Acoustic streaming coupling calculation method and device based on Galbrun equation

Abstract

The invention provides an acoustic streaming coupling calculation method and device based on a Galbrun equation. The method provided by the invention comprises the steps of obtaining a Galbrun equation based on an Euler equation of ideal fluid under the condition of no external force and in combination with Euler disturbance, Lagrange disturbance and a mixed disturbance relation; the Galbrun equation represents the coupling effect of the flow field and the sound field; based on a Navier-Stokes equation, in combination with a k-epsilon turbulence model in the RANS model, solving to obtain flow field data of the non-uniform flow field; a perfect matching layer is introduced for sound wave propagation simulation of an unbounded region, and boundary conditions suitable for the Galbrun equation are constructed through geometric transformation, complex variable definition and coordinate transformation; and carrying out weak transformation on the Galbrun equation, embedding the boundary condition, combining the flow field data, carrying out discretization processing by adopting a triangular finite element, constructing a global linear algebraic equation set, and solving to obtain a discrete solution of sound field parameters.

Description

Technical Field

[0001] This application relates to the field of acoustic-fluid coupling technology, and in particular to an acoustic-fluid coupling calculation method and apparatus based on the Galbrun equation. Background Technology

[0002] Against the backdrop of continuously growing global resource demands, the sustainable development and protection of the ocean, as a vital resource treasure trove and ecological barrier, has become a focus of international attention. Marine acoustic technology, with its unique advantage of sound waves propagating over long distances underwater, plays an indispensable role in fields such as marine communication and resource exploration. However, the non-uniform flow fields in complex marine environments significantly alter the propagation path, velocity, and energy distribution of sound waves, directly affecting the accuracy and reliability of acoustic detection. Therefore, conducting coupled calculations of underwater sound fields and non-uniform flow fields is of great significance.

[0003] Currently, various sound field calculation models have been proposed to address the sound propagation characteristics under different environmental conditions. These mainly include ray models, normal mode models, fast field models (FFM), parabolic equation models (PE), and hybrid models. These traditional models are mostly based on the Helmholtz equations and focus on sound field simulation calculations in homogeneous media. Existing sound field calculation models have the following problems: on the one hand, they are difficult to accurately handle the influence of flow velocity gradients on sound wave propagation in non-uniform flow fields, and cannot fully reflect the complex coupling between the flow field and the sound field; on the other hand, their calculation accuracy and applicability decrease significantly when facing nonlinear, high-frequency scenarios and complex seabed topography, making it difficult to meet the needs of marine engineering, military exploration, and underwater acoustic communication for high-precision sound field simulation.

[0004] Therefore, there is an urgent need for a method to accurately describe the influence of velocity gradient and seabed topography on sound propagation by coupling calculations of the underwater sound field and the non-uniform flow field. Summary of the Invention

[0005] In view of this, this application provides a method and apparatus for acoustic-fluid coupling calculation based on the Galbrun equation, which can accurately describe the influence of velocity gradient and seabed topography on sound propagation by coupling calculation of underwater sound field and non-uniform flow field.

[0006] Specifically, this application is implemented through the following technical solution:

[0007] The first aspect of this application provides a method for calculating acoustic-fluid coupling based on the Galbrun equation, the method comprising:

[0008] Based on the Euler equation for ideal fluid without external forces, and combining the relationships of Euler perturbation, Lagrange perturbation, and mixed perturbation, the Galbrun equation is obtained; the Galbrun equation characterizes the coupling effect between the flow field and the sound field.

[0009] Based on the Navier-Stokes equations and combined with the k-ε turbulence model in the RANS model, the flow field data of the non-uniform flow field are obtained.

[0010] For the simulation of sound wave propagation in unbounded regions, a perfectly matched layer is introduced. Boundary conditions suitable for the Galbrun equation are constructed through geometric transformation, complex variable definition, and coordinate transformation.

[0011] The Galbrun equation is weakly transformed, the boundary conditions are embedded, and the flow field data is combined with the flow field data. The equations are then discretized using triangular finite elements to construct and solve a global system of linear algebraic equations, yielding discrete solutions for the sound field parameters.

[0012] A second aspect of this application provides a computational device for acoustic-fluid coupling based on the Galbrun equation, the device comprising an acquisition module, a solution module, and a construction module;

[0013] The acquisition module is used to obtain the Galbrun equation based on the Euler equation under ideal fluid conditions without external forces, combined with the relationships of Euler perturbation, Lagrange perturbation and mixed perturbation; the Galbrun equation characterizes the coupling effect between the flow field and the sound field.

[0014] The solution module is used to solve for the flow field data of the non-uniform flow field based on the Navier-Stokes equations and combined with the k-ε turbulence model in the RANS model.

[0015] The construction module is used to introduce a perfectly matched layer for the simulation of sound wave propagation in unbounded regions. Through geometric transformation, complex variable definition and coordinate transformation, it constructs boundary conditions suitable for the Galbrun equation.

[0016] The solution module is also used to perform a weak transformation on the Galbrun equation, embed the boundary conditions, combine the flow field data, discretize the equations using triangular finite element method, construct a global linear algebraic equation system and solve it to obtain the discrete solution of the sound field parameters.

[0017] The acoustic-flow coupling calculation method and apparatus based on the Gablrun equation provided in this application, in the first aspect, form a complete closed loop through four closely related steps: "deriving the Gablrun equation → solving for flow field data → constructing adaptive boundary conditions → discretizing and solving for acoustic field parameters," which fully connects the physical mechanism to the numerical realization. The Gablrun equation is used as the core governing equation for acoustic-flow coupling. Real data of non-uniform flow fields are obtained as input by relying on the Navier-Stokes equations and the k-ε turbulence model. Adaptive boundary conditions are constructed through a perfectly matched layer to solve the simulation problem of unbounded regions. Finally, the acoustic field parameters are obtained by discretizing and solving the problem through finite element method. Each step is closely linked to ensure the integrity and coherence of acoustic-flow coupling calculation from theoretical basis to actual solution, and realizes the systematic simulation of the interaction between sound and flow in non-uniform flow fields. Secondly, in constructing the Galbrun equation, the Euler equation for ideal fluid without external forces is used as a foundation. Combined with Euler perturbations, Lagrange perturbations, and mixed perturbation relationships, the coupling physical mechanism between the flow field and the sound field is accurately captured. This equation naturally includes the influence of flow field motion on sound wave propagation (such as convection effects) and the feedback effect of sound wave convection, providing mathematically rigorous and physically meaningful control equations for sound-flow coupling calculations. This avoids distortion of the coupling mechanism that may result from simplified models, ensuring that the calculations truly reflect the essential laws of sound-flow interaction. Thirdly, a perfectly matched layer is introduced when constructing boundary conditions. Through geometric transformation to adapt to the flow field motion state, defining complex variables to set the boundary energy absorption coefficient, transferring attenuation characteristics through coordinate transformation, and integrating by direction to form a unified rule, this solves the boundary reflection problem caused by computational domain truncation in unbounded region sound wave propagation simulations. It also ensures that the boundary conditions adapt to the Galbrun equation and the characteristics of non-uniform flow fields, guaranteeing that the attenuation behavior of the sound field at the boundary conforms to physical reality, avoiding false reflections that interfere with sound-flow coupling calculations, and improving the accuracy and reliability of the simulation. Attached Figure Description

[0018] Figure 1 A flowchart of the acoustic-fluid coupling calculation method based on the Galbrun equation provided in Embodiment 1 of this application;

[0019] Figure 2 This is a schematic diagram of the acoustic-fluid coupling computing device based on the Galbrun equation provided in Embodiment 2 of this application. Detailed Implementation

[0020] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application.

[0021] The terminology used in this application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. The singular forms “a,” “the,” and “the” used herein are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used herein refers to and includes any and all possible combinations of one or more of the associated listed items.

[0022] It should be understood that although the terms first, second, third, etc., may be used in this application to describe various information, such information should not be limited to these terms. These terms are only used to distinguish information of the same type from one another. For example, without departing from the scope of this application, first information may also be referred to as second information, and similarly, second information may also be referred to as first information. Depending on the context, the word "if" as used herein may be interpreted as "when," "when," or "in response to determination."

[0023] The following specific embodiments are given to illustrate the technical solution of this application in detail.

[0024] Figure 1 This is a flowchart illustrating the acoustic-fluid coupling calculation method based on the Galbrun equation provided in Embodiment 1 of this application. Please refer to... Figure 1 The method provided in this embodiment may include:

[0025] S101. Based on the Euler equation for an ideal fluid without external forces, and combining the relationships between Euler perturbation, Lagrange perturbation, and mixed perturbation, the Galbrun equation is obtained.

[0026] The Galbrun equation characterizes the coupling effect between the flow field and the sound field.

