A method for low wave number prediction of fluctuating pressure in turbulent boundary layers

Abstract

This application discloses a method for predicting the low wavenumber of fluctuating pressure in turbulent boundary layers, relating to the field of hydrodynamic noise prediction. This method employs statistical methods to describe the relationship between the autospectral density of the excitation force of the resonant mode and the low wavenumber spectrum of the wavenumber-frequency spectrum of fluctuating pressure in turbulent boundary layers. It then combines the autospectral density of the resonant mode displacement and the resonant mode impedance of a rectangular elastic thin plate under the excitation of fluctuating pressure in turbulent boundary layers to predict the low wavenumber spectrum of the wavenumber-frequency spectrum of fluctuating pressure in turbulent boundary layers. This method utilizes the spatial filtering characteristics of elastic structures, using the elastic structure as a wavenumber filter sensor. Based on the modal resonance response of the elastic structure surface, it predicts the low wavenumber spectrum of the wavenumber-frequency spectrum of fluctuating pressure in turbulent boundary layers. This method is highly sensitive to the modal wavenumber of the elastic structure and can effectively suppress interference from other wavenumbers, improving the prediction accuracy of the low wavenumber spectrum of fluctuating pressure in turbulent boundary layers.

Description

Technical Field

[0001] This application relates to the field of hydrodynamic noise prediction, and in particular to a method for predicting low wavenumber of turbulent boundary layer pulsating pressure. Background Technology

[0002] When a fluid medium moves relative to the surface of an object, a boundary layer forms on the surface due to the viscosity of the fluid. When the Reynolds number (Re) is sufficiently high, the boundary layer, except for a very thin layer near the wall, is in a turbulent state, forming a fully developed turbulent boundary layer. Turbulence is an unstable flow, and the turbulent boundary layer has a complex internal structure, which determines the random temporal and spatial variation of the turbulent boundary layer pulsating pressure. As a surface-distributed random excitation force source, the turbulent boundary layer pulsating pressure excites the vibration of elastic structures such as plates and shells, radiating noise both outwards and inwards. Because the turbulent boundary layer pulsating pressure has migration characteristics, its distribution characteristics determine the energy input to the structure. The corresponding structural vibration and sound radiation are also random processes. These characteristics make the theoretical analysis of the turbulent boundary layer pulsating pressure and its excitation of the elastic structure's vibration response quite difficult, and statistical methods are usually used for research.

[0003] Starting with the Lighthill equations, the frequency-wavenumber spectrum is used to describe the fluctuating pressure in the turbulent boundary layer, opening up an important avenue for the theoretical analysis of this problem. The frequency-wavenumber spectrum of the fluctuating pressure in the turbulent boundary layer contains both high-wavenumber components reflecting the fluid dynamic pressure and low-wavenumber components representing acoustic properties. The high-wavenumber components attenuate as they propagate to the far field, while the low-wavenumber components generate acoustic radiation that can propagate over long distances. The characteristics of the frequency-wavenumber spectrum directly determine the selective excitation of the structural vibration modes by the fluctuating turbulent pressure.

[0004] In aerodynamic noise problems, the migration velocity is often on the order of the speed of sound, and the migration peak has a significant effect, thus models such as Corcos have good applicability. However, in hydrodynamic noise problems, the flow exhibits low Mach number characteristics. When the turbulent boundary layer pulsating pressure excites structural vibration, the structural response is mainly near the structural wavenumber, located in the low wavenumber region, far below the migration peak. Therefore, in hydrodynamic noise problems, special attention is paid to the spectral characteristics in the low wavenumber region.

[0005] In 1997, Graham WR. published in Journal of Sound and Vibration, 206(4), pp. 541-565 (A comparison of models for the wavenumber-frequency spectrum of turbulenceboundary layer pressures). This work showed that, based on a large number of experimental measurements, many scholars have established a variety of empirical models for the frequency-wavenumber spectrum of turbulent boundary layer pulsating pressures. These empirical models have good consistency in the high wavenumber region of the migration peak component, and the differences mainly exist in the low wavenumber component region far from the migration peak. A paper published in the 2025 issue of *Acta Acustica* (Vol. 50, No. 2, pp. 308-321) introduces the wavenumber spectrum prediction results of the paper's predictive model and the Chase II model at typical frequencies under different flow velocities. The results are compared with wind tunnel and underwater towing experiments. The results show that both the paper's predictive model and the Chase II model exhibit "white" wavenumber spectrum characteristics compared to the wind tunnel results, but the amplitudes differ significantly from the experimental values. There are also some differences between the two models. Compared with the underwater towing experiment results, the paper's predictive model and the Chase II model do not exhibit "white" wavenumber spectrum characteristics, and the amplitudes differ significantly from the experimental values. These results indicate differences in the low wavenumber component region far from the migration peak. In summary, many existing models show significant discrepancies in the low wavenumber region far from the migration peak, failing to accurately assess the low wavenumber white spectrum characteristics, resulting in insufficient prediction accuracy in hydrodynamic noise problems. Summary of the Invention

