Investigation on Gas-Liquid Two-Phase Flow-Induced Vibrations of a Horizontal Elastic Pipe

In recent years, flow-induced vibration (FIV) phenomenon caused by two-phase flow has gained significant attention in various engineering fields, such as the petroleum transportation and the nuclear power station [1–3]. Compared with single-phase flows, two-phase flows are much more unstable in terms of various hydrodynamic parameters, such as the local fluctuations of density, pressure, and velocity of the fluid. The hydrodynamic forces induced by the fluctuation characteristics of gas–liquid two-phase flows excite the wall of the pipe and lead to significant structural vibration responses [4]. Resonances may occur when the predominant fluctuation frequency of the flow coincides with the natural frequency of the pipe, leading to serious consequences such as structural fatigue and failure [5]. Understanding the vibration mechanisms of elastic pipes caused by two-phase flow is of great importance for designing a piping system that can operate safely and avoid structural damage.

In the past few decades, considerable works have been dedicated to investigating flow-induced vibrations of elastic pipes subjected to external flows, while the research efforts focused on the vibration behaivors of elastic pipes containing two-phase internal flow is rather limited [6]. Hara [7] analyzed the two-phase flow-induced vibration of a horizontal pipe. It was found that the structural vibrations of the pipe were excited by parametric excitation and resonance caused by the external force. An and Su [8] analyzed the dynamic behaviors of pipes conveying gas–liquid two-phase flow. The results indicated that the vibration amplitudes of the pipe increased as the void fraction increased, and the dynamic system lost its stability when the liquid flowrate reached a critical value. Al-Hashimy et al. [9] carried out experiments to study the vibration responses of a horizontal pipe excited by internal slug flow at a fixed air superficial velocity. It was found that the displacement increased gradually with an increase in the water superficial velocity. Two-phase flow-induced vibrations of a horizontal pipe were investigated experimentally by Enrique et al. [10]. They found that the peak frequency of vibration responses depended strongly on the void fraction, and the hydrodynamic mass parameter was proportional to mixture density. Wang et al. [11] investigated the dynamic behaviors of a pipeline-riser system. They found that severe slug flow could induce periodic vibrations of the pipeline. Few experimental and numerical studies were concerned with the influence of two-phase flow parameters, such as mixture velocity, void fraction and flow-pattern on the structural responses of piping systems [12,13].

Most of the existing works are devoted to investigating the dominant fluctuating forces and the effect of specific two-phase flow properties such as the volume fraction of each phase, the volume flowrate, and the flow pattern [14]. Tay and Thorpe [15] found that the viscosity and surface tension of the liquid phase had minimal effects on the fluctuating forces, especially in the slug flow regime. Hossain [16] investigated the behavior of multiphase flow-induced vibration of a bend pipe. They found that the predominant frequency of the excitation force decreased with an increase in the superficial gas velocity. Meanwhile, both the predominant frequency and the root-mean-square value of the force fluctuations increased with an increase in the superficial liquid velocity. Riverin and Pettigrew [17] found that the magnitude of the hydrodynamic force was correlated with local fluctuations of the void fraction within an elbow or a tee pipe. Riverin et al. [2] performed experiments to investigate the mechanism of hydrodynamic force excited by internal two-phase flow. The fluctuating forces consisted of narrow bands and periodic components, of which predominant frequencies increased proportionally with the flowrate. Liu et al. [18] performed experiments and studied the vibration responses of an elbow pipe subjected to internal two-phase flow excitations. It was found that the exciting force was related to the momentum flux of two-phase flow, and the predominant frequency of the fluctuating force peaked at slug flows for a fixed liquid flowrate.

