Comparative Analysis of Monitoring Methods for Vortex-induced Vibration of Multistage Pressure Reducing Valves

In this paper, a multistage pressure reducing valve is presented. The main frequency of vortex-induced vibration is evaluated by monitoring the lift coecient during vortex shedding and the pressure uctuation formed after vortex shedding in the ow eld. By comparative analysis of two different methods, the number of vortices is relatively small at small openings. Due to the limitations of the location and quantity of monitoring points, accurately locating the most active position where pressure uctuation occurs is dicult. Monitoring the lift coecient is more suitable to evaluate the main frequency of vortex-induced vibration. At medium and large openings, due to the increase in the number of vortices, the superposition effect of the pressure uctuation and the inuence of the ow channel shape is more obvious. Monitoring the pressure uctuation is more appropriate to evaluate the main frequency of vortex-induced vibration the valve. Therefore, a combination of the two methods can more accurately evaluate the vortex-induced vibration characteristics of the valve. When monitoring pressure uctuation, the position and number of monitoring points directly affect the evaluation accuracy. The pressure uctuations around the outlet and the multilayer sleeve are more active. It is more meaningful to monitor the pressure uctuation at these points. The main frequency of the pressure uctuation at these points better reects the vortex-induced vibration characteristics of the valve.


Introduction
The regulating valve is an indispensable component in modern control systems, as it can control ow rate and stabilize pressure [1][2][3]. This valve is widely used in petrochemical, water conservancy, energy and other industrial sectors. With the development of science and technology, to improve the reliability of control systems, higher requirements are proposed for the design rationality of throttle components in regulating valves. To realize the functions of throttling and reducing pressure, the pressure reducing components of a valve are designed as labyrinth disc, window and porous sleeve types. Some scholars [4][5] have studied the combined structure of labyrinth discs and multistage sleeves to achieve pressure and noise reduction functions. In actual control systems with high temperature, pressure and other factors, when a uid ows through the throttling components, vortices may form near the throttling components. This vortex shedding is periodic and has a main frequency. After vortex shedding, pressure uctuations form and are superimposed on each other which produces pulsating shocks. When the main frequency of vortex shedding or the pressure uctuation is equal to or close to the natural frequency of the valve, vortex-induced resonance will occur. This will cause large displacements and deformations in the valve trims, resulting in considerable vibration and noise that affects the safety of the control system. Some scholars have researched vortex-induced vibration phenomena in valves. Wang Hai-min [6][7] calculated the natural frequencies of a triple-eccentric butter y valve and combined the frequencies with the frequency calculation formula of the Karman vortex street to determine the conditions for no resonance.  studied the effects of hysteresis, surface roughness and other factors on the vortex-induced vibration of vertical pipes. Chizfahm A [10] studied the in uence of wind speed on the lift coe cient of turbines and concluded that vortex shedding and structural oscillation were synchronized. Youn [11] analyzed the ow of compressed air through ori ces and radial slits based on CFD and obtained pressure uctuation curves under different ow conditions. A multistage pressure reducing valve is presented in this paper. The pressure reducing components are shown to effectively prevent cavitation vibration in the valve caused by a sharp decrease in uid pressure. The simulation shows that many vortices appear around the multilayer sleeve. Two different methods are used to analyze the main frequency of the vortex-induced vibration of the valve. The rst method is to directly monitor the main frequency of vortex shedding by monitoring the lift coe cient, and the other method is to monitor the pressure uctuations at points in the valve and the uid-solid coupling surface.
However, there are differences in the monitoring results of these two methods. Therefore, by analyzing the monitoring mechanism, advantages, and disadvantages of the two monitoring methods, the vortexinduced vibration characteristics of the valve can be more accurately evaluated.