[0006] To address the aforementioned problems and technical requirements, this application proposes a method for predicting low wavenumber of fluctuating pressure in turbulent boundary layers. The technical solution of this application is as follows: A method for predicting low-wavenumber fluctuations in turbulent boundary layer pressure, comprising: Based on the boundary conditions, using the modal functions of the free vibration of the rectangular elastic thin plate, the forced vibration equation of the rectangular elastic thin plate under the action of sound pressure and excitation force is solved by the modal expansion method. Then, by utilizing the orthogonality and normalization of the modal functions, the equation is transformed to the modal domain to obtain the equation of the rectangular elastic thin plate under the action of sound pressure and excitation force. Modal vibration equations under first resonance modes; Based on rectangular elastic thin plates The modal vibration equation under the first resonance mode determines the rectangular elastic thin plate. First-order resonant mode impedance , Let be the angular frequency of the vibration of the rectangular elastic thin plate; Describing using statistical methods middle The autospectral density of the excitation force of the first resonant mode Low wavenumber spectrum of turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum The relationship between them was explored, and the application of a rectangular elastic thin plate under turbulent boundary layer pulsating pressure excitation was utilized. Autospectral density of the first resonant mode displacement The low wavenumber spectrum of the fluctuating pressure wavenumber-frequency spectrum in the turbulent boundary layer was predicted. .

[0007] A further technical solution involves constructing the forced vibration equation of a rectangular elastic thin plate under sound pressure and excitation force as follows:

[0008] in, The bending stiffness of a rectangular elastic thin plate and , The Young's modulus of the material. For the material's Poisson's ratio, For plate thickness; The damping coefficient is... is the surface density and , Density of the board material; , For the Laplace operator, ; The coordinates on a rectangular elastic thin plate in the time domain Surface vibration displacement at that location The coordinates acting on a rectangular elastic thin plate in the time domain The sound pressure at that location, The coordinates acting on a rectangular elastic thin plate in the time domain The motivational force of the location; Indicates time, Represents the coordinates along the length of the rectangular elastic thin plate. Represents the coordinates along the width of the rectangular elastic sheet, and the flow along the length of the rectangular elastic sheet; In time The equation for forced vibration under simple harmonic transformation is transformed as follows:

[0009] in, i The imaginary unit; The coordinates of a rectangular elastic thin plate in the frequency domain Surface vibration displacement at that location The coordinates acting on a rectangular elastic thin plate in the frequency domain The sound pressure at that location, The coordinates acting on a rectangular elastic thin plate in the frequency domain The motivational force of the location; The solution to the forced vibration equation is transformed to obtain the rectangular elastic thin plate under any given conditions. The modal vibration equation under the first mode is:

[0010] in, , For modal variables; It is a rectangular elastic thin plate. First-order modal displacement, It acts on a rectangular elastic thin plate. First-order modal sound pressure, It acts on a rectangular elastic thin plate. First-order modal excitation force;

[0011]

[0012]

[0013] in, The structural damping factor and ; for First modal frequency and , It is the modal wavenumber along the length of the rectangular elastic thin plate. It is the modal wavenumber in the width direction of the rectangular elastic thin plate. It is the length of the rectangular elastic thin plate. It is the width of the rectangular elastic sheet; It is a rectangular elastic thin plate vibrating freely. First-order mode function, under simply supported conditions .

[0014] Its further technical solution is to use statistical methods to describe The autospectral density of the excitation force of the first resonant mode Low wavenumber spectrum of turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum The relationship between them is:

[0015] The low wavenumber spectrum of the fluctuating pressure wavenumber-frequency spectrum in the turbulent boundary layer is predicted using the following formula. :

[0016] Among them, wavenumber vector , It is along the length of the rectangular elastic thin plate. unit vector, It is along the width direction of the rectangular elastic thin plate The unit vector; It is a rectangular elastic thin plate vibrating freely. The first-order resonant mode wave function.

[0017] Its further technical solution is to use statistical methods to describe The autospectral density of the excitation force of the first resonant mode Low wavenumber spectrum of turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum The relationships between them include: Representing the fluctuating pressure in the turbulent boundary layer using the wavenumber-frequency spectrum. Acting on a rectangular elastic thin plate First-order resonant mode excitation force for:

[0018] in, This represents the two-dimensional wavenumber-frequency spectrum of fluctuating pressure in the turbulent boundary layer; the superscript * indicates conjugation. for The first resonant mode wave function and its relationship with For Fourier transform pairs; get The autospectral density of the excitation force of the first resonant mode for:

[0019] Combining wavenumber-frequency spectrum characteristics The transformation yields:

[0020] By combining the white wavenumber spectrum characteristic transformation of the low wavenumber components of turbulent fluctuating pressure, we obtain The autospectral density of the excitation force of the first resonant mode Low wavenumber spectrum of turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum The relationship between them:

[0021] in, Represents the Dirac function, This represents the wavenumber-frequency spectrum of fluctuating pressure in the turbulent boundary layer. Represents the wavenumber vector. This represents the angular frequency of vibration of a rectangular elastic thin plate.

