Method for analyzing high-frequency aerodynamic noise field in closed space

Abstract

The invention provides a method for analyzing a high-frequency aerodynamic noise field in a closed space, which comprises the following steps of: 1) based on a CAD (Computer Aided Design) model, carrying out computational domain cutting and grid division on a flow field outside the model, dividing the flow field into a sound source area and a noise radiation area, and carrying out local grid encryption on the sound source area to ensure the subsequent turbulence resolution; 2) solving a flow field by adopting a DDES method, and capturing near-field sound source pulsation; 3, the generalized Green function analysis form in the limited space is introduced into an FW-H equation for integration, and far-field noise considering wall and ground boundary reflection is output. The concept is reasonable, the prediction accuracy, comparable with APE, in the tunnel can be obtained, and a new feasible way is provided for rapid evaluation and optimization of high-speed train tunnel acoustic performance.

Description

Technical Field

[0001] This invention relates to the field of aerodynamic noise analysis of high-speed trains and numerical simulation of sound fields in enclosed spaces, specifically to a method for analyzing high-frequency aerodynamic noise fields in enclosed spaces. Background Technology

[0002] Over the past two decades, with the simultaneous leaps in high-performance computing and numerical algorithms, computational fluid dynamics (CFD) and computational aeroacoustics (CAA) have become irreplaceable tools for the study of aerodynamic noise in high-speed trains. Especially in enclosed spaces such as tunnels, where testing conditions are complex, numerical simulation is highly anticipated to accurately predict noise levels during the vehicle design phase.

[0003] The most commonly used method for solving tunnel acoustics in academia and industry is based on large eddy simulation and acoustic perturbation equations (APE). [1] The APE (Audio Processing) model, which considers sound wave reflection at tunnel walls and wave convection effects when solving for noise fields within tunnels, is considered the "most physical" sound field model for tunnels because it can characterize multiple reflections of sound waves between tunnel walls and wave convection effects. However, APE is extremely sensitive to spatial resolution: to accurately capture mid-to-high frequency components above 0.5 kHz, the flow grid size needs to be refined to the millimeter or even sub-millimeter level, resulting in an exponential increase in the total number of grid cells and a sharp increase in computational cost, making it difficult to meet the needs of batch evaluation in engineering projects. Ironically, the area around 1 kHz is the frequency band most sensitive to the human ear and the core range that noise reduction control strategies must target, highlighting this contradiction.

[0004] Therefore, the Ffowcs Williams-Hawkings (FW-H) acoustic analogy method is preferred in engineering. FW-H is a generalization of the Lighthill equations, equating each sound source to a monopole source (pulsating mass source caused by changes in boundary volume or mass), a dipole source (pulsating dynamic source caused by the interaction of an object surface with the fluid), and a quadrupole source (turbulent stress source caused by turbulent interaction). Sound pressure information of the noise field is directly obtained through time-domain integration. Coarse or even mismatched meshes can be used outside the sound source region, improving computational efficiency by 1–2 orders of magnitude compared to APE. However, the classical FW-H equations are derived based on the Green's function in infinite space, implicitly assuming a "free field" and neglecting crucial wall reflections and low-frequency reverberation effects in closed or finite spaces. Furthermore, for low-frequency noise of 0.5 kHz and below, its wavelength (≈0.68 m) is comparable to the transverse cross-section of a tunnel and the sound wave propagation distance, making the equivalent sound source assumption difficult to hold. Therefore, the classic FW-H acoustic analogy method is not applicable to aerodynamic noise problems in enclosed or confined spaces.

[0005] In fact, for high-frequency noise of 1 kHz and above, its wavelength (≈0.34 m) is much smaller than the tunnel cross-section diameter, the convection effect of the wave is weak, and wall reflection becomes the dominant factor. If "wall mirror reflection" can be embedded into the FW-H integral as a boundary condition, the reflected sound field in the enclosed space can be reproduced while maintaining the low grid dependence advantage of FW-H. However, existing literature has not yet provided a systematic theoretical derivation and an engineering-ready implementation path.

