# An Approach to Evaluate the Lifetime of a High Head Francis Runner

Eduard Doujak, Markus Eichhorn

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Eduard Doujak, Markus Eichhorn. An Approach to Evaluate the Lifetime of a High Head Francis Runner. 16th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC 2016), Apr 2016, Honolulu, United States. hal-01516837

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# An Approach to Evaluate the Lifetime of a High Head Francis Runner

Eduard Doujak¹*, Markus Eichhorn¹

<figure><img src="image_2.png" /></figure>

ISROMAC 2016

International Symposium on Transport

Phenomena and Dynamics of

Rotating Machinery

Hawaii, Honolulu

April 10-15, 2016

**Abstract**

Nowadays, the electricity market is changing rapidly and the requirements for hydropower plant operators vary between base load electricity production and ancillary services with a high share of transient operational points. The institute for Energy Systems and Thermodynamics, together with partners from the industry, initiated a research project to investigate the effects of these new operational modes on the components of a hydraulic machine. Especially the runner is of high importance due to the long service time and unfavorable flow phenomena when operated at off design points. A method to evaluate the lifetime of a high head Francis runner should be developed to investigate the existing runner and to serve later on a possible procedure for the design stage. Therefore, prototype measurements with applied strain gauges have been carried out for validation purposes of the developed method. Unsteady Numerical flow simulations using the open-source software OpenFOAM and static as well as dynamic Finite Element calculations with Code Aster are performed to investigate the lifetime of the observed Francis runner. The research project is still ongoing but first results regarding the procedure and the methods will be shown in this paper.

**Keywords**

Francis Runner — Fluid-structure interaction — Strain gauge measurements — Lifetime investigation

¹Institute for Energy Systems and Thermodynamics, Research Group Fluid-Flow Machinery, TU Wien, Vienna, Austria

*Corresponding author: eduard.doujak@tuwien.ac.at

## INTRODUCTION

The history of the European electricity market shows over the past twenty years a more and more changing behavior. Drivers for this changing situation are manifold. At the beginning opening and liberalization of the market caused a lot of uncertainty and an investment stop into new hardware. This change had more impact on the economical than on the technical side. One of the next important key points in the development was the implementation of the Kyoto protocol by 2005 aiming to reduce CO₂ emissions around the world. At around that time all around Europe new incentives to promote renewable energy sources like wind or solar have been established. Many countries followed the fed-in tariff system to encourage investors to build new electricity supplier by using renewable energy sources. Additionally to these increasing installations of small power production units, Germany decided to shut down their nuclear power stations within the next twenty years. These activities had a big impact on technical as well as on economical aspects. Side effect of this electricity market change was paradoxically an increased investment in Hydro power as well, because load dispatchers needed storage and grid control capabilities. And the consequences of these market changes are even today remarkable.

Today we are dealing with faster response time at the electro-mechanical units in our Hydro power plants as well as some kind of standby operation, which means that the units are running for hours at very low load conditions and far away from the design point. Some resulting effects have already been published [1], showing the general effect of

increased fluctuating sources of energy. The requirements of Hydro power plants are nowadays more and more to deliver regulating power and ancillary services to the electrical grid, which results in higher dynamic loads and stresses on turbine components. These dynamic stresses can be of the steady-state or transient type, deterministic or stochastic.

The expected flexibility of turbine operation is characterized by more unit start-stop cycles, part load operation for power regulation and less steady state operation at the nominal point. All different modes have a strong influence on the flow conditions within the machine itself and cause a known variation of flow phenomena. Francis runners at part load operation are for example dominated by rotor-stator interaction (RSI) and by draft tube instabilities (DTI) having some harmonic excitation effects. Whereas vortex shedding effects (VSE) at low load conditions show a more stochastic pattern and non harmonic behavior. At start-stop cycles the machine unit runs through all these phenomena with additional effects of speed-no-load (SNL) condition. Sometimes, the hydraulic unit also runs in synchronous condenser operational mode in air, which means that the synchronized unit has to be dewatered with compressed air causing high stochastic stresses.

In the past, computational resources haven not been high enough to investigate all these transient phenomena on the total hydraulic unit. Singular parts have been numerically computed by means of Computational Fluid Dynamics (CFD). The numerical results have been evaluated with measurements but there was still a lack of uncertainties at the boundaries.
Most of the time the best information about the behavior of Francis runners have been gained by interpreting prototype measurements like published by [2]. Slowly, the computational resources raised to a range to calculate dynamic stresses on single parts of the hydraulic unit, mostly the Francis runner as shown in [3]. At that point, fluid-structure interaction (FSI) at the total hydraulic unit was big task. Major efforts have been made over the past five years to close this gap and to predict flow phenomena and pressure pulsations as basic for dynamic stress calculations by means of CFD. The way of investigating the impact of flexible operation on Francis runners was published at [4] but still relies on measurement basis. As prototype measurements are very expensive the goal of today is to provide basic information for a lifetime investigation of a runner by numerical simulations.