Flow patterns and superficial velocities have been reported to be of great importance in characterizing vibration responses of pipes. Liu and Wang [11] explored the natural frequency of a cantilevered horizontal pipe conveying gas–liquid slug flow. It was found that the intermittent feature of slug flows had a crucial impact on the natural frequency of the piping system, especially when both the superficial gas velocity and the pipe length were large enough. Wang et al. [19] performed experiments to investigate the dynamic responses of horizontal pipes caused by gas–liquid slug flow. The results showed that the maximum load acting on the pipe depended on the length of the liquid slug, and the recurrence frequency of the liquid slug determined the rate of change of the structural stiffness. Giraudeau et al. [20] studied the characteristics of hydrodynamic force with a U-bend tube carrying vertical upward air–water flow. It was found that the slug flow produced the highest hydrodynamic forces due to the large momentum variation between the Taylor bubbles and liquid slugs. The bubbly flow induced a lower and wider spectrum, which consisted of multiple peak frequencies. Miwa et al. [21] investigated the pulsating force induced by stratified wavy flow of a horizontal 90 deg pipe bend. The results showed that the momentum and pressure fluctuations as well as the collisional effects contributed mostly to the force fluctuations of the pipe.

In view of the above, it is clear that available literature on horizontal two-phase flow is much less than that of vertical two-phase flow. Existing studies have predominantly focused on the two-phase flow-induced fluctuating forces and structural vibration responses of the vertical pipes with smaller inner diameter. Therefore, the main objective of this work is to investigate the dynamic responses of a horizontal pipe with relatively large inner diameter, of which frequencies of the pressure fluctuations are lower than the natural frequencies of the structure. The flow characteristics under different flow regimes are evaluated in detail, and the effects of superficial velocities and flow patterns on vibration responses are discussed deeply. In this work, the combination of experiments and numerical simulations ensures a better understanding of the flow-induced vibration mechanisms, and helps develop a fluid–structure interaction model for experiments and simulations.

As shown in Fig. 1, experiments are conducted on an air–water loop system to investigate the two-phase flow-induced vibration responses of a horizontal pipe. Figure 1(a) represents the schematic diagram of the experiment system. Filtered and de-ionized water is used as liquid phase in the fluid system, and the density and the viscosity of the liquid are 998 kg/m³ and 1.003 × 10⁻³ Pa·s, respectively. The liquid phase is circulated from a 12 m³ stainless steel tank using a centrifugal pump with a flowrate ranging from 0 to 120 m³/h, which is controlled by an electric regulating valve and a number of ball valves. The gas phase is delivered by an air compressor with flowrate ranging from 0 to 200 m³/h, which is adjusted by changing the opening of the ball valves and the electric regulating valve. Two electromagnetic flowmeters with 0.1% accuracy are used for measuring the liquid flowrate with maximum capacities of 90 m³/h and 180 m³/h. Two Coriolis mass flowmeters with 0.2% accuracy are mounted for air flowrate measurement with maximum capacities of 100 kg/h and 500 kg/h, respectively. Two pressure transmitters with 0.5% accuracy are used to acquire the air density according to the state equation of the ideal gas. After the fluid passes through the SV-static mixer, a straight pipe with length of 4 m is adopted to ensure the full-developed flow patterns. During the experiment, identification of flow pattern for each condition is based on photos captured by a high-speed camera with a capture rate of 512 frames per second, and the viewing section is made of polymethyl methacrylate with inner diameter of 51 mm. To isolate the vibration transmission from adjacent pipes to the testing section, rubber tubes are placed closely on the test section.