Structural Description Of The Multistage Pressure Reducing Valve
The structure of the multistage pressure reducing valve is shown in Fig. 1. The valve consists of a valve body, valve seat, multilayer sleeve, valve plug, pressure cage, upper bonnet, and valve rod. The valve rod is xed with the valve plug, and the valve rod is connected with an actuator. The valve rod is driven by the external actuator and is able to drive the valve plug to move up and down. In this manner, the ori ces on the inner sleeve of the multilayer sleeve are exposed to form effective ow areas.
When a uid ows through the multilayer sleeve, the uid velocity increases sharply as the ow area decreases quickly, and the hydrostatic pressure of the uid drops. If the uid pressure suddenly drops to the saturated vapor pressure, cavitation vibration will occur [12]. If the pressure difference of each stage pressure drop is greater than the blocked ow critical pressure difference, blocked ow will occur. The multilayer sleeve can gradually reduce the pressure of the uid in the valve to prevent cavitation vibration and block ow. The structure of the outer sleeve and the middle sleeve can be designed by assigning the same ori ces. The inner sleeve not only reduces pressure but also controls the ow rate at each valve opening.
When the uid in the valve is liquid water, the diameter of the valve is Φ250 mm, the valve plug diameter is Φ165 mm, and the inlet and outlet pressures are 10 and 3 MPa, respectively. The valve is designed according to the linear ow characteristics (C v =360). The multilayer sleeve structure is designed as shown in Fig. 2. 3 Flow Field Simulation Of The Multistage Pressure Reducing Valve 3.1 Establishing the simulation model. Before simulation, a 3D model of the valve is established, as shown in Fig. 1. The structure of the simulation model is established according to the above design parameters. The pressure at the inlet of the valve is 10 MPa, and the pressure at the outlet is 3 MPa. The uid model corresponding to each opening is generated by reverse modeling. The uid in the valve is liquid water with a temperature of 473.15 K. To simulate the ow eld of the valve, the uid model is divided into tetrahedral and hexahedron hybrid meshes using the ANSYS meshing tool, and local re nement is used at the ori ces. The grid independence is checked, and the reference value of grid independence is based on the ow rate and average ow velocity at the valve outlet at a 100% opening. The test data is shown in Table 1: According to Table 1, when the number of grids increases from 5177818 to 6361212, the ow velocity at the outlet increases by 0.038%, and the ow rate decreases by 0.01%. When the number of grid cells is greater than 5177818, the change in ow velocity and ow rate at the outlet can be ignored. Further re nement of the nite element mesh will not signi cantly affect the simulation results. At this time, the grid independence requirements are satis ed. Figure 3 shows the uid mesh model.

Selection of turbulence model for transient ow eld
The RNG k-ε turbulence model is used for transient ow eld simulation in this paper. The RNG k-ε model is based on Standard k-ε and is used to simulate the transient ow eld. In complex ow elds, numerical simulation is more accurate when the RNG k-ε model is used but di cult to converge [13][14]. Variable parameters in the model are correlated with the uid model, uid ow state and corresponding space coordinate system functions. The RNG k-ε model can effectively manage the turbulent ow in the piping system of the regulator valve.
3.3 Numerical simulation of the ow eld 3.3.1 Numerical simulation of the ow rate.
The maximum stroke of the valve plug is 100 mm. The inlet of the valve is on the left side. The ow rate at the outlet of the valve is monitored. After iterative calculations, the simulation shows that the circulation volume of the valve (Cv) is 358.96. The relative ow coe cient obtained by simulation is compared with the standard theoretical value, as shown in Fig. 4.
C, the relative ow coe cient, is dimensionless and is the ratio of the ow rate at a certain opening to the circulation volume (Cv). The horizontal coordinates are the corresponding openings. The maximum error is at a 10% opening, and the error value is only 6.69%. The minimum error is at a 100% opening, and the error value is only 0.29%. This shows that the valve has good linear ow characteristics.