Lifetime investigation of hydraulic parts is not only dominated by calculating static mean stresses but is more and more a task of fatigue analysis. Especially dynamic loads seek for this kind of analysis as shares of low-cycle fatigue (LCF) and high-cycle fatigue (HCF) have to be managed. Some selected publications about this topic show the state of the art and problems in defining the life expectancy of a Francis runner. In 2005 a fatigue design of hydraulic turbine runners has been published by [5]. Five years later [6] determined the impact of start-up scheme on Francis runner life expectancy. Recent publications [7] or [8] show the importance of this approach for turbine runners.

The following paper gives an overview about a method to investigate the lifetime of a high head Francis runner by discussing all necessary steps and numerical tools on the one hand and processing of prototype measurements on the other hand.

# 1. METHOD

The following method was developed within a large research project to investigate a high head Francis turbine with a specific speed of $n_q \approx 24 \text{ min}^{-1}$ under different operating conditions. Goal of this study is the evaluation of the lifetime of the turbine runner using numerical calculations as well as site measurements. As mentioned above, the approach is nearly similar to already known procedures unless the operating point is far away from the nominal point and more or less at the region of irregular flow pattern causing stochastic dynamic stresses. Within the research project, prototype measurements using strain gauges applied to the runner have been performed to validate the numerical results.

Figure 1 shows the procedure for the runner lifetime investigation, which consists of the following numerical tasks:

* Transient CFD calculations

* Static Finite Element (FE) calculations

* Modal analysis

* Harmonic response analysis

At the measurement path the following tasks have to be considered:

* Prototype site measurements

* Stress extrapolation using FE-methods

* Rainflow counting analysis

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Numerical and measurement procedure for Francis runner lifetime investigation</figcaption></figure>

As University resources are mostly limited, one additional point of investigation is the usability of open source software. In this case OpenFOAM 2.3 was selected for the transient CFD calculations and Code Aster 11.7 for the subsequent tasks of the structural Finite Element calculations.

# 2. PROTOTYPE SITE MEASUREMENTS

## 2.1 Measurement Setup

To evaluate the static and dynamic stresses appearing on the Francis runner in different operating conditions, prototype site measurements have been performed. Therefore, eight strain gauges have been attached to the suction side (SS) and pressure side (PS) of the trailing edge of one turbine blade (see figure 2 left). Due to manufacturing tolerances, geometric differences appeared among the 15 runner blades. To acquire the maximum stresses, the blade with the thinnest trailing edge has been chosen for the strain gauge measurements. The exact geometry has been measured using a 3D measurement arm to increase the accuracy of the results for the numerical calculations. Thereby also the positions of the strain gauges have been included in the Computer Aided Design (CAD) model.
The data of the strain gauges during turbine operation has been recorded using a data logger, which has been mounted to the shaft inside the draft tube of the machine (see figure 2 right).

<figure><img src="image_4.png" /><figcaption>Figure 2. Positions of the strain gauges applied to one runner blade (left) and data logger (right)</figcaption></figure>

Additionally, to validate the CFD simulations, the operating parameters such as head, output power, guide vane opening, pressure in the spiral casing and draft tube, have been recorded during the measurements. The vibration of the shaft as well as the acceleration of the shaft bearing in radial direction have been recorded to further investigate the overall behavior of the machine. The data of the strain gauges have been measured using a sampling rate of 1 kHz, the other parameters with a sampling rate of 2.4 kHz.

## 2.2 Results

The measurements were performed in different operating points from the start-up in steps of 10 MW respectively 20 MW to the maximum output power of 180 MW and back again. At the end of the normal turbine operation, the water was drained from the machine to measure the strains on the runner due to the centrifugal forces without any hydraulic pressure (Rotation). These conditions are used for an evaluation of the Finite Element calculations. The results of the strain gauge measurements on the suction side are displayed in figure 3. The strains for SS1 are nearly constant with varying operating conditions, whereas the strains for SS2 and SS3 are changing from compression in low and part load to tension in high and full load conditions. The results for the strain gauge SS4, which was located close to the radius between the blade and the hub of the runner, show an opposing behavior.