As shown in Fig. 1(b), the test section is made of 304 stainless steel pipes (2 m long) with 51 mm inner diameter and 3 mm wall thickness. The test section is rigidly mounted on the ground by the steel fixed supports on both ends. After passing through the test section, the fluid enters a cyclone separator through another rubber tube. The air is vented to the atmosphere, while the water flows back to the storage tank and continues to be pumped. All outlet parts of the backwater pipes are immersed in water, which allows the water tank to be considered as a pressurizer with atmosphere pressure. In this paper, we mainly focus on the variation of the pressure fluctuation amplitude rather than static pressure. The dynamic pressure transducer is adopted to measure the pressure fluctuations of the two-phase flow, of which model type is PCB 113B28. As shown in Fig. 1(a), there is a small hole at the top of the pipe, and the data acquisition chip of the pressure transducer is aligned with the inner wall of the pipe, which can directly measure the pressure fluctuations of the two-phase flow. Figure 1(b) indicates that a total of three pressure transducers are used to measure the pressure fluctuations along the horizontal pipe. The pressure transducers are labeled from P1 to P3, which are calibrated with 0.1% accuracy and have a range of ±314 kPa. Three tri-axial accelerometers are used to monitor the vibration responses of the test section, of which labels are A1, A2, and A3, respectively. The direction of the acceleration transducer is shown in Fig. 2, and the Z-axis aligned in the negative direction of gravity. A total of 48 flow conditions are investigated, ranging from slug flows to bubbly flows. The gas superficial velocity (j_g) ranges from 2.44 m/s to 9.25 m/s, and the liquid superficial velocity (j_l) ranges from 1.23 m/s to 6.70 m/s. The superficial velocities of the gas and liquid phases are calculated, respectively, as: j_g = Q_g/A and j_l = Q_l/A, where Q_g and Q_l are the gas and liquid volume flow rates, respectively, and A is the cross-sectional area of the pipe. In this work, a Donghua high-speed data acquisition instrument with 32 analog signal input channels is employed to acquire the data from the metering system and test system. It should be noted that the data are not periodic and the acquisition time is selected to be sufficiently large to ensure independent results. In addition, there are errors in the data obtained through direct measurement due to the uncertainty of the measuring instruments. Thus, the uncertainty of the measured data is evaluated by using the method proposed by Moffat [22]. The uncertainty values of the measured parameters are presented in Table 1. Meanwhile, repeated experiments are carried out three times to ensure the reliability of the experimental system. Figure 3(a) depicts the standard deviation (STD) values of the pressure pulsations at point P2 under different superficial gas velocities, while the superficial liquid velocity j_l = 1.8 m/s. and the corresponding acceleration responses measured at point A2 are shown in Fig. 3(c). The maximum relative error observed in Figs. 3(a) and 3(c) for all flow conditions is about 5.1%, and the results of data verifications indicate a good agreement among the repeated experiments. Figures 3(b) and 3(d) depict the frequency responses of the pressure fluctuation and acceleration of the pipe with j_g = 2.5 m/s and j_l = 2.2 m/s, respectively. The peak frequency and amplitude of the results from three experiments are very similar.

In order to validate the operation of the experimental system, an initial analysis is carried out for the slug flow-induced vibration responses of a horizontal pipe based on the cosimulation of fluent and ansys. The finite volume method is employed for the spatial discretization of the fluid domain, while the finite element method is employed to establish the structural model of the pipe. Partitioned strong fluid–structure interaction algorithm is employed to solve the equations of the fluid and the structure as a coupled system, and a number of iterations are adopted within each time-step to ensure converged solutions.

where t is the time, u and ρ are the mixture velocity vector and mixture phase density, respectively, P is the pressure in the flow field, g is the gravity vector, μ and μ_t are the physical viscosity and turbulent viscosity, respectively. The surface tension force F is the contribution to the body force related to surface tension.

where subscript q represents each phase component, α_q and ρ_q are the volume fraction and the density of the fluid q present in the control volume, respectively.

where subscript l and g represents the liquid phase and the gas phase, respectively.

Turbulence is generated in both water and air phases because of the high-velocity gradient at the phase interface. The standard k − ε turbulence model is chosen to deal with the turbulence in the fluid because it presents a good balance between accuracy and processing time [25]. The standard k − ε model is defined by the following equations:

The constants used for the solution of the model are C_μ = 0.09, C₁ = 1.44, C₂ = 1.92, σ_k = 1, and σ_ε = 1.3. The turbulent viscosity μ_t is written as μ_t = C_μk²/ε and G_k represents the production of turbulent kinetic energy.

