flux is obtained.

[0044] The first-order and second-order flow field fluxes are expanded using the Chapman-Enskog method to reduce the flow field interface fluxes to the Navier-Stokes equations.

[0045] According to one aspect of this application, a two-phase flow gas kinetics algorithm optimization device is also disclosed, the device comprising:

[0046] The first construction module is used to construct a phase field gas kinetic scheme algorithm for phase field solving, wherein the phase field gas kinetic scheme algorithm includes a continuous Boltzmann-BGK equation with source terms.

[0047] The first analysis module is used to construct the phase field interface flux of the gas kinetic scheme algorithm for solving the phase field based on Chapman-Enskog analysis, wherein the phase field interface flux can be recovered to the Cahn-Hilliard equation.

[0048] The second construction module is used to construct a gas kinetic scheme algorithm for solving the flow field, wherein the gas kinetic scheme algorithm for solving the flow field includes a continuous Boltzmann-BGK equation with a source term;

[0049] The second analysis module is used to construct the flow field interface flux of the gas kinetic scheme algorithm for solving the flow field based on Chapman-Enskog analysis, wherein the flow field interface flux can be recovered to the Navier-Stokes equations.

[0050] The coupling module is used to implement a two-phase flow numerical algorithm based on the Cahn-Hilliard equation and the Navier-Stokes equation by coupling the phase field flow field.

[0051] According to another aspect of this application, an electronic device is also disclosed, the electronic device including a memory and at least one processor, the memory storing instructions; the at least one processor invokes the instructions in the memory to cause the electronic device to perform various steps of the two-phase flow gas kinetics algorithm optimization method as described above.

[0052] According to another aspect of this application, a computer-readable storage medium is also disclosed, wherein instructions are stored on the computer-readable storage medium, characterized in that, when executed by a processor, the instructions implement the various steps of the two-phase flow gas kinetics algorithm optimization method as described in any of the preceding claims.

[0053] The present invention includes, but is not limited to, the following beneficial effects: Starting from the Boltzmann-BGK equation, the present invention constructs a gas kinetic scheme algorithm for phase-field and flow-field coupling within a finite volume framework. Through Chapman-Enskog expansion, it can accurately recover the numerical flux of the macroscopic equation, simplifying the calculation process and significantly improving the calculation efficiency and accuracy. This method can effectively simulate the interface dynamics and flow field evolution in complex two-phase flow systems. Attached Figure Description

[0054] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below.

[0055] Figure 1 is a flowchart of a two-phase flow gas kinetics algorithm optimization method according to an embodiment of this application;

[0056] Figure 2 shows the density distribution along the center line of a stationary droplet;

[0057] Figure 3 shows the verification of Laplace's law for stationary droplets under different surface tensions.

[0058] Figure 4 shows the interface distribution of the three-dimensional Rayleigh-Taylor instability at different times (At = 0.5, Re = 128);

[0059] Figure 5. The positions of the bubble tip, saddle point, and spike in the three-dimensional Rayleigh-Taylor instability bubble change over time (At = 0.5, Re = 1024).

[0060] Figure 6 is a structural block diagram of the two-phase flow gas kinetics algorithm optimization device according to an embodiment of this application;

[0061] Figure 7 is a diagram of an electronic device. Detailed Implementation

[0062] The terms "first," "second," "third," "fourth," etc. (if present) in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" or "having" and any variations thereof are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0063] For ease of understanding, the specific process of the embodiments of the present invention is described below. Specifically, Figure 1 is a flowchart of a two-phase flow gas kinetics algorithm optimization method of the present application, the steps of which include:

[0064] S100. Construct a gas kinetic scheme algorithm for phase field solution, which includes a continuous Boltzmann-BGK equation with source terms.

[0065] S102. Phase field interface fluxes based on the gas kinetic scheme algorithm for phase field solution constructed by Chapman-Enskog analysis, wherein the phase field interface fluxes can be recovered to the Cahn-Hilliard equation.

[0066] S104. Construct a gas kinetic scheme algorithm for solving the flow field. The gas kinetic scheme algorithm for solving the flow field includes a continuous Boltzmann-BGK equation with source terms.