The Fourier transformation of the time signal for the strain gauge SS2 reveals a dynamic excitation of the runner at the first and second harmonic of the blade passing frequency $f_{BP}$ in each operating point (see figure 4). This excitation is induced by the rotor-stator interaction (RSI) between the runner (RN) and the guide vane (GV) domain and is the main impact to runner fatigue for high head Francis turbines (see [9] and [10]). The blade passing frequency is described by

$$f_{BP} = Z_g \cdot f_0 \quad (1)$$

with the number of guide vanes $Z_g = 20$ and the rotating frequency $f_0$ of the machine. In low load conditions the strains

<figure><img src="image_5.png" /><figcaption>Figure 3. Results for the strain gauges applied to the suction side of one runner blade</figcaption></figure>

and therefore the pressure pulsations show a more stochastic behavior, although the RSI remains the dominant dynamic influence. At the operating point with an output power of one third of the maximum power, a high peak appears in the low frequency regime which could be an indicator for different flow phenomena like draft tube vortex or trailing edge vortex shedding. The amplitudes at the rest of the signal are also even higher. This operating area will therefore be of special interest for future simulations but for the current presentation of the method of runner lifetime investigation the RSI will be discussed further.

<figure><img src="image_6.png" /><figcaption>Figure 4. Frequency spectrum of strain gauge SS2 in different operating conditions</figcaption></figure>

The next steps towards the runner lifetime analysis would be the stress extrapolation for the strain gauge measurement points to the location with the highest stresses and the rainflow counting analysis. These two points will not be discussed further here due to limited paper length and as they are general steps of a fatigue analysis and not specially associated with hydraulic runner fatigue assessment. The complementary parts of runner lifetime analysis will be discussed at chapter 4.
# 3. NUMERICAL INVESTIGATIONS

## 3.1 CFD calculations

### 3.1.1 Numerical model

To calculate the fluctuating pressure field induced by the rotor stator interaction, unsteady numerical flow simulations using the Reynolds averaged Navier-Stokes (URANS) equations are performed with OpenFOAM 2.3. This is a suitable approach for steady operating conditions with a slight amount of turbulence and vorticity (see [11]). In part load and especially no load conditions, stochastic flow phenomena like vortex shedding or draft tube instabilities are increasing (as shown in figure 4) and turbulence modeling has a large influence on the dynamic behavior (see [12] or [13]). Therefore, the application of the SAS turbulence model with OpenFOAM will be investigated in future publications for the CFD calculations in part load conditions.

The model for the CFD simulation consists of the spiral casing, 10 stay vanes, 20 guide vanes, the runner with 15 blades and the draft tube (see figure 5).

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Computational domains for CFD simulations (left: entire machine; right: guide vanes and runner)</figcaption></figure>

<table><caption>**Table 1.** Setup for the CFD calculations</caption><thead><tr><td>Parameters</td><td>Description</td></tr></thead><tbody><tr><td>Software</td><td>OpenFOAM 2.3</td></tr><tr><td>Analysis type</td><td>Transient rotor-stator</td></tr><tr><td>Time step</td><td>0.36 degrees (1.0e-4 s)</td></tr><tr><td>Simulation time</td><td>10 runner rotations (1.0 s)</td></tr><tr><td>Inlet BC</td><td>Mass flow rate</td></tr><tr><td>Outlet BC</td><td>Average static pressure</td></tr><tr><td>Reference pressure</td><td>0 kPa</td></tr><tr><td>Interfaces</td><td>Arbitrary mesh interface</td></tr><tr><td>Turbulence model</td><td>k-ω SST</td></tr><tr><td>Advection scheme</td><td>Second order</td></tr><tr><td>Transient scheme</td><td>Second order backward Euler</td></tr><tr><td>Max. Courant number</td><td>≤ 15</td></tr></tbody></table>

The calculations are performed in different operating points from the maximum output power of 180 MW to low load conditions, according to the prototype site measurements described in section 2. The parameters of the hydraulic

machine (e.g. head H, runner torque T, efficiency η) are compared to evaluate the accuracy of the used setup. The simulations for full and high load conditions are done using the two-equation turbulence model k-ω SST. The time step for the CFD simulations at the different operating conditions is choosen to 0.36 degrees, which yields a maximum Courant number of $Co_{max} \le 15$. To gain enough data for the Fourier transformation of the unsteady pressure field, at least ten runner rotations are simulated. The setup and the boundary conditions for the numerical flow simulations are summarized in table 1.

