# Modal Decomposition of the Precessing Vortex Core in a Hydro Turbine Model

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Set-Up and Measurements

#### 2.1. Test Section

#### 2.2. Wall Pressure Measurements

#### 2.3. Stereo-PIV Measurements

## 3. Post-Processing Routines

#### 3.1. Spatial Fourier Mode Decomposition

#### 3.2. Proper Orthogonal Decomposition (POD)

#### 3.3. Symmetric and Antisymmetric Decomposition of the Velocity Fields

#### 3.4. Outline of Applied Methods

- -
- Extract symmetry properties from the four signals of pressure fluctuations by using the spatial Fourier mode decomposition;
- -
- Find a mode coupling in the velocity data by classical POD analysis;
- -
- Extract symmetry properties from POD of the symmetric/antisymmetric decomposed velocity data;
- -
- Track modes as a function of operating condition and identify the onset of instabilities.

## 4. Results and Discussion

#### 4.1. Wall Pressure Fluctuations

#### 4.2. Mean Velocity Distributions

#### 4.3. Classic POD at Part-Load Regime

#### 4.4. Symmetric/Antisymmetric POD at Various Flow Regimes

#### 4.5. Identification of Stability

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Swirl Number Definition

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**Figure 3.**Pressure fluctuation spectra in terms of normalized PSD for the azimuthal modes $m=1$ and $m=2$ and different flow rates $Q/{Q}_{c}$.

**Figure 4.**Mean axial (

**a**,

**d**,

**g**) and mean tangential (

**b**,

**e**,

**h**) components of velocity, and TKE distributions (

**c**,

**f**,

**i**) for $0.3{Q}_{c}$, $0.5{Q}_{c}$, and ${Q}_{c}$ cases.

**Figure 5.**POD analysis for $0.5{Q}_{c}$: TKE content of POD modes (

**a**), Lissajous figures of ${a}_{1-4}$ (

**b**–

**e**), and the POD mode shapes (

**f**).

**Figure 6.**Comparative POD analysis of the flow snapshots decomposed in symmetric/antisymmetric parts (right column) and without decomposition (left column) for all three components of velocity fields: 1st and 2nd modes of classical POD analysis (

**a**); 3rd and 4th modes of classical POD analysis (

**c**); 1st and 2nd modes of antisymmetric POD (

**b**), 1st and 2nd modes of symmetric POD (

**d**), TKE content of POD modes (

**e**), Lissajous figure of ${a}_{1}^{antisymm}({a}_{2}^{antisymm})$ (

**f**) and Lissajous figure of ${a}_{1}^{symm}({a}_{1}^{antisymm})$ (

**g**). Flow rate is $0.5{Q}_{c}$.

**Figure 7.**(

**a**) Kinetic energy E and the wall pressure amplitudes of the $m=1$ and $m=2$ modes as a function of flow rate. The inset shows the linear fit to determine the bifurcation point of both modes. (

**b**) Swirl number and mode frequencies as a function of flow rate.

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**MDPI and ACS Style**

Litvinov, I.; Sharaborin, D.; Gorelikov, E.; Dulin, V.; Shtork, S.; Alekseenko, S.; Oberleithner, K.
Modal Decomposition of the Precessing Vortex Core in a Hydro Turbine Model. *Appl. Sci.* **2022**, *12*, 5127.
https://doi.org/10.3390/app12105127

**AMA Style**

Litvinov I, Sharaborin D, Gorelikov E, Dulin V, Shtork S, Alekseenko S, Oberleithner K.
Modal Decomposition of the Precessing Vortex Core in a Hydro Turbine Model. *Applied Sciences*. 2022; 12(10):5127.
https://doi.org/10.3390/app12105127

**Chicago/Turabian Style**

Litvinov, Ivan, Dmitriy Sharaborin, Evgeny Gorelikov, Vladimir Dulin, Sergey Shtork, Sergey Alekseenko, and Kilian Oberleithner.
2022. "Modal Decomposition of the Precessing Vortex Core in a Hydro Turbine Model" *Applied Sciences* 12, no. 10: 5127.
https://doi.org/10.3390/app12105127