A method and apparatus for calculating acoustic streaming based on the galbrun equation

Abstract

The application provides a sound flow coupling calculation method and device based on a Galbrun equation. The method provided by the application comprises the following steps: obtaining a Galbrun equation based on Euler equations of an ideal fluid under no external force, in combination with Euler disturbance, Lagrange disturbance and mixed disturbance relationship; the Galbrun equation represents the coupling effect of a flow field and a sound field; obtaining flow field data of a non-uniform flow field based on Navier-Stokes equations, in combination with a k-ε turbulence model in a RANS model; introducing a perfect matched layer for sound wave propagation simulation in an unbounded region, and constructing boundary conditions suitable for the Galbrun equation through geometric transformation, complex variable definition and coordinate conversion; performing weak transformation on the Galbrun equation, embedding the boundary conditions, combining the flow field data, adopting triangular finite elements for discretization processing, constructing a global linear algebraic equation group and solving, and obtaining discrete solutions of sound field parameters.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of acoustic streaming coupling, in particular to an acoustic streaming coupling calculation method and device based on Galbrun equation. BACKGROUND

[0002] Under the background of continuous growth of global resource demand, as an important resource treasure and ecological barrier, the sustainable development and protection of the ocean has become the focus of international attention. Ocean acoustic technology plays an indispensable role in the fields of ocean communication, resource detection, etc. due to the unique advantage of underwater long-distance propagation of sound waves. However, the non-uniform flow field in the complex ocean environment can significantly change the propagation path, speed and energy distribution of sound waves, directly affecting the accuracy and reliability of acoustic detection, so it is of great significance to carry out the coupling calculation of underwater acoustic field and non-uniform flow field.

[0003] At present, various acoustic field calculation models have been proposed for the acoustic propagation characteristics under different environmental conditions, mainly including ray model, normal mode model, fast field model (FFM), parabolic equation model (PE) and hybrid model, etc. These traditional models are mostly based on Helmholtz equation and focus on acoustic field simulation calculation in homogeneous medium. The existing acoustic field calculation models have the following problems: on the one hand, they are difficult to accurately process the influence of flow velocity gradient in non-uniform flow field on acoustic propagation, and cannot fully reflect the complex coupling between flow field and acoustic field; on the other hand, in the face of nonlinear, high-frequency scenarios and complex seabed topography, etc., the calculation accuracy and applicability are greatly reduced, and it is difficult to meet the demand for high-precision acoustic field simulation in the fields of ocean engineering, military detection and underwater acoustic communication, etc.

[0004] Therefore, there is an urgent need for a method to accurately describe the influence of flow velocity gradient and seabed topography on acoustic propagation through the coupling calculation of underwater acoustic field and non-uniform flow field. SUMMARY

[0005] Therefore, the present application provides an acoustic streaming coupling calculation method and device based on Galbrun equation, which accurately describes the influence of flow velocity gradient and seabed topography on acoustic propagation through the coupling calculation of underwater acoustic field and non-uniform flow field.

[0006] Specifically, the present application is realized by the following technical solutions:

[0007] The first aspect of the present application provides an acoustic streaming coupling calculation method based on Galbrun equation, which comprises:

[0008] Based on the Euler equation of ideal fluid under the action of no external force, combined with the Euler disturbance, Lagrange disturbance and mixed disturbance relationship, the Galbrun equation is obtained; the Galbrun equation represents the coupling between flow field and acoustic field;

[0009] solving the non-uniform flow field based on the Navier-Stokes equation combined with the k-ε turbulence model in the RANS model to obtain flow field data;

[0010] A perfect matched layer is introduced for the simulation of sound wave propagation in an unbounded region, and boundary conditions suitable for the Galbrun equation are constructed through geometric transformation, complex variable definition and coordinate conversion.

[0011] The Galbrun equation is weakly transformed, the boundary conditions are embedded, the flow field data is combined, triangular finite elements are used for discretization processing, a global linear algebraic equation system is constructed and solved to obtain discrete solutions of the sound field parameters.

[0012] The second aspect of the application provides a sound flow coupling calculation device based on the Galbrun equation, which comprises an acquisition module, a solving module and a construction module.

[0013] The acquisition module is used to obtain the Galbrun equation based on the Euler equation of an ideal fluid under the action of no external force, combined with Euler disturbance, Lagrange disturbance and mixed disturbance relationship; the Galbrun equation represents the coupling effect of the flow field and the sound field.

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

[0015] The construction module is used to introduce a perfect matched layer for the simulation of sound wave propagation in an unbounded region, and boundary conditions suitable for the Galbrun equation are constructed through geometric transformation, complex variable definition and coordinate conversion.

[0016] The solving module is also used to weakly transform the Galbrun equation, embed the boundary conditions, combine the flow field data, use triangular finite elements for discretization processing, construct a global linear algebraic equation system and solve it to obtain discrete solutions of the sound field parameters.

[0017] The application provides a sound flow coupling calculation method and device based on the Galbrun equation. In a first aspect, a complete closed loop is formed through four closely related steps of "deriving the Galbrun equation, solving flow field data, constructing adaptive boundary conditions, and discretely solving sound field parameters", which fully connects from the physical mechanism to the numerical implementation: the Galbrun equation is used as the core control equation of sound flow coupling, the real data of the non-uniform flow field is obtained by relying on the Navier-Stokes equation combined with the k-ε turbulence model as input, the adaptive boundary conditions are constructed by the perfect matched layer to solve the simulation problem of unbounded region, and finally the sound field parameters are obtained by discrete solution through the finite element method. The steps are closely related to each other, ensuring the integrity and continuity of sound flow coupling calculation from theoretical basis to actual solution, and realizing the system simulation of the interaction between sound and flow in the non-uniform flow field. In a second aspect, when constructing the Galbrun equation, the Euler equation of an ideal fluid without external force is used as the basis, combined with the Euler disturbance, Lagrange disturbance and mixed disturbance relationship, the coupling physical mechanism of the flow field and the sound field is accurately captured. The equation naturally contains the influence of flow field motion on sound wave propagation (such as convection effect) and the feedback of sound wave on flow, providing a mathematically rigorous and physically meaningful control equation for sound flow coupling calculation, avoiding the distortion of coupling mechanism caused by the simplified model, and ensuring that the calculation can truly reflect the essential law of sound flow interaction. In a third aspect, the perfect matched layer is introduced when constructing the boundary conditions. Through geometric transformation, the boundary energy absorption coefficient is set by complex variable definition, the attenuation characteristics are transmitted by coordinate conversion, and the unified rules are formed by directional integration, which not only solves the boundary reflection problem caused by the truncation of the calculation domain in the simulation of sound wave propagation in unbounded region, but also makes the boundary conditions adapt to the Galbrun equation and the characteristics of the non-uniform flow field, ensures that the attenuation behavior of the sound field at the boundary conforms to the physical reality, avoids the interference of false reflection on the sound flow coupling calculation, and improves the accuracy and reliability of the simulation. BRIEF DESCRIPTION OF DRAWINGS

[0018] Figure 1 A flow chart of the sound flow coupling calculation method based on the Galbrun equation provided by the first embodiment of the application is provided.

[0019] Figure 2 A structure schematic diagram of the sound flow coupling calculation device based on the Galbrun equation provided by the second embodiment of the application is provided. DETAILED DESCRIPTION

[0020] The exemplary embodiments will be described in detail herein with reference to the accompanying drawings. Unless otherwise indicated, the same numbers in different drawings indicate the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all the embodiments consistent with the present application.

[0021] The terminology used in this application is for the purpose of describing particular embodiments only and is not intended to be limiting thereof. As used in this application, the singular forms "a," "an," and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms "and / or," as used herein, refer to and encompass any or all possible combinations of one or more of the associated listed items.

[0022] It is to be understood that, although the terms first, second, third, etc. can be used herein to describe various information, the information should not be limited to these terms. These terms are only used to distinguish one piece of information from another. For example, a first information can also be termed a second information, and, similarly, a second information can also be termed a first information, without departing from the scope of the present application. Depending on the context, the word "if' as used herein can be interpreted as meaning "when" or "upon determination" or "in response to determining."

[0023] The specific embodiments are given as follows to introduce the technical scheme of the present application in detail.

[0024] Figure 1 A flow chart of the sound flow coupling calculation method based on the Galbrun equation provided by Embodiment One of the present application is given. Please refer to Figure 1 The method provided by the present embodiment can include:

[0025] S101, based on the Euler equation of ideal fluid under no external force, combined with Euler disturbance, Lagrange disturbance and mixed disturbance relationship, Galbrun equation is obtained.

[0026] The Galbrun equation represents the coupling effect of flow field and sound field.