[0022] Further technical solutions include: the low wavenumber prediction method for turbulent boundary layer fluctuating pressure also includes: The power spectral density of the vibration displacement on the surface of a rectangular elastic thin plate is determined as follows:

[0023] in, It is any free vibration of a rectangular elastic thin plate First-order mode function, It is any rectangular elastic thin plate First-order modal displacement; It is any free vibration of a rectangular elastic thin plate First-order mode function, It is any rectangular elastic thin plate First-order modal displacement; This represents the angular frequency of vibration of a rectangular elastic thin plate. This represents the Dirac function, with the superscript * indicating conjugate; The autospectral density of the vibration displacement on the surface of a rectangular elastic thin plate is obtained by utilizing the orthogonality and normalization of the modal functions. satisfy:

[0024] Define any Autospectral density of first-order modal displacements satisfy And obtained:

[0025] Autospectral density based on the vibration displacement of a rectangular elastic thin plate surface The characteristics obtained from the superposition of the autospectral densities of the displacements of each mode, combined with the characteristic that the resonant modal response dominates during the resonance of the elastic structure, determine the use of within the error range. Autospectral density of the first resonant mode displacement The autospectral density representing the vibrational displacement of a rectangular elastic thin plate surface It is used to predict the low wavenumber spectrum of the turbulent boundary layer fluctuation pressure wavenumber-frequency spectrum. .

[0026] A further technical solution is to use a rectangular elastic thin plate in any... Determination of modal vibration equations under first mode First-order modal impedance include: Determine the rectangular elastic thin plate that is unaffected by fluid load. First-order modal impedance ; Determine the coupling condition between the flow-induced vibration and the radiated sound field of a rectangular elastic thin plate when one side is affected by a fluid load and the other side is a vacuum. The first modal radiation sound pressure is subject to arbitrary Modal acoustic radiation impedance under first-order modal vibration And obtain a rectangular elastic thin plate affected by fluid load. First-order modal impedance .

[0027] A further technical solution involves determining the rectangular elastic thin plate affected by fluid load. First-order modal impedance ; in, Indicates the wet modal frequency. , Indicates modal radiation damping and , for The first-order modal radiated sound pressure is subject to its own Modal radiation efficiency of first-mode vibration. for The first-order modal radiated sound pressure is subject to its own Modal-added mass of first-order modal vibration, For the speed of sound in the medium, The density is the medium.

[0028] A further technical solution involves obtaining a rectangular elastic thin plate affected by fluid load. First-order modal impedance include: Determine the radiated sound pressure of the rectangular elastic thin plate reacting with the flow-induced vibration of the plate under fluid load. and substitute ,get First-order modal sound pressure and First-order modal displacement The relationship is:

[0029] in, The first modal radiation sound pressure is subject to arbitrary Modal acoustic radiation impedance generated by first-order modal vibration The expression is:

[0030] The superscript * indicates conjugate. First-order mode wave function and First-order mode function For the Fourier transform pair and:

[0031] modal acoustic radiation impedance Represented as Form and determination The first modal radiation sound pressure is subject to arbitrary Modal radiation efficiency of first-mode vibration Modal added mass ; modal acoustic radiation impedance Substitution First-order modal sound pressure With any First-order modal displacement In the relationship, and substitute The first-order modal vibration equation is obtained for a rectangular elastic thin plate affected by fluid load. First-order modal impedance .

[0032] Its further technical solution is to substitute... The first-order modal vibration equation is obtained for a rectangular elastic thin plate affected by fluid load. First-order modal impedance include: modal acoustic radiation impedance Substitution First-order modal sound pressure With any First-order modal displacement In the relationship, and substitute From the first-order modal vibration equations, we obtain a set of linear equations coupling the vibration of a rectangular elastic thin plate with the radiated sound field:

[0033] based on The first-order modal radiated sound pressure is subject to its own The modal self-radiation impedance under the influence of first-order modal vibration is greater than the modal mutual radiation impedance under the influence of other-order modal vibration, which can be ignored within the error range. The modal mutual radiation impedance of the first-order modal radiated sound pressure under the influence of other modal vibrations is obtained by considering only... The first-order modal radiated sound pressure is subject to its own Linear equations for the coupling between the vibration of a rectangular elastic thin plate and the radiated sound field under modal self-radiation impedance of first-order vibration:

[0034] The rectangular elastic thin plate affected by fluid load is obtained by processing. First-order modal impedance .