[0067] S106. The flow field interface flux is constructed based on the gas kinetic scheme algorithm for solving the flow field using Chapman-Enskog analysis, wherein the flow field interface flux can be recovered to the Navier-Stokes equation.

[0068] S108, based on the Cahn-Hilliard equation and the Navier-Stokes equation, implements a numerical algorithm for two-phase flow by coupling the phase field and the flow field.

[0069] It is understood that the gas kinetic scheme algorithm for phase-field coupling, which is constructed under the framework of finite volume based on the Boltzmann-BGK equation, can accurately recover the numerical flux of the macroscopic equation through Chapman-Enskog expansion, simplifying the calculation process and significantly improving the calculation efficiency and accuracy. This method can effectively simulate the interface dynamics and flow field evolution in complex two-phase flow systems.

[0070] The following section will provide a detailed explanation of the proposed solution using specific formulas:

[0071] Given that the method proposed in this invention is based on the two-phase flow GKS method using a phase-field model, this section first introduces the phase-field model, then describes how to construct the numerical fluxes of the phase field and flow field under the GKS scheme, and finally demonstrates the simulation of complex two-phase flow problems by coupling the phase field and flow field. The technical solution of this invention is described in detail below:

[0072] In the phase-field theory of two-phase systems, different fluid phases are determined by order parameters. To identify this, the order parameter is closely related to the system's free energy. For an isothermal binary fluid system, the total free energy consists of two parts: bulk free energy and interfacial free energy.

[0073] Where V is the control volume, and the bulk free energy can be expressed by the double potential well function ψ(φ) = β(φ - φ). A ) 2 (φ-φ B ) 2 This indicates that κ and β are parameters related to surface tension σ and interface thickness W, respectively, and their expressions are as follows:

[0074] In the formula φ A φ B These are the gas phase and liquid phase order coefficients, respectively. The phase interface can be represented as (φ). A -φ B The contour lines are horizontal, representing 1 / 2. The chemical potential μ is defined as the variational derivative of the system's free energy with respect to the order parameter.

[0075] When the system is in equilibrium, the chemical potential is constant, especially for planar interfaces, and the distribution of the order coefficients can be described as follows:

[0076] Where z is the coordinate perpendicular to the interface. The key to capturing the two-phase fluid interface lies in the calculation of the order coefficients, and the macroscopic governing equations for capturing the phase interface can be described by the Cahn-Hilliard equations.

[0077] Where u is the fluid velocity and M is the migration coefficient, the Cahn-Hilliard equations satisfy the properties of mass conservation and energy not increasing with time. In incompressible two-phase flow, the flow field variation can be described by the Navier-Stokes equations:

[0078]

[0079] In two-phase flow, the density of the fluid can be calculated using order parameters:

[0080] Where ρ1 and ρ2 are the densities of the two-phase fluid, p is the hydrodynamic pressure, and v is the kinematic viscosity. Let F be the Navier-Stokes strain rate tensor of the velocity gradient, and ξ be the volume viscosity coefficient. total It is the total force, including surface tension F. s Other possible volume forces. There are different forms of surface tension; here we adopt the form of a potential function, which can reduce the generation of spurious velocities.

[0081] Furthermore, this can be achieved by redefining the pressure p. h =p / ρ, thus obtaining the nonconservative momentum equation based on velocity.

[0082] In mesoscopic methods, the particle velocity distribution function is typically used to characterize the probability distribution of particle velocities at a given time position. This distribution satisfies the Boltzmann-BGK equation.

[0083] Where f is the velocity distribution function, f eq Let τ be the equilibrium velocity distribution function and τ be the relaxation time. The NS equations described above can be accurately recovered using Chapman-Enskog analysis. The technical solution proposed in this invention starts from the BGK equations containing external force terms and constructs numerical fluxes through Chapman-Enskog analysis, yielding the corresponding macroscopic equations, including the Cahn-Hilliard phase-field equations and the NS equations, i.e., the Low Mach Number Approximation (LMNA) method. The construction of numerical fluxes will be described below.

[0084] First, we construct an aerodynamic scheme for solving the phase field, starting with the continuous Boltzmann-BGK equations containing source terms.