Numerical simulation of the ow eld pressure and ow velocity.
The steady-state eld is taken as the initial value of the transient eld to accelerate convergence. The computational domain is the whole uid eld. The effect of gravity acceleration on uid ow is considered. The time step size is set to 0.00025 s, and the number of time steps is set to 8000. The ow pressure distributions of the ow eld at the 2nd second at openings of 10%, 50% and 100% are analyzed, as shown in Fig. 5.
According to Fig. 5, the maximum and minimum pressures are 10 and 3 MPa, respectively, and are distributed at the inlet and outlet of the valve. At a 10% opening, the lowest ori ces of the inner sleeve are in the ow state. The uid pressure drops signi cantly when the uid ows through the ori ces. At this time, the ow rate is relatively low, and the uid pressure does not change signi cantly as the uid ows through the middle sleeve and outer sleeve. At openings of 50% and 100%, when the uid passes through the multilayer sleeve, the static pressure of the uid drops signi cantly. The total pressure difference of the valve is separated into several small pressure differences, the pressure difference of each stage is greater than the blocked ow pressure difference, and the pressure is greater than the saturated vapor pressure after pressure reduction. This shows that the multilayer sleeve can effectively prevent cavitation and ash. The ori ces of the inner sleeve not only control the ow rate well but also have a pressure reducing function.
The ow velocity distribution and the velocity distribution of the vortex core in the internal ow eld at openings of 10%, 50% and 100% are shown in Figs. 6 and 7. From Fig. 6, the ow velocity increases signi cantly as the ow area at the multilayer sleeve decreases. The maximum velocity at three openings is distributed at the ori ces of the multilayer sleeve. Figure 6(a) shows the ow velocity distribution in the ow eld at a 10% opening. At this time, the maximum velocity is distributed at 151.7 m/s. The uid in most areas is stationary in the ow eld. At openings of 50% and 100%, as the throttle area increases, the maximum uid velocity decreases. However, the uid in most areas of the ow eld is in a ow state and a vortex is easily formed.
In Fig. 7, most vortex cores are distributed around the ori ces of the multilayer sleeve. As the opening increases, the number of vortices at the ori ces gradually increases, and the vortex core velocity gradually decreases. Some large vortices gradually become small vortices, and some vortices begin to shed, which can easily cause vortex-induced vibration in the valve. In addition, the pressure uctuations become more active. Therefore, the main frequency of the lift coe cient of the vortex shedding process and the main frequency of the pressure uctuations can be used to evaluate the main frequency of the vortex-induced vibration.

Frequency spectrum analysis of the vortex-induced vibration based on the lift coe cient
Vortex shedding is a main reason for lateral vortex-induced vibration. Transverse ow-induced vibration is the main form of vortex-induced vibration. When vortex shedding occurs, a drag force forms in the downstream direction, and a lift force forms in the cross ow direction. The average lift force is represented by the lift coe cient, which is dimensionless. The frequency of the peak value of the lift coe cient power spectral density can be considered a parameter for evaluating the main frequency of vortex shedding. The frequency of vortex shedding can be obtained by applying a fast Fourier transform (FFT) to the time-domain information of the lift coe cient obtained through simulation of the transient ow eld [15][16]. The lift coe cient expression is: where F L is the vortex-induced lift force; ρ is the density of the uid, kg/m 3 ; U is the ow velocity in the vortex core, m/s; and A is the cross-sectional area of the vortex.
In the transient ow eld, the lift coe cients of the vortices at different openings are monitored, and the corresponding time-domain information is obtained in 2 seconds, as shown in Fig. 8. A fast Fourier transform (FFT) is applied to the time-domain information of the lift coe cient, and the frequency-domain curve of the corresponding lift coe cient is obtained as shown in Fig. 9.
In Fig. 9, the ordinate is the power spectral density of the lift coe cient, and the frequency of the peak value for the lift coe cient power spectral density is the main frequency of vortex shedding. At a 10% opening, the peak value of the lift coe cient power spectral density is signi cantly higher than at other  Figure 10a shows the uid-solid coupling surface in the valve. In Fig. 10b, monitoring points P1~P9 are distributed on a symmetrical plane of the uid channel. Monitoring points S1~S2 and S3~S4 are located near the inner wall of the valve body at the inlet and outlet, respectively.