[0027] Specifically, the Euler equations are the fundamental governing equations describing the motion of an ideal fluid (inviscid and incompressible). Derived from physical laws such as the conservation of mass and momentum, they primarily characterize the changes in physical quantities such as velocity and pressure of a fluid at a fixed spatial point over time, without tracking the trajectory of a specific fluid element. They are the foundational equations in fluid mechanics used to analyze fluid motion. Euler perturbations, using a fixed spatial location as a reference, describe the deviation of physical quantities (such as velocity and pressure) in the flow field from their average state. That is, they observe the changes in fluid physical quantities at a fixed point before and after the perturbation, without considering the origin and motion history of the fluid element at that location. Lagrange perturbations, using a fluid element as a reference, track the deviation of physical quantities (such as displacement and velocity) of a specific fluid element from its initial or average state during motion. Lagrange perturbations focus on the changes of a specific fluid element over time, rather than the changes at a fixed spatial point. The hybrid perturbation relation is the mathematical relationship connecting Euler and Lagrange perturbations. Since Eulerian perturbations and Lagrange perturbations describe the changes in physical quantities from the perspectives of a fixed point in space and fluid micro-particles, respectively, the conversion between the two can be achieved through the hybrid perturbation relationship, thereby comprehensively utilizing the characteristics of the two perturbations to more fully characterize the interaction between the flow field and the sound field.

[0028] Furthermore, the Galbrun equation is derived from the Euler equation under ideal fluid conditions without external forces, combining the relationships between Euler perturbations, Lagrange perturbations, and their mixtures. Its core function is to characterize the coupling between the flow field and the sound field, simultaneously reflecting the influence of flow motion on sound wave propagation and the reaction of sound wave propagation on the flow field. Since traditional acoustic models (such as the Helmholtz equation) struggle to accurately describe the complex coupling between non-uniform flow fields and sound fields, the Galbrun equation, by integrating the characteristics of Euler and Lagrange perturbations, can more comprehensively and accurately characterize the influence of flow field properties such as velocity gradients on sound wave propagation. This provides a more reliable theoretical foundation for underwater sound field calculations under non-uniform flow fields, thus solving the problem of insufficient computational accuracy of traditional models in complex flow field environments.

[0029] In specific implementation, the Euler equation based on ideal fluid without external force, combined with the relationships of Euler perturbation, Lagrange perturbation, and mixed perturbation, yields the Galbrun equation, which includes:

[0030] (1) Select the Euler equations for an ideal fluid without external force.

[0031] Specifically, the Euler equations for an ideal fluid under no external force are selected, which include the mass conservation equation, momentum conservation equation, etc.

[0032] For example, in one embodiment, the Euler equations can be expressed as:

[0033] ;

[0034] in, , , and These are density, velocity, pressure, and entropy, respectively.

[0035] (2) Define the Euler perturbation of a physical quantity as the dynamic change of the physical quantity at a fixed spatial position, and define the Lagrangian perturbation of a physical quantity as the change of the physical quantity of a fluid particle on its trajectory.

[0036] Specifically, the two forms of disturbance of physical quantities are defined separately. Taking a fixed spatial position as a reference, the dynamic change of fluid physical quantities (such as velocity, pressure, and density) at that position over time is defined as Eulerian disturbance; taking a specific fluid particle as the tracking object, the change of physical quantities (such as velocity, pressure, and density) of that particle at different times on its trajectory relative to its initial state is defined as Lagrange disturbance.

[0037] For example, in one embodiment, the Euler perturbation can be expressed as:

[0038] ;

[0039] Among them, the For Eulerian perturbations; the In spatial location ,time The instantaneous actual value below; the In spatial location ,time The baseline reference value is as follows.

[0040] The Lagrange perturbation can be expressed as:

[0041] ;

[0042] Among them, the For Lagrange perturbations; the stated In spatial location ,time The instantaneous actual value below; the This is a reference value for the initial state of the fluid particles.

[0043] (3) Establish the hybrid perturbation relationship between Eulerian perturbation and Lagrange perturbation; the Lagrange perturbation is equal to the sum of the products of Eulerian perturbation, Lagrange displacement potential and steady-state uniform flow gradient.

[0044] Specifically, a mathematical correlation is constructed to connect Eulerian perturbations and Lagrange perturbations, defining the specific composition of the Lagrange perturbation, namely, the Lagrange perturbation is equal to the sum of two parts: the first part is the Eulerian perturbation of the corresponding physical quantity, and the second part is the product of the Lagrange displacement potential and the steady-state uniform flow gradient. The quantitative conversion between the two perturbation forms is realized through this relationship.

[0045] For example, in one embodiment, the hybrid perturbation relationship can be expressed as:

[0046] ;

[0047] Among them, the For Lagrange perturbations; the stated For Eulerian perturbations; the The Lagrange displacement potential; the The gradient represents the steady-state uniform flow rate.

[0048] (4) The flow field is defined as a steady flow field, and the Euler equations are linearized under the steady flow field; the steady-state uniform flow rate of the steady flow field is constant and does not change with time.

[0049] Specifically, the calculation object is limited to a stable flow field, where the steady-state uniform flow rate is a constant value that does not change with time. Then, based on this stable flow field condition, the selected Euler equations are linearized by ignoring the higher-order terms in the Euler equations to obtain the linearized Euler equations.

[0050] (5) Substitute the relationships of the Euler perturbation, Lagrange perturbation and mixed perturbation into the linearized Euler equations, and by eliminating pressure, velocity and density, obtain the Galbrun equation with Lagrange perturbation pressure and Lagrange displacement potential as the core variables.

[0051] Specifically, the step of substituting the Eulerian, Lagrange, and mixed perturbation relationships into the linearized Eulerian equations, and eliminating pressure, velocity, and density to obtain the Galbrun equations with Lagrange perturbation pressure and Lagrange displacement potential as core variables, includes: treating the Eulerian, Lagrange, and mixed perturbation relationships as a whole, and simultaneously substituting them into the linearized mass conservation equation, momentum conservation equation, and the equations relating pressure, density, and entropy to obtain a first set of equations containing perturbation variables; performing variable substitution on the first set of equations, expressing the perturbation terms of pressure, velocity, and density through Lagrange perturbation pressure and Lagrange displacement potential; eliminating the original variables and intermediate perturbation terms of pressure, velocity, and density by solving the simultaneous equations, retaining Lagrange perturbation pressure and Lagrange displacement potential as core variables; and rearranging the resulting equations to form the Galbrun equations that simultaneously contain the coupling relationship between stable flow field parameters and acoustic field parameters.

[0052] In practical implementation, predefined Eulerian and Lagrange perturbations, along with established mixed perturbation relationships, are substituted into the linearized mass conservation equation, momentum conservation equation, and pressure-density-entropy relationship equation. Through algebraic substitution, a first set of equations containing various perturbation variables (including perturbation pressure, perturbation velocity, and perturbation density) is obtained. For the pressure, velocity, and density perturbation terms in the first set of equations, based on physical relationships and predefined rules, they are re-expressed using Lagrange perturbation pressure and Lagrange displacement potential. For example, using derived formulas, the perturbation velocity... Transformed into The form in which the calculations are performed. Further, the system of equations after variable substitution is solved simultaneously, and mathematical elimination methods (such as substitution elimination, addition and subtraction elimination) are used to eliminate the original variables of pressure, velocity, and density (in their unperturbed form). , , All intermediate perturbation terms (transitional variables that have not yet been fully transformed) generated during the substitution and simultaneous equation-making process are eliminated, leaving only the two core variables, Lagrange perturbation pressure and Lagrange displacement potential, in the system of equations. The equations obtained after elimination are then rearranged according to mathematical norms (combining like terms and adjusting the order of coefficients) to finally form the Galbrun equations.

[0053] For example, in one embodiment, the Galbrun equation can be expressed as:

[0054] ;

[0055] Among them, the The fluid density under steady-state uniform flow field; The fluid pressure under steady-state uniform flow field; The Lagrange displacement potential; the The velocity of sound in a steady-state uniform flow field; The The fluid velocity under steady-state uniform flow field; Angular frequency; For time variables, It is the imaginary unit.

[0056] The method provided in this embodiment, when constructing the Galbrun equations, first selects the Euler equations for an ideal fluid without external forces as the basic framework describing fluid motion, providing a physical logical starting point for acoustic-fluid coupling; defines Euler and Lagrange perturbations, covering the perturbation forms of physical quantities in non-uniform flow fields from both spatially fixed and particle-tracking perspectives, then establishes hybrid perturbation relationships, opening up the conversion channel between the two, and fully capturing the perturbation details caused by flow field non-uniformity (such as velocity gradient); then, it defines a stable flow field (steady-state uniform flow rate is constant) and linearizes the Euler equations, simplifying calculations while preserving core physical laws, adapting to the engineering calculation needs of acoustic-fluid coupling in non-uniform flow fields; finally... Then, various disturbances are substituted and intermediate variables such as pressure, velocity, and density are eliminated. Focusing on the Lagrangian disturbance pressure and displacement potential, redundant variables are removed, and the interaction between fluid particle motion and sound field propagation is directly reflected. The influence of velocity gradient and seabed topography on sound propagation in non-uniform flow fields is accurately characterized. This provides a practical and reliable theoretical model for marine engineering, military exploration, and underwater acoustic communication, improving the practicality and accuracy of acoustic-flow coupling calculations. It helps solve the problem of acoustic technology application in complex marine environments. From laying the foundation of basic equations, comprehensively characterizing disturbances, simplifying and adapting equations to focusing on core variables, it progressively achieves accurate and efficient calculation of acoustic-flow coupling in non-uniform flow fields.

[0057] S102. Based on the Navier-Stokes equations and combined with the k-ε turbulence model in the RANS model, the flow field data of the non-uniform flow field are obtained by solving.

