# Cavitation Inception on Hydrokinetic Turbine Blades Shrouded by Diffuser

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Optimization for Hydrokinetic Blades Shrouded by Diffuser

#### 2.1. Cavitation Criterion on Hydrokinetic Rotors

#### 2.2. The Optimization Model

Algorithm 1 Chord and twist angle optimization |

Require:
r, $\mathsf{\Omega}$, ${C}_{L}\left({\alpha}_{opt}\right)$, ${C}_{D}\left({\alpha}_{opt}\right)$ and ${V}_{0}$for$i=1$ to $Ns$ (Number of sections) do Compute ${a}_{{b}_{opt}}$ and ${a}_{{b}_{opt}}^{\prime}$ using Equations (11) and (21), respectively; Compute ${C}_{n}={C}_{l}cos\varphi +{C}_{d}sin\varphi $, calculated for ${\alpha}_{opt}$ obtained from maximum ${C}_{l}/{C}_{d}$; Compute the relative velocity, W; Compute ${V}_{CAV}$, using Equation (4); if $W>{V}_{CAV}$ thenend ifend forCompute blade geometry. |

## 3. Computational Fluid Dynamics Methodology

#### 3.1. Diffuser Geometry

#### 3.2. Cavitating Flow Simulation

#### 3.3. Numerical Setup

## 4. Results and Discussion

#### 4.1. The Optimization Model

#### 4.2. CFD Simulations and Validation

#### 4.2.1. Verification of Optimal Point of Rotor Position and Mesh Independence Study

#### 4.2.2. Numerical Simulation of Cavitation Effect on Diffuser-Augmented Hydrokinetic Blades

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BET | Blade Element Theory |

CFD | Computational Fluid Dynamics |

DAHT | Diffuser-augmented Hydrokinetic Turbine |

DNS | Direct Numeric Simulation |

HT | Hydrokinetic Turbine |

MRF | Moving Reference Frame |

LES | Large Eddy Simulation |

RANS | Reynolds-Averaged Navier–Stokes |

VOF | Volume Of Fluid Technique |

Arabic Symbols | |

a, ${a}^{{}^{\prime}}$ | Streamtube average axial and tangential induction factors |

${a}_{b}$, ${a}_{b}^{{}^{\prime}}$ | Axial and tangential induction factors at the blade |

A | Area of the Disc (${\mathrm{m}}^{2}$) |

${A}_{3}$ | Cross section at the diffuser outlet (${\mathrm{m}}^{2}$) |

c | Chord (m) |

${c}^{uc}$, ${c}^{co}$ | Uncorrected and corrected chord (m) |

${C}_{l}$, ${C}_{d}$ | Lift and drag coefficients |

${C}_{P}$ | Power coefficient |

${c}_{p3}$ | Pressure coefficient at the diffuser outlet |

${c}_{pmin}$ | Minimum pressure coefficient |

${C}_{n}$ | Normal force coefficient |

${C}_{T}$, ${C}_{Td}$ | Thrust coefficient and diffuser thrust coefficient |

D | Turbine Diameter (m) |

${D}_{i}$, ${D}_{e}$ | Inlet and outlet diffuser diameters (m) |

$dP$ | Elementary power (W) |

f | Additional momentum source ($\mathrm{m}{\mathrm{s}}^{-2}$) |

${f}_{S}$ | Safety factor |

F | Prandtl’s tip loss factor |

${F}_{c}$ | Empirical constant of the cavitation model |

g | Gravity ($\mathrm{m}{\mathrm{s}}^{-2}$) |

h,H | Distance between free surface and turbine radial or center position (m) |

$\dot{{m}_{l}}$,$\dot{{m}_{v}}$ | Rate of change mass per unit of volume for liquid and vapor phases |

${L}_{1}$, ${L}_{2}$ | Upstream and downstream diffuser lengths relative to rotor center plane (m) |