Time domain and frequency domain analysis of the pressure uctuation.
To obtain the main frequency of the pressure uctuation in the ow eld, the monitoring points are divided into two categories: monitoring points (P1~P6, S1~S2) in front of the multilayer sleeve and monitoring points (P7~P9, S3~S4) behind the multilayer sleeve. The time domain information of the pressure uctuations is monitored, as shown in Figs. 11, 12 and 13. According to Figs. 14, 15 and 16, the ordinate is the pressure uctuation power spectrum density. The abscissa is the pressure uctuation frequency. Therefore, the frequency of the peak value of the pressure uctuation power spectrum density is the main frequency of vortex-induced vibration. At a 10% opening, the power spectral densities of the pressure uctuations do not have an obvious main frequency at monitoring points P1~P4 and S1~S2. The main frequencies of the pressure uctuations at monitoring points P5~P9 and S3~S4 uctuate within 20 Hz. At a 50% opening, the power spectral densities of pressure uctuations do not have an obvious main frequency at monitoring points P1~P3 and S1~S2.
The main frequencies of the pressure uctuations at monitoring points P4~P6 are within 30 Hz. The main frequencies of the pressure uctuations at monitoring points P7~P9 and S3~S4 are within 160 Hz. At a 100% opening, the power spectral densities of the pressure uctuations do not have an obvious main frequency at monitoring points P1~P3 and S1~S2. The main frequencies of the pressure uctuations at monitoring points P4~P5 are within 60 Hz. The main frequencies of the pressure uctuations at monitoring points P6~P7 are within 100 Hz. At monitoring points P8~P9 and S3, the main frequencies of the pressure uctuations are within 170 Hz. The main frequencies of the pressure uctuations at monitoring point S4 are within 180 Hz. On the uid-solid coupling surface, the main frequency of pressure uctuation is within 20 Hz for different openings. Combined with the three opening degrees, the main frequency range of the vortex-induced vibration is calculated to be 0~180 Hz. Figure 16 Frequency spectrums of pressure uctuation at 100% opening. a monitoring points in front of multilayer sleeve, b monitoring points behind multilayer sleeve and c uid solid coupling surface The pressure uctuations at the monitoring points in front of the multilayer sleeve do not have obvious main frequencies. At this time, the ow eld is relatively stable and approximate to laminar ow. A large number of vortices and vortex shedding form near and behind the multilayer sleeve, and the power spectrum density of the pressure uctuation uctuates signi cantly. Because the pressure uctuation on the uid-solid coupling surface is the average value of the main frequency and the power spectral density of the pressure uctuation uctuates slightly in front of the multilayer sleeve, the monitored information cannot be used to accurately evaluate the main frequency of vortex-induced vibration.
Behind the multilayer sleeve, the pressure uctuations are superimposed on each other, and the highest main frequency range of the pressure pulsations occurs. Due to the in uence of channel shape, the streamlines usually turn back at the inner wall of the valve body and overlap at monitoring point S4. Therefore, the widest main frequency range occurs, and the power spectrum density amplitude of the pressure uctuation is largest at monitoring point S4. This indicates that the opening and ow channel shape have an effect on the main frequency of vortex-induced vibration.
6 Thermal-uid-solid Coupling Modal Analysis Under the ANSYS Workbench platform, the uid module, temperature eld module, static module and modal analysis module are combined to calculate the prestress mode (wet mode) of the valve [19][20]. In the uid module, the pressure and temperature distribution can be obtained by calculating the steadystate ow eld of the valve. In the temperature module, the temperature distribution of the valve is obtained by transmitting the uid temperature to the valve through the coupling surface. In the static module, the uid pressure is transmitted to the valve through the coupling surface, and the temperature of the valve in the temperature eld is imported to the static module. Then, the calculations from the static module are imported to the modal module to complete the modal calculation (wet mode).
In the ANSYS static module, the pressure information and temperature information are loaded into the valve. A xed constraint is applied to the inlet of the valve, and a displacement constraint is applied to the outlet. The in uence of gravitational acceleration is considered. High-order modal frequencies can be considered combinations of several low-order modal frequencies. Therefore, a 1st-to 6th-order modal simulation analysis at 3 openings is carried out. Table 2 shows the material parameters of the main parts of the valve. Table 3 shows the 1st-to 6th-order natural frequencies of the valve under the condition of thermal-uid-solid coupling. In Table 3, the modal frequencies of each order are relatively smaller at a 100% opening. At each opening, the high-order modal frequencies of the valve are greater. When the valve is at a 100% opening, the lowest modal frequency of the valve occurs, which is 255.96 Hz.
By comparing the main frequency of vortex shedding with the modal frequency of the valve, there is not an equal or close frequency value at 3 openings, and vortex-induced resonance cannot occur in the valve.