The arbitrary Lagrange–Eulerian dynamic mesh technique is adopted to accommodate the deformation of the fluid domain caused by the vibration of the pipe. The pressure-based solver is employed for the simulation, and the pressure-implicit with splitting of operators algorithm [26] is used for pressure–velocity coupling to improve the efficiency of the iteration calculation. A body force weighted method is adopted for the pressure spatial discretization, while the gradient is calculated based on the Green-Gauss node-based method. Geometric reconstruction method [27] is employed for the volume fraction term, which provides the most accurate calculation near the interface. The second-order upwind scheme is utilized for the discretization of the momentum, the turbulent kinetic energy, and the specific dissipation rate. Because of the instability of the two-phase flow, a transient model with a time-step of 10⁻⁵ s is selected for the overall simulation process. A residual value of 10⁻⁵ is employed as the convergence criteria for the fluid variables.

where $U¨$⁠, $U˙$⁠, and U the displacement, velocity, and acceleration vectors of the pipe, respectively. M, C, and K are the mass matrix, damping matrix, and stiffness matrix, respectively. F_f and F_s are the external force vectors due to the fluid and structure, respectively. The Newmark method [28] is utilized as an implicit direct time integration method to solve Eq. (10) in time.

where subscript s and f represent the structure and the fluid component, respectively, n is the normal vector on the fluid-structure interface.

At each time-step of the simulation, the coupling between the fluid and the structure is carried out according to the following procedure. First, the structural solver obtains the convergence solutions and provides the value of the structural displacement. Second, the smart bucket mapping method is adopted to interpolate the structural displacement values at the coupling interface to the fluid mesh. Third, the fluid mesh is updated and deformed based on the obtained structural displacements, and the converged solutions of the fluid calculations provides the stress values, which are subsequently used as nodal forces acting on the structural interaction boundary. Finally, these external forces are interpolated and transferred to the structural mesh according to the General Grid Interface algorithm. The analyses are looped through repeatedly until overall equilibrium is reached between the structure and fluid solutions [27].

The geometry of the horizontal pipe and computational boundary conditions are displayed in Fig. 2. The internal diameter and wall thickness of the pipe are 51 mm and 3 mm, respectively. The pipe is made of stainless steel with the material parameters given as: density ρ = 7930 kg/m³, Young's elastic modulus E = 196 GPa and Poisson's ratio υ = 0.3. The fluid domain consists of three parts, i.e., full-developed section, test section, and outlet section, whose length are 4 m, 2 m, and 0.8 m, respectively. A total of three pressure monitoring points is located on the top of the horizontal pipe, whose labels are P1, P2, and P3, respectively. Three acceleration monitoring points are located on the side of the pipe, of which labels are A1, A2, and A3. Air and water are considered in the simulation of the transient and turbulent two-phase flow. The density and viscosity of fluids in the pipe are: ρ_l = 998 kg/m³, μ_l = 1.003 e⁻³ kg/m·s for water, and ρ_g = 1.29 kg/m³, μ_g = 1.8 e⁻⁵ kg/m·s for the air. The surface tension coefficient is taken as σ = 0.072 N/m. At the wall, the nonslip condition is adopted, and standard wall-function is employed in the k−ε model to avoid refining the mesh excessively. The effect of the gravity is also taken into account in this model, and the gravity is assumed along the negative direction of the Y axis. Clamped boundary conditions are applied at both ends of the test section.

Mesh dependency test has been carried out using three different meshes for the slug flow with j_l = 3.0 m/s and j_g = 1.34 m/s. As shown in Fig. 4, the block-structured meshes are applied to discretize the fluid domain and solid domain. The three meshes for fluid domain, with 456,960 elements, 783,360 elements, and 1033,600 elements, are labeled as coarse mesh, medium mesh, and fine mesh, respectively. Eight-node hexahedral solid elements are used for the discretization of the structure domain. The three meshes, with 97,920 elements, 130,560 elements, and 163,200 elements, are labeled as coarse mesh, medium mesh and fine mesh, respectively. The inlet of the fluid domain is divided into two parts to promote the development of flow patterns, with the top region used for air flow and the bottom region for water flow. The inlet velocity of each phase is specified at the inlet and the atmospheric pressure is imposed at the outlet. The inlet velocities of gas and liquid phases are calculated respectively as: v_g = j_gA/A_g and v_l = j_lA/A_l, where v_g and v_l are the gas and liquid inlet velocities and A_g and A_l are the gas and liquid inlet areas [30].

