# CFD Investigation of a High Head Francis Turbine at Speed No-Load Using Advanced URANS Models

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## Abstract

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## Featured Application

**The present numerical investigation gives some insights regarding the challenges to compute the flow in hydraulic Francis turbines at off-design operating points. However, such simulations are useful to design hydraulic turbines that will be operated with more flexibility. Therefore, the present results can be used to determine which turbulence models are useful to compute off-design operating points, keeping in mind limitations and future improvements.**

## Abstract

## 1. Introduction

## 2. Case Study

## 3. CFD Setup

#### 3.1. Governing Equations

- $\rho $ the fluid density.
- $\overline{p}$ the mean pressure.
- ${\overline{u}}_{i}$ the mean velocity vector.
- ${\sigma}_{ij}$ the viscous stress tensor computed using a Newtonian linear constitutive equation.
- ${\tau}_{ij}=-\rho \overline{{u}_{i}^{{}^{\prime}}{u}_{j}^{{}^{\prime}}}$ the Reynolds stress tensor (an instantaneous variable a is decomposed following the Reynolds decomposition in a mean part $\overline{a}$ and a fluctuating part ${a}^{{}^{\prime}}$).

- A rotation-curvature correction term, which allows the model to be sensitive to streamline curvature and system rotation [45].

#### 3.2. Mesh

#### 3.3. Numerical Setup

^{®}CFX v17.2 software [53]. The CFX solver uses an element-based finite volume method with a co-located grid. The Rhie–Chow discretization is used for the pressure-velocity coupling [56]. The system of equations is solved using a coupled formulation and a multigrid accelerated Incomplete Lower Upper (ILU) factorization technique. For the steady state simulations, a pseudo-time step is used to reach convergence, whereas for the unsteady simulations, the second order backward Euler scheme has been chosen. Regarding the convection terms, a high order scheme [57] is used for the mean transport equation and a first order scheme for the turbulence transport equations, except for the simulations with the SAS SST $k-\omega $ model, for which a high order scheme is specified. The fluxes at the interfaces between the stationary and rotating domains are computed using a General Grid Interface (GGI) algorithm. A frozen formulation is used for the steady simulations, whereas a fully-transient approach is used for the unsteady simulations.

- no-slip walls are specified at the solid walls.
- the mass flow rate or the total pressure is set at the inlet of the spiral case.
- the surface averaged pressure is set at the outlet through an opening boundary condition.
- the rotational speed is imposed in the rotating domain.

## 4. BEP Results

## 5. SNL Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Description | Unit |

${a}_{ij}$ | Anisotropy stress tensor | − |

D | Outlet runner diameter | m |

E | Specific energy | $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$ |

$\u03f5$ | Dissipation rate of turbulent kinetic energy | ${\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-3}$ |

k | Turbulent kinetic energy | ${\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}$ |

H | Net head | $\mathrm{m}$ |

${L}_{vK}$ | von Karman length scale | $\mathrm{m}$ |

${\mu}_{t}$ | Eddy viscosity | $\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ |

n | Runner rotational frequency | ${\mathrm{s}}^{-1}$ |

${n}_{ED}$ | Speed factor | − |

$\nu $ | Specific speed | − |

$\omega $ | Turbulent eddy frequency | ${\mathrm{s}}^{-1}$ |

$\Omega $ | Vorticity magnitude | ${\mathrm{s}}^{-1}$ |

p | Pressure | $\mathrm{Pa}$ |

${P}_{ED}$ | Power factor | − |

Q | Flow discharge | ${\mathrm{m}}^{3}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

${Q}_{ED}$ | Discharge factor | − |

$\rho $ | Density | $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ |

${\sigma}_{ij}$ | Viscous stress tensor | $\mathrm{Pa}$ |

${\tau}_{ij}$ | Reynolds stress tensor | $\mathrm{Pa}$ |

${u}_{i}$ | Velocity vector | $\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ |

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**Figure 1.**Evidence of harmful structural loading of the turbine runner blades during the normal start-up and shut-down procedures. Signals recorded with the onboard instrumentation [37].