[0027] Specifically, the Euler equation is a basic control equation for describing the motion of an ideal fluid (inviscid, incompressible), which is derived based on physical laws such as mass conservation and momentum conservation, and mainly depicts the variation of physical quantities such as velocity and pressure of the fluid at a fixed spatial point with time, without tracking the motion trajectory of a specific fluid micro-cluster. It is a basic equation in fluid mechanics for analyzing the motion of a fluid. The Euler perturbation describes the deviation of physical quantities (such as velocity and pressure) in the flow field from their average state with respect to a fixed spatial position. That is, it observes the changes of fluid physical quantities at a fixed point before and after the perturbation occurs, without considering the source and motion history of the fluid micro-cluster at that position. The Lagrangian perturbation tracks the deviation of physical quantities (such as displacement and velocity) of a specific fluid micro-cluster from its initial state or average state during the motion process, and focuses on the changes of a specific fluid micro-cluster with time rather than a fixed spatial point. The mixed perturbation relationship is a mathematical relationship connecting the Euler perturbation and the Lagrangian perturbation. Since the Euler perturbation and the Lagrangian perturbation describe the changes of physical quantities from the perspectives of a spatial fixed point and a fluid micro-cluster respectively, the conversion between the two can be realized through the mixed perturbation relationship, so as to comprehensively utilize the characteristics of the two kinds of perturbations to more comprehensively depict the interaction between the flow field and the acoustic field.

[0028] Further, the Galbrun equation is derived based on the Euler equation of an ideal fluid under the action of no external force, combined with the Euler perturbation, the Lagrangian perturbation and the mixed perturbation relationship of the two. Its core function is to represent the coupling between the flow field and the acoustic field, and can reflect the influence of the flow field motion on the sound wave propagation and the reaction of the sound wave propagation on the flow field. Since the traditional acoustic model (such as Helmholtz equation) cannot accurately describe the complex coupling relationship between the non-uniform flow field and the acoustic field, the Galbrun equation can more comprehensively and accurately depict the influence of flow velocity gradient and other flow field characteristics on the sound wave propagation by integrating the characteristics of the Euler perturbation and the Lagrangian perturbation, and provides a more reliable theoretical basis for underwater acoustic field calculation under non-uniform flow field, thereby solving the problem of insufficient calculation accuracy of the traditional model in complex flow field environment.

[0029] In specific implementation, the Galbrun equation is derived based on the Euler equation of an ideal fluid under the action of no external force, combined with the Euler perturbation, the Lagrangian perturbation and the mixed perturbation relationship, including:

[0030] (1) Selecting the Euler equation set of an ideal fluid under the action of no external force.

[0031] Specifically, the Euler equation set of an ideal fluid under the action of no external force is selected, which includes mass conservation equation, momentum conservation equation, etc.

[0032] For example, in an embodiment, the Euler equation set 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 connecting Euler disturbance and Lagrange disturbance is constructed, and the specific composition of Lagrange disturbance is defined, i.e., the Lagrange disturbance is equal to the sum of two parts, the first part is the Euler disturbance corresponding to the physical quantity, and the second part is the product of the Lagrange displacement potential and the steady uniform flow gradient, and the quantitative conversion between the two disturbance forms is realized through the relationship.

[0045] For example, in an embodiment, the mixed disturbance relationship can be expressed as:

[0046] ;

[0047] Wherein, the is the Lagrange disturbance; the is the Euler disturbance; the is the Lagrange displacement potential; and the is the steady uniform flow gradient.

[0048] (4) Limiting the flow field to be a steady flow field, and linearizing the Euler equation set under the steady flow field; the steady uniform flow of the steady flow field is a constant and does not change with time.

[0049] Specifically, the calculation object is limited to a steady flow field, and the steady uniform flow of the steady flow field is a constant value and does not change with time; then, based on the steady flow field condition, the selected Euler equation set is linearized, and the high-order small terms in the Euler equation set are ignored, to obtain the linearized Euler equation set.

[0050] (5) Substituting the Euler disturbance, the Lagrange disturbance and the mixed disturbance relationship into the linearized Euler equation set, and eliminating the pressure, the velocity and the density, to obtain the Galbrun equation taking the Lagrange disturbance pressure and the Lagrange displacement potential as the core variables.

[0051] Specifically, the substitution of the Euler disturbance, the Lagrange disturbance and the mixed disturbance relationship into the linearized Euler equation set, and the elimination of the pressure, the velocity and the density to obtain the Galbrun equation taking the Lagrange disturbance pressure and the Lagrange displacement potential as the core variables, includes: the Euler disturbance, the Lagrange disturbance and the mixed disturbance relationship are taken as a whole and are simultaneously substituted into the linearized mass conservation equation, the momentum conservation equation and the relationship equation of pressure and density, entropy, to obtain a first equation set containing disturbance variables; the first equation set is subjected to variable substitution, and the disturbance terms of pressure, velocity and density are expressed by the Lagrange disturbance pressure and the Lagrange displacement potential; the original variables of pressure, velocity and density and the intermediate disturbance terms are eliminated by simultaneously solving the equation set, and the Lagrange disturbance pressure and the Lagrange displacement potential are reserved as the core variables; the obtained equation is arranged to form the Galbrun equation containing the coupling relationship between the steady flow field parameters and the sound field parameters.

[0052] In a specific implementation, the predefined Euler perturbation, the Lagrange perturbation, and the established mixed perturbation relationship are substituted into the mass conservation equation, the momentum conservation equation, and the pressure-density-entropy relationship equation that have been linearized. Through algebraic substitution, a first equation set containing various perturbation variables (including perturbation pressure, perturbation velocity, perturbation density, etc.) is obtained. For the perturbation terms of pressure, velocity, and density in the first equation set, they are re-expressed in terms of the Lagrange perturbation pressure and the Lagrange displacement potential according to the physical relationship and the pre-set rules. For example, the perturbation velocity is converted into the form of Further, the equation set after variable substitution is solved simultaneously, and the original variables (non-perturbation form , , ) of pressure, velocity, and density and the intermediate perturbation terms (transition variables that have not been completely converted) generated in the substitution and simultaneous solving process are all eliminated, so that only the Lagrange perturbation pressure and the Lagrange displacement potential remain in the equation set. The equation obtained after elimination is arranged according to the mathematical norms (combining like terms and adjusting the order of coefficients), and the Galbrun equation is finally formed.

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

[0054] ;

[0055] wherein the is the fluid density in a steady-state uniform flow field; the is the fluid pressure in a steady-state uniform flow field; the is the Lagrange displacement potential; the is the sound speed in a steady-state uniform flow field; , the is the fluid velocity in a steady-state uniform flow field; the is the angular frequency; the is the time variable, is the imaginary unit.

[0056] The method provided by the embodiment, when constructing the Galbrun equation, first selects the Euler equation set of an ideal fluid without external force as a basic framework for describing fluid motion, provides a physical logic starting point for acoustic flow coupling, defines Euler perturbation and Lagrange perturbation, covers the perturbation form of non-uniform flow field physical quantities from the dual perspectives of spatial fixation and particle tracking, further establishes a mixed perturbation relationship, opens up the conversion channel between the two, and completely captures the perturbation details caused by the non-uniformity of the flow field (such as flow velocity gradient); then, the stable flow field (steady-state uniform flow constant) is limited, and the Euler equation set is linearized, thereby simplifying the calculation while retaining the core physical laws, adapting to the engineering calculation requirements of acoustic flow coupling in the non-uniform flow field; finally, various perturbations are substituted into and the intermediate variables such as pressure, velocity, and density are eliminated, the Lagrange perturbation pressure and displacement potential are focused on, redundant variables are removed, the interaction between fluid particle motion and sound field propagation is directly reflected, the influence of the flow velocity gradient and seabed topography on sound propagation in the non-uniform flow field is accurately described, a practical, accurate, and reliable theoretical model is provided for marine engineering, military detection, underwater acoustic communication, and the like, and the practicability and accuracy of acoustic flow coupling calculation are improved, thereby helping to solve the acoustic technology application problems in complex marine environments, and realizing accurate and efficient calculation of acoustic flow coupling in a non-uniform flow field from the foundation equation, comprehensive perturbation description, equation simplification adaptation, and core variable focusing.

[0057] S102, based on the Navier-Stokes equation, combined with the k-ε turbulence model in the RANS model, flow field data of the non-uniform flow field is solved.

[0058] Specifically, the Navier-Stokes equation is a basic formula for describing how a fluid (a fluid involved in acoustic flow coupling) flows, can reflect the relationship between key attributes such as velocity, pressure, and viscosity of the fluid and motion, and is the “bottom basis” for calculating the flow field; since there are often irregular “turbulent vortices” in fluid motion, it is too complex to directly calculate the Navier-Stokes equation, the RANS model first divides the fluid motion into “average flow” and “fluctuating vortices”, the k-ε model then specially calculates “turbulent kinetic energy ( )” and “vortex energy consumption rate ( )”, and through simplified calculation, the turbulent flow field can be actually calculated, which is a “simplification tool” for adapting to complex flow fields.