[0035] A further technical solution involves determining the radiated sound pressure of the rectangular elastic thin plate reacting with the radiated sound field generated by the flow-induced vibration of the rectangular elastic thin plate under the influence of fluid load. include: Determine the spatial coordinates of the flow-induced vibration of a rectangular elastic thin plate under fluid load. Radiated sound pressure at the location The wavenumber domain expression is:

[0036] Determine the time In the case of simple harmonic variation, the wave equation transforms into the Helmholtz equation:

[0037] in, The sound wave number of the medium, Along the thickness direction; Radiated sound pressure Substituting the wavenumber domain expression into the Helmholtz equation and solving, we obtain:

[0038] in, These are coefficients to be determined; Determine the surface of the rectangular elastic thin plate Boundary conditions at:

[0039] Determining undetermined coefficients using boundary conditions And obtained:

[0040] Performing an inverse spatial Fourier transform yields the radiated sound pressure generated by the flow-induced vibration of a rectangular elastic thin plate. for:

[0041] make The radiated sound pressure of the rectangular elastic thin plate reacting with the radiated sound field generated by the flow-induced vibration is:

[0042] in, , The coordinates on a rectangular elastic thin plate Surface vibration displacement at the location and .

[0043] The beneficial technical effects of this application are: This application discloses a method for predicting low wavenumber of fluctuating pressure in turbulent boundary layers. This method utilizes the spatial filtering characteristics of elastic structures, using the elastic structure as a wavenumber filter sensor. Based on the modal resonance response of the elastic structure surface, it predicts the low wavenumber spectrum of the wavenumber-frequency spectrum of fluctuating pressure in turbulent boundary layers. This method is very sensitive to the modal wavenumber of the elastic structure and can effectively suppress the interference of other wavenumbers, thereby improving the prediction accuracy of the low wavenumber spectrum of fluctuating pressure in turbulent boundary layers. This is beneficial to improving the accuracy and robustness of subsequent hydrodynamic noise prediction. Attached Figure Description

[0044] Figure 1 This is a flowchart of a method for predicting low wavenumber pressure in turbulent boundary layer according to an embodiment of this application.

[0045] Figure 2 This is a comparison of the low wavenumber spectrum prediction results of turbulent boundary layer pulsating pressure obtained by applying the method of this application with the wavenumber spectrum results of the calculated vibration response of a flow-induced rectangular elastic thin plate in an example.

[0046] Figure 3 This is a comparison of the low wavenumber spectrum prediction results of turbulent boundary layer pulsating pressure obtained by applying the method of this application with the wavenumber spectrum results of the calculated vibration response of a flow-induced rectangular elastic thin plate in another example.

[0047] Figure 4 This is a comparison of the low wavenumber spectrum prediction results of the turbulent boundary layer pulsating pressure obtained by applying the method of this application with the wavenumber spectrum results of the calculated vibration response of the flow-induced rectangular elastic thin plate in another example. Detailed Implementation

[0048] The specific embodiments of this application will be further described below with reference to the accompanying drawings.

[0049] This application discloses a method for predicting low wavenumber pressure in turbulent boundary layer fluctuations. The method, applied to a rectangular elastic thin plate, includes the following steps, referencing… Figure 1 The flowchart shown is as follows: Step 1: A theoretical model is constructed based on the small-amplitude bending vibration equation satisfying the vibration displacement of the rectangular elastic thin plate, yielding the forced vibration equation of the rectangular elastic thin plate under sound pressure and excitation force: (1) time In the case of simple harmonic transformation, equation (1) is transformed into: (2) in, The bending stiffness of a rectangular elastic thin plate and , The Young's modulus of the material. For the material's Poisson's ratio, The thickness is the plate thickness. The damping coefficient is... is the surface density and , The density is the density of the board material. , For the Laplace operator, . i The imaginary unit, Let be the angular frequency of the vibration of the rectangular elastic thin plate.

[0050] The coordinates on a rectangular elastic thin plate in the time domain Surface vibration displacement at that location The coordinates acting on a rectangular elastic thin plate in the time domain The sound pressure at that location, The coordinates acting on a rectangular elastic thin plate in the time domain The motivation of the place, Indicates time. The coordinates of a rectangular elastic thin plate in the frequency domain Surface vibration displacement at that location The coordinates acting on a rectangular elastic thin plate in the frequency domain The sound pressure at that location, The coordinates acting on a rectangular elastic thin plate in the frequency domain The incentive of the location. Represents the coordinates along the length of the rectangular elastic thin plate. This represents the coordinates along the width of the rectangular elastic sheet, while the flow is along the length.

[0051] Step 2: Based on the boundary conditions, using the modal functions of the free vibration of the rectangular elastic thin plate, solve the forced vibration equation (2) using the modal expansion method, and use the orthogonality normalization of the modal functions to transform to the modal domain to obtain the equation of the rectangular elastic thin plate under any given conditions. Modal vibration equations under first mode: (3) in, , The modal number. It is a rectangular elastic thin plate. First-order modal displacement, It acts on a rectangular elastic thin plate. First-order modal sound pressure, It acts on a rectangular elastic thin plate. The first modal excitation force, and: (4) (5) (6) in, The structural damping factor and . for First modal frequency and , It is the modal wavenumber along the length of the rectangular elastic thin plate. It is the modal wavenumber in the width direction of the rectangular elastic thin plate. It is the length of the rectangular elastic thin plate. It is the width of the rectangular elastic sheet. It is a rectangular elastic thin plate vibrating freely. First-order mode function, under simply supported conditions .

