Study on Acoustic–Vibration Characteristics and Noise Reduction Methods for Elbows

Abstract

Fluid pipelines with large flow changes often result in noise due to multi-physical interactions (fluid–structure and acoustic–vibration interactions) between the pulsating fluid and the pipe wall, especially at the elbows. Therefore, the acoustic–vibration characteristics and noise reduction methods of elbows are studied in this paper. Firstly, a two-way fluid–structure interaction (FSI) model is established to analyze the vibration characteristics of the elbow under water excitation. Maximum stress occurs at the elbow inlet, with maximum deformation in the elbow. Experimental validation confirms the model’s accuracy. Secondly, the effects of water and structural parameters on elbow vibration are studied, revealing that increased water pressure, pulsating frequency, and flow rate intensify pipe vibration. Finally, an acoustic–vibration coupled model is built; the simulations suggest that increasing wall thickness and elbow radius and reducing elbow angle effectively reduce the noise level of the elbow. Using elastic supports and damping materials can reduce elbow noise by at least 26.3%. This study provides guidance for the noise reduction and structural optimization of elbows by coupled multi-physics.

1. Introduction

Pipelines carry water in a cheap way, extensively used in industry [1]. Due to the periodic inflow and outflow of water in pumps, flow pulsations and pressure pulsations will be generated in pipelines. When the water flows through the elbow of a pipeline, interactions between the pulsating water and the pipe wall induce vibrations known as fluid–structure interactions (FSIs), becoming a source of noise within the pipe [2,3,4]. The vibration of the elbow may not only cause leaks in the ground pipeline, but also increase the risk of fatigue damage of the ground pipeline. The noise generated can interfere with workers’ health and reduce working efficiency [5]. Therefore, it is significant to analyze the FSI vibration characteristics of elbows under pulsating water. This will help to understand the factors that affect the vibration and noise of elbows and provides guidance for ground pipeline optimization and fluid flow control. At present, an increasing number of scholars have studied the aspects of the pipe noise, FSIs, and pipe vibration [6,7,8,9].

Pipeline vibrations result in annual losses of tens of billions of dollars, prompting increased focus on fluid–structure interactions (FSIs) in piping systems [10]. Paidoussis and Li [11], Gorman et al. [12], and Lee and Chung [13] have developed numerous FSI vibration models to clarify the nonlinear dynamics of liquid-carrying pipelines. Kochupillai et al. [14], Zanganeh et al. [15], and Xu et al. [16] used various numerical methods such as the Method of Characteristics (MOC), the Finite Element Method (FEM), MOC-FEM, and the Transfer Matrix Method (TMM) to analyze FSI vibration responses in both time and frequency domains. Scholars have extensively explored the effects of fluid and structural parameters on FSI vibration characteristics in piping systems using various theoretical models and numerical methods [17]. Zhang et al. [18] investigated the coupled vibration of deep-water risers, analyzing the influences of internal flow, velocity changes, and top stress amplitude. Tian et al. [19] studied the vibration characteristics of pipelines under the effect of gas pressure pulsation and obtained the relationship between the natural frequency of pipelines and structural parameters, vibration displacement, and fluid parameters. Liu and Li [20] investigated pipeline vibration characteristics with and without elastic constraints. While existing studies mainly focus on one-way FSIs, there is limited research on two-way FSI vibrations in water elbows, and the mechanism’s influence on structural noise generation remains unclear.

The study of acoustic field characteristics in pipes has received much attention in many fields for a long time [21]. Previous research has mainly focused on the characteristics of the flow-induced noise in elbows. Liu et al. analyzed elbow pipelines at natural gas gathering stations, finding lower sound pressure levels (SPL) in 45° elbows compared to 90° elbows [22]. Mori et al. studied the relationship between the acoustic and vibration characteristics of pipes [23]. The experimental results indicate that the acoustic characteristics of the acoustic field significantly impacted flow-induced noise in the pipes, with the frequency characteristics of the sound source depending on the inlet velocity. According to literature reviews, there are a few previous research studies regarding the characteristics of two-way fluid–structure interaction noises in water elbows. At the same time, many methods have investigated how to reduce the noise of pipelines. Zhang et al. investigated the structural vibration and flow noise caused by turbulent flow in a 90° elbow [24]. The guide vane used for the noise reduction within the elbow was investigated. However, the influence of fluid and pipe parameters on the acoustic field requires further exploration. The coupled behavior of Computational Fluid Dynamics (CFD) and structural acoustics in flow-induced elbows remains underexplored.