**Figure 2.**(

**a**) View of the CFD domain with the standard draft tube. (

**b**) View of the CFD domain with the extended draft tube.

**Figure 3.**(

**a**) Structured hexahedral mesh, meridional plane view. (

**b**) Structured hexahedral mesh, zero machine plane view. (

**c**) Unstructured mesh, meridional plane view. (

**d**) Unstructured mesh, zero machine plane view.

**Figure 4.**(

**a**) Dimensionless pressure contours on the meridional plane, CFD1 structured mesh, BEP. (

**b**) Dimensionless pressure contours on the meridional plane, CFD2 refined structured mesh, BEP. (

**c**) Dimensionless axial velocity contours on the meridional plane, CFD1 structured mesh, BEP. (

**d**) Dimensionless axial velocity contours on the meridional plane, CFD2 refined structured mesh, BEP.

**Figure 5.**Contours of the ${y}^{+}$ quantity, BEP. (

**a**) Pressure side of a blade, CFD1 structured mesh. (

**b**) Pressure side of a blade, CFD2 refined structured mesh. (

**c**) Suction side of a blade, CFD1 structured mesh. (

**d**) Suction side of a blade, CFD2 refined structured mesh.

**Figure 6.**(

**a**) Contours of the dimensionless magnitude of the relative velocity ${u}^{\ast}$ on the suction side of a blade, CFD1 structured mesh, BEP. (

**b**) Contours of the dimensionless magnitude of the relative velocity ${u}^{\ast}$ on the suction side of a blade, CFD2 refined structured mesh, BEP. (

**c**) Contours of the dimensionless pressure ${p}^{\ast}$ on the suction side of a blade, CFD1 structured mesh, BEP. (

**d**) Contours of the dimensionless pressure ${p}^{\ast}$ on the suction side of a blade, CFD2 refined structured mesh, BEP. (

**e**) Contours of the dimensionless eddy viscosity ${\nu}_{t}^{\ast}$ on the suction side of a blade, CFD1 structured mesh, BEP. (

**f**) Contours of the dimensionless eddy viscosity ${\nu}_{t}^{\ast}$ on the suction side of a blade, CFD2 refined structured mesh, BEP. (

**g**) Contours of the dimensionless wall shear ${\tau}_{w}^{\ast}$ on the suction side of a blade, CFD1 structured mesh, BEP. (

**h**) Contours of the dimensionless wall shear ${\tau}_{w}^{\ast}$ on the suction side of a blade, CFD2 refined structured mesh, BEP.

**Figure 7.**(

**a**) Contours of the dimensionless eddy viscosity ${\nu}_{t}^{\ast}$ in a cross-section of a blade channel, CFD1 structured mesh, BEP. (

**b**) Contours of the dimensionless eddy viscosity ${\nu}_{t}^{\ast}$ in a cross-section of a blade channel, CFD2 refined structured mesh, BEP. (

**c**) Contours of the dimensionless vorticity ${\Omega}^{\ast}$ in a cross-section of a blade channel, CFD1 structured mesh, BEP. (

**d**) Contours of the dimensionless vorticity ${\Omega}^{\ast}$ in a cross-section of a blade channel, CFD2 refined structured mesh, BEP. (

**e**) Position of the cross-section in a blade channel.

**Figure 8.**(

**a**) Time history of the discharge factor ${Q}_{ED}$, Speed-No-Load (SNL). (

**b**) Time history of the speed factor ${n}_{ED}$, SNL. CC, Curvature Correction; Plim, Production limiter; OES, Organised Eddy Simulation; SAS, Scale-Adaptive Simulation; BSL-EARSM, Baseline Explicit Algebraic Reynodlds Stress Model.