### 3.1.2 Discretization

The computational model is spatially discretized in a multi-block hexahedral grid using Ansys ICEM 15.0. The particular domains are meshed separately and the interfaces are connected using the arbitrary mesh interface (AMI) implemented in OpenFOAM 2.3 (see figure 6). The number of nodes and the quality of the mesh with a total number of about 5.9 million nodes are displayed in table 2.

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Discretization and interfaces between CFD domains</figcaption></figure>

### 3.1.3 Grid independence study

In order to estimate the uncertainties due to the discretization of the CFD model, a grid independence study is performed according to the approach of [14]. Therefore, the fluid domains of the Francis turbine, including the spiral casing, the stay vanes, the wicket gate, the runner and the draft tube are considered. The mesh is refined in three different stages (coarse - G3, medium - G2, fine - G1) regarding the grid size h, which is calculated by

$$h = \left[ \frac{1}{N} \sum_{i=1}^{N} (\Delta V_i) \right]^{1/3} \quad (2)$$

with the total number of cells N and the volume of the ith cell $\Delta V_i$. The refinement factor $r_{21} = h_{medium}/h_{fine}$ for the fine grid respectively $r_{32} = h_{coarse}/h_{medium}$ for the medium
<table><caption>**Table 2.** Parameters for the discretization</caption><thead><tr><th>Parameters</th><th>Spiral casing</th><th>Stay vanes</th><th>Guide vanes</th><th>Runner</th><th>Draft tube</th><th>Full mesh</th></tr></thead><tbody><tr><td>Number of cells N</td><td>599,556</td><td>723,444</td><td>1,688,000</td><td>2,568,360</td><td>298,128</td><td>5,877,488</td></tr><tr><td>Min. determinant</td><td>0.28</td><td>0.36</td><td>0.69</td><td>0.21</td><td>0.33</td><td>-</td></tr><tr><td>Min. angle</td><td>11</td><td>16.1</td><td>22.5</td><td>18.6</td><td>18.9</td><td>-</td></tr><tr><td>Min. aspect ratio</td><td>0.001</td><td>0.001</td><td>0.002</td><td>0.002</td><td>0.015</td><td>-</td></tr><tr><td>y<sup>+</sup><sub>mean</sub></td><td>95</td><td>50</td><td>65</td><td>70</td><td>60</td><td>-</td></tr></tbody></table>

grid is therefore considered to be greater than 1.3. The mesh refinement of the structured grid is done consistent in all three directions with a constant resolution of the wall boundary layer according to $30 \le y^+_{mean} \le 100$. The convergence of the grid independence study is verified by computing different parameters $\Phi$ of the Francis turbine at the operating point with maximum output power for all three meshes. These values are the head H, the runner torque T and the efficiency $\eta$. Therefore, the SIMPLE-algorithm of OpenFOAM, which calculates the steady-state pressure-velocity coupling, is used. The extrapolated values for the head, the torque and the efficiency for a mesh with grid size $h_n = 0$ are calculated by

$$ \Phi_{ext}^{21} = \frac{r_{21}^p \Phi_1 - \Phi_2}{r_{21}^p - 1} \qquad (3) $$

with the order of accuracy p computed according to [14]. The numerical uncertainty is considered by calculating the relative error $e_a^{21}$ for the fine grid by

$$ e_a^{21} = \left| \frac{\Phi_1 - \Phi_2}{\Phi_1} \right| \qquad (4) $$

and the grid convergence index GCI by

$$ GCI_{21} = \frac{1.25 \cdot e_a^{21}}{r_{21}^p - 1} \qquad (5) $$

These are similarly calculated for the medium and the extrapolated grid. The description and results for the grid independence study are displayed in table 3 and figure 7.

The results of the grid independence study reveal a monotonous convergence for all three parameters with a decreasing grid size $h_i$ and small deviations for the GCI from the asymptotic value. Considering the computational costs and the accuracy for the unsteady CFD simulations, the medium grid with a normalised size of $h_{2,n} = 1.32$ (as described in 3.1.2 and table 2) is used for further investigations.

### 3.1.4 CFD Results

The following results are performed for the operating point with a maximum output power of 180 MW. The deviations

<table><caption>**Table 3.** Mesh refinement and discretization uncertainties</caption><thead><tr><th>Parameters</th><th>Head H</th><th>Torque T</th><th>Efficiency η</th></tr></thead><tbody><tr><td>r<sub>21</sub></td><td>1.32</td><td>1.32</td><td>1.32</td></tr><tr><td>r<sub>32</sub></td><td>1.34</td><td>1.34</td><td>1.34</td></tr><tr><td>p</td><td>2.56</td><td>3.62</td><td>13.96</td></tr><tr><td>e<sup>21</sup><sub>a</sub> [%]</td><td>1.26</td><td>1.28</td><td>0.02</td></tr><tr><td>e<sup>21</sup><sub>ext</sub> [%]</td><td>1.19</td><td>0.72</td><td>0.01</td></tr><tr><td>GCI<sup>21</sup> [%]</td><td>1.50</td><td>0.91</td><td>0.01</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Grid convergence</figcaption></figure>

of the mean values for the head H, the runner torque T and the efficiency $\eta$ calculated with CFD from the measured values are less than 1% and are therefore in the range of the measurement uncertainties. Figure 8 shows the unsteady pressure distribution for the converged calculation.

