Numerical Investigation of Symmetrical and Asymmetrical Characteristics of a Preloading Spiral Case and Concrete during Load Rejection

Abstract

During the transient process of load rejection, the hydraulic pressure applied to the pump-turbine and plant concrete changes dramatically and induces high dynamic stress on the spiral case. The preloading spiral case has been widely used in large-scale pumped-storage power stations due to its excellent load-bearing capacity. However, studies on the impact of preloading pressure on the structural response during load rejection are still few in number. In this paper, 3D flow domain and structural models of a prototype pump-turbine are designed to analyze the hydraulic characteristics and flow-induced dynamic behavior of the preloading steel spiral case under different preloading pressures during load rejection. The results show that the asymmetric design of the logarithmic spiral lines ensures an axially symmetric potential flow within the spiral case domain with uniform pressure distribution. Higher preloading pressure provides larger preloading clearance, leading to greater flow-induced deformation and stress, with their maximum values located at the mandoor and the inner edge, respectively. The combined effect of the asymmetrical shape, internal hydraulic pressure and unbalanced hydraulic force leads to an asymmetrical preloading clearance distribution, resulting in an asymmetrical distribution along the axial direction but a symmetrical characteristic near the waistline of the structural response. Stress variations at sections and between sections share similar characteristics during load rejection. It follows the same trend as the hydraulic pressure under lower preloading pressures, while there is a delayed peak of stress due to the delayed contact phenomenon when the preloading pressure reaches the maximum static head. The conclusions provide scientific guidance for optimizing the preloading pressure selection and structural design for the stable operation of units.

1. Introduction

With the increasing demand for power regulation in recent decades, the development of pumped-storage power stations is facing the challenges of large-capacity and high-water-head conditions [1,2]. The flexible features of the pumped-storage power station include quick startup/shutdown and load adjustment. As a key component of stations, the pump-turbine can experience varying hydraulic pressure on its flow passage structures with complex turbulence [3]. High hydraulic excitation may lead to unexpected stress concentration and severe vibrations in the structures, greatly affecting the safe and stable operation of the stations.

To meet the needs of the power grid, pump-turbines often experience unstable operation processes, and load rejection is one of the transient processes. Drastic variation in the flow domain can create extremely unstable turbulence, resulting in higher hydraulic excitation of the structures compared to rated conditions [4,5,6,7]. Numerous studies [8,9,10] have confirmed the reliability of 1D-3D coupling simulation methods for pump-turbines by comparing the results with measured data. According to a literature review, researchers at home and abroad have studied the flow characteristics and pressure fluctuations of units during load rejection. Researchers [11,12,13] analyzed the mechanisms of pressure fluctuations induced by the inter-blade flow during load rejection. Some scholars [4,14] also studied the effects of complex flow patterns on variations in axial hydraulic thrust. Given that these studies primarily concentrated on flow domain analysis, the fluid–structure interaction (FSI) method was introduced to analyze structural behaviors induced by hydraulic excitation. There have been some studies [5,6,7] focusing on the internal flow and structural response of the runner. Also, the authors of [15,16,17] discussed the deformation and stress distribution of the stationary parts of the unit via the one-way FSI method. However, researchers have primarily concentrated on other components of the pump-turbine, such as the runner, headcover, bottom ring and so on, while studies on the structural response of the spiral case during load rejection are few in number.

As an essential component of the unit, the spiral case plays an important role in supporting the hydraulic pressure as well as the structural load from the upper structures. The embedment of the spiral case can be categorized into three structural forms: the padded spiral case, preloading spiral case and directly embedded spiral case. Among these, the preloading steel spiral case (PSSC) is widely used in large-scale units due to its good combined load-bearing performance [18,19]. During the construction of the PSSC, the first step is to place it on a concrete or steel pedestal and close off the inlet and outlet by the test head and sealing plate. The PSSC is filled with water until it stabilizes at the preloading pressure. After that, it takes several days to pour the concrete outside the PSSC and wait for its hardening. The internal pressure will then be unloaded with the removal of the test head, which changes the inlet condition of the PSSC and causes unbalanced hydraulic force during operation [20]. At that time, there is an initial preloading clearance between the PSSC and concrete due to the shrinkage of the PSSC. Therefore, the preloading clearance is strongly determined by the preloading pressure and plays a vital role in the transmission of the hydraulic load between the PSSC and the surrounding concrete.

Based on the isentropic spiral line theory, the asymmetrical design of the PSSC is intended to ensure the internal axisymmetric potential flow during operation, reducing energy dissipation caused by swirling and vortexing. The combined effect of the asymmetrical shape of the PSSC, internal hydraulic pressure and unbalanced hydraulic force leads to horizontal deflection, resulting in an asymmetric distribution of contact states. The preloading clearance can be reflected by the contact state between the PSSC and the concrete, which has an impact on the load transmission mechanism and, in turn, causes symmetric and asymmetric characteristics of deformation and stress distributions. It has been proven that the deformation and stress responses of the PSSC can be affected by preloading pressures [21,22,23]. As for lower preloading pressures that underutilize the tensile strength of the PSSC, additional steel bars may be necessary to prevent concrete cracking. Excessive preloading pressure with a large initial preloading clearance is likely to cause stress concentration and excessive vibration in the structures, especially under low-head operating conditions. However, the recommended range for preloading pressure values remains broad in engineering practice and relevant codes, typically ranging from 50% to 100% of the maximum static head [24].