The purpose of this study is to investigate the variation law of vibration and noise in elbows based on the following coupled multi-physics: two-way FSIs and acoustic–vibration simulations. Firstly, an ANSYS Workbench (2020R2 edition) was used to establish a finite element FSI model and assess FSI vibration characteristics, validated using experiments. Secondly, the impact of fluid and structural parameters on elbow vibrations was examined, revealing a significant reduction in natural frequency due to FSI effects. Finally, an acoustic–vibration coupled model was built, and the simulations demonstrate that increasing wall thickness and elbow radius and reducing elbow angle effectively reduce the noise levels of the elbow. Laying damping materials and installing elastic supports can reduce the noise level at the elbow. This study can effectively reduce the vibration and noise of elbows and improve the operation safety of pipelines. The overall research flowchart of this study is shown in Figure 1.

2. Materials and Methods

2.1. Two-Way FSI Mathematical Model

It is necessary to consider not only the influence of the fluid on the pipe structure, but also the influence of the pipe deformation on the fluid domain. Based on the above factors, a two-way FSI method is selected for analysis. The control equations of the two-way FSI problem include three parts: fluid domain, solid domain, and FSI interface.

For the fluid domain, the fluid flow calculation in the water elbow satisfies the law of conservation of mass and the law of momentum conservation, and the governing equation is as follows [25]:

$${\displaystyle \frac{\partial {\rho }_{\mathrm{f}}}{\partial t}}+\nabla \left({\rho }_{\mathrm{f}}v\right)=0$$

(1)

$${\displaystyle \frac{\partial {\rho }_{\mathrm{f}}v}{\partial t}}+\nabla \left({\rho }_{\mathrm{f}}vv-{\tau }_{\mathrm{f}}\right)=f_{\mathrm{f}}$$

(2)

In the pipe fluid, there exists a vortex flow, and the Reynolds number is large. It can be regarded as a turbulent flow, and the standard k-ε turbulence model is selected. Its expression is as follows [26]:

$$\rho {\displaystyle \frac{\mathrm{d}k}{\mathrm{d}t}}={\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\delta }_k}}\right){\displaystyle \frac{\partial k}{\partial x_i}}\right]+G_k+G_b-\rho \epsilon -Y_{\mathrm{M}}$$

(3)

$$\rho {\displaystyle \frac{\mathrm{d}\epsilon }{\mathrm{d}t}}={\displaystyle \frac{\partial }{\partial x_i}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\delta }_g}}\right){\displaystyle \frac{\partial \epsilon }{\partial x_i}}\right]+G_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}\left(G_k+G_{3\epsilon }{G}_b\right)-G_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{k}}$$

(4)

For the solid domain, considering that the pipe is affected by fluid, its overall structural dynamics equation is as follows [27]:

$$M\ddot{u}+C\dot{u}+Ku=F$$

(5)

Without considering thermal effects, the conservation of stress and displacement in the fluid domain and solid domain should be satisfied at the fluid–structure interaction interface. The following equations are satisfied:

$$d_f=d_s$$

(6)

$${\tau }_f\cdot n_f={\tau }_s\cdot n_s$$

(7)

2.2. Simulation Model and Meshing of Water Elbow

The governing equations were solved by using the commercial software ANSYS Workbench (2020R2 edition) [28]. This article takes the elbow of oilfield water injection as the research object. The Finite Element Model is shown in Figure 2. In the simulation, the end surfaces at the inlet and outlet are set as fully constrained, and the wall surface between the pipe and the fluid is set as the FSI’s surface. The pipe is placed horizontally, with a gravity acceleration of 9.8 m/s² in the -y direction.