[0058] Specifically, the Navier-Stokes equations are the fundamental formulas describing how fluids (fluids involved in acoustic-fluid coupling) flow. They reflect the relationship between key properties of fluids such as velocity, pressure, and viscosity and their motion, serving as the "underlying basis" for calculating flow fields. Because fluid motion often involves irregular "turbulent vortices," directly using the Navier-Stokes equations is too complex. The RANS model first decomposes fluid motion into "mean flow" and "fluctuating vortices," and the k-ε model then specifically calculates the "turbulent kinetic energy" (…). ")" and "vortex energy consumption rate ( These two quantities, by simplifying calculations, allow the turbulent flow field to be actually calculated, making them a "simplification tool" suitable for complex flow fields.

[0059] Furthermore, the flow field data for non-uniform flow fields refers to key data of "non-uniform flow fields" (flow fields where fluid velocity, density, etc., are not uniformly distributed), such as fluid velocity, pressure, and turbulence intensity at different locations. These are the core input data required for subsequent acoustic-fluid coupling calculations. In this embodiment, the Navier-Stokes equations are used as a foundation, and the k-ε turbulence model is relied upon to simplify the calculation of complex turbulence, ultimately calculating the velocity, pressure, and other flow field data of the non-uniform flow field. With this data, further analysis of how the flow field and sound field influence each other is conducted to complete the acoustic-fluid coupling calculation.

[0060] In specific implementation, the flow field data of the non-uniform flow field is obtained by solving the Navier-Stokes equations and combining them with the k-ε turbulence model in the RANS model, including:

[0061] (1) Select the Navier-Stokes equations to describe fluid motion.

[0062] Specifically, the core equations used to describe the motion of the target fluid are first determined. These equations are the fundamental equations in fluid mechanics that describe the motion of viscous fluids. They include the mass conservation equation (which reflects the conservation of fluid mass, meaning that mass at any position in the flow field is neither created nor destroyed) and the momentum conservation equation (which reflects the law of change of fluid momentum and relates the fluid velocity, pressure, viscous force and other parameters to external forces).

[0063] For example, in one embodiment, the Navier-Stokes equations can be expressed as:

[0064] ;

[0065] Among them, the For fluid in The velocity component in the direction; For fluid in The velocity component in the direction; For fluid in The velocity component in the direction.

[0066] ;

[0067] Among them, the The fluid density; For time variables; the , , For fluid micro-elements in , , The velocity components in three directions; It is the pressure on the fluid element: , and It is the viscous stress on the surface of the micro-element. The amount, , and It is the mass force on the infinitesimal element.

[0068] (2) Introduce the k-ε turbulence model in the RANS model to decompose the turbulent motion in the flow field into time-averaged motion and pulsating motion. By defining turbulent kinetic energy and turbulent kinetic energy dissipation rate, establish additional transport equations to describe the turbulent pulsating characteristics.

[0069] Specifically, turbulent motion is an irregular flow state of fluids, characterized by disordered trajectories of fluid particles and random fluctuations in physical quantities such as velocity (pressure, temperature) over time and space. In the RANS model framework, time-averaged motion is the "averaged and relatively stable" flow state obtained by averaging the instantaneous motion of fluid particles in turbulent flow (which constantly changes due to random fluctuations) over a period of time using mathematical methods. For example, measuring the instantaneous velocity at a point in turbulent flow through a pipe will result in random variations due to turbulent fluctuations, but averaging this instantaneous velocity over a longer period yields the time-averaged velocity, reflecting the overall flow trend of the fluid. Fluctuating motion is the component superimposed on time-averaged motion in turbulence; it represents the random fluctuations in instantaneous physical quantities (such as velocity and pressure) of fluid particles deviating from their time-averaged values. Taking pipe turbulence as an example, the difference between the instantaneous velocity and the time-averaged velocity at a certain point is the fluctuating velocity, which reflects the random and irregular characteristics of turbulence and is a key factor in turbulent energy dissipation and momentum exchange.

[0070] Furthermore, additional transport equations are introduced in the k-ε turbulence model to describe the fluctuating characteristics of turbulence (such as the generation and dissipation of fluctuating kinetic energy). These equations are used to "close" the Navier-Stokes equations after Reynolds averaging (because Reynolds averaging introduces new unknowns). Specifically, the k-ε turbulence model introduces two additional transport equations: the turbulent kinetic energy equation, which describes the generation (e.g., the energy per unit mass of fluid due to fluctuating motion), dissipation (due to fluid viscosity, the conversion of fluctuating kinetic energy into heat energy), and transport (diffusion with fluid flow) of turbulent fluctuating kinetic energy (i.e., the energy possessed by a unit mass of fluid due to fluctuating motion) in the flow field. The turbulent kinetic energy dissipation rate equation describes the rate at which turbulent kinetic energy is dissipated. Together with the turbulent kinetic energy equation, it determines the characteristic scales of turbulence (such as vortex size), allowing the model to calculate the impact of turbulence on the time-averaged flow field. These two equations, through the form of "transport" (similar to the transport of physical quantities in a fluid, including terms such as convection, diffusion, generation, and dissipation), mathematically represent the key characteristics of turbulent fluctuations and couple them with the Reynolds-averaged master equation to solve the turbulent flow field.

[0071] In specific implementation, the k-ε turbulence model introduced from the RANS model decomposes the turbulent motion in the flow field into time-averaged motion and fluctuating motion. By defining turbulent kinetic energy and turbulent kinetic energy dissipation rate, an additional transport equation describing the fluctuating characteristics of turbulence is established, including: decomposing the physical quantities in the flow field into time-averaged components and fluctuating components based on the Reynolds time-averaged method; the time-averaged component characterizes the macroscopic average motion, and the fluctuating component characterizes the random disturbance characteristics; defining turbulent kinetic energy as the turbulent fluctuating kinetic energy per unit mass of fluid, and defining the turbulent kinetic energy dissipation rate as the rate at which turbulent kinetic energy is dissipated due to viscosity; the magnitude of the turbulent kinetic energy is related to the square of the fluctuating velocity component, and the turbulent kinetic energy dissipation rate is the rate at which turbulent kinetic energy is dissipated due to viscosity. The kinetic energy dissipation rate is used to characterize the conversion process of turbulent energy into thermal energy. For non-uniform flow field characteristics, a first transport equation for turbulent kinetic energy and a second transport equation for the turbulent kinetic energy dissipation rate are determined. The second transport equation introduces empirical coefficients related to turbulent kinetic energy and characteristic length. Solving the first and second transport equations yields the distribution of turbulent kinetic energy and the turbulent kinetic energy dissipation rate. Based on this distribution, the turbulent viscosity coefficient is determined. The product of the turbulent viscosity coefficient and the turbulent kinetic energy dissipation rate is proportional to the square of the turbulent kinetic energy. The turbulent viscosity coefficient is incorporated as a correlation parameter into the coupling relationship between the first and second transport equations, forming an additional transport equation system.

[0072] Specifically, the Reynolds time-averaged method is used to decompose all physical quantities (including velocity, pressure, etc.) involved in the flow field description. Each physical quantity is averaged over a sufficiently long time interval, splitting it into a time-averaged component and a fluctuating component. The time-averaged component is obtained by averaging the instantaneous values ​​of the physical quantity over time, while the fluctuating component is obtained by calculating the difference between the instantaneous value and the time-averaged component. Furthermore, turbulent kinetic energy is defined as the fluctuating kinetic energy per unit mass of fluid, its magnitude determined by half the sum of the squares of the fluctuating velocity components; the turbulent kinetic energy dissipation rate is defined as the rate at which turbulent kinetic energy is converted into heat energy due to fluid viscosity. To address the characteristics of non-uniform flow fields, corresponding transport equations are constructed: A first transport equation for turbulent kinetic energy is established, comprising a convection term (describing the migration of turbulent kinetic energy with fluid flow), a diffusion term (describing molecular and turbulent diffusion of turbulent kinetic energy in the flow field), a generation term (mainly generated by the interaction of the time-averaged velocity gradient and Reynolds stress, reflecting the process of energy transfer from the time-averaged flow to turbulent fluctuations), and a dissipation term (directly related to the turbulent kinetic energy dissipation rate, reflecting the consumption of turbulent kinetic energy). A second transport equation for turbulent kinetic energy dissipation rate is established, also comprising a convection term, a diffusion term, a generation term (related to the generation term of turbulent kinetic energy and related to the characteristic scale of turbulence), and a dissipation term. Several empirical coefficients related to turbulent kinetic energy and the characteristic length of turbulence are introduced (these coefficients are derived from extensive experimental data and empirical summaries, used to accurately describe the variation of the dissipation rate under different flow conditions). The first and second transport equations are solved using numerical calculation methods to obtain specific distribution data of turbulent kinetic energy and turbulent kinetic energy dissipation rate at various locations in the flow field. Based on the distribution of turbulent kinetic energy and turbulent kinetic energy dissipation rate obtained from the solution, the turbulent viscosity coefficient is calculated and determined. The calculation follows the relationship that the turbulent viscosity coefficient is directly proportional to the square of the turbulent kinetic energy and inversely proportional to the turbulent kinetic energy dissipation rate. The calculated turbulent viscosity coefficient is used as a key correlation parameter and substituted into the first and second transport equations, thus establishing a coupled relationship between the two equations. This ultimately constitutes a complete system of additional transport equations capable of describing the characteristics of turbulent fluctuations.

