# Turbulent Eddy Generation for the CFD Analysis of Hydrokinetic Turbines

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## Abstract

**:**

## 1. Introduction

## 2. Review on Turbulence Generation Methods

#### 2.1. Precursor Methods (PM)

#### 2.2. Synthetic Methods (SM)

#### 2.3. Synthetic Volume Forcing Methods (SVFM)

## 3. Turbulence Production and Control Methodology

- ${\mathbf{f}}^{H}$, a spatially harmonic and constant in time deterministic sine distribution, with given wavelength $\lambda $,
- ${\mathbf{f}}^{RH}$, a randomly fluctuating harmonic distribution obtained by introducing at each time in ${\mathbf{f}}^{H}$ a random variation of the sine phase, and a spatially random perturbation of the intensity.

#### 3.1. Turbulent Flow Metrics

#### 3.2. Generated Turbulence Control Strategy

## 4. Computational Model

## 5. Numerical Application

#### 5.1. Computational Set-Up

#### 5.2. Flow Field Simulations

#### 5.3. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFL | Courant-Friedrichs-Lewy |

DES | Detached eddy simulation |

DNS | Direct numerical simulation |

LES | Large eddy simulation |

N-S | Navier–Stokes Equation |

PM | Precursor Methods |

PID | Proportional Integral Derivative |

PSD | Power Spectral Density |

Probability Density Function | |

SM | Synthetic Method |

SDFM | Synthetic Digital Filtering Method |

SVFM | Synthetic Volume forcing Metho |

SRFM | Synthetic Random Fourier Method |

## Appendix A. Summary of Statistical Relationships

## References

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**Figure 3.**Sample harmonic distribution of volume force terms with phase and intensity random noise from Equation (7) (${N}_{k}=1$).

**Figure 5.**Layout of the computational domain grid and details of the generation and control blocks (

**left**). The control block is split into four sub-blocks (

**right**).

**Figure 6.**Residuals of velocity components and pressure, CFL condition during iteration on physical time and pseudo-time loops (

**left**). Details of pseudo-time loop iteration (

**right**).

**Figure 7.**Effect of turbulence generation on the flow field downstream of the generation region. (

**Left**): vorticity magnitude. (

**Right**): ${u}_{1}$ velocity component. Representative time step ${t}_{1}>{T}_{conv}$.

**Figure 8.**Flow velocity components ${u}_{2}$ (

**left**) and ${u}_{3}$ (

**right**) at representative time step ${t}_{1}>{T}_{conv}$.

**Figure 9.**Iso-surfaces of the ${\lambda}_{2}$ quantity and colormaps of the vorticity magnitude on the generation block and on a portion of the control block (

**right**). Sketch of the considered control block portion (

**left**).

**Figure 10.**Control block split into four sub-blocks in the streamwise direction. Sub-block centers are denoted as points ${P}_{i}$ with i from 1 to 4.

**Figure 11.**PID control error for the three velocity components evaluated by Equation (15) and plotted as a function of time.

**Figure 12.**Probe no. 4: time series and time-bin average of the velocity components ${u}_{1}$ (

**left**), ${u}_{2}$ (

**center**), ${u}_{3}$ (

**right**), evaluated by Equation (20).

**Figure 13.**Comparison of non dimensional mean velocity $\overline{{u}_{i}}/{u}_{\infty}$ and variance ${\sigma}_{i}^{2}/{u}_{\infty}^{2}$$(i=1,2,3)$ for all velocity components by spatial averaging on sub-blocks 1 to 4 (

**top**) and by time averaging at probes ${P}_{1}$ to ${P}_{4}$ (

**bottom**).

**Figure 14.**Standard deviation of velocity components from time-bin averaging at sub-block probes and from spatial averaging over sub-blocks. Top: $\overline{u}\left(t\right),<\tilde{u}>$; bottom: ${\sigma}^{2}\left(\overline{u}\left(t\right)\right),<{\sigma}^{2}\left(\tilde{u}\right)>$. Left to right: velocity components ${u}_{1},{u}_{2},{u}_{3}$.

**Figure 15.**Probe no. 4: time auto-correlation ${R}_{ii}\left(\tau \right)$ of the velocity components ${u}_{1}$ (

**left**), ${u}_{2}$ (

**center**), ${u}_{3}$ (

**right**).

**Figure 16.**Kinetic energy Power Spectral Density (PSD) at probes ${P}_{i}$ with $i=$ 1 to 4, left to right, top to bottom.

**Figure 17.**Correlation maps between velocity component pairs $({u}_{1},{u}_{2})$ (

**left**) and $({u}_{1},{u}_{3})$ (

**right**).

**Figure 18.**Isotropy diagrams by using ${S}_{ij}$ tensor invariants $(\mathbb{II},\mathbb{III})$ from Equation (23). Top to bottom, left to right: Probe 1 to 4.

**Figure 19.**Probability Density Function (PDF) of velocity components ${u}_{1}$ (

**left**), ${u}_{2}$ (

**center**), and ${u}_{3}$ (

**right**) compared with normal Gaussian distributions. Top to bottom: probes ${P}_{i}$, with i from 1 to 4.

**Table 1.**Computational grid: main blocks size ${L}_{i}$ in the i-wise direction ($i=1,2,3$), and number of cells on the finest level.