[0085] Specifically, a detailed description of steps S100-S102 is provided below:

[0086] The gas kinetic scheme algorithm for phase field solving includes the continuous Boltzmann-BGK equations with source terms, based on the following formula:

[0087] Where c is the velocity of the particle distribution function g, and τ g Let G be the relaxation time, and G be the phase field source term.

[0088] Furthermore, in order to accurately recover the Cahn-Hilliard equations, the equilibrium distribution function and the moments of each phase source term satisfy the following conditions:

[0089] In the formula, η is the relaxation time adjustment coefficient, and the mobility can be expressed as: parameter Set it to 0, B α C0 and C0 are coefficients to be determined. First, we use... The operator yields an expression for the distribution function, which can be approximated by the Chapman-Enskog formula as the equilibrium state and the source term.

[0090] Then, taking the zeroth moment of the above equation, we get: In the formula F = <c α Let g> be the phase-field interface flux of GKS. Substituting the moments of the phase-field distribution function and the phase-field source term into the equation yields the first-order phase-field flux.

[0091] By performing a Chapman-Enskog expansion on the first-order phase flux in the above equations, the constructed expression for the first-order phase flux can be restored to the Cahn-Hilliard equations. The GKS flux equation can be obtained from equation (13).

[0092] Where n jk Let be the unit normal vector at the center jk of the interface between element j and element k. The distribution function is then subjected to a Taylor expansion with second-order time precision at t=0, yielding the position-time distribution function:

[0093] And perform Chapman-Enskog expansion based on the expression of the distribution function.

[0094] Then, combining the distribution function expression in the above formula, we can further obtain the following representation of the distribution function:

[0095] Integrating the flux over time, we can finally obtain the flux expression for the phase-field equation as follows:

[0096] Where φ h and These are all obtained by performing a half-time step evolution on the equation, and are approximations of... The time step Δt in GKS is given by the CFL condition:

[0097] Where Δx is the grid step size.

[0098] Since the diffusion term in the CH (Cahn-Hilliard) equation includes a gradient of chemical potential, it is necessary to calculate the fourth derivative of the order parameter. To characterize the influence of nearby units on the target unit in the diffusion term, this paper calculates this term using the following formula.

[0099] The above process details how to construct the interface flux of the GKS. Based on this flux formula, a numerical example of calculating the phase field component using the GKS algorithm can be presented. The gas kinetic scheme algorithm based on the finite volume scheme framework is as follows:

[0100] (1) Determine the computational domain, divide or read grid information, fluid parameters and control parameters.

[0101] (2) Based on the information read, an array is allocated, the corresponding parameters of the computational region are initialized, and after completion, the time step loop is entered. In the gas kinetic scheme, macroscopic quantities will be directly evolved.

[0102] (3) Set boundary conditions and assign values ​​to the corresponding macroscopic quantities of all boundaries in the flow field. This paper uses a virtual grid to process the boundaries of the flow field.

[0103] (4) Solve the Navier-Stokes equations for the flow field. First, calculate the pressure and velocity (or momentum) flux at the interface of the flow field unit. Then, update the macroscopic quantities within the unit using the flux information at the interface. If there is an external force, apply the external force through a splitting method and update the macroscopic quantities at the same time.

[0104] (5) Determine if the time step meets the loop condition, output the data, and end the loop.

[0105] The above process constructs a flux based on the GKS scheme based on the Cahn-Hilliard phase field theory, and an interface capture GKS method based on phase field solution can be realized based on this flux. The interface flux is given by the first-order Chapman-Enskog approximation, and the method has second-order time accuracy.

[0106] Next, starting from the continuous Boltzmann-BGK equations, a low Mach approximation numerical flux for the flow field will be constructed within a finite volume framework. This flux can be recovered to the Navier-Stokes equations through the Chapman-Enskog approximation expansion. Based on the assumption of near-incompressible flow at constant density, the momentum-based flux is simplified to a velocity-based flux form. Unlike the GKS method for handling high-speed compressible flows, constructing a simpler interface flux for low-speed two-phase flow problems simplifies the computational process and improves computational efficiency. Specifically, steps S104-S106 are described in detail: The expression for the continuous Boltzmann-BGK equations with source terms in the flow field is as follows:

[0107] In the formula, τ f S and S represent the relaxation time and source term of the flow field, respectively.