[0006] In summary, the current contradictions in numerical simulation of tunnel aerodynamic noise are: the APE method is physically complete but has an excessively heavy mesh dependency and the cost of high-frequency computation is unacceptable; the classic FW-H method is computationally efficient but lacks wall reflection and low-frequency reverberation effects, resulting in loss of accuracy in enclosed spaces. Summary of the Invention

[0007] To address the technical problems existing in the background art mentioned above, this invention proposes a high-frequency aerodynamic noise field analysis method in a closed space. Its concept is reasonable and can achieve high-frequency noise prediction accuracy in tunnels comparable to APE, providing a feasible new approach for rapid evaluation and optimization of the acoustic performance of high-speed train tunnels.

[0008] To solve the above-mentioned technical problems, the present invention provides a method for analyzing high-frequency aerodynamic noise fields in a confined space, characterized by comprising the following steps:

[0009] 1) Based on the CAD model, the computational domain of the external flow field of the model is cut and meshed, divided into the sound source region and the noise radiation region, and the local mesh is refined in the sound source region to ensure the subsequent turbulence resolution;

[0010] 2) The DDES method is used to solve the flow field and capture near-field sound source fluctuations;

[0011] 3) The analytical form of the generalized Green's function in the finite space is introduced into the FW-H equation for integration, and the output is far-field noise considering the reflections from the wall and ground boundaries.

[0012] The method for analyzing high-frequency aerodynamic noise fields in a confined space, wherein step 3) is specifically as follows:

[0013] 3.1) Construct the FW-H equations for solving the aerodynamic noise generated by high-speed trains during operation:

[0014] (1);

[0015] The first term on the right side of equation (1) above Represents a monopole source, the second term on the right-hand side Represents a dipole source, the third term on the right-hand side Indicates a quadrupole source; Speed ​​of sound; Represents the Lighthill tensor; This represents the Heaviside function; For Diraclet functions; This represents a closed control surface, which can be taken as a moving solid surface or a permeable curved surface. This represents the external flow field of a closed control surface. This represents the internal flow field of a closed control surface;

[0016] 3.2) The right-hand side of equation (1) in step (3.1) above is regarded as a sound source term. The equation of this sound source term is a typical sound wave equation, which can be solved by the Green function; where the time-domain Green function in free space is:

[0017] (2);

[0018] In equation (2) above, x is a point within the sound source region, and y is any point within the noise radiation region; and These represent the spatial locations of the receiving point and the sound source point, respectively. The distance between the receiving point and the sound source point; Indicates the delay time; In the subsonic case, the noise from the quadrupole source is very small and can be ignored. Therefore, the solution to the above equation (1) can be composed of two parts:

[0019] (3);

[0020] In the above formula (3), The solution corresponding to the monopole source is expressed as equation (4). The solution corresponding to the dipole source is expressed as equation (5):

[0021] (4);

[0022] In the above formula (4), ;

[0023] (5);

[0024] In the above formula (5), ;

[0025] In curvilinear coordinates, equations (4) and (5) above can be further solved to obtain the analytical form of the monopole component of the sound source:

[0026] (6);

[0027] In the above formula (6), , for arrive The unit radius component, For delay time; The control surface is the entire sound source control surface, and the integration is performed across it.

[0028] (7);

[0029] In the above formula (7), Let be the analytical form of the dipole component of the sound source, where Solving equations (6) and (7) above will yield the numerical solution of the sound field;

[0030] 3.3) Key aerodynamic sound source components of high-speed trains are affected by the walls or ground in tunnels; considering only the ground effect, a generalized Green's function of semi-free space is introduced, and the ground reflection effect is solved by the following equation (8):

[0031] (8);

[0032] In the above formula (8), for Regarding the mirror image of the ground; , Indicates the delay time. ;

[0033] 3.4) Considering the reflection effect of the tunnel wall, the tunnel cross-section is regarded as a two-dimensional closed semicircle. The reflection effect of the tunnel's curved wall is solved by combining the generalized Green's function in semi-free space and the generalized Green's function in the three-dimensional sphere. The generalized Green's function in the three-dimensional sphere is:

[0034] (9);

[0035] In the above formula (9), for The distance to the center of the sphere; Let be the radius of the sphere; for Regarding the inversion point of the sphere, it satisfies Treating the tunnel cross-section as a two-dimensional problem, therefore and The coordinates in the X direction are consistent;

[0036] 3.5) Combining the expression (8) of the generalized Green's function in semi-free space and the expression (9) of the generalized Green's function in a three-dimensional sphere, we obtain the generalized Green's function in a two-dimensional closed semicircle as follows:

[0037] + (10);