Set the elbow wall as the solid domain and the fluid as the fluid domain. The basic parameters and associated dimensions of the pipe are given in

Table 1. Dimensional parameters of elbow.

Parameter	Length L1 (mm)	Length L2 (mm)	Radius R (mm)	Angle θ (°)	Wall Thickness ∆d (mm)	Diameter d (mm)
Value	300	600	150	90	4	40

. The material properties of the pipe and water are shown in

Table 2. Physical parameters of elbow and water.

Parameter	Density (kg/m3)	Elastic Modulus (Pa)	Poisson’s Ratio (Pa)	Dynamic Viscosity (μPa·s)
Elbow	7850	2.1 × 1011	0.3	-
Water	1000	2.34 × 109	-	890.08

. The fluid domain solution uses the standard k-ε turbulence model. The fluid inlet (end d1) was set as the velocity inlet, and the fluid outlet (end a1) was set as the pressure outlet. The residual criterion for convergence is set to as low as 10⁻⁶ for all variables.

The fluid domain mesh is subdivided with an expansion layer at the fluid pipe wall boundary to closely simulate the actual model. Tetrahedral meshes are used for the remaining fluid elements. The inlet pipe velocity is set at 2.21 m/s and the outlet pressure is set at 1 MPa, ensuring mesh independence. Analysis of monitoring point v1 velocity in Figure 3a indicates no correlation between mesh number and velocity when the grids exceed 14,000. The solid domain of the elbow is structured meshes, with the stress at monitoring point s1 analyzed in Figure 3b. No correlation between mesh number and stress is observed when the meshes surpass 18,000. Hence, this study employs 14,000 meshes in the fluid domain and 18,000 meshes in the solid domain for subsequent calculations.

2.3. FSI Simulation Calculation of Water Elbow

In ground water injection pipelines, water pumps typically power water delivery. Assuming that the flow rate of the water pump is 10 m³/h, the average flow velocity v₀ in the pipe is 2.21 m/s. In actual flow, the pressure in the pipe can be regarded as the superposition of the average pressure (load pressure) and the pulsating pressure. Assuming that the pump speed is 500 r/min, the pulsation frequency $f$ of the output fluid is 50 Hz. The pulsation rate ${\delta }_q$ is 10%, and the pulsation amplitude $\xi$ is 5%. Set the pipe inlet velocity according to the pump output as follows:

$$v=v_0\times \left[1+\xi \sin (2\pi ft)\right]=2.21\times \left[1+0.05\sin (100\pi t)\right]$$

(8)

Pulsating pressure results from pump flow pulsation, ensuring frequency, and phase consistency in the pulsation curves of pressure and velocity, with differences in amplitude. Assuming that the average pressure at the pipe outlet is 1 MPa, the pipe pressure outlet boundary condition can be set as follows:

$$p=p_0\times [1+\xi \sin (2\pi ft)]=1\times [1+0.05\sin (100\pi t)]$$

(9)

Therefore, the inlet velocity and outlet pressure changes with time can be obtained, as shown in Figure 4. As shown in Figure 5, nine monitoring points are set up at sections b1, c1, and e1 to monitor the vibration acceleration and stress of the elbow wall.

Figure 6 gives the deformation (a) and stress results (b) of the elbow from simulations. In Figure 6a, the maximum deformation that occurs at point G is 4.72 μm. In Figure 6b, the maximum stress that occurs at the inlet of the elbow is 6.94 MPa, experiencing significant stresses due to complete constraints. Stress at point F inside the elbow exceeds that at point D outside the elbow. In Figure 7, vibration acceleration amplitudes at point D are notably larger in the z and y directions (a_z, a_y) compared to the x direction (a_x). The total vibration acceleration at the monitoring point can be calculated using the following formula:

$$a_{\mathrm{total}}=\sqrt{a_{\mathrm{x}}^2+a_{\mathrm{y}}^2+a_{\mathrm{z}}^2}$$

(10)

From Figure 7, the pulsation frequency of the total vibration acceleration at the monitoring point is twice that of each direction, amounting to 100 Hz.

Table 3. The stress and acceleration of different monitoring points.