The evaluation of the pressure signal of the monitor point in the rotating domain between the runner and guide vane blades reveals the blade passing frequency $f_{BP}$ as the dominant excitation (see figure 9). This corresponds to the frequency analysis of the strain gauge measurements, although the sensors therefore have been attached to the trailing edge of the runner. Thus, the impact of the rotor-stator interaction on the dynamic pressure field is acquired sufficiently with the used CFD methods. Two further amplitude peaks are revealed at frequency ratios of $f/f_0 = 10$ and $f/f_0 = 30$ which could be indicated by the pressure wakes of the stay vanes with a number of 10 blades. The time depending fluctuation of the pressure distribution is further transformed into the frequency
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Unsteady pressure distribution at maximum output power</figcaption></figure>

domain using the Fast Fourier method (see 3.2.3).

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Frequency spectrum of the pressure signal at monitor point</figcaption></figure>

## 3.2 Structural analysis

### 3.2.1 Steady calculations

To evaluate the overall load and the points with the highest stresses on the Francis runner, steady Finite Element analysis are performed using the open-source program Code Aster. Therefore, a cyclic sector consisting of one runner blade and the appropriate hub and shroud regions is considered in order to reduce simulation time and costs. The unstructured tetrahedral mesh with quadratic elements consists of about 200,000 nodes (see figure 10). The results from the unsteady CFD simulations are averaged over time and the pressure distribution is then applied to the FE model by an interpolation between the meshes of the fluid and the structural domain. Cyclic symmetry boundary conditions are used for the connection between the sectors of the runner, which is fixed at the bolt circle. The centrifugal forces induced by the rotation and the influence of gravitation are also considered. The pressure distributions in the side chambers of the runner are calculated and applied by an analytically approach using the pressure values at the inlet

and outlet of the runner or by an axial thrust calculation. The material for the Francis runner is a steel of type X5CrNi13-4 with an E-modulus of $E = 216 \text{ kN/mm}^2$, a Poisson ratio of $\nu = 0.30$ and a density of $\rho_R = 7.700 \text{ kg/m}^3$.

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Mesh and boundary conditions for steady structural analysis</figcaption></figure>

Figure 11 shows the Mises stress results of the static FE analysis for the operating point at 180 MW output power. The hot spot with the maximum load is at the trailing edge close to the hub of the runner. The method for the static structural investigation is validated in future publications by comparing the mean stresses at the positions of the strain gauges applied on the prototype Francis runner.

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Mises stress at maximum power output</figcaption></figure>

### 3.2.2 Modal analysis

At the whole procedure of runner lifetime prediction the question about the natural frequencies and mode shapes of the structure is evident. The following chapter will describe in brief the basics and challenges by performing these calculations.

It is common sense that the dynamic structure could be described by the following equation:

$$[M_S]\{\ddot{u}\} + [C_S]\{\dot{u}\} + [K_S]\{u\} = \{F_S\} \quad (6)$$

where $[M_S]$ is the structural mass matrix, $[C_S]$ the structural damping matrix, $[K_S]$ the structural stiffness matrix, $\{F_S\}$ the applied load vector and $\{u\}$ the nodal displacement vector.
As Francis runners operate in surrounding water the fluid- structure interaction has to be taken into account. The mathe- matical description of this problem including a comparison of some results between calculation and measurements is well described by [7]. It gets clear, that the natural frequencies differ between investigations in air and water, which were also described by [15] and [16]. Still unsolved is the question about the influence of the attached fluid part to the numerical model. Francis runners will never run in this big amount of surrounded water like shown at several publications. This question is part of our research project and will be published later on.

As a first step, the natural frequencies and mode shapes are investigated following the description mentioned above by using the model shown in figure 12. For the runner itself a mesh with about 1.2 million nodes consisting of tetrahedral quadratic solid elements is used. The added mass of the water around the runner for the coupled investigation is simulated by using acoustic fluid elements sharing the same node group at the interface with the structural elements. As Code Aster 11.7 provides no cyclic symmetry option regarding acoustic elements for sectoral investigations the total runner has to be modeled. The water properties under environmental temperature and atmospheric pressure are: Density ρ<sub>W</sub> = 1000 kg/m<sup>3</sup> and speed of sound in water c = 1483 m/s. The complete runner model is fixed at the bolt circle, whereas the outside of the surrounding water is fixed by a wall definition.