Since numerical methods can provide a more detailed representation of contact status and response behaviors compared to experiments, most research on the PSSC is performed by CFD and FEM methods. To simulate the initial preloading clearance, simplified methods were first introduced, in which the contact status was determined by judging the preloading pressure value with hydraulic pressure only [25,26]. It has been proven that this method is unreliable for safety assessments due to its over-simplification and tendency to underestimate stress values [22]. More methods focusing on the nonlinear contact behavior were proposed by modifying the grid node coordinates or contact element types. Li et al. [27] and Qian et al. [28] studied the bearing ratio and contact behaviors between the spiral case and concrete under different preloading pressures. Xu et al. [29] characterized the premature and delayed closure phenomena of the preloading clearance. Fu et al. [21] revealed that the structural responses were strongly determined by clearance closing characteristics. However, the simulation methods of modifying grid node coordinates in these studies may cause node penetration and stress concentration, and contact element types need complex procedures that fail to fully satisfy the simulation requirements [20]. As a contact algorithm with good stability and convergence, the augmented Lagrange method has been proven effective in studying the contact behavior between the spiral case and concrete [19,29]. To analyze the structural characteristics of the PSSC under operational conditions, some papers [30,31,32] applied a static load as hydraulic pressure on the PSSC without considering the effects of pressure variation in the flow domain. Some other researchers [33,34] introduced harmonic waves into the structural model for the simulation of hydraulic pressure. However, these simulation methods for hydraulic pressure fail to capture the pressure fluctuation characteristics present in the actual flow domain, which can significantly affect the accuracy of simulating flow-induced responses. Research on the flow-induced structural response of the PSSC during load rejection is critical for optimal preloading pressure selection and structural design. As far as the authors know, there is a lack of published studies that consider the combined effect of preloading pressure and hydrodynamic excitation during load rejection.

In this paper, a 3D fluid domain model of the pump-turbine is established at key time points during load rejection, which are selected through 1D pipeline calculations. A preloading spiral case model is proposed for the simulation of the initial preloading clearance. Based on the one-way FSI method and nonlinear contact algorithm, the 3D structural modeling of the unit and plant concrete is performed under different preloading pressures. The hydraulic characteristics of the unit are analyzed, and the symmetrical and asymmetrical flow-induced structural characteristics of the PSSC under the effect of preloading pressure are discussed in detail. The results in this paper are helpful for the in-depth understanding of the dynamic characteristics of the preloading spiral case during load rejection and have an important reference value for preloading pressure selection and structural optimization.

2. Numerical Theory and Method

The numerical calculation theory and methods in this paper include the 1D pressurized pipeline calculation, 3D flow simulation, FSI and nonlinear contact algorithm for hydraulic and structural analysis.

2.1. Governing Equations of Unsteady Flow in 1D Pipeline Calculations

The transient calculation of the water conveyance system is composed of the upper and lower reservoirs, pipelines, a surge shaft, valves and the pump-turbine unit, with the primary operational parameters changing over time. The hydraulic characteristic of pipelines can be regarded as a one-dimensional flow, for which the continuity equation and momentum equation are as follows:

$$V{\displaystyle \frac{\partial H}{\partial x}}+{\displaystyle \frac{\partial H}{\partial t}}-V\sin \beta +{\displaystyle \frac{c^2}{g}}{\displaystyle \frac{\partial V}{\partial x}}=0$$

(1)

$$g{\displaystyle \frac{\partial H}{\partial x}}+V{\displaystyle \frac{\partial V}{\partial x}}+{\displaystyle \frac{\partial V}{\partial t}}+{\displaystyle \frac{fV\left|V\right|}{2D}}=0$$

(2)

where $H$ is the water head, $V$ is the flow velocity, $\beta$ is the angle between the pipe axis and the horizontal plane, $c$ is the wave velocity, g is the local gravity acceleration, $f$ is the Darcy–Westbach friction coefficient, and $D$ is the inner diameter of the pipelines.

Based on the characteristic line method, the original equations can be transformed into differential equations:

$$C^{+}:H_i^{j+1}=C_P-B{Q}_i^{j+1}$$

(3)

$$C^{-}:H_i^{j+1}=C_M+B{Q}_i^{j+1}$$

(4)

$$C_P=H_{i-1}^j+\left(B+C\right)Q_{i-1}^j-R{Q}_{i-1}^j\left|Q_{i-1}^j\right|$$

(5)

$$C_M=H_{i+1}^j-\left(B-C\right)Q_{i+1}^j+R{Q}_{i+1}^j\left|Q_{i+1}^j\right|$$

(6)

where $C^{+}$ and $C^{-}$ are the characteristic line directions, $H_i^j$ is the water head at point $i$ when $t=j∆t$, $Q=VA$ is the flow rate, $A$ is the cross-sectional area, $∆t=∆x/c$ is the time step, $B=c/Ag$, $C=∆t\sin \beta /A$, and $R=f∆x/2gD{A}^2$.

The equations can be integrated by the variation in the water head and flow rate between two points in the form of differences as $H_i^{j+1}$ and $Q_i^{j+1}$. According to the given water head of the upper and lower reservoirs, the initial flow rate, the guide vane opening, etc., the 1D pipeline calculations can be solved by the above differential equations through iterations.