[0108] f eq =ρΓ(u,U) is the equilibrium distribution function, ρ is the density of the gas, ρ = <f>The momentum is ρu = <uf>.

[0109] The expression for Γ(u,U) is:

[0110] In some examples, the flow field source term can be expressed as:

[0111] when When this is achieved, the moments of the equilibrium distribution function of the flow field can be obtained as:

[0112] The moments of the flow field source terms are:

[0113] According to Chapman-Enskog analysis, the zeroth and first moments of the gas kinetic scheme algorithm equations for solving the flow field are as follows:

[0114] The above moment calculation results demonstrate that the kinetic model can be accurately reconstructed from the Navier-Stokes equations. To further derive the expression for the interfacial flux, the numerical flux of the flow field will be derived below based on the low Mach approximation assumption.

[0115] Define the new distribution function as:

[0116] Where Γ0(u) is:

[0117] The expression for the BGK equation based on the new distribution function is:

[0118] The corresponding expressions for the flow field equilibrium state and the flow field source term are:

[0119] The moments of the equilibrium distribution function of the flow field are as follows:

[0120] Moments of the flow field source term:

[0121] Substituting the moments of the flow field equilibrium distribution function and the flow field source term into the zero moment, we obtain the first-order flow field flux.

[0122] Substituting the moments of the flow field equilibrium distribution function and the flow field source term into the first moment, we obtain the second-order flow field flux.

[0123] By performing Chapman-Enskog expansions on the first-order and second-order flow fluxes, we can obtain the flux formulas for pressure p and momentum ρU:

[0124] For constant-density incompressible flow problems, a simplified velocity-based flux formula can be obtained:

[0125] In real two-phase flow problems, the Navier-Stokes equations are typically used to describe the evolution of the flow field. A key issue is how to couple the phase and flow fields within the framework of the gas kinetic scheme. The following section describes how to achieve phase-flow field coupling within the GKS framework, specifically step S108: The numerical algorithm steps for simulating complex two-phase flow using the gas kinetic scheme are presented.

[0126] The interfacial flux formula in the phase field is denoted as J. φ According to formula (19),

[0127] in, Let J be the velocity over half a time step of the flow field evolution. The pressure and velocity interface fluxes in the flow field are denoted as J, respectively. p and J u According to formula (34):

[0128] The gradient term in the source term can be calculated using a second-order isothermal central scheme:

[0129] Where Φ is an arbitrary scalar and the coefficient a = θ r θ l b=θ r +θ l c = θ r -θ l θ r =(x j+1 -x j ) / δ x θ l =(x j -x j-1 ) / δ x These represent the scaling factors for the backward step and the forward step, respectively.

[0130] It is understandable that the calculation process of the two-phase flow GKS algorithm over one time step is as follows:

[0131] (1) After initializing the macroscopic quantities in each cell, apply boundary conditions to the computational domain and calculate the flux J at the cell interface according to formulas (36) and (37). p and J u .

[0132] (2) In the phase field part, first, according to the flux J in formula (37), u Calculate the velocity u of the flow field evolution at half a time step h Then, calculate the interfacial flux J of the unit according to formula (35). φ Then, according to formula (35), the macroscopic quantity within the cell at the next moment is calculated, that is, the update sequence parameter φ. n+1 .

[0133] (3) In the flow field part, the flux J calculated in (1) p and J u By applying a source term to the flow field evolution over a time step using the splitting method, the macroscopic quantity p of the flow field can be obtained. n+1 and u n+1 .

[0134] (4) The two-phase flow process of the coupled velocity field can be calculated by solving the CH equation and NS equation using the GKS method.

[0135] Therefore, the steps can be summarized as follows:

[0136] Step 1: Determine the computational domain, divide or read the mesh information, fluid parameters, and control parameters, allocate arrays based on the read information, initialize the corresponding parameters of the computational domain, and then enter the time step loop to proceed to Step 2.

[0137] Step 2: Set boundary conditions, assign values ​​to all macroscopic quantities at the boundaries of the flow field, and proceed to Step 3.