<figure><img src="image_14.png" /><figcaption>Figure 12. Boundary conditions for modal analysis</figcaption></figure>

The results of the numerical simulation for the coupled system are shown in figure 13 with a mode shape of two nodal diameters (ND) at *f*/*f*<sub>0</sub> = 19.8 and a mode shape with three ND at *f*/*f*<sub>0</sub> = 37.6 for the investigated runner. Simulations have been performed up to *f*/*f*<sub>0</sub> = 40, which is twice the frequency of the rotor-stator interaction (compare figure 9) and therefore sufficient to evaluate, if there are natural modes in the range of the RSI frequency. Although the mode shape with two nodal diameters is very close to the blade passing frequency at *f*<sub>BP</sub> = 20.0, the runner can only be excited by the rotor-stator interaction, if the nodal diameter of the pressure field *k* matches the natural mode shape (see equation 7).

<figure><img src="image_15.png" /><figcaption>**Figure 13.** Mode with ND 2 *f*/*f*<sub>0</sub> = 19.8 (left) and ND 3 at *f*/*f*<sub>0</sub> = 37.6 (right)</figcaption></figure>

### 3.2.3 Harmonic response analysis

The dynamic pressure field excited by the rotor-stator interac- tion can be described by

$$m \cdot Z_r - n \cdot Z_g = k \quad (7)$$

with the number of runner blades *Z*<sub>*r*</sub> and the associated harmonic index *m*, the number of guide vanes *Z*<sub>*g*</sub> with its harmonic index *n* and the corresponding nodal diameter of the pressure field *k* (see [11]). The temporal pressure field observed in the rotor frame for any point *M* in a meridian section and any azimuth *θ* can be described in the Fourier domain by

$$p_{m,n}(M, \theta, t) = p_{m,n}(M, \theta)e^{-2i\pi \cdot n \cdot Z_g \cdot f_0 \cdot t} \quad (8)$$

The Fourier transformation at the blade passing frequency *f*<sub>BP</sub> of all discrete points in the runner domain is sufficient to describe the dynamic behavior due to the rotor stator interaction.

The critical nodal diameter of the pressure field for the investigated Francis turbine is *k* = 5 as described in table 4.

<table><caption>**Table 4.** Nodal diameter *k* of the dynamic pressure field</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="3">Harmonic index n of the guide vanes</th></tr><tr><th>0</th><th>1</th><th>2</th></tr></thead><tbody><tr><th rowspan="3">Harmonic index m of the runner</th><th>0</th><td>0</td><td>-20</td><td>-40</td></tr><tr><th>1</th><td>15</td><td>-5</td><td>-25</td></tr><tr><th>2</th><td>30</td><td>10</td><td>-10</td></tr></tbody></table>

The Fourier transformation of the pressure field obtained by the CFD simulations is displayed in figure 14. The nodal diameter *k* = 5 is clearly visible. The results for the different operating conditions are then applied to the entire runner structure for the harmonic response analysis using Code Aster, to evaluate the dynamic loads and further the lifetime of the turbine. The boundary conditions for the harmonic response analysis are the same as for the modal analysis described in the previous section.
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Phase (left) and amplitude (right) of the pressure field in the Fourier domain</figcaption></figure>

## 4. RUNNER FATIGUE ANALYSIS AND LIFETIME EXPECTATIONS

The last section of this paper is devoted to the big part of runner fatigue analysis. The aim today is the determination of an exact lifetime of a hydraulic runner at the design stage. At the present the hydropower community deals with either numerical or experimental investigations. Some single flow phenomena, like for example RSI behavior (see [10], [9], [17]), machine startup (see [6]) or off design operation (see [12]), are already researched and the impact onto the structure is extrapolated to the runner life time. Common to all of these investigations is the numerical background and starting point, whereas structural belongings, like for example pre-stressed material or residual stresses at the runner itself, are again investigated separately, see [18]. To include all phenomena into one bigger method is the challenge of today and the future. An interdisciplinary approach is therefore of high importance and suppliers, operators and universities are invited to work together on this exciting topic. And one of the biggest question at this stage is still the portability to other Francis runners as each prototype layout of a turbine is more or less unique.