**Figure 9.**Contours of the dimensionless mean axial velocity ${U}_{x\phantom{\rule{0.166667em}{0ex}}mean}^{\ast}$ on the meridional plane, SNL. (

**a**) SST $k-\omega $. (

**b**) SST-CC-Plim $k-\omega $. (

**c**) SST-OES $k-\omega $. (

**d**) SST-SAS $k-\omega $. (

**e**) BSL-EARSM $k-\omega $.

**Figure 10.**Contours of the dimensionless mean pressure ${p}_{mean}^{\ast}$ on the meridional plane, SNL. (

**a**) SST $k-\omega $. (

**b**) SST-CC-Plim $k-\omega $. (

**c**) SST-OES $k-\omega $. (

**d**) SST-SAS $k-\omega $. (

**e**) BSL-EARSM $k-\omega $.

**Figure 11.**Instantaneous contours of the dimensionless eddy viscosity ${\nu}_{t}^{\ast}$ on the meridional plane, SNL. (

**a**) SST $k-\omega $. (

**b**) SST-CC-Plim $k-\omega $. (

**c**) SST-OES $k-\omega $. (

**d**) SST-SAS $k-\omega $. (

**e**) BSL-EARSM $k-\omega $.

**Figure 12.**Instantaneous iso-surface of the Q-criterion and contours of the dimensionless pressure on the hub and one runner blade (downstream view), SNL. (

**a**) SST $k-\omega $. (

**b**) SST-CC-Plim $k-\omega $. (

**c**) SST-OES $k-\omega $. (

**d**) SST-SAS $k-\omega $. (

**e**) BSL-EARSM $k-\omega $.

**Figure 13.**(

**a**) Positions of the probes in one guide vane and blade channel. (

**b**) Positions of the probes on the suction side of a runner blade close to the trailing edge.

**Figure 14.**(

**a**) Time history of the dimensionless pressure, probe GV1. (

**b**) Time history of the dimensionless velocity magnitude, probe GV1. (

**c**) Time history of the dimensionless pressure, probe GV3. (

**d**) Time history of the dimensionless velocity magnitude, probe GV3.

**Figure 15.**(

**a**) Time history of the dimensionless pressure, probe Ru1. (

**b**) Time history of the dimensionless velocity magnitude, probe Ru1. (

**c**) Time history of the dimensionless pressure, probe Ru19. (

**d**) Time history of the dimensionless velocity magnitude, probe Ru19.

**Figure 16.**(

**a**) Time history of the dimensionless pressure, probe Ru26. (

**b**) Time history of the dimensionless velocity magnitude, probe Ru26. (

**c**) Time history of the dimensionless pressure, probe Ru35. (

**d**) Time history of the dimensionless velocity magnitude, probe Ru35.

**Figure 17.**Pressure spectrum at probe Ru26: (

**a**) SST $k-\omega $. (

**c**) SST-CC-Plim $k-\omega $. (

**e**) SST-OES $k-\omega $. (

**g**) SST-SAS $k-\omega $. (

**i**) BSL-EARSM $k-\omega $. Pressure spectrum at probe Ru35: (

**b**) SST $k-\omega $. (

**d**) SST-CC-Plim $k-\omega $. (

**f**) SST-OES $k-\omega $. (

**h**) SST-SAS $k-\omega $. (

**j**) BSL-EARSM $k-\omega $.

Sub-Domain | Structured Mesh | Unstructured Mesh | ||
---|---|---|---|---|

Nodes (${10}^{6}$) | Elements (${10}^{6}$) | Nodes (${10}^{6}$) | Elements (${10}^{6}$) | |

Spiral Case | 3.7 | 3.6 | ||

Stay Vanes | 2.9 | 2.8 | 2.8 | 8.9 |

Guide Vanes | 3.7 | 3.5 | ||

Runner | 2.8 | 2.6 | 2.2 | 6.6 |

Draft Tube | 1.6 | 1.5 | 0.8 | 2.7 |

Total | 14.7 | 14.0 | 5.8 | 18.2 |

Sub-Domain | Structured Mesh | Refined Structured Mesh | Unstructured Mesh | |||
---|---|---|---|---|---|---|