${L}_{d}$ | Diffuser total length (m) |

N | Number of blades |

${N}_{B}$ | Number of bubbles per unit of mixture volume |

p | Local pressure (Pa) |

${p}_{atm}$ | Atmospheric pressure (Pa) |

${p}_{0}$ | Pressure in the external flow (Pa) |

${p}_{2}$ | Pressure in the turbine upstream (Pa) |

${p}_{3}$ | Pressure in the diffuser outlet (Pa) |

${p}_{v}$ | Vapor pressure (Pa) |

r | Radial position at the rotor plane (m) |

R | Radius of the rotor (m) |

${r}_{*}$ | Dimensionless radial position |

${r}_{l}$,${r}_{v}$ | liquid and vapor volume fractions |

${r}_{nuc}$ | Nucleation volume fraction |

${R}_{B}$ | Bubble radius (m) |

${S}_{ij}$ | Symmetric part of the velocity gradient tensor |

${u}_{i}$,${u}_{i}^{{}^{\prime}}$ | Mean velocity and flutuations components ($\mathrm{m}{\mathrm{s}}^{-1}$) |

$\overline{{u}_{i}^{\prime}{u}_{j}^{\prime}}$ | Reynolds Stress Tensor (${\mathrm{m}}^{2}{\mathrm{s}}^{-2}$) |

${t}_{d}$ | Diffuser thickness (m) |

${V}_{x}$ | X-component of the flow velocity at the diffuser centerline ($\mathrm{m}{\mathrm{s}}^{-1}$) |

${V}_{0}$ | Freestream flow velocity ($\mathrm{m}{\mathrm{s}}^{-1}$) |

${V}_{1},{V}_{2}$ | Axial velocity at the rotor ($\mathrm{m}{\mathrm{s}}^{-1}$) |

${V}_{3}$,${V}_{4}$ | Axial velocity at the diffuser outlet and at the wake ($\mathrm{m}{\mathrm{s}}^{-1}$) |

${V}_{CAV}$ | Minimum cavitating flow velocity ($\mathrm{m}{\mathrm{s}}^{-1}$) |

X | Longitudinal position at the diffuser centerline ($\mathrm{m}$) |

W | Relative velocity of fluid |

Greek Symbols | |

$\alpha $ | Angle of attack ($\mathrm{rad}$) |

$\beta $ | Cross sectional area ratio |

${\Delta}_{y}$ | Wall distance ($\mathrm{m}$) |

${\epsilon}_{1}$ | Velocity ratio |

${\epsilon}_{4}$ | Far-wake velocity ratio |

${\eta}_{d}$ | Diffuser efficiency |

$\gamma $ | Diffuser speed-up ratio |

$\mu $ | Dynamic viscosity ($\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$) |

$\nu $ | Kinematic viscosity (${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$) |

$\mathsf{\Omega}$ | Angular velocity of turbine (${\mathrm{s}}^{-1}$) |

$\rho $ | Fluid density ($\mathrm{kg}{\mathrm{m}}^{-3}$) |

${\rho}_{l},{\rho}_{v},{\rho}_{m}$ | Liquid, vapor and mixture densities ($\mathrm{kg}{\mathrm{m}}^{-3}$) |

$\sigma $ | Cavitation number |

${\sigma}_{s}$ | Local solidity |

${\sigma}_{st}$ | Surface tension coefficient |

$\varphi $ | Flow angle ($\mathrm{rad}$) |

${\varphi}_{d}$ | Diffuser opening angle |

${\tau}_{ij}$ | Reynolds stress tensor (${\mathrm{m}}^{2}{\mathrm{s}}^{2}$) |

$\theta $ | Twist angle ($\mathrm{rad}$) |

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**Figure 2.**Simplified illustration of the flow velocities through an ideal DAHT. Control volume locations are: (0), free flow; (1) and (2), rotor plane; (3), diffuser end; and (4), far-wake.