Parameter	A	B	C	D	E	F	G	H	I
Stress (MPa)	3.56	3.98	3.93	3.82	4.22	3.94	3.56	4.03	3.80
Acceleration (mm/s2)	17.80	16.96	15.92	20.29	18.58	18.55	19.62	18.80	18.00

indicates stronger vibration at the middle of the elbow, with a greater vibration response observed outside the pipe than inside. Point D, exhibiting the maximum vibration acceleration, is selected as the monitoring point for subsequent simulation calculations of elbow vibration acceleration.

2.4. Validation of the Elbow Simulation Model

To validate the accuracy of the two-way FSI simulation model for the pipe, FSI vibration response calculations were conducted on an experimental elbow. The experimental setup includes pumps, control valves, a water tank, flowmeters, and the elbow, as depicted in Figure 8. The geometry of the experimental elbow is as follows: inlet straight pipe section length L₁ = 500 mm, outlet straight pipe section length L₂ = 1000 mm, elbow radius R = 195 mm, pipe inner diameter d = 65 mm, and pipe wall thickness $\Delta d$ = 5.5 mm. The pump’s flow rate is set to 20 m³/h, with a pulsation frequency of 5 Hz using the control valve 1. The reference pressure pulsation rate is 20%, and the load pressure is set to 1 MPa using the control valve 2. During the experiment, the pipe flow rate, pressure, and vibration signals of the vibration test points of the experimental elbow were collected. The vibration test point is located in the middle of the elbow, and a micro piezoelectric accelerometer KSI-138M050 from KingSci Instruments (Beijing, China) was used with analog signal data and a sampling rate of 1~5000 Hz, installed using magnetic force.

Using the geometric dimensions of the experimental pipe, the model was created in ANSYS Workbench (2020R2 edition). Monitoring points were established at the same locations in the simulation model as in the experimental pipe section. For fluid calculations, the boundary conditions matched the experimental conditions. Set the pipe inlet velocity to $v=1.7\times \left[1+0.1\sin (10\pi t)\right]$ m/s and the outlet pressure to $p=1\times \left[1+0.1\sin (10\pi t)\right]$ MPa. In Figure 9a, a comparison of the z-direction vibration acceleration at the monitoring point between the simulation and experiment shows that the vibration acceleration value changes periodically with time, and the trends of the two results are consistent.

To facilitate a more intuitive comparison of pipe vibration intensity, the root mean square (effective value or RMS value) value is obtained as the measurement parameter. Calculating the RMS value of vibration acceleration from discrete data points can be performed using the following formula:

$$X_{\mathrm{RMS}}=\sqrt{{\displaystyle \frac{X_{\mathrm{x}}^2+X_y^2+X_z^2}{3}}}$$

(11)

In Figure 9b, the vibration of the elbow is mainly concentrated in the z direction. Experimental monitoring points exhibit greater vibrations in all directions compared to the simulation results. This discrepancy arises from the fixed constraints set at the pipe inlet and outlet during simulation, limiting overall pipe deformation. In reality, the pipe is supported by clamps, which are not securely restrained, leading to increased pipe vibration. The effective value errors of vibration in all three directions are less than 10%, validating the correctness of the FSI Finite Element Model in this study.

3. Results and Discussion

3.1. Vibration Characteristics of Water Elbow

3.1.1. Effect of Fluid Parameters

The orthogonal test method was employed to investigate the influence of water pressure and pulsation frequency on pipe vibration. Water pressure was varied from 1 to 4 MPa, and pulsation frequency ranged from 50 to 200 Hz, while the other simulation conditions remained constant. The response of the effective value of vibration acceleration at point D to pulsating fluid pressure and frequency was calculated. In Figure 10a, higher pressure and pulsation frequency resulted in increased elbow vibration response. Compared to water pressure, the pulsation frequency has a more significant effect on the vibration of the elbow.