### 4.1 Operating conditions - Load spectrum

An important role to estimate the lifetime of a runner is the load spectrum as it should give some kind of real usage over a given time history of the component. Questions about the operational conditions, operation duration at off design point, start-up and shut down of the machine, load rejection and so on have to be taken into account. Each of these questions have to be investigated afterward in terms of load samples either by numerical investigation or better by measurement setup. Separating this huge amount of data into single segments will deliver a data pool for the operating profile. Subsequently, the combination of this data pool with the sequence of operational conditions should deliver a load spectrum or load-time history of the hydraulic runner. Figure 15 shows a general approach for the generation of a load spectrum and was adapted from [19]. The importance and correct prediction of this load-time history was also recognized by [20]. Therefore, it is mandatory that operators are involved in these tasks as they can provide the appropriate basic data.

<figure><img src="image_17.png" /><figcaption>**Figure 15.** General approach for the generation of standardized load-time histories (adapted from [19])</figcaption></figure>

### 4.2 Runner fatigue analysis

The most important part of this method is dedicated to the runner fatigue analysis itself as the output of this investigation should be the expected or, in terms of already operating runners, the residual lifetime.

Normally the points with the highest stresses in Francis turbines are at the trailing edge near the band and crown, which is the result of the FSI calculations before. The subsequent runner fatigue calculations rely on the based fatigue model (see for example [21] or [5]) due to the appearance of multi-axial stress state or not. This would impose that more sophisticated fatigue models should be used like for example the critical plane principle or the micro-crack approach. And as hydraulic runners are designed for a long service life the excitations from different flow phenomena result in a high number of fatigue cycles. Therefore, a high-cycle fatigue (HCF) approach is recommended. Material properties, manufacturing processes (welding) and stress state have to be combined together to resolve this part of the described method.

The applied method revealed for the investigated runner at the worst case a residual life time of about 60 years.

## 5. CONCLUSION AND OUTLOOK

At the presented paper we have described a method to evaluate the lifetime of a high head Francis runner. Single steps have been discussed and two approaches, numerical and experimental, have been shown. Goal of this research project is a lifetime calculation at the design process of a Francis runner prototype. Therefore, strain gauge measurements at an existing runner have been performed to gain validation parameters for the numerical methods.

Nevertheless, some special tasks still have to be discussed as an outlook for the future of this method and the research project. These are:

1. Comparison of open-source and commercial software for this kind of investigation regarding resources and calculation time.
2. Check the applicability of recently given turbulence models for special stochastic flow phenomena. Are the turbulence models able to solve these problems realistically?

3. Suitability of the harmonic response analysis approach for non-harmonic flow phenomena. Some of them can be considered like the rotor-stator interaction as shown but stochastic ones need a transient response analysis as shown for example by [22]. How can we perform any kind of signal decomposition?

4. Sensitivity analysis of the influence of modeling the added mass of water at the calculation of the natural frequencies. How much surrounding water is necessary and what is a realistic approach. This task could be combined with the question of appropriate system modeling and fixation of the runner.

5. Generation of the load-time history for the investigated runner to obtain a realistic load spectrum. What is the correct prediction and validation of the necessary assumptions?

## ACKNOWLEDGMENTS

The K-Project GSG-GreenStorageGrid is funded in the framework of COMET - Competence Centers for Excellent Technologies by the Federal Ministry of Transport, Innovation and Technology, the Federal Ministry of Science, Research and Economy, the Vienna Business Agency, the Federal Province of Lower Austria and by the Federal Province of Upper Austria. The program line COMET is administered by the Austrian Research Promotion Agency (FFG). The authors would like to thank all project partners for their contribution to the research project.

The computational results presented have been achieved (in part) using the Vienna Scientific Cluster (VSC).

## REFERENCES

[1] Eduard Doujak. Effects of increased solar and wind energy on hydro plant operation. *HRW*, 22:28–31, 2014.

[2] André Coutu, Hans Aunemo, Bruce Badding, and Omprakash Velagandula. Dynamic behaviour of high head francis turbines. In *Hydropower & Dams*, editor, *Conference Proceedings of HYDRO 2005*. Aqua Media International, 2005.

[3] André Coutu, Christine Monette, and Omprakash Velagandula. Francis runner dynamic stress calculations. In *Hydropower & Dams*, editor, *Conference Proceedings of HYDRO 2007*. Aqua Media International, 2007.

[4] André Coutu and Joel Chamberland-Lauzon. The impact of flexible operation on francis runners. *International Journal on Hydropower & Dams*, Issue Two:90–93, 2015.

