Study on FSI Analysis Method of a Large Hydropower House and Its Vortex-Induced Vibration Regularities

The working principle of a large hydropower station is to guide the high-pressure water ﬂow to impact the turbine to rotate and generate electricity. The high-pressure water ﬂow impacts the turbine blades, which forms complex high-speed eddy currents in the spiral case and the draft tube and causes complicated vortex-induced vibration problems. Traditionally used harmonic response methods and dynamic time-history analysis methods are diﬃcult to reﬂect this complex ﬂuid-solid dynamic coupling problem. In this paper, the bidirectional ﬂuid-structure interaction (FSI) simulation analysis theory for a large hydropower house is studied, and the analysis methods of geometric simulation, mechanical simulation, and vibration energy transmission path simulation are presented. A large-scale 3D ﬂuid-hydraulic machinery-concrete structure coupled model of a hydropower house is established to study the vortex-induced vibration mechanism and coupled vibration law during transient unit operation. A comparison of the ﬂuid results against the in-site data shows good agreement. Structural responses of vibration displacement, velocity, and acceleration reveal coupled regularity of hydraulic machinery-concrete structure-ﬂuid during blades rotating periods, and it comes to the conclusion that the turbine blade rotation is the main vibration source of the hydropower house. The research results can provide a scientiﬁc basis for the design and safe operation of the hydropower house.