With water pressure set at 1 MPa and a constant pulsation frequency of 50 Hz with a 10% pulsation rate, varying water flow rates were tested at 5 m³/h, 15 m³/h, 20 m³/h, and 25 m³/h. Utilizing Equation (8), the pipe’s flow velocity was obtained, and the vibration acceleration at point D was calculated for different flow rates. In Figure 10b, as the flow rate increases, the effective value of the vibration acceleration at point D also increases. The impact force on the pipe becomes greater and the vibration response becomes more severe.

3.1.2. Effect of Structural Parameters

Under fluid pressure of 1 MPa and an inlet flow velocity of 2.1 m/s,

Table 4. Natural frequency comparison of elbows (Unit: Hz).

Modal Order	1	2	3	4	5	6
Without FSIs	176.20	421.05	486.98	943.35	976.79	1359.90
Two-way FSIs	155.28	370.85	430.07	610.48	840.57	871.84

displays the natural frequencies obtained through modal analysis without FSI and with the two-way FSI method for the elbow. When considering the FSI effect, the first six-order frequencies of the pipe are significantly lower compared to ignoring the FSI effect. In particular, the fourth- and sixth-order natural frequencies of the pipe drop significantly. After considering the FSI, the mass of the fluid is attached to the pipe. The mass of the entire pipe system increases and the inertial force of the system increases, thereby reducing the natural frequency.

This section primarily analyzes the impact of parameters such as pipe wall thickness $\Delta d$, elbow diameter ratio R/d, and elbow angle α ($\alpha =180-\theta$) on the natural frequency and vibration of the pipe. Velocity inlet and pressure outlet conditions are set according to Equations (8) and (9). Assuming that the pipe wall thickness ranges from 2 to 6 mm, with other parameters consistent with Section 2.1, in Figure 11a, an increase in wall thickness results in a significant rise in the natural frequency of the elbow. This is due to the thickening of the pipe wall, which increases the stiffness of the pipe, thereby increasing the natural frequency. As the wall thickness increases, the vibration amplitude of the entire pipe decreases. Especially when the wall thickness increases from 2 mm to 4 mm, the vibration acceleration of the monitoring point decreases by 51.6% (Figure 11a). Therefore, to reduce pipe vibration, employing thick-walled pipe fittings is recommended.

Assuming an elbow diameter ratio (R/d) ranging from 1 to 5 with a wall thickness set at 4 mm, Figure 11b illustrates that larger R/d resulted in a significant increase in the first three-order natural frequencies of the elbow. The first fourth-order natural frequency increases slowly as the elbow radius increases. Therefore, the elbow radius under low-frequency vibration should be as large as possible. The vibration acceleration at the monitoring point shows no significant change as the elbow radius increases. Elbow angles are set at 30°, 45°, 60°, 75°, and 90°, respectively. In Figure 11c, the smaller the angle of the elbow, the smaller the probability of resonance and better stability. The vibration acceleration at point D increases with the elbow angle. This is attributed to the smoother fluid flow in smaller elbow angles, reducing the impact of the fluid on the pipe.

3.1.3. Harmonic Response Analysis Results

Utilizing the FSI modal analysis results from the elbow, a harmonic response analysis was conducted by employing the mode superposition method. The vibration characteristics of the pipeline considering the fluid–solid coupling effect were studied, and the values of the first six natural frequencies of the elbow were calculated, as shown in Table 4.

This paper only considers the excitation of the pipeline by the fluid, and regards the pressure of the pulsating fluid as the excitation source. The effect of the pulsating fluid pressure on the bend of the elbow is mainly calculated, and the exciting force at the elbow is analyzed, as shown in Figure 12. The inner diameter of the pipeline is D, and the pressure loss caused by the resistance along the way is ignored. The pressure at the inlet and outlet of the pipeline is equal to p, and the angle between the outlet section and the inlet section of the elbow is θ. Therefore, the exciting force F generated at the elbow is shown in Equation (12).

$$F=2\times \left({\displaystyle \frac{1}{4}}\pi D^{2}p\right)\sin \left({\displaystyle \frac{\theta }{2}}\right)$$

(12)