[0059] Further, the flow field data of the non-uniform flow field refers to the key data of the “non-uniform flow field” (the flow field in which the fluid velocity, density, etc. is not uniformly distributed), such as fluid velocity, pressure, and turbulent intensity at different positions, which are core input data required for subsequent acoustic flow coupling calculation. That is, the embodiment uses the Navier-Stokes equation as the basis, relies on the k-ε turbulence model to simplify the calculation of complex turbulence, and finally calculates the velocity, pressure, and other flow field data of the non-uniform flow field. With these data, how the flow field and the sound field interact with each other is further analyzed, and acoustic flow coupling calculation is completed.

[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 fluid, the motion trajectory of fluid micro-cluster is disorder, and physical quantities such as velocity, pressure and temperature randomly fluctuate with time and space. In the framework of RANS model, time-averaged motion is the "averaged, relatively stable" flow state obtained by the mathematical method of "time averaging" (taking the average of instantaneous physical quantity in a period of time) of the instantaneous motion of fluid micro-cluster in turbulent flow (constantly changing due to random fluctuations of turbulent flow). For example, the instantaneous flow velocity at a point in the turbulent flow of the pipeline will randomly change due to turbulent fluctuations, but the time-averaged flow velocity obtained by averaging the instantaneous flow velocity over a long period of time is the embodiment of time-averaged motion, which can reflect the overall flow trend of the fluid. Fluctuation motion is the part superimposed on the "time-averaged motion" in turbulent flow, which is the random fluctuation of the instantaneous physical quantity (such as velocity and pressure) of fluid micro-cluster deviating from the time-averaged value. Still taking the turbulent flow of the pipeline as an example, the difference between the instantaneous flow velocity at a point and the time-averaged flow velocity is the fluctuation flow velocity, which reflects the random and irregular characteristics of turbulent flow and is the key factor of turbulent energy dissipation and momentum exchange.

[0070] Further, the additional transport equation is an equation introduced in the k-ε turbulent model to describe the turbulent fluctuation characteristics (such as the generation and dissipation of fluctuation kinetic energy) to "close" the Navier-Stokes equation after Reynolds averaging (because Reynolds averaging will introduce new unknown quantities). Specifically, the k-ε turbulent model introduces two additional transport equations: the turbulent kinetic energy equation describes the generation (such as the generation of kinetic energy due to the increase of fluctuation caused by the time-averaged velocity gradient), dissipation (due to fluid viscosity, fluctuation kinetic energy is converted into heat energy) and transmission (diffusion with fluid flow) rules of turbulent fluctuation kinetic energy (i.e. the energy possessed by unit mass of fluid due to fluctuation motion) in the flow field. The turbulent kinetic energy dissipation rate equation describes the rate of turbulent kinetic energy dissipation, which, in cooperation with the turbulent kinetic energy equation, can determine the characteristic scale of turbulent flow (such as vortex size) to enable the model to calculate the influence of turbulent flow on the time-averaged flow field. These two equations mathematically describe the key characteristics of turbulent fluctuations through "transport" (similar to the transport of physical quantities in fluid, including convection, diffusion, generation, dissipation, etc.), coupled with the main equation after Reynolds averaging, to solve the turbulent flow field.

[0071] In the specific implementation, the k-ε turbulent flow model introduced in the RANS model decomposes the turbulent motion in the flow field into time-averaged motion and fluctuation motion, establishes an additional transport equation describing the turbulent fluctuation characteristics by defining turbulent kinetic energy and turbulent kinetic energy dissipation rate, including: based on the Reynolds time-averaged method, the physical quantity in the flow field is decomposed into time-averaged component and fluctuation component; the time-averaged component represents the macroscopic average motion, and the fluctuation component represents the random disturbance characteristics; the turbulent kinetic energy is defined as the turbulent fluctuation kinetic energy possessed by the unit mass of fluid, and the turbulent kinetic energy dissipation rate is defined as the rate of dissipation of turbulent kinetic energy due to viscous action; the size of the turbulent kinetic energy is related to the square of the fluctuation velocity component, and the turbulent kinetic energy dissipation rate is used to represent the conversion process of turbulent energy to heat energy; for the characteristics of non-uniform flow field, a first transport equation of turbulent kinetic energy is determined, and a second transport equation of turbulent kinetic energy dissipation rate is determined; the second transport equation introduces an empirical coefficient related to the turbulent kinetic energy and the characteristic length; the distribution of the turbulent kinetic energy and the turbulent kinetic energy dissipation rate is obtained by solving the first transport equation and the second transport equation, and the turbulent viscosity coefficient is determined based on the distribution; 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 integrated into the coupling relationship of the first transport equation and the second transport equation as a correlation parameter to form an additional transport equation system.

[0072] Specifically, the Reynolds time-averaged method is used to decompose all physical quantities (including velocity, pressure, etc.) participating in the flow field description. The average operation is performed on each physical quantity in a long enough time interval, which is divided into two parts: time-averaged component and fluctuation component. Among them, the time-averaged component is obtained by time averaging the instantaneous value of the physical quantity, and the fluctuation component is obtained by calculating the difference between the instantaneous value of the physical quantity and the time-averaged component. Further, the turbulent kinetic energy is defined as the turbulent kinetic energy of unit mass fluid, and its value is determined by half the sum of the squares of each component of the fluctuating velocity; the turbulent energy dissipation rate is defined as the rate of conversion of turbulent kinetic energy into heat energy due to the action of fluid viscosity. According to the characteristics of the non-uniform flow field, the corresponding transport equation is constructed: the first transport equation of turbulent kinetic energy is established, which includes the convection term (describing the migration of turbulent kinetic energy with the fluid flow), the diffusion term (describing the molecular diffusion and turbulent diffusion of turbulent kinetic energy in the flow field), the generation term (mainly generated by the interaction of time-averaged velocity gradient and Reynolds stress, reflecting the process of energy transfer from time-averaged flow to turbulent fluctuation) and the dissipation term (directly related to the turbulent energy dissipation rate, reflecting the consumption of turbulent kinetic energy). The second transport equation of turbulent energy dissipation rate is established, which also includes the convection term, the diffusion term, the generation term (related to the generation term of turbulent kinetic energy, and related to the turbulent characteristic length) and the dissipation term, and introduces a plurality of empirical coefficients related to turbulent kinetic energy and turbulent characteristic length (these coefficients are based on a large amount of experimental data and experience, and are used to accurately describe the variation law of dissipation rate under different flow conditions). The first transport equation and the second transport equation are solved by numerical calculation method to obtain the specific distribution data of turbulent kinetic energy and turbulent energy dissipation rate at each position in the flow field. According to the distribution of turbulent kinetic energy and turbulent energy dissipation rate obtained by solving, the turbulent viscosity coefficient is calculated and determined, and in the calculation, the relationship between the turbulent viscosity coefficient and the square of the turbulent kinetic energy is proportional, and the relationship between the turbulent viscosity coefficient and the turbulent energy dissipation rate is inversely proportional. The calculated turbulent viscosity coefficient is used as the key correlation parameter and substituted into the first transport equation and the second transport equation, so that the two equations form a mutually coupled relationship, and finally form a complete additional transport equation system that can describe the characteristics of turbulent fluctuation.

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

[0074] ;

[0075] Among them, the is the fluid density; the is the turbulent kinetic energy; the is the time variable; the is the spatial coordinate variable, indicating the position in the flow field; the is the time-averaged velocity component of the fluid in the direction; the is the molecular dynamic viscosity; the is the turbulent viscosity coefficient; the is the Prandtl number of the turbulent kinetic energy; the is the turbulent production term, expressed as , the , is the time-averaged velocity component of the fluid in the corresponding direction; the is the dissipation term of the turbulent kinetic energy, the is the dissipation rate of the turbulent kinetic energy.

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

[0077] ;

[0078] wherein, the is the fluid density; the is the time variable; the is the spatial coordinate variable; the is the time-averaged velocity component of the fluid in the direction; the is the dissipation rate of the turbulent kinetic energy; the is the molecular dynamic viscosity; the is the turbulent viscosity coefficient; the is the Prandtl number of the turbulent kinetic energy; the is the turbulent kinetic energy; the and are model constants; the is the turbulent production term.

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

[0080] ;

[0081] wherein, the is the turbulent viscosity coefficient; the is the fluid density; the is the turbulent kinetic energy; the is the dissipation rate of the turbulent kinetic energy; the is a model constant.

[0082] (3) coupling the additional transport equation with the Navier-Stokes equation set to form a complete control equation set suitable for a non-uniform flow field; the complete control equation set can reflect the average motion characteristics of the fluid and the influence of the turbulent fluctuation on the flow at the same time.