Introduction
e vibration problem during the operation of the hydropower station is one of the key issues in the design of the hydropower house, which mainly involves mechanical vibration induced by the operation of the hydraulic turbine, hydraulic vibration induced by the coupled interaction between the high-speed rotating flow and the structure, and concrete structural vibration through the transmission of the vibration source. With the increase of unit capacity and structural dimensions, strong vibrations may occur and affect the stable operation of the hydropower station [1,2]. Relevant cases can be referred to Yantan and Xiaolangdi hydropower stations in China, and Grand Coulee Project in the United States. Obvious vibrations have also been detected during the start-up and shutdown processes of Xiangjiaba hydropower house, which owns currently the largest unit capacity of 800 MW in China. In order to ensure the stable operation of the entire hydropower station and provide practical design and construction reference, it is of great significance for the vibration control and safe operation of the hydropower house to study the fundamental theory reflecting fluid-solid coupling features, to understand the rules of hydraulic pulsation, structural vibration, and their dynamic interaction in the entire hydropower house, as well as finding out the vibration source of the hydropower house.
Computational fluid dynamics (CFD) is considered a powerful method to provide insight into the flow features in hydropower components. Numerous simulations have been performed for the analysis of the fluid flow in both hydraulic turbines and pumps [3]. Li et al. [4] conducted CFD analysis of the pressure pulsations in a prototype pump turbine in turbine mode with different blades numbers. Kc et al. [5] investigated transient flow features inside a Francis turbine including the distribution of pressure, streamline, and vortex rope at full-load and part-load operation. Minakov et al. [6] presented numerical simulation results like vortex rope and pressure pulsation for two Francis turbines at different openings of the wicket gate. Wei et al. [7] performed a study on features of flow, performance, cavitation in the runner and vortex rope, and pressure fluctuations in the draft tube of the perforated and imperforated turbines. Trivedi et al. [8] focused on the experimental and numerical studies of the runaway condition and presented the flow field and its effects on the runner blades. Chalghoum et al. [9] analyzed the effect of the impeller diameter and number of blades on the pressure evolution in a centrifugal pump during the starting process. Li et al. [10] studied the hump characteristics of a pump turbine with the presentation of transient streamline and velocity in the tandem cascade passages at different discharges. In research of hydraulic machinery, a threedimensional CFD model was generally established with the inner characteristics as its main focus, such as variation of the vortex core, regularity of pressure pulsation in time and frequency domain, velocity, and streamline distribution in spatial domain.
Computational structural dynamics (CSD) is one of the most accurate and effective calculation tools for simulating complex structural vibrations. It can reveal the real-time vibration deformation, vibration velocity, and vibration stress of the combined structure under complex loads and help to understand the mechanism of structural vibration damage, which are very important for the vibration control and optimization design of the hydropower house. It is vital to conduct structural vibration analysis of the hydropower house induced by flow excitations during turbine operation. Numerous scholars have conducted relevant research using the quasi-static method, harmonic response method, and the dynamic time-history method [11,12]. Zhang and Zhang [13] studied the nonlinear dynamic response of the hydropower house under the frame of isotropic damage coupled with plasticity for concrete and found that tensile damage in the concrete surrounding the spiral case showed little impact on the superstructure but exerted a significant effect on the substructure. e dynamic time-history method was adopted in their study, in which time pressure distribution inside the concrete flow passage was artificially supposed, and the time-varying pressure values were exacted from nine measuring points.
Fluid-structure interaction (FSI) is the state-of-the-art technique for studying fluid and solid dynamic interactions, which combines CFD and CSD theory and has increased applications in complicated hydraulic mechanical engineering. One-way coupled FSI method, in which the steady-state CFD simulation provides the fluid pressure for FEA calculation as a load or a boundary condition, is generally adopted in previous research. Negru et al. [14] and Saeed et al. [15] analyzed the stresses induced in a Francis turbine runner by steady flow for different operating regimes and concluded that the sensitive areas to fatigue crack initiation was in the trailing edge between the blades and the crown. Xiao et al. [16] conducted a one-way FSI analysis of the flow field, the static and dynamic stress in the runner of a Francis turbine, and concluded that the dynamic stress was the main contributor to fatigue and cracks of the blade during off-design operation. Luna-Ramírez et al. [17] performed a failure analysis of runner blades based on the results of fluid pressure in CFD simulation and concluded that the most likely cause of failure was that turbine operation at low loads, accelerated by detected cavitation, and high water levels in the upper reservoir and tailwater. Although the one-way FSI method could economically describe structural responses under the fluid effect, it is known to be insufficient to evaluate structural integrity due to no feedback of the blade deflection on the flow. Two-way coupled FSI could fill this gap by conducting iterations between fluid and solid systems. Dompierre et al. [18] calculated the dynamic behavior of a Francis turbine runner under operating conditions based on the two-way FSI method introduced by ANSYS. Schmucker et al. [19] utilized two different methods to demonstrate the influence by means of two-way FSI simulation of the propeller turbine on the hydraulic performance. All their research highlights the importance of two-way coupled FSI simulation in the application of hydraulic engineering, but, without taking the concrete structure of the hydropower house into account, their concern is mainly the hydraulic machinery.
Zhang et al. [20] optimized C 2 interpolation method for data transfer of the fluid-structure coupling interface in the hydropower house and conducted a vibration analysis of the hydropower house. Most previous research rarely considered dynamic interaction between powerhouse structure and the entire water flow and rarely performed a 3D fully fluidsolid coupling simulation, in which the fluid analysis was conducted with the CFD method, and the entire structure of hydropower house was investigated by the two-way FSI approach. erefore, it is a tremendous need to apply the FSI method to the hydropower house research for detailed study.
In existing papers for studying the vibration of the hydropower house, the fluid model and hydraulic machinery are often omitted, or the machinery is simulated as a onedimensional numerical model. e main purpose of this paper is to perform the research on the bidirectional FSI simulation analysis theory applicable to large hydropower houses by combining CFD theory and CSD theory and to propose the analytical method of geometric simulation, mechanical simulation, and vibration energy transmission path simulation. CFD based on the finite volume method (FVM) is adopted in the fluid domain to explore hydrodynamic features of the turbulent flow. CSD based on the finite element method (FEM) is utilized in the solid domain to study structural dynamic responses of the hydropower house. A bidirectional iterative FSI method is adopted to solve the dynamic interaction between the fluid and solid domains.
e transient structural dynamics model is employed to study the vibration characteristics of the hydropower house structures. Based on the powerful FSI function of the commercial software ADINA, a fluid-hydraulic machinery-concrete structure coupled model for Xiangjiaba hydropower house is established, including the concrete structure of the hydropower house, the mechanical structure of the turbine, the generator, the upper and lower brackets, and the fluid inside the entire flow passage. e techniques of the sliding mesh and the upwind scheme are adopted to carry out the FSI bidirectional iterative calculation, and the high-nonlinear dynamic problems are effectively resolved. e vortex-induced vibration mechanism and coupled vibration regularity during transient operation of the unit are discussed. e main vibration source and the transmission path of the hydropower house vibration are clarified, which provides an important reference for the design and construction of the hydropower house.