2.2. Governing Equations of the 3D Flow Simulation

The results of the 1D pipeline calculations, including the spiral case inlet pressure, draft tube outlet pressure, flow rate, opening and rotational speed at different time steps, are set as the boundary conditions of the 3D flow simulation. Representative points on the curves of these parameters are selected as key time points to describe the transient process of load rejection. The Reynolds-averaged Navier–Stokes (RANS) method regards the incompressible unsteady turbulent flow in the pump-turbine as a time-averaged and instantaneous fluctuating flow, which averages the time by the turbulent control equation. The expressions of the continuity equation and the momentum equation (N-S equation) are

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial \overline{u_i}}{\partial x_i}}=0$$

(7)

$${\displaystyle \frac{\partial \left(\rho \overline{u_i}\right)}{\partial t}}+{\displaystyle \frac{\partial \left(\rho \overline{u_i}\overline{u_j}\right)}{\partial x_j}}={\displaystyle \frac{\partial }{\partial x_j}}\left[\mu \left({\displaystyle \frac{\partial \overline{u_i}}{\partial x_j}}+{\displaystyle \frac{\partial \overline{u_j}}{\partial x_i}}-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)\right]-{\displaystyle \frac{\partial \overline{p}}{\partial x_i}}+F_i$$

(8)

where $\rho$ is the density, the range of indicators i and j is (1, 2, 3), $\overline{u_i}$ denotes Reynolds-averaged velocity components along the Cartesian coordinate axes in the x-, y- and z-directions, $\overline{\rho u_i^{\prime }{u}_j^{\prime }}$ denotes the Reynolds stresses, $\mu$ is the dynamic viscosity, $\overline{p}$ is the average pressure, and $F_i$ denotes the body forces.

2.3. Governing Equations of the Fluid–Structure Interaction Analysis

It has been proven that structural deformation has a minimal impact on the fluid domain in pump-turbine research, so the one-way FSI method is introduced to calculate the flow characteristics and the flow-induced structural response. By applying the hydraulic pressure to the fluid–structure interfaces of the structure, the flow-induced structural response can be simulated following the governing equation of structural dynamics:

$${\mathit{M}}_s\ddot{\mathit{u}}+{\mathit{C}}_s\dot{\mathit{u}}+{\mathit{K}}_s\mathit{u}={\mathit{f}}_s\left(t\right)+{\mathit{f}}_{fs}\left(t\right)$$

(9)

where ${\mathit{M}}_s,{\mathit{C}}_s,{\mathit{K}}_s$ are the mass, damping matrix and stiffness matrix, respectively; $\ddot{\mathit{u}},\dot{\mathit{u}},\mathit{u}$ are the acceleration, velocity and displacement vectors, respectively, of the mesh node; ${\mathit{f}}_s\left(t\right)$ is the external excitation load vector acting on the structures; ${\mathit{f}}_{fs}\left(t\right)$ is the hydraulic pressure load vector at the interface; and $t$ is time.

The von Mises stress ${\sigma }_{vM}$ is used to describe the structural strength of the PSSC, while the maximum principal stress ${\sigma }_1$ is adopted to evaluate the tensile strength of the concrete.

$${\sigma }_{vM}=\sqrt{{\displaystyle \frac{1}{2}}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]}$$

(10)

where ${\sigma }_i(i=1,2,3)$ denotes the principal stresses.

2.4. Algorithm of Nonlinear Contact

There is nonlinear contact behavior occurring between the PSSC and concrete, which leads to the bonding and separation of the mesh nodes. The surface–surface contact type is applied to the structural model for a better simulation of the contact state and the free movement of the target surfaces. For good convergence and lower sensitivity to the magnitude of the contact stiffness, the augmented Lagrange method (ALM) is selected as the contact algorithm based on the augmented Lagrange function, and the expression is

$$\min \prod ^{\wedge }\left(U,\lambda \right)=\prod \left(U\right)+{\prod }^{\prime }\left(U,\lambda \right)+{\prod }_p$$

(11)

where $\lambda$ is the Lagrange multiplier, $\widehat{\mathsf{\Pi }}\left(U,\lambda \right)$ is the modified function, $\mathsf{\Pi }\left(U\right)$ is the total potential energy function, ${\mathsf{\Pi }}^{\prime }\left(U,\lambda \right)$ is the additional function of the Normal Lagrange, and ${\mathsf{\Pi }}_p$ is an additional function of the Pure Penalty.

$${\prod }^{\prime }=\lambda g\begin{array}{cc},& \end{array}{\prod }_p=\alpha g^2/2$$

(12)

The repeated iterations of ${\mathsf{\Pi }}_p$are calculated by the following expression:

$${\lambda }_{k+1}={\lambda }_k+\alpha \theta$$

(13)

where $\theta$ is the clearance.

According to the iterations of ${\mathsf{\Pi }}_p,$ the contact constraint is modified to find the exact $\lambda$ without increasing the unknowns. One iteration will continue until there is no change in the contact state.

3. The Numerical Simulation of the Pump-Turbine Model

The numerical simulation takes a prototype pumped-storage unit as the research object and includes the unit flow passage model, the preloading spiral case model and the pump-turbine and plant concrete structural model, which should be established in specific work steps, as shown in Figure 1.

As for the calculation of the flow domain, 1D pipeline calculations are performed to obtain the flow parameters that serve as boundary conditions for the 3D CFD of the unit, including the inlet pressure of the spiral case, the outlet pressure of the draft tube, the relative rotational speed, the relative flow rate and the relative guide vane opening. Based on the moments corresponding to the extreme points of each parameter, several key time points are selected to describe the flow domain variation during the load rejection transient process.