[0083] In a specific implementation, the established k-ε additional transport equation is mathematically coupled with the selected Navier-Stokes equation set: in the momentum equation of the Navier-Stokes equation set, the turbulent fluctuation generates "Reynolds stress" (an additional force of turbulent flow on average motion), and the k-ε model converts the Reynolds stress into an expression related to the average motion parameters through the calculation results of k and ε and substitutes it into the momentum equation; at the same time, k and ε in the additional transport equation are solved in dependence on the average velocity, pressure and other parameters calculated by the Navier-Stokes equation set, and through such interrelated mathematical processing, a complete set of control equations is finally formed.

[0084] (4) The coupled control equation set is discretized and processed, combined with the boundary conditions of the flow field calculation domain, and a numerical solution method is used to solve the equation set to obtain the flow field data of the non-uniform flow field.

[0085] In a specific implementation, the complete coupled control equation set is discretized and processed: the flow field calculation domain (i.e. the fluid space range to be analyzed, such as the fluid region in a specific device) is divided into a large number of discrete units (such as tetrahedral, hexahedral grids), and then the continuous partial differential equation is converted into an algebraic equation set on each discrete unit (i.e. the "equation on continuous space" is converted into "numerical relationship on a limited number of grid nodes") through a discrete method such as the finite volume method or the finite element method; subsequently, the boundary conditions of the flow field calculation domain (such as the fluid velocity and pressure at the inlet of the calculation domain, the pressure at the outlet, the fluid no-slip condition on the wall surface, etc., which need to be set in combination with the interaction between the fluid and the boundary in the actual scene) are determined and converted into algebraic equation constraints for the corresponding discrete units; finally, a numerical solution method (such as the SIMPLE algorithm, the PISO algorithm, etc.) is used to iteratively solve the discretized algebraic equation set, 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 condition (such as the difference between adjacent two iterations is less than a 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 is obtained.

[0086] The method provided by the embodiment is based on the Navier-Stokes equation, strictly follows the conservation of mass and momentum, accurately restores the velocity field and pressure field distribution of the fluid average motion, provides a real flow field basis for the fact that the fluid convection changes the sound propagation direction and speed in the sound flow coupling (for example, the speed difference of sound wave propagation along and against the flow), introduces the k-ε turbulence model, defines the turbulent kinetic energy and turbulent kinetic energy dissipation rate by decomposing the turbulence into time-average and fluctuation motion, can quantitatively capture the scattering and attenuation effect of the sound propagation caused by the turbulent fluctuation in the non-uniform flow field (for example, the sound energy dissipation caused by the turbulent vortex), and fills the key physical process omitted by the pure laminar flow model. The complete control equation set formed by the coupling of the two can output "average flow field information + turbulent fluctuation information" at the same time. This exactly matches the dual requirements of sound flow coupling: the average flow field supports the "first-order influence of flow on sound" (for example, the drag effect of the average flow speed on sound propagation in the pipeline flow); and the turbulent fluctuation data are used to calculate the "second-order influence of flow on sound" (for example, sound scattering and energy dissipation in a high-turbulence area), so that the sound flow coupling can comprehensively cover the effects of complex flow field characteristics on acoustic propagation. After discretization processing and numerical solution, the irregular geometric boundary of the non-uniform flow field can be adapted, the spatial gradients of the flow speed and k-ε (for example, the speed mutation in the boundary layer) can be captured by the grid, and the coupling error caused by insufficient resolution of the flow field data can be avoided; and the mature numerical solution framework guarantees the calculation efficiency and stability, and provides efficient and reliable pre-input for subsequent sound flow coupling (which requires the flow field data as the boundary condition and source term of the acoustic equation).

[0087] S103, introducing a perfect matched layer for sound wave propagation simulation in an unbounded region, constructing a boundary condition suitable for the Galbrun equation through geometric transformation, complex variable definition and coordinate transformation.

[0088] Specifically, the perfect matched layer (PML) is a virtual absorption boundary layer artificially set, which can almost absorb the sound waves incident to the boundary without reflection through special material parameter design and mathematical processing (such as geometric transformation, complex variable definition), simulates the effect of natural attenuation of sound waves propagating to a distant place in an infinite space, and solves the problem of "false reflection of sound waves on the boundary of the finite calculation domain" in numerical calculation. The boundary condition refers to the constraint or setting of a physical quantity (such as sound pressure, displacement potential) on the boundary of the calculation domain, which is used to describe the propagation behavior (such as reflection, transmission, absorption, etc.) of sound waves at the boundary, and is an indispensable condition 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). , ) defined for realizing "boundary absorbing sound wave" instead of traditional real number coordinate. Matrix conversion rule is a mathematical matrix (usually Jacobian matrix) used to quantitatively describe the "partial derivative relationship" between new coordinate parameters and transformed coordinate system coordinates. The core role is to establish the conversion channel of "sound wave change rate (differential operator)" under two kinds of coordinates. 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 parameter, which is used to "quantitatively adjust the sound wave energy attenuation rate of the calculation domain boundary area". The new coordinate system is a coordinate system with "spatial position description" and "boundary energy absorption" dual functions, which is constructed based on the above "new coordinate parameters" and "matrix conversion rule", and is the core carrier to realize "perfectly matched layer (PML)".

[0095] In the transformed coordinate system (X, Y, Z) , , , new coordinate parameters (such as ξ, η, ζ ) are defined, each of which contains a complex number form (such as ξ = x' + iα(x'), i is the imaginary unit, and α(x') is a real function). At the same time, the matrix conversion rule is established, such as using Jacobian matrix to describe the partial derivative relationship between new coordinates (ξ, η, ζ) and transformed coordinates (X, Y, Z) , , . By setting the real function α(x') in the complex parameter, a larger positive value is taken for α(x') in the calculation domain boundary area (such as the grid close to the boundary), so that the sound wave energy entering this area will be exponentially attenuated due to the imaginary part of the complex coordinate, forming a new coordinate system with energy attenuation characteristics.

[0096] (3) Based on the new coordinate parameters and matrix conversion rule, the new coordinate system is converted back to the space and time coordinates actually used for calculation, and the corresponding relationship between the differential operators describing the sound wave change under two kinds of coordinate systems is determined.

[0097] Specifically, the new coordinate system is converted back to the actual calculation coordinates because the new coordinates are abstract mathematical tools designed for energy absorption, and the actual calculation relies on real coordinates corresponding to the physical scene and grid division, otherwise the boundary absorption effect cannot be integrated into the Galbrun equation solving framework established based on the actual coordinates. The correspondence between the two coordinate systems mainly reflects the conversion relationship between the differential operators (such as spatial gradient, divergence, and time partial derivative) describing the change of sound waves. The spatial operator is associated through the partial derivative (Jacobian matrix) of the new coordinates and the actual coordinates and the chain rule of differentiation, and the time operator is associated through the partial derivative of the time coordinate conversion and the flow field coupling term. The calculation of this correspondence is to convert the energy attenuation mechanism designed in the new coordinates through complex numbers into a form recognizable by the Galbrun equation in the actual coordinates, ensuring that the equation maintains the consistency of physical meaning when numerically discretized and solved, and achieving sound wave non-reflective absorption in the boundary area to avoid false reflections affecting the simulation results.

[0098] In a specific implementation, the conversion of the new coordinate system back to the spatial and time coordinates used in actual calculation based on the new coordinate parameters and the matrix conversion rule determines the correspondence between the differential operators describing the change of sound waves in the two coordinate systems, including: taking the coordinate parameters in the new coordinate system containing complex numbers as input based on the matrix conversion rule as the operation framework, mapping them to the spatial and time coordinates used in actual calculation through coordinate inverse transformation operation, to obtain the correspondence between the two coordinate parameters; based on the correspondence between the coordinate parameters, determining the conversion mode of the differential operators between the new coordinate system and the actual calculation coordinate system, and in the conversion process, the energy absorption characteristics possessed by the new coordinate system due to the complex parameters are preserved in the differential operators of the actual calculation coordinate system through parameter transmission in the matrix conversion rule; the conversion mode obtained by calculation is substituted into a typical boundary condition for verification and adjustment, and the differential operators in the two coordinate systems after adjustment are consistent in describing the same sound wave change process.