The pressure fluctuations of two-phase flow measured at point P2 are shown in Fig. 5. It is observed from Fig. 5(a) that as the mesh resolutions is refined, the computed results of the pressure gradually converge. The pressure fluctuations of the two-phase flow exhibit stochastic behaviors, and therefore, the time-histories of the computed pressure fluctuations show significant discrepancy for the three different meshes. The STD values of pressure fluctuations at measuring point P2 are calculated to be 1.087 kPa, 0.741 kPa, and 0.725 kPa for three meshes, while the experimental value is 0.824 kPa. The peak frequencies in Fig. 5(b) corresponding to the three different meshes are 2.56 Hz, 5.48 Hz, and 5.49 Hz, respectively. The dominated frequency obtained by experiment is 6.98 Hz. The vibration amplitudes measured at middle of the pipe along the Y direction are shown in Fig. 5(c). The STD values of the pipe acceleration responses are calculated to be 0.133 m/s², 0.098 m/s², and 0.089 m/s² for three different mesh sizes. The experimental value is 0.103 m/s². As shown in Fig. 5(d), the dominant frequencies of the structural responses computed by the three meshes are 68.98 Hz, 72.61 Hz, and 71.06 Hz, respectively. The dominant frequency obtained by experiment is 70.32 Hz. The corresponding relative errors between the numerical solutions and the experimental result are 1.9%, 3.15%, and 1.04%. It is observed from the figure that the structural response obtained by the medium mesh agrees well with the experimental data.

A number of time snapshots of contour plots of air volume fraction are shown in Fig. 6. It is observed from Fig. 6 that the slug flows are characterized by large gas bubbles surrounding thin liquid films and circulating liquid structures. As expected in the slug flows, small gas bubbles are also entrained in the liquid structures. Figure 6(a) shows that the length of the slug bubble is between 0.54 m and 0.6 m, of which velocity ranges from 5.2 m/s to 5.29 m/s. The time duration for the bubbles passing through the measurement point P2 is about 0.1 s, which is found to be consistent with the time duration of the fluid pressure drop. The distance between two adjacent bubbles ranges from 0.79 m to 0.81 m, and the time interval through the same measuring point P2 is about 0.158 s. This time duration is in accordance with the peak frequency of the pressure fluctuation. The computed flow pattern is in good agreement with the experimental result. The results show that large interfaces between the two phases can be well captured by the VOF model, but small bubbles entrained within the liquid structures are not tracked in the simulation.

Modal tests are performed prior to the FIV experiments in order to determine the natural frequencies of the horizontal pipe filled with air or water. The multipoint excitation method is adopted in the modal tests, and the impact hammer is moved to different positions of the pipe. The vibration response signals of the pipe are collected by three tri-axial accelerometers. The frequency responses of the measuring point A1 along the Y direction are shown in Fig. 7. As shown in Fig. 7(a), the first two natural frequencies of the pipe with air are 81.3 Hz and 223.5 Hz, respectively. For the pipe filling with water, the first two natural frequencies of the pipe are 65.0 Hz and 171.8 Hz. Finite element models for computing the natural frequencies of the pipe containing fluid are established. For computing the free vibration of the pipe, the fluid in the pipe is assumed to be compressible and inviscid acoustic fluid, which is discretized by 366,912 eight-node fluid elements. The elastic pipe is discretized by 85,313 hexahedral solid elements. The natural frequencies of the pipe are computed by the block Unsymmetrical method, and the first two modes of the pipe filled with water are shown in Figs. 7(c) and 7(d). The computed results of the natural frequencies of the first two modes of the empty pipe are 79.4 Hz and 214.5 Hz. The relative discrepancies between the numerical and experimental results for the first and second natural frequency of the pipe are 2.3% and 4.0%, respectively. For the pipe filled with water, the first two mode frequencies of the pipe are 62.3 Hz and 169.8 Hz, and the corresponding relative errors are 4.1% and 1.2% with respect to the experimental results.