[0073] For example, in one embodiment, the first transport equation for turbulent kinetic energy can be expressed as:

[0074] ;

[0075] Among them, the The fluid density; It is turbulent kinetic energy; the For time variables; the These are spatial coordinate variables, representing positions within the flow field; For fluid in The time-averaged velocity component in the direction; The molecular dynamic viscosity; The turbulent viscosity coefficient; The Prandtl number represents the turbulent kinetic energy; The turbulence generation term is expressed as follows: The , The time-averaged velocity component of the fluid in the corresponding direction; The term representing the dissipation of turbulent kinetic energy is... The turbulent kinetic energy dissipation rate.

[0076] The second transport equation for the turbulent kinetic energy dissipation rate can be expressed as:

[0077] ;

[0078] Among them, the The fluid density; For time variables; the For spatial coordinate variables; the For fluid in The time-averaged velocity component in the direction; The turbulent kinetic energy dissipation rate; The molecular dynamic viscosity; The turbulent viscosity coefficient; The Prandtl number represents the turbulent kinetic energy; It is turbulent kinetic energy; the and These are model constants; This is a turbulence generation term.

[0079] The turbulent viscosity coefficient can be expressed as:

[0080] ;

[0081] Among them, the The turbulent viscosity coefficient; The fluid density; It is turbulent kinetic energy; the The turbulent kinetic energy dissipation rate; These are model constants.

[0082] (3) The additional transport equations are coupled with the Navier-Stokes equations to form a complete set of control equations applicable to non-uniform flow fields; the complete set of control equations can simultaneously reflect the average motion characteristics of the fluid and the influence of turbulent fluctuations on the flow.

[0083] In practical implementation, the established k-ε additional transport equations are mathematically coupled with the selected Navier-Stokes equations: In the momentum equations of the Navier-Stokes equations, turbulent fluctuations generate "Reynolds stress" (the additional force exerted by turbulence on the mean motion). The k-ε model uses the calculation results of k and ε to transform the Reynolds stress into an expression related to the mean motion parameters and substitutes it into the momentum equations. At the same time, k and ε in the additional transport equations depend on the average velocity, pressure, and other parameters calculated by the Navier-Stokes equations to solve. Through this interrelated mathematical processing, a complete set of governing equations is finally formed.

[0084] (4) Discretize the coupled control equations and solve the equations using numerical methods in combination with the boundary conditions of the flow field computation domain to obtain the flow field data of the non-uniform flow field.

[0085] In practice, the coupled complete set of governing equations is first discretized: the computational domain of the flow field (i.e., the fluid space to be analyzed, such as the fluid region within a specific device) is divided into a large number of tiny discrete units (such as tetrahedral or hexahedral meshes). Then, using discretization methods such as the finite volume method and the finite element method, the continuous partial differential equations are transformed into a set of algebraic equations on each discrete unit (i.e., transforming "equations in continuous space" into "numerical relationships on a finite number of mesh nodes"). Subsequently, the boundary conditions of the computational domain of the flow field are determined (such as the fluid velocity and pressure at the inlet of the computational domain, the pressure at the outlet, and the fluid no-slip condition on the wall). Conditions, etc., need to be set in combination with the interaction between fluid and boundary in the actual scenario), and the boundary conditions are transformed into algebraic equation constraints for the corresponding discrete elements; finally, numerical solution methods (such as SIMPLE algorithm, PISO algorithm, etc.) are used to iteratively solve the discrete algebraic equation system, and the fluid parameters (velocity, pressure, k, ε, etc.) on each grid node are repeatedly calculated and corrected until the calculation results meet the convergence conditions (such as the parameter difference between two adjacent iterations is less than the set threshold), and finally the flow field data (including velocity, pressure, turbulent kinetic energy, turbulent kinetic energy dissipation rate, etc.) of each discrete node in the non-uniform flow field are obtained.

[0086] The method provided in this embodiment uses the Navier-Stokes equations as its underlying framework, strictly adheres to the conservation of mass and momentum, and accurately reconstructs the velocity and pressure field distributions of the average fluid motion. This provides a realistic flow field basis for the "fluid convection changing the direction and velocity of sound propagation" (such as the velocity difference between upstream and downstream sound wave propagation) in acoustic-flow coupling. By introducing the k-ε turbulence model and decomposing turbulence into time-averaged and fluctuating motions, and defining turbulent kinetic energy and turbulent kinetic energy dissipation rate, it can quantitatively capture the scattering and attenuation effects of turbulent fluctuations on sound propagation in non-uniform flow fields (such as sound energy dissipation caused by turbulent vortices), filling the gaps in key physical processes missed by simple laminar flow models. The complete set of governing equations formed by the coupling of these two methods can simultaneously output "average flow field information + turbulent fluctuation information". This perfectly matches the dual requirements of acoustic-fluid coupling: the average flow field supports the "first-order influence of flow on sound" (such as the drag effect of average flow velocity on sound propagation in pipe flow); turbulent fluctuation data is used to calculate the "second-order influence of flow on sound" (such as sound scattering and energy dissipation in highly turbulent regions), allowing acoustic-fluid coupling to comprehensively cover the effect of complex flow field characteristics on acoustic propagation. Through discretization and numerical solution, it can adapt to the irregular geometric boundaries of non-uniform flow fields, and captures the spatial gradient of flow velocity and k-ε (such as velocity abrupt changes within the boundary layer) through fine meshing, avoiding coupling errors caused by insufficient flow field data resolution; and the mature numerical solution framework ensures computational efficiency and stability, providing efficient and reliable pre-input for subsequent acoustic-fluid coupling (which requires flow field data as boundary conditions and source terms of acoustic equations).

[0087] S103. For the simulation of sound wave propagation in unbounded regions, a perfectly matched layer is introduced. Through geometric transformation, definition of complex variables, and coordinate transformation, boundary conditions suitable for the Galbrun equation are constructed.

[0088] Specifically, a perfectly matched layer (PML) is an artificially created virtual absorbing boundary layer. Through special material parameter design and mathematical processing (such as geometric transformations and complex variable definitions), it can absorb incident sound waves at the boundary with almost no reflection, simulating the natural attenuation effect of sound waves propagating to a distant location in an infinite space. This solves the problem of "spurious reflection of sound waves at the boundary of a finite computational domain" in numerical calculations. Boundary conditions refer to the constraints or settings imposed on physical quantities (such as sound pressure and displacement potential) at the boundary of the computational domain. They are used to describe the propagation behavior of sound waves at the boundary (such as reflection, transmission, and absorption) and are essential conditions for solving wave equations (such as the Galbrun equation).

[0089] It should be noted that this embodiment takes into account that the Galbrun equation is a specific wave equation describing the coupling of the flow field and the sound field. Its variables (such as Lagrange perturbation pressure and displacement potential) and equation form are different from those of ordinary sound wave equations, and general boundary conditions cannot be directly adapted. However, the special boundary conditions constructed by PML can not only satisfy the mathematical characteristics of the Galbrun equation, but also eliminate spurious reflections in the simulation of unbounded regions by absorbing the boundary, ensuring the accuracy of the sound wave propagation simulation in the acoustic-flow coupling calculation, and making the numerical results closer to the interaction between sound waves and non-uniform flow fields in actual unbounded space.

[0090] In specific implementation, the acoustic wave propagation simulation for unbounded regions introduces a perfectly matched layer. Through geometric transformations, complex variable definitions, and coordinate transformations, boundary conditions suitable for the Galbrun equations are constructed, including:

[0091] (1) Based on the flow field velocity parameters in the flow field data, perform geometric transformation on the spatial and temporal coordinates of the computational domain to obtain a transformed coordinate system that adapts to the flow field motion state.

[0092] Specifically, flow field velocity parameters (such as time-averaged velocity components in each direction) are extracted from the solved flow field data, and the spatial coordinates of the computational domain are then used as a basis for calculation. , ) and time coordinates Perform geometric transformations. For example, if the flow field exists along... The velocity in the direction, then for Coordinates and time Perform a transformation so that the transformed coordinate system ( , , It can follow the overall motion trend of the flow field, eliminate the influence of the macroscopic motion of the flow field on the coordinate system, and obtain a transformed coordinate system that adapts to the motion state of the flow field.

[0093] (2) Based on the transformed coordinate system, define new coordinate parameters in complex form and corresponding matrix transformation rules, and set the energy absorption coefficient of the boundary region through complex parameters to form a new coordinate system with energy attenuation characteristics.

[0094] Specifically, the new coordinate parameters are in the transformed coordinate system (a coordinate system adapted to the flow field motion, such as the one obtained in the previous steps). , The new coordinate system is a complex spatial coordinate system defined to achieve "boundary sound wave absorption," rather than the traditional real coordinate system. The matrix transformation rule is a mathematical matrix (usually a Jacobian matrix) used to quantitatively describe the "partial derivative relationship" between the new coordinate parameters and the transformed coordinate system. Its core function is to establish a transformation channel for the "sound wave rate of change (differential operator)" under the two coordinate systems. The energy absorption coefficient is a parameter defined by the imaginary part function α(x') (or α(y'), α(z', corresponding to different spatial directions) of the new coordinate parameters, used to "quantitatively adjust the sound wave energy attenuation rate in the boundary region of the computational domain." The new coordinate system is a coordinate system built upon the aforementioned "new coordinate parameters" and "matrix transformation rules," possessing both "spatial location description" and "boundary energy absorption" functions. It is the core carrier for realizing the "Perfect Matching Layer (PML)."