Basic Equations.
In this section, the basic equations for the numerical simulation are introduced. e basic equation for the turbulence flow is given in the following.
In the fluid domain of the hydropower house, the continuity equation is adopted to satisfy the conservation of the mass, as shown in the following equation: where ρ is the fluid density and μ i are the velocity components.
As a built-in feature of the ADINA software, Spalart-Allmaras detached eddy simulation (SA-DES) turbulence equation was adopted to model the turbulence and produce the eddy viscosity for the flow. Derived from the SA oneequation eddy-viscosity RANS model, SA-DES combines the advantages of large eddy simulation (LES) and RANS method and could simulate the large-scale turbulent pulsations accompanied by the precession of vortex in hydraulic machinery economically and reliably [6,21]. e equation of the SA-DES turbulence model is expressed as follows [22][23][24]: where v is the modified turbulent viscosity linked to the turbulent viscosity v t and a wall function As the switch function of the LES-SGS and RANS turbulence model, d DES is described in the form of the following equation: where d is the nearest distance to the wall and Δ max is the maximum local grid size, C DES � 0.65. e basic equation for the structural vibration is given in the following. e control equation for the structure of the hydropower house can be written in the following equation [25]: where τ s , f s , € u, ρ s , and L T denote the solid stress tensor, the solid body force, the solid particle acceleration vector, the solid density, and the differential operator, respectively. conducted for the fluid model to obtain a reasonable number of the computational grids.

Numerical Simulation
In the fluid domain, FSI boundary condition was replaced by a no-slip wall boundary to perform the CFD test. With the steady-state analysis, head H in different grid size was evaluated by imposing velocity at the inlet and outflow at the outlet. Figure 1 shows the influence of grid size on the ratio of calculated head H to rated head H 0 . e fluid mesh was required to converge till obtaining the variable residual of 1e − 4. ere was no significant variation for H/H 0 values with nodes number beyond C. Based on comprehensive consideration of the simulation accuracy and intense computational resources required by the two-way iterative FSI solver, nodes number C with 442704 elements and 481906 grid nodes in the fluid domain was selected to perform transient state FSI analysis.

Computational Model.
e previously described coupling method is applied to the Xiangjiaba hydropower house owning currently the largest unit capacity of 800 MW in China. In order to represent realistic hydrostructural dynamic features, a more comprehensive model and delicately divided finite element mesh are obtained according to the actual construct of mechanical structures like the hydraulic turbine, generator, and upper and lower brackets. Figure 2 shows the computational model for fluid and solid domain. As shown in Figure 2  (2) Solid Model (Concrete and Mechanical Structure) Simulation. As the core component of the hydraulic turbine, the runner, motivated by high-speed flowing water, drives the generator rotor through the main shaft and transmits the energy of the vibration source upwards through its coupling with water. Vertical dynamic loads such as runner-axis-rotor coupled gravity and axial water pressure imposed on the runner are transferred to the base pier through thrust bearing and bracket. As the lower bracket, it bears the main vibration load generated during the operation of the unit. e upper bracket is connected to the stator and floor to transmit the load exerted by the weight of its own and other structures. Comprehensive components of concrete and mechanical structure in the hydropower house are modeled to obtain a realistic and accurate simulation of vibration energy transmission path. Figure 4 shows the grids for concrete structure, including turbine floor, dynamo floor, draft tube block structure, sidewall, and roof. e most concrete structure is simulated using a 3D solid element except a shell element for the roof. 3D solid elements are also adopted for modeling the brackets, beam, and floor to avoid serious stress concentration. Mesh of mechanical structure, including runner, axis and rotor, upper and lower bracket, and stay and guide vane, is presented in Figure 5. e solid domain is composed of 187636 elements and 136752 grid nodes. Table 1 shows the main computational parameters. In the current research, the turbine operates at best efficiency point (BEP) with a rated head of 100 m and a rated flow rate of 890 m 3 /s. For the fluid domain, density and viscosity are set as 998.2 kg/m 3 and 0.001003 Pa·s, respectively. e number of runner blades is 15, and the number of guide vanes is 24. As for the inlet, a uniform velocity of 7.613 m/s is applied in terms of the flow rate of 890 m 3 /s. e turbulent flow is supposed to be fully developed at the outlet with outflow set as the outlet boundary condition. e rotational speed of 75 rpm is assigned to the hydraulic turbine.

