# Efficiency Assessment for Rehabilitated Francis Turbines Using URANS Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methodology

_{Runner}= 350 mm). The full turbine computational domain is composed of three subdomains corresponding to the spiral case, the runner, and the draft tube. Two interfaces are used to connect the fixed and rotating subdomains, as shown in Figure 2.

#### 2.1. Numerical Parameters

^{6}in case 1 and 4.8 × 10

^{6}in case 2, at model test dimension. The turbulent flow is computed using the ANSYS CFX solver [26]. Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations, the k-ω SST turbulence model, and transient rotor-stator interfaces are used to conduct complete turbine simulations [26,27,28]. This interface type actualizes flow information between rotating and stationary domains at each time step without averaging flow quantities, which allows the wakes of older SV geometries to be well-resolved at the runner inlet. URANS simulations also improve the modeling of vortex phenomena in the draft tube flow [25].

^{−4}s) are performed, followed by one at 0.5° (~1.05 × 10

^{−4}s) to obtain the final solution. This unsteady strategy obtains good statistical convergence of the main hydraulic turbine quantities relative to their average value. The initial solution to these simulations is obtained from a steady computation using frozen rotor interfaces and the high-resolution advection scheme. The time step of the last rotation is comparable to the value used for the numerical analysis of rotor-stator interaction and draft tube pressure fluctuations in Francis turbines ([18,28,30]).

^{−4}at each time step. However, in some simulations, there may be very few cells (0.003% of the runner’s sub-domain volume) near the blade’s inlet, where convergence is not fully achieved even after three iterations. This is mostly due to the lack of control over tetrahedral cell quality, combined with the high-velocity gradients in this area.

#### 2.2. Boundary Conditions

#### 2.3. Evaluation of Turbine Parameters

_{n}) was evaluated according to IEC 60193 [32], Equation (2), where static pressure is the average value measured by pressure probes at a spiral case inlet (P

_{1}) or at a draft tube outlet (P

_{2}). The kinetic energy was computed using the turbine discharge amount Q and the flow area at a spiral case inlet A

_{1}or at a draft tube outlet A

_{2}.

#### 2.4. Mesh Generation

^{+}in all of the meshes was set to approximately 2.5 to take advantage of the wall-flow resolution of the k-ω SST turbulence model [29]. With this Y

^{+}, only a few cells are in the viscous sub-layer and a minimum of 20 cells are in the turbulent boundary layer when using a 1.2 expansion factor. The Y

^{+}on the SVs of case 2 is ~1 due to their rounded leading edges. Indeed, the SV chamfered edge of case 1 sets the boundary layer separation point position. Meshing parameters impose a minimum of 20 and a maximum of 60 prismatic tetrahedral cell layers on the walls to provide a smooth transition to tetrahedral cells. The hydraulic profile’s thin edges, the draft tube pier noses, and the runner’s cone are all spatially discretized with ordered tetrahedral cells. The volume mesh size was controlled by approximating the surface-to-volume cell size factor. According to these criteria, the spiral case meshes and full turbine simulations are similar to those shown in Figure 5 in the SV and GV zones.

#### 2.5. Grid Scaling Test

^{+}is the same for all grids of the spiral case and tandem cascade; it provides a similar flow resolution on walls. Three different grid densities (1 for fine, 2 for medium, and 3 for coarse density) for each case and geometry are used in the spiral case. Fine mesh of case 1 old geometry spiral case use significantly more nodes than new geometry to capture the velocity gradient on the sharp edge of the SVs.

_{Runner}below the runner. The flow imposed at the GV inlet is an average uniform flow with the SV’s outlet orientation.

_{Runner}below the runner and the same outlet as the complete turbine simulation domain. The velocity and turbulence fields imposed at the inlet come from an average flow of one runner rotation of a complete turbine simulation of case 1’s new geometry.

_{fine}

^{21}) is relatively small, with a maximum value of 0.1% and a minimum of 0.04% with finer meshes. The apparent order of the simulations varies between 11.4 and 21.2, with no possible values for case 2’s new SVs. Negative values of ε

_{32}/ε

_{21}reveal oscillatory convergence in three cases. These are due to the small difference in the efficiency loss between each mesh density. Oscillatory convergence could indicate a level of the mesh-independence of the efficiency losses, as mesh densities appear to be sufficient to model boundary layer separations and recirculation zones at SVs, or in their absence, in the new geometries. These results allow the use of the medium mesh density in the spiral case for the complete turbine simulation.

#### 2.6. Model Tests

## 3. Results

#### 3.1. Efficiency Increase

#### 3.2. Tandem Cascade

_{m}= 0) and a contour of the flow velocity on the tandem cascade horizontal plane. Boundary layer separation is accentuated at the tandem cascade horizontal plane from the higher radial component of the flow at this location caused by the secondary flows in the spiral case. These figures show that the modification eliminates the boundary layer separation over the entire height of the SVs in the original geometry. However, the thinner trailing edge of the SV in case 1 adds a small recirculation zone near the flange, starting from the exit of the flange and the SV’s curve that lead to a local adverse pressure gradient increase.