[0095] In specific implementation, in the transformed coordinate system ( , , Under this condition, define new coordinate parameters (such as ξ, η, ζ). Each parameter contains a complex form (e.g., ξ = x' + iα(x'), where i is the imaginary unit and α(x') is a real function). Simultaneously, matrix transformation rules are established, such as using Jacobian matrices to describe the new coordinates (ξ, η, ζ) and the transformed coordinates (ξ, η, ζ). , , The partial derivative relationship of ) is determined by setting the real function α(x') in the complex parameters. In the boundary region of the computational domain (such as the grid near the boundary), α(x') is made to take a larger positive value, so that the sound wave energy entering the region will be exponentially attenuated due to the imaginary part of the complex coordinates, thus forming a new coordinate system with energy attenuation characteristics.

[0096] (3) Based on the new coordinate parameters and matrix transformation rules, the new coordinate system is converted back to the spatial and temporal coordinates used in actual calculations, and the correspondence between the differential operators describing the changes of sound waves under the two coordinate systems is determined.

[0097] Specifically, the conversion from the new coordinate system back to the actual computational coordinate system is necessary because the new coordinate system is an abstract mathematical tool designed to achieve energy absorption, while actual calculations rely on real coordinates corresponding to the physical scene and mesh division. Otherwise, the boundary absorption effect cannot be integrated into the solution framework of the Galbrun equation based on the actual coordinates. The correspondence between the two coordinate systems is mainly reflected in the transformation relationship between the differential operators describing sound wave changes (such as spatial gradient, divergence, and time partial derivatives). Spatial operators are related through the partial derivatives (Jacobi matrix) of the new coordinates and the actual coordinates and the chain rule, while time operators are linked through the partial derivatives of the time coordinate transformation and the flow field coupling terms. Calculating this correspondence is to transform the energy attenuation mechanism designed using complex numbers in the new coordinate system into a form recognizable by the Galbrun equations in the actual coordinates. This ensures that the equations maintain physical consistency during numerical discretization and achieve non-reflective sound wave absorption in the boundary region, avoiding spurious reflections from affecting the simulation results.

[0098] In specific implementation, the step of converting the new coordinate system back to the spatial and temporal coordinates used in actual calculations based on the new coordinate parameters and matrix transformation rules, and determining the correspondence between the differential operators describing sound wave changes in the two coordinate systems, includes: using the matrix transformation rules as the operational framework, taking the coordinate parameters in complex form in the new coordinate system as input, and mapping them to the spatial and temporal coordinates used in actual calculations through inverse coordinate transformation operations to obtain the correspondence between the two coordinate parameters; based on the correspondence between the coordinate parameters, determining the conversion method of the differential operators between the new coordinate system and the actual calculation coordinate system; during the conversion process, retaining the energy absorption characteristics of the new coordinate system due to the complex parameters in the differential operators of the actual calculation coordinate system through parameter transfer in the matrix transformation rules; verifying and adjusting the calculated conversion method by substituting it into typical boundary conditions, and ensuring that the adjusted differential operators in the two coordinate systems produce consistent results when describing the same sound wave change process.

[0099] Specifically, using matrix transformation rules (such as the Jacobian matrix and its inverse matrix) as the basic operational framework, the complex coordinate parameters (such as ξ = x' + i•α(x'), η = y' + i•α(y'), ζ = z' + i•α(z')) in the new coordinate system are used as input data. Through inverse coordinate transformation (i.e., deducing in reverse based on the functional relationship between the new coordinate parameters and the transformed coordinates), these complex coordinate parameters are mapped to the spatial coordinates used in the actual calculation. , ) and time coordinates This yields a one-to-one correspondence between the new coordinate parameters and the actual coordinate parameters (e.g., ξ corresponds to...). η corresponds The specific function expression, etc., is derived using the chain rule based on the coordinate parameter correspondence obtained above. This includes deriving the differential operators (including the spatial gradient operator). divergence operator Laplace operator and the time partial derivative operator The transformation method between the new coordinate system and the actual computational coordinate system is described. During the transformation, the imaginary part information (i.e., the part related to the energy absorption coefficient α) in the complex coordinate parameters is passed through parameters in the matrix transformation rules (such as Jacobian matrix elements). This allows the differential operator of the actual computational coordinate system to retain the energy absorption characteristics of the new coordinate system due to the complex parameters (e.g., introducing an α-related attenuation term in the differential operator). The calculated differential operator transformation method is then substituted into typical boundary conditions (such as perpendicular and oblique incidence boundaries of plane sound waves) for verification. By comparing the results of the differential operators describing the same sound wave change process (such as the attenuation amplitude and phase change of the sound wave at the boundary) under the two coordinate systems, the parameters in the transformation method (such as the correction coefficients of the matrix elements and the distribution gradient of the energy absorption coefficient) are adjusted until the calculation results of the differential operators under the two coordinate systems are consistent, ensuring that the transformed actual coordinate differential operator accurately reflects the energy absorption effect of the new coordinate system.

[0100] (4) Based on the correspondence between differential operators, the transformation rules of the perfect matching layer are designed for different spatial directions, and the transformation rules of each direction are integrated into a unified total transformation mechanism to form the boundary absorption rule.

[0101] Specifically, the boundary absorption rule is a unified, computationally applicable mechanism that integrates PML transformation rules from various spatial directions. It enables sound waves to be absorbed at the boundary without reflection. This mechanism includes direction-specific transformation logic (such as complex absorption parameters and differential operator transformations) and a comprehensive transformation logic adapted to the entire boundary. The correspondence between differential operators forms the basis for constructing the boundary absorption rule. The direction-specific transformation rules rely on this to design the differential operator transformation logic, and the comprehensive transformation mechanism relies on it to ensure compatibility of rules across directions, guaranteeing that the absorption effect can be accurately embedded into actual calculations.

[0102] In specific implementation, based on the correspondence between differential operators, conversion rules for perfectly matched layers are designed for different spatial directions. These conversion rules are then integrated into a unified overall conversion mechanism to form boundary absorption rules. This includes: determining conversion rules adapted to each direction based on the correspondence between differential operators and the flow field velocity components and energy absorption coefficients in each direction; each direction's conversion rule reflects the influence of flow field characteristics on acoustic wave absorption in the corresponding dimension; substituting the conversion rules of each direction into matrix conversion rules, and integrating the conversion logic of different directions through matrix operations to form a general conversion mechanism containing information from all spatial directions; based on this general conversion mechanism, extracting the correlation between flow field parameters and energy absorption coefficients, and converting this correlation into a mathematical expression with the same dimension and variable type as the coupling terms in the Galbrun equation; and replacing intermediate parameters in the mathematical expression that are inconsistent with the variables in the Galbrun equation with the core variables of the Galbrun equation through variable substitution to form boundary absorption rules.

[0103] Specifically, first, identify the core input parameters for each spatial direction, and then extract the time-averaged velocity component for that direction from the flow field data (e.g., for the x-direction). At the same time, it calls the preset energy absorption coefficient in that direction from the previous steps (e.g., the energy absorption coefficient corresponding to the x-direction). Based on the mathematical basis of the correspondence between differential operators (such as the gradient operator transformation formula between actual coordinates and new coordinates), and combined with the flow field characteristics in this direction (such as...), (Caused by acoustic convection effect), deriving the PML transformation rule in this direction: for example, in the x-direction rule, it is necessary to add to the differential operator transformation formula. and Coupling terms (such as) , (This is the new coordinate in the x-direction). Similarly, this process is repeated for other directions to obtain independent transformation rules for each direction. The transformation rules for each direction are substituted into the previously established matrix transformation rules (such as the Jacobian matrix framework) according to their dimensional correspondence. Assuming the core of the matrix transformation rules is the Jacobian moment J, the differential operator coefficients in the x-direction transformation rules are filled into the (1,1) position of matrix J, the y-direction coefficients into the (2,2) position, and so on, forming a diagonalized Jacobian matrix (if there is no coupling between directions, it is a diagonal matrix; if there is coupling, off-diagonal elements are added). Through matrix multiplication, the transformation logic of each direction is merged: for example, in a 3D scene, the overall transformation mechanism is "actual coordinate differential operator vector = merged Jacobian matrix × new coordinate differential operator vector", which preserves the flow field characteristics and absorption effects of each direction while eliminating computational conflicts between directions, forming a unified overall transformation mechanism covering all spatial directions. Furthermore, the correlation between flow field parameters and energy absorption coefficient is deconstructed from the overall conversion mechanism—for example, by extracting the "" from the elements of the fused Jacobian matrix. and The relationships, such as "ratios," are considered. Referring to the characteristics of coupling terms in the Galbrun equation (e.g., coupling terms are often in the form of a three-dimensional vector of "velocity-pressure-spatial derivative," with variables being real physical quantities), the extracted relationships are transformed into mathematical expressions of the same dimension and variable type: if the Galbrun equation coupling terms are three-dimensional vectors, the relationships are also organized into three-dimensional vector form; if the coupling terms are in real differential form, the complex terms (including i) in the relationships are transformed into real expressions through real and imaginary solutions. The intermediate parameters in the above mathematical expressions (such as the new coordinate coefficients temporarily defined in the previous steps) are then analyzed. , ), referring to the core variables of the Galbrun equation (such as Lagrange perturbation pressure and displacement potential) Establish the substitution relationship between intermediate parameters and core variables—for example, if the intermediate parameter It is proportional to the spatial derivative of the displacement potential ( , If it is a constant, then the expression will contain... Replace with After replacing all non-core variables one by one, we obtain mathematical rules containing only the core variables of the Galbrun equation, namely the boundary absorption rules.