Calculation Method.
Geometric simulation, mechanical simulation, and transmission path simulation are performed to study the coupled dynamic model. As mentioned above, appropriate element types and well-designed mesh division are chosen to perform geometric and mechanical simulation, (3) Solve the solid solution vector X k s from the structural equation: (4) e fluid nodal displacements d k f are computed with the prescribed boundary conditions: (5) Both displacement and stress convergence conditions are checked according to stress and the displacement criteria. If the iteration has not converged yet, the program goes back to step 2 and continues for the next iteration unless a maximum number of FSI iterations have been reached.
Both kinematic condition and the dynamic condition are applied to the fluid-solid coupling interface to obtain compatible traction and displacement [26]. Corresponding equations are expressed as follows: where d f and d s are the fluid and solid displacements, respectively, and τ f and τ s are the fluid and solid stresses, respectively. Figure 6 displays the boundary conditions applied in the current simulation. Based on Arbitrary Lagrangian-Eulerian (ALE) theory, sliding mesh boundary conditions are employed to connect rotating domains and to ensure the compatibility, continuity, and completeness conditions along the nonconforming interfaces. Sliding mesh I is placed between the rotating region of the fluid around runner and the static mesh of the fluid around the guide vane. Sliding mesh II is placed between the rotating runner region and the draft tube. FSI boundary condition is imposed on the interfaces between the fluid and solid model, including the interfaces between the entire water flow and the surrounding concrete, water and the rotating blades, water and the stay vanes, water and the guide vanes. e SIMPLE algorithm is adopted to solve the pressure and velocity coupling. e upwind scheme is applied to the governing equations of the fluid part to ensure the stability of the calculation process. In equation, residual mass convergence criteria are adopted, while, in variable, residual both velocity and pressure convergence criteria are employed. e equations are required to be converged at every time step after iteration. e time step is 0.002222 seconds, corresponding to 1°of the runner rotation. e total calculation time is set as 8 seconds to acquire stable simulation results. Figure 7 depicts the pressure and streamlines distribution in the whole flow passage. Pressure gradually decreases along flow direction from the spiral case, to stay vane, guide vane, and runner, with strong regularity of pressure distribution in the entire fluid domain. e maximum pressure magnitude in the spiral case is approximately 0.9 MPa. As the water flows into the runner, the pressure sharply decreases due to the shrinking cross section and increasing flow acceleration. In the draft tube, most of the kinetic energy is converted into static pressure thereby delimiting outlet velocity to minimal. It can be inferred from Figure 7 that there is no flow collision in the Analysis of flow structure in the draft tube plays an important role in pressure pulsation prediction and turbine performance evaluation. e Q-criterion is considered as a criterion with explicit physical meaning and hereon is adopted to analyze the vortex structure in the flow domain. Q-criterion represents the relative strength of the rotation to axial deformation and locates regions where rotation dominates strain in the flow [27]. e equation form for the incompressible flow is given as follows [28]:

Transient Fluid Behavior.
where Q is the second invariant of ∇u, S and Ω denote the symmetric and antisymmetric parts of ∇u, respectively, and ‖ · ‖ is the Euclidean matrix norm. A coherent vortex is defined as a region where Q > 0. Figure 8 shows typical vortices in the flow behind runner calculated by Q-criterion and colored by static pressure. A short and straight vortex is obtained in the straight cone of the draft tube, and its shape and strength remain steady with runner rotation. It is well known that the shape of the vortex rope is usually related to the opening of the vane and greatly affects the strength of the pressure pulsation. In the current research, the normal operating process is investigated when the turbine operates at the rated head with the rated output, and the generated vortex rope is short and straight, which predicts lower pressure pulsation caused by the vortex in the draft tube during normal operation.