The pressure at the inlet and outlet of the pipeline can be seen to fluctuate around the mean value, $p=p_m+\Delta p$. $p_m$ is the mean pressure of the fluid in the pipeline and represents the pressure change. When $\Delta p=p_{\max }-p_m=p_m-p_{\min }$, the amplitude of the exciting force $\Delta F$ generated by the pulsating fluid at the bend of the elbow can be calculated using Equation (13):

$$\Delta F={\displaystyle \frac{\pi }{2}}{D}^2\Delta p\sin \left({\displaystyle \frac{\theta }{2}}\right)$$

(13)

During the harmonic response analysis, the excitation amplitude $\Delta p$ was set to 1 MPa. The phase angle θ was set to 0°, and the exciting force application point was set at the center D of the outer side of the inner wall surface of the bend. The frequency range for pipe excitation was established as being from 100 Hz to 1000 Hz, encompassing the first six natural frequencies of the elbow based on the results of a prior modal analysis.

The simulation obtained the curve of the deformation response of the outer wall of the bent pipe in the x, y, and z directions as the frequency changes, as shown in Figure 13. It can be seen from the figure that when the excitation frequency is between about 370 Hz and 840 Hz, the pipeline has a large deformation. By comparing the amplitudes of the deformation in various directions in the figure, it can be obtained that the deformations in the y and z directions are larger.

The deformation distribution of the elbow in all directions at 370 Hz is shown in Figure 14. shows. The maximum deformation in the x direction occurs at the bend of the elbow, the maximum deformation in the y direction occurs at the straight pipe of the outlet section, and the maximum deformation in the z direction occurs at the straight pipe of the inlet section. This may be due to the weak constraints in the middle of the pipeline, which is prone to deformation. Usually, when downloading long-distance pipeline transportation, it is necessary to add support constraints in the middle of the pipeline to avoid the large deformation of the pipeline.

The displacement will have peak values near the second-order natural frequency (about 370 Hz) and the fifth-order natural frequency (about 840 Hz). The difference between the frequency corresponding to the maximum peak and the natural frequency is mainly due to setting the interval to 15 Hz when extracting the output data. However, the errors between the response peak frequency and the second-order and fifth-order frequencies are 0.77% and 0.43%, respectively, which has little impact on the results. The vibration displacement in the z direction is the largest. Therefore, when the excitation frequency is the second- and fifth-order natural frequencies, the pipe vibrates violently.

3.2. Acoustic Characteristics and Noise Reduction in Water Elbow

3.2.1. Acoustic–Vibration Simulation Model of Elbow

Software Virtual.Lab 13.6 edition (Munich, German) is developed for noise and vibration analysis [29]. This software, known for its accuracy and time-saving capabilities, enables acoustic–vibration coupling response analyses based on the modules used. In a collaborative simulation, ANSYS Workbench 2020 R2 software and LMS Virtual.Lab 13.6 acoustic analysis software are employed. The flow field results post-FSI calculation in Workbench are exported to Virtual.Lab to compute the acoustic field noise level outside the elbow. The analysis process is illustrated in Figure 15.

In the structural domain, considering sound pressure as structural load, the equation of acoustic–vibration coupling can be expressed as follows:

$$\left(K+\mathrm{j}\omega C-{\omega }^{2}M\right)\left\{u\right\}+L_c\left\{p_{\mathrm{f}}\right\}=\left\{F_s\right\}$$

(14)

By solving the above equation, the structural vibration displacement and radiated sound field SPL can be obtained. The established acoustic–vibration model and sound field are depicted in Figure 16. The internal acoustic field mesh geometry aligns with the flow field model. For the external acoustic field, it is constructed based on the structural grid of the pipe wall. An AML (Automatic Matched Layer) is applied to the acoustic–vibration coupled surface, and the material properties of the external acoustic field and AML boundary are defined as air [29].