[0104] For example, in one embodiment, a geometric transformation of the two-dimensional numerical model is first performed:

[0105] ;

[0106] Among them, the , , The original physical coordinates and time; , , The intermediate coordinates and time after transformation; For reference speed of sound; the Let be the Mach number in the y-direction. It is the Mach number.

[0107] Next, we introduce new variables:

[0108] ;

[0109] Among them, the , , The final target variable after transformation; The correction coefficients related to the Mach number in the x-direction; This is an additional correction factor in the x-direction. The For reference speed of sound.

[0110] Then, applying the classic PML complex transformation, the Jacobian matrix is ​​given:

[0111] ;

[0112] Among them, the , , The final transformed target variable; Here is the PML correction factor in the x-direction. , The positive absorption coefficient, the ω is the angular frequency.

[0113] In the final step, return to the initial time and space coordinates ( , , ), so that:

[0114] ;

[0115] Among them, the , , The final transformed target variable; For the additional correction factor in the x-direction; the The Mach number in the y-direction; The correction coefficients related to the Mach number in the x-direction; , , These are the original physical coordinates and time.

[0116] PML layer ( The total transformation of () This can be obtained by multiplying with a matrix. The total transformation of the x-PML layer can be expressed as:

[0117] ;

[0118] Among them, the For the total transformation of the x-PML layer; the The PML correction factor in the x-direction; The Mach number in the y-direction; The This is an additional correction factor in the x-direction.

[0119] Applying this to the differential operator in spatial variables yields:

[0120] ;

[0121] ;

[0122] ;

[0123] Among them, the The modified gradient operator; The PML correction factor in the x-direction; For correction factors; the The Mach number in the y-direction; The modified divergence operator; The displacement potential gradient; , The displacement potential gradients in the x and y directions; This is the modified gradient operator (for vector fields).

[0124] Define the y-PML layer transformation using the same method:

[0125] ;

[0126] Among them, the For y-PML layer transformation; the The PML correction factor in the y-direction; Let be the Mach number in the x-direction; , , The For the additional correction coefficient in the y-direction, the For reference speed of sound; The positive absorption coefficient, the ω is the angular frequency.

[0127] Then it will be given:

[0128] ;

[0129] Among them, the The modified gradient / differential operator; For correction factors; the The Mach number in the x-direction; the y is the PML correction factor in the y direction.

[0130] Then combine the x-layer and y-layer transformations in the equation:

[0131] ;

[0132] Among them, the The transformation matrix; The PML correction factor in the x-direction; For correction factors; the The Mach number in the y-direction; For correction factors; the The Mach number in the x-direction; the y is the PML correction factor in the y direction.

[0133] Further results were obtained:

[0134] ;

[0135] ;

[0136] Among them, the For the modified gradient class operator; the The PML correction factor in the x-direction; For correction factors; the The Mach number in the y-direction; For correction factors; the The Mach number in the x-direction; the The PML correction factor in the y-direction; To correct the operator for vector fields The function; the described , Let be a vector field in the x and y directions.

[0137] Finally, a perfectly matched layer (PML) suitable for the hybrid Galbrun equation is obtained:

[0138] ;

[0139] Among them, the The Lagrange perturbation velocity vector; For Lagrange perturbation pressure; the To correct the gradient operator; the Density; For reference speed of sound.

[0140] The method provided in this embodiment uses flow field velocity to drive geometric transformation, ensuring that the transformed coordinate system conforms to the flow field motion. This incorporates the differential operator of the Galbrun equation into the flow field convection effect, guaranteeing the physical consistency of acoustic-flow coupling at the coordinate system level. It avoids computational distortion of sound wave propagation due to coordinate system misalignment and provides a mathematical framework that conforms to the flow field for subsequent coupling. A new coordinate system with a complex imaginary part is defined, and the energy absorption coefficient is encoded into the imaginary part. Combined with matrix transformation, a new coordinate system with attenuation is constructed, transforming open, unbounded sound propagation into a computable problem with a finite computational domain and an absorption layer. This avoids secondary coupling interference between boundary reflected sound waves and the flow field. To ensure the purity of the acoustic-flow coupling process and focus on real interactions, this embodiment derives the correspondence of differential operators through inverse coordinate transformation, and transmits the attenuation logic of the new coordinate system back to the actual calculation coordinates. This allows the Galbrun equation to reflect boundary absorption in real coordinates, ensuring that the absorption effect is consistent with actual physics, avoiding spurious effects introduced by coordinate transformation, and improving the reliability of the results. The PML transformation rules are designed separately for different spatial directions and then integrated into a general mechanism, enabling the equation to flexibly handle anisotropic boundaries in non-uniform flow fields, precisely control the details of acoustic-flow coupling in each direction, suppress reflected waves in specific directions while retaining the flow field's driving force on sound waves, and improve simulation accuracy under complex boundaries. Overall, this embodiment, from the dimensions of coordinate system adaptation, unbounded absorption, physical consistency, and boundary refinement, endows the Galbrun equation with the ability to adapt to non-uniform flow fields and accurately simulate acoustic-flow coupling. It overcomes the bottlenecks of boundary reflection interference in non-uniform flow fields and the difficulty in calculating unbounded regions, accurately analyzes the complex coupling process of flow field and sound waves, and provides a high-precision simulation tool for engineering problems such as aero-engine noise.

[0141] S104. The Galbrun equation is weakly transformed, the boundary conditions are embedded, and the flow field data is combined with the triangular finite element method for discretization. A global linear algebraic equation system is constructed and solved to obtain the discrete solution of the sound field parameters.

[0142] Specifically, the discrete solution of the sound field parameters transforms the sound field physical quantities, which are originally continuously distributed in space and time, into specific values ​​at the nodes of discrete triangular units. This "set of node values" approximates the distribution pattern of the actual continuous sound field, containing both the magnitude of the sound field parameters and reflecting their spatial distribution differences in the flow field (such as sound wave attenuation and phase changes in different flow field regions). The discrete solution of the sound field parameters is a key bridge and core intermediate result for realizing the acoustic-fluid coupling calculation of non-uniform flow fields. From the perspective of the "two-way interaction of acoustic-flow coupling," on the one hand, the solution process of the discrete solution incorporates flow field data (such as flow field velocity, density, and other parameters reflected through the coefficients and geometric transformations of the Galbrun equation). The results accurately reflect the influence of flow field motion on the sound field (such as the deflection of the sound wave propagation direction and the stretching of the amplitude by high-speed airflow), directly representing the numerical manifestation of "flow field driving sound field changes." On the other hand, this discrete solution provides fundamental data for subsequent analysis of the "feedback effect of sound field convection." Through the discrete solution, the pressure and velocity disturbance distribution of the sound field can be obtained, and the force exerted by the sound wave on the fluid (such as sound radiation force) can be calculated. This force is then fed back into the flow field control equations as a source term, completing the closed-loop calculation of "acoustic-flow interaction." In short, the discrete solution is both the calculation result of "flow field influencing sound field" and the input basis for "sound field reacting on flow field." Without this step, the two-way physical process of acoustic-flow coupling cannot be fully simulated, ultimately making it difficult to analyze the true laws governing the interaction between sound and flow in non-uniform flow fields (such as the generation, propagation, and coupling evolution of aerodynamic noise with the flow field).

[0143] In specific implementation, the Galbrun equation is weakly transformed, the boundary conditions are embedded, and the flow field data is combined with the triangular finite element method for discretization. A global linear algebraic equation system is constructed and solved to obtain the discrete solution of the sound field parameters. This includes: weakly transforming the Galbrun equation, introducing trial functions and integrating both sides of the equation to transform it into an integral form suitable for finite element discretization; embedding the boundary conditions into the boundary integral terms of the weak form equation; using the energy absorption characteristics at the boundary as the constraint condition for solving the equation; introducing the flow field data and substituting it as a known parameter into the weak form equation; using triangular elements to mesh the computational domain and representing the sound field parameters in each triangular element using an interpolation function; discretizing the weak form equation on each triangular element based on the discretized interpolation function to obtain a unit-level linear algebraic equation system, assembling the unit equation systems to form a global linear algebraic equation system; and solving the global linear algebraic equation system using numerical methods to obtain the sound field parameter values ​​at each discrete node.