[0138] Step 3: Solve the Navier-Stokes equations for the flow field. Calculate the pressure and velocity (or momentum) fluxes at the interface of the flow field unit according to the flow flux formula. Then update the macroscopic quantities of the flow field within the unit using the flux information at the interface. If there are external forces, apply them through a splitting method and update the macroscopic quantities. Proceed to Step 4.

[0139] Step 4: Calculate the macroscopic quantity information of the flow field flux evolution obtained in Step 3 for half a time step. Calculate the macroscopic quantity of the phase field passing through the interface within this time step, i.e., the sequence parameter, according to the phase field flux formula, and update the macroscopic quantity in the cell. Proceed to Step 5.

[0140] Step 5: Determine if the time step meets the loop condition. If it does, proceed to Step 2; otherwise, proceed to Step 6.

[0141] Step 6: Output the flow field data and end the simulation.

[0142] The mesh information in Step 1 above includes mesh length, width, number of meshes, unit mesh length, and mesh coordinates; fluid parameters include two-phase density, velocity, pressure, and viscosity parameters at each mesh point; control parameters include dimensionless parameters such as time step and CFL number.

[0143] The boundary conditions in Step 2 above are similar to those in the macroscopic method, where the boundary conditions are set directly by giving the macroscopic values ​​at the given boundary.

[0144] The Navier-Stokes equation in Step 3 above is:

[0145] In the above formula, ρ is the fluid density, t is time, u is the fluid velocity, p is the hydrodynamic pressure, S is the strain rate tensor, ξ is the bulk viscosity coefficient, and F... total It refers to the external force acting on the fluid. The interfacial flux in this step includes the flux of pressure and velocity (or momentum), and the macroscopic quantity information within the cell can be updated using the cell interfacial flux.

[0146] The splitting method in Step 3 is implemented as follows:

[0147] The velocity u can be obtained from the first formula in the above solution. * The velocity u of the next time step is obtained by applying an external force according to the splitting method. n+1 .

[0148] The interface flux in Step 4 is the flux of the phase field sequence parameter. To calculate this flux, we first need to obtain the value of the flow field velocity evolution at the interface for half a time step, and then calculate it using the flow field flux formula in Step 3. Next, we calculate it according to the phase field flux formula and update the sequence parameter information in the cell.

[0149] Understandably, GKS, as a mesoscopic method within a finite volume framework, offers several key advantages: flow field information evolves and updates through macroscopic quantities, allowing direct application of efficient macroscopic-based processing techniques found in traditional numerical methods, such as reconstruction of physical quantities within cells or at interfaces, boundary conditions, and acceleration techniques for steady flow. Furthermore, the gas kinematic scheme is constructed based on the continuous Boltzmann equations. It eliminates the need for velocity space discretization and storage of the distribution function, significantly reducing computational and storage requirements—far less than the overhead of directly solving the BGK equations. Constructing numerical fluxes within a finite volume framework from the continuous Boltzmann-BGK equations allows the fluxes constructed using the Chapman-Enskog expansion analysis to be reduced to macroscopic governing equations (Cahn-Hilliard and Navier-Stokes equations). Moreover, based on the assumption of near-incompressible flow with isodense density, momentum-based fluxes can be simplified to velocity-based fluxes. Once the flux form is obtained, numerical algorithms based on gas momentum schemes can be constructed. This algorithm directly evolves and updates macroscopic quantities in the flow field. The calculation process is simple and the calculation accuracy is high, enabling efficient numerical simulation of physical processes in low-speed flows.

[0150] Furthermore, this invention constructs a numerical flux based on the Cahn-Hilliard phase-field model within the GKS method, successfully achieving accurate capture of the phase interface in two-phase flow. The two-phase flow algorithm in this invention significantly improves the interface capture capability when dealing with complex two-phase flow problems. This algorithm can accurately simulate complex phenomena such as phase separation and interface movement, overcoming the shortcomings of the traditional GKS method in interface capture.

[0151] Furthermore, this invention achieves low Mach number adaptability. Specifically, by improving the traditional GKS method, this invention constructs a flow field numerical flux that can accurately recover the macroscopic Navier-Stokes equations based on the low Mach number approximation assumption. By coupling the phase field and flow field, an efficient and accurate two-phase flow algorithm is proposed. Compared with traditional methods, this invention not only simplifies the construction process of the numerical flux but also significantly improves computational efficiency and numerical stability, and can accurately simulate the complex phenomena of interface motion and flow field evolution in two-phase flow.