[0038] In the above formula (10), for The distance to the center of the semicircle; Let be the radius of the sphere; for Regarding the inversion point of the sphere, it satisfies In the formula for Regarding the mirror point on the ground, Inversion point of the sphere Regarding the mirror image of the ground;

[0039] 3.6) Combining equations (6), (7), and (10) above, solve the right side of equation (1) in the curvilinear coordinate system to obtain the sound pressure at the two-dimensional receiving point. :

[0040] (11);

[0041] In the above formula (11) This represents the solution corresponding to the monopole source at the two-dimensional receiving point; This represents the solution corresponding to the dipole source at the two-dimensional receiving point;

[0042] The solution corresponding to the monopole source is: ;

[0043] The solution corresponding to the dipole source is:

[0044] (13);

[0045] 3.7) Obtain the sound pressure level at the two-dimensional receiving point Then, based on the noise similarity rate formula:

[0046] (14);

[0047] In the above formula (14), For the incoming flow velocity, Let be the area of ​​the model perpendicular to the direction of the incoming flow. Distance between the sound source and the receiver; subscript Represents the normalized corresponding parameter, subscript These represent the actual parameters of the model;

[0048] Based on the distances between the two-dimensional and three-dimensional receiving points and the sound source, as well as the sound wave propagation delay, and considering the attenuation of noise sound pressure during propagation, the sound pressure at the three-dimensional receiving point is obtained. :

[0049] (15);

[0050] In the above formula (15), This represents the distance from the two-dimensional receiving point to the sound source. This represents the distance from the three-dimensional receiving point to the sound source. This is the delay time for sound wave propagation.

[0051] By adopting the above technical solution, the present invention has the following beneficial effects:

[0052] This invention addresses the unexplored scenario of "high-frequency sound field in enclosed spaces" by proposing a correction method that embeds wall reflections within the FW-H integral framework. The aim is to achieve high-frequency noise prediction accuracy in tunnels comparable to APE with computational costs close to FW-H, providing a feasible new approach for the rapid evaluation and optimization of the acoustic performance of high-speed train tunnels.

[0053] Since current numerical simulation methods cannot efficiently and accurately simulate high-frequency aerodynamic noise fields of 1 kHz and above in enclosed spaces, this invention will develop a high-frequency aerodynamic noise field analysis method based on the analytical form of the generalized Green's function in a finite space, considering the influence of walls / ground on the aerodynamic noise intensity and spatiotemporal distribution. This method will be incorporated into the existing FW-H integral framework. Specifically, the ground reflection effect is determined by correcting the FW-H integral equation based on the analytical form of the generalized Green's function in semi-free space; the wall reflection effect is derived by simplifying the analytical form of the generalized Green's function in a three-dimensional sphere to derive the generalized Green's function in a two-dimensional circle; combining the generalized Green's functions in semi-free space and the two-dimensional circle, the analytical form of the generalized Green's function in a two-dimensional semicircle is obtained and substituted into the FW-H integral equation. Attached Figure Description

[0054] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0055] Figure 1 Flowchart for implementing the analytical method;

[0056] Figure 2 For semi-free domains A diagram showing the mirror image of the ground;

[0057] Figure 3 For the sphere A schematic diagram of the inversion points of a sphere;

[0058] Figure 4 Within a two-dimensional closed semicircular region Inversion point diagram;

[0059] Figure 5 This is a schematic diagram of the receiving point and its projection point inside the tunnel. Detailed Implementation

[0060] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0061] The present invention will be further explained below with reference to specific embodiments.

[0062] This embodiment provides a method for analyzing high-frequency aerodynamic noise fields in a closed space. Its noise field solution framework follows the core idea of ​​the classic FW-H acoustic analogy, but pushes the "infinite domain" to the "finite space": by deriving the analytical form of the generalized Green's function in the finite space, the wall / ground reflection effect is directly embedded into the FW-H integral, thereby minimizing the simulation error without increasing the mesh burden.