[0144] Specifically, a weak form transformation is performed on the Galbrun equation: Trial functions (such as velocity and pressure trial functions) corresponding to sound field parameters (e.g., Lagrange perturbation velocity and pressure) are introduced. The original equation is multiplied by these trial functions, and then integrated over the entire computational domain. This transforms the strong-form differential equation into a weak-form integral equation containing spatial and boundary integrals, eliminating the smoothness requirements of higher-order derivative terms on the solution. The constructed boundary conditions are substituted into the boundary integral terms of the weak-form equation. By imposing constraints on the trial functions or sound field parameters in the boundary sub-region (e.g., specifying that the trial functions are 0 at the boundary), the energy absorption characteristics at the boundary (e.g., PML) are optimized. The complex attenuation coefficient is transformed into a constraint condition for solving the equation, ensuring that the discrete solution satisfies the physical boundary behavior. Known parameters from the flow field data (such as the time-averaged velocity, time-averaged density, time-averaged pressure, and sound speed of the background flow field) are substituted as coefficients into the integrals of the weakly form equations, enabling the equations to reflect the influence of flow field motion on the sound field (such as convection effects and modulation of density non-uniformity). The computational domain is discretized using triangular elements, dividing the continuous computational domain into a large number of non-overlapping triangular elements. Each element is connected by vertices (nodes), and the spatial coordinates of each node are recorded, forming a mesh system covering the entire computational domain. Within each triangular element, the sound field parameters are expressed as node parameter values ​​and interpolation functions (such as linear interpolation basis functions). The sum of the products of the numbers (i.e., the sound field parameter distribution at any point within the element is approximated by the interpolation function using the unknown parameter values ​​at the element nodes). Based on the interpolation function within the element, the weak form equations are integrated on each triangular element. By transforming the integration into a linear combination of the element node parameters, a system of linear algebraic equations (including the element mass matrix, stiffness matrix, damping matrix, and element load vector) is obtained for each element. According to the connection relationship (shared nodes) of each triangular element in the computational domain, the system of linear algebraic equations of all elements is superimposed and assembled according to the nodal degrees of freedom to form a global system of linear algebraic equations covering the entire computational domain. Numerical solution methods suitable for linear algebraic equations (such as LU decomposition, conjugate gradient method, etc.) are used to solve the global system of equations to obtain the sound field parameter values ​​at all discrete nodes in the computational domain (such as the Lagrange perturbation velocity vector components and Lagrange perturbation pressure vector values ​​of each node), i.e., the discrete solutions of the sound field parameters.

[0145] For example, in one embodiment, a trial function is multiplied on both sides of the Galbrun equation. and Applying boundary conditions and integrating into the frequency domain, we obtain the following equation:

[0146] ;

[0147] Among them, the , For trial functions; the Angular frequency; Density; The Lagrange perturbation velocity vector; The time-averaged velocity vector; For Lagrange perturbation pressure; the For reference speed of sound; the The average pressure over time; For the computational domain; The , For volume source terms; the , This refers to a subregion that represents the boundary of the computational domain.

[0148] When using the hybrid finite element method, "hybrid" refers to the general form of the linear algebraic system that is produced when the problem is discretized:

[0149] ;

[0150] in, and It is a matrix. , , and It is a vector. Choosing the triangular finite element method to solve the numerical problem, after assembling the matrix and applying boundary conditions, the formula for the global discrete variables can be written as follows:

[0151] ;

[0152] Among them, the Angular frequency; The mass matrix; , , The damping / convection matrix; , , , The stiffness matrix; The Lagrange perturbation velocity vector; For Lagrange perturbation pressure; the The load vector for velocity disturbance; The load vector represents the pressure disturbance.

[0153] Finally, we can obtain the following formula:

[0154] ;

[0155] Among them, the The stiffness matrix; For an unknown degree of freedom vector; the For load vectors; Depends on Choose sparse storage. For fixed... Final use Obtained by decomposition.

[0156] The method provided in this embodiment, firstly, forms a complete closed loop through four closely related steps: "deriving the Galbrun equation → solving for non-uniform flow field data → constructing adaptive PML boundary conditions → finite element discretization to solve for sound field parameters." This comprehensively supports the acoustic-flow coupling calculation of non-uniform flow fields from the physical mechanism to numerical implementation. The Galbrun equation serves as the core governing equation, ensuring a clear mathematical framework for acoustic-flow coupling. The flow field data is solved using the Navier-Stokes equations combined with the k-ε turbulence model, providing a realistic non-uniform flow field environment (including average flow field and turbulent fluctuation information) for coupling calculations. Adaptive boundary conditions are constructed using PML to solve the boundary reflection problem in unbounded region simulation. Finally, the continuous acoustic-flow coupling problem is transformed into a computable system of algebraic equations through triangular finite element discretization. Each step is interconnected, ensuring both the coherence of the physical logic (the bidirectional influence from the flow field to the sound field) and the operability for engineering applications (efficient and stable numerical solution), allowing for the systematic simulation of the complex interaction between sound and flow in non-uniform flow fields.

[0157] Secondly, in constructing the Galbrun equation, the Euler equation for an ideal fluid without external forces is used as the basis, taking into account the physical essence of mass and momentum conservation. At the same time, Euler perturbation (from the perspective of a fixed spatial point), Lagrange perturbation (from the perspective of fluid particles), and mixed perturbation relationships are introduced to open up the conversion channel between the two types of perturbations. This can fully capture the sound field details caused by the velocity gradient in a non-uniform flow field (such as the velocity difference of sound wave propagation in the upstream and downstream directions, and the scattering of sound by turbulence). By simplifying the calculation through linearization and focusing on the core coupling relationship, redundant variables are finally eliminated. With Lagrange perturbation pressure and displacement potential as the core variables, the interaction between fluid particle motion and sound field propagation is directly reflected. This avoids the distortion of the coupling mechanism caused by neglecting the non-uniformity of the flow field in traditional models, and provides a theoretical model for sound-flow coupling that is physically rigorous and adapted to non-uniform flow fields.

[0158] Thirdly, a perfectly matched layer is introduced when constructing boundary conditions. Geometric transformation driven by flow field velocity makes the coordinate system fit the flow field motion, ensuring that the boundary conditions can adapt to the flow field convection effect. A new coordinate system with a complex imaginary part is defined and an energy absorption coefficient is set to give the boundary region non-reflection absorption characteristics, solving the problem of spurious reflection caused by truncation of the computational domain in the simulation of unbounded regions. The attenuation characteristics of the new coordinate system are transferred to the actual computational coordinates through coordinate transformation, and physical authenticity is guaranteed by relying on the correspondence of differential operators. The transformation rules are designed and integrated in different directions to form boundary absorption rules, which accurately deal with the anisotropic boundaries of non-uniform flow fields (such as differences in flow velocity and turbulence intensity in different directions). This allows the boundary conditions to not only adapt to the mathematical characteristics of the Galbrun equation, but also to realistically simulate the attenuation behavior of sound waves at the boundary of non-uniform flow fields, avoiding spurious reflections that interfere with the acoustic-flow coupling calculation, and significantly improving the accuracy and reliability of the simulation.

[0159] Corresponding to the aforementioned embodiment of the acoustic-fluid coupling calculation method based on the Galbrun equation, this application also provides an embodiment of an acoustic-fluid coupling calculation device based on the Galbrun equation.

[0160] Figure 2 This is a schematic diagram of the acoustic-fluid coupling calculation device based on the Galbrun equation provided in Embodiment 2 of this application. Please refer to... Figure 2 The apparatus provided in this embodiment includes an acquisition module 210, a solution module 220, and a construction module 230;

[0161] The acquisition module 210 is used to obtain the Galbrun equation based on the Euler equation under ideal fluid conditions without external force, combined with the relationships of Euler perturbation, Lagrange perturbation and mixed perturbation; the Galbrun equation characterizes the coupling effect between the flow field and the sound field.

[0162] The solution module 220 is used to solve for the flow field data of the non-uniform flow field based on the Navier-Stokes equations and combined with the k-ε turbulence model in the RANS model.

[0163] The construction module 230 is used to introduce a perfectly matched layer for the simulation of sound wave propagation in unbounded regions, and to construct boundary conditions suitable for the Galbrun equation through geometric transformation, complex variable definition and coordinate transformation.

[0164] The solution module 220 is also used to perform a weak transformation on the Galbrun equation, embed the boundary conditions, combine the flow field data, discretize the equations using triangular finite element method, construct a global linear algebraic equation system and solve it to obtain the discrete solution of the sound field parameters.

[0165] The apparatus of this embodiment can be used to perform... Figure 1The steps of the method embodiment shown are similar in principle and process, and will not be repeated here.

[0166] The specific implementation process of the functions and roles of each unit in the above device can be found in the implementation process of the corresponding steps in the above method, and will not be repeated here.

[0167] For the device embodiments, since they basically correspond to the method embodiments, the relevant parts can be referred to in the description of the method embodiments. The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this application according to actual needs. Those skilled in the art can understand and implement this without creative effort.

[0168] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A method for calculating acoustic-fluid coupling based on the Galbrun equation, characterized in that, The method includes: Based on the Euler equation for ideal fluid without external forces, and combining the relationships of Euler perturbation, Lagrange perturbation, and mixed perturbation, the Galbrun equation is obtained; the Galbrun equation characterizes the coupling effect between the flow field and the sound field. Based on the Navier-Stokes equations and combined with the k-ε turbulence model in the RANS model, the flow field data of the non-uniform flow field are obtained. For the simulation of sound wave propagation in unbounded regions, a perfectly matched layer is introduced. Boundary conditions suitable for the Galbrun equation are constructed through geometric transformation, complex variable definition, and coordinate transformation. The Galbrun equation is weakly transformed, the boundary conditions are embedded, and the flow field data is combined with the triangular finite element method for discretization. A global linear algebraic equation system is constructed and solved to obtain the discrete solution of the sound field parameters.