[0152] The technical solution of this invention, through the aforementioned technological innovations, successfully overcomes the limitations of the traditional GKS method in handling variable-density low-speed flows and two-phase flows, providing an efficient and accurate mesoscopic numerical algorithm. By employing the low Mach number two-phase flow GKS method based on the phase field model proposed in this invention, the interfacial dynamics and complex flow field evolution in two-phase flow systems can be simulated efficiently and accurately.

[0153] To verify the correctness of the algorithm proposed in this invention, the following numerical examples are performed.

[0154] In the simulation of the stationary droplet example, a circular droplet of radius R is placed at the center of an L×L computational domain, with periodic boundary conditions set around it. The distribution of the order parameters is as follows:

[0155] Among them, (x c ,y c The coordinates (L / 2, L / 2) represent the location of the center of the computational region. The computational grid is 200×200, and the droplet radius is R=50. Other parameters of the model are set to ρ. A =500,ρ B =1,ν=0.2,W=4.0,σ=1.0×10 -5 M = 0.5, CFL = 0.5. The order parameter distribution of the stationary droplet in equilibrium, as shown in Figure 2, indicates that the density distribution along the horizontal centerline agrees well with the standard solution. When the system reaches equilibrium, the relationship between the droplet radius R and the pressure difference ΔP inside and outside the droplet satisfies Laplace's law, i.e., ΔP = σ / R. The formula for calculating pressure P is... in Figure 3 shows the fitting curves of the pressure difference between the inside and outside of the droplet and the reciprocal of the radius under three different surface tensions. The results show that the model conforms to Laplace's law.

[0156] In the simulation of Rayleigh-Taylor instability, the computational domain is a cube with a square cross-section and an aspect ratio of 4:1. The mesh size of the computational domain is 128×128×512. No-slip bounce boundary conditions are applied to the upper and lower boundaries, while periodic boundary conditions are applied to the four side boundaries. Other parameters are set as follows: W = 4, σ = 1×10⁻⁶. -4 CFL = 0.1, At = 0.5. The Peckley number is fixed at 744. The initial interface function and the initial order parameter distribution of the two-phase fluid are as follows:

[0157] The two dimensionless parameters closely related to this problem are the Atwood number and the Reynolds number, which are defined as follows:

[0158] Where g is the acceleration due to gravity, and the other parameters are fixed at L0 = 256. W = 5, σ = 5 × 10 -5 CFL = 0.25, characteristic time is Defined as The Pecklet number is calculated. Figure 4 shows the interface evolution at different times when the Reynolds number is 1024. Figure 5 shows the changes in the positions of the bubble tip, saddle point, and spike over time when the Reynolds number is 1024, which also agree well with the reference results. In summary, it can be demonstrated that this method has high accuracy and reliability in simulating complex two-phase flow problems.

[0159] According to another aspect of this application, a two-phase flow gas kinetics algorithm optimization device is also disclosed. Specifically, as shown in Figure 6, the device includes:

[0160] The first building module is used to build a gas kinetic scheme algorithm for phase field solving, which includes a continuous Boltzmann-BGK equation with source terms.

[0161] The first analysis module is used to construct the phase field interface flux of the gas kinetic scheme algorithm based on Chapman-Enskog analysis, where the phase field interface flux can be recovered to the Cahn-Hilliard equation.

[0162] The second building module is used to build a gas kinetic scheme algorithm for solving the flow field. The gas kinetic scheme algorithm for solving the flow field includes a continuous Boltzmann-BGK equation with source terms.

[0163] The second analysis module is used to calculate the flow field interface flux based on the Chapman-Enskog analysis and the kinetic scheme algorithm for solving the flow gas. The flow field interface flux can be recovered to the Navier-Stokes equations.

[0164] The coupling module is used to implement two-phase flow numerical algorithms based on the Cahn-Hilliard equation and the Navier-Stokes equation by coupling the phase field and the flow field.