Comparison with the Measurement Data.
In order to evaluate the pressure pulsation level in the flow-induced vibrations, four measuring points F01 ∼ F04 are chosen and placed at head cover, vaneless region, bottom ring, and draft tube, respectively, as shown in Figure 9. Figure 10 displays the time-history curves for pressure pulsation recorded during the numerical calculation. Note that the pressure pulsation is the realistic value that contains a constant value during the calculation process. e pressure coefficient may be more effective in presenting the pressure pulsations by excluding the constant pressure. But we want to show the actual pressure pulsation values, so the coefficient is not adopted here. Periodical behavior is observed at four positions with the occurrence of fifteen wave peaks and wave troughs during one blade rotational period of 0.8 s, which is consistent with fifteen blades rotation rule of the turbine and demonstrates evident contribution of rotor-stator-interaction to the pulsation. e wave shape of the pressure pulsation at the draft tube is largely different from that at the other three positions, which could be ascribed to the measuring positions relative to the runner. e approximate fluctuation amplitudes of the pressure at head cover, vaneless region, bottom ring, and 0.3D of the draft tube are 2.5 kPa, 8 kPa, 4 kPa, and 8 kPa, respectively. e peak calculation of the pressure pulsation and the swing for the in-site data is 97% confidence fusion 8-cycle averaging period method. To compare with the in-site data, the intensity of the simulated pressure oscillations is measured by the parameter of peak-to-peak value ΔH/H, where H is the rated head in the current operation point, and ΔH is the peak-to-peak value calculated by 97% confidence. Based on the probability and statistics method, 3% of untrusted data is eliminated from the original data to calculate the difference between the minimum and maximum value. Table 2 shows the comparison of the calculated and in-site peak to peak values of the relative pressure pulsations with 97% confidence for different measuring points. e in-site data could be referred to report [29]. e computed ΔH/H at head cover and bottom ring are slightly lower than in-site data, while the relative pressure pulsation is slightly overestimated at the vaneless region. Improved results are expected if LES turbulence model is applied together with finer grids. As a whole, amplitudes of the pulsations agree well with the in-site data, which represents the potential of the method to predict turbine performance and thus provides a reliable basis for the simulation of transient responses of the structure.