Min Angle | Max Aspect | Min Angle | Max Aspect | Min Angle | Max Aspect | |

(deg) | Ratio | (deg) | Ratio | (deg) | Ratio | |

Spiral Case | 20 | 2248 | 20 | 2248 | ||

Stay Vanes | 29 | 77 | 29 | 77 | 12 | 592 |

Guide Vanes | 23 | 62 | 23 | 118 | ||

Runner | 7 | 75 | 3 | 212 | 30 | 147 |

Draft Tube | 20 | 171 | 20 | 774 | 20 | 329 |

Sub-Domain | Structured Mesh | Refined Structured Mesh | Unstructured Mesh |
---|---|---|---|

Spiral Case | 720 | 720 | 90 |

Stay Vanes | 550 | 550 | 160 |

Guide Vanes | 850 | 285 | 300 |

Runner | 2700 | 690 | 250 |

Draft Tube | 330 | 310 | 90 |

Reference | Algorithm | Rotor/Stator Interface | Computational Domain | Mesh | Inlet Boundary Condition |
---|---|---|---|---|---|

CFD1 | Steady | Frozen | standard | structured | mass flow rate |

CFD2 | Steady | Frozen | standard | refined structured | mass flow rate |

CFD3 | Steady | Frozen | standard | unstructured | mass flow rate |

CFD4 | Steady | Frozen | extended | structured | mass flow rate |

CFD5 | Steady | Frozen | standard | structured | total pressure |

CFD6 | Unsteady | Transient | standard | structured | mass flow rate |

Reference | ${\mathit{Q}}_{\mathit{E}\mathit{D}}$ | ${\mathit{n}}_{\mathit{E}\mathit{D}}$ | ${\mathit{P}}_{\mathit{E}\mathit{D}}$ | ${\mathit{\eta}}^{\ast}$ |
---|---|---|---|---|

CFD1 | 0.187 | 0.268 | 0.175 | 1.00 |

CFD2 | 0.190 | 0.271 | 0.177 | 1.00 |

CFD3 | 0.188 | 0.269 | 0.177 | 1.01 |

CFD4 | 0.187 | 0.267 | 0.174 | 0.99 |

CFD5 | 0.186 | 0.275 | 0.174 | 1.00 |

CFD6 | 0.187 | 0.268 | 0.178 | 1.01 |

EXP | 0.183 | 0.271 | 0.171 | 1.00 |

**Table 6.**CFD results at the SNL operating point compared with the measurements. Averaged values over 6.25 runner revolutions.

Turbulence Model | ${\mathit{Q}}_{\mathit{E}\mathit{D}}$ | ${\mathit{n}}_{\mathit{E}\mathit{D}}$ |
---|---|---|

SST | 0.0176 | 0.280 |

SST-CC-Plim | 0.0176 | 0.281 |

SST-OES | 0.0170 | 0.272 |

SST-SAS | 0.0177 | 0.283 |

BSL-EARSM | 0.0178 | 0.284 |

EXP | 0.0170 | 0.270 |

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**MDPI and ACS Style**

Decaix, J.; Hasmatuchi, V.; Titzschkau, M.; Münch-Alligné, C. CFD Investigation of a High Head Francis Turbine at Speed No-Load Using Advanced URANS Models. *Appl. Sci.* **2018**, *8*, 2505.
https://doi.org/10.3390/app8122505

**AMA Style**

Decaix J, Hasmatuchi V, Titzschkau M, Münch-Alligné C. CFD Investigation of a High Head Francis Turbine at Speed No-Load Using Advanced URANS Models. *Applied Sciences*. 2018; 8(12):2505.
https://doi.org/10.3390/app8122505

**Chicago/Turabian Style**

Decaix, Jean, Vlad Hasmatuchi, Maximilian Titzschkau, and Cécile Münch-Alligné. 2018. "CFD Investigation of a High Head Francis Turbine at Speed No-Load Using Advanced URANS Models" *Applied Sciences* 8, no. 12: 2505.
https://doi.org/10.3390/app8122505