Two-phase flow visualization technology is carried out in this study to understand the physical behaviors of different flow patterns. Sample images for bubbly flow are presented in Figs. 8(a) and 8(b), and different flow conditions are investigated. The superficial liquid velocity is kept as j_l = 5.78 m/s, while the superficial gas velocity varies from 2.19 m/s to 4.47 m/s. For the bubbly flow investigated, a large number of small bubbles are dispersed in the continuous liquid phase, and the individual bubbles are difficult to distinguish. As shown in Figs. 8(c) and 8(d), the slug flows with superficial gas velocity j_g = 2.19 m/s are investigated while the superficial liquid velocity varies from 0.96 m/s to 1.34 m/s. In all conditions, the length of the air bubble extends beyond the observation window. Therefore, three images are combined in the figure to show the complete shape of the gas slug. It is observed from Fig. 8(c) that the tail region of the gas slug occupies more flow area and increases in length as the gas flowrate is increased. Moreover, few small bubbles are observed in the region of the liquid between consecutive gas slugs, which have higher velocities compared with the larger gas slugs.

The time domain signals and their FFT of the pressure fluctuations for slug flows are shown in Fig. 9. The superficial gas velocity is kept as j_g = 5.4 m/s, while the superficial liquid velocity varies from 1.47 m/s to 2.06 m/s. As shown in Fig. 9(a), the local pressure fluctuations for slug flow exhibit certain periodicities due to the intermittent flow of the liquid slugs and the bubbles. The maximum fluid pressure appears when the liquid slug passes through the measuring point. On the contrary, the minimum fluid pressure appears in the case of the measuring point occupied by air bubbles. It is observed from Fig. 9(c) that the highest amplitude of the fluid pressure between two flow conditions peaks at 0.47 Hz, which is found to be consistent with the time interval between two adjacent liquid slugs passing through the measuring point. As shown in Fig. 9(b), as the liquid flowrate is increased, the adjacent liquid slugs tend to interact with each other and the slug bubbles are destructed. The result in Fig. 9(d) indicates that the dominant frequency of the gas slugs distributes between 0 Hz and 5 Hz. In this case, the peak frequency of the fluid pressure increases with an increase in the liquid superficial velocity, and the frequency band of the fluid pressure is further broadened due to the liquid slug contains a large number of bubbles.

The time domain signals and their FFT of the pressure fluctuations for bubbly flows are shown in Fig. 10. The superficial gas velocity is kept as j_g = 5.4 m/s, while the superficial liquid velocity varies from 5.05 m/s to 6.28 m/s. As shown in Figs. 10(a) and 10(b), the fluid pressure of the bubbly flow can be characterized by a relatively broadband signal with small magnitude fluctuations. This is caused by the random motion of large quantity of bubbles, which are much smaller than the diameter of the pipe. It is observed from Fig. 10(b) that the maximum peak amplitude is one order of magnitude smaller compared to the slug flow and the frequency band is further broadened. A large peak near zero frequency exists in the frequency responses of the pressure fluctuations of the fluid. This may be due to the fluctuations of the inlet flowrate and the fluid pressure of the piping system. Figures 10(c) and 10(d) show that the average momentum of the two-phase flow increases with an increase in liquid flowrate, and the intensity of pressure fluctuations of the two-phase flow is much stronger.

The STD and the peak frequency of fluid pressure fluctuations for 27 two-phase flow conditions are shown in Figs. 11 and 12. These experiments are done for three cases of gas superficial velocity and various cases of liquid superficial velocity. The fluid pressure fluctuations are measured at the measuring point P2. As shown in Fig. 11, the maximum magnitude of the STD of the pressure fluctuation appears in slug flows, and the value of the STD is almost zero for bubbly flows. When superficial velocity of the liquid phase is small, the kinetic energy of the fluid increases with an increase in the liquid superficial velocity for a fixed gas flowrate, and the intensity of the pressure fluctuation increases. As the liquid flowrate increases further, the length of the liquid slug becomes larger and a large number of bubbles are involved in the liquid slug. In this case, the frequency band of the fluid pressure fluctuation becomes larger and the energy of the two-phase flow is distributed over a large frequency range, which results in a continuous decrease in STD of pressure fluctuations.