[5] Hans-Jörg Huth. *Fatigue Design of Hydraulic Turbine Runners*. PhD thesis, Norwegian University of Science and Technology (NTNU), Trondheim, February 2005.

[6] M. Gagnon, S. A. Tahan, P. Bocher, and D. Thibault. Impact of startup scheme on francis runner life expectancy. *IOP Conference Series: Earth and Environmental Science*, 12:012107, 2010.

[7] Miriam Flores, Gustavo Urquiza, and José María Rodríguez. A fatigue analysis of a hydraulic francis turbine runner. *World Journal of Mechanics*, 02(01):28–34, 2012.

[8] Andrea Carpinteri, Cristian Bagni, Daniela Scorza, and Sabrina Vantadori. High-cycle fatigue in a hydraulic turbine runner. In Andrea Carpinteri, Francesco Iacoviello, Les P. Pook, and Luca Susmel, editors, *Proceedings of Crack Paths (CP 2012), Gaeta, Italy 2012*, pages 287–294. IGF-Gruppo Italiano Frattura, 2012.

[9] R. Guillame and J. Deniau. Rotor-stator interaction in high head pump turbines: dynamic response of the structure. *Hydro 2011 (Prague, Czech Republic)*, 2011.

[10] U. Seidel, B. Hübner, J. Löfflad, and P. Faigle. Evaluation of rsi-induced stresses in francis runners. *26th IAHR Symposium on Hydraulic Machinery and Systems (Beijing, China)*, 2012.

[11] R. Guillaume, J. L. Deniau, D. Scolaro, and C. Colombet. Influence of the rotor-stator interaction on the dynamic stresses of francis runners. *IOP Conference Series: Earth and Environmental Science*, 15(5):052011, 2012.

[12] B. Nennemann, J. F. Morissette, J. Chamberland-Lauzon, C. Monette, O. Braun, M. Melot, A. Coutu, J. Nicolle, and A. M. Giroux. Challenges in dynamic pressure and stress predictions at no-load operation in hydraulic turbines. *IOP Conference Series: Earth and Environmental Science*, 22(3):032055, 2014.

[13] Marcelo Vinicius Magnoli. *Numerical simulation of pressure oscillations in large Francis turbines at partial and full load operating conditions and their effects on the runner structural behaviour and fatigue life*. PhD thesis, Technische Universität München, München, 10.06.2015.

[14] B. Celik, U. Ghia, P. Roache, Ch Freitas, H. Coleman, and P. Raad. Procedure for estimation and reporting of uncertainty due to discretization in cfd applications. *Journal of Fluids Engineering*, 130(7):078001, 2008.

[15] Q. W. Liang, Cristian G. Rodriguez, Eduard Egusquiza, Xavier Escaler, Mohamed Farhat, and François Avellan. Numerical simulation of fluid added mass effect on a francis turbine runner. *Computers & Fluids*, 36(6):1106–1118, 2007.

[16] C. G. Rodriguez, E. Egusquiza, X. Escaler, Q. W. Liang, and F. Avellan. Experimental investigation of added mass effects on a francis turbine runner in still water. *Journal of Fluids and Structures*, 22(5):699–712, 2006.

[17] B. Hübner, U. Seidel, and S. Roth. Application of fluid-structure coupling to predict the dynamic behavior of turbine components. *IOP Conference Series: Earth and Environmental Science*, 12:012009, 2010.
[18] M. Sabourin, D. Thibault, D. A. Bouffard, and M. Lévesque. New parameters influencing hydraulic runner lifetime. *IOP Conference Series: Earth and Environmental Science*, 12:012050, 2010.

[19] P. Heuler and H. Klätschke. Generation and use of standardised load spectra and load-time histories. *International Journal of Fatigue*, 27(8):974–990, 2005.

[20] Martin Gagnon, Antoine Tahan, Philippe Bocher, and Denis Thibault. Influence of load spectrum assumptions on the expected reliability of hydroelectric turbines: A case study. *Structural Safety*, 50:1–8, 2014.

[21] Lixia Zhang, Fuzhou Feng, Xinhai Fan, and Pengcheng Jiang. Reliability analysis of francis turbine blade against fatigue failure under stochastic loading. In Hong-Zhong Huang, Ming J. Zuo, and Yu Liu, editors, *The Proceedings of 2012 International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering: ICQR2MSE 2012*, pages 987–990, Piscataway and NJ, ©2012. IEEE.

[22] Martin Gagnon, Antoine Tahan, Philippe Bocher, and Denis Thibault. On the stochastic simulation of hydro-electric turbine blades transient response. *Mechanical Systems and Signal Processing*, 32:178–187, 2012.