[0063] The specific steps of the high-frequency aerodynamic noise field analysis method in a confined space according to the present invention are shown in Figure 1:

[0064] 1) Based on the CAD model, the computational domain of the external flow field of the model is cut and meshed, divided into the sound source region and the noise radiation region, and the local mesh is refined in the sound source region to ensure the subsequent turbulence resolution;

[0065] 2) The flow field is solved using the DDES (Delayed Detached-Eddy Simulation) method to capture near-field sound source fluctuations;

[0066] 3) Introduce the analytical form of the generalized Green's function within the confined space into the FW-H equation for integration, and output the far-field noise considering reflections from the wall and ground boundaries; the specific process includes:

[0067] 3.1) Construct the FW-H equations for solving the aerodynamic noise generated by high-speed trains during operation:

[0068] (1);

[0069] The first term on the right side of equation (1) above Represents a monopole source, the second term on the right-hand side Represents a dipole source, the third term on the right-hand side Indicates a quadrupole source; Speed ​​of sound; Represents the Lighthill tensor; This represents the Heaviside function; For Diraclet functions; This represents a closed control surface, which can be taken as a moving solid surface or a permeable curved surface. This represents the external flow field of a closed control surface. This represents the internal flow field of a closed control surface.

[0070] 3.2) The right-hand side of equation (1) in step (3.1) above is regarded as a sound source term. The equation of this sound source term is a typical sound wave equation, which can be solved by the Green function; where the time-domain Green function in free space is:

[0071] (2);

[0072] In equation (2) above, x is a point within the sound source region, and y is any point within the noise radiation region; and These represent the spatial locations of the receiving point and the sound source point, respectively. The distance between the receiving point and the sound source point; Indicates the delay time; In the subsonic case, the noise from the quadrupole source is very small and can be ignored. Therefore, the solution to the above equation (1) can be composed of two parts:

[0073] (3);

[0074] The aforementioned quadrupole sound source refers to the turbulent stress source caused by turbulent interaction. At low speeds, the intensity of the quadrupole sound source is lower than that of the monopole sound source (the pulsating mass source caused by changes in boundary volume or mass) and the dipole sound source (the pulsating force source caused by the interaction of the object surface with the fluid); in the above equation (3), The solution corresponding to the monopole source is expressed as equation (4). The solution corresponding to the dipole source is expressed as equation (5):

[0075] (4);

[0076] in, ;

[0077] (5);

[0078] In the above formula (5), ;

[0079] In curvilinear coordinates, equations (4) and (5) above can be further solved to obtain the analytical form of the monopole component of the sound source:

[0080] (6);

[0081] In the above formula (6), , for arrive The unit radius component, For delay time; The control surface is the entire sound source control surface, and the integration is performed across it.

[0082] (7);

[0083] In the above formula (7), Let be the analytical form of the dipole component of the sound source, where Solving equations (6) and (7) above will yield the numerical solution of the sound field.

[0084] 3.3) Key aerodynamic sound source components of high-speed trains are affected by the walls or ground in tunnels; considering only the ground effect, a generalized Green's function of semi-free space is introduced, and the ground reflection effect is solved by the following equation (8):

[0085] (8);

[0086] In the above formula (8), for Regarding the mirror image of the ground; , Indicates the delay time. ;

[0087] 3.4) Considering the reflection effect of the tunnel wall, the tunnel cross-section is regarded as a two-dimensional closed semicircle. The reflection effect of the tunnel's curved wall is solved by combining the generalized Green's function in semi-free space and the generalized Green's function in the three-dimensional sphere. The generalized Green's function in the three-dimensional sphere is:

[0088] (9);

[0089] In the above formula (9), for The distance to the center of the sphere; Let be the radius of the sphere; for Regarding the inversion point of the sphere, it satisfies Treating the tunnel cross-section as a two-dimensional problem, therefore and The coordinates in the X direction are consistent, as shown in Figure 3.

[0090] 3.5) Combining the expression (8) of the generalized Green's function in semi-free space and the expression (9) of the generalized Green's function in a three-dimensional sphere, we obtain the generalized Green's function in a two-dimensional closed semicircle as follows:

[0091] + (10);

[0092] In the above formula (10), for The distance to the center of the semicircle; Let be the radius of the sphere; for Regarding the inversion point of the sphere, it satisfies As shown in Figure 4; where for Regarding the mirror point on the ground, Inversion point of the sphere Regarding the mirror image of the ground.

[0093] 3.6) Combining equations (6), (7), and (10) above, solve the right side of equation (1) in the curvilinear coordinate system to obtain the sound pressure at the two-dimensional receiving point. :

[0094] (11);

[0095] In the above formula (11) This represents the solution corresponding to the monopole source at the two-dimensional receiving point; This represents the solution corresponding to the dipole source at the two-dimensional receiving point;

[0096] The solution corresponding to the monopole source is: ;

[0097] The solution corresponding to the dipole source is:

[0098] (13);

[0099] The two-dimensional receiving point is the projection point of the three-dimensional receiving point onto the plane (two-dimensional semi-circular cross-section) where the sound source is located, along the tunnel length direction (X direction), as shown in Figure 5.