[0099] Specifically, taking the matrix conversion rule (such as the Jacobian matrix and its inverse matrix) as the basic operation framework, taking the coordinate parameters (such as ξ = x' + i•α(x'), η = y' + i•α(y'), ζ = z' + i•α(z')) in the new coordinate system containing complex numbers as input data, and through coordinate inverse transformation operation (i.e. reverse derivation according to the functional relationship between the new coordinate parameters and the transformed coordinates), these complex coordinate parameters are mapped to the spatial coordinates (x, y, z) 、 ) and time coordinates used in actual calculation, to obtain a one-to-one correspondence between the new coordinate parameters and the actual coordinate parameters (such as ξ corresponding to x, η corresponding to y, and ζ corresponding to z). ​specific function expressions of the like). Based on the above-obtained coordinate parameter correspondence relationship, the conversion mode of differential operators (including the gradient operator ∇ , the divergence operator ∇ , the Laplace operator ∇ 2 , and the partial derivative operator with respect to time ) between the new coordinate system and the actual calculation coordinate system is derived by using the chain rule. In the conversion process, the imaginary part information (i.e., the part related to the energy absorption coefficient a) in the complex coordinate parameter is transmitted through the parameters in the matrix conversion rule (such as the elements of the Jacobian matrix), so that the differential operator of the actual calculation coordinate system retains the energy absorption characteristics possessed by the new coordinate system due to the complex parameter (such as introducing an a-related attenuation term in the differential operator). The conversion mode of the differential operator obtained by calculation is substituted into typical boundary conditions (such as the scenarios of vertical incidence boundary of plane sound wave, oblique incidence boundary, etc.), and the results of the differential operators in the two coordinate systems describing the same sound wave variation process (such as the attenuation amplitude and phase change of the sound wave at the boundary) are compared. The parameters in the conversion mode (such as the correction coefficient of the matrix element and the distribution gradient of the energy absorption coefficient) are adjusted until the calculation results of the differential operators in the two coordinate systems are consistent, so as to ensure that the actual coordinate differential operator after conversion can accurately reflect the energy absorption effect of the new coordinate system.

[0100] (4) Based on the correspondence relationship between the differential operators, the conversion rules of the PML in different spatial directions are designed respectively, the conversion rules in each direction are integrated into a unified total conversion mechanism, and the boundary absorption rule is formed.

[0101] Specifically, the boundary absorption rule is a unified and directly usable mechanism formed by integrating the PML conversion rules in each spatial direction, which can allow the sound wave to be absorbed without reflection at the boundary, and includes the total conversion logic suitable for the whole boundary and the direction-specific conversion logic (such as the complex absorption parameter in each direction and the differential operator conversion). The differential operator correspondence relationship is the basis for constructing the boundary absorption rule, the direction-specific conversion rule needs to rely on it to design the differential operator conversion logic, and the total conversion mechanism needs to rely on it to ensure the compatibility of the rules in each direction and to ensure that the absorption effect can be accurately embedded in the actual calculation.

[0102] In a specific implementation, the conversion rules of the PML for different spatial directions are designed based on the correspondence between differential operators, and the conversion rules for different directions are integrated into a unified total conversion mechanism to form the boundary absorption rule, including: based on the correspondence between differential operators, the flow field velocity component and the energy absorption coefficient of each direction are determined to adapt the conversion rule of the direction; the conversion rule of each direction reflects the influence of the flow field characteristics in the corresponding dimension on the sound wave absorption; the conversion rules of different directions are substituted into the matrix conversion rule, and the conversion logic of different directions is integrated through matrix operation to form a total conversion mechanism containing all spatial direction information; based on the total conversion mechanism, the correlation between the flow field parameters and the energy absorption coefficient is extracted, and the correlation is converted into a mathematical expression form with the same dimension and variable type as the coupling term in the Galbrun equation; through variable substitution, the intermediate parameters inconsistent with the variables of the Galbrun equation in the mathematical expression form are replaced by the core variables of the Galbrun equation to form the boundary absorption rule.

[0103] Specifically, the core input parameters of each spatial direction are first determined, and the time-averaged velocity component of the direction (for example, x-direction takes ) is extracted from the flow field data, and the energy absorption coefficient of the direction (for example, the x-direction corresponds ) is called from the preset in the previous step. Based on the correspondence between differential operators (for example, the gradient operator conversion formula of the actual coordinates and the new coordinates), the flow field characteristics of the direction (for example, the advection effect of the sound wave caused by ) are combined to derive the PML conversion rule of the direction: for example, in the x-direction rule, the coupling term of and (including , for the new coordinates of the x-direction) needs to be added to the differential operator conversion formula; similarly, the other directions repeat this process to obtain independent conversion rules for each direction. The conversion rule of each direction is substituted into the matrix conversion rule (for example, the Jacobian matrix framework) established in the previous step according to the dimension correspondence. Assuming that the core of the matrix conversion rule is the Jacobian matrix J, the differential operator coefficients in the x-direction conversion rule are filled into the (1, 1) position of the matrix J, the y-direction coefficients are filled into the (2, 2) position, and so on, to form a diagonalized Jacobian matrix (if there is no coupling between directions, it is a diagonal matrix, and if there is coupling, non-diagonal elements are supplemented). The conversion logic of different directions is integrated through matrix multiplication operation: for example, in a three-dimensional scene, the total conversion mechanism is "actual coordinate differential operator vector = fused Jacobian matrix × new coordinate differential operator vector", which not only retains the flow field characteristics and absorption effect of each direction, but also eliminates the calculation conflicts between directions, forming a unified total conversion mechanism covering all spatial directions. Further, the correlation between the flow field parameters and the energy absorption coefficient is extracted from the total conversion mechanism, for example, from the elements of the fused Jacobian matrix, with ratio” and so on. Refer to the characteristics of the coupling terms in Galbrun equation (e.g. the coupling terms are mostly in the form of three-dimensional vector of “velocity-pressure-space derivative”, and the variable type is real physical quantity), the extracted correlation is converted into a mathematical expression of the same dimension and variable type: if the coupling term of Galbrun equation is a three-dimensional vector, the correlation is also arranged in the form of a three-dimensional vector; if the coupling term is in the form of real differential, the complex terms (including i) in the correlation are decomposed into real expressions through the real part and the imaginary part. The intermediate parameters in the above mathematical expressions (such as the new coordinate coefficients defined temporarily in the previous step , ) are combed, and the replacement relationship between the intermediate parameters and the core variables of Galbrun equation (such as the Lagrange disturbance pressure, displacement potential ) is established — for example, if the intermediate parameter is proportional to the spatial derivative of the displacement potential ( , is a constant), then in the expression is replaced by . After replacing all non-core variables one by one, the mathematical rule containing only the core variables of Galbrun equation is obtained, that is, the boundary absorption rule.

[0104] For example, in an embodiment, first, the geometric transformation of the two-dimensional numerical model is carried out:

[0105] ;

[0106] wherein, the , , are original physical coordinates and time; the , , are transformed intermediate coordinates and time; the is a reference sound speed; the is a Mach number in the y direction, is a Mach number.

[0107] Then new variables are introduced:

[0108] ;

[0109] wherein, the , , are final transformed target variables; is a correction coefficient related to the Mach number in the x direction; the is an additional correction coefficient in the x direction, , the is the reference sound speed.

[0110] Then the classical PML complex change is applied to give the Jacobian matrix:

[0111] ;

[0112] wherein the , , are the target variables after the final transformation; the is the PML correction coefficient in x direction, , is the positive absorption coefficient, and the is the angular frequency.

[0113] In the last step, the original time and space coordinates (x, y, t) are turned back to make: , ,

[0114] ;

[0115] wherein the , , are the target variables after the final transformation; the is the additional correction coefficient in x direction; the is the Mach number in y direction; is the correction coefficient related to the Mach number in x direction; and the , , are the original physical coordinates and time.

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

[0117] ;

[0118] wherein the is the total transformation of the x-PML layer; the is the PML correction coefficient in x direction; the is the Mach number in y direction; , and the is the additional correction coefficient in x direction.

[0119] Applying this to the differential operator in the space variable gives:

[0120] ; ​​​

[0121] ;

[0122] ;

[0123] wherein the is the modified gradient operator; the is the PML correction coefficient in x direction; the is the modified coefficient; the is the Mach number in y direction; the is the modified divergence operator; the is the displacement potential gradient; the , are the displacement potential gradients in x, y direction; the is the modified gradient operator (for vector fields).

[0124] The y-PML layer transformation is defined using the same approach:

[0125] ;

[0126] wherein the is the y-PML layer transformation; the is the PML correction coefficient in y direction; the is the Mach number in x direction; , , the is an additional correction coefficient in y direction, the is the reference sound speed; is the positive absorption coefficient, the is the angular frequency.

[0127] Afterwards:

[0128] ;

[0129] wherein the is the modified gradient / differential operator; the is the correction coefficient; the is the Mach number in x direction; the is the PML correction coefficient in y direction.

[0130] The x-layer and y-layer transformations in the equation are combined again:

[0131] ;

[0132] wherein the is the transformation matrix; the is the PML correction coefficient in x direction; the is a correction coefficient; the is a Mach number in the y direction; the is a correction coefficient; the is a Mach number in the x direction; the is a PML correction coefficient in the y direction.