In the structural calculation, a preloading spiral case model is first proposed to calculate the initial preloading clearance distribution. After that, pump-turbine structural models considering preloading clearance under four different preloading pressures are established. The hydraulic pressure files at each key time point from the CFD model are sequentially exported, transferred and mapped to the corresponding structural components of the unit via the one-way FSI method. By introducing the nonlinear contact algorithm, the structural response and contact status of the PSSC and concrete can be simulated for the analysis of structural characteristics and preloading clearance distribution. An introduction to each model is described in detail in the subsequent sections.

3.1. Three-Dimensional Flow Calculation Model

The 3D modeling of the fluid domain in Figure 2 consists of a spiral case, a runner, guide vanes, stay vanes, a draft tube and two pressure balance pipes. The headcover and bottom ring, as the clearances in the fluid domain, are less than 1.9 mm, but they have a significant impact on the flow characteristics. There is a penstock setting in front of the spiral case for minimizing numerical errors of the spiral case pressure distribution, influenced by the inlet boundary condition. For the calculation of a high-Reynolds-number flow, the shear stress transport (SST) k-ω model is selected as the turbulence model to solve the N-S equation with a convergence criterion of 10⁻⁴ based on ANSYS CFX 2021. According to the calculation results from the 1D pipeline simulation, the total pressure type for the inlet and the static pressure type for the outlet are applied to the boundary conditions at the key time points. The surfaces of the headcover and bottom ring that are in contact with the runner domain remain stationary relative to it, so they are set as Rotating Walls, with the rotational speed set to the rated value. The rotor–stator interfaces between the runner and guide vanes, as well as between the runner and draft tube, are configured with the Frozen Rotor method. The wall boundary condition is set as a non-slip wall.

For the prototype pump-turbine in the model, the design parameters provided by the manufacturer’s drawings are as follows: the rated power is 306 MW, the rated head is 354 m, the rated rotational speed is 375 rpm (the rotating direction is +Z), the runner diameter is 4.33 m, and the rated flow rate is 95.97 m³/s. The number of stay vanes and guide vanes is 16. The runner has 10 blades, with 5 long blades and 5 short blades arranged alternately.

To ensure that the mesh quality meets the needs of both calculation accuracy and efficiency, a mesh independence analysis was carried out by comparing the normalized hydraulic torque using different mesh numbers. Four sets of meshes were established in ANSYS MESH 2021 to calculate the steady state at the key time point of load rejection when the spiral case is at the maximum pressure, and a mesh with 7,253,227 elements was adopted for the load rejection calculation, as shown in Figure 3. The mesh element types and the number of different flow domains are shown in

Table 1. Mesh element type and number.

Flow Domain	Element Type	Element Number
spiral case	tetrahedral	1,453,828
stay vanes	hexahedral	543,760
guide vanes	hexahedral	431,608
runner	tetrahedral	2,791,382
draft tube	tetrahedral	1,091,405
pressure balance pipe	tetrahedral	260,108
headcover	hexahedral	134,800
bottom ring	hexahedral	158,200
penstock	tetrahedral	388,136
Total		7,253,227

in detail.

To verify the accuracy of the CFD model simulation, the simulation results under four different conditions were compared with experimental data provided by the manufacturer. The conditions include the maximum head 100% $P_r$, intermediate head 50% $P_r$, and rated and minimum $P_r$ conditions. The calculation errors of the pressure fluctuation in the vaneless space are less than 5.37% (

Table 2. Comparison of simulation results and experimental data on pressure fluctuation in the vaneless space.

Condition	Head (m)	Opening (°)	Power (MW)	$\mathsf{\Delta }\mathit{H}/\mathit{H}$ (%)
				CFD	Experiment	Error
Maximum head 100% $P_r$ condition	387.28	19.57	306	2.73	2.64	3.41%
Intermediate head 50% $P_r$ condition	364	10.98	153	4.84	4.67	3.64%
Rated condition	354	24.53	306	3.51	3.40	3.23%
Minimum $P_r$ condition	326.6	8.85	65.04	15.31	14.53	5.37%

). Since the inlet and outlet are set as the boundary conditions of pressure, the calculated flow rate from the numerical simulation is compared with experimental data, showing good agreement, as shown in Figure 4. These validations confirm the reliability and credibility of the numerical methods used in this paper.

Water flows through the inlet of the spiral case and flows toward the vane region after moving along the equi-angular spiral. Based on velocity matrix calculations, it can be confirmed that the internal water flow is axisymmetric and potential. To analyze the pressure changes in the spiral case during load rejection, several pressure-monitoring points are set along the axis line of the spiral case, as shown in Figure 5. The structural responses at the sections and between sections are possibly different due to the structural shape. Pressure-monitoring points (A1, B1, C1, D1) are located in the section, and pressure-monitoring points (A2, B2, C2, D2) are distributed between the sections along the axis of the spiral case.

3.2. Numerical Simulation of the Initial Preloading Clearance

To calculate the initial preloading clearance between the PSSC and concrete, a preloading spiral case structural model is proposed, as shown in Figure 6, including the PSSC (with mandoor), stay ring, test head and steel pedestal. The mandoor, which remains closed during operation, is bonded to the PSSC. The test head is connected to the PSSC using joint contact. The sealing plate placed at the outlet of the stay ring is simplified as a fixed support on its inner ring surfaces, as the main function of this component is to prevent water leakage. Since the steel pedestal comes into contact with already-hardened bottom concrete in engineering practice, the ground of the steel pedestal is set as a fixed support. The structural components of the model share the same mesh setting with the pump-turbine and plant concrete structural model, which will be introduced in detail in the subsequent sections. Structural components not included in this model will be temporarily suppressed. Using the same mesh settings facilitates a one-to-one correspondence between the nodes of the PSSC and the concrete, enhancing the simulation accuracy of the preloading clearance and the contact state between them. The material properties of the structural components are configured according to

Table 3. The material and mechanical properties of the components.