[0052] Step 3: Determine the rectangular elastic thin plate based on the modal vibration equation (3). First-order modal impedance This step is determined in two cases: 1. When a rectangular elastic thin plate is not affected by fluid load, its First-order modal impedance We can obtain: (7) Furthermore, we can obtain: (8) 2. Considering the influence of fluid load on the rectangular elastic thin plate, and one side of the rectangular elastic thin plate ( Under conditions of fluid load on one side and vacuum on the other, arbitrary coupling between flow-induced vibration and radiated sound field in a rectangular elastic thin plate. The first modal radiation sound pressure is subject to arbitrary Modal acoustic radiation impedance generated by first-order modal vibration ,include: First, determine the radiated sound pressure of the rectangular elastic thin plate caused by the flow-induced vibration under the influence of fluid load. It includes the following steps: Radiated sound pressure generated by flow-induced vibration of a rectangular elastic thin plate under fluid load Perform a two-dimensional Fourier transform in space to obtain the wavenumber domain expression for the radiated sound pressure generated by the flow-induced vibration of a rectangular elastic thin plate under the influence of fluid load; (9) time In the case of simple harmonic variation, the wave equation transforms into the Helmohltz equation. (10) in, The sound wave number of the medium, The speed of sound in the medium. It acts on spatial coordinates Radiated sound pressure at that location. Along the thickness direction of the plate.

[0053] Determine the surface of the rectangular elastic thin plate The boundary conditions at the location are: (11) in, The density is the medium. The coordinates on a rectangular elastic thin plate Surface vibration displacement at that location.

[0054] Considering that there are no reflected sound waves in the semi-infinite space, substituting equation (9) into equation (10) and solving equation (10), we can obtain the sound field form solution as follows: (12) in, These are coefficients to be determined.

[0055] The undetermined coefficients can be determined using the boundary condition equation (11). And obtained: (13) From equation (9), we can see that and For the spatial Fourier transform pair, perform an inverse spatial Fourier transform on equation (13) to obtain the radiated sound pressure generated by the flow-induced vibration of the rectangular elastic thin plate. for: (14) here, and Let be a pair of spatial Fourier transforms, and we have: (15) make The radiated sound pressure generated by the flow-induced vibration of the rectangular elastic thin plate is obtained as a reaction force on the rectangular elastic thin plate. for: (16) Substituting equation (16) into equation (5), we get First-order modal sound pressure The expression is: (17) Surface vibration displacement of a rectangular elastic thin plate Expanding by modal functions, we get: (18) Substituting equation (18) into equation (15), and then into equation (17), we get First-order modal sound pressure With any First-order modal displacement The relationship is: (19) in, For any The first modal radiation sound pressure is subject to arbitrary The modal acoustic radiation impedance generated by first-order modal vibration is expressed as follows: (20) Among them, superscript The sign indicates conjugate. (Arbitrary) First-order mode wave function and First-order mode function For the Fourier transform pair and: (twenty one) will any The first modal radiation sound pressure is subject to arbitrary Modal acoustic radiation impedance generated by first-order modal vibration It shall be represented in the following form: (twenty two) in, For any The first modal radiation sound pressure is subject to arbitrary Modal radiation efficiency of first-mode vibration. For any The first modal radiation sound pressure is subject to arbitrary Modal added mass of first-order modal vibration.

[0056] Modal acoustic radiation impedance of equation (22) After substituting into equation (19), then substituting into the rectangular elastic thin plate... The first-order modal vibration equation (3) yields a set of linear equations for the coupling of vibration and radiated sound field in a rectangular elastic thin plate: (twenty three) Equation (23) is a set of linear equations for the coupling of vibration and radiated sound field of a rectangular elastic thin plate, which can be solved directly.

[0057] Modal acoustic radiation impedance of a rectangular elastic thin plate , containing representation The first-order modal radiated sound pressure is subject to its own Modal self-radiation impedance and characterization of first-order modal vibration The modal mutual radiation impedance of the first-order modal radiated sound pressure is affected by the vibration of other modal modes. Based on the characteristic that the modal mutual radiation impedance is less than the modal self-radiation impedance, the modal mutual radiation impedance in equation (23) can be neglected within the error range. The modal mutual radiation impedance of the first-order modal radiated sound pressure under the influence of other modal vibrations is obtained by considering only... The first-order modal radiated sound pressure is subject to its own Linear equations for the coupling between the vibration of a rectangular elastic thin plate and the radiated sound field under modal self-radiation impedance of first-order vibration: (twenty four) Furthermore, we can obtain: (25) Thus, a rectangular elastic thin plate affected by fluid load is obtained. First-order modal impedance .in, Indicates the wet modal frequency. , Indicates modal radiation damping and . for The first-order modal radiated sound pressure is subject to its own Modal radiation efficiency of modal vibration. for The first-order modal radiated sound pressure is subject to its own Modal added mass of first-order modal vibration.

[0058] In summary, any First-order modal impedance .