[0133] Further, we have:

[0134] ;

[0135] ;

[0136] wherein the is a corrected gradient-like operator; the is a PML correction coefficient in the x direction; the is a correction coefficient; the is a Mach number in the y direction; the is a correction coefficient; the is a Mach number in the x direction; the is a PML correction coefficient in the y direction; the is an action of the correction operator on a vector field ; the , are vector fields in the x and y directions.

[0137] Finally, we obtain a Perfectly Matched Layer (PML) applicable to the mixed Galbrun equation:

[0138] ;

[0139] wherein the is a Lagrangian perturbed velocity vector; the is a Lagrangian perturbed pressure; the is a corrected gradient operator; the is a density; the is a reference sound speed.

[0140] The method provided by the embodiment drives geometric transformation through flow field speed, makes the transformed coordinate system fit the flow field movement, makes the Galbrun equation differential operator integrate the convection effect of the flow field, guarantees the sound flow coupling physical consistency from the coordinate system level, avoids the sound wave propagation calculation distortion caused by the misalignment of the coordinate system, provides a mathematical framework fitting the flow field for subsequent coupling, defines a new coordinate containing a complex imaginary part, encodes the energy absorption coefficient into the coordinate imaginary part, constructs a new coordinate system with attenuation by cooperating with matrix conversion, converts the open and unbounded sound propagation into a calculable problem of a limited calculation domain + absorption layer, avoids the secondary coupling interference of the boundary reflected sound wave and the flow field, guarantees the purity of the sound flow coupling process and focuses on the real interaction, deduces the corresponding relationship of the differential operator through coordinate inverse conversion, transmits the attenuation logic of the new coordinate system back to the actual calculation coordinate, makes the Galbrun equation reflect the boundary absorption under the real coordinate, ensures that the absorption effect is consistent with the actual physics, avoids the introduction of false effects by coordinate transformation, and improves the result reliability, designs PML conversion rules in the spatial direction and integrates them into the total mechanism, so that the equation can flexibly cope with the anisotropic boundary of the non-uniform flow field, accurately control the sound flow coupling details in each direction, suppress the reflection wave in a specific direction while retaining the driving of the flow field on the sound wave, and improve the simulation accuracy under complex boundaries. Overall, the embodiment gives the Galbrun equation the ability to adapt to the non-uniform flow field and accurately simulate the sound flow coupling from the coordinate system adaptation, unbounded absorption, physical consistency, and boundary refinement dimensions, breaks through the boundary reflection interference of the non-uniform flow field and the calculation bottleneck of the unbounded region, accurately analyzes the complex coupling process of the flow field and the sound wave, and provides a high-precision simulation tool for engineering problems such as aircraft engine noise.

[0141] S104, the Galbrun equation is weakly transformed, the boundary condition is embedded, the flow field data is combined, triangular finite elements are used for discretization processing, a global linear algebraic equation group is constructed and solved, and a discrete solution of the sound field parameter is obtained.

[0142] Specifically, the discrete solution of the sound field parameter is to convert the originally continuous distribution of the sound field physical quantity in space and time into specific numerical values on the nodes of the discrete triangular element, to approximate the distribution rule of the actual continuous sound field in the form of "node value set", which contains not only the size information of the sound field parameter, but also reflects the spatial distribution difference of the sound field in the flow field (such as the sound wave attenuation and phase change in different flow field regions). The discrete solution of the sound field parameter is the key bridge and core intermediate result to realize the sound flow coupling calculation in the non-uniform flow field. From the "two-way action of sound flow coupling", on the one hand, the solving process of the discrete solution has been integrated into the flow field data (such as the flow field velocity, density and other parameters through the coefficients of the Galbrun equation and geometric transformation), and its result can accurately reflect the influence of the flow field movement on the sound field (such as the deflection of high-speed airflow to the sound wave propagation direction and the stretching of the amplitude), which is the direct numerical embodiment of "flow field driving sound field change"; on the other hand, the discrete solution also provides basic data for the subsequent analysis of "feedback effect of sound field on flow transmission", through which the pressure and velocity disturbance distribution of the sound field can be obtained, and then the sound wave acting force on the fluid (such as the acoustic radiation force) can be calculated and fed back to the flow field control equation as a source term to complete the closed-loop calculation of "sound flow interaction". In short, the discrete solution is not only the calculation result of "flow field influencing sound field", but also the input basis of "sound field acting on flow field", without this link, the two-way physical process of sound flow coupling cannot be simulated completely, and finally it is difficult to analyze the real law of sound and flow interaction in the non-uniform flow field (such as the generation, propagation and coupling evolution of aerodynamic noise).

[0143] In specific implementation, the Galbrun equation is weakly transformed, the boundary conditions are embedded, the flow field data are combined, the triangular finite element is discretized, the global linear algebraic equation set is constructed and solved, and the discrete solution of the sound field parameter is obtained, including: the Galbrun equation is weakly transformed, a trial function is introduced, and integral operation is performed on both sides of the equation, which is transformed into an integral form suitable for finite element discretization; the boundary conditions are embedded in the boundary integral term of the weak form equation; the energy absorption characteristics at the boundary are the constraint conditions for equation solving; the flow field data are introduced, and the flow field data are substituted into the weak form equation as known parameters; the triangular element is used to divide the calculation domain, and the sound field parameter in each triangular element is represented by an interpolation function; based on the discretized interpolation function, the weak form equation is discretized on each triangular element to obtain a linear algebraic equation set at the element level, and the element equation sets are assembled to form a global linear algebraic equation set; the global linear algebraic equation set is solved by a numerical solving method to obtain the sound field parameter values at each discrete node.

[0144] Specifically, the Galbrun equation is weakly transformed: the test functions (such as the velocity test function, the pressure test function) corresponding to the sound field parameters (such as the Lagrangian disturbance velocity, the disturbance pressure) are introduced, the original equation is multiplied by the test function on both sides, and then integrated on the entire calculation domain, the strong form differential equation is transformed into a weak form integral equation containing spatial integral and boundary integral, and the high-order derivative term in the equation is eliminated to eliminate the smoothness requirement of the solution. The boundary conditions constructed are substituted into the boundary integral term of the weak form equation, the test function or the sound field parameter is constrained on the boundary sub-region (such as the test function being 0 on the boundary), the energy absorption characteristics (such as the complex attenuation coefficient of PML) at the boundary are transformed into constraint conditions for equation solving, and the discrete solution is ensured to satisfy the physical boundary behavior. The known parameters (such as the time-averaged velocity, the time-averaged density, the time-averaged pressure, the sound speed, etc.) in the flow field data are substituted as coefficients into each integral of the weak form equation, so that the equation can reflect the influence of the flow field motion on the sound field (such as the convection effect, the modulation of density non-uniformity). The calculation domain is discretized by using triangular elements, the continuous calculation domain is divided into a large number of non-overlapping triangular elements, each element is connected by vertices (nodes), the spatial coordinates of each node are recorded, and a grid system covering the entire calculation domain is formed. In each triangular element, the sound field parameters are expressed as the sum of the product of the node parameter values and the interpolation function (such as the linear interpolation basis function), that is, the unknown parameter values on the element nodes are used to approximate the sound field parameter distribution at any point in the element by using the interpolation function. Based on the interpolation function in the element, the weak form equation is integrated on each triangular element, and by converting the integral into a linear combination of the element node parameters, a linear algebraic equation set (including element mass matrix, stiffness matrix, damping matrix and element load vector) corresponding to each element is obtained. According to the connection relationship (shared nodes) of each triangular element in the calculation domain, the linear algebraic equation sets of all elements are stacked and assembled according to the node degrees of freedom to form a global linear algebraic equation set covering the entire calculation domain. A numerical solving method suitable for linear algebraic equation set (such as LU decomposition, conjugate gradient method, etc.) is used to solve the global equation set, and the sound field parameter values (such as the Lagrangian disturbance velocity vector component of each node, the Lagrangian disturbance pressure vector value) at all discrete nodes in the calculation domain are obtained, that is, the discrete solution of the sound field parameters.

[0145] For example, in an embodiment, the Galbrun equation is multiplied by the test function on both sides and . After applying the boundary conditions and integrating into the frequency domain, the following equation is obtained:

[0146] ;

[0147] wherein the test function is , the test function is is the angular frequency; the is the density; the is the Lagrangian perturbation velocity vector; the is the time-averaged velocity vector; the is the Lagrangian perturbation pressure; the is the reference sound speed; the is the time-averaged pressure; the is the computational domain; ; the , is the volume source term; the , is a subregion of the computational domain boundary.