#### 3.3. Runner

#### 3.4. Draft Tube

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Notation | |

A | area (m^{2}) |

c | grid refinement factor |

e_{a} | approximate relative error |

e_{ext} | extrapolated relative error |

E | energy head (m) |

g | gravitational acceleration (ms^{−2}) |

GCI_{fine} | fine-grid convergence index |

H_{i} | runner internal head (m) |

H_{n} | turbine hydraulic head (m) |

h | representative cell size (m) |

N_{sq} | hydraulic turbine specific speed (s^{−1}) |

$\dot{m}$ | mass flow (kgs^{−1}) |

P | pressure (kgm^{−1}s^{−2}) |

p | GCI method apparent order |

Q | discharge (m^{3}s^{−1}) |

r | radius (m) |

T | torque (Nm) |

t | unit of time (s) |

u | velocity (ms^{−1}) |

u_{m} | flow velocity (ms^{−1}) |

u_{θ} | circumferential velocity (ms^{−1}) |

u_{r} | radial velocity (ms^{−1}) |

x | length (m) |

Y^{+} | dimensionless distance |

z | elevation (m) |

γ | guide vanes opening (°) |

ε | critical value difference |

η | efficiency |

φ | discharge coefficient |

ϕ | critical value |

ϕ_{ext} | extrapolated value |

ψ | energy coefficient |

Ω | rotational speed (s^{−1}) |

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**Figure 4.**Surfaces and pressure tap locations to evaluate turbine head and components efficiency losses.

**Figure 8.**Hydraulic turbine efficiency of case 1 and 2 with and without SV modifications. Efficiency is relative to the maximal efficiency of the new geometry for each case.

**Figure 9.**Efficiency loss coefficient in the tandem cascade in cases 1 and 2. Flow direction is from the right to the left in the figure. SV and GV position are marked with the vertical dashed lines.

**Figure 10.**Velocity at the tandem cascade horizontal plane in case 1 hydraulic turbine functions of the inlet area of the spiral case and the turbine discharge.

**Figure 11.**Velocity at the tandem cascade horizontal plane in case 2 hydraulic turbine functions of the inlet area of the spiral case and the turbine discharge.

**Figure 12.**Flow velocity at the tandem cascade horizontal plane in case 1 hydraulic turbine functions of the inlet area of the spiral case and the turbine discharge.

**Figure 13.**Flow velocity at the tandem cascade horizontal plane in case 2 hydraulic turbine functions of the inlet area of the spiral case and the turbine discharge.

**Figure 14.**Flow velocity at the tandem cascade horizontal plane functions of the inlet area of the spiral case and the turbine discharge and u

_{m}= 0 isosurface in case 1 hydraulic turbine.

**Figure 15.**Flow velocity at the tandem cascade horizontal plane functions of the inlet area of the spiral case and the turbine discharge and u

_{m}= 0 isosurface in case 2 hydraulic turbine.

**Figure 19.**Efficiency loss coefficient as function of relative position in the runner of the case 1 and 2.

**Figure 21.**Flow velocity in the draft tube of old and new geometries of case 1 functions of the inlet area of the draft tube and the turbine discharge.

**Figure 22.**Flow velocity in the draft tube of old and new geometries of case 2 functions of the inlet area of the draft tube and the turbine discharge.

**Figure 23.**Axial, circumferential and radial flows at runner outlet of case 1 hydraulic turbine. Axial velocity is function of the inlet area of the draft tube and the turbine discharge.

**Figure 24.**Axial, circumferential and radial flows at runner outlet of case 2 hydraulic turbine. Axial velocity is function of the inlet area of the draft tube and the turbine discharge.