Comparative Analysis Of Monitoring Methods For Vortex-induced Vibration
By analyzing the lift coe cient power spectral density, the main frequency of vortex-induced vibration in the valve is obtained, which is within 140 Hz. By analyzing the pressure uctuation power spectral density, the main frequency of vortex-induced vibration in the valve is evaluated, which is within 180 Hz.
The main frequency ranges of vortex-induced vibration at openings of 10%, 50%, and 100% are evaluated using two methods, as shown in Table 4.  The mechanism of monitoring the lift coe cient is to monitor the generation, development and shedding process of the vortex. However, factors such as the superposition effect of the pressure uctuations after vortex shedding and the ow channel shape are not considered when monitoring the lift coe cient spectrum. At this time, the number of vortices obviously increases, and the effect of the superposition of the pressure uctuations becomes increasingly signi cant. In addition, the increased number of vortices increases the likelihood of capturing active pressure uctuations. Therefore, the main frequency of vortex-induced vibration obtained by monitoring the pressure uctuation is more accurate at medium and large openings.

Conclusion
In this paper, vortex-induced vibration characteristics of multistage pressure reducing valves are analyzed using two different methods. By analyzing the lift coe cient power spectral density, the main frequency of vortex-induced vibration in the valve is obtained, which is within 140 Hz. By analyzing the pressure uctuation power spectral density, the main frequency of vortex-induced vibration in the valve is evaluated, which is within 180 Hz. The lowest modal frequency of the valve is 255.96 Hz. The valve has good vortex-induced vibration characteristics, and vortex-induced resonance does not occur. Monitoring the lift coe cient monitors all vortex shedding processes, but does not consider the superposition of pressure uctuations after vortex shedding and the in uence of ow channel shape. At a small opening, the number of vortices in the ow eld is small. The velocity of the ow eld varies dramatically. The superposition effect of the pressure uctuation and the in uence of the ow channel shape is not obvious. So monitoring the lift coe cient is more suitable for evaluation of the vortex-induced vibration characteristics of the valve at a small opening. Monitoring the pressure uctuation is seriously affected by the location and number of monitoring points. The selected location and quantity of set monitoring points are very important, and determine the evaluation accuracy. The main frequency of vortex-induced vibration is easier to detect when monitoring points are set near and behind the multilayer sleeve. The method of monitoring the pressure uctuation on the uid-solid coupling surface for evaluating the main frequency of vortex-induced vibration of the valve is unreliable. At medium and large openings, the number of vortices in the ow eld is larger. the in uence of the superposition effect of pressure uctuations and ow channel shape is more obvious. It is more appropriate to monitor the pressure uctuation spectrum at medium and large openings. So, a combination of these two methods can more accurately evaluate the vortex-induced vibration characteristics of the valve. Figure 1 The structure of multistage pressure reducing valve Figure 2 The multilayer sleeve. a inner sleeve, b middle sleeve and c outer sleeve    Pressure distribution of the ow eld at different openings. a 10% opening, b 50% opening and c 100% opening Figure 5 Velocity distribution of the ow eld at different openings. a 10% opening, b 50% opening and c 100% opening Figure 6 Velocity distribution of the vortex core at different openings. a 10% opening, b 50% opening and c 100% opening     Fluid-solid coupling surface and monitoring points. a uid-solid coupling surface and b monitoring points Figure 10 Time domain curves of pressure uctuation at 10% opening. a monitoring points in front of multilayer sleeve, b monitoring points behind multilayer sleeve and c uid solid coupling surface Time domain curves of pressure uctuation at 100% opening. a monitoring points in front of multilayer sleeve, b monitoring points behind multilayer sleeve and c uid solid coupling surface Figure 13 Frequency spectrums of pressure uctuation at 10% opening. a monitoring points in front of multilayer sleeve, b monitoring points behind multilayer sleeve and c uid solid coupling surface Figure 14 Frequency spectrums of pressure uctuation at 50% opening. a monitoring points in front of multilayer sleeve, b monitoring points behind multilayer sleeve and c uid solid coupling surface Figure 15 Frequency spectrums of pressure uctuation at 100% opening. a monitoring points in front of multilayer sleeve, b monitoring points behind multilayer sleeve and c uid solid coupling surface