Four monitoring points are established in the acoustic field outside the elbow, positioned 150 mm away from the outer pipe wall, as illustrated in Figure 17a. Based on the simulation results in Section 2.3, this paper uses a weighted sound pressure levels (SPL). The total SPL at P1, P2, P3, and P4 were 26.5 dB(A), 31.4 dB(A), 45.2 dB(A), and 48.6 dB(A), respectively. In Figure 17b, all monitoring points exhibit maximum sound pressure at 50 Hz, which corresponds to the pipe’s main vibration frequency induced by water pulsation at 50 Hz. Consequently, the sound pressure peaks at 50 Hz and its multiples. The total sound pressure levels at points P3 and P4 are similar and significantly greater than those at points P1 and P2. This is attributed to the point P4 on the outside of the elbow, where the pipe experiences a more intense vibration, resulting in a larger sound pressure response compared to other locations. Therefore, point P4 will be used as the monitoring point for subsequent simulations.

3.2.2. Effect of Fluid Parameters on the Acoustic Field

Set the water pressure to 1 MPa, 2 MPa, 3 MPa, and 4 MPa, and other simulation conditions remain unchanged. The SPLs at the monitoring point P4 outside the elbow are 48.6 dB(A), 55.1 dB(A), 57.6 dB(A), and 60.3 dB(A), respectively. In Figure 18a, as the water pressure increases, the radiation noise of the elbow increases. At medium and low frequencies, the fluid pressure has a significant impact on pipe noise radiation. Therefore, the greater the fluid pressure, the greater the noise level.

Set the fluid pressure pulsation frequency to 0 Hz, 50 Hz, 100 Hz, 150 Hz, and 200 Hz, and keep the other simulation conditions unchanged. The SPLs at the acoustic field monitoring point P4 outside the elbow are 29.6 dB(A), 48.6 dB(A), 51.7 dB(A), 52.3 dB(A), and 55.6 dB(A), respectively. In Figure 18b, as the pressure pulsation frequency increases, the radiation noise of the elbow increases, and the position of the noise peak corresponds to the frequency of the respective fluid pulsation. Compared to the stable flow condition (the pulsating pressure frequency is 0 Hz), pipes with varying fluid pulsation frequencies exhibit significantly increased noise levels. Therefore, fluid pulsation is identified as the primary source of radiated noise in pipe systems.

At inlet flow rates of 10 m³/h, 15 m³/h, and 20 m³/h, the total SPLs at the acoustic field monitoring point P4 outside the elbow are recorded as 48.6 dB(A), 51.0 dB(A), and 52.4 dB(A), respectively. In Figure 18c, the sound pressure curves at monitoring point P4 exhibit similar trends with varying flow rates. The frequencies at the peak positions of the sound pressure curves under different flow rates are relatively close. The greater the fluid flow in the pipe, the higher the SPL at monitoring point P4.

3.2.3. Effect of Structural Parameters on the Acoustic Field

Based on the FSI calculation results in Section 3.1, utilizing the wall pulsating pressure data as the sound source condition in Virtual.Lab calculations, simulations were conducted for varying wall thicknesses from 2 mm to 6 mm. The resulting SPLs at the monitoring point P4 outside the elbow wall were observed to be 51.1 dB(A), 48.6 dB(A), and 47.4 dB(A), respectively.

In Figure 19a, it is evident that the wall thickness of the elbow significantly influences the noise radiation level in the external sound field, especially for medium- and high-frequency noise. With an increase in wall thickness from 2 mm to 6 mm, the sound pressure decreases by 3.7 dB, indicating that augmenting the wall thickness effectively reduces pipe noise radiation.

The elbow radius R/d is set to 1, 3, and 5, respectively. The SPLs at the monitoring point P4 outside the elbow wall are 56.1 dB(A), 48.6 dB(A), and 46.7 dB(A) in sequence. In Figure 19b, the elbow radius has a significant impact on the noise in the middle- and low-frequency range. The larger the elbow radius of the pipe, the greater the intensity of the radiated noise. This is because the larger the elbow radius of the elbow, the slower the flow state of the fluid changes at the elbow, thereby reducing the radiated noise. But the effect on high-frequency noise is not obvious.

Assuming that the elbow angles are 90°, 60°, and 30°, respectively, the SPLs at the monitoring point P4 outside the elbow are 48.6 dB(A), 45.0 dB(A), and 40.1 dB(A), respectively. In Figure 19c, reducing the elbow angle significantly lowers the pipe’s noise level. The smaller the elbow angle, the less unstable flow caused by the flow diversion of the fluid, which reduces the noise in other frequencies. However, the influence of noise near the fluid pulsation frequency is not obvious.