As can be seen from Fig. 12, there is no dominant frequency in the spectrum of the pressure fluctuations for the bubbly flows, while the slug flows show obvious narrow-band spectrum with dominant frequencies less than 3 Hz. For a fixed gas flowrate, the dominant frequencies of pressure fluctuations increase proportionally with the liquid flowrate for slug flows. This can be explained by the mass conservation for the liquid phase. Assuming the length and velocity of each liquid slug are constant for a given gas flowrate, the frequency of the liquid slug should be proportional to the liquid flowrate in order to satisfy the continuity equation. As the liquid superficial velocity increases further, the gas slugs burst into a large number of small bubbles, and the peak frequency drops rapidly in the case of the transition regions of the slug flow and the bubbly flow.

The STD and the peak frequency of fluid pressure fluctuations for 21 two-phase flow conditions are shown in Figs. 13 and 14. Three gas superficial velocities are considered, and the pressure fluctuations are measured at the measuring point P2. It is observed from Fig. 13 that for a fixed liquid flowrate, the STD value of pressure fluctuation increases monotonically with the gas flowrate. The STD of pressure fluctuations increases monotonically with the gas flowrate. This can be explained by that the pressure fluctuations are largely determined by the increasing momentum flux of the gas phase, and the momentum flux is proportional to the velocity square. As the liquid superficial velocity decreases, the growing rate of the STD value of the pressure fluctuation increases with the gas flowrate.

It can be observed from Fig. 14 that the peak frequencies of pressure fluctuations vary from 0.75 Hz to 2.25 Hz. When the gas superficial velocities are small, the predominant frequencies of the pressure fluctuations are larger for slug flows, and the predominant frequencies become smaller with an increase in the gas flowrate as shorter gas slugs coalesce to form longer ones. When the gas superficial velocities reach around 5 m/s, the peak frequencies of slug flows increase slightly. This is due to the fact that the high-speed gas phase picks up the liquid phase at the bottom of the pipe, resulting in an increase in the number of liquid slugs. As the gas flowrate increases further, the peak frequency of the pressure fluctuations drops slightly, which can be explained by that the increase of the gas flowrate leads to the increase of the length of the gas slugs and the decrease of the frequency of the liquid slug. As the slug flow transits to annular flow, the predominant frequency increases slightly.

The time domain signals and frequency responses of the pipe acceleration measured at measuring point A2 are shown in Fig. 15. The superficial gas velocity is kept as j_g = 5.4 m/s, while the superficial liquid velocity varies from 2.06 m/s to 6.28 m/s. Figures 15(a) and 15(c) illustrate the influence of slug flows on structural vibration responses. As shown in Fig. 15(a), the two-phase internal flow mainly excites the bending vibration of the pipe, and the axial vibration responses of the pipe are rather small due to large axial stiffness of the pipe. The vibration responses of the pipe are strongly linked to the liquid slug frequencies, and the maximum vibration amplitude appears in either the vertical or the lateral direction. The pressure fluctuations of slug flows exhibit as a periodic impact excitation on the pipe, which contain a wide range of frequency components. Therefore, the pressure fluctuations of the fluid can excite the operating vibration modes of the pipe, as shown in Fig. 15(c). Figures 15(b) and 15(d) illustrate the influence of bubbly flows on structural vibration responses. The results in Fig. 15(b) show that the vibration responses of the pipe induced by the bubbly flows are quite similar to those of the pipe conveying single-phase flows. As shown in Fig. 15(d), the bubbly flows also excite the operating vibration modes of the pipe, and the amplitudes of acceleration are smaller than that of the slug flows.

Figure 16 depicts the time domain signals and their FFT of the vibration responses of three acceleration measuring points along the Y direction. The superficial gas velocity is kept as j_g = 5.4 m/s, while the superficial liquid velocities vary from 1.47 m/s to 5.05 m/s. Figure 16 indicates the influence of bubbly flows and slug flows on structural vibration responses, respectively. As can be seen from Figs. 16(a) and 16(b), the maximum displacement amplitudes of lateral and vertical vibration responses appear at the middle of the pipe. It is observed from Figs. 16(c) and 16(d) that two dominant frequencies exist in the structural responses of the pipe, and these frequencies are close to the first two natural frequencies of the pipe filled with water. The monitoring point A2 located at the middle of the pipe, which is a nodal point of the second-order vibration mode of the pipe. Therefore, the vibration responses of the pipe measured at point A2 does not contains the second-order operating mode frequency.