**Figure 6.**(

**a**) Outer semi domain grid. Dark region in the center of the figure corresponds to the diffuser and rotor location. (

**b**) Grid close to the diffuser (outer domain and MRF cylindrical meshes suppressed).

**Figure 7.**(

**a**) Plan view in the cylindrical MRF grid surrounding the rotor. (

**b**) 3-D close-up view near hub and blade.

**Figure 8.**(

**a**) View of the unstructured grid at a blade section. (

**b**) Close-up view of the hydrofoil profile near the leading edge.

**Figure 9.**(

**a**) Chord and (

**b**) twist angle distributions along the blade under the effect of a diffuser.

**Figure 11.**(

**a**) Cavitation effect on the chord distribution along the blade. (

**b**) Relative and cavitation velocities as functions of the radial position.

**Figure 12.**Comparative CFD-experimental velocity ratios on bare diffuser. Adapted from [24].

**Figure 14.**Volume fraction of water vapor at suction side: (

**a**) bare turbine; (

**b**) shrouded turbine; (

**c**) corrected blade.

**Figure 15.**Streamlines colored by velocity magnitude and pressure contour in uncorrected blades surfaces. (

**a**) Pressure side. (

**b**) Suction side.

**Figure 16.**Streamlines colored by velocity magnitude and pressure contour in corrected blades surfaces. (

**a**) Pressure side. (

**b**) Suction side.

**Figure 17.**Pressure contour at a blade radius 4.9m from the rotor center. (

**a**) Bare turbine. (

**b**) Shrouded Turbine. (

**c**) Shrouded optimized turbine.

**Figure 18.**Power coefficient of the optimized blade with diffuser. Adapted from [7].

Parameter | Value |
---|---|

Inlet diffuser diameter (${D}_{i}$) | $10.50$ m |

Outlet diffuser diameter (${D}_{e}$) | $12.60$ m |

Diffuser upstream length (${L}_{1}$) | $3.700$ m |

Diffuser downstream length (${L}_{2}$) | $11.263$ m |

Diffuser total length (${L}_{d}$) | $14.963$ m |

Diffuser thickness (${t}_{d}$) | $4.0$ mm |

Opening angle (${\varphi}_{d}$) | $4.0$ deg |

Quantity | Value |
---|---|

${F}_{c}$ | $0.01$ (Condensation) and 50 (Vaporization) |

${\rho}_{l}$ | 997 kg/m${}^{3}$ |

${\rho}_{v}$ | $0.02308$ kg/m${}^{3}$ |

${r}_{nuc}$ | $5\times {10}^{-4}$ |

Mean bubble diameter (${R}_{B}$) | $1\times {10}^{-6}$ m |

Pressure of vapor (${p}_{v}$) | 3.170 kPa |

Radial Distance $\left(\mathit{r}\right)$ | Span Station $(\mathit{r}/\mathit{R})$ | Chord Length | Twist | Twist Axis |
---|---|---|---|---|

[m] | $[-]$ | [m] | [deg.] | [% chord] |

0.793 | 0.148 | 0.27 | 24 | 30 |

0.949 | 0.189 | 0.56 | 20 | 30 |

1.185 | 0.237 | 0.55 | 16 | 30 |

1.400 | 0.280 | 0.51 | 13 | 30 |

1.635 | 0.327 | 0.47 | 11 | 30 |

1.860 | 0.372 | 0.43 | 10 | 30 |

2.086 | 0.417 | 0.39 | 8 | 30 |

2.311 | 0.462 | 0.35 | 7 | 30 |

2.536 | 0.507 | 0.33 | 6 | 30 |

2.761 | 0.552 | 0.30 | 5 | 30 |

2.985 | 0.597 | 0.28 | 5 | 30 |

3.210 | 0.642 | 0.26 | 4 | 30 |

3.432 | 0.686 | 0.26 | 4 | 30 |

3.657 | 0.731 | 0.26 | 3 | 30 |

3.880 | 0.776 | 0.26 | 3 | 30 |

4.101 | 0.820 | 0.25 | 2 | 30 |

4.328 | 0.865 | 0.26 | 2 | 30 |

4.550 | 0.910 | 0.26 | 2 | 30 |

4.776 | 0.955 | 0.26 | 2 | 30 |

5.000 | 1.000 | 0.11 | 1 | 30 |

Parameter | Value |
---|---|

Turbine diameter (D) | 10 m |

Hub diameter | $1.5$ m |

Number of blades (N) | 3 |

Free stream velocity $\left({V}_{0}\right)$ | $2.5$ m/s |

Water density $\left(\rho \right)$ at 25 ${}^{\circ}$C | 997 kg/m${}^{3}$ |

Submergence of the turbine (H) | 9 m |

${p}_{atm}$ | $1\times {10}^{5}$ Pa |

${p}_{v}$ | $3.17\times {10}^{3}$ Pa |

Gravity (g) | $9.81$ m/s${}^{2}$ |

Angular velocity ($\mathsf{\Omega}$) | 35 rpm |

Foil type | NACA ${65}_{3}$-618 |

Region | Condition |
---|---|

Inlet | ${V}_{0}=2.50$ m/s |

Outlet | $p=constant$ |

Blade surface | No-slip |

Rotor surface | No-slip |

Top, bottom and lateral surfaces | Slip |

Rotatory domain | Frozen rotor |

Turbulence intensity | $5\%$ |

Parameter | Value |
---|---|

$\beta $ | $0.7511$ |

${\eta}_{d}$ | $0.4712$ |

${C}_{Td}$ | $0.6458$ |

Mesh | Cells [×10${}^{6}$] | ${\mathit{y}}_{\mathbf{max}}^{+}$ | Power |
---|---|---|---|