	Steel Structural Component	Concrete
Material	HD610CF	C25W6
$\mathrm{Density}(\mathrm{kg}·$m−3)	7850	2300
Elastic modulus (MPa)	2 × 105	3 × 104
Poisson’s ratio	0.3	0.18
Ultimate Tensile Strength (MPa)	610	3
Ultimate Compressive Strength (MPa)	250	32.7

.

As is known, the PSSC expands outward under the preloading pressure during its construction, with deformation values in the millimeter range. After the surrounding concrete has been poured and hardened, the rejection of preloading pressure causes the spiral case to shrink, creating an initial preloading clearance between it and the concrete. Previous researchers have often simulated the preloading water process by applying a positive value of preloading pressure to the inner wall of the PSSC and then modifying the mesh node coordinates or connecting elements to calculate the preloading clearance. However, this may cause node penetration and stress concentration, which reduce simulation accuracy.

This paper proposes a simulation method that applies uniform negative pressure to the inner wall of the spiral case instead to calculate the initial preloading clearance distribution. This means that the magnitude of the pressure is set to a negative value, and its direction along the normal vector points from the exterior of the PSSC toward its center, with the other settings remaining at the default. The negative pressure should ensure that the shrinkage deformation of the PSSC is equal in absolute value to the expansion caused by the positive preloading pressure, ensuring that the load application method does not affect the clearance distribution. Given the small deformation compared with the size of the spiral case, the direction of deformation has a negligible impact on the structural response. As the inner surfaces of the concrete retain the original shape of the spiral case, a clearance exists between the deformed spiral case and the concrete, which is shrinkage deformation. Due to the minimal adhesion of the concrete to the spiral case during pressure unloading, the clearance is considered the initial preloading clearance and will be applied in the FSI model.

According to the pressure curve provided by the manufacturer, the maximum static head of the spiral case reaches 6.52 MPa. To analyze the impact of different preloading pressures on the structural response, preloading pressures of 3.20 MPa, 3.91 MPa, 4.56 MPa and 6.52 MPa were selected from a range based on engineering experience. The four values correspond to 0.5, 0.6, 0.7 and 1 times the maximum static water head, which are frequently used in engineering, as shown in

Table 4. Four different preloading schemes.

	Preloading Pressure (MPa)	Multiple ($\mathit{P}/{\mathit{P}}_{\mathit{m}\mathit{a}\mathit{x}\mathit{s}\mathit{t}\mathit{a}\mathit{t}\mathit{i}\mathit{c}}$)
1	3.20	0.5
2	3.91	0.6
3	4.56	0.7
4	6.52	1.0

. By applying different preloading pressures to the preloading spiral case model, the corresponding initial preloading clearance distributions can be obtained.

3.3. Pump-Turbine and Plant Concrete Structural Model

Based on the results of the CFD and preloading spiral case models, a pump-turbine and plant concrete structural model was established for the FSI analysis of load rejection. Figure 7 shows that the model is composed of the deformed PSSC (with mandoor), stay ring, headcover, bottom ring and plant concrete, where the deformed PSSC is obtained from the preloading spiral case model.

To balance the calculation time and accuracy, four sets of meshes for the structural model were established for the mesh independence analysis. The structural model is mainly meshed by tetrahedral mesh elements. To achieve better grid quality, the PSSC and stay ring are bolted together as a single structure. It can be seen that an insufficient mesh density can lead to stress singularities, so the mesh is refined at the typical stress concentration locations, including the contact region and the rounded corners of the stay vanes. Also, a two-layer grid is set on the spiral case in the radial direction considering the difference between the internal and external transmission of hydraulic pressure. Figure 8 shows that once the mesh element number reaches 2.996 × 10⁶, the calculation result of normalized stress no longer shows significant changes with further increases in mesh refinement. As a result, it is concluded that this mesh setting can be used for further finite element calculations of the FSI model.

The hydraulic pressure distributions at different time points during load rejection by CFD simulation are mapped onto the structural field. Standard Earth gravity is applied to the structures. The nonlinear contact and bond settings of the structural components and concrete are shown in Figure 9. Based on existing research and engineering experience, the contact between the PSSC and concrete was simulated using the Classical Coulomb Friction Model, with the empirical value of the friction coefficient set at 0.25 [35,36]. As the friction coefficient increases, the tendency for clearance between the PSSC and concrete to close becomes more pronounced. Considering the separation and sliding behaviors between the PSSC and concrete, the interface treatment is set to ‘adjust to touch’. The hydraulic pressure distributions obtained from the CFD model are applied to the respective structural components at the key time points. Each pressure file contains the 3D coordinates and corresponding pressure values for each mesh node on the fluid–structure interaction interface. Due to the restraint from shafting, the connecting face of the headcover is set as a remote displacement that restricts its movement in the XY plane. The external surfaces of the bottom ring are set as bond contact, while those of the headcover are set as free support. Because of the high stiffness of the mountain outside, the outer surfaces of the plant concrete are assigned as fixed support.