Figure 17 shows the STD values of the vibration responses of the pipe along the vertical direction measured at point A2. It is observed from Fig. 17(a) that the bubbly flows lead to the minimum vibration response, while the maximum vibration response occurs in the pipe conveying slug flows. The results in Fig. 17(a) show that for a fixed gas flowrate, the STD values of the vibration responses of the pipe increase as the liquid flowrate is increased and they decrease rapidly in the case of the transition regions of the slug flow and the bubbly flow. The results in Fig. 17(b) indicate that for a fixed liquid flowrate, the STD values of the pipe increase proportionally with gas flowrate due to the increase of the momentum. As shown in Fig. 17(b), as the liquid flowrate is small, the variation of the structural vibration amplitude gradually decreases with the increase of the gas flowrate. This can be explained by that the distance between the liquid slugs increases as the specific gravity of the air increases, and the slug flow transitions into the annular flow earlier. The effect of the gas flowrate on the vibration responses of the pipe is much higher than that of the liquid flowrate.

In this study, the two-phase flow induced vibration responses of a horizontal pipe have been studied experimentally and numerically. In the experiments, measurements of pressure fluctuations and vibration signals are analyzed simultaneously in order to evaluate the effect of two-phase flow behaviors on the vibration responses of the structure. For the numerical simulation, the VOF model and the finite element method are used. The results of numerical model for the flow pattern and the pressure fluctuations of the gas–liquid two-phase flow as well as the vibration responses of the pipe agree well with the experimental results. In the slug flow regime, the peak frequency of pressure fluctuation tends to proportionally increase rapidly with an increase in the superficial liquid velocity, and reaches to their maximum values in the region where a transition from bubbly flow to slug flow. Also, the STD of pressure fluctuation increases monotonically with the gas flowrate, and the maximum magnitude appears in slug flows. The two-phase flow mainly excites the operating modes of the pipe, and the peak frequencies of the vibration responses vary in the neighborhood of the natural frequencies. The effect of gas flowrate on vibration responses is much higher than that of liquid flowrate, and the maximum amplitude of vibration response appears at the transition boundary between the slug flow and annular flow or the bubbly flow.

• National Natural Science Foundation of China, China (Grant Nos. 11922208 and 11932011; Funder ID: 10.13039/501100001809).

• Natural Science Foundation of Shanghai, China (Grant No. 18ZR1421200; Funder ID: 10.13039/100007219).

• National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant No. 2019ZX06004001; Funder ID: 10.13039/501100002855).

Informed consent has been obtained from all individual participants included in the study, and the authors declared that they have no conflicts of interest to this work.

The authors attest that all data for this study are included in the paper.

English Symbols

C =

damping matrix

E =

Young's elastic modulus, Pa

F =

surface tension

F_f =

external force vector due to the fluid

F_s =

external force vector due to the structure

g =

gravity vector

G_k =

production of turbulent kinetic energy

j_g =

superficial gas velocity, m/s

j_l =

superficial liquid velocity, m/s

K =

stiffness matrix

M =

mass matrix

P =

pressure, Pa

Q_g =

gas volume flow rate, m³/s

Q_l =

liquid volume flow rate, m³/s

t =

time, s

u =

mixture velocity vector

U =

displacement vector

$U˙$ =

velocity vector

$U¨$ =

acceleration vector

Greek Symbols

k =

turbulence kinetic energy, m²/s²

α =

volume fraction

ε =

turbulence dissipation rate, m²/s³

κ_l =

interface curvature

μ =

physical viscosity, kg/m·s

μ_t =

turbulent viscosity, kg/m·s

ρ =

mixture phase density, kg/m³

σ =

surface tension coefficient, N/m

υ =

Poisson's ratio

Subscripts

f =

fluid

g =

gas phase

l =

liquid phase

q =

each phase component

s =

structure

t =

turbulence

Abbreviations

ALE =

arbitrary Lagrangian–Eulerian method

FSI =

fluid–structure interaction

GGI =

general grid interface

PISO =

pressure-implicit with splitting of operators

STD =

standard deviation

VOF =

volume of fluid