Mesh A | 16.2 | 1.32 | 388 kW |

Mesh B | 17.6 | 1.01 | 337 kW |

Mesh C | 18.9 | 1.15 | 333 kW |

Mesh D | 19.4 | 1.10 | 325 kW |

Mesh E | 20.5 | 1.11 | 324 kW |

Bare Turbine | Shrouded Turbine | |||||
---|---|---|---|---|---|---|

$r$ [m] | $W$ [m/s] | $\sigma $ | ${c}_{Pmin}$ | $W$ [m/s] | $\sigma $ | ${c}_{Pmin}$ |

1.00 | 4.4366 | 17.9746 | −4.4438 | 4.8882 | 14.8072 | −4.5960 |

1.25 | 5.2192 | 12.8085 | −3.8261 | 5.6089 | 11.0903 | −4.2369 |

1.50 | 6.0395 | 9.4309 | −3.1398 | 6.3809 | 8.4489 | −3.6099 |

1.75 | 6.8841 | 7.1554 | −2.8491 | 7.1868 | 6.5652 | −3.4404 |

2.00 | 7.7450 | 5.5713 | −2.4387 | 8.0165 | 5.2003 | −3.0272 |

2.25 | 8.6173 | 4.4344 | −2.2454 | 8.8635 | 4.1915 | −2.8229 |

2.50 | 9.4979 | 3.5959 | −2.1149 | 9.7230 | 3.4313 | −2.5876 |

2.75 | 10.3847 | 2.9625 | −2.0876 | 10.5926 | 2.8474 | −2.5156 |

3.00 | 11.2762 | 2.4740 | −1.9430 | 11.4700 | 2.3911 | −2.2933 |

3.25 | 12.1714 | 2.0904 | −1.7277 | 12.3537 | 2.0292 | −2.0685 |

3.50 | 13.0695 | 1.7843 | −1.7887 | 13.2425 | 1.7379 | −1.7737 |

3.75 | 13.9700 | 1.5365 | −1.5611 | 14.1354 | 1.5008 | −1.5322 |

4.00 | 14.8724 | 1.3336 | −1.3559 | 15.0299 | 1.3058 | −1.3339 |

4.25 | 15.7764 | 1.1654 | −1.1897 | 15.9229 | 1.1441 | −1.1691 |

4.50 | 16.6818 | 1.0247 | −1.0465 | 16.8117 | 1.0089 | −1.0314 |

4.75 | 17.5882 | 0.9060 | −0.9224 | 17.6833 | 0.8963 | −0.9166 |

5.00 | 18.4957 | 0.8049 | −0.8194 | 18.5221 | 0.8026 | −0.8180 |

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## Share and Cite

**MDPI and ACS Style**

Picanço, H.P.; Kleber Ferreira de Lima, A.; Dias do Rio Vaz, D.A.T.; Lins, E.F.; Pinheiro Vaz, J.R.
Cavitation Inception on Hydrokinetic Turbine Blades Shrouded by Diffuser. *Sustainability* **2022**, *14*, 7067.
https://doi.org/10.3390/su14127067

**AMA Style**

Picanço HP, Kleber Ferreira de Lima A, Dias do Rio Vaz DAT, Lins EF, Pinheiro Vaz JR.
Cavitation Inception on Hydrokinetic Turbine Blades Shrouded by Diffuser. *Sustainability*. 2022; 14(12):7067.
https://doi.org/10.3390/su14127067

**Chicago/Turabian Style**

Picanço, Hamilton Pessoa, Adry Kleber Ferreira de Lima, Déborah Aline Tavares Dias do Rio Vaz, Erb Ferreira Lins, and Jerson Rogério Pinheiro Vaz.
2022. "Cavitation Inception on Hydrokinetic Turbine Blades Shrouded by Diffuser" *Sustainability* 14, no. 12: 7067.
https://doi.org/10.3390/su14127067