[0059] Step 4: Based on the statistical stationarity of the turbulent boundary layer pulsating pressure, a statistical method is used to describe the turbulent boundary layer pulsating pressure and the vibration response caused by its excitation, and the relationship between the low wavenumber spectrum of the turbulent boundary layer pulsating pressure and the vibration displacement response of the rectangular elastic thin plate is constructed.

[0060] The power spectral density of the vibration displacement on the surface of a rectangular elastic thin plate is: (26) in, It is any free vibration of a rectangular elastic thin plate First-order mode function, It is any rectangular elastic thin plate First-order modal displacement. This represents the angular frequency of vibration of a rectangular elastic thin plate. This represents the Dirac function, and the superscript * indicates conjugate.

[0061] By utilizing the orthogonality and normalization of modal functions, the autospectral density of the vibration displacement on the surface of a rectangular elastic thin plate is obtained. satisfy: (27) make ,but yes The autospectral density of the first modal displacement can be obtained from equations (26) and (27): (28) From equation (28), we can see the autospectral density of the vibration displacement on the surface of the rectangular elastic thin plate. It is obtained by superimposing the autospectral density of each modal displacement.

[0062] As an elastic structure, the rectangular elastic thin plate exhibits a dominant resonant modal response at resonance. Therefore, at the resonant frequency, the response of a single resonant mode can be directly used to represent the infinite superposition of modal responses. Considering the orthogonality of modes, the interaction between vibrational displacements of different orders of modes cancels out along the plate surface, and modal coupling is neglected. Therefore, it can be determined that, within the error range, the response can be directly utilized... Autospectral density of the first resonant mode displacement The autospectral density representing the vibrational displacement of a rectangular elastic thin plate surface It is used to predict the low wavenumber spectrum of the turbulent boundary layer fluctuation pressure wavenumber-frequency spectrum. .

[0063] From equations (8) and (25), it can be seen that, Autospectral density of the first resonant mode displacement and The autospectral density of the excitation force of the first resonant mode satisfy: (29) in, For the rectangular elastic thin plate determined in step three The impedance of the first resonant mode, and has .

[0064] Self-spectral density of resonant modal excitation force When turbulent boundary layer pulsating pressure excites a rectangular elastic thin plate, the turbulent boundary layer pulsating pressure can be represented by the wavenumber-frequency spectrum of the turbulent boundary layer pulsating pressure. Acting on a rectangular elastic thin plate First-order resonant mode excitation force According to the definition, the excitation force of the resonant mode for: (30) in, This represents the two-dimensional wavenumber-frequency spectrum of fluctuating pressure in the turbulent boundary layer. (Wavenumber vector) , It is along the length of the rectangular elastic thin plate. unit vector, It is along the width direction of the rectangular elastic thin plate The unit vector. for The first resonant mode wave function and its relationship with First resonant mode function For Fourier transform pairs.

[0065] Substitute equation (30) into get: (31) Combining wavenumber-frequency spectrum characteristics Equation (31) is transformed into: (32) in, This represents the wavenumber-frequency spectrum of fluctuating pressure in the turbulent boundary layer.

[0066] From equation (32), it can be seen that the modal excitation force self-spectral density of the fluctuating pressure in the turbulent boundary layer is... With structural modal wave function related.

[0067] In most aqueous media, the low wavenumber component of the turbulent boundary layer fluctuation pressure is related to the structural modal wavefunction. The interaction of peak components is the main source of acoustic radiation from flow-induced structural vibration. Combining the "white wavenumber" spectral characteristics of the low wavenumber components of turbulent pulsating pressure, equation (32) can be expressed as: (33) in, This is the low wavenumber spectrum of the turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum.

[0068] Substituting equation (33) into equation (29) yields Autospectral density of the first resonant mode displacement : (34) Furthermore, we can obtain: (35) According to equation (35), the low wavenumber spectrum of turbulent boundary layer pulsating pressure can be predicted using the vibration modal resonance response of the elastic structure. .

[0069] In different examples, considering the influence of fluid load, the comparison between the low wavenumber spectrum prediction results of the turbulent boundary layer pulsating pressure of a rectangular elastic thin plate obtained using the prediction method of this application and the wavenumber spectrum of the calculated flow-induced vibration response of the rectangular elastic thin plate is shown in the figure. Figures 2-4 As shown, by Figures 2-4 The comparison of the forecast results shows that the low wavenumber spectrum forecast of the turbulent boundary layer pulsating pressure obtained by this application is in good agreement with the wavenumber spectrum of the calculated flow-induced vibration response of a rectangular elastic thin plate, indicating the effectiveness of the forecast method in this application.

[0070] The above descriptions are merely preferred embodiments of this application, and this application is not limited to the above embodiments. It is understood that other improvements and variations that can be directly derived or conceived by those skilled in the art without departing from the spirit and concept of this application should be considered to be included within the protection scope of this application.