Title | Case 1 | Case 2 | ||
---|---|---|---|---|

Geometry | Old | New | Old | New |

$\gamma \left(\xb0\right)$ | 25 | 25 | 24 | 24 |

$\phi /{\phi}_{opt}$ | 0.998 | 1.002 | 1.002 | 1.001 |

Title | Case 1 | Case 2 | ||
---|---|---|---|---|

Geometry | Old | New | Old | New |

Coarse | 24.6 | 15.1 | 9.7 | 10.8 |

Medium | 28.8 | 20.5 | 16.2 | 20.9 |

Fine | 68.4 | 22.9 | 25.7 | 27.9 |

Simulation | Runner | Draft Tube |
---|---|---|

Coarse | 0.46 | 2.67 |

Medium | 0.72 | 4.77 |

Fine | 1.12 | 6.59 |

**Table 4.**Spatial discretization error and numerical uncertainties in spiral case and tandem cascade.

Title | Case 1 | Case 2 | ||
---|---|---|---|---|

Geometry | Old | New | Old | New |

${c}_{21}$ | 1.36 | 1.04 | 1.17 | 1.10 |

${c}_{32}$ | 1.05 | 1.11 | 1.19 | 1.25 |

${\varphi}_{1}$ (m) | 0.818 | 0.474 | 0.487 | 0.446 |

${\varphi}_{2}$ (m) | 0.798 | 0.474 | 0.492 | 0.435 |

${\varphi}_{3}$ (m) | 0.783 | 0.471 | 0.491 | 0.437 |

${\epsilon}_{32}$ | 0.0198 | −0.00291 | 0.00023 | 0.00222 |

${\epsilon}_{21}$ | 0.0244 | −0.00033 | 0.00435 | −0.0112 |

${e}_{a}^{21}$ | 0.0244 | 0.000702 | 0.00892 | 0.0252 |

$GC{I}_{fine}^{21}$ | 0.000943 | 0.000859 | 0.000429 | - |

Sub-Domain | Runner | Draft Tube |
---|---|---|

${c}_{31}$ | 1.159 | 1.114 |

${c}_{32}$ | 1.161 | 1.213 |

${\varphi}_{1}$ (m) | 12.31 | 4.03 |

${\varphi}_{2}$ (m) | 12.83 | 3.58 |

${\varphi}_{3}$ (m) | 11.54 | 3.13 |

${\epsilon}_{32}$ | −1.290 | −0.450 |

${\epsilon}_{21}$ | 0.520 | −0.450 |

${e}_{a}^{21}$ | 0.042 | 0.112 |

$GC{I}_{fine}^{21}$ | 0.036 | 0.063 |

Case 1 | Case 2 | |||
---|---|---|---|---|

Sub Domain | Old | New | Old | New |

Spiral case | 39.7 | 28.2 | 24.9 | 28.5 |

Runner | 22 | 17.8 | ||

Draft tube | 6.7 | 5.6 |

**Table 7.**Efficiency losses coefficient reduction evaluate by complete turbine simulation. Model tests values are show as reference.

Title | $\mathbf{\Delta}\mathit{\psi}/{\mathit{\phi}}^{2}{}_{\mathit{O}\mathit{l}\mathit{d}}-\mathbf{\Delta}\mathit{\psi}/{\mathit{\phi}}^{2}{}_{\mathit{N}\mathit{e}\mathit{w}}$ | $\frac{{\left(\mathbf{\Delta}\mathit{\psi}/{\mathit{\phi}}^{2}{}_{\mathit{O}\mathit{l}\mathit{d}}-\mathbf{\Delta}\mathit{\psi}/{\mathit{\phi}}^{2}{}_{\mathit{N}\mathit{e}\mathit{w}}\right)}_{\mathit{C}\mathit{F}\mathit{D}}}{{\left(\mathbf{\Delta}\mathit{\psi}/{\mathit{\phi}}^{2}{}_{\mathit{O}\mathit{l}\mathit{d}}-\mathbf{\Delta}\mathit{\psi}/{\mathit{\phi}}^{2}{}_{\mathit{N}\mathit{e}\mathit{w}}\right)}_{\mathit{C}\mathit{F}\mathit{D}\mathit{T}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}}\left(\mathit{\%}\right)$ | ||
---|---|---|---|---|

Case | 1 | 2 | 1 | 2 |

Spiral case | 0.001 | −0.002 | ~0 | −4 |

Stay vanes | 0.137 | 0.032 | 57 | 72 |

Guide vanes | 0.081 | 0.017 | 34 | 39 |

Runner | 0.025 | −0.001 | 11 | −3 |

Draft tube | −0.004 | −0.002 | −2 | −4 |

Total | 0.240 | 0.044 | 100 | 100 |

Model tests | 0.367 | 0.111 | - | - |

Geometry | Old | New |
---|---|---|

Case 1 | 14.2° | 6.1° |

Case 2 | 23.8° | 4.2° |

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**MDPI and ACS Style**

Martineau Rousseau, P.; Soulaïmani, A.; Sabourin, M.
Efficiency Assessment for Rehabilitated Francis Turbines Using URANS Simulations. *Water* **2021**, *13*, 1883.
https://doi.org/10.3390/w13141883

**AMA Style**

Martineau Rousseau P, Soulaïmani A, Sabourin M.
Efficiency Assessment for Rehabilitated Francis Turbines Using URANS Simulations. *Water*. 2021; 13(14):1883.
https://doi.org/10.3390/w13141883

**Chicago/Turabian Style**

Martineau Rousseau, Philippe, Azzeddine Soulaïmani, and Michel Sabourin.
2021. "Efficiency Assessment for Rehabilitated Francis Turbines Using URANS Simulations" *Water* 13, no. 14: 1883.
https://doi.org/10.3390/w13141883