[0148] In using the mixed finite element method, "mixed" means that the linear algebra system of general form is generated when the problem is discretized:

[0149] ;

[0150] where, and are matrices, , , and are vectors. Triangular finite elements are chosen to solve the numerical problem, and after assembling the matrices and applying the boundary conditions, the formula of the global discrete variable can be written as follows:

[0151] ;

[0152] where, the is the angular frequency; the is the mass matrix; the , , is the damping / convection matrix; the , , , is the stiffness matrix; the is the Lagrangian perturbation velocity vector; the is the Lagrangian perturbation pressure; the is the load vector of velocity perturbation; the is the load vector of pressure perturbation.

[0153] Finally, the following formula can be obtained:

[0154] ;

[0155] where, the is the stiffness matrix; the is an unknown degree of freedom vector; the is a load vector; depends on sparse storage is selected. For fixed final use is obtained by decomposition.

[0156] The method provided by the embodiment, in the first aspect, forms a complete closed loop through four closely related steps of "deriving Galbrun equation, solving non-uniform flow field data, constructing adaptive PML boundary condition, and finite element discrete solving sound field parameters", and fully supports non-uniform flow field sound flow coupling calculation from physical mechanism to numerical implementation: taking the Galbrun equation as the core control equation, ensuring that the sound flow coupling has a clear mathematical carrier; relying on the Navier-Stokes equation combined with the k-ε turbulence model to solve the flow field data, providing a real non-uniform flow field environment (including average flow field and turbulence fluctuation information) for coupling calculation; constructing adaptive boundary conditions through PML to solve the boundary reflection problem of unbounded region simulation; finally, discrete solving through triangular finite element, converting the continuous sound flow coupling problem into a calculable algebraic equation set, each step is closely related, which not only guarantees the coherence of the physical logic (two-way influence from the flow field to the sound field), but also realizes the operability of the engineering application (efficient and stable numerical solving), so that the complex interaction between sound and flow in the non-uniform flow field can be systematically simulated.

[0157] In the second aspect, when constructing the Galbrun equation, the Euler equation of an ideal fluid without external force is taken as the basis, the physical nature of mass and momentum conservation is considered, and the Euler disturbance (fixed space point perspective), Lagrange disturbance (fluid particle perspective) and mixed disturbance relationship are introduced, the conversion channel of the two disturbances is opened, which can completely capture the details of the sound field caused by the flow velocity gradient in the non-uniform flow field (such as the speed difference of sound wave propagation in the same direction and opposite direction, and the scattering of turbulence on sound); through linearization processing, the calculation is simplified and the core coupling relationship is focused, and finally the redundant variables are eliminated, taking the Lagrange disturbance pressure and displacement potential as the core variables, directly reflecting the interaction between fluid particle motion and sound field propagation, avoiding the distortion of the coupling mechanism caused by ignoring the non-uniformity of the flow field in the traditional model, and providing a theoretical model with rigorous physical meaning and adaptive non-uniform flow field for sound flow coupling.

[0158] In a third aspect, a perfect matched layer is introduced when constructing the boundary condition, a coordinate system is fitted to the flow field motion through a geometric transformation driven by the flow field velocity, and the boundary condition is ensured to adapt to the convection effect of the flow field; a new coordinate with a complex imaginary part is defined, and an energy absorption coefficient is set to make the boundary region have a non-reflective absorption characteristic, thereby solving the false reflection problem caused by the truncation of the calculation domain in the simulation of the unbounded region; the attenuation characteristics of the new coordinate are transmitted to the actual calculation coordinate through coordinate conversion, and the physical reality is ensured by relying on the corresponding relationship of the differential operator; the conversion rules are designed and integrated in different directions to form the boundary absorption rules, so as to accurately cope with the anisotropic boundary of the non-uniform flow field (such as different flow velocities and differences in turbulence intensity in different directions), so that the boundary condition can not only adapt to the mathematical characteristics of the Galbrun equation, but also can truly simulate the attenuation behavior of sound waves at the boundary of the non-uniform flow field, avoid false reflection interference in the sound flow coupling calculation, and significantly improve the accuracy and reliability of the simulation.

[0159] Corresponding to the foregoing embodiment of the Galbrun equation-based sound flow coupling calculation method, the application also provides an embodiment of a Galbrun equation-based sound flow coupling calculation device.

[0160] Figure 2 A structural schematic diagram of a Galbrun equation-based sound flow coupling calculation device provided in Embodiment Two of the application is provided. Please refer to Figure 2 The device provided in this embodiment includes an acquisition module 210, a solving module 220, and a construction module 230.

[0161] The acquisition module 210 is configured to obtain a Galbrun equation based on the Euler equation of an ideal fluid under the action of no external force, in combination with the Euler disturbance, the Lagrange disturbance, and the mixed disturbance relationship; and the Galbrun equation represents the coupling effect between the flow field and the sound field.

[0162] The solving module 220 is configured to solve flow field data of a non-uniform flow field based on the Navier-Stokes equation in combination with the k-ε turbulence model in the RANS model.

[0163] The construction module 230 is configured to introduce a perfect matched layer for the simulation of sound wave propagation in an unbounded region, and to construct a boundary condition suitable for the Galbrun equation through geometric transformation, complex variable definition, and coordinate conversion.

[0164] The solving module 220 is further configured to weakly transform the Galbrun equation, embed the boundary condition, combine the flow field data, discretely process by using a triangular finite element, construct a global linear algebraic equation set, and solve to obtain a discrete solution of the sound field parameter.

[0165] The device of this embodiment can be used to perform Figure 1The steps of the method embodiment, the specific implementation principle and the implementation process are similar, and details are not repeated here.

[0166] The implementation process of the functions and roles of each unit in the above device is specifically described in the implementation process of the corresponding steps in the above method, and details are not repeated here.

[0167] For the device embodiment, since it basically corresponds to the method embodiment, the related part can be referred to the part of the method embodiment. The device embodiment described above is only schematic, and the units described as separate components can or can not be physically separate, and the components displayed as units can or can not be physical units, that is, they can be located in one place, or distributed on multiple network units. According to actual needs, part or all of the modules can be selected to achieve the purpose of the scheme of the present application. Those skilled in the art can understand and implement without creative labor.

[0168] The above is only the preferred embodiment of the present application, and is not used to limit the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the scope of protection of the present application.

Claims

1. A method for calculating acoustic streaming coupling based on the Galbrun equation, characterized in that, The method comprises: Based on the Euler equation of ideal fluid under no external force, combined with Euler disturbance, Lagrange disturbance and mixed disturbance relationship, Galbrun equation is obtained; the Galbrun equation represents the coupling effect of flow field and sound field; Based on the Navier-Stokes equation, combined with the k-ε turbulence model in the RANS model, the flow field data of the non-uniform flow field is solved; For the simulation of sound wave propagation in unbounded region, a perfect matched layer is introduced, and through geometric transformation, complex variable definition and coordinate transformation, a boundary condition suitable for the Galbrun equation is constructed; The Galbrun equation is weakly transformed, the boundary condition is embedded, combined with the flow field data, and the triangular finite element is used for discretization processing, the global linear algebraic equation group is constructed and solved, and the discrete solution of the sound field parameter is obtained; The Galbrun equation is obtained by combining the Euler equation of ideal fluid under no external force, combined with Euler disturbance, Lagrange disturbance and mixed disturbance relationship, including: Selecting the Euler equation group of ideal fluid under no external force; Define the Euler disturbance of physical quantity as the dynamic change of physical quantity at fixed spatial position, and define the Lagrange disturbance of physical quantity as the change of physical quantity of fluid particles on the motion trajectory; Establish the mixed disturbance relationship between Euler disturbance and Lagrange disturbance; the Lagrange disturbance is equal to the sum of the product of Euler disturbance and Lagrange displacement potential and the gradient of steady uniform flow; Limit the flow field to be a stable flow field, and linearize the Euler equation group under the stable flow field; the steady-state uniform flow of the stable flow field is constant and does not change with time; Substitute the Euler disturbance, Lagrange disturbance and mixed disturbance relationship into the linearized Euler equation group, and eliminate pressure, velocity and density to obtain the Galbrun equation with Lagrange disturbance pressure and Lagrange displacement potential as core variables.

2. The method of claim 1, wherein, The Galbrun equation is obtained by combining the Euler equation of ideal fluid under no external force, combined with Euler disturbance, Lagrange disturbance and mixed disturbance relationship, including: Substitute the Euler disturbance, Lagrange disturbance and mixed disturbance relationship as a whole into the linearized mass conservation equation, momentum conservation equation and pressure and density, entropy relationship equation to obtain the first equation group containing disturbance variables; Substitute the pressure, velocity and density disturbance terms by Lagrange disturbance pressure and Lagrange displacement potential for the first equation group; Eliminate the original variables of pressure, velocity and density and the intermediate disturbance terms by simultaneous equations, and retain Lagrange disturbance pressure and Lagrange displacement potential as core variables; Organize the obtained equation to form the Galbrun equation containing the coupling relationship of stable flow field parameters and sound field parameters.