Claims

1. A method for predicting low wavenumber pressure in turbulent boundary layer, characterized in that, The method for predicting low-wavenumber pressure in the turbulent boundary layer includes: Based on the boundary conditions, using the modal functions of the free vibration of the rectangular elastic thin plate, the forced vibration equation of the rectangular elastic thin plate under the action of sound pressure and excitation force is solved by the modal expansion method. Then, by utilizing the orthogonality and normalization of the modal functions, the equation is transformed to the modal domain to obtain the equation of the rectangular elastic thin plate under the action of sound pressure and excitation force. Modal vibration equations under first resonance modes; Based on rectangular elastic thin plates The modal vibration equation under the first resonance mode determines the rectangular elastic thin plate. First-order resonant mode impedance , Let be the angular frequency of the vibration of the rectangular elastic thin plate; Describing using statistical methods middle The autospectral density of the excitation force of the first resonant mode Low wavenumber spectrum of turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum The relationship between them was explored, and the application of a rectangular elastic thin plate under turbulent boundary layer pulsating pressure excitation was utilized. Autospectral density of the first resonant mode displacement The low wavenumber spectrum of the fluctuating pressure wavenumber-frequency spectrum in the turbulent boundary layer was predicted. .

2. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 1, characterized in that, The equation for the forced vibration of a rectangular elastic thin plate under sound pressure and excitation force is as follows: in, The bending stiffness of a rectangular elastic thin plate and , The Young's modulus of the material. For the material's Poisson's ratio, For plate thickness; The damping coefficient is... is the surface density and , Density of the board material; , For the Laplace operator, ; The coordinates on a rectangular elastic thin plate in the time domain Surface vibration displacement at that location The coordinates acting on a rectangular elastic thin plate in the time domain The sound pressure at that location, The coordinates acting on a rectangular elastic thin plate in the time domain The motivational force of the location; Indicates time, Represents the coordinates along the length of the rectangular elastic thin plate. Represents the coordinates along the width of the rectangular elastic sheet, and the flow along the length of the rectangular elastic sheet; In time The forced vibration equation under the simple harmonic transformation is transformed as follows: in, i The imaginary unit; The coordinates of a rectangular elastic thin plate in the frequency domain Surface vibration displacement at that location The coordinates acting on a rectangular elastic thin plate in the frequency domain The sound pressure at that location, The coordinates acting on a rectangular elastic thin plate in the frequency domain The motivational force of the location; Solving the forced vibration equation and transforming it, we obtain the rectangular elastic thin plate under any given conditions. The modal vibration equation under the first mode is: in, , For modal variables; It is a rectangular elastic thin plate. First-order modal displacement, It acts on a rectangular elastic thin plate. First-order modal sound pressure, It acts on a rectangular elastic thin plate. First-order modal excitation force; in, The structural damping factor and ; for First modal frequency and , It is the modal wavenumber along the length of the rectangular elastic thin plate. It is the modal wavenumber in the width direction of the rectangular elastic thin plate. It is the length of the rectangular elastic thin plate. It is the width of the rectangular elastic sheet; It is a rectangular elastic thin plate vibrating freely. First-order mode function, under simply supported conditions .

3. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 2, characterized in that, Describing using statistical methods The autospectral density of the excitation force of the first resonant mode Low wavenumber spectrum of turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum The relationship between them is: The low wavenumber spectrum of the fluctuating pressure wavenumber-frequency spectrum in the turbulent boundary layer is predicted using the following formula. : Among them, wavenumber vector , It is along the length of the rectangular elastic thin plate. unit vector, It is along the width direction of the rectangular elastic thin plate The unit vector; It is a rectangular elastic thin plate vibrating freely. The first-order resonant mode wave function.

4. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 3, characterized in that, Describing using statistical methods The autospectral density of the excitation force of the first resonant mode Low wavenumber spectrum of turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum The relationships between them include: Representing the fluctuating pressure in the turbulent boundary layer using the wavenumber-frequency spectrum. Acting on a rectangular elastic thin plate First-order resonant mode excitation force for: in, This represents the two-dimensional wavenumber-frequency spectrum of fluctuating pressure in the turbulent boundary layer; the superscript * indicates conjugation. for The first resonant mode wave function and its relationship with For Fourier transform pairs; get The autospectral density of the excitation force of the first resonant mode for: Combining wavenumber-frequency spectrum characteristics The transformation yields: By combining the white wavenumber spectrum characteristic transformation of the low wavenumber components of turbulent fluctuating pressure, we obtain The autospectral density of the excitation force of the first resonant mode Low wavenumber spectrum of turbulent boundary layer fluctuating pressure wavenumber-frequency spectrum The relationship between them: in, Represents the Dirac function, This represents the wavenumber-frequency spectrum of fluctuating pressure in the turbulent boundary layer. Represents the wavenumber vector. This represents the angular frequency of vibration of a rectangular elastic thin plate.

5. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 1, characterized in that, The method for predicting low-wavenumber pressure in turbulent boundary layer fluctuations also includes: The power spectral density of the vibration displacement on the surface of a rectangular elastic thin plate is determined as follows: in, It is any free vibration of a rectangular elastic thin plate First-order mode function, It is any rectangular elastic thin plate First-order modal displacement; It is any free vibration of a rectangular elastic thin plate First-order mode function, It is any rectangular elastic thin plate First-order modal displacement; This represents the angular frequency of vibration of a rectangular elastic thin plate. This represents the Dirac function, with the superscript * indicating conjugate; The autospectral density of the vibration displacement on the surface of a rectangular elastic thin plate is obtained by utilizing the orthogonality and normalization of the modal functions. satisfy: Define any Autospectral density of first-order modal displacements satisfy And obtained: Autospectral density based on the vibration displacement of a rectangular elastic thin plate surface The characteristics obtained from the superposition of the autospectral densities of the displacements of each mode, combined with the characteristic that the resonant modal response dominates during the resonance of the elastic structure, determine the use of within the error range. Autospectral density of the first resonant mode displacement The autospectral density representing the vibrational displacement of a rectangular elastic thin plate surface It is used to predict the low wavenumber spectrum of the turbulent boundary layer fluctuation pressure wavenumber-frequency spectrum. .

6. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 2, characterized in that, Based on rectangular elastic thin plates in any Determination of modal vibration equations under first mode First-order modal impedance include: Determine the rectangular elastic thin plate that is unaffected by fluid load. First-order modal impedance ; Determine the coupling condition between the flow-induced vibration and the radiated sound field of a rectangular elastic thin plate when one side is affected by a fluid load and the other side is a vacuum. The first modal radiation sound pressure is subject to arbitrary Modal acoustic radiation impedance under first-order modal vibration And obtain a rectangular elastic thin plate affected by fluid load. First-order modal impedance .

7. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 6, characterized in that, Determine the rectangular elastic thin plate affected by fluid load. First-order modal impedance ; in, Indicates the wet modal frequency. , Indicates modal radiation damping and , for The first-order modal radiated sound pressure is subject to its own Modal radiation efficiency of first-mode vibration. for The first-order modal radiated sound pressure is subject to its own Modal-added mass of first-order modal vibration, For the speed of sound in the medium, The density is the medium.

8. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 7, characterized in that, Obtaining a rectangular elastic thin plate affected by fluid load First-order modal impedance include: Determine the radiated sound pressure of the rectangular elastic thin plate reacting with the flow-induced vibration of the plate under fluid load. and substitute ,get First-order modal sound pressure and First-order modal displacement The relationship is: in, The first modal radiation sound pressure is subject to arbitrary Modal acoustic radiation impedance generated by first-order modal vibration The expression is: The superscript * indicates conjugate. First-order mode wave function and First-order mode function For the Fourier transform pair and: modal acoustic radiation impedance Represented as Form and determination The first modal radiation sound pressure is subject to arbitrary Modal radiation efficiency of first-mode vibration Modal added mass ; modal acoustic radiation impedance Substitution First-order modal sound pressure With any First-order modal displacement In the relationship, and substitute The first-order modal vibration equation is obtained for a rectangular elastic thin plate affected by fluid load. First-order modal impedance .

9. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 8, characterized in that, Substitution The first-order modal vibration equation is obtained for a rectangular elastic thin plate affected by fluid load. First-order modal impedance include: modal acoustic radiation impedance Substitution First-order modal sound pressure With any First-order modal displacement In the relationship, and substitute From the first-order modal vibration equations, we obtain a set of linear equations coupling the vibration of a rectangular elastic thin plate with the radiated sound field: based on The first-order modal radiated sound pressure is subject to its own The modal self-radiation impedance under the influence of first-order modal vibration is greater than the modal mutual radiation impedance under the influence of other-order modal vibration, which can be ignored within the error range. The modal mutual radiation impedance of the first-order modal radiated sound pressure under the influence of other modal vibrations is obtained by considering only... The first-order modal radiated sound pressure is subject to its own Linear equations for the coupling between the vibration of a rectangular elastic thin plate and the radiated sound field under modal self-radiation impedance of first-order vibration: The rectangular elastic thin plate affected by fluid load is obtained by processing. First-order modal impedance .

10. The method for predicting low wavenumber pressure in turbulent boundary layer according to claim 8, characterized in that, Determine the radiated sound pressure of the rectangular elastic thin plate reacting with the flow-induced vibration of the plate under fluid load. include: Determine the spatial coordinates of the flow-induced vibration of a rectangular elastic thin plate under fluid load. Radiated sound pressure at the location The wavenumber domain expression is: Determine the time In the case of simple harmonic variation, the wave equation transforms into the Helmholtz equation: in, The sound wave number of the medium, Along the thickness direction; Radiated sound pressure Substituting the wavenumber domain expression into the Helmholtz equation and solving, we obtain: in, These are coefficients to be determined; Determine the surface of the rectangular elastic thin plate Boundary conditions at: Determine the undetermined coefficients using the aforementioned boundary conditions. And obtained: Performing an inverse spatial Fourier transform yields the radiated sound pressure generated by the flow-induced vibration of a rectangular elastic thin plate. for: make The radiated sound pressure of the rectangular elastic thin plate reacting with the radiated sound field generated by the flow-induced vibration is: in, , The coordinates on a rectangular elastic thin plate Surface vibration displacement at the location and .

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