4. Results and Discussion

The normalized values of the basic parameters and pressure curves with time obtained by 1D pipeline calculations during load rejection are shown in Figure 10a. The guide vane opening decreases from 20° to a fully closed state over approximately 20 s. Before load rejection, the parameters remain stable and close to their rated values. As the opening begins to decrease linearly, both the hydraulic torque and flow rate drop rapidly, while the rotational speed increases to 1.4 times the rated value. At t = 18.61 s, these basic parameters approach the extreme points. Once the guide vane opening is fully closed, the hydraulic torque remains negative, the flow rate approaches zero, and the rotational speed decreases to the rated value.

Significant pressure fluctuations occur at the spiral case inlet and the draft tube outlet during load rejection (Figure 10b). As the guide vanes start to close, the pressure at the spiral case inlet initially increases and then decreases with periodic fluctuations. Conversely, the pressure at the draft tube outlet exhibits an anti-symmetric trend, which initially decreases and then increases. The pressure at the spiral case inlet fluctuates within an amplitude of 0.8 to 1.2 times the initial value, while the fluctuations at the draft tube outlet are more significant, ranging from 0.6 to 1.4 times the initial value. After the guide vanes reach a fully closed state, both of them gradually stabilize.

According to the parameter variation results of the 1D pipeline calculations, key time points are selected as operational conditions for the 3D flow domain simulation. Six points are chosen and marked with symbols in the figure. Given that the spiral case is the main research object of this paper, the extreme values of pressure at its inlet, as well as the inflection points of other parameters, are chosen to describe the flow characteristics during load rejection; these values occur at t = 1.03 s, 11.03 s, 18.61s, 25.65 s, 29.14 s and 30.54 s.

4.1. Hydraulic Characteristics of the Pump-Turbine

Based on the boundary conditions obtained from the 1D pipeline calculations, a 3D CFD of the unit was conducted to analyze the hydraulic pressure and streamline distribution characteristics.

4.1.1. Pressure Changes at the Monitoring Points

Significant pressure fluctuation occurs within the fluid domain of the spiral case, with variations ranging between 0.975 and 1.01 times the initial value in Figure 11. The pressure-monitoring points located in the front half of the spiral case (A, B and C) experience negligible pressure changes during the initial period (t = 1.03 s and t = 11.03 s). High rotational speed leads to an increase in flow velocity, resulting in violent rotor–stator interaction phenomena and pressure fluctuation at t = 18.61 s. As the guide vanes are fully closed, the flow rate diminishes to nearly zero, and the pressure fluctuations tend to stabilize. Points D1 and D2, which are close to the nose vanes, have high flow velocity due to the small sectional area. As the guide vanes approach closure, flow separation and vortices of the incoming flow from the nose vane result in significant energy dissipation, causing a sharp drop in pressure and extremely unstable fluctuations at these points.

Compared with these two pictures, it can be seen that the trends and magnitudes of pressure fluctuations show differences in various locations. The pressure fluctuations between sections (A2, B2 and C2) share similar characteristics with those in the boundary condition at the inlet with a smaller amplitude. As for the pressure at sections (A1, B1 and C1), there is a slight increase in pressure at t = 18.31 s, but the overall process shows a decreasing trend. This is attributed to the vortices from the incoming flow developing at the sharp corners of the sections, which affects the transmission of pressure fluctuations, leading to differences in the pressure variation both at the sections and between sections.

4.1.2. Pressure Changes in the Flow Passage

The relative pressure distributions of the flow passage during load rejection are shown in Figure 12, where they initially increase and then decrease. From t = 1.03 s to t = 11.03 s, the pressure gradient within the spiral case domain is considerable. Due to the rotor–stator interaction between the vane region and the runner with long and short blades, the pressure distribution within the spiral case decreases progressively from the inner to the outer edge, with high pressure concentrated between sections.

At t = 18.61 s, when the flow rate undergoes a sudden reduction, the pressure distribution within the spiral case domain is at a peak of fluctuation and becomes relatively uniform, while it gradually decreases until t = 29.14 s. An increase in rotational speed leads to more wake cutting in the guide vane domain, resulting in an unstable and non-circularly symmetric pressure distribution in the vaneless space. The pressure fluctuation instability has little impact on the spiral case domain due to the closing of the guide vanes. By t = 30.54 s, as the guide vanes approach a fully closed state, the pressure distribution shows a slight increase and reaches a lower peak of fluctuation during the stabilization process. Pressure concentration at the guide vane tips arises from the separation of high-speed annular flow passing through a narrow inter-vane clearance, resulting in a sharp drop in flow velocity and a local pressure rise as the flow energy is converted into impacts against the edges of the vanes.

4.1.3. Flow Pattern Change in the Flow Passage

The streamline and relative pressure distribution of the cross-section (z = 0) during load rejection are shown in Figure 13. At the initial stage, when t = 1.03 s and t = 11.03 s, the water flows smoothly through the straight-pipe section of the spiral case, with the spiral flow converging on the outer side of the elbow section. High-speed streamlines concentrate near the wall. The flow aligns well with the logarithmic spiral lines of the spiral case domain, which exhibits an axially symmetric flow characteristic without vortices as it enters the vane region.