2. The method according to claim 1, characterized in that, The Euler equations based on ideal fluid without external forces, combined with the relationships of Euler perturbations, Lagrange perturbations, and mixed perturbations, yield the Galbrun equations, including: Choose the Euler equations for an ideal fluid under no external forces; The Eulerian perturbation of a physical quantity is defined as the dynamic change of the physical quantity at a fixed spatial position, and the Lagrangian perturbation of a physical quantity is defined as the change of the physical quantity of a fluid particle along its trajectory. Establish a hybrid perturbation relationship between Eulerian and Lagrange perturbations; the Lagrange perturbation is equal to the sum of the products of the Eulerian perturbation, the Lagrange displacement potential, and the steady-state uniform flow gradient; The flow field is defined as a steady flow field, and the Euler equations are linearized under the steady flow field; the steady-state uniform flow rate of the steady flow field is constant and does not change with time; Substituting the relationships of Euler perturbation, Lagrange perturbation, and mixed perturbation into the linearized Euler equations, and eliminating pressure, velocity, and density, we obtain the Galbrun equation with Lagrange perturbation pressure and Lagrange displacement potential as the core variables.

3. The method according to claim 2, characterized in that, The process of substituting the relationships of Euler perturbation, Lagrange perturbation, and mixed perturbation into the linearized Euler equations, and eliminating pressure, velocity, and density, yields the Galbrun equation with Lagrange perturbation pressure and Lagrange displacement potential as core variables, including: By taking the relationships of Euler perturbation, Lagrange perturbation and mixed perturbation as a whole, and simultaneously substituting them into the linearized mass conservation equation, momentum conservation equation and the relationship equation between pressure, density and entropy, we obtain the first set of equations containing perturbation variables. By substituting variables into the first set of equations, the perturbation terms of pressure, velocity, and density are expressed by Lagrange perturbation pressure and Lagrange displacement potential; By eliminating the original variables and intermediate disturbance terms of pressure, velocity, and density through a system of simultaneous equations, the Lagrange disturbance pressure and Lagrange displacement potential are retained as core variables. The resulting equations are rearranged to form the Galbrun equations, which simultaneously incorporate the coupling relationship between steady flow field parameters and acoustic field parameters.

4. The method according to claim 1, characterized in that, The acoustic wave propagation simulation for unbounded regions introduces a perfectly matched layer. Through geometric transformations, complex variable definitions, and coordinate transformations, boundary conditions suitable for the Galbrun equations are constructed, including: Based on the flow field velocity parameters in the flow field data, the spatial and temporal coordinates of the computational domain are geometrically transformed to obtain a transformed coordinate system that adapts to the flow field motion state. Based on the transformed coordinate system, new coordinate parameters in complex form and corresponding matrix transformation rules are defined. The energy absorption coefficient of the boundary region is set by the complex parameters to form a new coordinate system with energy attenuation characteristics. Based on the new coordinate parameters and matrix transformation rules, the new coordinate system is converted back to the spatial and temporal coordinates used in actual calculations, and the correspondence between the differential operators describing the changes of sound waves under the two coordinate systems is determined. Based on the correspondence between differential operators, transformation rules for perfectly matching layers are designed for different spatial directions. The transformation rules for each direction are integrated into a unified overall transformation mechanism to form boundary absorption rules.

5. The method according to claim 4, characterized in that, Based on the correspondence between differential operators, transformation rules for perfectly matching layers are designed for different spatial directions. These transformation rules are then integrated into a unified overall transformation mechanism, forming boundary absorption rules, including: Based on the correspondence of differential operators, and based on the velocity components and energy absorption coefficients of the flow field in each direction, a conversion rule adapted to the direction is determined; the conversion rule for each direction reflects the influence of the flow field characteristics in the corresponding dimension on sound wave absorption. Substitute the transformation rules of each direction into the matrix transformation rules, and integrate the transformation logic of different directions through matrix operations to form a total transformation mechanism that contains information of all spatial directions; Based on the overall conversion mechanism, the correlation between flow field parameters and energy absorption coefficient is extracted, and the correlation is converted into a mathematical expression with the same dimension and variable type as the coupling terms in the Galbrun equation. By substitution of variables, intermediate parameters in the mathematical expression that are inconsistent with the variables of the Galbrun equation are replaced with the core variables of the Galbrun equation, thus forming a boundary absorption rule.

6. The method according to claim 4, characterized in that, Based on the new coordinate parameters and matrix transformation rules, the process of converting the new coordinate system back to the spatial and temporal coordinates used in actual calculations, and determining the correspondence between the differential operators describing the changes in sound waves under the two coordinate systems, includes: Using the matrix transformation rule as the operational framework, the coordinate parameters in the new coordinate system, which contain complex forms, are taken as input. Through inverse coordinate transformation, they are mapped to the spatial and temporal coordinates used in actual calculations, thus obtaining the correspondence between the two types of coordinate parameters. Based on the correspondence of coordinate parameters, the transformation method of differential operators between the new coordinate system and the actual computation coordinate system is determined. During the transformation process, the energy absorption characteristics of the new coordinate system due to complex parameters are retained in the differential operators of the actual computation coordinate system through parameter transfer in the matrix transformation rules. The calculated transformation method was substituted into typical boundary conditions for verification and adjustment. The results of the differential operators in the two coordinate systems after adjustment were consistent when describing the same acoustic wave change process.

7. The method according to claim 1, characterized in that, The flow field data for non-uniform flow fields, obtained by solving the Navier-Stokes equations and combining them with the k-ε turbulence model in the RANS model, includes: We select the Navier-Stokes equations to describe fluid motion; By introducing the k-ε turbulence model from the RANS model, the turbulent motion in the flow field is decomposed into time-averaged motion and fluctuating motion. By defining turbulent kinetic energy and turbulent kinetic energy dissipation rate, an additional transport equation describing the fluctuating characteristics of turbulence is established. The additional transport equations are coupled with the Navier-Stokes equations to form a complete set of governing equations applicable to non-uniform flow fields; the complete set of governing equations can simultaneously reflect the average kinematic characteristics of the fluid and the influence of turbulent fluctuations on the flow. The coupled control equations are discretized, and the equations are solved numerically by combining the boundary conditions of the flow field computational domain to obtain the flow field data of the non-uniform flow field.

8. The method according to claim 7, characterized in that, The k-ε turbulence model introduced into the RANS model decomposes the turbulent motion in the flow field into time-averaged motion and fluctuating motion. By defining turbulent kinetic energy and turbulent kinetic energy dissipation rate, additional transport equations describing the fluctuating characteristics of turbulence are established, including: Based on the Reynolds time-averaged method, the physical quantities in the flow field are decomposed into time-averaged components and fluctuating components; the time-averaged components represent macroscopic average motion, and the fluctuating components represent random disturbance characteristics. Turbulent kinetic energy is defined as the turbulent pulsating kinetic energy per unit mass of fluid, and the turbulent kinetic energy dissipation rate is defined as the rate at which turbulent kinetic energy is dissipated due to viscosity. The magnitude of the turbulent kinetic energy is related to the square of the pulsating velocity component, and the turbulent kinetic energy dissipation rate is used to characterize the process of turbulent energy conversion into thermal energy. For non-uniform flow field characteristics, a first transport equation for turbulent kinetic energy is determined, and a second transport equation for turbulent kinetic energy dissipation rate is determined; the second transport equation introduces empirical coefficients related to turbulent kinetic energy and characteristic length. Solving the first and second transport equations yields the distributions of turbulent kinetic energy and turbulent kinetic energy dissipation rate. Based on these distributions, the turbulent viscosity coefficient is determined. The product of the turbulent viscosity coefficient and the turbulent kinetic energy dissipation rate is proportional to the square of the turbulent kinetic energy. The turbulent viscosity coefficient is incorporated as a correlation parameter into the coupling relationship between the first and second transport equations to form an additional transport equation system.

9. The method according to claim 1, characterized in that, The Galbrun equations are weakly transformed, the boundary conditions are embedded, and the flow field data is combined with the triangular finite element method for discretization. A global linear algebraic equation system is constructed and solved to obtain the discrete solutions of the sound field parameters, including: The Galbrun equation is transformed into a weak form by introducing trial functions and integrating both sides of the equation, thus transforming it into an integral form suitable for finite element discretization. The boundary conditions are embedded in the boundary integral terms of the weak form equation; the energy absorption characteristics at the boundary are the constraints for solving the equation. The flow field data is introduced and substituted into the weak form equation as known parameters. The computational domain is meshed using triangular elements, and the acoustic field parameters are represented by interpolation functions within each triangular element. Based on the discretized interpolation function, the weak form equations are discretized on each triangular element to obtain a system of linear algebraic equations at the element level. The system of equations at each element level is then assembled to form a global system of linear algebraic equations. The global linear algebraic equations are solved using numerical methods to obtain the acoustic field parameter values ​​at each discrete node.

10. A computational device for acoustic-fluid coupling based on the Galbrun equation, characterized in that, The device includes an acquisition module, a solution module, and a construction module; The acquisition module is used to obtain the Galbrun equation based on the Euler equation under ideal fluid conditions without external forces, combined with the relationships of Euler perturbation, Lagrange perturbation and mixed perturbation; the Galbrun equation characterizes the coupling effect between the flow field and the sound field. The solution module is used to solve for the flow field data of the non-uniform flow field based on the Navier-Stokes equations and combined with the k-ε turbulence model in the RANS model. The construction module is used to introduce a perfectly matched layer for the simulation of sound wave propagation in unbounded regions. Through geometric transformation, complex variable definition and coordinate transformation, it constructs boundary conditions suitable for the Galbrun equation. The solution module is also used to perform a weak transformation on the Galbrun equation, embed the boundary conditions, combine the flow field data, discretize the equations using triangular finite element method, construct a global linear algebraic equation system and solve it to obtain the discrete solution of the sound field parameters.

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