[0100] 3.7) Obtain the sound pressure level at the two-dimensional receiving point Then, based on the noise similarity rate formula:

[0101] (14);

[0102] In the above formula (14), For the incoming flow velocity, Let be the area of ​​the model perpendicular to the direction of the incoming flow. Distance between the sound source and the receiver; subscript Represents the normalized corresponding parameter, subscript These represent the actual parameters of the model;

[0103] Based on the distances between the two-dimensional and three-dimensional receiving points and the sound source, as well as the sound wave propagation delay, and considering the attenuation of noise sound pressure during propagation, the sound pressure at the three-dimensional receiving point is obtained. :

[0104] (15);

[0105] In the above formula (15), This represents the distance from the two-dimensional receiving point to the sound source. This represents the distance from the three-dimensional receiving point to the sound source. This is the delay time for sound wave propagation.

[0106] The following is a further explanation of the DDES method in step 2) above:

[0107] Decoupled eddy simulation, or the Decoupled Eeddy Simulation (DES) method, was proposed by Spalart. Its approach is to use the Decoupled Eeddy Simulation (LES) method in the decoupled region and the Randomized Resonant Eeddy Simulation (RANS) method in other regions. This is achieved by introducing the length scale L of the DES method. DES To achieve automatic conversion from RANS to LES, L DES The definition is as follows:

[0108] (01);

[0109] Where Δ is the maximum value of the grid scale in the three directions:

[0110] (02);

[0111] Strelets introduce the length scale L of LES LES The length scale L of the RANS mode RANS Based on this, L was proposed DES The general form:

[0112] (03);

[0113] Where L RANS and L LES Defined as:

[0114] (04);

[0115] (05);

[0116] For SST mode, use a mixed length scale. To replace in equation (2.9) To complete the construction of the DES method, coefficient C DES Depend on and The coefficients of the two branches are weighted to obtain:

[0117] (06);

[0118] In equation (6) above, C DES , and C DES , It needs to be calibrated by isotropic turbulence.

[0119] Delayed detached eddy simulation, or DDES method, determines the transition from RANS to LES by the mesh scale. When the spanwise or flowwise mesh within the boundary layer is sufficiently fine, the LES... LES < L RANS In some cases, the LES method is used within the boundary layer; however, when the mesh is not dense enough to effectively resolve boundary layer turbulence, insufficient Reynolds stress modeling can occur, leading to non-physical mesh-induced separation. To avoid mesh-induced separation, it is necessary to delay the LES scale entering the boundary layer.

[0120] To achieve boundary layer protection, Menter utilizes transition functions F1 and F2 from the SST mode to improve the length scale; the delayed DES length scale is:

[0121] (07);

[0122] Among them, F SST F1 or F2, which are close to 1 near the wall, can ensure L within the boundary layer.DES The value is L RANS The value is 0 when it is far from the surface of the object, and equation (7) degenerates into equation (3). Spalart et al. introduced a flow-related delay function. Delay function See equation (9), which formally proposes the DDES method:

[0123] (08);

[0124] (09);

[0125] Among them, C d1 = 8, C d2 = 3, κ = 0.41;

[0126] (010).

[0127] This invention is well-conceived and can achieve high-frequency noise prediction accuracy in tunnels comparable to APE, providing a feasible new approach for the rapid evaluation and optimization of the acoustic performance of high-speed train tunnels.

[0128] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for analyzing high-frequency aerodynamic noise fields in a confined space, characterized in that, Includes the following steps: (1) Based on the CAD model, the computational domain of the external flow field of the model is cut and meshed, divided into the sound source region and the noise radiation region, and the local mesh is refined in the sound source region to ensure the subsequent turbulence resolution; (2) The DDES method is used to solve the flow field and capture the near-field sound source pulsation; (3) The analytical form of the generalized Green function in the finite space is introduced into the FW-H equation for integration, and the output is far-field noise considering the reflection of the wall and ground boundary.