[0165] The application of the relevant modules of the device in this example can be referred to the relevant introduction of the method principle above, and will not be repeated here.

[0166] According to another aspect of this application, this application also discloses an electronic device, which includes a memory and at least one processor, wherein the memory stores instructions; the at least one processor invokes the instructions in the memory to cause the electronic device to perform the various steps of the two-phase flow gas kinetics algorithm optimization method described above.

[0167] Figure 6 above describes the device in the embodiment of the present invention in detail from the perspective of modular functional entities. The following describes the electronic device in the embodiment of the present invention in detail from the perspective of hardware processing.

[0168] Figure 7 is a schematic diagram of an electronic device 700 provided in an embodiment of the present invention. The electronic device 700 can vary considerably depending on its configuration or performance, and may include one or more central processing units (CPUs) 710 (e.g., one or more processors) and a memory 720, and one or more storage media 730 (e.g., one or more mass storage devices) for storing application programs 733 or data 732. The memory 720 and storage media 730 can be temporary or persistent storage. The program stored in the storage media 730 may include one or more modules (not shown in the figure), each module including a series of instruction operations on the electronic device 700. Furthermore, the processor 710 may be configured to communicate with the storage media 730 and execute the series of instruction operations in the storage media 730 on the electronic device 700.

[0169] Electronic device 700 may also include one or more power supplies 740, one or more wired or wireless network interfaces 750, one or more input / output interfaces 750, and / or one or more operating systems 731, such as Windows Server, Mac OS X, Unix, Linux, FreeBSD, etc. Those skilled in the art will understand that the electronic device structure shown in FIG7 does not constitute a limitation on electronic devices and may include more or fewer components than shown, or combine certain components, or have different component arrangements.

[0170] The present invention also provides a computer-readable storage medium, which can be a non-volatile computer-readable storage medium or a volatile computer-readable storage medium, wherein the computer-readable storage medium stores instructions that, when executed on a computer, cause the computer to perform the steps of a logistics order control method.

[0171] Those skilled in the art will understand that, for the sake of convenience and brevity, the specific working process of the system, device, or unit described above can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.

[0172] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods of the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0173] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.< / uf> < / f> < / f>