The streamlines from the guide vane to the runner outlet encounter extremely unstable turbulence after the guide vanes start closing, which are no longer axially symmetric. The imbalance of the guide vane opening and flow direction at t = 18.61 s disturbs the flow from the spiral case to the vane space. A large number of swirl eddies with obvious flow separation and strong circulation on the suction side of the guide vane extend to the spiral case space. Turbulence appears on the leading edge of the nose vane, which is caused by the reverse refluxes of the inter-vane flow with the incoming flow, making the flow violently unstable with the pressure near the nose vane (points D1 and D2).

As the guide vanes continue to close (t = 25.65 s), a turbulent water ring appears between the vanes and between the guide vanes and the runner. Blocking from the water ring in the vaneless region prevents most of the flow from passing through the runner, resulting in the blade passage vortices within the runner. At the same time, the surrounding swirling flow encircling the runner domain obstructs the upstream flow from the guide vane domain. The combined effects of the water ring’s blockage and vortex core result in a highly unstable turbulent flow. By t = 29.14 s, the vortices near the nose vane of the spiral case have disappeared, and turbulence chiefly concentrates in the elbow section, with an overall reduction in pressure. The streamlines within the spiral case domain become relatively smooth again until the guide vanes are nearly fully closed (t = 30.54 s), but a significant amount of recirculation and vortices remain in the runner domain.

4.2. Effect of Preloading Clearance on Flow-Induced Structural Characteristic

To perform the FSI analysis, pressure distributions exerted on the structures at key time points during load rejection were exported from the 3D CFD results above. The node locations on the PSSC structure that map to the pressure-monitoring points of the fluid domain (A1, A2, B1, B2, C1, C2, D1 and D2) are designated as the structural monitoring points and are labeled with the same names. Also, FSI models under four preloading pressures were established to discuss the impact of preloading clearance on the flow-induced structural response characteristics. The deformations and stresses are normalized with reference to the maximum value of the PSSC when the preloading pressure is 6.52 MPa.

4.2.1. Deformation Distribution

The total deformation distribution of the PSSC decreases along the axial direction during load rejection, as shown in Figure 14. The huge hydraulic pressure acting on the inner surface of the PSSC results in high deformation concentrations in the front half of the PSSC with a large cross-sectional area. Under different preloading pressures, the maximum deformations are consistently located at the mandoor near the PSSC inlet, where there is no support provided by surrounding concrete in this region. The higher the preloading pressure, the greater the initial preloading clearance that forms between the PSSC and the concrete. By applying the same internal hydraulic pressure, a larger initial clearance leads to greater deformation of the PSSC and less constraint from the surrounding concrete. The increase in deformation near the tongue of the PSSC is because of the high hydraulic pressure caused by reverse refluxes near the nose vane in the flow domain calculations. The maximum deformation and the overall deformation distribution of the PSSC increase with higher preloading pressure.

In addition, the constraints from the stay ring and steel pedestal lead to local pressure concentration at the upper and lower inner edges of the PSSC. Due to the circular cross-section of the PSSC and the relatively uniform distribution of internal hydraulic pressure in the later stages of load rejection, the deformation distribution near the waistline is symmetrical about the horizontal plane. The intersection line between the central plane and the outer wall of the PSSC is referred to as the waistline of the PSSC. Given that the stay ring has a closed, symmetrical structure and the internal water pressure is relatively low, the flow-induced deformation remains at a low level.

It is concluded that the total deformation of the PSSC is determined by the combined effects of the initial preloading clearance, internal hydraulic pressure and structural characteristics. The deformation distribution of the PSSC is asymmetrical along the axial direction but shows a symmetrical characteristic near the waistline.

4.2.2. Stress Distribution

The stress distribution in the rear half of the PSSC shows similar characteristics under different preloading pressures, and no significant stress concentration appears near the tongue, as shown in Figure 15. The maximum stress value of the PSSC is around 150 MPa when the preloading pressure is 6.52 MPa, which indicates that the design of the PSSC under these four preloading pressures is safe during load rejection. Due to the removal of the test head near the inlet during operation, the boundary conditions of the PSSC have changed, which leads to an unbalanced force under the huge hydraulic pressure. This can cause the PSSC to deviate horizontally, affecting the contact state and clearance distribution between the PSSC and the concrete [20]. Compared to the normalized stress results, the PSSC with higher preloading pressure exhibits greater stress in the straight-pipe section. The maximum von Mises stress occurs on the inner edge of the PSSC near its tongue. Local stress concentrations are located near the sections, but the stress values are lower than those between the sections. The stress in the PSSC shows a radial trend of decreasing and then increasing, with high stress concentration near the inner edge and waistline.

Figure 16 shows the flow-induced stress comparison of the normalized values relative to the average of the structural monitoring points on the outer wall of the PSSC during load rejection. It can be seen that when the preloading pressure (3.20 MPa, 3.91 MPa, 4.56 MPa) is below the maximum hydraulic pressure, the stress variation at the section and between the sections are consistent at different time points and show similar characteristics to the pressure at monitoring points (A2, B2 and C2) in the flow domain. The von Mises stress of PSSC reaches its respective maximum at t = 18.61 s, as the internal hydraulic pressure increases sharply due to the closing of the guide vanes. The stress values at points (C1, D1) in the rear half of the PSSC are relatively higher, whereas the stress at other points in the front half adjusts with changes in preloading pressure. However, for a preloading pressure of 6.52 MPa, the stress variation at the section and between sections remains consistent, while the peak value shifts to a later time. At t = 18.61 s and t = 25.65 s, there are big differences from the stress variation at lower preloading pressures, with stress slightly decreasing at t = 18.61 s and reaching a respective maximum at t = 25.65 s. This pattern strongly correlates with the pressure changes at pressure-monitoring points (D1, D2) in the flow domain.