2. The method for analyzing high-frequency aerodynamic noise fields in a confined space as described in claim 1, characterized in that, The specific process of step 3) is as follows: (3.1) Construct the FW-H equations for solving the aerodynamic noise generated by high-speed trains during operation: （1）； The first term on the right side of equation (1) above Represents a monopole source, the second term on the right-hand side Represents a dipole source, the third term on the right-hand side Indicates a quadrupole source; Speed ​​of sound; Represents the Lighthill tensor; This represents the Heaviside function; For Diraclet functions; This represents a closed control surface, which can be taken as a moving solid surface or a permeable curved surface. This represents the external flow field of a closed control surface. This represents the internal flow field of a closed control surface; (3.2) The right-hand side of equation (1) in step (3.1) above is regarded as a sound source term. The equation of this sound source term is a typical sound wave equation, which can be solved by the Green function; where the time-domain Green function in free space is: （2）； In equation (2) above, x is a point within the sound source region, and y is any point within the noise radiation region; and These represent the spatial locations of the receiving point and the sound source point, respectively. The distance between the receiving point and the sound source point; Indicates the delay time; In the subsonic case, the noise from the quadrupole source is very small and can be ignored. Therefore, the solution to the above equation (1) can be composed of two parts: （3）； In the above formula (3), The solution corresponding to the monopole source is expressed as equation (4). The solution corresponding to the dipole source is expressed as equation (5): （4）； In the above formula (4), ; （5）； In the above formula (5), ; In curvilinear coordinates, equations (4) and (5) above can be further solved to obtain the analytical form of the monopole component of the sound source: （6）； In the above formula (6), , for arrive The unit radius component, For delay time; The control surface is the entire sound source control surface, and the integration is performed across it. （7）； In the above formula (7), Let be the analytical form of the dipole component of the sound source, where Solving equations (6) and (7) above will yield the numerical solution of the sound field; (3.3) Key aerodynamic sound source components of high-speed trains are affected by the walls or ground in tunnels; considering only the ground effect, a generalized Green's function of semi-free space is introduced, and the ground reflection effect is solved by the following equation (8): （8）； In the above formula (8), for Regarding the mirror image of the ground; , Indicates the delay time. ; (3.4) Considering the reflection effect of the tunnel wall, the tunnel cross-section is regarded as a two-dimensional closed semicircle. The reflection effect of the tunnel's curved wall is solved by combining the generalized Green's function in semi-free space and the generalized Green's function in the three-dimensional sphere. The generalized Green's function in the three-dimensional sphere is: （9）； In the above formula (9), for The distance to the center of the sphere; Let be the radius of the sphere; for Regarding the inversion point of the sphere, it satisfies Treating the tunnel cross-section as a two-dimensional problem, therefore and Their X-axis coordinates are consistent; (3.5) Combining the expression (8) of the generalized Green's function in semi-free space and the expression (9) of the generalized Green's function in a three-dimensional sphere, the generalized Green's function in a two-dimensional closed semicircle is obtained as follows: + （10）； In the above formula (10), for The distance to the center of the semicircle; Let be the radius of the sphere; for Regarding the inversion point of the sphere, it satisfies In the formula for Regarding the mirror point on the ground, Inversion point of the sphere Regarding the mirror image of the ground; (3.6) Combining equations (6), (7) and (10) above, we can solve the right side of equation (1) in the curvilinear coordinate system to obtain the sound pressure at the two-dimensional receiving point. : （11）； In the above formula (11) This represents the solution corresponding to the monopole source at the two-dimensional receiving point; This represents the solution corresponding to the dipole source at the two-dimensional receiving point; The solution corresponding to the monopole source is: ； The solution corresponding to the dipole source is: （13）； (3.7) Obtain the sound pressure at the two-dimensional receiving point Then, based on the noise similarity rate formula: （14）； In the above formula (14), For the incoming flow velocity, Let be the area of ​​the model perpendicular to the direction of the incoming flow. Distance between the sound source and the receiver; subscript Represents the normalized corresponding parameter, subscript These represent the actual parameters of the model; Based on the distances between the two-dimensional and three-dimensional receiving points and the sound source, as well as the sound wave propagation delay, and considering the attenuation of noise sound pressure during propagation, the sound pressure at the three-dimensional receiving point is obtained. : （15）； In the above formula (15), This represents the distance from the two-dimensional receiving point to the sound source. This represents the distance from the three-dimensional receiving point to the sound source. This is the delay time for sound wave propagation.

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