# Vortex Cascade Features of Turbulent Flow in Hydro-Turbine Blade Passage with Complex Geometry

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Setup

#### 2.1. Computation Domain and Grid

_{1}= 4.6875 m and 7 blades is taken as the simulating object. For the present study, the water head is H = 200 m, the rated flow of the hydro-turbine is 148 m

^{3}/s, and the rated rotation speed of the runner is n = 300 rpm. There is a tongue plate at the nose end of the spiral case. A circular arc guide ring is used to connect the spiral case with the guide vane apparatus. The inlet diameter of the spiral case is 3.131 m, and the height of the guide vane apparatus is B

_{0}= 0.6263 m, the distribution circle diameter of the movable guide vane is D

_{0}= 5.625 m, the rated opening of the guide vanes in rated power is $\alpha $ = 23, equivalent to 85% of the max opening. The outlet diameter of the runner is D

_{3}= 3 m. The outlet diameter of the draft tube is 6.2194 m.

#### 2.2. Governing Equations

#### 2.3. Calculation Method and Initial Condition

^{−3}, and the maximum number of iterations per time-step is set to 30. The stable calculation results are selected to analysis. Considering the calculation scale to be large, the simulations are conducted in the PowerCube-S01 cloud cube high performance computing system in Kunming University of Science and Technology for parallel computing. Considering that sometimes the large number of selected CPU cores might not be useful to computation efficiency for different scale parallel computing, 60 CPU cores in node 1 are therefore chosen for parallel computing in this numerical simulation. Each case consumed about 72–96 h, and the total computation time is about 720 h.

## 3. Results and Analysis

#### 3.1. Transitional Process from Rated Load to No-Load

#### 3.2. Different Forms of Channel Vortex

#### 3.3. Cascade Characteristics of Vortex Structures

#### 3.4. Similarity Analysis of Vortex Cascade and Dissipation Features

## 4. Conclusions

- (1)
- Due to the inconsistency between streamwise direction of the inlet flow and tangential direction of the bone line of the blade, angle of attack exists. Fluid flows through inner arc and back arc of the blade after the obstruction of the head of the blades. Under the influences of the viscous action of the curved blade surface and the inverse pressure gradient caused by the projecting objects, the flow separation is generated near the suction surface of the blade, and the flow state is unstable. The complex vortex structures caused by the dehydration are formed in the blade path, that is, the channel vortex. The vortex structure in the runner is different for the different angles of attack. When the angle of attack is small, the rupture of the tubular vortices will lead to developments of the spiral vortices into the horn vortex in the blade channel. When the angle of attack is large, it develops into a horseshoe vortex. When the angle of attack is medium, both horn vortex and horseshoe vortex are formed in the blade passage. The larger the angle of attack, the more complex the vortex system.
- (2)
- The “large scale” eddies at the blade passage have low survival rates, while “small scale” eddies have a high one. The volume ratio of the adjacent scale vortex is about 1.2–1.6.
- (3)
- The average vorticity near wall surfaces is large, and that of suction surface is higher than that of pressure surface. The flow separation near the surface is relatively significant. The average vorticity in the middle of the runner is secondary. The smaller the guide vane opening degree, the larger the energy loss of the vortex structure.
- (4)
- Turbulent kinetic energy near the outlet increases significantly. The existence of a large number of small-scale eddies which are formed in the middle of the channel increases turbulent viscosity in the flow field, causing the increase of the energy dissipation.
- (5)
- Large scale vortices evolve into large scale vortices, and small-scale vortices become more and more common in the blade passage along flow direction, especially in the rear segment of the bale passage.
- (6)
- Vorticity, eddy viscous, turbulent kinetic energy have similarities in their numerical values. The vortex cascade and dissipation features of the turbulent flow are also similar.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$n$—rotation speed of the runner (rpm) |

$t$—time (s) |

${x}_{i}$—co-ordinate component of Cartesian system (m) |

${u}_{i}$—component of flow velocity (m/s) |

${u}_{i}^{\prime}$$=\partial {u}_{i}/\partial t$ (m/s^{2}) |

${\overline{u}}_{i}$—filtered component of flow velocity (-) |

${\overline{\overline{u}}}_{i}$—test filtered component of flow velocity (-) |

$\nu $—kinetic viscosity of fluid (Pa·s) |

${\nu}_{t}$—eddy viscosity of fluid (Pa·s) |

$\overline{p}$—filtered pressure divided by mass density (-) |

${\lambda}_{p}$—pressure non-uniformity index (-) |

${\lambda}_{u}$—velocity non-uniformity index (-) |

$\overline{\nu}$—eddy viscosity mean values (Pa·s) |

${p}_{\mathrm{max}}$—the maximum pressure in the surviving space of a certain scale vortex (Pa) |

${p}_{\mathrm{min}}$—the minimum pressure in the surviving space of a certain scale vortex (Pa) |