3.2.4. Noise Reduction Method for Elbows

The damping material is laid on the outer surface of the pipe to absorb the noise radiation in the pipe and achieve the effect of noise reduction. The geometric dimensions of the pipe are shown in Figure 20a. The elbow model after laying damping materials and the elbow model with elastic support installed are established, as shown in Figure 20b. The elastic support has a certain degree of elasticity and can effectively weaken the impact on the pipe. These supports restrain the pipe, elevate its natural frequency, and increase the overall system’s stiffness. The length between the elastic support installation position and the elbow outlet is L₃ = 300 m. The elastic support stiffness is set to 2 N/m.

Young’s modulus, damping coefficient, and material thickness are important parameters of damping materials that affect the noise reduction effect of the bends. The results in Figure 21a show that the noise radiation level of the bent pipe decreases significantly with the increase in Young’s modulus. At Young’s moduli of 1.5 × 10⁹, 2.5 × 10⁹, and 4.5 × 10⁹, the sound pressure levels at the monitoring points were 35.8 dB(A), 32.2 dB(A), and 28.1 dB(A), respectively, indicating that larger Young’s modulus can more effectively control low-frequency noise. For the effect of the damping coefficient, the results in Figure 21b show that the noise control effect improves significantly with the increase in damping coefficient. At damping coefficients of 0.2, 0.4, and 0.6, the sound pressure levels are 35.8 dB(A), 32.6 dB(A), and 31.5 dB(A), respectively, and higher damping coefficients are able to control middle- and high-frequency noise effectively. As for the thickness, the increase in material thickness helps to further reduce the noise level. The results in Figure 21c show that the thicknesses of the damping materials are 10 mm, 20 mm, and 30 mm, and the sound pressure levels at the monitoring points are 35.8 dB(A), 29.5 dB(A), and 23.6 dB(A), respectively, which indicates that the thicker damping materials can significantly reduce the low-frequency noise. Therefore, the selection of damping materials with larger Young’s modulus, damping coefficient, and thickness can help to achieve the best noise control effect of the elbow.

The SPL at monitoring point P4 outside the original elbow is 48.6 dB(A); after installing the elastic support and damping material, it shows at least a 26.3% reduction (35.8 dB(A)). Figure 21 shows that different noise reduction methods have a more significant effect on low-frequency noise reduction. Laying damping materials outside the pipe is low-cost and easy to operate, and the elastic support equipment is easy to install, disassemble, and replace. It is suitable for noise reduction work in the water pipe system.

4. Conclusions

The main objective of this paper is to use fluid–structure interactions and acoustic–vibration coupled simulation to study vibration characteristics and noise reduction for water elbows.

A two-way FSI simulation model was established, identifying pulsating water as the primary cause of pipe vibration. Maximum stress occurs at the elbow inlet, with maximum deformation occurring in the elbow. Natural frequencies for each order considering the FSI effect are significantly lower than that without FSIs. For the water’s pressure, as pulsation frequency and flow rate increase, the vibration acceleration and displacement of the elbow increase. Increasing the wall thickness of the elbow increases the overall stiffness and the natural frequencies and reduces the vibration acceleration amplitude. Increasing the elbow radius increases the natural frequencies, particularly the first three, with no significant effect on vibration acceleration.

An acoustic–vibration coupled model was built, and the results show that as the water’s pressure increases, the elbow radiation noise increases, particularly in the mid and low frequencies. As the inlet flow rate increases, the SPL at the monitoring point outside the elbow increases. Increasing the wall thickness reduces the external noise, particularly the low-frequency noise. Increasing the elbow radius and reducing the elbow angle can effectively reduce external radiation noise. Using elastic supports and damping materials can reduce elbows’ noise by at least 26.3%, and the SPL decreased with the increasing Young modulus, damping coefficient, and material thickness of damping materials. Therefore, laying damping materials outside the pipe is a low-cost method to noise reduction.