3. The method of claim 1, wherein, The perfect matched layer is introduced for the simulation of sound wave propagation in unbounded region, and through geometric transformation, complex variable definition and coordinate transformation, a boundary condition suitable for the Galbrun equation is constructed, including: Based on the flow field velocity parameter in the flow field data, the spatial and time coordinates of the calculation domain are geometrically transformed to obtain a transformed coordinate system that adapts to the motion state of the flow field; On the basis of the transformed coordinate system, new coordinate parameters in complex form and corresponding matrix conversion rules are defined, the energy absorption coefficient of the boundary region is set by the complex parameters, and a new coordinate system with energy attenuation characteristics is formed; Based on the new coordinate parameters and the matrix conversion rules, the new coordinate system is converted back to the spatial and time coordinates used in actual calculation, and the corresponding relationship between the differential operators describing the sound wave changes in the two coordinate systems is determined; Based on the corresponding relationship between the differential operators, the conversion rules of the perfect matched layer are designed for different spatial directions, the conversion rules in each direction are integrated into a unified total conversion mechanism, and the boundary absorption rule is formed.

4. The method of claim 3, wherein, The conversion rules of the perfect matched layer are designed for different spatial directions based on the corresponding relationship between the differential operators, the conversion rules in each direction are integrated into a unified total conversion mechanism, and the boundary absorption rule is formed, including: Based on the corresponding relationship between the differential operators, the flow field velocity component and the energy absorption coefficient in each direction are determined to adapt to the conversion rule of the direction; the conversion rule of each direction reflects the influence of the flow field characteristics in the corresponding dimension on the sound wave absorption; The conversion rules in each direction are substituted into the matrix conversion rule, and the conversion logic in different directions is integrated through matrix operation to form a total conversion mechanism containing all spatial direction information; Based on the total conversion mechanism, the correlation between the flow field parameters and the energy absorption coefficient is extracted, and the correlation is converted into a mathematical expression form with the same dimension and variable type as the coupling term in the Galbrun equation; Through variable substitution, the intermediate parameters in the mathematical expression form that are inconsistent with the variables of the Galbrun equation are replaced by the core variables of the Galbrun equation to form the boundary absorption rule.

5. The method of claim 3, wherein, The conversion rules of the perfect matched layer are designed for different spatial directions based on the corresponding relationship between the differential operators, the conversion rules in each direction are integrated into a unified total conversion mechanism, and the boundary absorption rule is formed, including: Taking the matrix conversion rule as the operation framework, the coordinate parameters in complex form in the new coordinate system are taken as input, and through coordinate inverse transformation operation, they are mapped to the spatial and time coordinates used in actual calculation to obtain the corresponding relationship between the two coordinate parameters; Based on the corresponding relationship between the coordinates, the conversion mode of the differential operator between the new coordinate system and the actual calculation coordinate system is determined, and in the conversion process, the energy absorption characteristics of the new coordinate system due to the complex parameters are preserved in the differential operator of the actual calculation coordinate system through parameter transmission in the matrix conversion rule; The conversion mode obtained by calculation is verified and adjusted under typical boundary conditions, and the differential operators in the two coordinate systems are consistent in describing the same sound wave change process after adjustment.

6. The method of claim 1, wherein, Based on the Navier-Stokes equation, the k-ε turbulence model in the RANS model is combined to solve the flow field data of the non-uniform flow field, including: The Navier-Stokes equation set describing fluid motion is selected; The k-ε turbulence model in the RANS model is introduced to decompose the turbulent motion in the flow field into time-averaged motion and fluctuation motion, and an additional transport equation describing the fluctuation characteristics of the turbulence is established by defining turbulent kinetic energy and turbulent kinetic energy dissipation rate; The additional transport equation is coupled with the Navier-Stokes equation set to form a complete control equation set suitable for a non-uniform flow field; the complete control equation set can reflect the average motion characteristics of the fluid and the influence of the turbulent fluctuation on the flow at the same time; The discretization processing is performed on the coupled control equation set, the boundary conditions of the flow field calculation domain are combined, and a numerical solution method is used to solve the equation set to obtain the flow field data of the non-uniform flow field.

7. The method of claim 6, wherein, The k-ε turbulence model in the RANS model is introduced to decompose the turbulent motion in the flow field into time-averaged motion and fluctuation motion, and an additional transport equation describing the fluctuation characteristics of the turbulence is established by defining turbulent kinetic energy and turbulent kinetic energy dissipation rate, including: Based on the Reynolds time-averaged method, the physical quantities in the flow field are decomposed into time-averaged components and fluctuation components; the time-averaged components represent macroscopic average motion, and the fluctuation components represent random disturbance characteristics; The turbulent kinetic energy is defined as the turbulent fluctuation kinetic energy of unit mass of fluid, and the turbulent kinetic energy dissipation rate is defined as the rate of dissipation of turbulent kinetic energy due to viscosity; the size of the turbulent kinetic energy is related to the square of the fluctuation velocity component, and the turbulent kinetic energy dissipation rate is used to represent the conversion process of turbulent energy to heat energy; For the characteristics of the non-uniform flow field, a first transport equation of the turbulent kinetic energy is determined, and a second transport equation of the turbulent kinetic energy dissipation rate is determined; the second transport equation introduces an empirical coefficient related to the turbulent kinetic energy and the characteristic length; The distribution of the turbulent kinetic energy and the turbulent kinetic energy dissipation rate is obtained by solving the first transport equation and the second transport equation, and the turbulent viscosity coefficient is determined based on the distribution; 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 integrated into the coupling relationship of the first transport equation and the second transport equation as a correlation parameter to form an additional transport equation system.

8. The method of claim 1, wherein, The Galbrun equation is weakly transformed, the boundary conditions are embedded, the flow field data are combined, and the triangular finite element is discretized to construct a global linear algebraic equation set and solve it to obtain discrete solutions of the sound field parameters, including: The Galbrun equation is weakly transformed, a trial function is introduced, and integral operations are performed on both sides of the equation to transform it into an integral form suitable for finite element discretization; The boundary conditions are embedded in the boundary integral term of the weak form equation; the energy absorption characteristics at the boundary are the constraint conditions for equation solving; The flow field data are introduced, and the flow field data are substituted into the weak form equation as known parameters; The calculation domain is meshed by using triangular elements, and the sound field parameters in each triangular element are represented by an interpolation function; Based on the discretized interpolation function, the weak form equation is discretized on each triangular element to obtain a linear algebraic equation set at the element level, and the element equation sets are assembled to form a global linear algebraic equation set; A numerical solution method is used to solve the global linear algebraic equation set to obtain the sound field parameter values at each discrete node.

9. An apparatus for calculating acoustic streaming based on the Galbrun equation, characterized in that, The device comprises an acquisition module, a solving module and a construction module; The acquisition module is configured to obtain a Galbrun equation based on Euler equations of ideal fluid under no external force, combined with Euler disturbance, Lagrange disturbance and mixed disturbance relationship; the Galbrun equation represents the coupling effect of flow field and sound field; The solving module is configured to solve flow field data of non-uniform flow field based on Navier-Stokes equation, combined with k-ε turbulence model in RANS model; The construction module is configured to introduce a perfect matched layer for sound wave propagation simulation in unbounded region, and construct boundary conditions suitable for the Galbrun equation through geometric transformation, complex variable definition and coordinate conversion; The solving module is further configured to weakly transform the Galbrun equation, embed the boundary conditions, combine the flow field data, discretize by triangular finite element, construct and solve global linear algebraic equations to obtain discrete solution of sound field parameters; The Galbrun equation is obtained based on Euler equations of ideal fluid under no external force, combined with Euler disturbance, Lagrange disturbance and mixed disturbance relationship, comprising: Selecting Euler equation set of ideal fluid under no external force; Defining Euler disturbance of physical quantity as dynamic change of physical quantity at fixed spatial position, and defining Lagrange disturbance of physical quantity as physical quantity change of fluid particle on motion trajectory; Establishing mixed disturbance relationship between Euler disturbance and Lagrange disturbance; the Lagrange disturbance is equal to the sum of the product of Euler disturbance and Lagrange displacement potential and steady-state uniform flow gradient; Limiting flow field as steady flow field, linearizing the Euler equation set under the steady flow field; the steady-state uniform flow of the steady flow field is constant and does not change with time; Substituting the Euler disturbance, Lagrange disturbance and mixed disturbance relationship into the linearized Euler equation set, eliminating pressure, velocity and density, obtaining the Galbrun equation with Lagrange disturbance pressure and Lagrange displacement potential as core variables.

Images