Structural force analysis indicates that both the internal water pressure and the support from the concrete affect the stress distribution of the PSSC. It is concluded that when the preloading pressure is significantly greater than the internal hydraulic pressure, the large preloading clearance has a substantial impact on the mechanism characteristics of the PSSC during load rejection.

4.2.3. Contact Status

To further investigate the reasons for differences in stress variation under different preloading pressures, the impact of preloading clearance on the structural characteristics of the PSSC during load rejection will be analyzed from the perspective of the contact status. Since the normalized stress curves in Figure 15 show significant differences at t = 18.61 s and t = 25.65 s, the contact status between the PSSC and concrete under different preloading pressures at these two moments is shown in Figure 17 and Figure 18. The regions colored orange and red indicate contact between the two, while yellow, blue and pink represent no contact.

At t = 18.61 s, the PSSC shows uniform contact with the concrete by sliding and sticking from the inlet to the tongue under lower preloading pressures (3.20 MPa, 3.91 MPa, and 4.56 MPa), with a symmetrical contact state of the waistline about the horizontal plane. This indicates that the PSSC is fully in contact with the concrete under internal hydraulic pressure with a combined load-bearing state. However, for a preloading pressure of 6.52 MPa, it is obvious that only the inner edge and top of the PSSC are in contact, and the PSSC has an asymmetrical clearance distribution in the axial and radial directions. Due to the cross-sectional shape and unbalanced forces, the preloading clearance remains in the straight-pipe section and the waistline. The top and bottom of the PSSC are the first to come into contact with the surrounding concrete, causing radially asymmetric contact and resulting in high stress concentration at the inner edge. As a result, the pressure at the monitoring points near the waistline, which is not the primary load-bearing region, does not increase with hydraulic pressure at that time.

As the guide vanes continue to close at t = 25.65 s, the contact area with the concrete decreases as the hydraulic pressure begins to decrease under preloading pressures of 3.20 MPa, 3.91 MPa and 4.56 MPa. The regions that first separate from the concrete are near the waistline and the inlet due to greater deformation compared to their surroundings (Figure 13). In contrast, the PSSC with a preloading pressure of 6.52 MPa shows an increase in contact area, with only the preloading clearance remaining at each section. The delayed contact phenomenon is primarily caused by the asymmetric shape of the PSSC and unbalanced hydraulic forces. At that time, the PSSC near the waistline transitions to a combined load-bearing state with the concrete, resulting in a more uniform radial stress distribution of the overall structure. Thus, the pressure at the monitoring points reaches its maximum and shows a delayed peak during load rejection.

5. Conclusions

In this paper, the hydraulic characteristics and flow-induced structural responses of the PSSC and concrete under different preloading pressures during load rejection are studied based on FSI theory and the nonlinear contact model. The key conclusions drawn are as follows.

During load rejection, the streamlines are no longer axially symmetric due to the imbalance of the guide vane opening and flow direction, especially at the outlet of the spiral case, where swirl eddies with flow separation and strong circulation occur. The pressure distribution within the spiral case domain becomes relatively uniform after a sudden flow rate reduction at t = 18.61 s. The trends and magnitudes of the pressure variation show differences at the section and between sections, with all monitoring points reaching peak pressure simultaneously at t = 18.61 s.

The deformation of the PSSC shows an asymmetrical distribution along the axial direction but a symmetrical characteristic near the waistline due to the combined effects of the initial preloading clearance, internal hydraulic pressure and structural characteristics. Higher preloading pressures increase the preloading clearance, leading to greater deformation of the PSSC, and the maximum deformations consistently appear at the mandoor, where there is a lack of support from surrounding concrete.

The PSSC with higher preloading pressure shows greater stress in the front half. The maximum flow-induced stresses are located at the inner edge near the tongue. The design of the PSSC under the four selected preloading pressures is confirmed to be safe during load rejection. Under lower preloading pressures (3.20 MPa, 3.91 MPa and 4.56 MPa), the maximum stress is strongly correlated with pressure at the monitoring points (A2, B2 and C2) in the flow domain, and the peak occurs at t = 18.61 s. As the preloading pressure reaches the maximum static head (6.52 MPa), there is a great difference such that the peak value of stress shifts to a later time, t = 25.65 s, with a higher amplitude of about 0.85–1.42 times the normalized value.

The preloading clearance has a great impact on the distribution of and variation in the structural characteristics of the PSSC during load rejection. At t = 18.61 s, the PSSC is fully in contact with the concrete under lower preloading pressure, while only the inner edge and top region are in contact under 6.52 MPa. At t = 25.65 s, the contact area declines with the decreasing hydraulic pressure under lower preloading pressure, whereas the PSSC under 6.52 MPa nearly achieves full-contact status instead. Due to the asymmetrical shape and unbalanced hydraulic force, the preloading clearance distribution is asymmetrical in both the axial and radial directions, resulting in a delayed contact phenomenon and a delayed peak of stress under 6.52 MPa. The preloading pressure directly affects the extent and timing of the structural contact, which, in turn, influences stress changes.

The results of this paper can provide theoretical references for revealing the impact of preloading pressure on symmetrical and asymmetrical flow-induced structural behaviors of the PSSC. The numerical methods also offer scientific guidance for the optimal selection of preloading pressure and the structural design for the safe and stable operation of pump-turbine units.