Claims

A two-phase flow gas kinetics scheme algorithm optimization method, characterized in that, The method includes: A gas kinetic scheme algorithm for phase field solution is constructed, wherein the gas kinetic scheme algorithm for phase field solution includes a continuous Boltzmann-BGK equation with source terms; The phase field interface flux is constructed based on the gas kinetic scheme algorithm for solving the phase field using Chapman-Enskog analysis, wherein the phase field interface flux can be recovered to the Cahn-Hilliard phase field equation. A gas kinetic scheme algorithm for solving flow fields is constructed, which includes continuous Boltzmann-BGK equations with source terms; The flow field interface flux is constructed based on the gas kinetic scheme algorithm for solving the flow field using Chapman-Enskog analysis, wherein the flow field interface flux can be recovered to the Navier-Stokes equations. Based on the Cahn-Hilliard equation and the Navier-Stokes equation, a numerical algorithm for two-phase flow is realized by coupling the phase field and the flow field. The two-phase flow gas kinetics algorithm optimization method according to claim 1 is characterized in that, The constructed gas kinetic scheme algorithm for phase field solution includes a continuous Boltzmann-BGK equation with source terms, embodied in the following formula: In the formula, c is the velocity of the particle distribution function g, and τ g Let G be the relaxation time, and G be the phase field source term. The two-phase flow gas kinetics algorithm optimization method according to claim 1 is characterized in that, The phase field interface flux is constructed based on the gas kinetic scheme algorithm for solving the phase field using Chapman-Enskog analysis, wherein the phase field interface flux can be recovered to the Cahn-Hilliard phase field equations, including: Obtain the phase field equilibrium distribution function and phase field source term for each moment that satisfy the first condition; The equilibrium distribution function and the moments of the source term satisfy the following conditions: In the formula, η is the relaxation time adjustment coefficient, and the mobility can be expressed as: parameter Set it to 0, B α C0 and C0 are coefficients to be determined. First, we use... The operator is used to obtain the expression for the phase field equilibrium distribution function; The phase field equilibrium distribution function, based on the Chapman-Enskog representation, is as follows: Taking the zeroth moment of the phase field equilibrium distribution function, we obtain the zeroth-order flux equation; The zeroth-order flux equation is: In the formula, F = <c α ,g>,F is the phase field interface flux of the gas kinetic scheme algorithm for phase field solution; Substituting the phase field equilibrium distribution function and the moments of the phase field source terms into the zeroth-order flux equation yields the first-order phase field flux. The first-order phase field flux is: The first-order phase field flux is expanded using the Chapman-Enskog method to restore the phase field interface flux to the Cahn-Hilliard phase field equation. The two-phase flow gas kinetics algorithm optimization method according to claim 1 is characterized in that, The constructed gas kinetic scheme algorithm for solving the flow field includes a continuous Boltzmann-BGK equation with a source term, embodied in the following formula: In the formula, τ f Let S be the relaxation time, S be the source term of the flow field, and c be the velocity of the particle distribution function f. f eq =ρΓ(u,U) is the equilibrium distribution function; ρ is the density of the gas, ρ = <f> ；< / f> The expression for Γ(u,U) is: The two-phase flow gas kinetics algorithm optimization method according to claim 1 is characterized in that, The flow field source term is represented as: The flow field interfacial flux is constructed based on the gas kinetic scheme algorithm for solving the flow field using Chapman-Enskog analysis, wherein the flow field interfacial flux is restored to the Navier-Stokes equations by: Sure The moments of the equilibrium distribution function of the flow field and the moments of the source terms of the flow field at that time: The moments of the equilibrium distribution function of the flow field are: The moments of the flow field source terms are: The zeroth and first moments of the gas kinetic scheme algorithm for solving the flow field, based on Chapman-Enskog analysis, are respectively included; The zeroth moment of the gas kinetic scheme algorithm for solving the flow field is: The first moment of the gas kinetic scheme algorithm for solving the flow field is: Define a new distribution function: in, Based on the new distribution function, the expression for the Boltzmann-BGK, the moments of the equilibrium distribution function of the flow field, and the moments of the source term of the flow field are determined. The expression for Boltzmann-BGK is: The moments of the equilibrium distribution function of the flow field are: The moments of the flow field source terms are: Substituting the equilibrium distribution function of the flow field and the moments of the source term of the flow field into the zero moment, we obtain the first-order flow flux. Substituting the equilibrium distribution function of the flow field and the moments of the source term of the flow field into the first moment, the second-order flow flux is obtained. The first-order and second-order flow field fluxes are expanded using the Chapman-Enskog method to reduce the flow field interface fluxes to the Navier-Stokes equations. A two-phase flow gas kinetics algorithm optimization device, characterized in that, The device includes: The first construction module is used to construct a gas kinetic scheme algorithm for phase field solving, wherein the gas kinetic scheme algorithm for phase field solving includes a continuous Boltzmann-BGK equation with source terms; The first analysis module is used to construct the phase field interface flux of the gas kinetic scheme algorithm for solving the phase field based on Chapman-Enskog analysis, wherein the phase field interface flux can be recovered to the Cahn-Hilliard phase field equation. The second construction module is used to construct a gas kinetic scheme algorithm for solving the flow field, wherein the gas kinetic scheme algorithm for solving the flow field includes a continuous Boltzmann-BGK equation with a source term; The second analysis module is used to construct the flow field interface flux of the gas kinetic scheme algorithm for solving the flow field based on Chapman-Enskog analysis, wherein the flow field interface flux can be recovered to the Navier-Stokes equations. The coupling module is used to implement a two-phase flow numerical algorithm based on the Cahn-Hilliard equation and the Navier-Stokes equation by coupling the phase field flow field. An electronic device, characterized in that, The electronic device includes a memory and at least one processor, the memory storing instructions; the at least one processor invokes the instructions in the memory to cause the electronic device to perform the various steps of the two-phase flow gas kinetics algorithm optimization method as described in claims 1-5. A computer-readable storage medium storing instructions, characterized in that, When the instructions are executed by the processor, they implement each step of the two-phase flow gas kinetics algorithm optimization method as described in any one of claims 1-5.

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