${u}_{\mathrm{max}}$—the maximum velocity in blade passage (m/s) |

${u}_{\mathrm{min}}$—the minimum velocity in blade passage (m/s) |

${\nu}_{i}$—the eddy viscosity at the midline of blade passage (Pa·s) |

${p}_{0}$—the reference pressure (Pa) |

${u}_{0}$—the reference velocity (m/s) |

$\rho $—liquid mass density (kg/m^{3}) |

${\delta}_{ij}$—Kronecker’s delta (-) |

$\overline{{s}_{ij}}$—strain rate tensor (-) |

${\Delta}_{i}$—grid width in i direction (m) |

${\tau}_{ij}$—SGS stress (-) |

${L}_{ij}$—Leonard stress (-) |

${C}_{ij}$—Cross stress (-) |

${R}_{ij}$—Reynolds stresses (-) |

Pn—rated power (W) |

T—rotation period (s) |

$\theta $—attack angle (°) |

$N\left(A,B\right)$—survival volume ratio of adjacent scale vortex (-) |

$V(A)$—survival volume of the eddy of scale A in space (m^{3}) |

$V(B)$—survival volume of the eddy of scale B in space (m^{3}) |

$I$—dissipation coefficient of vortex energy (J·s·m^{4}) |

${w}_{e}$—vortex energy (J) |

$Q$—fluid flow (m^{3}/s) |

$H$—water head (m) |

$\alpha $—the guide vane opening degree (%) |

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**Figure 1.**Configuration of computational domains (in meter): (

**a**) Vertical and overhead views; (

**b**) Overall view.

Section | Radius (m) | Section | Radius (m) | Section | Radius (m) | Section | Radius (m) |
---|---|---|---|---|---|---|---|

S1 | 1.5009 | S10 | 1.0540 | S19 | 0.4173 | d7 | 2.1403 |

S2 | 1.4553 | S11 | 0.9967 | S20 | 0.4088 | d8 | 2.1202 |

S3 | 1.4096 | S12 | 0.9359 | S21 | 0.4066 | d9 | 2.0921 |

S4 | 1.3644 | S13 | 0.8734 | d1 | 1.9670 | d10 | 2.0470 |

S5 | 1.3176 | S14 | 0.8093 | d2 | 2.0064 | d11 | 2.0024 |

S6 | 1.2677 | S15 | 0.7391 | d3 | 2.0719 | d12 | 1.9654 |

S7 | 1.2161 | S16 | 0.6629 | d4 | 2.1124 | d13 | 1.9475 |

S8 | 1.1637 | S17 | 0.5825 | d5 | 2.1404 | d14 | 1.9475 |

S9 | 1.1103 | S18 | 0.4880 | d6 | 2.1469 | d15 | 3.1097 |

Condition | 100%Pn | 75%Pn | 50%Pn | 25%Pn | 1%Pn |
---|---|---|---|---|---|

The guide vane opening degree (%) | 85% | 67% | 53% | 38% | 15% |

Inlet velocity for the prototype (m/s) | 19.28 | 15.20 | 12.02 | 8.62 | 3.40 |

Adjacent Scale | 0.25–0.2 | 0.2–0.15 | 0.15–0.1 | 0.1–0.05 | 0.05–0.01 |
---|---|---|---|---|---|

The survival volume ratio of adjacent vortex | 1.359 | 1.402 | 1.549 | 1.404 | 1.232 |

Equal Points | P1 | P2 | P3 | P4 |
---|---|---|---|---|

Eddy viscosity for 100%Pn (Pa·s) | 0.1187 | 0.1434 | 0.2613 | 0.3413 |

Eddy viscosity for 75%Pn (Pa·s) | 0.1242 | 0.1513 | 0.2485 | 0.3373 |

Eddy viscosity for 50%Pn (Pa·s) | 0.1522 | 0.1619 | 0.2483 | 0.2840 |

Eddy viscosity for 25%Pn (Pa·s) | 0.1742 | 0.2384 | 0.2941 | 0.5403 |

Eddy viscosity for 1%Pn (Pa·s) | 1.4615 | 1.8220 | 2.7041 | 2.8693 |

Turbulent kinetic energy for 100%Pn (m^{2}·s^{−2}) | 8.59 | 21.03 | 39.312 | 45.467 |

Turbulent kinetic energy for 75%Pn (m^{2}·s^{−2}) | 9.68 | 22.63 | 24.09 | 25.92 |

Turbulent kinetic energy for 50%Pn (m^{2}·s^{−2}) | 6.19 | 14.27 | 16.03 | 16.86 |

Turbulent kinetic energy for 25%Pn (m^{2}·s^{−2}) | 3.46 | 9.45 | 19.66 | 20.97 |

Turbulent kinetic energy for 1%Pn (m^{2}·s^{−2}) | 22.34 | 19.22 | 58.56 | 51.66 |

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**MDPI and ACS Style**

Hu, X.; Zhang, L.
Vortex Cascade Features of Turbulent Flow in Hydro-Turbine Blade Passage with Complex Geometry. *Water* **2018**, *10*, 1859.
https://doi.org/10.3390/w10121859

**AMA Style**

Hu X, Zhang L.
Vortex Cascade Features of Turbulent Flow in Hydro-Turbine Blade Passage with Complex Geometry. *Water*. 2018; 10(12):1859.
https://doi.org/10.3390/w10121859

**Chicago/Turabian Style**

Hu, Xiucheng, and Lixiang Zhang.
2018. "Vortex Cascade Features of Turbulent Flow in Hydro-Turbine Blade Passage with Complex Geometry" *Water* 10, no. 12: 1859.
https://doi.org/10.3390/w10121859