Block | ${\mathit{L}}_{1}/\mathit{L}$ | ${\mathit{L}}_{2}/\mathit{L}$ | ${\mathit{L}}_{3}/\mathit{L}$ | Cells |
---|---|---|---|---|

Generation | 0.05 | 1.4 | 1.4 | $48\times 128\times 128$ |

Control | 2.0 | 1.3 | 1.3 | $700\times 128\times 128$ |

Background | 26.0 | 9.0 (radius) | – | $360\times 60\times 90$ |

**Table 2.**Non dimensional averaged mean value $<\overline{{u}_{i}}>/{u}_{\infty}$ and variance $<{\sigma}_{i}^{2}(\overline{{u}_{i}}/{u}_{\infty})>$ of the velocity components ${u}_{i}\phantom{\rule{3.33333pt}{0ex}}(i=1,2,3)$ in the control block and in the 4 sub-blocks.

$\mathit{i}=1$ | $\mathit{i}=2$ | $\mathit{i}=3$ | ||
---|---|---|---|---|

Control | $<\overline{{u}_{i}}>/{u}_{\infty}$ | 0.933 | −0.0356 | −0.0378 |

block | $<{\sigma}_{i}^{2}(\overline{{u}_{i}}/{u}_{\infty})>$ | 0.788 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.686 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.681 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

Sub | $<\overline{{u}_{i}}>/{u}_{\infty}$ | 0.931 | −0.0456 | −0.0478 |

block 1 | $<{\sigma}_{i}^{2}(\overline{{u}_{i}}/{u}_{\infty})>$ | 0.174 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.146 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.146 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

Sub | $<\overline{{u}_{i}}>/{u}_{\infty}$ | 0.931 | −0.0359 | −0.0379 |

block 2 | $<{\sigma}_{i}^{2}(\overline{{u}_{i}}/{u}_{\infty})>$ | 0.696 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.602 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.599 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

Sub | $<\overline{{u}_{i}}>/{u}_{\infty}$ | 0.935 | −0.0318 | −0.0339 |

block 3 | $<{\sigma}_{i}^{2}(\overline{{u}_{i}}/{u}_{\infty})>$ | 0.387 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.347 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.342 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

Sub | $<\overline{{u}_{i}}>/{u}_{\infty}$ | 0.938 | −0.0290 | −0.0315 |

block 4 | $<{\sigma}_{i}^{2}(\overline{{u}_{i}}/{u}_{\infty})>$ | 0.250 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.228 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.225 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

**Table 3.**Turbulence intensity by Equation (13) in the control block and in the four sub-blocks.

Control Block | Sub-Block 1 | Sub-Block 2 | Sub-Block 3 | Sub-Block 4 |
---|---|---|---|---|

14.68% | 21.59% | 13.77% | 10.37% | 8.38% |

**Table 4.**Non dimensional mean value $<\tilde{{u}_{i}}>/{u}_{\infty}$ and variance $<{\sigma}^{2}(\tilde{{u}_{i}}/{u}_{\infty})>$ of velocity components ${u}_{i}(i=1,2,3)$ from time series at probes ${P}_{j}$ ($j=$ 1 to 4) and $t>{T}_{conv}$.

$\mathit{i}=1$ | $\mathit{i}=2$ | $\mathit{i}=3$ | ||
---|---|---|---|---|

Probe ${P}_{1}$ | $<\tilde{{u}_{i}}>/{u}_{\infty}$ | 0.986 | −0.041 | −0.020 |

$<{\sigma}^{2}(\tilde{{u}_{i}}/{u}_{\infty})>$ | 0.193 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.156 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 0.156 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | |

Probe ${P}_{2}$ | $<\tilde{{u}_{i}}>/{u}_{\infty}$ | 0.956 | −0.029 | −0.031 |

$<{\sigma}^{2}(\tilde{{u}_{i}}/{u}_{\infty})>$ | 0.810 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.672 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.792 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |

Probe ${P}_{3}$ | $<\tilde{{u}_{i}}>/{u}_{\infty}$ | 0.952 | −0.028 | −0.027 |

$<{\sigma}^{2}(\tilde{{u}_{i}}/{u}_{\infty})>$ | 0.410 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.360 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.372 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | |

Probe ${P}_{4}$ | $<\tilde{{u}_{i}}>/{u}_{\infty}$ | 0.949 | −0.027 | −0.027 |

$<{\sigma}^{2}(\tilde{{u}_{i}}/{u}_{\infty})>$ | 0.260 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.292 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 0.221 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

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## Share and Cite

**MDPI and ACS Style**

Gregori, M.; Salvatore, F.; Camussi, R.
Turbulent Eddy Generation for the CFD Analysis of Hydrokinetic Turbines. *J. Mar. Sci. Eng.* **2022**, *10*, 1332.
https://doi.org/10.3390/jmse10101332

**AMA Style**

Gregori M, Salvatore F, Camussi R.
Turbulent Eddy Generation for the CFD Analysis of Hydrokinetic Turbines. *Journal of Marine Science and Engineering*. 2022; 10(10):1332.
https://doi.org/10.3390/jmse10101332

**Chicago/Turabian Style**

Gregori, Matteo, Francesco Salvatore, and Roberto Camussi.
2022. "Turbulent Eddy Generation for the CFD Analysis of Hydrokinetic Turbines" *Journal of Marine Science and Engineering* 10, no. 10: 1332.
https://doi.org/10.3390/jmse10101332