Research on Vibration Regularities of the Hydropower House
In order to characterize and understand the vibration features of the hydropower structure, several important indexes including dynamic displacement, velocity, and acceleration are adopted to study and evaluate the structural vibration. Figure 11 distributes the extreme value of the dynamic displacement of the hydropower house. In order to understand vibration values inside the powerhouse, Figure 11(b) displays the section diagram. Here is more illustration about the absence of upper structure like sidewall and tailwater platform. Distribution of their vibration values could make unclear the regularity of the main structural components due to their relatively higher vibration level.
us, the analysis of them is not included in the current research. As shown in Figure 11, dynamic displacement decreases gradually from a higher elevation and the unit center to downstream structures. It is reasonable because the vibration of the concrete structure surrounding units is directly affected by the fluctuating fluid pressure produced during the unit operation. e superstructure like generator layer has a relatively higher vibration level due to the displacement amplification effect. e extreme dynamic displacement appears at the junction of the generator floor and It is helpful to further understand the transient responses by analyzing the time-varying dynamic displacement. Extreme dynamic displacement point A of the lower block structure in Figure 11 is selected to conduct time and frequency analysis, and the result is demonstrated in Figure 12.
e dynamic displacement fluctuates between 0.192 mm to 0.200 mm periodically. Based on the Fast Fourier Transform (FFT) algorithm, frequency domain analysis is conducted and shown in Figure 12(b). Here, we define f n as the blade rotation frequency; the value of f n is 1.25 Hz. From Figure 12(b), we can see that the first dominant frequency is 30f n , two times the blades passing frequency 15f n , so the fluctuation of the dynamic displacement at the extreme point is mainly affected by blades passing frequency. Figure 13 is the distribution of extreme vibration velocity of hydropower house in three-dimensional and section view. e dynamic velocity represents a strong regularity with dynamic velocity diminishing from the unit center. e diffuse section of the draft tube, which is far away from the unit, owns a relatively lower vibration velocity level and the value is about 0.2 mm/s. Relatively drastic vibration velocity occurs at the entrance of the straight cone section of the draft tube, which is approximately below a 130-degree envelope angle of the spiral case. e extreme value of the vibration velocity is 1.13 mm/s, which conforms to the restrictive requirement of 5.00 mm/s in relevant vibration specifications. Figure 14 reveals the time history and frequency spectrum of the extreme velocity point B. e vibration velocity of the extreme point fluctuates between 0 and 1.20 mm/s periodically, and the wave is denser compared to timevarying dynamic displacement. Figure 14(b) shows the frequency domain graph of the extreme velocity point B. More frequency components are detected largely times the blades passing frequency 15f n . e first dominant frequency is 60 times rotational frequency f n , which indicates the velocity vibration is greatly affected by the rotation of runner.
e maximum acceleration distribution of the hydropower house is shown in Figure 15. Although it is easily detected that the distribution regularity of acceleration extreme value is similar to that of vibration velocity, we want to present not only the location of the maximum variable in the graph but also the general distribution of the variable values.
Significant vibration acceleration appears at the entrance of the straight cone section of the draft tube, and the value gradually decreases to concrete structures around. e vibration level in most of the areas is less than 0.20 m/s 2 . In the lower block structure, especially the diffuser of the draft tube, the vibration acceleration is at a much lower level. In order to see the distribution of vibration acceleration inside the   house, the section diagram is provided in Figure 15 Extreme acceleration point C is selected and the corresponding time-varying and frequency spectrum is shown in Figure 16. Figure 16(a) describes the variation range and regularity of the extreme vibration acceleration point during one blade rotational period. Acceleration fluctuates between 0 and 0.45 m/s 2 steadily and periodically. Figure 16(        Advances in Civil Engineering 11 acceleration is 60f n . Other main frequencies are largely multiple times the blade passing frequency 15f n . e blades rotation is the major contributor to structural vibration. e above analysis shows that the vibration caused by the rotation of the turbine blade is the main vibration source of the hydropower house.

Conclusion
is paper combined CFD theory and CSD theory, discussed the bidirectional FSI simulation analysis theory and calculation method for the large hydropower houses, and explored the general law of the vortex-induced vibration. Using the famous structural analysis software ADINA, a 3D FSI finite element model of the fluid-hydraulic mechanics-concrete structure of the Xiangjiaba hydropower house was established. e vortex-induced vibration mechanism and coupled vibration law were explored during transient unit operation. e following main conclusions were obtained: (1) e time-varying pressure pulsations at typical positions of head cover, vaneless region, bottom ring, and draft tube were analyzed and verified with the insite measuring data. e results showed that the pressure distribution was reasonable in the flow passage, and the streamlines were smooth in the entire flow passage under the design condition. e short and straight shape of the vortex generated behind the runner indicated the low intensity of pressure pulsations. A comparison of the in-site data with the calculated values showed that the calculated results could satisfactorily describe the behavior of the vortex features and pressure pulsations in the fluid domain during design operation, which laid the foundation for turbine's performance prediction and the structural vibration evaluation.
(2) e responses of the vibration displacement, vibration velocity, and vibration acceleration of the hydropower house were studied, and the vibration law of the hydropower house during the unit operation was discussed. e results showed that the extreme values of the vibration displacement of the hydropower house occurred on the generator floor. Both the extreme values of the vibration velocity and vibration acceleration appeared at the entrance of the straight cone of the draft tube. e extreme values were lower than the criterion value, which satisfied the vibration control requirements. e dominant frequencies of the extreme point of vibration displacement, vibration velocity, and vibration acceleration in lower block concrete structure were multiple of blades passing frequency, indicating the blades rotation was the main vibration source during operation.
In the current paper, our main focus is on the vibration features of the hydropower house in design operation. e vibration features of operation of the Francis turbines at offdesign conditions will be our further focus.

Data Availability
Some or all data, models, or codes generated or used during the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.

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Advances in Civil Engineering